A classification of homogeneous operators in the Cowen-Douglas class

TL;DR

This paper provides a comprehensive classification of homogeneous operators within the Cowen-Douglas class by explicitly realizing all relevant homogeneous vector bundles on the unit disc under a specific group action.

Contribution

It offers the first complete list of such operators, derived from explicit realizations of homogeneous Hermitian holomorphic vector bundles on the unit disc.

Findings

Complete classification of homogeneous operators in the Cowen-Douglas class

Explicit realization of all homogeneous Hermitian holomorphic vector bundles on the unit disc

Connection established via the universal covering group of the automorphism group

Abstract

A complete list of homogeneous operators in the Cowen-Douglas class $B_n(D)$ is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc.