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

Generalized affine Springer theory and Hilbert schemes on planar curves