We performed a study on the probable existence of long-lived charged sleptons in the so-called coannihilation region of parameter space in the framework of the MSSM with supergravity-inspired soft SUSY breaking. We show the possibility to get long-lived charged sleptons in a tiny thin band of mSUGRA parameters space which is consistent with the WMAP restrictions. We show also the strong dependence of this region on the values of tanβ. In this context, we analyze the production cross section of long-lived charged sleptons at LHC in annihilation channels and found that, for light long-lived charged ̃ τ, the production cross section may reach a few percent of pb. © 2008 Pleiades Publishing, Ltd.