# A topic on homogeneous vector bundles over elliptic orbits: A condition for the vector spaces of their cross-sections to be finite dimensional

## TL;DR

This paper establishes a sufficient condition, based on root systems, for the vector spaces of holomorphic cross-sections of homogeneous vector bundles over elliptic orbits to be finite dimensional.

## Contribution

It introduces a new criterion linking root systems to the finite dimensionality of cross-section spaces over elliptic orbits.

## Key findings

- Provided a root system-based condition for finite dimensionality
- Characterized when holomorphic cross-section spaces are finite dimensional
- Enhanced understanding of vector bundle sections over elliptic orbits

## Abstract

In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in view of root systems.