On Lie algebra modules which are modules over semisimple group schemes

Abstract

Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over $K:=\text{Frac}(R)$ with the property that the set of lattices of $V$ with respect to $R$ which are $G$-modules is as well the set of lattices of $V$ with respect to $R$ which are $\text{Lie}(G)$-modules. We apply this classification to get a general criterion of extensions of homomorphisms between reductive group schemes over $\text{Spec} K$ to homomorphisms between reductive group schemes over $\text{Spec} R$. We also show that for a simply connected semisimple group scheme over a reduced $\mathbb Q$--algebra, the category of its representations is equivalent to the category of representations of its Lie algebra.