Weakly symmetric spaces and spherical varieties
More than 40 years ago, \textit{A. Selberg} [J. Indian Math. Soc., New Ser. 20, 47-87 (1956; Zbl 0072.08201)] introduced weakly symmetric spaces \((M,g)\) in the following way. Let \((M,g)\) be a connected Riemannian manifold, \({\mathcal I}(M)\) its full isometry group, \(G\) a subgroup of \({\mathcal I}(M)\) acting transitively on \(M\), and \(u\) an element of \({\mathcal I}(M)\) satisfying (1) \(uGu^{-1}=G\), (2) \(u^2\in G\), (3) for any two points \(x,y\in M\) there exists \(g\in G\) such that \(gx=\mu y\), \(gy=\mu x\). All symmetric spaces are trivially weakly symmetric spaces but only few non-symmetric examples were known. Some years ago, J. Berndt and the reviewer proved that \((M,g)\) is weakly symmetric with respect to \(G={\mathcal I}(M)\) if and only if for each two points \(x,y\) of \(M\) there exists an isometry interchanging them [\textit{J. Berndt} and \textit{L. Vanhecke}, J. Math. Soc. Japan 48, 745-760 (1996; Zbl 0877.53027)]. Since then, the whole subject has been revitalized and many new nontrivial examples have been discovered. The paper under review is also devoted to the study of weakly symmetric homogeneous spaces. The authors start with the following Definition. Let \(G\) be a connected Lie group and \(\widetilde G=(G,u)\) an extension of index 2. Then (1) and (2) hold. Let \(H\) be a compact subgroup of \(G\). Then \(M=G/H\) is said to be weakly symmetric with respect to \(G\) and \(\mu\) if the action of \(G\) extends to an action of \(\widetilde G\) so that (3) holds. The main result in this paper shows that weakly symmetric homogeneous spaces of real reductive algebraic groups are closely related to spherical varieties. More specifically, they prove the following. Let \(H\) be a reductive algebraic subgroup of a connected algebraic group \(G\) and let \(X=G/H\). Then \(X\) is weakly symmetric with respect to \(G\) and some \(\mu\) if and only if \(X\) is a spherical variety. In his paper quoted above, A. Selberg proved that each weakly symmetric space \(G/H\) is commutative, i.e., the algebra \({\mathcal D} (G/H)\) of \(G\)-invariant differential operators is commutative. He also asked whether the converse holds. Recently, a negative answer to this question has been given by \textit{J. Lauret} [Bull. Lond. Math. Soc. 30, 29-36 (1998)] who provided a particular counterexample. In this paper, the authors prove that, in contrast to the general situation, the answer is positive inside the class considered in their main result.
weakly symmetric homogeneous spaces of real reductive algebraic groups
scientific article; zbMATH DE number 1286196
scientific article