A uniform classification of discrete series representations of affine Hecke algebras

© 2017, Mathematical Sciences Publishers. All rights reserved. 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 Q reg b := Qb \ Q sing b ∋ v → Ind D (b; v), such that ɛ(b; v)Ind D (b; v) (for some ɛ(b; v) ∈ {±1}) is an irreducible discrete series character of Hv . Here Q sing b ⊂ Q is a finite union of hyperplanes in Q . In the nonsimply laced cases we show that the families of virtual discrete series characters Ind D (b; v) are piecewise rational in the parameters v. Remarkably, the formal degree of Ind D (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 d b determining the formal degree in the family of discrete series characters ɛ(b; v)Ind D (b; v). We will compute the canonical constants d b , and the signs ɛ(b; v). For certain geometric parameters we will provide the comparison with the Kazhdan–Lusztig–Langlands classification.