  The growth of instabilities is key to trigger a supernova explosion during
the phase of stalled shock, immediately after the birth of a proto-neutron star
(PNS). We assess the effect of stellar rotation on neutrino-driven convection
and SASI when neutrino heating is taken into account. Rotation affects the
frequency of the mode m=2 detectable with gravitational waves (GW). We use a
linear stability analysis in the equatorial plane between the PNS and the
stationary shock and consider a large range of specific angular momenta,
neutrino luminosities and mass accretion rates. The nature of the dominant
instability depends on the convection parameter chi and the rotation rate.
Convective modes with chi>=5 are hampered by differential rotation. At smaller
chi, however, mixed SASI-convective modes with a large angular scale m=1,2,3
benefit from rotation and become dominant for relatively low rotation rates at
which centrifugal effects are small. For rotation rates >0.3 Keplerian rotation
at the PNS surface (KPNS), the growth rate of the dominant mode depends weakly
on neutrino heating which highlights a new instability regime. Its frequency is
surprisingly independent of the heating rate, with a strong prograde spiral m=2
dominating over a large parameter range, favourable to the production of GW. A
simple linear relation exists between the dominant oscillation frequency and
the specific angular momentum. Three regimes are distinguished. For rotation
rates <0.1KPNS, differential rotation has a quadratic effect on equatorial
purely convective modes and a linear destabilizing effect on SASI. Intermediate
rotation rates (0.1 to 0.3KPNS) lead to the emergence of mixed
SASI/convection/rotation modes involving large angular scales. Finally, strong
rotation erases the influence of buoyancy on the instability. This allows for a
reduction of the parameter space, which is valuable for GW analysis.
