  Broken Galilean invariance in a spin-orbit coupled system can amplify
many-body effects on its different responses. We study the anomalous Hall and
spin Hall conductivities of a magnetic two-dimensional electron gas with Rashba
spin-orbit coupling. We show that both of these conductivities in the intrinsic
limit are fully specified in terms of the longitudinal and transverse spin-spin
response functions. We include the effect of electron-electron interaction in
the spin-spin linear response functions, going beyond the random-phase
approximation. We do this by incorporating the local-field correction in the
response functions, which takes into account the many-body exchange-correlation
effects. We observe a significant enhancement of the static anomalous Hall
conductivity due to the electron-electron interaction. The many-body correction
on the spin Hall effects is more non-trivial, and strong electron-electron
interaction can even reverse the sign of the static spin Hall conductivity.
