Bundles of Verlinde spaces and group actions

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.