Elements of finite order in the normalizer of a maximal torus of a semisimple group

TL;DR

The paper characterizes elements of finite order in the normalizer of a maximal torus in a semisimple group, describing their geometric structure and relation to Weyl group actions.

Contribution

It provides a detailed geometric description of finite order elements in the normalizer of a maximal torus, linking their structure to Weyl group fixed points.

Findings

Finite order elements form finitely many irreducible subvarieties.

Dimensions of these subvarieties relate to fixed vectors under Weyl group elements.

Each subvariety is an orbit under conjugation by the torus.

Abstract

We prove that the set of elements of a given finite order in the connected component $N_w$ of the normalizer $N_G(T)$ of a maximal torus $T$ of a semisimple group $G$ is either empty or a disjoint union of finitely many irreducible subvarieties $C_i$. The dimension of each $C_i$ equals the dimension of the subspace of fixed vectors for the action of the element $w$ of the Weyl group $W$ corresponding to the component $N_w$. Moreover, each $C_i$ is an orbit of the action of the torus $T$ on the component $N_w$ by conjugation.