On support varieties and the Humphreys conjecture in type $A$

TL;DR

This paper verifies Humphreys' conjecture relating support varieties of tilting modules to nilpotent orbits for special linear groups over finite fields, specifically when the characteristic exceeds the group rank.

Contribution

It proves Humphreys' conjecture for the case of $SL_n$ when the prime characteristic is greater than $n+1$, confirming the predicted support variety and nilpotent orbit correspondence.

Findings

Support varieties of tilting modules match nilpotent orbits for $SL_n$

Verification of Humphreys' conjecture in new parameter range

Supports the Lusztig bijection in this context

Abstract

Let $G$ be a reductive algebraic group scheme defined over $\mathbb{F}_p$ and let $G_1$ denote the Frobenius kernel of $G$. To each finite-dimensional $G$-module $M$, one can define the support variety $V_{G_1}(M)$, which can be regarded as a $G$-stable closed subvariety of the nilpotent cone. A $G$-module is called a tilting module if it has both good and Weyl filtrations. In 1997, it was conjectured by J.E. Humphreys that when $p\geq h$, the support varieties of the indecomposable tilting modules coincide with the nilpotent orbits given by the Lusztig bijection. In this paper, we shall verify this conjecture when $G=SL_n$ and $p > n+1$.