Let G be a unitary group over a nonarchimedean local field of odd residual characteristic. This paper concerns the study of the "wild part" of the irreducible smooth representations of G, encoded in a so-called "semisimple character". We prove two fundamental results concerning them, which are crucial steps towards a classification of the cuspidal representations of G. First we introduce a geometric combinatoric condition under which we prove an "intertwining implies conjugacy" theorem for semisimple characters, both in G and in the ambient general linear group. Second, we prove a Skolem-Noether theorem for the action of G on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of G which have the same characteristic polynomial must be conjugate under an element of G if there are corresponding semisimple strata which are intertwined by an element of G.It turns out that this condition is also sufficient to obtain an "intertwining implies conjugacy" result:Theorem (Theorem 10.2). Let θ P CpΛ, m, βq and θ 1 P CpΛ 1 , m, β 1 q be semisimple characters which intertwine, let ζ : I Ñ I 1 be the matching given by Theorem 10.1, and suppose that the condition (1.1) holds. Then θ is conjugate to θ 1 by an element ofŨpΛq. Now we turn to our results for classical groups, so we assume that our underlying strata rΛ, q, r, βs are self-dual -that is, β is in the Lie algebra of G and Λ is in the building of the centralizer in G of β (see [BS09]). Our first main result here is a Skolem-Noether theorem for semisimple strata, which is crucial in the sequel [KSS16]:Theorem (Theorem 7.12). Let rΛ, q, r, βs and rΛ 1 , q, r, β 1 s be two self-dual semisimple strata which intertwine in G, and suppose that β and β 1 have the same characteristic polynomial. Then, there is an element g P G such that gβg´1 " β 1 .In order to prove this statement, in Section 4 we analyse the Witt groups W˚pEq of finite field extensions E of F and trace-like maps from W˚pEq and W˚pF q .Given a self-dual semisimple stratum rΛ, q, r, βs, the set C´pΛ, m, βq of semisimple characters for G is obtained by restricting the semisimple characters in CpΛ, m, βq. (Equivalently, one may just restrict those senisimple characters which are invariant under the involution defining G.) Our final result is an "intertwining implies conjugacy" theorem for semisimple characters for G.Theorem (Theorem 10.3). Let θ´P C´pΛ, m, βq and θ 1 P C´pΛ, m, β 1 q be two semisimple characters of G, which intertwine over G, and assume that their matching satisfies (1.1). Then, θ´and θ 1 are conjugate under UpΛq "ŨpΛq X G.

For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field $${\mathbf {C}}$$ C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal $${\mathbf {C}}$$ C -representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart’s notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

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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.