The Verlinde Formula for Parabolic Bundles

Let Σ be a compact Riemann surface of genus g, and G ̄SU(n). The central element c ̄ diag(e#id/n,... , e#id/n) for d coprime to n is introduced. The Verlinde formula is proved for the Riemann–Roch number of a line bundle over the moduli space g," (c,Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component, for which the loop around the boundary is constrained to lie in the conjugacy class of c exp(Λ) (for Λ ` t + ), and also for the moduli space g,b (c,Λ) of representations of the fundamental group of a Riemann surface of genus g with s­1 boundary components for which the loop around the 0th boundary component is sent to the central element c and the loop around the jth boundary component is constrained to lie in the conjugacy class of exp(Λ( j)) for Λ( j) ` t + . The proof is valid for Λ( j) in suitable neighbourhoods of 0.