Elements with finite Coxeter part in an affine Weyl group

TL;DR

This paper characterizes elements with finite Coxeter part in affine Weyl groups, showing their conjugacy classes correspond to Coxeter elements in maximal proper parabolic subgroups, extending to twisted classes.

Contribution

It provides a classification of conjugacy classes with finite Coxeter part in affine Weyl groups, identifying unique maximal parabolic subgroups associated with these classes.

Findings

Each conjugacy class with finite Coxeter part has a unique maximal proper parabolic subgroup.

Minimal length elements in these classes are exactly Coxeter elements of the subgroup.

Results extend to twisted conjugacy classes.

Abstract

Let $W_a$ be an affine Weyl group and $\eta:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $\eta(w)$ is conjugate to a Coxeter element of $W_0$. The elements with finite Coxeter part is a union of conjugacy classes of $W_a$. We show that for each conjugacy class $\mathcal{O}$ of $W_a$ with finite Coxeter part there exits a unique maximal proper parabolic subgroup $W_J$ of $W_a$, such that the set of minimal length elements in $\mathcal{O}$ is exactly the set of Coxeter elements in $W_J$. Similar results hold for twisted conjugacy classes.