Hecke Modifications, Wonderful Compactifications and Moduli of Principal Bundles

TL;DR

This paper develops parametrizations of the moduli space of principal bundles over Riemann surfaces using Hecke modifications and the De Concini--Procesi compactification, advancing understanding of their geometric structures.

Contribution

It introduces new parametrization methods for principal bundle moduli spaces via Hecke modifications and utilizes the wonderful compactification for deformation theory.

Findings

Parametrizations are constructed for specific structure groups.

Universal Hecke modifications are explicitly described.

The De Concini--Procesi compactification aids in deformation analysis.

Abstract

In this paper, we obtain parametrizations of the moduli space of principal bundles over a compact Riemann surface using spaces of Hecke modifications in several cases. We begin with a discussion of Hecke modifications for principal bundles and give constructions of "universal" Hecke modifications of a fixed bundle of fixed type. This is followed by an overview of the construction of the "wonderful," or De Concini--Procesi, compactification of a semi-simple algebraic group of adjoint type. The compactification plays an important role in the deformation theory used in constructing the parametrizations. A general outline to construct parametrizations is given and verifications for specific structure groups are carried out.