# Dual singularities in exceptional type nilpotent cones

### Paul Levy (University of Lancaster)

Wednesday 29th October, 2014 16:00-17:00 Maths 516

#### Abstract

It is well-known that nilpotent orbits in \sl_n(\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 \g is any simple Lie algebra over \C then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in \g to the set of special nilpotent orbits in the Langlands dual Lie algebra \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 \O_1<\O_2 are adjacent in the partial order then so are their duals \O_1^t>\O_2^t, and the isolated singularity attached to the pair (\O_1,\O_2) is dual to the singularity attached to (\O_2^t,\O_1^t): a Kleinian singularity of type A_k is swapped with the minimal nilpotent orbit closure in \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 \O_1<\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.

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