
Journal of Lie Theory 19 (2009), No. 1, 055078 Copyright Heldermann Verlag 2009 Buildings of Classical Groups and Centralizers of Lie Algebra Elements Paul Broussous Université de Poitiers, Dép. de Mathématiques, Bd. Marie et Pierre Curie, Téléport 2  BP 30179, 86962 Futuroscope Chasseneuil, France broussou@math.univpoitiers.fr Shaun Stevens School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom Shaun.Stevens@uea.ac.uk [Abstractpdf] \def\g{{\frak g}} \def\h{{\frak h}} Let $F_\circ$ be a nonarchimedean locally compact field of residual characteristic not $2$. Let $G$ be a classical group over $F_\circ$ (with no quaternionic algebra involved) which is not a general linear group. Let $\beta$ be an element of the Lie algebra $\g$ of $G$ that we assume semisimple for simplicity. Let $H$ be the centralizer of $\beta$ in $G$ and $\h$ its Lie algebra. Let $I$ and $I_{\beta}^{1}$ denote the (enlarged) BruhatTits buildings of $G$ and $H$ respectively. We prove that there is a natural set of maps $j_{\beta} \colon I_{\beta}^{1}\rightarrow I$ which enjoy the following properties: they are affine, $H$equivariant, map any apartment of $I_{\beta}^{1}$ into an apartment of $I$ and are compatible with the Lie algebra filtrations of $\g$ and $\h$. In a particular case, where this set is reduced to one element, we prove that $j_{\beta}$ is characterized by the last property in the list. We also prove a similar characterization result for the general linear group.\par\smallskip In this article, we work with Lie algebra filtrations defined by using lattice models of buildings. It is not clear that they coincide with the filtrations constructed by A. Moy and G. Prasad for a general reductive group. This fact is proved by B. Lemaire (see his article in this volume, pp. 2954). Keywords: BruhatTits building, classical groups over padic fields, Moy and Prasad filtrations, functoriality of affine buildings. MSC: 22F50 [ Fulltextpdf (295 KB)] for subscribers only. 