Hochschild homology of affine Hecke algebras

TL;DR

This paper computes the Hochschild and cyclic homology of affine Hecke algebras, showing they are independent of parameters and relate to geometric structures, with implications for representation theory.

Contribution

It provides explicit descriptions of Hochschild and cyclic homology for affine Hecke algebras, revealing their independence from parameters and connecting them to extended quotients and geometric data.

Findings

Hochschild and cyclic homology are independent of parameters q.

Homology descriptions relate to extended quotients of tori by Weyl groups.

Representation families depend analytically on complex parameters.

Abstract

Let H = H (R,q) be an affine Hecke algebra with complex, possibly unequal parameters q, which are not roots of unity. We compute the Hochschild and the cyclic homology of H. It turns out that these are independent of q and that they admit an easy description in terms of the extended quotient of a torus by a Weyl group, both of which are canonically associated to the root datum R. For q positive we also prove that the representations of the family of algebras H (R,q^\epsilon) come in families which depend analytically on the complex number \epsilon. Analogous results are obtained for graded Hecke algebras and for Schwartz completions of affine Hecke algebras. Correction: in 2021 some problems with the construction of families of representations surfaced. These are discussed in additional comments in Section 2.