Tuesday, June 27, 2023
11:00 a.m. (sharp)
Zoom Web Conference
PhD Candidate: Pavel Shlykov
Supervisor: Alexander Braverman
Thesis title: Certain cases of Hikita-Nakajima conjecture
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Let M0 be an affine Nakajima quiver variety, and M is the corresponding BFN Coulomb branch. Assume that M0 can be resolved by the (smooth) Nakajima quiver variety M. The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras H∗S (M, C) ≃ C[MC×s], where S ↷ M0 is a torus acting on M0 preserving the Poisson structure, Ms is the (Poisson) deformation of M over s = Lie S, C× is a generic one-dimensional torus acting on M, and C[MC×s] is the algebra of schematic C×-fixed points of Ms. In this thesis we prove the Hikita-Nakajima conjecture for M = C^2/Γ (Kleinian singularities) and M = M(n,r) Gieseker variety (ADHM space). In the latter case we produce the isomorphism explicitly on generators. We also describe the Hikita-Nakajima isomorphism above using the realization of Ms as the spectrum of the center of rational Cherednik algebra corresponding to Sn ⋉ (Z/rZ) n and identify all the algebras that appear in the isomorphism with the center of degenerate cyclotomic Hecke algebra.
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The draft of the thesis can be found here: thesis_template_shlykov
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