Infinitesimal change of stable basis

Abstract

The purpose of this note is to study the Maulik–Okounkov K-theoretic stable basis for the Hilbert scheme of points on the plane, which depends on a “slope” $$m \in {\mathbb {R}}$$ . When $$m = \frac{a}{b}$$ is rational, we study the change of stable matrix from slope $$m-\varepsilon $$ to $$m+\varepsilon $$ for small $$\varepsilon >0$$ , and conjecture that it is related to the Leclerc–Thibon conjugation in the q-Fock space for $$U_q\widehat{{\mathfrak {gl}}}_b$$ . This is part of a wide framework of connections involving derived categories of quantized Hilbert schemes, modules for rational Cherednik algebras and Hecke algebras at roots of unity.