  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.
