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**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 \(\mathrm{Spin}_8\).

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 \(\mathrm{Spin}_8\).

Reviewer: Alan Koch (Decatur)

### MSC:

14L15 | Group schemes |

20E42 | Groups with a \(BN\)-pair; buildings |

20G25 | Linear algebraic groups over local fields and their integers |