Homology of graded Hecke algebras

TL;DR

This paper proves that the Hochschild, cyclic, and periodic cyclic homologies of graded Hecke algebras are parameter-independent and computes them explicitly, revealing new insights into their structure and representation theory.

Contribution

It introduces a novel approach to periodic cyclic homology via sheaf cohomology, and establishes a basis for the representation ring of the Weyl group using tempered modules.

Findings

Hochschild, cyclic, and periodic cyclic homologies are parameter-independent.

Explicit computation of these homologies for graded Hecke algebras.

Tempered modules with real central character form a Q-basis of the representation ring.

Abstract

Let H be a graded Hecke algebra with complex deformation parameters and Weyl group W. We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that, if the deformation parameters are real, the collection of irreducible tempered H-modules with real central character forms a Q-basis of the representation ring of W. Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex affine variety.