Linearity for Actions on Vector Groups

Let k be an arbitrary field, let G be a (smooth) linear algebraic group over k, and let U be a vector group over k on which G acts by automorphisms of algebraic groups. The action of G on U is said to be linear if there is a G-equivariant isomorphism of algebraic groups U ' Lie(U). Suppose that G is connected and that the unipotent radical of G is defined over k. If the G-module Lie(U) is simple, we show that the action of G on U is linear. If G acts by automorphisms on a connected, split unipotent group U, we deduce that U has a filtration by G-invariant closed subgroups for which the successive factors are vector groups with a linear action of G. When G is connected and the unipotent radical of G is defined and split over k, this verifies an assumption made in earlier work of the author on the existence of Levi factors. On the other hand, for any field k of positive characteristic we show that if the category of representations of G is not semisimple, there is an action of G on a suitable vector group U which is not linear.