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

Abstract

Let $G/K$ be a Hermitian symmetric space and $V_{\ddot{I}}„$ an irreducible representation of $K$. We study the ring ${\mathcal{D}}^G(G/K,V_{\ddot{I}}„)$ of $G$-invariant differential operators on sections of vector bundles $G{\times }_{(K,\ddot{I}„)}{V}_{\ddot{I}}„$ over $G/K$ defined by a finite-dimensional representation $(V_{\ddot{I}}„,\ddot{I}„)$ of $K$. We classify irreducible representations $(V_{\ddot{I}}„,\ddot{I}„)$ such that ${\mathcal{D}}^G(G/K,V_{\ddot{I}}„)$ 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 $G$.

23 pages