Reduction to real infinitesimal character in affine Hecke algebras

Abstract

The main result of this paper is to show that the problem of the determination of the unramified unitary dual of a split p-adic group is equivalent to the problem of determining the unitary dual of the corresponding graded Hecke algebra. In [BM], the authors established this equivalence in the case of Iwahori spherical representations under a certain restriction; namely, it was essential for the infinitesimal character to be real (in the terminology of [BMJ). In terms of the Langlands-Deligne-Lusztig parameters (s, u, p) [KUL, the restriction is that S E LG be a purely hyperbolic element. The technique used in [BM] was to combine the notion of the signature of a K-character in [V] with some facts which follow from [KL], namely, that the 2w-characters of tempered representations are linearly independent. This is essentially true precisely when the infinitesimal character is real; for if not, s has an elliptic part Se such that