RéGA

Singularities and resolutions of nilpotent orbits in positive characteristic

par Dr Jason Kountouridis (Université-Paris-Saclay)

Europe/Paris
Pierre Grisvard (IHP)

Pierre Grisvard

IHP

Description
Let g be a simple Lie algebra and G its associated simply-connected Lie group. Under certain restrictions on the ground field characteristic, the adjoint quotient map χ:gg//G (induced from the adjoint G-action on g) enjoys various properties and is well-understood. Of particular interest is the (geometric) zero fiber of χ, i.e. the nilpotent cone of g, which itself splits into multiple nilpotent orbits under the G-action. It turns out that the closures of these nilpotent orbits provide a rich source of singularities that have been independently studied by representation theorists and algebraic geometers; as an example, the unique so-called "subregular" orbit exhibits all Gorenstein rational surface singularities, and map χ describes all possible miniversal deformations of these singularities. Furthermore, the Grothendieck—Springer resolution simultaneously resolves these fiber singularities and provides clean descriptions of each resolution as a vector bundle over a specific partial flag variety.
 

All these results are somewhat classically known over the complex numbers, yet can still be made to work in positive characteristic, other than a few exceptional primes. We will aim to give a brief survey of this circle of ideas, focusing on the explicit example of the special linear group, with the goal of giving some concrete motivation for the study of characteristic p singularities.