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Let $G$ an agebraic group and $X$ it's spherical building, that is, $X$ is the set of maximal proper parabolic subgroups of $G$ and the simplexes of $X$ are the finite set of $X$ of the form $S=\{P_{1},...,P_{r}\}$, where $P_{i}\in X$, such taht $P_{1}\cap ....\cap P_{r}$ is parapolic subgroup. My question is : what is the appartement of $X$

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I guess you mean "what ARE the appartments of X" ? I am not sure I know the abstract definition of what an appartment should be in an abstract building, but in this case, I think appartments are in bijection with maximal split tori, and the appartment associated to such a torus T consists of all parabolic subgroups that contain T. – Jef Mar 19 '12 at 8:44
Jef is right. There are nice books on buildings, Ronan's "Buildings" is my favorite. If you want to see a quick introduction, then you can read Brown's "What is a building?", Notices Amer. Math. Soc. 49 (2002), no. 10, 1244-1245, or, for more details, his "Five lectures on buildings." Group theory from a geometrical viewpoint (Trieste, 1990), 254–295, World Sci. Publishing, River Edge, NJ, 1991. – Misha Mar 19 '12 at 13:38
Ok, thank you Mr Jef. – Rajkarov Mar 20 '12 at 3:14

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