  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.
