# Jordan Decomposition for Formal G-Connections

## TL;DR

This paper provides straightforward proofs of the Jordan decomposition theorem for formal G-connections, extending classical results to the context of semisimple groups and emphasizing the analogy with linear algebra.

## Contribution

It offers simplified proofs of the Hukuhara-Levelt-Turrittin theorem and its generalization to formal G-connections, clarifying the conceptual parallels with linear algebra.

## Key findings

- Proofs of Jordan decomposition for formal G-connections
- Extension of classical theorems to semisimple group context
- Enhanced understanding of the analogy between linear and differential settings

## Abstract

A theorem of Hukuhara, Levelt, and Turrittin states that every formal differential operator has a Jordan decomposition. This theorem was generalised by Babbit and Varadarajan to the case of formal $G$-connections where $G$ is a semisimple group. In this paper, we provide straightforward proofs of these facts, highlighting the analogy between the linear and differential settings.