A Quasi-Coherent Description of the Category D-mod(Gr(GL(n)))

We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of SO(N − 1, C[[t]])-equivariant perverse sheaves on the affine Grassmannian of SON . We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh. ...

Added: October 22, 2022

Kazhdan–Lusztig conjecture via zastava spaces

Braverman A., Michael Finkelberg, Nakajima H., Journal fur die reine und angewandte Mathematik, Germany 2022 Vol. 2022 No. 787 P. 45–78

We deduce the Kazhdan–Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category O from the equivari-ant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and W-algebras. ...

Added: June 10, 2022

Line bundles over Coulomb branches

Braverman A., Michael Finkelberg, Nakajima H., Advances in Theoretical and Mathematical Physics 2021 Vol. 25 No. 4 P. 957–993

This is the third companion paper of [Part II]. When a gauge theory has a flavor symmetry group, we construct a partial resolution of the Coulomb branch as a variant of the definition. We identify the partial resolution with a partial resolution of a generalized slice in the affine Grassmannian, Hilbert scheme of points, and resolved Cherkis bow variety ...

Added: April 13, 2022

Almost dominant generalized slices and convolution diagrams over them

Krylov V., Perunov I., Advances in Mathematics 2021 Vol. 392 No. 3 Article 108034

Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $\La$ the coweight lattice of $T$. Let $\La^+ \subset \La$ be the submonoid of dominant coweights. For $\la \in \La^+,\,\mu \in \La,\,\mu \leqslant \la$, in "Coulomb branches of 3d $\mathcal{N}=4$ quiver gauge theories and slices in the ...

Added: January 10, 2022

Mirabolic Satake equivalence and supergroups

Braverman A., Michael Finkelberg, Ginzburg V. et al., Compositio Mathematica 2021 Vol. 157 No. 8 P. 1724–1765

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of GL(N − 1, C[[t]])-equivariant perverse sheaves on the affine Grassmannian of GLN . We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and ...

Added: July 22, 2021

Coherent IC-sheaves on type A_n affine Grassmannians and dual canonical basis of affine type A_1

Finkelberg Michael, Fujita R., Representation Theory 2021 Vol. 25 P. 67–89

The convolution ring of loop rotation equivariant K-homology of the affine Grassmannian of GL(n) was identified with a quantum unipotent cell of the loop group of SL(2) by Cautis and Williams. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell. ...

Added: January 29, 2021

Beilinson-Drinfeld Schubert varieties and global Demazure modules

Dumanski I., Feigin E., Finkelberg M. V., / Series math "arxiv.org". 2020. No. 2003.12930.

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson- Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules of the current Lie algebra. ...

Added: April 2, 2020

Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian and geometric Satake equivalence

Michael Finkelberg, Krylov V., Mirkovic I., Journal of Topology 2020 Vol. 13 No. 2 P. 683–729

Let G be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semi-infinite orbits in the affine Grassmannian Gr G . We prove Simon Schieder’s conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semi-infinite orbits with U (n ∨ ) (the universal enveloping ...

Added: March 19, 2020

Coulomb branches of 3-dimensional gauge theories and related structures

Braverman A., Michael Finkelberg, , in: Geometric Representation Theory and Gauge TheoryVol. 2248.: Switzerland: Springer, 2019. Ch. 1 P. 1–52.

These are (somewhat informal) lecture notes for the CIME summer school “Geometric Representation Theory and Gauge Theory” in June 2018. In these notes we review the constructions and results of Braverman et al. (Adv Theor Math Phys 22(5):1017–1147, 2018; Adv Theor Math Phys 23(1):75–166, 2019; Adv Theor Math Phys 23(2):253–344, 2019) where a mathematical definition of Coulomb branches of 3d ...

Added: December 24, 2019

Ring objects in the equivariant Satake category arising from Coulomb branches

Braverman A., Michael Finkelberg, Nakajima H., Advances in Theoretical and Mathematical Physics 2019 Vol. 23 No. 2 P. 253–344

We consider the morphism from the variety of triples introduced in our previous paper to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake category). We show that various constructions in our previous paper work for an arbitrary commutative ring object. The second purpose of this ...

Added: November 12, 2019

Coulomb branches of 3d N = 4 quiver gauge theories and slices in the affine Grassmannian

Braverman A., Michael Finkelberg, Nakajima H., Advances in Theoretical and Mathematical Physics 2019 Vol. 23 No. 1 P. 75–166

This is a companion paper of [Part II]. We study Coulomb branches of unframed and framed quiver gauge theories of type ADE. In the unframed case they are isomorphic to the moduli space of based rational maps from P^1 to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In ...

Added: September 28, 2019

A quasi-coherent description of the the category of D-mod(Gr_GL(n))

Finkelberg M. V., Braverman A., / Series arXiv "math". 2018.

In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr_GL(n)) of (all) D-modules on the affine Grassmannian of the group GL(n) as the category of ind-coherent sheaves on a certain stack (it is explained in loc. cit. that this conjecture "follows" naturally from some heuristic arguments involving 3-dimensional quantum field theory). In this paper we prove a ...

Added: December 3, 2018

Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian

Krylov V., Functional Analysis and Its Applications 2018 Vol. 52 No. 2 P. 113–133

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$. Let $\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct the integrable crystals $\mathbf{B}^{G}(\lambda),\ \lambda\in\Lambda^{+}_{G}$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $\mathbf{p}_{\lambda_{1},\lambda_{2}}\colon \mathbf{B}^{G}(\lambda_{1}) \otimes \mathbf{B}^{G}(\lambda_{2}) \rightarrow \mathbf{B}^{G}(\lambda_{1}+\lambda_{2})\cup\{0\}$ ...

Added: September 11, 2018

Comultiplication for shifted Yangians and quantum open Toda lattice

Michael Finkelberg, Kamnitzer J., Pham K. et al., Advances in Mathematics 2018 Vol. 327 P. 349–389

We study a coproduct in type A quantum open Toda lattice in terms of a coproduct in the shifted Yangian of sl2. At the classical level this corresponds to the multiplication of scattering matrices of euclidean SU(2) monopoles. We also study coproducts for shifted Yangians for any simply-laced Lie algebra. ...

Added: February 21, 2018