$\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G^\mathrm{I})$ is Compactly Generated

Abstract

Drinfeld and Gaitsgory proved that $\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G)$ is compactly generated. Let $\mathrm{Bun}_G^{\mathrm{I}}$ be the algebraic stack of principal $G$-bundles on $X$ together with Iwahori level structure at a fixed point $x \in X$. More generally, for a finite collection of points $x_1, ..., x_k \in X$, let $\mathrm{Bun}_G^{(\mathrm{I}; x_1, ..., x_k)}$ be the algebraic stack of principal $G$-bundles on $X$ together with Iwahori level structure at each point $x_j$. We will show that $\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G^{\mathrm{I}})$ and $\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G^{(\mathrm{I}; x_1, ..., x_k)})$ are compactly generated.