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We consider the cotangent bundle $T^*F_\lambda$ of a $GL_n$ partial flagvariety, $\lambda=(\lambda_1,...,\lambda_N)$, $|\lambda|=\sum_i\lambda_i=n$,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 Bethealgebra $B^q(K_T(T^*F_\lambda))$ by generators and relations in terms of adiscrete Wronski map. We prove that the limiting Bethe algebra$B^q(K_T(T^*F_\lambda))$, called the Gelfand-Zetlin algebra, coincides with thealgebra of multiplication operators of the algebra $K_T(T^*F_\lambda)$. Weconjecture that the Bethe algebra $B^q(K_T(T^*F_\lambda))$ coincides with thealgebra of quantum multiplication on $K_T(T^*F_\lambda)$ introduced by Giventaland Lee.  The stable envelope maps are defined with the help of Newton polygons ofLaurent polynomials representing elements of $K_T(T^*F_\lambda)$ and with thehelp of the trigonometric weight functions introduced in [TV1, TV3] toconstruct q-hypergeometric solutions of trigonometric qKZ equations.  The paper has five appendices. In particular, in Appendix 5 we describe theBethe algebra of the XXZ model by generators and relations.

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