Multiplicity Formulas for Orbifolds Multiplicity Formulas for Orbifolds

Given a symplectic space, equipped with a line bundle and a Hamiltonian group action satisfying certain compatibility conditions, it is a basic question to understand the decomposition of the quantization space in irreducible representations of the group. We derive weight multiplicity formulas for the quantization space in terms of data at the fixed points on the symplectic space, which apply to general situations when the underlying symplectic space is allowed to be an orbifold, the group acting is a compact connected semi-simple Lie group, and the fixed points of that action are not necessarily isolated. Our formulas extend the celebrated Kostant multiplicity formulas. Moreover, we show that in the semi-classical limit our formulas converge to the Duistermaat-Heckman measure, that is the push-forward of Lebesgue measure by the moment map. Thesis Supervisor: Victor W. Guillemin Title: Professor of Mathematics