A formula of Arthur and affine Hecke algebras

TL;DR

This paper derives a simple formula for the Euler--Poincaré pairing of tempered representations of affine Hecke algebras, linking it to an analytic R-group, and extends the approach to reductive groups over nonarchimedean fields.

Contribution

It introduces a new explicit formula for the Euler--Poincaré pairing involving an analytic R-group, applicable to affine Hecke algebras and reductive p-adic groups.

Findings

Derived a formula for Euler--Poincaré pairing using an analytic R-group

Extended the method to reductive groups over nonarchimedean fields

Unified approach applicable to affine Hecke algebras and p-adic groups

Abstract

Let $\pi, \pi'$ be tempered representations of an affine Hecke algebra with positive parameters. We study their Euler--Poincar\'e pairing $EP (\pi,\pi')$, the alternating sum of the dimensions of the Ext-groups. We show that $EP (\pi,\pi')$ can be expressed in a simple formula involving an analytic R-group, analogous to a formula of Arthur in the setting of reductive p-adic groups. Our proof applies equally well to affine Hecke algebras and to reductive groups over nonarchimedean local fields of arbitrary characteristic.