Lisa Jeffrey - The Verlinde formula for moduli spaces of parabolic bundles

The moduli space M(n,d) is an algebraic variety parametrizing those representations of the fundamental group of a punctured Riemann surface into the Lie group SU(n) for which a loop around the boundary is sent to a particular n-th root of unity multiplied by the identity matrix. If n and d are coprime it is in fact a Kaehler manifold. One may relax the constraint and study moduli spaces M(a)parametrizing those representations for which the loop around the boundary is sent to an element conjugate to a, if a is some element in SU(n), and these are also Kaehler manifolds for a suitable class of a.

The Verlinde formula calculates the dimension of the space of holomorphic sections of certain line bundles over the spaces M(n,d)and M(a): these dimensions are in a sense the dimensions of the quantizations of these spaces. We recall how a new proof of the Verlinde formula for M(n,d), given in joint work with F. Kirwan (Ann. Math., 1998) may be obtained, and show how to modify this proof to obtain a proof of the variant of the Verlinde formula which applies to M(a).