We study hermitian structures, with respect to the standard
neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional
Lie group $G$, which are left invariant with respect to the
Lie group structure on $T^*G$ induced by the coadjoint action.
These are in one-to-one correspondence with left invariant
generalized complex structures on $G$. Using this correspondence
and results of [8] and [10], it turns out that when $G$ is
nilpotent and four or six dimensional, the cotangent bundle
$T^*G$ always has a hermitian structure. However, we prove
that if $G$ is a four dimensional solvable Lie group admitting
neither complex nor symplectic structures, then $T^*G$ has
no hermitian structure or, equivalently, $G$ has no left invariant
generalized complex structure.