Flat orbits and kernels of irreducible representations of the group algebra of a completely solvable Lie group

We show that the kernel of an irreducible unitary representation π of the group algebra L1(G) of a completely solvable Lie group G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit Oπ of π if and only if this orbit Oπ is flat. This is a generalization of a result obtained before for nilpotent Lie groups.