# Green functions and Glauberman degree-divisibility

## TL;DR

This paper proves a key divisibility property for characters in finite groups of Lie type, completing a conjecture related to Green functions and the Glauberman correspondence, which enhances understanding of character degrees in group theory.

## Contribution

It provides a general proof that the degree-divisibility property holds universally for characters related by the Glauberman correspondence, confirming a conjecture from 1994.

## Key findings

- Degree-divisibility property holds for all relevant characters
- Completes the proof of a conjecture from Hartley and Turull
- Confirms the Green functions satisfy the necessary congruence condition

## Abstract

The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing the proof of that congruence condition. Consequently, the degree-divisibility property holds in complete generality.