Extensions of tempered representations

Eric Opdam, Maarten Solleveld

Published 2011-05-19, updated 2012-02-14Version 2

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.