Characterizing weakly symmetric spaces as Gelfand pairs

Abstract: Let $(M,G,\mu)$ be a Riemannian weakly symmetric space. Fix a base point $x_0 \in M$ and denote by $H$ to be the compact isotropy subgroup of $G$ at $x_0$. It is proven that $L^1(H\!\setminus\! G/H)$ is commutative, i.e., $(G,H)$ is a Gelfand pair. This extends E. Cartan's result for Riemannian symmetric spaces. Conversely, if $(G,H)$ is a Riemannian weakly symmetric pair, then $M=G/H$ can be made to be Riemannian weakly symmetric. An application of this result is presented.

Keywords: weakly symmetric spaces, isotropy subgroups, Gelfand pairs

Classification (MSC91): 53C35; 53C15

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