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 $T$ on $C$ a certain complex $S(C,T)$ of $G\times {\mathbb{G}}_m$-equivariant coherent sheaves on $\mathcal{N}$. 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 $H^{\bullet }(S(C,T))$ 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.

arXiv comment34 pages. v2: new introduction