On the descent of Levi factors

Abstract

Let G be a linear algebraic group over a field k of characteristic p > 0, and suppose that the unipotent radical R of G is defined and split over k. By a Levi factor of G, one means a closed subgroup M which is a complement to R in G. In this paper, we give two results related to the descent of Levi factors. First, suppose l is a finite Galois extension of k for which the extension degree [l : k] is relatively prime to p. If G /l has a Levi decomposition, we show that G has a Levi decomposition. Second, suppose that there is a G-equivariant isomorphism of algebraic groups $${R \simeq Lie(R)}$$ – i.e. R is a vector group with a linear action of the reductive quotient G/R. If $${G_{{/k}_{sep}}}$$ has a Levi decomposition for a separable closure k sep of k, then G has a Levi decomposition. Finally, we give an example of a disconnected, abelian, linear algebraic group G for which $${G_{{/k}_{sep}}}$$ has a Levi decomposition, but G itself has no Levi decomposition.