Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

Abstract

We consider the cotangent bundle $T^*F_λ$ of a $GL_n$ partial flag variety, $λ=(λ_1,...,λ_N)$, $|λ|=\sum_iλ_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^*F_λ)$. We introduce K-theoretic stable envelope maps $\Stab_σ: \oplus_{|λ|=n} K_T((T^*F_λ)^T)\to\oplus_{|λ|=n}K_T(T^*F_λ)$, where $σ\in S_n$. Using these maps we define a quantum loop algebra action on $\oplus_{|λ|=n}K_T(T^*F_λ)$. We describe the associated Bethe algebra $B^q(K_T(T^*F_λ))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^*F_λ))$, called the Gelfand-Zetlin algebra, coincides with the algebra of multiplication operators of the algebra $K_T(T^*F_λ)$. We conjecture that the Bethe algebra $B^q(K_T(T^*F_λ))$ coincides with the algebra of quantum multiplication on $K_T(T^*F_λ)$ introduced by Givental and Lee. The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of $K_T(T^*F_λ)$ and with the help of the trigonometric weight functions introduced in [TV1, TV3] to construct q-hypergeometric solutions of trigonometric qKZ equations. The paper has five appendices. In particular, in Appendix 5 we describe the Bethe algebra of the XXZ model by generators and relations.