Do Kien Hoang: Hikita conjecture for nilpotent orbits

Do Kien Hoang, Yale University

Let G be a simple algebraic group and let G^∨ be its Langlands dual group.  Barbasch and Vogan, based on earlier work of Lusztig and Spaltenstein, define a duality map D that sends nilpotent orbits 𝕆_e^∨ ⊂ 𝔤^∨ to special nilpotent orbits 𝕆_e ⊂ 𝔤.   In work of Losev, Mason-Brown and Matvieievskyi, an upgraded version D̃ of this duality is considered, called the refined BVLS duality.  D̃(𝕆_e^∨) is a G-equivariant cover 𝕆'_e of 𝕆_e.  Let S_e^∨ be the nilpotent Slodowy slice of the orbit 𝕆_e^∨.  The two varieties X^∨ = S_e^∨ and X= Spec(ℂ[𝕆'_e]) are expected to be symplectic dual to each other.  In this context, a version of the Hikita conjecture predicts an isomorphism between the cohomology ring of the Springer fiber ℬ_e^∨ and the ring of regular functions on the scheme-theoretic fixed point X^T for some torus T.   This conjecture holds when G is of type A.  In this talk, I will discuss the status of similar statements about the Hikita conjecture for general G.  Part of the result is based on joint work in preparation with Krylov and Matvieievskyi.