Geometry Seminar

Let $G$ and $G^{\vee}$ be Langlands dual semisimple complex Lie groups, and $\mathcal{N}$ and $\mathcal{N}^\vee$ be the respective nilpotent cones. It is expected that $\mathcal{N}$ and $\mathcal{N}^\vee$ form a pair, satisfying the properties of the conjectural symplectic duality. For such symplectic dual pair there are natural questions concerning the relation between symplectic leaves of the dual varieties. In the example of nilpotent cones, the answers are connected with certain duality maps, the best known one is the order reversing duality map (BVLS duality) between the sets of special orbits, studied by Barbasch and Vogan, and by Lusztig and Spaltenstein. Extending the image to non-special orbits, there are duality maps by Sommers and Achar, generalizing BVLS duality. In this talk I will explain how these duality maps (or their refined versions) fit into the context of symplectic duality between $\mathcal{N}$ and $\mathcal{N}^\vee$. It is based on the joint works with Ivan Losev, Lucas Mason-Brown, Shilin Yu and the ongoing project with Shilin Yu.