We study the homogeneous space of the quantum group U q (n) related to the subgroup U q (n − m) (m < n), classify its irreducible representations and get a formula for its invariant integral. We also study the double cosets U q (n − m)\U q (n)/U q (n − m) and the hypergroup structure associated with them.

We prove that a conditional expectation on a compact quantum group that satisfies certain conditions can be decomposed into a composition of two conditional expectations one of which is associated with quantum double cosets and the other preserves the counit.

We find an explicit formula for finding pairs of cocycles for the construction of examples of locally compact quantum groups by using the bicrossed product of Lie groups.

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