Yangians and quantizations of slices in the affine Grassmannian

Abstract

We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians --- these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for PGL(n) of Finkelberg-Rybnykov.

37 pages; v2, slightly strengthened Theorem 2.9

References in corpus (3)

• Quantizations of conical symplectic resolutions I: local and global structure
• Quiver varieties and Beilinson-Drinfeld Grassmannians of type A
• Shifted Yangians and finite W-algebras

Cited by in corpus (41)

• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• Quantizations of conical symplectic resolutions I: local and global structure
• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• Shifted Quiver Yangians and Representations from BPS Crystals
• Mirror symmetry and line operators
• Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras
• Coulomb branches of quiver gauge theories with symmetrizers
• Shifted quantum affine algebras: integral forms in type $A$ (with appendices by Alexander Tsymbaliuk and Alex Weekes)
• Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
• The $R$-matrix presentation for the Yangian of a simple Lie algebra
• Representations of twisted Yangians of types B, C, D: I
• Multiplicative Hitchin Systems and Supersymmetric Gauge Theory
• Finite W-superalgebras via super Yangians
• Highest weights for truncated shifted Yangians and product monomial crystals
• Generators for Coulomb branches of quiver gauge theories
• A quantum Mirković-Vybornov isomorphism
• Rational Lax matrices from antidominantly shifted extended Yangians: BCD types
• Gelfand-Tsetlin modules in the Coulomb context
• Coulomb Branches of Star-Shaped Quivers
• Bethe subalgebras in Yangians and the wonderful compactification
• Representations of twisted Yangians of types B, C, D: II
• Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type
• Q-operators are 't Hooft lines
• A Family of ${\mathrm{GL}}_r$ Multiplicative Higgs Bundles on Rational Base
• Geometry and categorification
• Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves
• Branes, Quivers, and the Affine Grassmannian
• Multiplicative slices, relativistic Toda and shifted quantum affine algebras
• Towards a cluster structure on trigonometric zastava
• Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties
• Towards the theory of Yangians
• Algebras, traces, and boundary correlators in $\mathcal{N}=4$ SYM
• Line Operators in 4d Chern-Simons Theory and Cherkis Bows
• On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products
• Theta series for quantum loop algebras and Yangians
• Computing fusion products of MV cycles using the Mirkovic-Vybornov isomorphism
• Twisted super Yangians of type AIII and their representations
• Poisson Geometry of Monic Matrix Polynomials
• The restricted quantum double of the Yangian
• Stable envelopes for slices of the affine Grassmannian
• Integrable system on minimal nilpotent orbit