Hom schemes for algebraic groups

TL;DR

This paper extends the representability of Hom schemes to non-reductive algebraic groups, simplifying proofs and exploring applications to parabolics, orbit structures, and connections with complete reducibility.

Contribution

It generalizes the representability results of Hom schemes beyond reductive groups with simpler proofs and explores their applications and examples.

Findings

Hom schemes are representable for non-reductive groups.

Simpler proofs are provided for these representability results.

Connections to Serre's complete reducibility and orbit structures are established.

Abstract

In SGA3, Demazure and Grothendieck showed that if $G$ and $H$ are smooth affine group schemes over a scheme $S$ and $G$ is reductive, then the functor of $S$-homomorphism $G \to H$ is representable. In this paper we extend this result to cover cases in which $G$ is not reductive, with much simpler proofs. Our results apply in particular to parabolics over any base, and they are essentially optimal over a field. We also relate the closed orbits in Hom schemes to Serre's theory of complete reducibility, answer a question of Furter--Kraft, and provide many examples.