## 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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### References:

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