TITLE: Representations of big groups: combinatorial and probabilistic aspects

BY: Grigori Olshanski, Moscow

ABSTRACT:

The model examples of big groups are the infinite symmetric group and the infinite-dimensional unitary group. The tentative plan of the talks is as follows:The talks are based on joint works with A.Borodin, S.Kerov, A.Okounkov, and A.Vershik.

- Introduction. Thoma's and Voiculescu's character formulas. Asymptotical theory of characters and spherical functions.
- Generalizations involving branching graphs with formal multiplicities of edges and multivariate orthogonal polynomials.
- What is harmonic analysis on a big group.
- Stochastic point processes (=measures on point configurations) arising in harmonic analysis and their correlation functions.

EXPECTED NUMBER OF TALKS: 4

- Invariant Differential Operators on Homogeneous Spaces.
- Invariant Differential Operators in Conformal Geometry.
- Invariant Differential Operators and the Translation Principle.

Each lecture would be for one hour (well 50 minutes allowing 10 minutes for questions) and here is an abstract for the whole series:

An invariant linear differential operator on a homogeneous space G/P is a linear differential operator between homogeneous vector bundles invariant under the action of G. It is well-known that such operators correspond to homomorphisms of induced modules and, in many cases, this allows a classification. Of particular interest is the n-sphere under the action of SO(n+1,1) since this is the flat model of conformal differential geometry. In this case, there are close links between the G-invariant differential operators on the sphere and those differential operators on a general conformal manifold defined intrinsically by the geometry. The construction of these conformally invariant differential operators may often be achieved by variations on the Jantzen-Zuckerman translation principle.

In this series of lectures, I shall illustrate the general theory by means of a series of examples. No prior knowledge will be assumed save for some familiarity with Lie algebras and elementary differential geometry.

TITLE: Dunkl Operators

BY: Eric M. Opdam, Leiden Univ.

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:

- DEFINITION AND BASIC ANALYTIC RESULTS. The Knizhnik-Zamolodchikov connection, the Harich-Chandra system, monodromy representation, the shifting principle, asymptotic expansions, the Gauss' summation formula.
- ALGEBRAIC PROPERTIES. Nonsymmetric orthogonal polynomials, the graded Hecke algebra, (affine) intertwiners, the recursion formula of Knop and Sahi.
- HARMONIC ANALYSIS. The Fourier transform for the Dunkl-Cherednik operators, the Paley Wiener theorem, the action of the affine Weyl group.
- RESIDUE CALCULUS FOR ROOT SYSTEMS. The Plancherel measure for the attractive case; classification of all square integrable eigenfunctions, and their explicit norms.

EXPECTED NUMBER OF TALKS: 4

TITLE: Special functions solving analytic difference equations

10/28-30 11:00 - 12:00 (3 lectures)

- 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 $|\Im (z)|~~0$). 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 $q \rightarrow 1$ ( or `nonrelativistic') limit $ _2 F_1 $.

III. The `$q=1$/nonrelativistic' Lam\'e 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.

TITLE:
"On the Knizhnik Zamolodchikov equations
for complex reflection groups"

AND "Symmetries for fake degrees of complex reflection groups"

The second talk is an application of the first, and in the first talk I can explain some topological results of Broue, Malle, and Rouquier for orbit spaces, and results/conjectures for Dunkl theory for complex reflection groups.

title : Invariant Differential Operators, Special Functions and Representation Theory

organizer : Toshio Oshima (University of Tokyo)

date : 1997/10/20 (Mon) -- 10/31 (Fri)

place : RIMS

10/20(Mon) 13:30 - 14:30 Grigori Olshanski(IPIT) "I: Representations of big groups: combinatorial and probabilistic aspects - Introduction. Thoma's and Voiculescu's character formulas. Asymptotical theory of characters and spherical functions. " 14:45 - 15:45 Toshiyuki Kobayashi(Univ. of Tokyo) "Conformal geometry and unipotent branching laws of O(p,q) attached to minimal nilpotent orbits" 10/21(Tue) 9:45 - 10:45 Roger Zierau(Oklahoma State Univ.) "Realization of some singular representations" 10:00 - 11:00 Michael Eastwood(Univ. Adelaide) "I: Invariant Differential Operators on Homogeneous Spaces" 13:30 - 14:30 Grigori I. Olshanski(IPIT) "II: Representations of big groups: combinatorial and probabilistic aspects - Generalizations involving branching graphs with formal multiplicities of edges and multivariate orthogonal polynomials" 14:45 - 15:45 Vera Serganova(UC Berkeley) "Blocks in categories of representations of Lie superalgebra" 16:00 - 17:00 Akihito Hora(Okayama Univ.) "Central limit theorem for the adjacency operators on the infinite symmetric group" 10/22(Wed) 9:15 - 10:15 Michael Eastwood(Univ. Adelaide) "II: Invariant Differential Operators in Conformal Geometry" 10:30 - 11:30 Grigori I. Olshanski(IPIT) "III: Representations of big groups: combinatorial and probabilistic aspects - What is harmonic analysis on a big group" 11:45 --> Excursion 10/23(Thur) 9:45 - 10:45 Hiroshi Yamashita(Hokkaido Univ.) "Associated variety and invariant differential operators: relation to the U(n)-action on Harish-Cahndar modules" 11:00 - 12:00 Michael Eastwood(Univ. Adelaide) "III: Invariant Differential Operators and the Translation Principle" 13:30 - 14:30 Grigori I. Olshanski(IPIT) "IV: Representations of big groups: combinatorial and probabilistic aspects - Stochastic point processes (=measures on point configurations) arising in harmonic analysis and their correlation functions." 14:45 - 15:45 Ivan B. Penkov(UC Riverside) "Weight modules of finite dimensional and infinite dimensional Lie algebras" 16:00 - 17:00 Hideko Sekiguchi "The Penrose transform and singular unitary representations" 10/24(Fri) 9:15 - 10:15 Patrick Delorme(Univ. Marseille) "Plancherel formula for real reductive symmetric spaces I" 10:30 - 11:30 Patrick Delorme(Univ. Marseille) "Plancherel formula for real reductive symmetric spaces II" 11:45 - 12:45 Zhao Yan Da(Hangzhou Univ., China) "The isotropy representation for Normal j-algebras" *** Lecture of Yurii A. Neretin(Moscow Institute of Electronics and Mathematics) is postponed to the next week. -------------------------------------------------------------------------- 10/27(mon) 13:30 - 14:30 Eric Opdam(Leiden Univ.) "I: Dunkl Operators - Definition and basic analytic results" 14:30 - 15:00 Discussion of the program of this week 15:00 - 16:00 Alberto Grunbaum(UC Berkeley) "The bispectral problem and connections with hypergeometric functions, monodromy and non-linear evolution equations" 10/28(Thus) 9:45 - 10:45 Peter Littelmann(Univ. Strasbourg) "Contracting modules and standard monomial theory" 11:00 - 12:00 Simon N. M. Ruijsenaars(CWI, Netherland) "I: Special functions solving analytic difference equations - generalized gamma functions" 13:30 - 14:30 Eric Opdam(Leiden Univ.) "II: Dunkl Operators - Algebraic properties" 14:45 - 15:45 Hiroyuki Ocihai(Rikkyo Univ.) "Spherical representation on GL(n+1)/GL(n)*GL(1) over F_q" 10/29(Wed) 9:45 - 10:45 van Diejen(CRM, Canada) "The compact Ruijsenaars-Schneider systems and discrete analysis on a Weyl group" 11:00 - 12:00 Simon N. M. Ruijsenaars(CWI, Netherland) "II: Special functions solving analytic difference equations - a generalized hypergeometric function" 13:30 - 14:30 Eric Opdam(Leiden Univ.) "III: Dunkl Operators - Harmonic analysis" 14:45 - 15:45 Masatoshi Noumi(Kobe Univ.) "Some topics on multivariable Askey-Wilson polynomials" 16:00 - 17:00 Yurii A. Neretin(Moscow Institute of Electronics and Mathematics) 10/30(Thur) 9:45 - 10:45 Takashi Takebe(Univ. of Tokyo) "On the Gaudin models" 11:00 - 12:00 Simon N. M. Ruijsenaars(CWI, Netherland) "III: Special functions solving analytic difference equations - generalized Lame functions" 13:30 - 14:30 Eric Opdam(Leiden Univ.) "IV: Dunkl Operators - Residue calculus for root systems" 14:45 - 15:45 Atsushi Mastuo "Axioms for a vertex algebra and the locality of quatum fields" 16:00 - 17:00 Koji Hasegawa (Tohoku Univ.) "On Ruijsenaars' elliptic integrable system" 10/31(Fri) 9:45 - 10:45 Kenji Taniguchi "Commutants of CMS type Hamiltonians" 11:00 - 12:00 Toshio Oshima "Differential equations characterizing degenerate principal series for classical Lie groups"

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