Hochschild homology of twisted crossed products and twisted graded Hecke algebras

TL;DR

This paper computes the Hochschild homology of twisted crossed products of certain algebras, including regular function rings and graded Hecke algebras, using algebraic families of representations.

Contribution

It provides explicit Hochschild homology formulas for twisted crossed products of regular function rings and graded Hecke algebras, advancing understanding of their algebraic structures.

Findings

Hochschild homology described as modules over the algebra's center

Results apply to twisted crossed products with 2-cocycles

Framework prepares for Hochschild homology computation of p-adic group Hecke algebras

Abstract

Let A be a \C-algebra with an action of a finite group G, let $\natural$ be a 2-cocycle on $G$ and consider the twisted crossed product $A \rtimes \C [G,\natural]$. We determine the Hochschild homology of $A \rtimes \C [G,\natural]$ for two classes of algebras A: - rings of regular functions on nonsingular affine varieties, - graded Hecke algebras. The results are achieved via algebraic families of (virtual) representations and include a description of the Hochschild homology as module over the centre of $A \rtimes \C [G,\natural]$. This paper prepares for a computation of the Hochschild homology of the Hecke algebra of a reductive p-adic group.