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In this paper we consider the cotangent bundles of partial flag varieties. Weconstruct the $K$-theoretic stable envelopes for them and also define a versionof the elliptic stable envelopes. We expect that our elliptic stable envelopescoincide with the elliptic stable envelopes defined by M. Aganagic and A.Okounkov. We give formulas for the $K$-theoretic stable envelopes and ourelliptic stable envelopes. We show that the $K$-theoretic stable envelopes aresuitable limits of our elliptic stable envelopes. That phenomenon was predictedby M. Aganagic and A. Okounkov. Our stable envelopes are constructed in termsof the elliptic and trigonometric weight functions which originally appeared inthe theory of integral representations of solutions of qKZ equations twentyyears ago. (More precisely, the elliptic weight functions had appeared earlieronly for the $\frak{gl}_2$ case.) We prove new properties of the trigonometricweight functions. Namely, we consider certain evaluations of the trigonometricweight functions, which are multivariable Laurent polynomials, and show thatthe Newton polytopes of the evaluations are embedded in the Newton polytopes ofthe corresponding diagonal evaluations. That property implies the fact that thetrigonometric weight functions project to the $K$-theoretic stable envelopes.

Elliptic and K-theoretic stable envelopes and Newton polytopes