UC Santa Cruz

In this work we aim to describe, in significant detail, certain filtrations and sequences of lattices inside of the split octonion algebra O, when O is constructed over the field Q_p. There are several reasons why one would like to consider such lattice filtrations, not the least of which is the connection that they have to certain filtrations of subgroups of the automorphism group G_2 = Aut(O).

It is an eventual goal to try to uncover previously unknown supercuspidal representations of G_2 by examining representations of the subgroups making up these subgroup filtrations, which are known as Moy-Prasad filtrations. When G_2 is constructed over Q_p, each of the filtration subgroups will be compact and open, and normal in the previous subgroup in the filtration, so that the respective quotients are all finite groups. A basic strategy then, is to identify representations of these finite quotients, extend them to representations of the filtration subgroups, and then induce them to representations of the whole group G_2.

To understand the lattice filtrations that identify the Moy-Prasad filtration subgroups, the work of W.T. Gan and J.K. Yu is indispensable. In their article, they draw connections between certain lattice filtrations, octonion orders, maximinorante norms, and points in the Bruhat-Tits building B(G_2). Their main idea was to use the norm-preserving quadratic form inherent in O, along with the natural 8-dimensional representation of G_2 = Aut(O), to create a canonical embedding of the building into B(SO(O)). Since the latter building had been previously described as the set of ``maximinorante norms'' on O, they arrived at an explicit description of B(G_2) in terms of certain maximinorante norms and orders in O.

The present work then attempts to describe all of these structures in detail, beginning with general composition algebras and the construction of the group G_2. Then we will construct the Bruhat-Tits building B(G_2) via the coroot lattice of type G_2, though we will mainly concern ourselves only with the standard apartment in B(G_2). Finally we will define our lattice filtrations and draw the connections between them and points B(G_2) , which will reveal the action of the group G_2 on its own building. Along the way, we identify many important structures and facts about the group itself.