Double affine Hecke algebras

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London Mathematical Society Lecture Note Series. 319

Double Affine Hecke Algebras IVAN CHEREDNIK University of North Carolina, Chapel Hill

   Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521609180 © Ivan Cherednik 2005 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 - -

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Dedicated to Ian Macdonald PREFACE

This book is based on a series of lectures delivered by the author in Kyoto in 1996–97 and at Harvard University in 2001. The first chapter was written in collaboration with T. Akasaka, E. Date, K. Iohara, M. Jimbo, M. Kashiwara, T. Miwa, M. Noumi, Y. Saito, and K. Takemura. V. Ostrik is the coauthor of the second chapter. The author owes them a lot, as well as P. Etingof, D. Kazhdan, M. Nazarov, and E. Opdam for help and encouragement. The book was supported in part by the National Science Foundation and the Clay Mathematics Institute. In many ways, this book began with one man, Ian Macdonald. I am deeply indebted to him. After a comprehensive introduction, the classical and quantum Knizhnik–Zamolodchikov equations attached to root systems are studied, and their relations to the affine Hecke algebras, Kac–Moody algebras, and harmonic analysis discussed. These equations are of key importance in the analytic theory of Coxeter groups. In Chapter 2, we switch to a systematic theory of the one-dimensional double affine Hecke algebra and its representations. It is the simplest case, but far from b 2 , the Heisenberg and Weyl trivial. This algebra is closely connected with sl2 , sl algebras, and has impressive applications. The third chapter is about DAHA in full generality, including the Macdonald polynomials, Fourier transform, Gauss–Selberg integrals, Verlinde algebras, Gaussian sums, and diagonal coinvariants. The transition to this abstract level will be smooth for readers familiar with root systems. Only reduced root systems are considered. This book is essentially self-contained. The chapters are relatively independent. I hope that it will be helpful for both mathematicians and physicists who want to master the new double Hecke algebra technique.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 Introduction 0.0 Universality of Hecke algebras . . . . . 0.0.1 Real and imaginary . . . . . . . 0.0.2 New vintage . . . . . . . . . . . 0.0.3 Hecke algebras . . . . . . . . . 0.1 KZ and Kac–Moody algebras . . . . . 0.1.1 Fusion procedure . . . . . . . . 0.1.2 Symmetric spaces . . . . . . . . 0.1.3 KZ and r–matrices . . . . . . . 0.1.4 Integral formulas for KZ . . . . 0.1.5 From KZ to spherical functions 0.2 Double Hecke algebras . . . . . . . . . 0.2.1 Missing link? . . . . . . . . . . 0.2.2 Gauss integrals and sums . . . . 0.2.3 Difference setup . . . . . . . . . 0.2.4 Other directions . . . . . . . . . 0.3 DAHA in harmonic analysis . . . . . . 0.3.1 Unitary theories . . . . . . . . . 0.3.2 From Lie groups to DAHA . . . 0.3.3 Elliptic theory . . . . . . . . . . 0.4 DAHA and Verlinde algebras . . . . . 0.4.1 Abstract Verlinde algebras . . . 0.4.2 Operator Verlinde algebras . . . 0.4.3 Double Hecke Algebra . . . . . 0.4.4 Nonsymmetric Verlinde algebras 0.4.5 Topological interpretation . . . 0.5 Applications . . . . . . . . . . . . . . . 0.5.1 Flat deformation . . . . . . . . 0.5.2 Rational degeneration . . . . . 0.5.3 Gaussian sums . . . . . . . . . 0.5.4 Classification . . . . . . . . . . vii

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CONTENTS 0.5.5 0.5.6

Weyl algebra . . . . . . . . . . . . . . . . . . . . . . . Diagonal coinvariants . . . . . . . . . . . . . . . . . . .

1 KZ and QMBP 1.0 Soliton connection . . . . . . . . . . . 1.0.1 Classical r–matrices . . . . . . . 1.0.2 Tau function and coinvariant . . 1.0.3 Structure of the chapter . . . . 1.1 Affine KZ equation . . . . . . . . . . . 1.1.1 Hypergeometric equation . . . . 1.1.2 AKZ equation of type GL . . . 1.1.3 Degenerate affine Hecke algebra 1.1.4 Examples . . . . . . . . . . . . 1.2 Isomorphism theorems for AKZ . . . . 1.2.1 Induced representations . . . . 1.2.2 Monodromy of AKZ . . . . . . 1.2.3 Lusztig’s isomorphisms . . . . . 1.2.4 AKZ is isomorphic to QMBP . 1.2.5 The GL–case . . . . . . . . . . 1.3 Isomorphisms for QAKZ . . . . . . . . 1.3.1 Affine Hecke algebras . . . . . . 1.3.2 Definition of QAKZ . . . . . . . 1.3.3 The monodromy cocycle . . . . 1.3.4 Macdonald’s eigenvalue problem 1.3.5 Macdonald’s operators . . . . . 1.3.6 Arbitrary root systems . . . . . 1.4 DAHA and Macdonald polynomials . . 1.4.1 Rogers’ polynomials . . . . . . 1.4.2 A Hecke algebra approach . . . 1.4.3 The GL–case . . . . . . . . . . 1.5 Abstract KZ and elliptic QMBP . . . . 1.5.1 Abstract r–matrices . . . . . . . 1.5.2 Degenerate DAHA . . . . . . . 1.5.3 Elliptic QMBP . . . . . . . . . 1.5.4 Double affine KZ . . . . . . . . 1.6 Harish-Chandra inversion . . . . . . . . 1.6.1 Affine Weyl groups . . . . . . . 1.6.2 Degenerate DAHA . . . . . . . 1.6.3 Differential representation . . . 1.6.4 Difference-rational case . . . . . 1.6.5 Opdam transform . . . . . . . . 1.6.6 Inverse transform . . . . . . . . 1.7 Factorization and r–matrices . . . . . .

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129 131 135 136 138 138 141 144 151

2 One-dimensional DAHA 2.0 Overview . . . . . . . . . . . . . . . . . 2.0.1 Classical origins . . . . . . . . . 2.0.2 Main results . . . . . . . . . . . 2.0.3 Other directions . . . . . . . . . 2.1 Euler’s integral and Gaussian sum . . . 2.1.1 Euler’s integral, Riemann’s zeta 2.1.2 Extension by q . . . . . . . . . 2.1.3 Mehta–Macdonald formula . . . 2.1.4 Hankel transform . . . . . . . . 2.1.5 Gaussian sums . . . . . . . . . 2.2 Imaginary integration . . . . . . . . . . 2.2.1 Macdonald’s measure . . . . . . 2.2.2 Meromorphic continuations . . 2.2.3 Using the constant term . . . . 2.2.4 Shift operator . . . . . . . . . . 2.2.5 Applications . . . . . . . . . . . 2.3 Jackson and Gaussian sums . . . . . . 2.3.1 Sharp integration . . . . . . . . 2.3.2 Sharp shift formula . . . . . . . 2.3.3 Roots of unity . . . . . . . . . . 2.3.4 Gaussian sums . . . . . . . . . 2.3.5 Etingof’s theorem . . . . . . . . 2.4 Nonsymmetric Hankel transform . . . . 2.4.1 Operator approach . . . . . . . 2.4.2 Nonsymmetric theory . . . . . . 2.4.3 Rational DAHA . . . . . . . . . 2.4.4 Finite dimensional modules . . 2.4.5 Truncated Hankel transform . . 2.5 Polynomial representation . . . . . . . 2.5.1 Rogers’ polynomials . . . . . . 2.5.2 Nonsymmetric polynomials . . . 2.5.3 Double affine Hecke algebra . .

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1.8

1.7.1 Basic trigonometric r–matrix 1.7.2 Factorization and r–matrices . 1.7.3 Two conjectures . . . . . . . . 1.7.4 Tau function . . . . . . . . . . Coinvariant, integral formulas . . . . 1.8.1 Coinvariant . . . . . . . . . . 1.8.2 Integral formulas . . . . . . . 1.8.3 Proof . . . . . . . . . . . . . . 1.8.4 Comment on KZB . . . . . .

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2.6

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2.11

2.12

2.5.4 Back to Rogers’ polynomials . . 2.5.5 Conjugated polynomials . . . . Four corollaries . . . . . . . . . . . . . 2.6.1 Basic definitions . . . . . . . . . 2.6.2 Creation operators . . . . . . . 2.6.3 Standard identities . . . . . . . 2.6.4 Changing k to k+1 . . . . . . . 2.6.5 Shift formula . . . . . . . . . . 2.6.6 Proof of the shift formula . . . DAHA–Fourier transforms . . . . . . . 2.7.1 Functional representation . . . 2.7.2 Proof of the master formulas . . 2.7.3 Topological interpretation . . . 2.7.4 Plancherel formulas . . . . . . . 2.7.5 Inverse transforms . . . . . . . Finite dimensional modules . . . . . . 2.8.1 Generic q, singular k . . . . . . 2.8.2 Additional series . . . . . . . . 2.8.3 Fourier transform . . . . . . . . 2.8.4 Roots of unity q, generic k . . . Classification, Verlinde algebras . . . . 2.9.1 The classification list . . . . . . 2.9.2 Special spherical representations 2.9.3 Perfect representations . . . . . Little double Hecke algebra . . . . . . 2.10.1 The case of odd N . . . . . . . 2.10.2 Little double H . . . . . . . . . 2.10.3 Half-integral k . . . . . . . . . . 2.10.4 The negative case . . . . . . . . 2.10.5 Deforming Verlinde algebras . . DAHA and p–adic theory . . . . . . . 2.11.1 Affine Weyl group . . . . . . . . 2.11.2 Affine Hecke algebra . . . . . . 2.11.3 Deforming p–adic formulas . . . 2.11.4 Fourier transform . . . . . . . . 2.11.5 One-dimensional case . . . . . . Degenerate DAHA . . . . . . . . . . . 2.12.1 Definition of DAHA . . . . . . 2.12.2 Polynomials, intertwiners . . . . 2.12.3 Trigonometric degeneration . . 2.12.4 Rational degeneration . . . . . 2.12.5 Diagonal coinvariants . . . . . .

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201 202 203 204 205 206 208 209 210 212 213 215 216 220 224 225 226 231 232 234 237 238 240 246 251 252 253 255 257 259 261 262 263 265 267 268 270 271 273 274 277 279

CONTENTS 3 General theory 3.0 Progenitors . . . . . . . . . . . . . . . . 3.0.1 Fourier theory . . . . . . . . . . . 3.0.2 Perfect representations . . . . . . 3.0.3 Affine Hecke algebras . . . . . . . 3.0.4 Gauss–Selberg integrals and sums 3.0.5 From generic q to roots of unity . 3.0.6 Structure of the chapter . . . . . 3.1 Affine Weyl groups . . . . . . . . . . . . 3.1.1 Affine roots . . . . . . . . . . . . 3.1.2 Affine length function . . . . . . . 3.1.3 Reduction modulo W . . . . . . . 3.1.4 Partial ordering in P . . . . . . . 3.1.5 Arrows in P . . . . . . . . . . . . 3.2 Double Hecke algebras . . . . . . . . . . 3.2.1 Main definition . . . . . . . . . . 3.2.2 Automorphisms . . . . . . . . . . 3.2.3 Demazure–Lusztig operators . . . 3.2.4 Filtrations . . . . . . . . . . . . . 3.3 Macdonald polynomials . . . . . . . . . . 3.3.1 Definitions . . . . . . . . . . . . . 3.3.2 Spherical polynomials . . . . . . 3.3.3 Intertwining operators . . . . . . 3.3.4 Some applications . . . . . . . . . 3.4 Polynomial Fourier transforms . . . . . . 3.4.1 Norm formulas . . . . . . . . . . 3.4.2 Discretization . . . . . . . . . . . 3.4.3 Basic transforms . . . . . . . . . 3.4.4 Gauss integrals . . . . . . . . . . 3.5 Jackson integrals . . . . . . . . . . . . . 3.5.1 Jackson transforms . . . . . . . . 3.5.2 Gauss–Jackson integrals . . . . . 3.5.3 Macdonald’s eta-identities . . . . 3.6 Semisimple representations . . . . . . . . 3.6.1 Eigenvectors and semisimplicity . 3.6.2 Main theorem . . . . . . . . . . . 3.6.3 Finite dimensional modules . . . 3.6.4 Roots of unity . . . . . . . . . . . 3.6.5 Comment on finite stabilizers . . 3.7 The GL–case . . . . . . . . . . . . . . . 3.7.1 Generic k . . . . . . . . . . . . . 3.7.2 Periodic skew diagrams . . . . . . 3.7.3 Partitions . . . . . . . . . . . . .

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281 281 281 286 288 289 290 292 293 294 296 298 302 304 305 305 307 310 311 313 314 317 320 323 325 325 326 328 331 334 335 337 339 342 343 349 353 355 356 357 358 360 362

xii

CONTENTS

3.8

3.9

3.10

3.11

3.7.4 Equivalence . . . . . . . . . . . . . 3.7.5 The classification . . . . . . . . . . 3.7.6 The column-row modules . . . . . . 3.7.7 General representations . . . . . . . Spherical representations . . . . . . . . . . 3.8.1 Spherical and cospherical modules . 3.8.2 Primitive modules . . . . . . . . . 3.8.3 Semisimple spherical modules . . . 3.8.4 Spherical modules at roots of unity Induced and cospherical . . . . . . . . . . 3.9.1 Notation . . . . . . . . . . . . . . . 3.9.2 When are induced cospherical? . . 3.9.3 Irreducible cospherical modules . . 3.9.4 Irreducibility of induced modules . Gaussian and self-duality . . . . . . . . . . 3.10.1 Gaussians . . . . . . . . . . . . . . 3.10.2 Perfect representations . . . . . . . 3.10.3 Generic q, singular k . . . . . . . . 3.10.4 Roots of unity . . . . . . . . . . . . DAHA and double polynomials . . . . . . 3.11.1 Good reductions . . . . . . . . . . 3.11.2 Main theorem . . . . . . . . . . . . 3.11.3 Weyl algebra . . . . . . . . . . . . 3.11.4 Universal DAHA . . . . . . . . . . 3.11.5 Universal Dunkl operators . . . . . 3.11.6 Double polynomials . . . . . . . . .

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364 365 367 369 371 371 373 376 378 382 382 384 387 390 392 393 394 399 403 407 408 409 410 413 415 416

Bibliography

418

Index

431

Chapter 0 Introduction 0.0 0.0.1

Universality of Hecke algebras Real and imaginary

Before a systematic exposition, I will try to outline the connections of the representation theory of Lie groups, Lie algebras, and Kac–Moody algebras with the Hecke algebras and the Macdonald theory. The development of mathematics may be illustrated by Figure 0.1.

Imaginary axis (conceptual mathematics)

Real axis (special functions, numbers) Figure 0.1: Real and Imaginary Mathematics is fast in the imaginary (conceptual) direction but, generally, slow in the real direction (fundamental objects). Mainly I mean modern mathematics, but it may be more universal. For instance, the ancient Greeks created a highly conceptual axiomatic geometry with modest “real output.” I do not think that the ratio 0, we have Z ∞i ∞ √ Y 1 − q j+k −x2 , 0. (0.2.3) (−i) q δ(x; q, k) dx = 2 aπ 1 − q j+2k −∞i j=0 Here both sides are well defined for all k except for the poles but coincide only when 0. One can make (0.2.3) entirely algebraic by replacing the Gaussian γ −1 = 2 q −x by its expansion +∞ X 2 −1 q n /4 q nx γ˜ = and using Const Term(

P

−∞

cn q nx ) = c0 instead of the imaginary integration:

Const Term (˜ γ −1 δ) = 2

∞ Y 1 − q j+k . j+2k 1 − q j=0

Jackson integrals. A promising feature of special q–functions is the possibility of replacing the integrals by sums over Zn , the so-called Jackson integrals. Technically, we switch from the imaginary integration to that for the path, which begins at z = ²i + ∞, moves to the left till ²i, then down through the origin to −²i, and then returns down the positive real axis to −²i + ∞ (for small ²). Then we apply Cauchy’s theorem under the assumption that |=k| < 2², 0. We obtain the following counterpart of (0.2.3): ∞ X

q

(k−j)2 4

j=0

q

k2 4

j 1 − q j+k Y 1 − q l+2k−1 = 1 − q k l=1 1 − q l

∞ Y (1 − q j/2 )(1 − q j+k )(1 + q j/2−1/4+k/2 )(1 + q j/2−1/4−k/2 ) j=1

(1 − q j )

,

16

CHAPTER 0. INTRODUCTION

which is convergent for all k. When q = exp(2πi/N ) and k is a positive integer ≤ N/2, we come to a Gauss–Selberg-type sum: NX −2k j=0

q

(k−j)2 4

j k 2N −1 X 1 − q j+k Y 1 − q l+2k−1 Y 2 j −1 = (1 − q ) q m /4 . k l 1 − q l=1 1 − q m=0 j=1

The left-hand side resembles the so-called modular Gauss–Selberg sums. However, the difference is dramatic. The modular sums are calculated in the finite fields and are embedded into roots of unity right before the final summation [Ev]. Our sums are defined entirely in cyclotomic fields. Substituting k = [N/2], we arrive at (0.2.2).

0.2.4

Other directions

There are other projects involving the double Hecke algebras. We will mention only some of them. (1) Macdonald’s q–conjectures [M2, M3, M4]. Namely, the constant term, norm, duality, and evaluation conjectures [BZ, Kad],[C14, C16, C19]. See also [A2, It, M6, C23] about the discrete variant of the constant term conjecture, the Aomoto conjecture. My proof of the norm formula was based on the shift operators and is similar to that from [O1] in the differential case (the duality and evaluation conjectures collapse as q → 1). I would add to this list the Pieri rules. As to the nonsymmetric Macdonald polynomials, the references are [O3, M5, C20]. See also [DS, Sa, M8], and the recent [St, Ra] about the case of C ∨ C, the Koornwinder polynomials and generalizations. (2) K–theoretic interpretation. See papers [KL1, KK], then [GG, GKV], and the important recent paper [Vas]. The latter leads to the Langlands-type description of irreducible representations of double Hecke algebras. The case of generic parameters q, t is directly connected with the affine theory [KL1]. For the special parameters, the corresponding geometry becomes significantly more complicated. The Fourier transform remains unclear in this approach. I also mention here the strong Macdonald conjecture (Hanlon) and the recent [FGT]. (3) Elementary methods. The theory of induced, semisimple, unitary, and spherical representations can be developed successfully without K–theory. The main tool is the technique of intertwiners from [C23], which is similar to that for the affine Hecke algebras. The nonsymmetric Macdonald polynomials generate the simplest spherical representation, with the intertwiners serving as creation operators (the case of GL is due to [KnS]). For GL, this technique gives a reasonably complete classification of irreducible representations similar to the theorem by Bernstein and Zelevinsky in the affine case. Relations to [HO2] are expected.

0.2. DOUBLE HECKE ALGEBRAS

17

(4) Radial parts via Dunkl operators. The main references are [Du1, H1] and [C11]. In the latter, it was observed that the trigonometric differential Dunkl operators can be obtained from the degenerate (graded) affine Hecke algebra from [Lus1] ([Dr1] for GLn ). The difference, elliptic, and differenceelliptic generalizations were introduced in [C12, C13, C17, C18]. The connections with the KZ equation play an important role here [Mat, C11, C22, C13]. The radial parts of the Laplace operators of symmetric spaces and their generalizations are symmetrizations of the Dunkl operators. The symmetric Macdonald polynomials are eigenfunctions of the difference radial parts. The nonsymmetric Opdam–Macdonald polynomials appear as eigenfunctions of the Dunkl operators. (5) Harmonic analysis. The Dunkl operators in the simplest rationaldifferential setup lead to the definition of the generalized Bessel functions and the generalized Hankel transform (see [O2, Du2, Je] and also [He3]). In contrast to the Harish-Chandra and the p–adic spherical transforms, it is self-dual. The self-duality resumes in the difference setting [C28, C21]. The Mehta–Macdonald conjecture, directly related to the transform of the Gaussian, was checked in [M2, O1] in the differential case, and extended in [C21] to the difference case. It was used there to introduce the q–spherical functions. Concerning the applications to the Harish-Chandra theory, see [HO1, O3, C24] and also [HS]. (6) Roots of unity. The construction from [C28] generalizes and, at the same time, simplifies the Verlinde algebras, including the projective action of P SL(2, Z) (cf. [Ka], Theorem 13.8, and [Ki1]), the inner product, and a new theory of Gaussian sums. In [C20], the nonsymmetric Verlinde algebras were considered. The symmetric elements of such algebras form the k–generalized Verlinde algebras. The simplest example is the classical N –dimensional representation of the Weyl algebra at q = exp(2πi/N ). Let me also mention the recent [Go, C29] about the Haiman conjecture [Ha1] on the structure of the so-called diagonal coinvariants, which appeared to be directly connected with rational DAHAs and DAHAs at roots of unity. (7) Topology. The group P SL(2, Z) acts projectively on the double Hecke algebra itself. The best explanation (and proof) is based on the interpretation of this algebra as a quotient of the group algebra of the π1 of the elliptic configuration space from [C13]. The calculation of π1 in the GL–case is essentially due to [Bi, Sc]. For arbitrary root systems, it is similar to that from [Le], but our configuration space is different. Such π1 governs the monodromy of the eigenvalue problem for the elliptic radial parts, the corresponding Dunkl operators, and the KZB equation. Switching to the roots of unity, the monodromy representation is the nonsymmetric Verlinde algebra; applications to the Witten–Reshetikhin–Turaev and Ohtshuki invariants are expected. (8) GL–Duality. The previous discussion was about arbitrary root sys-

18

CHAPTER 0. INTRODUCTION

tems. In the case of GL, the theorem from [VV1] establishes the duality between the double Hecke algebras (actually its extension) and the q–toroidal (double Kac–Moody) algebras in the sense of Ginzburg–Kapranov–Vasserot. It generalizes the classical Schur–Weyl duality, Jimbo’s q–duality for the nonaffine Hecke algebra [Ji], and the affine Hecke analogs from [Dr1, C5]. When the center charge is nontrivial the duality explains the results from [KMS] and [STU], which were extended by Uglov to irreducible representations of the Kac–Moody glˆN of arbitrary positive integral levels. (9) Rational degeneration.. The rational degeneration of the double affine Hecke algebra with trivial center charge (q = 1) is directly related to the Calogero–Moser varieties [EG]. The rational degenerations of the double affine Hecke algebra [CM, EG], in a sense, play the role of the Lie algebras of the q, t–DAHA. The trigonometric degeneration is also a sort of Lie algebra, but the rational one has the projective action of P SL(2, Z) and other symmetries that make it closer to the general q, t–DAHA. The theory of the rational DAHA and its connections appeared to be an interesting independent direction (see [BEG, GGOR, Go, C29]. Some of the latest developments. There are interesting papers [GK1, GK2] that continued earlier results by Kapranov [Kap] towards using DAHA in the so-called double arithmetic, started by Parshin quite a few years ago. The general problem is to associate a double p–adic Lie group with DAHA. Let me also mention the results by Ginzburg, Kapranov and Vasserot concerning the interpretation of DAHAs via “Hecke correspondences” over special algebraic surfaces. Presumably DAHAs are somehow related to the Nakajima surfaces [Nak]; at least, this is understood for the Calogero–Moser varieties [EG]. An important recent development is the establishment of the connection of DAHA with the Schur algebra in [GGOR] and then in [VV2]. It is proven in the latter paper that, under minor technical restrictions, DAHA of type A is Morita equivalent to the quantum affine Schur algebra at roots of unity. The latest project so far employs DAHAs for the global quantization of algebraic surfaces. Oblomkov used the rank one C ∨ C–DAHA to quantize the cubic surfaces. Then Etingof, Oblomkov, Rains [EOR] extended this approach to quantize the Del Pezzo surfaces via certain generalizations of DAHA. Relations to [Ra] are expected. Related directions. Let me also mention several directions that are not based on double Hecke algebras (and their variants) but have close relations to these algebras and to the theory of the symmetric Macdonald polynomials. Mainly they are about the GL–case and the classical root systems. (a) The spherical functions on q–symmetric spaces due to Noumi and others, which are related to the Macdonald polynomials for certain values of t, central elements in quantum groups (Etingof and others).

0.2. DOUBLE HECKE ALGEBRAS

19

(b) The so-called interpolation polynomials (Macdonald, Lassalle, Knop– Sahi, Okounkov–Olshansky, Rains), continuing the classical Lagrange polynomials, appeared to be connected to double Hecke algebras. (c) The interpretation of the Macdonald polynomials as traces of the vertex operators, including applications to the Verlinde algebras (Etingof, Kirillov Jr.). It is mainly in the GL–case and for integral k. See also [EVa]. (d) Various related results on KZB and its monodromy, (Etingof, Felder, Kirillov Jr., Varchenko). The monodromy always satisfies the DAHA grouptype relations, but the quadratic ones are valid in special cases only. See, e.g., [Ki2, TV, FTV, FV3]. (e) There are multiple relations of DAHA to the theory of the W –invariant differential operators, including recent developments due to Wallach, Levasseur, Stafford, and Joseph. In the theory of DAHA, a counterpart is concerning the centralizer of the nonaffine Hecke subalgebra. (f) There are direct links to the theory of quantum groups defined by Drinfeld and Jimbo and dynamical Yang–Baxter equations through the classical and quantum r–matrices; we refer to [Dr2] and the books [Lus2, ES] without going into detail. Strong connections with the affine Hecke algebra technique in the classical theory of GLN and Sn must be noted. I mean [C5, Na1, Na2, NT, LNT, C28] and promising recent results towards Kazhdan–Lusztig polynomials by Rouquier and others. The expectations are that Kazhdan–Lusztig polynomials and the canonical (crystal) bases in quantum groups are important for the theory of double Hecke algebras, although I don’t know a good definition of the ”double” Kazhdan–Lusztig polynomials. The coefficients of the symmetric Macdonald polynomials in the stable GL–case have interesting combinatorial properties (Macdonald, Stanley, Hanlon, Garsia, Haiman, and others). The most celebrated is the so-called n!– conjecture recently proved by Haiman. See [GH] and [Ha2]. Let me mention here that the Macdonald polynomials for the classical root systems appeared for the first time in a Kadell work. He also proved the Macdonald norm conjecture for the BC systems [Kad]. The constant term conjecture in the GLn –case was verified in [BZ]; see [C16] and [M8] for further references. The first proof of the norm formula for the GL is due to Macdonald. The elliptic counterparts of the Macdonald operators in the case of GLn are defined in [Ru]. Quite a few constructions can be extended to arbitrary finite groups generated by complex reflections. For instance, the Dunkl operators and the KZ connection exist in this generality (Dunkl, Opdam, Malle). One can also try the affine and even the hyperbolic groups (Saito’s root systems [Sai]).

20

0.3

CHAPTER 0. INTRODUCTION

DAHA in harmonic analysis

This section is an attempt to outline the core of the book in the special case of A1 and discuss connections between harmonic analysis and mathematical physics. There are quite a few projects where DAHA is involved (see above). However, I think the Verlinde algebra is the most convincing demonstration of the power of new methods. It is the main object of this section.

0.3.1

Unitary theories

Generally, the problem is that we do not have the Kac–Moody harmonic analysis. For instance, we do not have a good definition of the category of ” L2 ”–representations in this case. The theory of Kac–Moody algebras, playing a well-known role in modern theoretical physics, is too algebraic for this, and, presumably, these algebras must be developed to more analytic objects. Actually, the von-Neumann algebras are such objects, but they are, in a sense, too analytic. We need something in between. Hopefully the Verlinde algebras and DAHA can help. Physical connection. Concerning the classical roots of the harmonic analysis on symmetric spaces, the corresponding representation theory was greatly stimulated by (a) physics, (b) the theory of special functions, and (c) combinatorics, historically, in the opposite order. In my opinion, the demand from physics played the major role. (a) Harish-Chandra was Dirac’s assistant for some time and always expressed unreserved admiration for Dirac, according to Helgason’s interesting recent note ”Harish-Chandra.” The Lorentz group led him to the theory of infinite dimensional representations of semisimple Lie groups. However, later the mathematical goals like the Plancherel formula became preponderant in his research. (b) The theory of special and spherical functions was the main motivation for Gelfand in his studies of infinite dimensional representations, although he always emphasized the role of physics (and physicists). It was reflected in his program (1950s) aimed at “adding” the spherical and hypergeometric functions to the Lie theory. (c) Before the Lie theory, the symmetric group was the main “representation” tool in the theory of functions. It still remains of fundamental importance. However, using the symmetric group only is not sufficient to introduce and understand properly the differential equations and operators needed in the theory. As for (a), not all representation theories are of physical importance. It is my understanding, that “unitary” representations are of major importance, although modern theoretical physics uses all kinds of representations. For

0.3. DAHA IN HARMONIC ANALYSIS

21

instance,“massive” quantum theories must have a positive inner product. Mathematically, the unitarity is needed to decompose some natural spaces of functions, for instance, the spaces of square integrable functions, Lp –spaces, and so on and so forth. Main unitary lines. There are four main sources of unitary theories: (A) compact and finite groups, (B) noncompact Lie groups, (C) Heisenberg and Weyl algebras, (D) operator and von-Neumann algebras. The Clifford algebra, super Lie algebras, and “free fermions” are important too, but require a special discussion. The theory of automorphic forms and the corresponding representation theory, including that of the p–adic groups and Hecke algebras, definitely must be mentioned too. The arithmetic representation theory is an important part of modern mathematics attracting increasing attention in physics now. The above unitary theories have merits and demerits. The Heisenberg–Weyl algebra is heavily used in physics (“bosonization”), but it has essentially a unique unitary irreducible representation, the Fock representation. The von-Neumann factors actually have the same demerit. Only the pair ”factor–subfactor” appeared good enough for combinatorially rich theory. Note that the operator algebras give another approach to the Verlinde algebras. Spherical functions. The theory (A) is plain and square, but only finite dimensional representations can appear in this way. The representation theory of noncompact Lie groups is infinite dimensional (which is needed in modern physics), but the Harish-Chandra transform is not self-dual and its analytic theory is far from being complete. Also (B) is not very fruitful from the viewpoint of applications in the theory of special functions. For instance, the spherical functions in the so-called group case (k = 1) ”algebraically” coincide with the characters of compact Lie groups. The other two values k = 1/2, 2 in the Figure 0.5 below correspond, respectively, to the orthogonal case (SL(n, R)/SO(n)) and the symplectic case. The left column of the top block of Figure 0.5 shows the classical theory of characters and spherical functions of compact and noncompact Lie groups extended towards the orthogonal polynomials, Jack–Heckman–Opdam polynomials. The latter are generally beyond the Lie theory. We can define them as orthogonal polynomials (k must be assumed real positive or even “small” negative), or as eigenfunctions of the Sutherland–Heckman–Opdam operators, generalizing the radial parts of the Laplace operators on symmetric spaces. The latter definition works for arbitrary complex k (apart from a series of special values where the complete diagonalization is impossible).

22

0.3.2

CHAPTER 0. INTRODUCTION

From Lie groups to DAHA

Considering the left column of Figure 0.5 as a sample harmonic analysis program, not much is known in the Kac–Moody case (the right column). At the level of “Compact Characters” in the classical harmonic analysis, we have the theory of Kac–Moody characters which is reasonably complete, in spite of combinatorial difficulties with the so-called string functions. The next level is supposed to be the theory of KM spherical functions. There are several approaches (let me mention Dale Peterson, Lian and Zuckerman, and the book of Etingof, I.Frenkel, Kirillov Jr.), but we have no satisfactory general theory so far with a reservation about the group case, where the technique of vertex operators and conformal blocks can be used. Extending the theory from the group case (k = 1) to arbitrary root multiplicities is a problem. However, in my opinion, the key problem is that the Kac–Moody characters are not pairwise orthogonal functions in a way that one can expect taking the classical theory (the left column) as a sample. Analytically, they are given in terms of theta functions and cannot be integrated over noncompact regions in any direction unless special algebraic tools are used. It is exactly how Verlinde algebras enter the game. Verlinde algebras. We can identify the characters of integrable representations of the Kac–Moody algebra of a given level (central charge) c with the corresponding classical characters treated as functions at a certain alcove of P/N P for the weight lattice P and N = c + h∨ ; h∨ is the dual Coxeter number. The P/N P is naturally a set of vectors with the components in the N –the roots of unity. The images of the characters form a linear basis in the algebra of all functions on this alcove, called the Verlinde algebra. The fusion product corresponds to the pointwise multiplication, and the images of the characters become pairwise orthogonal with respect to the Verlinde inner product. The identification of the KM characters with the classical characters at roots of unity was used by Kac for the first time, when he calculated the action of the SL(2, Z) on the KM-characters. The automorphism τ 7→ −1/τ transforms the images of the characters into the delta functions of the points of the alcove. We mention that the interpretation of the Verlinde algebras in terms of quantum groups due to Kazhdan–Lusztig–Finkelberg leads to the classical characters at roots of unity as well. The general drawback of this approach is that the c, the levels, must be positive integers. The corresponding Verlinde algebras are totally disconnected for different levels, unless a special p–adic limiting procedure is used, similar to the one due to Ohtshuki in the theory of invariants of knots and links. It is why a uniform theory for all (unimodular) q is needed.

0.3. DAHA IN HARMONIC ANALYSIS

LIE GROUPS | Compact Characters | ⇓ | Spherical Functions | (k = 1, 2, 1/2) | ⇓ | Orthog Polynomials | (k is arbitrary) | − − − − − − − − −− DEGENERATE DAHA & Nonsym Harish-Chandra | H-Ch transform :

23

KAC-MOODY KM-Characters ⇓ Symmetric Spaces? ellip radial parts ! ⇓ Measure&Integrtn? V erlinde Algebras ! −−−−−−−−−− DUNKL OPERATORS Elliptic Nonsym Theory

↓ | ↓:

not finished yet

| Rational-Difference Nonsymmetric Theory | − − − − − − − − −− − − − − − − − − − − GENERAL q,t–DAHA & ACTION OF PSL2 (Z) q-Fourier Transform becomes self-dual ! Figure 0.5: Harmonic Analysis and DAHA Another drawback is of a technical nature. The Verlinde algebras are too combinatorial apart form the sl2 –case. For instance, there is a very difficult problem of finding the P SL(2, Z)–invariants in the tensor product of the Verlinde algebra and its dual. There is a solution of this problem, important in physical applications, only in the cases of sl2 and sl3 . The list of the “invariants” is sophisticated in the latter case. It is expected that the theory at generic q would be simpler to deal with. Many physicists now are working in the setting of unimodular q (|q| = 1) that are not roots of unity. Difference theory. The bottom block of Figure 0.5 shows the difference theory and the general double affine Hecke algebra (DAHA). In contrast to the previous “Harish-Chandra level,” the q–Fourier transform is self-dual, as holds for the classical Fourier transform and in the group case. Thus we are back to “normal” Fourier theory. This, however, does not mean that the analytic difficulties dissappear. Six different analytic settings are known in the q, t–case. Namely, there are one compact and two noncompact theories, and each exists in two variants: for real q and unimodular q, not counting the choice of the analytic spaces that can be used for the direct and inverse transforms.

24

CHAPTER 0. INTRODUCTION

There are two other theories of an algebraic nature. The seventh setup is the theory at the roots of unity. All irreducible representations are finite dimensional as q is a root of unity. The major example is the nonsymmetric Verlinde algebra, which is Fourier-invariant and, moreover, invariant with respect to the projective action of P SL(2, Z). The eighth is the theory of p–adic integration, where the connection with DAHA has not yet been established. The projective action of P SL(2, Z) is one of the most important parts of the theory of DAHA. The Fourier transform corresponds to the matrix ³ 0 1 ´ and is related to the transposition of the periods of an elliptic -1 0

curve. Special functions. There is a fundamental reason to expect much from DAHA in analysis: it is a direct generalization of the Heisenberg–Weyl algebra. DAHA was designed to fulfill a gap between the representation theory and classical theory of the special functions. It naturally incorporates the Bessel function, the hypergeometric and basic (difference) hypergeometric functions, and their multidimensional generalizations into the representation theory. This is especially true for the basic hypergeometric function. DAHA is heavily involved in the multidimensional theory of hypergeometric-type functions. This direction is relatively recent. The Dunkl operators and the rational DAHA are the main tools in the theory of the multidimensional Bessel functions. This theory is relatively recent; the first reference seems [O2]. Respectively, the degenerate DAHA and the trigonometric Dunkl operators serve the multidimensional hypergeometric functions. The one-dimensional DAHA is closely related to the super Lie algebra osp(2|1) and to its “even part,” sl(2) via the “exponential map” from DAHA to its rational degeneration. The one-dimensional Dunkl operator is the square root of the radial part of the Laplace operator in the rank one case, similar to the Dirac operator, although it is a special feature of the onedimensional case. For instance, the finite dimensional representations of osp(2|1) have a natural action of DAHA of type A1 . Their even parts, the classical finite dimensional representations of sl(2), are modules over the subalgebra of symmetric elements of DAHA: DAHA can be viewed as a natural successor of sl(2).

0.3.3

Elliptic theory

A solid part of the Kac–Moody harmonic analysis is the construction of the ”elliptic radial parts” due to Olshanetsky–Perelomov (the GLN –case), Ochiai–Oshima–Sekiguchi (the BC–type), and from [C17] for all reduced root systems. They exist only at the critical level, which does not make them too promising. We certainly need the theory for an arbitrary central charge. The

0.3. DAHA IN HARMONIC ANALYSIS

25

degenerate (“trigonometric,” to be more exact) double affine Hecke algebra presumably provides such a theory. Nonsymmetric theory. The most recognized applications of the degenerate DAHA so far are in classical harmonic analysis, namely, this algebra is very helpful in the Harish-Chandra theory of spherical functions and, more importantly, adds the nonsymmetric spherical functions and polynomials to the Lie–Harish-Chandra theory. The same degenerate DAHA also serves the elliptic nonsymmetric HarishChandra theory. The “nonsymmetric” is the key here, because the “elliptic” Dunkl operators (infinite trigonometric, to be more exact), commute at arbitrary levels, in contrast to the elliptic radial parts. The latter can be always defined as symmetrizations of the Dunkl operators, but commute only at the critical level [C17]. We also mention the recent [Ra] devoted to the interpolation and orthogonal “elliptic polynomials” in the C ∨ C–case. The connection with the DAHA of type C ∨ C (defined by Noumi–Sahi) has not been established yet. In the theory of DAHA, the “transforms” become homomorphisms of representations. For instance, the nonsymmetric Harish-Chandra transform appears as an analytic homomorphism from the trigonometric-differential polynomial representation of the degenerate DAHA (“Nonsym Harish-Chandra”) to its rational-difference polynomial representation. There exists a third elliptic-differential polynomial representation of DAHA (“Elliptic nonsym theory”) which is analytically isomorphic to the rationaldifference representation. The corresponding isomorphism can be called a nonsymmetric elliptic Harish-Chandra transform. See the middle block of Figure 0.5. The analytic part of the elliptic theory is not finished yet. The general q, t–DAHA governs the monodromy of the eigenfunctions of the elliptic Dunkl eigenvalue problem, corresponding to the action of the Weyl group and the translations by the periods of an elliptic curve. Therefore, besides the “elliptic transform,” shown in the figure as the arrow to the “difference-rational theory,” there is another arrow from the elliptic theory to the q, t–block. Similar to the Verlinde algebras, that describe the action of P SL(2, Z) on the Kac–Moody characters, the monodromy representation can be used to describe the projective action of the P SL(2, Z) on the eigenfunctions of the elliptic Dunkl eigenvalue problem. Deformation. A natural objective is to define the generalized Verlinde algebras for arbitrary unimodular q. Since DAHA describes the monodromy of the elliptic nonsymmetric theory (the previous block), such an extension of the Verlinde theory is granted. However, what can be achieved via DAHA appears more surprising. Under minor technical restrictions, DAHA gives a flat deformation of the

26

CHAPTER 0. INTRODUCTION

Verlinde algebra to arbitrary unimodular q. To be exact, little Verlinde algebras must be considered here, associated with the root lattices instead of the weight lattices. The dimension of the little Verlinde algebra remains the same under this deformation. All properties are preserved but the integrality and positivity of the structural constants, which are lost for generic q. These deformations were constructed in [C27, C28] using singular k, rational numbers with the Coxeter number as the denominator. See Section 0.5.1 below and the end of Chapter 3. It triggers the question about the classification of all representations of DAHA for such special fractional k. Indeed, if a “unitary” Kac–Moody representation theory exists, it could be associated with such representations, by analogy with the interpretation of the Verlinde algebras via the integrable representations of the Kac–Moody algebras. We also note a relation of the finite dimensional representations of DAHAs (for singular k) to the generalized Dedekind–Macdonald η–identities from [M1]; see [C28] and Chapter 3. There is an important application. We establish a connection between little Verlinde algebras of type A1 and finite dimensional representations of sl(2), since the deformation above followed by the limit q → 1 to the rational DAHA equip the little Verlinde algebras with the action of sl(2). Toward KM harmonic analysis. We conclude, that generalizations of the Verlinde algebra are reasonable candidates for categories of representations of unitary type, especially for those associated with the Kac–Moody algebras. If we expect the reduced category of integrable Kac–Moody representations to be a part of such a category, then it must be an extension of the Verlinde algebra, or, more likely, must have the reduced category as a quotient. I think that it is unlikely that the L2 –KM theory, if it exists, can be assumed to contain “ 1 ”, the basic representation, and other integrable representations from [Ka]. Using DAHA for this project, its unitary representations at |q| = 1 are natural candidates for such categories, especially for singular k, although generic q, k are interesting too. They are expected to have some extra structures. The most important are a) “fusion multiplication”, b) hopefully, the action of P SL(2, Z), and c)“good restrictions” to the affine Hecke subalgebras similar to the classical decomposition with respect to the maximal compact subgroup. The existence of multiplication indicates that the quotients (if we include 1) and constituents (generally) of the polynomial representation of DAHA or those for the functional variants of the polynomial representation must be examined first. All of them are commutative algebras. The most interesting “AHA–restriction theory” occurs at the singular k, when the polynomial representation becomes reducible.

0.4. DAHA AND VERLINDE ALGEBRAS

0.4

27

DAHA and Verlinde algebras

The Lie groups formalize the concept of symmetry in the theory of special functions, combinatorics, geometry, and, last but not least, physics. In a similar way, abstract Verlinde algebras “describe” Fourier transforms, especially the theories with the Gaussian, satisfying the following fundamental property. The Fourier image of the Gaussian must be to be proportional to its inverse or the Gaussian itself, depending on the setting, similar to the properties of the Laplace and Fourier transforms in the classical theory. Thus, the Verlinde algebras (are expected to) formalize an important portion of the classical Fourier analysis. Note that the Fourier transform has an interpretation in the Lie theory as a reflection from the Weyl group. There many such reflections apart from the sl(2)–case. This does not match well the natural expectation that there must be a unique Fourier transform, or, at least, the major one if there are several. It is really unique in the DAHA theory.

0.4.1

Abstract Verlinde algebras

In the finite dimensional semisimple variant, the abstract Verlinde algebra is the algebra of C–valued functions V =Funct( ./ ) on a finite set ./ equipped with a linear automorphism σ, the Fourier transform. The algebra V has a unit, which is 1 considered as a constant function. Note that σ is not supposed to preserve the (pointwise) multiplication. As a matter of fact, it never does. The space V has a natural basis of the characteristic functions χi (j) = δij , where i, j ∈ ./ , δij is the Kronecker delta. The first two assumptions are that (a) σ −1 (1) = χo for the zero-point o ∈ ./ , and def (b) the numbers µi == σ(χi )(o) are nonzero. Since the latter constants are interpreted as masses of particles in conformal field theory, let us impose further conditions µi > 0. def

Then the spherical functions are pi == σ(χi )µ−1 i for i ∈ ./ . In other words, pi is proportional to σ(χi ) and satisfies the normalizing condition pi (o) = 1. def P Introducing the inner product as hf, gi == i µi f (i)g(i), the correspond−1 ing delta functions are δi = µ−1 i χi = σ (pi ). Indeed, they are obviously dual to the characteristic functions with respect to the inner product. Concerning taking the weights µi = σ(χi )(o) in the inner product, such choice is equivalent to the following classical property, which holds in all variants of the Fourier theory: the Fourier-images of the delta functions are spherical functions. The next assumption is that (c) σ is unitary up to proportionality with respect to h , i.

28

CHAPTER 0. INTRODUCTION It readily gives the norm formula: hpi , pj i = δij µ−1 i h1, 1i.

In this approach, the latter formula is a result of a simple sequence of formal definitions. However, it is really fruitful. It leads to the best-known justification of the norm formula for the Macdonald polynomials, including the celebrated constant term conjecture, which is the formula for h1, 1i. To be more exact, it provides a deduction of the norm formula from the socalled evaluation formula for the values of the Macdonald polynomials at the “zero-point.” In its turn, the latter formula results from the self-duality of the double affine Hecke algebra (the existence of the self-dual Fourier transform). The self-duality and the evaluation formula are directly connected with the following important symmetry: (d) pi (j) = pj (i) for i ∈ ./ 3 j. Note that the theory of Macdonald polynomials requires an infinite dimensional variant of the definitions considered above. Namely, σ −1 becomes an isomorphism from the algebra of Laurent polynomials to its dual, the space of the corresponding delta functions. Hence it becomes a map from one algebra to another algebra. Nevertheless it is possible to deduce the major formulas for the Macdonald polynomials within the finite dimensional “self-dual” setting above, by using the consideration at the roots of unity. Without going into detail, it goes as follows. The number of Macdonald polynomials, which are well defined when q is a root of unity, grows together with the order of q. One checks, say, the Pieri rules for such Macdonald polynomials using the duality argument, then tends the order of q to ∞, and, finally, obtain the desired formula for arbitrary q and all Macdonald polynomials. The Pieri rules for multiplication of the Macdonald polynomials by the monomial symmetric functions were, indeed, justified in [C19, C20] by using the roots if unity. Concluding the discussion of abstract Verlinde algebras, the Gaussian appears as a function γ ∈ V such that (e) σ(γ) =const·γ −1 , γ(i) 6= 0 for i ∈ ./ . The constant here is the abstract Gauss–Selberg sum. It is necessary to fix the normalization of σ and γ to make this sum well defined. The normalization of σ has already been fixed by the condition σ(χo ) = 1. The natural normalization of the Gaussian is γ(o) = 1. The assumptions (a-e) are more than sufficient to make the Verlinde algebras rigid enough. However, in my opinion, the key axiom is PBW, the Poincare–Birkhoff–Witt property, which requires the operator approach, to be discussed next.

0.4. DAHA AND VERLINDE ALGEBRAS

0.4.2

29

Operator Verlinde algebras

The above discussion is actually about a “good self-dual” invertible linear operator acting in a commutative algebra. It is obviously too general, and more structures have to be added. We need to go to the operator level, switching from the characteristic functions and the spherical functions to the corresponding commutative algebras X , Y of the operators, which are diagonal at the corresponding sets of functions. The operator Verlinde algebra A (the main example will be the double affine Hecke algebra) is generated by commutative algebras X and Y, and the algebra H controlling the symmetries of the X–operators and the Y – operators. In the main examples, the Weyl groups (or somewhat more general groups) are the groups of symmetries upon degenerations; H are the corresponding Hecke algebras serving the non-degenerate q, t–case. The key and the most restrictive assumption is the PBW property, which states that (A) the natural map from the tensor product X ⊗ Y ⊗ H to A is an isomorphism of the linear spaces, as well as the other five maps corresponding to the other orderings of X , Y, H. The Fourier transform and the Gaussian are formalized as follows. The projective P SL2 (Z) must act in A by outer automorphisms, i.e., (B) τ+ , τ− act in A as algebra automorphisms and satisfy the Steinberg relation τ+ τ−−1 τ+ = τ−−1 τ+ τ−−1 ; def

(C) they preserve the elements from H, the element σ == τ+ τ−−1 τ+ maps X onto Y, and τ+ is the identity on X . Note that (B)–(C) provide that τ− = στ+−1 σ −1 is identical on Y. The automorphism τ+ becomes the multiplication by the Gaussian in Verlinde algebras, which connects (C) with the assumption (e) above. The remaining feature of abstract Verlinde algebras to be interpreted using the operator approach is the existence of the inner product. We will postulate the existence of the corresponding involution. We assume that: (D) A is a quotient of the group algebra of the group B such that the antiinvolution B 3 g 7→ g −1 of B becomes an anti-involution of A; (E) τ± and σ come from automorphisms of the group B and therefore commute with the anti-involution from (D). We note that the inner products can be associated with somewhat different involutions, for instance, in the case of real harmonic analysis (q > 0). The assumption that there is a system of unitary generators is a special feature of the unimodular theory (|q| = 1). The anti-involution in the Verlinde case and those in the main generalizations do satisfy (D)–(E). The Verlinde algebras can be now re-defined as σ–invariant unitary irreducible representations V of A that are X –spherical, i.e., are some quotients

30

CHAPTER 0. INTRODUCTION

of the (commutative) algebra X . A representation is called unitary if it has a hermitian inner product inducing the anti-involution of A from (D). Note that (D)–(E) automatically guarantee that the Fourier transform σ is “projectively unitary” in σ–invariant unitary irreducible representations. Thanks to irreducibility, both σ and τ+ are fixed uniquely in the group AutC (V )/C∗ (if they act in V ) and induce the corresponding automorphisms of A. Strictly speaking, assumptions (D)–(E) can be replaced by a more general property that A has an anti-involution which commutes with the τ± and σ. However the double Hecke algebras (the main examples so far) really appear as quotients of the group algebras and τ± and σ act in this group, the elliptic braid group. An important advantage of the operator approach is that we can relax the constraints for the abstract Verlinde algebras defined above by considering non-unitary and even non-semisimple projective P SL(2, Z)–invariant X–spherical irreducible representations of A. They do appear in applications. Another advantage is that it is not necessary to impose the Fourierinvariance. Generally, the main problem of the Fourier analysis is in calculating the Fourier images σ(V ) of arbitrary A–modules V and the corresponding transforms V 3 v 7→ σ(v) ∈ σ(V ), which induce σ in A. Similarly, τ+ becomes multiplication by the Gaussian given by a variant 2 of the classical formula ex in the main examples. The operator approach makes it possible to define the Gaussian for any irreducible A–module V ; it as an operator with values in τ+ (V ). Let us discuss now the DAHA of type A1 in detail, where the theory is already quite interesting. The transition to arbitrary root system is sufficiently smooth. See Chapter 2 and, especially, Chapter 3.

0.4.3

Double Hecke Algebra def

The most natural definition goes through the elliptic braid group Bq == hT, X, Y, q 1/4 i/ with the relations T XT = X −1 , T Y −1 T = Y, Y −1 X −1 Y XT 2 = q −1/2 . Here

B1 = π1orb ({E \ 0}/S2 )

∼ = π1 ({E × E \ diag}/S2 ), where E is an elliptic curve. Using the orbifold fundamental group here makes it possible to “divide” by the symmetric group S2 without removing the ramification completely, i.e., removing all four points of second order. Only one puncture is needed to obtain the above relations.

0.4. DAHA AND VERLINDE ALGEBRAS

31

Actually, in the case of An , it is sufficient to consider the product of n + 1 copies of E and remove the “diagonal” before dividing by Sn+1 instead of using the orbifold group. The corresponding braid group is isogenous to the one with the relations above; it was calculated by Birman and Scott. To complete the definition of DAHA, we impose the quadratic T –relation: def

HH == C[Bq ]/((T − t1/2 )(T + t−1/2 )). If t = 1, HH becomes the Weyl algebra extended by the reflection; T becomes s satisfying sXs = X −1 , sY s = Y −1 , s2 = 1. The Fourier transform, which plays a major role in the theory, is the following outer automorphism of DAHA: σ : X 7→ Y −1 , Y 7→ XT 2 , T 7→ T. Thus the DAHA Fourier transforms finds a conceptual interpretation as the transposition of the periods of the elliptic curve, This is not surprising from the viewpoint of CFT, KZB, and the Verlinde algebras, but such a connection with topology still remains challenging when we deal with the applications of DAHA in harmonic analysis. The representations where σ acts (i.e., becomes inner) are called Fourierinvariant or self-dual. The nonsymmetric Verlinde algebra and the Schwartz space are examples. More generally, the topological interpretation above readily gives that the group P SL(2, Z) acts projectively in HH: τ+ : Y 7→ q −1/4 XY, X 7→ X, T 7→ T, τ− : X 7→ q 1/4 Y X, Y 7→ Y, T 7→ T, σ = τ+ τ−−1 τ+ = τ−−1 τ+ τ−−1 , µ ¶ µ ¶ 10 11 7→ τ− . 7→ τ+ , where 11 01 We will use k such that t = q k and (sometimes) set HH(k) as t = q k . The algebra HH(k) acts in P = Laurent polynomials in terms of X = q x , namely, T 7→ t

1/2

t1/2 − t−1/2 s+ (s − 1), q 2x − 1 def

Y 7→ spT, sf (x) == f (−x), def

pf (x) == f (x + 1/2), t = q k . The operator Y is called the difference Dunkl operator. It is important to note that τ− preserves P. On the other hand, τ+ does not act there for a very simple reason. If it acts then it must be multiplication 2 by the Gaussian q x , which does not belong to P.

32

CHAPTER 0. INTRODUCTION

The “radial part” appears as follows. One checks that Y + Y −1 preserves def Psym == symmetric (even) Laurent polynomials (the Bernstein lemma in the theory of affine Hecke algebras). The restriction H = Y + Y −1 |sym is the q–radial part and can be readily calculated: t1/2 X −1 − t−1/2 X −1 t1/2 X − t−1/2 X −1 p+ H= p . X − X −1 X −1 − X

0.4.4

Nonsymmetric Verlinde algebras

The key new development in the theory of orthogonal polynomials, algebraic combinatorics, and related harmonic analysis is the definition of the nonsymmetric Opdam–Macdonald polynomials. The main references are [O3, M5, C20]. Opdam mentions in [O3] that this definition (in the differential setup) was given in Heckman’s unpublished lectures. These polynomials are expected to be, generally speaking, beyond quantum groups and Kac–Moody algebras because of the following metamathematical reason. Major special functions in the Lie and Kac–Moody theory, including the classical and Kac–Moody characters, spherical functions, and conformal blocks, are W –invariant. However, this definition is not quite new in representation theory. The limits of the nonsymmetric polynomials as q → ∞ are well known. They are the spherical functions due to Matsumoto. The nonsymmetric Verlinde algebras are directly connected to the nonsymmetric polynomials evaluated at the roots of unity, Let q = exp(2πi/N ), 0 < k < N/2, k ∈ Z. The nonsymmetric Verlinde algebra V is defined as the algebra of functions of the set ./ = {−

N −k+1 k+1 k k+1 N −k , ..., − ,− , , ..., }. 2 2 2 2 2

(0.4.1)

It has the unique structure of a HH –module that makes the map q x (z) = q z from P to V a HH–homomorphism. The above set is not s–invariant; it is sp–invariant. Nevertheless, the formula for T can be used in V, because the contributions of the “forbidden” points k/2 = s(−k/2) and (N − k + 1)/2 = s(−(N − k + 1)/2) come with zero coefficients. The operators X, Y, T are unitary in V with respect to the positive hermitian form which will not be discussed here (it generalizes the inner product for conformal blocks). See Chapter 2. The positivity requires choosing the “minimal” primitive N –root of unity q above. The whole P SL(2, Z) acts in V projectively as well as in the image Vsym of Psym . The latter image can of course be defined without any reference to the polynomial representation. A general definition is as follows: Vsym = {f | T f = t1/2 f }. Here it simply means that the function f (z) must be s–invariant

0.4. DAHA AND VERLINDE ALGEBRAS

33

(even) for the points z that do not leave the above set under the action of s (recall that it is not s–invariant). Therefore: dimC V = 2N − 4k, dimC Vsym = N − 2k + 1. We call V a perfect representation. By perfect, we mean that it has all major features of the irreducible representations of the Weyl algebras at roots of unity, i.e., cannot be better. Formally, it means that it has a perfect duality pairing, directly realted to the Fourier transform, and a projective action of P SL(2, Z). The nonsymmetric characters in V and the symmetric ones in Vsym are, respectively, the eigenfunctions of Y and Y + Y −1 . When k = 0, we come to the well-known definitions in the theory of Weyl k=1 is the usual Verlinde algebra [Ver]: algebra. As k = 1, Vsym τ+ becomes the Verlinde T –operator and σ becomes the S–operator.

0.4.5

Topological interpretation

Let E be an elliptic curve over C, i.e., E = C/Λ, where Λ = Z + Zı. Topologically, the lattice can be arbitrary. Let 0 ∈ E be the zero-point, and −1 the automorphism x 7→ −x of E. We are going to define the orbifold fundamental group of the space (E \ 0)/ ± 1 = P1C \ 0. Since this space is contractible, its usual fundamental group is trivial. One could take the quotient after removing all (four) ramification points of −1. However, it would enlarge the fundamental group dramatically. Therefore we need to understand this space in a more refined way. We take the base (starting) point ? = −ε − εı ∈ C for small ε > 0. The orbifold fundamental group π1orb ((E \ 0)/ ± 1) is defined following [C13]. We switch from E to its universal cover C and define the paths as curves γ ∈ C\Λ c = {±1}nΛ. from ? to w(?), b where w b∈W c : we add the image of the second The composition of the paths is via W path under w b to the first path if the latter ends at w(?). b The fundamental group π1orb (·) of the above paths modulo homotopy is isomorphic to B1 = hT, X, Y i/hT XT = X −1 , T Y −1 T = Y, Y −1 X −1 Y XT 2 = 1i, where T is the half-turn, i.e., the clockwise half-circle from ? to s(?), X, Y are, respectively, 1 and ı considered as vectors from ?. The definition we used is close to the calculation of the fundamental group of {E×E\ diagonal} divided by the transposition of the components, however,

34

CHAPTER 0. INTRODUCTION

there is no exact coincidence. Let me also mention the relation to the elliptic braid group due to v.d. Lek, although he removes all points of second order. The topological interpretation is the best way to understand why the group P SL2 (Z) acts in B1 projectively. Its elements act in C naturally, by the corresponding real linear transformations. On E, they commute with the reflection −1, preserve 0, and permute the other three points of second order. Given g ∈ SL2 (Z), we set g = exp(h), gt = exp(th) for the proper h ∈ sl2 (R), 0 ≤ t ≤ 1. The position of the base point ? will become g(?), so we need to go back, i.e., connect the image with the base point by a path. To be more exact, the g–image of γ ∈ π1orb will be the union of the paths {gt (?)} ∪ g(γ) ∪ {w(g b −t (?)}, where the path for γ goes from ? to the point w(?). b We can always choose the base point sufficiently close to 0 and connect it with its g–image in a small neighborhood of zero. This makes the corresponding automorphism of B1 unique up to powers of T 2 . All such automorphisms fix T, because they preserve zero and the orientation. Thus we have constructed a homomorphism α : SL2 (Z) → AutT (B1 )/T 2Z , where AutT (B1 ) is the group of automorphisms of B1 fixing T. The elements from T 2Z = {T 2n } are considered here as innerµ automorphisms. ¶ µ ¶ 1 0 1 1 and . Then Let τ+ , τ− be the α–images of the matrices 1 1 0 1 µ ¶ 0 1 σ = τ+ τ−−1 τ+ corresponds to . −1 0 Taking the “simplest” pullbacks for τ± , we arrive at the Steinberg relation: τ+ τ−−1 τ+ = τ−−1 τ+ τ−−1 .

Abstract construction. Generalizing, let E be an algebraic, or complex analytic, or symplectic, or real analytic manifold, or similar. It may be noncompact and singular. We assume that there is a continuous family of topological isomorphisms E → Et for manifolds Et as 0 ≤ t ≤ 1, and that E1 is isomorphic to E0 = E. The path {Et } in the moduli space M of E induces an outer automorphism ε of the fundamental group π1 (E, ?) defined as above. Namely, we take the image of γ ∈ π1 (E, ?) in π1 (E1 , ?1 ) for the image ?1 of the base point ? ∈ E and conjugate it by the path from ?1 to ?. We obtain that the fundamental group π1 (M) (whatever it is) acts in π1 (E) by outer automorphisms modulo inner automorphisms.

0.5. APPLICATIONS

35

The above considerations correspond to the case when a group G acts in E preserving a submanifold D. Then π1 (M) acts in π1orb ((E \ D)/G) by outer automorphisms. Another variant is with the Galois group taken instead of π1 (M), assuming that E is an algebraic variety over a field that is not algebraically closed. The action of π1 (M) on an individual π1 (E) extends, in a way, the celebrated Kodaira–Spencer map and is of obvious importance. However, calculating the fundamental groups of algebraic (or similar) varieties, generally, is difficult. The main examples are the products of algebraic curves and related configuration spaces. Not much can be extracted from the action above without an explicit description of the fundamental group.

0.5

Applications

The Verlinde algebras and, more generally, the finite dimensional representations of DAHA, have quite a few applications. We will mainly discuss the non-cyclotomic Gaussian sums and the diagonal coinvariants. Both constructions are based on the DAHA deformation–degeneration technique. We also discuss a classification of the Verlinde algebras.

0.5.1

Flat deformation

Recall that we set HH(k) when t = q k . We will need the little nonsymmetric Verlinde algebra Ve k which is the unique nonzero irreducible quotient of V k def f upon the restriction to the little DAHA H H == hT, X 2 , Y 2 i, corresponding to the subspace of Laurent polynomials in terms of X 2 . Its dimension is dimVe k = dimV k /2 = N − 2k. k=1 is a subalgebra of the Verlinde algebra defined for the Generally, Vesym radical weights (from the root lattice) instead of all weights. Let N = 2n + 1, q 1/2 = − exp( πi ). Such a choice of q is necessary for the N positivity of the corresponding inner product. We set m = n − k, dimC Ve k = N − 2k = 2m + 1. The aim is to deform Ve , Vesym , the projective action of P SL(2, Z), and all other structures making q arbitrary unimodular. The construction is as follows. ¯ For any q, k¯ = − 12 − m, m ∈ Z+ , P considered as an HH(k)–module has a unique irreducible quotient V 6= 0: V = Funct(

k¯ + 1 k¯ + 1 k¯ , ..., − , − ), 2 2 2

dimC V = 2m + 1, dimC V sym = m + 1. It is unitary as |q| = 1, 0 < arg(q) < π/m.

36

CHAPTER 0. INTRODUCTION The representation V , the desired deformation, becomes Ve k as 2πi 2π 1 q = e N , arg(q) = , k¯ = k − − n, i.e. , m = n − k. 2n + 1 2

2π Concerning the positivity of the inner product, note that 2n+1 < π/m, since k ≥ 0. The deformation construction makes it possible to connect the Verlinde algebras and their nonsymmetric k–generalizations with the classical representation theory. Since the parameter q is now generic, one may expect relations to sl(2). Indeed, the representation V is a q–deformation of an irreducible representation of osp(2|1) (see Chapter 2). Respectively, V sym deforms the irreducible representation of sl(2) of spin=m/2. We need the rational degeneration of DAHA to clarify it.

0.5.2

Rational degeneration

The rational DAHA is the following quasi-classical limit: √

Y = e−

κy/2

√

, X=e

κx

, q = eκ , κ → 0

of HH . Explicitly, def

HH00 == hx, y, si/relations: [y, x] = 1 + 2ks, s2 = 1, sxs = −x, sys = −y. Note that the notation HH0 is reserved for trigonometric degeneration (see Chapter 2). The algebra HH00 acts in the polynomial representation, which is d k C[x], s(x) = −x, y 7→ D = − (s − 1). dx x The square of the Dunkl operator D is the radial part of the Laplace operator: 2k d d2 + . 2 dx x dx The DAHA Fourier automorphism σ becomes the outer automorphism corresponding to the Hankel transform in the theory of Bessel functions. Let k¯ = − 21 − m. Then D2 |sym =

00 def

lim V = V == C[x]/(x2m+1 ).

κ→0

We call this module a perfect rational representation. The automorphism 00 σ acts there. Its symmetric part V sym is nothing else but the irreducible representation of sl(2) of dim= m + 1. The formulas for the generators of sl(2) in terms of x, y and the action of σ are h = (xy + yx)/2, e = x2 , f = −y 2 , σ becomes w0 .

0.5. APPLICATIONS

0.5.3

37

Gaussian sums

The new approach to the Gaussian sums based on the deformation construction above is as follows. The τ− acts in V and V sym , since it acts in the polynomial representation P. In contrast to the polynomial representation, τ+ also acts in V and its symmetric part. It is the multiplication by 2

def

q x == q

2 ¯ (±k+j) 4

,

where ± =plus for j > 0 and minus ³ otherwise. Similarly, without going ´ ¯ ±k+j into detail, σ is essentially the matrix chari ( 2 ) for the nonsymmetric characters chari (Y –eigenfunctions), where i and j belongs to the same set (0.4.1). The interpretation of the Gausssian as τ+ is of key importance for the calculation of the non-cyclotomic Gaussian sum. Generally speaking, a Gauss– Selberg sum is a summation of the Gaussian with respect to a certain “measure.” Here it becomes 2j m ¯ Y ¯ 2j+k X 1 − q l+2k−1 ¯ 1−q j 2 −kj q 1 − q k¯ l=1 1 − q l j=0 for any q, k¯ = −1/2 − m. It equals m ¯ Y 1 − q 2k+2j j=1

¯ 1 + q k+2j

.

Under the reduction considered above, N = 2n + 1, m = n − k, 0 ≤ k ≤ n, q 1/2 = − exp(πi/N ), V = Ve k , and the product can be somewhat simplified: m ¯ Y 1 − q 2k+2j j=1

1+

¯ q k+2j

=q

−

m(m+1) 4

m−1 Y

(1 − q n−j ).

j=0

Now let m=n, i.e., let m be the maximal possible. Then k = n − m = 0, ¯ k = −1/2 − n = 0 mod N, and the “measure” in the Gaussian sum becomes trivial. Setting l = n2 mod N, we arrive at the identity: N −1 X j=0

2

qj = ql

2

n Y j=1

(1 − q j ).

√ The product can be readily calculated using Galois theory. It equals N √ for n = 2l and i N otherwise. It gives a new proof of the classical Gauss formulas.

38

0.5.4

CHAPTER 0. INTRODUCTION

Classification

We are going to describe all nonsymmetric Verlinde algebras of type A1 , i.e., irreducible quotients of the polynomial representation P that are P SL(2, Z)– invariant. The assumption is that q 1/2 is a primitive 2N –th root of unity, t = q k . All σ–invariant quotients of P must be through the HH –module V4N = P/(X 2N + X −2N − (tN + t−N )). Indeed, the invariance gives that the element X 2N + X −2N , central in HH, must act as its σ–image Y 2N + Y −2N , which is tN + t−N in P. The module V4N is irreducible unless k is integral or half-integral. Note that for generic k, it is not σ–invariant. Instead, the involution X ↔ Y, T ↔ −T −1 , t1/2 7→ t1/2 , q 1/2 7→ q −1/2 is an inner automorphism of V4N . Let us discuss special (“singular”) k in detail. The module V4N becomes, respectively, V −2 = P/(X 2N + X −2N − 2) or V 2 = P/(X 2N + X −2N + 2), as k ∈ Z or k ∈ 1/2 + Z. Respectively, tN = q kN = ±1. The module V 2 is an def extension of V2N == P/(X N + X −N ) by its ςy –image ςy (V2N ). The representations V = V2N −4k of dimension 2N − 4k defined above for the integral 0 < k < N/2 is a quotient of V −2 . The representation V = V 2|k| of dimension 2|k| for the half-integral −N/2 < k = −1/2 − m < 0 (the notation was k¯ was such k) is a quotient of V 2 . The construction of V2N −4k holds without changes for the positive halfintegral k < N/2, but then V2N −4k becomes a quotient of V 2 . The same notation V2N −4k will be used. Let k ∈ Z/2, |k| < N/2. The substitution T 7→ −T, t1/2 7→ −t1/2 identifies the polynomial representations for t1/2 and −t1/2 . Thus it is sufficient to decompose P upon the transformation k 7→ k + N, and we can assume that N/2 ≤ k < N/2. We will also need the outer involutions of HH: ι : T 7→ −T, X 7→ X, Y 7→ Y, q 1/2 7→ q 1/2 , t1/2 7→ t−1/2 , ςx : T 7→ T, X 7→ −X, Y 7→ Y, q 1/2 7→ q 1/2 , t1/2 7→ t1/2 , ςy : T 7→ T, X 7→ X, Y 7→ −Y, q 1/2 7→ q 1/2 , t1/2 7→ t1/2 . def

def

+ − Let V2|k| == V 2|k| , V2|k| == ςx (V 2|k| ). Finally, up to ι, ς, there are three different series of nonsymmetric Verlinde algebras, namely, V2N −4k (integral N/2 > k > 0), V2|k| (half-integral −N/2 < k < 0), and V2N +4|k| for integral −N/2 < k < 0. The latter module is defined as a unique nonzero irreducible quotient of P for such k and is also isomorphic to the ιςy –image of the kernel of the map

0.5. APPLICATIONS

39

V −2 → V2N −4|k| . It is non-semisimple. The previous two (series of modules) are semisimple. All three are projective P SL(2, Z)–invariant. The modules V ±2 are decomposed as follows. There are four exact sequences 0 → ιςy (V2N +4k ) → V −2 → V2N −4k → 0 for k ∈ Z+ , + − ) → V2N → V2N −4k → 0 for k ∈ 1/2 + Z+ . 0 → ι(V2k ⊕ V2k The arrows must be reversed for k < 0: 0 → ιςy (V2N −4|k| ) → V −2 → V2N +4|k| → 0 for k ∈ −1 − Z+ , + − 0 → ι(V2N −4|k| ) → V2N → V2|k| ⊕ V2|k| → 0 for k ∈ −1/2 − Z+ . Otherwise V −2 and V2N are irreducible. Recently, a certain non-semisimple variant of the Verlinde algebra appeared in [FHST] in connection with the fusion procedure for the (1, p) Virasoro algebra, although this connection is still not justified in full. Generally speaking, the fusion procedure for the Virasoro-type algebras and the socalled W –algebras can lead to non-semisimple Verlinde algebras. There are no reasons to expect the existence of a positive hermitian inner product there like the Verlinde pairing for the conformal blocks, because the corresponding physics theories are massless. Surprisingly, the algebra from [FHST] is defined using the usual (massive) Verlinde algebra under certain degeneration. Presumably it coincides with the algebra of even elements of V2N +4|k| for k = −1 or, at least, is very close to this algebra. Mathematically, the module V2N +4|k| and its multidimensional generalizations could be expected to be connected with the important problem of describing the complete tensor category of the representations of Lusztig’s quantum group at roots of unity [Lus3]. The Verlinde algebra, the symmetric part of V2N −4|k| for k = 1, describes the so-called reduced category and, in a sense, corresponds to the Weyl chamber. The non-semisimple modules of type V2N +4|k| are expected to appear in the so-called case of the parallelogram.

0.5.5

Weyl algebra

Before turning to the diagonal invariants, let us first discuss the specialization t = 1. One has: HH(t=1) = WoS2 , T 7→ s ∈ S2 , s2 = 1, where the Weyl algebra, denoted here by W, is a quotient of the algebra of noncommutative polynomials C[X ±1 , Y ±1 ] with the relations sXs = X −1 , sY s = Y −1 , XY = Y Xq 1/2 .

40

CHAPTER 0. INTRODUCTION Given N ∈ N, we set q 1/2 = exp( 2πi ), N def

W • = W/(X N = 1 = Y N ), HH• == HH(t=1) /(X N = 1 = Y N ) = W • oS2 . The algebra W • has a unique irreducible representation def

V • == C[X, X −1 ]/(X N = 1), Y (X m ) = q −m/2 X m , m ∈ Z, which is also a unique irreducible HH• –module. Moreover, τ and σ act in V • . Recall that the action of σ in HH(t=1) is as follows: X 7→ Y −1 , Y 7→ X, s 7→ s. The problem is that the above description of V • doesn’t make the σ– invariance clear and is inconvenient when studying the σ–action, playing a major role in the theory of theta functions and automorphic forms. To make the X and the Y on equal footing, we can take a σ–eigenvector v ∈ V • and set V • = (W/Jv )(v) for the σ − invariant ideal Jv = {H | H(v) = 0, H ∈ W}. This presentation makes the σ–invariance of V • obvious, but finding “natural” σ–eigenvectors in V • with reasonably explicit Jv is not an easy problem. The simplest case is N = 3, dimV • = 3, where we can proceed as follows. The space {v ∈ V • | s(v) = −v} is one-dimensional for N = 3. It is nothing but the space of odd characteristics in the classical theory of onedimensional theta functions. It equals Cd for d = X − X −1 . Since σ(s) = s, this space must be σ–invariant. Then, calculating Jd is simple: Jd = WJo , where Jo = {H ∈ W • | sH = Hs, H(d) = 0 ∈ V • . One has: {Y + Y −1 }(d) = {(Y + Y −1 )(X − X −1 )}(1) = {q −1/2 (XY − X −1 Y −1 ) +q 1/2 (XY −1 − X −1 Y )}(1) = (q 1/2 + q −1/2 )(d) = −d, {X + X −1 }(d) = X 2 − X −2 = X −1 − X = −d. Thus, Y + Y −1 + 1, X + X −1 + 1 ∈ Jo , and we can continue: {Y X + Y −1 X −1 }(d) = q −1 X 2 − 1 + 1 − q −1 X −2 = −q −1 d. Actually, this calculation is not needed, since Y + Y −1 + 1 and X + X −1 + 1 algebraically generate Jo , (it is not true in the commutative polynomials!) and we can use the Weyl relations. Finally, we arrive at the equality Jd = WJo . The next section contains a generalization of this construction for HH .

0.5. APPLICATIONS

41

Using the non-commutativity of X and T in the latter argument is directly connected with the recent results on W –invariant polynomial differential operators due to Wallach, Levasseur, Stafford, and Joseph. They prove that the invariant operators are generated by the W –invariant polynomials and the W –invariant differential operators with constant coefficients, using that they do not commute. Of course it is not true for the polynomials in terms of two sets of variables.

0.5.6

Diagonal coinvariants

The previous construction can be naturally generalized to the case of an arbitrary Weyl group W acting in Rn . We set w−1 f (x, y) = f (wx, wy) for a function f in terms of x, y ∈ Rn , def

def

Io == Ker(C[x, y]sym 3 f 7→ f (0, 0)), and ∇ == C[x, y]/(C[x, y]Io ). Haiman conjectured that ∇ has a natural quotient V of dim= (h + 1)n for the Coxeter number h. It must be isomorphic to the space Funct(Q/(h + 1)Q) as a W –module for the root lattice Q and have a proper character with respect to the degree in terms of x and y. Note that Q/(h + 1)Q is isomorphic to P/(h + 1)P for the weight lattice P , since the order [P : Q] is relatively prime with (h + 1). He proved that ∇=V and dim∇ = (n+2)n in the case of An for W =Sn+1 . The coincidence is a very special feature of An . As far as I know, no general uniqueness claims about V as a graded vector space and as a W –module were made. One needs the double Hecke algebra to make this quotient “natural.” Haiman’s conjecture was recently justified by Gordon. He proves that 00 V = grV for k¯ = −1/h − 1, 00

00

where V is a W –generalization of the perfect rational representation V considered above as m = 1. By gr, we mean taking the graded vector space 00 of V with respect to the degree in terms of x and y; see Section 2.12.5 from Chapter 2 and Section 3.11 from Chapter 3. 00 Recall that h = 2 and, generally, dimC V = 2m+1 in the case of A1 . Here we make m = 1, so the representation considered by Gordon is of dimension 00 3 for A1 . Even in this case, the coincidence grV = ∇ is an instructional exercise. It is not immediate because ∇ is given in terms of double polynomials 00 and V was defined as a quotient of the space of single polynomials; cf. the previous section. Adding q to the construction, we obtain the following theorem, describing a W –generalization of the perfect module V considered above for the simplest nontrivial k¯ = −1 − 1/h.

42

CHAPTER 0. INTRODUCTION 00

(a) There exist Lusztig-type (exp-log) isomorphisms V ­ V ­ V • for ¯ k=−1/h − 1. Here V • is a unique HH• –quotient of P = C[X, X −1 ] for the reduction: • = {q 1/[P :Q] = exp(

2πi ¯ def ), N == h + 1, t = q k = 1}. N

(b) Concerning the coinvariants, V = HH/HHIo (∆), Io = Ker(HHsym 3 H 7→ H(∆) ∈ V ), where HHsym is a subalgebra of the elements of HH commuting with T. Here ∆ is the discriminant ∆ = t1/2 X − t−1/2 X −1 for n = 1; generally, it is a product of such binomials over positive roots. The second part demonstrates how double polynomials appear (the general definition of V is given in terms of single polynomials). Using that ∆ generates the one-dimensional “sign-representation” of the nonaffine Hecke algebra hT i, we can naturally identify V with quotients of the space of double Laurent polynomials. Since t = 1 for the •–reduction, one has: HH• =(Weyl algebra)oW. Thus V • is its unique irreducible representation under the relations X N =1=Y N imposed, namely, V • ' C[X, X −1 ]/(X N = 1). Generally, its dimension is dim=N rank (rank = the number of X–generators), exactly what Haiman conjectured. Note that given n, part (b) holds only thanks to a very special choice of N. In Gordon’s proof, the (h + 1)n –formula requires a construction of the 00 resolution of V and more. The theorem above gives an explanation of why the dimension is so simple. The dimension looks like that from the theory of Weyl algebra, although the definition of the space of coinvariants has nothing to do with the roots of unity, and Gordon’s theorem really belongs to this theory!

Chapter 1 KZ and QMBP 1.0

Soliton connection

The main object of this chapter is the r–matrix KZ equation for the usual classical r–matrices and their generalizations attached to the root systems from [C6]. The key observation in this paper was that the classical Yang– Baxter equation (YBE) for r is equivalent (under minor technical constraints) to the cross-derivative integrability condition for the generalized Knizhnik– Zamolodchikov (KZ) equation in terms of r. An immediate application was a new broad class of KZ-type equations with the trigonometric and elliptic dependence on the arguments. Another application is the W –invariant KZ, a W –equivariant system of the differential equations with values in the group algebra CW of the Weyl group. It was defined in [C6] as an example of r–matrix KZ, but, as a matter of fact, it is directly based on the so-called “reflection equations” and their generalizations from [C16], the YBE extended to arbitrary Weyl groups. The paper [C16] readily gives a justification of the cross-derivative conditions for this system, although a straightforward proof via the reduction to the rank two consideration is simple as well. Generally, the r–matrix must have some symmetries and satisfy the rank two commutator relations, that can be immediately verified (if they hold). The problem is with finding examples and the representation theory interpretation. In the case of the W –invariant KZ from [C6], the cross-derivative integrability conditions and the W –equivariance follow from the properties of the intertwining operators of the affine Hecke algebras; see [C10, C11] and the references therein. The W –invariant KZ is now the main tool in the analytic theory of the configuration spaces associated with the root systems. It was extended to the affine KZ equation (AKZ) for the affine Weyl group in [C7, C8], and then to the groups generated by complex reflections by Dunkl, Opdam and Malle. A significant part of this chapter is devoted to the calculation of the monodromy 43

44

CHAPTER 1. KZ AND QMBP

representation of AKZ. Note that one can formally associate Hecke algebras with abstract Coxeter groups and similar groups. The most universal proof of the fact, if it holds, that the resulting Hecke algebra satisfies the Poincare–Birkhoff–Witt property, i.e., that this algebra is a flat deformation of the group algebra, goes via the consideration of a KZ-type connection and obtaining the Hecke algebra as its monodromy. For instance, the monodromy of the W –invariant KZ identifies the corresponding Hecke algebra with CW for generic values of the parameter k in the KZ equation directly related to the parameter t of the Hecke algebra.

1.0.1

Classical r–matrices

The classical r–matrices were derived from the quantum inverse method due to Faddeev, Sklyanin, and Takhtajan (see [FT]). They allowed us to standardize the study of soliton equations, to come close to their classification, and to understand the genuine role of soliton theory in mathematics. Without going into detail, let me give the following general references [KuS, BD, Se, C3] and mention the paper [Sk], which influenced my research in soliton theory a great deal. In the definition of the r–matrix KZ, I combined the r–matrices with the theory of the τ –function from [DJKM1, DJKM2] and some previous papers of the same authors. As for the τ –function, let me also mention [Sat] and the exposition [Ve]. The third important ingredient was the approach to affine flag varieties from the papers [KP, PK]. Let g be a simple Lie algebra and r(λ) a function of λ ∈ C taking values in g ⊗ g. We assume that r(λ) = t/λP+ r˜(λ) for some analytic function r˜ in a neighborhood of λ = 0, where t = α Iα ⊗ Iα , {Iα } ⊂ g is an orthonormal basis with respect to the Killing form ( , )K on g. The notation, standard in the theory of r–matrices, will be used: 1

a = a⊗1⊗1⊗. . . , 2 a = 1⊗a⊗1⊗. . . , . . . , ij (a⊗b) = i a j b, ((a⊗b)c)K = a(b, c)K

for a, b, c being from the universal enveloping algebra U (g) of g (or from any of its quotient algebra). We also set {A ⊗, B} if the products of the entries ab in A ⊗ B are replaced everywhere by {a, b}. A classical r–matrix is a solution of the classical Yang–Baxter equation [ 12 r(λ), 13 r(λ + µ) + 23 r(µ)] + [ 13 r(λ + µ), 23 r(µ)] = 0.

(1.0.1)

Theorem 1.0.1. Assuming that 12 r(λ) + 21 r(−λ) = 0, the following three assertions are equivalent to (1.0.1) and to each other: (a) The relation {M (λ) ⊗, M (µ)} = [ r(µ − λ), 1 M (λ) + 2 M (µ)]

1.0. SOLITON CONNECTION

45

defines a Poisson bracket on the indeterminate coefficients of M (λ) that is the generic element of the Lie algebra e g of g–valued meromorphic functions. (b) The functions in the form X Mr (λ) = Res (12 r(λ − µ)2 M (µ))K dµ), (1.0.2) where the sum is over the set of poles of M ∈ e g, constitute a Lie subalgebra e gr ⊂ e g. (c) The equations X ij ∂G/∂λi = κ r(λi − λj )G (1.0.3) j6=i

for a function G of λ1 , λ2 , . . . taking values in U (g) ⊗ . . . ⊗ U (g) satisfy the cross-derivative integrability conditions. The references are [C3, C6]. It readily follows from (1.0.2) that the formal differentiations Dg F = (F gF −1 )0 F,

M0 = M − Mr ,

g∈e g

(1.0.4)

satisfy the relations [Dg , Df ] = D[g,f ] ,

g, f ∈ e g.

(1.0.5)

Here F = exp(M0 ) for the generic element M0 of the Lie subalgebra e g0 ⊂ e g of holomorphic functions; differentiations act in the algebra of functionals of the coefficients of M0 . These {Dg , g ∈ e g} are the generalized B¨acklund– Darboux infinitesimal transformations, while formula (1.0.5) and its group analog embrace many concrete results (from the 1850 to the present) on composing B¨acklund transforms for nonlinear differential equations; see [C1, C3] and the references therein. Now we can introduce the τ –function as a solution of the system (g ∈ e g, κ ∈ C) X (Dg τ )τ −1 = κ( Res (F −1 dF, g)K ). (1.0.6) Formula (1.0.2) induces the decomposition b g = g ⊕ Cc,

[x + ζc, y + ξc] = [x, y] +

X

Res (dx, y)K c.

In this decomposition, c acts as a multiplication by κ and e g0 ⊂ b g annihilates τ. If κ ∈ kN for an appropriate integer k > 0 (depending on the r–matrix under consideration; see [C3]), then the τ –function is “integrable,” for instance, satisfies infinitely many Hirota-type relations. All these are very close to the theory of modular forms and representations of ad`ele groups in arithmetic, with e gr ⊂ e g⊃e g0 playing the role of principal and, respectively, integer ad`eles, and τ being something like the Tamagawa measure.

46

1.0.2

CHAPTER 1. KZ AND QMBP

Tau function and coinvariant

System (1.0.3) for 1 ≤ i, j ≤ n is a natural generalization of the Aomoto– Kohno system with r(λ) = t/λ, which, in its turn, is a direct generalization of the Knizhnik–Zamolodchikov equation in the two-dimensional conformal field theory (with g = glN ). I came to (1.0.3) while performing the following calculation in [C6]. Let us generalize the definition of τ to allow arbitrary representations of b g (not only the representations with the trivial action of e g0 on τ ). We also allow e the elements from g to have poles at (pairwise distinct) points λ1 , . . . , λn . Given a finite dimensional representation V of g, e g0 naturally acts on ⊗n V by the projection onto the Lie algebra g × · · · × g (n times). Let us introduce the BGG–Verma module M as the universal b g–module generated ⊗n by V , where c acts by multiplication by κ and e g0 acts on V ⊗n as above. For every x ∈ M, there exists a unique element τ (x) ⊂ V ⊗n ⊂ M such that τ (x) − x ∈ e gr M. Here we use the decomposition b g = Cc ⊕ e g0 ⊕ e gr . This construction (from [C1]) gives an important interpretation of the τ – function as the Kac–Moody coinvariant. In soliton theory, we examine its transformations with respect to Kac–Moody commutative subalgebras (the Dg –flows for pairwise commutative g). In conformal field theory (CFT), the tau function appears as the n-point correlation function in a way similar to the above calculation. The dependence on the positions of the points, the celebrated Knizhnik–Zamolodchikov equation, is of key importance for CFT. As a matter of fact, the following reasoning is similar to the original deduction of KZ in [KZ]. We set formally X ˜ j ), ˜ −1−j )(Iα λ (li τ )(x) = τ (li x), li = (Iα λ i i α,j≥0

˜ i = λ − λi is a local parameter in a neighborhood of λi and li is the where λ Sugawara element of degree −1, namely, the generator L−1 of the Virasoro algebra embedded into the i–th component of b g. Then one can verify that X ij rV (ui − uj ))τ li τ + κ0 ∂τ /∂ui = (ρi + j6=i

for a proper κ0 and some ρ (trivial in the most interesting cases), where ij rV is the image of r in EndC (V ⊗n ). This leads to the compatibility condition for equations (1.0.3) and eventually provides the integral formulas for the τ –function. This derivation of the r–matrix KZ equation was used in [C9] to simplify the algebraic part of the Schechtman–Varchenko construction [SV] of the integral formulas for the rational KZ ([DJMM] for SL2 ) and generalize their formulas to the trigonometric case.

1.1. AFFINE KZ EQUATION

1.0.3

47

Structure of the chapter

We will reproduce the above calculation in detail and consider the integral formulas in the last two sections. The r–matrices there are “of type A,” i.e., depending on the differences of the arguments (ui − uj ) as above. In the first section, we introduce the affine KZ equations using a different approach, based on the properties of the Weyl groups and root systems; the differences (ui −uj ) are replaced by the roots. The r–matrices attached to the root systems and the corresponding KZ are studied in Section 1.5, including the elliptic case. The affine KZ is a special example of the general r–matrix trigonometric KZ. Its “integrability” reveals itself in explicit formulas in terms of the hypergeometric function for the monodromy representation. The later plays the key role when establishing an isomorphism with another well-known “integrable” system of differential equations, the QMBP, also called the Heckman–Opdam system (Section 1.2). In the next Sections 1.3–1.4, we switch to the quantum theory including QAKZ and Macdonald polynomials (in the GL–case). The exposition is based on the affine and double affine Hecke algebras. Section 1.6 contains applications of DAHA to the Harish-Chandra theory and QMBP, mainly, a new theory of the inverse Harish-Chandra transform. It has its own mini-introduction. Acknowledgments. This chapter is based on two series of lectures that I gave at IIAS and RIMS Kyoto University, and the paper [C22]. I thank the participants of the lectures for useful questions, comments, and discussion, which made for a highly stimulating atmosphere. I am very grateful to those who took notes and helped to prepare [C22] for publication. This chapter is a truly joint venture. I acknowledge my special indebtedness to M. Kashiwara and T. Miwa. I also used my notes of the lectures on the r–matrix KZ and the affine and double affine Hecke algebras delivered in Paris. I am thankful to P. Gerardin, R. Rentschler, and especially M. Duflo. I am grateful to M. Kashiwara, who helped me with [C24], which is the basis of the section on the Harish-Chandra inversion. This paper was inspired by [O3]. I thank E. Opdam for his comments.

1.1

Affine KZ equation

We discuss in this section the degenerate affine Hecke algebra and the corresponding affine Knizhnik–Zamolodchikov equation. It is demonstrated that the W –equivariance of the latter (we say W –invariance, abbusing a little the standard terminology) and the cross-derivative integrability conditions

48

CHAPTER 1. KZ AND QMBP

(“compatibility”) are equivalent to the definition of the degenerate affine Hecke algebra. This observation (see [C7, C11]) is an important link between the affine Hecke theory and the theory of special functions, especially, the theory of the hypergeometric function. We begin with the rank one and rank two examples.

1.1.1

Hypergeometric equation

We introduce the affine Knizhnik–Zamolodchikov equation (AKZ) associated with the root system of type A1 . First, we consider the differential equation: µ ¶ ∂Φ s = k u + x Φ. (1.1.1) ∂u e −1 Here Φ is a function of a complex variable u, k ∈ C is a parameter, and s and x are operators acting in the vector space or algebra, where Φ takes its value. Second, we impose the following two relations: s2 = 1, sx + xs = k.

(1.1.2) (1.1.3)

These relations are necessary and sufficient to make (1.1.1) invariant. By invariance, we mean that if Φ satisfies (1.1.1), then so does ˜ Φ(u) = sΦ(−u). Let us check it. We plug in µ ¶ ˜ ∂ Φ(u) s − = s k −u + x Φ(−u) ∂u e −1 ¶ µ s ˜ + sxs Φ(u) = k −u e −1 and use

Finally,

1 1 = u + 1. −u 1−e e −1 ˜ ∂ Φ(u) = ∂u

where ks − sxs = x.

µ

¶ s ˜ k u + ks − sxs Φ(u), e −1

(1.1.4)

1.1. AFFINE KZ EQUATION

49

Third, the affine Knizhnik–Zamolodchikov equation is an equation in 0 the form (1.1.1) with values in the degenerate affine Hecke algebra HA of 1 type A1 generated by the elements s and x subject to (1.1.2) and (1.1.3): def

0 HA == hs, xi/{(1.1.2), (1.1.3)}. 1

(1.1.5)

0 Let Φ(u) be a function of u with values in HA . Note that one can multiply 1 a solution Φ(u) of (1.1.1) by an arbitrary constant element on the right, i.e., 0 Φ(u)a for a ∈ HA is a solution as well. 1 We claim that (1.1.1) is integrable in terms of the classical hypergeometric function or elementary functions in arbitrary irreducible representations of 0 HA . 1 Let us start with a more general equation µ ¶ A B ∂Φ = + Φ. (1.1.6) ∂z 1−z z

It becomes (1.1.1) for z = e−u , A = −ks, and B = −x. Generally speaking, equation (1.1.6) is non-integrable. There is an interesting “new” theory of this equation started by Drinfeld with application in arithmetic, but we are not going to consider this direction. If A, B are 2 × 2 matrices acting on the 2–component vector-function Φ, then (1.1.6) is nothing but a variant of the hypergeometric differential 0 equation. This includes the case of irreducible representations of HA , because 1 the latter exist only in dimension 1 or 2. 0 Explicitly, the generic irreducible representation ρ of HA is given by 1 µ ¶¶ ¶ µ µ s 0 ζ 1 0 + . (1.1.7) , ρ(x) = k ρ(s) = ξ 0 0 −1 2 Because of the gauge transformation ζ → cζ, ξ → c−1 ξ, it is characterized by ζξ up to isomorphisms. It will be more convenient to switch to ¶1/2 µ 1 . (1.1.8) µ = ζξ + 4 ¡ 1¢ Equation (1.1.6) for A = −kρ(s), B = −ρ(x) has a solution Φ = Φ in Φ2 terms of the Gauss hypergeometric function with the first component Φ1 (u) = z −kµ (1 − z)k F (k(1 − 2µ), k, 1 − 2kµ; z),

(1.1.9)

where z = e−u and def

F (α, β, γ; z) ==

∞ X (α)n (β)n n=0

(γ)n n!

z n for (x)n = x(x + 1) · · · (x + n − 1).

If the first component is known, we automatically know the second. We note that the parameters α, β, γ in (1.1.9) obey the constraint α + 1 = β + γ. If ζξ = 0, then the representation ρ from (1.1.7) is reducible. In this case, the solutions are given in terms of elementary functions.

50

1.1.2

CHAPTER 1. KZ AND QMBP

AKZ equation of type GL

This equation is a natural abstract variant of the original Knizhnik–Zamolodchikov (KZ) equation from conformal field theory. Its consistency and invariance lead to the defining relations of the degenerate affine Hecke algebra 0 HGL introduced by Drinfeld [Dr1]. n The KZ equation reads as à ! X Ωij ∂Φ =k Φ (0 ≤ i ≤ n). (1.1.10) ∂zi z i − zj 0≤j≤n, j6=i In the least sophisticated case from [KZ], the Ωij are the permutation matrices in the tensor square of some CN , where N has nothing to do with n, the number of variables. Let us generalize it assuming that the Ωij are symbols, constant matrices, or operators and Ωij = Ωji . We will treat them formally and consider the function Φ(z) in terms of z = (z0 , . . . , zn ), taking values in the abstract algebra generated by the elements Ωij . The self-consistency of the system of equations (1.1.10) means that ∂Aj ∂Ai − = [Ai , Aj ], ∂zi ∂zj where Ai = k

X Ωij . z i − zj j6=i

(1.1.11)

(1.1.12)

It holds for any complex parameter k if and only if [Ωij , Ωkl ] = 0, [Ωij , Ωik + Ωjk ] = 0,

(1.1.13) (1.1.14)

where the indices i, j, k, l are pairwise distinct. The KZ in this abstract form is, essentialy, due to Aomoto. It was also studied by Kohno [Ko]. Comment. Generally, we need atrigonometric KZ equation to obtain the degenerate (graded) affine Hecke algebra. Namely, we must start with the r– matrix KZ for the simplest trigonometric classical r–matrices. The GLn –case is special, since there exists a substitution transforming the trigonometric KZ that is needed (there are many), to a rational Aomoto-type KZ, although the output is not exactly in the Aomoto–Kohno–Drinfeld form. The trigonometric KZs (and their elliptic generalizations) appeared for the first time in [C6]. The so-called KZB [Be] and more general KZ associated with the curves of arbitrary genus a not closed system in terms of the positions of the points. The reference [C6] is definitely the first concerning the elliptic and trigonometric r–matrix KZ, although it is possible that the trigonometric KZ-type formulas appeared before.

1.1. AFFINE KZ EQUATION

51

In the “standard” conformal field theory, the dependence on the positions of the points zi is rational. I remember that my physicists friends considered the trigonometric KZ a pure mathematical speculation (indeed, they were), when I demonstrated them. Now they are quite common for both mathematicians and physicists. ❑ Consider the group algebra CSn of the permutation group Sn of the set {1, . . . , n}. We denote the transposition of i and j by sij . Then, if we make Ωij = sij (0 ≤ i, j ≤ n), the relations (1.1.13)–(1.1.14) are satisfied. Let us treat the zero-index separately. Setting z0 = 0, Ωij = sij (i, j 6= 0), Ω0i = k −1 Ωi ,

(1.1.15) (1.1.16) (1.1.17)

equation (1.1.10) turns into " Ã ! # X ∂Φ sij Ωi = k + Φ (1 ≤ i ≤ n), ∂zi zi − zj zi 1≤j≤n, j6=i

(1.1.18)

and relations (1.1.13)–(1.1.14) read as follows: [sij , Ωi + Ωj ] = 0, [ksij + Ωi , Ωj ] = 0, [sij , Ωl ] = 0,

(1.1.19) (1.1.20) (1.1.21)

where the indices i, j, l are pairwise distinct. Substituting zi = evi , we come to ∂Φ = ∂vi

à k

X j6=i

(1.1.22)

sij + Ωi 1 − evj −vi

Using the elements yi = Ω i + k

X

! Φ.

(1.1.23)

sij ,

(1.1.24)

j>i

∂Φ = ∂vi

à k

X j>i

sij evi −vj − 1

−k

X j

sij evj −vi − 1

! + yi Φ.

The elements {y} are convenient here, since in the limit v1 À v2 À · · · À vn ,

(1.1.25)

52

CHAPTER 1. KZ AND QMBP

we obtain the system

∂Φ = yi Φ. (1.1.26) ∂vi The consistency of these equations is equivalent to the commutativity [yi , yj ] = 0,

(1.1.27)

which of course can be deduced algebraically from (1.1.19), and (1.1.20), and (1.1.21). We claim that (1.1.27), together with the relations [si , yj ] = 0 if j 6= i, i + 1, si yi − yi+1 si = k,

(1.1.28) (1.1.29)

where si = si i+1 (1 ≤ i ≤ n − 1), ensure (1.1.19)-(1.1.21); it is a simple algebraic calculation. Let us introduce the degenerate affine Hecke algebra of type GLn as an algebraic span of CSn and yi (1 ≤ i ≤ n) subject to (1.1.27), (1.1.28), and 0 (1.1.29). It will be denoted by HGL , or simply by Hn0 . We call the system n (1.1.25) with values in Hn0 the AKZ of type GLn . It is well defined, i.e., self-consistent, and Sn –invariant in the following sense. The group Sn acts on Cn naturally by v = (v1 , . . . , vn ) ∈ Cn 7→ w(v) = (vi1 , . . . , vin ) ∈ Cn for w−1 = (i1 , i2 , . . . , in ) ∈ Sn . Given a function Φ(v) of v ∈ Cn with the values in Hn0 , we define the action of w ∈ CSn on Φ(v) by ³ ´ (w(Φ)) (v) = w · Φ w−1 (v) . (1.1.30) Here the dot means the product in Hn0 . It follows from (1.1.28) and (1.1.29) that if Φ satisfies (1.1.25), then so does w(Φ). The invariance alone is equivalent to relations (1.1.28) and (1.1.29). We conclude that, by imposing the invariance, the “limiting self-consistency” (1.1.27) results in the general self-consistency of our system for all k. Example. The basic variant of the Knizhnik–Zamolodchikov equation is as follows. Let glN be the matrix algebra with the standard set of generators {elm , 1 ≤ l, m ≤ n} with the entries eab lm = δla δmb . Given any representations V0 , V1 . . . , Vn of glN , the tensor product V = V0 ⊗ V1 ⊗ · · · ⊗ Vn has the P (i) (j) (i) natural structure of a glN –module. We set Ωij = lm elm eml , where elm act in Vi , i 6= j. Then (1.1.14) and (1.1.13) are fulfilled and we get a selfconsistent KZ equation. This example was generalized by Kohno to arbitrary reductive Lie algebras and extensively used by Drinfeld in his theory of quasiHopf algebras.

1.1. AFFINE KZ EQUATION

53

If Vi coincide with the fundamental N –dimensional representation for all i > 0, then Ωij can be identified with the transposition sij for i, j > 0. The representation V0 still can be arbitrary. This results in an Sn –invariant selfconsistent equation in the form of (1.1.18). According to the above analysis we must obtain a representation of Hn0 in V . The formulas coincide with (1.1.25): X Ωij , 1 ≤ i, j ≤ n. sij 7→ Ωij , yi 7→ Ω0i + k j>i

Note that, given a weight of glN , the KZ equation under consideration can be restricted to the subspace of the corresponding highest vectors in V. Therefore the latter is a Hn0 –submodule.

1.1.3

Degenerate affine Hecke algebra

In this subsection we define the degenerate affine Hecke algebra and the corresponding affine KZ for an arbitrary root system. Let Σ be a root system in Rn with the inner product ( , ). We choose a system of simple roots α1 , . . . , αn of Σ and denote the set of positive roots by Σ+ . For a root α ∈ Σ, the corresponding coroot is α∨ =

2α , (α, α)

and the corresponding reflection is sα (u) = u − (α∨ , u)α (u ∈ Rn ). We will denote sαi simply by si . The fundamental coweights bi are as follows: (bi , αj ) = δij . Sometimes the notation ai = αi∨ will be used. For u ∈ Rn , the coordinates will be ui = (u, αi ). We also set uα = (u, α) for α ∈ Σ. Check that ∂uα = ναi = (bi , α) = the multiplicity of αi in α. ∂ui Let W be the Weyl group of Σ: W = hsα , α ∈ Σi = hs1 , . . . , sn i. It acts on functions on Rn by w

f (u) = f (w−1 (u)) (u ∈ Rn ).

One then has: w

uα = (w−1 (u), α) = uw(α) .

(1.1.31)

54

CHAPTER 1. KZ AND QMBP

0 is the associaDefinition 1.1.1. The degenerate affine Hecke algebra HΣ tive algebra generated by CW and x1 ,. . .,xn with the following relations:

[xi , xj ] = 0, ∀i, j, [si , xj ] = 0, if i 6= j, si xi − xˆi si = k.

(1.1.32) (1.1.33) (1.1.34)

Here k is a complex number and xˆi = xi −

n X

(αi∨ , αj )xj .

(1.1.35)

j=1

❑ This definition is due to Lusztig [Lus1]. He calls it the graded affine Hecke algebra because k is a formal parameter in his paper. In this chapter, we mainly treat k as a complex number. Drinfeld introduced this algebra in the GLn –case in [Dr1]. These algebras are natural degenerations of the affine Hecke algebras. Introducing xb =

n n X X k i xi (b, αi )xi = i=1

i=1

for b =

n X

ki bi ,

(1.1.36)

i=1

we can express the right-hand side of (1.1.35) as xsi (bi ) = xbi − xai . Therefore, in the degenerate Hecke algebra: si xb − xsi (b) si = xb si − si xsi (b) = k(b, αi ).

(1.1.37)

The AKZ. We will use the following partial derivatives: ∂b (uα ) = (α, b),

∂ = ∂i = ∂bi . ∂ui

(1.1.38)

Let us consider the following system of partial differential equations:   X ∂Φ  sα = k ναi uα (1 ≤ i ≤ n). (1.1.39) + xi  Φ ∂ui e − 1 α∈Σ +

Here k is a complex number. We denote the right-hand side of (1.1.39) by Ai Φ. Here we assume that Φ takes values in an associative algebra generated by CW and x1 ,. . ., xn . We call (1.1.39) self-consistent if [

∂ ∂ − Ai , − Aj ] = 0. ∂ui ∂uj

(1.1.40)

It is called invariant if, for any solution Φ of (1.1.39) and any element w of W, the function w(Φ) defined in (1.1.30) is a solution of (1.1.39) as well.

1.1. AFFINE KZ EQUATION

55

Theorem 1.1.2. The system (1.1.39) is self-consistent and invariant if and only if s1 , . . ., sn , x1 , . . ., xn satisfy (1.1.32), (1.1.33), and (1.1.34) from the 0 definition of HΣ . ❑ The AKZ equation associated with Σ is the system (1.1.39) for the 0 function Φ with values in HΣ . Using xb and ∂b from (1.1.36) and (1.1.38), the system (1.1.39) can be expressed as   X sα ∂b Φ = k (b, α) uα (1.1.41) + xb  Φ. e − 1 α∈Σ +

Comment. The parameter k may depend on the length of roots. Generally speaking, the AKZ equation is as follows:   ∂Φ  X sα = k|α| ναi uα (1.1.42) + xi  Φ. ∂ui e − 1 α∈Σ +

1.1.4

Examples

Let us write down the explicit forms of the AKZ equation in the simplest cases. When Σ = A1 , the AKZ equation is exactly (1.1.1). For A2 , the AKZ equation is ½ µ ¶ ¾ ∂Φ s12 s13 = k u1 + + x1 Φ, ∂u1 e − 1 eu1 +u2 − 1 ½ µ ¶ ¾ ∂Φ s23 s13 = k u2 + + x2 Φ, ∂u2 e − 1 eu1 +u2 − 1 where sij denotes the transposition of i and j. In this case, xˆ1 = x2 − x1 , xˆ2 = x1 − x2 . The root system B2 appears in the following way. Let ²1 and ²2 form an orthonormal basis of R2 . Then the set of positive roots consists of the following vectors: α1 = ²1 − ²2 , α2 = ²2 , α1 + α2 = ²1 , α1 + 2α2 = ²1 + ²2 . Let s = s1 and t = s2 . Then s and t satisfy tsts = stst (the Coxeter relation for WB2 = WC2 ) and s2 = 1, t2 = 1. One has xˆ1 = x2 − x1 , and xˆ2 = 2x1 − x2 .

56

CHAPTER 1. KZ AND QMBP

The AKZ equation reads as ½ µ ¶ ¾ s ∂Φ sts tst = k u1 + + + x1 Φ ∂u1 e − 1 eu1 +u2 − 1 eu1 +2u2 − 1 ½ µ ¶ ¾ t ∂Φ sts tst = k u2 + + 2 u1 +2u2 + x2 Φ. ∂u2 e − 1 eu1 +u2 − 1 e −1 Note the appearance of the multiplier 2 in the last equation. Generally, the multipliers can be from 1 to 6. The coefficient 6 occurs for E8 only. The An−1 –case. We will show how the AKZ equation of type GLn discussed in Section 1.1.2 reduces to the AKZ equation for the root system Σ ⊂ Rn−1 of type An−1 . First, x = y1 + . . . + yn (1.1.43) is central in the algebra Hn0 . Setting i x, (1.1.44) n 0 we have an embedding of HΣ , where Σ is the root system of type An−1 , into 0 Hn . Concerning the AKZ equations, let xi = y1 + . . . + yi −

ui = vi − vi+1

(1 ≤ i ≤ n − 1).

(1.1.45)

The space Rn−1 will be identified with the quotient space of Rn = ⊕ni=1 R²i : Rn−1 ' ⊕ni=1 R²i /R², where {²i }1≤i≤n is the orthonormal basis and ² = ²1 +. . .+²n . Due to (1.1.25), n X ∂Φ i=1

∂vi

= xΦ.

Therefore, the function 1

Φ0 (v) = e−x· n (v1 +···+vn ) Φ(v) satisfies the AKZ equation of type An−1 : ! Ã X sjl ∂Φ0 = k + xi Φ0 uj +···+ul−1 − 1 ∂ui e j≤ii) ³ X sij − 1 ´ n + 1 ´o Φ. (1.2.42) + y + k i − − i evj −vi − 1 2 j(i)

−

X

(evj −vi − 1)−1 (σij − 1) + i −

j( 1), etc., using (a) and (b). The pairwise commutativity of all Yi and other required relations formally follow from them. The first reads Y1 T1−1 Y1 T1−1 = T1−1 Y1 T1−1 Y1 . We plug in the formula for Y1 and move P to the left. The commutativity with “distant” T is obvious. ❑

1.3.2

Definition of QAKZ

In this subsection, we introduce the QAKZ equation.

78

CHAPTER 1. KZ AND QMBP

Definition 1.3.2. For u ∈ C, we define the intertwiners by t1/2 − t−1/2 eu − 1 Fi (u) = . 1/2 −1/2 t − t t1/2 + eu − 1 Ti +

(1.3.9)

❑ They satisfy Fi (u)Fi (−u) = 1, Fi (u)Fi+1 (u + v)Fi (v) = Fi+1 (v)Fi (u + v)Fi+1 (u).

(1.3.10) (1.3.11)

The second relation can be deduced from Lemma 1.2.10 in the same way as for the degenerate affine Hecke algebra. The affine quantum Knizhnik–Zamolodchikov (QAKZ) equation is the following system of difference equations for a function Φ(v) that takes values in Hnt (or any Hnt –module): Φ(v1 , . . . , vi + h, . . . , vn ) = Fi−1 (vi − vi+1 + h) . . . F1 (vi − v1 + h)T1 · · · Ti−1 Yi −1 Fn−1 (vi − vn ) · · · Fi (vi − vi+1 ) × Ti−1 · · · Tn−1 × Φ(v1 , . . . , vi , . . . , vn ) (i = 1, . . . , n).

(1.3.12)

Here h is a new parameter. Theorem 1.3.3. The QAKZ system (1.3.12) is self-consistent. This system is W –invariant in the following sense: if Φ(v) is a solution, then so is Fi (vi+1 − vi ) si Φ(v) = si (Fi (vi − vi+1 )Φ(v)) . This follows from (1.3.10), (1.3.11). Later we will make the invariance obvious “conceptually.” ❑ Let us discuss the quasi-classical limit of the QAKZ system. Setting def

t == ekh = q k ,

q = eh ,

let h → 0. The generators Ti , Yi are supposed to be in the form kh + ··· 2 = 1 + hyi + · · · ,

Ti = si + Yi

(s2i = 1),

where by “· · · ,” we mean here and in the following formulas the terms of order h2 .

1.3. ISOMORPHISMS FOR QAKZ

79

The relations of the degenerate affine Hecke algebra for si , yi can be readily verified. Using the formula t1/2 Ti−1 Fi (u) = 1 +

kh (si − 1) + · · · , −1

eu

we find that h−1 (Φ(. . . , vi + h, . . .) − Φ(. . . , vi , . . .)) © ¡ X sij − 1 = yi + k evi −vj − 1 −

X j(i)

sij − 1 n + 1 ¢ª Φ(. . . , vi , . . .) + · · · . + i − evj −vi − 1 2

Hence the AKZ equation (1.2.42) is the limit of the QAKZ equation as h → 0, which is called the semi-classical limit. To make the QAKZ equations more transparent, we need the affine Weyl group of type GLn . It is the semidirect product b Sn = Sn n Z n , where Zn =

n M

Zγi

i=1

is a free abelian group of rank n. Define the action of b Sn on a vector v = n (v1 , . . . , vn ) ∈ R by sij v = (v1 , . . . , vj , . . . , vi , . . . , vn ) = sji v, i < j, γi γi v = (v1 , . . . , vi + h, . . . , vn ), (vj ) = vj − hδij . Introducing π = γ1 s1 · · · sn−1 = s1 · · · sn−1 γn , its action on Rn and the coordinates read as πv = (vn + h, v1 , . . . , vn−1 ), π vn = v1 − h, π v1 = v2 , . . . . It is convenient to represent the elements γi , π graphically. Figure 1.3 shows the following reduced decomposition of γi : γi = si−1 · · · s1 πsn−1 · · · si .

80

CHAPTER 1. KZ AND QMBP

vn

vn−1

vn vn−1

.. . vi + h .. .

.. .

vi+1 vi vi−1

v1

.. .

v1 vn + h

γi

v1 π

Figure 1.3: Graphs for γ and π

Lemma 1.3.4. The affine Weyl group of the GLn –type can be presented as b Sn = hs1 , . . . , sn−1 , πi/ ∼, where the relations are s2i = 1,

si sj = sj si

(|i − j| > 1),

si si+1 si = si+1 si si+1 ,

(a) πsi−1 π −1 = si (1 < i < n), and (b) π n is central. ❑ For a function Ψ(v) with values in Hnt , let s˜i (Ψ) = Fi (vi+1 − vi ) si Ψ, π ˜ (Ψ) = P π Ψ.

(1.3.13) (1.3.14)

Theorem 1.3.5 ([C12]). Formulas (1.3.13) and (1.3.14) can be extended to an action of b Sn . We denote this action by b Sn 3 w : Ψ 7→ w(Ψ). ˜ For instance, γ˜i (Ψ)(v1 , . . . , vn ) = Fi−1 (vi−1 − vi )−1 · · · F1 (v1 − vi )−1 P ×Fn−1 (vi − vn − h) · · · Fi (vi − vi+1 − h)Ψ(v1 , . . . , vi − h, . . . , vn ). Hence the QAKZ equation simply means the invariance of Φ(v) with respect to the pairwise commuting elements γi : QAKZ ⇐⇒ γ˜i (Φ) = Φ (i = 1, . . . , n).

(1.3.15)

1.3. ISOMORPHISMS FOR QAKZ

81

Let us connect QAKZ with the QKZ introduced by Smirnov and Frenkel-Reshetikhin [Sm, FR]. We fix an N –dimensional complex vector space V and define the matrix T ∈ End(V ⊗ V ) by 1/2

T = (t

−1/2

−t

)

X i<j

Eii ⊗ Ejj +

X

1/2

Eij ⊗ Eji + t

N X

Eii ⊗ Eii ,

i=1

i6=j

following Baxter and Jimbo. The algebra Hnt acts on V ⊗n by Ti (a1 ⊗ · · · ⊗ an ) = a1 ⊗ · · · ⊗ T (ai ⊗ ai+1 ) ⊗ · · · ⊗ an , (1.3.16) P (a1 ⊗ · · · ⊗ an ) = Can ⊗ a1 ⊗ · · · ⊗ an−1 , (1.3.17) where ai ∈ V and C = diag(λ1 , . . . , λn ). One can check that this action is well defined by a direct calculation. Assuming that N, the dimension of V, coincides with n, the number of components, let ¡ ⊗n ¢ V = span{ew(1) ⊗ · · · ⊗ ew(n) | w ∈ Sn } 0 be the 0–weight subspace. Here e1 , . . . , en are the standard basic vectors of V. It is easy to see that this subspace is closed under the action of Hnt . We state the next proposition without proof. Proposition 1.3.6. If N = n and λ = (λ1 , . . . , λn ) is generic, then the Ht 0–weight space (V ⊗n )0 is isomorphic to Iλ = IndC[Yn 1 ,...,Yn ] (λ). Writing AQKZ in (V ⊗n )0 we obtain the QKZ for the GLn in its fundamental representation. Combining this observation with the isomorphism of AQKZ with the Macdonald eigenvalue problem (our next aim), we can readily explain why the Macdonald polynomials appear in many calculations involving the vertex operators. The QKZ was introduced as an equation for the n-point function in conformal field theory based on quantum groups. The most universal definition of the n-point function is in terms of the vertex operators.

1.3.3

The monodromy cocycle

Let Φ be a solution of the QAKZ equation. Thanks to (1.3.15), w(Φ) ˜ is also t ˜ a solution of the QAKZ equation for any w ∈ Sn . We define Tw ∈ Hn by w

Tw = Φ−1 w(Φ) ˜

˜n for w ∈ S

and call it the monodromy cocycle. It follows from (1.3.13) and (1.3.14) that Fi (vi − vi+1 )Φ = si ΦTi (1.3.18)

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and

−1

P Φ = π ΦTπ .

(1.3.19)

Here Ti stands for Tsi . Lemma 1.3.7. w2−1

(Tw1 )Tw2 = Tw1 w2

˜n. for w1 , w2 ∈ S

Indeed, Φw1 w2 Tw1 w2 = w ] ˜1 (w˜2 Φ) = w ˜1 (Φw2 Tw2 ) = Φw1 Tw1 w1 w2 Tw2 . 1 w2 (Φ) = w The QAKZ equation implies that Tγi = 1. Hence T w depends only on the image w ¯ of w in Sn . Let F(Cn , Hnt ) be the set of Hnt –valued functions on Cn . We obtain the ˜n: following two anti-actions of S σw (Ψ) =

w−1

Ψ,

(1.3.20)

σw0 (Ψ)

w−1

ΨTw ,

(1.3.21)

=

˜ n and Ψ ∈ F(Cn , Ht ). For instance, σγ (vi ) = vi + h = σ 0 (vi ). where w ∈ S γi n i Lemma 1.3.7 means exactly that σ 0 is an anti-action, i.e., σw0 1 w2 = σw0 2 σw0 1 . We note that, in the difference theory, the monodromy of KZ can be always made trivial. Indeed, the 1-cocycle {Tw , w ∈ W } is always a co-boundary because of the Hilbert Theorem 90. Hence, by conjugating solutions of AQKZ we can always get rid of the monodromy. So the above actions σ, σ 0 are not very much different, in contrast to the differential theory. One can try to apply the Hilbert Theorem 90 to the AQKZ itself, though the group Zn is infinite. It gives the following formal integration of QAKZ. Let Ψ ∈ F(Cn , Hnt ). Then the infinite sum X ˜b(Ψ), (1.3.22) b∈P ∨

˜ n , satisfies the AQKZ. where P ∨ = ⊕ni=1 Zγi ⊂ S P If Ψ is rapidly decreasing, then one can check that b∈P ∨ ˜b(Ψ) is convergent. Therefore, constructing some solutions Φ of QAKZ poses no problem. What is difficult is to ensure good analytic properties and a proper asymptotical behavior.

1.3.4

Macdonald’s eigenvalue problem

In this subsection, we introduce the Macdonald eigenvalue problem and prove its equivalence to the QAKZ equation. This is a q–analog of the relation between AKZ and QMBP discussed in Section 1.2.4.

1.3. ISOMORPHISMS FOR QAKZ

83

Let Φ be a solution of the QAKZ equation with values in End(V ) for an Hnt –module V. We assume that it is invertible for sufficiently general v. Setting σi0 = σs0 i , we obtain from (1.3.18) and (1.3.9): Fi (vi − vi+1 )Φ = σi0 (Φ), µ ¶ t1/2 − t−1/2 0 1/2 0 Ti Φ = t σi + vi −vi+1 (σ − 1) Φ. e −1 i 0 Let us introduce the operator Tˆi (1 ≤ i ≤ n) by 0 t1/2 − t−1/2 0 Tˆi = t1/2 σi0 + vi −vi+1 (σ − 1). e −1 i

(1.3.23) 0

0

Then Tˆi Φ = Ti Φ and σπ0 Φ = P Φ (see (1.3.19)). The operators Tˆi and σπ0 commute with the left multiplication by Tj , P, and any elements from Hnt . Using all these: −1 · · · T1−1 P Tn−1 · · · Ti+1 Ti Φ Yi Φ = Ti−1 = T −1 · · · T −1 P Tn−1 · · · Ti+1 Tˆ0 Φ i−1

1

i

−1 · · · T1−1 P Tn−1 · · · Ti+1 Φ = Tˆi0 Ti−1 ··· 0 0 = Tˆi0 · · · Tˆn−1 σπ0 (Tˆ10 )−1 · · · (Tˆi−1 )−1 Φ.

We come to the following definition: def 0 0 ∆0i == Tˆi0 · · · Tˆn−1 σπ0 (Tˆ10 )−1 · · · (Tˆi−1 )−1 , 1 ≤ i ≤ n.

(1.3.24)

Since Yi Φ = ∆0i Φ and the elements Yi commute with each other, we conclude that [∆0i , ∆0j ] = 0. Recall that the operators ∆0i act in End(V )–valued functions. If we understand them formally, then their commutativity follows algebraically from the relations σi0 vi = vi+1 σi0 , σi0 σγi = σγi+1 σi0 , σγ0 i = σγi . The last equality means that Tγi = 1. Let Q be a polynomial in n variables. Then Q(Y1 , . . . , Yn )Φ = Q(∆01 , . . . , ∆0n )Φ

(1.3.25) (1.3.26) (1.3.27)

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CHAPTER 1. KZ AND QMBP

and we can represent Q(∆01 , . . . , ∆0n ) =

X

Dw(Q) σw0 ,

(1.3.28)

w∈Sn (Q)

where the Dw are pure difference operators, which do not contain σw0 (w ∈ Sn ). For symmetric Q, we introduce a difference operator MQ of Macdonald type by X MQ = Dw(Q) . w∈Sn

Let ϕ be a C–valued function on Cn . The system MQ ϕ = Q(λ1 , . . . , λn )ϕ.

(1.3.29)

will be called the Macdonald eigenvalue problem. The Macdonald operators MQ can be calculated for σ instead of σ 0 . Similar to the differential case, the result is the same. Given λ = (λ1 , . . . , λn ) ∈ Cn , we need a (left) Hnt –module Vλ with the following properties: 1. For any symmetric polynomial Q in n variables and all a ∈ Vλ , Q(Y1 , . . . , Yn )a = Q(λ1 , . . . , λn )a. 2. There exists a C–linear map tr : Vλ → C such that tr((Ti − t1/2 )a) = 0 for all i and a ∈ Vλ . As in the previous sections, we fix a local invertible solution Φ(v) of the QAKZ equation with values in End(Vλ ). Note that all Vλ –valued solutions ϕ(v) of the QAKZ equation can be written in the form ϕ(v) = Φ(v)a(v) for a P ∨ –periodic Vλ –valued function a(v). The periodicity means that a(. . . , vi + h, . . .) = a(v) for i = 1, . . . , n. Theorem 1.3.8. Let Vλ be an Hnt –module with the above properties. We denote the space of solutions of the QAKZ equation with values in Vλ by SolQAKZ (Vλ ), and the space of solutions of the Macdonald eigenvalue problem (1.3.29) by SolM ac (λ). Then tr : SolQAKZ (Vλ ) → SolM ac (λ).

1.3. ISOMORPHISMS FOR QAKZ

85

Proof. Let ϕ(v) = Φ(v)a ∈ SolQAKZ (Vλ ). Then µ (σi0

− 1)Φ =

t

1/2

t1/2 − t−1/2 + v −v e i i+1 − 1

¶−1 (Ti − t1/2 )Φ.

For a reduced decomposition w = si1 · · · sil of w ∈ Sn , σw0 − 1 = σi0l · · · σi01 − 1 = σs0 i · · · σi02 (σi01 − 1) + σi0l · · · σi02 − 1 l

··· =

l X

σi0l · · · σi0k+1 (σi0k − 1).

k=1

Since σi0 commutes with the left action of {T }, we have (σw0

− 1)Φ =

l X

σi0l · · · σi0k+1 (σi0k − 1)Φ

k=1

= =

l X k=1 l X

σi0l · · · σi0k+1 (a scalar function)(Tik − t1/2 )Φ (a scalar function)(Tik − t1/2 )σi0l · · · σi0k+1 Φ.

k=1 (Q)

(Q)

Using the commutativity of Dw with Ti −t1/2 , we represent Dw (σw0 −1)Φ P in the form of a sum (Ti −t1/2 )Ψi for some Hnt –valued functions Ψi . Finally Q(λ1 , . . . , λn )Φ = Q(∆01 , . . . , ∆0n )Φ X = Dw(Q) σw0 Φ w∈Sn

=

X

Dw(Q) Φ +

w∈Sn

X

Dw(Q) (σw0 − 1)Φ

w∈Sn

X = MQ Φ + (Ti − t1/2 )Ψi . Applying this relation to a ∈ Vλ and taking tr, we conclude: Q(λ1 , . . . , λn ) tr(ϕ) = MQ tr(ϕ). Let us now consider Vλ = ential case. We start with

Jλ◦ .

❑ The definition is quite similar to the differHt

Jλ = IndHnnt (+)/Lλ .

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CHAPTER 1. KZ AND QMBP

Here Hnt = hT1 , . . . , Tn−1 i ⊂ Hnt , + : Hnt −→ C is the one-dimensional representation sending Ti to t1/2 , and Lλ is the ideal generated by p(Y1 , . . . , Yn ) − p(λ) (p ∈ C[x1 , . . . , xn ]Sn ). As in Section 1.2.1, Jλ◦ stands for the dual module defined via the anti-involution ◦ of Hnt : Yi◦ = Yi ,

Ti◦ = Ti .

The main result of this subsection is the following theorem from [C13, C15]. Theorem 1.3.9. If Vλ = Jλ◦ , then the map from SolQAKZ (Vλ ) to SolM ac (λ) is injective. The theorem results from the following two lemmas. Lemma 1.3.10. Let K be an Hnt –submodule of Jλ◦ . Then tr(K) = 0 implies K = 0. The proof repeats that for the differential case. See (1.2.38).

❑

Lemma 1.3.11. Let ϕ be a Vλ –valued solution of the QAKZ equation. Assume tr(ϕ) = 0. Then tr(Hnt ϕ) = 0. Proof. First tr(Ti ϕ) = t1/2 tr(ϕ) = 0 for all i. Then π = s1 s2 · · · sn−1 γn , 0 σπ0 − 1 = σγn σn−1 · · · σ10 − 1 Ã n−1 ! X 0 0 σn−1 · · · σk+1 (σk0 − 1) + σγn − 1, = σγn k=1

and tr(σγn ϕ) = tr(ϕ) = 0. Therefore, representing ϕ = Φa (a ∈ Vλ ), we have tr(P ϕ) − tr(ϕ) = tr ((σπ0 − 1)Φa) ! Ã n−1 X 0 0 (σk0 − 1)Φ)a · · · σk+1 σn−1 = tr σγn ( k=1

=

n−1 X

¢ ¡ 0 0 fi (v)(Tk − t1/2 )Φa · · · σk+1 tr σγn σn−1

k=1

=

n−1 X k=1

= 0,

¢ ¡ 0 0 fi (v)Φa · · · σk+1 tr (Tk − t1/2 )σγn σn−1

1.3. ISOMORPHISMS FOR QAKZ

87

where the fi (v) are C–valued functions. Hence tr(P ϕ) = 0 and −1 tr(Yn ϕ) = tr(Tn−1 · · · T1−1 P ϕ)

= t(1−n)/2 tr(P ϕ) = 0. Now we shall prove that tr(Yi ϕ) = 0 for all i by induction. Assume that −1 tr(Yi ϕ) = 0 for k + 1 ≤ i ≤ n. Since Yk = Tk−1 · · · T1−1 P Tn−1 · · · Tk , it is enough to see that tr(P Tn−1 · · · Tk ϕ) = 0. Using that ϕ is a solution of QAKZ, −1

−1 tr(Fk−1 · · · F1−1 P Fn−1 · · · Fk ϕ) = tr(γk ϕ) −1

= γk tr(ϕ) = 0. On the other hand, Fi (v) = ci (v)(Ti + fi (v)), where ci (v) and fi (v) are some scalar functions. Therefore 0 = tr(P Fn−1 · · · Fk ϕ) = tr(cn−1 · · · ck P (Tn−1 + fn−1 (v)) · · · (Tk + fk (v))ϕ) X = tr(cI (v)P Til · · · Ti1 ϕ), I=(i1 ,...,il )

where I = (i1 , . . . , il ) is a sequence of integers such that k ≤ i1 < i2 < · · · < il ≤ n − 1, and cI (v) are some scalar function. If I 6= I0 = (k, k + 1, . . . , n − 1), then we have the following possibilities: 1. il 6= n − 1, 2. il = n − 1 and there exists an m (1 ≤ m ≤ l) such that ij − ij−1 = 1 for any j = m + 1, m + 2, . . . , l and im − im−1 > 1, 3. Otherwise. Case (1): As il < n − 1, we have tr(P Til · · · Ti1 ϕ) = tr(Til +1 · · · Ti1 +1 P ϕ) = tl/2 tr(P ϕ) = 0. Case (2): Since[Ti , Tj ] = 0 for |i − j| > 1, tr(P (Til · · · Tim )(Tim−1 · · · Ti1 )ϕ) = tr(P (Tim−1 · · · Ti1 )(Til · · · Tim )ϕ) = tr(Tim−1 +1 · · · Ti1 +1 P Til · · · Tim ϕ) = t(m−1)/2 tr(P Til · · · Tim ϕ).

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CHAPTER 1. KZ AND QMBP

By the induction hypothesis, tr(P Til · · · Tim ϕ) = 0. Hence tr(P Til · · · Ti1 ϕ) = 0. Case (3): In this case I = (i1 , . . . , il ) must be of the form il = n−1, il−1 = n − 2, . . . , i1 = n − l > k. By induction, tr(P Til · · · Ti1 ϕ) = 0. So tr(Yi ϕ) = 0 for all i. Because of the relations between T and Y , it remains to check that ❑ tr(Yi1 · · · Yil ϕ) = 0 for any l. One can show this by induction on l.

1.3.5

Macdonald’s operators

We set t1/2 − t−1/2 Tˆi = t1/2 σi + v −v (σi − 1), e i i+1 − 1 t1/2 − t−1/2 Gij = t1/2 + vi −vj (1 − σij ), e −1 −1 . ∆i = Tˆi · · · Tˆn−1 σπ Tˆ1−1 · · · Tˆi−1

(1 ≤ i ≤ n − 1), (1.3.30) (1 ≤ i, j ≤ n),

(1.3.31) (1.3.32)

Here the σw are from (1.3.20), σij = σsij . Switching from {T } to {G}: Tˆi σi = Gii+1 , G−1 ij ∆i

t1/2 − t−1/2 = t − vi −vj (1 − σij ), e −1 −1 = Gii+1 · · · Gin σγi G−1 1i · · · Gi−1i . −1/2

Let em be the m–th elementary symmetric polynomial in n variables. We represent X Dw(m) σw , (1.3.33) em (∆1 , . . . , ∆n ) = w∈Sn (m)

for difference operators Dw

and define

Mm = Mem =

X

Dw(m) .

w∈Sn

All these operators are W –invariant, which results from the following lemmas. ˆ generated by Tˆi (1 ≤ i ≤ n − 1), Lemma 1.3.12. Consider the algebra H ˆ ∆j (1 ≤ j ≤ n). Then Ti 7→ Ti , Yj 7→ ∆j extends to an algebra isomorphism ∼ ˆ Moreover, if Q is a symmetric polynomial in n variables, then Hnt → H. ˆ Q(∆1 , . . . , ∆n ) is a central element in H.

1.3. ISOMORPHISMS FOR QAKZ

89

Actually, this observation is the key point (it can be checked directly or with some representation theory). We note that the formulas for T generalize the so-called Demazure operations and the Bernstein–Gelfand–Gelfand operations. They were also used by Lusztig and in a paper by Kostant–Kumar. ˆ From now on, we identify Hnt and its image H. Lemma 1.3.13. Let f (v1 , . . . , vn ) be a function on Cn . Then f is symmetric if and only if (Tˆi − t)f = 0 for all i. ❑ Lemma 1.3.14. Let Q be a symmetric polynomial in n variables. Then Q(∆1 , . . . , ∆n ) acts on the space of the symmetric polynomials in evi (1 ≤ i ≤ n). Proof. This follows immediately from Lemma 1.3.12 and Lemma 1.3.13. ❑ Let us calculate M1 . Since M1 is symmetric, it suffices to find the coefficient of σγ1 . Using the G–representation, it is easy to see that σγ1 does not Q −1/2 1/2 v −v appear in ∆2 , . . . , ∆n . The σγ1 –factor of ∆1 equals ni=2 t eev11 −vii−t σγ1 . −1 Upon the symmetrization: n Y 1/2 vi X t e − t−1/2 evj M1 = σγi . evi − evj i=1 j6=i

Similarly, Mm =

X I=(i1 ,...,im )

Y t1/2 evi − t−1/2 evj σγi1 · · · σγim , vi − evj e i∈I j6∈I

where I = (i1 , . . . , im ) is a sequence of integers such that 1 ≤ i1 < · · · < im < n. To recapitulate, let us consider the classical limit of the Macdonald operators. Setting q = eh and t = q k , h → 0, we have ∆i = 1 + hDi + O(h2 ), X ∂ M1 − n = h + O(h2 ), ∂vi n(n − 1) = h2 L2 + O(h3 ). 2 Comment. Take a solution Φ = Φ(v) of the QAKZ equation in an Ht – module V , assuming that Φ has the trivial monodromy. Then, for any polynomial p ∈ C[x1 , . . . , xn ], we have M2 − (n − 1)M1 +

p(Y1 , . . . , Yn )Φ = p(∆1 , . . . , ∆n )Φ,

(1.3.34)

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CHAPTER 1. KZ AND QMBP

where ∆i are the difference Dunkl operators defined previously. Note that ∆0i can be replaced by ∆i because the monodromy of Φ is trivial. We also need a linear functional pr : Vλ → C for a vector λ = (λ1 , . . . , λn ) ∈ Cn such that pr(Yi b) = λi pr(b) (i = 1, . . . , n)

(1.3.35)

for any b ∈ V . Given any element a ∈ V , let us define a scalar-valued function ϕ = ϕ(v) by setting ϕ(v) = pr(Φ(v)a) ∈ C. (1.3.36) Then formula (1.3.34) implies p(λ1 , . . . , λn )ϕ = p(∆1 , . . . , ∆n )ϕ.

(1.3.37)

Thus, the scalar-valued function ϕ = ϕ(v) solves the difference Dunkl eigenvalue problem.

1.3.6

Arbitrary root systems

Let Σ = {α} ∈ Rn be a reduced root system of rank n (of type A, B, C, D, E, F, or G) and Ht = hT1 , . . . , Tn , X1 , . . . , Xn i (1.3.38) be the corresponding affine Hecke algebra. The baxterization (a parametric deformation satisfying the Yang–Baxter relations) of Ti will be given by F i = Ti +

t1/2 − t−1/2 eui − 1

with ui = (u, αi )

(1.3.39)

for each i = 1, . . . , n. We also have to use the element T0 = Xθ∨ Tθ−1

(1.3.40)

corresponding to the simple affine root α0 = δ − θ for θ being the highest root. Its baxterization is quite similar: F 0 = T0 +

t1/2 − t−1/2 , eh−uθ − 1

(1.3.41)

where uθ = (u, θ). The functions F0 , F1 , . . . , Fn satisfy the Yang–Baxter equations associated with the extended Dynkin diagram. For example, in the case of ◦1 ⇒ ◦2 , we have F1 (v)F2 (u + v)F1 (2u + v)F2 (u) = F2 (u)F1 (2u + v)F2 (u + v)F1 (v). (1.3.42) The arguments of Fi have a nice geometric interpretation as the angles in the pictures of two lines with the reflection in the x–axis. Such a plane

1.3. ISOMORPHISMS FOR QAKZ

91

geometric interpretation holds for arbitrary classical root systems, including the C ∨ C. See [C12]. Using T0 , the affine Hecke algebra Ht has an alternative presentation: Ht = hT0 , T1 , . . . , Tn ; Πi,

(1.3.43)

where Π is a certain finite abelian group. The group Π is isomorphic to P ∨ /Q∨ . It is the set of all elements of the extended affine Weyl group c = W n P ∨, W

∨

P =

n M

Z bi ,

(1.3.44)

i=1

preserving the set {α0 , α1 , . . . , αn } of the simple affine roots. It gives the embedding of Π into the automorphism group of the extended Dynkin diagram. L n c The action of W on R Rδ is by the affine reflections and the corresponding shifts in the δ–direction for P ∨ : b(z + ζδ) = z + (ζ − (b, z))δ. c = hs0 , s1 , . . . , sn ; Πi for s0 = (θ∨ ) · sθ . Lemma 1.3.15. One has: W

❑

The group Π can be embedded into the affine Hecke algebra. The images Pπ of the elements π ∈ Π permute {Ti } in the same way as π permute {si }. The baxterization of the elements in Π is trivial: Fπ = Pπ for each π ∈ Π. Keeping the notations of the previous sections, we have the following theorem. Theorem 1.3.16. Given any Ht –valued function Ψ = Ψ(u), the formulas sei (Ψ) = si (Fi Ψ)

(1.3.45)

π e(Ψ) = Pπ πΨ

(1.3.46)

for all i = 0, 1, . . . , n and

c. for all π ∈ Π induce a representation of W

❑

The QAKZ equation for Σ is the invariance condition eb(Φ) = Φ for all b ∈ P ∨ . It can be shown that this equation is equivalent to the difference QMBP associated with the root system Σ defined via similar Dunkl operators. A conceptual proof of this isomorphism theorem is given by means of the intertwiners of double affine Hecke algebras (see [C13, C15]).

92

1.4 1.4.1

CHAPTER 1. KZ AND QMBP

DAHA and Macdonald polynomials Rogers’ polynomials

We will apply the Hecke algebra technique to the Macdonald polynomials. We will concentrate on the duality and the recurrence relations. The key notion will be the double affine Hecke algebra. Let us start with A1 . The corresponding L–operator in the differential case reads as follows: L(k) =

∂2 eu + e−u ∂ + 2k + k2, ∂u2 eu − e−u ∂u

(1.4.1)

where k is a complex parameter. There are two special values of k when the operator L(k) is very simple. For k = 0 we have L(0) = ∂ 2 /∂u2 . When k = 1, L(1) = d−1

∂2 d, ∂u2

with d = eu − e−u .

(1.4.2)

Similarly, we can conjugate by dk for any k: dk L(k) d−k =

∂2 4k(k − 1) − u . 2 ∂u (e − e−u )2

(1.4.3)

Sometimes the latter operator is more convenient to deal with than the original L. Let us now consider the eigenvalue problem for the operator L(k) : L(k) ϕ = λ2 ϕ.

(1.4.4)

If k = 1, there is an immediate solution: ϕ(u; λ) =

sinh(uλ) . sinh(u) sinh(λ)

(1.4.5)

In this normalization, this solution is symmetric with respect to u and λ. Without the sinh(λ) in the denominator, it extends the characters of finite dimensional representations of SL2 (C). If k = 1/2, this operator is the radial part of the Casimir operator for the symmetric space SL2 (R)/SO2 (R). Namely, it is the restriction of the Casimir operator C to the double coset space SO2 \SL2 /SO2 , which is identified with a domain in R∗ /S2 . If k = 1, 2, then L(k) corresponds to SL2 (C)/SU (2) and SL2 (K)/SU2 (K) for the quaternions K. For any k, there is a family of even (u → −u) solutions of (1.4.4) in the form pn = enu + e−nu + lower integral exponents. (1.4.6) The eigenvalues are L(k) pn = (n + k)2 pn for n = 0, 1, 2, . . . .

(1.4.7)

1.4. DAHA AND MACDONALD POLYNOMIALS

93

This family of Laurent polynomials satisfies the orthogonality relations Constant Term (pn pm d 2k ) = cn δnm .

(1.4.8)

They are called the ultraspherical polynomials. We can also consider the rational limit `(k) =

2k ∂ ∂2 + 2 ∂u u ∂u

(1.4.9)

of the operator L(k) , switching from the Sutherland model to the Calogero model. The solutions of the rational eigenvalue problem are expressed in terms of the Bessel function. In this case, the solutions can be normalized to ensure the symmetry between the variable and the eigenvalue. In the trigonometric case it is possible only for two special values, k = 0, 1. Apart from these special cases, there is no such symmetry, which is a serious demerit of the harmonic analysis on symmetric spaces. In the difference theory, this symmetry holds for any k and for all root systems, which is expected to help to renew the Harish-Chandra theory. The so-called group case (k = 1) is an intersection point of the differential (classical) and the difference (new) theories. Difference case. We set x = eu and introduce the “multiplicative difference” Γ acting as Γ(f (x) = f (q 1/2 x) and satisfying the obvious commutation relation Γx = q 1/2 xΓ with the coordinate. The Macdonald operator L is as follows: L=

t1/2 x−1 − t−1/2 x −1 t1/2 x − t−1/2 x−1 Γ + Γ . x − x−1 x−1 − x

(1.4.10)

The parameter k in the difference setup is determined from the relation t = q k . When t = q (or k = 1), the operator L is simple: L=

1 (Γ + Γ−1 )(x − x−1 ). x − x−1

(1.4.11)

Compare this formula with (1.4.2) in the differential case and notice that (1.4.11) is easier to check than (1.4.2). The eigenvalue problem Lϕ = (Λ + Λ−1 )ϕ

(1.4.12)

always has a self-dual family of solutions. When n = 0, 1, 2, . . ., there exists a unique family of Laurent polynomials, the so-called q, t–ultraspherical or Rogers polynomials, pn = xn + x−n + lower terms,

(1.4.13)

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which are symmetric with respect to the transformation x → x−1 and satisfy the equation Lpn = (t1/2 q n/2 + t−1/2 q −n/2 )pn . (1.4.14) The following duality formula is proved in the next subsection by using the double affine Hecke algebra. Theorem 1.4.1 (Duality). pn (t1/2 q m/2 )pm (t1/2 ) = pm (t1/2 q n/2 )pn (t1/2 ) for any m, n = 0, 1, 2, . . .. ❑ If we set πn (x) =

pn (x) , pn (t1/2 )

(1.4.15)

the duality can be rewritten as follows: πn (t1/2 q m/2 ) = πm (t1/2 q n/2 ) (m, n = 0, 1, 2, . . .).

(1.4.16)

The Rogers polynomials are the Macdonald polynomials of type A1 . Concerning the general Macdonald polynomials associated with arbitrary root systems, there are three main Macdonald conjectures (see [M4, M5] and the book [M7]): (1) the scalar product conjecture, (2) the evaluation conjecture, (3) the duality conjecture. One may also add the Pieri rules to the list. These conjectures were justified recently using the double affine Hecke algebras in [C14, C19]. The scalar product conjecture (the norm formula) was established by Opdam in the differential setup in complete generality in [O1]. He introduced the shift operators and deduced the norm formula for any k ∈ N from the trivial particular case k = 0. The passage to an arbitrary complex k is based on analytic continuation. I extended his approach to the q, t–case. The evaluation and the duality conjectures collapse in the differential case.

1.4.2

A Hecke algebra approach

The duality from Theorem 1.4.1 can be rephrased as the symmetry of a certain scalar product. This product is closely connected with the difference counterpart of the spherical Fourier transform. For any symmetric Laurent polynomials f, g ∈ C[x + x−1 ], we set {f, g} = (a(L)g)(t1/2 ),

(1.4.17)

where a is a polynomial such that f (x) = a(x+x−1 ). So we apply the operator a(L) to g and then evaluate the result at x = t1/2 .

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95

Theorem 1.4.2. {f, g} = {g, f } for any f, g ∈ C[x + x−1 ].

❑

Theorem 1.4.1 follows from Theorem 1.4.2. Indeed, if f = πm and g = πn , we can compute the scalar product as follows: {πm , πn } = (a(L)πn )(t1/2 )

(1.4.18)

= a(t1/2 q n/2 + t−1/2 q −n/2 )πn (t1/2 ) = πm (t1/2 q n/2 ). Use Lπn = (t1/2 q n/2 +t−1/2 q −n/2 )πn and the normalization πn (t1/2 ) = 1. Hence Theorem 1.4.2 implies πm (t1/2 q n/2 ) = πn (t1/2 q m/2 ). In fact, the theorems are equivalent, since the pn form a basis in the space of all symmetric Laurent polynomials. Definition 1.4.3. The double affine Hecke algebra HHq,t of type A1 is the quotient HHq,t = hX, Y, T i/ ∼, (1.4.19) by the relations for the generators X, Y, T T XT = X −1 , Y

−1

X

−1

T −1 Y T −1 = Y −1 , 2

Y XT = q

−1/2

,

(T − t

1/2

)(T + t

−1/2

(1.4.20) ) = 0.

(1.4.21) ❑

Now we consider q, t as numbers or parameters. The first point of the theory is the following statement of the PBW-type. Any element of H ∈ HH can be uniquely expressed in the form X ci²j X i T ² Y j (ci²j ∈ C, ² = 0, 1). (1.4.22) H= i,j∈Z

The second important fact is the symmetry of HHq,t with respect to X and Y. Theorem 1.4.4. There exists an anti-involution φ : HH → HH fixing q 1/2 , t1/2 such that φ(X) = Y −1 , φ(Y ) = X −1 , and φ(T ) = T . Proof. Indeed, φ transposes the first two relations and leaves the third one invariant. ❑ Third, we introduce the expectation value {H}0 ∈ C of an element H ∈ HHq,t by X ci²j t−i/2 t²/2 tj/2 , ² = 0, 1, (1.4.23) {H}0 = i,j∈Z

using the expansion (1.4.22). The definitions of φ and { }0 give that {φ(H)}0 = {H}0

for any H ∈ HHq,t .

(1.4.24)

96

CHAPTER 1. KZ AND QMBP Fourth, the operator counterpart of the pairing {f, g} on HHq,t × HHq,t is {A, B}0 = {φ(A)B}0

(1.4.25)

for any A, B ∈ HHq,t . The φ–invariance of the expectation value (1.4.24) ensures that it is symmetric: {A, B}0 = {B, A}0 . We also remark that this pairing is non-degenerate for generic q, t. Theorem 1.4.2 now readily results from the following lemma. Lemma 1.4.5. For any symmetric Laurent polynomials f (x),g(x)∈ C[x + x−1 ], {f (X), g(X)}0 = {f, g}. Proof. We need the polynomial representation of the double affine Hecke algebra HHq,t . We start with the one-dimensional representation of the Hecke algebra HY = hT, Y i sending T 7→ t1/2 and Y 7→ t1/2 , denoted simply by +. The corresponding induced representation of HHq,t is H V = IndH H/{HH(T − t1/2 ) + HH(Y − t1/2 )} ' C[x, x−1 ]. (1.4.26) HY (+) = H The last isomorphism is xn ↔ X n mod {HH(T −t1/2 )+HH(Y −t1/2 )}. Under the identification of V with the ring C[x, x−1 ] of Laurent polynomials, the element X acts on C[x, x−1 ] as multiplication by x; T and Y act by the operators t1/2 − t−1/2 (s − 1) and Yb = sΓTb, Tb = t1/2 s + x2 − 1

(1.4.27)

respectively. Here s(f )(x) = f (x−1 ). We use the fact that the equality Hf (x) = g(x) in V means that Hf (X) − g(X) belongs to HH(T − t1/2 ) + HH(Y − t1/2 ). The expectation value is the composition β α HH −→ V ∼ = C[x, x−1 ] −→ C,

(1.4.28)

where α is the residue modulo {HH(T − t1/2 ) + HH(Y − t1/2 )} and β(f ) = f (t−1/2 ) is the evaluation map at t−1/2 . Taking any f, g ∈ C[X, X −1 ], {f (X), g(X)}0 = {φ(f (X))g(X)}0 = {f (Y −1 )g(X)}0 = f (Yb −1 )(g)(t−1/2 ).

(1.4.29)

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97

The last equality follows from (1.4.28). If f and g are symmetric and f (X) = a(X + X −1 ), then {f (X), g(X)}0 = a(L)(g)(t1/2 ) = {f, g},

(1.4.30)

since the operator Yb + Yb −1 acts on the symmetric Laurent polynomials via L. It is straightforward. The lemma and the duality are established. ❑ This method of proving the duality theorem can be generalized to any root systems. We now discuss the application of the duality to the Pieri rules, which are the recurrence formulas for πn with respect to the index n. First, we discretize functions and operators. Recall that the π–polynomials πn (x), the renormalized Rogers polynomials, are characterized by the conditions Lπn = (t1/2 q n/2 + t−1/2 q −n/2 )πn ,

πn (t1/2 ) = 1,

(1.4.31)

where

t1/2 x − t−1/2 x−1 t1/2 x−1 − t−1/2 x −1 Γ + Γ . (1.4.32) x − x−1 x−1 − x As above, Γx = q 1/2 xΓ. Denote the set of C–valued functions on Z by Funct(Z, C). For any Laurent polynomial f ∈ C[x, x−1 ] or a more general rational function, we define fb ∈ Funct(Z, C) by setting L=

fb(m) = f (t1/2 q m/2 ) for all m ∈ Z.

(1.4.33)

Considering A = hC(x), Γi as an abstract algebra with the fundamental relation Γx = q 1/2 xΓ, the action of A on ϕ ∈ Funct(Z, C) is as follows: x bϕ(m) = t1/2 q m/2 ϕ(m),

b Γϕ(m) = ϕ(m + 1).

(1.4.34)

The correspondence f 7→ fb, whenever it is well defined (the functions f may have denominators), is an A–homomorphism C(x) → Funct(Z, C). Due to (1.4.31): bπn (m) = (t1/2 q n/2 + t−1/2 q −n/2 )b Lb πn (m).

(1.4.35)

The Pieri rules result from this equality as follows. The duality π bn (m) = π bm (n) implies: Lb πm (n) = (t1/2 q n/2 + t−1/2 q −n/2 )b πm (n) = (b x+x b−1 )b πm (n).

(1.4.36)

b acting on the indices m instead of n. Explicitly, Here L is L (b x+x b−1 )b πm (n) =

(1.4.37)

tq m/2 − t−1 q −m/2 q −m/2 − q m/2 π b (n) + π bm−1 (n). m+1 t1/2 q m/2 − t−1/2 q −m/2 t−1/2 q −m/2 − t1/2 q m/2

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For generic q, t, the mapping f → fb is injective. Hence one can pull (1.4.37) back, removing the hats: tq m/2 − t−1 q −m/2 πm+1 t1/2 q m/2 − t−1/2 q −m/2 q m/2 − q −m/2 + 1/2 m/2 πm−1 . t q − t−1/2 q −m/2

(x + x−1 )πm =

(1.4.38)

This is the Pieri formula in the case of A1 . See [AI]. We remark that this formula makes sense when m = 0, since the coefficient of πm−1 “automatically” vanishes at m = 0. Such vanishing conditions are automatically fulfilled for any generalizations of the Pieri rules in the difference theory (but not in the differential theory). See the example of GLn below. The Pieri rules obtained above can be used to prove the so-called evaluation conjecture describing the values of pn at x = t. Applying (1.4.38) repeatedly, we obtain the formula (x + x−1 )` πm = c`,m πm+` + lower terms

(1.4.39)

for each ` = 0, 1, 2, . . .. The leading coefficient c`,m can be readily calculated: c`,m =

`−1 Y i=0

tq (m+i)/2 − t−1 q −(m+i)/2 . t1/2 q (m+i)/2 − t−1/2 q −(m+i)/2

(1.4.40)

Let us apply (1.4.39) for m = 0: (x + x−1 )` = c`,0 π` + lower terms.

(1.4.41)

Comparing the coefficients of x` + x−` , we have 1 = c`,0 /p` (t1/2 ), since p` = x` + x−` + · · · . Hence p` (t

1/2

) = c`,0 =

`−1 Y i=0

tq i/2 − t−1 q −i/2 . t1/2 q i/2 − t−1/2 q −i/2

(1.4.42)

This value is easy to calculate directly (the formulas for pn are known). However, the method descibed in this subsection is applicable to arbitrary root systems. We need only the duality, which is the major advantage of the difference theory versus the differential case.

1.4.3

The GL–case

In this subsection, we will generalize the above considerations to the double affine Hecke algebra of GLn –type. We will focus on the main points and skip

1.4. DAHA AND MACDONALD POLYNOMIALS

99

the justification of the duality and the Pieri rules because they are very close to the case of A1 . In the GLn –case, the Macdonald operators M0 = 1, M1 , . . . , Mn are as follows: X

Mm =

Y t1/2 xi − t−1/2 xj Γi1 · · · Γim . xi − xj

(1.4.43)

I={i1 (λ1 − 1, λ2 , λ3 + 1 . . .) > · · · . (1.4.46) We will use the abbreviation tρ q λ = (t(n−1)/2 q λ1 , . . . , t(−n+1)/2 q λn ),

(1.4.47)

where 2ρ = (n − 1, n − 3, . . . , −n + 1). So Mm pλ = em (tρ q λ )pλ . Using k: t = q k and tρ q λ = q kρ+λ . Given a partition λ, we set πλ (x) =

pλ (x) pλ (x1 , . . . , xn ) = . ρ (n−1)/2 pλ (t ) pλ (t , t(n−3)/2 , . . . , t(−n+1)/2 )

(1.4.48)

Theorem 1.4.6 (Duality). For any partitions λ and µ, we have πλ (tρ q µ ) = πµ (tρ q λ ). This duality formula implies the following Pieri formula.

(1.4.49)

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CHAPTER 1. KZ AND QMBP

Theorem 1.4.7. Setting ²I = em (x)πλ (x) =

X

P

i∈I ²i ,

where ²i are unit vectors,

Y tj−i+1/2 q λi −λj − t−1/2 πλ+²I (x). tj−i q λi −λj − 1

(1.4.50)

|I|=m i∈I, j ∈I /

Here the summation is taken only over subsets I ⊂ {1, 2, . . . , n} such that |I| = m and λ + ²I remain partitions (in the case of arbitrary root systems, remain dominant). As a matter of fact, it happens automatically, since the coefficient of πλ+²I (x) on the right vanishes unless λ+²I is a partition (dominant). We can determine the value of the Macdonald polynomial pλ (x) at x = tρ exactly by the method used in the A1 –case. The corresponding evaluation formula was conjectured by Macdonald and proved by Koornwinder. The above theorems (for GLn ) are also due to Macdonald and Koornwinder; [EK3, EK2] contain the approach via the vertex operators, which also leads to the Pieri rules. For arbitrary roots, the above theorems were established in [C19]. Cf. Chapter 3. Double Hecke algebra. The double Hecke algebra is the best way to introduce the difference Dunkl operators. The operators Mm are the restrictions of the polynomials em in terms of the operators ∆i from (1.3.32) to the symmetric functions. The operators ∆i describe the action of the generators Yi of H the HH, which will be defined next, in the induced representation IndH HY (+), isomorphic to the algebra of Laurent polynomials C[X1±1 , . . . , Xn±1 ] and called the polynomial representation. So the analogy with (1.4.26) is complete. The double affine Hecke algebra (DAHA) HH = HHq,t for GLn is the algebra generated by the following two commutative algebras of Laurent polynomials in n variables: C[X1±1 , . . . , Xn±1 ] and C[Y1±1 , . . . , Yn±1 ],

(1.4.51)

and the nonaffine Hecke algebra of type An−1 : H = hT1 , . . . , Tn−1 i

(1.4.52)

with the standard braid and quadratic relations. The other relations are as follows: Ti Xi Ti = Xi+1 (i = 1, . . . , n − 1), Ti Xj = Xj Ti (j 6= i, i + 1), Ti−1 Yi Ti−1 = Yi+1 (i = 1, . . . , n − 1), Ti Yj = Yj Ti (j 6= i, i + 1), Y2−1 X1 Y2 X1−1 = T12 , Ye Xj = qXj Ye

e e j = q −1 Yj X. and XY

(1.4.53) (1.4.54) (1.4.55) (1.4.56)

1.4. DAHA AND MACDONALD POLYNOMIALS

101

e = Qn Xi and Ye = Qn Yi . They commute with {T1 , . . . , Tn−1 }, Here X i=1 i=1 thanks to (1.4.53) and (1.4.55). Actually, only one of the relations in (1.4.56) is sufficient. It follows from the {X, T, π}–presentation of HH considered below. When q = 1, t = 1, we come to the elliptic Weyl group or 2-extended Weyl group of type GLn due to Saito [Sai]. If q = 1 but there are no quadratic relations, the corresponding group is the elliptic braid group (π1 of the product of n elliptic curves without the diagonals divided by Sn ). It was calculated by Birman [Bi] and Scott [Sc]. Establishing the connection with (1.3.32), the formulas Xi = evi , Ti = Tˆi , and Yi = ∆i give the polynomial representation of HH . There is another version of this definition, using the element π, introduced by the formula Y1 = T1 · · · Tn−1 π −1 (1.4.57) and having the following commutation relations with Xi and Ti : πXi = Xi+1 π

(i = 1, . . . , n − 1),

πXn = q −1 X1 π,

(1.4.58)

and πTi = Ti+1 π

(i = 1, . . . , n − 2).

(1.4.59)

In the polynomial representation, this element coincides with π from Lemma 1.3.4. It acts on the functions Xi = evi through the action of π −1 on vectors v. Considered formally, π is the image of the element P from Lemma 1.3.1 with respect to the Kazhdan–Lusztig involution, sending T → T −1 , Y → Y −1 , and t1/2 → t−1/2 . We note that Ye = π −n . Since Ti−1 · · · T1 X1 T1 · · · Ti−1 = Xi ,

T1 · · · Ti−1 Yi Ti−1 · · · T1 = Y1 ,

(1.4.60)

we can reduce the list of generators. Namely, HH = hX1 , Y1 , T1 , . . . , Tn−1 i

(1.4.61)

HH = hX1 , π, T1 , . . . , Tn−1 i.

(1.4.62)

or In terms of {T, π, X}, the list of defining relations of HH is as follows: (a) Xi Xj = Xj Xi (1 ≤ i, j ≤ n), Ti Xi Ti = Xi+1 (1 ≤ i < n); (b) the braid relations and quadratic relations for T1 , . . . , Tn−1 ; (c) πXi = Xi+1 π (i = 1, . . . n − 1) and π n Xi = q −1 Xi π n (i = 1, . . . , n); (d) πTi = Ti+1 π( i = 1, . . . n − 2) and π n Ti = Ti π n (i = 1, . . . , n − 1).

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For instance, let us deduce (1.4.55) from these formulas. Substituting, the left-hand side equals: −1 (T1 πTn−1 · · · T2−1 )X1 (T2 · · · Tn−1 π −1 T1−1 )X1−1 −1 = T1 πTn−1 · · · T2−1 (T2 · · · Tn−1 )X1 π −1 (T1−1 X1−1 T1−1 )T1 = T1 (πX1 π −1 X2−1 )T1 = T12 .

The presentation via π, however, is not convenient from the viewpoint of the symmetry between Xi and Yi , which will be discussed now. It is better to use {Y } instead of π. Comment. (i) The definition of HH is adjusted to the formulas for ∆i from (1.3.32). In the next chapters, we will replace the Yi by their images with respect to the following anti-involution of HH : Ti 7→ Ti , Xi 7→ Xi , π 7→ π −1 , q 7→ q, t1/2 7→ t1/2 . Under this anti-involution: −1 −1 . . . T1−1 πTn−1 . . . Ti . Yi = Ti . . . Tn−1 π −1 T1−1 . . . Ti−1 7→ Ti−1

(ii) Concerning the case of SLn , we set π n = 1 = Ye . The corresponding double Hecke algebra HHAn is a variant of HH, where the C[X] is replaced by the subalgebra of e 1/n ] C[X1 , . . . , Xn , X formed by the polynomials of degree zero. Also q 1/n must be added. The X–generators corresponding to the fundamental weights and are as follows: e −i/n , 1 ≤ i ≤ n − 1. e i = X1 · · · Xi X X The anti-dual Y –generators are Yei = Y1 · · · Yi , where the definition of Yi remains unchanged. They satisfy Theorem 1.4.8 below. The particular case of SL2 , i.e., the case of A1 , was considered above. ❑ The algebra HH contains the following two affine Hecke algebras: t HX = hX1 , . . . , Xn , T1 , . . . , Tn−1 i,

HYt = hY1 , . . . , Yn , T1 , . . . , Tn−1 i.

(1.4.63)

They are isomorphic to each other by the correspondence Xi ↔ Yi−1 . This map can be extended to an anti-involution of HH. It is a general statement that holds for any root system. Theorem 1.4.8. There exists an anti-involution φ : HHq,t → HHq,t such that φ(Xi ) = Yi−1 , φ(Yi ) = Xi−1 for i = 1, . . . , n, and φ(Ti ) = Ti (i = 1, . . . , n − 1). It preserves q, t.

1.4. DAHA AND MACDONALD POLYNOMIALS

103

Proof. We need to check that relation (1.4.55) is self-dual with respect to φ. The other relations are obviously φ–invariant. One has: 1 = T1−2 Y2−1 X1 Y2 X1−1 = T1−1 {Y1−1 (T1 X1 T1−1 )Y1 T1−1 X1−1 } = Y1−1 X2 T1−2 Y1 (T1−1 X1−1 T1−1 ) = Y1−1 X2 T1−2 Y1 X2−1 . The latter can be rewritten as Y1 X2−1 Y1−1 X2 = T12 , which is the φ–image of (1.4.55). ❑ Using this involution we can establish the duality theorem for the Macdonald polynomials in the GLn –case in the same way as we did in the A1 –case. Generalizing the theory to the case of arbitrary roots, we can prove the Macdonald conjectures and much more. This gives a very convincing example of the power of the modern difference-operator methods. Some relations. Formula (1.4.55), the affine Hecke relations, and the duality result in a variety of identities of the commutator type. First, for n > j ≥ i ≥ 1, −1 −1 −1 · · · Tj−1 . (1.4.64) Xi Yj+1 Xi−1 = Tj · · · Ti+1 Ti2 Ti+1 Xi Yi+1 Xi−1 = Ti2 , Yj+1 Yi+1

By the duality, −1 −1 · · · Tj−1 . Yi Xj+1 Yi−1 = Tj · · · Ti+1 Ti−2 Ti+1 Xj+1

More generally, we can calculate the group commutators for products of different Yj and products of different Xi assuming that the Y –indices j and the X–indices i are disjoint. The commutators must be of type (1.4.64), i.e., either in the form (Y )−1 (X)(Y )(X)−1 or in the form (X)−1 (Y )(X)(Y )−1 . For instance, (Yj+m · · · Yj+1 )−1 Xi (Yj+m · · · Yj+1 )Xi−1 =(Tj · · · Tj+m−1 ) · · · (Ti+1 · · · Ti+m )(Ti · · · Ti+m−1 ) −1 −1 −1 · · · Tj−1 ) )(Tj+m−1 · · · Ti+1 × (Ti+m−1 · · · Ti )(Ti+m

(1.4.65)

−1 }{Tj · · · Tj+m−1 Tj+m−1 · · · Tj }{Tj−1 · · · Ti }, ={Ti−1 · · · Tj−1

where j ≥ i ≥ 1, n − j ≥ m ≥ 1, and the inverse powers of T appear only as j > i. The following identity contains only positive powers of T : (Yj+m · · · Yj+1 )−1 (Xj · · · Xi )(Yj+m · · · Yj+1 )(Xj · · · Xi )−1 =(Tj · · · Tj+m−1 )(Tj−1 · · · Tj+m−2 ) · · · (Ti · · · Ti+m−1 ) × (Ti+m−1 · · · Ti ) · · · (Tj+m−2 · · · Tj−1 )(Tj+m−1 · · · Tj ) for the same range of i, j, m.

(1.4.66)

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If there is a “gap” in the sequence of indices between the product of the consecutive Y and the consecutive X, we can shift the indices of Y using conjugations with the proper T : (Yj+m+l · · · Yj+l+1 )−1 (Xj · · · Xi )(Yj+m+l · · · Yj+l+1 )(Xj · · · Xi )−1 © ª = (Tj+l · · · Tj+m+l−1 ) · · · (Tj+1 · · · Tj+m ) × (Tj · · · Tj+m−1 )(Tj−1 · · · Tj+m−2 ) · · · (Ti · · · Ti+m−1 ) × (Ti+m−1 · · · Ti ) · · · (Tj+m−2 · · · Tj−1 )(Tj+m−1 · · · Tj ) © −1 ª −1 −1 −1 × (Tj+m · · · Tj+1 ) · · · (Tj+m+1−1 · · · Tj+l ) .

(1.4.67)

Note that all group commutators of considered type are elements of the pure braid group and obviously generate it. This group is formed by the products of Ti±1 with trivial images in the Sn . In the case j = 1, i = 1, j + m = n, we obtain 2 (Yn · · · Y2 )−1 X1 (Yn · · · Y2 )X1−1 = T1 T2 · · · Tn−1 · · · T2 T1 .

Therefore we can rewrite the first of the defining relations from (1.4.56) (recall that it results in the second) as follows: −2 X1 Y1 X1−1 Y1−1 = q −1 T1−1 · · · Tn−1 · · · T1−1 .

More generally, −2 · · · Ti−1 ). Xi Yi (Ti−1 · · · T12 · · · Ti−1 )Xi−1 Yi−1 = q −1 (Ti−1 · · · Tn−1

(1.4.68)

Denoting the T –products in the left-hand side by Li and those in the righthand side by Ri , L1 = 1 = Rn , Li Lj = Lj Li , and Ri Rj = Rj Ri for all indices. The commutativity is due to [C5]. The mapping Ti 7→ Ti (i < n), t Xi 7→ Li (i ≤ n) determines a surjective homomorphism of HX onto the nonaffine Hecke subalgebra. So do the Ri . Combined with (1.4.56), formula (1.4.67) readily gives the formulas for the group commutators of the fundamental X–weights and Y –weights. Namely, bi = X1 · · · Xi , Ybj = Y1 · · · Yj for 1 ≤ i ≤ n. Thanks to the duality one let X may assume that j ≥ i. Then b −1 bi Yb −1 X Ybj X i j i

(1.4.69) −1

−1

=q (Yn · · · Yj+1 ) (X1 · · · Xi )(Yn · · · Yj+1 ))(X1 · · · Xi )

=q i {(Tj · · · Tn−1 ) · · · (T1 · · · Tn−j )}{(Tn−j · · · T1 ) · · · −1 −1 −1 ) · · · (Tn−1 · · · Tj−1 )}. · · · Ti+1 × (Tn+i−j−1 · · · Ti )}{(Tn+i−j

1.5. ABSTRACT KZ AND ELLIPTIC QMBP

1.5

105

Abstract KZ and elliptic QMBP

We will introduce a general r–matrix KZ. It can be reduced to the affine Knizhnik–Zamolodchikov equation, which gives a simple proof of the selfconsistency of AKZ. Another application is the double KZ and the elliptic quantum many-body problem for arbitrary root systems, generalizing that due to Olshanetsky–Perelomov in the GLn –case.

1.5.1

Abstract r–matrices

We keep the notation from the previous sections. Let Σ = {α} ⊂ Rn be a finite root system of rank n of type An , Bn , . . . , G2 . We normalize the inner product of Rn setting (θ, θ) = 2, where θ ∈PΣ is the maximal root. Let {α1 , . . . αn } be the simple roots, Σ+ = {α = ni=1 ci αi ∈ Σ| ∀i, ci ≥ 0} the set of positive roots, {b1 , . . . , bn } ⊂ Rn the dual fundamental weights (i.e., (bi , αj ) = δij ), P ∨ the lattice spanned by {b1 , . . . , bn }, and W the Weyl group generated by the reflections sα (u) = u −

2(α, u) α, u ∈ Rn , α ∈ Σ, si = sαi . (α, α)

n For Σ, uα = (α, u) ∈ R and ui = (αi , u). Thus, if Pnu ∈ R and α ∈ P α = i=1 ci αi , then uα = ni=1 ci ui . We will use the derivations:

∂v (uα ) = (v, α), ∂i (uα ) = (bi , α), i = 1, . . . n, v ∈ Rn .

(1.5.1)

Let us begin with the formal theory of the following partial differential equations:   X (1.5.2) ναi rα  Φ(u). ∂i Φ(u) =  α∈Σ+

Here the values of Φ(u) are in a vector space V over C, and rα = rα (uα ) is a function of one variable uα with values in End(V ). We set: ναi = multαi (α) = (bi , α). Equivalently, for v ∈ Rn , 

 ∂v Φ(u) = 

X

(v, α)rα  Φ(u),

(1.5.3)

α∈Σ+

where v = bi in (1.5.2). Let X X (v, α)rα . ναi rα , Xv = Xi = α∈R+

α∈R+

(1.5.4)

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CHAPTER 1. KZ AND QMBP

The compatibility of system (1.5.2) (the cross-derivative integrability conditions) is equivalent to the purely algebraic commutativity of {Xi }: ⇔ [Xv , Xv0 ] = 0 ∀v, v 0 ∈ Rn . Indeed, ∂i (ναj rα ) = ∂j (ναi rα ) for all α. We are going to establish the necessary conditions for the compatibility. For α, β ∈ Σ ( β 6= ±α), let hα, βi be the subspace of Rn spanned by α and β. Then Σ ∩ hα, βi is a root system of rank 2. Proposition 1.5.1. If the operators X Xv(α,β) = (v, γ)rγ , γ ∈ Σ+ , γ ∈ Σ ∩ hα, βi (v ∈ hα, βi)

(1.5.5)

commute for all v ∈ hα, βi 3 v 0 , for instance, as v = α and v 0 = β, and for arbitrary pairs {α, β} ⊂ Σ+ , then [Xi , Xj ] = 0 (∀i, j = 1, . . . , n). That is, the commutativity of X for subsystems of rank 2 is sufficient for the total commutativity. ❑ By Proposition 1.5.1, it is sufficient to consider the rank 2 cases (Figure 1.4). One may assume that α, β are the standard simple roots from the figure, switching to proper linear combinations if necessary. The commutativity conditions are directly connected with the quantum Yang–Baxter equations from [C2]. The definition of quantum {Rα } and their relation to {rα } are as follows. α6

α

-β

¾

?

J ]J Á ­­ J ­ -β ¾ J­ ­J ­ J JJ ­ À­ ^

A1 × A1 -case

A2 -case

α

6

@ I @

µ ¡ ¡

@

¾

¡

@¡ ¡@

@

¡

¡ ¡ ª

-β

?

@ R @

B2 -case

α

6

Q k Q

3 ´ ] Á ­ ´´ QJ QJ ­´ β ¾ Q J­ ´ ´Q ´ ­J Q ´ ­ J Q ´ ­ À ^ J Q s Q ´ + ?

G2 -case

Figure 1.4: Root Systems of Rank 2 We set formally Rα = 1 + hrα + O(h2 ), α ∈ Σ+ .

(1.5.6)

The classical Yang–Baxter equations are the coefficient of h2 in the quantum Yang–Baxter equations for R: (0) A1 × A1 -case: Rα Rβ = Rβ Rα ⇒ [rα , rβ ] = 0.

(1.5.7)

1.5. ABSTRACT KZ AND ELLIPTIC QMBP

107

(1) A2 -case: Rα Rα+β Rβ = Rβ Rα+β Rα ⇒ [rα , rα+β + rβ ] + [rα+β , rβ ] = 0.

(1.5.8)

(2) B2 -case: Rα Rα+β Rα+2β Rβ = Rβ Rα+2β Rα+β Rα ⇒ [rα , rα+β + rα+2β + rβ ] + [rα+β , rα+2β + rβ ] + [rα+2β , rβ ] = 0.

(1.5.9)

(3) G2 -case: Rα Rα+β R2α+3β Rα+2β Rα+3β Rβ = Rβ Rα+3β Rα+2β R2α+3β Rα+β Rα ⇒ [rα , rα+β + r2α+3β + rα+2β + rα+3β + rβ ]+ +[rα+β , r2α+3β + rα+2β + rα+3β + rβ ]+ (1.5.10) +[r2α+3β , rα+2β + rα+3β + rβ ] + [rα+2β , rα+3β + rβ ] + [rα+3β , rβ ] = 0. The structure of the quantum Yang–Baxter equations is very simple: the product of all Rα0 for positive α0 clockwise coincides with the product taken counterclockwise. Respectively, the left-hand sides of the classical equations are the sums of the commutators [rα0 , rβ 0 ], where α0 , β 0 are ordered clockwise. Recall that all the roots above must be positive in Σ. Otherwise, the symbols Rα0 , rα0 are not defined. The classical Yang–Baxter equations are very close to the desired commutativity relations for Xv , but not exactly what we need. Indeed, the coefficients of all [rα , rβ ] in these equations are 1, but they are not always 1 when (a,b) we calculate the commutators of Xv . The simplest example is B2 , when [Xα , Xβ ] = 0 is not the classical B2 –YBE from (1.5.9). This discrepancy disappears in the theory of the quantum KZ. See [C12]. The limiting procedure from QKZ to KZ clarifies in full why we need to add some additional relations to the classical YBE to ensure the commutativity of X. However, it is not difficult to find these extra conditions without any reference to QKZ, as I did in my first paper devoted to the Knizhnik– Zamolodchikov equations [C6]. Proposition 1.5.2. For any pair {α, β} ∈ Σ+ such that the entire intersection of Σ and hα, βi is represented by Figure 1.4, we impose the conditions: (0) In the A1 × A1 –case: (1.5.7). (1) In the A2 –case: (1.5.8). (2) In the B2 –case: (1.5.9) and {[rα , rα+2β ] = 0}. (3) In the G2 –case: (1.5.10), { α0 ⊥ β 0 ⇒ [rα0 , rβ 0 ] = 0, α0 , β 0 ∈ hα, βi∩Σ+ }, and the A2 –relation (1.5.8) for the long roots. Then the operators X are pairwise commutative. ❑ The proposition gives the conditions necessary and sufficient for the X– commutativity, if we consider a bit more special situation. For the same root system Σ, we set X kα ναi rα , (1.5.11) xi = α∈Σ+

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CHAPTER 1. KZ AND QMBP

where kα = kβ if there exists w ∈ W such that w(α) = β (in other words, kα = kβ if (α, α) = (β, β)). We call such k invariant. Note that here we use the lower case x with k instead of the upper case X in the “abstract” theory of r–matrices. We call a two-dimensional subsystem of Σ with the generators α, β standard if it is described by one of the pictures in Figure 1.4. Respectively, α, β are called standard. They can generate a subsystem that is smaller than the complete intersection Σ ∩ hα, βi. Proposition 1.5.3. Let Σ be a root system of rank n. If a pair of positive roots α, β is standard, then we impose the conditions (0) A1 × A1 –case: (1.5.7); (1) A2 –case: (1.5.8); (2) B2 –case: (1.5.9); (3) G2 –case: (1.5.10). They are necessary and sufficient for the commutativity of the operators xi (i = 1, . . . , n) for arbitrary invariant k.

1.5.2

Degenerate DAHA

Sometimes the {rα } exist for all roots in Σ, positive and negative, and the relations (0–3) from Proposition 1.5.3 hold for all standard α, β ∈ Σ. Then we will call such a set, if it exists, an extension of {rα , α ∈ Σ+ }. Let us assume that r exists and satisfies the conditions of Proposition 1.5.3 for positive roots only. Then there are two natural extensions of r to all roots. The first is the extension by 0: r−α = 0 (α ∈ Σ+ ). The second is given by r−α = rα (α ∈ Σ+ ). The most useful extension is the third one, which will be called the invariant extension of a given r–matrix. It requires a W –invariance of {rα , α ∈ Σ+ }, which is defined as follows. We need to suppose that the Weyl group W acts on the algebra R containing {rα }, provided that we have the relations w(rα ) = rw(α) whenever {α, w(α) ∈ Σ+ , w ∈ W }. Such r are called W –invariant. Given a negative root β ∈ Σ, we set rβ = w(rα ) for any w ∈ W and α ∈ Σ+ satisfying w(α) = β. The definition does not depend on the particular choice of w and α. The group W acts on all {rα | α ∈ Σ} by the same formulas as for positive roots. We will call a W –invariant r satisfying the assumption of Proposition 1.5.3 for all roots, not only positive, an invariant r–matrix. Proposition 1.5.4. For an invariant r–matrix, we set X xb = kα (b, α)rα(u) ,

(1.5.12)

α∈Σ+

where u ∈ Rn and kα = kβ if ∃w ∈ W s.t. w(α) = β. Then [xb , xb0 ] = 0 (∀b, b0 ∈ Rn ), si (xb ) − xsi (b) = kαi (b, αi )(si (rαi ) + rαi ).

(1.5.13) (1.5.14)

1.5. ABSTRACT KZ AND ELLIPTIC QMBP

109

Proof. Relation (1.5.13) readily results from Proposition 1.5.3. As to (1.5.14), it follows from the relation si (Σ+ ) = (Σ+ \ {αi }) ∪ {−αi }. ❑ If W is a subgroup of the group of invertible elements of the ring R, and w(a) = waw−1 , for w ∈ W, a ∈ R, and, moreover, s(rαi ) + rαi = si , then we obtain si xb si − xsi (b) = kαi (b, αi )si . (1.5.15) 0 So we come back to the definition of the degenerate Hecke algebra HΣ from the previous sections. Let us recall it. 0 Definition 1.5.5. The degenerate Hecke algebra HΣ for the root system ∨ Σ is generated by {sα , xb , α ∈ R, b ∈ P } with the defining relations

[xb , xb0 ] = 0 (∀b, b0 ∈ P ∨ ), si xb − xsi (b) si = kαi (b, αi ), and those for the simple generators si of the Weyl group.

(1.5.16) (1.5.17) ❑

Proposition 1.5.4 looks like a formal manipulation with the definitions. However, it is very useful. Thanks to it, we are able to construct quite a few 0 representations of HΣ using various r–matrices. For instance, let us apply this machinery to establish the compatibility and W –invariance of AKZ. Let us double the set of variables by adding the pairwise commutative elements {va , a ∈ P ∨ }, provided we have that va+b = va + vb , a, b ∈ P ∨ . However, in contrast to the “old” set {ua }, they will not commute with W. We impose: wvα = vw(α) w (α ∈ P ∨ , w ∈ W ). Setting rαo = −(evα − 1)−1 sα ,

(1.5.18)

such a ro satisfies Proposition 1.5.3. The verification is simple. Here one can also use that this r–matrix is derived from the R–matrix Rα = t1/2 + (t1/2 − t−1/2 )(evα − 1)−1 (1 − sα ),

(1.5.19)

when h → 0 (t = 1 + h). Namely, ro is the coefficient of h up to a scalar term, that does not influence the commutators. It is easy to check that this R satisfies the quantum Yang–Baxter equations (0-3) from (1.5.7)–(1.5.10). The latter claim is due to Lusztig. There is a “conceptual” check of these relations (“without calculations”) based on the explicit formulas for the generators {T } in the polynomial representation of the affine Hecke algebra induced from the character {Ti 7→ t1/2 , 0 ≤ i ≤ n}.

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Proposition 1.5.6. The elements xob =

X

kα (b, α)rαo (b ∈ P ∨ )

(1.5.20)

α>0 0 satisfy the relations of the degenerate Hecke algebra HΣ . The representation 0 va ∨ o HΣ → End(C[e , a ∈ P ]) sending w 7→ w, xb 7→ xb is faithful. ❑

Now we will add {uα }: rˆα = rˆα (uα ) =

sα + rαo . −1

euα

(1.5.21)

It is an invariant r–matrix and, moreover, unitary: rˆα (uα ) + sα rˆα (−uα )sα = 0.

(1.5.22)

This r is nothing but the W –extension of the intertwining operators of the degenerate Hecke algebra, where the generators {T } are taken in the polynomial representation C[eva ]. Proposition 1.5.7. (i) Consider the following system of partial differential equations: X kα (b, α)ˆ rα Φ(u) (1.5.23) ∂b Φ(u) = α>0

for rˆα from (1.5.21), Φ(u) ∈ C[eva ], and for invariant {kα }. It is self-consistent and W –invariant. 0 (ii) The AKZ with values in HΣ : 

 X

∂Φ  sα kα ναi uα = + xi  Φ, ∂ui e − 1 α∈Σ

1 ≤ i ≤ n,

(1.5.24)

+

is self-consistent and W –invariant. Proof. Here the W –invariance of (1.5.23) follows from (1.5.22). Recall that the system ∂b Φ(u) = Ab Φ(u) (b ∈ P ∨ ) is said to be self-consistent if [∂b − Ab , ∂b0 − Ab0 ] = 0

(b, b0 ∈ P ∨ ).

(1.5.25)

The invariance means that if Φ(u) is a solution, so are the si Φ(si (u)) for all i. The second claim readily results from the first. Collecting all ro together and replacing them by {x}, we obtain a realization of AKZ in the C[eva ], which is a faithful module.

1.5. ABSTRACT KZ AND ELLIPTIC QMBP

1.5.3

111

Elliptic QMBP

Another application of the general theory of the classical r–matrices is a generalization of the elliptic quantum many-body problem from [OP] to arbitrary root systems. Olshanetsky and Perelomov introduced it for GLn . Ochiai, Oshima, and Sekiguchi in [OOS] generalized their construction to arbitrary classical root systems. More exactly, they considered the quantum Hamiltonian in the form H=

n X

∂ αi ∂ i +

i=1

X

V (uα )

(1.5.26)

α>0

and deduced from the existence of the higher conservation laws (differential operators commuting with H) that the potential V has to be the Weierstrass ℘–function or its degenerations. Arbitrary root systems were covered in [C17]. We will reproduce here the simplest variant of the construction from this paper. Let us first recall the construction of the Calogero–Sutherland operators. We keep the notation from the previous section. The root system Σ(⊂ Rn ) is arbitrary. We will use the same uα = (u, α) imposing the relation sα ub s−1 α = usα (b) , where sα ∈ W . So in this subsection {uα } they do not commute with W. Proposition 1.5.8. The trigonometric Dunkl operators X Db = ∂ b − kα (b, α)(euα − 1)−1 sα

(1.5.27)

α∈Σ+

are pairwise commutative for b ∈ P ∨ and satisfy the relations of the degenerate affine Hecke algebra: si Db − Dsi (b) si = ki (b, αi ).

(1.5.28)

Proof. Here k is invariant: kα = kβ if (∃w ∈ W s.t. w(α) = β). The statement is obvious, since rαo = −(euα − 1)−1 sα obey Proposition 1.5.3 and the relation (1.5.29) rαo + sα rαo sα = sα . ❑ Recall that we proceeded in Section 1.2.4 as follows. Let C[x1 , . . . , xn ]W be the algebra of W –invariant polynomials. We set Lp = p(Db1 , . . . , Dbn ),

(1.5.30)

where p ∈ C[x1 , . . . , xn ]W . Since wLp w−1 = Lp , the operators Lp = Lp |Sym (the restriction |Sym is to the symmetric functions) are pairwise commutative

112

CHAPTER 1. KZ AND QMBP

and W –invariant. If p2 = Lp2 =

Pn i=1

n X i=1

xαi xbi , then

∂αi ∂bi +

X kα (1 − kα )(α, α) . uα uα (e 2 − e− 2 )2 α∈Σ+

(1.5.31)

This operator is the Hamiltonian of the Sutherland model. We are going to “double” this construction, switching to the affine root system. Proposition 1.5.3 can be readily extended to affine root systems provided we have the convergence of the products of Di . The affine root system is the set Σa = {[α, j] ∈ Rn+1 | α ∈ Σ, j ∈ Z}, Σa+ = {[α, j]| α ∈ Σ, j ∈ Z>0 } ∪ {α = [α, 0]| α ∈ Σ+ }.

(1.5.32) (1.5.33)

f is generated by the reflections sα˜ (˜ The affine Weyl group W α = [α, j] ∈ Σa ) n+1 acting in R : (α, v) sα˜ ([v, ξ]) = [v, ξ] + 2 α ˜, (1.5.34) (α, α) where v ∈ Rn and ξ ∈ R. c is generated by W and the “transThe extended affine Weyl group W ∨ n lations” corresponding to b ∈ P = ⊕i=1 Zbi : b([v, ξ]) = [v, −(b, v) + ξ],

(1.5.35)

where v × ξ ∈ Rn+1 . On the space Rn+1 , W acts preserving ξ. The group c contains W f generated by W and Q∨ = ⊕n Zai for ai = α∨ = 2αi /(α, α). W i i=1 One then has: c = W n P ∨, W c =ΠnW f, W (1.5.36) for the group Π isomorphic to P ∨ /Q∨ . Concerning the abstract theory of affine r–matrices, one starts with an c W –invariant nonaffine r–matrix rα and assumes the following. The group W must act on the algebra R, containing {rα }, provided we have the relations c –invariant b(rα ) = rα whenever (b, α) = 0. This is sufficient to extend r to a W affine r–matrix: r[α,ξ] = b(rα ) for any b ∈ P ∨ such that (b, α) = −ξ. The relations from Proposition 1.5.3 and the W –invariance are necessary and sufficient to make this construction well defined. Note that it clarifies in full the additional relations, that were necessary in Proposition 1.5.2 to ensure the commutativity of {X}. Recall that the commutativity of r for orthogonal long roots in the case of B2 was added there, etc. They are exactly the relations which are necessary to ensure the

1.5. ABSTRACT KZ AND ELLIPTIC QMBP

113

existence of an affine extension. This connection is not by chance and results directly from the difference (quantum) theory. Infinite Dunkl operators. Once an affine r–matrix is given, we can introduce the operators X and x, that are the special case of the X–operators “with k” (in this section). They are now given by infinite sums. Formally they are pairwise commutative. However, the convergence of their products must be provided. We will manage the commutativity of the infinite Dunkl operators in this subsection and then will consider the W –symmetrizations. The analysis from [C17] shows that, generally, the symmetrizations are not pairwise commutative because infinite sums are involved. We will consider here only the setting when this problem does not occur. Technically, we use sα instead of the usual (sα − 1) in the definition of the Dunkl operators. The general claim is that the symmetrizations of the latter (“elliptic radial parts”) commute with each other only at the critical level; see [C17]. The infinite affine Dunkl operators will be differential operators with P n c c ˜ cw˜ ∈ C} the coefficients from Funct(C , C∞ W ), where C∞ W = { w∈ ˜ w, c cw ˜ W c. consists of infinite sums in contrast to the standard group algebra CW We fix a parameter η ∈ C, an arbitrary W –invariant function kα on Σ, and set α ˜ = [α, m] and euα˜ = euα +mη . Then for each b ∈ P ∨ , Db = ∂b −

X

kα (b, α)(euα˜ − 1)−1 sα˜ .

(1.5.37)

a α∈Σ ˜ +

Lemma 1.5.9. (i) If Reη > 0, then the products of the operators Dc are well defined and can be represented as infinite sums: def

Dc1 . . . Dck ==

X

Ψw˜ (u, η)w, ˜

(1.5.38)

c w∈ ˜ W

where Ψw˜ (u, η) are differential operators with the coefficients meromorphic in Cn 3 v. (ii) Moreover, the absolute values of the coefficients of Ψw˜ (u, η) are bounded ˜ pointwise (apart from the singularities) by a function C(u, η) ²(u, η)l(w) , where c 0 ≤ ²(u, η) < 1, C > 0, and l(w) ˜ is the length of w˜ in W with respect to the generators {si , 0 ≤ i ≤ n}. Proof. See [C17] for the proof and the exact definition of the length. The condition Reη > 0 is replaced by a somewhat weaker inequality in [C17], but the claim holds for all positive Reη. ❑

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CHAPTER 1. KZ AND QMBP

Theorem 1.5.10. If Reη > 0, then we have the following relations: [Db , Db0 ] = 0, si Db − Dsi (b) si = ki (b, αi ), s0 Db − Dsθ (b) s0 = −kθ (b, θ), c. where θ is the maximal root of Σ and s0 = s[−θ,1] ∈ W a ⊂ W

(1.5.39) (1.5.40) (1.5.41) ❑

For p ∈ C[x1 , . . . , xn ]W , we set Lp = p(Db1 , . . . , Dbn ).

(1.5.42)

def c ) and the operators Lp = Then wL b pw b−1 = Lp (w b∈W = Lp ||Sym are pairwise commutative, i.e., [Lp , Lp0 ] = 0 for all W –invariant p, p0 . c –invariant functions in contrast Here ||Sym is the affine restriction to the W to the nonaffine theory. Explicitly, if we have X Ψwb (u, η)w, b p(Db1 , . . . , Dbn ) = w b

where Ψwb (u, η) is a differential operator (and does not contain the elements c ), then we have from W X Ψwb (u, η). (1.5.43) Lp = w b

The convergence readily follows from the lemma. The coefficients of Lp c –equivariant, in particular, P ∨ –invariant. are W P For instance, if we put p2 = ni=1 xαi xbi , then Lp2 =

n X

∂αi ∂bi +

i=1

X

(α, α)kα (kα − 1)ζ˜0 (uα ),

(1.5.44)

α∈Σ+

where ˜ = ζ(z) Note that

∞ X

1

emη+z − 1 m=0

−

∞ X

˜ ˜0 (z) = dζ(z) . , ζ emη−z − 1 dt m=1

˜ + η) = ζ(z) ˜ + 1, ζ(z

1

˜ + ζ(−z) ˜ ζ(z) = −1, ½ ¾ X 1 1 1 0 −ζ˜ (z) + c = ℘(z; Ω) = 2 + − z (z − ω)2 ω 2 ω∈Ω\{0} √ for some constant c. Here Ω = {2π −1Z + ηZ} is a lattice in C.

(1.5.45)

(1.5.46) (1.5.47)

Comment. (i) The above construction gets more interesting when we consider the Dunkl operators with (sα˜ − 1) instead of sα˜ . Still they are

1.5. ABSTRACT KZ AND ELLIPTIC QMBP

115

pairwise commutative. However, relation (1.5.41) is more complicated and the reduction to the L–operators is governed by the double affine Hecke algebra with zero central charge. The resulting operators preserve the Looijenga space [Lo] of (formal) theta functions of “critical” level c = −kg for the dual Coxeter number g (assuming that we have one k). See [C17]. (ii) When k = 1 (i.e., in the group case), we come to the relation c + g = 0, which coincides with the the critical level condition for the Kac–Moody algebras (c is the central charge). I did not check the detail, but it is likely that there may be an explanation based on the duality from [VV1] in the case of GLn . The appearance of the Kac–Moody critical level condition as the commutativity condition for the elliptic L–operators has no explanation for other root systems at the moment. However, it obviously indicates that double Hecke algebras are on the right track.

1.5.4

Double affine KZ

Continuing the main theme of this chapter, let us establish the relations between the elliptic QMBP and the corresponding KZ equations, to be introduce. One can expect this KZ to be of elliptic type, but the most natural construction leads to a trigonometric KZ with infinite sums. The commutativity of the parameters uα with the extended affine Weyl c and HH00 , the degenerate double affine Hecke algebra of zero level, group W will be assumed (contrary to the previous subsection). The {ui } will be the arguments of the double affine KZ. We follow [C17]. Definition 1.5.11. The 0–level degenerate double affine Hecke algebra f HH00 is generated by pairwise commutative elements {xb } and the group W satisfying the “cross-relations” si xb − xsi (b) si = ki (b, αi ), s0 xb − xsθ (b) s0 = −kθ (b, θ),

(1.5.48) (1.5.49)

where s0 = s[−θ,1] ∈ W a . Theorem 1.5.12. The double affine KZ (DAKZ) ´ ³X kα (euα˜ − 1)−1 sα˜ + xb Φ ∂b (Φ) =

(1.5.50)

a α∈Σ ˜ +

c –invariant and self-consistent. Here α is W ˜ = [α, j], kα is an arbitrary W – ❑ invariant function on Σ, and euα˜ = euα +jη for a fixed η. Let us consider (1.5.50) in the representation of HH00 induced from the f . It is isomorphic to C[x1 , . . . , xn ]. We can reduce character {si 7→ 1} of W

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it further, since the symmetric polynomials p(x1 , . . . , xn ) (∈ C[x1 , . . . , xn ]W ) are central in HH00 . We fix λ = (λ1 , . . . , λn ) ∈ Cn and define AKZλ as the AKZ equation considered in the representation Jλo dual to Jλ = C[x1 , . . . , xn ]/((p(x)−p(λ)) ∀p). Here the p are symmetric. By dual, we mean Hom(J, C) with the action of HH00 via the anti-involution preserving the generators si , xb . It is similar to the considerations of Section 1.2.4. Because of the definition of Jλo , there is a natf –invariant map tr : J o → C dual to the embedding C → C[x1 , . . . , xn ]. ural W λ Theorem 1.5.13. The map Φ 7→ φ = tr(Φ) is an isomorphism of the space of all (local) solutions of DAKZλ and the space of solutions of the eigenvalue problem Lp φ = p(λ)φ for p ∈ C[x1 , . . . , xn ]W . The proof follows the same lines as in the nonaffine case. We note that the double affine KZ can be defined for the central extension HH0 (nonzero level) of HH00 as well as the infinite Dunkl operators. The algebra HH0 is much c –invariant L–operators can be constructed more interesting. However, the W 0 only for HH0 , which eventually leads to the relation generalizing the critical level condition in the Kac–Moody theory (see Chapter 0). We will consider the an interpretation and integration of (trigonometric) KZ equations via the Kac–Moody algebras in the next sections. Now we discuss the application to the Harish-Chandra transform.

1.6

Harish-Chandra inversion

In this section, we use the degenerate double Hecke algebras and Dunkl operators to calculate the images of the operators of multiplication by symmetric Laurent polynomials with respect to the Harish-Chandra transform. These formulas were known only in some cases, although they are directly connected with the important problem of making the convolution on the symmetric spaces as explicit as possible. We first solve a more general problem of calculating the transforms of the coordinates for the nonsymmetric HarishChandra transform defined by Opdam; this problem has a complete solution. Then we employ the symmetrization. The resulting formulas readily lead to a new simple proof of the Harish-Chandra inversion theorem (see [HC, He3]) and the corresponding theorem from [O3]. We will assume that k > 0 and restrict ourselves to compactly supported functions. In this case, we can borrow the growth estimates from [O3]. Hopefully this approach can be extended to any k and to the elliptic radial parts, which were discussed in the previous section. It is also expected to be helpful for developing the analytic theory of the direct and inverse transforms. For a maximal real split torus A of a semisimple Lie group G, the HarishChandra transform is the integration of symmetric (W –invariant) functions

1.6. HARISH-CHANDRA INVERSION

117

in terms of X ∈ A multiplied by the spherical zonal function φ(X, λ), where λ ∈ (LieA ⊗R C)∗ . The measure is the restriction of the invariant measure on G to the space of double cosets K\G/K ⊂ A/W for the maximal compact subgroup K ⊂ G and the restricted Weyl group W . The function φ, the kernel of the transform, is a W –invariant eigenfunction of the radial parts of the G–invariant differential operators on G/K; λ determines the set of eigenvalues. The parameter k is given by the root multiplicities (k = 1 in the group case). There is a generalization to arbitrary k due to Calogero, Sutherland, Koornwinder, Moser, Olshanetsky, Perelomov, Heckman, and Opdam; see the papers [HO1, H1, O1] for a systematic (“trigonometric”) theory. In the nonsymmetric variant due to Opdam [O3], the Dunkl-type operators from [C10] replace the radial parts of G–invariant operators and their k–generalizations. The analytic part of the theory is in extending the direct and inverse transforms to various classes of functions. Not much is known here. Hopefully the difference theory will be more promising analytically. In the papers [C28, C21], a difference counterpart of the Harish-Chandra transform was introduced, which is also a deformation of the Fourier transform in the p–adic theory of spherical functions. Its kernel (a q–generalization of φ) is defined as an eigenfunction of the q–difference “radial parts” (Macdonald’s “minuscule-weight” operators and their generalizations). There are “algebraic” applications in combinatorics (the Macdonald polynomials), representation theory (for instance, for quantum groups at roots of unity), and mathematical physics (see Chapter 0 and the previous sections). This section is toward analytic applications. The q–Fourier transform is self-dual, i.e., its kernel is x ↔ λ symmetric for X = q x . This holds in the differential theory only for either the so-called rational degeneration with the tangent space Te (G/K) instead of G/K (see [He3, Du2, Je]) or for a special k = 0, 1. For such a special k, the classical differential and the new difference transforms coincide up to a normalization. The differential case corresponds to the limit q → 1. At the moment, the analytic methods are not mature enough to manage the limiting procedure in detail, which would give a complete solution of the inversion problem. So we develop the corresponding technique without any reference to the q–Fourier transform and the general double affine Hecke algebra. We use only the theory of the degenerate one, which generalizes Lusztig’s graded affine Hecke algebra [Lus1] (the GLn –case is due to Drinfeld). Mainly we need the intertwining operators from [C23] (see [C13] and [KnS] for GLn ). The key result of the section is that the Opdam transforms of the operators of multiplication by the coordinates coincide with the operators from [C12] (see (1.6.17) below). Respectively, the Harish-Chandra transform sends the operators of multiplications by W –invariant Laurent polynomials to the

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difference operators from (1.6.20). It is not very surprising that the images of these important operators haven’t been found before. The calculation involves the following ingredients, which are new in the harmonic analysis on symmetric spaces: (a) the differential Dunkl-type operators from [C10, C11]; (b) the Opdam transform [O3]; (c) the difference Dunkl operators [C12, C13].

1.6.1

Affine Weyl groups

Recall the definition of the affine roots and the extended affine Weyl group. Let Σ = {α} ⊂ Rn be the root system of type A, B, ..., F, G, normalized by the standard condition (α, α) = 2 for long α; α1 , ..., αn simple roots; a1 = α1∨ , ..., an = αn∨ simple coroots, where α∨ = 2α/(α, α); b1 , ..., bn the dual fundamental weights determined by the relations (bi , αj ) = δij for the Kronecker delta. We will also use Q∨ = ⊕ni=1 Zai ⊂ P ∨ = ⊕ni=1 Zbi , P+∨ = ⊕ni=1 Z+ bi for Z+ = {m ≥ 0}. The vectors α ˜ = [α, j] ∈ Rn × R = Rn+1 for α ∈ Σ, j ∈ Z form the affine def root system Σa ⊃ Σ (z ∈ Rn are identified with [z, 0]). We add α0 == [−θ, 1] to the simple roots for the maximal root θ ∈ Σ. The corresponding set Σa+ of positive roots coincides with Σ+ ∪ {[α, j], α ∈ Σ, j > 0}. The set of indices of the orbit of the zero vertex in the affine Dynkin diagram by its automorphisms will be denoted by O (O = {0} for E8 , F4 , G2 ). Let O0 = {r ∈ O, r 6= 0}. We identify the indices with the corresponding simple affine roots. The elements br for r ∈ O0 are the so-called minuscule coweights, namely, (br , α) ≤ 1 for α ∈ Σ+ . Given α ˜ = [α, j] ∈ Σa , b ∈ P ∨ , let sα˜ (˜ z ) = z˜ − (z, α∨ )˜ α, b0 (˜ z ) = [z, ζ − (z, b)]

(1.6.1)

for z˜ = [z, ζ] ∈ Rn+1 . f is generated by all sα˜ (simple reflections si = The affine Weyl group W sαi for 0 ≤ i ≤ n are enough). It is the semidirect product W nQ∨ , where the nonaffine Weyl group W is the span of sα , α ∈ Σ+ . We will identify b ∈ P ∨ with the corresponding translations. For instance, α∨ = sα s[α,1] = s[−α,1] sα for α ∈ R.

(1.6.2)

c generated by W and P ∨ is isomorphic The extended Weyl group W ∨ to W nP : (wb)([z, ζ]) = [w(z), ζ − (z, b)] for w ∈ W, b ∈ P ∨ .

1.6. HARISH-CHANDRA INVERSION

119

For b+ ∈ P+∨ , let c , u i = ub , π i = π b , ub+ = w0 w0+ ∈ W, πb+ = b+ (ub+ )−1 ∈ W i i

(1.6.3)

where w0 (respectively, w0+ ) is the longest element in W (respectively, in Wb+ generated by si preserving b+ ) relative to the set of generators {si } for i > 0. def The elements πr == πbr , r ∈ O0 , and π0 = id leave {αi , i ≥ 0} invariant and form a group denoted by Π, which is isomorphic to P ∨ /Q∨ by the natural projection br 7→ πr . As with {ur }, they preserve the set {−θ, αi , i > 0}. The relations πr (α0 ) = αr = (ur )−1 (−θ) distinguish the indices r ∈ O0 . Moreover (see, e.g., [C12]): c = ΠnW f , where πr si π −1 = sj if πr (αi ) = αj , 0 ≤ i, j ≤ n. W r c is the length of any of the reduced decomThe length l(w) b of w b = πr w e∈W f with respect to {si , 0 ≤ i ≤ n}: positions of w e∈W def

l(w) b = |λ(w)| b for λ(w) b == Σa+ ∩ w b−1 (−Σa+ ) = {˜ α ∈ Σa+ , l(ws b α˜ ) < l(w)}. b

(1.6.4)

For instance, given b+ ∈ P+∨ , λ(b+ ) = {˜ α = [α, j], α ∈ Σ+ and (b+ , α) > j ≥ 0}, X α. l(b+ ) = 2(b+ , ρ), where ρ = (1/2)

(1.6.5)

α∈Σ+

We will later need the dominant affine Weyl chamber Ca = {z ∈ Rn such that (z, αi ) > 0 for i > 0, (z, θ) < 1}.

1.6.2

(1.6.6)

Degenerate DAHA

Let us fix a collection k = {kα } ⊂ C such that kw(α) = kα for w ∈ W, i.e., kP α depends only on the (α, α). We set ki = kαi and k0 = kθ . Let ρk = (1/2) α∈Σ+ kα α. One then has: (ρk , αi∨ ) = ki for 1 ≤ i ≤ n. It will be convenient in this section to switch to the parameters def

def

κα == να kα for να == (α, α)/2; κi = kαi , κ0 = k0 . P For instance, ρk = (1/2) α∈Σ+ κα α∨ and (ρk , αi ) = κi for 1 ≤ i ≤ n. Also, notice that we use the generators {y} in the definition of the degenerate affine Hecke algebra that satisfy the “cross-relation” with the opposite sign, in contrast to the {x} used above.

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Definition 1.6.1. The degenerate double affine Hecke algebra HH0 is c and the pairwise commutative yb for b ∈ P ∨ formed by the group algebra CW or in Rn satisfying the following relations: si yb − ysi (b) si = −κi (b, αi∨ ), πr y˜b = yπr (˜b) πr , def

for (b, α0 ) = −(b, θ), y[b,u] == yb + u, 0 ≤ i ≤ n, r ∈ O.

(1.6.7) ❑

Without s0 and πr , these are the defining relations of the graded affine Hecke algebra from [Lus1]. This algebra is a degeneration as q → 1, t = q κ of the general double affine Hecke algebra, which will be studied in the following chapters. Note that relations (1.6.7) are dual to those in terms of xb in this chapter. See, e.g., Definition 1.5.11. The trigonometric Dunkl operators will be changed correspondingly. We will need the intertwiners of HH0 : κi κ0 Ψ0i = si + , Ψ00 = Xθ sθ + , yαi 1 − yθ 0 Pr0 = Xr u−1 (1.6.8) r , for 1 ≤ i ≤ n, r ∈ O . They belong to the extension of HH0 by the field C(y) of rational functions in {yb }. The elements {Ψ0i , Pr0 } satisfy the homogeneous Coxeter relations for {si , πr }. So the elements c, Ψ0wb = Pr0 Ψ0il · · · Ψ0i1 , where w b = πr sil · · · si1 ∈ W

(1.6.9)

are well defined for reduced decompositions of w, b and Ψ0wb Ψ0uˆ = Ψ0wˆ bu whenever l(wˆ bu) = l(w) b + l(ˆ u). The following key property of {Ψ0 } fixes them uniquely up to left or right multiplications by functions of y: 0 c, Ψ0wb yb = yw(b) b∈W b Ψw b, w

(1.6.10)

def

where y[b,j] == yb + j, for instance, yα0 = 1 − yθ . The operator P10 in the case of GLn (it becomes of infinite order) plays the key role in [KnS]. The formulas for nonaffine Ψ0i when 1 ≤ i ≤ n are well known in the theory of degenerate (graded) affine Hecke algebras. See [Lus1, C10] and [O3], Definition 8.2. However, the applications to Jack polynomials and the Harish-Chandra theory do require the affine intertwiners Φ00 and Pr0 .

1.6.3

Differential representation

Introducing the variables Xb = exb for b ∈ P ∨ , we extend the following formulas to the derivations of C[X] = C[Xb ]: ∂a (Xb ) = (a, b)Xb , a, b ∈ P ∨ .

1.6. HARISH-CHANDRA INVERSION

121

Note that w(∂b ) = ∂w(b) , w ∈ W . The key measure-function in the Harish-Chandra theory of spherical functions and its k–generalizations is as follows: Y def τ == τ (x; k) = |2 sinh(xα /2)|2κα . (1.6.11) α∈Σ+

Proposition 1.6.2. (a) The following trigonometric Dunkl operators acting on the Laurent polynomials f ∈ C[X] = C[Xb ], def

Db ==∂b +

X κα (b, α∨ ) ¡ ¢ 1 − sα − (ρk , b) −1 (1 − Xα ) α∈Σ

(1.6.12)

+

are pairwise commutative, and y[b,u] = Db + u satisfy (1.6.7) for the following c: action of the group W wx (f ) = w(f ) for w ∈ W, bx (f ) = Xb f for b ∈ P ∨ . For instance, sx0 (f ) = Xθ sθ (f ), πrx (f ) = Xr u−1 r (f ). (b) The operators Db are formally self-adjoint with respect to the inner product Z def (1.6.13) {f, g}τ == f (x)g(−x)τ dx, i.e., τ −1 Db+ τ = Db for the anti-involution c. −∂b , where w b∈W

+

sending w bx 7→ (w bx )−1 and ∂b 7→ ❑

We skip the proof. Concerning the algebraic properties of the Dunkl operators, here and further see the previous sections or [C23, C11]. Comment. (i) We mention that Opdam uses a somewhat different pairing and a different involution in [O3]. We will discuss a q, t-generalization of the involution he considered, “with w0 ”, in Chapter 2 and, in full generality, in Chapter 3. (ii) Note that the trigonometric Dunkl operators used in the previous sections are somewhat different; see, e.g., Proposition 1.5.8. Namely, the term (1−sα ) replaces sα , respectively, the ρ–term appears to ensure the invariance. This renormalization is the most convenient in the Harish-Chandra theory.

1.6.4

Difference-rational case

We introduce the variables λa+b = λa + λb , λ[b,u] = λb + u, b ∈ P ∨ , u ∈ C,

(1.6.14)

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and define the rational Demazure–Lusztig operators from [C12] as follows: Si = sλi +

κi λ (s − 1), 0 ≤ i ≤ n, λαi i

(1.6.15)

where by w bλ we mean the action on {λb }: λ ∨ w bλ (λb ) = λw(b) b , b (λc ) = λc − (b, c) for b, c ∈ P .

For instance, S0 = sλ0 + We set

κ0 (sλ 1−λθ 0

− 1).

def

(1.6.16)

def

(1.6.17)

b = πr si1 . . . sil , Swb == πrλ Si1 . . . Sil for w ∆b == Sb for b ∈ P ∨ , ∆i = ∆bi .

The definition does not depend on the particular choice of the decomposition c , and the map w of w b∈W b 7→ Swb is a homomorphism. The operators ∆b are pairwise commutative. They are called difference Dunkl operators. The counterpart of τ is the asymmetric Harish–Chandra c¯ c–function: def

σ == σ(λ; k) =

Y Γ(λα + κa )Γ(−λα + κα + 1) , Γ(λα )Γ(−λα + 1) a∈Σ

(1.6.18)

+

where Γ is the classical Γ–function. Proposition 1.6.3. (a) The operators Swb are well defined and preserve the space of polynomials C[λ] in terms of λb . They form the degenerate double Hecke algebra, namely, the map w b 7→ Swb , yb 7→ λb is a faithful representation 0 of HH . (b) The operators Swb are formally unitary with respect to the inner product Z def (1.6.19) {f, g}σ == f (λ)g(λ)σdλ, i.e., σ −1 Sw+b σ = Sw−1 b for the anti-involution c where w b ∈ W.

+

sending λb 7→ λb and w b 7→ w b−1 , ❑

The operators Λp = p(∆1 , . . . , ∆n ) for p ∈ C[X1±1 , . . . , Xn±1 ]W

(1.6.20)

are W –invariant and preserve C[λ]W (usual symmetric polynomials in λ). So if we restrict them to W –invariant functions, we will get W –invariant difference operators. However, the formulas for the restrictions of Λp are simple enough in special cases only.

1.6. HARISH-CHANDRA INVERSION

123

P Proposition 1.6.4. Given r ∈ O0 , let mr = w∈W/Wr Xw(−br ) , where Wr is the stabilizer of br in W . Then λ(br ) belongs to Σ+ and the restriction of Λmr (∆b ) onto C[λ]W is as follows: Λr =

X

Y λw(α) + κα w(−br ), λw(α)

(1.6.21)

w∈W/Wr α∈λ(br )

where w(−br ) = −w(br ) ∈ P ∨ is considered as a difference operator: w(−br )(λc ) = λc + (w(br ), c). ❑ The formula for Λr is very close to the corresponding formulas for the general double Hecke algebra from [C16]. Note that in the differential case, the formulas for the invariant operators are much more complicated than in the difference theory. It was (and still is) a convincing argument in favor of the difference theory.

1.6.5

Opdam transform

From now on, we assume that R 3 κα > 0 for all α and keep the notation from the previous section; i is the imaginary unit; 0, g(λ) ≤ C(1 + |λ|)−N exp(A| 0 and σ has no singularities in U+a + iRn . Then we use the classical formulas for |Γ(x + iy)/Γ(x)| for real x, y, x > 0. It gives (cf. [He3, O3]): X |σ(λ)| ≤ C(1 + |λ|)K , where K = 2 κα , λ ∈ U+a + iRn , (1.6.29) α>0

for sufficiently large C > 0. Thus the products of PW-functions by σ tend to 0 for |λ| 7→ ∞, and we can switch to ξ. Actually, we can do this for any integrand analytic in U+a + iRn and approaching 0 at ∞. We come to the following. Proposition 1.6.7. The conditions of part (b), Proposition 1.6.3 are satisfied for Z def 0 g(λ)g 0 (λ)σdλ (1.6.30) {g, g }σ == iRn

in the class of PW-functions, i.e., the operators Swb are well defined on such functions and unitary. Proof. It is sufficient to check the unitarity for the generators Si = Ssi (0 ≤ i ≤ n) and πrλ (r ∈ O0 ). For instance, let us consider s0 . We will integrate over def ξ + iR, assuming that ξ 0 == sθ (ξ) + θ ∈ U+a and avoiding the wall (θ, ξ) = 1. We will apply formula (1.6.19), the formula Z Z Z s0 (g(λ)σ)dλ = g(λ)σdλ = g(λ)σdλ, ξ+iRn

ξ 0 +iRn

ξ+iRn

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and a similar formula for (1−λθ )−1 gσ, where g is of the PW-type in U+a +iRn . Note that (1 − λθ )−1 σ is regular in this domain. One then has: Z S0 (g(λ)) g 0 (λ)σdλ n Zξ+iR ¡ ¢ s0 + κ0 (1 − λθ )−1 (s0 − 1) (g(λ)) g 0 (λ)σdλ = n Zξ+iR ¡ ¢ = g(λ) s0 + κ0 (s0 − 1)(1 − λθ )−1 (g 0 (λ)σ)dλ n Zξ+iR = g(λ) S0−1 (g 0 (λ))σdλ. (1.6.31) ξ+iRn

Since (1.6.31) holds for one ξ, it is valid for all of them in U+a , including 0. The consideration of the other generators is the same. ❑ n n c, b ∈ P ∨ , f (x) ∈ C∞ Theorem 1.6.8. Given w b∈W c (R ), g(λ) ∈ P W (C ),

w bx (G(x, λ)) = Sw−1 b G(x, λ), e.g., Xb G = x F(w b (f (x))) = Swb F(f (x)), G(Swb (g(λ))) F(Xb f (x)) = ∆b (F(f (x))), G(∆b (g(λ))) F(Db (f (x))) = λb F(f (x)), G(λb g(λ)) =

∆−1 b (G), = w bx (F(f (x))), = Xb G(g(λ)), Db (G(g(λ))).

(1.6.32) (1.6.33) (1.6.34) (1.6.35)

Proof. Applying the intertwiners from (1.6.8) to G(x, λ), we see that κi −1 κi ) (si + )(G) = sλi (G), for 1 ≤ i ≤ n, λαi λαi κ0 −1 κ0 (1 + ) (Xθ sθ + )(G) = sλ0 (G), 1 − λθ 1 − λθ λ −1 0 Pr0 (G) = Xr u−1 r (G) = (πr ) (G) for r ∈ O . (1 +

(1.6.36)

Due to the main property of the intertwiners (1.6.10), we obtain these equalities up to λ–multipliers. The scalar factors on the left are necessary to preserve the normalization G(0, λ) = 1, so we can use the uniqueness of G(x, λ) from Theorem 1.6.5. Expressing sxj in terms of sλj (when applied to G!), we obtain (1.6.32) for w b = sj . It is obvious for w b = πr . Using the x c , we establish this relation in the commutativity of w b and Suˆ for w, b uˆ ∈ W general case. For w ∈ W it is due to Opdam. Formula (1.6.33) results directly from (1.6.32) because we already know that the w bx are unitary for {f, f 0 }τ and the Swb are unitary with respect to {g, g 0 }σ for the considered classes of functions. See Theorem 1.6.2, Theorem 1.6.3, and formulas (1.6.25) and (1.6.30).

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127

For instance, let us check (1.6.34) for F, which is a particular case of (1.6.33): Z F(Xb f (x)) = f (−x)Xb−1 G(x, λ)τ dx Rn Z = f (−x)∆b (G(x, λ))τ dx = ∆b (F(f (x))). (1.6.37) Rn

Thus the Opdam transforms of the operators of multiplication by Xb are the operators ∆b . Since Db and λb are self-adjoint for the corresponding inner products (see Theorem 1.6.2), we obtain (1.6.35), which is in fact the defining property of F and G. ❑ Corollary 1.6.9. The compositions GF : Cc∞ (Rn ) → Cc∞ (Rn ), and FG : P W (Cn ) → P W (Cn ) are multiplications by nonzero constants. The transforms F, G establish isomorphisms between the corresponding space identifying { , }τ and { , }σ up to proportionality. Proof. The first statement readily follows from Theorem 1.6.8. The comn position GF sends C∞ c (R ) into itself, is continuous (due to Opdam), and commutes with the operators Xb (multiplications by Xb ). Thus it is a multiplication by a function u(x) of C∞ –type. It must also commute with Db . Hence G(x, λ)u(x) is another solution of the eigenvalue problem (1.6.22), and we conclude that u(x) has to be a constant. Let us check that FG, which is a continuous operator on P W (Cn ) (for any fixed A) commuting with multiplications by any λb , has to be a multiplication by an analytic function v(λ). Indeed, the image of FG with respect to the standard Fourier transform (k = 0) is a continuous operator on Cc∞ (Rn ) commuting with the derivatives ∂/∂xi . Thus it is a convolution with some function, and its inverse Fourier transform is the multiplication by a certain v(λ). Since FG(g) = Const g for any g(λ) from F(Cc∞ (Rn )), v is constant. The claim about the inner products is obvious because {F (f ), g}σ = ❑ {f, G(g)}τ for f ∈ Cc∞ (Rn ), g ∈ P W (Cn ). The corollary is due to Opdam ([O3],Theorem 9.13 (1)). His proof is different. He uses Peetre’s characterization of differential operators (similar to what Van Den Ban and Schlichtkrull did in [BS]). In his approach, a certain nontrivial analytic argument is needed to check that GF is multiplication by a constant. The proof above is analytically elementary. The symmetric case. The transforms can be readily reduced to the symmetric level. If f is W –invariant, then Z F(f (x)) = f (x)F (−x, λ)τ dx (1.6.38) Rn

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for F from (1.6.23). Here we have applied the W –symmetrization to the integrand in (1.6.24) and used that τ is W –invariant. So F coincides with the k–deformation of the Harish-Chandra transform on W –invariant functions up to a minor technical detail. The W –invariance of F in λ results in the W –invariance of F(f ). As for G, we W –symmetrize the integrand in the definition with respect to x and λ. Namely, we replace G by F and σ by its W –symmetrization. The latter is the genuine Harish-Chandra “c¯ c− function” σ0 =

Y Γ(λα + κa )Γ(−λα + κα ) Γ(λα )Γ(−λα ) a∈Σ

(1.6.39)

+

up to a coefficient of proportionality. n Finally, given a W –invariant function f ∈ C∞ c (R ), Z g(λ)F (x, λ)σ 0 dλ for g = F(f ).

f (x) = Const

(1.6.40)

iRn

A similar formula holds for G. See [HC, He1] and [GV](Ch.6) for the classical theory. As an application, we are able to calculate the Fourier transforms of the operators p(X) ∈ C[Xb ]W (symmetric Laurent polynomials acting by multiplication) in the Harish-Chandra theory and its k–deformation. They are exactly the operators Λp above. In the minuscule case, we obtain formulas (1.6.21). We mention that in [O3] and other papers the pairings serving the Fourier transforms are hermitian. Complex conjugations can be added to ours. I hope that the method we used can be generalized to negative k, to other classes of functions, and to the p–adic theory (cf. [BS, HO2]). The relations (1.6.32) considered as difference equations for G(x, λ) with respect to λ may also help with the growth estimates via the theory of difference equations and the equivalence with the difference Knizhnik–Zamolodchikov equations. However, I believe that the main advantage here will be connected with the q–Fourier transform.

1.7

Factorization and r–matrices

We will define the “factorizable” Kac–Moody algebras following [C1] and introduce at the end of the section the corresponding τ –function as an infinite wedge product. It will be generalized in the next section using the notion of the coinvariant. The r–matrix KZ of trigonometric type will be defined, to be interpreted as the equation for the coinvariant in the next section.

1.7. FACTORIZATION AND R–MATRICES

1.7.1

129

Basic trigonometric r–matrix n

z }| { N N = C ⊗ ··· ⊗ C

Let us start with the simplest example. We set V = C⊗n N and ³x´ 1 1 X r(x) = coth P+ (elm ⊗ eml − eml ⊗ elm ) + D. 2 2 2 1≤l<m≤N

(1.7.1)

P Here coth(x) = (ex +e−x )/(ex −e−x ), P = 1≤l,m≤N elm ⊗eml for the standard N generators {eab P} of End(C ) with the entries δli δmj , and D is anyNdiagonal matrix (D = clm ell ⊗ emm ). So the values of r belong to End(C ⊗ CN ). Note that P is the permutation matrix: P (v1 ⊗ v2 ) = v2 ⊗ v1 . We put rij = r(ui − uj )(i,j) , where C (i,j) =

X

i

j

1 ⊗ · · · ⊗ a ⊗ · · · ⊗ b ⊗ . . . 1 if C =

X

a⊗b

(1.7.2)

for any matrices {a, b}. Note that r21 = (r(u2 − u1 ))(2,1) . The following relations can be verified directly: [rij , rik + rjk ] − [rik , rkj ] = 0, [rij , rkl ] = 0, rij + rji = D(i,j) + D(j,i)

(1.7.3) (1.7.4)

for pairwise distinct indices i, j, k, l (rkj is not a misprint!). When D(i,j) + D(j,i) = 0 we obtain the relations [rij , rik + rjk ] + [rik , rjk ] = 0,

(1.7.5)

which are nothing but the assumptions of Proposition 1.5.3 for the root system An−1 (the pairs {ij} can be identified with the roots). This r–matrix is invariant and unitary. So we can introduce the corresponding KZ equation. Identifying P (i,j) with the transpositions sij , we can represent it in the form (1.1.23) from Section 1.1.2. Note that in this section we use u instead of v. The formulas for the operators xi are straightforward. They involve D; so we arrive at a representation of the degenerate affine Hecke algebra Hn0 of type GLn in V . Any weight subspaces of V with respect to the standard action of glN are Hn0 –submodules. In this example, it is not necessary to impose the relation D(i,j) +D(j,i) = 0. One can define KZ for any D. Arbitrary g. Let us generalize this example to an arbitrary simple finite dimensional Lie algebra g of rank= l over C. Let Σ = {α} ⊂ Rl be the root system associated with g and ( , ) the W –invariant inner product on Rl normalized by the condition (θ, θ) = 2 for the maximal root θ with respect to the simple roots α1 , . . . , αl .

130

CHAPTER 1. KZ AND QMBP We choose nonzero eα ∈ gα , fα ∈ g−α , α ∈ Σ+ ,

(1.7.6)

provided we have the relations [hα , eβ ] = (α, β)eβ , [hα , fβ ] = −(α, β)fβ , (α, α) [eα , fβ ]. for hα = 2

(1.7.7) (1.7.8)

The elements {eα }, {fα } are linearly independent and generate the Borel subalgebras b± , respectively. The elementsPhm = hαm (m = 1, . . . , l) form a basis in the Cartan subalgebra h; hα = lm=1 (α, bm )hm for the fundamental coweights bm ((bm , αn ) = δmn ). Let us connect the above form on Rl with the the standard invariant form (f, f 0 ) on f ∈ g 3 f 0 . By the invariant form, we mean that ([f, g], h) = (f, [g, h]). All invariant forms are proportional to each other. The standard normalization is as follows: (hα , hβ ) = (α, β), 2 δαβ , (eα , fβ ) = (α, α) (eα , eβ ) = 0 = (fα , fβ ).

(1.7.9) (1.7.10) (1.7.11)

See [Ka]. In terms of this form, the definition of {hα } does not depend on the particular choice of eα , fα in the corresponding weight spaces: hα =

[eα , fα ] . (eα , fα )

(1.7.12) def

Let us connect the standard form with the Killing form (f, f 0 )K == Tr(adf adf 0 ) for f, f 0 ∈ g: (f, f 0 )K = (2g)(f, f 0 ), g = 1 + (ρ, θ) (the dual Coxeter number), P where ρ = 12 α∈Σ+ α. We set X Ω= Ia ⊗ Ia∗ ∈ g ⊗ g

(1.7.13)

(1.7.14)

a

for a basis {Ia (a = 1, . . . , dimg)} of g and its dual {Ia∗ }: (Ia , Ia∗0 ) = δa,a0 . The definition of Ω does not depend on the choice of the basis {Ia }. Using the canonical generators e, f, h, Ω=

l X m=1

hm ⊗ h∗m +

X α∈R+

1 (eα ⊗ fα + fα ⊗ eα ), (eα , fα )

(1.7.15)

1.7. FACTORIZATION AND R–MATRICES

131

where {h∗m = hbm } are dual to {hm }. Let A : h → h be a linear map. The basic trigonometric r–matrix is given by the formula: l ³x´ 1 1 X fα ⊗ eα − eα ⊗ fα 1 X r(x) = coth Ω+ + A(hi )⊗h∗i . (1.7.16) 2 2 2 (e , f ) 2 α α + m=1 α∈R

It takes values in g ⊗ g. We use the same notation as for vector tensor products: m

j

rij = r(ui − uj )(i,j) , (a ⊗ b)(i, j) = 1 ⊗ · · · ⊗ a ⊗ · · · ⊗ b ⊗ . . . 1 for U (g)⊗n ; U (g) is the universal enveloping algebra of g. Generalizing formula (1.7.1), we arrive at Proposition 1.7.1. [rij , rik + rjk ] − [rik , rkj ] = 0, [rij , rks ] = 0, rij + rji = Θij , Pl Θ = (1/2) m=1 (A(hm ) ⊗ h∗m + h∗m ⊗ A(hm )).

(1.7.17) (1.7.18)

Here the indices 1 ≤ i, j, k, s ≤ n are pairwise distinct.

1.7.2

Factorization and r–matrices

We will establish the one-to-one correspondence between the r–matrices and the Lie subalgebras complementary to the standard holomorphic subalgebra in a Kac–Moody algebra. In this subsection, we fix a simple Lie algebra g, an open neighborhood U0 (⊂ C) of 0, and the set of pairwise distinct points u1 , . . . , un ∈ U0 such that ui − uj ∈ U0 (1 ≤ i 6= j ≤ n). The following is a generalization of the classical Yang–Baxter equation (YBE). For unitary r, it is exactly YBE. Definition 1.7.2. Let r(x) be a holomorphic g ⊗ g–valued function of x ∈ U0 \ {0}. We call r(x) a quasi-unitary r–matrix if: (α) r(x) − Ωx is holomorphic at 0, (β) [r12 , r13 + r23 ] = [r12 , r32 ], (γ) r12 + r21 = Θ, d(Θ)/dx = 0. It is called unitary if r12 + r21 = 0. ❑

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Here rij = r(xi − xj )(i,j) for independent variables x1 , x2 , x3 , . . . , r = r12 , x = x1 − x2 . Note that r21 = (r(−x))(2,1) . The relations (α, β, γ) must be fulfilled whenever xi − xj ∈ U0 \ {0}. The element Ω is from (1.7.14). def From now on, we set xi == x − ui , X e gk xki | p ∈ Z, gk ∈ g}, (1.7.19) gi = g((xi )) = { k≥p

e gi0

X gk xki | gk ∈ g}, = g[[xi ]] = {

(1.7.20)

k≥0

e g=

n Y

e gi ,

e g0 =

n Y

e gi0 .

(1.7.21)

i=1

i=1

All these are Lie algebras: [f, g]eg = ([f 1 , g 1 ], . . . , [f n , g n ]) for f = (f 1 , . . . , f n ), g = (g 1 , . . . , g n ) ∈ e g,

(1.7.22)

where the components f i , g i of f, g are formal series in xi ; their commutators are understood coefficient-wise. The Lie algebras e g, e g0 are direct counterparts of the groups of all ad`eles and integral ad`eles in arithmetic. The central extension b g=e g ⊕ Cc (c is the center element) of e g is introduced as follows: µ ¶ df [f + ξc, g + ζc]bg = [f, g]eg + Res , g c, (1.7.23) dx µ ¶ X µ i ¶ n df df i Res ,g = Resxi , g dxi , (1.7.24) dx dxi i=1 P where Resxi ( k fk xki )dxi = f−1 . Definition 1.7.3. The Kac–Moody algebra b g is called factorized if it is endowed with a subspace, a factorizing subalgebra, e gr ⊂ e g such that (a) e gr ⊕ e g0 = e g, (b) e gr is a Lie subalgebra of e g, e b (c) gr is a Lie subalgebra of g. ❑ Given an r–matrix r(x) with the values in g ⊗ g, let us construct the corresponding factorizing subalgebra e gr . For f = (f 1 , . . . , f n ) ∈ e g, we define ˜ a function of x in a certain neighborhood of zero U0 ⊂ U0 : f (x) = Res(r(x − y), 1 ⊗ f )dy =

n X i=1

Resyi (r(xi − yi ), 1 ⊗ f i (yi ))dyi , (1.7.25)

1.7. FACTORIZATION AND R–MATRICES

133

where (a ⊗ b, 1 ⊗ c) = (b, c)a (a, b, c, ∈ g). Let fri = fri (xi ) be the expansion of f (x) at ui (xi = x − ui ). We set def

g = {f }, e g}. gr = {fr == (fr1 (x1 ), . . . , frn (xn ) | f ∈ e

(1.7.26)

Theorem 1.7.4. If r(x) is a quasi-unitary r–matrix, then g is a Lie algebra and e gr is a factorizing subalgebra of e g. Conversely, every factorizing subalgebra e gr that is invariant with respect to the differentiation d/dx is associated to a quasi-unitary r–matrix defined in some neighborhood U˜0 of 0. The corresponding r is unique: X µ Ia ¶ r(x − ui ) = (x) ⊗ Ia∗ , (1.7.27) x i a where i can be arbitrary; {Ia } is a basis of g. The expansion fr of f coincides with the first component of the factorization f = fr + f0 with respect to the decomposition e g=e gr ⊕ e g0 . The proof is based on the relation [fr , gr ] + [f, g]r = [fr , g]r + [f, gr ]r , f, g ∈ e g,

(1.7.28)

directly following from the condition (b) of Definition 1.7.3 by considering the principal parts of all four terms of the equality. The coincidence of the principal parts is sufficient because of the condition (a). Relation (1.7.28) results in the (equivalent) equality [f0 , g0 ] + [f, g]0 = [f0 , g]0 + [f, g0 ]0 . Expressing fr , gr in terms of r, we obtain the relation (β) from Definition 1.7.2. ❑ Comment. Here it is not necessary to assume that r depends on the difference of the arguments (see [C1, C3]). The interpretation of r as a projection was formalized by Manin (see [Dr2]) in the definition of the so-called Manin triple, and by Semenov- Tjan-Shanskii [Se]. They considered abstract (“constant”) r–matrices. In [C1], it was done for r depending on the parameter, which resulted in a fruitful direction, “the ad`elic approach to soliton theory”, with many examples. The paper [C1] was stimulated by [BD], where the unitary case and the corresponding classification of r–matrices were considered. Note that the definition of the classical r–matrix appeared for the first time in paper [KuS]. Let me also mention the book [ES], where one can find some related information and new developments, including the classification due to Belavin and Drinfeld, and a recent theorem of Etingof and Kazhdan about the Drinfeld conjecture concerning the lift of the classical r–matrices to their quantum counterparts. ❑ Arbitrary e gr satisfying condition (a) of Definition 1.7.3 can be obtained by the construction from (1.7.25) and (1.7.26) for a certain series r(x, y)

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defined on the square of a formal neighborhood of the set {ui }. If r(x, y) depends on the difference, i.e., r(x, y) = r(x − y), then condition (α) from Definition 1.7.2 is fulfilled. For such r, the relations (β) and (b) are equivalent. Moreover, imposing (α, β) (or (a,b)), the remaining relations (γ) and (c) are also equivalent. See [C3, C9]. Examples. The Yang r–matrix is given by the formula r(x) = Ωx for Ω defined in (1.7.14). It is unitary: r12 + r21 = 0. The corresponding e gr consists n 1 ¯ e of the sets (fr (x1 ), . . . , fr (xn )) ∈ g of expansions of f at {ui } from the Lie algebra g of rational g–valued functions on P1 with poles at {u1 . . . un } and the normalization condition f (∞) = 0. We then have µ ¶ g g (1.7.29) = xi x − ui for any i, g ∈ g, where (·) is defined in (1.7.25). Given a linear map A : h → h, let us describe the factorizing subalgebra associated with r from (1.7.16): l ³x´ 1 X fα ⊗ e α − e α ⊗ fα X 1 Ω+ r(x) = coth + A(hi ) ⊗ h∗i . (1.7.30) 2 2 2 (eα , fα ) + n=1 α∈R

First, µ ¶ exi /2 e−xi /2 eα = xi /2 f , = eα , α e − e−xi /2 xi exi /2 − e−xi /2 µ ¶ ³x ´ 1 h i h + A(h) (h ∈ h). = coth xi 2 2 µ

fα xi

¶

(1.7.31) (1.7.32)

gi g, where fri (xi ) ∈ e The Lie algebra e gr consists of (fr1 (x1 ), . . . , frn (xn )) ∈ e are the expansions of g–valued functions f¯(x) such that (1) they are rational in terms of v = ex on v ∈ P1 ; (2) they have poles at v1 = eu1 , . . . , vn = eun only; (3) f¯(v = 0) ∈ b+ , f¯(v = ∞) ∈ b− ; (4)(f¯(0) + f¯(∞))|h = A((f¯(0) − f¯(∞))|h ). Here b+ = heα , α ∈ Σ− i, b− = hfα , α ∈ Σ+ i, and h is the Cartan subalgebra. The simplest way to establish that r is a quasi-unitary solution of the classical YBE in the sense of Definition 1.7.2 is definitely by checking (1–4). Indeed, the functions satisfying these conditions obviously form a Lie algebra. So we arrive at (1.7.28), which readily results in (β). The r–matrix KZ. Given a quasi-unitary r, let X X %= ρa Ia ∈ U (g) for {Ω/x − r(x)}(x = 0) = ρa ⊗ Ia , (1.7.33) a

where U (g) is the enveloping algebra of g.

a

1.7. FACTORIZATION AND R–MATRICES

135

Theorem 1.7.5. Setting Ri = % i −

X

rji , 1 ≤ j ≤ n, for 1 ≤ i ≤ n,

(1.7.34)

j(6=i)

the following system of differential equations, the r–matrix KZ, for a U (g)– valued function Φ(u) is self-consistent: κ∂(Φ)/∂ui = Ri Φ, 1 ≤ i ≤ n.

(1.7.35)

for all κ = k −1 . The theorem can be checked by a straightforward calculation using the relation [r12 , %1 ] + [%2 , r21 ] = [r12 , r21 ]. (1.7.36) Recall that %i = %(i) is % considered in the i–th component of U (g)⊗n . ❑ The commutation relations are more transparent for the unitary r. The last relation becomes [r12 , %1 + %2 ] = 0 in this case. Considering the basic trigonometric r from (1.7.16), l 1X 1 X % = hρ − A(hm )h∗m , ρ = α. 2 m=1 2 α∈Σ

(1.7.37)

+

So % = (1/2)

1.7.3

P α>0

hα ∈ h when A = 0.

Two conjectures

If e g¡r satisfies conditions (a) and (b), then (c) is equivalent to the relation ¢ df Res dx , g = 0 for all f, g ∈ e gr . The suffix r stays for ”r–matrix” or “rational.” The latter does not mean that this subalgebra (a counterpart of the group of principal ad`eles in arithmetic) must be associated with an algebraic curve. However, in all known examples it is so. One can always find a Lie algebra of the rational g–valued functions on P1 or an elliptic curve (containing U0 as an open subset) such that e gr is formed by expansions of these functions at u1 , . . . , un . We conjecture that this is true for all r. More generally, let us suppose that the spaces e g/(e gr + e g0 ), e gr ∩ e g0

(1.7.38)

are finite dimensional. Such e gr is a Kac–Moody variant of a discrete subgroup of an ad`ele group in arithmetic with the quotient space of finite volume or compact.

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CHAPTER 1. KZ AND QMBP

Conjecture 1.7.6. There exists a complete algebraic curve C over C containing U0 such that for the Lie algebra e gC of the expansions at u1 , . . . , un of all g–valued rational functions on C with poles apart from this set both spaces (e gr + e gC )/e gr , (e gr + e gC )/e gC

(1.7.39)

are finite dimensional. Its counterpart in arithmetic was proved by Margulis. It holds when either the number of points u or the rank of the arithmetic group is larger than 1. It seems that we do not need this restriction in the Kac–Moody setup. A special unitary r–matrix case of the above general conjecture is proved in the paper [BD]. If r is not unitary and does not depend on the difference of the arguments, the conjecture does not look too difficult, but remains an open question. See [C1, C3]. I think that almost nothing is known about the general case (any curves). We note that the unitary r–matrices (r12 + r21 = 0) have the following interpretation in terms of e gr : {f ∈ e g | Res(f, e gr )dx = 0} = e gr . It looks similar to condition (c) from Definition 1.7.3, but of course does not coincide with it. In (c), f must be replaced by df /dx. An interesting problem concerns the independence of condition (c) in the definition of the factorizing subalgebra. Conjecture 1.7.7. The subspace e gr satisfying conditions (a) and (b) from Definition 1.7.3 is a Lie subalgebra of b g for a proper choice of the central extension. All central extensions are described by 2–cocycles on e g (see [Ka]). I expect that the conjecture is true even in the general setting of Conjecture 1.7.6. We will outline the proof in the r–matrix case in the following subsection.

1.7.4

Tau function

This subsection is based on the paper [C3] devoted to a generalization of the τ –function introduced by Date, Jimbo, Kashiwara, and Miwa [DJKM1] and a construction due to Kac, Peterson from [KP]. The definition of the r–matrix τ –function we give uses the fact that the group G˜0 =exp(e g0 ) has a natural structure of an infinite dimensional algebraic variety. Given f ∈ e g, let us define the corresponding vector field Df on this variety. Practically, we need a formula for Φ−1 Df (Φ) ∈ e g0 for the generic ˜ element Φ ∈ G0 .

1.7. FACTORIZATION AND R–MATRICES

137

The coefficients of Φ in the expansions with respect to x1 , . . . , xn are precisely the coordinates of G˜0 . Thus, equating both sides of the following relation, we obtain the complete list of Df –derivatives of these coefficients (= coordinates) and therefore determine uniquely the corresponding differentiation of Funct(G˜0 ): Φ−1 Df (Φ) = (Φf Φ−1 )0 , where e g 3 f = f0 + fr , f 0 ∈ e g0 , f r ∈ e gr .

(1.7.40)

The decomposition f = f0 + fr is the factorization with respect to e gr ; by Φf Φ−1 , we mean the adjoint action. Note that Φ−1 Dg (Φ) = g for g ∈ e g0 , so the corresponding Dg are lefte invariant fields on the group g0 . Q We will assume that r is quasi-unitary. Then % ∈ ni=1 g. Actually, % = 0 for the most interesting examples. See [C3]. A simple straightforward calculation based directly on the definition of the factorization (see (1.7.28)) gives that D[f,f 0 ] = [Df , Df 0 ] on Funct(G˜0 ).

(1.7.41)

We introduce the τ –function as the infinite wedge product τ = ∧g Dg of all vector fields Dg , where g runs over the natural basis {IAK = ((Ia1 xk11 ), . . . , (Ian xknn ))} of e g0 . Here {Ia } is a certain fixed basis of g, A = (a1 , . . . , an ), and K = (k1 , . . . , kn ), ki ≥ 0. Thus τ is defined as a section of the ∧top T for the tangent bundle T of G˜0 . def ˆ f (τ ) = Theorem 1.7.8. The commutators D = [Df , τ ] are well defined for f ∈e g and ¡ ¢ ˆ f (τ ) = Res(Φ−1 dΦ, f )K − Res(Φf Φ−1 , %)K dx τ. D (1.7.42)

Moreover, in the linear space Funct (G˜0 ) τ, µ ˆf, D ˆf0] = D ˆ [f,f 0 ] + (2g)Res [D for the dual Coxeter number g.

df 0 ,f dx

¶ (1.7.43) ❑

Thus the central extension of e g emerges naturally and automatically for top the adjoint action of {D} in ∧ T . Here the coefficient 2g appears because it is the ratio of the Killing form and the standard invariant form. From the viewpoint of the next section, τ is the coinvariant for the basic representation of b g with the central charge 2g. The basic representations are

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CHAPTER 1. KZ AND QMBP

irreducible quotients of the Weyl modules for the zero starting representations of g. Returning to a justification of (c) from Definition 1.7.3, this condition is not necessary in Theorem 1.7.8. Moreover, an arbitrary central extension of e g can be obtained for a proper choice of the basis in e g0 . We have arrived at the standard cocycle because our basis {IAK } was invariant with respect to the differentiation d/dx. So, given e gr , (c) is equivalent to the existence of a basis in e g0 that is “compatible” with the adjoint action of this Lie subalgebra. Then the commutators of the differentiations Df for the “rational” f will contain no central additions. I think such a basis always exists. As for the theory of KZ, such a construction would give a definition of the r–matrix KZ for the most general class of r–matrices, which are not unitary and are not supposed to depend on the differences.

1.8

Coinvariant, integral formulas

The main object of this section is the r–matrix coinvariant. We establish that it satisfies the r–matrix KZ when differentiated with respect to the Sugawara elements L−1 , and use this to justify the integral formulas for the basic trigonometric KZ. We also discuss KZB. The notation is from the previous section.

1.8.1

Coinvariant def

Given g–modules V1 , . . . , Vn , let V == V1 ⊗ · · · ⊗ Vn . It has a natural structure of a g–module and a e g0 –module: g(v1 ⊗ v2 ⊗ · · · ⊗ vn ) = g 1 (0)v1 ⊗ v2 ⊗ · · · ⊗ vn + v1 ⊗ g 2 (0)v2 ⊗ · · · ⊗ vn + · · · + v1 ⊗ v2 ⊗ · · · ⊗ g n (0)vn , X i m where g = (g 1 , . . . , g n ), g i = gm xi , g i (0) = g0i .

(1.8.1)

m≥0

Setting e g0 ⊕ Cc = b g0 , we define the induced module, the Weyl module, b

MVσ = Indbgg0 V,

(1.8.2)

where the central element c acts as σ ∈ C, i.e., c · v = σv ∀v. Because of the decomposition e gr ⊕ e g0 = e g, given any m ∈ MVσ , there exists a unique element π(m) ∈ V such that m − π(m) ∈ e gr MVσ . The linear map π : MVσ → V is called the coinvariant. Its defining property is π(gr m) = π(m) (∀g ∈ e gr , m ∈ MVσ ).

(1.8.3)

1.8. COINVARIANT, INTEGRAL FORMULAS

139

i gi = (Ia )i xki ∈ e Let {Ia } be a basis of g and {Ia∗ } the dual basis. We put Ia,k ∗i ∗ i k and Ia,k = (Ia ) xi . The Sugawara element of degree −1 at ui is given by the series XX i ∗i Li−1 = Ia,−1−k Ia,m , (1.8.4) m≥0

a

which belongs to a completion of U (b g) (to a completion of U (b gi ), to be exact). The definition does not depend on the choice of the basis {Ia }, and the action of the Sugawara elements in MVσ is well defined. They commute with g) are pairwise each other, [Li−1 , Lj−1 ] = 0, because different components of U (b commutative. We want to determine the dependence of the coinvariant on the positions of the points u1 , . . . , un . So we need to enlarge the algebras and modules under consideration assuming that the elements are functions on u. Let U be the algebra of C–functions of u1 , . . . , un ∈ U0 , b g(U0 ) = U ⊗ b g, MVσ (U0 ) = U ⊗ MVσ . The functions that we will consider will have singularities along the diagonals {ui = uj }. The definition of the coinvariant remains unchanged, but now the values of π on MVσ (U0 ) will belong to U ⊗ V. When calculating with the coinvariant, we always assume that the ui are pairwise distinct. Otherwise π is not well defined. We extend the derivatives ∂/∂ui from U to g(u) ∈ b g(U0 ) and m(u) ∈ σ MV (U0 ), setting ∂ j ∂ j Ia,k = 0, (I v) = 0, (1.8.5) ∂ui ∂ui a,k for all v ∈ V. Here we mean the original V, without replacing C by U. We note that the derivatives may be highly nontrivial at the elements in the form fr v,, where fr is the ”rational” component of f ∈ e g, i.e., the projection of f onto e gr . i For instance, let m(u) = (g i /xi )r v for v ∈ V, g ∈ g, g i = g (i) = 1 ⊗ · · · ⊗ g ⊗ · · · ⊗ 1 . Then n ∂m(u) ∂ Y = { (r(xj + uj − ui ), g)}v = ∂uk ∂uk j=1

={

n Y j=1

(δjk

∂ ∂ − δik )(r(xj + uj − ui ), g)}v, ∂xk ∂xj

(1.8.6) (1.8.7)

where (a ⊗ b, c) = (b, c)a for the standard form on g. This example demonstrates that the dependence of the coinvariant of u may be complicated, since the calculation of π is based on the factorization. It also shows that one can express the u–derivatives in terms of x–derivatives, which is important in the next theorem.

140

CHAPTER 1. KZ AND QMBP We will use the notation from (1.7.35) and (1.7.33) : X %= ρa Ia ∈ U (g) for a

(Ω/x − r(x))(x = 0) =

X

ρa ⊗ Ia , Ri = %i −

a

X

rji .

(1.8.8)

j(6=i)

Here %i = %(i) and the values of rij = r(ui −uj )(i,j) are considered as endomorphisms of V . Recall that −rji = rij for unitary r. It is called quasi-unitary if d(Θ)/dx = 0 for Θ = r12 + r21 . Theorem 1.8.1. Let r(x) be a quasi-unitary r–matrix, e gr the factorizing σ b subalgebra of g corresponding to r(x). For V , MV , π, and Li−1 defined above, µ ¶ ∂ i π κ m(u) + L−1 m(u) ∂ui à ! X ∂ = κ ri,j (ui − uj ) + ρi π(m(u)), + (1.8.9) ∂ui j6=i where κ = σ + g, g is the dual Coxeter number of g, m(u) ∈ MVσ (U0 ). Proof. We will consider here the case when κ = 0. Only this degeneration will be applied later to the integral formulas. See [C9] for the general case. One has X X Li−1 v = (Iai /xi )Ia∗i v = ((Iai /xi )r + ρia )Ia∗i v (1.8.10) a

a

for v ∈ V . The e gr –invariance of π (see (1.8.3)) gives: X X (Iai /xi )jr Ia∗i v, π(Li−1 v) = ρia Ia∗i v − a

(1.8.11)

a,j6=i

later is the expansion of fr (∈ e gr ) at uj with respect to where frj here and P ji j ∗i i xj = x − uj . Since a (Ia /xi )r Ia v = r v = r(uj − ui )(i,j) , thanks to relation (1.7.27), we obtain (1.8.9) for a constant m(u) = v.P Any element of MVσ can be represented as v + fr m for proper fr and m ∈ MVσ . Since the level is critical (κ = 0) the Sugawara elements commute with any f = (f 1 , . . . , f n ) ∈ e g. Indeed, the general relation is [Li−1 , f ] = −κ∂f i /∂xi .

(1.8.12)

See [Ka]. So π(Li−1 (fr m)) = π(fr (Li−1 m)) = 0, and (1.8.9) results from the above calculation with v. ❑

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141

As a by-product, we can establish the self-consistency of the r–matrix KZ from (1.7.35): κ∂(Φ)/∂ui = Ri Φ, 1 ≤ i ≤ n. (1.8.13) Moreover, we actually have a generic formula for solutions of this equation. Namely, n ³ X ¡ ¢´ i (xi /κ)L−1 m for arbitrary constant m ∈ MVσ Φ = exp − i=1

satisfies the KZ equation. Comment. This is analogous to the general claim that a τ –function is a universal solution of soliton equations. It is formal and cannot be used for constructing explicit solutions without special analytic or algebraic methods. In soliton theory, the main “constructive” applications of this general claim are (a) the Backlund–Darboux transformations of the solutions, and (b) the solutions in terms of θ–functions and degenerations. The integral formulas for KZ, which will be discussed next, have many points in common with the theory of the BD transforms as well as with the θ–solutions. This is not surprising, since the τ –function in soliton theory (see the previous section) and the coinvariant π coincide. The soliton equations are different from KZ because they correspond to different flows. In soliton theory, we consider mainly the flows for the vector fields Df (f ∈ e g) from (1.7.40). As for KZ, the flows correspond to the Sugawara elements.

1.8.2

Integral formulas

We keep the same notation as in the previous subsection. However, we assume now that every Vi (1 ≤ i ≤ n) is a highest weight module relative to b+ . So Vi is generated by the vacuum vector νi associated to a weight λi ∈ Cn (the highest weight): hα (vac) = (α, λ)vac. The above consideration will be applied to V 0 = V ⊗ Vn+1 ⊗ · · · Vn+m , where V = V1 ⊗ · · · ⊗ Vn is from the previous section, and the new b g–modules are one-dimensional: Vi = C = Cν0 (n + 1 ≤ i ≤ n + m) with zero action of g. The elements ν0 are the basic vectors in these modules. Respectively, we introduce the induced or Weyl b g–module MVσ 0 , where the central element c acts as the scalar σ, and with the coinvariant π : MVσ 0 → V 0 . Since the vacuum vectors ν0 in Vn+i (i > 0) are fixed, we can identify V and V 0 . Thus the values of the coinvariant will actually be in the same old space V . Note that the Weyl modules defined for zero g–modules are very nontrivial, and there are no obvious connections between MVσ and MVσ 0 .

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The points {u1 , . . . , un } will be called old points, and {un+1 , . . . , un+m } new points. The same names will be used for the corresponding indices. def

Recall that l == rank g. We fix a sequence of numbers (n + 1)0 , . . . , (n + m)0 ∈ {1, . . . , l}, which may coincide, and define ¶ µ ¶ µ f(n+m)0 f(n+1)0 ν0 ⊗ · · · ⊗ ν0 , (1.8.14) w = π(m), ˆ m ˆ = ν1 ⊗ · · · ⊗ νn ⊗ xn+1 xn+m m ˆ ∈ MVσ 0 , w ∈ V = V 0 . The definition of w is applicable to any r–matrices. However, w gets sufficiently simple and can be used for the integral formulas only when r are sufficiently special. Therefore we need to reduce the generality at this point. From now on, we will consider the basic trigonometric r–matrix from (1.7.16): l ³x´ 1 X fα ⊗ eα − eα ⊗ fα 1 X 1 Ω+ + A(hi )⊗h∗i , (1.8.15) r(x) = coth 2 2 2 (e , f ) 2 α α + m=1 α∈R

depending on an arbitrarily given endomorphism A : h → h. Recall that l 1 X 1X A(hk )h∗k , ρ = α. % = hρ − 2 k=1 2 α∈Σ

(1.8.16)

+

We identify the elements from h with the corresponding weights and the corresponding linear forms on the complexification Cl of h: hb =

l X

(b, bk )hk , hk = hαk , (hb , λ) = (b, λ) for b, λ ∈ Cl

(1.8.17)

k=1

Here the invariant bilinear form on h is extended to the complexification. Using the above sequence of numbers {i0 } for i > n, let Λi = λi for 1 ≤ i ≤ n, Λi = −αi0 for i > n, Λ =

n+m X

Λi .

(1.8.18)

i=1

Note that the Λi for new i > n have nothing to do with the corresponding weights, which are all zero. Such a uniform notation is convenient in the following definition: ωi =

X

1 ui − uk A∗ (Λ) (Λi , Λk ) coth( ) + (% − , Λi ), 2 2 2 1≤k≤n+m,k6=i

(1.8.19)

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143

where (Aa, b) = (a, A∗ b) on a, b ∈ h. These functions are logarithmic derivatives. Namely, ωi = ∂ω/∂ui for ω =

(1.8.20) Y Y ui ´ A (Λ) , Λi ) . = (e(ui −uj )/2 − e(uj −ui )/2 )(Λi ,Λj )/κ exp (% − 2 κ 1≤i<j≤n+m 1≤i≤n+m We will also use ωJ =

³

X

∗

ωj

(1.8.21)

j∈J

for subsets of new points(indices) J ⊂ {n + 1, . . . , n + m}. Definition 1.8.2. (i) An ordered sequence c = (j1 , . . . , js ; i) for the pairwise distinct new indices j1 , . . . , js ∈ {n+1, . . . , n+m} and one old index 1 ≤ i ≤ n is called a chain. (ii) An ordered sequence d = (c1 , . . . , cr ) of chains is called a diagram if every new n + 1 ≤ j ≤ n + m belongs to one and only one chain (no restrictions for the old indices). (iii) Given a linear ordering on the new indices Â, a diagram ³ ´ (1) (1) (2) (2) (r) (r) d = c1 = (j1 , j2 , . . .), c2 = (j1 , j2 , . . .), . . . , cr = (j1 , j2 , . . .) (1)

(2)

(r)

is called well–ordered if j1 Â j1 Â · · · Â j1 . (iv) For a diagram d, the component of an old i, denoted by compi (d), is a union of all new indices connected with i by a chain from d. Definition 1.8.3. (i) Given a chain c = (j1 , . . . , js ; i), we set Fc =

0 ], fjs0 ]i [···[fj10 , fj20 ], fj30 ], . . . , fjs−1

(euj1 −uj2 − 1)(euj2 −uj3 − 1) · · · (eujs −ui − 1)

.

(1.8.22)

(ii) Given a diagram d = (d1 , . . . , dr ), we set Fd = Fr · · · F2 F1 ∈ U (g)⊗n , where Fj = Fcj , F{∅;i} = 1.

(1.8.23) P

Theorem 1.8.4. We fix an ordering Â. Then w = π(m) ˆ = d Fd (vac), where vac = ν1 ⊗ · · · ⊗ νn ⊗ ν0 ⊗ · · · ⊗ ν0 and the summation is over all well-ordered diagrams. In particular, w does not depend on the central charge P σ. Conversely, d Fd (vac) does not depend on the ordering. The following sums do not depend on the particular choice of the ordering either: X X (1.8.24) Fd0 (vac), wi [J] = Fd (vac), wi (J) = d

d0

where the summation is over all well-ordered diagrams such that compi (d) = J (in the first sum) and J ⊂ compi (d0 ) (in the second sum) for any set of new points J.

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Theorem 1.8.5. For any old i (1 ≤ i ≤ n) and Ri = %i − Ri w = ωi w +

n+m X

ωj wi (j) = ωi w +

X

j=n+1

P j(6=i)

ωJ wi [J],

rji , (1.8.25)

J

where J runs over all new subsets (⊂ {n + 1, . . . , n + m}). Theorem 1.8.6. For any old i, (Ri − κ∂/∂ui )W = κ

n+m X

∂Wi (j)/∂uj = κ

j=n+1

X

∂Wi [J]/∂uj ,

(1.8.26)

new J

where W = wω and Wi (j) = wi (j)ω, Wi [J] = wi [J]ω for ω from (1.8.20). Proof. We will establish the first two theorems in the next subsection. Let us deduce the third one from them now. One has: κ∂w/∂ui = −κ

n+m X

∂wi (j)/∂uj ,

(1.8.27)

j=n+1

since ∂Fc /∂ui = ∂Fj1 /∂uj1 + . . . + ∂Fjs /∂ujs for any chain c = (j1 , . . . , js ; i). Indeed, the formula for Fc depends only on the differences ujp − ujq for 1 ≤ p, q ≤ s + 1, js+1 P = i. Hence, the same P holds for any Fd with compi (d) = n+m J. The relations {new J} ωJ wi [J] are obvious because j=n+1 ωj wi (j) = P ❑ wi (j) = J3j wi [J]. R To get the integral formulas for KZ, we set Φ = W dun+1 · · · dun+m , where the integration contours are taken to ensure that Z (∂Wi (j)/∂uj )dun+1 · · · dun+m = 0 for all new j. Then (κ∂/∂ui − Ri )Φ = 0. The proper choice of contours and the description of the spaces of corresponding solutions Φ can be a difficult problem, especially if the κ are not assumed to be generic (see [Va]). We discuss in this work the algebraic machinery only: the integrands, but not the integrals.

1.8.3

Proof

For the sake of simplicity we will consider here the Yang r–matrix only. We refer the reader to [C9] for the general case (somewhat more general than that considered above). Let us degenerate the main formulas, replacing the trigonometric formulas (hyperbolic, to be exact) by rational ones.

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145

From now on, r = Ω/x and X

(Λi , Λk ) , ωi = ∂ω/∂ui , u i − uk 1≤k≤n+m,k6=i Y ω= (ui − uj )(Λi ,Λj )/κ .

ωi =

(1.8.28)

1≤i<j≤n+m

Since r is unitary and % = 0, we can simplify Ri : Ri =

n X j=1

ij

r =

n X j=1

Ωij , j 6= i, 1 ≤ i ≤ n. ui − uj

(1.8.29)

Given a chain c = (j1 , . . . , js ; i), we set Fc =

0 ], fjs0 ]i [···[fj10 , fj20 ], fj30 ], . . . , fjs−1

(uj1 − uj2 )(uj2 − uj3 ) · · · (ujs − ui )

.

(1.8.30)

The definition of Fd for a diagram d = (c1 , . . . , cr ) is the same: Fd = Fr · · · F2 F1 ∈ U (g)⊗n for Fj = Fcj . (1.8.31) P Let us fix the ordering and check that w = π(m) ˆ = d Fd (vac), where vac = ν1 ⊗ · · · ⊗ νn ⊗ ν0 ⊗ · · · ⊗ ν0 and the summation is over all well-ordered diagrams. It is a straightforward calculation based directly on the defining property of the coinvariant, that is, π(fr v˜) = 0 for f ∈ e gr and v˜ ∈ MVσ 0 . Recall that µ ¶ µ ¶ g g g , = xi x − ui xi r µ ¶ g g ,..., = , (1.8.32) x 1 + u 1 − ui xn+m + un+m − ui where xi = x − ui , g ∈ g, and 1 ≤ i ≤ n + m. Here g/xi is considered as an element of the i–th component of e g (defined in a formal neighborhood of ui only) and g/(x − ui ) is a rational function of x (defined globally). So õ ¶ ! õ ¶ ! i i g g v˜ v˜ = π π xi xi r à µ ¶j ! X g = −π v˜ . (1.8.33) x j + uj − ui 1≤j≤n+m,j6=i If v˜ = v˜1 ⊗· · ·⊗˜ vn+m and v˜j ∈ Vj for some j 6= i, then further simplification is possible: µ ¶j ¶j µ g g v˜ = v˜1 ⊗ · · · v˜j · · · ⊗ v˜n+m . (1.8.34) x j + uj − ui uj − ui

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Moreover, if j is new, then (g/(uj − ui ))j v˜j = 0. If the point j is new (j > n) and v˜j = (f j /xj )ν0 for f ∈ g, then we can proceed as follows: µ µ ¶j ¶j j g f g v˜j = ν0 = x j + uj − ui x j + uj − ui xj ¶j j i hµ f g 1 [f, g]j , ν0 . (1.8.35) = ν0 = x j + u j − ui xj ui − uj x j Formula for the integrand. Now let m ˆ be from (1.8.14), w = π(m), ˆ µ ¶ µ ¶ f(n+1)0 f(n+m)0 m ˆ = ν1 ⊗ · · · ⊗ νn ⊗ ν0 ⊗ · · · ⊗ ν0 . (1.8.36) xn+1 xn+m We start with the new index i which is maximal with respect to the ordering Â, and then use (1.8.33) in combination with (1.8.35) to clear up the i–th component, by replacing (fii0 /xi )ν0 by the sum over all remaining components. Applying (1.8.33) again and again, we will eventually come to the element from V ⊗ ν0 ⊗ · · · ⊗ ν0 that contains pure ν0 at all new points and coincides with its coinvariant. Note that we have to apply this procedure to all terms of the sum obtained after the previous step. In this calculation, the terms are in one-to-one correspondence with the chains c = (j1 , . . . , js ; i) from Definition 1.8.2. The indices j1 , . . . , js must be pairwise distinct because gνj = 0 for any g ∈ g. There can be only one old i in the chain due to (1.8.34). The old point is always at the end of the above simplification process. If all the chains have reached their end points, we start the next chain, picking the maximal new point among the untouched ones and follow the same procedure. The new points that have already been cleared of the f –terms will not participate. We come to the above definition of the diagram and also establish the formula for w, which is the first claim of Theorem 1.8.4. We can take any ordering  in this calculation. The result will of course be the same. Note that if we did not have an interpretation of w in terms of the coinvariant, establishing the fact that different orderings result in the same formula is a mundane combinatorial reasoning (cf. [SV]). Let us prove that wi [J] =

X d

Fd (vac), wi (J) =

X

Fd0 (vac)

(1.8.37)

d0

do not depend on the ordering as well. Here the summation is over all wellordered diagrams such that compi (d) = J or compi (d0 ) ⊂ J. It is sufficient to examine wi [J] only, because the quantities {wi (J)} can be expressed in terms of {wi [J]}.

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147

We will deform the positions of the points u1 , . . . , un+m . Let u˜j = uj + δ for j ∈ {J ∪ i}, u˜j = uj for j 6∈ {J ∪ i}.

(1.8.38)

The terms Fd will remain unchanged iff they appear in the sum for wi [J]. Moreover, wi [J] = lim w, ˜ where w ˜ = w(˜ u1 , . . . , u˜n+m ). (1.8.39) δ→∞

Indeed, all Fd with compi (d) 6= J will contain at least one δ in the denominator. This representation does not depend on the ordering.P Thus we proved Theorem 1.8.4 and established that w = J wi [J] for any old i. Calculating Ri w. Let us establish the formulas X Ri w = ωi w + ωJ wi [J]. newJ

See Theorem 1.8.5. To simplify the index notation, we set i = n. So we need to check that à ! X X nj r (un − uj ) π(m) ˆ = ωJ wn [J] + ωn π(m). ˆ (1.8.40) 1≤j n are the same, but their meaning is different. In the second formula, we used that [hp , fj 0 ] = −(hp , αj 0 ) = (hp , Λj ) due to the definition of Λj at the new points. Let us now get rid of (eα /xn ). The calculation is more involved than that with h, because eα may “interact” with fj 0 at the new points. However it is still not too complicated (in the case of the basic trigonometric r–matrix). Since νj (1 ≤ j < n) are the highest weight vectors, we move eα only to the right (to the new points): µµµ ¶ ¶n ¶ eα fα π m ˆ = xn (eα , fα ) à ! µ ¶j µ ¶n X [eα , fj 0 ] fα =π m{j} ˆ . (1.8.46) (u − u )x (e , f ) n j j α α n+1≤j≤n+m Here m{j} ˆ denotes m ˆ without fj 0 /xj . Namely (see (1.8.36)): µ ¶ µ ¶ f(j−1)0 f(j+1)0 m{j} ˆ = ν1 ⊗ · · · ⊗ νn ⊗ · · · ν0 ⊗ ν0 ⊗ ν0 ⊗ · · · . (1.8.47) xj−1 xj+1 Next we move every ([eα , fj 0 ]/xj ) to all components ˜j 6= j to clear the j–th component of [e, f ]. Then we will get the triple commutators [[e, f ], f ] at the new ˜j. Next we will go to the components ˆj 6= j, ˜j and produce the four-term commutators. Then the five-term commutators will be produced, and so on. Note that the poles will always be simple after every step. We stop this process when the s–fold commutator is 0 or when we come to an old point (including the point un ) for the first time. Let us examine the resulting commutators. Since fj 0 are simple, any commutator given by this procedure is proportional to (a) (eβ /xk )k , (b) (hβ /xk )k , or (c) the central element c

(1.8.48)

for a certain β > 0 and a new index k. In case (a), we may continue and get another new point. In case (b), there will be one more (final) step when we replace (hβ /xk )k by the corresponding sum over all points (old and new). After this we will stop. In case (c), we simply plug in c = σ = −g.

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149

Note that since one never obtains f –terms in this procedure, the only way to reach the old points is via (b), but the point un must be excluded here. Thus the result is given in terms of the chains again. Namely, we arrive at the chains js = j, js−1 = ˜j, js−2 = ˆj, . . . , j1 = k. However, now we read {j} in the opposite order and the chain describe a different process, that is, the process of eliminating eα /xn . Note that β = αk0 for the last index k = j1 . Once we know all the indices, it is not difficult to determine the exact formula up to the contribution of the central element (the case (c)) and the chains that go back to the starting point un . The calculation is based on the identities: if [···[fj10 , fj20 ], . . . , fjs0 ] = cfα , then 0 ], . . . , fj10 ]) = (−1)s−1 c(fα , eα )hj10 . [···[eα , fjs0 ], fjs−1

Setting φ = (un − uj1 )(uj1 − uj2 ) · · · (ujs−1 − ujs ), we obtain µµµ ¶ ¶ ¶ eα fα π m ˆ = xn (eα , fα ) ´ ³¡ ¢ ˆ . =φ−1 π (fα )n ⊗ 1 ⊗ · · · (hαk0 /xk ) · · · ⊗ 1 m{J}

(1.8.49)

(1.8.50)

Here k = j1 , m{J} ˆ denotes m ˆ without (fj 0 /xj ) for all j ∈ J, namely, ¶ j1 z}|{ f10 ν0 ⊗ · · · ⊗ ν0 ⊗ m{J} ˆ = ν1 ⊗ · · · ⊗ νn ⊗ · · · x1 µ ¶ ¶ µ ¶ µ j2 js f()0 f()0 f(n+m)0 z}|{ z}|{ ··· ⊗ . ν0 · · · ν0 · · · ν0 · · · · · · ν0 · · · x() x() xn+m µ

(1.8.51)

Here the set {j} is not ordered. For instance, one can get j2 , . . . , j3 , . . . , j1 , . . . . Comparing (1.8.49) with the definition of Fc from (1.8.30) for the chain c = (j1 , · · · , jn ; n), represented in the form   k ´ ³ z}|{ ˆ , (1.8.52) Fc = φ−1 π (fα )n ⊗ 1 ⊗ · · · 1 · · · ⊗ 1 m{J} we see that these two expressions are different only due to the appearance of hαk0 /xk in (1.8.49). These h–terms readily result in a sum over all, old and new, indices i 6= k. Note the cancelation of (−1)s−1 from the denominator of the first formula with (−1)s−1 from (1.8.49).

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This procedure gives the first step. We still need to clear the new components i 6∈ J that contain {f }. We will follow exactly the process of calculating w and numerate the terms by the diagrams. The final formula will be the sum of Fd over all diagrams d multiplied by proper ω coming from hαk0 /xk at the first step. Here we need to use that wn [J] do not depend on the ordering, because when eliminating eα we cannot control the endpoint k = j1 of the (first) chain. So we may need to change the ordering of the indices to make k the maximal element. We arrive at the formula X Rn w − ωn w − ωJ wn [J] = S, (1.8.53) newJ

where S is a sum of the terms with denominators that may not contain the “chain products” in the form (un − uj1 )(uj1 − uj2 ) · · · (ujp − ui ) for any new j and old i < n. It is the contribution of the terms of type (c) and the chains ending at the n–th component. The latter can come from h (the case (b)) or from certain e if we go back and fuse them with (fα )νn . P Let us check that S = 0.P It can be represented as S = J Sn [J] exactly in the same manner as w = J wn [J]. We will prove that Sn [J] = 0 for any new set J. Consider the deformation u˜ from (1.8.38) for i = n and tend δ to P i ,Λn ) , exactly one index ∞. Then w → wn [J]. In the sum ωn + ωJ = i,j (Λ ui −uj from any pair {i, j} belongs to {J ∪ n}. So it goes to 0 as δ approaches ∞. Since all the terms of Rn contain the differences ui − un for some i < n in the denominators, the left-hand side of (1.8.53) identically equals zero. The right-hand side tends to Sn [J]. Thus the latter is zero. The proof of the theorem is completed. Comment. The direct (combinatorial) definition of the w and W in the rational case is due to Schechtman and Varchenko (Preprint MPI/8951, 1989). They established that W satisfies KZ up to exact derivatives in terms of the new {uj , j > n}. This theorem generalized the paper by Date, Jimbo, Matsuo, and Miwa [DJMM] (the SL2 –case) and that by Matsuo for SLn . There were also results of Dotsenko, Fateev, Aomoto, Christe, and Flume in this direction. An extended version of the MPI-preprint is [SV]. Paper [C9] (first published as preprint RIMS–699, 1990) contained the interpretation of the Schechtman–Varchenko via the coinvariant, the trigonometric generalization, and Theorem 1.8.5 with new explicit formulas for the exact derivatives. The approach via the Kac–Moody algebras dramatically simplified the formulas from [SV] and their justification. Theorem 1.8.5 has applications not only to KZ. Actually, it is a pure algebraic statement and must have algebraic corollaries. Indeed, let us choose the parameters {un+1 , . . . , un+m } to ensure the relations ωj = 0 for j > n. Then w is an eigenvector of the pairwise commutative matrices Ri (1 ≤ i ≤ n)

1.8. COINVARIANT, INTEGRAL FORMULAS

151

with the eigenvalues {ωi }. This eigenvalue problem is called the Gaudin model. The first results on the diagonalization of the “Hamiltonians” {Ri } were obtained by Babujan and Flume in [BF]. See also [FFR]. We will not discuss this direction here.

1.8.4

Comment on KZB

In the elliptic KZ theory, we must first examine the Baxter–Belavin r– matrix. It is unique among unitary elliptic classical r–matrices. There are also some other examples of non-unitary type. An important variant of Belavin’s r in infinite matrices was introduced by Shibukawa–Ueno. Theorem 1.8.1 holds for all such r. However, there are problems with the integral formulas. Our method can be generalized, but we need to assume that the rational extensions f¯ of the elements f = fα /xi and f = eα /xi are proportional to fα and eα , respectively, with certain scalar meromorphic functions as coefficients of proportionality. As was demonstrated in [FW1], the KZB equation due to Bernard [Be] leads to a Lie algebra of elliptic functions satisfying this very property (see also [Fe]). In this subsection, we will comment on it. The above approach to the integral formulas also can be extended to this case, but we are not going to discuss it here. Let E be an algebraic elliptic curve, 0 its zero point, and x a local parameter in a neighborhood U0 ⊂ E of 0. We fix pairwise distinct u1 , . . . , un ∈ U0 . Given µ ∈ C, a Baker function ϕ is a function on E, that is regular apart from −1 U0 and such that e−µx ϕ(x) is meromorphic with poles at points u1 , . . . , un . Thus Baker functions have an “exponential” singularity at 0. We denote the space of such functions by Bµ . Baker functions are completely determined byPtheirPprincipal parts, that may be arbitrary. More precisely, given p(x) = ni=1 1≤j0 This formula was conjectured by Mehta for the root system of type An . Bombieri readily deduced it from Selberg’s integral. Then Macdonald extended it in [M2] to all root systems and proved his formula for the classical systems B, C, D. Opdam found a uniform proof using the technique of shift operators [O1]. Example. Let us check the Mehta–Macdonald formula for the root system of type A1 . In this case n = 1 and there is only one positive root α : (α, α) = 2. Thus α ˜ = α∨ = α, ρ = α2 , and (ρ, α∨ ) = 1. Let u = x˜α . Then formula (2.1.4) reads as Z ∞ du Γ(2k + 1) 2 |u|2k e−u /4 √ = (2π)1/2 . Γ(k + 1) 2 −∞

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Changing the variables and using the doubling formula for the Γ–function √ Γ(2k + 1) 1 , 22k Γ(k + ) = π 2 Γ(k + 1) we obtain (2.1.1). Thus (2.1.4) is a generalization of formula (2.1.1) to higher rank root systems.

2.1.4

Hankel transform

For any complex number k 6∈ − 21 − Z+ , we define the function φ(k) (t) =

∞ X

t2m Γ(k + 21 ) 1 . ) m!Γ(k + m + 2 m=0

The convergence is for all t and is very fast. It is clear that φ(k) (0) = 1. The function φ(k) (t) is related to the classical Bessel function Jk (t) by the formula 1 φ(k) (t) = Γ(k + ) t−k+1/2 Jk−1/2 (2it). 2 Theorem 2.1.2. For any complex k with − 21 , Z ∞ 1 2 2 2 e−x φ(k) (λx)φ(k) (µx)|x|2k dx = φ(k) (λµ)eλ +µ Γ(k + ), 2 −∞

(2.1.5)

(2.1.6)

which reduces to (2.1.1) as λ = µ = 0. Definition 2.1.3. The real Hankel transform of an even function f (x) is given by the formula Z ∞ 1 k Hre (f )(λ) = φ(k) (xλ)f (x)|x|2k dx. Γ(k + 21 ) −∞ The Hankel transforms are even functions of λ; the transforms of odd f (x) are zero. Formula (2.1.6) states that 2

2

2

Hkre (φ(k) (µx)e−x ) = eµ (φ(k) (µλ)eλ ).

(2.1.7)

Definition 2.1.4. The imaginary Hankel transform of an even function f (x) is Z 1 k φ(k) (xλ)f (x)|x|2k dx. Him (f )(λ) = 1 iΓ(k + 2 ) iR 2

Let Vrek be the linear span of the functions φ(k) (λx)e−x treated as functions 2 k is the span of φ(k) (λx)eλ considered as functions of of x ∈ R. Similarly, Vim k k to Vrek . As an and Hkim maps Vim λ ∈ iR. It is clear that Hkre maps Vrek to Vim immediate consequence of (2.1.6), we obtain the inversion formula.

2.1. EULER’S INTEGRAL AND GAUSSIAN SUM

163

Corollary 2.1.5. The maps Hkre and Hkim are inverse to each other. k by their suitable completions. Here we can replace the spaces Vrek and Vim

Corollary 2.1.6. Let Ure (respectively, Uim ) be the linear span of functions 2 2 in the form p(x)e−x (respectively, p(λ)eλ ), where p runs through even polynomials. Then Hre maps Ure to Uim , Him maps Uim to Ure , Him Hre = id, and Hre Him = id. Moreover, ≤n ≤n ≤n ≤n , 7→ Ure , Him : Uim Hre : Ure 7→ Uim ≤n ≤n ⊂ Uim ) are the subspaces generated by ⊂ Ure (respectively, Uim where Ure 2 2 p(x)e−x (respectively, p(λ)eλ ) for all even polynomials p of degree ≤ n.

Proof. Differentiating formula (2.1.7) 2n times in terms of the variable µ 2 and setting µ = 0, we obtain that Hre (x2n ) is eλ times a polynomial in λ. A better and more constructive proof will be presented below.

2.1.5

Gaussian sums

Let N be a natural number and q a primitive N –th root of 1. We will also need q 1/4 , which will be picked in primitive 4N –th roots of 1. We will consider the so-called generalized Gaussian sums, τ=

2N −1 X

qj

2 /4

.

j=0 2

Note that q j /4 depends only on the residue of j modulo 2N, so we can assume that the summation index runs through Z mod 2N . A natural choice for q 1/4 is eπı/2N . In this case, we have the celebrated formula of Gauss: √ τ = (1 + ı) N . (2.1.8) Pn−1 j 2 q . Comment. The standard definition of the Gaussian sum is τ 0 = j=0 2πi/n Gauss proved that for q = e ,  √ if n mod 4 = 1,   √n  i n if n mod 4 = 3, √ τ0 = (1 + i) n if n mod 4 = 0,    0 if n mod 4 = 2. Formula (2.1.8) corresponds to n = 4N. It somewhat resembles Fresnel’s integral r Z ∞ π ix2 e dx = (1 + i) , 2 −∞

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√ x and which can be obtained from (2.1.2) by changing the variable x 7→ 1+i 2 √ √ shifting the contour of integration. In a sense, 2N is a substitute for π when we switch from the real theory to roots of unity. ❑ There are two levels in the theory of Gaussian sums: (a) checking that τ τ¯ = 2N, which is simple and entirely conceptual, (b) finding arg(τ ), which is not. Let us recall the proof of (a). We consider the space V of complex-valued functions on Z mod 2N, pick a primitive 2N –th root q 1/2 of 1, and introduce the Fourier transforms F+ and F− on V : 2N −1 X F± (f )(λ) = q ±xλ/2 f (x). x=0

Then (i) F− F+ = F+ F− = multiplication by 2N, (ii) F+ (γ) = τ γ −1 and F− (γ −1 ) = τ¯γ, 2 where γ(x) = q x /4 for a primitive 4N –th root of unity q 1/4 . Indeed, X q (x−y)λ/2 = 2N λ∈Z mod 2N for x = y and zero otherwise. We obtain (i). As for (ii), it suffices to examine X 2 q xλ/2 q x /4 F+ (γ) = x∈Z mod 2N X 2 2 2 = q (x+λ) /4 q −λ /4 = τ q −λ /4 . x∈Z mod 2N The relation τ τ¯ = 2N is immediate from (i) and (ii). By the way, it is easy to show that τ¯ = T r(F+ ), but it is not helpful for calculating arg(τ ). We are going to deduce (2.1.8) from a q–variant of (2.1.1). There will be two interesting corollaries: (1) a formula for τ for any primitive root q, (2) a generalization of (2.1.8) to an arbitrary integral k. The former is closely connected with known formulas, but the latter is new.

2.2

Imaginary integration

In this section, we will discuss the imaginary integration, which is called the compact case in the context of symmetric spaces. Let us first switch from R to iR. Recall our main formula: Z ∞ 1 2 e−x |x|2k dx = Γ(k + ), −1/2. 2 −∞

2.2. IMAGINARY INTEGRATION Changing the variable x 7→ ix, we obtain Z 1 1 2 ex |x|2k dx = Γ(k + ), −1/2. i iR 2

165

(2.2.1)

These two formulas are of course equivalent, but their q–counterparts are not. It is similar to the Harish-Chandra theory; real integration corresponds to the noncompact case, which is very different from the compact case (imaginary integration). Comment. (i) There is an improvement of (2.2.1): Z 1 2 (2.2.2) ex (−x2 )k dx = Γ(k + ) cos(πk), 2 −ε+iR which holds for all complex k provided that ε > 0. It can be readily deduced from the well-known classical definition of the Γ–function: Z (−z)k e−z 1 1 √ dz = Γ(k + ) cos(πk), 2 C i −z 2 by change the variable z 7→ −x2 . Here the path of the integration C begins at z = −εi + ∞, moves to the left down the positive real axis to −εi, then circles the origin, and returns along the positive real axis to √ εi + ∞. The branches k k log(−z) of the log in the expression (−z) = e and −z are standard, with the cutoff at -R+ . 2 (ii) The use of the Gaussian ex makes the classical formula in terms of e−z somewhat more elegant, but of course it is not significant. It is different in the q–theory, which does require the formula in terms of the Gaussian. This is directly connected with the appearance of the double Hecke algebra. The (formal) conjugation by the Gaussian is an automorphism of the latter. The substitution z = −x2 does not seem meaningful from the viewpoint of this algebra.

2.2.1

Macdonald’s measure

Let q be a real number, 0 < q < 1. We set q = e−1/a for a real number a > 0. The q–counterpart of x2k is the following function of a complex variable x: δk (x) =

∞ Y

(1 − q 2x+j )(1 − q −2x+j ) . 2x+j+k )(1 − q −2x+j+k ) (1 − q j=0

(2.2.3)

If k is a non-negative integer, formula (2.2.3) reads δk (x) =

k−1 Y

(1 − q 2x+j )(1 − q −2x+j ).

j=0

(2.2.4)

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Theorem 2.2.1. Assume that 0. Then δk (x) is regular for x ∈ iR and Z ∞ √ Y 1 1 − q k+j −x2 q δk (x)dx = 2 πa . (2.2.5) 2k+j i iR 1 − q j=0 Comment. The right- and left-hand sides of formula (2.2.5) are well defined for any number k such that 0 an odd integer, assuming that they are relatively prime: (m, n) = 1. We set Pn−1 mj © m ª def (n−1)(m−1)/2 == ı (−1) j=1 [ 2n ] , n n X 2 m def eπı j n +πıjm . G(m, n) ==

(2.3.10) (2.3.11)

j=1

Formula (2.3.11) is the classical definition of the generalized Gaussian sum (see, e.g., [Ch]). As we will ¡ ¢see in the next theorem, the first definition extends the Legendre symbol m . The latter is ±1 as m is a quadratic residue n (nonresidue) modulo n, where n is an odd prime number or its power. Theorem 2.3.9. (a) For odd m coprime with n, µ ¶1−n ©mª √ 1+ı G(m, n) = n √ . n 2 © ª ¡m¢ (b) Taking n = pa for odd prime p and odd a, m = n . n

(2.3.12)

Proof. The G(m, n) is the sum from (2.3.9) for q = eπı/n . The product on the right-hand side of this equality was calculated in Corollary 2.3.6 as m = 1. Indeed, G(1, n) is given by √ (2.3.12). (a) The equality |G(m, n)| = n is immediate: N −1 1Y N G(m, n)G(m, n) = (1 − q j ) = = n. 2 j=1 2

2.3. JACKSON AND GAUSSIAN SUMS

181

Here G(1, n) is sufficient to examine (apply the Galois automorphisms). The formula for the argument of G(m, n) is direct from formula (2.3.9). Using arg(1 − eiφ ) = φ/2 − π/2 for 0 < φ < 2π: · ¸ n−1 X πmj mj π ( arg(G(m, n)) = − π− ) 2n 2n 2 j=1 ¸ n−1 · X πm n(n − 1) π mj = − (n − 1) − π = 2n 2 2 2n j=1 ¸ n−1 · X π π mj = (m − 1)(n − 1) − (n − 1) + mod 2π. 4 4 2n j=1 (b) For m = 1, the coincidence is evident. Generally, G(m, n) = ±G(1, n) =

©mª G(1, n). n

Using the left-hand side of (2.3.9), the sign is plus if m is a quadratic residue modulo n and is constant on all nonresidues m. We use that Z∗n is cyclic. It cannot always be plus because it would give the invariance of G(1, n) under the Galois automorphisms q 7→ q m and would result in G(1, n) ∈ Q. Therefore the sign is minus at the nonresidues. ❑

2.3.5

Etingof ’s theorem

So far we have considered the imaginary integration, the Jackson summation, and itsR variant at roots of unity. However, the most natural choice is of 2 course iε+R q x δk (x)dx. The calculation of the latter integral was performed by P. Etingof. I am grateful for his permission to include his note. Here we use i (instead of ı) for the imaginary unit. Recall that q = exp (−1/a), ω = 2πia. Let k be a positive real number, and ² > 0 a small positive number. We will use the function Πk , the righthand side of (2.3.2), and its properties, including information about the zeros. Let Z 1 2 Ψk = q x δk (x)dx. gk i²+R It is clear that Ψk is well defined. Our goal is to calculate Ψk , which will be done in Theorem 2.7.4 below. To formulate the theorem, we need some definitions. First, the theta function is X θ(k, τ ) = (−1)n e2πink+πin(n−1)τ , =(τ ) > 0. n∈Z

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It is a periodic entire function with period 1 that satisfies the equation θ(k + τ, τ ) = −e−2πik θ(k, τ ). It is defined by this equation uniquely up to scaling. The zeros of θ(k, τ ) with respect to k are m + nτ , m, n ∈ Z (all of them are simple). Define the following (degenerate) Appell function: X e2πin2 τ +2πink A(k, τ ) = 2πi . e2πinτ − 1 n∈Z\0

It is not expressed via theta functions. Theorem 2.3.10. The function Ψk extends to an entire function, and one then has √ θ(k, ω) Ψk = 2 πa · e−πik q −k(k+1)/2 0 × θ (0, ω) µ 0 ¶ θ θ0 × (k, ω) − (1/2, ω) − A(k, ω) + A(1/2, ω) . θ θ We present the proof as a chain of lemmas. Lemma 2.3.11. The function Ψk has an analytic continuation to the region 0, |=(k)| < |ω|. Proof. The contour of integration can be replaced by 1 1 1 (−∞ + i², i²] ∪ [i², ω − i²] ∪ [ ω − i², ω − i² + ∞), 2 2 2 or by

1 1 1 ( ω − i² + ∞, ω − i²] ∪ [ ω − i², i²] ∪ [i², ∞ + i²). 2 2 2 This implies the statement.

❑

Lemma 2.3.12. Ψk+ω − Ψk = −2iΨ]k = −2iΠk . In particular, Ψk extends to a holomorphic function in the half-plane 0. Proof. This follows directly from Cauchy’s residue formula. Lemma 2.3.13. One has Ψk = −q k+1 Ψk+1 .

❑

2.3. JACKSON AND GAUSSIAN SUMS

183

Proof. This is proved by using the shift formula in the same way as relation (2.3.5). Actually, the sharp integration is a variant of the real integration, so they result in the same multiplier. However, their behavior in the imaginary direction is different. ❑ Lemma 2.3.14. Ψk extends to an entire function. Proof. This is immediate from Lemma 2.3.13. Let us now define the entire function

❑

Fk = q k(k+1)/2 eπik Ψk . Lemma 2.3.15. Fk is periodic with period 1 and e2πik Fk+ω + Fk = 2ieπik q k(k+1)/2 Πk ,

(2.3.13)

where Πk is the right-hand side of (2.3.2). Proof. The real periodicity is direct from Lemma 2.3.13. Equation (2.3.13) is checked following the same lines as formula (2.2.7). ❑ Now, we are going to construct at least one holomorphic solution of equation (2.3.13) and then correct it by adding a solution of the homogeneous equation, i.e., a multiple of θ(k, ω). To construct such a solution, let us express the right-hand side of (2.3.13) in terms of the theta function. Lemma 2.3.16. One has ieπik q k(k+1)/2 Πk = C θ(k + ω + 1/2, 2ω)

θ(k, ω) , θ0 (0, ω)

where C is a constant. Proof is a combination of the translational properties of Πk and the information about its zeros. ❑ Now consider the following two equations: Gk+ω − Gk = 1, and

(2.3.14)

1 Hk+ω − Hk = θ(k + ω + , 2ω) − 1. (2.3.15) 2 It is clear from Lemma 2.3.16 that if Gk solves (2.3.14) and Hk solves (2.3.15), then −2C θθ(k,ω) 0 (0,ω) (Gk + Hk ) solves (2.3.13). Concerning (2.3.14), it is satisfied by 1 θ0 Gk = − (k, ω). 2πi θ

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CHAPTER 2. ONE-DIMENSIONAL DAHA

On the other hand, Hk =

1 A(k, ω) 2πi

solves (2.3.15). This implies that Fk = −

C θ(k, ω) θ0 (A(k, ω) − (k, ω) − β) πi θ0 (0, ω) θ

for constants C, β (note that the right-hand side is an entire function). So we need to find these constants. √ Lemma 2.3.17. The function Fk satisfies the conditions F0 = 2 πa and F1/2 = 0. Proof. The first statement is clear (use the Gauss integral); the second one follows from the fact that gk has a pole at k = −1/2 and Fk is 1–periodic. ❑ √ 0 Now we see that β = A(1/2, ω) − θθ (1/2, ω) and C = 2πi πa. This completes the proof of the theorem. Comment. The function A(k, ω) has the following connection to Appell functions (see, e.g., [Po] and references therein). The Appell function κ(u, k, τ ) is defined as a unique holomorphic 1–periodic in k solution of the equation 1+τ κ(u, k + τ, τ ) = e2πiu κ(u, k, τ ) + θ(k + , τ ). 2 It has the Fourier series expansion X eπin2 τ +2πink κ(u, k, τ ) = . 2πinτ − e2πiu e n∈Z The Appell function has a pole at u = 0, so one can introduce its regular part: X eπin2 τ +2πink κ0 (k, τ ) = . e2πinτ − 1 n∈Z\0

Then A(k, ω) = 2πi(κ0 (k, 2ω) + κ0 (k + ω, 2ω)).

2.4

Nonsymmetric Hankel transform

Recall that we want to connect the Hankel transform and the Fourier transform on ZN in one theory.

2.4. NONSYMMETRIC HANKEL TRANSFORM

185

We will interpret the integral Z 2 e−x |x|2k dx = Γ(k + 1/2), −1/2, R

as the structural constant of the classical Hankel transform, and prove Theorem 2.1.2. Our approach is different from the usual treatment of similar integrals in classical works on Bessel functions. Then we will switch to the nonsymmetric theory and demonstrate that it provides significant simplifications.

2.4.1

Operator approach (k)

def

Recall that φλ (x) == φ(k) (λx) is given as follows: φ(k) (t) =

∞ X n=0

t2n Γ(k + 1/2) , k 6∈ −1/2 − Z+ . n!Γ(k + n + 1/2)

It is even in both x and λ: φ(k) (t) = φ(k) (−t). The connection with the classical Bessel function is established in (2.1.5). Our first step is to reprove the classical formula (see, e.g., [Lu]) for the Gauss integral in the presence of the Bessel functions, using the operator interpretation of the latter. Theorem 2.4.1. For arbitrary complex λ, µ and −1/2, Z 2 2 (k) (k) −x2 φλ (x)φ(k) |x|2k dx = Γ(k + 1/2)φλ (µ)eλ +µ . µ (x)e

(2.4.1)

R

Proof is based on the theory of the differential operator L=

d2 2k d + . dx2 x dx (k)

Lemma 2.4.2. (i) The function φλ (x) is a unique even solution of the eigenvalue problem Lφ(x) = 4λ2 φ(x) with the normalization φ(0) = 1. (ii) The operator L is self-adjoint with respect to the scalar product Z hf (x), g(x)i = f (x)g(x)|x|2k dx. R

Proof. Both statements are straightforward. Concerning (ii), we check k(1−k) d2 that L = |x|−k ◦ H ◦ |x|k , where R H = dx2 + x2 , and use that H is selfadjoint for the scalar product R f (x)g(x)dx. ❑

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Corollary 2.4.3. Asymptotically as |x| → ∞, (k)

φλ (x) ∼ C(λ)(e2λx + e−2λx )|x|−k for a constant C(λ). Proof. The eigenfunctions of H with the eigenvalue 4λ2 are asymptotically e and e−2λx . ❑ cx Let f (x) be an even function of x ∈ R such that f (x)e → 0 as x → ∞ for any c. The Hankel transform is defined as follows: Z 1 def (k) F(f )(λ) == f (x)φλ (x)|x|2k dx, −1/2. (2.4.2) Γ(k + 1/2) R 2λx

Let Fop be the corresponding transform on the operators: Fop (A) = F A F−1 . Lemma 2.4.4. (a) Fop (L) = 4λ2 , def d2 2k d (b) Fop (4x2 ) = Lλ == dλ 2 + λ dλ , d d op (c) F (4x dx ) = −4λ dλ − 4 − 8k. Proof. Formulas (a) and (b) hold because the operator L is self-adjoint (k) and φλ (x) is its eigenfunction. Formula (c) formally follows from (a), (b), and the identity d [L, x2 ] = 4x + 2 + 4k. dx ❑ (k) (k) ±x2 and We set φ±,λ = φλ (x) e 2

2

2

2

L+ = ex ◦ L ◦ e−x , L− = e−x ◦ L ◦ ex . Explicitly, L+ = L + 4x

d d + 2 + 4k + 4x2 , L− = L − 4x − 2 − 4k + 4x2 . dx dx

Using Lemma 2.4.4, Fop (L− ) = L+ . (k)

(k)

It is immediate from Lemma 2.4.2(i) that L± φ±,λ = 4λ2 φ±,λ and (k)

2

(k)

F(φ−,µ ) = Cµ eµ φ+,µ (λ), where Cµ does not depend on λ. Therefore Z 2 2 2 (k) φλ (x)φµ(k) (x)e−x |x|2k dx = Γ(k + 1/2)Cµ φµ(k) (λ)eλ +µ . R

2.4. NONSYMMETRIC HANKEL TRANSFORM

187

The left-hand side of this equality is invariant under the change λ ↔ µ, and so is the right-hand side. Hence Cµ = C0 = 1. The theorem is proved. ❑ The key step in this proof is the calculation of the Hankel transform for the algebra of symmetric (even) differential operators. Thanks to the formula d for [L, x2 ], which gives the answer for the Euler operator x dx , it is is simple. For arbitrary root systems, the L–operators and the symmetric (W –invariant) functions in terms of the coordinates denerate the algebra of all symmetric differential operators. However, this statement becomes much more involved technically. (k) The symmetry x ↔ λ of the function φλ (x), which was used in the proof, holds in general, but the justification is not straightforward. Also note that the calculation we performed was simple, but not completely sufficient to clarify why the final formula (2.4.1) is so nice. Certainly other methods are needed for the multidimensional counterparts of 2.4.1. The operator method we have used gives a really simple prove of formula (2.4.1) and its multidimensional generalizations in the nonsymmetric setting, which is based on the Dunkl operators instead of the L–operators.

2.4.2

Nonsymmetric theory

The Hankel transform sends even functions to even functions (and is zero when applied to odd functions), by construction. In this subsection we consider its nonsymmetric version. The reflection f (x) 7→ f (−x) will be denoted by s. d Key definition. (C. Dunkl) D = dx − xk (s − 1). ❑ The operator D is obviously odd, i.e., sDs = −D. The restriction of the d2 2k d even operator D2 to the space of even functions coincides with L = dx 2 + x dx : 2k 0 f for even f (x). x The following two lemmas are counterparts of Lemmas 2.4.2 and 2.4.4. D2 (f ) = D(f 0 ) = f 00 +

Lemma 2.4.5. (i) If λ 6= 0 or λ = 0 and k 6∈ −1/2 − Z+ , then the eigen(k) value problem Dψ = 2λψ has a unique analytic at 0 solution ψλ (x) with the (k) (k) normalization ψλ (0) = 1. Moreover, ψλ (x) = ψ (k) (λx) and ψ (k) (t) = φ(k) (t) +

(φ(k) (t))0 ψ (k) (t) + ψ (k) (−t) for φ(k) (t) = . 2 2

If λ = 0 and k = −1/2 − n, n ∈ Z+ , then ψ = 1 + Cx2n+1 for C ∈ C. (ii) Let D∗ be the adjoint of D with respect to the scalar product Z hf (x), g(x)i = f (x)g(x)|x|2k dx. R

Then D∗ = −D.

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Comment. For any k, there is an extra solution ψ = x/|x|2k+1 of the equation Dψ = 0, but it is analytic at 0 only as k ∈ − 21 − Z+ . Proof. (i) We set ψ = ψ0 + ψ1 , where ψ0 is even and ψ1 is odd. Then Dψ = 2λψ is equivalent to the system of equations ψ00 = 2λ ψ1 , ψ10 +

2k ψ1 = 2λψ0 . x

We obtain that ψ0 satisfies Lemma 2.4.2(i). d (ii) The operator |x|k ◦ D ◦ |x|−k =R dx − xk s is anti-self-adjoint with respect to the scalar product (f (x), g(x)) = R f (x)g(x)dx. ❑ Following Dunkl, we introduce the nonsymmetric Hankel transform: Z 1 (k) F(f )(λ) = f (x)ψλ (x)|x|2k dx; Γ(k + 1/2) R Fop is its action on operators. Lemma 2.4.6. (a) Fop (D) = −2λ, (b) Fop (s) = sλ , sλ f (λ) 7→ f (−λ), def d (c) Fop (2x) = Dλ == dλ − λk (sλ − 1).

❑

The main result of this subsection is the following theorem. Theorem 2.4.7. (Master formula) Z 2 2 2 (k) (k) ψλ (x)ψµ(k) (x)e−x |x|2k dx = Γ(k + 1/2)ψλ (µ)eλ +µ .

(2.4.3)

R

Proof is parallel to the proof of Theorem 2.4.1. However, the necessary calculations are simpler than in the symmetric case (even in the rank one 2 (k) (k) case). We introduce the functions ψ±,λ (x) = ψλ (x)e±x and operators D+ = 2 2 2 2 ex ◦ D ◦ e−x and D− = e−x ◦ D ◦ ex . Easy calculations show that (k)

(k)

D± ψ±,λ = 2λψ±,λ , D± = D ∓ 2x, Fop (D− ) = Dλ+ . Hence

(k)

2

(k)

F(ψ−,µ ) = Cµ eµ ψ+,µ (λ),

(2.4.4)

where Cµ does not depend on λ. Finally, Cµ = C0 = 1, thanks to the symmetry λ ↔ µ. ❑ Comment. Theorem 2.4.7 is equivalent to Theorem 2.4.1, which is a special feature of the one-dimensional setup. Generally, the nonsymmetric formula results in the symmetric one, but not the other way round. In the first place, Theorem 2.4.7 implies Theorem 2.4.1 since (k)

(k)

(k)

(k)

φλ (x) = (ψλ )0 (x) = (ψλ (x) + ψ−λ (x))/2, and for µ.

2.4. NONSYMMETRIC HANKEL TRANSFORM

189

To deduce Theorem 2.4.7 from Theorem 2.4.1 we need to show that Z 2 2 2 (k) (k) (ψλ )1 (x) (ψµ(k) )1 (x) e−x |x|2k dx = Γ(k + 1/2)(ψλ )1 (µ) eλ +µ , R

(k)

(k)

where (ψλ )1 (x) is the odd component of (ψλ )(x). We may ignore the “cross(k) (k) terms” (ψλ )1 (x)(ψµ )0 (x), since they are odd and their integrals are zero. To calculate the integral above, we can use the same Theorem 2.4.1 because of the following shift–formula: 1 (k) 2λ (k+1) (ψλ )1 = φ . x 1 + 2k λ This equivalence somewhat clarifies why the nonsymmetric Hankel transform did not appear (as far as we know) in classical works on Bessel functions. It adds nothing new to the symmetric (even) one. ❑ 2 Using the operator e−D /4 , we can rewrite formula (2.4.3) in the following form: Z 2 2 2 (k) ψλ (x)e−D /4 (f (x))e−x |x|2k dx = Γ(k + 1/2)f (λ)eλ , (2.4.5) R

where f (x) is a function from a suitable completion of the space linearly (k) generated by the functions ψµ (x). Indeed, Theorem 2.4.7 shows that (2.4.5) (k) holds for f (x) = ψµ (x). Formula (2.4.5) leads to the following entirely algebraic definition of the Hankel transform: 2

F = ex ◦ eD

2 /4

2

◦ ex .

We will use it later to introduce the truncated Hankel transform. Since D is nilpotent, (2.4.5) results in the following important corollaries. Corollary 2.4.8. (cf. Corollary 2.1.6) The nonsymmetric Hankel transform 2 2 restricts to a map F : C[x]e−x → C[λ]eλ and, moreover, preserves the filtration by degrees of polynomials. ❑ P P 2 2 (k) (k) We set Bx = µ C ψµ (x)e−x , Bλ = µ C ψµ (λ)e+λ . Corollary 2.4.9. (Inversion) Introducing the imaginary Hankel transform Z 1 def Fim (g)(x) == g(λ)ψx(k) (−λ)|λ|2k dλ, iΓ(k + 1/2) iR 2

2

F ◦ Fim = id and Fim ◦ F = id in the spaces Bx , Bλ or C[x]e−x , C[λ]eλ or their suitable completions.

190

CHAPTER 2. ONE-DIMENSIONAL DAHA Proof. This follows from the formula Z 1 2 2 2 (k) (k) ψλ (−x)ψµ(k) (x)ex |x|2k dx = Γ(k + 1/2)ψλ (µ)e−λ −µ , i iR

(2.4.6)

which results from (2.4.3) upon x 7→ ix, λ 7→ iλ, µ 7→ −iµ. The passage to 2 2 C[x]e−x and C[λ]eλ is either by means of (2.4.5) or via the completion. ❑ Corollary 2.4.10. (Plancherel formula) For functions f (x), g(x) in Bx or 2 those in C[x]e−x , we set fb = F(f ), gb = F(g). Then Z Z 1 2k f (x)g(x)|x| dx = fb(−λ)b g (λ)|λ|2k dλ. (2.4.7) i R iR Proof. The left-hand side of (2.4.7) can be readily calculated for f (x) = 2 (k) and g(x) = ψµ2 (x)e−x , thanks to formula (2.4.4). We substitute √ (k) 2x 7→ x and use that ψµ (x) depends on the product µx. Similarly, we calculate the right-hand side using (2.4.6). Completing or using (2.4.5), we 2 ❑ switch to the space C[x]e−x . Concerning completions, when k ∈ R, the last corollary determines the nonsymmetric Hankel transform on the space L2 (R, |x|2k ). Indeed, the latter 2 space is Rthe L2 –completion of the space C[x]e−x with respect to the scalar product R f (x)g(x)|x|2k dx. To be more exact, we first use this corollary for real-valued functions. The image of the corresponding L2 will be the R–subspace of L2 (ıR, |λ|2k ) formed by functions satisfying f (λ) = f (−λ). Then we extend the coefficients from R to C. 2 (k) ψµ1 (x)e−x

2.4.3

Rational DAHA

Concerning the notation, the rational DAHA is denoted by HH00 . The notation HH0 is reserved for the trigonometric limit which leads to the Harish-Chandra theory. In a sense, HH00 is a double degeneration of DAHA. This algebra is defined as follows: HH00 = hD, x, si/{sDs = −D, sxs = −x, [D, x] = 1 + 2ks, s2 = 1}; HH00 acts on the vector space C[x] via the Dunkl operator: s(f (x)) = f (−x), x(f (x)) = xf (x), D(f ) = (

d k − (s − 1))f. dx x

We will call this module the polynomial representation. Theorem 2.4.11. (a) The HH00 –module C[x] is faithful. (b) (PBW property) The elements xn Dm sε , n, m ∈ Z+ , ε = 0, 1, form a basis of HH00 .

2.4. NONSYMMETRIC HANKEL TRANSFORM

191

Proof. It is clear that any element of HH00 can be expressed in the following form: X H= cn,m,ε xn Dm sε , cn,m,ε ∈ C. n,m,ε

If cn,m,ε 6= 0 at least once, then the image of H is nonzero in the space of either even or odd functions. This gives both (a) and (b). ❑ Because the polynomial representation is always faithful, we will identify D considered as a generator of the rational DAHA with its image, the Dunkl operator. The algebra HH00 has an automorphism ω defined by ω(D) = −2x, ω(2x) = D, ω(s) = s.

(2.4.8)

It can be represented as follows: 2

ω = ex ◦ eD

2 /4

2

◦ ex = eD 2

2 /4

2

◦ ex ◦ eD

2 /4

.

(2.4.9)

2

Here we extend HH00 by adding ex , eD /4 , and treat the latter as inner auto00 c morphisms in the resulting greater algebra H H . It is simple to see that both preserve HH00 . The automorphism ω is nothing but an algebraic version of the Hankel transform F. The coincidence of the two representations of ω can be deduced from the defining relations or (much simpler) checked in the polynomial representation. It defines the action of the projective P SL(2, Z) (due to Steinberg) on HH00 . The polynomial representation is always faithful, but not always irreducible. The reducibility may occur only for special, singular, values of k. Theorem 2.4.12. The HH00 –module C[x] is irreducible if and only if k 6∈ −1/2 − Z+ . Proof. If k 6∈ −1/2 − Z+ , then the equation Dψ = 0 has a unique solution ψ = 1 in C[x]. Since the operator D is nilpotent in C[x], any submodule should contain 1 and therefore the whole C[x]. If k = −1/2 − n, then x2n+1 generates a nontrivial submodule of C[x], since D(x2n+1 ) = 0. ❑ 00 We see that HH can have finite dimensional irreducible representations for some k. It is not difficult to describe them all. Generally, the theory of finite dimensional representations of double Hecke algebras associated with root systems is far from being complete.

2.4.4

Finite dimensional modules

We will use that [h, x] = x, [h, D] = −D for h = (xD + Dx)/2.

(2.4.10)

192

CHAPTER 2. ONE-DIMENSIONAL DAHA

d Note that h is x dx + k + 1/2 in the polynomial representation. Since it is faithful, (2.4.10) is the claim that x, D are homogeneous operators of degrees ±1, which is obvious. These relations are the defining relations of osp(2|1), but we will not rely on the theory of this super Lie algebra. For our purpose, a reduction to sl2 is sufficient. Namely, we will use that the elements e = x2 , f = −D2 /4, and h satisfy the defining relations of sl2 (C). Indeed, [e, f ] = h because

[D2 , x2 ] = [D2 , x]x + x[D2 , x] = 2Dx + x(2D), and the relations [h, e] = 2e, [h, f ] = −2f readily result from (2.4.10). The Casimir operator C = h2 − 2h + 4ef becomes (h2 − 2h)(1) = h(h − 2)(1) = (k + 1/2)(k − 3/2) in C[x]. The module C[x] is the Verma module with the h–lowest weight 1/2 + k over U (osp(2|1)). It is the direct sum of the two Verma modules with the h–lowest weights 1/2 + k and 3/2 + k, formed by even and odd functions respectively, with respect to the action of U (sl2 ). Let k = −n − 21 for n ∈ Z+ . Then V2n+1 = C[x]/(x2n+1 ) is an irreducible representation of HH00 . The elements of V2n+1 can be identified with polynomials of degree smaller than 2n + 1. Theorem 2.4.13. Finite dimensional representations of HH00 exist only as k = −n − 1/2 or k = n + 1/2 for n ∈ Z+ . Given such k, the algebra HH00 has a unique finite dimensional irreducible representation up to isomorphisms. It is either V2n+1 for negative k or its image under the HH00 –automorphism: x 7→ x, D 7→ D, s 7→ −s, k 7→ −k,

(2.4.11)

in the case of positive k. Proof. Let V be an irreducible finite dimensional representation of HH00 . Then the subspaces V 0 , V 1 of V formed, respectively, either by s–invariant or s–anti-invariant vectors are preserved by e, f, and h. Indeed, sxs = −x, sDs = −D, and sh = hs. Note that s leaves all h–eigenspaces invariant. For instance, it commutes with the sl2 –action. One obtains Dx = h + k + 1/2, xD = h − k − 1/2 in V 0 , and the other way round in V 1 . Note that x and D send V 0 to V 1 . Let us check that ±k ∈ −1/2 − Z+ . All h–eigenvalues in V are integers thanks to the general theory of finite dimensional representations of sl2 (C). We pick a highest vector v, i.e., a nonzero h–eigenvector v ∈ V with the maximal possible eigenvalue m. Using the automorphism (2.4.11), we may

2.4. NONSYMMETRIC HANKEL TRANSFORM

193

assume (till the end of the proof) that it belongs to V 0 . Then m ∈ Z+ (the theory of sl2 ) and x(v) = 0 because the latter is an h–eigenvector with the eigenvalue m + 1. Hence Dx(v) = 0, m + k + 1/2 = 0, and k = −1/2 − m. Let U 0 be a nonzero irreducible sl2 (C)–submodule of V 0 . The spectrum of h in U 0 is { −n, −n + 2, . . . , n − 2, n } for an integer n ≥ 0. Let vl 6= 0 be an h–eigenvector with the eigenvalue l. If e(v) = 0, then v = cvn for a constant c, and if f (v) = 0, then v = cv−n . Let us check that Dx(vn ) = 0, xD(v−n ) = 0, and Dx(vl ) 6= 0 for l 6= n, xD(vl ) 6= 0 for l 6= −n. Both operators Dx and xD obviously preserve U 0 : Dx(vl ) = (l + k + 1/2)vl , xD(vl ) = (l − k − 1/2)vl . Hence, D2 x2 (vl ) = ((Dx)2 + (1 − 2k)(Dx))(vl ) = (l + k + 1/2)(l − k + 3/2)vl . Setting l = n, we obtain that (n + k + 1/2)(n − k + 3/2) = 0, and therefore we come to the formula k = −1/2 − n, since k < 0 and n − k + 3/2 > 0. Thus Dx(vn ) = 0. The case of xD is analogous. The next claim is that x(vn ) = 0, D(v−n ) = 0. Indeed, x(v 0 ) = 0 and D(v 0 ) = 0 for v 0 = x(vn ). Therefore v 0 ∈ V 1 , s(v 0 ) = −v 0 , and 0 = [D, x](v 0 ) = (1 + 2ks)(v 0 ) = (1 + 1 + 2n)v 0 = (2 + 2n)v 0 . Hence v 0 = x(vn ) = 0. A similar argument shows that D(v−n ) = 0. Now we use the formula D(x2 (vl )) = x(2 + xD)(vl ) = (2 + l − k − 1/2)x(vl ) = (2 + l + n)x(vl ) and obtain that x(vl ) ∈ D(U 0 ) for any −n ≤ l ≤ n. Hence U = U 0 + D(U 0 ) is x–invariant. It is obviously D–invariant and s–invariant (⇐ D(V 0 ) ⊂ V 1 ). Also, the sum is direct. Finally, U is an HH00 –module and has to coincide with V because the latter was assumed to be irreducible. The above formulas make the HH00 – isomorphism U ' V2n+1 explicit: the h–eigenvectors xi (v−n ) ∈ U go to the monomials xi ∈ V2n+1 . ❑

2.4.5

Truncated Hankel transform

Let us consider the representation V2n+1 closely. We endow it with the following two scalar products: def

hf, gi+ == Res (f (x)g(x)x−2n−1 )

194

CHAPTER 2. ONE-DIMENSIONAL DAHA

and

def

hf, gi− == Res (f (x)g(−x)x−2n−1 ), analogous to the standard scalar products considered above. Here by Res , we mean the coefficient of x−1 . One has: hsf, gi± = hf, sgi± , and hxf, gi± = ±hf, xgi± , hDf, gi± = ∓hf, Dgi± . P i Note that if f (x) = 2n i=0 ai x , then hf, f i± =

2n X

(2.4.12)

(±)i ai a2n−i ,

i=0

so these scalar products are far from being positive definite. In contrast to C[x], the Gaussians are well defined in V2n+1 : 2

e±x =

2n X

(±x2 )m /m!.

m=0

We may introduce the truncated Hankel transform on this space by the 2 2 2 2 2 old formula F+ = ex eD /4 ex . Multiplication by ex and eD /4 is well defined in the space V2n+1 because x and D are nilpotent. Similarly, F− (f (x)) = F+ (f (−x)). Another (equivalent) approach requires the truncated nonsymmetric Bessel functions, which will not be discussed here. See [CM]. Theorem 2.4.14. The truncated inversion reads as F− ◦ F+ = (−1)n id = F+ ◦ F− . Let fb = F+ (f ) and gb = F+ (g). We have the Plancherel formula hf, gi+ = (−1)n hfb, gbi− . Proof. Using (2.4.12), we conclude that F+ , inducing the automorphism ω on HH00 , sends the anti-involution corresponding to h·, ·i+ to that of h·, ·i− . We need to examine the generators s, x, D. The representation V2n+1 is irreducible, so the scalar products hf, gi+ and hfˆ, gˆi− have to be proportional. To find the proportionality coefficient, let us calculate the Hankel transform explicitly. First, F+ (1) has to be proportional to x2n , since 1 is a unique eigenvector of D and ω sends D to −2x. Hence F+ (xm ) is proportional to Dm F+ (1), that is, x2n−m . Second, 2

2

F+ (e−x ) = (ex eD

2 /4

2

2

2

ex )e−x = ex ; therefore m! F+ (x2m ) = (−1)m x2n−2m , and (n − m)! m! F+ (x2m+1 ) = (D/2)F+ (x2m ) = (−1)m x2n−2m−1 . (n − m − 1)!

2.5. POLYNOMIAL REPRESENTATION

195

2n

In particular, F+ (1) = xn! and F+ (x2n ) = (−1)n n!. We obtain the inversion formula. The Plancherel formula holds, since h1, x2n i+ = 1 and hF+ (1), F+ (x2n )i− = (−1)n hx2n , 1i− = (−1)n . ❑

2.5

Polynomial representation

Recall Theorem 2.2.1: Z ∞ √ Y 1 1 − q k+j −x2 q δk (x)dx = 2 πa , 0, i iR 1 − q 2k+j j=0 where δk (x) =

(2.5.1)

∞ Y

(1 − q j+2x )(1 − q j−2x ) (1 − q k+j+2x )(1 − q k+j−2x ) j=0

for q = e−1/a . P def nx in the variable q x , hf (x)i == c0 , For a Laurent series f (x) = ∞ n=−∞ cn q the constant term. We expand δk (x) in a Taylor series in terms of q k . It becomes a Laurent series in terms of q x with the coefficients from C[q k ][[q]]. Let ∞ X 2 def γc q nx+n /4 . − == n=−∞ 2

It is a definition, but the right-hand side does coincide with q −x in the space of distributions on periodic functions in the variable x with the period ω = 2πia. Formula (2.5.1) is equivalent to hc γ− δk (x)i = 2

2.5.1

∞ Y 1 − q k+j . 2k+j 1 − q j=0

(2.5.2)

Rogers’ polynomials

We are going to interpret the latter formula as a calculation of the structural constant of the difference spherical Fourier transform in the A1 –case. Generally speaking, the spherical transform is an integration with the spherical (or hypergeometric) functions. In this setup, the spherical functions are Rogers’ polynomials and the integration is the constant term functional. Definition 2.5.1. The Rogers polynomials pn (x) ∈ C(q, q k )[q x + q −x ] are uniquely defined by the properties: P (a) pn (x) = q nx + q −nx + |m| 0, p0 = 1; (b) hpn (x)q mx δk (x)i = 0 for all m with |m| < n.

196

CHAPTER 2. ONE-DIMENSIONAL DAHA The first two nontrivial Rogers’ polynomials are: p1 = q x + q −x , p2 = q 2x + q −2x +

(1 − q k )(1 + q) . 1 − q k+1

The formula for p1 is immediate from the following simple lemma. Lemma 2.5.2. Let pn (x) =

P

m cm q

mx

. Then cm 6= 0 only for even n − m.

Proof. The Laurent series δk (x) involves only even powers of q x . To calculate p2 , we use that δk (x) can be replaced in (b) by def

δk0 (x) ==

❑

δk (x) . hδk (x)i

We calculated it in Theorem 2.2.5 in terms of (q 2nx + q −2nx ): δk0 (x) = 1 +

(1 + q)(q k − 1) 2x (q + q −2x ) + . . . . 2(1 − q k+1 )

The constant term of p2 is the first coefficient of this expansion multiplied by −2. More generally, this argument gives that the coefficients of all pn (x) lie in Q(q, q k ), since so do the coefficients of δk0 (x). Rogers’ polynomials pn (x) play the role of the Bessel functions in the following theorem Theorem 2.5.3. hpn (x)pm (x)c γ− δk0 (x)i = 2 +n2 +2k(m+n))/4

= pn ((m + k)/2)pm (k/2)q (m

hc γ− δk0 (x)i. (2.5.3)

Proof will be given later. Obviously (2.5.3) results in the following m ↔ n –invariance:

❑

pn ((m + k)/2)pm (k/2) = pm ((n + k)/2)pn (k/2). Conversely, this formula can be readily deduced from this symmetry, as we will see. The most natural proof of this theorem goes via the q–counterparts of the nonsymmetric Bessel functions.

2.5. POLYNOMIAL REPRESENTATION

2.5.2

197

Nonsymmetric polynomials

Recall formulas (2.2.8) and (2.2.9): δk0 (x) = where

µ0k (x) + µ0k (−x) , 2

∞ Y

(1 − q j+2x )(1 − q j+1−2x ) µk (x) = , (1 − q k+j+2x )(1 − q k+j+1−2x ) j=0

q k − 1 2x (q + q 1−2x ) + · · · . 1 − q k+1 We use the following linear order on the set of monomials q nx : q nx  q mx if either |m| < |n| or n = −m and n is negative (so q −x  q x ). µ0k (x) = µk (x)/hµk (x)i = 1 +

Definition 2.5.4. The nonsymmetric polynomials en (x) ∈ C(q, q k )[q x , q −x ] for n ∈ Z are uniquely defined by the properties: (a) en (x) = q nx + lower terms with respect to Â, (b) hen (x)q −mx µ0k (x)i = 0 if q mx ≺ q nx (note the minus sign!). The definition and the beginning of the expansion of µ0k give that e0 = 1, e1 = q x , e−1 = q −x +

1 − qk x 1 − qk 2x q , e = q + q . 2 1 − q k+1 1 − q k+1

Similar to pn , each en (x) involves only monomials q mx with even n−m because µ0k (x) contains only even powers of q x . Similar to pn (x), en (x) ∈ Q(q, q k )[q x , q −x ], since µ0k (x) ∈ Q(q, q k )[[q x , q −x ]]. The main technical advantage of the theory of e–polynomials versus the p–polynomials is that we can construct them using the intertwining operators of the double Hecke algebra. The following creation operator is due to Knop and Sahi in the case of An . See [KnS] and [C16, C23] for the general theory. Proposition 2.5.5. Denoting π(f (x)) = f (1/2 − x), q (1−n)/2 en = q x π(e1−n (x)) for n > 0. Proof. It is clear that q (n−1)/2 q x π(e1−n (x)) = q nx + lower terms. So it suffices to check that hq x π(e1−n (x))q −mx µ0k (x)i = 0, m = n − 1, n − 2, . . . , 1 − n. The following two properties of π are evident: (a) hπ(f )i = hf i, (b) π(µ0k (x)) = µ0k (x).

(2.5.4)

198

CHAPTER 2. ONE-DIMENSIONAL DAHA

Using them, we see that (2.5.4) is true for m = n − 1, . . . , 2 − n. The only remaining case m = 1 − n is trivial: π(e1−n (x))q nx µ0k (x) involves only odd powers of q x . ❑ We can deduce (2.5.3) from its nonsymmetric analog. Setting εn (x) = en (x)/en (−k/2), the nonsymmetric variant of Theorem 2.5.3 reads as follows: 2 +n2 +2k(|m|+|n|))/4

hεn (x)εm (x)c γ− µ0k (x)i = hc γ− µ0k (x)i εn (m] ) q (m

,

(2.5.5)

def

where n] == (n + sgn(n)k)/2 and we set sgn(0) = −1. The nonsymmetric (and more general) formula is easier to prove than the symmetric one, similar to the case of the Hankel transform considered above. The coefficient of proportionality in formula (2.5.5) is a combination of formula (2.5.2) and the constant term conjecture (2.2.10): hc γ− µ0k (x)i =

∞ Y 1 − qj . j+k 1 − q j=0

Formula (2.5.5) implies that εn (m] ) is invariant under the change m ↔ n. This of course can be established directly. Let us do it.

2.5.3

Double affine Hecke algebra

We denote the double affine Hecke algebra by ”double H” HH. It depends on the two parameters q 1/2 and t1/2 . The generators are X ±1 , Y ±1 , T ; the relations are    T XT = X −1 , T Y −1 T = Y,  Y −1 X −1 Y XT 2 q 1/2 = 1, .   1/2 −1/2 (T − t )(T + t )=0 Comment. If t = 1, then HH is the extension of the Weyl algebra by the reflection S, i.e., hX, Y, Si/{Y −1 X −1 Y X = q −1/2 , S 2 = 1, SXS = X −1 , SY S = Y −1 }. ❑ Theorem 2.5.6. (a) (PBW property) The elements X n T ε Y m , n, m ∈ Z, ε = 0, 1, form a basis of HH. (b) Using s(f (x)) = f (−x) and π(f (x)) = f (1/2 − x), the formulas T 7→ t1/2 s +

t1/2 − t−1/2 (s − 1), X 7→ q x , Y 7→ πT q 2x − 1

(2.5.6)

define a representation of HH in the space C[q x , q −x ]. It is faithful for q apart from roots of unity.

2.5. POLYNOMIAL REPRESENTATION

199

Proof. We will start with the following lemma, which is simple to check. def

Lemma 2.5.7. Setting π == Y T −1 , the algebra HH can be alternatively described as follows: ½ ¾ T XT = X −1 , π 2 = 1, πXπ −1 = q 1/2 X −1 , ±1 HH = hT, X , πi/ . (T − t1/2 )(T + t−1/2 ) = 0 ❑ The lemma gives that the formulas from part (b) of Theorem 2.5.6 define a representation of HH . Indeed, the formula for T is well known in the theory of the affine Hecke algebra of type A1 . This operator does satisfy T XT = X −1 . So we need to check only the relations involving Y or, equivalently, π. The relations with π are simple. (a) Any element of HH is a linear combination of X n T ε Y m . These monomials are linearly independent in HH if they are independent for at least one pair of special values of q, t. Let us take q that is not a root of unity. Then the images of these monomials in C[q x , q −x ] are linearly independent. It is immediate for t1/2 = 1, i.e., for generic t. It is also true for arbitrary t, and not difficult to check. We obtain (a), and the remaining part of (b). ❑ Comment. In fact, the theorem can be reformulated as follows: H C[q x , q −x ] = IndH (C), where the subalgebra hT, Y i ⊂ HH acts on the one-dimensional space C via the character T 7→ t1/2 , Y 7→ t1/2 . Indeed, this relation includes the PBW theorem and readily results in the formulas from part (b). However, this coincidence does not include the fact that the polynomial representation is faithful for generic q. ❑ We introduce the conjugation f 7→ f¯ on the polynomial representation: q x = q −x , q 1/2 = q −1/2 , t1/2 = t−1/2 , and define the scalar product: def

hf, gi == hf g¯µ0k (x)i, where q k = t. Here it is necessary to change the field of coefficients C. It will be replaced by the ring C[q ±1/2 , t±1/2 ] or its field of rationals. We will do the same for HH . From now on q 1/2 and t1/2 will be considered as formal parameters, unless stated otherwise. Note that the operators T and π from (2.5.6) preserve the space C[q ±1/2 , t±1/2 , q ±x ]. Theorem 2.5.8. The operators T, X, Y, π, q, t are unitary with respect to the scalar product h , i.

200

CHAPTER 2. ONE-DIMENSIONAL DAHA

Proof. It is evident for X, q, t. It is also clear that π ∗ = π = π −1 . For T, 2x+k we use the formula µ0k (−x) = 1−q µ0 (x). It gives that the adjoint of s is q k −q 2x k (1 − q 2x t)(t − q 2x )s. Concerning Y, use Y = πT. ❑ Since the coefficints of the polynomials en (x) have denominators, the algebra HH and the polynomial representation will be, as a matter of fact, considered over the field C(q 1/2 , t1/2 ) in what follows, although we will continue to use the notation HH and C[q ±x ]. Moreover, it is sufficient to take Q instead of C. Note that the coefficients of en do not require the square roots; they are expressed in terms of q, t. However, the action of T and π does require the square roots of q, t. Theorem 2.5.9. Considering the polynomials en (x) as elements of the HH – module C[q ±x ], def

Y en (x) = q −n] en (x), n] == (n + sgn(n)k)/2, sgn(0) = −1, n ∈ Z. Proof. The operator Y is unitary and preserves the filtration induced by ≺, which is easy to check. Note that T does not preserve this filtration. It readily gives that en (x) are eigenvectors of Y . Use the definition. Examining the leading term q nx of en (x), one obtains the corresponding eigenvalue. ❑ def

Theorem 2.5.10. Setting εn (x) == en (x)/en (−k/2), we obtain the duality formula

εn (m] ) = εm (n] ). def

Equivalently, the pairing {f , g} == f (Y −1 )(g)(−k/2) is symmetric on C[q ±x ], where f (Y −1 ) = f (q x 7→ Y −1 ). Proof. First, we recall the definition of the anti-involution φ of HH: φ(X) = Y −1 , φ(Y ) = X −1 , φ(T ) = T, φ(q) = q, φ(t) = t.

(2.5.7)

Later it will be associated with the Fourier transform on the space of the generated functions. Second, we need the evaluation map : HH → C(q 1/2 , t1/2 ) defined by {X n T ε Y m } = t−n/2 tε/2 tm/2 . It is obviously φ–invariant. Explicitly, {H} is the evaluation of H(1) ∈ C[q ±x ] for H ∈ HH. The evaluation of a polynomial in q ±x is its value at the point x = −k/2.

2.5. POLYNOMIAL REPRESENTATION

201

Third, the scalar product {A, B} = {φ(A)B} on HH is symmetric, i.e., {A, B} = {B, A}. It follows from {φ(H)} = {H} for H ∈ HH. It coincides with {f , g} on Laurent polynomials upon the substitution X = q x . Finally, {en (X), em (X)} = {en (Y −1 )em (X)} = en (m] )em (−k/2),

(2.5.8)

where we used that Y −1 en (x) = q n] en (x) in the polynomial representation. ❑

2.5.4

Back to Rogers’ polynomials

The relation of the Macdonald polynomials to the double affine Hecke algebras was the first obvious confirmation of their importance. These algebras were designed for a somewhat different purpose: to connect the Knizhnik– Zamolodchikov equation with the quantum many-body (eigenvalue) problem in the q–case. The paper [C22] is devoted to it. It was an important step in the new theory of generalized q–hypergeometric functions, which appear as solutions of the QMBP, that is equivalent to the proper QKZ. However, the application to the celebrated Macdonald constant term conjecture and the norm conjecture was the first recognized success of the double Hecke algebras. In the A1 –case, symmetric means even. We will constantly use X = q x instead of x. Then the reflection s sends X 7→ X −1 . Concerning operators, we call them symmetric if they commute with s. Symmetric operators on the space C[X ±1 ] preserve the subspace C[X + X −1 ] of symmetric polynomials. For the operators we consider it is necessary and sufficient. Recall that p0 = 1 and the other pn are defined from the relations pn (x) = X + X −1 + lower terms, hpn (x)pm (x)δk0 (x)i = δmn Cn , where δk0 (x) = (µ0k (x) + µ0k (−x))/2 and Cn are certain constants. Theorem 2.5.11. (a) The operator Y + Y −1 commutes with T and is symdef def metric. Setting $(f )(x) == f (x + 1/2), L == Y + Y −1 | C[X+X −1 ] , L=

t1/2 X − t−1/2 X −1 t1/2 X −1 − t−1/2 X −1 $ + $ . X − X −1 X − X −1

(b) For any n > 0, µ pn = (1 + t

1/2

T )en = (1 + s)

¶ t − X2 en , 1 − X2

Lpn = (q n/2 t1/2 + q −n/2 t−1/2 )pn .

(2.5.9) (2.5.10)

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Proof. (a) The commutativity of T and Y + Y −1 immediately results from the defining relations. Generally, it is due to Bernstein and Zelevinsky. The symmetric polynomials f ∈ C[X + X −1 ] are exactly those satisfying T f = t1/2 f . Therefore Y + Y −1 is symmetric. The calculation of the restriction of Y + Y −1 to C[X + X −1 ] is simple, especially if one uses that it is symmetric. (b) Let us use (2.5.9) to introduce the polynomials p˜n . We must check that they coincide with pn . First, L˜ pn = (q n/2 t1/2 + q −n/2 t−1/2 )˜ pn , since T commutes with Y + Y −1 . Second, (Y + Y −1 )∗ = Y + Y −1 , where ∗ is with respect to the scalar product hf g¯µ0k (x)i because the operator Y is unitary. It readily results in L∗ = L, where L∗ is now defined with respect to the scalar product hf g¯δk0 (x)i, which is the symmetrization of the above pairing with µ. Note that here we can drop the conjugation and use hf gδk0 (x)i, because L is bar-invariant. Respectively, L has ∗–invariant eigenfunctions. For generic q and t, the eigenvalues of L in the space of even polynomials are pairwise distinct, and therefore the eigenvectors are pairwise orthogonal. We obtain the desired coincidence with p. ❑ 2x −2x Comment. When k = 1, we have δ1 (x) = (1 − q )(1 − q ) and it is (n+1)x −q −(n+1)x (k=1) (k=1) easy to show that pn (x) = q qx −q . Thus pn (x) is the character −x of the irreducible representation of sl2 of dimension n + 1. Is there any rea(k=1) or general sonable interpretation of the coefficients of the polynomilas en k en considered as rational functions or series in terms of q, t ? ❑

2.5.5

Conjugated polynomials

Let us examine the action of the bar-involution ¯ = X −1 , q 1/2 7→ q −1/2 , t1/2 7→ t−1/2 X on the ε–polynomials. It preserves pn and the renormalized polynomials pn /pn (−k/2), thanks to formula (2.5.10) and the relation L∗ = L. One can also see it using that δk0 is bar-invariant. We will need the following extension to HH of the Kazhdan–Lusztig involution (cf. [KL1]): 1

1

1

1

η(T ) = T −1 , η(π) = π, η(X) = X −1 , q 2 7→ q − 2 , t 2 7→ t− 2 .

(2.5.11)

Introducing T0 = πT π, we obtain η(T0 ) = T0−1 , so it coincides with the Kahdan-Lusztig involution on the affine Hecke algebra genearated by T1 = T, T0 . The importance of η in the theory of double affine Hecke algebras is due to the following proposition.

2.6. FOUR COROLLARIES

203

Proposition 2.5.12. For arbitrary f ∈ C(q 1/2 , t1/2 )[q ±x ], H(f¯) = η(H)f , H ∈ HH . For m ∈ Z, the polynomial em (x) is an eigenvector of η(Y ) with the eigenvalue q m] . Proof. It suffices to check that T is T −1 , which is straightforward.

❑

Proposition 2.5.13. For all m ∈ Z, (a) εm (x) = t−1/2 T (εm (x)), (b) εm (x) = t−1/2 X −1 ε1−m (x), (c) XT (εm (x)) = ε1−m (x). Proof. We will work backwards and prove part (c) first. Using Proposition 2.5.5, XT (em ) = XπY (em ) = q −m] Xπ(em ) = q −m] +m/2 e1−m = q −k/2 e1−m = t−1/2 e1−m for m > 0. Thus (c) results from e1−m (−k/2)/em (−k/2) = t1/2 , which is a simple corollary of the evaluation formula in the next section. To check it, one can also proceed as follows. In the relation Xπ(εm ) = Cε1−m , we need to find C. The right-hand side is C at −k/2. So we need to evaluate the left-hand side at −k/2. It suffices to know εm (1/2 + k/2). Using the duality: εm (1/2 + k/2) = ε1 (m] ) = q m] q k/2 . (b) Thanks to the previous proposition, εm (x) is an eigenvector of η(Y ) with eigenvalue q m] . Since η(Y ) = Y T −2 = πY −1 π, we obtain that εm (x) is proportional to πεm (x) and to X −1 ε1−m (x). Thus εm (x) = CX −1 ε1−m (x) for a constant C, which has to be t−1/2 due to εn (−k/2) = 1. (a) It is a combination of (b) and (c).

2.6

❑

Four corollaries

Let us emphasize the main points of the previous section. So far we have five key definitions and five key properties of the double Hecke algebra and the e–polynomials.

204

2.6.1

CHAPTER 2. ONE-DIMENSIONAL DAHA

Basic definitions

(i) Double affine Hecke algebra: ½ ¾ (T − t1/2 )(T + t−1/2 ) = 0, T XT = X −1 , HH = hX, Y, T i/ . Y −1 X −1 Y XT 2 q 1/2 = 1, T Y −1 T = Y (ii) There is also an alternative description in terms of the generators T, X, π = Y T −1 : ½ ¾ T XT = X −1 , πXπ −1 = q 1/2 X −1 , HH = hX, T, πi/ . π 2 = 1, (T − t1/2 )(T + t−1/2 ) = 0 (iii) The µ–function µ0k (x) is uniquely defined by the properties µ0k (x + 1/2) =

q 2x+k − 1 0 µk (x), and hµ0k (x)i = 1, 2x k q −q

where h i is the constant term. ¯ = (iv) The formal conjugation is the automorphism of C[X ±1 ] defined by X X −1 , q¯ = q −1 , t¯ = t−1 . We will constantly identify X with q x . (v) The linear order ≺ on the set of monomials X n : X n ≺ X m if either |n| < |m| or n = −m > 0. Let us summarize what has been proved. (a) (PBW property) The elements X n T ε Y m , n, m ∈ Z, ε = 0, 1, form a basis of HH. (b) (Polynomial representation) For s(X) = X −1 , $(X) = q 1/2 X, the formulas t1/2 − t−1/2 T 7→ t1/2 s + (s − 1), π 7→ s$, X 7→ X, Y 7→ s$T q 2x − 1 def

define a representation of HH in the space P == C[X ±1 ]. (c) (Unitary structure) The operators X, q, Y, π, T are unitary with respect to the scalar product. hf, gi = hf g¯µ0k (x)i on the space P. (d) (Nonsymmetric polynomials) They can be defined as follows: en (x) = X n mod {X m ≺ X n }, Y en = q −n] en , n] = (n + sgn(n)k)/2, 0] = −k/2. def

(2.6.1)

(e) (Duality) εn (m] ) = εm (n] ), where εn (x) == en (x)/en (−k/2). Let us discuss some applications. Our main instrument will be the duality and, in particular, the anti-involution φ : HH → HH from (2.5.7) defined on the generators by φ(X) = Y −1 , φ(Y ) = X −1 , φ(T ) = T, φ(q) = q, φ(t) = t.

2.6. FOUR COROLLARIES

2.6.2

205

Creation operators

In this subsection we will consider q and t = q k as numbers (not as formal variables). We will assume that q is not a root of unity. The coefficients of the polynomials en are rational functions of q and q k , so the polynomials en (x) are not well defined for some particular values of q and k. The theory of e–polynomials is (relatively) simple because they can be produced using the intertwining operators. The latter play the role of the creation (raising) operators in Lie theory. Corollary 2.6.1. The polynomials en (x), e1−n (x) for n > 1 are well defined for k 6∈ −{[n/2], . . . , n−1}. Explicitly, they can be obtained using the following operations: def (A) Introducing Π == Xπ, i.e., Π(X m ) = q m/2 X 1−m , e1−n = q −n/2 Πen for n ∈ Z. (B) Assuming that q 2n] 6= 1, e−n = q k/2 (T +

t1/2 − t−1/2 )en . q 2n] − 1

Proof. Recall that Y preserves the filtration on P = C[X ±1 ] induced by ≺, as well as the subspaces C[X ±2 ] and XC[X ±2 ]. The eigenvector of Y with an eigenvalue q −n] is clearly well defined if q −n] 6= q −m] for all m such that X m ≺ X n and n − m is even. Since q is not a root of unity, this condition is equivalent to n] 6= m] . It always holds when n, m > 0 or n, m ≤ 0. If n > 0, m ≤ 0, then the conditions (n + k)/2 6= (m − k)/2 or k 6= (n − m)/2 are sufficient. (A) Let us check that φ(Π) = Π: φ(Π) = φ(Xπ) = φ(Xπ −1 ) = φ(XT Y −1 ) = XT Y −1 = Π. Using ΠXΠ−1 = q 1/2 X −1 , we obtain that ΠY −1 Π−1 = q 1/2 Y. So we only need to calculate the coefficient of proportionality. Compare the proof of Proposition 2.5.5, which is based directly on the definition. (B) The relation T XT = X −1 implies µ

t1/2 − t−1/2 T+ X2 − 1

¶

µ X=X

−1

t1/2 − t−1/2 T+ X2 − 1

¶ .

Applying φ we come to ¶ µ ¶ µ t1/2 − t−1/2 t1/2 − t−1/2 −1 Y =Y T+ , T+ Y −2 − 1 Y −2 − 1

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CHAPTER 2. ONE-DIMENSIONAL DAHA

which gives the desired result up to a coefficient of proportionality. We obtain the latter coefficient from consideration of the action of T on the leading term of en . ❑ Comment. (i) We will see later that the polynomials en , e1−n , n > 1, are not well defined for k ∈ −{[n/2], . . . , n−1}. The polynomial en is well defined if and only if e1−n is well defined, thanks to (A). (ii) The corollary gives an inductive procedure for calculating the nonsymmetric polynomials, which will be used a great deal for the classification of the finite dimensional representations: A

A−1

B

A−2

B

2 0 1 e−2 −→ . . . . 1 = e0 −→ e1 −→ e−1 −→ e2 −→

2.6.3

Standard identities

Recall that εn (x) = en (x)/en (−k/2). We continue using X instead of q x in the formulas. Corollary 2.6.2. (a) (Pieri formula) Setting ν = 1 for m ≤ 0 and ν = −1 for m > 0, t1/2+ν q −m+1 − t−1/2 t1/2 − t−1/2 ε − ε1−m , m−1 tν q −m+1 − 1 tν q −m+1 − 1 t−1/2+ν q −m − t1/2 t−1/2 − t1/2 Xεm = ε − ε1−m . m+1 tν q −m − 1 tν q −m − 1 X −1 εm =

(2.6.2) (2.6.3)

(b) (evaluation formula (A1 )) We have the equalities em (−k/2) = t−|m|/2

Y 0<j 0 and |m|0 = 1 − m if m ≤ 0. (c) (norm formula) For m, n ∈ Z, hεm , εn i = δmn hεm εm µ0k (x)i = δmn

Y 0<j 0, one uses (2.6.8). ❑ Comment. (i) The existence of the three-term relation in the form of (a) can be seen directly from the orthogonality of polynomials en (x). The approach based on the duality is simpler and readily gives the exact coefficients. (ii) Formula (b) shows that the polynomial en is not well defined for values of k from Corollary 2.6.1(a), because otherwise en (−k/2) would be defined. So we have the complete list of singular k for each en . (iii) Formula (b) as k = 1 is a polynomial in the variable q with positive integral coefficients for either positive odd m or negative even m. These (k=1) coefficients have some combinatorial interpretation. Note that pn (1/2) is the so-called q–dimension of the representation of sl2 of dimension n+1. There is a partial interpretation of en (−k/2) and pn (k/2) considered as a series in terms of q, t1/2 . However, it is for A1 only and there is no known connection with representation theory.

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CHAPTER 2. ONE-DIMENSIONAL DAHA

2.6.4

Changing k to k+1

Now we are going to discuss the shift formula in the presence of the polyno(k) mials pn (x). The p–polynomials appear because of the following corollary. Corollary 2.6.3. Let A = (t1/2 X − t−1/2 X −1 ) and B = (t1/2 Y −1 − t−1/2 Y ). def The operator S == t−1/2 A−1 B preserves the space C[X + X −1 ]. Its restriction def $−1 −$ to this space is X−X −1 for $(f )(x) == f (x + 1/2) , and (k+1)

S(pn(k) (x)) = (q −n/2 − q n/2 )pn−1 for n > 0.

(2.6.9)

Proof. It is straightforward to check the following equality in HH: (T + t−1/2 )(t1/2 X − t−1/2 X −1 ) = (t1/2 X −1 − t−1/2 X)(T − t1/2 ). As a corollary, we obtain that the eigenspace of T in P with the eigenvalue −t−1/2 is AC[X + X −1 ]. Applying the anti-involution φ to this equality we obtain (t1/2 Y −1 − t−1/2 Y )(T + t−1/2 ) = (T − t1/2 )(t1/2 Y − t−1/2 Y −1 ). Therefore B maps C[X + X −1 ] to the T –eigenspace of P with eigenvalue −t−1/2 , that is, AC[X +X −1 ]. This shows that S is well defined on C[X +X −1 ] and, moreover, preserves this space. The formula for the restriction of S to C[X + X −1 ] is simple. Cf. Theorem 2.5.11. The formula for S (or, directly, the definition) givs that the left-hand side and the right-hand side of (2.6.9) coincide. Therefore it suffices to check 0 hS(pn(k) (x))g(x)δk+1 (x)i = 0

for any g(x) ∈ C[q x + q −x ] of degree m < n − 1. Let us prove that hS(pn(k) (x))g(x)µ0k+1 (x)i = 0 for any g(x) ∈ C[q ±x ] of degree m < n − 1. Symmetrizing the latter for even g, we obtain the former. First, µ0k+1 (x) =

1 − q k+1 (1 − q k+2x )(1 − q k+1−2x )µ0k (x) = (1 − q 2k+1 )(1 + q k+1 ) 0 ˜ = A(x)A(x)µ k (x),

where A(x) = t1/2 X −t−1/2 X −1 = q (k+2x)/2 −q −(k+2x)/2 and A˜ is a polynomial in q x of the first degree. Second, 0 ˜ hS(pn(k) (x))g(x)µ0k+1 (x)i = hBpn(k) (x)Ag(x)µ k (x)i.

˜ Here Ag(x) is a polynomial of degree m0 < n. Third, pn (x) is a linear (k) (k) combination of en (x) and e−n (x) by Theorem 2.5.11(b). The same holds for (k) Bpn (x). We get the desired result. ❑ (k)

2.6. FOUR COROLLARIES

2.6.5

209

Shift formula

Let us discuss an analytic interpretation of this corollary. We need to choose the space of functions F and the integration %. The following cases can be considered: (a) F is the space of even Laurent series in terms of q x with the constant term taken as the functional %; (b) F is the space of even analytic functions on the strip Z −1/2 − δ ≤ <x ≤ 1/2 + δ for δ > 0, and % = (·); iR

R (c) F is the space of continuous even functions on the real line and % = R ; (d) F is a space of even functions that are defined and continuous on the two lines z = ±εı + R for small ε > 0, i.e., on C ∪ {−C} for the the sharp integration path C from (2.3.1), and, moreover, they are analytic in the rectangle {x | −ε < =x < ε, −1/2 − δ ≤ <x ≤ 1/2 + δ}. In this case,

Z

Z

%=2

= C

b C

b = C ∪ {−C} , where C

for the complex conjugation (preserving C, but changing its orientation). In all cases, the integration % is invariant under the change of variable x 7→ −x, because the integration paths are s–invariant. Note that, topologically, b is the cross of the diagonals at the origin, both directed upwards. On even C R functions, 21 Cb coincides with the sharp integration from (2.3.1). Note that all the integrations defined above are zero for odd functions. The analytic properties of f are necessary to ensure the invariance of % under the change of variables x 7→ x ± 1/2. The corresponding domains of analyticity allow us to deform the contours after the shift by ±1/2. The real case is the most relaxed: no analiticity domains are necessary and there is no problem with the real translation. However, the q–Gauss integrals are the most difficult to calculate in this case. Corollary 2.6.4. Let F, % be one of the pairs above. Then, for any f ∈ F such that either the left-hand or the right-hand side below is well defined, the other is well defined too and (k+1)

%(S(f )pn−1 (x)δk+1 (x)) = q k (q (−k−n)/2 − q (k+n)/2 )%(f pn(k) (x)δk (x)). Proof will be given below. ❑ The corallary readily results in the main property of the q–Mellin transform, namely, Theorem 2.2.7. The proof given there was by straightforward integration.

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Q 1−q k+j Recall the notation gk = ∞ j=0 1−q 2k+j . The q–Mellin transform was defined as follows: 1 Ψk (f (x)) = %(f (x)δk (x)). gk Let us deduce from the corollary that Ψk (f (x)) = (1 − q k+1 )Ψk+1 (f (x)) + q k+3/2 Ψk+2 (S 2 (f (x))).

(2.6.10)

First, δk+1 (x) = (1 − tX 2 )(1 − tX −2 )δk (x) = µ µ ¶¶ (1 − t)(1 + q) (k) 2 = 1 + t − t p2 (x) − δk (x) = 1 − tq ¶ µ (1 + t)(1 − t2 q) (k) − tp2 (x) δk (x). = 1 − tq Using Corollary 2.6.4 twice, %(f (x)δk+1 (x)) (1 + t)(1 − t2 q) (k) = %(f (x)δk (x)) − t%(f (x)p2 (x)δk (x)) = 1 − tq (1 + t)(1 − t2 q) = %(f (x)δk (x))− 1 − tq t−1 q −1 %(f (x)δk+2 (x)), − (tq − t−1 q −1 )(tq 3/2 − t−1 q −3/2 ) which is equivalent to formula (2.6.10).

2.6.6

Proof of the shift formula

First, we switch from δ to µ: δk+1 (x) = (1 − q −2x )(1 − q k+2x )µk (x). Let us rewrite it in terms of A, q x = X and q k/2 = t1/2 : A2 µk (x) = −t−1/2

A δk+1 (x). X − X −1

Since f (x) is an even function, %(A2 f (x)µk (x)) = %(−t−1/2

A f (x)δk+1 (x)) = X − X −1

1 + t−1 %(f (x)δk+1 (x)). =− 2

(2.6.11)

2.6. FOUR COROLLARIES

211

A When replacing X−X −1 f (x) by its symmetrization, we used the invariance of % and δk+1 (x) under the change x 7→ −x. Second, we take an even function g(x) of the same type as f such that g(x) = g(x). For instance, it can be a Laurent polynomial or a convergent Laurent series with the coefficients invariant under q 7→ q −1 and t 7→ t−1 . It is (k) not restrictive because later g(x) will be one of pn , which are bar-invariant. One gets

1 + t−1 %(S(f (x))S(g(x))δk+1 (x)) 2 (1)

(2)

= −%(A2 S(f (x))S(g(x))µk (x)) = %(A2 S(f (x))S(g(x))µk (x)) (4)

(3)

= −%(B(f (x))B(g(x))µk (x)) = %(B 2 (f (x))g(x)µk (x)),

where the first equality follows from (2.6.11), the second follows from S = −S, and the third follows from A¯ = −A. Only the fourth equality requires some comment. In the algebraic variant, i.e., for the constant term integration (a), we can simply use that Y is unitary. See Theorem 2.5.8. The same argument is applied for the other three integrations, since % is invariant under x 7→ −x and x 7→ x ± 1/2. Note that the latter symmetry is necessary when collecting B together. def Third, we apply the t–symmetrizer P == (1 + t)−1 (1 + t1/2 T ). It projects C[X ±1 ] onto C[X + X −1 ]. One then has PB 2 P = −BB 0 P, where B 0 = t1/2 Y − t−1/2 Y −1 .

(2.6.12)

Indeed, BB 0 = −Y 2 − Y −2 + t + t−1 commutes with T and, in particular, is even. So it suffices to check that PB(B + B 0 )(f ) = 0 for any f ∈ C[X + X −1 ]. However, B + B 0 = (t1/2 − t−1/2 )(Y + Y −1 ) commutes with T. Hence f 0 = (B + B 0 )(f ) ∈ C[X + X −1 ] and T B(f 0 ) = −t−1/2 f 0 . Cf. the proof of Corollary 2.6.3 above. Formula (2.6.12) is now checked. Combining the previous formulas, we obtain 1/2(1 + t−1 )%(S(f (x))S(g(x))δk+1 (x)) = %(B 2 (f (x))g(x)µk (x)) = = %(B 2 (P(f (x)))P(g(x))µk (x)) = %(PB 2 P(f (x))g(x)µk (x)) = = −%(BB 0 (f (x))g(x)µk (x)) = −%(f (x)BB 0 (g(x))µk (x)) = = −1/2(1 + t)%(f (x)BB 0 (g(x))δk (x)). Thus the relation %(S(f (x))S(g(x))δk+1 (x)) = −t%(f (x)BB 0 (g(x))δk (x)) is proved, provided that f (x), g(x) ∈ C[q x + q −x ] and g(x) = g(x).

(2.6.13)

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CHAPTER 2. ONE-DIMENSIONAL DAHA (k)

Now we simply take g(x) = pn (x) in (2.6.13) for n ≥ 1. Using Theorem 2.5.11, n −n −1 BB 0 (p(k) t + t + t−1 )p(k) n (x)) = (−q t − q n (x)

= (tq n/2 − t−1 q −n/2 )(q −n/2 − q n/2 )p(k) n (x). Using Corollary 2.6.3, (k+1)

%(S(f )pn−1 δk+1 ) = −t(tq n/2 − t−1 q −n/2 )%(f (x)pn(k) δk ). Corollary 2.6.4 is proved.

2.7

DAHA–Fourier transforms

Recall the notation. We set 0] = −k/2 and n] = (n + sgn(n)k)/2 for integers n 6= 0, where t = q k . For generic q, t, the Laurent polynomials εn (x) are uniquely defined by the properties Y εn (x) = q −n] εn (x) and εn (0] ) = 1. We permanently use X = q x . P∞ n2 /4 n X . We will also use γ, a solution The Gaussian is γc − = n=−∞ q of the corresponding difference equation treated as a formal symbol. When 2 Jackson’s summation is considered, γ = q x . The scalar product hf, gi = hf g¯µ0k i is given in terms of µ0k (x) = 1 +

q k − 1 2x (q + q 1−2x ) + . . . , k+1 1−q

the constant term functional h·i, and the involution f 7→ f¯ of the space of Laurent polynomials P = C(q 1/2 , t1/2 )[X ±1 ], where ¯ = X −1 , q¯1/2 = q −1/2 , t¯1/2 = t−1/2 . X Our aim is the following theorem. Theorem 2.7.1. (Master formulas) For arbitrary m, n ∈ Z, hεn (x)εm (x)c γ− µ0k (x)i = q hεn (x)εm (x)c γ− µ0k (x)i = q

m2 +n2 +2k(|m|+|n|) 4 m2 +n2 +2k(|m|+|n|) 4

εm (n] )hc γ− µ0k (x)i,

(2.7.1)

εm (n] )hc γ− µ0k (x)i.

(2.7.2)

Before proving the theorem, we need to establish several general facts.

2.7. DAHA–FOURIER TRANSFORMS

2.7.1

213

Functional representation

def For Z] == {n] |n ∈ Z} ⊂ C, let Fˆ be the space of functions on Z] and let F ⊂ Fˆ be the subspace of functions with compact support. We have an ˆ evident discretization map χ : C[X ±1 ] → F.

Theorem 2.7.2. The space Fˆ has the natural structure of an HH–module making the map χ : P → Fˆ an HH –homomorphism; F ⊂ Fˆ is an HH – submodule. ˆ We define (Xf )(n] ) = q n] f (n] ), (πf )(n] ) = f (1/2−n] ). Proof. Let f ∈ F. Note that the set Z] is invariant under π : x 7→ 1/2 − x. It is not s–invariant. However, the operator (T f )(n] ) =

t1/2 q 2n] − t−1/2 t1/2 − t−1/2 f (−n ) − f (n] ) ] q 2n] − 1 q 2n] − 1

(2.7.3)

is well defined because −n] 6∈ Z] for n = 0, precisely when the first term of (2.7.3) vanishes. ❑ Definition 2.7.3. (a) The Fourier transform S acts from P to F and is given by the formula def

S(f )(n] ) == hf εn (x)µ0k (x)i. (b) Its antilinear counterpart, conjugating q and t, is introduced as follows: def E(f )(n] ) == hf¯εn (x)µ0k (x)i. ❑ The fact that the functions S(f ) and E(f ) have compact supports readily follows from orthogonality relations for the polynomials εn (x); any polynomial is their linear combination. Theorem 2.7.4. (a) The formulas ²(X) = Y, ²(Y ) = X, ²(T ) = T −1 , ²(q) = q −1 ; ²(t) = t−1 can be extended to an antilinear automorphism of HH. The formulas σ(X) = Y −1 , σ(T ) = T, σ(Y ) = q −1/2 Y −1 XY = XT 2 define a linear automorphism of HH , fixing q, t. The connection with the involution η from (2.5.11) is as follows: η = ²σ = σ −1 ². (b) For any H ∈ HH, f ∈ P, S(H(f )) = σ(H)(S(f )), E(H(f )) = ²(H)(E(f )).

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Proof. Claim (a) is straightforward. Let us check (b) for the operator E. It holds for Y because of the definition of εn and, therefore, for X, thanks to the duality (Theorem 2.5.8). Hence it holds for T 2 = q −1/2 X −1 Y −1 XY and for T = (t1/2 − t−1/2 )−1 (T 2 − 1). Here we use that X, T, Y are unitary for h f , g i. The special case t = 1 is not a problem because it is sufficient to check (b) for generic t. The automorphism corresponding to the operator S is ²η = σ, thanks to Proposition 2.5.12. ❑ We already defined automorphisms ², σ, η = ²σ of the algebra HH and the anti-involution φ. See (2.5.7). Completing the preparation to proving the master formulas, let us introduce two more automorphisms. Proposition 2.7.5. (a) The formulas τ+ : X 7→ X, T 7→ T, Y 7→ q −1/4 XY, τ− : Y 7→ Y, T 7→ T, X 7→ q 1/4 Y X can be extended to linear automorphisms of HH . (b) We have the following identities in Aut(HH): τ+ τ−−1 τ+ = σ = τ−−1 τ+ τ−−1 , σ 2 = T −1 (·)T, στ+ σ −1 = τ−−1 , ²τ+ ² = τ− , φτ+ φ = τ− . Proof. Straightforward. ❑ 2 Comment. It is immediate from (b) that σ commutes with τ+ and τ− . This also follows from σ 2 = T (·)T −1 . The automorphism τ− is inner in the representation P. This is not true for τ+ . ❑ The automorphism τ+ is directly related to the Gaussian γc − , namely, H γc c − = γ − τ+ (H) in the polynomial representation. Actually, what we need here is an abstract function γ satisfying the relation: γ(x + 1/2) = q 1/4 q x γ(x), which readily results in γHγ −1 = τ+ (H) in any functional space.

2.7. DAHA–FOURIER TRANSFORMS

2.7.2

215

Proof of the master formulas 2

x We start with S. Treating γ −1 as γc − and γ as q , we put def

def

+ ε− c − εm (x), εm (n] ) == (γεm )(n] ). m (x) == γ

Note that they belong to different spaces. Then (2.7.1) is equivalent to the formula 2 m2] −02] S(ε− εm (n] )q n] hc γ− µ0k (x)i m )(n] ) = q or, equivalently, to

2

2

m] −0] + εm hc γ− µ0k (x)i. S(ε− m) = q

Actually, it is sufficent to establish that + S(ε− m ) = Cm εm

(2.7.4)

for a constant Cm , since the left-hand side of (2.7.1) is m ↔ n–symmetric and this constant can only be 1. Introducing the operators def

def

Y + == γY γ −1 , Y − == γ −1 Y γ,

(2.7.5)

we claim that σ(Y − ) = Y + . Indeed, Y − = τ+−1 (Y ) = q 1/4 X −1 Y and σ(Y − ) = σ(q 1/4 X −1 Y ) = q −1/4 Y Y −1 XY = q −1/4 XY = τ+ (Y ). −m] ± εm . For generic q and k, the eigenvalues are Obviously Y ± (ε± m) = q − distinct, the HH –module P γc − is irreducible, and {εm } is its basis. Similarly, all eigenvectors of Y + in Fγ are simple. This gives (2.7.4) and therefore (2.7.1). Now we can make q, t arbitrary, provided that (2.7.1) is well defined. Switching to (2.7.2), its left-hand side is not m ↔ n–symmetric anymore. So we have to proceed in a slightly different way. def (−) def (+) First, we set ε¯m == εm (x)c γ− , ε¯m (n] ) == εm (n] )γ(n] ), and come to the relation 2 (−) (2.7.6) S(¯ εm )(n] ) = Cm εm (n] )q n] . def Indeed, for Ye == η(Y ),

Ye (εn ) = q n] εn (±)

and ε¯m are eigenvectors of the operators def def Ye + == γ Ye γ −1 = τ+ (Ye ), Ye − == γ −1 Ye γ = τ+−1 (Ye ).

Then we observe that σ(Ye − ) = Ye + , which results from the relation σ = τ+ τ−−1 τ+ : σ(Ye − ) = στ+−1 η(Y ) = στ+−1 ²σ(Y ) = τ+ (τ+−1 στ+−1 )²σ(Y ) = τ+ τ −1 ²σ(Y ) = τ+ ²στ −1 (Y ) = τ+ ²σ(Y ) = Ye + . −

−

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Formula (2.7.6) is now checked. Second, we establish that 2

m] E(ε− . n )(m] ) = Dn εn (m] )q

(2.7.7)

This formula follows from the identity ²(Y − ) = τ+ η(Y ) for the operator Y − from (2.7.5). Let us prove it: ²(Y − ) = ²τ+ (Y ) = τ+ η(σ −1 ²τ+−1 ²τ+ )(Y ) = τ+ η(σ −1 τ−−1 τ+ )(Y ) = τ+ ητ− (Y ) = τ+ η(Y ). 2

2

Third, formulas (2.7.6) and (2.7.7) result in Cm = q m] C and Dn = q n] C, where C does not depend on m and n, which concludes the proof of formula (2.7.2). ❑

2.7.3

Topological interpretation

We use that the relations of an HH are mainly of group nature and introduce the braid group (elliptic) À ¿ T XT = X −1 , T Y −1 T = Y, 1/4 . Bq = hT, X, Y, q i/ Y −1 X −1 Y XT 2 q 1/2 = 1 Now T, X, Y, q 1/4 are treated as group generators and q 1/4 is assumed central. The double affine Hecke algebra HH is the quotient of the group algebra of Bq by the quadratic Hecke relation. It is easy to see that the change of variables q 1/4 T 7→ T , q −1/4 X 7→ X, q 1/4 Y 7→ Y defines an isomorphism Bq ∼ = B1 × Z, where the generator of Z is q 1/4 . In this subsection, we give a topological interpretation of the group B1 = hT, X, Y i/hT XT = X −1 , T Y −1 T = Y, Y −1 X −1 Y XT 2 = 1i. Let E be an elliptic curve over C, i.e., E = C/Λ, where Λ = Z + Zı. Topologically, the lattice can be arbitrary. Let o ∈ E be the zero point, and −1 the automorphism x 7→ −x of E. We are going to calculate the fundamental group of the space (E \ o)/ ± 1 = P1C \ o. Since this space is contractible, its usual fundamental group is trivial. We can take the quotient after removing all (four) ramification points of −1. However, it would enlarge the fundamental group dramatically. Thus we need to understand this space in a more refined way. Let us fix the base (starting) point ? = −ε − εı ∈ C for small ε > 0. Proposition 2.7.6. We have an isomorphism B1 ∼ = π1orb ((E \ o)/ ± 1), where π1orb (·) is the orbifold fundamental group, which will be defined in the process of proving the proposition.

2.7. DAHA–FOURIER TRANSFORMS

217

Proof. The projection map E \ o → (E \ o)/ ± 1 = P1C \ o has three branching points, which come from the nonzero points of order 2 on E. So by definition, π1orb ((E \ o)/ ± 1) is generated by three involutions A, B, C, namely, the clockwise loops from ? around the branching points in P1C . There are no other relations. We claim that the assignment A = XT , B = T −1 Y , C = XT Y defines a homomorphism π1orb ((E \ o)/ ± 1) → B1 . Indeed, A and B are obviously involutive. Concerning C, the image of its square is XT Y XT Y = T −1 X −1 Y XT Y = T −1 Y T −1 Y = 1. This homomorphism is an isomorphism. The inversion is given by the formulas ACB = T , ABCA = X and AC = Y . ❑ This approach can hardly be generalized to arbitrary root systems; the following (equivalent) constructions can. The definition of the fundamental group is modified as follows. We follow [C13]. See also paper [Io] (in this paper the orbifold group is not involved and the construction due to v.d. Lek is used). We switch from E to its universal cover C and define the paths as curves c = {±1}nΛ. The generators T, X, Y γ ∈ C \ Λ from ? to w(?), b where w b∈W are shown as the arrows in Figure 2.1.

Y

P

u S

t a

n

c

t u

r e

r t

p

t .

T

X

Figure 2.1: Generators of B1 c : we add the image of the second The composition of the paths is via W path under w b to the first path if the latter ends at w(?). b The corresponding variant of Proposition 2.7.6 reads as follows.

218

CHAPTER 2. ONE-DIMENSIONAL DAHA

Proposition 2.7.7. The fundamental group of the above paths modulo homotopy is isomorphic to B1 , where T is the half-turn, i.e., the clockwise halfcircle from ? to s(?), and X, Y are 1 and ı considered as vectors in R2 (= C) originated at the base point ?. Proof is in Figures 2.1 and 2.2. Figure 2.1 gives that when we first use X then T and then again X (note that after T the direction of X will be opposit!), and then again T, the loop corresponding to the product T XT X will contain no punctures inside. Thus it equals id in the fundamental group. The reasoning for T Y −1 T Y −1 T is the same. Concerning the “commutator” relation, see Figure 2.2. ❑

X 2 - 1

T

- 1 Y

- 1

- 1 X

Y X

Y

Y T T

X

Figure 2.2: Relation T 2 Y −1 X −1 Y X = 1

Actually, this definition is close to the calculation of the fundamental group of {E × E\ diagonal} divided by the transposition of the components. See [Bi]. However, there is no exact coincidence. Let me mention the relation to the elliptic braid group due to v.d. Lek, although he removes all points of second order and his group is significantly larger. Noncommutative Kodaira–Spencer map. A topological interpretation is the best way to understand why the group SL2 (Z) acts in B1 projectively. Its elements act in C natuarally, by the corresponding real linear transformations. On E, they commute with the reflection −1, preserve o, and permute the other three points of second order. Given g ∈ SL2 (Z), we set g = exp(h), gt = exp(th) for the proper h ∈ sl2 (R), 0 ≤ t ≤ 1.

2.7. DAHA–FOURIER TRANSFORMS

219

The position of the base point ? will become g(?), so we need to go back, i.e., connect the image with the base point by a path. To be more exact, the g–image of γ ∈ π1orb will be the union of the paths {gt (?)} ∪ g(γ) ∪ {w(g b −t (?)}, where the path for γ goes from ? to the point w(?). b

rotation Y

s (Y)= T

-1

-1 X

T

=X

T

2

X

s (X

-1

)=Y

Figure 2.3: Relations σ(X) = Y −1 , σ(Y ) = XT 2

Figure 2.3 shows the action of the automorphism σ corresponding to the rotation of the periods and ? by 90◦ with the origin taken as the center. Here the dark straight arrows show the images of X, Y (straight white arrows) with respect to this rotation. The quarter of a turn from the point ? is the rotation path {gt (?)} of this point as 0 ≤ t ≤ 1. The other quarter of a turn is its X–image with the opposit orientation. We can always choose the base point sufficiently close to 0 and connect it with its g–image in a small neighborhood of zero. This makes the corresponding automorphism of B1 unique up to powers of T 2 . All such automorphisms fix T, because they preserve zero and the orientation. Thus we have constructed a homomorphism α : SL2 (Z) → AutT (B1 )/T 2Z , where AutT (B1 ) is the group of automorphisms of B1 fixing T. The elements from T 2Z = {T 2n } are identified with the corresponding inner automorphisms.

220

CHAPTER 2. ONE-DIMENSIONAL DAHA µ

1 1 0 1

¶

µ

1 0 1 1

¶

. Then and Let τ+ , τ− be the α–images of the matrices µ ¶ 0 1 , and σ 2 has to be the conjugation σ = τ+ τ−−1 τ+ corresponds to −1 0 by T 2l−1 for some l. Similarly, τ+ τ−−1 τ+ = T 2m τ−−1 τ+ τ−−1 . Using the rescaling τ± 7→ T 2m± τ± for m− + m+ = m, we can eliminate T 2m and make 0 ≤ l ≤ 5. Note that, generally, l mod 6 is the invariant of the action, that is due to Steinberg. Taking the “simplest” pullbacks for τ± , we easily check that l = 0 and calculate the images of the generators under τ± and σ. We arrive at the relations from Proposition 2.7.5. Abstract construction. Generalizing, let E be an algebraic, or complex analytic, or symplectic, or real analytic manifold, or similar. It may be noncompact and singular. We assume that there is a continuous family of topological isomorphisms E → Et for manifolds Et as 0 ≤ t ≤ 1, and that E1 is isomorphic to E0 = E. The path {Et } in the moduli space M of E induces an outer automorphism ε of the fundamental group π1 (E, ?) defined as above. The procedure is as follows. We take the image of γ ∈ π1 (E, ?) in π1 (E1 , ?1 ) for the image ?1 of the base point ? ∈ E and conjugate it by the path from ?1 to ?. Thus, the fundamental group π1 (M) (whatever it is) acts in π1 (E) by outer automorphisms modulo inner automorphisms. The above considerations correspond to the case when a group G acts in E preserving a submanifold D. Then π1 (M) acts in π1orb ((E \ D)/G) by outer automorphisms. Another variant is with a Galois group taken instead of π1 (M) assuming that E is an algebraic variety over a field that is not algebraically closed. The action of π1 (M) on an individual π1 (E) generalizes, in a way, the celebrated Kodaira–Spencer map and is of obvious importance. However, calculating the fundamental groups of algebraic (or similar) varieties, generally speaking, is difficult. The main examples are the products of algebraic curves and related configuration spaces. Not much can be extracted from the action above without an explicit description of the fundamental group.

2.7.4

Plancherel formulas

Recall that we have two representations of the double affine Hecke algebra HH , the polynomial representation P = C[X ±1 ] and the representation F in

2.7. DAHA–FOURIER TRANSFORMS

221

the space of finitely supported functions on the set Z] . We constructed two Fourier transforms S, E : P → F defined by S(f )(n] ) = hf εn (x)µ0k (x)i0 , E(f )(n] ) = hf εn (x)µ0k (x)i0 , where h·i0 denotes the constant term. def The space P is equiped with the scalar product hf, gi0 == hf g¯µ0k (x)i0 . Its F–counterpart is as follows: def

hf, gi1 == hf g¯µ1k i1 , where g¯(n] ), = g(n] ), X def def f (n] ), µk1 (z) == µk (z)/µk (0] ). hf i1 ==

(2.7.8)

n∈Z

A simple calculation for n ∈ Z>0 gives that µ1k (n] )

=

µ1k ((1

− n)] ) = q

−k(n−1)

n−1 Y j=1

1 − q 2k+j . 1 − qj

(2.7.9)

Note that (1 − n)] = 1/2 − n] and hence π(µ1k ) = µ1k . Also, µ1k = µ1k , which makes the form h·, ·i1 symmetric. Theorem 2.7.8. (Plancherel formula I). For any f, g ∈ P, hf, gi0 = hS(f ), S(g)i1 = hE(f ), E(g)i1 .

(2.7.10)

Proof. Let H 7→ H ? denote taking the adjoint with respect to the scalar product h·, ·i0 . From Theorem 2.5.8, X ? = X −1 , Y ? = Y −1 T ? = T −1 q ? = q −1 , t? = t−1 . It is an anti-automorphism of HH . The theorem follows formally from the following statements. (i) For automorphisms σ and ² from Theorem 2.7.4, σ ? σ −1 = ?, ² ? −1 ² = ?. Use that σ and ² are homomorpisms of the group Bq . Any group automorphisms commute with the inversion g 7→ g −1 . (ii) The representations P and F of the algebra HH are irreducible. This is true only for sufficiently generic q, t, but we may assume this because it sufices to check (2.7.10) for generic q, t. (iii) The relations (2.7.8) for f = g = 1, that guarantees that the coefficient of proportionality in the Plancherel formula is 1. ❑ ] ] We introduce the characteristic functions and delta functions χm , δm ∈ F by the formulas ] χ]m (n] ) = δmn , δm (n] ) = δmn /µ1k (n] ),

where δmn is the Kronecker delta.

(2.7.11)

222

CHAPTER 2. ONE-DIMENSIONAL DAHA

] , Theorem 2.7.9. (a) E(εm (x)) = δm

] ). (b) S(εm (x)) = t1/2 T −1 (δm

Proof. (a) By the definition of E and the norm formulas (2.6.5), ] E(εm (x)) = χ]m hεm (x), εm (x)i0 = δm .

Instead of using the norm formulas we can apply the Plancherel theorem: hεm (x), εm (x)i0 = hχ]m , χ]m i1 hεm (x), εm (x)i20 . Since hχ]m , χ]m i1 = µ1k (m] ), we obtain that hεm (x), εm (x)i0 = µ1k (m] )−1 . ] (b) From (a), S(εm (x)) = δm . Then we use part (a) of Proposition 2.5.13. ❑ def

Corollary 2.7.10. Let εb(m] ; x) == εm (x). For any g ∈ F, S−1 (g) = t−1/2 hg T (1) (b ε(m] ; x))µ1k i1 ,

(2.7.12)

where T (1) acts on the first argument of εb via (2.7.3). ] due to Theorem 2.7.9, Proof. Using that S(εm (x)) = t1/2 T −1 δm ] S( t−1/2 hg T (1) (b ε(m] ; x)) ) = t−1/2 hg T (t1/2 T −1 δm )µ1k i1 ] 1 = hg δm µk i1 = hg χ]m i1 = g(m] ).

(2.7.13) ❑

We are going to drop the bar-conjugation in the scalar products: def

def

hhf, gii0 == hf gµ0k (x)i0 on P, hhf, gii1 == hf gµ1k i1 on F. Theorem 2.7.9 and Theorem 2.7.2. result in the following corollary. Corollary 2.7.11. For arbitrary n, m ∈ Z, hεn (x)εm (x)µ0k (x)i0 = t1/2 T −1 δn] (m] ), where T (δn] ) =

(2.7.14)

t1/2 q 2n] − t−1/2 ] t1/2 − t−1/2 ] δ − δ . −n q 2n] − 1 q 2n] − 1 n

In particular, the inequality hεn (x)εm (x)µ0k (x)i0 6= 0 implies m = ±n.

❑

Theorem 2.7.12. (Plancherel formula II) For any f, g ∈ P, we have t−1/2 hhS(f ), T S(g)ii1 = hhf, gii0 = t−1/2 hhE(f ), T E(g)ii1 .

(2.7.15)

2.7. DAHA–FOURIER TRANSFORMS

223

Proof. Let ¦ denote the anti-involution corresponding to the scalar product hhf, gii0 . It equals ? ◦ η: X ¦ = X, T ¦ = T, π ¦ = π, Y ¦ = T Y T −1 . The same anti-involution serves the scalar product hhf, gii1 . Following Theorem 2.7.8 (see (i–iii)), we need to check that σ ¦ σ −1 = ² ¦ ²−1 = T −1 ¦ T. This is straightforward. ❑ Preparing for the inversion theorems, let us introduce the conjugated ¯ E: ¯ Fourier transforms S, ¯ : f 7→ hf εn (x)µ0 (x)i0 , S k ¯ E : f 7→ hf εn (x)µ0 (x)i0 .

(2.7.16)

k

Theorem 2.7.4 states that, for any H ∈ HH, f ∈ P, S(H(f )) = σ(H)(S(f )), E(H(f )) = ²(H)(E(f )). ¯ and E: ¯ It is not difficult to find the automorphisms corresponding to S ¯ ¯ )), E(H(f ¯ ¯ )). S(H(f )) = σ −1 (H)(S(f )) = η²(H)(E(f Following Theorem 2.7.1, we get another master formula: hεn (x)εm (x)c γ− µ0k (x)i =q

m2] +n2] −202]

t−1/2 (T εm (x))(n] )hc γ− µ0k (x)i.

Algebraically, it is equvalent to the formula σ −1 τ+−1 η(Y ) = τ+ (T (η(Y )T −1 ), resulting from σ −2 = T (·)T −1 and ητ± η = τ±−1 : σ −1 τ+−1 η = τ+ τ+−1 (σ −2 σ)τ+−1 η = τ+ σ −2 (τ+−1 (σ)τ+−1 )η = τ+ σ −2 τ−−1 η = τ+ σ −2 τ−−1 η = τ+ σ −2 ητ− .

(2.7.17)

224

CHAPTER 2. ONE-DIMENSIONAL DAHA

2.7.5

Inverse transforms

Let us summarize what we have done so far. We have defined four transforms acting from P to F: S0 (f ) = hf εn (x)µ0k (x)i0 , E0 (f ) = hf εn (x)µ0k (x)i0 , ¯ 0 (f ) = hf εn (x)µ0 (x)i0 , E ¯ 0 (f ) = hf εn (x)µ0 (x)i0 . S k k Let us introduce their counterparts acting in the opposite direction, from the space Fˆ of all functions on Z] to P. Replacing 0 ; 1 in these two formulas, h·, ·i0 ; h·, ·i1 , µ0k ; µ1k , S1 (f ) = hf εn (x)µ1k i1 , E1 (f ) = hf εn (x)µ1k i1 , ¯ 1 (f ) = hf εn (x)µ1 i1 , E ¯ 1 (f ) = hf εn (x)µ1 i1 . S k k Theorem 2.7.13. (Inversion theorem) We have the following identities: ¯ 1 ◦ S0 = id = S1 ◦ S ¯ 0, S ¯ 0 ◦ S1 = id = S0 ◦ S ¯ 1, S ¯1 ◦ E ¯ 0 , E0 ◦ E1 = id = E ¯0 ◦ E ¯ 1. E1 ◦ E0 = id = E ¯ E ¯ Proof. We know that the automorphisms corresponding to S, E, S, −1 are σ, ², σ , ησ, respectively. This gives that all maps in the theorem are homomorphisms of HH–modules. Now we use that the HH–modules P, F are irreducible and the inversion formulas are true for f = 1 ∈ P or f = δ0] ∈ F. ❑ Jackson master formulas. We can obtain the Jackson master formulas from the old ones by formal conjugating performed together with replacing x2 γc − by γ = q . The proofs of these formulas (including the convergence) are a straightforward copy of those in the compact case. For example, formula (2.7.17) results in 2

2

2

hεm (x)εn (x)γµ1k i1 = q 20] −m] −n] t1/2 T −1 (εm (x))(n] )hγµ1k i1 .

(2.7.18)

Formulas (2.7.1) and (2.7.2) read as follows: 2

2

2

hεm (x)εn (x)γµ1k i1 = q 20] −m] −n] εm (n] )hγµ1k i1 , 2 +n2 +2k(|m|+|n|)

where 202] − m2] − n2] = − m

4 2

(2.7.19)

, and, respectively, 2

2

hεm (x)εn (x)γµ1k i1 = q 20] −m] −n] εm (n] )hγµ1k i1 .

(2.7.20)

The symmetrization of these formulas (it does not matter which one is taken) gives a direct counterpart of Theorem 2.5.3, namely,

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225

hpn (x)pm (x)γδk1 (x)i1 = 2 +n2 +2k(m+n))/4

= pn ((m + k)/2)pm (k/2)q −(m

hγδk1 (x)i1 .

(2.7.21)

The Plancherel formulas can also be transformed to the Jackson case with ease using Theorem 2.7.13. We mention here the papers [KS1, KS2] devoted to a variant of the Fourier transform S1 and its inversion in the analytic setting. Note that formulas (2.7.18),(2.7.19), and (2.7.20) have counterparts for the integration over the contour i² + R, ² > 0 instead of the Jackson integration (which is essentially the sharp R integration in the A1 –case). Cf. case (c) in Section 2.6.5. The constant i²+R γµk must be replaced by the formula from Section 2.3.5. No other changes are necessary. ¯ 1 : F → P is the Comment. (i) The limit q → ∞ of the transform S p–adic spherical transform due to Matsumoto [Ma]. Its inverse S0 is also compatible with the limit. In the papers [O4, O5], Opdam developed the Matsumoto theory of “nonsymmetric” spherical functions towards the theory of nonsymmetric polynomials. The operator Tw0 appears there in the inverse transform in a way similar to what we are doing ([O4], Proposition 1.12). (ii) As q → 1, the transform S0 becomes the Hankel transform. One can also obtain the Harish-Chandra spherical transform under the same limit by switching from the variable x to the variable X = q x . That is, we need to rewrite all formulas in terms of X and then leave X untouched in the limit. Note that the self-duality of the Fourier transform and the Gaussian do not survive in this limit (and in the p–adic case). (iii) We can transform the norm formula (2.6.5) and its variant without conjugation (2.7.14) to the Jackson case: hεm (x)εn (x)µ1k i1 = µ1k (n] )−1 δmn , hεm (x)εn (x)µ1k i1 = t1/2 T −1 (δn] )(m] ). However, these formulas hold only for 0. tq m − 1 Because q is not a root of unity, the chain is infinite and contains only invertible intertwiners Bm (Am are always invertible), unless for some m ∈ Z+ , (a) either tq m = 1 (nonexistence), (b) or tq m = t−1 (noninvertibility). There is only one case which was not covered by the previous theorem, namely, when t = −q −n/2 for some integral n > 0.

(2.8.11)

Note that the case when t = −1 and q is generic is excluded from the consideration; we leave it as an exercise. Let as impose (2.8.11). Then (b) may hold for some m, but we have no problem with (a) and the polynomials em always exist. Indeed, the B– intertwiners always create the polynomials with the desired leading term, even if they are not invertible. Because e−n is proportional to Bn (en ), one has: T (e−n ) = t1/2 e−n and s(e−n ) = e−n . Thus e−n = X −n +constX −n+2 . . . + X n . Only the terms with X −n+2l can appear in the decomposition. Using the argument as above, we conclude that the space Ce−n ⊕ Cen+1 ⊕ Ce−n−1 ⊕ . . . is an ideal (e−n ), and, moreover, an HH –submodule. The following holds.

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Theorem 2.8.2. (i) Assuming that q, t are not roots of unity, the nonzero finite dimensional representations of HH that are not described in Theorem 2.8.1 exist only for t = −q ±n/2 as 0 6= n ∈ Z+ . Given q, t, such a representation is unique, has dimension 2n, and is irreducible. For t = −q −n/2 , it is def isomorphic to V2n == P/(e−n ). The automorphisms ςx and ςy do not change its isomorphism class; ι(V2n ) corresponding to t = −q n/2 is not a quotient of P for such t. (ii) The series µ0k and the pairing hf, gi on P are well defined for such t. The values em (t−1/2 ) of em and their scalar squares from (2.8.8) and (2.8.9) are nonzero precisely for m = 0, 1, −1, . . . , n. The radical Rad0 of the pairing hf, gi coincides with (e−n ), so its restriction to V2n is well defined. (iii) The discretization map identifies V2n with the space ³ ´ def −1/2 1−n −1/2 1/2 1/2 1/2 n 2 2 F2n == Funct t q ,...,t ,t q ,...,t q . (2.8.12) Its kernel coincides with (e−n ) and is the radical Rad of the pairing {f , g} = f (Y −1 )(g)(X 7→ t−1/2 ) from Theorem 2.5.10. Proof. Claim (i) has been mainly checked. The invariance of V2n with respect to the substitution X 7→ −X follows from the structure of e−n described above. The map Y 7→ −Y acts in V2n , since the Y –spectrum of V, which is the inverse of the set (2.8.12) in the multiplicative notation, that is, n o n−1 −n t1/2 q 2 , . . . , t1/2 , t−1/2 q −1/2 , . . . , t−1/2 q 2 , is invariant with respect to the multiplication by −1. Claim (ii) is straightforward. Claim (iii) readily folows from the descrition of the Y –spectrum of V2n . ❑

2.8.3

Fourier transform

To conclude the consideration of the case of generic q, let us describe the + action of the projective P SL(2, Z) on V2n+1 for k = −1/2 − n and V2n for −n/2 t = −q . We will begin with the case k = −1/2 − n. Generally, the σ–invariance (up to proportionality) of an HH –module V gives that if Y (v) = q λ v, then X(v 0 ) = q −λ v 0 for some v 0 . Applying this to + + the case under consideration, we conclude that V2n+1 and ςx ςy (V2n+1 ) are σ– + + invariants, but ςx (V2n+1 ) and ςy (V2n+1 ) are not; the latter two are transposed by σ. + We will study V2n+1 only. Applying ςx ςy , one can manage the module + + + ςx ςy (V2n+1 ). Actually, the σ–invariant module ςx (V2n+1 ) ⊕ ςx (V2n+1 ) must be considered too. This case follows along the same lines; we leave it as an exercise.

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233

+ It is convenient to switch from V2n+1 to its “functional realization” F2n+1 from Theorem 2.8.1(iv). This space has the following scalar product: X def hf, gi0 == hf g¯µ1 i0 , g¯(m] ) = g(m] ), hf i0 = f (m] ), (2.8.13) m] ∈./ 0 1

1

µ (m] ) = µ ((1 − m)] ) = q

−k(m−1)

m−1 Y i=1

1 − q 2k+i as m > 0. 1 − qi

Recall that (1 − m)] = 1/2 − m] , π(µ1 ) = µ0 , and π(./0 ) =./0 . Also, = µ1 and therefore the form h·, ·i0 is symmetric. The operators X, Y, T, q, t are unitary with respect to this scalar product. We come to the following truncation of Theorem 2.7.2. µ1

Theorem 2.8.3. (Truncated master formula) (i) We introduce the Gauss2 ian by the formula γ(m] ) = q m] . Then it induces τ+ upon the action H 7→ γHγ −1 on H ∈ HH . Respectively, the automorphisms of F2n+1 S(f )(m] ) = hf εm µ1 i0 , E(f )(m] ) = hf¯εm µ1 i0

(2.8.14)

induce σ and ² on HH . (ii) For l] , m] ∈ ./0 , hεl εm γµ1 i0 = q − 2n+1 X

=

j=n+1

q

(j−k)2 4

m2 +n2 +2k(|m|+|n|) 4

εl (m] )hγµ1 i0 , hγµ1 i0 =

(2.8.15)

j

n Y 1 − q j+k Y 1 − q i+2k−1 1/16 = q (1 − q 1/2−i ). 1 − q k i=1 1 − q i i=1

(2.8.16) ❑

+ Comment. In the simplest case k = −1/2, the module V2n+1 becomes one-dimensional. Formula (2.8.16) is then a trivial identity. Nevertheless, it is directly related to the A1 –case of the Dedekind–Macdonald η–type identity [M1]. The latter is closely connected with the Kac–Moody algebras [Ka]. The case of arbitrary n seems beyond the Kac–Moody theory. It is definitely more reasonable to discuss the connection with [M1, Ka] for arbitrary root systems, because there are many incidental coincidences in the rank one case. See [C28]. ❑

Now let us consider the additional series, namely, the modules V2n for t = −q , n ∈ N. The analysis is straightforward, so we will simply formulate the counterpart of the previous theorem in this case. We set q = ea , k = −n/2 + πia for a > 0, n ∈ N. (2.8.17) −n/2

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The formulas below are algebraic identities in terms of q, t and their fractional powers. This special setting is convenient for the continued use of the “exponential” notation. We use the standard m] and the functional realization F2n from Theorem 2.8.2 in the set: def

./0− ==

n1 − n − k 2

k k+1 n + ko ,...,− , ,..., . 2 2 2

(2.8.18)

2

The Gaussian will be γ(m] ) = q m] . For instance, γ(m] ) = exp(

(m − n/2)2 (m − n/2)πi π 2 a + − ) for m > 0. 4a 2 4

The εm are the spherical polynomials, normalized by εm (0] ) = 1; µ1 = µ/µ(0] ). Theorem 2.8.4. (Additional series) Provided (2.8.17), for l] , m] ∈ ./0− , hεl εm γµ1 i0− = q − hγµ1 i0−

=q

(j−k)2 4

m2 +n2 +2k(|m|+|n|) 4

+

n X j=1

q

(j−k)2 4

εl (m] )hγµ1 i0− ,

j 1 − q j+k Y 1 − q i+2k−1 , 1 − q k i=1 1 − q i

where the Gaussian sum is given by the formulas: hγµ1 i0− = e−

π2 a 4

l−1 Y

(1 + q −i ) for m = 2l, l ∈ N, and

(2.8.19)

i=0

hγµ1 i0−

l−1 Y 2 √ 1 − π4 a 16a = 2e (1 + q −i−1/2 ), m = 2l + 1, l ∈ Z+ .

(2.8.20)

i=0

❑

2.8.4

Roots of unity q, generic k

Let us study the case that is, in a sense, opposite to the previous one. We assume that q 1/2 is a primitive 2N –th root of unity for N ≥ 1 and consider generic k. In this subsection, k 6∈ Z/2. We will continue using the symbol C for the field of constants, although it now can be made Q(q 1/4 , t1/2 ). As above, by generic, we mean that all fractional powers of q must be defined in terms of the “highest” primitive root of unity.

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235

Theorem 2.8.5. Let k 6∈ Z/2. (i) The polynomials em (x), εm (x) are well defined and constitute a basis of P = C[X ±1 ]. In particular, e−N = X N +X −N , e−2N = X 2N +X −2N +2 = e2−N . The form h·, ·i0 is also well defined: hεl (x), εm (x)i0 = δlm (µ1 (m] ))−1 , where µ1 is given by formula (2.8.13). (ii) The vectors εm (x) (m ≤ −N ) and the vectors εm (x) (m ≥ N +1) form a basis of the radical Rad0 of the pairing h·, ·i0 on the space P. The HH –module def V2N == P/ Rad0 = ⊕N ≥n≥−N +1 Cεm (x) is irreducible of dimension 2N with the simple Y –spectrum: nk + N − 1 k k+1 k+No ,..., ,− ,...,− . 2 2 2 2 As an ideal, Rad0 = (e−N ), so V2N = P/(X N + X −N ). (iii) The polynomials εl − εm for m] = l] mod N linearly generate the radical Rad of the pairing {f , g} = f (Y −1 )(g)(−k/2) from Theorem 2.5.10. The quotient P/Rad is irreducible and isomorphic to the HH –module Funct def (./N ) for the set ./N == Z] mod N of cardinality 4N, under the action from Theorem 2.7.2. As an ideal, Rad = (X 2N + X −2N − tN − t−N ). (iv) Irreducible quotients of P are either def

V C == P/(X 2N + X −2N + C) for C 6= 2 or V2N = P/(X N + X −N ), which is a quotient of V 2 by the submodule isomorphic to the image of V2N under ςy . The dimension of V C is 4N and it is Y –semisimple with simple weights constituting the set − ./N . It is also X– semisimple (with simple weights) unless C = −2. The module P/Rad from (iii) equals V −Ct as Ct = tN +t−N = q kN +q −kN . Here Ct 6= ±2, since k 6∈ Z/2 by the asumption. Proof. (i) The existence of the series µ0 , the e–polynomials, and the ε– polynomials readily follows from Theorem 2.2.5, Corollary 2.6.1, and (2.8.8). To check this, one can also involve the Pieri rules from Corollary 2.6.2 or the chain (2.8.6) directly. Let us calculate e−N . We get Y (e−N ) = q N/2+k/2 e−N = −t1/2 e−N , where q N/2 = −1 because q 1/2 is a primitive root of unity. Then e−N = t1/2 (T + t−1/2 )eN , thanks to (2.8.6), so T (e−N ) = t1/2 e−N . The latter immediately gives that e−N is s–invariant (i.e., even) and that π(e−N ) = Y T −1 (e−N ) = −e−N , where π(f (x)) = f (1/2 − x);

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so e−N (x + 1/2) = −e−N . Then, e−N can contain only the monomials X N and X −N with some coefficients. Combining this with the s–invariance of e−N , we obtain the formula for e−N . Note the relation e−N (−k/2) = tN/2 + t−N/2 , that matches the evaluation formula (2.6.4). The calculation of e−2N is similar. It is s–invariant and equals X 2N + X −2N + C for a constant C which must be C = 2 due to (2.6.4). (ii) The norm formula (2.8.9) gives a desired description of Rad0 as a linear space. Since the radical is an ideal, Rad0 = (e−N ). As an immediate application, we obtain that (e−N ) is an HH –submodule. Note that it is not difficult to check directly the HH –invariance of (e−N ) using the Pieri formula for Xe−N together with the above formulas for the action of Y, T on e−N . (iii) The calculation of the radical Rad of the form { , } is similar to that for Rad from the previous theorem. The module P/Rad (of dimension 4N ) has to be irreducible because of the following argument. If it contains a nontrivial submodule W, then, using the corresponding chain of invertible intertwiners, we can find there a Y –eigenvector either of weight k/2 or of weight N/2 + k/2, i.e., either 1 or e−N . The former is impossible. The latter is impossible too because the generator X 2N + X −2N − tN − t−N of Rad is not divisible by e−N = X N + X −N : e−N (k/2) = q kN/2 + q −kN/2 = q −kN/2 (q kN + 1) = 0 ⇔ k ∈ 1/2 + Z. (iv) The element X 2N + X −2N is central in HH . Indeed, it commutes with π and with 1

1

t 2 − t− 2 Ti + for i = 0, 1, T1 = T, X1 = X, T0 = πT1 π, X0 = π(X), Xi2 − 1 generating a proper localization of HH . Therefore it is central in HH without the localization. Thus V C is an HH –module indeed. Using the chain of intertwiners (see (iii)), it is irreducible if and only if X 2N + X −2N + C is divisible by X N + X −N ; so it holds for C = 2 only. Otherwise the Y –spectrum of V C is simple. Concerning the X–spectrum, the polynomial X 2N + X −2N + C has simple roots unless C = ±2. The action of X is not semisimple in V −2 (use the affine Hecke algebra hT, Xi). ❑ Comment. The form h , i making the generators X, Y, T, q, t unitary and inducing ? on HH cannot be introduced on V C for C 6= 2. It follows from formula (6.26) (before Proposition 6.3) in [C28]. The form { , } inducing φ exists only in P/Rad from (iii). Note that if V2N had such a form, then ?·φ = ² would be an inner automorphism of V2N and result in SpecX = −SpecY , which is impossible. Conversely, the module P/Rad cannot have a ?–invariant form, because it would lead to C = 2.

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237

Similarly, τ− acts in P, but does not act in P/Rad for generic k. Technically, it is because τ− multiplies e−2N = X 2N + X −2N + 2 by the constant q −N k 6= 1. See the following corollary. ❑ Corollary 2.8.6. Let q be as above, but k is not supposed generic. (i) Assuming that en is well defined, τ− (en ) = q −n

2 /4−|n|k/2

en for n ∈ Z,

where the canonical action τ− in P is used. ˜ (ii) The module τ− (V C ) equals V C for C˜ − 2 = (C − 2)q −N k . Therefore it is τ− –invariant if and only if either C = 2 (and k is arbitrary) or q N k = 1. The module V −Ct is τ− –invariant for Ct = tN + t−N if and only if q 2kN = 1, that is, 2k ∈ Z. (iii) If the radical Rad of { , } is τ− –invariant, then the projective P SL2 (Z) generated by {τ± } acts in P/Rad by inner automorphisms. Proof. The first claim is straightforward. The action of τ− in P exists due to the definition of the polynomial representation. The natural normalization is τ− (1) = 1. Using that τ− (e−2N ) = q −N k e−2N , we arrive at (ii). If τ− preserves Rad, then it can be restricted to P/Rad. We introduce the action of τ+ from the relation {τ+ (v), w} = {v, τ− (w)}, that is necessary and sufficient to match the relation φτ− φ = τ+ . See Proposition 2.7.5. ❑

2.9

Classification, Verlinde algebras

In this section we continue to assume that q 1/2 is a primitive 2N –th root of unity, N ≥ 1. However, now k may be arbitrary (possibly special). Let us begin with some preliminary technical remarks. The next theorem is stated over C for the sake of simplicty; the actual field of constants is Q(q 1/4 , t1/2 ). Recall that the Y –weights λ are defined as follows: Y v = q λ v, v 6= 0. The classification below gives the list of all possible pairs {λ, k}. In the process of proving the next theorem, we give a complete description of the defining parameters of the corresponding irreducible representations. Note that consideration of the regular case can be simplified using the calculation of the center of DAHA. It makes the parametrization X ↔ Y – symmetric. Using the center makes sense for the special cases too, although the technique of intertwiners seems more convenient. There is a variant of the general theory for odd N when one picks q 1/2 in primitive N –th roots of unity. This variant will not be considered in this section, not will we consider the (equivalent) case of the little DAHA generated by hX 2 , T, Y 2 i ∈ HH (see below).

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2.9.1

The classification list def

Definition 2.9.1. (i) A number λ ∈ C is called regular if the orbit Oλ == ±λ + Z/2 mod N is simple, i.e., contains 4N elements. Equivalently, 2λ 6∈ Z/2 mod N. Otherwise it is called half–singular as 2λ ∈ 1/2 + Z mod N, and singular as 2λ ∈ Z mod N. (ii) An irreducible HH –module V is called a Y –principal module if its dimension is 4N and Y has a simple spectrum in it. Otherwise it is called a Y –special module. Note that the orbit Oλ can consist of 4N or 2N elements. In either case, (−q −2N λ )

2N −1 Y

(Y − q λ+j/2 )(Y −1 − q λ+j/2 )

j=0

=Y

2N

+ Y −2N − q 2N λ − q −2N λ

(2.9.1)

is a central element of HH . The Y –roots in this orbi-product are simple if |Oλ | = 4N and are of multiplicity two otherwise. This fact is not needed in the proof of the following theorem, but readily gives that Y –cyclic HH – modules are direct sums of Jordan Y –blocks of dimension no greater than two. We will use the automorphisms ι and ςy from (2.8.1,2.8.2) in the following theorem. Theorem 2.9.2. Let V be an irreducible finite dimensional HH –module with a Y –eigenvector of weight λ. The following list exhausts all possible pairs {λ, k}. (A) Let λ be regular. Then V is Y –principal if k 6∈ ±2λ + Z. Otherwise, k ∈ ±2λ + Z, so k 6∈ Z/2 and, up to ι, ςy and their product, V is one of the irreducible quotients of P described in Theorem 2.8.5. These quotients are Y –principal modules unless V = P/(X N + X −N ). (B) Let λ be half-singular, i.e., 2λ ∈ 1/2+Z, and either k 6∈ Z/2 or k ∈ Z. Then V is Y –semisimple. The dimension can be 2N (then the Y –spectrum is simple) or 4N (then all weights are of multiplicity 2). (C) Let λ be singular, i.e., 2λ ∈ Z, and k 6∈ Z. Then the dimension of V is 4N and it is not Y –semisimple, unless t = 1. (D) Let λ be either half-singular under k ∈ 1/2 + Z or λ singular under k ∈ Z. Then V is a quotient of P or its image under the automorphisms ι, ςy , ιςy . Proof. Given a Y –eigenvector v = v0 of weight λ = λ0 , we will apply the chain of intertwiners from (2.8.6): A

B

A−m+1

B

m 0 1 · · · v2N , v0 −→ v1 −→ · · · −→v−m+1 −→ vm −→

(2.9.2)

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239

to construct v1−m , vm for 1 ≤ m ≤ 2N. Recall that A−m+1 = q (m−1)/2 Xπ, Bm = t1/2 (T +

t1/2 − t−1/2 ), and q −2λm − 1

1 1 m λ1 = − − λ, λ−1 = λ, . . . , λm = − − λ = λ−m for m > 0. 2 2 2 We can also start here with the space V0 = V (λ) ⊂ V of all λ–eigenvectors. Then the composition def

K == B2N A1−2N . . . B2 A−1 B1 A0 preserves V0 = V (λ) if K is well defined. Similarly, the chain can be applied to the space Vb0 = Vb (λ) of the generalized eigenvectors, satisfying (Y − λ)M (u) = 0 for sufficiently large M. In this case, however, we need to go back to the formula t1/2 − t−1/2 1/2 Bm = t (T + ). Y −2 − 1 (A) All intertwiners exist and are invertible if k/2 6∈ Oλ . We can set K(v0 ) = q κ v0 for arbitrary κ ∈ C and uniquely extend it to the action of HH on ⊕2N m=1−2N Cvm . It is a simple exercise. This representation is principal. If k/2 belongs to the orbit Oλ , then we use the intertwiners to find a Y – eigenvector v0 of weight ±k/2 mod N/2 such that the B–intertwiner is zero on it. Indeed, if B(v0 ) 6= 0, then we take it as v0 . Using ι and ςy , we can assume that this weight is k/2. This means that V is a quotient of P. Now we can apply Theorem 2.8.5. (B) All intertwiners exist and are invertible in this case. We can make λ = −1/4. Then S = q −1/4 Xπ creates the weight −1/2 − λ = −1/2 and therefore preserves the space V0 of all eigenvectors of weight −1/4. The operators K, S act in V0 and satisfy the relation S 2 = 1, SKS = K −1 . The irreduciblity of V0 as a {S, K}–module is necessary and sufficient for the irreduciblity of V. Similar to (A), we can uniquely extend an arbitrary action of S, K on V0 to a structure of an HH –module on V. Thus V0 is either one-dimensional (K = ±1, S = ±1) or two-dimensional. We obtain the desired result. (C) The intertwiners are invertible provided we have their existence. We can find v0 with λ = 0. Then Hy v0 is a two-dimensional irreducible representation of the affine Hecke algebra HY = hT, Y i, which coincides with Vb0 = Vb (0) = {u | (Y − 1)2 (u) = 0}. The action of Y here is not semisimple. We take invertible def

L == A1−N . . . B2 A−1 B1 A0 instead of K. It sends λ = 0 to −N/2 and Vb0 to VbN . Recall that q N/2 = −1, so the intertwiner BN acting in VbN is singular.

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We obtain that (Y +1)2 = 0 on VbN and T preserves this space. If the action of the operator T in this space is known, then this information is sufficient to reconstruct uniquely and completely the HH –action in V. Indeed, we know how Y acts on VbN and know the action of Xπ, that sends VbN back to Vb1−N . The T –action depends on one parameter, because we can conjugate T by the matrices from the centralizer of Y (the Y –action is known). Any choice of this parameter results in the corresponding action of HH on V. Here we need to exclude the case k = 0, which requires a somewhat different consideration. (D) Similar to the second part of (A), we can find a Y –eigenvector v0 of weight ±k/2 mod N/2 such that the B–intertwiner is zero on it. Using ι and ςy if necessary, we obtain a surjection P → V. ❑ Actually, we have proved more than what was stated in the theorem. In cases (A–C), we gave the following complete description of all parameters that determine the irreducible representations: (1) Given a Y –weight λ, we need to know either the action of K for (A) or K and S for (B) in the corresponding Y –eigenspace. (2) In case (C), we need to know the action of T in {v | (Y + 1)2 (v) = 0} as k 6= 0; this action can be arbitrary compatible with the action of Y. (3) An explicit description of the irreducible modules in case (D) requires an extension of Theorem 2.8.5 to the integral and half-integral k, which we are going to discuss now.

2.9.2

Special spherical representations

First of all, the substitution T 7→ −T, t1/2 7→ −t1/2 identifies the polynomial representations for t1/2 and −t1/2 . It does not act on monomials X m , so the nonsymmetric polynomials em remain unchanged, as long as they are well defined. The spherical polynomials εm for even m do not change either; εm 7→ −εm for odd m. Thus it is sufficient to decompose P upon the transformation k 7→ k + N, and we can assume that N/2 ≤ k < N/2. We will also use the outer involutions ι, ς of HH from (2.8.1) and (2.8.2). Theorem 2.9.3. (i) Let k ∈ Z/2 mod 2N and −N/2 ≤ k < N/2. Then the representations V C = P/(X 2N + X −2N + C) of dimension 4N from Theorem 2.8.5 remain irreducible for C 6= ±2. However, now V C for such C becomes Y –non-semisimple for integral k 6= 0 and Y –semisimple with the two-fold spectrum otherwise. Moreover, V −2 is irreducible for the half-integral k, as well as the quotient V2N =P/(X N + X −N ) of V 2 for the integral k. Recall that the kernel of the map V 2 → V2N is isomorphic to the image of V2N under ςy sending Y to −Y.

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241

(ii) If 0 < 2k < N, then either V −2 for k ∈ Z or V2N for k ∈ 1/2+Z has a unique irreducible nonzero quotient, that is, V2n = P/(ε−n ) of dimension 2n for n = N − 2k. Its Y –spectrum is simple. The eigenvectors are the images of εm for −n + 1 ≤ m ≤ n, which are all well defined. This module coincides with P/Rad for the radical of the pairing { , } and T (e−n ) = −t−1/2 e−n = Y (e−n ).

(2.9.3)

These statements can be extended to k = 0 when t = 1 if we take ε−n = X − X −N in place of ε−n . (iii) Let k = −1/2 − n for integral 0 ≤ n < (N − 1)/2. Then the module V2N = P/(X N + X −N ) has two non-isomorphic irreducible Y –semisimple quotients, namely, the representations from (2.8.5): N

± ± − + V2n+1 = P/(εn+1 ± ε−n ), dim(V2n+1 ) = |2k| = 2n + 1, V2n+1 = ςx (V2n+1 ).

Here the εm are well defined when −2n ≤ m ≤ 2n + 1. The binomials X n+1 ± X −n must be taken in place of εn+1 ± ε−n to extend these statements to the boundary case N = 2n + 1. + The kernel (ε−2n−1 ) of the map from V2N to the direct sum of V2n+1 and − V2n+1 has dimension 2N − |4k| and is isomorphic to V2N −4|k| from (ii) under the involution ι sending k to −k, T 7→ −T. The vector e = ε−2n−1 satisfies T (e) = −t−1/2 e, Y (e) = t−1/2 e. (iv) In the last case, that is, k ∈ −1 − Z+ and −N/2 ≤ k < 0, the module −2 V has a unique irreducible nonzero quotient V2N +4|k| (of dimension 2N + 4|k|). It equals P/Rad for the radical of the pairing { , }, and is isomorphic to P/(e) for e = εN −ε−N −2|k| , satisfying relations (2.9.3), where the polynomials εN , ε−N −2|k| are well defined. This module is also isomorphic to the kernel of the map V −2 → V2N −4|k| from (ii) upon applying the outer involution ιςy . (v) The polynomials εm for {m = −2|k|, 2|k|+1, −2|k|−1, . . . , −N +1, N } exist and their images generate the Y –semisimple component of V2N +4|k| (of dimension 2N − 4|k|). The corresponding Y –weights are {λ =

N − 1 − |k| |k| − N |k| −|k| − 1 |k| + 1 , , ,... , }. 2 2 2 2 2

The Y –non-semisimple component of V2N +4|k| is the direct sum of 4|k| Jordan two-blocks of the total dimension 8|k| corresponding to the remaining 4|k| weights in the orbit Oλ = {λ = 0, −1/2 , 1/2 , −1 , . . . , (N − 1)/2, −N/2}. There are two sequences of such non-semisimple λ: {

−|k| |k| − 1 −1 0 N − k |k| − N − 1 N − 1 −N , ,..., , }, { , ,..., , }. 2 2 2 2 2 2 2 2

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Proof. (i) We use that the X–spectrum is simple in V C and the intertwiners are always invertible. This readily gives the irreducibility. The Jordan Y –blocks will appear if and only if the Y –spectrum contains 0 (with a reservation about the degenerate case k = 0, which must be considered separately). The space Vb (0) = {v | (Y − 1)2 v = 0} is an HY –module that is not Y –semisimple. So are all other generalized Y –eigenspaces. This happens precisely for integral k 6= 0. This argument has been used quite a few times. We obtain (i). Concerning the existence of the e–polynomials, the invertibility of the intertwiners is sufficient, but not necessary. The following lemma gives the general construction. In fact, it has already been used before. It will be applied a couple of times, especially in the parts (iv,v). It uses formula (2.9.1). Let Vb0 = Qq,t , Vb1 = Qq,t X, . . . , Vb−m = Bm Vbm , Vbm+1 = A−m Vb−m , . . . ,

(2.9.4)

where m > 0, A−m = q m/2 Xπ, and Bm denotes the restriction of the intertwiner t1/2 (T + (t1/2 − t−1/2 )/(Y −2 − 1)) to Vbm , provided that q 2λm 6= 1 for λm = −m/2 − k/2. If q 2λm = 1 and the denominator of Bm becomes infinity, then we set Bm = t1/2 T, Vb−m = Vbm + T Vbm . Lemma 2.9.4. (i) The space Vb±m is one-dimensional or two-dimensional. In the latter case, it is the Jordan 2–block satisfying (Y − q ±λm )2 Vb±m = {0}. If dimVb−m = 1 then dimVbm+1 = 1 and the generators are e−m = Bm−1 · · · B1 A0 (1), em+1 = A−m E−m . If dimVb−m = 2, then dimVbm+1 = 2 and the e–polynomials e−m , em+1 do not exist, although these spaces contain the e–polynomials of smaller degree. (ii) Let us assume that either q 2λm = t or q 2λm = t−1 . Then dimVb−m = 1 and this space is generated by e−m . If Vbm is one-dimensional then, respectively, (T + t−1/2 )e−m = 0 or (T − t1/2 )e−m = 0. If dimVbm = 2, then, respectively, (T + t−1/2 )e−m or (T − t1/2 )e−m is nonzero and proportional to the (unique) e–polynomial which is contained in the space Vbm . ❑ If V = P/J is a proper (neither {0} nor P) irreducible HH –quotient, then J is an ideal. We may set J = (e) for either (A) : e = X l + . . . + cX −l or (B) : e = X l+1 + . . . + cX −l ,

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where 2N > l > 0, c 6= 0, and, respectively, (A) : dim V = 2l or (B) : dim V = 2l + 1. In either case, e is an eigenvector of Y. Indeed, Y (e) ∈ J has the same type as e with the same degrees. A proper linear combination of e and Y (e) would be lower than e, unless they are proportional. We set e = e0 +e1 , where eα (−X) = (−1)α eα (X). Note that this decomposition is always nontrivial in case (B). Since Y commutes with the ιx sending X 7→ −X, then {e0 , e1 } are Y – eigenvectors of top X–degrees (A) : {l, m} as 0 < m < l for odd l − m, (B) : {l + 1, l}. The Y –weights of e0 , e1 must coincide mod N. They can be readily calculated: (A) : {(l + k)/2, ±(m + k)/2}, (B) : {−(l + 1 + k)/2, (l + k)/2}. Here it suffices to know the leading term of eα with respect to the ordering Â: X ±m ⇒ ∓(m + k)/2 for m > 0. The coincidence mod N immediately gives that only the plus–sign is possible in (A). We arrive at the following relations: (A) : k = −(l + m)/2 mod N, (B) : k = −1/2 − l

mod N.

In either case, k ∈ 1/2 + Z if the decomposition e = e0 + e1 is nontrivial. We obtain that (B) results in the claims from (iii) from the theorem. Indeed, 0 ≤ l < 2N, and l is nothing but n = k + 1/2; the representations ± V2n+1 exist and remain irreducible in this case. We now need to examine (A). We will impose this condition to the end of the proof. The argument above, which ensured the Y –invariance of e, can be used for T as well. It preserves the type of e (note that it does not hold in case (B)). We obtain that T (e) = ±t±1/2 e. Therefore e = e−l . Indeed, e − e−l must be proportional to em for positive m ≤ l. However, such small em are never eigenvectors of T , since the chain of intertwiners remains invertible in this range. We have four subcases of (A): (α) T (e) = t1/2 e = Y (e),

(β) T (e) = t1/2 e = −Y (e),

(γ) T (e) = −t−1/2 e = Y (e), (δ) T (e) = −t−1/2 e = −Y (e). Concerning (α, β), we obtain that s(e) = e, so e is even and the decomposition e = e0 + e1 is trivial. Then (l + k)/2 = k/2 mod N for (α) and

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(l + k)/2 = k/2 + N/2 mod N for (β). The former equality results in l = 0 mod 2N, which is impossible. As for (β): π(e) = Y T −1 (e) = −e ⇒ e(x + 1/2) = −e(x) ⇒ e = X N + X −N . Therefore (β) implies that the quotient P/(X N +X −N ) is an HH –module and is irreducible unless it is covered by (iii) of the theorem, which was already considered. (γ) We obtain (l + k)/2 = N/2 − k/2 mod N and l = N − 2k mod 2N. Recall that e = e−l . If 2k > 0, we arrive exactly at (ii) of the theorem. In this case, l = n. All polynomials em are well defined and have nonzero em (−k/2) in this case. Otherwise 2k < 0. (γ1 ) If 2k < 0, then k cannot be a half-integer. Indeed, e−l (−k/2) = 0 in this case (use the evaluation formula), as well as for the generator of the ideal + for V2n+1 in case (b). We see that these two ideals together do not generate + the whole P. Since V is irreducible, it has to coincide with V2n+1 , which is impossible because we assumed that e is in the form (a). Therefore k must be an integer. (γ2 ) We shall consider the integers −N/2 ≤ k < 0. Lemma 2.9.5. (a) For integral k such that −N/2 ≤ k < 0, the quotient def V2N +4|k| == V/Rad by the radical of the pairing { , } is an irreducible HH – module of dimension 2N + 4|k|. (b) The polynomials em exist and em (−k/2) 6= 0, i.e., εm exist, for the sequences: m = {0, 1, −1, . . . , −|k| + 1, |k|}, m = {−2|k|, 2|k| + 1, . . . , −N + 1, N }, m = {−N, N + 1, . . . , −N − |k| + 1, N + |k|}, respectively, with 2|k|, 2(N − 2|k|), and 2|k| elements. They do not exist for the following 2|k| + 2|k| indices: m = {−|k|, |k| + 1, . . . , −2|k| + 1, 2|k|}, m = {−N − |k|, N + |k| + 1, . . . , −N − 2|k| + 1, N + 2|k|}. (c) The Y –semisimple component of V2N +4|k| of dimension 2N − 4|k| is linearly generated by em for {m = −2|k|, 2|k| + 1, −2|k| − 1, . . . , −N + 1, N }. The corresponding Y –weights are {λ =

|k| −|k| − 1 |k| + 1 N − 1 − |k| |k| − N , , ,..., , }. 2 2 2 2 2

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245

(d) The rest of V2N +4|k| is the direct sum of 4|k| Jordan two-blocks of the total dimension 8|k|. There are two series of the corresponding (multiple) weights λ: {

−|k| |k| − 1 −1 0 N − |k| |k| − N − 1 N − 1 −N , ,..., , }, { , ,..., , }. 2 2 2 2 2 2 2 2

Proof. We will use the chain of the spaces of generalized eigenvectors Vb0 = Qq,t , Vb1 = Qq,t X, Vb−1 , . . . , Vbm , . . . from (2.9.4) and Lemma 2.9.4. Recall that m > 0. The following holds: (0) the spaces Vb±m are all one-dimensional from 0 to m = |k|, i.e., in the sequence V0 , . . . , V−|k|+1 , V|k| ; (1) the intertwiner Bm becomes infinity at m = |k|, i.e., B|k| = t1/2 T, and dimVbm = 2 in the range |k| < m ≤ 2|k|; (2) the intertwiner Bm kills 1 ∈ Vbm at m = 2|k|, and, after that, dimVbm = 1 for 2|k| < m ≤ N ; (3) Bm is proportional to (T + t−1/2 ) at m = N, also, e−N = X N + X −N and dimVbm = 1 as N < m ≤ N + |k|; (4) the intertwiner Bm becomes infinity again at m = N + |k| and then dimVbm = 2, provided that N + |k| < m ≤ N + 2|k|; (5) Bm kills e−N at m = N + 2|k| and Bm (Vbm ) is generated by e−N −2|k| , that has the same Y –eigenvalue as eN does. Concerning step (5), the polynomials ²−N −2|k| and ²N both exist and the difference e = εN − ε−N −2|k| belongs to the radical Rad, i.e., becomes zero in V2N +4|k| . Note that (T + t1/2 )e = 0, the fact which is important to know to continue the decomposition of V further. It follows along the same lines. We see that step (5) is the first one that produces no new elements in V2N +4|k| , namely: BN +2|k| (VbN +2|k| ) = Qq,t eN in V2N +4|k| , and we can stop at (5). The lemma gives that between (2) and (3), the polynomials em exist and their images linearly generate the Y –semisimple part of V. It is equivalent to the inequalities em (q −ρk ) 6= 0 because these em have different Y –eigenvalues. Apart from (2)–(3), there will be always Jordan two-blocks with respect to Y. Let us check this. First, we obtain a two-dimensional irreducible representation of HY = hT, Y, πi in the corresponding Vb –space at step (1). Then we apply the chain of invertible A, B–intertwiners to this space (the weights will go back) and eventually will obtain a two-dimensional Vb –space for the weight coinciding

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with the initial weight λ = −|k|/2. Note that the vector e0 = 1 is not from the Y –semisimple component of V2N +4|k| ; it belongs to a Jordan two-block. Second, the intertwiner from (2) makes the previously obtained Vb –space one-dimensional and make it Y –semisimple (the corresponding eigenvalue is simple in V2N +4|k| ). It will remain one-dimensional until (3). After step (3), we obtain the Jordan two-blocks. The steps (4)–(5) are parallel to (1)–(2). The above consideration readily results in the irreducibility of the module V2N +4|k| . Indeed, Lemma 2.9.4(ii) gives that if a submodule of V2N +4|k| contains at least one simple Y –eigenvector then it contains the image of 1 and the whole space. Step (5) guarantees that it is always the case, because we can obtain eN from an arbitrary Y –eigenvector. ❑ We arrive at (iv,v). Let us consider the last subcase. (δ) Now l = 2N −2k. If 2k > 0, then e−l belongs to the ideal (e−n ). Indeed, its image in V2n is zero since there are no eigenvectors of weight −k/2 in this module. This contradicts to the irreducibility. Negative k are impossible because l < 2N by assumption. Thus this case is empty. Concerning the “duality” claim involving ι from (iii), it holds because + − e = e−2n−1 , generating the kernel (e) of the map P → V2n+1 ⊕ V2n−1 , satisfies −1/2 the relation (δ): T (e) = −t e = −Y (e). Note here the reservation about the boundary value 2|k| = N. As for the duality (ii)↔(iv), the vector e = e−n = e−N −2k , generating the kernel of the map P → V2N −4k , satisfies the relation (γ): T (e) = −t−1/2 e = Y (e). The same relation holds for e = εN − ε−N −2|k| from (iv). Note here the reservation about the boundary value k = 0. ❑

2.9.3

Perfect representations

It is important to find out which irreducible modules are P SL(2, Z)–invariant. Using the classification, it is clear that the exceptional representations (ii)– (iv) from the theorem are invariant because they can be distinguished by the dimensions. It is also not difficult to describe P SL(2, Z)–invariant modules from scratch without using the classification. It is what we are going to do now. We continue to assume that q 1/2 is a primitive 2N –th root of unity and t = q k . It readily results from the description of the Y –spectrum of the polynomial representation that all σ–invariant quotients of P must be through V 0 = V −Ct = P/ ( X 2N + X −2N − tN − t−N ) = P/Rad from Theorem 2.8.5. Indeed, the invariance gives that the element X 2N + X −2N must act as its σ–image Y 2N + Y −2N , that is tN + t−N in P. Recall that X 2N + X −2N is central in HH, so V 0 is obviously an HH –module.

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247

The module V 0 is irreducible unless k is integral or half-integral. For generic t, it is not σ–invariant. Instead, the involution X ↔ Y, T ↔ −T −1 , t1/2 7→ t1/2 , q 1/2 7→ q −1/2 is its inner automorphism. The corresponding map sends the image 10 ∈ V 0 of 1 to the image eˆ0 of the polynomial eˆ = (X 2N + X −2N − tN − t−N )/(X −1 − t−1/2 ), satisfying X(ˆ e0 ) = t1/2 eˆ0 , T (ˆ e0 ) = −t−1/2 eˆ0 . Let us discuss the singular k in detail. The module V 0 becomes, respectively, V −2 = P/(X 2N + X −2N − 2), V 2 = P/(X 2N + X −2N + 2), as k ∈ Z, k ∈ 1/2 + Z, and tN = q kN = ±1. Recall that the module V 2 is an def extension of V2N == P/(X N + X −N ) by its ςy –image ςy (V2N ). Proposition 2.9.6. Let k ∈ Z/2, |k| < N/2. The notation is from Theorem 2.9.3. There are four exact sequences: 0 → ιςy (V2N +4k ) → V −2 → V2N −4k → 0 for k ∈ Z+ , + − 0 → ι(V2k ⊕ V2k ) → V2N → V2N −4k → 0 for k ∈ 1/2 + Z+ . The arrows must be reversed for k < 0: 0 → ιςy (V2N −4|k| ) → V −2 → V2N +4|k| → 0 for k ∈ −1 − Z+ , + − 0 → ι(V2N −4|k| ) → V2N → V2|k| ⊕ V2|k| → 0 for k ∈ −1/2 − Z+ . In other cases, V −2 and V2N are irreducible. ± The modules V2|k| , V2N −4|k| , and V2N +4|k| are irreducible and projective P SL(2, Z)–invariant. The modules V2N +4|k| are Y –non-semisimple. The others are semisimple. Proof. We use Theorem 2.9.3, the structure of the Y –spectrum in the polynomial representation, and that the dimensions of the Jordan Y –blocks are no greater than 2. Concerning the action of τ± and σ, see Corollary 2.8.6. ❑ We obtain that, up to ι and ς, there are three different series of projective P SL(2, Z)–invariant irreducible spherical representations at roots of unity, namely, V2N −4k (integral N/2 > k > 0), V2|k| (half-integral −N/2 < k < 0), and V2N +4|k| (integral −N/2 < k < 0).

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sym If k = 1, the subalgebra V2N −4 of dimension N − 1 is isomorphic to the c usual Verlinde algebra of sl2 of level N (central charge = N − 2). The (1) symmetric polynomials pm are the classical characters of finite dimensional sym representations of sl2 . In V2N −4 , these characters are considered as functions at roots of unity. Comment. (i) Recently a non-semisimple variant of the Verlinde algebra appeared [FHST] in connection with the fusion procedure for the (1, p) Virasoro algebra, although the connection with the fusion is still not justified. Generally speaking, the fusion procedure for the Virasoro-type algebras and the so-called W –algebras can lead to non-semisimple Verlinde algebras. At least, there are no reasons to expect the existence of a positive hermitian inner product there like the Verlinde pairing for the conformal blocks, because the corresponding physics theories are expected massless. Surprisingly, the algebra from [FHST] is defined using the usual (massive) Verlinde algebra sym under certain degeneration. Presumably it coincides with V2N +4|k| for k = −1 or is very close to it. (ii) The latter module and its multidimensional generalizations are also expected to be connected with the important problem of describing the complete tensor category of the representations of the Lusztig quantum group at roots of unity. The Verlinde algebra, the symmetric part of V2N −4|k| for k = 1, describes the so-called reduced category, in a sense, corresponding to the Weyl chamber. The nonsemisimple modules of type V2N +4|k| are expected to appear in the so-called case of the parallelogram. ❑

Motivated by the Verlinde algebras [Ver] (see also [KL2]) we will require more structures. Definition 2.9.7. We say that a finite dimensional irreducible HH–module V is perfect or a nonsymmetric Verlinde algebra if the following conditions hold: (a) V is spherical, i.e., there exists a surjection P → V ; (b) it is {τ± , ²}−invariant, which means that there are pullbacks of the HH–automorphisms τ± and ² to V satisfying the relations τ+ τ−−1 τ+ = τ−−1 τ+ τ−−1 , ²2 = 1, ²τ+ = τ− ²; (c) it is also X–pseudo-unitarity: there exists a nondegenerate form h·, ·i on the space V such that the corresponding anti-involution of HH is ? and he, ei 6= 0 for any X–eigenvector e ∈ V . Comment. Generally, a quasi-perfect module is defined as an HH – module equipped with a non-degenerate pairing inducing the anti-involution φ from (2.5.7). For instance, the polynomial representation is quasi-perfect if the radical of { , } is zero.

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249

If the pairing is perfect, i.e., establishes an identification of the module and its dual as a vector space, and, moreover, there is a projective action of P SL(2, Z), then the module is called perfect. In this chapter, we consider only finite dimensional spherical perfect representations. So far, the main considered examples of perfect and quasi-perfect modules are spherical, however, non-spherical representations are expected to appear when decomposing the polynomial representation (for special q, t) and in various applications. In a sense, “perfect representations” in this chapter mean ”could not be better.” They have all properties of the irreducible modules of the Weyl algebras and are really perfect from the viewpoint of Fourier analysis. ❑ By condition (a), the module V has a structure of a commutative algebra, since it is a quotient of the commutative algebra P. Obviously, def

V sym == {v ∈ V | T v = t1/2 v} is a subalgebra of V, a generalization of the Verlinde algebra. Due to condition (b), one can replace the X–eigenvectors by the Y – eigenvectors in (c). Thus such an X–semisimple V is Y –semisimple and the other way round. Also, (c) formally implies that V is X–semisimple. Indeed, if (X −q ξ )v = e for e satisfying (X − q ξ )e = 0, then he, ei = h(X − q ξ )v, ei = hv, (X −1 − q −ξ )ei = 0.

The following theorem directly results from the previous considerations. Actually, we do not need much theory for its justification. We will sketch a straightforward proof. Theorem 2.9.8. The perfect representations V are exactly the cases (a) : 2k ∈ Z+ , 0 < k < N/2, and (b) : k = −1/2 − n, n ∈ Z, 0 ≤ n < N/2 from Theorem 2.9.3. Both V are quotients of P by the radical Rad of the form {f, g} = f (Y −1 )(g)(−k/2). They are isomorphic to Funct( ./0 ): (a) ./0 =

nk − N + 1 k − N + 2 k k+1 N − ko , ,...,− , ,... , 2 2 2 2 2

(b) ./0 =

n 1/2 − n 3/2 − n 1 1 3 1/2 + n o , ,...,− , , ,..., . 2 2 4 4 4 2

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The action of HH is defined here by the formulas from Theorem 2.7.2. The invariant form is (2.8.13): X def hf, gi0 == hf g¯µ1 i0 , g¯(m] ) = g(m] ), hf i0 = f (m] ), (2.9.5) m] ∈./ 0 1

1

µ (m] ) = µ ((1 − m)] ) = q

−k(m−1)

m−1 Y i=1

1 − q 2k+i as m > 0. 1 − qi

Proof. We firstly check that a perfect V must coincide with P/ Rad . Indeed, using h , i and ² on V, we introduce the form {·, ·}² on P, which is the pullback of the form hf, ²gi. The corresponding anti-involution (in either direction) is ² ? = φ, i.e., the same as for the form {·, ·}. A proper linear combination {·, ·}† of {·, ·} and {·, ·}² satisfies {1, 1}† = 0. Therefore it vanishes identically and {·, ·}, {·, ·}² are proportional. This gives the desired result. Here we did not use that the form {·, ·}² is symmetric. It immediately follows from the proportionality. Thus hf, ²gi = h²f, gi. Second, conditions (a,b) follow from the existence of the form ( , ) from the claim (c) from the definition. One can also use the previous proposition. Third, using the chain of intertwiners from (2.8.6) and (2.9.2), we obtain that all εm for m] ∈ − ./0 are well defined and their images in V linearly generate an irreducible HH –submodule V 0 . Indeed, the polynomial e−n exists, e−n (−k/2) = 0, and therefore it belongs to Rad. Equivalently, one may check that the formulas from Theorem 2.7.2 define an HH –action on Funct( ./0 ). The orthogonal complement of V 0 in V with respect to h , i (or {·, ·}² ) intersects V by zero. Fourth, the operator X has a simple spectrum on V . The existence of the X–pseudo-unitary pairing guarantees that X is semisimple. Since V is a cyclic module over C[X ±1 ], each eigenvalue of X has a unique Jordan block. Therefore the spectrum of X in V is simple. Applying ², the same holds for Y. For instance, the image of 1 in V can belong either to V 0 or to its orthogonal complement constructed above. This is impossible because 1 is a generator of V. Fifth, we can define τ+ as the operator of multiplication by the restricted 2 Gaussian γ(m] ) = q m] in the realization V =Funct( ./0 ). The automorphism ² in V is the Fourier transform E: fb(m] ) = hf¯εm µ1 i0 from (2.8.14). ❑ sym Note that the action of σ in V2n (for n = N − 2k and the integral k) was introduced and calculated for the first time in [Ki1] on the basis of the interpretation of Rogers’ polynomials for the integral positive k discovered by Etingof and Kirillov. Verlinde considered only the case k = 1, when the classical SL2 –characters are sufficient. We will finish this section with the following generalization of Theorem 2.3.4. We follow Theorem 2.8.3, where negative half-integers k were considered. See also [C27].

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251

Theorem 2.9.9. Let 0 < 2k < N, l] , m] ∈ ./0 . Then hεl εm γµ1 i0 = q − hγµ1 i0 = 2N −1 X

qj

m2 +n2 +2k(|m|+|n|) 4

k Y

1 1 − qj j=1

2 /4

2N −1 X

2 /4

qj

εl (m] )hγµ1 i0 ,

(2.9.6)

, where k ∈ Z,

(2.9.7)

j=0

√ = (1 + ı) N as q 1/4 = exp(πı/2N ),

j=0 k−1/2 1 0

hγµ i = 2q

1/16

Y j=1

1 for k = 1/2 + Z. 1 − q 1/2−j

(2.9.8) ❑

We note an interesting connection of these formulas with Theorem 2.8.4, which describes the additional series for generic non-cyclotomic q and t = −q −n/2 . The formulas (2.8.20) and (2.8.19) there actually can be used to obtain (2.9.6), and the other way round. Indeed, setting in these formulas k = N/2 − n/2 for n ∈ N, we arrive at the relation t = −q −n/2 . If N is sufficiently big, then the corresponding q can be treated as “generic.” Then, to establish the connection, we need to rewrite the Gaussian sums using Corollary 2.3.6, which states that hγµ1 i0 = q n

2 /4

n Y

(1 − q j )

if N = 2n,

j=k+1

hγµ1 i0 = q n

2 /4

(1 + q (2n−1)/4 )

n Y

(1 − q j )

if N = 2n + 1.

j=k+1

Choosing sufficiently big N and providing the desired parity of N , the additional representation V2n becomes V2N −4k for such N. This is, essentially, what was observed in [C20](Appendix). ❑

2.10

Little double Hecke algebra

Concerning the above choice q 1/4 = exp(πı/2N ), the formulas for the Gaussian sums that do not explicitly contain the imaginary unit ı hold for any primitive q, since we can apply the Galois automorphisms. However, the “smallest” q is necessary to ensure the positivity of the inner product. See the formulas for µ1 (m] ). There are some other examples of unitary representations of HH when q 1/4 is not a primitive 4N –th root of unity; they will be the contents of this section.

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All representations considered in this section will be unitary, i.e., with positive invariant ∗–hermitian scalar products. Recall that the involution ∗ is the complex conjugation trivial on x, where X = q x and q x (m/2) = q m/2 for m ∈ Z. Let q 1/4 be arbitrary (maybe nonprimitive) 4N –th root of unity; q −1/4 = (q 1/4 )∗ = (q 1/4 )−1 . We will start with 0 < k < N/2, k ∈ Z. The field of constants will be Qq = Q(q 1/4 ). The definition of the functional representation remains the same. See Theorem 2.9.8. For −N < m ≤ N, def

./0 == {m] | µ1 (m] ) 6= 0} n −N + k + 1 k k+1 N − ko , ,...,− , ,..., = 2 2 2 2

(2.10.1)

where the measure is µ1 (m] ) = µ1 ((1 − m)] ) = q −k(m−1)

m−1 Y j=1

1 − q 2k+j for m > 0. 1 − qj

The space F 0 = F 0 (k) = Funct( ./0 , Qq ) is an HH –module; the discretization map P 3 f 7→ f (m/2) ∈ F 0 is an HH –homomorphism. Setting 0 def

hf i ==

N X

f (m] )µ1 (m] ),

m=−N +1

the form hf, gi0 = hf g ∗ i0 is ∗–hermitian on P. The quotient of P by the radical of this form coincides with the whole F 0 , except in the special case when N is odd and q 1/2 is an N –th root of unity. In this case, the image of P upon the discretization mapping is (two times) smaller than F 0 .

2.10.1

The case of odd N

Let N = 2n + 1, q 1/2 = − exp(πı/N ). When considering the Gaussian, we take q 1/4 = ı exp(πı/N ). Then there are two irreducible components of F 0 : F 0 = (F 0 )0 ⊕ (F 0 )1 , dim(F 0 )j = N − 2k, 0 j

0

(2.10.2)

j

(F ) = {f ∈ F | f (m] + N/2) = (−1) f (m] ), m] < 0}. The component (F 0 )0 is the image of P. The hermitian form introduced above is nondegenerate on F 0 , and, moreover, makes it a ∗–unitary representation. Actually, F 0 is unitary for either sign of q 1/2 because the inner product involves only q. ¯ 1 defined by the formulas The Fourier transforms S1 , S ¯ 1 (f )(m] ) = hf εm i0 S1 (f )(m] ) = hf εm i0 and S

(2.10.3)

2.10. LITTLE DOUBLE HECKE ALGEBRA

253

vanish identically on (F 0 )1 . They are zero on the Gaussian when N = 4d + 1, because it sits in (F 0 )1 for such N. For N = 4d + 3, the Gaussian belongs to (F 0 )0 (see (2.10.2)). Equivalently, as q 1/4 = ı exp(πı/2N ), N X

2 /4

qm

= 0 ⇔ N = 4d + 1.

m=−N +1

To make the above considerations more systematic and cover all classical formulas for the Gaussian sums, we need to diminish the double Hecke algebra and, respectively, the size of its irreducible representations.

2.10.2

Little double H

f Let H H be a subalgebra of HH generated by X 2 , T, and Y 2 . We will assume for def a while that q and t are arbitrary nonzero. Respectively, Qq,t == Q(q 1/4 , t1/2 ) def is the field of constants. The image of T0 == Y 2 T −1 in the polynomial representation P can be readily calculated: Tb0 = t1/2 s0 + (q 1−2x − 1)−1 (t1/2 − t−1/2 )(s0 − 1), s0 = s$2 ,

(2.10.4)

f where s(f (x)) = f (−x), $(f (x)) = f (x + 1/2). This gives that H H satisfies the PBW theorem, which of course can be checked directly using the abstract relations between the generators. f All the symmetries of HH hold for H H. Moreover, we may restrict its def 0 action on P to the even part P == Qq,t [X 2 , X −2 ], which is an irreducible def f H H–module for generic q, t. The odd part P 1 == XP 0 is a tilde-submodule as well, irreducible for generic q, t. Respectively, the spherical polynomials εm for even and odd m are even and odd and linearly generate P 0 and P 1 . Similarly, we may diminish F by introducing the following tilde-module: def Fe == Functf inite ((2Z)] , Qq,t ).

Correspondingly, the summation in the definition of h i1 will now be over the set (2Z)] . We write h i11 . Formulas (2.7.20) and (2.9.6) hold for h i11 with arbitrary l, m. The identity (2.3.3) now reads 2

hq x i11 = q k

2 /4

2j ∞ X 1 − q 2j+k Y 1 − q l+2k−1 j=0

= qk

2 /4

1 − qk

∞ ∞ Y 1 − q j+k X j=1

1−

qj

j=−∞

1 − ql

l=1

qj

2 −jk

.

qj

2 −jk

(2.10.5)

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Let us take q = exp(2πı/N ), where 0 < k ≤ n = [N/2], and pick q 1/2 = ± exp(πı/N ). The sign of q 1/2 can be arbitrary; all formulas hold for either choice. We obtain the identity 2j n−k X 1 − q 2j+k Y 1 − q l+2k−1 j 2 −jk q 1 − q k l=1 1 − q l j=0

=

k Y

N −1 X 1 2 q j −jk , j (1 − q ) j=0 j=1

(2.10.6)

which is nothing but (2.3.7). Note that this identity may be 0 = 0. It happens exactly when N = 2n and n − k is odd. Setting k = n, we come to the formulas N −1 X

q

j2

=q

d2

(1 − q j ) for N = 2n + 1, d = n/2 mod N,

j=1

j=0 n−1 X

n Y

j2

j

q (−1) =

j=0

n−1 Y

(1 − q j ) for N = 2n.

(2.10.7)

j=1

The sum in thePlast formula gives the Legendre symbol. −1 (j−k/2)2 does not depend on k, we may simplify (2.10.6) Using that N j=0 q under the assumption that n − k is even for N = 2n : 2j n−k X 1 − q 2j+k Y 1 − q l+2k−1 j 2 −jk q 1 − q k l=1 1 − q l j=0

=q

(d−h)(d+h)

n Y

(2.10.8)

(1 − q j ) for d ± h = (n ± k)/2 mod N.

j=k+1

Let us discuss the functional representation and, as a by-product, clarify why the case of even N and odd n−k is exceptional. We assume that k < N/2 and define the set ./00 : n −N + k k k N − ko if N = 2n, (2.10.9) + 1, . . . , − , + 1, . . . , = 2 2 2 2 n −N + k + 1 k k N − k − 1o = , . . . , − , + 1, . . . , if N = 2n + 1. 2 2 2 2 def

f The dimension of the H H–module F 00 == Funct( ./00 , Qq ) is N − 2k. It is f irreducible for odd N and has two irreducible H H–components otherwise: F 00 = (F 00 )0 ⊕ (F 00 )1 , dim(F 00 )j = n − k for N = 2n : (F 00 )j = Image(P j ) = {f ∈ F 00 | f (m] + n) = (−1)j f (m] )},

(2.10.10)

2.10. LITTLE DOUBLE HECKE ALGEBRA

255

where we take only negative m] . 2 The Gaussian q m] belongs to (F 00 )0 precisely when n−k is even. Otherwise it sits in (F 00 )1 . This explains the degeneration of (2.10.6) observed above. Note that F 00 for odd N is nothing but (F 0 )0 with q 1/2 = − exp(πı/N ). It f gives that (F 0 )0 from (2.10.2) remains irreducible upon restriction to H H. Let us adjust (2.9.6) from Theorem 2.9.9 to the case under consideration. We will need X def hf i00 == f (z)µ1 (z), z ∈ ./00 , and def Ck00 ==

q

d2

n Y

(1 − q j ), where

(2.10.11)

j=k+1

d = n/2 as N = 2n and N = 4d ± 1 for N = 2n + 1.

(2.10.12)

Now we involve the spherical polynomials. The images of εm in F 00 are linearly independent for m ≤ N − 2k and generate the whole space. Picking 1 ≤ l, m ≤ N − 2k, hεl εm γi00 = C q −

l2 +m2 +2k(|l|+|m|) 4

N (l2 +m2 +2k(|l|+|m|))

εl (m] ) Ck00 ,

C=ı if N = 2n + 1, C = 0 if k + l + m + n is odd for N = 2n, C = 1 otherwise.

(2.10.13) (2.10.14)

These formulas give a complete description of the Fourier transforms S1 and ¯ 1 . See (2.10.3). S

2.10.3

Half-integral k

Generally speaking, the identities we discuss here are beyond the classical theory because they depend on k and moreover involve the Macdonald polynomials. However, when k = 0, 1 and for other integers k > 1 there are strong links to the classical formulas. The transformation k 7→ k + 1 can be performed using the shift operator. The case of the half-integral k does not seem to have direct counterparts in the classical theory of Gaussian sums. The shift operator acts within the set of such k. Thus no immediate connections with the integral k can be expected. The formulas for the half-integral k are not too complicated to prove directly and sometimes even simpler than the “integral” ones. For instance, the Gaussian sum for k = 1/2 is of the collapsing type, in contrast to the classical sums for k = 0, 1, that give some nontrivial identities for the theta functions and eta functions.

256

CHAPTER 2. ONE-DIMENSIONAL DAHA

Note that the value k = 1/2 formally corresponds to the symmetric space SL2 (R)/SO2 . Thus, in the limit q → 1, the Fourier transform under consideration for the half-integral k is directly connected with the “orthogonal” Harish-Chandra transform. Let us note that there is a classical connection between the orthogonal and symplectic cases, the Howe duality. It could lead to a certain relation between the cases k = 1/2 and k = 2, classically and in the q–case. Due to the shift operators, it can lead to a connection between the integral and the half-integral cases too. This possibility has not been examined so far. The half-integers k can be taken positive or negative. At roots of unity, generally speaking, the positive and negative values cannot be distinguished. Moreover, in the case of odd N, half-integers are congruent to integers modulo N. A combination of these two remarks gives that the negative half-integral k are directly connected with the integral positive k for odd N, as we will later use to deform the little Verlinde algebras Let k ∈ 1/2 + Z. We follow the procedure which was already used above and begin with the formulas for the Gaussian sums with generic q. By generic, we mean that q a tb do not represent nontrivial roots of unity for a, b ∈ Z. Then we will “specialize” the formulas as q are roots of unity. The general identities we need are ∞ X

q

j 2 −kj 1

j=0

2j − q 2j+k Y 1 − q l+2k−1 1 − q k l=1 1 − q l

∞ ∞ Y (1 + q −k−1+2j ) Y (1 − q 2k+2j ) . = (1 + q k+2j ) j=1 (1 − q −1+2j ) j=1

(2.10.15)

We follow the same procedure and begin with the formulas for the Gaussian sums with generic q, assuming that the q a tb do not represent nontrivial roots of unity for a, b ∈ Z. Let us denote the double product from the last line by ΠΠ. It can be simplified. It equals, respectively: s Y 1 + q k−2j for k = 1/2 + s, s ∈ Z+ , ΠΠ = 1 − q 2k−2j j=0

ΠΠ =

s Y 1 − q 2k+2j j=1

1 + q k+2j

for k = −1/2 − s, s ∈ Z+ .

(2.10.16) (2.10.17)

Now let us consider q 1/2 = exp(πı/N ), provided that 0 < 2k < N. The analysis of F 00 in this case is similar to that for the integral k. However, the

2.10. LITTLE DOUBLE HECKE ALGEBRA

257 def

roles of even and odd N are inverse. We set ./00 == n −N + k k k N − ko + 1, . . . , − , + 1, . . . , if N = 2n + 1, 2 2 2 2 n −N + k + 1 k k N − k − 1o , . . . , − , + 1, . . . , if N = 2n. 2 2 2 2

(2.10.18)

f The dimension of the H H–module F 00 = Funct( ./00 , Qq ) remains N − 2k as in the integral case. It is irreducible for even N and has the following f H H–decomposition: F 00 = (F 00 )0 ⊕ (F 00 )1 , dim(F 00 )j = n − k for N = 2n + 1 :

(2.10.19)

(F 00 )j = Image(P j ) = {f ∈ F 00 | f (m] + N/2) = (−1)j f (m] )}, where m] < 0 are taken. The corresponding reduction of (2.10.15) reads [N/2−k]

X

qj

2 −kj

j=0

2j 1 − q 2j+k Y 1 − q l+2k−1 1 − q k l=1 1 − q l k−1/2

2k−N

= (1 + ı

) Ck00 ,

def Ck00 ==

Y 1 + q k−2j . 2k−2j 1 − q j=0

The main formula then becomes X def hεl εm γi00 == µ1 (z)εl (z) εm (z)γ(z)

(2.10.20)

(2.10.21)

z∈./00

= (1 + ı2k+|l|+|m|−N ) Ck00 q −

2.10.4

l2 +m2 +2k(|l|+|m|) 4

εl (m] ).

The negative case

Recall the corresponding formulas for HH . Taking def −N/2 ≤ k¯ == −1/2 − s < 0 for s ∈ Z+ and 0 ≤ s < N/2,

we set: n k¯ + 1 k¯ + 2 k¯ + 1 k¯ o ¯ == (2.10.22) ./ , ,...,− ,− 2 2 2 2 n k¯ + 1 k¯ + 2 k¯ + s + 1 k¯ k¯ + 2s + 1 o 1 1 = , , ... ,− , = ,...,− = 2 2 4 4 2 2 2 n 1/2 − s 3/2 − s o 1 1 3 1/2 + s = , ,...,− , , ,..., . 2 2 4 4 4 2 0 def

258

CHAPTER 2. ONE-DIMENSIONAL DAHA

¯ ¯ 0 , Qq ) is an irreducible HH(k)–module Then F¯ 0 = Funct( ./ of dimension ¯ instead of HH to show explicitly the dependence −2k¯ = 2s + 1. We put HH(k) ¯ on k. Let q = exp(2πı/N ). To more exact, we pick q 1/2 = − exp(πı/N ). The minus–sign in the case under consideration makes the inner product hf, gi0 ¯ 0 . All formupositive. Recall that it is the summation of f g ∗ µ1 over the set ./ las below will hold for either sign, as well as the analysis of the irreducibility. Only the positivity will be missing for the plus–sign. The module F¯ 0 is generated by the eigenfunctions of Y , which are the ¯ 0 , and images of εm for m] ∈ ./ hεl εm γi0 = q − Ck¯0

=

2s+1 X

q

¯ 2 (j−k) 4

j=s+1

¯ l2 +m2 +2k(|l|+|m|) 4

εl (m] ) Ck¯0 ,

(2.10.23)

s ¯ j ¯ Y 1 − q j+k Y 1 − q l+2k−1 1/16 =q (1 − q 1/2−j ). ¯ l k 1 − q l=1 1 − q j=1

f The case of H H. As above, k¯ = −s − 1/2 and q 1/2 = − exp(πı/N ). The latter ensures the positivity of the hermitian form. f The reduction of (2.10.15) to H H is straightforward: s X

qj

j=0

=

2 −kj ¯

¯ ¯ 2j 1 − q 2j+k Y 1 − q l+2k−1 1 − q k¯ l=1 1 − q l

s ¯ Y 1 − q 2k+2j j=1

¯ 1 + q k+2j

.

(2.10.24) ¯2

Actually, this formula coincides with the formula for Ck¯0 q −k /4 resulting from (2.10.23). We will skip the identification of the sums. Concerning the coincidence of the products (in the right-hand sides), it is an elementary algebraic exercise: q

¯2 /4 1/16−k

s Y j=1

(1 − q

1/2−j

)=

s ¯ Y 1 − q 2k+2j j=1

¯ 1 + q k+2j

.

(2.10.25)

Here q can be absolutely arbitrary. This formula is useful in the theory of η–like identities (in the limit s → ∞). ¯ –module F¯ 0 The coincidence has a simple explanation. Indeed, the HH(k) ¯ The f introduced above for k¯ remains irreducible upon the restriction to H H(k). ¯ and responsible for the summation in formula f ./–set constructed for H H(k) (2.10.24) is o n k¯ k¯ k¯ k¯ 00 def ¯ ./ == − − s, . . . , − , + 1, . . . , + s . 2 2 2 2

(2.10.26)

2.10. LITTLE DOUBLE HECKE ALGEBRA

259

¯ + 1, etc.) sit between consecutive Here the points from the second half (k/2 pairs of points from the first half. Ordering them naturally, we obtain n k¯ o 1/2 − s k¯ 3/2 − s − −s= , +1= , and so on , 2 2 2 2 ¯ 0 from (2.10.22). which is nothing but the third line for ./ The main formula is equivalent to (2.10.23) and reads as follows: 2j−1 s X Y 1 − q l+2k¯ 2j + k¯ 2j + k¯ ¯ ¯ j 2 +kj −(2j−1)k ) q ε ( ε ( (q ) ) l m 1 − ql 2 2 j=1 l=1

+

2j s ¯ X Y 2j + k¯ 2j + k¯ 1 − q l+2k j 2 +kj ¯ ¯ ) q ε (− ) ) ε (− (q −2j k l m l 1 − q 2 2 j=0 l=1

= q−

¯ l2 +m2 +2k(|l|+|m|) 4

εl (m] )

s ¯ Y 1 − q 2k+2j j=1

¯ 1 + q k+2j

.

(2.10.27)

¯ 00 . Here we take l] , m] ∈ ./

2.10.5

Deforming Verlinde algebras

We will conclude this section with the following observation. When N = def 2n+1, the case k¯ = −1/2−s is equivalent to the case of the integral kˆ == n−s. Here we may replace −1/2 by n modulo N because q 1/2 = − exp(πı/N ) is ˆ coincides with H ¯ for f f an N –th root of unity. To be more exact, H H(k) H(k) 1/2 such q , since the defining relations are the same. The functional represenˆ and F 00 (k) ¯ (see (2.10.9)) are isomorphic too. The dimensions tations F 00 (k) ¯ = −2k¯ = 2s + 1 = N − 2kˆ = dimF 00 (k). ˆ are obviously the same: dimF 00 (k) ˆ ˆ So we hit the main sector 0 < k < N/2, k ∈ Z. Recall that the irreˆ the positivity of the inner product, and all formulas were ducibility of F 00 (k), independent of the choice of the sign of q 1/2 for such k. Comparing (2.10.24) and (2.10.8), we arrive at the identity s s−1 Y Y 1 − q 1−2j −s(s+1)/4 = q (1 − q n−j ), n−s+2j 1+q j=1 j=0

(2.10.28)

which readily follows from (2.10.25). ¯ hold for any q. It is important that all claims and formulas about F 00 (k) ¯ the formula (2.10.27) For instance, its dimension is always 2s + 1 = −2k, works, and so on. The particular choice of q does not matter. The constraint k¯ = −1/2 − s is sufficient. For |q| 6= 1, the conjugation (q 1/4 )∗ = q −1/4 must be understood formally.

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CHAPTER 2. ONE-DIMENSIONAL DAHA

We conclude that F 00 (− 21 − s) for generic q is a flat q–deformation of F (n − s) as q 1/2 = − exp(πı/N ) for N = 2n + 1 and 0 ≤ s ≤ n. If one wishes to make the µ–form hermitian and positive, it is necessary to be more specific. The conditions q 1/2 = − exp(πıω) and 0 < ωs < 1/2 will do it. It is of course not necessary to assume that q is a root of unity. Note that 1 n 1 ω= and ωs ≤ < for q = −eπı/N . N 2n + 1 2 ¯ + as There is an application. Considering the T –plus component F 00 (k) a P SL(2, Z)–module, one can calculate its limit as q → 1. Upon proper renormalization, it will become nothing but the restriction of the irreducible representation of SL2 (C) of dimension s to its subgroup SL2 (Z). This readily ¯ + for generic q. This fact is unigives the P SL(2, Z)–irreducibility of F 00 (k) versal: if the dimension of the T –plus component is fixed, then there can be only finitely many N, q such that the action of P SL(2, Z) in this component is reducible. f Note that one can use HH instead of H H when discussing the correspon1/2 ¯ ˆ ˆ dence k ↔ k for odd N and q = − exp(πı/N ). Indeed, the module (F 0 )0 (k) 0 ¯ is isomorphic to F (k). We remind the reader that both modules remain irˆ and F 00 (k) ¯ upon the restriction to reducible and become, respectively, F 00 (k) ˆ = F 0 (k) ¯ becomes F 00 (k) ˆ = F 00 (k) ¯ over H f f H H . The identification F 0 (k) H . Thus the usage of HH here is possible; however, it does not add anything new. We see that the module F 00 (k) for integral 0 < k < N/2 allows a q– deformation when N = 2n + 1, together with all related structures. We can apply it to the little Verlinde algebra: the T –plus component (F 00 (1))+ of F 00 (k = 1). Generally speaking, little Verlinde algebras are introduced for the lattice Q of the radical weights. The standard Verlinde algebra is defined for all weights, i.e., for the lattice P. We obtain that (F 00 (1/2 − n))+ for generic q is a deformation of (F 00 (1))+ . The corresponding q–deformation of the “big” Verlinde algebra (for P ), which is (F 0 (1))+ in our notation, is not known. Comment. It is worth mentioning that the structural constants of the Verlinde multiplication are integral and positive in the basis formed by the (1) images of the SL2 –characters, i.e., by the discretizations of pm . This holds because multiplication can be interpreted as the restricted tensor (or fusion) product. This is the only property of the Verlinde algebras that will be lost upon the deformation above. At least we don’t know how to reformulate the integrality/positivity for generic q. Everything else will survive. ❑ Let us calculate the first nontrivial deformed little Verlinde algebra, which is V = (F 00 (−3/2))+ . It is the space of Q(q 1/2 )–valued functions on the set ¯ 00 is { −1/4, 3/4, 1/4 }. {1/4, 3/4}. Indeed, s = 1, k¯ = −3/2, and the set ./ Since we consider even functions only, the points 1/4, 3/4 are sufficient. 00

2.11. DAHA AND P–ADIC THEORY

261

The “restricted characters” are the images p00 , p02 of the Rogers polynomials p0 = 1, p2 (x) = q 2x + q −2x + 1 + (q + q 1/2 + 1 + q −1/2 + q −1 ). The restricted multiplication is pointwise. Since p00 = 1, there is only one nontrivial multiplication formula: ¢ ¡ (p02 )2 = q −3/2 (1 + q 1/2 )2 (1 + q)2 p02 − q −1 (1 + q 1/2 ) (1 + q 3/2 ) p00 . The inner product of functions f, g ∈ V is hf, gi = f g ∗ (3/4) + (1 − q 1/2 − q −1/2 ) f g ∗ (1/4). ¯

The normalized Gaussian is given by γ(3/4) = 1, γ(1/4) = q 1+k = q −1/2 . The main formula (the summation with the Gaussian) reads pl pm (3/4) + q −1/2 (1 − q 1/2 − q −1/2 ) pl pm (1/4) = (1 − q −1 )(1 + q 1/2 )−1 q

3(l+m)−l2 −m2 4

pl (m/2 − 3/4) pm (3/4).

(2.10.29)

Here q is arbitrary. If we take q 1/2 = − exp(πıω) and −2/3 < ω < 2/3, then the inner product is positive. Note that this inequality is weaker than 0 < ωs < 1/2 imposed above. f f The space V is not an H H –module, but the elements from H H that commute with T act there and make it irreducible. Actually, the action of X+X −1 and Y + Y −1 already results in its irreducibility (it is a general fact). The reduction to the corresponding Verlinde algebra V 0 is as follows: N = 5, n = 2, q 1/2 = − exp(πı/5), 1 − q 1/2 − q −1/2 = (q + q −1 )−2 . The latter number appears in the inner product. The polynomial p2 (x) becomes q 2x + q −2x + 1 with the restricted square (p02 )2 = p02 + p00 . Let me mention that some formulas in the theory of Verlinde algebras can be deformed using filtrations in the spaces of the coinvariants. See [FKLMM].

2.11

DAHA and p–adic theory

This section is devoted to double Hecke algebras in the general setting and their connection with the classical p–adic spherical transform. In contrast to previous sections, we state the results without proofs. The paper [C28] contains a complete theory.

262

CHAPTER 2. ONE-DIMENSIONAL DAHA

2.11.1

Affine Weyl group

Let R ⊂ Rn be a simple reduced root system, R+ ⊂ R the set of positive roots, {α1 , . . . , αn } ⊂ R+ the corresponding set of simple roots, and ϑ ∈ R+ the maximal coroot that is also the longest positive root in R∨ . We normalize the scalar product on Rn by the condition (ϑ, ϑ) = 2, so ϑ belongs to R as well and it is the longest short root there. For any α ∈ R, its dual P coroot 2α is α∨ = (α,α) = α/να . So να = (α, α)/2 = 1, 2, 3. We set ρ = (1/2) α α, P ρ∨ = (1/2) α α∨ . The affine roots are ˜ = {˜ R α = [α, να j]}, j ∈ Z. Note the appearance of να here. It is because of our nonstandard choice of ˜ ϑ. We identify nonaffine roots α with [α, 0] and set α0 = [−ϑ, 1]. For α ˜ ∈ R, sα˜ ∈ Aut(Rn+1 ) is the reflection (x, α) α ˜ , and (α, α) f = hsα˜ | α ˜ W = hsα | α ∈ Ri, W ˜ ∈ Ri.

sα˜ ([x, ζ]) = [x, ζ] − 2

(2.11.1)

f is a Coxeter group with the We set si = sαi . It is well known that W n generators {si }. Let ωi ⊂ R be the fundamental weights: (ωi , αj∨ ) = δij def

for 1 ≤ i, j ≤ n, P = ⊕ni=1 Zωi the weight lattice, and P+ == ⊕ni=1 Z+ ωi the def c= cone of dominant weights. We call W = W nP the extended affine Weyl group: wb([x, ζ]) = [w(x), ζ − (b, x)] for b ∈ P, x ∈ Rn . c → Z+ is given by the formula The length function l : W X X |(b, α∨ ) + 1|, |(b, α∨ )| + l(wb) = α ∈ R+ w(α∨ ) ∈ R+

(2.11.2)

α ∈ R+ w(α) ∈ −R+

f = where w ∈ W , b ∈ P . Let Q = ⊕ni=1 Zαi be the root lattice. Then W c is a normal subgroup and W c /W f = P/Q. Moreover, W c is the W nQ ⊂ W f , where semidirect product ΠnW c | l(π) = 0} = {π ∈ W c | π : {αi } 7→ {αi } }, 0 ≤ i ≤ n. Π = {π ∈ W It is isomorphic to P/Q and acts naturally on the affine Dynkin diagram for R∨ with the reversed arrows. It is not the standard Dynkin diagram for R because of our choice of ϑ. We set π(i) = j as π(αi ) = αj .

2.11. DAHA AND P–ADIC THEORY

2.11.2

263

Affine Hecke algebra

We denote the affine Hecke algebra by H. Its generators are Ti for i = 0, . . . , n and π ∈ Π; the relations are Ti Tj Ti . . . = Tj Ti Tj . . . , πTi π −1 = Tπ(i) , | {z } | {z } order of si sj order of si sj

(2.11.3)

(Ti − t1/2 )(Ti + t1/2 ) = 0 for 0 ≤ i ≤ n, π ∈ Π. c is a reduced expression, i.e., l(π −1 w) = l, we set If w ˆ = πsil . . . si1 ∈ W Twˆ = πTil . . . Ti1 . The elements Twˆ are well defined and form a basis of H. Let G be the adjoint split p–adic simple group corresponding to R. Then H is the convolution algebra of compactly supported functions on G that are left-right invariant with respect to the Iwahori subgroup B, due to Iwahori and Matsumoto. Namely, the Ti are the characteristic functions of the double cosets Bsi B, where we use a natural embedding W → G. Generally, it is not a homomorphism. To be more exact, the p–adic quadratic equations are represented in the form (Ti − 1)(Ti + ti ) = 0 for the standard normalization of the Haar measure. Here the ti may depend on the length of αi and are given in terms of the cardinality of the residue field. We will stick to our normalization of the T and assume that the parameters t coincide to simplify the formulas of this section. We will also use def ˆ δwˆ == t−l(w)/2 Twˆ , which satisfy the quadratic equation with 1. Let ∆ be the left regular representation of H. In the basis {δwˆ }, the representation ∆ is given by ½ t1/2 δsi wˆ if l(si w) ˆ = l(w) ˆ + 1, (2.11.4) Ti δwˆ = −1/2 1/2 −1/2 t δsi wˆ + (t − t )δwˆ if l(si w) ˆ = l(w) ˆ −1 c. and the obvious relations πδwˆ = δπwˆ , where π ∈ Π and w ˆ∈W The spherical representation appears as follows. Let X X def δ + == ( tl(w) )−1 tl(w) δw ∈ H. w∈W

w∈W

One readily checks that Ti δ + = t1/2 δ + for i = 1, . . . , n, and (δ + )2 = δ + . We call δ + the t–symmetrizer. Then def

∆+ == ∆δ + = ⊕b∈P Cδb+ , δw+ˆ = δwˆ δ + , is an H–submodule of ∆.

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CHAPTER 2. ONE-DIMENSIONAL DAHA

It is nothing but IndH H (Ct1/2 ), where H ⊂ H is the subalgebra generated by Ti , i = 1, . . . , n and Ct1/2 is the one-dimensional representation of H defined by Ti 7→ t1/2 . Due to Bernstein, Zelevinsky, and Lusztig (see, e.g., [Lus1]), we set Ya = Ta for a ∈ P+ and extend it to the whole P using Yb−a = Yb Ya−1 for dominant a, b. These elements are well defined and pairwise commutative. They form the subalgebra Y ∼ = C[P ] inside H. Using Y, one can omit T0 . Namely, the algebra H is generated by {Ti , i > 0, Yb } with the following relations: if (b, αi∨ ) = 1, Ti−1 Yb Ti−1 = Yb Yα−1 i Ti Yb = Yb Ti if (b, αi∨ ) = 0, 1 ≤ i ≤ n.

(2.11.5)

The PBW theorem for H gives that the spherical representation can be canonically identified with X , as δ + goes to 1. The problem is to calculate this isomorphism explicitly. It will be denoted by Φ. We come to the definition of Matsumoto’s spherical functions: φa (Λ) = Φ(δa+ )(Y 7→ Λ−1 ), a ∈ P. Here, by Y 7→ Λ−1 , we mean that Λ−1 substitutes for Yb . See [Ma]. In the b symmetric (W –invariant) case, the spherical functions are due to Macdonald. By construction, X −(ρ∨ ,a) −1 ∨ α∨ . φa = t−l(a) Λ−1 = t Λ for a ∈ P , ρ = (1/2) + a a α>0 ∨

The formula l(a) = (ρ , a) readily results from (2.11.2). So the actual problem is to calculate φa for non-dominant a. Example. Consider a root system of type A1 . In this case, P = Zω, where ω ∈ Z is the fundamental weight, Q = 2Z, and ω = πs for s = s1 . We identify def ∆+ and Y, so δ + = 1. Letting Y = Yω , T = T1 , we get Ym == Ymω = Y m and def

φm == φmω = t−m/2 Λ−m , for m ≥ 0. Note that T Y −1 T = Y and π = Y T −1 . Let us check that Λφ−m = t1/2 φ−m−1 − (t1/2 − t−1/2 )φm+1 , m > 0.

(2.11.6)

Indeed, φ−m = t−m/2 (T π)m (1) |Y 7→Λ−1 and Y −1 φ−m = t−m/2 (T −1 π)(T π)m (1) =t−m/2 (T − (t1/2 − t−1/2 ))π(T π)m (1) =t−m/2 (T π)m+1 (1) − −t−m/2 (t1/2 − t−1/2 )(πT )m π(1) =t1/2 φ−m−1 (Y −1 ) − t−m/2 (t1/2 − t−1/2 )(πT )m (t−1/2 πT )(1) =t1/2 φ−m−1 (Y −1 ) − (t1/2 − t−1/2 )φm+1 (Y −1 ).

(2.11.7)

2.11. DAHA AND P–ADIC THEORY

2.11.3

265

Deforming p–adic formulas

The following chain of theorems represents a new vintage of the classical theory. We are not going to prove them here. Actually, all claims that are beyond the classical theory of affine Hecke algebras can be checked by direct and not very difficult calculations, with a reservation about Theorems 2.11.5 and 2.11.6. Theorem 2.11.1. Let ξ ∈ Cn be a fixed vector and let q ∈ C∗ be a fixed def ξ scalar. We represent wˆ = bw, where w ∈ W, b ∈ P . In ∆ξq == ⊕w∈ c Cδw , the ˆ W formulas πδwξˆ = δπξ wˆ and ( t1/2 q(αi ,w(ξ)+b) −t−1/2 ξ 1/2 −t−1/2 δsi wˆ − q(αt i ,w(ξ)+b) δ ξˆ if i > 0, ξ q (αi ,w(ξ)+b) −1 −1 w Ti δwˆ = t1/2 q 1−(ϑ,w(ξ)+b) −t−1/2 ξ t1/2 −t−1/2 δs0 wˆ − q1−(ϑ,w(ξ)+b) δ ξˆ if i = 0 q 1−(ϑ,w(ξ)+b) −1 −1 w define a representation of the algebra H, provided that all denominators are nonzero, i.e., q (α,b+ξ) 6= 1 for all α ∈ R, b ∈ P. The regular representation ∆ with the basis δwˆ is the limit of representation ∆ξq as q → ∞, provided that ξ lies in the fundamental alcove: (ξ, αi ) > 0 for i = 1, . . . , n, (ξ, ϑ) < 1.

(2.11.8)

We see that the representation ∆ξq is a flat deformation of ∆ for such ξ. Moreover, ∆ξq ∼ = ∆. This will readily follow from the next theorem. Note that taking ξ in other alcoves, we get other limits of ∆ξq as q → ∞. They are isomorphic to the same regular representation; however, the formulas do depend on the particular alcove. We see that the regular representation has rather many remarkable systems of basic vectors. They are not quite new in the theory of affine Hecke algebras, but such systems were not studied systematically. Theorem 2.11.2. (i) We set Xb (δwξˆ ) = q (αi ,w(ξ)+b) δwξˆ for wˆ = bw, where we use the notation X[b,j] = q j Xb . These operators have a simple spectrum in ∆ξq under the conditions of the theorem and satisfy the relations dual to (2.11.5) with i = 0 added: Ti Xb Ti = Xb Xα−1 if (b, αi∨ ) = 1, i Ti Xb = Xb Ti if (b, αi∨ ) = 0, 0 ≤ i ≤ n, and moreover, πXb π −1 = Xπ(b) as π ∈ Π.

(2.11.9)

(ii) The double affine Hecke algebra HH is defined by imposing relations (2.11.3) and (2.11.9). Then the representation ∆ξq is nothing but the induced representation def ξ H IndH X (Cδid ), X == C[Xb ], which is isomorphic to ∆ as an H–module.

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Comment. The operators Xb are in a way the coordinates of the BruhatTits buildings corresponding to the p–adic group G. In the classical theory, we use only their combinatorial variants, namely, the distances between vertices, which are integers. The X–operators clarify dramatically the theory of the p–adic spherical Fourier transform, because, as we will see, they are the “missing” Fourier–images of the Y –operators. Obviously, the Xb do not survive in the limit q → ∞; however, they do not collapse completely. Unfortunately, 2 the Gaussian, which is q x /2 as Xb = q (x,b) , does. ❑ This theorem is expected to be connected with [HO2] and via this paper with [KL1]. Here we will stick to the spherical representations. The following theorem establishes a connection with ∆+ . Theorem 2.11.3. We set q ξ = t−ρ , i.e., q (ξ,b) 7→ t−(ρ,b) for all b ∈ P. The e It is well defined for generic corresponding representation will be denoted by ∆. q, t. For any b ∈ P, let πb be the minimal length representative in the set {bW }. It equals bu−1 for the length-minimum element ub ∈ W such that b def def e the space ∆] = = ⊕b∈P Cδb] is an HH – ub (b) ∈ −P+ . Setting δb] == δπb in ∆, e It is isomorphic to ∆+ as an H–module. submodule of ∆. The representation ∆] is described by the same formulas from Theorem 2.11.1, which vanish automatically on si πb not in the form πc , thanks to the special choice of q ξ . It results directly from the following: def

si πb = πc ⇔ (αi , b + d) 6= 0, where (αi , d) == δi0 . def

Here c = b − ((αi , b + d)αi∨ for α0∨ == −ϑ. c on Rn by the formulas wa((x)) = w(a + x). We define the action (( )) of W The above c is si ((b)). This action is constantly used in the theory of Kac– Moody algebras. It is very convenient when dealing with ∆] . Note that −1 πb ((c)) = bu−1 b ((c)) = ub (c) + b for b, c ∈ P.

Let us calculate the formulas from Theorem 2.11.1 upon q ξ 7→ t−ρ as q → ∞. This substitution changes the consideration, but not too much: ( t1/2 δs]i ((b)) if (αi , b + d) > 0, ] Ti δ b = (2.11.10) −1/2 ] 1/2 −1/2 ] t δsi ((b)) + (t − t )δb as (αi , b + d) < 0. ] Otherwise it is zero. The formulas πδb] = δπ((b)) hold for arbitrary π ∈ Π, b ∈ P. Since this calculation is different from that for generic ξ, it is not surprising that (2.11.10) does not coincide with (2.11.4) restricted to w ˆ = b and multiplied on the right by the t–symmetrizer δ + . The representations limq→∞ ∆] and ∆+ are equivalent, but the T –formulas with respect to the limit of the basis {δ ] } are different from those in terms of the classical basis {δb+ = δb δ + }.

2.11. DAHA AND P–ADIC THEORY

2.11.4

267

Fourier transform

In the first place, Macdonald’s nonsymmetric polynomials generalize the Matsumoto spherical functions. We use πb = bu−1 b = Min-length {bw, w ∈ W }. Theorem 2.11.4. (i) Let P be the representation of the double affine Hecke algebra HH in the space of Laurent polynomials P = C[Xb ]: 1/2

1/2

Ti = ti si + (ti

−1/2

− ti

)(Xαi − 1)−1 (si − 1), 0 ≤ i ≤ n,

(2.11.11)

j

Xb (Xc ) = Xb+c , π(Xb ) = Xπ(b) , π ∈ Π, where X[b,j] = q Xb . (ii) For generic q, t, the polynomials εb are uniquely defined from the relations: Ya (εb ) = t(u(ρ),a) q −(b,a) εb , where πb = bu for u ∈ W

(2.11.12)

εb (t−ρ ) = 1, where Xb (t−ρ ) = t−(b,ρ) . (iii) Setting Xb∗ = X−b , q ∗ = q −1 , and t∗ = t−1 , the limit of ε∗b (X 7→ Λ) as q → ∞ coincides with φb (Λ) for b ∈ P. Comment. Note that the ∆ξ –formulas from Theorem 2.11.1 are actually the evaluations of (2.11.11) at q ξ . To be more exact, there is an HH – homomorphism from P to the HH –module of functions on ∆ξ . For instance, Theorem 2.11.1 can be deduced from Theorem 2.11.4. The formulas for the polynomial representation of the double affine Hecke algebra HH are nothing but the Demazure–Lusztig operators in the affine setting. We are going to establish a Fourier-isomorphism ∆] → P, which is a generalization of the Macdonald–Matsumoto inversion formula. We use the constant term functional on Laurent series and polynomials denoted by h i. The first step is to make both representations unitary using µ =

Y 1 − Xα˜ , µ0 = µ/hµi, 1 − tXα˜ ˜

(2.11.13)

α∈ ˜ R

µ (πb ) = µ(t−πb ((ρ)) )/µ(t−ρ ), πb = bu−1 b . 1

Here we treat µ as a Laurent series to define µ0 . The coefficients of µ0 are rational functions in terms of q, t. The values µ1 (πb ) are rational functions in terms of q, t1/2 . The corresponding pairings are hf , gi pol = hf Tw0 w0 (g(X −1 ) µ0 i, f, g ∈ P, X ] X ] X h fb δb , gb δb i Del = (µ1 (πb ))−1 fb gb . Here w0 is the longest element in W. Note that the element Tw20 is central in the nonaffine Hecke algebra H generated by {Ti , i > 0}. Both pairings are

268

CHAPTER 2. ONE-DIMENSIONAL DAHA

well defined and symmetric. Let us give the formulas for the corresponding anti-involutions: Ti 7→ Ti , Xb 7→ Xb , T0 7→ T0 , Yb 7→ Tw0 Yw−1 T −1 0 (b) w0

in ∆+ ,

Ti 7→ Ti , Yb 7→ Yb , T0 7→ Ts−1 Yϑ , Xb 7→ Tw−1 Xw−10 (b) Tw0 in P, 0 ϑ where 1 ≤ i ≤ n, b ∈ P. P P Theorem 2.11.5. (i) Given f = b fb δb] ∈ ∆] , we set fb = b fb ε∗c ∈ P, where Xb∗ = Xb−1 , q ∗ = q −1 , t∗ = t−1 . The inversion of this transform is as follows: fb = tl(w0 )/2 (µ1 (πb ))−1 h fb, ε∗ i pol . (2.11.14) (ii) The Plancherel formula reads hf , gi Del = tl(w0 )/2 hfb, gbi pol .

(2.11.15)

Both pairings are positive definite over R if t = q k , q > 0, and k > −1/h for the Coxeter number h = (ρ, ϑ) + 1. P P f (iii) The transform f = b fb δb] 7→ fe = b fb∗ δb] is an involution: (fe) = f. To apply it for the second to replace ε∗b by the corresponding P ∗ time,1 we need ] δ–function, which is c εb (πc )µ (πc )δc . Recall that ε∗b becomes the Matsumoto spherical function φb in the limit q → ∞ upon the substitution X 7→ Λ. It is easy to calculate the limits of µ0 and µ1 (πb ). We come to a variant of the Macdonald–Matsumoto formula. Claim (iii) has no counterpart in the p–adic theory. Technically, it is because the conjugation ∗ sends q 7→ q −1 and is not compatible with the limit q → ∞. It is equivalent to the non−p–adic self-duality εb (πc ) = εc (πb ) of the nonsymmetric Macdonald polynomials. The following theorem also has no p–adic counterpart because the Gaussian is missing. ( πb ((kρ)) , πb ((kρ)) )/2 Theorem 2.11.6. We set , where t = q k , use να = P γ(πb ) = q (α, α)/2, and ρ = (1/2) α>0 α. For arbitrary b, c ∈ P,

hε∗b , ε∗c γiDel = γ(π0 )2 γ(πb )−1 γ(πc )−1 ε∗c (πb )h1 , γiDel , ∞ ³ X Y Y 1 − t(ρ,α) q jνα ´ h1 , γiDel = ( γ(πa )) . (ρ,α)−1 q jνα 1 − t j=1 a∈P α∈R

(2.11.16) (2.11.17)

+

2.11.5

One-dimensional case

Let us use the Pieri formula (2.6.2) from Corollary 2.6.2 for X −1 and m ≥ 0 upon the conjugation ∗ : Xε∗−m =

t−1/2−1 q −m−1 − t1/2 ∗ t−1/2 − t1/2 ∗ ε − ε . −m−1 t−1 q −m−1 − 1 t−1 q −m−1 − 1 1+m

(2.11.18)

2.11. DAHA AND P–ADIC THEORY

269

Under the limit q → ∞, we obtain exactly (2.11.6) for φ−m (Λ) 7→ ε∗−m (X): Xε∗−m = t1/2 ε∗−m−1 − (t1/2 − t−1/2 )ε∗m+1 .

(2.11.19)

The generalization is Theorem 2.11.4(iii). 1−X Also, limq→∞ µ0k (x) = 1−tX . So the limit of the pairing h , ipol reads as hf, gi∞ pol = hf T (g)

1−X i, f, g ∈ P. 1 − tX

] . In the limit q → ∞, Substituting m for mω in the indices: ∆] = ⊕m Cδm the operators T and π act here as follows: ] ] ] ] ] T δm = t1/2 δ−m , T δ−m = t−1/2 δm + (t1/2 − t−1/2 )δ−m , ] ] = δ1−m and πδm for m ≥ 0. Concerning h·, ·idel , we use (2.7.9) for m > 0: 1

1

µ (m] ) = µ ((1 − m)] ) = t

−(m−1)

m−1 Y j=1

=

m−1 Y j=1

1 − t2 q j 1 − qj

(2.11.20)

tq j/2 − t−1 q −j/2 → tm−1 as q → ∞. q j/2 − q −j/2

Here µ1 (m] ) = µ1 (πmω ). Therefore the limit of this scalar product is simply ½ 1−m t if m > 0, ] ] ∞ hδm , δn idel = δmn m t if m ≤ 0. Comment. The latter inner product is different from the classical one, which is calculated as follows. We define the pairing (Twˆ , Tuˆ ) = Constant Term (Twˆ Tuˆ−1 ) on the affine Hecke algebra hY ±1 , T i, where the constant term is with respect to the decomposition via Twˆ . It is simply δuˆ,wˆ . Then we switch to δwˆ = ˆ + + ❑ , δn ), which is δmn t−|m| /(1 + t). t−l(w)/2 Twˆ and finally calculate (δm b The Fourier transform f 7→ f from Theorem 2.11.5 is compatible with the limit q → ∞. The inversion formula (2.11.14) and the Plancherel formula (2.11.15) survive as well. We obtain a minor reformulation of the Matsumoto formulas. Upon symmetrization, T disappears from the inversion formula and we come to the Macdonald inversion. What is completely missing in the limit is (2.11.16). One of the main applications of the double Hecke algebra is adding the Gaussian to the classical p–adic theory. Technically, one does not need HH to do this. The ξ–deformation of the Iwahori-Matsumoto formulas (Theorem 2.11.1) is the main tool. Its justification is elementary. It is surprising that it had not been discovered well before the double Hecke algebras were introduced.

270

2.12

CHAPTER 2. ONE-DIMENSIONAL DAHA

Degenerate DAHA

The p–adic origin of the double affine Hecke algebra (DAHA) is the most natural to consider, however, the connections with real harmonic analysis, radial parts and Dunkl operators are equally important. They played a key role in the beginning of the theory of DAHA; the exact link to the p–adic theory is relatively recent. We now discuss the trigonometric and rational degenerate DAHAs, which govern the applications in real harmonic analysis. This section is a continuation of Section 1.6 from Chapter 1, where we interpreted the Harish-Chandra transform as a map from the trigonometricdifferential polynomial representation of the degenerate DAHA (in Laurent polynomials) to the difference-rational polynomial representation. Affine roots. Continuing from the previous section, let R = {α} ⊂ Rn be a root system of type A, B, ..., F, G with respect to a euclidean form (z, z 0 ) on Rn 3 z, z 0 , W the Weyl group generated by the reflections sα , R+ be the set of positive roots corresponding to (fixed) simple roots roots α1 , ..., αn , Γ the Dynkin diagram with {αi , 1 ≤ i ≤ n} as the vertices, R∨ = {α∨ = 2α/(α, α)} the dual root system, and Q = ⊕ni=1 Zαi ⊂ P = ⊕ni=1 Zωi , where {ωi } are the fundamental weights defined by (ωi , αj∨ ) = δij for the simple coroots αi∨ . Recall that the form is normalized by the condition (α, α) = 2 for the short roots. This normalization coincides with that from the tables in [Bo] for A, C, D, E, G. Hence να = (α, α)/2 can be 1, 2, or 3. We write νlng for long roots (νsht = 1) and also set nui = ναi . Let ϑ ∈ P R∨ be the maximal positive P coroot (it is maximal short in R) and ρ = (1/2) α∈R+ α = i ωi . The vectors α ˜ = [α, να j] ∈ Rn ×R ⊂ Rn+1 for α ∈ R, j ∈ Z form the affine def ˜ ⊃ R (z ∈ Rn are identified with [z, 0]). We add α0 = root system R = [−ϑ, 1] to ˜ of positive roots is R+ ∪ {[α, να j], α ∈ R, j > 0}. the simple roots. The set R Let α ˜∨ = α ˜ /να , so α0∨ = α0 . The Dynkin diagram Γ of R is completed by α0 (by −ϑ to be more exact). ˜ It is the completed (affine) Dynkin diagram for R∨ from The notation is Γ. [Bo] with the arrows reversed. ˜ The set of the indices of the images of α0 by all the automorphisms of Γ 0 will be denoted by O (O = {0} for E8 , F4 , G2 ). Let O = r ∈ O, r 6= 0. The elements ωr for r ∈ O0 are the so-called minuscule weights: (ωr , α∨ ) ≤ 1 for α ∈ R+ . f is generated by the affine reflections sα˜ . This The affine Weyl group W group is the semidirect product W nQ of its subgroups W and the lattice Q. c is generated by W and P. It is isomorphic to The extended Weyl group W f for the group Π formed by the elements W nP and, also, isomorphic to ΠnW c leaving Γ ˜ invariant. of W

2.12. DEGENERATE DAHA

271

The latter group is isomorphic to P/Q by the natural projection {ωr 7→ πr , r ∈ O}, where ωr = πr ur , ur ∈ W, π0 = id. The elements {ur } preserve the set {−ϑ, αi , i > 0}. c, πr ∈ Π, w f , the length l(w) Setting w b = πr w e∈W e∈W b is by definition the length of the reduced decomposition w e = sil ...si2 si1 in terms of the simple reflections si , 0 ≤ i ≤ n.

2.12.1

Definition of DAHA

By m, we denote the least natural number such that (P, P ) = (1/m)Z. Thus m = 2 for D2k , m = 1 for B2k and Ck , otherwise m = |Π|. The double affine Hecke algebra depends on the parameters q, tν , ν ∈ def {να }. The definition ring is Qq,t == Q[q ±1/m , t±1/2 ], formed by the polynomials ±1/2 in terms of q ±1/m and {tν }. We set tα˜ = tα = tνα , ti = tαi , qα˜ = q να , qi = q ναi , ˜ 0 ≤ i ≤ n. where α ˜ = [α, να j] ∈ R,

(2.12.1)

It will be convenient in many formulas to switch to the parameters {kν } instead of {tν }, setting tα = tν = qαkν for ν = να , and ρk = (1/2)

X

kα α.

α>0

The notation ki = kαi also will be used. For pairwise commutative X1 , . . . , Xn , X˜b =

n Y

b ˜b ) = Xw( Xili q j if ˜b = [b, j], w(X b ˜b) .

(2.12.2)

i=1

where b =

n X i=1

li ωi ∈ P, j ∈

1 c. Z, w b∈W m

We set (˜b, c˜) = (b, c) ignoring the affine extensions in the inner product unless (˜b, c˜ + d)= (b, c) + j is considered. Later Y˜b = Yb q −j will be needed. Note the negative sign of j. ∗ −1 We will also use that πr−1 is πr∗ and u−1 r is ur∗ for r ∈ O , ur = πr ωr . The reflection ∗ is induced by the standard nonaffine involution of the Dynkin diagram Γ.

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Definition 2.12.1. The double affine Hecke algebra HH is generated over Qq,t by the elements {Ti , 0 ≤ i ≤ n}, pairwise commutative {Xb , b ∈ P } satisfying (2.12.2), and the group Π, where the following relations are imposed: 1/2 −1/2 (o) (Ti − ti )(Ti + ti ) = 0, 0 ≤ i ≤ n; (i) Ti Tj Ti ... = Tj Ti Tj ..., mij factors on each side; (ii) πr Ti πr−1 = Tj if πr (αi ) = αj ; (iii) Ti Xb Ti = Xb Xα−1 if (b, αi∨ ) = 1, 0 ≤ i ≤ n; i (iv) Ti Xb = Xb Ti if (b, αi∨ ) = 0 for 0 ≤ i ≤ n; (v) πr Xb πr−1 = Xπr (b) = Xu−1 q (ωr∗ ,b) , r ∈ O0 . ❑ r (b) f, r ∈ O, the product Given w e∈W def

Tπr we == πr

l Y

e= Tik , where w

l Y

e sik , l = l(w),

(2.12.3)

k=1

k=1

does not depend on the choice of the reduced decomposition (because {T } satisfy the same “braid” relations as {s} do). Moreover, c. TvˆTwb = Tvˆwb whenever l(ˆ v w) b = l(ˆ v ) + l(w) b for vˆ, w b∈W

(2.12.4)

In particular, we arrive at the pairwise commutative elements from the previous section: n n Y X def Yb = Yili if b = li ωi ∈ P, where Yi == Tωi . (2.12.5) i=1

i=1

They satisfy the relations if (b, αi∨ ) = 1, Ti−1 Yb Ti−1 = Yb Yα−1 i Ti Yb = Yb Ti if (b, αi∨ ) = 0, 1 ≤ i ≤ n.

(2.12.6)

For arbitrary nonzero q, t, any element H ∈ HH has a unique decomposition in the form X H= gw fw Tw , gw ∈ Qq,t [X], fw ∈ Qq,t [Y ], (2.12.7) w∈W

and five more analogous decompositions corresponding to the other orderings of {T, X, Y }. It makes the polynomial representation (to be defined next) 1/2 the HH –module induced from the one-dimensional representation Ti 7→ ti , 1/2 Yi 7→ Yi of the affine Hecke subalgebra HY = hT, Y i. Automorphisms. The following maps can be uniquely extended to automorphisms of HH (see [C20],[C28]): −1 ε : Xi 7→ Yi , Yi 7→ Xi , Ti 7→ Ti−1 (i ≥ 1), tν 7→ t−1 ν , q 7→ q ,

(2.12.8)

(ω ,ω ) − r2 r

τ+ : Xb 7→ Xb , Yr 7→ Xr Yr q , Ti 7→ Ti (i ≥ 1), tν 7→ tν , q 7→ q, −1 −1 −1 (2.12.9) τ+ : Yϑ 7→ q Xϑ T0 Tsϑ , T0 7→ q Xϑ T0−1 , and def

def

τ− == ετ+ ε, and σ == τ+ τ−−1 τ+ = τ−−1 τ+ τ−−1 = εσ −1 ε,

(2.12.10)

2.12. DEGENERATE DAHA

273

where r ∈ O0 . In the definition of τ± and σ, we need to add q ±1/(2m) to Qq,t . Here the quadratic relation (o) from Definition 2.12.1 may be omitted. Only the group relations matter. Thus these automorphisms act in the corresponding braid groups extended by fractional powers of q treated as central elements. The elements τ± generate the projective P SL(2, Z), which is isomorphic to the braid group B3 due to Steinberg. Adding ε, we obtain the projective P GL(2, Z). These and the statements below are from [C16]. See also Chapter 3. Note that HH , its degenerations below, and the corresponding polynomial representations are actually defined over Z extended by the parameters of DAHA. We prefer to stick to the field Q, because the Lusztig isomorphisms will require Q.

2.12.2

Polynomials, intertwiners

The Demazure–Lusztig operators are defined as follows: 1/2

1/2

Ti = ti si + (ti

−1/2

− ti

)(Xαi − 1)−1 (si − 1), 0 ≤ i ≤ n,

(2.12.11)

and obviously preserve Q[q, t±1/2 ][X]. We note that only the formula for T0 involves q: 1/2

1/2

−1/2

T0 = t0 s0 + (t0 − t0

)(qXϑ−1 − 1)−1 (s0 − 1),

−(b,ϑ) (b,ϑ)

where s0 (Xb ) = Xb Xϑ

q

, α0 = [−ϑ, 1].

(2.12.12)

The map sending Tj to the formula in (2.12.11), Xb 7→ Xb (see (2.12.2)), and πr 7→ πr induces a Qq,t –linear homomorphism from HH to the algebra of linear endomorphisms of Qq,t [X]. This HH –module, which will be called the polynomial representation, is faithful and remains faithful when q, t take any nonzero complex values, assuming that q is not a root of unity. The images of the Yb are called the difference Dunkl operators. To be more exact, they must be called difference-trigonometric Dunkl operators, because there are also difference-rational Dunkl operators. Intertwining operators. The Y –intertwiners (see [C23]) are introduced as follows: 1/2

Ψi = Ti + (ti

−1/2

− ti

)(Yα−1 − 1)−1 for 1 ≤ i ≤ n, i

1/2

−1/2

Ψ0 = Xϑ Tsϑ − (t0 − t0 1/2

−1/2

= Y0 T0 X0 + (t0 − t0

1/2

Fi = Ψi (ψi )−1 , ψi = ti

)(Yα0 − 1)−1

)(Yα−1 − 1)−1 , for Y0 = Yα0 = q −1 Yϑ−1 , 0 1/2

+ (ti

−1/2

− ti

)(Yα−1 − 1)−1 . i

(2.12.13)

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CHAPTER 2. ONE-DIMENSIONAL DAHA

These formulas are the ε–images of the formulas for the X–intertwiners, which are a straightforward generalization of those in the affine Hecke theory. The intertwiners belong to HH extended by the rational functions in terms of {Y }. The F are called the normalized intertwiners. The elements def

, 0 ≤ i ≤ n, r ∈ O0 , Fi , Pr == Xr Tu−1 r satisfy the same relations as {si , πr } do, so the map c, b = πr sil · · · si1 ∈ W w b 7→ Fwb = Pr Fil · · · Fi1 , where w

(2.12.14)

c. is a well defined homomorphism from W The intertwining property is def

Fwb Yb Fwb−1 = Yw(b) where Y[b,j] == Yb q −j . b The P1 in the case of GL is due to Knop and Sahi. As to Ψi , they satisfy the homogeneous Coxeter relations and those with Πr . So we may set Ψwb = Pr Ψil · · · Ψi1 for the reduced decompositions. They intertwine Y as well. The formulas for Ψi when 1 ≤ i ≤ n are well known in the theory of affine c , are the raising Hecke algebras. The affine intertwiners, those for w b ∈ W operators for the Macdonald nonsymmetric polynomials, serve the HarishChandra spherical transform and Opdam’s nonsymmetric transform, and are a key tool in the theory of semisimple representations of DAHAs.

2.12.3

Trigonometric degeneration def

def

We set Qk == Q[kα ]. If the integral coefficients are needed, we take Zk == Z[kα , 1/m] as the definition ring. The degenerate (graded) double affine Hecke algebra HH0 is the span c and the pairwise commutative of the group algebra Qk W def

y˜b ==

n X (b, αi∨ )yi + u for ˜b = [b, u] ∈ P × Z, i=1

satisfying the following relations: def

sj yb − ysj (b) sj = −kj (b, αj ), (b, α0 ) == −(b, ϑ), πr y˜b = yπr (˜b) πr for 0 ≤ j ≤ n, r ∈ O.

(2.12.15)

Comment. Without s0 and πr , we arrive at the defining relations of the graded affine Hecke algebra from [Lus1]. The algebra HH0 has two natural

2.12. DEGENERATE DAHA

275

polynomial representations via the differential-trigonometric and differencerational Dunkl operators. There is also a third one, the representation in terms of infinite differential-trigonometric Dunkl operators, which generates (at trivial center charge) differential-elliptic W –invariant operators generalizing those due to Olshanetsky–Perelomov. See, e.g., [C23]. We will need in this section only the (most known) differential-trigonometric polynomial representations. ❑ Let us establish a connection with the general DAHA. We set q = exp(v), tj = qiki = q ναi ki , Yb = exp(−vyb ), v ∈ C. Using ε from (2.12.8), the algebra HH is generated by Yb , Ti for 1 ≤ i ≤ n, and , r ∈ O0 . ε(T0 ) = Xϑ Tsϑ , ε(πr ) = Xr Tu−1 r It is straightforward to see that the relations (2.12.15) for the yb , si (i > 0), s0 , and the πr are the leading coefficients of the v–expansions of the general relations for this system of generators. Thus HH0 is HH in the limit v → 0. When calculating the limits of the Yb in the polynomial representation, the “trigonometric” derivatives of Q[X] appear: ∂a (Xb ) = (a, b)Xb , a, b ∈ P, w(∂b ) = ∂w(b) , w ∈ W. The limits of the formulas for Yb acting in the polynomial representation are the trigonometric Dunkl operators def

Db == ∂b +

X α∈R+

¢ kα (b, α) ¡ − (ρk , b). 1 − s α (1 − Xα−1 )

(2.12.16)

They act on the Laurent polynomials f ∈ Qk [X], are pairwise commutative, c: and y[b,u] = Db + u satisfy (2.12.15) for the following action of the group W wx (f ) = w(f ) for w ∈ W, bx (f ) = Xb f for b ∈ P. For instance, sx0 (f ) = Xϑ sϑ (f ), and πrx (f ) = Xr u−1 r (f ). Degenerating {Ψ}, one obtains the intertwiners of HH0 : ¡ ¢ νi ki k0 Ψ0i = si + , 0 ≤ i ≤ n, Ψ00 = Xϑ sϑ + in Qk [X] , y 1 − yϑ ¢ ¡ α0i 0 −1 0 (2.12.17) Pr = πr , Pr = Xr ur in Qk [X] , r ∈ O . The operator P10 in the case of GL (it is of infinite order) plays the key role in [KnS]. Recall that the general normalized intertwiners are 1/2

Fi = Ψi ψi−1 , ψi = t1/2 + (ti

− t1/2 )(Yα−1 − 1)−1 . i

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CHAPTER 2. ONE-DIMENSIONAL DAHA

Their limits are Fi0 = Ψ0i (ψi0 )−1 , ψi0 = 1 +

νi ki . y αi

They satisfy the unitarity condition (Fi0 )2 = 1, and the products Fwb0 can be defined for any decompositions of w. b One then has: Fwb0 yb (Fwb0 )−1 = yw(b) b . Equating Fi = Fi0 for 0 ≤ i ≤ n, Pr = Pr0 for r ∈ O, we come to the formulas for Ti (0 ≤ i ≤ n), Xr (r ∈ O0 ) in terms of si , yb , and Yb = exp(−vyb ). These formulas determine the Lusztig homomorphism æ0 from HH to def the completion Zk,q,t HH0 [[vyb ]] for Zk,q,t == Zk Zq,t . See, e.g., [C23]. H0 , where the T –factor has to For instance, Xr ∈ HH becomes πr Tu−1 −1 in H r be further expressed in terms of s, y. In the degenerate polynomial representa)ur )−1 , not as straightforward multiplication tion, æ0 (Xr ) acts as Xr (æ0 (Tu−1 r by Xr . However, these two actions coincide in the limit v → 0, since Tw become w. Upon the v–completion, we obtain an isomorphism æ0 : Qk [[v]] ⊗ HH → Qk [[v]] ⊗ HH0 . We will use the notation (d, [α, j]) = j. For instance, (b+d, α0 ) = 1−(b, ϑ). Treating v as a nonzero number, an arbitrary HH0 –module V 0 that is a union of finite dimensional Y –modules has a natural structure of an HH– module provided that we have q (αi ,ξ+d) = ti ⇒ (αi , ξ + d) = νi ki ,

(2.12.18)

q (αi ,ξ+d) = 1 ⇒ (αi , ξ + d) = 0, where 0 ≤ i ≤ n, yb (v 0 ) = (b, ξ)v 0 for ξ ∈ Cn , 0 6= v 0 ∈ V 0 . For the modules of this type, the map æ0 is over the ring Qk,q,t extended by (α, ξ + d), q (α,ξ+d) for α ∈ R, and y–eigenvalues ξ. Moreover, we need to localize by (1 − q (α,ξ+d) ) 6= 0 and by (α, ξ + d) 6= 0. Upon such extension and localization, æ0 is defined over Zk,q,t if the module is y–semisimple. If there are nontrivial Jordan blocks, then the formulas will contain factorials in the denominators. For instance, let I 0 [ξ] be the HH0 –module induced from the one-dimensional y–module yb (v) = (b, ξ)v. Assuming that q is not a root of unity, the mapping æ0 supplies it with a structure of HH–module if q (α,ξ)+να j = tα implies (α, ξ) + να j = να kα

2.12. DEGENERATE DAHA

277

for every α ∈ R, j ∈ Z, and the corresponding implications hold for t replaced by 1. This means that (α, ξ) − να kα 6∈ να Z +

2πı (Z \ {0}) 63 (α, ξ) for all α ∈ R. v

(2.12.19)

Generalizing, we obtain that æ0 is well defined for any HH0 –module generated by its y–eigenvectors with the y–eigenvalues ξ satisfying this condition, assuming that v 6∈ πıQ. Comment. Actually, there are at least four different variants of æ0 because the normalization factors ψ, ψ 0 may be associated with different onedimensional characters of the affine Hecke algebra hT, Y i and its degeneration. It is also possible to multiply the normalized intertwiners by the characters c before equating. Note that if we divide the intertwiners Ψ and/or Ψ0 of W by ψ, ψ 0 on the left in the definition of F, F 0 , it corresponds to switching from −1/2 Ti 7→ ti to the character Ti 7→ −ti together with the multiplication by the f sign-character of W . ❑

2.12.4

Rational degeneration

The limit to the rational Dunkl operators is as follows. We set Xb = ewxb , db (xc ) = (b, c), so the above derivatives ∂b become ∂b = (1/w)db . In the limit w → 0, wDb tends to the operators def

Db ==db +

X kα (b, α) ¡ ¢ 1 − sα , xα α∈R

(2.12.20)

+

which were introduced by Dunkl and, as a matter of fact, gave birth to the DAHA direction, although KZ and the Macdonald theory must be mentioned too as origins of DAHAs. These operators are pairwise commutative and satisfy the cross-relations X Db xc − xc Db = (b, c) + kα (b, α)(c, α∨ )sα , for b, c ∈ P. (2.12.21) α>0

These relations, the commutativity of D, the commutativity of x, and the W –equivariance w xb w−1 = xw(b) , w Db w−1 = Db

for b ∈ P+ , w ∈ W,

are the defining relations of the rational DAHA HH00 . The references are [CM] (the case of A1 ) and [EG]; however, the key part of the definition is the commutativity of Db due to Dunkl [Du1]. The Dunkl operators and the operators of multiplication by the xb form the polynomial representation of HH00 , which is faithful. It readily justifies the PBW theorem for HH00 .

278

CHAPTER 2. ONE-DIMENSIONAL DAHA

Note that in contrast to the q, t–setting, the definition of the rational DAHA can be extended to finite groups generated by complex reflections (Dunkl, Opdam and Malle). There is also a generalization due to Etingof– Ginzburg from [EG] (the symplectic reflection algebras). Comment. Following [CO], there is a one-step limiting procedure from HH to HH00 . We set √ √ Yb = exp(− uDb ), Xb = e uxb , assuming that q = eu and u → 0. We come directly to the relations of the rational DAHA and the formulas for Db . The advantage of this direct construction is that the automorphisms τ± survive in the limit. Indeed, τ+ 2 in HH can beP interpreted as the formal conjugation by the q–Gaussian q x /2 , x2 /2 ∨ , where x2 = i xωi xαi . In the limit, it becomes the conjugation by e preserving w ∈ W, xb , and taking Db to Db − xb . Respectively, τ− preserves w and Db , and sends xb 7→ xb − Db . These automorphisms do not exist in the HH0 . ❑ The abstract Lusztig-type map from HH0 to HH00 is as follows. Let w 7→ w and Xb = ewxb . We expand Xα in terms of xα in the formulas for the trigonometric Dunkl operators Db : ∞ X X 1 Bm Db = Db − (ρk , b) + kα (b, α) (−wxα )m (1 − sα ) w m! m α∈R

(2.12.22)

+

for the Bernoulli numbers Bm . Then we use these formulas as abstract expressions for yb in terms of the generators of HH00 : yb = w1 Db + . . . . One obtains an isomorphism æ00 : Q[[w]]⊗HH0 → Q[[w]]⊗HH00 , which maps HH0 to the extension of HH00 by the formal series in terms of wxb . An arbitrary representation V 00 of HH00 that is a union of finite dimensional Qk [x]–modules becomes an HH0 –module provided that wζα 6∈ 2πı(Z \ {0}) for xb (v) = (ζ, b)v, 0 6= v ∈ V 00 .

(2.12.23)

Similar to (2.12.19), this constraint simply restricts choosing w 6= 0. The formulas for yb become locally finite in any representations of HH00 , where xb act locally nilpotent, for instance, in finite dimensional H00 –modules. In this case, there are no restrictions for w. Comment. Note that the “identification” of HH0 and HH00 has a common source with the method used in the so-called localization due to Opdam and Rouquier (see [GGOR], [VV2]). They separate the differentials db from the formula for the Dunkl operators to define a KZ-type connection with values in double Hecke modules. We equate these differentials in the rational and trigonometric formulas for the Dunkl operators to connect HH0 and HH00 . ❑

2.12. DEGENERATE DAHA

279

Finally, the composition def

æ == æ00 ◦ æ0 : HH[[v, w]] → HH00 [[v, w]] is an isomorphism. Without the completion, it makes an arbitrary finite dimensional HH00 –module V 00 a module over HH as q = ev , tα = q kα for sufficiently general (complex) nonzero numbers v, w. This isomorphism was discussed in [BEG] (Proposition 7.1). The finite dimensional representations are the most natural here because, on the one hand, æ00 lifts the modules that are unions of finite dimensional x–modules to those for X, on the other hand, æ0 maps the HH0 –modules that are unions of finite dimensional y–modules to those for Y. So one must impose these conditions for both x and y. We obtain a functor from the category of finite dimensional representations of HH00 to that for HH . Indeed, it is known now that there are finitely many irreducible objects in the former category. Therefore we can find v, w serving all irreducible representations and their extensions. This functor is faithful provided that q is generic in the following sense: q a tb = 1 ⇒ a + kb = 0 for a, b ∈ Q. This condition ensures the equivalence of the categories of finite dimensional representations for HH and HH00 . Using æ for infinite dimensional representations is an interesting problem. It makes the theory analytic. For instance, the triple composition æ00 ◦ G ◦ æ0 for the inverse Opdam transform G (see [O3] and formula (6.1) from [C24]) n H– embeds HH in HH00 and identifies the HH00 –module C∞ c (R ) with the H module of PW-functions under the condition −1/h. See [O3, C24] for more detail. The degenerations above play the role of the Lie algebras in the theory of DAHA. The Lusztig isomorphisms are certain counterparts of the exp-log maps. It is especially true for the composition map from HH to the rational degeneration, because the latter has the projective action of the P SL(2, Z) and some other features that make it close to the general HH. The algebra HH0 is not projective P SL(2, Z)–invariant. Note that there are some special “rational” symmetries and tools that have no q, t–counterparts. On the other hand, the semisimple representations have no immediate analogs in the rational theory and the perfect representations are simpler to deal with in the q, t–case. Thus it really resembles the relation between Lie groups and Lie algebras.

2.12.5

Diagonal coinvariants

It was conjectured by Haiman [Ha1] that the space of diagonal coinvariants for a root system R of rank n has a “natural” quotient of dimension (1 + h)n

280

CHAPTER 2. ONE-DIMENSIONAL DAHA

for the Coxeter number h. This space is the quotient C[x, y]/(C[x, y]C[x, y]W o ) for the algebra of polynomials C[x, y] with the diagonal action of the Weyl W of the W –invariant group on x ∈ Cn 3 y and the ideal C[x, y]W o ⊂ C[x, y] polynomials without the constant term. In [Go], such a quotient was constructed. It coincides with the whole space of the diagonal coinvariants in the An –case due to Haiman, but this does not hold for other root systems. Let ksht = −(1+1/h) = klng , h be the Coxeter number 1+(ρ, ϑ). The polynomial representation Q[x] of HH00 has a unique nonzero irreducible quotientmodule. It is of dimension (1 + h)n , which was checked in [BEG] and [Go], and also follows from [C28] via æ. See Chapter 3. The application of this representation to the coinvariants of the ring of commutative polynomials Q[x, y] with the diagonal action of W is as follows. The polynomial representation Q[x] is naturally a quotient of the linear space Q[x, y] considered as an induced HH00 –module from the one-dimensional W –module w 7→ 1. So is V 00 . The subalgebra (HH00 )W of the W –invariant def Q elements from HH00 preserves the image of Qδ in V 00 for δ == α>0 xα . Let Io ⊂ (H00 )W be the ideal of the elements vanishing at the image of δ in V 00 . Gordon proves that V 00 coincides with the quotient V˜ 00 of HH00 (δ) by the HH00 –submodule HH00 Io (δ). It is sufficient to check that V˜ 00 is irreducible. The graded space gr(V 00 ) of V 00 with respect to the total x, y–degree of the polynomials is isomorphic as a linear space to the quotient of Q[x, y]δ by the graded image of HH00 Io (δ). The latter contains Q[x, y]W o δ for the ideal W ⊂ Q[x, y] of the W –invariant polynomials without the constant Q[x, y]W o 00 term. Therefore V becomes a certain quotient of Q[x, y]/(Q[x, y]Q[x, y]W o ). See [Go],[Ha1] about the connection with the Haiman theorem in the An –case and related questions for other root systems. The irreducibility of the V˜ 00 above is the key fact. The proof from [Go] requires considering a KZ-type local systems and the technique from [GGOR]. We will demonstrate in the next chapter that the irreducibility can be readily proved in the q, t–case by using the passage to the roots of unity and therefore gives an entirely algebraic and simple proof of Gordon’s theorem via the æ– isomorphism.

Chapter 3 General theory 3.0

Progenitors

Following [C23], [C28], and the previous chapter, we will study DAHA, its representations, and the q–Fourier transform. The basic representations of this chapter are those in Laurent polynomials, Laurent polynomials multiplied by the Gaussian, and in various spaces of delta functions. We also consider induced, semisimple, spherical, and finite dimensional representations in this chapter. The q–Fourier transform generalizes the Harish-Chandra spherical transform, its p–adic counterpart due to Macdonald–Matsumoto, and the Fourier transforms for the Heisenberg and Weyl algebras. The applications of this transform to Verlinde algebras, Gauss–Selberg integrals and sums, Dedekind–Macdonald-type η–type identities, and the diagonal coinvariants will be discussed.

3.0.1

Fourier theory

There are two major directions in the q–Fourier analysis, compact and noncompact, generalizing the corresponding parts of the harmonic analysis on symmetric spaces. The compact direction is based on the imaginary integration and its variant, the constant term functional. The representation of DAHA in Laurent polynonials, called the polynomial representation, and in Laurent polynomials multiplied by the Gaussian, a counetrpart of the Schwartz space, are the key here. The theory of Fourier transforms of these spaces includes the Macdonald conjectures (now the theorems), the Mehta– Macdonald formula, and has other applications of combinatorial nature. Compact case. Concerning the Macdonald conjecture and the Mehta– Macdonald formula, we refer to [C16, C19, C20, C21, M8]. They will not be proven in the chapter, as well as their Jackson counterparts. They were justified using the shift operator. This proof does not add too much to the theory of DAHA, however, the formulas themselves are definitely useful; they help to 281

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CHAPTER 3. GENERAL THEORY

control the reducibility of the polynomial representation and the singularities of the Fourier transforms. As a matter of fact, the evaluation and duality formulas follow directly from the existence of the duality involution, an outer automorphism of DAHA. The latter readily gives the duality formula; the evaluation formula requires the Pieri rules, based on the duality involution too. It is very much similar to the case of A1 considered in Chapters 1 and 2 in detail. The norm formula can be readily deduced from the evaluation formula using the action of the intertwining operators on the nonsymmetric Macdonald polynomials; we discussed this approach in the chapter. By the way, it gives the final answer to an old question in the theory of Macdonald polynomials. It was always surprising that the formulas for the evalutaions and norms are given in terms of the Macdonald measure from the definition of the Macdonald polynomials. Now we understand that this reflects the self duality of the Fourier transform in the space of nonsymmetric polynomials. Thus, we arrive at a new proof of the Macdonald conjectures, reasonably simple and entirely conceptual, that does not require the shift operator and goes uniformly for arbitrary k. The k is a DAHA counterpart of the root multiplicity in the theory of symmetric spaces. The shift operator technique is induction with respect to integral positive k followed by the analytic continuation to arbitray k. Noncompact case. The noncompact direction is, generally, the theory of integration over Rn . This integration is not considered in this chapter; the theory is not finished at the moment. Instead, we deal with its discrete variant, the Jackson integration. The Jackson–Fourier transform is the inverse of the “compact” one mentioned above. The Jackson counterparts of the Macdonald norm formula and Mehta–Macdonald integral are needed for the Plancherel formulas. They still remain unknown for the real integration for general root systems (the A1 –case is considered in Section 2.3.5). Generally, there are eight different variants of the DAHA harmonic analysis (without taking into consideration the degenerations of DAHA and different functional spaces): We have one compact and two noncompact theories (the real and Jackson integrations). Then, the compact and noncompact directions exist in two different variants: (a) standard, as 0 < q < 1 or q > 1, and (b) unimodular for |q| = 1. This gives six analytic theories. The seventh theory is at the roots of unity, where the compact and noncompact directions coincide and all irreducible representations are finite dimensional. Concening the eighth theory, there is a challenging parallelism between our cyclotomic Gaussian sums and the so-called modular Gauss–Selberg sums

3.0. PROGENITORS

283

(see, for instance, [Ev]), although almost nothing has been done so far to establish an exact relation. In the modular case, Selberg-type kernels are calculated in finite fields and are embedded “into” the roots of unity right before the summation. Our sums are defined entirely in cyclotomic fields. Presumably they generalize the modular sums. If this is true, DAHAs could be used for the p–adic integration, the eighth theory, directly realted to the Gaussian sums. Connecting the p–adic Gamma with the q–Gamma function seems an important step in this direction. The analytic aspects are discussed here to put the q–Fourier transform into perspective, however, this chapter is mainly devoted to the algebraic aspects. The basic functional spaces considered in this chapter of of algebraic nature. We note that, generally, harmonic analysis is about the decomposition of the L2 –space and other standard functional spaces in terms of irreducible representations. It is somewhat different in DAHA theory; an irreducible representation of DAHA is, in a sense, already a “theory.” The standard functional spaces become irreducible DAHA–modules, unless q, k, or the corresponding weights are special. Calculating Fourier transforms of arbitrary irreducible representations seems one of the main objectives of the DAHA– Fourier analysis. The decomposition of the whole DAHA, treated as a regular representation, has not been discussed so far. Similar to the theory of Heisenberg and Weyl algebras, the Fourier transform is associated with a certain outer automorphism of the double Hecke algebra. This fix it uniquely up to proportionality in irreducible representations. Thus, the problem of harmonic analysis is to calculate it explicitly, i.e., to make the Fourier transform an inner homomorphism or automorphism, in a proper analytic setting. Representation theory. The semisimple and unitary representations are of major importance. Their description will be reduced below to a certain combinatorial problem, which can be managed in several cases, including the case of GLn (apart from the roots of unity). The classification depends on the structural constants q, t = q k of the double Hecke algebras. For generic q, k, the DAHA modules induced up from the irreducible representations of the affine Hecke algebra are irreducible and we can use the classification from [KL1] (see [C23]). The special, singular, k lead to a rich new theory. It is important itself and is a natural step toward the theory at roots of unity (as q N = 1). This of course does not mean that the “generic case” is not interesting. Even the simplest examples of induced representations give a lot, for instance, the polynomial representation with generic q, k. However, the theory for singular k, when the classification of the irreducible representations is beyond [KL1], seems the most challenging now. One certainly needs a rich “decom-

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position theory” with respect to the affine Hecke subalgebra. This exactly occurs at singular k. In the case of DAHAs of type GLn for singular k and generic q, the semisimple irreducible representations can be described in terms of periodic skew Young diagrams using the technique of intertwiners. It is similar to the classification due to Bernstein-Zelevinsky in the affine GLn –case and papers [C5, C4, Na1]. The non-semisimple case will be touched upon only a little. The technique of this chapter is mainly “semisimple.” Recently Vasserot extended the “geometric approach” from [KL1] (see also [CG]) to the double affine case. His results are for arbitrary root systems. He gives a general geometric classification of irreducible representations including an important interpretation of the case of singular k. Unfortunately, the corresponding algebraic varieties are complicated, and the geometric (K– theoretic, to be more exact) interpretation of the Fourier transform remains unknown. Arbitrary reduced root systems are considered in this chapter. The papers [No2, Di, Sa, St], the book [M8], and [Ra], are devoted to the so-called C ∨ C or Koornwinder case. The corresponding DAHA was defined and studied first by Noumi and Sahi. The prior results are due to Koornwinder. Let me also mention [C12], where the C ∨ C and the corresponding quantum KZ equation was introduced in terms of the pictures of lines in the plane with reflections in two lines. The root system C ∨ C unifies (and generalizes) the classical root systems and has all the important features of the case of reduced root systems considered in this chapter, including the self-duality and the Fourier-invariance of the Gaussians. It has a record number of defining parameters (five) and important applications. One-dimensional case. Modern methods are good enough to transfer smoothly the one-dimensional theory from Chapter 2 to arbitrary root systems. Note that DAHAs give quite a few new results and constructions in the rank one case, including new identities and new proofs of the classical formulas. The rank one theory is very rich, expecially in the C ∨ C case or if the roots of unity are considered. The most interesting new one-dimensional development is the theory of q–Fourier transform. It seems that this transform was not studied before [C19], although its kernel, the basic hypergeometric function, is a classical object. As far as I know, the key point of the new theory, the Fourier-invariance of the Gaussian, remained undiscovered in the works (classical and new) devoted to the basic hypergeometric function, although this fact is not far from some known identities. The symmetry of the basic hypergeometric function responsible for the self-duality of the difference transform was known in the one-dimensional case. The multidimensional Fourier theory is entirely based on DAHAs.

3.0. PROGENITORS

285

The DAHA theory is essentially algebraic now. We do not discuss the analytic aspects in this chapter. There are some results in this direction in [C21] (the construction of the general spherical functions), [C26] (analytic continuations in terms of k with applications to q–counterparts of Riemann’s zeta function), and [KS1, KS2]. The latter two papers are devoted to explicit analytic properties of the one-dimensional Fourier transforms in terms of the basic hypergeometric function, directly connected with our ones. Nonsymmetric polynomials. The key new development in the theory of orthogonal polynomials, related combinatorics, and related harmonic analysis is the definition of the nonsymmetric Opdam–Macdonald polynomials. The main references are [O3, M5, C20]. Opdam mentions in [O3] that this definition (in the differential setup) was given in Heckman’s unpublished lectures. These polynomials are expected to be, generally speaking, beyond quantum groups and Kac–Moody algebras because of the following metamathematical reason. Major special functions in the Lie and Kac–Moody theory, including the classical and Kac–Moody characters, spherical functions, and conformal blocks, are W –invariant. One can expect these functions to be W –symmetrizations of fundamental nonsymmetric functions, but it does not happen in the traditional theory, with a reservation about the Demazure character formula. One needs double Hecke algebras; the Macdonald symmetric polynomials are symmetrizations of their nonsymmetric counterparts, that are of fundamental importance in the DAHA theory. However, our considerations are not quite new in representation theory. We directly generalize the harmonic analysis on Heisenberg and Weyl algebras and borrow a lot from the affine Hecke algebra technique. The limits of the nonsymmetric Macdonald polynomials as q → ∞ are well known. They are the spherical functions due to Matsumoto. See [Ma] and [O4] about recent developments. The Eisenstein integrals and series in the theory of affine Hecke algebras are also related to the nonsymmetric polynomials. Number theory does require nonsymmetric functions. For instance, we need all one-dimensional theta functions, not just even ones. However, say, applications of the Kac character formulas to the Dedekind-type identities generally lead to symmetric (even) functions. The nonsymmetric polynomials and their various counterparts can make the connection between representation theory and number theory stronger. The intertwining operators of the double affine Hecke algebras are the basis of the new technique, at least as far as the semisimple representations are concerned. They were applied in [Kn], [KnS] and further papers to the combinatorial conjectures about the coefficients of Macdonald’s polynomials in the GL–case. In [O3] the nonaffine intertwiners were used to define a nonsymmetric counterpart of the Harish-Chandra theory. The affine intertwines made it possible to calculate the Harish-Chandra transform of the coordinates

286

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treated as multiplication operators [C24], which was an old open problem in the Harish-Chandra theory.

3.0.2

Perfect representations

A convincing demonstration of the new technique is the theory of perfect representations (nonsymmetric Verlinde algebras) with applications to the Verlinde algebras, Gaussian sums and so-called diagonal coinvariants. Double Hecke algebras generalize Heisenberg and Weyl algebras, so it is not surprising that they are helpful to study Fourier transforms. Perfect representations representations have the simplest possible Fourier theory. Generally, quasi-perfect representations are assumed to posses a non-degenerate pairing which induces the self-duality involution. If the pairing is perfect and there is a projective action of P SL(2, Z), then the modules are called perfect. In this chapter we consider only semisimple spherical perfect modules. In more detail, perfect modules of this chapter are (a) unitary (or pseudo-unitary), (b) Fourier-invariant, (c) spherical. The latter property makes them commutative algebras with a unit. We also suppose that they are (d) semisimple, and (e) finite dimensional. We note that the inner product in (a) may be non-positive. For the rational DAHA, it is always indefinite and the representations are not of semisimple type. There are examples when (general) perfect representations are not semisimple in the q, t–case as well. Nevertheless, the semisimplicity is an important part of the theory of perfect representations and Verlinde algebras. The perfect irreducible representations, i.e., those satisfying all five conditions (a), (b), (c), (d), (e), exist either when q is a root of unity or when q is generic but k is singular. The corresponding DAHA has, essentially, only one perfect representation up to isomorphisms and the choice of the character of the affine Hecke algebra; the latter is needed in the definition of spherical representations. This generalizes the well-known uniqueness of the irreducible representation of the Weyl algebra at the roots of unity. As q is a root of unity, perfect representations lead to a new class of cyclotomic Gauss–Selberg sums. These sums are defined as eigenvalues of the Gaussian with respect to the q–Fourier transform. Interesting non-cyclotomic sums appear for generic q and singular k. They are directly related to Dedekind–Macdonald η–type identities. In the rational setting, they describe the degenerate multidimensional Bessel functions. The subspaces of symmetric (W –invariant) elements of perfect representations are commutative algebras generalizing the Verlinde algebras. The usual Verlinde algebra appera as k = 1; it is formed by integrable representations

3.0. PROGENITORS

287

of the Kac–Moody algebras of fixed level with the fusion procedure as multiplication. It can be identified with the restricted category of representations of quantum groups at the roots of unity due to Kazhdan–Lusztig [KL2] and Finkelberg. The third approach to the Verlinde algebras is via factors and subfactors. The perfect representations “double-generalize” the Verlinde algebras and, more importantly, simplify the theory. First, the Macdonald symmetric polynomials replace the classical compact characters [Ki1, C19]. The parameter k, which is 1 in the Verlinde case [Ver], becomes an arbitrary positive integer (satisfying certain inequalities) or even fractional, namely, from (1/h)Z for the Coxeter number h subject to some inequalities. Second, the nonsymmetric polynomials replace the symmetric ones [C20]. It is much more comfortable to deal with the Fourier transform when all functions are available, not only the symmetric ones. It is exactly what perfect representations provide. See Section 3.10 of this chapter. The final construction is distant from the original Verlinde algebra (and from the Lie and Kac–Moody theories). The structural constants of multiplication are not positive integers any longer. However, all other important features, including the projective action of P GL(2, Z) and the positivity of the inner product, are saved. Perfect representations for generic q and k = −1 − 1/h solve the Haiman conjecture about the canonical quotients of the so-called space of the diagonal coinvariants for arbitrary root systems; see [Ha1, Go, C29]. This space is of fundamental importance for the combinatorics of two sets of variables started in [GH]. Macdonald’s η–identities. The approach via double Hecke algebras provides interesting q–deformations of the perfect representations at roots of unity and the corresponding cyclotomic Gauss–Selberg sums. We replace the roots of unity q by any q such that |q| = 1. The Gaussian sums remain finite, of the same length. We also deform other structures, for instance, the Verlinde algebra formed by W –invariants, the hermitian form, and the projective P SL2 (Z)–action. The deformation goes as follows. We start with a perfect representation of the little double affine Hecke algebras where q is generic and k is singular, special rational from (1/h)Z (see above). They remain irreducible as q N = 1 under certain conditions. Provided that (N, h) = 1, one can pick k such that k = 1 mod N if we want to deform the standard (little) Verlinde algebra. It gives the desired deformation. Considering little DAHA, for the root lattice instead of the weight lattice, is necessary here. The representation theory of double Hecke algebras at singular k appeared to be connected with the Macdonald η–identities [M1], that are product formulas for the Gauss-type sums over the root lattices shifted by (1/h)ρ. Here

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CHAPTER 3. GENERAL THEORY

ρ is the half-sum of all positive roots and h is the Coxeter number. The Dedekind–Macdonald identities are directly related to the one-dimensional representation of the double Hecke algebra. Replacing (1/h) by (m/h)ρ for integral m > 0 relatively prime to h, we come to new η–type identities associated with general finite dimensional perfect representations. Here the Fourier-invariance of the perfect representations is important, and also that they are unitary with respect to a hermitian form (not always definite). By unitary, we mean that the generators X, T, Y of the double Hecke algebra are unitary matrices in a given representation, provided that the conjugation sends q to q −1 . It is a the standard complex conjugation if |q| = 1. The form is positive definite as |q| = 1 and the angle of q is sufficiently small. It generalizes the Verlinde pairing, which plays an important role in conformal field theory. The Verlinde inner product is actually a straightforward variant of the classical paring which makes the characters of compact Lie groups pairwise orthogonal. It uses the same square of the discriminant and summation over a certain finite set (“alcove”) versus the integration in the classical compact theory. The inner products in q, t–theory are not that simple; they are given in terms of the “truncated” theta–functions.

3.0.3

Affine Hecke algebras

There is another important inner product that does not involve the conjugation of q and serves the case of real q, i.e., the real harmonic analysis. The operators Y are not normal with respect to this inner product, which diminishes the role of the Y –semisimplicity in the theory. Actually, we face the same problem as in the theory of representations of affine Hecke algebras. The Y –operators due to Bernstein and Zelevinsky are, generally, not normal with respect to the natural pairing. It is one of the reasons why advanced methods like K–theory [KL1] are needed. It seems better in the double affine case. Many important representations are either Y –semisimple or X–semisimple, although there are some that are neither. Both variants of the Fourier transform are considered in this chapter, with and without the conjugation q 7→ q −1 , t 7→ t−1 . The Fourier theory without the conjugation is actually close to that from [O3] and to the p–adic theory due to Macdonald and Matsumoto. In a sense, we replace w0 in Opdam’s paper by Tw0 . In papers [O4, O5], Opdam developed the Matsumoto theory of “nonsymmetric” spherical functions towards the theory of nonsymmetric polynomials. The operator Tw0 appears there in the inverse transform in a way similar to that here. Compare [CO] and [C28] with [O4](Proposition 1.12). The p–adic theory and Opdam’s construction are the limit of our theory when q → ∞,

3.0. PROGENITORS

289

with t being p or its proper power (in the rigorous p–adic setup). See Section 2.11 of the previous chapter. It is important to mention here that the double affine Hecke algebra is a quotient of the group algebra of a central extension of the orbifold fundamental group of the elliptic configuration space; see [C13]. The central extension is by certain fractional powers of the parameter q treated as a central element. The Fourier transform finds a topological interpretation as the transposition of the periods of the elliptic curve. This clarifies the true potential of DAHAs in representation theory: they provide an improvement of the classical harmonic analysis. The self-duality of the Fourier transform does not survive in the p–adic limit, as well as in the Harish-Chandra theory. It is not surprising because the topological interpretation of the affine Hecke algebra is in terms of C∗ , that has only one “period”. In the limit q → 1, t = q k , we come to the construction from [O3] and, then, via the symmetrization, to the k–extension of the Harish-Chandra theory. The nonsymmetric generalization of the Harish-Chandra transform in the space of compactly supported C ∞ –functions is analyzed in [O3] in detail. The representation of the degenerate double affine Hecke algebra in terms of the Dunkl operators plays the key role. The representation theory of this algebra is actually similar to that in the q, t–case, though the analytic properties of the corresponding Fourier transforms are very different. At the moment, we don’t understand how it happens and why the q, t–case is closer analytically to the p–adic theory than to the Harish-Chandra–Opdam theory.

3.0.4

Gauss–Selberg integrals and sums

Selberg’s integral was one of the major starting points for the theory of symmetric Macdonald polynomials. These polynomials appeared for the first time in Kadell’s (unpublished) work devoted to generalizations of this integral. The Mehta–Macdonald integrals is actually a particular case of Selberg’s integral for the classical root systems. Q Mehta suggested a formula for the integral of 1≤i<j≤n (xi − xj )2k with respect to the Gaussian measure (see [Meh]). The formula was readily deduced from Selberg’s integral by Bombieri. Macdonald extended Mehta’s integral in [M2] to arbitrary root systems and verified his conjectures for the classical systems using Selberg’s integral too. Later Opdam found a uniform proof for all root systems using the shift operators in [O1]. Note that neither alternative proof is known for the special root systems. In [C21], the q was added to this theory. The (classical) Mehta–Macdonald integral is the normalization constant for the generalized Hankel transform introduced by Dunkl [Du1]. The generalized Bessel functions [O2] multiplied by the Gaussian are eigenfunctions of

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this transform. The eigenvalues are given in terms of this constant [Du1, Je]. This theory can be extended to the q–case. The Hankel transform is a “rational” degeneration of the Harish-Chandra spherical transform. The symmetric space G/K is replaced by its tangent space Te (G/K) with the adjoint action of G; see [He3]. Here G is a semisimple Lie group and K is its maximal compact subgroup. Only particular k, given in terms of the root multiplicities, may appear in the Harish-Chandra theory. For instance, k = 1 is the so-called group case. The number k is arbitrary real in the quantum many-body problem and in [HO1]. Additional parameters k appear for the non-simply-laced root systems, up to five independent k in the so-called C ∨ C–case. The theory of DAHA unifies the Selberg–Mehta–Macdonald integrals with the classical Gaussian sums at the roots of unity. The latter can be obtained as limits of the q–analogs of the former. Roots of unity always played a fundamental role in mathematics. In the modern theory of Hecke algebras and quantum groups, the roots of unity are the focus of many projects. It does not mean that the case of generic q is not interesting, but quite a few q–developments aim at roots of unity. The reduction to the Gaussian sums consists of four steps: (a) the q–deformations of the Mehta–Macdonald integrals, (b) the transfer from the q–integrals to the Jackson sums, (c) calculating the limits of the latter at roots of unity, (d) establishing connections with the classical Gaussian sums. Let us follow these steps in the one-dimensional case. The aim is to go from the celebrated Euler integral Z ∞ 2 e−x x2k dx = Γ(k + 1/2), −1/2, (3.0.1) −∞

to the almost equally famous Gauss formula 2N −1 X

e

πm2 i 2N

√ = (1 + i) N , N ∈ N.

(3.0.2)

m=0

Connecting these formulas is of obvious importance, but the real objective will be to incorporate k into the last formula. Obtaining (3.0.2) from (3.0.1) is a natural problem. However, it was not solved and, it seems, never considered in the classical works on Gaussian sums.

3.0.5

From generic q to roots of unity

Obviously a trigonometric counterpart of (3.0.1) is needed. A natural candidate is the measure-function in the Harish-Chandra theory, where sinh(x)2k

3.0. PROGENITORS

291

substitutes for x2k . However, such a choice creates problems. Formula (3.0.1) has no straight sinh–counterpart (at least for generic k). We need the q– theory. The compact case (a). It was demonstrated in [C19, C20, C21] that the self-duality and other important features of the classical Fourier transform hold for the kernel def

δk (x; q) ==

∞ Y

(1 − q j+2x )(1 − q j−2x ) , 0 < q < 1, k ∈ C. (1 − q j+k+2x )(1 − q j+k−2x ) j=0

(3.0.3)

Actually, the self-duality of the corresponding transform can be expected a priori because the Macdonald truncated theta function δ is a unification of sinh(x)2k and the Harish-Chandra measure (A1 ) serving the inverse spherical transform. Setting q = exp(−1/a), a > 0, Z ∞i ∞ √ Y 1 − q j+k −x2 (−i) q δk dx = 2 aπ , 0. (3.0.4) j+2k 1 − q −∞i j=0 The limit of (3.0.4) multiplied by (a/4)k−1/2 as a → ∞ is (3.0.1) in the imaginary variant. It is the key step. The others are as follows. Jackson sums (b). Special q–functions have many interesting properties that have no classical counterparts. For instance, we can replace (some) integralsR by sums, the Jackson integrals. Let ] be the integration for the path that begins at z = ²i + ∞, moves to the left till ²i, then down through the origin to −²i, and finally returns down the positive real axis to −²i + ∞ (for small ² > 0). Then for |=k| < 2², 0, Z ∞ 1 aπ Y (1 − q j+k )(1 − q j−k ) 2 x2 × hq x i] , q δk dx = − j+2k j+1 2i ] 2 j=0 (1 − q )(1 − q ) x2

def

hq i] ==

∞ X

q

(k−j)2 4

j=0

= q

k2 4

∞ ∞ Y (1 − q j+k ) X j=1 ∞ Y

(1 − q j ) (1 − q

j=1 2

j/2

q

j 1 − q j+k Y 1 − q l+2k−1 1 − q k l=1 1 − q l

(k−j)2 4

=

(3.0.5)

j=−∞

)(1 − q j+k )(1 + q j/2−1/4+k/2 )(1 + q j/2−1/4−k/2 ) . (1 − q j )

(3.0.6)

The sum for hq x i] is the Jackson integral for a special choice of the starting point. Its convergence is for all k.

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Gauss–Selberg sums (c). When q = exp(2πi/N ) and k is a positive integer ≤ N/2 we come to the simplest Gauss–Selberg cyclotomic sum: NX −2k

q

(k−j)2 4

j=0

j k 2N −1 X 1 − q j+k Y 1 − q l+2k−1 Y 2 j −1 = (1 − q ) q m /4 . k l 1 − q l=1 1 − q m=0 j=1

(3.0.7)

Its modular counterpart is formula (1,2b) from [Ev]. Formula (3.0.7) can be deduced directly from (3.0.5) following the classical limiting procedure from [Ch]. Formulas (3.0.4)–(3.0.6) can be verified by elementary methods too (using the Ψ–summation). They are proved in [C21] for arbitrary root systems, as well as the generalizations of these formulas involving the Macdonald polynomials. The double Hecke algebra technique is needed even in the one-dimensional case when two Rogers’ polynomials are added to the integrand. The nonsymmetric formulas and the multidimensional degenerations are based on DAHA; see [C21]. New proof of the Gauss formula (d). Substituting k = [N/2] in the last formula (not k = 0, as one may expect, which makes (3.0.7) a trivial P −1 m2 /4 , which can be readily identity) we obtain the product formula for 2N m=0 q 2πi/N calculated as q = e and quickly results in (3.0.2). Let us demonstrate it in the case of N = 2k (odd N = 2k + 1 are quite similar). We obtain 2N −1 X

q

m2 4

= q

k2 4

Π for Π =

m=0

k Y

(1 − q j ).

j=1

¯ = (X N − 1)(X + 1)(X − 1)−1 (1) = 2N. Second, arg(1 − eiφ ) = First, ΠΠ φ/2 − π/2 when 0 < φ < 2π, and therefore arg Π =

π k(k + 1) πk π(1 − k) − = . N 2 2 4

Here we can switch to arbitrary primitive q = exp(2πil/N ) as (l, N ) = 1 if we can control the set of arg(q j ) for 1 ≤ j ≤ k. This can be used to justify the quadratic reciprocity (see [CO]). We do not discuss this direction here. Finally, k2

arg(q 4 Π) = π(1 − k)4 +

3.0.6

√ k2 πk π = , and q 4 Π = N (1 + i). 4 4

Structure of the chapter

The first three sections mainly contain the basic theory, including the scalar products with and without the conjugation q 7→ q −1 , and the case of roots of unity.

3.1. AFFINE WEYL GROUPS

293

Sections 3.4–3.5 are about the general theory of the Fourier transforms in the compact and noncompact cases. They act on polynomials, polynomials multiplied by the Gaussian, and in spaces of delta functions (called the functional representations). The application to the η–identities concludes Section 3.5. In Section 3.6, we find necessary and sufficient conditions for a representation with the cyclic vector to be semisimple and pseudo-unitary. The consideration of the case of GLn is the subject of the next Section 3.7. The main theorem of Section 3.6 is the key tool for the next Sections 3.8–3.10, which are devoted to spherical, cospherical, self-dual, and perfect representations. The application to the diagonal coinvariants and the universal Dunkl operators is discussed in the last Section 3.11. The exposition is essentially self-contained. The previous chapters and the one-dimensional papers [CM, C27, CO] combined with the second part of [C25] could be a reasonable introduction. The papers [C23] and [C21] contain additional results on the nonsymmetric polynomials and q–spherical functions. We also recommend the book [M8]. Acknowledgments. The author thanks V. Kac and I. Macdonald for useful comments on the material of this section He also thanks P. Etingof and E. Vasserot for discussing the case of GLn . I acknowledge my indebtedness to E. Opdam for reading the chapter and making important suggestions that especially influenced Section 3.8. I am also thankful to G. Heckman for discussing Section 3.9. A significant part of the chapter was prepared during my visit to Harvard University (spring 2001). I am grateful to D. Kazhdan and P. Etingof for inviting me.

3.1

Affine Weyl groups

Let R = {α} ⊂ Rn be a root system of type A, B, ..., F, G with respect to a euclidean form (z, z 0 ) on Rn 3 z, z 0 , W the Weyl group generated by the reflections sα , R+ the set of positive roots (also, R− = −R+ ) corresponding to the (fixed) simple roots α1 , ..., αn and Γ be the Dynkin diagram with {αi , 1 ≤ i ≤ n} as the vertices. We will also use the dual roots (coroots) and the dual root system: R∨ = {α∨ = 2α/(α, α)}. The root lattice and the weight lattice are Q = ⊕ni=1 Zαi ⊂ P = ⊕ni=1 Zωi , where {ωi } are fundamental weights: (ωi , αj∨ ) = δij for the simple coroots αi∨ .

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CHAPTER 3. GENERAL THEORY

Replacing Z by Z± = {m ∈ Z, ±m ≥ 0} we obtain Q± , P± . Note that Q ∩ P+ ⊂ Q+ , i.e., the coefficients of the fundamental weights in terms of the simple roots are non-negative. Moreover, each ωj has all nonzero coefficients (sometimes rational) when expressed in terms of {αi }; see [Bo]. The form will be normalized by the condition (α, α) = 2 for the short roots. This normalization coincides with that from the tables in [Bo] for def A, C, D, E, G. Thus, να == (α, α)/2 can be either 1 or in {1, 2} or in {1, 3}. This special normalization leads to the inclusions Q ⊂ Q∨ and P ⊂ P ∨ , where P ∨ is generated by the fundamental coweights ωi∨ . We will use the notation νlng for the long roots (νsht = 1). Setting νi = ναi , νR = {να , α ∈ R}, one has X X def ρν == (1/2) α = ωi , where α ∈ R+ , ν ∈ νR . (3.1.1) να =ν

νi =ν

Note that (ρν , αi∨ ) = 1 as νi = ν. We will call ρν the partial ρ.

3.1.1

Affine roots

Let ϑ ∈ R∨ be the maximal positive coroot. All simple coroots appear in its decomposition. Also note that 2 ≥ (ϑ, α∨ ) ≥ 0 for α > 0, (ϑ, α∨ ) = 2 only for α = ϑ, and sϑ (α) < 0 if (ϑ, α) > 0. See [Bo] to check that ϑ considered as a root (it belongs to R because of the choice of normalization) is maximal among all short positive roots of R. For the sake of completeness, let us prove another defining property of ϑ (see Proposition 3.1.7 below for a uniform proof). Proposition 3.1.1. The least nonzero element in Q+ + = Q+ ∩ P+ = Q ∩ P+ with respect to Q+ is ϑ, i.e., b − ϑ ∈ Q+ for all b ∈ Q+ +. Proof. If a ∈ Q+ + , then all coefficients of the decomposition of a in terms of αi are nonzero. Indeed, if αi does not appear in this decomposition, then its neighbors (in the Dynkin diagram) P have negative scalar products with a. This readily gives the claim when ϑ = ni=1 αi , i.e., for the systems A, B. In these cases, ϑ is ω1 + ωn for A and ω1 for B. Otherwise, it is ω2 (C, D, E6 ), ω1 (E7 ), ω8 (E8 ), ω4 (F4 ), and ω1 (G2 ), respectively (in the notation from [Bo]). Let us denote the corresponding subscripts of ω (1, 2, 1, 8, 4, 1) by o˜. They are the indices of the simple roots neighboring the root α0∨ in the affine (completed) Dynkin diagram for R∨ . The proposition holds for E8 , F4 , and G2 , because in these cases P = Q and ωj − ωo˜ ∈ Q+ (use the tables from [Bo]). So we need to examine C, D, and E6,7 . As to C, one verifies that ωj −ω2 ∈ (1/2)Q+ for j ≥ 2 and Q+ + is generated by ω2i and ω2i+1 +ω2j+1 for 1 ≤ i ≤ j ≤ [n/2]. Thus the relation 2ω1 −ω2 ∈ Q+

3.1. AFFINE WEYL GROUPS

295

proves the claim. In the D–case, a − ω2 for a ∈ Q+ + can be apart from (1/4)Q+ only if it is a linear combination of ω1 , ωn−1 , and ωn . However, such a combination “contains” either ωn−1 + ωn or 2ω1 , which is sufficient to make it greater than ω2 (with respect to Q+ ). For R = E6 , ωj − ω2 ∈ (1/3)Q+ when j ≥ 2 = o˜ except for j = 6. The intersection of Q and Z+ ω1 + Z+ ω6 is generated by 3ω1 , 3ω6 , ω1 + ω6 . Either weight is greater than ω2 . In the case of E7 , o˜ = 1, ωj − ω1 ∈ (1/2)Q+ if j ≤ 6, and 2ω7 is greater than ω1 . ❑ n n+1 The vectors α ˜ = [α, να j] ∈ R ×R ⊂ R for α ∈ R, j ∈ Z form the affine def ˜ ⊃ R (z ∈ Rn are identified with [z, 0]). We add α0 = = [−ϑ, 1] root system R ˜ to the simple roots for the maximal short root ϑ. The corresponding set R of positive roots coincides with R+ ∪ {[α, να j], α ∈ R, j > 0}. We complete the Dynkin diagram Γ of R by α0 (by −ϑ to be more exact). ˜ One can obtain it from the completed Dynkin diagram for The notation is Γ. R∨ from [Bo] by reversing the arrows. The number of laces between αi and ˜ is denoted by mij . αj in Γ ˜ will The set of indices of the images of α0 by all the automorphisms of Γ 0 be denoted by O (O = {0} for E8 , F4 , G2 ). Let O = r ∈ O, r 6= 0. The elements ωr for r ∈ O0 are the so-called minuscule weights: (ωr , α∨ ) ≤ 1 for α ∈ R+ . ˜ b ∈ B, let Given α ˜ = [α, να j] ∈ R, sα˜ (˜ z ) = z˜ − (z, α∨ )˜ α, b0 (˜ z ) = [z, ζ − (z, b)]

(3.1.2)

for z˜ = [z, ζ] ∈ Rn+1 . f is generated by all sα˜ (we write W f = hsα˜ , α The affine Weyl group W ˜∈ ˜ R+ i). One can take the simple reflections si = sαi (0 ≤ i ≤ n) as its generators and introduce the corresponding notion of the length. This group is the semidirect product W nQ0 of its subgroups W = hsα , α ∈ R+ i and Q0 = {a0 , a ∈ Q}, where α0 = sα s[α,να ] = s[−α,να ] sα for α ∈ R.

(3.1.3)

c generated by W and P 0 (instead of Q0 ) The extended Weyl group W is isomorphic to W nP 0 : (wb0 )([z, ζ]) = [w(z), ζ − (z, b)] for w ∈ W, b ∈ B.

(3.1.4)

From now on, b and b0 , P and P 0 will be identified. Given b ∈ P+ , let w0b be the longest element in the subgroup W0b ⊂ W of the elements preserving b. This subgroup is generated by simple reflections. We set c , u i = uω , π i = π ω , ub = w0 w0b ∈ W, πb = b(ub )−1 ∈ W i i

(3.1.5)

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where w0 is the longest element in W, 1 ≤ i ≤ n. def def The elements πr == πωr for r ∈ O0 and π0 == id form a group denoted by Π, which is isomorphic to P/Q by the natural projection {ωr 7→ πr }; these ˜ invariant. As to {ur }, they preserve the set {−ϑ, αi , i > 0}. elements leave Γ The relations πr (α0 ) = αr = (ur )−1 (−ϑ) distinguish the indices r ∈ O0 . Moreover (see, e.g., [C16]): c = ΠnW f , where πr si π −1 = sj if πr (αi ) = αj , 0 ≤ j ≤ n. W r

3.1.2

(3.1.6)

Affine length function

c for πr ∈ Π and w f, the length l(w) Setting w b = πr w e∈W e∈W b is, by definition, the length of the reduced decomposition w e = sil ...si2 si1 in terms of the simple reflections si , 0 ≤ i ≤ n. The number of si in this decomposition such that νi = ν is denoted by lν (w). b The length also can be defined as the cardinality |λ(w)| b of def ˜ ˜ − ) = {˜ ˜ + , w(˜ ˜ − }, w c. λ(w) b == R b−1 (R α∈R b α) ∈ R b∈W +∩w

Respectively, def

λ(w) b = ∪ν λν (w), b λν (w) b == {˜ α ∈ λ(w), b ν(˜ α) = ν}.

(3.1.7)

The coincidence with the previous definition is based on the equivalence of the length equality c (a) lν (wˆ bu) = lν (w) b + lν (ˆ u) for w, b uˆ ∈ W and the cocycle relation (b) λν (wˆ bu) = λν (ˆ u) ∪ uˆ−1 (λν (w)), b

(3.1.8)

which, in its turn, is equivalent to the positivity condition ˜+. (c) uˆ−1 (λν (w)) b ⊂R Formula (3.1.8) obviously includes the positivity condition (c). It also implies that ¡ ¢ u)) ∩ λν (w) b = ∅, λν (ˆ u) ∩ uˆ−1 (λν (w)) b = uˆ−1 uˆ(λν (ˆ thanks to the general formula λν (w b−1 ) = −w(λ b ν (w)). b Thus it results in the equality lν (wˆ bu) = lν (w) b + lν (ˆ u) and we have the implications (a) ⇐ (b) ⇒ (c).

3.1. AFFINE WEYL GROUPS

297

The remaining implications (a) ⇒ (b) ⇐ (c) are based on the following simple general fact: ˜ + for any uˆ, w. λν (wˆ bu) \ {λν (wˆ bu) ∩ λν (ˆ u)} = uˆ−1 (λν (w)) b ∩R b

(3.1.9)

For instance, the length equality (a) readily implies (3.1.8). For the sake of completeness, let us deduce (3.1.8) from the positivity condition (c) above. We follow [C16]. It suffices to check that λν (wˆ bu) ⊃ λν (ˆ u). If there exists a positive α ˜ ∈ ˜ λν (ˆ u) such that (wˆ bu)(˜ α) ∈ R+ , then ˜ − ⇒ −ˆ w(−ˆ b u(˜ α)) ∈ R u(˜ α) ∈ λν (w) b ⇒ −˜ α ∈ uˆ−1 (λν (w)). b We come to a contradiction with the positivity. Hence (a) ⇔ (b) ⇔ (c). Applying (3.1.8) to the reduced decomposition w b = πr sil · · · si2 si1 , λ(w) b ={˜ α1 = αi1 , α ˜ 2 = si1 (αi2 ), α ˜ 3 = si1 si2 (αi3 ), . . . ...,α ˜l = w e−1 sil (αil )}.

(3.1.10)

The cardinality l of the set λ(w) b equals l(w). b This set can be introduced for nonreduced decompositions as well. Let us ˜ w) denote it by λ( b to differentiate it from λ(w). b It always contains λ(w) b and, moreover, can be represented in the form ˜ w) ˜ + (w) ˜ + (w), λ( b = λ(w) b ∪λ b ∪ −λ b ˜ + (w) ˜ w)) ˜ + ∩ λ( where λ b = (R b \ λ(w). b

(3.1.11)

The coincidence with λ(w) b is for reduced decompositions only. Let us prove another standard property of the λ–sets: λν (w) b = {˜ α > 0, lν (ws b α˜ ) < lν (w)}. b

(3.1.12)

It suffices to consider λ(w). b Thanks to (3.1.10), {˜ α > 0, l(ws b α˜ ) < l(w)} b contains λ(w). b Obviously (3.1.12) holds for sα˜ = sαi (0 ≤ i ≤ n). An arbitrary sα˜ has a reduced decomposition in the form: sα˜ = si1 si2 · · · sip sm sip · · · si2 si1 , i• , m ≥ 0. Indeed, αi ∈ λ(˜ α) if and only if (˜ α, αi ) > 0 because {αi } are minimal positive affine roots. Given such αi , the reflection sα˜ is divisible by si on the right (i.e., has a reduced decomposition in the form · · · si ) and l(sβ˜) ≤ l(sα˜ ) for def β˜ == si (˜ α). If l(s ˜) = l(sα˜ ), then s ˜ = si sα˜ si = s−1 is divisible by si on the β

β

β˜

˜ αi ) = −(˜ left or on the right, which contradicts the inequality (β, α, αi ) < 0. Therefore l(sβ˜) < l(sα˜ ) and we can proceed by induction. We also have

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CHAPTER 3. GENERAL THEORY

obtained that the desired decomposition can be started with an arbitrary αi1 = αi ∈ λ(˜ α). Now, we will prove (3.1.12) by induction with respect to the length l(sα˜ ) for α ˜ such that l(ws b α˜ ) < l(w). b As we already noticed, it holds for simple reflections. Let αi ∈ λ(sα˜ ). If αi 6∈ λ(w), b then l((ws b i )(si sα˜ si )) < l(w) b + 1 = l(ws b i ), and the induction statement gives that def β˜ == si (˜ α) ∈ λ(ws b i ), si (˜ α) ∈ si (λ(w)), b and α ˜ ∈ λ(w). b

Otherwise αi ∈ λ(w) b and therefore w(α b i ) < 0. Since αi ∈ λ(sα˜ ) and (˜ α , αi ) > 0, we obtain: (ws b α˜ )(αi ) = w(α b i−2

(˜ α , αi ) α ˜ ) < 0 for w(˜ b α) > 0. (˜ α, α ˜)

The opposite case w(˜ b α) < 0 means that α ˜ ∈ λ(w), b which gives the desired fact. Thus it suffices to consider the case when αi ∈ λ(ws b α˜ ) for the above αi ∈ λ(˜ α), which gives: l((ws b i )sβ˜) = l(ws b α˜ si ) ≤ l(ws b α˜ ) − 1 < l(w) b − 1. By induction, β˜ ∈ λ(ws b i ) because now αi ∈ λ(ws b i ) (the only case left unchecked) and therefore l(ws b i ) = l(w) b − 1. Hence, β˜ ∈ si (λ(w)) b and, finally, α ˜ ∈ λ(w). b

3.1.3

Reduction modulo W

The following proposition is from [C20]. It generalizes the construction of the elements πb for b ∈ P+ . Proposition 3.1.2. Given b ∈ P , there exists a unique decomposition b = πb ub , ub ∈ W satisfying one of the following equivalent conditions: (i) l(πb ) + l(ub ) = l(b) and l(ub ) is the greatest possible, (ii) λ(πb ) ∩ R = ∅. The latter condition implies that l(πb ) + l(w) = l(πb w) for any w ∈ W. def In addition, the relation ub (b) == b− ∈ P− = −P+ , holds, which, in its turn, determines ub uniquely if one of the following equivalent conditions is imposed: (iii) l(ub ) is the smallest possible, (iv) if α ∈ λ(ub ), then (α, b) 6= 0. ❑

3.1. AFFINE WEYL GROUPS

299

Since πb = bu−1 = u−1 b b b− , the set πP = {πb , b ∈ P } can be described in terms of P− : πP = {u−1 b− for b− ∈ P− , u ∈ W such that α ∈ λ(u−1 ) ⇒ (α, b− ) 6= 0}.

(3.1.13)

Indeed, −1 α ∈ λ(u−1 b ) = −ub (λ(ub )) ⇒ (ub (α), b) = (α, ub (b)) = (α, b− ) 6= 0.

Using the longest element w00 in the centralizer W00 of b− , such u constitute the set {u, | l(u−1 w00 ) = l(w00 ) + l(u−1 )}. Their number is |W |/|W00 |. ˜ + , one has: For α ˜ = [α, να j] ∈ R λ(b) = {˜ α, (b, α∨ ) > j ≥ 0 if α ∈ R+ , (b, α∨ ) ≥ j > 0 if α ∈ R− }, λ(πb ) = {˜ α, α ∈ R− , (b− , α∨ ) > j > 0 if u−1 b (α) ∈ R+ , ∨ (b− , α ) ≥ j > 0 if u−1 b (α) ∈ R− }, −1 ∨ λ(πb ) = {˜ α, −(b, α ) > j ≥ 0}, λ(ub ) = {α ∈ R+ , (b, α∨ ) > 0}. P For instance, l(b) = l(b− ) = −2(ρ∨ , b− ) for 2ρ∨ = α>0 α∨ .

(3.1.14) (3.1.15) (3.1.16) (3.1.17)

Let us also calculate the set λ(sβ˜) for β˜ = [−β, lνβ ], where β ∈ R+ , l > 0. One has: (−lβ)0 = sβ sβ˜, λ(sβ ) = sβ (R− ) ∩ R+ , sβ˜(λ(sβ )) = {[−α, l(β, α)], α ∈ λ(sβ )}. Here (β, α) must be greater than zero. Therefore the above decomposition of −lβ ∈ Q is reduced, thanks to the positivity condition (c) above. Note that (β, α) > 0 does not guarantee that α ∈ λ(sβ ). However, for some elements in W it holds, for instance, for α = ϑ. Using (3.1.14), λ(−lβ) = {˜ α = [−α, να j] | l(α∨ , β) > j ≥ 0 as α ∈ R− , l(α∨ , β) ≥ j > 0 as α ∈ R+ }. Finally, ˜ + | l(α∨ , β) > j} ∪ λ(sβ˜) ={[−α, να j] ∈ R {[−α, l(β, α)] | α > 0 < sβ (α), (β, α) > 0}.

(3.1.18)

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CHAPTER 3. GENERAL THEORY

c on z ∈ Rn : We will need later the following affine action of W (wb)((z)) = w(b + z), w ∈ W, b ∈ P, ˜ sα˜ ((z)) = z − ((z, α) + j)α, α ˜ = [α, να j] ∈ R.

(3.1.19)

Also, (bw)((z)) = b + w(z), (bw)((0)) = b for any w ∈ W. The relation to the above action is given in terms of the affine pairing def ([z, l], z 0 + d) == (z, z 0 ) + l: c, (w([z, b l]), w((z b 0 )) + d) = ([z, l], z 0 + d) for w b∈W

(3.1.20)

where we treat d formally. Introducing the affine Weyl chamber C

a

=

n \

Lαi , L[α,να j] = {z ∈ Rn , (z, α) + j > 0},

i=0

we come to another interpretation of the λ–sets: ˜ + , Ca 6⊂ w((L λν (w) b = {˜ α∈R b α˜ )), να = ν}.

(3.1.21)

c preserving Ca with For instance, Π is the group of all elements of W respect to the affine action. Similarly, the elements πb for b ∈ P are exactly those sending the negative −Ca of Ca to the negative −C of the nonaffine def Weyl chamber C == {z ∈ Rn ,(z, αi ) > 0 as i > 0}. More generally, we have the following proposition. Proposition 3.1.3. Given two finite sets of positive affine roots {β˜ = [β, νβ i]} a c and {˜ γ = [γ, νγ j]}, let Lβ,γ be the closure of the union of w((−C b )) over w b∈W such that β˜ 6∈ λ(w) b 3 γ˜ . Then ˜ γ˜ }. Lβ,γ = {z ∈ Rn , (z, β) + i ≤ 0, (z, γ) + j ≥ 0 for all β, The same holds in the nonaffine variant for C in place of Ca .

❑

We will also need the following “affine” variant of Proposition 3.1.2. Given z ∈ Rn , there exists a unique element w ˜ = uz az with az ∈ Q and uz ∈ W satisfying the relations def z− == w((z)) ˜ ∈ −C¯a , (z− , ϑ) = −1 ⇒ u−1 z (ϑ) ∈ R− , and −1 (αi , z− ) = 0 ⇒ uz (αi ) ∈ R+ , i > 0,

(3.1.22)

3.1. AFFINE WEYL GROUPS

301

where −C¯a is the negative of the closure C¯a of Ca . The element b− = ub (b) is a unique element from P− which belongs to the orbit W (b). So the equality c− = b− means that b, c belong to the same orbit. def We will also use b+ == w0 (b+ ), a unique element in W (b) ∩ P+ . In terms of the elemnts πb , ub πb = b− , πb ub = b+ . Note that l(πb w) = l(πb ) + l(w) for all b ∈ P, w ∈ W. For instance, l(b− w) = l(b− ) + l(w), l(wb+ ) = l(b+ ) + l(w), l(ub πb w) = l(ub ) + l(πb ) + l(w) for b ∈ P, w ∈ W.

(3.1.23)

We will use these relations together with the following proposition when calculating the conjugations of the nonsymmetric Macdonald polynomials. Proposition 3.1.4. The set λ(w) b for w b = wb consists of α ˜ = [α, να j] with positive α if and only if (b, α∨ ) ≥ −1 and (b, α∨ ) = −1 ⇒ α ∈ λ(w)

(3.1.24)

for all α ∈ R+ , b ∈ P. The elements w b ∈ W · P+ are of this type. Proof. We use (3.1.11): ˜ + ∪ −λ ˜ + ), λ(wb) = λ(b) ∪ (−b λ(w)) \ (λ ˜+ = R ˜ − ∩ (−b λ(w)). where λ

(3.1.25)

The latter set is {[α, να (b, α∨ )]} such that (b, α∨ ) < 0, α ∈ λ(w). Let us calculate λ(wb) ∩ λ(b). We need to remove from λ(b) the roots −[α, να (b, α∨ )] for α > 0 such that (b, α∨ ) < 0, α ∈ λ(w). The roots in the form [−α, να j] belong to λ(b) exactly for 0 < j ≤ −(b, α∨ ). Therefore λ(wb) ∩ λ(b) = { [α, να j], 0 ≤ j < (b, α∨ ) as (b, α∨ ) > 0, [−α, να j], 0 < j < −(b, α∨ ) as (b, α∨ ) < 0 and α ∈ λ(w), [−α, να j], 0 < j ≤ −(b, α∨ ) as (b, α∨ ) < 0 and α 6∈ λ(w) }

(3.1.26)

for α ∈ R+ . Now let us assume that the nonaffine components of the roots from λ(wb) are all positive. Then (b, α∨ ) ≥ −1 for all α > 0 and λ(w) contains all α making −1 in the scalar product with b above. These conditions are also sufficient. ❑

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CHAPTER 3. GENERAL THEORY

3.1.4

Partial ordering in P

The main application of this section is the ordering needed in the theory of nonsymmetric polynomials; see [O3, M5]. This ordering was also used in [C16] in the process of calculating the coefficients of the Y –operators. The definition is as follows: b ≤ c, c ≥ b for b, c ∈ P if c − b ∈ Q+ , b ¹ c, c º b if b− < c− or {b− = c− and b ≤ c}.

(3.1.27) (3.1.28)

Recall that b− = c− means that b, c belong to the same W –orbit. We write , ≺, Â, respectively, if b 6= c. The following sets def

def

σ(b) == {c ∈ P, c º b}, σ∗ (b) == {c ∈ P, c  b}, def

def

σ− (b) == σ(b− ), σ+ (b) == σ∗ (b+ ) = {c ∈ P, c− > b− }

(3.1.29)

are convex. By convex, we mean that if c, d = c + rα ∈ σ for α ∈ R+ , r ∈ Z+ , then {c, c + α, ..., c + (r − 1)α, d} ⊂ σ.

(3.1.30)

The convexity of the intersections σ(b) ∩ W (b), σ∗ (b) ∩ W (b) is by construction. For the sake of completeness, let us check the convexity of the sets σ± (b). Both sets are W –invariant. Indeed, c− > b− if and only if b+ > w(c) > b− for all w ∈ W. The set σ− (b) is the union of σ+ and the orbit W (b). Here we use that b+ and b− are the greatest and the least elements of W (b) with respect to “>.” This is known (and can be readily checked by the induction with respect to the length; see, e.g., [C16]). If the end points of (3.1.30) are between b+ and b− , then it is true for the orbits of all inner points even if w ∈ W changes the sign of α (and the order of the end points). Also the elements from σ(b) strictly between c and d (i.e., c + qα, 0 < q < r) belong to σ+ (b). This gives the required fact. The next two propositions are essentially from [C23]. Proposition 3.1.5. (i) Let c = uˆ((0)), where uˆ is obtained by striking out any number of {sj } from a reduced decomposition of πb (b ∈ P ). Then c  b. Generally speaking, the converse is not true even if c ∈ W (b) (I. Macdonald). c when restricted to {πb , b ∈ P } is In other words, the Bruhat order of W stronger than  . (ii) Letting b = si ((c)) for 0 ≤ i ≤ n, the element si πc can be represented in the form πb for some b ∈ P if and only if (αj , c + d) 6= 0. More exactly, the following three conditions are equivalent: {c  b} ⇔ {(αi∨ , c + d) > 0} ⇔ {si πc = πb , l(πb ) = l(πc ) + 1}.

(3.1.31) ❑

3.1. AFFINE WEYL GROUPS

303

The following lemma from [C20] describes the case (αi∨ , c + d) = 0. Lemma 3.1.6. If (αi , c + d) = 0 for an index 0 ≤ i ≤ n, then uc (αi ) = αj as i > 0 and uc (−ϑ) = αj as i = 0 for proper j > 0. Equivalently, si πc 6∈ πP provided that l(si πc ) = 1 + l(πc ) if and only if k + (αi∨ , πc ((−kρ)) + d) = 0

for any k ∈ C, i ≥ 0.

(3.1.32)

Proof. If i > 0, then α = uc (αi ) > 0 and (α, c− ) = 0. If α = β + γ for positive roots β, γ, then (β, c− ) = 0 = (γ, c− ). Hence β 0 = u−1 c (β) > 0, (γ) > 0, and α = β + γ, which is impossible. Thus α is simple. γ 0 = u−1 i c Similarly, (ϑ, c) = 1 implies that α = uc (−ϑ) > 0, (α, c− ) = −1. Let α = β + γ, where β > 0 < γ. Since ϑ and α are short, we can assume that at least one of them is short, but it will follow automatically. Then, transposing β ↔ γ, if necessary, (β, c− ) = −1 and (γ, c− ) = 0. Since (β, c− ) = νβ (β ∨ , c− ), 0 we conclude that νβ = 1 and β is short. Hence β 0 = −u−1 c (β) > 0, γ = 0 0 0 ∨ 0 ∨ −u−1 c (γ) < 0 and ϑ = β + γ < β , which gives that ϑ < (β ) . The latter is ∨ . impossible because ϑ∨ is the maximal positive root in R+ −1 −1 Using πc = cuc and πc (z) = c + uc (z), we obtain that k + (αi∨ , πc ((−kρ)) + d) = k − k(uc (αi ), ρ) + (αi , c + d) and come to (3.1.32).

❑

Proposition 3.1.7. (i) Assuming (3.1.31), let i > 0. Then b = si (c), b− = c− , and ub = uc si . The set λ(πb ) is obtained from λ(πc ) by replacing the strict inequality (c− , α∨ ) > j > 0 for α = uc (αi ) (see (3.1.15)) with (c− , α∨ ) ≥ j > 0. Here α ∈ R− and (c− , α∨ ) = (c, αi∨ ) > 0. (ii) In the case i = 0, the following holds: b = ϑ + sϑ (c), the element c is from σ+ (b), b− = c− − uc (ϑ) ∈ P− , and ub = uc sϑ . For α = uc (−ϑ) = α∨ , the λ–inequality (c− , α∨ ) ≥ j > 0 is replaced with the strict inequality (c− , α∨ ) + 2 > j > 0. Here α ∈ R− and (c− , α∨ ) = −(c, ϑ) ≥ 0. (iii) For any c ∈ P, r ∈ O0 , πr πc = πb , where b = πr ((c)). Respectively, ub = uc ur , b = ωr + u−1 r (c), b− = c− + uc w0 (ωr ). In particular, the latter weight always belongs to P− . = si (c)(uc si )−1 . Proof. Let us check (i). First, πb = si πc = si cu−1 c The uniqueness of the latter decomposition gives the coincidence ub = uc si . Second, λ(πb ) is the union of λ(πc ) and (uc c−1 )(αi ) = [α, (c, αi )] for α = uc (αi ) ∈ R− (see (3.1.8)). Third, the inequality with α∨ in (3.1.16) is strict for c because u−1 c (α) = αi ∈ R+ and becomes non-strict for b, since −1 ub (α) = −αi ∈ R− . For (ii), it is the other way round. Namely, the extra affine root from λ(πb ) \ λ(πc ) is πc−1 (α0 ) = [uc (−ϑ), 1 − (c, ϑ)]. Therefore α = uc (−ϑ) ∈ R− −1 and the λ–inequality for πc is nonstrict. As for πb , u−1 b (α) = (uc sϑ ) (α) = ϑ ∈ R+ and the inequality becomes strict. Explicitly, (b− , α) = (b, −ϑ) = (sϑ (c − ϑ), −ϑ) = 2 − (c, ϑ). ❑

304

3.1.5

CHAPTER 3. GENERAL THEORY

Arrows in P

We write c → b or b ← c in the cases (i) or (ii) from the proposition above and use the left-right arrow c ↔ b for (iii) or when b and c coincide. By c →→ b, we mean that b can be obtained from c by a chain of (simple) right arrows. Respectively, c ↔→ b indicates that ↔ can be used in the chain. Actually, one left-right arrow of the type c ↔→ b is always sufficient because one element πr is sufficient in a reduced decomposition and it can be placed at the end of the decomposition. If such double arrows are excluded, then the ordering “→” is obviously stronger than the Bruhat order given by the procedure from Proposition 3.1.5(i), which, in its turn, is stronger than “Â.” b = πb πc−1 If l(πb ) = l(πc ) + l(πb πc−1 ), then the reduced decomposition of w readily produces a chain of simple arrows from c to b. The number of right arrows is precisely l(w) b (see (3.1.31)). Only transforms of types (ii) and (iii) will change the W –orbits, adding negative short roots to the corresponding c− for (ii) and the weights in the form w(ωr ) (w ∈ W ) for (iii). For instance, let c, b, b − c ∈ P− . Then πc = c, πb = b, w b = b − c is of length l(b) − l(c), and we need to decompose b − c. If c = 0, then b − ωr ∈ Q, the reduced decomposition of b begins with πr (r ∈ O0 ), and the first new c− is w0 (ωr ). When there is no πr and b ∈ Q ∩ P− , the chain always starts with −ϑ; it gives another proof of Proposition 3.1.1. Let us now examine the arrows c → b from the viewpoint of the W –orbits. The following proposition is useful for the classification and description of the perfect representation. Proposition 3.1.8. (i) Given c− ∈ P− , any element in the form c− +u(ωr ) ∈ P− can be represented as b− for proper b ↔ c. Respectively, c− + u(ϑ) ∈ P− for u ∈ W such that u(ϑ) ∈ R− can be represented as b− for proper b such that W (c− ) 3 c → b. (ii) In the case of A, D, E, any element b− such that b− ≺ c− (both belong to P− ) can be obtained from c− ∈ P− using consecutive arrows W (c− ) 3 c → b ∈ W (b− ). This cannot be true for all root systems because only short roots may be added to c− using such a construction. Proof. Let c0 = c− + u(ωr ) ∈ P− . Given a = u(ωr ), one can assume that u is the greatest possible, i.e., u(β) ∈ R− if β ∈ R+ and (β ∨ , ωr ) = 0. Explicitly, u = ua−1 w0 . We use the inequalities (c0 , α) ≤ 0 (α ∈ R+ ) for α such that (α, c− ) = 0 (if such α exist). This gives that (c0 , α) = (ωr , u−1 (α)) is either 0 or negative (actually only −1 may appear). In the latter case, u−1 (α) contains αr with a negative coefficient, and therefore u−1 (α) ∈ R− . If the scalar product is zero, then u−1 (α) is negative too because of the maximality of u.

3.2. DOUBLE HECKE ALGEBRAS

305

Thus w0 u−1 sends such α to R+ and therefore can be represented as u−1 c for proper c ∈ W (c− ) (see Proposition 3.1.2 and (3.1.13)). This gives the first part of (i). If c0 = c− + u(ϑ) ∈ P− , we may apply the same argument: (ϑ, u−1 (α)) < 0 implies that u−1 (α) is negative. So u can be represented as uc w0 and c0 = c− − uc (ϑ) for proper c. Then we observe that (c− , uc (ϑ)) = (c, ϑ) ≤ 0 because u(ϑ) was assumed to be negative. This proves (i). It is worth mentioning that the direct statement (that c0 ∈ P− for u from Proposition 3.1.7) also holds and can be readily checked. One needs to use that the weights {ωr } are minuscule and (ϑ, α∨ ) ≥ 2 (α ∈ R), with the equality exactly for α = ϑ (α ∈ R). Now let us assume that c, b ∈ P− and 0 6= x = b − c ∈ Q− in the simplylaced case A, D, E. We pick a connected component It of I = P {i | (x, αi ) ≤ 0} P in the Dynkin diagram Γ and set xt = i∈It li αi for x = ni=1 li αi . Then (xt , αi ) ≤ 0 for i ∈ It , since (αi , αj ) < 0 as i ∈ It 63 j. Obviously, xt ≥ x. Let ϑt be the maximal short root for the root subsystem Rt ⊂ R generated by {αi , i ∈ It } as simple roots. Then −ϑt ≥ xt , thanks to Proposition 3.1.1. Moreover, (ϑt , αj ) ≥ −1 for j 6∈ It . Here we use that the coefficient of αi in ϑc is 1 if αi is an end point of It and at the same time an inner point in Γ. This can be readily checked using the tables of [Bo]. Note that it is not necessary to assume that αi is inner in Γ if It is of type A, D. Here the value −1 is reached precisely for the neighbors of the end points of It in Γ \ It (otherwise the scalar product is zero). Such points αj do not belong to I because It is a connected component of I. Therefore (c, αj ) < 0, since (x, αj ) > 0 and (b + x, αj ) ≤ 0. We see that c > c0 = c − ϑt ≥ b (all three weights are from P− ) and can continue by induction. ❑

3.2

Double Hecke algebras

We keep the notations from the previous section. For the sake of uniformity, ([b, l], [b0 , l0 ]) = (b, b0 ), [α, να j]∨ = [α∨ , j], X0 = Xα0 = qϑ Xϑ−1 , (d, [α, j]) = j, for instance, (b + d, α0∨ ) = 1 − (b, ϑ). By m, we denote the least natural number such that (P, P ) = (1/m)Z; namely, m = 2 for D2k , m = 1 for B2k , Ck , otherwise m = |Π|.

3.2.1

Main definition

The double affine Hecke algebra depends on the parameters q, {tν , ν ∈ νR }. def The definition ring will be Qq,t == Q[q ±1/(2m) , t±1/2 ], formed by polynomials ±1/2 in terms of q ±1/(2m) and {tν }. If q, t are regarded to be complex numbers

306

CHAPTER 3. GENERAL THEORY

(e.g., when q is a root of unity), then the fractional powers of q, t have to be fixed somehow. We set tα˜ = tα = tνα , ti = tαi , qα˜ = q να , qi = q ναi , ˜ 0 ≤ i ≤ n, να = (α, α) . where α ˜ = [α, να j] ∈ R, 2

(3.2.1)

To simplify the formulas we will use the k–function, {kν }, together with the {tν }, setting {kα = kνα }, X X tα = tν = qαkν for ν = να , and ρk = kν ρ ν = kα α/2. (3.2.2) ν

α>0

We will call ρk the weighted ρ; see (3.1.1). Q (ρ ,α∨ ) ; here α ∈ R and therefore For instance, by q (ρk ,α) , we mean ν∈νR tν ν this product contains only integral powers of tsht and tlng . Note that (ρk , αi∨ ) = ki = kαi for i > 0. Let X1 , . . . , Xn be pairwise commutative and algebraically independent. We set X˜b =

n Y

Xili q j if ˜b = [b, j],

(3.2.3)

i=1

where b =

n X

li ωi ∈ P, j ∈

i=1

1 Z. m

c act in the ring Q[q ±1/m ][X] of polynomials in terms of The elements w b∈W Xb (b ∈ P ) and q ±1/m by the formulas: w(X b ˜b ) = Xw( b ˜b) .

(3.2.4)

In particular, def

πr (Xb ) = Xu−1 q (ωr∗ ,b) for αr∗ == πr−1 (α0 ), r ∈ O0 . r (b)

(3.2.5)

The involution r 7→ r∗ of the set O0 satisfies the relations ur ur∗ = 1 = πr πr∗ . Moreover, w0 (ωr ) = −ωr∗ for the longest element w0 ∈ W. Thus αr 7→ αr∗ is nothing but the automorphism of the nonaffine Dynkin diagram (preserving α0 ). This can be readily seen from the tables of [Bo], where only the case of Dn requires some consideration. Definition 3.2.1. The double affine Hecke algebra HH (see [C16]) is generated over the field Qq,t by the elements {Ti , 0 ≤ i ≤ n}, pairwise commutative {Xb , b ∈ P } satisfying (3.2.3), and the group Π, where the following relations are imposed: 1/2 −1/2 (o) (Ti − ti )(Ti + ti ) = 0, 0 ≤ i ≤ n;

3.2. DOUBLE HECKE ALGEBRAS

307

(i) Ti Tj Ti ... = Tj Ti Tj ..., mij factors on each side; (ii) πr Ti πr−1 = Tj if πr (αi ) = αj ; (iii) Ti Xb Ti = Xb Xα−1 if (b, αi∨ ) = 1, 0 ≤ i ≤ n; i (iv) Ti Xb = Xb Ti if (b, αi∨ ) = 0 for 0 ≤ i ≤ n; q (ωr∗ ,b) , r ∈ O0 . (v) πr Xb πr−1 = Xπr (b) = Xu−1 r (b)

❑

f, r ∈ O, the product Given w e∈W def

Tπr we == πr

l Y

e= Tik , where w

l Y

e sik , l = l(w),

(3.2.6)

k=1

k=1

does not depend on the choice of the reduced decomposition (because the {T } satisfy the same “braid” relations as do {s}). Moreover, c. TvˆTwb = Tvˆwb whenever l(ˆ v w) b = l(ˆ v ) + l(w) b for vˆ, w b∈W

(3.2.7)

In particular, we arrive at the pairwise commutative elements Yb =

n Y

Yili

if b =

n X

def

li ωi ∈ P, where Yi == Tωi ,

(3.2.8)

i=1

i=1

satisfying the relations if (b, αi∨ ) = 1, Ti−1 Yb Ti−1 = Yb Ya−1 i Ti Yb = Yb Ti if (b, αi∨ ) = 0, 1 ≤ i ≤ n.

3.2.2

(3.2.9)

Automorphisms

The following maps can be uniquely extended to automorphisms of HH (see [C23, C20]): −1 ε : Xi 7→ Yi , Yi 7→ Xi , Ti 7→ Ti−1 , tν 7→ t−1 ν , q 7→ q ,

(3.2.10)

τ+ : Xb 7→ Xb , Yr 7→ Xr Yr q −(ωr ,ωr )/2 , Ti 7→ Ti , tν 7→ tν , q 7→ q, def

τ+ : Y0 7→ Y0 T02 X0 = q −1 Ts−1 T0 X0 , Y0 = Yα0 == q −1 Yϑ−1 , ϑ

(3.2.11)

where 1 ≤ i ≤ n, r ∈ O0 . The formulas for the images of {Yr , Y0 } readily give: ε(T0 ) = (Y0 T0 X0 )−1 = Xϑ Tsϑ , τ+ (T0 ) = X0−1 T0−1 , , τ+ (πr ) = q −(ωr ,ωr )/2 Xr πr = q (ωr ,ωr )/2 πr Xr−1 ε(πr ) = Xr Tu−1 ∗ , r . for πr Xr∗ πr−1 = q (ωr ,ωr ) Xr−1 , Xr∗ Tur Xr = Tu−1 r∗ Theorem 2.3 from [C20] states that the mapping ³ 0 −1´ ³1 1´ 7→ ε, 7→ τ+ −1 0 0 1

(3.2.12)

(3.2.13)

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CHAPTER 3. GENERAL THEORY

can be extended to a homomorphism from GL2 (Z) to the group of automorphisms of HH modulo conjugations by the central elements from the group generated by T1 , . . . , Tn . In this statement, the quadratic relation (o) from Definition 3.2.1 may be omitted. Only the group relations matter. We will also need the following automorphisms of HH : def

def

τ− == ετ+ ε, and σ == τ+ τ−−1 τ+ = τ−−1 τ+ τ−−1 = εσ −1 ε.

(3.2.14)

They preserve T1 , . . . , Tn , t, q and are uniquely defined by the following relations: (ωr ,ωr )

τ− : Yb 7→ Yb , Xr 7→ Yr Xr q 2 , τ− : X0 7→ (T0 X0 Y0 T0 )−1 = qTsϑ X0−1 T0−1 ,

(3.2.15)

X0−1 Tsϑ , σ : Xb 7→ Yb−1 , Yr 7→ Yr−1 Xr Yr q −(ωr ,ωr ) , Y0 7→ Ts−1 ϑ where b ∈ P, r ∈ O0 . Besides, τ− (T0 ) = T0 , τ− (πr ) = πr (r ∈ O0 ), σ(T0 ) = Ts−1 Xϑ−1 , σ(πr ) = Yr−1 Xr πr q −(ωr ,ωr ) = Tu−1 Xr−1 ∗ . r ϑ Thus, τ− corresponds to (

(3.2.16)

0 1 1 0 ). Note that ), σ to ( −1 0 1 1

στ+−1 σ −1 = (τ+ τ−−1 τ+ )τ+−1 (τ− τ+−1 τ− ) = τ− .

(3.2.17)

The braid relation τ+ τ−−1 τ+ = τ−−1 τ+ τ−−1 is exactly the definition of the projective P SL(2, Z) due to Steinberg. It formally gives that σ 2 commutes with τ± . Generally speaking, σ 2 is not inner in HH . It is the conjugation by Tw−1 if w0 = −1; see below. Always, σ 4 is the conjugation by Tw−2 , which is 0 0 central in the nonaffine Hecke algebra H = hTi , 1 ≤ i ≤ ni. All automorphisms introduced above are unitary. An automorphism ω of HH is called unitary if ? ω ? = ω for the main anti-involution ? from [C16]: Xi? = Xi−1 , Yi? = Yi−1 , Ti? = Ti−1 , −1 ? ? ? tν 7→ t−1 ν , q 7→ q , 0 ≤ i ≤ n, (AB) = B A .

(3.2.18)

The commutativity with ? is obvious because this anti-involution is the inversion with respect to the multiplicative structure of HH . The following automorphism (involution) of HH[ will play an important def role in the theory: η == εσ. It conjugates q, t and is uniquely defined from the relations η : Ti 7→ Ti−1 , Xb 7→ Xb−1 , πr 7→ πr , where 0 ≤ i ≤ n, b ∈ P, r ∈ O0 .

(3.2.19)

3.2. DOUBLE HECKE ALGEBRAS

309

This involution extends the Kazhdan–Lusztig involution on the affine Hecke algebra generated by {Ti , i ≥ 0}. Here we have used (3.2.15) to calculate the images of X, Ti (i > 0), namely, applying (3.2.16), εσ(T0 ) = ε(Ts−1 Xϑ ) = Tsϑ Yϑ−1 = T0−1 , εσ(πr ) ϑ −1 =ε(Tu−1 Xr−1 = Tur∗ Yr−1 = πr−1 ∗ ) = Tu−1 Yr ∗ ∗ ∗ = πr . r r −1 Recall that πr∗ = πr−1 and ur∗ = u−1 r , where r ∈ O and αr∗ = πr (α0 ). ∗ The map r 7→ r corresponds to the standard automorphism of the nonaffine Dynkin diagram. It is straightforward to check that −1 η(Yr ) = Tw0 Yr−1 for r ∈ O0 . ∗ Tw 0

We are going to calculate the η–images of arbitrary Yb . We will use the same symbol ς for two different maps. The first on is b 7→ bς = −w0 (b), b ∈ P, si 7→ sς(i) , where (3.2.20) ς(αi ) =ας(i) = −w0 (αi ) for 1 ≤ i ≤ n, ς(0) = 0, ς(α0 ) = α0 , c . The second one is the corresponding auwhich is extended naturally to W tomorphism of HH : ς : Xb 7→ Xς(b) , Yb 7→ Yς(b) , Ti 7→ Tς(i) , πr 7→ πr∗ .

(3.2.21)

The automorphism from (3.2.21) commutes with all previously considered DAHA automorphisms. Proposition 3.2.2. η(Yb ) = Tw0 Yw0 (b) Tw−1 , σ(Yb ) = Tw−1 Xw0 (b) Tw0 , 0 0 2 −1 −1 2 −1 −1 σ (Xb ) = Tw0 Xw0 (b) Tw0 , σ (Yb ) = Tw0 Yw0 (b) Tw0 ,

(3.2.22)

σ 2 (T0 ) = Tw−1 T0 Tw0 , Tw−1 Yϑ Tw0 = Ts−1 Yϑ Tsϑ , 0 0 ϑ Tς(i) = Tw−1 Ti Tw0 for i > 0, Tw−1 Tsϑ Tw0 = Tsϑ , 0 0 2 −1 4 σ (H) = Tw0 ς(H)Tw0 , σ (H) = Tw−2 HTw20 for H ∈ HH. 0 Proof. The formulas for σ easily follow from those for η, so it suffices to −1 for b ∈ P+ coincides with check that η(Yb ) = T−b −1 −1 −1 −1 Tw0 Yw0 (b) Tw−1 = Tw0 Yς(b) Tw0 = Tw0 Tς(b) Tw0 , 0

. i.e., we need the relation T−b = Tw0 Tς(b) Tw−1 0 Since −b ∈ P− , T−b Tw0 = T(−b)·w0 . Here we apply (3.1.23). Similarly, ς(b) ∈ P+ and Tw0 Tς(b) = Tw0 ·(ς(b)) . However, w0 · (ς(b)) = w0 · w0 · (−b) · w0 = (−b) · w0 .

❑

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CHAPTER 3. GENERAL THEORY

3.2.3

Demazure–Lusztig operators

Following [Lus1, KK, C16], the operators 1/2 1/2 −1/2 Tˆi = ti si + (ti − ti )(Xαi − 1)−1 (si − 1) (0 ≤ i ≤ n)

(3.2.23)

are well defined in the space of Laurent polynomials Q[q, t±1/2 ][X]. We note that only T0 involves q: 1/2 1/2 −1/2 Tˆ0 = t0 s0 + (t0 − t0 )(qXϑ−1 − 1)−1 (s0 − 1), −(b,ϑ) (b,ϑ)

where s0 (Xb ) = Xb Xϑ

q

, α0 = [−ϑ, 1].

(3.2.24)

Theorem 3.2.1. (i) The map Tj 7→ Tˆj , Xb 7→ Xb (see (3.2.3)), πr 7→ πr (see (3.2.5)) induces a Qq,t –linear homomorphism from HH to the algebra of linear endomorphisms of Qq,t [X]. This representation, which will be called the polynomial representation, is faithful and remains faithful when q, t take any nonzero complex values, assuming that q is not a root of unity. (ii) For arbitrary nonzero q, t, any element H ∈ HH has a unique decomposition in the form X H= gw Tw fw , gw ∈ Qq,t [X], fw ∈ Qq,t [Y ], (3.2.25) w∈W

and five more analogous decompositions corresponding to the other orders of {T, X, Y }. ˆ of H ∈ HH is uniquely determined from the following (iii) The image H condition: ˆ (X)) = g(X) for H ∈ HH , if Hf (X) − g(X) ∈ H(f n X X 1/2 Hr (πr − 1), where Hi , Hr ∈ HH }. { Hi (Ti − ti ) + i=0

(3.2.26)

r∈O0

The automorphism τ− preserves the right-hand side of (3.2.26) and therefore acts in the polynomial representation. Proof. This theorem is essentially from [C16] and [C19]. One only needs to extend (ii) to roots of unity (it was not formulated in [C16] in the complete generality). In the first place, the existence of such a decomposition is true for all q, which follows directly from the defining relations. Secondly, the uniqueness holds for generic q because the polynomial representation is faithful (a similar argument was used in [C16]). Finally, the number of linearly independent expressions in the form (3.2.25) (when the degrees of f, g are bounded) may not become smaller for special q than for generic q (here by special we mean the roots of unity). ❑

3.2. DOUBLE HECKE ALGEBRAS

311

Intermediate subalgebras. We will also use the above statements for the subalgebras of HH with P replaced by its subgroups containing Q. Let B be any lattice between Q and P, respectively, Π[ be the preimage of B/Q c [ = Π[ · W f = B · W. in Π, and W The intermediate subalgebra HH[ is the subalgebra of HH generated by Xb (b ∈ B), π ∈ Π[ , and Ti (0 ≤ i ≤ n). Thus Yb ∈ HH[ when b ∈ B. Actually, the lattices for Xb and Yb can be different, but, when discussing the Fourier transform, it is technically convenient to impose the coincidence of these lattices. Proposition 3.2.3. The algebra HH[ satisfies all claims of Theorem 3.2.1 for B instead of P and the polynomial representations defined as Qq,t [Xb , b ∈ B]. The automorphisms ε, τ± , σ, η and the anti-involutions ?, φ preserve this subalgebra. Proof. The compatibility of the definition of HH[ with the automorphisms τ± follows directly from the formulas for their action on Xr , Yr (r ∈ O0 ) and T0 (see (3.2.12, 3.2.15)). Since HH[ is generated by Xb , Yb (b ∈ B) and {T1 , ..., Tn }, this holds for ε as well. Claim (i) from Theorem 3.2.1 remains true for the polynomials in Xb (b ∈ B) because the formulas for Tˆi involve Xα only. To check (ii) we need a more complete version of the relations (iii,iv) from Definition 3.2.1 (cf. [Lus1] and formula (3.2.23) above). Namely, for all b ∈ P, 1/2

Ti Xb − Xsi (b) Ti = (ti

−1/2

− ti

)

si (Xb ) − Xb , 0 ≤ i ≤ n. Xαi − 1

(3.2.27)

This relation and its dual counterpart for Y instead of X ensure the existence of the decompositions (3.2.25). Using the fact that the polynomial representation is faithful (for generic q) we establish the uniqueness (for all q 6= 0). Claim (iii) formally results from (ii). ❑ c[ = B · W ⊂ W c, Qq,t [Xb ] = Concerning the notations, we, naturally, set W Qq,t [Xb , b ∈ B], and replace m by the least m ˜ ∈ N such that m(B, ˜ B) ⊂ Z [ in the definition of the Qq,t , that is, the ring of constants of HH . The same notation Qq,t will be used in the [–case.

3.2.4

Filtrations

The following proposition is essentially from [C20] (Proposition 3.3). In the notation from (3.1.29), we set def

def

Σ(b) == ⊕c∈σ(b) Qq,t xc , Σ∗ (b) == ⊕c∈σ∗ (b) Qq,t xc Σ− (b) = Σ(b− ), Σ+ (b) = Σ∗ (b+ ), b ∈ B.

(3.2.28)

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CHAPTER 3. GENERAL THEORY

Let us denote the maps b 7→ −b by ı. Recall that b 7→ −w0 (b) is denoted by ς. Respectively, ı : Xb 7→ X−b = Xb−1 , ς : Xb 7→ Xbς = ς(Xb ) = X−w0 (b) . Then ς preserves the ordering and sends Σ(b) and Σ∗ (b) to Σ(ς(b)) and Σ∗ (ς(b)), respectively. ˜ with α > 0, the Proposition 3.2.4. (i) Given b ∈ B, α ˜ = [α, να j] ∈ R operator −1/2 Rα˜ = tα1/2 + (t1/2 )(Xα˜−1 − 1)−1 (1 − sα˜ ) α − tα

(3.2.29)

preserves Σ(b), Σ∗ (b), Σ± (b). Moreover, 1/2 −1/2 )sα˜ (Xb ) if (b, α) < 0, Rα˜ (Xb ) mod Σ+ (b) = t1/2 α Xb + (tα − tα

= t−1/2 Xb if (b, α) > 0, α = t1/2 α Xb if (b, α) = 0.

(3.2.30)

(ii) The operators w b−1 Twb preserve Σ(b), Σ∗ (b), and Σ± (b) provided that all affine roots α ˜ = [α, να j] ∈ λ(w) b have positive α. The elements w b ∈ W · B+ have this property. For instance, Yc leave the Σ–sets invariant for all c ∈ B and ı · Tw0 : Σ(b) → Σ(ς(b)), Σ∗ (b) → Σ∗ (ς(b)). (iii) Assuming (ii), w b−1 Twb (Xb ) mod Σ∗ (b) =

Y

t1/2 α

(α,b)≤0 ˜

Y

t−1/2 Xb α

(α,b)>0 ˜

multiplied over α ˜ = [α, να j] ∈ λ(w). b

(3.2.31)

In particular, −1

(3.2.32) Yc (Xb ) mod Σ∗ (b) = q (c,ub (ρk )−b) Xb , Y ı · Tw0 (Xb ) mod Σ∗ (b) = tνlν (w0 )/2−lν (ub ) Xς(b) mod Σ∗ (ς(b)). ν∈νR

❑ The operators {Tj , 0 ≤ j ≤ n} may not fix the spaces Σ(b), Σ∗ (b). However, they leave Σ− (b) and Σ+ (b) invariant. Hence the affine Hecke algebra HY[ generated by Ti (0 ≤ i ≤ n) and the group Π[ act in the space def V (b) == Σ− (b)/Σ+ (b) ∼ = Qq,t [Xc , c ∈ W (b)].

(3.2.33)

3.3. MACDONALD POLYNOMIALS

313

This representation is the (parabolically) induced HY[ –module generated by the image X+ of Xb+ in V (b) subject to 1/2

Ti (X+ ) = ti X+ if (b+ , αi ) = 0, 1 ≤ i ≤ n, Ya (X+ ) = q

(a,u−1 b (ρk )−b+ ) +

X+ , a ∈ B.

(3.2.34)

This is true for arbitrary nonzero q, t. Note the following explicit formulas: 1/2

si (Xc ) = (ti Ti (Xc )) if (c, αi ) > 0, 1/2

= (ti Ti (Xc ))−1 if (c, αi ) < 0, 1/2

Ti (Xc ) = ti Xc if (c, αi ) = 0, πr (Xc ) = Xπr (c) , c ∈ W (b), r ∈ O0 .

(3.2.35)

Here 0 ≤ i ≤ n, (c, α0 ) = −(c, ϑ).

3.3

Macdonald polynomials

Continuing from the previous section, we set µ =µ

(k)

=

∞ Y Y α∈R+ i=0

(1 − Xα qαi )(1 − Xα−1 qαi+1 ) . (1 − Xα tα qαi )(1 − Xα−1 tα qαi+1 )

(3.3.1)

It is considered as a Laurent series with coefficients in Q[tν ][[qν ]] for ν ∈ νR . def We denote the constant term of f (X) by hf i. Let µ◦ == µ/hµi, where hµi =

∞ Y Y α∈R+ i=1

(1 − q (ρk ,α)+i )2 . (ρk ,α)+i ) (1 − tα q (ρk ,α)+i )(1 − t−1 α q

(3.3.2)

(ρ ,α∨ )

Recall that q (ρk ,α) = qα k , tα = qαkα . This formula is from [C16]. It is nothing but the Macdonald constant term conjecture from [M2]. The coefficients of the Laurent series µ◦ are actually from the field of rationals def Q(q, t) == Q(qν , tν ), where ν ∈ νR . One also has µ∗◦ = µ◦ for the involution Xb∗ = X−b , t∗ = t−1 , q ∗ = q −1 . This involution is the restriction of the anti-involution ? from (3.2.18) to X– polynomials (and Laurent series). These two properties of µ◦ can be directly seen from the difference relations for µ. See the next two propositions.

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CHAPTER 3. GENERAL THEORY

3.3.1

Definitions

Setting def

hf, gi◦ == hµ◦ f g ∗ i = hg, f i∗◦ for f, g ∈ Q(q, t)[X],

(3.3.3) (k)

we introduce the nonsymmetric Macdonald polynomials Eb (x) = Eb Q(q, t)[X] for b ∈ P by means of the conditions Eb − Xb ∈ ⊕cÂb Q(q, t)Xc , hEb , Xc i◦ = 0 for P 3 c  b.

∈

(3.3.4)

They are well defined because the pairing is nondegenerate and form a basis in Q(q, t)[X]. This definition is due to Macdonald (for ksht = klng ∈ Z+ ), who extended Opdam’s nonsymmetric polynomials introduced in the differential case in [O3] (Opdam mentions Heckman’s contribution in [O3]). The general case was considered in [C20]. Another approach is based on the Y –operators. See formulas (3.2.34) and (3.2.32) above, and also Proposition 3.2.4. In the previous section we denoted Y, T acting in the polynomial representation by Yˆ , Tˆ. From now on, we will mainly omit the hat (if it does not lead to confusion). Proposition 3.3.1. The polynomial representation is ?–unitary: hH(f ), gi◦ = hf, H ? (g)i◦ for H ∈ HH, f ∈ Qq,t [X].

(3.3.5)

The polynomials {Eb , b ∈ P } are unique (up to proportionality) eigenfunctions def of the operators {Lf == f (Y1 , . . . , Yn ), f ∈ Q[X]} acting in Qq,t [X]: def

Lf (Eb ) = f (q −b] )Eb , where b] == b − u−1 b (ρk ),

(3.3.6)

Xa (q b ) = q (a,b) for a, b ∈ P, ub = πb−1 b is from Section 3.1.

(3.3.7) ❑

Thus we have two equivalent definitions of the nonsymmetric polynomials. Both are compatible with the transfer to the intermediate subalgebras HH[ def and subspaces Qq,t [Xb ] == Qq,t [Xb , b ∈ B] ⊂ Qq,t [X]. Let us check that the coefficients of µ◦ and the Macdonald polynomials are rational functions in terms of qν , tν . We need to make the construction of h , i◦ more abstract. A form (f , g) will be called ∗–bilinear if (rf , g) = r (f , g) = (f , r∗ g) for r ∈ Qq,t .

(3.3.8)

3.3. MACDONALD POLYNOMIALS

315

Proposition 3.3.2. (i) Forms hf , gi on Qq,t [X] satisfying (3.3.5) and which are ∗–bilinear are in one-to-one correspondence with Qq,t –linear maps $ : Qq,t [X] → Qq,t such that 1/2

$(Ti (u)) = ti

$(u), $(Ya (u)) = q (a , ρk ) $(u), i ≥ 0, a ∈ P.

(3.3.9)

Such a form is ∗–hermitian in the sense of (3.3.3) if and only if $(f ∗ ) = ($(f ))∗ . Given $, the corresponding form is hf , gi = $(f g ∗ ). e t) = Q(q 1/(2m) , t1/2 ) in (ii) Replacing Qq,t by the field of rationals Q(q, the definition of HH and in the polynomial representation, there is a unique nonzero $ up to proportionality. Namely, it is the restriction to e t) e t)[X] ⊂ HH ⊗Q Q(q, Q(q, e t) uniquely determined from the relation of the linear map $ext : HH → Q(q, def H − $ext (H)I ⊂ J == {

X¡

Hr (πr − 1) + (πr −

1)Hr0

r∈O0

¢

+

n X ¡

¢ 1/2 1/2 Hi (Ti − ti ) + (Ti − ti )Hi0 },

i=0

where Hi , Hi0 , Hr , Hr0 ∈ HH . Equivalently, f (X) = $(f ) +

X r∈O0

(πr − 1)(hr ) +

n X 1/2 (Tˆi − t )(hi ) i

(3.3.10)

i=0

e t)[X]. for hi , hr ∈ Q(q, Proof. In the first place, given a form hf, gi satisfying (3.3.5), the corresponding linear form is of course $(f ) = hf, 1i. It obviously satisfies (3.3.10). ˆ Its extension to HH is $ext (H) = $(H(1)). Now let $unv be the projection HH → HH /J. We set hA , Biunv = $unv (AB ? ) = hB , Ai∗unv , where the anti-involution ? acts naturally on HH /J thanks to the ?–invariance of J. By construction, hHA , Biunv = $unv (HAB ? ) = $unv (AB ? H) = hA , H ? Biunv for H = Ti , πr . Upon the restriction to Qq,t [X], def ? (g)i for hf , gi = \ ˆ ) , gi = hf , H hH(f = $unv (f (X)g(X)∗ )

(3.3.11)

ˆ is the action of H in the polynomial representation. and such H. Here H Obviously (3.3.11) holds when H is a multiplication by a polynomial. So

316

CHAPTER 3. GENERAL THEORY

it is true for all H ∈ HH . Thus the form hf , gi satisfies the same relations as hf , gi◦ . Namely, it is hermitian with respect to ∗ and HH –invariant with respect to ?. Recall that its values are in the vector space HH /J with a natural action of ?. The form hf , gi is universal among such forms. To be more exact, an arbitrary Qq,t –valued linear form on Qq,t [X] obeying the same relations as $unv is the composition of $unv and a homomorphism ω : HH /J → Qq,t . The latter has to satisfy the ?–invariance relations ω(H ? ) = ω(H)∗ for H ∈ HH to make the corresponding bi-form ∗–hermitian. Let us switch from Qq,t to the field of rationals Q(q, t). We already know that at least one form h , i exists for generic q, t with values in a proper completion of Q(q, t). It is given by (3.3.3) and is unique up to proportionality. Indeed, the polynomial representation is irreducible, the Y –operators are diagonalizable there, and the Y –spectrum is simple. Thus the space HH /J is one-dimensional upon proper completion, so it has to be one-dimensional over Q(q, t) as well. We obtain the rationality of the coefficients of µ◦ and therefore the rationality of the coefficients of the nonsymmetric Macdonald polynomials. ❑ Symmetric polynomials. Following Proposition 3.3.1, the symmetric (k) Macdonald polynomials Pb = Pb can be introduced as eigenfunctions of the W –invariant operators Lf = f (Y1 , . . . , Yn ) defined for symmetric, i.e., W –invariant, polynomials f : X Lf (Pb ) = f (q −b+ρk )Pb , b ∈ P− , Pb = Xc mod Σ+ (b). (3.3.12) c∈W (b)

Here P it suffices to take the monomial symmetric functions, namely, fb = c∈W (b) Xc for b ∈ P− . Actually, if a polynomial P is an eigenfunction of {Lf }, then the corresponding “eigenvalue” is represented in the formPof q −b+ρk for a certain b ∈ P− , which automatically results in Pb = Const c∈W (b) Xc mod Σ+ (b). Thus, it is sufficient to say that {P } are eigenfunctions of {L}. For minuscule b = −ωr and for b = −ϑ, the restrictions of the operators Lfb to symmetric polynomials are the Macdonald operators from [M3, M4, Ru]. The P –polynomials are pairwise orthogonal with respect to h , i◦ , the inner product that makes the nonsymmetric polynomials {E} pairwise orthogonal. However, since they are W –invariant, µ can be replaced by the measure–function due to Macdonald, the symmetric truncated theta function: δ = δ (k) =

∞ Y Y α∈R+ i=0

(1 − Xα qαi )(1 − Xα−1 qαi ) . (1 − Xα tα qαi )(1 − Xα−1 tα qαi )

(3.3.13)

3.3. MACDONALD POLYNOMIALS

317

The corresponding pairing remains ∗–hermitian because δ◦ is ∗–invariant. These polynomials were introduced in [M3, M4]. They were used for the first time in Kadell’s unpublished work (classical root systems). In the onedimensional case, they are due to Rogers (see [AI]). The connection between E and P is as follows: Pb− = Pb+ Eb+ , b− ∈ P− , b+ = w0 (b− ), X Y def Pb+ == tνlν (wc )/2 Tˆwc ,

(3.3.14)

c∈W (b+ ) ν

where wc ∈ W is the element of the least length such that c = wc (b+ ). See [O3, M5, C20]. There is another formula [M5, C20] Pb− =

X

Y

c∈W (b+ ) (α∨,c)>0,α∈R+

tα − Xα (q c] ) Ec . 1 − Xα (q c] )

(3.3.15)

The demonstration is based on the technique of intertwiners, that will be considered below. For instance, it gives that Pb− is Eb− modulo lower terms, that is, a part of the definition of Pb− .

3.3.2

Spherical polynomials

We will mainly use the following renormalization of the E–polynomials (see [C20]): def

Eb == Eb (X)(Eb (q −ρk ))−1 , where Y ³ 1 − q j tα Xα (q ρk ) ´ α , Eb (q −ρk ) = q (ρk ,b− ) j ρk ) X (q 1 − q α α [α,j]∈λ0 (π )

(3.3.16)

b

0

λ (πb ) = {[α, j] | [−α, να j] ∈ λ(πb )}.

(3.3.17)

We call them spherical polynomials. Explicitly (see (3.1.15)), λ0 (πb ) ={[α, j] | α ∈ R+ , − (b− , α∨ ) > j > 0 if u−1 b (α) ∈ R− , ∨ − (b− , α ) ≥ j > 0 if ub−1 (α) ∈ R+ }.

(3.3.18)

Formula (3.3.16) is the Macdonald evaluation conjecture in the nonsymmetric variant. Note that one has to consider only long α (resp., short) if ksht = 0 (resp., klng = 0) in the λ0 –set. All of the formulas below involving λ or λ0 have to be modified correspondingly in such a case.

318

CHAPTER 3. GENERAL THEORY We have the following duality formula: Eb (q c] ) = Ec (q b] ) for b, c ∈ P, b] = b − u−1 b (ρk ),

(3.3.19)

justifying the definition above; see [C20]. The proof is based on the following anti-involution def

φ == ε ? = ? ε : Xb 7→ Yb−1 , Ti 7→ Ti (1 ≤ i ≤ n).

(3.3.20)

In contrast to the previous chapter, the latter anti-involution will not be used too much in this chapter, because mainly we study the inner product based on the ?–involution, inducing the conjugation q 7→ q −1 , t 7→ t−1 . The following “derivative” ♥ = σ ·φ of φ is a counterpart of ? in the theory without the conjugation, i.e., for the harmonic analysis in the case of real q. It is defined as an anti-involution fixing q, t and their fractional powers, and sending ♥ : Ti 7→ Ti , πr 7→ πr , Yb 7→ Yb , Xb 7→ Tw−1 Xς(b) Tw0 0

(3.3.21)

for 0 ≤ i ≤ n, b ∈ P. Comment. The evaluation formula can be obtained from the duality via the Pieri rules. This derivation in the case of symmetric Macdonald polynomials of type A is due to Koornwinder. The norm formula, including the celebrated Macdonald constant term conjecture, follows from the duality and the evaluation formula by using the technique of intertwiners. See, e.g., [C23]. It is the simplest known approach. ❑ The symmetric duality and evaluation formulas from [C19] are Pb (q c−ρk )Pc (q −ρk ) = Pc (q b−ρk )Pb (q −ρk ), b, c ∈ P− , Pb (q

−ρk

) = Pb (q

+ρk

ρk

) = Xb (q )

Y α∈R+

−(b,α∨ )³

Y j=1

1 − qαj−1 tα Xα (q ρk ) ´ . 1 − qαj−1 Xα (q ρk )

(3.3.22) (3.3.23)

Conjugations. We now come to the formulas for the ∗–conjugations of the nonsymmetric polynomials, which are important in the theory of Fourier transforms. The conjugation { }∗ is well defined on the polynomials Eb because their coefficients are rational functions in terms of q, t. Note that, generally, conjugating infinite series in terms of q, t reverses their “direction” and creates problems. Proposition 3.3.3. The conjugation ∗ in the polynomial representation is induced by the involution η of HH[ . The paring hf gµ0 i induces on HH the anti-involution ? · η = η · ?. For b ∈ B, Y Eb∗ = tνlν (ub )−lν (w0 )/2 Tw0 (Eς(b) ), where ς(b) = −w0 (b). (3.3.24) ν∈νR

3.3. MACDONALD POLYNOMIALS

319

Proof. The conjugation ∗ sends Tˆi to Tˆi−1 for 0 ≤ i ≤ n, i.e., it coincides with the action of ? on Ti . This can be checked by a simple direct calculation. However, ∗ is an involution, in contrast to ? which is an anti-involution. By definition, the conjugation takes Xb to Xb−1 and fixes πr for r ∈ O. Using (3.2.19), we conclude that ∗ is induced by the involution η of HH[ . Applying (3.2.32) to ς(b), Y lν (w0 )/2−lν (uς(b) ) Tw0 (Eς(b) ) mod Σ∗ (b) = tν X−b mod ı(Σ∗ (b)). ν∈νR

Here ı sends Xb 7→ X−b and fixes q, t. Obviously ı(Σ∗ (b)) coincides with (Σ∗ (b))∗ and lν (uς(b) ) = lν (ub ). This gives the coefficient of proportionality in (3.3.24). Conjugating, η(Yc )(Eb∗ ) = q (c,b] ) Eb∗ = q (w0 (c),w0 (b] )) Eb∗ = q −(w0 (c),w0 (−b)] ) Eb∗ . The last transformation is possible because the automorphism ς = −w0 is compatible with the representations b = πb ub . Finally, η(Yc ) = Tw0 Yw0 (c) Tw−1 0 ∗ ) has to be propor(E due to Proposition 3.2.2 (formula (3.2.22)), and Tw−1 b 0 tional to Ew0 (−b) . ❑ ∗ For the symmetric polynomials, Pb = Pς(b) , b ∈ P− , which readily results from the orthogonality property and the second relation from (3.3.12). The conjugation formula (3.3.24) gets simpler in the spherical normalization. To recalculate it we use the equality Y q (ρk ,b− ) = tν−lν (b)/2 ν∈νR

and present the evaluation formula (3.3.16) in the following form: Y Eb (q −ρk ) = tν−lν (ub )/2 Mb , where

(3.3.25)

ν

Y

def

Mb ==

[α,j]∈λ0 (πb

³ t−1/2 − q j t1/2 X (q ρk ) ´ α α α α , j ρ k ). 1 − q X (q α α )

The factor Mb is “real,” i.e., ∗–invariant. Hence, Y Eb∗ = Mb−1 (Eb tνlν (ub )/2 )∗ =

Y

ν

tνlν (ub )/2−lν (w0 )/2 Tw0 (Eς(b) )

ν

=

Y

tν−lν (w0 )/2 Tw0 (Eς(b) ).

(3.3.26)

ν

We will give later a better proof of this relation without using (3.3.16). It will be based on the intertwining operators and the Fourier transform.

320

3.3.3

CHAPTER 3. GENERAL THEORY

Intertwining operators

The X–intertwiners (see, e.g., [C23]) are introduced as follows: 1/2

−1/2

Φi = Ti + (ti − ti )(Xαi − 1)−1 , Si = (φi )−1 Φi , Gi = Φi (φi )−1 , X0 = qXϑ−1 , 1/2

φi = ti

1/2

+ (ti

−1/2

− ti

)(Xαi − 1)−1 ,

(3.3.27)

for 0 ≤ i ≤ n. They belong to HH extended by the field Qq,t (X) of rational functions in {X}. The elements Si and Gi satisfy the same relations as {si , πr }. Hence the map c, b = πr sil · · · si1 ∈ W w b 7→ Swb = πr Sil · · · Si1 , where w

(3.3.28)

c. The same holds for Gwb and Swb . is a well defined homomorphism from W The latter are called the normalized intertwiners. It suffices to check it for Swb and in the polynomial representation, where Swb = w. b We do not need to assume here that the representation is reduced, i.e., l = l(w). b As to Φi , they satisfy the relations for {Ti }, i.e., the homogeneous Coxeter relations, and those with πr . So the decomposition in (3.3.28) has to be reduced to make Φwb well defined. A direct proof of the latter is based on the following intertwining property of {Φ}: c, Φwb Xb = Xw(b) b∈W b, w b Φw

(3.3.29)

which fixes Φwb uniquely up to left (or right) multiplications by functions of (1) (2) X. Therefore Φwb = Φwb Λ(X) for a Laurent polynomial Λ(X) if we obtain (1,2) Φwb using the formula Φwb = Φπr Φsil · · · Φsi1 for two different decompositions of w b (maybe nonreduced). The normalization of Φ guarantees that the difference Φwb − Twb is a combination of Twb0 for l(w b0 ) < l(w) b for any reduced decompositions of w. b Therefore the Φwb are all well defined (do not depend on the choice of the reduced decomposition) and the map Φ has the desired multiplicative property. We will also use that Φi , φi are self-adjoint with respect to the antiinvolution (3.2.18) and therefore c. b∈W Φ?wb = Φwb−1 , Sw?b = Gwb−1 , w

(3.3.30)

This gives that G and S are ?–unitary up to an X–functional coefficient of proportionality. Explicitly: ¢−1 ¡ = (3.3.31) (Gwb )? Gwb = (Swb )? Swb j −1/2 Y³ t1/2 Xα ´ α − qα t α , [α, να j] ∈ λ(w). b −1/2 j 1/2 t − q t X α α α α α,j

3.3. MACDONALD POLYNOMIALS

321

To define the Y –intertwiners we apply the involution ε to Φwb and to def −j c, where Y[b,j] = S, G. Thus ε(Φwb )Yb = Yw(b) b∈W = q Yb . b , for w b Φw The Y –intertwiners satisfy the same ?–relations (3.3.30). The formulas can be easily calculated using (3.2.12). In the case of GLn , one obtains the intertwiners from [KnS]. For i > 0, the Y –intertwiners ε(Φsi ) are the X– without touching T and t. Here intertwiners Φsi with Xαi replaced by Yα−1 i 1/2 −1/2 −1 we use that ε sends nonaffine Ti to Ti , tν to tν , and transposes X and Y. Using (3.2.12), ε(T0 ) = Xϑ Tsϑ = (Y0 T0 X0 )−1 and we obtain the relations 1/2

ε(Φi ) = Ti + (ti

−1/2

− ti

1/2

)(Yα−1 − 1)−1 for i > 0, i −1/2

ε(Φ0 ) = Xϑ Tsϑ − (t0 − t0 1/2

)(Yα0 − 1)−1

−1/2

= Y0 T0 X0 + (t0 − t0 )(Yα−1 − 1)−1 , for Y0 = q −1 Yϑ−1 , 0 ε(Si ) = ε((φi )−1 Φi ), ε(Gi ) = ε(Φi (φi )−1 ), 1/2

ε(φi ) = ti

1/2

+ (ti

−1/2

− ti

)(Yα−1 − 1)−1 , i ≥ 0. i

(3.3.32)

When applying the Y –intertwiners to the spherical polynomials (or any eigenfunctions of the Y –operators), it is more convenient to use τ+ instead of ε. Let us check that for i ≥ 0 and r ∈ O0 , 1/2

ti

−1/2

− ti def Ψi == τ+ (Ti ) + , and Pr == τ+ (πr ) −1 Yαi − 1 def

(3.3.33)

intertwine {Yb }. Recall that τ+ (T0 ) = X0−1 T0−1 = q −1 Xϑ T0−1 , τ+ (πr ) = q −(ωr ,ωr )/2 Xr πr . Thanks to the relations σ(Xb ) = Yb−1 , τ− (Yb ) = Yb : τ− σ(Φi Xb ) = τ− σ(Xc Φi ) ⇒ τ− σ(Φi )(Yb−1 ) = Yc−1 τ− σ(Φi ) for c = si (b). Thus τ− σ(Φi ) intertwine {Yb }. So do τ− σ(πr ). However, 1/2

τ− σ(Φi ) = τ+ (Ti ) + (ti

−1/2

− ti

)(Yα−1 − 1)−1 i

because τ− σ = τ+ τ−−1 and τ− preserves Ti . Similarly, τ− σ(πr ) = τ+ (πr ). The automorphism τ− acts in V and commutes with the Y –operators. The following proposition describes its action on the Ψ–intertwiners. Proposition 3.3.4. (i) For generic q, t or for arbitrary q, t provided that the polynomial Eb for b ∈ P is well defined, τ− (Eb ) = q −

(b− , b− ) +(b− 2

, ρk )

Eb for P− 3 b− ∈ W (b).

(3.3.34)

322

CHAPTER 3. GENERAL THEORY (ii) For any q, t, Y0 = q −1 Yϑ−1 , and Ψi from (3.3.33), τ− (Ti ) = Ti (i > 0), τ− (τ+ (T0 )) = τ+ (T0 )−1 Y0 , τ− (Ψi ) = Ψi (i > 0), τ− (Ψ0 ) = Ψ0 Y0 = Y0−1 Ψ0 ,

(3.3.35)

τ− (τ+ (πr )) = q (ωr ,ωr )/2 Yr τ+ (πr ) = q −(ωr ,ωr )/2 τ+ (πr )Yr−1 ∗ . Proof. The claims can be checked using directly the definitions. One can −1 also identify τ− in V with the operator of multiplication by τf γy − = γy (1) for the Y –Gaussian defined as follows: X X def q (b,b)/2 q (b,ρk ) . (3.3.36) q (b,b)/2 Yb , γy (1) = γy == b∈P

b∈P

Here we assume that 0 < q < 1 and use that V is a union of finite dimensional spaces preserved by the Y –operators. The irreducibility of V for generic q, t is sufficient to conclude that τf − = τ− , because the conjugation by τf indices τ in H H . This implies their coincidence − − for generic q (apart from roots of unity) and arbitrary tν , and, finally, results in (3.3.35) for arbitrary q, t. ❑ Setting 1/2

Φbi = Φi (q b] ) = Ti + (ti

−1/2

− ti

1/2

b] Gbi = (Φi φ−1 i )(q ) =

Ti + (ti 1/2

ti

−1/2

− ti

1/2

+ (ti

)(Xαi (q b] ) − 1)−1 , )(Xαi (q b] ) − 1)−1

−1/2

− ti

)(Xαi (q b] ) − 1)−1

(3.3.37) ,

(3.3.38)

we come to the Main theorem 5.1 from [C23]. Proposition 3.3.5. Given c ∈ P, 0 ≤ i ≤ n such that (αi , c + d) > 0, 1/2

Eb q −(b,b)/2 = ti τ+ (Φci )(Ec )q −(c,c)/2 for b = si ((c))

(3.3.39)

and the automorphism τ+ from (3.2.11). If (αi , c + d) = 0, then 1/2

τ+ (Ti )(Ec ) = ti Ec , 0 ≤ i ≤ n,

(3.3.40)

which results in the relations si (Ec ) = Ec as i > 0. For b = πr ((c)), where the indices r are from O0 , q −(b,b)/2+(c,c)/2 Eb = τ+ (πr )(Ec ) = Xωr q −(ωr ,ωr )/2 πr (Ec ).

(3.3.41) ❑

3.3. MACDONALD POLYNOMIALS

3.3.4

323

Some applications

We will reformulate the proposition using the spherical polynomials. One can also replace P by any lattice Q ⊂ B ⊂ P. The above considerations are compatible with the reduction P → B. Let def bb = E = τ+ (πr Gcill . . . Gci11 )(1), where

(3.3.42)

c1 = 0, c2 = si1 ((c1 )), . . . , cl = sil ((cl−1 )), for πb = πr sil . . . si1 . These polynomials do not depend on the particular choice of the decomposition of πb (not necessarily reduced) and are proportional to Eb for all b ∈ B: Eb q −(b,b)/2 =

Y¡

¢ 1/2 bb tip φip (q cp ) E

1≤p≤l

=

Y [α,j]∈λ0 (πb

³ 1 − q j t X (q ρk ) ´ α α α bb . E j ρk ) X (q 1 − q α α )

(3.3.43)

We conclude that bb . Eb = q (−ρk +b− ,b− )/2 E

(3.3.44)

Numerically, it is a straightforward combination of (3.3.43) and (3.3.16). As a matter of fact, it is a simple calculation with the intertwining operators (see [C23] and Corollary 3.3.6 below), so the evaluation formula is not needed here. As an application we obtain that, given b ∈ P, the spherical polynomial Eb is well defined for q, t ∈ C∗ if Y ¡ ¢ 1 − qαj tα Xα (q ρk ) 6= 0. (3.3.45) [α,j]∈λ0 (πb )

Similarly (see [C23], Corollary 5.3), the polynomial Eb is well defined if Y

¡

¢ 1 − qαj Xα (q ρk ) 6= 0.

[α,j]∈λ0 (πb )

If b = b− ∈ P− and the latter inequality holds for b+ = w0 (b− ) ∈ P+ , then the corresponding symmetric polynomial Pb− is well defined. Another application is the following variant of the evaluation formula (3.3.16), which requires only basic properties of the intertwining operators and readily leads to a justification of the evaluation formula. It can be used for arbitrary q, t. We follow Proposition 3.1.7.

324

CHAPTER 3. GENERAL THEORY

Corollary 3.3.6. Let us assume that Ec is well defined. We will introduce Eb using formula (3.3.39) provided that l(πb ) = 1 + l(πc ) for b = si ((c)), i ≥ 0, and using (3.3.41) for b = πr ((c)). Then (α ˜ ∨ , c− +d)

q

−(ρk ,b− )

Eb (q

−ρk

)=q

−(ρk ,c− )

Ec (q

−ρk

)

1 − qα

tα Xα (q ρk )

(α ˜ ∨ , c +d)

− 1 − qα Xα (q ρk ) α ˜ = uc (αi ), α = uc (αi ) for i > 0, α = uc (−ϑ) for i = 0,

q −(ρk ,b− ) Eb (q −ρk ) = q −(ρk ,c− ) Ec (q −ρk ) for b = πr ((c)).

,

(3.3.46)

(3.3.47) ❑

Symmetric polynomials. Recall that the connection of Pb− and {Eb } is given in (3.3.14) and (3.3.15). Using the G–intertwiners, the formulas read Pb− = Pb+ Eb+ = Gb+ Eb− , where X Y X Pb+ = tlνν (wc )/2 Twc , Gb+ = Guc . c∈W (b+ ) ν

(3.3.48)

c∈W (b+ )

These formulas are verifiedP as follows. def −1/2 on the left for One checks that G == c∈W Gw is divisible by Tj + tj −1/2 j > 0, i.e., G ∈ (Tj + tj ) · H for the nonaffine Hecke algebra H. Indeed, G is divisible by (sj + 1) on the leftP in the polynomial representation, where it becomes simply the symmetrizer w∈W w. Q l (w)/2 def P Tw . The The operator Pb+ for generic b+ becomes P == c∈W ν tνν −1/2 P is divisible by Tj + tj on the left. Moreover, due to the formula 1/2

−1/2 Tˆj + tj = (sj + 1)

−1/2

− tj tj Xα−1 j −1 Xα−1 j

,

the operator P is also divisible by (sj + 1) on the left in the polynomial representation. Recall that by Tˆj , we denote the action of Tj in V. See, e.g., [C16] (Corollary 4.7). This gives that P transfers any polynomials to symmetric ones. Thus P has the same divisibility property as G, and they must coincide up to proportionality as elements in H. Applying P or G to an arbitrary Eb with b ∈ W (b+ ), we obtain Pb− up to proportionality. The formula (3.3.48) reflects the normalization and that b can have a nontrivial W –stabilizer. These formulas can be used to deduce the evaluation formulas and the norm formulas (see Proposition 3.4.1 below and (3.4.3)) in the symmetric case from those for the E–polynomials. Actually, directly proving the symmetric formulas using the shift operator is not difficult either. We note that there is a similar expression for the anti-symmetric Macdonald polynomial Pb−− in [C20]. This polynomial belongs to the same space

3.4. POLYNOMIAL FOURIER TRANSFORMS

325

linearly generated by Ec (c ∈ W (b+ )) and satisfies the t–anti-symmetric re−1/2 lations (Tj + tj )Pb−− = 0 for all j > 0. The formula for Pb−− is useful in the theory of the shift operators.

3.4

Polynomial Fourier transforms

We will begin with the norm formula for the spherical polynomials. See [C23] (Theorem 5.6) and [M4, C16, M5, C20, M7].

3.4.1

Norm formulas

From here on, B will be any lattice between Q and P, {bi , 1 ≤ i ≤ n} a basis of B, and HH[ the corresponding intermediate subalgebra of HH . We also set c[ = B · W ⊂ W c, Qq,t [Xb ] = Qq,t [Xb , b ∈ B], and replace m by the least W m ˜ ∈ N such that m(B, ˜ B) ⊂ Z in the definition of the Qq,t . Proposition 3.4.1. For b, c ∈ B and the Kronecker delta δbc , bb , E bc i◦ = δbc µ−1 (q b] )µ(q −ρk ) = hEb , Ec i◦ = hE j −1/2 Y ³ t1/2 Xα (q ρk ) ´ α − qα t α δbc . 1/2 −1/2 − qαj tα Xα (q ρk ) [α,j]∈λ0 (πb ) tα

(3.4.1)

Proof. Using (3.3.42) and that τ+ is ?–unitary, bb , E bb i◦ = hτ+ (Gπ )(1) , τ+ (Gπ )(1)i◦ = hG?π Gπ (1) , 1i◦ . hE b b b b ❑ Hence we can apply (3.3.31), substituting Xα 7→ Xα (q −ρk ). Let us use the evaluation formulas to recalculate (3.4.1) in terms of the E–polynomials. One has: Y

hEb , Ec i◦ = δbc

[α,j]∈λ0 (πb

³ (1 − q j t−1 X (q ρk ))(t−1 − q j X (q ρk )) ´ α α α α α α . (3.4.2) j j ρ (1 − qα Xα (q k ))(1 − qα Xα (q ρk )) )

The corresponding formula for the symmetric polynomials, the Macdonald norm conjecture, proved in [C16] in full generality, reads as: hPb− , Pc− i◦ = δb− c−

(3.4.3) Y

−(α∨ ,b)−1

α>0

j=0

Y

ρk j ρk (1 − qαj+1 t−1 α Xα (q ))(1 − qα tα Xα (q )) . (1 − qαj Xα (q ρk ))(1 − qαj+1 Xα (q ρk ))

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CHAPTER 3. GENERAL THEORY

3.4.2

Discretization

This proposition will be interpreted as a calculation of the Fourier transform acting from the polynomial representation to the functional representation of HH[ . The functional representation depends on n independent parameters, the exponentials q ξi for {1 ≤ i ≤ n}. We set n n Y X ξ def li ξi Xa (q ) == q , where a = li bi ∈ B, i=1

i=1 b+w(ξ)

(a,b)

Xa (bw) = Xa (q ) = q Xw−1 (a) (q ξ ), uˆ(g)(bw) = g(ˆ u−1 bw), b ∈ B, w ∈ W.

(3.4.4)

Note that b + w(ξ) = (bw)((ξ)), using the affine action from (3.1.19). These formulas determine the discretization homomorphism g(X) 7→ δ(g)(w) b (depending on ξ) that maps the space of rational functions g ∈ Qq,t (X) or more general functions of X to def c [ , Qξ ), F[ξ] == Funct(W

c[ . formed by Qξ –valued functions on W def From here on, Qξ == Qq,t (q ξ1 , . . . , q ξn ). We also omit δ and write g instead of δ(g) when it is clear that F[ξ] is considered. This homomorphism can be naturally extended to the operator algebra def

u. A == ⊕uˆ∈W c [ Qq,t (X)ˆ ˆ of the operators H ˆ for H ∈ HH[ are For instance, the discretizations δ(H) [ well defined and we obtain the action of HH in Fξ . We will call the latter the functional representation. It contains the image of the polynomial representation as an HH[ –submodule. Explicitly, δ(Xa ) is (multiplication by) the discretization of Xa , δ(πr ) = πr , for πr ∈ Π[ , and 1/2

−1/2

t Xαi (q w(ξ) )q (αi ,b) − ti δ(Ti (g))(w)) b = i Xαi (q w(ξ) )q (αi ,b) − 1 1/2

g(si w) b

−1/2

ti − ti g(w) b for 0 ≤ i ≤ n, w b = bw. − Xαi (q w(ξ) )q (αi ,b) − 1

(3.4.5)

The map δ is an HH[ –homomorphism. The inner product corresponding to the anti-involution ? of HH[ is given by the discretization of the function µ• (X) = µ(X)/µ(q ξ ): µ• (w) b =

1/2 Y³ t−1/2 − qαj tα Xα (q ξ ) ´ α , [α, να j] ∈ λ(w). b 1/2 j −1/2 ξ) t − q t X (q α α α α α,j

(3.4.6)

3.4. POLYNOMIAL FOURIER TRANSFORMS

327

Actually, we will mainly use δ(µ• ), not µ• itself. The δ will be dropped in the formulas with δ(µ• ). def We put formally ξ ∗ = ξ. To be more exact, (q ξi )∗ == q −ξi for 1 ≤ i ≤ n. Then µ• (w) b ∗ = µ• (w). b The counterpart of h i is the sum X def f (w). b hf iξ == c[ w∈ b W

The corresponding pairing is as follows: X µ• (w)f b (w) b g(w) b ∗ = hg, f i∗• . hf, gi• = hf g ∗ µ• iξ =

(3.4.7)

c[ w∈ b W

Here f, g are from the HH[ –submodule of finitely supported functions F[ξ] ⊂ F[ξ]. Given H ∈ HH[ , hH(f ), gi• = hf, H ? (g)i• , hH(f ) gµ• iξ = hf η(H)? (g)µ• iξ .

(3.4.8)

To justify (3.4.8), one can follow the considerations in the case of h i or simply use the homomorphism δ. Also, (3.4.8) can be formally deduced from Proposition 3.3.1. Note that the operators Ti , Xb are unitary for hf, gi• and self-adjoint with respect to the pairing hf gµ• iξ . c[ ) The characteristic functions χwb and delta functions δwb in F[ξ](w b∈W are defined from the relations χwb (ˆ u) = δw,ˆ u) = µ• (w) b −1 χwb b u , δw b (ˆ for the Kronecker delta. We have the formulas: def c[ . Xb (χwb ) == δ(Xb )(χwb ) = Xb (w)χ b wb , b ∈ B, w b∈W

When considering concrete (non-generic) ξ ∈ Cn , one needs to check that all the above formulas are well defined (the denominators in (3.4.5) and (3.4.6) are nonzero). Note that µ• (w) b = 1 if q b+w(ξ) = q ξ , i.e., q (b+w(ξ),a) = q (ξ,a) for all a ∈ B, provided that the function µ• is well defined at q b+w(ξ) . However, it is not true, generally speaking, if (3.4.6) is used as a definition of µ• (w), b without any reference to the function µ• (X). The main example discussed in this chapter is the following specialization: ξ = −ρk . More precisely, q (ξ,b) 7→ q −(ρk ,b) for b ∈ B. The HH[ –module F(−ρk ) becomes reducible. Namely, def

def

I] == ⊕w6b∈πB Qq,t χwb ⊂ F(−ρk ) and F] == F(−ρk )/I]

(3.4.9)

328

CHAPTER 3. GENERAL THEORY def

are HH[ –modules. We denote h , i• by h , i] in this case. Respectively, h i] == h i−ρk . Since I] is the radical of this pairing, it is an HH[ –submodule, and so is F] . Here F] can be identified with the space of finitely supported Qq,t –valued def functions on πB == {πb , b ∈ B}. It is irreducible for generic q, t. The HH[ – module F] = Funct(πB , Qq,t ) is its natural completion. Note that Xa (πb ) = Xa (q b] ). Therefore πB can be naturally identified with the set B] = {b] = b − u−1 b (ρk )} for generic k. [ The HH –invariance of I] (see [C23] and [C20]) can be seen directly from (3.1.31) and the following general (any ξ) relations: 1/2

Ti (χwb ) =

(q w(ξ) )q −(αi ,b) − t−1/2 ti Xα−1 i χsi wb (q w(ξ) )q −(αi ,b) − 1 Xα−1 i 1/2

−1/2

ti − ti χwb for 0 ≤ i ≤ n, Xαi (q w(ξ) )q (αi ,b) − 1 c[ . for πr ∈ Π[ , w b = bw ∈W (3.4.10) −

πr (χwb ) = χπr wb ,

See Lemma 3.1.6 and formula (3.1.32) there to check that the expressions in the right-hand side are well defined. For further references, the delta-counterparts of these formulas are 1/2

t Xαi (q w(ξ) )q (αi ,b) − t−1/2 Ti (δwb ) = i δsi wb Xαi (q w(ξ) )q (αi ,b) − 1 1/2

−1/2

ti − ti δwb for 0 ≤ i ≤ n, − Xαi (q w(ξ) )q (αi ,b) − 1

(3.4.11)

and πr (δwb ) = δπr wb . The action of πr at the delta functions remains the same as for {χ} because µ• (πr w) b = µ• (w). b

3.4.3

Basic transforms

We introduce the Fourier transform ϕ◦ and the skew Fourier transform ψ◦ as follows: ϕ◦ (f )(πb ) = hf (X)Eb (X)µ◦ i, b ∈ B, ψ◦ (f )(πb ) = hf (X)∗ Eb (X)µ◦ i = hEb , f i◦ .

(3.4.12)

They are isomorphisms from the space of polynomials Qq,t [Xb ] 3 f onto F] . The ϕ◦ is Qq,t –linear; the ψ◦ conjugates q, t (sends q 7→ q −1 , t 7→ t−1 ). We will denote the characteristic and delta functions χwb , δwb ∈ F] for w b = πb by χπb and δbπ , respectively.

3.4. POLYNOMIAL FOURIER TRANSFORMS

329

Theorem 3.4.2. The Fourier transform ϕ◦ is unitary, i.e., it sends h , i◦ to h , i] , and induces the automorphism σ on HH[ : ϕ◦ (H(p)) = δ(σ(H))(ϕ◦ (p)) for p(X) ∈ Qq,t [Xb ], H ∈ HH[ . Respectively, ψ◦ is unitary too and induces the involution ε. Explicitly, X X ψ◦ : gb Eb (X) 7→ gb∗ δbπ ∈ F] , for gb ∈ Qq,t , (3.4.13) which is equivalent to the relations Eb (X) = ε(Gπb )(1) for b ∈ B,

(3.4.14)

and results in the formula X X Y π ϕ◦ : gb Eb (X) 7→ gb tlνν (w0 )/2 Tw−1 (δς(b) ) 0 ν

for gb ∈ Qq,t , ς(b) = −w0 (b).

(3.4.15)

Proof. Let us begin with ψ (the subscript “◦” will be dropped until the “noncompact” Fourier transforms appear). We use the definition of Eb , the duality formula (3.3.19), and the fact that the polynomial representation is ?–unitary (see (3.2.18),(3.3.5)). The duality relations are needed in the form of the Pieri rules from [C20], Theorem 5.4. Namely, we use that, given a ∈ B, Xa−1 Eb (X) is a transformation of Ya (δbπ ) when the delta functions δcπ are replaced by Ec (see (3.4.11)). This readily gives that ψ induces ε on the double Hecke algebra. Since ε is unitary and both the polynomial representation and F] are irreducible (for generic q, t), we obtain that ψ is unitary up to a constant. The constant is 1 because the image of E1 = 1 is δ0π = δid = χπ0 . Thus (3.4.13) is only a reformulation of (3.4.1). There is another way of checking it that is based on the technique of intertwiners. See [C23] (Corollary 5.2), Proposition 3.3.5 above, and formula (3.3.44). Turning to ϕ, we will use the involution η = εσ sending Ti 7→ Ti−1 (0 ≤ i ≤ n), Xb 7→ Xb−1 , and πr 7→ πr . It is an automorphism of HH[ , conjugating q, t. Recall that η extends the conjugation ∗ in the polynomial representation. Therefore the transformation of HH[ corresponding to ϕ is the composition εη = σ. This gives that ϕ is unitary. Concerning (3.4.14), we apply ψ to the relations δbπ = Gπb (δ0π ) for b ∈ B.

(3.4.16)

To obtain (3.4.15), we can combine (3.4.13) with (3.3.26), which states that Y (3.4.17) tν−lν (w0 )/2 Tw0 (Eς(b) ). Eb∗ = ν

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CHAPTER 3. GENERAL THEORY

Let us give another proof of (3.4.14) which does not require knowing the exact value of the coefficient of proportionality in Eb∗ = Cb Tw0 (Eς(b) ) and, moreover, automatically gives its value. What we have proved so far can be represented as follows: hEb Ec µ◦ i = Cb−1 h(Tw0 (Eb ))∗ Ec µ◦ i π (δς(b) )(πc ). = Cb−1 Tw−1 0

(3.4.18)

Here the left-hand side is b ↔ c –symmetric. On the other hand, Tw−1 (δbπ )(πc ) = hTw−1 (δbπ )δc𠵕 i] , 0 0

(3.4.19)

where the last expression is b ↔ c –symmetric too, thanks to formula (3.4.8). Therefore the multiplier Cb does not depend on b and must be exactly as in (3.4.17). ❑ Corollary 3.4.3. Given b, c ∈ P, the second norm formula, without the conjugation, reads as Y π hEb Ec µ◦ i = tlνν (w0 )/2 Tw−1 (δς(b) )(πc ). (3.4.20) 0 ν def Q −l (w )/2 The second inner product hhf , gii> == ν tν ν 0 hf , Tw0 (g ς )µ◦ i is symmetric for f, g ∈ Qq,t [X]. Here g ς (X) = g(w0 (X)−1 ). The corresponding antiinvolution ♥ is the composition

♥ = η · ? · Tw0 · ς = σ 2 · η · ? = σ · ε · ? = σ · φ

(3.4.21)

in the notation from (2.12.8), (2.12.10), (3.3.20), and (3.3.21), where by Tw0 . It sends we mean the conjugation Tw0 (·)Tw−1 0 Ti 7→ Ti (i ≥ 0), πr 7→ πr , Yb 7→ Yb , Xb 7→ Tw−1 Xς(b) Tw0 . 0

(3.4.22)

One then has: hhEb , Ec ii> = δbc µ−1 (q b] )µ(q −ρk ). Assuming that 0 < q < 1 and imposing the inequalities 1+(ρk , α∨ )+kα > 0 for all α ∈ R+ , this paring is positive and therefore the polynomial representation is irreducible. If ksht = k = klng , then the latter inequalities mean that k > −1/h for the Coxeter number h = (ρ, ϑ) + 1. ❑ We will introduce the conjugated Fourier transforms by “conjugating” (3.4.12): def

ϕ¯◦ (f )(πb ) == hf (X)Eb (X)∗ µ◦ i = hf , Eb i◦ , def ψ¯◦ (f )(πb ) == hf (X)∗ Eb (X)∗ µ◦ i, b ∈ B.

Let us “conjugate” Theorem 3.4.2.

(3.4.23)

3.4. POLYNOMIAL FOURIER TRANSFORMS

331

Theorem 3.4.4. The Fourier transforms ϕ, ¯ ψ¯◦ are unitary, i.e., transform h , i◦ to h , i] . They induce the automorphisms σ −1 and ησ on HH[ respectively, and send X

X

gb δbπ for b ∈ B, gb ∈ Qq,t , X X Y ψ¯◦ : gb Eb (X) 7→ gb∗ tν−lν (w0 )/2 Tw0 (δbπ ).

ϕ¯◦ :

gb Eb (X) 7→

(3.4.24)

ν

❑

3.4.4

Gauss integrals

We introduce the Gaussians γ ±1 as W –invariant solutions of the following system of difference equations: ωj (γ) = q (ωi ,ωi )/2 Xi−1 γ, ωi (γ −1 ) = q −(ωi ,ωi )/2 Xi γ −1

(3.4.25)

for 1 ≤ i ≤ n. In the current setting, we need the Gaussian as a Laurent series: X def q (b,b)/2 Xb . (3.4.26) γ˜ −1 == b∈B

Analytically, the values of this series at any q z (z ∈ C) are well defined if |q| < 1. However, we will not use this fact and instead treat γ˜ −1 as a formal series in this section. The multiplication by γ˜ −1 preserves the space of Laurent series with co˜ efficients from Q[[q 1/(2m) ]]. Recall that m ˜ ∈ N is the smallest positive integer −1 such that m(B, ˜ B) ⊂ Z. The γ˜ will be later multiplied by the q–expansions of Eb (X) and that of the µ◦ . The latter is a Laurent series with coefficients in Q[t][[q]], which makes the coefficients of γ˜ −1 µ◦ well defined. The q–expansions of the coefficients of Eb are rational functions in terms of q, t and their natural expansions belong to Q[t±1 ][[q]]. Theorem 3.4.5. Given b, c ∈ B and the corresponding spherical polynomials Eb , Ec , hEb Ec γ˜ −1 µ◦ i = q (b] ,b] )/2+(c] ,c] )/2−(ρk ,ρk ) Ec (q b] )h˜ γ −1 µ◦ i,

(3.4.27)

hEb Ec∗ γ˜ −1 µ◦ i = q (b] ,b] )/2+(c] ,c] )/2−(ρk ,ρk ) Ec∗ (q b] )h˜ γ −1 µ◦ i,

(3.4.28)

hEb∗ Ec∗ γ˜ −1 µ◦ i

(b] ,b] )/2+(c] ,c] )/2−(ρk ,ρk )

= q Y γ −1 µ◦ i, × tν−lν (w0 )/2 Tw0 (Ec∗ )(q b] )h˜ ν

(3.4.29)

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CHAPTER 3. GENERAL THEORY

where the coefficients of µ◦ , Eb , Ec , and Ec∗ are expanded in terms of positive powers of q. Here the coefficient of proportionality, a q–generalization of the Mehta–Macdonald integral, can be calculated explicitly: −1

h˜ γ µ◦ i =

∞ ³ (ρ ,α∨ )+j ´ Y Y 1 − t−1 qα k α

(ρ ,α∨ )+j

α∈R+ j=1

1 − qα k

.

(3.4.30)

ˆ γ˜ −1 = γ˜ −1 τ+ (H) ˆ in the polyThe proof is based on the following fact: H nomial representation extended by the Gaussian, where H ∈ HH and the ˆ is from Theorem 3.2.1. Indeed, the conjugation by γ˜ corremapping H 7→ H sponds to τ+ on the standard generators Xb (b ∈ B), Ti (0 ≤ i ≤ n), πr (r ∈ O0 ). ˆ γ˜ −1 coincides with γ˜ −1 (τ+ (H))ˆ for all H ∈ HH[ . It can To be more exact, H be readily deduced from the W –invariance of γ˜ and (3.4.25). The same holds in the functional representations F[ξ] if we take def

γ ±1 (bw) == q ±(b+w(ξ),b+w(ξ))/2 , b ∈ B, w ∈ W.

(3.4.31)

Here we need to extend the field of constants Qξ by q ±(ξ,ξ)/2 . We can avoid such an extension by removing ±(ξ, ξ)/2 from the exponent of (3.4.31) because we need the Gaussian only up to proportionality. However, we prefer to stick to the standard formula γ(q z ) = q (z,z)/2 in this section. Involving Theorem 3.4.2, we conclude that the map ϕγ : f (X) 7→ f 0 (πb ) = γ −1 (b] )hf Eb γ˜ −1 µ◦ i induces the involution τ+−1 στ+−1 = τ−−1 on HH[ . Cf. Proposition 3.3.4. Note that γ(b] ) is nothing but γ(πb ) evaluated at ξ = −ρk . The map ϕγ acts from Qq,t [Xb ] to the HH[ –module F] , where the field of constants is extended by q −(ξ,ξ)/2 . The automorphism τ− fixes the Y –operators. Hence the image of f = Ec is an eigenfunction of the discretizations δ(Ya ) of the Ya (a ∈ B) corresponding to the same set of eigenvalues as for Ec . Let us prove that ϕγ (Ec ) has to be proportional to the discretization δ(Ec ) of Ec . In the first place, it suffices to consider c = 0 because we may employ the Y –intertwiners (see (3.3.42)). Second, the images of the Y –intertwiners with respect to τ−−1 can be calculated exactly. We do not need explicit formulas here. It is sufficient to know that the intertwiners are invertible operators acting in F] . Third, the function g = ϕγ (1) has additional symmetries: −1/2 ti δ(Ti )(g) = g = δ(πr )(g). We have used that τ− fixes Ti and πr for all c [ –invariance of g, which means i, r (see (3.2.16)). This readily leads to the W that it has to be constant. Now, setting ϕγ (Ec ) = hc γ(πc )q −(ρk ,ρk ) δ(Ec ) for hc ∈ Qq,t , we need to check that hc = 1. It obviously holds for h0 . Since the left-hand and the right-hand

3.4. POLYNOMIAL FOURIER TRANSFORMS

333

sides of (3.4.27) are b ↔ c symmetric, hc = hb for all c, b ∈ B and hc = 1. Here we used the duality formula (3.3.19). The second formula can be easily deduced from the first thanks to (3.4.17): Y Ec∗ = tν−lν (w0 )/2 Tw0 (Eς(c) ), ς(c) = −w0 (c). ν

Indeed, Tw0 commutes with γ and its Fourier transform ϕγ (Tw0 ) is the discretization of σ(Tw0 ) = Tw0 , which leads to (3.4.29). The formula (3.4.17) and its conjugation will be important when we construct the inverse Fourier transform in the next section. A direct proof of (3.4.28), without the conjugation formula, is not difficult either. Here it is. First, Ec∗ are eigenfunctions of the operators η(Ya ), where η = εσ (see above). The images of these operators under ϕγ are τ−−1 (η(Ya ). One has: τ−−1 η = τ−−1 εσ = ετ+−1 σ = ετ−−1 τ+ = εστ− = ητ− . Therefore ϕγ fixes η(Ya ) and ϕγ (Ec∗ ) is proportional to δ(Ec∗ ) for any a, c ∈ B. ˆ c γ(πc )q −(ρk ,ρk ) δ(E ∗ ). However, ˆ from ϕγ (E ∗ ) = h Second, we introduce h c c now it is not obvious that the left-hand side of (3.4.28) is symmetric (a ↔ b), as it was for (3.4.27). We need to involve ψγ : f (X) 7→ γ −1 (b)hf ∗ Eb γ˜ −1 µ◦ i. It induces the automorphism ε˜ = τ+−1 ετ+ . Indeed, hf ∗ Eb γ˜ −1 µ◦ i = h(f γ˜ )∗ Eb µ◦ i. Since η(Ya ) = ετ−−1 τ+ τ−−1 (Ya ) = τ+−1 ετ+ (Ya ) = ε˜(Ya ), ψγ (Ec ) is proportional to δ(Ec∗ ). Therefore, letting ˜ c γ(πc )q −(ρk ,ρk ) δ(E ∗ ), ψγ (Ec∗ ) = h c ˆc = h ˜ b for all b, c ∈ B. Hence both functions h, ˆ h ˜ are we conclude that h constants and, moreover, equal 1, thanks to the normalization of Eb . The explicit formula for h˜ γ −1 µ◦ i was calculated in [C21] using the shift operators and the analytic continuation. ❑ Comment. (i) The product µ˜ γ −1 generalizes the (radial) Gaussian measure in the theory of Lie groups and symmetric spaces. To be more exact, the “noncompact” case, which will be considered next, is such a generalization. The above setup can be called “compact,” because taking the constant term corresponds to integration with respect to the imaginary period.

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CHAPTER 3. GENERAL THEORY

(ii) We would like to mention that the appearance of the nonsymmetric polynomials has no known counterparts in the classical representation theory even in the so-called group case when k = 1, i.e., t = q. In this case, the symmetric Macdonald polynomials become the characters of the compact Lie groups, and one can expect the nonsymmetric polynomials to be somehow connected with these representations. However, it does not happen. There are certain relations of the (degenerate) nonsymmetric polynomials with the Demazure character formulas (the Kac–Moody case, essentially basic representations). However, it looks more accidental than conceptual. It merely reflects the fact that both constructions are based on the Demazure operations. (iii) In the group case, the symmetrizations of (3.4.27) and (3.4.28) can be readily deduced from the Weyl character formula. It seems that these formulas were not used in harmonic analysis, possibly, because they cannot be extended to the Harish-Chandra zonal spherical functions on general symmetric spaces.

3.5

Jackson integrals

The formulas from the previous sections can be generalized for Jackson integrals taken instead of the constant term functional (corresponding to the imaginary integration over the period). It is a variant of the classical noncompact case. Another variant is a straightforward analytic integration in the real direction, which will not be discussed. We keep the same notation, expend all functions in terms of non-negative powers of q, and consider q (ξ,bi ) as independent parameters. One may also treat ξ as a complex vector in a general position, assuming that |q| < 1. The Jackson integral or Jackson sum of f (bw) ∈ F[ξ] is def

hf iξ ==

X

f (bw), where b ∈ B, w ∈ W.

Recall that Xa (q z ) = q (a,z) , Xa (bw) = q (a,w(ξ)+b) , γ(q z ) = q (z,z)/2 , and γ(bw) = γ(q w(ξ)+b ). Thus |W |−1 hγiξ =

X

q (ξ+a,ξ+a)/2 = γ˜ −1 (q ξ )q (ξ,ξ)/2

a∈B

for γ˜ from (3.4.26). The Jackson inner product already appeared in (3.4.7). It is hf, gi• = hf g ∗ µ• iξ for µ• from (3.4.6). To be more precise, here we need the discretization δ(µ• ).

3.5. JACKSON INTEGRALS

3.5.1

335

Jackson transforms

The Fourier–Jackson transform ϕ• , the skew Fourier–Jackson transform ψ• , and their bar-counterparts are as follows: ϕ• (f )(πb ) ψ• (f )(πb ) ϕ¯• (f )(πb ) ψ¯• (f )(πb )

= = = =

hf (w)E b b (w)µ b • iξ , b ∈ B, ∗ hf (w) b Eb (w)µ b • iξ = hEb , f i• , ∗ hf (w)E b b (w)µ b • (w)i b ξ , b ∈ B, ∗ ∗ hf (w) b Eb (w)µ b • (w)i b ξ = hf, Eb i• .

(3.5.1) (3.5.2)

These transforms act from subspaces of F[ξ] to proper completions of the space F] , provided that we have the convergence. The involution ∗ is the conjugation of the values of functions f ∈ F[ξ]. It is well defined because the values are rational functions in terms of q, t (and their certain fractional powers). One has: (f ∗ )(w) b = (δ(f ∗ ))(w) b = (δ(f )(w)) b ∗ = (f (w)) b ∗ . For instance, χ∗wb = χwb , and δw∗b = δwb since µ• (w) b ∗ = µ• (w). b Thus in (3.4.12) and (3.5.1), we ∗ ∗ can replace Eb (w) b by Eb (w) b . It is straightforward to check that the corresponding automorphisms of HH[ are the same as in Theorems 3.4.2 and 3.4.4. We simply replace h i by h iξ : ϕ ↔ σ, ϕ¯ ↔ σ −1 , ψ ↔ ε, ψ¯ ↔ ησ.

(3.5.3)

These automorphisms commute with the anti-involution ?, so all transforms are ?–unitary up to a coefficient of proportionality provided we have the convergence and the irreducibility of the corresponding HH[ –modules. Here ξ is generic. We can be more precise for the spherical specialization. Theorem 3.5.1. (i) For ξ = −ρk , the Fourier transforms from (3.4.12) and (3.5.1) act from the space of the delta functions F] to the discretization of the space Qq,t [Xb ] upon restriction to the set πB . They send the form h , i] to h , i◦ and satisfy the relations ϕ¯◦ · ϕ• ϕ¯• · ϕ◦ ψ◦ · ψ• ψ• · ψ◦

= id = ϕ◦ · ϕ¯• , = id = ϕ• · ϕ¯◦ , = id = ψ¯◦ · ψ¯• , = id = ψ¯• · ψ¯◦ .

(3.5.4)

(ii) Setting fb = ϕ◦ (f ) ∈ F] for f (X) ∈ Qq,t [Xb ], the inverse transform reads as Y (3.5.5) f (X) = tν−lν (w0 )/2 h fb Tw•0 (Eς(b) )µ• i] , ν

336

CHAPTER 3. GENERAL THEORY

where T • acts on Eb via formula (3.4.11) for δbπ = δπb : 1/2

Ti• (Eb ) =

1/2

−1/2

ti q (αi ,b] ) − t−1/2 ti − ti E Eb , si (b) − (α ,b ) (α ,b ) q i ] −1 q i ] −1

(3.5.6)

extended to W naturally. Similarly, we have the Plancherel formula hf gµ◦ i = tν−lν (w0 )/2 hfbTw0 (b g ς )µ• i] for f, g ∈ Qq,t [Xb ], h(w) b ς = h(ς(w)). b

(3.5.7)

Proof of (i) is based on (3.5.3) and the irreducibility of the polynomial representation and its delta-counterpart F] . The same argument gives the unitarity of the Fourier transforms under consideration up to proportionality. Thanks to the normalization of µ◦ and µ• , hδ0 , δ0 i] = 1 = h1, 1i◦ . So the coefficient of proportionality is 1. The explicit inversion formula from (ii) is straightforward: Y ϕ◦ ( tν−lν (w0 )/2 hfb Tw•0 (Eς(b) )µ• i] ) ν

=

Y

tν−lν (w0 )/2 hfb Tw•0 ϕ◦ (Eς(b) )µ• i]

ν

Y = hfb Tw•0 ( tν−lν (w0 )/2 ϕ◦ (Eς(b) ))µ• i] ν

= hfb Tw0 Tw−1 (δb )µ• i] = fb(πb ). 0

(3.5.8)

Here we use (3.4.15). Let us check the Plancherel formula. The anti-involution corresponding to the symmetric form hf gµ◦ i is def

¦ == η · ? = ? · η. The one for the form hfbTw0 (b g ς )µ• i] can be readily calculated as follows: h fb Tw0 (ϕ◦ (H(g))ς )µ• i] = h fb Tw0 σ(H ς )(b g ς )µ• i] =h (Tw σς(H)T −1 )¦ (fb) Tw (b g ς )µ• i] w0

0

0

=h (σ (H)) (fb) Tw0 (b g ς )µ• i] =h σ(H ¦ )(fb) Tw0 (b g ς )µ• i] = h (H ¦ (f ))b Tw0 (b g ς )µ• i] . −1

¦

Here we use that the Fourier transform ϕ induces σ on HH , the relations (3.2.22): Tw0 σς(H)Tw−1 = σ −1 (H), 0

3.5. JACKSON INTEGRALS

337

and (see the last line) the formula σ ¦ = ¦ · σ · ¦ = ηση = εσε = σ −1 . ❑ Part (ii) of the theorem generalizes the main theorem about the p–adic spherical transform due to Macdonald (symmetric case) and Matsumoto (see [Ma] and the recent [O4, O5]). The classical p–adic spherical transform acts from F] to the polynomials. So we need to reverse (ii). The Eb∗ generalize the p–adic (nonsymmetric) spherical functions due to Matsumoto. The limit q → ∞ of the following corollary is exactly the theory of the spherical Fourier transform. The case of generic ξ (in place of −ρk ) is presumably connected with the general (non-spherical) Fourier transform on affine Hecke algebras [KL1] and is expected to be directly related to [HO2]. Note that the latter paper is devoted to affine Hecke algebras with arbitrary labels. Only equal labels are considered in [KL1]. There are Lusztig’s papers towards nonequal labels. However, the general labels are difficult to interpret geometrically. Corollary 3.5.2. Setting fb = ϕ¯• (f ) ∈ Qq,t [Xb ] for f ∈ F] , its inversion and the Plancherel formula are as follows: Y ∗ f = tνlν (w0 )/2 h fb Tw−1 (Eς(b) )µ◦ i, (3.5.9) 0 ν

h f gµ• i] = tlνν (w0 )/2 h fb Tw−1 (b g ς )µ◦ i 0 for f, g ∈ Qq,t [Xb ], h(Xb )ς = h(Xw−10 (b) ).

3.5.2

(3.5.10)

Gauss–Jackson integrals

In a sense, these integrals are the missing part of the Harish-Chandra theory of spherical functions and the theory of the p–adic spherical transform. The exact formulas for the integrals of the Gaussians with respect to the Harish-Chandra (zonal) transform exist in the group case only (k = 1). The Gaussians cannot be added to the p–adic theory too. Let us transfer Theorem 3.4.5 from the compact case, with hi as the integration, to the Jackson case. The proof remains essentially unchanged. In fact, the formulas below are ∗–conjugations of (3.4.27) with a minor reservation about (3.4.30). See also Theorem 7.1 from [C21]. Theorem 3.5.1. Given b, c ∈ P and the corresponding spherical polynomials Eb , Ec , hEb Ec∗ 㵕 iξ = q −(b] ,b] )/2−(c] ,c] )/2+(ρk ,ρk ) Ec (q b] )h㵕 iξ ,

(3.5.11)

hEb∗

(3.5.12)

Ec∗

㵕 iξ =

q −(b] ,b] )/2−(c] ,c] )/2+(ρk ,ρk ) Ec∗ (q b] )h㵕 iξ ,

338

CHAPTER 3. GENERAL THEORY hEb Ec 㵕 iξ = q −(b] ,b] )/2−(c] ,c] )/2+(ρk ,ρk ) Y × tνlν (w0 )/2 Tw−1 (Ec )(q b] )h㵕 iξ , 0

(3.5.13)

ν ξ

−1

−1

h㵕 iξ =µ(q , t ) |W | hγiξ

∞ ³ −(ρk ,α∨ )+j ´ Y Y 1 − t−1 α qα α∈R+ j=0

−(ρk ,α∨ )+j

1 − qα

.

(3.5.14)

Here by µ(q ξ , t−1 ) we mean the right-hand side of (3.3.1) evaluated at X = q ξ ❑ where all tα are replaced by t−1 α . In these formulas, t is generic or complex, provided that the discretization δ(µ• ) = µ• (w) b is well defined and the polynomials Eb , Ec exist. Note that the right-hand side of (3.5.14) has to be replaced by the limit when kν ∈ Z+ . Let us state the inversion theorem and the corresponding Plancherel formulas. The pairings h f gµ◦ i and h f gµ• iξ can be naturally extended to the HH[ –modules Qq,t [Xb ]˜ γ −1 and the discretization δ(Qq,t [Xb ]γ) ⊂ F[ξ] of Qq,t [Xb ]γ. We will use these pairings in the Plancherel formulas. We set cξ = h㵕 iξ . The Fourier transforms ϕc = c−1 ¯c• = c−1 ¯• ξ ϕ• and ϕ ξ ϕ transfer δ(Qq,t [Xb ]γ) → Qq,t [Xb ]˜ γ −1 . Respectively, ψ•c , ψ¯•c preserve the first module. In the compact case, setting c = hγµ◦ i, ϕc◦ = c−1 ϕ◦ , ϕ¯c◦ = c−1 ϕ¯◦ , ψ◦c = c−1 ψ◦ , ψ¯◦c = c−1 ψ¯◦ , where the first two transforms act from the second module Qq,t [Xb ]˜ γ −1 to the first and the last two preserve Qq,t [Xb ]˜ γ −1 . They satisfy the counterparts of the inversion formulas (3.5.4): ϕ¯c◦ · ϕc• ϕ¯c• · ϕc◦ ψ◦c · ψ◦c ψ•c · ψ•c

= id = ϕc◦ · ϕ¯c• , = id = ϕc• · ϕ¯c◦ , = id = ψ¯◦c · ψ¯◦c , = id = ψ¯•c · ψ¯•c .

(3.5.15)

It is straightforward to reformulate part (ii) of Theorem 3.5.1 for ϕc and the pairings h f gµ◦ i on Qq,t [Xb ]˜ γ −1 and h f gµ• iξ on δ(Qq,t [Xb ]γ). Theorem 3.5.1 and the above facts remain valid when ξ = −ρk . In this case, µ = µ(k) =

∞ Y Y α∈R+ i=0

(1 − Xα qαi )(1 − Xα−1 qαi+1 ) , (1 − Xα tα qαi )(1 − Xα−1 tα qαi+1 )

3.5. JACKSON INTEGRALS µ(q

−ρk

−1

,t ) =

µ• (bw) = µ• (q

−(ρk ,α∨ )+i

∞ Y Y α∈R+ i=0 −w(ρk )+b

339

(1 − qα (1 −

−(ρk ,α∨ )+i )(1 t−1 α qα

Y

)=

(ρ ,α∨ )+i+1

)(1 − qα k −

)

(ρk ,α∨ )+i+1 ) t−1 α qα

³ t−1/2 − t1/2 q (α∨ ,ρk )+j ´

[α,j]∈λ0 (bw)

α 1/2 tα

−

α α −1/2 (α∨ ,ρk )+j t α qα

,

,

(3.5.16) (3.5.17)

˜ + ∩ (bw)−1 (R ˜ + ). The for λ0 (bw) = {[−α, j] | [α, να j] ∈ λ(bw)}, λ(bw) = R (for generic function µ• (bw) is well defined. It is nonzero only as πb = bu−1 b q, t). See (3.3.18) and (3.4.6). For w b = πb , all α in the product (3.5.17) are positive. Formula (3.5.14) now reads as follows: −1

h㵕 i] = |W | hγi]

∞ ³ (ρ ,α∨ )+j ´ Y Y 1 − qα k (ρ ,α∨ )+j

α∈R+ j=1

k 1 − t−1 α qα

.

(3.5.18)

Recall that h iξ for ξ = −ρk is denoted by h i] . The inversion formulas for ϕ are ϕ¯◦ · ϕ• = |W |−1 hγi] id = ϕ◦ · ϕ¯• , ϕ¯• · ϕ◦ = |W |−1 hγi] id = ϕ• · ϕ¯◦ .

3.5.3

(3.5.19)

Macdonald’s eta-identities

As a by-product, we can represent hγi] as a theta-like product times a certain finite sum, provided that the left-hand side of (3.5.18) contains finitely many nonzero terms. It happens when a certain Z+ –linear combination of ksht , klng is from −N. The main example (which will be used later) is as follows. Let q be generic such that |q| < 1. def We call a root α ∈ R+ extreme if hα (k) == (ρk , α∨ )+kα does not coincide with any (ρk , β ∨ ) for positive roots β, and strongly extreme if also (α, ωi ) > 0 for all 1 ≤ i ≤ n. Note that hϑ (k) is kh for the Coxeter number h = (ρ, ϑ)+1 if ksht = klng . Here the ksht , klng are treated as independent parameters. We omit the subscript of k in the simply-laced case: ρk = kρ, hα (k) = ((ρ, α) + 1)k. In this case, there is only one extreme root, namely, ϑ. Let us list the extreme roots for the other root systems (the notation is from [Bo]). All of them are short: Bn ) all short roots ²i (1 ≤ i ≤ n), where h²i (k) = 2ksht + 2(n − i)klng ; Cn ) ϑ = ²1 + ²2 with hϑ (k) = 2(n − 1)ksht + 2klng and α = ²1 − ²n with hα (k) = n ksht ; F4 ) 0011, 0121, 1121, ϑ = 1232 with h equal to 3ksht , 4ksht +2klng 4ksht +4klng , 6ksht + 6klng respectively; G2 ) α1 (hα1 (k) = 2ksht ), ϑ = 2α1 + α2 (hϑ (k) = 3(ksht + klng )).

340

CHAPTER 3. GENERAL THEORY

Thus ϑ is a unique strongly extreme root for all root systems except for F4 . The root 1121 for F4 is strongly extreme too. There are three special cases: (a) klng = ksht , (b) ksht = 0, (c) klng = 0. Under either constraint, ϑ becomes a unique strongly extreme root. The next theorem can be extended to these cases. Theorem 3.5.3. (i) Let us assume that q is generic, |q| < 1, and (ρk , α∨ ) 6∈ Z \ {0} for all α ∈ R+ .

(3.5.20)

These conditions result in (ρk , α∨ ) − kα 6∈ Z \ {0} for α ∈ R+ , and therefore are sufficient for the right-hand side of (3.5.18) and the sum in the definition of h㵕 i] to exist and be nonzero. The sum h㵕 i] is finite if and only if hα (k) ∈ −N for a strongly extreme root α. (ii) If hϑ (k) = −1 and (3.5.20) is satisfied (which automatically holds as ksht = klng ), the measure µ• (bw) takes exactly |Π[ | nonzero values. All of them equal 1 and ³X ´ X hγi] = q (b+ρk ,b+ρk )/2 = q (ωr +ρk ,ωr +ρk )/2 × b∈B

πr ∈Π[

∞ ³ (ρ Y Y 1 − t−1 qα k

,α∨ )+j

α

(ρ ,α∨ )+j

α∈R+ j=1

1 − qα k

´ , where ω0 = 0.

(3.5.21)

In the exceptional case {R = F4 , α = 1121, hα (k) = −1}, the number of nonzero values of µ• (bw) is greater than |Π| = 1. Proof. Concerning the first claim, (ρk , α∨ ) − kα is zero for simple α = αi . If α is not simple, then there exists a simple root αi of the same length as α > 0 such that α∨ −αi∨ is a positive coroot β ∨ . Then (ρk , β ∨ ) = (ρk , α∨ )−kα cannot be from Z \ {0} due to (3.5.20). Generally, there must be at leastPone root α satisfying hα (k) ∈ −N to (b] ,b] )/2 µ• (πb ) finite. Assumption make the Jackson sum h㵕 i] = b∈B q (3.5.20) gives that at least one such α has to be strongly extreme. Note that if there are two such roots α 6= β (this may happen for F4 only), then hβ (k) − hα (k) ∈ Z, which contradicts assumption (3.5.20). We obtain that the above sum is finite. Indeed, recall that µ• (id) = 1 and µ• (πb ) =

1/2 (α∨ ,ρ )+j Y³ t−1/2 − tα qα k ´ α

for α ∈ R+ , (3.5.22) 1/2 −1/2 (α∨ ,ρ )+j t α − t α qα k ∨ −(b− , α∨ ) > j > 0 if u−1 b (α) ∈ R− , −(b− , α ) ≥ j > 0 otherwise.

3.5. JACKSON INTEGRALS

341

See (3.5.17) and (3.3.18). Let us assume now that α = ϑ, hϑ (k) = −1, and b 6= 0. The case of F4 , α = 1121 is left to the reader. Then µ• (πb ) can be nonzero only for b from the W –orbits of the minuscule weights b− = −ωr . Indeed, otherwise −(b− , ϑ) ≥ 2, [ϑ, 1] ∈ λ0 (πb ), and the ∨ product (3.5.22) is zero. Moreover, u−1 b sends the roots {αi , i 6= r, i > 0} to −1 ∨ ∨ , since it is minimal such that u−1 R+ b (ωr ) = b− . Also, ub (−ϑ) ∈ R+ to make the product (3.5.22) nonzero. However, {−ϑ, αi∨ , i 6= r, i > 0} form a basis of R∨ . Therefore these conditions determine u−1 uniquely, and it has to coincide with u−1 b b+ . Thus b = b+ = w0 (−b− ) = ωr∗ , the λ0 –set of πb+ = πr∗ is empty, and µ• (πr∗ ) = 1. See (3.2.5). To conclude, we need to evaluate the Gaussian, which is performed as follows: −1 ((ωr )] , (ωr )] ) = (ωr − u−1 r (ρk ) , ωr − ur (ρk )) = (ωr∗ + ρk , ωr∗ + ρk ).

❑ Comment. (i) There is another method of proving (3.5.21). It is based on the symmetrization of (3.5.18) due to [C21], formula (1.11). Namely, we use the formulas: ³X ´ X (b− −ρk , b− −ρk )/2 (b+ρk , b+ρk )/2 q ∆• (b− ) = q (3.5.23) b− ∈B−

×

Y α∈R+

=

b∈B

∞ ³ Y j=1

(ρ ,α∨ )+j ´ 1 − qα k , (ρk ,α∨ )+j 1 − t−1 α qα

∆• (b− ) =

(α∨ , ρ −b ) 1/2 (α∨ , ρ )+j−1 Y³ (t−1/2 )(1 − qα k − ) ´ − tα qα k α 1/2

−1/2 (α∨ , ρk )+j )(1 qα

(tα − tα

(α∨ , ρk )

− qα

)

.

(3.5.24)

The latter product is over the set from (3.5.22) for b = b− , i.e., the relations −(b− , α∨ ) ≥ j > 0 for all α ∈ R+ must hold. In (3.5.24), only b− that equals −ωr have nonzero ∆• (b− ). Thus, it is sufficient to check that the latter expression is 1 for such a b− (which is true). This approach seems convenient for computing explicit formulas when more general hϑ (k) = −m (m > 1) are considered. Such formulas are generalizations of (3.5.21). (ii) We mention that when hα (k) = −m for an extreme but not strongly extreme root α, then the sum for h㵕 i] and, equivalently, the sum in (3.5.23) are not finite. However, the corresponding variant of the theorem allows us to reduce the summation dramatically. Also note that we have one free parameter ksht in the case of B, C, G, F, provided that it is in a general position. For instance, ϑ = ²1 for Bn and there is only one constraint hϑ (k) = 2ksht + 2(n − 1)klng = −1, so ksht can be arbitrary.

342

CHAPTER 3. GENERAL THEORY

(iii) Formula (3.5.21) is one of the modifications of the Macdonald identities [M1] closely related to the Kac–Moody algebras (cf. [Ka], Ch.12). In Section 3.10 of this chapter, we will connect it with one-dimensional representations of HH[ .

3.6

Semisimple representations

In this section we begin with representations HH[ with cyclic vectors and find out when they are semisimple and pseudo-unitary. The latter means the existence of a nondegenerate hermitian ?–invariant form. We add here “pseudo” because the form is not supposed to be positive, and, moreover, the involution ∗ acts on the constants via the true complex conjugation only as |q| = 1 for real k. Otherwise this conjugation is understood formally. By definite, we mean a form with nonzero squares of the eigenfunctions with respect to either X– or Y –operators. So it is a certain substitute for the positivity. We give necessary and sufficient conditions for induced modules to possess nonzero semisimple and pseudo-unitary quotients. We will begin this section with several general definitions, mainly concerning the semisimplicity. To put things in perspective, let us now comment on the relations to the theory of affine Hecke algebras. Comment. It is worth mentioning that the main theorem generalizes the classification of semisimple representations of affine Hecke algebra of type A for generic t. See, e.g., [C5](Section 3), references therein, and [Na1, Na2]. In the A–case, the classification of the semisimple representations is clarified in full, especially for the degenerate affine Hecke algebra. The theory is directly related to the classical Young’s bases. There are also partial results for other classical root systems in the author’s works and in [Ram] (A. Ram also considered some special systems, for instance, G2 , F4 ). Still there is no complete description of the semisimple representation apart from the A–case. Generally, one can use the classification of all irreducible representations from [KL1]. However, checking the semisimplicity is far from immediate if the “geometric” approach is used. Actually, the importance of semisimple representations is somewhat doubtful from the p–adic viewpoint, in spite of the interesting combinatorial theory. The main reason is that the Bernstein–Zelevinsky Y –operators are not normal with respect to the natural unitary structure that comes from the p–adic theory. The theory of double affine Hecke algebras, especially at |q| = 1, does require semisimple representations. In many interesting representations, either X–operators or Y –operators are normal, although there are important representations that are neither. The case of GLn will be considered below. ❑

3.6. SEMISIMPLE REPRESENTATIONS

343

The notation is from the previous sections. All HHb –modules (the lattice B is fixed) will be defined over the field Qq,t or over its extension Qξ = Qq,t (q ξ1 , . . . , q ξn ), where q ξi = q (bi ,ξ) for a basis {bi } of B. Here q, t, ξ can be generic as above or arbitrary complex numbers. In the latter case, Qξ becomes a proper subfield of C. We must provided the existence of the involution ∗ in the algebras Qq,t ⊂ Qξ taking ˜

˜

q 1/2m 7→ q −1/2m , t1/2 7→ t−1/2 , q ξi 7→ q −ξi . ν ν This involution is a restriction of the anti-involution (3.2.18) of HH[ .

3.6.1

Eigenvectors and semisimplicity

A vector v satisfying Xa (v) = q (a,ξ) v for all a ∈ B

(3.6.1)

is called an X–eigenvector of weight ξ. Respectively, a vector v satisfying Ya (v) = q (a,ξ) v for all a ∈ B

(3.6.2)

is called a Y –eigenvector of weight ξ. We set def

VXs (ξ) == {v ∈ V | (Xa − q (a,ξ) )s (v) = 0 for a ∈ B}, VX (ξ) = VX1 (ξ), VX∞ (ξ) = ∪s>0 VXs (ξ).

(3.6.3)

They are called the spaces of generalized eigenvectors. Taking Ya instead of Xa and (3.6.2) in place of (3.6.1), we introduce the spaces VYs (ξ). c on the weights is affine: w(ξ) The action of the group W b = w((ξ)). b See [ c –stabilizer of ξ is (3.1.19). The W def b , a) c [ [ξ] = c [ , q (w((ξ))−ξ W = {w b∈W = 1 for all a ∈ B}. 0

(3.6.4)

From now on weights ξ and ξ 0 will be identified if q (a,ξ −ξ) = 1 for all 0 a ∈ B. We put q ξ = q ξ if it is true. The category OX is formed by the modules V such that V = ⊕ξ VX∞ (ξ) and the latter spaces are finite dimensional, where the summation is over different q ξ . This is supposed to hold for a proper extension of the field of constants. In the definition of OY , we substitute Y for X. Using the decomposition (3.2.25) from the Poincare–Birkhoff–Witt (PBW) property from Theorem 3.2.1, one can check that HH[ –modules V from OX ∩ OY with finitely many generators are finite dimensional. Indeed, the space def U˜ == Qq,t [Yb ]U is finite dimensional for any finite dimensional subspace U of

344

CHAPTER 3. GENERAL THEORY

[ such a module. If U is preserved by the affine Hecke subalgebra HX generated by Qq,t [Xb ] and Ti (1 ≤ i ≤ n) then U˜ is a finite sum of the spaces HH[ u for proper u ∈ U. Therefore it is an HH[ –submodule of V. Since V is finitely generated, we obtain the required fact. As a matter of fact, it is not necessary to assume in this reasoning that the spaces VX∞ (ξ) and VY∞ (ξ) are finite dimensional. If we know that V is finitely generated, then it is sufficient to use that

⊕ξ VX∞ (ξ) = V = ⊕ξ VY∞ (ξ). Intertwiners. If the X–weight of v is ξ, then Φwb v is an X–eigenvector of weight w((ξ)) b for the X–intertwiners Φ from (3.3.27); we also can take Swb or Gwb instead of Φwb because both are proportional to Φwb . Here the denominators of Φ, S, or G have to be nonzero upon evaluation at q ξ if the latter is not supposed generic. Given a reduced decomposition w b = πr sil . . . si1 , Φwb v = πr Φil (q ξ{l−1} ) · · · Φi1 (q ξ{0} )(v) for ξ{0} = ξ, ξ{p} = sip ((ξ{p − 1})), 1 ≤ p ≤ l, 1/2

Φi (q ξ ) = Ti + (ti

−1/2

− ti

)(q (ξ+d , αi ) − 1)−1 , i ≥ 0.

(3.6.5)

See (3.3.37) and (3.3.42). In the Y –case, the Φi is replaced by ε(Φi ), respectively, the “eigenvalue” q ξ is replaced by its ε–image q −ξ . For instance, the ε(Φi (q ξ )) for i > 0 is given by the right-hand side of the last formula from (3.6.5) after the substitution q ξ 7→ q −ξ , without touching Ti , ti . This requires a simple algebraic manipulation. Only the ε(Φ0 (q ξ )) becomes really different. The process of producing ξ{p} in terms of the initial Y –weight ξ remains unchanged. There is another variant of the system of Y –intertwiners, technically more convenient. It has been applied in Proposition 3.3.5. ¡We will ¢use such intertwiners in the main theorem below. They are τ+ Φwb (q −ξ ) instead of ε(Φwb )(q ξ ). These operators intertwine the Y –eigenvectors as follows. ¡ ¢ Given an eigenvector v of Y –weight ξ, the vector v 0 = τ+ Φwb (q −ξ ) v is a Y –eigenvector of weight w((ξ)). b Notice the opposite sign of ξ here. Recall that τ+ can be interpreted as a formal conjugation by the Gaussian, fixing X, Ti (1 ≤ i ≤ n), and preserving the anti-involution ?. Explicitly, the formula for v 0 reads ¢ ¡ (3.6.6) v 0 = τ πr Φil (q −ξ{l−1} ) · · · Φi1 (q −ξ{0} ) (v), where the notation from (3.6.5) is used. Induced and semisimple modules. An HH[ –module V over Qξ is called X–cyclic of weight ξ if it is generated by an element v ∈ V satisfying (3.6.1).

3.6. SEMISIMPLE REPRESENTATIONS

345

The universal cyclic module of weight ξ is called induced module and is denoted by IX [ξ]. It is generated by a vector v satisfying (3.6.1), that is regarded as the defining relation of IX [ξ]. c [ ((ξ)) with respect to the affine The weights of IX [ξ] constitute the orbit W c . Respectively, the weights of any cyclic modules of weight ξ action of W belong to this orbit. The map H 7→ Hv naturally identifies IX [ξ] with the affine Hecke algebra HY[ ⊂ HH[ . It sends 1 to the cyclic vector v. Similarly, the Y –cyclic modules of weight ξ are introduced for Yb instead of Xb , using (3.6.2) in place of (3.6.1). Respectively, the Y –induced modules satisfy [ IY [ξ] ∼ = hXb , Ti | b ∈ B, i > 0i. = HX

By X–semisimple, we mean an HH[ –module V coinciding with ⊕ξ VX (ξ) over an algebraic closure of the field of constants. Here ξ constitute the set of different X–weights of V, denoted by SpecX (V ) and called the X–spectrum. If V is a cyclic module of weight ξ, then the spectrum is defined over the field Qξ . Indeed the spectrum of IX [ξ] is the orbit W [ ((ξ)). The definition of a Y –semisimple module is for Y taken instead of X. The functional representation F[ξ] introduced in Section 3.4 is X–cyclic c ((ξ)), satisfying for generic ξ. It is isomorphic to IX [ξo ] for the weight ξo ∈ W the inequalities ξo ˜+, t−1 ˜∈R ˜ (q ) 6= 1 for α α Xα

(3.6.7)

c [ [ξ] = {1}. It is proved in [C23] (after Corollary 6.5). provided that W The polynomial representation, which will be denoted by V, is Y –cyclic for generic q, t. Any Eb can be taken as its cyclic vector. One has SpecY (V) = {b] | b ∈ B}. Pseudo-unitary structure. A quadratic form ( , ) is called, respectively, ∗–bilinear and pseudo-hermitian if (ru, v) = r(u, v) = (u, r∗ v) for r ∈ Qq,t and (u, v) = (v, u)∗ . We will always consider nondegenerate forms unless stated otherwise. An HH[ –module V is pseudo-unitary if it is equipped with a ?–invariant pseudo-hermitian form: (H(u), v) = (u, H ? (v)) for all u, v ∈ V, H ∈ HH[ . We call it definite if (v, v) 6= 0 for all v, and X–definite if (v, v) 6= 0 for all X–eigenvectors v ∈ V , assuming that the field of constants contains all the eigenvalues. By X–unitary, we mean X–semisimple V with an X–definite

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form. Strictly speaking, they should be called X–pseudo-unitary, but we drop the “pseudo” in the presence of X. The definitions of Y –definite and Y –unitary are parallel to the above ones. We do not suppose X–definite or Y –definite forms to be positive (or negative) hermitian forms. Note that X–unitary V from the category OX is X–semisimple. Conversely, pseudo-unitary X–semisimple V with the simple X–spectrum is X–unitary. When Qξ ⊂ C, the involution ∗ is the restriction of the complex conjugation, and (u, u) > 0 for all 0 6= u ∈ V, then we call the form positive unitary without adding “pseudo,” X, or Y. Any irreducible HH[ –quotients IX [ξ] → V can be defined over C assuming that |q| = 1, kα ∈ R, ξ ∈ Rn in the following sense. We claim that there is a continuous deformation of any triple q, k, ξ to such a triple, which preserves the structure of IX [ξ], including quotients, submodules, semisimplicity, and the pseudo-hermitian invariant form. This simply means that the variety of all triples {q, k, ξ} has points satisfying the above reality conditions. Thus ∗ can be assumed to be a restriction of the complex conjugation to the field of constants upon such deformation. Semisimple range. Given a weight ξ, its plus–range Υ+ [ξ] is the set of c [ satisfying the following condition: all elements w b∈W α ˜ = [α, να j] ∈ λ(w) b ⇒ tα Xα˜ (q ξ ) = qα(α

∨

, ξ)+kα +j

6= 1.

(3.6.8)

Equivalently, given a reduced decomposition w b = πr sil · · · si1 of an arbitrary c w b ∈ W , we need to check the conditions (α∨ ip , ξ{p−1}+d)+kip

qi p

6= 1 for p = 1, . . . , l,

(3.6.9)

where ξ{0} = ξ, ξ{p} = sip ((ξ{p − 1})). Obviously the definition (in the second form) does not depend on πr , so πr Υ+ [ξ] = Υ+ [ξ]. Also, Υ+ [πr ((ξ))] = πr Υ+ [ξ]πr−1 = Υ+ [ξ]πr−1 . Reading w b in the opposite order, the following equivalent set of relations must be checked: (α∨ ip , ξ{p}+d)−kip

qip

6= 1 for p = l, . . . , 1.

(3.6.10)

Notice the opposite sign of kip . c by setting uˆ 7→ w Let us introduce the arrows in W b if (a) either w b = si uˆ for 0 ≤ i ≤ n provided that l(w) b = l(ˆ u) + 1 or (b) w b = πr uˆ for πr ∈ Π[ . ¨ + [ξ] of Υ+ [ξ] is defined as the set of all w The boundary Υ b 6∈ Υ+ [ξ] such that

3.6. SEMISIMPLE REPRESENTATIONS

347

uˆ 7→ w b for an element uˆ ∈ Υ+ [ξ]. It may happen in case (a) only. Actually, if w b = si uˆ and w b 6∈ Υ+ [ξ] 3 uˆ, then automatically l(w) b = l(ˆ u) + 1, because otherwise λ(w) b ⊂ λ(ˆ u), which is impossible. The minus–range Υ− (ξ) is defined for t−1 α in place of tα in (3.6.8) and with k replaced by −k in (3.6.9) and (3.6.10). The semisimple range Υ∗ [ξ] is the intersection Υ+ [ξ] ∩ Υ− [ξ]. The zero– range Υ0 [ξ] is introduced without tα in (3.6.8) and without k in the equivalent conditions (3.6.9)–(3.6.10). The minus-boundary and zero-boundary are defined respectively. Notice that Υ− [ξ] = Υ+ [−ξ], Υ0 [ξ] = Υ0 [−ξ], ¨ ∗ [ξ] ⊂ Υ ¨ + [ξ] ∪ Υ ¨ − [ξ]. Υ

(3.6.11)

Also, we will use the compatibility of the semisimple and zero ranges with the right multiplications: Υ∗ [w((ξ))] b = Υ∗ [ξ] w b−1 for every w b ∈ Υ∗ [ξ].

(3.6.12)

It readily results from (3.1.11). Let us prove it. It is suffices to consider w e = sil · · · si1 assuming that the decomposition is reduced. We can start with w((ξ)), e read w e in the opposit order following (3.6.10), and then continue following (3.6.9) for a reduced decomposition of uˆ ∈ Υ∗ [ξ]. Since we are doing the semisimple range, changing the sign of k in (3.6.10) does not matter, and all weights from the (opposit) ξ–sequence for w e and the (direct) ξ–sequence for uˆ will satisfy the required inequalities. ˜ ˜ instead of λ So λ(ˆ uw e−1 ) satisfies the relation (3.6.8) with t±1 . Here we use λ because the resulting decomposition of uˆw e−1 may be nonreduced. However, −1 ˜ uw the reduced set λ(ˆ uw e ), which is needed in the definition, belongs to λ(ˆ e−1 ) due to (3.1.11). We arrive at Υ∗ [ξ] w b−1 ⊂ Υ∗ [w((ξ))]. b The inclusion obviously can be reversed: Υ∗ [w((ξ))] b w b ⊂ Υ∗ [ξ]. This means the coincidence and proves (3.6.12). Note that this argument cannot be used separately for Υ+ , Υ− because of the change of sign of k in (3.6.10). However, it can be applied to Υ0 : Υ0 [w((ξ))] b = Υ0 [ξ] w b−1 for every w b ∈ Υ0 [ξ].

(3.6.13)

Semisimple stabilizer. The following semisimple stabilizer of ξ def c [ [ξ] = c [ [ξ] | w c [ [−ξ] W = {w b∈W b ∈ Υ∗ [ξ]} = W ∗ ∗

(3.6.14)

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CHAPTER 3. GENERAL THEORY

will play an important role in the main theorem of this section. It is a subgroup c [ [ξ]. Indeed, if uˆ, w c∗[ [ξ], then the set of W b∈W ˜ wˆ λ( bu) = λ(ˆ u) ∪ uˆ−1 (λ(w)) b satisfies (3.6.8) and its counterpart for Υ− [ξ] because uˆ does not change q ξ . ˜ wˆ However, λ(wˆ bu) ⊂ λ( bu) thanks to (3.1.11). The same argument proves that c∗[ [ξ]. Υ∗ [ξ]w b = Υ∗ [ξ] for every w b∈W

(3.6.15)

In contrast to (3.6.12), we do not need to use (3.6.10) here. Combining (3.6.15) with (3.6.12), we arrive at c [ [w((ξ))] c [ [ξ] w b−1 for all w b ∈ Υ∗ [ξ]. W =w bW ∗ ∗ b

(3.6.16)

c [ [ξ]. They preserve c [ [ξ] and W Similarly, one can introduce the groups W 0 ± the corresponding Υ[ξ] acting via the right multiplication, i.e., (3.6.15) holds for all three of them. In fact, this property can be used as another (equivalent) definition of the corresponding stabilizer. Note that the relation (3.6.16) holds, generally speaking, for Υ[ξ]∗ and Υ[ξ]0 only. Sharp case. When considering spherical and self-dual representations, the sharp range Υ+ [−ρk ] will be used, which is simply Υ+ [ξ] as ξ = −ρk . The other Υ are defined correspondingly. It follows from Lemma 3.1.6 that Υ+ [−ρk ] = πB for generic q and tν . Indeed, formula (3.1.32) there gives that kν + (αi∨ , πc ((−ρk )) + d) 6= 0 for ν = νi = ναi , i ≥ 0, if and only if si πc ∈ πB , assuming that l(si πc ) = 1 + l(πc ). It is equivalent to k +(α∨ i , πc ((−ρk ))+d)

qi i

6= 1,

which we are supposed to check to conclude that w b = si πc (under the same assumption) belongs to Υ+ [−ρk ] provided that πc is already in this set. The equivalence is of course for generic q, t only. c with those used in B. Recall that by Let us connect the arrows in W c →→ b, we mean that b ∈ B can be obtained from c ∈ B by a chain of transformations (i)–(ii) (the simple arrows) from Proposition 3.1.7. The c ↔ b indicates a simple transformation of type (iii), corresponding to the application of πr ∈ Π[ . It is always invertible. If such a transformation is involved, then we set c ↔→ b. The meaning of the simple arrows coincides with that of the uˆ 7→ w b introduced above upon the restriction to Υ+ (−ρk ). Indeed, the latter is the

3.6. SEMISIMPLE REPRESENTATIONS

349

set of all w b = πb satisfying (3.6.8). If πc ∈ Υ+ (−ρk ) and c → b for πb = si πc , then πb ∈ Υ+ (−ρk ) if and only if tα q (α , c− −ρk ) 6= 1 for i > 0, α = uc (αi ), tα q 1+(α , c− −ρk ) 6= 1 for i = 0, α = uc (−ϑ).

(3.6.17)

−1 Recall that c] = c − u−1 c (ρk ) = uc (−ρk + c− ). See (3.1.31). In the remaining case c ↔ b for πb = πr πc , πb ∈ Υ+ (−ρk ) if and only if πc ∈ Υ+ (−ρk ). Generally, in terms of λ(πc ) = {[α, να j]}, the inclusion πc ∈ Υ+ (−ρk ) is equivalent to the conditions

qαj+kα −(α

∨ ,ρ +d) k

6= 1 for such α ˜.

Concerning applications to the E–polynomials, these conditions ensure 1/2 that there will be no intertwiners proportional to (τ+ (Ti ) − ti ) in the prodbc from (3.3.42). The simple intertwiners uct formula for the polynomial E −1/2 proportional to (τ+ (Ti ) + ti ) cannot appear for generic q, t in the corresponding chain.

3.6.2

Main theorem

We turn to a general description of semisimple and pseudo-unitary representations. It will be reduced to a certain combinatorial problem that can be managed in several cases. We permanently identify the cyclic generator v ∈ IX [ξ] with 1 ∈ HY[ , which can be uniquely extended to an HY[ – isomorphism IX [ξ] ∼ = HY[ . Theorem 3.6.1. (i) Assuming that def

Υ∗ [ξ] ⊂ Υ0 [ξ] and Jξ ==

X

HY Φwb (q ξ ) 6= HY ,

(3.6.18) def

¨ ∗ [ξ], the quotient U = U ξ == H[ /Jξ is where the summation is over w b∈Υ Y X [ an X–semisimple HH –module with a basis of X–eigenvectors {Φwb (q ξ )} for w b ∈ Υ∗ [ξ]. Its X–spectrum is {w(ξ)} b for such w. b [ If V is an irreducible X–semisimple HH –module with a cyclic vector of weight ξ, then (3.6.18) holds and V is a quotient of U, provided that tsht 6= 1 6= tlng . (ii) The elements Φwb v 0 , Swb v 0 , and Gwb v 0 are well defined for all v 0 from the ξ–eigenspace U (ξ) of U and w b ∈ Υ∗ [ξ]. They induce isomorphisms U (ξ) ∼ = U (w(ξ)). b c [ [ξ] acts in U (ξ) by automorphisms via w b 7→ Swb . A quotient The group W ∗ [ V of U is an irreducible HH –module if and only if V (ξ) is an irreducible c [ [ξ]–module. Moreover, provided we have (3.6.18), an arbitrary irreducible W ∗

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CHAPTER 3. GENERAL THEORY

representation V 0 of this group can be uniquely extended to an irreducible X– semisimple HH[ –quotient V of IX [ξ] such that V 0 = V (ξ). Here G can be used instead of S. The above statements hold ¡ for I¢Y [ξ] and its semisimple quotients when the Φwb (q ξ ) are replaced by τ+ Φwb (q −ξ ) everywhere. (iii) Under the same constraint (3.6.18), the function def

µ• (w) b ==

Y

³ t−1/2 − q j t1/2 X (q ξ ) ´ α α α α 1/2

[α,να j]∈λ(w) b

−1/2

tα − qαj tα

Xα (q ξ )

(3.6.19)

has neither poles nor zeros at w b ∈ Υ∗ [ξ]. Moreover, c∗[ [ξ] µ• (ˆ u)µ• (w) b = µ• (ˆ uw) b whenever w b∈W

(3.6.20)

for uˆ, w b ∈ Υ∗ [ξ], An HH[ –quotient V of U is pseudo-unitary if and only if the eigenspace c [ [ξ]–module. The latter means that the form ( , ) V (ξ) is a pseudo-unitary W ∗ is nondegenerate on V (ξ) and def ξ c∗[ [ξ], S ξ = (Swξb )? Swξb = µ• (w) b for w b∈W b (q ). w b = Sw

(3.6.21)

(iv) A pseudo-hermitian form on e0 , e00 ∈ V (ξ), can be uniquely extended to a pseudo-hermitian invariant form on V : (Suˆ e0 , Swb e00 ) = δu0ˆ,wb µ• (ˆ u) (e0 , Suˆ−1 wb e00 ), where (3.6.22) c∗[ [ξ] and = 0 otherwise. uˆ, vˆ ∈ Υ∗ [ξ], δu0ˆ,wb = 1 for uˆ−1 w b∈W If (e0 , e0 ) 6= 0 as V (ξ) 3 e0 6= 0, then V is X–unitary. An arbitrary irreducible pseudo-unitary (X–unitary) HH[ –module with a cyclic vector of weight ξ can be obtained by this construction as tsht 6= 1 6= tlng . In the Y –case, ¡ ¢ def −ξ ˜ξ = G b (q ) w b = τ+ Gw is taken instead of Swb (q ξ ); these operators send V (ξ) to V (w((ξ))). b The extension from a pseudo-hermitian form on V (ξ) is ˜ ξ e0 , G ˜ ξ e00 ) = δ 0 µ• (ˆ ˜ ξ −1 e00 ). (G u)−1 (e0 , G u ˆ ,w b u ˆ w b u ˆ w b

(3.6.23)

c [ provided that Φwb (1) is well defined. Proof. We set ewb = Φwb (1) for w b∈W It is an X–eigenvector of weight w(ξ). b Recall that we permanently identify v and 1. Then HY ewb is an HH[ –submodule because HH[ = HY · Qq,t [Xb ]. Assuming (3.6.18), Jξ and U are HH[ –modules. Moreover, all ewb are invertible in HY[ and their images linearly generate U as w b ∈ Υ∗ [ξ]. Indeed,

3.6. SEMISIMPLE REPRESENTATIONS

351

if Φi ewb loses the invertibility, then si w b leaves Υ∗ [ξ], l(si w) b = l(w) b + 1, and ¨ ∗ [ξ]. By (3.6.18), Υ ¨ ∗ [ξ] ⊂ Υ0 [ξ]. We conclude that the element Ti (ewb ) si w b∈Υ is proportional to ewb in this case. This means that each Ti (ewb ) is a linear combination of euˆ for any n ≥ i ≥ 0, where w, b uˆ ∈ Υ∗ [ξ]. Of course it holds for πr ∈ Π[ too. Thus {ewb0 } linearly generate U. Moreover, given w b ∈ Υ∗ [ξ], the map v 0 7→ Φwb v 0 induces an isomorphism U (ξ) ∼ b = U (w(ξ)). Summarizing, (3.6.18) implies that {ewˆ bu } form a basis of U when c [ [ξ], (a) w b are representatives of all classes Υ∗ [ξ]/W c [ [ξ] and {euˆ } form a basis of U (ξ). (b) uˆ ∈ W Let us discuss the usage of S and G. The elements Swb (1) ∈ HY[ are well def P ¨ + [ξ], is an HH[ – defined for w b ∈ Υ+ [ξ]. The sum Jξ+ == HY Swb , w b ∈Υ submodule of IX [ξ]. Finally, Swb (1) are invertible for w b ∈ Υ∗ [ξ], which results directly from the definition of the latter set. The same holds for Υ− [ξ] with S being replaced by G. Turning to IY [ξ], G and S serve, respectively, Υ+ [ξ] and Υ− [ξ]. Now let us check that the existence of irreducible X–semisimple quotients V 6= {0} of IX [ξ] implies (3.6.18), and all such V are quotients of U. We suppose that tsht 6= 1 6= tlng . Actually, we will impose tsht 6= ±1 6= tlng in the argument below, leaving the consideration of t = −1 to the reader. This 1/2 allows us to separate Ti − ti and Ti + t−1 i . When ti = −1 the Ti is not semisimple, but the theorem still holds in this case. The images ewb of the vectors Φwb (1) in V are nonzero as w b ∈ Υ∗ [ξ]. If 0 0 0 Υ0 [ξ] 6⊃ Υ∗ [ξ], we can pick e = ewb such that Si (e ) = e and Xαi (e0 ) = e0 . This gives that Xαi is not semisimple in the two-dimensional module generated by i def e0 and e00 = Ti (e0 ). The dimension is two because the characters of HX == = 6 1. So it cannot be one. hTi , Xαi i send Xαi 7→ t± i ¨ + [ξ]. The case w ¨ − [ξ] Now let uˆ belong to Υ∗ [ξ] and w b = si uˆ ∈ Υ b = si uˆ ∈ Υ is completely analogous, so we will skip it. Then l(w) b = l(ˆ u) + 1. Both 1/2 0 00 0 0 e = Φuˆ (e) and e = Φi (e ) = (Ti − ti )(e ) are X–eigenvectors. Here by e, we mean the image of 1 in V. Let us prove that e00 = 0. Supposing that e00 6= 0, it cannot be proportional to e0 . Indeed, this may −1/2 only happen if (Ti + ti )(e0 ) = 0, which is impossible, since uˆ ∈ Υ∗ [ξ] ⊂ Υ− [ξ]. Note that if V is assumed to be X–unitary, then restricting its X–definite form to the linear span of e0 , e00 readily leads to a contradiction with the relation Ti? = Ti−1 . This argument does not require the irreducibility of V but involves the X–unitary structure. If e00 6= 0, then it generates V as an HH[ –module and as an HY[ –module,

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CHAPTER 3. GENERAL THEORY

thanks to the irreducibility. This means that 1/2

HY[ (Ti − ti )Φwb (1) + JV = HY[ , where V = HY[ /JV , for a proper ideal JV ⊂ HY[ that is an HH[ –submodule of IX [ξ]. Hence, JV −1/2 contains HY[ (Ti + ti )Φwb (1) and e00 has to be proportional to e0 . This is whereas Xαi (e0 ) 6= t±1 impossible because Xαi (e00 ) = t−1 i . i Summarizing, we have almost completed parts (i)–(ii) from the theorem. The remaining claims there are c [ [ξ]– (a) the equivalence of the irreducibility of the HH[ –module V and the W ∗ module V (ξ), c [ [ξ] (b) the possibility of taking an arbitrary irreducible representation of W ∗ as V (ξ). Applying the intertwining operators, both claims readily follow from the statements that have been checked. Let us turn to (iii) and (iv). We take e0 , e00 ∈ V (ξ) and denote e0wb = Swb e0 and e00wb = Swb e00 , where w b ∈ Υ∗ [ξ], for a certain pseudo-unitary irreducible [ quotient V of IX . They are nonzero X–eigenvectors. If the eigenvalues of e0uˆ c∗[ [ξ], then these vectors are orthogonal and e00wb are different, i.e., uˆ−1 w b 6∈ W with respect to the invariant form of V. Applying (3.3.31), (e0wb , e00wb ) = (Swb e0 , Swb e00 ) = (Sw?b Swb e0 , e00 ) = ξ ´ Y³ tα−1/2 − q j t1/2 α α Xα (q ) 0 00 (e , e ) , [α, να j] ∈ λ(w). b 1/2 j −1/2 Xα (q ξ ) α,j tα − qα tα

(3.6.24)

The latter product is just µ• (w) b from (3.6.19). Thus the extension of ( , ) from V (ξ) to V is unique if it exists. Explicitly: (Suˆ e0 , Swb e00 ) = δu0ˆ,wb µ• (ˆ u) (e0 , Suˆ−1 wb e00 ), where e0 , e00 ∈ V (ξ), c∗[ [ξ] and 0 otherwise. uˆ, vˆ ∈ Υ∗ [ξ], δu0ˆ,wb = 1 for uˆ−1 w b∈W

(3.6.25)

To check that this formula correctly extends an arbitrarily given definite form on V (ξ), it suffices to compare (Swb e00 , Suˆ e0 )∗ and (Suˆvˆ(Svˆ−1 e0 ) , Swb e00 ) c∗[ [ξ]. Using (3.6.25), we need and to show that they coincide whenever vˆ ∈ W the relations µ• (w b−1 uˆ)µ• (w) b = µ• (ˆ u), µ• (ˆ uvˆ)µ• (ˆ v −1 ) = µ• (ˆ u), which are nothing but (3.6.20). Claim iv) is verified.

3.6. SEMISIMPLE REPRESENTATIONS

353

Relation (3.6.20) obviously holds when l(ˆ uw) b = l(ˆ u) + l(w), b because the µ• –product for uˆw b is taken over λ(ˆ uw), b which is a union of λ(w) b and the −1 set w b (λ(ˆ u)). The µ• –product over the latter set coincides with the product over λ(ˆ u)). Thanks to (3.1.11), we can omit the constraint l(ˆ uw) b = l(ˆ u) + l(w) b in + ˜ this argument. Indeed, the extra positive affine roots α ˜ = [α, να j] ∈ λ (ˆ uw) b ˜ uw) ˜ ∈ −λ ˜ + (ˆ appear in λ(ˆ b together with −λ uw). b Here we take the product of the reduced decompositions of uˆ and w, b which may be nonreduced, to ˜ construct λ(ˆ uw). b However, the product of the µ• –factors corresponding to α ˜ = [α, να j] and −˜ α is ³ t−1/2 − q j t1/2 X (q ξ ) ´³ t−1/2 − q −j t1/2 X −1 (q ξ ) ´ α α α α α α α α 1/2

−1/2

tα − qαj tα

Xα (q ξ )

1/2

−1/2

tα − qα−j tα

Xα−1 (q ξ )

= 1.

This concludes (iii) and the proof of the theorem. ❑ Thanks to Proposition 3.1.3, there is a reasonably simple and explicit combinatorial reformulation of the first part of condition (3.6.18). However, it does not guarantee the second part. Proposition 3.6.2. The relation Υ∗ [ξ] ⊂ Υ0 [ξ] is equivalent to the following ¨ = {β˜ = [β, νβ j]} be the set of all affine positive roots such property of ξ. Let Υ ±1 ξ that tβ Xβ˜(q ) = 1. Then an arbitrary positive affine γ˜ = [γ, νγ i] such that ¨ with positive rational (not Xγ˜ (q ξ ) = 1 is a linear combination of some β˜ ∈ Υ a ¯ always integral) coefficients. Using the closure C of the affine Weyl chamber Ca from Section 3.1 of this chapter, ¨ for w ∪wb w((− b C¯a )) = {z ∈ Rn | (z, β) + j < 0, β˜ ∈ Υ} b ∈ Υ∗ [ξ].

(3.6.26) ❑

3.6.3

Finite dimensional modules

We will start the discussion of the theorem with the irreducible IX [ξ]. Followc to the c∗[ [ξ] is conjugated in W ing [C23], Theorem 6.1, let us assume that W span of a proper subset of {s0 , s1 , . . . , sn }, possibly ∅, but smaller than the whole set of affine simple reflections. It is possible only if q is not a root of c∗ [ξ] is infinite. By the way, this constraint results unity, because otherwise W in the implication X ¨ ∗ [ξ]. Υ∗ [ξ] ⊂ Υ0 [ξ] ⇒ Jξ = HY Φwb 6= HY , where w b∈Υ We do not prove/use this claim. c[ = Apart from roots of unity, IX [ξ] is irreducible if and only if Υ+ [ξ] = W Υ− [ξ]. Thus its semisimplicity is equivalent to the triviality of the stabilizer c [ [ξ]. W ∗

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The theorem can be applied to a general description of the finite-dimensignal semisimple representations. We continue to suppose that q is generic. If q, t are treated as complex numbers, then we assume that (q 1/(2m) )i (t1/2 )j do not represent nontrivial roots of unity for i, j ∈ Z. Proposition 3.6.3. i) Under condition (3.6.18), the induced module IX [ξ] has a nonzero finite dimensional X–semisimple quotient if and only if there exists a set of roots T ⊂ R such that for any b ∈ B there exist β ∈ T with (b, β) > 0, i.e., the β do not belong to any half-plane in Rn , and for every β ∈ T, (ξ, β ∨ ) + kβ ∈ −1 − Z+ or (ξ, β ∨ ) − kβ ∈ −1 − Z+ for β < 0, (ξ, β ∨ ) + kβ ∈ −Z+ or (ξ, β ∨ ) − kβ ∈ −Z+ for β > 0. (3.6.27)

Proof. Let us assume that for all α ∈ R such that (b, α) > 0, either (ξ, α∨ ) ± kα 6∈ −1 − Z+ for both signs of kα as α < 0 or the same inequalities hold with −Z+ as α > 0. Then λ(lb) for l ∈ Z+ consists of α ˜ = [α, νβ j] for ∨ such α with integers j satisfying inequalities 0 < j ≤ l(b, α ) as α < 0 and with 0 ≤ j otherwise. We take b from Υ∗ [ξ], so all these α ˜ satisfy (3.6.8) and −1 its counterpart with tα instead of tα . Since l is arbitrary positive, we obtain that Υ∗ [ξ] is infinite if the relations (3.6.27) don’t hold. Let us check that this condition guarantees that Υ∗ [ξ] is finite. First of all, any given infinite sequence of pairwise distinct bi ∈ B is a (finite) union of subsequences {˜bi } such that there exist β± ∈ T satisfying ±(˜bi , β± ) > 0. Taking one such infinite subsequence, we may assume that there exist β± ∈ T such that ±(bi , β± ) > 0. If the latter scalar products are bounded, we switch to a new sequence bi formed by proper differences of the old {bi }, ensuring that all (bi , β± ) = 0. Then we find the next pair β± 0 such that ±(bi , β± 0 ) > 0, form the next sequence of differences, and so on, until all the T are exhausted. We conclude that any sequence {bi } is a union of subsequences {˜bi } such that |(˜bi , β)| → ∞ as i → ∞ for certain β ∈ T. Following the proof of Proposition 3.1.4 (see (3.1.25), (3.1.26)), given w ∈ W, the sets λ(w˜bi ) contain the roots [β, jνβ ], where max{j} → ∞ as i → ∞. Note that the proof gets more transparent via the geometric interpretation from Proposition 3.1.3. ❑ The generalized Macdonald identities correspond to the following pair. The set T = {α1 , . . . , αn , −ϑ} must be taken together with ξ = −ρk . The sign of kβ in the relations (3.6.27) from the proposition is plus for ϑ and minus otherwise. See the end of Section 3.10 of this chapter. The constraint (3.6.18) will be fulfilled, for instance, if (ξ, α∨ ) 6∈ Z for all α ∈ R. The simplest example is as follows. Letting klng = k = ksht , ξ = −kρ,

3.6. SEMISIMPLE REPRESENTATIONS

355

we take k = −m/h for the Coxeter number h provided that (m, h) = 1. This representation is spherical. In the An –case, one can follow [C5] to describe all irreducible representations under consideration (for generic q). See the next section.

3.6.4

Roots of unity

We go back to arbitrary root systems. Let q be a primitive root of unity of c∗[ [ξ] always contains order N. Then W def

def

A(N ) == (N · A) ∩ B, where A == {a ∈ P ∨ | (a, B) ⊂ Z}, so A = Q∨ for B = P, and always A ⊃ Q∨ .

(3.6.28)

Recall that Q ⊂ Q∨ and P ⊂ P ∨ ; it may occur that A 6⊂ B and A(N ) 6= N ·A. ˜ From here on, q 0 = q 1/(2m) is assumed to be a primitive root of order 2mN ˜ unless otherwise stated. Recall that m ˜ ∈ N is the smallest positive integer such that m(B, ˜ B) ⊂ Z. If q 0 is a primitive root of order m0 N for m0 | 2m, ˜ 0 −1 then we shall replace A by m (2m) ˜ A in the definition of A(N ). Let us describe semisimple irreducible finite dimensional quotients V of c [ [ξ] = A(N ), i.e., the stabilizer of ξ is the smallest IX [ξ] assuming that W ∗ c [ = Υ− [ξ]. The main constraint (3.6.18) becomes possible, and Υ+ [ξ] = W c [ in this case. Υ0 [ξ] = W Theorem 3.6.1 states that the spectrum of {Xb } in any V has to be simple, the dimension is always |W | · |B/A(N )|, and the {V } are in one-to-one correspondence with the one-dimensional characters %(a) of the group A(N ), which determine the action via S of A(N ) on V (ξ). To be more exact, the V –quotients up to isomorphisms are in one-to-one correspondence with the c pairs { %, orbit q W ((ξ)) }. We denote them 0 def b c. V == V [ξ, %], V [ξ, %] ' V [ξ 0 , %] if q ξ = q w((ξ)) for w b∈W

(3.6.29)

The isomorphisms here are established using the intertwiners Φwb . Concerning the unitary structure, let us calculate µ• on A(N ). Setting Nα = N/(N, να ), the scalar products (a, α∨ ) = (a, α)/να are divisible by Nα as a ∈ A(N ), α ∈ R. On the other hand, qα = q να is a primitive root of order Nα . Using formula (3.1.14), def

µ• (a) ==

³ t−Nα /2 − tNα /2 q Nα (α,ξ) ´((a,α∨ )/Nα )

Y

α∈R|(a,α)>0

α N /2 tα α

−

α −Nα /2 Nα (α,ξ) tα q

N /2 α /2 Y ³ t−N − tα α q Nα (α,ξ) ´((a,α∨ )/Nα ) α N /2

α∈R+

tα α

−Nα /2 Nα (α,ξ) q

− tα

= (3.6.30)

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for a ∈ A(N ). By the way, it is clear from this formula that µ• is a homoN (ξ,α) ). morphism from A(N ) to the multiplicative group of the field Q(tN α ,q It was already checked in the theorem. We come to the following proposition. Proposition 3.6.4. Assuming that q is a primitive root of unity of degree N and c∗[ [ξ] = A(N ), Υ+ [ξ] = W c [ = Υ− [ξ], W X–semisimple irreducible quotients of IX [ξ] exist if and only if c [ i.e., Xα˜ (q ξ ) 6= 1 for all α ˜ Υ0 [ξ] = W ˜ ∈ R.

(3.6.31)

Such quotients V [ξ, %] are described by the characters % of A(N ). The pseudounitary structure exists if and only if %(a)∗ %(a) = µ• (a) for the generators of A(N ).

3.6.5

Comment on finite stabilizers

When q is a root of unity, the case of finite stabilizators is, in a sense, opposite c∗[ [ξ] is finite if and to the case of representations V [ξ, %]. Indeed, the group W only if the set Υ∗ [ξ] is finite. The check is simple. If the latter set is infinite, then it contains infinitely many elements in the form aw b for a fixed w b and c a ∈ A(N ), because the quotient W /A(N ) is finite. Note that A(N ) is normal c, so this quotient is a group. We conclude that W c [ [w((ξ))] is infinite. in W ∗ b [ c However, it is conjugated to W∗ [ξ] due to (3.6.16), so the latter has to be infinite as well. Recall that the existence of the pseudo-unitary structure on V reads as c [ [ξ] of finite order. Sw?b Swb = µ• (w) b in V (ξ) for w b∈W ∗

(3.6.32)

We are going to check that here µ• (w) b is ±1 and, moreover, automatically 1 in many case. c [ [ξ] of finite order, µ• (w) b = ±1 Proposition 3.6.5. For an element w b in W ∗ without any assumptions on q, t, ξ. Moreover, µ• (w) b = 1 if the order of w b is 2M + 1, i.e., odd, or 2(2M + 1). Proof. First we consider the case w b2 = 1. Then λ(w) b = −w(λ( b w)) b and each Xα˜ (q ξ ) appears in the product (3.6.19) together with Xα˜−1 (q ξ ), thanks to w(q b ξ ) = q ξ . If Xα˜ (q ξ ) 6= Xα˜−1 (q ξ ) then the corresponding binomials will annihilate each other, i.e., their contribution to the product will be 1. However, they can coincide. There are two possibilities: Xα˜ (q ξ ) = ±1. If Xα˜ (q ξ ) = −1, then ³ t−1/2 − q j t1/2 X (q ξ ) ´ α α α α = 1. 1/2 j −1/2 tα − qα tα Xα (q ξ )

3.7. THE GL–CASE

357

The equality Xα˜ (q ξ ) = 1 is impossible, thanks to the constraint (3.6.18). So we obtain the complete annihilation of all factors in µ• (w). b Now we assume that w b is of finite order. Then µ• (w) b has to be a root of unity. Under the reality conditions |q| = 1, kα ∈ R, ξ ∈ Rn , µ• (w) b is real and can be ± only. An arbitrary triple {q, k, ξ} can be continuously c∗[ [ξ] and the value deformed to the triple above, without changing the group W c [ [ξ] depends on certain multiplicative of µ• (w). b Indeed, the structure of W ∗ relations among q(α,ξ) and qαkα . Therefore we need to deform {q, k, ξ} within some subvariety, which is possible. We obtain the claim about ±1. It gives that the orders of µ• (w) b and µ• (w b2M +1 ) = µ• (w) b 2M +1 always coincide. If the order of w b is odd or 2(2M + 1), then µ• (w) b has to be 1, thanks to the claim about the elements of the second order. ❑

3.7

The GL–case

This section is somewhat exceptional. It does not conatin proofs, although the main result is not too difficult to justify using the previous theory. We give a description of the X–semisimple representations for the double Hecke algebra associated to GLn for fractional k in terms of the periodic skew Young diagrams. The DAHA of type GLn is defined as follows: def

HHn == hX1±1 , . . . , Xn±1 , π, T1 , . . . , Tn−1 i, where

(3.7.1)

(i) πXi = Xi+1 π (i = 1, . . . n − 1) and π n Xi = q −1 Xi π n (i = 1, . . . , n), (ii) πTi = Ti+1 π ( i = 1, . . . n − 2) and π n Ti = Ti π n (i = 1, . . . , n − 1), (iii) Xi Xj = Xj Xi (1 ≤ i, j ≤ n), Ti Xi Ti = Xi+1 , i < n, (iv) Ti Ti+1 Ti = Ti+1 Ti Ti+1 , Ti Tj = Tj Ti as i < n − 1, j > i + 1, (v) (Ti − t1/2 )(Ti + t−1/2 ) = 0 for T1 , . . . , Tn−1 , t = q k . See Chapter 1, Section 1.4.3. Note that the formulas for the generators Y in terms of π, T1 , . . . , Tn in this chapter are dual to those considered in Chapter 1. For instance, now Y1 = πTn−1 · · · T1 . Actually, we we do not need the formulas for Y in this section. The classification will be produced in terms of the X only. It triggers a very interesting question about the description of the Y –action in the representations considered below. For the sake of completeness, let us give the formulas for the automorphisms τ+ and σ from (3.2.10)–(3.2.16) in the case of GLn . Recall that they

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preserve T1 , . . . , Tn−1 and do not change q, t. Also, τ+ (Xi ) = Xi , σ(Xi ) = Yi−1 for 1 ≤ i ≤ n. The other formulas are as follows: τ+ (Y1 · · · Yi ) = q −i/2 (X1 · · · Xi )(Y1 · · · Yi ), σ(Yi ) = Tw−1 Xn−i+1 Tw0 . 0

(3.7.2)

We use that in a proper completion of HHn , τ+ (π) = q P

P

x2i /2

◦ π ◦ q−

P

x2i /2

= q −1/2 X1 π,

2

where Xi = q xi and q xi /2 is the Gaussian for GLn . Switching here from C[Xi±1 ] to the subalgebra C[Xi Xj−1 ] of Laurent polynomials of degree zero and setting π n = 1, we come to the definition of the double Hecke algebra HHon of type An−1 , i.e., the one for SLn . We use that π n commutes with the X–monomials of degree zero and with Ti , thanks to (i–ii). To be exact, this definition corresponds to the root lattice Q taken as B 3 b for the Xb in the definition of HH[ . In this section, we will deal with either HHn or HHon . e −1/n )±1 ] for When the subalgebra C[Xi Xj−1 ] ⊂ HHon is extended to C[(Xi X e = X1 · · · Xn , we arrive at the case B = P. The X e appears here because of X the formulas for the fundamental weights in the case of An−1 . Later on, we will always consider the representations where π is invertible and identify the representations under the automorphisms π 7→ cπ of HHn for c ∈ C∗ . Obviously all relations are preserved under this multiplication. The following construction is a “cylindrical” counterpart of that from [C5](Section 3) and [Na1]. It generalizes the corresponding affine classification due to Bernstein and Zelevinsky. It is a direct application of the Main theorem. Fortunately, the most technical part of the construction can be borrowed from the affine Hecke theory. We will publish the details elsewhere. Examples concerning the semisimple representations of HHn can be found in [SuV]. See also [Suz].

3.7.1

Generic k

Let ∆ = {δ 1 , . . . , δ m } be an unordered collection of the skew Young diagrams without empty rows of total degree n (i.e., with n boxes in their union), C = {c1 , . . . , cm } a set of complex numbers such that q ca −cb 6∈ tZ q Z as a 6= b. A diagram is a subset of Zn with the coordinate i used for the rows and the coordinate j used for the columns. We identify the points (i, j) with the centers of the corresponding unit boxes, so (il , jl ) are the coordinates of the center of box l. We set pl = p if this box is in δ p . The skew diagram δ is defined as follows. If box l belongs to the i–th row of δ, then mi < jl ≤ ni for the “end points” mi ≤ ni of the rows (the row is empty if they coincide). The end points satisfy the inequalities mi ≥ mi+1 , ni ≥ ni+1 , i ∈ (set of the row-numbers in δ).

(3.7.3)

3.7. THE GL–CASE

359

We always number the boxes of the collection of diagrams in the inverse order, i.e., from the last box of the last row of δ m through the first box of the first row of δ 1 , and set ξl = cpl + k(il − jl ), l = 1, . . . , n, where box l belongs to δ pl .

(3.7.4)

Concerning the notation pl , note that p1 = m, . . . , pn = 1. We say that the weight ξ is associated with the pair {∆, C}. The weight is the main numerical invariant of the pair. We will identify the diagrams that can be obtained from each other by translation by the diagonal vectors (c, c) (in the {i, j}–plane). This identification does not change the weight. More generally, translations of the diagrams with the corresponding adjustments of the c–numbers are considered identical transformations if they do not change the weight. Theorem 3.7.1. Let q i tj 6= 1 for any (i, j) ∈ Z2 \ (0, 0). Then condition (3.6.18) is satisfied if ξ is associated with a ∆–set of skew diagrams and the corresponding C–set of complex numbers. For such ξ, U = UXξ from the Main theorem is a unique irreducible X–unitary quotient of the induced module IX [ξ]. Its X–spectrum is simple. An arbitrary irreducible X–semisimple HHn –module can be constructed in this way for a proper pair {∆, C}. Isomorphic modules correspond to the pairs that can be obtained from each other by a permutation of the skew diagrams from the collection ∆ together with permuting the corresponding numbers from C, and by adding arbitrary integers to the c–numbers. Proof follows [C23]. Adding intergers to the c–numbers is allowed because of the above formulas for conjugation by the element π. ❑ Given ξ, the construction of the Main theorem restricted to the nonaffine Weyl group W results in the irreducible representation V ξ of the affine Hecke algebra, def

Hn == hX1±1 , . . . , Xn±1 , T1 , . . . , Tn−1 i, from [C23] and [C5]. It is a unique X–semisimple nonzero irreducible quotient of the induced Hn –module of weight ξ. Note that adding integers to the c– numbers alters the isomorphism class of the Hn –representation. If ξ corresponds to a single skew diagram, then the restriction of the corresponding Hn –module to the nonaffine Hecke algebra def

Hn == hT1 , . . . , Tn−1 i is isomorphic to Hn Pδ for the q–Young symmetrizer Pδ associated with the skew diagram δ in a way similar to the classical definition. It is Hn –irreducible if and only if δ is the standard Young diagram (recall that t is generic).

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CHAPTER 3. GENERAL THEORY

Given a pair {∆ = {δ p }, C = {cp }}, the corresponding irreducible Hn – module V ξ is parabolically induced from the modules associated to {δ 1 , c1 }, . . . , {δ m , cm }, and so is its restriction to Hn . Inducing the module V ξ up from Hn to HHn , we get exactly the UXξ from the theorem. The modules V ξ and UXξ remain irreducible upon restriction to the subalgebra Hno of Hn with C[Xi Xj−1 ] instead of C[Xi±1 ] and to HHon . Concerning the last statement of the theorem, this symmetry becomes stronger. Adding an arbitrary common complex constant to the c–numbers does not change the HHon –isomorphism classes, as well as translations of the diagrams in ∆ by a common integral vector.

3.7.2

Periodic skew diagrams

We assume that q is not a root of unity, and the products q a tb for a, b ∈ Z do not represent nontrivial roots of unity. Let tr = q s for integral r, s, where r 6= 0, s > 0, and (r, s) = 1. Setting t = q k , we obtain that k = s/r. We will begin with the definitions for r > 0. An infinite skew diagram δ without empty rows is called r–periodic of degree n if (a) it is invariant with respect to translation by a vector w = (v, v − r) in the (i, j)–plane for a proper positive integer v, and (b) the subdiagram of δ with the rows in the range 1 ≤ i ≤ v contains exactly n boxes. The latter subdiagram is called the basic subdiagram. It is skew. The subdiagrams with the rows subject to 1 + c ≤ i ≤ v + c are skew too. Note that v ≤ r for periodic skew diagrams. To visualize the diagrams, here and later, we identify the centers of boxes (i, j) with the points (x = j, y = −i) in the standard {x, y}–plane, i.e., the rows go down when the row-numbers increase and the column-numbers are the x-coordinates. Figure 3.1 demonstrates how we place a Young diagram in the (x, y)–plane and number the boxes in the basic and other skew subdiagrams of δ. This skew diagram is the basic subdiagram of the (portion) of the periodic skew diagram with n = 11, r = 7, v = 3, w = (3, 3 − 7) = (3, −4) shown at Figure 3.2. Any skew diagram, finite or infinite, is naturally a portion of the {x, y}– plane between its upper-left boundary and its lower-right boundary.

3.7. THE GL–CASE

361 j=1 j=2 j=3 j=4 j=5 j=6

i=1

11

10 4

i=2

7

6

5

i=3

3

2

1

9

8

Figure 3.1: Skew Diagram j=1 j=2 j=3 j=4 j=5 j=6 ··· ··· i=1

11

10 4

i=2

7

6

5

i=3

3

2

1

9

8

i=4 i=6

n=11

i=7

r=7 ··· ···

v=3

Figure 3.2: Periodic Skew Diagram In terms of the boxes, the upper-left boundary of a single unit box is a union of its top and left sides. The bottom and right sides represent the lower-right boundary. The general definition is analogous. The boundaries will be needed for periodic diagrams only. The intersections of any diagrams with the horizontal lines i = const are segments between the left and the right end points. For skew diagrams, their intersections with vertical lines are segments too, between the lower and the upper end points. The boundaries are infinite continuous piecewise linear paths in the coordinate net, i.e., they are “made” from the segments where either i ∈ Z or j ∈ Z. They satisfy the skew condition: if i0 > i, then j 0 ≤ j, if j 0 > j, then i0 ≤ i, for any pair (i, j), (i0 , j 0 ) of points. We will call such paths the skew paths. A subdiagram of δ between a given skew path Ω and its translation w + Ω is called a fundamental subdiagram, provided that (a) Ω coincides with the left bondary for sufficiently big i, (b) coincides with the right boundary for sufficiently small i, (c) the sets of boxes above Ω and below w + Ω do not intersect.

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CHAPTER 3. GENERAL THEORY

Such Ω is called a skew path associate with δ. Note that the i–coordinate of the Ω associated to an r–periodic δ is infinite, i.e., changes from −∞ to +∞. The j–coordinate can be bounded, which happens exactly for the infinite vertical rectangles. The fundamental subdiagram is a skew diagram that is a fundamental domian for the action of Zw in δ (so it contains n boxes). It may have empty rows. The basic subdiagram or, more generally, a portion of δ between the horizonta lines, i.e., with the boxes satisfying the inequalities c + 1 ≤ i ≤ c + v for a constant c ∈ Z, are examples of fundamental subdiagrams that do not have empty rows. Note that there are infinitely many different shapes, diagram types up to translations, of the fundamental subdiagrams unless δ is an infinite onecolumn or one-row, since fundamental subdiagrams can be collections of distant disconnected pieces. Similarly, a skew diagram δ without empty columns is called r–periodic for negative r if it is invariant with respect to w = (v + r, v), v ∈ N, and the basic fundamental diagram and any other fundamental diagrams, e.g., those formed by v consecutive columns, contain n boxes. The skew paths Ω in the definition of the fundamental diagram must coincide for r < 0 with the upper boundary for sufficiently big j and with the lower boundary for sufficiently small j. Respectively, the sets of boxes on the left of Ω and on the right of w + Ω do not intersect. The j coordinate changes from −∞ to +∞ for such Ω in the case of negative r. In short, skew paths associated to an r–periodic δ “cross” it from left to right as r > 0 and from bottom to top, as r < 0. Here we use the (x, y)–plane under the “visualization” (i, j) 7→ (x = j, y = −i).

3.7.3

Partitions

An s–increasing partition of a skew diagram δ of degree n is defined as follows. (a) It is a representation of δ as a disjoint union of u ≤ s skew nonempty subdiagrams {δ0 , . . . , δu } that are the portions of δ between a system of consecutive skew paths without crossings. (b) Different paths from such a system may have coinciding subpaths, but do not cross each other. The paths are numbered upwards from the lower boundary to the upper boundary. (c) Every δa is decorated with an integer 0 ≤ oa < s, called the o–number. They must form a strictly increasing sequence: o0 < o1 < · · · < ou . Note that the subdiagrams in the partition cannot be empty, but some may have empty rows. Let me comment on (b).

3.7. THE GL–CASE

363

The subdiagrams in a given partition are partially ordered naturally. Namely, δ 0 > δ 00 if at least one box of δ 0 is in the same column and higher than a box from δ 00 or is in the same row and to the left of a box from δ 00 . Using this ordering, property (b) means that, first, it is not allowed that δa > δb and at the same time δa < δb for different a, b, and, second, a > b if δa > δb . For instance, the representation of a skew diagram δ of degree n as a union of all its boxes counted from the last box in the last row through the first box in the first row (i.e., with respect to the inverse order) can be taken. These boxes can be ordered in other ways unless δ is a one-column or a one-row. Indeed, the boxes in the diagonals i + j = const are not ranked and can be placed in any order. Given such a “total” decomposition of δ, the corresponding o–numbers must increase for increasing numbers of δa . It is possible when s ≥ n. If s = n, then we must take the complete o–sequence {0, 1, 2, . . . , n − 1}. An opposit example is when we take the whole diagram and make u = 0, i.e., set δ0 = δ, and then assign an arbitrary number 0 ≤ o ≤ s − 1 to it. Such partitions exist for arbitrary s > 0. The increasing partitions of the fundamental subdiagrams will play the key role when decomposing the semisimple representations of HHn under the action of Hn . New (periodic) pairs. Extending the definitions above to a collection ∆, we call it r–periodic of degree n if all δ p are r–periodic and the total degree of the fundamental domains of {δ p } is n. The weights ξ associated with a new pair {∆, C} are those for all choices of the fundamental subdiagrams {δ˜p ⊂ δ p } calculated using the formulas above. Note that the fundamental diagrams are defined as subsets of {δ p }, i.e., as subsets of the corresponding copy of Z2 associated with p. The embeddings δ˜p ⊂ Z2 are needed when calculating the corresponding weights. The cp – numbers are taken from the corresponding δ p . We call the set of fundamental subdiagrams {δ˜1 , . . . , δ˜n } with the same set of c–numbers a fundamental subpair. The constraint for the c–numbers of the new (periodic) pairs is the same as for the old pairs. However, thanks to the condition tr = q s , it now reads as ca − cb 6∈ (1/r)Z for all 1 ≤ a, b ≤ m. Note that only the “diagonal” distances matter when calculating the weights of a pair. An s–partition of {∆, C} is a collection of fundamental diagrams {δ˜p } supplied with their (skew, increasing) s–partitions: n o δ˜ap , a = 0, 1, . . . , up ≤ s − 1, 1 ≤ p ≤ m , δ˜p = ∪a δ˜ap .

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Given p, we will stick to the notation δ˜p for a fundamental subdiagram of the component δ p of ∆, and will use a to number the components of a partition. Recall that the number of subdiagrams up can be smaller than s. The u–numbers may be different for different δ˜p . Partition weights. The main change in the definition of the weight associated to a partition will be that the integer oa must be added to the cp –number for every δ˜ap . The exact definition is as follows. Given a fundamental subpair {δ˜p }, we number its boxes in the totally inverse ordering that is a union of consecutive the inverse orderings (1) for the p–indices, (2) for the a–indices, (3) inside every δ˜ap , i.e., first, we take into consideration the p–indices, second, the a–indices, and, third, the (standard) numeration of the boxes inside δ˜ap . The partition weight is given by the formula ξl = cpl + oal + k(il − jl ) for δ˜p = ∪a δ˜ap , 0 ≤ a ≤ up ≤ s − 1,

(3.7.5)

where 1 ≤ l ≤ n, pl = p, al = a if box l belongs to δ˜ap . Let us consider the example of the “total partition” for a single periodic δ, i.e., when m = 1, {∆, C} = {δ, c}. We chose its skew fundamental subdiagram δ˜ and represented it as a union of all its n boxes counted from the last box through the first one. That is, δ˜ = ∪0≤a 0, the module U is isomorphic to a direct sum of the pairwise non-isomorphic irreducible Hn –modules associated with all increasing s–partitions of the fundamental subdiagrams {δ˜p }, where, recall, the integer oa is added to the weight of the partition component δ˜ap . We treat such a partition as a system of u1 +u2 +· · · +up skew diagrams of total degree n equipped with the c–numbers and o–numbers and define the Hn –modules correspondingly. (iv) The previous claim holds for the algebra HHon and the irreducible module U = UXξ defined for this algebra in terms of the restriction of the above weight ξ to C[Xi Xj−1 ] (this U is not a restriction of the above U to HHon ). The module U under the action of Hno is a direct sum of pairwise non-isomorphic irreducible modules over all classes of SL–equivalent s–partitions of fundamental subpairs. (v) The HHon –module UXξ is finite dimensional if and only if ∆ = {δ} and δ is either the infinite column as r > 0 or the infinite row as r < 0. In either case, r = ±n and s is an arbitrary natural number relatively prime to n. When s = 1, the dimension of this representation is 1. Generally, it equals sn−1 . ❑ An important part of the proof is that transposing the two consecutive segments of the weight associated with a pair {∆, C} coming, respectively, from p δ˜ap and δ˜a+1 gives another X–weight of UXξ . The corresponding X–eigenvector generates this module and can be obtained from the initial one by a chain of invertible intertwiners acting in UXξ . Part (v) readily follows from (ii) because the number of nonequivalent fundamental subdiagrams of a periodic diagram is infinite unless it is a column (infinite) or a row. Let us describe the s–partitions in this case and calculate the dimension formula.

3.7. THE GL–CASE

3.7.6

367

The column-row modules

An s–partition of the n–column or the n–row is the set n = {n0 , . . . , nu } such that n = n0 + · · · + nu for u < s, na ∈ Z+ . (3.7.6) The numbers na are identified with the consecutive segments of the column (or the row), counted from the last box through the first one. Recall that the δ˜ap –component of the weight is increased by oa (depending on p) by definition. The dimension of the corresponding irreducible Hno –module is n!/(n0 ! · · · nu !). Upon restriction to the Hn , it is the standard induced module corresponding to the decomposition n = n0 + · · · + nu . Given a decomposition (3.7.6), we need to assign an increasing sequence of the o–integers o = {0 ≤ o1 < o2 < · · · < ou ≤ s} to the n–numbers. Therefore the total contribution of the decomposition n to the dimension of U for all o–sequences will be the number of such o–sequences times n!/(n0 ! · · · nu !). Let us ignore the SL–equivalence for a while and calculate the desired dimension Dim. It is the sum of such products over all n–decompositions. There is a simple interpretation of this sum. One can naturally identify a pair n = n0 + · · · + nu for u < s, 0 ≤ o1 < o2 < · · · < ou ≤ s − 1, with the tensor product ⊗n1 ⊗nu 0 E(o, n) = e⊗n o0 ⊗ eo1 ⊗ · · · ⊗ eou

(3.7.7)

from (Cs )⊗n of the basic vectors e0 , e1 , . . . , es−1 ∈ Cs . Multiplication by n!/(n0 ! · · · nu !) simply means that the o–sequence is not assumed to be increasing, and, moreover, the o–numbers may coincide for different indices. Upon this relaxation, all tensor products of the basic vectors can be represented in the form (3.7.7). We obtain that the above Dim equals sn . The map to the linear space of the initial “o–increasing” tensor products is just the symmetrization homomorphism (Cs )⊗n → S n (Cs ) to the n–th symmetric power of Cs . The linear span of E(o, n) for increasing o = {o0 , . . . , os−1 } becomes isomorphic to S n (Cs ) under this mapping. The SL–equivalence means that we need to identify the vectors (3.7.7) by the folowing action of Zs = Z/(s) 3 c: ej 7→ ej 0 for j 0 = (j + c) mod s. This readily gives that dimUXξ = sn−1 in the “column-row” case. Actually, we have obtained more than just the dimension formula. This construction gives that the U –module from (v), when restricted to the nonaffine Hecke subalgebra Hn = hTi , 0 < i < ni, becomes isomorphic to the

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t–counterpart of the Sn –module C[Q/sQ] =Funct(Q/sQ) for the root lattice Q of type An−1 with the action of Sn : n Q = ⊕n−1 i=1 Z(ei − ei+1 ) ⊂ Z = ⊕i Zei .

Recall that (n, s) = 1, so here one can take the weight lattice P instead of Q. There is another, much simpler, proof of this fact together with the dimension formula without using the Hno –decomposition, which can be readily extended to perfect finite dimensional representations of the double affine Hecke algebras for arbitrary reduced root systems. See Theorem 3.5.3, formula (3.6.26), and Theorem 3.10.6 below. Comment. (i) Berest, Etingof, and Ginzburg verified that the relation k = ±s/n, provided that s ∈ Z+ is not divisible by n, is necessary for the existence of finite dimensional irreducible representations of HHon in the rational limit. They also found that if such a representation exists, then its dimension is divisible by sn−1 and its Sn –character is divisible by the one for C[Q/sQ] for the root lattice Q. Later they justified the coincidence. See [BEG]. The weaker statement above, with an unknown multiplier, requires relatively elementary algebraic considerations. In particular, it gives that the dimension of this representation must be greater than or equal sn−1 . Our q, t–representation U is in the space C[Q/sQ] almost by construction. It is not difficult to see that it has a rational limit. Thus the standard deformation argument combined with the inequality from [BEG] readily shows that our representation remains irreducible upon the rational degeneration and that the “unknown multiplier” in the rational theory is actually 1. (ii) Generally, to establish the connection between the DAHA representations in the q, t–case and the “rational” ones, there is a general method based on the existence of an isomorphism from HH to its rational degeneration HH00 upon proper completion (this was conjectured by Etingof). We√set q = eh , t = q k , and k 6= 0, and operate over the formal series in terms of h. The completion is not needed in the finite dimensional case. This connection was established in [C29] by combining the Lusztig-type isomorphism discussed in [C23](Appendix) and the straightforward formula connecting the trigonometric and rational differential Dunkl operators. The first map connects HH and its trigonometric degeneration HH0 ; the second one leads to an identification of HH0 and HH00 (upon proper completion). It is not just a formal isomorphism. It establishes an isomorphism between finite dimensional representations of HH for generic q and those for HH00 . By generic, we mean that q a tb do not represent nontrivial (6= 1) roots of unity for integral a, b. See Section 3.11 below. (iii) We also mention an important recent work [Vas] devoted to the K– theoretic classification of irreducible representations of HH for arbitrary roots systems (with coinciding k), including those subject to the constraint tr =

3.7. THE GL–CASE

369

q s from the theorem. In a sense, it is a continuation of [KL1]. However, it involves more sophisticated geometric methods and related combinatorial problems. In the case of GLn , Vasserot proves that irreducible HHn –modules are described in terms of cylic quivers. Combinatorially, his answer is equivalent to that from Theorem 3.7.3. There is an important continuation [VV2]. Under minor technical constraints, DAHA of type A is Morita-equivalent to the quantum affine Schur algebras at roots of unity. The latter are known to be also related to representation of the cyclic quivers. Vasserot also gives a geometric description of the perfect representations, including the “homological” deduction of the dimension formula for arbitrary root systems; see Theorem 3.10.6 below. ❑ The representations from (v) are unique finite dimensional representations among all irreducible representations, not only among the X–semisimple ones. To see this we will generalize the Bernstein-Zelevinsky classification. Here we need the considerations that are beyond the Main theorem (the semisimple case), although the technique essentially remains the same, the intertwiners.

3.7.7

General representations

The classification will be given in terms of periodic diagrams, but with the following relaxation of the conditions from (3.7.3): ni ≥ ni+1 or {ni = ni+1 − 1 and mi ≥ mi+1 − 1}.

(3.7.8)

We call such diagrams generalized. Respectively, we define the generalized pairs and then construct the corresponding weights using the same formulas as above. For generic q, t, an arbitrary irreducible representation of HHn is induced from an irreducible representation of Hn . This can be proved following [C23]. Therefore all of them are infinite dimensional for such q, t. This subsection does not contain justifications. It is somewhat beyond the framework of this chapter. We hope that the details will be published elsewhere. Theorem 3.7.3. Imposing the relation tr = q s for r > 0, s > 0, (r, s) = 1, all X–cyclic irreducible HHn –modules up to isomorphism and multiplication π by a constant are in one-to-one correspondence with the generalized new pairs {∆, C} up to the equivalence, defined in the same way as above. We can skip the case of r < 0 because it follows formally from the positive case thanks to the automorphism of HHn sending Xi 7→ Xi , π 7→ π, q 7→ q, Ti 7→ −Ti , t 7→ t−1 . ❑

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These modules are quotients of the induced modules I = I∆,C corresponding to the weights associated with {∆, C}. As above, the construction of such weights involves the basic fundamental subdiagrams formed by consecutive rows of {δ˜p }. The latter depends on the position of the zero row, but different choices can be compensated for proper changes of the c–numbers. The induced modules I are isomorphic to the affine Hecke Y –algebra P y def Hn == hπ, Ti , 1 ≤ i ≤ n − 1i. Let I 0 = I/( Hny (1 − si )) summed over all simple (nonaffine) reflections si corresponding to adjacent boxes in the rows of the basic fundamental subdiagrams of ∆. The I 0 is an HHn –module, and the module from the theorem is its unique irreducible nonzero quotient; cf. [C4] and the references therein. The w–periodicity and other definitions remain unchanged. The following reservation about the fundamental subdiagrams and their s–partitions is needed. We assume that the fundamental subdiagrams are intersections of a given periodic diagram with sufficiently big skew diagrams, i.e., they are the portions of a given δ between two skew paths interesecting δ from left to right. They give some (but not all!) submodules of the HHn –module under consideration, provided that they are generalized. Similarly, the partitions are portions of fundamental subdiagrams between consecutive skew paths intersecting δ “from left to right” (r > 0.) The s– partitions of fundamental subpairs of {∆, C} describe (some but not all) submodules of the corresponding irreducible representations upon the restriction to Hn . Here we consider the increasing partitions only. The partial ordering of the diagrams is as follows. We set δ > δ 0 if and only if it holds for two disjoint skew diagrams containing δ and δ 0 , respectively. These partial results on the restriction to the affine Hecke algebra Hn are obviously weaker, than what can be achieved in the semisimple case, however, they are sufficient to prove that finite dimensional irreducible representations are X–semisimple. Because of the duality X ↔ Y, they are also Y –semisimple. Indeed, it is clear that the irreducible X–cyclic representations of HHon have infinitely many non-isomorphic Hno –submodules, except in the case of the infinite column (recall that r > 0). Comment. (i) More explicitly, it is not difficult to construct an infinite chain of X–eigenvectors with pairwise distinct eigenvalues if ∆ is not a column. This argument can be applied to arbitrary root systems; however, the finite dimensional representations are not described yet in such generality. (ii) There is a general “geometric” approach, the Vasserot theory mentioned above. It is based on the advanced K–theory of affine flag varieties. There are technical difficulties when dealing with the corresponding Springer fibers and the corresponding geometric classification is far from simple. How-

3.8. SPHERICAL REPRESENTATIONS

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ever, we think that the K–theoretic methods are more promising for general representations, even in the GLn –case (where the intertwiners can be used). (iii) Our tools based on the technique of intertwiners are good enough for semisimple and spherical representations. However, they are not sufficient to describe more general representaions. In a sense, the technique of intertwiners is similar to the classical approach based on the highest vectors and the creation and annihilation e, f –operators (raising and lowering operators) in semisimple Lie algebras. The disadvantages are similar, indeed. ❑

3.8

Spherical representations

It is a continuation of Section 3.6 of this chapter. The parameters q, t, ξ are still arbitrary (i.e., nongeneric). We assume that tsht 6= ±1 6= tlng unless otherwise stated. Let us continue the list of basic definitions.

3.8.1

Spherical and cospherical modules

An HH[ –module V will be called X–spherical if it is generated by an element v, a spherical vector, such that Ti (v) = Xαi (q ρk )v = ti v for 1 ≤ i ≤ n, Xa (v) = Xa−1 (q ρk )v = q −(a , ρk ) v for a ∈ B.

(3.8.1)

It is an X–cospherical module if the space of linear form $ satisfying 1/2

(Ti (u)) = ti

$(u) for 1 ≤ i ≤ n,

$(Xa (u)) = q −(a , ρk ) $(u) for a ∈ B, u ∈ V,

(3.8.2)

is nonzero and at least one of the $, called a cospherical form, does not vanish identically on each HH[ –submodule of V. We will mainly call the vectors v satisfying (3.8.1) X–invariants and the cospherical forms X–coinvariants. It will be assumed that the spaces of invariants and coinvariants are finite dimensional. It is true for all modules in the category OX . Note that HH[ –quotient of a spherical representation is spherical, and nonzero submodules of a cospherical module are cospherical. An X–spherical module is X–cospherical and vice-versa for pseudo-unitary modules V in the category OX . The bi-form identifies the spaces of invariants v and coinvariants $. Recall that it is nondegenerate ?–invariant and ∗–hermitian. One can drop the last condition in this definition. The cospherical form is constructed as follows. If v is a given spherical vector, then the linear form (u, v) is a coinvariant and has no HH[ –submodules

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in its kernel because the latter have to be orthogonal to HH[ v = V. Therefore (u, v) is a cospherical form. To construct an X–spherical vector from an X–cospherical form, we use that the coinvariants vanish on VX∞ (ξ) (these spaces are finite dimensional) unless the X–character ξ is from the second line of (3.8.1). Indeed, they obviously vanish on VX (ξ) for such ξ, so we can go to VX∞ (ξ)/VX (ξ) and continue by induction. Therefore an arbitrary coinvariant $(u) can be represented as (u, v) for an invariant v. The latter has to be a spherical vector for a cospherical form $. Otherwise the orthogonal complement of HH[ v 6= V would belong to the kernel of $, which is impossible. This complement is nonzero since V ∈ OX . Concerning the coincidence of the space of invariants and coinvariants for V ∈ OX , it is not necessary to assume that the form is HH[ –invariant. It is [ sufficient to have a nondegenerate ∗–bilinear and HX –invariant form. Similarly, a Y –spherical vector, a cospherical form $, Y –invariants, and Y –coinvariants are defined by the relations: Ti (v) = ti v, πr (v) = v, for i > 0, πr ∈ Π[ , Ya (v) = q (a , ρk ) v for a ∈ B, $(Ti (u)) =

1/2 ti

(3.8.3)

$(u), $(Ya (u)) = q (a , ρk ) $(u).

Note the different signs of (a , ρk ) for X and Y. A module V with a one-dimensional space of invariants v has a unique nonzero spherical submodule. It is the HH[ –span of v. Dualizing, a module V with a one-dimensional space of coinvariants $ has a unique nonzero cospherical quotient. It is V divided by the sum of all HH[ –submodules of V inside the kernel of $. If V is already (co)spherical, then it has no proper (co)spherical submodules or, respectively, quotients. The main example is IX [ξ], which has a unique Y –cospherical quotient because the space of Y –coinvariants $ from (3.8.3) for the module IX [ξ] is one-dimensional. Cf. [C23](Lemma 6.2). def The polynomial representation V == Qq,t [Xb ] = Qq,t [Xb , b ∈ B] is Y –spherical and maps onto an arbitrary given Y –spherical module. The constant term is its Y –coinvariant. If q, t are generic, then V is irreducible and automatically Y –cospherical. When q is a root of unity there are infinitely many irreducible spherical modules. Comment. (i) We note that in the definition of spherical and cospherical modules, one may take any one-dimensional character of the affine Hecke subalgebra with respect to X or Y in place of (3.8.1), (3.8.2), and (3.8.3). The spherical representation for generic q, t remains isomorphic to the space of polynomials, however, with different formulas for the Tˆ–operators depending on the choice of the character. The functional representations, which are defined via the discretization of these operators, also change. We will stick to

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373

the “plus-character” in this section, leaving a straightforward generalization to the reader. (ii) It is worth mentioning that relations between the spherical and cospherical representations constructed for different one-dimensional characters play an important role in the theory of HH as well as for the classical representations of affine Hecke algebras. An example is a general theory of the shift operators (Opdam, Felder, Veselov, Macdonald, and the author). We will not consider these questions.

3.8.2

Primitive modules

The definition is of a general nature, but these modules are mainly needed for finite dimensional representations. Recall that finite dimensional HH[ – modules exist only when either q is a root of unity or q is generic and k is singular, a special rational number. By generic, we mean that the products (q 1/(2m) )i (t1/2 )j do not represent nontrivial roots of unity for i, j ∈ Z. We define Y –principal spherical HH[ –modules as Y –spherical HH[ –modules equipped with nondegenerate ?–invariant ∗–bilinear forms. So these modules are also Y –cospherical. Conversely, the modules that are spherical and at the same time cospherical have to be principal, thanks to Proposition 3.3.2. In this definition, we do not assume the pairing to be ∗–hermitian. We only need it to be nondegenerate. A Y –principal module is called Y –primitive if the dimension of the space of its Y –coinvariants is one. Using the nondegenerate form, we conclude that the dimension of the space of Y –invariants has to be one as well. Note that these dimensions always coincide for Y –principal spherical modules in the category OY , as was discussed above (for X instead of Y ). A primitive module has no nontrivial spherical submodules and cospherical quotients. The key property of such modules, a variant of the Schur lemma, is that a nonzero HH[ –homomorphism between two primitive modules V1 → V2 is an isomorphism. Indeed, the image of V1 is spherical and therefore must coincide with V2 . Hence V2 is a cospherical quotient of V1 , which is impossible unless the kernel is zero. We also have the following proposition. Proposition 3.8.1. Let V be a Y –principal module from OY embedded into a finite direct sum Vsum = ⊕Vi of Y –primitive modules such that its intersections with all Vi are nonzero. Then V coincides with Vsum . P Proof. The Y –invariants of Vsum are linear combinations ci vi of the spherical vectors vi ∈ Vi . The space of Y –invariants of V is smaller than the space of invariants of Vsum if V 6= Vsum . P Similarly, Y –coinvariants of Vsum are linear combinations $ = ci $i of the cospherical forms $i of Vi naturally extended to the sum. Since V has

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nonzero intersections with Vi , all such $ 6= 0 remain nonzero when restricted to V. Indeed, ci 6= 0 for certain i and if $(V ) = 0, then the cospherical form $i of Vi vanishes on V ∩ Vi , which is impossible. Using the coincidence of the dimensions of the spaces of invariants and coinvariants for Y –principal modules in OY , we conclude that V = Vsum . ❑ Later we will see that a Y –primitive X–semisimple module is irreducible. Also, a Y –principal X–cyclic module, i.e., generated by an X–eigenvector, is Y –primitive and a Y –spherical X–cyclic irreducible module which has at least one coinvariant is primitive (the next proposition). We note that a Y –spherical module such that V = ⊕ξ VX∞ (ξ) has to be finite dimensional (in particular, one can take V ∈ OX ). Indeed, V = ⊕ξ VY∞ (ξ) because it is a quotient of the Y –induced module with the “plus-character” as the weight. Finitely generated modules that have such decompositions for X and Y together are finite dimensional. See Section 3.6.1. Assuming that a finite dimensional Y –spherical module V possesses at least one Y –coinvariant $ 6= 0, let V˜ be the quotient of V with respect to the intersection of all radicals of all ∗–bilinear ?–invariant forms. Since V is finite dimensional there exists a Y –coinvariant with the kernel precisely coinciding with the intersection of the kernels of all coinvariants. This coinvariant makes V˜ cospherical and therefore Y –principal. Obviously it is the universal Y – principal quotient of V . Comment. It is worth mentioning that spherical (finite dimensional) representations in the classical theory of the affine Hecke algebra are defined with respect to its nonaffine Hecke subalgebra and always have isomorphic spaces of invariants and coinvariants provided that the latter algebra is semisimple. Therefore irreducible spherical modules are primitive in the above sense, which readily gives that (affine) primitive modules are simply irreducible quotients of the polynomial representation of the affine Hecke algebra. ❑ In the next proposition, the field of constants is assumed algebraically closed. Note that the definition of principal and primitive modules does not depend on the particular choice of the field of constants. They remain principal or primitive over any extension of Qq,t , and vice versa. If they are defined over Qq,t and become principal/primitive over its extension, then they are already principal/primitive over Qq,t . Recall that the ∗–bilinear forms above and below are not supposed to be ∗–hermitian. By the radicals, we mean the left radicals of such forms. Proposition 3.8.2. A Y –spherical module V ∈ OX is principal if and only if the dimensions of its X–eigenspaces V (ξ) are no greater than one and there exists a Y –coinvariant of V , that is nonzero at all nonzero X–eigenvectors. The sum Vsum = ⊕Vi of Y –principal modules from OX is Y –principal if and

3.8. SPHERICAL REPRESENTATIONS

375

only if SpecX (Vi ) ∩ SpecX (Vj ) = ∅ for i 6= j.

(3.8.4)

If Y –principal V ∈ OX is generated by its X–eigenvectors, then it is a direct sum of primitive modules. Proof. In the first place, given a Y –spherical vector v ∈ V (by definition, it generates V ), the coinvariants $ are in one-to-one correspondence with ?– invariant ∗–bilinear forms on V . Namely, any such form can be represented as $0 (f1 f2∗ ) for f1 , f2 ∈ V, where $0 is the pullback of $ with respect to the HH[ –homomorphism V → V, sending 1 to v. That is, (v1 , v2 )$ = $0 (f1 f2∗ ), where v1 = f1 (v) = f1 (X)v, v2 = f2 (v) = f2 (X)v. It follows from Proposition 3.3.2. The form may be degenerate and non-hermitian. Recall that hermitian forms (satisfying (u, v) = (v, u)∗ ) correspond to coinvariants of V such that $0 (f ∗ ) = $0 (f )∗ . We do not use hermitian forms in the proposition. The radical of ( , )$ is the greatest HH[ –submodule of V inside the kernel of $. Hence this form is nondegenerate if and only if the coinvariant $ is a cospherical vector. Let us check that in the category OX , a Y –coinvariant $ is a cospherical form if and only if $(e) 6= 0 for all X − eigenvectors e ∈ V.

(3.8.5)

These relations are obviously sufficient, since any submodule of V has at least one nonzero X–eigenvector. Let us verify that (3.8.5) is necessary. If $(e) = 0 for such e, then $(HY[ X (e)) = 0 = $(HH[ (e)), where by X we mean the algebra generated by Xb for b ∈ B. Now, if there exists a weight ξ such that dim V (ξ) > 1, then V cannot be cospherical because any given coinvariant vanishes on a proper linear combination of e1 and e2 . Thus the first claim holds as well as relations (3.8.4). Comment. Note that the same argument proves that the dimension of the space of Y –coinvariants of an X–cyclic module is no greater than one. In particular, Y –principal X–cyclic modules are Y –primitive. Indeed, otherwise HH[ v 6= V for an arbitrary X–eigenvector v of V because HH[ v belongs to the kernel of a proper coinvarinat of V vanishing at v, that always exists if there are at least two non-proportional coinvariants. ❑ Coming to the last claim, for each X–eigenvector e 6= 0 in V, let We be the hyperplane of all coinvariants of V vanishing at e in the space W of all coinvariants. The codimensions of We are one because V is principal. The intersection of all We consists of the coinvariants vanishing at all X– eigenvectors of V, so it is zero because V is generated by its X–eigenvectors.

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Since we have at least two different We , we may pick finitely many coinvariants $i (two are sufficient for the induction step) that do not belong (all of them) to any particular We and consider the direct sum of the quotients Vi of V by the corresponding radicals. The resulting map V → ⊕Vi is injective. Indeed, for any X–eigenvector eo 6= 0 in its kernel, the hyperplane Weo would contain all $i , which is impossible. Applying this procedure repeatedly to the resulting factors Vi (the quotients of V are generated by X–eigenvectors as well as V ) we can eventually make all Vi in the sum primitive. We may also assume that neither Vi can be removed from the sum ⊕Vi , i.e., that the intersections V ∩ Vi are all nonzero, and use the previous proposition. ❑

3.8.3

Semisimple spherical modules

Generally speaking, primitive modules can be reducible. However, in this chapter, we will mainly stick to the semisimple modules. Let us check that Y –primitive X–semisimple modules are irreducible. If such V has an HH[ –submodule U , then its orthogonal complement U ⊥ with respect to the (unique) ?–invariant ∗–bilinear form has zero intersection with U. Here we use that the X–spectrum of Y –principal module is simple. Hence U ⊕ U ⊥ = V. However, this is impossible because the space of Y – invariants of V is one-dimensional and generates V as an HH[ –module. Let V be an irreducible X–semisimple HH[ –module that is also Y –cosphe[ rical. Then V = IX [ξ]/J for proper P ξ and proper HH –submodule J such that Jξ ⊂ J ⊂ IX [ξ] for Jξ = HY Φwb , where the summation is over ¨ ∗ [ξ]. w b∈Υ It is Y –cospherical if and only if $+ (J ) = 0 for the plus-character of IX [ξ] ' HY[ : 1/2

$+ (Ti H) = ti $+ (H), $+ (πr H) = $+ (H), $+ (1) = 1.

(3.8.6)

The condition $+ (J ) = 0 readily results in Υ∗ [ξ] = Υ+ [ξ] since $+ (Φwb ) 6= 0 ¨ − [ξ]. We see that the condition Υ+ [ξ] ⊂ Υ0 [ξ] is fulfilled for ξ, for w b ∈ Υ and for any other X–weight ξ 0 , under the assumption that V is irreducible, X–semisimple, and Y –cospherical. In such V, the eigenspace V (ξ) is one-dimensional and Swb = 1 in V (ξ) c [ [ξ]. Indeed, $+ (Swb (1ξ )) = 1 in H[ due to the normalization of S for w b∈W Y from (3.3.27). Here 1ξ is 1 from HY[ with the action of {X} via ξ. Therefore the same holds in any quotients of HY[ by submodules belonging to Ker $+ . c [ [ξ], we obtain that Swb 7→ 1. Since V (ξ) is an irreducible representation of W Let us demonstrate that the conditions (a) Υ+ [ξ] ⊂ Υ0 [ξ], (b) dimC V (ξ) = c [ [ξ], are sufficient to make V irre1, and (c) Swb = 1 in V (ξ) whenever w b∈W ducible and Y –cospherical.

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377

Given a weight ξ, first we need to check that X ¨ + [ξ], Jξ = HY Φwb 6= HY , summed over Υ assuming that Φwb are well defined. It is obvious since $+ (Jξ ) = 0. Moreover, the latter relation gives that U = IX [ξ]/Jξ has a unique cospherical quotient. It is the quotient by the span of all HH[ –submodules in the kernel of $+ on U. Let us denote it by V. The restriction of $+ to the eigenspace V (ξ) is nonzero; otherwise it is identically zero on the whole V. The quotient of V by the HH[ –span of the kernel of $+ on V (ξ) is irreducible and cospherical. Hence it coincides with V due to the uniqueness of the cospherical quotients. c [ [ξ] is trivial. We come to the Moreover, dim V (ξ) = 1, and the action of W following theorem. Theorem 3.8.3. i) The induced representation IX [ξ] possesses a nonzero irreducible Y –cospherical X–semisimple quotient V if and only if Υ0 [ξ] ⊃ Υ+ [ξ] ⊂ Υ− [ξ].

(3.8.7)

Such a quotient is unique and has one-dimensional V (ξ) with the trivial (Swb c [ [ξ]. Its X–spectrum is simple. act as 1) action of the group W c∗[ [ξ]. If This V is X–unitary if and only if µ• (w) b = 1 for all w b ∈ W such a V is finite dimensional, then it is spherical and, moreover, primitive, i.e., it has a unique nonzero Y –invariant and a unique Y –coinvariant up to proportionality. Setting V = Vsph [ξ], the modules Vsph [ξ 0 ] and Vsph [ξ] are 0 b for w b ∈ Υ+ [ξ]. isomorphic if and only if q ξ = q w((ξ)) ii) Finite dimensional Vsph [ξ] can be identified with def def c [ [ξ]. F0 [ξ] == Funct(Υ0+ [ξ], Qξ ) where Υ0+ [ξ] == Υ+ [ξ]/W +

(3.8.8)

Here the action of HH[ is introduced via formulas (3.4.10) and is well defined on F0 [ξ]. The characteristic functions {χwb , w b ∈ Υ0+ [ξ]} form a basis of F0 [ξ] and are permuted by the intertwiners Swb , which become w b in this module. The pseudo-hermitian form of V is proportional to def

hf, gi0 ==

X

µ• (w)f b (w) b g(w) b ∗.

(3.8.9)

0 [ξ] w∈Υ b +

Proof. The “cospherical” part of (i) has been already checked. The spectrum of V is simple, thanks to Theorem 3.6.1. The condition (3.6.21), which is necessary and sufficient for the existence of an invariant X–hermitian form c [ [ξ]. See also (3.6.32). on V, means that µ• (w) b = 1 for all w b∈W ∗

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Since the space of Y –coinvariants of Vsph [ξ] is one-dimensional, then any ?–invariant ∗–bilinear form on V is hermitian (skew-symmetric) up to proportionality. This is because V has the “real” structure by the construction; we assume that ∗ fixes q ξ . Note that any pseudo-hermitian form on V is X–definite. The scalar squares of X–eigenvectors are nonzero because the X–eigenspaces are onedimensional. Indeed, if one of these squares vanishes, then the radical of the form is a nonzero HH[ –submodule of V, which contradicts its irreducibility. If such a V is also finite dimensional, then it is spherical and principal. The space of spherical vectors of V is one-dimensional because so is the space of $. See Proposition 3.8.2. By the way, we obtain a surjective HH[ –homomorphism V → V sending 1 to the spherical generator 1sph of V. Its kernel has to be an intersection of maximal ideals of V. This argument gives another demonstration of the simplicity of the X–spectrum of V. Part (ii) is nothing but a reformulation of (i). Really, we may introduce (the action of) si and w b in Vsph [ξ] as Si and Swb . Then the formulas from (ii) will follow from (i). ❑ As an immediate application of the proposition, we obtain that the irreducible representations V [ξ, %] from Proposition 3.6.4 cannot be, generally speaking, principal (spherical and cospherical). They are cospherical if and only if % = 1. For such %, µ• (a) have to be 1 on A(N ) to provide the existence of the X–unitary structure. Formulas (3.6.30) combined with the constraint (3.6.31) indicate that it may happen only for very special values of the parameters.

3.8.4

Spherical modules at roots of unity

Let us give a general description of spherical modules as q is a primitive root of unity of order N ≥ 0. We will use the lattices A(N ) = (N · A) ∩ B: A=

m0 {a ∈ P ∨ | (a, B) ⊂ Z}, m0 divides 2m. ˜ 2m ˜

(3.8.10)

˜ Here q 0 = q 1/(2m) is a primitive root of order m0 N. See formula (3.6.28) and the remark after it concerning m0 and q 0 . If (m, ˜ N ) = 1, then A(N ) = N (A ∩ B). Recall that Q ⊂ A = Q∨ ⊂ P ∨ for B = P. Also, A = P ∨ for B = Q. To start with, let us assume that tν are generic and establish that the b of V by the radical of the standard form of the polynomial repquotient V resentation is well defined, Y –spherical, Y –cospherical, Y –semisimple, X– semisimple, and irreducible.

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379

In the first place, all polynomials Eb are well defined, thanks to the intertwining formulas. See (3.3.45). Then eb = Eb (q −ρk )Eb are well defined because Eb (q −ρk ) 6= 0 due to (3.3.16). The polynomials Eb (b ∈ B) form a basis of V; so do Eb , and the formulas (3.4.1) for hEc , Ec i◦ , which were obtained for generic q, hold in the considered case as well, i.e., define the form h , i◦ on V. Its radical Vo =Rad h , i◦ can be readily calculated. b = V/V0 is well Proposition 3.8.4. (i) For generic tν , the HH[ –module V defined and def b ' ⊕b Qq,t Eb as − (αi∨ , b− ) ≤ Ni = V = N/(N, νi ), 1 ≤ i ≤ n, −1 {−(αi∨ , b− ) = Ni } ⇒ {u−1 b (αi ) ∈ R− i.e., αi ∈ λ(ub )}.

(3.8.11)

The other Eb lineraly generate Rad. (ii) Given b− from (3.8.11), let W 0 and W • be subgroups of W generated, respectively, by si such that (αi∨ , b− ) = 0 and −(αi∨ , b− ) = Ni . Their longest from (3.8.11) elements will be denoted by w00 and w0• . Then all possible u−1 b are represented in the form w w0• for w ∈ W such that l(w w0• w00 ) = l(w) + l(w0• ) + l(w00 ). b is simple, and this module is irreducible. The Y –spectrum of V Proof. The conditions (3.8.11) are necessary and sufficient to ensure that hEb , Eb i◦ 6= 0. Given such b− ∈ B− , formula (3.1.13) provides the following general description of all possible ub : α ∈ λ(u−1 ) ⇒ (α, b− ) 6= 0.

(3.8.12)

Equivalently, l(u−1 w00 ) = l(u−1 ) + l(w00 ) for the longest element w00 in the centralizer W00 of b− in W. −1 • If −(αi , b− ) = Ni , then αi ∈ λ(u−1 b ) for such i, i.e., ub is divisible by w0 −1 • −1 • • • in the following exact sense: l(ub ) = l(ub w0 ) + l(w0 ). Here w0 ∈ W0 is the longest element. This gives the description of ub from the proposition. Since {kν } is generic, Eb and Ec have coinciding q b] and q c] if and only if b− = c− mod A(N ) and ub = uc . Let us check this. Recall that A ⊂ B ⊂ P. So b− − c−P∈ N · P in this case. Provided (3.8.11) for b and c, we obtain that b− = c− + ±Nj ωj for some indices j and proper signs. Here (b− , αj∨ ) = Nj and (c− , αj∨ ) = 0 for j such that the term +Nj ωj appears in the sum, and vice versa for −Nj ωj . Let us assume that ub = u = uc . Transposing b− and c− , we can find j with the plus–sign of Nj ωj in this sum. Then u−1 = ub is divisible by sj . However, l(u−1 sj ) = l(u−1 ) + 1 because u = uc . This contradiction gives that ub and b is simple. uc cannot coincide for different b, c. Hence, the spectrum SpecY (V)

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b is spherical, i.e., generated by E0 = 1, and also Y –pseudo-unitary, it Since V must be irreducible. ❑ The next theorem gives a different (and more general) construction of spherical representations involving the central characters. Theorem 3.8.5. (i) For generic kν , the center Z of HH[ contains the finite sums X H= Fa Sa such that Fa ∈ Qq,t [Xb ], (3.8.13) a∈A(N )

b(Fa ) = Fa for b ∈ A(N ), Fw(a) = Fa for w ∈ W, and Y Fa · (tα1/2 qαj Xα − t−1/2 )−1 ∈ Qq,t [Xb ], [α, να j] ∈ λ(a). α [α,j]

(ii) The algebra ZX ∈ Qq,t [Xb ] consisting of W nA(N )–invariant polynomials is central as well as the algebra ZY of W nA(N )–invariant polynomials in terms of Y. The algebra HH[ is a free module of dimension |W | | W [ /A(N ) |2 over ZX · ZY . The elements (3.8.13) for rational Fa ∈ Qq,t (Xb ) constitute the center Zloc of the localization HH[loc of HH[ by nonzero elements of ZX . The algebra HH[loc is of dimension | W [ /A(N ) |2 over Zloc . (iii) Given a central X–character, a homomorphism of algebras ζ: ZX → Qq,t , let us denote its kernel by Ker ζ. Then def

Vζ == V / (V · Ker ζ)

(3.8.14)

is a Y –spherical module of dimension | W [ /A(N ) |. It is X–semisimple if and only if V · Ker ζ is an intersection of maximal ideals of V. (iv) The module Vζ is irreducible for generic ζ. If it is irreducible, then the space Vζ◦ = Hom(Vζ , Qξ ) is isomorphic to V [ξ, 1] from Proposition 3.6.4, where Ker ζ ∈ Ker ξ. The action of HH[ on Vζ◦ is via the Qq,t –linear X–trivial anti-involution H 7→ H ◦ fixing Ti (0 ≤ i ≤ n) , fixing Xb (b ∈ B), and sending πr 7→ πr−1 . Proof. Upon X–localization, the defining property of the intertwiners Swb readily gives that the elements (3.8.13) for rational Fa ∈ Qq,t (Xb ) form the center of the localization HH[loc . Any element of the latter can be represented c [ and arbitrary rational in this form with the summation over all w b ∈ W coefficients. Conjugating such sums by Xb and Swb , we get (3.8.13). The integrality condition is sufficient (but not necessary) to go back to HH[ . We readily obtain that ZX is central. The automorphisms of P GLc2 (Z) preserve the center of HH[ . In particular, ZY is central. The calculation of the rank of this algebra over ZX · ZY is a direct corollary of the PBW theorem (see (3.2.25)).

3.8. SPHERICAL REPRESENTATIONS

381

We leave the calculation of the rank of HH[loc over Zloc to the reader; it will not be used in the book. Note that the calculation of the rank of HH[ over the center without the X–localization is a difficult problem even for q = 1 = tsht = tlng . Switching to Vζ , its dimension is | W [ /A(N ) | because it is a quotient of [ HH by the left ideal generated by Ker ζ and Ker $+ ⊂ HY[ for $+ from (3.8.6). The X–semisimplicity of Vζ readily implies that the X–spectrum of this module is simple and ζ is associated with ξ. The module Vζ◦ = Hom(Vζ , Qξ ) is Y –cospherical and X–semisimple with the same spectrum. The action of HH[ via H 7→ H ◦ is well defined because this anti-involution preserves ζ. This module is irreducible if and only if Vζ is irreducible. We can apply Theorem 3.8.3 in this case. It results in Vζ◦ = V [ξ, 1]. ❑ An immediate application of this theorem is that the central elements are given by formulas that are uniform when represented in terms of the N –th powers of X and S. Let us formulate it letting (N, m) ˜ = 1 = (N, να ), α ∈ R+ , for the sake of simplicity. Then A(N ) = N (A ∩ B) and Nα = N/(N, να ) = N. Note that XbN Sa = Sa XbN , Xb SaN = SaN Xb for a ∈ A ∩ B, b ∈ B. We follow the calculation performed when checking formula (3.6.30). Corollary 3.8.6. (i) For a ∈ A ∩ B+ and b ∈ B, the elements X X N N N N Za,b = Sw(a) Xw(b) = Xw(b) Sw(a) w∈W

w∈W

linearly generate Zloc over the field of fractions of ZX . (ii) For the same a and b, the following elements belong to the center Z ⊂ HH[ : X Y N −N/2 (a,α∨ ) N w(XbN Sw(a) ) ). (tN/2 α Xα − tα w∈W

α∈R+

(iii) The action of Za,b in the polynomial representationPV is well defined N . The and is multiplication by the symmetric monomial function w∈W Xw(b) N symmetric monomial functions in terms of Y act in V as constants. ❑ Comment. (i) The centers of the counterpart of HH for C ∨ C1 at roots of unity were recently calculated by Oblomkov. The answer is N –uniform in the above sense and is given in terms of cubic surfaces.

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(ii) Similar to the theorem, one can describe the algebra of the elements of HH[ commuting with the nonaffine Hecke subalgebra upon a proper localization. Without the localization, it is a difficult problem even apart of roots of unity. This is connected with the results on the structure of the algebra of W –invariant polynomial differential operators due to Wallach, Levasseur, Stafford, and Joseph.

3.9

Induced and cospherical

In this section, we will figure out when the induced representations for generic q are irreducible and/or cospherical following the paper[C23]. These two problems are natural starting points in the classical theory of affine Hecke algebras. We assume that the products (q 1/(2m) )i (t1/2 )j for i, j ∈ Z do not represent nontrivial roots of unity.

3.9.1

Notation

Let q and {tν } be nonzero complex numbers and ξ ∈ Cn . We continue using notation from (3.4.4), that is, Xa˜ (bw) = q ζ Xa (q b+w(ξ) ) = q ζ+(a,b) Xw−1 (a) (q ξ ), where a ˜ = [a, ζ], a ∈ B.

(3.9.1)

The corresponding weight, a C–valued character of C[X], will be denoted by ξbw . It can be naturally identified with the characteristic function χbw def c[ . introduced in Section 3.4 of this chapter. We set uˆ(ξwb ) == ξuˆwb for uˆ, w b∈W c [ –orbit of ξ1 = ξ. This action is just the So such weights constitute the W c on the weights from the previous sections. Note that the affine action of W weights ξuˆ , ξwb with different indices uˆ, w b can coincide as characters of C[X] for certain q, ξ; we will identify them in this case. From now on, q is not a root of unity. This hypothesis is necessary and sufficient to make the stabilizers from (3.6.4) def c [ c [ (ˆ c [ , ξwˆ W b∈W ˆ ] = {w bu = ξu ˆ} ξ u) == W [ξu

c[ . finite for all uˆ ∈ W We also assume in this section that there exists a primitive weight ξo = ξuˆo def c [ def such that W o == W uo ) is generated by the elements from S o == W o ∩ ξ (ˆ {s0 , . . . , sn }. This means that W o is the Weyl group of the nonaffine Dynkin ˜ (not necessarily connected) with S o as the set of vertices. diagram Γo ⊂ Γ c [ –orbit of ξ always holds true The existence of the primitive weight in the W for sufficiently general q, t, say, for the degenerate double affine Hecke algebras

3.9. INDUCED AND COSPHERICAL

383

from Chapter 1. We denote the nonaffine Hecke algebra corresponding to Γo by Ho . Adding {Xαi as si ∈ S o } to Ho , one obtains the corresponding o affine Hecke algebra HX , which will be considered as a subalgebra of HH[ . c [ has a natural Recall that the space of finitely supported functions on W [ structure of an HH –module defined by formulas (3.4.10) in the basis of χ– functions. We call it the functional representation and denote it by the F[ξ]. It is well defined if the following holds: Xα (q ξ )qαj 6= 1 for all α ∈ R, j ∈ Z.

(3.9.2)

c [ (ˆ These conditions automatically imply that all W b are ξ u) are trivial and all ξw pairwise different for different indices and, moreover, linearly independent. Note that F[ξ] is isomorphic to F[ξ 0 ] for any weight ξ 0 = ξwb . The corresponding isomorphism is the right multiplication by w b (ξuˆ → ξuˆwb ). Similarly, one can introduce the delta–representation ∆[ξ] linearly generated by δwb with the action of HH[ via (3.4.11). We need to impose the same c [ –conjugated weights are condition (3.9.2). The delta-representations for W isomorphic too. There is a natural nondegenerate pairing between F[ξ] and ∆[ξ]: def

{g, δwb } == g(w), b {H(g), δwb } = {g, H ¦ (δwb )}, c[ for H ∈ HH[ , g ∈ F[ξ], w b∈W

(3.9.3)

for the following X–trivial anti-involution of HH[ : Tj¦ = Tj (0 ≤ j ≤ n), πr¦ = πr−1 (r ∈ O), Xi¦ = Xi (0 ≤ i ≤ n),

(3.9.4)

sending q, t to q, t (and AB to B ¦ A¦ ). It also gives a nondegenerate pairing between ∆[ξ] and the polynomial representation C[X] 3 g. Recall that the functional representation is the discretization of the polynomial representation. Generally speaking, F[ξ] and ∆[ξ] are not isomorphic. For instance, let us consider an important specialization ξ = −ρk assuming that t is generic. Then the module ∆ξ has a unique nonzero irreducible submodule ∆] that is isomorphic to the unique irreducible quotient F] of F [ξ] from (3.4.9). Recall that an HH[ –module I is called Y –cospherical if there exists at least one Y –cospherical form (the coinvariant) that does not vanish identically on every HH[ –submodule V 6= {0} of I. We will consider only Y –cospherical modules in this section, so Y will be dropped. The module ∆ξ is cospherical. Use the pairing {g, δwb } = g(w) b with the polynomial representation, which is spherical. By Y –cospherical forms, we mean the linear functions $ such that 1/2

$(Tj (v)) = tj v, $(πr (v)) = v for 0 ≤ j ≤ n, r ∈ O, v ∈ I.

(3.9.5)

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We will consider only the modules where the space of such forms {$} is of dimension no greater than one, adding this to the definition of cospherical modules in this section.

3.9.2

When are induced cospherical?

Let us figure out when the induced representations (X–induced, to be more precise) are irreducible and/or cospherical. We use the simplified notation Iξ instead of IX [ξ] for the induced representation of weight ξ, which is the universal HH[ –module generated by the element vξ such that Xa (vξ ) = q ξa vξ for all a ∈ B. As an HY[ –module, it is isomorphic to HY[ with the left regular action (vξ is identified with 1 ∈ HY ). Hence, Iξ posses only one HY[ –invariant 1/2 $ up to proportionality. It sends Tj → tj , πr → 1 upon identification vξ = 1. Theorem 3.9.1. (a) The module Iξ is irreducible if and only if ˜+. Xa˜ (q ξ ) 6= t±1 for all α ˜ = [α, να j] ∈ R α

(3.9.6)

c [ –orbit (and Irreducible Iξ are isomorphic for the weights ξ from the same W only for such weights). (b) The module Iξ is cospherical if and only if ˜+. Xa˜ (q ξ ) 6= t−1 for all α ˜∈R α

(3.9.7)

If (3.9.7) holds, then the induced modules associated with the weights in the form of ξwb and satisfying the same inequalities are isomorphic to Iξ . (c) Under the same conditions, Iξ contains a unique nonzero irreducible submodule Uξ . It is cospherical, i.e., it has a nonzero coinvariant $. Any other irreducible constituents of Iξ are not cospherical. An arbitrary irreducible c [ (ξ) cospherical module possessing an eigenvector of weight ξuˆ from the orbit W is isomorphic to Uξ . Proof. The statements are parallel to the corresponding affine ones. The first one is essentially due to Bernstein–Zelevinsky (the GL–case), the second is from [Kat]. There were previous papers on spherical Hecke modules and related p–adic theory by Muller, Casselman, Steinberg, and others. In the affine theory, both statements result from the irreducibility of the induced modules with the simplest weight trivial on the root lattice. There is a simple proof of this fact in [C10](Lemma 2.12), which uses the reduction to the degenerate (graded) affine Hecke algebra. Note that in the affine case, the definitions of spherical and cospherical modules coincide. Let us first renormalize the intertwining operators Φ from (3.3.27) to avoid the denominators: ˜ j = Tj (Xα − 1) + (t1/2 − t−1/2 ), 0 ≤ j ≤ n, Φ j j j −1/2 1/2 u ˆ c[ . ˜ = Φ ˜ j (ˆ Φ u) = Tj (Xαj (ˆ u) − 1) + (tj − tj ), uˆ ∈ W j

(3.9.8)

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385

˜ wb are well defined and enjoy the main property of the The corresponding Φ intertwiners (3.3.29). Recall that there are also the elements πr for r ∈ O in ˜ the product formulas for Φ. ˜ uˆ (w) The multiplication on the right by the element Φ b (which belongs to [ [ HY ) is an HH –homomorphism from the module Iwb ' HY[ into the module Iwˆ bu . Here Iw b is the induced representation corresponding to the weight (character) Xa → Xa (w b−1 ) = Xa (q ξwb−1 ). Similarly, if v is an X–eigenvector ˜ uˆ (v) is that associated with ξwˆ corresponding to ξuˆ , then Φ bu . w b We will use the spaces of generalized eigenvectors (3.6.3) in Iξ : Iξs (ξuˆ ) = {v ∈ Iξ , (Xi − Xi (ˆ u))s (v) = 0 for s ∈ N}, def

Iξ∞ (ξuˆ ) = Iξs (ˆ u) as s → ∞, Iξs (ˆ u) == Iξs (ξuˆ ).

(3.9.9)

u)} are finite dimensional and Iξ = ⊕Iξ∞ (ˆ u) for pairwise The spaces {Iξ∞ (ˆ different weights ξuˆ . Recall that they are identified if they induce the same characters. Generally, HH[ –modules I such that I ∞ (µ) are finite dimensional and I = ⊕I ∞ (µ) for all characters µ of C[x] constitute the category OX . All irreducible modules possessing an X–eigenvector belong to this category. If I ∞ (µ) 6= {0}, then I 1 (µ) 6= {0}. The immediate application of the generalized eigenvectors is that the set of all weights associated with X–eigenvectors of Iξ for any ξ is exactly the orbit {ξwb }. Indeed, if sufficiently general ξ 0 tends to ξ, then Iξ∞ (ξuˆ ) is the image of the direct sum of Iξ∞0 (ξuˆ0 ) over all ξuˆ0 → ξuˆ . If uˆo is primitive, the weight ξuˆo , to be more precise, then Iξ∞ (ˆ uo ) = Ho ⊂ HY[ .

(3.9.10)

o This space is the induced HX –module corresponding to the trivial character o {Xαi → 1, si ∈ S }. Moreover, it is irreducible and contains a unique X– eigenvector that is, 1 ∈ Ho (Lemma 2.12 from [C10]). Using this, we see that the dimension of the eigenspace Iξ1 (ˆ uo ) for primitive uˆo is no greater than one, as it is for any submodules and subquotients (called constituents) of Iξ . c[ ˜ , then there exists w b∈W Let us check (a). If Xa˜ (q ξ ) = t±1 α for some α such that the operator of the right multiplication by Φwb (1) has a nontrivial kernel. Hence I1 = Iξ is reducible. Let us assume that the inequalities (3.9.6) hold true. ˜ uˆ ∈ H[ are invertible. Applying them to vξ , we can All the elements Φ j Y obtain an eigenvector vo corresponding to ξuˆo . The latter generates the whole ˜ ujˆ )−1 in the opposite order. Hence we Iξ , since we can go back to vξ using (Φ can assume that uˆo = 1. If Iξ contains an HH[ –submodule V 6= {0}, then there exists at least one X–eigenvector v ∈ V (any vectors from Iξ belong to finite dimensional X– invariant subspaces). A proper chain of the intertwiners applied to v produces

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a nonzero eigenvector vξ0 corresponding to ξ1 = ξ. It is proportional to vξ , since it has to belong to Ho (see above). Hence v generates Iξ and V = Iξ . Now, the “only if” statement from (b). If Xa˜ (q ξ ) = t−1 α for a certain ˜ α ˜ ∈ R+ , then, using a chain of invertible intertwiners, we can replace vξ by an eigenvector v ∈ Iξ associated to ξwb such that Xj (w) b = t−1 j for some index −1 0 ≤ j ≤ n. Therefore we can assume that ξj = tj , which implies that $ = 0 on the nonzero HH[ –submodule ˜ 1 = (t Iξ Φ j j

1/2

−1/2

− tj

−1/2

)HY[ (1 − tj

Tj ),

so it is not cospherical. Let us suppose that ξ will satisfy (3.9.7). Applying the chains of intertwiners corresponding to the reduced decompositions of elements w b to 1 = vξ , we can obtain nonzero eigenvectors corresponding to all ξwb . Moreover, $ is nonzero at all of them, that can be obtained by this construction. Let us check this. One has that Xαj (ˆ u) = Xuˆ−1 (αj ) (q ξ ) 6= t−1 j in all the factors 1/2 −1/2 ˜ ujˆ = Tj (Xα (ˆ Φ u) − 1) + (tj − tj ) j

that may appear in the chains, since the roots uˆ−1 (αj ) are positive there. ˜ ujˆ only for such values. It However, $ vanishes after the application of such Φ gives the desired fact. Now, we can always pick a primitive weight ξo = ξuˆo satisfying the inequalities (3.9.7). It will be called plus-primitive, as are the corresponding eigenvectors and uˆo . The chain of intertwiners used above gives more if we start with ξ = ξo . Since the eigenvector 1 ∈ Iξo is of multiplicity one in Ho = Iξ∞o (ξ1 ), then ˜ wb (I ∞ (ξ1 )) = I ∞ (ξwb ). Φ ξo ξo Indeed, the dimensions are the same and the image of 1 is nonzero, since it belongs to any X–submodules of Iξ∞o (ξ1 ). Actually, this argument may be applied to any ξ (not only plus-primitive) if we know that the corresponding eigenvector is simple. ˜ Thus, all eigenvectors of Iξo are exactly the Φ–images of 1, in particular, they are simple and $ is nonzero at them. If Iξo contains a submodule where $ vanishes, then the latter possesses at least one eigenvector. However, it is impossible and Iξo is cospherical. To go from Iξo to Iξ , we need the following general lemma, where all the modules are from the category O (we will apply it to subquotients of induced representations). Actually, it was already used before. Lemma 3.9.2. Any submodule V of cospherical module I is cospherical. If here V 6= {0}, then I/V is not cospherical. There exists a unique irreducible

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nonzero submodule U ⊂ I. A module with at least one Y –cospherical form $ 6= 0 possesses a unique nonzero cospherical quotient. Proof. The first and second claims readily follow from the definition. If there are two irreducible submodules U, U 0 ⊂ I, then cospherical U is contained in the cospherical Uˆ = U ⊕ U 0 ⊂ I. Hence Uˆ /U ' U 0 could not be cospherical. The last statement is obvious as well. The kernel of the homomorphism to any cospherical module contains (the sum of) all submodules belonging to Ker($). It cannot be greater because of the second assertion. ❑ We are ready to check that Iξ satisfying the inequalities under consideration is cospherical. There exists a map Iξ → Iξo sending vξ = 1 to an eigenvector of Iξo with ξ as the weight. From now on, all maps will be HH[ – homomorphisms. Since Iξo is cospherical, the kernel of this map belongs to Ker($). For every ξwb , there exist at least one eigenvector in Iξ apart from Ker($) (we have already established this). Hence the image of Iξ contains all eigenvectors of Iξo and this map is surjective. Since the spaces I ∞ (µ) for fixed µ have the same dimensions in all induced modules with the weights from the same orbit, this map has to be an isomorphism. This also proves that all cospherical modules (with the weights from the same orbit) are isomorphic. Coming to (c), the lemma gives the uniqueness of the irreducible submodule Uξ and that there are no submodules in Iξ greater than Uξ that have cospherical quotients. Any cospherical irreducible module U containing an eigenvector of weight ξuˆ is the image of a surjective homomorphism Iuˆ → U . Here we use the universality of the induced representations. On the other hand, we can map Iuˆ to Iξ , since the latter contains an eigenvector corresponding to ξuˆ . The image will be cospherical. Therefore this map goes through U (the last claim of the lemma) and U is isomorphic to Uξ . ❑ Corollary 3.9.3. Imposing conditions (3.9.2), the module ∆[ξ] is isomorphic to the module Iξo for plus-primitive ξo . In particular, it is generated by any eigenvector with the weight satisfying (3.9.7). For instance, setting ξ = −ρ, the delta-representation ∆[−ρ] is generated by the eigenvector of weight ξw0 = tρ for the longest element w0 ∈ W, and ∆] (isomorphic to F] from (3.4.9)) is a unique nonzero irreducible submodule of ∆[−ρ], provided that t is generic. Proof. There exists a nonzero homomorphism Iξo → ∆[ξ]. Since both are cospherical, it has to be an isomorphism. ❑

3.9.3

Irreducible cospherical modules

There are examples when cospherical irreducible modules exist, but there are no cospherical induced representations at all. Therefore it is not always reasonable to represent irreducible cospherical modules as submodules of the

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induced ones even if the orbit contains a primitive weight, which is always imposed in this section. Somitemes it is better to describe them as quotients of induced modules. The following theorem makes the picture more complete and it will be used to endow irreducible cospherical representations with ?– invariant ∗–hermitian forms. c [ (ξ) Theorem 3.9.1. (a) Let us assume that there exists a weight ξ 0 ∈ W satisfying the condition dual to (3.9.7), that is, 0

Xa˜ (q ξ ) 6= tα for

˜+. all α ˜∈R

(3.9.11)

Then Iξ0 has a unique nonzero irreducible quotient Uξ0 . It is automatically cospherical. Other irreducible constituents (i.e., subquotients) of Iξ0 are not cospherical and do not contain eigenvectors associated with the weight ξ 0 . (b) All Iξ˜0 are isomorphic to Iξ0 for the characters ξ˜0 satisfying (3.9.11) c [ –conjugated to ξ 0 . If ξ from this and from the ξ 0 –orbit, i.e., when ξ˜0 is W orbit does not satisfy these inequalities, then Iξ˜ has an irreducible quotient that is not cospherical. Imposing (3.9.11), the eigenvectors of Iξ0 are simple. An eigenvector generates Iξ0 (and Uξ0 ) if and only if its weight satisfies these inequalities. A cosperical irreducible representation U with a weight from the orbit of ξ 0 is unique up to isomorphisms and is isomorphic to Uξ0 . Proof. Primitive weights satisfying (3.9.11) will be referred to as minusprimitive. They always exist in the orbit of the weight ξ 0 if we impose (3.9.11) and assume that ξ 0 is conjugated to a primitive one. Both conditions are permanently imposed in this section. The same terminology will be used for the corresponding eigenvectors. Let v be an X–eigenvector of an irreducible cospherical U with the weight ˜ ujˆ (v) with weight ξuˆ . Starting with v, we can construct the eigenvector v1 = Φ 1 u ˆ ˜ 1 (v1 ), and so on, until we obtain a minusξuˆ1 for uˆ1 = sj1 uˆ, then v2 = Φ j2 primitive eigenvector v 0 = vl associated with some ξo0 . As in Lemma 2.13 from [C10], the values ul−1 ) Xj1 (ˆ u), Xj2 (ˆ u1 ), . . . , Xjl (ˆ can be chosen avoiding t−1 jr (1 ≤ r ≤ l). Let us check that v 0 6= 0. Otherwise vr 6= 0 in this chain for some r < l and −1/2 0 = vr+1 = (tjr − 1)(Tjr + tjr )(vr ), where tjr 6= −1. However, U = HY[ (vr ), since vr is an eigenvector and U is irreducible. Hence, $(vr ) = 0 and $(U ) = 0, which is impossible. This gives a surjective map Iξo0 → U . It establishes an isomorphism o – between Ho = Iξ∞o0 (ξ1 ) and U ∞ (ξo0 ) because the first is an irreducible HX module. The surjectivity also holds because the eigenvector 1 is simple in

3.9. INDUCED AND COSPHERICAL

389

Ho . We see that the submodule that is the kernel of this map (and any of its subquotients) cannot contain ξo0 –eigenvectors. Hence U is a unique irreducible quotient of Iξo0 . By the way, here the combinatorial part can be simplified somewhat. We can finish this process with v 0 associated with an arbitrary primitive weight. Then (cf. the proof of Theorem 3.9.1(b)) it has to satisfy (3.9.11) automatically because otherwise we can construct a quotient of Iξo0 that contains no $ 6= 0 (use a proper intertwiner). Coming to the inverse statement, if ξo0 is minus-primitive, then Iξo0 has a unique nonzero irreducible quotient Uξo0 (it holds true for all primitive characters). On the other hand, it has a unique nonzero cospherical quotient V, which contains a cospherical irreducible submodule U . It is a quotient of a proper Iξ˜o0 . Due to [C10], Lemma 2.8, the latter is isomorphic to Iξo0 because all primitive eigenvectors are connected by invertible intertwiners. Hence we obtain a nonzero homomorphism from Iξo0 onto U , which results in the coincidence V = U and the isomorphism U ' Uξo0 . To establish the required isomorphism without any reference to [C10] and also check the remaining statements of the theorem, it is convenient to apply Theorem 3.9.1 for the character of HY[ −1/2

Ti → −ti

, πr → 1, 0 ≤ i ≤ n, r ∈ O,

replacing the character used in the definition of coinvariants $. The modules under consideration become cospherical for such a character. It gives, first, that all eigenvectors of Iξo0 are simple and can be obtained from 1 by the intertwiners. Here, if the intertwiner is not invertible, then its image belongs to Ker($). Second, all Iξ0 satisfying (3.9.11) are isomorphic to Iξo0 (and to each other). Moreover, all eigenvectors in Iξo0 corresponding to such weights ξ 0 are connected with 1 by invertible intertwiners (and only such eigenvectors). Third, if an irreducible constituent of Iξo0 contains an eigenvector associated with ξ 0 , then Iξo0 maps onto it through Iξ0 . It can happen for Uξo0 only. ❑ Plus-minus duality. Let us now assume that both plus-primitive and minus-primitive characters ξo , ξo0 can be found in the orbit of ξ. In this case, Uξ ' Uξ0 , since they are unique cospherical irreducible constituents, and, moreover, the first one contains the whole space Iξ∞ (ξ 0 ). The modules Iξ0 , Iξo0 are dual to each other in the following sense. Given a module I = ⊕I ∞ (µ) ∈ O, we combine the anti-involution ¦ from (3.9.4) with the natural anti-action of HH[ on the def

I ¦ == {f ∈ Hom(I, Qq,t ) s.t. f (I ∞ (µ)) = 0 for almost all µ}. It makes I ¦ an HH[ –module. We claim that Iξ¦o ' Iξo0 .

(3.9.12)

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First, Uξ¦o ' Uξo0 . Indeed, they have the same set of characters and coinciding dimensions of the spaces of generalized vectors. Use the duality. Hence Uξ¦o can be covered by Iξo0 . However, the latter has a unique irreducible quotient, which is Uξo0 . Second, the map ν : Iξo0 → Iξ¦o sending 1 to an eigenvector corresponding to ξo0 composed with the map Iξ¦o → Uξ¦o is obviously nonzero. This map is a dualization of the natural embedding Uξo ⊂ Iξo (Theorem 3.9.1). Third, the module Iξ¦o has a unique nonzero irreducible quotient. We conclude that ν is surjective and has to be an isomorphism because the dimensions of the spaces I ∞ (µ) are the same for Iξo0 and Iξo (their characters are from the same orbit). Invariant form. The following corollary shows that, generally speaking, induced modules have invariant ∗–hermitian forms. Corollary 3.9.4. In the setup of Theorem 3.9.1, the module Iξ0 possesses a unique up to proportionality ?–invariant ∗–hermitian form in the sense of Section 3.6 in this chapter. Its radical R coincides with the kernel K of the map Iξ0 → Uξ0 from Theorem 3.9.1. It induces a unique nondegenerate ?–invariant form on Uξ0 . Proof. Any Iξ0 can be considered as a limit of a one-parametric family of proper F [ξ˜0 ] satisfying the inequalities of the theorem for ξ˜0 . Therefore Iξ0 has a nonzero ∗–invariant form. If its radical R is smaller than K, then Iξ0 /R contains an irreducible submodule V 6' Uξ0 . All generalized X–eigenvectors with the characters not from V belong to its orthogonal compliment V 0 . Since V has no eigenvectors associated with ξ 0 , the image of 1 belongs to V 0 . A contradiction. The uniqueness of the ∗–invariant form on Uξ0 follows from the irreducibility. ❑ Provided we have (3.9.2), we claim that Iξo0 ' F[ξ] for the minus-primitive ξo0 weight from the orbit of ξ (cf. Corollary 3.9.3). Indeed, one can map Iξo0 into F[ξ]. We can use that ∆[ξ] is isomorphic to Iξo for the plus-primitive ξo . It results in the pairing { , } in Iξo0 × Iξo . If this pairing is degenerate, then its right radical (⊂ Iξo ) has to contain Uξo because the latter is the smallest irreducible submodule. However, this is impossible because the image of Iξo0 in F [ξ] contains Uξo0 ' Uξo as a constituent.

3.9.4

Irreducibility of induced modules

Let us generalize the first part of Theorem 3.9.1 and check that, generally speaking, cyclic irreducible modules of HH[ are induced from the irreducibles of HX . Proposition 3.9.5. (a) Given a finite dimensional irreducible HX –module U let as assume that 1 6= Xα˜ (q ξ ) 6= t±1 for α

˜ \ R, all α ˜∈R

(3.9.13)

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391

where ξ is any weight of U (it does not matter which because they are W – [ conjugated). Then the induced HH[ –module M = Ind HH (U ) is irreducible. U

[

HX

(b) Any irreducible HH –module M posessing an X–eigenvector with the c [ –conjugated to ξ from (3.9.13) is induced from its irreducible HX – weight W submodule U generated by an eigenvector with weight from W (ξ). Given the orbit W (ξ), such a submodule U is unique up to isomorphisms: MU ' MU 0 ⇒ U ' U 0. Proof. Let us check that the left-hand side inequality gives that M ∞ (ξ) = U ∞ (ξ) for M = MU , where (see (3.9.9)) by M ∞ (ξ), we mean the space of all generalized eigenvectors in M associated with a weight ξ. It holds for Iξ and for the induced HX –module IξX = IndC[X] HX (q ξ ) instead of M and U . However, Iξ and IξX cover M, U and, correspondingly, the spaces of generalized eigenvectors. Once the coincidence is checked at the ∞–level, it is valid for all levels of the spaces of generalized eigenvectors. In particular, M 1 (w(ξ)) = U 1 (w(ξ)) for any w ∈ W . We obtain that U = ⊕ζ∈W (ξ) MU∞ (ζ).

(3.9.14)

Given b ∈ B and w ∈ W, ˜ ζπ )−1 (M 1 (πb (ζ)) = M 1 (ζ), (Φ b ˜ ζπ = Φ ˜ π (ζ), ζ ∈ W (ξ). where Φ b b

(3.9.15)

˜ πw(ξ) in the We follow the notation from (3.9.8) and use the invertibility of Φ b 1 ∞ space M (w(ξ)) (and in M (w(ξ))), thanks to the right-hand side inequality. The elements {πb (ζ)} for the above b and ζ constitute the whole orc [ (ξ). Any nonzero irreducible submodule of M has at least one X– bit W c [ (ξ). Due to (3.9.15), it generates eigenvector for a weight from the orbit W the whole M . c [ –conjugated to ξ from An arbitrary irreducible M possessing a weight W (3.9.13) can be represented as MU , where U is any irreducible HX –submodule of M with an eigenvalue from W (ξ). The existence of U results from the same formula (3.9.15). Moreover, U can be reconstructed uniquely as a submodule of M by means of (3.9.14). Therefore M ' M 0 ⇒ U ' U 0 . ❑ Restriction to HX . Let us describe the structure of MU upon the restriction to HX , provided we have (3.9.13). In the first place, MU = ⊕Jc,ζ , where c ∈ B− , ζ runs over a fixed set of representatives of W (ξ) mod Wc for the centralizer Wc of c in W , ˜ ξ˜ ∈ W (c(ζ)). Jc,ζ = ⊕ξ˜MU∞ (ξ),

(3.9.16)

392

CHAPTER 3. GENERAL THEORY All {J} are HX –submodules. Their structure can be described as follows: ˜ ω (U˜c ), U˜c = Φ ˜ c (Uc ), Jc,ζ = ⊕ω Φ def

Uc == ⊕ξ0 U ∞ (ξ 0 )), for ξ 0 ∈ Wc (ζ).

(3.9.17)

Here {ω = u−1 b , b ∈ W (c)} = {ω ∈ W, λ(ω) ⊂ {α ∈ R+ s.t. (α, c) < 0}}. ˜ ω, Φ ˜ c are invertible because c = c− = πc and (α, c) 6= The intertwiners Φ 0 for all α ∈ λ($). The space Uc is a module over the subalgebra Pc of HX generated by {Ti , si (c) = c} and the whole Qq,t [X]. So is U˜c . Indeed, applying Φc means replacing the action of Xj by that of q (c,bj ) Xj for all 1 ≤ j ≤ n, sj (c) 6= c without touching {Ti }. We conclude that Jc,ζ is isomorphic to the representation of HX induced from the Pc –module U˜c . Its irreducibllity is equivalent to the irreducibility of the Pc –module Uc . The simplest application of this construction is decomposing the polynomial representation considered as an HY[ –module. See (3.2.33), (3.2.34), (3.2.35), and also [C20](formulas (3.15)–(3.17)). Summarizing, in the case of generic q, the classification of irreducible representations of HH[ is not too far from that in the affine Hecke case. If all tν coincide, then we can directly use the main theorem from [KL1]. Of course, there are representations that do not satisfy (3.9.13) and can be expected to be associated with more sophisticated ”Springer fibers.”

3.10

Gaussian and self-duality

We follow the notation from Section 3.8 of this chapter. An HH[ –module V is self-dual if the involution ε from (3.2.10) comes from a certain ∗–linear involution of V upon the lift to End(V ). A module is called P GLc (2, Z)– invariant if ε, τ± act there projectively, i.e., if there are automorphisms of V satisfying def

τ+ τ−−1 τ+ == σ = τ−−1 τ+ τ−−1 , τ± = ετ∓ ε−1 ,

(3.10.1)

and inducing the ε, τ± from Section 3.2 on End(V ). By P GLc (2, Z), we mean the central extension of P GL(2, Z) due to Steinberg, introduced by the relations (3.10.1). Neither functional nor polynomial representations are self-dual. Either τ− or τ+ act there respectively, but not all together. Actually, self-dual semisimple irreducible modules (from OX ) have to be finite dimensional. Indeed, they belong to both OX and OY , and one can readily check that they are finite dimensional.

3.10. GAUSSIAN AND SELF-DUALITY

393

def

The non-cyclic module V1 == γ˜ −1 V is self-dual. Namely, the following ∗–linear involution ψ(Eb γ˜ −1 ) = q −(b] , b] )/2+(ρk , ρk )/2 Eb γ˜ −1 for b ∈ B

(3.10.2)

corresponds to ε from the double Hecke algebra. It is formula (5.8) from [C21], which is equivalent to (3.4.28) above. Note that the module V1 is not P GLc (2, Z)–invariant because any extension of the automorphism τ+ has to be proportional to multiplication by γ˜ (X), that does not preserve V1 .

3.10.1

Gaussians

We turn to the Gaussians in Vsph [ξ], which will be called restricted Gaussians. Generally speaking, the problem is to make τ+ an inner automorphism. The analysis is not difficult, thanks to the Main theorem of Section 3.6. The Gaussian commutes with the X–operators, so it preserves the X–eigenspaces. This theorem provides their description. The Gaussian, if it exists in a spherical irreducible representation, has to be its element, not just an operator; the X–action makes such representations commutative algebras. Thus, the spherical case is especially interesting. According to the Main theorem, τ+ is inner in V = Vsph [ξ] if and only if the map c [ [ξ] and S ξ = Swb (q ξ ), b∈W Swξb 7→ τ+ (Swξb ), where w ∗ w b

(3.10.3)

is inner (i.e., multiplication by a matrix) in the eigenspace V (ξ). In the spherical case, the Swb become simply w b in the realization F0 [ξ], and the automorphisms τ+ (w) b can be readily calculated. We come to the following proposition. Proposition 3.10.1. The restricted Gaussian defined by the relations γ H γ −1 = τ+ (H) in V

(3.10.4)

exists in V = F0 [ξ] if and only if c [ [ξ] for b ∈ B, w ∈ W. q (b,b+2ξ)/2 = 1 whenever bw ∈ W ∗

(3.10.5)

def

Then it is proportional to γ∗ (bw) == q (b+2w(ξ),b)/2 . Proof. Let v be the cyclic generator of V (of weight ξ). We identify γ with the operator of multiplication by γ. If it exists, then γ(Swb (v)) = Swb (Sw−1 b ∈ Υ+ [ξ]. b )(v) for all w b γSw

(3.10.6)

−1 Here γw0b = Sw−1 is a function that can be readily calculated due to bγ b γSw the definition of τ+ . However, it is more convenient to involve F0 [ξ], where Swb

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is nothing but w. b Recall that τ+ is interpreted as the formal conjugation by γ(q z ) = q (z,z)/2 in any functional space. See (3.4.25) and (3.4.31). Therefore γw0b (ˆ u) = q (ξ

00 ,ξ 00 )/2−(ξ 0 ,ξ 0 )/2

for ξ 0 = uˆ((ξ)), ξ 00 = w((ξ)). b

(3.10.7)

Its action on v is γw0b (id) = γ∗ (w). b Thanks to the argument that has been used several times, formula (3.10.6) c∗[ [ξ]. This is exactly (3.10.5). defines γ correctly if and only if γw0b = 1 for w b∈W ❑

3.10.2

Perfect representations

Following the same lines, let us find out when V = Vsph [ξ] is self-dual i.e., ε can be made an inner involution in V. For the sake of simplicity, we use the functional realization of F0 [ξ] of V. So 1 is the spherical vector, δ0 = δid is the X–cyclic generator, and δ0 = δid . Recall that χwb (ˆ u) = δw,ˆ u) = µ• (w) b −1 χwb b u , δw b (ˆ

(3.10.8)

π for the Kronecker δw,ˆ b u . We also set δb = δπb in the case ξ = −ρk .

Theorem 3.10.2. (i) Provided that we have (3.8.7), the finite dimensional representation Vsph [ξ] is self-dual if and only if it is X–unitary and q −ρk = b q w((ξ)) for a certain w b ∈ Υ+ [ξ]. The representation Vsph [−ρk ], is HH[ –isomorphic to the irreducible X–cospherical Y –semisimple nonzero quotient of IY [ρk ], which is unique. The X–spectrum of Vsph [ξ], which is Υ+ [−ρk ], belongs to πB = {πb | b ∈ B} and is the negative of its Y –spectrum. The X–unitary structure of Vsph [−ρk ] is also Y –unitary. (ii) The module F0 [−ρk ] ' Vsph [−ρk ] has a basis formed by the discretizations of the spherical polynomials Eb : def

b k )) Eb0 (w) b == Eb (q −w((ρ ) for w b ∈ Υ+ [−ρk ].

(3.10.9)

They are well defined assuming that πb ∈ Υ+ [−ρk ]. Explicitly, F0 [−ρk ] = ⊕b• Qq,t Eb0• , where the summation is over the set of representatives def c [ [ρk ] = πb• ∈ Υ+ [−ρk ]/W = Υ0+ [−ρk ]. ∗

c [ [ρk ]. Moreover c [ [−ρk ] = W Here we use that W ∗ ∗ c [ [ρk ]. Eb0 = Ec0 whenever πb ((ρk )) = πc ((ρk )), i.e., πb−1 πc ∈ W ∗

(3.10.10)

3.10. GAUSSIAN AND SELF-DUALITY (iii) The map X X ψ0 : gb Eb0 7→ gb∗ δbπ for gb ∈ Qq,t , πb ∈ Υ+ [−ρk ],

395

(3.10.11)

is an HH[ –automorphism of F0 [−ρk ], inducing ε on HH[ . Here the δbπ depend only on the images of πb in Υ0+ [−ρk ]. Equivalently, h1i0 ψ 0 (f ) (πb ) = hEb0 , f i0 = h f ∗ Eb0 i0 , X def where hf i0 == µ• (w)f b (w). b

(3.10.12)

0 [−ρ ] w∈Υ b k +

b is necessary because ε sends Xb to Proof. The condition q −ρk = q w((ξ)) Yb , conjugating the eigenvalues, and the self-duality implies that SpecX = −SpecY . So if the involution of V, let us denote it by ψ 0 , exists, then ψ 0 (1) has to be proportional to δ0 . Its X–weight is −ρk . Here we use the F0 –realization of V. Recall that the Y –weight of 1 is ρk . Now let ξ = −ρk and

Υ0 [ρk ] ⊃ Υ− [ρk ] ⊂ Υ+ [ρk ].

(3.10.13)

It is a reformulation of (3.8.7) using the obvious relations Υ± [ξ] = Υ∓ [−ξ]. We know how to describe nonzero irreducible Y –semisimple quotients of IY [ρk ]. The latter module has such quotients thanks to (3.10.13) and Theorem 3.6.1. def [ The universal semisimple quotient is UY [ρk ] == HX /J for X ¡ ¢ def (3.10.14) J == HX · τ+ Φwb (q −ρk ) , ¨ + [−ρk ] and 1 is identified with the cyclic where the summation is over w b∈Υ generator of IY [ρk ]. Using Theorem 3.8.3, we obtain that UY [ρk ] has a unique nonzero irreducible X–cospherical quotient. This implies that the latter is a unique irreducible X–cospherical quotient of IY [ρk ]. Cf. Lemma 6.2 from Y [ρk ]. [C23]. We will denote it by Vsph Let us check that the module Vsph [−ρk ] ' F0 [−ρk ] is not only Y –cosherical but also X–cospherical. Since it is irreducible, we need to find a vector v satisfying (3.8.1): Ti (v) = ti v for i > 0, Xa (v) = q −(a , ρk ) v for a ∈ B.

(3.10.15)

Such a v is exactly the generator δ0 . Note that the first formula in (3.10.15) ¨ + [−ρk ]. holds because {s1 , . . . , sn } ⊂ Υ [ Summarizing, the HH –homomorphism IY [ρk ] → F0 [−ρk ] sending 1 7→ 1, which exists because the former module is induced, identifies the latter with Y [ρk ] constructed above. There are several immediate corollaries. the Vsph

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First, the operators Ya are diagonalizable in the module F0 [−ρk ]. Second, the Y –spectrum is simple there and coincides with the negative of the X– Y [ρk ] has spectrum. Third, the invariant Y –definite ∗–bilinear form on Vsph 0 to be proportional to the X–definite invariant form on F [−ρk ] upon this identification. Fourth, the polynomials Eb ∈ V are well defined for πb ∈ Υ+ [−ρk ]. Let us check the latter claim. Using the standard HH[ –homomorphism from V to F0 [−ρk ] and the simplicity of the Y –spectrum of the target, we diagonalize the Y –operators in the subspaces Σ(b) = ⊕cÂb Qq,t Xc ⊕ Qq,t Xb , for B 3 c  b,

(3.10.16)

starting with b = 0 and “decreasing” b as far as πb ∈ Υ+ [−ρk ]. See (3.3.4). The leading monomial Xb always contributes to Eb . Thus the Macdonald polynomials {Eb } are well defined in this range and their images Eb0 in F0 [−ρk ] are nonzero. Let us verify that Eb (q −ρk ) 6= 0 for such b. Following [C19, C20], we set {f, g} = {Lı(f ) (g(X))}(q −ρk ) for f, g ∈ V, ı(Xb ) = X−b = Xb−1 , ı(z) = z for z ∈ Qq,t ,

(3.10.17)

where Lf is from Proposition 3.3.1. Let us use the Qq,t –linear anti-involution of HH[ from (3.3.20): def

φ == ε ? = ? ε : Xb 7→ Yb−1 , Ti 7→ Ti (1 ≤ i ≤ n).

(3.10.18)

The basis of all duality formulas is the following lemma. Lemma 3.10.3. For arbitrary nonzero q, tsht , tlng , {f, g} = {g, f } and {H(f ), g} = {f, H φ (g)}, H ∈ HH[ .

(3.10.19)

The quotient V 0 of the polynomial representation V by the radical Rad{ , } of the pairing { , } is an HH[ –module such that (a) all Y –eigenspaces VY0 (ξ) of V 0 are zero or one-dimensional, (b) E(q −ρk ) 6= 0 if the image E 0 6= 0 of E in V 0 is a Y –eigenvector. Proof. Formulas (3.10.19) are from Theorem 2.2 of [C20]. See also [C23], Corollary 5.4. Thus Rad{ , } is a submodule and the form { , } is well defined and nondegenerate on V 0 . For any pullback E ∈ V of E 0 ∈ V 0 , E(q −ρk ) = {E, 1} = {E 0 , 10 }. If E 0 is a Y –eigenvector of weight ξ and E(q −ρk ) vanishes, then {Y [ (E 0 ), HY[ (10 )} = 0 = {E 0 , V · HY[ (10 )} for Y [ = Qq,t [Yb ]. Therefore {E 0 , V 0 } = 0, which is impossible.

3.10. GAUSSIAN AND SELF-DUALITY

397

We can use it to check that dim VY0 [ξ] ≤ 1. Indeed, there is always a linear combination of two eigenvectors in V 0 of the same weight with zero value at ❑ q −ρk , which has to be zero identically. The following fact is of similar nature. It will not be used in proving the theorem, but it is actually the key in the general theory of the pairing { , } and perfect representations. Cf. also the definition of the principal and primitive modules from Proposition 3.8.1. Proposition 3.10.4. If the quotient V 0 of the polynomial representation V by the radical Rad{ , } is finite dimensional and Rad is τ− –invariant for the canonical action of τ− in V, then V/Rad is an irreducible HH[ –module. Proof. Using φτ− φ = τ+ , the relation {τ+ f, g} = {f, τ− g} for f, g ∈ V 0 defines the action of τ+ and the action of σ in V 0 satisfying τ+ τ−−1 τ+ = σ = τ−−1 τ+ τ−−1 . def

The pairing {f, g}σ == {σf, g} = {f, σ −1 g} corresponds to the anti-involution ♥ = σ · φ = φ · σ −1 of HH[ from (3.3.21). It holds in either direction, from f to g and the other way round, but the form {f, g}σ , generally speaking, may be non-symmetric. Actually, it is symmetric, but we do not need it for the proof. Using this non-degenerate pairing, we proceed as follows. Any proper HH[ – submodule V 00 of V 0 contains at least one Y –eigenvector e00 , so we can assume that V 00 = HH[ e00 . The corresponding eigenvalue cannot coincide with that of 1, thanks to the previous lemma. Therefore {10 , V 00 }σ = 0 and the orthogonal complement of V 00 in V 0 is a proper HH[ –submodule of V 0 containing 10 , which is impossible. ❑ Let us check that the pairing {f, g}σ is symmetric, indeed. First of all, {τ+ (10 ), 10 } = {10 , τ− (10 )} = {10 , 10 } = 1 ⇒ {10 , 10 }σ = {σ(10 ), 10 } = {τ+ (10 ), τ− (10 )} = {τ+ (10 ), 10 } = 1. Then, {10 , f }σ − {f, 10 }σ = {(σ − σ −1 )(10 ), f } = {(1 − σ −2 )(10 ), f }σ . Recall that τ− (1) = 1 in V. However, σ −2 coincides with Two up to proportionality in irreducible HH –modules where σ acts. See Proposition 3.2.2. Thus (1 − σ −2 )(10 ) is proportional to 10 and must be zero in V 0 due to the calculation above. We obtain that 10 is in the radical of the pairing {f, g}σ − {g, f }σ , which makes this difference identically zero, since 10 is a generator. Comment. The radical is always τ− –invariant if q is not a root of unity. To see this, use that τ− in V is proportional to the multiplication by Pone can (b,b)/2 the series b∈B q Yb provided that |q| < 1. See Proposition 3.3.4.

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If q is a root of unity, then V 0 is not τ− –invariant for generic tν , i.e., in the setting of Proposition 3.8.4. b Concerning the previous considerations, note that the representation V from Proposition 3.8.4 which is the quotient of v by the radical of the pairing h , i◦ is not isomorphic to V 0 for generic k. Cf. also Theorem 2.8.5(iii) from the previuos chapter (the module V 0 was denoted P/Rad there). ❑ Let us go back to the proof of the theorem. Obviously the radical of the pairing { , } belongs to the kernel of the discretization map δ 0 : V → F0 [−ρk ]. Since the latter module is irreducible, the radical and the kernel coincide, and we have a nondegenerate pairing on F0 [−ρk ] inducing φ on HH[ . Since the image of Eb in this module is nonzero for πb ∈ Υ+ [−ρk ], we conclude that Eb (q −ρk ) 6= 0 and the spherical polynomials Eb = Eb /(Eb (q −ρk )) are well defined for such b. Moreover, if Eb and Ec are well defined and their eigenvalues coincide, then they can be different in V, but their images in F0 [−ρk ] have to coincide, thanks to the normalization Eb (q −ρk ) = 1. So far we have not used the X–unitary structure at all. If it exists, then Eb (q −ρk ) = 0 ⇒ ( Eb0 , δ0 ) = 0 ⇒ ( Ya (Eb0 ) , Ti Xc (δ0 ) ) = 0 for i > 0, a, c ∈ B ⇒ ( Eb0 , Ya−1 Ti Xc (δ0 ) ) = 0 ⇒ ( Eb0 , HH[ (δ0 ) ) = 0 ⇒ ( Eb0 , F0 [−ρk ] ) = 0 ⇒ Eb0 = 0.

(3.10.20)

The existence of an X–unitary structure, i.e., the relation µ• (w) b = 1 for 0 [ c w b ∈ W∗ [ρk ], is equivalent to the existence of ψ from the theorem. Indeed, the form (f, g) = {ψ(f ), ψ(g)} is obviously invariant pseudo-hermitian, assuming that ψ induces ε, and vice versa. Recall that by invariant pseudo-hermitian forms, we mean ?–invariant ∗–skew-symmetric nondegenerate forms. As we already used, such a form is automatically X–definite because the X– eigenspaces of F0 [−ρk ] are one-dimensional. It is equally applicable to Y instead of X. Recall that we do not suppose X–definite or Y –definite forms to be positive (or negative) hermitian forms. It only means that (v, v) 6= 0 for all the corresponding eigenvectors. c [ [ρk ], then the delta functions δ π depend only on the classes If µ• = 1 on W ∗ b of πb modulo the latter group. Hence, the map ψ 0 from (iii) is a well defined restriction of ψ◦ from formula (3.4.13). It induces ε on HH[ (Theorem 3.4.2) and Eb (X) = ε(Gπb )(1) for b ∈ B.

(3.10.21)

The latter holds until the elements πb sit in Υ+ [−ρk ]. ❑ Let us combine together all the structures discussed above. Our aim is to describe finite dimensional irreducible self-dual semisimple pseudo-unitary

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representations with the action of P GLc (2, Z). We call them perfect in the book. General quasi-perfect representations are HH[ –modules which have a nondegenerate form { , } inducing the duality anti-involution ψ. The greatest quasi-perfect quotient of the polynomial representation is V/Rad. Indeed, any such quotient V of V supplies it with a form {f, g}V = {f 0 , g 0 } for the images f 0 , g 0 of f, g in V. Then a proper linear combination { , }o of the canonical form { , } from (3.10.17) and { , }V will satisfy {1, 1}o = 0, which immediately makes it zero identically. Corollary 3.10.5. (i) A self-dual spherical X, Y –semisimple pseudo-unitary irreducible representation of HH[ is finite dimensional and possesses an X– eigenvector of weight −ρk and a Y –eigenvector of weight ρk . It exists if and only if Υ0 [−ρk ] ⊃ Υ+ [−ρk ] ⊂ Υ− [−ρk ], | Υ+ [−ρk ] | ≤ ∞, c∗[ [ρk ]. and µ• (w) b = 1 for w b∈W

(3.10.22)

There is only one such representation up to isomorphisms, namely, F0 [−ρk ]. (ii) It is P GLc (2, Z)–invariant if and only it contains the restricted Gaussian −1 def γ∗ (πb ) == q (b−2ub (ρk ) , b)/2 for b] = b − u−1 b (ρk ), which means the relations c [ [ρk ]. q (b,b−2ρk )/2 = 1 for all πb ∈ W ∗

(3.10.23)

In this case, τ+ corresponds to multiplication by γ∗ , and ψ 0 from (3.10.11) is proportional to the involution of F0 [−ρk ] sending Eb0 γ −1 7→ γ∗ (πb )−1 Eb0 γ∗−1 , where πb ∈ Υ+ [−ρk ] .

(3.10.24)

Proof. Only the last formula requires some comment. It is a straightforward specialization of (3.10.2). ❑

3.10.3

Generic q, singular k

The above corollary gives an approach to the classification of the triples {q, t, B ⊂ P } such that HH[ possesses a perfect representation, i.e., a finite dimensional nonzero irreducible self-dual semisimple pseudo-unitary representation with the action of P GLc (2, Z). If it exists, then this representation is isomorphic to F0 [−ρk ]. The importance of these representations is obvious. They carry all properties of the classical Fourier transform and directly generalize the truncated

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Bessel functions; see [CM],[CO], and Chapter 2. The Verlinde algebras and irreducible representations of the Weyl algebras are examples of such representations at the roots of unity Here we will discuss only two “main sectors” of perfect representations, negative k = −e/h for arbitrary q as (e, h) = 1 and, later, the case of positive integral hk < N as q is a primitive N –root of unity. Here h is the Coxeter number. Actually, the theorems below are somewhat more general because we do not assume that the k–parameters coincide. The root system A1 is considered in detail in Chapter 2 and also in [C27] and [CO], including the explicit formulas for the Gauss–Selberg sums. We begin with the case of generic q: (q 1/(2m) )i (t1/2 )j do not represent nontrivial roots of unity for i, j ∈ Z. Theorem 3.10.6. (i) In notation from Theorem 3.5.3, let us impose a somewhat stronger variant of condition (3.5.20), assuming that q is generic and (ρk , α∨ ) 6∈ Z \ {0} for all α ∈ R+ , (3.10.25) ∨ hα (k) = (ρk , α ) + kα 6∈ Z for all extreme α 6= ϑ, hϑ (k) ∈ −N. Then F0 [−ρk ] is perfect. It exists and remains perfect if q is an N –th root of unity, provided that hϑ (k) ≥ −N and all fractional powers of q that may appear in the formulas are primitive roots of unity of maximal possible order. def (ii) Setting e == −hϑ (k), c [ [ρk ] = {(eωr )u−1 | eωr ∈ B, r ∈ O}. W ∗ r Identifying b upon the symmetries −1 c[ πb 7→ πb · ur · (eωr ), i.e., b 7→ b + eu−1 b (ωr ) for (eωr ) · ur ∈ W∗ [ρk ],

F0 [−ρk ] =

X

Qq,t δbπ , where either b = 0,

(3.10.26)

b∈B

or (ϑ, ρk ) + kϑ − (b− , ϑ) < 0, or (ϑ, ρk ) + kϑ − (b− , ϑ) = 0 and u−1 b (ϑ) ∈ R− . (iii) For instance, we may pick klng = ksht = −e/h to ensure (3.10.25), where (e, h) = 1 for e ∈ N and h = (ρ, ϑ) + 1 is the Coxeter number. In this case, the index of P/Q is relatively prime to e, so c∗[ [ρk ] = {(eωr )u−1 W r , ωr ∈ B, r ∈ O}. For example, let ω = ω1 , k = klng , e = −2k ∈ N, and B = P in the case R = A1 . Then assumption (3.10.25) means that e is odd and c [ [ρk ] = {π0 = id, πeω }, F0 [−ρk ] = W ∗

e−1 X j=0

π Qq,t δ−jω .

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The proof is close to that of Theorem 3.5.3. The explicit description of the set Υ+ [−ρk ] is nothing but the definition (3.6.8). c∗[ [ρk ]. See (3.6.4) and (3.6.14). Then bw((ρk )) = w(ρk )+b = ρk , Let bw ∈ W w ∈ W, b ∈ B. Setting β = w−1 (αi ) for 1 ≤ i ≤ n, (w(ρk ) − ρk , αi∨ ) = (ρk , β ∨ ) − kβ = −(b, αi∨ ) ∈ Z. If β > 0, then there exists a simple root αj of the same length as β such that β ∨ − αj∨ is a positive coroot. Hence β = αj thanks to the first condition in (3.10.25). Similarly, β can be negative only for extreme −β. However, it contradicts the second condition unless β = −ϑ. We see that w preserves the set {−ϑ, α1 , . . . , αn } and is u−1 r for certain r ∈ O. We obtain that −1 b = ρk − u−1 r (ρk ) = eωr and πeωr = (eωr ) · ur .

Indeed, (b, αi ) = 0 ⇔ ur (αi ) 6= −ϑ ⇔ i 6= r, (b, αr ) = −hϑ (k). There is an additional condition to check, namely, (3.10.26) for b− = −eωr∗ and ur : (eωr∗ , ϑ) < e, or { (eωr∗ , ϑ) = e and u−1 r (ϑ) ∈ R− }. Since (ωr∗ , ϑ) = 1, only the second relation may happen. It holds because u−1 r (ϑ) = −αr . Now let us establish the existence of the pseudo-unitary structure, which is equivalent to the relations µ• (πeωr ) = 1. Thanks to Proposition 3.6.5, only A4l−1 and D2l+1 have to be examined. In the other cases, the orders of the c [ [ρk ] are not divisible by 4. The proof below is actually elements of bw ∈ W ∗ uniform, but the calculation is more relaxed in the simply-laced case. Letting k = klng , the integer e = −kh has to be relatively prime to h and, in particular, must be odd. Recall that r 7→ r∗ for r ∈ O describes the inversion in Π and corresponds to the automorphism of the nonaffine Dynkin diagram induced by ς = −w0 . We will use (3.5.22): µ• (πb ) =

Y³ tα−1/2 − tα1/2 qα(α∨ ,ρk )+j ´

for α ∈ R+ , (3.10.27) 1/2 −1/2 (α∨ ,ρ )+j tα − tα qα k ∨ −(b− , α∨ ) > j > 0 if u−1 b (α) ∈ R− , −(b− , α ) ≥ j > 0 otherwise. In the simply-laced case, tα = t, qα = q, and α∨ = α. Let b = eωr . One has −(b− , α∨ ) = e(ωr∗ , α) = e when α contains αr∗ , and = 0 otherwise. Using that the total number of factors here is even, namely, 2 ∗ (e − 1) ∗ (ρ, ωr∗ ), we may transpose the binomials in the numerator: 1<j<e

µ• (πeωr ) =

Y

(ωr∗ ,α)=1

q 2(α,ρk )+2j

³ t1/2 − t−1/2 q −(α,ρk )−j ´ t1/2 − t−1/2 q (α,ρk )+j

.

(3.10.28)

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Here j < e because j = e may appear only under the condition u−1 r (α) > 0, which never holds. Indeed, let α = αr∗ + β, where β does not contain αr∗ . 0 0 Then u−1 r (αr∗ + β) = −ϑ + β for β without αr . Since ϑ contains all simple 0 roots, we conclude that −ϑ + β is always negative. As a by-product, we have established that −u−1 r , sending def

Λ == {α > 0, (α, ωr∗ ) = 1} 3 α 7→ def

0 0 0 α0 = −u−1 r (α) ∈ {α > 0, (α , ωr ) = 1} == Λ ,

(3.10.29)

is an isomorphism. Its inverse takes (α, ρk ) 7→ (−ur (α), ρk ) = −(α, u−1 r (ρk )) = (α, eωr − ρk ) = e − (α, ρk ). Since the sets Λ and Λ0 are isomorphic under the automorphism ς, the sets {(α, ρk )} and {(α0 , ρk )} coincide for α ∈ Λ and α0 ∈ Λ0 . Finally, we conclude that (α, ρk ) 7→ e − (α, ρk ) is a symmetry of Λ = Λ0 . It ensures a complete cancelation of the binomials in (3.10.28) and results in X µ• (πeωr ) = q 2Σ , Σ = ((α, ρk ) + j) α∈Λ,1<j<e

= (e − 1)hk(ωr∗ , ρ) + (e(e − 1)/2)(2(ωr∗ , ρ)) = 0.

(3.10.30)

We have used that hk = e, |Λ| = 2(ωr∗ , ρ), and the following general formula X (a, α)(b, α∨ ) = h(a, b), a, b ∈ Cn . α>0

Therefore F0 [−ρk ] is pseudo-unitary. The last check is (3.10.23), ensuring the existence of the restricted Gaussian in this module: (ωr , eωr − 2ρk )/2 = (ωr , −u−1 r (ρk ) − ρk )/2 = −(ωr + ur (ωr ), ρk )/2 = −(ωr∗ − ωr , ρk )/2 = 0, c [ [ρk ]. where {(eωr ) · u−1 } = W r

(3.10.31)

∗

Once again we have used that the automorphism ς = −w0 transposes ωr and ωr∗ and preserves ρk . The analysis of the case of A1 is straightforward, as is the statement about the roots of unity. ❑ 0 Recall that F [−ρk ] is a unique nonzero irreducible quotient of the polynomial representation V because the latter is spherical and the multiplicity of the character $ of HY[ in V is one for generic q. It is instructional to calculate the dimension of F0 [−ρk ].

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For the sake of simplicity, let B = Q, klng = −e/h = ksht . We combine (3.10.26) with formula (3.1.22) applied to z ∈ (1/e)Q, which gives the existence and uniqueness of u ∈ W, a ∈ Q such that def

z− == ua((z)) ∈ (1/e)P− , (z− , ϑ) ≥ −1, (z− , ϑ) = −1 ⇒ u−1 (ϑ) ∈ R− , and (αi , z− ) = 0 ⇒ u−1 (αi ) ∈ R+ , i > 0.

(3.10.32)

Multiplying z and a by e, we obtain that the elements b from (3.10.26) are in one-to-one correspondence with the elements of Q/eQ. So the dimension is en . There is an immediate identification of the set of all b and Q/eQ and the dimension formula based on formula (3.6.26). Indeed, it gives that the dimension is the volume of the domain {z ∈ Rn } such that (z, αi ) ≤ 0 for i > 0 and (z, ϑ) ≥ −e divided by the volume of the affine Weyl chamber. More generally, the module F0 [−ρk ] for arbitrary B is naturally isomorphic c [ [ρk ] = {(eωr )u−1 , ωr ∈ B}. We to the space of functions on B/eB because W r ∗ see that its dimension remains en .

3.10.4

Roots of unity

Let q be a primitive N –th root of unity under assumption (3.8.10). See also (3.6.28). However, we not assume now that fractional powers of q that appear in the formulas are primitive roots of unity unless otherwise stated. We are going to describe the main “positive” sector of perfect HH[ –modules, generalizing the Verlinde algebras. The original Verlinde algebras are subalgebras of ”symmetric” elements in the perfect representations as k = 1, i.e., in the group case. The symmetric 1/2 elements in arbitrary HH[ –modules are defined as follows: Ti (v) = ti v for i > 0. For the main sector, klng and ksht are positive and rational. The perfect modules exist only if further constraints are imposed. We note that there exist other sectors of perfect representations, for instance, with negative hϑ (k) (see the above theorem) and with more special choices of the roots of unity, not only among the primitive ones. The complete list is unknown, although Corollary 3.10.5 seems sufficient for the complete classification. To simplify considerations, we assume that N is greater than the Coxeter number, but this is actually not necessary. The perfect representations considered below are well defined for small N. To consider small N, one can follow the proof of the previous theorem instead of using the affine Weyl chambers (see below).

404

CHAPTER 3. GENERAL THEORY We pick ˆ qˆ = q 1/(2m) for (B, B + 2ρk ) = m ˆ −1 Z, m ˆ ∈ N,

setting q (b,c+2ρk ) = qˆ(b,c+2ρk )mˆ unless stated otherwise. Note that q is primitive, but qˆ is not supposed to be primitive 2mN ˆ –th root of unity. Theorem 3.10.7. (i) We assume that klng , ksht > 0, N 3 hϑ (k) = (ρk , ϑ) + ksht < Nsht = N, and also either (a) (N, νlng ) = 1, or (b) N 3 hθ (k) = (ρk , θ∨ ) + klng < Nlng = N/(N, νlng ) for the longest root θ ∈ R+ (νlng = (θ, th)/2), or (c) (ρk , α∨ ) 6∈ Z+ for all long α ∈ R+ .

(3.10.33)

In case (c), we suppose that qˆ is a primitive root of unity. Then relation (3.8.7) holds, which guarantees the existence of the Y –cospherical X– semisimple finite dimensional irreducible module F0 [−ρk ]. It is nonzero if hϑ (k) − N ≤ Max (B− , ϑ). In case (b), we also add here the counterpart of this inequality for θ. (ii) We also impose either the condition {ρ ∈ B, hϑ (k) + h − 1 ≤ N, hθ (k) + h − 1 < Nlng } or {ρ 6∈ B, hϑ (k) + h − 1 ≤ N + 1, hθ (k) + h − 1 < Nlng + 1}.

(3.10.34)

Only the relations with ϑ are necessary under condition (a) or (c) from (i). def Then F0 [−ρk ] is pseudo-unitary and for e == N − hϑ (k), c [ [ρk ] = {(eωr )u−1 | eωr ∈ B, q N (ωr ,B) = 1}. W ∗ r

(3.10.35)

This module is positive unitary if q 1/2 = exp(πi/N ). The restricted Gaussian exists in F0 [−ρk ] if and only q mr = qˆ2mr mˆ = 1 for mr = (N − hϑ (k))N (ωr , ωr )/2 and ωr from (3.10.35). This may restrict the choices for qˆ. (iii) Provided that we have the conditions from (i) and (3.10.34) from (ii), X (3.10.36) F0 [−ρk ] = Qq,t ⊕ Qq,t δbπ , where 06=b∈B

(a) hϑ (k) − (b− , ϑ) < N, or {hϑ (k) − (b− , ϑ) = N and u−1 b (ϑ) ∈ R− }, and ∨ (b) hθ (k) − (b− , θ ) < Nlng , or {hθ (k) − (b− , θ∨ ) = Nlng and u−1 b (θ) ∈ R− }.

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The latter is necessary only under (b) from (i). Here we identify b modulo c∗[ [ρk ]: W c[ b 7→ b + u−1 b (c), b− 7→ uc (b− ) + c− for πc ∈ W∗ [ρk ]. The space without this identification is also an HH[ –module, which always contains the restricted Gaussian. (iv) For example, let R = A1 , ω = ω1 , k = klng , e = N − 2k ∈ N, B = P. Then F0 [−ρk ] =

e X

π for primitive q 1/2 Qq,t δjω

j=−e+1

is perfect. It is positive unitary for q 1/2 = ± exp(πi/N ) if k ∈ N; the sign has to be plus for a half-integral k. If k ∈ N and N is odd, we may pick q 1/2 in primitive N –th roots of unity. Then the summation above is over 1 ≤ j ≤ e. The Gaussian exists in F0 [−ρk ] in this case if either N = 4l + 3 for integral k or k is half-integral. This module is positive unitary for q 1/2 = − exp(πi/N ). Proof. The elements πb for b satisfying (3.10.36) constitute the set Υ+ [−ρk ] due to the positivity of kα . Here we use that ρk ∈ P+ , ϑ is the maximal coroot, and θ∨ is maximal among the short coroots. The positivity also results in the inequalities (ρk , α∨ ) − (b− , α) − υkα + ²bα < Nα = N/(να , N ) for the same b and all α > 0, where υ = 0, 1, ²bα = 1 if u−1 b (α) > 0 and 0 otherwise. These inequalities are necessary and sufficient for the existence of c [ [ρk ]. F0 [−ρk ]. Let us calculate the stabilizer W ∗ Lemma 3.10.8. Under the assumptions from (ii), the group of elements πc such that Υ+ [−ρk ]πc = Υ+ [−ρk ] is exactly Π = {(eωr )u−1 r , r ∈ O} for e = N − hϑ (k). Proof. By definition, πc = cu−1 c preserves the set of all element b satisfying (3.10.36) under the mapping b 7→ b0 , where πb πc = πb0 , b0 = b + u−1 b (c), ub0 = uc ub . N . In terms of b− : The corresponding set of b− will be denoted by B− −1 −1 −1 −1 πb πc = u−1 b b− uc c− = ub uc uc (b− )c− = ub0 · (uc (b− ) + c− ). N . Using the affine action The element b0− = uc (b− ) + c− has to be from B− (3.1.19):

b− 7→ b0− = c− uc ((b− )) = uc c((b− )).

(3.10.37)

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N (it happens Due to the conditions (3.10.34) from part (ii), either −ρ ∈ B− N when ρ ∈ B) or −ρ − ωr ∈ B− for proper minuscule ωr otherwise. The inner points of the segment connecting 0 and such b− in Rn are also inner in the ¯ N ∈ Rn , that is defined by the same inequalities but for real polyhedron B − ¯b− . The set of its inner points is either −eCa , where e = N − hϑ (k), Ca is the affine Weyl chamber, or its intersection with the counterpart for θ and Nlng − hθ (k) in case (b) from (i). See (3.1.21) for the definition of the affine Weyl chamber and its basic properties. Hence, there exist points that do not leave the negative Weyl chamber −Ca under z 7→ e−1 uc c(( ez )).

Thus, all the points do not leave this chamber. The stabilizer of −Ca = w0 (Ca ) is {(−ωr∗ )u−1 r∗ }. Therefore −1 c− uc = (−eωr∗ )u−1 r∗ for r ∈ O and c− = −eωr∗ , uc = ur∗ , −1 πc = u−1 c c− = ur∗ · (−eωr∗ ) = (eωr )ur , c = eωr .

❑ [ c The elements πc ∈ W∗ [ρk ] satisfy the conditions of the lemma. See (3.6.4) and (3.6.14). The definition is as follows: b k ))−ρk , b) c [ [ρ] = {w W b ∈ Υ+ [−ρk ], q (w((ρ = 1 for all b ∈ B}.

(3.10.38)

Setting w b = (eωr )u−1 r , w((ρ b k )) − ρk = u−1 r (ρk ) − ρk + eωr = hϑ (k)ωr + eωr = N ωr . Cf. the proof of the previous theorem. Hence q N (ωr ,b) has to be 1 for all b ∈ B. We arrive at the conditions from (ii). Now we can simply follow Theorem 3.10.6, formulas (3.10.30) and (3.10.31), to check the existence of the pseudo-unitary structure and the restricted Gaussian. The claim about the positivity of the pairing for the “smallest” root of unity q 1/2 = ± exp(πi/N ) is straightforward too. We represent the binomials in formula (3.10.28) for µ• in the form (eix − e−ix ) and use inequalities (3.10.36). In the numerator, the x do not reach π and eix − e−ix = ic for positive numbers c. Therefore the same holds true in the denominator. ❑ As an application, we obtain a counterpart of Theorem 3.5.1 at roots of unity, namely, hEb0 (Ec0 )∗ 㵕 i0] = q −(b] ,b] )/2−(c] ,c] )/2+(ρk ,ρk ) Ec (q b] )h㵕 i0] , hEb0

Ec0 㵕 i0]

(3.10.39)

−(b] ,b] )/2−(c] ,c] )/2+(ρk ,ρk )

= q Y (Ec )(q b] )h㵕 i0] . × tνlν (w0 )/2 Tw−1 0 ν

(3.10.40)

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407

P Here hf µ• i0] is the sum f µ• (πb ) over {πb } for b satisfying (3.10.36) upon c the identification modulo W∗[ [ρk ]. The functions Eb0 , Ec0 are the images of the polynomials Eb , Ec in F0 [−ρk ] for b, c satisfying the same constraint. They are well defined and nonzero. See Theorem 3.10.2. Concerning the generalized Gauss–Selberg sums h㵕 i0] , the formulas can be obtained using the shift operators, which we do not discuss. See [C25] about the integral k and [C27] with the complete list of formulas for A1 . Generally speaking, the shift operator can be used as follows. Given k, first we calculate hγµκ• i0] for κ = {κsht , κlng } taken from the sets ksht + Z and klng + Z with the simplest possible perfect representations. We denote the corresponding µ by µκ. If the k are fractional, we take κ negative and use the Macdonald identities from Section 3.5. Then we can apply the counterpart of (3.5.18): ∞ ³ (ρ ,α∨ )+j ´³ (ρκ ,α∨ )+j ´ Y Y hγµk• i0] 1 − qα k 1 − t−1 α qα . = ∨ 0 (ρ ,α∨ )+j −1 (ρk ,α )+j hγµκ 1 − qα κ • i] α∈R+ j=1 1 − tα qα

3.11

(3.10.41)

DAHA and double polynomials

In this section we will proof the q, t–variant of the Gordon theorem justifying the Haiman conjecture about the diagonal coinvariants from [Go]. See Section 2.12.5 in Chapter 2. The last two subsections are devoted to the universal Dunkl operators. We follow [C29]. In this section, we take B = P, i.e., HH[ = HH . def Let ksht = −(1 + 1/h) = klng for the Coxeter number h == 1 + (ρ, ϑ). The polynomial representation Q[x] of the rational DAHA HH00 is {D, W }– spherical by construction. Indeed, 1 is its generator and is a unique W – invariant polynomial killed by all Dunkl operators Db . Then, the general theory guarantees that the polynomial representation has a unique nonzero irreducible quotient module. Let us denote it V 00 . The subalgebra (HH00 )W of the W –invariant elements from HH00 obviously preserves the space Y def Qδ˜ ⊂ V 00 for δ == xα ; δ˜ = image of δ. α>0

Let Io ⊂ (H00 )W be the ideal of the elements from the rational DAHA vanishing at δ˜ in V 00 . Gordon proves that V 00 coincides with the quotient V˜ 00 of the module HH00 (δ) = Q[x] by the HH00 –submodule HH00 Io (δ), where the ˜ Here the key step is his proof of the irreducibility of map is naturally δ 7→ δ. 00 V˜ . We are going to generalize the equality V˜ 00 = V 00 to the q, t–case. The proof is simpler than in the rational case and leads to a new proof of Gordon’s

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theorem via the Lusztig isomorphisms from Section 2.12.5 (Chapter 2) and the classical facts about the Weyl algebras at roots of unity.

3.11.1

Good reductions

We will begin with a generalization of the deformations of the perfect representations, from the previous chapter (A1 ), also called there the nonsymmetric Verlinde algebras. Let ksht = −e/h = klng , provided that (e, h) = 1, e ∈ N. The notation will be k for both ksht and klng , since we assume that they coincide in this section. The HH is for the generic q (t = q k ). By generic, we mean that (q 1/(2m) )i (t1/2 )j do not represent nontrivial roots of unity for i, j ∈ Z. A good cyclotomic reduction of the pair {q 1/m , k} is defined as follows: (a) q and q 1/m are both primitive roots of unity of order N ∈ Z+ ; (b) (N, h) = 1 and e kh + e = N for the Coxeter number h. f The corresponding DAHA, subject to (a) and (b), will be denoted by H H e e f in this section. Note that k = k mod N and kh < N, so HH is really a specialization of HH . In contrast to Theorem 3.10.7, we choose q 1/m in the primitive N –th roots of unity. Recall that (P, P ) = (1/m)Z; hence, the condition (N, h) = 1 implies (m, h) = 1 and such a choice is possible. As a matter of fact, the theory for such q 1/m is completely equivalent to that of the HH[ for B = Q and a primitive N –th root q, i.e., is equivalent to the case of little DAHA. However, we prefer to stick to the “standard” DAHA, with B = P , and the above choice of q 1/m . The perfect representations for generic q and k = −e/h remain perfect for ˜ q, k satisfying (a) and (b). We do not claim that such N, q, k constitute the whole set of possibile “good reductions” of perfect representations, i.e., when the latter are well defined and remain irreducible at the roots of unity. Theorem 3.11.1. The perfect representation of HH for generic q and k = f −e/h as (e, h) = 1, e ∈ N becomes irreducible H H–modules when q 1/m is taken in the primitive N –th roots of unity and e k ∈ Z+ , e kh + e = N. Proof follows that of Theorem 3.10.6. We also apply Theorem 3.10.7. Concerning the use of Theorem 3.10.7 for such a choice of q 1/m , the perfect modules become smaller. They are now quotients of the polynomial representation for B = Q, that is, C[Xα , α ∈ R]. In all other respects the theory remains unchanged. We may identify the corresponding perfect representation with the space Funct(Q/eQ), which proves the theorem. ❑ The theorem is the main engine of our approach to the diagonal coinvariants, although we will not need this statement for the proof of the Main

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409

theorem below. The latter uses the good reduction only for trivial e k = 0 when e = h + 1.

3.11.2

Main theorem

Now ksht = −1 − (1/h) = klng and q is generic. We denote ksht and klng by k. It is the simplest nontrivial (non-one-dimensional) example of a perfect module for generic q. The t–counterpart of the element δ is Y Xα−1/2 ). ∆= (tα1/2 Xα1/2 − t−1/2 α α∈R+

It plays a major role in the definition of the t–shift operator (see [C16]). One has: −1/2 Ti (∆) = −ti ∆, 1 ≤ i ≤ n. −1/2

We extend $− (Ti ) = −ti to a one-dimensional representation of the nonaffine Hecke algebra H generated by Ti , 1 ≤ i ≤ n. We come to a q, t–generalization of Gordon’s theorem. The field of ratioe q,t . nals of Qq,t will be denoted by Q Theorem 3.11.2. (i) The polynomial representation Qq,t [X] of HH has a e q,t . It unique nonzero quotient V that is torsion-free and irreducible over Q n is of dimension (1 + h) . The action of X and Y is semisimple with simple e q,t , considered as an H –module, contains a unique spectra. The module V ⊗ Q submodule isomorphic to $− . (ii) The module V coincides with the quotient V˜ of Qq,t [X] by the HH– e q,t ∆) intersected with Qq,t [X], where Io is the kernel of the submodule HHIo (Q algebra homomorphism HHinv 3 H 7→ H(∆) ∈ V for the subalgebra HHinv of the elements of HH commuting with T1 , . . . , Tn . Proof. The existence of V and its isomorphism with the linear space Funct[P/(1 + h)P ] follow from the previous section. See Theorem 3.10.6 and formula (3.10.32). The statements there are for general k = −e/h as (e, h) = 1. Comment. Actually, we need here only the self-duality of V from the general theory of this chapter, that is, the action of the involution ε of HH from (3.2.10) in V. No other facts will be used, including the isomorphism with the Funct[P/(1 + h)P ]. The self-duality can be justified directly, without the general theory. It readily follows from the realization of V as the quotient of Qq,t [X] by the radical of the invariant bilinear form from Lemma 3.10.3. Moreover, the self-duality will be needed only upon the reduction • below, where it is simple to check directly. We mention that the automorphisms τ±

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can be defined in V as well, but we do not use it in this section. Thus the proof we give is essentially self-contained. ❑ The construction of V holds over Qq,t . Actually, it suffices to have the e q,t . Then one can use the basic definition of V and to prove the theorem over Q facts about the modules over PID. Note that the module Qq,t [X]/HHIo (∆), generally speaking, has torsion. e q,t results from the following fact. The The uniqueness of $− in V ⊗ Q orbit W (ρ) is a unique simple W –orbit in P/(1 + h)P. It will be checked below. ❑ Comment. There is another proof of the theorem based on the shift operator, which identifies the $− –component of V = V k with the $+ –component 1/2 of V k+1 defined for k + 1 = −1/h, where $+ : Ti 7→ ti . The V k+1 is onedimensional and coincides with its $+ –component. The shift operator here is the division by ∆. This description is helpful when calculating the ideal Io ; its intersections with Qq,t [X]inv and Qq,t [Y ]inv are not difficult to describe. This approach gives that the Io is ε–invariant and results in a “direct” verification of the self-duality of the V˜ . Note that the self-dualty of V combined with the uniqueness of $− in V formally imply that V˜ is self-dual too. Indeed, the character $− is ε– invariant. So are Qq,t ∆ ⊂ V, Io , and the kernel of the map Qq,t [X] → V˜ .

3.11.3

Weyl algebra

def

For N == 1 + h, we take q = exp(2πı/N ), making k = 0, t±1/2 = 1. Using that να and the index of P/Q are relatively prime to N, we will pick q 1/m in the roots of unity of the same order N. The bullet • will be used to denote this reduction. The algebra HH• is simply the semidirect product of the Weyl algebra generated by pairwise commutative Xa , Yb for a, b ∈ P and H• = Qq W. The relations are w Xa w−1 = Xw(a) , w Yb w−1 = Yw(b) , Xa Yb Xa−1 Yb−1 = q −(a,b) . We define V • as a unique nonzero irreducible quotient of Qq [X]. It is selfdual. It is straightforward to check it directly, as well as the semisimplicity of X and Y , in the •–case. Since all eigenvalues of Yb in V • (and in the whole Qq [X]) are N –th roots of unity, the same holds for Xb in V • , thanks to the self-duality. Thus XbN = 1 = YbN in V • for all b ∈ P. Theorem 3.10.6 guarantees that V remains irreducible under such specialization, so V • is a point of good reduction of V. It also can be seen from the dimension formula in the following lemma.

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411

def

Lemma 3.11.3. (i) The algebra HH•N == HH• /(X N = 1 = Y N ) has a unique irreducible nonzero representation V • up to isomorphisms. Its dimension is N n. (ii) It is isomorphic to Qq [P/N P ] as a W –module. The representation • $− : w 7→ sgn(w) has multiplicity one in V • . (iii) The quotient V˜ • of Qq [X] by HH• Io• (∆• ) is an HH•N –module and coincides with V • . Proof. In the first place, HH•N is a group algebra of a finite group, and therefore it is semisimple. Let us use the well-known fact that the Weyl algebra generated by Xa , Yb modulo the (central) relations X N = 1 = Y N has a unique irreducible representation up to isomorphisms and W acts in this representation. This representation equals Qq [X]/(X N = 1) and is just V •. The multiplicity one statement from (ii) follows from the uniqueness of a simple W –orbit in P/N P. Let us check it. We can assume that the orbit is in the form W (b) for bP∈ P such that 0 < (b, α∨ ) < 1 + h for all α ∈ R+ . Therefore b can be ρ = ni ωi or ρ + ωr for r ∈ O0 , i.e., for a minuscule weight ωr . Indeed, the coefficient of αi∨ in the decomposition of ϑ in terms of simple coroots is one only for r ∈ O0 . For b = ρ + ωr , let w = u−1 r for ur from ωr = πr ur . Then (w(ρ) − b, αr ) = −(ρ, ϑ) − (ωr , αr∨ ) − 1 = −(1 + h) = −N and (w(ρ) − b, αi ) = (ρ, αj ) − (ρ, αi ) = 0 for i 6= r, w−1 (αi ) = αj . Thus b and ρ generate the same W –orbit modulo N P. The module V • is obviously self-dual; so is V˜ • because the H–module Qq ∆• is of multiplicity one in V˜ • and therefore invariant with respect to ε. It gives that V˜ • is a finite dimensional HH•N –module. It has V • as a quotient • . and contains a unique W –submodule isomorphic to $− • • ˜ Supposing that the kernel K of the map V → V is nonzero, it must contain a nonzero HH•N –submodule. Hence K contains at least one copy of • in V˜ • cannot be one. ❑ V • (the uniqueness), and the multiplicity of $− Actually, the proof we give is similar to that of the standard fact that the discriminant is the W –anti-invariant of the lowest possible degree in the ring of polynomials from [Bo]. Comment. Note that claims (ii)–(iii) are somewhat unusual in the general theory of Weyl algebras. Given the rank, they hold only for special choices of the roots of unity. For instance, N must be 3 in the case of A1 . Otherwise the dimension of the space of “anti-invariants” (odd characteristics in the classical theory of theta functions) is greater than one. A general problem is to find “natural” σ–eigenfunctions in the module • V . Its self-duality does not garrantee that the solution is simple. Actually, it is a typical problem in the theory of automorphic functions. Recall that

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σ is a cyclotomic counterpart of the classical Fourier transform, where such questions are of fundamental importance. The above approach gives a simple answer, but only for very small N. ❑ Coming back to the general case, the coincidence statement of the theorem ˜ q,t , so it suffices to check it at one special point. The lemma is actually over Q gives it for the •–point. To be more exact, claim (iii) of the lemma gives the irreducibility of V˜ and therefore the coincidence V˜ = V at the common point, and proves the theorem. ❑ The application to the diagonal coinvariants goes via the universal Dunkl operators, that will be introduced later. The problem is to calculate the action of Yb in the linear space of Laurent polynomials Qq,t [X, Y ] identified with the HH –module induced from a one-dimensional character of H. We assume here that the X–monomials are placed before the Y –monomials. Then the action of Xb in the space Qq,t [X, Y ] is the “commutative” multiplication by Xb . The action of Yb is by the left multiplication of the monomials in the form X · Y · . Hence it requires reordering and leads to nontrivial formulas. The HH –module Qq,t [X, Y ] is obviously self-dual. However, since it is necessary to order X and Y after applying ε, the formulas for its action in Qq,t [X, Y ] are involved. Now, the module V is the quotient of the module Q− q,t [X, Y ] induced for the character $− of H by its submodule HHIo (1). It is a one-parametric deformation of the polynomial ring Qq,t [X, Y ] divided by the ideal Qq,t [X, Y ]W o = {g(X, Y )(f (X, Y ) − f (1, 1))} for W –invariant Laurent polynomials f and arbitrary g. Therefore it can be identified with a quotient of the space of diagonal coinvariants W Q[X, Y ]/(Q[X, Y ]Q[X, Y ]W o ) ' Q[x, y]/(Q[x, y]Q[x, y]o ),

where the ring of definition is extended to Qq,t . Concerning Gordon’s theorem, the degenerations V 0 , V 00 of V can be introduced as quotients of the polynomial representation by the radicals of the degenerations of the bilinear form from Lemma 3.10.3. They are irreducible modules for HH0 , HH00 , thanks to Lusztig’s isomorphisms from Section 2.12.5, and satisfy the same multiplicity one statement. The modules V˜ 0 , V˜ 00 are defined in terms of the ideal of the W –invariant elements of the double Hecke algebra vanishing at the discriminant subspace of V 0 , V 00 (corresponding to the sign-character of W ). The V˜ 0 , V˜ 00 are irreducible due to Lusztig’s isomorphisms, and therefore V 0 = V˜ 0 and V 00 = V˜ 00 . The latter is the (main part of the) Gordon theorem. Comment. One can use the specialization HH• to obtain a “Laurent” counterpart of the Gordon theorem, without any reference to the theory of DAHA. An obvious difficulty is that there is no grading with respect to the degree of polynomials in this case. We therefore need to proceed as follows.

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413

Let assume that N is a prime number p and make the field of constants Fp . Then we readily obtain that the space Fp [X, Y ]/(Fp [X, Y ]Fp [X, Y ]W o ) must • have a “natural” quotient-space isomorphic to V over Fp . We use that 1 − q 2πi belongs to the ideal in the ring of integers of the cyclotomic field Q[e p ], which contains (p) and has the residue field Fp . One can check that the formulas for the Lusztig homomorphism æ contain no denominators divisible by p, and therefore it is well defined over Fp . Applying æ, we interpret V • ⊗ Fp as a quotient of the space of the diagonal coinvariants considered over the field Fp . It coincides with the quotient due to Gordon taken modulo p.

3.11.4

Universal DAHA

First, we will give a X ↔ Y –symmetric presentation of HH . It goes via the b This group is defined to be generated universal double affine braid group B. by the pairwise commutative Xb , the pairwise commutative Yb , the elements b = {b Tbi , where b ∈ P, 0 ≤ i ≤ n, and the group Π πr , r ∈ O} ' Π with the following defining relations: (a) Tbi Tbj Tbi ... = Tbj Tbi Tbj ..., mij factors on each side, br (αi ) = αj ; br−1 = Tbj if π π br Tbi π b b , Tbi−1 Yb Tbi−1 = Yb Yα−1 (b) Ti Xb Ti = Xb Xα−1 i i ∨ if (b, αi ) = 1 for 0 ≤ i ≤ n; (c) Tbi Xb = Xb Tbi , Tbi Yb = Yb Tbi if (b, αi∨ ) = 0 for 0 ≤ i ≤ n; br Yb π br−1 = Yπbr (b) , r ∈ O0 . (d) π br Xb π br−1 = Xπbr (b) , π Note that no relations between X and Y are imposed. We continue using the notation X[b,j] = Xb q j , Y[b,j] = Yb q −j . The element q 1/m is treated as a generator that is central. Later, q 1/(2m) will be needed. The relations (a–d) are obviously invariant with respect to the involution br 7→ π br (r ∈ O). εb : Xb ↔ Yb , Tbi → 7 Tbi−1 (0 ≤ i ≤ n), π

(3.11.1)

Concerning (d), recall that πr−1 is πr∗ for r∗ ∈ O. The same holds for ur = πr−1 ωr and πbr . Theorem 3.11.4. The group B generated by X, T, Π, q 1/(2m) subject to the b by the relations (i–v) from Definition 3.2.1 coincides with the quotient of B relations: br = q (ωr ,ωr )/2 Yr Tbu−1 Xr−1 Tb0 = q −1 Xϑ Tbsϑ Yϑ−1 , π ∗ . r

(3.11.2)

The map is pr : Xb 7→ Xb , Yb 7→ Yb , Tbi → 7 Ti (i > 0), Tb0 → 7 q −1 Xϑ Tsϑ Yϑ−1 , 1/(2m) π br 7→ q (ωr ,ωr )/2 Yr Tu−1 Xr−1 7→ q 1/(2m) , ∗ , q r

(3.11.3)

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where the elements Yb ∈ HH are given by (3.2.8). The images of Tbi , π br in B coincide with τ+ (Ti ), τ+ (πr ) for τ+ from (3.2.11). The relations (3.11.2) are invariant with respect to the involution εb. The latter becomes ε from (3.2.10) in B. Proof. The key fact here is that ε(τ+ (Ti )) = (τ+ (Ti ))−1 (i ≥ 0), ε(τ+ (πr )) = τ+ (πr ), (3.11.4) −1 −1 where τ+ (Ti ) = Ti for i > 0, τ+ (T0 ) = q Xϑ Tsϑ Yϑ , τ+ (πr ) = q

(ωr ,ωr ) 2

Yr Tu−1 Xr−1 ∗ , r ∈ O. r

See formula (3.2.19) for the action of the involution η = ετ−−1 τ+ τ−−1 . We recall that this formula directly results in the invariance of the Gaussians with respect to the q–Fourier transform corresponding to the involution ε. The elements τ+ (Ti ), τ+ (πr ) are exactly the images of the elements Tbi , π br in B under pr. Relations (3.11.4) readily follow from the explicit formulas. ❑ b and the algebra H c There are important quotients of the group B H (defined below) obtained by imposing the commutativity of X with Y. They are essential in the theory of the q–Fourier transform. The {Xi } are treated as the coordinates, and the {Yi } play the role of spectral parameters. Note the immediate projection of these quotients to B, HH when we make Yb = Xb−1 . b can be used for a Comment. The realization of B as a quotient of B direct proof that ε can be extended to an involution of B. It can simplify the straightforward proof due to Macdonald and the author from [M8], but b has something in common with the difference is not very significant. Our B the ”triple affine Artin group” introduced recently by Ion and Sahi in [IS] for the purpose of interpreting the projective action of P SL(2, Z). Compare our relation (3.11.2) and formula (23) in [IS], which establishes the connection of their group with the double affine braid group from [C13]. ❑ Turning to the Hecke algebras, let us define the universal affine double b by the quadratic c Hecke algebra H H as the quotient of the group algebra Qq,t B

relations (o) from Definition 3.2.1 for Tbi . Here we do not assume that tlng = tsht . Recall that the elements Xb and Yb are entirely independent, so the b ∈H c counterpart of the PBW theorem is that an arbitrary element H H can be uniquely represented as X b= Qwb Twb , where Qwb are noncommutative polynomials in X, Y. H c w∈ b W

For applications to the Dunkl operators, this definition will be needed in c the following form. We claim that the algebra H H is generated over Qq,t by the affine Hecke algebra def b b= H = hTi , π br i, i ≥ 0, r ∈ O,

3.11. DAHA AND DOUBLE POLYNOMIALS

415

the pairwise commutative Xb (b ∈ P ) and the pairwise commutative Yb (b ∈ P ), satisfying the relations (d) above with the π br , and the Lusztig-type relations −1/2 Xs (b) − Xb 1/2 Tbi Xb − Xsi (b) Tbi = (ti − ti ) i , 0 ≤ i ≤ n, Xαi − 1 −1/2 Ys (b) − Yb 1/2 , 0 ≤ i ≤ n. Tbi Yb − Ysi (b) Tbi = (ti − ti ) i −1 Yαi − 1

(3.11.5) (3.11.6)

c Imposing (3.11.2), we represent HH as a quotient of H H. Note that this definition is compatible with the restriction to the lattices B between Q and P taken instead of P. We set def Ybb+ == π br Tbi1 · · · Tbil for b+ = πr si1 · · · sil as b+ ∈ P+ , l = l(b+ ).

for b+ , c+ ∈ P+ . More generally, Ybb+ −c+ = Ybb+ Ybc−1 +

3.11.5

Universal Dunkl operators

b the general universal Dunkl operators are Given a representation Vb of H, c the images of Ybb (b ∈ P ) and π br (r ∈ O) in the H H –module IVb induced from b V . As a linear space, it is isomorphic to the linear space of noncommutative polynomials of X, Y with the (right) coefficients in Vb : IVb = ∪l∈N Pl for X ª def © (3.11.7) Xb1 Yc1 · · · Xbl Ycl vbb,c , b ∈ P = P l 3 c, vbb,c ∈ Vb . Pl == b,c∈P

Here the sums in (3.11.7) represent different vectors in IVb for different v– coefficients if we assume that bi 6= 0 6= cj for the indices 1 < i ≤ l, 1 ≤ j < l as vbb,c 6= 0. The action of X and Y is by left multiplication. The subspaces Pl are b H–submodules and also X–submodules for l > 0. The action of Tbi (i ≥ 0) and π br in Pl can be calculated using formulas (3.11.5) and and (3.11.6), and the relations (d) above. In the first interesting case l = 1, it is as follows: Tbi (Xb Yc vb) = Xsi (b) Ysi (c) Tbi (b v )+ (3.11.8) ¡ ¢ Ys (c) − Yc 1/2 −1/2 Xsi (b) − Xb (ti − ti ) vb, 0 ≤ i ≤ n, Yc + Xsi (b) i −1 Xαi − 1 Yαi − 1 π br (Xb Yc vb) = Xπr (b) Yπr (c) π br (b v ) for b, c ∈ P, vb ∈ Vb . (3.11.9) When the initial representation Vb is the character 1/2 $ b + : Tbi 7→ ti , i ≥ 0, π br 7→ 1,

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we arrive at a double variant of formulas (3.2.23) and the corresponding Dunkl operators: x syi − 1 ¢ 1/2 1/2 −1/2 ¡ s − 1 . Tbi = ti sxi syi + (ti − ti ) i + sxi −1 Xαi − 1 Yαi − 1

(3.11.10)

Here sxi , syi act, respectively, on X and Y , and the differences are applied before the division in the divided differences. Similarly, π br = πrx πry .

3.11.6

Double polynomials

The above consideration leads to the formulas for the action of the hatoperators in the HH –module Qq,t [X, Y ] induced from the character $+ of the nonaffine Hecke subalgebra H. These operators are the images of {Tbi , π br } under the projection pr. They coincide with τ+ (Ti ), τ+ (πr ). See (3.11.4). We will use the same hat-notation for them, although now they are the elements of HH . The formulas are Tbi (Xb Yc ) = Xsi (b) Ysi (c) Tbi (1)+ Ys (c) − Yc ¢ −1/2 ¡ Xsi (b) − Xb 1/2 (ti − ti ) , Yc + Xsi (b) i −1 Xαi − 1 Yαi − 1 1/2 Tbi (1) = t (i > 0), Tb0 (1) = q −1 Xϑ Ts Y −1 (1) i

=q

−1

ϑ

1/2 −1/2 Xϑ (Yϑ Ts−1 (1) − (t0 − t0 )), ϑ P P 1−(ϑ, sht α∨ ) −(ϑ, lng α∨ )

Ts−1 (1) = tsht ϑ

tlng

ϑ

, α ∈ R+ .

(3.11.11)

The X–monomials act by left multiplication, the operators π br via the relations (d) and the formula for π br (1), similar to that from (3.11.11). Knowing the action of Xb , Tbi , and π br is sufficient for determining the structure of the HH –module. Recall that the involution ε acts naturally in Qq,t [X, Y ], sending Tbi 7→ Tbi−1 (all i) and π br 7→ π br . We note that the hat-operators can be used, for instance, to introduce the “double radial parts.” As usual, the simplest ones, corresponding to the minuscule symmetric monomial functions, can be calculated explicitly. To conclude, we note that there is an interesting group of automorphisms b c of B, H H generated by the tau-automorphisms def

τ±x , τ±y == εbτ∓x εb. They act, respectively, in hTb, π b, Xi fixing Y and in hTb, π b, Y i fixing X. This group is an extension of the P SL3 (Z). Hopefully it is “naturally” connected

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417

to the action of P SL3 (Z) on solutions of some KZB–type equations found by Felder and Varchenko. See [FV2]. The action of P SL3 (Z) is directly related to the multiple Gamma functions introduced and studied by Barns [Ba]. This 100 year-old theory is of great potential for the DAHAs and the modern theory of special functions.

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[A1]

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Index αi : simple roots, 1–417 additional series (A1 ), 231, 233, 251 affine r–matrix, 112 affine action (wb)((z)), 300 affine Hecke algebra H, 61, 312 affine KZ equation (A1 ), 48 affine pairing ([z, l], z 0 + d), 300 affine quantum KZ (GLn ), 78

C: complex numbers, 1–417 category O, 343 central extension b g, 132 central extension (P GL(2, Z)), 392 chain (integral formulas), 143 characteristic functions χm , 221 characteristic functions χwb , 327 coinvariant π (b g), 138 coinvariant π (KZB), 153 coinvariants $ (DAHA), 371, 372 commutativity relations (Hecke), 77 compact case, 164 component of a diagram, 143 composition of paths, 60, 217 conjugation on polynomials, 199 constant term CT , 168 constituents of Iξ (DAHA), 385 convex set of roots, 302 coroot α∨ , 262 cospherical forms $, 371, 372 cospherical module (DAHA), 371 Coxeter number h, 339 Coxeter relations (Hecke), 61, 77 creation operator (A1 ), 197 cross-relations (Hecke), 77 cutoffs, 68 cyclic module (X, Y ), 344, 345

affine root system, 118, 295 affine Weyl chamber Ca , 119, 300 affine Weyl chamber C¯a , 125 affine Weyl group b Sn , 79 c , 112, 262, 295 affine Weyl group W f , 112, 295 affine Weyl group W AKZ, 48, 52, 55 anti-involution φ, 95, 200, 318 anti-involution φ, GLn , 102 anti-involution ?, 308 anti-involution X–trivial, 380, 383 anti-involution trivial, 57 Appell function A(k, τ ), 182 AQKZ, 78 asymptotically free solution, 63 automorphism σ , 213, 272, 307 automorphisms τ± , 272, 307 Baker function ϕ, 151 basic hypergeometric function, 161 Baxter–Belavin r–matrix, 151 baxterization (affine), 90 bilinear form (∗–bilinear), 314 boundary of δ , 360 braid group (elliptic), 216 braid group (pure), 104 braid group (universal), 413

DAHA, 100, 120, 306 definite form for X, Y , 345, 346 degenerate DAHA HH0 , 120, 274 degenerate DAHA HH00 , 190, 277 degenerate DAHA HH00 , 115 degenerate Hecke algebra Hn0 , 52 0 degenerate Hecke algebra HA , 49 1 0 degenerate Hecke algebra H , 53, 109

431

432 delta functions δm (A1 ), 221 delta functions δwb , 327 delta-representation ∆, 383 Demazure–Lusztig operators Tˆi , 310 Demazure–Lusztig operators Si , 122 diagonal coinvariants, 279, 412 diagram δ (generalized), 369 diagram δ (periodic), 360, 362 diagram (integral formulas), 143 discretization δ (DAHA), 326 discretization (A1 ), 97, 213 double Hecke algebra, 100, 120, 306 double Hecke algebra (A1 ), 95, 198 double Hecke algebra (GLn ), 100, 357 double Hecke algebra (rational), 277 double Hecke universal, 414 duality formula (A1 ), 94, 200 duality formula (GLn ), 99 duality formula (DAHA), 318, 396 Dunkl operators (difference), 122, 273 Dunkl operators (infinite), 113 Dunkl operators (no shifts), 71, 111 Dunkl operators (rational), 190, 277 Dunkl operators (trig), 75, 121, 275 Dunkl operators (universal), 415

Eb , Eb : nonsym polynomials, 314 en , εn : nonsym polynomials, 197 ε involution, 213, 272, 307 η involution, 202, 308 eigenvectors v (X, Y ), 343 elliptic Weyl group (GLn ), 101 evaluation formula, 98, 100, 206 evaluation formula (DAHA), 317, 319 evaluation map (A1 ), 96, 200 expectation value {H}0 (A1 ), 95 extended Weyl group, 112, 262, 295

Φ: solution of KZ,QKZ, 1–153 ϕ: solution of QMBP, 1–153 Φwb , Swb , Gwb : intertwiners, 281–417 F, F, 326, 383 φ anti-involution, 95, 200, 318 factorized KM algebra (b g), 132

INDEX factorizing subalgebra e gr , 132 Fourier transform S (A1 ), 213 Fourier transform E (A1 ), 213 ¯ E ¯ (A1 ), 223 Fourier transforms S, Fourier transform ϕ◦ , 328 Fourier transform ψ◦ (skew), 328 Fourier transforms ϕ ¯◦ , ψ¯◦ , 330 Fourier–Jackson transform ϕ• , 335 Fourier–Jackson transform ψ• , 335 functional representation, 326, 383 Gaudin model, 151 Gaussian γ , 233, 331 Gaussian γ (restricted), 393 Gaussian sum τ (generalized), 163

HH, HHn , HH[ , 100, 306, 311 H, HX , HY , 61, 312 0 HΣ , H0 , 54, 109 0 HH , HH00 , 120, 190, 277 Hn : nonaffine Hecke, 100, 359, 367 H: nonaffine Hecke, 308, 383, 409 ♥ anti-involution, 318, 330, 397 Hankel transform, 186, 188, 194 Hankel transform Hre,im , 162 Harish-Chandra function σ , 122 Harish-Chandra transform, 116 Hecke algebra H, 308, 383, 409 Hecke algebra Hn , 100, 359, 367 Heckman–Opdam operators L0p , 123 hypergeometric function, 123 induced module IX [ξ] (DAHA), 344 0 induced module Iλ (HΣ ), 57 intermediate subalgebras HH[ , 311 intertwiners fw (degenerate), 58 intertwiners Fw (u) (affine, GLn ), 78 intertwiners Ψ0wb (degenerate), 120 intertwiners Φwb (DAHA), 320, 321 intertwiners normalized, 274, 320 intertwining operator Π (A1 ), 205 invariance (AKZ, Sn ), 52 invariance (AKZ, W ), 54 invariance (QAKZ, W ), 78

INDEX invariant form (?–invariant), 345 invariant module (P GLc (2, Z)), 392 invariants v (DAHA), 371, 372 inverse order in δ , 359 inversion (H0 ), 125 inversion (A1 ), 162, 189, 224 inversion (A1 , truncated), 194 inversion (DAHA), 268, 337–339 involution η , 202, 308 involution ε, 213, 272, 307 involution r 7→ r∗ , 271 Jackson master formula, 224 Jackson sum, 176, 334 Jucys–Murphy generators, 67

k–function, 108, 306 Kazhdan–Lusztig involution, 202, 308 Kodaira–Spencer map, 35, 220 KZ (r–matrix), 135 λ regular (A1 ), 238 λ half-singular (A1 ), 238 λ singular (A1 ) , 238 length l(w) b , 119, 262, 296 little Verlinde algebra, 260 Lusztig map, 276, 278 Macdonald operators, 84, 316 Macdonald polynomials Pb , 316 Macdonald polynomials pλ , 99 Macdonald problem, 84 maximal root θ, 118 maximal short root ϑ, 262, 295 Mehta–Macdonald intg, 289, 332 Mellin transform Ψ, 171, 174 monodromy (AKZ), 62, 69 monodromy cocycle (QAKZ), 81 monomial functions, 316

433 nonsymmetric Verlinde algebra, 248 norm formula, 206, 325, 330 old points (integral formulas), 142 Opdam transform F , 124 Opdam transform G (inverse), 125

Pb , pn : sym polynomials, 93, 316 P : weight lattice, 1–417 πr : elements from P ∨ /Q∨ , 1–417 P : polynomial rep, 1–280 pair {∆, C}, 359 pair {∆, C} (generalized), 369 pair {∆, C} (new), 363 pairs {∆, C} (equivalent), 364 partition of δ , 362 partition increasing, 362 partition of δ new, 363 partition ordered, 363 perfect module (DAHA), 399 perfect representation (A1 ), 248 Pieri formula (A1 ), 97, 98, 206 Pieri formula (GLn ), 99 Plancherel HH, 268, 336–338 Plancherel A1 , 190, 194, 221 polynomial rep, 273, 310, 372 polynomial rep A1 , 96, 190 polynomial rep (rational), 277 primitive module (DAHA), 373 primitive weight ξo , 382, 386, 388 principal module (DAHA), 373 principal-special (A1 ), 238 pseudo-hermitian form, 345 pseudo-unitary module, 248, 345

Q: rational numbers, 1–417 Q: root lattice, 1–417

N: natural numbers, 1–417

QMBP, 60, 74 quadratic relations (Hecke), 77 quantum many-body problem, 60

new points (integral formulas), 142 noncompact case, 165 nonsymmetric polynomials Eb , 314 nonsymmetric polynomials en , 197

R: real numbers, 1–417 R, r: r–matrices, 1–153 R: root system, 281–417

434 ρ, 294, 306 radial part L(k) (SO2 ), 92 range Υ+ [ξ] (plus), 346 ¨ + [ξ] (boundary), 346 range Υ range Υ− (ξ) (minus), 347 range Υ∗ [ξ] (semisimple), 347 range Υ0 [ξ] (zero), 347 range Υ+ [−ρk ] (sharp), 348 reflection r 7→ r∗ , 271 restricted category, 287 rho ρ, 294, 306 r–matrix (affine), 112 r–matrix (extension of), 108 r–matrix (invariant), 108 r–matrix (unitary), 131, 140 Rogers polynomials pn , 93, 195 root (extreme, η–identities), 339

Sn : permutation group, 1–417 Σ: root system, 1–153 si : simple reflections, 1–417 σ automorphism, 213, 272, 307 ? anti-involution, 308 self-consistent (AKZ), 54 self-dual module (DAHA), 392 semi-classical limit (QAKZ), 79 semisimple module (X, Y ), 345 shift formula, 189 shift operator S , 171 skew diagram, 358 skew paths in δ , 361 spectrum (X, Y ), 345 spherical irrep (A1 ), 248 spherical module (DAHA), 371 spherical polynomials Eb , 317 spherical rep ∆+ , 263

INDEX spherical vectors, 371, 372 c [ [ξ], 343, 347 stabilizer W standard invariant form (g), 130 standard roots α, β , 106 standard subsystem (rk=2), 108 subdiagram basic, 360 subdiagram fundamental, 361 subdiagram of δ , 360 subpair fundamental, 363 subpairs equivalent, 365 Sugawara elements Li−1 , 139 symmetrizer, 263

τ± automorphisms, 272, 307 τ –function, 45 τ –function (r–matrix), 136 trace on Jλ , 72 truncated θ–function, 169, 316 ultraspherical polynomials, 93 unitary automorphism, 308 unitary module for X, Y , 345, 346

V : polynomial rep, 372–417 vacuum vector, 141 Verlinde algebra (A1 ), 248 Verlinde algebra (DAHA), 403 Verlinde algebra (little), 260 Verma module, 46

f ,W c : affine Weyl groups, 295 W weights ξ (GL), 359 weights ξ (X, Y ), 343 well-ordered diagram, 143 Weyl group W , 53, 293 Weyl module MV (b g), 138

Z: integers, 1–417