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Embeddings of local fields in simple algebras and simplicial structures. (English) Zbl 1344.20068

Summary: 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 \(\mathfrak a\) of \(\mathrm M_m(D)\) and an unramified field extension \(E|F\) in \(\mathrm M_m(D)\) which is embeddable in \(D\) and which normalizes \(\mathfrak a\). Such a pair \((E,\mathfrak a)\) is called an embedding. The embedding types classify the \(\mathrm{GL}_m(D)\)-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 \((E,\mathfrak a)\) from the simplicial type of the image of the barycenter of \(\mathfrak a\) under the canonical isomorphism, from the set of \(E^\times\)-fixed points of the reduced building of \(\mathrm{GL}_m(D)\) to the reduced building of the centralizer of \(E^\times\) in \(\mathrm{GL}_m(D)\). Conversely the formula allows to calculate the simplicial type up to cyclic permutation of the Coxeter diagram.

MSC:

20G35 Linear algebraic groups over adèles and other rings and schemes
20E42 Groups with a \(BN\)-pair; buildings
16K20 Finite-dimensional division rings
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