Elliptic and K-theoretic stable envelopes and Newton polytopes

Abstract

In this paper we consider the cotangent bundles of partial flag varieties. We construct the K-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the Ktheoretic stable envelopes and our elliptic stable envelopes. We show that the K-theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the gl 2 case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the K-theoretic stable envelopes. ⋆

Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

“…Weight functions first arise as integrands in the integral presentations of solutions to qKZ equations, associated with certain Yangians of type A [43,40,41,42,12,11,39]. For us, the weight functions here are the elliptic version introduced in [34].…”

“…Here we list some of them which will be used below. A more detailed exposition can be found in [33,34]. Let us set P I (z 1 , .…”

Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

“…Comparing these properties with those of the elliptic stable envelopes in [1], we conjectured that the elliptic weight functions can be identified with the elliptic stable envelopes. Some of similar but slightly different results were presented in [46]. The purpose of this paper is to formulate a geometric representation of the higher rank dynamical elliptic quantum group associated with sl N .…”

“…In [46], a similar formula for elliptic weight functions of type sl N and their triangularity and the orthogonality properties are presented without derivation. There the triangular property agrees with ours but the orthogonality property seems wrong due to a lack of the dynamical shift.…”

“…There the triangular property agrees with ours but the orthogonality property seems wrong due to a lack of the dynamical shift. There are also no formulas for the shuffle algebra in [46]. In addition, it seems that in [46] a different formulation of elliptic stable envelopes from the one in [1] is presented.…”