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.

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