Optimal Subgroups and Applications to Nilpotent Elements

TL;DR

This paper investigates the optimal cocharacters associated with group actions on varieties, especially how these optimal classes behave under subgroup restrictions, with applications to nilpotent elements in Lie algebras.

Contribution

It establishes a connection between optimal cocharacters for a group and its G-completely reducible subgroups, providing new insights into nilpotent elements in Lie algebras.

Findings

Optimal cocharacters for subgroups are derived from those of the larger group.

The orbit of a point under a subgroup remains non-closed if it is non-closed under the larger group.

New information about cocharacters associated with nilpotent elements in Lie algebras is obtained.

Abstract

Let G be a reductive group acting on an affine variety X, let x in X be a point whose G-orbit is not closed, and let S be a G-stable closed subvariety of X which meets the closure of the G-orbit of x but does not contain x. In this paper, we study G.R. Kempf's optimal class Omega_G(x,S) of cocharacters of G attached to the point x; in particular, we consider how this optimality transfers to subgroups of G. Suppose K is a G-completely reducible subgroup of G which fixes x, and let H = C_G(K)^0. Our main result says that the H-orbit of x is also not closed, and the optimal class Omega_H(x,S) for H simply consists of the cocharacters in Omega_G(x,S) which evaluate in H. We apply this result in the case that G acts on its Lie algebra via the adjoint representation to obtain some new information about cocharacters associated with nilpotent elements in good characteristic.