Varieties of nilpotent elements for simple Lie algebras I : Good primes

Benson, David J. and Bergonio, Phil and Boe, Brian D. and Chastkofsky, Leonard and Cooper, Bobbe and Guy, G. Michael and Hyun, Jo Jang and Jungster, Jerome and Matthews, Graham and Mazza, Nadia and Nakano, Daniel K. and Platt, Kenyon (2004) Varieties of nilpotent elements for simple Lie algebras I : Good primes. Journal of Algebra, 280 (2). pp. 719-737. ISSN 0021-8693

Abstract

Let G be a simple algebraic group over k = ℂ, or F̄p where p is good. Set g = Lie G. Given r ∈ ℕ and a faithful (restricted) representation ρ: g → gl(V), one can define a variety of nilpotent elements Nr,ρ(g) = {x ∈ g: ρ(x)r = 0}. In this paper we determine this variety when ρ is an irreducible representation of minimal dimension or the adjoint representation.