Involutions of reductive Lie algebras

Let G be a reductive group over a field of characteristic ̸ = 2, let g = Lie(G), let θ be an involutive automorphism of G and let g = k ⊕p be the associated symmetric space decomposition. For k = C, Kostant and Rallis studied [17] properties of orbits, centralizers, and invariants related to the (−1) eigenspace p. In this paper, we generalise [17] to the case of good positive characteristic. Among other results, we prove that the variety N of nilpotent elements in p has a dense open orbit, and give the number of irreducible components of N for each class of involution of a simple algebraic group. We also show that every fibre of the quotient map π: p → p /G θ has a dense open orbit, and that the corresponding statement for G, conjectured by Richardson, is not true.