Buildings of classical groups and centralizers of Lie algebra elements

TL;DR

This paper constructs and characterizes natural affine, H-equivariant maps between Bruhat-Tits buildings of classical groups and their centralizers, revealing structural insights into their geometric and algebraic relationships.

Contribution

It introduces a canonical set of maps between buildings of classical groups and their centralizers, with a unique characterization in certain cases.

Findings

Existence of natural affine H-equivariant maps between buildings

Maps preserve apartment structures and are compatible with Lie algebra filtrations

Unique characterization of these maps in specific cases

Abstract

Let F_o be a non-archimedean locally compact field of residual characteristic not 2. Let G be a classical group over F_o (with no quaternionic algebra involved) which is not of type A_n for n>1. Let b be an element of the Lie algebra g of G that we assume semisimple for simplicity. Let H be the centralizer of b in G and h its Lie algebra. Let I and I_b denote the (enlarged) Bruhat-Tits buildings of G and H respectively. We prove that there is a natural set of maps j_b : I_b --> I which enjoy the following properties: they are affine, H-equivariant, map any apartment of I_b into an apartment of I and are compatible with the Lie algebra filtrations of g and h. In a particular case, where this set is reduced to one element, we prove that j_b is characterized by the last property in the list. We also prove a similar characterization result for the general linear group.