Frobenius kernels of algebraic supergroups and Steinberg's tensor product theorem

TL;DR

This paper investigates the structure and representation of Frobenius kernels in algebraic supergroups, providing conditions for unimodularity and extending Steinberg's tensor product theorem to this setting.

Contribution

It introduces a criterion for the unimodularity of Frobenius kernels and generalizes Steinberg's tensor product theorem to algebraic supergroups.

Findings

Frobenius kernels are unimodular under specific root system conditions

Steinberg's tensor product theorem is extended to algebraic supergroups

Provides structural insights into supergroup representations

Abstract

For a split quasireductive supergroup $G$ defined over a field, we study structure and representation of Frobenius kernels $G_r$ of $G$ and we give a necessary and sufficient condition for $G_r$ to be unimodular in terms of the root system of $G$. We also establish Steinberg's tensor product theorem for $G$ under some natural assumptions.