Holomorphic current groups -- Structure and Orbits

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.