In the classical theory , parabolic induction is used to construct the (reduced) dual of a (semi-simple) Lie Group. However, for this we need subgruops. Given that the theory of "quantum subgroups" of locally compact quantum groups (LCQG) is still being debated, let me provide two pictures (which are probably not exhaustive) of quantum subgroups. Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ it`s Lie algebra.
a) Joseph-Letzter`s Coideal Subalgebras - A subalgebra $B$ of $U := U_q(\mathfrak{g})$, is called a (left) coideal subalgebra if $\Delta(B) \subset B \otimes U$. Letzter in one of her papers does quantize some parabolic subgroups of $G$ from which parabolic induction is easily done.
b) Vaes` approach - $\mathbb{H}$ is a subgroup of $\mathbb{G}$ if there is a normal injective $^*$-morphism $\pi: \mathbb{H} \rightarrow \mathbb{G}$ and a another one $\gamma: L^\infty(\mathbb{\hat{H}}) \rightarrow L^\infty(\mathbb{\hat{G}})$, with a compatibility condition.
My questions are:
1) Is there progress in "topologizing" JL`s coideal subalgebras from picture a)? By which I mean, finding suitable operator algebraic analogs in the LCQG theory developed by Vaes.
2) Still in picture a), what is the relation between the structure of the the set of representations obtained from parabolic induction and the Hopf dual of $U$?
3) Has anyone attempted to develop a theory of parabolic induction in picture b)? Suppose it could be (or maybe was) done. What is the relation between the structure of the the set of representations obtained in this manner and the Pontryagin dual? Naively they seem to me to give essentially the same object.
Best regards, HTTT.
