Reduction to real infinitesimal character in affine Hecke algebras

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