holomorphic sections of $L$ are all trivial, the holomorphic Lefschetz fixed point formula expresses the characters of this representation in terms of characteristic classes of $M$ and $L$.

With the physical interpretation in mind, it is natural to consider the prequantum bundle automorphisms instead of the holomorphic automorphisms. These are the automorphisms of $L$ preserving the Chern connection and the metric but not necessarily the holomorphic structure. Whereas the group of holomorphic automorphism is finite dimensional, the group of the prequantum bundle automorphisms is infinite-dimensional. Its Lie algebra identifies with the Poisson algebra of $M$. Furthermore each Hamiltonian symplectomorphism of $M$ lifts to a prequantum bundle automorphism, and if $M$ is simply connected, each symplectomorphism isotopic to the identity is Hamiltonian. The goal of this paper is to define an asymptotic representation of these automorphisms such that a suitable version of the Lefschetz fixed point formula holds. Here the terms asymptotic refers to the semi-classical limit, obtained by replacing the prequantum bundle $L$ by its $k$-th tensor power with large value of $k$.

It is convenient to work with the metaplectic correction. Let $\delta
\rightarrow M$ be a half-form bundle, that is a square root of the canonical bundle of 