A uniform classification of discrete series representations of affine Hecke algebras

作者：Ciubotaru Dan; Opdam Eric

来源：ALGEBRA %26 NUMBER THEORY, 2017, 11(5): 1089-1134.

DOI：10.2140/ant.2017.11.1089

• Affine Hecke algebra
• graded affine Hecke algebra
• Dirac operator
• discrete series representation

We give a new and independent parametrization of the set of discrete series characters of an affine Hecke algebra H-v, in terms of a canonically defined basis 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 H, and to all v epsilon 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 epsilon B-gm a continuous (in the sense of Opdam and Solleveld (2010)) family Q(b)(reg) := Q(b) \ Q(b)(sing) (sic) v -> Ind(D) (b; v), such that epsilon(b; v)Ind(D)(b; v) (for some epsilon(b; v) is an element of {+/- 1}) is an irreducible discrete series character of H-v. Here Q(b)(sing) subset of 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 is an element of B-gm there exists a universal rational constant db determining the formal degree in the family of discrete series characters epsilon(b; v)Ind(D) (b; v). We will compute the canonical constants d(b), and the signs epsilon(b; v). For certain geometric parameters we will provide the comparison with the Kazhdan-Lusztig-Langlands classification.

• 出版日期2017