Invariant integrals on coideals and their Drinfeld doubles

Let A be a CQG Hopf ∗-algebra, i.e. a Hopf ∗-algebra with a positive invariant state. Given a unital right coideal ∗-subalgebra B of A, we provide conditions for the existence of a quasi-invariant integral on the stabilizer coideal B⊥ inside the dual discrete multiplier Hopf ∗-algebra of A. Given such a quasi-invariant integral, we show how it can be extended to a quasi-invariant integral on the Drinfeld double coideal. We moreover show that the representation theory of the Drinfeld double coideal has a monoidal structure. As an application, we determine the quasi-invariant integral for the coideal ∗-algebra Uq(sl(2,R)) constructed from the Podleś spheres.