Generalized affine Springer theory and Hilbert schemes on planar curves

We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine Springer theory for Braverman-Finkelberg-Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes, and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn-Kamnitzer-Weekes, which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of Hogancamp-Mellit, we also show the rational Cherednik algebra acts on the HOMFLY homologies of torus knots. This work was inspired in part by a construction in three-dimensional $\mathcal{N}=4$ gauge theory.