The Oblomkov-Rasmussen-Shende conjecture, combined with a number of other results, suggests that orbital integrals of regular semisimple elements for p-adic GL(n) are equal to Poincaré polynomials of HOMFLY homologies of algebraic links (or "knot superpolynomials"), at least in large enough residual characteristic and after ignoring some of the gradings. Certain results on the harmonic analysis side, such as the Shalika germ expansion, seem to translate to interesting facts on the knot homology side, and vice versa. I will explain the state of the art, including known results and a possible explanation through 3d mirror symmetry for Hilbert schemes.