Generalized Gelfand pairs: a survey

The group $G=SL(2,\mathbf R)$ of $2\times 2$ matrices with determinant 1, acts on the complex upper half plane by fractional linear transformations in a multiplicity free way: the $L^2$ space decomposes multiplicity free as a direct integral of irreducible spaces. This property was studied and extended by Gelfand a.o. to pairs $(G,K)$, where $G$ is a Lie group and $K$ a compact subgroup. The equivalent of the upper half plane is the space $G/K$. Pairs $(G,K)$ such that $L^2(G/K)$ splits multiplicity free are called Gelfand pairs. The most well-known examples are given by pairs $(G,K)$ where $G$ is a semi-simple Lie group and $K$ a maximal compact subgroup.
An extension of the notion of Gelfand pair for pairs $(G,K)$ where $K$ is a closed, non-necessarily compact subgroup of $G$ was discussed and several examples of (generalized) Gelfand pairs were be given.