Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

To any action of a compact quantum group on a von Neumann algebra
which is a direct sum of factors we associate an equivalence relation corresponding
to the partition of a space into orbits of the action. We show
that in case all factors are finite-dimensional (i.e., when the action is on a
discrete quantum space) the relation has finite orbits. We then apply this
to generalize the classical theory of Clifford, concerning the restrictions of
representations to normal subgroups, to the framework of quantum subgroups
of discrete quantum groups, itself extending the context of closed
normal quantum subgroups of compact quantum groups. Finally, a link
is made between our equivalence relation in question and another equivalence
relation defined by R. Vergnioux.