A Representation-Theoretic Criterion for Local Solvability of Left Invariant Differential Operators on Nilpotent Lie Groups

Let $L$ be a left invariant differential operator on the nilpotent Lie group $N$. It is shown that if $\pi (L)$ is invertible for all irreducible representations $\pi$ in general position (and if the inverses satisfy some mild technical conditions), then $L$ is locally solvable. This result generalizes a theorem of ${\text {L}}$. Rothschild.