Partially Symmetric Macdonald Theory

The modified Macdonald functions play an important role in the study of the K-theory of the Hilbert schemes of points in the plane. Haiman showed in his seminal work on the Macdonald positivity conjectures that the modified Macdonald functions correspond to the torus fixed point classes of the Hilbert schemes by the way of a derived equivalence. Carlsson-Gorsky-Mellit introduced a larger family of schemes called the parabolic flag Hilbert schemes related to Carlsson-Mellit's prove of the Shuffle Theorem. In this talk, I will discuss the partially-symmetric generalization of modified Macdonald polynomials, their relation to the K-theory of the parabolic flag Hilbert schemes, and their interactions with the stable-limit double affine Hecke algebra of Ion-Wu. This is joint work with Daniel Orr.