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Some functorial properties of the Bruhat-Tits building. (English) Zbl 0937.20026

The subject of this article is the question whether the Bruhat-Tits building \({\mathcal{BT}}(G,K)\) of a connected, reductive \(K\)-group over a local field \(K\) depends functorially on the group \(G\). It is in the case of Levi subgroups – and a bit more general – that Bruhat and Tits have given a positive answer. Now let \(L/K\) be a Galois extension of complete, discrete valuation fields with perfect residue class fields and let \(G\subset H\) be an inclusion of connected, reductive \(K\)-groups. The main theorem of this article states that we obtain a continuous, isometrical map \({\mathcal{BT}}(G,L)\to{\mathcal{BT}}(H,L)\) which is \(G(L)\)- and \(\text{Gal}(L/K)\)-equivariant. Moreover the set of all such maps is completely determined by the image of a given special point \(x\in{\mathcal{BT}}(G,L)\). Let us fix a point \(\widetilde x\) to which \(x\) can be mapped under a map with the above properties and denote by \(\mathcal Z\) the centralizer of \(G(L)\) in \(H(L)\). The second aim of this article is to show that there is a bounded subset \(\Omega\) of \({\mathcal{BT}}(H,L)\) containing \(\widetilde x\) such that \(\mathcal{Z}\Omega\) is the set of all points to which \(x\) can be mapped. Moreover if \(G\) and \(H\) are split, then we can replace \(\mathcal{Z}\Omega\) by the convex hull of \({\mathcal Z}\widetilde x\).
Reviewer: E.Landvogt (Köln)

MSC:

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