The modular decomposition conjecture for principal series representations

Let $S$ be a surface with holes, let $S'$ be obtained by cutting $S$ along a simple loop, and let $\gamma_\pm$ be the two new boundary components. Let ${\bf V}_{G,S;\lambda}$ be the principal series representation assigned to $S$ with central character $\lambda$, and let $\chi$ range over dominant characters of $H({\mathbb R}_{>0})$. Modular decomposition conjecture. There is a natural ${\Gamma}_{G,S'}$-equivariant decomposition

$${\bf V}_{G,S;\lambda}\stackrel{\sim}{\longrightarrow}\int_{\chi}{\bf V}_{G,S';\lambda,\chi,\chi^{-1}}\,d\mu_\chi,$$

where ${\bf V}_{G,S';\lambda,\chi,\chi^{-1}}$ is the principal series representation assigned to the cut surface with central character $(\lambda,\chi,\chi^{-1})$. This is the proposed cutting or gluing law underlying the modular-functor conjecture; the decomposition and its equivariance are not established in the supplied text.