Let $\mathbb{G}=(A,\Delta_A)$ be a C*-quantum group and $\mathbb{H}=(B,\Delta_B)$ be a closed quantum subgroup of $\mathbb{G}$. We say that $\mathbb{H}$ is a closed quantum subgroup of $\mathbb{G}$ if there is a unitary bicharacter $V\in\mathcal{UM}(A\otimes B)$ such that $V$ generates $\mathbb{H}$ as a quantum group. Here, $\mathcal{UM}(A\otimes B)$ denotes the set of unitary multipliers of $A\otimes B$.
Question: If $U\in\mathcal{UM}(K(\mathcal{L})\otimes B)$ be a representation of $\mathbb{H}$ on a Hilbert space $\mathcal{L}$: $$(Id_{K(\mathcal{L})}\otimes \Delta_{B})U=U_{12}U_{13},$$ then is there any way to induce representation of $\mathbb{G}$ on $\mathcal{L}$ from $U$? How does the theory of induced represntations work in case of C*-quantum groups?
