
LISA JEFFREY, Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada 
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 nth 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).