A sufficient condition for orbits of Hermann actions to be weakly reflective
type: Item
instance of: scholarly article
title: A sufficient condition for orbits of Hermann actions to be weakly reflective
zbMATH Open document ID: 1373.53071
author: Shinji Ohno
published in: Tokyo Journal of Mathematics
publication date: 2017-04-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1484903136
review text: 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 o…
reviewed by: Jacques Faraut
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 6701609
zbMATH Keywords: compact symmetric space
zbMATH Keywords: weakly reflective submanifold
MaRDI profile type: Publication
OpenAlex ID: W2572184064
DOI: 10.3836/TJM/1484903136
description: scientific article; zbMATH DE number 6701609