The closure of nilpotent orbits in the classical symmetric pairs and their singularities (Q1177407)

Let \(G\) be a complex reductive algebraic group with involution \(\Theta\), let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) be the corresponding decomposition of the Lie algebra, and \(K\) the fixed point subgroup of \(\Theta\). The author studies three problems on nilpotent orbits, i.e. \(K\)-orbits of nilpotent elements in \({\mathfrak p}\), for classical symmetric pairs \(({\mathfrak g}, {\mathfrak k})\). The first result characterizes the closure relation between nilpotent orbits (when is an orbit contained in the Zariski closure of another orbit?) by means of a certain ordering of the corresponding ``ab- diagrams''. The second result extends a theorem on singularities of the closure of a nilpotent orbit, proved by Kraft and Procesi for the case of a Lie algebra (instead of a general symmetric pair). The third result shows that the above closure relation is preserved by Sekiguchi's bijection, relating (by means of \(SL(2)\)-triples) nilpotent \(K\)-orbits in \({\mathfrak p}\) to nilpotent \(G_ R\)-orbits in \({\mathfrak g}_ R\), where \(G_ R\) is a real form of \(G\).