Classification of unipotent representations of simple p-adic groups

0.1. Let K be a nonarchimedean local field with a residue field of cardinal q. Let G(K) be the group of K-rational points of a connected, adjoint simple algebraic group G defined over K which becomes split over an unramified extension of K. Let U(G(K)) be the set of isomorphism classes of unipotent representations of G(K) (see [L4, 1.21]). Let G be a simply connected almost simple algebraic group over C of the type dual to that of G (in the sense of Langlands); let θ : G −→ G be the ”graph automorphism” of G associated to the K-rational structure of G as in [L4, 8.1]. One of the main results of this paper is the construction of a bijection between U(G(K)) and a set of parameters defined in terms of G and θ. (See 10.11, 10.12.) This result (or rather a close variant of it) was stated without proof in [L4, 8.1] and was proved in [L4] assuming that θ = 1; it supports the Langlands philosophy. See [L4, 0.3] for historical remarks concerning this bijection. One of the main observations of [L4] and the present paper is that the various affine Hecke algebras which arise in connection with unipotent representations of G(K) can be also found in a completely different way, in terms of G, θ and certain cuspidal local systems. Then the problem reduces to classifying the simple modules of these ”geometric affine Hecke algebras” with parameter equal to √ q. This last problem makes sense in the case where √ q is replaced by any v0 ∈ C. This problem was solved in [L4] assuming that θ = 1 and v0 ∈ R>0. In the present paper we treat more generally the case where θ is arbitrary and v0 is either 1 or is not a root of 1. Moreover, using results of [L5], we determine which representations are tempered or square integrable.