2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159995We 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 Cavalcanti-Gualtieri and Fernández-Gotay-Gray, 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.26 pages. Typos correctedDifferential Geometry17B30; 53C15; 22E25; 53C55; 53D17text