## 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)

### MSC:

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

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