Haglund--Haiman--Loehr formula via Carlsson--Mellit Algebra

We prove a Haglund–Haiman–Loehr type combinatorial formula for the torus fixed point classes $I_{\mu ,w}$ in the equivariant $K$-theory $K_{{\mathbb{C}}^{\ast }\times {\mathbb{C}}^{\ast }}({\mathrm{PFH}}_{n,n-k})$ of parabolic flag Hilbert schemes. Under the identification of Bechtloff Weising and Orr, our result gives an HHL type formula for modified partially symmetric Macdonald functions in terms of weighted partial Dyck paths. When $w=\mathrm{\varnothing }$, it specializes to the classical HHL formula. The proof relies essentially on the Carlsson–Gorsky–Mellit action of the Carlsson–Mellit algebra ${\mathbb{A}}_{q,t}$.