Character and multiplicity formulas for compact Hamiltonian GG-spaces

Let K⊂GK⊂G be compact connected Lie groups with common maximal torus TT. Let (M,ω)(M,ω) be a prequantisable compact connected symplectic manifold with a Hamiltonian GG-action. Geometric quantisation gives a virtual representation of GG; we give a formula for the character χχ of this virtual representation as a quotient of virtual characters of KK. When MM 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 λλ a dominant integral weight of KK, gives the multiplicity in χχ of the irreducible representation of KK of highest weight λλ.