Desingularizations of some unstable orbit closures (Q1893303)

For a simply connected reductive algebraic group \(G\), its semisimple automorphism \(\sigma\) is defined. Let \(G_\sigma\) be the group of \(\sigma\)-fixed points, and \(P\) the \(\sigma\)-stable parabolic subgroup of \(G\). It is proved that the group of \(\sigma\)-fixed points in any \(\sigma\)-stable parabolic subgroup of \(G\) is parabolic in \(G_\sigma\). Some examples of this theorem are also studied. Let us consider the \(G_\sigma\)-orbits on the space of nilpotent elements in eigenspace \(d\sigma\). The desingularization of the orbit closures is constructed. The relationship of \(G_\sigma\)-orbits is investigated as well.