Global nilpotent cone for local systems

Definition

Let $X$ be a smooth projective curve and $G$ a reductive group. A point of the scheme of singularities of $\operatorname{LocSys}_G(X)$ can be represented by a pair $(E,\phi)$, where $E$ is a $G$-local system and $\phi$ is a horizontal section of the coadjoint local system $\operatorname{ad}(E)^*$.

The global nilpotent cone

is the conical substack of pairs for which $\phi$ takes values in the nilpotent cone of $\mathfrak g^*$. For a reductive group in characteristic zero, an invariant nondegenerate form may be used to identify this condition with pointwise nilpotence in $\mathfrak g$.

It is the support condition defining $\operatorname{IndCoh}_{\mathcal N_G}(\operatorname{LocSys}_G)$.