On Lie algebra modules which are modules over semisimple group schemes

TL;DR

This paper classifies certain finite-dimensional representations of split semisimple group schemes over normal integral domains, linking lattice structures of group and Lie algebra modules, and explores extensions of homomorphisms and category equivalences.

Contribution

It provides a classification of representations with compatible lattice structures, and establishes criteria for extending homomorphisms and for category equivalences between group schemes and their Lie algebras.

Findings

Classified representations with compatible G- and Lie(G)-lattice structures.

Derived criteria for extending homomorphisms between reductive group schemes.

Proved equivalence of representation categories for simply connected semisimple group schemes over reduced Q-algebras.

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.