Linearity for actions on vector groups.
Let \(k\) be a field. The \textit{vector groups} in the title of this paper are linear algebraic groups over \(k\) which are isomorphic to a product of copies of the additive group \(\mathbb G_a\). Suppose \(U\) is such a vector group and \(G\) is a linear algebraic group over \(k\) acting on \(U\) by group automorphisms. Then the action of \(G\) on \(U\) gives rise to a linear action on the Lie algebra \(\mathrm{Lie}(U)\). The aim of this paper is to provide conditions on \(G\) which ensure that there is a \(G\)-equivariant isomorphism \(U\simeq\mathrm{Lie}(U)\); in such a situation we call the action of \(G\) on \(U\) \textit{linear}.NEWLINENEWLINE The main result of the paper is the following: assume that the geometric unipotent radical of \(G\) (that is, the unipotent radical of \(G\) of the algebraic closure of \(k\)) arises by base change from a subgroup \(R\) of \(G\), and that \(G\) acts by automorphisms on a vector group \(U\). If \(\mathrm{Lie}(U)\) is a simple module for the identity component \(G^0\), then the action of \(G\) on \(U\) is linear.NEWLINENEWLINE The assumption on the unipotent radical allows the author to avoid possible issues which can arise if the unipotent radical ``cannot be seen'' over \(k\). Under the stronger condition that the subgroup \(R\) in the theorem is also split over \(k\), the author is able to apply his result to the action of \(G\) on the group \(R\) itself to deduce the existence of a Levi factorisation of \(G\) over \(k\), and the conjugacy of any two Levi factors under the action of \(R(k)\), building on his earlier work [Transform. Groups 15, No. 4, 937-964 (2010; Zbl 1221.20034)].
scientific article; zbMATH DE number 6340907
scientific article