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## A Weyl Module Stratification of Integrable Representations

Cornell University
,
2017.
No. 1712.03508.

Loktev S., Kato S.

We construct a filtration on integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each Weyl module there is given by a corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as the graded dimension of the space of conformal coinvariants. In addition, as an application of the level one case of the main result, we realize global Weyl modules of current algebras of type ADE in terms of Schubert manifolds of thick affine Grassmanian, as predicted by Boris Feigin.

Feigin E., Makedonskyi I., Selecta Mathematica, New Series 2017 Vol. 23 No. 4 P. 2863-2897

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum ...

Added: October 10, 2017

Kato S., Loktev S., Communications in Mathematical Physics 2019 Vol. 368 No. 1 P. 113-141

We construct a filtration on an integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each global Weyl module there is given by the corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka ...

Added: October 4, 2019

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

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

Braverman A., Michael Finkelberg, Moscow Mathematical Journal 2013 Vol. 13 No. 2 P. 233-265

This is the third paper in a series which describes a conjectural analogue of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The present paper is dedicated to the description of the conjectural analogue of the convolution diagram for the double affine Grassmannian and affine zastava. ...

Added: September 18, 2013

Finkelberg M. V., Braverman A., / Cornell University. 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

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

Benett M., Berenstein A., Chari V. et al., Selecta Mathematica, New Series 2014 No. 2 P. 585-607

We study the category of graded representations with finite--dimensional graded pieces for the current algebra associated to a simple Lie algebra. This category has many similarities with the category O of modules for g and in this paper, we use the combinatorics of Macdonald polynomials to prove an analogue of the famous BGG duality in ...

Added: September 29, 2012

Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2017. No. 1709.05674.

The goal of this paper is twofold. First, we write down the semi-infinite Pl\"ucker relations, describing the Drinfeld-Pl\"ucker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, i.e. the quotient by the ideal generated by the semi-infinite Pl\"ucker relations. We establish the isomorphism with the ...

Added: September 19, 2017

Dumanski I., Feigin E., Finkelberg M. V., / Cornell University. 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

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

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

Chari V., Loktev S., Journal of Algebra 2012 Vol. 349 No. 1 P. 317-328

We identify the sl(n+1) isotypical components of the global Weyl modules W(kω1) with certain natural subspaces of the polynomial ring in k variables. We then apply the representation theory of current algebras to classical problems in invariant theory. ...

Added: September 29, 2012

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

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

Dumanski I., Feigin E., Finkelberg M., Forum of Mathematics, Sigma 2021 Vol. 9

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 over the current Lie algebra. ...

Added: September 8, 2021

Feigin E., Makedonskyi I., Mathematical Research Letters 2017 Vol. 24 No. 3 P. 741-766

The main goal of our paper is to establish a connection between the Weyl modules of the current Lie
superalgebras (twisted and untwisted) attached to $\mosp(1,2)$ and the nonsymmetric Macdonald polynomials of types $A_2^{(2)}$
and $A_2^{(2)\dagger}$. We compute the dimensions and construct bases of the Weyl modules.
We also derive explicit formulas for the $t=0$ and $t=\infty$ specializations
of ...

Added: September 2, 2017

Michael Finkelberg, Feigin E., Reineke M., Kyoto Journal of Mathematics 2017 Vol. 57 No. 2 P. 445-474

We study the connection between the affine degenerate Grassmannians in type A, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type GL(n) and identify it with semi-infinite orbit closure of type A_{2n-1}. We show that principal quiver Grassmannians for the ...

Added: May 10, 2017

191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...

Added: September 23, 2016

Losev A. S., Slizovskiy S., JETP Letters 2010 Vol. 91 P. 620-624

Added: February 27, 2013

Ilyashenko Y., Яковенко С. Ю., М. : МЦНМО, 2013

Предлагаемая книга—первый том двухтомной монографии, посвящённой аналитической теории дифференциальных уравнений.
В первой части этого тома излагается формальная и аналитическая теория нормальных форм и теорема о разрешении особенностей для векторных полей на плоскости.
Вторая часть посвящена алгебраически разрешимым локальным задачам теории аналитических дифференциальных уравнений , квадратичным векторным полям и проблеме локальной классификации ростков векторных полей в комплексной области ...

Added: February 5, 2014

Kalyagin V.A., Koldanov A.P., Koldanov P.A. et al., Physica A: Statistical Mechanics and its Applications 2014 Vol. 413 No. 1 P. 59-70

A general approach to measure statistical uncertainty of different filtration techniques for market network analysis is proposed. Two measures of statistical uncertainty are introduced and discussed. One is based on conditional risk for multiple decision statistical procedures and another one is based on average fraction of errors. It is shown that for some important cases ...

Added: July 19, 2014

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Maslov V., Теоретическая и математическая физика 2019 Т. 201 № 1 С. 65-83

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint
using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes
a boson at the instant of separating from a fermionic nucleus. We study the further transformations of
boson and fermion states of separation in a ...

Added: November 1, 2019