In this note we present a symbolic pseudo-differential calculus on the
Heisenberg group. We particularise to this group our general construction
[4,3,2] of pseudo-differential calculi on graded groups. The relation between
the Weyl quantization and the representations of the Heisenberg group enables
us to consider here scalar-valued symbols. We find that the conditions defining
the symbol classes are similar but different to the ones in [1]. Applications
are given to Schwartz hypoellipticity and to subelliptic estimates on the
Heisenberg group.Comment: 9 page