Cofree embeddings of algebraic tori preserving canonical sheaves

Abstract

Let $\varrho : G \to GL(V)$ be a finite dimensional rational representation of a diagonalizable algebraic group $G$ over an algebraically closed field $K$ of characteristic zero. Using a minimal paralleled linear hull $(W, w)$ of $\varrho$ defined in [N4], we show the existence of a cofree representation $\widetilde{G_w} \hookrightarrow GL(W)$ such that $\varrho(G_w) \subseteq \widetilde{G_w}$ and $W//G_w \to W// \widetilde{G_w}$ is divisorially unramified is equivalent to the Gorensteinness of $V//G$.