Homological algebra for affine Hecke algebras

TL;DR

This paper investigates the homological properties of modules over affine Hecke algebras, providing a comparison result for higher extensions and a construction applicable across all positive parameters, aiding in classifying discrete series representations.

Contribution

It introduces a self-contained method to compare higher extensions of modules over affine Hecke algebras and constructs bounded contractions applicable for all positive parameters.

Findings

Comparison result for higher extensions of tempered modules

Construction of bounded contractions for standard resolutions

Application to classification of discrete series characters

Abstract

In this paper we study homological properties of modules over an affine Hecke algebra H. In particular we prove a comparison result for higher extensions of tempered modules when passing to the Schwartz algebra S, a certain topological completion of the affine Hecke algebra. The proof is self-contained and based on a direct construction of a bounded contraction of certain standard resolutions of H-modules. This construction applies for all positive parameters of the affine Hecke algebra. This is an important feature since it is an ingredient to analyse how the irreducible discrete series representations of H arise in generic families over the parameter space of H. For irreducible non-simply laced affine Hecke algebras this will enable us to give a complete classification of the discrete series characters for all positive parameters (we will report on this application in a separate article).