Extensions of tempered representations

Abstract

Let $\pi ,{\pi }^{\prime }$ be irreducible tempered representations of an affine Hecke algebra H with positive parameters. We compute the higher extension groups $Ex{t}_H^n(\pi ,{\pi }^{\prime })$ explicitly in terms of the representations of analytic R-groups corresponding to $\pi$ and ${\pi }^{\prime }$. The result has immediate applications to the computation of the Euler-Poincaré pairing $EP(\pi ,{\pi }^{\prime })$, the alternating sum of the dimensions of the Ext-groups. The resulting formula for $EP(\pi ,{\pi }^{\prime })$ 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-Poincaré pairing of admissible characters.

This paper grew out of "A formula of Arthur and affine Hecke algebras" (arXiv:1011.0679). In the second version some minor points were improved

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