Poisson transforms on vector bundles

Let G G be a connected real semisimple Lie group with finite center, and K K a maximal compact subgroup of G G . Let ( τ , V ) (\tau ,V) be an irreducible unitary representation of K K , and G × K V G\times _K\,V the associated vector bundle. In the algebra of invariant differential operators on G × K V G\times _K\,V the center of the universal enveloping algebra of Lie ⁡ ( G ) \operatorname {Lie}(G) induces a certain commutative subalgebra Z τ Z_\tau . We are able to determine the characters of Z τ Z_\tau . Given such a character we define a Poisson transform from certain principal series representations to the corresponding space of joint eigensections. We prove that for most of the characters this map is a bijection, generalizing a famous conjecture by Helgason which corresponds to τ \tau the trivial representation.