Stable envelopes for slices of the affine Grassmannian

TL;DR

This paper investigates the cohomological stable envelopes of the affine Grassmannian, providing explicit recursive formulas for specific cases and an exact divisor multiplication formula, advancing understanding of these geometric structures.

Contribution

It introduces explicit recursive relations for stable envelopes in the affine Grassmannian, particularly for G=PSL_2, and computes the first-order correction for the general case.

Findings

Derived recursive relations for stable envelopes in the PSL_2 case

Computed the first-order correction in the general case

Obtained an exact formula for divisor multiplication

Abstract

The affine Grassmannian associated to a reductive group $\mathbf{G}$ is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. We study the cohomological stable envelopes of D. Maulik and A. Okounkov [arXiv:1211.1287] in this family. We construct an explicit recursive relation for the stable envelopes in the $\mathbf{G} = \mathbf{PSL}_2$ case and compute the first-order correction in the general case. This allows us to write an exact formula for multiplication by a divisor.