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In this note, we construct dual PBW bases of the positive and negative subalgebras of the two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ in classical types, as used in our earlier work arXiv:2407.01450. Following the ideas of Leclerc and Clark-Hill-Wang, we introduce the two-parameter shuffle algebra and relate it to the subalgebras above. We then use the combinatorics of dominant Lyndon words to establish the main results., Comment: v1: 36pp, comments are welcome!

We present a formula for trigonometric orthosymplectic $R$-matrices associated with any parity sequence, and establish their factorization into the ordered product of $q$-exponents parametrized by positive roots in the corresponding reduced root systems. The latter is crucially based on the construction of orthogonal bases of the positive subalgebra through $q$-bracketings and combinatorics of dominant Lyndon words, as developed in [Clark, Hill, Wang, "Quantum shuffles and quantum supergroups of basic type", Quantum Topol. 7 (2016), no.3, 553-638]. We further evaluate the affine orthosymplectic $R$-matrices, establishing their intertwining property as well as matching them with those obtained through the Yang-Baxterization technique of [Ge, Wu, Xue, "Explicit trigonometric Yang-Baxterization", Internat. J. Modern Phys. A 6 (1991), no.21, 3735-3779]. This reproduces the celebrated formulas of [Jimbo, "Quantum $R$ matrix for the generalized Toda system", Comm. Math. Phys. 102 (1986), no.4, 537-547] for classical BCD types and the formula of [Mehta, Dancer, Gould, Links, "Generalized Perk-Schultz models: solutions of the Yang-Baxter equation associated with quantized orthosymplectic superalgebras", J. Phys. A 39 (2006), no.1, 17-26] for the standard parity sequence., Comment: v1: 31pp, comments are welcome! v2: 64pp, significantly improved version (factorization added as a new Section 5; fixed treatment for n=m case adding all details in a new Appenix B)

Root vectors in quantum groups (of finite type) generalize to fused currents in quantum loop groups ([5]). In the present paper, we construct fused currents as duals to specialization maps of the corresponding shuffle algebras ([7,8,9]) in types ADE. Both root vectors and fused currents depend on a convex order of the positive roots, and the choice we make in the present paper is that of the Auslander-Reiten order ([24]) corresponding to an orientation of the type ADE Dynkin diagram.

We generalize the study of standard Lyndon loop words from [A.Negut, A.Tsymbaliuk, "Quantum loop groups and shuffle algebras via Lyndon words", Adv. Math. 439 (2024), Paper No. 109482] to a more general class of orders on the underlying alphabet, as suggested in Remark 3.15 of loc.cit. The main new ingredient is the exponent-tightness of these words, which also allows to generalize the construction of PBW bases of the untwisted quantum loop algebra via the combinatorics of loop words., Comment: v2: 32 pages, minor corrections, details added. v1: 30 pages, comments are welcome! arXiv admin note: text overlap with arXiv:2102.11269

We construct finite $R$-matrices for the first fundamental representation $V$ of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ for classical $\mathfrak{g}$, both through the decomposition of $V\otimes V$ into irreducibles $U_{r,s}(\mathfrak{g})$-submodules as well as by evaluating the universal $R$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $U^{\pm}_{r,s}(\mathfrak{g})$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine $R$-matrices, both through the Yang-Baxterization technique of [M.Ge, Y.Wu, K.Xue, "Explicit trigonometric Yang-Baxterization", Internat. J. Modern Phys. A 6 (1991), no.21, 3735-3779] and as the unique intertwiner between the tensor product of $V(u)$ and $V(v)$, viewed as modules over two-parameter quantum affine algebras $U_{r,s}(\widehat{\mathfrak{g}})$ for classical $\mathfrak{g}$. The latter generalizes the formulas of [M.Jimbo, "Quantum R matrix for the generalized Toda system", Comm. Math. Phys. 102 (1986), no. 4, 537-547] for one-parametric quantum affine algebras., Comment: v1: 45pp, comments are welcome!

We construct Lax matrices of superoscillator type that are solutions of the RTT-relation for the rational orthosymplectic $R$-matrix, generalizing orthogonal and symplectic oscillator type Lax matrices previously constructed by the authors in arXiv:2001.06825, arXiv:2104.14518, and arXiv:2112.12065. We further establish factorisation formulas among the presented solutions., Comment: v1: 25 pages, comments are welcome! v2: 26 pages, minor corrections, details added (including a new Appendix)

We generalize an algorithm of Leclerc describing explicitly the bijection of Lalonde-Ram from finite to affine Lie algebras. In type $A_n^{(1)}$, we compute all affine standard Lyndon words for any order of the simple roots, and establish some properties of the induced orders on the positive affine roots., Comment: v2: final version (minor corrections, proof of Theorem 4.7 expanded, proof of Proposition 5.8 added), 39 pages. v1: 32 pages, comments are welcome!

We construct a family of PBWD bases for the positive subalgebras of quantum loop algebras of type $B_n$ and $G_2$, as well as their Lusztig and RTT (for type $B_n$ only) integral forms, in the new Drinfeld realization. We also establish a shuffle algebra realization of these $\mathbb{Q}(v)$-algebras (proved earlier in arXiv:2102.11269 by completely different tools) and generalize the latter to the above $\mathbb{Z}[v,v^{-1}]$-forms. The rational counterparts provide shuffle algebra realizations of positive subalgebras of type $B_n$ and $G_2$ Yangians and their Drinfeld-Gavarini duals. All of this generalizes the type $A_n$ results of arXiv:1808.09536 by the second author., Comment: This is a majorly corrected version of arXiv:2206.12806 by the first author, with a new generalization to integral forms and Yangian counterparts. v1: 46 pages, comments are welcome! v2: 47 pages, minor corrections, details added

These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. These notes consist of three chapters, providing a separate treatment for: the quantum loop algebras of $\mathfrak{sl}_n$ (as well as their super- and 2-parameter generalizations), the quantum toroidal algebras of $\mathfrak{gl}_1$, and the quantum toroidal algebras of $\mathfrak{sl}_n$. We provide the shuffle realization of the corresponding ``positive'' subalgebras as well as of the commutative subalgebras and some combinatorial representations for the toroidal algebras. One of the key techniques involved is that of ``specialization maps''. Each chapter aims to emphasize a different aspect of the theory: in the first chapter we use shuffle algebras to construct a family of new PBWD bases for type $A$ quantum loop algebras and their integral forms; in the second chapter, we provide a geometric interpretation of the Fock modules and use shuffle description of a commutative subalgebra to construct an action of the Heisenberg algebra on the equivariant $K$-theory of the Hilbert schemes of points; in the last chapter, we relate vertex and combinatorial representations of quantum toroidal algebras of $\mathfrak{sl}_n$ using Miki's isomorphism and use shuffle realization to explicitly compute Bethe commutative subalgebras and their limits. The latter construction is inspired by Enriquez's work relating shuffle algebras to the correlation functions of quantum affinized algebras., Comment: v2: 124 pages, significantly improved version, typos fixed. v1: 126 pages

We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras, thus generalizing its rational version of arXiv:2104.14518 and the type A construction of arXiv:1811.12137. As an application, this allows us to construct large families of commuting and q-commuting difference operators, in particular, providing a convenient approach to the Q-systems where it proves a conjecture of arXiv:1704.00154., Comment: v2: 33 pages, few typos fixed, some details added. v1: 32 pages, comments are welcome!

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits., Comment: v2: 61 pages, minor corrections. v1: 61 pages; Comments are Welcome!

Generalizing our recent joint paper with Vasily Pestun (arXiv:2001.04929), we construct a family of $SO(2r),Sp(2r),SO(2r+1)$ rational Lax matrices, polynomial in the spectral parameter, parametrized by the divisors on the projective line with coefficients being dominant integral coweights of associated Lie algebras. To this end, we provide the RTT realization of the antidominantly shifted extended Drinfeld Yangians of $\mathfrak{so}_{2r}, \mathfrak{sp}_{2r}, \mathfrak{so}_{2r+1}$, and of their coproduct homomorphisms. This establishes some of the recent conjectures in the physics literature by Costello-Gaiotto-Yagi (arXiv:2103.01835) in the classical types., Comment: v3: 67 pages, minor corrections, details added. v2: 65 pages, minor corrections, some details added, Remark 4.37 added. v1: 64 pages, comments are welcome!

We study the regular surface defect in the Omega-deformed four-dimensional supersymmetric gauge theory with gauge group SU(N) with 2N hypermultiplets in fundamental representation. We prove its vacuum expectation value obeys the Knizhnik-Zamolodchikov equation for the 4-point conformal block of current algebra of a two-dimensional conformal field theory. The level and the vertex operators are determined by the parameters of the Omega-background and the masses of the hypermultiplets; the cross-ratio of the 4 points is determined by the complexified gauge coupling. We clarify that in a somewhat subtle way the branching rule is parametrized by the Coulomb moduli. This is an example of the BPS/CFT relation., Comment: v3 41 page, 2 figures; some typos fixed, added additional clarifications of the main calculation

We study PBW bases of the untwisted quantum loop group $U_q(L\mathfrak{g})$ (in the Drinfeld new presentation) using the combinatorics of loop words, by generalizing the treatment of [29,30,43] in the finite type case. As an application, we prove that Enriquez' homomorphism [11] from the positive half of the quantum loop group to the trigonometric degeneration of Feigin-Odesskii's elliptic algebra [15] associated to $\mathfrak{g}$ is an isomorphism.

We construct a family of $GL_n$ rational and trigonometric Lax matrices $T_D(z)$ parametrized by $\Lambda^+$-valued divisors $D$ on $\mathbb{P}^1$. To this end, we study the shifted Drinfeld Yangians $Y_\mu(\mathfrak{gl}_n)$ and quantum affine algebras $U_{\mu^+,\mu^-}(L\mathfrak{gl}_n)$, which slightly generalize their $\mathfrak{sl}_n$-counterparts. Our key observation is that both algebras admit the RTT type realization when $\mu$ (respectively, $\mu^+$ and $\mu^-$) are antidominant coweights. We prove that $T_D(z)$ are polynomial in $z$ (up to a rational factor) and obtain explicit simple formulas for those linear in $z$. This generalizes the recent construction by the first two authors of linear rational Lax matrices in both trigonometric and higher $z$-degree directions. Furthermore, we show that all $T_D(z)$ are normalized limits of those parametrized by $D$ supported away from $\{\infty\}$ (in the rational case) or $\{0,\infty\}$ (in the trigonometric case). The RTT approach provides conceptual and elementary proofs for the construction of the coproduct homomorphisms on shifted Yangians and quantum affine algebras of $\mathfrak{sl}_n$, previously established via rather tedious computations. Finally, we establish a close relation between a certain collection of explicit linear Lax matrices and the well-known parabolic Gelfand-Tsetlin formulas., Comment: v2: 57 pages, typos fixed, some details and references added. v1: 54 pages, comments are welcome!

We introduce super Yangians of $\mathfrak{gl}(V),\mathfrak{sl}(V)$ (in the new Drinfeld realization) associated to all Dynkin diagrams of $\mathfrak{gl}(V)$, where $V$ is a finite-dimensional super vector space. We show that all of them are isomorphic to the super Yangians introduced by Maxim Nazarov, by identifying them with the corresponding RTT super Yangians. However, their "positive halves" are not pairwise isomorphic, and we obtain the shuffle algebra realizations of those. We adapt the latter to the trigonometric setup by obtaining the shuffle algebra realizations of the "positive halves" of type $A$ quantum loop superalgebras associated to arbitrary Dynkin diagrams., Comment: v1: 18 pages, comments are welcome! This note is a companion to arXiv:1808.09536. v2: 20 pages, minor corrections, some details added. v3: 21 pages, minor corrections, some details and references added

We show that the Lusztig integral form is dual to the RTT integral form of the type A quantum loop algebra with respect to the new Drinfeld pairing, by utilizing the shuffle algebra realization of the former and the PBWD bases of the latter obtained in arXiv:1808.09536., Comment: v1: 12 pages, comments are welcome! v2: 12 pages, minor corrections. v3: 13 pages, minor corrections, Remark 4.38 (claiming the opposite duality) added

We define an integral form of shifted quantum affine algebras of type $A$ and construct Poincar\'e-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized $K$-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type $A$. Finally, we obtain the rational (homological) analogues of the above results (proved earlier in arXiv:1611.06775, arXiv:1806.07519 via different techniques)., Comment: v1: 65 pages. v2: 67 pages; added a dominance condition in Section 2(vii), another definition in Appendix A(viii), the injectivity of $\mathfrak{g}\to Y$ in Appendix B(ii). v3: 70 pages; table of contents added, Section 3(vi) updated and Remark 4.33 added. v4: 70 pages; this is a slight modification of the version published in the Arnold Mathematical Journal, the difference is in the introduction

We construct a family of PBWD (Poincar\'e-Birkhoff-Witt-Drinfeld) bases for the quantum loop algebras $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ in the new Drinfeld realizations. In the 2-parameter case, this proves Theorem 3.11 of [Hu-Rosso-Zhang] (stated in loc. cit. without a proof), while in the super case it proves a conjecture of [Zhang]. The main ingredient in our proofs is the interplay between those quantum loop algebras and the corresponding shuffle algebras, which are trigonometric counterparts of the elliptic shuffle algebras of Feigin-Odesskii. Our approach is similar to that of [Enriquez] in the formal setting, but the key novelty is an explicit shuffle algebra realization of the corresponding algebras, which is of independent interest. This also allows us to strengthen the above results by constructing a family of PBWD bases for the RTT forms of those quantum loop algebras as well as for the Lusztig form of $U_v(L\mathfrak{sl}_n)$. The rational counterparts provide shuffle algebra realizations of the type $A$ (super) Yangians and their Drinfeld-Gavarini dual subalgebras., Comment: v1: 14 pages, comments are welcome! v2: 16 pages; introduction expanded, Remarks 2.18, 4.7, 5.8 added. v3: 31 pages; corrected description of the integral forms of shuffle algebras (Section 3.3). v4: 37 pages; significantly improved and rewritten version; list of references expanded

We propose a natural generalization of the construction of the quantum difference Toda lattice (introduced independently by Etingof and Sevostyanov) associated to a simple Lie algebra $\mathfrak{g}$. Our construction depends on two orientations of the Dynkin diagram of $\mathfrak{g}$ and some other data (which we refer to as a pair of Sevostyanov triples). In types $A$ and $C$, we provide an alternative construction via Lax matrix formalism, generalizing the one of Kuznetsov-Tsyganov for the classical $q$-Toda. We also show that the generating function of the pairing of Whittaker vectors in the Verma modules is an eigenfunction of the corresponding modified quantum difference Toda system and derive fermionic formulas for the former in spirit of the work by Feigin-Feigin-Jimbo-Miwa-Mukhin. We give a geometric interpretation of all Whittaker vectors in type $A$ via line bundles on the Laumon moduli spaces and obtain an edge-weight path model for them, slightly generalizing the construction of Di Francesco-Kedem-Turmunkh., Comment: v1: 42 pages. v2: 43 pages, minor corrections, Conjecture 3.1 is replaced by Theorem 3.1 (whose proof is given in a separate appendix), added Remarks 3.27(b) and 3.33(b). v3: 46 pages, minor corrections, references updated, several proofs expanded

We introduce the shifted quantum affine algebras. They map homomorphically into the quantized $K$-theoretic Coulomb branches of $3d\ {\mathcal N}=4$ SUSY quiver gauge theories. In type $A$, they are endowed with a coproduct, and they act on the equivariant $K$-theory of parabolic Laumon spaces. In type $A_1$, they are closely related to the open relativistic quantum Toda lattice of type $A$., Comment: 125 pages. v2: references updated; in section 11 the third local Lax matrix is introduced. v3: references updated. v4=v5: 131 pages, minor corrections, table of contents added, Conjecture 10.25 is now replaced by Theorem 10.25 (whose proof is based on the shuffle approach and is presented in a new Appendix). v6: Final version as published, references updated, footnote 4 added

In this short note, we compute the classical limits of the quantum toroidal and the affine Yangian algebras of sl(n) by generalizing our arguments for gl(1) from arXiv:1404.5240. These results were mentioned as motivation in the recent study of toroidal algebras by [Feigin-Jimbo-Miwa-Mukhin], while they played a crucial role in the recent paper arXiv:1512.09109 of the author and Bershtein. An alternative proof of our key isomorphisms for n>2 was first established by [Varagnolo-Vasserot]. We also discuss some consequences of these results., Comment: v1: 10 pages; v2: 12 pages, minor corrections, some details added

In this paper, we relate the well-known Fock representations of the quantum toroidal algebra of sl(n) to the vertex, shuffle, and L-operator representations of the same algebra. These identifications generalize those for the quantum toroidal algebra of gl(1), which were recently established by Feigin-Jimbo-Miwa-Mukhin., Comment: v1: 24 pages. v2: Final version, 27 pages, references updated, several proofs expanded, a few mistakes corrected. To appear in Algebras and Representation Theory

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of $\mathfrak{sl}_n$, denoted by $\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$, respectively. Our motivation arises from the milestone work of Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra $U_q(L \mathfrak{g})$ and the Yangian $Y_h(\mathfrak{g})$ has been established by constructing an isomorphism of $\mathbb{C}[[\hbar]]$-algebras $\Phi:\widehat{U}_{\exp(\hbar)}(L\mathfrak{g})\to \widehat{Y}_\hbar(\mathfrak{g})$ (with $\ \widehat{}\ $ standing for the appropriate completions). These two completions model the behavior of the algebras in the formal neighborhood of $h=0$. The same construction can be applied to the toroidal setting with $q_i=\exp(\hbar_i)$ for $i=1,2,3$. In the current paper, we are interested in the more general relation: $\mathrm{q}_1=\omega_{mn}e^{h_1/m}, \mathrm{q}_2=e^{h_2/m}, \mathrm{q}_3=\omega_{mn}^{-1}e^{h_3/m}$, where $m,n\in \mathbb{N}$ and $\omega_{mn}$ is an $mn$-th root of $1$. Assuming $\omega_{mn}^m$ is a primitive $n$-th root of unity, we construct a homomorphism $\Phi^{\omega_{mn}}_{m,n}$ from the completion of the formal version of $\mathcal{U}^{(m)}_{\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3}$ to the completion of the formal version of $\mathcal{Y}^{(mn)}_{h_1/mn,h_2/mn,h_3/mn}$. We propose two proofs of this result: (1) by constructing the compatible isomorphism between the faithful representations of the algebras; (2) by combining the direct verification of Gautam and Toledano Laredo for the classical setting with the shuffle approach., Comment: v2: 30 pages, significant modifications from the previous version, minor mistakes corrected. v3: Published version, 30 pages, minor corrections, some details added

In this article, we construct certain commutative subalgebras of the big shuffle algebra of cyclic type. This can be considered as a generalization of the similar construction for the small shuffle algebra, obtained by Feigin-Hashizume-Hoshino-Shiraishi-Yanagida. We present a Bethe algebra realization of these subalgebras. The latter identifies them with the Bethe subalgebras of quantum affine gl(n)., Comment: v1: 21 pages; v2: 23 pages, minor corrections, some details added

The affine Yangian of $\mathfrak{gl}_1$ has recently appeared simultaneously in the work of Maulik-Okounkov and Schiffmann-Vasserot in connection with the Alday-Gaiotto-Tachikawa conjecture. While the former presentation is purely geometric, the latter algebraic presentation is quite involved. In this article, we provide a simple loop realization of this algebra which can be viewed as an "additivization" of the quantum toroidal algebra of $\mathfrak{gl}_1$ in the same way as the Yangian $Y_h(\mathfrak{g})$ is an "additivization" of the quantum loop algebra $U_q(L\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of $\mathfrak{gl}_1$ by generalizing the milestone result of Gautam and Toledano Laredo to the current settings., Comment: v2: 43 pages, minor corrections, some details added

In this article we classify all infinitesimal Hecke algebras of so_N. We establish isomorphism of their universal versions and the W-algebras of so_{N+2m+1} with a 1-block nilpotent element of the Jordan type (1,...,1,2m+1). This should be considered as a continuation of the recent joint paper with I. Losev, where the analogous results were obtained for the cases of gl_n and sp_{2n}., Comment: 14 pages; v2: minor corrections, some details added

In this article we establish an isomorphism between universal infinitesimal Cherednik algebras and W-algebras for Lie algebras of the same type and 1-block nilpotent elements. As a consequence we obtain some fundamental results about infinitesimal Cherednik algebras., Comment: 24 pages; v2 minor corrections

Infinitesimal Cherednik algebras, first introduced in [EGG], are continuous analogues of rational Cherednik algebras, and in the case of gl_n, are deformations of universal enveloping algebras of the Lie algebras sl_{n+1}. Despite these connections, infinitesimal Cherednik algebras are not widely-studied, and basic questions of intrinsic algebraic and representation theoretical nature remain open. In the first half of this paper, we construct the complete center of H_\zeta(gl_n) for the case of n=2 and give one particular generator of the center, the Casimir operator, for general n. We find the action of this Casimir operator on the highest weight modules to prove the formula for the Shapovalov determinant, providing a criterion for the irreducibility of Verma modules. We classify all irreducible finite dimensional representations and compute their characters. In the second half, we investigate Poisson-analogues of the infinitesimal Cherednik algebras and use them to gain insight on the center of H_\zeta(gl_n). Finally, we investigate H_\zeta(sp_{2n}) and extend various results from the theory of H_\zeta(gl_n), such as a generalization of Kostant's theorem., Comment: 21 pages, 2 figures. Changes: 1) to appear in J. Of Represent. Theory 2) multiple typos fixed 3) detailed Appendix with explicit calculations for the case of n=2 deleted

In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...]., Comment: some typos fixed, the last 2 subsections added, 20 pages

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL_n. We construct the action of the quantum loop algebra U_v(Lsl_n) in the equivariant K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra U^{tor}_v(Lsl}_n) in the equivariant K-theory of the affine version of Laumon spaces. We write down explicit formulae for this action in the affine Gelfand-Tsetlin base, corresponding to the fixed point base in the localized equivariant K-theory., Comment: v2: multiple typos fixed, proofs of Theorems 4.13 and 4.19 expanded, 23 pages. v3: formulas of Theorems 4.9 and 4.13 corrected, resulting minor changes added. arXiv admin note: text overlap with arXiv:0812.4656, arXiv:math/0503456, arXiv:0806.0072 by other authors