Dynamical Gelfand-Zetlin algebra and equivariant cohomology of Grassmannians

TL;DR

This paper constructs a new module structure on the equivariant cohomology of cotangent bundles of Grassmannians using dynamical quantum groups and stable envelope maps, linking algebraic and geometric structures.

Contribution

It introduces a novel $E_y(gl_2)$-module structure on equivariant cohomology of Grassmannian cotangent bundles via dynamical stable envelopes and Gelfand-Zetlin algebra actions.

Findings

Established the action of the Gelfand-Zetlin algebra as a shift operator in the cohomology

Connected dynamical stable envelopes with rational dynamical weight functions

Provided a geometric interpretation of dynamical classes as variants of Chern-Schwartz-MacPherson classes

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 $\lambda$. We consider the dynamical Gelfand-Zetlin algebra which is a commutative algebra of difference operators in $\lambda$. 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 $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))\otimes \C[\delta^{\pm1}]$ on $H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $\delta : \zeta(\lambda)\to\zeta(\lambda+y)$ 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.