Algebras of invariant functions on the Šilov boundary of generalized half-planes

Abstract:

Let $\mathcal {N}$ be the nilpotent Lie group identified to the Šilov boundary of a symmetric generalized half-plane $\mathcal {D}$ and $L$ a compact group acting on $\mathcal {N}$ by automorphisms, arising from the realization of $\mathcal {D}$ as hermitian symmetric space. Is then $(L \ltimes \mathcal {N},L)$ a Gelfand pair? We study the problem and resolve it in the case of classical families.