Equivariant Index of Twisted Dirac Operators and Semi-classical Limits

  • Paul-`Emile Paradan
  • Mich`ele Vergne
Part of the Progress in Mathematics book series (PM, volume 326)


Let G be a compact connected Lie group with Lie algebra g. Let M be a compact spin manifold with a G-action, and 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 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.


Dirac operator equivariant index semi-classical limits 

Mathematics Subject Classification (2010):

57S15 53C27 58J20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.IMAG, Universit`e de Montpellier, CNRSMontpellierFrance
  2. 2.IMJ-PRG, Université Paris 7, CNRSParisFrance

Personalised recommendations