An extension of the Duistermaat-Singer Theorem to the semi-classical Weyl algebra

Motivated by many recent works (by L. Charles, V. Guillemin, T. Paul, J. Sj\"ostrand, A. Uribe, S. Vu Ngoc, S. Zelditch and others) on the semi-classical Birkhoff normal forms, we investigate the structure of the group of automorphisms of the graded semi-classical Weyl algebra which is used to get the normal forms. The answer is quite similar to the Theorem of Duistermaat and Singer for the usual algebra of pseudo-differential operators where all automorphisms are given by conjugation by an elliptic Fourier Integral Operator (a FIO). Here what replaces general non-linear symplectic diffeomeorhisms is just linear complex symplectic maps, because everything is localized at a single point.