Towards a quantization of the double via the enhanced symplectic category

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".