Character and multiplicity formulas for compact Hamiltonian G-spaces

Abstract

Let K ⊂ G be compact connected Lie groups with common maximal torus T . Let ( M , ω ) 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 χ 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 λ a dominant integral weight of K , gives the multiplicity in χ of the irreducible representation of K of highest weight λ .