# A question on the representations of affine Hecke algebras

In the paper Proof of the Deligne-Langlands conjecture for Hecke algebras, Kazhdan and Lusztig give a classification of simple modules of the affine Hecke algebra associated to a connected reductive linear group with simply connected derived subgroup (requiring that the parameter $q$ is not a root of unity).

I wonder whether we have a classification of simple modules for affine Hecke algebras associated to general reductive linear groups without the restriction of simply connectedness.

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I don't have an expert viewpoint on this work, but the overall goal is to understand representations of reductive groups over local fields. For this the representations of affine Hecke algebras (inspired by Iwahori-Matsumoto) play a big role. Each simple Lie type determines a single well-defined affine Weyl group, but there is also an extended version using the full weight lattice rather than just the root lattice. So you have to specify carefully how the isogeny type of your group interacts with the version of affine Weyl group and Hecke algebra formalism used. – Jim Humphreys Jan 31 '11 at 20:53
Yes, Thanks for your reminding. In my question, given a connected reductive group $G$, the affine Weyl group associated to $G$ is assumed to be the semidirect product of $W_0$ and $X$, where $W_0$ is the Weyl group of $G$, and $X$ is the group of charaters of a maximal torus of $G$. The affine Hecke algebra associated to $G$ is defined similarly. – niesian Feb 1 '11 at 3:12

This is perhaps overkill for your question, but there is a classification for the simple modules of an interesting family of affine Hecke algebras even with (certain) unequal parameters due to Lusztig. The proof is different from the one in the original case by Kazhdan and Lusztig, using equivariant perverse sheaves and the graded affine Hecke algebra. The first paper associates to an affine Hecke algebra (attached to any root datum) a graded version, and shows how the classification of simple modules (when $q$ is not a root of unity) can be reduced to the same classification problem for the graded algebras (though for a single affine Hecke algebra, one needs to understand the irreducibles for possibly smaller graded Hecke algebras, as well as the irreducibles for it own associated graded Hecke algebra).
The problem of classifying simple modules for the graded algebras (with a certain family of parameters arising from the study of unipotent representations of $p$-adic groups) is then taken up in a series of papers Lusztig wrote on "Cuspidal local systems" which solves th problem by geometric means (perverse sheaves and equivariant cohomology). This is then used in two later papers to give a complete "Langlands style" classification of the unipotent representations of simple $p$-adic groups.