## Group schemes and buildings of exceptional groups over a local field. I: The group $$G_ 2$$.(English)Zbl 1060.14063

Let $$A$$ be a discrete valuation ring with field of fractions $$k$$, and perfect residue field of characteristic $$p$$. Let $$G$$ denote a simple algebraic group over $$k$$ of type $$G_2$$. Denote by $$\mathcal{B}(G)$$ the extended building of $$G$$. This paper gives a Bruhat-Tits theory for $$G$$, focusing mostly on the case where $$G$$ is a split form of $$G_2$$.
The group $$G$$ can be realized as the automorphism group of an octonion algebra $$V$$, thereby giving it an 8-dimensional rational representation, the so-called standard representation of $$G$$. This representation gives a map $$\iota :G\hookrightarrow \mathrm{SO}(V),$$ which gives rise to an embedding $$\mathcal{B}(G) \hookrightarrow \mathcal{B}(\mathrm{SO}(V))$$ which is denoted $$\iota _{\ast }$$. The building of $$\mathrm{SO}(V)$$, having been described by F. Bruhat and J. Tits [Bull. Soc. Math. France 115, 141–195 (1987; Zbl 0636.20027)], is the set of maximinorante norms on $$V$$, and thus the authors are able to obtain several results about $$\mathcal{B}(G)$$:
– the image of $$\iota _{\ast }$$ is described, thereby giving a description of $$\mathcal{B}(G)$$ in terms of maximinorante norms on $$V$$: namely, the norms which are algebra norms for the octonion multiplication.
– the description of the simplicial complex structure of $$\mathcal{B}(G)$$ in terms of orders in $$V$$ is obtained. This enables the authors to describe the parahoric subgroups of $$G(k),$$ the associated $$A$$-group schemes, and the structure of apartments in $$\mathcal{B}(G)$$.
– the action of $$S_{3}$$ on $$\mathrm{Spin}(V)$$ whose group of fixed points is $$G$$ gives an action of $$S_{3}$$ on $$\mathcal{B}(\mathrm{Spin}(V))$$, and it is shown that the set of fixed points under this action is $$\mathcal{B}(G)$$.
In addition, the techniques employed in this paper, the determination of $$i_{\ast },$$ can be useful in other situations. For example, it can be used in determining the building of a classical group as a subset of the building of the ambient general linear group. In addition it can be used in the study of the building of a general trialitarian $$\mathrm{Spin}_8$$.

### MSC:

 14L15 Group schemes 20E42 Groups with a $$BN$$-pair; buildings 20G25 Linear algebraic groups over local fields and their integers

### Keywords:

Bruhat-Tits theory; octonion algebra; parahoric subgroups

Zbl 0636.20027
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