Embeddings of local fields in simple algebras and simplicial structures

Abstract

We give a geometric interpretation of Broussous{Grabitz embedding types. We fix a central division algebra D of finite index over a non-Archimedean local field F and a positive integer m. Further we fix a hereditary order a of Mm(D) and an unramified field extension EjF in Mm(D) which is embeddable in D and which normalizes a. Such a pair (E, a) is called an embedding. The embedding types classify the GLm(D)-conjugation classes of these embeddings. Such a type is a class of matrices with non-negative integer entries. We give a formula which allows us to recover the embedding type of (E, a) from the simplicial type of the image of the barycenter of a under the canonical isomorphism, from the set of Ex-fixed points of the reduced building of GLm(D) to the reduced building of the centralizer of Ex in GLm(D). Conversely the formula allows to calculate the simplicial type up to cyclic permutation of the Coxeter diagram.