We present a compact formula computing rational trigonometric sums. Such sums appeared in the work of E. Verlinde on the dimension of conformal blocks in Wess-Zumino-Witten (WZW) theory. As an application, we show that a formula of J.-M. Bismut and F. Labourie for the Riemann-Roch numbers of moduli spaces of flat connections on a Riemann surface coincides with Verlinde's expression.
"Residue theorem for rational trigonometric sums and Verlinde's formula." Duke Math. J. 118 (2) 189 - 227, 1 June 2003. https://doi.org/10.1215/S0012-7094-03-11821-9