# Seminario: “Dual singularities in nilpotent cones”

## Giovedì 19 Aprile 2018, ore 14:30 - Aula 1BC45 - Paul Levy

ARGOMENTI: Seminars

Giovedì 19 Aprile 2018 alle ore 14:30 in Aula 1BC45, Paul Levy (Lancaster) terrà un seminario dla titolo “Dual singularities in nilpotent cones”.

Abstract
It is well-known that nilpotent orbits in $\mathfrak{sl}_n(\mathbb{C})$ correspond bijectively with the set of partitions of $n$, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type $A$ there is an order-reversing involution on the poset of nilpotent orbits. More generally, if $\mathfrak{g}$ is any simple Lie algebra over $\mathbb{C}$ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in $\mathfrak{g}$ to the set of special nilpotent orbits in the Langlands dual Lie algebra $\mathfrak{g}^L$. It was observed by Kraft and Procesi that the duality in type $A$ is manifested in the geometry of the nullcone. In particular, if two orbits $\mathcal{O}_1<\mathcal{O}_2$ are adjacent in the partial order then so are their duals $\mathcal{O}_1^t>\mathcal{O}_2^t$, and the isolated singularity attached to the pair $(\mathcal{O}_1,\mathcal{O}_2)$ is dual to the singularity attached to $(\mathcal{O}_2^t,\mathcal{O}_1^t)$: a Kleinian singularity of type $A_k$ is swapped with the minimal nilpotent orbit closure in $\mathfrak{sl}_{k+1}$ (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits.
In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs $\mathcal{O}_1<\mathcal{O}_2$ of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.