Endo-parameters for p-adic classical groups
aa r X i v : . [ m a t h . R T ] N ov Endo-classes for p -adic classical groups R. Kurinczuk ∗ , D. Skodlerack † , S. Stevens ‡ July 10, 2018
Abstract
For a unitary, symplectic, or special orthogonal group over a non-archimedean local fieldof odd residual characteristic, we prove that two intertwining cuspidal types are conjugatein the group. This completes work of the third author who showed that every irreduciblecuspidal representation of such a classical group is compactly induced from a cuspidal type,now giving a classification of irreducible cuspidal representations of classical groups in termsof cuspidal types. Our approach is to completely understand the intertwining of the so-called self dual semisimple characters, which form the fundamental step in the construction.To this aim, we generalise Bushnell–Henniart’s theory of endo-class for simple characters ofgeneral linear groups to a theory for self dual semisimple characters of classical groups, andintroduce (self dual) endo-parameters which parametrise intertwining classes of (self dual)semisimple characters.
Mathematics Subject Classification 2010:
Primary 22E50 (Representations of Lie andlinear algebraic groups over local fields); Secondary 11F70 (Representation-theoretic methods;automorphic representations over local and global fields)
Let G be a unitary, symplectic or special orthogonal group over a non-archimedean local field F of odd residual characteristic and let G + be either G in the unitary and symplectic case or theambient orthogonal group. In previous works, the third author has developed an approach tothe smooth dual of G via Bushnell–Kutzko types. In this article, we accomplish an importantpart of this programme and complete recent work ([12] and [18]) in refining the third author’sconstruction [22] of all cuspidal representations of G to a classification. Moreover, we develop anatural framework for our results on semisimple characters, the theory of endo-parameters forself dual semisimple characters of G + (and of G ), which contains in its core a generalisation ofBushnell–Henniart’s theory of endo-class ([5]).In [22], the third author explicitly constructs pairs ( J, λ ) consisting of a compact open sub-group J of G and an irreducible representation λ of J . If these pairs satisfy certain conditions ∗ Robert Kurinczuk, Department of Mathematics, Imperial College, London, SW7 2AZ, United King-dom. Email: [email protected] † Daniel Skodlerack, Department of Mathematics, Imperial College, London, SW7 2AZ, United King-dom. Email: [email protected] ‡ Shaun Stevens, School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United King-dom. Email: [email protected] cuspidal types . The main result of ibid. says that every irreducible cuspidalrepresentation π of G contains a cuspidal type ( J, λ ), and that in this case it is compactlyinduced π ≃ ind GJ λ . If π contains two cuspidal types then the cuspidal types necessarily inter-twine, and in the first main result of this paper we show: Theorem A (intertwining implies conjugacy).
Cuspidal types intertwine in G if and onlyif they are conjugate in G .As conjugate cuspidal types induce isomorphic representations, this completes a classificationby types of the irreducible cuspidal representations of G . This result was expected by analogywith other classifications of cuspidal representations via types for other connected reductivegroups (such as Bushnell–Kutzko for GL n [9], the third author and S´echerre for inner formsof GL n [16], and Hakim–Murnaghan for all connected reductive groups under tame conditions[11]), but our proof has required a substantial amount of work and relies on the main results ofa number of papers (recently, [12] and [18]). We expect this theorem to find many applicationsin arithmetic - and will be useful whenever detailed analysis of cuspidal representations of G isrequired.In the second theme of this paper, we generalise Bushnell and Henniart’s notions of potentialsimple character and endo-class, as defined in [5], to self dual potential semisimple charactersand endo-class for G . As well as appearing in an essential way in our proof of Theorem A, thistheory warrants independent study and allows a parametrisation of intertwining classes of (selfdual) semisimple characters of GL n , G + and of G via endo-parameters which we introduce atthe end of the paper.Potential applications include a local Langlands correspondence for endo-parameters or Ramifi-cation Theorem for classical groups (see [7, 6.1 Theorem] and, recently, [8] for the state of theart for GL F ( V )), and a decomposition via endo-parameters of the category of all smooth repre-sentations of G (see [6, Type Theorem] for simple characters of GL F ( V )). Both of which are thesubject of current work of the authors. Let W F denote the Weil group of F , and P F the wildinertia subgroup of W F . We expect the local Langlands correspondence for endo-parameters totake the following form: Conjectural form of the LLC for endo-parameters.
Suppose that the defining algebraicgroup for G is quasi-split.(i) There is a unique bijection, compatible with the local Langlands correspondence for G ,between the set of endo-parameters of self dual semisimple characters for G (forgettingWitt type data) and the set of W F -conjugacy classes of representations of P F which areextendible to a Langlands parameter W F × SL ( C ) → L G ( C ).(ii) Let φ : W F → L G ( C ) be a Langlands parameter for an L -packet Π( φ ) of G . Thereis a unique bijection, compatible with the local Langlands correspondence for G , be-tween endo-parameters of representations in Π( φ ) and the irreducible representations of C L G (im( φ | P F )) /Z L G ( C ).For symplectic groups Part (i) of the Conjecture follows from current work of the third authorwith Blondel and Henniart [2].Let F/F be a quadratic or trivial extension with Gal( F/F ) = h σ i . If V is an F -vector spacewith a ε -hermitian form h defining a unitary, symplectic or orthogonal group G + = U( V, h ),we let Σ denote the cyclic group of order two generated by the inverse of the adjoint anti-involuton of h so that G + = GL( V ) Σ . We fix a character of the additive group F + trivial2n F . Let E = F [ β ] be a sum of field extensions of F , and k be an integer satisfying certainbounds (see Section 8) - a semisimple pair . To a quadruple ( V, ϕ, Λ , r ) consisting of an F -vectorspace V , an embedding ϕ : E → End F ( V ), a ϕ ( o E )-lattice sequence in V , and an integer r closelyrelated to k , using work of Bushnell–Kutzko [9] and the third author [21], we can associate aset C (Λ , r, ϕ ( β )) of semisimple characters of a compact open subgroup of GL( V ) (whose grouplevel is controlled by k ). If in this quadruple V is equipped with a hermitian form h and therest of the data satisfies certain duality properties, Σ acts on this set with fixed points, and thesubset of characters fixed by Σ defines by unique restriction to G + the set C − (Λ , r, ϕ ( β )) of selfdual semisimple characters of G + . Moreover, there are natural transfer maps between the setsof characters defined by different quadruples.A pss-character (resp. self dual pss-character ) is a function from the class of all quadru-ples ( V, ϕ, Λ , r ) (resp. (( V, h ) , ϕ, Λ , r )) to the class of all semisimple characters (resp. class ofall self dual semisimple characters) whose values are related by transfer. For two pss-characters(resp. self dual pss-characters), we define comparison pairs in Section 8; these are essentiallypairs of quadruples with the same underlying space (resp. hermitian space), where we can testif the values of the pss-characters (resp. self dual pss-characters) intertwine. Note that, in thesymplectic case, we need a more restrictive notion of comparison pair a Witt comparison pair (see Section 8). We call pss-characters (resp. self dual pss-characters) endo-equivalent if when-ever we can compare their values they intertwine. Our second main theorem is the surprisingfact that this definition is equivalent to just having one such intertwining pair of values:
Theorem B (10.6).
Let Θ − and Θ ′− be two self dual pss-characters and Θ and Θ ′ their lifts.Then, the following assertions are equivalent:(i) The self dual pss-characters Θ − and Θ ′− are endo-equivalent;(ii) The lifts Θ and Θ ′ are endo-equivalent.(iii) There is a comparison pair such that the corresponding realisations of Θ − and Θ ′− inter-twine over G + .An important corollary of Theorem B is a transitivity of G + -intertwining of self dual semisimplecharacters statement in Corollary 10.7, which we extend to an analogous statement for G -intertwining of semisimple characters in Corollary 12.3. While intertwining of characters isobviously a reflexive and symmetric relation in general, it is clearly not necessarily transitive,thus the transitivity statement we obtain is a reflection of the structure in the collection of all(self dual) semisimple characters. These statements are key to our proof of intertwining impliesconjugacy for cuspidal types (Theorem A).By a deep result of Bushnell–Henniart [6, Intertwining Theorem], simple characters of generallinear groups intertwine if and only if they are endo-equivalent. This not only implies thatintertwining of simple characters with same group level is transitive, it also shows that endo-classes parametrise intertwining classes of simple characters of GL n . In the final section weprove a broad generalisation of this result to semisimple and self dual semisimple characters byintroducing endo-parameters .A semisimple character is called full if it lies in a set of semisimple characters C (Λ , , β ), andan endo-class is called full if it contains a pss-characters supported on a semisimple pair ( k, β )with k = 0. A GL -endo-parameter is a function from the set of all full simple endo-classes E tothe natural numbers (including zero) with finite support. Theorem C (14.7).
The set of intertwining classes of full semisimple characters for GL F ( V )3s in bijection with the set of those endo-parameters f which satisfy X c ∈E deg( c ) f ( c ) = dim F V. The definition of endo-parameters for classical groups is much more intricate. We associate toa self dual simple character a Witt tower of hermitian spaces over a self dual field extensionof F , and show that if two skew semisimple characters intertwine then blockwise their Witttowers match . We encode this information into an invariant of a skew semisimple characterwe call its Witt type , and denote the set of all Witt types for ( σ, ǫ ) by W σ,ǫ . A self dualsemisimple character is called elementary if it is skew or its associated indexing set contains twoelements. A ( σ, ǫ ) -endo-parameter is a map from the set of full elementary ( σ, ǫ )-endo-classes E − to N × W σ,ǫ with finite support. Our final main result shows that ( σ, ǫ )- endo-parameters parametrise intertwining classes of full self dual semisimple characters: Theorem D (14.14).
The set of intertwining classes of full self-dual semisimple charactersfor G + is in bijection with the set of ( σ, ǫ )-endo-parameters f = ( f , f ) which satisfy X c ∈E − deg( c )(2 f ( c ) + diman( f ( c ))) = dim F V, such that the sum of the Witt tower over F of the f ( c )s is equal to the Witt tower of h . Weextend this theorem to parametrise intertwining classes of self dual semisimple characters ofspecial orthogonal groups in Corollary 14.15.We now describe the structure of the article. In Section 2, we carefully recast definitions andkey results relevant to the sequel. In Section 3, we introduce comparison pairs, allowing us inSection 4 to define endo-equivalence for simple characters for unitary and special orthogonal,but not symplectic groups. The symplectic case requires quite a lot more work. We introducematching Witt towers in Section 5 and more restrictive comparison pairs where we requirethe corresponding Witt towers to match called Witt comparison pairs . The main result ofSection 5 shows that matching of Witt towers is inherited along defining sequences of simplestrata. The additional condition of Witt comparison pairs allows us in Section 7 to defineendo-equivalence for simple characters for symplectic groups following a study of intertwiningsimple characters for symplectic groups in Section 6. In Section 8, we recall definitions ofsemisimple strata and characters, and define potential semisimple characters. In Section 9,making significant use of recent work of the second and third authors [18], we investigateintertwining semisimple characters. In Section 10, we can finally introduce semisimple endo-equivalence and prove Theorem B. Section 11 generalises results on intertwining and conjugacyof skew characters of the second and third authors [18] to self dual characters, and Section 12investigates the difference between intertwining under orthogonal and special orthogonal groups.We then deduce Theorem A (intertwining implies conjugacy for cuspidal types) in Section 13,using the work of the previous sections to reduce to the level zero part where it is known thanksto recent results of the first and third authors [12]. Section 14 introduces endo-parameters andproves Theorems C & D on the parametrisation of intertwining classes of (self dual) semisimplecharacters by endo-parameters.
Acknowledgements.
This work was supported by the Engineering and Physical Sciences Re-search Council (EP/H00534X/1 and EP/M029719/1), the Heilbronn Institute for MathematicalResearch, and Imperial College London. We thank David Helm for his interest and fruitful con-versations. 4
Preliminaries
Let
F/F be a trivial or quadratic extension of (locally compact) non-archimedean local fields ofodd residual characteristic p , and σ be the generator of Gal( F/F ). If E is any non-archimedeanlocal field, we denote by o E the ring of integers of E , p E its maximal ideal, k E = o E / p E theresidue field, and ν E the additive valuation on E , which we will always normalise to haveimage Z ∪ {∞} . We fix a uniformiser ̟ F ∈ p F such that σ ( ̟ F ) = − ̟ F if F/F is ramified,and σ ( ̟ F ) = ̟ F otherwise. If E/F is an algebraic extension of non-archimedean local fields,we write E tr and E ur for the maximal tamely ramified and maximal unramified subextensionof E/F , respectively. Let ψ be a character of F , with conductor p F . We put ψ F = ψ ◦ Tr F/F .Let ǫ = ±
1, and V be a N -dimensional F -vector, space equipped with a non-degenerate ǫ -hermitian form h : V × V → F , defined with respect to the involution σ . Thus, h is a biadditiveform h : V × V → F , satisfying h ( xv, yw ) = σ ( x ) ǫσ ( h ( w, v )) y, for all v, w ∈ V , and x, y ∈ F . We let A = End F ( V ), e G = Aut F ( V ), and σ h denote the adjointanti-involution induced on A by h . Let σ denote both the involution σ ( g ) = σ h ( g ) − , g ∈ e G ,on e G and the involution σ ( a ) = − σ h ( a ), a ∈ A , on A . Let Σ = { , σ } , where 1 acts as theidentity on e G and A .We set G + = e G Σ = { g ∈ G : h ( gv, gw ) = h ( v, w ) , for all v, w ∈ V } , the F -points of a unitary group (if F/F is quadratic, ǫ = ± F = F , ǫ = − F = F , ǫ = 1), defined over F . If G + is not orthogonalwe set G = G + , and in the orthogonal case we let G denote the special orthogonal subgroup ofelements of determinant 1. We call G a classical group . For e H a σ -stable subgroup of e G , wewrite H + = e H ∩ G + and H = e H ∩ G .We denote set of σ -skew-symmetric elements of A by A − and, for any σ -stable subset X of A ,we write X − = X ∩ A − for the skew-symmetric elements and X + for the symmetric elements.For a symmetric or skew-symmetric element a of A we define the twist of h by a to be theform h a : V × V → F defined by h a ( v, w ) = h ( v, aw ) , for all v, w ∈ V . If a is invertible the adjoint anti-involutions are then conjugate by aσ h a ( x ) = a − σ h ( x ) a, for all x ∈ A .Let E = F [ β ] be a simple field extension of F . If σ extends to a unique involution σ E on E , wecall E a self dual extension and in this case we denote by E the fixed field of σ E . Let ( V, h )be an ε -hermitian ( F/F )-space. Let E be a self dual field extension of F equipped with an F -embedding ϕ : E ֒ → End F ( V ). We call ϕ self dual if σ h ◦ ϕ = ϕ ◦ σ E .An o F -lattice sequence in V is a map Λ : Z → { o F -lattices in V } which is decreasing (wehave Λ( k ) ⊆ Λ( k − k ∈ Z ) and periodic (there exists e (Λ) ∈ Z such that Λ( n + e (Λ)) =5 F (Λ( n )), for all n ∈ Z ). An o F -lattice sequence Λ in V defines an o F -lattice sequence in A , bysetting a n (Λ) = { a ∈ A : a Λ( i ) ⊆ Λ( i + n ) , i ∈ Z } , for n ∈ Z . The o F -lattice a (Λ) = a (Λ) is a hereditary o F -order in A with Jacobson radical a (Λ).If dim k F (Λ( i ) / Λ( i + 1)) is independent of i ∈ Z we say that Λ is regular ; or equivalently,that a (Λ) is a principal order . The normaliser in GL F ( V ) of Λ is a compact mod-centre sub-group n (Λ) = { g ∈ GL F ( V ) : there exists n ∈ Z , g (Λ( k )) = Λ( k + n ) , for k ∈ Z } . We have a group homomorphism ν Λ : n (Λ) → Z by setting ν Λ ( g ) = n , if g (Λ(0)) = Λ( n ).The kernel of ν Λ is a compact open subgroup P (Λ) of GL F ( V ) which coincides with the groupof units in a (Λ). This subgroup has a decreasing separable filtration by compact open pro- p subgroups P n (Λ) = 1 + a n (Λ), for n > L is an o F -lattice in V , we put L ♯ = { v ∈ V : h ( v, L ) ⊆ p F } . An o F -lattice sequence Λ in V is called self dual (with respect to the ǫ -hermitian form h on V ) if there exists d ∈ Z , suchthat Λ( d − k ) = Λ( k ) ♯ , for all k ∈ Z . If Λ is self dual, then P − (Λ) = P (Λ) ∩ G is a compactopen subgroup of G . Intersecting the decreasing separable filtration P n (Λ) of P (Λ) with G gives a decreasing separable filtration P − n (Λ) of P − (Λ) by compact open pro- p subgroups. Thequotient group M − (Λ) = P − (Λ) /P − (Λ) is the group of k F -points of a reductive group M − defined over k F . However, the algebraic group M − need not be connected. We let P ◦ (Λ)denote the inverse image in P − (Λ) of the k F -points of the connected component of M − , andcall P ◦ (Λ) a parahoric subgroup of G . First we recall the definition of the Witt group over local fields. Let a ∈ F be such that a = ǫσ ( a ). We write h a i for the one-dimensional E -vector space with ǫ -hermitian form h ( α, β ) = σ ( α ) aβ , for α, β ∈ F . We let H denote the hyperbolic plane , and, for m ∈ N , let m H denote theorthogonal sum of m copies of H . We will always choose a basis so that the Gram matrix of m H is anti-diagonal with ǫ ’s and 1’s on the anti-diagonal. Let V be an ǫ -hermitian F -space, by thiswe mean an F -vector space equipped with non-degenerate ǫ -hermitian form h with respect to σ .Then there exists m ∈ N and an anisotropic space V such that V is isometric to m H ⊕ V ;moreover, the integer m and the isometry class of V are unique. We call V the anisotropicpart of V (or of h ) and write diman( V ) = dim( V ), the anisotropic dimension of V .For an ǫ -hermitian form h , we call the class of all ǫ -hermitian spaces with anisotropic partisometric to the anisotropic part of h the Witt tower of h . The Witt group W σ,ǫ ( F ) is theabelian group formed from the abelian semigroup of Witt towers of ǫ -hermitian forms inducedby orthogonal sum of ǫ -hermitian spaces.(i) Unitary case: If F/F is quadratic and ǫ = ± W σ,ǫ ( F ) is of order 4 and is isomorphic tothe Klein group if − ∈ N F/F ( E × ), and is cyclic if not.(ii) Orthogonal case: If F/F is trivial and ǫ = 1, W id , ( F ) is of order 16.(iii) Symplectic case: If F/F is trivial and ǫ = − W id , − ( F ) is trivial.For ǫ -hermitian spaces ( V, h ) and ( V ′ , h ′ ), we will write h ≡ h ′ if they define the same elementof the Witt group, i.e. are in the same Witt tower.6et E = F [ β ] /F be a self dual field extension with the induced involution σ E , in particular β is zero or E = E , since σ E ( β ) = − β , and λ : E → F be a non-zero F -linear form on E which is ( σ E , σ )-equivariant. Such forms exist, indeed, we can choose such a linear form λ bysetting λ (1) = 1 and λ ( β ) = λ ( β ) = · · · = λ ( β n − ) = 0. We will denote this particular choiceby λ β . Let ( V, h ) be an ǫ -hermitian E -space. Then ( V, λ ◦ h ) is a non-degenerate ǫ -hermitian F -space, called the transfer of ( V, h ). It is easy to see that the transfer preserves orthogonal sums,isometries, and hyperbolic spaces. Hence it induces a group homomorphism λ ∗ : W σ E ,ǫ ( E ) → W σ,ǫ ( F ) . In general, it is easy to see that this homomorphism is not bijective (for example, by taking
E/F of even dimension). However, we have the following rather surprising theorem:
Theorem 2.1.
Assume that if
F/F is trivial then ǫ = 1 (the non-symplectic case). Thenrestriction of the transfer homomorphism λ ∗ : W σ E ,ǫ ( E ) → W σ,ǫ ( F ) to spaces of the sameparity of dimension is injective.Note that, the assertion of the theorem would not be true without the assumption that E = F [ β ]is a self dual field extension. Proof.
By twisting, and composing with a transfer W σ, ( F ) → W id , ( F ) if necessary, it issufficient to prove it in the orthogonal case F = F . For separable extensions the statementwith λ = Tr E/F is proved in [13]. In ibid. it is explained that the statement also holds forinseparable extensions (cf. [13, Remark 1.4]), while a different proof for inseparable extensionscan be found in the proof of [18, Theorem 4.4] where particular linear forms are chosen. However,the following simple observation shows that if the result holds for one ( σ E , σ )-equivariant linearform then it holds for all such forms: Lemma 2.2.
Let λ, λ ′ be non-zero ( σ E , σ )-equivariant F -linear forms on E , then there exists aunique γ ∈ E such that λ ( xγ ) = λ ′ ( x ) for all x ∈ E . Moreover, the image λ ∗ ( W σ E ,ǫ ( E )) of λ ∗ in W σ,ǫ ( F ) is independent of the choice of λ . Proof.
From the bijection of E onto its dual space, there exists a unique γ ∈ E such that λ ( xγ ) = λ ′ ( x ) for all x ∈ E . As λ, λ ′ are self dual, one finds λ ( σ E ( x ) σ E ( γ )) = λ ( σ E ( x ) γ ), which impliesthat γ = σ E ( γ ), i.e. γ ∈ E . (Note that, similarly, for any γ ∈ E , the form γ · λ ( x ) = λ ( xγ )is always self dual.) The second part follows as multiplication by E ( γ ) (given by the tensorproduct) defines an automorphism of the Witt ring W σ E ,ǫ ( E ).This completes the proof of the theorem. Proposition 2.3 ([18, Theorem 4.4]) . The map λ ∗ maps the Witt tower of maximal anisotropicdimension to the Witt tower of maximal anisotropic dimension.We will need more precise information on the image of the transfer homomorphism in particularinstances. Proposition 2.4.
Suppose ǫ = 1. Let E = F [ β ] be a self dual extension of F of degree n , ( V, h E )be an ǫ -hermitian E -space, and λ be any ( σ E , σ )-equivariant F -linear form on E .(i) Granted p = 2, we havedet( λ ∗ ( V )) = det( λ ∗ h i ) dim E ( V ) N E/F (det( V )) . W σ,ǫ ( F ), we have λ ∗ β ( h i ) ≡ ( h i if n is odd; h i ⊕ h ( − n +1 N E/F ( β ) i otherwise. Proof.
The analogue of these statements for transfer of quadratic forms are proved by Scharlauin [15, Lemma 5.8, Theorem 5.12]. The hermitian case follows mutatis. mutandis. being carefulof the extra signs which appear, and for this reason we sketch the proof of Part (ii). Suppose β has minimal polynomial X n + β n − X n − + · · · + β X + β . Then we can easily calculate theGram matrices of the form λ ∗ β ( h i ): J ( λ ∗ β ( h i )) = · · · · · · β ... − β ⋆... . . . ⋆ ⋆... − n − β ⋆ ⋆ ⋆ − n β ⋆ ⋆ ⋆ ⋆ , In particular, if n = [ E : F ] is odd, then λ ∗ β ( h i ) ≃ n − H ⊕ h i , as it is an orthogonal sum of h i with the subspace X generated by β, β , . . . , β n ; however, X has a totally isotropic subspace ofhalf its dimension generated by β, β , . . . β n − , hence is hyperbolic. Similarly, if n = [ E : F ] iseven, then λ ∗ β ( h i ) ≃ n − H ⊕ h i ⊕ h δ i , with δ = ( − n +1 N E/F ( β ).We will need two properties of the norm map: Lemma 2.5.
Let F = F and E/F be a finite field extension and σ E be a non-trivial involutionon E extending σ and α ∈ E .(i) Then N E/F ( α ) = N E /F ( α ).(ii) We have α ∈ N E/E ( E × ) if and only if N E /F ( α ) ∈ N F/F ( F × ). Proof.
Consider all field extensions in a given Galois closure F of F . Let m = [ E : F ] sep be the degree of the separable part of the extension and d = [ E : F ] rad be the degree of theinseparable part of the extension so that [ E : F ] = dm . Similarly, define d and m for theextension E /F . Then, as p = 2, we have d = d and m = m . Let τ , . . . , τ m denote the m distinct F -homomorphisms E ֒ → F . For each i , the homomorphism τ i | E : E ֒ → F is an F -homomorphism, and for i = j , we have τ i | E = τ j | E as E = F E and they are distinct as F -homomorphisms E ֒ → F . The first assertion now follows by definition of the norms. The secondassertion for purely inseparable extensions E/F follows directly by considering valuations, as inthis case [ E : F ] is necessarily odd. Thus, by transitivity of the norm, we can suppose that E/F and hence E /F is separable. By local class field theory, for any finite abelian extension K/k (contained in a given separable closure) of local fields (so, for example as p = 2, E/E or F/F ),we have an isomorphism Art K/k : k × /N K/k ( K × ) ≃ Gal(
K/k ) . Moreover, the base change property of class field theory applied to the field extensions above,gives on E × /N E/E ( E × ) Res EF ◦ Art
E/E = Art F/F ◦ N E /F .
8e have Gal(
E/E ) = { , σ E } and Gal( F/F ) = { , σ } , with Res EF ( σ E ) = σ , and the Artinreciprocity maps are isomorphisms. Hence Art E/E is trivial on the class of α if and onlyif Art F/F ◦ N E /F is also trivial on this class, and the second assertion follows. Remark 2.6.
In the quadratic unitary case:(i) As dimension and determinant modulo the norm group N F/F ( F × ) form a complete setof invariants, i.e. characterise ǫ -hermitian spaces up to isometry completely, Lemma 2.5and Proposition 2.4 combine to characterise the standard transfer λ ∗ β of the self dual fieldextension E = F [ β ] completely.(ii) Let x be a non-zero element of F :(a) If − F , then x is a norm with respect to N F/F if and only if x is asquare in F .(b) If − F , then x is a norm with respect to N F/F if and only ifeither ν F ( x ) is even and x is a square in F or ν F ( x ) is odd and x is not squarein F . Let ϕ : E ֒ → A be a self dual embedding. This gives V the structure of an E -vector space andwe write V ϕ when we want to make it clear that we are considering V as an E -vector spacevia ϕ . The F -linear map Hom E ( V ϕ , E ) → Hom F ( V, F ) ψ λ ◦ ψ is an isomorphism of F -vector spaces, and the ( F/F )-form h defines an isomorphism V → Hom F ( V, F ) by v h ( v, − ). We let ψ v ∈ Hom E ( V ϕ , E ) be the unique E -linear map suchthat h ( v, − ) = λ ◦ ψ v , then define h ϕ : V ϕ × V ϕ → E by h ϕ ( v, w ) = ψ v ( w ). Lemma 2.7 ([4, Lemma 5.3]) . The map h ϕ : V ϕ × V ϕ → E is a nondegenerate hermitian ( E/E )-form. Moreover, it is the unique hermitian ( E/E )-form on V ϕ such that h ( v, w ) = λ ( h E ( v, w ))for all v, w ∈ V .Sometimes, it will be more convenient to take an element β ∈ A generating a field, and considerthe embedding to be the inclusion of the subfield of A . We denote the form given by the lemmafor this embedding by h β .Let ϕ, ϕ ′ : E ֒ → A be self dual F -embeddings and ( V ϕ , h ϕ ) and ( V ϕ ′ , h ϕ ′ ) be the hermi-tian ( E/E )-spaces defined by ( V, h ) and a fixed ( σ E , σ )-invariant F -linear form as in Lemma 2.7.An observation of the second author in [17], we use later without reference, is the following usefulcorollary of Lemma 2.7. Corollary 2.8 ([17, Proposition 1.3]) . The hermitian (
E/E )-spaces ( V ϕ , h ϕ ) and ( V ϕ ′ , h ϕ ′ ) areisometric if and only if the embeddings ϕ and ϕ ′ are conjugate in U( V, h ). For the remainder, we fix our quadratic or trivial extension
F/F , σ the generator of Gal( F/F ),and a choice of sign ǫ . 9 stratum in A is a 4-tuple [Λ , n, r, β ] where Λ is an o F -lattice sequence, n, r ∈ Z , with n > r >
0, and β ∈ a − n (Λ). The fraction ne (Λ) is called the level of the stratum. We call thestratum [Λ , r, r,
0] a zero-stratum . A stratum [Λ , n, r, β ] is called pure if it is zero or E = F [ β ] is afield, Λ is an o E -lattice sequence, and ν Λ ( β ) = − n . Let n k ( β, Λ) = { x ∈ a (Λ) : βx − xβ ∈ a k (Λ) } and define the critical exponent k ( β, Λ) by k ( β, Λ) = max { ν Λ ( β ) , sup { k ∈ Z : n k ( β, Λ) a (Λ E ) + a (Λ) }} , for non-zero β and k (0 , Λ) = −∞ . A pure stratum [Λ , n, r, β ] is called simple if k ( β, Λ) < − r ,in particular if n = r then the stratum has to be zero. A stratum [Λ , n, r, β ] is called self dual if Λ is a self dual o F -lattice sequence and β ∈ A − .Let [Λ , n, r, β ] be a simple stratum in A with non-zero β . Associated to [Λ , n, r, β ] and ourinitial choice of ψ F , in the work of Bushnell and Kutzko [9] - extended to non-strict latticesequences by the third author in [21], are a compact open subgroup H r +1 ( β, Λ) of P (Λ) and aset of characters C (Λ , r, β ) of H r +1 ( β, Λ) called simple characters . For the stratum [Λ , r, r,
0] wedefine C (Λ , r,
0) to be the singleton consisting only of the trivial character on H r +1 (0 , Λ) whichis defined as P r +1 (Λ). This trivial character is also called a simple character.Let [Λ , n, r, β ] be a self dual simple stratum in A . Then H r +1 − ( β, Λ) = H r +1 ( β, Λ) ∩ G is a com-pact open subgroup of P − (Λ), and we can define a set of self dual simple characters C − (Λ , r, β )of H r +1 − ( β, Λ) by restriction from C (Λ , r, β ). This restriction of characters coincides with theGlauberman correspondence by [20, § θ ∈ C (Λ , r, β ) is called σ -invariant if it isfixed by the involution x σ ( x ) = σ h ( x ) − , for all x ∈ e G , and we write C (Λ , r, β ) σ h for thesubset of σ -invariant characters. By the Glauberman correspondence, if θ − ∈ C − (Λ , r, β ) is asimple character, then there is a unique θ ∈ C (Λ , r, β ) σ h whose restriction to H r +1 − ( β, Λ) is θ − .We call θ the lift of θ − .Let E = F [ β ] be a field extension and denote n F ( β ) = − ν E ( β ). Let Λ( E ) be the o F -latticesequence i p iE , i ∈ Z . This is the unique (up to translation) o E -lattice chain in E ,and ν Λ( E ) ( β ) = ν E ( β ). For any integer 0 k n F ( β ) −
1, the stratum [Λ( E ) , n F ( β ) , k, β ]in End F ( E ) is pure; and we set k F ( β ) = k ( β, Λ( E )) e ( E/F ) . Remark 2.9.
Note that our definition of k F ( β ) differs by a normalisation from the standardchoice of Bushnell–Henniart [5]; we make this change as with this normalisation k F ( β ) genera-lizes to the semisimple case.A simple pair over F is a pair [ k, β ] where E = F [ β ] is a finite field extension of F and k isan integer satisfying 0 k < − k F ( β ) e ( E/F ). A simple pair [ k, β ] over F is called self dual if E = F [ β ] is a self dual field extension. Given a simple pair [ k, β ], we consider quadru-ples ( V, ϕ, Λ , r ), consisting of: a finite dimensional F -vector space V ; an embedding ϕ : E ֒ → A ;a ϕ ( o E )-lattice sequence Λ in V (hence we have ϕ ( E × ) ⊆ K (Λ)); and an integer r such thatthe group level j re (Λ) k is equal to k . Given such a quadruple ( V, ϕ, Λ , r ), we obtain a simplestratum [Λ , n, r, ϕ ( β )] in A , with n = − ν E ( β ) e (Λ E ), which we call a realisation of the simplepair [ k, β ]. (It is simple as k F ( β ) = e (Λ E ) e ( E/F ) k ( β, Λ) by [9, 1.4.13]). Let [ k, β ] be a sim-ple pair and Q ( k, β ) denote the class of all such quadruples ( V, ϕ, Λ , r ). Suppose that [ k, β ] isself dual. Let Q − ( k, β ) denote the class of all quadruples (( V, h ) , ϕ, Λ , r ) where ( V, h ) is an ǫ -hermitian F -space; ( V, ϕ, Λ , r ) ∈ Q ( k, β ), and ϕ, Λ are self dual with respect to h . We notethat the set Q − ( k, β ) just depends on our initial choice of σ, ǫ . If (( V, h ) , ϕ, Λ , r ) ∈ Q − ( k, β ),then [Λ , n, r, ϕ ( β )] is a self dual simple stratum which we call a (self dual) realisation of the selfdual simple pair [ k, β ]. 10or realisations [Λ , n, r, ϕ ( β )] and [Λ ′ , n ′ , r ′ , ϕ ′ ( β )] of a simple pair [ k, β ] there is a canonicalbijection τ Λ , Λ ′ ,β : C (Λ , r, ϕ ( β )) → C (Λ ′ , r ′ , ϕ ′ ( β )) , by [9, 3.6.14], where despite the dependence of τ Λ , Λ ′ ,β on ( ϕ, ϕ ′ , r, r ′ ) we do not include it inour notation. Let [ k, β ] be a self dual simple pair. By [21, Proposition 2.12], if [Λ , n, r, ϕ ( β )]and [Λ ′ , n ′ , r ′ , ϕ ′ ( β )] are self dual realisations of [ k, β ], then τ Λ , Λ ′ ,β commutes with the involutionsdefined on C ( ϕ ( β ) , r, Λ) and C ( ϕ ′ ( β ) , r ′ , Λ ′ ) and restricts to give a bijection τ Λ , Λ ′ ,β : C − (Λ , r, ϕ ( β )) → C − (Λ ′ , r ′ , ϕ ′ ( β )) . We let C ( k, β ) denote the class of all simple characters defined by a realisation of a simplepair [ k, β ] and C − ( k, β ) denote the class of all self dual simple characters defined by a realisationof a self dual simple pair [ k, β ], i.e. C ( k, β ) = [ ( V,ϕ, Λ ,r ) ∈Q ( k,β ) C (Λ , r, ϕ ( β )) , C − ( k, β ) = [ ( V,ϕ, Λ ,r ) ∈Q − ( k,β ) C − (Λ , r, ϕ ( β )) . (2.10)Note that, as introduced by Bushnell and Henniart in [5], the standard notation for C ( k, β )is R ( k, β ).A potential simple character , or ps-character , supported on the simple pair [ k, β ] is a function,Θ : Q ( k, β ) → C ( k, β )such that Θ( V, ϕ, Λ , r ) ∈ C (Λ , r, ϕ ( β )), for ( V, ϕ, Λ , r ) ∈ Q ( k, β ), andΘ( V ′ , ϕ ′ , Λ ′ , r ′ ) = τ Λ , Λ ′ ,β (Θ( V, ϕ, Λ , r )) , for ( V, ϕ, Λ , r ) , ( V ′ , ϕ ′ , Λ ′ , r ′ ) ∈ Q ( k, β ). For ( V, ϕ, Λ , r ) ∈ Q ( k, β ), we call Θ( V, ϕ, Λ , r ) a re-alisation of Θ. Thus, a ps-character is determined by any one of its realisations. Let Θ bea ps-character supported on the simple pair [ k, β ] and Θ ′ be a ps-character supported on thesimple pair [ k ′ , β ′ ]. We say that Θ and Θ ′ are endo-equivalent , denoted Θ ≈ Θ ′ , if there existsa finite dimensional vector space V , which is part of a quadruple ( V, ϕ, Λ , r ) ∈ Q ( k, β ) and partof a quadruple ( V, ϕ ′ , Λ ′ , r ′ ) ∈ Q ( k ′ , β ′ ), such that Θ( V, ϕ, Λ , r ) and Θ ′ ( V, ϕ ′ , Λ ′ , r ′ ) intertwinein GL F ( V ), i.e. there exist realisations on a common vector space which intertwine. We havethe following key results for ps-characters Theorem 2.11 ([3, Theorem 1.11] and [5, Corollary 8.10]) . Suppose we are given two endo-equivalent ps-characters Θ and Θ ′ supported on [ k, β ] and [ k ′ , β ′ ], respectively, and let θ and θ ′ be realisations of Θ and Θ ′ , respectively, on the same vector space V . Then, θ and θ ′ intertwineover ˜ G . Theorem 2.12 ([3, Corollary 8.3]) . This defines an equivalence relation on the class of ps-characters.The equivalence classes are called simple GL -endo-classes . Theorem 2.13.
Let θ, θ ′ and θ ′′ be simple characters with the same group level. If θ intertwineswith θ ′ and θ ′ intertwines with θ ′′ over ˜ G , then θ intertwines with θ ′′ over ˜ G .The Theorem is proven for the case of strict lattice sequences in [6, Corollary 2].11 roof. The ps-characters Θ and Θ ′′ defined by θ and θ ′′ are endo-equivalent to each other byTheorem 2.13 and thus θ and θ ′′ intertwine by some element of ˜ G by Theorem 2.11.A ps-character supported on a self dual simple pair [ k, β ] is called σ -invariant if for any(or equivalently one) self dual realisation of [ k, β ] the value is σ -invariant. A self dual ps-character is a function, Θ − : Q − ( k, β ) → C − ( k, β ) such that Θ − (( V, h ) , ϕ, Λ , r ) ∈ C − (Λ , r, ϕ ( β )),for (( V, h ) , ϕ, Λ , r ) ∈ Q − ( k, β ), andΘ − (( V ′ , h ′ ) , ϕ ′ , Λ ′ , r ′ ) = τ Λ , Λ ′ ,β (Θ − (( V, h ) , ϕ, Λ , r )) , for (( V, h ) , ϕ, Λ , r ) , (( V ′ , h ′ ) , ϕ ′ , Λ ′ , r ′ ) ∈ Q − ( k, β ). Thus, again, a self dual ps-character is de-termined by any one of its realisations. By the Glauberman correspondence, every self dualps-character comes uniquely from the restriction of a σ -invariant ps-character. We call thevalues of Θ − realisations and the σ -invariant ps-character Θ : Q ( k, β ) → C ( k, β ) which restrictsto Θ − the lift of Θ − .We will need a proposition on minimal elements in tamely ramified field extensions: Proposition 2.14.
Suppose E /F and E /F are two finite field extensions with the first onebeing tamely ramified. Further, let b i be an element of E × i , i = 1 ,
2, such that(i) F [ b ] = E and ν E ( b ) is prime to e ( E /F ) which we denote by e ,(ii) ν E ( b ) /e is equal to ν E ( b ) /e ( E /F ), and(iii) b e ̟ − ν E ( b ) F + p E and b e ̟ − ν E ( b ) F + p E have the same minimal polynomial over k F , andthe first element generates the residue field k E over k F .Then, there is an F -embedding φ : E → E such that ν E ( φ ( b ) − b ) > ν E ( b ) . Proof.
Let P be the minimal polynomial of b e ̟ − ν E ( b ) F over k F , and choose a lift P ∈ o F [ X ]of P . Hensel’s lemma guaranties the existence of zeroes γ ∈ E ,ur and γ ∈ E ,ur of P congruent to b ei ̟ − ν E ( b ) F mod p E i , respectively. There is an F -monomorphism from E intoan algebraic closure of E which maps γ to γ , and thus we can assume that F = E ,ur and γ = γ . By Bezout’s lemma it is enough to restrict to the case where b is a uniformiser.Let e ̟ ∈ p k \ p k be an e th power in E . Then b e e ̟ − and b e e ̟ − are e -th powers. The latter isequal to b e e ̟ − and Hensel’s lemma provides e -th roots λ i of b ei e ̟ − , respectively, such that λ is equal to λ . The F -monomorphism φ which maps b λ − to b λ − satisfies that φ ( b ) b − isan element of 1 + p E . Remark 2.15.
An element b as in Proposition 2.14 is called a minimal element for E /F , see[9, 1.4.14]. Let [ k, β ] be a self dual simple pair. Let [Λ , n, r, β ] be a self dual simple stratum for [ k, β ] (inparticular the group level is k ) in an ǫ -hermitian ( F/F )-space V , and let θ − ∈ C − (Λ , r, β ).This gives V the structure of a ǫ -hermitian ( E/E )-space defining a class in W σ,ǫ ( E ), and we12ill say that the Witt group W σ,ǫ ( E ) is the Witt group of the stratum, of θ − , or of β , withrespect ǫ . Definition 3.1.
Let Θ − and Θ ′− be two self dual ps-characters supported on possibly dif-ferent simple pairs [ k, β ] and [ k, β ′ ] for the same k . We call two elements (( V, h ) , ϕ, Λ , r )and (( V ′ , h ′ ) , ϕ ′ , Λ ′ , r ′ ) in the domain of Θ − and Θ ′− , respectively, a comparison pair if V = V ′ , h = h ′ , e (Λ) = e (Λ ′ ) , r = r ′ . A comparison pair (((
V, h ) , ϕ, Λ , r ) , (( V, h ) , ϕ ′ , Λ ′ , r ′ )) is called strong if Λ = Λ ′ .We will formulate several statements like “realisations of Θ ′− and Θ ′− with respect to a givencomparison pair intertwine over G + ”. Here, we mean that G + is the classical group defined bythe comparison pair, i.e. it is U( V, h ) if the comparison pair is ((
V, h ) , Λ , ϕ, r ) , (( V, h ) , Λ ′ , ϕ ′ , r ′ )).We are mainly interested in the case of Witt groups of two given quadratic extensions E/E and E ′ /E ′ which have similar descriptions in terms of units and uniformisers, i.e either W σ,ǫ ( E )and W σ ′ ,ǫ ( E ′ ) are isomorphic to a cyclic group of order 4 (i.e. when − E nor E ′ ) or both are Kleinian four groups and there are generators h a i ≡ , h a i ≡ for W σ,ǫ ( E )and h a ′ i ≡ , h a ′ i ≡ for W σ,ǫ ( E ) such that ν E ( a i ) e ( E | F ) is equal to ν E ′ ( a ′ i ) e ( E ′ | F ) . the next lemma ensures thatthis is the case for intertwining positive level self dual simple characters. Lemma 3.2.
Suppose that two simple characters θ ∈ C (Λ , r, β ) σ h and θ ′ ∈ C (Λ ′ , r, β ′ ) σ h inter-twine over G + and both lattice sequences have the same period. Then, E/E is ramified if andonly if E ′ /E ′ is ramified. Proof.
Let e be the period of Λ, hence the period of Λ ′ by assumption. Adding 2 e shifts of Λ,we obtain the self dual principal lattice chainΛ † = (Λ − e + 1) ⊕ . . . ⊕ (Λ − ⊕ (Λ − ♯ ⊕ · · · ⊕ (Λ − e + 1) ♯ , with respect to ǫ -hermitian form h † = h. . .h (2 e -times). Put β † = L β (2 e -times).Similarly, define, Λ ′ † , β ′ † . Then Λ ′ † is also a self dual principal lattice chain in ( V † , h † ), withthe same jumps in the filtration as Λ † ; hence Λ † and Λ ′ † are conjugate in U( V † , h † ). Thecharacters θ † = τ Λ , Λ † ,β,β † ( θ ) and θ ′ † = τ Λ ′ , Λ ′ † ,β ′ ,β ′ † ( θ ′ ) are elements of C (Λ † , r, β † ) σ h † and C (Λ ′ † , r, β ′ † ) σ h † respectively, which intertwine over U( V † , h † ) (cf. [12, §
3] for more details onconstructions like this). Thus we can restrict to the case that both lattice sequences coincideby [17, Proposition 5.2] and that the characters θ † and θ ′ † coincide by intertwining impliesconjugacy [18, Theorem 10.3]. By [10, 5.2 (i)] the residue fields of F [ β ] and F [ β ′ ] coincidein a / a and thus the induced action of σ h on the residue fields also coincide, which finishes theproof. Here we assume that we are in non-symplectic case, i.e. if ǫ = − F = F . Proposition 4.1.
If two pure self dual strata [Λ , n, n − , β ] and [Λ ′ , n, n − , β ′ ] of the samelevel intertwine over e G then they intertwine over G + .13 roof. As they intertwine, by [18, Proposition 7.1] the stratum [Λ ⊕ Λ ′ , n, n − , β ⊕ β ′ ] isequivalent to a simple stratum, and thus by Proposition [20, 1.10] the stratum [Λ ⊕ Λ ′ , n, n − , β ⊕ β ′ ] is equivalent to a self dual simple stratum [Λ ⊕ Λ ′ , n, n − , γ ] with γ ∈ e G × e G . Wecan therefore replace β and β ′ by skew-symmetric elements with the same minimal polynomialover F , implying that they are conjugate by an element of G + by [18, Corollary 5.1] (here weuse that h is not symplectic). Thus the strata in the statement intertwine over G + . Proposition 4.2.
Let θ − ∈ C − (Λ , r, β ) and θ ′− ∈ C − (Λ , r, β ′ ) be two simple characters of G + .Then, their lifts (and hence θ − and θ ′− ) are conjugate over G + if and only if their lifts intertwineover e G . Proof.
Instead of θ − and θ ′− we consider lifts θ ∈ C (Λ , r, β ) σ h and θ ′ ∈ C (Λ , r, β ′ ) σ h . The proof isby induction on r . The minimal case is given by Proposition 4.1. By the induction hypothesis,[18, Theorem 10.3] and the translation principle [18, Theorem 9.18] we can assume that thereis a self dual simple stratum [Λ , n, r + 1 , γ ] equivalent to [Λ , n, r + 1 , β ] and [Λ , n, r + 1 , β ′ ] suchthat θ and θ ′ coincide on H r +1 ( γ, Λ). But then there is skew-symmetric element c of a − ( r +1) such that θ = θ ψ β − γ + c and θ ′ = θ ψ β ′ − γ for some θ ∈ C (Λ , r, γ ) σ h and [18, Proposition 9.11(i)], Proposition 4.1 and [18, Proposition 9.19 (ii)] imply that the θ and θ ′ intertwine over G + .The result now follows from [18, Theorem 10.3].We give a corollary which will be useful in the next section. Corollary 4.3.
Suppose that two simple characters θ ∈ C (Λ , r, β ) σ h and θ ′ ∈ C (Λ ′ , r, β ′ ) σ h intertwine over e G and both lattice sequences have the same period. Then, E/E is ramified ifand only if E ′ /E ′ is ramified. Proof.
As in the proof of Lemma 3.2, we can reduce to the case where Λ and Λ ′ coincide.Proposition 4.2 implies that the characters are conjugate over G + , and hence intertwine over G + .Hence we can conclude by Lemma 3.2.We now define an equivalence relation for self dual ps-characters in the non-symplectic case: Definition 4.4.
Two self dual ps-characters Θ − and Θ ′− for possibly different simple pairs arecalled endo-equivalent if for all strong comparison pairs their respective realisations intertwineover G + . Theorem 4.5.
Two self dual ps-characters Θ − and Θ ′− are endo-equivalent if and only if thereexists a strong comparison pair such that the realisations of Θ − and Θ ′− on this comparisonpair intertwine over G + . Proof.
This follows from Proposition 4.2 and Theorem 2.11.
To incorporate the symplectic case we need a new idea because, in contrast to the non-symplecticcase, two self dual embeddings of a field extension
E/F into (
A, σ h ) are not necessarily conjugateby an element of G + . In fact, we have two conjugacy classes if β is non-zero, i.e. E = E . Thuswe need more restrictive comparison pairs in the symplectic case.14iven two self dual field extensions ( E = F [ β ] /F, σ E ) and ( E ′ = F [ β ′ ] /F, σ E ′ ) with non-zeroskew-symmetric elements β and β ′ there is a unique bijection w β,β ′ : W σ E , − ( E ) → W σ E ′ , − ( E ′ )which sends the Witt tower of h β i to the Witt tower of h β ′ i and preserves the anisotropicdimension. Similarly, there is a unique bijection w β ,β ′ : W σ E , ( E ) → W σ E ′ , ( E ′ )which sends the Witt tower of h β i to the Witt tower of h β ′ i and preserves the anisotropicdimension. We first notice exactly when w β ,β ′ coincides with w , . Lemma 5.1.
We have w , = w β ,β ′ if and only if − ∈ E (2) ∩ E ′ (2) or − E (2) ∪ E ′ (2) granting β and β ′ to be non-zero. Proof. If − ∈ E (2) ∩ E ′ (2) then β and β ′ are norms and thus w , = w β ,β ′ . If − E (2) ∪ E ′ (2) then E/E and E ′ /E ′ are both ramified and β and β ′ have odd valuation. Thus ν E ( β )and ν E ′ ( β ′ ) are even, but not divisible by 4. As − E (2) ∪ E ′ (2) it follows that thesesquares are not norms and thus w , = w β ,β ′ . Analogously, if − ∈ E (2) and not − ∈ E ′ (2) then w , = w β ,β ′ Let w = ( w β,β ′ if ǫ = − w β ,β ′ if ǫ = 1 . Let ϕ, ϕ ′ be self dual embeddings of E, E ′ into A . Let h ϕ (resp. h ϕ ′ ) be the nondegenerate ǫ -hermitian ( E/E )-form (resp. ǫ -hermitian ( E ′ /E ′ )-form) defined as in Section 2.3 from theseembeddings using the standard equivariant linear forms λ β , λ β ′ defined by β, β ′ . For the caseof β = β ′ = 0 we define h := h and w to be the identity of W σ, − ( F ) ∪ W σ, ( F ). Definition 5.2.
Let F [ β ] and F [ β ′ ] be two self-dual field extension such that β and β ′ are si-multaneously zero or non-zero. We say that the Witt towers of h ϕ and h ϕ ′ match if w (( h ϕ ) ≡ ) =( h ϕ ′ ) ≡ , and similarly we call a comparison pair ((( V, h ) , ϕ, Λ , r, ) , (( V, h ) , ϕ ′ , Λ ′ , r ′ , )) Witt when-ever the Witt towers of h ϕ and h ϕ ′ match.If we are given a self dual simple stratum [Λ , n, r, β ] then we fix in this section a definingsequence [Λ , n, r + i, γ ( i ) ] for i = 0 , . . . , n − r of simple strata, as in [9, 2.4.2]. And analogously,for a self dual simple stratum [Λ ′ , n, r, β ′ ] we fix a defining sequence [Λ ′ , n, r + i, γ ′ ( i ) ]. The mainresult of this section is that matching of Witt towers is inherited along defining sequences. Proposition 5.3.
Let θ ∈ C (Λ , r, β ) σ h and θ ′ ∈ C (Λ , r, β ′ ) σ h be simple characters. Supposethat θ and θ ′ intertwine over e G , e (Λ) = e (Λ ′ ), and if G + is symplectic that the Witt towersof h β and h β ′ match. Then the Witt towers of h γ ( i ) and h γ ′ ( i ) match for all i .Let β and γ be two non-zero skew-symmetric elements of A generating field extensions E and K of F . Lemma 5.4.
Assume that we are in the non-symplectic case, i.e. ǫ = 1 or F = F . Sup-pose that dim E V and dim K V have the same parity and either this is even or w , = w β ,γ and dim F V is odd. Then the Witt towers of h β and h γ match.15 roof. By twisting with a skew element of F in the unitary case, if necessary, we can alwaysassume that ǫ = 1. If dim E V is even and if λ ∗ β ( h β ) is not hyperbolic then by injectivity of λ ∗ β onthe set of classes of even anisotropic dimension and as hyperbolic spaces transfer to hyperbolicspaces, the form h is not hyperbolic and thus λ ∗ γ ( h γ ) is not hyperbolic. Hence w β ,γ (( h β ) ≡ ) = ( h γ ) ≡ , Similarly, if λ ∗ γ ( h β ) is hyperbolic we find that w β ,γ (( h β ) ≡ ) = ( h γ ) ≡ .Next, if dim F V is odd, by assumption we have w , = w β ,γ . Thus we have to show that h β ≡h i in W σ E , ( E ) if and only if h γ ≡ h i in W σ K , ( K ). However, as dim F V is odd [ E : F ]and [ K : F ] are odd and, we have λ ∗ β ( h i ) ≡ λ ∗ γ ( h i ) ≡ h i , by Proposition 2.4. Finally, we conclude by injectivity of both transfers on their sets of classesof odd anisotropic dimension. Lemma 5.5.
Suppose ǫ = 1, F = F , both dim E V and dim K V are odd, w , = w β ,γ and there exists a tamely ramified σ h -invariant extension e F /F in E ∩ K not fixed by σ h .Then λ ∗ β ( h i ) = λ ∗ γ ( h i ), and in particular the Witt towers of h β and h γ match. Proof. As e F is not fixed by σ h , the F -dimension dim F V is even. By Proposition 2.4, we have a : ≡ λ ∗ β ( h i ) ≡ h i + h± N E/F ( β ) i , b : ≡ λ ∗ γ ( h i ) ≡ h i + h± N K/F ( γ ) i , where the signs are determined by Proposition 2.4, however the signs are not important forthis proof. If [ E : e F ] is even then the e F -dimension of V is even, so [ K : e F ] is even. Hencethe images of the maps λ ∗ β : W σ E , ( E ) → W id , ( F ) and λ ∗ γ : W σ K , ( K ) → W id , ( F ) coincide,consisting of only two elements, the class of hyperbolic F -spaces and the class of the space ofmaximal anisotropic dimension. If [ E : e F ] is odd, then the transfer λ ∗ β : W σ E , ( E ) → W σ e F , ( e F )is a bijection, hence the image of λ ∗ β : W σ E , ( E ) → W id , ( F ) is equal to the image of the tracetransfer Tr ∗ e F /F : W σ h , ( e F ) → W id , ( F ) (or any other transfer), by Lemma 2.2. Similarly, thesame holds for λ ∗ γ . Thus, the transfers λ ∗ β and λ ∗ γ have the same image. Denote by M thecommon image of λ ∗ β and λ ∗ γ in W id , ( F ).Now, assume for a contradiction that a and b are different. Then a − b is a non-zero elementof M of anisotropic dimension 2. In particular M must have 4 elements, because it containsalso the classes of the hyperbolic and the maximal anisotropic space, and hence M has order 4.As λ ∗ β and λ ∗ γ are injective and the image is order 4, the classes a and b must be non-zeroand a − b is either a or b , as in our image we only have two elements of anisotropic dimensionnot 0 or maximal. Therefore a = 2 b , as b is non-zero. Hence b has order 4 and M is cyclic. Inparticular, M has only one element of order 2. But this is impossible, because the class of thespace of maximal anisotropic dimension and a both have order 2. Lemma 5.6.
Let [Λ , n, n − , β ] and [Λ ′ , n, n − , β ′ ] be two G + -intertwining self dual pure strata.Then there is an element u ∈ G + such that E ∩ uE ′ u − contains a tamely ramified σ h -invariantextension e F of F not fixed by σ h . Moreover, the field e F can be chosen to be P − (Λ)-conjugate tothe maximal tamely ramified subextension of F [ γ ( n − − r ) ] /F , i.e. of a minimal stratum equivalentto [Λ , n, n − , β ], and if Λ = Λ ′ then u can be chosen in P (Λ). Proof.
Let e F denote the maximal tamely ramified subextension of F [ γ ( n − − r ) ] /F and anal-ogously e F ′ denote the maximal tamely ramified subextension F [ γ ′ ( n − − r ) ] /F . By Proposi-tion 2.14 and [18, Theorem 5.2] we can assume that e F is a subfield of E and that e F ′ is a subfield16f E ′ . By Proposition 2.14 there is an σ h -equivariant isomorphism φ between e F /F and e F ′ /F ,and they are conjugate by an element of G + by [18, Theorem 5.2]. Now assume Λ = Λ ′ , thenit follows from [17, Theorem 1.2] that e F and e F ′ are conjugate by an element of P − (Λ). Lemma 5.7.
Let [Λ , n, n − , β ] and [Λ ′ , n, n − , γ ] be e G -intertwining self dual pure strata.Suppose that F = F , dim E V and dim K V are odd, dim F V is even, e ( E/F ) = e ( K/F ),and e ( E/E ) = e ( K/K ). Then,(i) w , = w β ,γ ;(ii) λ ∗ β ( h i ) = λ ∗ γ ( h i );(iii) the Witt towers of h β and h γ match. Proof.
The conditions imply that ( −
1) is a square in E if and only if it is a square in K , andthus w , = w β ,γ by Lemma 5.1.The strata intertwine over G + by Proposition 4.1, so after conjugation with the element whichintertwines them, we can assume that they intertwine by the identity. By twisting with a skew-symmetric element of F if necessary we can also assume that ǫ = 1. With the same argumentas the end of the proof of Lemma 5.4, the first two assertions imply the third. Thus it remainsto prove the second assertion.Let π be a skew-symmetric uniformiser of F . Then we claim that π − β is a norm with respectto E/E if and only if π − γ is a norm with respect to K/K . Indeed, the strata intertwine by 1,so π − ν E ( β ) β e ( E/F ) ≡ π − ν K ( γ ) γ e ( K/F ) ( a + a ′ ) . Thus π − ν E ( β ) β e ( E/F ) + p E is a square in k E if and only if π − ν K ( γ ) γ e ( K/F ) + p K is a square k K .If e ( E/E ) = 2 then e ( E/F ) is odd as there is a skew-symmetric uniformiser of F which is apower of a skew-symmetric uniformiser of E , so the exponent has to be odd. Further, and ν E ( β )is odd as β is skew symmetric. Hence π − β is a square if and only π − γ is a square. Hence, π − β is a norm if and only π − γ is a norm by Remark (ii)(b). If e ( E/E ) = 1 then π − β is a normwith respect to E/E if and only if it has even valuation in E . Now, by our assumptions on thestrata and the ramification indices, the integers ν E ( π − β ) and ν K ( π − γ ) agree, and thus π − β is a norm with respect to E/E if and only if π − γ is a norm with respect to K/K . Thiscompletes the proof of the claim.By Lemma 2.5, N E/F ( π − β ) = π − [ E : F ] N E/F ( β ) is a norm with respect to F/F if and onlyif π − [ K : F ] N K/F ( γ ) is a norm with respect to F/F . But the conditions imply that [ E : F ]and [ K : F ] are even, hence − N E/F ( β ) is a norm with respect to F/F if and only if − N K/F ( γ )is also a norm with respect to F/F . Thus, by Proposition 2.4, λ ∗ β ( h i ) and λ ∗ γ ( h i ) coincide. Proposition 5.8.
Let θ ∈ C (Λ , r, β ) σ h and θ ′ ∈ C (Λ ′ , r, β ′ ) σ h be simple characters which inter-twine over e G . Suppose that e (Λ) = e (Λ ′ ), and if G is symplectic suppose further that θ and θ ′ intertwine over G + . Then, the Witt towers of h β and h β ′ match. Corollary 5.9.
Under the assumptions of Proposition 5.8 there is a stratum [Λ ′ , n, r, β ′′ ] suchthat β ′′ is conjugate to β by an element of G + and C (Λ ′ , r, β ′ ) is equal to C (Λ ′ , r, β ′′ ) and thereis a lattice sequence Λ ′′ normalised by E × and an element g in of G + such that g Λ ′ is equal toΛ ′′ . Proof.
As all of the arithmetical invariants are the same, we can build a lattice sequence Λ ′′ relative to β ′ such that there exists g ∈ G + with g Λ = Λ ′′ as follows: We can find an E - and17n E ′ -Witt basis ( v j ) and ( v ′ j ) for h β and h β ′ such that Gram matrices M and M ′ have forevery entry the equality of valuations ν E ( m l,k ) = ν E ′ ( m l,k ) , and such that ( v ′ j ) splits Λ ′ . The latter splitting does mean that for every integer t we have adecomposition Λ ′ t = ⊕ p µ j,t E ′ v j . We now define Λ ′′ via (just by replacing E ′ by E and v ′ by v )Λ ′′ t = ⊕ p µ j,t E v j . By [17, Proposition 5.2], there is a g ∈ G + such that g Λ ′ = Λ ′′ . We conjugate θ ′ to thetransfer of θ to Λ ′′ by an element g ′ of G + by [18, Theorem 10.3]. Then, C (Λ ′ , r, β ′ ) is equalto C (Λ ′ , r, g ′− βg ′ ). We define β ′′ := g ′− βg .During the proof we use that twisting in the unitary case is a bijection and the following twoelementary properties of twisting:( h β ) β = ( h β ) β , ( ) β ′ ◦ w β,β ′ ◦ ( ) β − = w β ,β ′ . In particular, thanks to the first property, we can write h ββ with no ambiguity. Proof.
From the intertwining of the two characters we find(i) [Λ , n, n − , β ] and [Λ ′ , n, n − , β ′ ] intertwine over G + by Proposition 4.1;(ii) e ( E/F ) = e ( E ′ /F ) and f ( E/F ) = f ( E ′ /F );(iii) e ( E/E ) = e ( E ′ /E ′ ), by Lemma 3.2 and Corollary 4.3.Now in the non-symplectic case the Lemmas 5.4, 5.5 and 5.7 imply that the Witt towers of h β and h β ′ match. We are thus left with the symplectic case. The characters intertwine over G ,so by conjugating if necessary we can assume that they intertwine by the identity. Thus,[Λ , n, n − , β ] and [Λ ′ , n, n − , β ′ ] intertwine by the identity. In particular, if we twist by β and β ′ ,then the orthogonal forms h β and h β ′ are isomorphic by an element of P (Λ ′ ) P (Λ) by [18,Corollary 3.2]. Hence, w β ,β ′ ( h ββ ≡ ) is equal to h β ′ β ′ ≡ by the previous cases and thus w β,β ′ ( h β ≡ ) = h β ′ ≡ . Definition 5.10 (Witt comparison pair) . A comparison pair, see Definition 3.1, is called
Witt ,if the corresponding Witt towers match.
Remark 5.11.
In the non-symplectic case by Proposition 5.8, for two endo-equivalent self dualps-characters every comparison pair is Witt.We need two further results for the symplectic case to prove Proposition 5.3.
Lemma 5.12.
Let [Λ , n, n − , β ] and [Λ ′ , n, n − , γ ] be self dual pure strata which intertwineover G + . Suppose that ǫ = − F = F , dim E V − dim K V is even, and either dim E V is even,or w , = w β ,γ . Then, the Witt towers of h β and h γ match.18 roof. We can assume that the strata intertwine by 1, and hence we have an isomorphismfrom h β to h γ given by an element of P (Λ ′ ) P (Λ) by [18, Corollary 3.2]. Thus, we can restrictto the orthogonal case, and conclude by Lemmas 5.4 and 5.5. Lemma 5.13.
Let [Λ , n, n − , β ] and [Λ , n, n − , γ ] be self dual pure strata which intertwineover G + . Suppose ǫ = − F = F , dim E V is odd, dim K V is even and γ is minimal. Then(i) If h β ≡ h β i , then h γ is hyperbolic if and only if − E .(ii) If h β
6≡ h β i , then h γ is hyperbolic if and only if − E . Proof.
By intertwining implies conjugacy for simple strata with same lattice sequence, by con-jugating if necessary, we can assume that the strata are equivalent. We show that if h β ≡ h β i and − E then h γ is hyperbolic. The other cases can be proved in a similar fash-ion. As h β ≡ h β i , by twisting we have h ββ = h β i and h β i = h i because − E .However, h β is isomorphic to h γ by an element u of P (Λ) and [Λ , n, n − , u − γu ] is a minimalself dual stratum with respect to h β equivalent to [Λ , n, n − , β ]. By Lemma 5.6 we can choose u such that E ∩ u − Ku contains a σ h -invariant, but non-fixed, tamely ramified extension e F /F .The conditions on the dimensions imply that [ E : e F ] is even and thus λ ∗ β ( h i ) = 0 because ofProposition 2.4 and Proposition 2.3, i.e. h β and h γ are hyperbolic. Then, h γγ is hyperbolic, byinjectivity of λ ∗ γ on classes of even anisotropic dimension, and thus h γ is hyperbolic.We now can finish the proof of Proposition 5.3. Proof of Proposition 5.3.
The non-symplectic statement is given by Proposition 5.8 and thesymplectic case follows from Lemmas 5.12 and 5.13 applied to the defining sequences togetherwith the equivalence relation property of matching Witt towers.
In this section we consider only the symplectic case, that is we fix σ = 1 and ǫ = −
1. We nowdefine the equivalence relation for the symplectic case using Witt comparison pairs.
Definition 6.1.
Two ps-characters Θ − and Θ ′− for possibly different simple pairs are called endo-equivalent if for all strong Witt comparison pairs for Θ − and Θ ′− the corresponding reali-sations intertwine over G .We now formulate an analogue of Theorem 4.5 for the symplectic case: Theorem 6.2.
Two self dual ps-characters Θ − and Θ ′− are endo-equivalent if and only if thereexists a strong Witt comparison pair with realisations that intertwine over G .To prove Theorem 6.2 we need an analogue of Proposition 4.1: Proposition 6.3.
If two non-zero minimal self dual strata [Λ , n, n − , β ] and [Λ ′ , n, n − , β ′ ]of the same level intertwine over e G and their Witt towers match then they intertwine over G .19 roof. As in the proof of Proposition 4.1, we replace β and β ′ with elements with the sameminimal polynomial, and Proposition 5.8 ensures that the Witt towers still match. Hence,without loss of generality, we may assume β and β ′ have the same minimal polynomial. As theWitt towers match, under the isomorphism φ : E → E ′ , which maps β to β ′ , h β and h β ′ match.Thus β is conjugate to β ′ by an element of G which finishes the proof. Proposition 6.4.
Let θ − ∈ C − (Λ , r, β ) and θ ′− ∈ C − (Λ , r, β ′ ) be two simple characters of G .Then, θ − and θ ′− intertwine over G if and only if their lifts are conjugate over e G and their Witttowers match. Proof. If θ − and θ ′− intertwine over G , then their lifts intertwine over e G by the Glaubermancorrespondence, and hence are conjugate by intertwining implies conjugacy for characters of e G .Moreover, by Proposition 5.8, their Witt towers match. For the converse, it follows fromProposition 5.3 that the Witt towers of the i -th members of the defining sequences of [Λ , n, r, β ]and [Λ , n, r, β ′ ] match, and the result follows mutatis mutandis the proof of Proposition 4.2. Notethat, in the argument of [18] we use in the proof of Proposition 4.2 there is a step to derivedcharacters, which is here fine because G γ is unitary, i.e. we do not need matching Witt towers forthe derived characters (in fact, we know that they still match because of Proposition 5.3).This proposition gives an analogue to Corollary 4.3 for the symplectic case with a similar proof. Corollary 6.5.
Suppose that two simple characters θ ∈ C (Λ , r, β ) σ h and θ ′ ∈ C (Λ ′ , r, β ′ ) σ h intertwine over e G , both lattice sequences have the same period, and the Witt towers of h β and h β ′ match. Then E/E is ramified if and only if E ′ /E ′ is ramified. Proof.
Similar to the proof of Corollary 4.3 using Proposition 6.4.
After Sections 5 and 6, we can improve Proposition 4.2 to allow for non-conjugate latticesequences, and prove the analogue in the symplectic case using matching Witt towers:
Proposition 7.1.
Let θ − ∈ C − (Λ , r, β ) and θ ′− ∈ C − (Λ ′ , r, β ′ ) be self dual simple charactersof G + , and suppose that e (Λ) = e (Λ ′ ).(i) In the non-symplectic case, the self dual simple characters θ − and θ ′− intertwine over G + if and only if their lifts intertwine over e G .(ii) In the symplectic case, the self dual simple characters θ − and θ ′− intertwine over G + ifand only if their lifts intertwine over e G and their Witt towers match. Proof.
Let θ ∈ C (Λ , r, β ) σ h and θ ′ ∈ C (Λ ′ , r, β ′ ) σ h be the unique lifts of the given self dual simplecharacters.Suppose we are in the non-symplectic case. From the e G -intertwining we obtain by Corollary 4.3and Proposition 5.8 that under a Witt basis the anisotropic part of h β has the same dimensionand the same number or diagonal elements of odd valuation as the anisotropic part of h β ′ . Thus,there is a self dual o E ′ -lattice sequence Λ ′′ which is G + -conjugate to Λ. Let θ ′′ = τ Λ ′ , Λ ′′ ,β ( θ ′ ) ∈C (Λ ′′ , r, β ′ ) σ h . The characters θ and θ ′′ intertwine by some element of ˜ G by Theorem 2.13, and20hus are conjugate by some element of G + by Proposition 4.2. Whence θ and θ ′ intertwineover G + . The converse is clear.The second statement in the symplectic case is completely analogous using Corollary 6.5 andProposition 6.4 to show that e G -intertwining and Witt towers matching implies G + -intertwining,and Proposition 5.8 for the converse.Using the proposition, we can now prove our main result on endo-equivalence of ps-characters: Theorem 7.2.
Let Θ − and Θ ′− be two self dual ps-characters and Θ and Θ ′ their lifts. Then,the following assertions are equivalent:(i) Θ − and Θ ′− are endo-equivalent;(ii) Θ and Θ ′ are endo-equivalent;(iii) There is a comparison pair such that the realisations of Θ − and Θ ′− intertwine over G + ;(iv) For all Witt comparison pairs, the realisations of Θ − and Θ ′− intertwine over G + .In the non-symplectic case these four assertions are equivalent to:(v) For all comparison pairs, the realisations of Θ − and Θ ′− intertwine over G + . Proof.
If Θ and Θ ′ are endo-equivalent, it follows from Theorem 2.11 and Proposition 7.1that Θ − and Θ ′− are endo-equivalent. By definition, this implies there is a (strong) comparisonpair such that the respective realisations of Θ − and Θ ′− intertwine over G + . And, if there isany comparison pair such that the respective realisations of Θ − and Θ ′− intertwine over G + ,then by the Glauberman correspondence, this implies Θ and Θ ′ are endo-equivalent. Hence thefirst three properties are equivalent. Property (iv) implies Property (iii) and follows from Prop-erty (ii) by Proposition 7.1. In the non-symplectic case Property (v) follows from the equivalentproperties (ii) and (iv) from Remark 5.11. Furthermore, Property (v) implies Property (iv). Remark 7.3.
As endo-equivalence of ps-characters for general linear groups is an equivalencerelation, see Theorem 2.13 (cf. [5, Corollary 8.10]), from Theorem 7.2 we deduce that endo-equivalence of self dual ps-characters for classical groups is an equivalence relation, and we callthe equivalence classes of self dual ps-characters for this relation simple ( σ, ǫ ) -endo-classes . or simple classical endo-classes with respect to ( σ, ǫ ).As already known for e G -intertwining of simple characters, see Theorem 2.13 (cf. [6, Corollary2]), we also obtain that G + -intertwining of self dual simple characters is a transitive relation,hence an equivalence relation: Corollary 7.4.
Let θ − , θ ′− and θ ′′− be self dual simple characters with the same group level.If θ − intertwines with θ ′− and θ ′− intertwines with θ ′′− over G + , then θ − intertwines with θ ′′− over G + . Proof.
The corresponding self dual ps-characters Θ − and Θ ′− and Θ ′′− and Θ ′− are endo-equiva-lent by Theorem 7.2. Hence Θ − and Θ ′′− are endo-equivalent as endo-equivalence is an equiva-lence relation, by Remark 7.3, and thus θ − and θ ′′− intertwine over G + , by Proposition 5.8 andagain Theorem 7.2. 21 Semisimple pairs
We recall definitions of the third author of semisimple strata and characters before definingsemisimple pairs and potential semisimple characters.
Let [Λ , n, r, β ] be a stratum in A . Suppose V = L i ∈ I V i is a decomposition of V into F -subspaces. We let Λ i = Λ ∩ V i and we let β i = e i β e i , where e i : V → V i is the projectionwith kernel L j = i V j . The decomposition V = L i ∈ I V i of V is called a splitting of [Λ , n, r, β ]if β = P i ∈ I β i and Λ( k ) = L i ∈ I Λ i ( k ), for all k ∈ Z . Definition 8.1.
A stratum [Λ , n, r, β ] in A is called semisimple if it is a zero-stratum orif ν Λ ( β ) = − n and there exists a splitting L i ∈ I V i for [Λ , n, r, β ] such that(i) for i ∈ I , the stratum [Λ i , n i , r, β i ] in End F ( V i ) is simple, where n i = − ν Λ i ( β i );(ii) for i, j ∈ I with i = j , the stratum [Λ i ⊕ Λ j , max { n i , n j } , r, β i + β j ] is not equivalent to asimple stratum in End F ( V i ⊕ V j ).We write E = F [ β ] and E i = F [ β i ], hence E = L i ∈ I E i is a sum of fields, and we write B β = C A ( β ). If [Λ , n, r, β ] is self dual with associated splitting V = L i ∈ I V i then, for each i ∈ I ,there exists a unique σ ( i ) = j ∈ I such that β i = − β j . We set I = { i ∈ I : σ ( i ) = i } andchoose a set of representatives I + for the orbits of σ in I \ I . Then we let I − = σ ( I + ) so thatwe have a disjoint union I = I + ∪ I ∪ I − . A semisimple stratum [Λ , n, r, β ] is called skew if itis self-dual and the associated splitting L i ∈ I V i is orthogonal with respect to the ǫ -hermitianform h , i.e. I = I in the notation above. By abuse of notation, we call a sum of o E i -latticesequences in V (such as Λ above), an o E -lattice sequence.Let [Λ , n, r, β ] be a semisimple stratum in A . Associated to [Λ , n, r, β ] (and our initial choiceof ψ F ) in [21, Section 3.2] are a compact open subgroup H r +1 ( β, Λ) of P (Λ), and a set ofcharacters C (Λ , r, β ) called semisimple characters . For each i ∈ I , there is a natural em-bedding H r +1 ( β i , Λ i ) ֒ → H r +1 ( β, Λ) and hence a map C (Λ , r, β ) → C (Λ i , r, β i ) which wewrite θ θ i . We call the θ i the simple block restrictions of θ .Suppose that [Λ , n, r, β ] is self dual. Then H r +1 ( β, Λ) is invariant under σ , and we define thegroup H r +1 − ( β, Λ) as the intersection of H r +1 ( β, Λ) with G . We call the restrictions of semisimplecharacters in C (Λ , r, β ) to H r +1 − ( β, Λ) self dual semisimple characters , and denote the set of allsuch characters by C − (Λ , r, β ); this restriction coincides with the Glauberman correspondence(cf. [21, Section 3.6]). As in the simple setting, we write C (Λ , r, β ) σ h for the subset of σ -invariant semisimple characters. If θ − ∈ C − (Λ , r, β ) is a self dual semisimple character, thenthere is a unique θ ∈ C (Λ , r, β ) σ h whose restriction to H r +1 − ( β, Λ) is θ − by the Glaubermancorrespondence; we call θ the lift of θ − . Definition 8.2.
Let [Λ , n, , β ] be a non-zero semisimple stratum. We let k ( β, Λ) = − min { r ∈ Z : [Λ , n, r, β ] is not semisimple } denote the critical exponent of [Λ , n, , β ] and k F ( β ) := e (Λ) k ( β, Λ); by [21, § k (0 , Λ) = k F (0) = −∞ .22 emark 8.3. It is possible to generalise the critical exponent to all pairs ( β, Λ) where β generates a product of fields, and Λ decomposes into o E i -lattice sequences, in the following way:Assume β is non-zero, then there is a negative integer l such that [Λ , − ν Λ ( β ) − le (Λ) , , ̟ lF β ]is a semisimple stratum. We can define k ( β, Λ) as k ( ̟ lF β, Λ) − le (Λ). This is not used in thesequel. Definition 8.4. A semisimple pair is a pair [ k, β ] where E = F [ β ] is a sum of pairwise non-isomorphic fields E = L i ∈ I E i , and k is an integer satisfying0 ke < − k F ( β ) , where e = lcm i ∈ I ( e ( E i /F )) is the lowest common multiple of the ramification indices. Asemisimple pair [ k, β ] is called self dual if we can extend σ to an involution σ E on E whichmaps β to β .Let [ k, β ] be a self dual semisimple pair, and write the minimal polynomial of β as Ψ( X ) = Q i ∈ I Ψ i ( X ) with Ψ i ( X ) irreducible, so E i ≃ F [ X ] / (Ψ i ( X )). The action of σ E on the primitiveidempotents defines an action on I . We let I = { i ∈ I : σ ( i ) = i } , and choose a set ofrepresentatives I + for the orbits of σ in I \ I . Then we let I − = σ ( I + ) so that we have a disjointunion I = I + ∪ I ∪ I − . A self dual semisimple pair [ k, β ] is called skew if I = I in the notationabove.Given a semisimple pair [ k, β ] with non-zero β , let Q ( k, β ) denote the class of quadruples( V, ϕ, Λ , r ) consisting of: a finite dimensional F -vector space V ; an embedding ϕ : E ֒ → A ;a ϕ ( o E )-lattice sequence Λ in V ; and an integer r such that the group level j re (Λ E ) k is equalto k , where e (Λ E ) is defined to be the greatest common divisor of all e ((Λ i ) E i ). Given such aquadruple ( V, ϕ, Λ , r ), we obtain a semisimple stratum [Λ , − ν Λ ( ϕ ( β )) , r, ϕ ( β )] with splitting V = L i ∈ I V i where V i = ker(Ψ i ( β )); we call this semisimple stratum a realisation of the semisimplepair [ k, β ]. For the semisimple pair [ k,
0] we take the notation of section 2.4.Given a self dual semisimple pair [ k, β ] with non-zero β , let Q − ( k, β ) denote the class consistingof all quadruples (( V, h ) , ϕ, Λ , r ) where ( V, ϕ, Λ , r ) is an element of Q ( k, β ) and V is equippedwith an ǫ -hermitian F -form; and ϕ, Λ are self dual with respect to this form. Given sucha quadruple ((
V, h ) , ϕ, Λ , r ), we obtain a self dual semisimple stratum [Λ , − ν Λ ( ϕ ( β )) , r, ϕ ( β )]with splitting V = L i ∈ I V i where V i = ker(Ψ i ( β )).For realisations [Λ , n, r, ϕ ( β )] and [Λ ′ , n ′ , r ′ , ϕ ′ ( β )] of a semisimple pair [ k, β ] there is a canonicalbijection τ Λ , Λ ′ ,β : C (Λ , r, ϕ ( β )) → C (Λ ′ , r ′ , ϕ ′ ( β )) , by [21, Proposition 3.26]. By [21, Proposition 3.32], if [Λ , n, r, ϕ ( β )] and [Λ ′ , n ′ , r ′ , ϕ ′ ( β )] areself dual realisations of a self dual [ k, β ], then τ Λ , Λ ′ ,β commutes with the involutions definedon C ( ϕ ( β ) , r, Λ) and C ( ϕ ′ ( β ) , r ′ , Λ ′ ) and restricts to give a bijection τ Λ , Λ ′ ,β : C − (Λ , r, ϕ ( β )) → C − (Λ ′ , r ′ , ϕ ′ ( β )) . Thanks to this result and with the definition of semisimple pairs, we can now define (self dual)potential semisimple characters. At first we define C ( k, β ) and C − ( k, β ) as in (2.10). Definition 8.5.
Let [ k, β ] be a semisimple pair.23i) A potential semisimple character , or pss-character , supported on [ k, β ] is a functionΘ : Q ( k, β ) → C ( k, β )such that Θ( V, ϕ, Λ , r ) ∈ C (Λ , r, ϕ ( β )), for all ( V, ϕ, Λ , r ) ∈ Q ( k, β ), andΘ( V ′ , ϕ ′ , Λ ′ , r ′ ) = τ Λ , Λ ′ ,β (Θ( V, ϕ, Λ , r )) , for all ( V, ϕ, Λ , r ) , ( V ′ , ϕ ′ , Λ ′ , r ′ ) ∈ Q ( k, β ).(ii) Suppose that [ k, β ] is self dual. A pss-character supported on a self dual simple pair [ k, β ]is called σ -invariant if for any (or equivalently one) self dual realisation of [ k, β ] the valueis σ -invariant.(iii) Suppose that [ k, β ] is self dual. A self dual potential semisimple character , or self dualpss-character , supported on the self dual semisimple pair [ k, β ] is a functionΘ − : Q − ( k, β ) → C − ( k, β )such that Θ − (( V, h ) , ϕ, Λ , r ) ∈ C − (Λ , r, ϕ ( β )), for all (( V, h ) , ϕ, Λ , r ) ∈ Q − ( k, β ), andΘ − (( V ′ , h ′ ) , ϕ ′ , Λ ′ , r ′ ) = τ Λ , Λ ′ ,β (Θ − (( V, h ) , ϕ, Λ , r )) , for all (( V, h ) , ϕ, Λ , r ) , (( V ′ , h ′ ) , ϕ ′ , Λ ′ , r ′ ) ∈ Q ( k, β ).As in the simple setting, every self dual pss-character comes uniquely from the restriction ofa σ -invariant pss-character, we call the values of a pss-character realisations , and we call theunique σ -invariant pss-character restricting to a self dual pss-character its lift .Let [ k, β ] be a self dual semisimple pair, whose indexing set splits as a disjoint union I = I + ∪ I ∪ I − as above, and (( V, h ) , ϕ, Λ , r ) ∈ Q − ( k, β ). Then we can write V = M i ∈ I + ( V i ⊕ V σ ( i ) ) ⊕ M i ∈ I V i . Moreover, writing M for the Levi subgroup of G + stabilising this decomposition, as in [14, § H r +1 − ( β, Λ) ∩ M ≃ Y i ∈ I + H r +1 ( β i , Λ i ) × Y i ∈ I H r +1 − ( β i , Λ i ) . Let Θ − be a self dual pss-character supported on [ k, β ]. Then, after identifying H r +1 − ( β, Λ) ∩ M with the decomposition above, we haveΘ − (( V, h ) , ϕ, Λ , r ) | H r +1 ( β, Λ) ∩ M = O i ∈ I + Θ i ( V i , ϕ i , Λ i , r ) ⊗ O i ∈ I Θ i, − (( V i , h i ) , ϕ i , Λ i , r )where(i) For i ∈ I + , Θ i is a ps-character supported on the simple pair [ k, β i ].(ii) For i ∈ I , Θ − ,i is a self dual ps-character supported on the self dual simple pair [ k, β i ].Moreover, as in [1, 4.3 Lemma 1], H r +1 ( β i , Λ i ) = H r +1 (2 β i , Λ i ) , Θ i ( V i , ϕ i , Λ i , r ) ∈ C (Λ i , r, β i ) , for i ∈ I + . We record these observations, along with the decomposition of a pss-character into ps-charactersin the following lemma: 24 emma 8.6.
Let [ k, β ] be a semisimple pair with index set I and β = L i ∈ I β i , and let Θ be apss-character supported on [ k, β ].(i) If β i is non-zero, the pair [ k, β i ] is a simple pair. Moreover, the map Θ i , defined by re-stricting Θ to its i -th component, is well defined and is a ps-character supported on [ k, β i ].(ii) Suppose [ k, β ] is self dual with I = I + ∪ I ∪ I − , and Θ − is a self dual pss-charactersupported on [ k, β ] with lift Θ.(a) For i ∈ I + , the pair [ k, β i ] is a simple pair, and the map Θ i , defined by restric-tion of Θ − to its ( i ∪ σ ( i ))-th component, is well defined and when the domain isidentified with the domain of the i -th component this restriction is the square of aps-character Θ i relative to [ k, β i ], and is a ps-character relative to [ k, β i ].(b) For i ∈ I , the pair [ k, β i ] is a self dual simple pair, and the map Θ − ,i , defined byrestriction of Θ − to its i -th component, is well defined and is a self dual ps-character. In this section we consider two semisimple strata [Λ , n, r, β ] and [Λ ′ , n, r, β ′ ] with the associatedsplittings V = L i ∈ I V i and V ′ = L i ∈ I ′ V ′ i , respectively. The starting point for this section isthe following result of the second and third authors: Theorem 9.1 ([18, Theorem 10.1]) . Let θ ∈ C (Λ , r, β ) and θ ′ ∈ C (Λ ′ , r, β ′ ) be intertwiningsemisimple characters. Then there is a unique bijection ζ : I → I ′ and an element g of e G suchthat, for all i ∈ I ,(i) we have gV i = V ′ ζ ( i ) ;(ii) the characters g θ i and θ ′ ζ ( i ) intertwine.Moreover, any element which satisfies the first property also satisfies the second property.A map between splittings of two semisimple strata which satisfies Properties (i), (ii) of thetheorem is called a matching . Let θ ∈ C (Λ , r, β ) and θ ′ ∈ C (Λ ′ , r, β ′ ) be semisimple characters.We denote by I ( θ, θ ′ ) the set of elements of e G which intertwine θ with θ ′ , i.e. those g ∈ e G suchthat Hom g H r +1 ( β, Λ) ∩ H r +1 ( β ′ , Λ ′ ) ( g θ, θ ′ ) = 0 . We put I G ( θ, θ ′ ) = I ( θ, θ ′ ) ∩ G . Given a semisimple character θ ∈ C (Λ , r, β ) and a sub-set J of the index set I we denote by θ J the restriction of θ to the intersection of its domainwith Aut F ( L i ∈ J V i ). Corollary 9.2.
Under the assumptions of Theorem 9.1, we have I ( θ, θ ′ ) = S (Λ ′ , β ′ ) Y i ∈ I I ( θ i , θ ′ ζ ( i ) ) ! S (Λ , β )where S (Λ , β ) is a subset of P (Λ) and of the normaliser of each element of C (Λ , r, β ) whichonly depends on Λ and β (cf. [9, (3.5.1)]). Assume further that for i ∈ I , V i and V ′ ζ ( i ) areisomorphic ǫ -hermitian spaces and both characters are self dual. Then, we have I G ( θ, θ ′ ) = ( S (Λ ′ , β ′ ) ∩ G + ) Y i ∈ I + ∪ I I ( θ { i,σ ( i ) } , θ ′{ ζ ( i ) ,σ ( ζ ( i )) } ) ∩ G + ( S (Λ , β ) ∩ G + ) . Proposition 9.3. (i) Let θ ′ ∈ C (Λ ′ , r, β ) be the transfer of θ ∈ C (Λ , r, β ) where both latticesequences have the same period. Then, I ( θ, θ ′ ) = S (Λ ′ , β ) B × β S (Λ , β ) . (ii) Let θ ′− ∈ C − (Λ ′ , r, β ) be the transfer of θ − ∈ C − (Λ , r, β ) where both lattice sequences havethe same period. Then, I ( θ, θ ′ ) ∩ G + = ( S (Λ ′ , β ) ∩ G + )(B × β ∩ G + )( S (Λ , β ) ∩ G + ) . See [21, Theorem 3.22] for the case Λ = Λ ′ . Proof.
This proof is anagolous to the proof of [12, Theorems 3.9 & 3.10].(i) Let us at first assume that both latices sequences are blockwise principal lattice chainsof the same block size. There is an element g in B × β such that g Λ is equal to Λ ′ and theconjugation with g realises the transfer from C (Λ , r, β ) to C (Λ ′ , r, β ). Thus we can reduceto the case where θ is equal to θ ′ which follows from [21, Theorem 3.22].We now consider the general case. Applying the † -construction of [12, § θ † ∈ C (Λ † , r, β ), θ ′ † ∈ C (Λ ′ † , r, β ), where Λ † and Λ ′ † are principallattice chains of the same block size in a direct sum of e -copies of V (where e = e (Λ) = e (Λ ′ )). Hence, as above we have the formula (i) for I ( θ † , θ ′ † ). As in [19, Corollary 4.14] wededuce that this formula behaves well under intersection with the Levi group M attachedto the † construction, i.e. I ( θ † , θ ′ † ) ∩ M = ( S (Λ ′ † , β † ) ∩ M )( B × β † ∩ M )( S (Λ † , β † ) ∩ M ) , by [12, Theorem 2.7 (ii) (b)], using the group Γ = {± } e , if we have the intersectionproperty S (Λ ′ † , β † ) xS (Λ † , β † ) ∩ B × β † = ( S (Λ ′ † , β † ) ∩ B × β † ) x ( S (Λ † , β † ) ∩ B × β † ) , for all x ∈ B × β † . The proof of the intersection property follows mutatis mutandis tothe proof of [12, Lemma 3.6]. We restrict to the first block of M to obtain the desireddescription of I ( θ, θ ′ ).(ii) This follows from (i) and a cohomology argument [12, Theorem 2.7 (ii) (b)]; see [12,Corollary 3.7]. Proof of Corollary 9.2.
By [9, 3.5.1], using the † -construction on the i -th component of Lem-ma 3.2, we obtain for every index i that E i /F and E ′ ζ ( i ) /F have the same ramification indexand inertia degree. Thus there is an o E -lattice sequence Λ ′′ which is ˜ G conjugate to Λ ′ byan element which maps Λ ′′ i to Λ ′ ζ ( i ) . Let θ ′′ = τ Λ ′′ , Λ ,β ( θ ) be the transfer of θ to C (Λ ′′ , r, β )which, by intertwining implies conjugacy [18, Theorem 10.2], is conjugate to θ ′ by an element26hich maps Λ ′′ i to Λ ′ ζ ( i ) . Thus we can assume that θ ′ and θ ′′ are equal. We can further assumethat β and β ′ have the same associated splitting and that the matching ζ is the identity, seeTheorem 9.1. By Proposition 9.3, I ( θ, θ ′ ) = S (Λ ′′ , β ) B × β S (Λ , β )= S (Λ ′′ , β ) n (Λ ′′ E ) B × β S (Λ , β )= S (Λ ′ , β ′ ) n (Λ ′ E ′ ) B × β S (Λ , β ) , the last equality as S (Λ ′′ , β ) n (Λ ′′ E ) and also S (Λ ′ , β ′ ) n (Λ ′ E ′ ) is the normaliser of every elementof C (Λ ′′ , r, β ). The second assertion follows from a cohomology argument as in [21, 4.14] (cf.also [12, 2.4]).We can now prove a conjecture of the second and third authors, cf. [18, Conjecture 10.4]. Theorem 9.4.
Let θ − ∈ C − (Λ , r, β ) and θ ′− ∈ C − (Λ , r, β ′ ) be intertwining self dual semisimplecharacters. Then, for i ∈ I , the spaces V i and V ′ ζ ( i ) are isometric and the characters θ − ,i and θ ′− ,ζ ( i ) intertwine by an isomorphism from ( V i , h i ) to ( V ′ ζ ( i ) , h ζ ( i ) ), where h i , h ζ ( i ) denotethe restrictions of h to V i , V ζ ( i ) , respectively. Moreover, the Witt towers of h i,β i and h ζ ( i ) ,β ′ ζ ( i ) match (via the isometry). Proof.
We work with the lifts of θ − and θ ′− . By Corollary 9.2, it is enough to show the firstassertion, and we proceed by induction. For minimal strata this result is known by [18, Propo-sition 7.10], and having done the proof for r − γ in the firstmember of a defining sequence, and that θ | H r +2 ( γ, Λ) and θ ′ | H r +2 ( γ, Λ) intertwine by 1, i.e. theyare transfers. Now a modification of [18, Proposition 9.19] for different lattice sequences showsthat there is a c ∈ a − ( r +1) ∩ Q i A ii such that ψ s ( β − γ + c ) intertwines with ψ s ( β ′ − γ ) via 1, andthus applying the result for strata of [18, Proposition 7.10] blockwise gives the result. The finalassertion follows from Proposition 7.1. Corollary 9.5.
Under the assumptions of Theorem 9.1, suppose that β and β ′ have the samecharacteristic polynomial, then β i and β ζ ( i ) have the same characteristic polynomial for all i ∈ I . Proof.
As in the proof of Lemma 3.2, we can restrict to the case where all Λ i are principal byadding up e (Λ) shifts of Λ and similarly for Λ ′ . Conjugate β to β ′ by an element g of e G , g Λ to Λ ′ by an element g of B × β ′ , and then g g .θ to θ ′ by an element of P (Λ) using [18, Theorem 10.2].In fact, the last conjugation can be done by an element u of P (Λ) which follows by inductionand [18, Proposition 9.11]. By [18, Proposition 9.7], knowing that S ( β ′ , Λ ′ ) is contained in thenormaliser of θ ′ , we can take u in Q i ′ A i ′ i ′ . The element ug does not permute the blocks of theassociated splitting of β ′ and therefore the matching from θ to θ ′ is induced by g .Finally, we can deduce a semisimple Skolem–Noether result:
Corollary 9.6.
Let θ ∈ C (Λ , r, β ) σ h and θ ′ ∈ C (Λ ′ , r, β ′ ) σ h be two G + -intertwining semisimplecharacters and suppose that β and β ′ have the same characteristic polynomial. Then β and β ′ are conjugate in G + . Proof.
By Theorem 9.4, we can assume that the matching ζ is the identity of I . Now, thecharacters θ i and θ ′ i intertwine over G + and the characteristic polynomials of β i and β ′ i areequal by Corollary 9.5. Hence, by [18, Theorem 5.2], β i and β ′ i are conjugate by an elementof U( V i , h i ). Thus β and β ′ are conjugate in G + .27 We continue with the notation of Section 8.
Definition 10.1.
We call semisimple pairs [ k, β ] and [ k, β ′ ] comparable if there are realisa-tions [Λ , n, r, ϕ ( β )] and [Λ ′ , n ′ , r ′ , ϕ ′ ( β ′ )] in A , and semisimple characters θ ∈ C (Λ , r, ϕ ( β ))and θ ′ ∈ C (Λ ′ , r, ϕ ′ ( β ′ )) which intertwine in e G .By Theorem 9.1, we then have a bijection ζ : I → I ′ between the indexing sets of the semisimplepairs provided by the matching from θ to θ ′ . Lemma 10.2.
This bijection is independent of the choice of semisimple characters.
Proof.
Having a second pair e θ and e θ ′ with e ζ then adding up shifts of lattice sequences andcutting down parts of block together with conjugation reduces the problem to the case θ, e θ ∈ C (Λ , r, β ) , θ ′ , e θ ′ ∈ C (Λ , r, β ′ ) . Further we can assume that we can conjugate θ ′ to θ by [18, Theorem 10.2], say via g ∈ P (Λ)and e θ ′ to e θ via an element ˜ g of P (Λ). Thus, C (Λ , r, gβ ′ g − ) = C (Λ , r, ˜ gβ ′ ˜ g − ) . Further we can chose the elements g and ˜ g such that they map the splitting of β ′ to the splittingof β , by [18, 9.8]. Now, Corollary 9.5, applied only for θ , states that ζ − is equal to ˜ ζ − .We can now define the generalisation of comparison pairs defined in the simple case (see Def-inition 3.1). In fact, we only need to add the dimensions of the spaces which occur in thesplitting. Definition 10.3. (i) Let Θ and Θ ′ be two pss-characters supported on comparable semisim-ple pairs [ k, β ] and [ k, β ′ ] with matching ζ : I → I ′ . We call a pair(( V, ϕ, Λ , r ) , ( V ′ , ϕ ′ , Λ ′ , r ′ ))of quadruples ( V, ϕ ′ , Λ ′ , r ′ ) ∈ Q ( k, β ) and ( V ′ , Λ ′ , r ′ , ϕ ′ ) ∈ Q ( k, β ′ ) a semisimple compari-son pair if V = V ′ , r = r ′ , e (Λ) = e (Λ ′ ) , dim F V i = dim F V ′ ζ ( i ) such that k = (cid:22) re (Λ E ) (cid:23) . (ii) A semisimple comparison pair is called strong if Λ = Λ ′ as o F -lattice sequences and, forall i ∈ I and all integers j , we haveΛ ij / Λ ij +1 ∼ = Λ ′ ζ ( i ) j / Λ ′ ζ ( i ) j +1 . (iii) Let Θ − and Θ ′− be self dual pss-characters supported on comparable self dual semisimplepairs. A comparison pair for Θ − and Θ ′− is a comparison pair((( V, h ) , ϕ, Λ , r ) , (( V, h ) , ϕ ′ , Λ ′ , r ))where V is equipped with an ǫ -hermitian form h which splits under both splittings of V and such that ( V i , h i ) is isometric to ( V ζ ( i ) , h ζ ( i ) ), for all i ∈ I .28iv) A comparison pair for self dual pss-characters is called Witt if it is blockwise Witt on I ,see Lemma 8.6.(v) The pss-characters Θ and Θ ′ are called endo-equivalent if for all comparison pairs thecorresponding realisations of Θ and Θ ′ intertwine in e G .(vi) The self dual pss-characters Θ − and Θ ′− are called endo-equivalent if for all Witt compar-ison pairs the corresponding realisations of Θ − and Θ ′− intertwine in G + . Proposition 10.4.
Two pss-characters Θ and Θ ′ supported on comparable semisimple pairs,say [ k, β ] and [ k, β ′ ], with matching ζ : I → I ′ , are endo-equivalent if and only if the ps-characters Θ i and Θ ′ ζ ( i ) defined by componentwise restriction are endo-equivalent, for all i ∈ I . Proof.
Suppose that the ps-characters Θ i and Θ ′ ζ ( i ) are endo-equivalent, for all i ∈ I for which β i is non-zero. We can assume by conjugation with an automorphism of V that V i is equal to V ζ ( i ) ,for all i . Taking a comparison pair for Θ and Θ ′ , by restriction to the i -th block we havea comparison pair for Θ i and Θ ′ ζ ( i ) , whose realisations intertwine as Θ i and Θ ′ ζ ( i ) are endo-equivalent (the case where β i is zero is trivial). We thus obtain a block diagonal intertwiner ofthe realisations of Θ and Θ ′ . For the converse, take intertwining realisations of Θ and Θ ′ . Itfollows from Corollary 9.2, that the simple characters realising Θ i and Θ ζ ( i ) intertwine, and weconclude by Theorem 7.2.We now prove transitivity of endo-equivalence of pss-characters, showing that endo-equivalenceof pss-characters is an equivalence relation. Proposition 10.5.
For pss-characters endo-equivalence is an equivalence relation, and twopss-characters are endo-equivalent if and only if there is a comparison pair such that the corre-sponding realisations of the pss-characters intertwine in e G . Proof.
This follows now directly from Proposition 10.4 and the fact that endo-equivalence is anequivalence relation for ps-characters, by Theorem 2.11.We call the equivalence classes GL-endo-classes. Since we consider the zero strata [Λ , r, r,
0] to besimple strata, this definition includes the so-called zero-endo-class , for each k >
0: this consistsof the unique ps(s)-character supported on [ k, Theorem 10.6.
Let Θ − and Θ ′− be two self dual pss-characters and Θ and Θ ′ their lifts. Then,the following assertions are equivalent:(i) The self dual pss-characters Θ − and Θ ′− are endo-equivalent;(ii) The lifts Θ and Θ ′ are endo-equivalent.(iii) There is a comparison pair such that the corresponding realisations of Θ − and Θ ′− inter-twine over G + . Proof.
As in Lemma 8.6, the pss-characters Θ − and Θ ′− decompose into ps-characters andself dual ps-characters. Suppose that there is a comparison pair such that the correspondingrealisations of Θ − and Θ ′− intertwine then Theorem 9.4 and Corollary 9.2 ensure that there29re comparison pairs such that the corresponding realisations of the ps-characters and self dualps-characters intertwine. Hence, via Theorem 7.2, point (iii) implies both (i) and (ii). Theother implications follow similarly using Theorem 7.2.From the Theorem 10.6 and Proposition 10.5, we obtain transitivity of endo-equivalence as aconsequence, for example the as an analogue to Corollary 7.4, we obtain the rather surprisingresult that intertwining of self dual semisimple characters with same group level is transitive,and hence equivalence relation: Corollary 10.7. (i) For self dual pss-characters endo-equivalence is an equivalence relation.(ii) Intertwining over G + is an equivalence relation on the class of self dual semisimple char-acters with the same group level.(iii) intertwining over e G is an equivalence relation on the class of semisimple characters withthe same group level. Proof.
The first and then last assertion follow directly from Theorem 10.6 and Proposition 10.5.The proof of the second assertion is analogous to the simple case, using Theorem 10.6 and thefirst assertion of this corollary in place of their simple counterparts.
Definition 10.8.
We call the equivalence classes of self dual pss-characters ( σ, ǫ ) -endo-classes or classical endo-classes .
11 The translation principle for self dual characters
This section is written for a generalisation of the “Intertwining and Conjugacy” Theorem forskew characters [18, Theorem 10.3] to the case of self dual characters. The strategy is completelythe same if one substitutes the following generalised version of the “Translation Principle” [18,Theorem 9.18].
Theorem 11.1.
Let [Λ , n, r + 1 , γ ] and [Λ , n, r + 1 , γ ′ ] be self dual semisimple strata with thesame associated splitting such that C (Λ , r + 1 , γ ) = C (Λ , r + 1 , γ ′ ) . Let [Λ , n, r, β ] be a self dual semisimple stratum, with splitting V = L i ∈ I V i , such that [Λ , n, r +1 , β ] is equivalent to [Λ , n, r + 1 , γ ] and γ is an element of Q i ∈ I A i,i . Then, there exists a selfdual-semisimple stratum [Λ , n, r, β ′ ], with splitting V = L i ′ ∈ I ′ V ′ i ′ , such that [Λ , n, r + 1 , β ′ ] isequivalent to [Λ , n, r + 1 , γ ′ ], with γ ′ ∈ Q i ′ ∈ I ′ A i ′ ,i ′ and C (Λ , r, β ) = C (Λ , r, β ′ ) . Proof.
Let θ ∈ C (Λ , r, β ) be a lift of an element of C − (Λ , r, β ). Then there is a skew-symmetricelement b of a − m such that θψ − b is an element of C − (Λ , m, γ ′ ). By the translation princi-ple for semisimple characters [18, 9.10] there is a semisimple stratum [Λ , n, r, β ′′ ] equivalentto [Λ , n, r, γ ′ + b ] such that C (Λ , r, β ′′ ) is equal to C (Λ , r, β ). By self duality the latter is alsoequal to C (Λ , m, − σ ( β ′′ )). Let I ′′ be the index set of β ′′ and σ ( I ′′ ) be the index set of σ ( β ′′ ). Thematching (coming from [18, Lemma 7.17], since the strata [Λ , n, r, β ′′ ] and [Λ , n, m, − σ ( β ′′ )] areequivalent) gives a map τ from I ′′ to σ ( I ′′ ) such that for every i the idempotent 1 i is congruentto 1 τ ( i ) modulo m − ( k + m ) . The indexing set I ′′ decomposes as the union of the two sets:30i) I ′′ which is the set of all the indexes i which satisfy σ ( i ) = τ ( i ) and(ii) I ′′± which is the complement of I ′′ in I ′′ .The action of τ − σ on I ′′± decomposes into orbits all of length two; we choose a representativefor each orbit and let I ′′ + be this set of representatives. We also write I ′′− = { τ − σ ( i ) | i ∈ I ′′ + } ,the complement of I ′′ + in I ′′ .We recall the following idempotent lifting lemma from [18]: Lemma 11.2 ([18, Lemma 7.13]) . Let ( k r ) r ≥ be a decreasing sequence of o F -lattices in A suchthat k r k s ⊆ k r + s , for all r, s ≥
0, and T r ≥ k r = { } . Suppose there is an element α of k whichsatisfies α − α ∈ k r . Then there is an idempotent ˜ α ∈ k such that ˜ α − α ∈ k r . Moreover,if σ ( α ) = α then we can choose ˜ α such that σ ( ˜ α ) = ˜ α , while if α + σ ( α ) = 1 then we canchoose ˜ α such that ˜ α + σ ( ˜ α ) = 1. Proof.
The only part not proved in [18, Lemma 7.13] is the very final assertion. If α + σ ( α ) = 1then α and σ ( α ) commute and it is easy to check that α ′ := 3 α − α also satisfies α ′ + σ ( α ′ ) = 1and the result follows in the same way as in loc. cit. We will apply Lemma 11.2 to find a decomposition1 = X i ∈ I ′′ e i as a sum of idempotents e i such that the set of all the e i is stable under the action of τ − σ and 1 i is congruent to e i modulo m − ( k + m ) , for each i ∈ I ′′ . We proceed as follows:(i) At first we apply the lemma find σ -fixed idempotents congruent to the 1 i with i ∈ I ′′ : Startby taking α = i + σ (1 i )2 to obtain e by Lemma 11.2, and then α := (1 − e ) + σ (1 )2 (1 − e )and continue inductively. Having found all e i , i ∈ I ′′ , we denote their sum as e .(ii) Then we find idempotents e + and e − such that their sum is 1 − e and such that e + is congruent to the sum 1 + of the 1 i where i runs through I ′′ + and similarly for e − . Weapply the Lemma to α = (1 − e ) (cid:16) + − σ (1 + )2 (cid:17) (1 − e ) to find e + and we define e − tobe 1 − e − e + .(iii) Similar to part (i) we obtain for all indexes i of I ′′ + idempotents e i , congruent to 1 i modulo m − ( k + m ) , which all sum up to e + and as a consequence the idempotents e − i := σ ( e i ) have the same properties with respect to e − .We are now ready to construct the element β ′ :(i) For i ∈ I ′′ , there is a simple self dual stratum [Λ i , n i , r, β ′ i ] equivalent to [Λ i , n i , r, e i β ′′ e i ].(ii) for i ∈ I ′′ + , we take for a simple stratum [Λ i , n i , r, β ′ i ] equivalent to [Λ i , n i , r, e i β ′′ e i ].(iii) for i ∈ I ′′ + , we define β ′ i to be − β ′ τ − ( σ ( i )) .Thus, we have the desired element β ′ = P i ∈ I ′′ β ′ i , since [Λ , n, r, β ′ ] is a self-dual semisimpleequivalent to [Λ , n, r, γ ′ + b ]. 31e will need later the following “Intertwining implies conjugacy” property for self dual non-skewcharacters: Theorem 11.3.
Let θ ∈ C − (Λ , r, β ) and θ ′ ∈ C − (Λ , r, β ′ ) be two semisimple self dual characterswhich intertwine over G + such that the matching ζ : I → I ′ satisfiesΛ ij / Λ ij +1 ∼ = Λ ′ ζ ( i ) j / Λ ′ ζ ( i ) j +1 , for all indexes i ∈ I and integers j . Then, there is an element of P − (Λ) which conjugates θ to θ ′ . Proof.
The proof is mutatis mutandis the proof of [18, Theorem 10.3] following a strata induc-tion, see [18, after Remark 7.2], if we know the result for minimal strata. Thus, we supposethat we have two strata [Λ , n, n − , β ] and [Λ ′ , n, n − , β ′ ] which intertwine over some elementof G + . We only need to consider the case that the first stratum is a self dual non-skew stratumwhose index set I has cardinality two. By the matching theorem Theorem 9.4 the index set I ′ of the second stratum has also two elements interchanged by the adjoint involution. By [18,Theorem 10.2] there is an isomorphism g from V to V ζ (1) such that g Λ is equal to Λ ′ ζ (1) and [Λ ′ ζ (1) , n, n − , gβ g − ] is equivalent to [Λ ′ ζ (1) , n, n − , β ′ ζ (1) ]. The self duality of the strataimply the equivalence of [Λ ′ ζ (2) , n, n − , σ ( g ) − β σ ( g )] and [Λ ′ ζ (2) , n, n − , β ′ ζ (2) ].
12 Intertwining under special orthogonal groups
Here we consider endo-equivalent pss-characters Θ − and Θ ′− for self-dual semisimple pairs [ k, β ]and [ k, β ′ ] in the orthogonal setting σ = 1, ǫ = 1. Let θ − ∈ C − (Λ , r, ϕ ( β )) and θ ′− ∈C − (Λ ′ , r, ϕ ′ ( β ′ )) be realisations of Θ − and Θ ′− respectively on an orthogonal space ( V, h ). In thissection we analyse the intertwining classes of an orthogonal endo-class. Let us assume that Λand Λ ′ have the same period, i.e. e (Λ) = e (Λ ′ ). Proposition 12.1. (i) Assume that all ϕ ( β i ) have the same valuation for the lattice se-quence Λ, i.e. ν Λ ( ϕ ( β i )) = − q for all indexes i . The realisations θ − and θ ′− intertwineover G if and only if the symplectic forms h ϕ ( β ) and h ϕ ′ ( β ′ ) are isometric by an automor-phism of V of determinant congruent 1 modulo p F .(ii) Assume that there is an index i such that β i is zero. Then, θ and θ ′ intertwine by anelement of G . Proof. (i) The assumption on the β i is also true for the β ′ i by the matching theorem, The-orem 9.1. As Θ − and Θ ′− are endo-equivalent, there is an element g of G + which inter-twines θ − and θ ′− . Then, the fundamental strata [Λ , n, n − , φ ( β )] and [Λ ′ , n, n − , φ ′ ( β ′ )]intertwine under g and the twists of h , h ϕ ( β ) and h ϕ ′ ( β ) , are isometric by an element ofthe form u ′ gu with elements u ∈ a (Λ) and u ′ ∈ a (Λ ′ ), by [18, Proposition 3.1].The element u ′ gu has determinant congruent to the determinant of g modulo p . The factthat an isometry of a symplectic space has determinant 1 finishes the proof.(ii) By Theorem 9.4 we can assume that both characters have the same associated splitting.The characters θ i and θ ′ i are trivial, and therefore intertwine under any element of thegroup U( V i , h i ), and in particular by an element of determinant −
1. Thus, there is anelement of G which intertwines the semisimple characters.32 heorem 12.2. (i) If there is an index i such that β i is zero, then θ − and θ ′− intertwineunder an element of G .(ii) If β has no zero component, then θ − and θ ′− intertwine under an element of G if and onlyif h ϕ ( β ) and h ϕ ′ ( β ) are isometric by an automorphism of V of determinant congruent 1modulo p . In this case every element of G + intertwining θ − and θ ′− is in G . Proof. (i) See Proposition 12.1.(ii) Without loss of generality we can assume that the matching is the identity, because thereis an element of G which maps the decomposition of V for ϕ ( β ) to the decomposition of V for ϕ ′ ( β ′ ). Write the index set I in the coarsest way as the union of subsets S such thatthe elements ϕ ( β s ) have all the same valuation with respect to Λ, for all s ∈ S . We nowapply Proposition 12.1 to all blocks S and we get the result.As a corollary we have for semisimple characters an analogue of Corollary 10.7. Corollary 12.3.
Let θ − , θ ′− and θ ′′− be self dual semisimple characters with the same grouplevel. If θ − intertwines with θ ′− and θ ′− intertwines with θ ′′− over G , then θ − intertwines with θ ′′− over G .The next theorem is about intertwining and conjugacy of semisimple characters over specialorthogonal groups. There are examples of θ and θ ′ which intertwine over G and are conjugateover G + , but are not conjugate under G . This, can not happen if β has no zero-component.But if β has a zero-component β i = 0 such that there is a lattice sequence in Λ i which hasno element of determinant − g i of determinant − h i ) and define θ ′′ := diag(id , g i ) .θ , then θ and θ ′′ are not conjugate by an element of G ,but intertwine by an element of G , by Theorem 12.2. We fix this problem in the following way: Theorem 12.4.
Suppose θ and θ ′ intertwine under an element g of G and there is an element g ′ of G + such that g ′ Λ i is equal to Λ ′ ζ ( i ) . We write g = u diag( g i | i ∈ I + ∩ I ) v using Corollary 9.2.Suppose one of the following three assertions:(i) The element β has a zero component, say β i , Λ i and θ i have the same normaliserin U( h i ), g ′ ∈ G and g − i g ′ i is an element of SU( h i ).(ii) The element β has a zero component, say β i , and the normaliser of Λ i contains anelement of U( h i ) with determinant − β has no zero component.Then θ is conjugate to θ ′ by an element of G ∩ P − (Λ). Remark 12.5.
Unfortunately the theorem is very unsatisfying, because in the first part wehave the condition that Λ i and θ i have the same normaliser in G + . In general this is a propercondition, i.e. there are many examples where this fails. As an example consider the latticesequence Λ on the Iwahori in O (1 , F ) but with three lattices in one period, such that foronly one index j the lattice Λ j is self dual. Then, a is the radical of a maximal order and thusthe trivial characters on P (Λ) has a normaliser in O (1 , F ) bigger than the normaliser of Λ.But, in spite of the example, the normaliser condition holds if r = 0, we call this later full , seeSection 14, because a is the radical of Λ and both have the same normaliser.33 roof of Theorem 12.4. (i) We can assume g ′ is the identity. Therefore ζ is the identityand Λ i and Λ ′ i equal. In particular, on the block for the zero component of β the charactersequal, i.e. θ i = θ ′ i . There is nothing to prove if Λ i has an element of determinant − h i ). Thus let us assume the opposite, which by assumption impliesthat the U( h i )-normaliser of θ i is also contained in SU( h i ). By Theorem 11.3, θ and θ ′ are thus conjugate by an element of P − (Λ). Corollary 9.2 and Theorem 12.2 now implythat this conjugating element must have determinant 1.(ii) This follows directly from Theorem 11.3.(iii) The determinant of the conjugating element and the intertwining element have to coincideby Proposition 12.1. Remark 12.6.
The second assumption of Theorem 12.4 is satisfied if the lattice sequence Λ i ,represents a vertex in the Bruhat Tits building of U( h i ) , i.e. (say for simplicity that h = h i )there is an index l such that ̟ F Λ l ⊂ Λ l ⊆ Λ l , and the image of Λ only consists of F × -multiples of Λ l and its dual (Λ l = Λ l is possible). Proposition 12.7.
Let G be an F = F -form of a special orthogonal group on an F -vectorspace V . Suppose Λ is a self dual lattice sequence which corresponds to a vertex in the BruhatTits building of G + . Then there is an element of G + \ G in the normaliser of Λ. Proof.
We have two cases. Let us at first assume that there is a positive anisotropic dimension.Then every lattice sequence Λ ′ is normalised by an orthogonal element of determinant −
1, be-cause we just have to take a Witt basis whose apartment contains the point corresponding to Λ ′ ,i.e. a Witt basis which splits the lattice sequence. Then the diagonal matrix diag(1 , . . . , , − P − (Λ ′ ). Let us now assume that the anisotropic dimension is zero. We takea Witt basis which splits Λ, i.e. a basis ( v j ) j ∈ J + ∪ J − with − J + = J − and h ( v j , v j ′ ) = δ j, − j ′ (Kronecker symbol) for all j ∈ J + . We denote M := Λ l , see Remark 12.6. Then, M = M j ∈ J + ∪ J − p ν j F v j , and we can rescale the basis the way that ν j = 0 for all j ∈ J + . Then the condition on M rephrases as ν j is 1 or 2, for j ∈ J − , such that at least one exponent has to be 1, say ν − = 1.We define an element of G + by g ( v j ) := v j , j = 1 , − , g ( v ) := v − ̟ F , g ( v − ) := v ̟ − F , and g normalises Λ and has determinant −
13 Intertwining implies conjugacy for cuspidal types
We recall definitions of the third author in [22] of cuspidal types in G , then prove our maintheorem that two intertwining cuspidal types are conjugate in G .A skew semisimple stratum [Λ , n, , β ] is called cuspidal if G E has compact centre and P ◦ (Λ E )is a maximal parahoric subgroup in G E . By [21, § , n, , β ] are com-pact open subgroups J ( β, Λ) ⊇ J ( β, Λ) ⊇ H ( β, Λ) of G . The quotient J ( β, Λ) /J ( β, Λ) ≃ (Λ E ) /P (Λ E ) is a finite reductive group, and we put J ◦ ( β, Λ) the preimage of its connectedcomponent in J ( β, Λ).Let [Λ , n, , β ] be a cuspidal skew semisimple stratum. Let θ ∈ C − (Λ , , β ) be a self dualsemisimple character. By [21, Corollary 3.29], there exists a unique irreducible representation η of J ( β, Λ) containing θ . The representation η extends to J ( β, Λ) and we choose a particulartype of extension κ as in [22, Theorem 4.1], called a β -extension . Let τ be an irreduciblerepresentation of J ( β, Λ) /J ( β, Λ) with cuspidal restriction to J ◦ ( β, Λ) /J ( β, Λ). Put J = J ( β, Λ) and λ = κ ⊗ τ . A pair ( J, λ ), constructed as above, is called a cuspidal type for G . Themain result of [22] (noting the correction of [14, Appendix A]) can be stated as follows: Theorem 13.1 ([22, Corollary 6.19 & Proposition 7.13]) . Let (
J, λ ) be a cuspidal type for G .Then the representation ind GJ ( λ ) is irreducible and cuspidal. Moreover, every irreducible cusp-idal representation of G appears in this way.Thus, it remains to determine when two cuspidal types ( J, λ ) , ( J ′ , λ ′ ) induce isomorphic cuspidalrepresentations. Notice that, if ind GJ ( λ ) ≃ ind GJ ′ ( λ ′ ) then ( J, λ ) , ( J ′ , λ ′ ) intertwine in G . Main Theorem (intertwining implies conjugacy).
Cuspidal types intertwine in G if andonly if they are conjugate in G . Proof.
As the cuspidal types intertwine so do the underlying skew semisimple characters θ, θ ′ .Suppose that θ ∈ C − (Λ , , β ) and θ ′ ∈ C − (Λ ′ , , β ′ ). By Theorem 9.4, there is a matching ζ : I → I ′ ; and there exists g ∈ G such that gV i = V ′ ζ ( i ) and the characters θ g i i, − and θ ′ ζ ( i ) , − intertwinein G ζ ( i ) . By conjugating by an element of G , if necessary, we can assume that V = V ′ , ζ = 1,and I G i ( θ i, − , θ ′ i, − ) = 0 . As all of the arithmetical invariants are the same, we can build a lattice sequence Λ ′′ relativeto β ′ such that there exists g ∈ G + with g Λ ′ = Λ ′′ respecting the block structure, as follows: Forevery index i there is an E ′ i -lattice sequence Λ ′′ i and an element g i ∈ U( h i ) such that g i Λ i = Λ ′′ i by Corollary 5.9 and [17, Proposition 5.2]. And at the end we define Λ ′′ as the sum of the Λ ′′ i and g := L i g i . If there is some β ′ i equal to zero and we are in the special orthogonal casethen there is an element of determinant − P − (Λ i ) by Proposition 12.7, and Theorem 12.4ensures the existence of an element g of G which conjugates Λ to Λ ′′ .We let θ ′′ = τ Λ ′ , Λ ′′ ,β ′ ( θ ′ ), which intertwines with θ by Corollary 10.7 and 12.3. By conjugating,see [18, Theorem 10.3] and Theorem 12.4, we can assume θ = θ ′′ , and hence θ is a semisimplecharacter for β ′ . We can now conclude by [12, Theorem 11.3]. Remark 13.2.
In fact, due to [12], we have the analogous result in the greater generality ofrepresentations with coefficient field any algebraically closed field of characteristic prime to p .
14 Parametrisation of the intertwining classes of semisimplecharacters
This section is about the classification of intertwining classes of (self dual) semisimple characters.But before we start we need the following definition of endo-equivalent semisimple characters.
Definition 14.1.
Two (self dual) semisimple characters are called endo-equivalent if theircorresponding (self dual) pss-characters are endo-equivalent.35 GL -endo-parameters In this section we parametrise the intertwining classes of semisimple characters of e G . We saythat a semisimple character is full if it lies in C (Λ , , β ), for some semisimple stratum [Λ , n, , β ].Similarly, a pss-character is full if it is supported on a semisimple pair of the form [0 , β ], whilean endo-class of pss-characters is full if it consists of full pss-characters; note that this includesthe full zero-endo-class . Definition 14.2.
We define E to be the set of all full simple endo-classes of ps-characters. AGL -endo-parameter for e G is a function from the set E to the set of non-negative integers withfinite support.The goal of this section is to prove that a natural set of endo-parameters parametrise e G -intertwining classes of semisimple characters of e G . Let E fin be the set of finite subsets of E . Proposition 14.3.
There is a bijection between E fin and the set of all full GL-endo-classes.For the proof we need the following lemmas: Lemma 14.4.
Let θ ∈ C (Λ , r, β ) and θ ′ ∈ C (Λ ′ , r, β ′ ) be two endo-equivalent simple characterswith the same period. Then, there is an element β ′′ with the same minimal polynomial as β and a simple character ˜ θ ∈ C (Λ ⊕ Λ ′ , r, β + β ′′ ) such that ˜ θ | H r +1 ( β, Λ) = θ and ˜ θ | H r +1 ( β ′′ , Λ ′ ) = θ ′ . Proof.
By endo-equivalence β and β ′ have equal inertial degrees and equal ramification indexes,and therefore there is an element β ′′ of End F ( V ′ ) which has the same minimal polynomial as β .The transfer of θ to C (Λ ′ , r, β ′′ ) is conjugate to θ ′ and thus we can choose β ′′ such that θ ′ is atransfer of θ . But then there is a simple character ˜ θ whose restrictions are θ and θ ′ . Lemma 14.5.
Let θ ∈ C (Λ , r, β ) be a semisimple character and θ ′ ∈ C (Λ ′ , r, β ′ ) be a simple char-acter, and suppose Λ , Λ ′ have the same period. Suppose moreover that θ ′ is not endo-equivalentto any simple block restriction θ i of θ . Then, there is a semisimple stratum [Λ ⊕ Λ ′ , n, r, β + β ′′ ]whose associated decomposition is a refinement of V ⊕ V ′ , and there is a semisimple charac-ter ˜ θ ∈ C (Λ ⊕ Λ ′ , r, β + β ′′ ) such that the restrictions to H r +1 ( β, Λ) and H r +1 ( β ′′ , Λ ′ ) are θ and θ ′ , respectively. In particular, the groups H r +1 ( β ′′ , Λ ′ ) and H r +1 ( β ′ , Λ ′ ) coincide. Remark 14.6.
If the stratum [Λ ⊕ Λ ′ , n, n − , β + β ′ ] is equivalent to a semisimple stratumwhose associated splitting is a refinement of V ⊕ V ′ , then there is already such a character ˜ θ in C (Λ ⊕ Λ ′ , r, β + β ′ ) by definition of the semisimple characters, because in this particular therestriction map from C (Λ ⊕ Λ ′ , r, β + β ′ ) to C (Λ , r, β ) × C (Λ ′ , r, β ′ ) is a bijection. Proof of Lemma 14.5.
Let ˜ q be the maximum of q = − ν Λ ( β ) and q ′ = − ν Λ ′ ( β ′ ), and we write ˜Λfor the direct sum of Λ and Λ ′ . We prove the result by induction on m . The case m = q ′′ − m is smaller than q ′′ −
1. We have to consider three cases:Case 1: The restrictions θ | r +2 := θ | H r +2 ( β, Λ) ∈ C (Λ , r + 1 , γ ) and θ ′ | r +2 := θ ′ | H r +2 ( β ′ , Λ ′ ) sat-isfy the conditions of the lemma. Then by induction hypothesis there is a semisimple char-acter ˜ θ | r +2 ∈ C ( ˜Λ , r + 1 , γ + γ ′′ ) whose restrictions are θ | r +2 and θ ′ | r +2 . By the translationprinciple [18, Theorem 9.10] we can assume that γ ′′ is a first member of the defining sequenceof [Λ ′ , − ν Λ ′ ( β ′ ) , r, β ′ ]. Now by the definition of the semisimple characters there is a semisimplecharacter ˜ θ , element of C ( ˜Λ , r, β + β ′ ), whose restrictions are θ and θ ′ .36ase 2: Let us assume that θ | r +2 is simple and that θ | r +2 and θ ′ | r +2 are endo-equivalent: Thus,by Lemma 14.4 we can find a simple stratum [Λ , n, r + 1 , γ ] split by V ⊕ V ′ such that γ ismod a − ( r +1) congruent to β on V and a simple character ˜ θ | r +2 in C ( ˜Λ , r + 1 , γ ) with restric-tions θ | r +2 and θ ′ | r +2 . By the translation principle we can assume that γ | V ′ is a member of adefining sequence for β ′ . If the stratum [ ˜Λ , n, r, β + β ′ ] is already semisimple then the induc-tive definition of semisimple characters provides the the desired character ˜ θ ∈ C (Λ , r, β + β ′ ).In the opposite case, there is an index i such that [Λ i , n i , r, β i ] and [Λ ′ , n, r, β ′ ] sum up toa stratum equivalent to a simple one. By transfer, we can assume without loss of generalitythat β ′ is equal to β i ,Λ i = Λ ′ and θ i | r +2 = θ ′ | r +2 . By the first part of Case 2 there is asemisimple character ˜ θ ∈ C (Λ , r, β ) whose restrictions are θ i for i = i and θ ′ for the restrictionto the i th block. But then there is an element b ∈ a − ( r +1) (Λ i ) such that θ is equal to ˜ θψ b . Butthen [ ˜Λ , n, r, β + ( β ′ + b )] is equivalent to a semisimple stratum [ ˜Λ , n, r, β + β ′′ ] because θ and ˜ θ do not intertwine. Thus, [ ˜Λ , n, r, β + β ′′ ] is semisimple and there is a semisimple character ˜ θ with the desired restrictions, by part 1 of Case 2.Case 3: Let us assume the general case: Let I be the index set of β and let J be the subset of I consisting of all indexes i such that θ i | r +2 is endo-equivalent to θ ′ | r +2 . We can assume that J isnon-empty because of Case 1. Then, by the same argument as in Case 2, there is a semisimplecharacter ˜ θ J, ′ ∈ C (Λ J + Λ ′ , r, β J + β ′′ ) such that the restrictions are θ J and θ ′ . Let [Λ , n, r + 1 , γ ]be the first element in a defining sequence of [Λ , n, r + 1 , β ] and analogously [Λ ′ , n, r + 1 , γ ′′ ]for [Λ ′ , n, r + 1 , β ′′ ] such that γ J has the same minimal polynomial as γ ′′ . Then, θ J | r +2 is atransfer of, ˜ θ J, ′ | r +2 , and the stratum [ ˜Λ , n, r + 1 , γ + γ ′′ ] is semisimple because the set J describesexactly one block of [Λ , n, r + 1 , γ ].Thus, we can take the transfer of θ | r +2 to ( ˜Λ , γ + γ ′′ ) to obtain a characters ˜ θ | r +2 whoserestrictions are θ I \ J | r +2 and ˜ θ J, ′ | r +2 . The inductive definition of semisimple characters ensuresnow the existence of the desired ˜ θ . Proof of Proposition 14.3.
By Proposition 10.4, given a semisimple endo-class, the restriction tothe blocks gives a tuple of simple endo-classes. Conversely, given a tuple of simple endo-classes,take from every class a value of a ps-character, so that one gets a tuple of simple characters. ByLemma 14.5 there is a semisimple character which has the given simple characters as restrictions.The corresponding pss-character defines a semisimple endo-class.Recall that the degree of a simple character θ ∈ C (Λ , r, β ) is defined to be [ F [ β ] : F ]. This isindependent of intertwining and transfer. Thus we define the degree of a simple endo-class c ∈ E to be the degree of the values of the ps-characters in c , and we denote it by deg( c ). Theorem 14.7.
The set of intertwining classes of full semisimple characters for e G = GL F ( V )is in bijection to the set of those endo-parameters f which satisfy X c ∈E deg( c ) f ( c ) = dim F V. (14.8) Proof.
For a full semisimple character θ ∈ C (Λ , r, β ) for e G , denote by c i the endo-classes of itssimple block restrictions θ i , and put f i = dim F V i / deg( c i ) (which are integers), for i ∈ I . Theendo-classes c i and the integers f i are invariant under intertwining, by Theorem 9.1. Thus weget an endo-parameter f given by f ( c ) = ( f i if c = c i , for some i ∈ I ,0 otherwise,37hich depends only on the intertwining class of θ and satisfies (14.8).Conversely, given an endo-parameter f , write supp( f ) = { c i | i ∈ I } . Then Proposition 14.3gives us a full semisimple endo-class corresponding to supp( f ). We pick any pss-character Θ inthis endo-class, with corresponding semisimple pair [0 , β ] and β = P i ∈ I β i numbered such thatthe simple block restriction Θ i has endo-class c i , for each i ∈ I . Since deg( c i ) = [ F [ β i ] : F ],we can choose a realisation of each β i on an F -vector space V i of dimension f ( c i ) deg( c i ). Thesum of these gives us a realisation of the semisimple pair [0 , β ] on (a space isomorphic to) V ,and the realisation of Θ on this space gives us a full semisimple character.It is straightforward to check that the two maps described are the inverses of each other. Here we want to describe the intertwining classes of G + and of G . We start at first with G + .Let us fix ( σ, ǫ ). We call a self-dual semisimple character elementary if it is simple (hence skewsimple) or non-skew and the index set contains two elements. If a self dual pss-character has anelementary value, then all its values are elementary and if there is a non-skew value then all ofits values are non-skew. Thus we call a self-dual pss-character elementary if one (equivalentlyall) of its values is elementary. Moreover, if a self dual pss-character is endo-equivalent to anelementary one then it too is elementary and either both are skew or both are non-skew. Thus,the notions of skew elementary and non-skew elementary depend only on the endo-class, and weapply them to endo-classes as well. Let E − = E σ,ǫ be the set of full elementary ( σ, ǫ )-endo-classes(including the ( σ, ǫ )-zero-endo-class).One invariant of an intertwining class of skew semisimple characters comes from the theory ofmatching Witt towers, i.e. if two skew semisimple characters θ and θ ′ intertwine by an elementof G + then blockwise the Witt towers match. We now encode this into the invariants in thefollowing way.We consider the class of pairs ( β, t ) where β is zero or β generates a self dual field extension F [ β ]and t is an element of W σ,ǫ ( F [ β ]). Two such pairs ( β , t ) and ( β , t ) are called equivalent if:(i) t and t are hyperbolic;(ii) t and t are not hyperbolic, the β i are non-zero, λ β ( t ) is equal to λ β ( t ), and the Witttowers match;(iii) t and t are not hyperbolic, β = β = 0 and t = t .The equivalence classes are called Witt types . The set of Witt types is denoted by W σ,ǫ and wedenote the class of hyperbolic Witt towers by 0. Definition 14.9.
A ( σ, ǫ ) -endo-parameter is a map f = ( f , f ) : E σ,ǫ → ( N × W σ,ǫ ) of finitesupport, such that for all non-skew elements c ∈ E σ,ǫ the Witt type f ( c ) is zero.The aim now is to prove that a natural set of ( σ, ǫ )-endo-parameters parametrises the G + -intertwining classes of self-dual semisimple characters of G + (and similarly for G ). As in thecase of ˜ G , we write E fin − for the set of finite subsets of E − , and we have: Proposition 14.10.
There is a bijection between E fin − and the set of all full ( σ, ǫ )-endo-classes.38 roof. The proof mimics that of Proposition 14.3. Using a completely analogous proof, or bythe Glauberman correspondence, we get the following two G + -versions of Lemmas 14.4 and 14.5. Lemma 14.11. (i) Let θ − ∈ C − (Λ , r, β ) and θ ′− ∈ C − (Λ ′ , r, β ′ ) be two endo-equivalent skewsimple characters and suppose Λ , Λ ′ have the same period. Then, there is a skew-symmetric element β ′′ with the same minimal polynomial as β and a simple character˜ θ − ∈ C − (Λ ⊕ Λ ′ , r, β + β ′′ ) such that ˜ θ − | H r +1 − ( β, Λ) = θ − and ˜ θ | H r +1 − ( β ′′ , Λ ′ ) = θ ′− .(ii) Let θ − ∈ C − (Λ , r, β ) and θ ′− ∈ C − (Λ ′ , r, β ′ ) be two endo-equivalent non-skew elementarycharacters and suppose Λ , Λ ′ have the same period. Then, there is a skew-symmetricelement β ′′ with the same minimal polynomial as β and a simple character ˜ θ − ∈ C − (Λ ⊕ Λ ′ , r, β + β ′′ ) such that ˜ θ − | H r +1 − ( β, Λ) = θ − and ˜ θ − | H r +1 − ( β ′′ , Λ ′ ) = θ ′− . Proof.
It is the second assertion of this lemma which needs a new idea in addition to theproof strategy of Lemma 14.4. So, we proof the second assertion: At first to save notationwe assume that both characters have the same index set I and that the matching ζ is theidentity. Then, by Lemma 14.4, there is an element ˜ β ′ with the same minimal polynomialas β and a character ˜ θ − ∈ C − (Λ + Λ ′ , r, β + ˜ β ′ ) such that the restrictions to C (Λ , r, β ) andto C (Λ ′ , m, ˜ β ′ ) are θ and θ ′ which gives a self dual character ˜ θ − ∈ C − (Λ+Λ ′ , r, β + ˜ β ′ ), where ˜ β ′ is the sum of ˜ β ′ and − σ ( ˜ β ′ ), with restrictions θ − on C − (Λ , r, β ), θ ′ on C (Λ ′ , r, β ′ ) and θ ′ on C (Λ ′ , r, β ′ ). We can now apply Theorem 11.3 to obtain that ˜ θ − | H r +1 (Λ ′ , ˜ β ′ ) is conjugate to θ ′− by an element of U( h ′ ) where h ′ is the given form for the character θ ′− . Lemma 14.12.
Let θ − ∈ C − (Λ , r, β ) be a self dual semisimple character and θ ′− ∈ C − (Λ ′ , r, β ′ )be a skew-simple character and suppose Λ , Λ ′ have the same period. Suppose moreover that theGlauberman lift of θ ′− is not endo-equivalent to any simple block restriction of the Glaubermanlift of θ − . Then, there is a self dual semisimple stratum [Λ ⊕ Λ ′ , n, r, β + β ′′ ] whose associateddecomposition is a refinement of V ⊕ V ′ , and there is a self dual semisimple character ˜ θ − ∈C (Λ ⊕ Λ ′ , r, β + β ′′ ) such that the restrictions to H r +1 − ( β, Λ) and H r +1 − ( β ′ , Λ ′ ) are θ − and θ ′− ,respectively.We need another lemma to add a non-skew elementary character to a self dual semisimplecharacter: Lemma 14.13.
Let θ − ∈ C − (Λ , r, β ) be a self dual semisimple characters and θ ′ ∈ C − (Λ ′ , r, β ′ )be a non-skew elementary self dual character and suppose Λ , Λ ′ have the same period. Supposemoreover that the block restrictions of the Glauberman lift of θ ′ are not endo-equivalent toany block restrictions of the Glauberman lift of θ − . Then, there is a self dual semisimplestratum [Λ ⊕ Λ ′ , n, r, β + β ′′ ] whose associated decomposition is a refinement of V ⊕ V ′ with twoblocks in V ′ , and there is a self dual semisimple character ˜ θ − ∈ C − (Λ ⊕ Λ ′ , r, β + β ′′ ) such thatthe restrictions to H r +1 − ( β, Λ) and H r +1 − ( β ′ , Λ ′ ) are θ − and θ ′− , respectively. Proof.
The proof is analogous to the proof of Lemma 14.5 using Lemma 14.11 instead ofLemma 14.4, and we use the self dual version of the translation principle, 11.1.This also completes the proof of Proposition 14.3, which follows directly from Lemmas 14.12and 14.13.To state the main theorem we need two more notions: For an elementary endo-class c wedefine deg( c ) to be the degree of the restriction to the first block. Secondly, we attach to a Witttype T := [( β, t )] the Witt tower W T F ( T ) := λ β ( t ) ( λ := id F ) of W σ,ǫ ( F ) .39 heorem 14.14. The set of intertwining classes of full self-dual semisimple characters for G + are in bijection with the set of ( σ, ǫ )-endo-parameters f = ( f , f ) which satisfy X c ∈E − deg( c )(2 f ( c ) + diman( f ( c ))) = dim F V and X c ∈E − W T F ( f ( c )) = h ≡ . Proof.
The proof is completely analogous to the proof of Theorem 14.7; we just need to keeptrack of Witt towers, using Theorem 9.4, while the multiplicities for the classical simple endo-classes have the meaning of the F [ β ]-Witt index instead of the F [ β ]-dimension of the block.We conclude the classification of the intertwining classes of semisimple characters for a specialorthogonal groups G . For that let us fix a symplectic form h sym on V if the F -dimension iseven. We take a system of representatives R of F × / ( ± p F ). Corollary 14.15.
Suppose σ = id and ǫ = 1, then the set of G -intertwining classes of maximalself dual semisimple characters is in bijection with the union of the two following sets:(i) The set of orthogonal endo-parameters whose support contains the orthogonal zero-endo-class, and(ii) The set of pairs ( f, y ), where the first coordinate is an orthogonal endo-parameter whosesupport does not contain the orthogonal zero-endo-class and where the second coordinateis either 1 or − θ − ∈ C − (Λ , , β ) be a self dual semisimple character in V . Let [ θ − ] + and [ θ − ] be the G + -and G -intertwining class of θ − , respectively. Then,(i) if β has a zero-component then [ θ − ] + = [ θ − ] and is mapped to Φ([ θ − ] + ), otherwise(ii) β has no zero-component and [ θ − ] is mapped to the pair (Φ([ θ − ] + ) , y ) where y equals 1 ifonly if there is an isometry from h sym to h β of determinant in (1 + p F ) R . Proof.
Theorem 14.14 and Theorem 12.2.
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