Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 132, 41 pages      arXiv:1702.08060      https://doi.org/10.3842/SIGMA.2018.132
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology

Giovanni Felder ^a, Richárd Rimányi ^b and Alexander Varchenko ^b
a) Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland
^b) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Received April 30, 2018, in final form December 12, 2018; Published online December 21, 2018

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.

Key words: elliptic cohomology; elliptic quantum group; elliptic stable envelope.

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