Singularities and resolutions of nilpotent orbits in positive characteristic

par Dr Jason Kountouridis (Université-Paris-Saclay)

mercredi 15 janv. 2025, 14:00 → 15:15 Europe/Paris

Pierre Grisvard (IHP)

Let $\mathfrak{g}$ 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 $\chi: \mathfrak{g} \to \mathfrak{g}/\!/G$ (induced from the adjoint $G$-action on $\mathfrak{g}$) enjoys various properties and is well-understood. Of particular interest is the (geometric) zero fiber of $\chi$, i.e. the nilpotent cone of $\mathfrak{g}$, 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 $\chi$ 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.