Hereditary orders and embeddings of local fields in simple algebras
Let \(F\) be a commutative locally compact non-archimedean field, \(D\) a central \(F\)-division algebra of index \(d^2\), and \(V\) a right \(D\)-vector space of finite dimension. Let \(G\) denote the group of units of \(A\), the \(D\)-endomorphism ring of \(V\). The smooth complex representations of \(G\) have been classified if \(A\) is either split or a division algebra. In each case, certain field extensions \(E/F\subset A\), called pure extensions, play a rôle. \textit{A. Fröhlich} [Proc. Lond. Math. Soc., III. Ser. 54, 247-266 (1987; Zbl 0615.12021)] pointed out that certain pure extensions are undesirable when \(A\) is neither split nor a division algebra. Fröhlich described extensions that he called sound extensions, and the current author generalizes this to the notion he calls good extensions. He then gives a complete study of good extensions. Each extension \(E/F\) determines a sequence of lattices in \(V_E\), and hence, a hereditary order. The condition for goodness is explicitly described in terms of an embedding of the residue field \(k_E\) into a \(k_F\) algebra of finite dimension.
scientific article; zbMATH DE number 1218700
scientific article