Strongly and Weyl transitive group actions on buildings arising from Chevalley groups

TL;DR

This paper investigates the properties of group actions on buildings derived from Chevalley groups, demonstrating the existence of subgroups with specific transitivity properties and providing new examples beyond classical cases.

Contribution

It proves the existence of elements in the Weyl group with finite order representatives, enabling the construction of subgroups acting Weyl transitively but not strongly transitively on affine buildings.

Findings

Existence of non-trivial Weyl group elements with finite order representatives.

Construction of subgroups acting Weyl transitively but not strongly transitively.

Examples extend known cases from SL_2(Q_p) to general Chevalley groups.

Abstract

Let K be a field and g(K) a Chevalley group (scheme) over K. Let (B,N) be the standard spherical BN-pair in g(K), with T=B\cap N and Weyl group W=N/T. We prove that there exist non-trivial elements w\in W such that all representatives of w in N have finite order. This allows us to exhibit examples of subgroups of g(Q_p) that act Weyl transitively but not strongly transitively on the affine building Delta associated with g(Q_p). Such examples were previously known only in the case when g(Q_p)=SL_2(Q_p) and Delta is a tree.