Toeplitz Operators on Contact Manifolds and Equivariant K-homology

Abstract

We present an equivariant generalization of Boutet de Monvel's index theorem for Toeplitz operators on contact manifolds. We prove that the Dirac operator and the Szeg\"o projection determine the same class in equivariant $K$-homology, generalizing a theorem of Baum-Douglas-Taylor. We do not assume that the contact manifold is the boundary of a strictly pseudoconvex domain. The proof proceeds by a deformation linking the principal symbols of the classical and Heisenberg pseudodifferential calculi. At the level of symbols, the projection defining the Dirac class deforms to the principal Heisenberg symbol of the Szeg\"o projection. This deformation implies equality of the corresponding classes in K-homology. This, in turn, gives an equivariant generalization of Boutet de Monvel's index formula for Toeplitz operators.