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, ℒk) be the equivariant index of the Dirac operator on M twisted by ℒk. Let mG(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 ⟨Θk,φ⟩ = kdim(G/T)/2 Σλ mG(λ,k)⟨βλ/k,φ⟩ has an asymptotic expansion when k tends to infinity of the form ⟨Θk,φ⟨ ≡ kdimM/2\( \sum\nolimits_{n = 0}^\infty {{k^{ - n}}} \left\langle {{\theta _n},\phi } \right\rangle \). Here φ is a test function on g* and ⟨βξ,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.
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Paradan, P`., Vergne, M. (2018). Equivariant Index of Twisted Dirac Operators and Semi-classical Limits. In: Kac, V., Popov, V. (eds) Lie Groups, Geometry, and Representation Theory. Progress in Mathematics, vol 326. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-02191-7_15
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DOI: https://doi.org/10.1007/978-3-030-02191-7_15
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-02190-0
Online ISBN: 978-3-030-02191-7
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