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

TL;DR

This paper introduces K-theoretic stable envelope maps for cotangent bundles of flag varieties, constructs a quantum loop algebra action, and describes the associated Bethe algebra, linking geometric representation theory with quantum integrable systems.

Contribution

It defines new K-theoretic stable envelope maps for flag varieties and establishes a quantum loop algebra action on their equivariant K-theory.

Findings

Bethe algebra coincides with multiplication operators in K-theory.

Gelfand-Zetlin algebra matches the limiting Bethe algebra.

Conjecture that Bethe algebra equals quantum multiplication algebra.

Abstract

We consider the cotangent bundle $T^*F_\lambda$ of a $GL_n$ partial flag variety, $\lambda=(\lambda_1,...,\lambda_N)$, $|\lambda|=\sum_i\lambda_i=n$, and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory algebra $K_T(T^*F_\lambda)$. We introduce K-theoretic stable envelope maps $\Stab_{\sigma}: \oplus_{|\lambda|=n} K_T((T^*F_\lambda)^T)\to\oplus_{|\lambda|=n}K_T(T^*F_\lambda)$, where $\sigma\in S_n$. Using these maps we define a quantum loop algebra action on $\oplus_{|\lambda|=n}K_T(T^*F_\lambda)$. We describe the associated Bethe algebra $B^q(K_T(T^*F_\lambda))$ by generators and relations in terms of a discrete Wronski map. We prove that the limiting Bethe algebra $B^q(K_T(T^*F_\lambda))$, called the Gelfand-Zetlin algebra, coincides with the algebra of multiplication operators of the algebra $K_T(T^*F_\lambda)$. We conjecture that the Bethe algebra $B^q(K_T(T^*F_\lambda))$ coincides with the algebra of quantum multiplication on $K_T(T^*F_\lambda)$ introduced by Givental and Lee. The stable envelope maps are defined with the help of Newton polygons of Laurent polynomials representing elements of $K_T(T^*F_\lambda)$ 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.