Embedding types and canonical affine maps between Bruhat-Tits buildings of classical groups (Thesis)

TL;DR

This thesis generalizes the construction of compatible maps between Bruhat-Tits buildings for classical groups over p-adic fields, and explores how these maps encode embedding types via geometric properties.

Contribution

It extends existing results on CLF-maps to quaternionic cases and introduces a method to decode embedding invariants through building geometry.

Findings

Existence of affine CLF-maps for quaternionic unitary groups.

Uniqueness of CLF-maps under certain equivariance conditions.

Decoding of embedding invariants via CLF-map geometry.

Abstract

P. Broussous and S. Stevens studied maps between enlarged Bruhat-Tits buildings to construct types for p-adic unitary groups. They needed maps which respect the Moy-Prasad filtrations. That property is called (CLF), i.e. compatibility with the Lie algebra filtrations. In the first part of this thesis we generalise their results on such maps to the Quaternion-algebra case. Let k0 be a p-adic field of residue characteristic not two. We consider a semisimple k0-rational Lie algebra element beta of a unitary group G:=U(h) defined over k0 with a signed hermitian form h. Let H be the centraliser of beta in G. We prove the existence of an affine H(k0)-equivariant CLF-map j from the enlarged Bruhat-Tits building B^1(H,k0) to B^1(G,k0). As conjectured by Broussous the CLF-property determines j, if none of the factors of H is k0-isomorphic to the isotropic orthogonal group of k0-rank one and all factors are unitary groups. Under the weaker assumption that the affine CLF-map j is only equivariant under the center of H^0(k0) it is uniquely determined up to a translation of B^1(H,k0). The second part is devoted to the decoding of embedding types by the geometry of a CLF-map. Embedding types have been studied by Broussous and M. Grabitz. We consider a division algebra D of finite index with a p-adic center F. The construction of simple types for GLn(D) in the Budhnell-Kutzko framework required an investigation of strata which had to fulfil a rigidity property. Giving a stratum especially means to fix a pair (E,a) consisting of a field extension E|F in Mn(D) and a hereditary order a which is stable under conjugation by E^x, in other words we fix an embedding of E^x into the normalizer of a. Broussous and Grabitz classified these pairs with invariants. We describe and prove a way to decode these invariants using the geometry of a CLF-map.