The multiplicities of the equivariant index of twisted Dirac operators
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 generality the quantisation-commutes-with-reduction principle holds. It was proved there in the Spin\(^c\) case for circle actions. The result by Paradan and Vergne in this paper shows that quantisation-commutes-with-reduction is a general principle in equivariant index theory of Spin\(^c\)-Dirac operators, and not just in geometric quantisation in the traditional sense. This leads to a broad scope for applications, for example to almost complex and Spin-manifolds. In the Spin case, the result turns out to imply Atiyah-Hirzebruch's famous vanishing result [\textit{M.\ Atiyah} and \textit{F.\ Hirzebruch}, Essays Topol. Relat. Top., Mém. dédiés à Georges de Rham 18--28 (1970; Zbl 0193.52401)].
scientific article; zbMATH DE number 6361365
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