$\mathrm{D}-\mathrm{mod}({\mathrm{Bun}}_G^{\mathrm{I}})$ 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.

30 pages, fixed minor typos, added comment in introduction