A uniform classification of discrete series representations of affine Hecke algebras

We give a new and independent parametrization of the set of discrete series characters of an affine Hecke algebraHv, in terms of a canonically defined basisBgm 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 forH. By analytic Dirac induction we define for each b ∈Bgm a continuous (in the sense of Opdam and Solleveld (2010)) familyQ reg b:=Qb \Q sing b ∋ v → IndD (b; v), such that ɛ(b; v)IndD (b; v) (for some ɛ(b; v) ∈ {±1}) is an irreducible discrete series character ofHv. HereQ 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 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 ɛ(b; v)IndD (b; v). We will compute the canonical constants db, and the signs ɛ(b; v). For certain geometric parameters we will provide the comparison with the Kazhdan–Lusztig–Langlands classification.