  We prove Hardy inequalities for the conformally invariant fractional powers
of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two
versions of such inequalities depending on whether the weights involved are
non-homogeneous or homogeneous. In the first case, the constant arising in the
Hardy inequality turns out to be optimal. In order to get our results, we will
use ground state representations. The key ingredients to obtain the latter are
some explicit integral representations for the fractional powers of the
sublaplacian and a generalized result by M. Cowling and U. Haagerup. The
approach to prove the integral representations is via the language of
semigroups. As a consequence of the Hardy inequalities we also obtain versions
of Heisenberg uncertainty inequality for the fractional sublaplacian.
