Vol. 11, No. 5, 2017

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
A uniform classification of discrete series representations of affine Hecke algebras

Dan Ciubotaru and Eric Opdam

Vol. 11 (2017), No. 5, 1089–1134
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 := QbQbsingv 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.

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