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Elliptic and K-theoretic stable envelopes and Newton polytopes

  • R. Rimányi
  • V. Tarasov
  • A. VarchenkoEmail author


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 \(K\)-theoretic 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 \({\mathfrak {gl}}_2\) 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.

Mathematics Subject Classification

55N34 14M15 17B37 



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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA
  2. 2.Faculty of Mathematics and MechanicsLomonosov Moscow State UniversityMoscow GSP-1Russia
  3. 3.Department of Mathematical SciencesIndiana University – Purdue University IndianapolisIndianapolisUSA
  4. 4.St. Petersburg Branch of Steklov Mathematical InstituteSt. PetersburgRussia

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