Homogeneous Hermitian holomorphic vector bundles and operators in the Cowen-Douglas class over the poly-disc

Abstract

In this article, we obtain two sets of results. The first set of results are for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc.A classification of irreducible hermitian holomorphic vector bundles over D2, homogeneous with respect to Möb×Möb, is obtained assuming that the associated representations are multiplicity-free. Among these the ones that give rise to an operator in the Cowen-Douglas class of D2 of rank 1,2 or 3 are determined.Any hermitian holomorphic vector bundle of rank 2 over Dn, homogeneous with respect to the n-fold direct product of the group Möb is shown to be a tensor product of n hermitian holomorphic vector bundles over D. Among them, n−1 are shown to be the line bundles and one is shown to be a rank 2 bundle. Also, each of the bundles are homogeneous with respect to Möb.The classification of irreducible homogeneous hermitian holomorphic vector bundles over D2 of rank 3 (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of Dn, n>2.It is shown that there is no irreducible n - tuple of operators in the Cowen-Douglas class B2(Dn) that is homogeneous with respect to Aut(Dn), n>1. Also, pairs of operators in B3(D2) homogeneous with respect to Aut(D2) are produced, while it is shown that no n - tuple of operators in B3(Dn) is homogeneous with respect to Aut(Dn), n>2.