  Halo is one of the most interesting phenomena in exotic nuclei especially for
$^{31}$Ne, which is deemed to be a halo nucleus formed by a $p-$wave resonance.
However, the theoretical calculations don't suggest a $p-$wave resonance using
the scattering phase shift approach or complex scaling method. Here, we apply
the complex momentum representation method to explore resonances in $^{31}$Ne.
We have calculated the single-particle energies for bound and resonant states
together with their evolutions with deformation. The results show that the
$p-$wave resonances appear clearly in the complex momentum plane accompanied
with the $p-f$ inversion in the single-particle levels. As it happens the $p-f$
inversion, the calculated energy, width, and occupation probabilities of major
components in the level occupied by valance neutron support a $p-$wave halo for
$^{31}$Ne.
