Partial flag varieties, stable envelopes, and weight functions

Abstract

We consider the cotangent bundle $T^{\ast }{\mathfrak{F}}_{\lambda }$ of a ${\mathrm{GL}}_n$ partial flag variety, $\lambda =({\lambda }_1,\dots ,{\lambda }_N),|\lambda |={\sum }_i{\lambda }_i=n$, and the torus $T=({\mathbb{C}}^{\times }{)}^{n+1}$ equivariant cohomology $H_T^{\ast }(T^{\ast }{\mathfrak{F}}_{\lambda })$. In [9], a Yangian module structure was introduced on ${⨁}_{|\lambda |=n}{H}_T^{\ast }(T^{\ast }{\mathfrak{F}}_{\lambda })$. We identify this Yangian module structure with the Yangian module structure introduced in [5]. This identifies the operators of quantum multiplication by divisors on $H_T^{\ast }(T^{\ast }{\mathfrak{F}}_{\lambda })$, described in [9], with the action of the dynamical Hamiltonians from [20], [10], [5]. To construct these identifications we provide a formula for the stable envelope maps, associated with the partial flag varieties and introduced in [9]. The formula is in terms of the Yangian weight functions introduced in [19], c.f. [21], [22], in order to construct q-hypergeometric solutions of qKZ equations.