Some questions about the index of quantized contact transformations

Abstract

An index formula is proposed for contact transformations between contact manifolds equipped with CR structures or with fillings by symplectic manifolds. The formula generalizes the Atiyah-Singer formula and gives a conjectured formula for the index of Fourier integral operators, as well as Epstein's relative index for CR structures.

This is an expanded version of a lecture given at the Symposium on Geometric Methods in Asymptotic Analysis, RIMS, Kyoto, May 20, 1997