Toeplitz Operators on CR Manifolds and Group Actions

Abstract

Let X be a connected orientable compact CR manifold with non-degenerate Levi form. In this paper, we study the algebra of Toeplitz operators on X and we establish star product for a certain class of symbols on X. In the second part of this paper, we consider a locally free action of a compact Lie group G on X and we investigate the associated algebra of G-invariant Toeplitz operators. The main application is a construction of deformation quantization of compact “quantizable” pseudo-Kähler manifolds.