Noncommutative residue invariants for CR and contact manifolds

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In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric Ψ H DO projections. In the CR setting these operators arise from the -complex and include the Szegö projections acting on forms. In the contact setting they stem from the generalized Szegö projections at arbitrary integer levels of Epstein-Melrose and from the contact complex of Rumin. In particular, we recover and extend recent results of Hirachi and Boutet de Monvel and we answer a question of Fefferman.