Affine Hecke algebras and their graded version
0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s satisfy (Ts + l)(Ts v2c(s)) = 0, where C(S) E N depend on s and are subject only to c(s) = c(s') whenever s, s are conjugate in the affine Weyl group.) Such Hecke algebras appear naturally in the representation theory of semisimple p-adic groups, and understanding their representation theory is a question of considerable interest. Consider the "special case" where c(s) is independent of s and the coroots generate a direct summand. In this "special case," the question above has been studied in [1] and a classification of the simple modules was obtained. The approach of [1] was based on equivariant K-theory. This approach can be attempted in the general case (some indications are given in [5, 0.3]), but there appear to be some serious difficulties in carrying it out.
Some examples of square integrable representations of semisimple p-adic groups
We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer's reflection representation corresponding to finite Weyl groups, and we show that in many cases they correspond to a square integrable representation of a simple p-adic group.
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