Bundles of Verlinde spaces and group actions

TL;DR

This paper explores the action of the Theta group on Verlinde spaces of higher level, providing a decomposition of these spaces over the moduli space of curves and methods to compute the ranks of components.

Contribution

It introduces a decomposition of Verlinde spaces under the Theta group action and details how to compute the ranks of the resulting isotypical components.

Findings

Decomposition of Verlinde spaces via Theta group action

Method to compute ranks of isotypical components

Insights into the structure of Verlinde bundles over moduli spaces

Abstract

A Verlinde space of level $k$ is the space of global sections of the $k$-th power of the determinant line bundle on the moduli space $\cSU_C(r)$ of semi-stable bundles of rank $r$ on a curve $C$. The aim of this note is to make accessible some remarks on the action of the Theta group on the Verlinde spaces of higher level. This gives a decomposition of the bundle of Verlinde spaces over the moduli space of curves and we indicate how to compute the rank of the isotypical components in the decomposition.