Lattice Filtrations for G_2 of a p-adic Field

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.