Involutions of reductive Lie algebras

We generalise Kostant-Rallis' study of involutions of Lie algebras to the case where the ground field is of odd characteristic. Let $G$ be a reductive group, let ${\mathfrak g}$ be the Lie algebra of $G$, let $\theta$ be an involutive automorphism of $G$ and let ${\mathfrak g}={\mathfrak k}\oplus{\mathfrak p}$ be the associated symmetric space decomposition. Among other results, we prove that the variety of nilpotent elements in the (-1) eigenspace ${\mathfrak p}$ has a dense open orbit, and give the number of its irreducible components for each class of involution of a reductive group. We also prove that each fibre of the geometric quotient morphism ${\mathfrak p}\to{\mathfrak p}/G^\theta$ has a dense open orbit, and show that the corresponding statement for $G$, conjectured by Richardson, is not true.