A classification of homogeneous operators in the Cowen-Douglas class

An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc $\mathbb{D}$ is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over $\mathbb{D}$, the ones corresponding to operators in the Cowen-Douglas class ${\mathrm B}_n(\mathbb D)$areidentified\mathrm{.}Theclassificationofhomogeneousoperatorsin${\mathrm B}_n(\mathbb D)$ is completed using an explicit realization of these operators. We also show how the homogeneous operators in ${\mathrm{B}}_n(\mathbb{D})$ split into similarity classes.