Homogeneous Hermitian Holomorphic Vector Bundles And Operators In The Cowen-Douglas Class Over The Poly-disc

TL;DR

This paper classifies homogeneous Hermitian holomorphic vector bundles over polydiscs, explores their relation to Cowen-Douglas operators, and extends known results from bi-disc to higher dimensions, revealing limitations on homogeneity.

Contribution

It provides a complete classification of certain homogeneous vector bundles over bi-disc and extends these classifications to higher dimensions, also analyzing the homogeneity of associated operators.

Findings

Classified irreducible homogeneous bundles over bi-disc with multiplicity-free representations.

Showed tensor product structure of rank 2 bundles over higher-dimensional polydiscs.

Demonstrated non-existence of certain homogeneous operator tuples in higher dimensions.

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 $\mbox{Aut}(\mathbb{D}^n)$, $n > 2$.