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 σ be the adjoint involution. Suppose we are given two σ-invariant but not σ-fixed field extensions E1 and E2 of k in EndD(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 and E × 2 in the reduced building of AutD(V ). Then E1 is conjugate to E2 under an element of the stabilizer of x in G if E1 and E2 are conjugate under an element of the stabilizer of x in AutD(V ) and a weak extra condition. In addition in many cases the conjugation by g from E1 to E2 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 E1-times fixed points of the building of G onto the building of the centralizer of E1 in G.Proposition 1.1 If two embeddings (E i , a) of A with k-algebra isomorphic fields and the same hereditary order Proposition 1.3 Let φ 1 and φ 2 be two σ ′ -σ-equivariant k-algebra homomorphisms from E|k into A. Then, φ 1 • φ −1 2 can be described as a conjugation under G if and only if (V, h φ1 ) is isomorphic to (V, h φ2 ) as signed Hermitian modules.

Endo-parameters for p-adic classical groups

“…An observation of the second author in [17], we use later without reference, is the following useful corollary of Lemma 2.7.…”

Endo-parameters for p-adic classical groups

“…This construction becomes particularly useful when applied to two selfdual lattice sequences , with e( ) = e( ) (which we can always ensure by an affine translation) but with , possibly not G-conjugate. The lattice sequences † , † , as they are self-dual, regular and strict of the same o Fperiod, are conjugate in G † by [38,Proposition 5.2].…”

“…Doing the same with E /E o , we obtain a space (V , h ) isometric to (V, h) and a regular self-dual o E -lattice sequence with e( |o F ) = e( |o F ) and (0) # h = (0). By [38,Proposition 5.2], there is an isometry from (V, h) to (V , h ) which sends to so we may assume…”

“…By a self-dual †-construction we can reduce to the case of standard strict regular lattice sequences of the same period. By [38,Proposition 5.2] there is then an element of G which maps to . Thus θ and θ intertwine by an element of G, in the symplectic case by assumption and in the non-symplectic case by Lemma 6.5.…”

“…Take an isometry g of (V, h) which sends V 0 to V 0 and V ⊥ 0 to V ⊥ 0 . By [38,Proposition 5.2] we can modify g such that g is an element of P − ( ). Conjugating θ by g, we may assume without loss of generality that V 0 and V 0 coincide.…”