We give a new and independent parametrization of the set of discrete series characters of an affine Hecke algebra , in terms of a canonically defined basis of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras , and to all , where denotes the vector group of positive real (possibly unequal) Hecke parameters for . By analytic Dirac induction we define for each a continuous (in the sense of Opdam and Solleveld (2010)) family , such that (for some ) is an irreducible discrete series character of . Here is a finite union of hyperplanes in .
In the nonsimply laced cases we show that the families of virtual discrete series characters are piecewise rational in the parameters . Remarkably, the formal degree of in such piecewise rational family turns out to be rational. This implies that for each there exists a universal rational constant determining the formal degree in the family of discrete series characters . We will compute the canonical constants , and the signs . For certain geometric parameters we will provide the comparison with the Kazhdan–Lusztig–Langlands classification.
"A uniform classification of discrete series representations of affine Hecke algebras." Algebra Number Theory 11 (5) 1089 - 1134, 2017. https://doi.org/10.2140/ant.2017.11.1089