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

TL;DR

This paper classifies when the algebra of invariant differential operators on vector bundles over Hermitian symmetric spaces is commutative, constructs eigenfunctions, and examines eigenvalue invariance under Weyl group actions.

Contribution

It provides a classification of irreducible representations leading to commutative invariant differential operator rings on Hermitian symmetric spaces.

Findings

Identified conditions for commutativity of the differential operator ring.

Constructed explicit eigenfunctions for these operators.

Analyzed the invariance of eigenvalues under Weyl group actions.

Abstract

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 $G$.