We define and study the quantum equivariant K-theory of cotangent bundles over Grassmannians. For every tautological bundle in the K-theory we define its one-parametric deformation, referred to as quantum tautological bundle. We prove that the spectrum of operators of quantum multiplication by these quantum classes is governed by the Bethe ansatz equations for the inhomogeneous XXZ spin chain. In addition, we prove that each such operator corresponds to the universal elements of quantum group U ( sl 2 ). In particular, we identify the Baxter operator for the XXZ spin chain with the operator of quantum multiplication by the exterior algebra tautological bundle. The explicit universal combinatorial formula for this operator is found. The relation between quantum line bundles and quantum dynamical Weyl group is shown.

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.

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