the prequantum bundle automorphism group, whereas with a general auxiliary line bundle we have to consider a $U(1)$-extension. Second, the formula giving the asymptotic of the trace is much more complicated for a general auxiliary line bundle. It involves the complex structure of $M$ in a essential way, whereas it depends only on the symplectic data in the half-form case.

The index $m_x$ appearing in the estimate of the trace (\[eq:trace\]) is one of the main point of this paper. It is similar to a Maslov or a Conley-Zehnder index. Our definition requires a choice of a complex polarization, whereas the already known definitions of indices for symplectomorphisms involved a choice of a real polarization. Besides the definition given in section \[sec:definition-index\], we propose a simple useful characterization in section \[sec:char-index\]. We also compute the index of the elements of the metalinear and unitary groups.

The results of the paper relies on the articles [@oim_qm] and [@oim_mc]. In [@oim_qm] we proposed an elementary definition of a Fourier integral operator in the context of geometric quantization of K[ä]{}hler manifolds. Previously, a definition was given by Zelditch [@Ze] using the general theory of Toeplitz operators of Boutet de Monvel and Guillemin [@BoGu]. 