Vol. 11, No. 5, 2017

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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

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.

Affine Hecke algebra, graded affine Hecke algebra, Dirac operator, discrete series representation
Mathematical Subject Classification 2010
Primary: 20C08
Secondary: 22D25, 43A30
Received: 21 April 2016
Revised: 6 September 2016
Accepted: 4 December 2016
Published: 12 July 2017
Dan Ciubotaru
Mathematical Institute
University of Oxford
Andrew Wiles Building
United Kingdom
Eric Opdam
Korteweg-de Vries Institute for Mathematics
Universiteit van Amsterdam
Science Park 105-107
1098 XG Amsterdam