  In this paper we obtain a martingale representation theorem in the
progressive enlargement $\mathbb{G}$ by a random time $\tau$ of the filtration
$\mathbb{F}^L$ generated by a L\'evy process $L$. The assumptions on the random
time are that $\mathbb{F}^ L$ is immersed in $\mathbb{G}$ and that $\tau$
avoids $\mathbb{F}^ L$ stopping times. We also study the multiplicity of a
progressively enlarged filtration.
