Stable-limit partially symmetric Macdonald functions and parabolic flag Hilbert schemes

The modified Macdonald functions $\widetilde{H}_{\mu}$ are fundamental objects in modern algebraic combinatorics. Haiman showed that there is a correspondence between the $(\mathbb{C}^{*})^2$-fixed points $I_{\mu}$ of the Hilbert schemes $\mathrm{Hilb}_{n}(\mathbb{C}^2)$ and the functions $\widetilde{H}_{\mu}$ realizing a derived equivalence between $(\mathbb{C}^{*})^2$-equivariant coherent sheaves on $\mathrm{Hilb}_{n}(\mathbb{C}^2)$ and $(\mathfrak{S}_n \times (\mathbb{C}^{*})^2)$-equivariant coherent sheaves on $(\mathbb{C}^2)^n.$ Carlsson--Gorsky--Mellit introduced a larger family of smooth projective varieties $\mathrm{PFH}_{n,n-k}$ called the parabolic flag Hilbert schemes. They showed that an algebra $\mathbb{B}_{q,t}$, directly related to the double Dyck path algebra $\mathbb{A}_{q,t}$ employed in Carlsson--Mellit's proof of the Shuffle Theorem, acts naturally on the $(\mathbb{C}^{*})^2$-equivariant K-theory $U_{\bullet}$ of these spaces and, moreover, there is a $\mathbb{B}_{q,t}$-isomorphism $\Phi: U_{\bullet} \rightarrow V_{\bullet}$ where $V_{\bullet}$ is the polynomial representation. The isomorphism $\Phi: U_{\bullet} \rightarrow V_{\bullet}$ is known to extend Haiman's correspondence. In this paper, we explicitly compute the images $\Phi(H_{\mu,w})$ of the normalized $(\mathbb{C}^{*})^2$-fixed point classes $H_{\mu,w}$ of the spaces $\mathrm{PFH}_{n,n-k}$ and show they agree with the modified partially symmetric Macdonald polynomials $\widetilde{H}_{(\lambda|\gamma)}$ introduced by Goodberry-Orr, confirming their prior conjecture. We use this result to give an explicit formula for the action of the involution $\mathcal{N}$ on $V_{\bullet}.$