Dynamical Gelfand-zetlin Algebra and Equivariant Cohomology of Grassmannians

  • Published 2015


We consider the rational dynamical quantum group Ey(gl2) and introduce an Ey(gl2)module structure on ⊕k=0H GLn×C×(T Grk(C)), where H∗ GLn×C×(T Grk(C)) is the equivariant cohomology algebra H∗ GLn×C×(T Grk(C)) of the cotangent bundle of the Grassmannian Grk(C) with coefficients extended by a suitable ring of rational functions in an additional variable λ. We consider the dynamical Gelfand-Zetlin algebra which is a commutative algebra of diffference operators in λ. We show that the action of the Gelfand-Zetlin algebra on H∗ GLn×C×(T Grk(C)) is the natural action of the algebra H∗ GLn×C×(T Grk(C)) ⊗ C[δ±1] on H∗ GLn×C×(T Grk(C)), where δ : ζ(λ)→ ζ(λ+ y) is the shift operator. The Ey(gl2)-module structure on ⊕k=0H GLn×C×(T Grk(C)) is introduced with the help of dynamical stable envelope maps which are dynamical analogs of the stable envelope maps introduced by Maulik and Okounkov, [MO]. The dynamical stable envelope maps are defined in terms of the rational dynamical weight functions introduced in [FTV] to construct q-hypergeometric solutions of rational qKZB equations. The cohomology classes in H∗ GLn×C×(T Grk(C)) induced by the weight functions are dynamical variants of Chern-Schwartz-MacPherson classes of Schubert cells.

Cite this paper

@inproceedings{RIMNYI2015DynamicalGA, title={Dynamical Gelfand-zetlin Algebra and Equivariant Cohomology of Grassmannians}, author={R. RIM{\'A}NYI and A. VARCHENKO}, year={2015} }