Homomorphisms of Algebraic Groups: Representability and Rigidity

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