  The antiperovskite-related nitride Cr$_3$GeN forms a tetragonal structure
with the space group $P\bar{4}2_1m$ at room temperature. It shows a tetragonal
($P\bar{4}2_1m$) to tetragonal ($I4/mcm$) structural transition with a large
hysteresis at 300--400 K. The magnetic susceptibility of Cr$_3$GeN shows
Curie-Weiss type temperature dependence at high temperature, but is almost
temperature-independent below room temperature. We carried out $\mu$SR and
$^{14}$N-NMR microscopy measurements to reveal the magnetic ground state of
Cr$_3$GeN. Gradual muon spin relaxation, which is nearly
temperature-independent below room temperature, was observed, indicating that
Cr$_3$GeN is magnetically inactive. In the $^{14}$N-NMR measurement, a
quadrupole-split spectrum was obtained at around $^{14}K = 0$. The temperature
dependence of $^{14}(1/T_1)$ satisfies the Korringa relation. These
experimental results indicate that the ground state of Cr$_3$GeN is Pauli
paramagnetic, without antiferromagnetic long-range order.
