  The infinitesimal generators of L\'evy processes in Euclidean space are
pseudo-differential operators with symbols given by the L\'evy-Khintchine
formula. This classical analysis relies heavily on Fourier analysis which in
the case when the state space is a Lie group becomes much more subtle. Still
the notion of pseudo-differential operators can be extended to connected,
simply connected nilpotent Lie groups by employing the Weyl functional
calculus. With respect to this definition, the generators of L\'evy processes
in the simplest step 3 nilpotent Lie group $G$ are pseudo-differential
operators which admit $C_c(G)$ as its core.
