Abstract
We show that the kernel of an irreducible unitary representation
π of the group algebra
L
1
(
G
)
of a completely solvable Lie group
G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit
O
π
of
π if and only if this orbit
O
π
is flat. This is a generalization of a result obtained before for nilpotent Lie groups.