Induction for representations of coideal doubles, with an application to quantum $SL(2,\mathbb{R})$

We investigate the theory of induction in the setting of doubles of coideal $\ast$-subalgebras of compact quantum group Hopf $\ast$-algebras. We then exemplify parts of this theory in the particular case of quantum $SL(2,\mathbb{R})$, and compute the decomposition of the regular representation for quantum $SL(2,\mathbb{R})$ into irreducibles.