A sufficient condition for orbits of Hermann actions to be weakly reflective
The author studies Hermann actions on compact symmetric spaces, and gives sufficient conditions for orbits of Hermann actions to be weakly reflective. Let \(G\) be a compact Lie group, \(K_1\) and \(K_2\) two symmetric subgroups, assumed to be connected. One studies the action of \(K_2\) on the compact symmetric space \(M_1=G/K_1\), and of \(K_1\times K_2\) on \(G\): \((k_1,k_2)\mapsto k_2gk_1^{-1}\). A submanifold \(M\subset M_1\) is said to be weakly reflective if, for each \(x\in M\), and \(\xi \in T_x^{\perp}M\), there exists an isometry \(\sigma \) of \(M_1\) such that \[ \sigma (x)=x,\;\sigma (M)=M,\;(d\sigma ) (\xi ) =-\xi . \] The Lie algebra \(\mathfrak{g}\) of \(G\) admits the decomposition \[ \mathfrak{g}=\mathfrak{k}_1\oplus \mathfrak{m}_1\;(\mathfrak{k}_1=\mathrm{Lie}(K_1)),\quad \mathfrak{g}=\mathfrak{k}_2\oplus \mathfrak{m}_2\;(\mathfrak{k}_2=\mathrm{Lie}(K_2)). \] One considers a maximal abelian subspace \(\mathfrak{a}\) in \(\mathfrak{m}_1\cap \mathfrak{m}_2\). The conditions are given in terms of symmetric triads \((\tilde \Sigma ,\Sigma ,W)\) in \(\mathfrak{a}\). In such a triad, \(\tilde \Sigma \) is an irreducible root system, \(\Sigma \) a root system, and \(W\) a finite system such that \(\tilde \Sigma =\Sigma \cup W\). Consider the orbit \(K_2gK_1\) in \(G\), with \(g=\exp H\), \(H\in \mathfrak{a}\). The first main result says: If \(\langle \lambda ,H\rangle \in {\pi \over 2}\mathbb{Z}\) for all \(\lambda \in \tilde \Sigma \), then \(K_2gK_1\) is weakly reflective. Associated to \(H\in \mathfrak{a}\), one defines \[ \begin{aligned} \Sigma _H=&\{\lambda \in \Sigma \mid \langle \lambda ,H\rangle \in \pi \mathbb{Z}\},\\ W_H=&\{\lambda \in W \mid \langle \alpha ,H\rangle \in {\pi \over 2}+\pi \mathbb{Z}\},\\ \tilde \Sigma _H=&\Sigma _H \cup W_H. \end{aligned} \] The second main result says: If \(\text{Span}(\tilde \Sigma _H)=\mathfrak{a}\), and \(-\text{id}_{\mathfrak{a}}\in W(\tilde \Sigma _H)\), where \(W(\tilde \Sigma _H)\) is the Weyl group of \(\tilde \Sigma _H\), then the orbits \(K_2gK_1\) in \(G\), \(K_2(gK_1)\) in \(G/K_1\), and \((K_2g)K_1\) in \(K_2\backslash G\) are weakly reflective. These results extend previous results by Ikawa. Further the author studies case by case the simple compact groups \(G\) and the possible symmetric subgroups \(K_1\) and \(K_2\). He determines, in each case, the corresponding triads, and construct new examples of weakly reflective submanifolds.
scientific article; zbMATH DE number 6701609