Algebra & Number Theory

A uniform classification of discrete series representations of affine Hecke algebras

Dan Ciubotaru and Eric Opdam

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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.

Article information

Algebra Number Theory, Volume 11, Number 5 (2017), 1089-1134.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20C08: Hecke algebras and their representations
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

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


Ciubotaru, Dan; Opdam, Eric. A uniform classification of discrete series representations of affine Hecke algebras. Algebra Number Theory 11 (2017), no. 5, 1089--1134. doi:10.2140/ant.2017.11.1089.

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