On local solvability for complex coefficient differential operators on the Heisenberg group

We discuss the problem of solvability for some classes of complex coefficient second order left-invariant operators on the Heisenberg group ℍn. We give several examples of operators that are not locally solvable for all choices of certain parameters, even if one allows the addition of lower order terms, in some cases also non-invariant ones. This is in striking contrast with the phenomenona known so far in the theory of local solvability of invariant second-order differential operators on nilpotent Lie groups. In order to disprove local solvability we use two different technical tools. The first one is a criterion by Hormander [Ho1]. The second one is an extension of a criterion for local non-solvability in [CR]. This extension, which is of interest in its own right, allows us to deal with non-homogeneous invariant differential operators. Our analysis of the differential operators is based on the classification of normal forms for involutive complex Hamiltonians under the action of the real symplectic group.