Hochschild cohomology of torus-equivariant D-modules

TL;DR

This paper explores the Hochschild cohomology of categories of D-modules on algebraic stacks, especially torus-equivariant cases, linking it to loop space cohomology and proposing methods for general stacks.

Contribution

It provides a description of Hochschild cohomology for torus-equivariant D-modules and introduces an approach using relative compactification for general stacks.

Findings

Hochschild cohomology described as cohomology on loop space

Explicit description for torus-equivariant D-modules

Proposed method for understanding support theory on stacks

Abstract

We discuss the Hochschild cohomology of the category of D-modules associated to an algebraic stack. In particular we describe the Hochschild cohomology of the category of torus-equivariant D-modules as the cohomology of a D-module on the loop space of the quotient stack. Finally, we give an approach for understanding the Hochschild cohomology of D-modules on general stacks via a relative compactification of the diagonal. This work is motivated by a desire to understand the support theory (in the sense of Benson-Iyengar-Krause) of D-modules on stacks.