  We investigate flavour violating decays of the long-lived lightest slepton
and the tau lepton in the coannihilation region of the Minimal Supersymmetric
Standard Model with a Seesaw mechanism to generate neutrino masses. We consider
a situation where the mass difference between the lightest neutralino, as the
Lightest Supersymmetric particle (LSP), and the lightest slepton, as the
Next-to-LSP, is smaller than the mass of tau lepton. In this situation, the
lifetime of the lightest slepton is very long and it is determined by lepton
flavour violating (LFV) couplings because the slepton mainly consists of the
lighter stau and the flavour conserving 2-body decay is kinematically
forbidden. We show that the lifetime can change many orders of magnitude by
varying the Yukawa couplings entering the Seesaw mechanism. We also show that
branching ratio of LFV tau decays are strongly correlated with the lightest
slepton lifetime. Therefore the branching ratios of LFV tau decays can be
determined or constrained by measuring the slepton lifetime at the LHC
experiment.
