1. A Theory of Minimal $K$-Types for Flat $G$-Bundles
- Author
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Daniel S. Sage and Christopher L. Bremer
- Subjects
Pure mathematics ,Degree (graph theory) ,General Mathematics ,Ramification (botany) ,010102 general mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Reductive group ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,14D24 (Primary), 22E67, 34Mxx (Secondary) ,0103 physical sciences ,FOS: Mathematics ,Filtration (mathematics) ,Point (geometry) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematical Physics ,Mathematics - Representation Theory ,Real number ,Mathematics - 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., Comment: 37 pages. A new trivialization-free definition of containment of a stratum in a flat G-bundle has been added. The paper has been reorganized so that the main definitions and results are given in Section 2
- Published
- 2017