  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 $m = \frac ab$ 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.
