An invariant complex structure on an even dimensional real Lie group is a complex structure on the underlying manifold in the usual sense for which left multiplication in the group is holomorphic. In this paper, invariant complex structures are studied for reductive groups, extending the work of H. C. Wang [Am. J. Math. 76, 1-32 (1954; Zbl 0055.166)], A. Morimoto [C. R. Acad. Sci., Paris 242, 1101-1103 (1956; Zbl 0070.260)], and T. Sasaki [Kumamoto J. Sci., Math. 14, 115-123 (1981; Zbl 0447.32007); ibid. 15, 59-72 (1982; Zbl 0509.32020)]. It is shown that in almost all cases, a reductive Lie group with an invariant complex structure can be fibered holomorphically over an open orbit in a homogeneous projective rational manifold where much is known [see, e.g., J. A. Wolf, Bull. Am. Math. Soc. 75, 1121-1237 (1969; Zbl 0183.509)].
A detailed classification of all regular invariant complex structures is then given and a moduli space is constructed for them. For many reductive groups, invariant complex structures are always regular. The only exceptions come from simple factors of the second category, e.g. \(SL(2m+1,{\mathbb{R}})\) or complex simple Lie groups. These latter groups are also the ’richest’ in terms of the variety of invariant complex structures they can posses. Examples are included to illustrate the classification and the construction of the moduli space.