Silting complexes of coherent sheaves and the Humphreys conjecture

TL;DR

This paper constructs specific complexes of coherent sheaves related to nilpotent orbits in algebraic groups and proves their connection to tilting module cohomology, confirming Humphreys' conjecture in certain cases.

Contribution

It introduces a new construction of complexes of coherent sheaves associated with nilpotent orbits and tilting bundles, and establishes their role in understanding tilting module cohomology and Humphreys' conjecture.

Findings

Constructed complexes $S(C,T)$ for nilpotent orbits and tilting bundles.

Proved these complexes form the indecomposable objects in a co-$t$-structure.

Confirmed Humphreys' conjecture for primes larger than the Coxeter number.

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ 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.