[RESCHEDULED] Marion Jeannin: On smooth integration of p-nilpotent Lie algebras in positive characteristic

The seminar is cancelled and rescheduled for Jun 18.

Abstract:

Let \(k\) be a field of characteristic zero, and let \(\mathfrak{u}\) be a nilpotent \(k\)-Lie algebra of finite dimension. The Baker–Campbell–Hausdorff formula, induced by the exponential map, defines a group law on the vector group \(V(\mathfrak{u})\), making it into a unipotent algebraic \(k\)-group. In other words there is an equivalence between the category of nilpotent \(k\)-Lie algebras of finite dimension and unipotent algebraic \(k\)-groups. On the other hand, the functor \(G \to \operatorname{Lie}(G)\) induces a quasi-inverse equivalence. If now \(k\) is of characteristic \(p > 0\), such a nice conversation between (unipotent algebraic) groups and (nilpotent) Lie algebras no longer exists in general, but one can still wonder whether under suitable assumptions it is still possible to associate a unipotent algebraic group to a “nilpotent” (this notion will need to be adapted to the context) Lie algebra. More precisely, in this talk we wonder whether, given a field of positive characteristic \(k\), a reductive \(k\)-group \(G\) and a restricted \(p\)-nil subalgebra \(\mathfrak{u}\) of the Lie algebra of \(G\), there exists a smooth unipotent subgroup \(U \subset G\) such that \(\operatorname{Lie}(U) = \mathfrak{u}\). Obstructions are both arithmetic and algebraic : what will play the role of the exponential here? This will lead us to discuss assumptions to ensure the existence of Springer isomorphisms, and more specifically generalised exponential maps; but also geometric: algebraic groups are no longer a priori smooth in positive characteristic, a way of controlling the lack of smoothness is to refine the notion of infinitesimal saturation, first introduced by Deligne.