Tempered modules in exotic Deligne-Langlands correspondence

The main purpose of this paper is to identify the tempered modules for the affine Hecke algebra of type $C_n^{(1)}$ with arbitrary, non-root of unity, unequal parameters, in the exotic Deligne-Langlands correspondence in the sense of Kato. Our classification has several applications to the Weyl group module structure of the tempered Hecke algebra modules. In particular, we provide a geometric and a combinatorial classification of discrete series which contain the sign representation of the Weyl group. This last combinatorial classification was expected from the work of Heckman-Opdam and Slooten.