The antiperovskite-related nitride Cr$_3$GeN forms a tetragonal structure with the space group $Pbar{4}2_1m$ at room temperature. It shows a tetragonal ($Pbar{4}2_1m$) 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.
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