# Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology

## TL;DR

This paper introduces an elliptic stable envelope for equivariant elliptic cohomology of cotangent bundles of Grassmannians, connecting elliptic quantum groups with geometric representation theory.

## Contribution

It constructs an elliptic stable envelope based on weight functions and shuffle products, and defines an action of the dynamical elliptic quantum group on these cohomologies.

## Key findings

- Defined an elliptic stable envelope for equivariant elliptic cohomology.
- Constructed an action of the elliptic quantum group on cohomology sections.
- Introduced the notion of admissible bundles and difference operators.

## Abstract

We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on weight functions and shuffle products. We construct an action of the dynamical elliptic quantum group associated with $\mathfrak{gl}_2$ on the equivariant elliptic cohomology of the union of cotangent bundles of Grassmannians. The generators of the elliptic quantum groups act as difference operators on sections of admissible bundles, a notion introduced in this paper.