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

TL;DR

This paper develops an effective inductive algorithm for computing characters of tempered modules and determines constants in the formal degrees of discrete series for affine Hecke algebras of classical types, advancing understanding in their representation theory.

Contribution

It introduces a new inductive algorithm for character computation and determines formal degree constants, differing from existing methods and simplifying analysis for affine Hecke algebras.

Findings

Effective algorithm for character computation of tempered modules.

Determination of formal degree constants in discrete series.

Comparison with existing algorithms and theoretical frameworks.

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 \cite{Re}). The former is completely different than the Lusztig-Shoji algorithm \cite{Sh, L}, 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 $C^{\ast}$-theoretic results of Opdam, Delorme, Slooten, Solleveld \cite{O, DO, Sl2, OSa, OS}, and the geometric construction from \cite{K1,K2,CK}.