A L\'evy process on the real line seen from its supremum and max-stable processes

We consider a process $Z$ on the real line composed from a L\'evy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$ and the time $T$ at which the supremum occurs. In fact, it is closely related to the laws of the original and the tilted L\'evy processes conditioned to stay negative and positive. The result is used to derive a new representation of stationary particle systems driven by L\'evy processes. In particular, this implies that a max-stable process arising from L\'evy processes admits a mixed moving maxima representation with spectral functions given by the conditioned L\'evy processes.