Elliptic classes of Schubert varieties via Bott-Samelson resolution

Abstract

Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety $G/B$. For this first we need to twist the notion of elliptic characteristic class of Borisov-Libgober by a line bundle, and thus allow the elliptic classes to depend on extra variables. Using the Bott-Samelson resolution of Schubert varieties we prove a BGG-type recursion for the elliptic classes, and study the Hecke algebra of our elliptic BGG operators. For $G=G{L}_n(C)$ we find representatives of the elliptic classes of Schubert varieties in natural presentations of the K theory ring of $G/B$, and identify them with the Tarasov-Varchenko weight function. As a byproduct we find another recursion, different from the known R-matrix recursion for the fixed point restrictions of weight functions. On the other hand the R-matrix recursion generalizes for arbitrary reductive group $G$.

the paper has been accepted for publication by the Journal of Topology; this version contains minor corrections

References in corpus (8)

• Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
• Motivic Chern classes and K-theoretic stable envelopes
• Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology
• Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety
• Chern-Schwartz-MacPherson classes of degeneracy loci
• Elliptic quantum groups and their finite-dimensional representations
• Elliptic Weight Functions and Elliptic q-KZ Equation
• Elliptic genus of singular algebraic varieties and quotients

Cited by in corpus (1)

• Geometric properties of the Kazhdan-Lusztig Schubert basis