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$\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G^\mathrm{I})$ is Compactly Generated
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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.
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Cited by 1 Pith paper
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Iwahori Fundamental Local Equivalence
The author establishes three equivalences of factorization module categories: the Iwahori-ramified versions of the Arkhipov-Bezrukavnikov, Bezrukavnikov, and Fundamental Local Equivalences.
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