We define a special nilpotent Lie group
N
N
to be one which has a
1
1
-dimensional center, dilations, square-integrable representations, and a maximal subordinate algebra common to almost all functionals on the Lie algebra
N
\mathfrak {N}
. Every nilpotent Lie group with dilations imbeds in such a special group so that the dilations extend. Let
L
L
be a homogeneous left invariant differential operator on
N
N
. We give a representation-theoretic condition on
L
L
which
L
L
must satisfy if it has a tempered fundamental solution and which implies global solvability of
L
L
. (The sufficiency is a corollary of a more general theorem, valid on all nilpotent
N
N
.) For the Heisenberg group, the condition is equivalent to having a tempered fundamental solution.