The Combinatorics of the Verlinde Formulas

We describe V G k for general groups in §2. These formulas were first written down by E. Verlinde [23] in the context of Conformal Field theory. The interest towards them in Algebraic Geometry stems form the fact that they give the Hilbert function of moduli spaces of principle bundles over projective curves. More precisely, let C be a smooth projective curve of genus g, and let MC be the moduli space of principal G-bundles over C (cf. e.g. [16] and references therein). Then there is an ample line bundle L over MC such that dimH(MC ,L ) = V G k+h(g), (1.2)