Equivariant covering spaces of quantum homogeneous spaces

We develop a fundamental theory of compact quantum group equivariant finite extensions of C*-algebras. In particular we focus on the case of quantum homogeneous spaces and give a Tannaka-Krein type result for equivariant correspondences. As its application, we show that every Jones' value appears as the index of an equivariant conditional expectation. In the latter half of this paper, we give an imprimitivity theorem in some cases: for general compact quantum groups under a finiteness conditions, and for the Drinfeld-Jimbo deformation $G_q$ of a simply-connected compact Lie group $G$. As an application, we give a complete classification of finite index discrete quantum subgroups of $\widehat{G_q}$.