A representation-theoretic characterization is given for hypoellipticity of homogeneous (with respect to dilations) left-invariant differential operators P on the Heisenberg group
H
n
{H_n}
; it is the precise analogue for
H
n
{H_n}
of the statement for
R
n
{{\mathbf {R}}^n}
that a homogeneous constant-coefficient differential operator is hypoelliptic if and only if it is elliptic. Under these representation-theoretic conditions a parametrix is constructed for P by means of the Plancherel formula. However, these conditions involve all the irreducible representations of
H
n
{H_n}
, whereas only the generic, infinite-dimensional representations occur in the Plancherel formula. A simple class of examples is discussed, namely
P
=
Σ
i
=
1
n
X
i
2
m
+
Y
i
2
m
P = \Sigma _{i = 1}^nX_i^{2m} + Y_i^{2m}
, where
X
i
,
Y
i
,
i
=
1
,
…
,
n
{X_i},{Y_i},i = 1, \ldots ,n
, and Z generate the Lie algebra of
H
n
{H_n}
via the commutation relations
[
X
i
,
Y
j
]
=
δ
i
j
Z
[{X_i},{Y_j}] = {\delta _{ij}}Z
, and where m is a positive integer. In the course of the proof a connection is made between homogeneous left-invariant operators on
H
n
{H_n}
and a class of degenerate-elliptic operators on
R
n
+
1
{{\mathbf {R}}^{n + 1}}
studied by Grušin. This connection is examined in the context of localization in enveloping algebras.