Homomorphisms of Algebraic Groups: Representability and Rigidity

Abstract

Given two algebraic groups G, H over a field k, we investigate the representability of the functor of morphisms (of schemes) $\mathbf{Hom}(\mathit{G},\mathit{H})$ and the subfunctor of homomorphisms (of algebraic groups) ${\mathbf{Hom}}_{\mathrm{gp}}(\mathit{G},\mathit{H})$. We show that $\mathbf{Hom}(\mathit{G},\mathit{H})$ is represented by a group scheme, locally of finite type, if the k-vector space $\mathcal{O}(\mathit{G})$ is finite-dimensional; the converse holds if H is not étale. When G is linearly reductive and H is smooth, we show that ${\mathbf{Hom}}_{\mathrm{gp}}(\mathit{G},\mathit{H})$ is represented by a smooth scheme M; moreover, every orbit of H acting by conjugation on M is open.