The centralizer of a classical group and Bruhat Tits buildings

TL;DR

This paper investigates the relationship between the Bruhat-Tits buildings of a classical group and its centralizer, establishing the existence and uniqueness of an affine, equivariant map compatible with Lie algebra filtrations, especially when elements are separable.

Contribution

It proves the existence of a compatible affine H-equivariant map between buildings of G and its centralizer H, and characterizes its uniqueness under certain conditions, extending previous work.

Findings

Existence of an affine H-equivariant map compatible with Lie algebra filtrations.

The map is toral if the element is separable.

Uniqueness of the map under specified conditions.

Abstract

This notes are additional remarks to an article of Broussous and Stevens [arXiv:math/0402228v1]. We consider a unitary group G over a non-Archimedean local field k_0 of residue characteristic different from two and an element \beta\ of the Lie algebra \mf{g} of G. Let H be the centralizer of \beta\ in G. We further assume k_0[\beta] to be semisimple. We prove that there is an affine H-equivariant map between the Bruhat-Tits buildings B(H)\ra B(G) which is compatible with the Lie-algebra filtrations (CLF) and maps apartments into apartments. The map is toral if \beta\ is separable. For simplicity let us now assume that \beta\ is separable, especially the centralizer bH of \beta\ in the reductive algebraic group defined by G is itself reductive, defined over k_0 and a product of Weil restrictions of classical groups. It will be proven that the map is unique by the CLF-property if no factor contains a split torus in the center. In general it is unique up to translation of B(H) if we assume CLF, affineness and the equivariance under the center of bH^0(k_0). The proofs are written for the general case where \beta\ is not separable.