On the unipotent characters of the Ree groups of type G_2

TL;DR

This paper investigates the unipotent characters of Ree groups of type G_2, determining associated roots of unity and verifying a conjecture about their Fourier matrices using Shintani descent.

Contribution

It identifies roots of unity linked to unipotent characters and confirms a conjecture on Fourier matrices for Ree groups of type G_2.

Findings

Roots of unity associated with unipotent characters are explicitly determined.

The Fourier matrix satisfies the Digne-Michel conjecture for Ree groups of type G_2.

Shintani descent is effectively used as a main tool.

Abstract

This note is concerned with the unipotent characters of the Ree groups of type G_2. We determine the roots of unity associated by Lusztig and Digne-Michel to unipotent characters of ^2G_2(3^{2n+1}) and we prove that the Fourier matrix of ^2G_2(3^{2n+1}) defined by Geck and Malle satisfies a conjecture of Digne-Michel. Our main tool is the Shintani descent of Ree groups of type G_2.