Dynamical Gelfand–Zetlin algebra and equivariant cohomology of grassmannians

Abstract

We consider the rational dynamical quantum group [Formula: see text] and introduce an [Formula: see text]-module structure on [Formula: see text], where [Formula: see text] is the equivariant cohomology algebra [Formula: see text] of the cotangent bundle of the Grassmannian [Formula: see text] with coefficients extended by a suitable ring of rational functions in an additional variable [Formula: see text]. We consider the dynamical Gelfand–Zetlin algebra which is a commutative algebra of difference operators in [Formula: see text]. We show that the action of the Gelfand–Zetlin algebra on [Formula: see text] is the natural action of the algebra [Formula: see text] on [Formula: see text], where [Formula: see text] is the shift operator. The [Formula: see text]-module structure on [Formula: see text] is introduced with the help of dynamical stable envelope maps which are dynamical analogs of the stable envelope maps introduced by Maulik and Okounkov [Quantum Groups and quantum cohomology, preprint (2012) 1–276, arXiv:1211.1287]. The dynamical stable envelope maps are defined in terms of the rational dynamical weight functions introduced in [G. Felder, V. Tarasov and A. Varchenko, Solutions of the elliptic QKZB equations and Bethe ansatz I, in Topics in Singularity Theory V. I. Arnold’s 60th Anniversary Collection, Advances in the Mathematical Sciences, AMS Translations, Series 2, Vol. 180 (AMS, 1997), pp. 45–76.] to construct q-hypergeometric solutions of rational qKZB equations. The cohomology classes in [Formula: see text] induced by the weight functions are dynamical variants of Chern–Schwartz–MacPherson classes of Schubert cells.