We consider a process $Z$ on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of the supremum $\overline Z$, its time $T$, and the process $Z(T+\cdot )-\overline Z$. This expression is in terms of the laws of the original and the tilted Lévy processes conditioned to stay negative and positive respectively. The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Lévy processes.
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