On automorphisms and adjoint orbits of real semisimple Lie algebras (Q1076786)

If \(G\) is the automorphism group and \(G_0\) the adjoint group of a real semisimple Lie algebra \({\mathfrak g}\), then there exists \(a\in G\) such that \(a(x)\in -G_0\cdot x\), for all \(x\in {\mathfrak g}\). The set of all \(a\) having this property is shown to be the same connected component \(G^*\) of \(G\), characterized differently by D. Vogan [see \textit{A. Borel} and \textit{N. Wallach}, ''Continuous cohomology, discrete subgroups, and representations of reductive groups'', Ann. Math. Stud. 94 (page 40) (1980; Zbl 0443.22010)]. The remainder of the paper is concerned with the problem of identifying \(G^*\) and with giving some information about the centralizer in \(G\) of a subalgebra \({\mathfrak s}\in {\mathfrak {sl}}(2,{\mathbb{R}})\) of \({\mathfrak g}\).