  Hydrodynamical instabilities, such as the standing accretion-shock
instability (SASI), play an essential role in the dynamics of core-collapse
supernovae, with observable imprints in the neutrino and gravitational wave
signals. Yet, the impact of stellar rotation on the development of SASI is
poorly explored. We investigate the conditions favoring the growth of SASI in
the presence of rotation through a perturbative analysis. The properties of
SASI are compared in two stationary configurations, cylindrical and spherical
equatorial, which mainly differ by their advection timescales from the shock to
the proto-neutron star surface. Without rotation, the mode $m=1$, corresponding
to a one-armed spiral SASI deformation, can be significantly more unstable in
the spherical equatorial configuration. In fact, the shorter advection time in
the spherical equatorial geometry allows for a larger contribution of the
entropic-acoustic coupling from the region of adiabatic compression near the
surface of the proto-neutron star. The angular momentum of the collapsing core
favors the growth of prograde spiral modes $m=1$ and $m = 2$ in both
geometries. Although the growth rate of the spiral instability is
systematically faster in spherical geometry, its oscillation frequency is
remarkably insensitive to the geometry. Such a contrast with non-rotating flows
calls for a deeper understanding of the role of advection in the mechanism of
spiral SASI. Our findings suggest that the resonant coupling of acoustic waves
with their corotation radius may play a major role in the instability mechanism
of collapsing cores with rotation. Elucidating this physical mechanism is
essential to interpret the signal from future multi-messenger supernova
observations.
