The closure of nilpotent orbits in the classical symmetric pairs and their singularities. (English) Zbl 0738.22007

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 \(G\).
Reviewer: F.Rouvière (Nice)


22E46 Semisimple Lie groups and their representations
17B20 Simple, semisimple, reductive (super)algebras
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