Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces

Let $G/K$ be a Hermitian symmetric space and $V_\tau$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_\tau)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, \tau)} V_\tau$ over $G/K$ defined by a finite-dimensional representation $(V_\tau, \tau)$ of $K$. We classify irreducible representations $(V_\tau, \tau)$ such that $\mathcal D^G(G/K, V_\tau)$ is commutative. We construct eigenfunctions for the differential operators and study the invariance property of the eigenvalues under the Weyl group for the restricted real root system of $G$.