For a Lie group \(G\) and a closed subgroup \(K\), both unimodular, the pair \((G,K)\) is said to be a generalized Gelfand pair if for any irreducible unitary representation of \(G\), the space of \(K\)-invariant distribution vectors has dimension 0 or 1. The authors consider the pairs \((G,K)\) with \(G=K\ltimes H_n\) where \(H_n\) is the Heisenberg group of dimension \(2n+1\) and \(K=\mathbb{R}_{>0}\times SO(n)\) for \(n\geq 2\), \(K=\mathbb{R}_{>0}\times O(1)\simeq \mathbb{R}^*\) for \(n=1\).
The main result is a description of the set of spherical distributions for these pairs. The proofs use a description of the irreducible unitary representations of \(G=K\ltimes H_n\) which is obtained from Mackey’s theory. They also use the following fact: consider the metaplectic representation \(\omega \) on \(L^2(\mathbb{R}^n)\); then \((G,K)\) is a generalized Gelfand pair if and only if the restriction of \(\omega \) to \(K\) is mutiplicity free.
The spherical distributions are eigendistributions of the \(K\)-invariant differential operators on \(H_n\), hence one first solves a system of differential equations. Then one determines the solutions which are of positive type by looking for which ones they are related to unitary representations of \(K\ltimes H_n\). Quite explicit formulae are given in the case \(n=1\) involving Bessel and confluent hypergeometric functions of the first and second kind.