Let \(N\) be a connected and simply connected nilpotent Lie group and \(K\) be a compact Lie group acting smoothly on \(N\) via automorphisms such that \((N,K)\) is a Gelfand pair. Let \(\Delta(N,K)\) denote the space of all bounded \(K\)-spherical functions on \(N\) with the topology of uniform convergence on compact sets and \(n^{\star}|K\) denote the space of \(K\)-orbits in the dual \(n^{\star}\) for the Lie algebra \(n\) of \(N\). It is well-known that there exists a map \(\Psi:\Delta(N,K)\to n^{\star}|K\) which is injective.
In one of their papers, the authors have conjectured that \(\Psi\) is a topological embedding. This conjecture has been proved when \(N\) is an abelian group or \(N\) is a Heisenberg group. In this paper, the goal of the authors is to prove the conjecture for nilpotent Gelfand pairs, where \(N\) is nor abelian nor a Heisenberg group.
As main result, they have proved that the conjecture holds for all irreducible nilpotent Gelfand pairs. In order to prove this result, they have established some results which have permitted them to reduce the proof to the verification of the conjecture for maximal irreducible nilpotent Gelfand pairs \((N,K)\) with \(K\) minimal and connected. Vinberg’s classification gives a list of all such pairs. Removing to the list Heisenberg groups, the authors have obtained twelve pairs and have classified them in a table. The verification of the conjecture for the six first entries is done in a previous paper. In this one, they verify the conjecture for the six remaining entries of the table.