Abstract

Let $A$ be a $ℂ$-algebra with an action of a finite group $G$, and consider a twisted crossed product $A⋊ℂ[G,♮]$. We find the Hochschild homology of $A⋊ℂ[G,♮]$ for two classes of algebras $A$: rings of regular functions on nonsingular affine varieties, and graded Hecke algebras. The results are achieved via algebraic families of (virtual) representations and include a description of the Hochschild homology as a module over the centre of $A⋊ℂ[G,♮]$. In noncommutative geometric terms, our results describe the differential forms on the space of irreducible representations of $A⋊ℂ[G,♮]$. This paper prepares for a computation of the Hochschild homology of the Hecke algebra of a reductive $p$-adic group.

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