On the Solvability of Homogeneous Left-Invariant Differential Operators on the Heisenberg Group

We discuss the problem of solvability for the class of homogenoeus left-invariant operators □S, αon the Heisenberg groupHnintroduced by F. De Mari, M. M. Peloso, and F. Ricci. (1995,J. Reine Angew. Math.464, 67–96), whereS∈Sp(C, n) has a degenerated semi-definite real part, andα=±(n+2l),l=0, 1, …. We completely characterize the solvable operators in this class. As a consequence we obtain that certain operators □S, αare solvable while their transposest□S, αare not. This is the case, for instance, for the operatorYZonH1, whereZ=X−iYis the unsolvable Lewy operator.