### The centralizer of a classical group and Bruhat-Tits buildings

Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G.$ We prove that the Bruhat-Tits building ${\U0001d505}^{1}\left(H\right)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building ${\U0001d505}^{1}\left(G\right)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and ${j}^{\prime}$ be maps from ${\U0001d505}^{1}\left(H\right)$ to ${\U0001d505}^{1}\left(G\right)$ which preserve the Moy–Prasad filtrations. We prove that...