Extensions of tempered representations

Abstract

Let π, π′ be irreducible tempered representations of an affine Hecke algebra $${\mathcal{H}}$$ with positive parameters. We compute the higher extension groups Ext $${{}_\mathcal{H}^n (\pi,\pi')}$$ 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.