Complex structures on affine motion groups

We study existence of complex structures on semidirect products \g \oplus_{\rho} \v where $\g$ is a real Lie algebra and $\rho$ is a representation of $\g$ on \v. Our first examples, the Euclidean algebra $\e(3)$ and the Poincar\'e algebra $\e(2,1)$, carry complex structures obtained by deformation of a regular complex structure on $\sl (2, \c)$. We also exhibit a complex structure on the Galilean algebra $\G(3,1)$. We construct next a complex structure on \g \oplus_{\rho} \v starting with one on $\g$ under certain compatibility assumptions on $\rho$. As an application of our results we obtain that there exists $k\in \{0,1\}$ such that $(S^1)^k \times E(n)$ admits a left invariant complex structure, where $S^1$ is the circle and E(n) denotes the Euclidean group. We also prove that the Poincar\'e group $P^{4k+3}$ has a natural left invariant complex structure. In case \dim \g= \dim \v, then there is an adapted complex structure on \g\oplus_{\rho} \v precisely when $\rho$ determines a flat, torsion-free connection on $\g$. If $\rho$ is self-dual, \g \oplus_{\rho}\v carries a natural symplectic structure as well. If, moreover, $\rho$ comes from a metric connection then \g\oplus_{\rho} \v possesses a pseudo-K\"ahler structure. We prove that the tangent bundle $TG$ of a Lie group $G$ carrying a flat torsion free connection $\nabla$ and a parallel complex structure possesses a hypercomplex structure. More generally, by an iterative procedure, we can obtain Lie groups carrying a family of left invariant complex structures which generate any prescribed real Clifford algebra.