Dynamical Gelfand-Zetlin algebra and equivariant cohomology of Grassmannians (Q2835380)

Maulik and Okounkov studied the classical and quantum equivariant cohomology of Nakajima quiver varieties for a quiver \(Q\) in [\textit{D. Maulik} and \textit{A. Okounkov}, ``Quantum groups and quantum cohomology'', Preprint, \url{arXiv:1211.1287}]. They constructed a Hopf algebra \(Y_Q\), called the Yangian of \(Q\), acting on the cohomology of these varieties, and showed that the Bethe algebra \(\mathcal{B}^q\) of this action, depending on some parameters \(q\) and acting on the cohomology of these varieties coincides with the algebra of quantum multiplication. If \(q\to \infty\), the limiting Bethe algebra \(\mathcal{B}^\infty\), called the Gelfand-Zetlin algebra, is isomorphic to the algebra of the standard multiplication on the cohomology. The construction of the Yangian and the Yangian action is based on the notion of the stable envelope maps introduced in [loc. cit.].NEWLINENEWLINELet \((\mathbb{C}^\times)^n\subset \mathrm{GL}_n\) be the torus of diagonal matrices. The groups \((\mathbb{C}^\times)^n\subset \mathrm{GL}_n\) act on \(\mathbb{C}^n\) and hence on the cotangent bundle \(T^*\!\text{Gr}_k(\mathbb{C}^n)\) of a Grassmannian. Extend these \((\mathbb{C}^\times)^n\subset \mathrm{GL}_n\) actions to the actions of \(T=(\mathbb{C}^\times)^n\times\mathbb{C}^\times\subset \mathrm{GL}_n\times\mathbb{C}^\times\) in such a way that the extra \(\mathbb{C}^\times\) acts on the fibers of \( T^*\!\text{Gr}_k(\mathbb{C}^n) \to \text{Gr}_k(\mathbb{C}^n)\) by multiplication. Consider the equivariant cohomology algebra \(H^*_T( T^*\!\text{Gr}_k(\mathbb{C}^n) )\). In this situation Maulik and Okounkov [loc. cit.] defined the stable envelope maps NEWLINE\[NEWLINE\text{Stab}_\sigma : \bigoplus_{k=0}^n H^*_T((T^*\!\text{Gr}_k(\mathbb{C}^n))^T) \to \bigoplus_{k=0}^n H^*_T(T^*\!\text{Gr}_k(\mathbb{C}^n)),NEWLINE\]NEWLINE where \((T^*\!\text{Gr}_k(\mathbb{C}^n))^T\subset T^*\!\text{Gr}_k(\mathbb{C}^n)\) is the fixed point set with respect to the action of \(T\) and \(\sigma\) is an element of the symmetric group \(S_n\). They described the composition maps \(\text{Stab}^{-1}_{\:\sigma'}\circ\text{Stab}_{\:\sigma}\) in terms of the standard \(\mathfrak{gl}_2\) rational R-matrix and these R-matrices give a Yangian \(Y(\mathfrak{gl}_2)\)-module structure on \(\bigoplus_{k=0}^n H^*_T(T^*\!\text{Gr}_k(\mathbb{C}^n))\). In the paper under review, the authors constructed the dynamical analog of the stable envelope maps for the equivariant cohomology algebras of the cotangent bundles of Grassmannians. They extend the coefficients \(H^*_T((T^*\!\text{Gr}_k(\mathbb{C}^n))^T) \subset H^*_T((T^*\!\text{Gr}_k(\mathbb{C}^n))^T)'\) and \(H^*_T(T^*\!\text{Gr}_k(\mathbb{C}^n)) \subset H^*_T(T^*\!\text{Gr}_k(\mathbb{C}^n))'\) by suitable rational functions in a new variable \(\lambda\). The authors defined dynamical stable envelope maps NEWLINE\[NEWLINE \text{Stab}_\sigma : \bigoplus_{k=0}^n H^*_T((T^*\!\text{Gr}_k(\mathbb{C}^n))^T)' \to \bigoplus_{k=0}^n H^*_T(T^*\!\text{Gr}_k(\mathbb{C}^n)))', \,\, \sigma\in S_n. NEWLINE\]NEWLINE They described the composition maps \(\text{Stab}^{-1}_{\:\sigma'}\circ\text{Stab}_{\:\sigma}\) in terms of the rational dynamical R-matrix NEWLINE\[NEWLINE\begin{aligned} R(\lambda,z,y)= \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & \frac{(\lambda + y)z}{\lambda(z-y)} & -\frac{(\lambda + z)y}{\lambda(z-y)} &0 \\ 0 & -\frac{(\lambda - z)y}{\lambda(z-y)} & \frac{(\lambda - y)z}{\lambda(z-y)}&0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\end{aligned}\tag{RM}NEWLINE\]NEWLINE and defined on \(\bigoplus_{k=0}^n H^*_T(T^*\!\text{Gr}_k(\mathbb{C}^n))'\) a module structure over the rational dynamical quantum group \({E_y(\mathfrak{gl}_2)}\).