Characters of the nullcone related to involutions of reductive groups. Issue 6 (3rd June 2022)

• Record Type: Journal Article
• Authors: Fox, Joseph A.
• Is Part Of: Communications in algebra. Volume 50:Issue 6(2022)
• Page Start: 2439
• Page End: 2450
• Publication Date: 2022-06-03
• Subjects: Algebraic groups -- good characteristic -- nullcone -- representation theory; 20G05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.2007531
• Languages: English
• ISSNs: 0092-7872

Abstract

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 .