Equivariant Spectral Flow and a Lefschetz Theorem on Odd Dimensional Spin Manifolds

As one of the most important theories in mathematics, Atiyah-Singer index theorems have various profound applications and consequences. At the same time, there are several ways to prove these theorems. Of particular interest is the heat kernel proof, which allows one to obtain refinements of the index theorems, i.e., the local index theorems for Dirac operators. Readers are referred to [BGV] for a comprehensive treatment of the heat kernel method on even dimensional manifolds. It is worthwhile to point out here that the heat kernel method also lead to direct analytic proofs of the equivariant index theorem for Dirac operators on even dimensional spin manifolds. Among the existing proofs we list Bismut [BV], Berline-Vergne [BV] and Lafferty-Yu-Zhang [LYZ]. The purpose of this paper is to present a heat kernel proof of an equivariant index theorem on odd dimensional spin manifolds, which is stated for Toeplitz operators. Recall that Baum-Douglas [BD] first stated and proved an odd index theorem for Toeplitz operators using the general Atiyah-Singer index theorem for elliptic pseudo-differential operators. It is known to experts that one can give a heat kernel proof of the above mentioned odd index theorem. However, let us still give a brief description of the basic ideas. The first step is to apply a result of BoossWojciechowski [BW] to identify the index of the Toeplitz operator to the spectral flow of a certain family of self-dual elliptic operator with positive order. The second step is then to use the well-known relationship between spectral flows and variations of η-invariants to evaluate this spectral flow (cf. [G]). Our proof of the equivariant odd index theorem follows the same strategy. For this purpose we need to introduce a concept of equivariant spectral flow and establish an equivariant version of the Booss-Wojciechowski theorem mentioned above. We then extend the relationship between the spectral flow and variations of η invariants to the equivariant setting. Finally, we use the local index techniques to evaluate these variations. Among the methods of Bismut [B], Berline-Vergne [BV] and Lafferty-Yu-Zhang [LYZ], for simplicity we will follow those of [LYZ] in this paper. There is, however, no difficulty applying other methods.