Embeddings of local fields in simple algebras and simplicial structures on the Bruhat-Tits building

Details Embeddings of local fields in simple algebras and simplicial structures on the Bruhat-Tits building Embeddings of local fields in simple algebras and simplicial structures on the Bruhat-Tits building

2009-03-26

arxiv.org/abs/0903.4634v1

Mathematics

Group Theory

Scientific paper

This article answers a question that naturally arises from the articles by M.Grabitz and P. Broussous (see \cite{broussousGrabitz:00}) and P. Broussous and B. Lemaire (see \cite{broussousLemaire:02}). For an Azumaya-Algebra $A$ over a non-archimedean local field $F,$ M. Grabitz and P. Broussous have introduced embedding invariants for field embeddings, that is for pairs $(E,\textfrak{a})$, where $E$ is a field extension of $F$ in $A$, and $\textfrak{a}$ is a hereditary order which is normalised by $E^{\times}.$ On the other hand if we take such a field extension $E$ and define $B$ to be the centralizer of $E$ in $A,$ then $G:=A^{\times}$ are $G_{E}:=B^{\times}$ are sets of rational points of reductive groups defined over $F$ and $E$ respectively. P. Broussous and B. Lemaire have defined a map $j_E: {\cal I}^{E^\times}\to {\cal I}_E$, where ${\cal I}$ is the g.r. (geometric realization) of the euclidean building of $G$, and ${\cal I}_E$ is the g.r. of the euclidean building of $G_{E}.$ The question which we address is to relate the embedding invariants to the behavior of the map $j_E$ with respect to the simplicial structures of ${\cal I}$ and ${\cal I}_E.$ I have to thank very much Prof. Zink from Homboldt University Berlin for his helpful remarks, the revision of the work and for giving my the interesting task.

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