Hecke modifications, wonderful compactifications and moduli of principal bundles (Q2844278)

The object of this paper (adapted from part of the author's doctoral thesis) is to parametrize the moduli space of principal bundles on a compact Riemann surface using Hecke modifications. This type of construction has been in use for a long time in connection with vector bundles, but, although there are many references in the literature to Hecke modifications of principal bundles, there seems to be no systematic account available. The author provides such an account in section 1 of the paper. Key elements are loop groups, affine Grassmannians and Bruhat cells. The section concludes with the construction of universal families of Hecke modifications of a fixed bundle.NEWLINENEWLINESection 2 contains an overview of wonderful compactifications, which are then used in section 3 to develop the deformation theory for the moduli of principal bundles. In section 4, the author outlines conditions for obtaining a parametrization of the moduli space. The idea is to carry out modifications to the fixed bundle to obtain a bundle whose structure group is reducible to a maximal torus. This bundle is not stable, but, if the family is of the right dimension, then there will be an open set of stable bundles near to it. This requires the vanishing of the first cohomology group of a certain vector bundle. In section 5, it is shown that the condition for vanishing is satisfied for certain groups (\(\mathrm{PGL}(4,\mathbb{C})\), \(\mathrm{PSp}(2l,\mathbb{C})\), \(\mathrm{PSO}(2l,\mathbb{C})\)) provided the genus is even.