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Periodic cyclic homology of Iwahori-Hecke algebras. (English) Zbl 1056.16005
Summary: We determine the periodic cyclic homology of the Iwahori-Hecke algebras $$H_q$$, for $$q\in\mathbb{C}^*$$ 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 and Lusztig and Lusztig show that, for the indicated values of $$q$$, there exists a weakly spectrum preserving morphism $$\phi_q\colon H_q\to J$$, to a fixed finite type algebra $$J$$. This proves that $$\phi_q$$ induces an isomorphism in periodic cyclic homology and, in particular, that all algebras $$H_q$$ have the same periodic cyclic homology, for the indicated values of $$q$$. The periodic cyclic homology groups of the algebra $$H_1$$ can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.

##### MSC:
 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 19D55 $$K$$-theory and homology; cyclic homology and cohomology 20C08 Hecke algebras and their representations 20F55 Reflection and Coxeter groups (group-theoretic aspects)
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