  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}.
