  Using an approach based on the techniques of FBI transforms, we give a new
simple proof of the global subelliptic estimates for non-selfadjoint
non-elliptic quadratic differential operators, under a natural averaging
condition on the Weyl symbols of the operators, established by the second
author. The loss of the derivatives in the subelliptic estimates depends
directly on algebraic properties of the Hamilton maps of the quadratic symbols.
Using the FBI point of view, we also give accurate smoothing estimates of
Gelfand-Shilov type for the associated heat semigroup in the limit of small
times.
