Flat G-bundles and regular strata for reductive groups

Let LG be an algebraic loop group associated to a reductive group G. A fundamental stratum is a triple consisting of a point x in the Bruhat-Tits building of LG, a nonnegative real number r, and a character of the corresponding depth r Moy-Prasad subgroup that satisfies a non-degeneracy condition. The authors have shown in previous work how to associate a fundamental stratum to a formal flat G-bundle and used this theory to define its slope. In this paper, the authors study fundamental strata that satisfy an additional regular semisimplicity condition. Flat G-bundles that contain regular strata have a natural reduction of structure to a (not necessarily split) maximal torus in LG, and the authors use this property to compute the corresponding moduli spaces. This theory generalizes a natural condition on algebraic connections (the GL_n case), which plays an important role in the global analysis of meromorphic connections and isomonodromic deformations.