Purity of generalized affine Springer fibers from generic planar curve singularities

We prove the cohomological purity of punctual Hilbert schemes of points on generic irreducible planar curve singularities, by constructing an explicit affine paving. Via their identification with generalized $GL_N$-affine Springer fibers attached to the direct sum of the adjoint and standard representations, this establishes a new case of the purity conjecture for generalized affine Springer fibers. The combinatorics of the paving - cell indices and dimensions - are controlled by $(dn,dm)$-Dyck paths extending results of Gorsky-Mazin-Oblomkov on compactified Jacobians. As a byproduct, we also give a simpler proof of their bijection between admissible $(dn,dm)$-invariant subsets and $(dn,dm)$-Dyck paths.