  We consider a $\mu-\tau$ symmetry in neutrino sectors realized at GUT scale
in the context of a seesaw model. In our scenario, the exact $\mu-\tau$
symmetry realized in the basis where the charged lepton and heavy Majorana
neutrino mass matrices are diagonal leads to vanishing lepton asymmetries. We
find that, in the minimal supersymmetric extension of the seesaw model with
large $\tan\beta$, the renormalization group (RG) evolution from GUT scale to
seesaw scale can induce a successful leptogenesis even without introducing any
symmetry breaking terms by hand, whereas such RG effects lead to tiny
deviations of $\theta_{23}$ and $\theta_{13}$ from $\pi/4$ and zero,
respectively. It is shown that the right amount of the baryon asymmetry
$\eta_B$ can be achieved via so-called resonant leptogenesis, which can be
realized at rather low seesaw scale with large $\tan\beta$ in our scenario so
that the well-known gravitino problem is safely avoided.
