The Griffiths bundle is generated by groups

Abstract: First the Griffiths line bundle of a $\mathbf Q$-VHS $\mathscr V$ is generalized to a Griffiths character ${\rm grif}(\mathbf G, \mu,r)$ associated to any triple $(\mathbf G, \mu, r)$, where $\mathbf G$ is a connected reductive group over an arbitrary field $F$, $\mu \in X_*(\mathbf G)$ is a cocharacter (over $\overline{F}$) and $r:\mathbf G \to GL(V)$ is an $F$-representation; the classical bundle studied by Griffiths is recovered by taking $F=\mathbf Q$, $\mathbf G$ the Mumford-Tate group of $\mathscr V$, $r:\mathbf G \to GL(V)$ the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to ${\rm grif}(\mathbf G, \mu, r)$. The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic $p$ given by a scheme mapping to a stack of $\mathbf G$-Zips. When $\mathbf G$ is $F$-simple, we show that, up to positive multiples, the Griffiths character ${\rm grif}(\mathbf G,\mu,r)$ (and thus also the Griffiths line bundle) is essentially independent of $r$ with central kernel, and up to some identifications is given explicitly by $-\mu$. As an application, we show that the Griffiths line bundle of a projective $\mathbf G{\rm -Zip}^{\mu}$-scheme is nef.