Discrete series characters for affine Hecke algebras and their formal degrees. (English) Zbl 1214.20008

Opdam, Eric; Solleveld, Maarten

Acta Math. 205, No. 1, 105-187 (2010).

The goal of this paper is to continue previous work of the authors and others to develop harmonic analysis on semisimple affine Hecke algebras (determined by a based root datum and a positive parameter function) modeled to an extent after harmonic analysis on Lie groups. The two main goals of the paper are to classify the irreducible discrete series characters and calculate their formal degrees. A key new tool used in this work is the idea of deformation of parameters. For a fixed root datum the authors consider a continuous field of affine Hecke algebras as the parameter is allowed to vary. This allows them to reduce the problem for arbitrary parameters to “generic” ones.
The results require the study of how the “central characters” behave under deformation, and also makes use of the classification of “generic residual points.” The authors achieve their second goal by obtaining a formula for the formal degree as a product of a (as yet unknown) rational number and an explicit rational function in the parameter. The main result of the paper is a classification of the irreducible discrete series characters for an affine Hecke algebra associated to an irreducible root datum (except type \(E\)) in terms of generic central characters. The classification is obtained by reducing the problem to degenerate affine Hecke algebras. The authors also discuss how to deal with the classification in the more general case of a semisimple root datum.

Reviewer: Christopher P. Bendel (Menomonie)

MSC:

20C08	Hecke algebras and their representations
22D25	\(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
22E35	Analysis on \(p\)-adic Lie groups
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods
43A99	Abstract harmonic analysis

Keywords:

affine Hecke algebras; discrete series characters; formal degrees; Schwartz completion; temperedness; Plancherel isomorphism; residual points; spectral diagrams

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