Support varieties for rational representations

TL;DR

This paper introduces support varieties for rational representations of algebraic groups over fields of positive characteristic, extending the concept to infinite-dimensional modules and relating them to Frobenius kernels.

Contribution

It defines support varieties for rational G-modules, establishing their properties and connections to Frobenius kernels, advancing the understanding of infinite-dimensional representations.

Findings

Support varieties are closed subspaces of the space of 1-parameter subgroups.

Support varieties satisfy standard properties analogous to finite group schemes.

There is a close relationship between support varieties of G and its Frobenius kernels.

Abstract

We introduce support varieties for rational representations of a linear algebraic group $G$ of exponential type over an algebraically closed field $k$ of characteristic $p > 0$. These varieties are closed subspaces of the space $V(G)$ of all 1-parameter subgroups of $G$. The functor $M \mapsto V(G)_M$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between $V(G)_M$ and the family of support varieties $V_r(G)_M$ obtained by restricting the $G$ action to Frobenius kernels $G_{(r)} \subset G$. These support varieties seem particularly appropriate for the investigation of infinite dimensional rational $G$-modules.