Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

We study the property of \emph{continuous Castelnuovo-Mumford regularity}, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in \cite{Kuronya:Mustopa:2020} by K\"{u}ronya and Mustopa. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety's endomorphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that we established in \cite{Grieve:R-R:abVars}. In a complementary direction, we explain how these topics pertain to the \emph{Index} and \emph{Generic Vanishing Theory} conditions for simple semihomogeneous vector bundles. In doing so, we refine results from \cite{Gulbrandsen:2008}, \cite{Grieve-cup-prod-ab-var} and \cite{Mum:Quad:Eqns}.