Dynamical Gelfand-Zetlin algebra and equivariant cohomology of Grassmannians

Abstract

We consider the rational dynamical quantum group $E_y(gl_2)$ and introduce an $E_y(gl_2)$-module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is the equivariant cohomology algebra $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))$ of the cotangent bundle of the Grassmannian $\Gr(k,n)$ 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 difference operators in $λ$. We show that the action of the Gelfand-Zetlin algebra on $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is the natural action of the algebra $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))\otimes \C[δ^{\pm1}]$ on $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $δ: ζ(λ)\toζ(λ+y)$ is the shift operator. The $E_y(gl_2)$-module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ is introduced with the help of dynamical stable envelope maps which are dynamical analogs of the stable envelope maps introduced by Maulik and Okounkov. 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 $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$ induced by the weight functions are dynamical variants of Chern-Schwartz-MacPherson classes of Schubert cells.