Quantum Differential Equation for Slices of the Affine Grassmannian

Abstract: The affine Grassmannian associated to a reductive group $\mathbf{G}$ is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. In this work, we study their quantum connection. We use the stable envelopes of D. Maulik and A. Okounkov[arXiv:1211.1287] to write an explicit formula for this connection. The classical part of the multiplication comes from [arXiv:2210.09967]. The computation of the purely quantum part is done based on the deformation approach of A. Braverman, D. Maulik and A. Okounkov[arXiv:1001.0056]. For the case of simply-laced $\mathbf{G}$, we identify the quantum connection with the trigonometric Knizhnik-Zamolodchikov equation for the Langlands dual group $\mathbf{G}^\vee$.