  $\beta$-decay half-lives of neutron-rich nuclei around $N=82$ are key data to
understand the $r$-process nucleosynthesis. We performed large-scale
shell-model calculations in this region using a newly constructed shell-model
Hamiltonian, and successfully described the low-lying spectra and half-lives of
neutron-rich $N=82$ and $N=81$ isotones with $Z=42-49$ in a unified way. We
found that their Gamow-Teller strength distributions have a peak in the
low-excitation energies, which significantly contributes to the half-lives.
This peak, dominated by $\nu 0g_{7/2} \to \pi 0g_{9/2}$ transitions, is
enhanced on the proton deficient side because the Pauli-blocking effect caused
by occupying the valence proton $0g_{9/2}$ orbit is weakened.
