Let \(G\) be a unimodular Lie group and \(K\) a subgroup of the automorphism group of \(G\). Given a unitary representation \((\pi,{\mathcal H})\) of \(G\), we denote by \({\mathcal H}^{-\infty}\) the dual space of the space of \(C^\infty\)-vectors of \(\pi\) and call its elements distribution vectors. The pair \((K, G)\) is said to be a generalized Gelfand pair if for each irreducible unitary representation \(\pi\) of \(G\), the space of \(K\)-fixed distribution vectors is at most one-dimensional.
Let us consider two real vector spaces \(V\) and \(Z\), endowed with inner products \(\langle\cdot,\cdot\rangle_V\) and \(\langle\cdot,\cdot\rangle_Z\) respectively and a non-degenerate skew-symmetric bilinear form \(\Psi: V\times V\to Z\). We define a Lie algebra \({\mathfrak n}= V\oplus Z\) by \([(v,z), (v',z')]= (0,\Psi(v, v'))\), and say that \({\mathfrak n}\) is of Heisenberg type if \(J_z: V\to V\) given by \(\langle J_z v,w\rangle_V= \langle z,\Psi(v, w)\rangle_Z\) is an orthogonal transformation for all \(z\in Z\) with \(|z|= 1\). A connected and simply connected Lie group \(N\) is of Heisenberg type if its Lie algebra \({\mathfrak n}\) is of Heisenberg type. Let \(A(N)\) be the group of automorphisms of \(N\) which act by orthogonal transformations on \({\mathfrak n}\).
In this paper, the author collects some known results and gives new examples concerning generalized Gelfand pairs \((K,N)\), where \(N\) is of Heisenberg type and \(K\) is a subgroup of \(A(N)\) [cf. A. Kaplan and F. Ricci, Harmonic analysis, Proc. Conf., Cortona/Italy 1982, Lect. Notes Math. 992, 416–435 (1983; Zbl 0521.43008); F. Levstein and L. Saal, J. Lie Theory 18, No. 3, 503–515 (2008; Zbl 1205.43007); F. Ricci, J. Lond. Math. Soc., II. Ser. 32, 265–271 (1985; Zbl 0592.22009)].