Harmonic analysis for certain representations of graded Hecke algebras
type: Item
instance of: scholarly article
title: Harmonic analysis for certain representations of graded Hecke algebras
zbMATH Open document ID: 0836.43017
DOI: 10.1007/BF02392487
published in: Acta Mathematica
publication date: 1996-04-29T00:00:00Z
review text: For a fixed positive root \(R_+ \subseteq R\) of a root system \(R \subset {\mathfrak a}^*\) of a Euclidean space \(\mathfrak a\) \textit{I. Cherednik} [Invent. Math. 106, 411-432 (1991; Zbl 0742.20019)] has defined differential operators \(D_\zeta\), \(\zeta \in {\mathfrak h}\) (= complexification of \(\mathfrak a\)), acting in the space of functions on \({\mathfrak h}\) which are invariant for the translations in the lattice \(2 \pi i ZR^\vee\) (\(R^\vee\) = the coroot system). These operators and the Weyl group generate an operator algebra, which is isomorphic to the graded Hecke algebra \(\mathcal H\) in the sense of \textit{G. Lusztig} [J. Am. Math. Soc. 2, 599-635 (1989; Zbl 0715.22020)]. In this paper the author analyses the properties of the spaces \(C^\infty_c ({\mathfrak a})\) and \(C^\infty (T)\) \((T = i{\mathfrak a}/2\pi iZ R^\vee\)), which are \(\mathcal H\)-modules with natural + and \(*\)-invariant inner products. Decompositions of the spaces into irreducible parts are given, the Paley- Wiener theorem is proved and the inversion and the Plancherel formula are established.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46H15
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 807538
zbMATH Keywords: hypergeometric functions
zbMATH Keywords: Weyl groups
zbMATH Keywords: root system
zbMATH Keywords: Weyl group
zbMATH Keywords: Hecke algebra
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: Plancherel formula
author: Eric Opdam
reviewed by: Ernest Płonka
MaRDI profile type: Publication
Wikidata QID: Q93274972
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cites work: Differential-Difference Operators Associated to Reflection Groups
cites work: Q3803293
cites work: An elementary approach to the hypergeometric shift operators of Opdam
cites work: Q3803292
cites work: Q3331943
cites work: The Dunkl transform
cites work: A new proof of a Paley-Wiener type theorem for the Jacobi transform
cites work: Affine Hecke Algebras and Their Graded Version
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cites work: Q4694963
cites work: An analogue of the Gauss summation formula for hypergeometric functions related to root systems
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description: scientific article; zbMATH DE number 807538