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

TL;DR

This paper explores the structure of homogeneous Hermitian holomorphic vector bundles over bounded symmetric domains, providing conditions for their decomposition and applications to Cowen-Douglas operators, especially in the unit ball in b2.

Contribution

It establishes a condition for when such bundles decompose into irreducible parts via differential operators and applies this to classify homogeneous Cowen-Douglas operator pairs.

Findings

The condition always holds for the unit ball in b2.

Homogeneous Cowen-Douglas pairs are similar to direct sums of basic pairs.

Abstract

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We give a condition under which the bundle and the direct sum of its irreducible constituents are intertwined by an equivariant constant coefficient differential operator. We show that in the case of the unit ball in $\mathbb C^2$ this condition is always satisfied. As an application we show that all homogeneous pairs of Cowen-Douglas operators are similar to direct sums of certain basic pairs.