  We compute statistical equilibrium states of rotating self-gravitating
systems enclosed within a box by maximizing the Fermi-Dirac entropy at fixed
mass, energy and angular momentum. We increase the rotation up to the Keplerian
limit and describe the flattening of the configuration until mass shedding
occurs. At the maximum rotation, the system develops a cusp at the equator. We
draw the equilibrium phase diagram of the rotating self-gravitating Fermi gas
and discuss the structure of the caloric curve as a function of degeneracy
parameter and angular velocity. We argue that systems described by the
Fermi-Dirac distribution in phase space do not bifurcate to non-axisymmetric
structures, in continuity with the case of polytropes with index n>0.808 (the
Fermi gas at T=0 corresponds to n=3/2). This contrasts with the study of
Votyakov et al. (2002) who consider a Fermi-Dirac distribution in configuration
space and find ``double star'' structures (their model at T=0 corresponds to
n=0). We also discuss the influence of rotation on the onset of the
gravothermal catastrophe for globular clusters. On general grounds, we complete
previous investigations concerning the nature of phase transitions in
self-gravitating systems. We emphasize the inequivalence of statistical
ensembles regarding the formation of binaries (or low-mass condensates) in the
microcanonical ensemble and Dirac peaks (or massive condensates) in the
canonical ensemble. We also describe an hysteretic cycle between the gaseous
phase and the condensed phase that are connected by a ``collapse'' or an
``explosion''. This notion of hysteresis in self-gravitating systems is new.
