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.

29 pages, final version accepted for publication, to appear in Manuscripta Matematica

References in corpus (2)

• On two theorems for flat, affine group schemes over a discrete valuation ring
• Extension theorems for reductive group schemes