Some remarks on monodromy

TL;DR

This work studies hypoelliptic symbols on a very regular Lie-group framework and analyzes monodromy for spectral stratifications via Nilsson–Bäcklund results. It develops a comprehensive system of conjugation, polar structures, and regularity notions (monogenity, interpolation, and continuity) to characterize how representations behave under constrained deformations, with Brelot mappings and rectifiable structures playing a central role. The approach culminates in linking monodromy to regular continuation, contraction to lower-dimensional subgroups, and the preservation of geometric properties (geodesic curvature, polar geometry) across stratifications, thereby clarifying when monogenic behavior yields well-behaved spectral projections. The results offer a rigorous pathway to understand spectral stratifications of hypoelliptic symbols on very regular groups and provide tools for recognizing when monodromy ensures nontrivial intersections and regular contractions at the boundary. Practical impact lies in a structured, geometry-driven framework for continuation and spectral analysis in non-elliptic settings.

Abstract

We consider hypoelliptic symbols over a very regular Lie group and discuss monodromy for a spectral stratification using results of Nilsson and Bäcklund.

Theorems & Definitions (19)

• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Example 3.3
• Lemma 4.1
• Lemma 6.1
• Lemma 7.1
• Proposition 7.2
• Lemma 9.1
• Example 9.1