Spectra for Gelfand pairs associated with the Heisenberg group

Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group ${\displaystyle H_n}$. We say that ${\displaystyle (K,H_n)}$ is a Gelfand pair when the set ${\displaystyle L^1\_K\left(H_n\right)}$ of integrable K-invariant functions on ${\displaystyle H_n}$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for ${\displaystyle L^1\_K\left(H_n\right)}$ can be identified with the set ${\displaystyle \Delta (K,H_n)}$ of bounded K-spherical functions on ${\displaystyle H_n}$. In this paper, we study the natural topology on ${\displaystyle \Delta (K,H_n)}$ given by uniform convergence on compact subsets in ${\displaystyle H_n}$. We show that ${\displaystyle \Delta (K,H_n)}$ is a complete metric space and that the 'type 1' K-spherical functions are dense in ${\displaystyle \Delta (K,H_n)}$. Our main result shows that one can embed ${\displaystyle \Delta (K,H_n)}$ quite explicitly in a Euclidean space by mapping a spherical function to its eigenvalues with respect to a certain finite set of (${\displaystyle K⋉H_n}$)-invariant differential operators on ${\displaystyle H_n}$. This viewpoint on the spectrum for ${\displaystyle \Delta (K,H_n)}$ was previously known for K=U(n) and is referred to as 'the Heisenberg fan'.