## Algebra & Number Theory

### A uniform classification of discrete series representations of affine Hecke algebras

#### Abstract

We give a new and independent parametrization of the set of discrete series characters of an affine Hecke algebra $ℋv$, in terms of a canonically defined basis $ℬgm$ 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 $v ∈Q$, where $Q$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $ℋ$. By analytic Dirac induction we define for each $b ∈ℬgm$ 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 $ℋv$. 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 $b ∈ℬgm$ 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

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

Dates
Revised: 6 September 2016
Accepted: 4 December 2016
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.ant/1513090723

Digital Object Identifier
doi:10.2140/ant.2017.11.1089

Mathematical Reviews number (MathSciNet)
MR3671432

Zentralblatt MATH identifier
1373.20003

#### Citation

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. https://projecteuclid.org/euclid.ant/1513090723

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