  Let K be a finite-dimensional, 1-connected complex Lie group, and let
\Sigma_k=\Sigma - {p_1,\ldots,p_k\} be a compact connected Riemann surface
\Sigma, from which we have extracted k > 0 distinct points. We study in this
article the regular Frechet-Lie group O(\Sigma_k,K) of holomorphic maps from
\Sigma_k to K and its central extension \widehat{O(\Sigma_k,K)}. We feature
especially the automorphism groups of these Lie groups as well as the coadjoint
orbits of \widehat{O(\Sigma_k,K)} which we link to flat K-bundles on \Sigma_k.
