  In continuation of a former work by the first author with F. Nier (2009) and
of a more recent work by the second author on the torus (2019), we consider the
Kramers-Fokker-Planck operator (KFP) with an external electromagnetic field on
R d. We show a maximal type estimate on this operator using a nilpotent
approach for vector field polynomial operators and induced representations of a
nilpotent graded Lie algebra. This estimate leads to an optimal
characterization of the domain of the closure of the KFP operator and a
criterion for the compactness of the resolvent.
