# Do Kien Hoang: Hikita conjecture for nilpotent orbits

Do Kien Hoang, Yale University

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Hikita conjecture for nilpotent orbits

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.