Pursuing quantum difference equations I: stable envelopes of subvarieties

TL;DR

This paper explores how K-theoretic stable envelopes of subvarieties within symplectic varieties can be derived from elliptic stable envelopes through limits, with detailed analysis on the Hilbert scheme of points.

Contribution

It introduces a method to obtain K-theoretic stable envelopes from elliptic ones via limits, specifically for subvarieties fixed by cyclic subgroups in symplectic varieties.

Findings

K-theoretic stable envelopes can be derived from elliptic stable envelopes via limits.

Application to Hilbert schemes of points in the complex plane.

Provides detailed example and methodology for subvarieties fixed by cyclic subgroups.

Abstract

Let $X$ be a symplectic variety equipped with an action of a torus $A$. Let $\nu \subset A$ be a finite cyclic subgroup. We show that K-theoretic stable envelope of subvarieties $X^{\nu}\subset X$ can be obtained via various limits of the elliptic stable envelopes of $X$. An example of $X$ given by the Hilbert scheme of points in the complex plane is considered in details.