Character and Multiplicity Formulas for Compact Hamiltonian G-spaces

TL;DR

This paper derives formulas for the character and multiplicities of the geometric quantization of compact Hamiltonian G-spaces, generalizing known formulas and providing explicit character quotient expressions.

Contribution

It introduces a new formula for the character of the quantization representation as a quotient of virtual characters of K, extending previous results to a broader setting.

Findings

Derived a character formula as a quotient of virtual characters of K

Generalized the Guillemin-Prato multiplicity formula

Confirmed agreement with the Gross-Kostant-Ramond-Sternberg formula for coadjoint orbits

Abstract

Let K $\subset$ G be compact connected Lie groups with common maximal torus T. Let (M, $\omega$) be a prequantisable compact connected symplectic manifold with a Hamiltonian G-action. Geometric quantisation gives a virtual representation of G; we give a formula for the character $\chi$ of this virtual representation as a quotient of virtual characters of K. When M is a generic coadjoint orbit our formula agrees with the Gross-Kostant-Ramond-Sternberg formula. We then derive a generalisation of the Guillemin-Prato multiplicity formula which, for $\lambda$ a dominant integral weight of K, gives the multiplicity in $\chi$ of the irreducible representation of K of highest weight $\lambda$.