Continuity of the stabilizer map and irreducible extensions

Let G be a locally compact group. For every G -flow X , one can consider the stabilizer map x \mapsto G_{x} , from X to the space \mathrm{Sub}(G) of closed subgroups of G . This map is not continuous in general. We prove that if one passes from X to the universal irreducible extension of X , the stabilizer map becomes continuous. This result provides, in particular, a common generalization of a theorem of Frolík (that the set of fixed points of a homeomorphism of an extremally disconnected compact space is open) and a theorem of Veech (that the action of a locally compact group on its greatest ambit is free). It also allows to naturally associate to every G -flow X a stabilizer G -flow \mathrm{S}_{G}(X) in the space \mathrm{Sub}(G) , which generalizes the notion of stabilizer uniformly recurrent subgroup associated to a minimal G -flow introduced by Glasner and Weiss.