zbMATH — the first resource for mathematics

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).

20E42 Groups with a \(BN\)-pair; buildings
20G25 Linear algebraic groups over local fields and their integers
Full Text: DOI arXiv