Support Varieties for Frobenius Kernels of Classical Groups

TL;DR

This paper characterizes the support varieties of simple modules over Frobenius kernels of classical algebraic groups, linking them to the first kernel and computing block varieties, advancing understanding of modular representation theory.

Contribution

It provides a new description of support varieties for simple modules over Frobenius kernels of classical groups, connecting higher kernels to the first kernel case.

Findings

Support varieties are described in terms of the first Frobenius kernel.

Block varieties of Frobenius kernels are explicitly computed.

The results deepen understanding of module structure over algebraic groups.

Abstract

Let G be a simple classical algebraic group over an algebraically closed field of positive characteristic. We describe the support variety of a simple G-module over the r-th Frobenius kernel of G, in terms of its calculation over the first Frobenius kernel. We then use this result to compute the block varieties of the Frobenius kernels of G.