CONTACT DEGREE AND THE INDEX OF FOURIER INTEGRAL …rbm/cdifiod.27.May.1998.bis.pdf ·...
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CONTACT DEGREE AND THE INDEX OF FOURIER INTEGRAL OPERATORS CHARLES EPSTEIN AND RICHARD B. MELROSE Date: Wed May 27 15:55:55 1998 ; Run: May 27, 1998 Abstract. An elliptic Fourier integral operator of order 0, associated to a homogeneous canonical diffeomorphism, on a compact manifold is Fredholm on L 2 . The index may be expressed as the sum of a term, which we call the contact degree, associated to the canonical diffeomorphism and a term, computable by the Atiyah-Singer theorem, associated to the symbol. The contact degree is shown to be defined for any oriented-contact diffeomorphism of a contact manifold and is then reduced to the index of a Dirac operator on the mapping torus, also computable by the theorem of Atiyah and Singer. In this case, of an operator on a fixed manifold, these results answer a question of Weinstein in a manner consistent with a more general conjecture of Atiyah. Introduction Let X be a compact contact manifold, with oriented contact line bundle L ⊂ T * X. Let M(X) be the contact mapping class group of X; that is the group of com- ponents of the space of contact diffeomorphisms. We construct a homomorphism which we call the contact degree c-deg : M(X) -→ Z. (1) This construction is based on the notion of a quantization of the contact structure introduced by Boutet de Monvel and Guillemin [5]. In particular if the hyperplane bundle, W = L ◦ on X, is given an almost complex structure which is positive with respect to the conformal symplectic structure and X is given a compatible partially Hermitian metric then in [5] generalized Szeg˝ o projections are shown to exist. Al- though the analysis in [5] is in terms of the Hermite calculus, these projections also lie in the Heisenberg calculus discussed by Beals and Greiner [1] and Taylor [18], and originally described by Dynin [6]. Extending an earlier idea of the second au- thor [8] (for the integrable, that is CR, case) we introduce below the relative index of two such generalized Szeg˝ o projections as the index of the Fredholm operator which is their composite acting between their ranges: ind(S 0 ,S 1 ) = ind(S 1 S 0 : Ran(S 0 ) -→ Ran(S 1 )). (2) We show below in Proposition 2 that this relative index faithfully labels the com- ponents of the space of generalized Szeg˝ o projections. The action of the group of contact diffeomorphisms, by conjugation, on the space of generalized Szeg˝ o projec- tions induces the homomorphism (1), c-deg(φ) = ind(S, S φ ),S φ =(φ * ) -1 Sφ * . (3) 1
CONTACT DEGREE AND THE INDEX OF FOURIER INTEGRAL …rbm/cdifiod.27.May.1998.bis.pdf · 1998-05-27 · 2 CHARLES EPSTEIN AND RICHARD MELROSE Let Z φ be the mapping torus of φ,i.e
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