**Journal of Lie Theory, Vol. 9, No. 2, pp. 285-291 (1999)
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Characterizing weakly symmetric spaces as Gelfand pairs

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Hieu Nguyen

Department of Mathematics, Rowan University, Glassboro NJ 08028, USA, nguyen@rowan.edu

**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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