A uniform classification of discrete series representations of affine Hecke algebras

Ciubotaru, Dan; Opdam, Eric

doi:10.2140/ant.2017.11.1089

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A uniform classification of discrete series representations of affine Hecke algebras

Dan Ciubotaru and Eric Opdam

Vol. 11 (2017), No. 5, 1089–1134

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 $ℋ_{v}$ , in terms of a canonically defined basis $ℬ_{gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $ℋ$ , and to all $v \in Q$ , where $Q$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $ℋ$ . By analytic Dirac induction we define for each $b \in ℬ_{gm}$ a continuous (in the sense of Opdam and Solleveld (2010)) family $Q_{b}^{reg} : = Q_{b} ∖ Q_{b}^{sing} ∋ v \to {Ind}_{D} (b; v)$ , such that $ϵ (b; v) {Ind}_{D} (b; v)$ (for some $ϵ (b; v) \in {\pm 1}$ ) is an irreducible discrete series character of $ℋ_{v}$ . Here $Q_{b}^{sing} \subset Q$ is a finite union of hyperplanes in $Q$ .

In the nonsimply laced cases we show that the families of virtual discrete series characters ${Ind}_{D} (b; v)$ are piecewise rational in the parameters $v$ . Remarkably, the formal degree of ${Ind}_{D} (b; v)$ in such piecewise rational family turns out to be rational. This implies that for each $b \in ℬ_{gm}$ there exists a universal rational constant $d_{b}$ determining the formal degree in the family of discrete series characters $ϵ (b; v) {Ind}_{D} (b; v)$ . We will compute the canonical constants $d_{b}$ , 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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Keywords

Affine Hecke algebra, graded affine Hecke algebra, Dirac operator, discrete series representation

Mathematical Subject Classification 2010

Primary: 20C08

Secondary: 22D25, 43A30

Milestones

Received: 21 April 2016

Revised: 6 September 2016

Accepted: 4 December 2016

Published: 12 July 2017

Authors

Dan Ciubotaru
Mathematical Institute University of Oxford Andrew Wiles Building Oxford OX2 6GG United Kingdom

Eric Opdam
Korteweg-de Vries Institute for Mathematics Universiteit van Amsterdam Science Park 105-107 1098 XG Amsterdam Netherlands

Vol. 11, No. 5, 2017