Poisson transforms on vector bundles

Let $G$ be a connected real semisimple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $(\tau,V)$ be an irreducible unitary representation of $K$, and $G\times _K\,V$ the associated vector bundle. In the algebra of invariant differential operators on $G\times _K\,V$ the center of the universal enveloping algebra of $\operatorname{Lie}(G)$ induces a certain commutative subalgebra $Z_\tau$. We are able to determine the characters of $Z_\tau$. 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 $\tau$ the trivial representation.