George McNinch

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 with the property that the quotient
mapping π:G → G/U determines upon restriction 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).