Algebra & Number Theory
- Algebra Number Theory
- Volume 11, Number 5 (2017), 1089-1134.
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 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.
Algebra Number Theory, Volume 11, Number 5 (2017), 1089-1134.
Received: 21 April 2016
Revised: 6 September 2016
Accepted: 4 December 2016
First available in Project Euclid: 12 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20C08: Hecke algebras and their representations
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
Ciubotaru, Dan; Opdam, Eric. A uniform classification of discrete series representations of affine Hecke algebras. Algebra Number Theory 11 (2017), no. 5, 1089--1134. doi:10.2140/ant.2017.11.1089. https://projecteuclid.org/euclid.ant/1513090723