Local orbit types of \(S\)-representations of symmetric \(R\)-spaces (Q1429184)

To describe the contents of the paper under review, we need the usual objects accompanying a compact semisimple symmetric space \(M=G/K\) with the base point \(o\in M\): the Lie algebras \({\mathfrak g}\) and \({\mathfrak k}\) of \(G\) and \(K\) respectively, a decomposition \({\mathfrak g}={\mathfrak k}\oplus{\mathfrak m}\) corresponding to an involutive automorphism \(\sigma\) of \({\mathfrak g}\) with \({\mathfrak k}=\text{Ker}\,(\sigma-\text{id}_{\mathfrak g})\) and \({\mathfrak m}=\text{Ker}\,(\sigma+\text{id}_{\mathfrak g})\), a maximal abelian subspace \({\mathfrak a}\) of \({\mathfrak m}\), and a restricted root system \(\Delta\subseteq{\mathfrak a}^*\). Now, for any \(H\in{\mathfrak a}\subseteq T_oM\), denote \(\Delta_H=\{\alpha\in\Delta\mid\alpha(H)=0\}\) and \({\mathfrak k}_H=\{X\in{\mathfrak k}\mid[X,H]=0\}\). Then the local orbit type of \(H\) (referred to in the title of the paper) is just the conjugacy class of \({\mathfrak k}_H\) under the automorphism group of \({\mathfrak k}\). The orbits of two points \(H,H'\in T_oM\) under the isotropy action of \(K\) are locally diffeomorphic if and only if \(\Delta_H\) and \(\Delta_{H'}\) are conjugate under the normalizer of \({\mathfrak a}\) in \(K\) (theorem~2.6). As a consequence, the number of local orbit types is at most \(2^r\), where \(r\) stands for the rank of the symmetric space \(M\) (corollary~2.8). Using these facts, the author is able to describe all local orbit types for several Riemannian symmetric spaces.