Holomorphic current groups -- Structure and Orbits

TL;DR

This paper investigates the structure and symmetries of holomorphic current groups formed by maps from punctured Riemann surfaces to complex Lie groups, focusing on their automorphisms and coadjoint orbits related to flat bundles.

Contribution

It introduces the structure of holomorphic current groups on punctured Riemann surfaces and analyzes their automorphism groups and coadjoint orbits, linking them to flat K-bundles.

Findings

Characterization of automorphism groups of holomorphic current groups

Description of coadjoint orbits in terms of flat K-bundles

Establishment of the relationship between group orbits and geometric structures

Abstract

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.