Spherical functions for Gelfand pairs associated with compact groupoids

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 study spherical functions on groupoids associated to compact Gelfand pairs.