  We present a detailed analysis to clarify what determines the growth of the
low-$T/|W|$ instability in the context of rapidly rotating core-collapse of
massive stars. To this end, we perform three-dimensional core-collapse
supernova (CCSN) simulations of a $27 M_{\odot}$ star including several updates
in the general relativistic correction to gravity, the multi-energy treatment
of heavy-lepton neutrinos, and the nuclear equation of state. Non-axisymmetric
deformations are analyzed from the point of view of the time evolution of the
pattern frequency and the corotation radius. The corotation radius is found to
coincide with the convective layer in the proto neutron star (PNS). We propose
a new mechanism to account for the growth of the low-$T/|W|$ instability in the
CCSN environment. Near the convective boundary where a small
Brunt-V\"ais\"al\"a frequency is expected, Rossby waves propagating in the
azimuthal direction at mid latitude induce non-axisymmetric unstable modes, in
both hemispheres. They merge with each other and finally become the spiral arm
in the equatorial plane. We also investigate how the growth of the low-$T/|W|$
instability impacts the neutrino and gravitational-wave signatures.
