A uniform classification of discrete series representations of affine Hecke algebras

Abstract

We give a new and independent parameterization of the set of discrete series characters of an affine Hecke algebra ${\mathcal{H}}_{\mathbf{v}}$, in terms of a canonically defined basis ${\mathcal{B}}_{gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $\mathcal{H}$, and to all $\mathbf{v}\in \mathcal{Q}$, where $\mathcal{Q}$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $\mathcal{H}$. By analytic Dirac induction we define for each $b\in {\mathcal{B}}_{gm}$ a continuous (in the sense of [OS2]) family ${\mathcal{Q}}_b^{reg}:={\mathcal{Q}}_b\backslash {\mathcal{Q}}_b^{sing}\ni \mathbf{v}\to {\mathrm{Ind}}_D(b;\mathbf{v})$, such that $\epsilon (b;\mathbf{v}){\mathrm{Ind}}_D(b;\mathbf{v})$ (for some $\epsilon (b;\mathbf{v})\in \{\pm 1\}$) is an irreducible discrete series character of ${\mathcal{H}}_{\mathbf{v}}$. Here ${\mathcal{Q}}_b^{sing}\subset \mathcal{Q}$ is a finite union of hyperplanes in $\mathcal{Q}$. In the non-simply laced cases we show that the families of virtual discrete series characters ${\mathrm{Ind}}_D(b;\mathbf{v})$ are piecewise rational in the parameters $\mathbf{v}$. Remarkably, the formal degree of ${\mathrm{Ind}}_D(b;\mathbf{v})$ in such piecewise rational family turns out to be rational. This implies that for each $b\in {\mathcal{B}}_{gm}$ there exists a universal rational constant $d_b$ determining the formal degree in the family of discrete series characters $\epsilon (b;\mathbf{v}){\mathrm{Ind}}_D(b;\mathbf{v})$. We will compute the canonical constants $d_b$, and the signs $\epsilon (b;\mathbf{v})$. For certain geometric parameters we will provide the comparison with the Kazhdan-Lusztig-Langlands classification.

31 pages, 2 tables

References in corpus (4)

• The principal series of $p$-adic groups with disconnected centre
• Cocenters and representations of affine $0$-Hecke algebras
• Formal degrees of unipotent discrete series representations and the exotic Fourier transform
• Spectral transfer morphisms for unipotent affine Hecke algebras

Cited by in corpus (2)

• On formal degrees of unipotent representations
• Topological K-theory of affine Hecke algebras