A finite dimensional proof of the Verlinde formula

We prove two recurrence relations among dimensions Dg(r,d,ω):=dimH0(UC,ω,⊖UC,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D_g}\left( {r,d,\omega } \right): = \dim \,{{\rm{H}}^0}\left( {{{\cal U}_{C,\omega }},{ \ominus _{{{\cal U}_{C,\,\omega }}}}} \right)$$\end{document} of spaces of generalized theta functions on the moduli spaces UC, ω. By using these recurrence relations, an explicit formula (the Verlinde formula) of Dg(r, d, ω) is proved (see Theorem 4.3).