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

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.