Hall polynomials are certain deformations of the irreducible characters of a semisimple algebraic group. One can extend their definition to the case of affine root systems, getting some analytic functions (no longer polynomials). It turns out that these functions have an algebro-geometric interpretation, as certain generating functions of Hodge polynomials of the moduli spaces of G-bundles on ruled algebraic surfaces. The talk will explain this interpretation as well as the analogy between affine Hall "polynomials" and Eisenstein series.