Document Type
Article
Publication Date
6-1-2018
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, β), where x is a point in the Bruhat-Tits building of the reductive group G, r is a nonnegative real number, and β 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 vector bundles. The slope can also be realized as the minimum depth of a stratum contained in the flat Gbundle, 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.
Publication Source (Journal or Book title)
International Mathematics Research Notices
First Page
3507
Last Page
3555
Recommended Citation
Bremer, C., & Sage, D. (2018). A theory of minimal K-types for flat G-bundles. International Mathematics Research Notices, 2018 (11), 3507-3555. https://doi.org/10.1093/imrn/rnw338
