  We describe boundedness and compactness properties for the operators obtained
by the Weyl-Pedersen calculus in the case of the irreducible unitary
representations of nilpotent Lie groups that are associated with flat coadjoint
orbits. We use spaces of smooth symbols satisfying appropriate growth
conditions expressed in terms of invariant differential operators on the
coadjoint orbit under consideration. Our method also provides conditions for
these operators to belong to one of the Schatten ideals of compact operators.
In the special case of the Schr\"odinger representation of the Heisenberg group
we recover some classical properties of the pseudo-differential Weyl calculus,
as the Calder\'on-Vaillancourt theorem, and the Beals characterization in terms
of commutators.
