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 $p > n+1$.
URI
http://purl.galileo.usg.edu/uga_etd/hardesty_william_d_201605_phdhttp://hdl.handle.net/10724/36129