KW-sections for exceptional type Vinberg's $\theta$-groups

TL;DR

This paper classifies automorphisms of certain algebraic groups of types F4, G2, D4, describes their structure, and confirms the existence of KW-sections, supporting Popov's conjecture for these cases.

Contribution

It provides a classification of positive rank automorphisms, their Kac diagrams, and the structure of little Weyl groups for specific Vinberg $ heta$-groups, confirming the existence of KW-sections.

Findings

All studied $ heta$-groups have KW-sections.

Classification of positive rank automorphisms and their Kac diagrams.

Description of the little Weyl group in each case.

Abstract

Let $k$ be an algebraically closed field of characteristic not equal to 2 or 3, let $G$ be an almost simple algebraic group of type $F_4$, $G_2$ or $D_4$ and let $\theta$ be an automorphism of $G$ of finite order, coprime to the characteristic. In this paper we consider the $\theta$-group (in the sense of Vinberg) associated to these choices; we classify the positive rank automorphisms and give their Kac diagrams and we describe the little Weyl group in each case. As a result we show that all such $\theta$-groups have KW-sections, confirming a conjecture of Popov in these cases.