Support varieties for rational representations

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.

Cited by in corpus (2)

• Graded analogues of one-parameter subgroups and applications to the cohomology of $G{L}_{m|n(r)}$
• On the Existence of Mock Injective Modules for Algebraic Groups