Abstract. We study the question of local solvability for second-order, leftinvariant differential operators on the Heisenberg group Hn, oftheform n∑ PΛ = λijXiYj = i,j=1 t XΛY, where Λ = (λij)isacomplexn×nmatrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that PΛ cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that Re Λ, Im Λ and their commutator are linearly independent, we show that PΛ is not locally solvable, even in the presence of lower-order terms, provided that n ≥ 7. In the case n =3weshow that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group H3 a phenomenon first observed by Karadzhov and Müller in the case of H2. It is interesting to notice that the analysis of the exceptional operators for the case n = 3 turns out to be more elementary than in the case n =2. When 3 ≤ n ≤ 6 the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time
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