  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.
