Lattice Filtrations for G_2 of a p-adic Field

TL;DR

This paper explores the structure of the group G_2 over p-adic fields by connecting its construction via octonion automorphisms and Lie algebra root diagrams, and describes its Bruhat-Tits building explicitly.

Contribution

It explicitly identifies Chevalley generators as octonion automorphisms and constructs the Bruhat-Tits building from the coroot diagram, linking algebraic and geometric perspectives.

Findings

Explicit identification of Chevalley generators as octonion automorphisms

Detailed description of norms and lattices in octonion algebra

Complete construction of the standard apartment of the Bruhat-Tits building

Abstract

The exceptional group $G_2$, when constructed over $Q_p$, may at the same time be considered as the group of automorphisms of an octonion algebra over $Q_p$, or alternatively it may be constructed as a Chevalley group from the root diagram of the appropriate Lie algebra, $g_2$. Each construction has its benefits and drawbacks, and the first part of this work focuses on drawing parallels between the two and describing certain group structures in terms of both. In particular, we explicitly identify the Chevalley generators of $G_2$ as automorphisms of octonions. In the second part of this work we describe the standard (affine) apartment of the Bruhat-Tits building of $G_2$, constructing it from the coroot diagram of the Lie algebra $g_2$. Previous work of W.T. Gan and J.K. Yu (2003) describes this Bruhat-Tits building, alternatively, in terms of certain "maximinorante norms" and lattices in the octonion algebra. Using our work from Part One, we describe these norms and lattices in detail, which allows us to act on our lattices using explicit octonion automorphisms and draw a very complete picture of the standard apartment.