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

SIGMA 14 (2018), 132, 41 pages      arXiv:1702.08060
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

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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