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Daniel Skodlerack
The centralizer of a classical group and Bruhat-Tits buildings
(Le centralisateur d’un groupe classique et l’immeuble de Bruhat-Tits)
Annales de l'institut Fourier, 63 no. 2 (2013), p. 515-546, doi: 10.5802/aif.2768
Article PDF | Reviews MR 3112840 | Zbl 06193039
Class. Math.: 11E57, 11E95, 14L35, 20E42, 20G25
Keywords: Building, classical group over a local field, centralizer

Résumé - Abstract

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 $\mathfrak{B}^1(H)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building $\mathfrak{B}^1(G)$ 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 $\mathfrak{B}^1(H)$ to $\mathfrak{B}^1(G)$ which preserve the Moy–Prasad filtrations. We prove that if there is no split torus in the center of the connected component of $H$ then $j$ and $j^{\prime}$ are equal, and in general if both maps are affine and satisfy a mild equivariance condition they differ up to a translation of $\mathfrak{B}^1(H).$

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