I will discuss how, via the corresponding generating series, the conjectures with E. Letellier and T. Hausel on the mixed Hodge structure of certain character varieties are related to the Frobenius series of appropriate modules in the theory of the Hilbert scheme of points on $\mathbb{C}^2$. These character varieties are themselves closely related to the moduli space of Higss bundles on a curve. In particular, the qt-Catalan polynomials of Garsia-Haiman have a natural formulation in our context. In fact, there appears to be natural a dictionary between the two theories translating questions of interest in one into the other