Silting complexes of coherent sheaves and the Humphreys conjecture

Abstract

Let G be a connected reductive algebraic group over an algebraically closed field $\mathbb{k}$ of characteristic $p\ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent G-orbit C and each indecomposable tilting vector bundle $\mathcal{T}$ on C a certain complex $\mathcal{S}(C,\mathcal{T})\in D^{\mathrm{b}}{\mathsf{Coh}}^{G\times {\mathbb{G}}_{\mathrm{m}}}(\mathcal{N})$. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-t-structure.

We then show that if p is larger than the Coxeter number, then the hypercohomology $H^{\bullet }(\mathcal{N},\mathcal{S}(C,\mathcal{T}))$ is identified with the cohomology of a tilting module for G. This confirms a conjecture of Humphreys on the support of the cohomology of tilting modules.