Lecture notes on Dunkl operators for real and complex reflection groups
type: Item
instance of: scholarly article
title: Lecture notes on Dunkl operators for real and complex reflection groups
zbMATH Open document ID: 0984.33001
author: Eric Opdam
published in: MSJ Memoirs
publication date: 2001-02-15T00:00:00Z
full work available at URL: http://projecteuclid.org/euclid.msjm/1389985754
review text: These memoirs are based on a series of lectures presented by the author at RIMS, Kyoto University, in 1997. It is divided in two parts. The first part deals with the spectral theory of Dunkl-Cherednik operators and outlines the most recent developments in the theory of hypergeometric functions associated with root systems. The second part studies the fake degrees of complex reflection groups and a topological cyclotomic Hecke algebra constructed by the monodromy representation of a certain system of differential operators. In the late 80ies, G.J. Heckman and E.M. Opdam defined the hypergeometric functions associated with root systems as analytic continuation of the Harish-Chandra's spherical functions on Riemannian symmetric spaces. See \textit{G. J. Heckman} and \textit{E. M. Opdam} [Compos. Math. 64, 329-352 (1987; Zbl 0656.17006)], \textit{G. J. Heckman} [ibid. 64, 353-373 (1987; Zbl 0656.17007)] and \textit{E. M. Opdam} [ibid. 67, 21-49, 191-209 (1988; Zbl 0669.33007, Zbl 0669.33008)]. The analytically continued parameters are the root multiplicities. The definition of Heckman-Odpam's hypergeometric functions is based on a generalization of the system of differential equations satisfied by Harish-Chandra's spherical functions. The system represents the common spectral theory of a commuting family of differential operators, the so-called hypergeometric differential operators. In the symmetric case, they agree with the radial components on a maximal split Cartan subgroup of the invariant differential operators on the given Riemannian symmetric space. The construction of the hypergeometric differential operators became of completely algebraic nature when \textit{G. J. Heckman} [Invent. Math. 103, 341-350 (1991; Zbl 0721.33009)] obtained them using a certain family of first-order non-commuting differential-reflection operators as building blocks. Heckman's algebraic construction was later improved by \textit{I. Cherednik} [Invent. Math. 106, No. 2, 411-432 (1991; Zb…
reviewed by: Angela Pasquale
Mathematics Subject Classification ID: 33-02
Mathematics Subject Classification ID: 33C67
Mathematics Subject Classification ID: 43A32
zbMATH DE Number: 1565321
zbMATH Keywords: hypergeometric functions associated with root systems
zbMATH Keywords: non-symmetric hypergeometric functions associated with root systems
zbMATH Keywords: Knizhnik-Zamalodchikov connection
zbMATH Keywords: Dunkl-Cherednik operators
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: Plancherel formula
zbMATH Keywords: complex reflection groups
zbMATH Keywords: fake degrees
zbMATH Keywords: topological cyclotomic Hecke algebra
MaRDI profile type: Publication
OpenAlex ID: W1919028956
DOI: 10.2969/MSJMEMOIRS/008010000
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description: scientific article; zbMATH DE number 1565321