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

Abstract

The modified Macdonald functions ${\tilde{H}}_{\mu }$ are fundamental objects in modern algebraic combinatorics. Haiman showed that there is a correspondence between the $({\mathbb{C}}^{\ast }{)}^2$-fixed points $I_{\mu }$ of the Hilbert schemes ${\mathrm{Hilb}}_n({\mathbb{C}}^2)$ and the functions ${\tilde{H}}_{\mu }$ realizing a derived equivalence between $({\mathbb{C}}^{\ast }{)}^2$-equivariant coherent sheaves on ${\mathrm{Hilb}}_n({\mathbb{C}}^2)$ and $({\mathfrak{S}}_n\times ({\mathbb{C}}^{\ast }{)}^2)$-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 $({\mathbb{C}}^{\ast }{)}^2$-equivariant K-theory $U_{\bullet }$ of these spaces and, moreover, there is a ${\mathbb{B}}_{q,t}$-isomorphism $\Phi :U_{\bullet }\to V_{\bullet }$ where $V_{\bullet }$ is the polynomial representation. The isomorphism $\Phi :U_{\bullet }\to V_{\bullet }$ is known to extend Haiman's correspondence. In this paper, we explicitly compute the images $\Phi (H_{\mu ,w})$ of the normalized $({\mathbb{C}}^{\ast }{)}^2$-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 ${\tilde{H}}_{(\lambda |\gamma )}$ 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 $V_{\bullet }.$