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 \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.