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We obtain explicit formulas for the K-theoretic capped descendent vertex functions of $${\text {Hilb}}^n(\mathbb {C}^2)$$ for descendents given by the exterior algebra of the tautological bundle. This formula provides a one-parametric deformation of the generating function for normalized Macdonald polynomials. In particular, we show that the capped vertex functions are rational functions of the quantum parameter. 

In this note we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, $$X=T^{*}\Gr(k,n)$$. This integral representation can be used to compute the $$\hbar\to \infty$$ limit of the vertex function, where $$\hbar$$ denotes the equivariant parameter of a torus acting on $$X$$ by dilating the cotangent fibers. We show that in this limit the integral turns into the standard mirror integral representation of the $$A$$-series of the Grassmannian $$\Gr(k,n)$$ with the Laurent polynomial Landau-Ginzburg superpotential of Eguchi, Hori and Xiong. 

Abstract Let $$\textsf {X}$$ and $$\textsf {X}^{!}$$ be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The $$K$$-theoretic limit of the elliptic duality interface is an equivariant $$K$$-theory class $$\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$$. We show that this class provides correspondences $$ \begin{align*} & \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end{align*}$$mapping the $$K$$-theoretic stable envelopes to the $$K$$-theoretic stable envelopes. This construction allows us to relate various representation theoretic objects of $$K(\textsf {X})$$, such as action of quantum groups, quantum dynamical Weyl groups, $$R$$-matrices, etc., to those for $$K(\textsf {X}^{!})$$. In particular, we relate the wall $$R$$-matrices of $$\textsf {X}$$ to the $$R$$-matrices of the dual variety $$\textsf {X}^{!}$$. As an example, we apply our results to $$\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$$—the Hilbert scheme of $$n$$ points in the complex plane. In this case, we arrive at the conjectures of Gorsky and Negut from [10]. 

Abstract We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice. 

Abstract We consider a pair of quiver varieties $$(X;X^{\prime})$$ related by 3D mirror symmetry, where $$X =T^*{Gr}(k,n)$$ is the cotangent bundle of the Grassmannian of $$k$$-planes of $$n$$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $$X \times X^{\prime} $$ (the mother function) whose restrictions to $$X$$ and $$X^{\prime} $$ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $$X$$ and $$X^{\prime}$$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.