Spherical analysis on homogeneous vector bundles

Abstract

Given a Lie group G, a compact subgroup K and a representation τ∈Kˆ, we assume that the algebra of End(Vτ)-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 Eτ over G/K. In particular, we observe that, under the above assumptions, (G,K) is a Gelfand pair and show that the Gelfand spectrum for the triple (G,K,τ) admits homeomorphic embeddings in some Cn.In the second part, we develop in greater detail the spherical analysis for G=K⋉H with H nilpotent. In particular, for H=Rn and K⊂SO(n) and for H equal to the Heisenberg group Hn and K⊂U(n), we characterize the representations τ∈Kˆ giving a commutative algebra.