On support varieties for algebraic groups

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. In this thesis we will analyze the support varieties for two types of modules: higher line bundle cohomology groups and tilting modules. In Part 1 we will compute the support varieties of the higher line bundle cohomology groups for $G=SL_3$.

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 $pgeq h$, the support varieties of the indecomposable tilting modules align with the nilpotent orbits given by the Lusztig bijection. Part 2 will be devoted to proving this conjecture when $G=SL_n$ and $p > n+1$.