Local formula for the index of a fourier integral operator

Abstract

Let X and Y be two closed connected Riemannian manifolds of the same dimension and ϕ : S*X → S*Y a contact diffeomorphism. We show that the index of an elliptic Fourier operator Φ associated with ϕ is given by ∫B*(X) eθ0 Â(T*X) − ∫B*(Y) eθ0 Â(T*Y) where θ0 is a certain characteristic class depending on the principal symbol of Φ and, B*(X) and B*(Y) are the unit ball bundles of the manifolds X and Y . The proof uses the algebraic index theorem of Nest-Tsygan for symplectic Lie Algebroids and an idea of Paul Bressler to express the index of Φ as a trace of 1 in an appropriate deformed algebra. In the special case when X = Y we obtain a different proof of a theorem of Epstein-Melrose conjectured by Atiyah and Weinstein. © Applied Probability Trust 2001.