Topic #12 ------------ OP-SF NET 4.6 ------------ November 15, 1997 ~~~~~~~~~~~~~ From: OP-SF NET Editor (thk@wins.uva.nl) Subject: Recent International Workshop at RIMS, Kyoto At RIMS (Research Institute of Mathematical Science, Kyoto, Japan) an international workshop on "Invariant Differential Operators, Special Functions and Representation Theory" was held during October 20-31, 1997. The organizer was Toshio Oshima (University of Tokyo). I take the following information from the URL http://w3rep.math.h.kyoto-u.ac.jp/projecte.html#meeting. In the next Topic Mathijs Dijkhuizen, who participated, will give a report. One half of the workshop (the second week) was devoted to "Integrable systems of difference and differential equations" The main speakers, with series of 3 or 4 lectures each, were Eric M. Opdam (Leiden Univ.): Dunkl Operators Abstract: In these lectures I will give an overview of results on Dunkl's "differential-reflection" operators, up to the most recent developments. Mainly I will concentrate on the (differential) trigonometric case, the case of the Dunkl-Cherednik operators, because in this case the theory has reached the most mature level at present. And also there are several older theorems and applications whose proofs can be polished by modern methods, but many of these things were never written. So I feel that giving such a series of lectures can be rewarding, and I am happy to embark on such a project. Roughly, I have in mind to treat the following subjects: 1. DEFINITION AND BASIC ANALYTIC RESULTS. The Knizhnik-Zamolodchikov connection, the Harish-Chandra system, monodromy representation, the shifting principle, asymptotic expansions, the Gauss' summation formula. 2. ALGEBRAIC PROPERTIES. Nonsymmetric orthogonal polynomials, the graded Hecke algebra, (affine) intertwiners, the recursion formula of Knop and Sahi. 3. HARMONIC ANALYSIS. The Fourier transform for the Dunkl-Cherednik operators, the Paley Wiener theorem, the action of the affine Weyl group. 4. RESIDUE CALCULUS FOR ROOT SYSTEMS. The Plancherel measure for the attractive case; classification of all square integrable eigenfunctions, and their explicit norms. S. Ruijsenaars (CWI, Amsterdam): Special functions solving analytic difference equations Abstract: * I Generalized gamma functions * II A generalized hypergeometric function * III Generalized Lame functions I. We discuss a new solution method for difference equations of the form F(z+ia/2)/F(z-ia/2) = Phi(z), with Phi(z) meromorphic and free of zeros and poles in a strip |Im(z)| < C. The method gives rise to generalized gamma functions of hyperbolic, elliptic and trigonometric type (Euler's gamma function being of rational type), whose properties we sketch. II. The hyperbolic gamma function can be used as a building block to construct a novel generalization of the hypergeometric function _2 F_1 . The new function is a simultaneous eigenfunction of four independent hyperbolic difference operators of Askey-Wilson type. The integral representation through which this joint eigenfunction is defined generalizes the Barnes representation for _2 F_1. It is meromorphic and has various remarkable symmetry properties that are not preserved for its q -> 1 ( or `nonrelativistic') limit _2 F_1. III. The `q=1/nonrelativistic' Lame differential operator can be generalized to a `q \ne 1/relativistic' difference operator. (The latter may be viewed as the Hamiltonian defining the elliptic relativistic Calogero-Moser N-particle system for N=2.) We present eigenfunctions of this operator. They are in fact joint eigenfunctions of three independent difference operators. The functions are used to define the Hamiltonian as a self-adjoint operator on a Hilbert space. Their asymptotics is governed by a c-function that is a quotient of two elliptic gamma functions. Topic #13 ------------ OP-SF NET 4.6 ------------ November 15, 1997 ~~~~~~~~~~~~~ From: Mathijs S. Dijkhuizen (msdz@math.s.kobe-u.ac.jp) Subject: Report on RIMS workshop, Kyoto SOME IMPRESSIONS FROM THE WORKSHOP AT RIMS The international workshop at RIMS late October was part of the scientific activities organized in the framework of a special Research Project on Representation Theory. Most of the activities are concentrated in the autumn. The project includes the invitation of a number of foreign researchers to RIMS for a more or less extended period. The two distinguished visitors this autumn are Prof. Grigori Olshanski (Moscow) and Prof. Eric Opdam (Leiden), who are staying here for four months. The workshop was split into two parts. The first week was devoted to representation theory and featured two series of lectures by Olshanski (on combinatorial and probabilistic aspects of harmonic analysis on big, i.e. infinite-dimensional, groups) and Prof. Michael Eastwood (Adelaide, on invariant differential operators on homogeneous spaces) plus a number of other talks. During the second week the topic was integrable systems of difference and differential equations. Since I did not attend the first part of the workshop, I will restrict myself to some comments about the second part. As suggested by the topic, most of the talks were somehow concerned with systems of commuting operators. The two main speakers were Eric Opdam and Prof. Simon Ruijsenaars (Amsterdam), who each gave three or four one-hour talks. Opdam talked about trigonometric Dunkl operators and their use in the study of multivariable hypergeometric functions associated with root systems. Hypergeometric functions in one variable have been known for a very long time; the earliest indications how to generalize them to many variables came from representation theory, where they arise as zonal spherical functions on Riemannian symmetric spaces. The notion of Dunkl operator, however, is something completely new which is not all hinted at by the connection with Riemannian symmetric spaces. Dunkl wrote down his original differential-reflection operators with rational coefficients around 1989. One of their main properties is that they commute with each other. A trigonometric version of these operators was introduced by Cherednik who related them to the degenerate affine Hecke algebra. The importance of Dunkl operators for the theory of hypergeometric functions is explained by the fact that they allow one to give an elementary algebraic construction of the commuting system of hypergeometric differential operators for arbitrary values of the coupling constants. The existence of this hypergeometric system was established earlier by Heckman and Opdam using analytic methods. Over the last couple of years a whole body of theory has developed around Dunkl operators. Opdam's talk was phrased in the elegant language of arbitrary reduced root systems and Weyl groups. This was in quite some contrast with Ruijsenaars' series of lectures, which, though certainly no less interesting, was characterized by a rather down-to-earth approach to a one-variable problem, namely the study of meromorphic solutions of certain types of analytic difference equations. Ruijsenaars actually started by remarking that an analyst from the late nineteenth century would have had no problem following at least the first part of his talks. Analytic difference operators were studied by several distinguished mathematicians until less than a hundred years ago, but later they failed to attract much interest. This seems to be changing now. Due to a notable lack of general theory about solutions of analytic difference equations most results have to be proved "by hand". One striking feature is the (rather obvious) fact that the solution space is usually infinite-dimensional. By imposing certain conditions on the asymptotic behaviour of the solution one can, however, arrive at certain uniqueness results. Ruijsenaars' motivation for studying these analytic difference equations partly comes from relativistic analogues of the quantum integrable Calogero-Moser-Sutherland models for N interacting particles on the real line. The trigonometric versions of these relativistic quantum models may be regarded as a q-analogue of the hypergeometric system discussed by Opdam. As shown by Cherednik, their algebraic properties are also amenable to a Dunkl operator approach. The polynomial solutions of these systems have been studied by Macdonald and others (for reduced root systems) and Koornwinder, Van Diejen, Noumi and others in the BC_n case (in the one-variable case they reduce to well-known families of q-hypergeometric polynomials of Askey-Wilson type). These polynomials are also known to occur in connection with quantum groups. As for non-polynomial solutions of these systems, not much is known at this time. As is apparent from Ruijsenaars' talks, a lot of interesting work in this direction is still waiting to be done. In short, this was a very stimulating workshop with some very interesting mathematics. Mathijs S. Dijkhuizen

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