On Gel′fand pairs associated with solvable Lie groups

Abstract:

Let $G$ be a locally compact group, and let $K$ be a compact subgroup of ${\operatorname {Aut}}(G)$, the group of automorphisms of $G$. There is a natural action of $K$ on the convolution algebra ${L^1}(G)$, and we denote by $L_K^1(G)$ the subalgebra of those elements in ${L^1}(G)$ that are invariant under this action. The pair $(K,G)$ is called a Gelfand pair if $L_K^1(G)$ is commutative. In this paper we consider the case where $G$ is a connected, simply connected solvable Lie group and $K \subseteq {\operatorname {Aut}}(G)$ is a compact, connected group. We characterize such Gelfand pairs $(K,G)$, and determine a moduli space for the associated $K$-spherical functions.