On the Humphreys conjecture on support varieties of tilting modules

TL;DR

This paper proves Humphreys' conjecture on support varieties of tilting modules for algebraic groups in large characteristic, showing the support variety always contains and sometimes equals the predicted variety.

Contribution

It establishes the Humphreys conjecture for all simply-connected semisimple algebraic groups when the characteristic is large enough.

Findings

Support variety of tilting modules contains the Humphreys predicted variety.

Support varieties coincide with the Humphreys prediction for large characteristic.

Variants involving relative support varieties are also proven.

Abstract

Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when $p$ is sufficiently large. We also prove variants of these statements involving "relative support varieties."