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2017 A uniform classification of discrete series representations of affine Hecke algebras
Dan Ciubotaru, Eric Opdam
Algebra Number Theory 11(5): 1089-1134 (2017). DOI: 10.2140/ant.2017.11.1089

Abstract

We give a new and independent parametrization of the set of discrete series characters of an affine Hecke algebra v, in terms of a canonically defined basis gm of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras , and to all v Q, where Q denotes the vector group of positive real (possibly unequal) Hecke parameters for . By analytic Dirac induction we define for each b gm a continuous (in the sense of Opdam and Solleveld (2010)) family Qbreg := Qb Qbsing v IndD(b;v), such that ϵ(b;v)IndD(b;v) (for some ϵ(b;v) {±1}) is an irreducible discrete series character of v. Here Qbsing Q is a finite union of hyperplanes in Q.

In the nonsimply laced cases we show that the families of virtual discrete series characters IndD(b;v) are piecewise rational in the parameters v. Remarkably, the formal degree of IndD(b;v) in such piecewise rational family turns out to be rational. This implies that for each b gm there exists a universal rational constant db determining the formal degree in the family of discrete series characters ϵ(b;v)IndD(b;v). We will compute the canonical constants db, and the signs ϵ(b;v). For certain geometric parameters we will provide the comparison with the Kazhdan–Lusztig–Langlands classification.

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Dan Ciubotaru. Eric Opdam. "A uniform classification of discrete series representations of affine Hecke algebras." Algebra Number Theory 11 (5) 1089 - 1134, 2017. https://doi.org/10.2140/ant.2017.11.1089

Information

Received: 21 April 2016; Revised: 6 September 2016; Accepted: 4 December 2016; Published: 2017
First available in Project Euclid: 12 December 2017

zbMATH: 1373.20003
MathSciNet: MR3671432
Digital Object Identifier: 10.2140/ant.2017.11.1089

Subjects:
Primary: 20C08
Secondary: 22D25 , 43A30

Keywords: affine Hecke algebra , Dirac operator , discrete series representation , graded affine Hecke algebra

Rights: Copyright © 2017 Mathematical Sciences Publishers

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