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Remarks on real nilpotent orbits of a symmetric pair. (English) Zbl 0627.22008
Let $${\mathfrak g}$$ be a real semisimple Lie algebra, $$\sigma$$ an involution of $${\mathfrak g}$$ and $${\mathfrak g}={\mathfrak h}\oplus {\mathfrak q}$$ its decomposition into $$+1$$ and -1 eigenspaces. Then ($${\mathfrak g,h})$$ is called a semisimple symmetric pair. Let H be the analytic subgroup of $$G=Int({\mathfrak g})$$ with Lie algebra $${\mathfrak h}$$, and let $${\mathcal N}({\mathfrak q})$$ denote the set of $$ad_{{\mathfrak g}}$$-nilpotent elements contained in $${\mathfrak q}$$. The author studies the structure of the set [$${\mathcal N}({\mathfrak q})]$$ of H-conjugacy classes in $${\mathcal N}({\mathfrak q}).$$
To $$\sigma$$ and a commuting Cartan involution $$\theta$$ (which always exists), one can associate other semisimple symmetric pairs: the associated pair ($${\mathfrak g}^ a,{\mathfrak h}^ a)$$ and the dual pair ($${\mathfrak g}^ d,{\mathfrak h}^ d)$$. The main theorem asserts that there exist bijections [$${\mathcal N}({\mathfrak q})]\simeq [{\mathcal N}({\mathfrak q}^ a)]\simeq [{\mathcal N}({\mathfrak q}^ d)]$$. This generalizes the following (unpublished) result of B. Kostant. Let $${\mathfrak g}={\mathfrak k}\oplus {\mathfrak p}$$ be the Cartan decomposition for $$\theta$$. Then the $$K_{{\mathbb{C}}}$$- conjugacy classes in $${\mathcal N}({\mathfrak p}_{{\mathbb{C}}})$$ correspond 1-1 with the G-conjugacy classes in $${\mathcal N}({\mathfrak g}).$$
For irreducible pairs ($${\mathfrak g,h})$$ of split rank 1 the set [$${\mathcal N}({\mathfrak q})]$$ is explicitly determined in terms of roots.
Reviewer: E.P.van den Ban

##### MSC:
 22E15 General properties and structure of real Lie groups 17B20 Simple, semisimple, reductive (super)algebras
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