The affine Springer fibers from the title are homeomorphic to the compactified Jacobians JC_(m,n) of curves x^m = y^n, gcd(m,n) = 1. In elementary terms, JC_(m,n) is the space of subspaces L \subset [[t]] of codimension (m-1)(n-1) that are preserved by multiplication on t^m and t^n. Together with Zhiwei Yun we described an action of the spherical rational Cherednik algebra eH_(m/n)(S_n)e on H^*(JC_(m,n)) and the ring structure of the cohomology. The ring structure is very similar to the ring structure of the finite dimensional Grassmannians, and in my talk I will discuss this analogy and connections with q,t Catalan numbers.