Quantizations of conical symplectic resolutions I: local and global structure

Abstract

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing the braid group action on category O via twisting functors. Our primary goal is to apply these results to other quantized symplectic resolutions, including quiver varieties and hypertoric varieties. This provides a new context for known results about Lie algebras, Cherednik algebras, finite W-algebras, and hypertoric enveloping algebras, while also pointing to the study of new algebras arising from more general resolutions.

v6: correcting a couple of small errors. See https://uwaterloo.ca/scholar/sites/ca.scholar/files/b2webste/files/qcsr-corrigendum.pdf for more details

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Cited by in corpus (40)

• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• Deformation quantization and superconformal symmetry in three dimensions
• Yangians and quantizations of slices in the affine Grassmannian
• Etingof conjecture for quantized quiver varieties
• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• Twisted Indices of 3d $\mathcal{N}=4$ Gauge Theories and Enumerative Geometry of Quasi-Maps
• On categories O for quantized symplectic resolutions
• Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
• Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O
• Mirror symmetry of 3d $\mathcal{N}=4$ gauge theories and supersymmetric indices
• A categorical action on quantized quiver varieties
• Etingof conjecture for quantized quiver varieties II: affine quivers
• Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface
• Quantizations of multiplicative hypertoric varieties at a root of unity
• A finiteness theorem on symplectic singularities
• Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems
• Quantizations of regular functions on nilpotent orbits
• The universal Poisson deformation of hypertoric varieties and some classification results
• Masses, Sheets and Rigid SCFTs
• Birational geometry for the covering of a nilpotent orbit closure
• From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron
• Morse decomposition for D-module categories on stacks
• On modular categories O for quantized symplectic resolutions
• Note on some properties of generalized affine Grassmannian slices
• Springer theory for symplectic Galois groups
• Affine Springer Fibers, Procesi bundles, and Cherednik algebras
• The Auslander-Gorenstein property for Z-algebras
• Symplectic resolutions for Higgs moduli spaces
• On Localization for Quantum Hamiltonian Reductions in Arbitrary Characteristic
• Representing the big tilting sheaves as holomorphic Morse branes
• String Theory and Math: Why This Marriage May Last
• Hikita conjecture for the minimal nilpotent orbit
• On the Localisation Theorem for rational Cherednik algebra modules
• Abelian localization for cyclotomic Cherednik algebras
• Tensor product algebras in type A are Koszul
• The universal covers of hypertoric varieties and Bogomolov's decomposition
• Equivariant deformation quantization and coadjoint orbit method
• Algebra and geometry of link homology
• Localization theorems for quantized symplectic resolutions
• Harish-Chandra bimodules for type A rational Cherednik algebras