  In this paper we find a criterion for the Gauss map of an immersed smooth
submanifold in some Lie group with left invariant metric to be harmonic. Using
the obtained expression we prove some necessary and sufficient conditions for
the harmonicity of this map in the case of totally geodesic submanifolds in Lie
groups admitting biinvariant metrics. We show that, depending on the structure
of the tangent space of a submanifold, the Gauss map can be harmonic in all
biinvariant metrics or non-harmonic in some metric. For 2-step nilpotent groups
we prove that the Gauss map of a geodesic is harmonic if and only if it is
constant.
