Extensions of tempered representations

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

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