We investigated properties of the Riesz transform and large time
behaviour of derivatives of the heat kernels corresponding
to left-invariant operators on Lie groups. We proved that
higher order Riesz transforms for such operators are bounded if and only if
the corresponding Lie group is a direct product of a nilpotent group and
a compact
group. We also obtained results concerning connections between boundedness
of the Riesz transform and the behaviour of derivatives of the heat
kernels. The finite speed propagation techniques are important part of
the proof.