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

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.

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Correspondence to Daniel Skodlerack.

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Skodlerack, D. Field embeddings which are conjugate under a p-adic classical group. manuscripta math. 144, 277–301 (2014). https://doi.org/10.1007/s00229-013-0654-6

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Mathematics Subject Classification

  • 51F25
  • 20E42
  • 51E24