  It is known that the variety parametrizing pairs of commuting nilpotent
matrices is irreducible and that this provides a proof of the irreducibility of
the punctual Hilbert scheme in the plane. We extend this link to the nilpotent
commuting variety of parabolic subalgebras of $M\_n(\K)$ and to the punctual
nested Hilbert scheme. By this method, we obtain a lower bound on the dimension
of these moduli spaces. We characterize the numerical conditions under which
they are irreducible. In some reducible cases, we describe the irreducible
components and their dimension.
