Representatives of elliptic Weyl group elements in algebraic groups

TL;DR

This paper investigates the properties of elliptic elements in Weyl groups and determines the order of their representatives in associated semisimple algebraic groups, revealing specific order relations in most cases.

Contribution

It provides a comprehensive analysis of the order of representatives of elliptic Weyl group elements in semisimple algebraic groups, extending understanding of their algebraic structure.

Findings

For elliptic elements w of order d, their representatives g in G typically have order d.

In simple groups not of type C_n or F_4, the order of g matches that of w.

The paper clarifies the order behavior of elliptic elements' representatives across different group types.

Abstract

An element w of a Weyl group W is called elliptic if it has no eigenvalue 1 in the standard reflection representation. We determine the order of any representative g in a semisimple algebraic group G of an elliptic element w in the corresponding Weyl group W. In particular if w has order d and G is simple of type different from C_n or F_4, then g has order d in G.