SYMMETRIES AND LAPLACIANS introduction to Harmonic Analysis, Group Representations and Applications
NORTH-HOLLAND MATHEMATICS STUDIES 174 (Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U S A .
NORTH-HOLLAND - AMSTERDAM
LONDON
NEW YORK
TOKYO
SYMMETRIES AND LAPLACIANS Introduction to Harmonic Analysis, Group Representations and Applications
David GURARIE Department of Mathematics and StaYjstics Case Western University Cleveland, OH, U.S.A.
1992
NORTH-HOLLAND - AMSTERDAM
LONDON
NEW YORK
TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN: 0 444 88612 5
0 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, RO.Box 521, 1000 A M Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U S A . All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher, Elsevier Science Publishers B.V. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
In the memory of my parents, whom I owe so much to Valentina
Eli and Mark
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TABLE OF CONTENTS
Introduction .................................................................................................................. 1 Chapter 1. Basics of representation theory
51.1. Groups and group actions ............................................................................. $1.2. Regular and induced representations; Haar measure and convolution algebras .................................................................................... 51.3. Irreducibility and decomposition .................................................................. 51.4. Lie groups and algebras; the infinitesimal method ........................................
13 24 37 46
Chapter 2. Commutative Harmonic analysis
$2.1. Fourier transform: inversion and Plancherel formula .................................... 62 52.2* Fourier transform on function-spaces ........................................................... 71 52.3. Some applications of Fourier analysis ........................................................... 83 52.4. Laplacian and related differential equations .................................................. 92 52.5* The Radon transform ................................................................................. 120 Chapter 3 . Representations of compact and finite groups
53.1. The Peter-Weyl theory ............................................................................... 53.2. Induced representations and Frobenius reciprocity ..................................... 53.3* Semidirect products ...................................................................................
125 137 147
Chapter 4 . Lie groups SU(2) and Sq3) 54.1. Lie groups and SU(2) and Sq? and their Lie algebras ................................ 161 54.2. Irreducible representations of U(2)............................................................................. 164 &3* Matrix entries and characters of irreducible representations: Legendre and Jacobi polynomials .............................................................. 171 54.4. Representations of SO(3): angular momentum and spherical harmonics ..... 174 $4.5* Laplacian on the n-sphere .......................................................................... 184 Chapter 5 . Classical compact Lie groups and algebras
$5.1. Simple and semisimple Lie algebras; Weyl “unitary trick” .......................... 55.2. Cartan subalgebra, root system, Weyl group .............................................. $5.3. Highest weight representations ................................................................... 95.4* Tensors and Young tableaux ...................................................................... $5.5. Haar measure on compact semisimple Lie groups ....................................... $5.6. The Weyl character formulae ..................................................................... $5.7* Laplacians on symmetric spaces .................................................................
191 197 206 217 227 231 241
Chapter 6. The Heisenberg group and semidirect products
$6.1. Induced representations and the Mackey’s group extension theory .............. 257
...
Vlll
56.2. The Heisenberg group and the oscillator representation .............................. $6.3* The Kirillov orbit method .......................................................................... Chapter 7. Representations of SL,
274 290
.
305 57.1. Principal. complementary and discrete series .............................................. 57.2. Characters of irreducible representations .................................................... 313 57.3. The Plancherel formula on SL. (W) .............................................................. 317 67.4. Infinitesimal representations of SL. spherical functions and characters ....... 328 §7.5* Selberg trace formula ................................................................................. 334 $7.6* Laplacians on hyperbolic surfaces W/r....................................................... 348 57.7* SL. (C) and the Lorentz group .................................................................... 361 Chapter 8. Lie groups and hamiltonian mechanics 58.1. Minimal action principle; Euler-Lagrange equations; canonical formalism ... 369 58.2. Noether Theorem, conservation laws and Marsden-Weinstein reduction ..... 378 385 $8.3. Classical examples ...................................................................................... 58.4, Integrable systems related to classical Lie algebras..................................... 393 $8.5* The Kepler Problem and the Hydrogen atom ............................................. 408 Appendices: A: Spectral decomposition of selfadjoint operators .............................................. 423 B: Integral operators .......................................................................................... 427 C: A primer on Riemannian geometry: geodesics. connection. curvature ............. 430 Rderences ................................................................................................................. List of frequently used notations ................................................................................ Index .........................................................................................................................
439 447 449
Introduction Throughout this book word Group will be synonymous with transformations and symmetries. Groups are omnipresent in the mathematical and (possibly) physical universe. In Mathematics symmetries play several important functions. On the one hand they allow to reduce the “number of variables”, and often render problem soluble (like reducing the order, or separating variables in differential equations). On the other hand they allow to analyze and synthesize complex objects in terms of simple (elemental) blocks/constituents (eigenvalues of matrices and operators, expansion in Fourier modes, harmonic analysis in general). The prime example came from the very onset of the subject, the Galois theory of algebraic equations. The thrust of this early work was precisely to relate the symmetry (Galois) group of the equation to its solvability’ in radicals. Then Sophus Lie set up the task to develop the “Galois theory” for differential equations. He naturally came up with the concept of a continuous Lie group and its infinitesimal versions Lie algebra. Since its inception in the mid XIX-century the Lie theory rapidly grew and spread across the broad range of subjects, to occupy its present central position at the crossroads of Geometry, Analysis, Differential equations, Classical mechanics and Physics in general. Symmetries in physical systems2 are inextricably linked to conservation laws (via Noether’s Theorem), and the Galois chain repeats: “symmetries of hamiltonians”
+ “conserved integrals” + “solvability (via symmetry-reductions)”.
Our book sets more limited goals. The main objective is to introduce the reader to a wide range of concepts, ideas, results and techniques, that evolve around
. 1-
[symmetry-groups\, Irepresentations], and
We strived to stress the diversity
and versatility of the subject, but at the same time to unravel its common roots. More specifically, our main interest lies in geometrical objects and structures {X}, discrete or continuous, that possess sufficiently large symmetry-group G, like regular graphs (Platonic solids); lattices; symmetric Riemannian manifolds, and other. All such objects have a natural Laplacian A, a linear operator on functions over X, invariant under the group action. There are many problems associated with Laplacians on X. Typically one is interested in certain “continuous or discrete-time” evolutions, on
X, random walks, diffusion processes, wave-propagation. All these problems require to ‘Thence came the terminology of solvable groups (abelian ext,ensions of abelian groups), as the abelian ones were directly associated with solvability. ‘Symmetries acquire yet greater significance in the murky world of subatomic physics, where nothing could be observed, measured, sensed and compared to the everyday experiences of the macroworld. Here symmetries and conservation laws provide the only guiding light, and Lie groups enter in the very formulation of fundamental models of particles, fields and forces.
2
introduction
compute certain “functions of A”, L = f ( A ) , like powers { A ” } ; the semigroup, generated by A {e-“}; function {cos(tJ-6)}, etc. To explain and motivate the problems and the methods, based on symmetries, we shall start with a few simple prototypical examples. Fibonacci sequence is defined by a 2-term recurrence relation: Ixn+l = x, (or more general, xnS1 = a x ,
+b
+ .z,-~]
~ ~ - and ~ ) the , two initial values {zo;xl}.One is asked
to find all {z”}. To solve the problem, one first converts the 2-term numerical recurrence to a one-term recurrence for vectors: y, = Frobenius-type matrix L =
and we need to compute all iterates {L“}. The latter could be accomplished by diagonalizing matrix L , i.e. solving the eigenvalue problem: finding spectrum {A,$,}, and the matrix of eigenvectors U = {&;$,}.
The latter serves to diagonalize L,
L = U A U - I , where A=diag{A,;X,}. Once L is brought into the diagonal form, its iterates {Ln= UA”U-’} come right away, as well as more general “function of L”, f(L)= U f ( A ) U - ’ . So we immediately get an explicit solution
’.
Of course, no group-analysis was needed here, as the “eigenvalue problem” had an easy direct solution. However, similar “finite” problems could quickly become intractable, as we shall demonstrate in our next example.
Random w&
on graphs
I‘. The process consists of random jumps from any site
(vertex) of the graph to any one of its nearest neighbors with equal probabilities. Starting with the initial position at {xo}, or the initial probability distribution Po(,) (ZE
r),one is asked to find the probability distribution after the R time-steps, { P n ( x ) } .
Once again vector {P,(z)} is completely determined by the initial state Po and the transition-matrix L , which could be called the “Laplacian” of the graph, that sends any vectorffunction $(x) on
I‘ into L!b(x) = ?&b(Y),
sum over all m nearest vertices {y} of
Y
2.
Since the jumps are statistically independent
(the process has no memory of the past, when deciding to go from {x} to {y}), and
stationary in time (the “jump-rules” remains the same for all times n = 0; 1;...), we get a sequence of iterates, 3The classical Fibonacci numbers are given by
=c
I+&”
+
]-fin
, the generalized ones, z, = clX1” where coefficients, {cl;ez} depend on the initial data {zo;zl}. 2,
, ( ~ ) c2(-7_!
+ C,X,~;
3
Introduction
P, = Ln[P,],at time n. To compute powers
{L”}we
(2)
need once again to diagonalize
L, i.e. to solve the
eigenvalue problem. But this time the difficulty increases drastically with the size of
r,
if one tries a direct approach (imagine diagonalizing a 20 x 20 “dodecahedral” matrix!). There is not much to do in the general case, so we turn to a special, but important, class of regular graphs (like polygons, Platonic solids, a “Knight on a periodic
r are characterized by the abundance of symmetries. One can associate with any such r group G of vertex-transformations, that preserve all links (the incidence relations) between vertices. For regular r, group G takes any vertex x into any other4, so r becomes a homogeneow space o f G, r = H\G, where H denotes the stabilizer subgroup of some xo E r, H = {g:xog = xo}. For instance, the chessboard”, etc.). Regular graphs
regular m-gone is itself a finite abelian group Z , = {integers modulo m} (its symmetries are made of all finite rotations and reflections), whereas Platonic solids (cube, octahedron, dodecahedron, etc.) are homogeneous spaces H\G
of some well-known
symmetric or alternating groups (see 31.1 and 33.2). Given a homogeneous space X, group G acts on X by translations, g:x-+xg,and this action gives rise to the regular representation R = R X of G on the function-space
L = L(X ) over X , R g f ( 4= f ( 4 .
(3)
One easily verifies that map g+Rg, from G into operators on L, takes group multiplication into the product of operators, in other words defines a group-
L commutes with all operators {Rg},group G leaves L invariant, and we have to analyze the resulting “reduced
representation. Since Laplacian eigenspaces { E x } of
L
in
representations” of G on eigenspaces { E x } . Among all representations of the group one distinguishes the minimal ones, called irreducible, the ones that do not allow further reduction to smaller invariant subspaces (they play the role of “joint eigenvalues” for a g E G}).The typical problems in representation theory include: family of operators {Rg: (i) characterization all ineducible {n’s}for a given group G (analysis-problem); and
(ii) decomposition of an arbitrary (natural) representation R into the direct sum (or integral, to be explained below) of irreducibles (synthesis problem). The harmonic analysis on X suggests a possible approach to the spectral problem 41n fact, in many cases G takes any connected pair of vertices {qy} to any other such pair. Such graphs T’s could be called 2-point symmetric spaces, by analogy with their continuous counterparts.
4
Introduction
for operator L. Namely, we need
(I) to study the structure of the symmetry-group G; (11) to determine its elemental (irreducible) representations
{T};
(111) to interpret Laplacian L in terms of group-elements of G, to be able to assign operators { T ~ to} it for various irreducible { T } (IV) finally, to decompose the regular representation into the direct sum of irreducible components: RX = @ d .
i
The latter means that space L ( X ) is split into the direct sum of subspaces @ Ei, 3
invariant under all operators { R,:g E G}, irreducible relative to the G-action. Such decomposition, generalizes the notion of spectral/resolution (diagonalization) of a single operator. Steps (I-IV), when successfully accomplished, would lead to the desired spectral resolution of L. Namely, the reduced Laplacian L I E j r L becomes scalar5, N
So the “eigenwalue spectrum {X,(L)]” could be identified the “representation spectrum {d}of R”, the “eigenspaces” being “irreducible components” of R. Once L is diagonalized we immediately obtain all iterates {L”), and more general functions {f(L)}, needed to solve (3). { T = ~ X,(L);
for each irreducible
T}.
The details of the scheme could be found in 52.3 (Finite Fourier transform) for polygons, and in 53.3 for Laplacians on Platonic solids. After a brief excursion through the finite/discrete cases (graphs) we shall turn now to a completely different geometric setup: smooth Riemannian manifolds, Lie groups and their quotients, symmetric spaces X . All these objects will be introduced and studied at length in the due course (chapters 1,4,5,6,7).
Here for the sake of
introduction we consider 3 familiar examples: the Euclidian space R”, the n-sphere
S” c Rn+’, and the hyperbolic (Poincare-Lobachevski) plane‘ W2 = { z = z+iy:y 2 0). To motivate the continuous case we shall mention 3 typical problems in the integral geometry with some physical context. The setup for all 3 is basically the same. We are given a solid body D of constant (or variable) density p ( z ) . In the first case one considers all lines {y}, passing through
D,and computes integrals along
y: ?(y)
=
I,
pds. In the
’There is a general reason, why Laplacian L goes into the scalar operator nL under any irreducible A, known as Schur’s lemma. ‘the latter could be also interpreted, as a continuous limit of properly scaled i n space and time random walks on trees.
5
Introduction
second solid D (assumed to be convex, symmetric) is illuminated from all possible directions, and we measure the area of the shadows, cast by D. In the third case (somewhat simpler than the second) we take all cross-sectional areas of D by the family of planes, passing through its center. In all 3 cases one would like to recover either density p (case l), or the geometric shape of D (case 2 and 3) from such data. The first case is essentially the celebrated X-ray transform, used in tomography. The 2-nd and 3-rd give somewhat different versions of the Radon transform on the 2-sphere. Indeed, a (convex) body D can be represented as a graph of function f(z) on the 2-sphere, S2 = { I z I = 1). The cross-sectional areas of D are obtained by integrating :fz along all
great circles {y} in S2. So in all cases we are dealing with certain integral transforms of functions, either on the Euclidian space, or the sphere.
W have the natural Riemannian metric (§1.1), and possess G of distance-preserving transformations (isometries). The
All 3 spaces, R", S", large symmetry-group
Euclidian isometries consists of rigid motions: group En, generated by all translations and rotations in R"; the n-sphere S" has the orthogonal symmetry-group SO(n+l), made of all (n+l) x ( n + l ) - orthogonal matrices in Rn+' 3 S"; symmetries of
W coincide
with the ubiquitous unimodular group SL,(R) - all 2 x 2 real matrices of detg = 1, acting az+b by the fractional-linear transformations, g: z-+Cz+d. In fact, all 3 examples represent the of group G modulo the maximal compact
so called symmetric spaces, quotients K\G
subgroup K , that fixes a particular point in X , K = {g:z,g examples:
Ii' = S q n ) for Rn and S",
while the hyperbolic
' A
= z,}.
In the above
= SO(2).
The analog of the discrete-time random-walk on space X is played either by
spherical means:
(average value of function
f over the sphere of radius r , centered at
{z}), or by more
general integral kernel,
I
u(z)+K[u]= Ii'(z;y)u(y)dy. Transition-operator7 K takes any distribution p,(y) on X into pl(z) = li'[po], at a single time-step. As before we are interested in iterates
{I{"},
and once again have to
face the "diagonalization problem" for Ii'. In general, the problem is hardly tractable, so we turn to a special class of integral kernels, K ( z ; y ) ,that commute with the G-action, 7Strictly speaking, density K ( z ;y)dy represents transitional probabilities t o j u m p from point {y} to {z}, provided jK(z;y)dy = 1.
6
Introduction
(3) on X . Any G-invariant operator has kernel K , that depends only on the distance’ T = d(z;y), between z and y, 1-(z;y) = K [ T ) .Spherical means give one such example, the Radon and X-ray transforms also obey this condition, but the foremost of all Ginvariant operators is the natural Laplacian’ A (the Laplace-Beltrami operator) on X . Associated to A there are various evolution processes on X (continuous analogs of the “discrete-time random walks”), described by partial differential equations (pde). One of them is the standard heatdiffusion problem, ut
- AU= 0; ~ ( 0 =) f;
(4)
whose formal solution of (4) is given by the heat-semigroup of A, u = et”[fl.
Another important model is the wave equation: utt - Au = ...,whose solutions can also be represented by “functions of A”, {u = e *it-;
or cos t a ; sin tJ--Zij}.
Since A commutes with the G-action on X , all its functions { f ( A ) }do the same, including the heat and wave-propagators (another names for the “transition-matrices”, I( = etA; e i t f i ) . But any such I( has a radially symmetric kernel (I = K ( r ) . So a pdeproblem for I(, is often reduced to an ordinary differential equation (see chapters 2;4;5). Those in many cases, including R”, S”, W, could be solved explicitly in terms of the well-known special functions. Once again we are lead to analyze the G-action on X , and the resulting regular representation R,f(z) = f(zg),on suitable function-spaces over X.
So spectral decomposition of any radial operator K is reduced to the harmonic analysis of representation R. The latter involves among other a good understanding of the symmetry-group G itself, its ‘elemental (irreducible) representations” nature of the direct sum and/or integral decomposition,
R=
@ x j ; or 3
xSdp(s).
{T},
and the
(5)
To give some feeling of the issues involved, let us take the Euclidian space Rn, as an abelian group with translations only. Irreducible representations of R” are all 1‘This result is a direct consequence of the “double-transitive” action of G on rank-one symmetric spaces X,i.e. any pair {z;y} C X , is taken any other equidistant pair. ’In fact, any spherically symmetric kernel K on X (in either one of 3 examples) is given by a “function of the Laplacian”, K = f(A). This result, far from obvious, comes from the harmonic analysis of rank-one symmetric spaces (chapters 4-5). It has to do with the “multiplicity-free spectrum” of regular representation R, equivalently, with the fact that the commutator of R (all operators, that commute with it) forms a commutative algebra. “Functions of operators” {f(L)}will be introduced and discussed in chapter 2 and Appendix A via spectral resolution of operator L.
7
Introduction
dimensional (characters!), they consist of the family of exponentials {eit. €:[ E R”}. Decomposition (5), amounts to the Fourier-transform of f(z), expansion in the Fourierseries or integral. In chapter 2 we explore in detail various aspects of the commutative Fourier analysis and apply it to differential equations. The relevant analysis on the n-sphere, however, has a very different flavor. Here one has to start with “irreducible representations”
S q n t l ) , that would play the role of exponentials
{T}
of the orthogonal group
{ e i ” * < } on
R”. This, by itself a
challenging task, requires to develop a fair amount of the linear and multilinear (tensorial) algebra, as well as the relevant structure theory of Lie groups and algebras (chapters 4-5). Irreducibles
{T}
of S q n t l ) turned out to be finite dimensional, and the
decomposition of R has purely “discrete spectrum”lo,
R=
co @ rk.
k=O The corresponding irreducible subspaces { 36k} of L2(Sn), called spherical harmonics, have many remarkable features. Aside of being eigenspaces of the spherical
Laplacian, they proved to be closely connected to harmonic polynomials on Fin+’, and the classical Legendre and Jacobi functions. Our last example
W2 reveals even more striking difference. SL, is a
simple, not
compact group (in fact, the “smallest” of them). Unlike R” or S q n ) , its irreducible
unitary representations are infinite-dimensional, they make up certain continuous and discrete families {#:s E R} and {rrn:mE Z}. The decomposition problem for regular representations { R X } on homogeneous spaces, strongly depends on the “geometric nature” of quotient
X
continuous (direct integral), R = discrete subgroup of
+
= H\G. In some cases (Poincare half-plane W) it becomes purely
G) -
TSdp(s);in other cases (compact quotients r\G,
purely discrete (direct sum) R =
%0 r S j .It
r-
could also be a
combination of both, like the regular representation R on the entire group
G! The
relevant harmonic analysis becomes quite involved (see chapter 7). Its most spectacular application to the spectral theory of Laplacians appears in the context of quotients X = r\G, modulo discrete subgroup r c G. It turns out that “spectrum of such X” (either Laplacian A,, or representation Rx) is intimately connected with some fine “arithmetic properties” of via the celebrated Selberg-trace Theorem (a noncommutative version of the “Poisson summation” on R”). As an off-short we
r,
-Both results are general and hold for arbitrary compact (finite in size) groups (chapter 3). But specific examples, like S q n ) will carry a great deal more structure and information about {m} and
R.
8
Introduction
establish some interesting links between the “eigenvaluespectrum {A,}
of A,”, and the
“geometric length-spectrum of X” (length of all closed geodesics in X ) . We elaborate some aspects of spectral theory on compact quotients r \ G in 57.6-7.7. Our discussion of the “continuous cases” vs. “discrete cases”, although very different in technical terms, clearly demonstrates similarities in the basic procedures. Namely,
I) investigation of the group-structure of G, and its “elemental” irreducible representations { T } ;
11) study of G-invariant objects “Laplacians” on G, or on its quotients X = H\G; 111) decomposition of natural G-actions, regular representations R X , RG; its L on G and X ;
connection to Laplacians
IV) application of parts (1)-(111), particularly, the correspondence between “spectrum of L” and “spectrum of R” to various “junctions of L”,like iterates {L”}, semigroups { e-“; e - t f i } , and solutions of the associated differential equations. The development in the book largely will largely follow the general scheme (IIV), although not always step by step, as in the model examples. We may or may not start with a simple model problem. Often a significant effort has to be spent first to study the relevant groups, and their representation theory: problems of irreducibility and decomposition (cf. chapters 5-7). As we mentioned our main interest lies in the natural G-actions and representations, regular RG; RX,and their generalizations called
induced representations. The latter extend G-actions on scalar function { f(s)}, to vector-functions and sections of vector-bundles over X (so G-action combines translations in the X-space with “twisting” in fiber-spaces). The decomposition of the regular and induced representations makes up the content of the so called Plancherel Theorem. Although we follow steps (I-IV) throughout the book their relative “size” and “weight”, as well as the amount and depth of applications, varies from chapter to chapter. Thus chapter 2 is largely devoted to applications of the commutative Fourier transform, while chapters 5 or 7, deal mostly with the analysis of groups and represent at ions. An introductory chapter 1 brings in the principal players: basic examples of
9
Introduction
groups and geometric structures (51.1). Then we proceed to develop some fundamental concepts and results of the representation theory (§§1.2-3), and the Lie theory ($1.4). In chapters 2-7 we take on the main themes of the book with many facets of the general scheme (I-IV). Each chapter has its own LLfavorite” Laplacians and symmetry-groups: 0
standard Laplacian A on commutative groups Wn;Un and certain domains
D c Rn, in chapter 2 0
polyhedral Laplacians and the relevant finite groups in chapter 3
0
the spherical Laplacian A on S2 and S”, whose theory is based on Lie groups
SU(2);SO(3), in chapter 4 0Laplacians on more general compact Lie groups and symmetric spaces in chapter 5 0
the harmonic oscillator H = - A
+ I x I ’,and the Heisenberg group in chapter 6
0Laplacians on the Poincare plane
W, SL,,
and compact Riemann surfaces r \ H
in chapter 7. Chapter 8 stands somewhat aside from the mainstream, as it takes on the subject of “symmetries in nonlinear(!) hamiltonian systems”, and their role in integrability. But the very end of chapter 8 (58.5) brings us back to the main issues in grouprepresentations. We discuss the quantization problem of classical hamiltonians, particularly, the “quantized Kepler problem” - the hydrogen atom, one of the best studied models in quantum mechanics (see [LL]). The “Laplacian” pops in here in a quite interesting and unexpected form. A remarkable feature of the Kepler (hydrogen) in @, is that the “discrete hamiltonian, Schrodinger operator H = - A -1 Izl
(negative) part of H” is equivalent to the inverse Laplacian A - ’ on the 3-sphere. Along with the main themes (I-IV) a number of side issues and problems come to the discussion in different places. The reader will learn, for instance, why the geodesic flow is integrable on Sz (chapter 8), and ergodic on compact negative-curved Riemann surfaces X (chapter 7); what group-representations have to do with a topological problem of “counting linearly independent vector fields on spheres” (chapter 3); how Lie groups and representations explain some peculiar properties and rela.t,ionsof special functions (Legendre, Jacobi, Hermite, etc.). The book was designed as an introduction to harmonic analysis and group representations for graduate students in Mathematics and applications, or anybody
10
Introduction
(non-expert), interested in the subject, who would like to gain a broad perspective, but also learn some basic techniques and ideas. In the words of H. Weyl
“... it
is primarily
meant f o r the humble, who want to learn as new the things set forth therein, rather than f o r the proud and learned who are already familiar with the subject and merely look for
quick and ezact information...” The material of the book is based on the lectures and seminars, given by the author over the passed few years at UC Irvine, Caltech, and CWRU. Student comments and suggestions were helpful in preparing the manuscript. Our goal was to cover a wide range of topics, rather than to delve deeply into any particular one. The exposition is largely based on examples and applications, which either precede or follow the general theory. Some are important on their own, others serve to elucidate and motivate general concepts and statements. Of course, if the general approach seemed conceptually easy and directly leads to the main point, we do not hesitate to bring it forth (like the “Peter-Weyl theory” of chapter 3). But we never engage in “abstract” studies for their own sake. In order to keep the minimal prerequisites, and to shorten the background preparations, the book often appeals to intuition, example and analogy, rather than formal derivations. So the reader versed to some degree in the basic Riemannian geometry, functional analysis and operator theory, should be able to go through most of the topics without difficulty. The less prepared reader would be granted a paragraph (or two) of a footnote/appendix style explanations, to enable him to grasp quickly a new concept (or idea), and to follow the rest. Certain parts of the book (sections/paragraphs) are addressed to a better prepared reader, without detracting from the main themes.
As we wanted not to rely heavily on the standard (4-8 semester) staple of real analysis, algebra, topology, the book provides a number of ‘shortcuts’. So some basic concepts in algebra, geometry, topology are introduces “on fly”, as the need arises. We were somewhat more patient and systematic with the analysis of operators and differential equations: chapter 2 could serve as a brief introduction to PDE’s, mostly from the classical standpoint (cf. [CHI; [WW]). We also provided 3 appendices: on spectral decomposition of self-adjoint operators (A); on integral operators (B), and on basic Riemannian geometry (C). Finally, to cover a sizable material in a moderate-size volume required a departure from certain standards of mathematical exposition. We found it impossible
11
Introduction
(and undesirable) to try to maintain a uniform level of rigor and detail throughout the text. So some results are provided with fairly complete arguments, others are only outlined, relegated to problems, or just stated. The role of formal proofs is in general downplayed. The book contains sufficient material for a 1 or 2-semester course in the Harmonic analysis and Group representations. The instructor could make several choices, and follow different path in selecting topics. Aside of chapter 1, that provides a general core and background, all other parts are relatively independent. So the reader could start at practically any place in the book, going back and forth, as deemed necessary. The only exception are chapter 4 and 5, which should proceed in their natural order. In each chapter we marked with
* more
advanced topics, that could be
omitted in the first reading. In writing an introductory text on a fairly broad subject, one inevitably has to make certain choices, and put aside some important topics. Our selection and style reflected largely on the author’s personal experience and prejudices, rather than anything else. Among a few important topics, left outside the scope, let us mention the representation theory of infinite-dimensional Lie groups and algebras (Kac-Moody), which was actively pursued over the passed 20 years, and recently came to the focus in connection with the String theory (monographs [Kac] and [GSW] present the mathematical and the physical view on the subject). Another important topic is related to symmetries of differential equations, dynamical systems, integrable hamiltonians. Although we do touch upon integrability in chapter 8, our analysis is limited to finitedimensional systems. The exciting developments in the field of “infinite integrable hamiltonians” over the past 30 years were also left out (see [Per]; [Olv] for references). The book was composed on the IEXP-2 word processing system, with an additional help of the Microsoft Windows PBRUSH-graphics, responsible for the figures (the author takes entire responsibility for errors, misprints, omissions). After a somewhat bumpy initiation to the world of the modern information technology, the author had a highly rewarding experience working with both programs. Both became the most indispensable tools in the arduous enterprise of writing and organizing the manuscript.
He thought he saw a Garden-Door That opened with a key He looked again, and found it was A Double Rule of Three: ‘And all its mystery”, he said, “Is clear as day to m e !” He thought he saw an Argument That proved he was the Pope He look again, and found it was A Bar of Mottled Soap. “A fact so dread, ” he faintly said, “Eztinguishes all hope !” Lewis Carroll, “The Mad Gardener’s Song”
Chapter 1. Basics of representation theory. 51.1. Groups and group actions. We introduce basic examples of discrete and continuous transformation groups; classical matrix Lie groups; isornetries of symmetric spaces; rigid motions of the Euclidian, spherical and hyperbolic geometry, as well as symmetries of regular polyhedra.
1.1. Geometric transformations. Groups, discrete and continuous, typically arise as symmetries of “geometric” structures of different kinds, which includes both discrete objects (graphs, polyhedra), and the continuous, like manifolds, symmetric spaces. Important examples of discrete groups include:
i) permutations of a finite sets A = {l;...n}, G(A) = W,; ii) isomorphisms of graphs, lattices, regular polyhedra;
iii) matrix groups over finite fields. Some of them will be described at the end of the section (example 1.1). Throughout this book we shall be mostly interested in the continuous (Lie) groups. The latter typically arise as transformations (linear or nonlinear) of vector spaces,
or
manifolds
A, equipped
with
certain
geometric
structure.
Such
transformations q5:A+A preserve the structure (or transform it in a prescribed manner), in other words they represent symmetries of A. For instance, classical (matrix) Lie groups over vector spaces R”, c”, or quaternionic1 Q”, consists of linear transformations, that preserve certain bilinear/quadratic forms, like the general linear
group made of all non-singular n x n - matrices, Gf, = {A:detA# 0 } , its subgroup
Sl, = { A :detA = l}, called special linear (or unimodular) group, as well as their numerous subgroups (orthogonal, unitary, symplectic etc.). Lie groups also arise naturally as isometries of certain Riemannian (pseudoRiemannian) manifolds, or more general conformal transformations2. The foremost cases are symmetric spaces, Riemannian manifolds which possess large isometry groups. Here we shall briefly discuss 3 basic examples of symmetric spaces (flat, spherical and hyperbolic), and the related symmetry groups. a) Euclidian space
Rn, respectively Minkowski Mn,equipped with either positive-
+ + +
‘Quaternions Q = {( = a b i c j dk a, b, c,d € R} form a noncommutative field (division algebra), generated by 3 imaginary units:i2 = j 2 = k2 = -1; i j = - - j k k ; j k = -kj=i; t i = -ik = j . They can be also represented by complex 2-vectors: ( = z wk ( z , w E C), where k2 = -1, with the multiplication rule: kzu = ii~ t. Quat,ernions along with R and C are known to form the only 3 possible division algebras over reals (see J3.1 for further details).
+
§l,l.Groups and group actions.
14
definite product: z . y = Czjyj, or indefinite product3: (z I y) = zoyo-
n-1
zcjyj. The
isometries of the Euclidian space form a group E, of rigid motions of W", generated by
+ a}, and rotations (orthogonal matrices) {U:'UU = I } . So each the form 4(z) = Uz + b. The proof is outlined in problem 1. The tricky part is to show that any rigid motion q5 is affine, d(z) = A z + b, with some
all translations {a:z+z transformation
4 E En is of
matrix A. Then orthogonality of A follows fairly straightforward. Thus group becomes a subgroup of a larger affine group,
En
+
A f f , = {4(z) = AX b: A E G L ; b E W"}. The En-linear factors { A } are either general orthogonal matrices, A E q n ) , or special orthogonal, A E S q n ) = {U:detU = l}, if 4 preserves the orientation.
Mn the role of the orthogonal affine factors is taken by the
In Minkowski space
Lorentz (pseudoorthogonaZ) group Sql;n - l), which c nsists of matrices B, that preserve the indefinite product (z I y) = Jz y, where J = - matrix of Minkowski form,
41-11)
{ A : ( A zI A d = (z I Y)). So symmetries of the Minkowski space form the Poincare group P, of Special Relativity. The latter is generated by all translations and all Lorentz transformations, 4(z) = Az
+ b; b E Rn;A E S q 1 ; n ) .
Let us notice that in all three cases (affine, Euclidian, Poincare) elements
4 are
2We recall that a Riemannian manifold Ab carries a positive definite metric g = Cgijdtidzj, i.e. an inner product (( I tJg = CgijO, on tangent spaces of Ab {( E Tz}.Pseudo-riemannian refers t o the indefinite metric g, usually of the type (+; - ;- ;...). Any diffeomorphism (change of variables) z = #(y), on a Riemannian (pseudo-Riemannian) manifold Ab transforms the metric: g+T = g# = #'*(go #) #', the new entries being N
gkm(Y)
= C (9; j 4) 'I' '
azjwhere
aykaYm9
4' = (%) denotes the Jacobian (matrix) of #. aY
Map # defines an isometry of Ab, if the transformed metric g4 is equal to g. Such maps # obviously preserve the length of any path y = { y ( t ) : 0 5 1 5 T } ,
IY['
T
=
{ Cgij(~(t))i.ii.jdt,
hence the distance between points, d(+;y) = rnin{L[y]: y(0) = 2 ; y ( T ) = y}. Conversely, for any distance preserving diffeomorphism #:Ab-+Jb, the differential 42 (Jacobian matrix of 4), considered as the map of tangent spaces, #':T,+Ty (y = d(z)), preserves the metric (norm) on T,, (#:([) I#:(()) = ((I (), for any tangent vector (. Isometries of Ab clearly form a group. Conformal maps # do not preserve metric, but multiply it with a scalar (conformal) factor p(z), i.e. g4(z) = p(z)g. So they form a larger symmetry group of Ab. 31n Special Relativity R4 = {zo;
...;z3} represents a simplified version of space-time,
serves as time variable, while (zl; z2;z3)represent space coordinates.
where
I,,
51.1. Groups and group actions
15
identified with pairs (A; b), and the group multiplication takes the form
( A ;b) * (A';b') = (AA';Ab' t b).
(1.1)
It is easy to check that translations {b} form a normal subgroup
H
N
Rn of G,
while linear factors { A } , or { U } , form a subgroup I(
N GL,, or S q n ) ; S q 1 ; n ) . So group G is decomposed into a semidirect product4, G = H D K . The representation theory of
semidirect products will be analyzed in chapters 3 and 6. Two other examples of geometric symmetries arise on the sphere and the hyperbolic (Poincare-Lobachevski) space, the prototypes of the spherical and hyperbolic geometries. They also serve as the simplest prototypes of symmetric spaces of the compact and non-compact (hyperbolic) type. b) Sphere: Sn-' = {3::113:11~= l} in Rn with the natural (Euclidian) metric has the isometry group G = S q n ) - all orthogonal transformations (rotations) in R". In fact,
Sn-' can be identified
with the quotient (coset) space of group G = S q n ) , modulo the
stabilizer (isotropy subgroup)'
K = S q n - 1) of
fixed point zo in
a
Sn-' (the
North
Pole!), Sn N G / K . In other words (definition of stabilizer), I( = { U : U ( z 0 )= x0}. c) Hyperbolic space W, can be realized either as Poincare-Lobachevski complex half-plane { z = 3:
+ iy: y > 0 } , with metric
or as complex disk D = {IZI
< 1 ) with metric, ds2 =
the Mobius transformation,
One can easily check that
0
rn. dzdf
Both spaces are related by
takes W onto D, and transforms their Poincare
metrics, one into the other. The isometry group of 04 in both realizations is made of the fractional-linear transformations: az+c @z+w = bz+d'
U:Z-+X",
4A semidirect product G = H D U , of groups H and U , where U acts by automorphisms on H , (zE H ; u E U ) , consists of all pairs g = (z,u) witbthe multiplication rule (2,u)* (y, w) = (2 * y" ;uw). It is easily seen that H = {(z;e)} forms a normal subgroup of G, while K = {(e;u)} a
subgroup, so that H
nK = { e } , and the whole group G = H . K .
'Stabilizer (isotropy subgroup) of point c in a G-space X , consist of all elements {g E C}, that leave z fixed, K , = {g: zg = z}.
31.1. Groups and group actions.
16
5
1
In the half-plane case matrices {A = fi]:ad- bc = 1 belong to SL,(R) - the real unimodular group. In the disk realization they are given by complex matrices
detA=iaiZ-IPlZ=l A that preserve the indefinite hermitian form: J ( z ) = I z1 1' - / z 2 on C2 = {(q; 2,)). This group is called conformal and denoted SU(1;l), by analogy with the standard unitary group SU(2), that preserves the positive
in
other
words
matrices
I,,
hermitian product, l z l z = 1z11' +1zzl2. Once again the hyperbolic space coincides with the quotient of G = SL, (or SU(l;l)),modulo stabilizer I< of a fixed point z,, = i E W. The stabilizer I< is easily seen to coincide with an orthogonal group in R2,
I< = {A:
= i} =
SO(^), so w N SL,/SO(~).
Remark: The hyperbolic nature of the Poincare-Lobachevski geometry stems from its close connection to a hyperboloid r = { z 2 - (z2+y2) = 1) in R3. The former represents an orbit of the Lorentz group Sq2;1), acting by linear transformations of @ N h.03. So the Lorentz metric ds2 = ds2+dyZ - dz2, restricted on I?, remains invariant under Sq2;1). Furthermore, its restriction, ds2 I r becomes positive-definite (since all rtangent vectors [ = (z;y;z) are space-Me, 2 + y 2 - z 2 > O!). So r turns into a Riemannian manifold with a large symmetry group (symmetric space). In fact,
r N Sq2;1)/S0(2), where Sq2) acts as a stabilizer of the vertex (0;O;l)of I?. The connection between I' and D is established via a stereographic map @ (see fig.1). We parametrize
r and D by polar coordinates in the sy-plane, { ( r ; 8 ) : z= 2 1 +r2)for l?,and { ( p , 8 ) ) for D.
Then @:p+r = 2'2P So 1-P
the metric on dr' dsz _-_ -$
is taken into the Poincare metric in D,
r:
l+r2
ds2 = 4
dp2
r2dO2,
+ p2d@
(1 - p y
.
Fig. 1. Sfereographic m a p 9 takes a I': z2 - (z2 + yz) = 1, in hyperboloid R3 -M3, i n f o f h e unit disk D p2 = x 2 + y z < 1, in the xy-plane, and transforms the nafural ( S O ( 2 ; 1)-invariant metric on r into the Poincare metric on D.
51.1. Groups and group actions
17
The relation between I?, D and W suggests that their groups of isometries should be identical. Indeed, we shall see (chapters 5,7) that SL,@) 21 Swl;1) makes a two-fold cover of SO(2;1)(problem 5). Thus we have exhibited 3 classes of manifolds with rich symmetry groups: Euclidian/Minkowski spaces, spheres and the hyperbolic plane. In all 3 cases group G acts transitively on A (each point is moved to any other by an element of G). Hence space A is identified with the quotient G / K , where K denotes a stabilizer of a point xo E A, namely Rfl
21 E,/SO(n);
(I = S q n ) - stabilizer of {O};
Mn 21 P,/SO(l;n-l) Sn-'
N
- stabilizer of {0} in P,
SO(n)/So(n - 1) - stabilizer of xo = (1; 0; ... )
W IISL,(R)/So(2);
or D = SU(1;1)/SO(2); K = So(2)-stabilizer of {i} (or (0))
In fact in all three cases group G acts in a stronger double-transitive manner on A, meaning that any pair of points (2; y } can be transformed into any other equidistant pair {x';y'}, d(e; y ) = d ( d ; y'). This fact has important implications for the analysis on such manifolds. Let us remark that all the Riemannian examples above (except Minkowski) belong to a wide class of symmetric spaces. These manifolds can be generally described as quotients of Lie groups modulo maximal compact subgroups, A N G / K , and we
shall see many more examples in subsequent sections.
1.2. Finite groups. The easiest to describe are commutative finite group. The complete list includes cyclic groups: Z, 21 Z/nZ, which could also be written in the - the n-th primitive root of unity), and complex form: {eiEw:O 5 k 5 n - l}, (w = their direct sums: G = Zfll x ... x Z ,.
2) Symmetries of regular polyhedra and finite subgroups of SO(3). We shall start with regular polygons in the plane.
i)
Symmetries of the regular n-gon consist of rotations by angles EZ , - a finite subgroup of all planar rotations SO(2), or the larger dihedral group, D, = Z, Z, c O(2) - a semidirect product of rotations and a reflection
{gk:k = 0; l...}
about any symmetry axis. D, can be thought of as the symmetry group of the dihedral A,, a solid in R3, built of 2 pyramids based on a regular polygon R, in W2 and a pair of opposite vertices that project onto the center of R, (see figure ). Then Z, implements
18
$1.1. Groups and group actions.
axisymmetric rotations of the dihedral in the plane, while a generator of
Z, flips 2
opposite (spatial) vertices. In 3-D one also has 5 Platonic perfect solids, each one with a symmetry group of orthogonal transformations in
W3 that
preserve its vertices:
ii) tetrahedral symmetries: A, - alternating group (even permutations) of order 4. The generators of A, are cyclic permutations of 3 vertices in each tetrahedral face, that leave the opposite vertex fixed. iii) cubo-octahedral symmetries: the symmetric (permutation) group W,. Cube contains two opposite regular inscribed tetrahedra (fig.2), so its group contains an A, plus an element u that transposes both tetrahedra. Hence Gcube= A,
UaA, = W,.
Fig.:! shows cube with 2 opposit inscribed tetrahedra.
A, - alternating group of order 5 [Cox]. Indeed, a dodecahedron contains 5 inscribed tetrahedra {TI;...;T5}(the set of 20 vertices is evenly split into 5 quadruples). A rotation of order 3 about any pair of opposit verteces leaves a pair of tetrahedra (say T,;T,) fixed and cyclicly permutes the remaining triple (2'5 T,;T,). Obviously, even permutations, {(123);(234);(345);...} generate As! iv) Icoso-dodecahedral group:
Fig.3 demonstrates one of 5 regular tetrahedrons, inscribed inside the dodecahedron. Any pair of opposit vertices ( A ;A') selects a pair of tetrahedra, and any rotation of order 3 about the AA'-aixis, leaves the pair fixed, and cyclicly permutes the remaining triple.
We shall see now that (i-iv) completely describe all finite subgroups of the orthogonal group S q 3 ) . Classification Theorem 1: A n y finite subgroup G of S q 3 ) coincides with one of the
above polyhedral groups (i)-(iv).
19
51.1. Groups and group actions The proof involves several steps. 1) We observe that any rotation U E S q 3 ) has two fixed points on the unit sphere
S2 = {It t 11 = l}, indeed the eigenvalues of an orthogonal U in R2 are X = e
* ip; l!
2) Take set X of all G-fixed points on S2. Set X is G-invariant, hence splits it into the union of G-orbits:
X = w1
G, = { U : U z = z}, and call
u...Uwm.
For each point z E X we consider its stabilizer
I G, I = n2 - the
degree of
(2).
same orbit have equal degrees (stabilizers are conjugate!),
80
Obviously, all points on the nz
= n(w 1.) = n3., for I E w j .
3) Next we count the number of pairs { ( z ; U ) :z E X ; U E Gz} (the “total degree of C”) in two ways: #{fixed points of all U E G\{e}} = “sum of degrees of all {z}”.This yields,
c
2(IGI-1)=
c rn
(IG,l
-l)=
lwjl(nj-1).
(1.3)
+EX 1 4) Dividing both sides of (1.3) by I G I we derive the equation relating the order of G t o degrees of its fixed points:
I
I
The rest of analysis closely resembles a classification of Platonic solids. Namely, one can show that m in (1.4) could take on two values m = 2;3 only! Each case is analyzed separately. 5) In case
m,one easily shows n1 = n2 = n = I G I. So G has two 1-point orbits of
degree n, which implies G = Zn- cyclic!
6) In case1 Case
there are 3 possible subcases:
PI = n2 = 2; n3 = nl. Here G has two “n-point” orbits of deg = 2 and a “2-point”
orbit of deg = n, which implies G = Dn- the dihedral group!
-1
Case
yields 3 possibilities for
n3
3; I G I = 12; Gtetr ( 3 orbits of degree: 2;3;3) 4; I G I = 24; GCube(3 orbits of degree: 2;3;4) 5; I G I = 60; Gdodec(3 orbits of degree:2;3;5)
In a similar vein one can describe symmetries of regular polytopes in higher dimensions [Cox].
3. Other interesting classes arise as automorphisms of finite groups, and regular graphs. Among them we shall mention finite groups of Lie type: G = GLn(F), and
SLn(IF),made of n x n-matrices with entries in
a
finite field IF. There many similarities
51.1. Groups and group actions.
20
in the analysis and representation theory of classical and finite-type Lie groups, but our attention will be focused mostly on the continuous case.
1.3. Compact groups. These are topological groups with compact space G. The E [O;l]}ci Rn/Zn; the classical most important examples include t o w 8" = {(tl;.,.tn):tj compact Lie groups, like orthogonal - S q n ) ; unitary - SU(n); symplectic - Sp(n), et al. The general theory of the compact and finite groups will be developed in chapter 3. The classical compact Lie theory will be covered in detail in chapters 4-5. Other interesting examples arise as matrix groups over p-odic numbers (pprime): Q,, or integers Z,, i.e. the closure of rationals Q, or integers Z, in the padic norm: II a llP = p-". Here n denotes the largest power (positive or negative) of prime p in fraction a = p"a'.
The corresponding p - a d i c Lie groups: GLn(Qp); SLn(Qp), and their compact subgroups
GL,(Z,); SL,(Z,), consists of all Qp(Zp)-valued n x n matrices (respectively matrices of det
= 1). These groups find applications in the number theory and algebraic geometry,
but we shall not venture into the subject of the padic analysis (see [GGP]; [JL]).
1.4. Lie groups form the most important and interesting class, which plays the fundamental role in many areas of Mathematics and Physics: analysis, differential equations, geometry; classical, quantum, statistical mechanics. In general, Lie groups are defined as manifolds with smooth (differential) group structure: multiplication and inversion operations. A brief introduction to the Lie theory is provided in $1.4. The main bulk of the book will deal with the analysis and representations of Lie groups, emphasizing both general aspects of the Lie group theory as well as many specific examples and applications. Lie groups could be divided into two large and distinct classes: solvable and nilpotent (a subclass of solvable); simple and semisimple. The definitions of both classes will be given in 51.4, and the detailed analysis conducted in chapter 6 (nilpotent and solvable groups), and chapters 4,5,7 (simple and semisimple groups). Although both classes differ substantially in their structure, there are abundant connections, and parallels in the harmonic analysis and representation theory. The first class is exemplified by
+
b}- all affine transformations of W. i) 1-D affine group: A f f = {da,*:x-iax This group can be also realized by 2 x 2 matrices of the form { A = b E W;a E R*}. ii) the celebrated Heisenberg group 3 x 3 matrices
W, (and its higher-D
(" i);
cousins Hn), realized by
21
31.1. Groups and group actions
iii) The group B, of upper/lower triangular matrices in GL, (called often the
Bore1 subgroup),
{
B,= A =
r
A, b
1
Xj;a,b,cER,orC
Groups Aff, and B, are solvable, whereas
.
W, - nilpotent.
In $1.4 we shall see that any Lie group can be decomposed into a semidirect
product B D H , of the solvable normal subgroup B, and a semisimple subgroup H . Any semisimple group H in turn can be decomposed into the direct product of sample groups. The latter were completely classified in celebrated works of E. Cartan. They comprise 4 series of classical Lie groups, listed below, and a few exceptional groups. Many classical Lie groups arise as linear transformations of vector spaces over 02, C or quaternions Q, that preserve certain bilinear/quadratic forms, in other words as subgroups of GL, or SL,. These include,
i) Orthogonal groups: q n ) , S q n ) , preserve Euclidian inner product x .y , q n ) = {U:Ux . Uy = z . y } , and S q n ) = {U E q n ) :det U = 1). There are also indefinite orthogonal groups q p ; q ) and S q p ;q), that preserve indefinite products: P P+Q (z y ) = f: z j y j - C z j y j in P+l
I
so that
RPtQ,
or
CP+q,
( U z I Uy) = (z I y ) , for all z,y. The principal difference between the definite-type and indefinite-type groups in the real case is that the former are compact, while the latter are not, as exemplified by the S q n + l ) and the Lorentz S q l ; n ) , mentioned earlier. But in the complex case the difference disappears, so S q p ;q ) N Sqp+q), for p , q. ii) Unitary groups: U(n), U(k;m), preserve definite (or indefinite) hermitian inner product in C“, z . F , or k
( z I w) = ?zjwj
- c zjwj; in cktrn. ktm
ktl
A particular example is the conformal group SU(1;1). iii) SympIectic grou s: Sp(n) consist of all 2n x 2n matrices that preserves a skewsymmetric form J =
[-I IF (z
I z’) = a:.y’-z’.y
= J z .z‘; z = ( 2 , y ) .
$1.1. Groups and group actions.
22
In all cases the corresponding group consists of matrices, that satisfy
U*JU = J , where U* denotes the transpose (or hermitian adjoint) of U , and
J is either identity (for
the orthogonal and unitary groups), or the symmetric form Xkm with k pluses and m minuses on the main (for the indefinite orthogonal and unitary groups), or the skew symmetric form J in the symplectic case. The above list contains most of the classical examples, but it does not represent their classification scheme (see chapter 5 ) , our emphasis was mainly on construction. The reader will find (problems 5,6), that 3 series overlap, particularly in low dimensions. More examples of this nature will come in $1.4 and subsequent chapters. 1.5. Discrete groups. Those typically arise as discrete subgroups of continuous
(Lie) groups, like lattices Z' 2 Z" in W", or their noncommutative counterparts, lattices r c Gf,; SL,, and other Lie groups. Important examples of the noncommutative lattices are i) the unimodular group I' = SL,(Z), and some related subgroups of SL, ii) discrete Heisenberg group: matrices (1.5) with integer a , b,c E Z.
iii) discrete subgroups
r of the Euclidian motion group En.
The latter are called crystallographic gToups, as they describe all possible crystalline arrangements in Euclidian spaces. They were thoroughly investigated, and classified in dimensions 2 and 3. As E, itself such groups are decomposed into a
r =A
U , where A 21 Zn forms a lattice in R", while a finite group U c Gf, acts by automorphisms (linear transformations) of A. semidirect product,
In 2-Dthe complete list contains 17 groups, whose U-components could take only 10 possible values: {Zm;Dm:m = 1;2;3;4;6}. The complete list in 3-D includes 219 nonisomorphic and 230 nonconjugate crystallographic groups. It was worked out by Fedorov, Schoenfliess and Barlow at the turn of the last century. As a step towards classification one needs all finite orthogonal symmetries, derived in Theorem 1. For further details and relates issues we refer to [BH]; [HC]; [Sch].
$1.1. Groups and group actions
23
Problems and Exercises: 1. Show that any rigid motion on R" (Euclidian or Minkowski) is linear, d(z) = A z for some a E R", and matrix A. Follow steps:
i) The Jacobian q5'(z) is an orthogonal matrix, i.e.
+ a,
aid. ajq5 = 6ij
ii) Differentiate the orthogonality relation (i) in the k-th variable, ak(...), then dot,. Show the resulting 4-tensor tLJ = (8,,,+4 Bid) to be symmetric multiply with 84 in both row-indexes: tik = tki; tm3= tJm; and antisymmetric in column-indexes (changing i - j or k-m changes its sign). iii) Show that any such t must be at once symmetric and antisymmetric in the pair of indexes {ij}. Hence t = 0, for all quadruples ijkm, which implies 8'4 = 0, i.e. q5 is linear! 2. The isometry group of S"-' consist of rotations {V E Sqn)}. Hint: any isometry of s n - 1 extends to an isometry of R"\{O} N S"-'xR+. But the latter are linear by problem l!. 3. Show that fractional-linear transformations (1.2) of SL, on H, or SU(1;I) on D, are isometries. 4. Show that all finite subgroups of S q 3 ) are either polyhedral, or polygonal groups, described in Example 1.1.
5. Establish and find explicit form of the isomorphism SL,(R)+SU(l;I), using their fractional-linear actions on H and D, and the Caley transformation 4: z - + G , from M z+z
to D. Notice that both are subgroups of the larger complex unimodular group SL,(C), with a Caley element u E SL,(C). Find and can be obtained by conjugation, U+U-'UU, U!
6. Show that the symplectic group Sp(1) coincides with SL,. 7. Show that complex groups S O ( p ; q ) and SO(p+q) are isomorphic (Hint: the indefinite product ( z 1 w ) in Cp+* is equivalent to the definite product z . w by conjugation with a complex diagonal matrix).
$1.2. Regular and induced representation
24
51.2. Regular and induced representations; Haar measure and Convolution algebras. We introduce two basic concepts of regular and induced representations, discuss continuity and unitarity and develop some basic algebraic constructions: direct sum, direct integral, tensor product. In the process we introduce the (invariant) Haar measure on groups and homogeneous (coset) spaces H\G, define convolution (group) algebras L’(G), and find the links between representations of the group and those of its groupalgebra.
2.1. Regular representations: In the last section we have described some examples of transformation groups acting on various geometric structures (graphs, manifolds, symmetric spaces), including the action of group on itself by the right/left multiplication (translation), g: x-+xg; or x+g-’x. With any such action of group G on space X (denoted by x - + x g ) , we can associate a
linear
G-action on functions over
homomorphism of G into linear operators over
X, {f(x)} = e = e ( X ) , i.e. a
e,
In particular, the right/left translations, x-+xg; x+g-lx,
on G give
IR,f(x) = f ( g - ’ z ) (left), or f ( x g ) (right), on e(G)l Homomorphisms g+T,, vector space
Y,are
(2.2)
of group G into linear transformations/matrices on a
called representations, and formulae (2.1)-(2.2) provide important
examples of regular representations of G on X, or on itself. In the analysis of group actions and associated representations on various function-spaces, it is often necessary to “integrate” over G or X . So we need some measures on X and G . 2.2. The Haar measure. It turns out that all discrete and continuous (locally compact) groups G , as well as large classes of homogeneous spaces6 X = H\G have an invariant measure, dp(xg) = dp(x), for all x E X, g E G, called the Haar measure. The general proof will be outlined below. More important, however, will be to compute the Haar measure in a suitably chosen coordinate system on group G or space X. Here we shall list a few examples of Haar measures on groups and homogeneous spaces (many more will appear throughout the book). ‘homogeneous (quotient) space X = H\G, or G / H , consists of all right/left cosets {z = Hgo:gOE C } , or {z goH:go E C}. Group G acts on X by right/left translations, g:z+zg = ( H g o ) g , or 2-9(z).So X could be viewed as a G-space, with a transitive G-action, and subgroup H coincides with the stabilizer of a fixed point, H = {g:rog= zo}.
$1.2. Regular and induced representation
a) finite/discrete group G has dp(z) =
25
C C ~ ~sum ( Xof) &-functions over
all
Y
y E G.
b) for commutative groups Wn;Tn, dp coincides with the standard Lebesgue measure (volume element), dz = dxldz, ...dx,. We have shown in $1.1that Rn could be regarded as a homogeneous space of the Euclidian motion group G = En. The Lebesgue measure is clearly invariant under all translations and rotations on
W", hence it also
forms a G-invariant measure on homogeneous space W" = E n / S q n ) . c) on compact Lie groups S q n ) , SU(n), the Haar measure can be explicitly calculated in terms of suitably chosen coordinates, like Euler angles (chapters 3-4). The same could be done on non-compact (Lorentz, conformal, etc.) groups, like S q 1 ; n ) or
SU(1;n ) , by a combination of spherical and hyperbolic angles.
W is a homogeneous space of dz d y measure d p = -is G-invariant.
d) the hyperbolic (Poincare-Lobachevski) half plane
G = SL,(R). One can check (problem l), that
Y2
e) GLn(W) has a natural set of coordinates, matrix entries {zjk}'&l
of g. One can
show (problem 6) the invariant measure on GL,
dPk)=
&=
ndXjk. Jk
For more examples see problems 5-8.
Existence Theorem: i) On any compact (and more general locally compact) group
there ezists a Bore1 measure dp(x), positive on all open subsets, finite on all compact subsets and invariant under all right (or left) translations: EX) = p ( E ) , for all subsets E c G and elements x E G. ii) A right and left Haar measures: d,x; d,x are unique up to a constant factor. iii) On a compact group G the left and right Haar measures are equal, furthermore dp is invariant under the group inversion and conjugation,
dp(x-') = dp(x), dp(g-'sg) = dp(z). Let us briefly outline the proof.
Existence: To construct a right-invariant measure dp, we pick a small neighborhood U of the identity e E G to serve as a gauge. Given a (open, closed) subset E optimal (least) covers of E and G by translates of U , m n E C u U z j , G = U U yj*
j=1
j=l
C G, we choose
51.3. Regular and induced representation
26 The ratio
represents an approximate relative size of E in G in the ”U-gauge”.
m ( E ; U ) as U-.{e}, (by standard arguments such limit always exists!), one Taking limit n(C;U ) gets an honest Borel measure on G. The limiting measure, dp, thus constructed is easily seen to be right-invariant. Also p ( E ) > 0 for any open E, and p(G)
< 00 for a compact
G , since G is covered by finitely many translates of E. Uniqueness: If dji is another right-invariant measure, then for small U ,
m ( E , U ) ef-i ( E ) while n ( G ; U )e m(U)’
subsets E.
fi(U)’
whence p ( E ) = lim
U-lel .
$ = Const f i ( E ) , for
all
I
iii) T o show that the right and the left-invariant Haar measures on a compact G are equal, we take a left translate, d j i ( z ) = dp(az), of a right-invariant measure d p ( z ) . Of course, dji is also right-invariant. Hence dji = p ( o ) d p , by the uniqueness of dp. The map a+p(a) is a homomorphism of
G into the multiplicative group R+. But there are no such
nontrivial continuous homomorphisms on a compact group (continuous functions on compact sets are always bounded, whereas nontrivial homomorphisms p: G-R,
are
unbounded!). Hence, p = 1, and d p is also right-invariant. Two other invariance properties of d p easily follow now. Indeed, change of variable 2-z-l
takes a right-invariant measure into a left-invariant measure, the conjugate-
invariance immediately follows from the bi-invariance: d p ( a r b ) = d p ( z ) for all a, b. This completes the proof. R e m a r k In many cases (e.g. Lie groups) the right and left invariant measures are dlb) absolutely continuous one relative t o the other. The corresponding density -= A(g) is 4(9) a character (homomorphism into the numbers) of G with positive real values
A:G+R,; A(gh) = A(g)A(h), called the modular function. Groups with equal right and left Haar measures, A(g) = 1, are called unimodular. Large classes of groups are known t o be unimodular, for instance, all compact groups, semisimple and nilpotent Lie groups. Nonunimodular examples include: a f i n e groups and more general Borel groups of all upper/lower triangular matrices (problem 8).
The invariant (Haar) measure, topology/metric, and the differential structure on group G allow to int,roduce a variety of function-spaces. The most important among those are
C(G) - continuous (bounded) functions {f(z)}on G, with norm
II f II 00 = S U P I f(z) I ZEG
7
$1.2. Regular and induced representation and its subspaces functions; 0
e,, -
LP-spaces on
27
functions vanishing at {cm},and C, - compactly supported
G with respect
to the Haar measure dz,
in particular Hilbert space LZ(G)with the standard inner product
0 For Lie groups G (smooth manifold) one can consider a variety of differentiable function-spaces: Y ( G ) - Um-smoothfunctions”, C”(G) - infinitely smooth functions; their subspaces of functions vanishing at {m}, or compactly supported ep, with norm
er;
sum over all
llfllrn = C S U P Jaaf(41, XEG “partial derivatives” of order I a I 5 m (the meaning
: f l * f z ) ( z= )
J
fl(zy-’) f z ~ d )y = G
J
of “derivatives”) on
~ I ( Y )f z ( ~ - ’ z d5 )
(2.3)
G
integration with respect to the Haar measure d y on
G. On
finite/discrete groups
integration in (2.3) is replaced by the sum,
f*h =
c
f(zy-’)h(y).
YEG
The meaning of convolution, as an extension of the group multiplication to functions, becomes transparent here, as each group element {z} is identified with the delta-function 6,, so that
31.2. Regular and induced representation
28
6,* 6, = SZy;for all x,y E G. For continuous groups
G convolution-integral (2.3) is well defined on compactly
supported functions (ec;ep),then it extends to larger classes by the standard density arguments7. Spaces closed under convolution forms convolutzon/group algebras. One important example is space L'(G) of all integrable functions on G. Here,
Ilf*hli O
(moreover, 36, contGns yet another dense subspace A of analytic vectors!). All subspaces are invariant under the group U , , and of course, U,136,
is strongly continuous in the rn-
norm.
Let us illustrate the general concepts of smooth vectors; closedness; domains; “strong” vs. “uniform” continuity of operator-groups with a few simple examples. 4.6. Examples: 1) Multiplication operator, A f ( z ) = zf(z), on L2-spaces: L2(W); ~ ~ ( bl); [ a ;or more general
L’(w; dp), generates a unitary group: U,f = e i t t f ( z ) , also
acting by multiplications. i) On finite intervals [a;b],or for measures dm of “finite support”, the group is uniformly (norm) continuous, and generator A is bounded, 11 A 11 5 C ,if supp I c [ - C;C ] . The two conditions (norm-continuity and boundedness) always come together. Furthermore, all vectors {f} in L2 are “00-smooth” (even real analytic), so
36 = 36, = A = L*.
ii) On infinite intervals (e.g. W), A is no more bounded, and U , is only strongly continuous: 1 U ,f - fIP0, as t+O, for any f E L2. Not all vectors are smooth, however, the space of “1-smooth vectors”: 361 = P(A) = {f: similarly “m-smooth vectors”:
I f 111 = I (I+ I 2 I 1f(z)llL2
36, = 9(Arn) = { f :llfllrn =Il(ltI z I
)“f(.)IIL2
< CQ}’ < 00}, etc.
Vector-function ei‘”f(z) could be differentiated at t = 0, only for 1-smooth vectors { f } , to get Af = zf!
2) The translation-group W: U,f(z) = f ( z - t ), acing on L2 (or other LP-spaces) is strongly continuous. Its generator A = d is an unbounded operator with a dense domain made of “1-smooth” vectors/functions, f E L2,so that f’ is also L2,
{
1 1
36, = f:1 f 1 1 = 1 f 1 L~ t f’
L2}
- the so called first-Sobolev space.
$1.4. Lie groups and algebras; The infinitesimal method
58
Similarly, LLm-smoothvectors” coincide with um-smooth (Sobolev) functions” (see
36, =
{f : l l f l l m
= IlfllL2
-+ ... +I1 f(m)IIL2} - m-th Sobolev space25.
The last example explains the terminology “smooth vectors”. It could be extended to more general one-parameter group actions, e.g. flows on
manifolds,
4,:
z+y(z;t) on A. Any such flow is generated by a vector field
as a solution of an ordinary differential system:
X = Cajaj,
+ = X(y); y(0) = z (so we could write:
dt = exptX). Associated to any flow dt is a 1-parameter group of linear transformation on function-spaces over A, e.g. L’(A),
U , f ( z ) = ~ ( I # J , ( z )or ) ; more general V,f(z) = a ( z ; t ) f od,(z), where
(I
satisfies the cocycle condition: a(+;t + s ) = a(+;t ) a ( d , ( z ) ; s ) The . generator of
X considered as a 1-st order differential operator C ajaj on (it plays the role of D = in example 2). Smoofh vectors consist of dz
group U,, is the vector-field functions over A
functions differentiable along the flow. In the second case, the generator is the 1-st order operator of the form: A = X + q , where q is a multiplication operator with function q(z) = crt(z;O)(at- derivative of
(I
at t = 0). If function q is sufficiently smooth, one can
check that Um-smoofh vectors” of A are the same as for X.
Once the notion of generator of one-parameter groups was made clear, we can proceed to more general Lie group representations.
Theorem 5: Any representation g+T, of Lie group G in space 36 gives rise to a representation X +Tx of its Lie algebra (li: (4.14)
In the 00-D c u e generators { T x } are unbounded operators defined a joint dense core of “smooth vectors” 36, = {.$(Tg(I q ) E C?(G);for all q E 36). Furthermore, for unitary T operators { T x } are skew-symmetric. Conversely, a representation of Lie algebra 0 by (closed, unbounded) generators { T x } can be ezponentiated (lifted) to a representation of its simply connected Lie group,
T,
= exp T x .
(4.15)
Proof: On the purely algebraic level (finite-D, uniformly continuous case) the result becomes a simple consequence of the exp and log maps (defined via Taylor expansions)
1 + I D I )“f
-One can show (52.2) that the m-th Sobolev norm is equivalent to (1 I D I = ( D * D ) 1 / 2 is the modulus of the differentiation operator D = L. dz
lL2;where
51.4. Lie groups and algebras; The infinitesimal method
59
and the Campbell-Hausdorff formula (4.2). On the functional-analytic (unitary) level some technical complications appear, as generators { TX} could be unbounded (densely defined) operators, each with its own domain !D(TX).However, all of them have a joint dense core of I-smooth uecfors 36,. As in the one-parameter case, 36, contains the range
{( = T f ( q ) :q E 36) of any mollifier T with f E e,!(G). In fact, there exists the whole
f
scale of
3 6 , 3
m-smooth
spaces: 36 3 36, 3 ...36,
3
...36,
3A
(analytic vectors), with
{ T f ( q ) :E f ey(G)}. All of them are dense and invariant under the group action
{Tg}(for analytic vectors this result is due t o L.G$rding). On each space 36, one can define the m-fold products and powers of generators {TX...T X :XjE a}, furthermore 00
1
,
on analytic vectors {( E A } the Taylor series:F$TF(E), converges. Thus we get the group action on analytic vectors { T gI d:g = e x p X } , which then extends rough the whole space 36. An alternative way t o obtain the group representation in the unitary case (skewsymmetric generators) is via spectral resolution:
T X = JXdE(X), E(X)-spectral measure, whence the corresponding unitary group: exp(t T
~= T~~~~~ ) = Je’”dE(X),
as in Proposition 4. This completes the proof.
Let us remark that many natural properties of representations, including irreducibility, decompositiona6, etc. can be transferred from Lie algebra 8 to Lie group
G. However, in the absence of simple connectivity of G local representation { T e X p x } given by (4.15) may not be extendible through the whole group. Indeed, only “half’ of irreducible representations of simply connected group SU(2) extend through the representations of Sq3)u SU(2)/{i I} (chapter 4), and a similar pattern holds for all other non-simply-connected groups. Clearly, the dual object of any factor-group G / H consists of all i~ E G, that annihilate H , i~ I H = 1.
26with some obvious qualification: for any invariant subspace of the Lie algebra generators So irreducibility of { T x } means any {Tx}-invariant subspace 36 is dense in 36.
ITx} we take its closure in the %norm.
60
51.4. Lie groups and algebras; The infinitesimal method
Problems and Exercises: 1. Prove Lemma 3 (Hint: some statements, like (ii), are easy to check first for diagonalizable mairices A = U-'DU (D - diagonal, U E GL,), and then to use density of diagonalizable {A) in Mat,!). 2. (i) Prove formula (4.5) for matrix subgroups, using the Taylor expansion of { e x p t A } . It is known as Trotter product formula (4.5), and has numerous extensions and applications to groups (and semigroups) of operators in spaces of finite and infinite dimensions. (ii) Show that formula (4.6) defines a 1-parameter group with generator A. 3. Show that subspaces d,; so(n) and
4.)of g/, = Mat,
form Lie algebras.
4. A (strongly continuous) one-parameter group of unitary Hilbert-space operators { U t } has a selfadjoint generator A = A*, i.e. U , = exp(iAt). Show U , is uniformly (norm)
continuous iff operator A is bounded. Steps: i) Use spectral decomposition of a selfadjoint (not necessarily bounded!) operator: A is unitarily equivalent to a multiplication, r : f + r f ( z ) , in LZ(R;dp) or in the direct sum
e,3 L2(R;dpj).
ii) Show: A is bounded iff s u p p t d p } or U s u p p { d p j } is compact in R. iii) Unitary group { U t : f-e""f(z)} compact.
in L2(dp) is norm-continuous iff s u p p { d p } is
5. Show that all upper triangular matrices over R or C form a solvable Lie algebra 23,. Find the derived series for B,, and the corresponding Lie group. Do the same for the the algebra U, of all upper triangular matrices with 0 on the main diagonal (show that U is nilpotent, find its lower derived series, and its Lie group).
6. Find block-structure (as explained in example 3) of Lie algebras: 4 n ) , su(n), s o ( p ; q ) , w ( p ; q ) , sp(n), and use it to compute dimensions of the corresponding Lie groups.
7. Show that (4.13) defines Lie bracket on the semidirect product % @ 8 of Lie algebras.
r
8. Show that a normal discrete subgroup of a Lie group G belongs in the center Z ( G ) (Hint: conjugate each y E r by elements g E G, y+g-'-yg, and observe that all {g} in a vicinity of {e} fix y).
Chapter 2. Commutative Harmonic Analysis. Schur's Lemma implies that irreducible representations of a commutative group G consists of I-D characters, i.e. homomorphisms { x:G+C* numbers)}, or {x:G-+T
(multiplicative complex
= {e"} - unitary characters}. We shall mostly deal with the
latter. The characters can be multiplied and inverted: (x1;xz)
+
x1x2;
x
x-l =
-+
1 Xb)' -
so they also form a commutative group, called the dual group
e. Dual group of a locally
compact G , is itself locally compact [Po,], hence carriea an invariant (Haar) measure. Three commutative groups lie a t the heart of the classical harmonic analysis: R";
Z" C R" and 1"N R"/Z". The characters of all three consist of exponentials X,(z)
= e i t . '.
In the case of R", parameter ( runs over R",
80
the group becomes isomorphic t o its dual.
For Z", identified with an integral lattice {m = (ml; ...mn)] in R", characters
x = xO(m)= e " . m ,
...
are labeled by the dual torus, 0 = (Ol; On), 0 5 O 5 2u, so
A
Z" u T". The characters of torus T" = {(el;...On) : 0 5 Oj 5 2 x 1 , or more general (0 5 Oj 5 Tj], are in turn labeled by the lattice points,
x = Xm(4
eim .O.
, m E Z",
so 1"N Z". Let remark that the mutual duality of groups 1" and
Z" exemplifies the
general Pontrjagin Duality principle for locally compact Abelian groups [Pon]: the second dual group (characters of G) is isomorphic t o G itself, via the map: u-G(x) = x(a), a E G (elements a E G define characters on G). So the classification problem (I) of
chapter 1 for commutative groups G amounts t o characterization of
6,
while the
decomposition problem (11) for the regular representation R will lead to study the Fourier transform 9 on L2-spaces and other spaces. This marks the starting point of the Fourier analysis. Chapter 2 gives the basic overview of the Commutative Fourier analysis, and some of its applications, the main emphasis being on differential equations.
$2.1. Inversion and Plancherel Formula.
62
$2.1.Fourier transform: Inversion and Plancherel Formula. In this section we shall introduce the Fourier transform/series expansion on R"; 2"; T", discuss their basic properties (with respect to translation, differentiation, dilation, rotations. Then we establish the key result of the Fourier analysis, Plancherel/Inversion formula, in 2 different ways. One exploits Gaussians and Schwartz functions, another Poisson summation.
1.1. Fourier transform 4 on group G = R";T" is defined by integrating functions {f} against characters {x = e i t e Z } ,
while on
Znintegration becomes summation,
Sometimes we shall indicate the
"I
to
transform by TZjt.
From the
definition one could easily check a few basic properties of 4. 1) Convolution-+product, 4:f*g---+?(t)G(t) 2) Translation-tmultiplication with character, 4:R, f = f ( z - a)-+eia't?(t)
3) Differentiationjmultiplication, 4:8,f +itj?(t); hence 4:aaf+(it)"Y Here
{aj} and (8%= ( a l...; a,,)}denote partial differentiations in I . The latter
relation is easily verified via integration by parts:
/(a"f)g = / f ( - a ) a g ;
with g = e T i Z . t .
We introduce an involution on functions { f ( s ) }on G,as a linear map,
-
f(-.),
*: f(.)-f*(.) = with the obvious properties: ( f * g ) * = g**f*; (f*)* = f . Clearly, 4) Involution-tcomplex conjugation, 9:
f*(z)+?(t).
5)A linear change of variable, A:z+Az
( A E GL,)
on Rn,
defines a
transformation of functions,
f + f A ( z )= f o A = ~ ( A I ) . Fourier transform takes such linear change into its inverse transpose, precisely 4 : f AdetA + ~ ~('AA-'). o
$2.1. Inversion and Plancherel Formula
63
Finally, relation (3) differentiation-+multiplication, can be reversed into
6) m ultiplication-tdifferentiation, 9:xjf(x)+$3,,f^([); As we mentioned earlier characters
+~~f-(ia,)~?
{x} describe all irreducible representations of
the commutative group G. On the other hand they can be interpreted as joint eigenfunctions of all translations { R,:a E G } , respectively, of their infinitesimal
generators {aj} (on 53"; U"), as well as convolution operators: R f [ g ]= ( f * g ) ( x ) . Formulae (1-3) can be rephrased now in the language of the representation theory by saying: map 9 diagonalizes regular representation R of G,
9R9-
N
"direct sum/integral of characters".
Precisely, any function f (in a suitable space) can be expanded into the direct sum/integral of characters
x = eix
*
€,
C
1
f(x) = bEeix* t; or f(z) = b( [) eiz ' '%(, (1.3) with "Fourier coefficients/transform" b(() = ?([), and the regular representation turns into multiplication on the space of "Fourier coefficients",
Later on (chapters 3-7) we shall establish various noncommutative versions of this result.
1.2. Plancherel and Inversion formula. The fundamental property of exponentials { e i < ' + } is their orthogonality and completeness. In case of compact G (e.g. U"), orthogonality can be understood literally',
as all
{x,}
are L2-functions on G. Completeness means that any f can be
approximated, hence uniquely expanded into the Fourier series. An elementary proof for
T"exploits the
Cezaro means (problem 3).
More general explanation has to do with compactness of convolution-operators, R f:u (z )d f*u (+) , in L2 (LP)-spaces on 1" (Appendix B). Compact self-adjoint (f = f*)
operators, are well known to have a complete orthogonal system of eigenvectors, and exponentials {xm = ei+ * m} are precisely the eigenvectors of R
f , f*eiz
*
= F(m)eic ' €.
'The same holds on an arbitrary compact group G (see [HR]; [Loo]),namely,
(X, I xa) = I&',&d+
= I G I 6km;
I G I = Ud(G);6km-
Kronecker 6.
62.1. Inversion and Plancherel Formula.
64
Similar arguments apply to all compact groups (chapter 3).
For non-compact groups W"; Z" orthogonaZity should be understood in a generalized (distributional) sense,
l(eit.z I ,ir)
.z)=
c
,i(t-r))
*
zdz
(1.5)
In other words, the role of Kronecker 6,k = S(m - k), being played by the Dirac 6-function. The integral (1.5) is strictly speaking divergent, but it makes sense as a distribution. Namely, for any nice (testing) function f(q) on R",
We shall see that in such form orthogonality is equivalent to the
Inversion/Plancherel formula for 9. Namely the transform,
4':h(q)-tL(x) = ( h I eWiV.') = Ih(q)eiV*'dq, takes
f^
(1.6)
back into Const x f. Therefore, (1.6) defines the inverse Fourier
transform of h. The precise result is usually stated for a suitable class of
functions on G. We shall treat separately compact (T";Z")-cases and noncompact Rn- case.
1.3. Compact case. A suitable class of functions on torus, called Fourier algebra, consists of absolutely convergent Fourier series, A = {f(x) = CakeiZ * k: Clearly, algebra Ace(U"),
c I ak I < m}.
is made of continuous functions on
U", and
contains sufficiently many smooth functions, A 3 em,for m > n. The latter is easy verified using differentiation formula 3). Indeed, Fourier coefficients of a partial derivative
aaf = C(ik)aakeiz'k , by (iii). So if f E em, then its
Fourier coefficients ak = o( I k I,),-
which
guarantees convergence of
c l%I. Uniqueness and Plancherel Theorem on T": (i) Any function f E A(Tn) is uniquely determined by its Fourier coefficients, {ak}*f(z)
(ii) For any function f =
=
Cakeiz' '.
CakeiZ " E L',
In other words 9:L2(T")+12(Z"), is a unitary map. Both results are fairly
52.1. Inversion and Plancherel Formula obvious corollaries of the completeness and orthogonality' (problem 3).
65 of {eiz * k: k E Z"}
1.4. Continuous case W". Here a suitable class consists of all functions { f},
integrable along with their 9-transforms,
{f E ~
T E L I } = L' nA.
1 :
As above, A denotes the Fourier algebra {y(z): f E L'}, of absolutely convergent Fourier integrals. As with T" algebra A is shown to consist of continuous functions, decaying at {oo}, a consequence of the Riemann-Lebesgue p ' - compactly supported mTheorem (problem 1 of 52.2). Furthermore, A 3 C smooth functions3, so
C'y c A c C',,
for m > n.
Inversion/Plaucherel Theorem on R": (I) A n y function f E A n L', can be uniquely
recovered from
1 b y the Inverse Fourier transform
(11) transformation 4,normalized b y factor L2(Rn) onto L"), i.e.
1becomes a unitary map from (27r)"/2 '
The Fourier transform of an Lz-function in (1.9) should be understood in the "square-mean" sense, i.e. by approximating f with nice (L'; compactly supported) functions. Let us make a few comments: 0
the general inversion formula (1.8) is equivalent to a special case z=O (and is
often stated in that form), G
multiplication of ?(() by e i z * < corresponds to a shift f(O)+f(z), under the inverse transform 9':
?+f.
'Indeed, for any complete orthogonal system (basis) { $ J ~in } Hilbert space 36, any f E 36, is uniquely decomposed into a series f = C akdlr,and 11 f 11 = C 1 ak I Gk (Parceval identity). Clearly, factor ( 2 ~ ) "in (1.9) represents L2-norms of Fourier harmonics {gk=
1) 1 '
3The inclusion
er c A
j( 0, into homogeneous functions of degree (s-n): f(t() = t""f((). Apply it to the Riesz potential to show, R,( I z I ) = C(s) I z I Compute constant
'-".
C(S).
10. Show that the Lm-norm in the definition of Schwartz functions Y can be replaced with any other Lp-norm. Show also that operations (1+ I z I 2, and (1 - A ) can be interchanged in the definition of 9. (Hint: apply a version of the Hardy-LittlewoodSobolev inequality (2.17) for LP-Sobolev spaces: 36: = (l-A)-'I2LP, to show that 36; is embedded in LQ, for = - & and
II f 11LQII (1-A)"'2f
IILP'
$2.3.Applications o f Fourier analysis
83
v.3.Some applications of Fourier anal+. Here we picked up a few selected topics and applications of the commutative Fourier analysis: Central limit Theorem of probability; the Heisenberg uncertainty principle; Finite Fourier transforms, Bochner’s Theorem and the Mellin transform. The latter has application to special functions and differential equations in R“, and also in the harmonic analysis on SL, (chapter 7).
3.1. The uncertainty principle. The Heisenberg uncertainty principle of the Quantum mechanics will be discussed in 56.2 of chapter 6. It has a simple Fourier-
for a n y function $ in
Iz I
and
I< I
Rn, such that $ and
4 are both square-integrable
with weights
respectively.
Rephrasing (3.1) in terms of the derivative (gradient) a$, we get
In this form it becomes an easy consequence of the standard integration by parts formula, /z$# =
/z(i I $ I
2)1
=
-$I 14 I ’,
(1-D); or/V$.z$ = -
I31
$ I ’(Vex) =
-51
I $ I 2, (n-D),
and the Cauchy-Schwartz inequality applied to the LHS,
I/z$.v+(/
lv$,~yz~
Izdi~)li2(l
Relation (3.2) implies that one can simultaneously localize functions $ and
4 at
I $ 11L2
(0) maintaining the Lz-norm = 1 (LHS), in other words, localizing supp($) in zspace will ‘‘stretch” its Fourier transform $, hence increase gradient, and vice versa. In quantum mechanics operators z: $+$, and &$+a$, represent the “position and momentum” of the quantum particle, while integrals
(1 I4
V d x ) and (IlW 1 2 4 ) estimate errors in their measurements. So the Heisenberg principle prevents
simultaneous precise determination of both operators to any degree of accuracy.
3.2. Finite Fourier transform. Here we shall briefly discuss the Fourier transform on finite commutative groups. The simplest of them is a cyclic group Z,. Its characters
52.3. Applications of Fourier analysis
84
coincide with the n-th roots of unity ( x Q $ j ) = w J } , where w = e z p ( F ) . So the dual group Z, N if,, and t,he Fourier transform on Z, becomes
c5
s:f(j)+?(e) =
f(j)w
3e.
(3.3) Characters { x Q= (wQJ)} form an orthogonal system of eigenvectors of the o53
n-1
translation operator R:f(j)+f( jtl),on space L*(Z,)
N
C"
, with norms
(x I x) = IIXIIZ = n. The inverse transform takes the form,
c
+i(j)= A
4-1:qe)
o
while the Plancherel formula,
a(e)w~l,
5 Q 5 n-1
llfllZ= ~lf(dlZ=AE I?(e,12=11?112, both being obvious consequences of the orthogonality of {xl}.
":I
We shall study now the czrculunt mat& A, that appears in many applications of finite groups (in linear algebra, probability, etc.),
[ :; :
... A = ... ... ... ... ...On-1 an a2
1 . -
each subsequent row of A is obtained by a cyclic permutation of the n-tuple (ul; ..a,,). Notice that matrix A is a convolution with function u(k) on Z,,
A f = a* f. So Fourier transform (3.3) applies to diagonalize A (find its eigenvalues and eigenvectors), compute inverse, determinant, etc. A specific model of such A is given by the
random walk on
Z,, where
az =a,
=f
(probabilities to j u m p from point (1) t o its neighbors (2) and {n) in a unit time), and the rest aJ = 0. More generally, we consider a stationary stochastic matriz':
C a , = 1 (aJ measures probability
aJ
2 0;
to j u m p from cite (1) t o { j ) ) .
Characters { A @ } form a complete set of eigenfunctions of A with eigenvalues,
c~,~J~,
A, = q e ) = - the l-th Fourier coefficient of function { ~ ( j ) l} ,= 0;1;...n-1. Diagonalizing A, via 4, 'Stationarity means that all rows of A are permutations of a single row, so the process is independent of the starting point: probability of "i+j"-jump depends only on the difference i - j E 2,: a . . = a(i-j). 13
92.3. Applications of Fourier analysis
85
allows to compute d e t A ; A-', all iterates A"', and yet more general "functions of A", B = f ( A ) . Indeed, any such B is itself a convolution with function b ( j ) , whose Fourier transform, b = f i?, so A
0
b(j)=T'[foi?] =
Sc e
foZ(e)wje.
As an application to random walk on Z,, we immediately find the probability distribution of the process after m steps (units of time), i.e. find entries of the m-th iterate Am. If p$:)+k denotes the probability of jump from j to j+k after m steps, then p$:)+k = s-'[xT].
Hence, pi:) =
ax
and P$:3)+k = 4
cosm (T); 2=e
e
c
cosm(@) cos(@).
e
Remark: More general finite groups G are known t o decompose into the direct product of cyclic groups, G cz
n Z,.
Hence, G =
nZ,
u G, and the Fourier transform reduces to
the Fourier transforms (1.3) on cyclic components of G. One interesting application of such finite Fourier transform is the random walk on the n-cube (problem 9).
3.3. Central Limit Theorem. Another interesting application of the Fourier Analysis is to r a n d o m walks on groups W , Z,or more generally, to sums of independent r a n d o m variables, S , = X,+X,+ ...+X,. Here we shall establish the Central Limit Theorem, that describes asymptotic distribution of S,, as n + w .
Any real random variable X defines a probability measure (distribution), d p X on R. Independence of two random variables X , Y means that their joint distribution d p x , y ( z , y ) on RZ is the product d p x ( z ) d p r ( y ) . So the sum of two independent random variables X + Y has distribution,
+x+,
= dPX*dClY (3.4) W. Another simple transformation of X ,
- convolution of two probability measures' on
scaling X + o X , results in the dilation of the measure dPx+dPx(3
(3.5)
We consider the sum of n independent identically distributed r a n d o m variables,
s,
x,,
= XI+ ...+
with equal probability distributions, d P x . = 4% 3
and ask about the asymptotic behavior of sums {S,} at large n. 'Convolution of L'-functions f*g on R" (or any group G) can be extended t o bounded measures {dm}, and other (compactly supported) distributions/generalized functions (see 92.2), via pairing to continuous test-functions
If).
$2.3.Applications of Fourier analysis
86
A typical example will be random walk on the lattice Z. Here variable X takes on values f 1 (jumps from
k t o k f 1) with equal probabilities
f, so dp = 36,+6-,).
Then the sum
of n variables, S,, measures the position of the walk after n time-steps, relative to the initial point.
It turns out that the limiting (large n ) behavior of sums {S,} carries some universal features, independent of a particular distribution dp. In what follows, we always talk about convergence of random variables {Y,} in the distributional (weak) s e m e , namely,
Yn+Y, if
I
fdpy
n-I
f d p y , for any continuous f(z)on W.
Theorem 1: L e t { X n } r be a sequence of independent identically distributed r a n d o m variables with mean: E = xdp(x), and the dispersion
I
0=
J
I(x-E)'dp = z'dp - E2 > 0.
Then n
(i) S, = EXi has n o limiting distribution, as n+w. j=1
.
(ii) $ , 4 , n
(iii) L E ( X I . - E ) + N ( E ; U ) = 0.
normal distribution with dispersion
J;;1
Rephrasing Theorem 1 we can say that the mean value of sum: E(S,) increases linearly with n, while its dispersion o(Sn-nE) u f i , grows as f i .
-
= nE,
The proof exploits the notion of the characteristic function of variable X ,
the Fourier transform of measure p. The characteristic function has the following properties:
A ) d(() is continuous and positive-definite 7, for all tuples
{t1;... tn} in R and {al; ... an] in C; also d(0) = 1; Id(€) I 51.
B) if measure dp has all moments to order k, i.e. 'The converse result is also true (see Bochner's Theorem below). is also true. Namely (problem 12): any positive-definite continuous function on group R; R", or more general commutative G , is the Fourier transform of a positive (probability) measure.
52.3.Applications of Fourier analysis then function
I
I z J I dp < co, j=O;
d( 0,
is positive for all functions f E 2'
nA, hence defines an inner product on A n L', II f Ib = (043I 3).
A
,
.
The regular representation, R,f = f( 0},
and assume operator Z to be reflection-invariant: (z;t)+(z;
= { ( z , t ) : z E Rn;
-t),
for instance
L = 8: + A - the half-space Laplacian. The corresponding Green’s function depends on the type of boundary condition. For Dirichlet Problems the Green’s function KD
= K ( z ; t , s )= K,(r;t - s) - I f o ( z ; t + s),
represents the difference between the “sourcesolution” K , ( y ; s) (e.g. Newton’s potential) and the
“reflected source” K , ( y ;
- s)
(fig.3). The reflected source gives the requisite correction u in
(4.37). So the half-space Dirichlet Green’s function represents a convolution kernel in 2-variable, K ( z - y ; t , s) of the form
If = &{( I z 1 2
+ ( t - s)2)
n-1 -2
-( Iz
12
+ ( t +.)2) -F}.
This yields, in particular, the half-space Poisson kernel P ( z - y ; t ) , derived earlier in by the Fourier-transform methods
(4.38)
Let us observe, that solution of the Dirichlet problem: Au = F ; u I r: = f; in any region R with boundary C = 80, is represented by the Green’s identity, as
4.) = where
an denotes
/
nK(z;Y)F(Y)dY
+ / za,K(z;Y)f(Y)dS(Y),
the normal derivative of the Dirichlet Green’s function in y E C. So the
Green’s function K picks up the “continuously distributed sources” F over R, while its normal derivative P = 8°K - the Poisson kernel gives the contribution of the “boundary sourcesn.
Fig.%: Free-space Green’s functions at a source {y] and the reflected source {b} cancel each other on the boundary r, separating two half-spaces (Dirichlet problem). In the Neumann problem 2 “sources” add up, so the normal derivative vanishes on the boundary.
106
$2.4. Laplacian and related differential equations. For Neumann Problem the difference of two source8 is replaced with the sum
K N = Ko(z;t-s) +Ko(z;t+s), (so that normal derivative8 would cancel each other). As application we obtain the conjugate
Poisson kernel Q in the half-plane14,
found earlier by the Fourier transform methods. In a similar vein one constructs Green’s functions for other problems, e.g. the half-space Dirichlet/Neumann “heat problem” has K(z,y; t ) = G(2 - y;t)
- the difference/sum {y’
C(z - y’; t )
of two Gaussians with “sources” a t {y = (yl; ...y,))
and the reflected point:
= ( - yl; ...yn)}. Quadrants and other product-type regions are treated in a similar fashion.
Thus the quadrant {(z1;z2) > 0) Green’s function consists of 4 reflected sources: K(z,Y ...) = Ko(z - y) - K o ( z - y’)
- Ko(2 - Y”)
+ KO(Z + y);
where y’; y” represent reflections of {y) relative to the 1-st, respectively 2-nd coordinate axis. 4.11. Green’s functions in finite regions. Green’s functions of elliptic differential operators in
bounded regions S2 (compact manifolds) can always be expanded in terms of eigenfunctions {$k} of L, subject to a proper boundary condition. We leave t o the reader as as exercise t o write expansions for the inverse operator L-’, the heat semigroup e-tL; Poisson kernel exp(-t&); wave-propagators, etc. Let us remark that the Green’s function of any regular S-L problem on interval [O;T]can be represented in two different ways: via the eigenfunction expansion 15 K(z;y) = Cfdk(Z)iJ,(Y), (4.39) k or in terms of a fundamental pair of solutions: L[ul,,] = 0, one of which satisfies the zero
boundary condition on the left (ul I t=O = 0)’ and the other on the right (uz I T = 0). Given a fundamental pair {ul; uz}, the Green’s function is constructed as, (4.40)
For constant coefficient operators L =
- 8 + q,
two representations are related by another
version of the Mtthod of images. Namely, we take source y E [O;TJ,and “reflect it” infinitely 141ndeed, solution of the Neumann problem: Au = F; a,u identity, as
I = f; is represented
via Green’s
where K = K N is the Neumann Green’s function, so Q(z;y) = K ( z ;y), for z E 0 , y E E. 151f one of eigenvalues A, = 0, then kernel K (4.5) (sum over nonzeros A’s defines the so called modified Green’s ,function, i.e. inverse of L on the subspace spanned by nonzeros eigenfunctions {dk},so the product of operators L K = b(z - y) - v,bo(z)$o(y).
52.4. Laplacian and related differential equations
107
many times by the lattice Z = {mT}. So the free-space Green's function of Z on R.
is shifted and summed over the
corresponding Green's function
K on [O;T]has different representations depending on the type of boundary condition. Namely, the periodic bounda y condition (torus), gives ~ ( z , y= )
C~
+
~- y ( mT) z
= +C'ezp(i%$z - y)); (4.41) k with eigenvalues: {Ak = ( r k / T ) ,+ q: k E Z}, while for the Dirichlet/Neumann condition the m
source a t {y} is reflected to {-y},
and then the process is repeated periodically with period 2T.
So one has
The reader should recognize relations (4.41) and (4.42), as special cases of the Poisson summation for function f ( t ) = KO(I z - y I
+ t ) on the lattice Z = {mT}. Similar considerations
apply to the heat, wave, KG and other equations, and to equations on multi-dimensional tori. Two such examples are
4.12. Dirichlet Laplacians on triangles.
Rightisoseeles triangle makes up
i of the square. The (rx r)-square eigenmodes are made of
products of the even and odd Fourier modes { c k = cos(k +:)z}, and
y).
{ C k ( t ) s,(y);
So
consist
of
products
{srn= sin2mz) (in variables z
{ c k ( ~ ) c , ( ~ ) ) , { s k ( ~ ) s , o ) and
s k ( z ) c , ( y ) } . The former two (cc- and ss-types) correspond to different families
+i)2+
+
(m +$)'}, and {4(k2 m')}, so they never mix. The latter ( c s - and types) produce eigenmodes Gkrn of the form Ck(z)6rn(Y) 2 L,(')'k(Y), that vanish on
of eigenvalues {(k
sc-
{4krn}
diagonals z = f y, hence yield the triangle Dirichlet eigenfunctions. Equilateral triangle T makes lattice
r c C, spanned by
i of the diamond,
the fundamental region of the hexagonal
{ l ;a = e i S / , } (fig.4). Its Dirichlet eigenfunctions fall into 3 families,
according to the natural action of dihedral symmetry group D, = Z, D Z, on T. The latter has 3 irreducible representations: trivial
xo = 1, x1 = f 1 (trivial on rotations { 1;u;CJ:u= eizXl3}
and
equal -1 on any reflection), and a 2-D representation u, uu:((;q)+(u(;CJq) (for rotations {I;wp}),
and u,((;q) = ( q , ( ) , for reflection r about the z-axis (see 53.3). So family (I) consists
of eigenfunctions16, symmetric relative to the D,-action
, family
(11) are rotation-invariant, but
"Dirichlet eigenfunctions are uniquely determined by the values of their normal derivatives on , on all 3 sides of T. the boundary, so the description here refers to the values of a+
108
$2.4. Laplacian and related differential equations. mapped t o -(1, by reflections, finally (111) gives a pair of conjugate values { @ ; a }{-l;-l},{Sp} , on each of 3 sides of T (see fig.5~). Our goal is to link eigenvalues of A, to the well known eigenvalues of the f-periodic A on torus
c/r, A,
= I i k + a m I = k2 +m2- km.
Figure 4 demonstrates how each eigenfunction of family (I), (11) and (111) extends t o a fperiodic eigenfunction of A: for (I)-(11) is equal to the size of the triangle (the plane is cut into the union of copies of T, and 11, is taken into all images {T‘} of T with signs f , depending on the orientation of T‘. That same reflection procedure applied to a 111-type eigenfunction requires to triple the period. Since each periodic eigenfunction is a combination of Fourier modes {exp[i(kz+m%(E::y)]:k,m E Z}, we obtain the eigenvalue-spectrum of A,,
t o be made of 2
sequences:
1 Atm
= k2 + m2 - km (for 1-11) A’km = g(k2 + rn2 - km) (for 111)
I
Fig.6 below demonstrates a 11-type eigenfunction as a combination of rotated and reflected exponents, (1, = XI - x2
+ x3 - x4 + x 5 - xs. Fig.4 shows cubic roots {w,G = w z } of 1, along with their negatives { a = - w-; E = - w } . Numbers { 1;w;G } give 3 irreducible characters of subgroup Z, c Q,, while reflection (complex conjugation) defines a 2-D representation of D,. Fig.5 illustrates a periodic extension of eigenfunctions of types I, 11, and 111. A Dirichlet eigenfunction (1, in T is determined by its normal derivatives {fi =a,$, on the i-th side Si, i = 1,2,3}.
We schematically represent each I f i } by a curve along Si; the portion of the curve inside the triangle corresponds t o a positive derivative a,f, while the outside portion t o the negative derivative. For type-I all functions { f i } are equal and symmetric with respect t o the center of the side.
So given a type-I (1, in a “+”-triangle, we reflect it t o a neighboring “”triangle, changing the sign to+-$, and repeat the process. The resulting fperiodic function on C is clearly an eigenfunction of A. For type-I1 11, (b) boundary function f is odd on
109
$2.4. Laplacian and related differential equations
each side Si and repeats itself under rotational symmetries with { 1 ; q Z }. Once again reflections +-$, from T + t o T - across {S,;S,;S,} produces a r-periodic eigenfunction of A in C. The type-111 (c) is somewhat more complicated. Here we pick a pair of complex conjugate functions {+;$}. Then one ca? easily verify that the boundary functions {fi;f i} have absolute values I f l I = I f z I = I f JI = p (an even positive function), and differ one from the other by a_ factor w , f 2 = wfl; f, = 5 f l (and similarly for f .). If we choose fl.(left side) to have phase a= then f 2 (horizontal) have phase - 1 , while f, = E p. The reflection $+-4 about side S, takes triple { a ; - l ; Z } into { u = - E ; + l ; G = a } , and the process has to continued 3-times along the (horizontal) line t o get the original triple { a ; - ; E } . So type-I11 $ yields a 3r-periodic eigenfunction of the Laplacian in C.
&,
Fig.6 shows a type-I11 eigenfunction constructed a combination of 6 exponentials: 4 = x1 - XZ x3 - x4 x 5 - x6.
as
+
+
4.13. Conformal symmetry: Green's function and Poisson kernel in the ball. The group-symmetries used so far in our discussion involved space-translations (continuous and discrete); rotations (spherical and hyperbolic) and reflections. Our last example will exploit yet another transformation, the conformal symmetry of the Laplacian, to study harmonic functions in the balls B , = {
12
I 5 R } c W".
The Green's function
A'
and
Poisson kernel P of the unit disk D c R2 = 43 are well known from the complex analysis, and can be derived in many different ways. We just state the result, referring to problem 6 for details, (4.43)
where z = reit; w = pei6 are complex points in D ( p = 1 for P). Our goal is to extend
(4.43) to higher dimensions. Namely, the Poisson kernel in the n-ball is given by
(4.44)
52.4. Laplacian and related differential equations.
110
2~r 4 2 - volume of the where 12 1 = r; I y I = 1; e = angle between z and y, and w ~ =- r(n/2) unit sphere in Wn. The derivation of (4.44) will utilizes a particular conformal transformation in R", the inversion, u: z+z*
=A *
(4.45)
IZl2'
We recall that name conformal refers to diffeomorphisms {$} of R" (or any Riemannian manifold) that preserve angles between vectors, but not necessarily norms. T h e Jacobian of such m a p $, A = $', is a conformal matriz, product 'scalar x orthogonal", i.e. t A A = AZ (problem 8). Conformal m a p u of (4.45) takes interior of the unit ball in R" into the exterior, and vice versa,
I I = 1). T o construct the Green's function of A on B (Newton's potential) K O = C, I z - y I - " (for n > 2),
and acts identically on the unit sphere { we take the free-space Green's function and &ln
Iz-y I
in R2. It satisfies the differential equation: A,K = 6(z - y), but not the
boundary condition, Ko(z - y) function u&z) in
I an= h,(z) > 0, 80
n, that satisfies A,[u,]
we need t o correct KO by a regular harmonic
I
= 0; ).(,it
an= h , ( z ) ,
K ( r ; y )= Ko(z - Y) - )(., Such uy can be constructed by inversion of KO.We shall use the following identity valid for any function u, A(
I2
I 2-nu(+)) IZI
= I z I -"(Au)
(-1,
(4.46)
IZI
The proof of invariance-relation17 (4.46) is outlined in problem 8. Its immediate corollary is
Proposition: Function u is harmonic iffv(x) = Applying Proposition t o the Newton potential
12
I "-'u(z/ I z I ')
is harmonic.
KO = C I z - y I - " we get the requisite
harmonic correction: u
Cn
(2)=
l(z/Izl-
IzlY)l"-2'
Obviously, uy is harmonic and coincides with K o ( r - y) on the unit sphere. Thus the Green's function of the unit ball (4.47)
Finally, t o compute the Poisson kernel we take the normal derivative 8°K = ny.V,K,and observe that for y on the boundary ( I y I = l), ny = y. Hence,
It remains to note that both denominators in (4.48) are equal (see fig.7),
50
we get Poisson
17The R"-Laplacian is not invariant under the conformal map u, but comes 'very near" t o it. !), Indeed, (4.30) means that the unitary operator U , : f ( z ) + I z I -"f(z*); ( I z I -" = ,/intertwines operators A I z I and A, (A I z I ' ) U , = U,A.
32.4. Laplacian and related differential equations
111
kernel (4.44),
From the Poisson kernel on the unit ball we can easily derive P for any radius p,
(4.49) Fig.?
demonstrates equality of two denominators of (4.49), that measure the distances between pairs {z;y} and
I./
I2
I ; I 2 I Yl.
Formula (4.49) has many applications in the theory of harmonic functions. It follows from (4.49)that any harmonic u(x) is real analytic, since P ( q ...) is real-analytic in x for each y on the boundary. In fact, one could estimate the radius of convergence of the Taylor series ~ ~ ~ ( ~ ) ( ofz harmonic - - x ~ )function ~ u ( z ) at each interior point {xo}
C. Furthermore, (4.49) gives the minimal and maximal value of the Poisson kernel for each 0 < r < p, they correspond to in terms of the distance from {xo} to the boundary
case=
fl,
(4.50) As a consequence of (4.50)we get an important result in the theory of harmonic
functions.
Harnack inequality: The ratio between the maximal and minimal values of a harmonic positive function u(x) 2 0 an the ball of radius r is estimated by max {u(x):I z I = r } min {u(x):I z I = r ) -
(4.51)
4.13. Green’s functions and Poiason kernels in arbitrary domains D are constructed, as in
solid sphere case, from the free-space Green’s function (Newton/Bessel potential) KO,and an auxiliary (harmonic) function u(x; y)
(2,y
E D), that satisfies the homogeneous
equation: A,u = 0; and the boundary condition: u( ...;y) or a,u
I aD = -a,Ko(x-y)
I,
= -Ko(z-y) (Dirichlet);
(Neumann). Then, the Green’s function of D,
62.4. Ladacian and related differential equations.
112
K(z;y) = Ko(z-y)--(z;y); while the Poisson kernel, P ( z ;y) = -B,K(z; y)(Dirichlet); and P(z;y) = -a,K(z;y) (Neumann). Solution of any inhomogeneous problem: Au = F; u I
=I
= f ; is given by
,my ) ~ ( y ) d y+ r aDp(z;y ) f ( y ) d w .
Another construction of P(z;y) in terms of the solid-angle form is outlined in problem 12.
Additional comments and results. Our discussion of differential operators was limited to specific examples and methods. Here we give a cursory introduction to the general elliptic theory, based on symbolic calculus. For detailed exposition see [Hor];[Tal]. 1. Symbolic calculus. We write differential operators on R", as L = Cu,(z)D", where (I
= (el;...a,) means a multi-index, Do- the corresponding partial derivative ( D =
-
partial divided by i). We define a symbol u = u L ( z ; 0 or a(+;() < 0, for all < # 0.
Hence,
c1It I I a(+;€) 5 C, I €I *; for all (,
(4.57)
the standard notion of ellipticity. Estimate (4.57) plays the crucial role in the approximate inversion of elliptic operators. Indeed, inverse $do A - (if it exists) has the principal symbol given by the product-rule, uA-l
-1 (l.o.t), +... a(+; 0
and 1 is also an elliptic $do or order -m. As a consequence, one can construct an
4%; C)
approximate inverse B N A - ’ for any elliptic &lo A. Namely, one defines a $do B by (4.37) with symbol 1 , then shows via the product-rule (4.39) and norm estimates 4 r ;C)
(4.40), that
A - B = Z + R (remainder), where remainder R has negative order -c, hence operator R is smoothing in the Sobolev scale (4.38),
R: 36,+36,+,.
c R”, any such R becomes a compact operator, due to compactness of the Sobolev embedding: 36,(R) c X t ( Q ) , for any pair s > t (Theorem 1 of 52.2). In other words approximate inverse B = &(+;D) becomes the When restricted on a compact domain R
Fredholm inverse of A (inverse modulo compact operators). This result along with the
basic spectral theory of compact operators (Appendix A) has important consequences for the theory of elliptic equations. Before we state the general results, let us mention that the notions of symbol, &lo, Sobolev space, etc., could be extended from R“ to manifolds and domains18, i.e. boundary value problems (see [Hijr];[Tal]). Theorem: Let A be an elliptic operator on a compact manifold, o r domain (with proper boundary conditions). Then (i) spectrum of A consists of a discrete set of eigenualues
{Xk-+m};
each eigenspace Ek ( o r root subspace f o r non-self-adjoini A ) being finite-dimensional;
T -o construct &lo’s on manifolds { A } one exploits local formula (4.36), then patches together “local pieces”. Such process, however, defines a +do A only approximately to the leading order. So { u L ( z ; ( ) }on rnanifolds could be understood only as principle symbols, a consistent choice of other lower-order terms, requires additional structures. In some cases, groupstructure or geometry provide necessary tools to build a “complete symbolic calculus on manifolds” ((Be];[Un];[Ur];[Wi]).
52.4. Laplacian and related differential equations (ii)
for
any
X
4 spec(A),
operaior
(A - A)
is
inueriible
and
115 (A - A ) -':36,*36,+,;
(m = order(A)). Hence, differential equation: (A - X)u = f , with any f E
K,, has solution
In particular, f E Coo, yields oo-smooth soluiions. Also
u E 36,+,.
(iii) for all X E C operaior (A - A) is Fredholm-invertible. So equation (A - X)u = f, has solutions for all f , orthogonal l o the null-space {$}, of ihe adjoini operator: ( A * - X ) $ = 0 (Fredholm alternative); (iv) eigenfunctions: A 4 = Ak$, are oo-smooih. In case of (real) analytic A 'COO" could be replaced by (real) analyticity. Further analysis of operators {A}, via $do techniques, reveals much more. One can find for instance, asymptotic distribution of eigenvalues {Xk}, given by the celebrated Weyl (wolumecounting) principle:
5 A},
N ( X ; A )= #{Xk(A) 5 A} -+'d{(z;():a(~; 0;
I
ii) 4 - 4 t ) LP(&) I IIf I L P iii) u(z;t)-f(z), as t-0, in LP-norm
Use the convolution representation of solutions, u = 4jt*f, with the Gaussian or Poisson kernel, 4 j t = t - W ( I z I /t),and show that both kernels form an approximate unity as defined in problem 2 of 92.1. In fact, one could show that u(z;t)+f(z), pointwise almost everywhere inside any cone: I z - zo I < c t , but this would require more powerful tools of Fourier analysis [SW]. 3. Use the Fourier transform to solve Dirichlet problem for the Laplaces equation in the strip: 0 < y < b, Au = 0; u I = = f(z); u I = b = 0.
4. Calculate wave propagators U ;V; W in higher dimensions, using the relation:
c
5. Obtain the fundamental solution of the beabproblem: Kt-AK=O - 2 )- Gaussian, * K ( z ; t ) = ( 4 ~ t ) - " / ~ e z p (41
I t = 0 = 6(z)
by symmetry reduction, using non-isotropic dilations in the
(2;
Itspace:
D,: (2, t)-(az; a%), a 2 0. (i) Show, if u ( t , t ) solves the homogeneous heat-equation, then u* = u ( a 2 t ; a z )does so. (ii) Write the fundamental solution K * = K ( a 2 t ; a z ) as a multiple of K ( t ; z ) , K * = c(a) K , compute coefficient .(a) from the "6 sourcen condition at t = 0, and prove K ( t ;z) = a" K ( a 2 t ; a z ) .Thus K is reduced to a single variable function K ( r ) ,
K = t-"/'K(rt-'/'). (iii) Verify that K ( r ) solves an ODE,
K'/ Change variable: r-$
2
+ (+ + 5)K' + ;K
= 0;
= z, and show that in the new variable K satisfies an ODE: 2r(K'+iK)'+ n(K'+:K) = 0;
(iv) Obtain the general solution of the latter,
K = e-'l2(Cl
+ C,
ez/2z-n/2dz),
show that the second (singular term) must vanish. Hence, K = e-'I2 = e-r2/4,
QED.
6. (i) Derive the Poisson kernel of the Laplacian in the unit disk D as Fourier series expansion 00 qr; 8 -t )= p I L. I - 0.
c
--m
(ii) Sum the resulting series to get
,w
52.4. Laplacian and related differential equations
117
in complex variables z = reie. (iii) Show that P- C( 0- t ) , coordinates).
as r+l,
and verify the equation A P = O (use polar
(iv) Expand the Green's function K(z;w ) ( z = reie; w = peiQ) into the Fourier series (4.59)
Km(r;p)eim(e-Q),
and demonstrate that coefficients {Km} are Green's functions of the S - L Problems:
+
+2~ = a(r - p), 2
KI' ~ K I
(4.60)
(v) Find a fundamental pair {ul(r);u2(r)} of each S-L problem (4.60) and compute functions { Km} using (4.40). (vi) Sum series (4.59) to get
(m
~ ( zw ); = k / n I *-l/tul ). Interpret this form of K in terms of reflected sources and P is equal t o the normal (radial) derivative of K on the boundary { I w I = 1).
7. Verify the Laplace equation A,P = 0, and the boundary condition, P ( z ;y) = 6(z - y), on aB - the unit sphere, for the Poiason kernel (4.44).
8. Prove the identity (4.46) for any function u on R". Steps: (i) Use a general coordinate-change formula for the Laplacian A: if map Q:Z-y = (...Qi(z)...), has Jacobian matrix A = Q', with determinant J = detA, then A,+V
. J ( = A A ) - 'V.
Written explicitly this yields,
A ( . 0 Q)=
{c ( c ajdi a k d i ) a p k u + c A Q aiu} ~ ik i
0
Q
I
(ii) Check that inversion u:z+z/ I z I ', gives a conformal map, so Jacobian = (alu; anu) satisfies tu' u' = p(z) I , with factor p = I z I - '. Hence,
ul
...
A[(*)]=
06)
II: 2).
IZI
(iii) Combine (4.61) with the product rule: A ( f g ) = A(f)g to f = I z I and g = u(z/ I z I 2), to get
A( I z I au
x
IzI -'(du-2(n-2)+.Vu
= I z I a-'{a(a
(4.61)
+ 2V f .V g + f Ag,
+ n - 2). - 2(a+n-2)+.
121
applied
V u + Au o u,
}
whence follows (4.46) for a = 2-n.
9. Mean-value property for harmonic functions: Show that any harmonic function u(z) on R" satisfies u(2) = 1 J u(z - ry)dSy - mean over the sphere of radius r; p-1 wn-l lyl = I
u(z) = 2J u(z - y ) d y - mean over the ball B ( r ) of radius r.
uolB(r) B(r)
Hint: Apply the Green's identity: JuAu - uAu = J (& - &)dS; aD an an D in region D c R" - normal derivative on OD), to the pair: { u ( z ) ; u = K o ( z - y)Newton's potential}, integrated over the ball D = {y: I z - y I _< r } .
(a an
10. Apply (4.51) to show that there are no non-constant positive harmonic functions on
118
$2.4. Laplacian and related differential equations. R". More elementary fact: there are no bounded harmonic functions on R", follows from the mean-value property (problem 9).
11. Use the mean-value property (problem 9) t o prove that a nontrivial harmonic function u(z) on R" is unbounded. Hint: w u m e I u(z) I 5 C ; pick a pair of points z,y; write u(z), u(y) by their mean values in large balls {B(r)}; estimate I u(z) - u(y) I 5 ...( r), and let r+m.
12. The Poisson kernel P(z;y), r E D , y E C = B D , in any (convex) region D, can be constructed in terms of the solid-angle kernel, T ( z , y ) = 1 - cos(I.
as
c,,-~as - ~ , - ~ ~ n - 1 '
where a is the angle between normal nu and vector (z-y),(see fig.8). Geometrically kernel T ( z ; y ) represents the density of the solid angle with vertex a t {z}, subtended by a surface area element a t {y}. It can be obtained by restricting the (n-1)-differential form (solid-angle form),
8 = 1 ( z l d z z A ...Adz, - zzdzl A d z 3 A ... A d z , r"
+ ...),
on the boundary surface C = a D . i) Show that T = 81,E (parametrize C by map @:R"-'+R"; vector surface-area element ndS = @91 A A @yn-ldn-l Y I
z = @(y)), and show that
...
where {By} are partial derivatives of @ in variables {y}, n-unit normal t o C. ii) Show u(z) = J'T(z;y)f(y)dS(y), boundary condition,
solves the Laplaces equation, Au = 0; but not the u I C = if
+ Q [ fI,
where operator Q is obtained by restricting kernel T ( z ; y ) on C. iii) Integral operator Q is compact on Lz(C) (see Appendix B), since Q has integrable So , operator ?jZ Q is Fredholm-invertible singularity, I Q(z;y) I 5 C 12-y I (modulo possible finitedimensional eigensubspace A = -?), and )the Poisson kernel,
'-".
I-
+
P = T($Z+ Q)- * = 2(T - 2TQ + 4TQ2- ...). Fig.8 Geometric view of the solid angle form, that gives a Poisson kernel
where 8 = 8(x;y) - solid angle, centered at {z}, as a function y E aD; S = S(y) - surface area element.
$2.4. Laplacian and related differential equations ~~~~
Table of basic differential equations and their Greens functions in
Equatwn
Fonnal solution
Laplace:
-Au = 6
Resolvent:
(A
Heat:
ut - A u = 0 L o =6
Schriidinger:
- A). = 6
Conj. Poisson:
RA = (A - A) - 1
Tt = e tA
ut-iAu=O UlkO
Poisson:
K=A-'6
=6
+
utt Au = 0 4t=0 =6 utt
+ Au = 0
Utlt=o
=6
p t
?t
- t o -e
=&
-
Wave (initial velocity): utt - c2Au = 0
u I = O;utlt=O= 6
I t
sin t f i
--
fi
Wave (initial disturbance): utt - czAu = 0 u 10 = 6; U t l t = O = 0
r, = cos t J E
1
6
119 ~~~
R".
~
62.5. Radon transform
120
$2.5. The Radon trandorm. The Radon transform appears in many guises and different geometric settings. The standard R2-transform takes a func_tion f in R2, and integrates it along all lines { l } , %:f-f(t). Similarly, the R"-functions can integrated along lines (X-ray transform), hyperplanes (Radon), or all intermediate p-planes (1 5 p 5 n-1). One can similarly define the Radon transform on n-spheres, integrating either over great circles (geodesics) or hyperspheres, or intermediate p-spheres. The basic problem is to invert W, i.e. recover function f from its integrals over a suitable family of lines, surfaces, etc. It turns out that the Radon transform is closely connected to the Fourier analysis in the Euclidian setup, and to harmonic analysis of relevant groups (orthogonal, Lorentz, etc.) in other cases. In this section we shall give a brief introduction to the Radon transforms on Euclidian spaces, and its connections to the Fourier transform, then mention how these results extend to other rank-one symmetric spaces.
5.1. General framework for the Radon transform is a dual pair of manifolds:
{t}c Z with a natural measure [(I = {z} c 96, with measure
where each paint z E 96 is assigned a subset [z] = d,((). Similarly, each point E Z corresponds to a subset
t
d&z). The generalized Radon transform 5 takes %-functions into Z-functions, and the dual transforms,
5:f (+3(t)
=
J "1 f (4d&);
%f(t)-.j(.) = /[,lf(Od,(t). In general, little can be said about the product and inversion of %-transforms,
5% - ? 3% ? jL-'-?
(5.1)
The situation becomes more manageable when both manifolds are homogeneous spaces of some (Lie) group G, and G-actions on 96 and Z are consistent: [tg]
= {zg:z E
[t]}, and [zg] = {(g:t
E [z]}, for all z;t.
We also assume that measures {d,([)} on leaves [z]c Z, and { d [ ( z ) } on leaves
[(I
c 96, are transformed accordingly, 9:d p, furthermore
C is divided into two opposite halves: positive roots
C+ and negative roots C - (the splitting depends, of course, on a particular choice of ordering in C). Having chosen some ordering we call a positive root a simple, if a can not be decomposed into the sum of other positive roots p y. Simple roots are linearly
+
independent and form a basis of
8,dim8 = #{simple
roots} is called the rank of Lie
algebra 0.We shall not provide the general argument (see [Ser]; [Jac]; [Hell), but rather illustrate all concepts with the example of Lie algebra
1) Cartan subalgebra trh =
6
q n ):
consists of diagonal matrices: h = d k g ( ...Itj...);
C h j = 0.
2 ) Root system C is labeled by pairs of indices: jk ( j# k), where ( h ) = h 3’ - h k . 3k
55.2. Cartan subalgebra. Root system. Weyl group.
198
3 ) Positive roots: { a j k } correspond to j
< k , negative
to j
> k.
This comes from the natural lexicographical ordering of Cartan elements (real
> H‘ = (hi;...), if t h e first j , where hi # h>, has two roots ajk > aj,k/iff j 5 f, and (in case j = j ’ ) k > k’.
diagonal matrices): H = (h,;h,; ...)
hj > h). It follows that
4) Simple roots are of the form {aii+l = aj}. k-1
Any other positive roots are sums of simple roots: a j k = C a .; root crln is the i
highest, while anl- the lowest (negative!).
’
5 ) Root vectors { X , = X j k } are Kronecker 6 -matrices with 1 a t the j k t h place and 0 at the rest. We shall denote by { X j k } ( j < k ) positive root vectors, and by
{Yjk = X k j } ( j < k ) negative root vectors, so jk will always mean pair j < k. 7 ) The commutation relations: [h;X,] = cr(h)X,; [h;Y,]= - a(h)Y,; [X,; Y,] E 8; hold for any h E 8; and root a = jk. Returning to the general situation, Cartan algebra product, the Killing form of (5:
( h I h’) = tr(adhadh,) =
8 c (5
(2.2)
has a natural inner
C a(h)a(h’)- sum over all roots,
(2.3)
aEC
in terms of its root-system. Product (2.3) allows us to identify each root a (a linear functional on
8)with
an element H , E 8, turning C into a system of vectors { H a } in
8. Let us compute roots {ITii} for Lie algebra sl,. Take a set of basic diagonal matrices { E i j = diag(... 1; ... -1;...)}, with 1 on the i-th, (-1) on the j-th place, and the rest zeros, and compute the product (problem 2 )
(Eji 1 h) = %(hi - hi).
(2.4)
So roots H i j itre diagonal matrices E i j divided by 2 (we ignore the unessential factor n in (2.4). In special cases 4 2 ) and 4 3 ) , the positive root system is made of matrices, q 2 ) : H = dia& -$); 4 3 ) : HI = diag(f; -3;o); H , = (0;:; -;); HI, = ($;o;- f ) = H,+H,; here {H,; H,} form a basis of simple roots of 43). Root system { H,} along with all (positive/negative) root vectors { X,; Y,} forms the so called Cartan basis of Lie algebra
(5.
One can easily verify the following
55.2. Cartan subalgebra. Root system. Weyl group. commutation relations for the Cartan basis of
199
43):
x,,
[H,; x121 = ; [H,; y121 = - y1*; [X,, ; y121 = 2 H1; which generalize the commutation formulae (2.10)for 4 2 ) in chapter 4.
(2.5)
Similarly relations are verified for q n ) ,
Let us now compute the (Killing) inner product on the Cartan subalgebra 4j
c 431, ( h I H,)= Wl - b);( h I H,)= % - h3); ( h I HlZ) = %, - h3),
(2.6)
for h = (h,; h,; h3). Introducing new coordinates on h = t,H, t,H,, we find
8: t, = ;(hl - h,), t , = f(h, - h3),
i.e. writing
+
11 H I.II = %;( H , I H,)=
-
i; (H,I H,,)
=
( h I H,)=3 4 ; ( h I H,J = 3(t,+t,); etc.
The inner-product structure on the real vector-space 4j allows one to associate to any root h, E C a reflection :s, 4j -+$, w.r. to the plane orthogonal to ha,
It turns out that all reflections {s,} map the root system C into itself. Thus the family {s,} generates a finite group W of isometries of 8 called the Weyl group of 0.
For algebras 4 .) reflections {s,:a = jlc}, can be explicitly calculated by (2.6) - ( 2 . 7 ) (problems 1,2). One can show that sij transposes the ith and j t h entry of h. Therefore the Weyl group of 4 .) coincides with the permutation group of n elements
w = w,. 2.2. Root systems for simple and semisimple Lie algebras. An alternative way to C = { a } in space 4j = Wm, equipped with an inner product ( I ), a finite system of vectors, invariant under all reflections: s,: p-ip - 2 introduce simple and semisimple Lie algebras is to start with a root system
Symmetries (2.8) generate the Weyl group of root-system: W = W(C). Under some additional (minor) constraints such systems could be completely classified and give rise to root systems of simple and semisimple Lie algebras. The former are called
indecomposible root-systems (i.e. C can not be broken into orthogonal pieces: C'UC'' with (a I p) = 0, for all a E C'; PEE'), while the latter (semisimple) are made of
$5.2. Cartan subalgebra. Root system. Weyl group,
200
orthogonal simple root systems, CIUC,U ...U C,, according to a decomposition of 0 into the direct sum of simple components
Geometrically, all root systems of low
&@k. 1
ranks: r = 1;2;3 are sketched below (fig.2;3). For higher ranks we adopt a description of [SerZ] (chapter 5 ) , in terms of an orthonormal basis {el;...en} c W", and the lattice
A = A, spanned by { e,}. ntl
to be a hyperplane in Rn+l orthogonal to C e,. 1 Then C consists of a.11 vectors in @flA,+l of norm 2, i.e. all {e, - ek: j # k}, the basis Series A, (n 2 1): We take
could be chosen as {e, - e3+1:1
5 j 5 n}, and the Weyl group W =
is made of all
permutations of {1;2;...;n+l}. Series B, (n
:> 1): In space @ = W" we consider all lattice points of norm 1 and
&; C = { a E A,: ( a I a ) = 1, or 2). Clearly, C is made of { fe, fe,: i # j } and { fe,} the basis of C: {el - q;q - q;... ; en-l - en; en}; and the Weyl group W is generated by all permutations of {1;2; ...n}, and all sign changes (multiplications): e,+ fe,. So W = (Z, x ... x Z, ) D W, - semidirect product.
-
For n = 1 algebras A, and B, are isomorphic, i.e. 4 2 ) N 4 3 ) (chapter 4). Series C , (n 2 1): The root system of C,-type is dual to the B,-root system, i.e.
C(C) = {a* = 2 A ; a E C(B)}.So C(C) consists of { fei fe3: i # j} and
).I
(a
{ f2e,},
has a basis {el - q ;e2 - q;... ; en-l - en; 2en}; and the same Weyl group as B,.
D, (n 2 2): The D,-root system consists of all a E A, of ( a I a ) = 2, i.e. all i # j } with the basis {el - q ;q - q; ... ; en-l -en; en-l +en}. The Weyl
Series
{ f e, fe,:
group is made of all permutations and all sign changes: e,+
{ - }, the latter group being isomorphic to 2, W = z, D W,.
N
(ZJn-'
fe,; with even number of
c (Z,)".
So Weyl group
Algebra G,: This root-system was sketched in fig.2 (iv). It could be described as the set of algebraic integers { z = a Q(w)
+ bw + cw2: a, b,c E Z} of
c C, generated by the cubic root of identity
the cyclotomic field
w = ei2*/3, of norm I Z I = 1, or 3.
For other exceptional algebras we refer to [Ser]. In low ranks there are many overlaps between 4 series:
B,: sp(1)N 4 2 ) N 4 3 ) (chapters 1,4);
El]Cl
N
A,
l=1C2
N
B:!:sp(2) N 4 5 ) ; and D,
N
A:%:4 6 ) N 44) (problem 6).
D,
N
N
A, @ A,: 4 4 ) N 4 2 ) @ 4 2 ) (problem 5)
201
55.2. Cartan subalgebra. Root system. Weyl group.
Given a root system C = { a } in Wn = !$ with a chosen basis { a l ...a,,} ; one can construct the corresponding semisimple Lie algebra. We denote by C * the set of positive/negative roots, and set
@ =f fa> O@(Off$@-,) with 1-dimensional subspaces: 6, = span{Xff}; 6-,= span{Yff= X - m } . One has the following commutation relations,
(i) [ h ; X f f=] a ( h ) X f f where ; a ( h ) = (a* I h) = 2-;(ff I 4 (ff Iff) and a* = 2 A = H , denotes the dual vector to a.
(ff Iff)
The root-vectors could be normalized so that if a+/3 E C 0; otherwise The coefficients { N a p } depend on the choice of basis in C. (ii) [ X , ; Y , ] = H,; [ X , ; X p ]=
The construction can also be implemented in terms of generators of
...an} in chosen basis {a1;
8, respectively
(5,
i.e. a
the dual basis {Hl;...H n } , and the
corresponding root vectors {XI;...Xn} (positive), and { Y l ;...Y n } (negative). The generators satisfy Weyl commutation relations:
(2.9)
(2.10) Here numbers { n2.3. = n(ai;aj)}denote pairings of basic roots {a1; ...a n } ,
.(a;@) = 2-(a I a) (a
Iff)'
The set of numbers { n ( i ; j ) }forms the Cartan matrii of C . Numbers n(a;@) are known to take on integer values (0; f 1;f 2 ; f 3). They have a simple geometric interpretation, .(@;a)= 2
hence
I I m
os6;
n(a;P)n(@;a) =4 ~ 0 ~ ~ 8 ; where I9 denotes the angle between roots a and
p.
In particular, angle 19 could take on
only 4 possible sets of values: I9 = { ~ } ; { ~ ; ~ } ; { ~ ; ~ accordingly } ; { ~ ; ~ } we , get 7 possible
202
$5.2. Cartan subalgebra. Root system. Weyl group.
configurations of (non-colinear) pairs {a$} (see fig.1 and the table).
Table 1. 1) n(a;P)= 0
@;a) = 0
9=;
orthogonal roots
2) .(a$) = 1
n(P;a)= 1
9=;
[PI = la1
3) n(a;P)=-1
n(P;a)= -1
9 = &3
IPI=IaI
4) n(a;P)= 1
.(p;a)
9=f
IPI=
5) n(a;P)= -1
n(P;a)= - 2
9=
6) .(a$) = 1
n(P;a)= 3
9=;
lPl= d3laI
7) n(a;P)= -1
n(p;a)= - 3
9=k 6
lPl=
=2
f
4lal
= 4lal
&la1
Let us remark that basis {a1; ...an}in any root system C is chosen in such a way, that the Cartan numbers {n(<j)} are negative, so all angles Bij between pairs {ai;aj} are > This also explains the choice of negative exponentials -n(z;j) in (2.10). Finally, any set of Weyl generators (2.9)-(2.10) yields a semisimple Lie algebra with root system
5.
c. B
ka
&a
B
@
\La
&; By +a
-a
Fig.1: illustrates all possible angles between pairs of root vectors {qp} and their relative length.
85.2.Cartan subalgebra.
Root system. Weyl group.
203
Fig. 2 shows all root systems in rank 2. There are 4 different cases:
(i) A,: algebra 4 3 ) has 6 roots in hexagonal arrangement; roots a,@, a+p correspond t o diagonal triples: (1; -1; 0); (0;l;-1) and (1;O;-1);
iE3
a+ZE
(ii) B, and C,: algebras 4 5 ) and sp(2) are isomorphic, as evidenced from their root diagrams; those become identical when roots a and p are interchanged.
(iii) D,: algebra 4 4 ) is not simple, but breaks into the direct sum of 2 orthogonal A,- diagrams. We have already mentioned that 4 4 ) 4'4@ 4-21, hence complex 4 4 ) N 42)%3 42).
=
( i v ) Ezceptional Lie algebra G, has 12 roots arranged in a king-David star.
55.2. Cartan subalgebra. Root system. Weyl group.
204
Fig.l(a): Root system A, = D, (d4N 4 6 ) ) consists of 12 vertices of cubo-octahedron. The figure shows 6 positive roots, spanned b y a iriple of simple roots: a;o;y. All roots have equal length.
4 (algebra sp(3)) consists of 12 vertices and 6 ceders of square faces. We have shown 9 positive roots, spanned by simple roots: a;P;y.
Fig.3 (b): Root system
The C, - root system is made of the same 1.2 poinis, but ihe “long roots” (vertices) and short roots (ceders) interchange. So a becomes long, while /3,r short.
Dynkin diagrams. Root systems C are conveniently labeled by certain graphs, called Dynkin diagrams. The vertices of diagrams are simple (positive) roots of a fixed Cartan basis. Two roots a, p are connected by a single, double or triple bar, if they make angles
%
and
f
respectively (the orthogonal roots are disconnected).
Furthermore, the lines between uneven roots are equipped with arrows from the short to the long root. Figure 4 below sketches Dynkin diagrams of all simple Lie algebras, classical and exceptional.
a-8
O2
Fig.4: Dynkin diagrams in ranks 2 and 9.
205
55.2. Cartan subalgebra. Root system. Weyl group. Problems and Exercises: 1. Use formula (2.6) to compute reflections { s j k } of 4 3 ) . Show that the Weyl group of 4 3 ) is W,.
1 1
I
2. Check the product formula (2.4); calculate E j k and ( E j k E .,k,) for the roots E j k = diag( 1;... - l...)of 4 n ) . Find angles between roots. Show t i a t the Weyl group of sl(n) is W,.
...
3. Find the Cartan subalgebra, root system and the Weyl group of the Lie algebra 4 4 ) . 4. Check that a Weyl reflection
u,: h - h - 2
o_ (a14
defined on the Cartan subalgebra algebra 0, by u,: X p -X
c 0, extends to an automorphisms
of the whole Lie
.,(P)'
5. Show the real orthogonal group S q 4 ; R )
N S y 2 ) x SU(2)/Z2. i) Use the representation of 3-sphere by unit quaternions, $4.1 (chapter 4),
S32:Q;={ m i
so one has m, = mi; ...mj-l = m>-,; but m j > m:.
ii) we call elements {s = qp:q E Q;p E P } simple, and say that s,u E W,
are
simply related, s u, if s = u(qp),for a simple qp. We shall see (Lemma 3, corollary 5 ) , that such relation defines an equivalence in W,. N
Theorem 2: i) Any symmetrizer x = xaU, considered as an element of group algebra e(W,) has x*x = p x , with constant p = p ( a ) , depending on a only, so ix becomes an idempotent (projection) in e(VW,). ii) Two symmetrizers
x = xas
and
x' = xatU
are mutually disjoint
x*x'
= 0, if
a # a' ( a > a'), or for a = a', i f u and s are not simply related, u # s ( q p ) . So mutually disjoint projections x;x' correspond to different diagrams a, a', or to unrelated s and u. iii) If a = a' and u;s are simply related, u = s(qp), then x*x' = ps(C(-l)'qp)u-'. The proof is based on a simple combinatorial Lemma.
Lemma 3: If (a,.) and (a',u) are two Young diagrams with either a > a', or a = a', but s and u non-simply related, then there ezist two indices j , k that belong to the same row in (a,s)and the same column in ( c Y ' , ~ ) . Proof of the Lemma: We take indices in the first (longest !) row of (a+)and call them il,i2,... If the first row of a', mi
< mlr then at least two of i's
must necessarily lie in the
same column of ( a ' , ~ )so, the Lemma's conclusion would hold. Assuming mi = ml and
222
$5.4. Tensors and Young tableaux. no i's belong to the same column in a', we can bring all of them by a vertical permutation q1 of a' to the first row, and within the first row we rearrange them by horizontal p 1 in the same order as in (a+).Thus 1" rows of (0,s) and (a',u) become identica1,i.e.s = a q l p l (modulo remaining rows). We continue the process with the remaining rows, each time seeking the Lemma's conclusion to fail. This yields a t the end a = a' and s = u q l p l q 2 p z . .qkpk. Since each p j is a permutation of the j t h row, it
commutes with all remaining terms of the product, and the latter becomes u ( q l q z . . . ) ( p l p z . . .). So the only way for the Lemma's conclusion to fail is when a s
= a' and
being simply related to u, QED.
Corollary 4 Given any pair {(a,s);(a',a)} with either a >a', or a = a', but s unrelated to 0 , there exists a row transposition ~ , E P and , a column transposition qo E Q, so that sp,s-' = aq,a-'. In particular, for each non-simple s # q p , there exist a pair p,, q,, s.t. s p , = qos. Another result could be deduced along the lines of Lemma 3.
Corollary 5: F o r a n y p a i r of elements p E P, q E Q in a fixed Y o u n g tableau a , there exists another p a i r p' E P ; q' E Q, so that simple element qp = p'q'. Hence, simple relation, s N u, is indeed an equivalence, although product QP is not a subgroup of W ,! Proof of Theorem 2: We observe that symmetrizer
x
of Young tableau (a+)has certain
invariance properties w.r. to the left multiplication with elements q E sQs-'and right multiplication with p E SPS-', X(QUP) = PXP
The same holds for
x2 = X*X,
or
= (-1)Qx(.), all
9, P.
x*x',
p X ' ( q 1 p ) = ( - l ) q x * x ' ( t ) , for all q E sQs-';
p E UPU-'.
(4.5)
Now we take a pair qorpo of the corollary and get: X2(qotpo) = - x 2 ( t ) , by (4.5), and x2(qotp,) = x 2 ( t ) , whenever 1 is non-simple by the corollary. Thus
0; for non-simple 1; = x2(e); for simple t = q i Similarly one verifies other statements of the Theorem. As a corollary of Theorem 2 we get a family of idempotents (projections) { x = xos} in
the group algebra e(W,).
Each of them is minimal, so
x
projects group algebra e(W,)
onto an irreducible subspace V = Vas = x * e . Indeed, the argument of Theorem 2 shows
55.4. Tensors and Young tableaux.
223
that x*f*x = p(f)x, for any f in the group algebra e(W,).
x
subprojection $ of
satisfies x*$*x=+=px,
hence $ = p x
In particular, any (a multiple of
x).
Furthermore, all irreducible representations { r = ra*' = R I V) are equivalent, so r = ra in R, d ( a ) , and the norming
(depends only on signature a). The multiplicity of
x = xas are related by p(u)d(a) = m! = I W, I.
coefficients p = p(a) of symmetrizer
Indeed, trace of the convolution operator
x:+x*+,
in the group algebra C(W,),
trx = x(e)rn! = m!, is equal to p . d ( r ) , but the degree of each irreducible
T
coincides by the Frobenius
reciprocity with its multiplicity m ( r ) in the regular representation R, d ( r ) = m(r)= #{disjoint projections
x ( , ~ )of
Minimal projections
algebra
e
x ( , ~ )of signature a).
give rise to central projections
xu on
primary
components 5, of the regular representation,
4EcW, x ( , , I ) = ' l i c s ( c ( - l ) Q Q P ) s - l .
xa =I Projections
{x"}
P
a
are orthogonal, and the family
{x,},
is complete in the center Z(W,)
of the group algebra. Indeed, their number is equal to the number of partitions of rn,
which is the same
BS
the number of conjugacy classes in W,
irreducible representations of W,!
or equivalence classes of
Thus we get at once a complete characterization and
explicit construction of irreducible representations { 7") of W,.
Next we shall apply Young symmetrizers to our main problem: decomposition of tensor products a("') of S f ( n ) .
Theorem 6 (i) Each operator x = x a s projects T"'(C") onto an irreducible subspace TaSof ?r(m)l and the restriction d"')I "range X " N ?ral an irreducible representation of SL(n) of signature a. (ii) The multiplicity of ?rd in d"') is equal to d ( a ) = degree of P. Moreover, the central projection x" of W, projects Tm onto the primary subspace 5, of ?ral invariant under both groups. (iii) The joint action of W, x SL(n) on 5, factors into the product of irreducible representations r" 8 T", and T, N Space(P) 8 Space(?r"). In other words, if r("') denotes the action of W, on 5, ( T ( ~ ) =R @ I , - the n-th multiple of the regular representation), and d"')the action of SL(n) on F , then the tensor-product action of W, x SL(n) is decomposed ag
65.4. Tensors and Young tableaux.
224
~ ( ~~1 r . ( ~R )@=r("') sum over all partitions a: m =
N
p T~ 8 K",
C mj.
The last statement means that groups SL(n) and W,
form a dual (mazimal
commuting) pair in P,by analogy with pairs: {So(3);SL2},or {So(n);SL,} in L2(Wn),
studied in chapter 4. So the algebras spanned by SL, and W, commut ant s:
Com(r(m)1 SL(n))= Alg (W,) Proof: i) Irreducibility of subspaces 05,,
in
'dn)are
mutual
= Alg (T"')).
and Corn(W,)
is established by finding all Weyl ( N - )
invariants in Ttrsand identifying the unique highest weight vector. Let Young symmetrizer of partition a. I claim that
x
x = xas be
a
maps all but one element of the Weyl
basis of invariants {was = s up1wp2...wLk] into 0,
li
0; if a # a' or a = a' but u Xas(Wa'u) = was; if'a=a' and (J = s
#s
Indeed, the basis of Weyl invariants written in the tensorial notation consists of WcI
= (el h %..
or their permutations
elements {el;
11-1
.h ek)
@
swa, s E W.,
p1
.
8 (el h . . h e k - 1 ) '8
,'. ..8
? I:
'l,
The corresponding Young tableau is filled by
...,; e k ] , according to the following allocation scheme, ml times
m, times
[q mk times Obviously, all
IOW
permutations p E P of diagram a
leave w a fixed while each q
multiplies it by (-1)q. Thus xa(wa) = wa, as claimed. For any diagram a' different from a, or unrelated s8 and u there will be a pair of basic elements { e j ; e j ] , that lie in the same
column of diagram (a',s-'u). So the corresponding determinant (wedge product) will be
0. We have thus shown each subspace 5,, = xas[Tm] to have a unique highest weight vector, hence follows irreducibility. ii) Next we take the primary subspace 05, = xa[Sm] and want to establish irreducibility of the joint action of W,xSL(n) on 5,. This would imply the proper factorization of space 5, and representation minimal projection
(dm).dm)) ITa into
xa E e(W,),
the product
7a@7ra.
But each
maps Tm onto an irreducible subspace of G = SL(n).
From the general result for commuting pairs (W,;G)
we get a tensor product
decomposition of part (iii), QED.
h a r k 7 Irreducible representations of G = W,
can be realized in a more direct way, using partitions (Young tableaux) a:m = E m j (ml 2 m, 2 ...). With any a
225
55.4. Tensors and Young tableaux.
we associate a subgroup H = H, 21 Wml x ... x Wmkof G , and a consider two natural 1D representations of H,: trivial 1, and signature Xg=sgns. Let T , and Tb, be the corresponding induced representations of G: T = ind(1 I H;G); T’ = ind(sgn I H ; G ) . All partitions { a } split into dual pairs: atta*, by interchanging rows and columns (so mf = #{mk 2 j } ) (fig.6). a
Fig.6 Dual Young tableau2 {a;,*}.
The key result (Weyl; von Neumann) gives the intertwining numbers of representations {T,} and {Tb,}:
Hence, T , and Th. have a joint irreducible component 7, of multiplicity 1. The degree of P can be derived by careful analysis of intertwining operators, and is equal to m ! n ( t i- e j ) i<j
el!. ..ek!
where ti = mi t i-1.
Remark 8 In the previous section we realized irreducible representations of SL(n) in function spaces TA (of small degree). Tensors and Young tableau give another realization in “large spaces” Tm.There exists an intermediate construction, where the role of W, is played by the group
SL(k) (see [Ze]). Namely, we take a vector space 96 of all k x n
matrices with the natural actions of two groups H = SL(k), and G = SL(n) by the left/right multiplication:
( h , g ) : X = (zij)+h-’Xg = TAJX). This action extends to all polynomial functions
Ym = U ( X ) = C c p X P : P = (Pij), CPij = m), of degree m, and gives rise to a representation
“ ; g ) f ( X ) = f(h-’Xg). The polynomial space 9 , Ym(96) ~ is the mth symmetric tensor power of 96, so zm is the mth symmetric power of
T’.
The reader is asked to show that representation
T”
of G x H
226
55.4. Tensors and Younn tableaux. is irreducible, groups G and R form a dual (maximal) pair in any space Trn(problem 5), hence there is 1-1 correspondence between signatures of I€ and G.
Problems and Exercises: 1. Show that both are irreducible subrepreaentations of r(')(Aint: identify Yz with the space of symmetric matrices and A' with antisymmetric, with the action of G on both spaces by g: X-+TgXg. Find all eigenvectors of the diagonal subgroup). 2. Show that T3 has a decomposition with signatures a = (3,0,. ..) (symmetric); a = (l,l,l,O ...) (antisymmetric); and 2 copies of a = (2,1,0...) (Hint: count Weyl invariants, observe that w 2 ( ( , ( ) w l ( 9 ) - w2(q,()wl(() = w2((,q)wl(() for any triple of vectors (,q,( in C"!).
3. Compute symmertizers of Young tableaux:
(1 ')
and
')
4. Show that xrn splits into the direct sum of irreducible products ra @ xa, where ra and
are irreducible representations of H and G respectively of the same signature a = (ml 2 m2 2 2 mj 2 ...). (Hint: use a version of Gauss decomposition for k x m matrices and show that the highest weight vectors for both groups are the familiar products of minors:
...
rn
rn
M12...p(X PM12...p-1(X) P-'...Ml(X)rnl? constructed from the first p columns of X, then first p-1 columns, etc.).
5. Show that the natural action of groups SL(n)xSL(m) in space 96= Matnxrn is irreducible; groups SL(n) and SL(m) form a maximal dual pair in any polynomial (symmetric tensor) space Pk(96).
6. Use Young tableau and formula (4.5) to classify irreducible representations of symmetric groups W,; W,; W,. Find their characters and degrees. Compare the results for W, with f3.3 (example 3.7), identify Young tableau of { 1; sgn; x';7r3+;ir3-}.
55.5. Haar measure on compact semisimple Lie groups.
227
$5.5. Haar measure on compact semisimple Lie groups. We compute the Haar measure on compact Lie group, reduced to conjugacy classes of C,taking as a model group Syn). This result will lay the ground for the celebrated Weyl character formulae in 55.6, that lie at the heart of the representation theory of classical compact groups ($5.6).
We shall assume G to be a subgroup of GLn(C), so its Lie algebra (5 = T , tangent space at {e}, could be identified with a subalgebra of M a t , = gqn). The tangent space at any point g E G is then T, = go. Each element X E 8 (tangent vector at {e}), extends to a left-invariant vector field < ( g ) = gX (matrix product). To construct an invariant volume element on G, we can choose a basis of left invariant vector fields, or the corresponding left-invariant differential 1-forms: w j = C a j k ( s ) d s k ,where {xl;...;xN: N = dirnG} are local coordinates on G (vector fields could be identified with 1-forms via a left-invariant metric). Then we define an invariant volume element, as a differential N -form W
=
N 1 Wk
= det(ajk)dzlA
... A d X N .
A canonical way to construct {wk} is to consider a Lie-algebra-valued differential 1-form, called gauge potential = g-'dg = g-'ak( g)dZk. At each point g in the group, R takes on values in the cotangent space at { g } , and is clearly left-invariant, (gog)-ld(g0g) = g-ldg. Choosing a basis {el;... e N } in 8 and expanding R in this basis, we get a basis of scalar G-invariant 1-forms, = CWkek; needed for the volume element (5.1). It is an easy exercise to directly verify (via the group identity g*g = I) that for unitary/orthogonal groups Syn), Sqn) matrixdifferential g-'dg is skew-symmetric, dgjk = - dijkj, so dg takes on values in the Lie algebra s 4 n ) or d n ) . The above procedure provides a general construction of the Haar measure dg on matrix Lie groups. However, for the purposes of analysis we often need an explicit form of dg (or w ) in a suitably chosen coordinate system on G. Here we shall do it for the unitary groups yn),Syn), and more general compact Lie groups G, based on the structure theory of 55.2. These results will be needed in the section on Weyl character formulae for irreducible representations of compact groups G.
228
$5.5. Haar measure on compact semisimple Lie groups. ~~
Since charact,ers depend only on conjugacy classes, it suffices to compute the reduced Haar measure on conjugacy classes of G. We denote by H the space of conjugacy classes: K , = {g-'hg:g E G}, and choose coordinates h = (hl; ...;hm) on H, as well as transversal variables y = (yl;...) on each class K , 11 G/Gh (G modulo the stabilizer of h, G, = {g:g-lhg = h}). We would like to decompose the Haar measure on
G, as . d g = p(h)dhd,Y, (5.2) where dh(y) denotes a G-invariant measure on K,; dh - some natural measure on H , and the density factor p ( h ) measures the relative size of conjugacy classes {K,:h E H}. Our goal is to compute factor p ( h ) in (5.2). We shall do it first for group SU(n). Each unitary matrix u is conjugate to a diagonal (Cartan) matrix
['
i0,
h=
eie2 eie,
1,
u = vhv-'.
(5.3)
Matrix h is unique, modulo all permutations of diagonal entries, i.e the W e y l group action on SU(n). Thus a generic conjugacy class is identified with a point in a "positive cone" in the torus, H = { 8 , > 8, > ... > 8,) = U"/modulo W,, the so called Weyl chambe?. Factor v in (5.3) is also non unique, v E SU(n)/Un, since stabilizer of a generic diagonal element h is T". So one can think of G (or rather a dense open subset in G), as decomposed into the product: G N H x G/Tn. To find invariant integral in coordinates (h,v) we differentiate the relation: u = v-lhv, d u = ( - v - v v v-')hv and multiply it on the right with u-' = v-'h-'v du u-1 = v-'(dh
h-1
+ v-'dh v + v-lh
+ h dv v-lh-1-
dv,
dv v-I)..
Introducing right-invariant cotangent vectors 6u = duu-', written, as 6 u = v-'{6h ( A d , - 1)6v}v.
+
the latter can be
(5.4)
@'
Equation (5.4) represents the Jacobian map of the coordinate transformation: We need to compute the determinant of @'. Modulo unessential v-
@:(h;v)-u.
21n genera1 a Weyl chamber A c 5, in a semisimple algebra 8 with a fixed system of positive roots C, = {a},consists of all { H E a:(aI H ) > 0, all a} (fig.7).
55.5. Haar measure on compact semisimple Lie groups.
229
conjugation (which has no effect on detdj’), the Jacobian map consists of an identity block in h-variables (dim = n - l ) , and a h-dependent linear block Adh - I , in the complementary variables 6v = { 6 ~ ; ~ ) . Variables {vij} can be chosen so that the diagonal part 6vjj = 0, which follows from the decomposition of the Lie algebra into the sum of its “diagonal” (Cartan) . matrix entries { b U j k : j 5 k}, as part, and “off-diagonal” (root) parts $ @ 8 Regarding generators of independent coordinates on G,and writing hk = eiBk,we get the formula for the Haar measure on G
4.)
Here dv represents the invariant volume on the quotient-space SU(n)/Un, the determinant of (Ad, - I) being squared, since each off-diagonal entry v j k represents a complex variable. Finally, remembering that all off-diagonal elements { 6ui = 6vij} are eigenvectors of the adjoint map Ad, (root vectors in @),
(Ad,-I)bvjk=
(hj/hk-1)6vjk
= ( e i(ej- ‘k)
- lYvjk;
we can write the determinant factor in (5.5) explicitly in terms of parameters 8, /(i(’j-’k) j 0.
Problems and Exercises: 1. a) Compute the Haar measure for other classical compact groups: S q n ) and Sp(n). b) Compare (5.6) with formulae of 84.1 for groups SU(2) and Sq3).
55.6.The Weyl character formulae.
231
$5.6. The Weyl character formulae. Our main goal in this section is to derive the celebrated Weyl character formulae for irreducible representations of compact semisimple Lie groups. Most of the discussion below will concentrate on groups SU(n);U(n) (respectively SL, and GL,, but whenever possible we shall indicate connections and extensions to other compact groups.
6.1. First Weyl formula. Irreducible representations { T " } of group G = SU(n), or U ( n ) were described in 55.3 in terms of their signatures (highest weights) a = ( m l ; m 2... ; m,). The latter means an ordered n-tuple of integers4: m, 2 m, 2 ... 2 m,. We denote the character of P by xa = ha. Since function x = xa is conjugate-invariant on G, and any element U E S y n ) is conjugate to a diagonal matrix h = diag(h,;. ..;h,), it suffices to evaluate xa on the diagonal (Cartan) subgroup D N I F 1 , or T" (for SU(n) and U(n),respectively). We shall start with two special cases: the symmetric Yk(r)and antisymmetric A k ( r ) tensor powers of the natural representation r in C". Their characters will be denoted pk and q k respectively. In both cases we know explicitly all weights { p } and weight vectors: 0
For symmetric space Lpk(C") realized by polynomials of degree k in n variables,
the weight vectors are monomials { z p :/3 = (k,;...;k,)}, and the corresponding weights are tuples of multi-indices /3 of norm l/3l=k. Each has multiplicity 1, hence Tk(zl;...;z,),
Similarly for antisymmetric A'((c") weights are labeled by ordered tuples i = (i, < i, < ... < ik), ha = h . ...h. , with the corresponding weight vectors: e . A ... A e. . ak 21 zk Hence, Qk(h)= = . hi,.. .hik. 0
Chi a
21
l n
1 from k t h column we express 1 - hlzh
55.6.The Weyl character formulae.
236
Step 2'. Now we subtract the 1'' column from all the rest. The resulting j k i h entry is then 'k - '1 (1 - h j ~ k ) ( l -
hjzl)'
Pulling out common factors: (zk - zl) from the kth column and row, we reduce the n - t h determinant to the (n - 1)-st,
det,
1 from 1 - hjzl
the j f h
ll ( h j - h l ) k F l ( ' ~-21) del, j=1 n (1 - h j z l ) i l ( l - ' 1 ' ~
= I,>
whence the result easily follows.
The Cauchy determinant K(h;t) = det 1 can be expanded in two different [1-hj.x] ways. On the one hand we shall see that
XI
1I
1,
(6.13) ~ ( h2);= j$. ..$en 2'1 ...Zen summation over all tuples of indices l,>e2> ...>en. On the other hand the RHS of (6.12) will be expanded into a similar series in "determinantal powers of {te}", but with different coefficients: symmetric characters { P k ( h ) } (6.1). Precisely, the RHS of (6.12), divided by A(h),will be shown to consist of (6.14) as above the summation extends over all symbolic notations p'e;p'e - ... for rewrite it in the following compact form
.t = (el>&> ...>en). Using (6.14) (k = 0,1,. . .) we can
(6.15)
Now it remains to compare the coefficients of t-terms in expansions (6.13) and (6.15), and to remember the 1-st character formula (6.10), to get
(6.16) a n n x n determinant whose entries are symmetric characters:
p k ( h ) = c hP, and signature a = (ml; ...mn) IPkk is related to a n n-tuple l = (el;...en) through the weight p = ( n - 1;n - 2; ...0) -
55.6. The Weyl character formulae.
f x sum of positive roots: el=ml t n-1;
rn, = ez+n-2;.
237
..;mn=en.
Warning! Some indices m = Q,-n+j in (6.16) could turn negative. The convention here is that the negative-order symmetric characters p , (rn < 0) are assumed to be 0, so they won't enter expansion (6.16). Formula (6.16) blends once again two fundamental tensorial operations of symmetrization and anti-symmetrization to produce in an intricate and somewhat mysterious pattern all irreducible characters of SU(n). Curiously the role of two operations is now interchanged compared to the previous sections (55.4-5). There we built all irreducible {T*}, based on symmetrization of all antisymmetric tensor-powers {.rr(') in Ak(C"):1 5 Ic 5 n}, as generators (spaces Sx). Here irreducible { T * } arise from "anti-symmetrizers" (6.16) of various symmetric tensor-representations. To show (6.13) we note that each column of matrix expansion in powers of
Zk,
1
has a geometric series -hjz*]
whose coefficients are powers 6.9) of column h
](i+z,i+z:iz+
I).+
...I; (i+rzi+@ +...);... ; ( i + z n i
Then we expand (6.17) in multi-powers of
(6.17)
...zmm) and collect terms with the identical
(tl;
h-determinants. This results in a double-determinant expansion (6.13). Since h-terms of
(6.13) are precisely the numerators in the 1-st character formula, the Cauchy determinant
1"
could be called a generating function of irreducible characters
{xa}.
To derive
6.15) we shall expand the RHS of the Cauchy identity (6.12). First observe that for each
t
= t k the product (6.18)
with symmetric characters {pm(h)} (6.1). Multiplying expansions (6.18) for different
z = zl;zz;
...;t , we get
,ji+=
c
c 2m;
PmlPmZ***Pm,
(6.19)
T€W
)k=l
where m = (ml; ...; mn) denotes an ordered tuple of integers (signature), and summation extends over all such tuples. We multiply (6.19) by the vandermondian difference-product
A(z) =
c
(-1)' z ' ( ~ ) ,
S€W
with the ''&xm positive weight" p = (n-1;. ..;O) to bring the RHS into the form
(6.20) Inner sum (6.20) consists of alternating orbital terms corresponding to various (non ordered) weights
e = (el;.,.;en)in the form e = p + mu. Notice that only non-degenerate
$5.6. The Weyl character formulae.
238 P s (i.e. t j #
ek, all j,k) will give a nonzeros contribution
to (6.20). Using the symmetry
of each product pml...pmn with respect to permutations r: m+m7 = r ( m ) , we can bring (6.20) in the form (6.21)
Here we abbreviated the product notation pml.. .pm as p(,,,). The outer sum in (6.21) n
extends over all (non-ordered) &tuples, that are related to m-tuples via p and
T:
..en) the unique highest tuple in the orbit of e , (i.e.
e = mr+p.
We denote by C' = (el;.
el>...>&),
and by u the permutation that takes C to
e',
and rewrite (6.21) as the sum
over all ordered t-tuples (6.22)
Here we utilized once again the total W- symmetry of the inner sum. Each of two inner sums (6.22) collapses into a n x n determinant in variables {rk} and { h k } (or p k ( h ) ) , whence follows (6.15).
Remark 6.2: Characters have simple transformation properties under the = xT operations of the direct sum and tensor product of representations: xT = xTxs. The second Weyl formula gives a formal expression of arbitrary xT
+ xs; xa in
@
terms of products of symmetric characters pk. This expansion can be "lifted" to representations. In ot,her words we get an expansion of any irreducible representation ?ra of SL(n) in terms of symmetric representations {rk = Yk(r)}E0. Namely, (6.23) each entry in the formal determinant (6.23) stands for a symmetric representation rk (in tensors of rank k) of signature (k0;...0), and f signs mean the direct sumldifference of representations.
Example 8.3: A representation of signature a = (4,1,0) of SU(3), respectively .t = (6,2,0) can be represented by the tensorial determinant
l o 0 *"I
(6.24)
Here ?yo denotes the trivial representation, two slots (3,l) and (3,2) in the matrix (6.24) are filled with zeros, since they have negative orders: -2,-1. Notice that 7r5 c s4@ d,since Y5(Cn) is naturally embedded in J4 my1. The degree of aa can be computed in 2 different ways, by the 1-st character formula (6.11),
239
55.6. The Weyl character formulae. (6 - 2)(6 - 0)(2 - 0) d(sQ) = (2 1)(2 - 0)(1 - 0)
-
= 24;
on the other hand from 2-nd formula (6.24)we get
d(ra)= d ( x 4 ) d ( d )- d(r5)= 15.3 - 21 = 24.
Additional comments. Our exposition in 55.6 followed [We31 and [Ze]. We shall conclude this section with yet another interesting formula for irreducible characters and degrees {d(cr)} (see [Ze], [Ta2], [Sim]). It can be stated in terms of the highest weight matrix entry,
I u,),
+,(s)= +(,
where u, is the highest-weight vector. Functions {+,}
have the property,
= +u+o,
+a+o
a consequence of the tensor multiplication rule: 77, @
= xu+@
@
... (lower weight T’s).
If we choose a system of basic roots {el; ...en}, n = rk(O), call 4 . = +a ., and write any ’
(I
J
as a linear combination of {ej} with integral coefficients, (I
then clearly,
=
Crnjej;
*, n+Tj. =
The function
+,
plays a role in the study of quantum partition function for hamiltonians,
associated to Yang-Mills potentials. But it also has an interesting connection to the character
x,.
Namely, x,(z) = d ( a )
+,(s- l”9) dg;
G
also the degree of ra is given in terms of
+, (6.25)
There is no direct link of (6.25) to the classical l-st and 2-nd Weyl formulae, so its meaning remains obscure.
$5.6. The Weyl character formulae.
240
Problems and Exercises: 1. Verify Weyl formula for characters p k and qk of the symmetric and antisymmetric tensor powers, Y k ( i r ) and A k ( ~ ) .
2. Write the 1-st and 2-nd Weyl character formulae for the orthogonal and symplectic groups W n ) ,
Mn).
3. Compare the 1-st and 2-nd Weyl formulae with the results of chapter 4 for characters of SU(2) and Sq3). Verify formula (6.23) for Sl42), using the Clebsch-Gordan decomposition of f4.4. (Caution: in chapter 4 we used a slightly different labeling of SU(2) representations, so the present "integral" signature ck = (ml;m2) would correspond to weight m = $ml-mz) of chapter 4). 4. Find characters of all irreducible representations of JIn;C), considered as real Lie algebra (any such is the product of a holornorphic and antiholomorphic representations ira
55.7. Laplacians on symmetric spaces.
241
55.7. LaplacMns on symmetric spaces. Symmetric spaces are defined as Riemannian manifolds, whose curvature tensor is invariant under all parallel transports (see Appendix C). E. Cartan posed the problem to classify all such spaces. By two ingenious arguments he gave the problem a group theoretic formulation. His first approach was baaed on the holonomy group of A. The holonomy group K a t point z E A consists of the parallel transport operators on tangent space T, along all closed path {y} through 2. Clearly, different points z E A give isomorphic groups K. The Riemannian metric is always invariant under K, and for locally symmetric A, the curvature tensor is also K-invariant. Hence, it follows from the Cartan structural equations (Appendix C), that each holonomy u E K induces a local isometry in a neighborhood of z,that leaves z fmed. This yields some algebraic relations between Lie algebra 0 of group K, and the metric and curvature tensors, g and R on A, g(AX;Y)+g(XAY)=O, for all A E R ; X , Y E T z . [A;R,(X,Y)]
= R(AX,Y) + R(X,AY); and R ( X , Y )E R.
Cartan showed that any R satisfying such relations and the standard symmetry condition of the Riemann tensor, defines a locally symmetric space. Thus he waa able to reduce the problem to (i) classification of all possible holonomy groups of symmetric spaces; and (ii) derivation of the curvature tensor on symmetric spaces in terms of their holonomy groups. The second Cartan’s method is based on another important observation: the curvature tensor is invariant under all parallel transports iff each geodesic symmetry y(s)-i7( - s), at 2 E A defines a local isometry on A. Here y = y(s;() denotes a geodesic through z in the direction ( E T,. The latter gave meaning to the terminology “locally symmetric”. This result becomes particularly significant for “globally symmetric spaces”, where each local symmetry extends through the global symmetry of A. Hence, any globally symmetric space possesses a transitive group G of isometries, and could be represented as a quotient K\G. Subgroup K c G stabilizes point z in A, and itself consists of fixed points of an involutive automorphism 0 (that results from the geodesic reflection). Group G becomes semisimple, when one drops a “trivial (Euclidian) factor”. The problem is then reduced to the study of involutive automorphisms of semisimple Lie algebras.
7.1. Compact symmetric spaces. In this section we shall be mostly interested in compact symmetric spaces A = K\G i.e. quotients of compact semisimple Lie groups
G, modulo a subgroup K , which stabilizes an involutive (Cartan) automorphism 0 of G, K = { u E G:0(u) = u } , 8’ = 1. Automorphism 0 splits Lie algebra (5 of group G into the direct (orthogonal) sum of the subalgebra Si of K , and a subspace !+ identified I, with the tangent space at zo = { K } , other tangent spaces5 { T z : z= xi}, being given by the adjoint action of G, T, N Adg(?), applied to 8. Furthermore, automorphism 0 takes on value f 1 (0 R N I and 0 @ 21 - I),so the resulting Lie brackets between the Si and !$ take the form:
242
55.7. Laplacians on symmetric spaces.
Space !# contains a maximal abelian (Curtan) subalgebras Z 2: Fir, whose dimension n = dim% is called the rank of A. The image of Z under the exponential map O-G, forms a maximal geodesicdy flat torus 7 = expZ 2: T" in A. Space On of all flat n-tori ( 7 ) is itself a smooth manifold, whose dimension depends on dimG and its rank. Group G acts on On,turning it into a homogeneous space K,\G, KO- stabilizer of 2 in G.
7.2. Restricted root system. The abelian part Z of !# could be embedded into a larger Cartan subalgebra $ of (5, in fact, $ can be chosen as the sum Z@1I(Cartan parts of !# and A). Algebra Z breaks space (5 into the sum of eigensubspaces @@, which lie in !# @ A (but not in either of two summonds). We want to define a restricted root system C = {a}in the !#-component of 0. To this end we square adjoint operators {ad&:HE %}, and note that squares map !# into itself by (7.2). Hence, !# is broken into ( p a@!#-,), where adh I !#*, = a(H)'. As in the case of entire O, the direct sum: @ a the restricted root system C(!#) can be ordered, i.e. split into its positive and negative parts C+lJC-. It has a basis of positive roots {a1; ....on}(n=rk!# = dim%), the (restricted) Weyl group W(!#), etc. In one respect, however, root systems C(O) and C(Sp) differ significantly, the former was shown in 55.2 to be multiplicity free, i.e. dim@, = 1, for all a # 0, whereas the latter has typically higher multiplicities { m a2 1). In fact, each root a of !# can be extended through a root of (5, and m, measures the number of such expansions. Examples below will illustrate the general concepts.
7.3. Classification. Symmetric spaces (compact and noncompact) were completely classified by Cartan (see [Hell), who discovered their close connection with semisimple Lie algebras and groups. Here we shall list the basic examples of so called irreducible globally symmetric compact spaces.
'One can associate with any compact pair {@R} a noncompact (dual) pair {a*$} (in the terminology of [Hell), where 8 = R 8 8, 8*= R @ iq c BC - a complexification of 8. The standard example of a dual pair in Lie algebras 8 = 8 (n) 3 R = 0 fl (n), and @* = al(n; R) 3 8 fl (n). The @Killing form on 9 (better to say, on its dual iq c 8 ), defines a G-invariant Riemannian metric on A, whose curvature tensor is obtained from the Lietracket,
R(X,Y).Z= -[[X;Y];Z], for X,Y,ZE $ ! = 8,. (7.2) Notice that the curvature tensor is always anti-symmetric relative to the metric {g(z)}. So operator R(X,Y)belongs to the Lie algebra of the holonomy group K of {z}, and the triple bracket (7.2) does obey the basic properties of the curvature tensor (Appedix C). Indeed, [[!&!$];!$I c !$, and operators {adi,,,l:X,Y E q} belong to the adjoint Lie algebra R, acting orthogonally on 8.
65.7. Laplacians on symmetric spaces.
243
Table of symmetric spacea.
I Space
lnvolu tion
Rank
6) SYn)/SO(n)
& X + X (complex conjugation)
T
(ii) SYZn)/Sp(n)
r=n-1 where J, is symplectic matrix [-I in C Z ~
e ( X )= J , X J ,
l,
=n -1
1'
the matrix of the indefinite P P+cl (p;q)-form: C z .y . - C z .y . 1 p+l " in c P + ~ ;
e(x)= Ip,XI,,,
same I,, as
r = rnin(p;q)
above in WP'q;
The are a few other examples, related to exceptional simple Lie algebras. More general symmetric spaces can be decomposed into products of the irreducibles ones: ALl x AL, x ... The simplest example is the product of 2-spheres (S2x SZ)/Z2,a quotient So(4)/So(2) x Sq2) (problem 5).
7.4. Laplaciana on manifold and symmetric spaces. We recall the definition of the Laplace-Beltrami operator A on a Riemannian manifold 96. Let (gjj) denote the Riemann metric tensor on tangent spaces, {Tz},and (g'j) - the dual metric on .. cotangent spaces {Ti}, matrix j = (ga3)= (gij)-'. So the metric (arc-length) element,
C
ds2 = . . gijdz'dzj, a3
in local coordinates z = ( x i ) on 96, while the volume element of g has the form,
d m . 4GiJ The g-gradient and g-divergence operations on functions { f} and vector-fields d T / = A = d m d z = Jdx, where J =
244
55.7. Laplacians on symmetric spaces.
{ F } are given by
of = ( C g ' j a j f ) = B * a f ; v . F = ) C a k ( J f k ) = ~ . ( J F ) . k
j
One can easily check that operations V and V - are dual (adjoint) one to the other relative to the L*-inner product with the volume element, (Of IF)= j V f . F d V = - / f V - F d V , (7.3) the dot-product in (7.3) clearly means the (gij)-product on tangent-spaces {Tz}. Let us remark that any coordinate change, z=$(y), takes tensor (gij) into g4 = 'AgA, and +A-'~J(~A-'), where A = 4' denotes the Jacobian of map 4. Hence one could check that all the above definition do not depend on a particular choice of local coordinates (z). The g-Laplacian is a product of operations g-div and g-grad,
Once again the definition is coordihate-free. For the purpose of analysis on symmetric spaces we shall need the so called multi-polar (or generalized radial) coordinates on 96. A natural framework for
introducing such coordinates is a manifold 96 with a compact Lie group K of isometries, like symmetric space 96 = K\G. We choose a transversal manifold 4 c 96, that cuts across each K-orbit S at a single point t, and assume (without loss of generality) that 4 is orthogonal to all orbits {S= S,}. Manifold T will play the role of multi-radial directions, while orbits {S,} will form a family of spheres of multi-radii t. The following examples will illustrate the concept: 1) R" with the natural SO(n)-action, where 4 = R+ (half-line), while S = S, are standard spheres of radii T 2 0; with S q n ) acting by mi-symmetric rotations. Here 4 is a 2) sphere S" c great circle (in fact, semicircle) through the "North pole" of S", while family {S,} is made of horizontal (transverse) spheres of "spherical radii" r(t) = t - angle, respectively Euclidian radii r ( t ) == sint (see fig.8). 3) Hyperbolic space
W", is usually realized
as a one-sheet hyperboloid
{ ( r' zI . ) : ~ :C.3 = I} in Wntl. Once again Sqn) acts on W" by mi-symmetric rotations, and foliates it into a one-parameter family of orbits {S,} of radii r = sinh t.
245
55.7. Laplacians on symmetric spaces.
Fig.8: Orbit-spaces of the n-sphere and n-hyperboloid relative lo So(,)-action.
We remark that all 3 represent model examples of rank-one symmetric spaces with symmetry-groups: M, = R” D S q n ) ; S q n t l ) ; S q l ; n ) , and the maximal compact subgroup K = S q n ) . Here radial manifold T is one-dimensional. But
S = K\G
higher rank symmetric spaces provide examples of multi-D radial part.
4) Group S q p ) x Sqq), p 5 q, acting on the flat space‘ 96 = Matp q, (u, v): X-bu- lXV, has the radial (transverse) manifold, made of diagonal matrices
{x, =[
11
...
]
t~
t = (t,;...t p ) ;t , > t , > ... > t ,
]
(7.4)
Given a manifold 96 with compact Lie group K , acting by isometries, and a transverse radial manifold 4, we pass to new variable { ( t ; O ) } on T x S , (orbit
S N K,\K), and find, ds2 = dt2
+ C g i j ( t ; O )d0’dOj;
where g1 denotes the restricted metric on K-orbits {S = S t } . Hence, the volume density
J =
,/-,
will depend on t only (due to K-invariance of the metric g1 on St). In
fact, factor J ( t ) will represent the Riemannian rn-volume (rn = dims)of the orbit. The Laplacian in “multi-polar” coordinates,
+ a,. g’(t; o p e ;
A = +atJa,
(7.5)
and its radial part,
‘Let us notice that space %VMat,-, space SO(p+dlSo(p) x So(d of type IV.
represents a linearization (tangent) of the symmetric
246
$5.7. Laplacians on symmetric spaces.
Here A , denotes the natural Laplacian on 2. We shall demonstrate both formulae with a few examples. 7.5. Rank-one spaces: Rn; Sn;H". Here the polar variable r measures the geodesic distance from the given point (pole) to z = (r;O).The metric tensor in polar coordinates is dr2 + r2de2;
R"
dr2+sinh2rd02; H"
Hence the density factors and the radial Laplacians become,
J ( r )=
i
r"-'; on Rn
sin"-'r; on S" sinh"-lr; on H"
; A, =
i
8:
+ +r;
on R"
+ (n-1)cot ra,; on S" 8: + (n-1)coth dr; on H" 8:
7.6. Higher rank spaces. Turning to a general symmetric space 96 = K\G, we choose a maximal abelian subalgebra Z in 'Ip (dim% = n), pick a system of positive roots in 2,{ a E C+(?$)}, or zE+(v), in the compact case, and consider a Weyl chamber in Z (fig.71,
A
= {H: (aI H) > 0, for all a E C,}
Exponential map, exp:'Ip+%, takes A into a flat (totally geodesic) manifold 4 c 96, that cuts transversely through all K-orbits. So here parameters { t = (tl;...;t,)} are coordinates of element H E A in a fixed basis of simple roots { a l ... ; an}, t j = ( H I aj).
To understand the role of the Weyl chamber, as "radial parameters", we first turn to the linearized case, i.e. space 'IpcC, with the adjoint action of I O}, and point (0). Irreducible representations Tpt" act on spaces L*(S;Y), where S = Sn-' z w; Y = Yu - the space of g E 0,(stabilizer U , N Sqn-l)), Subgroup
.
u
T ( a , u ) w= elPZ *
u)[4(2")1.
(1.19)
Here z is a unit vector on S, u E S q n ) , a E W", and x+xu denotes the orthogonal transformation u applied to the unit x. So Tr)" is induced by the representation ~
eirz a of
W" x Sqn-1).
The trivial orbit (0) c W" has stabilizer K = S q n ) . So the
corresponding representations are those of the compact factor-group S q n ) = E,/W". One can easily compute integral kernels {Kf(z;y)} of the group-algebra operators Ti and the irreducible characters.
In the simplest case E, we get a one-parameter family of co-D (induced) representations { T P : r> O}, acting in space L2(S'), and a sequence of l-D representations of Sq2) IE,/W2. The characters of {T'} are computed to be xr(a; u ) = 2 4 u ) J o ( r I a where J o is the Bessel function of order 0 (problem 2).
I 1,
(1.20)
The Plancherel/Inversion formula on E, has the form,
(1.21)
266
56.1. Induced representations and Mackey's group extension theory.
with the Plancherel measure dp(r) = rdr, supported on an object
pz= R+UZ.
m-D %+-part" of the dual
En we get
Similarly for
(1.22)
The proof of both formulae is left to the reader (problem 2). The Poincarc: group P, of special relativity is a semidirect product of the Minkowski space M",
with indefinite metric (+---), and the Lorentz group
(I = Sql;n - 1) of isometries of M". The K-orbits in MI" fall into 4 classes: point: (0) two-sheet hyperboloids: wr = {x:2. z = zo2- Exi' = r
> 0} < 0)
one-sheet hyperboloids: wr = { x:x. x = zo2- C :x = -r the light-cone: wo = {x:r = x . z = xo2-
E :x
= 0)
Fig.1: Orbits of the Poincare group in M", {O}, light-cone and 2 kinds of hyperboloids.
The stabilizer of (0) is group Sq1;n-1) itself. Its representation theory, in case n=3, will be developed in the next chapter 7. Two-sheet hyperboloids w have the compact stabilizer K ON S q n - l ) , whose representations were studied in chapter 5. They are labeled by highest weights a. The corresponding irreducible representations of
Pn consists of { T r 7 a } ,r > 0 (orbit parameter). They act in space L2(w)@ Ya and are induced by characters xp(x) = e
irz
0,
tensored with a. Such representations were first
introduced by E.Wigner [Wig2], and proved to play an important role in the quantum field theory. The one-sheet hyperboloids have non-compact stabilizers, KO= Sql;n-2). Their representations are still induced by characters x - , ( z ) = eirzl, tensored with irreducible representations
(0)
of
KO= S q l ; n - 2 ) . In
the simplest case, n = 3,
56.1. Induced representations and Mackey’s group extension theory. stabilizer K O= S q 1 ; l ) is W, so
0
267
E W, but in higher dimensions it involves the
machinery of chapter 7. The stabilizer of the light-cone is the motion group
K ON
= p a - 2 D S q n - 2 ) (problem 4). Its irreducible representations have form ( p > 0; a E Sqn-2)), and the corresponding representations T of Pn are o= induced by such d s . A
1.6. Holomorphically
induced
The
representations.
standard
induced
representations T = ind(S I H ; G ) ,were realized in several different ways: on sections of G-invariant vector bundles ($;Y),over quotients 96 = H\G,
where each stabilizer
{ H , = gi*Hg,),acts the fiber {Yz:z E 96). Alternatively, they were realized in certain subspaces of functions on group G itself (1.7), namely, C(G;S) = { f ( h z )= S(h)-’f(s):h E H ; z E G} C C(G).
For Lie groups G such subspace is characterized in terms of generators { X } of the Lie subalgebra 8,acting on Cm(G;S) by 1-st order differential operators,
+
(8, S,)f = 0; where a, denotes a left-invariant vector field of element X on
(1.23)
G.
The action of H by left translations on G foliates the latter into a union of fibers (H-cosets) and the representation space of T consists of all functions on certain differential system (1.23) along the H-fibers.
G,that
satisfy
One would like to extend this construction to situations when (1.23) becomes a system of Cauchy-Riemann equations:
& f = 0,
on certain complex lane bundles (i.e.
manifolds with analytic local structure holomorphic transition functions between fibers).
So the representation-space will consists of holomorphic sections. More generally, we shall see that homogeneous spaces 96 = H\G of Lie groups can combine both “real” and “complex” parts, so the corresponding functions/sections { f (z,...zk;zl...zm)) will depend on Ic real and m complex variables. We consider a complex Lie algebra gCwith a real form 0,and denote by z+z, conjugation in 0, relative to 0,. Assume that 0, contains a complex subalgebra 9,
8 c 0,with the following properties: 0 9na = Gc - a complexification of real 8,and the
and a real subalgebra
{adX(Y):XE 0
%+
!iicmc0.
adjoint action of
8 on 0,,
a} leaves 3 (and g ) invariant;
5 = !Dlc
- complex subalgebra, whose real part IUI = IUI,
c 0. Clearly,
268
56.1. Induced representations and Mackey’s group extension theory. Hence we get similar inclusions for the corresponding (real) L’ie groups:
HcMcG. Proposition 4 Homogeneous space H\G has the structure of a fiber bundle with complez fibers N H\M (of complez dim = m), over the base 3 = M\G (of dim = k). Clearly, quotient H\G is naturally projected onto M\G = SJ,
p : ‘H-cosetn--rUM-cosetn. 21 H\M,
and claim that the latter is isomorphic to a complex quotient N\M,, of two complex subgroups N M , c G,. It suffices to compare the tangent spaces of both. We observe that !llc = 9 !ll,and 9 n n = 8, We take a fiber space, p-l(z)
+
hence !Ill/ N @ tangent(H\M) 1? !ll,/9. Let us illustrate the above constructions for the unitary algebra 6 = su(n). Its complexification 0, = q n ;C) has the (Cartan) conjugation:X+ - X*. We pick a pair of Borel subalgebras 9; 3,made of upper/lower triangular matrices:
(1.24)
whose sum !XIc = m+g, comprises entire 0, (hence, !ll= ..in)!),
and whose
= R e ( 9 n 3 ) = { h = diag(ial; ...zan)}. So subgroup H i s the maximal torus Tn-’c G = SU(n), and the quotient
intersection is the diagonal (Cartan) subalgebra,
H\G = H \ M = Tn-l\Syn), acquires the natural complex structure, inherited from space SL(n)/N, called flag manifold (problem 3). In the simplest cave, 0 = 4 2 ) , the complexification Oc = 4 2 ; C) contains the upper and lower-triangular (Borel) subalgebras: %={[
q};3
={[ ;
-a]};
their sum 9l + 3 == 9R, = 8,; while the “real part” of the intersection,
A similar construction can be given for any simple/semisimple compact algebra 0 (see chapter 5). Here 9; 3 are two Borel subalgebras of 6, made of all
56.1. Induced representations and Mackey's group extension theory.
269
positive/negative root vectors. Their intersection is once again a Cartan subalgebra $ of G, and the quotient T\G (of compact Lie group modulo maximal torus), has a natural complex structure (problem 3). Compact Lie groups, and their quotients T\G provide one extreme, when the entire homogeneous space turns into a single complex leaf. Another extreme arises when algebra % = 8 = !lJlc, hence 8 = 3, = !Ill,so H = M c G, and we don't get any complex leaves, the entire quotient being the standard H\G! In the context of Proposition 4 we consider functions/sections {f(z;z ) } on space 96 = H\G, holomorphic in variables z = ( z j ) , and define induced representations in those. Precisely, a character (representation) S of the real subgroup H, extends to a holomorphic representation u(X) of the complex algebra %. Then we replace condition (1.24) with
[az + a(Z)]f = 0; for all Z E 3. and call the corresponding function-space C(G; u I N ) . Here az indicates the Cauchy-Riemann a -derivative
(1.25)
in the complex (fiber) variables. Indeed, in the simplest case: 0 = W2; GC= Cz; % = Span(X+iY), {X;Y}-basis in W2, and a(X+iY) = A, space H\G = {O}\R2 N C, and (1.25) turns into the standard 3 - equation,
B f = ;(az + 3,)f= - Xf, whose solution is f = e-"
x holomorphic function.
The resulting representations of G in spaces C(G;u I N ) are called (partially) holornorphically-induced, and denoted by ind(S 1 H ; a I N;G). Next one needs to introduce a G-invariant product in representation spaces C(G;a I N ) . In some cases it proves to be possible, for instance, if the complex component of H\G is compact (like SU(n)), or is equivalent to a bounded domain in C". In such cases, Hilbert space L(G;u I N ) turns into a closed subspace of L2(G;Y',)the representation space of ind(a I H,G). We shall not delve deeply into the subject of holomorphically induced representations here, but refer to [Kirl] for further details. Let us mention, however, an important result, due to Borel-Weil-Bott. They proved, that the holomorphic induction on compact Lie groups G, by characters u of a Bore1 subgroup N (1.24), yields all
56.1. Induced representations and Mackey’s group extension theory.
270
irreducible all irreducible representation
?r
E G.
To give a precise statement, we recall that irreducible ?r E G are labeled by weights (characters) {A} of the Cartan subgroup H N T ” c G , the highest weight X is unique, i.e. the corresponding eigenspace YX is 1-dimensional. Weight A, rather eigenspace YX, gives rise to a holomorphic line bundle 8 =$(A) over the complex manifold 96 = T”\G with fibers N Y,. Hence, we get a finite-dimensional vector space L(96;%)of holomorphic sections of 8. It turns out that dime = deg(nX).Furthermore, Theorem (Borel-Weil-Bott): The naturd (induced) action of G on 96 and B(X), considered on space P. of holomorphic sections is equivalent to 77.’ Proof exploits the highest-weight theory of chapter 5 and consists of several steps. 1) The Weyl “unitary trick” allows to identify representations
T
of G with “holomorphic
representations” of GC - its complexification, so that matrix-entries of
T
on GC become
analytic continuations of entries {f(z) = (T,( 1 7 ) : ~E G}. 2) We realize
T
on by functions (matrix-entries) on GC (or its real non-compact form
GR), that satisfy
f(Cz) = f(z); for all C 6 N f(hz)= X ( h ) f ( z ) ;for all h E H
(1.26)
where H is the Cartan (diagonal) subgroup of GC (GR), while N - is made of negative root-vectors (lower-triangular matrices). The embedding of space ‘V = Yx into functions (1.26) is given by the lowest-weight vector ‘lo E ‘V,
t-f&)
= ( T Z ( 0 I ‘lo).
Condition (1.26) can be written in the infinitesimal form as
= 0, for all negative root-vectors Y = Y , E 9a H f ( z )= (A I H ) f ( z ) ; for H in the Cartan subalgebra where
aY;
denote the right-invariant vector fields on GR (7),
b
(1.27)
(generators of left
translations). 3) An elegant observation of Borel-Weil-Bott was to note that condition (1.27) has an
equivalent form in terms of holomorphic differentiation a j n the complex domain GC where
,
z ,= X , + iY, (a- root of 8).Namely, a-Z f = 0; for all Z = Z = i(X I H); for all H E is, (compact Cartan part) (I
a;Hf
(1.28)
-
56.1. Induced representations and Mackeyi group extension theory.
271
One first observes that space (1.28) is finite-dimensional (it sufices to prove it for the restrictions {f} on the compact quotient T\G
c E\Gc,
then apply a unique analytic
continuation4 from the former to the latter). Hence, space (1.28) has the highest weight function 4(z). Since,
a- 4 = 0, for all z
implies lay4 = 0, for all Y (negative root-vectors)l
on GR, hence (analytic continuation) on Cc, and 4 is an eigenfunction of Cartan 8, it follows that
4 is equal to the standard matrix-entry 4o = (irZto I vo), that couples the
highest and the lowest weight A. Thus we get “space(1.28)” = “space(1.27)”, and two (induced) representations are equivalent. Finally, it remains to observe that space (1.28), when restricted on compact G c GC, defines a holomorphic line-bundle $
= $(A),
and all analytic sections of $,
80
r x becomes a holomorphically induced
representation in “sections” &($), QED.
We shall illustrate the holomorphic induction and the Borel-Weil-Bott Theorem for group SU(2). Here a unitary character S(0) = eime extends through a holomorphic representation/character (1.29)
It turns out that character o = u, gives rise to a complex line-bundle 1, over the quotient-space 96 = SU(Z)/T, whose first Chern-class’ is equal to m. The space of holomorphic sections of L , has dim = m, and the resulting holomorphically-induced 4The simplest prototype of the triple GC;GR;G are, multiplicative groups: C’; R,
and
T = {e”} (unit circle). Representations (characters) ( ~ “ ( 6 ’ ) = erme} of T have a unique extension to
holomorphic characters
{x,(z)
= z”} of real group R, and complex C*.
’Chern classes describe the degree of ‘twisting” of a line (or more general vector) bundle L over 96. The topological structure of any vector bundle is determined by a family of transition coefficients {g,v(r):U,V-a pair of neighborhoods of z in 96). Coefficients {guv} take on values in a etc.), which depends on specific geometric features of L structure group G of L (GL, SL, (Riemannian, Hermitian, holomorphic, etc.). They satisfy the cocycle condition: g~v(z)gvw(z)gwu(r)= 1, for all triples U , V , W3 r; and gvu = g d v
so, su,
So { g u v } defines a G-valued 2-nd cohomology class on 96. But for line bundles (dim[fiber] = l), group G = C’, 80 multiplicative G-cocycle can be turned into the standard (additive) 2-cocycle: -v v
The resulting a E IfZ($$) is the first Chern class of L. In fact, Chern classes always define integral cohomologies, a E If2($$), in other words, differential 2-form a = a; dz;A d r j on 96, integrated over any 2-cycle, (closed 2-D submanifold) S C 96, yields integral values, #sa E Z. The second cohomology group of the sphere S2 = su(2)/T is well known to consist of integers, hence a = m.
272
56.1. induced representations and M a c k e y i group extension theory.
representation is equivalent to the familiar ?-spin representation nmI2 of $4.2. The details will be outlined in 56.3 (example 3).
In conclusion let us remark that the induction procedure in any of its modifications: standard, holomorphic, or, more general, partially holomorphic (“combined”) seems to provide a universal method of constructing irreducible representations. We shall further explore its meaning and significance in 56.3, based on Kirillov’s orbit method and geometric quantization.
$6.1. Induced representations and M a c k e y i group extension theory.
Problems and Exercises: 1. Establish formula (1.24) for characters of irreducible representations (Hint: let go E G and zo be its fixed point. Show that the map (z;h)+s(z)-'hs(z), from $ x H - + G is a diffeomorphism, that takes invariant measure d z d h on S x H, into the measure dp(g) on G , whose density relative to the Haar measure dg at the point go is equal to I det(1 -g;) 1. Use also formula (1.15) for induced representations and the known relation, that gives Jacobian g,: as the ratio of two modular functions on G and H,
-Ad*) - det(gg). AH(h)
The modular function on a locally compact group G is a ratio of the left-to-right invariant Haar measures: On Lie group G, A(g) = det(Ad,). 2. i) Derive the character formulae for representations {T'} of E, and {TP'u}of En. ii) Derive the Plancherel-inversion formulae (1.21)-( 1.22) for motion groups.
3. Establish the complex structure of the quotient-space 96 = SU(n)/T"-*, parametrization by the flag-manifold Sa!(n; C ) / B , .
using a
4. Find stabilizers of orbits of the Lorentz group SO(1;n-1) in M". Show that stabilizer K O of the light-cone coincides with the motion group En-l (Hint: write stabilizer algebra St, of point (1;-1;O;...) by block matrices,
the off-diagonal 2 x ( n - 2 ) blocks {!a}; {Ta;Ta} forms a p-component of the Cartan decomposition of St, (95.7); show that the commutator [Q;'p] is zero!).
273
36.2. The Heisenberg group and
274
the oscillator representation
$6.2. The Heisenberg group and the asdlator representation. The Heisenberg group plays the fundamental role in many areas of harmonic analysis, differential equations, number theory and quantum Physics. It gives a mathematical formulation of the Heisenberg uncertainty principle of quantum mechanics, and reveals close connections to the harmonic oscillator. The latter in turn gives rise to the fundamental creation-annihilation structure of the quantum field theory. In this section we shall develop the representation theory of the Heisenberg group, based on Stone-von Neumann Theorem. Then we apply it to spectral theory of the harmonic oscillator in R", establish connections of the Heisenberg group to symplectic/metaplectic groups and the Weyl algebra, and construct the oscillator representation.
2.1. The Heisenberg group with multiplication given by
W,
consists of all triples {g = (x,y,t):z,y E Rn; t E R}
gag'= (z,y;t)(z',y';t')
Its Lie algebra $,
N
(2.1)
= (xtz';y+y';ttt'tz.y').
R*"+' is also made of triples
{X = (x,y;t)}
with the Lie
bracket
[X; X']= (0,O;2.y'-y
*
x').
Group W, (respectively, algebra 8,) has a 1-D center Z = { ( O , O ; t ) } N W, and the commutative factor-group G/Z N W2", so H, forms a 2-step nilpotent group: [G; GI c 2.
In fact, W, is the simplest among all noncommutative (nilpotent) groups, the
cocycZe6
$(g; 9') = x y', from G x G-tZ, (2.2) measuring a deviation from commutativity. Group W, can be viewed as a semidirect product, G = H D N , of two commutative subgroups: H = ((0, y; t ) }N RnS1, and
N = {(z,O;O)}N W", with N acting on H by unipotent automorpisms,
6A function 4(z; y): G x G-R, on group G with values in R (or in more general commutative group 2) is called a 2-cocycle, if it satisfies: 4 ( a ; b ) - 4 ( a ; b c ) + 4 ( a b ; c ) - q 5 ( b ; c )= 0 , for all a , b , c EG. Here we use an additive convention { f } for the group operation in Z, and multiplicative for G. In other words, the coboundary, &(a;b;c) = 0 , identically. A 2-cocycle is trivial (coboundary), if it can be expressed through a single-variable function $(a), 4(a; b) = $(a) - $(ab) $(b) = a$(a; b), for all a, b E G.
+
Any 2-cocycle gives rise to a central eztension of G by Z, i.e. group H with center Z ( H ) N Z, so that H I 2 N G. The elements of H are pairs { ( z ; t ) : zE G;t E Z } , and the multiplication is defined by $1 (2;t ) * (Y;s) = (zy; t+s+d(z; Y)). The cocycle condition ensures associativity of the group multiplication in H, while trivial cocycle 4 yields the trivial central extensions, direct product, H N Z x G (problem 1).
56.2.The Heisenberg group and
the oscillator representation
275
It is often convenient to write H, in the complex form: C" x R with the product ( 2;
Here cocycle
4
t ),(w; T ) = ( z t w ; t t 7 t s 2 .m).
of (2.2) is replaced by an equivalent (antisymmetric) cocycle
(problem l), qY(a;b)= sz - m = '2;( z
-Z
.w).
(2.4)
One might wonder, to what extent the Heisenberg group is unique among all 2-step nilpotent groups with 1-D center. The answer proves to be positive, within the class of bilinear central eztensions, i.e. extensions defined by bilinear forms, g(a; b) = Qa .b, a, b
E R". Trivial
C$
clearly correspond to symmetric (quadratic) forms Q. So nontrivial
possible central extensions, should correspond to the quotient-space of "all bilinear forms
{Q}", modulo "symmetric {Q}".In other words central (bilinear) extensions of R" are labeled by all antisymmetric matrices
{Q}.But any such Q can be brought by the change
of basis into the form,
or the direct sum of such matrices. This means that space R" is decomposed into the sum, RZm@ Rk, where Q is nondegenerate on the former (like
4' of (2.4)), and annihilates
the latter. The corresponding Q-central extensions is obviously isomorphic to the sum
H, x Rk (Heisenberg plus abelian).
Sometimes one considers the factor-group W,,
modulo a discrete subgroup
Z = ((0,O;k))of the center 2, and calls it the Heisenberg group. In this form space W, is identified with C" x T, and the multiplication is given by (z;t).(w;s) = (z+w;tseiS'"); Finally, let us remark that
t , s - unit complex numbers.
W, can be realized by 3 x 3 upper triangular matrices
IL
with the standard matrix multiplication. Similarly, W, is realized by ( n + 2 ) x ( n + 2 ) matrices, with row-vector I ; column-vector y' and the n x n-identity diagonal block in the middle. Lie algebra 8, has 3 types of generators:
pi = (ei,O;O);q j = (O,ej;O); Z = (O,O;l),
276
56.2.The Heisenberg group
and the oscillator representation
where ei denotes the i-th basic vector in R". They satisfy the canonical (Heisenberg) commutation relations (CCR),
2.2. Canonical commutation relations. An important example of the Heisenbergpair is given by operators of multiplication and differentiation,
Q:f(z)+izf(z); P:f+X(af)(z); in R,
Qj: f(z)+izjf(z); Pk:f+X(akf)(z);in W",
(2.6)
which obey the relations,
[P;Q] = iX, or [Pj;Qk]= i M j k .
(2.7)
We shall see that (2.6), in fact, serves a model example for the Heisenberg relations. Namely, any CCR (2.7) can be realized by a pair of operators (2.6), by a Theorem of Weyl and von Neumann. Historical Remarks: Heisenberg commutation relations first appeared in the context of Quantum mechanics. The states of a quantum system are usually described by Hilbert space vectors { q ~E 36, e.g. 36= L2(R)}, while the observables are given by certain (typically symmetric, but often unbounded!) operators in 36. Examples include the so called position and momentum operators,
Pj:$,-.ihaj$, on 36
Qi:$(z)+zi$(z);
Lz(R"),
angular momentum operator (see J4.4),
M..= z.a .-z.a. $3
8
3
3
8'
and the most important of all the energy (Hamiltonian) operator 7 (e.g. Schrijdinger operator with potential V),
H = - h2 T A + V(z) = :Pz
+V(Q).
An expected value of an observable A at a state $ E 36 is given by the quadratic form
2 = (A)$
( A $ I $1,
while the "mean quadratic deviation",
-4
q A )=
I
= ( A-
mJIll
(2.8)
measures the error in observation. So the precise knowledge of observable A at state 3 (zerwerror!) is attainable only for special states: the eigenuectors of A. In the physical parlor they are called bound states of hamiltonian A. Thus measurability
(or
observability) of 14 becomes paramount to its diagonalization. 'Hamiltonian H completely determines the evolution of quantum system, namely the initial state $, E 36, evolves at time t into a state $ ( t ) = eitH[$,,].In other words, quantum evolution is given by a unitary group, generated by H.
86.2. The Heisenberg group and the oscillator Obviously,
277
representation
any pair of commuting observables { A ; E } can be simultaneously
diagonalized, i.e. observed to any degree of precision. However, noncommuting observables, like position {Qi} and momenta { P i } , cannot be diagonalized. We shall see that a Heisenberg canonical pair { X ; Y } has no nontrivial 1-D, even finite-D representations (problem 2)! It was observed experimentally, that the position and momentum of an electron can not
he accurately measured at once, the product of errors always remained greater than the Plank constant h. This led Heisenberg to state his famous Uncerfainty principle of the quantum theory in the form of the commutation relation,
[ P ;Q ] = ih.
(2.9)
Indeed, one can easily check that for any pair of observables { A ; E } errors (2.8) satisfy,
, any state 4. c ( A ) c ( E ) % ( [ A ; B ] )in
So the Heisenberg relation (2.9) provided a mathematical formulation of experimentally observed uncertainty between P and Q:
(2.10)
c ( P I 4 Q ) L h.
Relation (2.10) has a simple reformulation in terms of the Fourier transform (see $2.1): for any function
4 in the weighted Sobolev space:
I
I z$ I 2dz < 00;
I
I($ I 'd< < 00,
In fact, for any constants a, b we have
2.3. Representations of 4;Stonevon Neumann Theorem. Irreducible representations of W, fall into two classes: 1-D characters of the commutative factor group G / Z N W2",
and a one-parameter family of
m-D representations T A (A E W)
realized in Hilbert space
% L2(Rn) by operators
(2.12) One can check (problem 3), that''2
are induced by 1-D representations (characters)
x x ( b ; t ) = eiXt, of subgroup H = {(O,b;t)}, as explained in $6.1, T X= ind (G;xX H).
$6.2.The Heisenberg group and
278
the oscillator representation
Let us remark that, the structure and realization of { ~ ' s }(2.11), , (2.12), in the form of induced representations, can be easily derived from a semidirect product decomposition of
H,, and the Mackey's theory of 56.1 (problem 3).
However, we shall produce a direct argument based on the Stone-von Neumann Theorem. Let us observe that infinitesimal generators of {T'} (representation of the Lie algebra &) consists of multiplications, and differentiations:
; = P 3.; Z+iXI, (2.13) q,.+TX(q 3.) = iXxj = Q ~pj-+a. 3 the center being represented by scalar operator Z+iXI. Conversely, an infinitesimal representation (2.13) of Lie algebra 8, integrates through the Lie group-representation (2.12).
Theorem (Stone-von Neunmann): A pair of antisymmetric (unbounded) operators P and Q in Hilbert space 36 satisfying the Heisenberg commutation relations: [P;Q] = iXI, can be realized as a multiplication and a difierentiation:
Q$(z)= ~XZ$(Z), P11,(x)= a $ ( ~ ) , on scalar or vector-valued functions 11, E L2(W)(or L2 @ Y). In other words there ezists a unitary intertwining map W: 36-+L2, that takes Q into iAz and P into 8,. The irreducible action (representation) of P,Q an 36 corresponds to dimT = 1 (scalar functions), so any action is equivalent to a "dimY-multiple" of an irreducible action. The proof exploits spectral decomposition of a n antisymmetric operator Any such
Q (Appendix A).
Q can be realized by multiplication in the direct integral space: 36 = R ~ L d = z {Y, -valued L2-functions +(z) on R},
4
Q:+(z)+iz .).(ol Here we assumed X 1, without loss of generality. Operator P generates a one-parameter unitary group U t = e P t . The Heisenberg commutation relations imply
U,QU;'
=Q +itl.
(2.14)
The latter means that conjugation with U t shifts "spectrum of Q" (spectral subspaces) 8 by the amount { t } . Consequently, all spaces {Yz:z E R}, labeling values of {$(z)}, become isomorphic (equidimensional),
TIN Y, and operators {a,} define
a family of
Y-
unitary maps (cocycle) { u ( z ; t ) } ,50 that
(Ut+)(.) = 4 2 ;t)[+(z+t)l. '%f Q had discrete spectrum {Ak}, the relation (2.14) would mean that the eigensubspace E X is shifted by U t onto EX+1. In fact, continuity of t implies that Q has continuous, indeed absolutely continuous (Lebeague) spectrum!
$6.2. The Heisenberg group and the oscillator representation
279
Cocycle a is easily verified to be trivial: a(z;t)= u(z)-'a(z+t), which allows to transform operators { U t } into shifts of V-valued Lz-functions on R, V,rO(Z)
= rO(z - t ) .
via map W : $ ~ ( z ) - + u ( z ) [ $ ( But z ) ] . the latter is obviously generated by operator P = a,,
QED. We have stated Theorem 1 for a single Heisenberg pair {P;Q}.The result easily extends to n-tuples {P,; ... P,; Q,; ...&,}, satisfying CCR (2.5). Here one simultaneously diagonalizes all { Q j } ,and analyzes the action of n-parameter unitary group
{V, = ezp(t,P,+ ...t,P,): t = (tl;A,,)E R"} on joint "spectral subspaced' of Q's. Hilbert space 36 then turns into {V-valued Lz-functions on R"; with translationinvariant (Lebesgue!) measure}, operators { Qj } become multiplications by independent variables { z j } on L2(R"),while { P j } turn into differentiations {aj}. To apply Theorem 1 to irreducible representations of W,,
we observe that any
such T restricted on center 2 must be scalar, T 12 = eiXt. If X = 0, then T factors through the representation of the commutative quotient-group G / Z , so it becomes a character (2.11). For nonzeros X generators (Lie algebra) of
CCR. So Theorem 1 applies, and we get subgroups translations and multiplications on L2(Rn),
W,
obey the Heisenberg
{(a;O;O)} and
{(O;b;t)},acting by
TO$ = $(z+a); Tb$ = eWJ* "+t)$(z), whence follows (2.12). Figure 1 below illustrates the dual object (set of all irreducible representations) of W,: it consists of a series of infinite-D representations {Tx : X E R\O}, and the hyperplane of 1-D representations (characters) of W,/Z { p ( g ) = e2''"W;g = (w;t)}.
I -A
Fig.2: The dual object of the Heisenberg group i s made of 1-parameter family of co- D representations {Tx}, and a twoparameter (or Cn-parameter) family of characters { p } .
2.4. Characters of TXand the Plancherel formula: Representations TXextend in the usual way to any of convolution algebras on G = W,: e,(G), Ch(G), L'(G), etc.,
280
56.2. The Heisenberg group
and the oscillator representation
I
f+Ti
= Cf(g)Twg.
Formula (2.12) implies that operators { T i } are given by integral kernels on R",
q"; Y) = 7 (Y-z; xz; 4, where
(2.15)
f denotes the Fourier transform of f(a,b;t)in the second and third variables, N
f (...; [;A) =
f (...;b; t)e'(€ ' b+At)dbdt.
(2.16)
Rn+1
Evidently smooth compactly supported functions {f} on G yield nice (compact; Hilbert-Schmidt; trace-class) kernels Kf (Appendix B). As before we define the character of T x as a distribution on G, xx = trTA, via pairing to nice test-functions f,
(xAI f ) = tr T Y f ) . The latter is computed by (2.15)-(2.16),
I
t r K f = K f ( z ;z)dz = where
I
f (0;xz; X)dz = (?)"?(O;
RnN
0; A),
denotes the 1-variable Fourier transform in t:
f+T( ...;A) = If(...;t)e%t.
Thus we get t,he character formula for irreducible representations { T x }of
W,, (2.17)
We can interpret (2.17) by saying that distribution xx is supported on a (oneparameter subgroup) center 2, and is equal to ''&function of 2" x "G-invariant density { ( ~ ) n e i X t } ' Integmting r. (2.17) in X we immediately derive the Plancherel/inversion formula on W,, , all f(e) = tr T X ( ~ )d p ( ~ ) for
f E CF(G),
(2.18)
with the Plancherel measure (2.19) supported on the set of infinite-D irreducible representations {TA:X E W\{O}}. The inversion formula (2.18) yields as usual the Plancherel formula
where ?(A) = T ; ( f ) means the noncommutative (operator-valued) ''Fourier transform" of f , and tr (?(X)*?(X)) - its Hilbert-Schmidt norm.
1 ?(X)Ibs=
36.2.The Heisenberg group and the oscillator representation
281
We remark that all integral-operators {Kf}(2.15), with f E e,"(G), belong to the Hilbert-Schmidt class for all A. Indeed,
I K flhs=
7 (y-s;Xs;A)I
2
d s d y < oo!
2.5. The harmonic oscillator. The quantum-mechanical harmonic oscillator is a
Schrodinger operator with quadratic potential. We shall take it in the simplest form (from which more general cases could be easily deduced):
H = :(-A
+ Iz I
2),
+ s2), in 1-D.
in L2(Wn) or H = $(-a2
One is interested is spectral theory of H : its eigenvalues and eigenfunctions. The harmonic oscillator happened to belong to a rear and beautiful species, called solvable models. In other words one can write down explicitly all eigenvalues and eigenfunctions of H . Theorem 2: Spectrum of operator H is purely discrete. In 1-D it consists of a sequence of eigenvalues: A, = k ;: k = 0;1; ...; the corresponding eigenfinctiona being the classical Hermite functions on W, (2.21) Here {hk} denote the classical Hermite polynomialsg, h,(") = 1 e 2 P / 2 ( e - 2 2 ) ( k ) , (2.22)
+
fi
W" has eigenvalues {Ak}, labeled b y n-tuples of integers, = (k,+ ... + k,) +i;and the corresponding eigenfinctions are
The multi-D oscillator on
k = (k,;...k,),
products of 1-variable Hermite functions, $,.(x) = $k,(q)...$kn(zn).
Normalizing factors
{&}
render system {&}
orthonormal in L2(W),
11 G k llL2 = lWe shall establish Theorem 1, aa a simple application of the Heisenberg CCR. But this time we choose a different realization of generators { p ; q } of H,, namely by the so called creation/annihilation pair10,
(2.23) or similarly defined creation/annihilation n-tuples a.=.l(aj+zi);a~=-L(-ak+zk);
J &
Clearly, daggered operator {a:}
J;
15 j ; k < n .
(2.24)
are adjoint to {ak] in L2(R"). One can easily verify that
'Hermite polynomials form one of three known families of classical orthogonal polynomials, along with Laguerre (chapter 8), and Jacobi (chapter 4) (see [Erd];[Leb]).
86.2. The Heisenberg group and the oscillator representation
282 pair {a;.’}
obey the Heisenberg CCR, [a; at]
= ;[a + 2;a - 21 = 1;
or [ a j ; a l ]= 6 j k I .
Q = 2;P = iV, operators a and
But unlike, the position and momentum,
af
adjoint in L2.The harmonic oscillator can be represented in terms of the pair
H = %-a2 + 22) = at,
+ ;= .at- f.
are not self{a;at},
(2.25)
Finally, triple {a; at; If} obeys the commutation relations,
[ H ;at] = at; [R;a] = -a. In this regard
{at;,}
(2.26)
behave, like the raising/lowering elements { X ; Y }of the Lie algebra
4 2 ) (chapter 4). The oscillator, however, is not a Cartan (diagonal) element of pair {at;.},
since [at;a] = I , rather than :If, as in 42)!. The nature of the triple {at;a; H} in
relation t o 4 2 ) will be elucidated below. Commutation relations (2.26) readily yield all spectral results of Theorem 2. Indeed, if X is an eigenvalue of H and cl, - the corresponding eigenvector, then all $k = ( a t ) k [ $ ] ,and
+-,,
If[$] = w,
= a”[?f~]are also eigenvectors,
Since operator H is positive,
14) = it has the lowest eigenvalue A,
1 V+ 1 2 + 2 2 14 12)dz > 0, for all 4 E L Z , 2 0, and eigenfunction $,
called the ground-slate, and the
ground-state energy, i.e. the lowest-energy states of the quantum system. These must be
annihilated by the lowering operator, a[cl,o]=
(a + z)$~= 0;
which immediately yields the ground-state 2
&(z) = e-z 12- the Gaussian.
‘“The terminology came from the quantum-field theory, whose main task is to account for the creation-annihilation of quantum particles (or better to say, “particlestates” of quantum-fields) in subatomic interactions. The mathematical structure for the creation-annihilation processes is based on the notion of (multi-particle) Fock-space, discussed below. I t typically has a (unique) vacuum-state (vector) w, and the corresponding space 36, = span{w}, as well “1-particle”, “2-particlen, ... spaces: 36,; X2;... The creation operators {at} send 36,-;.36,; 36,+36,; ... ; the particles being “created from the vacuum”, while annihilation operators go in the opposite direction, a:36n+36n-1+...+360, thus diminishing the particle number. The simplest model that accommodates such features are raising/lowering element,s {X; Y} of 4 2 ) (chapter 4).
56.2,The Heisenberg group and Substitution of
283
the oscillator representation
do in H (2.25) yields the lowest eigenvalue, H[+J = (a,+
+ i)+o= :lo*
m\
hence all other eigenvalues,
Ak = ;+ t ; t = 0; 1;2; ... Let
+k
= (at)k[$J denote the t-th eigenfunction of H . To get a Aermite-representationof
(lk we apply yet another identity,
= 6 - z = e"2/2(j
-,t
[e-Z2/q,
in other words the raising operator at is conjugated to a derivative-operator 8, via multiplication with the Gaussian. Hence, (J)k
= ,z2/2 ( - qk[e-z2/2].
Applying the latter to the ground-state +o we get the Rodriguez formula (2.22) for +kr
QED.
2.6. The oscillator representation and the metaplectic group. The Heisenberg Lie algebra $, acts naturally on Wn by differentiations and multiplications (positionmomentum operators) (Theorem l), and thus gives rise to the Weyl algebra W = W , (an associative hull of made of all differential operators with polynomial coefficients: A= a,px4aor. (2.27) la+PI < m Algebra W is graded according to the total degree m in variables {z} and derivatives of polynomials A = a ( x ; a )in (2.27),
en),
c
{a}
W o = {const} c W'
c W 2 c ... c W m = { A degA < m} c ...
One can easily check (problem 4) the product and the commutation formulae, W P Wc ~ W P + ~[ ;W P ; W ~ cIW P + ~ -for ~ ; all p , q 2 0.
(2.28)
The lost of 2 degrees in the commutator [ A , B ](A E WP, B E W q ) ,results from the basic relation = const, extended to other generators {x( ;a Y P} of W (problem
[.;a]
4). Hence follows (i) 2-nd degree operators {A} ( m = 2 ) form a Lie subalgebra W 2 of W , with respect to the natural commutator bracket: [A;B ] = AB - BA. In fact, (ii) A subspace of first degree operators {A} (rn = 1) is an ideal of W 2 , isomorphic to Heisenberg algebra $ =
a;,
86.2. The Heisenberg group and
284
the oscillator representation
(iji) W 2 contains a symplectic subalgebra 1' 1 N operators {a$; riaj; zizj}.
sdn), spanned
by all 2-nd order
So Lie algebra W 2 factors into a semidirect product s~'11,with 1' 1 acting by derivation on 8.Consequently, Lie group of W 2 also breaks into a semidirect product:
W,pG, where G denotes a simply connected cover of the symplectic group Sdn). So group S d n ) , and its cover G,act by automorphisms of Lie algebra Q,, and group W,. The latter could be also verified by the direct computation. Namely any
Sdn) 3 g
=[:
~]:(z,y;t)-(~z+cy;~z+dy;l).
(2.29)
respects the Heisenberg Lie bracket on triples {(z;y;t):z,y E R"; t E R} (problem 5).
In the previous subsection we have constructed an irreducible representation of
W, in space L2(Rn),of the form T j f ( z )= ei X ( z. bSc) f
(z+a); g = ( a ,b; c ) E
w,;
whose generators were given by operators,
p j 43.; q3.-Axj; Z+iX.
(2.30)
Formulae (2.30) extends through the representation of the Weyl algebra, in particular, its 2-nd order (Lie) part
W2.So we get
sp(n)-generators acting in L2(Rn) by
operators, q . .+z.d .; qiqj--iiXz.z .. Pipj +La?.; 2~ $3 8, 1 3 2 3
The latter can be lifted to a simply connected cover-group
(2.31)
gdn),and yields the
celebrated oscillator-representation (also known as spinor, metaplectic, Borel-Shale-
Wed), that appears in many different places and finds numerous applications. Another way to introduce the oscillator representation comes from the action of S d n ) by automorphisms of W, (2.29). Let us observe that symplectic automorphisms { u}preserve the center of W,, zu = z, for all z E Z. Hence two representations: Ti, and T i (g E W,), are equivalent, any T Xbeing uniquely determined by its value on 2,
T? = As a consequence we get a family of intertwining operators, { T u x E S d n ) } , determined modulo scalars (like in the Mackey's theory, 56.1). They define a projective representation of G = Sp(n) in L2(R"),
T,, = a(u,u)T,T,; for all u,u E Sdn).
(2.32)
with cocycle a. Any a-projective representation of any group G was shown in $3.2 to correspond to an "honest" representation of a central extension of G,(group G, with a
s6.2.The Heisenberg group and
285
the oscillator representation
center 2 c T, so that Ga/Z E G). In fact, Heisenberg representation (2.12) arises that way (problem 6). Let us also remark that a symplectic extension (2.32) of T’ depends only on sign of A, T = T It remains to compute cocycle a. We shall see that (Y is not trivial on S d n ) , but trivializes on a finite cover of Sdn). For the sake of presentation we consider here only 1-dimensional case Sp(1) SL2(R) (see problem 7 for the general case).
*.
Proposition 3: Cocycle a on SL, is 2-valuedJ a(.) = f I , so the Corresponding central eztension of SL, f o m a 2-fold cover, called the metaplectic group Mp(1).
”=
l1 1
denote the generator of rotations S q 2 ) C SL,. It corresponds to a
quadratic Let e ement i ( p z + q z ) in the Weyl algebra W 2 , which is taken by representation T (2.31) to the harmonic oscillator,
A = ;(-a2
+ z’),
The analysis of the oscillator in the previous part showed its spectrum to consist of halfintegers: {;$:;
...}. Hence, a unitary group, generated by A via (2.31), takes orthogonal
rotations
L
into unitary operators
J
1 The resulting representation, 4+T
44)’
becomes single-valued on a 2-fold cover of S0(2),
isomorphic to Su(2), so cocycle a becomes trivial on Su(2). The corresponding 2-fold cover of SL, “resolves” (trivializes) cocycle a on a subgroup So(2),
a(u;w ) =*- ’(uw)
P(u)P(w)’
for all u;w E sq2).
If G denotes the corresponding 2-fold cover of
(2.33)
SL,, then (2.33), along with the Cartan
decomposition, g = uhv (u,w E K ; h E H) on G , yields the trivial cocycle a on the entire group G, &ED.
2.7. Symmetries and spectral multiplicities of the oscillator. The symplectic group and its spinor representation allow to explain spectral multiplicities of the multiD harmonic oscillator H = ;(-A I z I ’) in Rn. The oscillator has an obvious SO(n)-
+
symmetry, since both the Laplacian A and potential However, the multiplicity of X k = Ic,
Iz I
commute with rotations.
56.2.The Heisenberg group and the oscillator representation
286
#(Ak)
= {(i in):(il+;)+ ...+(i " + f ) = k } = ( k p ) , is much higher than could predicted, based on SO(n)-symmetry (compare the obvious cases of
W2 and W3). This suggests that H might possess a larger symmetry-group. This,
indeed, proves to be the case.
Theorem 4 The symmetry group of the oscillator H in Sp(n) coincides with the unitary group SU(n), and the restriction of S y n ) on eigenspaces is irreducible. Let us remark that both S q n ) and S w n ) are subgroups of Sp(n), the former is given by all block-diagonal matrices
Sqn)= whil: J
=
{[
u
1-
~u u = I } ,
the-latter is made of all 2 n x 2 n orthogonal matrices that commute with I
J.
We associate with any matrix A on the (Heisenberg) phase-space { ( z ; p ) } = R'", a quadratic form
c
fa(";P) = ( A z I P) =
OijziPj9
and the corresponding differential operator LA, given by any possible convention: left {ziaj); right {a,zj} or symmetric (Weyl) {i(ziaj
LA =
Coijriaj
+ ajzi)}, so
(in the left convention).
The oscillator clearly corresponds to the identity matrix,
H=:(-A+ ~
'I'
Z ~ ~ ) C * T
(2.34)
Next we observe that the natural linear action o group GL(2n) on R2" is transformed under the map A+LA to conjugation of matrices, g : A+TgAg.
On the other hand the oscillator representation assigns to any symplectic g a unitary operator T, in L2(Rn), so that
T,-'L
T -L
; g E S d n ) ; A E g42n). ('gAg) So the commutator of H consists precisely of symplectic matrices, that preserve the A g-
Euclidian inner-product form (2.34), { g : T g . g = I } , i.e. group s q 2 n ) n S d n ) = Sqn) (the maximal compact subgroup of S d n ) , see 55.7). Conversely, the commutator of S q n ) can be shown to coincide with {H}. Thus { S q n ) ) and {H} form a maximal commuting pair in the sense of 54.4-4.5 (like the Laplacian
A
on S" and the orthogonal group
S q n + l ) ) . Since the metaplectic representation is irreducible in L2(R") any pair should break it into the sum of "joint irreducible componentsn, $gk. Hence, restrictions k H 18, = lk, and S y n ) I k,-irreducible, QED.
Remark: In the differential-operator form (2.31) Lie algebra w(n) can be represented in
56.2. The Heisenberg group and the oscillator representation
287
terms of the creation-annihilation operators, { a i ; a j t } of (2.24). We set X i j = aiajt, and have the reader verify that { X i j } satisfy the commutation relations of the rujn)-basis, and commute with H . [xij;xkJ = 6 i m X k j + 6 j k X ; , ;
[ H ; X1.1. ]= 0.
After the eigenspaces of H are shown to be irreducible under Syn) the natural question arises, what are these irreducibles {r'}),in terms of their weights, as described in $5.3. It would be difficult to derive the weights directly from the L2(Wn)-realizationof metaplectic T, as the latter does not correspond to any natural (regular/induced) action of SU(n) on its quotients. As we shall see the proper way to interpret TI SU(n) is in terms of the holomorphic-induction of $6.1. This would require yet another realization of T in the complex domain, that we shall explain.
2.8. The Bagman-%gal representation. We shall conclude this section with yet another realization of irreducible representations ITA}of W, and the Weyl algebra in spaces of holomorphic functions on C and Cn, 36=I(C)={F(z):llFI12=
I
'1
IF(Z)~~~-I'I dz .
(2.35)
{z"}r
Polynomials are easily verified (using polar coordinates) to form an orthogonal system in 36, with norms, II 2" 112 = nn!.
1
So we get an orthonormal basis (&zn:n =0;1;-. . Two operators, multiplication z: F(z)+zF(z), and complex differentiation a:F+F ( z ) , are adjoint one to the other, relative to the product (2.35), and obey the Heisenberg CCR, [.;a,] = 1.
So they behave like the creation/annihilation pair { u ; u t } (2.24), while their real/imaginary parts become the position/momentum operators: p = J1( l . t a); q = +( 2-8); z = i A.
d-.
Infinitesimal representation (2.36) of Lie algebra @ could be (exponentiated) through a unitary representation of Lie group W in space %(a)),
(2.36) lifted
In fact, the equivalence of all generators and representations: 2'' on L2(R) (2.12) and U Aon X(C) (2.37), can be established via an intertwining map W:Lz+J6, (2.38) J
-00
288
56.2. The Heisenberg group and the oscillator representation
We shall leave the details to the reader (see problem 8), and just mention that Bargman-Segal representation and spaces { 36(C)} have many remarkable features. One of them is the reproducing kernel { K ( z ; w ) }on CxC, which gives the value of F E f16 at any point z in terms of integrals over C,
F ( z ) = (K(z; ...) 1 J'(...)),2
K(a;w)F(w)e-
I I
2
d2w. C The reproducing kernel on space X(C) is equal to K(z;w) = ex* ' iij. =
Finally, we can go back to the decomposition problem for the metaplectic T , restricted on subgroup SU(n). Let us observe that SU(n) acts on C" N W2" in the natural way, by unitary linear maps, z+z". Hence, its action on space 36 consists of coordinate transformations,
T,: F(z)+F(z"), on polynomials { F ( z ) } . Space of polynomials 9 is made of homogeneous components of various degrees,
each 9k identified with symmetric tensors, and T 19, is precisely the k-th symmetric tensor power of the natural representation x in C". So it has signature (weight) a = (k;O; ...0) (see 55.3)! Further results and details could be found in [How]; [Ta2]; [GSl].
289
56.2.The Heisenberg group and the oscillator representation Problems and Exercises: 1. i) Show that any Z-valued 2-cocycle 4(a; b) on group G defines a central extension H of G by Z, via multiplication formula: (2; 1 ) .(y; s) = (zy; t+s+$(z; y)), 2, y E G; t , s E Z. ii) The extension H is trivial, iff the cocycle 4 is trivial: 4(a;b) = +(a) -+(cab) ++(a) = B+(a;b),for some $(a); iii) Two extensions H,; H,, defined by cocycles 9,; 42 are equivalent (isomorphic), iff 4, and d2 differ by a trivial cocycle a+ (Hint: construct an isomorphism u:H1+H2, via map, (a;O)+(a; +(a)). iv) Apply the above results to the Heisenberg group to show that cocycle (2.2) is nontrivial, and two cocycles: 4 , ( z ; z') = z y'; q&(z; w ) = 3 ( z E ). Hence two multiplication formulae are equivalent.
-
2. Show that the Heisenberg algebra has no nontrivial finitedimensional representations (Hint: commutator [Tx;Ty]has trace 0, so it cannot be X I with X # O!). 3. Apply the Mackey's groupextension theory for semi-direct products to derive all irreducible representations (2.11)-(2.12) of H,.
c WP+q-', for the Weyl algebra (Do it first for generators { z a ; a } of W, starting with the basic relations [ti$,] = 6 i j ) .
4. (i) Check the commutati n relation [Wp;W.]
B
(ii) conclude that the first-degree part W' is isomorphic to the Heisenberg algebra via map, P = a . a + b . z +c+(a; b ; c ) E 8, (a;b E R"; c E R).
s,,
(ii) Show that the 2-nd order operators of the form:
P = C a 13. .a?. + C bijziaj + C c i j t i z j = A a . a + B Z .a + Cz.t, :I
-. aE
with symmetric matrices { A ; B } and an arbitrary C E gqn), form a symplectic Lie algebra via identification:
C
B
'dn)*
5. Show that (2.29) defines an automorp ism of ,, that preserves its center, and any such automorphism is given by an element of Sdn). 6. Show that representation TX(2.12) of H, comes from a projective representation of the commutative group R2" N C", defined by the cocycle
a(z;w ) = e iX3(r.
= eiX(z * Y' - Y '"); where = t+iy;
7. Show that metaplectic group
-
= z'+iy'.
Mdn) is a 2"-fold cover of Sdn), with center ZN
Use commuting oscillators: A 3. = $(-a:+z:);
z2x
...x z 2
j = 1; ...n, in the Weyl algebra W,.
8. i) Check that map (2.38) intertwines representations TXand U x of H, in spaces L2(R") and X(C") (use Lie algebra generators). ii) Find the unitary group ( S q 2 ) c Sdl)), generated by the oscillator $(p2 %realization of the metaplectic extension of TX,and show that
T : ~ ) F ( ~=)eidI2F(zei$.
+ q 2 ) in the (2.39)
iii) Show that the standard Fourier transform T:L2(R)+L2(R) corresponds to 4 = (2.39).
4in
56.3.The Kirillov orbit
290
method
s6.3. The Kirillov orbit method. Many results of the representation theory of H,, outlined in the preceding section can be extended to arbitrary nilpotent, solvable, exponential and more general Lie groups [Kirl,Z], [Pu]. The key idea, crystallized in the work of Kirillov, Souriau, Kostant et al., w a to associate irreducible representations of G to orbits 0 of the co-adjoint action of G in the dual space (5' to Lie algebra (s. We shall outline the construction of representations To,then derive the character formulae, and the Plancherel measure, based on the orbit method.
3.1. Construction of representations 9. A co-adjoint orbit 0 c 8*always carries a natural sympkctic structure, nonsingular skew-symmetric bilinear form on tangent spaces: B = B 6 ( X ; Y ) ;X , Y E T6 - tangent vectors at point t E 0. Equivalently, there exists a nonsingular differential %form:
R = RB = Cbijdt' A d t j (in local coordinates { ( j ) on 0). The latter defines a Poisson-Lie bracket on functions (observables)
(3.1)
{f} on
0 (see
chapter 8):
So the space of smooth functions em(0) turns into an m-dimensional Lie algebra. The tangent space of 0 at point {t}is identified with the quotient 8/8(- Lie algebra modulo the stabilizer of t, G6= { Y : a d ; ( t ) = 0). By definition form B, at a point
t E 0 is equal to B&X;Y )= ( [ X Y , ]I t),for any pair of elements X , Y E 8.
(3.2)
Clearly, B6 depends only on the images (projections) of vectors X,Y on the We shall list a few basic properties of form B: tangent space T6= (5/8(.
B6 annihilates X , Y E G6,so it depends only on classes of X , Y in the tangent space T6 8/8( 0
0
B 6 is nonsingular on TC(symplectic)
0
2-form RB (3.1) is closed, i.e. differential
don,7
1 0, fibered over the circle S, = { I ( I = r } , and l-parameter family of degenerate (O-D) orbits { (s;0): s E W} (fig.5). The canonical 2-form on cylinders 0, is
of radius r
0, = rdd A ds. No quantization condition (3.5) arises here, as $Qr = 0, for all 2-cycles y c 0 7 (there are no nontrivial 2-cycles!). The reduced space (base) 9J-T has fundamental 1 )Z, whose characters (dual group) form a torus r = T. group 'I = ~ ~ ( 9 = We fix a point ( = ( 0 ; t )€ 0 , and find an admissible subalgebra $ = W 2 of (. According to the general theory, character x t ( u ) = eY ' a (u E$j), extends through a representation of the subgroup H = ezp$,
in f-different ways, labeled by angle
q5 E [0;27r]. So we get a 2-parameter family of characters { ~ ~ ; >~ O;q5 : r E [ 0 ; 2 ~ ] }of the admissible subgroup H , that induce irreducible representations: T r ; 4 = i n d ( ~ , . ;I ~H ;G ) . The appearance of a second parameter
4 seems to contradict our previous description of
the dual object of M, in J6.1. There we have shown M, to consist of a l-parameter family
{T'}, plus a discrete (quantized) set of characters IT"' = eims} of the quotient So(2). The contradiction is resolved by remembering that the general results (Theorem 1) apply N
to a simply connected cover M , which represents a central extension of M, via group I . The condition Tr;$I I = Z (trivial) selects a single member T' of the family {TrT9:4}.
56.3. The Kirillov orbit method
296
The quantization of degenerate (one-point) orbits {(s;O):s = 2x772) results from compactness of the admissible quotient H / [ H ; H ]N Sq2),H = MI,. Let us remark that all higher-D motion groups En (n 2 3) are themselves simply connected, so the quantization rules of Theorem 1 apply directly to them (problem 2). Co-orbits of examples 3.1-3.2 had real polarizations { H } , so the resulting irreducible representations {To} were obtained by the usual induction procedure. Our next example, compact group SU(2), lies at the opposite extreme. Its irreducible representations are all finite-dimensional (chapters 4-5), so they can not be induced in the usual sense. However, we shall see all of them realized as "holomorphically induced representations", according to the Borel-Weil-Bott prescription of 56.1. 3.4. Quantization of SU(2) and the Borel-Weil-Bott Theorem. Here
(5
= @*
N
R3
and co-adjoint orbits are spheres 0, = {tr(X2)= -?}
N
SU(Z)/U(1).
We identify each orbit with a complex projective space CP' via the map
Iu wl-+
G 3 -~i7 where matrices
{[' *]:
(u;w)/{(e"u; e"w)}
N
C x C/mod(C);
X = e"} make up the diagonal subgroup U(1) c SU(2).
The projective space ONCP' has standard homogeneous coordinate z in two neighborhoods 0 *, that cover it: z=$inO+={(u;w):w#O}, and z = $ i n O - =
{(u;w):u#O}.
Furthermore, space 0 is equipped with a family of symplectic structures which come from different orbits { O r } ,
One can show that R, is the only G-invariant 2-form on CP' (problem 3). The argument exploits the G-action on CP' by fractional-linear transformations:
Parameter (Y is simply related to the radius r of 0. The quantization condition of Theorem 1 requires (Y to be an integer m. The corresponding representation space is formed by sections of the holomorphic line bundle L, whose l-st Chern class is given by 0,. Following the general prescription of geometric quantization we express form R (3.9) as the exterior derivative of a l-form12 w = w (on each hemisphere 0 ). So we
297
56.3. The Kirillov orbit method write R = dw+ = dw-. The difference w+
- w- = d4 = 1 'dg, 2ri ,
where g = g(z) is a holomorphic transition function on the overlap 0+n0- N C\{O}. But any holomorphic nonvanishing function g ( z ) in C\{O} can be written as f+(z)z"f-(!), for some integer m, with holomorphic functions f* on C. This shows that any holomorphic line-bundle over CP' is equivalent to the one with transition function g = z". Taking such g(z,) we calculate 2-form R, on the 2-cycle 0 N S*, and find dw- =
-
4
g - ' d g = m.
equator
So the requirement that L have a proper line-bundle structure sets the integral over 0 (cohomology class) of R to an integral value rn! Holomorphic sections of L consist of functions {f,} transition function g: f+(Z+)
on 0, related by the
= g(z+)f-(z-) = Z"f-(Z-).
So if one asks both functions be analytic, both must be polynomials in z of degree m. Thus we get (mt1)-dimensional space of polynomials. To establish the Borel-WeilBott Theorem for SU(2) we only need to compute the SU(2)-action on sections of L. Remembering the fractional-linear action of G on CP', we get the familiar form irreducible representations of chapter 4 (§4.2),
Remark The above construction of irreducible representations, based on co-orbits in 0*, is a special case of the more general Geomefric quantization procedure on an arbitrary symplectic manifold (classical-mechanical phase-space) 9.The latter is typically equipped with a canonical 2-form, and a Poisson-Lie algebra
(9
of so-called primary obseruables,
functions of 9. In some cases, e.g. Lie group G = e z p ( 0 ) acting transitively on 9,the phase-space can be identified with a co-orbit 9 N 0 c 0*, via the momentum-map (see 8.1), so the above theory applies here. We refer to [Kirl]; [Kos]; [Woo]; [Hurl; [Sni] for
further details.
w can be interpreted as a connection form of a line-bundle L over CP'. An easy -Form computation shows, m Zdz * z t = 2*i(l+ 27 ); =
*
56.3. The Kirillov orbit method
298
3.5. The character formula and the Plancherel measure. We shall conclude this chapter with a universal character- and Plancherel-formula on Lie group G in terms of the orbit structure of 6*,due to A. Kirillov. These results apply to large classes of groups and representations, including compact Lie groups; SL,(W); exponential groups'' G (whose adX-map has no purely imaginary eigenvalues, so ezp:O-+Gbecomes a diffeomorphism); principal series representations of noncompact semisimple groups, and many more. Characters {xT = tr(T)} of irreducible representations of G are given by certain conjugate-invariant distributions on G , x ( g - ' s g ) = ~ ( z for ) , all z , g E G. We pick a pair of neighborhoods U c G and V c (5, related by the exponential map, ezp:V-U, and define an invariant distribution associated to an orbit 0 c 6*, ($0
If ) = J
J
0 u
f ( e z p X ) e i ( X I 4 I X dP&;
(3.10)
for any test-function f on 6. Here d X denotes the usual Lebesgue measure on (5, while /?= Po is a 2n-form (volume element) on 0,obtained by taking n = g i m 0 wedgefactors of 0 = nu,
p=$
(3.11)
flA...Afl *-
So distribution q50 can be thought of as the Fourier transform '3t+x of an adGinvariant measure d p on 0, pulled back to the group by the ezp-map. Since exp respects adjoint actionJconjugation on G and 6 , g-'(ezpX)g = ezp(adgX), the resulting distribution do is clearly conjugate-invariant. Distribution related to character x of representation T 0.
do is closely
Proposition: There ezists a conjugate-invariant function p = p0 on G, equal 1 at the identity {e}, and different from zero o n U c G, so that 1x0 = &4ul
(3.12)
Clearly (3.12) is equivalent to (3.13) The correspondence between characters {xu}, given by (3.13), and representations {To}, based on co-orbits, has not been proven yet in a complete generality, although the result is believed to be true. "This class includes fairly many nilpotent and solvable groups, affine, Heisenberg, etc.
56.3.The Kirillov orbit
299
method
Let us remark that formula (3.12) reveals the nature of distribution xo, that is closely connected with the geometry of orbit 0. Thus compactness of 0 (e.g. compact G) implies that xo is a regular function (hence To- finite-dimensional!). If 0 is a cylinder (example 2), so 0 is made of subspaces {to L}, then distribution xo contains &type factors. A particular case: L = 8 - annihilator of a subalgebra 8 c (5, implies that xo is supported on a subgroup H = e x p 8 c G.
+
For orbits of maximal dimension density pa in the denominator (3.13)proves to be a universal function: sinh(tl2) -
p ( e z p X ) = detF(adX);where F ( t ) = --
tl2
7O0
(t/2)2k
(3.14)
Next we proceed to the Plancherel formula for Lie groups. We remind that the Plancherel measure d p on any unimodular14 group has the form
(3.15) for a suitable class of test-functions {f}on G, e.g. f E L' nL2(G). One could understand (3.15) as a decomposition of &function on G into the the sum/integral of irreducible characters,
f ( e ) = /;r(Tt)dP(T);
or 6 ( e ) =
J G_XTdP(T).
(3.16)
In case when the character-formula (3.12) holds the meaning of (3.16) becomes transparent: it gives a decomposition of the Lebesgue measure on 0' into canonical measures on orbits. Indeed, let us assume that there exists a 1-1 correspondence between orbits and representations, and that the orbits of maximal dimension have the universal density function po of (3.14). The Lebesgue measure d( on @* is clearly adkinvariant, hence it can be decomposed (resolved) in the sum/integral of the canonical measures
{Po} (3.11)
on orbits,
(3.17)
Fourier-transforming (3.17) we get
G is called unimodular, if the right- and left-invariant Haar measures on G are equal. -Group Most examples considered in the book are unimodular: all compact groups; simple and semisimple noncompact groups; nilpotent (Heisenberg) groups; motion groups, etc.
56.3. The Kirillov orbit method
300
Now we apply the character-formula (3.12), remembering that p g ( e ) = 1, and find 6(z) =
J
x&)
dP(%
orbit-space which yields the requisite decomposition (3.16).
To compute dp explicitly we need a set of adG-invariant coordinates on 6*.Let us assume that maximal-D orbits are joint level-sets of the family of functions Xl;...Xk (k = codima), so {Aj} parametrize the orbit-space. Let us also introduce coordinates on each orbit 0 (2n=dimO). The orbit-space 4 = 6 * / a d f = can then be identified with the level-set: cpl = ... = cpZn = 0. To proceed further we need a notion of Pfaffian of a skew-symmetric matrix A = (akj). Any such A corresponds in a 1-1 way to an exterior %form: w A = z a k j e kA ej. The n-th exterior power of wA is a constant multiple of the only highest rank 2n-form, W AA . . . A W A= Ce, A...At+,,.
Constant C, represents a polynomial of degree n in variables {akj}, called the Pfaffian of A, and denoted by PfA. Pfaffian has many interesting properties, for instance, (PfA)’ = detA (problem 4). Returning to the Plancherel measure we can state the following general result. Let ((A) be a point on an orbit 0 with coordinates {Al;...Ak}. We denote by A the matrix of Poisson brackets at ((A), aij = {pi;pj}(t(’))*
Theorem 3: The Plancherel measure on group G is given by the formula Idp(A) = J(A;O)PfA(X)dA,...dXh1
(3.18)
where J(A;cp)is the Jacobian of the coordinate change: (-i(X;cp),
Formula (3.18) takes on a particularly simple form, when coordinate functions {Xi;cpj} are linear, i.e. elements of algebra 6 . Then
CC~’,;
where
{cc}
aij(X) = (((A) I [ c ~ i ; ’ ~= jl) are structure constants of the algebra 0.Now the Plancherel memure turns
$6.3, The Kirillov orbit method
301
Idp = P(A)dA,..A,
(3.19)
into
with a homogeneous polynomial P(A) of degree n = gimO(A) in {Aj}, with coefficients depending on the structure constants of 0. It is worth to remark that formal application (3.19) yields the correct Plancherel measure for complex semisimple groups (e.g. SL,(C)), and compact Lie groups. In the latter case integration in variables A’s must be replaced by summation over the discrete set of the “properly (Borel-Weil-Bott) quantized” orbits (problem 5)! In the real semisimple case (e.g. SL,(R)) it predicts the correct answer only for the so called discrete-series representations, as we shall demonstrate in the next chapter.
56.3.The Kirillov orbit method
302
Problems and Exercises. 1. Show that closedness of the canonical 2-form Sag (3.3) on a co-adjoint orbit 0 C (9' follows from the Jacobi identity for the Lie bracket on (9. i) Use the general formula for differential of a 2-form Sa on a manifold A given any triple of (tangent) vector fields X ;Y ;Z on A,
+
+
dR(X;Y ;Z ) = X [ R ( Y ;Z ) ] - Y [ R ( X Y ; ) ] Z[Sa(X;Y ) ]
+ R ( [ X Y ] ; Z ) - R ( [ X ; Z ] ; +YR ) ([Y;Z];X); Here X[. .I means (Lie) derivative of function (e.g. R ( Y ; Z ) )along vector field X , and [ X ;Y ] , etc. - Lie commutators of vector fields. ii) Consider functions { f( 2), and form a conjugate pair of unit complex numbers { e *id} for elliptic elements g ( t r g < 2). Since characters are conjugate-invariant functions on G, one expects them to depend on
{A$ '} only. Indeed,
Theorem 1: (i) T h e character
xs
of the principal series representation TSf is
equal t o e (Ag); for hyperbolic g for elliptic g
(ii) T h e character of the discrete series representation T
is equal t o
(iii) The complementary series character x p looks like the principal, with the real parameter p E [-1;1]in place of purely imaginary is, so
(Ag); for hyperbolic g
e
Xp(d =
for elliptic g
Proof: (i) The principal series operator T;* can be thought of as a distributional kernel on W of the form,
K g ( z ; y )= I bx+d
I
f
( . . . )az+c b ( m- 9).
So its trace is formally given by trT;* where a(t)denotes
I t I , e+(t)
= I(aiS-'ef)(bx+d)s(s9-
x)dx.
= 1, e - ( t ) = s g n t.
We apply the general change of variable formula for the &function to integral (24,
In our case, @(x) = sg - 2, so @ has real zeros (fixed points of g:z+sg) only for hyperbolic 9,and these are -d f Xg
51,z
=b .
Evaluating integrand ais-'e = I bx+d
I -,
and derivative
@' = (bx+d)-2
- 1 at
5 7.2.
we get ( b z t d )= *Ag,
z=
315
Characters of irreducible representations.
and @'(z)=
As2- 1. Substituting
the latter in (2.2)
yields the first character formula. (ii) The proof is more involved for the discrete series, due to the fact that the latter were realized in spaces of holomorphic functions on P or D (rather than L2(R)), so trace formulae of type (2.2) are not available here. The relevant formulae involve more complicated reproducing kernels. However, for elliptic g we can easily compute using the conformal SU(l,1 version of operators T t n , h =
[i
Sh(W),acting on %,(D), and
x*,
the fact that the
are diagonalized in the basis { z k } p = ;, with eigenvalues
{ f (2k+n+l)}. It follows immediately then that
To compute the hyperbolic part of the trace formulae we shall use yet another realization of T*" in L2-spaces on the half-line, L2(W+;t-"dt). Observe that the Fourier/Laplace transform 00
~ : f ( t ) - /f(t)eitzdt = ~ ( z ) , 0
takes unitarily L2(R+; P d t ) into holomorphic functions in the upper half-plane
P = {Zmz> 0}, square-integrable with weight y"-*dydz. Indeed, F(z+iy) = '3(e-tYf), so by the W"-Plancherel formula, 00
I f ( t ) I 2e-2Ytdt = 27r
6
J
I F(z+iy) I 2dz.
-00
Multiplying both sides with yn-ldy and integrating from 0 to
L H S = -r(n) p-11fIk2(t-ndt)
00,
we get
I F ( z )I 2yn-1dydz = 2?rIIFIf36,'
= RHS = 2.1
We shall transform the representation operators T * , from spaces 36,
to
L2(W+;...) by conjugating with 4. Since the inverse transform 9-1: F-
k
I
am2
e-iztF(z)dz,
= yo
we find that the transformed operators are given by integral kernels
Kn(t;v)=
i(r)zg-€z)
dz
(2.3) (bz+d)"+" It is convenient to change variables: z-t = b z t d . Then we compute: z = dz = zg - ( z = d() - (t r) ( t ) ] , and after substitution in (2.3), kernel K , takes
a; b v
i[(q+
the form,
Kn(t;v)= &e
y;
+
i(v)l
Smt = to
-i(r)lt + €t)/b&
p+l'
316
3 7.2.
Characters of irreducible representations.
Another change of variable, t+t-', brings the integral over the line Imt = ... into the integral over the circle C = { I t+i I = i} Kn(t
;rl
- h i ( - ) / b ,f ,-i(t~+C/t)lbtn-ldt,
) -2nb
C
+
Remembering, that exponential {t' - (a+d)t 1) in characteristic polynomial of g, the inner integral in (2.4) yields
(2.4) represents
a
b bt i [ ( t + l / t ) - (a+d)l= (t-Ag)(t-A;l).
Thus we derive the following trace-formula
The contour integral (2.5) is easily evaluated: for hyperbolic g the smallest of two eigenvalues (AS' < 1) is inside the circle. Taking the appropriate residue in (2.5) we finally get
and similar derivation applies for negative n, QED.
57.3.The Plancherel formula for SL,(W).
317
$7.3.The Plancherd formula for SL,(R). In this section we shall derive the Plancherel/inversion formula for group G = SL,(R), and get the direct-integral decomposition (see 56.1) for the regular representation of G on L*(G), into the sum of primary components. An interesting feature of SL, come8 from the fact that its Plancherel formula combines both the continuous (principal series), and discrete (discrete series) contributions, 80 SL, behaves like compact and noncompact group at once. We find the explicit Plancherel measure on SL,, including the density of the continuous branch, and multiplicities of discrete components.
3.1. The general noncommutative Plancherel formula was discussed in 56.1. It extends the classical Fourier-inversion formula (§2.1),
jG
f(0)= _?(E)dE,
(3.1)
d[ - properly normalized Haax measure on G,and yields a similar representation for any function f E eF(G), in terms of irreducible characters on G.Namely,
where (xT I f ) = trf^(r),and
I(*) =
L
f(z)r;'dz.
Integration in (3.2) extends over the (unitary) dual object 6 of G, and d p ( n ) denotes the Plancherel meaure of G. From the inversion formula (3.2) one easily derives the Plancherel Theorem for L2-functions on G. Indeed, applying (3.2) to a convolution f*f*,where f*(z) = f(z-'), the integrand becomes
)I l H S
tr[f^(r)3(ir)*] = ?(r)
2
(Hilbert-Schmidt norm)
so (3.2) turns into
(3.3) Let us also remark that inversion formula (3.2) at a particular group element go = e yields values f(g) at all other points of G
1-
(3.4)
Finally, from (3.2) and (3.3) we get a decomposition of the regular representation on L2(G)into the direct integral of irreducibles,
s7.3.The Plancherel formula for SL,(R).
318
(3.5) with multiplicity (finite or infinite) equal to the degree d ( r ) . Our goal here is to prove formula (3.2), and to compute the Plancherel measure dp for SL,(R).
*
*
Plancherel Theorem: Let {x, }, denote the principal and and {x =:}" discrete series irreducible characters of SL2yW). Then for any function f E CF(G) one has
Formula (3.6) shows that the Plancherel measure d p is supported on the union of the principle parts G,+UG,(G,* N R), and the discrete part G, N Z\{O},of the dual A
object
h
A
A
6, and is equal to
In the rest of this section we shall outline the proof of the Plancherel formula for
SL,(R).
3.2. Decompositions of G, Conjugacy classes, and the Haar measure. There different ways to decompose G, i.e. choose coordinates in G. Typically they are defined in terms of three special subgroups of G: maximal compact group
{
I{= u = u diagonal (Cartan) group
A={.=
.,I
cossin* 8 - [-sin
1
0 0) for SL,, or (f(0): 0 5 0 < r}, for SY2), then writing the convolution formulae for reduced functions (The general argument based on Cartan automorphism 0 for pairs G,K was outlined in problem 1 of f5.7). 2. Show that element A (4.5) commutes with all B E 8, hence all of N(0) (check it for the Cartan basis {W;X;Y}). Furthermore, the center of 9l consists of polynomials in A: f(A) = C a k A k . (The proof is somewhat involved: each element A E can be uniquely represented by a polynomial p = xakmnWkXmY"in 3 (noncommuting!) variables {w;x;Y} (in the given order). One has to write the commutation relations: ad&) = 0; adx(P)= O;ady(P)= 0 3. i) Show that a bi-invariant metric ( ( 1 ~ ) ~ on Lie group G , restricted on the tangent space a t {e}, T,(G) N 8, is invariant under the adjoint action,
(Ad,(
I Ad,9)
= (( I v), for all
LV E 0,g E G.
Conversely, any Adg-invariant metric on 8 2: T,(G) extends to a bi-invariant metric on
G. ii) All Ad-invariant products can be easily described for semisimple Lie algebras. Killing form, (X I Y)o, is one of them; for simple 8 Killing is the only one; if 0 = @ 8,- direct sum of simple ideals, and ( I ) denotes the Killing form on a, then any Ad-invariant form on 8, is given by fa,( I), (Hint: any bilinear form is obtained from a nondegenerate Killing form by an operator B, (X 1 Y)= ( B ( X )1 Y);form ( I ) is Adinvariant iff operator B commutes with Ad,!). 4. Verify directly that Laplacian (4.7) on H is invariant under all fractional-linear transformations g : z s , in SL,. What happens to A under the GL2-action?
5. Compute the infinitesimal characters of Proposition 3.
87.5. Selberg trace formula.
334
$7.5. Selberg trace fbrmuh. In this section we shall study the regular and induced representations R of group G = SL, on compact quotient-spaces Ab = G / r , where r is a discrete subgroup of G. We shall show that representation R breaks into the diserete sum of irreducibles, each entering R with a finite multiplicity. We prove a reciprocity Theorem (similar to Frobenius reciprocity of chapter 3), and establish the Trace-formula for operators R, ( f E €$') - a noncommutative analog of the Poisson summation. Then we shall proceed to evaluate the contribution of various uparts of f" to the traceformula. This eventually lead us to the celebrated Selberg trace formula on compact quotient-spaces of SL,. After proving the general result we shall outline its ramifications and special e88e8, and find interesting connections to spectral theory of Laplacians on hyperbolic Riemann surfaces H/r, Poincare half-plane modulo a discrete (Kleinian) subgroup r c SL,. We also give an application of the representation theory of SL, to geodesic flows on negatively curved Riemann surfaces.
Let
r
5.1. Induced representations on G / E reciprocity and the general trace-formula. be a discrete subgroup of G , a be a representation of We denote by
r.
R = Ra =ind(aI r ; G ) - an induced representation of G . It is an interesting and challenging problem to find a decomposition of Ra into irreducible components of G . A simple commutative prototype of such setup consists of a lattice I' c Wn (rN Z"), and a representation/character 4 7 ) = cia*-', y E r. The induced representation Ra acts on scalar/vector functions {f(z) on W"}, that satisfy the a-Floquet condition (a generalization of the periodic condition), f(z t 7)= eZa 'Yf(z); for d l z E
w"; 7 E r.
Representation R" is easily seen to decompose into the direct sum of characters,
(5.1) sum over the dual lattice
I"', in a special periodic case R, 2 @ , i m * z,. 2 E W".
rn E r' Both results are simple corollaries of the Poisson summation formula of $2.1. In
particular we get spectrum of the Laplacian A, differential operator) on
(or any other constant-coefficient
W"/rN T", with periodic/Floquet boundary conditions, Spec(A,) = {A, = (rntcr),: m E r'}.
(5.2)
Our main goal in this section is to derive a noncommutative version of (5.1) for quotients SL,/r, and then to link such decomposition to spectral theory of Laplacians
27.5. Selberx trace formula. on spaces
W/r. Throughout this section quotient-spaces
335
J b = G / F will be assumed
r
compact. The corresponding are called co-compact (or uniform lattices), and SL, can be shown to have a plenty of those. Uniform lattices are characterized by the following properties (see [GGP], chapter 1): i) r has finitely many generators {71;...7m},and a finite number of relations among them; ii) the G-conjugacy class {g-l7g:g E G} of any 7 E r is closed in G; iii) l' contains only elliptic and hyperbolic elements.
A hyperbolic element 7 is called primitive, if y # y?, for some yoE T,and k > 1. Each elliptic element 7 has a finite order ym = e. Let us remark that (iii) easily follows from (ii), since a conjugacy class C ( 7 )= {g-lyg} of any parabolic element 7 contains limit points fI, not in C(y). So parabolic C can not be a closed subset.
r
First we shall state the general trace formula for an arbitrary pair c G, of a (locally compact) group G, and discrete subgroup T,with compact quotient G / r . Let Ra be induced by a finite-D representation a! of r. Operators {RF}are given by integral kernels,
q"; Y) = c f(gz17gy)47); 7Ef
where gz;gy E G are coset representatives of points { z ; ~ }in A. Clearly, compactly supported (or rapidly convergent) functions {f} give rise to compact operators {Rf}.In fact, operators {Rf}belong in the trace class (see Appendix B), and (5.3) X X On the other hand, any representation T , which contains a compact operator { T f }has "discrete spectrum", in the sense that T is decomposed into the direct sum (vs. direct integral) of irreducible components:
T N TTs@m(s), (5.4) each one entering T with a finite multiplicity {m(s)}(problem 1). If xs denotes the character of T S ,then combining (5.4) with (5.3) we get the general trace-formula,
57.5. Selberg trace formula.
336
for a suitable class functions f on G (e.g. e$). In what follows, however, it would be more convenient to write the RHS of (5.5) as
where T,;G, are stabilizers (centralizers) of an element 7 in r and G, respectively, and d g denotes the invariant (Haar)measure on the quotient space G,\G. 5.2. Tram-formula on SL,. Our main goal is to compute (5.6) explicitly for group
SL,(W), then to deduce all possible information about the LHS of (5.5): irreducible components (2') of Ra,as well as their multiplicities. To state the general result we remind the reader 3 character formulae for irreducible representations of SL,, obtained in '$7.3. All three X > 1, for hyperbolic g
-
terms of the eigenvalue {A = X ( g ) } ; and complex, X = eie, for elliptic g
-
complementary: x s ( g ) =
I i s t I I -3. Ix-x-ll
-.A-m
COSS sin0
(5.7)
'
hyperbolic g
.
m
L. hyperbolic g
discrete: x S m ( g ) =
-.
elliptic g
elliptic g
Given a function f E CT(G),we introduced its generalized Fourier transform/ coefficients {T(s f ); T(s); fn)}, by integration of f against irreducible characters,
T(
j(s)
-
= / f ( g ) x , ( g ) d g = tl.(Tj);s = 2sf;s; f n ,
so ?(is+> = f ( g ) x i s + ( g ) d g , etc*
Following [GGP]we have to compute the contribution of different conjugacy classes of to tr (RY).Those are conveniently divided into 4 groups: hyperbolic classes, elliptic classes, and 2 special elements { e } and { - e } . Contribution of hyperbolic elements. We shall split all hyperbolic conjugacy classes y = {g-lyg} c T,into primitive classes Pr, = {S :7 # yi}, and their iterates (9 : y = 7;; 7,-primitive}. We have to evaluate each term of (5.6) for a hyperbolic 0
$7.5. Selberg trace formula.
337
-[
element y X-l]. The RHS of (5.6) involves the conjugacy-class H-transform of f , which was essentially evaluated in $7.4.Namely,
I
W
JD\G
f(s--’rs)ds= FA(r)= 4r I 1 - A-l I -oo { T(is+)+ &-)
sgn A}
I A I %s,
(5.9)
in other words the hyperbolic H-transform of f is equal to the inverse Mellin transform (52.3) of the even and odd principal series generalized Fourier transfom of f. Next we compute volume of the quotient-space A
wol(ry\Gy) =
Jq= In A,
(5.10)
1
since G, I IW (multiplicative group of reals), while r, = {ym: m E Z} forms a discrete lattice in G,. Combining (5.10) and (5.9) we come up with the total hyperbolic contribution to the trace,
Here the outer summation extends over all hyperbolic primitive classes {y }, while the inner consists of their iterates {y ’}); symbol 1; if a(-.) = I € a = { -1; if a(-.) = -I‘ Note that hyperbolic terms involve only the principal-series characters.
contribution of elliptic elements. We consider the set of primitive elliptic elements, and notice that each of them has a finite even orde2: y2m= e. We need to evaluate an elliptic H-transform of f in the RHS of (5.6). For any elliptic element u = u * - [ case sine -sine
case
].
do
I&+
-do
ch (e -
T)-?(is-)
Chy
sh (e - y )
shy
The first half of formula (5.12) (sum) consists of the discrete series “generalized Fourier coefficients” of f,while the second half (integral) is made of two principal series “generalized Fourier transforms”. The derivation essentially follows arguments of 57.4, where (5.12) was established at 6 = 0 (problem 2 provides further details). ’Indeed, if y had an odd order p , then -7 would have an even order 2 p , so y = (-y)p+’ not be primitive. Hence each elliptic y is equivalent to a rotation by angle 0 =
5.
could
$7.5. Selberg trace formula.
338
The volume of r7\G7 is easily found to be &, where y 2 m = e (7-primitive). Indeed, G , = Sq2)N [0;2r],while ry makes a finite subgroup of order 2m in Gy. Combining contribution of all elliptic classes, we get
(5.13) The outer summation extends over all conjugacy classes of primitive elliptic elements {y }, the inner over all iterates (7': 1 5 k 5 m-1}, from k=l through $ x order of y.
contribution of elements {e} and {-e}. Clearly, terms of (5.5), which correspond to { fe} are
4 a )v o V \ G ) { f ( e ) + d ( - e ) } , where symbol
1; if a(-.) = I c = -1;if a(-e) = -I*
{
It remains to express f( fe) (or &functions at { fe } ) , through irreducible characters (generalized 9-transforms) on G. But those are precisely the Plancherel formulae obtained in the previous section
A similar relation holds for class {-e} f(-e) =
-$jj=1 (-1)n-'n(T(tn)
t ?(-.mI)
+;I
t (l+c))(is+)sth
-m
We shall now summarize all 4 contributions into the final result. Selberg trace formula: Let r be a uniform lattice in G = SL,, a ( y ) - a finite-D representation of r of degree d(cr), and R = R" - the induced representation ind(a1 r ; G ) of G. Representation R is decomposed into the direct sum of irreducibles (principal, complementary and discrete series), each entering R with a finite multiplicity, ~ N e ~ ' ~ k * gT '~ ~@ &Ne ,.C B T * ~ B N $ . (5.14) k
-
j
d
principal
complementary
discrete
Operators { R f :f E eF(G)} belong to the trace-class, and trace of
Rf can be
67.5. Selbern trace formula.
339
ezpanded an two different ways, according to (5.5), the LHS being
c
f
k
+c j
?(sj)N8i
+ c ?( f
(5.15)
;
while the RHS is made of contributions of variow conjugacy-classes of G,
00
+ i/ (l+c)J(is+)
s th
y + (l-~))(is-)
scth?
-00
The first line of (5.16) contains the hyperbolic contributions (7 E Conjh(G)), lines 2-3 elliptic part (7 E Conj,,,(G)), lines 4-5 come from the classes { fe}. Looking at a cumbersome 5-line expression one can't help wondering about its meaning and utility? Here we shall give some partial answers to this daunting question, derive a few Corollaries of (5.16), and then mention some interesting connections of the Selberg-trace formula to spectral theory of Laplacians on hyperbolic surfaces.
5.3. Discrete series multiplicities. The first simple consequence of Selberg is the formula for multiplicities of irreducible components of Ra. For discrete series {T "} those are precisely the coefficients N , f of {)( fn ) } in (5.15). Hence, N;t = ('i(""l€)(dla)~~~(r\Crn - m-1 c7e*n= tra(-,') . r k ], (5.17)
*
k=14ktSanm
summation over elliptic classes
{7 },
and { fe}, as only those carry the discrete-series
components. A similar relation holds for N;. In special cases, when discrete subgroup has no elliptic elements, formula (5.17) simplifies to d (a)vol(r\c) Nn+ = N , = (l+(-l)n-b) "* n. The reason for the splitting of the discrete-series components in both sides of the trace formula (5.15)-(5.16) is fairly general and simple. It has to do with orthogonality
I'
57.5. Selberg trace formula.
340
properties of the discreteseries matrix entries and characters. Roughly speaking they behave like the characters on compact group (see. chapter 3). The latter, as we already know, have a unique decomposition for any conjugate-invariant function q4 on compact G, q5
=
asxs =
b
s
X
s for~ all irreducible ~ ~ {s),
due to their completeness and orthogonality in L2(G). For non-compact G , like SL,, characters (even discrete-series) are no more L2-functions. However, discrete-series matrixentries f ( g ) = ( T i $ I + ) do belong in L2, {T”} being embedded in the regular representation R I L2(G). Moreover, one can show that entry f can be chosen in space
L1(G), e.g. the highest/lowest entry fo = (T:+o I $o) (problem 3). Such f can then be paired with any (principal/complementary) matrix entry fl(g), the result being
(f I f J = 0, for all fl E SP4(T811 111)). Hence, operators T i = 0, for all principal/complementary series {Ts), as well as their traces,
I
tr T ; = f ( s ) x s o d s = 0. The latter provides the requisite orthogonality relation between the discrete and other series matrix entries and characters. It remains now to insert such f in both sides (5.15)-
(5.16) of the Selberg trace-formula, and observe that all non-discrete terms vanish. Along with splitting into the “discrete” and “continuous” parts trace-formula allows yet another splitting into the positive (even) and negative (odd) components”. “Odd and even” functions, representations, etc., arise on SL, due to the element {-e}, f(-z) = ff(z);T-,= fI; etc. Clearly, principal {is+}, as well as complementary {s}-series, hence their characters, are even, while the principal {+}-series is odd. Separating the even and odd terms in both sides of the trace formula we get a complete splitting, Discrete
m, C I(fn)N$
n=O
=
d(a)uol(r\G)
t ?(-n))t
=,
“No simple orthogonality relation would help this time, 84 principal/complementary characters form an “over-determined system” on G. Indeed, the trace-formula underpin the overdeterminancy of { x , } , as its the LHS (5.15) exhibits a discrete sum of characters, N: x i , N,X, while the other side (5.16) is represented by various combinations of “discrete sums” (over {y )) and “continuous integrals” of that same set of characters,
c
* +c
Notice, that the RHS of Selberg formula (5.16) contains only the principal series characters!
57.5. Selberg trace formula.
341
5.4. Recipricity Theorem. Next we turn to the principal/complementary part of
spectrum of Ra. Our goal is to find all irreducibles { i s k f } and { s j } , as well as their multiplicities {N} in (5.15). There are two possible approaches to the problem. One of them, based on a Frobenius-type reciprocity principle, allows to reduce a decomposition
of induced R = ind(a I F;G), to the study of certain r-invariants of irreducible representations {T'} of G. We remind the reader the classical Frobenius reciprocity Theorem for compact G (33.2): irreducible n
I 'I
if
E
enfers an induced representation R = ind(a I GG), iff the resfriction
contains a copy of a, n I f 3 a @ N,. Furthermore multiplicity N , ( R ) (of n in R )
is equal t o multiplicity N , ( r ) of a in
T I r. For trivial a = I ,
it says: n C R , iff R I f
confains r-invariants, and multiplicity N,(R) is equal to dimension of r-invariants in n.
The main difficulty with non-compact G has to do with the fact, that unitary representations in Hilbert (L2-type) spaces, like h,
c L2(H),
have typically no -'l
invariants. The latter appear in larger (Loo-type) spaces, associated to
{P}, which have
a dense (core) intersection with L2.Although topologically, representations in L2- and Lco-type spaces may look different, their joint core makes them essentially equivalent (they are also equivalent in the sense of characters, and/or infinitesimal characters!).
For group SL, such "expanded" spaces {h,} could be realized" in L 2 ( H ) .In 57.5
57.5. Selberg trace formula.
342
we described them, as eigenspaces of the invariant Laplacian A on M, u
X,={f(z): Af
=qf}.
We look for (finite-D) subspaces A, c f,, which "transform according to a" under the action of In other words, there exists a basis of functions { fl(z); ...fd(z)} in A , ( d = d(a)),so that
r.
(5.19)
3,
where zg means the usual fractional-linear action, of SL, on W. In the case of trivial a = I , we look for r-invariant eigenfunctions of the Laplacian,
Af =& f ; and f(zy) = f(z); all -y E r, the so called automorphic f o m on W . Functions (5.19) could be thought of as generalized (vector-valued) automorphic forms of weight a. The main reciprocity result for pairs (SL,;r) can now be stated as follows. Reciprocity Theorem: Irreducible representation T of the principle/ complementary series enters an induced representation Ra = ind(a I r;G)with multiplicity N,, iff the ezpanded space %, contains a subspace A, of automorphic forms of weight a. Furthermore, multiplicity of a an
T I r).
dimA, N, is equal to (i.e. multiplicity 4,)
For 1-D representations a (characters of r),reciprocity Theorem states that N , is equal to the number (dimension) of automorphic form of weight a. The proof of reciprocity can be obtained by a careful analysis of automorphic forms (eigenfunctions of the Laplacian), for all 3 series of SL,, as in [GGP] (chapter 1). But there is also a fairly general argument (due to Piatetski-Shapiro), valid for all semi-simple Lie groups G . We shall skip further details and refer the reader to [GGR]. Reciprocity Theorem reduces our "spectral problem" for Ra to the study of automorphic forms, a difficult and largely unresolved question by itself. So its utility in this regard is somewhat limited (as compared to, say, Frobenius reciprocity for Laplacians on spheres and other compact symmetric spaces). Another approach to the Spectral decomposition problem for Ra is closely related (in fact, includes as a special case) Spectral theory of Laplacians on Riemannian surfaces "We have done it in $7.1 for the discrete series, and for principal/complementary series it follows from the results of $7.5: existence of K-invariants in {T'}.
57.5. Selberg trace formula.
N(A) N g4") v o l ( r \ G ) X 2 , as X+w
343
(5.20)
Then the LHS of (5.18) becomes exactly N(A), while the leading asymptotic in the RHS is given by the last integral-tFrm of (5.18),
0 Strictly speaking functions { f} with the Heaviside-type "Fourier transform" are not allowed in the Selberg-trace formula, they can not be C r , and the corresponding operators { T ; } do not belong to the trace-class. However, any such {f} could be approximated by a family of nice (regular) functions {fc}, which suffices for the proof of (5.20) (see [GGP]).
Formula (5.20) means, in particular, that the number of complementary series constituents of Ra is always finite, all of them being located on interval [0;1], whereas no accumulation of Isj+} and {sk} is allowed at any finite point A, of [O;m). We shall indicate the connection of representation R" to Laplacians on Riemann surfaces F\H. To simplify matters let us take trivial cr = I. The representation space L2(F\G) contains a subspace 'V of K-invariants (made of all K-invariants of the evenprincipal and complementary components of R"). Space 'V can be identified with L2(r\H) = L2(r\G/K). Furthermore, the Casimir (central) element
57.5. Selberg trace formula.
344
A, = a{Wz- 2 ( X Y + Y X ) } E Z(!!l) turns into an invariant Laplacian on H. Taking quotient Al, = r \ H reduces A, to the natural Laplace-Beltrami operator on the Riemann surface AL (group plays the role of "periodic boundary conditions" for A,, the same way as discrete lattice Znc W" yields the "periodic" torus Laplacian A,, from the 03"-Laplacian).
r
We observe that the Casimir becomes scalar on each primary subspace L, c L2(r\G) (L,N 36,@ N , - a multiple of irreducible T', contained in R ) ,
Furthermore, each primary (spectral) subspace L, contains exactly N , "Kinvariants" (since each irreducible M, has a unique K-invariant, described in 57.5). We denote by T, a subspace of K-invariants in L,. Then the entire space Y of K-invariants is decomposed into the direct sum of eigenspaces of R, I T CY A,,
y,
T = $T,;
the eigenvalue: A, = on Ts has multipicity N , = dimT,. Here {s} varies over the spectrum (principal; complementary; discrete) of the induced representation R. Hence, solution to the spectral problem for R yields a complete solution of the eigenvalue problem for the Laplacians A , on Riemann surfaces r\H, and vice versa. Both problems, however, are equally difficult to solve exactly. In the next section we shall explore some aspects of asymptotic spectral analysis of A, and establish interesting links to geometry of A, particularly the relation between eigenvalues of A , and the length of closed (periodic) geodesics on A. 5.6. Geodesic flows on negatively curved Riemann surfaces. Finally, we shall give a simple application of spectral decomposition L 2 ( A )= $IT,, to a geodesic flow on A = r \ H . The geodesic flow (Appendix C) is a 1-parameter family of transformations on the unit cotangent bundle
S * ( A ) = { ( z ; ( ) : xE A;l/(ll= 1) = 0, generated by the vector field E = 8,a. 8, - a,a. a,, where a(x;() = root of the metric tensor form on covectors (.
/=
- square
Theorem 3: Geodesic jlow o n any compact R i e m a n n surface A = r\G i s ergodic, in the s e w e that a n y function f E LZ(R), invariant under the flow, is constant. Indeed, vector field S on J b is the image of geodesic generator on the PoincareLobachevski plane (or disk) D. But S*(D) is naturally identified with the group G = SU(1;l) itself, and since field E on S*(D) commutes with the G-action (by
$7.5.Selberg trace formula. isometries) it must be a left-invariant field, given by a Lie algebra element X E 4 1 ; 1).
To find X we just take a geodesics through {0} in the direction of z-axis, and find the 1parameter group of X to be,
hence X = an operator
[ ].
Any (geodesically) invariant function f on 0 must be a null-vector of
{T$}, for
some u in the spectrum of RA. But we know all generators {rx}
for (principal, discrete, complimentary series
T),
neither one has a nontrivial kernel for
s # 0. Hence, kernel(Rx) = 0, and the flow exp(t2) is ergodic.
345
5 7.5. Selberg trace formula.
346
Problems and Exercises: 1. Consider a unitary group representation T with sufficiently many compact operators {T ), (for a suitable class of test-functions {j)in the groupalgebra, and show that suci T has always a discrete (direct sum) decomposition into irreducible components: T 2: @ TP@m(p), of finite multiplicity, m(p) < m. Steps: P
i) take a symmetric element f = 'f on G , and the corresponding self-adjoint compact operator T * show that a non-zero eigensubspace 6 (A # 0) of T intersects G-invariant f subspaces of T; choose minimal among such% (always exists, due to fin of Bx!); and show that T 136 is irreducible.
(k}
ii) derive the direct sum decomposition of T, and show that all multiplicities must be finite. 2. Derive formula (5.12) for the H-transform of an elliptic element u = u8. Steps: i) Show:
7
--*
FK(~e)q5(~e)d8 = IG,lf(g)d(g)
I eit-e-it I- dg,
integration over all elliptic conjugacy classes in G ,where {e it} denote eigenvalues of matrix g E Gel;q5 - an arbitrary test-function, constant on conjugacy classes. ii) Apply (i) with q5 = e-int(e-i'
7
--*
FK(sle)e--ine(e-i6-eie) d8 =
- e i t ) to get
-inl
Jcelf(g) eif -e-ild
9;
aEd compare it with formulae (5.16) for characters and the ensuing transforms of f, {f(s)= (f Ixb): s = +is; s; or i n}, to show
iii) Deduce from (5.21) the formula for the elliptic Harish-Chandra transform o f f
-
,
Fh
iv) It remains to express the_ last sum Fh in (5.22) through the continuous series characters and transforms { f ( i sz t )}. Use the Fourier expansion of the conjugate Poisson kernel in D, A(,i8 - ,-i8 A-n(eine - , - i d *toget = 1 2Acos 8 A'' 1
-
+
v) Use the formulae for the hyperbolic H-transform (cf 57.4),
57.5. Selberg trace formula.
347
.
vi) Formula (v) simplifies, using the relations f(-is f ) = j ( i s f ), and the well known Fourier integrals:
Xis
71 -2XcosB+ 0
m
d
Xis 1 - 2Xcos 0
+
X2
r s i n h s(0-r) dX = sin 0 sinhrs ; O < O < r ;
dX = -
r s i n h s(O+r). -r sin 0 sinhrs '
< 0 < 0.
vii) Complete the derivation.
3. Take a discrete-series representation T", pick its highest/lowest weight-vector $ J ~ (T:$, = erne; for u = u E K), and show i) matrix entry fo(g) = $T;$,I $o) is in L'(G) (use Iwasawa decomposition); ii) show that operators: Ti = 0, for all principal and complementary-series representations {P} (Pair fo tooa spherical function 40(g) = (TdqoI qo) (Tlqo = qo; for u = u0 E K); check that (fo I q50) = 0, by comparing K-actions on fo;do;then use irreducibility of T'!).
348
57.6.Laplacians on hyperbolic surfaces W/r $7.6. Laplacians on hyperbolic surfaces H/r. Section 97.7 develops the Spectral theory of hyperbolic Laplacians A A on Riemann surfacea A = H/r, from a different prospective. It doea not directly involve the representation theory of the preceding sections (f7.1-7.6),but exploits the heat-kernel method, following McKean's approach [Mcl.
The hyperbolic space can be realized either as Poincare-Lobachevski half-plane W,or disk D. The distance between two points z = x1 ix, and y = y1 iy, in the half plane W is given by
+
+
while in the disk realization (problem 4) it corresponxi to
The geodesics in D are all circles perpendicular to the boundary, while in also include all lines parallel to the y-axis (see fig.3 and Appndix C).
W they
Fig.): illustrates geodesics in the hyperbolic geomeiry of the half-plane (left) and the disc (righi). We have shown 2 families of geodesics. One consists of concentric circles, ceniered at {0} in H, which are MSbius transformed into the family of vertical circles, centered on the horizonial axis of D (the left and righi poles in D correspond t o poinis (0) and {co} in M). The dashed circles connecting two poles of D represeni radial (dashed) rays in H. The second family of geodesics is made of parallel lines in H, ihat get iransformed into a family of circles converging io the right pole (00) in D. The hyperbolic geomeiry clearly violates the Euclid parallel lines aziom.
6.1. Fundamental regions in H. A discrete subgroup 'I of SL, divides space W into a union of fundamental regions: U A,,, non-overlapping geodesic polygons {A,,= 7 ( h ) :-y E each bounded by a finite number of geodesic arcs {Cj}.Such tessellations of W are similar to partition of the Euclidian plane (or Wn-space) into the union of fundamental rectangles by a discrete lattice I' c W2. Each fundamental region B c W2 is then mapped onto the quotient space, B+RZ/r N Y2 (2-torus).
r},
Throughout this section we shall assume that group
I'has no fixed points inside
$7.6. Laplacians on hyperbolic surfaces W/r.
349
W,i.e. no 7 E r fixes a point zo of Sz, > 0. This means l‘ has no elliptic elements (fixed points in W turn into “sharp vertices” of the quotient-space, where A loses its smooth structure, a situation reminiscent to quotients of sphere S2, modulo any of discrete “Platonid’ symmetries, chapter 1). Compactness of W / r (hence, of G / r ) also implies the absence of parabolic elements. Thus is made entirely of hyperbolic elements.
r
To construct r-fundamental polygons in H we pick any point zo E W, 7 E
r, and
consider a subset A = { z E W: d(z;z,)
5 d(z;7j(zo));j
= 1;2; ...}.
Such A forms a geodesic 4N-gone in M, the boundary arcs transformed one into the other by elements ( 7 ) . Boundary arcs
{Cj} being
{Cj}are
naturally
divided into opposite pairs, and under proper identification we get a smooth (analytic) Riemann surface A = W/r of genus N . Figures 5 below illustrates a deformation of octagon ( N = 2), into a surface of genus 2.
r then turns into the fundamental group12 of A, in other words each r corresponds to a class of equivalent closed loops { w y}. Furthermore, fundamental class { w } of r N x 1 ( A ) , contains a minimal length path yo, which Group
element 7 E any
N
coincides with the closed geodesics of the given homotopy type. We shall illustrate the foregoing with a few examples of fundamental regions. Examples of fundamental regions H/E 1. The modular group
r = SL,(Z)
is generated by 2 elements: translation by 1 in the
horizontal direction h: z+z+l, and reflection u: z-+l/z,given
Two types of fundamental regions of
by matrices
r are shown on fig.4:
12Two path yl; y2 on a manifold Jb are called (homotopy) equivalent, if y1 can be continuously deformed into y2 inside Jb. Any pair {yl;y2} of closed path (loops) passing through a fixed point zo E Jb can be formally mulfiplied by combining them into a single path y l q 2 (=yln followed by “y2”). Such multiplication is easily verified to respect the (homotopy) equivalence on space Q(z,) of closed path, through {z,}.
So the set of equivalence classes of {y E Q(zo)} acquires the group structure: the identity consisting of all path contractible to {z,}, while the inverse {y-’}, being given by y, traversed in the opposite direction). The resulting group is called the fundamental (or 1-32 homotopy) group of Jb, and denoted by nl(Jb). It carries some important topological information about Jb. For instance, the fundamental group = nl(Jb) of a 2-D surface A, when quotient modulo its commutator = yields the so-called homology group H l ( A ) = r/r’. The latter forms a commutative group, isomorphic to Z N x C (direct sum of the “free and torsion components”: ZNand C), and dimension N of the free component is precisely the genus of Jb!
r
r’ [r;r],
350
57.6. Laplacians on hyperbolic surfaces W/l"
3;
(I) vertical strips (light shading) are shifts of the basic strip {-$I %r 5 I z
I > 1);
(11) Fundamental triangles (dark shading), if fact, quadrilateral, since point i E C3 should be considered a vertex, obtained by shifting the basic triangle {O; f i ~ ]The . latter
3+ &
is bounded by geodesic arcs: C,,,=(I 63 1 =l}; and C3 = { I z 1 = 1). The first type regions are obtained by fixing point { 2 i } , shifting it by h:2i-2i f 1, and inverting by 0:2i+$, then writing the corresponding geodesic bisectors: d ( z ; 2 i )= d(r; f 1+2i) (vertical lines); and d ( z ; 2 i ) = d ( z ; q i ) (circle I z I =l).
Obviously, the fundamental triangle is obtained by a-inverting the fundamental strip. Let us remark that the fundamental region Jb of the modular group SL2(Z) is not compact, it has cusp (0) a t {co)(i.e. at the "infinite boundary" of H, aH = R
U (m}). But Ab has
finite volume (problem 2 ) . Topologically manifold Jb looks like a sphere with an (infinitely) long spike (Fig.5). Fig.4: Two types of fundamental regions of ihe modular group
r =sr,~).
Figd. Riemann surface H/SL,(Z)
with a
cusp (spike) at {m}.
2. Our next example (Fig.6) demonstrates a typical fundamental polygon in H, whose vertices
are (finite order) fixed points of certain elliptic elements (yj E Z'}.
$7.6.Laplacians on hyperbolic surfaces W/r.
351
Fig.6 A typical fundamental polygon in H.
Topologically quotients A = D/r, drawn in fig.6, turn into Riemann surfaces of higher genus g. Fig. 7 below demonstrates a transformation of an octagon with properly identified sides into a genus 2 Riemann surface. Let us notice that images of the fundamental region J b under all elements
{r E r } form a regular
tessellation of D into
4g-gons, an impossible fit in the Euclidian flat-land, for any g 2 1. A remarkable feature of the hyperbolic plane is the existence of regular tessellations of any order, hence of higher genus hyperbolic surfaces Jb/r of constant negative curvature!
Fig.7. (i) A Riemann surface of genus 2 (A) is cut into 2 truncated tori (B);
(ii) The tori are then unwrapped into squares (with holes in the middle, which correspond to the original cut) via 2 siandard fundamental cuts;
a
b
d
(iii) The inner circle is stretched through a vertex to turn each square into a pentagon C
a
(D)?
s7.6.Laplacians on hyperbolic surfaces H/I'
352
b
(iv) Finally two pentagons a r e glued together along the diagonal (the original cut) to form a n octagon (E). finally two pentagons a r e glued together along the diagonal (the original cut) to form a n of octagon (E). The fundamental group genus-2 surface has 4 generators {a; b;c;d}, a n d a single relation: aba-'b-'& - ' d - = e, which corresponds to traversing the octagon clockwise.
r
b
(v) The resulting fundamental group r of genus-2 surface has 4 generators {a; b; c; d}, and a single relation: = e, which aba-'b-'cdc-'d-' corresponds to traversing the octagon clockwise.
d"
d
6.2. Kernels, traces and heat-invariants.We are interested in eigenvalues {A,}
of the Laplacian A on hyperbolic surfaces A = W/F.These are usually impossible to compute exactly (in any closed form), so one would like to get some approximate or asymptotic expansions of {A,}. The latter often involves the study of certain "means" of {A's}, (cf. chapter 2), and the related transform (Laplace, Fourier, Stieltjes, etc.) of the counting (spectral) function N(A) = #{k:Ak _< A}. We shall mention a few of them Heat-kernel (Laplace):
m
o(t)= x e - t A k = je-t+iN(A) Z(s) =
Zeta-function:
cm-
Resolvent (Cauchy-Stieltjes): R(C)= - J7
= tr(etA);
0
CXks= JA-SdN(A) = tr(A-"); 0
c&
dN(A)
o(C-4
Wave-trace (Fourier):
m
mdN(X)
= J-
0C-A
- tr(C - A)-'; -
or
= tr(C - A)-"';
X ( t ) = C eW
G = JeiMdN(X) = t r ( e w ) .
The first two are called theta and zeta-functions of A, by analogy with the classical (Jacobi, Riemann) theta and zeta functions, which correspond to an "integral spectrum" { A, = k } . All transforms represent traces (regular or generalized) of various Green-functions of A: theta gives the trace of the heat-kernel, tr(eVtA), (the
$7.6. Laplacians on hyperbolic surfaces
W/r.
353
fundamental solution of the heat problem: ut = Au), Cauchy-Stieltjes transforms R corresponds to resolvent of A (and its powers), while di~tribution'~ X ( t ) on R yields the wave-trace, t r ( e i t d ) (Green's function of the wave equation: utt - Au = 0) (see $2.4).
A11 transforms are related one to the other, for instance, zeta is obtained from theta via T-integral, 00
Z(s) = &j! a consequence of the obvious relation
@(Ws-lat,
This often allows to link large-) asymptotics of spectrum { A k ] , to asymptotics of zeta, theta, etc. functions, via the so called AbelIan/Tauberian Theorems. If for instance, sequence Ak
-
kp, as k+m, for some p > 0, then its theta-function @(t) =
is easily seen to obey
-
e(t)
e-"k
-+re-tAApdA,
r(p+l)t-p-';
a t small t ,
and similar relations could be derived for other "kernels of A". The converse, however, is not true, typically means, like 8,Z, etc., behave much better, than irregularly distributed sequence of eigenvalues. For instance, the heat-trace admits an asymptotic expansion, due to Munakshisundaram-Plejel, tr etA
-
t-"/'{b,
+ b l t + b,t2 + ...},
(6.2) whose coefficients { b k } , called heat-invariants, carry some important geometrictopological information about manifold m. For instance, b, is proportional to vol(rn), with Const, depending on n; 6 , =
I
Kdz
-
integral of scalar curvature14, K ( x ) , over
the natural (Riemannian) volume elEment on m. Higher invariants involve certain universal polynomial expressions in curvature and its covariant derivatives, integrated over m. All of them could be computed in principle, but the formulae quickly become unmanageable [Gill. So to get a better insight into the structure of spec(A) one need to is no difficult to show that heat-semigroup and resolvent (( - A)-', of Laplacians tIon compact Riemannian manifolds m, are given by nice integral kernels: K t ( z ; y ) and R ( z ; y ) (cf. chapter 2 ) . Hence both are compact, and belong to the trace-class. But the wave-kernef (made of unitary operators { W ( t )= e z p ( i t 6 ) ) ) could be "traced" only in a generalized sense, as a distribution on R (see chapters 1-2).
14Scalar (Gauss) curvature K(z,,) at a point xo on a surface e measures the deviation of surface area of the geodesic sphere: S, = {d(z;z,) < 0}, from the Euclidian (flat) area: I S, I = ?TC2Kc4 + For an n-D manifold we take all geodesic 2-planes {e,} through 2, (images under the e x p map of all planes in the tangent space T20) ' and average the Gauss curvature of e, at zo over all {e,}. The formal definition of K involves the trace of the Ricci tensor.
... .
57.6.Laplacians on hyperbolic surfaces H / r
354 *
look for a different set of "geometric spectral data". One such class consists of the length of all c l d path (geodesies) in A.
In this section we shall compute the heat-trace (theta-function) for Laplacians on A, and will link it to closed geodesics in AL. A simple prototype of the main result and the basic approach below is the flat (torus) case ? N R"/r, discussed in chapter 2. Any lattice (discrete subgroup) r CR" can identified with the image of the standard (unit-cell) lattice Z" under a linear map A:R"+W". So the Laplacian on W/r, pulled back to the standard n-cell To= [0;1] x... x [0;1], turns into a 2-nd order constant-coefficient elliptic operator: L = Bd .d = C bijd:j, where matrix B = (tAA)-
'.
The eigenvalues of L in To (respectively, A in T) are readily computed while closed path in
= 4T'Bk. k; k = (k1;...;k,) E Zn. T,labeled by n-tuples m = (ml;...mn),have length em = I Am 1 = B-'m*m.
To get the heat-kernel Lon To, we take the free-space Gaussian, e w t z ,
averaged it over the lattice I',
Gk)+
C
G(z
+ m),
m E r
then applies the Poisson summation formula. This yields ~ ~ - t ( 4 a ' B k . k--) (4rt)"" m k
7
whose LHS gives the theta-function, x e - t A k , while the length of all closed path,
t-+
c
2
e-A@m
(6.3)
RHS involves exponentials of
.
In the hyperbolic setup the role of lattice r is played by a discrete subgroup of SL,, and "tori" become fundamental regions W/r. We shall take as above the free space heat-kernel K t on W, which depends only on the distance d(x;y), between {x} and {y}15, K = K(d(x;y)), average it via the r-action, and analyze all terms of the resulting 15The latter follows from the symmetry properties of the Laplacian A on H Laplacian is invariant under the SL2-action, hence any "function of A", e.g. K , = e-tA, is also C-invariant. But a G-invariant integral kernel F(z;y) must depend on the distance d(z;y) only, due to a double-transitive action of Sb on H: any pair (2;y) is taken to any other equidistant pair ( 2 ' ; ~ ' ) by an element g E SL, (problem 3). This is a general feature of so called rank-one symmetric spaces (see J5.7).
57.6.Laplacians on hyperbolic surfaces H/r.
355
series. The analysis could be carried out for any radial function K ( r ) , r = d(x;y), decaying sufficiently fast at {m}, to assure convergence of the r-series below. Any such K defines an SL.+variant integral kernel K(z;y) = K(d(z;y)) on 04, which can then be reduced (quotient) to the manifold A. The resulting kernel
The trace formula below gives trKA in terms of length of primitive inconjugate elements { p E r}.A hyperbolic element 7 is called primitive, if 7 is not an iterate of another element, i.e. 7 # for m 2 2,and r0 E r. We shall introduce the length of qEG, as
7r,
One can check (problem4) that e does, indeed, represent the length of the smallest closed path (geodesics) in the fundamental class of q E r . For hyperbolic q - one finds
- I"
m2dy
e(q) = J
= 1og(m*),hence e(qn) = 210g(mn).
1
Now we can state the main result of the section.
6.3. Trace formula: Given a radial function K ( r ) , r = chd(x;y),on W,the reduced kernel K A (6.4) on a compact quotient space AL = W/r belongs to the trace-class. Its trace, t r K A = K,(x;x)dx, can be expanded in a double series, over integers n = 1;2; ... a n A v e r all inconjugate, primitive elements { p E r},
Here pn denotes the n-th iterate of the primitive path p , and l(...) - the length of the path. The proof involves several steps, some of them similar to the derivation of the trace formulae in the preceding section. Namely, 1) Group
r
is split into the union of conjugacy classes of powers of inconjugate primitive
elements:
r = {e}U{y-'p"r:
yE
r rP},
57.6.Laplacians on hyperbolic surfaces M/r.
356
where r pdenotes the centralizer (commutator) of element p in f. 2) We observe that integrals J A K ( z g ; z ) d z , over a fundamental region A, are equal, for all
conjugate elements {g = y-'pny:y E r}.SO
where [Rfp] denotes the index of subgroup clasa {e},
r pc r. The first integral corresponds to the trivial
while the remaining terms give nontrivial classes {p"}.
[r:r,] = # {cosets in r
Notice that index
fp)is always finite
3) Let also remark that K(zg;z)dz,
is identified with an integral of K over yet larger fundamental region A,
N
H/fp,of subgroup
r pc r, A, is made of [I? rp]non overlapping copies of the original A! 4) Next we note that f,=
{pk:k
E 2 ) coincides with the subgroup generated by {p}. A
hyperbolic primitive element p E r is conjugate equivalent to a diagonal matrix q = Such q acts on H by dilations with m', strips { 1 5 I Sz I
60
a fundamental region of q can chosen to consist of two
5 mz} (fig.8). By the distance formula (6.1) the argument of K becomes
(I - mZn)' I z I 2mZnxZ2 Then integration over the fundamental region Aqyields, chd(x9"; z) = 1 +
/%r
'
m2
K= AAb,
dzlK(l+~(m"-m-n)'(1+~~)2~)
122-w
Changing variables: z 4 #z2;t =
2),we get
l o g m 2 7 K ( 1 + f ( m " - m - " 2 ) (1 -00
+ t2))dt.
Remembering that e(q) = logm', e(q") = logm'", the argument of (6.8) turns into
=
+ 2 s hne(9) 2 7
n+ 1
t2).
We call the new variable u, and make yet another change, t-u, to brings (6.8) to the form
which completes the proof of (6.7).
7.6. Laplacians on hyperbolic surfaces W/r.
357
Fig.8. Fundamental region of a hyperbolic element h can be chosen either as a strip {A 5 1922 I 5 m} ( A ) , or annulus {f 5 I z I 5 m} ( B ) .
The heat-trace. Now it remains to specify (6.8) to the heat-kernel Kt(z;y) on A. But first let us find the heat-kernel on W. We have already observed that K depends only on the hyperbolic distance r = d(z;y) between points z,y E W. There is a wellknown formula for K , in terms of ch r (see problem 51,
Next we shall compute the contributions'of various conjugacy classes in (6.7),
while the pn-term contributes
The inner integral gives the standard Beta-factor B(i;i)= a, while the outer integration results in e-e2/4t (4*t)'/2
Substitution in (6.9) yields the following version of the Selberg truce f o m d a for the heat-kernel @(t)= t r ( e t A ) = e-tAk, of the hyperbolic Laplacian A = A ,
with coefficient
358
57.6.Laplacians on
hyperbolic surfaces W/r.
Remarks: Formula (6.10) has many i:teresting
interpretations and applications in
Spectral theory of differential operators, and in quantum mechanics. It links directly the “Laplace-transformed” eigenvalues {A,}
of Ah on the one hand, and the length (period)
spectrum of manifold A {l(y): over all closed geodesics y
c A}. In
the picturesque
language of Mark Kac [Ka]: One can hear ihe length specirum of A. This allows to address the Inverse spectral problem: Can one hear (uniquely deiermine) the geometry
(metric) of Ab from specAA? The length spectrum
reduces the Inverse spectral question to a Problem
in
geometry/dynamics on manifolds: given iwo metrics on A, whose geodesic flows have
identical period/length spectra can one conclude thai metrics are isometrically equivalent? In the classical case of hyperbolic surfaces A = H/r of constant negative curvature, McKean [Mc] showed that there are at most finitely many isospectral surfaces { A } .He utilized the Selberg trace formula (6.10), and the Klein-Fricke double and triple traces {tr(ykY,,,); tr(yjykym): y k E
r}.The latter
served to identify elements {yk} up to
r-
conjugacy, hence the length of closed geodesics {e(y)}, and their orientations. The work of McKean [Mc] still left a possibility of a unique solution of the Inverse spectral Problem, until Vigneras [Vi] found in 1980 finite (but arbitrary large!) families of distinct isospectral surfaces of any genus. These examples, however turned out to be rather exceptional. Wolpert [Wo] showed that a generic hyperbolic surface is uniquely determined by spec(A)! For more general hyperbolic manifolds of non-constant curvature Guillemin and Kazhdan [GK] established “infinitesimal rigidity”. They also used a reduction to a dynamical Problem. In the study of dynamical systems it was found that uniqueness usually requires an additional ”discrete set of data”, for instance association of homotopy classes to the periods {L,,,}, [KB]. The latter was previously used by McKean [Mc] in the constant curvature case. Under such hypotheses, the geometric Problem was shown to have a unique solution [KB]. Though the length spectrum brings one tantalizingly close to settling the inverse spectral question (at least on hyperbolic surfaces), there remains the frustrating “labeling problem”, whose spectral content is unclear. In the quantum-mechanical context Laplacian A A represents the hamiltonian (energy-
operator), which describes the evolution of a quantum system (particle), constraint to move on a surface/manifold A. The underlying classical mechanical sysiem coincides
57.6.Laplacians on hyperbolic surfaces W/r.
359
with the geodesic flow on the phase-space p(A)- cotangent bundle of A, given by the corresponding classical hamiltonian/energy function: h ( z ;() =
gjj(z)(j(j
-
metric
tensor, as a function of phaoe-variables: z E A, ( E T,*. Formula (6.10) gives then a correspondence between “eigenualues of A ” (quantum energy levels), and the “length spectrum” (energies/actions) of classical trajectories (geodesics). Such relations exemplify the Correspondence principle of Quantum theory, which asserts close links between the classical and quantum systems. Usually, the correspondence holds only approximately/asymptotically at “large energies” (or for small Plank parameter), where the quantum dynamics becomes quasiclassicel. The remarkable feature of the flat (Tn) and hyperbolic Laplacians is that the quasiclassical expansions (6.3) and (6.10)
turns out to be exact! In the next chapter we shall explore in the greater detail the structure of classical hamiltonian systems, the quantization procedure and the role of groupsymmetries in the classical and quantum mechanics.
57.6.Laplacians on hyperbolic surfaces H/r.
360
Problems and Exercises: 1. Find the distance between two points: z = z1 +&and y = y 1 + iy2; in the Poincare half-plane H (6.1) (hint: it is easy to compute the distance between purely imaginary points d(ia;ib)= ln:. Use it and find a fractional linear map g:H+H, that takes 2 4 ; y-tib, and compute b). 2. Find the hyperbolic volume of the fundamental region Ab = H / C
r = SL,@) (fig.4).
3. Show that any 9,-invariant integral kernel F ( z ;y ) depends only on the distance d ( r ;y). Hint: F(zg;yg)= F ( r ;y ) , for all z , y E H, g E G; move 2 4 , and use the fact that stabilizer of { i } , K = S q 2 ) , acts transitively on all spheres S,(i) = { y : d ( i ; y )= r } , centered a t { i } , of radius r.
4. Show that the hyperbolic distance between points (6.6). Use the disk realization of H, and show that and d(O;Oq) = ln(
d ( 0 ; r ) =$In(*); l-lzl
(2)
and
{zq} (q
I I + I I ), where q = (I
E SL,) is given by
P [P
E SU(l;l)!
5. Derive the heat-kernel Kt(chr) (6.9) on the Poincare half-plane, in terms of hyperbolic radius r = d ( r ; y ) . Steps: (i) the Laplacian in hyperbolic polar coordinates (r;O) of D is given by A = 6’: + cth ra, + sh-2rag2. Changing r-m = ch r , bring it into the Legendretype differential operator,
Z = a(z2-1)8 +*02, 2
on the half-line [l;oo) X 1;
-1
ii) Laplace-transform the heat-kernel 00 K(1;...)-Q(r;A) = e e - ” K t ( r ) d t , 0
and check the resulting function &(.;A)
to solve the equation:
(Z+ AN91 = a(r),
(6.11)
i.e. Q(r;A) gives the resolvent-kernel of Z. But for radial function Q = Q ( r ) equation (6.11) reduces to the Legendre ODE: [(z2-1)Q‘]’ + A & = 0, with a suitable “source condition” a t z=O (cf. J2.3, chapter 2). Its solution is known to be a Legendre function of 2-nd kind and order v, Q = Qv(z), where A = v 2 f .
+
iii) The Legendre function Qv(r) has an integral representation ([Erd];[Leb]),
iv) Use this representation of Q, and compare it with the Laplace transform of the integrand of (6.9), ~[t-~/’ezp(-t/4-r~/4t}]. YThe later yields a modified Bessel (Kelvin) function K1/,(r A + 1/4), via an integral representation of Kelvin functions, 00 Y %ezp{-a(at 0 +:)}t-’-’dt = 2(;)”/’K A. ab). But in our
case
i,
order s = the corresponding Kelvig function becomes classical, ~
Complete the derivation!
~
= ~
~
(
= z
)$ e - z !
361
$7.7. SL,(C) and the Lorentzgroup $7.7. Sb(C) and Lorentzgmps. In the last section of ch.7 we shall briefly review the representation theory of the complex group SL,(C), the related Lorentz group of special relativity, and also higher-dimensional (pseudo-orthogonal) Lorentz group S q l ;n). Most results will be stated without proof.
7.1. The Lorentz group S0(1;3) of special relativity preserves the Minkowski (1;3)-form in
[email protected] 2-fold cover is the complex unimodular group SL(2;C). To
show
the
correspondence
SL,-Sq1;3)
we
note
that
the
real
4-space
= {(zo;z1;zz;z3)}can be identified with the space of complex hermitian matrices, ~~~~
'p=
{x = [
c'E --2
'
ro;zl
]
= z2+iz3}1
where the determinant-form d e t ( X ) = (X I X) = z2-Czjz, defines the Minkowski product on
3, and conjugations, g:
x+g*xg,
preserve hermitian symmetry, as well as the Minkowski norm, det(g*Xg) = det(X). So we get a map (isomorphism) of
SL,/{ f I } onto the Lorentz group Sq1;3).
Irreducible representations of consist of 2 series: principal and complementary. Principal aeries representations are induced from the Bore1 subgroup B of all uppertriangular matrices,
here C* denotes the multiplicative group of complex numbers, C* N T x R*. The characters of B are labeled by pairs { (s;m):s E R; m E Z},
The quotient-space 96 = B\G can be identified with the complex plane C (via the Gauss decomposition g = n-hn+), and G acts on C by fractional-linear transformations, as in $7.1,
So principal series representations {Ts9m}can be realized in space LZ(C;d2r),by operators ,
-'
Here d2z denotes the standard area element &fz A dZ in C, factor I bztd I serves to unitarize Tslm, and the complex argument ~br+dl bz+d plays the role of "sgn(bz+d)" in the real case ($7.1). So representation TgVm is induced by the character
$7.7.SL,(C) and
362
the Lorentz group
of B. An alternative realization of TSfmon the 2-sphere will be explained below in the context of the general Lorentz group So(1;n). We remark that the Riemann sphere S2 gives a 1-point compactification of the complex plane, Sz= C U {co}. 1rs7m
Irreducibility of Ts9m can be now established either by the Mackey's test (Theorem 1 of $7.1), or directly (see problem 2 of 57.1). The principal series representations have characters, given by
xs,m(g) =
I I istm ~
~
+- I m A, I -(is+m)
I Ag - A g - l l
A
m 8
. 7
(7.3)
where {Ag&'} denotes the eigenvalue of matrix g E SL,(C). Representations T S * m and T-S,-m are equivalent, which can be verified either directly (by constructing intertwining operator W), or as a consequence of (7.3). Complementary series correspond to the imaginary s = ia, -2 < a < 2 , and m = 0 in (7.3). As in the real case 57.1 they can be realized in the Hilbert space,
determined by the Riesz potential in C,
1 f Ip =
1I I z
w I "-"(z)fod2zd2w.
Their characters are analytically continued in s functions more details we refer to [GGV], chapter 2).
{ x ~ ,of~ (7.3) } (for
Group SL,(C) has no discrete series as will be explained below. The discrete series of semisimple Lie groups were directly linked by Harish-Chandra [Har 5) to the existence of compact the maximal abelian (Cartan) subalgebras, and SL,(C), as any other complex G, has none. 7.2. The Plancherel-inversion formula for SL,(C) has the standard formulation: for any smooth rapidly decaying function f on G,
where { x ~ , denote ~ } the principal-series characters. So the Plancherel measure dp is supported on principal-series part (a countable union of half-lines), and has density, dp(s, m) = (s2
+ m2)ds.
Gelfand, Graev, Piatetski-Shapiro [GGP] derive a general Plancherel formula on SL,(F) over any locally compact field F (reds, complex, p-adics). Their result for complex C states,
57.7.SL,(C) and where the Plancherel density,
363
the Lorentzgroup
Gprine
and { T = rstrn}varies over all characters (7.2) of B . Singular integral (7.4) is understood in the regularized sense, whence follows,
+
p ( r ) = Const(s2 m2), with Const = 1 . 32r4
Selberg-trace formula for compact quotients r\G of Sb(C) looks similar to 57.5,
but many simplifications arise due to the absence of the discrete series and the “elliptic conjugacy classes” (any g in SL,(C) is conjugate to a diagonal matrix!). Chapter I of [GGP] provides the complete details.
7.3. Lorentz groups SO(1;n) preserves the Minkowski product,
“,yo-
e x i y j in 1
MIn+’. The preceding discussion of SL, has prepared the reader for the introduction to representations of more general semisimple groups. Here we shall briefly discuss one such class higher-dimensional Lorentz groups. The key to the analysis lies in two decompositions, Cartan and Iwasawa (see 555.7 and 7.1). Group G contains a maximal
compact subgroup K = S q n ) , and an abelian subgroup of hyperbolic rotations,
[ ].
generated by a (relativistic-boost) operator H = The commutator of H in G is he product A . M , where compact subgroup M = Sqn-1) c K gives the commutator (centralizer) of A in K ( M consists of orthogonal rotations in the span{z,; ...zntl}!). Element H (its adjoint map adH) splits Lie algebra (5 = so(1;n) into the direct sum of eigenspaces: 0-1@ 0,@ 0,
(7.5) of eigenvalues {-l;O; 1). Here subalgebra (5, = ‘u @ !Dl - (Lie algebras of A and M ) , while are made of matrices of the type, subspaces (5
*
where (n-1)-vectors u = (v2;. . . v ~ + ~u)= ; (uz;...untl) satisfy the relations u = u for g1, u = -u for (5-l. Indeed, element H acts by left multiplications on row-matrices [i].
57.7.SL,(C) and the Lorentz group
364
Decomposition (7.5) clearly obeys the Lie-bracket relations, for p,X = 0; f 1, [O,;OA] c so 0 form two commutative subalgebras R f of 0, and the sum I @ R+ becomes a 2-step solvable subalgebra of O (51.4). Now the entire space 0 can be split into the direct sum of subalgebras,
O = si @?I@ R. With some additional effort (cf. [Hell) one can show that group G can be similarly factored into the Iwasawa product, G =KAN. We define the Borel subalgebra of 0, as B = !Ill@ I @ 9,using the fact that the centralizer !Ill of '(I leaves subalgebra 9 invariant. The corresponding Borel subgroup is also factored in the product, B =MAN. The quotient-space B\G can be identified either with the n-sphere S" = M\h', or with a "closure" of N , using a different (Gauss) factorization G = B N ,
Principal series representations of S q l ; n) are induced by irreducible representations of the Borel subgroup B, so T = T S Y mare labeled by the pairs16 { ( s , m ) : sE W;mE M } . Let 'T = Ym denote the space of m, and 7r = R"'' = xs 8 am the corresponding (inducing) representation of B, A
7r;,m = &sa T m. ",
b=U.a.h, ( u E M , u E A , ~ E N ) .
Then T s ~ m = i n d ( a S ~B;G) m I acts in the usual way on the product-space L2(Sn)8 'Tm, and can be shown to be irreducible (problem 1). A different realization of TSjmcould be given in spaces L 2 ( N...) using a Gauss-type decomposition G = B.N . Let us compare these constructions with special cases of S0(1;2) (or Sf,@)) and
S0(1;3)= Sf,(C)/Z,. Examples. 0
Sf,(R).Here the abelian subgroup, A=
{[
A A - 1 1I #
> 0) (positive diagonal); K
=
-The dual object of the orthogonal group Sqn-1) was described in J5.3. It is made of all ordered p-tuples {ml 2 m 2 2 ... 2 f mp; p = [;I}.
57.7. SL,(C) and the Lorentzgroup
M = { f1);N =
{[ :}I 1
365
upper-triangular.
Bore1 B made of all upper-triangular matrices. The principal series {TS* * } were realized in L2(R),where W N B\G, or L2(T), another (projective) form of B\G.
SL,(C) has A - positive real diagonal matrices; K = SU(2), or S q 3 ) for the Lorentz S q 1 ; 3); 0
M - unitary diagonal
{Ieie 1 ,-ie
(maximal torus in K);
N , B consist of upper-triangular matrices; the quotient-space B\G = M\K N S2 (from the Iwasawa factorization), or C = R2 (from the Gauss G = Be N ) . In (7.2) we were able to write down explicitly operators using the fractional-linear (conformal) action of SL,(C) on the quotient B\G
N
{T2m},
C.
It is interesting to note that the action of higher Lorentz groups on the homogeneous space S” N B\G implemented
by
’
a
is also conformal. The conformal action could be stereographic
map
from
the
hyperboloid
Wn = {q,’- I z I = l} N K\G (where G acts isometrically), onto the sphere Sn= (202 + I z I = 1). So right translations of G on the quotient B\G are equivalent to its conformal action on S” (problem 2). As their low “SLz-cousins”the higher Lorentz groups have complementary series. But the discrete series appear in only half of them {Sq1;2n)} with even space-like part
(like SL,(R)), and are absent in the odd case {Sq1;2n+l), e.g. SL,(C)}. The Plancherel formula includes only the principal and discrete series (when present). The detailed analysis of SO(1;n) is contained in the last chapter of [Wal,73], and the original papers
[Hir].
$7.7. SL,(C) and the Lorentzgroup
366
Problems and Exercises: 1. Establish irreducibility of the principal series representations of S q 1 ; n ) by the Mackey's method.
+
'
2. We define the stereographic map fig 9 from the unit sphere S" = {zoz I t I = I} to It 1' il)}, parametrizing both surfaces by a hyperplane the hyperboloid H" = {z R" = { t= ( 0 ; z ) )in Rntf Here maps @:S"-R" and P:H"-.R" are given by
'
b:y = (yo; y)-z = L; and 3:z = '-Yo
r'
x-
where yo = f
1- I y I denotes a point in S",
Y X I 1
Fig.9. Two stereographic maps 8, P fake fhe unit sphere {y} and the hyperboloid { z } onto R", hence define a conformal map @ - l o # from H" into S".
(i) Show that both maps are conformal from S", H" into R". Hint: use the polar coordinates on all three surfaces: {(r;O)} in R", {(q5,0):yo = sind} in S", and { ( t ; O ) : z o = s i n h t } in H". Note that 8 and P transform only the radial variable r = cot$ = cothi.
(ii) Combine 8 and P we get a conformal map 8 y-x:
1
rd
Y
'
o
P from H" into S",
D = 1 r J%
(iii) Since G = S q 1 ; n ) acts by isometries on the symmetric space H" = K\G (K = stabilizer of the pole N = (l;O)), the stereographic map 8-' o P takes it into the conformal group of the sphere S". One can show ([Tay]} chapter lo), that S q l ; n ) comprises the entire conformal group of S". (iv) Compute the conformal factor p ( z ; g ) for any g E S q l ; n) on z E S".
Additional results and historical comments. A. Kirillov [Kirl] traces the onset of the representation theory to the late
XIX century,
and divides its history into 3 periods. First connected with the names of Frobenius, Schur, Burnside dealt mostly with algebraic aspects: finite group and algebras, characters, projective representations, as was outlined in chapters 1-3. Second period brought in compact groups with the contributions of Haar and von Neumann (invariant integration), Peter-Weyl (completeness of finite-D representations). At the same time E. Cartan and
H. Weyl [We31 (1939) unveiled the structure and built the representations theory of simple and semisimple Lie algebras (chapters 4-5). These results not only strike with their profound inner beauty, but find deep applications in a wide range of mathematical and
57.7. SL,(C) and the Lorentzgroup
367
physical subjects: geometry, differential equations, analysis on symmetric spaces (the orginal Cartan’s motivation), quantum mechanics and particle Physics. Soon the need came to study noncompact groups and their infinite-D representations. The first result along these lines, the celebrated Stonevon Neumann Theorem, essentially amounted to classification of unitary irreducible representations of the Heisenberg group. In 1939 E. Wigner [WigZ] made the first attempt to build the theory of elementary particles, based on the infinite-D representations. However, more systematic study (third period) began in late ~ O ’ S , when first classification Theorems for the classid complex Lie group: SL(n;C);
Sqn;C); Sp(n;C), were obtained by Gelfand-Naimark [GN], and Bargman [Bar]. Since then the development of theory went a t an ever increasing pace. For general surveys on the role of the harmonic analysis and group representations we recommend surveys [Mac2,5] and [Gr].
Semisimple gmupe: The basic features of the representation theory of complex semisimple groups were unveiled in the monograph [GN]. It was shown that the principal part here is played by the so-called parabolic subgroups: P C G. These are characterized by the property, that P contains the maximal solvable (Borel) subgroup (e.g. all uppertriangular matrices in SL,), equivalently, the quotient-space P\G
- compact.
Zelobenko
and Naimark [Ze] proved that all irreducible (even nonunitary) representations of G are elernenlaryl7, i.e. induced by 1-D representations of the parabolic subgroup. Monograph [GN] derived characters of elementary representations, and established the Plancherel formula for classical complex groups. It was also found that semisimple groups have complementary series, that do not “belong” to the regular representation RG. Then Harish-Chandra [Harl] extended the “classical results” to arbitrary complex semisimple groups. He also found the general Plancherel formula for semisimple groups in terms of his celebrated c-function ([Hara]), dp(X) = Const I c(X) I -,dX. This yields an explicit expression of d p on an arbitrary group G , or a symmetric space
X = G / K , in terms of the (principal series) weight A, and the root system C = {a} of G, or the restricted root system of space X (55.7). Namely ([Har2];[He12]),
n
dp(X) = Const aEC+
0I 4
where function q5a depends on multiplicity ma of root a,and is given by 17The study of elementary representations is often facilitated by the fact that the double conjugacy claases P,\G/P, are finite for any pair of parabolics P , ; P , (this result h known as Bruhat’s Lemma). In particular, for the Borel (maximal solvable) subgroup B, they coincide with the elements of the Weyl group W of G!
$7.7. SL,(C) and the Lorentz group
368
I
All 4 cases are determined by the relative values of multiplicities m, and m2,, according to the table.
m = 2,4,6,8
... ...
and m2, = 0 m = 1,3,5,7 and m2, = 0 m = 4,8,12, and m2, = 1,3,7 m = 2,6,10,14, ... and m2, = 1
...
The classification of all nonunitary (Banach-space) irreducible representations of semisimple groups was given by Berezin [Ber] (1962), based on his study of Laplacians (Casimirs operators) on G. But the problem of selecting unitary representations among all elementary ones turned out to be technically difficult, and has not been yet completely solved. Significant progreas has been made in the past decades, that has brought about a complete classification for a growing list of groups: S q 2 ; 3); Su(2; 2); Sp(1; n); SL,(F), for any F = RC;Q. For surveys of this important work we refer to [Kn];[KS];[Sp];[Vo]. The representation theory of real semisimple groups added further difficulties. Here elementary representations proved insufficient to build a complete system, even if one allows the holomorphic induction ($6.1). First example of non-elementary “strange series” appeared in the work of Gelfand-Graev for group S y l ; 2 ) . But only after the Langlands’ work [JL], it became clear that “strange series” can not be realized in “function-spaces” (0-forms), but require higher-rank differential forms. Langlands conjectured that all of them could be realized in higher “Z’-cohomologies”, by a combination of the regular and holomorphic induction (see $6.1). The most significant contribution to representations of real semisimple groups came from the works of Harish-Chandra. His papers [Har] gave, in particular, a complete classification of
80
called discrete series, i.e. {T} with square-
integrable matrix-entries, equivalently {T}, embedded in the regular representation RG. Harish-Chandra associated discrete series to compact Cartan subalgebras in 8. So certain semisimple algebras have them, like 4 l ; n ) , while others, e.g. s(n),n 2 3, or any complex
a),
do not. Harish-Chandra characterized discrete series by means of their
characters. This left an open problem to explicitly construct such {T}. The latter was accomplished by Parthasarathy [Par]; Schmid [Sch] and Atiyah-Schmid [AS]. Our main sources in 57.1-7.5; 7.7 were books [GGP];[GGV];[GMS];[Lan];[Ta].
Chapter 8. Lie groups and Hamiltonian mechanics. We shall outlines some interesting applicatione of Lie groups and symmetries to hamiltonian mechanics. Our main emphasis will be on integrable systems and systems that possess large symmetry groups. Although most of the discuasion does not directly involve the representation theory of chapters 1-7 (save for the last section, §8.5), the Lie structure theory of groups and algebras will enter many times. To make our presentation self-contained we included in the first section, 58.1, some basics of the hamiltonian mechanics: Lagrangian formulation, Minimal action principle; EulerLagrange equations; Canonical formalism; sympIectic/F‘oisson structure; conserved integrals, integrability and Darbeaux Theorem.
$8.1. Minimal action principle; Euler-Lagrange equation; Canonical formalism.
1.1. Hamilton’s Minimal Action principle. The state of a classical mechanical system of n degrees of freedom is described by its position vector: q = q ( t ) = ( q l ; ...qn), varying over W”, or more general Riemannian manifold At, called configuration space, and velocity (tangent) vector: q = (ql; ...qn). Its dynamical evolution is determined by the action functional,
s=
li
P.(q;q)dt,
t0
whose integrand .t(q;q), called Lagrangian, depends on position and velocity, and has physical dimensionality of energy. In many cases of interest Lagrangian represents the difference of Kinetic and Potential energies, P. = K - P , where h’ = 1 . 2 . 2Q
7
or more generally, is given by a Riemannian metric-tensor { g i j ( q ) }on At,
K = !jC g i , qiqj; while P = V(q)- a potential function. According to the Hamilton’s principle of Minimal Action: a trajectory (evolution) of the classical system must minimize (or give a stationary path) of the actionfunctional. So it satisfies the Euler-Lagrange equation:
a 6q(t) =P q -d(P..e=o. dt .P case of
(1.1)
Equation (1.1) represents a 2-nd order OD system in n variables. In the classical P. = “kinetic” - “potential”, (1.1) turns into the Newton’s equation,
9’’
-aV(q) = F- force.
The canonical formalism reduces the 2-nd order Euler-Lagrange equations to a 1 - st order system of size 2n. We introduce a new set of variables: pi = a. .P.(q;q)- conjugate momenta, ‘11
(1.2)
$8.1. Minimal action principle; Euler- Lagrange equation;
370 and
H ( q ;p ) = p
- L, hamiltonian/energy function.
Solving a system of equations (1.2) for q-variables’, we get q = Q(q;p),and these are substituted in the hamiltonian H ( q ; p ) .Then the Euler-Lagrange equations (1.1) are shown to be equivalent to a hamiltonian system
@ - a PH ;
Ap= - a p .
(1.3)
We shall first demonstrate the canonical formalism in the case of N-particle systems. The corresponding Lagrangians are N
L = 3’ - V ( q ) ,or 3Cmjq: - V, where ( m j } denotes masses of particles. Then the conjugate momenta: p j = r n j q j , and the hamiltonian,
+
+-
H
= c l 2mj p . J 2 V(q);“kinetic” “potential”, a familiar expression from elementary calculus/mechanics. For
more general (“kinetic - potential”) Lagrangians on manifolds A, the Euler-Lagrange (1.1) takes the form & ~ g i j q j ) - a q i v = 0,
while the canonical variables:
..
P; = E g i j Q j ; H = + C g ” P i P j
+ V(q),
( g i j ) denotes the inverse matrix (tensor) to ( g i j ) , and the hamiltonian system
becomes \pi = - a,v. So geometrically, momentum variables { p } can be identified with cotangent vectors on A, and the (position-momentum) phase space becomes a cotangent bundle I*(A).The change of variables (q;q ; L)+(q; p ; H ) , called the Legendre transform, can be interpreted as a map from the velocity phase space, tangent bundle T ( A )= { ( q ; q ) } , to the momentum phase space, cotangent bundle T * ( A )= { ( q ; p ) } , that takes solutions (trajectories) { q ( t ) ; q ( t ) }of the Euler-Lagrange equations (1.1) to those of Hamilton system (1.3) (problem 2). ‘provided it could be solved, i.e. the Hessian of & in q-variables is nonsingular, det(-) a2L # 0. aCr;aqj The latter is always the case with the classical Lagrangians, L = “kinetic” - “potential”, on A, whose Hessian turns into the metric tensor { g j k ( q ) } .
$8.1. Minimal action principle; Euler-Lagrange equation;
371
1.2. Symplectic structure and Poisaon bracket. Phase space 9 = T * ( A ) is equipped with the natural symplectic/Poisson structure, given by a differential (canonical) 2-f0rm2 on 9:
R = C d p i Adqi, (1.4) i in standard (dual) local coordinates { ql...qn;pl...pn}.In other words, we take a basis in the tangent space T q ( A )and the dual basis { d q l ;...dqn} in the cotangent space T z ( A ) , so each point x = ( q ; t ) E T * ( A ) , q E A, ( = C p j d x j E T i , can be represented by a 2n-tuple { q i ; p j } .
{aql;...aqn}
Symplectic structure on a general manifold 9,with local coordinates { x l ;...zm}, is given by a non-degenerate closed %form,
0 = X U j k d X j dXk, (1.5) with the usual proviso that R be independent of a particular choice of { x l ;...xm} (i.e. coefficients { a j k } transform as a tensor on under coordinate changes on 9).Clearly, non-degeneracy of R constraints dim9 to be even. In addition, one requires closedness of R, in the sense that its differential &?= E E i j k a i ( a j k ) d Z i A d x j I\ d X k = 0.
Here Eijk denotes a completely antisymmetric symbol (tensor) in 3 indices, normalized by elZ3 = 1. An alternative way to describe a symplectic structure on 9 is in terms of a skew-symmetric bilinear form 1, on tangent spaces {T,(T):xE T},
(1t17) = z b j k t j 9 k ; (77 E T Z . Two structures are related one to the other by R = j - ' d x A dx, in other words matrix (a.J k ) = ' ( a $ ) - ' . Clearly, the standard 2-form 0 yields the standard symplectic matrix, L
J
Examples of symplectic manifolds include: 1) The standard (flat) phase spaces: W2 = { ( x ; p ) }with R = dx A d p , and W2" with 0 = C d x j A dpi- Those are often convenient to write in the complex form: z = x -ti p E C", then R = A d?i.
3.z
2) Cotangent bundles "*(A) over manifolds A, with R = E d q j A d p j ; 'Let us remark that (1.6) is independent of the choice of local coordinates on A. Indeed, if ,denotes a coordinate change, then differentials { d q j ] are transformed by the Jacobian map f' = 8 9 while differentials of co-vectors by inverse transpose of f'. So the product C d p j pI d q j remain8 invariant. f:q+q'
(r),
58.1. Minimal action principle; Euler-Lagrange equation;
372
3) The 2-sphere S2= {(c$;O)} with R = sinc$dc$A do.
4) Co-adjoint orbits of Lie groups: O C ~ (chapter * 4). The tangent space T , (2 E 0 ) is identified with the quotient 0/6, - Lie algebra modulo stabilizer subalgebra of z OZ = { ( : a d i ( z ) = 0). As a bilinear form on tangent spaces,
I [(;171)
Q(f;rl)= .(
= (ad;(.)
Id; t77l E (5.
5 ) Finally, we shall mention the so called KiiMer manifolds, complex manifolds A with a hermitian metric-form: ds2 =
C b,,dz,dZv;
-
b,,,, = b,,,;
whose imaginary part is a closed 2-form7
R=
b,,dz,,
(1.7)
A dTv.
To check skew-symmetry of R one notes that in real coordinates {z,, = Sz,,; y,, = St,,},
R = Adz A d z + 2Bdz A dy + Cdy A dy, where A = C and B are real (antisymmetric and symmetric) matrices S(b,,,,) %(b,,,),
i.e. (b,,”)
= B + iA. Hence
and
defines a symplectic structure on A.
Symplectic structure on any phase-space 9 (cotangent bundle, co-adjoint orbit, etc.), allows to assign certain vector fields, called hamiltonian fields, to functions (observables) F on 9, F 4 Z F = j(aF).
In other words we take a gradient vector field of F and “twist” it by a skewsymmetric linear map 1 on tangent spaces {Tz(9’)}.The standard symplectic structure (1.7) on phase-space R” x R”, yields hamiltonian vector field, z, = a p F . - a q F .
aq
ap.
Each vector hamiltonian field generates a hamiltonian flow, {ezp(tE,)}-a fundamental solution of ODS,
5 = B(aF(z)), (1.8) which generalizes the canonical system (1.3). Hamiltonian flows possess many special features, for instance, all of them preserve the canonical (Liouville) phase-volume, d”q
-
d”p on 9,since all respect the canonical/symplectic form 1, or R (problem 3), and “Liouville volume” = R A ... A 0.
Symplectic structure also defines a Lie-Poisson bracket on the vector space of observables { F ( z ) }on 9, namely
{F;G) = (r(aF)I aG),
58.1. Minimal action principle; Euler-Lagrange equation;
373
which in special cases, W2" or T * ( h ) ,turns into,
{ F ; G }= a , F - d , G - a , F . a , F . The reader can verify directly all properties of Lie bracket (skew-symmetry and Jacobi identity) for {F;G}3. In fact, the Poisson bracket of any two observables corresponds to the standard Lie bracket of their hamiltonian vector fields,
-- --
{ & G } ~ E { C G= I [ZF;5,] = = F = G - Z G C F . Thus the space of observables { F ( z ) } on 9 acquires a structure of an m-D Lie algebra, a subalgebra of all vector fields bD(9).The corresponding Lie group consists of all canonical transformations on 9, i.e. diffeomorphisms {q5} that preserve the symplectic structure,
Tq5'%#) = 8.; equivalently coordinate changes, y = d(z), that preserve the canonical 2-form,
We shall list a few examples of canonical transformations: i) symplectic matrices { A E S p ( n ) } in R2" = { ( q ; p ) } ; they clearly preserve the standard 2-form: R = C d p j A dqj. ii) any coordinate change (diffeomorphism) y = @(z), from manifold .A to N, induces a canonical map,
YO);
q5:(x;o+(@(x);TAwhere A denotes the Jacobian matrix of @ at {z}, A = @: (problem 4). The reader has probably noticed some coincidence in terminology: symplectic Lie groups/algebras on the one hand, and symplectic structure/geornetn'es on the other. The relation between two becomes apparent now: Jacobian matrices of canonical (symplectic) maps are symplectic matrices4! 1.3. Conserved integrals; action-angle variables and the harmonic oscillator. The hamiltonian evolution (1.8) of the position and momenta variables { q ; p } gives rise to 3The Jacobi identity on general symplectic manifolds results from closeness of the canonical 2form 0. 41n this regard symplectic groups plays the same role in the symplectic geometry, as orthogonal groups in the Riemannian geometry. There exists, however, a striking difference between two kinds of geometries. The Riemannian geometry is fairly rigid in the sense that isometries of m (even in the best case of symmetric spaces) form only a finite-dimensional Lie group, whereas "symplectic isometries" (canonical maps) are always infinite-dimensional!
374
$8.1. Minimal action principle; Euler-Lagrange equation;
evolution of any other observable (function) F on T * ( A ) ,
P = {F;H}. Functions that remain constant along trajectories of the hamiltonian flow, { F , H } = 0, are called first integrals of (1.8). Indeed, the Legendre “back-transform” (q;p)+(q;q), takes such F into a function F ( q ; p ( q ; q ) ) ,constant along trajectories (solutions) of the E-L equation (1.1) i.e. gives a 1-st integral of (1.1). Hamiltonian H is itself an integral, as {H;H} = 0. In dimension n=l, it is the only integral. So 2-nd order E-L equation, Lq- $L4 = 0, is reduced to a 1-st order ODE: H(q;p( ...;q ) ) = E -const.
In the classical (Newton) case this yields I *+v,I.2 zq +V = E, ZP
whence we get an implicit solution in the form of integral
5 J;i&
= t - to.
(1.9)
More generally, each Poisson integral F of the hamiltonian system (1.3) reduces its total order 2n by 2. The proof is based on the following general result.
Dubeaux Theorem: Any set of functions (obseruables) {F,; ...Fk;G1;...G,} on a phase space (symplectic manifold) 0 that satisfy the canonical commutation (Poisson bracket) relations { F j ; G i }= Jij, can be locally eztended to a canonical coordinate system on 0,{pl;...p,;q,;...q,}, p; = pi (i=l, ...k) and qj=Gj (j=l;...m).
In particular, a Poisson integral F can be made the 1-st momentum variable p , of the new coordinates. Then hamiltonian H ( p ; q ) becomes independent on the corresponding canonically conjugate position ql, and p , is constant along the flow,
p1 =
Q1
= { p , ; H } = 0.
Setting pl= El-const we reduce H to a hamiltonian H ( E , ; p , ;...p,;q,;...q,) in ( n 1) degrees of freedom, so the system becomes:
p . = d H;
(4i‘=
‘li
-dPiH;
t
i=2; ...n; and p , = El; q1 = ql(0)
+ IH(...)ds. n
A hamiltbnian system (1.3) in n degrees of freedom is called completely integrable (in the Liouville sense), if it has n functionally independent, Poisson commuting integrals {Fl;Fz;...F,}, { H ; F i } = 0; and { F i ; F j }= 0, for all i , j . (1.10) Functional independence means gradient-vectors { VF,} are linearly independent
58.1. Minimal action principle; Euler-Lagrange equation;
375
at each point t = ( q ; p ) . In the standard terminology, commuting integrals (1.10) are said to be in involution. Let us remark that any set of n functionally independent integrals {Fl;...Fn} allows to reduce the total order of the system by n, i.e. to bring (1.3) to a first order system in n variables: Fl(G4)= El F,(q;q)'= En' Complete integrability yields, however, more than a mere reduction of order by n, as each subsequent integral F , respects the joint level sets and evolutions of all preceding variables in the canonical reduction. Hence, the system could (in principle) be reduced to order 0, i.e. solved completely! The procedure is best illustrated by the harmonic oscillator
{
H=izpf+wfqionW2".
As first integrals of H one could take all coordinate oscillators,
{ H i = ;(pi
+ wiqi): 1 5 i 5
72).
zHi.
Obviously, H= The joint level sets of { H i = E j } form a n-parameter family of invariant tori in WZn, and the hamiltonian dynamics consists in a uniform motion in some direction along the torus. Introducing polar coordinates (P; 8 ) in the i-th phase plane { q j ; p j } after , rescaling, qi+wiqj, the j-th hamiltonian becomes H 1. = L( ? + q?) = LP?. 2Pc I 2 1 In polar coordinates the flow of each
Hiis given by an OD system
!= O, i.e. P = E -con~t, 8 = 0, + Et. (4 = r The Poisson bracket { r ; e }= i,hence {$';fI} = 1, and the entire set of variables { H,; ...H,; 8,;...en} satisfies the canonical commutation relations: {H..B I' 3.}=6.. 1J' Returning to a general completely integrable hamiltonian system with n commuting integrals { F,; ...F,} (called actions), the Darbeaux Theorem implies the existence of a canonically conjugate set of angle variables: {el;...en}, that satisfy the canonical relations The hamiltonian H in new coordinates { F; 6) becomes a function of actions only, H = f(Fl;...F,J, since aej-- Fj = 0. The joint level sets { F j = Ei} form a foliation of the phase-space into invariant tori Tn (or products TkxWn-'),
and the dynamics
376
$8.1. Minimal action principle; Euler-Lagrange equation;
resembles the oscillator case. Precisely, if
4j
denotes the canonical coordinate change:
(P;q)-+(F;e>,and
+
F~ = E ~e,(q ; = ej(o) i g 8H F i ( ~... l ;E"), the hamiltonian flow in the action-angle coordinates, then solution in the original coordinates becomes (1.11) ( p ; q ) ( t=) @-I(... E ~...; e,(t) ...). Let us remark, that solution (l.ll), although written explicitly, may be of limited utility unless one is able to compute the canonical map 4j.
Our next goal is to establish integrability, find l-st integrals and, if possible, explicit solutions of different hamiltonian systems. We shall review a number of the classical models, and also discuss some newly discovered examples. Our principal tools, in the study of conserved integrals, will be symmetries of the problem, and the Noether Theorem, to which we turn in now.
$8.1. Minimal action principle; Euler-Lagrange equation;
Problems and Exercises. 1. Check the equivalence of the Euler-Lagrange equations (1.1), and the hamiltonian system (1.3). 2. (i) Demonstrate that the "squareroot-kinetic-energy" functional of a Riemannian metric {gij(z)), L[z] = ( cgij(z)2i2j)1/2, yields geodesics, as extremal curves. So the EulerLagrange equation for L is precisely the equation of geodesics (see Appendix C). The geometric meaning of this result is quite transparent: the Lagrangian density Ldt represents the arc-length element of the metric g! (ii) Show that the Legendre transform kinetic energy:,functional K = + c g i j ( z ) i i 2 j of metric g, becomes the hamiltonian H = i c g r 3 ( z ) p i p j , of the dual metric (gr3) = (gij)-l, on the cotangent space.
3. Check that the Jacobian map ];o :[f-
a hamiltonian vector field
2, = ( a ( p ; q ) ; b ( p ; q ) ) a; = apF; b = -8,F; has determinant 1, hence the hamiltonian flow of F preserves the Liouville volume d"p A d"q. 4. i) Any coordinate change, q+(q),
on configuration space Ab defines a canonical
transformation,
Ad: ( q ; P)+(d(d;=dC YPN; on the phase-space T*(Ab)(Check that A preserves the canonical 2-form dq A dp).
4
ii) Apply part (i) to transform the standard canonical (q; pkvariables in the phase-space R2x W2 to polar { ( r ;O ) } , and spherical {(r;d;fl)}-coordinates. Compute the hamiltonian H = p2 V ( q )on R2 in polar and spherical coordinates, and show
+
2
Hpolor= P,
1 2 + -P r2 + V ; Haher= p j +$p$
+ sin24b2)+ V ;
iii) Do the same for elliptical coordinates: where {rI;-P2) are distances fig.1). Show
of focal points: {( f a;O)} (see
Fig.1: Elliptical coordinates in R2 are made of confocal ellipsi: ( = r1 + -P2 = Const; and hyperbolae: q = -Pl - r2 = Const. Here z = a ch t c o d ; y = a s h t sine; and parameters ( = 2 a c h t ; q = 2acose. Elliptical coordinate change can also be viewed as a conformal (analytic) map, w =t + i b z =z+iy, given by z = $ e W + e-"').
377
378
58.2. Noether Theorem, conservation laws and Marsden- Weinstein reduction. 58.2. Noether Theorem, wnservation lam and Marsden- Weinstein reduction. Conserved integrals of hamiltonian systems are often derived from the oneparameter groups of symmetries via the celebrated Noether Theorem. In particular all basic conservation laws of Physics: energy, momentum, angular momentum, arise this way from the translational and rotational symmetries of the system. Going further in this direction, we consider systems with ‘‘large” Lie groups, or algebras of symmetries (not necessarily commuting). Such symmetries allow to reduce the system to fewer variables, via the Weinstein-Marsden reduction process.
Given a configuration space of a classical mechanical system with Lagrangian
L = L ( q ; q ) , we consider a one-parameter group of transformations
CC:(q;t)+(q*;t*) on
A x [O;co)(point transformations along with time reparametrizations),
L*
+
+ ...
= q e$(q;t) = 11,,(q;t) (2.1) = t + q q q ; t ) + = &(q;t)* In other words vector field $J = Sq and scalar field q5 = S t , represent infinitesimal 9*
...
generators of the family Cc. Obviously any group of transformations (2.1) of A generates transformations on the path-space of AL (trajectories of the system):
(40 t ) +uq*(t*); t*)= ( q + 6%t + S t ) = 11,€(Q A).
(2.2)
O
2.1. Noether Theorem: A n y one-parameter group of transformations (2.2) with
generators (11, = Sq;$ = St), that leaves invariant S = L(t;q;q)dt,gives rise to a conserved integral
I
J=p.bq-HSt=p.$J-Hq5
the
action-functional
Const
Here p = a. L, and H = p . q - P. are the canonical variables (conjugate momenta Q and the hamiltonian) of L. The proof follows from the general variational formula for a functional L (problem l), when we allow free motion of the end points as well as all reparametrizations of the timevariable: t+t* = t+6t, ‘1
6s = ( p a 6 9 - H a t ) [
to
‘1
+ J(Lq-$Li*469
- 4 61)dt.
(2.3)
to
The first factor in the integral represent the E L equation that vanishes along any critical path. If furthermore functional S is invariant under (2.2), then
6s = 0, and we get
J ( t l ) = ( p . 6 9 - H 6 t ) I t , = J ( t , ) - constant along any critical path! QED.
Remark: Conservation of Noether integral J = p a 11, - Hq5 can be recast in the hamiltonian formulation. Indeed, function J = J(q,p; t ) , considered on the phase space
58.2. Noether Theorem and the Marsden- Weinstein reduction.
379
of variables (q; p ) clearly satisfies the equation,
J,+(J;H}=O. Thus each symmetry of the Lagrangian L produces a Poisson commuting integral (symmetry) of the corresponding hamiltonian H. So any Lie group/algebra of symmetries of the Lagrangian gets represented by the Lie group/algebra of hamiltonians. 2.2. Conservation laws. As an application of Noether's Theorem we shall derive the basic conservation laws of classical mechanics.
Energy conservation. If Lagrangian is time independent, L L ( q ; q ) ,then time shifts: t-+t+, form a translational symmetry group of L, whose generators: II, 0, #=I. Hence the Noether integral becomes hamiltonian/energy function, 0
IJ
= H ( q ;;p ) = Const].
.Momentum conservation. We assume now that the space shifts q+q+cu in certain directions u leave Lagrangian invariant. Then the generators 4 = 0; II, = u yield
J, =p . ~ , the u-component of the momentum remains constant. The standard example is a N particle system in R3 with pair interactions,
e = ;xq; - V , where potential V = C v(qi - q j ) . Obviously, simultaneous shifts, do not change V , hence system,
(Q1 ... QN)+(Ql+";... nNt"), 21 E R3, L. Thus we get conservation of the total momentum of the
1-1
.Angular momentum conservation. The source of the angular momentum conservation are rotational symmetries of the Lagrangian, like the central potential (Kepler) problem: V = V ( I q I ) in L = ;q2 - V . Let us assume that P. is invariant under rotations in the ij-th coordinate plane. The corresponding symmetry generator is a linear vector field
+ and the Noether integral becomes
.ij=[
-1
]:[I' 0
J i j = Piq j - P jqi
380
58.2. Noether Theorem, conservation laws and Marsden- Weinstein reduction.
- the ij-component of the angular momentum. In the N-body system with potential V = x u ( I qi - q j I ), only simultaneous rotations of ( q1...qN) by u E Sq3) leave L invariant. So the total angular momentum is conserved,
Remark. A relativistic particle Lagrangian has the form
so the corresponding symmetry group consists of Lorentz transformations S0(1;3), whose
generators include rotations in spatial directions,
-
-
I
J i j =I:[ 1 5 i;j 5 3, i.e. Lie algebra R = 4 3 ) c so(l;3), as well as relafiuislic boosts:
generators of hyperbolic rotations,
C 4 1 ; 3 ) . The corresponding Noether integrals are JOi
= PoQi + P i 9 0
2.3. The momentum map and Weinstein-Marsden reduction. Let 8 = span{F,; ...Fm)be a Lie algebra of hamiltonians on the phase space A. Then Lie group G of O acts on 311 by canonical transformations { e z p ( t Z F ) F : E (5). Any such algebra O defines a map J from JL into the dual space 0*,by evaluating hamiltonians F E O at a point z E A (“point evaluations” are clearly linear functionals on C ( A ) ! ) ,
( J , I F ) = F ( z ) ,for all F E 0. The resulting momentum map, intertwines two actions of group G: canonical action on A, and the co-adjoint action on 6*:if g = g(t) = ezp(tF), denote denote a Lie group element (of generator F E 0), and #g = e z p ( t Z F ) - the corresponding canonical transformation, then
J(dg(z))= Ad;(J(z)),for all g E G; z E A.
(2.5)
Furthermore, map J preserves the Poisson structures on JL and O*. We remind the reader that the dual space of a Lie algebra has a natural Poisson bracket (chapter 6 and f8.l), defined in terms of the Lie bracket on 0.Namely, for any pair of functions Fl(z); F z ( z ) on @*, gradients { a F .} belong to the Liealgebra itself, so one can 3
Poisson-Lie algebras
(5
often arise as symmetries of hamiltonians
H on A. The
58.2. Noether Theorem and the Marsden- Weinstein reduction.
38 1
momentum and angular momentum operators are obvious examples of the momentum map,
P, = u . p (u-fixed direction), and J = {J. = ( p A z) . 3E 3)
k -pkzj}.
= p .z 3
The former corresponds to translational symmetries (in the u-th direction)
O 21 Wm, the latter to rotational symmetries, 6 N 4 m ) . We shall see that symmetries of hamiltonians { H } play a double role. On the one hand they apply to reduce the number of variables (degrees of freedom). On the other hand some, seemingly complicated, systems could be “lifted” to larger, but “simpler” systems, typically based on Lie groups and symmetric spaces. We shall analyze a few important examples of this sort in the next section. But here we shall concentrate on the “reduction part”. Given a Lie algebra O of symmetries of hamiltonian H , we consider a joint level set of all observables { F E O} (it suffices to pick a basis Fl;F2;...Fm in 0 ) ,at a “level toE O*”,
A. = J-*(to). Subset A, is invariant under the flow of e z p ( t H ) . Furthermore, A, is invariant under the stabilizer of to,subgroup Go = {g E G:Ad,(to) = 0}, an obvious consequence of (2.5). The action of group Go on A, splits it into the union of orbits (a fiber bundle), and the reduced space we are interested in is the quotient of A. modulo Go, the orbit space5 R.
In our setup space R, c R = Jb0/G,, also has a natural symplectic structure: the Poisson bracket on A. restricted to Go-invariant functions. This Poisson bracket turned out to be non-degenerate, hence yields a symplectic structure on Ro (a Theorem due to Weinstein and Marsden). We shall illustrate the reduction procedure by 3 examples. 2.4. Examples.
Radial (spherically symmetric) hamiltonian H = H ( I q I ; I p I ) in W3 x R3, a generalization of the classical Kepler central-force problem: H = i p z V ( I q 1 ). Any such H has an Sq3)(angular momentum) symmetry, the momentum map being,
+
J : (%P)+9 x P
E43).
Here we identify 3-vectors [ = ( a ; b ; c )with antisymmetric matrices,
51t should be mentioned here that orbit-spaces are typically non-smooth manifolds with corners and edges (take, for instance, R”, modulo Sqn)). But there is always a dense open set 0,of ‘generic (maximal-D) orbits” in Q, that possesses a smooth (differentiable) structure!
'-
M -
c -c
-L b -a
-b a
;
-
2
so q x p becomes a element of 4 3 ) . We fix a joint level set
4 = { ( % P ) : P x q = Jol, of the (constant) angular momentum (Noether’s Theorem). Without loss of generality vector J , could be taken, as (O;O;jo),where j , = I Jo I . Then vectors ( q ; p ) belong in the plane orthogonal to the (constant) angular momentum, which reflects the well-known property of central potential forces: planar motion!
A 3-D manifold 4 = { ( q ; p )E R2x W2; q A p = j,} is foliated into orbits of the stabilizer subgroup GoN Sq2)of J , E d 11 4 3 ) . Clearly, any u E Sq2)takes 21:
k;p)+(qU;pU),
Hence, the reduced dynamics in the phase-space N=W2 is given by the hamiltonian.
where j o = IJoIis the total (fixed) angular momentum. Thus we have reduced he number of variables to 1, which shows complete integrability of rotationally symmetric hamiltonians (problem 2).
Example: Integrable hamiltonians H. A commuting family of integrals { J,} maps phase-space 9 into the Lie algebra R". Here group G N U" (or R"' x Un-"') is generated by the flow {ezp(t,Z, + ...+t,Z,)} of hamiltonian fields {Z,} of {Jk};the coorbits { t } are points in w", stabilizers Gc = U", and inverse images J - ' ( f ) coincide with invariant tori in 9. So the reduced space J-'(()/T" is trivial. 0
0 Example: Left-invariant metrics on Lie groups. Tangent/cotangent spaces of Lie group G at each z E G can be obtained from a single space (Lie algebra), 6 21 T,, and its dual O* N T,*,by left translations,
q : t E T e - - + q - t E T q andpETB-+q*-'*pET;; , qEG. For the sake of presentation we shall consider the matrix group and algebra G
58.2. Noether
Theorem and the Marsden- Weinstein reduction.
383
and 0.Then q .( will be the matrix multiplication. The left-invariaot fields on G are of the form: ( ( q ) = q - 6 (( E O ) , and any left-invariant metric is uniquely determined by its restriction at {e}, a symmetric bilinear form B on 8,
(Bq-'
I q-' v); (;v E T,. L = $(B... I ...), and the *(
*
Thus we get a Lagrangian corresponding hamiltonian H = $(B-'... I ...) on phase-space T*(G), which possess a large symmetry group G of right translations {z+z.q; s;q E G } . Those clearly commute with left-invariant vector fields {( = ( ( q ) = q . ( } . The corresponding family of right-invariant hamiltonians (integrals of H ) consists of {J&; P') = (t * Q I I)'):
t E w' E q , on T*(G).
Identifying co-vector p' at point { q } with a co-vector p = q - * . p' E @* (via left shift with q - ' ) , we can write such functions J's on G x @* N T*(G)as, J&
P) =
I
I
- '& PI ) = (ad,(n classical
mechanics [AM]; [Am]. The same applies to $3.2 (the Marsden-Weinstein reduction appeared first in [MW). Classical problem from the standpoint of Lie symmetries are treated in many sources ([Am], [Oh]). The recent book [Per] by A. Perelomov contains a comprehensive survey of the classical results, as well the recent developments in “integrable hamiltonians” (see also [FM]). This book along with the review article [OP] was our main source in 58.4. The “hydrogen quantization problem” goes to the very onset of quantum mechanics. W. Pauli (1926) and V. Fock (1935) first discovered the So(4)symmetry of the hydrogen hamiltonian on the Lie algebra level (see [LL]). J. Souriau
[Sou] and J. Moser [Ma] reviewed the classical Kepler problem, and applied the “stereographic projection” method to it. The higher S q 2 ; 4)-symmetries were analyzed in
88.5. The Kepler problem and the Hydrogen atom
422
[Simm];[Sou];[Kum];[Hur];[GSp], and more recently in [GS2]. The general review of geometric quantization and the references could be found in [Sn], [Hurl.
Problems and Exercises: 1. Verify that commutation relations (5.2) with constant E define one of 6-dimensional Lie algebras: 4 4 ) ; 4 3 ; l ) or e3= R3 D 4 3 ) , depending on sign of H. 2. Check that the "combined angular momentum"-"Runge-LenZ" quantum vectors (5.9) obey the commutation relations (5.2), and both Operators commute with H.
3. Laguerre polpomiab and the hydrogen bamilbnian. Generalized Laguerre polynomial L g ( z ) , of degree n and order a, can be be defined 88 a regular solution of the ODE, zy"
+(a+l-
z)y'
+ ny = 0.
They form an orthogonal family on Rt with weight w(z) = z"e-=, and can be generated by the Rodrigues formula, ~ g ( z= ) Const z-aeZ[za+ne-7(n).
Show that the reduced hydrogen hamiltonian, R"+:R'+[($-;)--]R
= 0.
m(m+l)
could be brought to the Laguerre form. Steps:
(5.22)
r2
1' Verify the following commutation relations for differential operators: L = 0' and M = z2a2 bza c (Euler-type);
+
+
+ + + +
+
= eAz[(a A)2 6(a A) c] = eAZ[L 2A8+ ( 6 1 ( i ) L[e (ii) M[Z....I = Z.[M + 2sza + s2+(6-1)s]
+ 60 + c,
+ A2)]
2' Take the reduced "hydrogen operator" (5.22), and apply (i) to the product R = e-r/2u(r), to get an ODE, m(mt1) )u = 0. M[u] = a%+ ($- 1)au - (++r2
Break M into the sum of 2 Euler-type operators, M , (M,
+ ~ , ) [ r ' v ] = r8{a2+
(v -
1)8 -
+ M,, and show that + s ( s t l ) -r2m ( m t l ) "1-
For the singular term (...)r-2 in the potential to cancel out index s must be equal rn, and the resulting ODE becomes a generalized Laguerre equation! 4. i) Verify relations (5.18) for { h ; J ; L } ; ii) show that the canonical 2-form on 9, c T * ( p )is taken into the form W , (5.19) by map I; iii) check that variables {zi}, and {yj} Poisson-commute, {zi;yj} = 0. Find the commutation relations among {zi}.
Appendix A: Spectral decomposition of self adjoint operators.
423
Appendix A Spectral decompasition of setfadjoint operators.
Any selfadjoint operator A in Hilbert space 36 (bounded or unbounded) admits a spectral decomposition, which generalizes the notion of an eigen-expansion of a symmetric matrix,
A
N
r 1
...
J
Here {A,; ... A,} = spec(A) denotes the eigenvalues of A , and the k-th diagonal entry/block corresponds to the k-th eigensubspace of A. In other words space 36 is decomposed into the direct sum of E,; and operator A I E,= AJ, i.e. (A.1) where Pk means the orthogonal projection from 36 to E,. Diagonal matrices can be thought of as multiplication operators A: (fk)+ ( X k f k ) , on spaces of (scalar/vector) ntuples {(fk):l _< k _< n}. In other words there exists a unitary map %: 3 6 4 @ Ek, that conjugates A to a multiplication operator,
I %A%-’
= A:
f(A)+Af(A)
I
(A.2)
Both results: spectral decomposition (A.l) and diagonalization (canonical form) (A.2), can be generalized for all selfadjoint Hilbert space operators. Precisely, spectrum of a selfadjoint operator, spec(A), is a closed subset of R. The role of eigen-projections {P,} is played in general by the family of spectral projections { P ( A ) } , associated to closed subsets of A c R, or equivalently by the spectral measure (resolution) dP(A). Projections { P(A)} form a commuting, monotone (P(A,) 0); -Poimare disk { I t I < 1) - Minkowski space - rank-one hyperbolic space of dim = n - projective spaces of dim 1,k; also - projective spaces over reals, complex spaces - the n-sphere {IIZII = 1) in Wn+' - root system (positive roots) of a semisimple Lie algebra - Weyl chamber
SP-
- vector/Hilbert
SMn
space - tensors of rank m over r;also T €4 N('T). - symmetric tensors of rank m - polynomials of degree m - algebra/space of d x d, n x m matrices - harmonic polynomials/sphericd harmonics of degree m - m-th anti-symmetric tensor power of space Y - symmetric n x n matrices
G;21
- group and its dual group (dual object)
Tn
- torus - permutation group of n elements;
Matd;Mat,
xrn
nrnm
Groups and algebras
Wn An
w
- alternating group (of even permutations) -Weyl group of a semisimple Lie algebra (root system) The standard notations in the upper-case bold are reserved for the
classical Lie groups: Sq3) S q p ;q); S q n ) ;SU(n);9 4 2 ) ; SL,(C); SL(n); SL(3);GL, etc., and the corresponding lower-case bold denote their Lie algebra En = Wn b S q n ) - Euclidian motion group; (3, - its Lie algebra Wn - Heisenberg group - Cartan (diagonal) subgroup/subalgebra of a semisimple Lie group H;8
448 B*;B* (li = st @
List of notations - Bore1 subgroup/subalgebra of upper/lower triangular matrices in SL, - Cartan decomposition
Function spaces; algebras; distributions - space of continuous functions e(A) P(31L); em(&)- rn-smooth (m-smooth) functions - compactly supported functions - Lebesgue spaces of integrable functions - Sobolev space of order s - distributions - algebra of all (bounded) operators of a Hilbert/Banach space 'Y - Weyl algebra on Wn, generated by all multiplications differentiations Transforms; Operators - Fourier transform and its inverse
- Radon transform and its dual - operation of symmetrization,
Tm('Y)+!fm('Y)
- wedge product of vectors, tensors, differential forms - Laplacian (Laplace-Beltrami operator)
- spherical mean of function f at {z}, of radius r Representations - intertwining space of representations S and T - the commutator algebra of T - tensor (Kronecker) product of representations T and S - direct sum of representations { T 3 } - direct integral of representations (2'': z E %} - adjoint representation of Lie algebra/group
and
Index.
A ............................... ............................................
Action-angle variables 373 Action functional (Lagrangian) .................369 46 Adjoint action Algebra: Lie ....................................................... 46 249; 256 enveloping (of Lie algebra) group (convolution) algebra 24 commutator algebra ............................. 37 Angular momentum (operator) 176. 276. 379
.......... .................. ..
Classical mechanics system .................369. 385 Clebsh-Gordan coefficients ........... 130.181. 240 151 Coboundary .............................................. Cocycle ...................30,58,138,150,275,285,306 150,271,292,302 Cohomology ........................... Commutator 38,40,45,138,258 algebra, Com(T) 48,52, 154 (derived) group G' ...................... (derived) Lie algebra ................ 53,192.207 47.50, 283 (Lie bracket) [A; B ]..................... Compact form (Lie algebra) .................. 193,208,248 19,25,41,63,126,157,170 group 39,63,425 operator Completely integrable hamiltonian system 374 Conformal map ................................... 13,109,365,377 group/algebra ...................... 15,21,305,366 432 Connection (affine) ................................... form ............................................. 296. 435 Conservation laws ..................................... 378 Conserved integrals ................373,378,385.393 Continuity (of representation): .................... 28 27 Convolution ................................................ Convolution algebra .................................... 28 389,411 Coulomb potential .............................. 353.432 Covariant derivative ........................... 241,353,358,434 Curvature ..............................
............
................... .....................................
B Bessel function .................... 73.78.82.89.96. 265 89. 103 differential equation ........................ potential .................................... 73.74. 111 Bore1 subgroup ............ 20.141.210.269.305. 361 subalgebra .............................. 207.268. 364 Bundle. vector. G-invariant .......... 138.149.258. 267 115.227.344. 359 tangent/cotangent holomorphic line .............. 267.271.293.296 Burnside-von Neumann Theorem ................ 39
...........
C Canonical commutation relations (CCR) ...... 275.375 variables ................................. 291.370.374 295.302. 371 2-form .................................... 373.398 transformation .............................. 372 volume ............................................... Cartan automorphism 241 167.176.189.204. 310 basis .......................... .192. 242 classification ................................ .318. 322 decomposition ............................. subgroup/subalgebra ....... 174.197.206. 242 Casimir (central element) ...... 250.330.344.413 52 Central ideal ............................................... extension (of group. algebra) ..150.274.285 131.223 projection ..................................... potential (force)............................ 379.385 Central Limit Theorem ............................... 85 Character (dual) group ........................... 37.61.69. 83 of representation ........................ 38.40. 125 infinitesimal 250.332 formula ..................... 261.273.280.298. 314 Characteristic ideal (of Lie algebra) Classical Lie groups: .................... 21.41.54. 191
....................................
.................................
D Decomposition: 40. 43 primary (of repn) ............................ spectral (operator) ........... 184.342.412.423 of representation ................ 39.130.142. 175 130. 181 tensor (Kronecker) product 39.225.235. 249 Degree (of representation) Derived subgroup. subalgebra. 53.191. 202 series ....................................... Derivative: covariant ...................................... 431. 437 296 exterior ............................................... fractional .............................................. 78 47. 302 Lie ................................................. Direct sum/integral ................................ 33. 40 Discrete groups .................................................. 22 308.314.339.362 series representation 17.147.149 Dihedral group Dual object (of group) 41.59.134.152.260. 317 (commuting) pair of algebras ........177.225 120 pair of manifolds ................................. pair of Lie algebras/symmetric spaces 241
...........
........
......... ............................... ...
.
Index.
450
E Envelope of Lie algebra ................. 249.256. 330 Equivalence (of representations) 42 Euler angles 161. 171 -Lagrange equations ...................... 369. 430 386 rigid body problem 49 Exponential map
.................. ........................................... ............................. ........................................
F
.................................. .....................................
Factor-algebra/group 53 62. 79 Fourier analysis algebra A ............................................. 64 64. 69 integral/series .................................. 62 transform ............................................. Frobenius reciprocity ............. 142.223.252. 34 1 Function 81.111.117. 184 harmonic positive-definite 86.88. 424 characteristic (of random variable) ....... 86 173.250.328 spherical ................................. Function spaces: L2;Lp; e; e, e, em;e" 27 Sobolev 36, 36. .......................... 27.57. 73 Fundamental solution (Green's function) 94 348 Fundamental region .................................. Fundamental (homotopy) group .... 51.349.358
........................... .........................
..
.....................
....
G Gauss d e c o m p ition ........................... .31. 211 Gaussian function (normal distribution) 66.71.87 kernel (semigroup) ............. 96.116.188. 283 Generator (infinitesimal) of group action ....48 Generating function (for polynomials) 189.237 49.120.188.350.377. 430 Geodesics Geodesic flow 344.358.393.408. 415 Green's function (see Fund solution) Group: 61. 65 abelian ............................................ 17.133.141.147 alternating ......................... affine ........................ 14.20.31.147.153. 264 classical (Lie. matrix) ............ 21.41.54. 191 crystallographic .................................... 22 17 cyclic .................................................... 17.147.149 dihedral Euclidian motion ........ 14.35.41.95.265. 294 finite 17.83. 199 Lorentz ............... 14.2 1.4 1.147.266.305. 361 20.53.191.274.292. 303 nilpotent
.....
..................... ...................
.
.
................................... ......................................... ................
one parameter (of operators) .................49 orthogonal (see Orthogonal) Poincare (see Poincare) 18.143 polyhedral ...................................... 54.191.363. 367 simple; semisimple 13.46.54.192.305 special linear. SL symmetric (permutation) .....13.34.199. 218 .........21.55.193.243. 284 symplectic. 20.53.191.274.292. 303 solvable transformation ...................................... 13 unitary (see Unitary) Group algebra (see Convolution)
............. ...........
wn).
..................
H Hamiltonian: classical ................................. 291.359.370 quantum 115.276.291.358 370.372 vector field/flow ........................... Hamilton's principle of minimal action 369 71 Hausdorff-Young inequality ......................... Harmonic 281.285.373 oscillator ................................ function (see Function) 175.184.253 polynomial ............................. Heat equation ............................................. 92 semigroup/kernel ..................... 96.185.353 invariants ..................................... 352.357 Heisenberg commutation relations (CCR) 276.278.411 20.41.53.154.274. 294 group ........................ uncertainty principle ...................... 83.272 Hermite polynomials/functions .................. 281 Huygens principle ................................ 101. 187 Hydrogen atom ................................... 411.420 Hyperbolic (Poincare) plane/space .. 15.25.123.244. 348 elements (in SL, ) ................... 313.321.335 (wave) differential equation ........... 93. 104
..........................
......
I Imprimitivity system (Mackey) .... 149.303.307 Indefinite (Minkowski) product ................... 14 Independent Random variables ................... 85 Induced action/representation .............. 30. 137 holomorphically induced ........ 267.296. 309 Inertia tensor/moments ............................. 387 Integral (conserved) 383.378.388.394 428 operator 71 Interpolation (Riesz) ................................... Intertwining: operator; space Int(T;S); .......42
......................
..............................................
Index . number ................................................. 42 Invariant (Haar) measure .24.162.227.308.319 subspace ............................................... 37 vector field (right/left) 48 polynomials ............................ 121.251.256 Inversion/Plancherel formula. measure commutative groups 63.65 compact ............................................. 131 general ......................................... 297.300
...........
.........................
........................
942);S q 3 ) ...........................................
........................................ ............................... .......................................... ......................................... ......... Sl42);S q 3 ) ...........................................
302
affine group 265 Euclidian motions 266 Heisenberg 280 SL, 318.326.362 Irreducible representation ............................ 37 of compact/finite groups 126.132.144
165 compact Lie groups ............................ 206 Heisenberg and semidirect prod ....259.278 SL, ......................................... 307.311.364 Iwasawa decomposition .................312.318.363
J Jacobi identity (in Lie algebras) ...................... polynomials ........................................
46 172
451 functions .......................... 173.178.180.360 transform (canonical variables) 370.377 Levi-Malcev decomposition ..................55.193 Lie group/algebra .................................. 20.46 classical 21.54.393 bracket 46.49.435 derivative (of v. field) ...................302.432 Theorem ............................................. 207 Log (inverse of expmap) 49 Lorentz group 14.21.41.266.305.361
.....
..................................... ......................................
............................. .................
M ........ ......................
Matrix entry (of repn) 29.40.126.171. 340 Mellin transform 88.306.327.337 Metric: G-invariant ........... 227.241.245.331. 388437 hyperbolic 13.147.266.394 Riemannian 13.15.430.434.353. 372 Pseudo-Riem . (Minkowski) ...... 13.147.266 26.273 Modular function ................................. Moment urn angular .......................... 176. 183.276. 379 380.389 map .............................................. operator ........................... 430.276.287.381 variable ................................. 370.379.381 Multiplicity (of irreducible i~ in T) ............... 40 Multi-polar coordinates ............................. 244
.......................... ............
N
K Kernel: reproducing ........................................ 288 integral kernel (of operator) 126.428 Poisson ........................... 81.97.98.106.346 Kronecker (tensor) product ................................... 33
..........
Nilpotent algebra/group (see Group) Noether Theorem ...................................... 378 Norm: Hilbert-Schmidt 33.128.428 trace-class 33.36.126.428 Sobolev (see Sobolev)
....................... ............................
0
L ............................................... .......... ........................................ ............................... .......... .................. .............................................
Lagrangian 369 Laguerre polynomials/functions 413.420 Laplacian: 73.105 on R";T" spheres S2;S" 176.184 symmetric spaces.................... 244.247.255 hyperbolic (Poincare) plane 332.344 Riemann surfaces r\H 352.358 Laplace-Beltrami operator .................. 243.332 Lax pair 394.400 Legendre: differential equation ...................... 173.178 polynomials .................................. 172.189
Operator compact 39.74.426.428.346 differential ........58.92.112.121.171.176. 432 elliptic 92.98.113.261.354 Fredholm ............................................ 114 Hilbert-Schmidt (see Norm) integral (see Integral) intertwining (see Intertwining) pseudo-differential ............................... 112 SchrGdinger..................93.225.276.281.411 trace-class (see Norm) unbounded (closed) 56.58.332 Orbit
........................ ..........................
..........
Index.
452 co-adjoint ............................... 290.294.297 method ............................................... 290 Orthogonal groupsla1gebras:qn); S q n ) ; S q p ; q )
Quaternions ................................... 13.161.163 Quaternionic-type representation .......... 38.157
.................. 14.21.54. 161.174.196.227.259. 434
R
Orthogonality relations (for matrix entries/ characters) 126.134.149.173.232.340
Radical (of Lie algebra) ........................ 55.193 Radon transform ................................. 120.182 Random variable 85 walk 84.85.87.145 Rank of Lie group/algebra ............................... 197 symmetric space 121.241.246 tensor ........................................... 217.368 Reducibility (complete. of representation) .193 Representation adjoint .................................. 34.46.52.192 co-adjoint ................................... ..256.293 contragredient (dual) ............................ 34 induced .................................... 30.137.257 holomorphically induced 269 irreducible ............................................. 37 primary (see Primary) projective (see Projective) regular 24.29.33 tensor (Kronecker) product of ...............33 unitary ................................................. 29 Root (of Lie algebra); ................................ 197 system ................................................ 197 vector ................................................. 197 Runge-Lenr symmetry ................. 389.390.408
.................
......................................... .....................................
P
.................................
Paley-Winer Theorem 79 Peter-Weyl Theory ................................... 126 Plancherel formula/measure (see Inversion) Phase-space 370.374.380 Poincare group P. ................. 14.41.55.147. 266 (Lobachevski) half plane ........ 15.25.51.430 disk ...................................................... 16 Poisson kernel (see Kernel) summation formula ..... 66.109.187.326.354 (Lie) bracket (see Symplectic) Polynomials (orthogonal) Legendre; Jacobi (see Legendre. Jacobi) Gegenbauer 185.188 Hermite 281 Laguerre (see Laguerre) Potential Bessel (see Beasel) energy .......................................... 369.379 Riesz ............................. 78.82.122.311.362 of Schriidinger operator (see Operator) 393 N-body Primary decomposition (see Decomposition) projection ............................................. 40 subspace ................. 40.43.130.223.329.344 representation ................................. 40.130 Product: direct of groups/algebra 21.43 semidirect (of groups/algebras)
..................................
.................................. .............................................
....................
.....................
.........................................
...............................................
...................
............ 15.21.35.46.52.55.133.147.200.260. 411
tensor (see Tensor) Kronecker (see Kronecker) Projective space representation
............................ .........................
........... ....................................... .............................. ......................................
121.141.296 150.284.289
115.276.385.409 Quantum hamiltonian state 276.291.409 observable 276.291.409 Quantization (procedure) .291.296.409.411.414 condition 292.293
S Selberg (trace formula) .............................. 338 Schur's Lemma ........................................... 37 Type-criterion ..................................... 157 Semisimple Lie group/algebra .............. 54.192 Semidirect product (of groups) ............. 15.147 (of algebras) .................................. 55.193 Signature (of representation) ..................... 212 Simple Lie group/algebra 54.192 Schriidinger operator (see Operator) Solvable group/algebra ........................ 53.191 Space: homogeneous 24 intertwining (see Intertwining) Sobolev 27.57.73 symmetric (see Symmetric) Spectral decomposition (unitary. s-a operator).424 subspace/projection .................38.279.424 Spherical functions 173.250.328 harmonics 143.174.184.253
.....................
........................................ ........................................
.
....................... ........................
Index. Subspace: cyclic ............................................... 40. 44 37 invariant of smooth vectors ................................. 56 primary (see Primary) spectral (see Spectral) 15 Stabilizer (isotropy) subgroup ..................... 290 subalgebra .......................................... Stereographic map ......................... 16.365. 415 92.112. 183 Symbol (of differential operator) Symmetric space 13.243. 437 Symmetric (permutation) group .13.44.200.218 Symmetry group of regular polyhedra 17.18.144 Euclidian space (see Group. Euclidian) Minkowski space (see Poincare) hyperbolic space (see Lorentz) Symplectic group/transform (see Group) (Poisson) structure .......... 290.296.371. 373 manifold .......................... 291.297.371. 409
..............................................
............................
....
......................
T Tensors: 159.164.175.214 symmetric/anti.sym 217 mixed ................................................. Tensor product of 428 matrices/operators vector spaces; algebraa .........33,45,175,428 34,43,152,181,218,223 representations 109,187,352 Theta-function
........
.............................. ....... ..............................
U Unitary operator/matrix/group .......... 16.20. 2 1. 161 representation (see Representation) Unitary trick (see Weyl)
V
.....................
Variation (of functional) Vector bundle (see Bundle)
378. 384
W Wave equation ............................................... 93 kernel/propagator 96.99 Weight in function-spaces ........................... 32.73 for orthogonal polynomials ..... 172.206. 420 (highest) of representation ............ 206. 231
............................
453 diagram ........................................ 206. 208 Weyl algebra W ........................ 174,183,284,410 chamber ................................. 228,230,246 character formulae 181,284 group 199,206,242 177,183,218,224 invariants 114 principle (volume counting) unitary trick ................................. 167,193
........................ ..................................... ........................ ................
Y Young inequality ......................................... Young tableau/symmetrizer ......................
72 219
2 Zeta function .............................................. 89 (of operator) .................................. 90.352
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