Characters of the nullcone related to involutions of reductive groups. Issue 6 (3rd June 2022) - Fox, Joseph A.
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 K = ( G θ ) °, and let p be the −1-eigenspace of d θ in Lie ( G ) . Then the adjoint action of G on Lie ( G ) restricts to an action of K on N, the variety of nilpotent elements in p . We describe the structure of N 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 F [ N ] . The multiplicity formula gives complete information about F [ N ] when F = C but only partial information when char ( F ) > 0 . This result is an adaptation of analogous results of Hesselink when F = C and Friedlander and Parshall when ch ( F ) > 0 .