Periodic cyclic homology of Iwahori-Hecke algebras

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We determine the periodic cyclic homology of the Iwahori-Hecke algebras, for q ∈ not a ``proper root of unity.'' (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a ``weakly spectrum preserving'' morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan--Lusztig and Lusztig show that, for the indicated values of q, there exists a weakly spectrum preserving morphism ϕq: → J, to a fixed finite type algebra J. This proves that ϕq induces an isomorphism in periodic cyclic homology and, in particular, that all algebras have the same periodic cyclic homology, for the indicated values of q. The periodic cyclic homology groups of the algebra ₁ can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.