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## Extremal part of the PBW-filtration and E-polynomials

Cornell University
,
2013.
No. arXiv:1306.3146.

Feigin E., Cherednik I.

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials, also called E-polynomails, in the limit t=infinity and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebra for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for G_2, and also establish the relations of our extremal degrees to minimal q-degrees of the extremal terms of the Kostant q-partition function; they coincide with the latter only for some root systems.

Feigin E., Makedonskyi I., / Cornell University. Series arXiv "math". 2015. No. 1507.01362.

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 osp(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 ...

Added: July 8, 2015

Khoroshkin A., International Mathematics Research Notices 2022 Vol. 2022 No. 4 P. 3106-3143

Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad.
In case $\mathcal{P}$ is Koszul a criterion ...

Added: February 13, 2021

Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2014. No. 1407.6316.

The Cherednik-Orr conjecture expresses the t\to\infty limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases. ...

Added: August 10, 2014

Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2015. No. 1512.03254.

The 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 ...

Added: December 15, 2015

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

Feigin E., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1606.05219.

We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and connection to the theory of nonsymmetric Macdonald polynomials. As an application we compute the dimension of the classical Weyl modules in the remaining unknown case. ...

Added: June 17, 2016

Feigin E., Kato S., Makedonskyi I., Journal fuer die reine und angewandte Mathematik 2020 Vol. 764 P. 181-216

We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on ...

Added: August 12, 2020

Feigin E., Makedonskyi I., Orr D., Advances in Mathematics 2018 Vol. 330 P. 997-1033

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules. ...

Added: September 13, 2018

Feigin E., Makedonskyi I., Orr D., / Cornell University. Series math "arxiv.org". 2016. No. 1605.01560.

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules. ...

Added: May 6, 2016

Feigin E., Kato S., Makedonskyi I., / Cornell University. Series math "arxiv.org". 2017. No. 1703.04108.

We study the nonsymmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the nonsymmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on ...

Added: March 20, 2017

Bezrukavnikov R., Finkelberg M. V., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 20, 2014

Finkelberg M., Braverman A., Shiraishi J., Providence : American Mathematical Society, 2014

Let G be an almost simple simply connected complex Lie group, and let G/U be its base affine space. In this paper we formulate a conjecture which provides a new geometric interpretation of the Macdonald polynomials associated to G via perverse coherent sheaves on the scheme of formal arcs in the affinizationof G/U. We prove ...

Added: March 5, 2015

Feigin E., Makedonskyi I., Journal of Combinatorial Theory, Series A 2015 P. 60-84

The Cherednik–Orr conjecture expresses the t →∞limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases. ...

Added: May 20, 2015

Gorsky E., Carlsson E., Mellit A., Mathematische Annalen 2019

The earlier work of the first and the third named authors introduced the algebra A_q,t and its polynomial representation. In this paper we construct an action of this algebra on the equivariant K-theory of certain smooth strata in the flag Hilbert schemes of points on the plane. In this presentation, the fixed points of torus action ...

Added: September 3, 2019

Olshanski G., Working papers by Cornell University. Series math "arxiv.org" 2020

Using Okounkov's q-integral representation of Macdonald polynomials we construct an infinite sequence Ω1,Ω2,Ω3,… of countable sets linked by transition probabilities from ΩN to ΩN−1 for each N=2,3,…. The elements of the sets ΩN are the vertices of the extended Gelfand-Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, q and t. These ...

Added: January 19, 2021

Finkelberg M. V., MATHEMATICAL SCIENCES 2013 Vol. 51 No. 596 P. 46-51

This is a survey of the author's and his collaboratots' recent works on the quasiflags' moduli spaces introduced by Gerard Laumon some 25 years ago. These spaces are used in the study of geometric Eisenstein series, quantum cohomology and K-theory of the flag varieties, Weyl modules, Nekrasov partition function of N=2 supersymmetric gauge quantum field ...

Added: February 14, 2013

Olshanski G., Communications in Mathematical Physics 2021 Vol. 385 P. 595-631

We introduce and study a family of (q, t)-deformed discrete N-particle beta ensembles, where q and t are the parameters of Macdonald polynomials. The main result is the existence of a large-N limit transition leading to random point processes with infinitely many particles. ...

Added: June 22, 2021

Khoroshkin A., / arXiv.org. Series 1312.7053 "1312". 2013. No. 1312.7053.

The aim of this paper is to introduce the categorical setup which helps us to relate the theory of Macdonald polynomials and the theory of Weyl modules for current Lie algebras discovered by V.\,Chari and collaborators. We identify Macdonald pairing with the homological pairing on the ring of characters of the Lie algebra of currents. ...

Added: February 14, 2014

Olshanski G., Selecta Mathematica, New Series 2021 Vol. 27 Article 41

Using Okounkov’s q-integral representation of Macdonald polynomials we construct an infinite sequence Ω1,Ω2,Ω3,… of countable sets linked by transition probabilities from Ω𝑁 to Ω𝑁−1 for each 𝑁=2,3,…. The elements of the sets Ω𝑁 are the vertices of the extended Gelfand–Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, q and t. These ...

Added: June 4, 2021

Belavin A. A., Bershtein M., Tarnopolsky G. M., 2013 No. 3 P. 1-36

A bstract We continue our study of the AGT correspondence between instanton counting on
${{{{{\ mathbb {C}}^ 2}}}\ left/{{{{\ mathbb {Z}} _p}}}\ right.} $ and Conformal field theories with
the symmetry algebra $\ mathcal {A}\ left ({r, p}\ right) $. In the cases r= 1, p= 2 and r= 2, p= 2
this algebra specialized to: $\ mathcal ...

Added: September 9, 2014

Vologodsky V., Finkelberg M. V., Bezrukavnikov R., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 17, 2015

Cherednik I., Feigin E., Advances in Mathematics 2015 Vol. 282 P. 220-264

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the ...

Added: September 3, 2015

Окубо Ю. undefined., Journal of Physics: Conference Series 2017 Vol. 804 No. 012036 P. 1-8

We investigate the existence and the orthogonality of the generalized Jack symmetric functions which play an important role in the AGT relations. We show their orthogonality by deforming them to the generalized Macdonald symmetric functions. ...

Added: October 26, 2017

Feigin E., Fourier G., Littelmann P., International Mathematics Research Notices 2011 Vol. 2011 No. 24 P. 5760-5784

We study the PBW filtration on the highest weight representations V(λ) of sp2n. This
filtration is induced by the standard degree filtration on U(n−). We give a description
of the associated graded S(n−)-module gr V(λ) in terms of generators and relations. We
also construct a basis of gr V(λ). As an application we derive a graded combinatorial
formula for ...

Added: October 12, 2012