Kashiwara's Lemma for Potent D-modules

Source abstract

On stacks there is a richer version of $\mathcal{D}$-modules---called $\textit{potent }\mathcal{D}\textit{-modules}$---which decompletes at equivariant parameters, in the same sense that genuine equivariant $K$-theory decompletes equivariant cohomology. In this paper, we prove a ``genuine'' version of Kashiwara's lemma for $\mathcal{D}$-modules. Namely, we show that for a broad class of geometric stacks $X$, if $Z \subset X$ is a closed substack, the category $\mathcal{D}^{\mathrm{pot}}(Z)$ of potent $\mathcal{D}$-modules on $Z$ is equivalent under pushforward to the category $\mathcal{D}^{\mathrm{pot}} _Z(X)$ of potent $\mathcal{D}$-modules on $X$ with support contained in $Z$.