Gorsky–Neguț quantum affine action conjecture for Hilbert-scheme stable bases

Let $K=\bigoplus_{n=0}^{\infty}K_{\mathbb{C}^*\times\mathbb{C}^*}(\operatorname{Hilb}_n)$, let $m=\frac{a}{b}$ with $\gcd(a,b)=1$, and let $\mathcal{A}^{(m)}$ be the slope-$m$ Heisenberg subalgebra of the spherical double affine Hecke algebra acting on $K$. Let $s^{m-\varepsilon}$ and $s^{m+\varepsilon}$ denote the stable bases on the two sides of the wall. Gorsky–Neguț quantum affine action conjecture. There exists an action $U_q\widehat{\mathfrak{gl}}_b\curvearrowright K$ such that $K$ is a level-1 vacuum module, isomorphic to the Fock space; $\mathcal{A}^{(m)}$ embeds as the standard diagonal $q$-Heisenberg subalgebra and this embedding intertwines its known action on $K$; and $s^{m-\varepsilon}$ and $s^{m+\varepsilon}$ are respectively the standard and costandard bases for this action, up to renormalization. This gives a representation-theoretic explanation of wall crossing and extends the Leclerc–Thibon description; the conjecture remains open in the source.