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, studying 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 features 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”.

本文考虑了用于量化拟哈密尔顿 ( G ) 空间的增强辛“范畴” ，其中 ( G ) 是一个紧凑的简单李群。我们的出发点是哈密​​顿几何中的余切丛\(T^*G\)与拟哈密尔顿几何中的内部融合双\(D(G)=G\times G\)之间广为人知的类比。Guillemin 和 Sternberg 考虑前者，研究其所谓特征拉格朗日函数\(\Lambda _{{\mathcal {O}}}\subseteq T^*G\) 的半密度和相位函数。我们的准哈密顿对应物用通用中心化器\(\Lambda _{{\mathcal {C}}}\longrightarrow {\mathcal {C}}\)替换了这些字符拉格朗日常规的\(\frac{1}{k}\) -积分共轭类\({\mathcal {C}}\subseteq G\)。我们将每个通用扶正器展示为D ( G ) 中的“准哈密尔顿拉格朗日”，并配备了半密度和相位函数。同时，我们考虑缺乏自然哈密顿类似物的 Dehn 扭曲诱导自同构\(R:D(G)\longrightarrow D(G)\)。每个准哈密尔顿拉格朗日算子\(R(\Lambda _{{\mathcal {C}}})\)被证明与每个\(\Lambda _{{\mathcal {C}}}'}\)有一个干净的交集，并配备了自己的半密度和相位功能。这使我们考虑在\(R(\Lambda _{{\mathcal {C}}})\)和\(\Lambda _{{\数学 {C}}'}\)。我们构建了这样的配对并研究了它的特性。这得益于\(R(\Lambda _{{\mathcal {C}}})\cap \Lambda _{{\mathcal {C}}'}\)的良好几何特征和经典 BKS 的重新表述配对。我们的工作也许是通过增强的辛“类别”实现D（G）的k级量化的第一步。