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 complete results are exclusively 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 $\mathbb{D}^2$, homogeneous with respect to $\mbox{M\"ob}\times \mbox{M\"ob}$, is obtained assuming that the associated representations are \textit{multiplicity-free}. Among these the ones that give rise to an operator in the Cowen-Douglas class of $\mathbb{D}^2$ of rank $1,2$ or $3$ is determined. * Any hermitian holomorphic vector bundle of rank $2$ over $\mathbb{D}^n$, homogeneous with respect to the $n$-fold product of the group $\mbox{M\"ob}$ is shown to be a tensor product of $n-1$ hermitian holomorphic line bundles, each of which is homogeneous with respect to $\mbox{M\"ob}$ and a hermitian holomorphic vector bundle of rank $2$, homogeneous with respect to $\mbox{M\"ob}$. * The classification of irreducible homogeneous hermitian holomorphic vector buldles over $\mathbb{D}^2$ of rank $3$ (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of $\mathbb{D}^n$, $n>2$. * It is shown that there is no irreducible $n$ - tuple of operators in the Cowen-Douglas class $\mathrm B_2(\mathbb{D}^n)$ that is homogeneous with respect $\mbox{Aut}(\mathbb{D}^n)$, $n >1$. Also, pairs of operators in $\mathrm B_3(\mathbb{D}^2)$ homogeneous with respect to $\mbox{Aut}(\mathbb{D}^2)$ are produced, while it is shown that no $n$ - tuple of operators in $\mathrm B_3(\mathbb{D}^n)$ is homogeneous with respect to $\mbox{Aut}(\mathbb{D}^n)$, $n > 2$.