  This paper considers the enhanced symplectic "category" for purposes of
quantizing quasi-Hamiltonian $G$-spaces, where $G$ is a compact simple Lie
group. Our starting point is the well-acknowledged analogy between the
cotangent bundle $T^*G$ in Hamiltonian geometry and the internally fused double
$D(G)=G\times G$ in quasi-Hamiltonian geometry. Guillemin and Sternberg
consider the former, studing half-densities and phase functions on its
so-called character Lagrangians $\Lambda_{\mathcal{O}}\subseteq T^*G$. Our
quasi-Hamiltonian counterpart replaces these character Lagrangians with the
universal centralizers $\Lambda_{\mathcal{C}}\longrightarrow\mathcal{C}$ of
regular, $\frac{1}{k}$-integral conjugacy classes $\mathcal{C}\subseteq G$. We
show each universal centralizer to be a "quasi-Hamiltonian Lagrangian" in
$D(G)$, and to come equipped with a half-density and phase function.
  At the same time, we consider a Dehn twist-induced automorphism
$R:D(G)\longrightarrow D(G)$ that lacks a natural Hamiltonian analogue. Each
quasi-Hamiltonian Lagrangian $R(\Lambda_{\mathcal{C}})$ is shown to have a
clean intersection with every $\Lambda_{\mathcal{C}'}$, and to come equipped
with a half-density and phase function of its own. This leads us to consider
the possibility of a well-behaved, quasi-Hamiltonian notion of the BKS pairing
between $R(\Lambda_{\mathcal{C}})$ and $\Lambda_{\mathcal{C}'}$. We construct
such a pairing and study its properties. This is facilitated by the nice
geometric fearures of $R(\Lambda_{\mathcal{C}})\cap\Lambda_{\mathcal{C}'}$ and
a reformulation of the classical BKS pairing. Our work is perhaps the first
step towards a level-$k$ quantization of $D(G)$ via the enhanced symplectic
"category".
