The monodromy of quantum difference equations is closely related to elliptic stable envelopes invented by M.Aganagic and

A. Okounkov. In the talk I will explain how to extract these equations from the monodromy

using the geometry of the variety X and of its symplectic dual Y. In particular, I will discuss how to extend the action of representation-theoretic objects on K(X), such as quantum groups, quantum Weyl groups, R-matrices, etc, to their action on K(Y). As an application, we will consider the example of the Hilbert scheme of points in the complex plane, where

these results allow us to prove the conjectures of E.Gorsky and A.Negut about the infinitesimal change of the stable basis. Based on joint work with A.Smirnov.

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