
Journal of Lie Theory 12 (2002), No. 1, 245257 Copyright Heldermann Verlag 2002 An Invariant Symmetric NonSelfadjoint Differential Operator Erik G. F. Thomas Mathematisch Instituut, Universiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands [Abstractpdf] Let $D$ be a symmetric left invariant differential operator on a unimodular Lie group $G$ of type $I$. Then we show that $D$ is essentially selfadjoint if and only if for almost all $\pi \in \widehat{G}$, with respect to the Plancherel measure, the operator $\pi(D)$ is essentially selfadjoint. This, in particular, allows one to exhibit a left invariant symmetric differential operator on the Heisenberg group, which is not essentially selfadjoint. [ Fulltextpdf (204 KB)] 