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On finite group actions on reductive groups and buildings. (English) Zbl 1020.22003
Let $$H$$ be a connected reductive group over a non-Archimedean local field $$k$$, and let $$F\subset \text{Aut}_k(H)$$ be a finite group of order $$f$$ not divisible by the residual characteristic of $$k$$. Let $$G$$ be the identity component of the subgroup of $$H$$ consisting of points fixed by $$F$$. The authors prove that $$G$$ is reductive (this is actually proved for any field $$k$$, provided $$\operatorname {char} k$$ does not divide $$f$$), and that the Bruhat-Tits building $$\mathcal B(G)$$ of $$G$$ can be identified with the set of $$F$$-fixed points of $$\mathcal B(H)$$. The last result was recently proved by several authors for a number of special cases. A similar problem for spherical buildings is also considered.

##### MSC:
 2.2e+21 General properties and structure of other Lie groups 2e+43 Groups with a $$BN$$-pair; buildings
##### Keywords:
reductive group; building; spherical building; automorphism
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