**Next:**Askold Khovansky - Algebraic

**Up:**Algebraic Geometry / Géométrie

**Previous:**Andrew Hwang - Construction

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

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 *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*).

**Next:**Askold Khovansky - Algebraic

**Up:**Algebraic Geometry / Géométrie

**Previous:**Andrew Hwang - Construction