Periodic Cyclic Homology of Iwahori–Hecke Algebras

Abstract

We determine the periodic cyclic homology of the Iwahori-Hecke algebras $\Hecke_q$, for $q \in \CC^*$ not a ``proper root of unity.'' (In this paper, by a {\em 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 $\phi_q : \Hecke_q \to J$, 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 $\Hecke_q$ have the same periodic cyclic homology, for the indicated values of $q$. The periodic cyclic homology groups of the algebra $\Hecke_1$ can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.