The equivariant index and Kirillov’s character formula. (English) Zbl 0604.58046

Berline, Nicole; Vergne, Michele

Am. J. Math. 107, 1159-1190 (1985).

Let M be a compact, oriented Riemannian manifold of even dimension, G a connected group of isometries of M. Let V be a G-equivariant Clifford module over M, \(D=D^ 0_ V\) the Dirac operator. The ”equivariant index formula” mentioned in the title states that for sufficiently small \(X\in g\), \[ tr_{Ker D}(\exp X)-tr_{Co\ker D}(\exp)=\int_{M}Ch(X,W) J^{-1/2} (X,TM). \] Here Ch(X,W) is an equivariant Chern character constructed using an auxiliary bundle W, \(J^{-1/2}(X,TM)\) is associated to the function \(\det^{1/2}(e^{A/2}-e^{-A/2}/A)\) by the equivariant Chern-Weil homomorphism. In case \(X=0\), this formula is the Atiyah-Singer index formula.
The proof of the theorem relies on a ”localization theorem” in equivariant cohomology, interesting in its own right, which is closely related to a classical result of R. Bott [Mich. Math. J. 14, 231- 244 (1967; Zbl 0145.438)], to results of J. J. Duistermaat and G. J. Heckman [Invent. Math. 69, 259-268 (1982; Zbl 0503.58015); ibid. 72, 153-158 (1983; Zbl 0503.58016] and to results of M. F. Atiyah and R. Bott [Topology 23, 1-28 (1984; Zbl 0521.58025)].
As an application, the authors derive a Kirillov-type character formula for compact groups and indicate its extension to the discrete series of non-compact groups.

Reviewer: W.Rossmann

MSC:

58J22	Exotic index theories on manifolds
58J70	Invariance and symmetry properties for PDEs on manifolds
55R05	Fiber spaces in algebraic topology
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods

Keywords:

Dirac operator; equivariant index formula; Chern character; character formula

Citations:

Zbl 0145.438; Zbl 0503.58015; Zbl 0503.58016; Zbl 0521.58025