Lectures on Gaussian Integral Operators and Classical Groups (EMS Series of Lectures in Mathematics)

EMS Series of Lectures in Mathematics Edited by Andrew Ranicki (University of Edinburgh, U.K.) EMS Series of Lectures i...

Author: Yurii A. Neretin

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EMS Series of Lectures in Mathematics Edited by Andrew Ranicki (University of Edinburgh, U.K.) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series: Katrin Wehrheim, Uhlenbeck Compactness Torsten Ekedahl, One Semester of Elliptic Curves Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp, María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie, Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups

Yurii A. Neretin

Lectures on Gaussian Integral Operators and Classical Groups

Author: Yurii A. Neretin Faculty of Mathematics University of Vienna Nordbergstrasse 15 1090 Vienna Austria

and Institute for Theoretical and Experimental Physics Bolshaya Cheremushkinskaya, 25 117218 Moscow Russia

and Faculty of Mechanics and Mathematics Moscow State University 119899 Moscow Russia

E-mail: [email protected]

2010 Mathematics Subject Classification: 22-02; 22E46, 53C35, 22E50, 47G10, 32M15, 51N30, 14K25, 47B50 Key words: Semisimple Lie groups and their representations, classical groups, symmetric spaces, Fourier transform, p-adic groups, integral operators, Hilbert spaces of holomorphic functions, holomorphic discrete series, reproducing kernels, theta-functions, Cartier model, Weil representation, Segal–Bargmann transform, central extensions, pseudo-Euclidean geometry

ISBN 978-3-03719-080-7 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2011 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Druckhaus Thomas Müntzer GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface The carefully chosen title of this book should alert the reader to a multiplicity of purposes to be served. The first purpose is to provide an exposition of Gaussian integral operators, i.e., operators of the form Z ² X ³ X 1 1X Sf .x/ D exp akl xk xl C bkl xk yl C ckl yk yl f .y/ dy: 2 2 Rn k;l

k;l

k;l

Such operators appear in analysis, probability theory, and mathematical physics in numerous contexts; the most classical examples are the Fourier transform, the Poisson formula for a solution of the heat equation, and the Mehler formula for the time evolution of a harmonic oscillator. Beyond these classical integral operators, we treat “Gaussian operators” in greater generality; for instance, we discuss Gaussian operators in Fock spaces, the Segal– Bargmann transform, the Zak transform, operators with theta-kernels, Gaussian p-adic operators, the real-adelic correspondence, etc. The second purpose is to present a non-orthodox introduction to the classical groups. Basically, this topic is covered in Chapters 2–3, which form an independent but highly relevant part of the book; also Chapter 10 contains a discussion of the p-adic case. The above two purposes can be pursued independently, but they are not as different as one might think. Gaussian operators are important in the representation theory of infinite-dimensional groups; in a certain sense they replace parabolic induction which is the main tool for construction of representations of finite-dimensional groups. Infinitedimensional groups provide an additional point of view to that of classical groups, which produces new phenomena and new problems. I completely remove infinitedimensional groups from consideration but leave Gaussian operators, so this is some kind of view to classical groups from infinity. On the other hand our detailed analysis of general Gaussian operators is more based on methods of classical groups than on the analytic machinery. The third purpose is to present an exposition of the “Weil representation“, which is closely related to Gaussian integral operators. Note that from a historical point of view, it is, actually, the “Friedrichs representation” or the “Friedrichs–Segal–Berezin– Shale–Weil” representation; it is interesting that the first four authors were motivated by Physics or Mathematical Physics. Here I must say something about the style of the book. Representation theory of real semisimple Lie groups and noncommutative harmonic analysis constituted an important and dynamic branch of mathematics during the 1940– 70s. Nowadays we have an obvious crisis of comprehensibility. As a result, we observe

vi

Preface

an isolation of that field of study. This is perhaps not so sad, since during the period mentioned above the theory deeply influenced other branches of mathematics (and some of these branches are still dynamic, such as the theory of special functions of several variables, integrable systems, infinite dimensional groups, etc.). It is not clear how to solve this old problem. In this book I expose a piece of the theory that preserves links with numerous branches of pure and applied mathematics and mathematical physics. The topic and the style of this book were determined by the modern crisis mentioned above. I began with the intention of writing a monograph or a textbook1 but ended up choosing the genre of an expository “reading-book”. In particular, I tried to avoid any representation-theoretical machinery and to make the chapters maximally selfcontained. In short, I have tried to produce a “democratic” book on the topic, accessible to any mathematician or mathematical physicist. The subject allows numerous possibilities for excursions in lateral directions and I have tried not to miss any of them. In such situations, I warn the reader that this is an “excursion”. On the other hand, our subject may grow without limitation from an arbitrary point. For this reason, I do not even think about being complete at any specific place. For instance, the Gaussian operators in spaces of functions of an infinite number of variables are more important in probability and mathematical physics than are “finite-dimensional” operators. Nevertheless they are not discussed at all. Also, the Howe duality is not mentioned. The present book has obvious intersections with the books by Lion and Vergne [122] and Folland [51]. I tried to avoid intersections with my own book [145], but Chapter 5 is one such. The reader is supposed to be familiar with standard university courses of linear algebra, functional analysis, and complex analysis. Some familiarity with Lie groups and Lie algebras would be also useful, although the latter will be avoided as far as it is possible. I am trying to keep the exposition at this level. However, in some excursions or isolated topics we shall need a slightly wider background (such as elements of differential geometry or topology). There are numerous problems in the book with varying level of difficulty; hard ones in the context of the book are marked with a star. This book is based on my lectures given at the University of Vienna in 2004–2006. I am grateful to Dean Harald Rindler and to Professor Peter Michor for this opportunity. This work was supported by the Austrian fund FWF, project 19064, and also by the Russian Agency for Nuclear Energy, the Dutch fund NWO, grant 047.017.015, and the Japan–Russian grant JSPS–RFBR 07-01-91209. Part of this work was done at White Sea Biological Station (WSBS) of Moscow State University. I thank this polar scientific center for hospitality. 1 Several textbooks of different types and levels have been published relatively recently, see e.g., [49], [62], [73], [80], [109], [184], [185], [221].

Preface

vii

I am grateful to Dmitry Leites, Alexei Rosly, and Dmitry Alekseevsky who read several chapters of the book and gave valuable comments. Vienna – Moscow, 2005–2010

Contents

Preface 1

2

Gaussian integral operators 1.1 Preliminaries and notation . . . . . . . . . . . . . . . . . . . . 1.2 The Heisenberg group and the Weil representation of Sp.2n; R/ . 1.3 Gaussian operators . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The product of Gaussian operators . . . . . . . . . . . . . . . . 1.5 A little linear symplectic geometry . . . . . . . . . . . . . . . . 1.6 Gaussian operators and Lagrangian subspaces . . . . . . . . . . 1.7 Linear relations. Emulation of basic definitions of matrix theory 1.8 The symplectic category . . . . . . . . . . . . . . . . . . . . . 1.9 The symplectic category. Details . . . . . . . . . . . . . . . . . 1.10 Proof of boundedness. Canonical forms of Gaussian operators . 1.11 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . .

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Pseudo-Euclidean geometry and groups U.p; q/ 2.1 The geometry of indefinite Hermitian forms . . . . . . . . . . . . . . 2.2 Pseudo-unitary groups U.p; q/ . . . . . . . . . . . . . . . . . . . . . 2.3 Cartan matrix balls . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The space U.n/. Cell decomposition . . . . . . . . . . . . . . . . . . 2.5 Jordan angles and the Hua double ratio . . . . . . . . . . . . . . . . . 2.6 Classification of pseudo-unitary and pseudo-self-adjoint operators . . 2.7 Indefinite contractions . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Potapov coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 The Krein–Shmul’yan category. The compression property . . . . . . 2.10 Isotropic category. Inverse limits . . . . . . . . . . . . . . . . . . . . 2.11 Kähler metrics on matrix balls. Some matrix tricks . . . . . . . . . . 2.12 Matrix balls as symmetric spaces . . . . . . . . . . . . . . . . . . . . Addendum 1 to Section 2. Blaschke–Potapov factorizations and operator colligations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˛. Potapov multiplicative integral . . . . . . . . . . . . . . . . . . . . . ˇ. Operator colligations and the spectral theory of almost unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The space of rational curves in Grassmannians . . . . . . . . . . . . . Addendum 2 to Section 2. The triangle inequality and the Klyachko theorem

1 1 12 19 26 29 33 37 44 46 52 55 57 57 67 76 86 92 100 109 117 122 127 137 142 147 147 151 154 155

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Contents

Linear symplectic geometry 3.1 Symplectic groups . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The group Sp.2n; R/ in a complex model . . . . . . . . . . . . . 3.3 Matrix balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conjugacy classes in Sp.2n; R/ . . . . . . . . . . . . . . . . . . . 3.5 Symplectic contractions . . . . . . . . . . . . . . . . . . . . . . . 3.6 The symplectic category and the isotropic category . . . . . . . . 3.7 Central extensions of groups Sp.2n; R/. Berezin’s formula . . . . 3.8 Central extensions. The Krein–Shmul’yan category . . . . . . . . 3.9 Geodesic triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Digression. Central extensions of groups of symplectomorphisms 3.11 Central extensions. The Maslov index . . . . . . . . . . . . . . . Addendum to Section 3.3. Lie semigroups and convex cones . . . . . . ˛. Invariant convex cones in classical Lie algebras . . . . . . . . . . ˇ. Lie semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . Causal structures (orders) on homogeneous spaces . . . . . . . . . ı. Total positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Bibliographical remarks to Chapters 2 and 3 . . . . . . . . . . . .

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159 159 163 167 173 182 184 184 190 192 198 201 209 209 211 214 220 220

4 The Segal–Bargmann transform 4.1 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Segal–Bargmann transform . . . . . . . . . . . . . . . . . . 4.3 Spectral analysis of signals . . . . . . . . . . . . . . . . . . . . . 4.4 Spectral analysis of singularities . . . . . . . . . . . . . . . . . . 4.5 Symbols of operators . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The Perelomov problem . . . . . . . . . . . . . . . . . . . . . . . 4.7 Preliminaries on -functions . . . . . . . . . . . . . . . . . . . . 4.8 Interpolation and the Lagrange formula . . . . . . . . . . . . . . 4.9 Inversion of the Segal–Bargmann transform from points of a lattice 4.10 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . .

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226 226 234 240 249 258 266 270 275 278 284

5

6

Gaussian operators in Fock spaces 5.1 Gaussian operators . . . . . . . . . . . . . . . . . . . . 5.2 Proof of the formula for products . . . . . . . . . . . . . 5.3 Boundedness. Spectra of self-adjoint Gaussian operators 5.4 Bibliographical remarks . . . . . . . . . . . . . . . . . .

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286 286 291 296 300

Gaussian operators. Details 6.1 Canonical forms and invariants . . . 6.2 Norms . . . . . . . . . . . . . . . . 6.3 Spectra and eigenvectors . . . . . . 6.4 Quadratic operators and exponentials

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301 301 305 308 312

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Contents

6.5 Matching functions and matrix elements of Gaussian operators . . . . 315 6.6 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . 320 6.7 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . . 321 7

Hilbert spaces of holomorphic functions in matrix balls 7.1 Reproducing kernels . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Highest weight representations of SU.1; 1/ . . . . . . . . . . . . . . 7.3 Hilbert spaces of holomorphic functions . . . . . . . . . . . . . . . . 7.4 Multipliers and invariant kernels . . . . . . . . . . . . . . . . . . . . 7.5 The Berezin scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Hardy spaces on Siegel wedges . . . . . . . . . . . . . . . . . . . . . 7.7 Realization of the Weil representation in holomorphic functions on a matrix ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Determinantal systems of differential equations . . . . . . . . . . . . 7.9 The symplectic category and the spaces H˛ . . . . . . . . . . . . . . 7.10 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . .

8 The Cartier model 8.1 The Zak transform . . . . . . . . . . . . 8.2 The pushforward of the Fourier transform 8.3 Action of Sp.2n; Z/ . . . . . . . . . . . . 8.4 Theta-functions and theta-distributions . . 8.5 Theta-kernels . . . . . . . . . . . . . . . 8.6 Bibliographical remarks . . . . . . . . . . 9

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322 322 331 337 343 347 365 370 374 377 379

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381 381 386 390 399 402 407

Gaussian operators over finite fields 9.1 Classification of quadratic forms . . . 9.2 The Fourier transform and Gauss sums 9.3 Gaussian quadratic sums . . . . . . . 9.4 Gaussian operators . . . . . . . . . . 9.5 Fast Fourier transform . . . . . . . . 9.6 Bibliographic remarks . . . . . . . .

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408 408 411 417 418 422 426

10 Classical p-adic groups. Introduction 10.1 p-adic numbers . . . . . . . . . . . . 10.2 Classification of quadratic forms . . . 10.3 Lattices . . . . . . . . . . . . . . . . 10.4 Bruhat–Tits trees . . . . . . . . . . . 10.5 Bruhat–Tits buildings . . . . . . . . . 10.6 Buildings related to symplectic groups 10.7 The Nazarov category . . . . . . . . . 10.8 Buildings. General comments . . . . ˛. Coxeter groups . . . . . . . . . . . .

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427 427 433 441 447 449 457 460 464 464

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Contents

ˇ. Axioms and basic properties of buildings . . . . . . . . . . . . . . . . 466 . Real symmetric spaces and the Tits metric . . . . . . . . . . . . . . . 472 11 Weil representation over a p-adic field 11.1 Gaussian integrals over a p-adic field . . . . . . . . . . 11.2 Weil representation, p-adic case . . . . . . . . . . . . . 11.3 Adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Adelic Weil representation . . . . . . . . . . . . . . . . 11.5 The group Sp.2n; Q/ and the real-adelic correspondence 11.6 Constructions of modular forms . . . . . . . . . . . . . 11.7 Bibliographical remarks to Chapters 10 and 11 . . . . . Addendum A. Classical groups. Notation . . . . . B. Representations. Several definitions C. Lie groups. Several logical arrows . D. Classical symmetric spaces. Tables

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476 476 482 488 492 495 499 504

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507 507 511 515 521

Bibliography

537

Notation

549

Index

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1 Gaussian integral operators

In this chapter, we investigate Gaussian integral operators in L2 .Rn /. As usual, §1 contains preliminaries (and as usual, the author strongly recommends that you do not read it). §1.2–1.6 contain an initial discussion of the Gaussian operators. The main statements of the chapter (a description of the multiplicative structure of Gaussian operators and the boundedness conditions) are formulated in §1.8 and proved in §§ 1.9– 1.10. However, Chapter 5 contains an independent (and more instructive) proof of the same theorems. On the other hand, an understanding of proofs (and even of statements) requires some linear algebra; the reader who is not sufficiently familiar with it can find an exposition of all necessary topics in Chapters 2 and 3 (I hope that it is also possible to read this book in the following order: Chapter 2 ! Chapter 3 ! Chapter 1.)

1.1 Preliminaries and notation This section is an unordered collection of definitions used below in various places. It provides an easy source for consultation should the reader meet an unfamiliar functional-analytical or matrix term. 1.1 Functional spaces. Recall the standard notation for some functional spaces. All details can be found in the books by Gelfand, Shilov [66], Hörmander [89], Vol. 1, or Reed and Simon [178], [179]. a) The space of smooth functions. C 1 .Rn / is the space of all infinitely differentiable functions on Rn . A sequence fj 2 C 1 .Rn / converges to f in the C 1 -topology if for each multi-index ˛ D .˛1 ; : : : ; ˛n / and each r > 0, ˇ ˇ max ˇ@˛ .fj .x/ f .x//ˇ D 0; (1.1) lim j !1

where1 jxj WD

qP

xW jxj6r

xi2 and

@˛ f .x/ WD

@ @x1

˛1

:::

@ @xn

˛n

f .x1 ; : : : ; xn /:

1 The ALGOL-58 symbol WD means that the right-hand side is the definition of the left-hand side; the meaning of the symbol DW is similar.

2

Chapter 1. Gaussian integral operators

b) The space of compactly supported smooth functions. D.Rn / is the space of all compactly supported functions f 2 C 1 .Rn /. A function f is compactly supported if f D 0 outside some compact set. A sequence fj converges to f in D.Rn /-topology if 1. one can find r > 0 such that fj D 0 for jxj > r, 2. the condition (1.1) is satisfied. c) The Schwartz space. .Rn / denotes the Schwartz space; it consists of functions f rapidly decreasing with all their derivatives, i.e., for each partial derivative @˛ f and each integer M > 0, we have M lim jxj C 1 j@˛ f .x/j D 0:

jxj!1

A sequence fj converges to f if for all M and all multi-indices ˛, ˇ ˇ lim maxn jxjM ˇ@˛ .fj .x/ f .x//ˇ D 0: j !1

x2R

(1.2)

d) The space L2 . The Hilbert space L2 .Rn / consists of Lebesgue measurable functions f on Rn satisfying the condition Z jf .x/j2 dx < 1: (1.3) Rn

This space is endowed with the inner product Z hf; gi D f .x/g.x/ dx:

(1.4)

Rn

If f g D 0 outside a set of measure zero, we say that f and g coincide as elements of L2 .Rn /. We recall the definition of a pre-Hilbert space. It is a linear space V endowed with an inner product h; i such that hv1 C v2 ; wi D hv1 ; wi C hv2 ; wi; hv; wi D hv; wi;

hv; wi D hw; vi; hv; vi > 0

for any v, w 2 V , 2 C. We denote by L the subspace of vectors v satisfying hv; vi D 0. The quotient space V =L is a Euclidean space; we complete it with respect to the norm kvk WD hv; vi1=2 and obtain the Hilbert space associated with the preHilbert space V . For instance, formulas (1.3) and (1.4) determine a structure of pre-Hilbert space and L2 is the associated Hilbert space.

1.1. Preliminaries and notation

3

e) Distributions. By D 0 .Rn / we denote the space of all distributions on Rn . Recall that a distribution is a continuous linear functional on D.Rn /. We denote pairing of a distribution and a function f in two ways, namely,2 Z hh; f ii or f .x/.x/: x2Rn

A linear functional on D.R / is said to be continuous if n

hh; fj ii ! hh; f ii

whenever fj converges to f :

A sequence j 2 D 0 .Rn / converges to if for each f 2 D.Rn /, lim hhj ; f ii D hh; f ii:

j !1

Recall that a measurable function F on Rn is called locally integrable if Z jF .x/j dx < 1 for all r: jxj6r

A locally integrable function F determines the distribution F .x/dx by the formula Z ' 7! F .x/'.x/ dx; ' 2 D.Rn /: (1.5) Rn

As usual, ı.x/ denotes the delta-function, hhı; 'ii D ı.0/: f) Tempered distributions. By 0 .Rn / D 0 .Rn / we denote the space of all tempered distributions on Rn . By definition, this space is the space of all continuous linear functionals on the Schwartz space .Rn /. If a locally integrable function F has polynomial growth, then it determines an element of 0 by (1.5). g) There are the following obvious inclusions: L2 D; 0

0

L D: 2

(1.6) (1.7)

set. We denote by `2 .„/ the Hilbert space h) The space `2 . Let „ be a countableP of functions on „ satisfying the condition 2„ jf ./j2 < 1. The inner product on `2 .„/ is defined to be X f ./g./: hf; gi WD 2„

2

There is a problem “to write, or not to write” dx in the next formula. From the point of view of a lawyer, the expression dx in an integral is a volume form. A volume form itself is a distribution. For this reason, we do “not write”.

4


1.2 Integral operators and the kernel theorem. Recall some versions of the Schwartz kernel theorem, see Hörmander [89], Gelfand, Shilov[66]. A. A distribution K.x; y/ 2 D 0 .Rm Rn / determines a bounded operator A W D.Rn / ! D 0 .Rm / as follows. For any f 2 D.Rn /, we define a linear functional Af on D.Rm /, i.e., an element of D 0 .Rm / by setting Af .g/ WD hhK.x; y/; g.x/f .y/ii: By the kernel theorem, every bounded (D continuous) linear operator A W D.Rn / ! D .Rm / has such a form. We symbolically write A as an integral operator Z Af .x/ D K.x; y/f .y/; (1.8) 0

y2Rn

where K.x; y/ 2 D 0 .Rm Rn /. In the same way, a distribution K.x; y/ 2 0 .Rm Rn / determines an integral operator .Rn / ! 0 .Rm / and each bounded operator .Rn / ! 0 Rm / has such a form. B. Let K.x; y/ 2 .Rn Rm /. Then the integral operator Z Af .x/ D K.x; y/f .y/ y2Rn

is a bounded operator 0 .Rn / ! .Rm /. Indeed, for a fixed x, the expression K.x; y/ is a function depending on y, hence we can consider its pairing with a distribution f .y/. Each bounded operator 0 .Rn / ! .Rm / has this form. C. Let A W L2 .Rn / ! L2 .Rm / be a bounded operator. The inclusions (1.6), (1.7) show that A is also an operator .Rn / ! 0 .Rm /. Therefore, the restriction of A to .Rn / is defined by formula (1.8) with a certain kernel K 2 D.Rm Rn /. But the operator A is bounded in L2 . Hence for any sequence fj 2 .Rn / that converges to f 2 L2 .Rn / in the L2 -sense, we have Z Af .x/ D lim K.x; y/ fj .y/ in L2 -sense: j !1 Rn

In this sense, all bounded operators L2 .Rn / ! L2 .Rm / are integral operators. It may be difficult to establish L2 -boundedness/unboundedness of the operator A determined by a given kernel K.x; y/. D. Hilbert–Schmidt operators. Let K.x; y/ 2 L2 .Rm Rn /, i.e., Z jK.x; y/j2 dx dy < 1: Rm Rn

5


Then the corresponding integral operator Z Af .x/ D K.x; y/ f .y/ dy Rn

is a bounded operator L2 .Rn / ! L2 .Rm /. Moreover, A is a Hilbert–Schmidt operator (and each Hilbert–Schmidt operator can be represented in this form). Let H , H 0 be Hilbert spaces. Let A W H ! H 0 be a compact operator, let j be its singular values3 . We recall that A is a Hilbert–Schmidt operator (see [66], [178]) if X j2 < 1: Let ek 2 H , el0 2 H 0 be orthogonal bases. We denote by akl D hAek ; el0 i the matrix elements of A. The Hilbert–Schmidt condition is equivalent to the convergence of the series X tr A A D jakl j2 < 1: k;l

1.3 The Fourier transform. The Fourier transform F (see e.g., [179]) P is the operator in space of functions on Rn defined by the kernel K.x; y/ D expfi xk yk g, i.e., Z o n X F f .x/ D fO.x/ D .2/n=2 (1.9) f .y/ exp i xk yk dy: Rn

We recall some properties of the Fourier transform. 1) F is a continuous bijection .Rn / ! .Rn /. 2) Therefore, F is a continuous bijection 0 .Rn / ! 0 .Rn /. 3) F is a unitary operator L2 .Rn / ! L2 .Rn /. In other words, for any f , g 2 L .Rn /, the Plancherel formula holds: Z Z f .x/ g.x/ dx D F f .y/ F g.y/ dy: 2

Rn

Rn

4) The inverse operator is defined by the kernel K.x; y/ D expfi

P

xk yk g.

5) Therefore,F f .x/ D f .x/; hence F D 1. 2

4

6) Consider the operators Ta f .x/ D f .x C a/;

o n X Sb f .x/ D exp i bk xk f .x/:

Then F Ta D Sa F ; 3

F Sb D Tb F :

See Subsection 1.1.7, for compact operators this definition makes sense, see [178].

(1.10)

6


The main topics of this book are several kinds of Fourier-type operators, see Subsections 1.3.3, 4.2.1, 5.1.3, 7.7.2, 8.1.1, 8.5.1, 9.2.3, 9.4.1, 11.2.3, 11.2.5, 11.5.4. 1.4 The weak convergence in Hilbert spaces. Let H be a Hilbert space4 . A sequence hn 2 H weakly converges to h 2 H if hhn ; i converges to hh; i 2 H (see [178]). Problem 1.1. a) Let en be an orthonormal system in a Hilbert space H . Show that the sequence en weakly converges to 0. P P b) The weak closure of the sphere xj2 D 1 is the ball xj2 6 1. Now we formulate a constructive criterion for weak convergence of vectors. First, recall that a family of vectors ˛ in a Hilbert space (or in a topological vector space) is called total if their finite linear combinations are dense in the whole space. Theorem 1.1. Let ˛ 2 H be a total family of vectors. Then the following two assertions are equivalent: 1) a sequence hn weakly converges to h; 2) for each ˛ , the sequence hhn ; ˛ i converges to hh; ˛ i as n ! 1 and supn khn k < 1. In particular, we can take an orthonormal basis as a system ˛ . 1.5 Convergence of operators in Hilbert space. For details, see [178]. a) A sequence of bounded operators Aj in a Hilbert space H uniformly converges to a bounded operator A if kAj Ak ! 0 as j ! 1. b) A sequence of bounded operators Aj strongly converges to A if for each h 2 H one has kAj h Ahk ! 0 as j ! 1. c) A sequence of bounded operators Aj weakly converges to A if for each pair of vectors , 2 H the sequence hAj ; i converges to hA; i. One has the following obvious implications: uniform convergence H) strong convergence H) weak convergence Problem 1.2. Consider the shift operators in L2 .R/, A.t /f .x/ D f .x C t /

for any t 2 R:

1B . The map t 7! A.t / is not uniformly continuous (at any point), because p kA.t / A.s/k D 2 for t ¤ s. 2B . The same map is strongly continuous. 3B . A.t/ tends weakly to 0 as t ! 1. However A.t / has no strong limit as t ! 1. 4

We consider only separable Hilbert spaces.


7

Theorem 1.2. Let ˛ 2 H be a total family of vectors. The following two conditions are equivalent: 1) a sequence Aj weakly converges to A; 2) for each pair ˛ , ˇ , the sequence hAj ˛ ; ˇ i weakly converges to h˛ ; ˇ i as j ! 1 and supj kAj k < 1. 1.6 Tensor products of Hilbert spaces. Let V , W be Hilbert spaces. Let ek be an orthonormal basis in V and fj an orthonormal basis in W . We define the tensor product V ˝W as a Hilbert space with orthonormal basis constituted by vectors (formal expressions) ek ˝ fjP . P For vectors v D ˛k ek 2 V and w D ˇj fj 2 W , we define X ˛k ˇj vk ˝ fj 2 V ˝ W: v ˝ w WD k;j

be linear operators. Their tensor product is the linear Let A W V ! Vz , B W W ! W determined by the condition .A ˝ B/.v ˝ w/ D operator A ˝ B W V ˝ W ! Vz ˝ W Av ˝ Bw. If M1 , M2 are measure spaces, then L2 .M1 / ˝ L2 .M2 / ' L2 .M1 M2 /: 1.7 Matrix notation. The purpose of this subsection is to fix some terminology and some notation. a) Unit matrices. We denote by 1n the unit n n matrix. Sometimes we omit the index n and merely write 1. The identity operator in a space V is denoted by 1V . b) Conditions of symmetry. Let A D faij g be a matrix. – At is the transposed matrix (its ij -th matrix element is aj i ), – AN is the element-wise complex conjugate matrix; its matrix elements are aN ij , – A D ANt is the adjoint matrix (its ij -th matrix elements is aNj i ). A matrix A is – symmetric if A D At , – skew-symmetric if A D At , – Hermitian or self-adjoint if A D A , – anti-Hermitian if A D A . We denote by Matp;q the space of all p q matrices; for n n matrices we write Mat n . We denote by Symmn ;

ASymmn ;

Hermn ;

AHermn

8


the spaces of symmetric, skew-symmetric, Hermitian, and anti-Hermitian nn matrices respectively. c) Let S be a complex matrix. Represent it in the form S D A C iB, where A and B are real matrices. We denote these summands by A D Re S and B D Im S . d) Matrices and forms. For a Hermitian n n-matrix R, we define the Hermitian form on Cn by 0 1 yN1 B:C X xRy WD x1 : : : xn R @ :: A D rkl xk yNl for any x; y 2 Cn : k;l yNn e) Positive definite matrices. A Hermitian matrix A is positive definite (notation: A > 0) if the corresponding Hermitian form is positive on all non-zero vectors x 2 Cn : xAx > 0

for any x ¤ 0:

(1.11)

A Hermitian matrix is positive if either one of the following conditions holds: – all eigenvalues are positive; – all principal minors are positive (the Sylvester criterion). We say that A is non-negative definite or positive semi-definite if we have > in (1.11). In the same way, we define negative definite and non-positive definite (negative semi-definite) matrices. f) Polar decomposition Theorem 1.3. Each invertible operator A in a Euclidean space admits a unique decomposition (‘polar decomposition’) A D US , where U is unitary and S is positive self-adjoint. Proof. The operator A A is self-adjoint because .A A/ D A A D A A. It is positive because hA Av; vi D hAv; Avi > 0. Hence A A can be represented in the form A A D RƒR1 , where p R is unitary p and ƒ is a diagonal matrix with positive eigenvalues. We define S WD A A WD R ƒR1 . Set U WD AS 1 . Then U U D AS 2 A D A.A A/1 A D 1, i.e., U is unitary. We use the notation jAj WD

p A A:

g) Norm. Let V , W be Euclidean spaces, A W V ! W a linear operator. The norm of A is sup jhAx; yiW j: kAk D sup kAxkW D kxkV D1

kxkV D1; kykW D1

9


For a self-adjoint operator A W V ! V with eigenvalues j , we have kAk D max jj j: h) Singular values. The singular values j of a p q-matrix A are the eigenvalues p of A A, see [178]. The singular values are invariants of the matrix with respect to the transformations A 7! U1 AU2 , where U1 , U2 are unitary matrices. Each (rectangular) matrix admits a unique representation in the form 1 0

1 0 : : : C B A D U1 @ 0 2 : : :A U2 ; where 1 > 2 > > 0 and U1 , U2 are unitaryI :: :: : : : : : the numbers j coincide with the singular values. We have kAk D max j ; p kAk D kA Ak: i) Determinants of block matrices. The following theorem is a corollary of formula (2.14) given below: Theorem 1.4. Let g D ac db be a .p C q/ .q C p/-matrix with an invertible block a. Then a b (1.12) det g D det D det a det.d ca1 b/: c d j) Jordan normal form and root subspaces. Let A be an operator in a finitedimensional linear space V over C. Let 2 C be an eigenvalue of A. Then ker.A / ker.A /2 ker.A /3 :

(1.13)

The root subspace V of the operator A is the union of the chain (1.13). L Theorem 1.5. V D V , where the summation is taken over all eigenvalues of A. Recall that a Jordan block is a matrix of the form 0 1 0 1 1 0 0 1 0 B0 1 0C 1 C ; ; @0 1A ; B @0 0 1A ; 0 0 0 0 0 0

etc.

(1.14)

An operator is a Jordan block if its matrix is a Jordan block in some basis. Theorem 1.6 (Jordan normal form). Each operator in a complex linear space is a direct sum of Jordan blocks.

10


k) Cyclic vectors. Let A be an operator in a finite-dimensional linear space V . A vector v is said to be cyclic if vectors v, Av, A2 v, …span the whole space. Problem 1.3. a) Let A be a diagonal matrix with distinct eigenvalues. Describe all cyclic vectors. b) Describe all cyclic vectors for a Jordan box. c) When does an operator in a finite-dimensional space have a cyclic vector? A vector v is said to be cyclic for a bounded operator A in a Hilbert space H if v, Av, A2 v, …is a total subset in V . l) Nilpotent and unipotent operators. An operator A in a finite-dimensional linear space is said to be nilpotent, if AN D 0 for some N . Problem 1.4. An operator A is nilpotent if and only if all its eigenvalues equal 0. An operator B is said to be unipotent if all its eigenvalues equal 1. Problem 1.5. a) An operator B is unipotent if and only if B 1 is nilpotent. b) For any nilpotent operator A, its exponential is unipotent. Moreover, any unipotent operator B has a unique representation B D exp.A/ with nilpotent A. Problem 1.6 (Jordan decomposition). a) Any operator L in a finite-dimensional complex linear space admits a unique decomposition L D T C A, where the operator T is diagonalizible, A is nilpotent, and AT D TA. b) Any invertible operator L in a finite-dimensional complex linear space admits a unique decomposition L D SB, where the operator S is diagonalizible, B is unipotent, and SB D BS. 1.8 The Gauss integral Theorem 1.7. Let K be a symmetric matrix, let the matrix Re K be positive definite. Let b 2 Cn be a row vector. Then Z ² ³ ² ³ .2/n=2 1 1 1 t exp xKx t C i bx t dx D exp b : (1.15) bK 2 2 det K 1=2 Rn First, we explain the meaning of the factor det K 1=2 . We give the following three equivalent definitions: 1) If K is real, then det K > 0 and hence .det K/1=2 is a well-defined positive function. The square root has a unique holomorphic continuation on the domain K > 0 in Cn.nC1/=2 . Indeed, this domain is convex. The function det K is non-zero in this domain5 , hence the square root has no ramification points. 5

Since Re xKx t > 0 for x ¤ 0, we have xK ¤ 0, i.e., the matrix K is invertible.

11


2) Let be an eigenvalue of K, let v be a corresponding eigenvector. Then 0 < v.Re K/v D Re.vKv / D Re. vv / D .Re / vv : Thus Re > 0. Define det K 1=2 WD

Q

j1=2 ; where j are the eigenvalues of K.

3) Let C be a contour in the domain Re z > 0 surrounding the spectrum fj g of K. Then Z 1 1=2 K WD .z K/1 z 1=2 dz: 2 i C Proof of Theorem 1.7. Step 1. Let K be real and b D 0. Represent K in the form K D UƒU t , where U is an orthogonal matrix (i.e., U U t D 1), and ƒ is a diagonal matrix with entries k > 0, Z Z ² ³ ² ³ 1 1 exp xKx t dx D exp .xU /ƒ.xU /t d.xU t / 2 2 Rn Rn Z Z ² ³ ² ³ n Y 1 1 D exp xƒx t dx D exp k xk2 dxk 2 2 Rn R kD1 Z n ² ³ n Y 1 2 1=2 exp dx x D k 2 R kD1 Z n ² ³ 1 2 1=2 D .det K/ exp x dx : 2 R Next, Z

²

³

1 exp x 2 dx 2 R

2

Z

²

³

1 D exp .x 2 C y 2 / dx dy 2 R2 Z 2 Z 1 D d' expfr 2 =2g r dr D 2: 0

0

This completes our evaluation of the integral in this special case. Step 2. Let K, i b DW c be real, Z ² ³ 1 exp xKx t C cx t dx 2 Rn ² ³Z ² ³ 1 1 1 t exp .x C cK 1 /K.x C cK 1 /t dx D exp cK c 2 2 Rn and after the substitution y D x C cK 1 we obtain the desired statement. Step 3. The identity (1.15) is proved if a matrix K and a vector i b are real. It remains to observe that both sides of (1.15) holomorphically depend on K and b.

12


1.2 The Heisenberg group and the Weil representation of Sp.2 n; R/ The material of this section is rather standard. We define the “Weil representation” following the original ideas of Friedrichs and Segal. In this chapter and in Chapter 5, we make the construction completely explicit and essentially extend it. 2.1 Representations. Let G be a group, let V be a complex linear space. A linear representation of the group G in the space V is a map that takes each g 2 G to an invertible linear operator .g/ such that .g1 /.g2 / D .g1 g2 /

for all g1 , g2 2 G:

(2.1)

This condition also implies .e/ D 1I

.g 1 / D .g/1 ;

where e is the unit of G:

We can also say that a representation is a homomorphism of G to the group GL.V / of all invertible linear operators in V . 2.2 Unitary representations. A representation of a topological group G in a Hilbert space H is said to be unitary if a) the operators .g/ are unitary; b) the function g 7! .g/ is continuous with respect to the weak operator topology6 . In other words, for all v, w 2 H , the matrix elements .g/ D h.g/v; wi of the representation are continuous functions7 on G. 2.3 Projective representations. Let G be a group. A projective representation of G in a linear space V is a map that takes each g 2 G into an invertible operator .g/ in V such that .g1 /.g2 / D .g1 ; g2 /.g1 g2 /; where .g1 ; g2 / 2 C . Example. Let G be the additive group R2 . For each .a; b/ 2 R2 , define the operator .a; b/ in L2 .R/ by .a; b/ D f .x C a/ expfi bxg: The operators determine a unitary projective representation of R2 . The same example is discussed in the next subsection (the group R2 is the quotient of the Heisenberg group by the center, see below). 6

See 1.5 and Problem 1.2. Officially, this requirement is necessary. Nevertheless, for locally compact groups, measurability of matrix elements implies their continuity. Since we can not invent a non-measurable function (see e.g., [92]), the condition b) holds automatically for any representation that can be described. However, one can not prove abstract theorems without this condition. 7

1.2. The Heisenberg group and the Weil representation of Sp.2n; R/

13

Some other definitions of representation theory are contained in Addenda B and C. 2.4 The Heisenberg group. The Heisenberg group is the group of operators in L2 .Rn / generated by the following transformations: T .a/f .x/ D f .x C a/;

S.b/f .x/ D f .x/ expfixb t g;

(2.2)

where, a, b 2 Rn . We wish to give a more formal definition. Let vectors vC and v range in Rn . The Heisenberg group Heisn D Heisn .R/ is the group of block .1 C n C 1/ .1 C n C 1/matrices of the structure 0 1 1 vC

t A H vC ˚ v I D @0 1 v : (2.3) 0 0 1 The product of two such matrices is given by 0 10 1 0 1 vC 1 wC 1 v C C wC t A@ t A @0 1 v 1 0 1 w D @0 0 0 1 0 0 1 0 0

1 t C C v C w t t A; v C w 1

or

t : (2.4) H vC ˚v I H wC ˚w I D H .v CwC /˚.v CwC /I C CvC w Hence Heisn is indeed a group. The matrices of the form H.0 ˚ 0; / commute with all matrices H.v ; vC ; /, i.e., such matrices form the center Z ' R of the Heisenberg group. The quotient group Heisn =Z is isomorphic to the additive group R2n (see (2.4)). For any matrix (2.3), we define the operator

t U vC ˚ v ; D f .x C v / exp.ixvC C i / in the space L2 .R/. The following statement is obvious. Theorem 2.1. U is a unitary representation of Heisn . 2.5 Irreducibility Theorem 2.2. a) Let A be a bounded operator L2 .Rn / ! L2 .Rn / satisfying AU.g/ D U.g/A for all g 2 Heisn :

(2.5)

Then A is a scalar operator. b) Let A W .Rn / ! .Rn / be a bounded operator satisfying the same equation (2.5). Then A is a scalar operator.

14


Problem 2.1. a) Let B be a diagonal matrix with pairwise distinct entries. If AB D BA, then A is a diagonal matrix. b) Solve the equation XB D BX for an arbitrary diagonal matrix B; for a Jordan block B. The first proof of the theorem. In both the cases, our operator A is determined by some kernel K 2 0 .Rn Rn /, see Subsection 1.2. The kernel of AS.b/ is K.x; y/ expfiyb t g, the kernel of S.b/A is expfixb t gK.x; y/. Hence, expfixb t gK.x; y/ D K.x; y/ expfiyb t g () K.x; y/ expfixb t g expfiyb t g D 0

for any b 2 Rn :

(2.6)

Therefore, K.x; y/ is supported by the set of .x; y/ satisfying the condition expfixb t g expfiyb t g D 0 In other words, moreover

1 .x 2

for all b:

y/b t 2 Z. Thus K.x; y/ is supported8 by the diagonal x D y, K.x; y/ D

.x/ı.x y/:

Therefore, the operator A has the form Af .x/ D

.x/f .x/:

Next, we write the condition AT .c/f .x/ D T .c/Af .x/ or .x/f .x C c/ D i.e.,

.x C c/f .x C c/;

D const.

The second proof of Theorem 2.2.a. Actually, we wish to prove again the following lemma, which (as we have seen above) immediately implies the desired statement: Lemma 2.3. Let A be a bounded operator in L2 .Rn / commuting with all operators S.b/f .x/ D exp.ixb t /. Then A is an operator of multiplication by a function. Proof. The operator A must commute with all linear combinations of the operators S.b/. Therefore, it must commute with all operators S.'/ of the form Z Z S.b/'.b/ db f .x/ D e ibx '.b/ db f .x/; S.'/f .x/ WD Rn 8

Rn

For a definition of the support of a distribution, see [89] or Section 4.4.


15

where ' is an integrable function on Rn . In particular, A commutes with any operator f .x/ 7! r.x/f .x/; where r 2 .Rn / (see 1.3). Hence, A.rf / D r.Af /: Thus if we know Af , then we know A.rf / for all r. For definiteness, assume f .x/ D exp.jxj2 /. Let g 2 D.Rn / be a compactly supported smooth function. Then g g Af Ag D A f D Af D g: f f f Thus the operator A coincides with the operator of multiplication by Af =f on the dense subspace D L2 . Since A is continuous in L2 .Rn /, it follows that A is a multiplication by a function. Corollary 2.4. The representation U is irreducible, i.e., there is no proper closed subspace in L2 .R/ invariant with respect to all operators U.g/. Proof. Let V L2 .R/ be a proper invariant subspace. Since the operators U.g/ are unitary, it follows that the orthogonal complement V ? to V is also invariant. The operator of orthogonal projection onto V (along V ? ) commutes with U.g/. It remains to apply Theorem 2.2. 2.6 Automorphisms of the Heisenberg group. We slightly modify the coordinates on Heisn . We write 0 1 t 1 vC ˛ C 12 vC v

A 2 Heisn : (2.7) R vC ˚ v I ˛ WD @0 1 v 0 0 1 Then

R vC ˚ v I ˛ R wC ˚ w I ˇ

t t D R .vC C wC / ˚ .v C w /I ˛ C ˇ C 12 .v wC vC w /: Let us change our language. Consider the real linear space V2n WD Rn ˚ Rn : Let us write its elements as v D vC ˚ v . We define a skew-symmetric bilinear form on V2n by t t fv; wg D vC w v wC : In this notation, the space Heisn is V2n ˚ R; the multiplication is given by

vI ˛ wI ˇ D v C wI ˛ C ˇ C 12 fv; wg :

(2.8)

16


The operators of the representation U in new coordinates are

˚

t t U vC ˚ v I D f .x C v / exp ixvC C i C 2i vC v :

(2.9)

Denote by Sp.2n; R/ (the real symplectic group) the group of all operators in V2n preserving the form f; g (for a detailed discussion of this group, see Chapter A 3). We B write out elements of Sp.2n; R/ as block .n C n/ .n C n/-matrices g D C D . By (2.8), it follows that the transformations

.g/ W vI ˛ 7! vgI ˛ ; where g 2 Sp.2n; R/; are automorphisms of the Heisenberg group, i.e., they send products in Heisn to products. Therefore, for a fixed g 2 Sp.2n; R/, the operators

Ug vI ˛ WD U vgI ˛ form a unitary representation of the Heisenberg group (since the composition of an automorphism and a representation is a representation). The key fact is equivalence of representations Ug for all g. Theorem 2.5. a) For each g 2 Sp.2n; R/, there is a unique, to within a scalar factor, bounded invertible operator W.g/ W L2 .R/ ! L2 .R/ such that

U vgI ˛ D W.g/1 U vI ˛ W.g/ for each vI ˛ 2 Heisn . (2.10) b) The operators W.g/ are unitary up to scalar factors and satisfy W.g1 /W.g2 / D c.g1 ; g2 /W.g1 g2 /;

where c.g1 ; g2 / 2 C .

(2.11)

In other words, W.g/ is a unitary projective representation of Sp.2n; R/. c) The operators W.g/ are bounded in .Rn /. Remark. By the definition, the operators W.g/ are defined to within scalars, therefore the appearance of a scalar factor in (2.11) is not surprising. Changing a normalization of W.g/, we change c.g1 ; g2 /; there is a (noncanonical) way to make c.g1 ; g2 / D ˙1 for all g1 , g2 . But it is impossible to obtain c.g1 ; g2 / D 1. For a detailed discussion, see §§ 3.7–3.11. The representation g 7! W.g/ has several names, the most usual terms are Weil representation, the harmonic representation, the oscillator representation, the metaplectic representation, the Shale–Weil representation. The construction was discovered by K. O. Friedrichs around 1950, see [55].


17

2.7 Refinement of Theorem 2.5 9 Lemma 2.6. a) Matrices of the form 1 B 1 A 0 ; t1 ; 0 1 C 0 A

0 ; 1

where B D B t , C D C t ;

generate the symplectic group Sp.2n; R/. b) Matrices of the form 1 B 0 1 A 0 ; ; ; 0 1 1 0 0 At1

where B D B t ;

(2.12)

(2.13)

also generate the symplectic group Sp.2n; R/. Proof. Let g D ac db 2 Sp.2n; R/. Suppose the block a is invertible. We wish to factorize g as 1 0 y 0 1 u a b y yu gD H) D : x 1 0 z 0 1 c d xy z C xyu We get the system of equations y D a;

yu D b;

Solving it, we obtain 1 a b D ca1 c d

xy D c;

z C xyu D d:

0 a 0 1 a1 b : 1 0 d ca1 b 0 1

(2.14)

Problem 2.2. a) The matrices (2.12) are contained in Sp.2n; R/. b) If g 2 Sp.2n; R/, then the three factors in (2.14) are contained in Sp.2n; R/ i.e., they have the form (2.12). See also (2.8.2). Thus, any element g 2 Sp.2n; R/ satisfying det a ¤ 0 can be represented as a product of matrices (2.12). Products of such elements cover the whole group. This proves a). The statement b) follows from a) and the fact that 1 0 1 1 0 0 1 1 C D : C 1 1 0 0 1 1 0 9

This theorem is a trivial fact of representation theory. By the Stone–von Neumann theorem (see Addendum B), the representations U and Ug of the Heisenberg group are equivalent. By the Schur Lemma (see Addendum B), there exists a unique operator W.g/ intertwining these two irreducible representations. Also, a product of intertwining operators is an intertwining operator. This implies the remaining part of the theorem. The reader can consult with Addendum B and convince himself that this footnote is a complete proof. Our Subsection 2.8 is equivalent to a reference to the Schur Lemma. Also, Theorem 2.7 provides us with a constructive proof of equivalence of the representations U and Ug avoiding a reference to the Stone–von Neumann Theorem.

18


Theorem 2.7. For generators (2.13) of Sp.2n; R/, the operators W./ are given by 1 A 0 f .x/ D p f .xA1 /; (2.15) W 0 At1 j det Aj ni o 1 B W f .x/ D f .x/ exp xBx t ; (2.16) 0 1 2 Z ˚

0 1 W f .x/ D f .y/ exp ixy t dy: (2.17) 1 0 Rn A verification of the relations (2.10) is trivial and we leave it as an exercise; for the Fourier transform, this reduces to (1.10). Due to a scalar factor j det Aj1=2 the operator (2.15) is unitary. 2.8 Proof of Theorem 2.5. Assume that one can find two operators W.g/, W0 .g/ satisfying (2.10). Then

W.g/1 U vI ˛ W.g/ D W0 .g/1 U vI ˛ W0 .g/:

Therefore W.g/W0 .g/1 commutes with U vI ˛ . By Theorem 2.2, it follows that W.g/1 W0 .g/ is a scalar. This proves the uniqueness of W.g/. Now, let W.g1 / and W.g2 / be solutions of the defining equation (2.10). Then

U vg1 g2 I ˛ D W.g2 /1 U vg1 I ˛ W.g2 /

D W.g2 /1 W.g1 /1 U vI ˛ W.g1 /W.g2 / 1

D W.g1 /W.g2 / U vI ˛ W.g1 /W.g2 /: But the last row is the defining relation for W.g1 /W.g2 /. Thus we obtain the following lemma: Lemma 2.8. If W.g1 / and W.g2 / are the solutions of the defining equation (2.10) for g1 , g2 , then W.g1 /W.g2 / is the solution of the defining equation for g1 g2 . By Theorem 2.7, it follows that the function W./ is defined on the generators of Sp.2n; R/, and hence W./ is defined everywhere. The identity (2.11) follows from the last lemma and the uniqueness of W./. c) The operators (2.15)–(2.17) are bounded and invertible in .Rn /. 2.9 Digression. Reformulation. Consider the operators Pk , Qk in .Rn / given by the formula Pk f .x/ D i

@ f .x/; @xk

Qk f .x/ D xk f .x/:

They satisfy the commutation relations10 ŒPk ; Qk D i; 10

all other commutators are zero:

Here ŒA; B WD AB BA is the commutator of operators.

19

1.3. Gaussian operators Consider their linear combinations a.v/ O D a.v O C ˚ v / WD

X

.k/ Qk C v

X

.k/ vC Pk D

X

k

k

.k/ v xk C i

k

X k

.k/ vC

@ : @xk

Then t t Œa.v/; O a.w/ O D i.vC w v wC / 1:

Therefore, for each g 2 Sp.2n; R/, we have Œa.vg/; O a.wg/ O D Œa.v/; O a.w/: O Theorem 2.9. a) For each g 2 Sp.2n; R/, there is a unique (to within a scalar factor) bounded linear operator Y .g/ W .Rn / ! .Rn / such that a.vg/ O D Y .g/1 a.v/Y.g/: O

(2.18)

b) Y.g/ D W.g/ for some 2 C. Proof. Uniqueness. We refer to considerations of Subsections 2.7, 2.8; for this, we need the following lemma: Lemma 2.10. Any operator in .Rn / commuting with operators Pk and Qk is a scalar operator. Proof of the lemma. We repeat the first proof of Theorem 2.2. Our operators commute with Qk . Instead of (2.6), we obtain the equation K.x; y/.x y/ D 0. Hence the kernel K.x; y/ is a delta-function supported by the diagonal x D y and we repeat the final part of the proof of Theorem 2.2 literally. Existence. First, the operators (2.15)–(2.17) satisfy the desired commutation relations. Lemma 2.8 and all the arguments of Subsection 2.8 survive literally. Problem 2.3. a) Show that exp a.v/ O D U.v; 0/. b) Show an a priori equivalence of (2.18) and (2.10).

1.3 Gaussian operators Here we define Gaussian operators and propose an initial collection of examples. 3.1 Gaussian vectors. Let T be an n n complex symmetric matrix, Re T > 0. We define a Gaussian vector bŒT 2 L2 .Rn / by ²

³

1 bŒT .x/ D exp xT x t : 2 By Theorem 1.7, x 1=2 : hbŒP ; bŒQiL2 .Rn / D .2/n=2 det.P C Q/

20


Vectors of the form bŒT , where 2 C, are also said to be Gaussian. 3.2 Gaussian distributions. By definition, the set of Gaussian vectors forms a cone. This cone is closed in the space L2 .Rn /, but is not closed in the space 0 .Rn / of tempered distributions. Proposition 3.1. The family '˛ of Gaussian functions ²

x2 '˛ .x/ D p exp 2˛ 2 ˛ 1

³

tends to the delta-function ı.x/ as ˛ ! 0. By the definition of a convergence of distributions (see Subsection 1.1), this means that Z 1 ² ³ 1 x2 lim p exp f .x/ dx D f .0/ (3.1) ˛!C0 2˛ 2 ˛ 1 for any f 2 .R/. Proof. The reason for this phenomenon is the following. First, Z '˛ .x/ dx D 1

(3.2)

R

for all ˛. Secondly '˛ is concentrated near zero for small ˛, see Figure 1.1. We present the following rough calculation: Z 1 Z " '˛ .x/f .x/ dx '˛ .x/f .x/ dx 1 " Z " Z 1 f .0/ '˛ .x/ dx f .0/ '˛ .x/ dx D f .0/: "

Figure 1.1. The graph of the function

1

p1 2 ˛

exp.x 2 =2˛/ for small ˛.

Let us repeat the same arguments more formally. Fix small " > 0, Z Z '˛ .x/ f .x/ dx D C jxj 0. A Gaussian operator is an integral operator of the form t ³ ² Z A B 1 x x y BŒS f .x/ D exp f .y/ dy: t t C y B 2 Rn

(3.5)

Here f is a function on Rn and BŒSf is a function on Rm (the case n D m is particularly interesting). 3.4 Gaussian operators. Definition. More generally, let .x; y/ be a Gaussian distribution on Rm Rn . We define a Gaussian operator as an integral operator of the form Z .x; y/f .y/: Bf .x/ D y2Rn

22


By Subsection 1.2, it follows that B is a well-defined operator B W .Rn / ! 0 .Rm /: A detailed discussion of L2 -boundedness of Gaussian operators is given below. 3.5 Example: the identity operator. The identity operator E W f 7! f can be represented in the form Z Ef .x/ D ı.x y/f .y/: (3.6) y2Rm

By definition, ı.x y/ is a Gaussian distribution on Rn Rn . 3.6 Example: the Fourier transform. The Fourier transform Z F f .x/ D f .y/e ixy dy R

is a Gaussian operator. We write Qf .x/ D xf .x/; Recall that

F 1 P F D Q;

Pf .x/ D i

d f .x/: dx

F 1 QF D P:

(3.7) (3.8)

Below we extend these formulas to arbitrary Gaussian operators. 3.7 Example. The heat equation. The Poisson formula. Let t 2 C satisfy Re t > 0. Consider the operators Z ² ³ 1 .x y/2 V .t/f .x/ WD p exp f .y/ dy: (3.9) 2t 2 t R For any f 2 0 .R/, we write: ‰.x; t / WD V .t /f .x/: Theorem 3.4. a) V .t/V .s/ D V .s C t /. b) For Re t > 0, we have 1 @2 @ ‰.x; t /: ‰.x; t / D @t 2 @x 2 c) For each f 2 0 .R/, we have

(3.10)

‰.x; 0/ WD lim V .t/f .x/ D f .x/; t!C0

i.e., V .0/ is the identity operator. d) The operators V .i s/, where s 2 R, are unitary in L2 .R/. e) The map t 7! V .t/ is a weakly continuous map from the half-plane Re.t / > 0 to the space of bounded operators in L2 .R/.

23

1.3. Gaussian operators

This theorem is nothing but the Poisson formula for a solution ‰.x; t / of the heat equation (3.10) with a given initial data f .x/. We can also express V .t / as follows: ²

³

t @2 V .t/ D exp : 2 @x 2 The proof is trivial. The statement a) can be verified by a straightforward calculation. The statement b) is equivalent to the identity

²

³

.x y/2 @ 1 1 @2 D 0: exp p @t 2 @x 2 2t 2 t

For a continuous f , the statement c) is (3.1). We leave the general case f 2 0 .R/ as an exercise. To prove d), we write out Z 1 ² ³ ² ³ ² ³ 1 x2 y2 1=2 V .is/f .x/ D exp .2 i s/ exp dy: xy f .y/ exp 2i s is 2i s 1 Therefore, the operator V .i s/ is the product of the following three unitary operators: 2

y – The operator of multiplication by the function '.y/ WD expf 2is g. This function satisfies j'.y/j D 1; therefore the operator is unitary.

– The Fourier transform. To be precise, we have the factor expf si xyg instead of the usual factor expfixyg in the formula for the Fourier transform. This operator became unitary after multiplication by .i s/1=2 . 2

y – The operator of multiplication by the same function expf 2is g.

Thus V .is/ is unitary. We leave e) as an exercise in functional analysis (one of the possible proofs: in the notation of Theorem 1.2, we set ˛ .x/ D exp.x 2 =2 C i ˛x/). Problem 3.2. V .i s/ D W previous section.

1 0

, where W./ is the Weil representation defined in the

s 1

3.8 Example. The harmonic oscillator. The Mehler formula. Let Re t 6 0. We define the operators " cosh t # 1 sinh t 1 sinh t B U.t / D p : (3.11) 1 t cosh 2 sinh t sinh t sinh t For any f 2 0 .R/ we write ‰.x; t / WD U.t /f .x/:

24


Theorem 3.5. a) U.t /U.s/ D U.s C t /. b) For Re t > 0,

@2 1 @ 2 C x 2 ‰.x; t /: ‰.x; t / D @t 2 @x

(3.12)

c) For each f 2 0 .R/, ‰.x; 0/ WD lim U.t /f .x/ D f .x/; t!C0

i.e., U.0/ is the identity operator. Also, U.2 ik/ D .1/k . d) U. i/f .x/ WD lim U. i C "/f .x/ D e i=2 f .x/. "!C0

e) U. i=2/ WD lim U. i=2 C "/ D e i=4 F ; "!C0

where F is the Fourier transform. f) The operators U.i s/, where s 2 R, are unitary in L2 .R/. g) The map t 7! U.t / is a weakly continuous map from the half-plane Re.t / > 0 to the space of bounded operators in L2 .R/. Proof. We repeat the arguments from the proof of the previous theorem.

Remark. The partial differential equation (3.12) is the well-known harmonic oscillator equation from quantum mechanics. Theorem 3.5 is the famous Mehler formula (Mehler, [133], 1866, Myller-Lebedeff, [137], 1907) for the solution of the oscillator equation. We can also say that we have evaluated the exponential ²

³

d2 t C x 2 D U.t /: 2 dx 2 I hope that the considerations given below make Theorem 3.5 obvious. exp

Proposition 3.6. The following identities hold:

i@ xU.t /f D U.t / cosh t y sinh t f; @y i@ i@ f: U.t /f D U.t / sinh t y C cosh t @x @y The proof is a direct verification. These formulas extend the similar identities (3.8) for the Fourier transform.

(3.13) (3.14)

Corollary 3.7. Let W./ denote the Weil representation as above. For all ' 2 R, the following identity holds: cos ' sin ' W D U.i'/: sin ' cos '


25

3.9 Example. Multiplication operator. Let Re t > 0. The operator A.t /f .x/ D expftx 2 =2gf .x/

(3.15)

is a Gaussian operator whose kernel is the Gaussian distribution expftx 2 =2gı.x y/: We have encountered operators A.i t / in the previous section. 3.10 Example. Linear change of variables. Let t 2 R. Consider the operator B.t /f .x/ D f .e t x/: This operator is a Gaussian operator defined by the kernel ı.x e t y/. Also, the operator B.t/ can be presented as ²

³

d B.t / D exp tx : dx More generally, let g W Rn ! Rn be an invertible linear transformation. Let us denote by xg the product of a row-vector x 2 Rn and the matrix g. Then the operator C.g/f .x/ D f .xg/ is Gaussian; its kernel is ı.xg y/. Such operators also appeared in the previous section. 3.11 Example. Projection onto Gaussian vectors. The operator Z 1 1 x 2 =2 2 1 0 f .y/e y =2 dy f .x/ D p e p B 0 1 2 2 R is the orthogonal projection onto the function e x

2 =2

.

More generally, Z 2 2 0 B f .x/ D e x =2 f .y/e y =2 dy 0 R is a rank 1 operator. Remark. In the notation of Subsection 3.8, p 1 0 2 sinh t U.t /: B D lim 0 1 t!C1

26


3.12 Example. Gaussian vectors and covectors. Consider the space L2 .R0 / D C. By definition, any Gaussian operator L2 .R0 / ! L2 .Rn / has the form ²

³

1 s 7! s exp xKx t ; 2

s 2 C:

(3.16)

The Gaussian operators L2 .Rn / ! L2 .R0 / are linear functionals of the form Z o n 1 (3.17) f .x/ exp xKx t dx: f 7! 2 Rn 3.13 Example. Tensor products of Gaussian operators. Consider two Gaussian operators K1 L1 W L2 .Rn1 / ! L2 .Rm1 /; A1 D B Lt1 M1 K2 L2 A2 D B W L2 .Rn2 / ! L2 .Rm2 /: Lt2 M2 Their tensor product (see 1.6) A1 ˝ A2 W L2 .Rn1 Cn2 / ! L2 .Rm1 Cm2 / is a Gaussian operator 2

K1 60 A1 ˝ A2 D B 6 4 Lt1 0

0 K2 0 Lt2

L1 0 M1 0

3 0 L2 7 7: 0 5 M2

1.4 The product of Gaussian operators Here we derive formulas (4.1)–(4.2) for the product of Gaussian operators. It seems rather complicated, but it is clarified in § 1.8. B 4.1 The formula for products. Let a symmetric .mCn/.mCn/ matrix S D BAt C be the same as above (3.4). Proposition 4.1. Let Re S > 0. Then a) BŒS W 0 .Rn / ! .Rm / is a bounded operator . b) BŒS W L2 .Rn / ! L2 .Rm / is a Hilbert–Schmidt operator. Proof. The first statement is a special case of the kernel theorem, see Subsection 1.2; the second statement is obvious, see Subsection 1.2.

27

1.4. The product of Gaussian operators

Theorem 4.2. Consider Gaussian operators K L W 0 .Rm / ! .Rk /; BŒS1 D B Lt M P Q BŒS2 D B W 0 .Rn / ! .Rm / Qt R such that Re S1 > 0; Then BŒS1 BŒS2 D where

Re S2 > 0:

.2/m=2 BŒS1 S2 ; p det M C P

S1 S2 D

K C L.M C P /1 Lt Qt .M C P /1 Lt

L.M C P /1 Q : R C Qt .M C P /1 Q

(4.1)

(4.2)

The proof is a straightforward calculation. We must evaluate the convolution of kernels of integral operators (see [178]), i.e., the integral t ³ t ³ Z ² ² K L P Q 1 1 x y x y y z exp exp dy t t t M y R zt L Q m 2 2 R ² ³Z ² ³ 1 1 1 exp y.M C P /y t y.Lt x t C Qz t / dy: D exp xKx t zRz t 2 2 2 Applying the Gauss integral (1.15), we obtain ²

³

1 1 exp xKx t zRz t .2/m=2 det.M C P /1=2 2 ² 2 ³ 1 t 1 t t t exp .xL C zQ /.M C P / .L x C Qz / 2 and this is the required statement.

Observation 4.3. The -product of matrices given by formula (4.2) is associative. Indeed, the product of linear operators is an associative operation. 4.2 Gaussian operators and Gaussian vectors. L Corollary 4.4. Let S D LKt M be an .m C n/ .m C n/ symmetric matrix and kS k < 1. Let bŒP 2 L2 .Rn / be a Gaussian vector. Then

K L bŒP D b K C L.M C P /1 Lt : (4.3) B t L M

28


Proof. We can evaluate a Gaussian integral again, but (see Subsection 3.12) a Gaussian vector represents an operator L2 .R0 / ! L2 .Rn / and we apply the previous theorem. 4.3 Categories. In fact, we obtain a category whose objects are the spaces L2 .Rn / and morphisms are Gaussian operators. Recall the definition of abstract categories. To define a category K, we need the following data: – a set11 Ob.K/ of objects of K; – for any two objects V , W 2 Ob.K/, a set Mor K .V; W / is defined; its elements are called morphisms from V to W ; – for any pair P 2 Mor K .V; W / and Q 2 Mor K .W; Y /, their product QP 2 Mor K .V; Y / is defined; the product is associative, i.e., for any V , W , Y , Z 2 Ob.K/ and any P 2 Mor K .V; W /, Q 2 Mor K .W; Y /, R 2 Mor K .Y; Z/; we have R.QP / D .RQ/P: Example 1. The objects are finite-dimensional linear spaces Cn , morphisms are linear operators. In a similar fashion, we have a category of groups and their homomorphisms; a category of smooth manifolds and smooth maps, etc., etc. Example 2. The objects are the spaces L2 .Rn / and the morphisms L2 .Rn / ! L2 .Rm / are Gaussian operators BŒS with Re S > 0. Example 3. The objects are nonnegativeintegers n!m n D 0, 1, 2,…, morphisms K L t are .m C n/ .m C n/-matrices S D Lt M such that S D S and Re S > 0. The product of two morphisms is given by formula (4.2). This -multiplication is associative. This is obvious, since the product of linear operators is associative. Problem 4.1. Show the associativity of the -product directly from (4.2).

We recall several other definitions. An endomorphism of an object V is an element of the semigroup Mor K .V; V /. We denote this semigroup by EndK .V /. A unit 1V 2 EndK .V / is an element satisfying the conditions: 11 For amateurs of mathematical logic: not a ‘set’, but a ‘class’. But our categories are as explicit as groups or fields.

29

1.5. A little linear symplectic geometry

– for any object W and any morphism Q 2 Mor K .V; W /, we have Q 1V D Q; – for any object Y and any morphism P 2 Mor K .Y; V /, we have 1V P D P . An element R 2 EndK .V / is an invertible element or an automorphism, if one can find an element S D R1 such that RS D SR D 1V . Automorphisms of V form a group AutK .V /. Below (Section 1.8) we show that the group of invertible Gaussian operators in L2 .Rn / is the extended real symplectic group Sp.2n; R/.

1.5 A little linear symplectic geometry In the rest of this chapter, we explain the -multiplication (4.2). We need several simple notions and constructions of linear algebra. The necessary background is contained in Chapters 2 and 3 (written for the reader who does not have sufficient experience in linear algebra). In this section, we discuss structures that arise when we complexify a space endowed with a skew-symmetric bilinear form. For a detailed and self-closed discussion see § 3.2. 5.1 Standard symplectic spaces. Fix n D 0, 1, 2, …. Denote by V2n .R/ the space12 V2n .R/ WD VnC .Rn / ˚ Vn .Rn / D Rn ˚ Rn equipped with the standard skew symmetric bilinear form C

C

fv ˚ v ; w ˚ w g WD

n X j D1

vjC wj

n X j D1

vj wjC

DW v

C

v

0 1 1 0

.w C /t : .w /t

(5.1) Denote by V2n the complexification13 of the space V2n .R/. We identify it with Cn ˚ Cn . Write VnC WD Cn ˚ 0; Vn D 0 ˚ Cn : Next, we are going to extend the form f; g to the space V2n . There are two ways to do this. 1B : We can extend f; g as a bilinear form. Let v D y C iz, v 0 D y 0 C iz 0 , where y, y 0 , z, z 0 2 V2n .R/. We set fv; v 0 g WD fy C iz; y 0 C iz 0 g D fy; y 0 g fz; z 0 g C i fy; z 0 g C i fz; y 0 g:

(5.2)

12 We permit the 0-dimensional space. Its geometry is not too interesting but it is an important object of the category Sp defined below. 13 For a definition of complexification, see Subsection 3.2.1.

30


This form is given by the same formula (5.1). It is more convenient to change the notation and introduce the form C t 0 i C .w / 0 0 v : (5.3) ƒ.v; v / WD ifv; v g D v .w /t i 0 2B : We can extend f; g as a sesquilinear form (see § 2.1) hv; v 0 i D hy C iz; y 0 C iz 0 i D fy; y 0 g C fz; z 0 g i fy; z 0 g C i fz; y 0 g:

(5.4)

This sesquilinear form satisfies the condition hv; wi D hw; vi; i.e., it is anti-Hermitian. It is more pleasant to change the notation and to define the Hermitian form C 0 i .w / : M.v; w/ WD i hv; wi D v C v i 0 .w / Remark. The inertia indices of the form M are .n; n/. Indeed, Vn˙ are n-dimensional isotropic subspaces. 3B : In addition, our space V2n is equipped with the operator J.x C iy/ D x iy;

where x; y 2 V2n .R/;

(5.5)

of complex conjugation. Obviously, this operator is an involution, i.e., J 2 D 1:

(5.6)

Also, J is antilinear, i.e., J.˛v/ D ˛J N v;

J.v C w/ D J v C J w:

The real subspace V2n .R/ consists of fixed points of the involution J , i.e., of vectors satisfying J v D v. The operator J is compatible with our forms in the following sense: ƒ.J v; J w/ D ƒ.v; w/;

M.J v; J w/ D M.v; w/:

(5.7)

ƒ.v; w/ D M.v; J w/:

(5.8)

Also, M.v; w/ D ƒ.v; J w/;

5.2 A formalization. Let Y be a 2n-dimensional linear space endowed with a nondegenerate skew-symmetric bilinear form ƒ, a nondegenerate Hermitian form M , and an antilinear involution J . Let J be compatible with these forms (i.e., (5.7) holds) and let M.v; w/ D ƒ.v; J w/.

1.5. A little linear symplectic geometry

31

Proposition 5.1. Y ' V2n . Problem 5.1. Let I be a linear operator in a real finite-dimensional linear space V such that I 2 D 1V . Let H˙ be the eigenspaces of I corresponding to the eigenvalues ˙1. Then V D HC ˚ H . Proof of the proposition. Consider the real subspace Y.R/ Y consisting of fixed points of the involution J . For any y 2 Y.R/, we have J.iy/ D i.Jy/ D iy: By the previous problem, Y D Y.R/˚i Y.R/ and we can regard Y as a complexification of Y.R/. By (5.7), it follows that the form ƒ is purely imaginary on Y.R/. By (5.8), ƒ D M on Y.R/. We define the R-valued form on Y.R/ by setting f; g D 1i ƒ. We omit a verification of the remaining details. Our complex linear space is equipped with three additional structures, i.e., a skewsymmetric form, a Hermitian form, and an antilinear involution. Actually, each pair of these structures determines the third structure canonically. Let us discuss this phenomenon. a) Given a space with a nondegenerate skew-symmetric bilinear form ƒ and an antilinear involution J preserving ƒ (as in 5.7) we introduce a Hermitian form M by (5.8) and obtain a standard symplectic space. b) Given a nondegenerate Hermitian form M and an antilinear involution J compatible with M , we again can introduce a form ƒ by (5.8). c) Finally, let us be given a nondegenerate Hermitian form M and a nondegenerate skew-symmetric form ƒ. We can reconstruct an antilinear operator K from the condition ƒ.v; w/ D M.v; Kw/. Incidentally, K 2 ¤ 1. Bur the pair of forms ƒ, M is not in general position; this implies some strange properties of these forms, see the next subsection. 5.3 The geometry of the pair of forms. Remarks. Here we collect several technical lemmas, which are used in several proofs in Section 1.9. Given two forms ƒ and M on V2n , we have two different orthocomplements14 . We denote them by P ?ƒ ; P ?M : Lemma 5.2. For an arbitrary subspace R, we have J.R?ƒ / D R?M ;

J.R?M / D R?ƒ :

Proof. By (5.7) it follows that a vector w is M -orthogonal to P if and only if J w is ƒ-orthogonal to P . 14

See § 2.1.

32


Lemma 5.3. Let P be a ƒ-Lagrangian15 subspace. Then a) P ?M is also ƒ-Lagrangian; b) P ?M D JP: Proof. By definition, P is the ƒ-orthogonal complement of itself. Hence JP is the M -orthogonal complement of P . Also, for any v, v 0 2 P , we have 0 D ƒ.v; v 0 / D ƒ.J v; J v 0 /

and hence JP is ƒ-Lagrangian.

Corollary 5.4. Let Q be a ƒ-isotropic16 subspace. Then JQ is ƒ-isotropic and M orthogonal to Q. Proof. We embed Q into a ƒ-Lagrangian subspace P and apply the previous lemma. Lemma 5.5. Let R be an M -positive subspace. Then JR is M -negative. Proof. Indeed, M.J v; J v/ D M.v; v/.

Lemma 5.6. Let R be a J -invariant subspace. Then R?ƒ D R?M . Proof. Let w 2 R, v 2 R?ƒ . Then 0 D ƒ.v; J w/ D M.v; w/.

5.4 An illustration. Apparently, it may be difficult to understand this list of lemmas in the previous subsection. To convince the reader, we propose the following problem: 0 Problem 5.2. a) Consider the plane R2 , the operator Ja WD 10 1 , and the two symmetric bilinear forms: ƒa .x; y/ D x1 y1 x2 y2 ;

Ma .x; y/ D x1 y1 C x2 y2 :

Formulate and prove analogs of the above lemmas for the forms ƒa , Ma . 0 1 b) Do the same for Jb WD 1 0 and the bilinear forms: ƒb .x; y/ D x1 y2 x2 y1 ;

Mb .x; y/ D x1 y1 C x2 y2 :

Remark. (This is important but it is not used in what follows). Recall that a semi-involution in a complex (real, quaternionic) linear space is a linear or anti-linear map J such that J 2 is a scalar operator. The effects of Subsections 5.3 and 5.4 are produced by a (symmetric, skew-symmetric, Hermitian, anti-Hermitian) form ƒ. ; / and a semi-involution compatible with ƒ. Namely, we define the second form M.v; w/ WD ƒ.v; J w/ and obtain a similar picture (for tables of such triplets, see Addendum D, Table 5, page 528. 15 16

See § 3.1. See § 2.1.

33

1.6. Gaussian operators and Lagrangian subspaces

R

P D P ?ƒ

P

R?ƒ J

J

R?M a)

P ?M D JP

b)

P ?M D JP

Figure 1.2. a) Reference Problem 5.2.a. The involution J is the symmetry with respect to the x-axis, the form M is the usual inner product in R2 . The line P is ƒ-isotropic, the line R is in general position. b) Reference Problem 5.2.b. The semi-involution J is a rotation by 90B ; M is the usual inner product. All lines through the origin are ƒ-isotropic.

1.6 Gaussian operators and Lagrangian subspaces 6.1 Symplectic spaces and differential operators. For any v D v C ˚ v 2 V2n , we define the differential operator on Rn by the formula a.v/f O .x/ WD

X

vj xj C i

X

vjC

@ f .x/: @xj

(6.1)

Commutators of the operators a.v/ O satisfy the following identities: Œa.v/; O a.w/ O D ƒ.v; w/; D M.v; w/; Œa.v/; O a.w/ O D a.J O v/: a.v/ O

(6.2) (6.3) (6.4)

6.2 The annihilators of Gaussian vectors. Let bŒT 2 .Rn / be a Gaussian vector. Consider the set P D P ŒT of all v 2 V2n such that a.v/bŒT O D 0: Theorem 6.1. a) The subspace P ŒT V2n has the following properties: 1. dim P ŒT D n; 2. P ŒT is a Lagrangian subspace with respect to the symplectic form ƒ; 3. the Hermitian form M is negative definite on P ŒT . b) If T ¤ T 0 , then P ŒT ¤ P ŒT 0 .

(6.5)

34


c) If 2 D 0 .Rn / is a non-zero distribution satisfying a.v/ O D 0 for any v 2 P ŒT ; then D bŒT for some 2 C . d) Each subspace P V2n with properties 1–3 is of the form P ŒT for some Gaussian vector bŒT . Thus we obtain a one-to-one correspondence between the set of Gaussian vectors (defined to within a scalar factor) and a certain domain in the complex Lagrangian Grassmannian. Proof. a) We have n @ i exp @xj h X n X i D vj i v˛C t˛j xj exp

a.v/bŒT O .x/ D

hX

j

vj xj C i

X

vjC

˛

o 1X tkl xk xl 2 k;l o 1 xT x t : 2

Therefore, a vector v D v C ˚ v 2 V2n is contained in P ŒT if and only if v D ivC T:

(6.6)

In particular, dim P ŒT D n. Furthermore, the condition T D T t means that the subspace P ŒT is Lagrangian (see Theorem 3.1.4 below) and Re T > 0 means the P ŒT is M -negative (see Proposition 2.3.10 below). b) This is evident. For different vectors, the matrices T are different, therefore the conditions (6.6) are different. c) Represent the distribution in the form ˚

D exp 12 xT x t ; where 2 D 0 .Rn /. For any v 2 P ŒT , n 1X o hX X @ i exp 0 D a.v/ O D vj xj C i vjC tkl xk xl @xj 2 k;l n 1 o X h o n 1 X X i @ D exp xT x t i vjC C xj vj i v˛C t˛j exp xT x t : 2 @xj 2 ˛ j

The second summand on the right-hand side is zero because the factor in the square brackets is zero (as we have seen in the proof of a)). Hence the first summand is also 0. Therefore, @ D 0 for all j : @xj

1.6. Gaussian operators and Lagrangian subspaces

35

Thus D const. d) Since P is M -negative and V is M -isotropic, we have P \ V D 0. Since dim P D n, the subspace P is a graph of an operator iT W V C ! V . Since P is Lagrangian, the matrix T is symmetric (see Theorem 3.1.4). Since P is negative, we have Re T > 0, see Proposition 2.3.10. There is also another version of the same theorem: Theorem 6.2. a) Let be a non-zero Gaussian distribution on Rn . Denote by P ./ the set of all v 2 V2n such that a.v/ O D 0: The subspace P ./ possesses properties 1–2 of the previous theorem (i.e., P ./ is a Lagrangian subspace) and also satisfies the following condition: 30 . The Hermitian form M is negative semi-definite on P ./. b) If P ./ D P .0 /, then 0 D s , for some s 2 C . c) If a distribution satisfies a.v/ O D 0 for all v 2 P ./; then D const . d) Each subspace in V2n with the properties 1, 2, 30 has the form P ./. Again, we obtain the one-to-one correspondence between the set of Gaussian distributions (defined to within a scalar factor) and the set of Lagrangian semi-negative subspaces. Remark. This theorem implies Proposition 3.3 on the closure of the cone of Gaussian distributions. Proof. a), b) Without loss of generality, we can consider a Gaussian distribution on Rn of the form Z m o n 1 X hh; f ii D exp tkl xk xl f .x1 ; : : : ; xm ; : : : ; xn / dx1 : : : dxm : 2 Rm k;lD1

In this case, xmC1 D 0;

:::

xn D 0

(6.7)

and P ./ consists of vectors C C ; vmC1 ; : : : ; vnC / ˚ .v1 ; : : : ; vm ; 0; 0; : : : ; 0/ v C ˚ v D .v1C ; : : : ; vm

satisfying the condition vjC

Di

m X kD1

vk tkj

for all j 6 m:

(6.8)

36


Now the statement becomes more-or-less obvious. c) Equations (6.7) imply that has the form .x1 ; : : : ; xn / ı.xmC1 / : : : ı.xn /; where

is a distribution supported by Rm . We represent

as

m o n 1 X exp tk;l xk xl 2 k;lD1

and repeat the calculation given in the proof of Theorem 6.1.c. We leave d) as an exercise in linear algebra for beginners.

6.3 The annihilators of Gaussian operators. Let be a Gaussian distribution on RmCn . Consider the corresponding integral operator B./ W .Rn / ! 0 .Rm /. Consider the subspace L./ V2n ˚ V2m (6.9) consisting of vectors v ˚ w such that a.w/B./ O D B./a.v/: O

(6.10)

Example. To simplify the notation, let m D n D 1, and consider a nonsingular Gaussian distribution o n 1 p q x K.x; y/ D exp x y : (6.11) q r y 2 Our equations (6.10) have the form Z h iZ h @ i C @ K.x; y/f .y/ dy D K.x; y/ v C y C iv f .y/ dy: w x C iw @x R @y R Integrating by parts on the right-hand side and equating the kernels, we obtain h

w C x C iw

h @ i @ i K.x; y/ D v C y iv K.x; y/: @x @y

Substituting the explicit expression (6.11) for K.x; y/, we obtain C

w x iw .px C qy/ K.x; y/ D v C y C iv .qx C ry/ K.x; y/: Collecting the terms with x and y, we obtain

wC

vC D i w

v

p q

q : r

(6.12)

1.7. Linear relations. Emulation of basic definitions of matrix theory

37

Thus we observe that the problems (6.5) and (6.10) lead to similar systems of partial differential equations and similar systems (6.6), (6.12) of linear equations. They only differ in signs. Let us formulate our observation more precisely. Let 2 0 .Rm ˚ Rn / be a Gaussian distribution. Consider the subspace P ./ V2mC2n defined in the previous subsection. We denote its vectors by C C ˚ VnC / ˚ .Vm ˚ Vn / D VmCn ˚ VnCm : .v C ˚ w C / ˚ .v ˚ w / 2 .Vm

Consider the subspace L./ V2m ˚ V2n defined by (6.9)–(6.10). We denote its elements by C .w C ˚ w / ˚ .v C ˚ v / 2 .Vm ˚ Vm / ˚ .VnC ˚ Vn / D V2m ˚ V2n :

Theorem 6.3. .v C ˚ w C / ˚ .v ˚ w / 2 P ./ () .w C ˚ w / ˚ ..v C / ˚ v / 2 L./: Below we formulate this result in a more closed form. 6.4 A preliminary remark on products of two Gaussian operators. Let us return to the formula (4.1) for products of Gaussian operators. Proposition 6.4. Let v ˚ w 2 LŒS2 and w ˚ y 2 LŒS1 . Then v ˚ y 2 LŒS1 S2 . Proof. BŒS1 BŒS2 a.v/ O D BŒS1 a.w/BŒS O O 2 D a.y/BŒS 1 BŒS2 :

This statement claims that the product of Gaussian operators corresponds to the product of linear relations, see next two sections.

1.7 Linear relations. Emulation of basic definitions of matrix theory This is an important section completely independent of the previous material. 7.1 Linear relations. Let V , W be linear spaces. A linear relation P W V W is a linear subspace P V ˚ W . Remark. Let A W V ! W be a linear operator. Its graph, graph.A/ V ˚ W , consists of all vectors v ˚ Av. By definition, graph.A/ is a linear relation, and dim graph.A/ D dim V:

38


Remark. Not all linear relations P W V W are graphs of operators. There are two obstacles. 1) graph.A/ \ .0 ˚ W / D 0. 2) The projection of graph.A/ on V ˚ 0 is the whole space V . Neither 1) nor 2) hold for every linear relation P W V W . Remark. Denote by Z the set of all linear relations P W V W of dimension dim V . The set of all graphs of operators is an open dense subset in Z. Our next purpose is to extend several definitions from linear operators to linear relations. 7.2 The product of linear relations. Let P W V W and Q W W Y be linear relations. Informally, the product of linear relations is the product of multivalued maps. If P takes a vector v to a vector w and Q takes the vector w to a vector y, then QP takes v to y. More precisely, the product QP is a linear relation QP W V W consisting of all v ˚ y 2 V ˚ Y such that there exists w 2 W satisfying the condition v ˚ w 2 P;

w ˚ y 2 Q:

Example. Let that P be the graph of an operator A W V ! W and Q the graph of an operator B W W ! Y . Then QP is the graph of the operator BA. Clearly, the product of linear relations is an associative operation. Remark. Thus the product of linear relations is an operation extending the product of linear operators. Below we reduce the product of linear relations to matrix operations but a reduction is not obvious, see Subsection 9.4. 7.3 Some definitions. Let P W V W be a linear relation. 1B . The kernel ker P consists of all v 2 V such that v ˚ 0 2 P . In other words, ker P D P \ .V ˚ 0/: Remark. If P D graph.A/ is a graph, then ker P D ker A. 2B . The image im P W is the projection of P on 0 ˚ W . Remark. If P D graph A, then im P D im A 3B . The domain of definiteness dom P V of P is the projection of P on V ˚ 0. 4B . The indefinity indef P W of P is P \ .0 ˚ W /.


39

Remark. Informally, indef P is the image of 0 under P . Remark. Obviously, P is the graph of an operator V ! W if and only if indef P D 0 and dom P D V . If P is the graph of an operator W ! V , then ker P D 0 and im P D W . For an arbitrary pair of linear relations P W V W and Q W W Y , ker QP ker P I dom QP dom P I indef QP indef QI im QP im Q:

(7.1) (7.2)

5B . We also define the rank of P : rk P D dim P dim ker P dim indef P D dim dom P dim ker P D dim im P dim indef P: This definition extends the definition of the rank of a linear operator. Obviously, (see (7.1), (7.2), rk QP 6 min.rk Q; rk P /: 6B . The last definition of this series is a pseudoinverse linear relation P . It is the same subspace P V ˚ W considered as a linear relation W V . Remark. If A W V ! W is an invertible linear operator, then .graph A/ D graph A1 : Generally, PP ¤ 1. Obviously,

P Q D .QP / :

(7.3)

Remark. The usual definition of a linear operator in functional analysis can be formulated in the following form: a linear operator is a linear relation P such that indef P D 0 and dom P is dense. 7.4 Dimensions of products Theorem 7.1. Consider linear relations P W V W and Q W W Y . a) Assume indef P \ ker Q D 0; im P C dom Q D W:

(7.4) (7.5)

dim QP D dim Q C dim P dim W:

(7.6)

Then b) The product of linear relations is a continuous operation on the domain determined by the conditions (7.4), (7.5).

40


Problem 7.1. a) The group GL.V / GL.W / GL.Y / acts in a natural way on the space of pairs .P; Q/ of linear relations. Find canonical forms of such pairs under this action. b) Prove the theorem. We prefer another (more complicated) way in the next subsection. 7.5 Reformulation of the definition of the product of linear relations. Consider the space V ˚ W ˚ W ˚ Y and the following subspaces: – consisting of vectors of the form v ˚ w ˚ w ˚ y, – „ consisting of vectors of the form 0 ˚ w ˚ w ˚ 0. Consider the natural projection W ! V ˚ Y along „. Lemma 7.2. Let P W V ! W and Q W W ! Y be linear relations. Then QP V ˚Y is the image of .P ˚ Q/ \ under the projection . We leave a proof as an exercise for the reader. Proof of Theorem 7.1. The condition (7.5) implies C .P ˚ Q/ D V ˚ W ˚ W ˚ Y: For arbitrary two subspaces, L and M , we have dim.L C M / C dim.L \ M / D dim L C dim M: Applying this formula to L D and M D P ˚ Q, we obtain that dim \ .P ˚ Q/ equals (7.6). The condition (7.4) implies .P ˚ Q/ \ „ D 0, and hence the projection is injective on \ .P C Q/. This completes the proof of a) and also proves b). Remark. Our proof gives the following formula, which is valid for all P and Q:

dim QP D dim Q C dim P dim indef P \ ker Q dim im P C dom Q : (7.7) 7.6 The action of linear relations on Grassmannians. Let P W V W be a linear relation, T V a subspace. We define the subspace P T V as the set of all w 2 W such that there exists v 2 T satisfying v ˚ w 2 P . This operation is a special case of the product of linear relations. Indeed, we can regard T as a linear relation T W 0 V , then P T coincides with the product of P and T . In particular, formula (7.7) implies dim P T D dim P C dim T dim T \ ker P dim T C dom P : (7.8)


41

7.7 Isotropic linear relations and unitary operators. Let V be a Euclidean space endowed with an inner product H.; /. Consider the following indefinite Hermitian form on the space V ˚ V : HV˚V .v ˚ w; v 0 ˚ w 0 / D HV .v; v 0 / HV .w; w 0 /: Theorem 7.3. The following conditions are equivalent: – g W V ! V is a unitary operator; – graph.g/ is a maximal isotropic17 subspace in V ˚ V . Proof. Let g be a unitary operator. For any two points v ˚ gv and v 0 ˚ gv 0 in graph g, H .v ˚ gv; v 0 ˚ gv 0 / D HV .v; v 0 / HV .gv; gv 0 /:

(7.9)

Since g is unitary, the right-hand side is zero. Conversely, let Z V ˚ V be a maximal isotropic subspace. The form H is zero on Z and is negative definite on 0˚V . Hence Z \Œ0˚V D 0. Since dim Z D dim V , the subspace Z is the graph of an operator V ! V . Since Z is isotropic, the left-hand side of (7.9) is zero. Hence g is unitary. Problem 7.2. Consider a linear space V equipped with a skew-symmetric bilinear form H .; /. Repeating the same arguments as above, show that an operator g W V ! V is symplectic if and only if its graph is a Lagrangian subspace. 7.8 Isotropic linear relations and self-adjoint operators. Let V be a finite-dimensional complex Euclidean space, H.; / an inner product. Define the indefinite Hermitian form H Ÿ on V ˚ V by H Ÿ .v ˚ w; v 0 ˚ w 0 / D i H.v; w 0 / H.w; v 0 / (we write i to obtain a Hermitian form, otherwise this expression is anti-Hermitian). Theorem 7.4. a) For any operator A W V ! V , we have graph.A/? D graph.A /: b) The operator A is self-adjoint if and only if graph.A/ is a maximal isotropic subspace. Proof. Let A and B be operators, let v ˚ Av and w ˚ Bw be points of their graphs. Then H Ÿ .v ˚ Av; w ˚ Bw/ D i H.v; Bw/ H.Av; w/ : If A D B , then graph.A/ is orthogonal to graph.B/. 17

See Subsection 2.1.9.

42


Remark. However, this construction is not a one-to-one correspondence between selfadjoint operators and maximal isotropic subspaces P . It can occur that indef P ¤ 0 and in this case P is not the graph of any operator. 7.9 Digression. The Cayley transform. Recall that the Cayley transform18 is given by 1g 1 iA iA D () g D : 1Cg 1 C iA Theorem 7.5. A matrix g is unitary if and only if A is self-adjoint. Proof. Let A be self-adjoint. Then gg D

1 iA 1 C iA D 1: 1 C iA 1 iA

If g is unitary, then

11g 1 1g 1 1 g 1 g1 D C C AA D 1 i 1Cg i 1Cg i 1Cg gC1

D 0:

Previous considerations offer another explanation of this theorem. In Subsection 7.7 we have constructed a bijection between the unitary group U.n/ and the Grassmannian of maximal isotropic subspaces. In Subsection 7.8 we obtained a correspondence between the same Grassmannian and the space Hermn of self-adjoint operators. The Cayley transform U.n/ $ Hermn is the composed correspondence. 7.10 Contractive operators and linear relations Theorem 7.6. The following conditions for a linear operator g are equivalent: – kgk 6 1; – the form H is non-negative on graph.g/. Proof. First, let us verify ‘(’. We have H .v ˚ gv; v ˚ gv/ D kvk2 kgvk2 :

(7.10)

If kgk 6 1, then this expression is positive. Now, let the form H be positive on graph.g/. Since the form is negative on 0˚V , the intersection graph.g/ \ .0 ˚ H / is zero. Hence g is the graph of an operator. Since (7.10) is non-negative for all v, we have kvk 6 1. 7.11 Digression. The normal forms of linear relations. Let P W V V be a linear relation in a complex linear space V . We are going to formulate an analog of the Jordan normal form for 18

In § 2.10 we use another normalization of the Cayley transform.


43

a linear relation. In other words, we wish to classify linear relations P W V V with respect to the equivalence P g 1 P g; g 2 GL.V /: We say that a linear relation P is decomposable if there is a decomposition V D V1 ˚V2 and linear relations P1 W V1 V1 , P2 W V2 V2 such that P D P1 ˚P2 . Clearly, the classification of linear relations reduces to the classification of indecomposable relations. Theorem 7.7. There are the following five types of indecomposable linear relations: Let e1 ; : : : ; ek be a basis in a linear space V . 1 : P W V V is the graph of a Jordan block 0 1 B B0 Jn ./ D B B: : :: @ :: 0 0

::: :: : :: : :::

0

1

C 0C C :: C :A

with ¤ 0. In other words, we get a linear relation with a basis e1 ˚ .e1 C e2 /; e2 ˚ .e2 C e3 /; : : : ; en ˚ en : 2 : The graph of the Jordan block J.0/. Of course, this is nothing but a version of a). A basis of the linear relation is e1 ˚ e2 ; e2 ˚ e3 ; e3 ˚ e4 ; : : : ; en ˚ 0: 3 : The linear relation J.0/ pseudoinverse for the Jordan block J.0/. Its basis is e2 ˚ e1 ; e3 ˚ e2 ; : : : ; en ˚ en1 ; 0 ˚ en : 4 : The .n 1/-dimensional linear relation spanned by e1 ˚ e2 ; e2 ˚ e3 ; : : : en1 ˚ en : 5 . The .n C 1/-dimensional linear relation spanned by e1 ˚ 0; e2 ˚ e1 ; e3 ˚ e2 ; : : : en ˚ en1 ; 0 ˚ en : Problem 7.3. Prove the theorem in the following way. For a linear relation P V ˚ V consider two projection operators A W P ! V ˚ 0;

B W P ! 0 ˚ V:

We get a pair of operators P ! V . Next, apply the Kronecker theorem (see [60]) on pencils of linear operators. Problem 7.4. Fix k ¤ dim V . Show that the group GL.V / acts with an open dense orbit on the set of all linear relations V V of dimension D k.

44


e1

en

e1

en

e1

en

e1

en

e1

en

e1

en

e1

en

e1

en

a)

b)

c)

d)

Figure 1.3. Reference Subsection 7.11. The indecomposable linear relations P W V V . The lower fat points correspond to the basis elements ek ˚ 0 of the first summand, the upper fat points correspond to the basis elements 0 ˚ ek of the second summand. If ek ˚ el 2 P , then we connect ek and el with a segment. If ek ˚ 0 2 P or 0 ˚ el 2 P , then we draw a circle. a) The Jordan block Jn .0/ (type 2*). b) The pseudo-inverse Jn .0/ of the Jordan block (type 3*). c) The linear relation of dimension .n 1/, type 4*. d) The linear relation of dimension .n C 1/, type 5*.

1.8 The symplectic category Here we describe the algebraic structure formed by Gaussian operators. Recall that the standard symplectic spaces V2n are defined above in Subsection 5.1. Recall that they are equipped with two forms, namely a symplectic form ƒ and a Hermitian form M . Also, they are equipped with an operator J of complex conjugation. Recall that the spaces V2n parameterize certain families of partial differential operators in .Rn /, see Subsection 6.1. 8.1 Contractive Lagrangian relations. Consider two standard linear spaces V2n and V2m . Endow their direct sum V2n ˚ V2m with two forms, namely the skew-symmetric bilinear form ƒ given by ƒ .v ˚ w; v 0 ˚ w 0 / D ƒ.v; v 0 / ƒ.w; w 0 /

(8.1)

and the indefinite Hermitian form M .v ˚ w; v 0 ˚ w 0 / D M.v; v 0 / M.w; w 0 /:

(8.2)

In addition, we define the operator J of complex conjugation in V2n ˚ V2m , by setting JV2n ˚V2m D JV2n ˚ JV2m : We say that a linear relation P W V2n V2m is contractive Lagrangian if 1 ) P is ƒ -Lagrangian; in particular, dim P D m C n; 2 ) the form M is non-negative definite on P .

45

1.8. The symplectic category

Theorem 8.1. a) For any contractive Lagrangian linear relation P W V2n V2m , there is a unique up to a scalar factor non-zero linear operator B D Be.P / W .Rn / ! 0 .Rm / such that a.w/B O D B a.v/ O for all v ˚ w 2 P :

(8.3)

b) The operator Be.P / is a Gaussian operator. c) If P ¤ P 0 , then Be.P / ¤ Be.P 0 /. d) Every Gaussian operator has the form Be.P / for some contractive linear relation P . Thus we have a one-to-one correspondence between the set of Lagrangian linear relations and the set of Gaussian operators defined to within a scalar factor.

Proof. This is a paraphrasing of Theorems 6.2, 6.3.

Remark. We use the dual notation for the Gaussian operators, namely BŒS and Be.P /. In the first case, S is a matrix; in the second case, P is a linear relation. 8.2 The boundedness conditions. It is easy to produce examples of Gaussian operators that are not bounded as operators L2 ! L2 and ! . Problem 8.1. Examine for boundedness the operators with the following kernels: ı.x/ ı.y/;

ı.x/;

ı.y/;

exp.ix 2 C iy 2 /;

exp.x 2 iy 2 /

in L2 .R/ and in .R/. Nevertheless, the set of all unbounded Gaussian operators is small. They form a submanifold lying on the boundary of the ‘ball’ of all Gaussian operators. Theorem 8.2. Let P W V2n V2m be a contractive Lagrangian linear relation. a) The operator Be.P / is a bounded operator L2 .R2n / ! L2 .R2m / if and only if the subspace indef P V2m is purely negative with respect to the Hermitian form M and the subspace ker P V2n is purely positive. b) Under these conditions, the operator Be.P / is also bounded as an operator .R2n / ! .R2m / and as an operator 0 .R2n / ! 0 .R2m /. We present two proofs of this theorem. The first one (see Section 1.10) is based on certain classification results in linear algebra. For a more conceptual proof based on the fixed point principle, see § 5.3. In the next subsection, we produce a definition from Theorem 8.2. 8.3 The symplectic category Sp. The objects of the category Sp are the standard symplectic spaces V2n , where n D 0, 1, 2, ….

46


The morphisms V2n ! V2m are contractive Lagrangian linear relations P W V2n V2m satisfying the additional conditions 3 a) the subspace ker P V2n is M -positive, 3 b) the subspace indef P V2m is M -negative. Remark. Note that the positivity condition 2 implies the semi-positivity of ker P and the semi-negativity of indef P . Thus 3 is a minor refinement of 2 . Theorem 8.3. The category Sp is well defined, i.e., the product of two morphisms is a morphism. The proof is given in Subsection 9.1. Theorem 8.4. The group Aut Sp .V2n / of automorphisms of V2n is isomorphic to Sp.2n; R/. The proof is given in Subsection 9.2. 8.4 The formula for products Theorem 8.5. Let P W V2n V2m and Q W V2m ! V2k be morphisms of the category Sp. Then Be.Q/ Be.P / D c.Q; P / Be.QP /; (8.4) where c.Q; P / is a non-zero complex constant. The proof is given in the next section. Remark. Theorem 8.1 defines the operators Be.P / to within a scalar factor. Hence the presence of a constant factor in formula (8.4) is not surprising. We discuss this factor in § 3.8 of Chapter 3.

1.9 The symplectic category. Details In Sections 1.9 and 1.10 we prove the theorems formulated in the previous section. Note that the considerations of Section 1.9 are semi-trivial from the point of view of Chapter 2, in Chapter 5 we propose a more natural proof of the boundedness of Gaussian operators. 9.1 The symplectic category Sp is well defined. For morphisms P W V2n V2m and Q W V2m ! V2k , we must verify that QP satisfies the conditions 1 – 3 . A. The subspace QP is ƒ -isotropic. Let v ˚ w, v 0 ˚ w 0 2 P and w ˚ y, 0 w ˚ y 0 2 Q. Then P is ƒ -isotropic H) ƒ.v; v 0 / D ƒ.w; w 0 /; Q is ƒ -isotropic H) ƒ.w; w 0 / D ƒ.y; y 0 /:

1.9. The symplectic category. Details

47

Therefore, for any v ˚ y, v 0 ˚ y 0 2 QP , we get ƒ.v; v 0 / D ƒ.y; y 0 /. Hence QP is ƒ -isotropic. The subspace QP is M -non-negative. Let v, w, y be as above. Then P is M -non-negative H) M.v; v/ > M.w; w/; Q is M -non-negative H) M.w; w/ > M.y; y/: Therefore, M.v; v/ > M.y; y/ and hence QP is non-negative. For the proof of Theorem 8.3 we need the following two lemmas: Lemma 9.1. Let W be a linear space endowed with a nondegenerate skew-symmetric form f; g. Let P be a Lagrangian subspace, Y a subspace on which the form f; g is nondegenerate. Then the orthocomplement of P \Y in Y coincides with the projection of P on Y . Proof of the lemma. Denote this orthocomplement by R, R D Y \ .Y \ P /? : Let us write vectors in W as y ˚ z, where y 2 Y and z 2 Y ? . For any h 2 P \ Y , we have h ˚ 0 2 P . Let y ˚ z 2 P . Then 0 D fh ˚ 0; y ˚ zg D fh; yg; and therefore the vector y (which is the projection of y ˚ z on Y ) is contained in .P \ Y /? \ Y . Let us verify the inverse inclusion. Consider a non-zero vector in the orthocomplement of our projection in Y . For any y ˚ 2 P , fy ˚ z; ˚ 0g D fy; g D 0: Thus ˚ 0 is orthogonal to the Lagrangian subspace P , and hence it is contained in P . Therefore, 2 .P \ Y /? \ Y . Lemma 9.2. For a morphism P W V2n V2m of symplectic category, we have .ker P /?ƒ D dom P I

.indef P /?ƒ D im P:

This follows from the previous lemma, W D V2n ˚ V2m , Y D V2n , P D P , P \ Y D ker P , and the projection of P on V2n is dom P . Proof of Theorem 8.3. The subspace QP is ƒ -Lagrangian. Since QP is isotropic, it suffices to verify that dim QP D n C m. To apply Theorem 7.1 on the dimension of a product we must verify the conditions (7.4) and (7.5). First, ker Q \ indef P D 0,

48


because one subspace is M -positive and the other subspace is M -negative. By the lemma, .ker Q/?ƒ D dom Q; .indef P /?ƒ D im P: Hence, .dom Q C im P /?ƒ D .dom Q/? \ .im P /?ƒ D ker Q \ indef P D 0 and (7.5) is also valid. Problem 9.1. We used superfluous arguments. The product of any two Lagrangian relations is Lagrangian. Apply Lemma 9.2 and formula (7.7). The subspace indef QP is strictly negative. Let y 2 indef QP , consider w 2 W such that 0 ˚ w 2 P and w ˚ y 2 Q. If w ¤ 0, then, by 3 a (see p. 46), M.w; w/ < 0. By M -contractivity, it follows that M.y; y/ 6 M.w; w/ < 0. If w D 0, then, by 3 a, we have M.y; y/ < 0. In both cases, M.y; y/ < 0. 9.2 Groups of automorphisms are Sp.2 n; R/. The symplectic group Sp.2n; R/ acts on the real symplectic space V2n .R/ preserving the form f; g. Hence it acts (via the same matrices) on the complex space V2n preserving the bilinear form ƒ and the Hermitian form M . For any g 2 Sp.2n; R/, its graph, graph.g/ V2n ˚ V2n is ƒ -isotropic and M -isotropic (see Subsection 7.7). In particular, graph.g/ is a morphism of the category Sp19 . Now let us show that all automorphisms of the category Sp have this form. Proof of Theorem 8.4. Let R be an automorphism. Then RR1 D R1 R D 1. By (7.1), (7.2), it follows that ker R D 0 and indef R D 0. Since dim R D 2n D dim V2n , the subspace R is the graph of an invertible operator g W V2n ! V2n . Since R is ƒ -Lagrangian, the operator g preserves the bilinear form ƒ (see Subsection 7.7). Since R is M -positive, the operator g satisfies the condition M.gv; gv/ 6 M.v; v/: Since R is invertible in the category Sp, we have also M.g 1 w; g 1 w/ 6 M.w; w/: Assuming w D gv, we obtain M.gv; gv/ D M.v; v/. 19 Below we say that g itself is a morphism of the category Sp. It is not an inaccuracy of our language, since the official set-theoretical definition of a map in the Zermelo axiom system (see e.g., K. Kuratowski, A. Mostowski “Set Theory”) is given in terms of its graph.


49

Thus, g preserves both the forms ƒ and M . By (5.8), the operator g also commutes with the complex conjugation J . Indeed, for all v, v 2 V2n , we have M.v; J v 0 / D ƒ.v; v 0 / D ƒ.gv; gv 0 / D M.gv; Jgv 0 / D M.v; g 1 Jgv/; and hence J D g 1 Jg. Hence, the matrix of g is real. Therefore, g 2 Sp.2n; R/. Theorem 9.3. a) For any g 2 Sp.2n; R/, the corresponding operator Be.P / in L2 .Rn / is unitary up to a scalar factor. b) This operator is bounded as an operator .Rn / ! .Rn /. Proof. The statement a) is a rephrasing of Theorem 2.5. To prove b), we note that the operators (2.15)–(2.17) are bounded in . 9.3 Remark. Operators contained in End.V2n / Proposition 9.4. An operator g W V2n ! V2n is a morphism of the category Sp if and only if M.gv; gv/ 6 M.v; v/ and ƒ.gv; gw/ D ƒ.v; w/: (9.1) The statement is obvious. The second condition (9.1) means that g is an element of the complex symplectic group Sp.2n; C/. For a discussion of the first condition, see § 2.7. 9.4 A preliminary discussion of the formula for products of Gaussian operators Quasi-proof of Theorem 8.5. This statement is obvious modulo the boundedness Theorem 8.2 (which is not proved yet). Indeed, for any v ˚ w 2 P and w ˚ y 2 Q, Be.Q/ Be.P /a.v/ O D Be.Q/a.w/ O Be.P / D a.y/ O Be.Q/ Be.P /:

(9.2)

Both sides make sense since all the present operators are bounded in the Schwartz space . But Be.QP /a.v/ O D a.y/ O Be.QP /: Bearing in mind Theorem 8.1.b, we obtain Be.QP / D c Be.Q/ Be.P /.

Thus Theorem 8.5 is proved modulo boundedness Theorem 8.2. Nevertheless the following special cases of Theorem 8.5 are actually proved: Lemma 9.5. The formula (8.4) for products holds if at least one of the following conditions is satisfied: – the operator Be.P / W .R2n / ! .R2m / is bounded; – the operator Be.Q/ W 0 .R2m / ! 0 .R2k / is bounded.

50


9.5 Images of Gaussian vectors. Let R V2n be a non-positive Lagrangian subspace. Denote by be.R/ 2 L2 .Rn / the corresponding Gaussian distribution, i.e., be.R/ a nonzero distribution satisfying the partial differential equations a.v/ O be.R/ D 0

for all v 2 R:

If the subspace R is strictly negative, then beŒP is a Gaussian vector. Theorem 9.6. The image of a Gaussian vector with respect to a Gaussian operator is a non-zero Gaussian distribution. More precisely, we have Be.P / be.R/ D be.PR/;

for some 2 C :

(9.3)

Proof. Since R W 0 V2n is a contractive linear relation 0 V2n , we apply Lemma 9.5. It remains to show that ¤ 0. Lemma 9.7. Let be a Gaussian distribution on Rk , and be.R/ 2 L2 .Rk / a Gaussian vector. Then hh; be.R/ii ¤ 0: Proof of the lemma. By definition (3.3) of a Gaussian distribution, the expression hh; be.R/ii is a Gaussian integral of the form Z ² ³ 1 .2/dim L=2 exp xAx t dx D ¤ 0: 2 det A1=2 L Now we return to Theorem 8.5. Let K.x; y/ be the kernel of Be.P /. Let be.R/ D expf 12 yBy t g, let expf 12 xC x t g 2 .Rm / be a Gaussian vector in the target space. Then ˝˝ ˚

˚

˛˛ Be.P / exp 12 yBy t ; exp 12 xC x t ˝˝ ˚

˚

˛˛ D K.x; y/; exp 12 yBy t exp 12 xC x t ¤ 0: Corollary 9.8. The product of two Gaussian operators is non-zero. Proof. Indeed, a Gaussian operator sends a non-zero Gaussian vector to a non-zero vector. 9.6 Explicit expression for Be.P/. In fact, an explicit formula was obtained above, see Section 1.6 and Theorem 6.3 Here we present the same result more precisely. Let P W V2n V2m be a morphism of the category Sp. Let P be the graph of an operator K L C W Vm ˚ Vn ! Vm ˚ VnC : (9.4) S WD Lt M


51

Remark. Thus, we get a map that sends a morphism in general position to a symmetric matrix. Actually, we get a coordinate system on the set Mor Sp .V2n ; V2m /. Below, we call S the Potapov transform of P . Theorem 9.9. Under this condition, Z ² 1 x Be.P /f .x/ D exp 2 Rn

K y Lt

L M

t ³ x f .y/ dy: yt

Proof. This is a rephrasing of Theorems 6.2 and 6.3.

(9.5)

Remark. Theorems 6.2 and 6.3 allow us to write explicit formulas for Gaussian kernels in the remaining cases. In Chapter 5, we achieve the same purpose using a more convenient language. 9.7 An addendum to the construction. An involution in the category Sp and adjoint operators. For a morphism P W V2n V2m , consider the complex conjugate subspace Px WD JP D P ?M (we apply Lemma 5.3). Write

P WD Px ;

where P is the pseudo-inverse to P (see Subsection 7.3). Proposition 9.10. a) P is a morphism V2m V2n . b) .PQ/ D Q P . c) P D P . Proof. b), c) are obvious. To prove a), we refer to Subsection 5.3. By Lemma 5.3, it follows that P is Lagrangian. By Lemma 5.5, Px D P ?M is non-positive in V2n ˚ V2m . The transposition of summands changes a sign of M . Hence P is non-negative. Lemma 9.11. ker P D .im P /?M and indef P D .dom P /?M . Proposition 9.12. For Be.P / W L2 .Rn / ! L2 .Rm /, we have Be.P / D Be.P /:

Proof. We take the adjoints of both sides of the defining equation (8.3) for Be.P /. We obtain a. O v/B N D B a. O w/: x This is the defining equation for Be.P /.

52


9.8 An addendum to the construction. The tensor products. Now let us use coordinate-free terms as in Subsection 5.2. Let Y, Z 2 Ob.Sp/. We define their direct sum Y ˚ Z as the linear space Y ˚ Z endowed with two forms: ƒ˚ .v ˚ w; v 0 ˚ w 0 / D ƒ.v; v 0 / C ƒ.w; w 0 /; M ˚ .v ˚ w; v 0 ˚ w 0 / D M.v; v 0 / C M.w; w 0 / and by the antilinear operator J ˚ WD J ˚ J . We have the natural identifications ˙ V˛˙ ˚ Vˇ˙ ' V˛Cˇ :

V2˛ ˚ V2ˇ ' V2.˛Cˇ / ; In particular, for the morphisms P W V2˛ V2ˇ ;

R W V2 V2ı ;

we have a well-defined morphism P ˚ R W V2.˛C/ V2.ˇ Cı/ : Proposition 9.13. Be.P ˚ R/ D const Be.P / ˝ Be.R/: This is obvious, see also Subsection 3.13.

1.10 Proof of boundedness. Canonical forms of Gaussian operators In what follows, we derive the conditions of L2 -boundedness of Gaussian operators (see Theorem 8.2). For this purpose, we reduce the boundedness of Gaussian operators to canonical forms of morphisms of the category Sp. The last problem reduces to canonical forms of symplectic contractive operators. This way is simple but its exposition is long. A more conceptual proof is given in Chapter 5. 10.1 Canonical forms of Gaussian operators. The formulation of the problem. We wish to examine the conditions of boundedness of a Gaussian operator Be.P / as an operator L2 .Rm / ! L2 .Rn /. Let g1 2 Sp.2m; R/, g2 2 Sp.2n; R/. Then the operators We.g1 /, We.g2 / are unitary up to scalar factors. Therefore, Be.P / and We.g2 / Be.P / We.g1 / are bounded or unbounded simultaneously. Therefore, it is reasonable – to find canonical forms of Gaussian operators under the equivalence Be.P / We.g2 /B Be.P / We.g1 /, where 2 C . The solution is given in the following theorem:

1.10. Proof of boundedness. Canonical forms of Gaussian operators

53

Theorem 10.1. Each Gaussian operator that is bounded in L2 -sense is a tensor product of the operators of the following five types. 1. An operator L2 .R0 / ! L2 .R/ of the type “Gaussian vector”, see (3.16). 2. An operator L2 .R/ ! L2 .R0 / of the type “Gaussian linear functional”; see (3.17). 3. The identity operator L2 .R/ ! L2 .R/. 4. The Mehler operators (3.11). 5. The operator L2 .R/ ! L2 .R/ of multiplication by a Gaussian function expfx 2 =2g. Problem 10.1. Describe canonical forms of bounded Gaussian operators L2 .R0 / ! L2 .R/. Lemma 10.2. If an operator Be.P / is bounded in L2 , then the linear relation P is a morphism of the category Sp. Proof. We must verify only the conditions 3.a and 3.b in the definition of Sp (see Subsection 8.3). Consider a contractive Lagrangian linear relation P W V2n V2m and the corresponding Gaussian operator Be.P / W .Rn / ! 0 .Rm /. Consider an arbitrary Gaussian vector be.R/ 2 L2 .Rn /. Then Be.P / be.R/ 2 2 L .Rm / is a Gaussian vector (see Theorem 9.6). By (8.3), for each w 2 indef.P /, we have a.w/ O Be.P / D 0 H) a.w/ O Be.P / be.R/ D 0: By Theorem 6.1, it follows that M.w; w/ > 0. Thus, indef P is negative. Consider the adjoint operator (see Proposition 9.12) Be.P / D Be.P / W L2 .Rm / ! L2 .Rn /:

But ker P D J indef P . The subspace indef P is negative, so by Lemma 5.5, ker P is positive. Therefore, we must describe canonical forms of morphisms P of the category Sp under the transformations P 7! gP h1 ;

where g 2 Sp.2m; R/, h 2 Sp.2n; R/:

We say that a morphism P W Y Z is decomposable if there are decompositions Y D Y1 ˚ Y2 , Z D Z1 ˚ Z2 and morphisms P1 W Y1 Z1 , P2 W Y2 Z2 such that P D P1 ˚ P2 .

54


Now we can formulate the problem of canonical forms: – describe (up to the equivalence) all indecomposable morphisms of the category Sp. 10.2 A classification of indecomposable morphisms Theorem 10.3. The five types of indecomposable morphisms listed below exhaust all the indecomposable morphisms of the category Sp. In particular, each morphism of the category Sp is a direct sum of morphisms listed below. Actually, all these operators were already discussed in § 1.3. 1 . A morphism V0 ! V2 . All such morphisms are equivalent. The corresponding operator L2 .R0 / D R ! L2 .R1 / is a “Gaussian vector”, see Subsection 3.12. 2 . A morphism V2 ! V0 . All such morphisms are equivalent. The Gaussian operator L2 .R1 / ! L2 .R0 / D R is a ‘Gaussian linear functional’, see Subsection 3.12. 3 . The identity operator V2 ! V2 . The Gaussian operator is the identity operator L2 .R1 / ! L2 .R1 /, see (3.6); 4 . The operator g W V2 ! V2 with the matrix cosh t sinh t gD ; sinh t cosh t where t > 0. The Gaussian operators are the Mehler operators U.t / given by (3.11). 5 . This type can be represented by linear transformations g W V2 ! V2 with matrices 1 t 1 0 or, equivalently, ; where t > 0: 0 1 t 1 All such morphisms are equivalent. The corresponding Gaussian operators are respectively the heat operators V .t/ given by the Poisson formula (3.9), or the multiplication operators A.t/ (see 3.15). 10.3 Reduction of Theorem 10.3 to a standard problem Proposition 10.4. Each morphism of the category Sp admits a decomposition into the direct sum of morphisms of the following three types: – morphisms X 0; – morphisms 0 Y; – invertible operators g W Z ! Z such that ƒ.gv; gv/ D ƒ.v; v/I

M.gv; gv/ 6 .v; v/:

(10.1)

1.11. Bibliographical remarks

55

Thus, our problem reduces to canonical forms of symplectic contractive operators, i.e., elements of the complex symplectic group Sp.2n; C/ contracting the form M , see § 3.5). All possible indecomposable symplectic contractive operators are operators 3 –5 from the previous subsection acting on the 2-dimensional space. Proof. Let P W Y Z be a morphism, and ker P ¤ 0 . The form M is positive on ker P . By (5.7), the form M is negative on J ker P . Hence ker P \ J ker P D 0. Let us show that Y1 WD ker P ˚ J ker P

is a standard symplectic space:

Indeed, the space ker P is ƒ-isotropic; therefore J ker P is M -orthogonal to ker P . By Corollary 5.4, the form M on ker P ˚ J ker P is nondegenerate. Write (see Lemma 5.6) Y1 WD ker P ˚ J ker P;

Y2 WD Y1?ƒ D Y1?M :

We define the linear relations P1 W Y1 0 and P2 W Y2 Z to be P1 WD ker P; P2 WD P \ Y2 ˚ Z/: Lemma 10.5. a) The subspace P2 is a morphism Y2 ! Z. b) P D P1 ˚ P2 . The lemma yields the proposition because we can split the indefinity in the same way. Proof of the lemma. Let Q be the M -complement of ker P in P . The subspace Q is ƒ- and M -orthogonal to ker P , hence Q is M - and ƒ- orthogonal to J ker P . Thus Q P2 . Since the form M is nondegenerate on ker P , we get P D ker P ˚ Q; this implies our statement. 10.4 The boundedness. Proof of the sufficiency. For all the indecomposable morphisms 1 5 the boundedness is clear. Direct sums of morphisms correspond to tensor products, see Proposition 9.13. Thus the boundedness in L2 and in became obvious.

1.11 Bibliographical remarks 11.1 The Weil representation. Apparently, the Weil representation must be attributed to Kurt O. Friedrichs (~ 1950). His book [55] does not contain the “Weil representation” of Sp.2n; R/, but clearly he thought that there is nothing to discuss here; he tried to extend the Weil representation

56


to the infinite-dimensional case. His book contains correct conjectures concerning this topic supported by ‘arguments at the physical level’. However, in the finite-dimensional case his argument (a reference to the Stone–von Neumann theorem, see Addendum B) is definitely sufficient. His conjectures for the infinite-dimensional case were proved by D. Shale [200] and Felix A. Berezin (in several short published papers 1960-63, which are not cited in our bibliography, and in the book [14]). The construction exposed in § 1.2 belongs to I. Segal. A. Weil, whose name is attached to the construction, extended it to finite and p-adic fields [214]. 11.2 The Mehler formula was obtained by F. G. Mehler [133] in 1866, see also W. MyllerLebedeff [137]. 11.3 The semigroup of compressive symplectic operators. As we have seen in Proposition 9.4, the Weil representation of Sp.2n; R/ admits a holomorphic continuation to a certain subsemigroup in Sp.2n; C/. This fact is a special case of G. I. Olshanski’s theorem [161]: – A highest weight unitary representation of a semisimple group G admits holomorphic continuation on a certain semigroup GC . For the groups G D U.p; q/, this phenomenon was observed by M. I. Graev in 1958 [74]; in this way he calculated characters of the Harish-Chandra holomorphic discrete series. 11.4 The semigroup of Gaussian operators and the strange multiplication (4.2). Apparently, many authors independently discovered the formula (4.2) and the corresponding strange algebraic structure. I know of the independent works of R. Howe and G. I. Olshanski. Both of them found the condition of boundedness and also described canonical forms. For references to earlier papers, see [94]; see also [32]. Howe’s paper [94] is also exposed in G. Folland’s book [51]. 11.5 The multiplication theorem. Theorem 8.5 was obtained in [140], [141]. 11.6 Some references on Gaussian operators in the real settings: [48], [121], [90]. 11.7 The Jordan normal form of linear relations. Apparently, it was discovered independently several times, e.g., see [65]. 11.8 Pairs of forms. On realizations of classical groups as groups preserving pairs of forms, see [147] and Addendum D, Table 5, page 528.

2 Pseudo-Euclidean geometry and groups U.p; q/ The standard inner product in an n-dimensional Euclidean space V D Rn is given by hx; yi D

n X

xj yj :

iD1

Changing this formula to hx; yi D

X i6p

xj yj C

X

xj yj ;

j >p

we get another geometry, namely the pseudo-Euclidean geometry. It is similar to Euclidean geometry, but also it differs in numerous important details. The basic theorems of Euclidean geometry (such as the reduction of quadratic forms or the classification of isometries) survive, but not literally; generalizations are not obvious. Also, there arise new phenomena that do not exist in Euclidean geometry. There are seven geometries of this type (see Addendum A.1), all of which appear often in various areas of mathematics. We consider in detail two geometries, namely, the complex pseudo-Euclidean geometry (in this chapter) and the real symplectic geometry (in the next chapter). The pseudo-Euclidean geometry was a subject of some works by L. S. Pontryagin, M. G. Krein, M. S. Livshits and V. P. Potapov on operator theory in the 1940-50s. Our point of view is partially influenced by these papers. Throughout this chapter, a ‘linear space’ is a finite-dimensional linear space over C.

2.1 The geometry of indefinite Hermitian forms 1.1 The definition of Hermitian forms. Let V be a complex finite-dimensional linear space. A sesquilinear form on V is a C-valued function B.v; w/ on V V such that B.v1 C v2 ; w/ D B.v1 ; w/ C B.v2 ; w/; B.v; w1 C w2 / D B.v; w1 / C B.v; w2 /; N B.v; w/ D B.v; w/; B.v; w/ D B.v; w/: A sesquilinear form is Hermitian if B.w; v/ D B.v; w/: In particular, B.v; v/ 2 R; indeed, B.v; v/ D B.v; v/.

58

Chapter 2. Pseudo-Euclidean geometry and groups U.p; q/

Remark. Adding the condition B.v; v/ > 0 for any v ¤ 0, we obtain the definition of an inner product in a complex Euclidean space. We say that a vector v is – B-positive if B.v; v/ > 0, – B-negative if B.v; v/ < 0, – B-isotropic if B.v; v/ D 0. Two vectors v and w are orthogonal if B.v; w/ D 0. Proposition 1.1. a) Each Hermitian form on Cn can be expressed as B.v; w/ D vQw ; where v, w 2 Cn are row vectors and Q is an n n Hermitian matrix. b) Change coordinates v 7! vg, where g is a linear transformation. Then the matrix Q changes according to the rule Q 7! gQg : Recall that A denotes the adjoint matrix, see Subsection 1.1.7; in particular v is a column vector. Proof. Denote by ej the standard basis in Cn . We have X X X vj w wk ek D xk B.ej ; ek /: B.v; w/ D B vj ej ;

1.2 Kernels of forms. The kernel ker B of a given form B is the subspace W V consisting of vectors w such that B.w; v/ D 0 for all v. The rank of a form B is the codimension of ker B. A form B is said to be nondegenerate if its kernel is zero. Lemma 1.2. Any form B induces a nondegenerate Hermitian form on the quotient space V = ker B. Proof. The space V = ker B consists of vectors defined up to the equivalence v v 0 () v v 0 2 ker B: Let v 0 D v C and w 0 D w C , where , 2 ker B. Then B.v 0 ; w 0 / D B.v C ; w C / D B.v; w/ C 0 C 0 C 0: Therefore, we actually get a form on the quotient space. Problem 1.1. Show that this form on V = ker B is nondegenerate.

2.1. The geometry of indefinite Hermitian forms

59

A pseudo-Euclidean space is a complex linear space equipped with a nondegenerate Hermitian form; in this case, we usually denote this form by h; i and call it a scalar product. 1.3 Several preliminary lemmas Lemma 1.3. Let h 2 V be an isotropic vector, v … ker B. Then one can find a vector g such that B.h; h/ D B.g; g/ D 0; B.h; g/ D 1: (1.1) Proof. Consider an arbitrary vector r 2 V such that B.h; r/ D ¤ 0. Setting r 0 WD N 1 r, we get B.h; r 0 / D 1. Put g WD r 0 C sh, where s ranges in R. Then B.h; g/ D 1;

B.g; g/ D B.r 0 ; r 0 / C 2s:

We take s WD 12 B.r 0 ; r 0 / and get the desired g.

Lemma 1.4. If B ¤ 0, then one can find a vector v such that B.v; v/ D ˙1 Proof. Take an arbitrary vector h. If it is not isotropic, set v WD jhv; vij1=2 h. Other wise, we choose a vector g satisfying (1.1) and put v WD p1 .h C g/. 2

Lemma 1.5. If one can find a positive vector and a negative vector, then there exists a non-zero isotropic vector. Proof. Let B.e; e/ D C1 and B.f; f / D 1. By the intermediate value theorem, one of the points of the segment t e C .1 t /f is isotropic.

hv; vi D 0 hv; vi D 1 hv; vi D 1

f h

e

g hv; vi D 0

Figure 2.1. The space R1;1 with the scalar product h.v1 ; v2 /; .w1 ; w2 /i D v1 w1 C v2 w2 . The level curves hv; vi D const are hyperbolas; isotropic vectors v are contained in a pair of lines. We present one of the possible bases e, f as in (1.3) and one of the bases h, g as in (1.1).

60

Chapter 2. Pseudo-Euclidean geometry and groups U.p; q/ l hv; vi D 1 A O m

C B

Figure 2.2. One of interpretations of the orthocomplement. Consider the two-dimensional real plane endowed with a positive inner product. The level curve hv; vi D 1 is an ellipse. Let l be a line. Draw a parallel line to l. Let A and B be the points of the intersection of the latter line and the ellipse. Consider the midpoint C of the segment ŒA; B. Then the line OC is the orthocomplement of l. L? M M? f e

L

N D N? S D S?

Figure 2.3. Four examples of orthocomplement in the pseudo-Euclidean plane R1;1 .

1.4 The classification of Hermitian forms. For a subspace S V , define its orthocomplement (or orthogonal complement) Z ? as the set of all vectors v 2 V such that B.v; z/ D 0 for all z 2 S . Remark. If the form B is degenerate, then S ? ker B for any S . Theorem 1.6 (The “inertia law”). a) Any Hermitian form on an n-dimensional space can be reduced to a “sum of squares” in some basis. More precisely, X X B.x; y/ D xj yNj C xj yNj : (1.2) j 6p

p<j 6pCq


61

b) The numbers p and q do not depend on the choice of basis. c) rk B D p C q. The collection .p; q; dim ker B/ is called the inertia of B. The numbers p and q are called negative and positive inertia indices and the number q p is called a signature. For instance, in an n-dimensional pseudo-Euclidean space one can choose a basis e1 ; : : : ; ep , f1 ; : : : ; fq (where p C q D n) such that he1 ; e1 i D D hep ; ep i D 1;

hf1 ; f1 i D D hfq ; fq i D C1

(1.3)

and all other scalar products of basis elements equal to 0. We denote by Cp;q the pseudo-Euclidean space equipped with such a basis. z

l L

y O x

Figure 2.4. An example of an orthocomplement in R2;1 . The cone is the cone of isotropic vectors. The orthocomplement L D l ? of an isotropic line l is the plane tangent to the cone along l.

Proof of the theorem. a) Consider an arbitrary complementary subspace Y to ker B. Obviously, the form is nondegenerate on Y , therefore the statement reduces to the case of nondegenerate forms. So we assume that B is nondegenerate. By Lemma 1.4, one can find a vector e1 such that B.e1 ; e1 / D ˙1. Lemma 1.7. V D Ce1 ˚ .Ce1 /? .

62


a)

b)

Figure 2.5. Let us write a scalar product in R1;1 as h.v1 ; v2 /; .w1 ; w2 /i D v1 w2 C v2 w1 . We exemplify a pseudo-orthogonal transformation of R1;1 , namely a hyperbolic rotation .v1 ; v2 / 7! .2v1 ; v2 =2/: Figure a) shows images of some points and their trajectories with respect to the ‘rotations’ R t W .v1 ; v2 / 7! .2t v1 ; 2t v2 / for t such that 0 6 t 6 1. On b) we show a transformation of a geometric configuration.

Proof of the lemma. Let be a vector noncollinear with e1 . Then the vector WD

h; e1 i e1 he1 ; e1 i

is orthogonal to e1 . Therefore, 2 Ce ˚ .Ce/? .

Let us continue the proof of the theorem. We take a vector e2 2 .Ce1 /? such that B.e2 ; e2 / D ˙1 and consider its orthocomplement in .Ce1 /? , etc. c) is obvious. b) is a corollary of Lemma 1.8 given below, which is important by itself.

1.5 Several definitions. A form B is positive-definite if B.v; v/ > 0 for all non-zero v 2 V . If the form is positive-definite, we say that the space V is Euclidean. In this case, (1.2) reduces to n X B.x; y/ D xj yNj : j D1

Recall also a collection of similar definitions. A form is: – negative definite if B.v; v/ < 0 for all v ¤ 0; – non-negative definite (or positive semi-definite) if and only if B.v; v/ > 0; – non-positive definite (or negative semi-definite) if B.v; v/ 6 0; – indefinite if it takes values of different signs. Let B be an arbitrary Hermitian form in V . We say that a subspace W V is positive if the form B is positive definite on W . In a similar way, we define nonnegative, negative, etc. subspaces. A subspace W is said to be regular if B is nondegenerate on W . Otherwise, W is said to be singular. By zero.W / we denote the kernel of the form B restricted to W .


63

M2

M1

L2 L1

L3

Figure 2.6. Reference the Witt theorem. Consider the space R2;2 and its projectivization, i.e., the 3-dimensional real projective space RP 3 . The cone of isotropic vectors in R2;2 corresponds to a (toroidal) quadric in RP 3 , i.e., to a one-sheeted hyperboloid. Lines in RP 3 correspond to two-dimensional planes in R2;2 . 1. L1 corresponds to a positive plane. 2. L2 corresponds to an indefinite plane; points of intersection of L2 and the hyperboloid correspond to the pair of isotropic lines in the indefinite plane. 3. L3 corresponds to a negative plane. 4. M1 corresponds to a plane of signature (1,0); the point of tangency indicates the kernel of the scalar product on the plane. 5. M2 corresponds to a degenerate semi-positive plane. 6. Linear elements of the hyperboloid correspond to isotropic planes in R2;2 . In particular, we observe that the Grassmannian of maximal isotropic subspaces in R2;2 is homeomorphic to a pair of disjoint circles. Denote by S.i; j / the set of all two-dimensional planes with inertia indices .i; j /. The scheme of adjacency of orbits is

S.2; 0/ S.1; 1/ S.0; 2/

S.1; 0/ S.0; 1/

S.0; 0/:

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Lemma 1.8. The negative (resp. positive) inertia index of the form B coincides with the maximal dimension of a negative (resp. positive) subspace. Proof. Consider the form B given by formula (1.2) Assume that there exists a negative subspace X, whose dimension is > p. Denote by Z the linear span of the vectors epC1 , epC2 ,…. Then dim X C dim Z > dim V , therefore X and Z have a nontrivial intersection. But X is negative and Z is non-negative. 1.6 Another canonical form Theorem 1.9. Let B be a Hermitian form with inertia indices p Then its matrix can be reduced to 0 1 0 0 1p 0 0 0 0 1p B1p 0 C B 0 1qp 0 0 0 B C B @ 0 0 1qp 0 A or @1p 0 0 0 0 0 0k 0 0 0

6 q, dim ker B D k. 1 0 0C C: 0A 0k

(1.4)

Without loss of generality, we can assume that B is nondegenerate. First proof. We consider a basis ei , fk as in (1.3). Define the new basis 1 hj D p .ej C fj / 2

and

1 gj D p .ej fj / 2

for j 6 p;

fk

for k > p:

This yields the desired canonical form. Second proof. Consider an isotropic vector h1 and a vector g1 satisfying (1.1). Then V D .Ch1 ˚ Cg1 / ˚ .Ch1 ˚ Cg1 /? :

(1.5)

Indeed, for a vector 2 V , we set WD h; g1 if1 h; f1 ig1 : Then 2 .Ch1 ˚ Cg1 /? and this yields (1.5). Next, we consider vectors h2 , g2 2 .Ch1 ˚ Cg1 /? in the same position as above, etc. As a result, we obtain a decomposition of the form hM i .Chj ˚ Cgj / ˚ R; j

where R does not contain an isotropic vector. By Lemma 1.5, the subspace R is sign-definite.

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1.7 Linear functionals. Let V be a pseudo-Euclidean space. We denote by V B the dual space, i.e., the space of all linear functionals on V . For a vector v 2 V , we define the linear functional `v on V by `v .w/ D hw; vi: Proposition 1.10. The map i W v 7! `v is an antilinear bijection from V onto V B . Proof. The injectivity is obvious. Since the dimensions of V and V B are the same, it follows that the map is bijective. Constructive proof of the bijectivity. For a linear functional `, let `.ej / D ˛j . Then `.v/ D hv; hi, where X X ˛Nj ej C ˛Nj ej : hD j 6p

j >p

1.8 Orthogonal complements Theorem 1.11. For any subspace S in a pseudo-Euclidean space V , a) .S ? /? D S; b) dim S C dim S ? D dim V: Lemma 1.12. Let W be a finite-dimensional linear space, let `1 ; : : : ; `k be linearly independent functionals on W . Then the subspace .ker `1 / \ \ .ker `k /

(1.6)

has codimension k in W . Proof of the lemma. Extend the collection `j to a basis in the dual space. Consider the dual basis e1 ; : : : ; en 2 W , i.e., ì .ej / D ıij : The subspace (1.6) is spanned by the vectors ekC1 ; : : : ; en .

Problem 1.2. Give a proof of the lemma avoiding the dual basis. For instance, the lemma remains valid in Banach spaces. Proof of the theorem. b) Let e1 ; : : : ; ekTbe a basis in S . Consider the functionals `j .v/ WD hv; ej i. By definition, S ? D ker `j , therefore codim S ? D dim S a) Obviously, .S ? /? S. But these spaces have the same dimension. Proposition 1.13. In any pseudo-Euclidean space, the following statements hold: a) S \ S ? D zero.S / D zero.S ? /. b) In particular, if S is regular, then V D S ˚ S ? . c) .S \ T /? D S ? C T ? . d) .S C T /? D S ? \ T ? . e) If S T , then S ? T ? .

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Proof. Obvious.

1.9 Isotropic subspaces. Consider a pseudo-Euclidean space V equipped with a form h; i. A subspace W V is called isotropic if hv; wi D 0

for all v, w 2 W :

Equivalently, W is isotropic if W W ? . Theorem 1.14. a) The maximal possible dimension of isotropic subspaces equals min.p; q/, where .p; q/ are the inertia indices. b) Any isotropic subspace is contained in an isotropic subspace of maximal possible dimension. Proof. a) For definiteness, assume p 6 q. Our second canonical form (Theorem 1.9) implies the existence of a p-dimensional isotropic subspace, it suffices to consider the linear span of the first p basis vectors. Assume that there exists an isotropic subspace of dimension > p. Then its intersection with a q-dimensional positive subspace is non-zero. This is a contradiction. b) Let T be an m-dimensional isotropic subspace. Bearing in mind T ? T and .T / D T , we observe that T D zero.T ? /. Denote by Z a complementary subspace to T in T ? ; then dim Z D dim V 2m and V D Z ˚ Z ? . Consider two cases. First, let Z be indefinite, h 2 Z an isotropic vector. Then T ˚ Ch is an isotropic subspace of dimension m C 1. Second, let Z be sign-definite. Then dim Z ? D 2m, T Z ? is an isotropic subspace. Hence the inertia indices of our form on Z ? are .m; m/, otherwise an mdimensional isotropic subspace in Z ? can not exist. Therefore, the inertia indices of our form on V are .m; m C dim Z/ or .m C dim Z; m/ (according to the sign of Z). Thus dim T is maximal possible. ? ?

Another proof of b). Let us repeat the second proof of Theorem 1.9, choosing h1 ; : : : ; hm 2 T . This yields a decomposition m i hM .Chj ˚ Cgj / ˚ R: j D1

If R is sign-definite, then the isotropic subspace T has the maximal possible dimension. Otherwise, we consider an isotropic vector 2 R; then T ˚C is an isotropic subspace. Actually our second proof implies the following statement (which is a stronger form of Theorem 1.9):

2.2. Pseudo-unitary groups U.p; q/

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Proposition 1.15. For an isotropic subspace T in a pseudo-Euclidean space V , one can find a decomposition V D T ˚ S ˚ Z;

where S is isotropic, dim S D dim T , and .T ˚ S /?Z:

Moreover, for each basis h1 ; : : : ; hm 2 T , there is a unique basis g1 ; : : : ; gm 2 S such that hhi ; gj i D ıij . 1.10 Duality of isotropic subspaces Problem 1.3. Let p D q, let W1 and W2 be complementary isotropic subspaces. For a vector h 2 W1 , we define a linear functional on W2 by `h .v/ D hv; hi. Then the map h 7! `h is a bijection of W1 onto the space of linear functionals on W2 . 1.11 Digression. Similar geometries Problem 1.4. a) Convince yourself that you can replicate the considerations of this section for symmetric bilinear forms over R. For instance, any nondegenerate symmetric bilinear form can be reduced to X X hx; yi D xi y i C xj yj : i6p

j >p

We denote by Rp;q the space equipped with this form. Figures 2.1–2.6 are drawn in the spaces R1;1 , R1;2 , and the projective space .R2;2 n 0/=R . b) Solve the same problem for symmetric bilinear forms over C. For instance, any nondegenerate form of this type can be reduced to X hx; yi D xk yk : The maximal possible dimension of an isotropic subspace in C2n equals n; in C2nC1 it also equals n.

2.2 Pseudo-unitary groups U.p; q/ 2.1 The pseudo-unitary groups U.p; q/. Let V ' Cp;q be a pseudo-Euclidean space. We say that an operator g in the space V is pseudo-unitary if it preserves our form, i.e.,1 hvg; wgi D hv; wi: (2.1) The pseudo-unitary group U.p; q/ or

UŒV

is the group of all operators satisfying this condition. 1 We can apply an operator g to a row-matrix v, then we write vg, we can also apply g to a column vector u, then we write gu.

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If the scalar product is positive definite (i.e., p D 0), then U.0; q/ is the usual unitary group U.q/, i.e., the group of unitary operators in a q-dimensional complex Euclidean space. This group is compact2 . Also, U.p; 0/ ' U.p/. Proposition 2.1. If an operator g satisfies hvg; vgi D hv; vi for all v; then g is pseudo-unitary. Proof. 2 Rehv; wi D hv C w; v C wi hv; vi hw; wiI 2i Imhv; wi D hv iw; v iwi hv; vi hw; wi: Thus operators preserving scalar squares preserve scalar products.

2.2 Equations. If hv; wi D vQw , then the condition (2.1) takes the form gQg D Q:

(2.2)

In particular, assuming that h; i is in the canonical form (1.2), and representing g as a block matrix g D ac db , we get the equation a b 1p 0 a b 1p 0 ; (2.3) D c d 0 1q 0 1q c d or

aa bb D 1;

cc d d D 1;

ac bd D 0:

(2.4)

We can also write out these conditions as a system of equations for matrix elements. Denote by rj the j -th row of the matrix g. Then the conditions are: ´ 1 if j 6 p; hrj ; rj i D 1 if j > pI hri ; rj i D 0

for i ¤ j :

In particular (if p ¤ 0, q ¤ 0), taking i D j D 1 we get ja11 j2 ja1p j2 C jb11 j2 C C jb1q j2 D 1I taking i D 1, j D p C 1, we come to a11 cN11 a1p cN1p C b11 dN11 C C b1q dN1q D 0: 2

because the equation gg D 1 implies that all matrix elements gij satisfy jgij j 6 1.


69

Remark. The formula (2.2) also implies the invertibility of a pseudounitary matrix g, more precisely g 1 D Qg Q1 . Lemma 2.2. For the standard scalar product, the conditions g 2 U.p; q/ and g 2 U.p; q/ are equivalent, i.e., the equation (2.3) is equivalent to 1 0 a b 1 0 a b D : (2.5) 0 1 c d 0 1 c d Proof. We write the condition (2.3) as ³ ² ³ ² 1 0 1 0 a b 1 0 a b D : 0 1 0 1 c d 0 1 c d Next, we permute the two factors on the left-hand side and reduce our condition to the desired form. The conditions (2.5) can be written out as a a c c D 1;

b b d d D 1;

a b c d D 0:

(2.6)

Lemma 2.3. Let S be a subspace invariant with respect to a pseudo-unitary operator g. Then the subspace S ? is also g-invariant. We leave this as an exercise for the reader. Problem 2.1. a) If g 2 U.p; q/, then j det gj D 1. b) U.p; q/ ' U.q; p/. 2.3 Self-adjoint operators. Consider a linear operator A in a pseudo-Euclidean space V . An operator A~ is adjoint to A if hvA; wi D hv; wA~ i

for all v, w 2 V .

Remark. We use the notation A~ keeping the symbol A for adjoint operators in Euclidean spaces and for adjoint matrices in the usual sense. Problem 2.2. Prove the existence and uniqueness of the adjoint operator. An operator A is said to be self-adjoint if A D A~ . Problem 2.3. Let a scalar product be given by the standard formula. Then ˛ ˇ is self-adjoint () ˛ D ˛ ; ı D ı ; ˇ D : ı

(2.7)

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Problem 2.4. Let S be a subspace invariant with respect to a self-adjoint operator A. Then S ? is also A-invariant. 2.4 The Lie algebra of U.p; q/. Now we are going to describe the Lie algebra u D u.p; q/ of the group U.p; q/. Let X 2 u. By the definition of a Lie algebra (see Addendum C.3), exp."X / 2 U.p; q/ for real ": Hence,

˝

˛ v exp."X /; w exp."X / D hv; wi:

The left-hand side equals ˝ ˛ ˝ ˛ ˝ ˛ ˝ ˛ v.1 C "X C O."/2 /; w.1 C "X C O."/2 / D v; w C " vX; w C " v; wX C O."/2 : Therefore, X satisfies the condition hvX; wi C hv; wXi D 0 or

X D X ~ ;

(2.8)

i.e., the operator X is anti-self-adjoint. Equivalently, 1i X is self-adjoint. Conversely, let X satisfy (2.8). Then hvX n ; wi D hv; w.X /n i, and hence E X 1 ˝ ˛ D "j X j v exp."X /; w exp."X / D v exp."X /; w jŠ D X .1/j E D v exp."X / "j X j ; w jŠ ˝ ˛ D v exp."X / exp."X /; w D hv; wi: Thus we obtain the following theorems. Theorem 2.4. The Lie algebra u of UŒV consists of operators X satisfying the antiself-adjointness condition (2.8). Theorem 2.5. Let X be an anti-self-adjoint operator in a pseudo-Euclidean space. Then its exponential is pseudo-unitary. Problem 2.5. Let g 2 U.p; q/ and kg 1k < 1, where k k denotes the usual matrix norm. Show directly that X D ln g WD

1 X .1/j 1 j D0

satisfies (2.8).

j

.g 1/j

(2.9)

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If the Hermitian form is written as hv; wi D vQw , then the condition (2.8) of anti-self-adjointness is XQ C QX D 0: (2.10) 1 0 Next, let the form be standard, i.e., Q D 0 1 . Represent X 2 u.p; q/ in the Q ˛ ˇ block form X D ı . The condition (2.10) takes the form

˛

ˇ ı

1 0 1 0 ˛ C ˇ 0 1 0 1

ı

D

0 0 : 0 0

Thus the Lie algebra u.p; q/ consists of the matrices ˛ ˇ XD ; where ˛ C ˛ D 0; D ˇ ; ı C ı D 0: ı

(2.11)

Problem 2.6. Verify directly that for each X , Y 2 u.p; q/, their commutator ŒX; Y WD XY YX is also an element of u.p; q/. Problem 2.7. Write similar conditions for the Hermitian form defined by the matrix 1 0 0 0 1p @ 0 1qp 0 A : 0 0 1p Consider individually the special case p D q. 2.5 The Witt theorem Theorem 2.6 (Witt theorem). Let V be a pseudo-Euclidean space, T , S two subspaces of the same dimension. Let A W T 7! S be a bijective operator preserving the scalar product, i.e., hAv; Awi D hv; wi for all v, w 2 T . Then there exists a pseudo-unitary operator AQ W V ! V such that the restriction of AQ on T is A. Problem 2.8. Show that the Witt theorem is not valid for spaces endowed with degenerate Hermitian forms. First proof. We consider three cases. Case 1. Let T be regular and let its inertia indices be .k; l/. Then both the spaces T ? and S ? have inertia indices .p k; q l/. Therefore, one can find a pseudo-unitary bijection B W T ? ! S ? , and we put AQ D A ˚ B. Case 2. Let T be singular and non-isotropic. Denote by Z some complementary subspace to zero.T / in T . Consider an isometry R W V ! V such that Rz D Az for all z 2 Z (the existence of R was proved in Case 1 ). Consider a new pair of subspaces T and S 0 D R1 S and the operator A0 WD 1 R A W T ! S 0 . By construction, Az D z for any z 2 Z.

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Now the extension problem reduces to the same problem in the space Z ? of a smaller dimension. Case 3. T is isotropic. Here we can refer to Proposition 1.15. Another proof for Case 3. Fix a non-zero linear functional ` on T and take a vector 2 V such that `.v/ D hv; i for all v 2 V . Consider the space Tz D T ˚ C. Then zero.Tz / D ker `. Take a vector h 2 T such that h … zero.Tz /. Next, we apply Lemma 1.3 to the subspace Tz and choose a vector g D gT 2 Tz such that hg; gi D 0, hg; hi D 1. Next, we repeat the same arguments for the subspace S , the linear functional `S WD ` B A1 , and the vector hS WD Ah. We construct a vector gS in the same way and Q T WD hS . extend the operator A putting Ah Now we pass to Case 2. Thus the Witt theorem is a rather simple statement. Nevertheless, below (in Chapters 9 and 10) we need a proof that is valid ‘for arbitrary fields’. Our considerations in Case 1 (and therefore in Case 2) essentially use the classification of Hermitian forms and we intend to avoid it. 2.6 Another proof of the Witt theorem. We need the following lemma, which is also a special case of the Witt theorem: Lemma 2.7. Fix ˛ ¤ 0 The group U.p; q/ acts transitively on the set of vectors v satisfying hv; vi D ˛. Proof of the lemma.3 Let hv; vi D hw; wi D ˛. We are going to construct an operator A, which takes v to w. If w D v, then we can take A as a multiplication by a scalar. Thus, let v and w span a plane. There are two cases. First, let the plane Cv ˚ Cw be regular. Then we define the following reflection operator A: Av D w;

Aw D v;

Ah D h for any h 2 .Cv ˚ Cw/? :

Second, let this plane be singular. Then hv; vi hv; wi det D 0: hw; vi hw; wi Hence hv; wi D ˙˛. Since v and .v/ are contained in one U.p; q/-orbit, we can set hv; wi D C˛. Then zero.Cv ˚ Cw/ D C.v w/. Choose an isotropic vector h 2 C.v C w/? such that hh; v wi ¤ 0. Then the 3-dimensional space R spanned by v, w, h is regular. We define A by Av D w;

Aw D v;

Ah D h;

A D for any 2 R?

3 Recall that we intend to avoid a reference to the classification of Hermitian forms. Otherwise, there is nothing to prove.


73

Let us continue the proof of the Witt theorem. First, let the subspaces T and S be non-isotropic. Let r 2 T be a non-isotropic vector. Take an operator U W V ! V such that U r D Ar. Let us consider the space S 0 D U 1 S and the operator A0 W T ! S 0 given by A0 D U 1 A. Then r 2 T; S 0 and A0 r D r. Now we can work in .Cr/? ; the dimension of this space is dim W 1. If the subspaces T and S are isotropic, we use the same argument as in the previous proof. Problem 2.9. Prove the Witt theorem for symmetric bilinear forms over an arbitrary field of characteristic ¤ 2. Where is “¤ 2” essential? 2.7 Grassmannians. Recall that a Grassmannian Gr k;nk .V / is the space of all k-dimensional subspaces in an n-dimensional linear space V . Theorem 2.8. a) Gr k;l is a compact kl-dimensional complex manifold. b) Gr k;l .Cn / is the homogeneous space Gr k;l ' U.k C l/=.U.k/ U.l//: Proof. Let us endow the space CkCl with the standard inner product. Obviously, Gr k;l is U.k C l/-homogeneous. If g stabilizes a subspace L 2 Gr k;l , then it must also stabilize its orthocomplement L? . Thus g 2 UŒL UŒL? . This proves b). Any homogeneous space G=H of a Lie group G with respect to a closed subgroup H is a smooth submanifold whose dimension is .dim G dim H /. Therefore the real dimension of Gr k;l is dim U.k C l/ dim U.k/ dim U.l/ D .k C l/2 k 2 l 2 D 2kl: Since U.k C l/ is compact, the U.k C l/-homogeneous space Gr k;l also is compact. The structure of a complex manifold is not visible in this reasoning. However, the space Gr k;l admits an action of a larger group, GL.k C l; C/. Consider the subspace M Cn spanned by the first k basis vectors. Its stabilizer P consists of matrices of the form A B : 0 D Both the Lie groups G and P are complex, therefore Gr k;l D GL.k C l; C/=P is a complex manifold. Another proof. The coordinates on Gr k;l . Decompose CkCl into the direct sum Cn D T ˚ S of k-dimensional and l-dimensional subspaces. Consider the set S of all k-dimensional subspaces L having trivial intersection with S . Each L 2 S is the graph of a linear operator AL W T ! S . Thus, there is the canonical one-to-one correspondence between S and the space of linear operators T ! S . Considering all possible pairs T , S, we obtain an atlas on the complex manifold Gr k;l .

74


Now let us try to visualize the topology in Gr k;l : A) We can treat points of the Grassmannian as .k 1/-dimensional subspaces in the projective space CP kCl1 . For a given subspace N CP kCl1 consider some of its open neighborhoods O CP kCl1 . Then all subspaces L O form a neighborhood of the point N , see Figure 2.7.a. B) Denote by B .v/ the ball in CkCl of radius and center v. A sequence Lj 2 Gr k;l converges to L if, for each point v 2 L, each ball B" .v/ has a nonempty intersection with all Lj starting with a certain number ˛. See Figure 2.7.b. Lj

N 0 a)

L

b)

Figure 2.7. On the definitions of topology in the Grassmannian. a) A neighborhood of a line N on the projective plane and some lines closed to N . b) Convergence of planes in R3 .

C) Certainly, the descriptions of a manifold structure on Gr k;l provides us descriptions of the topology. For instance, a sequence Lj converges to L if there is a sequence of operators gj convergent to 1 such that gj L D Lj . This is a rephrasing of the definition of the quotient topology. 2.8 The orbits of U.p; q/ on Grassmannians. A description of orbits of U.p; q/ on the Grassmannians is given by the Witt theorem. Problem 2.10. a) Consider a subspace with the inertia indices .r; s/. Find the closure of its U.p; q/-orbit. b) Find all closed orbits of U.p; q/ in the Grassmannians. Now we wish to discuss stabilizers of subspaces in the most important cases. A) The stabilizer of any regular k-dimensional subspace L is UŒLUŒL? . Indeed, any element of U.p; q/ stabilizing L stabilizes L? as well. Therefore, the set of all regular k-dimensional subspaces in Cp;q is a disjoint union of the homogeneous spaces a ı U.p; q/ U.r1 ; s1 / U.r2 ; s2 / : r1 Cs1 Dk; r1 Cr2 Dp; s1 Cs2 Dq

One of these spaces, namely U.p; q/=U.p/ U.q/, is the subject of our discussion in the next section and an important character of the whole book.


75

B) Now let us describe the stabilizer P of an isotropic subspace T . Decompose our space V D T ˚ Z ˚ S as in Proposition 1.15 where S is isotropic, Z is regular and Z D .T ˚ S/? . An operator g 2 U.p; q/ preserving T , preserves T ? D T ˚ Z as well. As we have seen in Proposition 1.15, the spaces T and S are dual. Thus the stabilizer P of L consists of pseudo-unitary matrices having the following block structure with respect to the decomposition T ˚ Z ˚ S: 1 0 A B C E A ; where A 2 GLŒL, D 2 UŒM : g D @0 D 0 0 A1 2.9 Exercises Problem 2.11. Convince yourself that you can extend all theorems given above to symmetric bilinear forms over R. 2.10 Miscellanea. A) It follows from abstract considerations that U.p; q/ is a smooth manifold, dim U.p; q/ D .p C q/2 , see Addendum C.3. B) The space U.p; q/ is homeomorphic to U.p/ U.q/ R2pq . Moreover, there is a canonical diffeomorphism (Cartan coordinates) between these spaces, see Theorem 3.4. In particular, U.p; q/ is connected and its fundamental group 1 .U.p; q// is isomorphic to Z ˚ Z, see Proposition 3.7.2. C) The center Z of U.p; q/ consists of matrices e i 1. D) There are the following two close relatives of U.p; q/ (but we prefer the group U.p; q/ itself as a basic object): – The special pseudo-unitary group SU.p; q/ is defined as the group of pseudo-unitary matrices whose determinant is 1. – The projective pseudo-unitary group is PU.p; q/ WD U.p; q/=Z. Note that the group effectively acting on the Grassmannians is PU.p; q/. Problem 2.12. a) The group SU.2/ consists of matrices a b ; where jaj2 C jbj2 D 1: gD bN aN b) The group SU.1; 1/ consists of matrices a b gD N ; where jaj2 jbj2 D 1: b aN 0 i . In this model, the c) Define the scalar product in C1;1 by the Hermitian matrix i 0 group SU.1; 1/ is the group of matrices a b ; where a; b; c; d 2 R; det g D 1: gD c d That is, SU.1; 1/ ' SL.2; R/.

76


2.3 Cartan matrix balls To be definite, let p 6 q. 3.1 The set of negative subspaces. Consider the pseudo-Euclidean space V ' Cp;q . Fix an orthogonal decomposition V D V ˚ VC ;

(3.1)

where a subspace V is negative and VC is positive. It is convenient (but not necessary) to think that V D Cp ˚ 0; VC D 0 ˚ Cq : Denote by Mp;q the set of all p-dimensional negative subspaces in V . By the Witt theorem, the set Mp;q is U.p; q/-homogeneous. If g 2 U.p; q/ stabilizes the subspace V D Cp ˚ 0, then it also stabilizes its orthogonal complement VC D 0 ˚ Cq . Hence the stabilizer K of V is isomorphic to U.p/ U.q/. Thus, Mp;q is a homogeneous space, Mp;q D G=K D U.p; q/=U.p/ U.q/: Remark. Apparently, it is correct to write U.p; q/=.U.p/ U.q//: However we prefer to omit brackets. 3.2 Coordinates in Mp;q . Denote by Bp;q (matrix ball ) the space of all p q matrices z with norm < 1. Equivalently, Bp;q is the set of all operators V ! VC with norm < 1. The space VC is Euclidean by definition. The space V is Euclidean with respect to the inner product .h; i/. Thus, the norm of any operator V ! VC is well defined. Theorem 3.1. a) For z 2 Bp;q , its graph graph.z/ V ˚ VC is a negative p-dimensional subspace. b) Each negative p-dimensional subspace in V is the graph of an operator z W V ! VC with z 2 Bp;q . Proof. a) Let z be in Bp;q . Its graph consists of all vectors of the form v ˚ v z. Then hv ˚ v z; v ˚ v ziV D hv ˚ 0; v ˚ 0iV C h0 ˚ v z; 0 ˚ v ziV D kv k2V C kv zk2VC

(3.2)

Since kzk < 1, the latter expression is negative. b) Let P 2 Mp;q . Since P is negative, its intersection with the positive subspace VC is zero. Hence P is the graph of an operator z W V ! VC . Formula (3.2) implies the required statement.

77

2.3. Cartan matrix balls

The group U.p; q/ acts on Mp;q and therefore it acts on Bp;q . Theorem 3.2. For any g D ac db , the corresponding transformation of Bp;q is z 7! z Œg WD .a C zc/1 .b C zd /:

(3.3)

Proof. Let w 2 V . The operator g takes a (negative) vector .w ˚ wz/ 2 graph.z/ to a b h WD w ˚ wz D w.a C zc/ ˚ w.b C zd /; c d which must also be negative. First of all w.aCzc/ ¤ 0. Otherwise, h D 0˚w.bCzd / and this vector is non-negative. Next, write y D w.a C zc/ and rewrite w.a C zc/ ˚ w.b C zd / D y ˚ y.a C zc/1 .b C zd /: Thus the subspace graph.z/ g is the graph of (3.3).

Problem 3.1. Derive the invertibility of .a C zc/ from (2.4) and (2.6).

z VC 1

VC

V

V

Figure 2.8. ‘Matrix balls’ for R1;2 and R2;1 . In the first case, negative subspaces are lines inside the isotropic cone. The corresponding ‘ball’ of 1 2 matrices is a disk. In the second case negative subspaces are planes outside the cone. Another example is given in Figure 2.6; points of the matrix ball for R2;2 are lines lying inside the hyperboloid.

78


Example. Let p D q D 1. The Grassmannian in C1;1 is nothing but the complex projective line CP 1 D C [ 1. The space B1;1 is the unit disk jzj < 1 on C. The group U.1; 1/ (for its description, see Problem 2.12) acts on B1;1 by the Möbius linear. The action is not effective, since the center of fractional transformations z 7! dzCb azCc U.1; 1/ acts trivially. Example. Let p D 1. The Grassmannian of 1-dimensional subspaces in C1;q is the complex projective space CP q . The space B1;q is the usual ball jzj < 1 in Cq CP q . Problem 3.2. Certainly Bp;q ' Bq;p . The transposition sends Bp;q ! Bq;p . Interpret this in terms of Grassmannians. 3.3 Digression. Different forms of linear-fractional transformations. Consider the Grassmannian Grp;q of all p-dimensional subspaces in CpCq . A subspace P 2 Grp;q of general position is the graph of an operator z W Cp ˚ 0 ! 0 ˚ Cq . Clearly, the action of GL.p C q; C/ on the Grassmannian can be written by the same formula (3.3). Note that .a C zc/1 can be non-invertible, therefore the formula is discontinuous. Problem 3.3. a) Write g 1 D Then

a c

b d

1

DW

P R

Q : T

.a C zc/1 .b C zd / D .P z C Q/.Rz C T /1 :

(3.4)

b) For any g 2 U.p; q/, simplify the right-hand side of (3.4) as .a C zc/1 .b C zd / D .a z C c /.b z C d /1 :

(3.5)

c) In Theorem 3.2, we begin with the right action of the group on row vectors. Repeat the same arguments for the left action of the group U.p; q/ on column vectors. There is a third way (due to Krein and Shmul’yan) to write linear fractional transformations, namely z 7! K C L.1 zN /1 zM D K C Lz.1 N z/1 M: Indeed, we write out .a C zc/1 .b C zd / D a1 .1 C zca1 /1 Œb C zd

D a1 .1 C zca1 /1 .1 C zca1 /b C z.d ca1 b/ D a1 b C a1 .1 C z ca1 /1 z .d ca1 b/: In §2.8, we will observe that the latter form is more general than the two previous forms.

3.4 The transitivity again. As we observe above, the transitivity of the action U.p; q/ on Bp;q follows from the Witt theorem. Here we present another proof of the transitivity. For any matrix C with norm < 1 and ˛ 2 C, we define .1 C C /˛ D 1 C

˛.˛ 1/ 2 ˛ CC C C : 1Š 2Š

(3.6)


79

Problem 3.4. a) If kAk and kBk < 1, then .1 AB/˛ A D A.1 BA/˛

(3.7)

(expand both sides in Taylor series). b) If ˛ is a positive integer, then (3.7) holds for all A and B. If ˛ is a negative integer, then (3.7) is valid if det.1 AB/ ¤ 0 (apply the analytic continuation). Write:

.1 zz /1=2 z : .1 z z/1=2

.1 zz /1=2 S.z/ D z .1 zz /1=2

(3.8)

Proposition 3.3. a) S.z/ 2 U.p; q/. b) The transformation S.z/ takes 0 to z. c) S.z/ D S.z/. d) S.z/ is positive definite. Proof. The statements a), b) can be easily checked by straightforward calculations, c) is obvious. To prove d), we observe that S.z/ D

.1 zz /1=4 0 0 .1 z z/1=4

1 z z 1

0 .1 zz /1=4 : 0 .1 z z/1=4

Since kzk < 1, the middle factor is positive definite.

Problem 3.5. a) Solve directly the system of equations g 2 U.p; q/;

0Œg D z

and show that gz is one of its solutions. b) Find a (unique) solution satisfying g 2 D 1. 3.5 The Cartan decomposition Theorem 3.4. a) Every r 2 U.p; q/ admits a unique decomposition r D hS;

where h 2 U.p/ U.q/ and S D S > 0:

(3.9)

b) Every r 2 U.p; q/ admits a unique decomposition r D h expfX g;

where h 2 U.p/ U.q/, X D X 2 u.p; q/:

Remark. By (2.11), X has the form XD

0 ˇ

ˇ : 0

(3.10)

80


Actually we will prove a stronger version of the theorem. Theorem 3.5. Every r 2 U.p; q/ admits a unique decomposition r D hS.z/;

where h 2 U.p/ U.q/, z 2 Bp;q ;

(3.11)

and where S.z/ has the form (3.8). Proof of Theorem 3.5. Set z D 0Œr , then h WD S.z/1 r stabilizes the point 0, hence h 2 U.p/ U.q/. Proof of Theorem 3.4. a) Consider the polar decomposition r D hS in the usual sense, see Theorem 1.1.3 and its proof. We have h 2 U.p C q/ and S D S > 0. The polar decomposition is unique, but a priori, it is not obvious that h, S 2 U.p; q/. On the other hand, Theorem 3.5 provides such a decomposition with positive S.z/. Therefore, h in (3.11) and (3.9) are the same. Thus S D S.z/. b) Any positive self-adjoint matrix S has a canonically defined logarithm ln S, 8 0 9 1 1 0 1 0 : : : 0 ::: ln 1 ˆ > < = B C B 1 ln 2 : : :C ln S D ln U @ 0 2 : : :A U 1 WD U @ 0 AU ; ˆ > : : : : : : : ; :: :: :: :: :: :: where U is a unitary matrix. Our S.z/ 2 U.p; q/ is positive. If it is sufficiently close to 1, then ln S 2 u.p; q/. Since logarithm is analytic, we get ln S 2 u.p; q/ for all positive self-adjoint S 2 U.p; q/. Problem 3.6. It is interesting to prove Theorem 3.4 without using Theorem 3.5. By Lemma 2.2, we have r 2 U.p; q/.p Since r r > 0, therefore r r has positive eigenvalues and no Jordan blocks. Why is r r 2 U.p; q/? See the discussion of conjugacy classes in Section 2.6. Problem 3.7. Each element of U.p; q/ can be represented in the form g D h1 A.t /h2 ; where h1 , h2 2 U.p/ U.q/ and 0 0 … cosh t1 0 cosh t … B 2 B :: :: :: B : : : B B sinh t 0 … 1 B B 0 sinh t2 … A.t/ WD B :: :: B :: B : : : B B 0 0 … B 0 0 … @ :: :: :: : : :

sinh t1 0 :: :

0 sinh t2 :: :

… … :: :

0 0 :: :

0 0 :: : : : :

cosh t1 0 :: : 0 0 :: :

0 cosh t2 :: : 0 0 :: :

… … :: : … … :: :

0 0 :: : 1 0 :: :

0 0 :: : : : : 0 1 :: :

1 … … C C C C … C C … C C; C C C … C C … A :: :

81


where t1 > t2 > > tp > 0: The numbers tj are uniquely determined by the matrix g. Are h1 , h2 unique? It is more pleasant to write the Hermitian form as (1.4), then the operators A.t / become diagonal. xp;q the set of all semi-negative 3.6 The boundary of Bp;q . Let p 6 q. Denote by M p;q subspaces in C . Denote by N˛ the set of semi-negative subspaces P such that the rank of h; i on P equals ˛. By definition, Np D Mp;q . xp;q . By the Witt theorem, the sets N˛ are precisely the orbits of U.p; q/ on M x Obviously, the closure N˛ of an orbit N˛ is [ Nˇ : Nx˛ D ˇ 6˛

xp;q the set of complex p q matrices with the norm 6 1. In SubsecDenote by B tion 3.2 we have constructed the bijection Mp;q $ Bp;q . Obviously, it can be extended xp;q $ B xp;q . to a bijection M v

u

w

Figure 2.9. Boundaries of matrix balls have a rather complicated structure. Unfortunately, it is difficult to visualize it because dim Bp;q D 2pq (and the simplest nontrivial case is p D q D 2). v On the figure we draw the set of 2 2 real symmetric matrices z D vu w satisfying the condition kzk 6 1. The conic surfaces are the sets of matrices with an eigenvalue 1 or .1/ respectively. The vertices of the cones correspond to the matrices z D ˙ 10 01 . The circle in the intersection of cones corresponds to matrices z whose eigenvalues are 1 and .1/.

xp;q defined by the condition Denote by O˛ the subset in B z 2 O˛

if rk.1 zz / D ˛:

(3.12)

82


xp;q . In Proposition 3.6. The U.p; q/-orbits N˛ correspond to the subsets O˛ B particular, the sets Op;q are U.p; q/-invariant. Proof. Let h 2 Cp . Since the operator .1 zz / is semipositive, h.1 zz / D 0 () 0 D hh; h.1 zz /iCp D hh; hiCp hzh; zhiCq () h ˚ hz 2 Cp;q is isotropic: Let P be the corresponding semi-positive subspace. Since h is isotropic, it follows that h 2 zero.P /. Thus we get a bijection between ker.1 zz / and zero.P /. Problem 3.8. Prove directly, starting from formula (3.3), that the sets rk.1 zz / D ˛ are U.p; q/-invariant. xp;q , is an invariant Problem 3.9 (exotic). a) Show directly that rk.z u/, where z, u 2 B Œg Œg of U.p; q/-action, i.e., rk.z u / D rk.z u/. b) The integer distance .z; u/ D rk.z u/ satisfies the axioms of metric spaces. xp;q . c) Interpret this metric in terms of M 3.7 Digression. Biholomorphic automorphisms of Bp;q Theorem 3.7 (Elie Cartan). If p ¤ q, then the group of biholomorphic4 transformations of Bp;q coincides with U.p; q/. If p D q, then each biholomorphic transformation is an element of U.p; q/ or a composition of such a map with z 7! z t . Sketch of proof. Let us denote by the group of all biholomorphic transformations of Bp;q . Let 0 be the stabilizer of the point 0. It suffices to show that 0 coincides with U.p/ U.q/. A) Each partial derivative @˛ .0/ is uniformly bounded for all 2 0 (because it can be evaluated by the Cauchy integral formula, see any textbook in complex analysis). B) Let 2 0 . Denote by R the transformation z 7! e i z. If is not a linear transformation, then WD R B B R1 ¤ . Write WD B 1 . C) Expand .z/ in Taylor series:

.z/ D z C 2 .z/ C 3 .z/ C ; here j is a homogeneous form of order j . Then

B ƒ‚ ::: B…

.z/ D z C n 2 .z/ C : „ n times

By A), it follows that 2 .z/ D 0. Writing out B : : : B again, we obtain 3 .z/ D 0 and so on. Therefore, .z/ D z and is a linear transformation. xp;q . D) It remains to describe linear transformations of Cpq preserving the compact set B There are many ways to do this; our approach is based on examination of the boundary of Bp;q . E) The cases p D 1 or q D 1 are trivial (because B1;q is the usual ball). Thus, we assume xp;q , see (3.12). p, q > 1. Consider the U.p; q/-orbit O1 B 4 Let CN be an open domain. A map W ! is said to be biholomorphic if and bijective (and hence 1 is also holomorphic).

is holomorphic


83

xp;q is a disjoint union of flat 1-dimensional (open) complex disks Lemma 3.8. The orbit O1 B lying on affine lines of the form a C H Cpq , where H is a rank 1 matrix, a 2 O1 , and runs over C. Moreover, all rank 1 matrices H 2 Matp;q arise in this context. Proof. Let z 2 O1 . Then there are bases ek 2 Cp and fl 2 Cq such that ze1 D f1 ;

zem D fm for m > 1I

the corresponding line consists of operators of the same form with arbitrary ; the complex disk is j j < 1 (its boundary is contained in O0 ). Our linear transformation must send lines described in the lemma to lines of the same form. F) Thus, our linear transformation preserves the cone of rank 1 matrices. Lemma 3.9. If p ¤ q, then all linear transformations of Matp;q preserving the cone of rank 1 matrices have the form z 7! azb, where a 2 GL.p; C/ and b 2 GL.q; C/. If p D q, then there also exists a map z 7! az t b. Proof. Denote our cone by „. Each point of the cone is of the form s t , where s is a columnmatrix, t is a row-matrix. Consider the projectivized cone P „ (that is, we consider matrices in „ up to proportionality). Obviously, P „ is the product of projective spaces, P „ D CP p1 CP q1 : The group of biholomorphic maps of CP k1 is PGL.k; C/. We leave the remaining details for an enquiring reader.

G) The set of limit points of O1 is O0 ; the latter is the set of isometric embeddings Cp ! Cq . The transformation z 7! azd must preserve O0 . It is more-or-less obvious that a 2 U.p/ and b 2 U.q/. Problem 3.10. Each U.p; q/-orbit O˛ is a disjoint union of flat pieces. Describe them. All these pieces are, actually, copies of the ball B˛;qpC˛ . Problem 3.11. Invent another end of the proof (instead of E)–G) or F)–G)). We must describe the group G of linear transformations preserving Bp;q . Some of the non-obvious possible arguments are – von Staudt type theorems, see [43]; – the group U.p/ U.q/ G is a maximal subgroup in U.pq/. Remark. By the Riemann theorem, the group of biholomorphic automorphisms of an arbitrary simply connected domain C is the Möbius group SU.1; 1/. For a “generic” bounded domain CN , where N > 2, the automorphism group is trivial. Let us present a heuristic explanation. A small piece of the boundary of is the graph of a function of .2N 1/-variables. On the other hand, a biholomorphic diffeomorphism is determined by N functions of N variables. If N > 1, then 2N 1 > N . Therefore, the “space of boundaries” is essentially larger than the “space of diffeomorphisms”. For this reason, usually the group of biholomorphic automorphism is trivial.

84


3.8 Siegel matrix wedges5 . For p D q, there is another nice way to introduce coordinates on the set Mq;q . Consider the pseudo-Euclidean space H WD Cq;q . Take a basis e1 ; : : : ; eq , f1 ; : : : ; fq such that (3.13) hei ; ej i D hfi ; fj i D 0; hei ; fj i D ıij : We denote by H (resp. H C ) the space spanned by ej (resp. by fj ). By definition, these two spaces are Aisotropic. B is an element of UŒH D U.q; q/ if A matrix g D C D

A C

B D

0 1 A 1 0 C

B D

D

0 1 : 1 0

(3.14)

Proposition 3.10. A subspace P is an element of Mq;q if and only if it is the graph of an operator T W H ! H C satisfying the condition .T C T / > 0: Proof. Since H C is isotropic and P is negative, it follows that H C \P D 0. Therefore, P is the graph of an operator T W H ! H C . Consider a point v ˚ v T 2 P . Then v T v D v .T C T /v : hv ˚ v T; v ˚ v T i D v T v

This expression is negative if and only if T C T > 0.

We denote by Wq the set (wedge) of all q q matrices T satisfying Re T > 0. The group U.q; q/ acts on Wq by the transformations T 7! .A C T C /1 .B C TD/: Since Bq;q and Wq are identified with the same set Mq;q , one can find a (noncanonical) identification Bq;q ' Wq . One of the possible maps is the Cayley transform u D .1 T /=.1 C T / () T D .1 u/=.1 C u/;

where T 2 Wq , u 2 Bq;q :

S q of Wq is non-compact, and hence the map Mq;q ! Wq can Remark. The closure W x q;q ! W Sq. not be extended to a bijection6 M Problem 3.12. Thus U.q; q/ acts on the boundary of Wq discontinuously. In spite of this, describe the ‘orbits’. The usual term is the Siegel half-plane. Actually Gaussian vectors in § 1.3 are enumerated by points of the matrix wedge and remaining Gaussian distributions correspond to remaining points of the boundary of the matrix ball. 5 6


85

3.9 Digression. Embeddings Bp;q ! U.p; q/ A) The map g 7! g is an anti-involution7 of the group U.p; q/. Therefore, the group U.p; q/ acts on itself by the transformations g W r 7! g rg. Problem 3.13. The stabilizer of the point r D 1 is the group U.p/ U.q/ U.p; q/. Thus the U.p; q/-orbit R of the point 1 is the homogeneous space U.p; q/=U.p/ U.q/. Hence there exists a U.p; q/-equivariant map W Bp;q ! U.p; q/. Problem 3.14. a) Show that the map Bp;q ! U.p; q/ given by z 7! S.z/2 commutes with the action of U.p; q/ (for instance, see proof of Theorem 3.4). b) Show that its image consists of pseudo-unitary matrices that are self-adjoint positive as operators in the Euclidean CpCq . c) Show that this map coincides with . B) Next, let U.p; q/ act on itself by conjugations r 7! g 1 rg. Consider the manifold S U.p; q/ whose elements are matrices satisfying g 2 D 1. The eigenvalues of such matrices are ˙1. Lemma 3.11. The space S is in a one-to-one correspondence with the set of all regular subspaces W Cp;q . Proof. For a given W , define the map g (reflection in W ) given by g.w C y/ D w y;

where w 2 W , y 2 W ? :

Conversely, for a given g 2 S we consider the subspaces W WD ker.g 1/ and W ? D ker.g C 1/. Now we are ready to construct another embedding Mp;q ! U.p; q/. Let W be a negative subspace of maximal possible dimension (dim W D p). Then the corresponding g 2 U.p; q/ is the reflection in W . Considering arbitrary regular subspaces, we observe that the manifold S is the disjoint union of the homogeneous spaces a ı SD U.p; q/ U.k; l/ U.p k; q l/: k6p; l6q

All these spaces are interesting (and probably very interesting) objects8 but our character U.p; q/=U.p/ U.q/ is distinguished and is the most interesting among them. Problem 3.15. Find a relation between the constructions A) and B) (in fact, they more-or-less coincide). 7 An anti-involution in a group is a map G ! G such that g D g and .hg/ D g h for any g, h 2 G. 8 They are examples of pseudo-Riemannian symmetric spaces, see Subsection 12.1 and Addendum D.

86


2.4 The space U.n/. Cell decomposition 4.1 Coordinates on the Grassmannian of maximal isotropic subspaces. First, let p D q. We keep the notation of Subsection 3.8. Recall that H C and H are complementary maximal isotropic subspaces in Cp;p . Let R be a p-dimensional isotropic subspace. Assume that R \ H C D 0; then R is the graph of an operator T W H ! H C. Theorem 4.1. The following conditions are equivalent: 1) R is isotropic; 2) T D T (i.e., T is anti-Hermitian).

Proof. See the proof of Proposition 3.10.

4.2 Isotropic subspaces and unitary matrices. Next, we decompose the space Cp;p into the orthogonal sum of a negative subspace V and a positive subspace VC . Theorem 4.2. The following conditions are equivalent: 1) R is a p-dimensional isotropic subspace; 2) R is the graph of an operator z W V ! VC satisfying the condition zz D 1 (in other words, z 2 U.p/). We formulate this theorem here due to its importance, but it is already proved in Subsection 1.7.7. Corollary 4.3. The group U.p; p/ acts on the space U.p/ by the transformations g W z 7! z Œg D .a C zc/1 .b C zd /;

where z 2 U.n/, and g satisfies (2.4):

Remark. In this context, the groups U.p/ and SU.p/ behave in a completely different manner. Certainly, the group SU.p/ SU.p/ acts on SU.p/; the first copy of SU.p/ acts by left multiplications and the second one by right multiplications. But there is not an analog of the action of U.p; p/. Moreover, SU.p/ SU.p/ is a maximal Lie group acting on SU.p/. 4.3 The Stiefel manifolds and their groups of symmetries. Let p 6 q. Recall that the complex Stiefel manifold Sti.p; q/ is the space whose points are orthonormal frames e1 ; : : : ; ep in Cq . We are going to describe three groups acting transitively on this space. A) The group U.q/ acts on Cq , therefore it also acts on Sti.p; q/. Obviously, the action is transitive and the stabilizer of any point is isomorphic to U.q p/, i.e., Sti.p; q/ ' U.q/=U.q p/:

2.4. The space U.n/. Cell decomposition

87

B) Equivalently, Sti.p; q/ is the space of isometric embeddings z W Cp ! Cq . The group U.p/ acts on Cp and the group U.q/ acts on Cq . Therefore, U.p/ U.q/ acts on Sti.p; q/. Thus, Sti.p; q/ is a homogeneous space,

ı

Sti.p; q/ D U.p/ U.q/ U.p/ U.q p/ ; where the stabilizer U.p/ U.q p/ consists of pairs ² ³ h 0 h; 2 U.p/ U.q/; where h 2 U.p/, r 2 U.q p/: 0 r C) Thus, we have enlarged the group of symmetries of Sti.p; q/. But our previous arguments show that Sti.p; q/ admits a group of symmetries that is larger than U.p/ U.q/. Indeed, z 2 Sti.p; q/ satisfies zz D 1. A matrix z is a point of the U.p; q/-orbit O0 on the boundary in the matrix ball, see Proposition 3.6. Thus, there is a one-to-one correspondence Sti.p; q/

! set of all maximal isotropic subspaces in Cp;q .

(4.1)

Proposition 4.4. The group U.p; q/ acts on the Stiefel manifold Sti.p; q/ by the transformations z 7! .a C zc/1 .b C zd /. 4.4 Digression. The Dynkin cell decomposition of U.n/. In Theorem 4.1 we defined a coordinate system (a chart) on U.n/, but this system does not cover the whole group. Clearly, we also get an atlas on U.n/ whose charts are enumerated by pairs of transversal maximal isotropic subspaces in Cn;n . Nevertheless it remains the problem of description (and parametrization) of the complement of our chart. In what follows we briefly discuss this question. 4.4.a. Complex reflections. Let Cn be the standard complex Euclidean space with the standard basis e1 ; : : : ; en . Denote by Ck the subspace spanned by the first k basis vectors e1 ; : : : ; ek . We say that a complex reflection9 in a vector v with parameter ' is a linear map in Cn of the form S ŒvI e i' x D x C .e i' 1/hx; vi v; (4.2) where the vector v satisfies hv; vi D 1 and ' 2 .0; 2/; note that we forbid e i' D 1 for which the map is identical. Remark. Let us complete the vector v to an orthonormal basis v, f2 ; : : : ; fn . Then S ŒvI e i' v D e i' v;

and

S Œv; e i' fj D fj

for j > 2:

Thus, a reflection is a unitary operator with a unique eigenvalue ¤ 1. 9 For instance, any ’reflection’ z 7! e i' z in C1 is a rotation. But the term ‘reflection’ is generally accepted.

88


Problem 4.1. a) Let hx; xi D hy; yi D 1, let y ¤ x. Then there is a unique reflection S such that S x D y. b) The operator inverse to a reflection is a reflection. 4.4.b. The space of reflections. Denote by R the set of all reflections. Let 2 C, jj D 1. Clearly, S Œ v; e i' D S Œv; e i' . Therefore, R is diffeomorphic to the space R ' CP n1 .0; 2/; where CP n1 is the complex projective space. Denote by Rk the set of all reflections in vectors v 2 Ck n Ck1 . By definition, R is a disjoint union n a Rj : RD j D1

Each set Rj is (canonically) diffeomorphic to Rj ' CP j 1 n CP j 2 .0; 2/ ' Cj 1 .0; 2/: 4.4.c. Decompositions of unitary matrices into products of reflections. Now, we are going to show that each g 2 U.n/ admits a canonical decomposition into the product of 6 n reflections. Let g 2 U.n/. Denote gn WD g. First, suppose gen ¤ en . Consider the unique reflection Sn such that Sn en D gn en . Set gn1 WD Sn1 gn . If gn en D en , we simply put gn1 WD gn . Thus gn1 en D en . Next, we consider gn1 en1 . If gn1 en1 ¤ en1 , then we take the reflection Sn1 such that Sn1 en1 D gn1 en1 . It is important that Sn1 en D en (because en is orthogonal to 1 both en1 and gn1 en1 ). Put gn2 WD Sn2 g1 . If gn1 en1 D en1 , we set gn2 WD gn1 , etc. For an element g 2 U.n/ in general position, we get g D S1 S2 : : : Sn : Moreover, Sj 2 Rj . For certain exceptional elements g, some factors Sj can be omitted. More precisely, we obtain the following theorem: Theorem 4.5. For each g 2 U.n/ there is a unique collection k1 < k2 < < kl 6 n and unique Skj 2 Rkj such that g D Sk1 Sk2 : : : Skl (4.3) (the empty collection f¿g corresponds to g D 1). 4.4.d. The cell decomposition. Thus we represent U.n/ as the disjoint union of sets a a U.n/ D Rk1 Rkl DW Dk1 ;:::;kl : (4.4) k1 k > 0 are uniquely determined by L1 and L2 . We say that the numbers 'j .L1 ; L2 / D arccos j .L1 ; L2 / are angles between L1 and L2 . Remark. There are a lot of terms for this geometric structure, namely Jordan angles, stationary angles, complex distance, compound distance. Unfortunately, it is almost impossible to visualize this geometric structure, because a pair of subspaces in R3 has only one nontrivial angle. The first nontrivial case is a pair of two-dimensional planes in R4 . We denote by O.n/ the group of all orthogonal transformations of the real Euclidean space Rn . Theorem 5.2. a) For any g 2 O.n/, we have j .gL1 ; gL2 / D j .L1 ; L2 /: b) If L1 , L2 and M1 , M2 are k-dimensional subspaces, and j .L1 ; L2 / D j .M1 ; M2 / for all j ; then there is an orthogonal transformation g such that gL1 D M1 ;

gL2 D M2 :

2.5. Jordan angles and the Hua double ratio

93

Proof of Theorem 5.1. Consider an orthonormal basis u˛ 2 L1 and an orthonormal basis vˇ 2 L2 . Consider the matrix Aij D hui ; vj i. We may choose both the bases, therefore the matrix A is defined up to the equivalence A A ;

, 2 O.k/:

(5.1)

Using these transformations, we can reduce the matrix A to the diagonal form 0 1 1 0 : : : B C A D @ 0 2 : : :A ; where j > 0: :: :: : : : : : The numbers j are the singular numbers13 of the matrix A, therefore they are uniquely determined. Finally, the matrix A coincides with the matrix of the orthogonal projection L1 ! L2 . Hence kAk 6 1 and l 6 1. Proof of Theorem 5.2. Choose a system of vectors e1 ; : : : ; ek , f1 ; : : : ; fk for L1 , L2 . Choose an orthonormal basis r in .L1 CL2 /? . Choose a similar system e˛0 , fˇ0 , r0 for M1 , M2 . It remains to take the orthogonal operator g such that ge˛ D e˛0 , gfˇ D fˇ0 , gr D r0 . Remark. There is a minor difference between the cases 2k 6 n and 2k > n. In the first case, for subspaces in general position, we have j ¤ 0. In the second case, L1 \ L2 ¤ 0; therefore, at least 2k n angles equal to 0 by definition. Corollary 5.3. Let 2k 6 n. Then there are k pairwise-orthogonal two-dimensional planes W1 ; : : : ; Wk such that each Wj intersects both L1 , L2 non-trivially and is perpendicular to both of them. Problem 5.1. a) j .L1 ; L2 / D j .L2 ; L1 /. ? b) The non-zero angles 'j .L? 1 ; L2 / coincide with the non-zero angles 'i .L1 ; L2 /. c) Describe the possible stabilizers O.n/ of pairs L1 and L2 . 5.2 Reformulations of the definition of angles. Denote by P1 (resp. P2 ) the orthogonal projection onto L1 (resp., L2 ). Proposition 5.4. a) The numbers cos 'j are the singular values of the operator ˇ P1 ˇL W L2 ! L1 : 2

b) The eigenvalues of the operator P1 P2 P1 are cos2 '1 ; : : : ; cos2 'k ; 0; 0; : : : : 13


94


The proof is an exercise. Problem 5.2. Show that the definition of angles survives for any pair of subspaces, even having different dimensions. L2

L2

L1

L1

P1

Figure 2.12. Reference Subsection 5.2. The definition of angles in terms of projectors.

5.3 The minimax principle Theorem 5.5. Let A be a self-adjoint operator in an n-dimensional Euclidean space W and denote by 1 > 2 > > n its eigenvalues. Then

®

¯ min hAv; vi ; dim V Dl v2V; kvkD1 ¯ ® l D min max hAv; vi ;

l D max

dim V DnlC1

v2V; kvkD1

(5.2) (5.3)

where V ranges in the set of all subspaces of W . Proof. Prove the first identity. Let e1 , e2 , …, be an eigenbasis for A. Consider the subspace H spanned by el , elC1 , …(its dimension is .n l C 1/). We have hAv; vi 6 l hv; vi

for all v 2 H .

Let V be an arbitrary l-dimensional subspace of W . The intersection V \ H is not zero (because dim V C dim H > n). Let h be a unit vector in V \ H . We have hAh; hi 6 l , hence min hAv; vi 6 l : v2V; kvkD1

Therefore, in (5.2) maxf: : : g 6 l . On the other hand, maxf: : : g > l because we can choose for V the subspace spanned by the vectors e1 ,…,el . Changing A to .A/, we obtain (5.3). Problem 5.3. (Rayleigh Theorem)14 Consider an ellipsoid in R3 with semi-axes a1 > a2 > a3 . Consider its elliptic section by a plane through 0. Show that semiaxes of this section satisfy the condition of interlacing a1 > b1 > a2 > b2 > a3 : 14

Originally it was a statement about the behavior of frequencies of an oscillating system with constraints.


95

Problem 5.4. Let V and W be Euclidean spaces. Let B W V ! W be an operator, denote by 1 > 2 > its singular values. Then ˚

l D max min kAvk : (5.4) dim SDl

v2S; kvkD1

Now we formulate the corresponding principle for the angles (it is similar to the common definitions of angles in R3 .) Proposition 5.6. Let L1 and L2 be k-dimensional subspaces in a Euclidean space. Then ˚

'k .L1 ; L2 / D min †.h; L1 / ; max V L2 ; dim V Dk

h2V

where †.h; L1 / is the angle between a vector h and the subspace L1 15 . This follows from the previous problem and Proposition 5.4a. Remark. In what follows we also use a more complicated form of the same statement, ˚ 'k D min max †.u1 ; u2 / ; max (5.5) V L2 ; dim V Dk

u2 2V

u1 2L1

where †.u1 ; u2 / is the angle between h and u. 5.4 The angles in pseudo-Euclidean spaces. Let p 6 q. Consider the space Cp;q and two points of L1 , L2 2 Mp;q (i.e., maximal negative16 subspaces, see § 2.3). We can regard such subspaces as Euclidean spaces with respect to the inner product h; i. Theorem 5.7. a) There are orthonormal bases ei 2 L1 , fj 2 L2 such that hek ; el i D ıkl ; hfk ; fl i D ıkl ; hei ; fj i D i ıij :

(5.6)

The numbers 1 > 2 > > 1 are uniquely determined by the subspaces L1 and L2 . b) For negative subspaces L1 , L2 , M1 , M2 , the following conditions are equivalent: – j .L1 ; L2 / D j .M1 ; M2 / for all j , – there is g 2 U.p; q/ such that gL1 D M1 and gL2 D M2 . The proof is the same as for Theorem 5.1, we must only show that the singular values j are > 1. For this purpose, consider the subspace H˛ WD L1 CCf˛ . It is the orthogonal direct sum of the .k1/-dimensional subspace ˚j ¤˛ Cej and the plane (or line17 ) Ce˛ CCf˛ . 15

I.e., the angle between h and its projection onto L1 . Observe that we consider only negative subspaces. 17 If e˛ D f˛ . In this case, ˛ D 1. 16

96


If 0 6 ˛ < 1, then Ce˛ C Cf˛ is a strictly negative two-dimensional plane. Hence H˛ is a .p C 1/-dimensional negative subspace in Cp;q . This contradiction concludes the proof. 18 Further, we define the numbers (hyperbolic angles) as q j2 1 : j D arccosh.j / D ln j C Problem 5.5. Convince yourself that all the considerations of previous subsections survive for hyperbolic angles. 5.5 Angles and coordinates. Now identify the set Mp;q of negative subspaces with the matrix ball Bp;q as above. Theorem 5.8. Let L, M 2 Mp;q , let z, u be their respective matrix coordinates. a) The collection cosh j .L; M / coincides with the collection of singular values of the matrix ƒ.z; u/ D .1 zz /1=2 .1 zu /.1 uu /1=2 : (5.7) b) The collection cosh2 the matrix

j .L; M /

coincides with the collection of eigenvalues of

„.z; u/ D .1 zz /1 .1 zu /.1 uu /1 .1 uz /:

(5.8)

Proof. a) Write V WD Cp ˚0 as above (Subsection 3.1). Consider the map Rz W V ! Cp;q given by v 7! v.1 zz /1=2 ˚ v.1 zz /1=2 z: We claim that Rz is an isometric embedding V ! Cp;q . Indeed, hvRz ; vRz i D v.1 zz /1 v C v.1 zz /1 zz .1 zz /1 v D vv : The image of this map is the subspace L. Therefore, for an orthonormal basis ej 2 V , the system of vectors aj WD ej Rz is an orthonormal basis in L. In the same way, the system of vectors bj WD ej Ru is an orthonormal basis in M . We write out the matrix hai ; bj i as in the proof of Theorem 5.1 and get hai ; bj i D ai .1 zz /1=2 .1 uu /1=2 bj C ai .1 zz /1=2 zu .1 uu /1=2 bj D ai .1 zz /1=2 .1 zu /.1 uu /1=2 bj : This is the statement a). 18 In elementary geometry an angle is the length of an arc of the unit circle. Now consider the hyperbolic plane R1;1 ; by definition, it is equipped with pseudo-Riemannian metric dx 2 dy 2 . Let v and w be vectors such that hv; vi D 1, hw; wi D 1, hv; wi > 0. Then the hyperbolic angle †.v; w/ is the length of the arc of the hyperbola hx; xi D 1 connecting v and w.

97


b) The squares of singular values of ƒ.z; u/ are the eigenvalues of the matrix ƒ.z; u/ƒ.z; u/ D .1 zz /1=2 .1 zu /.1 uu /1 .1 uz /.1 zz /1=2 D .1 zz /1=2 „.z; u/.1 zz /1=2 : We observe that the matrices ƒƒ and „ are conjugate. Therefore, they have the same eigenvalues. Problem 5.6. By construction, the matrices „.z; u/ and „.z Œg ; uŒg / have the same eigenvalues for any g 2 U.p; q/. Show directly that these two matrices are conjugate. 5.6 The classical double ratio. Let t1 , t2 , t3 , t4 be four distinct points on the projective line. Recall that their double ratio is the number .t1 ; t2 ; t3 ; t4 / D

t1 t3 t2 t3 W : t1 t 4 t2 t 4

For two of its geometric definitions, see Figure 2.13. Problem 5.7. There are 24 permutations of the points t1 , t2 , t3 , t4 . a) The permutations .12/.34/, .13/.24/, .14/.23/ do not change the double ratio. b) There are six numbers that can be obtained from D .t1 ; t2 ; t3 ; t4 / by permutations of the arguments tj , namely ;

`2

1 ;

1 ;

`4

`2 `3

a)

`1

1 1 ;

.1 /1 ;

`2

`4 `3 `1

.1 /1 :

`4 W y D x

`2

`3 W y D x `1

`4 `3

b)

`1

Figure 2.13. Reference Subsection 5.6. Given four lines on the plane. a) Let us apply linear transformations of the plane. We send `1 to the horizontal axis and `2 to the vertical axis. Using hyperbolic rotations (they preserve the axes), we send `3 to the line y D x. Now the position of the line `4 W y D x is fixed and D .`1 ; `2 ; `3 ; `4 / is the double ratio. b) Let us consider the pair of lines (`1 , `2 ) as coordinate axes, then `3 , `4 are graphs of certain operators B; C W `1 ! `2 . We consider the operator B 1 C W `1 ! `1 . It is a multiplication by a canonically defined constant .

5.7 The Hua double ratio. Now, let W be a .p Cq/-dimensional linear space. Let K1 , K2 be subspaces of dimension q; let L1 , L2 be subspaces of dimension p. Suppose this quadruple of subspaces is in general position. For us the “general position” means that K2 \ L1 D 0; K1 \ L2 D 0: (5.9)

98


Then K2 is a graph of an operator P W K1 ! L1 and L2 is a graph of an operator Q W L1 ! K1 : The Hua double ratio „.K1 ; L1 ; K2 ; L2 / of the quadruple of subspaces is the linear operator „.K1 ; L1 ; K2 ; L2 / WD QP W K1 ! K1 : Theorem 5.9. a) The eigenvalues 1 ; : : : ; p of „ are invariants of the quadruple of subspaces under linear transformations of W . b) Let p 6 q. Consider two quadruples .K1 ; L1 ; K2 ; L2 / and .K10 ; L01 ; K20 ; L02 /. Let the corresponding collections of eigenvalues .1 ; : : : ; p /, .01 ; : : : ; p0 / coincide and let j be pairwise distinct non-zero numbers.19 Then one can find an invertible operator g in the ambient space W that sends one quadruple to another. Remark. If p > q, then rk „./ 6 q, and therefore „./ has at least .p q/ zero eigenvalues. Proof. a) is obvious, since the operator „./ is canonically defined. b) Consider an eigenbasis e1 , e2 ; : : : ; ep of the operator „.K1 ; L1 ; K2 ; L2 /. Set fj WD P ej . Denote by Yj the plane spanned by ej and fj . In this plane, we have four distinguished lines: `1 D Cej D Yj \ K1 ; `3 D C.ej C fj / D Yj \ K2 ;

`2 D Cfj D Yj \ L1 ; `4 D C.fj C ej / D Yj \ L2 :

We have W D Y1 ˚ ˚ Yp ˚ .L1 \ L2 /: For the other quadruple, one can find a similar decomposition with the same picture in each summand. Corollary 5.10. Consider four n-dimensional subspaces V1 , V2 , V3 , V4 in general position in a 2n-dimensional L linear space W . There are n two-dimensional planes Y1 ,…,Yn such that W D Yj and each Yj intersects all the Vi , i D 1; : : : ; 4, nontrivially. Example. In C4 , consider four two-dimensional planes in general position. Equivalently, consider four pairwise skew lines `1 , `2 , `3 , `4 in the projective space CP 3 . For each point z 2 CP 3 , one can find a unique line m D mz containing z and intersecting 19

This is a property that holds in general position.


99

`1 and `2 . All lines intersecting `1 , `2 , `3 sweep a quadric in CP 3 (in our real20 life it is called a hyperboloid). A generic line `4 intersects this quadric at two points. Thus, there are two lines m1 , m2 intersecting the four lines `1 , `2 , `3 , `4 . We have a configuration of four points in each of the lines m1 , m2 (these points are the intersections m1 \ `j , m2 \ `j ). Thus, we have the double ratio on each of the lines m1 , m2 . This pair is the Hua double ratio. Problem 5.8. Interpret in terms of the Hua ratio the following positions of four lines: – four skew lines in a hyperboloid; – a given line `4 is tangent to the hyperboloid containing the lines `1 , `2 , `3 . 5.8 Matrix expressions for double ratio. Fix a decomposition of the ambient space W into the direct sum W D X ˚ Y of a p-dimensional subspace X and a q-dimensional subspace Y . The subspace K1 (resp. K2 ) is a graph of a certain linear operator A1 W X ! Y (resp. A2 W X ! Y ). In other words, ˚

˚

K1 D x ˚ A1 x; where x 2 X I K2 D x ˚ A2 x; where x 2 X : Likewise, the subspaces L1 , L2 are graphs of operators B1 , B2 W Y ! X : ˚

˚

L1 D B1 y ˚ y; where y 2 Y I L2 D B2 y ˚ y; where y 2 Y : Theorem 5.11. The double ratio of this quadruple is „.K1 ; L1 ; K2 ; L2 / D .1 B2 A1 /1 .B2 B1 /.1 A1 B2 /1 .A2 A1 /:

(5.10)

In this formula, we identify X and K1 by bijection A1 W X ! K1 ; the operator „ is written as an operator X ! X . Proof. We must write out the operators P and Q from the previous subsection. Let us decompose any vector h ˚ A2 h 2 K2 as the sum of vectors lying in K1 and L1 :

h ˚ A2 h D x ˚ A1 x C B1 y ˚ y ; i.e., h D x C B1 y;

A2 h D A1 x C y:

Excluding h, we obtain an expression for the operator P , y D .1 A2 B1 /1 .A2 A1 /x: Decomposing a vector B2 ˚ 2 L2 as a sum of vectors lying in K1 and L1 , we obtain a formula for the operator Q: x D .1 B2 A1 /1 .B2 B1 /y: The last two formulas imply the desired result. 20

In R3 .

100


Now, let the dimension of the ambient space W be 2n and let all subspaces from our quadruple K1 , K2 , K3 , K4 be of the same dimension n. Decompose W into the direct sum of n-dimensional subspaces: W D X ˚ Y . Now we can write out all the subspaces Kj in a uniform way. Thus, let Kj consist of vectors x ˚ Sj x, where j D 1, 2, 3, 4. Then „.K1 ; K2 ; K3 ; K4 / D .S4 S1 /1 .S4 S2 /.S3 S2 /1 .S3 S1 /I

(5.11)

we merely substitute A1 7! S1 , B1 7! S21 , A2 7! S2 , B2 7! S41 into (5.10). Proposition 5.12. Denote by j the eigenvalues of the matrix (5.11). Permutating S1 , S2 , S3 , S4 we can obtain one of the following six collections of numbers: ˚ ˚ 1 ˚

˚

˚

˚

j ; j ; 1 j ; 1 j1 ; .1 j /1 ; j .1 j /1 : Proof. A straightforward verification of the behavior of (5.11) under permutations of arguments is a pleasant exercise. But the statement is trivial; it follows from Corollary 5.10 and Problem 5.7, 5.9 The Jordan angles and the Hua ratio. Let us return to the situation discussed at the beginning of this section. We have two subspaces L1 and L2 in a Euclidean space ? with angles '1 ; : : : ; 'l . Also, we have a quadruple L1 , L2 , L? 1 , L2 . Theorem 5.13. The Hua double ratio of this quadruple coincides with the collection of numbers cos2 '1 ; : : : ; cos2 'k . We leave this as an exercise for the reader. 5.10 Digression. An example of visualization of an angle-type structure Problem 5.9. Consider the 3-dimensional sphere S 3 WD R3 [ 1 with the action of the conformal group O.4; 1/. Find invariants of pairs of circles C1 , C2 S 3 under this action. Hint. Points of S 3 are in a one-to-one correspondence with isotropic lines in the pseudoEuclidean space R4;1 . Circles are in a one-to-one correspondence with 3-dimensional planes whose inertia is .2; 1/. Such a pair of planes together with their orthocomplements have Hua double ratio. Interpret this double ratio in terms of geometry of our R3 . See [39].

2.6 Classification of pseudo-unitary and pseudo-self-adjoint operators 6.1 Formulation of problems. We intend to solve the following three problems: 1B . Find canonical forms of self-adjoint operators in pseudo-Euclidean spaces. More precisely, the group U.p; q/ acts on the set of self-adjoint operators (we also

2.6. Classification of pseudo-unitary and pseudo-self-adjoint operators

101

use the term pseudo-Hermitian operators) by conjugations g W A 7! g 1 Ag. Classify orbits and find canonical forms. 2B . Find canonical forms of pseudo-unitary operators. In other words, we intend to classify conjugacy classes in the group U.p; q/. 3B . Find canonical forms of pairs of Hermitian forms on a given linear space V under the action of the general linear group GLŒV . 6.2 The self-adjoint operators in general position

Figure 2.14. The spectrum of a pseudo-Hermitian operator is symmetric about the real axis. The spectrum of a pseudo-unitary operator is symmetric with respect to the circle jj D 1, i.e., the N spectrum is stable under the inversion 7! 1=.

Theorem 6.1. Let A be a self-adjoint operator in V with distinct eigenvalues. a) The spectrum of A is symmetric about the real axis, i.e., it has the form 1 ; 2 ; : : : ; 1 ; N 1 ; 2 ; N 2 ; : : : ; b) There is a basis e1 ; e2 ; : : : ,

where j 2 R, k 2 C n R:

f1 ; f10 ; f2 ; f20 ; : : :

hej ; ej i D ˙1;

in V such that

hfk ; fk0 i D 1

(6.1)

(all other pairwise scalar products being 0) and Aej D j ej ; Afk D k fk ;

Afk0 D N k fk0 :

Proof. Denote by v the eigenvector corresponding to an eigenvalue . We have hAv ; v i D hv ; Av i H) hv ; v i D hv N ; v i: Therefore, we have the alternative: either

hv ; v i D 0

or

D : N

In particular, if … R, then hv ; v i D 0. Since our form is nondegenerate, for each v , one can find v such that hv ; v i ¤ 0. The only possibility is D . N For real, this implies hv ; v i ¤ 0. If is not real, then D N must be an eigenvalue.

102


Problem 6.1. Let A 2 U.p; q/ be a pseudo-unitary operator with a simple spectrum. a) Show that the spectrum is symmetric with respect to the circle jj D 1, i.e., 2 Spec.A/ () N 1 2 Spec.A/. b) Show that there exists a basis e1 ; e2 ; : : : , f1 ; f10 ; f2 ; f20 ; : : : in V as above (6.1) such that A is Aej D j ej ; Afk D k ;

jj j D 1; Afk0 D Nk1 fk0 ;

jk j ¤ 1:

6.3 A reformulation of the problem. Often (and even usually) Theorem 6.1 suffices to understand a relevant situation. However, we start an examination of arbitrary selfadjoint operators in V . We say that a self-adjoint operator A in V is decomposable if there are A-invariant subspaces W and Y such that W ˚ Y D V and W D V ? . In particular, V and W must be regular. Problem 6.2. An operator A is decomposable if and only if there exists a proper A-invariant regular subspace W . Example. The self-adjoint operator 0i 0i in C1;1 is indecomposable. Its eigenvalues are ˙i , but the eigenvectors .1; ˙1/ are isotropic and non orthogonal. Thus the problem of description of canonical forms can be formulated as follows: Describe all indecomposable self-adjoint operators in all pseudo-Euclidean spaces. 6.4 The classification theorem Theorem 6.2. There are two types of indecomposable self-adjoint operators, In˙ ./, and IIn ./: – The type In˙ ./, where 2 R, n 2 N. There is a basis e1 ; : : : ; en with scalar products 21 hek ; el i D ˙ıkCl;nC1 : (6.2) The operator A is the following Jordan block: Aej D ej C ej C1 ; Aen D en :

for j ¤ n;

– The type IIn ./, where Im > 0, n 2 N. There is a basis f1 ; : : : ; fk , f10 ; : : : ; fk0 with scalar products hfi ; fj i D hfi0 ; fj0 i D 0I 21

hfi ; fj0 i D ıiCj;nC1 :

ıkl denotes the Kronecker symbol, it equals 1 if k D l and zero otherwise.


103

The operator A is the following “pair of dual Jordan blocks”: Afj D fj C fj C1 ; where j ¤ n; Afn D fn ;

0 Afk0 D f N k0 C fk1 ; 0 0 Af1 D f1 :

where k ¤ 1;

Since the description seems intricate, let us clarify it. 1) An indecomposable operator with a real eigenvalue is a Jordan block. Moreover, each Jordan block A with a real eigenvalue admits a unique to within a scalar factor pseudo-Euclidean scalar product making A self-adjoint. 2) Let us give a verbal description of an operator of type II. Consider the subspaces W and W 0 spanned respectively by f1 ; : : : ; fn and f10 ; : : : ; fn0 . They are isotropic and dual to each other (see Subsection 1.10). Further, W , W 0 are invariant and the restrictions of A to W and W 0 are Jordan blocks. These Jordan blocks are dual operators in the dual spaces W and W 0 . The theorem claims that nothing else can occur for nonreal eigenvalues. 3) Now let us discuss ˙ in (6.2). We have seen this sign above as an invariant of operators with simple spectrum (see Theorem 6.1) It is easy to distinguish InC ./ and In ./ if n D 2k C1, because the ambient pseudoEuclidean spaces have different inertia indices(namely, .k C 1; k/ and .k; k C 1/). Generally, consider a cyclic vector22 h of the Jordan block A . Then the sign of h.A /n1 h; hi is an invariant of a Jordan block A. 6.5 The symmetry of the spectrum Lemma 6.3. The spectrum of a self-adjoint operator is symmetric about the real axis. That is, if 2 Spec A, then N is also an element of the spectrum of the same multiplicity. Proof. The condition A~ D A of self-adjointness can expressed as JA D A J , 1 be 0 where A denotes the usual adjoint matrix and J D 0 1 . Let be an eigenvalue of A. Then

0 D det.A / D det.J 1 / det.A / det.J / D det.J 1 AJ / D det.A /: Therefore, the spectrum of A coincides with the spectrum of A .

6.6 The root decomposition. Consider a self-adjoint operator A in a pseudo-Euclidean space V . Denote by V the root subspace corresponding to an eigenvalue , V D ker.A /N

for sufficiently large N :

Lemma 6.4. The space V is the orthogonal direct sum of regular subspaces of the following form: 22


104


– V with 2 R; – V C VN with 2 C n R; the summands V and VN are isotropic and have the same dimension. Proof. Consider v 2 V and w 2 V . Then 0 D h.A /N v; wi D hv; .A / N N wi: The operator .A / N N is invertible in all root spaces V , except VN . Hence, V ?V

for ¤ : N

Therefore, V is an orthogonal direct sum of subspaces of the forms V and V ˚VN ; recall that 2 R and … R. Since the form h; i is nondegenerate, it follows that all summands of the orthogonal decomposition must be regular. The spaces V and VN are isotropic. Since their sum is regular, they have the same dimensions (because the dimension of an isotropic subspace is 6 one-half of the dimension of the ambient space).23 Remark. Let yk be a basis in V and ykB the dual basis in VN . Then the matrix of the operator A in V ˚ VN has the form H0 H0 . Thus, Theorem 6.2 on canonical forms reduces to the following two cases: a) an operator having a unique real eigenvalue , b) an operator with two complex conjugate eigenvalues. These cases are examined in Subsections 6.7–6.9. Problem 6.3. Prove a counterpart of Lemma 6.4 for pseudo-unitary operators. 6.7 Indecomposable operators. Complex eigenvalues. Let 2 C n R. Consider a self-adjoint operator whose eigenvalues are f; g. N Recall that the subspaces V and VN are isotropic and dual with respect to the scalar product. Assume that the restriction of A to V is not a Jordan block, i.e., V can be decomposed into a direct sum Y ˚ Z of two A-invariant subspaces. Choose a basis yj 2 Y and a basis zk 2 Z. Denote by yjB , zkB the dual basis in V . Then the vectors of the form yj , yiB span a regular A-invariant subspace. Thus an indecomposable operator A has a unique Jordan block in V . Choosing a Jordan basis in V and the dual basis in VN , we obtain the desired canonical form. 6.8 Indecomposable operators. Real eigenvalues Lemma 6.5. Any indecomposable self-adjoint operator with a real eigenvalue is a Jordan block. 23

This also gives another proof of Lemma 6.3.

105


Let A be a self-adjoint operator in a space V , let 2 R be the only eigenvalue of V . To simplify the notation, we set D 0; otherwise, we pass to the operator AQ WD A . First, we formulate two simple problems. Problem 6.4. Let A be a nilpotent24 Jordan block in a space W , h a cyclic vector. Then all A-invariant subspaces have the form Wk D C Ak h ˚ C AkC1 h ˚ ; or, equivalently, Wk D Ak W . Problem 6.5. Let Z be a space equipped with a degenerate Hermitian form S , and A be a linear operator self-adjoint with respect to S . Show that the kernel of S is A-invariant. Proof of Lemma 6.5. Denote by N the maximal exponent such that AN ¤ 0. Problem 6.6. .ker AN /? D im AN .

A ?

r r ' r r? r r ? r r? r ? r &

r r r r r

N r V = ker A r r r r r r r r r r? r r ? r r r r r r @ @ im AN

$ r r

ker AN %

Figure 2.15. Thick dots correspond to elements of a Jordan basis. The operator A is the operator of the shift down. Any vertical chain spans an invariant subspace.

Let us define the form B by B.v; w/ WD hAN v; wi

for any v; w 2 V :

(6.3)

Since A is self-adjoint, the form B is Hermitian. Obviously, ker B D ker AN . Thus, we can regard B as a nondegenerate Hermitian form on V = ker AN . In particular, there exists a vector h 2 V such that B.h; h/ ¤ 0:

(6.4)

Lemma 6.6. The subspace W spanned by the vectors h, Ah; : : : ; AN h is regular. 24


106


Proof. By construction, the operator A restricted to W is a Jordan block. Suppose W is singular. By Problem 6.5, the kernel zero.W / is an A-invariant subspace. By Problem 6.4 zero.W / contains AN h. But this contradicts the choice of the vector h, see (6.4). This completes the proof of the last lemma. Thus we have constructed a regular invariant subspace for A. This completes the proof of Lemma 6.5. 6.9 The end of the proof of Theorem 6.2. Nilpotent Jordan blocks. Due to Lemma 6.5, it remains to examine scalar products compatible with nilpotent Jordan blocks. Thus, let A be a self-adjoint Jordan block such that AN ¤ 0 and AN C1 D 0, let e0 be an A-cyclic vector. We write ek WD Ak e0 and ckl WD hek ; el i; see Figure 2.16. In particular, ˚

k > N or l > N ) ckl D 0

(6.5)

(because eN C1 D 0). By the self-adjointness of A, ck;lC1 D hek ; elC1 i D hek ; Ael i D hAek ; el i D ckC1;l :

(6.6)

Therefore c0m D c1;m1 D c2;m2 D :

l 6 0 0 0 0 0 s c c c c s@@s c c c s s@ s c c @ s s s @s c s s s s@s

l 6

0 0 0 0 0

-k

0 0 0 0 0 s c c c c c@@cs c c c c c@ cs c c @ c c c @cs c c c c c@cs

0 0 0 0 0

-k

Figure 2.16. The picture on the left-hand side: The coefficients ckl are constant along the lines k C l D const. For positions marked by white circles, ckl D 0. All coefficients on the black positions, except those on the marked diagonal, can be ‘killed’. We obtain a picture on the right-hand side.


107

We write

kCl WD ckl : By (6.5), it follows that p D 0 for p > N . Also, ckl D hek ; el i D hel ; ek i D cNlk : Hence kCl 2 R. A cyclic vector e0 is not unique, we can replace it by an arbitrary vector of the form eQ0 D s0 e0 C s1 e1 C s2 e2 C ;

where si 2 C, s0 ¤ 0:

Since N is real, choosing eQ0 D s0 e0 , we may assume N D ˙1. Next, choose eQ0 D e0 C s1 e1 . Then

Q N 1 D heQ0 ; eQN 1 i D he0 C s1 e1 ; eN 1 C s1 eN i D N 1 C .s1 C sN1 / N : N 1 Since n1 ; n 2 R, we choose s1 WD 2 . We get Q N 1 D 0. N Thus, assume N 1 D 0 and vary e0 as eQ0 D e0 C s2 e2 . Then

Q N 2 D he0 C s2 e2 ; eN 2 C s2 eN i D N 2 C .s2 C sN2 / N : Choosing s2 in an appropriate way, we ‘kill’ N 2 . Etc. This completes the proof of Theorem 6.2. 6.10 The conjugacy classes in the group U.p; q/. We say that g 2 UŒV is decomposable if V can be decomposed into the direct sum of two orthogonal invariant subspaces. In other words, an operator is decomposable if there is a g-invariant regular subspace W V . t sinh t Example. The operator cosh sinh t cosh t 2 U.1; 1/ is indecomposable. Its eigenvalues are e ˙t , but the eigenvectors .1; ˙1/ are isotropic. The classification of conjugacy classes reduces to a description of all indecomposable pseudo-unitary operators. Theorem 6.7. There are two types of indecomposable pseudo-unitary operators: – exp.iA/, where A is of the form In˙ ./ and 2 Œ0; 2/; – exp.iB/, where B is of the form IIn ./, where 0 6 Re < 2. Proof. 1. The spectrum is symmetric with respect to the circle jj D 1, i.e., 2 Spec.g/ implies N 1 2 Spec g and these eigenvalues have the same multiplicity. 2. Denote by V the root space corresponding to an eigenvalue . The space V is the orthogonal direct sum of subspaces of the following form: • V , where jj D 1;

108


• V ˚ VN 1 , where j j ¤ 1. 3. Let jj ¤ 1. The arguments of Subsection 6.7 can be applied in our case. We get a Jordan block J in the space V and the operator J 1 in the dual subspace VN 1 . 4. Consider the case jj D 1. We may assume that V D V . The operator g 0 D 1 g is also pseudo-unitary. Its unique eigenvalue is 1. We define its logarithm as X 1 ln g 0 WD .1 g 0 /j : j j D0

Obviously, this sum is finite. Problem 6.7. Show that

1 i

ln g 0 is a self-adjoint operator.

Comment: this is not totally trivial.

This problem completes the proof.

6.11 A pair of Hermitian forms. Now let B1 and B2 be a pair of Hermitian forms on a complex linear space V . We wish to find canonical forms of such pairs with respect to the action of the general linear group GLŒV . Assume that one of the forms, say B1 , is nondegenerate. Proposition 6.8. There is a unique operator T such that B2 .v; w/ D B1 .v; T w/: Moreover, T is B1 -pseudo-Hermitian. Thus our problem is equivalent to the description of canonical forms of pseudoHermitian operators. The latter problem is solved above. 6.12 Invariants of a pair of Hermitian forms. Now let Q1 and Q2 be matrices of the forms B1 , B2 . Changing a basis, we transform these two matrices as Q1 7! g Q1 g;

Q2 7! g Q2 g:

(6.7)

Theorem 6.9. The roots of the equation det.Q1 Q2 / D 0 are invariant under the transformation (6.7). Proof. det.g Q1 g g Q2 g/ D det.g / det.Q1 Q2 / det.g/. 6.13 Some problems Problem 6.8. Convince yourself that you can solve (some of) the following problems: a) Classification of conjugacy classes in the complex orthogonal group O.n; C/.

2.7. Indefinite contractions

109

b) Canonical forms of a pair of symmetric bilinear forms on a complex linear space (if one of these forms is nondegenerate). c) Canonical forms of a pair f symmetric bilinear form, skew-symmetric bilinear form g on a complex linear space (if one of these forms is nondegenerate). Problem 6.9. a) Let G D U.2/. For a given g 2 G find all elements X of the corresponding Lie algebra g such that exp.tX / D g. b) Solve the same problem for G D SU.1; 1/. When does a solution not exist? When is the solution not unique?

2.7 Indefinite contractions 7.0 Contractive operators in Euclidean spaces. First, we recall several definitions and elementary statements about operators in Euclidean spaces. An invertible operator g in a complex Euclidean space is said to be a contraction if hvg; vgi 6 hv; vi

for all v 2 V

or, equivalently, kgk 6 1. An operator X in a Euclidean space is said to be dissipative if RehXv; vi 6 0. Proposition 7.1. An operator X is dissipative if and only if it can be represented as X D A C B, where A D A is anti-Hermitian, and B D B is Hermitian and negative semi-definite. Theorem 7.2. The following conditions for an operator X are equivalent: – X is dissipative; – the semigroup exp.tX /, where t > 0, consists of contractions. Theorem 7.3. Each contractive invertible operator g admits a unique decomposition g D U exp.X /, where U is unitary and X is self-adjoint and semi-positive. Problem 7.1. Verify these statements. For instance, to prove Theorem 7.3, we consider the polar decomposition g D US and set X WD ln S. Since S is positive definite, it follows that its logarithm is well defined. Since kSk 6 1, the eigenvalues of S satisfy 0 < j 6 1. Therefore, ln j 6 0, and hence X is negative semi-definite. Remark. Let g be an arbitrary operator. Then s g is contractive for some s ¤ 0. This is not the case for pseudo-Euclidean contractions defined below. 7.1 Indefinite contractions. Let V ' Cp;q be a pseudo-Euclidean space. An invertible operator g W V ! V is an indefinite contraction if hvg; vgi 6 hv; vi

for all v 2 V :

110


We denote the set of all indefinite contractions in V by UŒV or U.p; q/. Obviously, U.p; q/ is a semigroup, i.e., for each g1 , g2 2 U.p; q/, we have g1 g2 2 U.p; q/. We say that g is a strict contraction if hvg; vgi < hv; vi

for all non-zero v 2 V :

Let the inner product in V be given by the formula hv; wi WD vQw . Proposition 7.4. g 2 U.p; q/ if and only if Q gQg > 0.

Proof. Obvious. 7.2 Several examples Example 1. Any pseudo-unitary operator is a contraction.

Example 2. Let A W Cp ! Cp and B W Cq ! Cq be invertible linear operators, 1 kAk 6 1, kBk 6 1. Then A0 B0 is an indefinite contraction in Cp;q . Example 3. Consider a two-dimensional space with a basis e1 , e2 such that he1 ; e1 i D he2 ; e2 i D 0;

he1 ; e2 i D 1

(7.1)

and the operator (a Jordan block) A t e1 D e1 t e2 ;

A t e2 D e2 ;

where t > 0

(7.2)

(this operator is I2C .1/ in the classification of Subsection 6.4). For v D ˛e1 C ˇe2 , we have hv; vi hA t v; A t vi D h˛e1 C ˇe2 ; ˛e1 C ˇe2 i h˛e1 C .ˇ t ˛/e2 ; ˛e1 C .ˇ t ˛/e2 i D 2t j˛j2 > 0: Thus, A t is a contraction, but not a strict contraction. Remark. Theorem 7.7 obtained below, shows that the operators of these three types generate the semigroup U.p; q/. Remark. Denote by C V the cone consisting of non-positive vectors v. A contractive operator g sends C to C . For a converse statement, see Theorem 7.13. The next example illustrates our argument that is used below for proofs of noncontractivity.


111

e2 e1

Figure 2.17. Reference Example 3. We draw R1;1 instead of C1;1 . The axes are isotropic, and the level lines hv; vi D const are hyperbolas. The signs ˙ indicate quadrants with positive/negative scalar squares. The operator A t fixes the vertical axis and shifts the lines x D c by .ct / (see the left-hand figure). The contractivity of A t is seen from the right-hand figure.

Example 4. In the notation of Theorem 6.2, consider the indecomposable pseudo-selfadjoint operator of type I4C ./. In other words, we have a basis e1 ; : : : ; e4 with scalar products hek ; el i D ıkCl;5 and the matrix 0 1 0 0 0 B1 0 0C C ADB @0 1 0A : 0 0 1 P Let v D ˛j ej . Then hv; vi hAv; Avi D 2 Re.˛1 ˛N 4 C ˛2 ˛N 3 / 2 Re.˛1 .˛N 4 C ˛N 3 / C .˛2 C ˛1 /.˛N 3 C ˛N 2 // 80 1 0 19 0 1 0 0 0 1 0 1 2 2 > ˆ ˆ > ˛N 1 ˆ ˆ > : ; 1 0 0 0 2 0 0 0 ˛N 4 The matrix in the curly brackets is not semi-positive. Indeed, for a Hermitian semipositive matrix R, the following inequality for the minors holds: ri i rij D ri i rjj jrij j2 > 0: det rj i rjj If rjj D 0, then rij D 0. Thus our matrix has too many zeros on the main diagonal therefore it is not semi-positive. Therefore, A is not contractive.

112


7.3 Dissipative operators. We say that an operator X in a pseudo-Euclidean space is (pseudo)-dissipative, if RehvX; vi 6 0 for all v or, equivalently, if

hv.X C X ~ /; vi 6 0:

Obviously, the set of all dissipative operators is a convex cone (wedge). We denote it by Diss D Diss.p; q/. Represent X in the form X D A C B;

where A D 12 .X X ~ /; B D 12 .X C X ~ /:

Then A D A~ 2 u.p; q/ and B ~ D B is self-adjoint dissipative, i.e., hvB; vi 6 0. We denote the cone of pseudo-self-adjoint dissipative operators by SDiss.p; q/, Diss.p; q/ D u.p; q/ ˚ SDiss.p; q/: 7.4 Exponentials Theorem 7.5. The following conditions are equivalent: – expftX g 2 U.p; q/ for all t > 0, – X 2 Diss.p; q/. Proof. Let expftX g 2 U.p; q/. Then 0 > hv expf"X g; v expf"X gi hv; vi D " hvX; vi C hv; vXi C O."2 /:

(7.3)

The left-hand side is non-positive, hence X is dissipative. Conversely, let X be dissipative. Then hv expftXg; v expftXgi hv; vi D

N 1 h X

˝

v exp

˚ t.j C1/ ˚ t.j C1/ ˛ ˝ ˚ tj ˚ tj ˛i X ; v exp X v exp X ; v exp X N N N N

j D0

D

N 1 h X

˝

vj exp

˚

t N

i

˚

˛ X ; vj exp Nt X hvj ; vj i ;

where vj D v exp

˚ tj N

X :

j D0

By (7.3), each summand 6 const.X; t; v/N 2 . Passing to the limit as N ! 1, we see that our expression is non-positive. Proposition 7.6. Consider a smooth curve .t / 2 GL.p C q; C/, where t > 0 and .0/ D 1. a) If .t/ 2 U.p; q/, then 0 .0/ is dissipative. b) If .t/1 0 .t / is dissipative for all t > 0, then .t / 2 U.p; q/.


Proof. We repeat arguments of the previous proof.

113

The statement a) means that the wedge of dissipative operators is the tangent cone to the semigroup U.p; q/ at the unit. Problem 7.2. The curves .t / as in Proposition 7.6 sweep out the whole semigroup U.p; q/ (see Theorem 7.7 below). 7.5 The Potapov–Olshanski decomposition Problem 7.3. Let g be an invertible operator in a pseudo-Euclidean space. Is it correct that g admits a unique decomposition g D UR, where U is pseudo-unitary and R is pseudo-Hermitian? Answer: no. Theorem 7.7. Each element g 2 U.p; q/ admits a unique decomposition g D exp.X / h;

(7.4)

where h 2 U.p; q/ and X 2 SDiss.p; q/. In particular, we get the following canonical isomorphism of topological spaces: U.p; q/ ' U.p; q/ exp.SDiss.p; q//: We use the following lemma that will become obvious in the next section, see Theorem 8.2. Lemma 7.8. The semigroup U.p; q/ is connected. Proof of the theorem. Let us try to repeat the proof of Theorem 1.1.3 about polar decomposition. We have hv; vi > hvg; vgi D hv; vgg ~ i:

(7.5)

The element gg ~pis pseudo-self-adjoint (.gg ~ /~ D gg ~ ). We must show the existence of a square root gg ~ . Denote by R the set of all (invertible) self-adjoint operators in a pseudo-Euclidean space satisfying the condition hvr; vi 6 hv; vi. We wish to list indecomposable elements of this set. Lemma 7.9. Indecomposable operators r 2 R exist only in the following spaces W of dimension 1 and 2 and: 1B . W D C0;1 ; wr D w, jj 6 1, ¤ 0; 2B . W D C1;0 ; wr D w, > 1; 3B . W D C1;1 ; the operator is the Jordan block .˙A t /, where A t is given by25 (7.1), (7.2) and ˙t > 0. 25

For all t > 0, the operators A t are equivalent.

114


Proof. Let us filtrate canonical forms of pseudo-self-adjoint operators. For P instance, consider the operator I3C ./ (in the notation of Theorem 6.2). For v D ˛j ej , we have 80 1 0 19 0 1 0 0 1 0 1 < = ˛1 hv; vi hvr; vi D ˛1 ˛2 ˛3 @0 1 0A @ 1 0 A @˛2 A : : ; 1 0 0 0 0 ˛3 The matrix in the curly brackets is not positive definite (indeed, it has too many zeros on the main diagonal), see Example 4 from Subsection 7.1. The last argument is also valid in other cases. For operators of type IIn the diagonal consists of zeros. For operators of type In˙ our argument is valid for all n > 3. The case n D 2 follows from a straightforward calculation (see Example 3 in the beginning of this section). Lemma 7.9 itself does not allow us to take a canonical square root, because r admits negative eigenvalues. Lemma 7.10. If r D gg ~ , then all eigenvalues of r are positive. Proof. All eigenvalues of an operator r 2 R are real. Therefore, the set R is disconnected. But the semigroup U.p; q/ is connected, hence the image of the map g 7! gg ~ is connected. Since 1 1~ has positive eigenvalues, it follows that an arbitrary matrix gg ~ also has positive eigenvalues. End of the proof of Theorem 7.7. Thus the operator gg ~ is a direct sum of several operators with positive eigenvalues (i.e., > 0 and .A t / is forbidden) listed in Lemma 7.9. In all these cases, there is a canonical square p root and a canonicallogarithm; . for the two-dimensional Jordan block A t , we have A t D A t=2 , ln A t D 00 t 0 p ~ Hence, there is a canonical square root y WD gg , which is pseudo-contractive and has positive eigenvalues: y ~ D y;

y 2 U.p; q/;

hvy; vi 6 hv; vi:

It remains to repeat the proof of Theorem 1.3 about polar decomposition.

Another (semi-) proof of Theorem 7.7. For a strict contraction g, the theorem admits a simple proof without a reference to canonical forms. By the Krein Theorem 7.12, one can find a negative r-invariant p-dimensional subspace H . Since r is self-adjoint, the positive subspace H ? is also r-invariant. A 0 Therefore, r has the form 0 B , where A and B are self-adjoint in the usual sense (both the spaces H and H ? are Euclidean). Now A > 1, 1 < B < 1. We again apply the argument with connectedness and take a square root.


115

Remark. It seems that after the last considerations we can apply a continuity arguments. This way is possible but not simple. The behavior of the matrix square root under a limit passage is bad (for instance, the set of solutions of the equation A2 D 1 is unbounded). We also get the following corollary of Theorem 7.7. Proposition 7.11. Each element g 2 U.p; q/, admits a decomposition g D h1 Bh2 , where h1 , h2 2 U.p; q/ and B is a block-diagonal matrix, whose blocks (of size 1 1 or 2 2) are matrices with positive eigenvalues listed in Lemma 7.9. For a strict contraction g, the operator B is diagonal; the eigenvalues i corresponding to negative eigenvectors are > 1, the eigenvalues j corresponding to positive eigenvectors satisfy 0 < j < 1. Problem 7.4. Equip Cp;q with the standard scalar product. Then g 2 U.p; q/ implies g 2 U.p; q/. Problem 7.5. a) The semigroup of contractions of Rp;q has four connected components. b) In spite of this obstacle, Theorem 7.7 survives in this case. 7.6 The action of indefinite contractions on matrix balls. Let S 2 Mp;q be a negative subspace, let g 2 U.p; q/. Obviously, the subspace Sg is also negative. Thus the semigroup U.p; q/ acts on Mp;q and on the matrix ball Bp;q ; the latter action is linear fractional and is given by the usual formula (3.3). Theorem 7.12 (Krein). Each element g 2 U.p; q/ has a semi-negative invariant subspace of dimension p. If g is strictly contractive, then it has a negative invariant subspace. xp;q . By the Proof. We consider the transformation z 7! z Œg of the closed matrix ball B Brouwer fixed point theorem26 , this transformation has a fixed point. If g is a strict contraction, then the image of a semi-positive subspace under g is a negative subspace. Hence a fixed point corresponds to a negative subspace. 7.7 A reformulation of contractivity. Denote by C Cp;q the cone of non-positive vectors. Theorem 7.13. Let AC C . Then there is a positive constant s such that s A is a contraction. Lemma 7.14. Under the conditions of the theorem, let e be a negative vector such that he; ei D 1 and let f be a positive vector such that hf; f i D C1. Then hAe; Aei > hAf; Af i: 26

“Any continuous map from the closed m-dimensional ball to itself has a fixed point.”

116


a)

b)

Figure 2.18. a) The operator .x; y/ 7! .2x; y/ in R1;1 . b) A contractive operator in R2;1 .

Proof of Lemma 7.14. The vectors e ˙ f are isotropic, therefore A.e ˙ f / 2 C . Hence, 0 > hA.e C f /; A.e C f /i C hA.e f /; A.e f /i D 2.hAe; Aei C hAf; Af i/: Proof of Theorem 7.13. We consider a constant separating all .hAe; Aei/ from all hAf; Af i D 1. Next, consider 1 A. 7.8 Digression. The cone of self-adjoint dissipative operators Proposition 7.15. a) Each indecomposable self-adjoint dissipative matrix has one of the following forms (we present the space, the inner product, and the operator): – the space is C0;1 with a basis e such that he; ei D 1;

Ae D s e;

with s > 0I

– the space is C1;0 with a basis e such that he; ei D 1;

Ae D s e;

with s > 0I

– the space is C1;1 with a basis such that e, f , he; f i D 1;

e and f are isotropic, A t e D tf , A t f D 0, where t > 0

(operators A t with different t are equivalent). b) Each operator B 2 SDiss is a direct sum of indecomposable operators. Proof. a) We must merely sort out the canonical forms of self-adjoint operators A from Theorem 6.2. For type IIn , the Hermitian form hAv; vi has the matrix of the shape J0 J0 . It is nonnegative semi-definite, since it has zeros on the diagonal. ˙ For the type In , the form hAv; vi is 1 0 0 0 ::: 1 B0 0 : : : 0C C B C B ˙ B ::: ::: : : : ::: ::: C : C B @1 : : : 0 0A

0

:::

0

0

117

2.8. Potapov coordinates

It has too many zeros on the main diagonal. The only cases when it can be non-negative definite are n D 1 or n D 2, D 0. Problem 7.6. a) Describe the boundary of SDiss.p; q/. b) Find the extreme points of the cone SDiss.p; q/.

2.8 Potapov coordinates 8.1 The Potapov transform. Let V D V1 ˚ V2 be a linear space, dim V1 D p and dim V2 D q. Consider a linear transformation in V , w1 a b v1 D : (8.1) w2 v2 c d Write out these relations in the form w1 D av1 C bv2 ; w2 D cv1 C dv2 : Let us regard these equations as equations for the variables v1 , w2 . The first equation implies v1 D a1 w1 a1 bv2 : Substituting this v1 in the second equation, we obtain v1 a1 a1 b v2 D : 1 1 w2 d ca b ca w1 We say that the matrix a1 b ….g/ D d ca1 b

a1 ca1

(8.2)

is the Potapov transform of

a b : c d

(8.3)

Remark. The domain of the Potapov transform is det a ¤ 0. Remark. The size of ….g/ is .p C q/ .q C p/, i.e., the blocks a1 and d ca1 b on the anti-diagonal are square, i.e., p p, q q. 8.2 The inverse transform. It can be readily checked that 1 ˛ ˇ ˇ ˇ 1 ˛ ….g/ D H) g D : ı ıˇ 1 ıˇ 1 ˛ We will refer to this operation as inverse Potapov transform …1 .g/.

(8.4)

118


Formulas for the direct Potapov transform (8.3) and for the inverse Potapov transform (8.4) are suspiciously similar. An explanation of this similarity is very simple. Let us represent the system (8.2) in the form 1 v1 a1 b a w1 D (8.5) ca1 d ca1 b w2 v2 z and denote the matrix on the right-hand side by ….g/. Then z ….g/ z … D g; z means that we solve the system (8.5) with resince the second transformation … spect to the variables w1 , w2 considering v1 , v2 as parameters. Obviously, we return to (8.1). 8.3 Coordinates-I. We realize the group U.p; q/ in the standard way (2.3). Theorem 8.1. a) If g 2 U.p; q/, then ….g/ is well defined. Moreover, ….g/ 2 U.p C q/. b) The image ….U.p; q// consists of matrices r D ˛ ˇı 2 U.p C q/ satisfying the following four equivalent conditions: 1. ˇ is invertibleI 2. is invertibleI

3. k˛k < 1I 4. kık < 1:

Proof. a) Consider a pseudo-unitary transformation g given by a b v1 w1 D : w2 v2 c d The pseudo-unitary condition means kw1 k2 C kw2 k2 D kv1 k2 C kv2 k2 ;

(8.6)

kv1 k2 C kw2 k2 D kv2 k2 C kw1 k2 :

(8.7)

therefore, But this is precisely the unitary property of ….g/. Problem 8.1. Show that for ac db 2 U.p; q/ the block a is invertible. For this purpose, we can refer to the defining relations (2.4). We can also refer to (8.6). It suffices to take v1 2 ker a, v2 D 0 and to look on the left-hand side. b) First, we must show that …1 .r/ 2 U.p; q/. The inverse transform is defined if ˇ is invertible. It remains to note that (8.7) implies (8.6).

119


Second, we must verify the equivalence of the conditions 1-4. A unitary matrix preserves the norm of any vector, in particular, it preserves the norm of x ˚ 0 2 CpCq : k˛x ˚ xk2 D kxk2 H) k˛xk2 C kxk2 D kxk2 :

(8.8)

If k˛k D 1, then there is a vector x such that k˛xk2 D kxk2 . Hence, kxk2 D 0 and is not invertible. This consideration proves 2 , 3. Repeating the same trick to 0 ˚ y, we obtain 1 , 4. Applying the same considerations to row vectors, we obtain the remaining equivalences. Problem 8.2. Thus the proof seems trivial. However, prove the statement a) directly starting from the explicit formula (8.3) and the defining relation (2.4) for U.p; q/. Theorem 8.2. a) If g 2 U.p; q/, then k….g/k 6 1. b) The image ….U.p; q// consists of matrices r D the conditions 1–4 of the previous theorem.

˛ˇ ı

satisfying krk 6 1 and

c) Moreover, for a matrix r with krk 6 1, each of the conditions 1 or 2 implies xpCq;pCq and the existence of ˇ 1 both 3 and 4. Therefore it suffices to require r 2 B 1 and . d) g 2 U.p; q/ is a strict contraction if and only if k….g/k < 1. The proof of the previous theorem survives literally. To prove a) we observe that kw1 k2 C kw2 k2 6 kv1 k2 C kv2 k2 () kv1 k2 C kw2 k2 6 kv2 k2 C kw1 k2 : The former condition is the condition of pseudo-contractivity, the latter is the condition of the usual (Euclidean) contractivity. Further, look at (8.3). By construction, ˇ D a1 , therefore it is invertible. Also, g is invertible. By Theorem 1.1.4, a b 0 ¤ det.g/ D det D det a det.d ca1 b/ H) is invertible. c d Also, k˛x ˚ xk2 6 kxk2 H) k˛xk2 C kxk2 6 kxk2 I if k˛k D 1, then is not invertible.

Problem 8.3. Reconstruct the remaining details of the above proof. Corollary 8.3. The semigroup U.p; q/ is connected. Proof. The set described in Theorem 8.2.b is connected.

120


8.4 Products of matrices in the Potapov coordinates Theorem 8.4. Let g1 and g2 be .p C q/ .p C q/-matrices. Let ˛ ˇ ' ….g1 / D ; ….g2 / D : ı ~ Then

˛ C ˇ.1 'ı/1 ' ….g1 g2 / D .1 ı'/1

ˇ.1 'ı/1 ~ C ı.1 'ı/1

:

(8.9)

Remark. Apparently, this formula is not the simplest way to multiply matrices. However, in the next section, we observe that this formula corresponds to a more general operation. Proof. We write out w1 ˛ D v2

ˇ ı

w2 ; v1

v1 u2

' D

v2 u1

~

or, w1 D ˛ w2 C ˇ v1 ; v1 D ' v2 C u1 ; v2 D w2 C ı v1 ; u2 D v2 C ~ u1 : We wish to express the variables w1 , u2 in terms of the variables w2 , u1 . For this purpose, we exclude v1 , v2 from the second and third equations, v1 D .1 'ı/1 .' w2 C

u1 /;

v2 D .1 ı'/1 .' w2 C

and substitute these v1 , v2 in the first and forth equation. 8.5 Krein–Shmul’yan maps of matrix balls Theorem 8.5. For g 2 U.p; q/, denote by ….g/ D Let z 2 Bp;q . Then g sends z to

g W z 7! ˛ C ˇ.1 zı/1 z:

u1 /;

˛ˇ ı

its Potapov transform.

(8.10)

Such maps are called Krein–Shmul’yan maps. Proof. Now we have the equations w1 D zw2 ; v1 D ˛v2 C ˇw1 ; w2 D v2 C ıw1 :

(8.11) (8.12) (8.13)

121


First, we substitute w1 from (8.11) into (8.13) and get w2 D v2 C ızw2 H) w2 D .1 ız/1 v2 : Using (8.11) again, we get w1 D z.1 ız/1 v2 : Substituting this w1 in (8.12), we obtain v1 D ˛ C ˇz.1 ız/1 v2 :

8.6 Digression. Some remarks on formulas (8.9) and (8.10). It is interesting to compare the formula (8.10) for Krein–Shmul’yan transformations and the formula for -product (8.9). First, the expression (8.10) is present in the upper left block of (8.9). Secondly, the expression (8.9) for a product can be represented in the form 1 ı 0 0 1 1 0 ˇ 0 ˛ 0 0 0 1 C (8.14) 0 ' 1 0 0 1 0 0 ~ 0 1 0 and in the form ˇ ˛ 0 C 0 0 0

0 1

'

~

1 0

ı 0 0 1

' 0 0

1 ~

0

0 : 1

(8.15)

These formulas look like a special case of the Krein–Shmul’yan map (8.10). Thus, both the formulas (8.9) and (8.10) seem to be particular cases of one another. We return to this analogy in Subsection 9.4.

8.7 Coordinates-II. Let p D q. We will realize the U.q; q/ as the group of group A B 0 1q matrices g D C D preserving the Hermitian form 1q 0 , see (3.14). We introduce the following transform: w2 A B w1 w1 AC 1 B C AC 1 D w1 D () D : C 1 D C 1 C D w2 w2 v1 v2 For a matrix g on the left-hand side, we denote the resulting matrix by …B .g/. This matrix differs from ….g/ by a permutation of blocks and changing signs. A B B Theorem 8.6. a) Let g D C D 2 U.q; q/, let C be invertible. Then … .g/ is an anti-Hermitian .q C q/ .q C q/-matrix. b) The image …B .U.q; q// consists of anti-Hermitian matrices ˛ ˇı such that the block is invertible. Thus we obtain an open chart on U.q; q/.

Problem 8.4. Verify the identity …B .g/ D …B .g/ directly from the defining equations (3.14).

122


A B B Theorem 8.7. a) Let g D C D 2 U.q; q/, let C be invertible. Then Re … .g/ 6 0. b) The map …B is a bijection of the semigroup of strict contractions into the subset in the open wedge Re r > 0 distinguished by the constraint: ˇ and are invertible. We omit the proofs which are similar to proofs of Theorems 8.1, 8.2.

2.9 The Krein–Shmul’yan category. The compression property Starting from this section, we use linear relations; see § 1.7, which is independent of other sections of Chapter 1. 9.1 Contractive linear relations. Let V and W be pseudo-Euclidean spaces with inertia indices .p; q/, .r; s/ (in our notation, p, r are negative indices). A linear relation P W V W is called contractive if the following conditions hold: a) If v ˚ w 2 P , then hv; viV > hw; wiW . b) dim P D q C r, i.e., this is the maximal possible dimension of a linear relation satisfying the condition a). Let us say the same in other words. Consider the space V ˚ W equipped with the Hermitian form hv ˚ w; v 0 ˚ w 0 iV ˚W WD hv; v 0 iV hw; w 0 iW :

(9.1)

A contractive linear relation V W is a maximal semi-positive subspace in V ˚ W . Theorem 9.1. The product of contractive linear relations is a contractive linear relation. Proof. Let P W V W and Q W W Y be contractive. Let v ˚ y 2 QP . Then one can find w 2 W such that v ˚ w 2 P and w ˚ y 2 Q. Hence, hv; viV > hw; wiW > hy; yiY : Therefore, the condition a) is valid for QP . Lemma 9.2. For a contractive linear relation P W V Y , we have ? ? dom P zero.ker P / ; im P zero.indef P / : Proof of Lemma 9.2. Let 2 zero.ker P /. Since the isotropic vector ˚0 is contained in a semi-positive subspace P , it follows that ˚ 0 2 zero.P /. Let v ˚ w 2 P . Then 0 D h ˚ 0; v ˚ viV ˚W D h; viV : Thus v?.

2.9. The Krein–Shmul’yan category. The compression property

123

End of the proof of Theorem 9.1. By the lemma, im P .zero.indef P //? ; Hence,

dom Q .zero.ker Q//? :

? dom Q C im P zero.ker Q/ \ zero.indef P / :

By (1.7.7),

dim QP D dim P C dim Q dim ker Q \ indef P dim im P C dom Q :

Since and indef P is non-positive, it follows that their intersection ker Q is non-negative is zero.ker Q/ \ .zero indef P /. Thus, dim QP D dim P C dim Q dim zero.ker Q/ \ zero.indef P / dim im P C dom Q > dim Q C dim P dim W: The last number is the maximal possible dimension of a semi-positive subspace in V ˚Y. 9.2 The Krein–Shmul’yan category. In spite of Theorem 9.1, it is natural to reduce27 the category of contractive linear relations. We define the Krein–Shmul’yan category as a category whose objects are pseudoEuclidean spaces and morphisms are contractive linear relations P W V W satisfying the two additional conditions: c) ker P V is strictly positive, c)0 indef P W is strictly negative. Remark. The condition a) in 9.1 implies semi-positivity of ker P and semi-negativity of indef P . As above, we denote by Mor.V; W / the set of morphisms V ! W . Now let V D V1 ˚ V2 and W1 ˚ W2 be objects of the Krein–Shmul’yan category, let the subspaces V1 , W1 be negative and the subspaces V2 , W2 be positive. Then by Theorem 3.1 a morphism P W V W is a graph of a contractive operator in Euclidean spaces ˛ ˇ ….P / D W V2 ˚ W1 ! V1 ˚ W2 ı (here we change signs of scalar products on V1 and W1 and regard both spaces as being as Euclidean). 27 This reduction is a matter of taste, but there are some natural reasons for it. First of all, the product of contractive linear relations does not depend continuously on the factors; in either case the continuity can be saved by a single condition c/ or by a single condition c/0 . A more important reason is the L2 -boundedness conditions for Gaussian operators from § 1.8. Another reason: the central extension of the Krein–Shmul’yan category (see Subsection 3.8.3) can not be extended to the category of contractive linear relations.

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Theorem 9.3. P 2 Mor.V; W / if k….P /k 6 1 and k˛k < 1, kık < 1. Problem 9.1. Prove this theorem. It is a rephrasing of Theorem 3.1. Thus, we get a parametrization of the set Mor.V; W /; also we can say that we have a coordinate system on Mor.V; W /. Theorem 9.4. The product in these coordinates is given by the formula (8.9). Proof. The proof of Theorem 8.4 survives literally.

Remark. Certainly, the formula (8.9) is not the best way to multiply block matrices g1 and g2 . Nevertheless, the product of linear relations is a more general operation than is matrix multiplication; our formula allows us to express this product in terms of matrix operations. 9.3 Digression. The involution in Krein–Shmul’yan category. Let P W V W be a morphism of the Krein–Shmul’yan category. We define the linear relation P W W V by setting P WD .P ? / ; (9.2) where P is the pseudo-inverse linear relation. Proposition 9.5. a) P is a morphism W V . b) For each pair P W V W and Q W W Y , we have .QP / D P Q . We omit a proof (see parallel considerations in Subsection 1.9.7). The operation (9.2) is an imitation of the operation A 7! A~ .

9.4 The Krein–Shmul’yan functor. Now, let R V be a negative subspace, let z be its coordinate as in§ 2.3. Let P W V W be a morphism of the Krein–Shmul’yan category, denoted by ˛ ˇı its Potapov transform. Theorem 9.6. a) The map28 R 7! RP is given by the formula P W z 7! ˛ C ˇ.1 zı/1 z: In particular,

k˛ C ˇ.1 zı/1 zk < 1:

(9.3)

b) This is a functor, i.e., P Q D PQ :

(9.4)

Proof. This is a particular case of the formula (8.9). Indeed, R can be considered as a morphism 0 ! V . 28

We apply a linear relation to a linear subspace.

2.9. The Krein–Shmul’yan category. The compression property

125

Denote by .p; q/ the inertia indices of V and by .r; s/ the inertia indices of W . We get a map P W Bp;q ! Br;s , or P W U.p; q/=U.p/ U.q/ ! U.r; s/=U.r/ U.s/: The maps of this form are called Krein–Shmul’yan maps. 9.5 Digression. The Krein–Shmul’yan maps and -products (8.9). The previous considerations explain a half of the strange observation from Subsection 8.6; the formula for Krein– Shmul’yan maps is a special case of the product-formula in the Potapov coordinates. We are also ready to discuss another half. Let V , W , Y be arbitrary pseudo-Euclidean spaces, let P W V W and Q W W Y be linear relations. Define the linear relation z WD 1 ˚ Q W V ˚ W V ˚ Y: Q z to the subspace P V ˚W , we obtain the subspace QP z V ˚Y , Applying the linear relation Q which is nothing but QP . This explains formula (8.15). Formula (8.14) arises from another rephrasing of the product of linear relations. Consider the linear relation P ˚ QW W ˚ W V ˚ Y and the diagonal S W ˚ W . Applying P ˚ Q to S we obtain a subspace in V ˚ Y , which is nothing but QP .

9.6 Compression of angles Theorem 9.7. Consider a Krein–Shmul’yan map W Bp;q ! Br;s . For any z, u 2 Bp;q , denote by '1 > '2 > the hyperbolic angles between z and u. Let 1 > 2 > be the hyperbolic angles between .z/ and .u/. Then '1 >

1;

'2 >

2;

'3 >

3;

::::

(9.5)

The proof is given in Subsection 9.8. 9.7 Example. Contractive maps of the unit disk. Consider the space C1;1 . The ‘matrix ball’ B1;1 is the unit disk D: jzj < 1 on C. The semigroup U.1; 1/ acts on D by the (injective) linear fractional transformations z 7!

b C zd : a C zc

(9.6)

This semigroup contains the Möbius group SU.1; 1/ of biholomorphic transformations N jaj2 jbj2 D 1, see of the disk; elements of the Möbius group satisfy d D a, N c D b, Subsection 2.10. The Krein–Shmul’yan maps D1 ! D1 are z 7! ˛ C

ˇz : 1 ız

126


If ˇ ¤ 0, then this transformation can be reduced to (9.6). Otherwise, we get Krein– Shmul’yan maps that are not linear fractional, namely z 7! ˛. The disk D is endowed with a U.1; 1/-invariant Riemannian metric dz d zN : .1 jzj2 / The geodesic distance is given by the formula .1 z u/ N 2 .z; u/ D '.z; u/ D arccosh p ; .1 z zN /.1 uu/ N where ' is the hyperbolic angle, see (5.7). Lemma 9.8. The transformations g 2 U.1; 1/ are contractive, i.e., .z Œg ; uŒg / 6 .z; u/:

(9.7)

Proof. It suffices to prove the statement for strict contractions. By the Potapov– Olshanski Theorem 7.7, it follows that each strict contraction is the product of the form g D h1 r t h2 , where h1 , h2 are in the Möbius group, and r t is a homothety z 7! t z;

where 0 < t < 1:

The following problem finishes the proof.

Problem 9.2. Show that r t contracts the Riemannian metric. Moreover, .r t z; r t u/ 6 t.z; u/. 9.8 Proof of Theorem 9.7. We must look at the minimax expression (5.5) for angles. Let L1 , L2 2 Mp;q be negative subspaces, let P be a linear relation. Then a1 2 L1 ; a2 2 L2 ; a1 ˚ b1 2 P; a2 ˚ b2 2 P H) b1 2 PL1 ; b2 2 PL2 : The angles 'j between L1 , L2 and the angles j between PL1 , PL2 are given by (5.5). We have cosh j > 1. For this reason, in the minimax expression for angles we watch only pairs b1 , b2 generating indefinite planes. Hence, a1 , a2 must also generate an indefinite plane (since Lj are negative, it follows that the vectors aj are negative; there is a logical possibility that a1 , a2 generate a negative plane, but after the contraction P it can not became indefinite). By Lemma 9.8, we have †.b1 ; b2 / 6 †.a1 ; a2 /. Looking to (5.5) we obtain the desired inequality. 9.9 Remarks. For maps B1;1 ! B1;1 , Theorem 9.7 claims only the contractivity in the usual sense, i.e., any Krein–Shmul’yan map is a contraction of the geodesic distance. Certainly, one can find a vast number of contractive maps of the disk into itself.

2.10. Isotropic category. Inverse limits

127

The same holds for maps of the usual ball B1;q into itself. Nevertheless for p > 1, q > 1, the picture changes drastically; the compressivity becomes a very rigid property. There are several versions of the inverse theorem, for instance, Let a holomorphic map W Bp;q ! Br;s satisfy the conclusion of Theorem 9.7. If there are two points z and u such that29 2 . .z/; .u// ¤ 0, then .z/ is either a Krein–Shmul’yan map or a map of the form z ! .z t /, where is a Krein–Shmul’yan map.

2.10 Isotropic category. Inverse limits 10.1 Isotropic relations. Fix m D 0; 1; 2; : : : . Let us consider pseudo-Euclidean spaces of the form Cp;pCm . Let V D Cp;pCm and W D Cq;qCm . We say that a linear relation P W Cp;pCm q;qCm C is form-preserving if the following conditions hold: a) For any v ˚ w, v 0 ˚ w 0 2 P , we have hv; v 0 iV D hw; w 0 iW I b) dim P D p C q C m, i.e., it is the maximal possible dimension of a subspace satisfying the condition a). Another definition. Introduce the Hermitian form in V ˚ W as above (9.1), then a form-preserving linear relation is nothing but a maximal isotropic subspace. In particular, a form-preserving linear relation is a contractive linear relation in the sense of the previous section. Theorem 10.1. Let P W V W and Q W W Y be form-preserving linear relations. Then QP W V Y is form-preserving. Proof. Let v ˚ y, v 0 ˚ y 0 2 QP . Then one can find w, w 0 2 W such that v ˚ w 2 P; v 0 ˚ w 0 2 P I Hence,

w ˚ y 2 Q; w 0 ˚ y 0 2 Q:

hv; v 0 i D hw; w 0 i D hy; y 0 i:

This yields the condition a). But P and Q are also contractive linear relations in the sense of the previous section. By Theorem 9.1, the dimension of the product is maximal possible. Thus, for any non-negative integer m, we get a category (isotropic category) whose objects are spaces Cp;pCm and morphisms are isotropic linear relations30 . 29 This condition means that the image of the map is not contained in a totally geodesic subball of type B1;k ; omitting this constraint, we obtain a vast number of contractive maps of Bp;q into a small subballs. 30 The multiplication in isotropic category is a discontinuous operation; apparently it is better to correct it as it was done in [145].

128


The Potapov transform of an isotropic linear relation is a unitary matrix ˛ ˇ SD ı

(10.1)

of size Œp C .q C m/ Œ.p C m/ C q. There are no additional constraints on the matrix S . 10.2 Formulas. Since isotropic linear relations are also contractive, we can use the formulas of the previous section. 10.3 The action of the isotropic category on the Stiefel manifolds. As we have seen in Subsection 4.3, the Stiefel manifold Sti.p; q/ is in a one-to-one correspondence with the set IGr.Cp;q / of all maximal isotropic subspaces in Cp;q . Let V ' Cp;pCm , W ' Cq;qCm , let P W V W be a form-preserving linear relation. Let R range in IGr.Cp;pCm /. Then R 7! PR is a well-defined map IGr.Cp;pCm / ! IGr.Cq;qCm /. The corresponding Krein–Shmul’yan map Sti.q; q C m/ ! Sti.p; p C m/ is given by the usual formula z 7! ˛ C ˇ.1 zı/1 z D ˛ C ˇz.1 ız/1 (this map has points of discontinuity). In particular, for m D 0, we get a family of natural maps31 U.q/ ! U.p/. 10.4 The Livshits map. Let m D 0, let us write q D p C j . Take the following Œp C .p C j / Œp C .p C j / matrix S: 0 1 0 1p 0 0 A: S WD @1p 0 (10.2) 0 0 1j The corresponding Krein–Shmul’yan map is

a b c d

³1 ² a b 0 0p a b 1p 1p 0 7 0p C 1p 0 ! 0 1j 0 1j c d c d 0 1 a b 0 1p 1p D 1p 0 c 1j C d c d 0 1 a b 0 1p 1p D 1p 0 1 1 .1j C d / c .1j C d / c d 0 a b.1j C d /1 c 1p D 1p 0 0

D a b.1 C d /1 c: 31

These maps are not group homomorphisms.


We define the Livshits map ‡ j W U.p C j / ! U.p/ by a b ‡j D a b.1 C d /1 c: c d

129

(10.3)

Our considerations imply the following statement: Theorem 10.2. ‡ j is a map U.p C j / ! U.p/. Second proof for Theorem 10.2. We write the equation a b D x c d x or, equivalently, D a C bx;

x D c C dx:

Excluding x, we get x D .1 C d /1 c or D .a b.1 C d /1 c/ : On the other hand, the matrix ac db is unitary; therefore,

(10.4)

kk2 C k xk2 D k k2 C kxk2 H) kk2 D k k2 : Therefore, the operator (10.4) is unitary.

Problem 10.1 (Livshits characteristic function, see Addendum 1 to this chapter). For a b 2 U.p C j /, consider the following matrix-valued function of the complex c d variable z: .z/ D a zb.1 C zd /1 c: a) k.z/k 6 1 for jzj < 1, k.z/k > 1 for jzj > 1. b) .z/ is unitary if jzj D 1. c) (The symmetry with respect to the inversion) .Nz 1 / D .z/1 . d) .z/ has poles as a matrix-valued map. But it is a well-defined holomorphic x D C [ 1 to the Grassmannian. map from the Riemann sphere C 10.5 Equivariance Proposition 10.3. For any h1 , h2 2 U.p/, we have ² ³ a b h2 0 h1 0 a b ‡j D h1 ‡ j h2 : 0 1 c d 0 1 c d

(10.5)

130


The statement is obvious.

This shows that the Livshits map is distinguished among relatively numerous32 Krein–Shmul’yan maps U.p C j / ! U.p/. Now let G be a group, let M , N be two G-spaces, i.e., spaces with action of G. A map W M ! N is a morphism of G-spaces if .xg/ D

.x/g;

where x 2 M , g 2 G.

Problem 10.2. a) Consider the action of the group G of rotations of a cube on the set of its vertices (edges, faces). Describe all quotient spaces33 . b) Describe all epimorphisms, and monomorphisms, and morphisms of a G-homogeneous space into G-spaces. We can reformulate Proposition 10.3 as: Observation 10.4. The map ‡ j W U.p C j / ! U.p/ is a morphism in the category of U.p/ U.p/-spaces. 10.6 Consistency Proposition 10.5. We have ‡ j B ‡ j D ‡ iCj : More precisely, the composition of the maps ‡ i W U.n/ ! U.n i / and ‡ j W U.n i / ! U.n i j / is ‡ iCj W U.n/ ! U.n i j /. This can be proved by a straightforward calculation, by multiplication of linear relations, or by a calculation of the “strange product” (8.9) for two matrices of type (10.2). 10.7 The Livshits map and the Cayley transform. Recall that the Cayley transform Cayn .g/ WD .1 g/.1 C g/1 D 1 C 2.1 C g/1 is a bijection of the space U.n/ onto the space AHermn of anti-Hermitian matrices. Define the “cut-off map” „ W AHermpCj ! AHermp by the formula j r11 r12 D r11 : „ r21 r22 32 33

They are parametrized by U.2p C j /. The standard term for quotients of homogeneous spaces in algebra textbooks is “imprimitivity system”.

131


Theorem 10.6. For g 2 U.p C j / in a general position, Cayp B ‡ j .g/ D „j B CaypCj .g/: Equivalently,

‡ j .g/ D Cayp1 B „j B CaypCj :

First proof. Obviously, for any pair v and u, we have u v D g.u C v/ () u D .1 g/1 .1 C g/v D Cay.g/v:

(10.6)

The following line can be considered as a definition of the map „: r r v1 u1 u1 D r11 v1 () D 11 12 for some u2 : u2 r21 r22 0 We are going to evaluate „.Cay.g// using these paraphrases of the Cayley transform and the cut-off operation. Thus, let u1 D „.Cay.g//v1 . Then there is u2 such that i u1 a b h v1 u v1 D C 1 u2 0 c d 0 u2 or .u1 v1 / D a.u1 C v1 / C bu2 ; u2 D c.u1 C v1 / C du2 : The second equation implies u2 D .1 C d /1 c.u1 C v1 /; substituting u2 in the first equation, we obtain .u1 v1 / D a b.1 C d /1 c v1 C u1 : But this is an equation of the form (10.6). Thus „.Cay.g// is the Cayley transform of ‡.g/. Proof by straightforward calculation. Recall the Frobenius trick for an inversion of a block matrix. First, 1 ˇı 1 ˛ ˇ 1 0 ˛ ˇı 1 0 D ; ı 0 ı ı 1 1 0 1 which is a paraphrasing of (1.2.14). Hence, 1 ˛ ˇ 1 0 .˛ ˇı 1 /1 D 1 ı 0 ı 1 1 1 .˛ ˇı / ::: D ::: :::

0

ı 1

1 ˇı 1 0 1

132


(only the upper left block is interesting for us). Further, calculate the Cayley transform .1 g/=.1 C g/ D 1 C 2.1 C g/1 of a block matrix g D ac db ,

h 1 C 2.1 C a b.1 C d /1 c/1 1 0 1 0 a b i1 D C2 C ::: 0 1 0 1 c d

::: : :::

By definition, the upper left block is „.Cay.g//. But this block is also the Cayley transform of ‡.g/. Problem 10.3. The group U.n/ U.n/ acts on itself by left and right multiplications g 7! u1 gv, where g 2 U.n/ and .u; v/ 2 U.n/ U.n/. Show that the corresponding transformation in the Cayley coordinates is the linear fractional transformation r 7! .˛ C r/1 .ˇ C rı/, where 1 vCu vu ˛ ˇ : (10.7) S WD D ı 2 vu vCu 10.8 Inverse limits of the spaces U.n/ Theorem 10.7. The pushforward of the Haar measure pCj on U.p C j / under the Livshits map is the Haar measure p on U.p/. Recall the definition of a direct image (pushforward) of a measure. Let M be a space equipped with a measure . Let f W M ! N be a map. The measure on the target space N is defined by the condition: for each C N , we set .C / WD .f 1 .C //. Proof. The Haar measure on U.p C j / is invariant with respect to left and right shifts. In particular, it is invariant with respect to the left-right action of U.p/ U.p/. Proposition 10.3 implies the invariance of the pushforward of the Haar measure. Thus we get a U.p/ U.p/-invariant measure on U.p/; by definition it is the Haar measure on U.p/. Normalize each Haar measure in such a way that the measure of the whole group is 1. We have the chain of the Livshits maps

U.p/

U.p C 1/

U.p C 2/

:

(10.8)

On each step of the chain, the pushforward of the Haar measure is the Haar measure. By the Kolmogorov Theorem about projective limits34 , there exists a space U equipped 34

Another term is the Kolmogorov extension theorem.


133

with a probability measure 1 and maps n W U ! U.n/ consistent with arrows (10.8) such that the pushforward of 1 under n is the Haar measure on35 U.n/. Certainly, U is not a group, however we are going to show that U inherits a certain algebraic structure. Denote by U.1/ the group of all infinite unitary matrices g such that g 1 has only a finite number of non-zero matrix elements. Theorem 10.8. The group U.1/U.1/ acts on U by the left and right multiplications. This action preserves the measure 1 . Proof. We use Observation 10.4. The Livshits map ‡ 1 W U.p C k C 1/ ! U.p C k/ is U.p C k/ U.p C k/-equivariant. In particular, it is U.p/ U.p/-equivariant. Therefore, the projective limit of the chain (10.8) inherits the action of U.p/ U.p/. The preserving of the measure is obvious. But this reasoning is valid for arbitrary p and we get action of U.1/ U.1/. Thus there arises the problem of harmonic analysis on the space U. 10.9 Another point of view to the inverse limits. Again, consider the Cayley transform U.n/ ! AHermn . It can be shown36 , that the pushforward of the Haar measure is dn WD const det.1 T 2 /n d T:

(10.9)

Theorem 10.9 (Hua Loo Keng). The pushforward of the measure dpCj with respect to cut-off map „j is dp . The statement is obvious because it is a paraphrasing of Theorem 10.7. But actually the Hua Theorem was obtained37 as a result of heavy calculations, which repelled specialists in semi-simple groups during several decades. 10.10 Digression. Hua-type measures. The projectivity theorem in the Hua Loo Keng’s form suggests that the Haar measure must be replaced by the measures det.1 C T /nCa det.1 T /nCb d T . Theorem 10.10. Fix a, b 2 C such that Re.a C b/ > 1. Consider the complex-valued measures (charges) da;b D det.1 C g/ det.1 C g/ dg (10.10) n is a positive measure. on U.n/; if b D a, N then a;b n 35 For a reader familiar with measure theory: .U; 1 / is a Lebesgue measure space and the -algebra of U is generated by n -pullbacks of the -algebras on U.n/. 36 We omit a formal proof. The tools of the next section and Problem 10.3 allow us to verify the invariance of the expression (10.9) under the linear fractional action of U.n/ U.n/. After this we can refer to the uniqueness of the Haar measure. 37 It was not formulated in this form in Hua’s book [95], but in fact it was used in calculations.

134


a) The total charge of U.n/ is H.a; b; n/ D

n Y .k/.k C a C b/ : .k C a/.k C b/

kD1

In particular, the measures a N d na;aN .g/ D H.a; a; N n/1 da; n .g/

(10.11)

are probabilistic. b) The system na;aN is consistent with the Livshits maps (i.e., the pushforward of na;aN under a;a N ). ‡ is n1 1

Lemma 10.11. Denote by D the disk jzj < 1. Consider the map U.n/ ! U.n 1/ D given by g 7! .‡ 1 .g/; gnn /; (10.12) where gnn in the right-most lowest matrix element of g. The pushforward of the Haar measure under this map is 1 .n 1/ .1 jzj2 /n2 n1 .h/ dzdz: Proof of the lemma. Since ‡ 1 is U.n 1/ U.n 1/-equivariant, the desired measure has the form ~.z/ n1 .h/ dz dz; where ~ is a function. To find ~, consider the pushforward of the latter measure under the projection U.n 1/ D onto D. Clearly, it is ~.z/ dz dz. Let us find ~.z/ dz dz directly as a pushforward of n under the map U.n/ ! D that takes g to gnn . In other words, we must find the distribution of the matrix element gnn on the unitary group. Denote by S 2n1 the sphere in Cn , by en the last basis element in Cn . Consider the map U.n/ ! S 2n1 given by g 7! gen

or

g 7! .g1n ; : : : ; gnn /:

Evidently, the pushforward of the Haar measure is the uniform measure on the sphere. Now we must find a distribution of the last coordinate on the sphere. We leave this problem as an exercise for the reader. a b Proof of the theorem. For g D c d we write out det.1 C g/ as the determinant of a block matrix, see (1.12), det.1 C g/ D .1 C d / det.1 C a b.1 C d /1 c/ D .1 C d / det.1 C ‡ 1 .g//: Thus the pushforward of a;b under the map (10.12) is n 1 .n 1/.1 C d / .1 C dN / .1 jd j2 /n2 a;b n1 : Considering the pushforward of this measure to U.n 1/ we get C.a; b; n/ a;b n1 :

135

2.10. Isotropic category. Inverse limits It remains to evaluate the constant C.a; b; n/ D

n1

Z .1 jzj2 /n2 .1 C z/a .1 C zN /b dz dz: jzj0

It can be evaluated in a straightforward way, we write u D x C iy and integrate dy. This is the Cauchy beta-integral, see [7], after the integration we get the usual beta-integral in x. We omit details38 . The measures na;aN also form a projective system (for fixed a 2 C) and it is more natural to consider the infinite-dimensional harmonic analysis in this generality. 10.11 Digression. Inverse limits of the symmetric groups. Denote by Sn the usual symmetric group. 1 2 :::Define the map Ъ W Sn ! Sn1 according to the following rule. Consider a substitution n i1 i2 ::: in and remove both the entries of n in the first and the second row. For instance, let n D 6. Consider the following substitutions: 1 2 3 4 5 1 2 3 4 5 6 ; 7! 2 3 1 5 4 2 3 1 5 4 6 ! 1 2 3 4 5 6 1 2 3 4 5 1 2 3 5 4 7! 7! : 2 3 1 6 4 5 2 3 1 4 5 2 3 1 4 5 In the first case, we immediately get an element of S5 . In the second case we do not know an image of “4” and a preimage of “5”. Therefore, set 4 ! 5. The Haar measure on Sn is defined by the condition: the measure of each point is 1=nŠ. Problem 10.4. a) Show that the map Ъ W Sn ! Sn1 commutes with the left-right action of Sn1 Sn1 . b) The image of the Haar measure under Ъ is the Haar measure. Problem 10.5. Find all morphisms Sn ! Sn1 in the category of Sn1 Sn1 -spaces. Next, we define the Ewens measures (the analog of Hua measures). Fix t > 0, set ~.t / WD t.t C 1/ : : : .t C n 1/. Denote by f g the number of independent cycles in 2 Sn . Assume that the measure of a point is ~.t /1 t fg . Problem 10.6. Show that the total measure of Sn is 1; moreover, this system of measures is Ъ-projective. 38 On the other hand, (10.13) looks like an L2 -inner product of two functions with respect to the measure .Re z/n2 dz dz. We can immediately write the final result applying the Plancherel formula for a weighted Laplace transform, see Theorem 7.5.16 below.

136


10.12 Chinese restaurant. A Chinese restaurant is the set S whose points are the following collections of data: P 1. A countable (non-ordered) set of positive numbers l1 , l2 ,…, where lj D 1. 2. A countable collection of oriented circles Tj (“tables”) with lengths l1 , l2 ,…. Origins on the circles are not fixed. 3. An injective map E W N ! [Tj . We define the map Ъ W S ! Sn in the following way. Consider the image of the set f1; 2; : : : ; ng under the map E and forget distances; we get a product of independent cycles, see Figure 2.19.

4

2

1

3 6

7 4

5

2

1

3 7

6

5

Figure 2.19. The projection S ! S7 . In the figure, we get a substitution 1 7! 4 7! 7 ! 1, 2 7! 6 7! 2, 5 7! 5, 3 7! 3. The projection S ! S6 gives 1 7! 4 7! 1, 2 7! 6 7! 2, 5 7! 5, 3 7! 3 (we simply remove 7 from the cycle decomposition). In this way, we obtain a family of consistent surjective maps S ! Sn for all Sn . A role of lengths is not immediately obvious, at first glance they are artificial. Now consider a sequence of substitutions n 2 Sn such that Ъ n D n1 . Let l1 .n/ > l2 .n/ > be the lengths of independent cycles of n . Theorem 10.12. For almost all sequences n all limits lj .1/ WD lim lj .n/=n exist and P lj .1/ D 1. Next, let integers ˛ and ˇ be contained in the same cycle in m . Then ˛ and ˇ are contained in the same cycle in each n for n > m. Denote by l ˛;ˇ .n/ the length of the chain ˛ 7! 1 7! 7! h 7! ˇ in n . Theorem 10.13. For almost all chains n for all “tables” and for all ˛, ˇ being in the same “table”, the limit limn!1 l ˛;ˇ .n/=n exists and is positive. This defines relative positions of points f1g, f2g, …on the “tables”. Also, this shows that the image of the map E W N ! T is dense (almost sure).

2.11. Kähler metrics on matrix balls. Some matrix tricks

137

Problem 10.7. Denote by S1 the group of all finitely supported substitutions of N. Describe the left-right action of S1 S1 on S. Problem 10.8. Devise a reasonable description for the inverse limit of the spaces U.n/ (the author does not know a solution).

2.11 Kähler metrics on matrix balls. Some matrix tricks In the next chapter, we discuss different approaches to central extensions of the groups Sp.2n; R/ and U.p; q/. One of the approaches (see § 3.9) requires the Kähler structure on the matrix ball. Also, in Chapter 7 (and in some “digressions”) we occasionally refer to some formulas of the present section. 11.1 The differential of a linear fractional map. Let z range in Matp;q . Œg WD .a C zc/1 .b C zd /, where g WD Lemma 11.1. a) Consider a map z 7! z a b 2 GL.p C q; C/. Its differential is c d

7! .a C zc/1 .cz Œg C d /:

(11.1)

b) If g 2 U.p; q/, then the differential also equals 7! .a C zc/1 .b z C d /1 :

(11.2)

Proof. a) Let us calculate .z C "/Œg mod ."2 /, 1 1 b C .z C "/d D .a C zc/ C "z .b C zd / C "d : a C .z C "/c We rearrange the first factor, 1 D .aCzc/1 1C" c.aCzc/1 D .aCzc/1 1" c.aCzc/1 CO."2 /; and obtain .a C zc/1 .b C zd / ".a C zc/1 c.a C zc/1 .b C zd / C ".a C zc/1 d C O."2 / D z Œg C ".a C zc/1 c.a C zc/1 .b C zd / C d C O."2 /: b) We transform the term z Œg in formula (11.1) by (3.5) and obtain cz Œg C d D c.a C zc/1 .b C zd / C d D c.a z C c /.b z C d /1 C d

D c.a z C c / C d.b z C d / .b z C d /1

D .cc C d d / C .ca C db /z .b z C d /1 : By the defining equations (2.4) for U.p; q/, the expression in the square brackets is 1.

138


11.2 The Jacobian of a linear fractional map Proposition 11.2. The complex Jacobian of a transformation z 7! z Œg , where g 2 GL.p C q; C/, equals det.a C zc/pq det.g/q . Lemma 11.3. Consider a linear map Matp;q ! Matp;q given by z 7! azb;

where a 2 GL.p/, b 2 GL.q/:

(11.3)

Its determinant is det.a/q det.b/p . Proof of the lemma. It suffices to consider the map z 7! az. We represent Matp;q D Cp ˚ ˚ Cp (q times); our operator acts as a ˚ a ˚ ˚ a. Proof of the proposition. The differential is of the form (11.3). It remains to evaluate det.cz Œg C d /. By formula (1.1.12) for the determinant of a block matrix, det.a C zc/ det c.a C zc/1 .b C zd / C d a C zc b C zd 1 z a b D det D det det c d 0 1 c d a b D det : c d 11.3 A useful formula Lemma 11.4. For g D

a b c d

2 U.p; q/,

1 z Œg .uŒg / D .a C zc/1 .1 zu /.a C uc/1 :

(11.4)

Proof. The left-hand side is 1 .a C zc/1 .b C zd /.b C d u /.a C c u /1

D .a C zc/1 .a C zc/.a C c u / .b C zd /.b C d u / .a C c u /1 D .a C zc/1 .aa bb / C z.ca db /

C .ac bd /u C z.cc d d /u .a C c u /1 : We apply defining equations (2.4) for U.p; q/ and obtain .1 zu / in the square brackets. Problem 11.1. Solve Problem 5.6. Problem 11.2. For each g 2 U.p; q/, 1 .z Œg / uŒg D .b C d z/1 .1 z u/.b C d u/1 :

(11.5)


139

11.4 The invariant Hermitian metric. Let T and S be tangent vectors to Bp;q at a point z (in other words, T and S are p q matrices). Define the Hermitian metric39 on Bp;q by .T; S / D tr.1 zz /1 T .1 z z/1 S : (11.6) Theorem 11.5. The Hermitian metric is U.p; q/-invariant Proof. The rules of transformation of all the factors are (11.2), (11.4), (11.5), .1 zz /1 7! .a C zc/ .1 zz /1 .a C zc/; T

7! .a C zc/1 T .b z C d /1 ;

.1 z z/1 7! .b z C d /.1 z z/1 .b z C d / ; S

7! .b z C d /1 S .a C zc/1 :

Hence, h i .1zz /1 T .1z z/1 S 7! .aCzc/ .1zz /1 T .1z z/1 S .aCzc/1

and the traces of the left and right expressions coincide.

Problem 11.3. a) Denote by „ the the coordinate expression (5.8) for angles. Then „.z; z C "T / D 1 C "2 .1 zz /1 T .1 z z/1 T C O."2 /: b) Describe the invariants of a pair fpoint z 2 Bp;q , tangent vector at zg under the action of U.p; q/. 11.5 The invariant measure on Bp;q Theorem 11.6. There is a unique (to within a scalar factor) U.p; q/-invariant measure on Bp;q , namely d.z/ D det.1 zz /pq

Y ij

d Re zij

Y

d Im zij :

(11.7)

ij

Proof. Due to Proposition 11.2 Q 11.4 we know the rules of the transforQ and Lemma mation of det.1 zz / and ij d Re zij ij d Im zij under the action of U.p; q/. This easily implies the invariance of (11.7). Uniqueness is more-or-less obvious, see Addendum C.10. 39 A Hermitian metric on a complex manifold is a Euclidean inner product on each tangent space; the product must depend on a point smoothly.

140


11.6 The Kähler potential Theorem 11.7. The Hermitian metric is Kähler. Its Kähler potential is ln det.1 zz / D tr ln.1 zz /: We define the logarithm of a matrix in the usual way, ln.1 C X / D

1 X .1/k1 k X k

if kX k < 1:

(11.8)

kD1

Recall that a Hermitian metric is said to be a Kähler metric40 if the 2-form Im is closed. In particular, our Bp;q is a symplectic manifold. A function K is called a Kähler potential of a Kähler metric if .dz; d z/ N D

X @2 K.z; z/ N k;l

where

1 @ @ @ D i ; 2 @xk @yk @zk dzk D dxk C idyk ;

@zk @zNl

dzk ^ d zN l ;

@ 1 @ @ D Ci ; @zNk 2 @xk @yk d zNk D dxk idyk :

Recall that a Kähler potential is not canonically determined by a metric. Proof (straightforward calculation). We must find the term "2 T S in the Taylor expansion of tr ln.1 .z C "T /.z C "S / /: tr ln 1 .z C "T /.z C "S / D tr ln 1 zz ".zS C T z / "2 T S D tr ln.1 zz / C ln 1 ".1 zz /1 .T z C zS /

"2 .1 zz /1 .T S / D tr ln.1 zz / C tr ".1 zz /1 .T z C zS / 12 "2 .1 zz /1 .T z C zS /.1 zz /1 .T z C zS /

"2 .1 zz /1 T S C O."3 /: We must only watch the terms with both T and S : "2 tr 12 .1 zz /1 T z .1 zz /1 zS

12 .1 zz /1 zS .1 zz /1 T z .1 zz /1 T S :

40

On Kähler metrics see e.g., [21], [5].

141


The traces of the first two summands coincide (since tr AB D tr BA), we get

"2 tr.1 zz /1 T z .1 zz /1 zS C T S

D "2 tr.1 zz /1 T z .1 zz /1 z C 1 S D "2 tr.1 zz /1 T .1 z z/1 S ;

and obtain the desired expression.

Another proof. We are going to watch the behavior of tr ln.1 zz / under the transformation z 7! z Œg . By (11.4), tr ln.1 z Œg .z Œg / / D tr ln .a C zc/1 .1 zz /.a C zc/1 D tr ln a1 .1 C zca1 /1 .1 zz /.1 C zca1 /1 a1 D tr ln a1 a1 .1 C zca1 /1 .1 zz /.1 C zca1 /1

D tr ln.1 zz / tr ln.aa / tr ln.1 C zca

1

/ tr ln.1 C zca

(11.9) (11.10) (11.11) (11.12) 1

/ : (11.13)

We explain what has happened. We intend to apply the formula tr ln.X Y / D tr ln X C tr ln Y:

(11.14)

The formula is dangerous, because the logarithm is a multivalued function. The initial expression (11.9) is well defined. Nevertheless, we can not immediately transform (11.10) because we do not know the meaning of the expression ln.a C zc/. The passage from (11.11) to (11.12) is an application of tr ln X Y D tr ln YX . At the last moment, we apply (11.14), because all the summands in (11.13) are well defined. Thus tr ln.1 z Œg .z Œg / / is equal to ² ³ ² ³ ² ³ holomorphic antiholomorphic C : tr ln.1 zz / C constant C summand summand After differentiating @@N the three additional terms vanish. Thus, – Kähler metrics defined by our potential are U.p; q/-invariant. The metric is also invariant. Therefore, it suffices to compare two metrics at the point 0. 11.7 Exercises on matrix differentials and Jacobians Problem 11.4. For the spaces Symmn and ASymmn of real symmetric and skew-symmetric matrices, consider the linear maps z 7! at za. Show that their determinants are respectively .det a/nC1 and .det a/n1 ; see also Lemma 7.5.2. Problem 11.5. Prove the Hua formula (10.9) for the Haar measure in Cayley coordinates. One of the possible ways is to verify the invariance of this expression with respect to linear fractional transformations from Problem 10.3.

142


Problem 11.6. a) Find the differential of the map X 7! e X on the space Mat n as follows. For a matrix T , define the linear maps L D LT , R D RT W Mat n ! Matn by LT Y D T Y;

RT Y D Y T:

Show that the differential of the map X 7! exp.X / is 7!

1 X 1 Ln R n exp L exp R WD : LR nŠ L R nD1

b) Show that the Jacobian of the map X 7! e X is Y e i e j Y D i j

i>j

i¤j

sinh.i j /=2 .i j /=2

2 ;

(11.15)

where j are the eigenvalues of X . Hint. First, assume that X is diagonalizable; next, apply continuity arguments. P c) Extend a), b) to arbitrary maps X 7! k>0 ck X k , where ck are scalars. Let G be a Lie group admitting a two-side-invariant Haar measure (see Addendum C.10), g a Lie algebra of G. The exponential map g ! G determines a local coordinate system on g. Problem 11.7. Show that the density of the Haar measure in the exponential coordinates is const

Y sinh j =2 j

j =2

;

(11.16)

where j are the eigenvalues of ad.X /41 . Hint. By the Ado theorem, g gl.n/. Let h WD exp.P /. Find the Jacobian of the map X 7! g 1 exp X at a point P .

2.12 Matrix balls as symmetric spaces We occasionally refer to this section below and above, but it is not crucial to our arguments. 12.1 Riemannian metrics on matrix balls. In the previous section, we have defined the invariant Hermitian metric on Bp;q . Its real part M.T; S/ WD Re tr.1 zz /1 T .1 z z/1 S is a U.p; q/-invariant Riemannian metric. 41

where ad is the adjoint map, see Addendum C.9.

2.12. Matrix balls as symmetric spaces

143

12.2 Symmetric spaces. Let M be a complete Riemannian manifold42 . Let x 2 M . A geodesic symmetry x is the map from a neighborhood of x to itself defined in the following way: x .y/ D z

if x is the midpoint of the geodesics connecting y and z:

The space M is called symmetric if all geodesic symmetries are isometries. y2 y1

z1

x z2

Figure 2.20. A geodesic symmetry.

Theorem 12.1. The matrix ball Bp;q is a symmetric space. Proof. The map z 7! .z/ is a geodesic symmetry in 0. Since the group of isometries acts transitively, now it follows that all geodesic symmetries are isometries. Let us present a coordinate-free description of geodesic symmetries in Bp;q . Recall that we have identified Bp;q with a certain domain Mp;q in a Grassmannian. Namely, for a negative subspace P 2 Mp;q , the map P is the reflection in P . 12.3 Description of geodesics Theorem 12.2. a) The submanifold in „ Bp;q consisting of matrices 1 0 1 0 : : : B 0 2 : : :C A ; where j are real, jj j < 1; @ :: :: : : : : :

(12.1)

is totally geodesic; the Riemannian metric is flat on „. Moreover, the coordinates xj D arctanh j on (12.1) are flat. b) Given j 2 R, the curve 1 0 0 ::: tanh. 1 t / B 0 tanh. 2 t / : : :C (12.2) .t / WD @ A :: :: :: : : : is a geodesic. 42 The definition given below can be extended to pseudo-Riemannian spaces in a trivial way. For a representative collection of examples of pseudo-Riemannian symmetric spaces, see Problem 3.3.6; for tables see Addendum D.

144


c) Each geodesic can be reduced to this form by an element of U.p; q/. d) The geodesic distance in Bp;q is given by the formula rX 2 .L1 ; L2 / D const j .L1 ; L2 / ; j

where

j

(12.3)

are the hyperbolic angles.

Recall that a submanifold in a Riemannian manifold is said to be totally geodesic if the short geodesic joining any pair of close points of the submanifold is contained in the submanifold. Lemma 12.3. Any connected component of the set of fixed points of any isometry is a totally geodesic subspace. Proof of the lemma. Denote the isometry by , by Fix. / the set of its fixed points. Given close points x, y 2 Fix. /, denote by the short geodesics connecting these points. Then is also a geodesic having the same length. Since a short geodesic is unique, it follows that D , and Fix. /. Proof of Theorem 12.2. a) Consider an isometry Bp;q ! Bp;q of the form 1 0 ƒ ; z 7! ƒz 0 ‚ where ƒ 2 U.p/, ‚ 2 U.q p/ are diagonal matrices with pairwise distinct entries ¤ 1. The set of its fixed points consists of matrices of the form (12.1) with j 2 C. Therefore, the set of matrices having the form (12.1) with complex entries43 j is totally geodesic. Next, we consider the isometry z 7! zN and apply the lemma again. Thus „ is totally geodesic. The Riemannian metric on „ is p X j D1

d j2 .1

j2 /2

D

p X

d arctanh j

2

:

j D1

Therefore, the coordinates xj D arctanh j are flat. b) This curve is a straight line in the flat coordinates. c) We move a geodesic by an element of U.p; q/ and put one of its points to 0. Next, we move the geodesic by an element of the group U.p/ U.q/. In this way, we can diagonalize a given point of the geodesics. Now two points of the geodesics are contained in the submanifold „ and we know the geodesic connecting these points. d) We merely evaluate the length of a segment of a geodesic. 43

It is the product of p copies of the Lobachevsky plane.

2.12. Matrix balls as symmetric spaces

145

12.4 Parallel transport in Bp;q . Let W Œa; b ! M be a smooth curve in a Riemannian manifold. Recall that it determines a canonical map (parallel transport) of the tangent spaces T.a/ ! T.b/ . Theorem 12.4. a) For M D Bp;q , one can find an isometry g 2 U.p; q/ taking .a/ to .b/ and inducing the parallel transport map T.a/ ! T.b/ . b) The same property holds for any symmetric space. To prove this theorem, we reformulate the definition of the parallel Denote transport. 2kC1 by t the natural parameter on , put a D 0. Given N , let x.k/ WD a C 2N C1 .b a/ (i.e., we divide the curve into 2N C 1 segments of equal lengths). Next, consider the product x.2N / B x.2N 1/ B B x.2/ B x.1/ and the induced map T.a/ ! T.b/ . Now let N ! 1. The limit is the parallel transport map. After this reformulation, the theorem becomes obvious.

Figure 2.21. A sequence of geodesic symmetries. 12.5 Digression. What is the geometry of symmetric spaces? Thus the standard objects of differential geometry as geodesics, parallel transport, Riemann curvature tensor, etc., admit a rather transparent description for Bp;q (and more generally, for all Riemannian symmetric spaces). But the main subjects of geometry of symmetric spaces are far beyond such questions. Some important classical topics are: – the topology of compact symmetric spaces (as Grassmannians, the group U.n/, the space U.n/=O.n/ discussed in the next chapter, etc.); – geometry of non-compact spaces at infinity; – discrete groups acting on symmetric spaces. In spite of (more than) a hundred years of discussion, it seems that the geometry of symmetric spaces is not well-understood up to now. It seems that the main geometric structure is the angles. In this context, we have no geometrical intuition. Only recently the triangle inequality was obtained. Recall this problem. Let L1 , L2 , L3 2 Bp;q . Then we have three collections of pairwise angles, say j , j , and ~j . Describe the inequalities for , , ~. We return to this problem in Addendum 2 to this chapter. 12.6 Digression. Problems of harmonic analysis on Bp;q . For definiteness, set p 6 q. The first problem of harmonic analysis on Bp;q is the decomposition of the space L2 .Bp;q ; d/ into an integral of irreducible representations, see, for instance [86].

146


There are two other types of inner products, which lead to a richer theory. A) Berezin kernels on Bp;q are defined by det.1 zz / det.1 uu / ˛ : K˛ .z; u/ WD j det.1 zu /j2 Let ız denote the delta-function supported by z 2 Bp;q . We introduce the inner products of finite linear combinations of delta-functions by hız ; ıu i˛ D K˛ .z; u/: Theorem 12.5. This inner product is non-negative definite if and only if ˛ D 0, 1; : : : ; p 1 or ˛ > p 1. This statement is a rephrasing of the Berezin–Gindikin–Vergne–Rossi–Wallach Theorem, see Section 7.5. Therefore, our inner product defines a structure of a pre-Hilbert space and we consider the associated Hilbert space H˛ (for a detailed discussion of constructions of this type see Section 7.1). It turns out that the harmonic analysis in H˛ is a quite rich subject; the classical L2 -analysis is its limit as ˛ ! 1. B) Stein–Sahi kernels. Define the inner product on the space of smooth compactly supported functions on Bp;q by the singular integral “ hf; gi˛ D det ˛ .1 zz /1 .z u/ det.1 u u/1 .z u / Bp;q Bp;q

f .z/g.u/ d.z/ d.u/: This integral converges for ˛ < 1; further we consider its meromorphic continuation in the parameter ˛. Conditions of positivity of this inner product are not completely investigated. It is known that for p D q the inner product is positive for j˛ pj < 1=2; for ˛ D p, we obtain the L2 -inner product. For certain p ¤ q there are some small islands of positivity. C) Recall that one has several ways to write the Hua double ratio. Both kernels (Berezin and Stein–Sahi) are powers of the determinant of variants of the double ratio. It is interesting that there is a third variant, det ˛ .1 u z/1 .u z /.1 zu /1 .u z/ : As far as I know, it is senseless. 12.7 Other Riemannian symmetric spaces. Matrix ball models Theorem 12.6. The submanifold in Bn;n consisting of symmetric matrices is totally geodesic. Corollary 12.7. It is a symmetric space. Proof. The map z 7! z t is an isometry (see Lemma 12.3).

Addendum 1 to Section 2. Blaschke–Potapov factorizations and operator colligations 147 This manifold Sp.2n; R/=U.n/ is an important character of the next chapter. We can also consider – the space of skew-symmetric complex matrices (SO .2n/=U.n/) and the space of Hermitian matrices (GL.n; C/=U.n/); – real symmetric and skew-symmetric matrices from which we get GL.n; R/=O.n/, O.n; C/=O.n/; real p q-matrices in Bp;q from which we get O.p; q/=O.p/ O.q/; – quaternions as complex 22-matrices; this allows us to realize the space Sp.p; q/=Sp.p/ Sp.q/ as a totally geodesic subspace Bp;q .H/ in B2p;2q ; – the totally geodesic subspaces GL.n; H/=Sp.n/, Sp.2n; C/=Sp.n/ in Bn;n .H/, consisting of quaternion Hermitian and anti-Hermitian matrices respectively. Modulo minor details, this is a complete list of classical noncompact Riemannian symmetric spaces.44

Addendum 1 to Chapter 2. Blaschke–Potapov factorizations and operator colligations This addendum consists of three parts: ˛) An introduction (without proofs) to V. P. Potapov’s work [175] about factorizations of matrix-valued contractive functions. ˇ) A few words on the original motivation of M. S. Livshic and V. P. Potapov (the spectral theory of non-self-adjoint operators). However, there are numerous reasons for interest in Theorem 3 independent of the original motivation. We use a language that was invented later for a clarification of the Livshits–Potapov theory. ) A few words on spaces of rational curves in Grassmannians. ˛. Potapov multiplicative integral 1. Blaschke products. Consider the unit disc Q D: jzj < 1 on the complex plane. Consider a sequence k 2 D such that the product k j k j converges; we admit finite (and empty) sequences j . The Blaschke product is defined by !.z/ D e i

Y k z j k j : 1 z Nk k

.1/

k

Note that a Blaschke factor b .z/ WD

z j j 1 z N

.2/

44 For a further discussion see Subsection D.1. In many cases, this remark allows us to transfer information from one series of Riemannian symmetric spaces to another. For instance, we immediately get explicit realizations, formulas for geodesics and geodesic distance, expressions for angles, compressivity theorem,etc.

148


satisfies the properties jb .e i' /j D 1;

b . / D 0;

and

jb .z/j < 1 for jzj < 1;

and the normalizing condition b .0/ 2 R. Theorem 1. a) The Blaschke product converges uniformly in each disc jzj < 1 "; in particular, !.z/ is holomorphic in the disc D. b) The product converges a.s on the circle jzj D 1. Moreover, j!.z/j D 1 a.s. on the circle, and the non-tangent limit lim !.z/ D !.e i' / z!e i'

exists a.s. on the circle.

Recall the definition of a non-tangent limit. Let f .z/ be holomorphic in the disc jzj < 1. Then a is a non-tangent limit if for any domain S of the form given in Figure 2.22, lim

z!e i' ; z2S

f .z/ D a:

Figure 2.22. Reference the definition of a non-tangent limit. We pass to a point of the circle from an arbitrary angle inside a disc. 2. Factorization of bounded holomorphic functions Theorem 2 (Riesz–Herglotz). a) Let f .z/ be holomorphic in the disc jzj < 1, let jf .z/j < 1. Then f .z/ can be represented as a product f .z/ D !.z/g.z/; where !.z/ is a Blaschke product, g.z/ is holomorphic and has no zeros in D, and jg.e i /j D jf .e i /j. b) Let g.z/ be a holomorphic function in D, jg.z/j < 1, and g have no zeros in the disc. Then g.z/ can be represented in the form ² Z 2 ³ z C e i g.z/ D exp d . / ; .3/ z e i 0 where . / is a positive finite measure on Œ0; 2/. c) jg.e i' /j D 1 a.s. if and only if the measure is singular45 . 45

I.e., it is supported by a set of zero measure.

Addendum 1 to Section 2. Blaschke–Potapov factorizations and operator colligations 149 We outline a proof; its actual realization is nontrivial. We write the Blaschke product !.z/ f .z/ whose zeros coincide with zeros of f .z/ and set g.z/ D !.z/ . The function g.z/ has no zeros, therefore ln g.z/ is well defined; moreover Re ln g.z/ > 0. A positive harmonic function can be represented as a Poisson integral Z 2 z C e i Re ln g.z/ D d . /; where jzj < 1 Re z e i 0 with a positive measure . Finally, we restore the imaginary part of ln g.z/. i

is pure imaginary on the circle. Its real part is negative Remark. Evidently the expression zCe ze i inside a disc. n o i Problem 1. Interpret the expression exp zCe as a limit of Blaschke products. i ze 3. Matrix-valued contractive functions. Now we consider a meromorphic function f .z/ on the disc D taking values in the semigroup U.p; q/ of indefinite contractions. We also assume that det f .z/ is not the identically zero. Such matrix-valued functions are said to be contractive. We can also say that f .z/ is a holomorphic function taking values in the semigroup of contractive linear relations in Cp;q . Poles of such matrix-valued functions correspond to linear relations that are not linear operators. We wish to extend the Riesz–Herglotz Theorem to the matrix case. 4. Blaschke factors. First, we must find analogs b˙ .z/ of the Blaschke factor b .z/. Values of our functions are linear relations. A linear relation can have a kernel and also can have an indefinity. Therefore we need two types of Blaschke factors. 1) For a given 2 D, we wish to find a function bC .z/ such that: – bC .z/ is an indefinite contraction for all z 2 D; – bC .e i' / is pseudounitary; – dim ker bC . / D 1; – ker bC .z/ D 0 for z ¤ ; – indef bC .z/ D 0 for all z 2 D. For D 0 a solution is obvious: 0

1p b0C .z/ D @ 0 0

0 z 0

0 0

1 A:

1q1

Next, we write a solution for arbitrary 2 D as follows: 0 1p z j j B C C b .z/ WD b0 D@ 0 1 z N 0

0 z 1z N

0

0

jj

0

1 C A

1q1

This solution is not unique; if S1 , S2 are pseudounitary matrices, then S1 b C .z/S21 also satisfies the desired conditions.

150


2) The function

0 B b .z/ WD @

1z N z

0 0

j j

0 1p1 0

1 0 C 0 A 1q

is contractive and satisfies the conditions: dim indef b ./ D 1;

indef b .z/ D 0 for ¤ z;

ker b .z/ D 0 for all z 2 D;

b .e i / 2 U.p; q/:

5. Blaschke products. We define a matrix Blaschke product as a product of the form Y Y Sj bCj .z/Sj1 Rk b k .z/Rk1 ; where Sj , Rk 2 U.p; q/:

.4/

k

j

These products are noncommutative. It is convenient to think that the sequences j j j and jk j are increasing. The condition of convergence of the Blaschke product (4) is: Q Q – products j j j j, k jk j converge; – the Blaschke product converges at one (arbitrary) point z D that is not contained in the sequences j , k . 6. The Potapov multiplicative integral. Next, we wish to write an analog of the multiplicative integral (3) in the Riesz–Herglotz theorem. Let t 7! H.t / be a “monotonic” function on a segment Œ0; a taking values in Hermitian matrices, i.e., H.t / > H.s/ for t > s and tr H.t / D t: 1p 0 Denote J WD 0 1q . Let .t / be an R-valued monotonic nondecreasing function on Œ0; 2. We define the multiplicative integral as Z Õ zCe i.t/ dH.t/J e zei.t/ Œ0;a

WD

lim

max.tj tj 1 /!0

n Y j D1

n z C e i.j /

exp

z e i.j /

o J H.tj / H.tj 1 / ;

(5)

the product is ordered and 0 D t0 6 1 6 t1 6 2 6 t2 6 6 tn1 6 n 6 tn D a: 7. The Potapov theorem Theorem 3. Any contractive holomorphic function f .z/, where z 2 D, can be represented as a product f .z/ D B.z/g.z/; where B.z/ is a Blaschke product (4), and g.z/ is a multiplicative integral (5).

Addendum 1 to Section 2. Blaschke–Potapov factorizations and operator colligations 151 ˇ. Operator colligations and the spectral theory of almost unitary operators 8. Operator colligations and characteristic functions. We say that an operator D in a Hilbert space W is almost unitary if it can be represented in the form D WD U.1 C L/, where U is a unitary operator and L is an operator of finite rank.46 We say that an operator is almost selfadjoint if it is a finite-dimensional perturbation X CH of a self-adjoint operator W . Our purpose is a spectral theory of such operators. Until Subsection 11 we consider only contractive operators D, i.e., kDxk 6 kxk for all x 2W. For a contractive almost unitary D, we build an operator A B D WD W V ˚W !V ˚W .6/ C D in a larger Hilbert space V ˚ W in such a way that D is unitary. Problem 2. a) Find such an extension for an operator D in a finite-dimensional space. Describe all such extensions. b) For an almost unitary contractive operator D one can find an extension such that V is finite-dimensional. Next, we write the characteristic function, D .z/ D A zB.1 C zD/1 C;

where z 2 C, jzj < 1;

it is a holomorphic matrix-valued function. Another definition. We write the following “perverse equation for eigenvalues”: A B D : zw C D w

.7/

Then D D .z/ if there is w satisfying the “perverse” equation (7). Problem 3. a) k.z/k 6 1 for jzj < 1. Hint. See the proof of Theorem 10.2. b) If the Hilbert space W is finite-dimensional, then the operators .e i' / are unitary.

c) The function .z/ has non-tangent limits a.s. on the circle jzj D 1.

Theorem 4. The following conditions are equivalent. – The sequences D n and .D /n strongly converge to 0, — .z/ is unitary on the circle jzj D 1. 46

In particular, D is a Fredholm operator of zero index.

152


We define an operator colligation as a block unitary operator (6) defined up to the following equivalence: 1 0 A B 0 1 A B .8/ C D 0 U C D 0 U 1 where U is a unitary operator in W . Observation 5. For equivalent operator colligations, their characteristic functions coincide. Thus the correspondence D 7! D is well defined on operator colligations. 9. Reconstruction of an operator colligation from its characteristic function. Let R be a unitary operator. Evidently, the characteristic functions of the colligation 0 1 A B 0 @ C D 0 A .9/ 0 0 R coincides with the characteristic function of the colligation (6). Problem 4 (Kolmogorov–Wold decomposition). A contractive operator Q in a Hilbert space is said to be pure non-unitary if the restriction of Q to any Q-invariant subspace is non-unitary. Let D be a contractive linear operator in a Hilbert space W . There exist a unique orthogonal decomposition W D W 0 ˚ Y such that subspaces W 0 , Y are D-invariant, the restriction of D to Y is unitary, and the restriction to W 0 is pure non-unitary. Problem 5. Any operator colligation admits a unique splitting of the form (9) such that D is pure non-unitary and R is unitary. We say that an operator colligation is pure if D is pure non-unitary. Theorem 6. A pure operator colligation can be reconstructed from its characteristic function. In particular “remembers” the operator D up to the unitary equivalence D UDU 1 . We present the inverse construction. Special case. Let .z/ is unitary on the circle jzj D 1. Consider the space L2 .S 1 ; V / consisting of L2 functions on the circle jzj D 1 taking values in the (finite-dimensional) Hilbert space V . In other words, our space consists of series X X f .z/ D z j vj ; where vj 2 V and kf k2 WD kvj k2 < 1: j 2Z

j 2Z

Consider the subspace H 2 .D; V / L2 .S 1 ; V / (vector-valued Hardy space) consisting of functions admitting a holomorphic continuation inside the disc jzj < 1. In other words, the Hardy space consists of series X f .z/ D z j vj 2 L2 .S 1 ; V /: j >0

Addendum 1 to Section 2. Blaschke–Potapov factorizations and operator colligations 153 We define the shift operator in L2 .S 1 ; V / by f .z/ D z 1 f .z/: Let ‚.z/ be a contractive matrix-valued function, whose non-tangent boundary values are unitary a.s. Note that the map f .z/ 7! ‚.z/f .z/ is a unitary operator in L2 .S 1 ; V / and a (proper) isometric embedding H 2 ! H 2 . Consider the following two chains of subspaces in L2 .S 1 ; V /: L2 H 2 ‚H 2 z‚H 2 ; L2 z 1 H 2 H 2 ‚H 2 : Let us regard the shift operator as an operator S W H 2 z‚H 2 ! z 1 H 2 ‚H 2 and consider the following block decomposition of S : S W ‚H 2 z‚H 2 ˚ H 2 ‚H 2 ! z 1 H 2 H 2 ˚ H 2 ‚H 2 : Next, note that

‚H 2 z‚H 2 ' z 1 H 2 H 2 ' V: It can be easily shown that S is the desired operator colligation. Problem 6. Find the characteristic function of S. General case. Now let ‚ be arbitrary, k‚.z/k 6 1 in the disk jzj < 1. Consider the operator f .e i' / D .1 j‚.e i' /j2 /1=2 f .e i' / in L2 .S 1 ; V /. Denote by M L2 .S 1 ; V / the closure of the image of . Problem 7. Describe M in terms of ‚. Denote by „ the (isometric) operator H 2 .D; V / ! H 2 .D; V / ˚ M given by „f D ‚f ˚ f: Now we have two chains of subspaces in L2 ˚ M of the form H 2 ˚ M „H 2 z„H 2 ; z 1 .H 2 ˚ M / .H 2 ˚ M / „H 2 ; and repeat the previous construction. 10. Multiplication of operator colligations. multiplication on operator colligations: A B P Q DBT D B C D R T 0 10 A B 0 P ´ @ C D 0 A@ 0 0 R 0 1

We define the following strange operation of

0 1 0

1 0 Q A 0 A D @ CP R T

The size of the final matrix is dim V C 1 C 1 D dim V C 1:

B D 0

1 AQ CQ A : T

(10)

154


Problem 8. a) This operation is well defined as an operation on equivalence classes.

b) This operation is associative.

Problem 9. Recall that the colligations D, T are unitary. Show that their B-product is unitary. Theorem 7. DBT .z/ D D .z/T .z/.

Problem 10. Prove Theorem 7.

11. The spectral theory of almost unitary contractive operators. Now let us return to the spectral theory of an individual pure almost unitary contraction K. We built it into an operator colligation K and write the corresponding characteristic function. Assume that we can decompose K as a product, K .z/ D f .z/ g.z/: Then we build operator colligations T , D such that f .z/ D T .z/;

g.z/ D D .z/

as it was explained in Subsection 9. Then we have K.z/ D D.z/ B T .z/: In particular,

D KD 0

CQ : T

Thus having a factorization of a characteristic function of K we get an invariant subspace of the operator K. Thus we come to a problem of multiplicative decompositions of a characteristic function. Theorem 3 produces numerous invariant subspaces of an operator and also produces a triangle representation of an operator K. Remark. If dim V D 1, then the Riesz–Herglotz theorem gives all factorizations .z/ D '.z/ .z/ of a characteristic function. If dim V > 1, then the product is noncommutative and the Potapov theorem gives many factorizations of a characteristic function but not all factorizations. 12. Non-contractive almost unitary operators. If an almost unitary operator D is not contractive we can not build it into a unitary operator colligation (6). But we can build it into a pseudounitary colligation. The program described above survives, but it requires more care in formulations and proofs. The Potapov theorem produces a triangular representation of an operator in this case also. . Rational curves in Grassmannians 13. Space of rational curves in Grassmannians. There is another way of looking at the same topics. Consider the Grassmannian Gr n;n of n-dimensional subspaces in C2n . Denote by Mn x to Gr n;n . Note that the onethe space of all holomorphic maps from the Riemann sphere C dimensional homology of the Grassmannian is Z (see the Schubert cell decomposition, § 2.4).

Addendum 2 to Section 2. The triangle inequality and the Klyachko theorem

155

Therefore a curve has a well-defined degree. We denote by Mn .k/ the space of curves of degree k. A B Now consider an .n C k/ .n C k/-matrix S WD C D . As above, we consider the x ! Matn given by characteristic function C S .z/ D A zB.1 C zD/1 C;

x where z 2 C.

x ! Gr n;n ; poles of are removable singularities On the other hand, we can consider as a map C of this map. Thus, we get a map (defined almost everywhere) from Mat nCk to the space Mn . But an actual argument of the map is an operator colligation, i.e., a matrix defined up to the equivalence 1 0 A B 1 0 A B ; where R 2 GL.k; C/: C D 0 R C D 0 R1 Theorem 8. The map S 7! S is a bijection between an open dense subset in the space of operator colligations of size .n C k/ .n C k/ and an open dense subset in Mm .k/. Moreover, an operator colligation in a general position has a canonical form with diagonal matrix D, and we get an explicit parametrization of an open dense subset in Mn .k/. We have also a natural pointwise multiplication on Mn ; it corresponds to the multiplication of operators colligations.

Addendum 2 to Chapter 2. The triangle inequality and the Klyachko theorem Here we discuss the triangle inequality in the simplest case. Namely: Horn problem. Let A and B be Hermitian matrices such that tr A D 0;

tr B D 0

with eigenvalues ˛1 > > ˛n ;

ˇ1 > > ˇn

respectively. What can we say about the spectrum 1 > > n of the sum A C B? Actually (but this is non-obvious), the set H.˛; ˇ/ of all possible spectra sweep a convex polyhedron (the so-called Horn polyhedron) Hn .˛; ˇ/ in the plane X X X ˇj D j D 0: ˛j D Equivalently, we can define the Horn cone, Hn WD

[ ˛;ˇ

Hn .˛; ˇ/:

156


We wish to present an explicit description of this cone (without proofs and with minimal comments). The final solution of the problem was found by A. Klyachko in 1996; his work initiated a chain reaction in this subject. 1. Lidskii–Horn Lemma Theorem 1 (Lidskii–Horn Lemma). Let matrices A and B 2 Hermn correspond to a point of the boundary of the Horn polyhedron Hn .˛; ˇ/. Then the operators A and B admit a simultaneous splitting B1 0 A1 0 ; BD AD 0 A2 0 B2 in some basis. This is a nontrivial differential geometrical statement, but a proof (it was published by Horn [91] in 1962) is relatively straightforward. Corollary 2. The boundary of H.˛; ˇ/ is contained in the union of hyperplanes i1 C C ik D ˛j1 C C ˛jk C ˇl1 C C ˇlk :

.11/

Theorem 3. The Horn polyhedron H.˛; ˇ/ is convex. This immediately follows from convexity of the image of the moment map. This topic and further explanations are beyond the scope of this book47 . 2. Discrete convex functions. Consider an equilateral lattice triangle whose side is n, see Figure 2.23.a. Consider a function h.i; j / on the set of vertices of the triangle lattice; i.e., on the set i , j 2 Z: i > 0, j > 0, i C j 6 n. We draw the graph of h.i; j /; it is a finite set of points .i; j; h.i; j // 2 R3 : Next, we draw triangles over small lattice triangles. Thus we get a “roof” built from triangles, see Figure 2.23.b. We say that the function h.i; j / is discrete convex, if the roof is convex. The formal condition of convexity is h.i; j C 1/ C h.i C 1; j / > h.i; j / C h.i C 1; j C 1/; h.i; j / C h.i; j C 1/ > h.i C 1; j / C h.i 1; j C 1/; h.i; j / C h.i C 1; j / > h.i; j C 1/ C h.i C 1; j 1/I we write an inequality for each edge. Denote by DCF.n/ the set of all discretely convex functions. Obviously, DCF.n/ is a convex cone in R.nC1/.nC2/=2 . 3. Traces of discrete convex functions. Now we restrict a discrete convex function to the boundary of the triangle. Evidently, we get a convex function on each side of the triangle. Consider differences along the sides of the triangle: ˛i WD h.i; 0/h.i 1; 0/;

ˇj WD h.j 1; 0/h.j; 0/;

k WD h.k; nk/h.k1; nkC1/:

47 In a strange way, this argument does not hold for real symmetric matrices; but the triangle inequality in this case is the same.

Addendum 2 to Section 2. The triangle inequality and the Klyachko theorem

157

j

i

a)

b)

c)

Figure 2.23. a) A lattice triangle. b) A roof. c) Pieces of puzzles.

These three finite sequences are monotonic. Thus we get a convex cone Hn0 R3n . Theorem 4. Hn0 coincides with the Horn cone Hn . This theorem provides one of dozens of descriptions of the Horn cone. As we have seen, the cone is determined by a collection of inequalities of the form i1 C C ik 6 ˛j1 C C ˛jk C ˇl1 C C ˇlk : Such inequality is determined by a collection of triples I WD fi1 ; : : : ; ik g;

J WD fj1 ; : : : ; jk g;

L WD fl1 ; : : : ; lk g:

A nontrivial problem is to describe the set of admissible triples I , J , L explicitly. 4. Labyrinths and puzzles. A puzzle of a lattice triangle is the following collection of data: – coloring of some edges of the lattice in black (labyrinths); – a partition (puzzle) of the big triangle into triangles and rhombi of size 1; we allow only the following pieces: “black” triangles (all the sides are black), white triangles (all the sides are non-colored) and rhombi of the types drawn in Figure 2.23.c. Theorem 5. a) Let us draw a puzzle of the lattice triangle and look at black segments on its boundary. The collection of their numbers fi1 ; : : : ; ik g;

fj1 ; : : : ; jk g;

L WD fl1 ; : : : ; lk g

produces an inequality for a face of the Horn cone. b) This collection of inequalities is minimal. 5. Digression. Idempotent convolution. Now we define the operation of idempotent convolution:48 DCF.n/ DCF.m/ ! DCF.n C m/: Namely, for f 2 DCF.n/, g 2 DCF.m/, we define the function f • g 2 DCF.n C m/ by 0 0 h.i; j / D f • g.i; j / D max f .i ; j / C g.i 00 ; j 00 / : i 0 Ci 00 Di;j 0 Cj 00 Dj

48

We use Maslov’s terminology; currently the term “tropical” is more common.

158


Figure 2.24. Two examples of labyrinths. The puzzles can be reconstructed in a unique way.

Figure 2.25. The continuation of the previous picture. Structure of puzzles. The black regions are unions of black triangles, the white regions are unions of white triangles. Parallelograms are unions of rhombi. Note that we can compose a triangle from all white regions and another triangle from all black regions.

Proposition 6. a) The idempotent convolution of discretely convex functions is discretely convex. b) The collection of differences of f • g along any side of the triangle is the union of collections of boundary differences of f and g along the same side. Assume that a discretely convex function f corresponds to a triple A, B, A C B of matrices and g corresponds to A0 , B 0 , A0 C B 0 . Then the function f • g corresponds to B 0 ACB 0 A 0 ; ; : 0 B0 0 A0 C B 0 0 A0 6. The triangle inequality in the symmetric space GL.n; C/=U.n/. We realize this space as the space of Hermitian positive matrices; the complex distance is the collection of roots of the equation det.A e B/ D 0. The triangle inequality for this space is the same.

3 Linear symplectic geometry

The symplectic groups Sp.2n; R/, which are the topic of this chapter, and the pseudounitary groups U.p; q/ considered above are immediate relatives. In §§ 3.1–3.6 we replicate the considerations of Chapter 2 with minor distinctions (the understanding of Chapter 2 is necessary, we strongly refer here to the tricks and ideas developed above). A selection of nontrivial additional material is contained in §§ 3.7–3.11, which form sui generis a ‘subchapter’. There we discuss the central extension of Sp.2n; R/ and several similar constructions.

3.1 Symplectic groups In this section, we discuss the symplectic groups Sp.2n; R/ and Sp.2n; C/ in parallel ways. In further sections we concentrate our attention on Sp.2n; R/. 1.1 Skew-symmetric bilinear forms. Let K D R or C. Let V be a linear space over K. A bilinear form over K is a K-valued function B.v; w/ on V V satisfying the conditions: B.v1 C v2 ; w/ D B.v1 ; w/ C B.v2 ; w/; B.v; w/ D B.v; w/;

B.v; w1 C w2 / D B.v; w1 / C B.v; w2 /; B.v; w/ D B.v; w/:

A bilinear form B is said to be symmetric if B.w; v/ D B.v; w/ and skew-symmetric if B.w; v/ D B.v; w/:

(1.1)

The latter condition implies B.v; v/ D 0: Any skew-symmetric form can be expressed as B.v; w/ D vQw t ;

where Q D Qt , i.e., Q is a skew-symmetric matrix:

We define the kernel of a form, non-degenerate forms, the rank of a form, and the orthocomplement of a form in the same way as above, see Subsection 2.1.2.

160

Chapter 3. Linear symplectic geometry

Problem 1.1. Prove an analog of Theorem 2.1.11 (properties of the orthocomplement) for skew-symmetric forms. 1.2 Classification of skew-symmetric forms Theorem 1.1. Any skew-symmetric bilinear form can be reduced to the following form in some basis: 1 0 0k 0 0 : : : B 0 J 0 : : :C C B B.v; w/ D vRw t ; where R D B 0 0 J : : :C ; A @ :: :: :: : : : : : : where J is the 2 2-matrix

0 1 1 0

and 0k is the zero k k matrix.

After a rearrangement of a basis, we can write the matrix of our form as 1 0 0 1m 0 0 A: R0 D @1m 0 0 0 0k Corollary 1.2. a) All skew-symmetric forms on odd-dimensional spaces are degenerate. b) All nondegenerate skew-symmetric forms on an even-dimensional space are equivalent. Thus, for a 2n-dimensional space endowed with a non-degenerate skew-symmetric form f; g, one can find a basis e1 ; : : : ; en , f1 ; : : : ; fn such that fei ; ej g D ffi ; fj g D 0;

fei ; fj g D 1:

(1.2)

A basis satisfying these conditions is called canonical. We use the term symplectic space for linear spaces equipped with a skew-symmetric non-degenerate form. Proof of Theorem 1.1. Consider an arbitrary vector e1 … ker B. One can find a vector h such that B.e1 ; h/ D ¤ 0. Set f1 D 1 h. Then B.e1 ; e1 / D B.f1 ; f1 / D 0;

B.e1 ; f1 / D 1:

The form B is nondegenerate on the plane Ke1 ˚ Kf1 and hence V D .Ke1 ˚ Kf1 / ˚ .Ke1 ˚ Kf1 /? : Next, we consider a vector e2 2 .Ke1 ˚ Kf1 /? , etc.

3.1. Symplectic groups

161

1.3 Isotropic subspaces. Let V be a 2n-dimensional space endowed with a nondegenerate skew symmetric form f; g. A subspace W is said to be regular if the restriction of the form to W is nondegenerate. A subspace S V is said to be isotropic if fv; wg D 0 for all v, w 2 S . Theorem 1.3. a) The maximal possible dimension of an isotropic subspace is n. b) Each isotropic subspace is contained in an n-dimensional isotropic subspace. Isotropic subspaces of maximal possible dimension are called Lagrangian subspaces. The set Lagr n D Lagr n .K/ of all Lagrangian subspaces is called a Lagrangian Grassmannian. Example. The subspace spanned by e1 ; : : : ; en is Lagrangian. Proof. a) If S is isotropic, then S ? S. Since dim S ? C dim S D 2n, it follows that dim S 6 n. b) If dim S < n, then S is a proper subspace in S ? . For each h 2 S , the subspace S ˚ Kh is isotropic. 1.4 Coordinates on the Lagrangian Grassmannian. Fix a canonical basis (1.2) in a symplectic space W . Denote by W the linear span of vectors e1 ; : : : ; en , by WC the linear span of vectors f1 ; : : : ; fn . Let S W be an n-dimensional subspace. Assume S \ WC D 0. In this case, S is the graph of an operator A D AS W W ! WC . Theorem 1.4. The following two conditions are equivalent: 1. The subspace S is Lagrangian; 2. A D At . Proof. For any v ˚ vC , w ˚ wC 2 W ˚ WC D W , t t fv ˚ vC ; w ˚ wC g D v wC vC w :

In particular, for v ˚ v A, w ˚ w A 2 S, t t t fv ˚ v A; w ˚ w Ag D v At w v Aw D v .At A/w

and now the statement becomes obvious.

Problem 1.2. Let I be a subset of f1; 2; : : : ; ng. Denote by I B its complement. Denote by W .I / the (Lagrangian) subspace spanned by the vectors ek , k 2 I , and fl , l 2 I B . Show that each P 2 Lagr n is a graph of an operator W .I / ! W .I B / for some I . Thus we obtain an atlas on the manifold Lagr n . Show that there is no unnecessary chart in this atlas.

162


1.5 The symplectic groups. A symplectic group Sp.2n; K/, where K D R, C, is the group of all operators preserving a nondegenerate skew-symmetric bilinear form. In a canonical basis (1.2), an element g D ac db 2 Sp.2n; R/ satisfies the equation t 0 1 a b 0 1 a b D ; (1.3) 1 0 c d 1 0 c d or ab t D bat ;

cd t D dc t ;

ad t bc t D 1:

(1.4)

Problem 1.3. a) Show that det g D 1. b) If g 2 Sp.2n; K/, then g t 2 Sp.2n; K/ (see Lemma 2.2.2). c) Show that Sp.2; R/ ' SL.2; R/ and Sp.2n; C/ ' SL.2; C/. 1.6 The Witt theorem Theorem 1.5. Let W1 and W2 be subspaces in a symplectic space V . Let A W W1 ! W2 be a bijection preserving the symplectic form. Then there exists g 2 Sp.2n; K/ such that g D A on W1 . We leave it as exercise, see Theorem 2.2.6. Problem 1.4. a) Show that the stabilizer of any 2m-dimensional regular subspace is isomorphic to Sp.2m; K/ Sp.2.n m/; K/. b) Describe the stabilizer of the Lagrangian subspace spanned by e1 ; : : : ; en . c) The stabilizer of any pair of transversal Lagrangian subspaces is isomorphic to GL.n; K/. Problem 1.5. a) Find orbits of Sp.2n; K/ on the set of pairs of Lagrangian subspaces. Show that a unique invariant is the dimension of the intersection. b) The Grassmannians Lagr n .R/ and Lagr n .C/ are connected. Problem 1.6. Describe orbits of Sp.2n; R/ on triples of Lagrangian subspaces (a nonobvious invariant appears in Section 3.11). 1.7 Lie algebras. The Lie algebra sp.2n; K/ of the group Sp.2n; K/ consists of real matrices ˛ ˇ XD ; where ı D ˛ t ; ˇ D ˇ t ; D t : ı In other words, exp.tX / 2 Sp.2n; K/ for all t 2 R if and only if X has such a form. We leave this as an exercise for the reader; see Subsection 2.2.4. 1.8 Exercises Problem 1.7. Convince yourself that all the considerations of this section (except Subsection 1.7) survive for an arbitrary field of characteristic1 ¤ 2. 1

We meet in Subsection 8.3.1 the case of characteristic 2 in a discussion of real analysis.

3.2. The group Sp.2n; R/ in a complex model

163

3.2 The group Sp.2n; R/ in a complex model This section is purely scholastic. It is natural to develop the usual linear algebra in complex linear spaces; various facts about real linear spaces can be established by a reduction to the complex case. We apply this approach to Sp.2n; R/. Partially, we replicate the considerations of § 1.5. 2.1 Complexification of linear space. Let V be a real linear space. Its complexification VC is a complex linear space defined as VC D V ˝R C: Let us repeat the same definition less Pformally. Given a basis ek in V , we can treat V as the space of linearP combinations uk ek , where uk 2 R. The space VC consists of linear combinations uk ek , where uk 2 C. Next, let us state the same definition in coordinate-free terms. Let Vz be the second copy of the same space. We define VC as the real linear space V ˚ Vz and endow it with the multiplication by i in the following way: i Œv ˚ w WD .w/ ˚ v: Thus, VC D V ˚ iV . We define the operator J of complex conjugation in VC as X X J uk ek D uN k ek or (equivalently) J v ˚ w WD v ˚ .w/: Obviously, J 2 D 1 and J is an antilinear operator, i.e., J.x C y/ D J x C Jy;

J.˛x/ D ˛J N x:

The initial space V VC consists of fixed points of the operator J : x 2 V ” J x D x;

x 2 iV ” J x D x:

Proposition 2.1. Let W be a complex linear space endowed with an antilinear operator J satisfying J 2 D 1. Then W D VC , where V is the set of fixed points of J . Proof. Consider W as a real linear space W R of double dimension. Then J is a linear operator in W R . Since J 2 D 1, it follows that the eigenvalues of J are ˙1. Also, J has no Jordan blocks. Let V WD ker.J 1/ and Y D ker.J C 1/. For any v 2 V , we have J.iv/ D iJ v D iv, therefore iv 2 Y . In the same way, iY V . Thus our space is V ˚ iV .

164


2.2 Operators in the complexification. Any R-linear operator Q W V ! V admits a canonical extension to a C-linear operator QC W VC ! VC , QC .v C iw/ WD Qv C iQwI usually, we write Q instead of QC . Problem 2.1. A C-linear operator g in VC has the form QC if and only if g preserves the R-linear subspace V . Next, consider an even-dimensional real linear space V with a basis e1 ; : : : ; en , f1 ; : : : ; fn . Consider the basis 1 pk D p .ek C ifk /; 2

1 qk D p .ek ifk / 2

(2.1)

in VC . Note that Jpk D qk . Lemma 2.2. Consider a C-linear operator g W VC ! VC . It is a complexification of an operator V ! V if and only if the matrix of g in the basis (2.1) has the form ˆ ‰ (2.2) g D x x W Vn ˚ VnC ! Vn ˚ VnC : ‰ ˆ Proof. Apply the matrix g to an element of the R-space V , h A B Ah C B hN D : C D hN C h C D hN N D C h C D h, N i.e., The vector on the right-hand side is in V if and only if .Ah C B h/ x x A D D and C D B. 2.3 Complexifications of bilinear forms. Let V be a real linear space endowed with a nondegenerate skew-symmetric bilinear from f; g. There are two natural ways to extend our form to a form on VC . First, consider the complex bilinear form ƒ on VC defined by ˚

˚

˚

˚

fv C iw; v 0 C iw 0 g WD v; v 0 C i v; w 0 C i w; v 0 w; w 0 : For some aesthetical reasons, we prefer another expression: ƒ.v C iw; v 0 C iw 0 / WD

1 fv C iw; v 0 C iw 0 g: i

Secondly, we can extend f; g as a sesquilinear form, ˚

˚

˚

Œv C iw; v 0 C iw 0 WD v; v 0 C i fw; vg i v; w 0 C w; w 0 :

3.2. The group Sp.2n; R/ in a complex model

165

This form is anti-Hermitian, i.e., Œv; w D Œw; v. It is more convenient to introduce the Hermitian form M.v C iw; v 0 C iw 0 / WD 1i Œv C iw; v 0 C iw 0 : We recall that VC is also equipped with the operator of complex conjugation J . Obviously, M.v; w/ D ƒ.v; J w/: This identity allows us to reconstruct the form ƒ (resp. M ) if we know M (resp. ƒ) and J . Also, this identity determines J if we know both the forms ƒ and M . 2.4 The standard basis in V2n . Consider a canonical basis ek , fk of a real linear symplectic space W , see (1.2). We define a new basis 1 pk D p .ek C ifk /; 2

1 qk D p .ek ifk / 2

(2.3)

in WC . We have ƒ.pk ; pl / D ƒ.qk ; ql / D 0I 0 1 i.e., the matrix of the form ƒ is 1 0 . Also,

ƒ.pk ; ql / D ıkl ;

(2.4)

M.pk ; ql / D 0;

M.pk ; pl / D ıkl ; M.qk ; ql / D ıkl ; (2.5) 1 0 i.e., the Hermitian form M in this basis is 0 1 . The operator J of complex conjugation is X X Jpk D qk ; J qk D pk or, equivalently J .uk pk Cvk qk / D .uN k qk CvN k pk /: (2.6) In particular, the initial real space W consists of vectors X v ˚ vN WD .vk pk C vN k qk /: We denote the space WC equipped with these structures by V D V2n , denote by2 the subspace spanned by the vectors p1 ; : : : ; pk and by Vn the subspace generated by q1 ; : : : ; qn . And we write V2n .R/ WD W .

VnC

2.5 The group Sp.2 n; R/. The complex model. Consider a matrix of g 2 Sp.2n; R/ ˆ ‰ in the standard basis (2.3). By Lemma 2.2, g has the block structure ‰ x ˆ x . Since g preserves the forms ƒ and M , it follows that ˆ ‰ ˆ ‰ 1 0 1 0 D ; (2.7) x ˆ x x ˆ x 0 1 0 1 ‰ ‰ t t ˆ ‰ ˆ ‰ 0 1 0 1 (2.8) x ˆ x x ˆ x D 1 0 : 1 0 ‰ ‰ 2

Here our notation differs from Subsection 1.5.

166


Corollary 2.3. The group Sp.2n; R/ contains a subgroup isomorphic to the unitary 0 group U.n/. It consists of matrices of the form ˆ , where ˆ is a unitary matrix in x 0 ˆ the usual sense. Remark. Let us describe the embedding U.n/ ! Sp.2n; R/ in another way. Consider the standard Euclidean space Cn . It is a symplectic space because Imh; i is a skewsymmetric bilinear from. The group U.n/ preserves this form, therefore U.n/ Sp.2n; R/. N g t 2 Sp.2n; R/. Problem 2.2. If g 2 Sp.2n; R/, then g , g, 2.6 Overgroups. Consider the following groups acting on the linear space V2n : – the group Sp.2n; C/ preserving the form ƒ; – the group U.n; n/ preserving the form M ; – the group GL.2n; R/ commuting with J . We have Sp.2n; R/ D U.n; n/ \ Sp.2n; C/ D Sp.2n; C/ \ GL.2n; R/ D U.n; n/ \ GL.2n; R/: 2.7 Digression. Overgroups. Zoo Problem 2.3. a) Let V be a real linear space endowed with a nondegenerate symmetric bilinear form with inertia indices .p; q/. Extending the bilinear form to the complexification, we obtain bilinear and Hermitian forms on VC and representation of O.p; q/ as intersections O.p; q/ D O.p C q; C/ \ U.p; q; C/ D O.p C q; C/ \ GL.p C q; R/ D U.p; q/ \ GL.p C q; R/: b) Consider the group U.p; q/ as a group of real linear operators in R2pC2q . Then it preserves the bilinear forms Reh; i and Imh; i. Therefore, U.p; q/ D O.2p; 2q/ \ Sp.2.p C q/; R/ D O.2p; 2q/ \ GL.p C q; C/ D Sp.2.p C q/; R/ \ GL.p C q; C/: c) Consider thestandard real symplectic spaceR2n . Endow it with the symmetric 0 , I 2 D 1. The group of all bilinear form 10 01 and with the operator I WD 10 1

3.3. Matrix balls

167

operators commuting with I0 isGL.n; R/GL.n; R/. Symplectic operators commuting with I have the form A0 At1 with A 2 GL.n; R/. Thus,

GL.n; R/ D Sp.2n; R/ \ GL.n; R/ GL.n; R/

D O.n; n/ \ GL.n; R/ GL.n; R/ D O.n; n/ \ Sp.2n; R/:

(2.9)

d) We encountered a similar phenomenon in Problem 2.10.3 (matrix products in the Cayley coordinates). MatricesS in formula (2.10.7) are unitary in the usualsense and preserve the Hermitian form 01 10 . They also commute with the operator 01 10 . Thus, U.n/ U.n/ D U.2n/ \ U.n; n/

D U.2n/ \ GL.n; C/ GL.n; C/

D U.n; n/ \ GL.n; C/ GL.n; C/ :

3.3 Matrix balls Here we replicate considerations of § 2.3. 3.1 Orbits of Sp.2 n; R/ on the complex Lagrangian Grassmannian. Recall that Lagr n .C/ denotes the Grassmannian of all ƒ-Lagrangian subspaces in V2n . We are going to describe orbits of the real symplectic group Sp.2n; R/ on this space. Theorem 3.1. a) The only invariants of Y 2 Lagr n .C/ under the action of Sp.2n; R/ are the rank and the inertia indices of the Hermitian form M restricted to Y . b) The set of M -regular ƒ-Lagrangian subspaces is a union of .n C 1/ open Sp.2n; R/-orbits of the form Sp.2n; R/=U.˛; n ˛/;

where ˛ D 0; 1; : : : ; n:

Proof. The statement a) is proved in Subsection 3.3. b) Let Y be an M -regular ƒ-Lagrangian subspace. We are going to show that Y is Sp.2n; R/-equivalent to a subspace spanned by the elements p1 ; : : : ; p˛ , q˛C1 ; : : : ; qn of the standard basis (2.3). Choose an M -orthogonal basis P1 ; : : : ; P˛ ; Q˛C1 ; : : : ; Qn 2 Y such that M.Pk ; Pk / D C1;

M.Ql ; Ql / D 1:

168


Consider the subspace J Y . Since Y is Lagrangian, it follows that J Y is also ƒLagrangian, and moreover J Y D Y ?M (see Lemma 1.5.3). Let Qk WD JPk ;

Pl WD JQl ;

where k 6 ˛, l > ˛. By (1.5.7), (1.5.8), it follows that M.Pl ; Pl / D C1 and M.Qk ; Qk / D 1. Therefore, P1 ; : : : ; Pn , Q1 ; : : : ; Qn form a basis satisfying the conditions (2.4)–(2.6). Consider the operator A such that APm D pm , AQm D qm . It puts the subspace Y in a canonical position. It remains to describe the stabilizer of Y . An element g of the stabilizer induces a pseudounitary transformation of Y . Since g.J v/ D J.gv/ and J v 2 Y ?M , it follows that each pseudounitary transformation of Y admits a unique extension to a (real) symplectic operator on the whole space. Therefore, the stabilizer is UŒY ' U.˛; n ˛/. 3.2 Real Lagrangian Grassmannians in the complex model. Obviously, the complexification of a real Lagrangian subspace is both ƒ-Lagrangian and M -isotropic. The converse is also true. Proposition 3.2. Each ƒ-Lagrangian M -isotropic subspace in V2n has the form YC , where Y 2 Lagr n .R/. Problem 3.1. Let W be a real linear space. A complex subspace Z WC is a complexification of a real subspace in W if and only if Z is J -invariant. Proof. We must show that Y is J -invariant. For any y, v 2 Y , we have 0 D ƒ.y; v/ D M.Jy; v/. Therefore, Jy 2 Y ?M D Y . Note that the proposition just proved is a special case of Theorem 3.1. 3.3 End of the proof of Theorem 3.1. Now, let Y be a ƒ-Lagrangian M -singular subspace. Lemma 3.3. The subspace zeroM .Y / is J -invariant. Proof of the lemma. The subspace zeroM .Y / is M -orthogonal to Y , so J zeroM .Y / is ƒ-orthogonal to Y (see Lemma 1.5.2). Since Y is ƒ-Lagrangian, J zeroM .Y / Y . On the other side, zeroM .Y / is ƒ-orthogonal to Y . Therefore, J zeroM .Y / is M -orthogonal to Y . Thus J zeroM .Y / D zeroM .Y /. Proof of Theorem 3.1. Denote by Z Y an arbitrary complement to zeroM .Y /. We decompose V2n as V2n D .Z ˚ J Z/ ˚ .Z ˚ J Z/? : The first summand contains Z, the second summand contains zeroM .Y /. Both the summands are spaces equipped with non-degenerate forms ƒ and M and an operator J . In the first summand, we observe a picture described in the statement b) of the theorem; the second summand corresponds to the picture described in Proposition 3.2.

3.3. Matrix balls

169

3.4 Matrix coordinates-I. As it was shown above, a pair of transversal subspaces determines a coordinate system on a Grassmannian. In the present context, we have additional structures in the linear space (i.e., the forms ƒ, M and the operator J ), therefore we can specify this pair of subspaces in several ways. The two most reasonable ways produce the Cartan ball and the Siegel wedge. The former variant is discussed immediately, for the latter variant, see Subsection 3.8. In this subsection, we consider the pair Vn and VnC of ƒ-Lagrangian subspaces. In coordinate-free terms, we can say that one subspace is M -negative and ƒ-Lagrangian; another one is its M -orthogonal complement (therefore it is M -positive and ƒ-Lagrangian). By Theorem 1.4, a generic Y 2 Lagr n .C/ is a graph of an operator z W Vn ! VnC satisfying z D zt : As in Theorem 2.3.2, the group Sp.2n; R/ acts by linear-fractional transformations x x 1 .‰ C z ˆ/ z 7! .ˆ C z ‰/

(3.1)

on the space of all matrices. Since Sp.2n; R/ preserves the Lagrangian Grassmannian, these transformations preserve the condition z D z t ; we can regard (3.1) as (discontinuous) transformations of the space Symmn .C/. Next, we reformulate Theorem 3.1 (about orbit structure) in coordinate terms. Theorem 3.4. a) The hypersurface det.1 zz / D 0 separates the space Symmn into .n C 1/ open pieces according to the number ˛ of eigenvalues of zz that are less than 1. b) The non-open orbits are classified according to rk.1 zz / and to the number of positive and negative eigenvalues of 1 zz . c) The set zz D 1 is a compact Sp.2n; R/-orbit; it is contained in the closure of any orbit. The reader can prove this or take it on trust (a proof is a rearrangement of arguments of Section 2.3). The object that is actually important for us is the open orbit with ˛ D n. This set is the subject of the next subsection. 3.5 Matrix balls. Denote by Bn (matrix ball) the set of n n-matrices satisfying the conditions z D z t , kzk < 1. Theorem 3.5. a) Each M -negative ƒ-Lagrangian subspace in V2n is a graph of an operator z W Vn ! VnC , where z 2 Bn . b) Each z 2 Bn originates from a ƒ-Lagrangian M -negative subspace. c) The group Sp.2n; R/ acts on Bn by (continuous) linear fractional transformations x 1 .‰ C z ˆ/: x z 7! .ˆ C z ‰/

(3.2)

170


d) The action of Sp.2n; R/ on Bn is transitive. The stabilizer of the point 0 2 Bn ˆ 0 consists of matrices 0 ˆ x , where ˆ 2 U.n/. In other words, Bn ' Sp.2n; R/=U.n/: All these statements are already proved. More precisely, the statements a), b) follow from Theorem 2.3.1 and Theorem 1.4. Formula (3.2) is a special case of Theorem 2.3.2 and d) is a special case of Theorem 3.1. The following problem provides another proof of the transitivity. Problem 3.2. Fix z 2 Bn . Show that .1 z zN /1=2 gz WD z.1 N z z/ N 1=2

z.1 zN z/1=2 .1 zz/ N 1=2

2 Sp.2n; R/

(3.3)

and g takes 0 to z (see also Proposition 2.3.3). x n the set of all symmetric matrices with norm 6 1. 3.6 The boundary. Denote by B x n is in a one-to-one correspondence with the set of all ƒ-Lagrangian Obviously, B M -semi-positive subspaces. x n are enumerated by the number j D Problem 3.3. Orbits Oj of Sp.2n; R/ on B rk.1 zz /. This j corresponds to the rank of the form M on a ƒ-Lagrangian subspace. We formulate the statement for j D 0 individually as a theorem. Theorem 3.6. a) The set O0 of matrices z such that z D zt ;

zz D 1

is in a one-to-one correspondence with the set of subspaces in V2n that are ƒ-isotropic and M -isotropic. b) O0 is Sp.2n; R/-homogeneous, moreover O0 ' Lagr n .R/. c) O0 is also homogeneous under the subgroup U.n/ Sp.2n; R/ acting by N moreover, h W z 7! ht zh (in (3.2), ‰ D 0 and ˆ D h); O0 D U.n/=O.n/: Proof. First, we note that this statement is a version of Theorem 2.4.2 and its Corollary 2.4.3. The statement a) follows from Theorem 2.4.2 and Theorem 1.4. The statement b) is a repetition of Proposition 3.2. c) The group U.n/ acts on O0 and the stabilizer of z D 1 is O.n/. Thus we get an embedding U.n/=O.n/ ! O0 . Next, dim U.n/=O.n/ D dim U.n/ dim O.n/ D n2 n.n 1/=2 D n.n C 1/=2 D dim Lagr n .R/ D dim O0 : But Lagr n .R/ is connected and this completes the proof.

3.3. Matrix balls

171

Another proof of the statement c). Lemma 3.7. Each complex symmetric matrix z can be reduced by transformations z 7! z 7! hzht , where h 2 U.n/, to a diagonal form with nonnegative entries. The statement c) of Theorem 3.6 follows from the lemma, because for unitary z, a matrix hzht is also unitary; the positive matrix elements of a unitary diagonal matrix can be only C1. Proof of the lemma. We must classify quadratic forms Q.v/ on V WD Cn under unitary transformations. Consider the real Euclidean space R2n ' Cn and reduce the form Re Q.v/ to principal axes. Denote by j the eigenvalues and by Vj the eigenspaces. Evidently, Re Q.iv/ D Re Q.v/. Therefore, the operator i sends principal axes to principal axes; furthermore, iVj D Vj . We get the decomposition V D

M j >0

Vj ˚ Vj ;

whose summands Œ: : : are invariant under the multiplication by i, i.e., the summands are complex subspaces. This is our statement. 3.7 The Cartan decomposition. Consider the complex realization of Sp.2n; R/. Theorem 3.8. a) For each g 2 Sp.2n; R/, there exists a unique decomposition g D hr, where h 2 U.n/ Sp.2n; R/ and r 2 Sp.2n; R/ satisfies r D r > 0. b) There exists a unique decomposition g D h exp.r/; where h 2 U.n/ and r is element of the Lie algebra sp.2n; R/ satisfying r D r , an 0 ˇ i.e., r has the form r D ˇN 0 , ˇ D ˇ t . This is a copy of Theorem 2.3.4. Both the proofs of Theorem 2.3.4 survive literally. In the last theorem, we explore the complex model of the group Sp.2n; R/. The map g 7! g is an anti-involution of this group. In the real model, this anti-involution is g 7! g t . For the real model, the theorem must be reformulated according to this remark. in the real model, the subgroup U.n/ Sp.2n; R/ consists of matrices APrecisely, B such that A C iB is a unitary matrix in the usual sense. The matrix r now has B A the form C D rD ; where C D C t , D D D t . D C 3.8 Matrix coordinates-II. Siegel wedges. As we noted above, there are many ways to introduce coordinates on Sp.2n; R/=U.n/. For instance, consider the symplectic space

172


P P R2n endowed with a canonical basis ek , fk . Set W C WD Ce: , W WD Cfk . Then each M -negative ƒ-Lagrangian subspace Y is a graph of an operator z W W C ! W . Since z is Lagrangian, it follows that z D z t . Moreover, z is negative; therefore Im z > 0. The group of real matrices Sp.2n; R/ acts on this wedge via linear fractional transformations. 3.9 Digression. A transposition of characters. In our previous considerations, the roles of the forms ƒ and M are different. Let us transpose their roles. Problem 3.4. Consider the action of Sp.2n; R/ on the Grassmannian of maximal M -isotropic subspaces in V2n . a) Show that orbits are classified according to the rank of ƒ on a subspace. b) Show that for n D 2m the unique open orbit is Sp.4m; R/=Sp.2m; C/. 3.10 Digression. Maximal compact subgroups. Our matrix balls G=K D U.p; q/=U.p/ U.q/, Sp.2n; R/=U.n/ are distinguished among other G-homogeneous spaces by the following formal property: K is a maximal compact subgroup in G. More precisely: Theorem 3.9. Let G D U.p; q/ or Sp.2n; R/. For each compact subgroup L G there exists g 2 G such that gLg 1 K Problem 3.5. Prove the last statement for Sp.2n; R/ in the following two ways: a) Consider the standard Euclidean Cn equipped with the inner product h ; i. Realize Sp.2n; R/ as the group of R-linear transformations of the skew-symmetric form Imh ; i. Consider the average of the inner product over the Haar measure on L, i.e., Z Œv; w WD hvg; wgi dg: L

Then ImŒv; w D Imhv; wi and also L preserves Œ; . b) A matrix ball is a manifold of non-positive curvature. Apply the following Cartan theorem: a compact group acting on a simply connected complete Riemannian manifold with a non-positive curvature has a fixed point. The Cartan theorem itself is a nice geometric exercise3 . 3.11 Symmetric subgroups and overgroups. An automorphism of a group G is said to be an involution if 2 D 1. A subgroup H G is said to be symmetric if it is the set of fixed points of an involution.4 . Problem 3.6. a) There are the following symmetric subgroups in the groups U.p; q/: U.p; q/ O.p; q/;

U.2p; 2q/ Sp.p; q/;

U.n; n/ GL.n; C/;

U.2n; 2n/ Sp.2n; R/;

Find them (or some of them). 3 4

See the proof of Theorem 10.8.12. See the list of symmetric subgroups in Subsection D.6.

U.p C k; q C m/ U.p; q/ U.k; m/; U.2n; 2n/ SO .2n/:

3.4. Conjugacy classes in Sp.2n; R/

173

b) The groups U.p; q/ are symmetric subgroups in the following overgroups: U.p; q/ O.2p; 2q/;

U.p; q/ Sp.p; q/;

U.p; q/ U.p; q/ U.p; q/;

U.p; q/ Sp.2.p C q/; R/; U.p; q/ GL.p C q; C/;

U.p; q/ SO .2.p C q//:

Find them. The spaces of the form G=H , where G arises from the problem are examples of so-called pseudo-Riemannian symmetric spaces., see Subsection 2.12.2 and Addendum D.

3.4 Conjugacy classes in Sp.2n; R/ Basically, this section replicates Section 2.6. But again, there is a simple and nice answer for classes in general position and a tedious analysis of Jordan blocks. 4.1 Symmetries of spectra Theorem 4.1. Assume that the spectrum of g 2 Sp.2n; R/ is simple. Then it is invariant under the transformations ! N and ! 1 . Corollary 4.2. For an arbitrary g 2 Sp.2n; R/, the spectrum is invariant (taking into account the multiplicities of the eigenvalues) under the same transformations. Proof of the corollary. The spectrum Spec.g/ is a continuous set-valued function of g. Problem 4.1. a) Recover details of the proof of the corollary. b) Give another proof with arguments from Lemma 2.6.3. Proof of Theorem 4.1. Consider the complexification V of a real symplectic space and the forms ƒ and M as above. Denote by v an eigenvector corresponding to an eigenvalue . Obviously, the N This complex conjugate vector vN WD J v is an eigenvector with the eigenvalue . N implies the symmetry $ . Secondly, ƒ.v ; v / D ƒ.gv ; gv / D ƒ.v ; v /: Therefore, ƒ.v ; v / D 0

or

D 1:

(4.1)

Thus v is orthogonal to all the vectors v , except v1= . Since our form is nondegenerate, it follows that a vector v1= exists. This implies the second symmetry $ 1 and also completes the proof of the theorem.

174


But we present one more argument, M.v ; v / D M.gv ; gv / D N M.v ; v /:

(4.2)

Hence, M.v ; v / D 0

or

N D 1:

The form M is nondegenerate, hence the existence of v implies the existence of v1=N . N This gives the symmetry $ .

a)

b)

c)

d)

e)

N N 1 Figure 3.1. Five types of spectra of indecomposable symplectic operators: a) , 1 , , for a generic 2 C; b) , 1 for 2 R; c) , N for jj D 1; d) D 1 (twice), e) D 1 (twice).

We present two minor additions to Theorem 4.1. and its Corollary 4.2. Observation 4.3. For jj D 1, there are two types of invariant subspaces Cv ˚Cv1= , according to sgn M.v ; v /. Indeed, in this case, M.v ; v / D 0 for all ¤ . Hence M.v ; v / ¤ 0 (otherwise, the form M is degenerate). We can set M.v ; v / D C1 or 1, but we can not change the sign. Observation 4.4. For any g 2 Sp.2n; R/ the multiplicities of the eigenvalues ˙1 are even. Indeed, an eigenvalue ¤ ˙1 is present together with 1 . This produces an even-dimensional subspace and the determinant of g on this subspace is 1. Since the ambient space is even-dimensional, it follows that the eigenvalues D 1 and D 1 together produce an even-dimensional subspace. But det g D 1, therefore both the eigenspaces have even dimension. Problem 4.2. Find symmetries of spectra for the groups O.n; C/ and O.p; q/. 4.2 Indecomposable elements. We propose three equivalent versions of a definition. 1) An operator g 2 Sp.2n; R/ is said to be decomposable if there is an orthogonal (with respect to the symplectic form) decomposition R2n D Y ˚ Z into a direct sum of g-invariant subspaces.


175

2) A shorter form of the definition: g is decomposable if there exists a proper ginvariant regular subspace H in R2n . Then H ? is also invariant and R2n D H ˚ H ? . 3) The most intricate and the most useful version of the definition: g is decomposable if there is a decomposition V2n D Y ˚ Z into a sum of two g-invariant subspaces Y and Z that are invariant under J and satisfy Z D Y ?ƒ D Y ?M : We are going to describe all indecomposable operators. 4.3 The root decomposition. Consider the complexified symplectic space V D V2n and an operator g 2 Sp.2n; R/ in V . Consider the root decomposition M [ VD V ; V WD ker.g /N : N

Lemma 4.5. a) V ?ƒ V for ¤ 1=. The form ƒ defines a duality between V and V1= . b) V ?M V for ¤ 1=. N The form M defines a duality between V and V1=N . In particular, for jj D 1 the space V D V1=N is M -regular. c) The operator J is a bijection V to VN . We leave this statement as an exercise for the reader, see Lemma 2.6.4. Given 2 C, consider the sum VŒ WD V C VN C V1 C VN 1 :

(4.3)

If 2 R or jj D 1, then some of the summands coincide, the actual number of distinct summands can be 1, 2, or 4 (according to ). By Lemma 4.5 all summands have the same dimension. Theorem 4.6. The forms ƒ and M are nondegenerate on each VŒ . The subspaces VŒ are pairwise orthogonal with respect to both the forms and J -invariant. This statement immediately follows from Lemma 4.5. An indecomposable element g has only one summand VŒ . In the following five subsections we examine all possible positions of . Only in the case D ˙1 do we meet some difficulties. In all the remaining cases we easily show that an indecomposable operator has only one Jordan block in each root subspace V , V1= , VN , V1=N . 4.4 Case I. Real ¤ ˙1. By Lemma 4.5, both the subspaces V and V1= are ƒisotropic and M -isotropic. Since 2 R, the subspaces V and V1= are J -invariant. So let us consider a real symplectic space and let us forget about M (because on the real subspace V2k .R/ the forms ƒ and M coincide).

176


The form ƒ defines a bilinear duality between V and V1= . Choose a basis in V and the dual basis in V1= . Since our operator g preserves the pairing, it follows that its matrix has the form A 0 g WD : 0 At1 Suppose the operator A is decomposable, i.e., V is a direct sum of two g-invariant subspaces Y ˚ Z. Consider the dual decomposition V1= D Y 0 ˚ Z 0 and split VŒ into a direct sum of Y ˚ Y 0 and Z ˚ Z 0 . Thus the operator g is indecomposable if and only if A is a Jordan block. 4.5 Case II. Eigenvalues on the unit circle. Now jj D 1, ¤ ˙1. The subspaces V and V1= are M -regular and M -orthogonal; they are also ƒ-Lagrangian and ƒ-dual. Consider an arbitrary basis ek 2 V and its J -image fk WD Jek 2 .V /? D V1= . Then ƒ.ek ; fl / D ık;l ; M.fk ; fl / D M.ek ; el /: In this basis, g has the form

A 0 gD ; 0 AN

where A is an M -unitary operator in V. If A is a decomposable pseudo-unitary operator, then the whole aggregate g is decomposable. In § 2.6 we have classified all indecomposable pseudo-unitary operators. Thus, A is a Jordan block. We also recall that such an operator has an additional invariant taking values ˙1. 4.6 Case III. Generic 2 C. Now Re ¤ 0, jj ¤ 1; therefore the sum in (4.3) is a direct sum of four summands. These summands are in the dualities described in Lemma 4.5. For a basis uj 2 V consider the following bases: a) the ƒ-dual basis uBk 2 V1 ; b) the complex conjugate basis uN k WD J uk 2 VN ; c) the basis uN Bk WD J uBk 2 VN 1 . It can readily be checked that this basis is M -dual to uk . The matrix of g in the compound basis has the form 0 1 A 0 0 0 B 0 At1 0 0 C C: gDB @0 0 AN 0 A 0 0 0 A1 This aggregate is decomposable if and only if A is decomposable. Thus for an indecomposable g, the matrix A is a Jordan block.


177

4.7 Case IV: D ˙1. Statement of the result. It suffices to consider D 1; otherwise, we take .g/ 2 Sp.2n; R/. Since g has a unique eigenvalue 5 D 1; it follows that .g 1/r D 0 for sufficiently large r. Let us discuss such operators in real symplectic spaces. For this reason, we forget about M and J . Proposition 4.7. There are only two types of irreducible symplectic unipotent operators. a) The first type. Linked boxes. Let k be odd. Consider the 2k-dimensional symplectic spaces equipped with a canonical basis. Our g is of the form A 0 gD ; 0 At1 where A is a unipotent Jordan block. Remark. For an even k, we can define an operator G in the same way. But it is decomposable. b) The second type. A self-dual box. Fix D ˙1. Consider a 2m-dimensional real space with a basis u1 ; : : : ; u2m such that ƒ.u˛ ; uˇ / D .1/˛ ı˛Cˇ;2mC1

(4.4)

(i.e., the matrix of the form consists of interlacing C1 and 1 on the anti-diagonal). We define g as the standard Jordan block ge˛ D e˛ C e˛C1 ;

ge2k D e2k :

4.8 The case D 1. Proof. The proof repeats the arguments of Subsection 2.6.8 with a minor variation. For an indecomposable symplectic unipotent operator g consider its logarithm X1 S WD ln g D ln 1 C .g 1/ D

kD0

.1/k .g 1/k k

(4.5)

(the sum is finite). Obviously, S r D 0 for sufficiently large r. It can easily be checked that S is symplectically skew-symmetric ƒ.S u; v/ D ƒ.u; S v/;

(4.6)

i.e., S is an element of the Lie algebra sp.2n; R/. Denote by N the maximal degree such that S N ¤ 0. As in Subsection 2.6.8, we introduce the bilinear form B.v; w/ WD ƒ.S N v; w/: 5

Operators whose unique eigenvalue is 1 are called unipotent.

(4.7)

178


Lemma 4.8. B determines a nondegenerate form on V = ker S N . Proof. By definition, ker S N ker B. On the other hand, if S N v ¤ 0, then there exists a vector w such that ƒ.S N v; w/ ¤ 0. Thus ker B coincides with ker S N . Since S is symplectically skew-symmetric, see (4.6), it follows that the bilinear form B is symmetric for even N and skew-symmetric for odd N . This gives two cases. Let N be even, B be symmetric. We choose a vector h such that B.h; h/ D ˙1. The form ƒ is nondegenerate on the S -cyclic span of h (see the arguments from Subsection 2.6.8) and this subspace splits off as a direct summand. Thus S is a single Jordan block. This is the type b) described in the previous subsection. Let N be odd, the form B be skew-symmetric. Then we choose a pair of vectors h and such that B.h; / D 1. Their cyclic span is a ƒ-regular subspace. But again, we can split it off. This is the type a) of the previous subsection. We are not yet finished. We must find canonical forms for even and odd N . For this purpose, we apply the cleaning procedure from Subsection 2.6.9. For the case of symmetric forms our considerations can be repeated one-to-one, in another case we must slightly modify them. 4.9 An illustration. The group Sp.2 ; R/. Let us examine in more detail the group Sp.2; R/ D SL.2; R/. As we have seen in Problem 2.2.12, there exists an isomorphism SU.1; 1/ ' SL.2; R/. Therefore, our discussion is also an illustration for conjugacy classes in U.p; q/. . xv

The Lie algebra sp.2; R/ D sl.2; R/ is the Lie algebra of real matrices of the form A D Represent such matrices as x yCz : AD yz x

u x /.

Then det A D x 2 y 2 C z 2 and the eigenvalues are

(4.8)

q p ˙ det.A/ D ˙ x 2 C y 2 z 2 :

The Lie group SL.2; R/ acts on the Lie algebra by conjugations A 7! gAg 1 . The level surfaces of the function det A are invariant under the group SL.2; R/, see Figure 3.2.a. Therefore we get the following types of SL.2; R/-orbits in sl.2; R/. 1) One-sheeted hyperboloids x 2 y 2 C z 2 D s 2 correspond to matrices A with real eigenvalues ˙s ¤ 0. 2) A two-sheeted hyperboloid x 2 y 2 C z 2 D t 2 is a union of two orbits. Indeed, the group SL.2; R/) is connected, therefore orbits must be connected. They correspond to matrices with the eigenvalues ˙i t , t 2 R. 3) The cone x 2 y 2 C z 2 D 0 is a union of three orbits: the point 0, the upper sheet, and the lower sheet. The point 0 corresponds to the zero matrix and the two other orbits correspond to Jordan blocks with eigenvalues D 0.


179

'

x

a)

_

b)

Figure 3.2. a) Conjugacy classes in the Lie algebra sl.2; R/. b) Conjugacy classes in the group SL.2; R/.

Thus signs that appear in the classification of conjugacy classes are observable in our case. The Lie group SL.2; R/ itself. According to the classification, there are three types of conjugacy classes. a) “Hyperbolic classes” consist of matrices with eigenvalues f, 1 g with 2 R, ¤ ˙1. b)“Elliptic classes” consist of matrices with eigenvalues e i' , e i' , where 0 < ' < . This set consists of two SL.2; R/-orbits that are separated according to the following alternative: For all v 2 R2 , the pair fv, Avg is either a left basis or a right basis: c) “Parabolic classes”. Consider the variety of all matrices whose eigenvalues are 1, 1. Then we have a one-point orbit corresponding to the unit matrix and two orbits corresponding to Jordan blocks. But again, we have a sign; it can be interpreted as above. c0 ) For the eigenvalues 1, 1, there is the same picture. The group SL.2; R/ in Figure 3.2.b. For any g 2 SL.2; R/ consider its characteristic polynomial 2 tr g C det g D 2 tr g C 1: Evidently, g is hyperbolic, elliptic, or parabolic if j tr gj is > 2, < 2, or D 2 respectively. Now let us draw a figure. By Theorem 3.8 (or Theorem 1.1.3), each g 2 SL.2; R/ has a unique representation6 gD

cos ' sin '

sin ' x exp cos ' y

y : x

6 We represent g as a product of a rotation and contraction–dilatation with respect to a pair of orthogonal axes.

180


Therefore, the topological space SL.2; R/ is the product of a circle S 1 and R2 . So we can visualize this space as a subset (infinite slab) in R3 of the form 6 ' 6 and .x; y/ 2 R2 I here the planes ' D and ' D are identified. Problem 4.3. Show that tr g D 2 cos ' cosh

q x2 C y2:

Apparently a direct calculation is not extremely aesthetic, however it is easy to observe that tr.g/ p depends only on ' and x 2 C y 2 . Conjugacy classes are shown in Figure 3.2. This figure is a vertical section of the slab x 2 R, y 2 R, ' 2 Œ; . The whole picture is obtained by the revolution of the strip around the vertical axis O'. Thick dots correspond to the matrices ˙1. The curves (actually surfaces of revolution) passing to thick dots are parabolic classes. A picture in a small neighborhood of 0 is similar to Figure 3.2a. The domain tr g > 2 is light-gray, the domain tr g < 2 is dark-gray. Two white domains correspond to elliptic classes. 4.10 Digression. Gaussian operators, quadratic operators, linear Hamiltonian systems. As we have seen in §§ 1.8–1.9 unitary Gaussian integral operators in L2 .Rn / are in a one-to-one correspondence with elements of the group Sp.2n; R/. Therefore, Observation 4.9. The classification of conjugacy classes in Sp.2n; R/ is also a classification of unitary Gaussian integral operators (up to a conjugation by a unitary Gaussian operator). We omit the corresponding listing of canonical forms. Observation 4.10. The classification of conjugacy classes in the Lie algebra sp.2n; R/ is also a classification of the second-order partial differential equations on Rn of the form 1 X X @f .x; t / 1X @2 @ Ci bkl xk C D akl ckl xk xl f .x; t /; @t 2 @xk @xl @xl 2 where akl , bkl , ckl are real; 2 C. Indeed, the differential operators on the right-hand side are operators of Lie algebra sp.2n; R/ in the Weil representation, see § 6.4. In particular, we have an “automatic way” to solve explicitly the Cauchy problem for such partial differential equations. Problem 4.4. Let .p; q/ range in the standard symplectic R2n . Consider a Hamiltonian of the form X 1X 1X H.p; q/ WD bkl qk pl C akl qk ql C ckl pk pl 2 2 and the corresponding Hamiltonian system dp @H.p; q/ D ; dt @q

dq @H.p; q/ D : dt @p

Classify such systems with respect to the natural action of Sp.2n; R/ on R2n .


181

4.11 Digression. The Jordan decomposition Problem 4.5. Show that g 2 Sp.2n; R/ admits a unique decomposition g D ru, where r 2 Sp.2n; R/ is diagonalizable, u is unipotent and ru D ur (this is more-or-less Theorem 4.6 on a root decomposition). 4.12 Digression. Similar problems. Here we discuss some problems of the same type, they can be solved by the same tools. Problem 4.6. a) Convince yourself that you can find conjugacy classes in O.p; q/ and in O.n; C/. b) Describe conjugacy classes in GL.n; R/. In fact, we must classify pairs g, J , where g is a linear operator in Cn , J is an antilinear involution, and gJ D Jg. c) A more exotic question. Describe conjugacy classes in GL.n; H/. We regard the quaternionic space Hn as C2n endowed with an antilinear operator J such that J 2 D 1. Now we must describe pairs g, J such that: – g is a C-linear operator, – J is an antilinear operator satisfying J 2 D 1, – gJ D gJ . A classification of conjugacy classes in Sp.p; q/ and SO .2n/ is a similar problem. Consider the symplectic space R2n . We say that an operator A in R2n is symplectically symmetric if fAv; wg D fv; Awg for all v and w. In the canonical (real) basis they have the form ˛ ˇ ; where ˇ D ˇ t , D t . ˛ t The symplectically skew-symmetric operators were defined above (4.6). We mention some classification problems, which seem slightly different. Namely, how to classify: A) symplectic anti-symmetric operators A (D elements of Lie algebra sp.2n; R/) up to a conjugation, X g 1 Xg, where g 2 Sp.2n; R/; B) symplectic symmetric operators; C) a pair of non-degenerate symplectic forms; D) a pair fa non-degenerate symplectic form, a symmetric bilinear formg; The problem A) reduces to the classification of conjugacy classes in Sp.2n; R/. (because X 2 sp.2n; R/ implies exp X 2 Sp.2n; R/). Also, A) is equivalent to D) because Q.v; w/ D fAv; wg is a symmetric bilinear form. For the same reason, the problems B) and C) are equivalent. 4.13 A pair of degenerate forms. In a strange way, a classification of a pair of degenerate forms is also solvable, moreover, it admits a nice final answer. For instance, for a pair of skew-symmetric bilinear forms ƒ1 and ƒ2 , the only additional indecomposable canonical form appears, namely the space is .2n C 1/-dimensional and ƒ1 .ek ; em / D .1/k ıkCm;2n ; ƒ2 .ek ; em / D .1/ ıkCm;2nC2 : k

(4.9) (4.10)

182


I have never seen a nice proof (and could not invent it). A start of the proof looks natural: a pair of forms determines the pair of operators from the space to the dual space. Apply Kronecker’s classification of pencils... Problem 4.7. Find kernels of forms a1 ƒ1 C a2 ƒ2 for .a1 ; a2 / ¤ .0; 0/. 4.14 Double cosets. A natural generalization of most of the classification problems discussed above (this section, §2.6, §2.5) is as follows: – Describe double cosets H nG=K, where G is a classical group and H and K are symmetric subgroups. Problem 4.8. Identify the following sets: ˚

˚

a) nondegenerate skew-symmetric bilinear forms on R2n () GL.2n; R/=Sp.2n; R/ : ˚

b) pairs of nondegenerate skew-symmetric bilinear forms on R2n ˚

() Sp.2n; R/ n GL.2n; R/=Sp.2n; R/ : ³ ² pairs ¹ nondegenerate skew symmetric form º and ¹ symmetric bilinear form º : c) on R2n modulo GL.2n; R/ [˚

() O.p; 2n p/ n GL.2n; R/=Sp.2n; R/ : p

n

˚

double cosets G n .G G/=G, o () conjugacy classes in G : G G G is the diagonal subgroup ˚

˚

˚

e) Orbits of H on G=K () Orbits of K on H n G () Double cosets H n G=K : d)

Interpret angles and the Hua double ratio on these terms.

3.5 Symplectic contractions We replicate the considerations of Section 2.7. This is slightly tedious, however these results are necessary in several places of the book. 5.1 Symplectic contractions. Consider the space V2n endowed with two forms ƒ and M as above. We say that an invertible operator g is a symplectic contraction if it preserves the symplectic form ƒ and contracts the indefinite Hermitian form M . We denote by Sp.2n/ the semigroup of all symplectic contractions. Sp.2n/ D Sp.2n; C/ \ U.n; n/: Example 1. Elements g 2 Sp.2n; R/ are symplectic contractions. Example 2. Consider the standard space V2 endowed with the standard basis. The operator is given by 0 ; where 0 < 6 1: (5.1) g. / WD 0 1

3.5. Symplectic contractions

Example 3. Consider V2 with standard basis. Consider the operator 1Cs s h.s/ WD ; where s > 0: s 1s

183

(5.2)

For different s ¤ 0, these operators are conjugate. It is more pleasant to write such an operator in the canonical basis e, f in R2 . Then 1 s h.s/ D : 0 1 5.2 Symplectic dissipative operators. We say that an element X of a Lie algebra sp.2n; C/ is symplectic dissipative if it is dissipative with respect to the Hermitian form M , i.e., Re M.Xv; v/ 6 0. We denote the wedge of all symplectic dissipative operators by DissSp.2n/. We also denote by PosSp.2n/ the cone of all X 2 sp.2n; R/ such that iX is dissipative. We have DissSp.2n/ D sp.2n; R/ ˚ i PosSp.2n/: Proposition 5.1. Each element of i PosSp.2n/ is Sp.2n; R/-conjugate to a direct sum of the following 2 2 blocks: 0 1 1 ; > 0; ; s > 0: (5.3) 0 1 1 We omit a proof, see Proposition 2.7.15. Remark. The cone PosSp.2/ is the upper cone in Figure 3.2.a. The matrices of the first type in (5.3) correspond to upper sheets of the two-sheeted hyperboloids; the matrices of the second type correspond to the upper sheet of the cone. 5.3 The Potapov–Olshanski decomposition Theorem 5.2. a) Each element g 2 Sp.2n/ admits a unique decomposition g D h exp.r/;

r 2 PosSp.2n/:

b) Each element g 2 Sp.2n; R/ admits a representation g D u1 T u2 ; where u1 , u2 2 Sp.2n; R/ and T is a direct sum of operators of the form (5.1)–(5.2) Proof. This is a replica of Theorem 2.7.7. We only note what has happened with nontrivial elements of the proof. Let g 2 Sp.2n; R/, p g ~ its M -adjoint. It can be ~ easily shown that g 2 Sp.2n; C/. We are interested in gg ~ . Since g 2 U.n; n/, we can refer to Theorem 2.7.7. Hence gg ~ has only positive real eigenvalues and the only admissible Jordan blocks are 2 2-boxes with the eigenvalue D 1. Therefore, we do not need a separate proof of Lemmas 2.7.9 and 2.7.10. Modulo these statements, the theorem is obvious.

184


3.6 The symplectic category and the isotropic category 6.1 The symplectic analog of the Krein–Shmul’yan category. This analog is the symplectic category Sp, which was defined in § 1.8. We recall that objects of Sp are the spaces V2n and morphisms are linear relations V2n ! V2m that are Lagrangian and are morphisms of the Krein–Shmul’yan category (see § 2.9) with respect to the form M . We can translate basic facts from Section 2.9 to our case literally. This category is discussed in more detail above in § 1.8. On the other hand, I hope that for a reader familiar with Chapters 2–3 this discussion is trivial. 6.2 The symplectic analog of isotropic category. We consider real symplectic spaces. Morphisms are Lagrangian linear relations R2n ! R2m . Such a linear relation determines a map Lagr n .R/ ! Lagr n .R/ or, equivalently, U.n/=O.n/ ! U.m/=O.m/. The Livshits map (2.10.3) is given by the same formula a b ‡ D a b.1 C d /1 I c d we apply this map to a symmetric unitary matrix and obtain a symmetric unitary matrix again. This allows us to produce the inverse limit of Lagrangian Grassmannians. The symplectic isotropic category does not appear again in this book.

3.7 Central extensions of groups Sp.2n; R/. Berezin’s formula In the rest of this chapter, we discuss central extensions. The central extension of the group Sp.2n; R/ exists and is unique (in a reasonable sense); this fact is rather trivial from the point of view of the general theory of Lie groups (see Addendum C). However, its tame construction is not obvious. The most simple and convenient variant is discussed in this section. This construction is sufficient for this book. Two more intricate constructions are exposed in §§ 3.9, 3.11. In §§ 3.8, 3.10 we get central extensions of the Krein–Shmul’yan category and of groups of symplectomorphisms. 7.1 Topology of Sp.2 n; R/ Proposition 7.1. The topological space Sp.2n; R/ is homeomorphic to the direct product U.n/ Rn.nC1/ . Proof. This follows from Theorem 3.8 (Cartan decomposition). Thus the topology of Sp.2n; R/ reduces to the topology of U.n/, see § 2.4.

3.7. Central extensions of groups Sp.2n; R/. Berezin’s formula

Proposition 7.2. The fundamental group 1 .U.n// of U.n/ is Z. The loop 0 i' 1 e 0 ::: B C G.'/ D @ 0 1 : : :A ; where ' 2 Œ0; 2, :: :: : : : : :

185

(7.1)

is a generator of 1 .U.n//. Corollary 7.3. The fundamental group 1 .Sp.2n; R// is Z. It is generated by the loop G.'/ 0 H.'/ D : (7.2) 0 G.'/ Proof of Proposition 7.2. Fix a unit vector e 2 Cn . Define the map from U.n/ to the sphere S 2n1 Cn given by g 7! ge. We obtain a fiber bundle whose fibers are homeomorphic to U.n 1/. We write out the exact sequence of bundle (see, for instance, [82]) ! 2 .S 2n1 / ! 1 .U.n 1// ! 1 .U.n// ! 1 .S 2n1 / ! ; where 2 ./ denotes a second homotopic group. Now, since 2 .S 2n1 / D 0 and 1 .S 2n1 / D 0 for n > 1, it follows that the standard embedding U.n 1/ ! U.n/ induces an isomorphism of the fundamental groups for n > 2. Clearly, 1 .U.1// D Z. Problem 7.1. Show that 1 .U.p; q// D Z ˚ Z (for p > 0, q > 0). 7.2 The universal covering group of U.n/. For a connected Lie group G, denote by G its universal covering group, see Addendum C.7. The map G ! G is a surjective homomorphism whose kernel is a discrete central subgroup. Let G D U.n/. Consider the group U.n/ R and its subgroup G U.n/ R consisting of elements .h; x/;

where h 2 U.n/, x 2 R, and det g D e ix :

Proposition 7.4. G is the universal covering group of U.n/. We leave this statement as an exercise for the reader. 7.3 Central extensions. Let G be a group, A an (additive) Abelian group. A central extension of G by A (or A-extension of G) is a group G B such that A is a subgroup in G B , A is contained in the center of G, and G B =A ' G. Example. Consider the group B of quaternion units, ˙i , ˙j , ˙k, ˙1. It is a central extension of the group Z2 ˚ Z2 .

186


Example. Let G be a Lie group. Its universal covering group G is a central extension of G. Example. The Heisenberg group Heisn is a central extension of the additive group R2n : 7.4 Cocycles. Denote by ' the homomorphism G B ! G B =A D G. Choose a section W G ! G B , '..g// D g. It is convenient to assume that takes the unit of G to the unit of G B . Each element of G B admits a unique representation q D a .g/;

a 2 A; g 2 G:

Thus we identify (non-canonically) the group G B with the set G A. We transport the multiplication from G B to G A and get the operation of the form .g1 ; a1 / .g2 ; a2 / D g1 g2 ; a1 C a2 C c.g1 ; g2 / ; (7.3) where c.; / is a function G G ! A. Lemma 7.5. The function c W G G ! A satisfies the identities c.e; g/ D c.g; e/ D 0 for all g, c.g1 ; g2 / C c.g1 g2 ; g3 / D c.g1 ; g2 g3 / C c.g2 ; g3 /;

(7.4) (7.5)

where e is the unit of G. The first identity means that .e; 0/ is the unit of G B . The second identity is equivalent to the associativity of the multiplication (7.3). Functions satisfying the conditions (7.4)–(7.5) are called 2-cocycles on G taking values in A. Conversely, if c.; / satisfies these conditions, then (7.3) defines a structure of a group on the set G A. Lemma 7.6. If we change a section G ! G B , then c.g1 ; g2 / transforms according to the rule (7.6) c.g1 ; g2 / 7! c.g1 ; g2 / .g1 / .g2 / C .g1 g2 /; where W G ! A is a function such that .e/ D 0. Proof. Indeed, let 0 be the second section G 7! G B . Then .g/ D 0 .g/ .g/.

A central extension is said to be trivial if c.g1 ; g2 / can be transformed to 0 by the operation (7.6), i.e., c.g1 ; g2 / D .g1 / C .g2 / .g1 g2 / for some : In this case, G B ' G A; we also say that .g/ is a trivializer of the cocycle c.

(7.7)

3.7. Central extensions of groups Sp.2n; R/. Berezin’s formula

187

Lemma 7.7. If and 0 are two trivializers of the same cocycle c, then 0 is a homomorphism G ! A. This follows from (7.7). The second cohomology group H 2 .G; A/ is the group of all A-valued 2-cocycles on G defined up to equivalence. ˆ ‰ 7.5 The Berezin cocycle on Sp.2 n; R/. For a matrix g D ‰ x ˆ x 2 Sp.2n; R/, we denote by ˆ.g/ the block ˆ. Theorem 7.8. a) The function Sp.2n; R/ Sp.2n; R/ ! R given by

c.g1 ; g2 / D Im tr ln ˆ.g1 /1 ˆ.g1 g2 /ˆ.g2 /1

(7.8)

is well defined. More precisely, kˆ.g1 /1 ˆ.g1 g2 /ˆ.g2 /1 1k < 1; and this allows us to define a matrix logarithm in the usual way (2.2.9). b) The function c.g1 ; g2 / is an R-valued 2-cocycle. Remark. At first glance, one can write out

Im tr ln ˆ.g1 /1 ˆ.g1 g2 /ˆ.g2 //1 D Im tr ln ˆ.g1 / C Im tr ln ˆ.g1 g2 / Im tr ln ˆ.g2 /:

(7.9)

We get an expression of the form (7.7), i.e., a trivial cocycle. Nevertheless, the summands tr ln ˆ on the right-hand side are not well defined (because the logarithm is a multi-valued function). On the other hand, the expression

a.g1 ; g2 / D Re tr ln ˆ.g1 /1 ˆ.g1 g2 /.ˆ.g2 //1 is also a 2-cocycle on Sp.2n; R/. However, the latter cocycle is trivial, the trivializer .g/ D Re tr ln ˆ.g/ WD ln j det.ˆ.g//j is well defined. Proof. a) First,

x 2 /; ˆ.g1 g2 / D ˆ.g1 /ˆ.g2 / C ‰.g1 /‰.g

therefore x 2 /ˆ.g2 /1 : ˆ.g1 /1 ˆ.g1 g2 /ˆ.g2 /1 D 1 C ˆ.g1 /1 ‰.g1 / ‰.g x determining the action of x 1 .‰ C z ˆ/ Secondly, we had the expression .ˆ C z ‰/ Sp.2n; R/ on the matrix ball, see (3.2). Substituting z D 0, we observe that ˆ must be

188


ˆ ‰ invertible, and, moreover, ˆ1 ‰ 2 Bn . Applying this remark to the matrix ‰ (see x ˆ x 1 1 1 x x Problem 2.2), we get ‰ˆ 2 Bn . Therefore, kˆ.g1 / ‰.g1 / ‰.g2 /ˆ.g2 / k < 1. b) Let A, B be n n matrices and kA 1k, kB 1k be sufficiently small. Then tr ln.AB/ D tr ln A C tr ln B: Hence for any pair g1 , g2 lying in a small neighborhood of the unit, we can write

tr ln ˆ.g1 /1 ˆ.g1 g2 /ˆ.g2 /1 D tr ln ˆ.g1 / C tr ln ˆ.g1 g2 / tr ln ˆ.g2 /: Now the cocycle identity (7.5) becomes trivial for any g1 , g2 , g3 lying in a sufficiently small neighborhood of the unit. But all our expressions are real analytic and the group Sp.2n; R/ is connected. Therefore, the cocycle identity (7.5) holds for all g1 , g2 , g3 2 Sp.2n; R/. 7.6 A realization of the universal covering of Sp.2 n; R/. Denote by G the group determined by the Berezin cocycle. In other words, consider the space G D Sp.2n; R/R endowed with the multiplication .g1 ; x1 / .g2 ; x2 / D .g1 g2 ; x1 C x2 C c.g1 ; g2 //:

(7.10)

Define the subset G Sp.2n; R/ R consisting of pairs .g; x/ such that

det ˆ.g/=j det ˆ.g/j D e ix :

Proposition 7.9. The group G is the universal covering group of Sp.2n; R/. Proof. Step 1, G is a group. Given .g1 ; x1 /, .g2 ; x2 / 2 G , we must show that the right-hand side of (7.10) is in G . For this aim, we must evaluate

˚

˚ exp i.x1 C x2 C c.g1 ; g2 / D expfix1 g expfix2 g exp i c.g1 ; g2 / ˚ det.ˆ.g1 // det.ˆ.g2 / D exp i Im tr ln ˆ.g1 /1 ˆ.g1 g2 /ˆ.g2 /1 : j det.ˆ.g1 /j j det.ˆ.g2 /j (7.11) Further, exp.tr ln.1 C Q// D det.1 C Q/;

exp.Re tr ln.1 C Q// D j det.1 C Q/j:

Therefore, exp.i Im tr ln.1 C Q// D det.1 C Q/=j det.1 C Q/j: Applying the last formula we get the desired expression det ˆ.g1 g2 /=j det ˆ.g1 g2 /j on the right-hand side of (7.11).

(7.12)

3.7. Central extensions of groups Sp.2 n; R/. Berezin’s formula

189

Step 2. G is a covering. Consider the natural projection … W G ! Sp.2n; R/ given by .g; x/ 7! g. Its kernel consists of elements .1; 2k/. Since the kernel of … is a discrete group, it follows that the projection … is a covering map. Step 3. G is the universal covering. Consider the generator WD H.'/ of 1 .U.n// given by (7.2). Its lift to G is the non-closed path Q WD .H.'/; '/;

where ' 2 Œ0; 2:

(7.13)

The lifts of loops k are given by the same formula (7.13) but ' ranges in Œ0; 2k. Such a lift starts at .1; 0/ 2 G and finishes in .1; 2 i k/. Thus, we can walk from one branch to another and hence G is connected. All these lifts are non-closed paths, therefore, G is the universal covering group. Problem 7.2. Describe the universal covering of SL.2; R/ D Sp.2n; R/. 7.7 The Z-central extension of Sp.2 n; R/. By definition, the universal covering Sp.2n; R/ is a Z-extension of Sp.2n; R/. Let us describe this extension. Set .g/ WD Im tr ln ˆ.g/ WD arg ln det ˆ.g/; (7.14) where 0 6 arg z < 2. Consider the 2-cocycle 1 c.g1 ; g2 / C .g1 / C .g2 / .g1 g2 / : ˛.g1 ; g2 / WD 2 Lemma 7.10. ˛.g1 ; g2 / 2 Z. Proof. Applying (7.12), we obtain exp.2 i ˛.g1 ; g2 // D 1.

We can regard the Z-cocycle ˛ as an R-cocycle. In this context, ˛ is equivalent to the Berezin cocycle. b.g/ 7.8 Central extensions of U.p; q/. We represent g 2 U.p; q/ as g D a.g/ c.g/ d.g/ and write two cocycles

cC .g1 ; g2 / D Im tr ln a.g1 /1 a.g1 g2 /a.g2 /1 I

c .g1 ; g2 / D Im tr ln d.g1 /1 d.g1 g2 /d.g2 /1 : They determine two different central extensions of U.p; q/. Remark. The cocycle cC C c is trivial on the subgroup SU.p; q/ U.p; q/. We realize the universal covering group U .p; q/ as set of triplies .g; ; / 2 U.p; q/ R R such that det a.g/ D e i ; j det a.g/j with multiplication

det d.g/ D e i ; j det d.g/j

.g1 ; 1 ; 1 / .g2 ; 2 ; 2 / D g1 g2 ; 1 C 2 C cC .g1 ; g2 /; 1 C 2 C c .g1 ; g2 / :

190


3.8 Central extensions. The Krein–Shmul’yan category Here we extend the Berezin cocycle to the Krein–Shmul’yan category U. Since Sp is a subcategory in U, the construction is valid also for Sp. In fact, this extension arises in a natural way in representations of U and Sp, see § 5.1, § 7.9. The last two subsections contain an emulation of Harish-Chandra’s discrete series (see Subsection 7.5.15 on page 359) on the level of the Krein–Shmul’yan category. 8.1 The 2-cocycle on the Krein–Shmul’yan category. Let P W V W and Q W W Y be morphisms of the Krein–Shmul’yan category. Denote by ' ˛ ˇ ….P / D ; ….Q/ D (8.1) ~ ı their Potapov transforms. We define the number c.P; Q/ D tr ln.1 ı'/:

(8.2)

Theorem 8.1. For any morphisms P W V W , Q W W Y , R W Y Z of the Krein–Shmul’yan category, c.P; Q/ C c.QP; R/ D c.Q; R/ C c.P; RQ/: (8.3) Proof. Let ….P / and ….Q/ be the same as above, let ….R/ D . By formula (2.8.9) for the multiplication in the Potapov coordinates, we get ' C .1 ~/1 : : : ::: ::: ; ….RQ/ D ….QP / D ::: ::: : : : ~ C ı.1 'ı/1 (other blocks do not concern us). Then

c.P; Q/ C c.QP; R/ D tr ln.1 'ı/ C tr ln 1 Œ~ C ı.1 'ı/1 ; (8.4) c.Q; R/ C c.P; RQ/ D tr ln.1 ~/ C tr ln 1 ıŒ' C .1 ~/1 : (8.5)

We rearrange the last summand in (8.4) as

1 Œ~ C ı.1 'ı/1 D .1 ~/ 1 C .1 ~/1 ı.1 'ı/1 :

Therefore, the right-hand sides of (8.4), (8.4) are respectively: tr ln.1 'ı/ C tr ln.1 ~/ C tr ln 1 C .1 ~/1 ı.1 'ı/1 ; tr ln.1 ~/ C tr ln.1 ı'/ C tr ln 1 C .1 ı'/1 ı .1 ~/1 : To identify these two expressions, we use the identity tr ln.1 AB/ D tr ln.1 BA/ (the identity easily follows from the definition of the logarithm (2.2.9)).

191

3.8. Central extensions. The Krein–Shmul’yan category

8.2 Central extensions of categories. Now we formulate the definition of a central extension of a category. Let K be a category, A an Abelian semigroup. Assume that for each object V , W , Y of K there is a function c W Mor.V; W / Mor.W; Y / ! A y whose objects are satisfying the cocycle identity (8.3). Then we define the category K the objects of K, Mor Ky .V; W / D Mor K .V; W / A; and for .P; u/ 2 Mor Ky .V; W /, .Q; v/ 2 Mor Ky .W; Y / the product is given by .P; u/ B .Q; v/ D .P B Q; u C v C c.P; Q//: The identity (8.3) provides the associativity of the “new” multiplication. Theorem 8.1 shows that we obtain a central extension of the Krein–Shmul’yan category. 8.3 Digression. Functors on the Krein–Shmul’yan category. Fix s 2 C. For an object V D Cp;q of the Krein–Shmul’yan category, denote by H ŒV D H .Bp;q / the space of holomorphic functions on the matrix ball Bp;q . Let P W V W be a morphism of the Krein–Shmul’yan category, transform. We define the operator s .P / W H ŒW ! H ŒV by

˛ ˇ ı

its Potapov

1 s .P /f .z/ D f ˛ C ˇz.1 ız / det.1 ız/s : Theorem 8.2. For any objects V , W , Y of the Krein–Shmul’yan category and morphisms P W V W , QW W Y , s .Q/s .P / D e s c.P;Q/ s .QP /: Proof. By (2.9.4), it suffices to verify the equality of the det-factors, i.e., s det.1 'ı/s det 1 zŒ~ C ı.1 'ı/1 s D det.1 ~z/s det 1 ıŒ' C .1 z~/1 z :

But this is the cocycle identity. 8.4 Digression. Two more functors. For a morphism P W V W write a.P; z/ WD ˇ.1 zı/1 I

b.P; z/ WD .1 ız/1 :

Lemma 8.3. Let P W V W , Q W W Y , and let P , Q be the corresponding Krein– Shmul’yan maps. Then a.QP; z/ D a P; Q .z/ a.Q; z/; b.QP; z/ D b.Q; z/ b P; Q .z/ :

192


Proof. Let the Potapov transforms be given by (8.1). Then a P; Q .z/ a.Q; z/ D ˇ 1 ' C

1 .1 z~/1 z ı .1 z~/1

1 D ˇ.1 'ı/1 1 .1 z~/1 zı.1 'ı/1 .1 z~/1 :

Applying A.1 BA/1 D .1 BA/1 B, we come to ˇ.1 'ı/1 D ˇ.1 'ı/1

1 1 .1 z~/1 zı.1 'ı/1 .1 z~/1

1 .1 z~/ zı.1 'ı/1 D a.QP; z/:

Next, for each object V D Cp;q of the Krein–Shmul’yan category, we assign the space A.V / of holomorphic functions on Bp;q taking values in Cq . For each morphism P W Cp;q ! Cr;s , we assign the operator A.V / ! A.W / given by A.P /f .z/ D a.P; z/f .˛ C ˇ.1 zı/ /: In a similar way, consider the space B.V / consisting of Cp -valued holomorphic functions and the operator B.V / ! B.W / given by B.P /f .z/ D b.P; z/t f .˛ C ˇ.1 zı/ /: Theorem 8.4. A. / and B. / are representations of the Krein–Shmul’yan category, i.e., for any objects V , W , Y of the Krein–Shmul’yan category and any morphisms P W V W , Q W W Y , we have A.P / A.Q/ D A.PQ/;

B.P / B.Q/ D B.PQ/:

This follows immediately from the previous lemma.

The last two Theorems 8.2 and 8.4 allow us to produce more (projective) representations of the Krein–Shmul’yan category by tensor operations.

3.9 Geodesic triangles In this section, we propose another construction of the central extension of Sp.2n; R/. It can be regarded as a generalization of the Berezin formula or as a version of it. 9.1 A function . Let z, u, w range in Bn . We define the function .z; u; w/ 2 R by .z; u; v/ D Im tr ln.1 zu / C ln.1 uv / C ln.1 vz / : (9.1) Theorem 9.1. a) The function is Sp.2n; R/-invariant, i.e., for each g 2 Sp.2n; R/, .z Œg ; uŒg ; v Œg / D .z; u; v/:

(9.2)

193

3.9. Geodesic triangles

b) The function is skew-symmetric, i.e., .z; u; v/ D .u; v; z/;

.z; v; u/ D .z; u; v/:

(9.3)

In particular, .z; z; u/ D 0:

(9.4)

c) satisfies the quadruple identity .z; u; v/ .u; v; w/ C .v; w; z/ .w; z; u/ D 0: w

w v

z

w v

z

u

(9.5)

u

v

z u

Figure 3.3. The identity .z; u; v/ C .v; w; z/ D .u; v; w/ C .w; z; u/ and a geodesic tetrahedron.

Lemma 9.2. a) For g D

ˆ ‰ x ˆ x ‰

2 Sp.2n; R/,

Im tr ln.1 z Œg .uŒg / / x 1 / C Im tr ln.1 C u‰ˆ x 1 /: D Im tr ln.1 zu / Im tr ln.1 C z ‰ˆ

(9.6)

b) Im tr ln.1 zu / D Im tr ln.1 uz /. Proof of the lemma. b)

.1 zu / D .1 uz / H) ln.1 zu / D ln.1 uz /

H) tr ln.1 zu / D tr ln.1 uz /

and this implies the required statement. The calculation that proves a) is done in (2.11.9)–(2.11.13). Also, Im ln.ˆˆ / D 0. Proof of the theorem. a) We apply the statement a) of the lemma. All the additional summands cancel. b) follows from the second statement of the lemma. c) The expression is a sum of 12 summands Im tr ln.: : : /. We collect these summands into six brackets of the form Im tr ln.1 zu / Im tr ln.1 zu / C D 0 C 0 C 0 C 0 C 0 C 0 D 0:

194


9.2 The central extension of Sp.2 n; R/ again. Fix z 2 Bn . Define the function cz on Sp.2n; R/ Sp.2n; R/ by cz .g1 ; g2 / D .z; z Œg1 ; z Œg1 g2 / D .z

Œg11

; z; z Œg2 /:

(9.7) (9.8)

Proposition 9.3. a) The function cz is a 2-cocycle on Sp.2n; R/. b) The cocycle .c0 .; // is the Berezin cocycle (7.8). c) All cocycles cz are equivalent. Proof. a) We must verify the cocycle identity (7.5), i.e., .z; z Œg1 ; z Œg1 g2 / C .z; z Œg1 g2 ; z Œg1 g2 g3 / D .z; z Œg2 ; z Œg2 g3 / C .z; z Œg1 ; z Œg1 g2 g3 /:

(9.9)

By the invariance (9.2), we transform the first term as .z; z Œg2 ; z Œg2 g3 / D .z Œg1 ; z Œg1 g2 ; z Œg1 g2 g3 /: Now (9.9) converts to the quadruple identity (9.5) for the points z, z Œg1 , z Œg1 g2 , z Œg1 g2 g3 . b) We use (9.8), two terms Im tr ln.1 0/ in (9.1) vanish, and we get 1 c0 .g1 ; g2 / D Im ln 1 0Œg2 0Œg1 : We claim that this is precisely the Berezin expression (7.8). To verify this, we need the 1 expression for .0Œg1 / . Since 1 0 ˆ.g1 / ‰.g1 /t 1 0 D ; g1 g11 D 0 1 ‰.g1 / ˆ.g1 /t 0 1 1

it follows that 0Œg1 D ˆ.g1 /1 ‰.g1 /t and we come to the Berezin cocycle (7.8). c) can be verified by formal manipulations, however in the next subsection we present a more aesthetic proof. 9.3 Geodesic triangles and the function . Recall (see 2.11.6)7 that Bn is a symplectic manifold endowed with the closed 2-form ! defined by !.T; S/ D Im tr.1 zz /1 T .1 z z/1 S ;

(9.10)

here T and S are tangent vectors at a point z. For points z, u, v, consider the segments of the geodesics Œz; u, Œu; v, Œv; z connecting these points. Let „ be an arbitrary (oriented) 2-dimensional submanifold „ in Bn whose boundary is the contour R Œz; u–Œu; v–Œv; z. Since the form ! is closed and Bn is contractible, it follows that „ ! does not depend on „. 7

In § 2.11 we consider Cartan balls Bp;q , but nothing changes by a transition to Bn .

195

3.9. Geodesic triangles uŒg1 g2

z Œg1 g2

uŒg1

u

z

Figure 3.4. Reference the proof of Proposition 9.3.c.

Theorem 9.4.

Z !D „

1 .z; u; v/: 2

(9.11)

Remark. This explains the quadruple identity (9.5). Its left-hand side is nothing but an integral of ! over the surface of a geodesic tetrahedron, see Figure 3.3. Since the surface is closed, it follows that the integral is zero. Derivation of Proposition 9.3.c from Theorem 9.4. We must show that the following cocycle is trivial: cz .g1 ; g2 / cu .g1 ; g2 / D .z; z Œg1 ; z Œg1 g2 / .u; uŒg1 ; uŒg1 g2 /:

(9.12)

Consider the prism in Figure 3.4. The integral of the closed form ! over the complete surface of the prism is 0; the expression (9.12) is the integral over bases of the prism, hence we can replace it by the integral over the lateral surface (with a minus), i.e., Z Z Z !C !C !: uzz Œg1 uŒg1 u

uŒg1 z Œg1 z Œg1 g2 uŒg1 g2 uŒg1

Writing

uŒg1 g2 z Œg1 g2 zuuŒg1 g2

Z .h/ WD

! uzz Œh uŒh u

and applying the invariance of !, we write the sum of integrals as .g1 / C .g2 / .g1 g2 /:

Therefore, our cocycles are equivalent, see (7.6). 9.4 Proof of Theorem 9.4 Lemma 9.5. .z; z C "T; z C "S / D "2 !z .T; S / C O."3 /;

" ! 0:

Proof of the lemma. .z; z C "T; z C "S / D Im tr ln.1 z.z C "T / / C Im tr ln.1 .z C "T /.z C "S / / C Im tr ln.1 .z C "S /z /:

196


We intend to write out the Taylor expansion of the right-hand side. By (9.4), it follows that the left-hand side is zero if T D 0 or S D 0. Hence the terms with "T , "S, "T 2 , "S 2 are absent. Therefore, it suffices to watch the term "2 T S . It can arise only from the second summand, the necessary calculation was done in Theorem 2.11.7. Lemma 9.6. Let z1 , z2 , z3 2 Bn , let uij be the midpoints of the segments Œzi ; zj . Then .z1 ; z2 ; z3 / D .z1 ; u12 ; u13 /C.u12 ; z2 ; u23 /C .u13 ; u23 ; z3 /C .u12 ; u23 ; u31 /: Proof of the lemma. Let v1 , v2 , v3 lie on a geodesic .t /. Without loss of generality, we can think that a geodesic has the canonical form described in Theorem 2.12.2. Obviously, .v1 ; v2 ; v3 / D 0, because all Im./ D 0. The proof of the lemma is completed by Figure 3.5a). Indeed, we apply the quadruple identity to the points z1 , z2 , z3 , u12 . The term .z1 ; z2 ; u12 / vanishes and we write .z1 ; z2 ; z3 / D .z1 ; u12 ; z3 / C .u12 ; z2 ; z3 /: This explains the first step on the figure. The last surgery is provided by the quadruple identity for z3 and three midpoints.

b)

a)

Figure 3.5. a) Proof of Lemma 9.6. b) Reference Theorem 9.4.

Proof of the theorem. By Lemma 9.5, for a geodesic triangle with sides O."/, the identity (9.11) holds modulo O."3 / (the O is uniform in each compact subset in Bn ). Let us divide a geodesic triangle into 22N geodesic triangles as in Figure 3.5.b. Now we get O."2 / pieces and a mistake in each piece is O."3 /. This completes the proof of the theorem. 9.5 Projective representations and central extensions. In this subsection, we refer to Chapter 5, but our discussion makes sense without this reference. Denote by T the group of complex numbers with the absolute value 1. Consider a unitary projective representation of a group G, .g1 /.g2 / D .g1 ; g2 /.g1 g2 /;

.g1 ; g2 / 2 T :

Due to the associativity, .g1 /..g2 /.g3 // D ..g1 /.g2 //.g3 /, we have .g1 g2 /.g3 / D .g1 /.g2 g3 /:

(9.13)

3.9. Geodesic triangles

197

Therefore8 , is a T -valued 2-cocycle on G and it determines a central extension G B of G by T . Problem 9.1. Consider the group G of all operators of the form .g/, where ranges in T and g ranges in G. When does G B D G ? As we have seen above, cocycles are defined to within the correction (7.6). Sometimes it is possible to normalize operators .g/ canonically and to obtain a canonical expression for the cocycle. Assume that there is a vector h such that h.g/h; hi ¤ 0 for all g 2 G: We define (non-unitary) operators ? .g/ WD h.g/h; hi1 .g/: By definition, these operators satisfy h? .g/h; hi D 1

for all g 2 G:

(9.14)

After that, we obtain a canonically defined C -valued cocycle ? D h by ? .g1 /? .g2 / D ? .g1 ; g2 /? .g1 g2 /: The Berezin formula and the triangle formula (9.11) can be produced in such a way from the Weil representation of Sp.2n; R/. Precisely, let h be a Gaussian vector bŒA in the notation of Subsection 5.1.1, then our remark produces the C -valued cocycle ˚

1 ? .g1 ; g2 / D exp 12 T .AŒg1 ; A; AŒg2 ; where

T .z; u; v/ D tr ln.1 zu / C ln.1 uv / C ln.1 vz / :

(9.15)

The expression T .; ; / slightly differs from .; ; /. To achieve better correspondence, we slightly modify our considerations. Set another normalization of our representation B .g/ WD

jh.g/h; hij .g/: h.g/h; hi

In other words, we choose a correcting scalar factor such that hB .g/h; hi > 0 8

and B .g/ is unitary:

Here we use the multiplicative notation for the operation in the Abelian group C .

198


This normalization produces the cocycle B .g1 ; g2 / 2 T by B .g1 /B .g2 / D B .g1 ; g2 /B .g1 g2 /: Actually B is nothing but ? =j ? j. Respectively, the remark on Gaussian vectors produces cocycles9 ˚

1 B .g1 ; g2 / D exp 12 .z Œg1 ; z; z Œg2 : The Gaussian vector bŒ0 produces the Berezin cocycle. Thus all our calculations are only a straightforward verification of facts that are (almost) self-evident (and Berezin himself wrote his formula in such a way).

3.10 Digression. Central extensions of groups of symplectomorphisms The material of this section will not used in what follows. Recall that a symplectic manifold M D M 2m is a smooth 2m-dimensional manifold equipped with a non-degenerate closed differential 2-form !. A symplectomorphism is a diffeomorphism preserving form !. We denote the group of all symplectomorphisms by Symp.M 2m ; !/, by Sympcomp .M 2m ; !/ we denote the group of symplectomorphisms with compact supports (i.e., gx D x outside some compact set). We present two explicit constructions of central extensions of groups of symplectomorphisms. 10.1 The Ismagilov formula. In addition, assume that M is simply connected and there is a global 1-form on ! such that ! D d . In particular, M 2m has to be non-compact10 . Fix a point a 2 M . For any g1 , g2 2 Symp.M 2m ; !/ define Z g2 a C.g1 ; g2 / WD .g1 /: (10.1) a

Lemma 10.1. The integral does not depend on a path connecting a and a g2 . Proof. The integrand is a closed form. Indeed, d.g1 / D g1 ! ! D 0.

Theorem 10.2. C.g1 ; g2 / is a 2-cocycle on the group Symp.M n ; !/. 9 Let A ! B be a homomorphism of Abelian groups. It induces the (obvious) homomorphism of cohomology W H 2 .G; A/ ! H 2 .G; B/. In our case, we have the map exp W R ! T . 10 Assume the converse. The form ! 2m is a non-vanishing differential form of maximal degree, hence it represents a non-zero (de Rham) cohomology class. If ! D d , then ! is a zero element of the second cohomology. Therefore ! n is a zero element of cohomology.

3.10. Digression. Central extensions of groups of symplectomorphisms

199

Proof. We must verify the cocycle identity, namely, ag Zag2 Zag3 Z2 g3 Zag3 .g2 / C .g1 / .g1 / .g1 g2 / D 0: a

a

a

We combine the second and the third terms Z ag2 g3 Z .g1 / D ag2

a

ag3

.g1 g2 g2 /:

a

After that, all the terms cancel.

Problem 10.1. Show that the cocycle C is a well-defined element of the cohomology group H 2 .Symp.M n ; !/; R/, i.e., it does not depend on the choice of a base point a and a potential . Problem 10.2. Let M be R2 endowed with the 2-form dx ^dy. Choose D 12 .xdy ydx/. Restrict the cocycle C to the group R2 of all translations and show that this produces the Heisenberg group. Now, let M D Bn , let ! be the invariant symplectic form on Bn . Consider the group Symp.Bn ; !/ of all symplectomorphisms of Bn . Proposition 10.3. The restriction of the cocycle C.; / to Sp.2n; R/ is equivalent to the Berezin cocycle. Proof. First, we write out a 1-form . Denote by @ the Dolbaux differential, recall that the de Rham differential is d D N and recall that @2 D 0. @ C @, In Section 2.11 we showed that K.u/ D ln.1 uu / is a Kähler potential on Bn . Put X @K.u/ dukl : WD Im @K D Im @ukl Then N D Im @@K N d D Im.@ C @/ D !: Since g 2 Sp.2n; R/ preserves the complex structure, it follows that g D Im @.K.uŒg / K.u//: By (2.11.9)–(2.11.13), x 1 / tr ln.1 C u‰ˆ x 1 / C tr ln.ˆˆ /: K.uŒg / K.u/ D tr ln.1 C u‰ˆ Therefore,

x 1 / ; @.K.uŒg / K.u// D @ tr ln.1 C u‰ˆ

200 and


x 1 / x 1 / D Im d tr ln.1 C u‰ˆ g D Im @ tr ln.1 C u‰ˆ

x 1 / D 0). Taking a WD 0 in (10.1), we get the Berezin cocycle. (since @N tr ln.1 C u‰ˆ 10.2 Another central extension. Now we consider an arbitrary symplectic manifold and the group Sympcomp .M; !/ of symplectomorphisms with compact supports. Equip the space R2n with the standard symplectic structure dx ^ dy. One can find an open set R2n and a symplectic embedding W ! M such that the measure of M n ./ is zero. Remark. a) We admit disconnected sets . b) It is pleasant (but not necessary) to think that M n./ is a union of submanifolds. Any element q 2 Sympcomp .M / induces a transformation q D 1 g of defined almost everywhere. Given q 2 Symp.M; !/ and x 2 , we denote by q0 .x/ the Jacobi matrix of q at the point x. Obviously, q.x/ 2 Sp.2n; R/. Let ˆ.g/ be the same as in Subsection 7.5. We define the 2-cocycle C.q1 ; q2 /, where q1 , q2 2 Sympcomp .M; !/, by C.q1 ; q2 / Z n

o D Im tr ln ˆ1 q01 .q 2 .m/ ˆ q01 .q 2 .m// q02 .m/ ˆ1 q02 .m/ dx Z

D c q01 .q 2 .m//; q02 .m/ dx

(10.2)

(here c is the Berezin cocycle). Theorem 10.4. The expression C.q1 ; q2 / defines an element of the second cohomology group H 2 .Symp.M /; R/; moreover C does not depend on the choice of a domain and a map . Proof. We must verify the identity Z Z 0 0 c q 2 .q 3 /; q 3 dx C c q01 .q 2 .q 3 //; q02 .q 3 / q03 dx Z Z 0 0 D c q1 .q 2 /; q 2 dx C c q01 .q 2 .q 3 // q02 .q 3 /; q03 dx: For this aim, we substitute x ! q3 .x/ to the first summand of the second row. Since q3 is symplectic, the Jacobian of this transformation is 1. After that, we get in the integrand the cocycle identity for the triple, q01 .q 2 .q 3 //; of elements of Sp.2n; R/. We omit a proof of the “independence”.

q02 .q 3 /;

q03

3.11. Central extensions. The Maslov index

201

3.11 Central extensions. The Maslov index Here we discuss an alternative approach to central extensions. This method is not too convenient for representation theory; however it is more natural for the theory of Fourier integral operators. Below we do not use the material of this section. Also, we notice that the common term “Maslov index” is used in several related but not identical senses. 11.1 Kashiwara’s definition of the Maslov index. Consider a triple L1 , L2 , L3 of Lagrangian subspaces in R2n . First, suppose that their pairwise L intersections are 0. Applying an element of Sp.2n; R/, we can put L equal to Cek and L3 equal to 1 L Cfm . Since L2 \ L3 D 0, the subspace L2 is a graph of a symmetric operator z W L1 ! L3 . We choose an arbitrary basis ek 2 L1 , then the dual basis fk 2 L2 is canonically determined by ek . If we change the basis ek , then the (real symmetric) matrix z transforms as z 7! gzg t , where g is the transition matrix for bases in L1 . Therefore, one can reduce z to a diagonal form with entries ˙1. Denote by p the number of .1/’s and q the number of .C1/’s. Obviously, the .L1 ; L2 ; L3 / WD .q p/=2:

(11.1)

is an invariant of our triple .L1 ; L2 ; L3 /. This invariant can be extended to arbitrary triples of Lagrangian subspaces as follows. Consider the abstract direct sum L1 ˚ L2 ˚ L3 and the quadratic form on this space defined by B.v1 ˚ v2 ˚ v3 / D fv1 ; v2 g C fv2 ; v3 g C fv3 ; v1 g:

(11.2)

Let r, s be its inertia indices. We define the Maslov index by K .L1 ; L2 ; L3 / WD .r s/=2:

(11.3)

The number si WD r s is called the signature of a quadratic form. Observation 11.1. Suppose that L1 , L2 , L3 2 Lagr n .R/ are in general position. Then K .L1 ; L2 ; L3 / coincides with the invariant (11.1) defined above. Proof. Without loss of generality, we can set L L L Cek ; L3 D Cfk ; L2 D C.ek C .1/k fk /; L1 D where k D 0, 1. Then the matrix of the form B is a direct sum of n items of 3 3 matrices of the form 1 0 0 1 1 @1 0 .1/k A : k 1 .1/ 0

202


Such a matrix has inertia .2; 1/ or .1; 2/ according to the sign .1/k . The consistency of the two definitions becomes clear. Theorem 11.2. a) The expression K .L1 ; L2 ; L3 / is skew symmetric under permutations of Lj . b) The expression K satisfies the quadruple identity (9.5). The statement a) is a simple exercise and the quadruple identity is proved in the next subsection. Corollary 11.3. For a fixed Lagrangian subspace L, the function c.g1 ; g2 / D K .L; L g1 ; L g1 g2 / is a 2-cocycle on the group Sp.2n; R/. In Proposition 9.3 we observed that the cocycle property follows from the quadruple identity. 11.2 A reformulation. Recall that each pair of transversal Lagrangian subspaces determines a coordinate system in Lagr n .R/, see Subsection 1.4. Proposition 11.4. Consider a triple L1 , L2 , L3 of Lagrangian subspaces with (welldefined) coordinates Z, U , W . Then K .Z; U; V / D

si.Z U / C si.U W / C si.W Z/ : 2

(11.4)

Proof. Our subspaces consist of vectors a ˚ aZ, b ˚ bU , c ˚ cW , where a, b, c range in the horizontal subspace Rn ˚ 0 R2n . We regard .a; b; c/ as coordinates in the linear space L1 ˚ L2 ˚ L3 . In these coordinates, the matrix of the bilinear form B is 0 1 0 ZU U V @Z U 0 V ZA : U V V Z 0 We note that any vector of the form .˛; ˛; 0/ is B-orthogonal to any vector .0; ˇ; ˇ/. Indeed, 0 10 1 0 Z U U V 0 @ A @ ˛ ˛ 0 ZU 0 V Z ˇt A U V V Z 0 ˇt t ZU U V ˇ D ˛ ˛ 0 V Z ˇt D ˛ .Z U / C .U V / C .V Z/ ˇ t D 0:


203

More formally, we define the subspaces H12 , H23 H13 consisting of vectors of the form .˛; ˛; 0/, .0; ˇ; ˇ/, .; 0; / respectively. Then L1 ˚ L2 ˚ L3 is a B-orthogonal direct sum H12 ˚ H23 ˚ H31 . Consider the restriction of B to H12 , 0 1 0 t1 0 Z U U V ˛ @ A @ ˛ ˛ 0 ZU 0 V Z ˛ t A D ˛ 2.Z U / ˛ t : U V V Z 0 0 Thus, the restrictions of B to H12 , H23 , H13 are given by the matrices 2.Z U /, 2.U V /, 2.V Z/ respectively. This implies the desired statement. Proof of Theorem 11.2 b (quadruple identity). Given four Lagrangian subspaces, we choose a coordinate system on the Lagrangian Grassmannian, in which our subspaces have well-defined coordinates Z, U , V , W respectively. Write out the quadruple identity for K using (11.4): si.Z U / C si.U V / C si.V Z/ C si.V W / C si.W Z/ C si.Z V / ‹ D si.U V / C si.V W / C si.W U / C si.W Z/ C si.Z U / C si.U W / : The boxed terms in the first row cancel, in the second row they also cancel; thus we observe an evident identity. Remark. The identity (11.4) is a quadruple identity for the vertical subspace and three subspaces whose coordinates are Z, U , V . Theorem 3.6 states that Lagr n .R/ is in a one-to-one correspondence with the set O0 of symmetric unitary matrices. We claim that The Kashiwara function K .z; u; v/ is the function .z; u; v/ defined above (9.1) for unitary symmetric matrices z, u, v. However, this claim is not trivial (and even is yet not formulated!). The point is xn B x n if and xn B that the function is a continuous function at a point .z; u; v/ 2 B only if det.1 zu / ¤ 0; det.1 uv / ¤ 0; det.1 vz / ¤ 0: If z, u, v are unitary, this condition means that the corresponding Lagrangian subspaces have pairwise zero intersections. Under this condition, a verification of the identity K D is a simple exercise. Now our aim is to define values of .z; u; v/ at points of discontinuity. Preparation for this definition occupies the following three subsections.

204


11.3 The case U.1/=O.1/ is demonstrated in Figure 3.6. We realize the Lobachevsky plane Sp.2; R/=U.1/ as an upper half-plane; the absolute U.1/=O.1/ is the real axis. Figure 3.6 suggests that in the general case we must pass to the manifold U.n/=O.n/ from normal directions. 1

1

1

u

v v

a)

u 0

"

v

u " 0

b)

0

Figure 3.6. This is the Poincaré upper half-plane. We watch the behavior of the area of a triangle z 1 u as z, u tend to the same point 0 of the boundary. a) Consider the triangle with vertices z D 1, u D "e i' , v D "e i . Its area is S D fsum of angles of triangleg. Therefore S D ' does not depend on ". In particular, the limit as " ! C0 is '. Thus we get different limits of S as u and v come to 0 in different ways. b) If z D 1 and u, v tend to 0 from the vertical direction, then the limit of the area is 0 (because the sum of the angles is ).

11.4 Curves totally normal to the boundary. We regard the set Matn;n as a real Euclidean space with respect to the inner product hz; ui D Re tr zu : Problem 11.1. Show that the orthocomplement of the space of Hermitian matrices is the space of anti-Hermitian matrices. We are going to examine the behavior of function tr ln.1 zu / defined on the matrix ball Bn;n Matn;n near , 2 U.n/. Later we shall return to the symplectic case, i.e., to Bn . For any z 2 U.n/, denote by Tz the tangent space to the submanifold U.n/ Matn;n at a point z. Let 2 Tz . Then one can find a curve ."/ D z C " C O."2 / 2 U.n/, i.e., 1 D ."/ ."/ D .z C" CO."2 // .z C" CO."2 // D z z C". z Cz /CO."2 /: Therefore, 2 Tz () z C z D 0 () z 1 D .z 1 / ; i.e., z 1 is anti-Hermitian: For each z 2 U.n/ Bn;n , denote by Nz Matn;n the space of normal vectors to U.n/ at z. The following description of Nz is obvious.


205

Lemma 11.5. a) Nz consists of matrices such that z 1 is Hermitian. b) Nz D iTz . Corollary 11.6. Transformations g 2 U.n; n/ send normal vectors to normal vectors. Proof. Indeed, these transformations send tangent vectors to tangent vectors and also they are complex-analytic. Hence they send iTz to iTz Œg . x n;n be a smooth curve defined for " 2 Œ0; ı, set M.0/ D z and Let M."/ 2 B M .0/ ¤ 0. We say that the curve M."/ is totally normal to U.n/ if the vector M 0 .0/ is normal to U.n/ and moreover, 0

z 1 M 0 .0/ is positive self-adjoint:

(11.5)

The last condition means that the normal vector M 0 .0/ is directed strictly inside Bn;n . x n;n be a curve totally normal to U.n/. Then for any Lemma 11.7. a) Let M."/ 2 B g 2 U.n; n/, the curve M."/Œg is also totally normal to the boundary. b) Let M."/ and N."/ be totally normal to U.n/. Then their matrix product M."/N."/ is also totally normal to U.n/. Proof. a) is a rephrasing of Corollary 11.6. b) We write out M."/ D M.0/ C "M 0 .0/ C O."2 /;

N."/ D N.0/ C "N 0 .0/ C O."2 /;

where M.0/1 M 0 .0/, N.0/1 N 0 .0/ are negative self-adjoint matrices. Then M."/N."/ D M.0/N.0/ C "M.0/N 0 .0/ C "N.0/M 0 .0/ C O."2 /: The vector N 0 .0/ is normal to U.n/ and is directed strictly inside the matrix ball; its shift M.0/N 0 .0/ satisfies the same property. Also, N 1 .0/M 0 .0/ is directed strictly inside Bn;n . Hence their sum also is directed inside Bn;n . Problem 11.2. Let .t / be a geodesic in Bn and lim t!C1 .t / 2 U.n/. Then the curve is totally orthogonal to the boundary. 11.5 The behavior of eigenvalues along totally normal curves. Let M."/ be a totally normal curve, M.0/ D z. Let j ."/ be the eigenvalues of the matrix M."/. A priori, there are n functions j ."/, which are real analytic on small intervals .0; ıj / and admit expansions in Puiseux series11 X j ."/ D cj "˛j ; where ˛j are rational, ˛j C1 > ˛j , and ˛j ! C1: ˛j 11 Theorem (I. Newton). Consider a plane curve C determined by an analytic equation f .x; y/ D 0, let .0; 0/ 2 C . Then there are positive integers Nj such that near .0; 0/ the curve is a union of curves having the form of the form y D j .x 1=Nj /; here j are analytic. We apply this to the curve f .; "/ D det.M."/ / D 0. It is more pleasant (but not necessary) to require analyticity of the curve M."/.

206


Lemma 11.8. For a totally normal curve, all j0 .0/ are well defined. Moreover, j0 .0/ ¤ 0. The curves j ."/ are orthogonal to the circle jj D 1.

Figure 3.7. Reference Lemma 11.8. Positions of curves j ."/ in the circle jj < 1.

Problem 11.3. Prove the following theorem on perturbations of spectra. Let ƒ be a diagonal matrix with pairwise distinct entries j . Consider a smooth perturbation M."/ D ƒ C "S C "2 T C : Denote by j ."/ the eigenvalues of M."/. Then j0 .0/ D sjj . Hint. Write out the determinant. Proof. Without loss of generality, we can assume that z D ƒ is a diagonal matrix with entries j , let M."/ D ƒ C "S C : Let 1 ."/ D 1 C "˛ C ;

where ˛ > 0; ¤ 0

(11.6)

be an eigenvalue of M."/, also 1 .0/ D 1 . Represent ƒ as a block matrix ƒ D 1 1 0 0 ƒ22 , where 1 1 denotes a scalar matrix and ƒ22 does not contain entries S12 . Since ƒ1 S is negative definite, it 1 . Respectively, represent S D SS11 21 S22 follows that its block 1 1 S11 is also negative definite. In particular, the block S11 is nondegenerate. Let v."/ be an eigenvector of M."/ with the eigenvalue 1 ."/, v1 C "ˇ w1 C ; where ˇ > 0, .w1 ; w2 / ¤ .0; 0/: v."/ D 0 C "ˇ w2 C We write out the equation M."/v."/ D 1 ."/v."/ in the block form, S11 S12 v1 C "ˇ w1 C 0 1 C" C S21 S22 0 ƒ22 0 C "ˇ w2 C v C "ˇ w1 C D .1 C "˛ C / 1 : 0 C "ˇ w2 C


207

Equating the series in the upper left blocks, we get 1 .v1 C "ˇ w1 C / C "S11 .v1 C "ˇ w1 C / C "S12 ."ˇ w2 C / C D 1 .v1 C "ˇ w1 C / C "˛ .v1 C "ˇ w1 C / C : After the cancelation, we come to "S11 v1 C O."/ D "˛ v1 C O."˛ /:

(11.7)

Since S11 is non-degenerate, S11 v ¤ 0. Also, ¤ 0 because of the assumption (11.6). Hence v1 ¤ 0. Therefore, the leading terms on the left- and right-hand sides of (11.7) are equal. Thus ˛ D 1 and .S11 /v1 D 0 H) .1 S11 1 /v1 D 0: Hence 1 is an eigenvalue of a negative definite matrix 1 1 S11 . Therefore, 0 > 1 D .0/1 0 .0/

and this is the desired statement.

Problem 11.4. Let K."/ D A C "B C be a curve in the wedge of dissipative matrices, let A be anti-Hermitian and B D B > 0. Let j ."/ be the eigenvalues of K."/. Then the curves j ."/ are contained in the half-plane Re > 0 and are orthogonal to the real axis. Given z 2 U.n/, denote by 1 D D m D 1, mC1 , mC2 , … its eigenvalues (we assume j ¤ 1 for j > m). Represent k D e i.'k / ;

< 'k < :

Lemma 11.9. For a totally normal curve M."/, passing to z, we have 1X 'k : lim Im tr ln.1 M."// D "!C0 2 We observe that the limit depends only on the eigenvalues of M.0/ WD z. Proof. Im tr ln.1 M."// D

X

Im ln.1 j ."//

and the question reduces to the behavior of the function f .w/ D ln.1 w/, w 2 C, on the circle jwj D 1. This function is continuous on the circle punctured at w D 1, Im ln.1 e i.'/ / D

1 '; 2

< ' <

and lim Im ln.1 .1 "// D 0:

"!C0

208


11.6 The limit of . Let the function W Bn Bn Bn ! R be the same as above (9.1). Recall that we identify the set U.n/=O.n/ of symmetric unitary matrices and the Lagrangian Grassmannian Lagr n .R/, see Theorem 3.6. Theorem 11.10. Let z, u, v 2 U.n/=O.n/. a) Let Mz ."/, Mu ."/, Mv ."/ 2 Bn be curves totally orthogonal to U.n/=O.n/ and coming to z, u, v respectively. Then the limit .z; u; v/ WD lim Mz ."/; Mu ."/; Mv ."/ (11.8) "!C0

exists and does not depend on the choice of curves Mz , Mu , Mv . b) .z; u; v/ 2

Z. 2

c) .z; u; v/, satisfies the quadruple identity (9.5). Proof. The statement a) follows from Lemma 11.9. We obtain c) passing to the limit in the quadruple identity in Bn . The statement b) follows from the next lemma. Lemma 11.11. For each z, u, v 2 U.n/=O.n/, one has expf2i .z; u; v/g D ˙1. Proof. Under the conditions of Lemma 11.9, ˚

Y ' exp 2i lim tr Im ln.1 M."// D e j D ˙ det M.0/1 I "!0

j

the sign depends on the multiplicity of the eigenvalue 1 in M.0/. Therefore, ˚

det z det u det v ˙ ˙ D ˙1: exp 2i .z; u; v/ D ˙ det u det v det z

Theorem 11.12. For z, u, v 2 U.n/=O.n/, the function .z; u; v/ coincides with the function K .z; u; v/ of Kashiwara. Proof. The statement is valid for n D 1. We can consider direct sums. This shows that and K coincide for triples L1 , L2 , L3 2 Lagr n .R/ such that L3 is transversal to L1 and L2 . Consider arbitrary triple L1 , L2 , L3 and a subspace L4 transversal to L1 , L2 , L3 . We have K D for the triples ˚

˚

˚

L1 , L2 , L4 , L1 , L3 , L4 , L2 , L3 , L4 . We write out the quadruple identity for L1 , L2 , L3 , L4 and observe that .L1 ; L2 ; L3 / D K .L1 ; L2 ; L3 /:

Addendum to Section 3.3. Lie semigroups and convex cones

209

Addendum to Section 3.3. Lie semigroups and convex cones This addendum consists of four subdivisions: ˛) In §§ 3.5, 2.7 we met invariant convex cones (of dissipative operators) in the Lie algebras sp.2n; R/, u.p; q/. Here we discuss the problem of uniqueness of such objects. ˇ) In §§ 3.5, 2.7 we discussed Lie semigroups of contractions of matrix balls. Here we briefly discuss other similar objects (so-called “Olshanski semigroups”) and generalities on subsemigroups in Lie groups. ) Causal structures on homogeneous spaces in the sense of Irving Segal. ı) There are interesting and important subsemigroups in Lie groups outside the formalism of Lie cones and Lie wedges. Here we briefly discuss semigroups of totally positive matrices. In subdivisions ˛- we are trying to explain examples and “ideology” avoiding “classifications”. For the latter topic, see Addendum D. ˛. Invariant convex cones in classical Lie algebras 1. Invariant convex cones in sp.2 n; R/. In Subsection 3.5.2 we introduced the cone PosSp sp.2n; R/; recall that X 2 PosSp if iX is indefinite dissipative. Theorem 1. Any convex Sp.2n; R/-invariant cone in the Lie algebra12 sp.2n; R/ is PosSp or .PosSp/. We need the following standard theorem. Theorem 2. There are no Sp.2n; R/-invariant subspaces in sp.2n; R/. Equivalently, the Lie algebra sp.2n; R/ is simple, i.e., it has no ideals. The reader can find this statement in any textbook on Lie algebras. To be complete, we outline a proof. Consider the real model of Sp.2n; R/. The group Sp.2n; R/ contains the n-dimensional Abelian subgroup (a Cartan subgroup) H consisting of diagonal matrices; collections of entries of such matrices have the form 1 1 ; : : : ; n ; 1 1 ; : : : ; n :

Problem 1. a) Find all H -invariant subspaces in sp.2n; R/. b) Prove Theorem 2. Corollary 3. Let L sp.2n; R/ be a convex Sp.2n; R/-invariant cone. Then L is sharp and contains an interior point. Proof. Indeed, the linear span of L is an invariant subspace. Also, the edge of L is invariant. We say that an element of sp.2n; R/ is regular if its eigenvalues are pairwise distinct. Lemma 4. Non-regular elements form a proper submanifold in sp.2n; R/. 12

Recall that the group Sp.2n; R/ acts on the Lie algebra by conjugations X 7! g 1 Xg.

210


Proof. Consider the discriminant13 D.X / of the characteristic polynomial of X 2 sp.2n; R/. Matrices X 2 sp.2n; R/ with pairwise distinct eigenvalues exist, therefore D.X / is a nonzero polynomial. Therefore set D.X / D 0 is a proper (singular) submanifold. Since the cone L has interior points, it contains a regular element. Theorem 1 follows from the next lemma:1 Lemma 5. For each regular element X 2 sp.2n; R/, the closed cone K spanned by its orbit O is PosSp, or .PosSp/, or the whole space. Proof of the lemma. The proof is based on an examination of the asymptotic cone of the real algebraic manifold O. Assume that K is not the whole space. Step 1. Decompose the symplectic space into a direct sum of indecomposable X-invariant subspaces, V2n D H1 ˚ H2 ˚ ; X D X1 ˚ X2 ˚ : There are three types of such summands according to the spectrum of Xj : Type I. The spectrum is i , i , where 2 R. In this case, there is an additional invariant, namely sgn M.vi ; vi /, where vi is the eigenvector corresponding to i . Type II. The spectrum is , , where 2 R. Type III. The spectrum is , , N , , N where , i … R. Since X is regular, it follows that the dimensions of summands of types I, II are 2, and dimensions of summands of type III are 4. Step 2. Fix one of the summands, say H1 , and consider the subgroup G Sp.2n; R/ whose elements act trivially on 0 ˚ H2 ˚ H3 ˚ , actually G ' Sp.2; R/ or Sp.4; R/. We emphasize that in all cases the orbit G X is not bounded. For types I and II, the reader can look at Figure 3.2. The case III is also evident. Consider a curve .t / in G X passing to infinity as t ! 1. Normalize it by a scalar h.t / in such a way that h.t / .t / is bounded (and does not tend to 0). Therefore, h.t / ! 0. Hence lim h.t / .t/ is zero on 0 ˚ H2 ˚ H3 ˚ and is a non-zero nilpotent on H1 . Again, we can look at Figure 3.2.a, the asymptotic cone for hyperboloids consists of nilpotent matrices. Thus, K contains a matrix of the form J ˚ 0 ˚ , where J is a non-zero nilpotent matrix. Next, if X1 is a block of type II or III, then X1 is G -conjugate to .X1 / (and this is not the case for type I, see Figure 3.2.a). Therefore, .J / ˚ 0 ˚ 2 K. Hence the cone is not sharp and we come to a contradiction. Step 3. Thus, X is a direct sum of blocks of type I. If X has blocks of different signs, then K contains both J ˚ 0 ˚ and 0 ˚ ˚ .J / ˚ 0 ˚ . Therefore the cone contains .J / ˚ 0 ˚ (we can transpose two summands, the transposition can be made by a symplectic matrix). Again the cone is not sharp. Step 4. The Sp.2; R/-orbit of J is a sheet of the cone in Figure 3.2.a. The convex hull of this sheet contains the whole conic body, i.e., the union of all elliptic classes of given sign in sl2 .R/. Q Let p.x/ be a polynomial, xj be its roots. Then the discriminant of p is i>j .xi xj /2 . The discriminant is a polynomial in coefficients of p. Also it is the resultant of p.x/ and p 0 .x/. 13


211

Thus, K contains all the operators of the form A1 ˚ A2 ˚ , where Aj are elliptic blocks of a given sign. The union of their Sp.2n; R/-orbits is PosSp or .PosSp/. This completes the proof of the lemma. 2. Invariant convex cones in the Lie algebra su.p; q/. Let us examine the same problem (the description of closed convex invariant cones) for the Lie algebra su.p; q/. First, we present two examples of invariant convex cones su.p; q/. Example 1. The cone Cmin consists of operators of the form X , where iX is pseudo-dissipative. Example 2. The cone Cmax consists of X such that for some real constant s D s.X /, the operator iX C s 1 is pseudo-dissipative. See Theorem 2.7.13. Problem 2. Any proper invariant closed convex cone in su.p; q/ satisfies Cmin C Cmax

or

Cmin .C / Cmax :

Intersect a convex invariant cone C with a set h of diagonal matrices. Example 10 . For Cmin , the intersection „min WD Cmin \ b consists of matrices whose entries a1 ; : : : ap , b1 ; : : : ; bq satisfy the following conditions: for all j , aj > 0;

and

for all k, bk 6 0:

Example 20 . For Cmax , the intersection „max WD Cmax \ b is determined by the following inequalities: for all j , k, aj > bk : Problem 3. The intersection C \ h completely determines a cone C . Precisely, SU.p; q/ .C \ h/ D C: The following theorem gives a complete classification of invariant convex cones in su.p; q/. Theorem 6 (Olshanski–Paneitz). Consider a proper closed convex cone „ h such that – „min „ „max I – „ is invariant with respect to all permutations of a1 ; : : : ; ap and all permutations of b1 ; : : : ; bq . Then „ D C \ h for some closed invariant convex cone in su.p; q/. The classification is nontrivial, but the actual difference between Cmin and Cmax corresponds to variations of understanding of “indefinite dissipativity”. ˇ. Lie semigroups 3. Convex cones in Lie algebras and local Lie semigroups. There are a lot of subsemigroups in Lie groups (see Figure 3.8), certainly most of them are uninteresting. However we met interesting “Lie semigroups” above (namely, indefinite contractions and symplectic contractions). Since there are some other interesting objects of this kind, we wish to discuss further examples and generalities.

212


Figure 3.8. Examples of subsemigroups in the additive group R2 .

Let g be a Lie algebra, G the corresponding Lie group. Consider a ball14 B g such that the exponential map exp W g ! G is injective on B, and let B D exp.B/. Let C g be a closed cone. A causal curve is an (absolutely continuous) curve .t / G, such that .0/ D 1 and .t /1 0 .t / 2 C . Denote by the set of all points g 2 G that can be achieved along such curves .t / B. It can occur that the tangent cone15 Cz to is larger than the initial cone C , see Figure 3.9.

.t / .t /

Figure 3.9. Example. The Lie group R2 , a cone C , its shifts, and a causal curve .t /. In the second case the cone is non-convex (therefore it is not a Lie wedge); a causal curve can achieve any point of the first quadrant.

A Lie wedge (D Lie cone) is a cone C satisfying the following conditions: 1) the cone C is convex; 2) its edge m WD C \ .C / is a subalgebra in g; denote by M its Lie group; 3) the cone C is invariant under conjugations by elements of M . Theorem 7 (Olshanski). The tangent cone to is C if and only if the conditions 1)–3) are satisfied. Problem 4. a) Show that the convexity of C is necessary. b) Show that the condition 2) is necessary. Hint. See Subsection C.3. 14 15

With respect to an arbitrary norm in g; this is not important. See Subsection C.3.


213

Objects of this kind are called local Lie semigroups. Evidently, there is a sufficiently small neighborhood U of a unit such that for g1 , g2 2 U \ , we have g1 g2 2 . 4. Examples of Lie wedges. 1) Consider the Lie algebra g WD gl.n; R/ and the subalgebra m WD o.n; R/ consisting of skew-symmetric matrices. The quotient space p WD g=m is the space of symmetric matrices. We have the evident convex cone Posn p consisting of positive (semi)-definite matrices. Thus we get a Lie wedge C gl.n; R/ consisting of matrices of the form A C B, where A is skew-symmetric and B is symmetric and non-negative definite. The corresponding semigroup is the semigroup of contractions in Euclidean Rn . 2) More generally, consider a pseudo-Euclidean space Rp;q and consider the semigroup of indefinite contractions. Now G D GL.p C q; R/, g D gl.p C q; R/, and m D o.p; q/. 3) Let C be a (sharp) G-invariant cone in a real Lie algebra g. Then g ˚ iC is a Lie wedge in the complexification gC WD g ˚ i g. 5. (Global) Lie semigroups. Let us return to a general picture (Subsection 3). Consider the set ? consisting of all possible products g1 g2 : : : gk , where gj 2 . Certainly, ? is a subsemigroup in G. However, generally ? \ B is larger than . If ? \ B D ; we say that C is the Lie cone of ? . Example. Consider the group R=Z. Its Lie algebra is R. The positive half-line is a cone in R. But the corresponding local semigroup generates the whole group R=Z. Problem 5. The Lie algebra sl.2; R/ contains a convex SL.2; R/-invariant cone. Show that the corresponding semigroup B is the whole SL.2; R/. The picture changes if we pass to the universal covering SL.2; R/ . Problem 6. Show that the convex cone in sl.2; R/ generates a global Lie subsemigroup in SL.2; R/ . Hint. See Figure 3.2. Theorem 8 (Vinberg). The cone PosSp in sp.2n; R/ is a Lie cone of a Lie subsemigroup in the universal covering group of Sp.2n; R/. The proof is given in Subsection 16. 6. The main zoo. Olshanski semigroups. As we noted in Subsection 2.12.7, our characters Bp;q D U.p; q/=U.p/ U.q/;

Bn D Sp.2n; R/=U.n/

are representatives of a wider zoo of matrix balls G=K (for tables, see Subsection D.1). These matrix balls are open domains in certain Grassmannians and the group G B of symmetries of the Grassmannian is wider than G. In all these cases there is a semigroup of maps 2 G B sending the matrix ball to itself. On the level of Lie algebras we have the pair gB g and the orthocomplement p to g in gB . There is also a convex G-invariant cone C p and the Lie wedge g ˚ C gB :

214


Problem 7. Consider the matrix ball GL.n; R/=O.n/ consisting of real symmetric matrices with norm < 1. Remark. For G D U.p; q/, Sp.2n; R/ the group G B D GL.p C q; C/, Sp.2n; C/ is the complexification GC of G, gB D g ˚ i g. Thus p D ig, and the cone C is contained in i g ' g. These phenomena are related to the existence of a Hermitian metric on G=K. Generally, this is not the case (if G D GL.n; R/, then G B D Sp.2n; R/). For list of Olshanski semigroups, see Addendum D. 7. Digression. The most interesting animal in the zoo. Next, we propose an example from another branch of mathematics. Denote by Diff the group of all orientation preserving diffeomorphisms of the circle. Denote by Vect the Lie algebra of vector fields on the circle. Observation 9. The set C of vector fields a.'/

@ ; @'

where a.'/ > 0;

is a convex Diff-invariant cone in Vect. One can easily construct the corresponding semigroup in the universal covering group Diff (but without exact wording; in the infinite-dimensional situation there is no formal Lie group – Lie algebra correspondence). Indeed, we realize Diff as the group of diffeomorphisms of R satisfying the condition q.x C 2/ D q.x/ C 2: Then the corresponding semigroup consists of diffeomorphisms satisfying the condition q.x/ > x. However, the object that is really interesting is the cone Vect ˚ iC VectC ; which produces an existing subsemigroup in the nonexisting group Diff C and also produces a conformal field theory from a classical mathematical standpoint. . Causal structures (orders) on homogeneous spaces 8. Wedges and causal structures (orders) on homogeneous spaces. Consider a Lie wedge in a Lie algebra g as in Subsection 3, let m be its edge, C the convex cone in g=m. Consider the homogeneous space G=M . The tangent space at the initial point of G=M is g=m. By invariance, we get a convex cone at any tangent space to G=M . A curve .t/ is said to be causal if for each t the velocity vector 0 .t / is contained in the corresponding cone at .t /. We define local causal structure as local partial order16 on G=M given by: x2 > x2 if there exists a causal curve .t / such that .t1 / D x1 , .t2 / D x2 , t2 > t1 . We say that a causal structure is global if this partial order is well defined on the whole manifold G=M . Precisely, extending the partial order to the whole manifold by transitivity, we get the same local order. 16

I.e., we define the relation x1 > x2 only on sufficiently small sets.


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Example. The circle R=Z has an obvious local causal structure but it is not global. 9. Initial examples of causal structures. a) The one-sheeted hyperboloid in R3 has two (essentially) different local17 causal structures. One of them determines a global partial ordering of the hyperboloid. For another one, there are closed causal curves, see Figures 3.10 and 3.11.

Figure 3.10. Two local causal structures on the one-sheeted hyperboloid.

Figure 3.11. The one-sheeted hyperboloid H can be regarded as the torus without diagonal. Indeed, denote by S the absolute (the circle at infinity) of H . For any point x 2 H , consider two generatrices passing through x and their intersections with S. Thus we get a map H ! S S . a) The torus without the diagonal and generatrices. b) We can change coordinates on the torus and put the diagonal to a parallel circle. Cutting the torus along the diagonal we get the cylinder. c) The universal cover over the cylinder is a strip. We also draw a domain x > a for the “nonglobal" order on the initial hyperboloid. b) For the universal covering space of the one-sheeted hyperboloid, both the causal structures determine global orders, see Figure 3.11. c) The hyperboloid x02 C x12 C x22 C x32 D 1 in R4 has a unique (up to the reversion) causal structure. It determines a global order. d) Consider the space-time R1;3 with indefinite scalar product h; i. There is a global partial order on R1;3 , namely, x>y

if hx y; x yi 6 0

and x0 > y0 :

This order is invariant with respect to shifts and with respect to the group SO0 .1; 3/. The common term for causal curves is a “time-like curve”. 17

That are not reduced one to another by the reversion of the order.

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10. Example. The conformal compactification of space-time. We add the indefinite inversion x 7! x=hx; xi to the group of symmetries of R1;3 , this transformation being conformal with respect to the pseudo-Riemannian metric dx02 C dx12 C dx22 C dx32 . In particular, it regards the isotropic cones in the tangent spaces. The indefinite inversion and the Poincaré group together generate the group O.2; 4/ of all conformal transformations of R1;3 . To make conformal transformations continuous, we complete R1;3 (i.e., add some points at infinity) in a natural way. Observation 10. There is a natural open dense embedding of R1;3 to the space M of isotropic lines in R2;4 Indeed, consider the two-dimensional pseudo-Euclidean space V with inner product 12 01 10 . To any point x 2 R1;3 we assign the isotropic vector .1; x02 x12 x22 x32 I x/ 2 V ˚ R1;3 ' R2;4 : Problem 8. Identify M with S 3 S 1 =Z2 . This space M inherits the local SO0 .2; 4/-invariant causal structure. In M, the time axis turns to a circle; therefore the order is not global. Problem 9. The universal covering of M admits a global partial order. 11. The basic example: the causal structure on Hermn . We define a natural (global) partial order on the space Hermn of Hermitian matrices by the rule A > B;

if A B is non-negative definite:

We have at least three points of view about this structure. a) The group GL.n; C/ acts on our space A 7! gAg . The additive group of Hermn acts on our space by additions. Thus, we get the action of the semidirect product GL.n; C/ Ë Hermn . Evidently, our order is invariant with respect to this group. b) Let us reduce the group of symmetries. Orbits of GL.n; C/ on Hermn are (symmetric spaces) GL.n; C/=U.p; n p/; these orbits inherit the causal structure. c) Extension of the group of symmetries. Consider the “inversion” A 7! A1 . Observation 11. If A is invertible, then for B being in a small neighborhood of A, A > B implies A1 > B 1 : Proof. The differential of our map is dA 7! A1 dA A1 . The condition det.dA/ D 0 determines a conic surface CA in the tangent space TA to Hermn . Evidently, the inversion sends the surface CA to CA1 . The surface CA separates the tangent space into n C 1 pieces according to the inertia of dA. Evidently, the differential regards this separation. It remains to watch the image of the positive piece. Without loss of generality we can take a diagonal matrix A. Next, we take a small diagonal matrix A > 0; then A C A > A. Evidently .A C A/1 C A1 > 0. This completes the proof.


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Problem 10. The transformation A 7! A1 does not preserve the global ordering on Hermn . The transformation group generated by GL.n; C/ Ë Hermn , and the inversion is the group U.n; n/. Thus: Observation 12. The group U.n; n/ acts on Hermn by discontinuous transformations preserving the local causal structure. Observation 13. The group U .n; n/ acts on U .n/ preserving the causal structure. We realize the group U .n; n/ as in Subsection 7.8. Let a b ; ; 2 U .p; q/; zQ D .z; / 2 U .n/: gQ D c d We define the action of U .n; n/ on U .n/ by gQ yQ D .a C zc/1 .b C zd /;

C Im tr ln.1 C zca1 / C Im tr ln.1 C bd 1 z 1 / :

Remark. The origin of the expression on the right-hand side of the last equation is the following identity: det.z/ det.d / det.1 C bd 1 z 1 / : det .a C zc/1 .b C zd / D det.a/ det.1 C zca1 / We write Im ln for the right-hand side: X X Y Im ln det. / D Im ln det. / D Im tr ln. /: The expressions ln.1 C bd 1 z 1 /, ln.1 C zca1 / are well defined, and we get the remaining summands from the identities det z D e i ;

det a D e i ; j det aj

det d D e i : j det d j

12. Another point of view. The local causal structure on U.n/. Now we consider the Cayley transform and pass to the unitary group. Let z1 , z2 be unitary matrices. We write the equation det.z1 e i' z2 / D 0; and let 0 6 '1 < 2;

:::;

0 6 'n < 2

.1/

be its roots. In these terms, the U.n; n/-invariant local order on U.n/ is z1 > z2

if

j

> 0 for all j .

13. Global partial order on the universal covering of U.n/. We realize the group U.n/ as above, i.e., as the set of pairs .z; / 2 U.n/ R, such that det z D e i .

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Let us describe the analog of the structure (1) in this case. For two pairs .z1 ; 1 /, .z2 ; 2 / we consider the roots '1 , …,'n of the equation det.z1 e i' z2 / D 0 and WD 1 2 . Now D '1 C C 'n and the collection '1 ; : : : ; 'n is defined up to the equivalence .'1 ; : : : ; 'n / .'1 C 2k1 ; : : : ; 'n C 2kn /;

where kj 2 Z,

X

kj D 0:

Now we define the global order by .z1 ; 1 / > .z2 ; 2 /

if we can make 'j > 0 simultaneously.

This order is invariant with respect to the group U.n/ U.n/ . 14. Causal structure on the Lagrangian Grassmannian. We have identified the real Lagrangian Grassmannian Lagr n .R/ D U.n/=O.n/ with the set of unitary symmetric matrices (see Theorem 3.6). Consider the identical embedding z 7! z of U.n/=O.n/ to U.n/. This produces a local Sp.2n; R/-invariant causal structure on the Lagrangian Grassmannian; also this produces a global U.n/ -invariant causal structure on the universal covering space U.n/=O.n/ . This structure also is Sp.2n; R/ -invariant. Problem 11. Invent a causal structure on U.2n/=Sp.n/ (this space can be realized as the space of skew-symmetric unitary matrices). Find the action of SO .2n/ on this space. 15. Induction of causal structures. Examples. Now we can embed various homogeneous spaces in U.n/ and induce causal structures. We present some examples a) In Section 3.5 we considered the wedge of symplectic dissipative matrices sp.2n; R/. We take g D sp.2n; C/ D sp.2n; R/ ˚ i sp.2n; R/, m D sp.2n; R/. By our arguments, it must determine a causal structure on the space Sp.2n; C/=Sp.2n; R/. We realize this space in the following way. Fix a skew-symmetric bilinear form ƒ on C2n . We realize the space Sp.2n; C/=Sp.2n; R/ as the space of all Hermitian forms M compatible with ƒ in the sense of §3.2. We assume M > M 0 if M M 0 is non-negative definite. The inertia of a form M is .n; n/. In other words, we get an embedding Sp.2n; C/=Sp.2n; R/ ! GL.2n; C/=U.n; n/; and consider the pull-back of the causal structure constructed in Subsection 11.b.

b) Consider the symmetric space Sp.2n; R/=GL.n; R/. It can be realized as a space of pairs of transversal Lagrangian subspaces in R2n . Therefore Sp.2n; R/=GL.n; R/ is an open dense subset in Lagr n .R/ Lagr n .R/. In each factor Lagr n .R/, we can reverse the order. Therefore, we get four local partial orders on Sp.2n; R/=GL.n; R/. Thus we have two essentially different causal structures on the space Sp.2n; R/=GL.n; R/. The reader can find numerous examples of open embeddings of symmetric spaces to compact causal Grassmannians U.n/; U.n/=O.n/; U.2n/=Sp.n/ in Tables 5, 6 in Addendum D.


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16. The causal structure on Sp.2 n; R/. We have a convex invariant cone in the Lie algebra sp.2n; R/, its translations produce a local causal structure on the Lie group Sp.2n; R/. Let us explain it in other terms. The graph of a symplectic operator is a Lagrangian subspace. Thus we get an open embedding Sp.2n; R/ ! Lagr 4n .R/ and the induced local causal structure on Sp.2n; R/. Observation 14. The map Sp.2n; R/ ! Lag4n .R/ can be lifted to an embedding of covers Sp.2n; R/ ! Lag4n .R/ . If we regard Lagr 4n .R/ as a subset in U.2n/, then Sp.2n; R/ is the set of .n C n/ .n C n/matrices K L SD 2 U.2n/; where S D S t , L is invertible Lt M (this topic is discussed in detail in Chapter 5). Next, consider the loop (7.2) generating the fundamental group of Sp.2n; R/. Its image in U.2n/ is the loop 0 G.'/ : G.'/ 0 Evidently, this loop winds off in the universal covering of U.2n/. Therefore the map Sp.2n; R/ 7! U.2n/ is an embedding. Thus, we get a global causal structure on Sp.2n; R/ . This also proves Theorem 8. The subsemigroup in Sp.2n; R/ consists of g > 1.

17. Convex cones and local Lie semigroups. The completion of the discussion. In spite of the long discussion, the list of natural objects and structures that appear in this context is observable. Basically, there are: A) Contractive semigroups of matrix balls (Olshanski semigroups). We have G GB;

.2/

where the subgroup G is the group of automorphisms of the matrix ball, and the overgroup G B is the group of automorphisms of the corresponding Grassmannian. The space gB =g contains a G-invariant convex cone C . B) Causal structures on pseudo-Riemannian symmetric spaces. The list of such objects is the duplicated list of Olshanski semigroups. Namely, we start with an Olshanski semigroup and produce a locally causal symmetric space in the following two ways: – We consider the spaces G B =G and the G-invariant cones in gB =g. – Consider the Lie algebras gB g, and the orthocomplement r to g in gB . We consider the new Lie algebra gO WD g ˚ i r. Then the space G O =G is locally causal. Causal structures on symmetric spaces admit a simple uniform description, see Subsection D.5. Also there is a “cloud” of variations of these constructions, namely some exceptional constructions for groups of small ranks; variations of notion “dissipative” (intermediate cones); coverings, quotients, degenerations, etc.

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Chapter 3. Linear symplectic geometry ı. Total positivity

18. Classical total positivity. Above we handled semigroups of pseudo-Euclidean and symplectic contractions. Classical groups contain other interesting subsemigroups. For instance, all matrices with positive elements form a semigroup. This object is rather important in probability; here we discuss a more refined example. An invertible square matrix is said to be totally positive if all its minors are non-negative. Observation 15. The product of totally positive matrices is totally positive. Theorem 16. The semigroup of totally positive matrices is generated following type (their elements are positive): 1 0 1 0 0 1 0 0 0 ::: u1 0 1 t1 0 0 : : : Bs1 1 0 0 : : :C B 0 u2 B0 1 t2 0 : : :C C B C B B B C B B0 0 1 t3 : : :C 0 C ; B 0 s2 1 0 : : :C ; B 0 B B 0 0 s3 1 : : :C B0 B0 0 0 1 : : :C 0 A @ A @ @ :: : : :: : : :: :: :: :: :: :: :: :: : : : : : : : : : : : :

by the matrices of the 0 0 u3 0 :: :

0 0 0 u4 :: :

1 ::: : : :C C : : :C C : .3/ : : :C A :: :

The set of all totally positive matrices has a non-empty interior. However, the tangent cone at 1 has dimension n C .n 1/ C .n 1/ and consists of 3-diagonal matrices. Problem 12. The semigroup of totally positive matrices is not generated by its tangent cone. 19. Symplectic total positivity. Now consider the symplectic linear space endowed with a basis e1 ; : : : ; e2n such that fei ; ej g D .1/i ıiCj;2nC1 . Define totally positive symplectic matrices as matrices g 2 Sp.2n; R/ that are totally positive in the usual sense. Theorem 17. The semigroup of totally positive symplectic matrices is generated by the symplectic matrices of the form (3).

3.12 Bibliographical remarks to Chapters 2 and 3 La vérité, ce n’est point ce qui se démontre, c’est ce qui simplifie. ............................................ Mais la vérité, vous le savez, c’est ce qui simplifie et non ce qui crée le chaos. Antoine de Saint-Exupéry, Terre des hommes 13.1 The unpleasant problem of generality. The following words were written in 1975: It is not easy to get into a representation theory,..., for a number of reasons. First, the general theorems on higher dimensional groups require massive doses of Lie theory. Second, one needs a good background in standard and also in not so standard analysis on a fairly broad scale. Third, the experts have been writing for each other for so long a time that the literature is somewhat labyrinthine. S. Lang, SL2 .R/

3.12. Bibliographical remarks to Chapters 2 and 3

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Many well-written books were published during the last 30 years and ... the situation became essentially worse. A structure and nature of the crisis is not clear and the author avoids analyzing this; however one of elements of the crisis is a too perfect language making the subject unobservable from the outside. All the classical groups (see Addendum A) are immediate relatives; various constructions usually (not always18 ) can be relatively easily transferred from one series of groups to another. For instance, the classical books by H. Weyl [216], I. M. Gelfand, M. A. Naimark [63], and D. P. Zhelobenko [220] are written from this point of view. All the phenomena discussed in Chapters 2–3 can be easily extended to arbitrary classical groups. Some initial data for such transferring (tables of matrix balls and symmetric spaces, representations of G as intersections of two overgroups, etc.) are contained in Addendum D to this book; a table of matrix wedges is contained in [148]. But if we want “general theorems”, then we can not avoid some repetitions. Also, an examination of one series G.n/ of classical groups quite often involves other series. As we have seen, U.n; n/ and Sp.2n; C/ are necessary for understanding of Sp.2n; R/); more generally, symmetric subgroups and symmetric overgroups of G.n/ usually are involved in considerations, see Problem 3.3.6. A generally accepted way to avoid such repetitions is to treat real classical groups as a particular case of semisimple groups (as in [213], [108], [73], [49]). The latter class includes 10 series of classical groups and 23 exceptional groups in dimensions 6 248 2 (basic facts of the theory of semisimple Lie algebras and their finite dimensional representations were obtained by W. K. J. Killing and E. Cartan in 1888–1910 modulo numerous aesthetic additions invented later). However it seems that this approach is one of the reasons of the crisis which I mentioned in the preface. Rather often simple (and even semi-trivial) facts became non-understandable for non-experts due to the well-developed theory and a perfect language. Further, numerous properties of classical groups can not be extended to exceptional groups or can be extended only partially19 . For this reason, we obtain a purer picture.20 A third possible way was outlined in the book by J. A. Dieudonné [43], who proposed an abstract definition of a classical group over an arbitrary field and developed foundations of an abstract theory. Nevertheless, a successive development of this approach met aesthetic difficulties, because classical groups preserve their individuality; this necessitated numerous insertions of a case-by-case analysis. Also, the existence of individualities rather often produces similar difficulties in “the unified semi-simple approach”. 18 Complex groups are simpler than real groups; on the other hand, the series GL are simpler than other series. 19 Some examples: categories of linear relations, inverse limits, the Howe duality, the Schur–Weyl–Brauer duality, the Gelfand–Tsetlin bases, the Berezin kernels, the Stein–Sahi kernels, relations of representation theory with asymptotic combinatorics. In fact, structures of subgroups in classical and exceptional groups are similar, but structures of overgroups are seriously different. Some important examples of the unification are the Cartan highest weight theorem, the Young–Weyl character formula, the Gindikin–Karpelevich formula, the Gelfand–Naimark and Harish-Chandra parabolic induction, Study–Semple–Satake–De Concini–Procesi–Oshima–Sekiguchi boundary, Langlands classification, triangle inequality. 20 On the other hand, the exceptional groups are strong individualities and experts in finite sporadic groups know and explore numerous nonclassical properties of these objects (see e.g., [4], [33], [75], [112]). Unfortunately, it is very difficult to expose the beauty of such objects.

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In any case, all the approaches have their own difficulties. In particular, I understand the negative sides of the point of view proposed in this book.21 С началом Крымской войны отец был вновь призван на военную службу и определён во вторую лёгкую батарею 13 артиллерийской бригады, на вакансию, оставшуюся свободной после Л. Н. Толстого, переведённого в другую бригаду. Л. Н. Толстой уже тогда хотел извести в батарее матерную ругань и увещевал солдат: Ну к чему такие слова говорить, ведь ты этого не делал, что говоришь, просто, значит бессмыслицу говоришь, ну скажи, например, “ёлки тебе палки”, “эх ты ендондер, пуп”, “эх ты, ерфиндер”, и т.п. Солдаты поняли это по-своему: – Вот бьл у нас офицер, его сиятельство граф Толстой, вот уж матершинник был, так загибает, что и не выговоришь. акад. А. Н. Крылов “Мои воспоминания” However, a reform is necessary. In the opposite case, the literature on real semisimple Lie groups will be a collection of “Hittite texts”. It is the author’s opinion that this is sad. However, an opposite thinking now is more usual. On the level of special functions there exists an effective unified approach of I. G. Macdonald [128], [129], G. Heckman–E. Opdam [85], [84], and I. Cherednik [36]; however unifying objects of Cherednik are far beyond semisimple groups. 13.2 Canonical forms in the linear algebra. The Jordan normal form was discovered by C. Guderman; pairs of non-degenerate complex quadratic forms were classified by his student K. Weierstrass, 1867. There is a long list of slightly more complicated problems, some of them have been discussed in §§ 2.6, 3.4. Pairs of operators A; B W Cn ! Cm were classified by L. Kronecker, 1890; he also classified pairs of degenerate complex quadratic forms, see [60], XII.6; for pairs of degenerate skewsymmetric forms, see G. B. Gurevich, [79]. T. Matsuki [131]–[130] considered general problems of classification of double cosets H1 n G=H2 , where H1 , H2 are symmetric subgroups in a semisimple Lie group, and also H n G=P , where H is a symmetric subgroup and P is a parabolic. However his results do not contain a complete classification. Arbitrary quadruples of linear subspaces in a given space were classified by I. M. Gelfand, V. A. Ponomarev, [65], this work generated a wide literature. 21

In a well-known story about Leo Tolstoi (quoted below from the book “My recollections” by the famous engineer-shipbuilder A. N. Krylov) who, while serving as an army officer, tried to persuade the soldiers to avoid using obscenities. To that end he invented some new, meaningless words and suggested the soldiers use them instead when they can not restrain themselves. However, the only result was that long after he left the battery, he was still remembered as one who loved to use some particularly intricate obsceneties... Sorry, a complete translation is far beyond my abilities. I am afraid that a perfect English translation is impossible because of some distinctions in the structure of Russian and English (more precisely, a translation must be accompanied by a deep philological treatise.). However, the author believes that translation problems between different branches of mathematics are artificial and do not require “treatises” of such sorts.


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13.3 Matrix balls and matrix wedges were introduced in the 1930s by E. Cartan and C. Siegel (however, geometry of matrix spaces was discussed in various contexts earlier, for instance by G. Voronoy). As far as I know, matrix expressions for complex distance appeared in their works; however angles were known to C. Jordan. The matrix double ratio was introduced by Hua Loo Keng [96]. 13.4 Topology. In Subsection 2.4.4, we follow E. B. Dynkin [45]. More on the topology of Lie groups and compact symmetric spaces, see in [24], [56], [82], [167]. For more fresh results and points of view, see e.g. [37], [58]. 13.5 The harmonic analysis on symmetric spaces. A standard reference for the L2 -analysis is S. Helgason [86]. For L2 -analysis on pseudo-Riemannian spaces and discrete series, see M. Flensted-Jensen [50]. For Theorem 12.5, see [49], see also Chapter 7. For the analysis of the Berezin kernels, see [148]–[150]. On the Stein–Sahi kernels, see [156]. 13.6 Indefinite contractions. Investigation of Hilbert spaces with indefinite inner products was initiated by L. S. Pontrjagin in 1940s (see the book by T. Ya. Azizov and I. S. Iokhvidov [9] on this topic). Apparently, indefinite contractions were introduced by M. G. Krein [114] (maybe by M. V. Keldysh). Lemma 7.7 was obtained by V. P. Potapov [175]. Symplectic contractions were considered by M. G. Krein andYu. L. Shmul’yan in the 1960s. M. I. Graev [74] explored semigroups of indefinite contractions for evaluation of characters of Harish-Chandra holomorphic discrete series. Contractive semigroups in a wider generality were discussed by T. Nagano [138] (1965), S. M. Paneitz [169], G. I. Olshanski [161]. The classification of invariant convex cones in semisimple Lie algebras was obtained by S. M. Paneitz and G. I. Olshanski, see [161]. Causality in homogeneous spaces was introduced by I. Segal [195]. The classification of causal structures on pseudo-Riemannian symmetric spaces was done in [162]; Theorem 7 was announced in [162], for a proof see, e.g., [88]. For Theorem 8, see [208]. An analog of Olshanski semigroups for the Virasoro algebra was obtained in [141]. 13.7 Total positivity. Apparently, total positivity (in GL.n; R/) was defined by F. R. Gantmacher and M. G. Krein in 1937, see an introduction to the topic in Gantmacher’s book [60] and a nice exposition in T. Ando [6]. Total positivity in split real forms of semisimple groups was defined by G. Lusztig [126], in particular subsemigroups of totally positive matrices exist in groups Sp.2n; R/, O.n; n/, O.n C 1; n/; for further results, see the paper by S. Fomin and A. Zelevinsky [53]. Theorem 17 was obtained by the author, proof is not published. Apparently similar simple characterizations have also the subsemigroups in O.n; n/, O.nC1; n/, however they must include Pfaffians of Potapov transforms (see [145], Chapter 2). 13.8 Systems of coordinates on spaces of matrices. Several such systems were discussed above, the known zoo is rather wide. For instance, the well-known Gauss algorithm for a solution of a system of linear equations can be easily interpreted (this is an exercise) as an explicit cell decomposition of GL.n/ with an explicit coordinate system on each cell. The result is called Bruhat cell decomposition (see Problem 10.5.4). Several other examples of coordinates on various matrix spaces (usually, they are called “decompositions”) can be found in books on semisimple Lie groups [95], [220], [108]. An essentially different way of coordinatization is to use the “Jacobi elliptic coordinates”. For a real symmetric matrix A, consider its upper left j j -blocks for all j . Consider all

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eigenvalues of all such blocks (the total number of eigenvalues equals the dimension of the matrix space...). For an application of this point of view, see calculation of spherical functions of GL.n; C/ in the book by I. M. Gelfand and M. A. Naimark [63] (apparently, it is the most complicated way for such calculation, but it is interesting), for further results, see [160], [154]. 13.9 Categories of linear relations. The Krein–Shmul’yan category was defined by M. G. Krein and Yu. L. Shmul’yan [115]. This category and other categories of linear relations were introduced into representation theory by the author (see [145]). 13.10 Inverse limits. The map Ъ W SnC1 ! Sn and Ewens measures appeared in population genetics (in examination of distribution of alleles, W. J. Ewens) in the 1970s. For the harmonic analysis on inverse limits of symmetric groups (and also for references), see [105]. For Theorem 10.12, see [104]. As far as I know, the operator expression ./ D a C b.1 d /c appeared in the works of M. S. Livshits, see [124], [125] (later it was rediscovered several times, e.g., by J. W. Helton and also by C. Foias and B. Sz.-Nagy). Projective systems of measures on spaces of Hermitian matrices were present in an implicit form in the book by Hua Loo Keng [95]; this book is a nice collection of horrible calculations and (numerous) readers usually read final formulas, calculations seem incredible. Many years later the inverse limits of complex Grassmannians were discovered by D. Pickrell [172]. Construction 2.10.8 and the measures (2.10.10) were obtained in [153] (this work was partially based on unpublished notes of G. I. Olshanski, see also H. Shimomura [202]); similar constructions exist for all the 10 series of compact classical symmetric spaces. Now there exists a well-developed harmonic analysis on the inverse limits of U.n/ including the Plancherel formula, see A. Borodin, G. Olshanski [26]. 13.11 Geometry of symmetric spaces. For basic facts, see (non-specific) books [5], [21]. 13.12 Operator colligations and non-contractive matrix functions. The proof of the theorem of V. P. Potapov is contained in his paper [175], we do not know of any re-exposition. For the case of contractions in the Euclidean sense, see the book by I. Gohberg, S. Goldberg, M. A. Kaashoek [70], Chapter XXVIII. On extensions and other links in the operator theory and the function theory, see the book by Nikolskii [159] and the survey by Yu. P. Ginzburg, L. V. Shevchuk, see [69]. Operator colligation in the form exposed above appeared in a note of V. M. Brodsky, 1971 (with a reference to a non-published work of M. G. Krein). They are also a tool of representation theory, see G. I. Olshanski [162], Yu. A. Neretin [145]. There is a “big business” related to matrix-valued rational functions, see [116] 13.13 The triangle inequality. The first (and serious) attempt to solve “Horn problem” was done by V. B. Lidskii in 1950. The next big attack was done by F. A. Berezin and I. M. Gelfand in 1956. After Alfred Horn’s paper [91] 1962, the problem came to a position of “near complete solution” and was in this “stable” location for a long time (during some time intervals it was considered as solved). In 1995 U. Helmcke and J. Rosenthal [87] proved all necessary inequalities. It is interesting that arguments used in their proof (but not the proof) were known in 1955; and the final result was obtained by A. Klyachko in 1996. After this, numerous authors solved similar problems in numerous contexts; now the problem is solved for all non-compact and compact Riemannian symmetric spaces and all buildings; see the survey by W. Fulton [58], 2000 (now it is “old”), see also a nice geometric approach for non-compact symmetric spaces in M. Kapovich, B. Leeb, J. J. Millson, [101]. However now all


225

solutions seem complicated (or use complicated tools), and there is the problem of reasonable exposition. In this book, I followed the point of view of V. I. Danilov, G. A. Koshevoy, [42]. However, their proof also contains technically intricate pieces. 13.14 Central extensions. The existence of a covering of Sp.2n; R/ is a trivial fact of the general theory of Lie groups (see Subsection C.7), however an aesthetic formula for the cocycle is not obvious. The Berezin formula appeared in [14], 1965 (actually he wrote a T -valued cocycle), later it was rediscovered in [77]. Apparently, the construction with geodesic triangle must be attributed to J. L. Dupont, [44] (I am not completely sure, since it was rediscovered many times). On the Maslov index in the most common sense, see Hörmander [89], Vol. 4. For Subsection 3.10.1, see [98], but this is a simplified version of the construction of B. Kostant [111]. For Subsection 3.10.2 (also for non-triviality results), see [158].

4 The Segal–Bargmann transform

This chapter is a union of three subchapters. The principal topics are contained in §§ 4.1–4.2 (only these two sections are used below). In §§ 4.3–4.5 we briefly discuss the Segal–Bargmann transform as a tool of the time-frequency analysis and the microlocal analysis. Our purpose is only a demonstration of logical links. In the remaining §§ 4.6–4.9 we discuss some exotic inversion formulas.

4.1 Fock space 1.1 The Fock space Fn . Let z D .z1 ; : : : ; zn / 2 Cn . Let zk D xk C iyk . We denote by .z/ the Lebesgue measure on Cn normalized by Yn dxj dyj : d .z/ D n j D1

Consider the space Fn consisting of holomorphic functions f .z/ on Cn satisfying the condition Z 2 jf .z/j2 e jzj d .z/ < 1: (1.1) Cn

We define the inner product in Fn by Z 2 f .z/g.z/e jzj d .z/: hf; gi D

(1.2)

Cn

Theorem 1.1. a) The space Fn is a Hilbert space, i.e., it is complete with respect to the inner product (1.2). b) Monomials z1k1 : : : znkn form an orthogonal basis in Fn and kz1k1 : : : znkn k2 D k1 Š : : : kn Š : Proof. To simplify the notation, we consider the case n D 1. 1) Let us verify the orthogonality of the monomials, Z Z Z 1 2 1 k ik' m im' r 2 z k zN m d .z/ D r e r e e rdr d' 0 C 0 Z 1 Z 2 1 2 i.km/' D e d' r kCm e r dr 2 : 2 0 0

227

4.1. Fock space

If m ¤ k, then the first factor is 0. For m D k, we get Z 1 Z 1 2 r 2k e r dr 2 D k e d D kŠ : 0

0

2) Next, we are going to show that each function f .z/ 2 F1 orthogonal to all z k is 0 (and hence z k is a basis). P k Expand f 2 F1 into a Taylor series,1 f .z/ D 1 kD0 ck z . Consider its Fourier l n coefficient hf; z i. Since both the functions f and z are contained in L2 , this integral absolutely converges, Z Z X X 2 2 ck z k zN l e jzj d .z/ D lim ck z k zN l e jzj d .z/: hf; z l i D R!1 jzj6R

C

Since the Taylor series converges uniformly on disks jzj 6 R, we can interchange the summation and the integration, X Z 2 ck z k zN l e jzj d .z/ lim R!1

jzj6R

Z

D lim

R!1

0 C C 0 C cn

R

2l r 2

r e

dr C 0 C 2

0

(to evaluate integrals, we pass to the polar coordinates as above). Finally, hf; z l i D cl lŠ. Hence each function f orthogonal to the vectors z l is zero. 3) Denote by F1 the completion of F1 . Since z k are pairwise orthogonal, X X jak j2 kŠ < 1: ak z k 2 F1 () P But the last condition provides a convergence of the series ak z k for all z 2 C, that is, F1 D F1 . The space Fn is called boson Fock space with n degrees of freedom. Since the fermion Fock space is not discussed in this book, we say simply Fock space. 1.2 The reproducing property. Denote by .; / the standard inner product in Cn , P i.e., .u; v/ D uj vNj . For each u 2 Cn , define the function o nX (1.3) zk uN k : 'u .z/ D exp.z; u/ D exp Theorem 1.2. a) 'u 2 Fn and h'u ; 'v i D exp.v; u/:

(1.4)

b) For each f 2 Fn , hf; 'u i D f .u/ (reproducing property): 1

k

Certainly, this is also the expansion of f with respect to the basis z , but we must prove this.

(1.5)

228

Chapter 4. The Segal–Bargmann transform

Proof. For simplicity, set n D 1. Taking into account the orthogonality of z k , we obtain h'u ; 'v i D

D X uN k kŠ

zk ;

X vN k kŠ

E X zk D

1 .v u/ N k hz k ; z k i D exp.v; u/: .kŠ/2

Hence k'u k2 D exp.juj2 /, therefore 'u 2 F1 . Next, hf; 'u i D

DX

ck z k ;

E X c uk X k ck uk D f .u/: zk D hz k ; z k i D kŠ kŠ

X uN k

Actually a) is a special case of b).

Remark. We stress the fundamental equation (1.5). It shows that 'u is something similar to a delta-function. There are several terms for 'u , namely supercomplete system, overfilled basis, system of coherent states; for this formalism, see § 7.1. 1.3 Growth of functions f 2 Fn Proposition 1.3. a) For any f 2 Fn , 2 =2

jf .u/j 6 kf k e juj

:

(1.6)

b) If an entire function on Cn satisfies the condition 2 =2

jf .z/j 6 const .1 C jzj/n1 e jzj

;

(1.7)

then f 2 Fn : Proof. a) Indeed, f .u/ D hf; 'u i 6 kf k k'u k. b) immediately follows from the definition (1.1).

1.4 Operators. For a bounded operator A W Fn ! Fm ; we define its kernel (or Berezin symbol)2 by N D hA'v ; 'u iFm : KA .u; v/

(1.8)

Remark. The kernel is a function of the variables u 2 Cm and v 2 Cn . The symbol vN indicates that K is antiholomorphic with respect to the second variable. Also, K is holomorphic with respect to the first variable. 2

For a more general construction, see § 7.1.

229

4.1. Fock space

Remark. Consider an integral operator in a space of function on Rn , Z Af .x/ D K.x; y/f .y/ dy: Then we can formally write K.y; z/ D hhAız ; ıy ii D Aız .y/; where ıy is the delta function supported by y. We observe a formal analogy with (1.8) and the next formula (1.9). We can also represent the kernel in the forms KA .u; v/ N D A'v .u/ D A 'u .v/: Theorem 1.4. For each f 2 Fn , Af .u/ D

Z

(1.9)

2

K.u; v/f N .v/e jvj d .v/:

(1.10)

The integral on the right-hand side absolutely converges. Proof.

Af .u/ D hAf; 'u i D hf; A 'u i:

But

A 'u .v/ D hA 'u ; 'v i D h'u ; A'v i D KA .u; v/: N

Therefore, Af .u/ is the inner product of f D f .v/ and the function su .v/ WD KA .u; v/. N But this is precisely formula (1.10). Corollary 1.5. The kernel KA of a bounded operator A satisfies jK.u; v/j N 6 kAke juj

2 =2

e jvj

2 =2

:

(1.11)

Proof. By (1.8), K.u; v/ D hA'v ; 'u i 6 kAk k'u k k'v k:

Proposition 1.6. The kernel of an operator A is a generating function for its matrix coefficients; precisely KA .u; v/ N D

ˇ X u˛1 vN ˇm D Y ˇj Y ˛i E u˛nn vN 1 1 1 ::: ::: 1 A zj ; Zi ; (1.12) ˛n Š ˇ1 Š ˛1 Š ˇm Š

X

˛1 ;:::;˛m ˇ1 ;:::;ˇn

where

Q j

ˇ

Zj j and

Q

˛

i

j

zi i are the standard bases in Fn and Fm .

i

230


Proof. We expand the defining equation (1.8) into a series in u, v. N

Problem 1.1. a) Verify (1.10) using the last formula. b) Find kernels of the operators Af .z/ D f .z/, Bf .z/ D f .2=z/, Cf .z/ D f .0/. Remark. The last proposition allows us to define the kernels in a more general situation than our basic definition (1.8). Indeed, the formal series (1.12) makes sense for an arbitrary operator acting from the space polynomials in z1 ; : : : ; zn to the space of formal series in Z1 ; : : : ; Zm . 1.5 Products of operators Theorem 1.7. Let A W Fn ! Fm and B W Fm ! Fk be bounded operators. Then Z 2 KB .u; w/K x A .w; z/e N jwj d .w/: (1.13) KBA .u; zN / D Cm

In particular, the integral is absolutely convergent for all pairs u and z. Proof. By definition, KBA .u; zN / D hBA'z ; 'u i D hA'z ; B 'u i: By the reproducing property, A'z .w/ D hA'z ; 'w i D KA .w; z/;

B 'u .w/ D hB 'u ; 'w i D h'u ; B'w i D hB'w ; 'u i D KB .u; w/: x

(1.14) (1.15)

But the right-hand side of the desired formula (1.13) is the inner product of (1.14) and (1.15). Problem 1.2. Prove the formula for -product starting from Proposition 1.6. 1.6 Standard orthogonal decomposition of Fn . Denote by Sjn the subspace in Fn consisting of homogeneous polynomials of power j . Obviously, we get orthogonal decomposition 1 M Sjn : Fn D j D0

1.7 Operators of change of variables. Let L be an operator Cm ! Cn with norm 6 1 (we consider Cn and Cm as Euclidean spaces). Then we have a well-defined operator Fn ! Fm given by R.L/f .z/ D f .zL/: Obviously, R.L/Sjn Sjm : The kernel of R.L/ is

K.z; u/ N D expfzLu g:

231

4.1. Fock space

Problem 1.3. Show that kR.L/k 6 1 and the vector f .z/ D 1 is a maximizer. Find the spectrum of an operator R.L/ W Fn ! Fn . 1.8 Adjoint operators. The following statement is obvious. Proposition 1.8. KA .z; u/ N D KA .u; N z/: 1.9 Creation-annihilation operators. We define the creation operators aOj and the annihilation operators aOj in Fn by aOj f .z/ D zj f .z/I Obviously,

aOj D

@ f .z/: @zj

(1.16)

ŒaO k ; aO l D ıkl : P Remark. These operators are unbounded. Indeed, for f .z/ D ck z k 2 F1 , X X X kŠjck j2 ; kaf O k2 D kŠj.kC1/ck j2 ; kaO f k2 D .k1/Šjkck j2 : kf k2 D ŒaO k ; aO l D 0I

ŒaO k ; aO l D 0I

k>0

Proposition 1.9. The operator aj is adjoint to aj . Proof. It suffices to write the matrices of aj and al in the standard basis. However, the following formal calculation explains the phenomenon better: Z Z @ @ z zN z zN f .z/ g.z/e d .z/ D f .z/ g.z/e d .z/ @zj Cn @zj Cn Z @ z zN D f .z/g.z/ e d .z/ (1.17) @zj Cn Z D f .z/g.z/zj e z zN d .z/: Cn

Next, we formulate several simple remarks. Demote by KB the kernel of the operator B. Then Z 2 aOj Bf .z/ D zj K.z; u/f N .u/e juj d .u/; (1.18) n ZC @K.z; u/ N 2 aOj Bf .z/ D f .u/e juj d .u/: (1.19) n @z j C Integrating by parts we obtain Z @K.z; u/ N 2 B aOj f .z/ D f .u/e juj d .u/; n @ u N j ZC 2 B aOj f .z/ D uNj K.z; u/f N .u/e juj d .u/: Cn

(1.20) (1.21)

232


1.10 The action of the Heisenberg group. For each a 2 Cn , define the operator T .a/f .z/ D f .z C a/ expf.z; a/ .a; a/=2g:

(1.22)

Proposition 1.10. The operators T .a/ are unitary and satisfy the identity T .a/T .b/ D expfi Im.a; b/gT .a C b/:

(1.23)

Proof. Let us verify the first statement, Z ˚

hT .a/f; T .a/gi D f .z C a/g.z C a/ exp .z; a/ .a; z/ .a; a/ .z; z/ d.z/ Z N a/ N D f .z C a/g.z C a/e .zCa/.zC d.z C a/ D hf; gi: Next, T .a/T .b/f .z/ D T .b/f .z C b/ expf.z; b/ .b; b/=2g D f .z C a C b/ expf.z C a; b/ .b; b/=2g expf.z; a/ .a; a/=2g

D f .z C a C b/ expf.z; a C b/g expf .a; a/ .b; b/ 2.a; b/ =2g: Remark. Thus the proof is trivial, however we wish to comment on it. Consider the 2 2 measure e jzj d .z/ on Cn and the shifted measure e jzCaj d .z C a/. Denote by a .z/ the density of the latter measure with respect to the former. The operators T .a/ are unitary, because the correcting factor in (1.22) satisfies p j expf.z; a/ .a; a/=2gj D a .z/: The operators

p T .a/f .z/ D f .z C a/ a .z/ 2 are unitary in L2 Cn ; e jzj and satisfy T .a/T .b/ D T .a C b/. Nevertheless, they are not well defined in Fn , because the square root is not holomorphic; we have to choose a holomorphic function with a given absolute value. A price for our choice is an appearance of the additional factor in (1.23).

By (1.23), T .a/ is a projective unitary representation of the additive group of Cn . Also, we can treat T as a linear representation of a central extension of Cn . More precisely, consider the space Cn ˚ R equipped with the multiplication .a; x/ B .b; y/ D .a C b; x C y C Im.a; b//: The corresponding operators of linear representation are .a; x/ 7! e ix T .a/:

(1.24)

233

4.1. Fock space

This group is nothing but the Heisenberg group defined in Section 1.2. Our construction produces not only the same group, but also the same representation; the intertwining operator is the Segal–Bargmann transform discussed in the next section. Problem 1.4. Recall that aO k , aO k denote the creation-annihilation operators (1.16) Find expfaO k aO k g and expfi.aO k C aO k /g. Show that for any anti-self-adjoint linear combination of aO 1 , aO 1 , …, aO n , aO n , its exponential has the form (1.24). 1.11 Wick symbols. To simplify the notation, consider the case n D 1. Let X L.z; / D ckl z k l kl

be a formal series of two variables. Define the operator A W F1 ! F1 by

A D L z;

@ @z

WD

X

ckl z k

kl

@l : @z l

(1.25)

We say that L.z; / is the Wick symbol of the operator A. Problem 1.5. The operator X1 .1/k @k zk k j D0 kŠ @z is the projection onto the one-dimensional space consisting of constants. Proposition 1.11. The kernel (Berezin symbol) of the operator (1.25) is KA .z; v/ N D L.z; v/ N exp.z; v/: N Proof. By (1.9), it follows that the kernel of (1.25) is X kl

ckl z k

X @l z vN k l e e z vN D L.z; v/e D c z v N N z vN : kl @z l

kl

Thus Wick symbols can be reduced to kernels. On the other hand, the Wick symbols exist only for operators acting from Fn to itself. For this reason, we prefer kernels. Problem 1.6. For any polynomial f , n @ o f .zL/ D exp lij zi f .z/ @zj (but this is not the Wick form).

(1.26)

234


1.12 Weak convergence in Fn Proposition 1.12. For any fj 2 Fn , f 2 Fn , the following conditions are equivalent: A. fj weakly converges to f . B. The norms kfj k are uniformly bounded and fj .z/ converges to f .z/ for each z. C. The norms kfj k are uniformly bounded and fj .z/ converges to f .z/ uniformly on each ball jzj < R Proof. A () B. We apply the criterion of weak convergence (Theorem 1.1.1) to the system 'z . B H) C. Since the norms kfj k are uniformly bounded, then the family of functions fj .z/ is uniformly bounded on any ball jzj < R, see (1.6). But a point-wise convergence of a uniformly bounded sequence of holomorphic functions is the uniform convergence. Problem 1.7. For any f 2 Fn , ˝ ˛ f .n/ .a/ D f .z/; z n exp.z; a/ : 1.13 Weak convergence of operators Proposition 1.13. Let Aj , A be bounded operators in Fn . The following conditions are equivalent: A) A sequence Aj of operators weakly converges to A. N converges to K.z; u/ B) The norms of Aj are uniformly bounded and KAj .z; u/ pointwise. N converges to K.z; u/ C) The norms of Aj are uniformly bounded and KAj .z; u/ uniformly on each domain jzj < R1 , juj < R2 . Proof. We apply the criterion of weak operator convergence (Theorem 1.1.2) to the family of vectors 'z .

4.2 The Segal–Bargmann transform Here we discuss a natural unitary operator identifying L2 .Rn / and the Fock space. 2.1 The definition. For any f 2 L2 .Rn / we define the holomorphic function Bf .z/ (the Segal–Bargmann transform of f ) on Cn by Z p ˚

n=4 Bf .z/ D f .x/ exp z 2 =2 C 2zx x 2 =2 dx; (2.1) here z 2 WD

P

zj2 , xz WD

P

Rn

xj zj .

4.2. The Segal–Bargmann transform

235

2.2 Unitarity property. Set

p ˚

ˆz .x/ D n=4 exp zN 2 =2 C 2zx N x 2 =2 :

(2.2)

In this notation, Bf .z/ D hf; ˆz iL2 .Rn / : Theorem 2.1. hû ; ˆv i D exp.v; u/: Proof.

Z

p ˚

exp uN 2 =2 C 2ux N x 2 =2 Rn p ˚

exp v 2 =2 C 2vx x 2 =2 dx Z p

˚ n=2 D expfv ug N exp 12 .v C u/ N 2 C 2x.v C u/ N x 2 dx n ZR p 2 ˚ D n=2 expfv ug N exp x .v C u/= N 2 dx Rn Z D .2/n=2 expfv ug N expf 12 y 2 g dy D expfv ug: N

hû ; ˆv i D

n=2

(2.3)

Rn

Problem 2.1. a) Let A be a set. Let H and H 0 be Hilbert spaces. Let ˛ 2 H and ˛0 2 H 0 be total systems of vectors enumerated by points ˛ 2 A. Set h˛ ; ˇ i D h˛0 ; ˇ0 i

for all ˛, ˇ 2 A:

Then there exists a unique unitary operator U W H ! H 0 such that U ˛ D ˛0 . b) Show that the systems û 2 L2 .Rn / and 'u 2 Fn are total in the corresponding spaces. Theorem 2.2. a) The Segal–Bargmann transform B is a unitary operator operator L2 .Rn / ! Fn . b) B ˆz D 'z . Proof. Comparing (1.4) and (2.3), we get hû ; ˆv iL2 .Rn / D h'u ; 'v iFn : Thus there is a unitary operator U W L2 .Rn / ! Fn ; Next,

such that U û D 'u :

Uf .z/ D hUf; 'z iFn D hf; U 1 'z iL2 .Rn / D hf; ˆz iL2 .Rn / :

But the last expression is the Segal–Bargmann transform.

236


Problem 2.2. Let us consider the operators (1.2.7) of the Heisenberg group acting on L2 .Rn /. Find their pushforwards under B and identify them with operators (1.24). In fact, the Segal–Bargmann transform was originally introduced from this reasoning. 2.3 Differential operators. Obviously, we get the following correspondence of differential operators under the Segal–Bargmann transform: 8 8 @ @ 1 1 ˆ ˆ ˆ ˆ ; ; C ! ! x z x z p p j j j < < j @xj @zj 2 2 (2.4) ˆ ˆ @ @ 1 @ 1 @ ˆ ˆ : : ; : ! p xj C ! p zj C @zj @xj @xj @zj 2 2 For instance, the formula for @x@j follows from the differentiation by parts in (2.1), the formula for xj follows from the differentiation of the integral (2.1) with respect to parameter zj . Corollary 2.3. The map D 7! BDB 1 is a bijection of the ring of polynomial differential operators in L2 .Rn / onto the ring of polynomial differential operators in Fn . Next, B sends the function e x

2 =2

to 1.

Corollary 2.4. The Segal–Bargmann transform is a bijection of the space of functions 2 p.x/e x =2 , where p is a polynomial, onto the space of all polynomials on Cn . 2.4 The images of the Schwartz space under the Segal–Bargmann transform B. The Segal–Bargmann transform makes sense for any tempered distribution f 2 0 .Rn /. Denote by Fn the image of the Schwartz space .Rn / under the Segal–Bargmann transform. Denote by 0 F the B-image of the space 0 .Rn / of tempered distributions. P Proposition 2.5. a) An entire function f .z/ D c˛ z ˛ is contained in Fn if and only @ if L.z; @z /g 2 F1 for each polynomial differential operator L. b) The space Fn consists of entire functions satisfying the condition: Y p ˛jM ˛j Š < 1: (2.5) for each M , sup jc˛ j ˛

Proof. a) To simplify the notation, set n D 1. First, we characterize the Schwartz space. A function h 2 L2 .Rn / is contained in .R1 / if and only if for each poly@ / the distribution Dh is contained in L2 . nomial differential operator D D D.x; @x Corollary 2.3 translates this characterization to F1 . @ p f .z/ 2 F1 i.e., b) In particular, f 2 F1 must satisfy z @z X

jc˛ j2 ˛ 2p ˛Š < 1 for all p;

237

4.2. The Segal–Bargmann transform

the terms of the series must pass to 0, and this is the necessity of (2.5). Conversely, the condition (2.5) implies f 2 F1 , the condition (2.5) is also stable @ /; therefore (2.5) is sufficient. under an application of any operator L.z; @z P Proposition 2.6. a) The space 0 Fn consists of entire functions g.z/ D c˛ z ˛ ad@ mitting a representation g D L.z; @z /f , where f ranges in Fn and L is a polynomial differential operator. P b) The space 0 Fn consists of entire functions f .z/ D c˛ z ˛ satisfying Y p (2.6) ˛jM ˛j Š < 1 for some M : sup jc˛ j ˛

j

Proof. a) The space 0 .R1 / consists of functions that can be represented in the form @ D.x; @x /h, where h 2 L2 .R1 / and D is a polynomial differential operator. It remains to refer to Corollary 2.3. @ b) Obviously, functions f D L.z; @z /g, where g 2 F1 , satisfy (2.6). Further, for f satisfying (2.6), the function X c˛ fŒp WD z˛ .˛ C 1/p @ p is contained in F1 for sufficiently large p. On the other hand, .1 C z @z / fŒp D f ; therefore f 2 F1 .

Corollary 2.7. Both the spaces Fn and 0 Fn are invariant with respect to the operators of a change of variable f .z/ 7! f .zU / with unitary U . Proof. The characterizations of Fn and 0 Fn given in the statements a) of Propositions 2.5 and 2.6 regard the action of the unitary group U.n/. 2.5 The Mehler semigroup again. The relations (2.4) yield the following correspondence of operators under B:

1 @2 C x2 2 @x 2 Therefore,

²

³

! z ²

@ 1 C : @z 2

³

@ 1 t @2 exp C x2 ! exp t z : C 2 @x 2 @z 2 We have evaluated the left-hand side expression in Theorem 1.3.5. An evaluation of the exponential on the right-hand side is trivial, ²

@ 1 C exp t z @z 2 Thus we proved the following theorem:

³

f .z/ D e t=2 f .e t z/:

238


Theorem 2.8. a) The Mehler operators U.t / in L2 .Rn / (see (3.11)) correspond to the following operators in Fn : Uz .t/f .z/ D e t=2 f .e t z/:

(2.7)

b) In particular, the Fourier transform e i=4 U. i=2/ corresponds to the “rotation” f .z/ ! f .iz/. Problem 2.3. Deduce the Mehler theorem calculating B 1 Uz .t /B. 2.6 The Hermite polynomials. We define the Hermite functions Hn by n=2

Hn .x/ D 2 Obviously, Hn .x/ D e x

2 =2

n

@ x @x

e x

2 =2

:

˚

a polynomial of degree n :

Such polynomials are called the Hermite polynomials3 . Theorem 2.9. a) .BHn /.z/ D z n . b) The system Hn .x/ is an orthogonal basis in L2 .Rn / and kHn k2 D nŠ. @ @ Hn .x/ D nHn1 , 21=2 x @x Hn .x/ D HnC1 : c) 21=2 x C @x @ d) 21=2 xHn D nHn1 HnC1 and 21=2 @x Hn .x/ D nHn1 .x/ Hn .x/.

e) The Hermite polynomials are the eigenfunctions of the oscillator operator,

@2 1 1 2 C x 2 Hn .x/ D n C Hn .x/: 2 @x 2 p ˚ P 1 k f) uN Hk D û .x/ D exp uN 2 =2 C 2x uN x 2 =2g. kŠ g) The following Mehler bilinear formula holds: 1 X 1 .kC1=2/t Hk .x/Hk .y/ e kŠ

kD0

²

1 x Dp exp 2 2 sinh t 1

h) Bf .z/ D 3

P1

zk kD0 kŠ

R R

y

t cosh sinh t 1 sinh t

1 sinh t t cosh sinh t

! ³ x : y

(2.8)

f .x/Hk .x/ dx.

There are several normalizations of these polynomials. Our normalization is not the most common.

239

4.2. The Segal–Bargmann transform 2

Proof. a) The Segal–Bargmann transform sends the function ˆ0 D e x =2 to '0 D 1. Applying the operator aO n to 1 and keeping in mind the correspondence formula (2.4), we obtain our statement. b) corresponds to the orthogonality relations for z k . c) corresponds to aO z n D nz n1 , az O n D z nC1 and d) follows from c). @ n e) corresponds to z @z z D nz n : P 1 k k N . f) corresponds to e z uN D 1 kD0 kŠ z u z g) The operator U .t/ given by (2.7) can be represented as

Uy .t/f .z/ D

X 1 e .kC1=2/t hf; z k iz k : kŠ

Hence the kernel of the corresponding operator U.t / in L2 .Rn / is the left-hand side of (2.8). But the explicit expression for this kernel obtained in Theorem 1.3.5 gives the right-hand side of (2.8). P 1 h) is a reformulation of the expansion f D hf; z n iz n . nŠ P Problem 2.4. p Show that the Schwartz space .R/ consists of series ck Hk .x/, where max jck j k M kŠ < 1 for all M . 2.7 Inversion formula. I. Obviously, for X ck1 ;:::;kn z1k1 : : : znkn g.z/ D k1 ;:::;kn

we have

B 1 g.x/ D

X

ck1 ;:::;kn Hk1 .x1 / : : : Hkn .xn /

k1 ;:::;kn

(the series on the right-hand side converges in the L2 -sense). 2.8 Inversion formula. II. Since B is a unitary operator, it follows that B 1 D B . 2 Further, we can regard B as an isometric embedding L2 .Rn / ! L2 .Cn ; e jzj d .z//. 2 We automatically write the kernel of the adjoint operator L2 .Cn ; e jzj d .z// ! L2 .Rn /. Thus, Z p 2 1 g.z/ expfzN 2 =2 C 2zx N x 2 =2ge jzj d .z/: (2.9) B g.x/ D Cn

Setting x .z/ WD ˆz .x/; we write this formula as Bg.x/ D hg; x iFn :

(2.10)

240


Since x … Fn , the formula (2.9) – (2.10) is well defined on a dense subspace in Fn . The function a is the image of ı.x a/ under B, hence a 2 0 Fn . Therefore, the inversion formula (2.9) – (2.10) holds for at least g 2 Fn , see (2.5). 2.9 Inversion formula. III. Represent formula (2.1) as Z h p i 2 2 g.z/ D n=4 2n=2 e z =2 e yz e y =4 f .y= 2/ dy: Rn

Substituting z D i t and applying the inversion formula for the Fourier transform, we obtain Z 1 f .x/ D 3n=4 2n=2 e x

2 =2

g.i t /e i

p 2txt 2 =2

dt:

1

In the same way, we can write the inversion formula from any line z D a C i t . Apparently, many other inversion formulas can be written, one exotic formula is discussed in § 4.9.

4.3 Spectral analysis of signals Sections 4.3–4.5 form an independent subchapter; our purpose is to show links of our topics with the spectral analysis of signals, the microlocal analysis, and the machinery of symbols of operators. The material of this subchapter is not used in the remaining part of the book. 3.1 Recognition of signals. Statement of the problem. Given a signal .t / (a sound, a radio signal, etc.) that is a superposition of waves, .t / D

N X

˛j .t /e ij t :

(3.1)

j D1

We wish to reconstruct the frequencies j and the amplitudes ˛j .t / starting from .t /. One of the variants of this problem is to separate a noise and a useful signal. The problem in this form is not well posed; for instance e it C e it D 2 cos

. /t i.C/t=2 : e 2

This identity shows that it is impossible to distinguish completely frequencies and modulation. On the other hand (as it is explained in school textbooks) this phenomenon is basic for radio communication. 3.2 Some remarks on Fourier transform. The idea to apply the Fourier transform for separation of waves is based on simple and well-known arguments. We are going to refresh them for the reader because we must keep in mind proofs rather than statements.

241

4.3. Spectral analysis of signals

Theorem 3.1 (Riemann Lemma). a) Let f be a compactly supported continuous function on R. Then Z 1 Z 1 Z 1 f .t / sin t dt; f .t / cos t dt; f .t /e i t dt 1

1

1

tend to 0 as ! ˙1. b) If f has N continuous derivatives, then these integrals are o. N / as ! 1. c) If f .t/ > 0, then the maximum of ˇZ ˇ ˇ ˇ v. / WD ˇˇ f .t /e i t dt ˇˇ is achieved at D 0. Proof. a) See Figure 4.1, Z

1

f .t/ sin. t / dt D

XZ

1

XZ

.2kC2/=

f .t / sin. t / dt C

2k=

k

D

Z

.2kC1/=

f .t / sin. t / dt .2kC1/=

.2kC1/=

f .t / f .t C = / sin t dt:

2k=

k

We estimate this expression as ˇ ˇ X ˇ: : : ˇ 6 k

Z

.2kC1/= ˇ

2k=

ˇ ˇ ˇ ˇf .t / f .t C = /ˇ dt:

Since f is continuous and is large, it follows that the integrands are uniformly small.

2k

2.k/=

f .t /

f .t / Figure 4.1. Reference the proof of the Riemann Lemma.

b) We integrate by parts Z

1

f .t/e i t dt D 1

1 i

Z

1

f 0 .t /e i t dt D D .1/N

1

1 .i /N

By the statement a), it follows that each integral tends to 0 as ! 1. Also, recall what happens if f .t / has a singularity.

Z

1

f .N / .t /e i t dt: 1

242


Example. Let f be a function supported by a segment Œ0; M and smooth everywhere with the exception t D 0. Assume f has the following asymptotic expansion: f .t/ D at ˛ C bt ˇ C C zt ! C o.t N /;

where 1 < ˛ < ˇ <

(3.2)

in the right half-neighborhood of 0. Consider the function fQ.t / D f .t / at ˛ e t D f .t / at ˛ .1 1=t C /: It has a decomposition of the same type (3.2), but the term at ˛ is replaced by a

X k>1; Re ˛Ck6N

.1/k ˛Ck t : kŠ

Then we repeat the same operation over and over again. As a result, we come to a representation of the form X f .t / D Aj t rj e t C f B .t /; where Aj , rj are some explicit constants, the sum is finite, rj 6 N , and f B has N continuous derivatives at 0. New f B is not compactly supported but the conclusion of the Riemann Lemma holds. Further, Z 1 .r C 1/ t r e t e it dt D : .1 i /rC1 0 Therefore, we get Z 1 X .rj C 1/ f .t /e i t dt D Aj C O. /N ; where ! 1: .1 i /rj C1 0 The parameters ˛; ˇ; : : : , a; b; : : : can be reconstructed from Aj , rj . Thus the Fourier transform “sees” (or “hears”) the singularities of our function.

3.3 The Fourier transform and the problem of separation of waves. Evidently, the global Fourier transform is not a perfect tool for a solution of our problem. We illustrate this with some examples. Let D Ae i˛t C Be iˇ t . Then the Fourier transform O ./ is the distribution Aı. ˛/ C Bı. ˇ/; thus the harmonics of are observable. Now let amplitudes depend on time, for instance let D A.t /e i˛t C B.t /e iˇ t , where A.t / and B.t / are positive compactly supported functions (whose supports are long as compared with 1=˛, 1=ˇ). Then R j.t /j has R two peaks near ˛ and ˇ, the heights of these peaks are approximately A.t / dt, B.t / dt. However it is not clear how to restore A.t / and B.t/. See Figure 4.2. We explain this in another way. The Fourier transform represents a sound as a superposition Z .t / D

c.s/e ist dt

of globally existing waves e ist . But an actual signal (for instance, a sound, radio signal, radar-location, seismic oscillation, etc.) usually is a superposition of shortly living waves; in this context, the Fourier transform seems to be a violence.

243


Figure 4.2. Reference Subsection 3.3. Consider the function f .x/ D exp.x 2 =2 C 8ix/ C exp..x 3/2 =2 C 12ix/;

Re F (x): 0

3

Its Fourier transform is fO./ D exp.. 8/2 =2/ C exp.. 12/2 =2/ exp.3.12 /i /: We present graphs of Re fO./ and jfO./j:

8

12

8

12

Certainly, they have peaks near D 8 and D 12 and this indicates the presence of harmonics exp.8ix/ and exp.12ix/ in f .x/. However, one does not observe the intervals of the line x 2 R, where these harmonics are present. A height of a peak is ' the product of the intensity and the duration of a harmonic. Figure 4.3 Consider a function (a signal) .t / of the form .t / D A.t / exp.i !t / C B.t / exp.i t /; where A.t /, B.t/ are supported by segments Œa; b, Œc; d respectively; Re..t / is a sum of two summands of approximately the following form. .t/: a

c

b

d

Consider r.t/ of a form r.t / 2 It that jG.; /j is more-or less clear

has ridges along the following two segments in R : .a; !/; .b; !/ , .c; /; .d; / and we can reconstruct our signal. Now we know frequencies of the signals, their intervals, and intensities.

244


3.4 Short time Fourier transforms. Fix a function r.t / supported by a finite interval ŒA; A. For a function .t /, define a function of two variables (short time Fourier transform) Z 1 .t /r.t /e it dt; 2 R; 2 R: G .; / D 1

y ./ of the function For a given , the function Gr .; / is the Fourier transform ‰ ‰ .t / WD .t /r.t /: The function ‰ .t / is supported by the interval ŒA C ; A C ; the global Fourier transform allows us to observe harmonics and their total intensities for ‰ . Now we also “see” a given harmonic present on a given interval ŒA C ; A C ; also we see its total intensity on the interval. Example. New possibilities are explained in Figures 4.4–4.5. An idealized picture for jG.; /j is a topographic map with a family of ridges in longitude directions. Latitudes of the ridges indicate the frequencies, longitudes indicate intervals of the existence of signals, and the altitudes of ridges show intensities.

3

1

2 a1

a2

b1

a3

b3

b2

Figure 4.4. An idealized picture of level lines of jG.; /j. Signals correspond to ridges whose longitudes indicate durations of signals. The latitudes are frequencies, altitudes are intensities.

Now we must choose r.t /. The most common way is r.t / D e t

2 =2

;

or, more generally, r.t / D expft 2 =2g: This produces the transform Z G Bf .; / D

1

f .t/e 1

.t/2 =2 it

e

dt D e

2

Z

1

f .t /e t

2 =2t.Ci/

dt:

1

However, this is a slightly modified Segal–Bargmann transform: we set and write an additional factor in front of the integral.

p

2z WD Ci

245


Figure 4.5 a) Set r.t/ WD e

t 2 =2

. Consider the short time Fourier transform G f .; / of

f .t / D exp.t 2 =8 C 8i t / C exp..t 3/2 =18 C 12i t /: On the topographic map we present the line curves of jG .t; /j. -4

-2

0

2

4

6

8

-6 -8 y

-10 -12 -14

The functions Re G f .; / Im G f .; / are oscillating (they are not depicted). Instead of smooth ridges, they have a sequence of sharp peaks and deep depressions. Here the frequencies D 8 and D 12 are relatively far from one another. b) We depict jG f .; /j for f .t / D exp.t 2 =600 C 11i t / C exp..t 30/2 =600 12i t /: Now we have close frequencies D 12 and D 11 on two intersecting intervals. -20

0

20

40

60

-9 -10 -11 y -12 -13 -14

Due to a difference of phases, a meeting of two close ridges produces a sequence of peaks. Two additional remarks. The same phenomenon of interference is present in Figure a) in the valley between ridges. Next, the west and east ends of the mountain system in b) are smooth because near each end only one frequency is present.

3.5 Choice of r.t/ and the Heisenberg uncertainty First remark. The immediate idea to take ´ 1 if 1 < t < 1; r.t / D 0 otherwise;

246


must be immediately rejected. Indeed, the functions ‰ .t / D .t /r.t / have y ./ at two singularities (jumps), this implies slow decreasing of Fourier transforms ‰ infinity ( 1=). Therefore, contributions of actually existing high frequencies will be confused with a strong noise produced by the artificial singularities. This argument shows that we need functions with rapidly decreasing Fourier transform. Second remark. We promised a compactly supported function r.t / and finally im2 posed e t =2 . However, expf202 g D 0 in any realistic computations. Actually our previous reasonings do not require a compact support, we need only rapidly decreasing r.t/. 2

After all, it seems that e x =2 is the best of functions. Indeed, a function f and its 2 Fourier transform fO can not simultaneously decrease too rapidly; the function e x =2 is singled out from this point of view. Recall two classical theorems. Theorem 3.2 (Heisenberg inequalities). a) Let A and B be self-adjoint matrices (or operators in a Hilbert space). Then, for each vector v, ˇ ˇ (3.3) kAvk kBvk > ˇh.AB BA/v; viˇ: The equality holds if and only if .A C iB/v D 0 for some real and . b) Let f 2 L2 .R/, fO be its Fourier transform. Then Z

Z

1

1

x jf .x/j dx 2

2

1

x jfO.x/j2 dx >

Z

1

The equality holds if and only if f .x/ D e x

2 =2

2

1

jf .x/j dx

2

2

:

(3.4)

1

for some > 0.

Proof. a) Let , 2 R. Then 0 6 k.A C iB/vk2 D h.A C iB/v; .A C iB/vi D 2 hAv; Avi C hi.BA AB/v; vi C 2 hBv; Bvi: The inequality (3.3) is precisely the condition of positivity of the quadratic trinomial. The equality holds if and only if k k D 0. @ b) We put Af .x/ D xf .x/ and Bf .x/ D i @x f .x/, hence AB BA D i. Next, 2 we rewrite kBf xk by the Plancherel formula for the Fourier transform, Z 1 Z 1ˇ ˇ 0 2 x 2 jfO.x/j2 dx: ˇf .x/j dx D 1

1

The equality holds for solutions of differential equations

@ x C f .x/ D 0: @x

247


Problem 3.1. Under the same conditions, kAvk kBvk > jh.AB C BA/v; vi. Remark. It is no wonder that the Heisenberg inequality, which is known in quantum mechanics, appears in our situation. Indeed, in both cases there arises the problem of simultaneous localization of a function and its Fourier transform. Theorem 3.3 (Hardy). Let f be a function, fO its Fourier transform. a)

f .x/ D O.e x

2 =2

2 2 /; fO.x/ D O.e x =2 / H) f .x/ D const e x =2 :

b) f .x/ D O.x N e x

2 =2

˚

2 2 /; fO.x/ D O.x N e x =2 / H) f .x/ D polynomial e x =2 :

For a proof, see [7], Ex. 6.51. 3.6 An extension of the Segal–Bargmann transform. Obviously, a short-time Fourier transform with a fixed r.t / does not allow us to recognize too long waves and also waves of too short duration, see Figure 4.6. r.t /: This r.t/ does not allow separation of waves of low frequency,

r.t / and this r.t/ does not allow us to localize signals of short duration. Figure 4.6.

We define the Bros–Iagolnitzer transform as Z

1

²

³

.t i /2 B f .; ; / D f .t/ exp dt 2 1 Z 1 ² ³ .t /2 it= 2 D 1=2 e =2 e i f .t / exp e dt; 2 1 1=2

(3.5)

where > 0 and , 2 R. It is more-or-less the short time Fourier transform, we only write an additional factor in the front of the integral and change ! . Also, we consider all the functions 2 r.t/ D e t =2 together.

248


Remark. The function g WD B f satisfies the heat equation

@2 @ 2 g.; z/ D 0; @ @z

z WD C i :

(3.6)

Moreover, the expression (3.5) is the Poisson formula (1.3.9). We only permit the space z coordinate to be complex.4 Problem 3.2. Let "

.t/ WD Ae i!t C Be .i!C"/t D e i.!C 2 /t .Ae i"t=2 C Be i"t=2 /: 2

Let H.; ; / WD 1=2 e =2 B.; ; /. Convince yourself in the following behavior of H on the planes D const. 1B . If is very large with respect to "1 , then the graph of jH.; ; /j looks like two infinite parallel ridges along the lines D ! and D .! C "/. 2B . If > 0 is very small, then H ' . 3B . An interesting phenomenon appears for (appropriate) intermediate values of ; level lines of jH.; ; /j are shown on Figure 4.7.b. We have an infinite chain of ridges supported near line D .! C "=2/ and their outlines are controlled by the function Ae i"t=2 C Be i"t=2 .

a)

1="

1=!

b)

! ! "

c)

! "=2

Figure 4.7. Reference Problem 3.2. a) The function .t /; b) jH j for very large ; c) H for intermediate values of .

The reader can observe the results of the solution of Problem 3B in our every-day life. First, a radio-receiver produces a short-time Fourier transform of electro-magnetic waves near a certain frequency D ! with a certain ; a value D ./ is chosen by engineers (sometimes we ourselves choose by tuning). We receive an “outline of ridges” as sound waves; next, our ears and brain produce a new short-time Fourier transform. 4 We can also admit complex with Re > 0, but this is not useful in the context of this and the next sections.

4.4. Spectral analysis of singularities

249

3.7 Inversion formulas Problem 3.3. Verify the following inversion formulas for a short time Fourier transform: Z 1 1 .t /r.t / D Gr .; /e i d ; 2 1 Z 1 Z 1 Z 1 1 .t / r.t / dt D Gr .; /e i d d ; 2 1 1 1 Z 1 Z 1 O ./ r.t / dt D Gr .; /e i d : 1

1

Problem 3.4. R R Let r1 and r2 be even and supported by intervals jt j 6 ˛1 , jt j 6 ˛2 respectively, r1 dt D r2 dt D 1. Representing f1 f2 in the form Z 1 Z 1 f1 .t /f2 .t /r1 .t /r2 .t / d d f1 .t /f2 .t / D 1

show that

b

1

“

f1 f2 ./ D ; W j j6˛1 C˛2

Gr1 .; / Gr2 . ; / d d ;

(3.7)

where denotes the convolution with respect to . This is not the shortest way of evaluation of the Fourier transform. However, sometimes this is a good method to calculate products of functions f1 f2 (see the next section).

4.4 Spectral analysis of singularities In this section, we discuss wave fronts of distributions and visualizations of wave fronts in the terms of short time Fourier transforms and the Bros–Iagolnitzer transform. 4.1 Lemmas on convolutions. Let f and g be functions on Rn . Recall that their convolution f g is the function defined by Z Z f .y1 /g.y2 / dy1 WD f .y1 /g.x y1 / dy1 : (4.1) .f g/.x/ D y1 Cy2 Dx

Rn

This integral can be divergent, it can be readily checked that it converges a.s when f and g are integrable on Rn . In this subsection, we discuss convolutions of functions admitting growth at infinity. Recall that the convolution is a commutative associative operation and the Fourier transform sends the convolution to the product:

1

f g D fO g: O A cone Rn is a subset invariant under homotheties, i.e., x 2 implies tx 2 x we denote the closure of the cone . for all t > 0. By For two cones 1 and 2 , define their sum as the set of vectors x1 C x2 , where x1 ranges in 1 and x2 ranges in 2 .

250


Problem 4.1. If 1 and 2 are closed cones, then 1 C 2 is a closed cone. If they are also convex, then 1 C 2 is the closed convex hull of 1 [ 2 . We say that a smooth function f on Rn is tempered if all its partial derivatives (including f itself) admit estimates ˇ ˛ ˇ ˇ @ ˇ N ˇ ˇ ˇ ˛ f .x/ˇ 6 C.1 C jxj/

@x

for some N D N.˛/ > 0 and C D C.˛/:

In this section, we consider only tempered functions. A tempered function rapidly decreases in a closed cone if for each multiindex ˛ and for each N > 0 there is a constant C such that ˇ ˛ ˇ ˇ @ ˇ ˇ ˇ 6 C.1 C jxj/N ; f .x/ ˇ ˛ ˇ

@x

where x 2 :

(4.2)

We say that a tempered function f rapidly decreases in an open cone if it rapidly decreases in any closed cone . For non-zero y 2 Rn consider the corresponding ray, i.e., the set ft yg, where t ranges over positive numbers. A tempered function rapidly decreases along a ray if there is an open cone containing the ray such that f rapidly decreases in this cone. For a function f , denote by .f / the (closed) cone that is the complement of the set of all directions of rapid decreasing. Problem 4.2. In any closed sub-cone in the complement of .f /, the estimate (4.2) holds. Hint. Apply the Heine–Borel Lemma about finite subcover. Lemma 4.1. a) The convolution of rapidly decreasing functions rapidly decreases. b) If f rapidly decreases in an open cone and g rapidly decreases in Rn , then f g rapidly decreases in . In other words, .f g/ .f /. x x D c) Let f and g be supported by open cones and „ respectively and let \. „/ 0. Then f g is a tempered function supported by the cone C „. d) Let f and g be tempered functions satisfying .f / \ ..g// D 0. Then the convolution f g is well defined. Moreover, it is tempered and .f g/ .f / C .g/: Proof. a) We must estimate the convolution (4.1). For this aim, we write jf .y1 /j 6 const .1 C jy1 j/N M ; and estimate (4.1) as jf g.x/j 6 const

Z y1 Cy2 Dx

jg.y2 /j 6 const .1 C jy2 j/M

.1 C jy1 j/N M .1 C jy2 j/M dy1 :

251


Since y1 C y2 D x, it follows that either jy1 j > jxj=2 or jy2 j > jxj=2. Therefore, .1 C jy1 j/M .1 C jy2 j/M 6 .1 C jxj=2/M (we “forget” one of the factors). ˇZ ˇ jf g.x/j D ˇ

Thus, Z ˇ ˇ M ˇ 6 .1 C jxj=2/

y1 Cy2 Dx

Rn

.1 C jy1 j/N dy1 :

If N > n C 1, then the last integral converges. Thus the decreasing of f g is proved. Partial differentiations commute with the convolution, @˛ .f g/ D .@˛ f / g and this implies the decreasing of derivatives. b) Consider a closed subcone in our open cone . If x 2 and y … , then for some (small) constants ", ı independent of x, y, jxj 6 "1 jx yj;

y

x

b)

(4.3)

„

x

a) 0

jyj 6 ı 1 jx yj:

y2 y1 0

Figure 4.8. a) Reference proof of Lemma 4.1.b. The cone and its subcone . b) Reference x \ ./ x ¤ 0, then this domain is proof of Lemma 4.1.c. The domain of integration. If „ non-compact.

Now we are ready to estimate (4.1) for x 2 . We split the integral Z f .y1 /g.y2 / dy1 y1 Cy2 Dx

into two summands corresponding to y1 2 and y1 … . Obviously (we repeat the proof of a), the integral over the domain y1 2 rapidly decreases as a function of x. Thus, let y1 … . Applying estimates (4.3), we get jf .y1 /j 6 const.1 C jy1 j/k ; jg.y2 /j 6 const.1 C jy2 j/N M 6 const.1 C ıjy1 j/N .1 C "jxj/M : Now rapid decreasing of (4.1) becomes obvious. N \ .„/ x D 0, the domain of integration in (4.1) is compact (see Figc) Since N y2 2 „, x then for some constant ", ure 4.8.b). Further, if x D y1 C y2 with y1 2 , jy1 j 6 "1 jxj;

jy2 j 6 "1 jxj

252


(this is an exercise in elementary topology). Hence the domain of the integration in (4.1) is contained in the ball jy1 j 6 "1 jxj and its volume 6 const "n jxjn . Also, jf .y1 /j 6 .1 C jy1 j/m 6 .1 C "1 jxj/m ;

jg.y2 /j 6 .1 C jy2 j/k 6 .1 C "1 jxj/k :

Therefore, (4.1) is 6 const jxjnCmCk . d) We decompose f D f1 C f2 and g D g1 C g2 with f1 , g1 supported by respective cones and f2 , g2 being rapidly decreasing. Further, we apply a), b), c). 4.2 The Localization Lemma Theorem 4.2 (Localization Lemma). Let u be a distribution on Rn with a compact support5 . Let ' be a smooth function. Then .c ' u/ .u/: O

Proof. This is a rephrasing of Lemma 4.1.b.

4.3 Wave fronts. Let u be a distribution on Rn , let x 2 Rn . Choose a sequence of compactly supported smooth functions 'j such that 'j .x/ ¤ 0 and the diameters of their supports tend to 0. Then the wave front of a distribution u at a point x of the distribution u at x is \ .'j u/: WFx .u/ WD j

b

It is convenient to exclude the point x D 0 from consideration and to regard the wave front as a conic subset in Rn n 0. Lemma 4.3. The result does not depend on the choice of a sequence 'j . Proof. Let k be another such sequence. Fix 'j . Starting with a certain l, one has l D 'j j l , where j l are smooth compactly supported functions. By the Localization Lemma, . l u/ .'j u/. Now the statement becomes obvious.

b

b

The total wave front WF.u/ is the subset in Rn .Rn n 0/ defined by .x; / 2 WF.u/ Lemma 4.4. .u/ O D

S

whenever 2 WFx .u/:

WFx .u/, where x ranges in the support 6 supp.u/ of u.

And hence uO is a holomorphic function in Cn . Recall the definition. A point x is not contained in the support supp.u/ of a distribution u if there is a neighborhood U of x such that hhu; f ii D 0 for each f supported by U . 5 6

253


Proof. By definition, WFx .u/ .u/. O Further, let 2 .u/. O Take small > 0 and a collection of smooth Ppositive functions '1 ; : : : ; 'N , whose supports are contained in balls of radii 6 , and 'j D 1 on supp.u/. Then 2 .'k u/ for some k. Next, we consider smaller and repeat the same operation with the “new” distribution 'k u, etc. In this way, we obtain a sequence ˛ u of distributions such that:

b

1. diameters of their supports tend to 0; 2. supp.˛ u/ supp.˛C1 u/. Let x be the unique point of intersection of the supports. Then 2 WFx .u/.

We can also reformulate our definition as follows: – a point .a; / … WF.u/ if there is a smooth compactly supported function ' such that '.a/ ¤ 0 and the ray is a ray of rapid decreasing of 'c u. We define WF.u/ as the complement of a union of open sets. Therefore, the wave front is a closed subset in Rn .Rn n 0/. Problem 4.3. a) Find the wave fronts of the following distributions on R: Z hhu; 'ii D '.0/;

hhu; 'ii D

Z

1

hhu; 'ii D lim

x f .x/ dx;

"!0 jxj>"

0

Z

1

hhu˙ ; 'ii D lim

"!˙0 1

Z

'.x/ dx ; x C i" Z

1

0

x f .x/ dx C B

hhu; 'ii D A 0

'.x/ dx; x

.x/ f .x/ dx: 1

b) Find the wave fronts of the following 2-dimensional distributions: Z

Z

2

hhu; 'ii D

'.cos t; sin t / dt;

1

hhu; 'ii D

'.t; t / dt: 1

0

c) Find the wave fronts of the following distributions on the space of Hermitian matrices: Z Z hhu; 'ii D det.X C i "/ '.X / dX: det.X / '.X / dX; hhu; 'ii D lim X>0

"!C0

Problem 4.4. Show that the projection of WF.u/ on the coordinate space Rn is the singular support of u. Recall the definition: x 2 Rn is not contained in the singular support of u if one can find a smooth function h such that x … supp.u h/.

254


4.4 Wave fronts and short time Fourier transforms. Let r be a smooth non-negative function with compact support and let r.0/ > 0. Consider the short time Fourier transform of a distribution u, Z x x u.x/r e ix WD hhu.x/; r e ix ii; where > 0: G .; ; /u D x2Rn Here we introduce an additional parameter as in Subsection 3.6. Fix . Define the set D .f / Rn .Rn n 0/ as: – .; / 2 if the function ./ WD G f .; ; / rapidly decreases along the direction . The set is open in Rn .Rn n 0/. If 0 < , then 0 . T Observation 4.5. WF.u/ D . This is nothing but a rephrasing of the previous definition. Theorem 4.9 extends this construction to functions r that are not compactly supported. However, the generalization is not straightforward. 4.5 Digression. Products of distributions. Now let u and v be compactly supported distributions satisfying the condition if

.x; / 2 WF.u/;

then

.x; / … WF.v/:

(4.4)

Then we can define their product u v as follows. Fix small " > 0. Choose a finite collection of non-negative smooth functions 'j such that: 1. 'j are supported by balls of radii 6 "; P 2. 'j D 1 on the supports of u and v. Lemma 4.6. Under (4.4), for sufficiently small " > 0,

b

supp 'k \ supp 'l ¤ 0 H) .'k u/ \ .'c l v/ D 0:

(4.5)

We omit a proof (it is a topological manipulation with Lemma 4.4). Next, we write formally uv D

X

X X 'k u 'l v D .'k u/ .'l v/: k;l

It is natural to think that the product of distributions is zero when their supports have empty intersection. We write X uv D .'k u/ .'l v/: k;lW supp 'k \supp 'l ¤¿

For all the summands, we can calculate their Fourier transforms by Lemma 4.1:

b

F .'k u 'l v/ WD 'k u 'c lv and we define .'k u/ .'l v/ as the inverse Fourier transform. Thus we know u v.

255


Problem 4.5. a) The result of the calculation does not depend on the choice of a collection 'k (recall that 'k must satisfy (4.5)). b) If .x; / 2 WF.u v/, then 2 WFx .u/ C WFx .v/ [ WFx .u/ [ WFx .v/ (the last expression is the closure of WFx .u/ C WFx .v/; we assumed that 0 … WFx . /, for this reason the sum of cones is not closed). 4.6 Digression. A rephrasing of the definition of products. R Let u and v be the same as above, let r be a smooth function supported by a ball jxj 6 " and r dx D 1, r.0/ ¤ 0. We write Z Z u D u.x/r.x / d ; v D v.x/r.x / d ; Z Z uv D

u.x/r.x / v.x/r.x / d d :

We assume that the integrand is 0 outside the strip j j < 2" (because r.x /r.x / D 0 outside the strip). Therefore we assume “

u v WD u.x/r.x / v.x/r.x / d d : jj62"

Further, we take the Fourier transform of both sides; on the right-hand side we represent Fourier transforms of products as convolutions “ u v./ D Gr .; / Gr . ; / d d :

b

jj62"

b

If " is sufficiently small, then the convolutions are well defined and we can evaluate u v and u v.

4.7 Wave fronts and the Bros–Iagolnitzer transform. Let us apply the Bros– Iagolnitzer transform B , see Subsection 3.6, to a compactly supported distribution u.x/ on Rn , ˚

B u.; z/ WD n=2 hhu.y/; exp .y z/2 =2 ii; z D x C i 2 Cn : We get a function defined in the domain > 0, z 2 Cn . Recall that it is a solution of the heat equation (3.6). Our point of our interest is its asymptotic behavior in the domain 0 < < ". Lemma 4.7. Fix x. If … WFx .u/, then for in a neighborhood of , B u.; x; / D e j j

2 =2

O.N / for all N ; ! C0;

(4.6)

and lying in the moving domain .x; / 2 V;

a1 1 < j j < a1 :

(4.7)

256


Proof. We write n=2 2 =2

e

Z e

ix=

Bu.; x; / D

u.y/e .yx/

2 =2

e iy= :

(4.8)

y2Rn

We are going to examine the behavior of the last integral. For this aim, we represent u D v C w, where supp w does not contain x and .v/ O does not contain . Write our integral as Z Z 2 .yx/2 =2 iy= v.y/e e C w.y/e .yx/ =2 e iy= : (4.9) y2Rn

y2Rn

The second summand of (4.9). Since w is a continuous linear functional in some C N -topology, it follows that X X C˛ max j@˛ 'j for some constants C˛ : jhhw; 'iij 6 j 2 =2

For our ' D e .yx/

j˛j6N

supp.w/

e iy= , partial derivatives have the form

polynomial.x; y; 1 / e .yx/

2 =2

: 1

Since x … supp.w/, the integral can be estimated as 6 const e ı . The first summand of (4.9). Applying the Plancherel formula for the Fourier transform, we get Z ² ³ v. / O exp . =/2 C ix. =/ d : 2 Rn This expression is the convolution of v. / O and e is taken at a point D =. B

2 =2ix

, the value of the convolution

1 C B

Figure 4.9. Reference the proof of Lemma 4.7. A cone of rapid decreasing of v. / O around the ray RC . Since 1 >> 1=2 , a shifted ball is contained in the cone for small . 2

The function vO rapidly decreases in the direction . The function e =2ix is very small (and very rapidly decreases) outside some ball B W j j 6 R 1=2 . Hence O therefore the shifted ball 1 C B is contained in the cone of rapid decreasing of v. /, the convolution is approximately equal to the integral over the shifted ball, hence it is O.N /. We omit precise estimates.

257


Theorem 4.8 (A. Cordoba, C. Fefferman). Let u be a tempered distribution. Then .x; / … WF.u/ if and only if the estimate (4.6) holds. We omit a proof. Next, we formulate (also without a proof) a more general statement (see Folland, [51]), which clarifies the Cordoba–Feferman theorem. Let r 2 .Rn / be an arbitrary non-zero even function, r.0/ ¤ 0. For a tempered distribution u, we write the expression Z p n=4 e isy r s.x y/ u.y/: P .sI ; x/ WD s y2Rn

Theorem 4.9. .0 ; x0 / … WF.u/ if and only if in some neighborhood of .0 ; x0 / the following estimate holds: P .sI ; x/ D O.s N /;

s ! 1;

for all N . In the previous theorem, '.x/ D e x from another normalization.

2 =2

and s D 1 . The factor e j j

2 =2

arises

4.8 The Bros–Iagolnitzer transform and analyticity Theorem 4.10. Let u be a compactly supported distribution. Then u is a real analytic function in a neighborhood of x0 if and only if for some " > 0, K > 0, and C , jB u.; x; /j 6 C expfjj2 =2g expf"=g;

jj > K:

Proof. We only prove “)”, which is the simple part of the theorem. To simplify the notation, set n D 1. We must examine the behavior of the integral Z 2 u.y/e s.yx/ =2 e isy y2R

as s ! 1. For definiteness, assume > 0. Let u be real analytic on a segment Œx0 ; x0 C . Let us choose an appropriate smooth non-negative function .y/ such that 1. .y/ D 0 outside our segment Œx0 ; x0 C ; 2. .y/ is a small constant on some interval Œx0 ı; x0 C ı. Moving a contour of integration y 7! y C i .y/, we come to Z 2 u.y C i .y//e s.yCi .y/x/ =2 e is.yCi .y// .1 C 0 .y//: y2R

Next, represent u.y C i y/ D v.y/ C w.y/, where the support of w does not contain x0 and v is supported by Œx0 ı; x0 C ı. Respectively, split the integral into the two summands. For the summand with w we get exponential decreasing as we mentioned in the proof of Lemma 4.7. For the summand with v, we get a smooth integrand with exponential decreasing in s (because 2 < ).

258


This theorem is an origin of the following definition of analytic wave fronts of distributions (we only present the definition and stop at this point). Let u be a compactly supported distribution on Rn . A point .x; / is not a point of the analytic wave front of u if there are a neighborhood O of .x; / and positive constants C , K and ı such that B u.; x; / 6 C expf.jj2 ı/=2/;

for jj > K, .x; / 2 O.

4.5 Symbols of operators A given function (distribution) of two variables can be considered as a kernel of an integral operator (see § 1.1). However there are many other natural correspondences between functions and operators. Such correspondences are power tools for investigation of certain classes of operators. 5.1 qp-symbols of polynomial differential operators. As above, define operators Qk f .x/ D xk f .x/;

Pk f .x/ D i

@ f .x/ @xk

in the space of functions on Rn . To each polynomial X

.x; / D c˛ˇ x ˛ ˇ

(5.1)

on Rn Rn , we assign the polynomial differential operator X

O .P; Q/ D c˛ˇ P ˛ Qˇ :

(5.2)

We say that is the qp-symbol of the differential operator O . Conversely, for a polynomial differential operator Dx in x we write ˇ

.x; / D Dx e i.yx/ ˇyDx :

(5.3)

Theorem 5.1. a) The symbol of the adjoint operator O .P; Q/ is ²

exp i

n X kD1

³

@2

.x; /: @xk @k

(5.4)

b) The symbol 3 .x; / of a product O 1 O 2 is given by the formula ²

3 .x; / D exp i

n X kD1

³

ˇ

ˇ @2

1 .x; / 2 .y; /ˇˇ : @yk @ k yDx; D

(5.5)

4.5. Symbols of operators

259

Proof. In both cases the question reduces to the change of order of P and Q. More precisely, we must find an explicit formula for Y

i

@ @xk

˛k Y

ˇ

xk k f .x/:

Since Pi and Qj commute for i ¤ j , the problem reduces to the case of one variable. By the Leibnitz formula,

i

@ @x D

m

x l f .x/ X

.i /k

k>0

l.l 1/ : : : .l k C 1/ m.m 1/ : : : .m k C 1/ kŠ

x

mk

@ i @x

mk

f .x/:

Thus the symbol of the operator P m Ql is ²

exp i

³

@2 .x l m /: @x @

Both the statements easily follow from this observation.

5.2 pq-symbols. Let us change the rule associating a differential operator to a given polynomial .x; /, see (5.1) and (5.2). The first variant is to change the order of P and Q, X

L .P; Q/ WD c˛ˇ P ˇ Q˛ : We say that is the pq-symbol of the operator L . Clearly,

L .P; Q/ D O .P; Q/ : This reduces pq-symbols to qp-symbols. 5.3 Weyl symbols. For any collection of linear operators X1 ; : : : ; Xm , define its symmetric product ŒX1 : : : Xm sym WD

1 X X.1/ : : : X.m/ ; mŠ 2Sm

where the summation is taken over all permutations of the set f1; : : : ; mg. Denote by ˛

ŒX1˛1 : : : Xl l sym the symmetric product of ˛1 copies of X1 , ˛2 copies of X2 , etc.

260


It is useful to keep in mind the following formulas:

ˇˇ ˚ @m exp s1 X1 C C sm Xm ˇ ; s1 DDsm D0 @s1 : : : @sm Y ˛j oˇ nX @ ˇ ˛m ŒX1˛1 : : : Xm sym D exp s X : j j ˇ ˛j sj D0 @sj

ŒX1 : : : Xm sym D

(5.6) (5.7)

Further, for a polynomial .x; / we define the operator X X

˛m ˇ1 M .Q; P / WD c˛ˇ ŒQ˛ P ˇ sym WD c˛ˇ Q1˛1 : : : Qm P1 : : : Pmˇm sym : ˛ˇ

˛ˇ

We say that is the Weil symbol of the operator . M For k ¤ l, the operators Pk and Ql commute, therefore we can write ˛1 ˛m ˇ m

˛m ˇ1 Q1 : : : Qm P1 : : : Pmˇm sym D Q1˛1 P1ˇ1 sym : : : Qm Pm sym : Theorem 5.2. a) The Weyl symbol .x; / and qp-symbol .x; / of the same operator are connected by ²

.x; / D exp

³ n i X @2

.x; /I 2 @xk @k

²

(5.8)

kD1

.x; / D exp

³ n i X @2 .x; /: 2 @xk @k

(5.9)

kD1

b) If .x; / is the Weyl symbol of an operator A, then the Weyl symbol of A is .x; /. c) The product of operators with the symbols 1 and 2 has the symbol 3 .x; / D exp

² X i

2

k

@2 @2 @xk @ k @yk @k

³

ˇ ˇ

1 .x; /2 .y; /ˇˇ

:

(5.10)

yDx; D

Lemma 5.3. Set n D 1. The qp-symbol of ŒQm P l sym is ²

exp

³

i @2 xm l : 2 @x@

Proof of the lemma. Formula (5.7) suggests that we evaluate the following exponential: ²

³

@ exp sx i t f .x/ @x

²

D e sxist=2 f .x i st / D e ist=2 expfsxg exp i t

³

@ f .x/: @x

(5.11)

261


To avoid questions about convergence, we consider these operators in the space of all entire functions on C. As soon as a formula for the exponential is written, it can be easily verified. In fact, this is a calculation in the Heisenberg group. It remains to extract the Taylor coefficient at s m t l from (5.11) and compare with the desired formula. Proof of the theorem. a) follows from the lemma, b) is obvious. To derive c), we 1) evaluate qp-symbols of our operators by formula (5.9); 2) evaluate qp-symbol of the product by formula (5.5); 3) return to the Weyl symbols again (see formula (5.8). The operations 1)–2) give ²

exp i

X

²

k

@2 @yk @k

³ ³

²

ˇ

³

ˇ i X @2 i X @2 exp exp 1 .x; /2 .y; /ˇˇ : 2 @xk @k 2 @yk @ k yDx; D k

k

(5.12) To apply formula (5.8), we must differentiate the result of the substitution (5.5). For this aim, we write out the usual Leibnitz formula as @m f .x/g.x/ D @m x

@ @ C @x @y

m

ˇ ˇ f .x/g.y/ˇˇ

:

yDx

The expression (5.12) is a (long) finite linear combination of the form ˇ X ˇ F .x; / D c f .x; / g .y; /ˇ : yDx; D

We write X @m @m F .x; / D c @xkm @km

@ @ C @x @y

m

@ @ C @ @

m

ˇ ˇ f .x; / g .y; /ˇˇ

:

yDx; D

Finally, the operations 1)–3) give exp

² X i @

2

k

@ C @xk @yk

@ @ C @k @ k

³

²

exp i

X k

@2 @yk @k

³

ˇ ³ ² ³ ˇ i X @2 i X @2 exp exp 1 .x; /2 .y; /ˇˇ : 2 @xk @k 2 @yk @ k yDx; D ²

k

k

(5.13) It remains to multiply the exponentials in the last expression. Since all the operators present in the formula commute, this step is trivial. We come to (5.10).

262


5.4 qp-symbols. General case. Now consider an arbitrary integral operator .Rn / ! 0 .Rn /, Z Af .x; y/ D K.x; y/f .y/: (5.14) y2Rn

We define its qp-symbol as Z

t

.x; / D

K.x; y/e i.yx/ :

(5.15)

y2Rn

Proposition 5.4. a) is a tempered distribution. b) If K 2 .Rn Rn /, then 2 .Rn Rn /. c) The inversion formula is Z 1 t K.x; y/ D

.x; /e i.xy/ : n .2/ 2Rn

(5.16)

d) For a polynomial differential operator .x; / coincides with the qp-symbol defined above. e) The symbol of A is “ 1 t t

.y; /e i.xy/. / : (5.17) n .2/ y2Rn ; 2Rn Proof. a), b), c) We write the formula for symbols as Z t ix t

.x; / D e K.x; y/e iy : y2Rn

The integral is the Fourier transform with respect to the variable y only; it sends .Rn Rn / ! .Rn Rn / and therefore 0 .Rn Rn / ! 0 .Rn Rn / Also, the inversion formula becomes obvious. d) is formula (5.3). e) is a straightforward corollary of the definition and the inversion formula. We stress that formulas (5.15)–(5.17) determine Gaussian integral operators in the sense of Chapter 1. All these operators are bounded as operators .Rn Rn / ! .Rn Rn /;

0 .Rn Rn / ! 0 .Rn Rn /;

L2 .Rn Rn / ! L2 .Rn Rn /: Consider the convolution of kernels of two integral operators: Z K3 .x; z/ D K1 .x; y/K2 .y; z/: y2Rn

(5.18)


263

We interpret it in the following way. Consider the Gaussian integral operator .R4n / ! .R2n / given by Z Jf .x; z/ D f .x; y; y; z/ dy: Rn

The operator J sends the function K1 .x; y/K2 .y 0 ; z/ to the function K3 given by formula (5.18). Since the composition of Gaussian operators is a Gaussian operator (and we know how to calculate it), a formula for symbols of products must be written as a Gaussian integral operator. For this reason, we present the following result without a proof. Theorem 5.5. The symbol of a product is given by Z t t

1 .x; / 2 .y; /e i.xy/. / :

3 .x; / D

(5.19)

y2Rn ; 2Rn

The formula holds at least for 1 and 2 being in the Schwartz space. We have discussed two formulas (5.5), (5.19) for symbols of products. It turns out to be that the first formula7 survives in essential wider generality. This phenomenon is an initial point of the theory of pseudodifferential operators. However, we stop our discussion here. Problem 5.1. Derive the following formulas (which hold at least for symbols being the Schwartz space): Z tr A D

.x; /; x2Rn ;2Rn Z tr A1 A2 D

1 .x; / 2 .x; /: x2Rn ;2Rn

5.5 The Weyl symbols. General case. For an integral operator A given by (5.14), define its Weyl symbol by Z K.x C =2; y =2/e ix : (5.20) .p; q/ D 2Rn

This formula extends the construction of Weyl symbols given above. We omit a detailed discussion and only note that formula (5.20) determines a Gaussian integral operator. It sends kernels in Schwartz’s sense to Weyl symbols. Since we can manipulate Gaussian operators, we terminate our discussion with the following statement. 7 It was derived for polynomial differential operators. Obviously, it holds for arbitrary differential operators.

264


Theorem 5.6. The Weyl symbol for the product is given by the formula 8 19 0 Z 1 1 1 = < 3 .p; q/ D .p1 ; q1 / .p2 ; q2 / exp 2i det @ q q1 q2 A ; : p p1 p2 p1 ;q1 ;p2 ;q2 2Rn Z ˚

D .p1 ; q1 / .p2 ; q2 / exp 2i.qp1 C q1 p2 C q2 p pq1 p1 q2 p2 q/ : p1 ;q1 ;p2 ;q2 2Rn

Weyl symbols provide an alternative formalism for pseudo-differential operators. 5.6 Wick symbols. Define the differential operators

1 @ 1 @ aO k f .x/ D p xk f .x/; aO k f .x/ D p xk C f .x/: @xk @xk 2 2 P For a polynomial W .z; u/ N D c˛ˇ z ˛ uN ˇ we define the differential operator X y WD W c˛ˇ aO ˛ .aO /ˇ : y We say that W is the Wick symbol of the operator W Remark. At first glance, there is no serious difference between the qp-symbols and the Wick symbols; for instance the algebraic considerations of Subsection 5.1 survive literally. Nevertheless, these theories drastically diverge from the analytic point of view. We reformulate the construction of Wick symbols in the following way. Take an operator A in L2 .R/. 1. We pass to the Fock space Fn and write the kernel (Berezin symbol) of AQ WD BAB 1 W Fn ! Fn . 2. We write the Wick symbol for the operator with the given Berezin symbol. Let us repeat this. First, we write the Berezin symbol L.u; v/ of A by definition, L.u; v/ N D hA'v ; 'u iFn D hAˆv ; û iL2 .Rn / ; where 'u is the supercomplete system in Fn and û is the corresponding system in L2 .Rn /, see formulas (1.3), (2.2). Next, we pass to the Wick symbol as it was explained in Subsection 1.11. Finally, we come to a general formula expressing Wick symbols in the terms of Schwartz kernels: “ p ˚ W .z; u/ N D K.x; y/ exp 12 x 2 C y 2 2 2.x uN C yv/ C .u C v/ N 2 : x;y2Rn

Observation 5.7. For each operator .Rn / ! 0 .Rn /, its Wick symbol is an entire function on Cn Cn .


265

Recall that qp-symbols and Weyl symbols are distributions. We stop our discussion here because all the statements about kernels (Berezin symbols) from Subsection 1.4 can be easily translated to the language of Wick symbols. P 5.7 Anti-Wick symbols. Now, for a polynomial W .s; t / D c˛ˇ s ˛ t ˇ , consider the differential operator X y B WD W c˛ˇ .aO /˛ aO ˇ : (5.21) y. We say that W is the anti-Wick symbol of W More generally, for a function W .z; u/ N we define the operator in Fn by Z 2 y B f .z/ D W .u; N u/e z uN f .u/e juj d .u/: W

(5.22)

Cn

Problem 5.2. If W is a polynomial, then we get the same operator (5.21). 2

Problem 5.3. Show that the operator in L2 .Cn ; e jzj / given by Z 2 e z uN g.u/e juj d .u/ P g.z/ D

(5.23)

Cn

is the orthogonal projection onto Fn . Now we can reformulate the definition. We fix a distribution on Cn . Take f 2 Fn , consider the distribution f and apply the projection P given by (5.23) to f . Finally, y f D P .f /: W y is well defined. There are at least two cases when the operator W A) If is a bounded measurable function, then the corresponding operator is bounded (such operators are called Töplitz (or Berezin–Töplitz) operators). B) Let be a compactly supported distribution. Obviously P .f / is a well-defined holomorphic function. y is the projection Problem 5.4. a) Let be the delta-function supported by 0. Then W to the function f D 1. y. b) Let be a partial derivative (of arbitrary order) of the ı-function. Describe W y W Fn ! Fn c) Let be a compactly supported measure on Cn . Then the operator W is bounded. y W Fn ! Fn is d) If is a compactly supported distribution, then the operator W bounded.

266


5.8 Remarks on abstract functions of noncommutative variables. Let g be a Lie algebra, let X1 ; : : : ; Xn be its basis. To each monomial z ˛ WD z1˛1 : : : zn˛n we assign the expression Q ˛j Š X Xk1 : : : Xkj˛j ; j˛jŠ P where j˛j D ˛j and the summation is taken over all product containing ˛1 entries of X1 , ˛2 entries of X2 , etc. Thus we identify the algebra of noncommutative polynomials in Xj (i.e., the universal enveloping algebra) with the space of polynomials in n variables (the “Poincaré– Birkhoff–Witt theorem”). Multiplication in the universal enveloping algebra induces a product } in the algebra of polynomials. For X , Y 2 g being in a neighborhood of 0, define ˆ.X; Y / WD ln.e X e Y /: Problem 5.5. Prove the following formula: z˛ } zˇ D

ˇ X 1 @ ˛ @ ˇ X ˇ ˆ tj Xj ; sj Yj ˇ : ˛ ˇ sD0;tD0 ˛ŠˇŠ @t @s

For a solvable Lie algebra g, the function ˆ can be written explicitly. Problem 5.6. Write a formula for }-product for the 2-dimensional Lie algebra, ŒX; Y D Y . However, for a semisimple (in particular, classical) g this way looks hopeless. Only for g D sl.2/ does the function ˆ admit an explicit (horrible...) expression in terms of elementary functions. However, there are numerous explicit calculations in enveloping algebras of semisimple g. Problem 5.7. In the notation of the previous problem, express X n } Y in terms of ŒX; Y , ŒX; ŒX; Y , ŒX; ŒX; ŒX; Y , …in an arbitrary Lie algebra (the final formula includes Bernoulli numbers). Next, for each noncommutative monomial X1 : : : Xn we write ‰.X1 : : : Xn / D

1 X1 ; ŒX2 ; : : : ŒXn1 ; Xn : : : : n

Extend ‰ as a linear map to the set of all noncommutative polynomials in variables X , Y . Theorem 5.8 (Campbell–Hausdorff–Dynkin formula). The following identity of formal series holds: ln.e X e Y / D ‰.ln.e X e Y //:

4.6 The Perelomov problem Sections 4.6–4.9 form a closed subchapter, below its material is not used (modulo some local references). From here on, until the end of the chapter, we discuss the following three problems.

4.6. The Perelomov problem

267

6.1 Perelomov problems. Let L be a closed subset in C. Denote by f jL the restriction of f 2 F1 to L. Problem A. Is f 2 F1 determined by f jL ? Equivalently, let f jL D 0. Is it correct that f D 0? Certainly, Problem A is interesting only for countable subsets L C without limit points; otherwise we have the immediate affirmative answer. Due to the reproducing property (1.5), we can reformulate Problem A as follows: – Consider the set of functions 'u , where u ranges in L. Is it total in F1 ? Problem B. Is it possible to reconstruct explicitly f from f jL ? Problem C. Let g 2 L2 .R/ and f D Bg its Segal–Bargmann transform. Is it possible to reconstruct g explicitly from f jL ? Problem 6.1. Reconstruct g 2 L2 .R/ from BgjR . Hint. First, reconstruct g, O see Theorem 2.8. 6.2 Perelomov Theorem. Our basic example of L C is a lattice. More precisely, fix !1 , !2 2 C such that !2 =!1 62 R. A lattice L.!1 ; !2 / C is the set fm!1 Cn!2 g, where m and n range in Z. Denote by S the area of the parallelogram with sides !1 and !2 . Theorem 6.1. a) If S < , then L.!1 ; !2 / is a set of uniqueness for F1 . b) If S > , then L.!1 ; !2 / is not a set of uniqueness. c) If S D , then L.!1 ; !2 / is a set of uniqueness for F1 . It remains to be a set of uniqueness if we remove an arbitrary point from L and loses this property if we remove an arbitrary pair of points. A proof is given in Subsection 7.8. We also solve Problems B and C for L D L.!1 ; !2 / (Theorems 8.3 and 9.1). We return to a general discussion. From this moment, L C is a sequence a1 , a2 ; : : : tending to 1. The Problem A takes the form: – Does there exist a non-zero function f 2 F1 such that f .aj / D 0. The last problem can be reduced to the general theory of entire functions and we briefly discuss this reduction. Theorems 6.2–6.5 formulated below can be found in textbooks on complex analysis, for an advanced exposition, see monographs [117], [23], [118]. 6.3 Entire functions with prescribed zeros. Consider a sequence aj 2 C tending to 1. It is convenient to put terms equal to 0 at the beginning and to write the sequence in the form 0; : : : ; 0, a1 ; a2 ; : : : : „ ƒ‚ … p times

268


We wish to find an entire function with zeros at these points. The immediate idea is to write Y1 z f .z/ WD z p 1 : j D1 aj P However, this product absolutely converges if and only if j azj j < 1. In other words, P our formula is satisfactory if and only if jaj j1 < 1. We can also write another function f .z/ WD z

p

Y1 j D1

z 1 aj

²

³

z exp : aj

Problem 6.2. Show that this formula is satisfactory if and only if

P

jaj j2 < 1:

Theorem 6.2 (Weierstrass). a) There is an entire function having zeros precisely at points of a given sequence aj ! 1. b) If F1 .z/ and F2 .z/ are two such functions, then F2 .z/ D F1 .z/ exp r.z/ , where r.z/ is an entire function. We recall the method of proof (for details, see standard textbooks). Write Yk .z/ WD

k X zk j D1

k

:

This expression is the terminated Taylor expansion of ln.1 z/; therefore,

²

³

z z 1 exp Yn a a

D 1 C c z nC1 C :

One can choose a sequence nj such that the product F .z/ D z p

1 Y j D1

1

z aj

²

exp Ynj

z aj

³

converges (this claim requires a proof). Then F .z/ has the desired property. To prove b), we note that F1 =F2 is a holomorphic function without zeros. Therefore, its logarithm ln.F2 =F1 / is well defined. 6.4 Growth of an entire function and the density of its zeros. We begin with two definitions. For a sequence aj in C, we define the exponent of convergence as ˚

P WD infimum of such that jaj j < 1 :

269

4.6. The Perelomov problem

Problem 6.3. For a sequence ak 2 C, denote by h.R/ the number of its terms in the disk jzj 6 R. Denote by Q the infimum of s such that h.R/ 6 const Rs . Show that Q D . The order of an entire function f .z/ is ˚ WD infimum of such that

jf .z/j 6 const e jzj :

Theorem 6.3 (Hadamard). For an entire function f , denote by its order and by the exponent of convergence of the sequence of its zeros. Then > . Moreover, if the order is non-integer, then D . Remark. If P .z/ is a polynomial of degree k, then e P .z/ is a function of integer order k without zeros. Theorem 6.4 (E. Borel). Let aj be a sequence with a finite exponent of convergence , let m WD bc be the integer part of . Then the function f .z/ D z p

1 Y

1

j D1

z aj

²

exp Ym

z aj

³

has the order . Theorem 6.5 (Hadamard). Let f be a function of finite growth ; let aj be its non-zero roots; let m D bc. Then f admits the representation f .z/ D z p expfH.z/g

1 Y j D1

1

z aj

²

exp Ym

z aj

³

;

where H.z/ is a polynomial of degree 6 m. 6.5 Corollaries. We know that a function f 2 F1 satisfies jf .z/j 6 const e jzj On the other hand, all entire functions satisfying

2 =2

.

2 =2

jf .z/j 6 const .1 C jzj/2 e jzj

are contained in F1 . Thus Theorems 6.3–6.5 imply the following corollary: Corollary 6.6. Let be the exponent of convergence of a sequence aj . a) If > 2, then a function f 2 F1 is uniquely determined by8 its values at aj . b) If < 2, then there is a non-zero function g 2 F1 such that g.aj / D 0. For the lattice L W fm!1 C n!2 g, Theorems 6.3–6.5 are not sufficient. Indeed, the exponent is D 2, hence a minimal order of a function with zeros at n!1 C m!2 is 2. Such a function can be or not be an element of F1 . There are finer general theorems that give (incomplete) information in this case. However, we prefer another way. 8

If some aj has a multiplicity s, we assume that f .aj /, f 0 .aj /; : : : ; f .s1/ .aj / are given.

270


4.7 Preliminaries on -functions This section is preparation for the next two sections. Also, we prove the Perelomov theorem. 7.1 Definition of the theta-function. Quasiperiodicity. Fix q 2 C such that 0 < jqj < 1. Consider the Laurent series (the theta-function) 9 #.uI q/ D

1 X

.1/n un q n.n1/=2 :

nD1

Obviously, this function satisfies the following quasiperiodicity condition: #.quI q/ D u1 #.uI q/:

(7.1)

Iterating this identity, we get (7.2) #.q n uI q/ D .u/n q n.n1/=2 #.uI q/: P k Problem 7.1. If a Laurent series T .u/ D 1 kD1 ck u satisfies the quasiperiodicity condition (7.1), then T .u/ D const #.u; q/. 7.2 Zeros Lemma 7.1. The function #.q; u/ has zeros precisely at the points u D q k . Proof. Obviously, u D 1 is zero. By the quasiperiodicity, it follows that points z D q k are also zeros. For the same reason, zeros must form several geometric progressions fAq k g, fBq k g, etc. Take a circle jzj D R free of zeros. Consider the ring qR 6 jzj 6 R. By the Argument Principle, the number of zeros of # inside the ring is

˚

˚ index of path #.Re i' I q/ index of path #.Rqe i' I q/ ; where ' 2 Œ0; 2: By the quasiperiodicity (7.1), the second path is R1 e i' #.q; Re i' /. Therefore, the difference of indices D 1 and there is a unique zero in the ring. This completes the proof. 7.3 The Jacobi triple identity Theorem 7.2 (Jacobi). #.uI q/ WD .1 u/

1 Y

.1 q n /.1 uq n /.1 u1 q n /:

nD1 9

Here and below (Subsection 7.6) we do not try to be consistent with the standard notation.

(7.3)

271

4.7. Preliminaries on -functions

Proof. Since #.uI q/ has zeros at u D q k , it is natural to compare it with the product having the same zeros, T .u/ WD .1 u/

1 Y

.1 uq /.1 u n

1 Y

1 n

q /D

.1 q jmj usgn m /:

(7.4)

mD1

nD1

We must compare and T . First, we observe that T .u/ satisfies the same quasiperiodicity condition (7.1). By Problem 7.1, we get #.uI q/ D AT .u/. It remains to find the constant A. Next, we are going to evaluate the partial products T˛;ˇ .u/ WD

ˇ Y

.1 q jmj usgn m / DW

mD˛

Evidently,

X

c˛;ˇ Œkuk :

(7.5)

k

´ T˛;ˇ .qu/ WD

if sgn ˛ D sgn ˇ; T˛C1;ˇ C1 .u/ 1 u T˛C1;ˇ C1 .u/ if sgn ˛ ¤ sgn ˇ.

(7.6)

Theorem 7.3 (q-binomial formula). T0;h .u/ is h Y

.1 q k u/ D

kD0

h X

.1/mC1 q m.mC1/=2

mD0

.1 q mC1 / : : : .1 q h / um : .1 q/.1 q 2 / : : : .1 q hm /

Remark. Passing to the limit q ! 1 we come to the usual Newton binomial formula. Proof of Theorem 7.3. Writing out (7.6) for T0;h , we get .1 u/T0;h .qu/ D T0;hC1 .u/ D .1 q hC1 u/T0;h .u/: Equating coefficients in un , we obtain the recurrence relation c0;h Œk.q k 1/ D c0;h Œk 1.q hC1 q k1 /: The definition (7.5) yields that the coefficient at the leading term is c0;h Œh C 1 D .1/hC1 uh.hC1/ : Further, we calculate all the c0;h Œk from the recurrence relation.

End of proof of the triple formula. Since we know T0;ˇ , formula (7.6) provides us with all the T˛;ˇ . In particular, the coefficient at u0 in Tl;l .u/ is .1 q lC2 / : : : .1 q 2lC1 / : .1 q/.1 q 2 / : : : .1 q l / Passing to the limit as l ! 1, we the obtain the desired coefficient A.

272


7.4 Derivatives at zeros. If we differentiate the triple identity, we get # 0 .1I q/ D

Y ˇ d #.uI q/ˇuD1 D .1 q n /3 : du

(7.7)

Differentiating (7.2) and substituting u D 1, we come to # 0 .q n I q/ D .1/nC1 q n.n1/=2 # 0 .1I q/:

(7.8)

7.5 Growth of #.uI q/. The function f .u/ D j#.uI q/j satisfies the quasiperiodicity condition f .qu/ D juj1 f .u/. It can be easily checked that the function ²

.u/ WD exp

³

1 1 ln2 juj C ln juj 2 ln jqj 2

(7.9)

satisfies the same quasiperiodicity condition, .qu/ D juj1 .u/: Therefore, their ratio

WD = is periodic. We get the following statement:

Observation 7.4. The function j#.uI q/j can be represented in the form j#.uI q/j D .u/ .u/;

where

.qu/ D

.u/:

(7.10)

7.6 Weierstrass functions. Fix a lattice L W fm!1 C n!2 g C. For definiteness, we assume Im !1 =!2 > 0 (otherwise, we change !2 7! .!2 /). Denote by S WD

1 .!N 1 !2 !N 2 !1 / > 0 2i

the area of the parallelogram spanned by the vectors !1 and !2 . Define the following Weierstrass function:10 n !N i o 1 2 &.z/ D &.zI !1 ; !2 / D exp z z #.e 2 iz=!1 ; e 2 i!2 =!1 /: 2S!1 !1

(7.11)

Proposition 7.5. a) The function &.zI !1 ; !2 / has zeros precisely at points z D m!1 C n!2 . b) The function &.z/ satisfies the quasi-periodicity conditions n o &.z C !j / D &.z/ exp (7.12) .2z !Nj C !j !Nj / ; where j D 1, 2: 2S 10 It is similar to the Weierstrass functions (see [1]), but our factor expf: : : g is not standard; we wish to have uniform growth in all directions (Lemma 7.6). Also, our factor expf: : : g provides a symmetry in !1 $ !2 in (7.12).

4.7. Preliminaries on -functions

273

Proof. a) is a rephrasing of the theorem about zeros of #. b) For the shift z 7! z C !1 , only the exponential factor in (7.11) changes. For the shift z 7! z C !2 we apply (7.1)). The calculation is straightforward. 7.7 Growth of the Weierstrass function Lemma 7.6. a) The function j&.zI !1 ; !2 /j can be represented in the form o n jzj2 '.z/; j&.zI !1 ; !2 /j D exp 2S

(7.13)

where '.z/ is a doubly periodic function, '.z C !1 / D '.z C !2 / D '.z/: b) For D m!1 C n!2 (i.e., for zeros of & )

o n jj2 : 2S

j& 0 .I !1 ; !2 /j D const exp

(7.14)

Proof. a) By (7.12),

n o N j C !j !Nj / .z !Nj C z! j&.z C !j /=&.z/j D exp n 2S oı n o D exp jz C !j j2 exp jzj2 2S 2S and this implies the desired statement. b) Certainly, the values & 0 at points of L can be evaluated explicitly by (7.7)–(7.8). But we simply differentiate (7.12) and obtain n o .2!Nj C !j !Nj / : (7.15) & 0 . C !j / D & 0 ./ exp 2S Hence, oı n o n (7.16) j C !j j2 exp jj2 : j& 0 . C !j /=& 0 ./j D exp 2S 2S Corollary 7.7. There is a constant ˇ such that o n ˇ ˇ ˇ&.z/ˇ 6 ˇ exp jzj2 for all z: 2S Denote by E the union of all lines ak .t / WD .k C 1=2/!1 C t !2 ;

bl .s/ WD s!1 C .l C 1=2/!2

(see thin lines in Figure 4.10). Consider some periodic closed neighborhood of E that does not contain points of the lattice L (for instance, is the complement of grey disks in the figure).

274


Lemma 7.8. There is a constant ˛ > 0 such that n o ˇ ˇ ˇ&.z/ˇ > ˛ exp jzj2 for z 2 : 2S

(7.17)

Proof. Denote by … the parallelogram with vertices ˙!1 =2 ˙ !2 =2 (the small parallelogram containing 0 in Figure 4.10). Let ˛ WD min

z2…\

j&.z/j ˚

: exp 2S jzj2

Now, the estimate of the type (7.17) holds in … \ . By the quasi-periodicity (Lemma 7.6), it follows that the estimate holds in the whole .

Figure 4.10. Reference Lemma 7.8 and Subsection 7.8. The set E is the grid of thin lines. Zeros of & are black points.

7.8 Proof of Perelomov Theorem 6.1. According to arguments of Subsection 6.2 we must verify the existence resp. nonexistence of a function g 2 F1 vanishing at points of the lattice L. The case S > . By (7.13) and Proposition 1.3 b), it follows that the function &.zI !1 ; !2 / is contained in F1 . The case S < . We must show that any function Q.z/ vanishing at points of L is not contained in F1 . First of all, &.zI !1 ; !2 / … F1 . Next, Q can be represented as Q.z/ D h.z/&.z/, where h.z/ is an entire function. Denote by Dk the boundary of the parallelogram with the vertices ˙.k C 1=2/!1 ˙ .k C 1=2/!2 ;

4.8. Interpolation and the Lagrange formula

275

see Figure 4.10. The estimate (7.17) holds on Dk . Therefore, ˇ oˇ n ˇ ˇ jzj2 ˇ > ˛ max jh.z/j: max ˇ&.z/h.z/= exp z2Dk z2Dk 2S If &h 2 F1 , then the left-hand side tends to 0 as k ! 1. Hence h.z/ D 0. The critical value S D . The proof is the same. Let a and b be points of the lattice. Then &.z/= .z a/.z b/ is contained in F1 (because of Proposition 1.3). Further, consider &.z/=.z a/. Let E be the same as in Lemma 7.8. Then j&.z/j > ˛ expfjzj2 =2g for z 2 . Hence, ˇ ˇ 2 ˇ &.z/ ˇ ˇ ˇ > ˛ expfjzj =2g ; ˇ ˇ

za

and the integral

Z

jz aj

z 2 ;

ˇ ˇ ˇ &.z/ ˇ2 jzj2 ˇ ˇ e d .z/ ˇ ˇ

Cn z a diverges. Therefore our function is not contained in the Fock space, see the definition (1.1).

4.8 Interpolation and the Lagrange formula 8.1 The Lagrange interpolation formula. Consider an open domain C and a sequence aj that has no limit points in . Let f .z/ be a function holomorphic in . Is it possible to reconstruct11 f from its values f .aj /? Let .z/ be a holomorphic function whose zeros are aj . Consider the function f .z/ WD

1 X j D0

f .aj / .z/ : .z aj / 0 .aj /

(8.1)

This expression is nothing but the Lagrange interpolation formula written for an infinite collection of nodes of interpolation. Formal substitution gives f .aj / D f .aj /. Indeed, f .ak / D

X j ¤k

All the summands of l’Hospital rule.

f .aj / .z/ ˇˇ

.z/ C f .ak / lim 0 : z!ak .ak /.z ak / .z aj / 0 .aj / zDak

P

.: : : / are zeros (because .ak / D 0) and lim.: : : / is 1 by the

11 Such reconstructions are used rather often in explicit calculations, see e.g., the textbook by Andrews, Askey, Roy on special functions [7](“Carlson theorem” and its numerous applications).

276


For two reasons, our success is doubtful. First, we are not sure that the series in (8.1) is convergent. Secondly, f .aj / D f .aj /, nevertheless we are not sure that f .z/ D f .z/. We only know that f .z/ f .z/ has zeros at points ak ; therefore, f .z/ f .z/ D .z/q.z/;

where q is holomorphic in .

8.2 An example of interpolation. The Paley–Wiener space. We begin our discussion with a toy example. Consider the tautological embedding L2 Œ; ! L2 .R/; namely, we extend a function f defined on Œ; to the whole line assuming that f D 0 outside Œ; . The Paley–Wiener space PW is the pushforward of L2 Œ; L2 .R/ under the Fourier transform. Since a function f 2 L2 Œ; has a compact support, it follows that the Fourier transform fO is an entire function. Recall (without proof, see [168], [89], Vol. 1) the following theorem. Theorem 8.1 (Paley–Wiener). An entire function g is an element of PW if and only if jg.z/j 6 const expfj Im zjg and the functions h˙ .t / WD g.˙i C t / are contained in L2 .R/. Problem 8.1. Write 'p .z/ WD

sin .x p/ ; xp

where p ranges in C:

Then the following reproducing property holds: hg; 'p iPW D g.p/

for any g 2 PW.

Theorem 8.2 (Whittaker–Kotelnikov–Shannon). For any function g 2 PW, g.z/ D

1 X

g.n/

nD1

1 X sin .z n/ sin z .1/n g.n/ D : zn zn nD1

(8.2)

Remark. This formula is the Lagrange interpolation formula for entire functions f 2 PW with nodes an WD n and .z/ WD sin z. Remark. This statement can also be represented as 1 X f .z/ 1 f .z/ D res ; sin z z n zDn sin z nD1

where “res” denotes the residue.

P Proof. Let g D fO. Expand f .x/ D cn e i nx . Then cn D fO.n/ and sin.xn/ is the xn i nx 2 Fourier transform of e 2 L Œ; . Thus (8.2) is simply the Fourier-transform pushforward of the expansion into a Fourier series.

277

4.8. Interpolation and the Lagrange formula

8.3 Interpolation in the Fock space Theorem 8.3 (Lyubarsky). Let m!1 C n!2 be a lattice in C, let S < . Then, for each f 2 F1 , f .z/ D &.zI !1 ; !2 /

X m;n

f .m!1 C n!2 / ; .z m!1 n!2 /& 0 .m!1 C n!2 I !1 ; !2 /

(8.3)

where & is the Weierstrass function defined in the previous section. Proof. Let f 2 F1 . We write out f by the Lagrange interpolation formula (8.1) and get the right-hand side of (8.3). Since f .z/ f .z/ D 0 at points of the lattice, f .z/ f .z/ D &.z/h.z/; where h.z/ is an entire function. Dividing both sides by &.z/, we get X m;n

f .z/ f .m!1 C n!2 / D h.z/: 0 .z m!1 n!2 /& .m!1 C n!2 / &.z/

(8.4)

Consider the contour Dk as above (see Figure 4.10) and let k tend to 1. We are going to show that both the summands on the left-hand side satisfy ˇ ˇ lim max ˇ: : : ˇ D 0: (8.5) k!1 z2Dk

If we believe this, then lim max jh.z/j D 0;

k!1 z2Dk

and the maximum principle implies h.z/ D 0; therefore f D f . Return to (8.4). For the second summand, (8.5) is obvious; indeed by (1.6), (7.17), we get ²

³

1 jf .z/j 6 exp jzj2 I 2

and

&.z/ > const exp

± ° jzj2 ; 2S

for z 2 Dk :

Since S < , we get the desired decrease. To examine the first summand, we observe that (7.16) implies ˇ ˇ ˇ f .m!1 C n!2 / ˇ ˇ 6 const expfıjm! C n! j2 g ˇ 1 2 ˇ ˇ 0

& .m!1 C n!2 /

Therefore,

for some ı > 0:

ˇX ˇ X expfıjm!1 C n!2 j2 g ˇ ˇ : : : ˇ 6 const : ˇ jz m!1 n!2 j m;n m;n

(8.6)

278


Next, we wish to estimate the last sum for z being in the grid E (see Figure 4.10). For this purpose, fix a contour Dp and consider separately points of the lattice lying inside and outside the contour. We obtain ˇX ˇ X 1 ˇ ˇ : : : ˇ 6 const ˇ jz m!1 n!2 j m;n m;nW m!1 Cn!2 is inside Dp (8.7) X C const expfıjm!1 C n!2 j2 g: m;nW m!1 Cn!2 is outside Dp

For the first sum we have noted that expfg 6 1; for the second sum we have observed that a distance between points of E and points n!1 C m!2 is bounded from below. The second sum in (8.7) is small for large p. The first sum (with a fixed p) tends to 0 as jzj ! 1. Thus (8.5) actually holds and this completes the proof.

4.9 Inversion of the Segal–Bargmann transform from points of a lattice Here we recover a functionpf 2 L2 .Rn / from values of Bf at points of a lattice k C i m. To avoid writing 2, we slightly change the normalization of B. 9.1 Inversion formula. Fix such that Re > 0 (it is pleasant but not necessary to think that 2 R). For a function f 2 L2 .R/, we define the coefficients Z 1 2 e ikx mx f .x/e x =4 dx; m;k D 1

where m and k range in Z. We wish to reconstruct f from m;k . By Perelomov Theorem 6.1, this is impossible for Re > ; for Re 6 , the problem is overdetermined. There are many ways for reconstruction of f . For instance, see Janssen [100]); we can also apply Lyubarsky Theorem 8.3. Here we discuss a formula that is relatively simple and relatively closed. Thus, suppose 0 < Re < : Write

q WD e 2 :

Define the coefficients .1/m q m.m1/=2 X .1/j q j.j C2mC1/=2 : Em . / D Q1 l /3 .1 q lD1 j >0

(9.1)

4.9. Inversion of the Segal–Bargmann transform from points of a lattice

279

Theorem 9.1. For any f 2 L2 .R/, f .x/ D

o X 1 x 2 =4 Xn m;k e ikx : Em . /e mx e 2 m

(9.2)

k

The inner sum is an L -sum of a Fourier series; the outer sum is an a.s. convergent series. P Remark. The series j >0 .: : : / in (9.1) is a terminated series for #.q mC1 I q/. Recall P that the sum of the complete series 1 1 .: : : / is 0. 2

9.2 Reduction to the interpolation problem. Write 1 2 f .x/e x =4 : 2 We apply the Poisson summation formula g.x/ WD

1 X

1 X

h.x C 2k/ D

kD1

O C l/ h.x

lD1

(see Theorem 8.2.1 below) to the function g.x/e mx and get 1 X

1 X

m;k e ikx D e mx

g.x C 2j /e 2 mj :

(9.3)

j D1

kD1

Write Am D Am .x/ D e mx

1 X

m;k e ikx :

kD1

We know k;m , therefore we know Am .x/. Fix x 2 R. Consider the function Gx .z/ WD

1 X

g.x C 2j /z j

j D1

defined in the domain C n 0. In this notation, formula (9.3) takes the form Gx .q m / D Am : We wish to find Gx .z/ from these data. For this goal, we apply the Lagrange interpolation formula (8.1). The function with zeros at points z D q n is the theta-function #.qI z/. Write Gx .z/

1 X

1 X #.zI q/ .1/nC1 q n.n1/=2 #.zI q/ Q D An A D : n .1 q j /3 .z q n /# 0 .q n I q/ .z q n / nD1 nD1 (9.4)

280


By construction,

Gx .q n / D Gx .q n /:

Therefore,

Gx .z/ D Gx .z/ C #.zI q/˛.z/

(9.5)

for a certain function ˛.z/ holomorphic in C n 0. Equivalently, Gx .z/ G .z/ x D ˛.z/: #.zI q/ #.zI q/

(9.6)

Lemma 9.2. The series (9.4) converges uniformly on compact sets in C n 0. Moreover, ˛.z/ D 0, i.e., Gx .z/ D Gx .z/. Theorem 9.1 is a corollary of this lemma. Indeed, g.x/ is the Laurent coefficient of Gx .z/ D Gx .z/ in z 0 ; it remains to evaluate the Laurent expansion of Gx .z/. For this purpose, we must write the Laurent series for 1 X #.zI q/ n 1 D .z q / .1/l z l q l.l1/=2 : z qn lD1

Assuming jzj > jqjn , we get .z

1

Cz

2 n

q Cz

3 2n

q

C /

1 X

.1/l z l q l.l1/=2 :

lD1

Finally, we obtain (9.1) as a coefficient in the front of z 0 . Remark. The function #.z; q/=.z q n / has a removable singularity at z D q n . Therefore, writing the Laurent expansion in the domain jzj < jqjn , we must obtain the same result. 9.3 Reduction of Lemma 9.2. For a function ˆ.z/ we write Mk Œˆ WD Lemma 9.3. a) limk!˙1 Mk b) limk!C1 Mk

h

.z/ Gx #.zIq/

i

h

Gx .z/ #.zIq/

i

max

jzjDq kC1=2

jˆ.z/j:

D 0.

D 0 and this sequence is bounded as k ! 1.

Assume that the last lemma is proved. Look at (9.6). By the Maximum Principle, the function ˛.z/ is bounded, therefore ˛.z/ has a removable singularity at 0. By the Liouville Theorem ˛.z/ D const. Since ˛ ! 0 at 1, it follows that ˛ D 0. Thus it remains to prove Lemma 9.3.


a)

281

b)

Figure 4.11. Reference Subsection 9.3. a) Zeros u D q k of the function .uI q/ and the contours juj D p kC1=2 . b) The distance between a circle and a point.

9.4 Preparation for the proof of Lemma 9.3. Estimation of coefficients in the interpolation formula Lemma 9.4. For any f 2 L2 .R/, the quantity X1 jf .x C 2j /j2 Vx WD j D1

is finite for almost all x and Z

Z

2

Vx dx D 0

1

jf .x/j2 dx: 1

Proof. We must justify the identity Z Z 2 X 1 jf .x/j2 dx D R

0

j D1

jf .x C 2j /j2 dx:

For this purpose, we identify the spaces Œ0; 2 Z and R by .a; n/ 7! .a C 2 n/. Applying the Fubini Theorem for a Lebesgue integral, we reduce an integral over R to an iterated integral over Œ0; 2 Z. Lemma 9.5. The following upper estimate holds: 2 2 2 1=2 jGx .z/j 6 Vx1=2 e x # jzj2 e 2x2 I e 4 : Proof. By the Schwartz inequality, ˇ ˇX 2 ˇ ˇ jGx .z/j D ˇ f .x C 2j /e .xC2j / =4 z j ˇ X 1=2 hX i1=2 2 6 jf .x C 2j /j2 e .xC2j / =2 jzj2j :

282


We rearrange the second factor as X

2 2 j 2 e 4 j.j 1/=2 jzj2 e 2x2 : : : D e x =2 De

x 2

j

2 2 # jzj2 e 2x2 I e 4 :

Corollary 9.6. For almost all x, the function jGx .z/j admits an upper estimate of the form ² ³ 1 2 jGx .z/j 6 exp ln jzj C O.ln jzj/ C O.1/ : (9.7) 4 2 Proof. We refer to the asymptotic behavior of the theta-function (7.9)–(7.10). Now we are ready to estimate coefficients in the interpolation formula (9.4). Lemma 9.7. For some " > 0 and some C , ˇ ˇ ˇ ˇ

ˇ

˚ Am ˇˇ 2 ˇ 6 C exp "m : 0 m # .q I q/

Proof. By the previous corollary, ²

jAm j D jGx .q m /j 6 exp

³

ln2 jqj 2 m C O.m/ C O.1/ : 4 2

By (7.8), ²

³

1 # .q I q/ D exp m2 ln jqj C O.m/ C O.1/ : 2 0

m

Since . ln jqj/ D 2 < 2 2 , we obtain our statement.

9.5 Proof of Lemma 9.3. Let z lie on some contour jzj D p kC1=2 . Statement a. We estimate Gx .z/=#.zI q/. By (7.9)–(7.10), ²

j#.z/j > exp

³

1 ln2 jzj C O.ln jzj/ C O.1/ ; 2 ln q

where jzj D p kC1=2 , k 2 Z:

Comparing with (9.7), we get

Mk

˚ Gx .z/ D O exp "jkj2 : #.z; q/

This proves the statement a) of Lemma 9.4. Statement b. Now let us estimate Gx .z/=#.zI q/ on the same contours:

Mk

X 2 X Gx .z/ Am e "m 1 ˇ ˇ: D Mk 6C ˇjqjkC1=2 jqjm ˇ #.zI q/ # 0 .q m I q/ z q m m m


283

ˇ ˇ Here we applied Lemma 9.7, and jz q m j > ˇjqjkC1=2 jqjm ˇ follows from Figure 4.11b. Next, we estimate the denominators: ´ ˇ ˇ mˇ mCkC1=2 ˇ ˇ kC1=2 ˇ 1 jqj > jqjm .1 jqj1=2 /; jqj m ˇjqj ˇ ˇ jqj ˇ D jqjkC1=2 ˇ1 jqjmk1=2 ˇ > jqjkC1=2 .1 jqj1=2 /: Applying the first estimate, we get Mk Œ: : : 6 C

1 X

2

e "m : jqjm .1 jqj1=2 / mD1

The series is convergent and independent in k. Therefore, the sequence Mk Œ : : : is bounded. Applying the second estimate, we obtain Mk Œ : : : 6 C

1 X

2

e "m : kC1=2 jqj .1 jqj1=2 / mD1

The right-hand side tends to zero as k ! 1. This completes the proof of Lemma 9.3 and the inversion formula. 9.6 Digression. The Post–Widder inversion formula for the Laplace transform. We cite the following result due to its exoticism. Consider the Laplace transform Z 1 g.t / D Lf .t / WD e ts f .s/ ds: 0

Then f can be reconstructed from the asymptotic behavior of g.t / for large real t in the following way: .1/k k kC1 .k/ k .1/k k 1=2 e k .k/ k 1 g g D p : (9.8) lim f .s/ D lim k!1 kŠ s s s s kC1 2 k!1 To prove this, we write

Z

1

.1/n g .n/ .nt / D

e nts s n f .s/ ds D

0

Z

1

e n.t sln s/ f .s/ ds:

0

Let us investigate the behavior of the last integral as b ! 1 by the Laplace method. The function ˛.s/ WD st ln s has a unique minimum at s0 D 1=t. Hence we obtain the asymptotics Z 1 Z 1 1 00 2 e n.tsln s/ f .s/ ds e n.˛.s0 /C 2 ˛ .s0 /.ss0 / / f .s/ ds 0

0

2 1=2 2 1=2 n˛.s0 / f .s /e D f .1=t /e n t n : 0 n˛ 00 .s0 / nt 2

Thus (at least for f with a continuous derivative) we get 1 lim .1/n .et /n t n1=2 g .n/ .nt /: f .1=t / D p 2 n!1

284


4.10 Bibliographical remarks 10.1 The (boson)L Fock space was introduced by V. A. Fock in 1929 as the direct sum of symmetric powers j1D0 Sn .H / of an infinite-dimensional Hilbert space H , see our 1.6, [14], [176], [145]. Several functional models of the Fock space are known. In particular: – L2 on a Hilbert space. This model was introduced by I. Segal, a nice readable text is the book by G. E. Shilov and Fan Dyk Tin [201]; unfortunately, it exists only in Russian. – Bargmann–Segal holomorphic model, i.e., our Section 4.1; it was introduced near 1960 by I. Segal, V. Bargmann, F. A. Berezin; see also [14], [145]. – The P. Cartier model [34], see Chapter 8. – L2 on the Poisson process (introduced independently by G. A. Goldin–J. Grodnik– R. T. Powers–D. H. Sharp, A. M. Vershik–I. M. Gelfand–M. I. Graev, and R. S. Ismagilov in 1974–1975, see [144]). – The fermion Fock space admits a canonical identification with our (boson) Fock space (T. H. R. Skyrme,1971), see [176], [152]. – The space of symmetric functions; I do not know who observed this correspondence, see [176], [152]. – The space of holomorphic functionals on the space of univalent functions, see [141]. The last three models exist only for an infinite-dimensional Hilbert space H , i.e., they are characteristic phenomena of analysis in an infinite number of variables. The Cartier model is known (or well-understood) only for finite-dimensional H . 10.2 The Segal–Bargmann transform was introduced (dependently) by I. Segal and V. Bargmann around 1960. 10.3 Short-time Fourier transforms usually are attributed to Gabor (1950s). I am not sure that this is precise since the radio communication existed earlier. On this subject, see e.g., [76]. 10.4 Wave fronts and the microlocal analysis, see the books by L. Hörmander [89], Vol. 1, 3, and G. B. Folland [51]. See also [40], [97]. @ 10.5 Symbolic calculus. Functions in x, @x were widely discussed in the 1920–30s in quantum mechanics, in the related mathematics (e.g., J. von Neumann, H. Weyl), and also in abstract algebra (D. E. Littlewood [123]). Pseudo-differential operators were introduced by J. J. Kohn and L. Nirenberg in 1965. On pseudo-differential operators (and on pseudo-differential Weyl symbols) see, for instance, L. Hörmander [89], Vol. 3). Expressions in creation-annihilation operators were used in field-theoretic physics, the mathematical theory appeared in [14]. On symbolic calculus, see papers by F. A. Berezin [16], [17] and the book by G. Folland [51]. 10.6 Campbell–Hausdorff–Dynkin formula. For further discussion, see F. A. Berezin [15], C. Reutenauer [180], A. Alekseev, E. Meinricken [3]. On samples of calculations in enveloping algebras, see [135], [204]. 10.7 Theta-functions. See a nice exposition in the book by Akhiezer [1]. 10.8 Entire functions and interpolation. For a general discussion of distribution of zeros of holomorphic functions and interpolation, see [23], [117], [118], [127], [196]; for § 4.6, see [100], [157].


285

10.9 E. Post–D. V. Widder formula. See Widder’s book [217]. Widder also refers to the J. S. Stieltjes–Ch. Hermite private correspondence, 29 August, 1893.

5 Gaussian operators in Fock spaces

The Segal–Bargmann transform B identifies L2 .Rn / with the Fock space Fn . Gaussian operators in Fn are B-pushforwards of Gaussian operators in L2 .Rn /. It is easy to translate the main results of Chapter 1 to the language of Fock spaces. However, we give independent proofs. The results are formulated in §1, the remaining two sections contain proofs.

5.1 Gaussian operators 1.1 Gaussian vectors. For a symmetric n n matrix R denote by bŒR the Gaussian function on Cn given by1

˚ bŒR.z/ D exp 12 zRz t : Proposition 1.1. a) bŒR is an element of the Fock space Fn if and only if kRk < 1. b) bŒR is a B-image of a tempered distribution if and only if kRk 6 1. c) Inner products of Gaussian vectors are given by ˛ ˝ x 1=2 : bŒT ; bŒR D det.1 T R/ (1.1) Note that the vectors bŒR are parameterized by points T of the Cartan matrix ball Bn , see Chapter 3. Proof. The statement a) follows from the estimates of growth of functions f 2 Fn , see Proposition 4.1.3. b) For the space F1 , we must examine functions expfz 2 g; we can refer to Proposition 4.2.6. For a general n, consider transformations Fn ! Fn given by f .z/ 7! f .zU /, where U is a unitary matrix. Then Gaussian vectors change as bŒT 7! bŒU T U t . Under transformations T 7! U T U t a complex symmetric matrix can be reduced to the diagonal form, see Lemma 3.3.7. Thus, without loss of generality, we can assume o nX j zj2 ; where j > 0: bŒT .z/ D exp Now the statement becomes obvious. c) This integral is evaluated in the next lemma. 1

This notation is similar to the parallel notation from Chapter 1, p. 19, but it is not identical.


Problem 1.1. Find the B-preimage of expf 12

P

287

j zj2 g.

Problem 1.2. a) Gaussian vectors in Fn are precisely the images of Gaussian vectors in L2 .Rn / under B. b) The B-images of Gaussian distributions 2 0 .Rn / are precisely the vectors bŒT with kT k 6 1. Hint. Without loss of generality, we can consider Gaussian distributions of the form expf 12 . 1 x12 C C k xk2 /gı.xkC1 / : : : ı.xn /, where Re k > 0. c) Prove Proposition 1.3.3 (“the cone of Gaussian distributions is closed in 0 .Rn /”). 1.2 Gaussian vectors. Extended variant. Consider more general vectors

˚ bŒT j WD exp 12 zT z t C z t ; where kT k < 1:

(1.2)

Lemma 1.2. a) A vector bŒT j is an element of Fn . ² ³ ˝ ˛ 1 T 1 t 1=2 x b) bŒT j ; bŒRj D det.1 T R/ exp N : (1.3) x 1 Nt R Proof. The statement a) is obvious, because bŒT j satisfies the estimate (4.1.6). Let us evaluate the inner products, Z ˝ ˛ ˚

® ¯ xzN t C NzN t exp¹z zN ºd .z/ bŒT j ; bŒRj D exp 12 zT z t C z t exp 12 zN R n ³ t ZC ² zt T 1 1 z N z zN D exp d .z/ C x zN t zN t 1 R 2 Cn t t Z ² 1 i 1 T 1 x 1 i x y D exp t x y 1 i 1 i 1 R 2 Cn t ³ 1 i dx dy x C N ; t 1 i y n and then apply the usual Gauss integral.

˚ Problem 1.3. When is exp 12 z 2 C z 2 0 F1 ? 1.3 Gaussian operators. Let K L SD ; Lt M

where K D K t , M D M t

be a symmetric .m C n/ .m C n/-matrix. We define the corresponding Gaussian operator BŒS W Fn ! Fm by t³ Z ² K L 1 2 z z uN BŒSf .x/ D exp f .u/e juj d .u/: (1.4) t t M u N L n 2 C

288

Chapter 5. Gaussian operators in Fock spaces

Proposition 1.3. If BŒS W Fn ! Fm is bounded, then 1. kS k 6 1, 2. kKk < 1 and kM k < 1. Proof. The first statement follows from the a priori estimate (4.1.11) of kernels of bounded operators. x Therefore, bŒM 2 Fm and Further, bŒS 1 D bŒM and bŒS 1 D bŒK. x bŒK 2 Fn and we refer to Proposition 1.1. Theorem 1.4 (Olshanski). Under the conditions 1 and 2 of Proposition 1.3, the operator BŒS W Fn ! Fm is bounded. Actually Gaussian operators in Fn are B-pushforwards of Gaussian operators in L2 .Rn /. Thus, the last theorem is equivalent to the boundedness Theorem 1.8.2, see p. 45. However, in this chapter, we present an independent proof. In Subsection 1.9.7 we defined the involution P 7! P in the category Sp. In our coordinates it is given by x L K L M ! 7 : Lt N LN Kx 1.4 Creation-annihilation operators. Let W2n D WnC ˚ Wn D Cn ˚ Cn be the complexification of a real symplectic space (see § 3.2). Recall that this space is endowed with: 0 1 – the symmetric bilinear form ƒ with the matrix 1 0 ; 1 0 – the Hermitian form M with the matrix 0 1 ; – the operator J W v C ˚ v 7! vN ˚ vN C of complex conjugation. Recall that these spaces are objects of the symplectic category Sp, see § 1.8. Consider the creation–annihilation operators a.w/ O WD

n X j D1

wjC zj

n X j D1

wj

@ : @zj

Obviously, D a.J O w/I a.w/ O Œa.v/; O a.w/ O D ƒ.v; w/ 1I D M.v; w/ 1: Œa.v/; O a.w/ O

289


Certainly, these relations are equivalent to (1.6.4)–(1.6.3); the explicit correspondence of operators is given by formulas (4.2.4). 1.5 Main construction. Let P W W2n W2m be a morphism of the category Sp. L Let S.P / D LKt M be its Potapov transform, i.e., P consists of vectors v ˚ w 2 W2n ˚ W2m such that K L wC v D w vC : (1.5) Lt M We define the Gaussian operator2 Be.P / D BŒS.P / W Fn ! Fm :

Theorem 1.5. a) Be.P / is bounded as an operator Fn ! Fm . b) Be.P / is bounded as an operator Fn ! Fm . c) For each v ˚ w 2 P , a.w/ O Be.P / D Be.P /a.v/: O

(1.6)

The operator Be.P / is a unique (up to a scalar factor) operator satisfying these conditions. d) Let P W W2n W 2m and Q W W2m ! W2k be morphisms of the category Sp, K L z L Q K let Lt M and Q t z be their Potapov transforms. Then3 L M

z K/1=2 Be.QP / Be.Q/ Be.P / D det.1 M or, equivalently Kz LQ K B Qt B t z L L M

L M

(1.7)

z K/ D det.1 M

1=2

B

Qt z LK.1 Q z K/1 L KC M t 1 t Q z L .1M K/ L

Q z /1 L L.1K M t zL z M CL .1M K/1 M

(1.8) :

e) The operators Be.P / W Fn ! Fm defined now coincide with the B-images of the operators Be.P / W L2 .Rn / ! L2 .Rm / defined above (Section 1.8). ˆ ‰ f) Let g D ‰ x ˆ x 2 Sp.2n; R/. Then the operators O

1=2

Be .g/ WD ˙ det.ˆ/ 2 3

1=2

Be.g/ D ˙ det.ˆ/

ˆ1 ‰ ˆ1 B x 1 ‰ˆ ˆt1

Thus we use the dual notation BŒS and Be.P / for the same operators. The scalar factor in (1.7) is well defined by (2.3.6).

(1.9)

290


are unitary (there is no canonical choice of sign) and BeO .g1 / BeO .g2 / D ˙ BeO .g1 g2 /:

(1.10)

g) The operators BeO .g/ are bijections Fn ! Fn . h) Be.P / D Be.P / The theorem is proved in the next two sections. h i 1 ‰ ˆ1 Remark. The matrix ˆ from formula (1.9) is unitary; indeed, g 2 t1 1 x ‰ˆ ˆ Sp.2n; R/ U.n; n/ and we apply Theorem 2.8.1. 1.6 Digression. Normalization of Gaussian operators. The relation (1.6) determines the operators Be.P / to within a scalar. However, formula (1.4) defines the operators Be.P / canonically. In fact, our operators Be.P / W Fn ! Fm satisfy the following normalization condition: hBe.P / 1Fn ; 1Fm iFm D 1:

(1.11)

This normalization produces the scalar factor ²

³

z K/1=2 D exp 1 tr ln.1 M z K/ det.1 M 2 ˚

in (1.7). This factor is exp 12 c.P; Q/ , where c.P; Q/ is the cocycle given by (3.8.2). For a further discussion, see Chapter 3. If we change a normalization of Be.P /, then the scalar factor in (1.7) changes. Observation 1.6. A canonical choice of the sign in the formula (1.9) is impossible; precisely, we can not get BeO .g1 / BeO .g2 / D BeO .g1 g2 /

(1.12)

by changing a normalization of BeO .g/. Proof. Let g 2 Sp.2n; R/ be in a small neighborhood U of 1. Then ˆ is close to 1; therefore we have a canonical square root .det ˆ/1=2 in a neighborhood of 1. Also, in a neighborhood U we have (1.12). If .g/ BeO .g/ satisfies the same identity (1.12), then .g/ is a local homomorphism U ! f˙1g (see Lemma 3.7.7). Since there is no such homomorphism, we get .g/ D 1. Now let g 2 U.n/ Sp.2n; R/, i.e., ‰ D 0. Then we must continue the function 0 ˆ1 det ˆ1=2 B t1 ˆ 0 to the whole group U.n/. Obviously, this is impossible.

5.2. Proof of the formula for products

291

Observation 1.7. We can extend the function BeO .g/ to the semigroup Spn (see Section 3.5) by 1 a b a1 a b ; here g D : BeO .g/ WD ˙ det a1=2 B c d at1 ca1 Then the identity (1.9) survives. This is obvious. A further (non-zero) continuation of BeO is impossible. Moreover, we can not reduce the cocycle to a smaller subgroup. Precisely: Observation 1.8. Fix some C -valued function r on the set of endomorphisms of an object of the category Sp. Set Be4 .P / WD r.P / Be.P /. Let Be4 .P / Be4 .Q/ D c 4 .P; Q/ Be4 .PQ/:

Then the numbers c 4 .P; Q/ generate the whole group C . Proof. K 0 Let C be a proper subgroup. Let us consider only rank 1 operators B 0 M . Then 0 0 K 0 K K 0 0 1=2 B B B D det.1 MK / : 0 M0 0 M 0 M0

We can change the cocycle by the rule det.1 MK 0 /1=2 7! det.1 MK 0 /1=2

‹ r.K; M 0 / 2 : 0 0 r.K; M /r.K ; M /

Assume that the last expression is contained in . Substitute K D K 0 . We get det.1 MK/1=2 2 : r.K; M 0 / This is impossible; precisely, we can break this inclusion by changing M .

5.2 Proof of the formula for products This section contains the proof of the formula (1.7)–(1.8) for products of Gaussian operators. The formula itself is more-or-less trivial, we simply calculate the convolution of kernels by (4.1.13). Nevertheless, this is a proof only modulo the boundedness Theorem 1.4; on the other hand, we need the formula (1.7) for the proof of the boundedness. We also prove all the statements of Theorem 1.5 except the boundedness.

292


2.1 Formal definition of BŒS Observation 2.1. For each f 2 Fn , the expression BŒS f is a well-defined entire function on Cm . Proof. Indeed, ³Z ² ³ 1 1 2 t t t BŒSf .z/ D exp exp uM uN C zLuN f .u/ e juj d .u/ zKz n 2 2 ² ³ C 1 x jzLiFn : D exp zKz t hf; bŒM 2 ²

(2.1)

x jzL 2 Fn . The inner product is well defined because f , bŒM

Thus we have defined operators BŒS as operators from Fn to the space of entire functions. 2.2 The Hilbert–Schmidt condition for Gaussian operators Lemma 2.2. Assume kS k < 1. a) The operator BŒS W Fn ! Fm is a Hilbert–Schmidt operator. In particular, BŒS is bounded. b) The operator BŒS sends Fn ! Fm . Proof. a) In this case, the kernel ²

1 z S.z; u/ N D exp 2 is contained in

K uN Lt

L M

t³ z uN t

2 2 L2 Cn ˚ Cm ; e jzj juj d .z/ d .u/ :

Therefore (see Subsection 1.1.2) this kernel determines a Hilbert–Schmidt operator 2 2 BŒS W L2 Cn ; e juj d .u/ ! L2 Cm ; e jzj d .z/ : Our operator BŒS is the restriction of BŒS to the subspace Fn . Therefore BŒS is a Hilbert–Schmidt operator. b) is proved in the next subsection. 2.3 Products of bounded operators Lemma 2.3. For bounded Gaussian operators BŒS , BŒSz, the formula (1.8) holds.

293


Proof. We must evaluate Z ² Kz 1 z uN exp Qt L 2 Cm

LQ z M

²

t³ z uN t

²

exp

K 1 u N Lt 2

L M

t ³ 2 u e juj d .u/ Nt

³

˛ 1 z t 1 N Nt ˝ z Q ; b Kx j L Nt : z Kz C M b M j zL 2 2 We apply formula (1.3). D exp

In particular, the product formula holds for operators BŒS1 , BŒS2 such that kS1 k < 1, kS2 k < 1. It suggests that the formula can be extended to the general case by continuity considerations. This actually is done in Subsection 2.6 after some preparations. Proof of Lemma 2.2.b. We decompose BŒS as K L 0 1" .1 "/2 K DB B B t .1 "/1 Lt L M 1" 0

.1 "/1 L : M

If " > 0 is sufficiently small, then the second factor satisfies the condition of the lemma; therefore it is a bounded operator Fn ! Fm . The first factor equals 0 1" B f .z/ D f .1 "/z 1" 0 and it sends Fn to Fm (see the characterization of Fn in Proposition 4.2.5).

2.4 Action on Gaussian vectors L Proposition 2.4. Let LKt M satisfy the conditions of Proposition 1.3, i.e., the hypothetical conditions of boundedness. Let kT k < 1. Then K L B bŒT j Lt M (2.2) ˇ

1=2 1 t ˇ 1 t D det.1 M T / b K C LT .1 M T / L L.1 TM / : Moreover, the vector bŒ on the right-hand side is contained in Fm . Proof. K B Lt D

L bŒT j .u/ M expf 12 zKz t g

Z

Cn

˚

˚

exp 12 uM N uN t C zLuN t exp 12 uT ut C u t e uuN d .u/:

The last integral is an inner product of the form (1.3). This implies the explicit formula (2.2). Finally, we must show that kK C LT .1 M T /1 Lt k < 1, this is necessary for bŒ 2 Fn . This is not obvious, however we can refer to our discussion of the Krein–Shmul’yan functor, see (2.9.3).

294


In particular, K L B bŒT D det.1 M T /1=2 b K C LT .1 M T /1 Lt : t L M

(2.3)

Observation 2.5. An operator Be.P / sends a Gaussian vector bŒT to a vector of the form const bŒT 0 , where T 0 is the image of T under the Krein–Shmul’yan map corresponding to P . 2.5 An invariant dense subspace. Denote by Fn the subspace in Fn consisting of vectors of the form p.z/ expf 12 zT z t C z t g;

where p.z/ is a polynomial and kT k < 1:

Proposition 2.6. For any f 2 Fn and each BŒS satisfying the conditions of the boundedness theorem, the vector BŒS f is contained in Fm . Proof. Let " WD ."1 ; : : : ; "n / 2 Cn . Consider the following f 2 Fn :

Then B

K

Lt

f .z/ D z1p1 : : : znpn expf 12 zT z t C z t g ˇ d p1 d pn t t t ˇ 1 D p1 : : : pn expf 2 zT z C z C "z g "D0 : d "1 d "n

L f is M

(2.4)

ˇ

ˇ d p1 d pn 1 t ˇ 1 t ˇ p1 : : : pn b KCLT .1M T / L L.1TM / . C"/ "D0 d "1 d "n (2.5) and this vector is contained in Fn . It is also necessary to watch the differentiation with respect to parameters in (2.1). det.1M T /1=2

2.6 Proof of the product formula (1.7). General case Lemma 2.7. Let Be.Q/, Be.P / satisfy the conditions of the boundedness theorem. Then, for any f 2 Fn , Be.Q/ Be.P /f D Be.QP /f

(2.6)

and is the same as in (1.7). Proof. This is obvious for the following reason. Let S , Sz be Potapov transforms of P and Q and let Y p 1 f .z/ D zj j expf zT z t C z t g: 2 z If kSk < 1, kSk < 1, then our operators are bounded and we can refer to Lemma 2.2.


295

Further, both the sides of (2.6) are explicit expressions depending real-analytically on4 S , Sz, T , . z < 1, then Since the left-hand side and the right-hand side coincide if kSk < 1, kSk they coincide always. 2.7 Verification of the commutation relations (1.6) Lemma 2.8. Let P W W2n W2m be a morphism of the category Sp. For each f 2 Fn and v ˚ w 2 P , a.w/ O Be.P /f D Be.P /a.v/f: O

(2.7)

Proof. First of all, by Observation 2.1 both sides are well defined. Denote the kernel of the operator Be.P / by S.z; u/, N t³ ² K L 1 z z uN : S.z; u/ N D exp t M u Nt L 2

(2.8)

Let us apply formulas (4.1.18)–(4.1.21). The kernel of the operator a.w/ O Be.P / is X

wjC zj

X

wj

h i @ S.z; u/ N D w C z t w Kz t w LuN t S.z; u/: N @zj

(2.9)

O is The kernel of the operator Be.P /a.v/ X X

@ C vj uN S.z; u/ N D v uN t v C K uN t v C Lt z t S.z; u/: N (2.10) vjC @uNj The factors in the square brackets in (2.9) and (2.10) coincide if and only if v ˚ w 2 P , see (1.5). 2.8 Adjoint operators. The statement h) of Theorem 1.5 is obvious. 2.9 Unitary operators. Now we are going to prove statement f). For each g 2 Sp.2n; R/, we have g g D 1. By Subsection 2.8, Be.g/ Be.g/ D Be.gg / D 1;

2 C :

The formula (1.7) for products provides p D det ˆˆ : p Since D j det ˆj, it follows that the operators BeO .g/ WD ˙ det.ˆ/1=2 Be.g/ are unitary. 4 It is not too difficult (and not too pleasant) to write these expressions explicitly; however this is not necessary for the proof.

296


It remains to evaluate products of the operators BeO.g/. Return to the notation ˆ.g/ ‰.g/ . The formula for products of § 3.7. For each g 2 Sp.2n; R/ let g D ‰.g/ x x ˆ.g/ provides

Be.g1 / Be.g2 / D det 1 C ˆ.g1 /1 ‰.g1 /‰.g2 //ˆ.g2 /1

1=2

Be.g1 g2 /:

We write out the determinant factor as 1=2 ˙ det ˆ.g1 /1=2 det ˆ.g1 /ˆ.g2 / C ‰.g1 /‰.g2 / ˆ.g2 /1=2 D ˙ det ˆ.g1 /1=2 det ˆ.g1 g2 /1=2 ˆ.g2 /1=2 I

(2.11)

we have to write ˙ since det ˆ.g/1=2 is not canonically defined. In other words, BeO .g1 /BeO .g2 / D ˙BeO .g1 g2 /. Thus all the statements of Theorem 1.5, except the boundedness, are proved. 2.10 Reduction of boundedness to a self-adjoint case. An operator Be.P / is bounded if and only if the following function is bounded on the unit ball kf k 6 1 in Fn : .f / D hBe.P /f; Be.P /f i D hf; Be.P / Be.P /f i D const hf; Be.P P /f i: Thus the problem reduces to boundedness of the symmetric operator Be.P P /.

5.3 Boundedness. Spectra of self-adjoint Gaussian operators Here we prove Boundedness Theorem 1.5. Recall that we have reduced the problem to self-adjoint operators. 3.1 Fix-point argument. Let P W W2n W2n be a morphism of the category Sp and let K L S.P / D (3.1) Lt M be its Potapov transform. Theorem 3.1. a) There exists a function bŒT with kT k 6 1 such that Be.P /bŒT D bŒT :

Moreover, D det.1 M T /1=2 : b) If kS.P /k < 1, then kT k < 1, i.e., bŒT 2 Fn .

(3.2)

5.3. Boundedness. Spectra of self-adjoint Gaussian operators

297

x n the closed Cartan matrix ball kT k 6 1. Let ~.P / W B xn ! B xn Proof. Denote by B be the Krein–Shmul’yan map corresponding to P . By the Brouwer theorem, it has a fixed point T . By (2.3), bŒT is an eigenfunction of Be.P /. x n to Bn . Therefore, a fixed point 2 Bn . If kSk < 1, then ~.P / sends B See also Theorem 2.7.12 about invariant subspaces. 3.2 Application to the self-adjoint case. Let P W W2n W2n be a self-adjoint morphism of the category Sp. Its Potapov transform satisfies K L x L D L : ; M D K; (3.3) S.P / D Lt M Theorem 3.2. Suppose Be.P / has a Gaussian eigenvector bŒT 2 Fn (this holds automatically if kS.P /k < 1). Denote by its eigenvalue. Then a) bŒT is a maximizer, i.e., k Be.P /k D . b) D det.1 M T /1=2 . c) The spectrum of Be.P / has the form n o n Y l jj ; j D1

where the j are fixed, 1 6 j 6 1, and the lj range in f0; 1; 2; 3; : : : g. Proof. Step 1. We can assume T D 0. Indeed, by Observation 2.5, for g 2 Sp.2n; R/ and T 2 Bn , Be.g/bŒT D const bŒT Œg ;

x 1 .‰ C z ˆ/: x where T Œg WD .ˆ C z ‰/

Since the action Sp.2n; R/ on the matrix ball is transitive, there is g 2 Sp.2n; R/ such that Be.g/bŒT D bŒ0. Further, bŒ0 is an eigenvector of Be.gP g 1 /; indeed Be.gP g 1 /bŒ0 D const BeO .g/ Be.P / BeO .g/1 bŒ0

D const0 BeO .g/ Be.P /bŒT D const00 BeO .g/bŒT D const000 bŒ0: 1 Thus passing from P to can assume T D 0. gPB g B we K L B Step 2. Let S D .LB /t M B be the Potapov transform of gP g 1 . Since bŒ0 is an eigenvector, formula (2.3) implies

0 D K B C LB 0.1 M B 0/1 .LB /t ; i.e., K B D 0. Since P is self-adjoint, it follows that M B D 0 and LB D .LB / (see (3.3).

298


Thus Be.P / can be represented in the form 0 Be.P / D const BeO .g/ B .LB /t

LB BeO .g/1 : 0

(3.4)

The middle factor is an operator of change of variable (see Subsection 4.1.7) 0 LB f .z/ D f .zL/: (3.5) B .LB /t 0 Step 3. We can reduce the self-adjoint matrix LB to the diagonal form, LB D UU 1 . Here the matrix U is unitary and is a diagonal matrix; denote by j the diagonal entries of . Q D , otherwise, we change coordinates in Without loss of generality, we can let L Cn by the matrix U . Thus the middle factor in (3.4) has the form 0 f .z1 ; : : : ; zn / D f .1 z1 ; : : : ; n zn /: B 0 Q l Q l The monomials zjj are the eigenvectors, the eigenvalues are jj , and f .z/ D 1 is a maximizer (since kLk 6 1). The initial operator Be.P / is conjugate to the operator of a change of variable (3.5) to within a scalar. Further, BeO .g/1 bŒ0 D const bŒT :

Therefore, bŒT is the maximizer for Be.P /. Hence the constant in (3.4) coincides with the eigenvalue of bŒT , i.e., D det.1 M T /1=2 . 3.3 An estimate of a norm Lemma 3.3. Let A be a square matrix with kAk < 1. Then: a) j det .1 A/j > det .1 jAj/. b) For each matrix R with kRk < 1, we have j det.1 RA/j > det.1 jAj/. Proof. a) Let aj be the eigenvalues of A, ˛i be the singular values. We must verify the inequality Y Y .1 jaj j/ > .1 ˛j /; j j P P or, equivalently, j ln.1 jaj j/ > j ln.1 ˛j /, or 1 1 X X 1 X 1 X jaj jn 6 j˛j jn : n n nD1 nD1 j

j

5.3. Boundedness. Spectra of self-adjoint Gaussian operators

299

The last inequality follows from the inequality X X jaj jn 6 ˛jn ; j

j

which is a special case of the Weyl inequalities for eigenvalues and singular numbers, see [216]. b) Let ˇj be the singular values of RA. By the mini-max characterization of singular values (2.5.4), it follows that ˇj 6 ˛j . Theorem 3.4. For a self-adjoint Gaussian operator BŒS , K L 1=2 B t : L M 6 det.1 jM j/ Proof. First, let kSk < 1. In this case, the eigenfunction bŒT is contained in Fn and the norm of the operator is det.1 M T /1=2 6 det.1 jM j/1=2 . Next, let us apply limit considerations. Consider a matrix .1 "/S,

B .1 "/S 6 det 1 .1 "/jM j 1=2 6 det.1 jM j/1=2 :

The kernels of B .1 "/S converge pointwise to the kernel of BŒS and norms of BŒ.1 "/S are uniformly bounded. By Proposition 4.1.13, it follows that the family BŒ.1 "/S weakly converges to BŒS , the operator BŒS is bounded, and its norm is majorized by the same constant. This also completes the proof of the boundedness of operators BŒS in Hilbert spaces Fn . 3.4 Operators BŒS in Fn . It remains to prove the statement b) of Theorem 1.5, i.e., we must show that our operators are bounded in the sense of Fn . Lemma 3.5. a) For each g 2 Sp.2n; R/, BeO .g/a.v/ O BeO .g/1 D a.vg/I O

(3.6)

in particular, domains of definiteness of these operators in Fn coincide. b) The operator BeO .g/ preserves the space Fn . Proof. a) By Lemma 2.8, the identity holds on a dense subspace Fn . The operator

BeO .g/ is unitary, therefore the conjugation by BeO .g/ sends closed operators to closed

operators. b) We use a characterization of Fn mentioned in the proof of Proposition 4.2.5.a. By the statement a), operators BeO .g/ regard this characterization. Since both BeO .g/, BeO .g 1 / send the space Fn to itself, it follows that BeO .g/ is a bijection Fn ! Fn ; this is the statement g) of Theorem 1.5. At this moment, we have not yet proved the statement b) of Theorem 1.5 (the operators BŒS are bounded in Fn . Here we refer to Theorem 1.3) which makes our statement trivial.

300


5.4 Bibliographical remarks The representation of the symplectic group Sp.2n; R/ in the Fock space (mainly for n D 1) was considered by F. A. Berezin in his seminal book “Method of second quantization” [14]. The boundedness theorem was obtained by G. I. Olshanski around 1983. Gaussian operators in the Fock space are exposed in a wider generality in [145].

6 Gaussian operators. Details

This chapter contains more detailed discussion of Gaussian integral operators (spectra, eigenvectors, norms, canonical forms, one-parametric groups and semigroups, explicit expression for matrices, etc.). As the preface warns, the author risks going too deeply in too many directions. As a compromise, we consider here only operators BŒS W Fn ! Fn .

6.1 Canonical forms and invariants 1.1 Tensor products Observation 1.1. a) Fn ˝ Fk ' FnCk . b) If operators A W Fm ! Fn and A0 W Fm0 ! Fn0 have kernels K.z; u/ and 0 0 K .z ; u0 / respectively, then the operator A ˝ A0 has the kernel K.z; u/K.z 0 ; u0 /. See the definition of products of Hilbert spaces, Subsection 1.1.6. The canonical operator Fn ˝ Fk ! FnCk sends a vector f ˝ f 0 to f .z/f 0 .z 0 /. Observation 1.2. B

K Lt

L K0 ˝B M .L0 /t

2 K 60 L0 D B6 4Lt M0 0

0 K0 0 .L0 /t

L 0 M 0

3 0 L0 7 7: 0 5 M0

1.2 Canonical forms of Gaussian operators Theorem 1.3. a) Each bounded Gaussian operator BŒS in Fn admits a decomposition BŒS D U1 BŒR U2 ;

(1.1)

where 2 C , U1 , U2 are unitary Gaussian operators (corresponding to elements of Sp.2n; R/) and BŒR is a tensor product of operators F1 ! F1 of the following two types: 0 sj ; where 0 6 sj 6 1; I: B (1.2) sj 0 1 k

k II: B ; where 0 < k < 1: (1.3)

k 1 k

302

Chapter 6. Gaussian operators. Details

b) The collection sj is uniquely determined by the operator BŒS . Also, the number of blocks of each type is uniquely determined by the operator BŒS . c) If kSk < 1, then there are only blocks of the first type with 0 < sj < 1. Remark. The numbers j are not invariants. Moreover, all operators in F1 of type II are conjugate by elements of Sp.2; R/. Indeed, these operators correspond to the operators A. /f .x/ D e x

2 =2

f .x/

in L2 .R/. The operators A. / are conjugate under the dilatations H.a/f .x/ D a1=2 f .ax/: Passing to the limit as ! 0, we get the identity operator; in our classification it is the operator of type I with s D 1). Proof. This is a rephrasing of canonical forms of symplectic contractions, see Sections 1.10, 3.5. 1.3 Invariants sj of Gaussian operators. We say that the numbers fs1 ; s2 ; : : : ; sn g are the invariants of the Gaussian operator BŒS W Fn ! Fn ; we assign s WD 1 to matrices of type II. Theorem 1.4. The roots of the equation ² ³ 1 0 1 0 det .1 S S/ S D0 S 0 1 0 1

(1.4)

are Sp.2n; R/ Sp.2n; R/-invariants of the operator BŒS . In addition, D ˙.1 sj2 /=.1 C sj2 /:

(1.5)

Proof. Let W2n be the complexification of a real symplectic space, as in Subsection 5.1.5. We have a contractive linear relation P W W2n W2n corresponding to the matrix S as in equation (5.1.5). Define the following two Hermitian forms on W2n ˚ W2n : M .v ˚ w; v 0 ˚ w 0 / WD M.v; v 0 / M.w; w 0 /; M ˚ .v ˚ w; v 0 ˚ w 0 / WD M.v; v 0 / C M.w; w 0 /: The group Sp.2n; R/ Sp.2n; R/ acts on W2n ˚ W2n in the natural way and preserves both forms.

6.1. Canonical forms and invariants

303

Further, restricting these forms to the subspace P , we get a pair of Hermitian forms on this subspace. Writing out these forms in the coordinates .v ; wC /, we get the Hermitian matrices 1 0 1 0 S S: .1 S S/; 0 1 0 1 We write invariants of a pair of Hermitian matrices as it was explained in Theorem 2.6.9. This implies the first statement of the theorem. 0 † Next, we evaluate the invariants j for the matrix S WD † 0 , where † is the diagonal matrix with entries sj . Comparing the results, we come to (1.5). 1.4 Compactness of operators BŒS Observation 1.5. If kSk < 1, then the operator BŒS is compact. Moreover, it is contained in all Schatten classes Lp . Remark. Recall that a compact operator A is P said to be an element of the Schatten class Lp if the singular values j of A satisfy j jp < 1. For instance, the class L2 is the class of Hilbert–Schmidt operators and L1 are operators of trace class. Other Schatten classes do not appear in this book. Proof. By Theorem, 1.3 the singular values of BŒS have the form s1j1 : : : snjn ; Another proof. We represent K L K D B B .1 "/1 Lt Lt M

where j˛ 2 ZC :

.1 "/1 L 0 1" B ; .1 "/2 M 1" 0

where the first factor on the right-hand side is bounded. The second factor is contained in all Schatten ideals. Problem 1.1. Consider an operator BŒR in L2 .Rn /, let Re R < 0. Let f be a tempered distribution. Then BŒRf 2 .Rn /; moreover, BŒRf is an entire function on Cn Rn . 1.5 Canonical forms of self-adjoint Gaussian operators Proposition 1.6. Each self-adjoint operator BŒS in Fn admits a representation BŒS D U BŒRU 1 ;

where 2 C , U is a unitary Gaussian operator, and BŒR is a tensor product of operators F1 ! F1 of the following two types: 0 tj ; where 1 6 tj 6 1; (1.6) I: B tj 0 ˙.1 k / k II: B ; where 0 < k < 1: (1.7) k ˙.1 k /

304


We omit a proof, because the statement is simply a symplectic variant of Lemma 2.7.9. Problem 1.2. Find the spectra of the operators (1.6)–(1.7). Find (generalized) eigenvectors. We say that the numbers t1 , t2 ; : : : are the invariants of BŒS . To each matrix of type II we assign t D ˙1 according to the sign in the matrix. 1.6 Invariants tj of self-adjoint operators. Consider a self-adjoint operator K L x. ; where L D L , K D M BŒS D B Lt M For simplicity, assume kS k < 1 and det L ¤ 0. In other words, the corresponding linear relation P is a graph of a (pseudo)-self-adjoint strictly contractive element g 2 Sp.2n; C/. Proposition 1.7. Under these conditions, the roots of the equation det Lt C tM.1 tL/1 K t D 0

(1.8)

lying in the interval .1; 1/ coincide with the invariants tj . Proof. Consider the corresponding element 1 L L1 K gD 2 Sp.2n; C/ ML1 Lt ML1 K (this is formula (2.8.4) for the inverse Potapov transform). Since g is symplectic, its spectrum is invariant under the transformation 7! 1 . Since g is pseudo-self-adjoint and contractive, its spectrum is real (see Lemma 2.7.9). We write the characteristic equation det.g / D 0 and apply formula (1.1.12) for a determinant of a block matrix, 1 L L1 K 0 D det ML1 Lt ML1 K (1.9) D det.L1 / det Lt ML1 K C ML1 .L1 /1 L1 K : Next, we rearrange the following subexpression of the last formula: ML1 K C ML1 .L1 /1 L1 K D M.L1 C L1 .L1 /1 L1 /K D M.L1 C L1 .1 L/1 /K D M.1 L/1 K:

305

6.2. Norms

Finally, we obtain 0 D det.L1 / det Lt C M.1 L/1 K :

(1.10)

By construction, the roots of this equation are invariants of BŒS . Note, that the righthand side of (1.10) is a product of meromorphic functions in 2 C. Usually, a root 0 of det.L1 / D 0 is a pole of the second factor det.: : : / and therefore 0 is not a root of the characteristic 0 „ equation (1.10). Next, set S WD „ 0 , where „ is a diagonal matrix with entries tj . We write out equation (1.10), 0 D det.„1 / det.„t /: The roots are precisely tj˙1 . Further, the collection of roots (1.10) is invariant under the transformation 7! 1 . Roots of the first factor satisfy jj j > 1, therefore all eigenvalues in the interval .1; 1/ are roots of the second factor. Remark. Nevertheless, the total collection of roots of (1.8) is not an invariant of BŒS . Occasionally, a root det.L1 / D 0 can be a root of the characteristic equation (1.10); for instance if K D M D 0. Respectively, we “lose” a root of (1.8). But its absolute value > 1. Our considerations also imply the following statement: Lemma 1.8. Suppose kSk < 1 (and BŒS is not necessarily self-adjoint). Then the collection of roots of the equation (1.8) lying inside the circle jj < 1 does not change under conjugations of BŒS by unitary Gaussian operators.

6.2 Norms 2.1 Norms of self-adjoint operators. Recall that a formula for a norm of a selfadjoint Gaussian operator was proposed in Theorem 5.3.2. Here we present formulas of another type. L Theorem 2.1. Let kS k < 1 and let BŒS be self-adjoint, i.e., S D LKt M satisfies x K D M and L D L . Let tj be the invariants of the operator BŒS . Then kBŒS k D det

1L K M 1 Lt

D det

1=2 Y n

1CL K M 1 C Lt

.1 tj /

(2.1)

j D1

1=2 Y n

.1 C tj /:

j D1

(2.2)

306


Proof. Recall that BŒS is a trace class operator, see Subsection 1.4. Let us calculate the trace of BŒS in two ways. First, the trace is equal to the integral of the kernel of the operator over the diagonal: t ³ Z ² K L 1 z z zN expfjzj2 g dz tr BŒS D exp t M z Nt L 2 (2.3) 1=2 1L K D det : M 1 Lt Secondly, the trace is the sum of eigenvalues. We know the spectrum of BŒS (see Theorem 5.3.2). Thus, tr BŒS D kBŒS k

X

t1k1

: : : tnkn

D kBŒS k

n Y

.1 tj /1 :

j D1

Comparing the two expressions, we obtain the first formula (2.1). 0 1

To derive the second formula (2.1), consider the operator B 1 0 f .z/ D f .z/; it commutes with any BŒS : h1 0 1 0i 0 1 0 1 S : DB B BŒS D BŒS B 0 1 0 1 1 0 1 0 0 1

Calculating tr B 1 0 BŒS in the same two ways, we come to (2.2).

Problem 2.1. Justify (2.3). 2.2 Norms of arbitrary Gaussian operators BŒS . Write J D

1 0 0 1

.

Theorem 2.2. a) Let kSk < 1. Then kBŒS k D det.1 S S/1=4

n Y

.1 sj2 /1=2

(2.4)

j D1

D det.1 S JSJ /1=4

n Y

.1 C sj2 /1=2 :

(2.5)

j D1

b) The second formula holds for an arbitrary bounded Gaussian operator BŒS . Proof. We calculate tr.BŒS BŒS / in two ways. First, we reduce the operator to the canonical form 0 „ U2 BŒS D U1 B „ 0

307

6.2. Norms

with unitary Gaussian operators U1 , U2 and diagonal n n matrix „. Then

2

0 „ BŒS BŒS D jj U1 B „ 0

2

U11 :

Hence the collection of eigenvalues of BŒS BŒS has the form ˚

jj2 s12k1 : : : sn2kn ;

where kj ranges in f0; 1; 2; : : : g

(see also Theorem 3.2). The maximal eigenvalue jj2 coincides with kBŒS BŒS k D kBŒSk2 . Thus, tr BŒS BŒS D kBŒS k2

X

s12k1 sn2kn D kBŒS k2

n Y

.1 sj2 /1 :

(2.6)

j D1

Problem 2.2. Let A W Fn ! Fn be a Hilbert–Schmidt operator, let K be its kernel. Then Z Z ˇ ˇ ˇK.z; u/ˇ2 e jzj2 e juj2 d .z/ d .u/: tr A A D Cn

Cn

In our case,

Z

Z

tr.BŒS BŒS / D Cn

Cn

ˇ ˚

ˇ2 2 2 ˇexp 1 .z u/S.z N u/ N t ˇ e jzj e juj d .z/ d .u/: 2

To simplify the notation, we write w WD z uN 2 Cn ˚ Cn and come to ³ Z ² S 1 1 w w w x exp d .w/ w x 1 S 2 C2n 1=2 S 1 D det D det.1 S S /1=2 : 1 S

(2.7)

Comparing the two (2.6) expressions 0 1

and (2.7), we obtain the first formula for norm. Calculating tr BŒS B 1 0 BŒS in the two ways, we get (2.5). b) The right-hand side in (2.5) has no indeterminacy as a function in S . After that, it remains to perform some exercises in functional analysis. We omit details. 2.3 One more formula Theorem 2.3. Let S D

K

L Lt M

K B t L

, let L be invertible and kS k < 1. Then

n Y L D j det.L/j1=2 jsj j1=2 : M j D1

(2.8)

308


Proof. Denote by the product of matrices in Potapov coordinates, see (2.8.9) or (5.1.8). By virtue of Theorem 1.3, the matrix S can be represented as a -product 0 † G H P Q ; SD † 0 Ht F Qt R where – † is a diagonal matrix with eigenvalues s1 ; : : : ; sn ; P Q – Qt R and HGt H are unitary matrices (see remarks to Theorem 5.1.5). F By Theorem 5.1.5.f), the operators P Q ; j det.Q/j1=2 B Qt R

j det.H /j1=2 B

G Ht

H F

0 † D 1, we find that the norm of are unitary. Taking into account the fact that B † 0 the operator P Q 0 † G H 1=2 1=2 B B X D j det.Q/j j det.H /j B Qt R † 0 Ht F is equal to 1. Using formula (5.1.7) to calculate the right-hand side, we obtain K L 1=2 1=2 1=2 : B t X D j det.Q/j j det.H /j det.1 †R†G/ L M Also,

L D Q†.1 †R†G/1 H:

Since kXk D 1, comparing the two last expressions, we obtain K L 1=2 B t j det.H /j1=2 j j det.1 †R†G/j1=2 L M D j det.Q/j D j det.L/j1=2 j det.†/j1=2

and this implies the desired statement.

6.3 Spectra and eigenvectors 3.1 Spectra of compact Gaussian operators Theorem 3.1. a) Let kS k < 1. Then the eigenvalues of BŒS have the form C

n Y j D1

m

j j ;

(3.1)

309

6.3. Spectra and eigenvectors

where C is a constant and 1 ; : : : ; n are fixed complex numbers, jk j < 1; the exponents mj range in the set 0; 1; 2; : : : . b) The numbers j are the solutions of the equation (3.2) det Lt C M.1 L/1 K D 0 that are contained inside the disk jj < 1. c) There is a unique eigenvector of BŒS of the form bŒT D expf 12 zT z t g with kT k < 1. The matrix T is the solution of the equation T D K C LT .1 M T /1 Lt :

(3.3)

The corresponding eigenvalue of BŒS is C . Moreover, C D det.1 M T /1=2 : d) All eigenvectors of BŒS have the form p.z/ expf 12 zT z t g;

where p.z/ are polynomials.

(3.4)

In the case of multiple eigenvalues, root subspaces consist of vectors of the form (3.4). 3.2 Proof of the statement c). Uniqueness of the Gaussian eigenvector. The statement c) is more-or-less Theorem 5.3.1. We must only prove the uniqueness of the solution of equation (3.3). Lemma 3.2. Let kSk < 1. Then the Krein–Shmul’yan map W T 7! K C LT .1 M T /1 Lt is contractive on the matrix ball Bn (with respect to the metric (2.12.3); therefore it has a unique fixed point. Proof of the lemma. We refer to Theorem 2.9.7 (about compression of angles) and get .T /; .T 0 / 6 .T; T 0 /: (3.5) Next, we represent our map as a product of maps W T 7! .1 "/T I

M L T 1 T Q W T 7! K C p 1" 1"

1

p

Lt 1"

such that the matrix Sz corresponding to Q still satisfies kSzk < 1; therefore (3.5) is Q satisfied for . It is sufficient to prove the following lemma:

310


Lemma 3.3. For any T; R 2 Bn , .1 "/T; .1 "/R/ 6 .1 "/ .T; R/: Proof of Lemma 3.3. Our distance is a geodesic distance (see Theorem 2.12.2) with respect to the Riemannian metric .1 T T /1 d T .1 T T / d T : It is sufficient to prove the contractivity on the level of Riemannian metrics, i.e., .1 .1 "/2 T T /1 d.1 "/T .1 .1 "/2 T T /1 d.1 "/T 6 .1 "/2 .1 T T /1 d T .1 T T / d T : Without loss of generality we can assume that T is diagonal; let xj be its diagonal values. Now our inequality converts to X k;l

X dtkl d tNkl dtkl d tNkl ; 6 2 2 2 2 .1 xk2 /.1 xl2 / 1 .1 "/ xk 1 .1 "/ xl

k;l

here 0 6 xj 6 1. The last inequality is obvious.

Problem 3.1. Show that our arguments are superfluous; the uniqueness of a Gaussian eigenvector follows from the remaining part of the proof of Theorem 3.1. 3.3 Proof of Theorem 3.1. Proposition 5.2.6 implies the following statement (we set D 0 in (5.2.5)): Observation 3.4. Let BŒS be a Gaussian operator and let BŒS bŒT D bŒR. Then, for each polynomial p.z/, the BŒS sends p.z/ expf 12 zT z t g to a function of the form q.z/ expf 12 zRz t g, where q.z/ is a polynomial; moreover, the degree of q does not exceed the degree of p. We need a more precise result. Without loss of generality, we can assume that the Gaussian eigenvector is bŒ0. Otherwise, we conjugate BŒS by a unitary Gaussian operator such that U bŒT D bŒ0. The invariance of roots jj j < 1 of the equation (3.2) was claimed in Lemma 1.8. Thus we assume BŒS 1 D 1: Therefore,

0 SD Lt

L : M

This reduces the equation (3.2) to the form det.Lt / D 0. That is, j are the eigenvalues of L.

6.3. Spectra and eigenvectors

Lemma 3.5. For any polynomial f 2 Fn , ² X ³ ²X ³ @ 1 @2 0 L l z exp m f .z/: f .z/ D exp B ij ij i Lt M 2 @zi @zj @zj

311

(3.6)

ij

Proof of Lemma 3.5. We write out the left-hand side and integrate by parts (see Proposition 4.1.9 and the calculation (4.1.17)), 0 L f .z/ B Lt M Z ² X ³ o nX 1 2 lij zi uNj exp mij uN i uNj f .u/ e juj d .u/ D exp 2 Cn

Z

D

² X ³ o nX 1 @2 2 exp lij zi uNj exp mij f .u/ e juj d .u/; n 2 @u @u i j C ij

and apply (4.1.26).

Denote by Pol6k the space of polynomials of degree 6 k, by PolDk the space of homogeneous polynomials of degree k. There is an obvious isomorphism Pol6k =Pol6k1 ' PolDk :

(3.7)

Further, the operator (3.6) is block-triangle with respect to the filtration Pol6k Pol6kC1 ; i.e., BŒS Pol6k Pol6k . Let us examine the diagonal blocks, which are transformations PolDk ! PolDk . For any f 2 Pol6k , exp

² X 1

2

For any g, exp

³

mij

ij

²X

@2 f f 2 Pol6k2 : @zi @zj ³

@ lij zi g.z/ D g.zL/: @zj

Thus in the diagonal blocks of BŒS we have the transformations g.z/ 7! g.zL/. Q k Hence the eigenvalues of diagonal blocks of (3.6) are P kj Dn j j . Now all the statements about eigenvalues, eigenvectors, and root vectors become obvious. Remark. The eigenvectors can be written explicitly in terms of matching functions, which are defined in § 6.5.

312


6.4 Quadratic operators and exponentials 4.1 A normalization of Gaussian operators. Let g D ac db 2 Sp.2n; R/ be a symplectic contraction sufficiently close to 1. Define the operator BeO .g/ in the Fock space Fn by 1 a b a1 O 1=2 Be .g/ WD det a B : (4.1) at1 ca1 Since g is near 1, the square root in this formula is well defined. Lemma 4.1. For any pair of symplectic contractive matrices g1 , g2 sufficiently close to 1, the following identity holds: BeO .g1 / BeO .g2 / D BeO .g1 g2 /:

Proof. This is more-or-less the calculation (5.2.11). ˇ 4.2 Quadratic operators. Let X WD ˛ ˛ be a dissipative element of the Lie t algebra sp.2n; C/. By definition, ³ ² ˛ ˇ a.s/ b.s/ DW ; where s > 0 exp s ˛ t c.s/ d.s/ is a symplectic contraction. The corresponding operator in Fn is a.s/1 b.s/ a.s/1 O a.s/ b.s/ 1=2 D det a.s/ : Be B c.s/ d.s/ a.s/t1 c.s/a.s/1 Denote by Ks .z; u/ the kernel of this operator. Expanding formally Ks in s, we get ³ ² ˇ ˛ s s z z u N C O.s 2 /: N 1C Ks .z; u/ D tr ˛ C expfz ug uN ˛ t 2 2 Passing to Wick symbols, see Proposition 4.1.11, we come to ˛ ˇ O a.s/ b.s/ Be D1CsW C O.s 2 /; c.s/ d.s/ ˛ t where ˛ W

ˇ ˛ t

X 1 1X @ @2 1X WD tr ˛ ˇkl zk zl ˛kl zk C kl : (4.2) 2 2 @zl 2 @zk @zl k;l

k;l

Such operators are called quadratic operators.

k;l

6.4. Quadratic operators and exponentials

313

4.3 Commutators of quadratic operators Theorem 4.2. For any R, Q 2 sp.2n; C/ and for any f 2 Fn , ŒW.R/; W.Q/f D W.ŒR; Q/f; i.e., W is a representation of the Lie algebra sp.2n; C/. This can be easily verified by a direct calculation. 4.4 Correspondence Lie semigroup – Lie algebra. The main statement of this section is: Theorem 4.3. For a symplectic pseudo-dissipative matrix R, expfs W.R/g D BeO expfsRg ; where s > 0:

(4.3)

Remark. At first glance, the theorem is trivial. For a matrix A of finite size, ˇ d exp "Aˇ"D0 D A: d" Formally, our equality is the same. However, Theorem 4.3 is not yet precisely formulated, because the operators W.R/ are unbounded; in particular, an operator exp.X / depends on the domain of definiteness of an operator X , see e.g. [178], VIII.2–4. Changing Dom.X / we can get another exp.X /. 4.5 Self-adjointness. We suppose that the reader is familiar with unbounded selfadjoint and dissipative operators (for definitions and criterions, see e.g., [178]–[179]). Theorem 4.4. Let an operator (4.2) be formally self-adjoint, i.e., ˛ D ˛ and D ˇ . Then it is essentially self-adjoint on the space Poln Fn of polynomials. We omit a proof; it is contained in many texts in a wider generality, see e.g., [14], [145]. 4.6 Formal dissipativity. Let S be an operator in a Hilbert space defined on a dense subspace V . We say that S is formally dissipative if RehSv; vi 6 0

for each v 2 V .

Let us consider quadratic operators defined on the domain Fn . Lemma 4.5. For each g 2 Sp.2n; R/, BeO .g/W.R/ BeO .g/1 D W.gRg 1 /:

Proof. In other words, ˇ ˇ d d BeO g expf"Rgg 1 ˇ"D0 D BeO .g/ BeO .expf"Rg/ˇ"D0 BeO .g/1 : d" d" This is obvious.

314


Theorem 4.6. The following conditions are equivalent: 1) the matrix R 2 sp.2n; C/ is pseudo-dissipative; 2) the quadratic operator W.R/ defined on Fn is formally dissipative; 3) there exists a constant h such that the quadratic operator h C W.R/ is formally dissipative. Proof. First of all, expand R D H C iT , where H , T 2 sp.2n; R/. Since W.H / D W.H /, the dissipativity of W.R/ is equivalent to the dissipativity of the self-adjoint operator W.iT /. Denote by C1 , C2 , C3 sp.2n; R/ the cones of all matrices R satisfying the conditions 1), 2), 3) of the theorem respectively. The cone C1 is convex and Sp.2n; R/invariant, see Section 3.5. Obviously, the cones C2 and C3 are convex. By Lemma 4.5 they are Sp.2n; R/-invariant. Evidently, C2 C3 . Lemma 4.7. C1 C2 . Proof of the lemma. We must prove the implication ˚

˚

iR is pseudo-dissipative H) W.iR/ is formally dissipative in Fn : Taking into account Lemma 4.5, we observe that it suffices to prove this for R being in the canonical form, see Theorem 3.5.2. Corresponding quadratic operators in Fn are

X j 6p

j zj

@ @zj

C

X 1 @ 2 C zj C 2 @zj

2

j >p

with j > 0. Such operators are formally dissipative.

However, C1 is a unique convex Sp.2n; R/-invariant cone in sp.2n; R/ (see Addendum to Chapter 3, Theorem 1). Therefore, C1 D C2 D C3 and this completes the proof of the theorem. 4.7 Dissipativity and exponentials. Let B.t /, t > 0 be a one-parameter semigroup of contractive operators in a Hilbert space, i.e., B.t1 /B.t2 / D B.t1 C t2 /: Its generator S is the operator S defined by 1 .B."/ 1/v: "!C0 "

Sv WD lim

(4.4)

By definition the domain dom.S / consists of all vectors v, for which the limit exists. The operator S is called the generator of the one-parameter semigroup B.t /; by definition, dissipative operators are operators that can be obtained in this way.

6.5. Matching functions and matrix elements of Gaussian operators

315

We also write B.t / D exp.tS /. But there is the following condition of dissipativity, see [71], 9.20: – If a formally dissipative operator has a dense set of analytic vectors1 , then its closure is a dissipative operator. Theorem 4.8. The closure of a formally dissipative quadratic operator2 is a dissipative operator and, moreover, (4.3) holds. x t g 2 Fn are analytic. Proof. It suffices to prove that the vectors 'w .z/ D expfz w Indeed, let us evaluate explicitly 1 a b a1 O 1=2 ' .z/ Be .g/'w .z/ ´ det a B at1 ca1 w ˚

D det a1=2 exp 12 wca1 w t b a1 bjwa1 ; where bŒ denotes a Gaussian vector. If g is in a small neighborhood of 1, then our vector is in Fn . Thus the map w W g 7! BeO .g/'w is a well-defined map ˚

neighborhood of 1 in Sp.2n; C/ ! Fn :

It can be easily shown that this map is analytic.

6.5 Matching functions and matrix elements of Gaussian operators 5.1 Configurations. Let H be a finite set, h1 ; : : : ; hk be its elements. A configuration over H is a pair .!; /, where ! is a finite set and is a map ! ! H . By ˛j we denote the number of points in the set 1 .hj /. Two configurations .!; / and .! 0 ; 0 / over H are equivalent if there is a bijection ~ W ! ! ! 0 such that D 0 B . Remark. See Figure 6.1. The set H is the horizontal chain of black circles. The set ! consists of white circles, it is organized as collections of columns over elements of H . Remark. In fact, this definition is nothing but a formalization of the following notion: a configuration is a set with multiplicities; we think that each point hj has ˛j copies. Two configurations are equivalent if and only if their multiplicities coincide. A vector v is an analytic vector of an (unbounded) operator A if the series for some t > 0. 2 Recall that it is defined on Fn . 1

P

t n kAn vk=nŠ is convergent

316


!

H

Figure 6.1. A configuration. A matching of configurations.

We denote the set of all configurations on A by .A/. 5.2 Matchings. Let ! be a configuration. A matching of ! is a partition of ! into two-point and one-point sets. If we draw ! as a collection of points, we can regard a matching as a graph such that each vertex is an end of one or zero edges. By „Œ! we denote the set of all matchings of the set !. For a matching , we denote by two./ the set of all two-point subsets of the matching, by one./ the set of all one-point subsets, see Figure 6.1. For a matching , we denote by sij the number of arcs between fibers 1 .hi / and 1 .hj /. Denote by tj the number of one-point subsets in a fiber 1 .hj /. Obviously, these numbers satisfy the identities X sij C 2si i C ti D ˛i ; (5.1) sij D sj i ; j ¤i

see Figure 6.2 Lemma 5.1. Let ! be a configuration with multiplicities ˛j . The number of its matchings with given sij , tj is Q ˛i Š Q s Q i Q : (5.2) i i i 2 i6j sij Š i ti Š Proof. Denote by Sn the group of permutations of a set with n elements. Denote by Q D Q.sij ; tj / the set of matchings with given sij , tj . Consider the group G of all permutations of the set ! preserving the fibers of the map (see Figure 6.1, we consider permutations of vertical columns). By the definition, G D S˛1 S˛2 : Obviously, G acts transitively on Q. It remains to describe the stabilizer Z of a point of Q.


317

tj

sij

tj

1

0

0

1

0

0

†D3

0

0

0

0

0

0

†D0

0

0

0

0

0

1

†D1

1

0

0

0

2

1

†D4

0

0

0

2

0

0

†D2

0

0

1

1

0

†D3 †D0 †D1 †D4 †D2

Figure 6.2. The numbers sij , tj corresponding to Figure 6.1.

An element of Z can – rearrange one-point sets inside a fiber; – permute arcs connecting i-th fiber and j -th fiber (in particular, for i D j ); – change ends of an arc connecting elements of one fiber. Thus the stabilizer is Z'

Y j

S tj

Y i<j

Ssij

Y

.Z=2Z/sjj :

i

This gives the denominator in (5.2).

5.3 Matching functions. Let .!; / be a configuration over a set H with multiplicities ˛j . Let A.hi ; hj / D aij be a symmetric function on H H , aij D aj i ; let bj D b.hj / be a function on H . We define the matching function M.A; bj˛/ by Y Y X A..u/; .v// b..w//: (5.3) M.A; bj˛/ WD 2„.!/ fu;vg2two./

fwg2one./

In other words, for a given matching, we write out a product in the following way. Given an arc connecting the i-th column and the j -th column, we write the factor aij . Given an isolated point in the j -th column, we write the factor bj . Then we write the sum over all matchings of a given configuration.

318


Proposition 5.2. X

M.A; bj˛/ WD

Q

sij ; ti W sij Dsj i P j ¤i sij C2si i Cti D˛i

i

2si i

Q Y s Y t ˛i Š Qi Q aijij bi i : i6j sij Š i ti Š i6j

j

Proof. This is an immediate corollary of the definition and Lemma 5.1. 5.4 Generating functions Theorem 5.3. Let aij D aj i . Then exp

² X 1

2

aij zi zj C

i;j

X

³

X

bj zj D

M.A; bj˛1 ; ˛2 ; : : : /

˛1 >0; ˛2 >0;:::

j

z1˛1 z2˛2 : : : : (5.4) ˛2 Š ˛2 Š

Proof. This is obvious. Indeed, Y Y ˚ Y exp : : : D expfaij zi zj g expf 12 ajj zj2 g expfbj zj g i<j

j

k

³ Y²X ³ ² ³ Y²X 1 1 l 2l Y X 1 m m k k k D a z a z z b z : kŠ ij i j mŠ j j 2l lŠ jj j m i<j

j

k

l

k

Opening brackets we obtain the desired expression 5.5 Partial differential equations for matching functions Corollary 5.4. a) The functions F˛ .A; b/ WD M.A; bj˛/ satisfy the equations

@ @2 F .A; b/ D 0; @aij @bi @bj

where i ¤ j I

(5.5)

1 @2 @ F .A; b/ D 0: @ajj 2 @bj2

(5.6)

b) Consider the following matrix (recall that aij and aj i denote the same variable) 1 0 @=@a13 : : : 2@=@a11 @=@a12 B @=@a21 2@=@a22 @=@a23 : : :C C B : (5.7) B @=@a31 @=@a32 2@=@a33 : : :C A @ :: :: :: :: : : : : For each 2 2 minor of this matrix, the following identity holds: F .A; b/ D 0:

(5.8)


319

Proof. Indeed, the left-hand side of (5.4) satisfies equations (5.5) and (5.6), hence the Taylor coefficients of the right-hand side satisfy the same equations. The equations (5.8) follow from (5.5) and (5.6). Problem 5.1. a) The functions F˛ .A; b/ WD M.A; bj˛/ form a basis in the space of all polynomial solutions of the system (5.5)–(5.6). b) A polynomial solution F .A; b/ of the system (5.5)–(5.6) is determined by F .0; b/. 5.6 Matrix elements of Gaussian operators. By Proposition 4.1.6, matrix elements of an operator in a Fock space are Taylor coefficients of its kernel. Thus, by Theorem 5.3, Observation 5.5. Matrix elements hBŒS z ˛ ; z ˇ i of Gaussian operators are matching functions. 5.7 Digression. Fock spaces as `2 . Consider the countable space n WD Zn C and denote its elements by ˛ D .˛1 ; : : : ; ˛nQ /. Introduce a measure on the set n from the assumption: the measure of a point ˛ is ˛Š WD ˛j Š. We can also regard n as the space of all configurations over the set f1; 2; : : : ; ng. Consider the Hilbert space `2 .n / whose inner product is given by the formula X hf; gi D f.˛/g.˛/˛Š: ˛2n

Define a canonical unitary operator U W Fn ! `2 .n /. To each function X f .z/ D f.˛/z ˛ 2 Fn ; ˛

we assign the collection f.˛/ 2 `2 .n / of its Taylor coefficients. By Theorem 5.3, a function ²

exp

³

X 1X bj zj 2 Fn aij zi zj C 2 j

corresponds to the collection b.˛/ D M.A; bj˛/=˛Š. Taking into account the identity (5.1.3), we get ² ³ X 1 1 A b : M.A; bj˛/M.P; qj˛/ D det.1 APx /1=2 exp b qN qN Px 1 ˛Š ˛

Certainly, this appears to be trivial. However, the last identity lose its self-evidence if we forget the method of its derivation.

320


5.8 Remarks on determinantal systems and matching functions Problem 5.2. a) Let be a tempered distribution on Rn . Show that the function ² ³ 1X f .a/ D ; exp akl xk xl 2 satisfies the system (5.7)–(5.8). b) Evaluate a Gaussian integral over an octant, Z ² ³ 1X exp aij xi xj dx 2 x1 >0;:::;xn >0 Z ² ³ o n X 1X D exp ajj xj2 exp aij xi xj dx 2 x1 >0;:::;xn >0 j

i >j

expanding the second factor into the Taylor series. Represent the integral as a series in matching polynomials. Under what conditions on aij is this way correct? c) Consider a simplicial cone in Rn . Show that a calculation of its solid angle is equivalent to a calculation of a certain Gaussian integral over an octant. Problem 5.3. Let xi be an infinite collection of formal variables. Denote by hk .x/ the complete symmetric function, X hk .x/ D of all monomials of degree k: Expand h1 :˚:P : hN in a sum of monomials. Compare the result with coefficients of the Taylor series of exp kl zk ul .

6.6 Analytic continuation 6.1 Analytic continuation. We realize the group Sp.2n; C/ and its subgroup Sp.2n; R/ as in § 3.2. In Theorem 5.1.5, for each g 2 Sp.2n; R/ we have assigned the operator in Fn by 1 ˆ ‰ ˆ1 ˆ ‰ x 1 : x ˆ x 7! B ˆt1 ‰ˆ ‰ We can extend this formula to the complex group Sp.2n; C/ as 1 a1 a b a b : 7! B at1 ca1 c d Certainly, these operators are unbounded. However, a matrix of such an “operator” in the standard basis z1k1 : : : znkn is well defined whenever a is invertible (see formulas for matrix elements in the previous section). Observation 6.1. Denote by the set (submanifold) of matrices g 2 Sp.2n; C/ such that det.a/ D 0. The Weil representation of Sp.2n; R/ admits a holomorphic continuation to Sp.2n; C/ n . The manifold is a common pole for all matrix elements.


321

6.7 Bibliographical remarks 7.1 Norms. The calculation of norms is based on G. I. Olshanski’s paper [166]. All remaining material of this chapter is well known or semi-well known and semi-obvious. 7.2 Analytic continuation. Observation 6.1 is a special case of a general phenomenon discovered by W. Casselman. An analytic continuation of K-finite matrix coefficients exists for all irreducible representations of all semisimple Lie groups G; however usually we obtain functions ramified on a certain (singular) submanifold „ in GC . For G D Sp.2n; R/, the manifold „ is reducible, is one of components of „. In our case, we must write the analytic continuation of formula (5.1.8) and get the expression (4.1)). It is ramified on Sp.2n; C/ due the factor det.a/1=2 . Forgetting the factor, we come to a meromorphic function.

7 Hilbert spaces of holomorphic functions in matrix balls

Here we discuss a certain natural one-parametric family of representations of Sp.2n; R/, which includes the Weil representation. The main construction is exposed in § 7.5. The previous sections are occupied by preparations for this construction and the subsequent sections contain discussion of various details.

7.1 Reproducing kernels Text-books in functional analysis propose orthogonal bases as a tool for explicit calculations. As Berezin claimed, one can use an arbitrary total system x of vectors instead of a basis; we need only an explicit formula for inner products of x . In numerous contexts (mainly, for spaces of functions of a large or infinite number of variables), such systems of coherent states are more convenient and flexible tool than explicit orthogonal bases. For the following discussion, the Fock space Fn is a good initial example. Numerous natural examples arise in this chapter. 1.1 Gram matrix Theorem 1.1. Let R be an n n Hermitian matrix. The following conditions are equivalent: i) R is non-negative definite; ii) there is a Euclidean space and vectors 1 ; : : : ; n such that hi ; j i D rij ; iii) for each collection 1 6 i1 < < ik 6 n, the corresponding principal minor of R is non-negative, det fri˛ iˇ g > 0: ˛;ˇ

P Proof. ii H) i. We must verify the inequality rij ai aNj > 0. Indeed, X X X X ai aNj hi ; j i D h ai i ; aj j i > 0: rij ai aNj D i H) ii. Consider an n-dimensional linear space V and a basis ej 2 V . Set hei ; ej i WD rij . We get a positive semi-definite Hermitian form on V . Let H be

323

7.1. Reproducing kernels

its kernel. Then the space V =H is Euclidean; we define j 2 V =H to be the images of ej . i () iii. This is the Sylvester criterion. Problem 1.1 (Menger theorem). Let R be a symmetric n n matrix with nonnegative elements. The following conditions are equivalent: – there are points 1 ; : : : ; n in some real Euclidean space such that k i j k2 D rij I – the matrix R is conditionally positive definite, i.e., X X rij ai aj > 0 for all real vectors a D .a1 ; a2 ; : : : / satisfying aj D 0: 1.2 Positive definite kernels. Let X be a set. Consider a function L on X X such that L.x; y/ D L.y; x/: It is called a positive-definite kernel if for each finite collection x1 ; : : : ; xn 2 X the matrix fL.xi ; xj /g is positive semi-definite. Example. Let H be a Hilbert space. Let x be a system of vectors in H indexed by x 2 X. Then L.x; y/ WD hx ; y i is a positive definite kernel on X. 1.3 The construction of Hilbert spaces from positive definite kernels Theorem 1.2. Let X be a separable1 metric space. Assume that a positive definite kernel L is continuous as a function on X X. a) There is a separable Hilbert space H and a total system of vectors x indexed by elements x 2 X such that hx ; y i D L.x; y/. b) A space H is unique in the following sense: if H0 is another Hilbert space and 0 a 2 H is another system of vectors with the same properties, then there is a unitary operator U W H ! H0 such that U x D x0 . Proof. a) Consider a countable dense set fy1 ; y2 ; : : : g contained in X. For each finite set fy1 ; : : : ; yk g, consider the corresponding Euclidean space Wk and the total system of vectors 1 ; : : : ; k 2 Wk such that h i ; j i D L.yi ; yj / (see Theorem 1.1). We have the canonical isometric embeddings: ! Wk1 ! Wk ! WkC1 ! ; which take k 2 Wk to k 2 WkC1 . Consider the pre-Hilbert space associated Hilbert space H. We claim that H is the desired object. 1

A metric space is said to be separable if it contains a countable dense subset.

S k

Wk and the

324

Chapter 7. Hilbert spaces of holomorphic functions in matrix balls

Indeed, for each x 2 X, choose a subsequence y˛1 , y˛2 , …convergent to x. To simplify S the notation denote it by w1 , w2 , etc. Consider the corresponding sequence m 2 Wk H. Then k m k k2 D L.wm ; wm / C L.wk ; wk / L.wm ; wk / L.wk ; wm /: Since L is continuous, this expression tends to 0 as m; k ! 1. Hence there exists the limit x WD limm!1 m . By continuity considerations, hx ; x 0 i D L.x; x 0 /. b) The construction of Wk is canonical. Problem 1.2. Extend the Menger theorem in a similar way. The following statement is a version of the previous Theorem 1.2. Theorem 1.3. Let X be an arbitrary set. a) For each positive definite kernel L on X, there is a (generally speaking, nonseparable) Hilbert space H and a total system of vectors x indexed by elements x 2 X such that hx ; x i D L.x; y/. b) A space H is unique in the same sense as above. P Proof. a) Consider the linear space W whose elements are finite sums i ai xi , where ai 2 C and x are formal symbols enumerated by elements of X. We define the inner product in W by DX E X X ai xi ; bj yj WD L.xi ; yj /ai bNj : i

j

i;j

In particular, DX i

ai xi ;

X i

E

ai xi WD

X i;j

L.xi ; xj /ai aNj > 0;

because the kernel L is positive-definite. Thus W is a pre-Hilbert space. Our space H is the associated Hilbert space. b) The space of linear combinations of vectors x0 2 H0 is canonically isometric to W . We denote the Hilbert space associated with a kernel L by H D H.XI L/

and say that x is an overfilled system. In the literature, fx g is also called a system of coherent states or a supercomplete basis. 1.4 Functional realization of H. Let L be a positive definite kernel. For each vector v 2 H.XI L/, define the function fv .x/ WD hv; x i


325

on the set X. Thus we obtain a certain linear space HB D HB .XI L/ consisting of functions fv on X. By construction, HB .XI L/ is in a one-to-one correspondence with H.XI L/, therefore HB .XI L/ inherits the structure of a Hilbert space. Theorem 1.4. a) The function in HB corresponding to a vector a is 'a .x/ WD L.a; x/: b) For each f 2 HB .XI L/, hf; 'a iH B D f .a/ (reproducing property):

(1.1)

c) If f .j / 2 HB .XI L/ weakly converges to f , then f .j / converges to f point-wise. d) Linear combinations of functions 'a are dense in HB .XI L/. Proof. a) is obvious. b) Let f D fv . Then hf; 'a iHB D hv; a iH D fv .a/: c) Let f .j / D fvj , f D fv . Then hvj ; a i converges to hv; a i. But this is precisely the convergence f .j / .a/ to f .a/. d) By the definition, linear combinations of a are dense in H.XI L/. Due to the reproducing property, the kernel L is also called the reproducing kernel of the space HB .XI L/. Remark. The reproducing property is an implicit description of the inner product in HB .XI L/. Also, Theorem 1.4 provides us with a (non-perfect) tool for watching a convergence. Example. Let X be a circle S 1 with the standard angle coordinate ˛. Set L.˛; ˇ/ D cos

˛ˇ : 2

The space HB .S 1 ; L/ is two-dimensional. The vectors ˛ 2 R2 can be chosen as ˛ WD .cos ˛; sin ˛/. This example shows that the space HB .XI L/ can drastically differ from the space of “all functions” on X; thus a reproducing kernel L “knows” not only an inner product but also a functional space itself. Proposition 1.5. Let X be a metric space, L be continuous on X X. a) If a sequence f .j / 2 HB .XI L/ converges to f in the norm topology, then f .j / converges to f uniformly on each compact subset X. b) For each point b 2 X there is a ball B.b; r/ with center b and radius r > 0 such that every sequence f .j / 2 HB .X; L/ that converges in the norm topology converges uniformly on B.b; r/.

326


Proof. a), b) Let y range in a set C . Let f .j / D fvj , f D fv . Then jf .j / .y/ f .y/j D jhvj ; y i hv; y ij D jh.vj v/; y ij 6 kvj vk ky k 6 kvj vk sup L.y; y/1=2 : y2C

If a set C is compact, then the supremum is finite. Also, for each point b 2 X there is neighborhood C , in which the function L.x; x/ is bounded. Since kvj vk ! 0, in both cases we get uniform convergence on C . Corollary 1.6. If a reproducing kernel L is continuous, then elements of HB .XI L/ are continuous functions. Theorem 1.7. Let X be a complex manifold. If a kernel L.z; u/ is holomorphic in z and antiholomorphic in u, then elements of HB .XI L/ are holomorphic functions. Proof. We apply Proposition 1.5. If a sequence fj of holomorphic functions converges uniformly on compact sets, then its limit is a holomorphic function. Problem 1.3. a) If a kernel L is C 1 -smooth, then elements of HB .XI L/ are smooth functions. P b) If a kernel is a finite sum of the form jkD1 pj .x/qj .y/, then the corresponding space HB is finite-dimensional. 1.5 Construction of a kernel from a functional space Theorem 1.8. Let M be a Hilbert space whose elements are functions on a set X. Assume for each a 2 X the evaluation linear functional f 7! f .a/ is continuous on2 M . Then M is determined by some reproducing kernel. Proof. Since our functional is continuous, for each a there is a function 'a such that hf; 'a i D f .a/. We take L.a; b/ WD h'a ; 'b i. Then L.a; b/ is a positive definite kernel and 'a is an overfilled system. 1.6 The dual functional realization. Now let be a smooth manifold and let E 0 ./ be the space of compactly supported distributions on . Let L.x; y/ be a smooth reproducing kernel in . Define a Hermitian form in E 0 ./ by B ? .1 ; 2 i/ WD hhL; 1 ˝ N 2 ii; where 1 ˝ 2 WDW .x/ .y/. Lemma 1.9. This inner product is positive semi-definite. 2 This proviso is necessary de jure. However, existence of a discontinuous linear functional defined on the whole Hilbert space is one of the forms of the axiom of choice, see [92].


327

We denote the corresponding Hilbert space by H? .X; L/ and put off a proof. Lemma 1.10. The form B ? is jointly continuous as a function on E 0 E 0 .

Proof of Lemma 1.10. Obvious.

Next, consider the subspace D E 0 consisting of finite linear combinations of delta-functions ıx . Endow D with the form B ? . Then the map ıx 7! x induces an isometric embedding D ! H.X; L/. Hence B ? is positive semi-definite on D. Proof of Lemma 1.9. The subspace D is dense in E 0 . Since the form B ? is positive semidefinite on D and jointly continuous, it follows that it is positive semi-definite on E 0 . Moreover, D is dense E 0 with respect to the inner product B ? . In other words, we obtain the following theorem: Theorem 1.11. There is a canonical isomorphism H? .I L/ ' H.I L/. Precisely, the map x 7! ıx produces a unitary operator H.I L/ ! H? .I L/. Next we combine the last isomorphism with the isomorphism H.I L/ ! HB .I L/. We get the map ‰ W E 0 ./ ! HB .I L/ given by Z 7! ˆ .x/ WD 'a .x/.a/: a2

0

Corollary 1.12. a) For any 2 E ./, the function ˆ is contained in HB .I L/. “

L.a; b/.a/0 .b/:

b) hˆ ; ˆ0 i D a2 ;b2

An important special case is being a derivative of a delta-function. Corollary 1.13. For sake of simplicity, assume is an open subset in RN . Then for each a 2 , ˇ @˛ 'aŒ˛ WD ˛ 'z .u/ˇzDa 2 HB .; L/ @z and @˛Cˇ h'aŒ˛ ; 'bŒˇ i D ˛ ˇ L.a; b/: @a @b Remark. 1) Usually, ˆ.E 0 / is a proper dense subspace in H? .X/. In many interesting cases, one can represent arbitrary elements of H? .X/ by distributions supported by some “boundary” @ of ; however I do not know a general theorem of such a kind. 2) Usually, the map E 0 ! H? has a nontrivial kernel. In other words, a vector in ? H .X/ can be represented by distributions in numerous ways.3 3

See a highly nontrivial exploring of this possibility in the work by M. Karasev and M. Kozlov, [102].

328


1.7 Reproducing kernels and orthogonal bases Theorem 1.14. Let ej be an arbitrary orthonormal basis in HB .X; L/. Then X L.x; y/ D ej .x/ej .y/:

(1.2)

j

Proof. Expanding 'a in the orthogonal basis ej , we get X X 'a .y/ D h'a ; ej iej .x/ D ej .a/ej .x/: j

j

Remark. This statement can be modified as follows. Let pj , qj be two (non-orthogonal) dual bases, i.e., hpi ; qj i D ıij . Then X L.x; y/ D pj .x/qj .y/: (1.3) Sense of the words “non-orthogonal basis” is not rather clear and we leave the last identity as an heuristic fact. For some applications of the identities (1.2)–(1.3), see for instance [129]. Problem 1.4. For the standard basis in the Fock space F1 we get e zu D

1 X

.nŠ/1=2 z n .nŠ/1=2 un :

nD0

Writing the same formula for the Paley–Wiener space (see Subsection 4.8.2) and its standard basis we come to the identity 1 X

1 D .cot u cot z/ .z n/.u n/ zu nD1 1.8 Operations with positive definite kernels Theorem 1.15. Let L and M be positive definite kernels on X. Then L C M and L M are positive definite kernels. Proof. Let H and V be Hilbert spaces, x 2 H , y 2 V be overfilled systems hx ; y iH D L.x; y/;

h x ; y iV D M.x; y/:

We consider the spaces H ˚ V and H ˝ V equipped with the systems x ˚ x , x ˝ x respectively. Then hx ˚ x ; y ˚ y iH ˚V D L.x; y/ C M.x; y/; hx ˝ x ; y ˝ y iH ˝V D L.x; y/ M.x; y/ and this implies our statement.


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Theorem 1.16. Let Lj be a sequence of positive definite kernels on X, let L be its pointwise limit. Then L is positive definite. Proof. This is a straightforward corollary of the definition.

Proposition 1.17. If a kernel L.x; y/ is positive definite, then for each function h.x/ the kernel h.x/h.y/L.x; y/ is positive definite. Proof. We consider the new overfilled system h.x/x .

1.9 Example. The kernel L0 .z; u/ D 1 on C is positive definite. Therefore, kernels j j positive definite. Hence, for an arbitrary sequence n > 0, the Lj .z; u/ PnD z uN j are kernel j D1 j z uN j is positive definite on C. Denote by r 2 the radius of convergence P of the series j w j . Then the series K.z; u/ N WD

1 X

j z j uN j

j D0

is a positive definite kernel on the disk jzj < r. Next, we apply Corollary 1.13. Differentiating K j times in the variable u, N we get

j z j 2 HB , i.e., z j 2 HB whenever j ¤ 0. Moreover, hz j ; z k i D j1 ıj k :

(1.4)

Thus fz j g is an orthogonal system. P It can be readily checked that the series K.z; a/ D

j z j aN j converges in the B j Hilbert space H ; therefore the system z is an orthogonal basis. Also, we get the following illustration for Theorem 1.14. 1.10 The Bochner theorem Theorem 1.18 (Bochner theorem). A function ‰.x y/ is a positive definite kernel on Rn if and only if ‰ is a Fourier transform of a finite positive Borel measure on Rm , i.e., Z ‰.x/ D

e itx d.t /:

(1.5)

Rm

Problem 1.5. a) Show that K.x; y/ D e it.xy/ is a positive definite kernel on R. Describe the corresponding Hilbert space. b) Show directly that ‰.x y/ defined by (1.5) is positive definite. c) In spite of evidence of b), reduce it to Theorem 1.16.

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Proof of Theorem 1.18. The additive group Rn acts on the space H? .Rn ; ‰/ by unitary operators T .a/f .x/ D f .x C a/: Further, hT .a/ı.x/; ı.x/iH? D ‰.a/:

(1.6)

By the definition of H , the functions ı.x a/ form a total family. In other words, ı.x/ is the cyclic vector of the group Rn . We apply the multiparametric version of the Stone Theorem. Consider a unitary representation T of the additive group Rn ; assume that T has a cyclic vector v. Then T can be realized in L2 on Rn with respect to a finite measure on Rn by the operators ?

Tz .a/f .t/ D e ita f .t /I the cyclic vector v corresponds to f D 1. In our case, v is ı.x/ and Z z hT .a/ 1; 1iL2 D e ita 1 1 d.t /:

(1.7)

Comparing (1.6) and (1.7), we observe that is the desired measure. B

B

1.11 Digression. Berezin symbols. For a bounded operator A W H .Y; M / ! H .X; L/ define its Berezin symbol KA .x; y/ as above (4.1.8), namely KA .x; y/ WD hA'y ; 'x iHB .X/ : Theorem 1.19. a) Af .c/ D hf .y/; K.y; c/iHB .Y/ ; where K.y; c/ is regarded as a function of the variable y. b) Let A and B be bounded operators, B W HB .ZI N / ! HB .YI M /;

A W HB .Y; M / ! HB .X; L/:

Then the symbol KAB of the product AB is KAB .u; v/ D hKA .u; y/; KB .y; v/iHB .Y/ : Here KA , KB on the right-hand side are functions of y 2 Y depending on the parameters u and v respectively. Proofs are the same as in § 4.1. 1.12 Digression. Berezin symbols and orthogonal bases. Let ej .x/ be an orthogonal basis in H B .X; L/, let hej ; ej i D j . Expand the overfilled system 'a .z/ in basis ej : X cj .a/ej .z/: 'a .z/ D j B

Let A be a bounded operator H .X; L/ ! H B .X; L/. Then its kernel is X ˝ X ˛ X ci .y/ei ; cj .x/ej D ci .y/cj .x/hAei ; ej i: KA .x; y/ D hA'y ; 'x i D A j

This is an extension of formula (1.2) and also of (4.1.12).

7.2. Highest weight representations of SU.1; 1/

331

7.2 Highest weight representations of SU.1; 1/ Here we discuss some of the simplest examples of Hilbert spaces of holomorphic functions and one of the simplest constructions of representation theory. A higher analog of this construction is discussed in § 7.5. 2.1 Example. The Bergman spaces in the disk. Denote by D the unit disk jzj < 1 on C. The Bergman space H2 .D/ is the space of holomorphic square integrable functions on the unit disk D. The inner product in the Bergman space is given by Z 1 hf; gi D f .z/g.z/ d .z/; (2.1) jzj0

ck : kC1

Therefore, our orthogonal system is complete. Proposition 2.1. Let jaj < 1. The function N 2 D 'a .z/ WD .1 z a/

1 X

.k C 1/z k aN k

kD0

satisfies the reproducing property hf; 'a i D f .a/. In other words, the Bergman space is determined by the reproducing kernel .1 z u/ N 2 . P Proof. For f .z/ D ck z k , X X hf; 'a i D ck ak .k C 1/hz k ; z k i D ck ak D f .a/: k

k

2.2 Example. The Hardy space4 H1 .D/ is the space Pspace inkthe disk. The P Hardy 2 of Taylor series f .z/ D k>0 ck z such that jck j < 1. The inner product is given by Z 2 1 f .e i' /g.e i' / d': (2.2) hf; gi D 2 0 4 The usual notation is H 2 .D/; in classical function theory, the symbol H p .D/ denotes the space of holomorphic functions in D whose boundary values are contained in Lp on the circle. Here we consider another scale of spaces.

332


Functions f .e i' /, g.e i' / are L2 -convergent Fourier series, therefore the integral makes sense. Also, E DX X X ck z k ; bk z k D ck bNk : Problem 2.1. The Hardy space is determined by the reproducing kernel .1 z u/ N 1 . 2.3 The Bargmann–Pukanszky scale. Fix s > 0. Consider the reproducing kernel on D given by .1 z u/ N s D 1 C

s s.s C 1/ 2 2 z uN C z uN C : 1Š 2Š

Since all Taylor coefficients are positive, it follows that the kernel is positive definite (see Subsection 1.9). Denote by Hs .D/ the corresponding Hilbert space. For s D 0, the kernel D 1, the corresponding Hilbert space is one-dimensional and consists of constant functions. Theorem 2.2. Let s > 0. a) The space Hs .D/ consists of functions holomorphic in D. b) For s > 1, the inner product in Hs .D/ can be written as Z s1 hf; gis D f .z/g.z/.1 jzj2 /s2 d .z/: jzj0

(2.3)

X jŠ j 1s jcj j2 < 1 (2.4) jcj j2 < 1; or, equivalently, s.s C 1/ : : : .s C j 1/ j >0

and the inner product is DX

cn z n ;

X

E

cn0 z n D

X j >0

jŠ cj cN 0 : s.s C 1/ : : : .s C j 1/ j

(2.5)

Proof. a) We refer to Theorem 1.7. b) is a straightforward calculation (as for the Bergman space). c) is a rephrasing of (1.4). In (2.4), we apply a standard asymptotic expansion for the -function, .x C a/= .x C b/ x ab . 2.4 Remark. Boundary values. We stress that formula (2.3) breaks down if s 6 1. Recall the following well-known fact (see textbooks on distributions): Theorem 2.3. a) Let ck , where k 2 Z, be a bilateral sequence such that jck j 6 C .jkj C 1/M

for some M and C :

(2.6)

7.2. Highest weight representations of SU.1; 1/

333

P Then the Fourier series ck e ik' converges in the sense of distributions. Moreover, for any distribution f on the circle, X1 hhf .'/; e ik' ii e ik' ; f D kD1

and the coefficients hh: : :ii satisfy (2.6). b) The sum of a Fourier series is a smooth function if and only if jcn j 6 const .1 C jnj/M

for all M :

c) A distribution f is contained in the Sobolev space5 W ˛ if and only if X jcn j2 .jnj C 1/2˛ < 1: P For a function f .z/ D ck z k on P the disk, define its boundary value at jzj D 1 as the sum of Fourier series f .e i' / D ck e ik' in the sense of distributions6 ; this makes sense, if the coefficients cn satisfy (2.6). Equivalently, f .e i' / WD lim f .re i' / r!10

in the sense of distributions:

Observation 2.4. Let s > 0. The space Hs .D/ consists of holomorphic functions whose boundary values are contained in the Sobolev space W s=2 .

This is a rephrasing of (2.4). 2.5 Analytic continuation of the integral formula

Observation 2.5. Let f , g 2 Hs .D/ be smooth up to the boundary. a) The integral (2.3) converges for Re s > 1 and determines a function I.s/ holomorphic in s. b) The series (2.5) converges for Re s > 0 and determines a function G.s/ holomorphic in s. This is an exercise in differentiation of series and integrals in parameters. However, I.s/ D G.s/ for real s > 1. In other words, the integral (2.3) admits an analytic continuation to the domain Re s > 0. We recall a more standard way7 of holomorphic continuations of the integral (2.3). Pass to the polar coordinates. Our problem reduces to continuation of the integral Z 2 Z 1 F .'; r/.1 r 2 /s2 r dr d': (2.7) .s 1/ 0 5

0

The reader unfamiliar with Sobolev spaces can consider this statement as a definition. There are numerous ways to define boundary values pointwise; however this leads to highly nontrivial phenomena; for instance, see e.g. a recent work [113]. 7 See e.g. [66]. 6

334


We let G.'; r/ WD F .'; r/.1 C r/s2 r and write the inner integral as

Z

1

::: D

G.'; r/.1 r/s2 dr: 0

Next, integrating by parts (for Re s > 1), we get

1 ::: D s1

Z 0

1

Gr0 .'; r/.1 r/s1 dr G.'; 0/:

We come to the integral that converges for Re s > 0. The last expression has a pole at s D 1, however in the integral formula (2.3) this pole is killed by the factor .s 1/. Therefore, the whole expression (2.3) is holomorphic in Re s > 0. Separately, let us discuss the case s D 1. Observation 2.6. Let h be a smooth function in the disk jzj 6 1. Then s1 lim s!1C0

“ h.z; zN /.1 jzj /

2 s2

jzj 0, the operators Ts .g/ are unitary in the space Hs . Proof. First, let s > 1. Then hTs .g/f1 ; Ts .g/f2 is “ ˇ C z ˛N ˇ C z ˛N s1 s N N s .1 jzj2 /s2 d .z/: D .˛ C z ˇ/ f2 .˛ C z ˇ/ f1 N N ˛ C zˇ ˛ C zˇ N and come to hf1 ; f2 i (this Problem 2.2. Change the variable u WD .ˇ C z ˛/=.˛ N C z ˇ/ is done in a wider generality in § 7.5). Thus for all s > 1 the operators are unitary. Taking into account the analytic continuation, we obtain the statement for all s > 0. The representations Ts .g/ are called unitary highest weight representation of SU.1; 1/. 2.7 Dual functional realization of Hs .D/. Now let us consider the “dual functional model” (see Subsection 1.6) for Hs . Let 1 , 2 be compactly supported distributions on D. Their inner product is given by Z Z h1 ; 2 i D .1 zu/ N s 1 .z/.u/: z2D

u2D

Lemma 2.9. For a distribution , denote by V . / its pushforward under the map z 7! z, where 0 < < 1. Then the function . / D hV . /; V . /i increases.

336


Proof. Indeed,

Z

Z

.1 2 zu/ N s .z/.u/ Z Z X s.s C 1/ : : : .s C k 1/ D 2k zN k .z/uk .u/ kŠ z2D u2D k>0 ˇZ ˇ2 X s.s C 1/ : : : .s C k 1/ ˇ ˇ z k .z/ˇ : 2k ˇ D kŠ z2D

. / D

z2D

u2D

k>0

Now consider distributions supported by the circle jzj D 1. We denote by Ms the space of all distributions satisfying the condition Z Z kk2 D lim .1 2 zu/ N s .z/.u/ < 1 !10 z2D

and define the inner product in Ms by Z h1 ; 2 i D lim

u2D

Z

!10 z2D

.1 2 z u/ N s 1 .z/2 .u/: u2D

Observation 2.10. 1 X 2 X s.s C 1/ : : : .s C k 1/ jck1 j2 : ck e ik' D kŠ kD1 k60 P Thus, the kernel Ks of our inner product consists of series k>0 ck e i k' . Problem 2.3. a) The space Ms =Ks is complete. b) Each element Hs admits a representation in the form Z f .z/ D lim .1 z u/ N s .u/; !10 u2D

where 2 Ms . c) Which elements f 2 Hs admit representations Z f .z/ D .1 z u/ N s .u/; u2D

where u is a compactly supported distribution on D? 2.8 Digression. A strange problem. There is a natural isomorphism SU.1; 1/ SL.2; R/. Consider the subgroup SL.2; Z/ SL.2; R/. Theorem 2.11. Let s D 26. There exists a function f 2 H26 such that the functions T26 .g/f;

where g ranges in SL.2; Z/;

form an orthogonal basis in Hs . A proof is far beyond our possibilities. This is a rephrasing of the following a priori theorem (see [72]): of T26 to PSL.2; Z/ is equivalent to the left representation of SL.2; Z/ in The restriction `2 SL.2; Z/ . The function f is not canonically defined; nobody knows how to find it.

7.3. Hilbert spaces of holomorphic functions

337

7.3 Hilbert spaces of holomorphic functions This section contains simple generalities on Hilbert spaces of holomorphic functions. 3.1 Spaces of holomorphic functions Theorem 3.1. Let be a domain in Cn . Let L.z; u/ N be a positive definite kernel holomorphic in z and anti-holomorphic in u. Then the Hilbert space HB .I L/ consists of functions holomorphic on . Proof. By Hartog’s Theorem (see, for instance, [199]), a function that is separately holomorphic in each argument is jointly holomorphic. Hence K.z; u/ is continuous on . Therefore, the convergence in our space implies uniform convergence on compact subsets. Functions 'a .z/ WD K.z; a/ are holomorphic. Their finite linear combinations are dense in HB . But a uniform limit of holomorphic functions is holomorphic. 3.2 Weighted Bergman spaces. Let .z/ be a non-negative function on . Denote by B.; / the space of holomorphic functions on satisfying the condition Z jf .z/j2 .z/ d .z/ < 1;

where d.z/ denotes the Lebesgue measure on . Define the L2 -inner product Z hf; gi D f .z/g.z/ .z/ d .z/ (3.1)

in the space B.; /. Theorem 3.2. Let be continuous and strictly positive on . Then the space B.; / is closed in L2 .; /. Moreover, this space is determined by some reproducing kernel K.z; u/ N holomorphic in z and anti-holomorphic in u. N Lemma 3.3. The theorem holds for the polydisk jzj j < 1 and the weight D 1. Proof of Qthe lemma. See Subsection 2.1. Our space is determined by the reproducing kernel .1 zj uNj /2 . Proof of the theorem. Consider an arbitrary point w 2 and a small polydisk D n with a center at w. For any f , Z Z 2 jf .z/j .z/ d .z/ > minn .z/ jf .z/j2 d .z/: (3.2)

z2D

Dn

Now, let fj 2 B.; / converge in L2 -sense to f . Then it converges in L2 .D n /. By the lemma, the limit f is holomorphic in D. Hence it is holomorphic in . By the same lemma, fj .w/ converges to f .w/. Therefore the evaluation functional is continuous and we apply Theorem 1.8.

338


3.3 Digression. Versions of Theorem 3.2. The condition .z/ > 0 of this theorem is too restrictive for some applications; we discuss how to weaken it. A. Denote by D n the polydisk jz1 j < 1; : : : ; jzn j < 1. Let .z/ be a continuous strictly positive continuous function on the torus T n W jz1 j D D jzn j D 1. Consider the space A.D n / of all functions in D n admitting a holomorphic extension to a larger polydisk jzj j < 1Cı. Equip A.D n / with the inner product Z hf; gi D f .z/g.z/ .z/ d .z/; Tn

where d is the Lebesgue measure on the torus. x n / of A.D n / in L2 .D n ; / consists of functions holomorphic Proposition 3.4. The closure A.D n n x in D . Moreover, A.D / is determined by some reproducing kernel. Proof. Let us show that for each a 2 D n , the evaluation linear functional à .f / D f .a/ on A.D n / is continuous. Represent f 2 A.D n / by the Cauchy integral formula Z Y dzj 1 f .z/ n .2 i / jz1 jDjz2 jDD1 zj aj Z o n Y 1 1 1 .z/ d .z/: D f .z/

.z/ .z a / j j .2/n jz1 jDjz2 jDD1

f .a/ D

Denote by a .z/ the expression in the curly brackets. The last integral is the L2 .T n ; /-inner product hf; Na i. Therefore, à is a bounded linear functional. Moreover, for a ranging in a compact set C D n , the linear functionals à are uniformly bounded. Let fj 2 A.D n / be a sequence convergent to a function f 2 L2 .T n ; / in the L2 .T n ; /sense. Due to the continuity of evaluation functionals, this sequence converges pointwise. Moreover, this sequence is L2 -bounded, therefore it is uniformly bounded in each compact subset C D n . Next, a uniformly bounded pointwise convergent sequence of holomorphic functions converges uniformly on compact sets. Therefore, f is holomorphic. x be the open B. Let † Cn be a smooth closed hypersurface. Let Cn (respectively ) (respectively closed) domain bounded by †. Let .z/ be a strictly positive continuous function on †. Consider the space A./ of functions admitting a holomorphic continuation to some x Equip A./ with the inner product neighborhood of . Z hf; gi D f .z/g.z/ .z/ dS.z/; †

where dS.z/ is the surface Lebesgue measure on „. x Denote by A./ the closure of A./ in L2 .†; /. x Proposition 3.5. The space A./ consists of functions holomorphic in ; this space is determined by a reproducing kernel.


339

Proof. Denote by ! the following .2n 1/-differential form on Cn : !.z/ WD

n X

zNj d zN 1 ^ ^ d zNj 1 ^ dzj C1 ^ d zNn ^ dz1 ^ ^ dzn :

j D1

By the Martinelli–Bochner formula (see [199]), Z !.u z/ .n 1/Š f .z/ f .u/ D n .2 i / ju zj2n z2† and we repeat the arguments given in the previous proof.

3.4 Examples and counterexamples. In the three previous statements, we have required the strict positivity of a weight function. A. Consider the disk D W jzj < 1 on C. Denote by A.D/ the space of functions holomorphic in a neighborhood of jzj 6 1. Observation 3.6. There exists a weight function .z/, z 2 D that is smooth everywhere and strictly positive outside the segment Œ1; 0 such that the closure of A.D/ in L2 .D; / contains a function non-holomorphic in D. Proof. Let U C be a simply connected domain. Any function f holomorphic in U C is a limit of a sequence of polynomials that uniformly converges on all compact subsets in U (see, for instance, [199]). p We take U to be the circle jzj < 1 C ı cut by the segment p Œ1 ı; 0. Let f .z/ D z in U . Let hj .z/ be a sequence of polynomials convergent to z uniformly on all compact subsets in U . It is more-or-less clear that if .z/ sufficiently rapidly decreases near the cut, then this uniform convergence p implies the convergence in L2 .D; /. Hence the L2 -closure of A.D; / contains the function z discontinuous along the cut. B. However, the function .z/ in the previous counterexample must be flat along Œ1; 0 (i.e., its partial derivatives of all orders are 0 on Œ1; 0). Now let .z/ be a smooth non-negative function on the circle T W jzj D 1 having a unique zero of order 2k at z D 1. Let A.D/ be the same as above. Define the inner product in A.D/ by Z hf; gi D

f .z/g.z/ .z/ d .z/: jzjD1

Observation 3.7. The closure of A.D/ in L2 .T ; / contains only functions holomorphic in the disk jzj < 1. Proof. Denote by A .D/ A.D/ the set of functions of the form f .z/ D h.z/.z 1/k , where h.z/ 2 A.D/. Consider the closure of A .D/ in L2 .D; /. The convergence of fj in the sense of L2 .T ; / is equivalent to the convergence of hj in the sense of L2 .T ; .z/j1 zj2k /. In the latter case, the limit of hj is holomorphic (because of Proposition 3.4).

340


u/ N C. Consider the space of entire functions on C determined by the reproducing kernel sin.z , zu N i.e., the Paley–Wiener space PW, which was discussed in Subsection 4.8.2. The inner product in PW admits the integral representation Z 1 hf; giPW D f .x/g.x/ dx 1

(by the definition). Also, PW is a closed subspace in L2 .R/. By the Whittaker–Kotelnikov–Shannon Theorem, the same inner product admits another “integral” representation 1 X f .n/g.n/ hf; gi D nD1

(moreover, there are other integral representations of such a type, see [28]). 3.5 Boundary values. Trace theorems. Here we discuss restrictions of holomorphic functions f 2 HB .; K/ to submanifolds in the boundary of . The problem is delicate, because generally functions f 2 HB .; K/ are discontinuous; therefore they have no values at individual points of @. x D \ @ be its closure. We Let Cn be an open domain, let @ be its boundary, say that is a complete circular domain, if – for each z 2 , and 2 C with jj 6 1, we have z 2 ; – if z 2 @, and jj < 1, then z 2 . Denote by T the one-dimensional group of numbers e i' . Let be a complete circular domain, let L.z; u/ N be a T -invariant reproducing kernel in , i.e., L.e i' z; e i' u/ N D L.z; u/: N B Due to the i' T -invariance of L, the group T acts on H .; L/ by unitary operators f .z/ 7! f e z . Let be a distribution supported by . We decompose it into the “Fourier series”,

.z/ D

1 X

e i n' Œk .z/ WD

kD0

1 X kD0

hŒk ; Œl i D 0

Z

2

e i n'

ze i n' d';

0

for k ¤ l:

(3.3)

Problem 3.1. Consider the corresponding holomorphic functions 2 HB .; L/, Z Z L.z; u/.u/; N F Œk .z/ D L.z; u/ N Œk .u/: F .z/ D u2

u2

Then the expansion F .z/ D

X

F Œk .z/

k>0

is the expansion of a holomorphic function in homogeneous polynomials.


341

Now let be a distribution supported by the boundary @. Next, for t being in the interval .0; 1/, denote by .t z/ the distribution (supported by ) obtained by a homothety, Z Z .z/f .z/ WD .z/f .t z/: z2

z2

By (3.3), k.t z/k2H? .;L/ D

X k>0

jt j2k kŒk k2H? .;L/ :

Observation 3.8. a) For 0 < t < 1, the value k.t z/k2 increases. b) If k.t z/k2 is bounded for t 2 .0; 1/, then represents an element of H? .; L/ by .z/ WD

lim .t z/;

t!10

where the limit is a limit in the Hilbert space H? .; L/. Moreover, X k.z/k2H? .;L/ D kŒk k2H? .;L/ : k>0

Now let M be a subset in @ (for instance, let M be a submanifold). Let be a measure supported by M . Theorem 3.9. Let – the limit

lim L.t z; t u/ DW L .z; u/

t!10

exist almost everywhere on M M in the measure ; – the limit be dominated, i.e., there exists a function h.z; u/ N 2 L1 .M M / such that for all t 2 .0; 1/ the following inequality holds: jL.t z; t u/j N 6 h.z; u/: N 1

a) Then for any f 2 L .M /, the distribution f represents an element of H? .; L/. b) The map I W f 7! f is a continuous operator L1 .M; / ! H ? .; L/ in the following sense: if a sequence fj 2 L1 is uniformly bounded and fj converges to f in measure, then the sequence Ifj converges to If in H ? .; L/. c) The inner product on H ? .; L/ induces the following inner product8 on L1 .M; /: “ ? hf1 ; f2 iH .;L/ D L .z; u/f N 1 .z/f2 .u/d.z/d.u/: M M

Proof. By definition, kfk2H? .;L/ D

“ L.t z; t u/f N .z/f .u/d.z/d.u/:

lim

t!10

M M

By Lebesgue’s dominated convergence theorem, the limit is finite and “ kf k2H? .;L/ D L .z; u/f N .z/f .u/d.z/d.u/: M M

8

Certainly, L1 .M; / with the Euclidean topology is incomplete.

342


Theorem 3.10. Under the same conditions, each holomorphic function f 2 H? .; L/ has a limit Jf .z/ D lim f .t z/ t!10

almost everywhere on M ; moreover Ff 2 L1 .M; /. In fact J W HB .; L/ ! L1 .M; / is the operator dual to I W H.M; / ! L1 .M /. We omit an examination of the details.

3.6 Positive definite kernels in tube domains. Below we discuss positive definite kernels in Siegel wedges. For this aim, we need some preparations and some generalities. Let be a nonempty open sharp9 convex cone in Rm . Let B Rm be the dual x B the closure of cone, i.e., B is the set of all vectors such that hx; yi > 0. Denote by B the cone . A tube domain in Cm is a domain of the form D Rm ˚ i , where is an open convex cone. We say that a holomorphic function f in a tube domain has polynomial growth if for some c 2 the following estimate holds for all z 2 c C : jf .z/j 6 C.1 C jzj/N

for some N and C :

Theorem 3.11. A holomorphic function f in the tube domain Rm ˚i has polynomial growth if and only if f is a Fourier (Laplace) transform10 of a tempered distribution x B. supported by the closed cone This is a version of the Paley–Wiener–Schwartz Theorem, see e.g. [89], Vol. 1, Theorem 7.4.3. Theorem 3.12. Let ‰ be a holomorphic function of polynomial growth in the tube N on is a positive definite kernel domain D Rm ˚ i . The function ‰.z u/ if and only if ‰ is a Fourier transform of a positive measure supported by the closed xB cone Proof. By Theorem 3.11,

Z ‰.z/ D

e itz R.t /;

(3.4)

t2Rn

x B . Fix c 2 . Let where R is a tempered distribution supported by ‰.x C ic/. Let Z e it.xCic/ R.t /: c .x/ D ‰.x C i c/ D t2Rn

9 10

A convex cone is sharp if it does not contain a line. The discussed situation is a multivariate generalization of the Laplace transform.

c .x/

WD

343

7.4. Multipliers and invariant kernels

The function c .x y/ is a positive definite kernel on Rn and hence transform of a positive measure, Z e itx d.t /: ‰.x C i c/ D

c

is a Fourier

t2Rn

Comparing the last expression and (3.4), we obtain d.t / D e cx R.t /, therefore the distribution R.t / is a positive measure. zuN s Problem 3.2. Let s > 0. Then kernel 2i on the upper half-plane Im z > 0 is positive definite.

7.4 Multipliers and invariant kernels 4.1 Multipliers. Let a group G act on a space X. Let .g; x/ be a C -valued function on G X. Consider the following operators in the space of functions on X : T .g/f .x/ D f .xg/ .g; x/:

(4.1)

Observation 4.1. T .g/ is a representation of the group G, i.e., T .g1 /T .g2 / D T .g1 g2 / if and only if .g; x/ satisfies the multiplier identity .g1 g2 ; x/ D .g1 ; xg2 / .g2 ; x/:

(4.2)

Such functions .g; x/ are called multipliers. Initial examples are given below: Example 1. The real Jacobian JR .g; x/ satisfies this identity. Now we think that X is a domain in Rn , or, more generally, a manifold equipped with a smooth measure. For each ˛ 2 C, the function .g; x/ WD JR .g; x/˛ also satisfies the multiplier identity. Example 2. Let X be a domain in Cn . The complex Jacobian JC .g; z/ satisfies (4.2). Certainly, integer powers JC .g; z/n of the complex Jacobian also satisfy our identity. But complex powers JC .g; z/˛ of complex functions, generally, are not well defined. In any case, JC .g; z/˛ JC .g; z/ ˇ is well defined if ˛ ˇ 2 Z. Example 3. See highest weight representations of SU.1; 1/ in Subsection 2.6 and the discussion of the difficulties with complex Jacobians.

344


Example 4. See the formula for the action of the Heisenberg group in the Fock space (Subsection 4.1.10). Problem 4.1. Let .g; x/ WD JR .g; x/1=2Cis , where s 2 R. Then the representation (4.1) is unitary in L2 .X /. 4.2 Invariant reproducing kernels. Now let a group G act on a space of functions on via transformations T .g/f .x/ D f .xg/ .g; x/: Let L.x; y/ be a reproducing kernel on . Theorem 4.2. a) The operators T .g/ are unitary in HB .; L/ if and only if L is invariant in the following sense: L.xg; yg/ D L.x; y/ .g; x/1 .g; y/ 1 :

(4.3)

b) Let CN be G-homogeneous. Then an invariant kernel K.x; y/ that is holomorphic in x and antiholomorphic in y is uniquely determined by the multiplier .g; x/ (we do not claim existence). Proof. a) Let unitary, hence, hf; T .g 1 /

x

be the corresponding overfilled system. The operators T .g/ are

x iHB

D hT .g/f;

xi

D fvalue of T .g/f at xg D f .xg/ .g; x/:

Thus, T .g 1 /

x

D .g; x/

xg :

Next, L.xg; yg/ D h

yg iHB

xg ;

D h.g; x/1 T .g 1 /

1 1 / yi x ; .g; y/ T .g

D .g; x/1 .g; y/ 1 hT .g 1 / 1

D .g; x/

.g; y/

1

h

x;

x ; T .g

1

/

yi

y i:

b) We know L.x0 ; x0 /, and, by (4.3), we know L.x; x/ for all x. Finally, by the uniqueness theorem for holomorphic functions, we know L.x; y/. Example. See Subsection 2.6 on the highest weight representations of SU.1; 1/. Example. See Subsection 4.1.10 on the representation of the Heisenberg group in the Fock space.

345

7.4. Multipliers and invariant kernels

Now we continue a list of examples; however, they are not necessary for us. 4.3 Digression. Unitary representations of the universal covering of SU.1; 1/. A. The non-unitary principal series. Fix p, q 2 C. Denote by S 1 the circle jzj D 1 on C. For g 2 SU.1; 1/, define the operator Tpjq .g/ in the space C 1 .S 1 / by the formula b C az N a b N p .a C bz/ N q : Tpjq N .a C bz/ f .z/ D f (4.4) N b aN a C bz Observation 4.3. Tpjq is a representation of SU.1; 1/ . These representations are called representations of non-unitary principal series. Proof. Recall, that the function ln a is well defined on SU.1; 1/ ; therefore we can define N q WD e .pq/ ln a .1 C a1 bz/ N p .1 C a1 bz/ N q : N p .a C bz/ .a C bz/ Next, the (complex) Jacobian of the transformation z 7! is a multiplier.

bCaz N N aCbz

N is .a C bz/

2

(4.5)

. Therefore (4.5)

Observation 4.4. TpC1jq1 is equivalent to Tpjq . The intertwining operator is the multiplication by z. Next, consider the integral operator Ipjq in C 1 .S 1 / given by Z 1 dz .1 z u/ N p1 .1 zu/ N q1 f .z/ : Ipjq f .u/ D 2 i .p C q 1/ jzjD1 z We define the integral in (4.6) as Z

(4.6)

Z WD lim

jzjD1

"!C0 jzjD1"

:

Theorem 4.5. The operator Ipjq intertwines Tpjq and T1qj1p . Problem 4.2. Prove this. B. Unitary principal series Observation 4.6. A representation Tpjq is unitary in L2 .S 1 / if and only if Im p D Im q;

Re p C Re q D 1:

These representations are called “representations of unitary principal series”. C. Highest weight representations. If q D 0, then the formula (4.4) reduces to the formula (2.8) for the highest weight representations. To be precise, note that a highest weight representation is realized in the subspace consisting of holomorphic functions11 . P If p D 0, we have an invariant subspace that consists of functions of the form k60 . The quotient is the highest weight representation again. D. Unitary lowest weight representations. They are twins of highest weight representations. Now we P take p D 0, q > 0 and consider the representation T0jq in the space of functions of the form k>0 ck z k . 11

In particular, Tpj0 is reducible.

346


Problem 4.3. (For a reader who is slightly familiar with representation theory.) a) A representation Tpjq is irreducible if only if p … Z, q … Z. b) Describe all subquotients of the representations Tp;q . Where are finite-dimensional representations? c) Which representations are well defined on the group SU.1; 1/? On its two-sheet covering SL.2; R/? E. Complementary series. Now let p, q 2 R, 0 < p < 1;

0 < q < 1:

(4.7)

We consider the inner product in C 1 .S 1 / given by hf1 ; f2 ipjq D

1 (4.8) .2 i /2 .p C q 1/ Z Z dz du lim .1 t 2 z u/ N p1 .1 t 2 zu/ N q1 f1 .z/f2 .u/ : t!C0 jzjD1 jujD1 z u

Theorem 4.7. a) Under the condition (4.7), the inner product (4.8) is positive definite. b) The operators Tpjq .g/ are unitary with respect to the inner product (4.8). These representations are said to be representations of “complementary series”. We leave a proof as an exercise for the reader. In fact, hz n ; z m i D

.1 p/.p/ : : : .p C n/ 1 ın;m : .p/.q/ q.q C 1/ : : : .q C n 1/

F. Bargmann–Pukanszky classification Theorem 4.8 (Bargmann–Pukanszky). The following unitary representations exhaust all irreducible unitary representations of SU.1; 1/ : – unitary principal series; – complementary series; – highest weight representations; – lowest weight representations; – trivial (one-dimensional) representation. In our notation, the segment p C q D 1, 0 < p < 1 is contained in both the principal and the complementary series. Also, T1j0 is a representation of principal series; but it is reducible and splits into a sum of the highest weight representation T1 and the lowest weight representation Tx1 . 4.4 Digression. Principal series of representations of SL.n; C/. Denote by Fln the space of all complete flags in Cn , see Subsection 4.6. Denote by Grp;np the Grassmannians of p-dimensional subspaces in Cn . We have an obvious map p W Fln ! Grp;np . As usual, we identify the space Matp;np .C/ with an open subset in Grp;np . The group SL.n; C/ acts on Grp;np and therefore it acts on Matp;np .C/ by linear fractional transformations. Denote by Jp .g; z/ the corresponding complex Jacobians.

347

7.5. The Berezin scale

Now fix ˛1 ; : : : ; ˛n1 , ˇ1 ; : : : ; ˇn1 2 C, such that ˛j ˇj 2 Z. We write a representation T˛jˇ of GL.n; C/ in C 1 .Fln / by the formula T˛jˇ f .V/ D f .gV /

n1 Y

Jp .g; p .V//˛p Jp .g; p .V //ˇp ;

where V 2 Fln :

pD1

Such representations are said to be representations of (non-unitary) principal series of SL.n; C/.

7.5 The Berezin scale We extend the construction of § 7.2 to arbitrary symplectic groups Sp.2n; R/. Digressions contain some more general constructions. 5.1 Action of Sp.2 n; R/ on the Siegel wedge. Consider the linear space Symmn .C/ of symmetric complex n n-matrices. Let Pn Symmn .C/ the Siegel wedge, i.e., the set of matrices satisfying Im z > 0, see Subsection3.3.8. We realize the group a b Sp.2n; R/ as the group of real symplectic matrices g D c d . Recall that this group acts on the Pn by the linear fractional transformations: z 7! z Œg WD .a C zc/1 .b C zd /: There is an alternative expression for z Œg : z Œg D .d t z C b t /.c t z C at /1 : To establish the equivalence of these two expressions, we must verify the identity .b C zd /.c t z C at / D .a C zc/.d t z C b t /: We open brackets and refer to the defining relations (3.1.4) for the symplectic group. Lemma 5.1. The complex Jacobian of the map z 7! z Œg is JC .g; z/ D det.a C zc/.nC1/=2 : Proof. By Lemma 2.11.1, the differential of our transformation is dz 7! .a C zc/1 dz .cz Œg C d /:

(5.1)

Next, cz Œg C d D c.d t z C b t /.c t z C at /1 C d

D c.d t z C b t / C d.c t z C at / .c t z C at /1

D .cd t C dc t /z C .cb t C dat / .c t z C at /1 D .c t z C at /1 :

348


Here we again have applied the defining relations (3.1.4) for the symplectic group and obtain 1 in the square brackets. Thus the differential is dz 7! .a C zc/1 dz .c t z C at /1 : Now the statement follows from the next lemma. Lemma 5.2. Consider the linear operator x 7! ˛x˛ t in the space Symmn .C/. Its determinant is det.˛/nC1 . Proof of Lemma 5.2. Denote the determinant of our operator by .˛/. Obviously, .˛˛ 0 / D .˛/.˛ 0 /: Represent ˛ D AB, where A, B 2 SU.n/ and is a diagonal matrix. First, the determinant of the map x 7! BxB t is 1, because the group SU.n/ has no nontrivial homomorphisms to C . It remains to evaluate the determinant of the map x 7! x. Denote by Eij the matrix units. The matrices Ekk and Eij C Ej i , where i > j , form a basis of the space Symmn .C/. These vectors are the eigenvectors of our linear operator. The eigenvalues are 2k , j j . The product of eigenvalues is Y jnC1 D det nC1 D det ˛ nC1 : 5.2 Actions of Sp.2 n; R/ in the space of holomorphic functions. For each ˛ 2 C, define the following operators T˛ .g/ in the space of holomorphic functions on Pn : T˛ .g/f .z/ D f z Œg det.a C zc/˛ : (5.2) Proposition 5.3. T˛ is a projective representation of Sp.2n; R/. If ˛ is integer, then this representation is linear. Proof. The multiplier identity can be easily verified by a straightforward calculation. However we prefer a longer way. First of all, det.a C zc/ D JC .g; z/1=n ¤ 0 for z 2 Pn because the Jacobian of a biholomorphic map does not vanish. Hence the multivalued function det.a C zc/˛ splits into a countable number of holomorphic branches, these branches differ by scalar factors e 2 i˛ . Since the Jacobian satisfies the multiplier identity (4.2), i.e., JC .g1 g2 ; z/ D JC .g2 ; z/JC .g1 ; z Œg2 /; it follows that the powers of the Jacobian satisfy the same identity up to scalar factors (that appear because powers are multi-valued).

349


Proposition 5.4. T˛ .g/ determines a representation of the universal covering group Sp.2n; R/ . Proof. Here we can refer to a general theorem: semisimple Lie groups have no central extensions except coverings; in particular projective representations of semisimple groups are representations of coverings. But we also present a direct proof. The expression .g/ WD det.a C i c/˛ does not vanish. Therefore this function is well defined on the simply connected space Sp.2n; R/ . Now we can choose a holomorphic branch of det.aCzc/˛ in a canonical way setting ˇ det.a C zc/˛ ˇzDi D det.a C i c/˛ : 5.3 Berezin kernels on matrix wedges. Statement of the main result Theorem 5.5. The kernel on Pn ,

z u K˛ .z; u/ D const det 2i

˛

;

(5.3)

is invariant under the operators T˛ .g/. Moreover, it is unique up to a scalar invariant kernel. Proof. We refer to Theorem 4.2. An invariant kernel is unique because the action is transitive. The invariance of K˛ .z; u/ N is equivalent to the identity

det

z Œg .uŒg / 2i

D det.a C zc/1 det.c t u C at /1 det

z u : 2i

This follows from12 z Œg uŒg D .a C zc/1 z u .c t u C at /1 :

(5.4)

To prove the last identity, we write z Œg uŒg D .a C zc/1 .b C zd / .d t u C b t /.c t u C at /1 D .a C zc/1 .b C zd /.c t u C at / .a C zc/.d t u C b t / .c t u C at /1 : Applying the defining relations (3.1.4) for Sp.2n; R/, we get z u in the big brackets. Theorem 5.6. The Berezin kernel K˛ .z; u/ is positive definite if and only if ˛ is contained in the Berezin–Wallach set ˛ D 0;

1 n1 n1 ; 1; : : : ; , or ˛ > : 2 2 2

(5.5)

350


trivial representation Weil representation 0

standard Hardy space nC1 2

n1 2

1 2

n

Hardy spaces

degenerate representation

˛

weighted spaces

Figure 7.1. Reference Theorems 5.6 and 6.3. The Berezin–Wallach set and exotic Hardy spaces.

The proof is given in Subsections 5.5–5.11. We denote the corresponding Hilbert space HB .Pn ; K˛ / of holomorphic functions by H˛ . In the next subsection, we give another description of H˛ for ˛ > n. This gives a partial proof of the theorem, however it is not used in the main proof. 5.4 Weighted spaces. Consider the space H˛0 of holomorphic functions f on the wedge Pn satisfying Z

˛n1 jf .z/j2 det Im.z/ d .z/ < 1; (5.6) Pn

where d.z/ is the Lebesgue measure on the space Pn . Equip this space with the inner product Z hf; gi˛ D C.˛/

f .z/g.z/ det Im.z/˛n1 d .z/;

(5.7)

Pn

where C.˛/ is a positive constant. Its actual form is not too important for us; the most natural choice is Z ˇ ˇ2˛ ˇ ˇ ˇ det 1 iz ˇ C.˛/1 D d .z/ ˇ ˇ 2 Pn (this integral is evaluated in the next section). Theorem 5.7. a) For ˛ > n the Hilbert space H˛0 is nonempty and the representation T˛ .g/ of Sp.2n; R/ is unitary in H˛0 . N and b) The space H˛0 is determined by the reproducing kernel K˛ .z; u/ H˛ D H˛0 for ˛ > n: c) For ˛ 6 n, the space H˛0 is empty. 12

In turn, this is a version of (11.4).


351

Since an invariant reproducing kernel is unique, the statement b) is a corollary of a). Incomplete proof. Substitute z D uŒg into the integral (5.7). The real Jacobian of the map z 7! z Œg is d.z Œg / D JR .g; u/ D jJC .g; u/j2 D j det.a C zc/j2.nC1/ : d.z/ Next, we use formula (5.4)

det.Im z

Œg

z Œg z Œg / D det 2i

D det.Im z/j det.a C zc/j2 :

Hence the right-hand part of (5.7) equals hf1 ; f2 i˛ Z D f1 .uŒg /f2 .uŒg / det j.a C uc/j2˛ det Im.u/˛n1 d .u/ Pn Z D f1 .uŒg / det.a C uc/˛ f2 .uŒg / det.a C uc/˛ det Im.u/˛n1 d .u/ Pn

D hT˛ .g/f1 ; T˛ .g/f2 i˛ : Thus the operators T˛ .g/ are unitary. For ˛ < n, the space H˛0 is empty. Indeed, the boundary @Pn of Pn consists of matrices x C iy;

where x 2 Symmn .R/, y 2 Symmn .R/, y > 0, det y D 0.

The boundary @Pn is a singular real manifold, a point x C iy 2 @P is nonsingular if and only if rk y D n 1. Near a nonsingular piece of @Pn the weight function det Im.z/˛n1 D det y ˛n1 has the usual power singularity. It is locally integrable if and only if the power is > 1. If ˛ > n C 1, then H˛0 is nonempty. Indeed, in this case, the function Im.z/˛n1 is continuous on the wedge; it remains to find a function that sufficiently rapidly decreases in Pn (we need the convergence of the integral (5.6)). We can set f .z/ WD N det .z C i/=2i for large N . The case n < ˛ < nC1 is not completely obvious (because the function det y ˛n1 has rather complicated singularities at the manifold rk y 6 n 2.) The statement follows from the next lemma. Lemma 5.8. The function det.i C z/˛ is contained in H˛0 for all ˛ > n. In particular, H˛0 is nonempty.

352


The L2 -norm of det.i C z/˛ can be evaluated explicitly. We can propose this as a (non-obvious) exercise; however a similar (and more complicated) calculation is given in Subsections 6.3–6.6. 5.5 The Laplace transform and the Berezin kernels. We wish to apply the Bochnertype Theorem 3.12. Consider the space Symmn .R/ ' Rn.nC1/=2 of real symmetric n n-matrices. Endow it with the inner product hx; yiSymmn .R/ D tr xy: Consider the cone Posn Symmn .R/ of positive (semi)-definite matrices. Problem 5.1. Show that the cone dual to Posn is Posn . N with The Berezin kernel K˛ is of the form ‰˛ .z u/ ‰˛ .z/ WD det.z= i /˛ I ‰˛ is a holomorphic function of polynomial growth in the tube Pn D Symmn .R/ C i Posn Symmn .C/: Therefore, K˛ is positive definite if and only if ‰˛ is the Fourier transform of a certain positive measure ˛ supported by the closed cone Posn . Next, ‰˛ .z/ satisfies the following homogeneity conditions: ‰˛ .gzg t / D j det.g/j2˛ ‰˛ .z/;

where g 2 GL.n; R/:

Hence the distribution ˛ on Posn satisfies the homogeneity condition

˛ .gxg t / D det.g/2˛ ˛ .x/:

(5.8)

Now we can find ˛ from this homogeneity. The immediate reaction is to write: Conjecture:

˛ .x/ D det.x/˛.nC1/=2 dx:

(5.9)

This is almost true, however the situation is more delicate. 5.6 Description of the measures ˛ . The regular case ˛ > .n 1/=2 and for z 2 Pn , Theorem 5.9. a) For ˛ > n1 2 Z ˚

det.x/˛.nC1/=2 exp i tr xz dx D ŒnI ˛ det.z= i /˛ ; x2Posn

where dx WD

Y k6l

dxkl

(5.10)

353


and the constant ŒnI ˛ is the Whishart–Siegel integral Z ˚

ŒnI ˛ WD det x ˛.nC1/=2 exp tr x dx:

(5.11)

Posn

b) The Whishart–Siegel integral converges if and only if ˛ > ŒnI ˛ WD

n.n1/=4

n1 2

and equals

n Y ˛ .k 1/=2 :

(5.12)

kD1

Recall that the -function is defined by Z 1 x ˇ 1 e x dx: .ˇ/ WD

(5.13)

0

It is a meromorphic function with poles at ˇ D 0, 1, 2,…. Corollary 5.10. For ˛ > measure ˛ is given by

n1 , 2

the Berezin kernel K˛ .z; u/ N is positive definite and the

˛ WD det.x/˛.nC1/=2 dx;

where ˛ > .n 1/=2.

(5.14)

Proof of the corollary. This measure is positive on the cone Posn . Let ˛ > nC1 . Then the density of the measure ˛ is continuous, therefore ˛ is a 2 tempered distribution. Let n1 < ˛ < nC1 . Then the local integrability of the density is not self-evident. 2 2 However, expf tr xg is a strictly positive function with a convergent integral with respect to ˛ . This implies the local integrability; thus the measure ˛ is a tempered distribution. Observation 5.11. For ˛ 6 .n 1/=2 the measure (5.14) is not a distribution. Proof of the observation. Let ˛ 6 n1 . The boundary of the cone Posn is a singular 2 manifold det.x/ D 0, x > 0. Nonsingular points of the boundary satisfy rk x D n 1. For ˛ 6 .n1/=2, the density of the measure is notRlocally integrable near a nonsingular point; therefore generally speaking the integrals '.x/ d ˛ .x/ are divergent, i.e., ˛ is not a distribution. Derivation of the formula (5.10). Firstly, set z D i r, where r 2 Posn . Then p p i tr zx D tr rx D tr rx r: We write the integral as Z ˚ p p p p p p det.r/˛CnC1 det. rx r/˛n1 exp tr rx r det.r/n1 d. rx r/:

354


p p After the substitution x 0 D rx r, we come to (5.10). Next, both sides of (5.10) are holomorphic in z 2 Pn , hence the identity holds for all z 2 Pn . This is almost sufficient for a proof of Corollary 5.10, however it remains a difficulty nC1 with the interval ˛ 2 n1 mentioned above and we need an explicit evaluation ; 2 2 of the Whishart–Siegel integral. 5.7 The evaluation of the Whishart–Siegel integral. Represent the matrix x as a block .1 C n/ .1 C n/ matrix and write out its determinant by formula (1.1.12), p q xD ; det x D det.r/ .p qr 1 q t /: qt r Then ŒnI ˛ Z ˚

WD det x ˛.nC1/=2 exp tr x dx Z x>0 ˚

D det.r/˛.nC1/=2 .p qr 1 q t /˛.nC1/=2 exp p tr r dp dq dr: r>0 pqr 1 q t >0

Firstly, we integrate dp dq with a fixed r > 0, Z .p qrq t /˛.nC1/=2 e p dp dq t p>qrq Z D .p qr 1=2 r 1=2 q t /˛.nC1/=2 e p det r 1=2 dp d.qr 1=2 /: p>qrq t

We pass to the new variable h WD qr 1=2 2 Rn1 and come to Z 1=2 .p hht /˛.nC1/=2 e p dp dh: det r p>hht

Next, we introduce a new variable y WD p hht instead of p and obtain Z t 1=2 y ˛.nC1/=2 e yhh dy dh det r n1 y>0; h2R Z Z t 1=2 ˛.nC1/=2 y D det r y e dy e hh dh Rn1

y>0

D det r

1=2

.˛

n1 / .n1/=2 : 2

355


Thus, we have an induction step: Z Œn; ˛ D .˛

n1 / .n1/=2 2

D Œn 1I ˛.˛

det.r/˛n=2 e tr r dr

r>0 n1 / .n1/=2 2

and this implies (5.12).

5.8 Construction of the measures ˛ . The singular case. Denote by Oj Posn the xj the closure of Oj , set of all positive semi-definite matrices of rank j . Denote by O S x i.e., Oj D k6j Ok . The sets Oj are precisely GL.n; R/-orbits on Posn ; the orbit On is the open cone Posn . All the orbits Oj , except On , are also SL.n; R/-homogeneous. Theorem 5.12. a) For each j D 0; 1; : : : ; n 1 there is a unique to within scalar factor SL.n; R/-invariant measure13 d j=2 on the orbit Oj . For each g 2 GL.n; R/, d j=2 .gxg t / D j det.g/jj d j=2 .x/:

(5.15)

b) This measure represents a tempered distribution on Rn.nC1/=2 . c) The Fourier transform of the measure j=2 is const det.z=2i /j=2 . Lemma 5.13. a) Oj ' GL.n; R/=H , where the subgroup H consists of matrices

a 0 ; b c

a 2 O.j /; c 2 GL.n j; R/:

(5.16)

b) Oj ' SL˙ .n; R/=H 0 , where H consists of matrices of the same form (5.16) satisfying the additional condition c 2 SL˙ .n j; R/. Proof of the lemma. Consider the symmetric matrix 0 1j 2 Oj Ij D 0 0nj or, equivalently, consider the degenerate quadratic form 12 C C j2 on Rn . Its kernel ker Ij is spanned by the last .nj / basis vectors. A linear transformation g 2 GL.n; R/ preserving this form must preserve its kernel. On the other hand, g must also induce an orthogonal transformation on Rn = ker Ij . Therefore, the stabilizer H GL.n; R/ of this quadratic form is (5.16). 13 We write measures on Oj as j=2 to keep the correspondence with the definition of measures ˛ given above.

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Proof of Theorem 5.12.a. We wish to apply Theorem C.11 in theAddendum, see p. 520. The tangent space to Oj D GL.n; R/=H at the point Ij is the space of j .n j /matrices. The group H acts on this space by the transformations 7! ac t . The determinant of this map is det.a/nj det.c/j (see Lemma 2.11.3). For elements of H 0 we get ˙1. We apply Theorem C.11 and obtain the existence and uniqueness of SL˙ .n; R/-invariant measure j=2 . Lemma 5.14. There is a constant J such that for each g 2 GL.n; R/, d .gxg t / D j det.g/jJ d .x/:

(5.17)

Proof of Lemma 5.14. Consider a scalar matrix ƒ WD 1n 2 GL.n; R/. It can be readily checked that the measure j=2 .ƒxƒ/ is SL.n; R/-invariant. Therefore,

j=2 .ƒxƒ/ D ~ j=2 .x/: Obviously, the constant ~ is multiplicative with respect to , i.e., ~0 D ~ ~0 : Hence ~ D jja for some a. Since each g 2 GL.n; R/ admits a representation g D ƒ h with h 2 SL˙ .n; R/, we have

j=2 .gxg t / D C j=2 .x/;

C D j det.g/jJ

with the exponent J WD a=n.

End of proof of Theorem 5.12.a. It remains to find the exponent J in (5.15). We consider g of the form (5.16). Then the factor det.g/J coincides with the absolute value of the determinant of the differential and we know this determinant. Statement c) of Theorem 5.12 follows from a) and b). The proof of b) is contained in the next subsection. 5.9 Convolutions and a more explicit description of measures j=2 . Firstly, consider O1 . Each x 2 O1 has the form x D v t v, where v 2 Rn is a row-matrix. This representation of x is unique up to the change of sign, v 7! v. Thus we get a coordinate system on O1 . The action of GL.n; R/ in these coordinates is the usual linear action v 7! vg. The n SL.n; R/-invariant measure is the Lebesgue measure dv on R . Let ' be in the Schwartz space Symmn .R/ . Then Z Z '.x/ d 1=2 .x/ D const '.v t v/ dx; O1

Rn

and the convergence of the integral on the right-hand side is obvious. Thus 1=2 determines a tempered distribution.


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We construct all other measures 1=2 as convolutions

1 D 1=2 1=2 ;

3=2 D 1=2 1=2 1=2 ; : : : :

Since all the j=2 are supported by the convex cone Posn , the convolutions are welldefined tempered distributions (see Subsection 4.4.1). Next, 1=2 is SL.n; R/-invariant, therefore all the measures j=2 are SL.n; R/-invariant. For this reason, j=2 are indeed the measures j=2 from the previous subsection. Problem 5.2. The measures j=2 can be described as follows. Equip the space Matj;n .R/ with the Lebesgue measure. Consider the map Matj;n .R/ ! Symmn .R/ given by u 7! uut . The measure j=2 coincides (up to a scalar factor) with the pushforward of the Lebesgue measure under this map. In other words, Z Z '.x/ d j=2 .x/ D const '.uut / du: Oj

Matj;n .R/

5.10 End of the proof of Theorem 5.6. Thus we have proved the affirmative part of the theorem. Let us establish the absence of positivity for all remaining ˛. To apply Bochner-type Theorem 3.12, we must describe all positive measures on the cone Posn homogeneous in the sense (5.8). Firstly, any such determines a homogeneous measure 0 on the open cone Posn ' GL.n; R/=O.n; R/. The usual arguments (see Subsections C.10–C.11) show that 0 has the form described above. xn1 . Consider the measure 0 , which is zero on Posn , i.e., it is supported by O The measure 0 determines an SL.n; R/-invariant measure 00 on On1 . Such a measure is unique up to a factor, therefore it coincides with .n1/=2 to within a factor. Next, we consider the measure 0 00 . This measure is supported by the closure of On2 and we repeat the same argument, etc. Thus the only positive homogeneous measures on Posn are the measures constructed above. 1 2 n 5.11 Another proof of Theorem 5.6 for ˛ 2 2 Z. Let 'z 2 L .R / be given by ˚

'z .x/ D exp 2i xzx t ; where z 2 Pn , x 2 Rn . Then

1=2 h'z ; 'u iL2 .Rn / D .2/n=2 det i.z u/ N :

N is positive definite. Therefore, Kj=2 .z; u/ N D K1=2 .z; u/ N j Thus the kernel K1=2 .z; u/ are positive definite for positive integer j . 5.12 Irreducibility Proposition 5.15. The representations T˛ of the group Sp.2n; R/ in H˛ are irreducible.

358


Proof. Let W be an invariant subspace. It is determined by a reproducing kernel. However, an invariant reproducing kernel is unique. 5.13 Weighted Laplace transforms. Let ˛ be the measure defined above. Consider the weighted Laplace transform Z L˛ f .z/ D f .x/e i tr xz d ˛ .x/: Posn

Theorem 5.16. The operator L˛ is a unitary operator from L2 .Posn ; ˛ / to the space H˛ .Pn / of holomorphic functions. Proof. Consider the function 'z .x/ WD e i tr xz on the cone Posn ,

Z

h'z ; 'u iL2 D

˛ N e tr x.izi u/ d˛ .x/ D det .z u/=2i D K˛ .z; u/: N

Posn B

Thus H˛ .Pn / D H .Pn ; K˛ / and the weighted Laplace transform is the standard map H.Pn ; K˛ / ! HB .Pn ; K˛ /. 5.14 Rephrasing of Theorem 5.6. Let Bn be the matrix ball, see Subsection 3.3.5. We realize the symplectic group Sp.2n; R/ as in Subsection 3.2.5. Fix ˛ 2 C. The group Sp.2n; R/ acts on the space of holomorphic functions on Bn by the formula ˆ ‰ x det.ˆ C z ‰/ x ˛ : x 1 .‰ C z ˆ/ T˛ x x f .z/ D f .ˆ C z ‰/ (5.18) ‰ ˆ Theorem 5.17. a) The kernel L˛ .z; u/ D det.1 zu /˛ on Bn is positive definite if and only if ˛ satisfies the Berezin–Wallach conditions (5.5). b) Under the conditions of positivity, the representation T˛ is unitary in the Hilbert N space H˛ .Bn / determined by the kernel L˛ .z; u/. This is the same Theorem 5.6, we simply identify the wedge Pn and Bn by the Cayley transform. For a holomorphic function g on Bn we define the holomorphic function C˛ g.z/ on Pn as i z C˛ f .z/ D f (5.19) det.i C z/˛ : i Cz

359

7.5. The Berezin scale Extensions, variations, comments

The construction of this section is a very special example of unitary representations. However, it was an initial point for several branches of mathematical evolution. In what follows (Subsections 5.15–5.20) we outline representation-theoretical elements of a landscape without proofs and details. 5.15 Vector-valued holomorphic functions. Let be an irreducible holomorphic finitedimensional representation of GL.n; C/ in a Euclidean space V ; “holomorphic” means that all matrix elements of are holomorphic functions, we also assume .g / D .g/ . We consider the space of V -valued holomorphic functions on Pn . The group Sp.2n; R/ acts on this space by the operators14 T f .z/ D .a C zc/f .a C zc/1 .b C zd / : The construction of this section corresponds to 1-dimensional representations, .h/ D det.h/˛ . Problem 5.3. T is a representation. Next, we need an analog of reproducing kernels. Let CN . Consider a function K.z; u/ N on whose values are operators in V . We assume that K is holomorphic in z and anti-holomorphic in u. N We say that K.z; u/ N is a matrix-valued positive-definite kernel if hK.z; u/; N iV is a positive-definite kernel on V ; here z, uN 2 , , 2 V : Problem 5.4. The invariant reproducing kernel (if it exists) has the form K.z; u/ N D .1 zu /:

Now recall (see e.g. [220]) that representations of GL.n; C/ are enumerated by signatures (5.20) D .1 ; 2 ; : : : ; n /; where j j C1 2 ZC . For us only the case j 2 R is interesting. Denote by the vector .1; 1; : : : ; 1/. Then Cs .h/ D det.h/s .h/: Theorem 5.18. Represent in the form D .n; : : : ; n; „ ƒ‚ … q C 1 times

a; : : : ; a ; b; : : : / C s .1; 1; : : : ; 1/; „ ƒ‚ …

where n > a > b.

r q C 1 times

The representation T is unitary if and only if s< 14

1 1 1 1 1 1 .r C 1/ or s D .r C 1/, .r C 1/ C , .r C 1/ C 1; : : : ; .q C r/. 2 2 2 2 2 2

Such representations are called highest weight representations.

360


5.16 Similar groups. The construction of this section exists for classical groups of the series G D Sp.2n; R/;

U.p; q/;

SO .2n/;

SO0 .n; 2/:

(5.21)

For the first three series the constructions are parallel; the last series is slightly different, see Subsection D.2. 5.17 Restrictions to symmetric subgroups. An important problem is to restrict a highest z (see (5.21) to a symmetric subgroup15 G. It proved to be that weight representation of a group G there are two completely different situations; to explain this, we need an additional notation. z Kz and G=K (see SubsecConsider the Riemannian noncompact symmetric spaces G= tion D.1). There is an alternative: z Kz or G=K is a totally real submanifold in G= z K. z – G=K is a complex submanifold in G= Example 1. We embed Bp;q D U.p; q/=U.p/ U.q/ to BpCq D Sp.2.p C q/; R/=U.n/ by z 7! z0t z0 : The image is a complex manifold. Example 2. The symmetric space GL.n; R/=O.n/ can be realized as a space Bn .R/ of real symmetric matrices with norm < 1 (see Subsection 2.12.7). Evidently, Bn .R/ is a totally real submanifold in the matrix ball Bn . This alternative leads to two series of problems of completely different nature. The first type. If G=K is a totally real submanifold16 , we come to the analysis of Berezin kernels mentioned in Subsection 2.12.6. z Kz is a complex submanifold, then the restriction of a highest The second type. If G=K G= z to G is a countable direct sum of highest weight representations of G.17 weight representation of G The restriction problem can be reduced to a finite-dimensional problem by a simple trick (H. Jacobsen, M. Vergne), we explain this in the simplest case. Example (the second type). Consider a tensor product Ts ˝ Ts 0 of two highest weight representations18 of SU .1; 1/. We get a representation in the space of holomorphic functions on the bidisk jzj < 1, juj < 1, ˛ ˇ N C ˛/s 0 : N C ˛/s .ˇu 0 f .z; u/ D f z Œg ; uŒg .ˇz Ts ˝ Ts N ˇ ˛N Denote by Wk the space of functions having zero of order > k along the diagonal z D u, W0 W1 W2 : Observation 5.19. The subspaces Wk are SU.1; 1/-invariant. The representation of SU.1; 1/ in a subquotient Wk =WkC1 is TsCs 0 Ck . Thus, M Ts ˝ Ts 0 D TsCs 0 Ck : k>0 15

For a list of symmetric subgroups, see Subsection D.6. For a list, see Subsection D.3. 17 See list in Subsection D.4. 18 This is also a restriction problem. Indeed, Ts ˝ Ts 0 can be regarded as a representation of the group SU .1; 1/ SU .1; 1/. We restrict Ts ˝ Ts 0 to the diagonal subgroup ' SU .1; 1/. 16


361

This case seems completely trivial; but a careful examination of tensor products Ts ˝ Ts 0 produces theory of all classical orthogonal polynomials of one discrete variable (the Kravchuk, Chebyshev–Hahn, Meixner, Rakah systems; the Charlier system is too degenerate). Example (the first type). Consider the tensor product Ts ˝ TB of a highest weight and a lowest weight representation of SU.1; 1/ . We get a representation in the space of holomorphic functions on the bidisk jzj < 1, juj < 1, ˛ ˇ N N C ˛/s .ˇu C ˛/ N : .ˇz Ts ˝ TB N f .z; u/ D f z Œg ; uŒg

ˇ ˛N The corresponding Hilbert space Hs; of holomorphic functions is determined by the reproducing kernel Ks; .z; uI zN 0 ; uN 0 / D .1 z zN 0 /s .1 uuN 0 / : Consider the antidiagonal a W z D uN of the bidisk. It is a totally real submanifold invariant with respect to the action of the group, a D SU.1; 1/=U.1/. Consider the intersection M of a with the Shilov boundary jzj D 1, juj D 1 of the bidisk. Consider the operator of restriction of holomorphic functions to the boundary described in Subsection 3.5. Observation 5.20. If s C < 1, then the restriction of a function f 2 Hs; to M is well defined. We get a representation of the complementary series in a space of functions on M . The proof is reduced to simple estimates of integrability of the kernel Ks; on M M , see Theorem 3.9. Thus we found one summand in the decomposition of Ts ˝ TB . The remaining spectrum is continuous and is beyond the methods presented here. 5.18 Stein–Sahi representations. Our next topic is a certain “perturbation” of highest weight representations. Consider the space Ln of unitary symmetric matrices z, zz D 1;

z D zt :

Recall that the group U.n/ acts on this space as z 7! hzht and the action is transitive, Ln ' U.n/=O.n/. Our space lies on the boundary of the matrix ball Bn ; the group Sp.2n; R/ acts on Ln by the same transformations (5.18). Let , 2 C, Re 6 0, Re 6 0. Consider the function ` on Ln given by

` .z/ D lim det 1 .1 "/z : "!C0

We define the Stein–Sahi kernel L; .z; u/ WD ` .zu /` .uz / DW det.1 zu /˛ det.1 uz / ; where z, u 2 Ln . Next, we define a sesquilinear form on C 1 .Ln / by “ hf; gi; WD L; .z; u/f1 .z/f2 .z/ d.z/ d.u/; U.n/U.n/

(5.22)

362


where denotes the U.n/-invariant measure on Ln . If ; 2 R, then this form is Hermitian. ˆ ‰ 1 For g D ‰ .U.n// by x ˆ x 2 Sp.2n; R/, define the operators T; .g/ in C x 1 .‰Cz ˆ/ x det.ˆCz ‰/ x .nC1/=2 det.ˆ C z ‰/ x .nC1/=2 : T; .g/f .z/ D f .ˆCz ‰/ Proposition 5.21. The sesquilinear form h; i; is invariant with respect to the operators T; . This is an exercise in Jacobians and formula (2.11.4).

Observation 5.22. Representations T; and TC1; 1 are equivalent. The intertwining operator is Af .z/ D det.z/f .z/.

Proposition 5.23. For given f1 , f2 2 C 1 .Ln /, the expression hf1 ; f2 i; admits a meromorphic continuation to the whole complex plane . ; / 2 C2 , possible poles are located on the hyperplanes Re , Re D 1, 3=2, 2; : : : . See Subsection 2.5, but the integration by part argument used there is not sufficient here. Observation 5.24. The holomorphic representations T˛ discussed above (5.2) coincide with the representations T˛;0 defined now. More precisely, for ˛ satisfying the conditions (5.5) of positivity, the form h; i˛;0 is positive semidefinite (it is degenerate); therefore we get a Hilbert space H˛;0 and a unitary representation of Sp.2n; R/. The operator Z Jf .z/ D const L; .z; u/f .u/ d.u/; z 2 Bn ; u 2 Ln L

is a unitary intertwining operator H˛;0 ! H˛ .19 Theorem 5.25. Let n > 1. a) Let , be real … 12 Z. The form h; i; is positive definite if and only if . ; / is contained in the chain of squares in Figure 7.2. b) For C D .n C 1/=2 our inner product coincides with the L2 inner product (i.e., the distribution ` .z/`.nC1/=2 .z / is the delta-function supported by z D 1). A proof is based on the explicit expansion of the distribution ` .z/` .z / in U.n/-harmonics on Ln . This is the basic nontrivial point in our considerations (and we omit a proof). 5.19 Unipotent representations. The analogs of singular holomorphic representations T˛ also exist. Theorem 5.26. Let n > 1. a) Fix , D 0; 1=2; : : : ; n=2, C D =2 6 n=2. Choose t , s 2 R satisfying t C s ¤ 0. Then Us;t . ; / WD lim h; i C"s; C"t (5.23) "!0

19

This is nothing but the construction of Subsection 1.6 (the “dual functional model”); however now distributions are supported by the boundary Ln of the domain Bn and some additional substantiation is necessary. See the case n D 1 in Subsection 2.7 and a general discussion in Subsection 3.5.

363

7.5. The Berezin scale L2

n=2 .n C 1/=2

Figure 7.2. The conditions of positivity of Hermitian forms h; i; .

is finite; Us;t is a T; -invariant Hermitian form on C 1 .Ln /. b) Denote by K the common kernel of all forms U t;s . Then the representation T; in C 1 .L/=K is a direct sum of . C 1/ unitary representations of Sp.2n; R/. The inner products can be explicitly expressed as linear combinations of Us;t . Remark. Note, that in formula (5.23) a meromorphic function of two variables has different limits at a point from different directions. This is similar to the standard exercise f .x; y/ D xy in the calculus. From the point of view of an algebraic geometer, we make a blow-up. x 2 Cy 2 Unitary representations from this theorem are called “unipotent representations”. Our next aim is to describe restrictions of unipotent representations to U.n/. First, we replace operators T j .g/ by .nC1/=2C Tj .g/: Tzj .g/ D det.ˆ/.nC1/=2C det ˆ

The operators Tz .g/ determine a projective representation of Sp.2n; R/. The group U.n/ Sp.2n; R/ now acts as ˆ 0 t z Tj x f .z/ D f .ˆzˆ /: 0 ˆ Denote by the representation of U.n/ with a signature , see (5.20). In this notation20 , M : (5.24) C 1 U.n/=O.n/ ' W all j are even integers 20

This statement is a “Theorem about spherical representations”, see e.g. [86].

364


For a generic Stein–Sahi representation Tzj (i.e., for interiors of squares in Figure 7.2), the U.n/–spectrum contains all representations with even signature. To describe U.n/-spectra of the unipotent representations, we change the notation and set j C j 1: 2 All mj are pairwise different, therefore a signature fmj g now is an n-point subset in Z; in Japanese literature, such subsets are called Maya diagrams, see Figure 7.3. mj D

m3 m 2

mn

m1

Figure 7.3. A Maya diagram for a signature .m1 ; : : : ; mn /.

0

a) all boxes are empty

obligatory part 0

b)

1

1

2

3

4

all boxes are empty

Figure 7.4. Maya diagrams for U.n/-spectra of a holomorphic representations T˛ . a) The regular case, ˛ > n1 . We have arbitrary n-point subsets in ZC . b) The singular case, 2 1 n1 ˛ D 0; 2 ; : : : ; 2 . obligatory part 0

1

1

Figure 7.5. Maya diagram for U.n/-spectra of the unipotent representations.

Theorem 5.27. Fix D 0; 1; : : : ; n. Denote by T 0 ; : : : ; T

the corresponding unipotent representations. a) The U.n/-spectrum of ˚j T j consists of signatures, whose Maya diagrams contain points 0; 1; : : : ; . The spectrum of T j contains Maya diagrams with j negative entries. b) The representation T 0 coincides with the singular highest weight representation T . Respectively, T

is the dual lowest weight representation.

7.6. Hardy spaces on Siegel wedges

365

In particular, for D 0 we get a canonical Sp.2n; R/-invariant orthogonal decomposition of the space L2 .U.n/=O.n//. 5.20 Restrictions to symmetric subgroups. Now we can restrict the representations T; and the unipotent representations to symmetric subgroups; this leads to the Stein–Sahi kernels on symmetric spaces mentioned in § 2.12.

7.6 Hardy spaces on Siegel wedges The boundary of the matrix wedge Pn is stratified. For this reason, there are several Hardy spaces of holomorphic functions in Pn . 6.1 The standard Hardy space Theorem 6.1. The space H.nC1/=2 is a Hardy space, i.e., the inner product in H.nC1/=2 is given by the formula Z hf; giH.nC1/=2 D f .x/g.x/ dx: Symmn .R/

Proof. This is a corollary of Theorem 5.16. The measure d .nC1/=2 on the cone Posn is the Lebesgue measure. Hence we have the usual Laplace transform and apply the usual Plancherel formula. Since f is the usual Laplace transform of an L2 -function, it follows that f .x/ is well defined as an L2 -function on the edge Symmn .R/ of the wedge Pn . 6.2 Exotic Hardy spaces. Set j D 0; 1; : : : ; n 1

˛ WD .n C 1 C j /=2 :

xn as z D x C iy, where x 2 Symmn .R/ and y 2 Posn . Denote by Represent z 2 P ˛ the measure on Pn given by d˛ .z/ D dx d j=2 .y/; where j=2 is the measure on the boundary of the cone Posn discussed in the previous section. Our next aim is the following statement. – For j D 0; 1; : : : ; n 1, the space H.nCj C1/=2 .Pn / is the space of holomorphic functions on Pn with the inner product Z Z Z hf; gi D f .z/g.z/ d˛ .z/ D f .x C iy/g.x C iy/ d j=2 .y/ dx: (6.1) x Pn

Symmn .R/ Oj

366


In a certain sense, the statement is trivial. We simply write the reproducing kernel of the space determined by this inner product from reasoning of invariance, see Subsection 6.8; we immediately get the Berezin kernels K˛ .z; u/. N But this way meets some logical difficulties; we must be sure that our spaces are nonempty, also we must be sure that the space of (boundary values of) holomorphic functions is closed in L2 .Pn ; ˛ /; this is not immediately provided by our a priori theorems from § 7.3. For this reason, we choose a roundabout way. It involves some additional tools (Hua and Selberg integrals, integration in eigenvalues), which are standard for this “science”. Let us formulate the statement more carefully. Consider the space of polynomials on the matrix ball Bn and consider the image W˛ .Pn / of this space under the twisted Cayley transform C˛ , see (5.19). Lemma 6.2. W˛ L2 .Pn I d˛ /. Theorem 6.3. The closure of W˛ in L2 .Pn I dj / coincides with H˛ .Pn /. Moreover, the L2 .Pn ; ˛ /-inner product in H˛ .Pn / coincides with the standard inner product. The proof occupies the rest of this section. 6.3 Reduction of Lemma 6.2 to Hua integrals. We consider the function G.z/ D det.i C z/˛ , which is the image of the function g.z/ D 1 under the Cayley transform C˛ ( see 5.19). Since a polynomial f is bounded on Bn , we have jCf .z/j 6 const jG.z/j, the constant depending on f . Hence it suffices to show that G 2 L2 .Pn ; ˛ /: We represent z D x C iy, where x 2 Symmn .R/, y 2 Posn , and write jG.z/j2 D j det.i C z/j2˛ D j det.x C i.1 C y/j2˛ 2˛ D det.1 C y/2˛ det i C .1 C y/1=2 x.1 C y/1=2 : We must integrate this expression with respect to d˛ D dx d j=2 . First, we integrate the “boxed” factor with respect to x (with fixed y), Z ˇ ˇ ˇdet i C .1 C y/1=2 x.1 C y/1=2 ˇ2˛ dx Symmn .R/

Z

D det.1 C y/

.nC1/=2

ˇ ˇ ˇdet i C .1 C y/1=2 x.1 C y/1=2 ˇ2˛ d.1 C y/1=2 x.1 C y/1=2 :

We substitute xQ D .1 C y/1=2 x.1 C y/1=2 , see Lemma 5.1, and come to Z Z ˇ ˇ2˛ .nC1/=2 .nC1/=2 ˇ ˇ det.i C x/ Q det.1 C xQ 2 /˛ d x: d xQ D det.1 C y/ Q det.1 C y/

7.6. Hardy spaces on Siegel wedges

367

2 Thus kGkL 2 splits into a product of two integrals, 2 kGkL 2 .P ;dx d n

j=2 /

where

Z

det.1 C x 2 /˛ dx;

I ´

(6.2)

Symmn .R/

Z J ´

Oj

D I J;

Z

det.1 C y/.nC1/=22˛ d j=2 .y/ D

Oj

det.1 C y/.nC1/=2j d j=2 .y/ (6.3)

By Problem 5.2, the latter integral (6.3) can be transformed to the form Z const det.1 C vv t /.nC1/=2j dv:

(6.4)

Matj;n .R/

Now Lemma 6.2 is reduced to: Lemma 6.4. The integrals (6.2) and (6.4) are convergent. These integrals were evaluated by Hua Loo Keng [95]; we present another calculation. For this purpose we need two common mathematical facts formulated (without proofs) in the following two subsections. 6.4 Integration with respect to eigenvalues. Denote by n the wedge in Rn determined by the inequalities 1 > 2 > > n : Consider the map L W Symmn .R/ ! n that takes a matrix x to the collection of its eigenvalues. Theorem 6.5. The image of the Lebesgue measure under L is Y Y .k l / d m : const m

k 2 > > j > 0: Let j 6 n. Consider the map M W Matj;n .R/ ! „ that takes v 2 Matj;n .R/ to the collection of its singular values. Theorem 6.6. The pushforward of the Lebesgue measure under M is Y Y nj Y const .2k 2l / k dk : k 0, Re a > 0, Re b > 0. Then21 Z

Z

1

1

::: 0

0

p Y

tja1 .1

Y

ab2c.p1/1

C tk /

kD1

jtk tl j

16k 0, Re a C Re b > .2p 2/ Re c C 1. Then 1 .2/n

Z

Z

1

1

::: 1

1

p Y

Y

.1 C i tk /a .1 i tk /b

kD1

jtk tl j2c

D2

dtj

j D1

16k 2, let f 2 Q.R2n /. Assume that the set Sf defined by f .x; / D 0 is a surface of codimension 2 with singularities at a surface of codimension > 3. The surface Sf has a canonical orientation at nonsingular points (because R2n and the target space C are oriented). Proposition 1.7. a) The Z-homological class of Sf in the torus T 2n does not depend on the choice of f . P b) This class coincides with the sum jnD1 ~j , where ~j is the .2n2/-dimensional subtorus defined by the equations xj D const, j D const. We leave this statement as an exercise in topology.

8.2 The pushforward of the Fourier transform 2.1 The Poisson summation formula. Denote by the distribution on Rn given by X Y .x/ D ı.xj kj /: .k1 ;:::;kn /2Zn j

Normalize the usual Fourier transform1 by Z fO.x/ D F f .x/ D

t

f .y/e 2 ixy dy:

Rn

Theorem 2.1. F D . In other words, for each f 2 .Rn /, X X fO.l/ (the Poisson summation formula): f .k/ D k2Zn 1

l2Zn

This normalization differs from (1.1.9) used above.

(2.1)

8.2. The pushforward of the Fourier transform

387

It suffices to prove the statement for n D 1. First proof. Consider the operators T and S in 0 .R/ given by Tf .x/ D f .x C 1/;

Sf .x/ D e 2 ix f .x/:

Obviously, F T D S 1 F ;

FS D TF:

(2.2)

Lemma 2.2. The is a unique to within a scalar factor distribution 2 0 .Rn / invariant with respect to the transformations S and T . Proof of Lemma 2.2. The S-invariance is equivalent to .e 2 ix 1/.x/ D 0: Hence the distribution is supported by zeros of .e 2 ix 1/, i.e., by Z. The derivative .e 2 ix 1/0 D 2 i e 2 ix does not vanish at integer points, therefore is a sum of ı-functions supported by x 2 Z. By T -invariance, is periodic, .x C 1/ D .x/. Hence D c. By (2.2), F is invariant with respect to S and T . Therefore, F D c . Hence, F 2 D c F D c 2 : On the other hand, F 2 g.x/ D g.x/, therefore .F 2 /.x/ D .x/ D .x/. Hence c 2 D 1. For .x/ WD exp.x 2 /, we have F D . Hence, hhF ; ii D hh; F

ii D hh; ii:

Thus c D C1.

The second proof. We verify (2.1). For any f 2 .R/, consider the periodic function '.x/ given by 1 X '.x/ D f .x C m/: mD1

Expand it into a Fourier series: '.x/ D

1 X kD1

ck e 2 ikx ;

388

Chapter 8. The Cartier model

where

Z

1

ck D

'.x/e 2kix dx D 0

XZ

D

XZ m

mC1

Z

1

f .x/e 2kix dx D

f .x/e 2kix dx D fO.k/:

1

X

Hence,

f .x C m/e 2kix dx

0

m

m

1

fO.k/ D

X

ck D '.0/ D

X

f .m/:

m

k

Corollary 2.3. Let Œa; jx WD

1 X

ı.x a k/e 2ki :

kD1

Then

F Œa; D e 2 ia Œ; a:

Proof. We use the standard properties of the Fourier transform (1.1.10), X X Œ0; I x D ı.x k/e 2 ik D e 2 ix ı.x k/ D e 2 ix Œ0; 0I x: Therefore, the Fourier transform of Œ0; I x is Œ; 0I x. Then we shift x 7! x a and come to the required formula. 2.2 The pushforward of the Fourier transform Theorem 2.4.

ZF Z1 g.x; / D e 2 ix g.; x/:

(2.3)

Proof. This is a rephrasing of the previous corollary because Zf .a/ D hhŒ; aI x; f ii:

Another proof. Let f 2 .R/, g WD Zf , Z 1h X Z 1 1 i f .y/e 2 iy dy D f .y C k/e 2 i .yCk/ dy F f . / D 1

Z

1

D

e 2 iy 0

Z D

0 1 h X

Z i f .y C k/e 2 ik dy D

kD1 1

kD1

1

e 2 iy g.y; / dy

(2.4)

0

e 2 iy g.y; / dy:

0

However, the last expression looks like the inverse Zak transform of a function h.x; / WD e 2 ix g.; x/. It is important that h 2 Q (to achieve this, we do the substitution y 7! y).

8.2. The pushforward of the Fourier transform

389

Remark. The two-step evaluation of the Fourier transform in (2.4) is an important trick for numerical computations, see § 9.5. 2.3 Another normalization of the Zak transform. Formula (2.3) looks like a rotation by 90B . We want to renormalize the Zak transform and to remove the exponential factor. Namely, for a function f on Rn , we define the modified Zak transform g D ZB f by ZB f .x; / WD e ix Zf .x; /: (2.5) Observation 2.5. ZB F .ZB /1 g.x; / D g.; x/. Observation 2.6. The quasiperiodicity condition for a function g D ZB f is g.x C p; C q/ D g.x; / expf i.xq t p t /g expf ipq t g;

(2.6)

where x, 2 Rn and p, q 2 Zn . Note that: – these conditions are symmetric under the transposition x $ ; – expf ipq t g D ˙1 for all p, q 2 Zn ; – the expression xq t p t is a skew-symmetric scalar product in R2n . Denote by QB .R2n / the space of C 1 -functions g on R2n satisfying the quasiperiodicity conditions (2.6); this space is the image of .R2n / under ZB . Since je ix j D 1, it follows that the modified transform ZB is a unitary operator 2 L .Rn / ! L2 Œ0; 12n . 2.4 Two Heisenberg groups. We need some preparation before writing of actions of the groups Sp.2n; Z/ and Sp.2n; Q/ on the space QB .R2n /. Consider the Heisenberg group Heis2n of double size. It acts on the space C 1 .R2n / in the usual way, see § 1.2, i.e., via the operators o ni ˚

.aˇCb˛/ : U a˚˛˚b˚ˇ g.x; / D g .xCa/˚.C˛/ exp i.xˇCb/ exp 2 These operators satisfy the usual relations U a ˚ ˛ ˚ b ˚ ˇ U a0 ˚ ˛ 0 ˚ b 0 ˚ ˇ 0 ni o D exp .aˇ 0 a0 ˇ C b 0 ˛ b˛ 0 / 2 U .a C a0 / ˚ .˛ C ˛ 0 / ˚ .b C b 0 / ˚ .ˇ C ˇ 0 / : This group of operators does not appear in our considerations. However it contains two bas subgroups Heisaux n and Heisn , which are an important tool in the next section. These two subgroups are singled out by the equations ´ ´ a D b; a D b; and ˛ D ˇ; ˛ D ˇ:

390


Both the subgroups are isomorphic to Heisn . For further references, we present explicit formulas for actions of both groups. The 2 basic Heisenberg group Heisbas n acts by T .a ˚ ˛/g.x; / D g.x C a; C ˛/ expf i.x˛ t at /g;

(2.7)

where a, ˛ 2 Rn . These operators satisfy T .a ˚ ˛/T .b ˚ ˇ/ D e i.aˇ

t b˛ t /

T .a C b/ ˚ .˛ C ˇ/ :

(2.8)

The auxiliary Heisenberg group Heisaux n acts by T .u ˚ v/g.x; / D g.x C u; C v/e i.xv Then

t ut /

t

e i uv :

t T .a ˚ b/T .u ˚ v/ D e 2 ibu T .a C u/ ˚ .b C v/ :

(2.9) (2.10)

In formula (2.9), we modify a normalization of operators; the purpose of this modification will be clear immediately (Observation 2.8). bas Observation 2.7. The groups Heisaux n and Heisn commute with each other.

This follows from the expression for product in Heis2n . In other words, these subgroups correspond to a pair of complementary orthogonal subspaces in the symplectic space R4n . Observation 2.8. The subspace QB .R2n / is precisely the space of functions fixed by n all operators T .p; q/ 2 Heisaux n , where p and q range in Z . This is a reformulation of quasiperiodicity. These two statements imply Corollary 2.9. The space QB .R2n / is invariant under the group Heisbas n . Observation 2.10. The transform ZB intertwines the standard representation of Heisn in .Rn / and the representation T .a ˚ ˛/ (the “basic Heisenberg group”). This is a rephrasing of (1.8), (1.9)

8.3 Action of Sp.2n; Z/ Our next purpose is to push forward the action of the symplectic group Sp.2n; R/ and the symplectic category from L2 .Rn / to the spaces QB .R2n /. This will be done in 2

The notation a ˚ ˛ is equivalent to .a; ˛/.

8.3. Action of Sp.2n; Z/

391

full generality in § 8.5. Here we transport the action of the subgroup in Sp.2n; R/ consisting of matrices 1 A B .2/ 0 .2/ 0 A 2B D ; gD C D 0 .2/1 D 0 .2/1 .2/1 C where the matrices A, B, C , D have integer entries. 3.1 The group Sp.2 n; Z/ and its Igusa subgroup 1;2 . Denote by Sp.2n; Z/ the subgroup of Sp.2n; Z/ consisting of matrices with integer entries. Lemma 3.1. Sp.2n; Z/ is a group. Proof. Indeed, for a symplectic matrix g, 0 1 t 0 1 g 1 D : g 1 0 1 0 Therefore, g 2 Sp.2n; Z/ implies g 1 2 Sp.2n; Z/.

Fix an integer positive N . Define the principal congruence subgroup N Sp.2n; Z/ consisting of matrices g such that N divides gij ıij . Problem 3.1. Denote by ZN the quotient-ring Z=N Z. To a matrix g 2 Sp.2n; Z/ we assign the matrix ŒgN over ZN whose matrix elements are gij mod .N /. Then the map g 7! ŒgN is a homomorphism Sp.2n; Z/ ! Sp.2n; ZN /. Its kernel is N . General values of N appear in our considerations below, in § 11.6. Now consider the case N D 2. We are going to define a certain intermediate subgroup 1;2 such that 2 1;2 Sp.2n; Z/: Denote by F2 the field of two elements. a b Define the group Sp.2n; F2 / as the group of .n C n/ .n C n/ matrices g D c d over the field F2 preserving the bilinear form B.vC ˚ v ; wC ˚ w / D vC .w /t v .wC /t D vC .w /t C v .wC /t : This form is skew-symmetric, but it is also symmetric, since 1 D C1 in F2 . The matrix g satisfies the following conditions (see (3.1.4)): ad t C bc t D 1;

ab t and cd t are symmetric:

(3.1)

Consider the quadratic form on F22n given by t Q.vC ˚ v / D vC v :

Clearly, B.v; w/ D Q.v C w/ Q.v/ Q.w/: Denote by O.2n; F2 / the group of matrices over F2 preserving the quadratic form Q. By the last identity, O.2n; F2 / Sp.2n; F2 /:

392


Lemma 3.2. A matrix g 2 Sp.2n; F2 / is contained in O.2n; F2 / if and only if the matrices ab t and cd t have zeros on the diagonals. Proof. A matrix g preserves Q if a b Q.vC ˚ v / D Q .vC ˚ v / c d () vC .v /t D .vC a C v c/ .vC b C v d /t ˚

t t () vC .v /t D vC .ad t C bc t /v ; 0 D vC .ab t /vC ; 0 D v .cd t /v : Compare the last row with the conditions (3.1) for each g 2 Sp.2n; F2 /. The first identity holds automatically. Next, a priori, ab t and cd t are symmetric. Let R be an arbitrary symmetric matrix over F2 . Consider the corresponding quadratic form: X X X X rij xi xj D ri i xi2 C .rij C rj i /xi xj D ri i xi2 : xRx t D i

i>j

i

This expression is zero if and only if ri i D 0.

a b

We define the Igusa subgroup 1;2 2 of Sp.2n; Z/ consisting of g D c d such that the matrices at c and b t d have even elements on the diagonals. In other words, 1;2 is the pullback of O.2n; F2 / Sp.2n; F2 / under the homomorphism Sp.2n; Z/ ! Sp.2n; F2 /.

3.2 The affine action of Sp.2 n; F2 /. Recall that an affine transformation of a linear space V is a map of the form h 7! hACv, where h ranges in V , v 2 V , and A W V 7! V is a linear transformation. In other words, an affine transformation is a composition of a linear transformation and a shift. Consider the linear space V WD .F2 /2n endowed with a bilinear form B and quadratic form Q as above. Denote by V 0 the space of linear functionals on V . Theorem 3.3. The group Sp.2n; F2 / acts on V 0 ' F22n by the affine transformations U.g/`.v/ D `.vg/ C Q.vg/ Q.v/;

` 2 V 0 ; v 2 V:

(3.2)

Lemma 3.4. For each g 2 Sp.2n; F2 /, the function '.v/ WD Q.vg/ Q.v/ is linear on V . Proof of the lemma. We must watch the proof of Lemma 3.2, X X ˇj .vj /2 ; Q.vg/ Q.v/ D ˛j .vjC /2 C where ˛j , ˇj are the diagonal values of matrices ab t , cd t . Since y 2 D y for all y 2 F2 , the last expression is a linear functional.


393

Proof of the theorem. Consider the space W consisting of F2 -valued functions on V of the form .v/ D Q.v/ C `.v/;

where ` is a linear functional on V :

By the lemma, the formula Z.g/ .v/ D .vg/ defines an action Z of Sp.2n; F2 / on the set W. We identify V 0 $ W by ` 7! ` C Q. The induced action of Sp.2n; F2 / on V 0 is (3.2). Remark. An affine transformation U.g/ is linear if and only if g 2 O.2n; F2 /. 3.3 Action of 1;2 in Q.R2n /. Recall that in the previous section we introduced the B group Heisbas n acting on Q by operators T given by (2.7). A B B 2n Theorem 3.5. a) For h D C D 2 1;2 and f 2 Q .R /, the function W .h/f .x; / WD f

x

A C

B D

D g.xA C C; xB C D/

(3.3)

is contained in QB .R2n /. b) W .h1 /W .h2 / D W .h1 h2 /. c) W .h/T .a ˚ ˛/ D T .a ˚ ˛/h W .h/ . Define two bilinear forms. The first one is the symplectic form on R2n , ƒ.x ˚ ; y ˚ / D x t y t : The second form is the F2 -valued quadratic form „ on Z2n given by „.p ˚ q/ D pq t

mod 2:

In this notation, the quasiperiodicity condition (2.6) is ˚ ˚

g.x C p; C q/ D g.x; / exp i ƒ x ˚ ; p ˚ q exp i „.p ˚ q/ : Proof of Theorem 3.5. a) Recall that the matrix h 2 1;2 preserves the integer lattice Z2n and both the forms ƒ and „. Therefore it regards the quasiperiodicity. b) is obvious, c) follows from a straightforward calculation. As a corollary, we obtain the following theorem:

394


Theorem 3.6. Let g D

A

We

2 1;2 . Then the operators of the Weil representation 1 # 2 0 2 0 g 0 .2/1 0 .2/1

B C D

"

defined by (1.2.10) correspond to the operators W .g/ under the Zak transform. Proof. Both operators are determined by the commutation relations with the Heisenberg group, see Theorem 3.5 and Theorem 1.2.5. The Zak transform intertwines the actions of the Heisenberg group in .Rn / and QB .R2n /. The factor 2 appears due to the different normalizations in formulas (1.2.8) and (2.8). Corollary 3.7. The restriction of the Weil representation to 1;2 is equivalent to a linear representation. Proof. Indeed, we have no scalar factor in Theorem 3.5.b). 3.4 An example: matrix 10 11 . Our next purpose is to write out the action of the whole subgroup Sp.2n; Z/ in QB . First, we discuss an example. Consider the operator R W .R/ ! .R/ given by ˚

1 2 Rf .x/ WD We f .x/ D exp ix 2 f .x/: 0 1 Proposition 3.8. The corresponding operator in QB .R2 / is z Rg.x; / D e ix=2 g.x; C x C 1=2/: Remark. In comparison to the previous Theorem 3.5, we have here an additional shift of the argument. Proof. Consider the distribution B Œa; I x of the variable x given by X B Œa; I x D e i a ı.x a k/e ik : k B

Let g WD Z f . Then g.a; / D hhf .x/; B Œa; I xii;

z Rg.a; / D hhf .x/; expf ix 2 gB Œa; I xii:

We evaluate

X ˚

ı.x a k/e 2k expf ix 2 gB .a; I x/ D exp ix 2 e i a X

k

De

i a

ı.x a k/e 2k e i.aCk/

De

i.aCa2 /

k

X k

2

2

ı.x a k/e 2 ik.aC/ e i k :


395

2

Since k is integer, e i k D e i k . We obtain X ı.xak/e 2 ik.aCC1=2/ D e ia=2 B Œa; CaC1=2I x: e ia=2 e i a.CaC1=2/ k

This is equivalent to the desired formula.

3.5 Action of Sp.2 n; Z/. Let us try to define the action of Sp.2n; Z/ by the same formula (3.3) as for 2;1 . A B Lemma 3.9. For any f 2 QB .R2n / and h D C D 2 Sp.2n; Z/, the function .h/f .x; / WD f .x ˚ /h g.x; / WD W (3.4) satisfies the quasiperiodicity conditions

˚ g.x C p; C q/ D g.x; / exp i.xq t p t C pq t / ˚

exp i.„..p ˚ q/h/ „.p ˚ q//

(3.5)

for p, q 2 Zn . Proof. See the proof of Theorem 3.5.

In the last formula (3.5) the first line corresponds to the quasiperiodicity conditions for QB . Nevertheless we also have an additional factor ˙1 in the second line of (3.5). The previous subsection suggests that our formulas must be corrected by halfinteger shifts. These correcting operators T ./ are contained in the Heisenberg group Heisaux n defined in the previous section. Now let u, v 2 Zn . Consider the operators T .u=2 ˚ v=2/ W C 1 .R2n / ! 1 C .R2n /, see (2.9). Denote by T# .u=2 ˚ v=2/ W QB .R2n / ! C 1 .R2n / the same operator restricted to the subspace QB .R2n /. Lemma 3.10. Let p, q 2 Zn . Then T# .u=2 C p/ ˚ .v=2 C q/ D ˙T# u=2 ˚ v=2 : Consider the standard map w 7! wNN from Z2n to F22n . The lemma claims that the operator T# is determined by uNN ˚ vNN 2 F22n canonically to within a factor ˙1. Proof of the lemma. By formula (2.9),

T .u ˚ v/T .p ˚ q/ D ˙T .p C u/ ˚ .q C v/ :

Next, we apply Observation 2.8. For any g 2 QB .R2n /, T .p C u/ ˚ .q C v/ g D ˙T .u ˚ v/T .p ˚ q/g D ˙T .u ˚ v/g:

396


Lemma 3.11. The image of T# .u ˚ v/ consists of functions satisfying h.x C p; C q/ D e 2 i.vp

t uq t /

h.x; /e i.xq

t p t /C ipq t

(3.6)

or, equivalently, T .p ˚ q/h.x; / D e 2 i.vp

t uq t /

h.x; /:

(3.7)

Proof. Straightforward calculation. t

t

We stress that the correcting exponential factor e 2 i.vp uq / in (3.7), is ˙1. Now we are ready to define operators W .h/ for each h 2 Sp.2n; Z/. In Subsection 3.1 we defined the “symplectic” form ƒ and the quadratic form Q on the space F22n . Referring to Theorem 3.3, for each h 2 Sp.2n; Z/, we consider an element uNN ˚ vNN 2 F22n such that NN NN pNN ˚ q/ NN Q .pNN ˚ q/h Q pNN ˚ qNN D ƒ.uNN ˚ v; NN qNN 2 F n . Put for all p, 2 .h/; W .h/g WD ˙T# .u ˚ v/1 W is given by (3.4). where the operator W Theorem 3.12. a) The operators W .h/g preserve the space QB .R2n /. b) The operators W .h/ satisfy the commutation relation W .h/T a ˚ ˛ D T .a ˚ ˛/s W .h/; where T are the operators of the Heisenberg group Heisbas n . c) W .h1 /W .h2 / D ˙W .h1 h2 /. d) Operators W .g/ correspond to operators We.g/ under the Zak transform. breaks the Proof. a) This follows from Lemmas 3.11 and 3.9. The operator W quasiperiodicity conditions and T# returns them again. satisfy the condition b) The operators W .h/T a ˚ ˛ D T .a ˚ ˛/h W .h/ W (see Theorem 3.5). By Observation 2.7, the operators T ./ and T ./ commute and this implies the desired relations. c), d) By b), the operators W .h/ are operators of the Weil representation.


397

3.6 Digression. Rational dilatations Proposition 3.13. a) Let r be a positive integer. Consider the operator Ur in .R/ given by x Ur f .x/ D f : r Then the corresponding operator Uzr D ZB Ur .ZB /1 in QB .R2 / is given by Uzr g.x; / D

r1 X

e i g

x C

D0

r

; r :

(3.8)

b) The operator Uzr satisfies the condition T .a ˚ ˛/Uzr D Uzr T .a=r ˚ ˛r/: c) Consider the operator

Ur1 .x/ D f .xr/:

The corresponding operator Uzr1 is given by r1 1 X i x C Uzr1 g.x; / D e g rx; : r r D0

Proof. a) For g.x; / D e ix

X

f .x C k/e 2 i k ;

k

we have Uzr g.x; / D e ix

X x C k f e 2 i k : r k

Representing k D lr C , we get Uzr g.x; / D e ix

r1 X

1 X

f

D0 lD1

D

r1 X D0

e i

1 X lD1

e i

xC r

xC C l e 2 i.lrC / r

r

f

r1 X xC xC e i g C l e 2 i l.r / D ; r r r D0

and this gives (3.8). The statement b) follows from a) or can be verified by a straightforward calculation. Let us prove c). r1 1 X X X 1 Uzr1 g.x; / D e ix f r.xCk/ e 2 ik D e ix f .rxCl/e 2 i l. C /=r : r k

lD1 D0

Here we apply the standard property of roots of unity, ´ r1 X r if l=r is integer, 2 il=r e D 0 otherwise. D0

398


Figure 8.2. Reference Proposition 3.13, r D 2. To evaluate a value of U2 f .x; / we need values f .x=2; 2/ and f ..x C 1/=2; 2/.

Next, we rearrange our expression to the form r1 r1 1 1 X i x i rx. C/=r X 1 X i x C 2 il. C /=r e e f .rx Cl/e D e g rx; r r r D0

and this proves c).

lD1

D0

3.7 Action of Sp.2 n; Q/ Problem 3.2. Show that each matrix g 2 GL.n; Q/ can be represented in the form g D ˛Dˇ, where ˛, ˇ 2 GL.n; Z/, and D is a diagonal matrix. Hint. We have a rational matrix g. We want to reduce it to a canonical form by transformations of the following types: – transpose rows or transpose columns; – add one row to another; add one column to another. Problem 3.3. Each matrix g 2 Sp.2n; Q/ can be represented in the form g D ˛Dˇ, where ˛, ˇ 2 Sp.2n; Z/, and D is a diagonal matrix. In fact the last problem follows from Theorem 11.3.5; I am not sure that this way is the shortest one. Now we can write W .g/ WD W .˛/W .D/W .ˇ/ for an arbitrary matrix g 2 Sp.2n; Q/; this operator corresponds to the operator We.g/ in .Rn /. We know W .˛/ and W .ˇ/ by Theorem 3.12 and we know W .D/ due to Proposition 3.13. Therefore we know (at least, potentially) W .g/. In the next section we present a more simple approach to this problem. However, now we describe the approximate structure of the final answer. We consider the graph of an operator g W R2n ! R2n and its shifts by elements of the lattice 4n Z . In this way, we get a countable locally finite collection of 2n-dimensional planes. Then we shift this picture by an appropriate rational vector. The kernel of W .g/ is a delta-distribution supported by the intersection of such a collection with the cube Œ0; 14n . 3.8 Digression. The restriction of the Weil representation to Sp.2 n; Z/. Recall that the Weil representation of Sp.2n; R/ is a direct sum of two irreducible representations (see § 7.7). They remain irreducible after the restriction to Sp.2n; Z/. Restrictions of the representations T˛ (see Section 7.5) to Sp.2n; Z/ are irreducible if ˛ 6 n (and terribly reducible if ˛ > n. See [41].

399

8.4. Theta-functions and theta-distributions

8.4 Theta-functions and theta-distributions 4.1 Theta-functions. Let T be an n n matrix with positive real part. We define the theta-function3 ŒT I x; D

X k2Z

²

³

1 exp .x C k/T .x t C k t /t e 2 ik 2 n

being the image of the Gaussian expf 12 xT x t g under the Zak transform. Theorem 4.1. h ŒT I x; ; ŒSI x; iL2 Œ0;22n D .2/n=2 det.S C T /1=2 :

(4.1)

Proof. Since the Zak transform is unitary, the left-hand side of (4.1) equals Z ² ³ ˛ ˝ 1 exp x.T C S /x t dx expf 12 xT x t g; expf 12 xSx t g L2 .Rn / D 2 Rn

and we apply(1.15).

4.2 Theta-distributions. Denote by Q0 .R2n / the space dual to Q.R2n /. Recall that a Gaussian distribution on Rn is a distribution of the form

˚ t D exp 12 xT x t ıL ;

(4.2)

where L is a linear subspace in Rn (in particular, we admit L D 0 and L D Rn ), ıL is its delta-function, and T is a matrix with a non-negative definite real part. We define -distributions as images of Gaussian distributions under the Zak transform. There are several more direct ways to define theta-distributions. 4.3 Theta-distributions as limits of theta-functions. First, the Zak transform is a continuous bijection 0 .Rn / ! Q0 .R2n /. In particular, Zn establishes a continuous one-to-one correspondence between the cone of Gaussian distributions 0 .Rn / and the cone of theta-distributions Q0 .R2n /. For instance, let T be a matrix with a positive semi-definite real part. Then we define the corresponding -distribution as “ hh ŒT ; g.x; /ii D lim

"!0C

3

ŒT C "I x; g.x; / dx d : Œ0;12n

In the standard language, they are called theta-functions with characteristics.

400


4.4 Representations of theta-distributions as sums of series Proposition 4.2. Let a Gaussian distribution t on Rn be given by (4.2). Assume that L Rn is not contained in coordinate hyperplanes. Then the corresponding linear functional on Q.R2n / is hh‚Œt; gii X Z D k2Znx2Œ0;1n

²

²

1 ıL .x k/ exp .x k/T .x k/t 2

(4.3)

³ Z

e

2 ik

³

g.x; / d :

Œ0;1n

The iterated integral and the series are absolutely convergent.4 Proof. By definition,

Z

hht; f iiRn

²

³

1 D f .x/ exp xT x t ıL .x/: 2 x2Rn x2 1

L

1 1

a)

b)

0

1 x1

Figure 8.3. Reference Proposition 4.2, n D 2. a) Reference formula (4.4). b) Reference formula (4.3). The coordinates 1 and 2 are absent on this figure. We show domain of integration with respect to x1 and x2 . If the angular coefficient of L is rational, then we obtain an integration (in x) with respect to a finite family of segments. Otherwise, we get an integration with respect to a dense winding.

Since f 2 .Rn /, the integral absolutely converges. We write it as ² ³ X Z 1 f .x/ exp xT x t ıL .x/ n 2 n x2kCŒ0;1

(4.4)

k2Z

(if L is contained in a face of the cube, then a sense of integrals is not clear; however this is forbidden). The series remains absolutely convergent. Applying the inversion formula (1.5), we come to Z ² ³ XZ 1 exp xT x t ıL .x/ g.x; / d : (4.5) 2 x2kCŒ0;1n 2Œ0;1n k

Next, we shift the argument x 7! x C k and apply the quasiperiodicity (1.3). 4

But

PR

j j diverges.

401

8.4. Theta-functions and theta-distributions

4.5 Example. Rational matrices. For simplicity, let L be the whole Rn . Let T D 2 iR,

where R has rational coefficients.

By Proposition 4.2, the corresponding -distribution is a sum of the series X ˚

exp i.x k/R.x k/t 2 i k t k2Zn

˚

X

˚

˚ D exp ixRx t exp 2 i.xR C /k t exp i kRk t : k2Zn

The last sum admits an explicit summation. Indeed, the factor expf i kRk t g is periodic and has only a finite number of values. More precisely, there is a lattice M Zn such that expf i.k C m/R.k C m/t g D expf i kRk t g; m 2 M: Thus we write out the sum as X X h X ˚

i D expf i ŒjRj t C 2.xR C /j t g exp 2 i.xR C /mt : j 2Zn =M

m2M

Denote by M ˙ the dual lattice, i.e., the set of all points l 2 Rn such that lmt 2 Z for all m 2 M t . Then X X t e 2 im D ı.x l/: l2M ˙

m2M

Finally, we obtain that our initial sum has the form X ı.xR C l/; ˆ.x; /

(4.6)

`2M ˙

where ˆ is an explicit linear combination of Gaussians. Recall that we must restrict the expression (4.6) to the cube Œ0; 12n . 4.6 Example. Rational subspaces Problem 4.1. Let n D 2. Find Z2 -images of the distributions ı.x1 /;

ı.2x1 3x2 /;

2

e x1 =2 ı.2x1 3x2 /;

ı.x1

p

2x2 /:

Now let L Rn be a subspace determined by linear equations with rational coefficients. Let M WD L \ Zn ; obviously M is a lattice. We look to the formula (4.3); precisely, to the term ıL .x k/. Here k ranges in Zn , but actually ıL .x k/ depend only on the class of k in Zn =M . In fact, we get an integration over a certain subtorus in .R=Z/n .

402


If Re T > 0, we can represent (4.3) as Z Z n X ıL .x /g.x; / 2 Zn are representatives x2Œ0;1n of classes in Zn =M

Œ0;1n

hX

m2M

o i o n 1 exp .x m /T .x m /t e 2 i.mC/ d : 2

Thus, we get an integration over a subtorus and a theta-like function on the subtorus. ˚

4.7 Partial differential equations for -functions. A Gaussian vector exp 12 xS x t satisfies the system of partial differential equations

²

³

X 1 @ C skj xk exp xS x t D 0: @xj 2 k

By (1.4), -functions satisfy the partial differential equations

X 1 @ @ C skj xk C @xj 2 i @k

ŒS I x; D 0:

k

Observation 4.3. The differential equations and the quasiperiodicity conditions determine the -function uniquely to within a scalar factor. Proof. Otherwise, the differential equations for a Gaussian vector have an additional solution. Similar systems can be easily written for -distributions (see Chapter 1, Theorem 6.2).

8.5 Theta-kernels In this section, we evaluate images of the Gaussian integral operators under the Zak transform. 5.1 Theta-kernels. Formulation of results. For a symmetric .m C n/ .m C n/ matrix S with positive definite real part denote, define the theta-kernel KŒS .x; I y; / as a function on Rm Rm Rn Rn given by

KŒS.x; I y; / WD S I .x; y/; .; / ³ t X X ² 1 x C kt xCk yCl S e 2 ik e 2 il ; D yt C l t 2 m n k2Z

l2Z

where .x; / 2 R2n and .y; / 2 R2m .

403

8.5. Theta-kernels

Define the corresponding integral operator Wn;m ŒS W Q.R2n / ! Q.R2m / by 1 .Wn;m ŒS g/.x; / D .2/n

Z T 2n

KŒS .x; I y; /g.y; /dy d ;

(5.1)

where g 2 Q.R2n /. Observation 5.1. K satisfies the quasiperiodicity conditions KŒS .x C k; I y C l; / D e 2 i.kCi l/ KŒS .x; I y; /I KŒS .x; C kI y; C l/ D KŒS .x; I y; /:

(5.2) (5.3)

Indeed, these formulas are nothing but the quasiperiodicity conditions for -functions. Observation 5.2. K is a kernel of an operator Q.Rn / ! Q.Rm /. More precisely, the conditions (5.2) and (5.3) allow us to write our integral (5.1) as an integral over an arbitrary shifted cube .x0 ; 0 I y0 ; 0 / C Œ0; 14n ; the result does not depend on a choice of .a; ˛/. In the next subsection, this allows us to integrate (5.1) by part without appearance of boundary summands. Also, these conditions provide the quasiperiodicity of the left-hand side of (5.1). Theorem 5.3. Let P1 W V2k V2m , P2 W V2m V2n be morphisms of the category Sp (see Section 1.8). Let A B K L S.P1 / D i ; S.P2 / D i (5.4) Bt C Lt M be their Potapov transforms in the sense of Subsection 1.9.6. Then Wq;m ŒS.P1 / Wn;q ŒS.P2 / D

.2/m=2 Wn;m ŒS.P1 P2 /: det.C C K/1=2

(5.5)

In particular, we obtain a projective representation of the category Sp. Recall that the Gaussian operators BŒS W L2 .Rn / ! L2 .Rm /, where S D were defined in § 1.3; namely BŒS is determined by the kernel t ³ ² A B 1 x x y K.x; y/ WD exp : yt Bt C 2 Theorem 5.4. Zm BŒS Z1 n D Wn;m ŒS :

A B Bt C

,

404


Consider a general Gauss operator B W S.R2n / ! S 0 .R2m / with the kernel ³ ² 1 t x x yt S ıL .x; y/: exp y 2 Let ‚L;S .x; ; y; / be the corresponding -distribution in Q0 .R2nC2m /. Theorem 5.5. The operator 2n 2m Zm BZ1 / n W Q.R / ! Q.R

has the kernel .2/n ‚L;S .x; ; y; . This follows from the previous theorem by continuity. In particular, this statement gives an explicit formula for the Weil representation of the group Sp.2n; R/ in Q.R2n /. Remark. For each g 2 Sp.2n; Q/, we get an elementary expression for the kernel, see Subsection 4.5. The rest of this section is a proof of Theorems 5.3–5.5. 5.2 Almost proof of Theorems 5.3 and 5.4. We are going to prove that the operator 2n 2m W WD Zm BŒS Z1 / n W Q.R / ! Q.R

is equal to the operator with the kernel .2/n KŒS to within a scalar factor (Theorem 5.4 claims that this factor is 1). Since we know the rules of multiplication of Gaussian operators, this statement also implies formula (5.5) for products to within a scalar. We replicate the usual arguments with commutation relations. For each ˛ C , ˛ 2 n C and ˇ C , ˇ 2 Cm satisfying A B C C ˇ D ˛ ˇ ˛ ; B t C we have m X

˛jC xj

j D1

C

˛j

X d d K.x; y/ D ˇjC yj ˇj K.x; y/ dx d yj n

j D1

or, equivalently, m n X X d d ˛jC xj C ˛j BŒS D BŒS ˇjC yj C ˇj : dx d yj

j D1

j D1

(5.6)

405

8.5. Theta-kernels

Hence, by Theorem 1.4, m X

˛jC xj C

j D1

@ 1 @ C ˛j Zn BŒS Z1 m 2 i @j @xj

D Zn BŒS Z1 m

n X

1 @ @ C ˇj : 2 i @ j @yj

ˇjC yj C

j D1

We rewrite these equations as equations for the kernel of the operator Zn BŒS Z1 m and obtain the partial differential system for a -kernel, which has a unique solution to within a constant factor, see Observation 4.3 5.3 Composition formula. Proof of Theorem 5.3. Let W1 WD Wq;m ŒS1 and W2 WD Wn;q ŒS2 be our operators. Denote their kernels by K1 and K2 respectively. We must evaluate the convolution of kernels, i.e., Z K1 .x; I s; /K2 .s; I y; / ds d : Œ0;22q‹

Expanding the theta-functions into series, we obtain ³ t Z ² XX A B 1 x C kt xCk sCp exp e 2 ik e 2 ip st C pt Bt C 2 Œ0;22q

k

p

t ³ K L 1 s C rt sCr yCl exp e 2 ir e 2 il dsd yt C l t Lt M 2 r l Z ² X XX 1 xCk sCp e ik e il exp D 2 p k l t ³ Œ0;22q x C kt A B e 2 ip st C pt Bt C ²

XX

X r

²

1 sCr exp 2

K yCl Lt

L M

st C r t yt C l t

³ e 2 ir dsd :

For fixed x, y, , the integral is the L2 Œ0; 12q -inner product of two theta-functions. These theta-functions are of the Zak transforms of two Gaussian functions, namely of t ³ ² A B 1 x C kt xCk s ; b1 .s/ WD exp st Bt C 2 ³ ² N Kx L 1 st s yCl b2 .s/ WD exp : x Nt M yt C l t L 2

406


Since the Zak transform is a unitary operator, we can write the last integral as an inner product in L2 .Rq /, i.e., Z b1 .s/b2 .s/ ds: Rq

The last expression is a Gaussian integral. We easily evaluate it and obtain a Gaussian in x, y. We omit a calculation (actually, it was done in the proof of Theorem 1.4.2). After summation in k and l, we get a -kernel. 5.4 End of the proof of Theorem 5.3. In § 5.2 we have evaluated the kernel of the operator Zm BŒS Z1 n to within a scalar, i.e., Zm BŒS Z1 n D .m; nI S /Wn;m ŒS ;

(5.7)

where .m; nI S/ 2 C. We want to prove that

.m; nI S/ D 1

for all m; n; S:

(5.8)

A priori we know the following facts about the function . Lemma 5.6. a) .m; nI S/ is a holomorphic function in the variable S. b) .0; mI S/ D 1. c) m; nI S.P1 / k; mI S.P2 / D .k; nI S.P1 P2 //. Proof. a) Both sides of (5.7) are holomorphic in S . b) Indeed, the Gaussian operators corresponding to elements Mor Sp .V0 ; Vm / are Gaussian vectors, see Subsection 3.12. By definition, images of Gaussians under the Zak transform are theta-functions, see Subsection 4.1. c) We observe (see (1.4.1), (5.5)) that in the identities BŒS.P1 /BŒS.P2 / D .P1 ; P2 / BŒS.P1 P2 /;

Wm;n ŒS.P1 /Wk;m .S.P2 // D .P1 ; P2 / Wk;n .S.P1 P2 //; the scalar factors .P1 ; P2 / coincide. This implies c).

Corollary 5.7. .n; nI P / D 1. Proof. We use statement c) of the lemma. Let P range in the semigroup Sp.2n; R/ of strict symplectic contractions of C2n . This semigroup is open in Sp.2n; C/. Obviously, each homomorphism W Sp.2n; R/ ! C admits a unique extension to a homomorphism Sp.2n; C/ ! C and hence D 1. The semigroup Sp.2n; R/ is dense in Mor.V2n ; V2n / and this implies the statement. Corollary 5.8. .n; mI P / does not depend on P .


407

Proof. Let R range in Sp.2n; R/ and Q range in Sp.2m; R/. By the previous corollary, .n; mI QPR/ D .n; mI P /. Proof of Theorem 5.4. By the previous corollary, .n; mI P / WD .n; m/ D const. On the other hand, .0; m/ D .0; n/ .m; n/ and .0; k/ D 1.

8.6 Bibliographical remarks 6.1 The Cartier model for representation of the Heisenberg group was proposed by P. Cartier [34]. Perhaps the “Zak transform” (J. Zak, [218]) also must be attributed to him. Another term for the Zak transform is A. Weil–J. Brezin transform, see [29]. See also the book by G. Lion and M. Vergne [122] and the paper of A. Janssen [99]. 6.2 Action of Sp.2 n; Q/. Maybe Theorems 3.5, 3.12 are new. Some statements are contained in the books of Mumford [136] and Gröchenig [76]. 6.3 Theta-kernels. We follow [54] but note that the statement was earlier conjectured by R. Howe.

9 Gaussian operators over finite fields

This chapter is very simple. In § 9.4 we construct a canonical correspondence between Gaussian operators and linear relations over finite fields. This is the main result of the chapter, it is almost independent of §§ 9.1–9.3; only some details require manipulations with Gauss sums (Section 9.2). However, §§ 9.1–9.3 are important for the next chapter. In the last section we discuss the numerical Fourier transform. In this chapter, p is a prime, p ¤ 2 .

9.1 Classification of quadratic forms 1.1 Notation. Legendre symbol. Let p be a prime integer, p ¤ 2. Denote by Fp the field of p elements. We identify Fp with the field of residues modulo p. By Fp D Fp n0 we denote the multiplicative group of Fp . Recall that this group is cyclic, Fp ' Zp1 : Consider the subgroup .Fp /2 Fp of squares, i.e., elements of the form x 2 , where x 2 Fp . Since .p 1/ is even, it follows that the quotient group Fp =.Fp /2 consists of two elements. Recall that the Legendre symbol is defined by ! ´ y 1 if y D x 2 ; .y/ D WD (1.1) p 1 if y is not a square: Note that we use the dual notation: ./ and p . If .y/ D 1, we say that y is a quadratic residue, otherwise, it is a quadratic nonresidue. Theorem 1.1. a) The Legendre symbol is a homomorphism Fp ! Z2 , ! ! ! y1 y2 y1 y2 D p p p and an isomorphism Fp =.Fp /2 ! Z2 . b) ! x D x .p1/=2 : p

(1.2)

(1.3)

409

9.1. Classification of quadratic forms

In particular,

! ² ³ 1 1 if p D 4k C 1 D D .1/.p1/=2 : 1 if p D 4k C 3 p

(1.4)

Problem 1.1. Prove the theorem. Sometimes, it is convenient to extend to the whole field Fp by the assumption .0/ D 0. 1.2 Quadratic forms. Preliminary remarks. Let V be a finite-dimensional linear space other Fp . We define symmetric bilinear forms on V in the usual way (see Subsection 3.1.1). For a symmetric bilinear form H.x; y/, define the quadratic form B.x/ WD H.x; x/. The form H can be reconstructed1 from B by H.x; y/ WD

1 .B.x C y; x C y/ B.x; x/ B.y; y//: 2

Lemma 1.2. a) Each quadratic form can be written as X B.x/ D aj xj2

(1.5)

(1.6)

in some basis. b) Let B.z; z/ D a ¤ 0. Then B can be reduced to the form X bkl xk xl : B.x/ D ax12 C k;l>2

c) Let B.z; z/ D 0 for some z ¤ 0. Then B can be reduced to the form X B.x/ D 2x1 x2 C bkl xk xl : k;l>3

Proofs are standard. 1.3 Quadratic forms. Classification. Consider a nondegenerate quadratic form X xQx t D qkl xk xl : We define its discriminant Dis.Q/ as the determinant det.Q/ regarded as an element of Fp =.Fp /2 , ! det Q Dis Q WD : p 1 For p D 2, we meet an obstacle, because C1 D 1; hence symmetric and skew symmetric forms are the same. However quadratic forms and symmetric bilinear forms are different objects, as we have seen in Subsections 8.3.1–8.3.2.

410

Chapter 9. Gaussian operators over finite fields

Observation 1.3. The discriminant is an invariant of a quadratic form. Proof. A linear transformation to x 7! xU sends Q to Q ! UQU t .

The following theorem claims that the discriminant is a unique invariant of quadratic forms. Theorem 1.4. Each nondegenerate quadratic form can be reduced to one of the following forms: 2 x12 C C xn1 C xn2 I

x12

C C

2 xn1

C

rxn2

;

(1.7) (1.8)

where r is a quadratic nonresidue. Proof. Substitute xj 7! ˛j xj in (1.6), where ˛j are arbitrary constants. Then the coefficients aj change to aj ˛j2 . Therefore, our form can be reduced to X X xj2 C r xj2 : (1.9) j 6t

j >t

Now the theorem follows from the next lemma: Lemma 1.5. The binary forms T1 .x/ D x12 C x22 ;

T2 .x/ D rx12 C rx22

are equivalent. Proof. First, 1 C x22 has .p C 1/=2 values. Also, r C rx22 has .p C 1/=2 values. Hence there are non-zero vectors u, v 2 Fp2 such that T1 .u/ D T2 .v/. There are two cases: 1) A joint value is 0. By Lemma 1.2c), both forms can be reduced to 2y1 y2 . 2) A joint value is a ¤ 0. By Lemma 1.2b), our forms can be reduced to T1 .z/ D ay12 C b1 y22 ;

T2 .z/ D ay12 C b2 y22 :

Since Dis T1 D Dis T2 , it follows that b1 D b2 as elements of Fp =.Fp /2 .

1.4 Another canonical form Theorem 1.6. Each nondegenerate quadratic form of .2k C 1/ variables can be reduced to 2 2x1 x2 C C 2x2k1 x2k C s x2kC1 ; where s is a parameter (quadratic residue or nonresidue). Each form of 2k variables can be reduced to 2 2 2x1 x2 C C 2x2k3 x2k2 C x2k1 C s x2k :

9.2. The Fourier transform and Gauss sums

411

Lemma 1.7. a) Let n > 3. Then each nondegenerate form on Fpn has an isotropic vector. b) Let n D 2. Then the set of values of a nondegenerate form includes all non-zero numbers. Proof of the lemma. a) Let '.x/ D a1 x12 C a2 x22 C a3 x32 ; where aj are 1 or a quadratic nonresidue r. First remark. Let ai C aj D 0 for some i , j ; say a1 C a2 D 0. Then the vector .1; 1; 0/ is isotropic. The case p D 4k C 1. Two of numbers ai , aj must be equal, say a1 D a2 DW c. Since .1/ is a quadratic residue, c.x12 C x22 / C a3 x32 c.x12 x22 / C a3 x32 and the vector .1; 1; 0/ is isotropic. The case p D 4k C 3. Then .1/ is a quadratic nonresidue. If both 1 and 1 are present among aj 2 Fp =.Fp /2 , then, obviously, there is an isotropic vector. Otherwise, by Lemma 1.5, a.x12 C x22 C x32 / a.x12 x22 C x32 / and we again get an isotropic vector .0; 1; 1/. b) It suffices to show that the equation a1 x12 C a2 y22 D c has a solution for each c ¤ 0. In other words, we must show that the form a1 x12 C a2 x22 cy 2 has an isotropic vector .; ; / with ¤ 0. By the statement a), this form has an isotropic vector. If D 0, then the form a1 x12 C a2 x22 itself has an isotropic vector. Hence (see Lemma 2.1.3) it can be reduced to the form 2x1 x2 and the latter form represents all values. Proof of Theorem 1.6. If n > 3, then there is an isotropic vector. We apply Lemma 2.1.3 and repeat the second proof of Theorem 2.1.9.

9.2 The Fourier transform and Gauss sums Here we discuss Gauss sums, which are counterparts of the classical -function. Precisely, consider the integral Z 1 Z 1 Z 1 s1 at s s1 at s t e dt D a .at / e d.at / D a x s1 e x dx D .s/as : 0

0

0

(2.1)

412


The factor e at corresponds to the additive characters of Fp , analogs of the factor t s1 are the multiplicative characters of Fp . The analogs of the integral (2.1) are `2 .Fp /-inner products of additive and multiplicative characters. 2.1 Characters of cyclic groups. Consider the finite (additive) cyclic group Zn with a generator z. Lemma 2.1. All characters2 ˆ W Zn ! C have the form ˆm .kz/ D e 2 imk=n ;

where m D 0; 1; 2; : : : ; n 1:

Proof. Let ˆ.z/ D . Then ˆ.kz/ D k . Since nz D 0, we get n D 1. Hence D e 2 im=n . 2.2 Additive characters of a finite field. Starting from this place until the end of the section we write ² ³ 2 i WD exp : (2.2) p Consider the functions m on Fp given by ²

m .x/ D exp

³

2 i mx D mx ; p

where m D 0; 1; : : : ; p 1 and x 2 Fp :

These functions are nothing but the characters of the additive group of Fp . We define the `2 -inner product in the space of functions on Fp , X hf; gi D f .x/g.x/: x2Fp

Lemma 2.2. The functions p 1=2 k constitute an orthonormal basis in `2 .Fp /. Recall the following lemma: Lemma 2.3. Let be a root of unity, n D 1, ¤ 1. Then 1 C C 2 C C n1 D 0: Proof of Lemma 2.3. Multiplying the left-hand side by , we obtain the same expression. Proof of Lemma 2.2. We must evaluate sums hk ; m i D

p1 X

kx N mx D

xD0

If k ¤ m, then all the powers P p1 x xD0 D 0, i.e., 0.

.km/x

2

See Subsection B.2.

p1 X

.km/x :

kD0

are pairwise different. Therefore we get


413

2.3 The finite Fourier transform. We define the Fourier transform F W `2 .Fp / ! `2 .Fp / by

² ³ 1 X 2 ixy F f .y/ D p f .y/ exp : p p

(2.3)

x2Fp

In other words the Fourier transform is the operator with the matrix 1 0 1 1 1 ::: 1 B1 2 : : : p1 C C 2 4 1 B B : : : 2.p1/ C F D p B1 C: :: :: :: C p B :: :: @: : : : : A p1 2.p1/ .p1/2 1 ::: Theorem 2.4. a) The operator F W `2 .Fp / ! `2 .Fp / is unitary. b) The inversion formula is n 2 ixy o 1 X F f .y/ exp : f .x/ D p p p

(2.4)

(2.5)

y2Fp

Proof. This is completely trivial. The rows of the matrix are pairwise orthogonal, therefore F is unitary. Since F is unitary, F 1 D F . Problem 2.1. Show that F 2 f .x/ D f .x/. Hence F 4 D 1. 2.4 Multiplicative characters. We write ²

WD exp

2 i p1

³

:

Let be a generator of the group Fp . Consider the functions ‰l W Fp ! C given by ‰l . h / D hl : These functions are nothing but the multiplicative characters of Fp , ‰l .uv/ D ‰l .u/‰l .v/: Lemma 2.5. The system of functions ‰l .x/ constitutes an orthogonal basis in `2 .Fp /. See the proof of Lemma 2.2. Next, the Legendre symbol is a multiplicative character. Its position in our picture is described by the following lemma:

414


Lemma 2.6.

! x ‰.p1/=2 .x/ D : p

(2.6)

Proof. By definition, .p1/=2 D 1. Hence, ‰.p1/=2 .j / D .p1/j=2 D .1/j : By definition, ./ D 1. Therefore .j / is .1/j .

We also need the following remark. Lemma 2.7. ‰k .1/ D .1/k . Proof. First, ‰k .1/.1/ D ‰k .1/ D 1. Hence ‰k .1/ D ˙1. If ‰1 .1/ D 1, then ‰k .1/ D 1 for all k. On the other hand, ‰k .1/ D 1. But functions ‰k form a basis in `2 .F /, therefore there exists a function ‰k with different values at .1/ and 1. This contradiction finishes the proof. 2.5 Gauss sums. Gauss sums are sums of the form X Skm D S.k ; ‰m / WD k .u/‰m .u/: u2Fp

Theorem 2.8. a) Let m D 1; : : : ; p 2. Extend each character ‰m to the whole Fp by the assumption ‰m .0/ D 0. Then x m; F ‰m D Fp .m/ ‰ where

1 X Fp .m/ WD p 1 .y/‰m .y/: p y2Fp

b) The factor Fp satisfies the following properties: jFp .m/j D 1;

(2.7)

Fp .m/ D .1/ Fp .m/: m

(2.8)

Proof. a) We imitate the substitution s D at to the integral (2.1): X p pF ‰m .x/ D 1 .xy/‰m .y/ y¤0

D

X

z¤0

1 .z/‰m .x 1 z/ D ‰m .x/1

X z¤0

1 .z/‰m .z/:


415

b) Since F is unitary, we have k‰k k`2 D kF ‰k k`2 D jFp .k/j k‰k k`2 and this implies (2.7). Further, X p pF ‰k .x/ D 1 .xy/‰k .y/ y

D

X

1 .xy/‰k .y/ D

y

y

D

X

X

1 ..x/y/‰k .y/ D

1 .xy/‰k .y/ p pF ‰k .x/

y

p D .1/k pF ‰k .x/:

2.6 Fourier transform of the Legendre character Theorem 2.9 (Gauss).

Fp

p1 2

´

Di

.p1/2 =4

1 if p D 4k C 1; i if p D 4k C 3:

D

Proof. First, we determine

(2.9)

² ³ ! X 2 iy y p1 D exp (2.10) Fp y¤0 2 p p ˇ ˇ ˇ D 1. to within a sign. By (2.7), ˇFp p1 2 If p D 4k C 1, then .y/ D .y/. Hence (2.10) is real. If p D 4k C 3, then .y/ D .y/. Therefore, (2.10) is imaginary. Thus, in these two cases, our sum is ˙1 or ˙i respectively. Evaluation of the sign is more complicated. Let us evaluate the determinant of the Fourier transform (2.4) in two ways. A priori, det F D ˙i or ˙ D 1, because F is unitary and F 4 D 1. Firstly, det.F / is the Vandermonde determinant Y .k m /: det.F / D p p=2

k>m

Let D expf i=pg, hence D . Then Y Y . 2k 2m / D p p=2

kCm . km mk / det F D p p=2 2

k>m

Dp

p=2

k>m

p.p1/=2 p.p 2 1/=2

.2i /

Y

k>m

sin

.k m/ : p

416


Since 2p D 1, we get p.p

2 1/=2

D 1. Thus we come to Y

det F D p p=2 .2i /p.p1/=2

k>m

sin

.k n/ : p

(2.11)

Therefore, det F D i p.p1/=2 ; Y .k n/ sin D p p=2 2p.p1/=2 : p

(2.12)

k>m

The second identity is amusing, however it does not matter for us. Now let us define another basis. We consider the multiplicative characters ‰1 , ‰2 ; : : : ; ‰p2 (here ‰0 D 1 is omitted) extended to Fp by the assumption ‰.0/ D 0. We also consider the function c.x/ WD 1 and the function ı0;x (the last two functions are not orthogonal). We use Theorem 2.8 about Gauss sums to write out the matrix of F in this basis. We obtain a block diagonal matrix having only 2 2 and 1 1 blocks. The list of blocks follows. 1. 2 2-blocks corresponding to vectors ‰k and ‰k : 0 Fp .k/ : Fp .k/ 0

(2.13)

Here k D 2; 3; : : : .p 3/=2. 2. The block corresponding to c.x/ and ı0;x ,

0

p 1=2 3. The 1 1 block Fp

p 1=2 : 0

p1 2

.

The determinant of (2.13) is Fp .k/Fp .k/ D .1/k Fp .k/Fp .k/ D .1/k : Multiplying together these quantities, we obtain .1/Œ.p3/=2C.p5/.p3/=8 . Thus, p1 .p1/.p3/=8 .1/Fp : det.F / D .1/ 2 It remains to compare (2.12) and (2.14).

(2.14)

417

9.3. Gaussian quadratic sums

9.3 Gaussian quadratic sums Here we discuss the analog of the Gaussian integral. 3.1 Gaussian quadratic sums Theorem 3.1. Let Q be a nondegenerate symmetric n n matrix over Fp . Then3 .Q/ WD

o p 1 n n 2 i exp : xQx t D p n=2 det Q Fp p 2 n

X

(3.1)

x2Fp

Proof. Let Q1 , Q2 be symmetric k k and l l matrices respectively. Then o n 2 i X Q1 0 D .xQ1 x t C yQ2 y t exp 0 Q2 p x2Fpk ; y2Fpl

D

o X o n 2 i n 2 i exp exp xQ1 x t yQ2 y t D .Q1 /.Q2 /: p p k l

X

x2Fp

y2Fp

The right-hand side of (3.1) is multiplicative in the same sense. Since we can reduce a form to a sum of squares, it suffices to verify the identity for n D 1. Let r 2 Fp be a quadratic nonresidue. Write H1 WD

n 2 i o exp x2 ; p

X x2Fp

H2 WD

n 2 i

X

exp

x2Fp

p

o rx 2 :

We have H1 D 1 C 2

X

n 2 i o exp y ; p 2

y2.Fp /

H2 D 1 C 2

X y2Fp n.Fp /2

n 2 i o y : p

exp

Hence, n 2 i o exp y D 0; p y2Fp p 1 n 2 i o X p H1 H2 D 2 exp y .y/ D 2 p Fp : p 2

H1 C H2 D 2

X

y2Fp

Therefore, H1 D 3

p 1 p p Fp D H2 : 2

Comparative to the real case, we do not write 1=2 in the exponent. Otherwise, we must add a factor 2 .2/n on the right-hand side, .2/ D .1/.p1/ =8 .

418


3.2 Nonhomogeneous forms Proposition 3.2. Let Q be a nondegenerate n n symmetric matrix, let b 2 Fpn . Then ²

X

³

³

²

2 i 1 2 i bQ1 b t : exp .xQx t C bx t / D .Q/ exp p 4 p n

x2Fp

Proof. We rearrange the sum to the form ²

³

²

³

2 i 1 2 i exp bQ1 b t exp .x C b=2/Q.x t C b t =2/ p 4 p and pass to the variable y D x C b=2.

3.3 Degenerate forms. It is easy to extend these results to degenerate forms. Let B be a quadratic form on a linear space V and let W D ker B. Let Bz be the form induced by B on V =W , let Bz B be the dual form on the dual space .V =W /B . Let ` be a linear functional on V . Then ® ¯ ² ³ ´ dim W z i 1 zB X p .B/ exp 2 4 B .`/ if ` D 0 on W ; 2 i p exp .B.x/C`.x// D p 0 otherwise: x2V Indeed, we decompose W D ker B ˚ Y , where Y is a complementary subspace. Now the statement becomes obvious.

9.4 Gaussian operators This section contains the main construction of the chapter, it is almost independent of the previous two sections. The construction and proofs are very simple imitations of the real case. 4.1 Gaussian operators. Let L F m ˚ F k be a subspace. Let Q be a quadratic form on L. A Gaussian operator is an operator BŒQI L W `2 .Fpk / ! `2 .Fpm / given by the formula BŒQI L f .x/ D

X y2Fpk W .x;y/2L

²

exp

³

i Q.x ˚ y/ f .y/; p

where 2 C :

Theorem 4.1. a) The product of Gaussian operators is a Gaussian operator. b) The category of Gaussian operators defined to within scalars is equivalent to the category of Lagrangian linear relations over Fp .

419


The proof of the theorem is a straightforward emulation of the corresponding “real” theorem (without difficulties related to the boundedness of Gaussian operators); the proof occupies the rest of this section. 4.2 The Heisenberg group over Fp . Let V2n be a 2n-dimensional space over Fp endowed with a nondegenerate skew-symmetric bilinear form S.; /. We define a product on V2n ˚ Fp by

u; ˛ B v; ˇ D u C v; ˛ C ˇ C 12 S.u; v/ : (4.1) Problem 4.1. Show that with the product defined in (4.1), V2n ˚ Fp becomes a group. This group is called the Heisenberg group Heisn .Fp / over the field Fp . The subgroup Z WD 0 ˚ Fp is the center of Heisn . The quotient group Heisn =Z is the additive group of V2n . 4.3 The standard representation of Heisn . Let V2n WD VnC ˚ Vn WD Fpn ˚ Fpn 0 1 equipped with the form 1 0 .

C For an element .v ˚ v ; h/ 2 Heisn .Fp / consider the operator T v C ˚ v I ˛ in `2 .Fpn / given by ²

³

²

2 i t 2 i 1 T v C ˚ v I ˛ f .x/ D f .x C v C / exp ˛ C vC .v /t v x exp p p 2

³

:

A straightforward calculation shows that such operators form the Heisenberg group. 4.4 Symplectic groups. We define skew-symmetric bilinear forms over Fp in the usual way. As above (Theorem 3.1.1), nondegenerate forms exist only on even-dimensional spaces; 0 1n all nondegenerate forms on a 2n-dimensional space are equivalent to the form 1n 0 . We denote by Sp.2n; Fp / (a symplectic group) the group of all linear transformations preserving this form. Problem 4.2. Construct the Weil representation of Sp.2n; Fp / as it was done in Theorem 1.2.7 4.5 Gaussian vectors and their stabilizers. Let M be a subspace in Fpn , let R be a quadratic form on M . Let 2 C. A Gaussian vector in `2 .Fpn / is a function of the form ´ ® ¯ exp pi R.x/ if x 2 M ; bŒRI M D 0 otherwise: For a Gaussian vector bŒRI M consider its stabilizer in the Heisenberg group, i.e., consider the subspace in W ŒRI M V2n consisting of v such that

T vI 0 bŒRI M D bŒRI M : (4.2)

420


Theorem 4.2. a) W is a Lagrangian subspace in V2n . b) The vector bŒRI M is a unique vector f satisfying

T vI 0 f D f for all v 2 W : c) We get a bijection

Lagrangian Grassmannian in V2m

!

The set of Gaussian vectors : defined up to a scalar factor

Proof. First, let M D Fpn . Let v D vC ˚ v . Then ²

³

i T vC ˚ v I 0 exp xRx t p ² ³ ² ² ³ o 2 i i i t t .x C vC /R.x C vC / exp xv exp vC v D exp p p p ² ³ ² ³

2 i t i t D exp : xRx t exp xv xRvC p p The condition (4.2) is satisfied if and only if v D vC R: By Theorem 3.1.4, this constraint determines a Lagrangian subspace. Obviously, this subspace “remembers” a Gaussian vector. Next, let M be a proper subspace. Without loss of generality (see Problem 4.2), we can assume that M is a coordinate subspace v1 D D vm D 0 in Fpn . Our Gaussian vector is fixed under the operators o nX vj xj f .x/ 7! f .x/ exp j 6m

and these operators are contained in the Heisenberg group. Problem 4.3. Complete the calculation of a stabilizer. To prove b), we repeat the arguments of Theorem 1.6.1.b.

4.6 Gaussian operators and Lagrangian linear relations. Equip V2k ˚ V2m with the difference of skew-symmetric bilinear forms as above, see Subsection 1.8.1. Theorem 4.3. a) Let BŒQI L be a Gaussian operator `2 .Fk / ! `2 .Fm /. Consider the set W D W ŒQI L V2k ˚ V2m

421


consisting of vectors u ˚ v satisfying

T vI 0 BŒQI L D BŒQI LT uI 0 :

(4.3)

Then W is a Lagrangian subspace in V2k ˚ V2m . b) The operator BŒQI L is a unique to within a scalar operator A W `2 .Fpk / ! `2 .Fpm / satisfying

T vI 0 A D AT uI 0 for all u ˚ v 2 W : (4.4) c) The map BŒQI L 7! W is a bijection ²

The set of Gaussian operators defined to within scalars

³

²

!

³

Lagrangian Grassmannian : in V2k ˚ V2m

d) Let BŒQI L W `2 .Fpk / ! `2 .Fpm /;

BŒQ0 I L0 W `2 .Fpm / ! `2 .Fpn /

be Gaussian operators. Then the product BŒQ0 I L0 BŒQI L is a non-zero Gaussian operator; it corresponds to the product W ŒQ0 I L0 W ŒQI L of linear relations Proof. A matrix of an operator A W `2 .Fpk / ! `2 .Fpm / is a function on Fpk ˚ Fpm . The equation (4.4) is nothing but the equation (4.2) for a Gaussian vector; only several signs are changed. Hence the statements a)–c) are nothing but a rephrasing of Theorem 4.2. To prove d), we repeat the arguments of Subsection 1.9.4. Let u ˚ v 2 W ŒQI L/ and v ˚ y 2 W ŒQ0 I L0 . Then

BŒQ0 I L0 BŒQI L T uI 0 D BŒQ0 I L0 T vI 0 BŒQI L D T yI 0 BŒQ0 I L0 BŒQI L: Thus BŒQ0 I L0 BŒQI L satisfies the relations (4.4) with W D W ŒQ0 I Y 0 W ŒQI Y . A reference to the statement b) almost proves d). It remains to show that the product of non-zero Gaussian operators is non-zero. Observation 4.4. For any quadratic form Q on Fpn , X

²

³

i exp Q.x/ ¤ 0: p n

x2Fp

Indeed, we know the explicit formula. Observation 4.5. Inner products of Gaussian vectors are non-zero. Proof. We must evaluate ˝

˛

bŒR1 I M1 ; bŒR2 I M2 D

X x2M1 \M2

²

exp

³

x2 .x/ ¤ 0: R1 .x/ C R i

422


Denote by ı.x/ D ıFpk .x/ 2 `2 .Fpk / the delta-function supported by 0. Observation 4.6.

ˇ

B QI L ı D b QˇV

2m \L

I V2m \ L :

End of the proof of Theorem 4.3. Consider a product B1 B2 of Gaussian operators. Then hB1 B2 ı; ıi D hB2 ı; B1 ıi ¤ 0: 4.7 Digression. Suprunenko gradations of semisimple Lie algebras Problem 4.4. Consider the action of Heisn .Fp / on the Lie algebra g WD sl.p n ; C/ by automorphisms

1

X 7! T vI ˛ X T vI ˛ :

Since T 0I ˛ are scalar operators, we obtain an action of the Abelian group V2n ' Fp2n . For a character W Fp2n ! C , consider the eigenspace g defined by 1

X T vI 0 D .v/X: X 2 g if T vI 0 a) For a nontrivial character the subspace g is one-dimensional. For the trivial character .v/ D 1, this subspace is trivial. b) Show that ´ g' if ' ¤ 1; Œg ; g' D 0 otherwise. Thus we get a gradation of the Lie algebra sl.p n ; C/. c) Describe this gradation explicitly for n D 1.

9.5 Fast Fourier transform Numerical evaluation of the Fourier transform appears often in applied problems. Here we discuss methods of fast calculations. Let N be a large (generally not prime) positive integer. By ZN D Z=ZN we denote the cyclic group of order N . The Fourier transform on this group is defined in the usual way, ² ³ N 1 X 2 i exp xy f .y/: FN f .x/ D N kD0

At first sight, this calculation requires N 2 multiplications4 . Several tricks discussed below allow us to evaluate FN f using 6 const N log2 N 4

This calculation also requires: – N 2 additions (an addition is a “short” computer operation compared to a multiplication); – calculation of N exponentials (which can be tabulated);

(5.1)

9.5. Fast Fourier transform

423

operations5 . This phenomenon drastically expands possibilities of numerical calculations of Fourier transforms. Denote ² ³ 2 i WD exp ; aj WD f .j /; bm WD F f .m/: N 5.1 The Cooley–Tukey trick. Let N D 2n . We must evaluate all numbers bm D a0 C m a1 C 2m a2 C 3m a3 C

D a0 C 2m a2 C 4m a4 C C m a1 C 2m a3 C 4m a5 C : Next, 2 is a primitive root of unity of degree N=2. Therefore, the expressions in the square brackets are Fourier transforms of vectors fa2j g and fa2j C1 g on the group ZN=2 . Now we must evaluate two Fourier transforms on the group ZN=2 instead of one Fourier transform on ZN . This requires .N=2/2 C .N=2/2 multiplications instead of N 2 . Thus the number of required multiplications is reduced by two times. But we can evaluate the Fourier transforms on ZN=2 in the same way. Etc. This trick allows us to evaluate the Fourier transform on ZN D Z2n using 6

3 N log2 N 2

multiplications. 5.2 A variation of the same trick. Let N D qr be a composite number. First, assume that q and r are relatively prime. Then ZN ' Zq Zr and we can decompose the Fourier transform FN D Fq B Fr : To evaluate FN , we must evaluate q Fourier transforms on sets j Zr and r Fourier transforms on the sets Zq k.6 Now let N D qr, where q and r are arbitrary. We write bm D

q1 X

mj aj C mq aj Cq C 2mq aj C2q C :

j D0

Next, we represent m D lr C k;

(5.2)

– some auxiliary “short” operations. In recent time, the complexity of a computer calculation was estimated as a number of multiplications. Such differences between operations are no longer as important as they once were. In any case, the total number of operations is O.N 2 /. 5 6 8N log2 N multiplications. 6 Also, we must spend some time for reordering of the set ZN .

424


where 0 6 k < r. Then bkClr D

q1 X

.r /lj kj aj C kq aj Cq C 2kq aj C2q C :

j D0

Note that q is a primitive root of unity of degree r and r is a primitive root of degree q. Let S.j; k/ D aj C kq aj Cq C 2kq aj C2q C C .r1/kq aj C.r1/q :

(5.3)

To find S.j; k/, we must evaluate Fr -transforms of q vectors of the form aj ; aj Cq ; : : : ; aj C.r1/q ; where j D 0, 1; : : : ; q 1: After that, the evaluation of .bk ; brCk ; b2rCk ; : : : ; bkC.q1/r / is nothing but an evaluation of the Fq -transform of the vector 0k S.0; k/; 1k S.1; k/; 2k S.2; k/; : : : ; .q1/k S.q 1; k/ : Thus to evaluate FN , we must evaluate q items of Fr transform and r items of Fq transforms. Remark. Actually, this calculation is completely parallel to the calculation (8.2.4). In the latter case, we evaluate the Zak transform and reduce the Fourier transform to the Zak transform. 5.3 Fast convolution. The convolution on the group ZN is defined in the usual way, X f g.k/ D f .j /g.k j /: j 2ZN

As usual, the Fourier transform sends convolutions to products. Having a fast way to evaluate the Fourier transform on ZN , we have a fast way to evaluate convolutions: we simply evaluate FN1 .FN f FN g/. Remark. In the multiplicative notation, the convolution is given by X X f g.x/ D f .x=y/g.y/ D f .xy/g.y 1 /: y

y

5.4 Prime N . Now let N D p be a prime. We must evaluate the vectors cm WD

p1 X j D0

mj aj ;

425

9.5. Fast Fourier transform

where m D 0; 1; : : : ; p 1. It suffices to find a fast way to evaluate the numbers hm WD

p1 X

where m ¤ 0:

mj aj ;

j D1

Now the summation is taken over the multiplicative group Fp ' Zp1 . The function h.m/ is nothing but a convolution of the function g.k/ WD ak and the function f .l/ WD l on the multiplicative group Fp . It remains to note that the number p 1 is composite and we can repeat the Cooley– Tukey trick7 . 5.5 Another way. For definiteness, assume that N is even (otherwise, the considerations given below must be slightly modified), bm D

N 1 X

2 =2

mj aj D m

j D0

N 1 X

2 =2

.mj /

j

2 =2

aj I

j D0

here we assume 1=2 WD expf i=N g. We observe that the vector wm WD m

2 =2

bm

is a convolution of the vectors uj WD j Since N is even,

2 =2

aj

and vk WD k

2 =2

:

o o ni .k 2 C 2kN C N 2 / D exp k 2 D vk ; N N therefore vk and uk are well-defined functions on ZN . Let M > 2N . We wish to evaluate the same convolution as a convolution on the group ZM . For this goal, consider the functions on ZM given by 8 ´ ˆ if 0 6 j < N ; 2N . Now we can choose M in the form M D 2n . If we fight for an improvement of the constant in (5.1) we can choose an appropriate number of the form 2n , 3 2n , 32 2n , 33 2n , 5 2n and apply the trick given above. 7

However, it is also necessary to find a generator of Fp and identify its power with residues modulo p.

426


9.6 Bibliographic remarks 6.1 Gauss sums were investigated by K. F. Gauss himself and the topic is covered in a seemingly infinite number of textbooks, for instance [25], [197], [119]; our presentation is far from complete. 6.2 Gaussian operators. The Weil representation of Sp.2n; Fp / was introduced by A. Weil. Theorem 4.3 was announced in [140]. 6.3 Exotic gradations of semisimple Lie algebras. As far as I know they were introduced by D. A. Suprunenko in the 1960s. For further extensions, see [4], [112]. 6.4 Fast Fourier transform. See e.g., M. Clausen, U. Baum [38] and further references in this book. The trick of Subsection 5.1 usually is attributed to J. W. Cooley and J. W. Tukey, but apparently it was known in the time of hand computations (there are such references in programmer’s literature). I met up also an attribution of this trick to Gauss, but I myself have not seen this Gauss’s work. The construction in 5.4 is attributed to C. M. Rader and the one in 5.5 to L. I. Bluestein.

10 Classical p-adic groups. Introduction

In § 10.1 we present a brief informal introduction to p-adic numbers. I am not sure that this is sufficient; in any case, this simple and nice topic requires some acclimatization. See also systematic books [110], [35], [25], [197] and a nonsystematic book [64]. Sections 10.2–10.7 contain a (not too advanced) discussion of p-adic classical groups. Our main purpose here is to explore these new phenomena as they compare with the real case. Both distinctions and analogies between the real and the p-adic cases look equally mysterious.

10.1 p-adic numbers 1.1 10-adic numbers. We know the rules of addition and multiplication of finite decimal fractions. For instance, C

8765:32 3526:67 12291:99

3:45 2:13 :1035 :345 6:90 7:3485

A usual decimal fraction is a sequence of digits infinite to the right side (and finite to the left side). The 10-adic numbers are decimal fractions that are infinite to the left side and finite to the right side. For instance, :::12121212121212121:3456789: There is no sign in the front of a 10-adic number. A 10-adic integer is a sequence of digits infinite to the left side and finished by a comma1 . For instance, : : : 1111111111111000000: We apply the usual rules of addition and multiplication; for instance, C

1

: : : 8765:32 : : : 3526:67 : : : 2291:99

: : : 3:45 : : : 2:13 : : : 7:3485

According to traditions of programming, we write commas in numbers as points.

428

Chapter 10. Classical p-adic groups. Introduction

Problem 1.1. a) Find 1. b) 10-adic numbers form a commutative ring. Integer p-adic numbers constitute a subring. c) Evaluate 1=2, 1=1024, 1=3. d) Show that for positive p, q 2 Z, the equation qx D p has a unique solution. Problem 1.2. Show that the equation x 2 D x has four solutions, namely, x1 D 0, x2 D 1, x3 D :::890625, x4 D 1 x3 D :::109376. Since a quadratic equation has > 2 solutions, the ring of 10-adic numbers is not a field. In fact, x3 x4 D 0. Define the norm jrj of a 10-adic number r as 10j 1 , where j is the number of the last non-zero digit; for instance, j:::11:j D 1;

j::111000:j D 103 ;

j::1111:1111j D 104 ;

j0j D 101 D 0:

Problem 1.3. a) Show that ja C bj 6 max.jaj; jbj/. b) Show that .a; b/ WD ja bj is a metric on 10-adic numbers. c) Show that the set of 10-adic numbers is a complete metric space. d) Denote by D10 the set f0; 1; 2; : : : ; 9g endowed with the discrete topology. The space of integer 10-adic numbers is homeomorphic to the infinite product D10 D10 : : : . In particular, a sequence of 10-adic numbers converges if and only if the sequence in each digit stops starting with some moment. For instance, the sequence a1 D : : : 00000001: a2 D : : : 00000011: a3 D : : : 00000111: a4 D : : : 00001111: :::::::::::::::::::: converges to : : : 11111111: Also, 10n tends to 0 as n ! 1. e) Show that the set of 10-adic integers is the closure of the set of the usual integers. In particular, we can consider 10-adic numbers as sums of convergent series of the form 1 X ˛j 10j ; ˛j ranges in f0; 1; : : : ; 9g: j Dk

1.2 p-adic numbers. Let p be a prime integer. We define p-adic numbers in the same way, only consider the p number system instead of the decimal one. Theorem 1.1. The ring of p-adic numbers is a field (in other words, a division is well defined).

10.1. p-adic numbers

429

All other properties of 10-adic numbers survive literally. Denote by Qp the field of p-adic numbers, by Op the ring of p-adic integers. By Qp , we denote the multiplicative group of the field Qp . We also use the notation r1 r2 0 .mod p n / () jr1 r2 j 6 p n : Proof of the theorem. First, we can represent a p-adic number in the form p k r, where jrj D 1. It suffices to show that for ˛0 ¤ 0, the number r D ˛0 C ˛1 p C ˛2 p 2 C is invertible. In other words, we must solve the equation .˛0 C ˛1 p C ˛2 p 2 C / .x0 C x1 p C x2 p 2 C / D 1; where unknowns xj range in 0; 1; : : : ; p 1. Assume that we have a solution mod .p n /, i.e., .˛0 C˛1 pC C˛n1 p n1 C /.x0 Cx1 pC Cxn1 p n1 C / D 1Ccp n C : Then .˛0 C ˛1 p C C ˛n p n C / .x0 C x1 p C C xn p n C /

D 1 C ˛0 xn C C˛1 xn1 C C ˛n x0 p n C : Since ˛0 ¤ 0, we can obtain Œ: : : D 0 for an appropriate choice of xn .

Problem 1.4. Why is this proof not correct for 10-adic numbers? Problem 1.5. a) Let p ¤ 2. Show that for r D 1 C ˛1 p C ˛2 p 2 C , the equation y 2 D r has a solution. b) Let p ¤ 2. Let r D p k .˛0 C ˛1 p C ˛2 p 2C / and ˛0 ¤ 0. The equation 2 y D p has a solution if and only if k is even and ˛p0 D 1. c) Show that the equation x p1 D 1 has .p 1/ solutions x1 ; : : : ; xp1 ; they satisfy xj D j mod p respectively. d) Investigate the equation y n D r. Thus, we have

Qp =.Qp /2 ' Z2 ˚ Z2 ;

for p ¤ 2:

Problem 1.6. a) The usual integers Z are dense in Op . b) All closed subgroups of the additive group of Qp have the form p j Op . c) The additive group Qp =p j Op is isomorphic to the inductive limit Zp1 WD lim Zpk ; k!1

(1.1)

where Zpk is regarded as a subgroup of ZpkC1 . In other words, we can say that Qp =p j Op is isomorphic to the multiplicative group of all complex numbers such N that p D 1 for sufficiently large N .

430


Problem 1.7. a) Show that pOp is an ideal in Op ; the quotient ring Op =pOp is the field Fp . b) For k > 0, the quotient ring Op =Opk is isomorphic to the residue ring Z=p k Z. Problem 1.8. Since Qp is a field, it contains all rational numbers. Show that numbers of the form a p k , where a 2 Z, have the form : : : 000000000˛m ˛m1 ˛m2 : : : in our notation. 1.3 A more formal definition. Represent t 2 Q as u t D p k ; where u, v 2 Z; p is not a divisor of u, v: v We define the norm of t as jt jp D p k . We have jt C sjp 6 max.jtjp ; jsjp /;

jt sjp D jt jp jsjp :

Also, .t; s/ WD jt sjp is a metric on Q. The field Qp is the completion of Q with respect to this metric. Problem 1.9. Show that the field defined in the previous subsection is complete and contains Q as a dense set. This implies the equivalence of the two definitions. 1.4 A still more formal definition. Consider the following natural chain of homomorphisms of rings: Z=pZ Z=p 2 Z Z=p 3 Z : In each step, we send the unit to the unit. We define the ring Op of p-adic integers as the inverse limit of this chain; Qp is the quotient field of this ring. Problem 1.10. Convince yourself that we obtain the same object. Problem 1.11. Show that O10 D O5 O2 , Q10 D Q5 Q2 .

1.5 Balls. We define a closed ball B.a; p k / Qp in the usual way, namely z 2 B.a; p k / if and only if jz aj 6 p k . Problem 1.12. a) If c 2 B.a; p k /, then B.c; p k / D B.a; p k /. Thus a ball has a well-defined radius, but each point of a ball is its center. b) For any pair of balls B1 , B2 , either B1 \ B2 D ¿, or B1 B2 , or B2 B1 . c) Closed balls are open sets. d) Each ball consists of points of the form k X j D1

xj p j C

l X

˛i p i ;

iDkC1

where ˛i are fixed and xj are arbitrary. e) Each ball of radius r D p k is a disjoint union of p balls of radius p k1 .

431

10.1. p-adic numbers

B.0; 1/ B.0; 1=2/; B.1; 1=2/ B.0; 1=4/; B.1; 1=4/; B.2; 1=4/; B.3; 1=4/ Figure 10.1. Reference Problem 1.12. Partitions of balls into smaller balls in Q2 . The complete picture is an infinite 3-valent tree.

1.6 Additive characters. A character of the additive group Qp is a continuous map Qp ! C satisfying .u C v/ D .u/.v/: We define the standard character standard by ˚

standard .z/ WD exp 2 iz : Precisely,

n nX

exp

o n o X ˛j p j WD exp 2 i ˛j p j :

j D1

All values of this character have the form

p

pm

j 0

nŠ

;

ln.1 C z/ WD

X .1/n z n j >0

n

;

where z, exp z, ln.1 C z/ 2 Qp . Problem 1.14. Let p ¤ 2. a) Both the series converge in the ball jzj 6 p 1 . b) The maps ln.1 C z/ and exp.z/ 1 sends this ball to itself bijectively.

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c) The exponential and the logarithm have the usual properties. d) The set 1 C pOp is a multiplicative subgroup in Qp . This group is isomorphic to the additive group pOp . As a corollary to this problem, we obtain the following theorem: Theorem 1.2. Let p ¤ 2. Then the group Qp is the product of the following three subgroups: Qp ' Z Fp Op : Precisely, – the subgroup Z consists of numbers p k , – the subgroup Fp ' Zp1 consists of solutions of the equation x p1 D 1, – the last factor consists of elements expf1 C pzg, where z ranges in Op ; equivalently, it consists of elements having the form 1 C pz, where z ranges in Op . In other words, each element z 2 Qp admits a unique representation z D p k expfpug;

where p1 D 1, u 2 Op :

Problem 1.15. Complete the proof of Theorem 1.2. Corollary 1.3. Let p ¤ 2. Then Qp =.Qp /2 ' Z2 ˚ Z2 . The generators of the quotient group are p and arbitrary such that p1 D 1, .p1/=2 ¤ 1. 1.8 Multiplicative group of Q2 . The case p D 2 slightly differs from the general case and below we (unsuccessfully) try to exclude it from our considerations. Theorem 1.4. Each element of Q 2 admits a unique representation in the form ˚ z D p k .1/ exp 4u ;

u 2 O2 :

We omit a proof. In this case, the functions exp z 1 and ln.1 C z/ define a bijection of the set 4O2 to itself. 2 =.Qp / is Z2 ˚ Z2 ˚ Z2 . Corollary 1.5. The quotient group Qp

Problem 1.16. An element 2m .1 C a1 2 C a2 22 C a2 23 C / is a square if and only if m is even and a1 D a2 D 0.

1.9 Invariant measure. We define the standard measure vol./ on Qp by the following conditions: – vol.a C U / D vol.U / for each a 2 Qp and each ball U , – vol.Op / D 1. Then the measure of any ball of radius p k is p k .


433

Problem 1.17. a) Prove the last statement. b) For any measurable subset U Qp and each non-zero 2 Qp , we have vol. U / D jj vol.U /. We also define the measure vol./ on the linear space Qpn by the conditions – vol./ is shift-invariant, – vol.Opn / D 1. Problem 1.18. For any U Qpn and each g 2 GL.n; Qp /, vol.g U / D j det gj vol.U /:

10.2 Classification of quadratic forms 2.1 Quadratic forms. Discriminant Lemma 2.1. Any symmetric bilinear form B.v; w/ over Qp is uniquely determined by the quadratic form B.v; v/ This statement is valid for arbitrary fields of characteristic ¤ 2, see (9.1.5). The following statement is also abstract and general (see Subsection 2.2.1). Proposition 2.2. Each nondegenerate quadratic form can be reduced to X aj zj2 ; aj ¤ 0:

(2.1)

There arises a natural question: When are the forms

X j

aj xj2 and

X

bj xj2 equivalent?

j

We can multiply xj by constants, xj 7! j xj ; then aj goes to j2 aj . Hence the answer is “definitely yes” if aj D bj as elements of the group Qp =.Qp /2 . For a nondegenerate quadratic form zAz t , we define its discriminant Dis.A/ being the determinant det.A/ considered as an element of the group Qp =.Qp /2 . As we have seen above (Observation 9.1.3) the discriminant is an invariant of a form. In the p-adic case, there is an additional (non-obvious) invariant. Problem 2.1. Prove the Witt Theorem 2.2.6 for quadratic forms over Qp . One of the proofs given above survives for arbitrary fields of characteristic ¤ 2.

434


2.2 Notation. In what follows, p ¤ 2 and the quadratic forms under discussion are nondegenerate. We define two homomorphisms , W Qp =.Qp /2 ! Z2 . Let z D p k .˛0 C ˛1 p C ˛2 p 2 C /; We write

where ˛0 ¤ 0.

! ˛0 I .z/ WD .˛0 / D p

.z/ WD k mod 2 (the Legendre symbol was defined in Chapter 9). Next, we define the function on Qp =.Qp /2 given by 8 ˆ if .z/ D 0; 3

(see Lemma 2.1.3). We take x2 D 1, x3 D x4 D D 0. P Observation 2.5. Let ˆ.x/ D aj xj2 . If a1 D a2 in Qp =.Qp /2 , then ˆ.x/ has an isotropic vector. Proof. Indeed, .1; 1; 0; 0; : : : / is isotropic.

Observation 2.6. If Ran.a1 x12 Ca2 x22 Ca3 x32 / D Qp , then .a3 / 2 Ran.a1 x12 Ca2 x22 /. Proof. Choose a non-zero .z1 ; z2 ; z3 / such that a1 z12 C a2 z22 C a3 z32 D 0. If z3 D 0, then a1 z12 C a2 z22 D 0, and the range of this form is Qp . Otherwise, a1 .z1 =z3 /2 C a2 .z2 =z3 /2 D a3 . Observation 2.7. There exists a form on Qp3 whose range is not Qp .

436


Proof. Set ˆ.x/ D x12 C rx22 C px32 . Show that ˆ.x/ ¤ 0 for any non-zero x. Indeed, the norms of the first two summands are even and the norm of the last summand is odd. The only chance to obtain zero is jx12 j D jrx22 j > jpx32 j. But the range of the form x1 C rx22 over Fp does not contain 0. 2.6 Addition of cosets in Q=.Qp /2 Theorem 2.8. For p D 4k C 1,

´

if u D v; Qp ¯ hui C hvi D ® hui; hvi otherwise. For p D 4k C 3,

8 ˆ if u D rv; 5 variables has an isotropic vector. P Proof. Let jnD1 aj xj2 have no isotropic vectors. If p D 4k C 1, then all haj i must be pairwise distinct and hence n 6 4. If p D 4k C 3, then coincidences hai i D hraj i and hai i D haj i D hak i are forbidden. Hence n 6 4.


437

Problem 2.4. Find a (unique) form on Qp4 without isotropic vectors. 2.7 Classification of binary forms. Recall that the term a “binary form” means a form of two variables. Observation 2.10. a) Forms u.x12 x22 / are pairwise equivalent for all u. b) Forms ux12 C vx22 and vx12 C ux22 are equivalent. c) There are no other equivalences between binary forms. Proof. Let ˆ D a1 x12 C a2 x22 , ‰ D b1 x12 C b2 x22 be equivalent forms. Then Dis ˆ D Dis ‰, i.e., a1 a2 D b1 b2 in Qp =.Qp /2 . Equivalently, a1 =b1 D a2 =b2 . Thus ‰ D ˆ. It remains to look for a possibility of the equivalence ˆ ˆ. If ˆ D x12 x22 , we refer to Observation 2.4 (this is variant a). If D a2 =a1 , then we come to variant b). Evidently, in all the remaining situations Ran ˆ ¤ Ran ˆ because Ran “remembers” coefficients mod Qp2 of a form. 2.8 Invariants of binary forms Observation 2.11. „ is an invariant of binary forms. Proof. If ˆ.x/ D a.x12 x22 /, then „ D 1.

Lemma 2.12. Discriminant and „ constitute a complete system of invariants of binary forms. Proof. We refer to the proof of Observation 2.10. We must show that „ separates nonequivalent ˆ and ˆ. For definiteness, consider p D 4k C 1. We must watch that „ changes under the following transformations (which preserve discriminants but change classes of the equivalence): x12 C rx22 ! p.x12 C rx22 /; x12 C px22 ! r.x12 C px22 /; x12 C prx22 ! r.x12 C prx22 /: However, this is obvious.

2.9 Transformations of orthogonal bases. Let ˆ be a nondegenerate orthogonal form. Let U D fu1 ; : : : ; un g and V D fv1 ; : : : ; vn g be orthogonal bases. We say that they are adjacent if they have 6 2 different basis elements. Lemma 2.13. Fix a form ˆ. For any pair of ˆ-orthogonal bases, say U , V , there is a finite sequence W1 ; : : : ; WM of orthogonal bases such that W0 D U , WM D V and Wj C1 is adjacent with Wj for all j .

438


Proof of Lemma 2.13. It suffices to join the basis fu1 ; : : : ; un g with some basis of the form fv1 ; z2 ; : : : ; zn g. After that, we pass to the orthogonal complement v1? and come to the same problem P for bases fz2 ; : : : ; zn g and fv2 ; : : : ; vn g. Expand v1 D j D1 sj uj . Case 1. If only one coefficient sj is non-zero, then there is nothing to discuss. Case 2. Let precisely two coefficients, say s1 say s2 , be non-zero. In other words, v1 D s1 u1 C s2 u2 . Consider a vector h D t1 u1 C t2 u2 orthogonal to v1 . Then v1 , h, u3 , : : : , un is a desired basis. Case 3. Let m > 2 coefficients sj be non-zero. To be definite, assume s1 , s2 , s3 ¤ 0. Consider the projections of v1 on planes spanned by the vectors fu1 ; u2 g, fu1 ; u3 g, fu2 ; u3 g: r11 D s1 u1 C s2 u2 ;

r13 D s1 u1 C s3 u3 ;

r23 D s2 u2 C s3 u3 :

At least one of these vectors, say r11 , is not isotropic. Otherwise, we obtain a system of equations of the form a1 s12 C a2 s22 D 0;

a1 s12 C a3 s32 D 0;

a2 s22 C a3 s32 D 0;

which has only a zero solution. Thus r11 is non-isotropic. Let h be a vector in the plane fu1 ; u2 g orthogonal to r11 . Consider the new basis fr11 ; h; u3 ; : : : ; un g. Now X v1 D r11 C sj uj : j >3

Only .m 1/ coefficients of this sum are non-zero. If m 1 D 2, then we come to Case 2. If m 1 > 2, we repeat the considerations of Case 3. 2.10 The invariance of „. By Lemma 2.12, „ is an invariant of binary forms. By Lemma 2.13, it is an invariant in the general case. Indeed, „ is the same for adjacent bases. 2.11 Completeness of the system of invariants. Let ˆ and ‰ be quadratic forms over Qpn with Dis.ˆ/ D Dis.‰/; „.ˆ/ D „.‰/: If n D 2, then these forms are equivalent. Let n > 3. Consider the form ˆ ‰ on Qpn ˚ Qpn , ˆ ‰.x ˚ y/ D ˆ.x/ ‰.y/: By Corollary 2.9, this form has an isotropic vector. In other words, there are vectors x and y such that ˆ.x/ D ˆ.y/.


439

z be the restriction of ˆ to x ? and ‰ z be the a) Assume ˆ.x/ D ‰.y/ ¤ 0. Let ˆ ? restriction of ‰ to yN . Obviously, x D Dis.‰/; x Dis.ˆ/

x D „.‰/ x „.ˆ/

and we reduce the problem to .n 1/-dimensional spaces. b) Assume ˆ.x/ D ˆ.y/ D 0. Then Ran ˆ D Ran ‰ D Qp . We take nonisotropic x 0 , y 0 such that ˆ.x 0 / D ‰.y 0 / and repeat the same argument. This completes the proof of Theorem 2.3. 2.12 Classical groups. We define the following classical groups: – GL.n; Qp / is the group of all invertible matrices over Qp ; – Sp.2n; Qp / is the group of matrices preserving a nondegenerate skew symmetric bilinear form; – O.n; Qp I ˆ/ is the group of matrices preserving a nondegenerate quadratic form ˆ on an n-dimensional Qp -linear space2 . For n > 5 all orthogonal groups are noncompact 3 . In the real case, there are 10 series of classical groups. In the p-adic case there are a lot of classical groups, because we have the possibility to consider finite algebraic extensions K of Qp and also central division algebras L over K. Immediately there arises a huge number of groups GL.n; K/ and GL.n; L/ (as analogs of GL.n; R/, GL.n; C/, and GL.n; H/). There are also symplectic and orthogonal groups over commutative fields K. Below this zoo does not appear. There is another distinction between the real and the p-adic cases that is more important for us. In the real case, any classical group has a maximal compact subgroup, which is also a classical group. In the p-adic case, maximal compact subgroups have another nature. We define GL.n; Op / as the group of matrices g such that g and g 1 have integer elements. This group is compact, moreover GL.n; Op / is a maximal compact subgroup in GL.n; Qp /. Problem 2.5. g 2 GL.n; Op / if and only if g has integer elements and j det gj D 1. For j D 1, 2, …, we define the congruence subgroup GLj consisting of matrices of the form 1 C p j A, where A is a matrix with integer elements. Problem 2.6. a) GLj are normal subgroups in GL.n; Op /. b) GL.n; Op /=GL1 ' GL.n; Fp /. c) GLj =GLj C1 is the Abelian group ' Zpn . 2

2 For n > 4, there are precisely eight nonequivalent quadratic forms (16 for p D 2). However, forms ˆ and ˆ produce the same orthogonal groups. For instance, for odd n, we get only two different orthogonal groups. 3 Proof. By Corollary 2.9, the form has an isotropic vector. We represent ˆ as (2.6) and observe that the group of isometries contains hyperbolic rotations, x1 7! x1 , x2 7! 1 x2 .

440


Problem 2.7. Consider the subgroup H GL.2; Qp / consisting of matrices of the form a pb : p 1 c d The group H is compact and is not conjugate to the subgroup GL.2; Op /. 2.13 Digression. The case p D 2. Let a D 2k .1 C a1 2 C a2 22 C /. Define ´ 1 p .1 C .1/a1 i / if k is even; 2 .a/ D 12 a a p .1 C i /i 1 .1/ 2 if k is odd: 2

Our main Theorem 2.3 survives literally, but a proof is more tedious. 2.14 The Hilbert symbol. We want to link our considerations with the classical approach to the problem. Let a, b 2 Qp . The Hilbert symbol is ´ 1 if ax 2 C by 2 z 2 has an isotropic vector; .a; b/p WD 1 otherwise: Equivalently, .a; b/p D 1 if and only if the form ax 2 C by 2 “represents a square”, i.e., ax C by 2 D c 2 for some x, y. Theorem 2.8 describes ranges of all binary forms, therefore we know values of the Hilbert symbol for all a, b. This allows us to verify the following statements in a straightforward way. 2

Theorem 2.14. a) For p ¤ 2, .a; b/p D .1/.a/.b/.p1/=2 .a/.b/ .b/.a/ : b) For p D 2, decompose a D 2s m, b D 2s l. Write ".u/ WD .u 1/=2;

!.u/ WD .u2 1/=8:

Then .a; b/2 D .1/".u/".v/C!.u/".v/C!.v/".u/ : Corollary 2.15. The Hilbert symbol is bilinear in the following sense: .a1 a2 ; b/p D .a1 ; b/p .a2 ; b/p ;

.a; b1 b2 /p D .a; b1 /p .a; b2 /p :

(2.9)

Corollary 2.16. .a; b/p D

.a/.b/ : .ab/

(2.10)

2.15 Hasse–Minkowski invariant. Let ˆ be a nondegenerate quadratic form, let ˆ.x/ D P n 2 j D1 aj xj . The Hasse–Minkowski invariant cp .ˆ/ is given by cp .ˆ/ D

Y i<j

.ai ; aj /p :

(2.11)

441

10.3. Lattices Theorem 2.17. a) cp does not depend on a choice of orthogonal basis. b) Dis.ˆ/ and cp .ˆ/ is a complete system of invariants of a quadratic form.

The following lemma shows that the Hasse–Minkowski invariant is equivalent to the invariant „ defined above. In particular, Theorems 2.3 and 2.17 are equivalent. Lemma 2.18. cp .ˆ/ D

„.ˆ/ : Œdet.ˆ/

Proof of the lemma. We apply (2.10), „.ˆ/ D .a1 /.a2 /.a3 /.a4 / : : : D .a1 ; a2 /p .a1 a2 /.a3 /.a4 / : : : D .a1 ; a2 /p .a1 a2 ; a3 /p .a1 a2 a3 /.a4 / : : : D .a1 ; a2 /p .a1 a2 ; a3 /p : : : .a1 a2 : : : an1 ; an /p .a1 a2 : : : an /: Since the Hilbert symbol is bilinear (2.9), the last expression is cp .ˆ/.a1 : : : an /.

2.16 Digression. Hasse–Minkowski principle. Recall (without a proof) theP famous Hasse– Minkowski principle. Consider a nondegenerate real quadratic form ˆQ .x/ D cij xi xj with rational coefficients cij . We can regard this expression as a quadratic form ˆR on the linear n space Rn or quadratic forms ˆQp on the spaces Qp . Theorem 2.19. Let ˆQ and ‰Q be nondegenerate quadratic forms on Qn . These forms are Q-equivalent if and only if ˆR ‰R and ˆQp ‰Qp ;

for all p:

10.3 Lattices Here we start a discussion of p-adic analogs of matrix balls. In what follows the term “linear space” means a finite-dimensional linear space over Qp . The set Qp is totally disconnected. Hence linear spaces V ' Qpn over Qp are also L totally disconnected. For each basis ej 2 V the subset jnD1 Op ej is open and closed L in V . The subsets p k jnD1 Op ej form a fundamental system of neighborhoods of 0. 3.1 Op -Submodules. An Op -submodule M in a p-adic linear space V is a subset satisfying the conditions: – if v, w 2 M , then v C w 2 M ; – if v 2 M and 2 Op , then v 2 M .

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Theorem 3.1. Each Op -submodule admits a representation Qp f1 ˚ ˚ Qp fk ˚ Op g1 ˚ ˚ Op gl ; where f1 ; : : : ; fk , g1 ; : : : ; gl are linearly independent vectors. Problem 3.1. Each closed subgroup (D closed Z-submodule) in Rn has the form Zf1 ˚ ˚ Zfk ˚ Rg1 ˚ ˚ Rgl ; where f1 ; : : : ; fk , g1 ; : : : ; gl are linearly independent vectors. Proof of the theorem. Let L V be an Op -submodule. Without loss of generality, we can assume that the linear span of L is the whole space M . Otherwise, we pass to a subspace spanned by L. Next, consider a maximal linear subspace Y L. Let W V be a complementary subspace. Let M WD L \ W . Obviously, L D W ˚ M . Thus Theorem 3.1 is equivalent to the following special case: Theorem 3.2. Let the Op -submodule M in a linear space W satisfy the following conditions: – M does not contain lines; – the linear span of M is the whole W . L Then M has the form Op ej , where ej is a basis in W . Proof of Theorem 3.2. Consider the quotient module M=pM . The subring pOp Op acts on this module trivially. Therefore, the quotient ring Op =pOp D Fp acts on M=pM . In other words, M=pM is an Fp -linear space. Consider a basis e1 ; : : : ; en in the Fp -linear space M=pM and choose representatives e1 ; : : : ; en in M . L We are going to show that ej is a basis in W and M D Op ej (this implies the theorem). 1) The elements ej generate M as an Op -module. Any h 2 M can be represented as hD

X

aj.1/ ej C h1 ;

Next, p 1 h 2 M and we represent it as X .2/ aj e2 C h2 ; p 1 h1 D

where aj.1/ 2 Op , h1 2 pM :

where aj.1/ 2 Op , h2 2 pM :

Repeating the same arguments, we get X .m/ X .2/ X .1/ aj ej ChmC1 ; where hmC1 2 p mC1 M : aj ej C Cp m hD aj ej Cp

10.3. Lattices

443

Passing to the limit, we represent h as a sum of a convergent series: hD

k X 1 X

p m1 aj.m/ ej :

j D1 mD1

Thus, h is contained in the Op -span of ej . 2) The vectors ej generate the whole linear space W . Indeed, they generate M and M generates W . 3) The vectors ej form L a basis in W . Choose a basis 1 ; : : : ; k of W consisting of elements of M . Set Y WD Op j . We claim that for sufficiently large ˛ and ˇ, p ˛ Y M p ˇ Y: Indeed, we can express i via ej and ej via i . Therefore, the group M=pM is a subquotient of the Abelian p-group: p ˇ Y =p ˛C1 Y ' .Zp˛CˇC1 /dim W : Hence the number of generators of W =pW does not exceed dim W , therefore ej is a basis in W . Observation 3.3. A subset L in a linear space is an Op -submodule if and only if L is a closed subgroup. Proof. “H)” follows from Theorem 3.1. On the other hand, for each v lying in a subgroup L, the element nv is also contained in L for any integer n. But Z is dense in Op , hence the set fnvg is dense in Op v. Therefore, L is an Op -module. 3.2 Lattices. A subset L in a linear space V is said to be a lattice if it can be represented in the form Op f1 ˚ ˚ Op fn ; where fn is a basis in V : For instance Opn Qpn is a lattice. Obviously, lattices are open compact subgroups in the additive group of a linear space. The previous considerations imply the converse statement: Proposition 3.4. Any open compact subgroup in a linear space is a lattice. 3.3 Bases in lattices. Let V be an n-dimensional linear space over Qp . In the proof of Theorem 3.1 we have noted the following fact: Observation 3.5. For a lattice L, the quotient L=pL is an n-dimensional Fp -linear space. Theorem 3.6. Vectors f1 ; : : :; fn of a lattice n L generate L if and only if the images of the fj in the Fp -linear space Op =pOp ' Fpn constitute a basis. Proof. This was proved during the proof of Theorem 3.1.

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3.4 The space of lattices Theorem 3.7. a) The group of linear operators preserving the lattice Opn Qpn is the group GL.n; Op /. b) The set of all lattices in Qpn is the homogeneous space Lat n D GL.n; Qp /=GL.n; Op /: c) The space Latn is discrete countable. Proof. a) Denote by ej the standard basis in Qpn . Let g 2 GL.n; Qp / preserve Opn . Then gej 2 Opn , i.e., all matrix elements of g are integer. The same holds for g 1 . b) The homogeneity is a corollary of Theorem 3.1. c) The subgroup GL.n; Op / GL.n; Qp / is open, therefore points of Latn D GL.n; Qp /=GL.n; Op / are open, i.e., the quotient space is discrete. Since the group GL.n; Qp / is separable, the space Latn is separable; hence it is countable. In fact, the spaces Lat n are p-adic analogs of matrix balls. They are the main objects of our interest in this chapter. 3.5 Lattices and norms. A norm k k on a linear space V over Qp is a function taking values p j , where j 2 Z, and 0 and satisfying the conditions: kv C wk 6 max.kvk; kwk/;

kvk D jj kvk;

kvk ¤ 0 whenever v ¤ 0:

Problem 3.2. The set of all v with kvk 6 1 is a lattice. Example. The function kvk D maxj jvj j is a norm on Qpn . The corresponding lattice is Opn . More generally, for fixed a1 ; : : : ; an , kvk D max p aj jvj j j

is a norm. The corresponding lattice is ˚p aj Op . More generally, for any collection `1 ; : : : ; `j of linearly independent linear functionals, the following function is a norm: kvk D max j`j .v/j: j

(3.1)

Proposition 3.8. For any lattice M V , the function kvk D

min

kW p k M 3v

pk

is a norm. Proof. Without loss of generality, we can set M D Opn .

10.3. Lattices

445

Problem 3.3. Each norm has the form (3.1). 3.6 Pairs of lattices Theorem 3.9. Let M and L be two lattices in an n-dimensional space. Then there is a basis f1 ; : : : ; fn such that M M M D Op fj ; L D p ˛j Op fj : The collection ˛1 6 ˛2 6 6 ˛n

(3.2)

is uniquely determined by the lattices L and M .4 We call the collection of numbers ˛j D ˛j .L; M / complex distance. Proof. Without loss of generality, we can assume M L (otherwise, we consider a lattice p ˛ M with large ˛. Then M=L is a finite Abelian p-group with n generators. We represent M=L in the form M M=L D Zpmj : Clearly, the numbers mj are invariants of the pair .M; L/. Next, consider the group M M=pL ' Zpmj C1 and choose a standard system of its generators fj . Then p mj C1 fj D 0 and p mj fj generate the subgroup L=pL. Take representatives fjB of fj in M . By Theorem 3.6, fjB generate M and p mj fjB generate L. Another proof of Theorem 3.9. Consider the space Latn D GL.n; Qp /=GL.n; Op /. Put L at the reference point Opn . Now we can move M by elements of GL.n; Op /. We must describe orbits of GL.n; Op / on Latn . In other words (see Subsection 3.4.14, p. 182), our problem can be reformulated as follows: – Describe double cosets GL.n; Op / n GL.n; Qp /=GL.n; Op /. Let g 2 GL.n; Qp /. We can transform it as g 7! h1 gh2 , where h1 and h2 range in GL.n; Op /. In particular, we can do the following operations: a) Left and right multiplications by diagonal matrices whose entries are in Op npOp . Equivalently, we can multiply rows and columns of matrices by numbers 2 Op n pOp . 4

A basis fj is not canonical.

446


b) Left multiplication by matrices of the form5 1 C zEij , where i ¤ j and z ranges in Op . This is equivalent to the usual operation of addition of rows, for instance, 0 10 1 0 1 1 0 ::: g11 g21 : : : g21 ::: g11 Bz 1 : : :C Bg21 g22 : : :C Bg21 C zg11 g22 C zg21 : : :C @ A@ AD@ A: :: :: : : :: :: :: :: :: :: : : : : : : : : : c) Right multiplications by matrices of the same form. This corresponds to addition of columns. d)–e) Left and right multiplications by operators of permutation of the standard basis in Qp . This corresponds to permutations of rows or columns respectively. Now we imitate the “Gauss algorithm” for solving linear systems. Choose a matrix element gij having a maximal absolute value jgij j. Move it to the position f11g by operations d)–e). Annihilate all elements of the first column but g11 by the operation b). Annihilate all elements of the first row but g11 by the operation c). Make g11 D p l by the operation a). l We get a matrix p0 g0Q . Apply the same algorithm to the .n 1/ .n 1/ matrix g. Q Problem 3.4. Show that ˛j .M; L/ D ˛nj C1 .M; L/. Problem 3.5. Find the number of lattices M such that the complex distance between M and On is a prescribed collection (3.2); this is an analog of Theorems 7.6.5–7.6.6 about distributions of eigenvalues. 3.7 Minimax characterization of invariants of a pair of lattices. The following statement is an analog of the minimax characterization of the eigenvalues and angles, see § 2.5. Theorem 3.10. Let W range in the set of all j -dimensional subspaces in Qpn . Then p ˛j D p ˛nj C1 D

min

kwkL ; kwkM kwkL min : w2W kwkM max

W Wdim W Dj w2W

max

W Wdim W Dj

(3.3) (3.4)

Proof. Prove the first statement. In the notation of Theorem 3.9, consider the subspace Y WD Qp fj ˚ ˚ Qp fn . For each y 2 Y , we have kykL =kykM > p ˛j : Since W \ Y ¤ 0, we get max kwkL =kwkM > p ˛j : w2W

The minimal value p 5

˛j

of max.: : : / is achieved on the W B WD Qp f1 ˚ ˚Qp fj .

Here Eij is a matrix with a unique non-zero entry equal to 1 at the position ij .

10.4. Bruhat–Tits trees

447

3.8 Dual lattices. Let V be a linear space, let V B be the dual linear space, i.e., the space of linear functionals on V . For a lattice L V define the dual lattice L˙ V B consisting of linear functionals ` such that `.v/ 2 Op

for all v 2 L:

Problem 3.6. a) Show that L˙ is a lattice. b) Find the complex distance between L˙ and M ˙ .

10.4 Bruhat–Tits trees Matrix balls are rich differential geometric objects. On the other hand, the space of lattices is discrete, at first glance it can not be interesting from a geometric point of view. However it admits a strange geometry discovered by Bruhat and Tits. The present section is some kind of a psychological preparation for their construction6 . 4.1 Bruhat–Tits trees. Denote by Lat2 the space of all lattices in Qp2 defined up to the equivalence L L, where 2 Qp ( actually can be chosen in the form D p ˇ ). Evidently7 , Lat 2 D PGL.2; Qp /=PGL.2; Op /: We define a graph Jp . Its vertices are enumerated by points of Lat2 . Two vertices L and M are connected by an edge if, for some k, p k L M;

p k L=M ' Zp :

In this case, we also say that the vertices L and M are neighbors. Problem 4.1. a) This relation is symmetric. b) Two lattices are neighbors if the complex distance has the form .˛ C 1; ˛/. Fix a vertex (a lattice) L. Let M be its neighbor. We can assume that p 1 L M L and M=L ' Zp . The group p 1 L=L is the linear space Fp2 and M=L is a line in Fp 2 . We get the following statement: Observation 4.1. The set of neighbors of a given lattice is in a one-to-one correspondence with the projective line PFp1 over the field Fp . Theorem 4.2. The graph Jp is a tree. Each vertex of this tree is adjacent to .p C 1/ edges. Such trees are called Bruhat–Tits trees.

448


Figure 10.2. A piece of the tree J2 .

Since PGL.2; Qp / acts on Lat2 , it acts on the Bruhat–Tits tree Jp . 4.2 Another definition of Bruhat-Tits trees. Now we are going to construct the same tree again. Until the identification of the constructions, we denote a “new” tree by Ip . Vertices of Ip are enumerated by balls B.a; p k /. We connect two balls B.a; p k / and B.c; p k1 / with an edge if and only if c 2 B.a; p k /. It is completely evident that we get a tree (this was noted above in Subsection 1.5). 4.3 A rephrasing. Let us describe the second construction in a more intricate method. We say that a quasiball in Qp is a ball B.a; p k / or the complement Qp n B.a; p k / to a ball. It is reasonable to regard quasiballs as subsets in the p-adic projective line PQp1 Qp . The group PGL.2; Qp / acts on the projective line by projective transformations z 7! .a C zc/1 .b C zd /. Lemma 4.3. Projective transformations send quasiballs to quasiballs. Proof. The group of projective transformations is generated by affine maps z 7! bCzd and the “inversion” z 7! z 1 . Obviously, affine maps send quasiballs to quasiballs. Examine the inversion. Let a ball B contain 0, i.e., B D B.0; p k /. Then inversion sends it to the complement of the ball B.0; p k /. If 0 … B, we represent B as a.1 C p m Op /, where a ¤ 0 is a point of the ball, and m > 0. The image of B under the inversion is a1 .1 C p m Op /, i.e., it is a ball again. We paraphrase the definition of Ip in terms of quasiballs. 6 Actually, Bruhat–Tits trees are interesting and important per se, however we have no possibility to discuss them here. 7 We use a common notation. Let K be a field; the group PGL.n; K/ is the quotient group of GL.n; K/ with respect to the subgroup of scalar matrices.

10.5. Bruhat–Tits buildings

449

Vertices. Consider a ball B.a; p k /, i.e., a vertex of the tree Ip . Consider p subballs of B.cj ; p k1 / B.a; p k / of radius p k1 and also the complement Qp n B.a; p k /. Thus to each vertex of Ip we assign a partition of the PQp into a disjoint union of .p C 1/ quasiballs. Problem 4.2. Each partition of PQp into a disjoint union of .p C 1/ quasiballs has such a form. Edges. For each ball B consider the following two subsets of a p-adic line: B and Qp n B. Such partitions are in a one-to-one correspondence with edges of Ip . Thus we have described the tree Ip in terms of partitions of PQp1 into quasiballs. Observation 4.4. The group PGL.2; Qp / acts on Ip . Proof. We refer to Lemma 4.3.

4.4 Identification of two constructions. Fix a lattice M . Recall that M=pM ' Fp2 . Let ` 2 PQp1 be a line in Qp2 . The image of ` \ M in M=pM ' Fp2 is a line in 2 Fp . Thus we get a map ıM W PQp1 ! PFp1 . 1 ./ of points 2 PFp1 are quasiballs PQpn . Observation 4.5. Preimages ıM

Proof. Since our considerations are PGL.2; Qp /-invariant, it suffices to consider the lattice M D Op2 . Thus each lattice determines a partition of a projective line into a disjoint union of .p C 1/ quasiballs. The coincidence of the two constructions is observed. This also proves Theorem 4.2. 4.5 Digression. Absolute of a tree. We say that a ray on Jp is a sequence of vertices L1 , L2 , …such that Li and LiC1 are neighbors and Li ¤ Li C2 . We say that two rays fLi g, fMj g are equivalent if Li D MiC˛ for some ˛ starting with a certain moment i D i0 . The absolute is the space whose points are rays defined up to equivalence. 1 Problem 4.3. a) Convince yourself that the absolute of Jp is the p-adic projective line PQp (see Figure 10.1). 1 b) Devise a definition of a convergence of lattices 2 Lat 2 to a point of PQp .

10.5 Bruhat–Tits buildings Now consider an n-dimensional p-adic linear space and the set Lat n WD GL.n; Qp /=GL.n; Op / of lattices defined to within a dilatation, L L. We construct an .n 1/-dimensional simplicial complex whose vertices are points of Latn .

450


5.1 Construction of Bruhat–Tits buildings. Step 1. Link of a vertex. Fix a lattice M and consider all lattices N such that M N p 1 M: Such lattices are in a one-to-one correspondence with linear subspaces n WD N=M in Fpn ' p 1 M=M Consider a nonempty subset I W i1 < i2 < < ik in the set f1; 2; : : : ; n 1g. Consider a flag of subspaces n1 nk Fpn ;

dim n˛ D i˛ :

(5.1)

For such a flag we “draw” a k-dimensional simplex (a facet) M .n1 ; : : : ; nk /, whose vertices correspond to the lattices N1 ; : : : ; Nk , and M . To each subflag nj1 njˇ of (5.1) our rule assigns the facet M .nj1 ; : : : ; njˇ /. We regard it as a face of the simplex M .n1 ; : : : ; nk /. Thus, we get a finite simplicial complex and all the simplices have a common vertex, namely M . 5.2 Construction of Bruhat–Tits buildings. We repeat the same operation for each lattice M 2 Latn . As a result we obtain an infinite simplicial complex BT ŒPGL.n; Qp / called a Bruhat–Tits building. Maximal simplices (chambers) correspond to complete flags in Fpn . Remark. We can state the definition as follows. If M N p 1 M , then we connect M and N by an edge. If for some collection N1 ; : : : ; Nk of vertices all the edges ŒN˛ ; Nˇ exist, then we “draw” the corresponding .k 1/-dimensional simplex with given vertices. This definition is shorter, however starting from this definition, we must describe all simplices. Proposition 5.1. The definition of the building is self-consistent. Specifically, let ŒM; N be an edge. Then we can draw simplices containing ŒM; N referring to M or referring to N . In both cases, we obtain the same result. Proof. Consider a simplex drawn with reference to M , i.e., a flag of lattices M U1 U˛ N V1 Vˇ p 1 M: We multiply M and Uj by p 1 and get the new chain N V1 Vˇ p 1 M p 1 U1 p 1 U˛ : It is the same simplex. But it is “drawn” with reference to N .

451


By definition, the group GL.n; Qp / acts on the building preserving its simplicial structure. 5.3 Fine arts. We present several sketches to portray the building BT ŒPGL.3; Q2 /. It is assembled from equilateral8 triangles. A) Each edge is contained in three triangles, see Figure 10.3.a). B) Immediate neighbors of a given triangle are shown in Figure 10.3.b). C) Fix a basis e1 , e2 , e3 2 Q32 . Consider all lattices of the form 2l1 O2 e1 ˚ 2l2 O2 e2 ˚ 2l4 O2 e3 : Drawing edges we get the “sheet”9 shown in Figure 10.3.c).

a)

b)

c)

Figure 10.3. Sketches for building of PGL.3; Q2 /:

By definition, the whole building BT ŒPGL.3; Q2 / can be assembled by gluing of a countable collection of such sheets. An intersection of two sheets forms a convex hexagon on each sheet (this is not obvious). D) Let us draw the link of a vertex. There are seven lines and seven planes in F23 . Also, there are 21 flags. On Figure 10.4.a), the referring vertex is O. Lines in F23 are shown by black vertices, planes by gray vertices. Two-dimensional simplices are 14 lateral faces of the pyramid and seven “diagonal” sections through O and dotted segments in the base. The group PGL.3; F2 / of order .8 1/.8 2/.8 4/ D 168 acts on this picture. Only a Sylow 7-subgroup of PGL.3; F2 / is visually observable on the figure10 . The same combinatorial picture11 is shown in Figure 10.4.b). Now the referring vertex is in the center of the antiprism, the lines in F23 correspond to vertices on the upper base, the planes correspond to vertices on the lower base. E) Now the reader can try to imagine the whole building12 . It is necessary to keep in mind that the number of vertices in a ball of radius R has exponential growth. 8 In the moment the term “equilateral” makes no sense. However the group GL.3; Qp / acts on the set of edges transitively. For n > 3 this is not true. 9 An apartment, see below. 10 The combinatorial group of symmetries of this graph is even larger than PGL.3; F2 /, because PGL.3; F2 / preserves black and gray vertices; but they have the same rights on the figure. See Subsection 5.10. 11 It is more appropriate from the point of view of conveying an impression. 12 The author is unable to do this.

452


a)

b) Figure 10.4. Sketches for building of PGL.3; Q2 /:


453

Other attempts to draw buildings are contained in Garrett’s book [61]. 5.4 Iwahoric and parahoric subgroups. Here we define important subgroups in PGL.n; Qp / having no analogs in the real case. The standard Iwahori subgroup Iw.n; Qp / in PGL.n; Qp / is the group of invertible matrices of the form 1 0 z12 z13 : : : z11 Bpz21 z22 z23 : : :C C B (5.2) Bpz31 pz32 z33 : : :C ; where zij 2 Op , zi i 2 Op n pOp . A @ :: :: :: :: : : : : As above, denote by GL1 the congruence subgroup consisting of matrices 1 0 p 12 p 13 ::: 1 C p 11 B p 21 p 23 : : :C 1 C p 22 C B ; where ij 2 Op : B p 31 p 32 1 C p 33 : : :C A @ :: :: :: :: : : : : Observation 5.2. GL1 Iw.n/, and the quotient group Iw.n/=GL1 is the uppertriangular matrix group T .n; Fp / over Fp . Standard parahoric subgroups are intermediate subgroups H : PGL.n; Op / H Iw.n; Qp / (in particular, the group PGL.n; Op / itself and Iw.n; Qp / are parahoric subgroups). A standard parahoric subgroup consists of block .k1 C k2 C / .k1 C k2 C / matrices of the form 1 0 Z12 Z13 : : : Z11 BpZ21 Z22 Z23 : : :C C B BpZ31 pZ32 Z33 : : :C ; A @ :: :: :: :: : : : : where Zij are integer matrices, and Zi1 i are also integer. A subgroup in PGL.n; Qp / is called iwahoric (resp. parahoric) if it is conjugate to the standard Iwahori subgroup (resp., a standard parahoric) subgroup. Proposition 5.3. a) Fixed points Lj 2 Latn of an iwahoric subgroup form a flag of lattices L D L0 L1 Ln1 p 1 L; such that all inclusions are proper. b) The stabilizer of each flag of such type is an iwahoric subgroup.

454


Proof. a) Since all iwahoric subgroups are conjugate, it suffices to verify our statement for the standard Iwahori subgroup Iw.n/. Let K Qpn be a lattice fixed by Iw.n/. Since our lattices are defined to within dilatations, we can assume K p 1 On but K 6 On . Let u … Opn be an element of K. The GL1 -orbit of u consists of vectors u C pAu, where A is an arbitrary integer matrix. Therefore, this orbit is the set u C Opn and hence K Opn . Thus, Opn K p 1 Opn : Such lattices are enumerated by linear subspaces in Fpn . Since the group GL1 stabilizes all lattices of this form, we actually get an action of the quotient group Iw.n/=GL1 . The latter group is the upper triangular matrix group over Fp ; it stabilizes only the flag 0 Fp1 Fp2 of the coordinate subspaces. b) Indeed, all such flags are PGL.n; Qp /-equivalent. 5.5 Definition of buildings in terms of iwahoric subgroups. Now we present an alternative definition of the same simplicial complex. Vertices of the complex are enumerated by subgroups in PGL.n; Qp / of the form g 1 PGL.n; Op / g. Simplices H are enumerated by parahoric subgroups H , moreover H H 0 whenever H H 0 . In particular, .n1/-dimensional simplices (chambers) are enumerated by iwahoric subgroups. Lemma 5.4. The two definitions of the Bruhat–Tits building are equivalent.

Proof. This follows from Proposition 5.3.

5.6 Apartments. Fix a basis e1 ; : : : ; en 2 Qpn and consider the space of all lattices L k of the form p j Op ej . Consider all simplices having vertices at these lattices. We obtain a simplicial subcomplex „. Such subcomplexes are called apartments13 . Vertices of the apartment are enumerated by integer vectors .k1 ; : : : ; kn / modulo the equivalence relation .k1 ; : : : ; kn / .k1 C l; : : : ; kn C l/. It is convenient to realize the apartment in a hyperplane t1 C C tn D 0

(5.3)

in Rn . Projecting integer points .k1 ; : : : ; kn / on this hyperplane, .k1 ; : : : ; kn / 7! .k1 ; : : : ; kn /

1X kj .1; : : : ; 1/; n

we get vertices of the complex. The hyperplanes ti tj D n divide „ into equal simplices. 13

Apartments are analogs of flat subspaces in matrix balls, see Theorem 2.12.2.


455

Problem 5.1. a) Prove the last statement. b) Show that the picture is symmetric with respect to each plane ti tj D n. c) Find the dihedral angles of the simplices. For n D 2 the apartment is drawn in Figure 10.3 c). For n D 3 a chamber (a simplex) is shown in Figure 10.9 a). Remark. We can also realize the apartment as the quotient of Rn modulo the equivalence .1 ; : : : ; n / .1 C ; : : : ; n C /: (5.4) Again, the vertices correspond to the integer points and the hyperplanes of symmetries are i j D n. 5.7 The group acting on an apartment. Changing a basis ej 2 Qpn we (generally speaking) get another apartment. Observation 5.5. a) Apartments in the building are enumerated by non-ordered collections `1 D Qp e1 ,…,`n D Qp en of lines spanning the space Qpn . b) The stabilizer G.„/ PGL.n; Qp / of an apartment „ is a semidirect product Sn Ë .Qp /n , where Sn is the symmetric group. Let us discuss in more detail transformations of the apartment „ induced by its stabilizer G.„/. Problem 5.2. Denote by D G.„/ the group of diagonal matrices with entries in Op n pOp . Show that D acts trivially on the apartment. PGL.n; Qp / the group of matrices such that Denote by W a) each column and each row contains precisely one non-zero element; b) all non-zero elements have form p k . Obviously, our group is a semidirect product D Sn Ë Zn1 ; W where the Zn1 consists of diagonal matrices with elements p k , and the symmetric group Sn consists of matrices whose entries are 0 or 1. preserves the standard apartment. In the coordinates (5.4), Evidently, the group W the subgroup Zn1 acts by shifts .1 ; : : : ; n / 7! .1 ; : : : ; n / C .k1 ; : : : ; kn /: The group Sn acts by permutations of .1 ; : : : ; n /.

456


acts transitively on the set of chambers. Problem 5.3. a) The group W b) The stabilizer of any chamber is a cyclic group Zn (see Figure 10.3.c), Figure 10.9.a)14 ). contains all reflections with respect to “walls” i j D k. c) The group W 5.8 The Euclidean metric on a building Observation 5.6. An apartment has the natural structure of a Euclidean space. We endow an apartment „ with the Euclidean distance r X 0 d„ .t; t / WD .tj tj0 /2 : -invariant Euclidean metric on „. It is a unique up to a scalar factor W Lemma 5.7. Let „ and „0 be apartments, let a, b 2 „\„0 . Then there is an isometric simplicial map „ ! „0 fixing both points. Proof. Denote by A, B minimal facets containing a, b. Denote by ˛k and ˇj their vertices. Then d„ .˛k ; ˇj / D d„0 .˛k ; ˇj / because this value is expressed in terms of the complex distance on Latn . 5.9 Canonical forms of pairs of chambers Theorem 5.8. a) For any pair of chambers, there is an apartment containing both of them. b) Equivalently, for each pair C and D of chambers, there is an element g 2 PGL.n; Qp / such that gC and gD are contained in the standard apartment. This is the problem of a description of orbits of PGL.n; Qp / on the space of pairs of chambers, i.e., on PGL.n/=Iw.n/ PGL.n/=Iw.n/: We use a standard algebraic remark: – The set of orbits of a group G on G=H G=H is in a one-to-one correspondence with double cosets H n G=H . Thus it suffices to describe the double cosets Iw.n/ n PGL.n/=Iw.n/. This description is given in the following theorem, which also implies Theorem 5.8. Theorem 5.9. Each double coset in Iw.n/ n PGL.n; Qp /=Iw contains an element of . the group W 14

This corresponds to cyclic permutations of a complete flag of lattices.

10.6. Buildings related to symplectic groups

457

Proof. We wish to find a canonical form of g 2 GL.n; Qp / under the transformations g 7! h1 gh2 , where h1 , h2 2 Iw.n/. A similar problem for GL.n; Op / n GL.n; Qp /=GL.n; Op / was solved in Theorem 3.9 and we refer to its proof. Now we can apply the following operations: a) Left and right multiplications by diagonal matrices whose entries are in Op npOp . b) Left multiplication by matrices of the form15 1 C zEij , i ¤ j . If i > j , then arbitrary z 2 Op is allowed; if i < j , then z 2 pOp . c) Right multiplications by matrices of the same form. For a given g 2 GL.n/, consider all the matrix elements with maximal possible jgij j. Take an element gkl of this kind, for which k C l is maximal. The operation b) allows us to annihilate all matrix elements in the l-th row but gkl . Applying c) we annihilate all elements in the k-th column. Further, we forget about the k-th column and l-th row and repeat the same operation. As a result, we come to a matrix having a unique non-zero element in each column and each row. Multiplying it by a diagonal matrix, we obtain a desired representative of the given double coset. Problem 5.4 (Bruhat cell decomposition). a) Denote by T .n/ GL.n; C/ the group of upper triangular matrices (with arbitrary elements on the diagonal). Show that double cosets T .n/ n GL.n; C/=T .n/ are in a one-to-one correspondence with the group Sn . b) Denote by CŒŒ" the field of formal Laurent series in the variable ". Consider the group GL.n; CŒŒ"/ and its “Iwahori” subgroup Iw.n/ consisting of matrices of the form R C O."/, where R 2 T .n/ GL.n; C/. Show that Iw.n/ n yn GL.n; CŒŒ"/=Iw.n/ ' W 5.10 Automorphisms of buildings. a) The group of automorphisms of the Bruhat–Tits tree is greatly larger than PGL.n; Qp /. Indeed, by construction, the set Xv of edges entering a given vertex is a one-to-one correspondence with the projective line PFp1 . The group PGL.2; Qp / regards this structure (i.e., it induces projective maps Xv ! Xw ). This is a very rigid restriction on an automorphism. b) It can be shown that, for n > 2, the group PGL.n; Qp / has index 2 in the group of automorphisms of the building. To observe an additional automorphism, consider an arbitrary n nondegenerate bilinear form on Qp ; it identifies V with its dual space. We consider the map ˙ that assigns the dual lattice L to each lattice L.

10.6 Buildings related to symplectic groups 6.1 Self-dual modules. Consider a space V2n ' Qp2n endowed with a nondegenerate skew-symmetric bilinear form B.; /. Such a form identifies the V2n and the dual 15

Here Eij is a matrix with a unique non-zero entry equal to 1 at the position ij .

458


space; namely to each vector v we assign the linear functional `v .w/ D B.v; w/. For each Op -submodule L we define the dual submodule L˙ as the set of vectors w such that B.v; w/ 2 Op for all w 2 L: An Op -submodule Y V2n is called self-dual if Y D Y ˙ . Problem 6.1. a) For each self-dual submodule Y there exists a canonical basis16 e1 ; : : : ; e˛ , E˛C1 ; : : : ; En , f1 ; : : : ; f˛ , F˛C1 ; : : : ; Fn such that Y D

˛ M

n M

Qp Ej ˚ Op ej ˚ Op fj :

j D1

j D˛C1

b) For each self-dual module, there is a decomposition of V2n into an orthogonal direct sum V2n D W1 ˚ W2 such that Y D L ˚ R, where L W1 is an isotropic subspace, and R W2 is a self-dual lattice. 6.2 Pairs of self-dual lattices Theorem 6.1. a) Let R and S be self-dual lattices in V2n . Then there is a canonical basis ej , fj in V2n such that RD

n M

Op ej ˚ Op fj ;

SD

j D1

n M

p kj Op ej ˚ p kj Op fj :

j D1

b) The collection kj .R; S / is uniquely determined modulo permutations and transformations kj 7! ˙kj . Problem 6.2. Adapt the second proof of Theorem 3.9 for our case. We say that the collection k1 .R; S / > k2 .R; S / > > kn .R; S / > 0

(6.1)

is the complex distance between R and S . 6.3 Almost self-dual lattices. We say that a lattice L V2n is almost self-dual if L contains a self-dual lattice M and for each v, w 2 L, B.v; w/ 2 p 1 Op . Let us describe such objects more explicitly. Fix a self-dual lattice M . Consider the quotient p 1 M=M ' Fp2n : If v, w 2 p 1 M , then B.v; w/ 2 p 2 O; if also v 2 M , then B.v; w/ 2 p 1 O. 16

See (3.1.2).

10.6. Buildings related to symplectic groups

459

Observation 6.2. We get a well-defined form bM .v; w/ on p 1 M=M taking values in p 2 Op =p 1 Op ' Fp . Lemma 6.3. Let M be a self-dual lattice. A lattice L M is almost self-dual if and only if L p 1 M and L=M p 1 M=M ' Fp2n is bM -isotropic. The volume of a self-dual lattice is 1; possible volumes of almost self-dual lattices take values 1, p; : : : ; p n . 6.4 Symplectic buildings. Now, we wish to construct a simplicial Bruhat–Tits

complex BT Sp.2n; Qp / related to the group Sp.2n; Qp /. Vertices of the complex are enumerated by almost self-dual lattices. Two vertices L1 and L2 are connected by an edge if and only if L1 L2 or L2 L1 . Simplices (facets) correspond to flags Li1 Lk of almost self-dual lattices. Maximal simplices (chambers) are enumerated by maximal flags of almost self-dual lattices M D L0 L1 Ln ;

M is self-dual, vol.Lj / D p j :

(6.2)

6.5 The definition of buildings in terms of iwahoric subgroups. If a symplectic transformation g stabilizes an almost self-dual lattice L, then g also stabilizes the dual lattice L˙ . If g stabilizes a flag (6.2) of almost self-dual lattices, then g stabilizes also the following long flag of lattices: ˙ ˙ ˙ L˙ n Ln1 L1 L0 D M D L0 L1 Ln :

(6.3)

An iwahoric subgroup is a subgroup stabilizing a maximal flag (6.2) of almost self-dual lattices. Equivalently, it stabilizes the long flag (6.3). Consider a basis ej 2 V2n satisfying B.ei ; e2ni / D 1 for i 6 n; all other scalar products are 0. The standard Iwahori subgroup is the group of symplectic matrices g that have the structure (5.2) in our basis, i.e., X X gei D zik ek C ui i ei C p vil el ; where zik , vil , ui i , u1 i i 2 Op : k

l>i

A parahoric subgroup is the stabilizer of a noncomplete flag of almost self-dual matrices. Now we present the definition of the building in terms of iwahoric subgroups (as

in Subsection 5.5). Vertices of the complex BT Sp.2n; Qp / are enumerated by maximal parahoric subgroups (i.e., stabilizers of almost self-dual lattices). A collection of vertices is united into a simplex (a facet) if there is a parahoric subgroup that fixes all these vertices. Chambers are enumerated by iwahoric subgroups.

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Parahoric subgroups and flags of almost self-dual lattices are in one-to-one correspondence; therefore our new definition is only a rephrasing of the previous one. 6.6 Apartments. Fix a canonical basis ej , fj in V2n . Consider the set of all almost self-dual lattices of the form M M p kj Op fj p lj Op ej ˚ (actually, lj C kj is 0 or .1/). All simplices of the building with vertices at these points form a subcomplex called an apartment. Problem 6.3. Show that we obtain a Euclidean space equipped with the Coxeter tiling (for instance, see Subsection 8.1) of type Czn . It is reasonable to argue as follows. Consider admissible points .k1 ; : : : ; kn ; l1 ; : : : ; ln / 2 R2n and draw the corresponding simplicial n-dimensional surface in R2n . Project this surface orthogonally to the plane H W x1 C xnC1 D 0, x2 C xnC2 D 0, … in Rn ; see Figure 10.5. For n D 2 the picture is present in Figure 10.8.a. (we mark different types of almost self-dual lattices by different colors), see Figure 10.9.b for n D 3. x2 x1 x1 D x2

x1 D x2

Figure 10.5. Reference Problem 6.3, n D 1.

10.7 The Nazarov category The Nazarov category is the p-adic analog of the symplectic category Sp. It is also equivalent to the category of p-adic Gaussian operators discussed in the next chapter. 7.1 Nazarov category. An object of the Nazarov category Sp.Qp / is a p-adic linear space endowed with a nondegenerate skew-symmetric bilinear form. For any pair of objects Y and Z, we define the skew-symmetric bilinear form B .y ˚ z; y 0 ˚ z 0 / WD BY .y ˚ y 0 / BZ .z; z 0 /: Morphisms Y ! Z are B -self-dual Op -submodules P Y ˚ Z such that the submodules ker P WD P \ Y and indef P WD P \ Z are compact: The product of morphisms is a product of relations.

(7.1)

10.7. The Nazarov category

461

Theorem 7.1. a) The product of morphisms is a morphism. b) The group of automorphisms of V2n is Sp.2n; Qp /. c) For any morphism P , ker P D .dom P /˙ ;

indef P D .im P /˙ :

(7.2)

Theorem 7.2. Let P W Y Z be a morphism. For a self-dual submodule L, the submodule PL is self-dual. 7.2 A preliminary lemma. Let R be a submodule in some V2k . Assume R R˙ . By definition, if v 2 R˙ and w 2 R, then B.v; w/ 2 Op . Hence we can regard B as a Qp =Op -valued form B on R=R˙ , B .x1 C x2 ; y/ D B .x1 ; y/ C B .x2 ; y/; B .x; y/ D B .y; x/; B .x; y/ D B .x; y/; 2 Op n pOp : Lemma 7.3. Let L be a self-dual Op -module in a p-adic linear space. Let R be an Op -submodule such that R˙ R. Then .L \ R/=.L \ R˙ / is a B -self-dual submodule of R=R˙ . Proof. We must show that .L \ R/ C R˙ is self-dual in the usual sense. Since R˙ R, .L \ R/ C R˙ D .L C R˙ / \ R: Next,

˙ .L C R˙ / \ R D .L C R˙ /˙ C R˙ D .L \ R/ C R˙

and this is equivalent to self-duality.

7.3 Proof of Theorem 7.2. The statement (7.2) is similar to Lemma 2.9.2 and we omit a proof. Next, P dom P ˚ im P; P ker P ˚ indef P: Denote by Pz the image of P in the quotient group dom P = ker P ˚ im P = indef P . Lemma 7.4. a) The groups dom P = ker P and im P = indef P are isomorphic. Moreover, the group Pz is a graph of an isomorphism, say P . b) The isomorphism P is an isometry of forms B . This statement is obvious. Proof of the theorem. Let L Y be an Op -submodule, P W Y Z a morphism of the Nazarov category. The Op -submodule PL can be obtained by the following chain of operations:

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1 . We project L \ dom P to the quotient group dom P = ker P . Denote by L1 the result. By Lemma 7.3, L1 is B -self-dual. 2 . We push forward L1 by the map P . Denote by L2 Im P = indef P the result. By Lemma 7.4, L2 is B -self-dual. 3 . Consider the preimage L3 of L2 under the map Im P = indef P . Obviously, L3 Z is self-dual. However, L3 D PL. 7.4 Proof of Theorem 7.1. The only non-obvious statement is a). It is a special case of Theorem 7.2. Indeed, let R W X Y and T W Y Z be morphisms. Let R be the pseudoinverse relation17 . Consider the relation R ˚ T W Y ˚ Y X ˚ Z (compare with Subsection 2.9.4). Clearly, R ˚ T is a morphism of Sp.Qp /. Take L Y ˚ Y as the graph of the identity operator. Then the relation TR X ˚ Z is a result of application of the linear relation P WD R ˚ T to the self-dual submodule L. It remains to apply Theorem 7.2 7.5 Action of Nazarov category on buildings. Compressivity Proposition 7.5. Let P W Y Z be a morphism of Sp.Qp /, L Y be an almost self-dual lattice. Then the lattice PL Z is almost self-dual. Proof. Let M L be a self-dual lattice, p 1 M L. Then L=M ' .Zp /j . Following the operations 1 –3 in the proof of Theorem 7.2, we obtain that PL=PM ' .Zp /k , where k 6 j . In particular, p 1 PM PL. The lattice PM is self-dual by Theorem 7.2. Further, let y, y 0 2 L, y ˚ z, 0 y ˚ z 0 2 P . Then B.y; y 0 / B.z; z 0 / 2 O; Hence PL is almost self-dual.

B.y; y 0 / 2 p 1 O H) B.z; z 0 / 2 p 1 O:

Therefore, a morphism of the Nazarov category sends vertices of buildings to vertices of buildings. Let B1 and B2 be simplicial complexes, let Vert.B1 /, Vert.B2 / be sets of their vertices. Consider a map W Vert.B1 / ! Vert.B2 /. We say that is simplicial if for a collection of vertices x1 ; : : : ; xk of any simplex of Vert.B1 /, their images .x1 /; : : : ; .xk / are vertices of some simplex in B2 (we allow .xi / D .xj /). We say that is strictly simplicial if .x1 /; : : : ; .xk / are pairwise distinct. A simplicial map admits a canonical extension to a map B1 ! B2 that is affine on each simplex of B1 . 17

I.e., the same subspace in Y ˚ X.

10.7. The Nazarov category

463

Proposition 7.6. Let P W Y Z be a morphism of the Nazarov category. Let L range in the set of almost self-dual lattices. Then the map L 7! PL is a simplicial map of the corresponding buildings. Proof. If L M , then PL PM , therefore P takes flags to flags.

Problem 7.1. a) Let L, M Y be self-dual lattices, let P W Y Z be a morphism. Then complex distance (see (6.1)) satisfies kj .L; M / > kj .PL; PM / for all j : 7.6 Canonical forms. We want to classify morphisms P W Y Z of the category Sp.Qp / modulo the equivalence P hP g;

h 2 Sp.X /; g 2 Sp.Y /

(this is used in the proof of Theorem 11.2.10). We say that a morphism P W Y Z is decomposable if there are decompositions Y D Y .1/ ˚ Y .2/ , Z D Z .1/ ˚ Z .2/ into orthogonal sums and morphisms P1 W Y .1/ ! Z .1/ , P2 W Z .2/ ! Z .2/ such that P D P1 ˚ P2 . Theorem 7.7. There are only the following five types of indecomposable morphisms (modulo equivalence): 1 : Y D 0, Z D V2 , and L Z is a self-dual lattice in the two-dimensional space V2 (we can set P D Op e ˚ Op f , where e, f is a canonical basis). 2 . Y D V2 , Z D 0, and L Y is a self-dual lattice. 3 . Y D V2 , Z D V2 and L is the graph of the identical operator. 4 . Y D V2 , Z D V2 ; let e, f be a canonical basis in Y and e 0 , f 0 be a canonical basis in Z; let k > 0. We take P WD Op .e C e 0 / ˚ p k Op .f C f 0 / ˚ p k Op e 0 ˚ Op f:

(7.3)

5 . Y and Z are the same, P WD Op .e C e 0 / ˚ Qp .f C f 0 / ˚ Op f:

(7.4)

We omit a proof, see [139]. Remark. a) We can regard the canonical form 5 as a form 4 with k D 1. On the other hand, the form 4 with k D 0 is a sum of 1 and 2 . b) The canonical form 4 admits the following transparent description. We consider a basis e, f 2 Y such that B.e; f / D p k and similar basis e 0 , f 0 2 Z. In this notation P WD Op e ˚ Op e 0 ˚ Op f ˚ Op f 0 C p k Op .e C e 0 / C Op .f C f 0 / p k Op e ˚ Op e 0 ˚ Op f ˚ Op f 0 :

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The corresponding map dom P = ker P ! im P = indef P (see Lemma 7.4) is the identical map .Zpk /2 ! .Zpk /2 . 7.7 Relation with the symplectic category over Fp Problem 7.2. Equip each object V2k of the category Sp.Qp / with a basis ej , fj satisfying B.fi ; fj / D B.ei ; ej / D 0; B.ei ; fj / D p 1 ıij : Consider the subcategory Sp.Fp / of Sp.Qp / consisting of morphisms P satisfying the conditions: ker P .Op /2n ;

dom P p 1 .Op /2n ;

indef P .Op /2n ;

im P p 1 .Op /2n :

Then Sp.Fp / is the category of symplectic linear relations over field Fp (see § 9.3).

10.8 Buildings. General comments This section is not used below, but it provides unexpected links between different topics of this book. ˛. Coxeter groups 8.1 Coxeter groups. Consider a simplex C in a k-dimensional Euclidean space. Reflect C with respect to all .k 1/-dimensional faces. Repeat the same operation for all “new” simplices, etc. The simplex C is called a Euclidean Coxeter simplex if this procedure produces a tiling of the Euclidean space by non-overlapping simplices. A Coxeter group is a group generated by reflections with respect to all faces of a Coxeter simplex. Problem 8.1. A simplex is Coxeter if and only if all its dihedral angles ˛j have form =mj , where mj are integer. Problem 8.2. A simplex is determined by its dihedral angles uniquely to within a similarity. In the same way, we define spherical Coxeter simplices; now we consider the sphere S n1 instead of Rn . Equivalently, we can regard spherical simplices as simplicial cones in Rn . 8.2 The classification of Coxeter simplices. See, for instance, N. Bourbaki [27]). The list of spherical simplices is given in Figure 10.6, the list of Euclidean simplices is in Figure 10.7. Notation for Figure 10.7. For each Coxeter simplex C , we draw a graph. Its vertices are enumerated by faces of C ; we connect two vertices if the angle between them is not 90B . We draw one segment if the angle is 60B , two segments if the angle is 45B , and three segments for an angle D 30B ; we draw a serrated segment and write a number k if the angle is =k with

465

10.8. Buildings. General comments

An :

BCn :

Dn :

E7 :

E6 : E8 : F4 : H3 :

G2 :

5

H4 :

5

I2 :

m

Figure 10.6. Spherical Coxeter simplices.

AQn :

Czn :

Bzn :

zn: D

Ez6 :

Ez7 :

Ez8 : Fz4 :

z2 : G Figure 10.7. List of Euclidean Coxeter simplices.

k ¤ 2; 3; 4; 6. The subscript of A, B; : : : ; I coincides with the dimension n mentioned above. Finally, A, B, … is a generally accepted notation. 8.3 Coxeter groups. Each Coxeter simplex generates a Coxeter tiling of the Euclidean space or the sphere by simplices. Reflections with respect to faces of the simplex generate a Coxeter group of isometries of the sphere or the Euclidean space. In the spherical case, this group is finite. In the Euclidean case Rn , the group consists of affine isometries of Euclidean spaces. Linear parts of such isometries range in a finite (spherical) Coxeter group having the same notation in our list without . Also, this group contains a subgroup ' Zn consisting of translations. z3 are shown in FigExample. Tilings of R2 corresponding to the Coxeter groups AQ3 , Cz3 , G ure 10.3.c and 10.8. Three-dimensional Coxeter simplices are in Figure 10.9.

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z2 . Figure 10.8. Coxeter tilings of the plane of types Cz2 and G

a)

b)

Figure 10.9. Fundamental simplices for the Euclidean Coxeter groups of types AQ3 (a)), Cz3 (a small simplex in b)), Bz3 (two-pieces simplex in b)). The 4-pieces simplex in b) is the simplex of type AQ3 .

Problem 8.3. Draw all Coxeter tilings of the 2-dimensional sphere. Example 1. For GL.n; Qp /, an apartment of the building is the Euclidean space Rn1 endowed n defined in Subsection 5.7 contains the with the Coxeter tiling of type AQn1 . The group W 18 Q Coxeter group An1 as a subgroup of index n. Example 2. For Sp.2n; Qp / an apartment is Rn equipped with a Coxeter tiling of type Czn . ˇ. Axioms and basic properties of buildings 8.4 Axioms of buildings. Let us regard a Euclidean space or a sphere with a Coxeter tiling as a simplicial complex. Let us call such objects Coxeter complexes. We call simplices of the tiling chambers. A facet is a face (of dimension 0,1, …, n) of a chamber. Fix a Coxeter group Gn . An abstract building is a simplicial complex B such that: B0. B is a union of Coxeter complexes; Coxeter subcomplexes of B are called apartments. B1. For any two chambers there is an apartment containing both of them. B2. Let „ and „0 be apartments containing facets A and B. Then there is an isomorphism „ ! „0 of simplicial complexes fixing A and B pointwise. 18

n contains a transformation preserving a simplex. Because W


467

According to the two types of Coxeter groups, we obtain the two types of buildings, namely spherical buildings and Euclidean buildings. The axiom B2 can be replaced by: B20 . Let apartments „ and „0 contain a chamber A and a facet B. Then there is an isomorphism „ ! „0 fixing A and B pointwise. Lemma 8.1. Axioms B0, B1, B20 imply B2. Proof. Take a chamber of „ containing A and a chamber D of „0 containing B. There is an apartment „00 containing C and D. We consider isometries „ ! „00 ! „0 . Proposition 8.2. The building BT ŒPGL.n; Qp / constructed above is a building in the sense of the new definition, i.e., it satisfies the axioms B0, B1 and B2 of an abstract building. Proof. Axiom B1 is Theorem 5.8. Let us prove that B20 . Euclidean distances between vertices of an apartment are determined by the complex distance. Therefore, pairwise distances between vertices of A and B are the same in „ and „0 . Thus A [ B „ and A [ B „0 are isometric figures in the Euclidean spaces „ and 0 „ . Therefore, there is an isometry „ ! „0 fixing A [ B. Since dim A D n, this isometry is unique. It is more-or-less evident that this isometry is a simplicial map. A Euclidean building can be constructed for an arbitrary classical group over a field of discrete valuation (our example BT ŒSp.2n; Qp is well-representative). For n > 8 all Euclidean buildings have such a form. For n 6 8 there are also buildings related to exceptional semisimple groups over fields of discrete valuations; for n 6 3 there are a lot of exotic constructions; for n D 2 there are also buildings related to reflection groups on the Lobachevski plane. 8.5 Some properties of Euclidean buildings. For a given apartment „ and a chamber C „, we construct the standard retraction19 '„;C W B ! „. For x 2 B, consider an apartment „0 containing x and C , consider also the unique isometric map x W „0 ! „ fixing the chamber C . We set '.x/ WD x .x/. Thus we obtain a strictly simplicial map B ! „ in the sense of Subsection 7.5. „0 x C .x/

„

Figure 10.10. Retraction to an apartment. Axiom B2 provides that the image of a chamber is independent of the choice of „0 . We stress the following fact: 19 Let X be a topological space, let Y X be a closed subspace. A retraction X ! Y is a continuous map that fixes X pointwise.

468


Observation 8.3. The preimage of the chamber C under '„;C is the chamber C itself. Since any two points x, y 2 B are contained in one apartment, we can measure the distance d.x; y/ as a distance in an apartment. By virtue of Axiom B2, the value d.x; y/ does not depend on „. Theorem 8.4. A Euclidean building is a metric space. Theorem 8.5. Intersection of two apartments „ [ „0 is a convex set. The last theorem implies the following corollary: Corollary 8.6. For any pair of points there is a unique segment connecting these points. Proposition 8.7. Standard retractions diminish the distance, i.e., d.'.x/; '.y// 6 d.x; y/: Proofs. 1) Contractivity. Consider two points x, y 2 B, an apartment containing them, and the segment connecting x and y in the apartment. The image of Œx; y under any standard retraction is a polygonal path whose pieces have the same lengths as pieces of Œx; y in the Coxeter tiling of †, see Figure 10.11. This proves contractivity.

Figure 10.11. A segment and its (possible) image under a retraction. We have a “ray” that can intersect a mirror or can be reflected from a mirror. 2) A building is a metric space. Indeed, let x, y, z be points of a building. Consider a chamber C containing x and apartments „, „0 connecting x with y and z. Consider an apartment † containing x, y. Consider the retraction '†;C . The segment Œx; y remains fixed. The image of Œx; z is a segment and hence, d.x; y/ C d.y; z/ > d.x; y/ C d.y; '.z// > d.x; '.z// D d.x; z/: 3) Convexity. Let x, y lie in an intersection „ \ „0 of two apartments. Consider the corresponding segments Œx; y„ and Œx; y„0 . By Axiom B2, their lengths are equal. Consider the first moment when Œx; y„0 leaves „. Let C „ be a chamber visited by Œx; y„ in the next moment. By Observation 8.3, the image of Œx; y„0 under '„;C differs from Œx; y„ . Thus the retraction '„;C strictly diminishes the distance between points x, y; and this contradiction completes the proofs.

Problem 8.4. For the building BT PGL.3; Qp / consider the apartments associated with bases e1 , e2 , e3 and e1 C p 10 e2 , e2 , e3 . Find their intersection.

469

10.8. Buildings. General comments 8.6 Non-positive curvature. We continue the discussion of Euclidean buildings.

Theorem 8.8. Let x, y, z 2 B, let 0 < t < 1. Denote by u t the point dividing the segment Œx; y in the ratio .1 t / W t . Then d 2 .z; u t /2 6 .1 t /d 2 .z; x/ C t d 2 .z; y/ .1 t /t d 2 .x; y/:

(8.1)

Remark. If x, y and z are points in Euclidean space, then we have D in the last row. Thus the theorem claims that the segment Œx; u t is shorter than in Euclidean geometry, see Figure 10.12. Inequality (8.1) is more pleasant for t D 1=2: 4d 2 .z; u t /2 6 2d 2 .z; x/ C 2d 2 .z; y/ d 2 .x; y/: C0

C

A

D

(8.2)

A0

B

D0

B0

Figure 10.12. Reference the non-positive curvature property. If AB D A0 B 0 , AC D A0 C 0 , BC D B 0 C 0 , AD=DB D A0 D 0 =D 0 B 0 , then C 0 D 0 6 CD.

Proof. Consider an apartment including Œx; y and a chamber C including u t . Consider the corresponding retraction. It preserves the distances xy, xu t , yu t , u t z and diminishes xz, yz. The points '.x/ D x, '.y/ D y, '.z/ are vertices of a Euclidean triangle, '.u t / D u t . Theorem 8.9. Euclidean buildings are contractible spaces. Proof. Indeed, fix a point a 2 B. For x 2 B and 0 6 t 6 1, we define the point t .x/ by t .x/ 2 Œa; x;

d.a; t .x// D t d.a; x/:

Thus we obtain a family of maps t W B ! B, 1 .x/ D x, 0 .x/ D a. It remains to look for the continuity of maps t . See Figure 10.13. The segment Œy; t .x/ is shorter than the Euclidean geometry forecasts. Therefore, the Œ t .x/; t .x/ is also shorter than the corresponding segment in the Euclidean geometry.

t .y/

y

a t .x/

x

Figure 10.13. Reference continuity of the homotopy.

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Theorem 8.10. A Euclidean building is a complete metric space. Sketch of proof. Let D be a simplex of the Coxeter tiling of Rn . All chambers of a building B are canonically isometric to D; this defines a canonical continuous map W B ! D. Obviously, this map diminishes distances (since the image of a segment is a billiard trajectory in C , see Figure 10.14). It is easy to show that the preimage of a point 2 D is a discrete subset in B. Consider a fundamental sequence xj 2 B. Then .xj / converges to some y 2 D. For each j choose a point yj such that: – .yj / D y; – yj , xj lie in one chamber. Then yj is a fundamental sequence (in a discrete set), therefore yj is eventually constant, hence it has a limit. But d.xj ; yj / ! 0 and hence the limit of xj is the same.

Figure 10.14. Reference proof of Theorem 8.10. The image of a segment under the map W B ! C. 8.7 Fixed point theorem. Let Q be a subset of B. We say that a is a circumcenter of Q if Q is contained in a closed ball of radius r with center at a and is not contained in any ball of a smaller radius < r. Lemma 8.11. Each compact subset of a Euclidean building has a unique circumcenter. Proof. The existence is a standard general topological exercise. Let a and b be two circumcenters (therefore, Q is contained in two balls of the same radius, say r). Let c be the midpoint of the segment Œa; b. By virtue of the inequality (8.2), we observe that Q is contained in a ball with a center at c and radius < r. Theorem 8.12 (Bruhat–Tits fixed-point theorem). A compact group acting on a building has a fixed point. Proof. Consider an arbitrary orbit of the group. Its circumcenter is a fixed point.

Let be a fixed point of a compact group K. Then K sends the minimal facet containing to itself. Observation 8.13. A classical p-adic group can have non-conjugate maximal compact subgroups.


471

For instance, the group PGL.n; Op / is a maximal compact subgroup in PGL.n; Qp /. However the stabilizer of a fixed chamber is also a maximal compact subgroup (an element stabilizing a chamber can move vertices of the chamber preserving their cyclic order). 8.8 Complex distance in Euclidean buildings and the triangle inequality. First, consider a . Put the point of origin 0 at a vertex space Rn equipped with an action of a Coxeter group W of the tiling. Denote by W the stabilizer of the vertex 0; recall that W is a finite Coxeter group. Mirrors of reflections r 2 H separate Rn into simplicial cones. Fix one of such cones, say C . For x 2 Rn , we define the complex distance d.0; x/ 2 C being the point of C lying in the orbit W x (see Figure 10.15). We define D.x; y/ WD D.0; y x/. D.0; x/ x

C

Figure 10.15. Reference the definition of complex distance.

Observation 8.14. The complex distance is invariant with respect to the Coxeter group. For points x, y of the building B, consider an apartment connecting x, y and denote by D.x; y/ the distance between x, y in the sense of the apartment. Clearly, D.x; y/ does not depend on the choice of apartment. Observation 8.15. D.x; y/ is invariant under the group of automorphisms of the building. As in § 2.12, we can formulate the triangle inequality problem. n it is equivalent to the following problem: For the space of lattices in Qp

– Let A be a finite Abelian p-group, B A a subgroup, C WD A=B the quotient group, M M M A' p mj ; B ' p kj ; C ' p lj : Find the domain of the possible values of triples fmj g, fkj g, flj g. Indeed, for a triple of lattices L, M , N 2 Latn , we can consider another triple L0 WD p ˛ L, M WD p ˇ M , N 0 WD N such that p ˛ L p ˇ M N (evidently, the problem does not change) and the quotient groups 0

A WD N 0 =L0 I

B WD M 0 =L0 ;

C WD N 0 =M 0 D A=B:

This problem for Abelian groups was solved in the 1960s (J. A. Green, T. Klein), for an exposition, see I. G. Macdonald, [129].

472


The triangle inequality in other buildings also is known; it is interesting that it is the same as in Riemannian symmetric spaces (and it depends only on the Coxeter group), see [101]. 8.9 Digression. Problems of harmonic analysis. Harmonic analysis on buildings has mysterious parallels with harmonic analysis on Riemannian symmetric spaces. This analogy, discovered by Macdonald [128], 1972, was an important initial point for further development of the theory of multi-variant special functions. We cite several problems in the harmonic analysis on buildings. 1) Decompose the space l2 on the set of vertices with respect to action of the group G (this is the subject of Macdonald’s book). 2) Decompose the space l2 on the set of simplices of a given type. As far as I know, this problem is not solved. 3) There is an analog of Berezin kernels (see 2.12.6 and Chapter 7) for the space of lattices GL.n; Qp /=GL.n; Op / (here it is reasonable to consider GL instead of PGL). Fix ˛ > 0. Consider a Hilbert space H˛ with a system of vectors eL enumerated by self-dual lattices; let their inner products be20 heL ; eM i D

vol.L \ M /2 ˛=2 vol L vol M

:

(8.3)

Theorem 8.16. A Hilbert space H˛ exists if and only if ˛ ranges in a set of the form ˛ D 0; 1; : : : ; n 1;

or ˛ > n 1:

n n Example. Consider the space L2 .Qp /. For each lattice M Qp denote by eM the function 1=2 that is equal to .vol M / on M and 0 outside M . Then the vectors eL satisfy the condition (8.3) with ˛ D 1. By Theorem 7.1.15, Hilbert spaces H˛ exist for all positive integers ˛.

The problem is – to decompose H˛ with respect to action of the group GL.n; Qp /. In this case, the problem is solved; however, for other series of groups a solution is unknown. . Real symmetric spaces and the Tits metric 8.10 Classical groups and spherical buildings. Now we wish to construct a spherical building of type An1 . Consider a field k and a linear space k n . Vertices of the building are enumerated by proper subspaces in k n . A collection of vertices assembles a simplex if these subspaces are elements of a flag. Thus a simplicial complex is constructed and it remains to present an apartment. Fix a basis e1 ; : : : ; en ; consider all proper subspaces spanned by collections ej1 ; : : : ; ejk of basis vectors; consider all simplices having such vertices. Problem 8.5. Verify axioms of buildings. 20

In fact, this is a special case of a Hilbert space determined by a reproducing kernel.


473

For k D Fp we already have seen this structure when we described a link of a vertex in the building BT .PGL.n; Qp //. For k D R, we will observe this structure in the following two subsections. 8.11 Tilings of symmetric spaces by Weyl chambers. Let Pn D SL.n; R/=SO.n; R/ be the space of real positive definite matrices with the determinant D 1. The invariant Riemannian metric on this space is given by the formula ds 2 D tr.T 1 d T T 1 d T /: We define the standard Cartan subspace X as the space of diagonal matrices 1 0 t1 0 : : : C B X.t / D @ 0 t2 : : :A ; where tj 2 R: :: : : :: : : :

(8.4)

This subspace is flat with respect to a Riemannian metric (see proof of Theorem 2.12.2). Cartan subspaces in P are images of the standard Cartan subspaces with respect to isometries of Pn . Denote by E the set of all Cartan subspaces containing the point 1 2 P . The group SO.n/ fixes 1 and hence it acts on E. The following remarks are obvious: Observation 8.17. The set E is in a one-to-one correspondence with the set of n-plets of pairwise orthogonal lines in Rn . Observation 8.18. If a symmetric matrix has pairwise distinct eigenvalues, then it is contained in a unique Cartan subspace. Observation 8.19. For any Y 2 E, the intersection Y \X X is contained in some hyperplane ti D tj . Problem 8.6. Moreover, Y \ X X is of the form ti1 D D ti˛ . The hyperplanes ti D tj separate the space X ' Rn1 into nŠ pieces, which are called Weyl chambers. Theorem 8.20. There is a canonical one-to-one correspondence between the set of all Weyl chambers in all Cartan subspaces Y 2 E and the set of all complete flags 0 V1 Vn1 Rn ;

dim Vj D j:

Proof. Consider a Cartan subspace Y and a Weyl chamber in it. Consider an interior point T 2 . It has pairwise distinct eigenvalues t1 > t2 > > tn . Let e1 ; : : : ; en be the corresponding eigenbasis, it is uniquely defined to within transformations ej 7! ˙ej . We set Vj D Ce1 ˚ ˚ Cej :

Thus we get a tiling of SL.n; R/=O.n/ by Weyl chambers. We discuss this tiling in more detail. Consider an intersection † of several hyperplanes ti D tj in a Cartan subspace Y (let us call such intersections facets). Consider a generic element T of this intersection. Denote by s1 > s2 > > s

474


pairwise distinct eigenvalues of T . Let E1 ; : : : ; Ek be the corresponding eigenspaces. We construct an incomplete flag 0 E1 E1 ˚ E2 E1 ˚ E2 ˚ E3 : Thus we obtain a stronger version of Theorem 8.20. Theorem 8.21. a) There is a canonical one-to-one correspondence ³ ² ³ ² the set of all facets the set of all : ! in all Cartan subspaces flags in Rn b) The intersection of two facets corresponds to the intersection of two flags (i.e., we select subspaces contained in both the flags). Thus we have represented our space SL.n; R/=O.n; R/ as a complex assembled from simplicial cones (facets). Observation 8.22. Intersecting this picture with a sphere centered at the origin, we get a spherical building drawn on the sphere. 8.12 Tits metric at infinity. Consider a Riemannian manifold M of non-positive curvature (this property is strongly used in further considerations). Denote by †.1/ its sphere at infinity. Recall this concept. Fix a point a 2 M . The sphere †a .1/ is in a one-to-one correspondence with the set of all oriented geodesics (parameterized by the lengths) starting at a (more formally, †a .1/ is the set of geodesics). For a point x 2 M denote by Œa ! x a unique geodesic connecting a and x. We say that a sequence xj 2 M converges to 2 †a .1/ if dist.a; xj / tends to 1 and Œa ! x tends to . Theorem 8.23. For any a, b 2 M there is a canonical bijection †a .1/ ' †b .1/. Precisely, we identify geodesic rays Œa ! x and Œb ! y if the distance between them at infinity is finite. Denote by †a .R/ the sphere of radius R with center at a. Consider geodesics 1 , 2 starting at a and coming to 1 , 2 2 †.1/. Denote by 1 ŒR, 2 ŒR the intersections of these geodesics with a sphere †a .R/. The Tits metrics at †.1/ is defined as dist. 1 ; 2 / D lim

R!1

1 1 ŒR; 2 ŒR : R

(8.5)

Example. a) If M D R2 is the Euclidean plane, then we get the standard metric on the circle embedded in the plane: the distance between two points is 2 sin '2 . b) If M is the Lobachevsky plane, then the distance between any two distinct points is 2. I.e., we get a discrete metric space. Now, let M D SL.n; R/=SO.n/. We present the final result without proof. Theorem 8.24. The Tits metric dist. ; / on SL.n; R/=SO.n/ is the intrinsic metric of the spherical building constructed in the previous subsection.

475


1 .R/

1

x

1 .R/

1

2 .R/

2

x 2 .R/

2

Figure 10.16. The Tits metric. Difference between the Euclidean and the Lobachevsky plane. In the second case the distance between x and the segment Œ1 .R/; 2 .R/ remains bounded as R ! 1. Hence the length of the segment Œ1 .R/; 2 .R/ is ' 2R for large R.

Remark. In particular, a distant geodesics connecting 1 .R/ and 2 .R/ is visible from a as a polygonal curve on the sphere whose segments are located in Weyl chambers. Remark. We stress that the Tits metrics is nonseparable. 8.13 Tits metric for Bruhat–Tits buildings. The previous construction can be adapted to Bruhat–Tits buildings. Fix a vertex a of a Euclidean building. Denote by †a .R/ the sphere of center a and radius R. For R0 > R there is a natural projection †a .R0 / ! †a .R/; namely, for x 2 †a .R0 /, we take the intersection of Œa; x with †a .R/. We define the sphere †.1/ at infinity as the inverse limit of the spheres †a .R/ and define the Tits metric on †.1/ by the same formula (8.5). Theorem 8.25. For the Euclidean building BT ŒPGL.n; Qp / we get the spherical building of field Qp .

11 Weil representation over a p-adic field

Here we show that the category of Gaussian operators over a p-adic field is equivalent to the Nazarov category defined above. We also discuss briefly the adelic version of our construction. At the end we discuss a funny “integral” operator connecting functions of the real and of the p-adic variables.

11.1 Gaussian integrals over a p-adic field For simplicity, we assume p ¤ 2. We are interested in integrals of the form Z ˚

exp 2 i.zAz t C bz t / dz;

(1.1)

rCL

where z 2 Qpn is a row-matrix, A is a symmetric n n-matrix, b 2 Qn is a row-matrix, L is an Op -submodule in Qpn , and r 2 Qpn is a vector. Applying the shift z 7! z r, we can reduce such integrals to the case r D 0. Also, if A is invertible, then the shift z 7! z C 12 bA1 allows us to “kill” b (but we can not remove both b and r). Recall that the exponential expf2 izg is constant on each a C Op , therefore our integrand is a locally constant function. Hence our integrals are sums; if the submodule L is compact, then the sum is finite. 1.1 One-dimensional Gaussian integrals over compact sets. Each one-dimensional integral of the type (1.1) can be reduced to one of the following Propositions 1.1, 1.3, 1.4, 1.5. Proposition 1.1.

´

Z expf2 i bzg dz D Op

1 0

if jbj 6 1; if jbj > 1.

(1.2)

Note that the first variant holds if and only if the integrand is a constant function on Op . Proof. If jbj 6 1, then the integrand D 1. Let b D p , jj D 1, > 0. We divide the ball jzj 6 1 into subballs k Cp Op of radii p , where k ranges in f0; 1; : : : ; p 1g. The integrand is constant on each

11.1. Gaussian integrals over a p-adic field

subball. Thus we come to the sum X

p

06k6p 1

²

exp 2 i

477

³

k : p

Since jj D 1, the map k 7! k induces a bijection of the group Zp to itself; thus our sum is the sum of all roots of unity of power p , i.e., 0. Corollary 1.2. Functions expf2 i bzg, where b ranges in Qp =Op , form an orthonormal basis in L2 .Op /. Problem 1.1. Show that this orthogonal system is complete. Remark. Certainly, the last corollary (and hence also Proposition 1.1) is a special case of the following statement: characters of a compact Abelian group A constitute a basis in L2 .A/. Proposition 1.3. Let the ball C p Op not contain 0 (i.e., jj > p ). Then ´ Z

˚ p exp¹2 i a 2 º if jaj 6 p ; 2 exp 2 i az dz D 0 otherwise. z2Cp Op Note that the first variant holds if and only if the integrand is constant. Proof. After a shift we get Z p Op

˚ exp 2 i a. C z/2 dz:

Dividing the ball p Op into p subballs kp C p C1 Op , we get D

p1 XZ kD0

p C1 Op

˚

exp 2 i a. C kp C v/2 dv:

(1.3)

Write out the expressions in curly brackets, f: : : g D 2 i a 2 C 2akp C akp kp C 2av C av 2 C 2akp v : (1.4) The first term in the brackets is constant. Since jj > jp j > jvj, the term 2akp has the maximal norm among nonconstant terms. Consider three cases. Case 1. Let jap j 6 1. Then all summands in (1.4), except a 2 , are integer. Therefore, expf: : : g D expf2 i a 2 g (for all k and v), i.e., the integrand in the whole integral is constant. Case 2. If jap j D p, then ˚

˚ exp 2a. C kp C1 C v/2 D exp 2a. 2 C 2kp /

478

Chapter 11. Weil representation over a p-adic field

(because other summands in(1.4) are integer). Hence the sum (1.3) is ³

²

X

p1 ˚ k exp 2 i.ap C1 / : exp 2 i a 2 p kD0

Since jap C1 j D 1, we get a sum of roots of unity, i.e., 0. Case 3. If jap j D p s > p 2 , then we divide our ball into p s1 smaller balls. In each subball Case 2 is realized. Proposition 1.4. Let jaj D p k . Then 8 ˆ if k 6 0; ˆ Z 2. We carry out the same transformation, Z D jzj61

p1 X

Z

mD0 mCOp

Z D

: Op

We leave only one term, because in the other terms the integrands are non-constant and hence the integrals are equal to 0 by Proposition 1.3. After the substitution z D up we come to Z expf2 i ap 2 u2 g du: p 1 juj61

We get an integral of the same type but a is changed to ap 2 . Thus we come again to one of the cases 1, 2 or 3 (with smaller k).


479

1.2 Gaussian integral over Qp Proposition 1.5. Z ² ³ b2 2 1=2 expf2 i.az C bz/g dz D .a/jaj exp 2 i : 4a Qp

(1.5)

Here the (divergent) integral is defined as Z expf2 i.az 2 C bz/g dz lim

(1.6)

and .a/ was defined above, (10.2.4). In particular Z expf2 i az 2 g dz D .a/jaj1=2 :

(1.7)

N !1 jzj6p N

Qp

Proof. RFirst, let b D 0. Proposition 1.4 provides us with an explicit expression for cn WD jzj6pN and shows that this sequence is eventually constant. If b ¤ 0, then we consider the shift z 7! z b=2a. Problem 1.2. Why is a shift of the arguments in the regularized integral (1.6) admissible? 1.3 Gaussian integrals over Qpn Theorem 1.6. Let A be a nondegenerate symmetric matrix. Then Z ˚

˚

exp 2 i.zAz t C bz t / dz D „.A/j det.A/j1=2 exp 2 i 14 bAb t ;

(1.8)

where „ is the invariant (10.2.5) of a quadratic form. In particular, Z ˚

exp 2 izAz t dz D „.A/j det.A/j1=2 :

(1.9)

n Qp

n Qp

Proof. Without loss of generality we can assume b D 0, otherwise we consider a shift z 7! z 12 bA1 . Next, assume that for some A the equality (1.9) holds. Substituting z D uR we get Z ˚

exp 2 izRARt z t dz D j det.R/j1 „.A/j det.A/j1=2 D „.A/j det.RARt /j1=2 ; i.e., the identity holds for RARt . Therefore, it suffices to prove the identity for a diagonal matrix A. In this case, our integral is a product of one-dimensional integrals evaluated in the previous theorem.

480


Remark. This theorem explains the invariant „./ of quadratic forms, see Theorem 10.2.3. 1.4 Analogy with R. Let A be a real nondegenerate symmetric matrix with p positive and q negative eigenvalues. Define Z Z ² ² ³ ³ i 1 I.A/ D x.iA C "/x t dx: exp xAx t dx WD lim "!C0 Rn 2 2 Rn Observation 1.7. I.A/ D e i.pq/=2 j det Aj1=2 : In this sense, e i.qp/=2 is an analog of the modified Hasse–Minkowski invariant „. Proof. Let A D hRht , where h is an orthogonal matrix and R is diagonal with elements rk . Then .2/n=2 I.A/ D lim det." C iA/1=2 D lim det." C ihRht /1=2 "!C0 "!C0 Y Y 1=2 ." C i rk / D lim D e isgn.rk /=2 rk1=2 "!C0

D e i.pq/=2 j det Aj1=2 :

1.5 One two-dimensional Gaussian integral. The following integral concludes the list of the simplest Gaussian integrals. Proposition 1.8. Z Qp

´ ˚

exp¹2 i bcº if jbj 6 p ; exp 2 i.xy C bx C cy/ dy dx D 0 otherwise: p Op

Z

Proof. We write the iterated integral Z Z expf2 i bxg Qp

p Op

˚

exp 2 i.x C c/y/ dy dx

and integrate in y with (1.2); the result is non-zero if and only if the integrand is constant, i.e., x C c 2 p Op . We come to Z Z p expf2 i bxg dx D p expf2 i bcg expf2 i bxg dx: cCp Op

It remains to apply (1.2) again.

p Op


481

1.6 Quadratic forms and lattices. Recall that p ¤ 2. We are going to present a classification of p-adic quadratic forms under the action of the group GL.n; Op /. First, reformulate the problem. Let V be a linear space over Qp , B be a symmetric ˇ bilinear form on V , L be a lattice. For a subspace W V , denote by B ˇW the restriction of B to W . We say that a triple .V; B; L/ is decomposable if there is a decomposition V D V1 ˚ V2 such that ˇ ˇ B D B ˇV ˚ B ˇV ; L D .L \ V1 / ˚ .L \ V2 /: 1

2

Theorem 1.9. The only indecomposable triples are one-dimensional triples. In other words, each quadratic form can be diagonalized by an operator 2 GL.n; Op /. A more precise version of the theorem is the following. Theorem 1.10. Fix r 2 Qp satisfying jrj D 1, .r/ D 1. Each nondegenerate quadratic form on Qpn is GL.n; Op /-equivalent to a unique form of the type p m1 .x12 C C x˛21 1 C "1 x˛21 / C p m2 .x˛21 C1 C C x˛22 1 C "2 x˛22 / C ; (1.10) where m1 > m2 > : : : and "1 , "2 ,…take values 1 or r. The statement is not difficult, see the book by J. Cassels [35]. 1.7 Quadratic forms and Op -submodules. Now, let M V D Qpn be a Op submodule such that Qp M D V . Let B be a symmetric bilinear form on V (it can be degenerate). Again, consider indecomposable triples .V; B; M /. Theorem 1.11. The only indecomposable triples are one-dimensional triples and the following two-dimensional objects: V D Qp e1 ˚ Qp e2 ; M D Qp e1 ˚ p m Op e2 ; B.xe1 ˚ ye2 ; x 0 e1 ˚ y 0 e2 / D xy 0 C yx 0 : The author does not know an appropriate reference, however the proof is straightforward and need not be carried out here. 1.8 General multivariate Gaussian integrals. Consider an arbitrary integral of the form Z ˚

exp 2 i.zAz t C bz t / dz; (1.11) rCL

where L V is an Op -submodule. Without loss of generality, we can assume that r D 0. Also, we can assume that Qp L ' V . The matrix A determines a (perhaps degenerate) quadratic form on V . We decompose the module L into a direct sum of one-dimensional and two-dimensional pairwise orthogonal submodules as it was done in Theorem 1.11 Now the integral reduces to a

482


product of one-dimensional and two-dimensional Gaussian integrals that were evaluated above. Thus, ideologically, our problem is simple. I do not know a nice presentation of the final result.

11.2 Weil representation, p-adic case We prefer to think that p ¤ 2. However, the main theorems and their proofs survive for p D 2. 2.1 Test-functions and distributions. Let V ' Qpn be a p-adic linear space. For a subset S V denote by IS .z/ the indicator function of S , ´ 1; z 2 S; IS .z/ D 0; z 62 S: Fix a point a 2 Qpn and k 2 Z. Consider a “ball” B.a; p k /: z W jzj aj j 6 p k : A test function on V is a finite linear combination of indicator functions of such balls. The Bruhat space B.V / is the space of all test functions. Problem 2.1. f 2 B.V / if and only if f is a locally constant compactly supported function.1 Let M L V be lattices. Denote by B.LjM / the space of functions supported by L and invariant under shifts z 7! z C h, where h 2 M . Obviously, dim B.LjM / D order of the group L=M . Observation 2.1. B.V / D

[

B.LjM /:

M LV

A p-adic distribution is a linear functional on the space B.V /. By B 0 .V / we denote the space of all distributions on V . In what follows we write distributions as integrals. Let '.x/ be a distribution on V , .y/ be a distribution on W . We define their tensor product '.x/ .y/ as a distribution on V ˚ W given by ²Z ³ Z Z f .x; y/ WD '.x/ .y/f .x; y/ : f 7! x˚y2V ˚W

x2V

y2W

1 A function f is called locally constant if for each z there is an open neighborhood U of z such that f .z/ D const in U .

11.2. Weil representation, p-adic case

483

We denote such a tensor product by '.x/ .y/. 2.2 The kernel theorem. For a distribution (kernel) K.x; y/ on V ˚ W , consider the bilinear form S on B.V / B.W / given by Z K.x; y/f .x/g.y/: S.f; g/ WD .x;y/2V ˚W

For a fixed f 2 B.W /, we define a linear functional g 7! B.f; g/. In other words, we get a map B.W / 7! B 0 .V /; we denote it by Z Ag.x/ D K.x; y/g.y/: (2.1) y2W

The following analog of the kernel theorem is trivial. Theorem 2.2. Each operator B.W / ! B 0 .V / is of the form (2.1). 2.3 Fourier transform. For a function f on Qpn , we define its Fourier transform by Z F f .x/ D f .y/ expf2 ixy t g dy: n Qp

Theorem 2.3. a) If f 2 B.Qpn /, then F f 2 B.Qpn /. b) For any f1 , f2 2 B.Qpn /, the following identity holds: Z Z f1 .x/f2 .x/ dx D .F f1 /.x/F f2 .x/ dx (the Plancherel formula): (2.2) n Qp

n Qp

c) The Fourier transform B.Qpn / ! B.Qpn / can be extended by continuity to a unitary operator L2 .Qpn / ! L2 .Qpn /. Proof. To simplify the notation, assume n D 1. Fix an integer k. Consider all balls of the form jx aj 6 p k , enumerated by elements of the quotient-group Qp =p k Op . Their indicator functions Ijxaj6pk are pairwise orthogonal. Further, Z F Ijzaj6pk .x/ D expf2 ixzg dz jzaj6p k Z D expf2 i axg expf2 i uxg du juj6p k Z D expf2 i axg expf2 ivxp k g p k dv jvj61

Dp

k

expf2 i axgIjzj6pk .x/

(here we substitute u D z a, v D up k ) and apply (1.2). The right-hand side is an element of B.Qp / and this proves a).

484


b) The identity (2.2) for f1 D Ijza1 j6pk ;

f2 D Ijza2 j6pk ;

reduces to the integral (1.2). P Given arbitrary f1 , f2 2 B.Qp /, we represent them as linear combinations of cj Ijzaj j6pN with sufficiently large N . It remains to refer to the previous paragraph.

c) follows from b).

2.4 The Weil representation of the group Sp.2 n; Qp /. We replicate the considerations of § 1.2 and follow the notation of that section. The p-adic Heisenberg group Heisn .Qp / is the group of .1 C n C 1/ .1 C n C 1/-matrices 0 1 vC R vC ˚ v I ˛ D R vI ˛ D @0 1 0 0

1 t ˛ C 12 vC v A v 1

with coefficients in Qp ; here v˙ 2 Qpn and ˛ 2 Qp . This group acts on L2 .Qn / by the unitary transformations

˚

t t C ˛ C 12 vC v / : U vC ˚ v I ˛ D f .x C vC / exp 2 i.xv For each g 2 Sp.2n; R/, we define the automorphism of Heisn .Qp / by

g W vI ˛ 7! vgI ˛ : Theorem 2.4. a) For each g 2 Sp.2n; Qp / there is a unique up to a scalar factor bounded invertible operator We.g/ W L2 .R/ ! L2 .R/ such that

U vgI ˛ D We.g/1 U vI ˛ We.g/

(2.3)

for each vI ˛ 2 Heisn .Qp /. b) Operator We.g/ is unitary up to a scalar factor. c) Operators We.g/ satisfy We.g1 / We.g2 / D c.g1 ; g2 / We.g1 g2 /;

where c.g1 ; g2 / 2 C . d) The operators We.g/ preserve the subspace B.Qpn / L2 .Qpn /.

(2.4)


485

The proof of Theorem 1.2.5 survives in our case literally; we simply write analogs of formulas (2.15)–(2.17) (p. 18): A 0 We (2.5) f .x/ D f .xA1 /j det Aj1=2 ; 0 At1 ˚

1 B We f .x/ D f .x/ exp ixBx t ; (2.6) 0 1 Z 0 1 We f .y/ expf2 ixy t g dy: (2.7) f .x/ D 1 0 n R Problem 2.2. a) Let p ¤ 2. For each g 2 Sp.2n; Op /, We.g/IOpn D IOpn ;

2 C :

There are (at least) two methods of proof, to use the definition (2.3) or to examine the action of generators of Sp.2n; Op /, see (1.2.14). b) For each g 2 Sp.2n; Op /, we can normalize We.g/ by the assumption D 1. Automatically, the factor c.g1 ; g2 / in (2.4) is 1, i.e., we obtain a linear representation of Sp.2n; Op /. Is it correct for p D 2? 2.5 Gaussian distributions. A Gaussian distribution on a p-adic linear space X D Qpn is a linear functional on B.X / given by Z expf2 ixAx t g'.x/ dx; ' 7! L

where L is an Op -submodule in X. Proposition 2.5. The Fourier transform of a Gaussian distribution is a Gaussian distribution Proof. Let be a Gaussian distribution. By virtue of Theorem 1.11, we can decompose into a tensor product of one-dimensional and two-dimensional distributions. Now we must verify the statement in one-dimensional cases and in the indecomposable two-dimensional case. The corresponding Gaussian integrals were evaluated in Propositions 1.1–1.5, 1.8. Straightforward calculations are rather simple. Only, for the function (2.8) f .x/ D expf2 i ax 2 gIjxj61 ; where jaj > 1, a final result is not immediately visible. We note that the Fourier transform sends products to convolutions, i.e., we get the function Z 1 O f .y/ D .a/jaj expf2 i a1 x 2 =4g dx: (2.9) yCOp

We have ja1 j < 1, therefore, for y D 0 we get and get a Gaussian supported by jyj 6 jaj.

R

D 1. Next, we refer to Proposition 1.3

486


2.6 Stabilizers of a Gaussian distribution in the Heisenberg group. Consider the space V2n WD Qpn ˚ Qpn equipped with the standard symplectic form. Consider the corresponding Heisenberg group. For a Gaussian distribution .x/ denote by P ./ the set of all v 2 V such that U ŒhI 0 D : Theorem 2.6. a) P ./ is a self-dual Op -submodule. b) The map 7! P ./ is a bijection between Gaussian distributions defined to within a scalar factor and self-dual Op -submodules in V2n . Proof. a) It suffices to verify the statement for indecomposable distributions (see Theorem 1.11), i.e., for the one-dimensional distributions 1 .x/ D ı.x/; 3 .x/ D expf2 i ax g; 2

2 .x/ D 1; 4 .x/ D expf2 i ax 2 gIjzj6pk .x/

and the two-dimensional distribution 5 .x1 ; x2 / D expf2 ixygIjzj6pk .x/: In all cases the result is sufficiently obvious. Problem 2.3. Examine the case 4 . For the proof of b), we need the following lemma: Lemma 2.7. Operators of the Weil representation send Gaussian distributions to Gaussian distributions Proof of the lemma. We must verify the statement for the generators (2.5)–(2.7) of the symplectic group. In the first two cases, the statement is obvious; the Fourier transform was examined in Proposition 2.5. Proof of Theorem 2.6.b. Now let L V2n be a self-dual module. We reduce it to a canonical form; after this the statement becomes obvious. We omit trivial details. Problem 2.4. Let be a Gaussian distribution. Show that 2 B.Qpn / () submodule V ./ is compact () 2 L2 .Qpn /: 2.7 The Weil representation of the Nazarov category. Now, let V2n ' Qp2n and V2m ' Qp2m be p-adic symplectic spaces. Consider their direct sum V2m ˚ V2n and equip it with the difference of the symplectic forms. Let L be a self-dual module in V2m ˚ V2n . Let L be the corresponding Gaussian distribution on V2m ˚ V2n . Define a Gaussian operator We.L/ W B.Qpn / 7! B 0 .Qpm /


by

487

Z We.L/f .x/ D

n x2Qp

.x; y/f .x/:

Theorem 2.8. The operator We.L/ satisfies the identity

U hI 0 We.L/ D We.L/U rI 0

(2.10)

for all r ˚ h 2 L. Moreover, We.L/ is determined by this identity to within a scalar factor.

Proof. This is a reformulation of Theorem 2.6.

Proposition 2.9. Let M V2n be a compact self-dual submodule. Let we.M / 2 B.Qpn / be the corresponding Gaussian distribution. Then We.L/ we.M / D .L; M / we.LM /;

where .L; M / 2 C :

We leave this statement as an exercise, see Theorem 1.9.6. Theorem 2.10. a) An operator We.L/ is a bounded operator L2 .Qpn / ! L2 .Qpm / if and only if the modules ker.L/ D L \ V2n ;

indef.L/ D L \ V2m

are compact. In other words,in this case the self-dual module L is a morphism of the Nazarov category. b) Under the same conditions We.L/ is an operator B.Qpn / ! B.Qpm /. c) For morphisms L W V2n V2m , M W V2m V2k , we have We.M / We.L/ D .M; L/ We.LM /;

where .L; M / 2 C .

Proof. a) Let We.L/ be bounded. Consider a self-dual submodule K V2n . By Proposition 2.9, We.L/ we.K/ D we.LK/. If indef.L/ is non-compact, then, by Problem 2.4, we.LK/ … L2 .Qpn /. Applying the same reasoning to the adjoint operator We.L/ , we obtain that ker L is compact. Thus L is a morphism of the Nazarov category. Conversely, let L be a morphism of the Nazarov category. We can reduce it to the canonical form given in Theorem 10.7.7 by transformations L 7! gLh;

where h 2 Sp.2n; Qp /, g 2 Sp.2m; Qp /:

Observation 2.11. For morphisms L1 W X1 Y1 and L2 W X2 Y2 of the Nazarov category, consider their direct sum L1 ˚ L2 W X1 ˚ X2 Y1 ˚ Y2 . We have We.L1 ˚ L2 / D We.L1 / ˝ We.L2 /:

488


Let us continue the proof of Theorem 2.10. Thus we must watch the boundedness for the canonical morphisms 1 –5 from Theorem 10.7.7. For the cases 1 –3 , there is nothing to discuss. For the cases 4 –5 , we get respectively Z f .y/ dy; Af .x/ D Ijxj61 .x/ xCp k Op

Z Bf .x/ D

f .y/ dy: jyxj61

In both cases the boundedness is obvious. This completes the proof of a). The same examination of canonical forms proves b). The statement c) follows automatically from the defining equation (2.10), see the proof of Theorem 9.4.3. 2.8 Central extension Theorem 2.12. The operators of the Weil representation of Sp.2n; Qp / admit a normalization such that We.g1 / We.g2 / D ˙ We.g1 g2 /: See [214], [122]. On the other hand, for an arbitrary morphism L W V2n V2m , we can normalize the operators We.L/ by hWe.L/IOpn ; IOpm i D 1: Then in the formula We.L/ We.M / D .L; M / We.LM /, the factor .L; M / is canonically defined. This cocycle was investigated in [139].

11.3 Adeles 3.1 Adeles. An adele is a sequence a D .a1 ; a2 ; a3 ; a5 ; a7 ; a11 ; : : : /; where a1 2 R and ap 2 Qp for p prime, and jap jp 6 1 for all but a finite number of p’s. We prefer to omit the term a1 . Our basic object is the space A of finite adeles (below we refer to them as adeles). A (finite) adele is a sequence a D .a2 ; a3 ; a5 ; a7 ; a11 ; : : : /; where ap 2 Qp , and jap jp 6 1 for all but a finite number of p’s.

11.3. Adeles

489

The space A is a commutative ring with respect to the coordinate-wise addition and multiplication. We define also the ring of integer adeles: Y OA WD Op A: p is prime

3.2 Adelic convergence. A sequence a.j / 2 A converges to a if 1) there is a finite set S of primes such that jap.j / jp 6 1 for all the p … S ; 2) for each p, the sequence ap.j / converges to ap . A neighborhood of a point a D fap g 2 A consists of all b 2 A such that jap bp jp 6 p , where j.p/ D 0 for all but a finite number of primes p. j.p/

Remark. For the ring OA of integer adeles we get the usual Tikhonov producttopology. We also obtain the product-topology on each subring Y Y Qp Op A; where the set S is finite: p2S

p…S

However, the topology on the whole space A is not a product-topology. 3.3 The image of Q in A. Each rational number r 2 Q can be regarded as an element of Qp . Therefore, we get the diagonal embedding Q 7! A, r 7! r WD .r; r; r; : : : /: Proposition 3.1. The map r 7! r induces an isomorphism of the quotient groups Q=Z ' A=OA. Proof. Each rational number r admits a unique representation as a finite sum r DnC

X

mp ; Np p p is prime

where n and mp are integer and 0 6 mp < p Np :

The image of r in Qp =Op is p Np mp , because all the other summands are p-adic integers. Therefore, we get the isomorphism M Q=Z ! Zp1 D A=Op A; p

see (1.1). Proposition 3.2. a) The image of Q is dense in A. b) The image of Z is dense in OA.

490


Proof. b) We want to approximate an element a D .a2 ; a3 ; : : : / 2 OA by a sequence n.j / 2 Z. Consider n.1/ 2 Z such that ja2 n.1/ j2 6 1=2. Next, consider integer ˛, ˇ 2 Z such that j˛ a2 j2 6 1=4; jˇ a3 j3 6 1=9: By the Chinese residue theorem, there exists n.2/ such that n.2/ ˛ mod 22 ;

n.2/ ˇ mod 32 ;

jn.2/ a2 j2 6 1=4;

jn.2/ a3 j3 6 1=9:

hence, By the same arguments, there is n.3/ such that jn.3/ a2 j2 6 23 ;

jn.3/ a3 j3 6 33 ;

jn.3/ a5 j5 6 53 ;

etc. In this way, we obtain a sequence n.j / convergent to a. a) An element b 2 A has the form a= l, where a 2 OA and l 2 Z. If n.j / converges to a, then n.j / = l converges to b. Theorem 3.3. Consider the map Q ! R A given by r 7! r D .r; r; r; : : : /. a) Its image is a discrete set. b) Moreover, .R A/=Q is compact. Proof. a) Let a non-eventually Q constant sequence rj D uj =vj 2 Q converge in R. Then its denominators vj D p Np .j / tend to 1. Hence either the number of distinct prime factors in all uj is infinite or for some p the sequence Np .j / is unbounded. In both cases, there is no adelic convergence. b) Let us show that each coset of .R A/=Q contains an element of Œ0; 1 OA. Indeed, let aN D .a1 ; a2 ; a3 ; a5 ; : : : / 2 R A. Let S be the set of all primes, for which jap j > 1. For each p 2PS consider the fraction bp D mp =p Np 2 Q such that jap bp j 6 1. Consider r WD bp . Then all Qp -entries of .aN r/ are p-adic integers. We correct r ! r C n by n 2 Z and put the R-entry into Œ0; 1/. However, the space Œ0; 1 OA is compact, therefore every sequence contains a convergent subsequence. 3.4 Rational lattices. Consider the linear space Qn over Q. Recall that a lattice in Qn is an additive subgroup of the form Zf1 ˚ ˚ Zfn , where fj is a basis in Qn . n n Observation L 3.4. Let L and L M be lattices in Q . Then there is a basis vj in Q such that L D Qvj , M D rj Qvj with rj 2 Q.

Indeed, choosing an appropriate a 2 Z, we can achieve aM L. Now the question reduces to canonical forms of subgroups in finitely generated free Abelian groups (see Algebra textbooks).

491

11.3. Adeles

3.5 Adelic lattices. Denote by An the product of n copies of A endowed with a product topology. This space is a module over A, the multiplication of a vector by a scalar is a continuous operation. We say that a lattice in An is a set L2 L3 L5 ; where Lp is a lattice in Qpn and Lp D Opn for all but a finite number of p’s. We denote the set of all lattices in An , Qn , Qpn by Latn .A/;

Latn .Q/;

Latn .Qp /

respectively. The group GL.n; Q/ acts on the space Latn .Q/. Clearly, Latn .Q/ D GL.n; Q/=GL.n; Z/; where GL.n; Z/ D SL˙ .n; Z/ is the group of matrices g such that both g and g 1 have integer matrix elements. The determinant of such a matrix is ˙1. We claim that the space Latn .Q/ coincides with Lat n .A/ D GL.n; A/=GL.n; OA/: N An is a lattice in An . Theorem 3.5. a) For a lattice L Qn , its closure L N is a bijection Latn .Q/ ! Latn .A/. b) The map L 7! L N M x are c) For any two lattices L M in Qn , the quotient groups L=M and L= isomorphic. A proof is given in Subsection 3.7. 3.6 p-adic closures of rational lattices. Denote by Z.p 1 / the ring of p-rational numbers, i.e., the numbers u=p N , where u 2 Z. By GL.n; Z.p 1 // we denote the group of matrices g with p-rational entries such that det g D p k . In other words, g and g 1 have p-rational entries.

n A lattice in Qn is said to be p-rational if it is contained in Z.p 1 / and its covolume2 has the form p k . By Lat n .Z.p 1 // we denote the set of all p-rational lattices, Latn .Z.p 1 // D GL.n; Z.p 1 //=GL.n; Z/: For a p-rational lattice L, consider its closure LN Qpn . N is a bijection Latn .Z.p 1 // ! Latn .Qp /. Theorem 3.6. The map L 7! L A covolume of a lattice L Rn is the volume of the quotient Rn =L. If vj D L, then the covolume is detfsij g. 2

P

sij ej is a basis of

492


Proof. If g 2 GL.n; Z.p 1 // is not contained in GL.n; Z/, then g … GL.n; Op /. Hence, gZn D gZn D gOpn ¤ Opn D Zn : Since our map is GL.n; Z.p 1 //-equivariant, it is injective. Next, let us show that each p-adic lattice K is a closure of a p-rational lattice. Choose a basis v1 ; : : : ; vn in K. Take ˛ such that p ˛ Op K. We choose p-rational vectors wj such that wj D vj mod p ˛C1 Opn : Then wj form a basis in K, see Theorem 10.3.6. On the other hand, p-rational vectors wj 2 Qn generate a lattice L in Qn ; by construction, LN D K. However, covolume of L is not necessarily p k . Consider a basis hj 2 Zn such L athe j that L D Zhj . We write L in the form bj LD

L aj0 bj0

p sj Zhj ;

where aj0 , bj0 are relatively prime with p:

Consider the p-rational lattice L0 WD of L and L0 coincide.

L

p sj hj . Since aj0 =bj0 2 Op npOp , the closures

3.7 Proof of Theorem 3.5. a) By Proposition 3.2, the closure of Zn is OAn . For each g 2 GL.n; Q/, we have gZn D gZn , and this proves a). b) The injectivity follows from the same argument as in Theorem 3.6. Let us prove the surjectivity. Consider a lattice K An , i.e., a collection of lattices Kp Qpn such that Lp D Op for all p but a finite set S . Take p 2 S and choose matrix g 2 GL.n; Z.p 1 // such that gOp D Kp . Consider the lattice g 1 K. For all primes q ¤ p, we have g 2 GL.n; Oq /, therefore, g 1 fixes Onq . Thus the new set S for g 1 K is smaller than for K. We repeat the same trick for the “next” p. Etc. c) The statement is obvious if n D 1. For the general case we refer to Observation 3.4.

11.4 Adelic Weil representation 4.1 Adelic measure. We define the measure da on An by the assumptions: Q 1. On the space OAn ' p Opn , our measure is the product measure. In particular, we require .OAn / D 1. 2. The measure is translation-invariant. Recall that An is a union of a countable collection of disjoint copies of OAn ; therefore is -finite.

11.4. Adelic Weil representation

493

4.2 Tensor products of Hilbert spaces. Recall the definition of the tensor product of a countable family of Hilbert spaces. Let Hj be Hilbert spaces, .j / 2 Hj be distinguished unit vectors. Choose an orthonormal basis e1.j / D .j / , e2.j / ; : : : , in each Hj . Consider the Hilbert space O Hj ; .j / with an orthonormal basis ek.1/ ˝ ek.2/ ˝ ek.3/ ˝ ; 1 2 3 where ek.jj / D .j / for all but a finite number3 of j ’s. Let v .j / 2 Hj be vectors satisfying Xˇ ˇ ˇhv .j / ; v .j / iH 1ˇ < 1; j

Xˇ ˇ ˇhv .j / ; .j / iH 1ˇ < 1: j

(4.1)

We define ˝v .j / 2 ˝Hj in the evident way (however, the conditions (4.1) are necessary for such definition). operators Aj W Hj ! Hj such that the product Q Consider a family of bounded kAj k absolutely converges4 and v .j / WD Aj .j / satisfy the condition (4.1). Then we define the tensor product of operators ˝Aj W ˝ Hj ; .j / ! ˝ Hj ; .j / by

˝Aj ˝v .j / D ˝ Aj v .j / :

4.3 Adelic L2 . We define the space L2 .An / in the usual way. Obviously, O L2 .An / D L2 .Qpn /; IOpn : 4.4 Test functions and distributions. A test function on An is a finite linear combination of functions f .x/ D IL .x C a/, where L is a lattice and a 2 An . We denote the space of all test functions by B.An /. The following statement is obvious. Proposition 4.1. a) A function f is a test function if and only if it is a compactly supported locally constant function. 3 This is a crucial phrase. If we omit it, then we get a set of basis vectors with cardinality the continuum and a (semi)-pathological object. We also stress the importance of j in this definition. However, the .j / .j / construction does not dependPon ˇ the choice ˇ of the remaining basis vectors e2 , e2 , … . 4 ˇ ˇ Equivalently, the series kAk 1 converges.

494


b) For lattices M L An , we denote by B.LjM / the set of all M -invariant functions supported by L. Then [ B.An / D B.LjM /: M LAn

4.5 The Fourier transform. Define the C -valued adelic exponential by Y expf2 i ag D expf2 i ap g p

(all the factors but a finite number are 1). Problem 4.1. a) The group of characters of the additive group Q coincides with the group .R A/=Q mentioned in Theorem 3.3. We fix .a1 ; a2 ; a3 ; : : : / 2 R A and set ˚

Y ˚ .r/ D exp 2 i a1 r exp 2 i ap rg; r 2 Q: p

b) The group of characters of Q=Z is the group of integer adeles.

The Fourier transform is given by the usual formula Z F f .x/ D expf2 ixyg f .y/ dy: An

It has the usual properties. Proposition 4.2. a) F is a bijection B.An / ! B.An /. b) F is a unitary operator L2 .An / ! L2 .An /. Actually, F is the tensor product of p-adic Fourier transforms, hence there is nothing to prove. 4.6 Adelic Weil representation. We define the adelic symplectic group Sp.2n; A/ as the group of sequences g D .g2 ; g3 ; g5 ; : : : /; where gp 2 Sp.2n; Qp / and for all gp 2 Sp.2n; Op / but a finite number of primes p. The product of sequences is component-wise. For each p, we normalize the operators We.g/, where g 2 Sp.2n; Op /, by the condition (see Problem 2.2), We.g/IOpn D IOpn : Now, we can define the Weil representation of Sp.2n; A/ as O We.g/ D We.gp /: p

11.5. The group Sp.2n; Q/ and the real-adelic correspondence

495

11.5 The group Sp.2n; Q/ and the real-adelic correspondence 5.1 The space M.Qn /. Let K L Qn be lattices. Denote by M.LjK/ D M.LjK; Q/ the space of functions on Qn that are K-invariant and supported by L. Obviously, dim M.LjK/ equals the order of the group L=K. Next, we define the space [ M.LjK; Q/: M.Qn / WD KLQn

Remark. The space M.Qn / is spanned by functions '.x/ D IK .x C a/, where K Qn is a lattice, a 2 Qn . We define the rational Heisenberg group Heisn .Q/ as the group of matrices 0 1 t 1 vC ˛ C 12 vC v

t A; R vC ˚ v I ˛ D R vI ˛ D @0 1 (5.1) v 0 0 1 where vC , v 2 Qn , ˛ 2 Q. As above, we define the operators

˚

t U vC ˚ v I ˛ f .x/ D f .x C vC / exp 2 i.xv C C ˛ C 12 vC v / :

Proposition 5.1. a) The operators U vC ˚ v I ˛ preserve the space M.Qn /. b) The representation of Heisn .Q/ in M.Qn / is irreducible. n n c) Each linear

operator S W M.Q / ! M.Q / commuting with all the operators U vC ˚ v I ˛ is a scalar operator. Proof. The statement a) is obvious. To show b) and c) we first need some auxiliary results. For a lattice K, denote by L˙ its dual lattice 5 . The following statement is obvious. Lemma 5.2. a) The space M.LjK/ is precisely the space of functions fixed by all operators: A.w/f .x/ D f .x C w/;

where w 2 K;

B.r/f .x/ D f .x/ expf2 ixr g; where r 2 L˙ : t

b) The space M.LjK/ is invariant with respect to the operators: A.w/f .x/ D f .x C w/;

where w 2 L;

B.r/f .x/ D f .x/ expf2 ixr t g; where r 2 K ˙ : 5

The lattice L˙ consists of y 2 Qn such that wy t 2 Z for each w 2 L.

496


Therefore, for an element w z 2 L=K, we have a well-defined operator A.w/ z in M.LjK/; for r 2 K ˙ =L˙ , we have a well-defined operator B.r/. Q The operators A.w/ z and B.r/ Q generate a certain finite group; we denote it by Heis.LjK/. The group Heis.LjK/ is a straightforward analog of the groups Heisn .k/ over a field k; now we consider the Abelian group L=K instead of k n . Lemma 5.3. a) The representation of Heis.LjK/ in M.LjK/ is irreducible. b) There is no proper Heis.LjK/-invariant subspace in M.LjK/. This is a simplified version of Theorem 1.2.2. We leave the proofs of both statements as exercises. Continuation of the proof of Proposition 5.1. b) Consider two chains of lattices L1 L2 L3 ;

K1 K2 K3

such that [M.Lj jKj / D M.Qn /. By definition, M.Lj C1 jKj C1 / M.Lj jKj /: We apply Lemma 5.3. If V M.Qn / is a Heis.MjL/-invariant subspace, then each intersection V \ M.LjK/ is 0 or the whole space M.LjK/. This implies V D 0 or M.Qn /. c) Let f 2 M.LjK/. Then for any w 2 K and r 2 L˙ , A.w/Sf D HA.w/f D Sf;

B.r/Sf D HB.r/f D Sf:

Therefore, S sends M.LjK/ to itself. By the last lemma, S is a scalar operator on M.LjK/. 5.2 Adelization of M.Qn / Proposition 5.4. a) Each function f 2 M.Qn / admits a unique continuous extension fN to a function defined on the space An . b) The map f 7! fN is a bijection M.Qn / ! B.An /. This map also induces a N K/. x bijection M.LjK/ ! B.Lj c) The map f 7! fN commutes with the action of the group Heisn .Q/. Proof. Let K Qn be a lattice, a 2 Qn ; let Kx be the adelic closure of K. Then the continuous extension of IKCa is IKCa . x n All functions f 2 B.Q / are linear combinations of such functions. 5.3 Poisson distributions and realification of M. For a function f 2 M.Qn / consider the distribution on Rn given by X p f .h/ı.x 2h/: h2Qn

11.5. The group Sp.2n; Q/ and the real-adelic correspondence

497

Let us call such objects Poisson distributions. We denote the linear space of all Poisson distributions by P .Rn /. The space P .Rn / is spanned by the functions X p X ı x 2 kj aj b ; k1 ;:::;kn 2Z

where a1 ; : : : ; an is a basis in Qn and b 2 Qn . Let us consider the standard action of the Heisenberg group Heisn .R/ in the space .Rn /. Precisely, for a matrix (5.1) define the operators p

˚p

t / : U vC ˚ v I ˛ f .x/ D f .x C 2vC / exp 2ixv C C 2 i.˛ C 21 vC v (We rather change formula (1.2.9); the purpose of coefficients 2 will be clear immediately). Recall that the symplectic group Sp.2n; R/ acts on the spaces .Rn / and 0 .Rn / via the Weil representation. Theorem 5.5. a) The space P .Rn / is invariant under the action of the subgroup Heisn .Q/ Heisn .R/. b) The space P .Rn / is invariant under the action of the subgroup Sp.2n; Q/ Sp.2n; R/. Proof. a) To simplify the notation, consider the case n D 1. Obviously, P .R/ is invariant with respect to shifts by rational vectors. Next, consider X p p exp. 2iwx/ ı x 2.ka c/ k2Z

D expf2 iwcg

X

p expf2 i kawgı x 2.ka c//:

k2Z

Since wa is rational, the factor expf2 i kawg is periodic in k. Now the statement becomes obvious. b) In Subsection 1.2.7, we discussed the generators of Sp.2n; R/, see (1.2.13). Matrices of the same types with rational entries generate Sp.2n; Q/. Thus we must verify the statement for the operators (1.2.15)–(1.2.17) with rational matrices A and B. 1) For the operators f .x/ 7! f .xA/;

where A 2 GL.n; Q/.

the statement is obvious. 2) The Fourier transform. We refer to the Poisson summation formula, see Corollary 8.2.3.

498


3) Examine the operators ˚

f .x/ 7! exp 2i xC x t f .x/: To simplify the notation, assume n D 1, ˚i

exp

2

cx 2

p

X ı x 2.ka b/ k2Z

D exp. i cb 2 /

X

p ˚

exp i.k 2 ca2 C 2k bac/ ı x 2.ka b/ :

k

The factor expf i.k 2 ca2 C 2k bac/g is periodic in k. This completes the proof.

5.4 Canonical embedding of the Schwartz space into the space of adelic distributions. We have the canonical bijections P .Rn /

! M.Qn /

! B.An /:

Therefore we get a through map from the space of Poisson distributions on R to the space of adelic test functions. Theorem 5.6. The canonical bijection P .Rn / $ B.An / a) commutes with the action of the Heisenberg group Heisn .Q/; b) commutes with the action of Sp.2n; Q/. Proof. The first statement is evident. We have two representations of Sp.2n; Q/ in M.Qn /, one is transferred from the space P .Rn / of Poisson distributions, the second representation is transferred from B.An /. They satisfy the same commutation relations U ŒvgI ˛ D We1 .g/1 U ŒvI ˛ We1 .g/;

U ŒvgI ˛ D We2 .g/1 U ŒvI ˛ We2 .g/:

Hence, We2 .g/1 We1 .g/

is an operator in M.Qn / commuting with the Heisenberg group Heisn .Q/. By Proposition 5.1, this operator is scalar. By duality, we get Sp.2n; Q/-equivariant embeddings .Rn / 7! B 0 .An /;

B.An / 7! 0 .Rn /:

11.6. Constructions of modular forms

499

5.5 Weil distribution Theorem 5.7. For any f 2 .Rn /, the corresponding distribution on An is given by X f .r/ı.a r/; r2Qn

where r WD .r; r; : : : / 2 An and ı is a delta-function on the adeles. We leave this statement as an exercise. We define “test functions” on R A as functions of the form '.x; a/ D

N X

fj .x/IL.j / Cbj .a/;

j D1

where fj 2 .Rn / and L.j / are adelic lattices. The Weil distribution W on R A is given by X hhW; 'ii D '.r; r/: p=q2Q

Then our map is nothing but the map .Rn / ! B 0 .An / whose kernel is the Weil distribution. 5.6 p-adic variant of the same construction. Let us return to the situation discussed in Subsection 3.6. Fix prime p. Write [ Mp .Qn / D M.LjK/: K L are p-rational lattices

Denote by Pp .R / the corresponding subspace in P .Rn /. n On the other hand, each function f 2 M.Qn / admits a unique continuous extension to Qp n and we obtain the bijection Mp .Qn / with the Bruhat space B.Qp /. n

n /, which commutes with actions Observation 5.8. We obtain an embedding .Rn / ! B 0 .Qp of Heisn Z.p 1 / and Sp 2n; Z.p 1 / .

11.6 Constructions of modular forms In this section, we explain the classical construction of modular forms through thetafunctions. 6.1 Congruence subgroups. Fix an integer positive N . A principal congruence subgroup N Sp.2n; Z/ consists of matrices g such that N divides all entries gij ıij .

500


A congruence subgroup in Sp.2n; Z/ is a subgroup containing some N . Denote by lB the subgroup in Sp.2n; R/ generated by matrices a 0 1 b 1 0 ; ; ; 0 at1 0 1 c 1

(6.1)

where l divides all entries of the matrices a 1, b, c. We give the following theorem (see [206]) without a proof: Theorem 6.1. The group lB is a congruence subgroup. 6.2 The definition of modular forms. First, we give a preparatory definition. Denote by Pn the Siegel wedge of symmetric matrices with positive imaginary part. A holomorphic function on Pn is a modular function of a weight k with respect to a congruence subgroup if a b f .a C zc/1 .b C zd / det.a C zc/k D f .z/ for all 2 : c d Remark. In Chapter 7 we examined certain representations Tk of Sp.2n; R/ in the space Hol.Pk / of holomorphic functions on Pk . Now we consider -invariant holomorphic functions. We stress that these functions are not elements of the Hilbert spaces Hk defined in Section 7.7.5. Next, we define modular forms. If n > 2, then a modular form is a modular function. For n D 1 a definition is more intricate, extra conditions are necessary. In this case, P1 is the usual upper half-plane on C. A modular function f is a modular form if a) f is bounded in some domain Im z > ; b) For each rational point 2 R there is h > 0 such that jf .z/j 6 const jz jk

for jz C ihj < h:

(6.2)

Let us discuss these conditions. 1) Our group contains some subgroup of shifts z 7! z C j l, where j ranges in Z and l is a constant depending on . Our function f is invariant under P such shifts. The condition a) claims that f can be expanded in a Taylor series f .u/ D k>0 ck uk in the variable u WD expf2 iz= lg. Let ac db be an element of SL.2; Z/ D Sp.2; Z/ taking r to 1. Then condition b) is condition a) written for the transformed function. 2) If D SL.2; Z/, then the condition b) follows from a) by the last remark. Generally, the group has several orbits on the set Q [ 1 and for each orbit we have its own condition b).


501

3) For n > 1 the following Koecher effect holds: a modular function is automatically bounded in each domain of the form z0 C Pn , where z0 is a fixed interior point of the Siegel wedge. 6.3 Linearization of the Weil representation on 1;2 . By 1;2 D 1;2 .n/ Sp.2n; R/ we denote the Igusa subgroup defined above in 8.3.1–8.3.2. For a proof of the following theorem, see Mumford [136]. Theorem 6.2. The Igusa subgroup 1;2 is generated by elements a 0 1 b 1 0 ; ; ; 0 at1 0 1 c 1

(6.3)

where matrices a, at1 , b, c are integer, b, c are symmetric and have even diagonals. Let .x/ D

X Y

ı.x

p

2kj /:

k1 ;:::;kn j

Proposition 6.3. The restriction of the Weil representation of Sp.2n; R/ to 1;2 is equivalent to a linear representation. Moreover, we can choose the operators We.g/, g 2 1;2 such that We.g/ D : (6.4) Proof. First, change the collection of generators (6.3) by (1.2.13). By Theorem 8.2.1, is invariant under the Fourier transform. Obviously, is an eigenfunction of operators a 0 f .x/ D f .xa/j det aj1=2 : We 0 at1 Next, ²

³ ° X ±Y X p i t exp xbx .x/ D exp i b k k ı.x 2kj / : 2 n j

k2Z

P

Since the diagonal of b is even, the sum b k k is even, hence expf: : : g D 1. Since is an eigenfunction for all generators, it is also an eigenfunction for each g 2 1;2 , i.e., We.g/ D .g/; 2 C : We can replace We.g/ 7! .g/1 We.g/. Then we have (6.4) and the representation becomes linear. 6.4 Congruence subgroups and the space of Poisson distributions. Now, we consider the linear action of 1;2 in the space P .Rn / of Poisson distributions.

502


Theorem 6.4. The stabilizer of any element of P .Rn / is a congruence subgroup. Proof. Fix an integer N . Considerp the subspace HN P .Rn / consisting of Poisp 1 n n son distributions supported by N 2Z and invariant under N 2Z . Clearly, S HN D P .Rn /. The theorem follows from the next lemma. B Lemma 6.5. The space HN is invariant with respect to the group 2N 2. B Proof. The group 2N 2 is generated by matrices of the form 0 1 C 2N 2 ˛ ; S.˛/ D 0 .1 C 2N 2 ˛/1 1 2N 2 ˇ T .ˇ/ D ; 0 1 1 1 0 0 1 1 2N 2 0 1 R. / D D : 2N 2 1 1 0 0 1 1 0

(6.5)

Let f 2 HN . 1) First, We S.˛/ f .x/ D f .x C2N 2 ˛x/. Since x 2 N 1 Zn , we have 2N 2 x˛ 2 .2N /Zn , hence f .x C 2N 2 x˛/ D f .x/: 2) The function expf2 iN 2 xˇx t g is constant on N 1 Zn , hence, ˚

We T .ˇ/ f .x/ D exp 2 iN 2 xˇx t f .x/ D f .x/: 0 1 3) The Fourier transform We 1 0 leaves the space HN invariant. 4) By (6.5), We R. / f D f .

6.5 Modular forms of weight 1=2. Now we apply the construction of § 7.7. Namely, we consider the operator Z ² ³ 1 J.z/ D exp xzx t .x/; where z 2 Pn , (6.6) 2 x2Rn from 0 .Rn / to the space Hol.Pn / of holomorphic functions on the Siegel wedge. Recall that the group Sp.2n; R/ acts on Hol.Pn / by transformations A B (6.7) f .x/ D f .A C zC /1 .B C zD/ det.A C zC /1=2 : T1=2 C D By Proposition 6.3, the action of the subgroup 1;2 .n/ Sp.2n; R/ is equivalent to a linear representation; we denote the corresponding operators by 0 T1=2 .g/ D .g/T1=2 .g/


503

(for an explicit expression of the factor .g/, see [136], it ranges in 8-th roots of unity). The operator J intertwines actions of Sp.2n; R/ in 0 .Rn / and Hol.Pn /. Therefore, it takes distributions invariant under a congruence subgroup to -invariant holomorphic functions. By virtue of Theorem 6.4, we get the following statement: Theorem 6.6. For any Poisson distribution 2 P .Rn /, the function J is a modular form with respect to some congruence subgroup. Observation 6.7. Jf is a -function in the sense of Section 8.4. 6.6 Equivariant embedding Pn ! Pnm . Representation theory provides tools for making numerous versions of the previous construction. We consider only the simplest product of this industry. Define the embedding Sp.2n; R/ ! Sp.2nm; R/ by 0 1 a 0 ::: b 0 ::: B0 a : : : 0 b : : :C B C B :: :: : : :: :: : : C C B: : : : : : a b C m W 7! B B c 0 : : : d 0 : : :C : c d B C B 0 c : : : 0 d : : :C @ A :: :: : : : : :: : : :: :: : : Define also the following embedding m W Pn 0 z B0 m W z 7! @ :: :

7! Pnm , 1 0 ::: z : : :C A: :: : : : :

Consider the action of Sp.2nm; R/ on the Siegel wedge Pnm and the corresponding action T1=2 in the space Hol.Pmn /. The image m .Pn / is a Sp.2n; R/-invariant submanifold in Pmn ; therefore we get an action of Sp.2n; R/ in the space of holomorphic functions on m .Pn / ' Pn . It is easy to see that this action is given by the formula a b f .x/ D f .a C zc/1 .b C zd / det.a C zc/m=2 ; Tm=2 c d i.e., this is an action discussed in Chapter 7. For the Igusa subgroup 1;2 .n/ Sp.2n; R/, there are the corresponding corrected operators 0 .g/ D m .g/Tm=2 .g/: Tm=2

504


6.7 Modular forms of arbitrary weights. Consider the through map ² ³ Space of holomorphic Space of holomorphic 0 .Rmn / ! ! : functions on Pmn functions on Pn The first arrow is the map J defined above (6.6). The second map is the restriction of a holomorphic function to .Pn /. The through map is given by the formula Z

˚ Jm .z/ D exp 12 x m .z/x t .x/; where z 2 Pn : x2Matn;m .R/

Theorem 6.8. For any Poisson distribution 2 P Matn;m .R/ , there is a congruence subgroup 1;2 .n/ that fixes Jm ./. Proof. Obvious.

11.7 Bibliographical remarks to Chapters 10 and 11 7.1 Historical remarks. The p-adic numbers were discovered by K. Hensel in 1897. The classification of quadratic forms over Q was obtained by H. Minkowski, 1885. Its nice form (Hasse–Minkowski principle) was discovered by H. Hasse in 1920; his approach admits wide generalizations. Adeles were introduced by C. Chevalley. Investigation of unitary representations of p-adic groups was initiated at the end of the 1950s by F. Mautner and F. Bruhat. The Bruhat– Tits buildings were invented by J. Tits and F. Bruhat in 1960s in a series of papers that are hard reading, see [31]. The p-adic Weil representation was introduced by A. Weil [214]. 7.2 p-adic numbers. For a systematic exposition, see e.g. the books by Z. I. Borevich, I. .R. Shafarevich [25], J. W. S. Cassels [35], J.-P. Serre [197], A. Weil, [215]. In several topics, we avoided the case p D 2; from the point of view of the Hasse–Minkowski principle and the adelic ideology this is inadmissible. The case p D 2 is not difficult, however theorems and proofs become longer. See the books cited above; for Gaussian integrals with p D 2, see V. S. Vladimirov, I. V. Volovich [209], E. I. Zelenov, [209]. 7.3 Buildings. See the books by M. A. Ronan [183], K. S. Brown [30], P. Garrett [61], see also surveys of R. Scharlau [189] and J. Rohlfs and T. A. Springer [182], and original works of J. Tits and F. Buhat [31]. On buildings at infinity, see P. B. Eberlein [46]. For a nice introduction to trees, see the book of J.-P. Serre [198]. The triangle inequality for p-groups was found by Ph. Hall, see I. G. Macdonald [129]. 7.4 Exotic buildings. There is also research on abstract buildings of non-p-adic origin, see e.g. M. A. Ronan [183], S. Barre [11]. 7.5 Harmonic analysis on buildings. Basic works were I. G. Macdonald [128], H. Matsumoto [132], see also I. G. Macdonald [129], but buildings remain invisible in this book. The analog of Berezin kernels was introduced in [151].


505

7.6 Nazarov semigroup and its Weil representation were constructed in M. L. Nazarov [139]. 7.7 Constructions of modular forms through theta-functions and the Weil representation are very old (older than Weil representation, I do not know the origin), see the books by B. Schoeneberg [191], D. Mumford [136], G. Lion, M. Vergne [122].

Addendum

A. Classical groups. Notation Let R be the real numbers, C the complex numbers, H the quaternions. We denote by R and C the multiplicative groups of real and complex numbers, by T C the group of complex numbers with absolute value D 1. A.1. Forms. We shall regard the following seven types of forms on linear spaces over R, C, H: 1) symmetric bilinear forms over R; 2) skew-symmetric bilinear forms over R; 3) symmetric bilinear forms over C; 4) skew-symmetric bilinear forms over C; 5) Hermitian forms over C; 6) Hermitian forms over H; 7) anti-Hermitian forms over H. We recall that a sesquilinear form on a left H-module is an H-valued function h; i satisfying the conditions: hv; w1 C w2 i D hv; w1 i C hv; w2 i; hv1 C v2 ; wi D hv1 ; wi C hv2 ; wi; hv; wi D hv; wi: N In coordinates, we can write out X hx; yi D xk akl yNl ;

where xk ; yl ; akl 2 H:

k;l

A form is Hermitian if hv; wi D hw; vi. A form is anti-Hermitian if hv; wi D hw; vi. Examples of Hermitian and anti-Hermitian forms are respectively hv; wi D

X

vk w xk ;

hv; wi D

X

vk i w xk :

Below the term “form” means a nondegenerate form. Remark. Bilinear forms on left modules over P H do not exist. Indeed let us write the simplest “bilinear” expression .a; b/ D aj bj . Then .a; b/ ¤ .a; b/. In fact .a; b/ is a pairing of a left module Hn and a right module Hn .

508

Addendum

A.2. Classical groups. We define the following 10 series of groups: 1) The (real) general linear group GL.n; R/ is the group of all invertible real matrices of order n. 2) GL.n; C/ is the group of all invertible complex matrices. 3) GL.n; H/ is the group of all invertible quaternionic matrices. 4) The pseudo-orthogonal group O.p; q/ is the group of real matrices preserving a symmetric bilinear form with the inertia .p; q/. 5) The real symplectic group Sp.2n; R/ is the group of matrices preserving a skewsymmetric bilinear form on R2n . 6) The complex orthogonal group O.n; C/ is the group preserving a symmetric bilinear form on Cn . 7) The complex symplectic group Sp.2n; C/ is the group of matrices preserving a skew-symmetric bilinear form on C2n . 8) The pseudo-unitary group U.p; q/ is the group of matrices preserving a Hermitian form over C with inertia indices .p; q/. 9) The pseudo-symplectic group or pseudo-unitary quaternionic group Sp.p; q/ is the group of matrices preserving a Hermitian form over H with inertia indices .p; q/ 10) The nameless classical group SO .2n/ is the group of matrices preserving an anti-Hermitian form on an n-dimensional space over H. Remark. The standard notation for U.n; 0/, O.n; 0/, Sp.n; 0/ is U.n/ (unitary group), O.n/ (real orthogonal group), Sp.n/ (unitary symplectic group). A.3. Standard realizations. Below we choose the forms in canonical ways and write equations for elements g of a group. A field is the same as above. 1p 0 1p 0 O.p; q/ W g gt D ; 0 1q 0 1q 0 1n t 0 1n Sp.2n; R/ W g g D ; 1n 0 1n 0 O.n; C/ W Sp.2n; C/ W U.p; q/ W Sp.p; q/ W SO .2n/ W

gg t D 1; 0 g 1n 1p g 0 1p g 0

1n t 0 1n g D ; 0 1n 0 0 1p 0 g D ; 1q 0 1q 0 1p 0 g D ; 1q 0 1q

gig D 1:

509

A. Classical groups. Notation

A.4. Lie algebras. In all the cases listed above, we can easily describe the Lie algebra as it was explained in Subsection 2.2.4. We obtain the following conditions for a matrix X 2 g: 1p 0 1p 0 o.p; q/ W X C X t D 0; 0 1q 0 1q 0 1n 0 1n sp.2n; R/ W X C X t D 0; 1n 0 1n 0 o.n; C/ W sp.2n; C/ W u.p; q/ W sp.p; q/ W so .2n/ W

X C X t D 0; 0 1n 0 X C 1n 0 1n 1p 0 1p X C 0 1q 0 1p 0 1p X C 0 1q 0

1n X t D 0; 0 0 X D 0; 1q 0 X D 0; 1q

X i C iX D 0:

A.5. Homomorphisms to Abelian groups and connected components 1) GL.n; R/ has a homomorphism g 7! det.g/ to R . The kernel is said to be a (real) special linear group SL.n; R/. The group GL.n; R/ consists of two connected components separated by the conditions det g > 0 and det g < 0. Also, SL˙ .n; R/ denotes the subgroup in GL.n; R/ singled out by the condition det.g/ D ˙1. 2) GL.n; C/ has a homomorphism g 7! det.g/ to C . The notation for the kernel is SL.n; C/. 3) GL.n; H/ has a homomorphism g 7! det.g/ to the multiplicative group of positive real numbers. The notation for the kernel is SL.n; H/. Recall the definition of the determinant of a quaternionic operator. We regard an operator g in Hn as a real operator gR in R4n . By definition, the determinant of g is p det g WD 4 det.gR / 2 R: 4) O.n; C/ has a homomorphism g 7! det.g/ D ˙1 to Z2 . The kernel is said to be the special orthogonal group SO.n; C/. The group O.n; C/ consists of two connected components. 5) O.p; q/ has a homomorphism to Z2 Z2 . For g D ac db we define .g/ WD .sgn.det a/; sgn.det d //. Also, det.g/ D sgn det.a/ sgn det.d /. The kernel of det is denoted by SO.p; q/, the notation for the kernel of is SO0 .p; q/. The last group is connected. 6) U.p; q/ has a homomorphism det.g/ to T . The kernel is said to be the special pseudo-unitary group SU.p; q/.

510

Addendum

To obtain the Lie algebras sl.n; R/, sl.n; C/ and su.p; q/ of the groups SL.n; R/, SL.n; C/ and SU.p; q/ we add the condition tr X D 0. For sl.n; H/ we require Re tr X D 0. A.6. Letter P. If G is a classical group then P G denotes the quotient of G with respect to the center. For instance, PGL.n; C/ WD GL.n; C/=C ; PO.n; C/ WD O.n; C/=f˙1g; PU.p; q/ WD U.p; q/=T : They are called projective linear group, projective orthogonal group, projective pseudounitary group, etc. A.7. The alternative notation GL.n; K/ DW GLn .K/;

O.n; C/ DW On .C/; : : : :

Some authors use SL.n; H/ DW SU .n/;

Sp.p; q/ DW U.p; qI H/;

Sp.n/ DW SpU.n/:

A.8. Notation for covers. Spin.n/ is a two-sheet cover of SO.n/. A relatively common notation for the universal covering group of a connected group G is G . A.9. Tables of real classical groups and symmetric spaces are given below. For categories of linear relations, see [145], Addendum A. A.10. The definition of classical groups over arbitrary fields. Let K be a commutative or noncommutative field of characteristic ¤ 2. Let a 7! aB be an anti-involution1 of K, i.e., aBB D a and .ab/B D b B aB . A sesquilinear form on a left K-module V is a map V V ! K (the notation: h; i) such that all maps v 7! hv; ai;

w 7! hb; wiB

are linear. A sesquilinear form is Hermitian if hv; wi D hw; viB and anti-Hermitian if hv; wi D hw; viB . A classical group in a narrow sense is a general linear group over K or the group of all isometries of a Hermitian or anti-Hermitian form over K.

1 For K D C one has two variants, aB WD a and aB WD a; N for the quaternions, aB D a; N for real numbers, aB D a.

B. Representations. Several definitions

511

B. Representations. Several definitions B.1. Subrepresentations. Let be a finite-dimensional representation of a group G in a complex linear space V . A subrepresentation is a subspace invariant with respect to G. A representation is reducible if it contains a proper subrepresentation2 . Otherwise, it is called irreducible. A subrepresentation of a unitary representation is a closed invariant subspace. If W V is a subrepresentation of a unitary representation, then its orthocomplement is a subrepresentation. B.2. Characters. A character3 of a group G is a one-dimensional unitary representation. In other words, a character is a map W G ! T such that .g1 g2 / D .g1 /.g2 /: B.3. Equivalence of representations. Let Œ1 ; V1 , Œ2 ; V2 be two representations of a group G. They are equivalent if there exists an invertible operator A W V1 ! V2 such that A1 .g/A1 D 2 .g/: Remark. For unitary representations we can require A to be unitary. But this is tantamount to a definition. B.4. Morphisms and the Schur Lemma. Let Œ1 ; V1 and Œ2 ; V2 be two representations of a group G. An operator A W V1 ! V2 is said to be intertwining if A1 .g/ D 2 .g/A: In fact, intertwining operators are morphisms in the category of representations. The following four theorems are called the Schur Lemma: Theorem B.1. If representations 1 and 2 are irreducible finite-dimensional and nonequivalent, then A D 0. Proof. Let v 2 ker A. Then A1 .g/v D 2 .g/Av D 0 H) Av 2 ker A: Next, let w 2 im A, w D Ah. Then 2 .g/Ah D A.1 .g/h/ H) 2 .g/w 2 im A: Therefore, ker.A/ and im.A/ are subrepresentations. Since 1 and 2 are irreducible, A D 0. 2 3

I.e., the subspace must be ¤ 0, ¤ V . This term also has other meanings.

512

Addendum

Theorem B.2. Let .; V / be an irreducible finite-dimensional representation4 . Any intertwining operator A W V ! V is of the form const 1. Proof. Let be an eigenvalue of A. The subspace ker.A / is a non-zero subrepresentation, hence ker.A / D V . Theorem B.3. Let .; V / be an irreducible unitary representation. Any intertwining operator A W V ! V has the form const 1. Proof. The previous proof breaks down. Note first A.g/ D .g/A H) .A.g// D ..g/A/ H) A .g 1 / D .g 1 /A : Therefore, A is also an intertwining operator. Hence the same is valid for A C A and i.AA /. The latter operators are self-adjoint. By the Spectral Theorem (see [178]), an operator commuting with a self-adjoint operator commutes with all spectral projectors. Hence the spectral projectors are intertwining operators. Therefore, the images of the spectral projectors are subrepresentations. Since is irreducible, all spectral projectors are 0 or 1. Hence both A A and i.A A / are scalar operators. Theorem B.4. If Œ1 ; V1 and Œ2 ; V2 are unitary irreducible and nonequivalent, then any intertwining operator A W V1 ! V2 is zero. Proof. The operator A A W V1 ! V1 is an intertwining operator. By the previous theorem A A D 1. For the same reason, AA D 1. Hence 1=2 A is unitary and, in particular, invertible. Thus representations are equivalent. Proposition B.5. An operator intertwining two irreducible unitary representations is unitary within to a scalar factor. Proof. Indeed, A A and AA must be scalar operators.

B.5. Several remarks Proposition B.6. Let be a unitary irreducible representation of a group G. Let be an element of the center of G. Then ./ is scalar. Proof. Indeed ./ is an intertwining operator.

Proposition B.7. Any irreducible unitary representation of an Abelian group is onedimensional. Proof. By the previous proposition, it follows that all operators .g/ are scalar.

For a group G denote by Z.G/ its center. 4 Recall that we consider complex linear spaces. For correct versions of Schur Lemma in real linear spaces, see [106], 8.2.

B. Representations. Several definitions

513

Observation B.8. Let be an irreducible unitary representation of G. Then for any z 2 Z.G/ the operator .z/ is scalar. In particular, a unitary representation of G determines a character of its center. Proof. Operators .g/ are intertwining.

B.6. Matrix elements Proposition B.9. Let and 0 be irreducible unitary representations of a group G in Hilbert spaces V1 and V2 respectively. Assume that for some v 2 V and v 0 2 V 0 , h.g/v; viV D h0 .g/v 0 ; v 0 iV 0 : Then and 0 are equivalent. Proof. Since h.g/v; .h/vi D h.h/ .g/v; vi D h.h/1 .g/v; vi D h.h1 g/v; vi; it follows that

h.g/v; .h/vi D h0 .g/v 0 ; .h/v 0 i:

Hence there exists a unique unitary operator U W V ! V 0 such that U.g/v D 0 .g/v 0 for all g. B.7. Stone–von Neumann theorem. Consider the Heisenberg group Heisn defined in Subsection 1.2.6 and the representation U given by formula (1.2.9). For each non-zero s 2 R, we define a representation of Heisn in L2 .R/ given by

˚

t t C C 12 vC v / : Us vC ˚ v ; D f .x C v / exp i s.xvC In other words, we consider the automorphism s W vC ˚ v I 7! svC ˚ v I s of Heisn and set Us WD U B s : All the representations Us are twins, but officially they are not equivalent. Theorem B.10 (Stone–von Neumann). Each irreducible unitary representation of the Heisenberg group of dimension > 1 has the form Us . The most natural way of proof is a reference to Mackey’s theorems on induced representations (see [106], 13.2). Here we outline the original proof of von Neumann, [211]. Sketch of proof. Let U be a unitary irreducible representation of Heisn . Without loss of generality, we assume U 0I D e i . Thus,

U vI 0 U wI 0 D exp 2i fv; wg U v C wI 0 ; where f; g is the bilinear form from Subsection 1.2.6.

514

Addendum

Let us consider the operator A WD

1 .2/n

Z

e jvj

2 =2

U vI 0 dv:

R2n

Lemma B.11. a) A ¤ 0.

2 b) AU cI 0 A D e jcj =2 A. c) A is self-adjoint and A2 D A. Proof of the lemma. The claim a) is proved by reductio ad absurdum. If A D 0 , then Z

1 2 0 D U cI 0 AU cI 0 D e ifc;vg=2 e jvj =2 U vI 0 dv: n .2/ R2n Considering linear combinations of such operators and their limits, one can easily show that Z

f .v/U vI 0 dv D 0 for all f .v/. b) can be verified by a straightforward calculation. Substituting c D 0, we obtain 2 A D A. Proof of the theorem. Let h be a unit vector satisfying Ah D h. Let us evaluate its matrix element, 2 2 hU cI 0h; hi D hU cI 0Ah; Ahi D hAU cI 0Ah; hi D e jcj =2 hAh; hi D e jcj =2 : But a matrix element determines an irreducible unitary representation.

Problem B.1. a) Recover details. b) Determine how it was possible to write the operator A. Apparently, this is an 2 2 emulation of the Weyl symbol calculus (see § 4.5). The function e .x Cy /=2 is the Weyl 2 symbol of the orthogonal projection onto the Gaussian e x =2 . Formal verification of A2 D A coincide (as a calculation) with evaluation of the Weyl symbol of the product.

C. Lie groups. Several logical arrows

515

C. Lie groups. Several logical arrows Here we present sketches of proof of several basic theorems of the theory of Lie groups. This section is not a regular exposition. C.1. Lie algebras. A Lie algebra g is a linear space endowed with an operation .X; Y / 7! ŒX; Y (commutator) such that ŒX; Y D ŒY; X ; ŒX1 C X2 ; Y D ŒX1 ; Y C ŒX2 ; Y ; ŒX; Y D ŒX; Y ; ŒŒX; Y ; Z C ŒŒY; Z; X C ŒŒZ; X ; Y D 0 (Jacobi identity): The initial example is the algebra gl.n/ of all n n matrices with the commutator ŒX; Y D X Y YX: Theorem C.1 (Ado). Each finite-dimensional Lie algebra is a subalgebra in gl.n/. A natural proof is in [155]. C.2. Representations of Lie algebras. A finite-dimensional representation of a Lie algebra g is a map W g ! gl.n/ satisfying Œ .X /; .Y / D

.ŒX; Y /:

C.3. Lie algebra of a matrix group Theorem C.2. Each closed subgroup G in GL.n; R/ is a smooth manifold. Let Rn be an arbitrary set. We say that a vector v at a point a 2 is tangent if one can find a sequence of points xj 2 converging to a such that directions of segments Œa; xj converge to v. All tangent vectors at a given point a form a closed cone (the tangent cone). Lemma C.3. The tangent cone g to G at 1 is a linear subspace. For each X 2 g, all points exp.tX/, where t 2 R, are contained in G. Moreover, all elements of G, lying in a small neighborhood of 1, have this form. Proof of the lemma. Consider a small neighborhood U of 1 in the group GL.n; R/. We represent g 2 U in the form g D exp.X /, where X is close to 0. We regard G \ U as the set in the definition of tangent vector given above. It is almost obvious that for each tangent vector Y 2 g, we have exp.t Y / 2 G; see Figure 1 a), but formally some proof is necessary. Further, let X , Y 2 g be tangent vectors. Then the group G contains expf"tX g expf"sY g D expf"tX C "sY g C O."2 / and hence tX C sy 2 g. Thus g is a linear space.

516

Addendum Zj

a)

1

b)

Figure 1. Reference proof of Lemma C.3.

It remains to prove the last statement of the lemma. Let exp.Zj / 2 G be a sequence convergent to 1 and let Zj … g. Then we can choose Xj 2 g such that directions of the vectors Xj C Zj are convergent to a line 6 g, see Figure 1b). This contradiction completes our quasi-proof. Proof of the theorem. From the lemma, a neighborhood of the unit in G is a submanifold in GL.n; C/. Further, the map h 7! hg identifies a neighborhood of the unit and a neighborhood of g. Therefore, all points of G have equal rights and G is a smooth manifold. Theorem C.4. If X , Y 2 g, then ŒX; Y 2 g Proof. Consider the curve ."/ D expf"X g expf"Y g expf"X g expf"Y g 2 G: Expanding ."/ in a Taylor series, after simple calculations, we obtain 1 ."/ D 1 C "2 .X Y YX / 2 and hence ŒX; Y 2 g.

ŒX; Y exp."X / exp."Y / 1

exp."X /

Figure 2. Reference Theorem C.4.

Thus the tangent space g has a canonical structure of a Lie algebra and g is said to be the Lie algebra of the Lie group g. C.4. Lie groups. A Lie group G is a manifold endowed with a smooth multiplication G G ! G satisfying the axioms of groups.


517

Now let describe briefly how to define the Lie algebra of G in this situation. Denote by g the tangent space to G at 1. A one-parameter subgroup in G is a smooth homomorphism of the additive group R to G, i.e., .˛/.ˇ/ D .˛ C ˇ/; where ˛; ˇ 2 R: Then X WD 0 .0/ is an element of g. Moreover, this is a one-to-one correspondence between the set of all one-parameter subgroups and g. We say that X is the generator of the one-parameter subgroup and denote by .˛/ DW exp.˛X /. Now we define a structure of Lie algebra on g by ŒX; Y WD

d2 expf"X g expf"Y g expf"X g expf"Y g: d "2

C.5. Inverse construction Theorem C.5. Each finite-dimensional Lie algebra g is a Lie algebra of some Lie group. Proof. By the Ado theorem, g gl.N; R/. We regard g as a subspace in the tangent space to GL.N; R/ at 1. Denote by Tg the tangent space to GL.N; R/ at a point g. For each g 2 GL.N; R/, we construct a subspace Ag Tg as follows. We consider right shifts Rg W h 7! hg on GL.N; R/ and Ag Tg is the pushforward of g with respect to Rg . It can be easily checked that Ag is an integrable distribution in the sense of Frobenius. Consider the integral manifold containing 1. It is not hard to prove that this manifold is an (incidentally, non-closed) subgroup in GL.N; R/. It is the desired Lie group. C.6. Functoriality Theorem C.6. Let G and G 0 be Lie groups. Let W G ! G 0 be a smooth homomorphism. Then its differential at the unit is a homomorphism g ! g0 of corresponding Lie algebras. This immediately follows from the definition of commutator. Theorem C.7. Let G and G 0 be connected Lie groups; denote by g, g0 their Lie algebras. Suppose G is simply connected. Then each homomorphism g ! g0 is a differential of a unique homomorphism G ! G 0 . Proof. Let W g ! g0 be a homomorphism. The graph q of is a subalgebra in g ˚ g0 . Next we consider the group G G 0 and the subspace q in the tangent space at 1. Further, construct a distribution of subspaces as in the proof of Theorem C.5. Again,

518

Addendum

this distribution is integrable in the Frobenius sense. Denote by Q the integral manifold containing the unit. It can be readily checked that Q GG 0 is a subgroup. Obviously, the projections W Q ! G and 0 W Q ! G 0 are homomorphisms. Moreover, the map W Q ! G is covering; since G is simply connected, it follows that the map is a bijection. Thus 0 B 1 W G ! G 0 is the desired homomorphism. C.7. Universal covering group. Let G be a connected Lie group. Consider the universal covering manifold G of G. Let us define a structure of a Lie group on G . Denote by W G ! G the natural projection map. For the unit e 2 G, fix an element e 2 G such that .e / D e and declare that e is a unit. Define the product of elements u and v. For this, consider smooth paths .t /,

.t/ W Œ0; 1 ! G , connecting e DW .0/ D .0/ with u DW .1/ and v DW .1/ respectively. We consider the path ..t / .t // 2 G and its lift ~.t / 2 G such that ~.0/ D e . We assume uv WD ~.1/. C.8. Groups with a given Lie algebra Theorem C.8. a) For a given finite-dimensional Lie algebra g, there exists a unique connected simply connected Lie group G whose Lie algebra is g. b) Moreover, each connected Lie group G B , whose Lie algebra is g, is a quotient G=, where is a discrete subgroup in the center of G. Proof. As we have seen above, a group G B with a given Lie algebra exists; we define G as the universal covering group of G B . Further, we refer to the proof of Theorem C.7. There is a homomorphism W G ! G B , which is bijective in a neighborhood of the unit. Obviously, this map is a covering. The preimage of 1 must be a discrete subgroup. Also, must be normal; this easily implies the last claim of the theorem. C.9. The adjoint representation. Denote by ad.X / the linear operator in a Lie algebra g given by ad.X / D ŒX; Y : By the Jacobi identity, ŒadX; adY D adŒX; Y : Hence ad is a representation of g on itself, it is named the adjoint representation. The adjoint representation Ad.g/ of a Lie group G is defined in the following way: Ad.g/ W g ! g is the differential of the map h 7! ghg 1 at the unit of the group G. For matrix groups it is given by Ad.g/X D gXg 1 : C.10. The Haar measure. By a measure on a manifold we mean a positive Borel measure, i.e., a measure defined on all Borel subsets.


519

S Let M be a manifold, M D Vn , where Vn are open sets diffeomorphic to the open ball in Rn . We say Rthat a measure R on M is smooth if for each function supported by some Vj , we have M f d D Vj f .x/j .x/ dx, where dx is the Lebesgue measure on the ball and j .x/ is a smooth non-negative function. Actually j .x/ dx can be considered as a differential form of a maximal order. If M is orientable, then the collection j .x/dx determines a globally defined differential form on M , otherwise j .x/ dx D ˙k .y/ dy on the intersection Vk \ Vl . Let G be a Lie group. Theorem C.9. a) There is a unique to within proportionality (“Haar”) measure on G invariant under all left shifts g 7! hg. b) The Haar measure is smooth. c) If det Ad.h/ D 1 for all h 2 G, then the Haar measure is also invariant under all right shifts g 7! gh. Proof. a)–b). The existence. We consider an exterior form of maximal degree on g (it is unique to within a scalar factor) and consider its left shifts to tangent spaces at all other points of the group. We get a non-zero invariant differential form (denote it by dg) of a maximal degree on G. The same argument also proves the uniqueness of the invariant measure in the class of smooth measures. Next, we must show that an invariant measure is smooth. Let be a left-invariant BorelR measure. Let ' be a positive compactly supported smooth function on G such that ' dg D 1. Define the following measure5 ' : Z Z Z f .g/ d' .g/ WD f .hg/'.h/ d.g/ dh: G

G

G

It is easy to show that this measure is smooth, therefore it is const dg. Consider a sequence 'j whose supports tend to e. Then 'j tends to weakly, hence itself is const dg. c) Let .g/ be a left-invariant volume form on G. Then its right shift .gh/ is also a left-invariant volume form, hence .gh/ D .h/.g/ with some scalar factor . Since .g/ D .h1 g/, we have .h1 gh/ D .h/.g/. We compare these forms at g D 1. Obviously, the factor is det Ad.h/. There are numerous explicit expressions for Haar measures on given Lie groups, for instance, in this book in (2.10.9), (2.11.16). C.11. Invariant measures on homogeneous spaces. Let G be a (real) Lie group and H be its closed subgroup. Let g h be their Lie algebras. Observation C.10. The homogeneous space G=H admits a canonical structure of a smooth manifold. The tangent space to G=H at the initial point is g=h. 5

It is the convolution of and '.

520

Addendum

Indeed, the set of manifolds gH , where g ranges in G, is a smooth foliation with closed fibres and we get a natural smooth structure on the quotient space. For each h 2 H , the adjoint transformation Ad.h/ W g ! g preserves the subspace h g and hence it induces an operator AdB .h/ W g=h ! g=h. Theorem C.11. a) A G-invariant measure on a homogeneous space G=H exists if and only if j det AdB .h/j D 1 for all h 2 H . b) An invariant measure is smooth. c) G-invariant measure on G=H is unique to within a constant factor. The proof is the same as for the previous theorem. We must only remember that an invariant differential form on G=H is determined modulo a sign.

521

Classical symmetric spaces. Tables

D. Classical symmetric spaces. Tables In what follows, we consider symmetric spaces G=H , where G, H are from the list of Subsection A.2. This means that we consider groups U.p; q/ but not SU.p; q/; GL.n; R/, but not SL.n; R/ and PGL.n; R/, etc.6 By definition, a symmetric subgroup H G is the set of fixed points of an involution g 7! g of a group G. D.1. Riemannian symmetric spaces. The list of classical Riemannian symmetric spaces is given in Table 1. The table can be easily produced from simple arguments of Subsection 2.12.7. The last row of the table is an exception, its meaning is explained in Subsection D.2. 1) Column 4 contains the list of all real classical groups G. 2) Column 6 contains the list of maximal compact subgroups K G. 3) G=K are all classical Riemannian non-compact symmetric spaces. 4) Gcomp =K are all classical Riemannian compact symmetric spaces.7 5) The spaces G=K and Gcomp =K are dual in the following sense. Consider the Lie algebras g k of G K. Denote by p the orthocomplement 8 of k in g, i.e., g D k ˚ p. Then gcomp D k ˚ i p. 6) The spaces Gcomp =K are Grassmannians or isotropic Grassmannians. The overgroup G g Gcomp is the natural group of symmetries of the Grassmannian. 7) The spaces G=K are open domains in the same Grassmannians. The group G g G acts on G=K locally. 8) The contractive (Olshanski) semigroup of G=K satisfies G G g . 9) A realization of G=K. We consider the matrix ball B consisting of matrices z with norm < 1 over K D R, C, H (Column 1) of size p q or n n (Column 2); the matrices satisfy the symmetry condition (Column 3). The set B is the non-compact Riemannian symmetric space G=K; the group G acts on B by linear-fractional transformations; the compact subgroup K is the stabilizer of the point z D 0. 10) A chart on Gcomp =K. We consider the set of all matrices over K of the size indicated in Column 2 satisfying the symmetry condition. Such matrices form an open dense chart in the space Gcomp =K. The group Gcomp and the overgroup G g act on this chart by linear-fractional transformations. In fact, the list of all possible “charts” is Matp;q .R/;

Symmn .R/;

ASymmn .R/;

Matp;q .C/;

Symmn .C/;

ASymmn .C/;

Hermn .C/;

Matp;q .H/;

Hermn .H/;

AHermn .C/:

This list contains all possible sizes of matrices and all natural conditions of symmetry of matrices. 11) The last column GC =KC contains common complexification of G=K and Gcomp =K. 6

See Subsection 2.4.2. In row 3, K D O.n/ O.n/ O.n/ Gcomp is the diagonal subgroup. The space Gcomp =K is the space O.n/. The group O.n/ O.n/ acts on O.n/ by the transformations h 7! g1 hg21 . 8 With respect to the Killing form K.x; y/ D tr xy. 7

522

Addendum

12) Tables 3–4 are continuations of Table 1. Tables 5, 8 contain additional information about Riemannian symmetric spaces. D.2. The semi-exceptional series O.2 ; q/=O.2/ O.2 ; q/. See the last row of Table 1. At first glance, this series is a subseries of O.p; q/=O.p/ O.q/. On the other hand, for this subseries there exists another overgroup G g G, namely O.2 C q; C/. Consider the space C2Cq equipped with a nondegenerate symmetric bilinear form. Denote by Q the space of all isotropic lines in C2Cq . Equivalently, we can regard Q as a quadric in the projective space CP 1Cq . Observation D.1. Q ' O.2 C q/=O.2/ O.q/. Proof. Denote by h; i the standard inner product in R2Cq . We extend it as a bilinear form to C2Cq . Let v C iw 2 C2Cq be an isotropic vector, v, w 2 R2Cn . Then hv; vi D hw; wi;

hv; wi D 0:

Elements of Q are complex lines; i.e., pairs v, w are defined up to the equivalence v C iw e i' .v C iw/;

where > 0, ' 2 R:

Obviously, the group O.2 C q/ acts on Q transitively and the stabilizer of the pair .1; 0; 0; : : : /, .0; 1; 0; : : : / is O.2/ O.q/. The most usual realization of SO0 .2; q/=SO.2/ SO.q/ is as follows (see [95], [177]). We consider the wedge Wq Cq defined by inequalities .Im z0 /2 > .Im z1 /2 C C .Im zq1 /2 I

Im z0 > 0:

There exist the following biholomorphic transformations of the wedge Wq : – shifts z 7! z C i a, where a 2 Rq ; – homotheties z 7! t z, where t > 0; – linear transformations z 7! gz, where z 2 SO0 .1; q 1/; – the indefinite inversion z 7!

z . 2 z02 z12 zq1

These transformations of Wq generate the group SO.2; q/. It acts by quadratic fractional maps of the form 2 ˛.z0 ; : : : ; zq1 ; z02 z12 zq1 / z 7! ; 2 ˇ.z0 ; : : : ; zq1 ; z02 z12 zq1 / where ˛ and ˇ are linear expressions. Observation D.2. The space SO0 .2; q/=SO.2/ SO.q/ can be realized as an open subset in Q. Proof. We consider a complex projective space with homogeneous coordinates . W W z0 W z1 W W zp1 /. Equip it with a quadric 2 D 0: z02 C z12 C C zq1 2 . The map Wq ! Q is given by D 1, D z02 z12 zq1

Size

pq

nn

nn

pq

nn

nn

nn

pq

nn

nn

K

R

R

R

C

C

C

C

H

H

H

…

1

2

3

4

5

6

7

8

9

10

1C

…

2

1

Sp.2n; C/

z D z SO0 .2; q/

GL.n; H/

z D z

…

Sp.p; q/

—

GL.n; C/

z D z

Sp.n/ SO.2/ SO.q/

SO.2 C q/

Sp.n/

Sp.p/ Sp.q/

U.n/

U.n/

U.n/

U.p/ U.q/

O.n/

O.n/

O.p/ O.q/

K

6

Sp.n/ Sp.n/

U.2n/

Sp.p C q/

U.n/ U.n/

O.2n/

SO .2n/

z D z t

U.p C q/

O.n/ O.n/

U.n/

O.p C q/

Gcomp

5

Sp.n/

U.p; q/

O.n; C/

GL.n; R/

O.p; q/

G

4

Sp.2n; R/

z D zt

—

z D z t

z D zt

—

3

SO.q C 2; C/

SO.2 C q; C/=SO.2; C/ SO.q; C/

Sp.2n; C/ Sp.2n; C/=Sp.2n; C/

GL.2n; C/=Sp.2n; C/

SO .4n/ Sp.n; n/

Sp.2.p C q/; C/= Sp.2p; C/ Sp.2q; C/

GL.n; C/ GL.n; C/=GL.n; C/

O.2n; C/=GL.n; C/

Sp.2n; C/=GL.n; C/

GL.p C q; C/=GL.p; C/ GL.q; C/

O.n; C/ O.n; C/=O.n; C/

GL.n; C/=O.n; C/

GL.p C q; H/

U.n; n/

O.2n; C/

Sp.2n; C/

GL.p C q; C/

O.n; n/

Sp.2n; R/

O.p C q; C/=O.p; C/ O.q; C/

GC =KC

Gg GL.p C q; R/

8

7

Table 1. Matrix ball. Riemannian symmetric spaces.


523

524

Addendum

For small q the spaces SO0 .2; q/=SO.2/ SO.q/ admit other realizations as Hermitian symmetric spaces G=K with G given in the list q D 1:

q D 4:

SU.1; 1/;

SU.2; 2/;

q D 2:

SU.1; 1/ SU.1; 1/;

q D 5:

Sp.4; R/;

q D 3:

SU.2; 1/;

q D 6:

SO .8/:

D.3. Hermitian spaces and Hermitizations. A Riemannian symmetric space is said to be Hermitian if it admits an invariant Hermitian metric. The list of (irreducible) classical Hermitian symmetric spaces is U.p; q/=U.p/ U.q/; U.p C q/=U.p/ U.q/;

Sp.2n; R/=U.n/; Sp.n/=U.n/;

SO .2n/=U.n/; O.2n/=U.n/;

SO0 .2; q/=SO.2/ SO.q/; SO.2 C q/=SO.2/ SO.q/:

The picture described in Chapter 7 admits an extension to arbitrary noncompact symmetric Hermitian spaces. Next, any classical symmetric space G=K admits a canonical embedding to a Hermitian space G =K (the Hermitization of G=K)9 as a totally real submanifold of half-dimension. It is rather simple; we look at Column 3 of Table 1. The equation z D ˙z determines a real linear subspace in the space of matrices; we simply complexify it. The final result is given in Table 4. For Riemannian spaces G=K and Gcomp =K we present their Hermitizations G =K and Gcomp =K. We also present the overgroup G g of the Hermitization, in fact G g D .G g /C . D.4. Canonical holomorphic embeddings of matrix balls. Now we consider the following z H z be a non-compact Hermitian symmetric space, let situation (see Subsection 7.5.17). Let G= z be a symmetric subgroup. We enumerate the cases when G=K is a complex submanifold GG z Kz (in particular, G=K itself is Hermitian). in G= Table 2. Hermitian subballs of Hermitian balls. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

9


z G U.p; q/ U.p; q/ U.p C r; q C s/ U.n; n/ U.n; n/ Sp.2n; R/ Sp.2n; R/ Sp.2.p C q/; R/ Sp.2.p C q/; R/ SO .2n/ SO .2n/ SO .2.p C q// SO .2.p C q// SO0 .n; 2/ SO0 .n; 2/ SO0 .n C 1; 2/

G U.p; q/ U.p; q/ U.r; s/ Sp.2n; R/ SO .2n/ Sp.2n; R/ U.p; q/ Sp.2p; R/ Sp.2q; R/ SO .2n/ U.p; q/ SO .2p/ SO .2q/ SO0 .n; 2/ SO0 .n; 2/

525


D.5. Causal symmetric spaces. The list of causal symmetric spaces is given in Table 3. It is produced from Table 1 by a procedure explained in the Addendum to Chapter 3, Subsection 17. Table 3 contains the list of causal symmetric spaces G g =G and G gB =G. The groups G g and G are the same as in Table 1. Denote by r the orthocomplement of g in gg ; then ggB D g ˚ i r: The last column indicates the (common) overgroup G gg acting on G g =G and G gB =G. In some cases, G g =G D G gB =G. This means that this space has two essentially different causal structures (see Addendum to Chapter 3, Subsection 15). In the next subsection, we construct embeddings of pseudo-Riemannian symmetric spaces to Grassmannians; causal spaces are mapped to causal Grassmannians (Grassmannians of maximal isotropic subspaces with respect to anti-Hermitian forms over R, C, H), and we can induce the causal structure. Table 3. Causal symmetric spaces. 1

2

3

4

Gg

G gB

G

G gg

1

GL.p C q; R/

U.p; q/

O.p; q/

Sp.2.p C q/; R/

2

Sp.2n; R/

Sp.2n; R/

GL.n; R/

Sp.2n; R/ Sp.2n; R/

3

O.n; n/

SO .2n/

O.n; C/

U.n; n/

4

GL.p C q; C/

U.p; q/ U.p; q/

U.p; q/

U.p C q; p C q/

5

Sp.2n; C/

Sp.2n; R/ Sp.2n; R/

Sp.2n; R/

Sp.4n; R/

6

O.2n; C/

SO .2n/ SO .2n/

SO .2n/

SO .4n/ U.n; n/ U.n; n/

7

U.n; n/

U.n; n/

GL.n; C/

8

GL.p C q; H/

U.2p; 2q/

Sp.p; q/

SO .4.p C q//

9

SO .4n/

SO .4n/

GL.n; H/

SO .4n/ SO .4n/ U.2n; 2n/

10

Sp.n; n/

Sp.4n; R/

Sp.2n; C/

11

SO.n C 2; C/

SO0 .2; n/ SO0 .2; n/

SO0 .2; n/

12

SO0 .p C 1; q C 1/

SO0 .2; p C q/

SO0 .1; p/ SO0 .1; q/

D.6. Models of pseudo-Riemannian symmetric spaces. There are 54 series of classical symmetric spaces and 131 exceptional spaces (see M. Berger, [20])10 . Our purpose is to give initial data, which allows us to handle these spaces as it was done for matrix balls. In what follows, we present a list of classical symmetric spaces and also give uniform models for them. In fact we realize all such spaces as open domains in Grassmannians. ˛. Preparation. List of semi-involutions. Consider a space (a right module) V over K D R, C, H. A semi-involution is a linear or antilinear operator in Kn satisfying J 2 D ˙1 (see also Subsection 1.5.4). There exist the following semi-involutions (we write canonical forms): 10 The list is rather strange. It contains numerous interesting objects; reasons for such interest seem to be quite different. On the other hand, some rows of the list seem exotic; I think that some of the rows never were subjects of a “personal” interest. All these spaces are relatives as abstract mathematical objects. But the individuality of some series and the “identity” of some taxons are strong.

526

Addendum

1) K D R, J 2 D 1. Then J D direct sum of two subspaces.

1

2) K D R, J 2 D 1. Then J D

0 0 1

0 1 1 0

. The operator J determines a splitting of V into a

, i.e., J is an operator of complex structure. 0 3) K D C, J is linear, J D 1. Then J D 10 1 . 2

4) K D C, J is antilinear, J 2 D 1. Then J v D vN is an operator of complex conjugation. 5) K D C, J is antilinear, J 2 D 1. Then J is an operator of quaternionic structure, J.w1 ˚ w2 / D .w x1 . I.e., the operators x2 / ˚ w ˛ C ˇi C J C ıiJ;

where ˛, ˇ, , ı 2 R;

form the algebra H. Therefore our space is an H-module. 0 6) K D H, J is linear, J 2 D 1. Then J D 10 1 . 7) K D H, J is linear, J 2 D 1. Then J v D iv, where i is the quaternionic imaginary unit. 8) Quaternionic antilinear involutions J v D ˙vN are not interesting for us. ˇ. The first part of the list. These constructions imitate the construction of the space Sp.2n; R/=U.n/ discussed in detail in § 3.2–3.3 (apparently it is impossible to understand the table without this example). The spaces Sp.2.p C q/; R/=U.p; q/

Sp.4n; R/=Sp.2n; C/

or

were touched upon in Subsections 3.3.1, 3.3.9 respectively. Below the term form means one of the forms mentioned in Subsection A.1. We have the following initial data: – a space V WD K2n , where K D R, C, H (the dimension of the space is even); – a pair forms, basic form ƒ and managing form M on the space V D Kn ; – a control semi-involution J in K2n that “regards” the form ƒ in the following sense: ƒ.J v; J w/ D ˙ƒ.v; w/ or

ƒ.J v; J w/ D ˙ƒ.v; w/:

– M.v; w/ D ƒ.J v; w/. Points of a symmetric space G=H are subspaces P V such that – P is maximal ƒ-isotropic and dim P D n D

1 2

dim V ;

– “the managing” form M is nondegenerate on P ; an equivalent condition is P \ JP D 0. In the following table we give a list of all possible situations of such kind. In each block 1–30 we present: 1) the symmetric space G=H ; the group G preserves both the forms ƒ, M and commutes with J ; 2) the overgroup G g that is the group preserving the basic form ƒ; this group acts locally on G=H ; 3) the group GM of operators in V preserving the managing form M ; 4) the group G J of linear operators in V commuting with J ;

1C2

1C1

10

9

8

7

6

5

4

3

2

1

1

SO0 .1/ SO0 .p/ SO0 .1/ SO0 .q/

SO0 .1 C p/ SO0 .1 C q/

SO0 .2/ SO0 .q/

Sp.n/

Sp.n/

Sp.p/ Sp.q/

U.n/

U.n/

U.n/

U.p/ U.q/

O.n/

O.n/

O.p/ O.q/

K

2

SO0 .1; p/ SO0 .1; q/

SO0 .2 C q/

SO0 .2; q/

Sp.n/ Sp.n/

Sp.2n; C/

U.2n/

GL.n; H/

Sp.p C q/

Sp.p; q/

U.n/ U.n/

GL.n; C/

O.2n/

SO .2n/

Sp.n/

Sp.2n; R/

U.p C q/

U.p; q/

O.n/ O.n/

O.n; C/

U.n/

GL.n; R/

O.p C q/

O.p; q/

Gcomp

G

SO0 .2; p C q/

SO0 .q C 2; C/

Sp.n; n/

SO .4n/

GL.p C q; H/

U.n; n/

O.2n; C/

Sp.2n; C/

GL.p C q; C/

O.n; n/

Sp.2n; R/

GL.p C q; R/

Gg

3

SO0 .2 C p C q/

SO0 .2; p C q/

SO0 .2 C q/ SO0 .2 C q/

SO0 .2; q/ SO0 .2; q/

Sp.2n/

Sp.4n; R/

O.4n/

SO .4n/

U.2p C 2q/

U.2p; 2q/

U.2n/

U.n; n/

O.2n/ O.2n/

SO .2n/ SO .2n/

Sp.2n/ Sp.2n/

Sp.2n; R/ Sp.2n; R/

U.p C q/ U.p C q/

U.p; q/ U.p; q/

O.2n/

SO .2n/

Sp.n/

Sp.2n; R/

U.p C q/

U.p; q/

Gcomp

G

4

Table 4. Hermitizations of symmetric spaces.

SO0 .2/ SO0 .p C q/

SO0 .2/ SO0 .q/

SO0 .2/ SO0 .q/

U.2n/

U.2n/

U.2p/ U.2q/

U.n/ U.n/

U.n/ U.n/

U.n/ U.n/

U.p/ U.q/

U.p/ U.q/

U.n/

U.n/

U.p/ U.q/

K

5

SO0 .2 C p C q; C/

SO0 .2 C q; C/

SO0 .2 C q; C/

Sp.4n; C/

O.4n; C/

GL.2.p C q/; C/

GL.n; C/

O.2n; C/ O.2n; C/

Sp.2n; C/

Sp.2n; C/

GL.p C q; C/

GL.p C q; C/

O.2n; C/

Sp.2n; C/

GL.p C q; C/

Gg

6


527

528

Addendum

5) the space V and the type of semi-involution; 6) we indicate cases when the series contains Riemannian symmetric spaces; 7) sometimes ƒ-isotropic M -nondegenerate subspaces constitute a union of several symmetric spaces G=Hj (as in Subsection 3.3.1); we indicate such cases. In Table 5, the term “Grassmannian” means ƒ-Lagrangian Grassmannian. Finally, note that G D G g \ GM D G g \ G J D GM \ G J :

Table 5. Pseudo-Riemannian symmetric spaces. List 1. 1 O.p; q/ O.p; q/=O.p; q/ G g D O.p C q; p C q/ G J D GL.p C q; R/ GL.p C q; R/ GM D O.2p; 2q/ V D R2.pCq/ , ƒ is symmetric, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ O.p/ O.p/=O.p/ is a compact Riemannian symmetric space 2 Sp.2n; R/ Sp.2n; R/=Sp.2n; R/ G g D Sp.4n; R/

G J D GL.2n; R/ GL.2n; R/

GM D Sp.4n; R/

V D R4n , ƒ is skew symm., J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ 3 GL.n; R/=O.p; n p/ G g D Sp.2n; R/

G J D GL.n; R/ GL.n; R/

GM D O.n; n/

V D R2n ,ƒ is skew symm., J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ Open orbits of G in Grassmannians are GL.n; R/=O.p; n p/ GL.n; R/=O.n/ is a non-compact Riemannian symmetric space 4 GL.2n; R/=Sp.2n; R/ G g D O.2n; 2n/ V DR

4n

G J D GL.2n; R/ GL.2n; R/

GM D Sp.2n; R/

, ƒ is symmetric, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

5 O.n; C/=O.p; n p/ G g D O.n; n/ V DR

2n

G J D GL.n; C/

GM D O.n; n/

, ƒ is symmetric, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

Open orbits of G in Grassmannians are O.n; C/=O.p; n p/ O.n; C/=O.n/ is a non-compact Riemannian symmetric space 6 Sp.2n; C/=Sp.2n; R/ G g D Sp.4n; R/ V DR

4n

G J D GL.2n; C/

GM D Sp.4n; R/

ƒ is skew symm., J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

7 U.n; n/=Sp.2n; R/ G g D O.2n; 2n/ V DR

4n

G J D GL.2n; C/

ƒ is symmetric, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

GM D Sp.4n; R/

529


8 U.p; q/=O.p; q/ G g D Sp.2.p C q/; R/ V DR

2.pCq/

G J D GL.p C q; C/

GM D O.2p; 2q/

, ƒ is skew symm., J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/

U.p/=O.p/ is a compact Riemannian symmetric space 9 O.n; C/ O.n; C/=O.n; C/ G g D O.2n; C/ V DC

2n

G J D GL.n; C/ GL.n; C/

GM D O.2n; C/

, ƒ is symmetric, J is linear, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

10 Sp.2n; C/ Sp.2n; C/=Sp.2n; C/ G g D Sp.4n; C/ V DC

4n

G J D GL.2n; C/ GL.2n; C/

GM D Sp.4n; C/

, ƒ is skew symm., J is linear, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

11 U.p; q/ U.p; q/=U.p; q/ G g D U.p C q; p C q/ G J D GL.p C q; C/ GL.p C q; C/ GM D U.2p; 2q/ V D C2.pCq/ , ƒ is Hermitian, J is linear, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ U.p/ U.p/=U.p/ is a compact Hermitian symmetric space 12 GL.2n; C/=Sp.2n; C/ G g D O.4n; C/ V DC

4n

G J D GL.2n; C/ GL.2n; C/

GM D Sp.4n; C/

, ƒ is symmetric, J is linear, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/

13 GL.n; C/=O.n; C/ G g D Sp.2n; C/ V DC

2n


GM D O.2n; C/

, ƒ is skew symm., J is linear, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/

14 GL.n; C/=U.p; n p/ G g D U.n; n/ V DC

2n


GM D U.n; n/

, ƒ is Hermitian, J is linear, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/

Open orbits of G in Grassmannians are GL.n; C/=U.p; n p/ GL.n; C/=U.n/ is a non-compact Riemannian symmetric space 15 Sp.2n; R/=U.p; n p/ G g D Sp.2n; C/ V DC

2n

G J D GL.2n; R/

GM D U.n; n/

, ƒ is skew symm., J is antilin., J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/

Open orbits of G in Grassmannians are Sp.2n; R/=U.p; n p/ Sp.2n; R/=U.n/ is a non-compact Hermitian symmetric space 16 O.2p; 2q/=U.p; q/ G g D O.2.p C q/; C/ V DC

2.pCq/

G J D GL.2.p C q/; R/

GM D U.2p; 2q/

, ƒ is symmetric, J is antilin., J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/

O.2p/=U.p/ is a compact Hermitian symmetric space 17 O.n; n/=O.n; C/ G g D U.n; n/ V DC

2n

G J D GL.2n; R/

GM D O.2n; C/

, ƒ is Hermitian, J is antilin., J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/

530

Addendum

18 Sp.4n; R/=Sp.2n; C/ G g D U.2n; 2n/

G J D GL.4n; R/

GM D Sp.4n; C/

V D C4n , ƒ is Hermitian, J is antilin., J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ 19 Sp.p; q/=U.p; q/ G g D Sp.2.p C q/; C/ V DC

2.pCq/

G J D GL.p C q; H/

GM D U.2p; 2q/

, ƒ is skew symm., J is antilin., J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

Sp.p/=U.p/ is a compact Hermitian symmetric space 20 SO .2n/=U.p; n p/ G g D O.2n; C/ V DC

2n

G J D GL.n; H/

GM D U.n; n/

, ƒ is symmetric, J is antilin., J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

Open orbits of G in Grassmannians are SO .2n/=U.p; n p/ SO .2n/=U.n/ is a non-compact Hermitian symmetric space 21 Sp.n; n/=Sp.2n; C/ G g D U.2n; 2n/ V DC

4n

G J D GL.2n; H/

GM D Sp.4n; C/

, ƒ is Hermitian, J is antilin., J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

22 SO .2n/=O.n; C/ G g D U.n; n/

G D GL.n; H/

GM D O.2n; C/

V D C2n , ƒ is Hermitian, J is antilin., J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ 23 Sp.p; q/ Sp.p; q/=Sp.p; q/ G g D Sp.p C q; p C q/ G J D GL.p C q; H/ GL.p C q; H/ GM D Sp.2p; 2q/ V D H2.pCq/ , ƒ is Hermitian, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ Sp.p/ Sp.p/=Sp.p/ is a compact Riemannian symmetric space 24 SO .2n/ SO .2n/=SO .2n/ G g D SO .4n/ V DH

2n

G J D GL.n; H/ GL.n; H/

GM D SO .4n/

, ƒ is anti-Hermitian, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

25 GL.n; H/=SO .2n/ G g D Sp.n; n/ V DH

2n


GM D SO .4n/

, ƒ is Hermitian, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

26 GL.n; H/=Sp.p; n p/ G g D SO .4n/ V DH

2n


GM D Sp.n; n/

, ƒ is anti-Hermitian, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/

Open orbits of G in Grassmannians are GL.n; H/=Sp.p; n p/ GL.n; H/=Sp.n/ is a non-compact Riemannian symmetric space 27 U.2p; 2q/=Sp.p; q/ G g D SO .4.p C q// V DH

2.pCq/

G J D GL.2.p C q/; C/

GM D Sp.2p; 2q/

, ƒ is anti-Hermitian, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

U.2p/=Sp.p/ is a compact Riemannian symmetric space

531


28 U.n; n/=SO .2n/ G g D Sp.n; n/ V DH

2n

G J D GL.2n; C/,

GM D SO .4n/

, ƒ is Hermitian, J D 1, ƒ.J v; J w/ D ƒ.v; w/ 2

29 O.2n; C/=SO .2n/ G g D SO .4n/

G J D GL.2n; C/

GM D SO .4n/

V D H2n , ƒ is anti-Hermitian, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ 30 Sp.2n; C/=Sp.p; n p/ G g D Sp.n; n/

G J D GL.2n; C/

GM D Sp.n; n/

V D H2n , ƒ is Hermitian, J 2 D 1, ƒ.J v; J w/ D ƒ.v; w/ Open orbits of G in Grassmannians are Sp.2n; C/=Sp.p; n p/ Sp.2n; C/=Sp.n/ is a non-compact Riemannian symmetric space

. The second part of the list. Now we have a space V D K2n equipped with a “basic” form ƒ. A point of a symmetric space G=H is a pair of complementary ƒ-isotropic subspaces. The group G is the group of isometries of the form ƒ. In Table 6 we also present the overgroup G g acting locally on G=H . However, in these cases g G D G G. Table 6. Pseudo-Riemannian symmetric spaces. List 2. 31 O.n; n/=GL.n; R/

G g D O.n; n/ O.n; n/

32 Sp.2n; R/=GL.n; R/

G g D Sp.2n; R/ Sp.2n; R/

33 O.2n; C/=GL.n; C/

G g D O.2n; C/ O.2n; C/

34 Sp.2n; C/=GL.n; C/

G g D Sp.2n; C/ Sp.2n; C/

35 U.n; n/=GL.n; C/

G g D U.n; n/ U.n; n/

36 Sp.n; n/=GL.n; H/

G g D Sp.n; n/ Sp.n; n/

37 SO .4n/=GL.n; H/

G g D SO .4n/ SO .4n/

ı. The third part of the list. Now we have a space V D K2n and a “control” semiinvolution J . A symmetric space G=H consists of subspaces P V such that V D P ˚ JP . The group G is the centralizer of the semi-involution. The overgroup G g D GL.V / acts locally on G=H . Table 7. Pseudo-Riemannian symmetric spaces. List 3. 38 GL.n; R/ GL.n; R/=GL.n; R/

G g D GL.2n; R/

V D R2n , J 2 D 1 39 GL.n; C/=GL.n; R/ V DR

2n

G g D GL.2n; R/

, J D 1 2

40 GL.n; C/ GL.n; C/=GL.n; C/ G g D GL.2n; C/

532

Addendum V D C2n , J 2 D 1, J is linear G g D GL.2n; C/

41 GL.2n; R/=GL.n; C/

V D C2n J 2 D 1, J is antilinear G g D GL.2n; C/

42 GL.n; H/=GL.n; C/ V DC

2n

, J D 1, J is antilinear 2

43 GL.n; H/ GL.n; H/=GL.n; H/ G g D GL.2n; H/ V D H2n , J 2 D 1, J is linear G g D GL.2n; H/

44 GL.2n; C/=GL.n; H/

V D H2n , J 2 D 1, J is linear ". Forth part of the list. Now we have only a form M in Kn . The space G=H consists of M -regular subspaces of given dimension (and given inertia, if the form M is Hermitian). Table 8. Pseudo-Riemannian symmetric spaces. List 4. 45 O.p; q/=O.r; s/ O.p r; q s/ S Set of regular subspaces is

G g D GL.p C q; R/ O.p; q/=O.r; s/ O.p r; q s/

r;sW rCsDm;r6p;s6q

O.p; q/=O.p/ O.q/ is a non-compact Riemannian symmetric space O.p/=O.r/ O.p r/ is a compact Riemannian symmetric space SO0 .2; q/=SO0 .2/ SO0 .q/ is a non-compact Hermitian symmetric space SO0 .2 C q/=SO0 .2/ SO0 .q/ is a compact Hermitian symmetric space 46 Sp.2.k C l/; R/=Sp.2k; R/ Sp.2l; R/ G g D GL.2.k C l/; R/ 47 O.n C m; C/=O.n; C/ O.m; C/

G g D GL.n C m; C/

48 Sp.2.k C l/; C/=Sp.2k; C/ Sp.2l; C/ G g D GL.2.k C l/; C/ 49 U.p; q/=U.r; s/ U.p r; q s/ S Set of regular subspaces is

G g D GL.p C q; C/ U.p; q/=U.r; s/ U.p r; q s/

r;sW rCsDm;r6p;s6q

U.p; q/=U.p/ U.q/ is a non-compact Hermitian symmetric space U.p/=U.r/ U.p r/ is a compact Hermitian symmetric space 50 Sp.p; q/=Sp.r; s/ Sp.p r; q s/ S Set of regular subspaces is

G g D GL.p C q; H/ Sp.p; q/=Sp.r; s/ Sp.p r; q s/

r;sW rCsDm;r6p;s6q

Sp.p; q/=Sp.p/ Sp.q/ is a non-compact Riemannian symmetric space Sp.p/=Sp.r/ Sp.p r/ is a compact Riemannian symmetric space 51 SO .2.m C n//=SO .2m/ SO .2n/ G g D GL.m C n; H/

. Fifth part of the list. Now a point of a space G=H is a pair of transversal p-dimensional and q-dimensional spaces. By definition, the space G=H is an open dense subset in the product


533

of two Grassmannians. Table 9. Pseudo-Riemannian symmetric spaces. List 5. 52 GL.p C q; R/=GL.p; R/ GL.q; R/

G g D GL.p C q; R/ GL.p C q; R/

53 GL.p C q; C/=GL.p; C/ GL.q; C/

G g D GL.p C q; C/ GL.p C q; C/

54 GL.p C q; H/=GL.p; H/ GL.q; H/

G g D GL.p C q; H/ GL.p C q; H/

. Taxonomy There are several natural classes of pseudo-Riemannian symmetric spaces. We briefly discuss them. Classical groups. They are located in Tables 5, 7. The semi-involution J is linear and preserves the ”basic” form ƒ, J 2 D 1. These properties characterize the taxon of the classical groups in our table. Causal spaces. See Subsection D.5. They are located in Tables 5, 6. In this case, the overgroup G g is a “Hermitian group” U.p; q/, Sp.2n; R/, SO .2n/. In this case Grassmannians are causal (see Addendum to Section 3, Subsections 13–14), therefore open domains in Grassmannians are causal. Spaces GC =GR . Here GC is a complex group, GR is its real form (they are relatively simple objects of harmonic analysis). Complex groups belong to this class. In this case, J is linear, J 2 D 1 , ƒ.J v; J w/ D ƒ.v; w/. In the complex space, this is equivalent to J 2 D C1, ƒ.J v; J w/ D Cƒ.v; w/). This characterizes the taxon in our table. Complex spaces. A complex structure on G=H is produced by a torus T lying in the center of H .11 In fact this class splits into the following two (intersecting) taxons: complex algebraic symmetric spaces and pseudo-Hermitian spaces. Complex affine algebraic varieties. In this case, both the groups G, H are complex; the list is given in the last column of Table 1. In this case, we have a quadratic form on each complex tangent space. The overgroup is complex; the forms ƒ, M are bilinear (if they are present) and the semiinvolution J is linear (if it is present). Pseudo-Kähler (pseudo-Hermitian) spaces. There are 10 complete series of pseudo-Hermitian spaces. Namely, we have 5 (D 3 2 1) series containing Hermitian spaces (in these cases, H D U.p; q/), two series GL.2n; R/=GL.n; C/;

GL.n; H/=GL.n; C/:

and three series Sp.2n; C/=GL.n; C/; 11

O.2n; C/=GL.n; C/;

GL.p C q; C/=GL.p; C/ GL.q; C/:

If it does not cancel with the center of G, as for GL.n; C/=U.n/.

534

Addendum

For instance, for the latter series the tangent space consists of pairs .z; u/ of p q and q p matrices. We consider the conjugate complex structure on the space q p matrices. Then the form ˝ ˛ .z; u/; .Qz; v/ Q WD tr z uQ C tr zu Q is Hermitian. Also there are two subseries SO0 .p C 2; q/=SO0 .p; q/ SO0 .2/;

SO .2n C 2/=SO .2n/ SO .2/:

Para-Hermitian (para-Kähler) spaces. A para-Kähler space is a symplectic space having two transversal (invariant) Lagrangian foliations. Tables 6 and 9 consist of 10 series of paraHermitian spaces. In these cases we get an embedding of G=H to a product of two Grassmannian, say Gr Gr. The latter space has two foliations fxg Gr and Gr fyg. We take their pullbacks. Also there are two subseries SO0 .p C 1; q C 1/=SO0 .1; 1/ SO0 .p; q/;

SO.n C 2; C/=SO.2; C/ SO.n; C/:

Spaces of forms. The space of forms of a given type on a given linear space V (see Subsection A.1) is a symmetric space G=H ; we have G D GL.V / and H is the group of symmetries of a form. Problem D.1. Tables 5–9 correspond to our way of “industrial production” of classical symmetric spaces. Certainly, they are taxons. Find a law of this classification. . Involutions of the list. Now let g be a classical Lie algebra, h a symmetric subalgebra, k a compact subalgebra (i.e., the subalgebra of the maximal compact subgroup). Let be the involution determining h and the involution determining k. We can assume D (this is a theorem obvious in each explicit case). Denote by p the orthocomplements to h, by q the orthocomplement to k. Then g D h ˚ p D k ˚ q D .h \ k/ ˚ .h \ q/ ˚ .p \ k/ ˚ .p \ q/: In fact, we get a Z2 ˚ Z2 gradation of g. Duality. The dual symmetric pair is .gB ; h/ where gB WD h ˚ i p. This is a kind of analytic continuation (as the Lobachevsky plane is an analytic continuation of the sphere). Association. The associated symmetric pair is .g; ha /, where ha D .h \ k/ ˚ .p \ q/: In other words, ha consists of fixed points of the involution . The duality and association are involutions, they do not commute. More operations. Next, we have a Z2 ˚Z2 ˚Z2 gradation on the Lie algebra gC D g˚i g. Therefore we can produce many symmetric pairs from a given one. Problem D.2. Investigate the space G=H D GL.n; R/=O.p; q/. Problem D.3. Explain the appearance of “10 rows”, which is present in several tables. Also explain why for some tables rows add up according to 7 C 3 D 10.


535

. Topology. Let H and K be in the same position as in Subdivision . Then: Theorem D.3. a) K=K \ H is a subsymmetric space12 in G=H . b) G=H is a fiber bundle over K=K \ H ; fibers are p \ q. Flensted-Jensen taxon. It is determined by the condition rk G=H D rk K=K \ H I the rank is the maximal possible dimension of a flat subsymmetric space.13 Under this condition there are discrete series in L2 .G=H /. ~. Bibliographical notes. Semisimple Lie algebras were introduced by Wilhelm Killing (1888-90), the symmetric spaces by Petr A. Shirokov (1923) and Harry Levy (1925). Root systems were classified by Killing14 , semisimple groups and Riemannian symmetric spaces by Eli Cartan15 . Other classifications related to pseudo-Riemannian symmetric spaces. The classification of spaces: A. G. Fedenko (1956), M. Berger (1957). Contractive semigroups: T. Nagano (1965), G. I. Olshanski (1983). Embeddings to Grassmannians: T. Nagano (1965), B. O. Makarevich (1973-79). Pseudo-Hermitian spaces: R. Shapiro (1971). Hermitizations: H. Jaffee (1975). Causal structures: Olshanski (1984). Para-Kähler spaces: S. Kaneyuki, M. Kozai (1985). Involutions of the list of symmetric spaces. In a special case, the duality was conjectured by Johann Lambert (1766) and later it was used by Nikolay Lobachevsky and by Ernst Minding (1840). Mysteriously, Lobachevsky could not prove it16 . A subsymmetric space in G=H is a totally geodesic submanifold with nondegenerate pseudo 0 and Riemannian metric. For instance, consider the group SL.2; R/. Then the subgroups 0 1 1 t cos ' sin ' sin ' cos ' are subsymmetric spaces. But the subgroup 0 1 is a geodesic in an isotropic direction and not a subsymmetric space. Generally, geodesics on groups are one-parameter subgroups and their shift. Subsymmetric spaces are semisimple subgroups. 13 A good representative example of flat subsymmetric spaces (D Cartan subspaces) is Theorem 2.12.2. Problem: classify flat subsymmetric subspaces in SL.n; R/. For the case n D 2, see the previous footnote. 14 With a minor mistake: two 4-dimensional root systems in his list actually coincide. He also constructed “by hand” the exceptional Lie algebra G2 and claimed the existence of other exceptional algebras. 15 After Killing, the structure of the exceptional Lie algebras g D F4 , E6 , E7 , E8 was clear and Cartan constructed them in his thesis (1894). There were some doubts on completeness of proofs (electronic computers yet were not invented). Later Cartan proposed an alternative construction (and proof). He takes an appropriate maximal parabolic subalgebra p g. Denote by n its nilpotent part. We can identify a tangent space to G=P with n . Next, n is non-Abelian (for all classical Lie algebras there are parabolics with Abelian nil-radical). Then the annihilator of Œn; n in n determines a non-integrable distribution (in Frobenius sense) of subspaces on G=P . Cartan constructs this distribution locally “by hand”. The Lie algebra g is the algebra of infinitesimal automorphisms of the distributions. 16 He wrote various equations in polar coordinates .r; '/. It was sufficient to observe that the equations of geodesics on the plane and on the “fictitious plane” differ by a change of variable in r (after this, the consistency of “fictitious geometry” and the analytic continuation sphere – Lobachevsky plane become an “end-game for a grand master”). It is really mysterious that Lobachevsky did not observe this. This step was done by Giuseppe Battaglini (1867). The analytic continuation of trigonometric formulas (as the cosine theorem) and formulas for geodesics were standpoints for the nice paper by Eugenio Beltrami [13], 1868. 12

536

Addendum

I could not find an origin of the association. In any case, this operation and its relatives were applied by Mogens Flensted-Jensen (1980) in the harmonic analysis. He also proposed a taxonomy of symmetric spaces into nine disjoint classes, see [50].

Bibliography

[1] Akhiezer, N. I., Elements of the theory of elliptic functions. Translated from the Russian, Transl. Math. Monogr. 79, Amer. Math. Soc., Providence, R.I., 1990. 272, 284 [2] Akivis, M. A., On the differential geometry of a Grassmann manifold. Tensor (N. S.) 38 (1982), 232–282. [3] Alekseev, A., Meinrenken, E., On the Kashiwara-Vergne conjecture. Invent. Math. 164 (2006), no. 3, 615–634. 284 [4] Alekseevskij, A. V., Finite commutative Jordan subgroups of complex simple Lie groups. Funct. Anal. Appl. 8 (1974), 277–279). 221, 426 [5] Alekseevski, D. V., Vinogradov, A. M., Lychagin, V. V., Basic ideas and concepts of differential geometry. Geometry I, translated from the Russian, Encyclopaedia Math. Sci. 28, Springer-Verlag, Berlin 1991. 140, 224 [6] Ando, T., Totally positive matrices. Linear Algebra Appl. 90 (1987), 165–219. 223 [7] Andrews, G. E., Askey, R., Roy, R., Special functions. Encyclopedia Math. Appl. 71, Cambridge University Press, Cambridge 2000. 135, 247, 275, 368 [8] Arazy, J., Upmeier, H., Boundary measures for symmetric domains and integral formulas for the discrete Wallach points. Integral Equations Operator Theory 47 (2003), no. 4, 375–434 380 [9] Azizov, T. Ya., Iokhvidov I. S., Foundations of the theory of linear operators in spaces with indefinite metric. Nauka, Moscow, 1986; English transl.: Pure Appl. Math. (N. Y.), John Wiley and Sons, Chichester 1989. 223 [10] Bargmann, V., On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14 (1961), 187–214. [11] Barre, S., Immeubles de Tits triangulaires exotiques. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 4, 575–603. 504 [12] Barut, A. O., Raczka, R., Theory of group representations and applications. Second edition, World Scientific, Singapore 1986. [13] Beltrami, E., Saggio di interpretazione della geometria non-Euclidea. Giornale di Matematiche 6 (1868), 284–312; reprinted in E. Beltrami Opere Matematiche di Eugenio Beltrami, Vol. 1, U. Hoepli, Milano 1902. 535 [14] Berezin, F. A., The method of second quantization. Translated from the Russian, Academic Press, New York, London 1966. 56, 225, 284, 300, 313 [15] Berezin, F. A., Several remarks about the associative envelope of a Lie algebra. Funct. Anal. Appl. 1 (1967), 91–102. 284 [16] Berezin, F. A., Wick and anti-Wick symbols of operators. Math. USSR Sb. 15 (1972), 577–606. 284 [17] Berezin F. A., Non-Wiener path integrals. Teoret. Mat. Fiz. 6 (1971), no. 2, 194–212; English transl.: Theoret. and Math. Phys. 6 (1971), no. 2, 141–155. 284

538

Bibliography

[18] Berezin, F. A., Quantization in complex symmetric spaces. Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 363–402; English transl.: Math. USSR Izv. 9 (1976), 341–379. 380 [19] Berezin, F. A., The connection between covariant and contravariant symbols of operators on classical complex symmetric spaces. Soviet Math. Dokl. 19 (1978), 786–789. [20] Berger, M., Les espaces symmetriques noncompacts. Ann. Sci. Ecole. Norm. Sup. 74 (1957), 85–177. 525 [21] Besse, A. L., Einstein manifolds. Springer-Verlag, Berlin, 1987. 140, 224 [22] Bhatia, R., Matrix analysis. Grad. Texts in Math. 169, Springer-Verlag, New York, 1997. [23] Boas, R. P., Jr., Entire functions. Academic Press, New York 1954. 267, 284 [24] Borel, A., Hirzebruch, F., Characteristic classes and homogeneous spaces. I–III. Amer. J. Math. 80 (1958), 458–538; ibid. 81 (1959), 315–382; ibid. 82 (1960), 491–504. 223 [25] Borevich, Z. I., Shafarevich, I. R., Number theory. Translated from the Russian, Pure and Appl. Math. 20, Academic Press, 1966. 426, 427, 504 [26] Borodin, A., Olshanski, G., Harmonic analysis on the infinite- dimensional unitary group and determinantal point processes. Ann. of Math. (2) 161 (2005), 1319–1422. 224 [27] Bourbaki, N., Elements de mathematique. Fasc. XXXIV, Groupes et algebres de Lie, Chapitres IV, V et VI: Groupes de Coxeter et systemes de Tits, Groupes engendres par des reflexions, Systemes de racines, Hermann, Paris 1968. 464 [28] de Branges, L., Hilbert spaces of entire functions. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. 340 [29] Brezin, J., Harmonic analysis on nilmanifolds. Trans. Amer. Math. Soc. 150 (1970), 611–618. 407 [30] Brown, K. S., Buildings. Springer Monogr. Math., Springer-Verlag, New York 1998. 504 [31] Bruhat, F., Tits, J., Groupes reductifs sur un corps local. Inst. Hautes Etudes Sci. Publ. Math. 41 (1972), 5–251. 504 [32] Brunet, M., Kramer, P., Complex extension of the representation of the symplectic group associated with the canonical commutation relations. In Group theoretical methods in physics, Lecture Notes in Phys. 50, Springer-Verlag, Berlin 1976, 441–449. 56 [33] Burichenko, V. P., Tiep, Pham Huu, Invariant lattices of type F4 and E6 : the automorphism groups. Comm. Algebra 21 (1993), no. 12, 4641–4677. 221 [34] Cartier, P., Quantum mechanical commutation relations and theta functions. In Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence, R.I., 1966, 361–383. 284, 407 [35] Cassels, J. W. S., Rational quadratic forms. Academic Press, 1978. 427, 481, 504 [36] Cherednik, I., Double affine Hecke algebras. Cambridge University Press, Cambridge 2005 222 [37] Chriss, N., Ginzburg, V., Representation theory and complex geometry. Birkhäuser, Boston 1997. 223 [38] Clausen, M., Baum, U., Fast Fourier transforms. B.I. Wissenschaftsverlag, Mannheim 1993. 426

Bibliography

539

[39] Coolidge, J. L., A treatise on the circle and the sphere. Clarendon Press, Oxford 1916; Reprint, Chelsea Publishing Co., Bronx, N.Y., 1971. 100 [40] Cordoba, A., Fefferman, Ch., Wave packets and Fourier integral operators. Comm. Partial Differential Equations 3 (1978), no. 11, 979–1005. 284 [41] Cowling, M., Steger, T., The irreduciblity of restrictions of unitary representations to lattices. J. Reine Angew. Math. 420 (1991), 85–98. 398 [42] Danilov, V. I., Koshevoy, G. A., Discrete convexity and Hermitian matrices. Proc. Steklov Inst. Math. 241 (2003), no. 2, 58–78. 225 [43] Dieudonne, J. A., La geometrie des groupes classiques. Troisieme edition, Ergeb. Math. Grenzgeb. 5, Springer-Verlag, Berlin 1971. 83, 221 [44] Dupont, J. L., Guichardet, A., A propos de l’article: “Sur la cohomologie reelle des groupes de Lie simples reels” par Guichardet et D. Wigner.1 Ann. Sci. Ecole Norm. Sup. (4) 11 (1978), no. 2, 293–295. 225 [45] Dynkin, E. B. Homologies of compact Lie groups. Uspehi Matem. Nauk (N.S.) 8 (1953), no. 5 (57), 73–120; English transl.: Amer. Math. Soc. Transl. (2) 12 (1959), 251–300. 223 [46] Eberlein, P. B. Geometry of nonpositively curved manifolds. The University of Chicago Press, Chicago, Ill., 1996. 504 [47] Enright, Th., Howe, R., Wallach, N., A classification of unitary highest weight modules. In Representation theory of reductive groups, Progr. Math. 40, Birkhäuser, Boston 1983, 97–143. 380 [48] Epperson, J. B., The hypercontractive approach to exactly bounding an operator with complex Gaussian kernel. J. Funct. Anal. 87 (1989), 1–30. 56 [49] Faraut, J., Koranyi, A., Analysis on symmetric cones. Clarendon Press, Oxford 1994. vi, 221, 223, 380 [50] Flensted-Jensen, M., Analysis on non-Riemannian symmetric spaces. CBMS Regional Conf. Ser. in Math. 61, Amer. Math. Soc., Providence, R.I., 1986. 223, 536 [51] Folland, G. B., Harmonic analysis in phase space. Princeton University Press, Princeton, N.J., 1988. vi, 56, 257, 284 [52] Folland, G. B., Sittaram, A., The uncertainty principle: A mathematical survey. J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. [53] Fomin, S., Zelevinsky, A., Total positivity: tests and parametrizations. Math. Intelligencer 22 (2000), no. 1, 23–33. 223 [54] Foth, T., Neretin, Yu. A., Zak transform, Weil representation, and integral operators with theta-kernels. Internat. Math. Res. Notices 2004 (2004), no. 43, 2305–2327. 407 [55] Friedrichs, K. O., Mathematical aspects of the quantum theory of fields. Interscience Publishers, New York 1953. 16, 55 [56] Fuchs, D. B., Classical manifolds. In Topology. II, translated from the Russian, Encyclopaedia Math. Sci. 24, Springer-Verlag, Berlin 2004, 197–252. 223 1

See [77].

540

Bibliography

[57] Fulton, W., Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209–24. [58] Fulton, W., On the quantum cohomology of homogeneous varieties. In The legacy of Niels Henrik Abel, Springer-Verlag, Berlin 2004, 729–736. 223, 224 [59] Fulton, W., Harris, J., Representation theory. A first course. Grad. Texts in Math. 129, Springer-Verlag, New York 1991. [60] Gantmacher, F. R., The theory of matrices. Vols. 1, 2, translated from Russian, Chelsea Publishing Co., New York 1959. 43, 222, 223, 543 [61] Garrett, P., Buildings and classical groups. Chapman & Hall, London 1997. 453, 504 [62] Gelfand, M. I., Graev, M. I., Gindikin. S. G., Selected topics in integral geometry. Translated from the Russian, Transl. Math. Monogr. 220, Amer. Math. Soc., Providence, R.I., 2003. vi [63] Gelfand, I. M., Na˘ımark, M. A., Unitary representations of the classical groups. Trudy Mat. Inst. Steklov. 36, Izdat. Nauk SSSR, Moscow, Leningrad, 1950 (in Russian); German transl. Unitäre Darstellungen der klassischen Gruppen. Akademie-Verlag, Berlin 1957. 91, 221, 224 [64] Gelfand, I. M., Graev, M. I., Pyatetskii-Shapiro, I. I., Representation theory and automorphic functions. Translated from the Russian, Generalized Functions 6, Academic Press, Boston 1990. 427 [65] Gelfand, I. M., Ponomarev, V. A., Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space. In Hilbert space operators and operator algebras, Colloq. Math. Soc. Janos Bolyai 5, North-Holland, Amsterdam 1972, 163–237. 56, 222 [66] Gelfand, I. M., Shilov, G. E., Generalized functions. Vol. 1. Properties and operations. Translated from the Russian, Academic Press, New York, London 1964. 1, 4, 5, 333 [67] Gindikin, S. G., Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (118) (1964) no. 4, 3–92; English transl.: Russian Math. Surveys 19 (1964), 1–89. 380 [68] Gindikin, S. G., Invariant distributions in homogeneous domains. Funct. Anal. Appl. 9 (1975), no. 1, 50–52. 380 [69] Ginzburg,Yu. P., Shevchuk, L. V., On the Potapov theory of multiplicative representations. In Matrix and operator-valued functions. The Vladimir Petrovich Potapov memorial volume, I. Gohberg, L. A. Sakhnovich, eds., Oper. Theory Adv. Appl. 72, Birkhäuser, Basel 1994, 28–47. 224 [70] Gohberg, I., Goldberg, S., Kaashoek, M. A., Classes of linear operators. Vol. II., Oper. Theory Adv. Appl. 63, Birkhäuser Verlag, Basel 1993. 224 [71] Goldshtein, J., Semigroups of linear operators and applications. Oxford Math. Monogr., Oxford University Press, New York 1985. 315 [72] Goodman, F. M., de la Harpe, P., Jones, V. F. R., Coxeter graphs and towers of algebras. Math. Sci. Res. Inst. Publ. 14, Springer-Verlag, New York 1989. 336 [73] Goodman, R., Wallach, N. R., Representations and invariants of the classical groups. Cambridge University Press, Cambridge 1999. vi, 91, 221

Bibliography

541

[74] Graev, M. I., Unitary representations of real simple Lie groups. Trudy Moskov. Mat. Obšˇc. 7 (1958), 335–389; English transl.: Amer. Math. Soc. Transl. (2) 16 (1960), 393–396. 56, 223 [75] Griess, R. L., Jr., Ryba, A. J. E., Finite simple groups which projectively embed in an exceptional Lie group are classified! Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 75–93. 221 [76] Gröchenig, K., Foundations of time-frequency analysis. Appl. Numer. Harmon. Anal. Birkhäuser, Boston 2001. 284, 407 [77] Guichardet, A., Wigner, D., Sur la cohomologie reelle des groupes de Lie simples reels. Ann. Sci. Ec. Norm. Sup. Ser. IV 11 (1978), no. 2, 277–292. 225, 539 [78] Guivarc’h, Y., Ji, Lizhen, Taylor, J. C., Compactifications of symmetric spaces. Progr. Math. 156, Birkhäuser, Boston 1998. [79] Gurevich, G. B., Canonization of a pair of bivectors. Tr. Sem. Vektor. Tenzor. Analizu. 8 (1950), 355–363 (in Russian). 222 [80] Hall, B. C., Lie groups, Lie algebras, and representations. Grad. Texts in Math. 222, Springer-Verlag, New York 2003. vi [81] Harish-Chandra, Representations of semisimple Lie groups. IV. Amer. J. Math. 77 (1955), 743–777. 379 [82] Hatcher, A., Algebraic topology. Cambridge University Press, Cambridge 2002. 88, 92, 185, 223 [83] Havin, V. P., Joricke, B., The uncertainty principle in harmonic analysis. In Commutative harmonic analysis, III, Encyclopaedia Math. Sci. 72, Springer-Verlag, Berlin 1995, 177–259. [84] Heckman, G., Schlichtkrull, H., Harmonic analysis and special functions on symmetric spaces. Academic Press, Inc., 1994. 222 [85] Heckman, G. J., Opdam, E. M., Root systems and hypergeometric functions. I. Compositio Math. 64 (1987), no. 3, 329–352. 222 [86] Helgason, S., Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. Math. Surveys Monogr. 83, Amer. Math. Soc., Providence, R.I., 2000. 145, 223, 363 [87] Helmcke, U., Rosenthal, J., Eigenvalue inequalities and Schubert calculus. Math. Nachr. 172 (1995), 207–225. 224 [88] Hilgert, J., Hofman, K. H., Lawson, J. D., Lie groups, convex cones, and semigroups. Oxford Math. Monogr., Oxford University Press, New York 1989. 223 [89] Hörmander, L., The analysis of linear partial differential operators. Vol. 1, Distribution theory and Fourier analysis, Springer-Verlag, Berlin 1983; Vol. 2, Differential operators with constant coefficients, Springer-Verlag, Berlin 1983; Vol. 3, Pseudodifferential operators, Springer-Verlag, Berlin 1985; Vol. 4, Fourier integral operators, Springer-Verlag, Berlin 1985. 1, 4, 14, 225, 276, 284, 342 [90] Hörmander, L., Symplectic classification of quadratic forms, and general Mehler formulas. Math. Z. 219 (1995), no. 3, 413–449. 56

542

Bibliography

[91] Horn, A., Eigenvalues of sums of Hermitian matrices. Pacific J. Math. 12 (1962), 225–241. 156, 224 [92] Howard, P., Rubin, J. E., Consequences of the axiom of choice. Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, R.I., 1998. 12, 326 [93] Howe, R., Remarks on classical invariant theory. Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. [94] Howe, R., The oscillator semigroup. In The mathematical heritage of Hermann Weyl, Proc. Sympos. Pure Math. 48, Amer. Math. Soc., Providence, R.I., 1988, 61–132. 56 [95] Hua, L. K., Harmonic analysis of functions of several complex variables in the classical domains. Science Press, Peking 1958 (in Chinese), Izdat. Inostr. Lit., Moscow 1959 (in Russian), Amer. Math. Soc., Providence, R.I., 1963. 133, 223, 224, 367, 368, 522 [96] Hua, Loo-Keng, Geometries of matrices. I. Arithmetical construction. Trans. Amer. Math. Soc. 57 (1945), 482–490; III. Fundamental theorems in the geometries of symmetric matrices. Trans. Amer. Math. Soc. 61 (1947), 229–255; reprinted in Hua, Loo Keng Selected papers. Springer-Verlag, Berlin 1983. 223 [97] Iagolnitzer, D., Appendix: Microlocal essential support of a distribution and decomposition theorems – an introduction. In Hyperfunctions and theoretical physics. Lecture Notes in Math. 449, Springer-Verlag, Berlin 1975, 121–132. 284 [98] Ismagilov, R. S., Losik M., Michor, P. W., A 2-cocycle on a group of symplectomorphisms. Moscow Math. J. 6 (2006), no. 2, 307–315. 225 [99] Janssen, A. J. E. M., Bargmann transform, Zak transform, and coherent states. J. Math. Phys. 23 (1982), no. 5, 720–731. 407 [100] Janssen A. J. E. M., Signal analytic proofs of two basic results on lattice expansions. Appl. Comput. Harmon. Anal. 1 (1994), no. 4, 350–354. 278, 284 [101] Kapovich, M., Leeb, B., Millson, J. J., The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra. Mem. Amer. Math. Soc. 192 (2008), no. 896. 224, 472 [102] Karasev, M. B., Kozlov, M. V., Floquet solutions for Hamiltonians over su.2/ from the viewpoint of symplectic geometry and coherent states. St. Petersburg Math. J. 6 (1995), no. 5, 1091–1107. 327 [103] Kashiwara, M., Vergne, M., On the Segal-Shale-Weil representations and harmonic polynomials. Invent. Math. 44 (1978), no. 1, 1–47. [104] Kerov, S. V., Subordinators and permutation actions with quasi-invariant measure. J. Math. Sci. (New York) 87 (1997), no. 6, 4094–4117. 224 [105] Kerov, S., Olshanski, G., Vershik, A., Harmonic analysis on the infinite symmetric group. Invent. Math. 158 (2004), no. 3, 551–642. 224 [106] Kirillov, A. A., Elements of the theory of representations. Translated from the Russian, Grundlehren Math. Wiss. 220, Springer-Verlag, 1976. 512, 513 [107] Kirillov, A. A., Gvishiani, A. D., Theorems and problems in functional analysis. Translated from the Russian, Problem Books in Math., Springer-Verlag, Berlin 1982.

Bibliography

543

[108] Knapp, A. W., Representation theory of semisimple groups. An overview based on examples. Princeton University Press, Princeton, N.J., 1984 91, 221, 223 [109] Kobayashi, T., Oshima, T., Lie groups and representations. Iwanami, 2005 (in Japanese). vi [110] Koblitz, N., p-adic numbers, p-adic analysis, and zeta-functions. Grad. Texts in Math. 58, Springer-Verlag, New York, Heidelberg 1977. 427 [111] Kostant, B., Quantization and unitary representations. I: Prequantization. In Lectures in modern analysis and applications. III. Lecture Notes in Math. 170, Springer-Verlag, Berlin 1970, 87–208. 225 [112] Kostrikin, A. I., Pham Hu’u Tiep, Orthogonal decompositions and integral lattices. De Gruyter Exp. Math. 15, Walter de Gruyter & Co., Berlin 1994. 221, 426 [113] Kozma, G., Olevski˘ı, A., Analytic representation of functions and a new quasi-analyticity threshold. Ann. of Math. (2) 164 (2006), no. 3, 1033–1064. 333 [114] Krein, M. G., On an application of the fixed point principle in the theory of linear transformations of spaces with an indefinite metric. Uspehi Mat. Nauk (N.S.) 5 (1950), no. 2 (36), 180–190; English transl.: Amer. Math. Soc. Transl. (2) 1 (1955), 27–35. 223 [115] Krein, M. G., Shmul’yan, Yu. L., Fractional linear transformations with operator coefficients, Mat. Issled 2 (1967), vyp. 3, 64–96.; English transl.: Amer. Math. Soc. Transl. Ser. 2 103 (1974), 125–152. 224 [116] Lancaster, P., Rodman, L., Algebraic Riccati equation. Oxford Sci. Publ., Clarendon Press, New York 1995. 224 [117] Levin B. Ya., Distribution of zeros of entire functions. Translated from Russian, Amer. Math. Soc., Providence, R.I., 1980 267, 284 [118] Levin, B. Ya., Lectures on entire functions. Transl. Math. Monogr. 150, Amer. Math. Soc., Providence, R.I., 1996. 267, 284 [119] Lidl, R., Niederreiter, H., Finite fields.. Addison-Wesley Publishing Company, Reading, Mass., 1983. 426 [120] Lidskii V. B., Inequalities for eigenvalues and singular values. Addendum to 2nd–5th Russian editions of [60]. [121] Lieb, E. H., Gaussian kernels have only Gaussian maximizers. Invent. Math. 102 (1990), no. 1, 179–208. 56 [122] Lion, G., Vergne M., The Weil representation, Maslov index and theta series. Progr. Math. 6, Birkhäuser, Boston 1980. vi, 407, 488, 505 [123] Littlewood, D. E., On the classification of algebras. Proc. London Math. Soc. (2) 35 (1933), 200–240. 284 [124] Livšic, M. S., Isometric operators with equal deficiency indices, quasi-unitary operators. Mat. Sb. (N.S.) 26 (68) (1950), 247–264; English transl.: Amer. Math. Soc. Transl. (2) 113 (1960), 85–103 224 [125] Livšic, M. S., On spectral decomposition of linear nonself-adjoint operators. Mat. Sb. (N.S.) 34 (76) (1954), 145–199; English transl.: Amer. Math. Soc. Transl. (2) 5 (1957), 67–117. 224

544

Bibliography

[126] Lusztig, G., Total positivity in reductive groups. In Lie theory and geometry, Progr. Math. 123, Birkhäuser, Boston 1994, 531–568. 223 [127] Lyubarskii, Yu. I., Frames in the Bargmann space of entire functions. In Entire and subharmonic functions, Adv. Soviet Math. 11, Amer. Math. Soc., Providence, R.I., 1992, 167–180. 284 [128] Macdonald, I. G., Spherical functions on a group of p-adic type. University of Madras, Madras, 1971. 222, 472, 504 [129] Macdonald, I. G., Symmetric functions and Hall polynomials. The Clarendon Press, Oxford University Press, New York 1995. 222, 328, 471, 504 [130] Matsuki, T., Double coset decompositions of reductive Lie groups arising from two involutions. J. Algebra 197 (1997), no. 1, 49–91. 222 [131] Matsuki, T., Orbits on flag manifolds. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo 1991, 807–813. 222 [132] Matsumoto, H., Analyse harmonique dans les systemes de Tits bornologiques de type affine. Lecture Notes in Math. 590, Springer-Verlag, Berlin 1977. 504 [133] Mehler, F. G., Über die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung. Crelle’s Journal 66 (1866), 161–176. 24, 56 [134] Miwa, T., Jimbo, M., Date, E., Solitons. Cambridge University Press, Cambridge 2000. [135] Molev, A., Nazarov, M., Capelli identities for classical Lie algebras. Math. Ann. 313 (1999), no. 2, 315–357. 284 [136] Mumford, D., Tata lectures on theta. I. Progr. Math. 28, Boston Birkhäuser, 1983; Tata lectures on theta. II. Jacobian theta functions and differential equations. Progr. Math. 43, Birkhäuser, Boston 1984; Tata lectures on theta. III. Progr. Math. 97, Birkhäuser, Boston 1991. 407, 501, 503, 505 [137] Myller-Lebedeff, W., Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen. Math. Ann. 64 (1907), 388–416. 24, 56 [138] Nagano, T., Transformation groups on compact symmetric spaces. Trans. Amer. Math. Soc. 118 (1965), 428–453. 223 [139] Nazarov, M., Oscillator semigroup over a non-Archimedean field. J. Funct. Anal. 128 (1995), no. 2, 384–438. 463, 488, 505 [140] Nazarov, M., Neretin, Yu., Olshanski, G., Semi-groupes engendres par la representation de Weil du groupe symplectique de dimension infinie. C. R. Acad. Sci. Paris Ser. I Math. 309 (1989), no. 7, 443–446. 56, 426 [141] Neretin, Yu. A., Holomorphic continuations of representations of the group of diffeomorphisms of the circle. Mat. Sb. 180 (1989), no. 5, 635–657, 720; translation in Math. USSR-Sb. 67 (1990), no. 1, 75–97. 56, 223, 284 [142] Neretin, Yu. A., On a semigroup of operators in the boson Fock space. Funct. Anal. Appl. 24 (1990), no. 2, 135–144. [143] Neretin, Yu. A., Integral operators with Gaussian kernels and symmetries of canonical commutation relations. In Contemporary mathematical physics, Amer. Math. Soc. Transl. Ser. (2) 175, Amer. Math. Soc., Providence, R.I., 1996, 97–135.

Bibliography

545

[144] Neretin, Yu. A., On the correspondence between the boson Fock space and the space L2 with respect to Poisson measure. Sb. Math. 188 (1997), no. 11, 1587–1616. 284 [145] Neretin,Yu. A., Categories of symmetries and infinite-dimensional groups. The Clarendon Press, New York 1996. vi, 127, 223, 224, 284, 300, 313, 510 [146] Neretin, Yu. A., Hinges and the Study-Semple-Satake-Furstenberg-De Concini- ProcesiOshima boundary. In Kirillov’s seminar on representation theory,Amer. Math. Soc. Transl. Ser. (2) 181, Amer. Math. Soc., Providence, R.I., 1998, 165–230. [147] Neretin, Yu. A., Pseudo-Riemannian symmetric spaces: Uniform realizations and open embeddings into Grassmannians. J. Math. Sci. (New York) 107 (2001), no. 5, 4248–4264; translation from Zap. Nauchn. Semin. POMI 256 (1999), 145–167. 56 [148] Neretin, Yu. A., Matrix analogues of the beta function and the Plancherel formula for Berezin kernel representations. Sb. Math. 191 (2000), no. 5, 683–715. 221, 223 [149] Neretin, Yu. A., Plancherel formula for Berezin deformation of L2 on Riemannian symmetric space. J. Funct. Anal. 189 (2002), no. 2, 336–408. [150] Neretin, Yu. A., Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants. Moscow Math. J. 1 (2001), no. 2, 157–220. 223 [151] Neretin, Yu. A., The beta function of the Bruhat-Tits building and the deformation of the space l 2 on the set of p-adic lattices. Sb. Math. 194 (2003), no. 11–12, 1775–1805. 504 [152] Neretin, Yu. A., Structures of boson and fermion Fock spaces in the space of symmetric functions. Acta Appl. Math. 81 (2004), no. 1–3, 233–268. 284 [153] Neretin,Yu. A., Hua-type integrals over unitary groups and over projective limits of unitary groups. Duke Math. J. 114 (2002), no. 2, 239–266. 224 [154] Neretin, Yu. A., Rayleigh triangles and nonmatrix interpolation of matrix beta integrals. Sb. Math. 194 (2003), no. 4, 515–540. 224, 368 [155] Neretin,Yu. A., A construction of finite-dimensional faithful representation of Lie algebra. In Proceedings of the 22nd Winter School “Geometry and Physics” (Srni, 2002). Rend. Circ. Mat. Palermo (2) Suppl. no. 71 (2003), 159–161. 515 [156] Neretin, Yu. A., Stein-Sahi complementary series and their degenerations. Preprint, available at arXiv:0804.1990v2 [math.RT]. 223, 379, 380 [157] Neretin,Yu.A., The Perelomov problem of the inversion of the Bargmann-Segal transform. Funct. Anal. Appl. 40 (2006), no. 4, 330–333. 284 [158] Neretin, Yu. A., Central extensions of groups of symplectomorphisms. Moscow Math. J. 6 (2006), no. 4, 703–729. 225 [159] Nikolskii, N. K., Treatise on the shift operator. Spectral function theory. Translated from the Russian, Grundlehren Math. Wiss. 273, Springer-Verlag, Berlin 1986. 224 [160] Okounkov, A., Olshanski, G., Shifted Jack polynomials, binomial formula, and applications. Math. Res. Lett. 4 (1997), no. 1, 69-78. 224 [161] Olshanskii, G. I, Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series. Funct. Anal. Appl. 15 (1981), no. 4, 275–285. 56, 223

546

Bibliography

[162] Olshanskii, G. I., Convex cones in symmetric Lie algebras, Lie semigroups, and invariant causal structures (orders) on pseudo-Riemannian symmetric spaces. Soviet Math. Dokl. 26 (1982), 97–101. 223, 224 [163] Olshanskii G. I., Infinite–dimensional classical groups of finite R-rank: description of representations and asymptotic theory. Funct. Anal. Appl. 18 (1984), 22–34. [164] Olshanskii, G. I., Unitary representations of infinite-dimensional pairs .G; K/ and the formalism of R. Howe. In Representation of Lie groups and related topics, Adv. Stud. Contemp. Math. 7, A. M. Vershik, D. P. Zhelobenko, eds., Gordon and Breach Science Publishers, New York 1990, 269–463. [165] Olshanskii, G. I., On semigroups related to infinite-dimensional groups. In Topics in representation theory, A. A. Kirillov, ed., Adv. Soviet Math. 2, Amer. Math. Soc., Providence, R.I., 1991, 67–101. [166] Olshanskii, G. I., The Weyl representation and the norms of Gaussian operators. Funct. Anal. Appl. 28 (1994), no. 1, 42–54. 321 [167] Onishchik, A. L., Topology of transitive transformation groups. Translated from the Russian, Johann Ambrosius Barth, Leipzig 1994. 223 [168] Paley, R. E. A. C., Wiener, N., Fourier transforms in the complex domain. Amer. Math. Soc. Colloq. Publ. Amer. Math. Soc., New York 1934. 276 [169] Paneitz S. M., Invariant convex cones and causality in semisimple Lie algebras and Lie groups. J. Funct. Anal. 43 (1981), 313–359. 223 [170] Perelomov, A. M., Remark on the completeness of the coherent state system. Teoret. Mat. Fiz. 6 (1971), no. 2, 213–224. [171] Perelomov, A., Generalized coherent states and their applications. Texts Monogr. Phys., Springer-Verlag, Berlin 1986. [172] Pickrell, D., Measures on infinite dimensional Grassmann manifold. J. Funct. Anal. 70 (1987), 323–356. 224 [173] Pontryagin, L. S., Hermitian operators in a space with an indefinite metric. Izvestia Akad. Nauk SSSP 8 (6) (1944), 243–280; English transl.: L. S. Pontryagin. Selected works, Vol. 1, Gordon and Breach, New York 1986, 303–334. [174] Popov, V. L., Contraction of actions of reductive Lie groups. Math. USSR-Sb. 58 (1987), N2, 311–335. [175] Potapov V. P., The multiplicative structure of J -contractive matrix functions. Trudy Moskov. Mat. Obshch. 4 (1955), 125–236; English transl.: Amer. Math. Soc. Transl. 15 (1960), 131–243. 147, 223, 224 [176] Pressley, A., Segal, Gr., Loop groups. Oxford Math. Monogr., Oxford University Press, New York 1986. 284 [177] Piatecki-Shapiro, I. I., Geometry of classical domains and theory of automorphic functions. Fizmatlit, Moscow 1961; English transl.:2 Automorphic functions and the geometry of classical domains. Gordon and Breach, New York 1969. 522 2

The Russian and the English editions are different books.

Bibliography

547

[178] Reed, M., Simon, B., Methods of modern mathematical physics. I. Functional analysis. Academic Press, New York, London 1972. 1, 5, 6, 9, 27, 313, 512 [179] Reed, M., Simon, B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York, London, 1975. 1, 5, 313 [180] Reutenauer, C., Free Lie algebras. The Clarendon Press, Oxford University Press, New York 1993. 284 [181] Ricci, F., Stein, E. M., Homogeneous distributions on spaces of Hermitean matrices. J. Reine Angew. Math. 368 (1986), 142–164. 380 [182] Rohlfs, J., Springer, T. A., Applications of buildings. In Handbook of incidence geometry, North-Holland, Amsterdam 1995, 1085–1114. 504 [183] Ronan, M., Lectures on buildings. Perspect. Math. 7, Academic Press, Boston 1989. 504 [184] Rosenfeld, B., Geometry of Lie groups. Math. Appl. 393, Kluwer Academic Publishers, Dordrecht 1997. vi [185] Rossmann, W., Lie groups. An introduction through linear groups. Oxf. Grad. Texts Math. 5, Oxford University Press, Oxford 2002. vi [186] Sahi, S., Jordan algebras and degenerate principal series. J. Reine Angew. Math. 462 (1995), 1–18. 380 [187] Sahi, S., Explicit Hilbert spaces for certain unipotent representations, Invent. Math. 110 (1992), no. 2, 409–418. 380 [188] Sally, P., Analytic continuations of irreducible unitary representations of universal covering group of SL2 .R/. Mem. Amer. Math. Soc. 69, Amer. Math. Soc., Providence, R.I., 1967. 379 [189] Scharlau, R., Buildings. In Handbook of incidence geometry, North-Holland, Amsterdam 1995, 477–645. 504 [190] Sato, M., Shintani, T., On zeta functions associated with prehomogeneous vector spaces. Ann. of Math. (2) 100 (1974), 131–170. 380 [191] Schoeneberg, B., Elliptic modular functions. An introduction. Translated from the German, Grundlehren Math. Wiss. 203, Springer-Verlag, Berlin 1974. 505 [192] Schoenberg, I. J., Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44 (1938), 522–536. 379 [193] Segal, I., Tensor algebras over Hilbert spaces, II. Ann. of Math. 63 (1956), 160–175. [194] Segal, I. E., Foundations of the theory of dynamical systems of infinitely many degrees of freedom. I. Mat.-Fys. Medd. Danske Vid. Selsk. 31 (1959), no. 12. [195] Segal, I. E., Mathematical cosmology and extragalactic astronomy. Pure Appl. Math. 68, Academic Press, New York 1976. 223 [196] Seip, K., Interpolation and sampling in spaces of analytic functions. Univ. Lecture Ser. 33, Amer. Math. Soc., Providence, R.I., 2004. 284 [197] Serre, J.-P., Cours d’arithmetique. Presses Universitaires de France, Paris 1970. 426, 427, 504

548

Bibliography

[198] Serre, J.-P., Trees. Translated from the French, Springer Monogr. Math., Springer-Verlag, Berlin 2003. 504 [199] Shabat, B. V., Introduction to complex analysis. Part II. Functions of several variables. Translated from the Russian, Transl. Math. Monogr. 110, Amer. Math. Soc., Providence, R.I., 1992. 337, 339 [200] Shale, D., Linear symmetries of free boson fields. Trans. Amer. Math. Soc. 103 (1962), 149–167. 56 [201] Shilov, G. E., Fan Dyk Tin, Integral, measure and derivative on linear spaces. Izdat. “Nauka”, Moscow 1967 (in Russian). 284 [202] Shimomura, H., On the construction of invariant measure over the orthogonal group on the Hilbert space by the method of Cayley transformation. Publ. Res. Inst. Math. Sci. 10 (1974/75), 413–424. 224 [203] Stein, E. M., Analysis in matrix spaces and some new representations of SL.N; C /. Ann. of Math. (2) 86 (1967), 461–490. 380 [204] Tolstoy, V. N., Fortieth anniversary of extremal projector method for Lie symmetries. In Noncommutative geometry and representation theory in mathematical physics, Contemp. Math. 391, Amer. Math. Soc., Providence, R.I., 2005, 371–384. 284 [205] Unterberger, A., Upmeier, H., The Berezin transform and invariant differential operators. Comm. Math. Phys. 164 (1994), 563–597. [206] Ventkatarama, T. N., On a system of generators of arithmetic subgroups of higher rank groups. Pacific J. Math. 166 (1994), 193–212. 500 [207] Vergne, M., Rossi, H, Analytic continuation of the holomorphic discrete series of a semisimple Lie group. Acta Math. 136 (1976), no. 1–2, 1–59. 380 [208] Vinberg, E. B., Invariant convex cones and orderings in Lie groups. Funct. Anal. Appl. 14 (1980), 1–10. 223 [209] Vladimirov, V. S., Volovich, I. V., Zelenov, E. I., p-adic analysis and mathematical physics. World Scientific, River Edge, N.J., 1994. 504 [210] Vogan, D. A., The unitary dual of GL.n/ over an Archimedian field. Invent. Math. 83 (1986), 449–505. 379, 380 [211] Von Neumann, J., Die Eindeutigkeit der Schödingerschen Operatoren. Math. Ann. 104 (1931), 570–578. 513 [212] Wallach, N. R., Analytic continuation of discrete series. Trans. Amer. Math. Soc. 251 (1979), 19–37. 380 [213] Wallach, N. R., Real reductive groups. I, II. Pure Appl. Math. 132, 132-II, Academic Press, Boston 1988, 1992. 221 [214] Weil, A., Sur certains groupes d’operateurs unitaires. Acta Math. 111 (1964) 143–211. 56, 488, 504 [215] Weil, A., Basic number theory. Classics Math., Springer-Verlag, Berlin 1995. 504 [216] Weyl, H., Inequalities between two kinds of eigenvalues of linear transformation. Proc. Nat. Acad. Sci. 35 (1949), 408–411. 221, 299

Bibliography

549

[217] Widder, D. V., The Laplace transform. Princeton University Press, Princeton, N.J., 1941. 285 [218] Zak, J., Finite translations in solid-state physics. Phys. Rev. Lett. 19 (1967), 1385–1387. 407 [219] Zhang, G., Berezin transform on real bounded symmetric domains. Trans. Amer. Math. Soc. 353 (2001), no. 9, 3769–3787. [220] Zhelobenko, D. P., Compact Lie groups and their representations. Nauka, Moscow 1970; English transl.: Transl. Math. Monogr. 40, Amer. Math. Soc., Providence, R.I., 1973. 221, 223, 359 [221] Zhelobenko, D. P., Principal structures and methods of representation theory. Translated from the Russian, Transl. Math. Monogr. 228, Amer. Math. Soc., Providence, R.I., 2006. vi

Notation

Symbols WD, DW hh; ii ˝ Œ; V ? , P ?ƒ ƒ h; i ƒ˚ z Œg bac L˙ P

the right-hand part is a definition of the left-hand part; vice versa, 1 pairing of a distribution and a test function, 3 tensor products, 7 commutator, ŒA; B D AB BA, 18, 515 orthogonal complement, 60, 31 44 scalar products, 59 52 linear fractional transformation, 77 integer part, 269 dual lattice, 447, 458 pseudoinverse linear relation, 39

Fields and rings Fp , Fp Qp , Qp , Op R, C, H, C , R , T

finite field and its multiplicative group, 408 p-adic field, its multiplicative group, p-adic integers, 429 507

Matrices 1, 1n At AN A A>0 graph./ A~

the unit matrix, 7 transposed matrix, 7 complex conjugate matrix, 7 adjoint matrix, 7 positive definite matrix, 8 graph of an operator, 37 pseudo-adjoint operator, 69

Linear relations PWV W ker./ im./

linear relation, 37 kernel of a linear relation, of a form, 38, 58 image of a linear relation, 38

Notation

dom./ indef./

551

domain of a linear relation, 38 indefinity of a linear relation, 38

Groups, semigroups, Lie algebras Heisn .R/, etc. U.p; q/, UŒV U.n/ u.p; q/ SU.p; q/ PU.p; q/ U.p; q/ O.n/ Sp.2n; R/, Sp.2n; C/ sp.2n; R/, sp.2n; C/ Sp.2n; R/ Sp.2n; Z/ 1;2 GL.n; Qp /, Sp.2n; Qp / GL.n; Op / Iw.n/ GL.n; R/, GL.n; C/, GL.n; H /, O.p; q/, U.p; q/, Sp.p; q/, Sp.2n; R/, Sp.2n; C/, O.n; C/, SO .2n/ U.n/, O.n/, Sp.n/ SL.n; R/, SL˙ .n; R/, SL.n; C/, SL.n; H/, SO.n; C/, SU.p; q/, SO.p; q/, SO0 .p; q/, PGL.n; R/, … gl.n; R/, gl.n; C/, gl.n; H/, o.n; C/, o.n; R/, o.p; q/, u.n/, u.p; q/, sp.p; q/, sp.n; R/, sp.2n; C/, sp.2n; R/, so .2n/, sl.n; R/, sl.n; C/, sl.n; H/, su.p; q/, u.n/

Heisenberg group, 13, 419, 484, 495 pseudo-unitary groups, 67 unitary groups, 68 Lie algebra of U.p; q/, 70 special pseudounitary groups, 75 projective pseudounitary groups, 75 semigroup of indefinite contractions, 110 orthogonal groups, 92 symplectic groups, 162 symplectic Lie algebras, 162 semigroup of symplectic contractions, 182 group of symplectic matrices with integer elements, 391 Igusa congruence subgroup, 392 classical p-adic groups, 439 group of matrices with integer elements, 439 Iwahori subgroup, 453

real classical groups, 508–508 real classical groups, 508

real classical groups, 509–510

classical real Lie algebras, 509–510.

Categories Mor.; / End./

set of morphisms of a category, 28 semigroups of endomorphisms, 28

552 Aut./ Sp

Notation

groups of automorphisms, 29 symplectic category, 45

Functional spaces C1 D L2 D0 0 `2 Fn Fn H.XI L/, HB .XI L/, H? .XI L/ B.V /, B 0 .V / B.LjM /

space of smooth functions, 1 space of compactly supported smooth functions, 2 Schwartz space, 2 space of functions with integrable square, 2 space of distributions, 3 space of tempered distributions, P3 space of sequences, satisfying xj2 < 1, 3 Fock space, 226 236, space determined by a reproducing kernel, 324, 325, 327 space of locally constant functions and the space of p-adic distributions, 482, 482 482

Geometrical spaces Matp;q , Matn Symmn , ASymmn , Hermn , AHermn V2n Cp;q , Rp;q Gr k .V / xp;q Mp;q , M Bp;q , Bn Wq Sti.p; q/ CP k Diss.p; q/, DissC .p; q/ VC V2n Lagr n Latn BT

space of matrices, 7 spaces of symmetric, skew-symmetric, Hermitian, anti-Hermitian matrices, 7 standard symplectic space equipped with the bilinear form ƒ.; /, 29, and with the Hermitian form M , 30 pseudo-Euclidean spaces, 61, 67 Grassmannian, 73 space of positive subspaces, 76, 81 matrix balls, 76, 169 84 Stiefel manifolds, 86 projective spaces, 88 the cone of pseudo-dissipative operators, 112 complexification of a real linear space, 163 standard symplectic space, 165 Lagrangian Grassmannian, 161 space of lattices, 447, 491 Bruhat–Tits buildings, 450

Notation

Operators F , gO bŒ , bŒ BŒ , BŒ , Be./, BeŒ a.v/ O a.v/, O Bf Z, ZB

Fourier transform, 5, 483 Gaussian vectors, 19, 286, 287 Gaussian operators, 21, 45, 287, 45, 289 19 creation–annihilation operators, 231 Segal–Bargmann transform, 234 Zak transform, 381, 389

Functions and vectors ı.x/ bŒ , bŒ 'u #.uI q/ &.zI !1 ; !2 / .x/

delta-function, 3 Gaussian vectors, 19, 286,287 overfilled system, 227 theta-function 270 Weierstrass function, 272 -function, 353

Miscellanea zero./

ıkl WF x p

, .x/ Dis.Q/ vol./ ad, Ad

kernel of a form on a subspace, 62 Kronecker symbol, 102 wave fronts, 252 Legendre symbol, 408 discriminant of a quadratic form, 409 volume of a lattice, 432 adjoint representation, 518

553

Index

absolute of tree, 449 adeles, 488 finite, 488 integer, 489 adjoint representation, 518 Ado theorem, 515 almost self-dual lattice, 458 analytic vector, 315 angles, 92 annihilation operator, 231 anti-Wick symbol, 265 antiinvolution, 85 apartment, 454, 460, 466 automorphism, 29 Berezin kernels, 146, 352 Berezin symbol, 228, 330 Berezin–Töplitz operators, 265 Berezin–Wallach set, 349 Bergman space, 331 biholomorphic map, 82 binary form, 437 Blaschke product, 147 matrix, 150 Bochner theorem, 329 Borel theorem, 269 boson Fock space, 227 Bros–Iagolnitzer transform, 247 Brouwer fixed point theorem, 115 Bruhat cell decomposition, 457 Bruhat space, 482 Bruhat–Tits buildings, 450, 459 Bruhat–Tits fixed point theorem, 470 Bruhat–Tits tree, 447 building abstract, 466 Euclidean, 467 spherical, 467

Campbell–Hausdorff–Dynkin formula, 266 canonical basis, 160 Cartan decomposition, 79, 171 Cartan fixed-point theorem, 172 Cartan matrix ball, 76, 169 Cartan subgroup, 209 Cartan subspace, 473, 535 category, 28 causal curve, 212, 214 causal structure, 214 Cayley transform, 42, 84, 130 central extension, 185 of category, 191 trivial, 186 chamber, 450, 459, 466 character, 511 characteristic function, 129, 151, 155 Chinese restaurant, 136 cocycle, 186 coherent states, 228 cohomology group, 187 commutator, 18, 515 commutator of operators, 18 complex distance, 92, 458 p-adic, 445 complexification, 163 compound distance, 92 cone, 249 configuration, 315 congruence subgroup, 439, 500 principal, 391, 499 contractive matrix-valued function, 149 contractive operator, 109 convergence of distributions, 3 of operators strong, 6

Index

uniform, 6 weak, 6 weak (in Hilbert spaces), 6 convolution, 249 idempotent, 157 Cooley–Tukey trick, 423 covolume of lattice, 491 Coxeter complex, 466 Coxeter group, 464, 465 Coxeter simplex Euclidean, 464 spherical, 464 creation operator, 231 cross-ratio, 97 cyclic vector, 10 decomposable element, 43, 53, 102, 104, 174 discrete convex function, 156 discriminant of quadratic form, 409, 433 distribution, 3 p-adic, 482 tempered, 3 double ratio, 97, 98 dual lattice, 401, 447, 458, 495 dual submodule, 458 Dynkin cells, 87 endomorphism, 28 Ewens measures, 135 exponent of convergence, 268 exponential p-adic, 431 C -valued, 431 facet, 450, 459, 466 flag, 90 complete, 90 Fock space, 226, 227 form, 507 anti-Hermitian, 507, 510 bilinear, 159

555

Hermitian, 57, 507, 510 indefinite, 62 negative definite, 62 negative semi-definite, 62 non-negative definite, 62 non-positive definite, 62 nondegenerate, 58 positive definite, 62 positive semi-definite, 62 sesquilinear, 57 over arbitrary field, 510 quaternionic, 507 skew-symmetric, 159 symmetric, 159 Fourier transform, 5 p-adic, 483 adelic, 494 finite, 413 Frobenius trick, 131 function compactly supported, 2 indicator, 482 locally integrable, 3 -function, 353 Gauss integral, 10 sum, 414 theorem, 415 Gaussian distribution, 21 p-adic, 485 operator, 21, 287 over finite field, 418 vector, 19, 286, 287 over finite field, 419 generator of one-parameter subgroup, 517 generator of one-parameter semigroup, 314 geodesic symmetry, 143 graph, 37 Grassmannian, 73 Lagrangian, 161

556 group classical, 510 p-adic, 439 nameless, 508 complex orthogonal, 508 general linear, 508 projective linear, 510 orthogonal, 510 pseudo-unitary, 75, 510 pseudo-orthogonal, 508 pseudo-symplectic, 508 pseudo-unitary, 67 quaternionic, 508 real orthogonal, 508 special linear, 509 orthogonal, 509 pseudo-unitary, 75, 509 symplectic, 162 adelic, 494 unitary, 68, 508 symplectic, 508 Haar measure, 518 Hadamard theorem, 269 Hardy space, 331, 365 vector-valued, 152 harmonic oscillator, 24 harmonic representation, 16 Hasse–Minkowski invariant, 440 principle, 441 heat equation, 23 Heisenberg group, 13 p-adic, 484 over a finite field, 419 rational, 495 Heisenberg inequality, 246 Heisenberg uncertainty, 246 Hermite polynomials, 238 Hermitian metric, 139

Index

Hermitization, 524 highest weight representations, 359 Hilbert symbol, 440 Hilbert–Schmidt operator, 5 Horn cone, 155 Horn polyhedron, 155 Horn problem, 155 Hua double ratio, 98 hyperbolic angles, 96 idempotent convolution, 157 Igusa subgroup, 392 image of linear relation, 38 indecomposable element, 43, 53, 102, 104, 174 indefinite contraction, 109 indefinity, 38 inertia indices, 61 integer distance, 82 integral operator, 4 intertwining operator, 511 involution, 30, 172 isotropic category, 127 Iwahori subgroup, 453, 459 iwahoric subgroup, 453, 459 Jacobi identity, 515 Jacobi triple identity, 270 Jordan angles, 92 Jordan block, 9 Jordan decomposition, 10, 181 Jordan normal form, 9 Kähler metric, 140 Kähler potential, 140 kernel of form, 58 of linear relation, 38 of operator, 228 kernel theorem, 4 p-adic, 483 Killing form, 521

Index

Kolmogorov–Wold decomposition, 152 Krein fixed-point theorem, 115 Krein–Shmul’yan category, 123 Krein–Shmul’yan functor, 124 Krein–Shmul’yan map, 120, 125 Lagrange interpolation formula, 275 lattice p-adic, 443 p-rational, 491 adelic, 491 rational, 490 Legendre symbol, 408 Lidskii–Horn Lemma, 156 Lie algebra, 515 of Lie group, 516 examples, 71, 162, 509 Lie cone, 212, 213 Lie group, 516 Lie wedge, 212 linear relation, 37 contractive, 122 domain, 38 form preserving, 127 image, 38 indefinity, 38 kernel, 38 pseudoinverse, 39 rank, 39 linear relations product of, 38 linear representation, 12 Livshits characteristic function, 129 Livshits map, 129, 184 local Lie semigroup, 213 Localization Lemma, 252 locally constant function, 482 logarithm p-adic, 431 logarithm of matrix, 140 Lyubarsky theorem, 277

557

Maslov index, 201 matching, 316 function, 317 matrix adjoint, 7 anti-Hermitian, 7 conditionally positive definite, 323 Hermitian, 7 negative definite, 8 negative semi-definite, 8 non-negative definite, 8 non-positive definite, 8 positive definite, 8 positive semi-definite, 8 self-adjoint, 7 skew-symmetric, 7 symmetric, 7 totally positive, 220 symplectic, 220 matrix ball, 76, 146, 169 matrix element of representation, 12 Maya diagram, 364 Mehler bilinear formula, 238 Mehler formula, 24 Menger theorem, 323 metaplectic representation, 16 modular form, 500 morphism of G-spaces, 130 morphism of category, 28 multiplier, 343 Nazarov category, 460 nilpotent operator, 10, 105 norm p-adic, 430, 444 of operator, 8 object of category, 28 Olshanski semigroups, 213 Olshanski theorem, 288 Olshanski–Paneitz theorem, 211 one-parameter subgroup, 517

558 operator (pseudo)dissipative, 112 adjoint, 69 antilinear, 30 contractive, 109 dissipative, 109, 314 formally dissipative, 313 nilpotent, 10 pseudo-Hermitian, 101 pseudo-unitary, 67 self-adjoint, 69 symplectic contractive, 182 symplectic dissipative, 183 unipotent, 10, 177 operator colligation, 152 operator colligations multiplication, 153 order of entire function, 269 orthocomplement, 60 orthogonal complement, 60 orthogonal group, 508 oscillator representation, 16 overfilled basis, 228 overfilled system, 324 p-adic ball, 430 p-adic distribution, 482 p-adic integers, 429 p-adic numbers, 428 Paley–Wiener space, 276 Paley–Wiener theorem, 276 parabolic subgroup, 91 minimal, 91 parahoric subgroup, 453, 459 Perelomov problem, 267 Perelomov theorem, 267 Plancherel formula, 5 p-adic, 483 Poisson distribution, 497 Poisson formula, 23 Poisson summation formula, 386 polar decomposition, 8

Index

positive definite kernel, 323 matrix-valued, 359 Post–Widder formula, 283 Potapov multiplicative integral, 150 Potapov theorem, 150 Potapov transform, 51, 117 Potapov–Olshanski decomposition, 113, 183 pq-symbol, 259 pre-Hilbert space, 2 pseudo-Euclidean space, 59 pseudo-unitary group, 508 pseudoinverse, 39 Puiseux series, 205 puzzle, 157 q-binomial formula, 271 qp-symbols, 258, 262 quadratic operator, 312 quadratic residue, 408 quaternionic determinant, 509 rank of form, 58 of linear relation, 39 rapidly decreasing function, 250 Rayleigh theorem, 94 reflection complex, 87 representation adjoint, 518 irreducible, 511 linear, 12 of Lie algebra, 515 projective, 12 reducible, 511 unipotent, 362, 363 unitary, 12 reproducing kernel, 325 property, 227, 325 reproducing property, 276

Index

retraction, 467 Riemann Lemma, 241 Riesz–Herglotz theorem, 148 root subspace, 9 scalar product, 59 Schubert cell, 90 Schur Lemma, 511 Schwartz kernel theorem, 4 Schwartz space, 2 Segal–Bargmann transform, 234 Selberg integral, 368 self-dual lattice, 458 self-dual submodule, 458 semi-involution, 32, 525 Shale–Weil representation, 16 sharp cone, 342 shift operator, 153 short time Fourier transform, 244 Siegel half-plane, 84, 171 Siegel matrix wedge, 171 Siegel wedge, 84 signature, 61, 201 simplicial map, 462 singular support of distribution, 253 singular values, 9 smooth measure, 519 space Euclidean, 62 pseudo-Euclidean, 59 symplectic, 160 sphere at infinity, 474 stationary angles, 92 Stein–Sahi kernels, 146, 361 Stiefel manifold, 86 Stone–von Neumann theorem, 513 strictly simplicial map, 462 subgroup iwahoric, 453 parahoric, 453, 459 submodule, 441 subrepresentation, 511

subspace isotropic, 66, 161 Lagrangian, 161 negative, 62 positive, 62 regular, 62, 161 singular, 62 subsymmetric space, 535 supercomplete basis, 324 supercomplete system, 228 support of distribution, 252 singular, 253 Suprunenko gradations, 422 symmetric space, 143 association, 534 duality, 534 Hermitian, 524 pseudo-Riemannian, 143, 173 symmetric subgroup, 172 symplectic category, 45 group over finite field, 419 manifold, 198 symplectic group, 162 symplectomorphism, 198 system of coherent states, 324 tangent cone, 515 tangent vector, 515 tempered function, 250 tensor product of Hilbert spaces, 7, 493 of linear operators, 7 test function p-adic, 482 adelic, 493 theta-distribution, 399 theta-function, 270 multivarate, 399 time-like curve, 215

559

560 Tits metric, 474 total family of vectors, 6 totally geodesic submanifold, 144 totally positive matrix, 220 trivializer of central extension, 186 tube domain, 342 unipotent operator, 10, 177 unipotent representation, 362, 363 unitary group, 508 unitary representations of SU.1; 1/, 345 universal covering group, 518 upper-triangular subgroup, 91 vector cyclic, 103 isotropic, 58 negative, 58 positive, 58

Index

wave front, 252 analytic, 258 weak convergence, 6 of operators, 6 Weierstrass function, 272 Weierstrass theorem, 268 Weil distribution, 499 Weil representation, 16 adelic, 494 Weil–Brezin transform, 407 Weyl chamber, 473 Weyl symbol, 260, 263 Whishart–Siegel integral, 353 Whittaker–Kotelnikov–Shannon theorem, 276 Wick symbol, 233, 264 Witt theorem, 71, 162 Zak transform, 381 modified, 389

Lectures on Gaussian Integral Operators and Classical Groups (EMS Series of Lectures in Mathematics)

Lectures on Gaussian Integral Operators and Classical Groups

Lectures on Algebraic Topology (EMS Series of Lectures in Mathematics)

Lectures on Differential Geometry (Ems Series of Lectures in Mathematics)

Lectures on singular integral operators

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Lectures on Gaussian Processes (SpringerBriefs in Mathematics)

Lectures on Gaussian Processes

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Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry (EMS Series of Lectures in Mathematics)

Mathematical Foundations of Supersymmetry (Ems Series of Lectures in Mathematics)

Uhlenbeck Compactness (EMS Series of Lectures in Mathematics)

Basic Noncommutative Geometry (Ems Series of Lectures in Mathematics)

Finite Generalized Quadrangles (Ems Series of Lectures in Mathematics)

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Four Lectures on Mathematics

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Lectures on Gaussian Integral Operators and Classical Groups (EMS Series of Lectures in Mathematics)