The multiplicities of the equivariant index of twisted Dirac operators
type: Item
instance of: scholarly article
title: The multiplicities of the equivariant index of twisted Dirac operators
zbMATH Open document ID: 1319.53045
author: Michèle Vergne
author: Paul-Émile Paradan
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1404.2003
reviewed by: Peter Hochs
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 53D50
zbMATH DE Number: 6361365
zbMATH Keywords: \(\mathrm{Spin}^c\) Dirac operators
zbMATH Keywords: geometric quantisation
zbMATH Keywords: index theory
MaRDI profile type: Publication
OpenAlex ID: W2964256232
arXiv ID: 1404.2003
cites work: Quantization of presymplectic manifolds and circle actions
cites work: Q3672192
cites work: Bott towers, complete integrability, and the extended character of representations
cites work: Equivariant index and the moment map for completely integrable torus actions
cites work: Geometric quantization and multiplicities of group representations
cites work: The moment map and line bundles over presymplectic toric manifolds
cites work: Symplectic surgery and the Spin\(^c\)-Dirac operator
cites work: Singular reduction and quantization
cites work: Localization of the Riemann-Roch character
cites work: Spin-quantization commutes with reduction
cites work: An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg
DOI: 10.1016/J.CRMA.2014.05.001
Recommended article: Quantising proper actions on \(\text{Spin}^c\)-manifolds
Recommended article: \(K\)-theory and the quantization commutes with reduction problem
Recommended article: Equivariant indices of \(\operatorname{Spin}^c\)-Dirac operators for proper moment maps
Recommended article: An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg
Recommended article: On the Vergne conjecture
Recommended article: Localization of the Riemann-Roch character
Recommended article: Quantization of presymplectic manifolds and circle actions
Recommended article: Q5494153
Recommended article: Symplectic surgery and the Spin\(^c\)-Dirac operator
Recommended article: Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups
review text: The quantisation-commutes-with-reduction principle of \textit{V.\ Guillemin} and \textit{S.\ Sternberg} [Invent. Math. 67, 515--538 (1982; Zbl 0503.58018)] was first proved for compact Lie groups acting on compact symplectic manifolds by \textit{E.\ Meinrenken} [Adv. Math. 134, No. 2, 240--277 (1998; Zbl 0929.53045)] and \textit{E.\ Meinrenken} and \textit{R.\ Sjamaar} [Topology 38, No.4, 699--762 (1999; Zbl 0928.37013)]. In this note, the authors announce an extension of this principle from symplectic to Spin\(^c\) manifolds. NEWLINENEWLINENEWLINEChallenges the authors need to overcome include: {\parindent=0.5cm\begin{itemize}\item[1.] defining a suitable analogue or generalisation of the moment map from symplectic geometry in the Spin\(^c\) case;\item[2.] defining the quantisation of reduced spaces for such moment maps; \item[3.] dealing with the fact that different (admissible) coadjoint orbits may have the same irreducible representation as their quantisations. NEWLINENEWLINE\end{itemize}} The authors solve the first problem by applying Kostant's formula to a connection on the determinant line bundle of a given Spin\(^c\)-structure. They solve the second issue by first considering torus actions. Then reduced spaces at regular values of the moment map are Spin\(^c\)-orbifolds, and their quantisation can be defined as the index of a Spin\(^c\)-Dirac operator. This definition can be extended to singular values. For nonabelian groups, the authors reduce the problem to the torus case using slices.NEWLINENEWLINENEWLINEBecause of the third point, the multiplicity of an irreducible representation in the equivariant index of a Spin\(^c\)-Dirac operator is a finite sum of quantisations of reduced spaces in general. Under certain assumptions, such as the generic stabiliser being abelian, this sum reduces to one term. NEWLINENEWLINENEWLINEIn [\textit{A. Cannas da Silva} et al., Trans. Am. Math. Soc. 352, No. 2, 525-552 (2000; Zbl 0977.53080)], the question was asked in what…
description: scientific article; zbMATH DE number 6361365