  For a connected Lie group G, we show that a complex structure on the total
space TG of the tangent bundle of G that is left invariant and has the property
that each left translation G-orbit is a totally real submanifold is induced
from a smooth immersion of TG into the complexification of G. For G compact and
connected, we then characterize left invariant and biinvariant complex
structures on the total space T*G of the cotangent bundle of G which combine
with the tautological symplectic structure to a Kaehler structure.
