Rational Cherednik Algebras and Torus Knots

We show that the triply-graded Khovanov-Rozansky homology of the torus knot T_{m,n} can be recovered from the finite-dimensional representation L_{m/n} of the rational Cherednik algebra at slope m/n, endowed with the Hodge filtration coming from the cuspidal mirabolic D-module N_{m/n}. Our approach involves expressing the associated graded of N_{m/n} in terms of a DG module closely related to the action of the elliptic Hall algebra on the equivariant K-theory of the Hilbert scheme of points on the plane, thereby proving the rational master conjecture. In addition, we prove that the Hodge, inductive and algebraic filtrations on L_{m/n} coincide.