An Equivariant Index Formula for Almost CR Manifolds

Abstract

We consider a consider the case of a compact manifold M , together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J (E, X) whose properties we describe.When E is equipped with a complex structure, we define a class of symbol mappings σ in terms of the resulting almost-CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only J (E, X) and standard equivariant characteristic classes. This formula generalizes the formula given in [9] for the case of a contact manifold.

On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

“…In [Fit09b] we concentrated mainly on the case of almost CR structures. Recall (see [Bog91] or [DT06], for example) that an almost CR structure on a manifold M is a constant rank…”

“…Given the action of a group G on M, transverse to a subbundle E, we may, provided that E is cooriented, construct a natural equivariant differential form with generalized coefficients, as described in [Fit09b]. (An equivariant differential form with generalized coefficients, which we denote by α(X) ∈ A −∞ (M, g), is defined in [KV93] to be a G-equivariant map from g to the space of differential forms on M that can be integrated against a G-invariant test function on g to produce a smooth differential form on M. That is, if α(X) ∈ A −∞ (M, g), and ψ(X) is a smooth G-invariant function with compact support in g, then g α(X)ψ(X)dX is a differential form on M.) Let ι : E 0 ֒→ T * M denote the inclusion mapping, and let p : E 0 → M denote projection onto the base.…”

“…We note that the condition that the group action be transverse to E is necessary, as this ensures that zero is not in the image of the momentum map µ E : E 0 → g * given by µ E , X = ι * θ(X M ). The differential form (33) is treated in detail in [Fit09b] and the author's thesis [Fit09a], and so we will only outline some of its properties here. In the case that E 0 is trivial, and we are given a trivializing frame {η i }, let η : E 0 × g → R k be the map given by…”

“…In [Fit09a] we showed that the principal symbols of the operators defined in Section 3 are the same as the symbol mappings considered in [Fit09b] for the case of almost CR manifolds. In [Fit09b], we saw that for such symbols, is possible to integrate over the fibres of T * M (beginning with either the Berline-Vergne or the Paradan-Vergne formula) to obtain a formula involving the integral of equivariant characteristic forms over the compact manifold M. If T M = E ⊕ T denotes the splitting of T M by an f -structure ϕ, and D is an operator of the type defined in Section 3, then…”

“…* Research supported by an NSERC postdoctoral fellowship f -structure is equivalent to an almost CR structure together with a choice of complement to the Levi distribution.In [Fit09b], we used almost CR structures to construct new examples of transversally elliptic symbols (in the sense of Atiyah [Ati74]), and gave a formula for their (cohomological) equivariant index. In this paper we will give a construction of a first-order differential operator whose principal symbol is of the type considered in [Fit09b]. Such an operator was introduced in the contact setting in [Fit09c], and the general approach first appeared in the author's thesis [Fit09a].…”