On generalized Gelfand pairs (Q760628)

Let G be a unimodular locally compact group, and H a closed unimodular subgroup. Let \(\pi\) be an irreducible unitary representation of G. The author gives a survey of some results concerning the following questions: can \(\pi\) be realized on a Hilbert space \({\mathcal H}\) of the space \({\mathcal D}'(G/H)\) of the Bruhat distributions on G/H ? What can be said about the uniqueness of such a realization ? In case of unique realization does \({\mathcal H}\) consist of functions ? The answer to the first question is yes if and only if there exists for \(\pi\) a non zero H-invariant distribution vector, and the realization is essentially unique if the space of these vectors has dimension one. If (G,H) is a Gelfand pair, that is if H is compact and if the convolution algebra of the H-biinvariant integrable functions on G is commutative, this space has dimension 0 or 1. This leads to the definition of a generalized Gelfand pair when H is no more compact.