GAAG seminar - Levi factors of linear algebraic groups

Let G be a linear algebraic group over the field k, and suppose that the unipotent radical U of G is defined and split over k (this condition always holds when k is perfect). A Levi factor of G is a k-subgroup M for which the quotient mapping π:G → G/U determines an isomorphism π:M → G/U. For any field k of positive characteristic, there are linear algebraic k-groups without Levi factors. We are interested in *descent* of Levi factors: if ℓ is a finite separable field extension of k and if the group G_ℓ obtained by base-change has a Levi factor, does G itself already have a Levi factor? In the talk, we will describe examples of *disconnected* G where this sort of descent fails. On the other hand, we’ll recall an older result which shows that descent holds if ℓ is Galois over k with [ℓ:k] invertible in k. Finally, we describe some sufficient conditions for descent using vanishing of the non-abelian cohomology set H^1(M,U).