A theory of minimal K-types for flat G-bundles

TL;DR

This paper extends the concept of minimal K-types to formal flat G-bundles, introducing the slope as a new invariant that characterizes irregular singularities and generalizes classical notions from flat connections.

Contribution

It develops a theory of minimal K-types for flat G-bundles, establishing the existence of fundamental strata and defining the slope as a new invariant.

Findings

Every flat G-bundle contains a fundamental stratum.

All fundamental strata in a bundle have the same rational depth.

Positive slope indicates irregular singularity.

Abstract

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.