Extensions of tempered representations

TL;DR

This paper explicitly computes higher extension groups for irreducible tempered representations of affine Hecke algebras, linking them to analytic R-groups and providing new insights into Arthur's elliptic pairing and Kazhdan's orthogonality conjecture.

Contribution

It introduces explicit calculations of Ext-groups for tempered representations, connecting them to R-group representations and extending results to reductive p-adic groups.

Findings

Explicit formulas for Ext-groups in terms of R-group representations

Verification of Arthur's formula for Euler-Poincaré pairing

New proof of Kazhdan's orthogonality conjecture

Abstract

Let $\pi, \pi'$ be irreducible tempered representations of an affine Hecke algebra H with positive parameters. We compute the higher extension groups $Ext_H^n (\pi,\pi')$ explicitly in terms of the representations of analytic R-groups corresponding to $\pi$ and $\pi'$. The result has immediate applications to the computation of the Euler-Poincar\'e pairing $EP(\pi,\pi')$, the alternating sum of the dimensions of the Ext-groups. The resulting formula for $EP(\pi,\pi')$ is equal to Arthur's formula for the elliptic pairing of tempered characters in the setting of reductive p-adic groups. Our proof applies equally well to affine Hecke algebras and to reductive groups over non-archimedean local fields of arbitrary characteristic. This sheds new light on the formula of Arthur and gives a new proof of Kazhdan's orthogonality conjecture for the Euler-Poincar\'e pairing of admissible characters.