Let (M, ω) be a symplectic manifold and τ. G × M → M a Hamiltonian action on M of a compact Lie group G. We will assume that this action can be quantized in the sense that by putting some additional structure on M (a spin structure, a polarization, and so on) one can fabricate from τ a unitary representation of G on a Hubert space (or “virtual” Hilbert space) Q(M). I won’t be too concerned in this monograph about how to do this, but instead will take the principle of “invariance of polarization” to mean that all methods of quantization give the same answer. Granting this assumption, one can view the multiplicities with which the irreducible representations of G occur in Q(M) as symplectic invariants of the triple (M, ω, τ). The purpose of the following chapters will be to discuss how to compute these multiplicities. Two methods are known for doing this. The first consists of identifying the irreducible representations ρ of G with the coadjoint orbits O = Gα, where α is the highest weight of the representation ρ. The multiplicity with which ρ occurs in Q(M) can then be expressed as a Riemann-Roch invariant of the reduced space M O . (The details will be explained in the third chapter.) To describe the second method, let’s assume for simplicity that G is Abelian, M is compact, and the fixed point set of τ, M G , is finite. Under these hypotheses, the multiplicity with which the one-dimensional representation ρ of G with weight a occurs in Q(M) can be expressed as a sum. ∑(-1)wp Np(α) p∈MG, where N p is a “partition function” associated with the isotropy representation of G at p and w p is an orientation index. (For the precise definitions, see Chapter 4.)
KeywordsManifold Editing Kirwan
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