Spherical analysis on homogeneous bundles

Given a Lie group G, a compact subgroup K and a representation , we assume that the algebra of -valued, bi-τ-equivariant, integrable functions on G is commutative. We present the basic facts of the related spherical analysis, putting particular emphasis on the rôle of the algebra of G-invariant differential operators on the homogeneous bundle over . In particular, we observe that, under the above assumptions, is a Gelfand pair and show that the Gelfand spectrum for the triple admits homeomorphic embeddings in some . In the second part, we develop in greater detail the spherical analysis for with H nilpotent. In particular, for and and for H equal to the Heisenberg group and , we characterize the representations giving a commutative algebra.