Equivariant Index of Twisted Dirac Operators and Semi-classical Limits

Abstract

Let G be a compact connected Lie group with Lie algebra g. Let M be a compact spin manifold with a G-action, and ℒ be a G-equivariant line bundle on M. Consider an integer k, and let $\vartheta _G^{{\rm{spin}}}$ (M, ℒ^k) be the equivariant index of the Dirac operator on M twisted by ℒ^k. Let m_G(λ, k) be the multiplicity in $\vartheta _G^{{\rm{spin}}}$ (M, ℒk) of the irreducible representation of G attached to the admissible coadjoint orbit Gλ. We prove that the distribution ⟨Θ_k,φ⟩ = k^dim(G/T)/2 Σλ mG(λ,k)⟨β_λ/k,φ⟩ has an asymptotic expansion when k tends to infinity of the form ⟨Θ_k,φ⟨ ≡ k^dimM/2$\sum\nolimits_{n = 0}^\infty {{k^{ - n}}} \left\langle {{\theta _n},\phi } \right\rangle$. Here φ is a test function on g* and ⟨β_ξ,φ⟩ is the integral of φ on the coadjoint orbit Gφ with respect to the canonical Liouville measure. We compute explicitly the distribution θn in terms of the graded  class of M and the equivariant curvature of ℒ.

If M is noncompact, we use these asymptotic techniques to give another proof of the fact that the formal geometric quantization of a manifold with a spinc structure is functorial with respect to restriction to subgroups.