  We prove global subelliptic estimates for quadratic differential operators.
Quadratic differential operators are operators defined in the Weyl quantization
by complex-valued quadratic symbols. In a previous joint work with M. Hitrik,
we pointed out the existence of a particular linear subvector space in the
phase space intrinsically associated to their Weyl symbols, called singular
space, which rules spectral properties of non-elliptic quadratic operators. The
purpose of the present paper is to prove that quadratic operators whose
singular spaces are reduced to zero, are subelliptic with a loss of
"derivatives" depending directly on particular algebraic properties of the
Hamilton maps of their Weyl symbols. More generally, when singular spaces are
symplectic spaces, we prove that quadratic operators are subelliptic in any
direction of the symplectic orthogonal complements of their singular spaces.
