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.
Categories: math.AG