NAMIKAWA-WEYL GROUPS OF AFFINIZATIONS OF SMOOTH NAKAJIMA QUIVER VARIETIES

Article Mathematics, Applied

Deformations of symplectic singularities and orbit method for semisimple Lie algebras

Ivan Losev

Summary: In this paper, we first classify filtered quantizations of conical symplectic singularities and use this classification to prove that all filtered quantizations of symplectic quotient singularities are spherical symplectic reflection algebras. We then apply this classification and Namikawa's classification of filtered Poisson deformations to establish a version of the Orbit method for semisimple Lie algebras. Along the way, we also obtain several new results on the Lusztig-Spaltenstein induction for coadjoint orbits.

SELECTA MATHEMATICA-NEW SERIES (2022)

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Article Mathematics

Etingof's conjecture for quantized quiver varieties

Roman Bezrukavnikov et al.

Summary: In this study, the number of finite dimensional irreducible modules for algebras quantizing Nakajima quiver varieties is computed. A lower bound is obtained for all quivers and framing vectors, with an exact count provided in certain cases. Different techniques, including categorical Kac-Moody actions and wall-crossing functors, are used to achieve these results.

INVENTIONES MATHEMATICAE (2021)

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Article Mathematics, Applied

Symplectic resolutions of quiver varieties

Gwyn Bellamy et al.

Summary: In this article, Nakajima quiver varieties are considered from the perspective of symplectic algebraic geometry. The study proves that they are all symplectic singularities according to Beauville and completely classifies which ones admit symplectic resolutions. Additionally, it is shown that the smooth locus aligns with the locus of canonically theta-polystable points, expanding on a previous result by Le Bruyn. The study also examines the etale local structure of these varieties and identifies their symplectic leaves, with an interesting finding that not all symplectic resolutions of quiver varieties seem to originate from variation of GIT.

SELECTA MATHEMATICA-NEW SERIES (2021)

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Article Mathematics