Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

TL;DR

This paper studies homogeneous Hermitian holomorphic vector bundles over bounded symmetric domains, constructing explicit intertwiners, and characterizes associated operator tuples in the Cowen-Douglas class.

Contribution

It provides an explicit differential operator intertwining bundles with their irreducible factors and characterizes homogeneous operator tuples in the Cowen-Douglas class.

Findings

Constructed explicit differential operators for bundle decomposition

Described Hilbert spaces of sections of these bundles

Provided a similarity theorem for Cowen-Douglas class operators

Abstract

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous $n$-tuples of operators in the Cowen-Douglas class of the Euclidean unit ball in $\mathbb C^n$.