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Compactifications of Bruhat-Tits buildings associated to linear representations. (English) Zbl 1131.20019
Let $$G$$ be a connected semisimple group over a non-Archimedean local field $$K$$. It is shown that every faithful geometrically irreducible representation $$\rho\colon G\to\text{GL}(V)$$ on a finite-dimensional $$K$$-vector space $$V$$ gives rise to a compactification $$\overline X(G)_\rho$$ of the Bruhat-Tits building for $$G$$ (Theorem 4.3). The compactifications given by two representations $$\rho$$ and $$\sigma$$ coincide if and only if the highest weights of $$\rho$$ and $$\sigma$$ lie in the same chamber face (Theorem 4.5).

##### MSC:
 20E42 Groups with a $$BN$$-pair; buildings 20G25 Linear algebraic groups over local fields and their integers
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