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

The modified Macdonald functions are fundamental objects in modern algebraic combinatorics. Haiman showed that there is a correspondence between the ‐fixed points of the Hilbert schemes and the functions realizing a derived equivalence between ‐equivariant coherent sheaves on and ‐equivariant coherent sheaves on . Carlsson–Gorsky–Mellit introduced a larger family of smooth varieties called the parabolic flag Hilbert schemes. They showed that an algebra , directly related to the double Dyck path algebra employed in Carlsson–Mellit's proof of the Shuffle Theorem, acts naturally on the ‐equivariant K‐theory of these spaces, and moreover, there is a ‐isomorphism where is the polynomial representation. The isomorphism is known to extend Haiman's correspondence. In this paper, we explicitly compute the images of the normalized ‐fixed point classes of the spaces and show that they agree with the modified partially symmetric Macdonald polynomials introduced by Goodberry–Orr, confirming their prior conjecture. We use this result to give an explicit formula for the action of the involution on .