The centralizer of a classical group and Bruhat Tits buildings

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 β of the Lie algebra \mf{g} of G. Let H be the centralizer of β in G. We further assume k_0[β] 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 β is separable. For simplicity let us now assume that β is separable, especially the centralizer bH of β 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 β is not separable.