Yangians and quantizations of slices in the affine Grassmannian

TL;DR

This paper introduces a new class of shifted Yangians for arbitrary types to quantize slices in the affine Grassmannian, connecting quantum groups with geometric structures like Zastava spaces.

Contribution

It generalizes the Brundan-Kleshchev shifted Yangian to arbitrary types and links these to the quantization of slices and Zastava spaces in the affine Grassmannian.

Findings

Constructed shifted Yangians for arbitrary types.

Proved a quotient of the shifted Yangian quantizes a scheme on slices.

Formulated a conjecture on the ideal defining these slices.

Abstract

We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians --- these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for PGL(n) of Finkelberg-Rybnykov.