On Gelfand pairs associated to transitive groupoids

Abstract

Let $G$ be a topological locally compact, Hausdorff and second countable groupoid with a Haar system and $K$ a compact subgroupoid of $G$ with a Haar system too. $(G, K)$ is a Gelfand pair if the algebra of bi-$K$-invariant functions is commutative under convolution. In this paper, we give a characterization of Gelfand pairs associated to transitive groupoids which generalize a well-known result in the groups case. Using this result, we prove that the study of Gelfand pairs associated to transitive groupoids is equivalent to that of Gelfand pairs associated to its isotropy groups.