On the Descent of Levi Factors

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 ` is a finite Galois extension of k for which the extension degree [` : k] is relatively prime to p. If G/` 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 ' Lie(R) – i.e. R is a vector group with a linear action of the reductive quotient G/R. If G/ksep has a Levi decomposition for a separable closure ksep of k, then G has a Levi decomposition. Finally, we give an example of a disconnected, abelian, linear algebraic group G for which G/ksep has a Levi decomposition, but G itself has no Levi decomposition.