# Elliptic classes of Schubert varieties via Bott-Samelson resolution

## TL;DR

This paper introduces a new elliptic Schubert calculus approach using Bott-Samelson resolutions, extending elliptic characteristic classes with line bundle twists, and connects these to Hecke algebras and weight functions.

## Contribution

It develops a novel method for elliptic Schubert calculus by twisting elliptic classes, establishing a BGG-type recursion, and linking to Tarasov-Varchenko weight functions.

## Key findings

- Proves a BGG-type recursion for elliptic classes of Schubert varieties.
- Identifies elliptic classes with Tarasov-Varchenko weight functions for GL_n(C).
- Derives a new recursion different from the R-matrix recursion.

## 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 $G=GL_n(C)$ 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 $G$.