#### 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
##### Abstract

We give a new and independent parametrization of the set of discrete series characters of an affine Hecke algebra ${\mathsc{ℋ}}_{v}$, in terms of a canonically defined basis ${\mathsc{ℬ}}_{gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $\mathsc{ℋ}$, and to all $v\in \mathsc{Q}$, where $\mathsc{Q}$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $\mathsc{ℋ}$. By analytic Dirac induction we define for each $b\in {\mathsc{ℬ}}_{gm}$ a continuous (in the sense of Opdam and Solleveld (2010)) family ${\mathsc{Q}}_{b}^{reg}:={\mathsc{Q}}_{b}\setminus {\mathsc{Q}}_{b}^{sing}\ni v\to {Ind}_{D}\left(b;v\right)$, such that $ϵ\left(b;v\right){Ind}_{D}\left(b;v\right)$ (for some $ϵ\left(b;v\right)\in \left\{±1\right\}$) is an irreducible discrete series character of ${\mathsc{ℋ}}_{v}$. Here ${\mathsc{Q}}_{b}^{sing}\subset \mathsc{Q}$ is a finite union of hyperplanes in $\mathsc{Q}$.

In the nonsimply laced cases we show that the families of virtual discrete series characters ${Ind}_{D}\left(b;v\right)$ are piecewise rational in the parameters $v$. Remarkably, the formal degree of ${Ind}_{D}\left(b;v\right)$ in such piecewise rational family turns out to be rational. This implies that for each $b\in {\mathsc{ℬ}}_{gm}$ there exists a universal rational constant ${d}_{b}$ determining the formal degree in the family of discrete series characters $ϵ\left(b;v\right){Ind}_{D}\left(b;v\right)$. We will compute the canonical constants ${d}_{b}$, and the signs $ϵ\left(b;v\right)$. 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