Ind-coherent sheaves with nilpotent singular support

Definition

Let $X$ be a smooth projective curve and let $\widehat G$ be a reductive group. For the quasi-smooth derived stack $\operatorname{LocSys}_{\widehat G}(X)$ , let $\mathcal N_{\widehat G}$ denote its global nilpotent cone in the scheme of singularities. The category

\operatorname{IndCoh}_{\mathcal N_{\widehat G}} \bigl(\operatorname{LocSys}_{\widehat G}(X)\bigr)

is the full subcategory of ind-coherent sheaves $\mathcal F$ satisfying

\operatorname{SS}(\mathcal F)\subseteq\mathcal N_{\widehat G},

where $\operatorname{SS}$ is coherent singular support.

This is the spectral category in the modern de Rham geometric Langlands correspondence for a general reductive group.

Why QCoh is not enough

On a smooth stack, quasi-coherent and ind-coherent theories are closely related. The derived stack of local systems is singular, and additional directions detected by singular support are needed to match all automorphic $D$ -modules. The zero-section condition recovers the quasi-coherent subcategory; allowing the global nilpotent cone is the required enlargement.

Scope

The support condition is a condition on the derived singularities of the entire moduli stack, not ordinary set-theoretic support on $\operatorname{LocSys}_{\widehat G}(X)$ .

References

Dima Arinkin and Dennis Gaitsgory, “Singular support of coherent sheaves, and the geometric Langlands conjecture,” Selecta Mathematica 21 (2015), 1–199. arXiv.