Holomorphic current groups -- Structure and Orbits

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

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