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