Characters of the nullcone related to involutions of reductive groups

Abstract

Let G be a reductive algebraic group over an algebraically closed field F of characteristic other than 2, let θ be an involution of G, let and let be the −1-eigenspace of in Then the adjoint action of G on restricts to an action of K on the variety of nilpotent elements in We describe the structure of as a K-module, providing a formula for the multiplicities of the simple highest weight K-modules as composition factors of the homogeneous parts of the coordinate ring The multiplicity formula gives complete information about when but only partial information when This result is an adaptation of analogous results of Hesselink when and Friedlander and Parshall when