Factorization of rank two theta functions. II: Proof of the Verlinde formula (Q1906503)

The authors extend the arguments used previously in ibid. 297, No. 3, 417-466 (1993; Zbl 0788.32012) to get a general result concerning the factorization of holomorphic sections of line bundles on the moduli spaces of rank two parabolic bundles on Riemann surfaces. The main result is the proof of the Verlinde formula for \(SU(2)\): For all \(g \geq 2\) and all integers \(n_1, \dots, n_m\), one has the equality \(D^{g,k}_{n_1, \dots, n_m} = N^{g, k}_{n_1, \dots, n_m}\), where \[ D^{g,k}_{n_1, \dots, n_m} = \begin{cases} \dim \Hom_{\mathcal G} (V_{n_1} \otimes \cdots \otimes V_{n_m}, H^0 ({\mathcal A}, \Delta^{\oplus k}), \quad & \text{if } n_1, \dots, n_m \geq 0 \\ 0, \quad & \text{otherwise} \end{cases} \] \[ \text{and } N^{g,k}_{n_1, \dots, n_m} = \sum^k_{l = 0} (S_{l0})^{2 - 2g} {S_{ln_1} \over S_{l0}} \cdots {S_{ln_m} \over S_{l0}}. \]