Remarks on real nilpotent orbits of a symmetric pair. (English) Zbl 0627.22008

Sekiguchi, Jiro

J. Math. Soc. Japan 39, 127-138 (1987).

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 This generalizes the following (unpublished) result of B. Kostant.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

Keywords:

nilpotent orbits; real semisimple Lie algebra; involution; semisimple symmetric pair; analytic subgroup; dual pair; Cartan decomposition