On characters and formal degrees of discrete series of affine Hecke algebras of classical types

Abstract

We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder (J. Reine Angew. Math. 520:37-93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239-267, 1983; Lusztig in Ann. Math. 131:355-408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C *-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531-648, 2004; arXiv:0909. 1227; Int. Math. Res. Not., 2008; Adv. Math. 220:1549-1601, 2009; Acta Math. 205:105-187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305-371, 2009; Am. J. Math. 133:518-553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538-1590, 2011). © 2011 The Author(s).