Abstract
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.
Citation
Dan Ciubotaru. Eric Opdam. "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
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