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.