DESINGULARIZATIONS OF SOME UNSTABLE ORBIT CLOSURES

Let sigma be a semisimple automorphism of a connected reductive group G, and let G(sigma) be the fixed points of sigma. We consider the G(si gma)-orbits on the space of nilpotent elements in an eigenspace of d s igma. We give a desingularization of the orbit closures and relate the G(sigma)-orbits to the G-orbits. Along the way, we describe the fixed points of sigma on a flag variety G/P where P is a sigma-stable parab olic subgroup of G.