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Manuscripta Mathematica

, Volume 144, Issue 1–2, pp 277–301 | Cite as

Field embeddings which are conjugate under a p-adic classical group

  • Daniel Skodlerack
Article

Abstract

Let (V, h) be a Hermitian space over a division algebra D which is of index at most two over a non-Archimedean local field k of residue characteristic not 2. Let G be the unitary group defined by h and let \({\sigma}\) be the adjoint involution. Suppose we are given two \({\sigma}\)-invariant but not \({\sigma}\)-fixed field extensions E 1 and E 2 of k in End D (V) which are isomorphic under conjugation by an element g of G and suppose that there is a point x in the Bruhat–Tits building of G which is fixed by \({E_1^{\times}}\) and \({E_2^{\times}}\) in the reduced building of Aut D (V). Then E 1 is conjugate to E 2 under an element of the stabilizer of x in G if E 1 and E 2 are conjugate under an element of the stabilizer of x in Aut D (V) and a weak extra condition holds. In addition, in many cases the conjugation by g from E 1 to E 2 can be realized as conjugation by an element of the stabilizer of x in G. Further we give a concrete description of the canonical isomorphism from the set of \({E_1^\times}\) fixed points of the building of G onto the building of the centralizer of E 1 in G.

Mathematics Subject Classification

51F25 20E42 51E24 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität MünsterMunsterGermany

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