  The theory of minimal K-types for p-adic reductive groups was developed in
part to classify irreducible admissible representations with wild ramification.
An important observation was that minimal K-types associated to such
representations correspond to fundamental strata. These latter objects are
triples (x, r, beta), where x is a point in the Bruhat-Tits building of the
reductive group G, r is a nonnegative real number, and beta is a semistable
functional on the degree r associated graded piece of the Moy-Prasad filtration
corresponding to x.
  Recent work on the wild ramification case of the geometric Langlands
conjectures suggests that fundamental strata also play a role in the geometric
setting. In this paper, we develop a theory of minimal K-types for formal flat
G-bundles. We show that any formal flat G-bundle contains a fundamental
stratum; moreover, all such strata have the same rational depth. We thus obtain
a new invariant of a flat G-bundle called the slope, generalizing the classical
definition for flat connections. The slope can also be realized as the minimum
depth of a stratum contained in the flat G-bundle, and in the case of positive
slope, all such minimal depth strata are fundamental. Finally, we show that a
flat G-bundle is irregular singular if and only if it has positive slope.
