Intertwining Semisimple Characters for -Adic Classical Groups

Abstract

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.

Endo-parameters for p-adic classical groups

“…Now (i) follows since γ * is an isomorphism. On the other hand, (ii) is given by [39,Theorem 4.4] for a particular linear form, and follows in general by the proof of (i) since γ * maps the maximal element to itself.…”

“…A special case of Theorem 11.9, where the self-dual semisimple characters underlying the cuspidal types are assumed to be closely related, is proved in [26]. The proof of Theorem 11.9 combines the work of this paper to control the choice in arithmetic data in the construction of cuspidal representations, together with an intertwining implies conjugacy result for semisimple characters of [39], to show that it is always possible to arrange for this to be the case. study and forms a key part of our parametrization of intertwining classes of (self-dual) semisimple characters via endo-parameters which we introduce at the end of the paper.…”

“…Suppose we have semisimple characters θ ∈ C ( , r, ϕ(β)) and θ ∈ C ( , r , ϕ (β )) which intertwine in G = Aut F (V). The matching theorem of the second and third authors [39,Theorem 10.1], tells us that there exists a unique bijection ζ : I → I and g ∈ G such that gV i = V ζ(i) and g θ i and θ ζ(i) intertwine in Aut F (V ζ(i) ). In this case, we say that θ intertwines with θ with matching ζ .…”

“…Given endo-equivalent pss-characters as in Theorem 9.9, we call the map ζ of (i) a matching. The condition, dim F (V i ) = dim F (V ζ (i) ) for all i ∈ I, in (ii) is necessary, as follows from [39,Theorem 10.1]. In the special case of ps-characters, of course, this condition is automatic.…”

“…It follows from [39,Theorem 5.2] that − ((V, h), ϕ, , r ) intertwines with − ((V, h), ϕ , , r ) in G if and only if ϕ and ϕ are conjugate in G. However, to develop endo-equivalence of self-dual pss-characters-where we may be dealing with embeddings of non-isomorphic fields-we need a more general notion than conjugacy. With this in mind, in Sect.…”