The interest of half-form bundle for these Fourier integral operators was understood in [@oim_mc] where the spaces $U( \Phi_L, \Psi)$ were introduced. The estimate of the trace with the definition of the index is new. Of course, it is very similar to the known formula for the usual Fourier integral operators. It was one of our goal to obtain the closest formula to the usual case.

In a sequel of this paper we will apply our result to the quantum representations of the mapping class group defined in topological quantum field theory. Estimating the character of these representations with our formula, we will obtain the leading order behavior of the quantum invariant of some 3-dimensional manifolds in the large level limit. These asymptotics were initially obtained by Witten in [@Wi] by doing the perturbative theory of some Feynman path integral and have been rigorously proved only in few cases. The article [@oim_MG] will be devoted to ${\operatorname{Sl}}(2, {{\mathbb{Z}}})$ and [@oim_MCG] to the mapping class group of surfaces with genus greater than 2.

The layout of this paper is as follows. Sections \[sec:half-forms\], \[sec:metaplectic-group\] and \[sec:gener-half-form\] are devoted to linear algebra preliminaries. In section \[sec:half-forms\], we define the half-form morphisms and 