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+arXiv:2301.12143v1  [math.NT]  28 Jan 2023
+The endoscopic classification of representations of non-quasi-split odd
+special orthogonal groups
+Hiroshi Ishimoto∗
+[January 31, 2023]
+Abstract
+In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and
+symplectic groups was established. Later Mok gave that of quasi-split unitary groups. After that, Kaletha, Minguez,
+Shin, and White gave that of non-quasi-split unitary groups for generic parameters. In this paper we prove the
+endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters,
+following Kaletha, Minguez, Shin, and White.
+Contents
+1
+Introduction
+1
+2
+Odd special orthogonal groups
+5
+3
+Notions around endoscopy and parameters, and Main theorem
+9
+4
+Local intertwining relation
+22
+5
+The decomposition into near equivalence classes and the standard model
+49
+6
+Globalizations and the proof of local classification
+54
+7
+The proof of the global theorem
+66
+1
+Introduction
+Background & Main result
+In a 2013 book [7], Arthur established the endoscopic classification of irreducible representations of quasi-split
+special orthogonal and symplectic groups over local fields of characteristic zero and of automorphic representations of
+those groups over global fields of characteristic zero. To local A-parameters he attached local A-packets characterized
+by the endoscopic character relation (which we shall call ECR for short), and proved that the local A-packets for
+generic parameters give the local Langlands correspondence (which we shall call LLC for short), and that automorphic
+representations are classified in terms of automorphic cuspidal representations of general linear groups and local
+A-packets.
+Let us roughly recall his result. Let first F be a local field of characteristic zero, and G a split odd special orthogonal
+group over F for simplicity. A local A-parameter for G is a homomorphism
+ψ : LF × SU(2, R) → LG,
+∗ishimoto.hiroshi.55m@gmail.com
+1
+
+with some conditions, where LF and LG denote the Langlands group of F and the L-group of G, respectively. For every
+A-parameter ψ, Arthur first constructed a multiset Πψ(G) over the set Πunit(G) of equivalence classes of irreducible
+unitary representations of G(F), and a mapping Πψ(G) → Irr(π0(Sψ)), where Sψ denotes the centralizer of ψ in
+the Langlands dual group “
+G of G, and then he proved that they satisfy ECR. An A-parameter is called a bounded
+L-parameter if its restriction to SU(2, R) is trivial. He also proved LLC by showing that if ψ is a bounded L-parameter
+then the packet Πψ(G) gives the L-packet, which has been expected to exist. Let us recall here the basic form of LLC,
+which is a conjecture in general.
+Conjecture 1.1 (LLC). Let G be a connected reductive algebraic group over a local field F. Let Πtemp(G) be the set
+of equivalence classes of irreducible tempered representations of G(F), and Φbdd(G) the set of equivalence classes of
+bounded L-parameters. There exists a canonical map
+LL : Πtemp(G) −→ Φbdd(G),
+such that for each φ ∈ Φbdd(G), the fiber Πφ(G) = LL−1(φ) is a finite set and there is an injective map ιφ : Πφ(G) →
+Irr(π0(Sφ)). These two correspondences satisfy some interesting properties. The finite set Πφ(G) is called the L-packet
+of φ.
+We refer the reader to [17] for details.
+On the other hand let next F be a number field, and G a split odd special orthogonal group over F. A global
+A-parameter for G is a formal finite unordered sum
+ψ =⊞
+i
+πi ⊠ νi,
+with some conditions, where πi is an irreducible cuspidal automorphic representation of a general linear group, and
+νi is an irreducible finite dimensional representation of SU(2, R).
+For all place v of F, we have the localization
+ψv = �
+i φi,v ⊠ νi, where φi,v is the unique L-parameter for a general linear group over Fv corresponding to πi,v
+via LLC. Then ψv is a local A-parameter for Gv over Fv. (Strictly speaking, it may not be an A-parameter, but a
+packet for it is defined.) Arthur determined an appropriate subset Πψ(εψ) of {�
+v πv | πv ∈ Πψv(Gv)} and showed
+the decomposition
+L2
+disc(G(F)\G(AF )) =
+�
+ψ
+L2
+disc,ψ(G(F)\G(AF )),
+L2
+disc,ψ(G(F)\G(AF )) =
+�
+π∈Πψ(εψ)
+π,
+(1.1)
+of the discrete spectrum into near equivalence classes, and then into irreducible automorphic representations. We shall
+call a decomposition like (1.1) ”Arthur’s multiplicity formula”, and abbreviate it as AMF.
+Later, Mok [30] proved ECR, LLC, and AMF for quasi-split unitary groups by the similar argument, and Kaletha-
+Minguez-Shin-White [18] partially proved those for non-quasi-split unitary groups. In this paper, following [18], we
+shall partially prove the analogous classification (ECR, LLC, and AMF) for non-quasi-split odd special orthogonal
+groups by the similar argument. In other words, our main theorem (Theorem 3.14) is the following:
+Theorem 1.2. Over a local field of characteristic zero, LLC holds for any non-quasi-split odd special orthogonal
+group, and the L-packets equipped with mappings ιφ satisfy ECR. Over a global field of characteristic zero, AMF holds
+for any non-quasi-split odd special orthogonal group, except the irreducible decompositions of L2
+disc,ψ(G(F)\G(AF )) for
+non-generic parameters ψ.
+The most important difference between this paper and [18] is the proof of the local intertwining relation. The local
+intertwining relation ([18, Theorem* 2.6.2], Theorem 4.13 in this paper), for which we shall write LIR for short, is a
+key theorem in the proof of the local main theorems ECR and LLC. By the similar idea to [18], we can reduce the
+proof of LIR to that for two special cases: the real special orthogonal groups SO(1, 4) and SO(2, 5) relative to Levi
+subgroups isomorphic to GL1 × SO(0, 3) and GL2 × SO(0, 3) respectively, and the special parameters. In the case of
+SO(1, 4), the special representation of the Levi subgroup under consideration is the trivial representation, and hence
+we can prove LIR by the similar argument to §2.9 in loc.cit. However, in the case of SO(2, 5), the situation is too
+complicated to calculate similarly to loc. cit., since the representation of the Levi subgroup is infinite dimensional.
+For this reason in this paper, we shall prove them by a completely different argument. We choose a test function using
+2
+
+the Iwasawa decomposition, while the relative Bruhat decomposition was used in loc. cit. The argument will appear
+in §4.9.
+We remark that LLC of non-quasi-split odd special orthogonal groups has already studied by Mœglin-Renard [28],
+but their result does not contain LIR. Thus our local theorem is differentiated from their work.
+Application
+In [7, Chapter 9], Arthur formulated the classification for non-quasi-split symmetric and orthogonal groups, and he
+has the intention of proving it. The results of this paper are included in his project, but we have a different motivation,
+the representation theory of the metaplectic groups. It is one of the most important application of this paper. Let us
+recall the results on the metaplectic groups by Adams, Barbasch, Gan, Savin, and Ichino.
+Let F be a local field. The metaplectic group, denoted by Mp2n(F), is a unique nonlinear two-fold cover of Sp2n(F)
+except F = C, in which case Mp2n(C) = Sp2n(C) × {±1}. We identify Mp2n(F) with Sp2n(F) × {±1} as sets, and
+a representation π of Mp2n(F) is said to be genuine if π((1, −1)) is not trivial. Let Πtemp(Mp2n) denote the set of
+equivalence classes of genuine tempered irreducible representations of Mp2n(F). Adams-Barbasch [2, 3], Adams [1]
+(archimedean case), and Gan-Savin [12] (non-archimedean case) showed that the local theta correspondence gives a
+canonical bijection
+Πtemp(Mp2n) ←→
+�
+V
+Πtemp(SO(V )),
+where V runs over all (2n + 1)-dimensional quadratic space of discriminant 1 over F. Thanks to their results, LLC
+for Mp2n is implied by that for all SO(V ). In addition, there is an article [16] which proved LIR for Mp2n assuming
+that for all SO(V ) over a p-adic field.
+Let next F be a number field. The metaplectic group Mp2n(AF ) is a nontrivial two-fold cover of Sp2n(AF ), and there
+is a canonical injective homomorphism Sp2n(F) ֒→ Mp2n(AF ). Hence the notions of ”automorphic representations”
+and ”discrete spectrum” are defined in a canonical way. AMF for the metaplectic group was studied by Gan-Ichino
+[11]. They proved the decomposition of the discrete spectrum of Mp2n into near equivalence classes without any
+assumption, and proved the decomposition into irreducible automorphic representations of the generic part assuming
+that for all non-quasi-split odd special orthogonal groups.
+Those theories will be completed by this paper. Namely, LLC and LIR for the metaplectic groups will hold true,
+and the result of Gan-Ichino [11] on the generic part of AMF will be unconditional.
+Organization
+§2 is the preliminary section, where some notations for odd special orthogonal groups are established.
+In §3 we first recall the notions of endoscopic triple, transfer factor, local and global parameters, and the canonical
+correspondence (e, ψe) ↔ (ψ, s). Next we shall state LLC, ECR, and AMF more precisely. We will recall the result of
+Arthur [7] and state our main theorem (Theorem 3.14).
+§4 is the most important section in this paper. We define the local intertwining operator, state LIR, and reduce
+its proof to the case when the parameter is discrete for M ∗ and elliptic or exceptional for G∗, following [18, 7, 30].
+Then in §4.9 we give a proof of LIR for the special case explained above.
+In §5, we will recall the global theory on the trace formula, and obtain some lemmas. Proofs of some lemmas are
+omitted since they are quite similar to those in [18, §3].
+In §6, we complete the proof of the local main theorem by the argument involving globalizations and trace Paley-
+Wiener theorem.
+In §7, we complete the proof of the global main theorem.
+Convention & Notation
+We marked some theorems, lemmas, and propositions with symbol * to indicate that they are only proved for
+generic parameters in this paper.
+(We omit * when we refer to them.)
+The author expect that their proofs for
+non-generic parameters will be completed by an analogous argument to a sequel of [18].
+We do not use ”S” in this paper. Instead, we shall use ”S” to denote the component groups.
+3
+
+In this paper every field is assumed to be of characteristic zero. In particular, a local field is either R, C, or a finite
+extension of Qp for some prime number p ∈ Z, and a global field is a number field, i.e., a finite extension of Q. For
+a field F, we write F for its algebraic closure, and Γ = ΓF for its absolute Galois group Gal(F/F). For a connected
+reductive algebraic group G over F, we write e(G) for the Kottwitz sign ([21]) of G, and “
+G for the dual group over
+C. If moreover F is a local or global field, we write WF for the absolute Weil group of F, and the Weil form of the
+L-group is defined by LG = “
+G ⋊ WF .
+If F is a number field, we write AF for the ring of adeles of F. We often fix a nontrivial additive character of
+F\AF , which always denoted by ψF . For each place v of F, we write ψF,v for the local component of ψF at v. We
+abbreviate ΓFv as Γv. Following [7] and [18], we do not use a symbol �′
+v for a restricted tensor product. We simply
+write �
+v. Similarly, we write �
+v in place of �′
+v. Unless otherwise specified, �
+v, �
+v, �
+v, and �
+v denote the certain
+operations taken over all places v of F, respectively.
+If F is a local field, the Langlands group is defied as follows.
+LF =
+®WF × SU(2, R),
+if F is non-archimedean,
+WF ,
+if F is archimedean.
+As in the global case, we often fix a nontrivial additive character of F, which always denoted by ψF .
+For an algebraic or abstract group G, its center is denoted by Z(G).
+In addition, when X is a subgroup or
+an element of G, we write Cent(X, G), ZX(G), or Z(X, G) (resp.
+NG(X) or N(X, G)) for the centralizer (resp.
+normalizer) of X in G.
+For a topological group G, its connected component of the identity element is denoted by G◦, and we put π0(G) =
+G/G◦.
+For an algebraic group G over a field F, put X∗(G) = Hom(G, GL1) and X∗(G) = Hom(GL1, G), which are
+equipped with the F-structure. Put moreover
+aG = Hom(X∗(G)F , R),
+a∗
+G = X∗(G)F ⊗Z R,
+aG,C = Hom(X∗(G)F , C),
+a∗
+G,C = X∗(G)F ⊗Z C.
+For any representation π, let π∨ denote its contragredient representation. If π is a representation of a topological
+group G with a Haar measure dg, for a function f on G, let fG(π) denote its character i.e.,
+fG(π) = tr(π(f)),
+where
+π(f) =
+�
+G
+f(g)π(g)dg.
+We also write f(π) if there is no danger of confusion.
+In this paper we shall write Ei,j for the (i, j)-th matrix unit. For a positive integer N, we put
+J = JN =
+
+
+
+
+
+
+
+
+
+1
+−1
+1
+...
+(−1)N−2
+(−1)N−1
+
+
+
+
+
+
+
+
+
+∈ GLN,
+and define an automorphism θN of GLN by θN(g) = J tg−1J−1. Then the standard pinning (TN, BN, {Ei,i+1}N−1
+i=1 ),
+where TN is the maximal torus consisting of diagonal matrices and BN is the Borel subgroup consisting of upper
+triangular matrices, is θN-stable. The dual group for GLN is GL(N, C), and the automorphism �θN of GL(N, C),
+which is dual to θN, is given by �θN(g) = J tg−1J−1. As GLN is split, the Galois action on GL(N, C) is trivial.
+Acknowledgment
+The author would like to thank his doctoral advisor Atsushi Ichino for his helpful advice. He also thanks the
+co-advisor Wen-Wei Li for his helpful comments. In addition, he also thanks Masao Oi and Hirotaka Kakuhama for
+sincere and useful comments. This work was partially supported by JSPS Research Fellowships for Young Scientists
+KAKENHI Grant Number 20J10875 and JSPS KAKENHI Grant Number 22K20333. The author also would like to
+appreciate Naoki Imai for his great support by JSPS KAKENHI Grant Number 22H00093.
+4
+
+2
+Odd special orthogonal groups
+In this section, we establish some notations for the odd special orthogonal groups, and recall the Kottwitz map.
+In the third subsection, we shall describe the real case as a preparation for the proof of Lemma 4.2.
+2.1
+Split odd special orthogonal groups SO2n+1
+Let F be any field of characteristic zero, and n a non-negative integer. We shall write SO2n+1 for the split odd
+special orthogonal group over F of size (2n + 1), which is defined by
+SO2n+1 =
+
+
+ g ∈ GL2n+1
+������
+tg
+Ñ
+1n
+2
+1n
+é
+g =
+Ñ
+1n
+2
+1n
+é 
+
+ .
+We put 2 at the center to make root vectors simple. Its Lie algebra is realized as
+so2n+1 =
+
+
+ X ∈ M2n+1
+������
+tX
+Ñ
+1n
+2
+1n
+é
++
+Ñ
+1n
+2
+1n
+é
+X = 0
+
+
+ ,
+over F. Let us fix the standard Borel subgroup and the standard maximal torus
+B∗ =
+
+
+
+Ñ
+a
+∗
+∗
+1
+∗
+ta−1
+é
+∈ SO2n+1
+������
+a ∈ GLn, upper triangular
+
+
+ ,
+T ∗ = � t = diag(t1, . . . , tn, 1, t−1
+1 , . . . , t−1
+n )
+�� ti ∈ GL1
+� ,
+and let χi denote the element of X∗(T ∗) such that χi(t) = ti, for i = 1, . . . , n. We shall define simple roots αi and
+simple root vectors Xαi so that
+R(T ∗, SO2n+1) = { ±(χi − χj), ±(χi + χj) }1≤i<j≤n ∪ { ±χi }1≤i≤n ,
+∆(B∗) = { α1, . . . , αn }
+= { χ1 − χ2, χ2 − χ3, . . . , χn−1 − χn, χn } ,
+Xαi = Ei,i+1 − En+1+i+1,n+1+i,
+1 ≤ i ≤ n − 1,
+Xαn = 2En,n+1 − En+1,2n+1,
+where R(T ∗, SO2n+1) and ∆(B∗) denote the root system of T ∗ in SO2n+1 and the set of simple roots with respect to
+B∗, respectively. Here we write the group law on X∗(T ∗) additively. Put
+Hi = Ei,i − Ei+1,i+1 − En+1+i,n+1+i + En+1+i+1,n+1+i+1,
+for 1 ≤ i ≤ n − 1,
+Hn = 2En,n − 2E2n+1,2n+1,
+for i = n,
+and
+X−αi = Ei+1,i − En+1+i,n+1+i+1,
+for 1 ≤ i ≤ n − 1,
+X−αn = En+1,n − 2E2n+1,n+1,
+for i = n.
+It can be seen that X−αi is a root vector corresponding to a negative root −αi. Also one has dαi(Hi) = 2 and
+[Xαi, X−αi] = Hi, so Hi is the coroot for αi and X−αi is the opposite root vector to Xαi. In this paper, by the
+standard pinning of SO2n+1, we mean the pinning (T ∗, B∗, {Xα}α∈∆(B∗)).
+The dual group of SO2n+1 is the symplectic group Sp(2n, C). We shall choose a standard pinning of Sp(2n, C)
+dual to the standard pinning of SO2n+1, to be the same as in [16, §7.1].
+5
+
+2.2
+Inner forms of odd special orthogonal groups
+Let F be any field of characteristic zero, and G∗ = SO2n+1 the split special orthogonal group over F. Since any
+automorphism of G∗ is an inner automorphism, the isomorphism classes of inner forms of G∗ and those of inner twists
+of G∗ are bijectively corresponding to each other in the natural way. Moreover, since G∗
+ad = G∗ and Z(G∗) = {1}, the
+isomorphism classes of inner twists, of pure inner twists, and of extended pure inner twists of G∗ are also bijectively
+corresponding to each other in the natural way.
+We refer the reader to [18, §0.3.1] for the definitions and basic
+properties of inner twists and so on.
+Hence there are bijective correspondences between the sets of isomorphism
+classes of inner forms, inner twists, and pure inner twists of G∗, and the Galois cohomology set H1(F, G∗):
+{inner forms} / ≃
+←→
+{inner twists} / ≃
+←→
+{pure inner twists} / ≃
+←→
+H1(F, G∗),
+G
+�→
+ξ : G∗ → G
+�→
+(ξ, z) : G∗ → G
+�→
+z.
+By abuse of terminology we often identify them. We remark that for any (2n + 1)-dimensional quadratic space V over
+F, its special orthogonal group SO(V ) is an inner form of G∗, and every inner form of G∗ is isomorphic to SO(V ) for
+some quadratic space V over F.
+Let G be an inner form of G∗, and M ∗ ⊂ G∗ a Levi subgroup. We say that M ∗ transfers to G if there exists an
+inner twist ξ : G∗ → G such that M := ξ(M ∗) ⊂ G is a Levi subgroup over F and the restriction ξ|M∗ : M ∗ → M is
+an inner twist over F. When M ∗ transfers to G, we will always choose an inner twist ξ as above.
+Let F be a local field. Kottwitz [22, Theorem 1.2] gave a canonical map of pointed sets
+α : H1(F, G∗) → π0(Z(”
+G∗)Γ)D,
+(2.1)
+extending the Tate-Nakayama isomorphism, where D denotes the Pontryagin dual. For any pure inner twist (ξ, z); G∗ →
+G, which we regard as an element of H1(F, G∗), put χG = α(G). We shall also write ⟨−, z⟩ or ⟨−, ξ⟩ for it. Note that
+in our case we have π0(Z(”
+G∗)Γ)D = Z(Sp(2n, C))D ≃ Z/2Z.
+If F is non-archimedean, then there are only two inner forms of G∗. One is non-quasi-split, and the other is split.
+In this case the map α is the unique isomorphism of pointed sets. If F = C, then H1(C, G∗) is singleton, and hence
+α is trivial. Suppose that F = R. For non-negative integers p and q with p + q = 2n + 1, put
+SO(p, q) =
+ß
+g ∈ GL2n+1(R)
+����
+tg
+Å1p
+−1q
+ã
+g =
+Å1p
+−1q
+ã ™
+.
+Then SO(p, q) is an inner form of G∗, and every inner form is isomorphic to SO(p, q) for some (p, q). Since SO(p1, q1)
+and SO(p2, q2) are isomorphic if and only if (p1, q1) = (p2, q2) or (q2, p2), we can naturally identify H1(R, SO2n+1)
+with {(p, q) | p, q ∈ Z≥0, p + q = 2n + 1}/(p, q) ∼ (q, p). Moreover, It is known that the Kottwitz map is characterized
+by
+ker α = { SO(p, q) | p − q ≡ 1, 7
+mod 8 } .
+Let F be a global field. For each place v of F there is a localization map H1(F, G∗) → H1(Fv, G∗). Let αv denote
+the Kottwitz map (2.1) from H1(Fv, G∗) to π0(Z(”
+G∗)Γv)D = π0(Z(”
+G∗)Γ)D. By [22, Proposition 2.6] we have an exact
+sequence
+H1(F, G∗) −→
+�
+v
+H1(Fv, G∗)
+�
+v αv
+−→ π0(Z(”
+G∗)Γ)D.
+(2.2)
+Remark. Such an exact sequence holds for general G∗.
+2.3
+On the real odd special orthogonal groups SO(p, q)
+Let us describe the real special orthogonal groups SO(p, q) and their Lie algebras so(p, q) more explicitly. Let p > q
+be non-negative integers with p + q = 2n + 1, and put r = p − n − 1 = n − q. Put
+S0 =
+Ñ 1n
+1n
+2
+1n
+−1n
+é
+.
+6
+
+Then an assignment g �→ S0gS−1
+0
+determines an isomorphism from SO2n+1 to SO(n + 1, n) over R. Put next
+S′
+p,q =
+Ñ 1n+1
+−i1r
+1q
+é
+.
+Then an assignment g �→ S′
+p,qgS′
+p,q
+−1 determines an isomorphism from SO(n + 1, n) to SO(p, q) over C. We thus
+obtain an inner twist ξ = ξp,q : G∗ = SO2n+1 → SO(p, q) given by g �→ Sp,qgS−1
+p,q, where Sp,q = S′
+p,qS0.
+The standard pinning (T ∗, B∗, {Xα}α) of G∗ gives a Chevalley basis of g∗ = so2n+1:
+[Xβ, Xγ] = ±(b + 1)Xβ+γ,
+where b is the greatest positive integer such that γ − bβ is a root. Explicitly,
+Xχi−χj = Ei,j − En+1+j,n+1+i,
+for 1 ≤ i < j ≤ n,
+Xχi+χj = Ei,n+1+j − Ej,n+1+i,
+for 1 ≤ i < j ≤ n,
+Xχi = 2Ei,n+1 − En+1,n+1+i,
+for 1 ≤ i ≤ n,
+Hχi−χj = Ei,i − Ej,j − En+1+i,n+1+i + En+1+j,n+1+j,
+for 1 ≤ i < j ≤ n,
+Hχi+χj = Ei,i + Ej,j − En+1+i,n+1+i − En+1+j,n+1+j,
+for 1 ≤ i < j ≤ n,
+Hχi = 2Ei,i − 2En+1+i,n+1+i,
+for 1 ≤ i ≤ n,
+X−(χi−χj) = Ej,i − En+1+i,n+1+j,
+for 1 ≤ i < j ≤ n,
+X−(χi+χj) = En+1+j,i − En+1+i,j,
+for 1 ≤ i < j ≤ n,
+X−χi = En+1,i − 2En+1+i,n+1,
+for 1 ≤ i ≤ n.
+As usual the Lie algebra of SO(p, q) has the real structure
+so(p, q)R =
+ß
+X ∈ M2n+1(R)
+����
+tX
+Å1p
+−1q
+ã
++
+Å1p
+−1q
+ã
+X = 0
+™
+,
+and the inner twist ξ = ξp,q : G∗ = SO2n+1 → SO(p, q) gives an isomorphism ξ : g∗ ∋ X �→ Sp,qXS−1
+p,q ∈ so(p, q) of the
+complex Lie algebras.
+The inner twist ξp,q sends an element
+diag(. . . , 1,
+i
+t, 1, . . . , 1,
+n+1+i
+t−1 , 1, . . .) ∈ T ∗,
+to
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+à 1i−1
+cos z
+− sin z
+1n
+sin z
+cos z
+1n−i
+í
+,
+for i ≤ r,
+à 1i−1
+t+t−1
+2
+t−t−1
+2
+1n
+t−t−1
+2
+t+t−1
+2
+1n−i
+í
+,
+for r < i,
+where z is a complex number such that t = e
+√−1z. (i ≤ r is non-split part, i > r is split part.)
+The images of the root vectors and the coroots are given as follows:
+ξ(Xχi−χj) = 1
+2×
+7
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ä
+Ei,j +
+√
+−1Ei,n+1+j − Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j + En+1+i,n+1+j −
+√
+−1En+1+j,i − En+1+j,n+1+i
+ä
+,
+for i < j ≤ r,
+Ä
+Ei,j + Ei,n+1+j − Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j −
+√
+−1En+1+i,n+1+j + En+1+j,i −
+√
+−1En+1+j,n+1+i
+ä
+,
+for i ≤ r < j,
+(Ei,j + Ei,n+1+j − Ej,i + Ej,n+1+i
++En+1+i,j + En+1+i,n+1+j + En+1+j,i − En+1+j,n+1+i) ,
+for r < i < j,
+ξ(Hχi−χj) =
+
+
+
+
+
+√
+−1 (Ei,n+1+i − Ej,n+1+j − En+1+i,i + Ej,n+1+j) ,
+for i < j ≤ r,
+√
+−1Ei,n+1+i − Ej,n+1+j −
+√
+−1En+1+i,i − Ej,n+1+j,
+for i ≤ r < j,
+Ei,n+1+i − Ej,n+1+j + En+1+i,i − Ej,n+1+j,
+for r < i < j,
+ξ(X−(χi−χj)) = 1
+2×
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ä
+−Ei,j +
+√
+−1Ei,n+1+j + Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j − En+1+i,n+1+j −
+√
+−1En+1+j,i + En+1+j,n+1+i
+ä
+,
+for i < j ≤ r,
+Ä
+−Ei,j + Ei,n+1+j + Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j +
+√
+−1En+1+i,n+1+j + En+1+j,i +
+√
+−1En+1+j,n+1+i
+ä
+,
+for i ≤ r < j,
+(−Ei,j + Ei,n+1+j + Ej,i + Ej,n+1+i
++En+1+i,j − En+1+i,n+1+j + En+1+j,i + En+1+j,n+1+i) ,
+for r < i < j,
+ξ(Xχi+χj) = 1
+2×
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ä
+Ei,j −
+√
+−1Ei,n+1+j − Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j − En+1+i,n+1+j +
+√
+−1En+1+j,i + En+1+j,n+1+i
+ä
+,
+for i < j ≤ r,
+Ä
+Ei,j − Ei,n+1+j − Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j +
+√
+−1En+1+i,n+1+j − En+1+j,i +
+√
+−1En+1+j,n+1+i
+ä
+,
+for i ≤ r < j,
+(Ei,j − Ei,n+1+j − Ej,i + Ej,n+1+i
++En+1+i,j − En+1+i,n+1+j − En+1+j,i + En+1+j,n+1+i) ,
+for r < i < j,
+ξ(Hχi+χj) =
+
+
+
+
+
+√
+−1 (Ei,n+1+i + Ej,n+1+j − En+1+i,i − Ej,n+1+j) ,
+for i < j ≤ r,
+√
+−1Ei,n+1+i + Ej,n+1+j −
+√
+−1En+1+i,i + Ej,n+1+j,
+for i ≤ r < j,
+Ei,n+1+i + Ej,n+1+j + En+1+i,i + Ej,n+1+j,
+for r < i < j,
+ξ(X−(χi+χj)) = 1
+2×
+8
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ä
+−Ei,j −
+√
+−1Ei,n+1+j + Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j + En+1+i,n+1+j +
+√
+−1En+1+j,i − En+1+j,n+1+i
+ä
+,
+for i < j ≤ r,
+Ä
+−Ei,j − Ei,n+1+j + Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j −
+√
+−1En+1+i,n+1+j − En+1+j,i −
+√
+−1En+1+j,n+1+i
+ä
+,
+for i ≤ r < j,
+(−Ei,j − Ei,n+1+j + Ej,i + Ej,n+1+i
++En+1+i,j + En+1+i,n+1+j − En+1+j,i − En+1+j,n+1+i) ,
+for r < i < j,
+ξ(Xχi) =
+®
+Ei,n+1 − En+1,i +
+√
+−1En+1,n+1+i −
+√
+−1En+1+i,n+1,
+for i ≤ r,
+Ei,n+1 − En+1,i + En+1,n+1+i + En+1+i,n+1,
+for r < i,
+ξ(Hχi) = 2 ×
+®√
+−1 (Ei,n+1+i − En+1+i,i) ,
+for i ≤ r,
+(Ei,n+1+i + En+1+i,i) ,
+for r < i,
+ξ(X−χi) =
+®
+−Ei,n+1 + En+1,i +
+√
+−1En+1,n+1+i −
+√
+−1En+1+i,n+1,
+for i ≤ r,
+−Ei,n+1 + En+1,i + En+1,n+1+i + En+1+i,n+1,
+for r < i.
+In particular we have
+tξ(Xα) = ξ(X−α) for any α ∈ R(T ∗, G∗).
+Put T = ξ(T ∗). The real form T (R) consists of the elements of the form ξ(diag(t1, . . . , tr, e
+√−1θr+1, . . . , e
+√−1θn, 1, . . .)),
+where ti ∈ R× and θi ∈ R/2πZ. For an element t ∈ T (R) of such form and a positive root α ∈ R(T, SO(p, q)), which
+is equal to R(T ∗, G∗) as sets, we have
+Ad(t)(ξ(Xα)) = α(t)ξ(Xα),
+where
+α(t) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+e
+√−1(θi±θj),
+for α = χi ± χj, i < j ≤ r,
+e
+√−1θit±1
+j ,
+for α = χi ± χj, i ≤ r < j,
+tit±1
+j ,
+for α = χi ± χj, r < i < j,
+e
+√−1θi,
+for α = χi, i ≤ r,
+ti,
+for α = χi, r < i.
+This means that as roots of T in G, α = ±(χi ± χj) (i < j ≤ r), ±χi (i ≤ r) are imaginary roots (i.e., α = −α),
+α = ±(χi ± χj) (i ≤ r < j) are complex roots (i.e., α ̸= ±α), and α = ±(χi ± χj) (r < i < j), ±χi (r < i) are real
+roots (i.e., α = α).
+3
+Notions around endoscopy and parameters, and Main theorem
+In this section, we first establish some notations for the endoscopy groups, the transfer factors, and L- or A-
+parameters. Afterwards, we shall recall Arthur’s work and state our main theorem.
+3.1
+Endoscopy
+In this subsection, we recall notion of an endoscopic triple and the transfer factor from [18, §1.1].
+3.1.1
+Endoscopic triples
+Let F be a global or local field. Consider a pair (G∗, θ∗) consisting of a connected quasi-split reductive group G∗
+defined over F with a fixed F-pinning, and a pinned automorphism θ∗ of G∗. Here, an automorphism θ∗ is said to
+9
+
+be pinned if it preserves the fixed pinning (i.e., the maximal torus, the Borel subgroup, and the set of simple root
+vectors are θ∗-stable). Then we have an automorphism “
+θ∗ of ”
+G∗, which preserves a Γ-invariant pinning for ”
+G∗. Put
+Lθ∗ = “
+θ∗ ⋊ idWF , which is an L-automorphism of LG∗.
+Definition 3.1. An endoscopic triple for (G∗, θ∗) is a triple e = (Ge, se, ηe) of a connected quasi-split reductive group
+Ge defined over F, a semisimple element se ∈ ”
+G∗, and an L-homomorphism ηe : LGe → LG∗ such that
+• Ad(se) ◦ Lθ∗ ◦ ηe = ηe;
+• ηe(�
+Ge) is the connected component of the subgroup of Ad(se) ◦ “
+θ∗-fixed elements in ”
+G∗.
+When η(Z(�
+Ge)Γ)◦ ⊂ Z(”
+G∗), the endoscopic triple e is said to be elliptic. When θ∗ is trivial (resp. not trivial), every
+endoscopic triple for (G∗, θ∗) is said to be ordinary (resp. twisted).
+Definition 3.2. Two endoscopic triples e1 = (Ge
+1, se
+1, ηe
+1) and e2 = (Ge
+2, se
+2, ηe
+2) for G∗ are said to be isomorphic (resp.
+strictly isomorphic) if there exists an element g ∈ ”
+G∗ such that
+• gηe
+1(LGe
+1)g−1 = ηe
+2(LGe
+2);
+• zgse
+1“
+θ∗(g)−1 = se
+2 for some z ∈ Z(”
+G∗), (resp. gse
+1“
+θ∗(g)−1 = se
+2).
+Then the element g is called an isomorphism (resp. strict isomorphism) from e1 to e2. If g is an isomorphism (resp.
+strict isomorphism) from e to e itself, then it is said to be an automorphism (resp. strict automorphism) of e, and we
+shall write AutG∗(e) (resp. AutG∗(e)) for the set of automorphisms (resp. strict automorphisms) of e.
+Let ξ : G∗ → G be an inner twist, and put θ = ξ ◦ θ∗ ◦ ξ−1. Then θ is an automorphism of G defined over F.
+Definition 3.3. By an endoscopic triple for (G, θ), we mean that for (G∗, θ∗). The notion of elliptic endoscopic triple,
+isomorphism and strict isomorphism of endoscopic triples are also the same as those for (G∗, θ∗).
+We shall write E(G⋊θ) (resp. E(G⋊θ)) for the set of isomorphism (resp. strict isomorphism) classes of endoscopic
+triples for (G, θ). The corresponding subsets of elliptic endoscopic triples will be indicated by the lower right index
+ell:
+E(G ⋊ θ)
+� � E(G ⋊ θ)
+Eell(G ⋊ θ)
+� �
+�
+Eell(G ⋊ θ).
+�
+When θ is trivial, then we may write E(G), E(G), Eell(G), and Eell(G).
+Put �E(N) = E(GLN, θN).
+If e = (Ge, se, ηe) ∈ �E(N), the group Ge is a direct product of finite numbers of
+symplectic or special orthogonal groups. We shall say e is simple if the number of factors is one. Define �Esim(N) to be
+the set of strictly isomorphism classes of simple endoscopic triples of (GLN, θN).
+3.1.2
+Normalized transfer factors
+Next, we shall recall some properties of transfer factors. Let G∗ be a quasi-split connected reductive group over
+a local or global field F with a fixed F-pinning, and θ∗ an automorphism of G∗ preserving the pinning. We assume
+that Z(G∗) is connected. Note that if G∗ is a direct product of a finite number of SO2m+1 or GLm (m ≥ 1), then
+G∗ is quasi-split connected reductive and its center is connected. Let (ξ, z) : G∗ → G be a pure inner twist, and put
+θ = ξ ◦θ∗ ◦ξ−1. Then θ is an automorphism of G defined over F. Not all twisted groups (G, θ) arise in this way, which
+is why an element gθ ∈ G∗ such that θ∗ = Ad(gθ) ◦ ξ−1 ◦ θ ◦ ξ was introduced in [23, §1.2]. In this paper, however, it
+suffices to assume gθ of [23] is 1. We assume that θ∗(z) = z, which is enough for our purpose. Let e = (Ge, se, ηe) be
+an endoscopic triple for (G, θ).
+Let first F be a local field, and ψF : F → C1 a nontrivial additive character. Let further δ ∈ Gθ−srss(F) and
+γ ∈ Ge(F), where Gθ−srss(F) denotes the set of θ-strongly regular and θ-semisimple elements in G(F). Then we have
+the transfer factor:
+∆[e, ξ, z](γ, δ) =
+
+
+
+ǫ(1
+2, V, ψF )∆new
+I
+(γ, δ)−1∆II(γ, δ)∆III(γ, δ)−1∆IV (γ, δ),
+if γ is a norm of δ,
+0,
+otherwise,
+10
+
+where V , ∆new
+I
+, ∆II, ∆III, and ∆IV are defined as in [18, pp.38-39]. Although they treated an extended pure inner
+twist in loc. cit., the similar arguments also works for a pure inner twist. The main difference is that we utilize an
+ordinary cocycle, instead of a basic cocycle. See also [23, §4.3-4.5, Appendices A-C] and [24, §3.4].
+Remark. In the case G∗ = SO2n+1 and θ∗ = 1, the transfer factor ∆[e, ξ, z] is determined by the isomorphism class
+of G, not by the choice of (ξ, z).
+Proposition 3.1. Let γ1, γ2 ∈ Ge(F) be norms of δ1, δ2 ∈ Gθ−srss(F), respectively. Then we have
+∆[e, ξ, z](γ1, δ1)
+∆[e, ξ, z](γ2, δ2) = ∆′(γ1, δ1; γ2, δ2),
+where the right hand side is the relative transfer factor of [24].
+Proof. The proof is similar to that of [18, Proposition 1.1.1].
+Lemma 3.2. Let se denote the image of se in (”
+G∗/”
+G∗der)Γ
+�
+θ∗,free, which is the set of Γ-fixed points in the torsion free
+quotient of the “
+θ∗-covariants of ”
+G∗/”
+G∗der. For x ∈ Z(”
+G∗)Γ and y ∈ Z1(F, (Z(G∗)θ∗)◦), we have
+∆[xe, ξ, z] = ⟨x, z⟩∆[e, ξ, z],
+and
+∆[e, ξ, yz] = ⟨se, y⟩∆[e, ξ, z],
+where xe denotes the endoscopic triple (Ge, xse, ηe).
+Proof. The proof is similar to that of [18, Lemma 1.1.2].
+We shall now recall the notion of matching functions from [23, §5.5]. For δ ∈ Gθ−srss(F) and f ∈ H(G), the orbital
+integral is defined as
+Oδθ(f) =
+�
+Gδθ(F )\G(F )
+f(g−1δθ(g))dg,
+where Gδθ = Centθ(δ, G) = {x ∈ G | δθ(x) = xδ}. If θ is trivial, it may be simply written Oδ(f). For γ ∈ Ge
+srss(F)
+and f e ∈ H(Ge), the stable orbital integral is defined as
+SOγ(f) =
+�
+γ′
+Oγ′(f e),
+where the sum is taken over a set of representatives for the conjugacy classes in the stable conjugacy class of γ.
+Definition 3.4. Two functions f e ∈ H(Ge) and f ∈ H(G) are called ∆[e, ξ, z]-matching (or matching when there is
+no danger of confusion) if for any strongly G-regular semisimple element γ ∈ Ge(F) we have
+SOγ(f e) =
+�
+δ
+∆[e, ξ, z](γ, δ)Oδθ(f),
+where the sum is taken over a set of representatives for the θ-conjugacy classes under G(F) of Gθ−srss(F). We also
+say that f and f e have ∆[e, ξ, z]-matching orbital integrals.
+Remark. If we do not assume θ∗(z) = z, then we need θe on Ge as in [23, §5.4].
+Next, let F be a global field, and ψF : AF /F → C1 a nontrivial additive character. For δ ∈ Gθ−srss(AF ) and
+γ ∈ Ge(AF ), the global absolute transfer factor is defined by
+∆A[e, ξ](γ, δ) =
+
+
+
+�
+v
+∆[ev, ξv, zv](γv, δv),
+if γ is a norm of δ,
+0,
+otherwise.
+Note that the product is well-defined because the almost all factors are 1. Moreover, Lemma 3.2 and the exact sequence
+(2.2) imply that ∆A[e, ξ] is independent of z.
+11
+
+Proposition 3.3. The factor ∆A[e, ξ] coincides with the inverse of the canonical adelic transfer given in [23].
+Proof. The proof is similar to that of [18, Proposition 1.1.3].
+We shall finish this subsubsection stating a lemma regarding local transfer factors and Levi subgroups. Let again
+F be a local field, and ψF : F → C1 a nontrivial additive character. Let M ∗ ⊂ G∗ be a standard Levi subgroup
+such that M = ξ(M ∗) ⊂ G is a Levi subgroup defined over F. Assume that θ is trivial. Let e = (Gese, ηe) and
+eM = (M e, se, ηe|LMe) be endoscopic triples for G and M respectively, such that M e ⊂ Ge is a Levi subgroup.
+Lemma 3.4.
+1. For each δ ∈ M(F) and γ ∈ M e(F), we have
+∆[eM, ξ|M∗, z](γ, δ) = ∆[e, ξ, z](γ, δ)
+Ç
+|DGe
+Me(γ)|
+|DG
+M(δ)|
+å 1
+2
+,
+where DG
+M and DGe
+Me(γ) are the relative Weyl discriminants.
+2. If f ∈ H(G) and f e ∈ H(Ge) are ∆[e, ξ, z]-matching, then their constant terms fM ∈ H(M) and f e
+Me ∈ H(M e)
+are ∆[eM, ξ|M∗, z]-matching.
+Proof. The proof is similar to that of [18, Lemma 1.1.4].
+3.1.3
+Endoscopic triples for odd special orthogonal groups
+Here we will explicate the set of representatives of isomorphism or strict isomorphism classes of ordinary endoscopic
+triples for odd special orthogonal groups. Let F be a local or global field, G∗ = SO2n+1 the split special orthogonal
+group of size 2n + 1 defined over F, and G an inner form of G∗. Then the Galois group Γ = ΓF acts on “
+G = Sp(2n, C)
+trivially, and thus we have LG = Sp(2n, C) × WF and Z(“
+G)Γ = {±1}.
+Every semisimple element s ∈ Sp(2n, C) is Sp(2n, C)-conjugate to an element of the form
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+a11m1
+...
+ar1mr
+1n1
+−1n2
+a−1
+1 1m1
+...
+a−1
+r 1mr
+1n1
+−1n2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+(3.1)
+where r ≥ 0, m1, . . . , mr ≥ 1, and n1, n2 ≥ 0 are integers such that m1 + · · · mr + n1 + n2 = n, and a1, . . . , ar ∈ C \
+{0, 1, −1} are complex numbers other than 0, 1, −1 that are different to each other. Thus its centralizer Cent(s, Sp(2n, C))
+is isomorphic to
+GL(m1, C) × · · · × GL(mr, C) × Sp(2n1, C) × Sp(2n2, C),
+(3.2)
+which is the dual group of
+GLm1 × · · · × GLmr × SO2n1+1 × SO2n2+1 .
+(3.3)
+Therefore, each endoscopic triple is isomorphic to a triple of a group (3.3), a semisimple element (3.1), and a natural
+inclusion map. Any description of complete systems of representatives of isomorphism or strict isomorphism classes
+of them is so complicated that we do not write down it here. Elliptic ones are simpler.
+For two nonnegative integers n1 and n2 such that n1 + n2 = n, put en1,n2 = (Ge
+n1,n2, sn1,n2, ηn1,n2), where
+• Ge
+n1,n2 = SO2n1+1 × SO2n2+1,
+12
+
+• sn1,n2 =
+Ü 1n1
+−1n2
+1n1
+−1n2
+ê
+,
+• ηn1,n2 denotes the direct product of the inclusion map Sp(2n1, C) × Sp(n2, C) ֒→ Sp(2n, C) given by
+ÅÅ A1
+B1
+C1
+D1
+ã
+,
+Å A2
+B2
+C2
+D2
+ãã
+�→
+Ü
+A1
+B1
+A2
+B2
+C1
+D1
+C2
+D2
+ê
+,
+and the identity map of WF .
+Then en1,n2 is an elliptic endoscopic triple for G, and four triples ±en1,n2, ±en2,n1 are isomorphic to each other. Here
+−(Ge, se, ηe) denotes (Ge, −se, ηe). Therefore, a set
+{ en1,n2 | n1 ≥ n2 ≥ 0, n1 + n2 = n }
+is a complete system of representatives of isomorphism classes of elliptic endoscopic triples for G. The set Eell(G) may
+be identified with it. In addition, en1,n2 is strictly isomorphic to −en2,n1, but not strictly isomorphic to en2,n1 unless
+n1 = n2. Therefore, a set
+{ en1,n2 | n1, n2 ≥ 0, n1 + n2 = n }
+is a complete system of representatives of strictly isomorphism classes of elliptic endoscopic triples for G. The set
+Eell(G) may be identified with it.
+3.2
+Local parameters
+In this subsection, we recall the notion of local parameters. Let F be a local field. Let G be a quasi-split connected
+reductive group over F. We say that a continuous homomorphism φ : LF → LG is an L-parameter for G if it commutes
+with the canonical projections LF ։ WF and LG ։ WF , and sends semisimple elements to semisimple elements.
+Two L-parameters are said to be equivalent if they are conjugate by an element in “
+G. We write Φ(G) for the set
+of equivalence classes of L-parameters for G. An L-parameter φ is called bounded (or tempered) if the image of the
+projection of φ(LF ) to “
+G is bounded. We write Φbdd(G) for the subset of Φ(G) consisting of bounded ones.
+We say that a continuous homomorphism ψ : LF × SU(2, R) → LG is a local A-parameter for G if the restriction
+ψ|LF to LF is a bounded L-parameter. Two local A-parameters are said to be equivalent if they are conjugate by an
+element in “
+G. We write Ψ(G) for the set of equivalence classes of local A-parameters for G. Let Ψ+(G) denote the
+set of equivalence classes of a continuous homomorphism ψ : LF × SU(2, R) → LG whose restriction ψ|LF to LF is an
+L-parameter. A parameter ψ ∈ Ψ+(G) is called generic if the restriction ψ|SU(2,R) (to the second SU(2, R)) is trivial.
+The subset of Ψ+(G) (resp. Ψ(G)) consisting of generic elements can be identified with Φ(G) (resp. Φbdd(G)). We say
+that ψ ∈ Ψ+(G) is discrete if there is no proper parabolic subgroup of LG which contains the image of ψ, and write
+Ψ+
+2 (G) for the subset of Ψ+(G) consisting of discrete elements. Put Ψ2(G) = Ψ(G)∩Ψ+
+2 (G), Φ2(G) = Φ(G)∩Ψ+
+2 (G),
+and Φ2,bdd(G) = Φbdd(G) ∩ Ψ+
+2 (G).
+We shall write Π(G) for the set of isomorphism classes of irreducible smooth representations of G(F). Let Πtemp(G)
+(resp. Π2(G)) to be the subsets of Π(G) consisting of the tempered (resp. essentially square integrable) representations.
+Put Π2,temp(G) = Π2(G) ∩ Πtemp(G).
+For ψ ∈ Ψ+(G), put Sψ = Cent(Im ψ, “
+G), Sψ = Sψ/Z(“
+G)Γ, Sψ = π0(Sψ) = Sψ/S◦
+ψ, Sψ = π0(Sψ) =
+Sψ/S◦
+ψZ(“
+G)Γ = Sψ/Z(“
+G)Γ, Srad
+ψ
+= Cent(Im ψ, “
+Gder)◦, and S♮
+ψ = Sψ/Srad
+ψ . When we emphasize that the param-
+eter is for G, we write Sψ(G), Sψ(G), Sψ(G), etc. The associated L-parameter φψ of ψ is defined by
+φψ(w) = ψ(w,
+Ç
+|w|
+1
+2
+0
+0
+|w|− 1
+2
+å
+),
+13
+
+for w ∈ LF . Finally, we put
+sψ = ψ(1,
+Å−1
+0
+0
+−1
+ã
+).
+Recall that when G is a general linear group GLN, then the parameters can be regarded as an N-dimensional
+representations of LF or LF × SU(2, R), and the equivalence of parameters is isomorphism of representations. A
+parameter which is irreducible as a representation, is said to be simple. We write Ψ+
+sim(GLN) for the subset of Ψ+(GLN)
+consisting of simple parameters, and put Ψsim(GLN) = Ψ(GLN) ∩ Ψ+
+sim(GLN), Φsim(GLN) = Φ(GLN) ∩ Ψ+
+sim(GLN),
+and Φsim,♥(GLN) = Φ♥(GLN) ∩ Φsim(GLN), for ♥ = 2, bdd. The canonical bijection between Π(GLN) and Φ(GLN),
+which is called the local Langlands correspondence for GLN, is known by Langlands [26] in the archimedean case
+(for any G in fact), by Harris-Taylor [13] and Henniart [14] in the non-archimedean case. As in [7], we shall write
+Π(N) = Π(GLN), Φ(N) = Φ(GLN), and so on. For any φ ∈ Φ(N), we write πφ for the unique corresponding element
+in Π(N). Moreover, for any ψ ∈ Ψ+(N), we define the corresponding (not necessarily irreducible) representation πψ
+as [18, (1.2.4)]. Put Ψ+
+unit(N) = {ψ ∈ Ψ+(N) | πψ is irreducible and unitary.}. Note that Ψ(N) ⊂ Ψ+
+unit(N). When
+ψ ∈ Ψ(N), then πψ = πφψ. A parameter ψ for GLN is said to be self-dual if it is self-dual as a representation. We shall
+write �Φ(N) (resp. �Φsim(N), resp. �Φsim,bdd(N)) for the subset of Φ(N) (resp. Φsim(N), resp. Φsim,bdd(N)) consisting
+of self-dual ones.
+Suppose that G is a simple twisted endoscopy group of (GLN, θN). We say a parameter ψ ∈ Ψ+(G) is simple if
+η ◦ ψ ∈ Ψ+(GLN) is simple, where η : LG → LGLN is the L-homomorphism attached to the endoscopy group G. We
+write Ψ+
+sim(G) for the subset of Ψ+(G) consisting of simple parameters, and put Ψsim(G), Φsim(G), and so on in the
+similar way. Moreover, put ‹Ψ+(G) to be the subset of Ψ+(GLN) consisting of the parameters of the form η◦ψ for some
+ψ ∈ Ψ+(G). Note that ‹Ψ+(G) = Ψ+(G) unless G is an even special orthogonal group, in which case ‹Ψ+(G) coincides
+with Ψ+(G)/ ∼ǫ in the sense of [8]. Put ‹Ψ(G) = ‹Ψ+(G) ∩ Ψ(N), �Φ(G) = ‹Ψ+(G) ∩ Φ(N), �Φsim(G) = �Φ(G) ∩ Φsim(N),
+and �Φsim,bdd(G) = �Φsim(G) ∩ Φbdd(N). Let �Φ2(G) be the subset of �Φ(G) consisting of parameters of multiplicity free
+as representations. Put �Φ2,bdd(G) = �Φ2(G) ∩ Φbdd(N).
+Now let us consider the case when G is an inner form of the split odd special orthogonal group G∗ = SO2n+1, which
+is not necessarily quasi-split. Then the parameters for G∗ are also called parameters for G, since “
+G = ”
+G∗. We omit
+the definition of ”G-relevant”, which is given in [18, §0.4] in a general setting. In this paper G-relevant parameters
+are simply said to be relevant, when there is no danger of confusion. We remark that ”parameter for G” is different
+from ”G-relevant parameter”. Although we use a phrase ”parameter for G”, we never write as Ψ(G). Moreover, we do
+not fix a symbol for the set of G-relevant parameters. Recall from [10, §4] that the parameters can be regarded as an
+2n-dimensional symplectic representations of LF or LF × SU(2, R), and the equivalence of parameters is isomorphism
+of representations. A parameter ψ ∈ Ψ+(G∗) can be written as
+ψ =
+�
+i∈I+
+ψ
+ℓiψi ⊕
+�
+i∈I−
+ψ
+ℓiψi ⊕
+�
+j∈Jψ
+ℓj(ψj ⊕ ψ∨
+j ),
+where ℓi, ℓj are positive integers, I±
+ψ , Jψ indexing sets for mutually inequivalent irreducible representations ψi of
+LF × SU(2, R) such that
+• ψi is symplectic for i ∈ I+
+ψ ;
+• ψi is orthogonal and ℓi is even for i ∈ I−
+ψ ;
+• ψj is not self-dual for j ∈ Jψ.
+Thus we have
+Sψ ∼=
+�
+i∈I+
+ψ
+O(ℓi, C) ×
+�
+i∈I−
+ψ
+Sp(ℓi, C) ×
+�
+j∈Jψ
+GL(ℓj, C),
+and its component group is a free (Z/2Z)-module
+Sψ ∼=
+�
+i∈I+
+ψ
+(Z/2Z)ai,
+14
+
+with a formal basis {ai}, where ai corresponds to the nontrivial coset O(ℓi, C) \ SO(ℓi, C). Note that “
+G = Sp(2n, C) is
+perfect, and thus Srad
+ψ
+= S◦
+ψ and S♮
+ψ = Sψ. It can be easily seen that the parameter ψ is discrete if and only if ℓi = 1
+for all i ∈ I+
+ψ and I−
+ψ = Jψ = ∅. Hence, we have
+Ψ+
+2 (G∗) = { ψ ∈ Ψ+(G∗) | Sψ is finite }
+= { ψ = ψ1 ⊕ · · · ⊕ ψr | ψi are symplectic, irreducible, and mutually distinct, r ≥ 1 } .
+A parameter ψ ∈ Ψ+(G∗) is called to be elliptic if there exists a semisimple element in Sψ whose centralizer in Sψ(G∗)
+is finite. We shall write Ψ+
+ell(G∗) for the set of such parameters. Then we have
+Ψ+
+ell(G∗) = { ψ ∈ Ψ+(G∗) | Cent(s, Sψ) is finite for some s ∈ Sψ,ss }
+=
+ß
+ψ = 2ψ1 ⊕ · · �� 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr
+����
+r ≥ 1, r ≥ q ≥ 0,
+ψi are symplectic, irreducible, and mutually distinct
+™
+,
+where Sψ,ss denotes the set of semisimple elements in Sψ. We write Ψell(G∗) (resp. Ψell(G∗)) for the subset of Ψ(G∗)
+consisting of elliptic (resp. non-elliptic) ones. We have Ψ2(G∗) ⊂ Ψell(G∗). Put
+Ψ2
+ell(G∗) = Ψell(G∗) \ Ψ2(G∗)
+=
+ß
+ψ = 2ψ1 ⊕ · · · 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr
+����
+r ≥ q ≥ 1,
+ψi are symplectic, irreducible, and mutually distinct
+™
+,
+Φell(G∗) = Ψell(G∗) ∩ Φ(G∗), and Φ2
+ell(G∗) = Ψ2
+ell(G∗) ∩ Φ(G∗).
+Let M ∗ be a standard Levi subgroup of G∗ = SO2n+1 isomorphic to
+GLn1 × · · · × GLnk × SO2n0+1,
+(3.4)
+where n1 + · · · nk + n0 = n. Then its dual group �
+M ∗ is isomorphic to
+GL(n1, C) × · · · × GL(nk, C) × Sp(2n0, C),
+and we regard it as a Levi subgroup of ”
+G∗. We have
+Ψ+(M ∗) = { ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 | ψ1 ∈ Ψ+(n1), . . . , ψk ∈ Ψ+(nk), ψ0 ∈ Ψ+(SO2n0+1) } .
+The canonical injection �
+M ∗ ֒→ ”
+G∗ induces the canonical mapping
+ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 �→ ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 ⊕ ψ∨
+1 ⊕ · · · ⊕ ψ∨
+k
+from Ψ+(M ∗) to Ψ+(G∗).
+3.3
+Global parameters
+In this subsection we recall from [7] the notion of global parameters. Let F be a global field, Γ = ΓF its absolute
+Galois group, and WF its absolute Weil group. For a connected reductive group G over F, as in [7, p.19] we put
+G(AF )1 = { g ∈ G(AF ) | |χ(g)| = 1, ∀χ ∈ X∗(G)F } ,
+where X∗(G)F = HomF (G, GL1). The quotient set G(F)\G(AF )1 has finite volume, and there is a sequence
+L2
+cusp(G(F)\G(AF )1) ⊂ L2
+disc(G(F)\G(AF )1) ⊂ L2(G(F)\G(AF )1),
+where L2
+cusp(G(F)\G(AF )1) is the subspace consisting of cuspidal functions, and L2
+disc(G(F)\G(AF )1) the subspace
+that can be decomposed into a direct sum of irreducible representations of G(AF )1. We have
+Acusp(G) ⊂ A2(G) ⊂ A(G),
+15
+
+where Acusp(G), A2(G), and A(G) denote the subsets of irreducible unitary representations of G(AF ) whose restrictions
+to G(AF )1 are constituents of the respective spaces L2
+cusp(G(F)\G(AF )1), L2
+disc(G(F)\G(AF )1), and L2(G(F)\G(AF )1).
+We also write A+
+cusp(G) and A+
+2 (G) for the analogues of Acusp(G) and A2(G) defined without the unitarity.
+For a finite set S of places of F such that v is non-archimedean and Gv is unramified over Fv for any place v /∈ S,
+put CS(G) to be the set consisting of families cS = (cv)v /∈S where each cv is a “
+G-conjugacy class in LGv = “
+G ⋊ WFv
+represented by an element of the form tv ⋊ Frobv with a semisimple element tv ∈ “
+G. For S ⊂ S′ there is a natural
+map cS = (cv)v /∈S �→ (cv)v /∈S′ from CS(G) to CS′(G). Let us define C(G) to be the direct limit of {CS(G)}S.
+Let π = ⊗vπv be an irreducible automorphic representation of G(AF ) and S a finite set of places such that πv is
+unramified for all v /∈ S. The Satake isomorphism associates a “
+G-conjugacy class c(πv) ∈ LGv to π. We can therefore
+associate cS(π) = (c(πv))v /∈S ∈ CS(G) to π. This leads to a mapping π �→ c(π) from a set of equivalence classes of
+irreducible automorphic representations of G to C(G). We shall write Caut(G) for its image.
+Let us next review some fact on automorphic representations of GLN(AF ). Let N be a positive integer. Following
+[7], we write A(N) = A(GLN), and similarly A♥(N) = A♥(GLN) and A+
+♥(N) = A+
+♥(GLN) for ♥ ∈ {2, cusp}. Put
+A+
+iso(N) =
+�
+π = π1 ⊞ · · · ⊞ πr
+�� r ∈ Z≥1, Ni ∈ Z≥1, N1 + · · · + Nr = N, πi ∈ A+
+cusp(Ni),
+∀i = 1, . . . , r
+�
+,
+where ⊞ denotes the isobaric sum. It is known by Mœglin and Waldspurger [29] that any π ∈ A2(N) is isomorphic to
+µ| det |
+n−1
+2
+⊞ µ| det |
+n−3
+2
+⊞ · · · ⊞ µ| det |− n−1
+2 ,
+for some m, n ∈ Z≥1 and µ ∈ Acusp(m) with N = mn. Moreover, (m, n, µ) is uniquely determined by π, and any
+representation of such form is an element of A2(N). In other words, A2(N) is in bijection with {(µ, m) | 1 ≤ m|N, µ ∈
+Acusp(N/m)}. We have
+Acusp(N) ⊂ A2(N) ⊂ A(N).
+(3.5)
+Put Ψcusp(N) = Acusp(N), which is trivially in bijection with Acusp(N). Put Ψsim(N) to be the set of ψ = µ ⊠ ν
+for an irreducible finite dimensional representation ν of SU(2, R) and µ ∈ A2(N/ dim ν). It is in bijection with A2(N)
+by the fact above, for which we shall write ψ �→ πψ. An element in Ψsim(N) is said to be simple. Put Ψ(N) to be
+the set of ψ = ℓ1ψ1 ⊞ · · · ⊞ ℓrψr for some r ∈ Z≥1, ℓi, Ni ∈ Z≥1 with ℓ1N1 + · · · + ℓrNr = N, and mutually distinct
+elements ψi ∈ Ψsim(Ni). To such ψ, we associate πψ = π⊞ℓ1
+ψ1 ⊞ · · · ⊞ π⊞ℓr
+ψr , so that a mapping ψ �→ πψ gives a bijection
+from Ψ(N) to A(N).We have a chain
+Ψcusp(N) ⊂ Ψsim(N) ⊂ Ψ(N),
+which is compatible with the chain (3.5). We say that a parameter ψ = ℓ1(µ1 ��� ν1) ⊞ · · · ⊞ ℓr(µr ⊠ νr) is generic if
+ν1, . . . , νr are trivial. Let Φ(N) be the subset of Ψ(N) consisting of generic parameters.
+Let v be a place of F. By LLC for GLN, for any µ ∈ Ψcusp(N), we have an L-parameter φv ∈ Φ(GLN /Fv)
+corresponding to µv ∈ Π(GLN /Fv). By abuse of notation, let us write µv for φv. Then we obtain the localization
+map Ψsim(N) → Ψ+
+v (N), ψ = µ ⊠ ν �→ ψv = µv ⊠ ν, where Ψ+
+v (N) = Ψ+(GLN /Fv). This leads the localization map
+Ψ(N) → Ψ+
+v (N),
+ψ = ℓ1ψ1 ⊞ · · · ⊞ ℓrψr, �→ ψv = ℓ1ψ1,v ⊕ · · · ⊕ ℓrψr,v.
+Let us now recall from [7] the definition of A-parameters for odd special orthogonal groups and other classical
+groups. An irreducible self-dual cuspidal automorphic representation φ of GLN(AF ) is said to be symplectic (resp.
+orthogonal) if the exterior (resp. symmetric) square L-function L(s, φ, ∧2) (resp. L(s, φ, Sym2)) has a pole at s = 1. By
+the theory of Rankin-Selberg L-function, any irreducible self-dual cuspidal automorphic representation φ of GLN(AF )
+is either symplectic or orthogonal, and this is mutually exclusive. Theorems 1.4.1 and 1.5.3 of [7] tell us the following
+proposition.
+Proposition 3.5 (1st seed theorem). For each irreducible self-dual cuspidal automorphic representation φ of GLN(AF ),
+there exists a simple twisted endoscopic triple eφ = (Gφ, sφ, ηφ) of (GLN, θN) and a representation π ∈ A2(Gφ)
+such that c(φ) = ηφ(c(π)). The triple eφ is unique up to isomorphism. If φ is symplectic (hence N is even), then
+Gφ = SON+1 and ηφ is regarded as the unique (conjugacy class of) embedding Sp(N, C) ֒→ GL(N, C). If φ is orthog-
+onal, then LGφ = SO(N, C) ⋊ WF .
+16
+
+Let e = (G∗, s, η) ∈ �Esim(N) be a simple twisted endoscopic triple of (GLN, θN), i.e., G∗ is the split symplectic
+group, the split odd special orthogonal group, or a quasi-split even special orthogonal group. Set �Φsim(G∗) to be the
+set of irreducible self-dual cuspidal automorphic representations φ of GLN(AF ) such that the corresponding endoscopic
+triple eφ is isomorphic to e. Here note that we cannot write Φsim(G∗) because G∗ may be an even special orthogonal
+group.
+An element ψ = µ ⊠ ν in Ψsim(N) is said to be self-dual if ψ = ψ∨, where ψ∨ = µ∨ ⊠ ν. We write ‹Ψsim(N) for the
+subset of Ψsim(N) consisting of such elements, and put �Φsim(N) = ‹Ψsim(N)∩Φ(N). Moreover, an element ψ = ⊞iℓiψi
+in Ψ(N) is said to be self-dual if ψ = ψ∨, where ψ∨ = ⊞iℓiψ∨
+i . Write ‹Ψ(N) for the subset of Ψ(N) consisting of such
+elements, and put �Φ(N) = ‹Ψ(N) ∩ Φ(N).
+For any ψ ∈ ‹Ψ(N), the substitute Lψ for the global Langlands group attached to ψ is defined as follows. We can
+write
+ψ = ⊞
+i∈Iψ
+ℓiψi ⊞ ⊞
+j∈Jψ
+ℓj(ψj ⊞ ψ∨
+j ),
+(3.6)
+where ψi = µi ⊠ νi ∈ ‹Ψsim(N) and ψj = µj ⊠ νj ∈ Ψsim(N) \ ‹Ψsim(N) for any i ∈ Iψ, j ∈ Jψ, and they are mutually
+distinct. Put Kψ = Iψ ⊔ Jψ and let mk be a positive integer such that µk ∈ Ψcusp(mk) for k ∈ Kψ. For i ∈ Iψ, put
+Hi = Gµi of Proposition 3.5 and let ‹
+µi denote for an embedding
+ηµi : LHi = LGµi ֒→ LGLmi,
+of Proposition 3.5. For j ∈ Jψ, put Hj = GLmj and let �
+µj be an embedding LHj ֒→ LGL2mj given by
+GL(mj, C) × WF → GL(2mj, C) × WF ,
+(h, w) �→ (diag(h, �θmj(h)), w),
+where �θmj is the twist given in the introduction. The substitute Lψ is the fiber product of {Hk}k∈Kψ over WF :
+Lψ =
+�
+k∈Kψ
+ÄLHk → WF
+ä
+.
+The associated L-embedding �ψ from Lψ × SU(2, R) to LGLN = GL(N, C) × WF is defined as
+�ψ =
+�
+i∈Iψ
+ℓi(‹
+µi ⊗ νi) ⊕
+�
+j∈Jψ
+ℓj(�
+µj ⊗ νj).
+Now we shall recall the definition of the A-parameters for odd special orthogonal groups. Let G∗ = SO2n+1. An
+A-parameter for G∗ = SO2n+1 is a parameter ψ ∈ ‹Ψ(2n) such that �ψ factors through LG∗ = Sp(2n, C) × WF . We
+shall write Ψ(G∗) for the A-parameters for G∗. Although Arthur[7] defined a set ‹Ψ(G∗) instead of Ψ(G∗), it is not
+needed now since NGL(2n,C)(Sp(2n, C)) = Sp(2n, C), cf. [7, p.31]. Thus for ψ ∈ Ψ(SO2n+1), we have the associated
+L-embedding from Lψ × SU(2, R) to LSO2n+1 = Sp(2n, C) × WF , which we shall also write �ψ. We will write Im ψ for
+the image of �ψ, in order to treat local and global matters uniformly. Let us write Φ(G∗) for the set of equivalence
+classes of generic A-parameters for G, i.e., Φ(G∗) = Ψ(G∗) ∩ �Φ(N).
+We have another characterization of Ψ(G∗) without Lψ. Recall that an irreducible finite dimensional representation
+ν of SU(2, R) is symplectic (resp. orthogonal) if dim ν is even (resp. odd). A self-dual simple parameter µ⊠ν ∈ ‹Ψsim(N)
+is said to be symplectic if one of µ and ν is symplectic and the other is orthogonal, and to be orthogonal otherwise.
+Let ψ be a parameter (3.6) in ‹Ψ(N). Then ψ is in Ψ(G∗) if and only if ℓi is even for i ∈ Iψ with orthogonal ψi.
+For ψ ∈ Ψ(G∗), we put Sψ = Cent(Im �ψ, “
+G), Sψ = Sψ/Z(“
+G)Γ, Sψ = π0(Sψ), and Sψ = π0(Sψ). Note that the
+Galois action is trivial in this case. We also write Sψ(G) and so on if the group G is to be emphasized. Take a partition
+Iψ = I+
+ψ ⊔ I−
+ψ such that I+
+ψ (resp. I−
+ψ ) is the set of i ∈ Iψ such that ψi is symplectic (resp. orthogonal). Then we have
+Sψ ∼=
+�
+i∈I+
+ψ
+O(ℓi, C) ×
+�
+i∈I−
+ψ
+Sp(ℓi, C) ×
+�
+j∈Jψ
+GL(ℓj, C),
+and its component group is a free (Z/2Z)-module
+Sψ ∼=
+�
+i∈I+
+ψ
+(Z/2Z)ai,
+17
+
+with a formal basis {ai}, where ai corresponds to ψi.
+As in [7, §4.1], we put
+Ψsim(G∗) = � ψ ∈ Ψ(G∗)
+�� |Sψ| = 1 � ,
+Ψ2(G∗) = � ψ ∈ Ψ(G∗)
+�� ��Sψ
+�� < ∞ � ,
+Ψell(G∗) =
+�
+ψ ∈ Ψ(G∗)
+�� ��Sψ,s
+�� < ∞, ∃s ∈ Sψ,ss
+�
+,
+and
+Ψdisc(G∗) = � ψ ∈ Ψ(G∗)
+�� ��Z(Sψ)
+�� < ∞ � ,
+where Sψ,s = Cent(s, Sψ). Put also Ψ2
+ell(G∗) = Ψell(G∗) \ Ψ2(G∗). Since we have an explicit description of Sψ, we
+can get a more explicit characterization of these sets. For instance, Ψ2(G∗) is a set of parameters (3.6) in ‹Ψ(N) such
+that ℓi = 1 for i ∈ I+
+ψ and I−
+ψ ⊔ Jψ = ∅. Let us put Φ♥(G∗) = Φ(G∗) ∩ Ψ♥(G∗), for ♥ ∈ {2, sim, ell, disc}. Put also
+Φ2
+ell(G∗) = Φell(G∗) \ Φ2(G∗).
+Consider a parameter ψ ∈ Ψ(G∗) ⊂ ‹Ψ(N) of the form (3.6). Let v be a place of F. The localization of ψ at v is
+defined as
+ψv = ⊞
+i∈Iψ
+ℓiψi,v ⊞ ⊞
+j∈Jψ
+ℓj(ψj,v ⊞ ψ∨
+j,v),
+where ψk,v = µk,v ⊠νk, and µk,v stands for the L-parameter for the local component µk,v of µk at v, for k ∈ Kψ. Since
+µk,v is a map from LFv to GL(mk, C), one can see that ψv is a map from LFv × SU(2, R) to GL(2n, C). By Theorem
+1.4.2 of [7] and our 1st seed theorem (Proposition 3.5), one can see that the image of ψv is contained in Sp(2n, C).
+Thus we obtain our 2nd seed theorem:
+Proposition 3.6 (2nd seed theorem). For an A-parameter ψ ∈ Ψ(G∗) and a place v, the equivalence class of ψv is
+uniquely determined. Hence, we have the localization ψv ∈ Ψ+(G∗
+v).
+Then there is a natural map Sψ → Sψv, which induces natural maps
+Sψ → Sψv,
+Sψ → Sψv,
+and
+Sψ → Sψv.
+If (Ge, se, ηe) ∈ E(G∗) is an endoscopic triple for G∗ = SO2n+1, the endoscopic group Ge is of the form GLn1 × · · ·×
+GLnr × SO2m1+1 × SO2m2+1, where r ∈ Z≥0, n1 + · · · + nr + m1 + m2 = n. Then the set of equivalence classes of
+A-parameters for Ge is defined as
+Ψ(Ge) = Ψ(n1) × · · · × Ψ(nr) × Ψ(SO2m1+1) × Ψ(SO2m2+1),
+or equivalently the set of pairs (ψ, �ψe) of a parameter ψ ∈ Ψ(G∗) and an L-embedding �ψe : Lψ × SU(2, R) → LGe with
+ηe ◦ �ψe = �ψ. We have a canonical mapping (ψ, �ψe) �→ ψ from Ψ(Ge) to Ψ(G∗), for which we will write ψe �→ ηe ◦ ψe.
+The subset of generic parameters is
+Φ(Ge) = Φ(n1) × · · · × Φ(nr) × Φ(SO2m1+1) × Φ(SO2m2+1),
+and we define Ψ♥(Ge) and Φ♥(Ge) (♥ = 2, disc, . . .) similarly. The component groups and localization are defined
+just like the case of GLN or SO2n+1.
+Now we consider the notion of parameters for non-quasi-split odd special orthogonal groups. Let G be an inner
+form of G∗ = SO2n+1. The notion of parameters for G is the same as that for G∗. In addition, a parameter ψ ∈ Ψ(G∗)
+is said to be G-relevant if it is relevant locally everywhere in the sense of [18, §0.4.4].
+Note that an inner twist
+ξ : G∗ → G is uniquely determined by G up to isomorphism, in our case. When there is no danger of confusion, we
+shall simply call relevant. In this paper, as in the local case, we never use a symbol Ψ(G). As in [18, §1.3.7], we have
+the following lemma.
+Lemma 3.7. Let ψ ∈ Ψ(G∗) be a G-relevant parameter and M ∗ ⊂ G∗ a Levi subgroup. If ψ comes from a parameter
+for M ∗, then M ∗ transfers to G.
+18
+
+3.4
+The bijective correspondence
+Here we recall the bijective correspondence (e, ψe) ←→ (ψ, s) from discussions in [7, §1.4] and [18, §1.4] in the case
+of ordinary endoscopy for SO2n+1.
+Let G∗ be a connected reductive split group over a local field F. Then Γ acts trivially on ”
+G∗. Two ordinary
+endoscopic triples e1 = (Ge
+1, se
+1, ηe
+1) and e2 = (Ge
+2, se
+2, ηe
+2) for G∗ are considered strictly equivalent (resp. equivalent) if
+se
+1 = se
+2 (resp. se
+1 = zse
+2 for some z ∈ Z(”
+G∗)) and ηe
+1(LGe
+1) = ηe
+2(LGe
+2). Note that the equivalence here is different from
+the isomorphism.
+Let E(G∗) (resp. E(G∗)) be a complete system of representatives of strict equivalence (resp. equivalence) classes
+of endoscopic data for G∗. We may identify E(G∗) (resp. E(G∗)) with the set of strict equivalence (resp. equivalence)
+classes of endoscopic triples for G∗. (The set E(G) in [7, p.246] does not correspond to E(G∗) here, but to E(G∗).)
+Define F(G∗) to be the set of the parameters ψ : LF × SU(2) → LG∗. (Here we mean the actual L-homomorphisms,
+not the equivalence classes.) For each ψ ∈ F(G∗), we shall write Sψ,ss (resp. Sψ,ss) for the set of semisimple elements
+in Sψ (resp. Sψ). Define
+X(G∗) = { (e, ψe) | e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ F(Ge) } ,
+Y (G∗) = { (ψ, s) | ψ ∈ F(G∗), s ∈ Sψ,ss } .
+They are left ”
+G∗ × Z(”
+G∗)-sets by the following actions: for (g, z) ∈ ”
+G∗ × Z(”
+G∗),
+(g, z) · ((Ge, se, ηe), ψe) = ((Ge
+1, se
+1, ηe
+1), (ηe
+1)−1 ◦ Ad(g) ◦ ηe ◦ ψe),
+(g, z) · (ψ, s) = (Ad(g) ◦ ψ, zgsg−1),
+where (Ge
+1, se
+1, ηe
+1) is an element of E(G∗) that is strictly equivalent to (Ge, zgseg−1, Ad(g) ◦ ηe).
+Put
+X(G∗) = � (e, ψe)
+�� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ F(Ge) � ,
+Y (G∗) = � (ψ, s)
+�� ψ ∈ F(G∗), s ∈ Sψ,ss
+� ,
+i.e., X(G∗) (resp. Y (G∗)) is the quotient set Z(”
+G∗)\\X(G∗) (resp. Z(”
+G∗)\\Y (G∗)) of X(G∗) (resp. Y (G∗)) by
+Z(”
+G∗) = {1} × Z(”
+G∗). Let us put
+X(G∗) = ”
+G∗\\X(G∗),
+Y(G∗) = ”
+G∗\\Y (G∗),
+standing for the quotient sets of X(G∗) and Y (G∗) by ”
+G∗ = ”
+G∗ × {1}, respectively. Likewise X (G∗) = ”
+G∗\\X(G∗),
+Y(G∗) = ”
+G∗\\Y (G∗). Note that E(G∗) = ”
+G∗\\E(G∗) and E(G∗) = ”
+G∗\\E(G∗) = ”
+G∗ × Z(”
+G∗)\\E(G∗). We can also
+see that
+X(G∗) =
+¶
+(e, ψe)
+��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)
+©
+,
+Y(G∗) =
+¶
+(ψ, s)
+��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”
+G∗ − conj.)
+©
+,
+X (G∗) =
+¶
+(e, ψe)
+��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)
+©
+,
+Y(G∗) =
+¶
+(ψ, s)
+��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”
+G∗ − conj.)
+©
+,
+where ‹Ψ(Ge) is the quotient set of Ψ(Ge) modulo the action of AutG∗(e). We use the symbol ‹Ψ here, because the idea
+is similar to that of ‹Ψ(G∗). (See [7, p.31].)
+Lemma 3.8. Let G∗ be SO2n+1 defined over a local field F. Then the natural map
+X(G∗) → Y (G∗),
+(e, ψe) �→ (ηe ◦ ψe, se),
+is a ”
+G∗ ×Z(G∗)-equivariant bijection. Here se and ηe are the second and third elements of e respectively, Furthermore,
+this induces a ”
+G∗-equivariant bijection X(G∗) ≃ Y (G∗), a Z(”
+G∗)- equivariant bijection X(G∗) ≃ Y(G∗), and a
+bijection X (G∗) ≃ Y(G∗).
+19
+
+Proof. Recall that E(G∗) is a complete system of representatives, not just a quotient set. The map X(G∗) → Y (G∗)
+is trivially well-defined. The equivariance under the actions of ”
+G∗ × Z(G∗) follows immediately from the definition of
+the actions. The injectivity also follows easily from the definition of strict equivalence of endoscopic triples. Let us
+show the surjectivity. Let (ψ, s) ∈ Y (G∗). Put Ge = Cent(s, ”
+G∗) · ψ(LF × SU(2)). We know that Cent(s, ”
+G∗) is of the
+form (3.2). Let G′ be the group (3.3). Then its dual group �
+G′ is isomorphic to Cent(s, ”
+G∗), with the trivial Galois
+action. Thus we have Ge ≃ LG′. Let us write η′ for an injective map LG′ ≃ Ge ֒→ LG∗, to obtain an endoscopic
+triple (G′, s, η′). Let e = (Ge, se, ηe) ∈ E(G∗) be the unique element that is strictly equivalent to (G′, s, η′). Since
+Im ψ ⊂ Ge = η′(LG′) = ηe(LGe), there exists a parameter ψe ∈ F(Ge) such that ψ = ηe ◦ ψe. Then we obtain
+(e, ψe) ∈ X(G∗), and get a map from Y (G∗) to X(G∗). This shows the surjectivity.
+The latter assertion follows from the former.
+Let F be a number field, and G∗ = SO2n+1. For an endoscopic triple e for G∗, every element in Ψ(Ge) can
+be regarded as an element in Ψ(G∗) in the natural way. For two parameters ψ1, ψ2 ∈ Ψ(Ge), we write ψ1 ∼ ψ2 if
+ηe ◦ ψ1 = ηe ◦ ψ2, i.e., they are same in Ψ(G∗). Let ‹Ψ(Ge) = Ψ(Ge)/ ∼ be the quotient set. The reason why we use
+the symbol ‹Ψ is because the idea is similar to that of ‹Ψ(G∗) (see [7, p.31]). Define
+X(G∗) =
+¶
+(e, ψe)
+��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)
+©
+,
+Y(G∗) =
+¶
+(ψ, s)
+��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”
+G∗ − conj.)
+©
+.
+As in the local case, we have the following lemma.
+Lemma 3.9. Let G∗ = SO2n+1 be the global split odd special orthogonal group over a global field F. Then the natural
+map
+X(G∗) → Y(G∗),
+(e, ψe) �→ (ψe, se)
+is bijective.
+Proof. The proof is similar to that of Lemma 3.8, with �ψ(Lψ × SU(2)) in place of ψ(LF × SU(2)).
+3.5
+Main theorems
+In this subsection we recall the statements of the endoscopic classification of representations of the odd special
+orthogonal groups, which will be proved for the generic part in this paper, and was already proven for the quasi-split
+case by Arthur [7].
+Let first F be a local field. Fix a nontrivial additive character ψF : F → C1. Let G∗ = SO2n+1 be the split special
+orthogonal group of size 2n + 1 defined over F, equipped with the fixed pinning. It can be regarded as a twisted
+endoscopic group of (GL2n, θ2n). Arthur proved the existence of the stable linear form satisfying the twisted ECR:
+Proposition 3.10 ([7, Theorem 2.2.1.(a)]). Let ψ ∈ Ψ(G∗). Then there is a unique stable linear form
+H(G∗) ∋ f �→ f G∗(ψ) ∈ C,
+satisfying the twisted endoscopic character relation [7, (2.2.3)].
+We sometimes write f(ψ) for f G∗(ψ) if there is no danger of confusion.
+Let (ξ, z) : G∗ → G be a pure inner twist of G∗. We now state the main local classification theorem.
+Theorem* 3.11.
+1. Let ψ ∈ Ψ(G∗). There is a finite multiset Πψ(G) of irreducible unitary representations of
+G(F), (i.e., a finite set over Πunit(G) in Arthur’s terminology,) satisfying the local and global theorems below.
+We sometimes write Πψ = Πψ(G) if there is no danger of confusion. If ψ = φ ∈ Φbdd(G∗), i.e., ψ is generic,
+then the set Πφ is empty if and only if φ is not G-relevant. In general, Πψ is empty if ψ is not G-relevant. The
+set is called the local A-packet or simply the packet for ψ. When ψ = φ is generic, it is also called the L-packet
+of φ.
+20
+
+2. The local A-packet is equipped with a map
+Πψ(G) → Irr(Sψ, χG),
+π �→ ⟨−, π⟩,
+satisfying the local and global theorems below. The packets Πψ(G) and the pairing ⟨−, π⟩ are independent of the
+choice of a inner twist (ξ, z), and determined only by the isomorphism classes of G.
+3. (ECR) Let e = (Ge, se, ηe) be an endoscopic triple for G with se ∈ Sψ, and ψe ∈ Ψ(Ge) a parameter for Ge such
+that ψ = ηe ◦ ψe is a parameter for G. If f ∈ H(G) and f e ∈ H(Ge) are matching, then we have
+f e(ψe) = e(G)
+�
+π∈Πψ(G)
+⟨sψse, π⟩f(π),
+(3.7)
+where the left hand side is the stable linear form given by Proposition 3.10.
+4. Let ψ = φ ∈ Φbdd(G∗) be a generic parameter.
+Then the packet Πφ(G) is multiplicity free, and Πφ(G) ⊂
+Πtemp(G).
+Moreover, if F is non-archimedean (resp.
+archimedean), then the map Πφ(G) → Irr(Sφ, χG) is
+bijective (resp. injective).
+5. (LLC) As φ runs over Φbdd(G∗) the sets Πφ(G) are disjoint, and we have
+�
+φ∈Φ2,bdd(G∗)
+Πφ(G) = Π2,temp(G),
+�
+φ∈Φbdd(G∗)
+Πφ(G) = Πtemp(G).
+Let ψ ∈ Ψ+(G∗) ∩ Ψ+
+unit(2n). If it is not G-relevant, define Πψ(G) to be empty. Suppose it is G-relevant. Then
+one can choose an inner twist ξ : G∗ → G, standard parabolic subgroups P ∗ ⊂ G∗ and P = ξ(P ∗) ⊂ G over F, the
+Levi subgroups M ∗ ⊂ P ∗ and M = ξ(M ∗) ⊂ P over F, the open chamber U ⊂ a∗
+M associated to P, a point λ ∈ U,
+and a parameter ψM∗ ∈ Ψ(M ∗), such that ψ is the image of the twist ψM∗,λ of ψM∗ by λ. Suppose that the packet
+ΠψM∗(M) and a map ΠψM∗(M) → Irr(SψM∗, χM) is given. Then we can define the packet for ψ and the associated
+map by
+Πψ(G) = { IP (πM,λ) | πM ∈ ΠψM∗(M) } ,
+⟨−, IP (πM,λ)⟩ = ⟨−, πM⟩.
+Note that Sψ = SψM∗ and χG = χM. See [7, pp.44-46] for more details.
+Let next F be a global field. Fix a nontrivial additive character ψF : F\AF → C1. Let G∗ = SO2n+1 be the split
+special orthogonal group of size 2n + 1 defined over F, equipped with the fixed pinning. Let (ξ, z) : G∗ → G be a pure
+inner twist of G∗.
+Theorem 3.12. There exists a decomposition
+L2
+disc(G(F)\G(AF )) =
+�
+ψ∈Ψ(G∗)
+L2
+disc,ψ(G(F)\G(AF )),
+where L2
+disc,ψ(G(F)\G(AF )) is a full near equivalence class of irreducible representations π = ⊗vπv in the discrete
+spectrum such that the L-parameter of πv is φψv for almost all v.
+This theorem will be proved in §5.1.
+For any global A-parameter ψ ∈ Ψ(G∗) and a place v of F, we have the localization ψv ∈ Ψ+(G∗
+v) ∩ Ψ+
+unit(2n)v
+thanks to Proposition 3.6. Then we have the packet Πψv(Gv) and the map Πψv(Gv) → Irr(Sψv, χGv). The global
+packet for ψ is defined as
+Πψ(G) =
+�
+π =
+�
+v
+πv
+����� πv ∈ Πψv(Gv),
+⟨−, πv⟩ = 1 for almost all v
+�
+.
+If a local packet Πψv(Gv) is empty for some v, then the global packet is defined to be empty. For any π = ⊗vπv ∈
+Πψ(G), we define a character ⟨−, π⟩ of Sψ by
+⟨x, π⟩ =
+�
+v
+⟨xv, πv⟩,
+x ∈ Sψ,
+21
+
+where xv denotes the image of x under the natural map Sψ → Sψv. The restriction of this character to Z(“
+G) is
+�
+v χGv, which is trivial by the product formula (2.2). Thus we may regard it as a character of Sψ. We have the
+character εψ : Sψ → {±1} defined by Arthur [7, p.48]. When we emphasize that the parameter ψ is for G∗ (or
+equivalently for G), we shall write εG∗
+ψ
+or εG
+ψ. Note that εψ is trivial if ψ is generic. Put
+Πψ(G, εψ) = { π ∈ Πψ(G) | ⟨−, π⟩ = εψ } .
+We now state the main global classification theorem.
+Theorem* 3.13 (AMF). For ψ ∈ Ψ(G∗), we have
+L2
+disc,ψ(G(F)\G(AF )) =
+
+
+
+
+
+�
+π∈Πψ(G,εψ)
+π,
+if ψ ∈ Ψ2(G∗),
+0,
+if ψ /�� Ψ2(G∗).
+Arthur [7] proved Theorems 3.11, 3.12, and 3.13 for the case G = G∗. In this paper, following Kaletha-Minguez-
+Shin-White [18], we will give a proof of Theorems 3.11, 3.12, and 3.13 for the case ψ is generic. The main theorem of
+this paper is stated as follows:
+Theorem 3.14 (main theorem). Let F be a local or global field and G an inner form of G∗ = SO2n+1 over F.
+1. Let F be local. Let ψ = φ ∈ Φbdd(G∗) be a generic parameter. Then part 1, 2, 3 of Theorem 3.11 hold true for
+φ. Moreover, part 4 and 5 also hold true.
+2. Let F be global. Then Theorem 3.12 holds. Let moreover ψ = φ ∈ Φ(G∗) be a generic parameter. Then Theorem
+3.13 holds true for φ.
+As stated in Introduction, Arthur [7] also established the endoscopic classification of representations of symplectic
+groups and quasi-split even special orthogonal groups. In the case of symplectic groups, he proved theorems similar
+to Theorems 3.11, 3.12, and 3.13. On the other hand, in the case of quasi-split even special orthogonal groups, he
+proved the similar theorems up to outer automorphisms. For a quasi-split even orthogonal group G over a local field
+F, two representations π and π′ are said to be ǫ-equivalence if there exists ρ ∈ AutF (G) such that π ≃ π′ ◦ ρ, and
+the sets of ǫ-equivalence classes is denoted by �Π instead of Π. Arthur defined a weak local A-packet Πψ(G) for each
+ǫ-equivalence class of A-parameters ψ ∈ ‹Ψ(G), and then he proved a theorem similar to but weaker than Theorem
+3.11 in that Ψ and Π are replaced by ‹Ψ and �Π. For a quasi-split even orthogonal group G over a number field F, two
+representations π = �
+v πv and π′ = �
+v π′
+v are said to be ǫ-equivalence if πv and π′
+v are so for all places v. The weak
+global packet �Πψ(G) for ψ ∈ ‹Ψ(G) is defined as the set of π = �
+v πv such that πv ∈ �Πψv(Gv) for all v and ⟨−, πv⟩
+is trivial for almost all v. As in the local case, he proved theorems similar to but weaker than Theorems 3.12 and
+3.13 in that Ψ and Π are replaced by ‹Ψ and �Π. See also [8] for the weak theorems for even orthogonal groups. These
+classification theorems for symplectic and even special orthogonal groups are needed for the globalization in §6.2
+In the rest of this paper, we shall proof the main theorem 3.14 by a long induction argument following [18]. Let
+n be a positive integer. As an induction hypothesis, we assume that the Theorem 3.14 (or Theorems 3.11, 3.12, and
+3.13) holds for any positive integer n0 < n, and hence for any proper Levi subgroup M ∗ ⊊ G∗ = SO2n+1.
+4
+Local intertwining relation
+In this section, we define the normalized local intertwining operator and state the local intertwining relation, which
+is a key theorem in the theory of the endoscopic classification of representations. After that, we reduce the proof LIR,
+and give a proof of LIR in the special (real and low rank) cases.
+4.1
+The diagram
+We shall describe explicitly the basic commutative diagram ([7, (2.4.3)], [18, (2.1.1)]) for odd special orthogonal
+groups. See [18, §2.1] for the diagram for general connected reductive groups.
+Let F be a local or global field, G∗ = SO2n+1, and M ∗ ⊂ G∗ a standard Levi subgroup. Let ξ : G∗ → G be an
+inner twist over F with a Levi subgroup M = ξ(M ∗) ⊂ G which is an inner form of M ∗ over F. Let ψ ∈ Ψ(M ∗) be a
+22
+
+parameter, and ψG ∈ Ψ(G∗) the image of ψ under the natural map induced by �
+M ֒→ “
+G. Recall that “
+G = Sp(2n, C).
+The parameter ψG ∈ Ψ(G∗) and its centralizer group Sψ(G) = SψG(G) can be written as
+ψG =
+�
+i∈I+
+ψ
+ℓiψi ⊕
+�
+i∈I−
+ψ
+ℓiψi ⊕
+�
+j∈Jψ
+ℓj(ψj ⊕ ψ∨
+j ),
+Sψ(G) = Cent(Im ψ, Sp(2n, C)) ≃
+�
+i∈I+
+ψ
+O(ℓi, C) ×
+�
+i∈I−
+ψ
+Sp(ℓi, C) ×
+�
+j∈Jψ
+GL(ℓj, C).
+Moreover, we can also write as
+ψG =
+r
+�
+t=1
+et
+
+�
+i∈I+
+t
+ℓt
+iψi ⊕
+�
+i∈I−
+t
+ℓt
+iψi ⊕
+�
+j∈Jt
+(ℓt
+jψj ⊕ ℓt∨
+j ψ∨
+j )
+
+
+⊕
+
+�
+i∈I+
+0
+ℓ0
+i ψi ⊕
+�
+i∈I−
+0
+ℓ0
+i ψi ⊕
+�
+j∈J0
+ℓ0
+j(ψj ⊕ ψ∨
+j )
+
+
+⊕
+r
+�
+t=1
+et
+
+�
+i∈I+
+t
+ℓt
+iψi ⊕
+�
+i∈I−
+t
+ℓt
+iψi ⊕
+�
+j∈Jt
+(ℓt∨
+j ψj ⊕ ℓt
+jψ∨
+j )
+
+ ,
+�
+M ≃
+r
+�
+t=1
+GL(kt, C)et × Sp(2n0, C),
+where et, kt are positive integers,
+
+
+
+
+
+I+
+t ⊂ I+
+ψ ,
+I−
+t ⊂ I−
+ψ ,
+Jt ⊂ Jψ,
+for t = 0, 1, . . ., r,
+are subsets of indices, and ℓt
+i, ℓt
+j are positive integers such that
+
+
+
+ℓt
+i ≤ ℓi
+2 ,
+ℓt
+j, ℓt∨
+j
+≤ ℓj,
+ℓt
+j + ℓt∨
+j
+≤ ℓj,
+for t = 0, 1, . . ., r,
+and
+¶
+((ℓt
+i)i∈I+
+t , (ℓt
+i)i∈I−
+t , (ℓt
+j)j∈Jt, (ℓt∨
+j )j∈Jt)
+��� t = 1, . . . , r
+©
+is mutually distinct. Put
+ψt =
+�
+i∈I+
+t
+ℓt
+iψi ⊕
+�
+i∈I−
+t
+ℓt
+iψi ⊕
+�
+j∈Jt
+(ℓt
+jψj ⊕ ℓt∨
+j ψ∨
+j ),
+for t = 1, . . . , r,
+ψ0 =
+�
+i∈I+
+0
+ℓ0
+i ψi ⊕
+�
+i∈I−
+0
+ℓ0
+i ψi ⊕
+�
+j∈J0
+ℓ0
+j(ψj ⊕ ψ∨
+j ).
+Then we have
+ψG =
+r
+�
+t=1
+etψt ⊕ ψ0 ⊕
+r
+�
+t=1
+etψ∨
+t ,
+ψ =
+r
+�
+t=1
+etψt ⊕ ψ0,
+and each ψt corresponds to GL(kt, C) ⊂ �
+M, and ψ0 corresponds to Sp(2n0, C) ⊂ �
+M. Here we note that
+• k1, . . . , kr are not necessarily mutually distinct;
+23
+
+• ψ1, . . . , ψr are mutually distinct;
+• ψ1, . . . , ψr, ψ∨
+1 , . . . , ψ∨
+r are not necessarily mutually distinct.
+Let A �
+M be the maximal central split torus of �
+M. Then �
+M = Cent(A �
+M, “
+G). One has
+A �
+M ≃
+r
+�
+t=1
+(C×)et.
+Put
+Sψ(G) = Cent(Im ψ, “
+G),
+Sψ(M) = Cent(Im ψ, �
+M),
+Nψ(M, G) = Sψ(G) ∩ N(A �
+M, “
+G) = N(A �
+M, Sψ(G)).
+Put moreover
+Wψ(M, G) = N(A �
+M, Sψ(G))
+Z(A �
+M, Sψ(G)) ,
+W ◦
+ψ(M, G) = N(A �
+M, Sψ(G)◦)
+Z(A �
+M, Sψ(G)◦) ,
+Nψ(M, G) = N(A �
+M, Sψ(G))
+Z(A �
+M, Sψ(G)◦),
+Sψ(M, G) = N(A �
+M, Sψ(G))
+N(A �
+M, Sψ(G)◦),
+Rψ(M, G) =
+N(A �
+M, Sψ(G))
+N(A �
+M, Sψ(G)◦) · Z(A �
+M, Sψ(G)),
+and
+Sψ(G) = π0(Sψ(G)) ≃
+�
+i∈I+
+ψ
+O(ℓi, C)/ SO(ℓi, C) ≃
+�
+i∈I+
+ψ
+(Z/2Z)ai,
+Sψ(M) = π0(Sψ(M)) ≃
+�
+i∈I+
+0
+O(ℓ0
+i , C)/ SO(ℓ0
+i , C) ≃
+�
+i∈I+
+0
+(Z/2Z)ai,
+where {ai}i is a formal basis. Direct calculations show that
+Z(A �
+M, Sψ(G)◦) ≃
+r
+�
+t=1
+
+ �
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+
+
+et
+×
+
+ �
+i∈I+
+0
+SO(ℓ0
+i , C) ×
+�
+i∈I−
+0
+Sp(ℓ0
+i , C) ×
+�
+j∈J0
+GL(ℓ0
+j, C)
+
+ ,
+Z(A �
+M, Sψ(G)) ≃
+r
+�
+t=1
+
+ �
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+
+
+et
+×
+
+ �
+i∈I+
+0
+O(ℓ0
+i , C) ×
+�
+i∈I−
+0
+Sp(ℓ0
+i , C) ×
+�
+j∈J0
+GL(ℓ0
+j, C)
+
+ .
+In particular, the connected component of the identity in Z(A �
+M, Sψ(G)) is equal to Z(A �
+M, Sψ(G)◦). Thus, we have
+�
+M ∩ Sψ(G)◦ = Sψ(M)◦,
+(4.1)
+Z(A �
+M, Sψ(G))
+Z(A �
+M, Sψ(G)◦) ≃ Sψ(M) ≃
+�
+i∈I+
+0
+(Z/2Z)ai.
+(4.2)
+24
+
+We now consider the group N(A �
+M, Sψ(G)). Take a partition
+{ 1, . . ., r } = T1 ⊔ T2 ⊔ T3 ⊔ T4
+such that
+T1 = { t | ψt ≃ ψ∨
+t } ,
+T2 = { t | ψt ̸≃ ψ∨
+t′ for any 1 ≤ t′ ≤ r } ,
+T3 = { t | ψt ≃ ψ∨
+t′ for a unique t′ ∈ T4 } ,
+T4 = { t | ψt ≃ ψ∨
+t′ for a unique t′ ∈ T3 } ,
+and for each t ∈ T3, put e∨
+t = et′, where t′ is the element of T4 such that ψt ≃ ψ∨
+t′.
+Before the description of N(A �
+M, Sψ(G)), we must fix notation. For a finite set X, let SX denote its symmetric
+group, and put Sm = S{1,2,...,m} for a positive integer m. For any positive integers e, e∨, let
+W(e, e∨)
+be a subgroup of Se+e∨ ⋉ (Z/2Z)e+e∨ generated by
+S{ 1,...,e }, S{ e+1,...,e+e∨ }, and
+®
+(h, h∨) ⋉ (· · · , 0,
+h
+1, 0, · · · , 0,
+h∨
+1 , 0, · · · )
+���� 1 ≤ h ≤ e, e + 1 ≤ h∨ ≤ e + e∨
+´
+.
+Then we obtain an explicit description of the group N(A �
+M, Sψ(G)). We have
+N(A �
+M, Sψ(G)) ≃ c′ ×
+Ñ
+�
+i∈I+
+0
+O(ℓ0
+i , C), ×
+�
+i∈I−
+0
+Sp(ℓ0
+i , C) ×
+�
+j∈J0
+GL(ℓ0
+j, C)
+é
+,
+where a subgroup c′ of Sp(2(n − n0), C) can be written as
+c′ ≃
+�
+t∈T1
+
+
+�
+wt∈Set⋉(Z/2Z)et
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet
+
+×
+�
+t∈T2
+
+
+�
+wt∈Set
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet
+
+×
+�
+t∈T3
+
+
+�
+wt∈W(et,e∨
+t )
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet+e∨
+t 
+ .
+Therefore we have natural isomorphisms
+Wψ(M, G) ≃
+�
+t∈T1
+(Set ⋉ (Z/2Z)et) ×
+�
+t∈T2
+Set ×
+�
+t∈T3
+W(et, e∨
+t ),
+(4.3)
+Nψ(M, G) ≃ Sψ(M) × Wψ(M, G).
+(4.4)
+We finally consider N(A �
+M, Sψ(G)◦). It is too complicated to write explicitly, but we can describe the natural
+surjection
+Nψ(M, G) = N(A �
+M, Sψ(G))
+Z(A �
+M, Sψ(G)◦) −→ Sψ(M, G) = N(A �
+M, Sψ(G))
+N(A �
+M, Sψ(G)◦),
+to understand the group N(A �
+M, Sψ(G)◦) in terms of its kernel. For e ≥ 1, let us write x for a group homomorphism
+Se ⋉ (Z/2Z)e −→ Z/2Z,
+σ ⋉ (dh)e
+h=1 �→
+e
+�
+h=1
+dh.
+25
+
+For each i ∈ I+
+ψ , we shall define a corresponding function
+xi : Nψ(M, G) −→ Z/2Z
+as follows. By the equations (4.4), (4.3), and (4.2), we may identify Nψ(M, G) with
+�
+i∈I+
+0
+(Z/2Z)ai ×
+�
+t∈T1
+(Set ⋉ (Z/2Z)et) ×
+�
+t∈T2
+Set ×
+�
+t∈T3
+W(et, e∨
+t ).
+(4.5)
+For any element
+u =
+Ñ
+�
+i∈I+
+0
+ciai, (wt)t∈T1, (wt)t∈T2, (wt)t∈T3
+é
+of (4.5), we define
+xi(u) = ci +
+�
+t∈T1⊔T3
+ℓt
+ix(wt),
+where we put ci = 0 for i /∈ I+
+0 , and ℓt
+i = 0 for i /∈ I+
+t . Now, we can describe the natural surjection. Let us define a
+homomorphism xψ : Nψ(M, G) → �
+i∈I+
+ψ (Z/2Z)ai = Sψ(G) by
+xψ(u) =
+�
+i∈I+
+ψ
+xi(u)ai
+for u ∈ Nψ(M, G). Then Sψ(M, G) and W ◦
+ψ(M, G) can be identified with the image and the kernel of xψ, respectively.
+Put O+(l, C) = SO(l, C) and O−(l, C) = O(l, C) \ SO(l, C). The homomorphism xψ gives a description:
+N(A �
+M, Sψ(G)◦) ≃
+�
+u
+{
+�
+t∈T1
+
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet
+
+×
+�
+t∈T2
+
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet
+
+×
+�
+t∈T3
+
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet+e∨
+t 
+
+×
+
+ �
+i∈I+
+0
+Oεi(ℓ0
+i , C), ×
+�
+i∈I−
+0
+Sp(ℓ0
+i , C) ×
+�
+j∈J0
+GL(ℓ0
+j, C)
+
+},
+where u =
+�
+i∈I+
+0 ciai, (wt)t∈T1, (wt)t∈T2, (wt)t∈T3
+ä
+runs over ker(xψ), and εi = (−1)ci.
+We have obtained the explicit description of the following commutative diagram with exact rows and columns.
+1
+1
+�
+�
+W ◦
+ψ(M, G)
+W ◦
+ψ(M, G)
+�
+�
+1 −−−−→ Sψ(M) −−−−→ Nψ(M, G) −−−−→ Wψ(M, G) −−−−→ 1
+���
+xψ
+�
+�
+1 −−−−→ Sψ(M) −−−−→ Sψ(M, G) −−−−→ Rψ(M, G) −−−−→ 1
+�
+�
+1
+1
+(4.6)
+26
+
+Lemma 4.1. If ψ ∈ Ψ2(M ∗), then xψ is surjective. In particular, we have Sψ(M, G) = Sψ(G).
+Proof. Suppose that ψ ∈ Ψ2(M ∗). Then Kt := I+
+t ⊔I−
+t ⊔Jt is a singleton for t = 1, . . . , r, and ℓt
+i = 1 for t = 0, 1, . . ., r.
+Let i ∈ I+
+0 . If we put u = (ai, 1, 1, 1) ∈ Nψ(M, G), then xψ(u) = ai. Let i ∈ I+
+t (t = 1, . . . , r). Then t ∈ T1, as ψt = ψi
+is self-dual. If we put u = (0, (ws)s∈T1, 1, 1) ∈ Nψ(M, G), where wt = 1 ⋉ (1, 0, . . . , 0) and wsis trivial if s ̸= t. Then
+xψ(u) = ai. Since I+
+ψ is the union of I+
+0 , I+
+1 , . . . , T +
+r , this completes the proof.
+Dividing the diagram (4.6) by Z(“
+G)Γ = {±1}, we obtain another diagram
+1
+1
+�
+�
+W ◦
+ψ(M, G)
+W ◦
+ψ(M, G)
+�
+�
+1 −−−−→ Sψ(M) −−−−→ Nψ(M, G) −−−−→ Wψ(M, G) −−−−→ 1
+���
+�
+�
+1 −−−−→ Sψ(M) −−−−→ Sψ(M, G) −−−−→ Rψ(M, G) −−−−→ 1
+�
+�
+1
+1
+where
+Sψ(M) = Sψ(M)
+Z(“
+G)Γ ,
+Nψ(M, G) = Nψ(M, G)
+Z(“
+G)Γ
+,
+Sψ(M, G) = Sψ(M, G)
+Z(“
+G)Γ
+.
+4.2
+The first intertwining operator
+Let F be a local field, ψF : F → C1 a nontrivial additive character, G∗ = SO2n+1, and M ∗ ⊊ G∗ a proper standard
+Levi subgroup. Let ξ : G∗ → G be an inner twist over F which restrict to an inner twist ξ|M∗ : M ∗ → M over F. Let
+ψ ∈ Ψ(M ∗) be a local parameter. Assume that ψ is M-relevant, and the corresponding packet Πψ(M) is not empty.
+Let π ∈ Πψ(M). We shall write Vπ for the representation space of π.
+Let P and P ′ be parabolic subgroups of G defined over F with common Levi factor M. We write (IP (π), HP (π))
+for the representation parabolically representation by (π, Vπ) from P to G.
+A function HM : M(F) → aM is defined by
+exp(⟨HM(m), χ⟩) = |χ(m)|F ,
+for any χ ∈ X∗(M)F , m ∈ M(F). Then each λ ∈ a∗
+M,C gives a character
+M(F) → C×,
+m �→ exp(⟨HM(m), λ⟩).
+We shall write πλ for the tensor product of π and this character.
+Then the unnormalized intertwining operator
+JP ′|P (ξ, ψF ) : HP (πλ) → HP ′(πλ) is given by
+[JP ′|P (ξ, ψF )f](g) =
+�
+N(F )∩N ′(F )\N ′(F )
+f(n′g)dn′,
+where N and N ′ are the unipotent radical of P and P ′, respectively. It is known that the integral converges absolutely
+when the real part of λ lies in a certain open cone. The Haar measure dn′ on N(F) ∩ N ′(F)\N ′(F) that we use here
+is the measure [18, §2.2] defined. Although the case of unitary groups is considered in loc. cit., we apply the same
+definition with the same notation.
+27
+
+We write WF ∋ w �→ |w|λ for the L-parameter of the character M(F) ∋ m �→ exp(⟨HM(m), λ⟩) attached to
+λ ∈ a∗
+M,C. Then ψλ = ψ| − |λ is a parameter of πλ. Let ρP ′|P be the adjoint representation of �
+M on �n′ ∩ �n\�n′. Put
+rP ′|P (ξ, ψλ, ψF ) =
+L(0, ρ∨
+P ′|P ◦ φψλ)
+L(1, ρ∨
+P ′|P ◦ φψλ)
+ǫ( 1
+2, ρ∨
+P ′|P ◦ φψλ, ψF )
+ǫ(0, ρ∨
+P ′|P ◦ φψλ, ψF ) ,
+where the factors are the Artin L- and ǫ- factors. The normalized intertwining operator is given by
+RP ′|P (ξ, ψλ) = rP ′|P (ξ, ψλ, ψF )−1JP ′|P (ξ, ψF ).
+It is known that RP ′|P (ξ, ψλ) is independent of the choice of ψF , and that λ �→ RP ′|P (ξ, ψλ) has meromorphic
+continuation to whole a∗
+M,C.
+Lemma 4.2. Assume that F = R and ψ = φ ∈ Φ(M ∗). Then the function λ �→ RP ′|P (ξ, φλ) has neither a zero nor
+a pole at λ = 0. Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over F with Levi factor M. Then we have
+RP ′′|P (ξ, φ) = RP ′′|P ′(ξ, φ) ◦ RP ′|P (ξ, φ).
+Proof. The idea of the proof is the same as that of [18, Lemma 2.2.1]. So it suffices to show that the measure used in
+the definition of JP ′|P (ξ, ψF ) coincides with the measure introduced by Arthur [5, §3].
+Let us calculate the measure given in loc.
+cit.
+Since F = R, there exist nonnegative integers p and q with
+p + q = 2n + 1 such that the inner twist ξ : G∗ → G is realized by ξp,q : SO2n+1 → SO(p, q) given in §2.3. So,
+we will use the notation introduced there. Let B be a SO(p, q)-invariant bilinear form on gC = so(p, q)C given by
+B(X, Y ) = 1
+2 tr(XY ). Then the quadratic form
+X �→ −B(X, θX)
+is positive definite on gR = so(p, q)R, where θ is the Cartan involution defined by θ(X) = − tX. The straightforward
+calculation shows that the constant αP ′|P defined in [5, §3] is equal to 2
+k
+2 , where k is the number of roots of the form
+χi whose restriction to aM are roots of both (P ′, AM) and (P, AM). Here, AM and P denote the maximal central
+split torus in M and the parabolic subgroup opposite to P containing M, respectively.
+Next, we calculate the Euclidean measure dX defined by the quadratic form X �→ −B(X, θX) ([5, §3]). Put
+T = ξ(T ∗), and write t for its Lie algebra. Then we have a decomposition
+gC = tC ⊕
+�
+a∈R(T,G)/Γ
+ga,C,
+where Γ is the Galois group of F/F = C/R. Note that each ga,C = ⊕α∈agα is defined over R, but gα is not necessarily
+defined over R. Let a ∈ R(T, G)/Γ and α ∈ a. If α is a complex root, then a = {α, σα}, and dim ga = 2, where
+σ ∈ Γ is the complex conjugation. In this case, ga has an R-basis {X1, X2}, where X1 = ξ(Xα) + σ(ξ(Xα)) and
+X2 = −√−1(ξ(Xα) − σ(ξ(Xα))). By a straightforward calculation, we have −B(X1, θX1) = −B(X2, θX2) = 2. If
+gα ⊂ n ∩ n′, the contribution of a to the measure dX is |d(2− 1
+2 X1 ∧ 2− 1
+2 X2)| = |d(ξ(Xα ∧ Xσα))|, which is equal to
+the contribution to our measure. If α is a real root, then a = {α}, and dim ga = 1. In this case, ga has an R-basis
+{X1}, where X1 = ξ(Xα). By a straightforward calculation, we have −B(X1, θX1) = 1 if α is of the form χi ± χj, and
+−B(X1, θX1) = 2 if α is of the form χi. If gα ⊂ n ∩ n′, the contribution of a to the measure dX is |dX1| = |d(ξ(Xα))|
+if α is of the form χi ± χj, and |d(2− 1
+2 X1)| = 2− 1
+2 |d(ξ(X1))| if α is of the form χi. The product of coefficients 2− 1
+2 for
+all α of the form χi and αP ′|P = 2
+k
+2 cancel. If α is an imaginary root, then gα ⊈ n ∩ n′. This completes the proof.
+Lemma 4.3. Assume that F is a p-adic field and ψ = φ ∈ Φ(M ∗) an L-parameter.
+Then the function λ �→
+RP ′|P (ξ, φλ) has neither a zero nor a pole at λ = 0. Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over F with
+Levi factor M. Then we have
+RP ′′|P (ξ, φ) = RP ′′|P ′(ξ, φ) ◦ RP ′|P (ξ, φ).
+Proof. Note that the proofs of Lemmas 5.2, 5.3, and 6.6 are independent from this lemma. In particular, we may use
+them here.
+Let π ∈ Πφ(M) be an arbitrary representation of M corresponding to φ. The operator RP ′|P (ξ, φλ) is an inter-
+twining operator from HP (πλ) to HP ′(πλ). By the reduction steps described in [5, §2] (more precisely, from the last
+28
+
+line in p.29 of loc. cit.), we may assume that π is supercuspidal. Let ˙F, u, v2, ˙G∗, ˙G, ˙ξ, ˙M ∗, ˙M, and ˙π be as in Lemma
+6.6. Furthermore, let ˙P, ˙P ′, and ˙P ′′ be the parabolic subgroups of ˙G such that ˙Pu = P, ˙P ′
+u = P ′, and ˙P ′′
+u = P ′′.
+Since we can choose v2 ̸= u arbitrarily, we may assume that v2 is a real place. By the induction hypothesis, we can
+take the A-parameter of ˙π, write ˙φ for it. Note that ˙φ is a generic parameter because ˙φu = φ.
+The global intertwining operator R ˙P ′| ˙P ( ˙πλ, ˙φλ) has neither a zero nor a pole at λ = 0, and is multiplicative in ˙P ′,
+˙P by Lemma 5.3. For any place v ̸= u, v2, ˙Gv splits over ˙Fv. For u = v2, we have ˙Fv2 = R. Thus for any v ̸= u, the
+local normalized intertwining operator R ˙P ′v| ˙Pv( ˙ξv, ˙φλ,v) has neither a zero nor a pole at λ = 0, and is multiplicative by
+Lemma 4.2 and the assumption from the split case. Now the assertion follows from Lemma 5.2.
+The following lemma is now proven for generic parameters. In general, we expect it can be proven as [18, Lemma
+2.2.4].
+Lemma* 4.4. Let F be a local field. Let ψ ∈ Ψ(M ∗) be a general local A-parameter for M. Then the function
+λ �→ RP ′|P (ξ, ψλ) has neither a zero nor a pole at λ = 0. Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over
+F with Levi factor M. Then we have
+RP ′′|P (ξ, ψ) = RP ′′|P ′(ξ, ψ) ◦ RP ′|P (ξ, ψ).
+4.3
+The second intertwining operator
+We maintain the assumptions of the previous subsection. For every finite separable extension E/F, Langlands [25,
+Theorem 1] defined a complex number λ(E/F, ψF ). By the definition, we have λ(F/F, ψF ) = 1.
+For w ∈ W(T ∗, G∗), Keys-Shahidi [19, (4.1)] defined the constant λ(w, ψF ). In this subsection, we shall construct
+a local intertwining operator denoted by ℓP ′
+P (w, ξ, ψ, ψF ). In the case of unitary groups, its construction involves the
+λ-factor as in [18, §2.3]. However, in the case of odd special orthogonal groups, we do not need the λ-factor since it
+always vanishes:
+Lemma 4.5. For any w ∈ W(T ∗, G∗), we have
+λ(w, ψF ) = 1.
+Proof. By the definition of λ(w, ψF ), it suffices to show that
+λ(Fa/F, ψF ) = 1,
+for any root a ∈ R(T ∗, G∗), where Fa/F is a finite extension such that G∗
+a,sc = ResFa/F (SL2). Here, G∗
+a,sc denotes
+the simply connected cover of the derived group G∗
+a,der of the Levi subgroup G∗
+a of semisimple rank 1 attached to a.
+Since we know that λ(F/F, ψF ) = 1, it suffices to show that Fa = F for all root a ∈ R(T ∗, G∗).
+If a is a long root (i.e., of the form ±χi ± χj), then G∗
+a is isomorphic to a product of GL2 and a finite number of
+GL1. Thus we have G∗
+a,der ∼= SL2 and G∗
+a,sc ∼= SL2, which means that Fa = F. If a is a short root (i.e., of the form
+±χi), then G∗
+a is isomorphic to a product of SO3 and a finite number of GL1. Thus we have G∗
+a,der ∼= SO3 ∼= PGL2
+and G∗
+a,sc ∼= SL2, which means that Fa = F.
+Let ψ ∈ Ψ(M ∗) be a local parameter such that ψ is M-relevant and the corresponding packet Πψ(M) is not empty.
+Let π ∈ Πψ(M). Moreover, let P and P ′ be parabolic subgroups of G defined over F with common Levi factor
+M. For an element w ∈ W(M ∗, G∗) ∼= W(M, G), we write �w ∈ N(M ∗, G∗) for its Langlands-Shelstad lift, and put
+˘w = ξ( �w). By replacing ξ by an equivalent inner twist if necessary, we assume that ˘w ∈ G(F). This gives us another
+representation
+[ ˘wπ](m) = π( ˘w−1m ˘w),
+m ∈ M(F),
+of M(F) on the same vector space as π, which corresponds to a parameter wψ := Ad(w) ◦ ψ.
+First, we define an unnormalized intertwining operator ℓP ′( ˘w) from HP ′w(π) to HP ′( ˘wπ) by
+[ℓP ′( ˘w)f](g) = f( ˘w−1g),
+where P ′w denotes w−1P ′w. Next, our normalizing factor is defined by
+ǫP (w, ψ, ψF ) = ǫ(1
+2, ρ∨
+P w|P ◦ φψ, ψF ),
+and we define
+ℓP ′
+P (w, ξ, ψ, ψF ) = ǫP (w, ψ, ψF )ℓP ′( ˘w).
+29
+
+Lemma 4.6. For any w1, w2 ∈ W(M ∗, G∗) ∼= W(M, G), we have
+ℓP
+P (w2w1, ξ, ψ, ψF ) = ℓP
+P (w2, ξ, w1ψ, ψF ) ◦ ℓP w2
+P
+(w1, ξ, ψ, ψF ).
+Proof. The proof is similar to that of [7, Lemma 2.3.4] or [18, Lemma 2.3.1]. Our case is simpler since the λ-factor is
+trivial and the Galois action on the dual group “
+G is also trivial.
+Lemma 4.7. We have
+ℓP ′
+P (w, ξ, ψ, ψF ) ◦ RP ′w|P w(ξ, ψ) = RP ′|P (ξ, wψ) ◦ ℓP
+P (w, ξ, ψ, ψF ).
+Proof. The assertion follows from two equalities ℓP ′
+P (w, ξ, ψ, ψF )◦RP ′w|P w(ξ, ψ) = RP ′|P (Ad( ˘w)◦ξ, ψ)◦ℓP
+P (w, ξ, ψ, ψF )
+and RP ′|P (Ad( ˘w) ◦ ξ, ψ) = RP ′|P (ξ, wψ). We can show them by a similar way to the second and the third assertions
+in [18, Lemma 2.2.5].
+4.4
+The third intertwining operator
+We maintain the assumptions of the previous subsection. In addition, we assume that we are given z ∈ Z1(F, M ∗) ⊂
+Z1(F, G∗), such that (ξ, z) is a pure inner twist; z commutes with the Langlands-Shelstad lift of every element of
+W(M ∗, G∗); and if we decompose z as z = z+ ×z− according to the decomposition M ∗ = M ∗
++ ×M ∗
+− then z+ takes the
+constant value 1, where M ∗
++ is a product of general linear groups and M ∗
+− is a special orthogonal group. Let u ∈ N( �T, “
+G)
+be such that Ad(u) ◦ ψ = ψ and u preserves the positive roots in �
+M. Let u♮ ∈ Nψ(M, G) and w ∈ W(�
+M, “
+G) be its
+images. We shall regard w as an element of W(M ∗, G∗) or W(M, G) via the natural isomorphisms. Although our
+group G∗ is a special orthogonal group and (ξ, z) is a pure inner twist, by the similar argument as [18, §2.4, §2.4.1],
+we can define
+• the operator π( ˘w)ξ : ( ˘wπ, Vπ) → (π, Vπ);
+• the sophisticated splittings s′ : Wψ(M, G) → Nψ(M, G) and s : Nψ(M, G) → Sψ(M) of the exact sequence
+1 → Sψ(M) → Nψ(M, G) → Wψ(M, G) → 1;
+• the constant ⟨u♮, π⟩ξ,z = ⟨s(u♮)−1, π⟩ξ|M∗,z ∈ C×.
+Then we define the operator π(u♮)ξ,z : ( ˘wπ, Vπ) → (π, Vπ) by π(u♮)ξ,z = ⟨u♮, π⟩ξ,zπ( ˘w)ξ. The assignment u♮ �→ π(u♮)ξ,z
+is multiplicative.
+Next we shall give a more abstract characterization (i.e., another equivalent definition) of the operator π(u♮)ξ,z :
+( ˘wπ, Vπ) → (π, Vπ), following [18, §2.4.3]. Let us put u′ = u−1 ∈ Nψ(M, G), and let ‹
+u′ ∈ N( �T, “
+G) be its Langlands-
+Shelstad lift. Put s = u′‹
+u′−1 ∈ �
+M. Then by the assumption that u preserves the positive roots in �
+M, we have s ∈ �T.
+Put �θ = Ad(‹
+u′), which is an automorphism of �
+M preserving the standard splitting inherited from “
+G. It is the dual of
+an automorphism θ∗ = Ad( �w) of M ∗. Put
+”
+M ′ =
+¶
+x ∈ �
+M
+��� Ad(sψs) ◦ �θ(x) = x
+©
+.
+Let M ′ be the subgroup of M such that ”
+M ′ is the dual of M ′. Note that Im ψ ⊂ ”
+M ′. Then (M ′, sψs, ”
+M ′ ⊂ �
+M) is a
+twisted endoscopic triple of (M ∗, θ∗), and hence of (M, θ), where θ = Ad( ˘w). We shall write eM,ψ for it. The operator
+π(u♮)ξ,z can be characterized as follows:
+Lemma 4.8. For each π ∈ Πψ(M), there exists a unique isomorphism π(u♮)ξ,z : ( ˘wπ, Vπ) → (π, Vπ) such that
+f eM,ψ(ψ) = e(M θ)
+�
+π∈Πψ(M)
+tr(π(u♮)ξ,z ◦ π(f)),
+for all ∆[eM,ψ, ξ|M∗, z]-matching functions f ∈ H(M) and f eM,ψ ∈ H(M ′). Moreover, the operator π(u♮)ξ,z coincides
+with the one constructed above.
+Proof. The proof is similar to that of [18, Lemma 2.4.1] for the case of the quasi-split unitary group UE/F(N).
+30
+
+4.5
+The compound operator
+We shall define the normalized self-intertwining operator RP (u♮, π, ψ, ψF ) ∈ EndG(HP (π)) as a composition
+RP (u♮, π, ψ, ψF ) = IP (π(u♮)ξ,z) ◦ ℓP
+P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ).
+Recall that the definitions of the Haar measures on the unipotent radicals and the representative ˘w which were used
+to define RP w|P (ξ, ψ) and ℓP
+P (w, ξ, ψ, ψF ), respectively, depend on ξ. Note also that the definition of IP (π(u♮)ξ,z)
+depends on (ξ, z). If the parameter ψ is not generic, we shall assume from now on that Lemma 4.4 holds true.
+Lemma 4.9. The operator RP (u♮, π, ψ, ψF ) does not depend on the choice of the pure inner twist (ξ, z).
+Proof. Suppose that (ξ, z) ≃ (ξ′, z′) as inner twists of G∗ = SO2n+1 over F. Since every automorphism is inner, there
+exists b ∈ G∗(F) such that ξ′ = ξ ◦ Ad(b). The proof is similar to that of [18, Lemma 2.5.1] except that our z is not a
+basic cocycle, but an ordinary 1-cocycle.
+The Kottwitz map (2.1) provides a pairing ⟨−, −⟩ on Z(“
+G) × H1(F, G∗). Note that in our case Z(“
+G) ≃ {±1}.
+Lemma 4.10.
+1. Let x ∈ Z(“
+G). Then RP (xu♮, π, ψ, ψF ) = ⟨x, z⟩−1RP (u♮, π, ψ, ψF ).
+2. Let y ∈ Sψ(M). Then RP (yu♮, π, ψ, ψF ) = ⟨y, π⟩ξ|M∗,zRP (u♮, π, ψ, ψF ).
+Proof. The proof is similar to that of [18, Lemma 2.5.2].
+Lemma 4.11. The operator RP (u♮, π, ψ, ψF ) is multiplicative in u♮.
+Proof. Let u♮
+1 and u♮
+2 be elements in Nψ(M, G), and w1 and w2 their images in Wψ(M, G) respectively. Since the
+assignment u♮ �→ π(u♮)ξ,z is multiplicative, we have
+IP (π(u♮
+2u♮
+1)ξ,z) = IP (π(u♮
+2)ξ,z ◦ π(u♮
+1)ξ,z) = IP (π(u♮
+2)ξ,z) ◦ IP (π(u♮
+1)ξ,z).
+Combining this with Lemmas 4.3, 4.6, and 4.7, we get
+RP (u♮
+2u♮
+1, π, ψ, ψF )
+= IP (π(u♮
+2)ξ,z) ◦ IP (π(u♮
+1)ξ,z) ◦ ℓP
+P (w2, ξ, w1ψ, ψF ) ◦ ℓP w2
+P
+(w1, ξ, ψ, ψF ) ◦ RP w2w1|P w1 (ξ, ψ) ◦ RP w1|P (ξ, ψ)
+= IP (π(u♮
+2)ξ,z) ◦ IP (π(u♮
+1)ξ,z) ◦ ℓP
+P (w2, ξ, ψ, ψF ) ◦ RP w2 |P (ξ, ψ) ◦ ℓP
+P (w1, ξ, ψ, ψF ) ◦ RP w1|P (ξ, ψ).
+We know that the operator IP (π(u♮
+1)ξ,z) commutes with ℓP
+P (w2, ξ, w1ψ, ψF )◦RP w2|P (ξ, ψ), as the operator IP (π(u♮
+1)ξ,z)
+acts on the values of the functions that comprise HP (π), while the operators ℓP
+P (w2, ξ, ψ, ψF ) and RP w2 |P (ξ, ψ) act on
+their variables. Thus we have
+IP (π(u♮
+2)ξ,z) ◦ IP (π(u♮
+1)ξ,z) ◦ ℓP
+P (w2, ξ, ψ, ψF ) ◦ RP w2 |P (ξ, ψ) ◦ ℓP
+P (w1, ξ, ψ, ψF ) ◦ RP w1 |P (ξ, ψ)
+= IP (π(u♮
+2)ξ,z) ◦ ℓP
+P (w2, ξ, ψ, ψF ) ◦ RP w2|P (ξ, ψ) ◦ IP (π(u♮
+1)ξ,z) ◦ ℓP
+P (w1, ξ, ψ, ψF ) ◦ RP w1|P (ξ, ψ)
+= RP (u♮
+2, π, ψ, ψF ) ◦ RP (u♮
+1, π, ψ, ψF ).
+4.6
+The two linear forms, the local intertwining relation, and the construction of the
+non-discrete packets
+Let F be a local field, ψF : F → C1 a nontrivial additive character, G∗ = SO2n+1, and M ∗ ⊊ G∗ a proper standard
+Levi subgroup. Let ξ : G∗ → G be an inner twist over F, and put M = ξ(M ∗). Let ψ ∈ Ψ(M ∗) be a local parameter,
+and u♮ an element of Nψ(M, G). When M ∗ transfers to G, we assume that M ⊊ G is a proper Levi subgroup over F
+and ξ|M∗ : M ∗ → M is an inner twist over F.
+We shall define the first linear form f �→ fG(ψ, u♮) on H(G) as follows:
+fG(ψ, u♮) =
+
+
+
+
+
+�
+π∈Πψ(M)
+tr(RP (u♮, π, ψ, ψF ) ◦ IP (π, f)),
+if ψ is relevant,
+0,
+if ψ is not relevant.
+31
+
+Next we shall give the definition of the second linear form f �→ f ′
+G(ψ, s) on H(G). For f ∈ H(G) and s ∈ Sψ,ss, let
+(e, ψe) ∈ X(G∗) be the element corresponding to (s, ψ) ∈ Y (G∗) under the bijective map given in Lemma 3.8. Then
+we write f ′
+G(ψ, s) for the value f e(ψe) in the endoscopic character relation (3.7). It does not depend on the choice of
+ξ.
+Lemma 4.12. Let s1, s2 ∈ Sψ,ss be two elements such that they have the same image in Sψ(G) and the image belongs
+to Sψ(M, G). Then f ′
+G(ψ, s1) = f ′
+G(ψ, s).
+Proof. The proof is similar to that of [18, Lemma 2.6.1].
+Theorem* 4.13 (LIR). Let u ∈ Nψ(M, G) and f ∈ H(G). Then
+1. If u♮ is in W ◦
+ψ(M, G), then we have RP (u♮, π, ψ, ψF ) = 1 for all π ∈ Πψ(M).
+2. The value fG(ψ, u♮) depends on the image of u♮ in Sψ(M, G).
+3. We have
+f ′
+G(ψ, sψu−1) = e(G)fG(ψ, u♮),
+(4.7)
+where e(G) denotes the Kottwitz sign of G. This equation is called the local intertwining relation.
+Lemma 4.14. For any y ∈ Z(“
+G∗), we have f ′
+G(ψ, sψys) = ⟨y, z⟩f ′
+G(ψ, sψs) and fG(ψ, yu♮) = ⟨y, z⟩−1fG(ψ, u♮).
+Proof. They follow from Lemmas 3.2 and 4.10.
+In this paper, Theorem 4.13 will be completely proved in the case when ψ = φ ∈ Φ(M ∗) is an L-parameter. The
+proof is inductive, and thus we assume that Theorem 4.13 holds for the case the degree is smaller than n, from now
+on. The case of a general parameter ψ ∈ Ψ(M ∗) is expected to be treated as a sequel of [18].
+We will now review the construction of local non-discrete A-packets which is an application of the local intertwining
+relation (Theorem 4.13). Suppose that the theorem is true. Assume that ψM∗ ∈ Ψ2(M ∗) and M ∗ ⊊ G∗, hence ψM∗
+is a discrete parameter for M ∗ and not discrete for G∗. We shall write ψ for the image of ψM∗ under the natural map
+Ψ(M ∗) → Ψ(G∗). We apologize for this change of convention. If ψ is not G-relevant, then we simply set Πψ(G) = ∅.
+Suppose that ψ is G-relevant. Then we may assume that a Levi subgroup M = ξ(M ∗) ⊊ G is defined over F, and
+by the assumption Theorem 4.13 holds true for ψM∗ and M. According to Lemma 4.1, we have SψM∗(M, G) = Sψ(G),
+hence the natural map NψM∗(M, G) → Sψ(G) is surjective. For any πM ∈ ΠψM∗(M), where note that the packet for
+M is given by the induction hypothesis, the map
+NψM∗(M, G) × G(F) → AutG(F )(HP (πM)),
+(u♮, g) �→ RP (u♮, πM, ψM∗, ψF ) ◦ IP (πM, g),
+is a representation of NψM∗(M, G) × G(F). Let ρ(πM) denote this representation. Put
+Π1
+ψ(G) =
+�
+πM∈ΠψM∗ (M)
+ρ(πM).
+Then we have
+tr �Π1
+ψ(G)(u♮, f)� =
+�
+πM∈ΠψM∗ (M)
+tr �ρ(πM)(u♮, f)�
+= fG(ψM∗, u♮),
+for u♮ ∈ NψM∗(M, G) and f ∈ H(G). Thus, Theorem 4.13 (and the character theory of representations of finite groups)
+implies that Π1
+ψ(G) can be regarded as a representation of Sψ(G)×G(F). By Lemma 4.10, we have Π1
+ψ(G)|Z( “
+G)×{1} =
+χ−1
+G . As πM is unitary, Π1
+ψ(G) is also unitary, in particular completely reducible. We define the packet Πψ(G) to be
+the multiset of irreducible representations of G(F) that appear in the direct summand of Π1
+ψ(G). Then we obtain
+Π1
+ψ(G) =
+�
+π∈Πψ(G)
+⟨−, π⟩−1 ⊗ π,
+32
+
+where ⟨−, π⟩ are characters of Sψ(G) whose restriction to Z(“
+G)Γ coincide χG. We equip the packet Πψ(G) with the
+map
+Πψ(G) → Irr(Sψ, χG),
+π �→ ⟨−, π⟩.
+The endoscopic character relation (3.7) follows directly from the local intertwining relation (4.7) and the definition of
+the pairing ⟨−, π⟩ above.
+4.7
+Reduction of LIR to discrete (relative to M) parameters
+In this subsection we review from [18, §2.7] the twisted local intertwining relation and the reduction of the proof of
+the part 1 and 2 of Theorem 4.13 to the case of discrete parameters. Concretely speaking, this section will be devoted
+to the proofs of Lemmas 4.15 and 4.16 below.
+Let us now consider two nested proper standard Levi subgroups
+M ∗
+0 ⊂ M ∗ ⊂ G∗. Let ψ0 ∈ Ψ(M ∗
+0 ) be a local parameter, and ψ ∈ Ψ(M ∗) its image under the natural map from
+Ψ(M ∗
+0 ) to Ψ(M ∗). Note that ψ0 is relevant if and only if ψ is relevant. When ψ and ψ0 are relevant, we assume that
+M = ξ(M ∗) and M0 = ξ(M ∗
+0 ) are proper Levi subgroups of G defined over F.
+Lemma 4.15. For any u ∈ Nψ0(M0, G) ∩ Nψ(M, G), we have
+fG(ψ0, u♮) = fG(ψ, u♮),
+where u♮ in the left (resp. right) hand side is the image of u in Nψ(M0, G) (resp. Nψ(M, G)).
+Note that the lemma is trivial if the parameter ψ0 is not relevant.
+Let P (resp. P0, resp. Q) be the standard parabolic subgroup of G (resp. G, resp. M), with the Levi sub-
+group M (resp. M0, resp. M0). Note that Nψ(M0, M) ⊂ Nψ(M0, G). Note also that the equation (4.1) implies
+Z(A�
+M0, Sψ0(G)◦) = Z(A�
+M0, Sψ0(M)◦), and hence Nψ(M0, M) ⊂ Nψ(M0, G).
+Lemma 4.16. Assume that ψ0 is relevant. For any u ∈ Nψ0(M0, M) and π0 ∈ Πψ0(M0), we have
+RP0(u♮, π0, ψ0, ψF ) = IP (RP0(u♮, π0, ψ0, ψF )).
+Moreover, we have
+fG(ψ0, u♮) = fM(ψ0, u♮).
+Before the proofs of these lemmas, we shall now record a consequence of Lemma 4.15, which is a key ingredient of
+the proof of the local intertwining relation.
+Proposition 4.17. Assume that the second and the third statements of Theorem 4.13 hold for a standard parabolic
+pair (M ∗
+0 , P ∗
+0 ) and a discrete parameter ψ0 ∈ Ψ2(M ∗
+0 ). Then they hold for every standard parabolic pair (M ∗, P ∗) and
+a parameter ψ ∈ Ψ(M ∗) such that M ∗
+0 ⊂ M ∗, where ψ is the image of ψ0 under the canonical map Ψ(M ∗
+0 ) → Ψ(M ∗).
+Proof. The proof is similar to that of [18, Proposition 2.7.3] with Sψ(G)◦ in place of Srad
+ψ . (Precisely speaking, we
+have Srad
+ψ
+= Sψ(G)◦ because G∗ = SO2n+1.)
+The third statement of Theorem 4.13 will also be reduced to the case of discrete parameters, in §6.4 Lemma 6.21.
+As a preparation of the proofs of Lemmas 4.15 and 4.16, we now review the twisted local intertwining relation.
+See [18, §2.7] and [7, pp.115-119] for detail.
+Let N and k be positive integers.
+Put H = GLk
+N.
+The standard
+pinning of GLN gives rise to a pinning of H, which we write (TH, BH, {XαH}αH) and call standard. Let θH be the
+automorphism of H given by θH(h1, . . . , hk) = (θ(hk), h1, . . . , hk−1), where θ is either the identity automorphism of
+GLN, or the automorphism θN. Let (MH, PH) be a standard parabolic pair of H. Recall that any packet for a direct
+product of a finite number of general linear groups, is a singleton. Let ψ ∈ Ψ(MH) and let π be the corresponding
+representation of MH(F). Put Sψ(H, θH) = Cent(Im ψ, “
+H ⋊ ”
+θH), which is not a group. Put also Nψ(MH, H ⋊ θH) =
+N(A ‘
+MH, Sψ(H, θ−1
+H )), W(‘
+MH, “
+H ⋊”
+θH
+−1) = N(A ‘
+MH , “
+H ⋊”
+θH
+−1)/‘
+MH, and W(MH, H ⋊θH) = N(AMH, H ⋊θH)/MH.
+We identify the Weyl sets W(‘
+MH, “
+H ⋊ ”
+θH
+−1) and W(MH, H ⋊ θH) in the standard way. Let u ∈ Nψ(MH, H ⋊ θH)
+and let w ∈ W(‘
+MH, “
+H ⋊ ”
+θH
+−1) ≃ W(MH, H ⋊ θH) be the image of u.
+33
+
+As in §4.2, let us define the first operator
+RP w
+H |PH(πλ) = rP w
+H |PH(πλ, ψF )−1JP w
+H |PH(πλ, ψF ),
+(4.8)
+where JP w
+H |PH(πλ, ψF ) : HPH(πλ) → HP w
+H (πλ) is the unnormalized intertwining operator defined by
+[JP w
+H |PH(πλ, ψF )f](h) =
+�
+NH(F )∩N w
+H(F )\N w
+H(F )
+f(nh)dn,
+and rP w
+H |PH(πλ, ψF ) is the normalizing factor given by
+rP w
+H |PH(πλ, ψF ) =
+L(0, ρ∨
+P w
+H |PH ◦ φψλ)
+L(1, ρ∨
+P w
+H |PH ◦ φψλ)
+ǫ( 1
+2, ρ∨
+P w
+H |PH ◦ φψλ, ψF )
+ǫ(0, ρ∨
+P w
+H |PH ◦ φψλ, ψF ) .
+Then (4.8) is holomorphic at λ = 0, thus we put RP w
+H |PH(π) = RP w
+H |PH(π0).
+In order to define the second operator, we have to fix a representative of w. Let ˙w be the representative of w in
+the Weyl set W(TH, H ⋊ θH) that stabilizes the simple positive roots inside MH. Since both ˙w and θH preserve TH,
+we have ˙w = ˙w0 ⋊ θH for some ˙w0 ∈ W(TH, H). Then we can take the Langlands-Shelstad lift �w0 ∈ N(TH, H) of ˙w0.
+Put �w = �w0 ⋊ θH, which is the representative of w we need. As in §4.3, let us define the second operator
+ℓPH
+PH(w, π, ψF ) = ǫPH(w, ψ, ψF )ℓP H( �w),
+where the normalizing factor is
+ǫPH(w, ψ, ψF ) = ǫ(1
+2, ρ∨
+P w
+H |PH ◦ φψ, ψF ),
+and the operator ℓP H( �w) from (IP w
+H (π), HP w
+H (π)) to (IPH( �wπ) ◦ θH, HPH( �wπ)) is defined by
+[ℓPH( �w)f](h) = f( �w−1 · h ⋊ θH).
+Here the identity component ‹
+H◦ = H ⋊ 1 of the twisted group ‹
+H = H ⋊ ⟨θH⟩ is identified with H.
+Note that �wπ is isomorphic to π. Let us define the third operator
+IPH(π( �w)) : HPH( �wπ) → HPH(π),
+where π( �w) : �wπ → π is the Whittaker normalized isomorphism. Here we use the Whittaker datum corresponding to
+ψF and the standard pinning.
+Now we obtain the normalized intertwining operator
+RPH(w, ψ, ψF ) = IPH(π( �w)) ◦ ℓPH
+PH(w, π, ψF ) ◦ RP w
+H |PH(π),
+from (IPH (π), HPH(π)) to (IPH(π) ◦ θH, HPH(π)), and the twisted first linear form
+H(H) ∋ f �→ fH(ψ, w) := tr (RPH(w, ψ, ψF ) ◦ IPH(π, f)) .
+On the other hand, for any s ∈ Sψ(H, θH), we have a pair (e, ψe) of a twisted endoscopic triple e ∈ E(H ⋊ θH) and
+its parameter ψe ∈ Ψ(He), corresponding to the pair (ψ, s). For any f ∈ H(H), choose f e ∈ H(He) such that f and
+f e are matching. Then we write f ′
+H(ψ, s) for the value f e(ψe). This is the twisted second linear form, which we need.
+We call the following formula the twisted local intertwining relation, or twisted LIR for short.
+Proposition 4.18 (twisted LIR). We have
+fH(ψ, w) = f ′
+H(ψ, sψu−1).
+Proof. The case k = 1 is guaranteed by [7, Corollary 2.5.4]. Then the proof is similar to that of [18, Proposition 2.7.4,
+Lemma 2.7.6].
+34
+
+Now we return to the notation used in Lemmas 4.15 and 4.16.
+If ψ0 and ψ are not relevant, then Lemma
+4.15 is trivial. Assume that ψ0 and ψ are relevant. Hence M0, M, P0, P, and Q are defined over F.
+Let u ∈
+Nψ0(M0, G) ∩ Nψ(M, G), and u♮ ∈ Nψ0(M0, G) be its image. We will also write u♮ for the image in Nψ(M, G), by
+abuse of notation. Let f ∈ H(G). By definition we have
+fG(ψ0, u♮) = tr
+Ñ
+�
+π0∈Πψ0(M0)
+RP0(u♮, π0, ψ0, ψF ) ◦ IP0(π0, f)
+é
+.
+The calculation similar to that in [18, pp.123-126] shows that
+�
+π0∈Πψ0(M0)
+RP0(u♮, π0, ψ0, ψF ) ◦ IP0(π0)(f)
+= ℓP
+P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦
+Ñ
+�
+π0∈Πψ0(M0)
+IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)
+é
+.
+ECR and LIR for inner form of SO2n0+1 for n0 < n, which is assumed by induction, and Lemma 4.18 imply ECR and
+(twisted) LIR for M. In particular, the operator RQ(u♮, π0, ψ0, ψF ) acts on each irreducible summand of IQ(π0) by
+scalar. Combined with the theory of induced characters ([9]) and Lemma 4.8, we have
+tr
+Ñ
+�
+π0∈Πψ0(M0)
+IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)
+é
+= tr
+Ñ
+�
+π0∈Πψ0(M0)
+RQ(u♮, π0, ψ0, ψF ) ◦ IQ(π0)(fM)
+é
+= tr
+Ñ
+�
+π∈Πψ(M)
+π(u♮) ◦ π(fM)
+é
+= tr
+Ñ
+�
+π∈Πψ(M)
+IP (π(u♮)) ◦ IP (π)(f)
+é
+,
+where fM denotes the constant term of f along P. By the linear independence of characters, this means that
+�
+π0∈Πψ0(M0)
+IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP0(π0)(f) =
+�
+π∈Πψ(M)
+IP (π(u♮)) ◦ IP (π)(f).
+Therefore, we have
+ℓP
+P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦
+Ñ
+�
+π0∈Πψ0(M0)
+IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)
+é
+= ℓP
+P(w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦
+Ñ
+�
+π∈Πψ(M)
+IP (π(u♮)) ◦ IP (π)(f)
+é
+=
+�
+π∈Πψ(M)
+IP (π(u♮)) ◦ ℓP
+P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦ IP (π)(f)
+=
+�
+π∈Πψ(M)
+RP (u♮, π, ψ, ψF ) ◦ IP (π, f),
+and hence
+fG(ψ0, u♮) = fG(ψ, u♮).
+35
+
+Thus Lemma 2.7.1 follows.
+Next, let u ∈ Nψ0(M0, M). Then u is trivial in W(�
+M, “
+G), and the calculations similar to those in [18, p.127] prove
+both two equations in Lemma 4.16.
+4.8
+Reduction of LIR to elliptic or exceptional (relative to G) parameters
+In this subsection, we review the reduction of the proof of LIR to the case of elliptic or exceptional parameters.
+Recall from §3.2 that Ψ2
+ell(G∗) is the set of the equivalence classes of parameters ψ of the form
+ψ = 2ψ1 ⊕ · · · 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr,
+where ψ1, . . . , ψr are irreducible, symplectic, and mutually distinct, and r ≥ q ≥ 1. Note that then we have
+Sψ ∼= O(2, C)q × O(1, C)r−q.
+Let now Ψexc1(G∗) and Ψexc2(G∗) be the subsets of Ψ(G∗) consisting of the parameters ψ of the form
+(exc1) ψ = 2ψ1 ⊕ ψ2 ⊕ · · · ⊕ ψr, where ψ1 is irreducible and orthogonal, and ψ2, . . . , ψr are irreducible, symplectic, and
+mutually distinct,
+(exc2) ψ = 3ψ1 ⊕ ψ2 ⊕ · · · ⊕ ψr, where ψ1, . . . , ψr are irreducible, symplectic, and mutually distinct,
+respectively. They are disjoint. We then have
+(exc1) Sψ ∼= Sp(2, C) × O(1, C)r−1,
+(exc2) Sψ ∼= O(3, C) × O(1, C)r−1,
+respectively. We put Ψexc(G∗) = Ψexc1(G∗) ⊔ Ψexc2(G∗), and we shall say that ψ is exceptional (resp. of type (exc1),
+resp. of type (exc2)) if ψ ∈ Ψexc(G∗) (resp. Ψexc1(G∗), resp. Ψexc2(G∗)). One can see that Ψexc(G∗) and Ψell(G∗)
+are disjoint. Put Ψell,exc(G∗) = Ψ(G∗) \ (Ψell(G∗) ⊔ Ψexc(G∗)).
+Let M ∗ ⊊ G∗ be a proper standard Levi subgroup, and ξ : G∗ → G an inner twist over F, such that M = ξ(M ∗) ⊊
+G is defined over F if M ∗ transfers to G. In view of §4.7, it is enough to consider discrete parameters for M. Let
+ψM∗ ∈ Ψ2(M ∗) be a discrete local A-parameter, and we shall write ψ for its image in Ψ(G∗) in this subsection. Recall
+from Lemma 4.1 that Sψ(M, G) = Sψ(G). Here we do not assume that M is defined over F, nor do that ψ is relevant.
+Let Tψ be the maximal central torus A �
+M of �
+M. Since a Levi subgroup for which the parameter is discrete is unique,
+the torus Tψ is determined by ψ and is a maximal torus of Sψ = Sψ(G). One can easily see that
+Wψ = Wψ(M, G) = W(Tψ, Sψ),
+W ◦
+ψ = W ◦
+ψ(M, G) = W(Tψ, S◦
+ψ).
+Therefore, if ψ is elliptic then W ◦
+ψ is trivial, as stated in [18, Lemma 2.8.1]. Similarly, let Bψ be the standard Borel
+subgroup of Sψ with a maximal torus Tψ. Then (Tψ, Bψ) is a Borel pair of S◦
+ψ. For any x ∈ Sψ, following [7, p.204],
+put
+Tψ,x = Cent(sx, Tψ)◦,
+where sx ∈ Sψ is a representative of x such that Ad(sx) stabilizes (Tψ, Bψ). Since sx is determined up to a Tψ-translate,
+Tψ,x is uniquely determined by x.
+The following lemma is the key lemma in this subsection.
+Lemma 4.19. If ψ ∈ Ψell,exc(G∗), then
+(1) every simple reflection w ∈ W ◦
+ψ centralizes a torus of positive dimension in Tψ;
+(2) dim Tψ,x ≥ 1, ∀x ∈ Sψ.
+Proof. The proof is similar to that of [18, Lemma 2.8.4].
+36
+
+We now define
+Sψ,ell = { s ∈ Sψ,ss | Z(Cent(s, S◦
+ψ)) is finite } ,
+Sψ,ell = { s ∈ Sψ,ss | Z(Cent(s, S
+◦
+ψ)) is finite } ,
+Sψ,ell = Sψ,ell/(Sψ,ell ∩ S◦
+ψ),
+Sψ,ell = Sψ,ell/(Sψ,ell ∩ S
+◦
+ψ).
+One can see that Sψ,ell (resp. Sψ,ell) is the image of Sψ,ell (resp. Sψ,ell) under the natural surjection Sψ → Sψ (resp.
+Sψ → Sψ).
+Lemma 4.20.
+1. The natural surjection Sψ → Sψ carries Sψ,ell onto Sψ,ell.
+2. The natural surjection Sψ → Sψ carries Sψ,ell onto Sψ,ell.
+3. If ψ ∈ Ψexc(G∗) then Sψ,ell = Sψ and Sψ,ell = Sψ.
+Proof. The first part can be seen easily. The proof of latter two is similar to that of [18, Lemma 2.8.6].
+Lemma 4.21. Assume that either
+(1) ψ is elliptic, or
+(2) every simple reflection w ∈ W ◦
+ψ centralizes a torus of positive dimension in Tψ.
+Then the parts 1 and 2 of Theorem 4.7 hold for ψ (and for all u and f).
+Proof. The proof is similar to that of [18, Lemma 2.8.7]. We appeal to Lemmas 4.11 and 4.16 instead of Lemmas 2.5.3
+and 2.7.2 of loc. cit.
+Lemma 4.22. Let x ∈ Sψ(M, G). Assume that either
+(1) ψ is elliptic and x /∈ Sψ,ell, or
+(2) ψ is not elliptic, every simple reflection w ∈ W ◦
+ψ centralizes a torus of positive dimension in Tψ, and dim Tψ,x ≥ 1.
+Then f ′
+G(ψ, sψs−1) = e(G)fG(ψ, u♮) whenever both u♮ ∈ Nψ(M, G) and s ∈ Sψ,ss map to x.
+Proof. Let sx ∈ Sψ,ss be a representative of x such that Ad(sx) stabilizes (Tψ, Bψ).
+Since Tψ = A �
+M, we have
+sx ∈ Nψ(M, G). Lemmas 4.1, 4.12, and 4.21 tell us that f ′
+G(ψ, sψs−1) = f ′
+G(ψ, sψs−1
+x ) and fG(ψ, u♮) = fG(ψ, s♮
+x).
+We need to show that f ′
+G(ψ, sψs−1
+x ) = e(G)fG(ψ, s♮
+x). Put �
+Mx = Z “
+G(Tψ,x). Let M ∗
+x ⊂ G∗ be the Levi subgroup
+corresponding to �
+Mx. If the assumption (1) holds, then we have |Tψ,x| = ∞ and hence �
+Mx ⊊ “
+G. If the assumption
+(2) holds, then we have dim Tψ,x ≥ 1. This implies |Tψ,x| = ∞ and hence �
+Mx ⊊ “
+G. Therefore, M ∗
+x is a proper Levi
+subgroup of G∗. The rest of the proof is similar to that of [18, Lemma 2.8.8]. We appeal to Lemmas 4.12 and 4.16
+instead of Lemmas 2.6.1 and 2.7.2 of loc. cit.
+Lemmas 4.19, 4.21, and 4.22 imply the following corollary.
+Corollary 4.23. Let x ∈ Sψ(M, G). Then Theorem 4.13 holds for all u♮ ∈ Nψ(M, G) mapping to x unless either
+(1) ψ is elliptic and x ∈ Sψ,ell, or
+(2) ψ is exceptional.
+Let us now define
+Wψ,reg(M, G) = { w ∈ Wψ(M, G) | w has finitely many fixed points on Tψ } .
+We write Nψ,reg(M, G) for the preimage of Wψ,reg(M, G) under Nψ(M, G) → Wψ(M, G).
+Lemma 4.24. Assume that ψ ∈ Ψexc(G∗) and u♮ /∈ Nψ,reg(M, G). Let s be an element of Sψ,ss whose image in
+Sψ(M, G) is the same as that of u♮. Then we have f ′
+G(ψ, sψs−1) = e(G)fG(ψ, u♮).
+37
+
+Proof. The proof is similar to that of [18, Lemma 2.8.10]. We appeal to Lemmas 4.12 and 4.22 instead of Lemmas
+2.6.1 and 2.8.8 of loc. cit.
+Consequently, if ψ ∈ Ψell,exc(G∗) then all the three statements of Theorem 4.13 hold (under the induction hypoth-
+esis). If ψ ∈ Ψ2
+ell(G∗) then the first and second part of Theorem 4.13 hold. In particular, when ψ is not exceptional,
+fG(ψ, x) is well-defined for x ∈ Sψ(M, G). Moreover, the third part of the theorem holds if the image of u in Sψ is
+not contained in Sψ,ell. When ψ is exceptional, for now, the result is Lemma 4.24, i.e., only a part of the third part.
+Let x ∈ Sψ(M, G). Since ψ is exceptional, we have |W ◦
+ψ| = 2, and there is exactly two elements in the fiber of x in
+Nψ(M, G). One is regular in Wψ(M, G), and the other is not. Put ux to be the former one, and define fG(ψ, x) to be
+fG(ψ, ux). Now we have defined fG(ψ, x) for x ∈ Sψ(M, G) for every ψ ∈ Ψ(G∗)
+Put Φ♥(G∗) = Ψ♥(G∗) ∩ Φ(G∗) and Φbdd,♥(G∗) = Ψ♥(G∗) ∩ Φbdd(G∗) for ♥ ∈ {exc, exc1, exc2}.
+4.9
+LIR for special cases
+In this subsection we shall prove the two special cases of Theorem 4.13. Later in §§6.2-6.4, we will reduce the proof
+of Theorem 4.13 for φ ∈ Φell,exc(G∗) to the special cases, which we shall treat in this subsection.
+Before we treat the two cases, we recall LLC for GL1(R) and GL2(R), and fix some notation. We shall realize the
+Weil group WR as C× ⊔ jC×, where j2 = −1 and jz = zj for any z ∈ C×. Recall from [32, §1] that the norm map
+on WR is given by |j| = 1 and |z| = zz for z ∈ C ⊂ WR. We define the (isomorphism classes of) finite dimensional
+representations σ ωt, and τ(l,t) of WR by
+σ : WR → C×,
+®
+C× ∋ z �→ 1,
+j �→ −1,
+ωt : WR → C×,
+®C× ∋ z �→ |z|t = (zz)t,
+j �→ 1,
+τ(l,t) : WR → GL(2, C),
+
+
+
+
+
+
+
+
+
+C× ∋ re
+√−1θ �→ r2t
+Ç
+e
+√−1lθ
+e−√−1lθ
+å
+,
+j �→
+Å
+1
+(−1)l
+ã
+,
+for t ∈ C and l ∈ Z>0, and put τl = τ(l,0). Then the local Langlands correspondence over R says that σεωt (ε = 0, 1)
+is corresponding to the 1-dimensional representation | · |t
+R sgnε, and τ(l,t) = τl ⊗ ωt to the 2-dimensional representation
+Dl ⊗ | det |t
+R, where Dl denotes the discrete series representation of GL2(R) of weight l + 1. Fix the standard additive
+character ψR(x) = exp(2π√−1x) of R. Put ΓR(s) := π− s
+2 Γ( s
+2) and ΓC(s) := 2(2π)−sΓ(s). The L-functions and the
+ǫ-factors are given by the following table. See [20] for more detail, but note that (+, t) and (−, t) in [20, (3.2)] should
+be (+, t
+2) and (−, t
+2) respectively.
+ϕ
+L(s, ϕ)
+ǫ(s, ϕ, ψR)
+ωt
+ΓR(s + t)
+1
+σωt
+ΓR(s + t + 1)
+√−1
+τ(l,t)
+ΓC(s + t + l
+2)
+√−1
+l+1
+For a symmetric matrix Q and an alternative matrix A of degree N, we shall write
+SO(Q) = { g ∈ GLN | tgQg = Q, det g = 1 } ,
+Sp(A) = { g ∈ GLN | tgAg = A } .
+4.9.1
+The first special case
+The first special case we are concerned with is the following. Let F = R, n = 2, and
+G∗ = SO5 = SO
+Ñ
+12
+2
+12
+é
+,
+38
+
+M ∗ =
+
+
+
+
+
+
+
+Ü t
+A
+b
+t−1
+β
+c
+ê ��������
+t ∈ GL1, A ∈ M2×2,
+Å A
+b
+β
+c
+ã
+∈ SO
+Ñ
+1
+2
+1
+é 
+
+
+
+
+
+
+∼= GL1 × SO3 .
+Let φ ∈ Φexc1(G∗) be an L-parameter of type (exc1) of the form
+φ = 2ω0 ⊕ τ1,
+and φM = ω0 ⊕ τ1 ∈ Φ2(M ∗) so that φ is the natural image of φM. More precisely, we write as
+“
+G = SpC
+Ü
+1
+1
+−1
+−1
+ê
+,
+�
+M =
+
+
+
+Ñ a
+X
+a−1
+é
+∈ “
+G
+������
+a ∈ GL(1, C), X ∈ SpC
+Å
+1
+−1
+ã
+= SL(2, C)
+
+
+ ⊂ “
+G,
+φ = φM =
+Ñ
+ω0
+τ1
+ω0
+é
+.
+Up to isomorphism, there exists only one nontrivial inner twist of G∗ for which φ is relevant. Let
+G = SO(1, 4) = SO
+Å 1
+−14
+ã
+.
+Put
+α =
+1
+√
+2
+à 1
+1
+√−1
+√−1
+2√−1
+1
+−1
+1
+−1
+í
+.
+Let z ∈ Z1(R, G∗) be a 1-cocycle such that
+zρ = α−1ρ(α) =
+à 1
+0
+−1
+−1
+1
+−1
+0
+í
+,
+where ρ is the nontrivial element in ΓF = Gal(C/R), i.e., the complex conjugation. Let ξ = Ad(α) : G∗ → G. Put
+M = ξ(M ∗)
+= { m(t, B) | t ∈ GL1, B ∈ SO(−13) = SO(3) } ,
+where
+m(t, B) =
+Ñ c(t)
+s(t)
+B
+s(t)
+c(t)
+é
+,
+c(t) = t + t−1
+2
+,
+s(t) = t − t−1
+2
+.
+39
+
+Then (ξ, z) is a pure inner twist G∗ → G which restricts to a pure inner twist M ∗ → M such that z+ ∈ Z1(R, GL1)
+is trivial. We have Πω0(GL1(R)) = {1}, Πτ1(SO(3)) = {1}, and hence ΠφM (M) = {1}, where 1 denotes the trivial
+representation of each group. Let us write π for the unique element in ΠφM (M), i.e., the trivial representation of
+M(R).
+Proposition 4.25. Theorem 4.13 is valid for G, M, and φ.
+Proof. The centralizer Sφ(G) is
+
+
+
+Ñ a
+b
+ε12
+c
+d
+é ������
+Å a
+b
+c
+d
+ã
+∈ SL(2, C), ε ∈ {±1}
+
+
+
+∼= SL(2, C) × O(1, C).
+Thanks to this explicit structure, one can easily understand the diagram (4.6). The diagram has the form
+1
+1
+�
+�
+{ ±1 } × 1
+{ ±1 } × 1
+�
+�
+1 −−−−→ 1 × { ±1 } −−−−→ { ±1 } × { ±1 } −−−−→ { ±1 } × 1 −−−−→ 1
+���
+�
+�
+1 −−−−→ 1 × { ±1 } −−−−→
+1 × { ±1 }
+−−−−→
+1
+−−−−→ 1
+�
+�
+1
+1
+where (−1, 1) and (1, −1) are represented by
+Ü
+1
+1
+1
+−1
+ê
+∈ N(A �
+M, Sφ(G)◦),
+Ü
+1
+−1
+−1
+1
+ê
+∈ Sφ(M),
+respectively. Note that (−1, 1) vanishes in Sφ(G). Let us write w for the image of (−1, 1) in Wφ(M, G) = Wφ(M, G)◦ ∼=
+W(M ∗, G∗) ∼= W(M, G). One can calculate the Langlands-Shelstad lift in “
+G = Sp(4, C) of w as in [16]. It is
+Ü
+−1
+−1
+−1
+1
+ê
+,
+and thus the first sophisticated splitting s′ : Wψ(M, G) → Nψ(M, G) sends (−1, 1) to (−1, −1). Therefore, the other
+sophisticated splitting s : Nψ(M, G) → Sψ(M) sends (−1, −1) to 1, and (−1, 1) to (1, −1).
+Let P ∗ be the standard parabolic subgroup of G∗ with Levi subgroup M ∗, and P = ξ(P ∗). Now in the same way
+as [18, pp.136-137], we know that the induced representation (IP (π), HP (π)) is irreducible and is the unique element
+of Πφ(G), and that the operator RP ((−1, 1), π, φ, ψR) is a scalar. Moreover, in order to prove the proposition, it is
+sufficient to show
+RP ((−1, 1), π, φ, ψR)f = f,
+40
+
+for a nonzero element f in HP (π). Recall that
+RP ((−1, 1), π, φ, ψR) = IP (π((−1, 1))ξ,z) ◦ ℓP
+P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ).
+We focus first on IP (π((−1, 1))ξ,z) = IP (⟨(−1, 1), π⟩ξ,zπ( ˘w)ξ).
+Since π is the trivial representation on the 1-
+dimensional vector space C, the operator π( ˘w)ξ is the identity map. On the other hand, we have ⟨(−1, 1), π⟩ξ,z =
+⟨s(−1, 1), π⟩M = ⟨−1, π⟩M = −1. We obtain IP (π((−1, 1))ξ,z) = −1.
+Next we consider ℓP
+P (w, ξ, φ, ψR)◦RP w|P (ξ, φ). Let �w be the Langlands-Shelstad lift of w in G∗(R). The calculation
+of �w has already been done in [16, §8]:
+�w =
+à
+1
+−1
+−1
+1
+−1
+í
+.
+We have ˘w = ξ( �w) =
+Å1
+−14
+ã
+∈ G(R). Let us define a connected compact subgroup K ⊂ G(R) as
+K =
+ß Å 1
+κ
+ã ���� κ ∈ SO(4)
+™
+.
+By abuse of notation, we shall write κ for the element
+Å1
+κ
+ã
+. Although K is not maximal, the Iwasawa decomposition
+tells us that G(R) = P(R)K. Let N ∗ be the unipotent radical of P ∗. Put
+n∗(b) =
+à
+1
+b1
+b2
+−b1b3 − b2
+2
+4
+b3
+1
+−b3
+1
+− b2
+2
+1
+−b1
+1
+í
+,
+for b = (b1, b2, b3) ∈ C3, so that N ∗(R) = {n∗(b) | b ∈ R3}. A direct calculation following §4.2 show that the Haar
+measure on N ∗(R) is d(n∗(b)) =
+1
+2db1db2db3. Put P = ξ(P ∗) and N = ξ(N ∗), which are the standard parabolic
+subgroup with Levi subgroup M and its unipotent radical. Put also
+n(x) =
+Ö
+1 + ∥x∥2
+2
+x
+− ∥x∥2
+2
+tx
+13
+− tx
+∥x∥2
+2
+x
+1 − ∥x∥2
+2
+è
+=
+â
+1 + ∥x∥2
+2
+x1
+x2
+x3
+− ∥x∥2
+2
+x1
+1
+−x1
+x2
+1
+−x2
+x3
+1
+−x3
+∥x∥2
+2
+x1
+x2
+x3
+1 − ∥x∥2
+2
+ì
+,
+for x = (x1, x2, x3) ∈ C3, where ∥x∥2 = x2
+1 + x2
+2 + x2
+3 so that N(R) = {n(x) | x ∈ R3}. A straightforward calculation
+shows that ξ(n∗(b)) = n(x) where
+x1 = −b1 + b3
+2
+√
+−1,
+x2 = −b2
+2
+√
+−1,
+x3 = b1 − b3
+2
+,
+and hence the measure on N(R) is d(n(x)) = 2dx1dx2dx3.
+For λ ∈ C, put φλ = ωλ ⊕ τ1 ⊕ ω−λ and πλ = | · |λ ⊠ 1, so that Πφλ(M) = {πλ} and Πφλ(G) = {IP (πλ)}. Define a
+function ϕ(λ) ∈ HP (πλ) by
+ϕ(λ)(m(t, B)n(x)κ) = |t|λ+ 3
+2 ,
+which is holomorphic in λ ∈ C. Assume that Re(λ) > 0. We have 0 ̸= ϕ(0) ∈ HP (π) and
+î
+ℓP
+P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ)ϕ(0)ó
+(g)
+41
+
+= lim
+λ→+0
+î
+ℓP
+P (w, ξ, φλ, ψR) ◦ RP w|P (ξ, φλ)ϕ(λ)ó
+(g)
+= lim
+λ→+0 ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ)
+�
+N(R)
+ϕ(λ)( ˘w−1ng)dn.
+A direct calculation implies that ρ∨
+P w|P ◦ φλ is isomorphic to
+τ(1,λ) ⊕ ω2λ.
+Therefore, we have
+ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ) = −ΓC(λ + 3
+2)
+ΓC(λ + 1
+2)
+ΓR(2λ + 1)
+ΓR(2λ)
+.
+We now turn to the integral. Put
+[MSO(1,4)ϕ(λ)](g) =
+�
+N(R)
+ϕ(λ)( ˘w−1ng)dn.
+In Lemma 4.26 below, we will show that
+MSO(1,4)ϕ(λ) = 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2)ϕ(−λ).
+This leads to
+ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ)
+�
+N(R)
+ϕ(λ)( ˘w−1n·)dn
+= −ΓC(λ + 3
+2)
+ΓC(λ + 1
+2)
+ΓR(2λ + 1)
+ΓR(2λ)
+· 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2)ϕ(−λ)
+= −2− 1
+2
+ΓC(λ)
+ΓC(λ + 1
+2)
+ΓR(2λ + 1)
+ΓR(2λ)
+ϕ(−λ),
+whose limit as λ approaching 0 from the right is −ϕ(0). Since ϕ(0) ̸= 0, this completes the proof.
+To finish the proof of Proposition 4.25, it remains to show the following lemma.
+Lemma 4.26. For Re(λ) > 0, we have
+MSO(1,4)ϕ(λ) = 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2)ϕ(−λ).
+Proof. By definition, MSO(1,4)ϕ(λ) is right K-invariant and left N(R)-invariant. Moreover, we have
+[MSO(1,4)ϕ(λ)](m(t, B)g) = |t|−λ+ 3
+2 [MSO(1,4)ϕ(λ)](g).
+Indeed, a direct calculation shows that
+n(x)m(t, B) = m(t, B)n(t−1xB),
+˘w−1m(t, B) = m(t−1, B) ˘w−1.
+Thus we have
+[MSO(1,4)ϕ(λ)](m(t, B)g) =
+�
+x∈R3 ϕ(λ)( ˘w−1n(x)m(t, B)g)d(n(x))
+=
+�
+x∈R3 ϕ(λ)(m(t−1, B) ˘w−1n(t−1xB)g) · 2dx1dx2dx3
+42
+
+= |t|−λ− 3
+2
+�
+y∈R3 ϕ(λ)( ˘w−1n(y)g) · 2|t|3dy1dy2dy3
+= |t|−λ+ 3
+2 [MSO(1,4)ϕ(λ)](g),
+where we have changed the variables y = t−1xB. Combining these properties with ϕ(λ)(1) = 1, we now have
+MSO(1,4)ϕ(λ) = MSO(1,4)ϕ(λ)(1) · ϕ(−λ).
+We must then show that
+MSO(1,4)ϕ(λ)(1) = 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2).
+For x, y ∈ R3, t ∈ R×, B ∈ SO(3), and κ ∈ SO(4) ∼= K, a direct calculation shows that
+˘w−1n(x) =
+Ö
+1 + ∥x∥2
+2
+x
+− ∥x∥2
+2
+− tx
+−13
+tx
+− ∥x∥2
+2
+−x
+−1 + ∥x∥2
+2
+è
+,
+m(t, B)n(y)κ =
+Ö
+c(t) + t∥y∥2
+2
+∗
+∗
+B ty
+∗
+∗
+s(t) + t∥y∥2
+2
+∗
+∗
+è
+,
+so if we write ˘w−1n(x) = m(t, B)n(y)κ then we have
+
+
+
+
+
+
+
+
+
+
+
+1 + ∥x∥2
+2
+= c(t) + t∥y∥2
+2
+,
+− tx
+= B ty,
+−∥x∥2
+2
+= s(t) + t∥y∥2
+2
+,
+which leads t = (1 + ∥x∥2)−1. Now the value MSO(1,4)ϕ(λ)(1) is equal to
+�
+x∈R3 ϕ(λ)( ˘w−1n(x))d(n(x))
+=
+�
+x∈R3
+��1 + ∥x∥2��−λ− 3
+2 · 2dx1dx2dx3
+=
+� ∞
+r=0
+� π
+θ=0
+� 2π
+α=0
+2r2 sin θ
+(1 + r2)λ+ 3
+2 dαdθdr
+= 8π
+� ∞
+0
+r2
+(1 + r2)λ+ 3
+2 dr,
+where we have changed the variables x1 = r cos α sin θ, x2 = r sin α sin θ, and x3 = r cos θ. Since Re(λ) > 0, the
+last integral converges absolutely, and the gamma function Γ(λ + 3
+2) has the integral expression which also converges
+absolutely. Therefore, the product of them is
+8π
+� ∞
+0
+r2
+(1 + r2)λ+ 3
+2 dr × Γ(λ + 3
+2)
+= 8π
+� ∞
+0
+� ∞
+0
+Å
+t
+1 + r2
+ãλ+ 3
+2
+r2e−tdrd×t,
+where d×t = t−1dt. First we fix r and change the variables u = (1 + r2)−1t, and then fix u and change the variables
+s = √ur, to obtain
+8π
+� ∞
+0
+� ∞
+0
+Å
+t
+1 + r2
+ãλ+ 3
+2
+r2e−tdrd×t
+43
+
+= 8π
+� ∞
+0
+uλe−ud×u
+� ∞
+0
+s2e−s2ds
+= 2π
+3
+2 Γ(λ).
+We now have
+MSO(1,4)ϕ(λ)(1) = 8π
+� ∞
+0
+r2
+(1 + r2)λ+ 3
+2 dr
+= 2π
+3
+2
+Γ(λ)
+Γ(λ + 3
+2) = 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2).
+4.9.2
+The second special case
+The second special case we are concerned with is the following. Let F = R, n = 3, and
+G∗ = SO7 = SO
+Ñ
+13
+2
+13
+é
+,
+M ∗ =
+
+
+
+
+
+
+
+Ü A
+X
+b
+tA−1
+β
+c
+ê ��������
+A ∈ GL2, X ∈ M2×2,
+Å X
+b
+β
+c
+ã
+∈ SO
+Ñ
+1
+2
+1
+é 
+
+
+
+
+
+
+∼= GL2 × SO3 .
+Let φ ∈ Φexc2(G∗) be an L-parameter of type (exc2) of the form
+φ = 3τ1,
+and φM = 2τ1 ∈ Φ2(M ∗) (as equivalent classes) so that φ is the natural image of φM. More precisely, we define the
+parameter as
+“
+G = SpC
+Ü
+12
+1
+−1
+−12
+ê
+,
+�
+M =
+
+
+
+Ñ A
+X
+tA−1
+é
+∈ “
+G
+������
+A ∈ GL(2, C), X ∈ SpC
+Å
+1
+−1
+ã
+= SL(2, C)
+
+
+ ⊂ “
+G,
+φ = φM =
+Ñ τ1
+τ1
+tτ1−1
+é
+.
+Up to isomorphism, there exists only one nontrivial inner twist of G∗ for which φ is relevant. Let
+G = SO(2, 5) = SO
+Å 12
+−15
+ã
+.
+Put
+α =
+1
+√
+2
+à 12
+12
+√−1
+√−1
+2√−1
+1
+−1
+12
+−12
+í
+.
+44
+
+Let z ∈ Z1(R, G∗) be a 1-cocycle such that
+zρ = α−1ρ(α) =
+à
+12
+0
+−1
+−1
+12
+−1
+0
+í
+,
+where ρ is the nontrivial element in ΓF = Gal(C/R). Put ξ = Ad(α) : G∗ → G and
+M = ξ(M ∗)
+= { m(A, B) | A ∈ GL2, B ∈ SO(−13) = SO(3) } ,
+where
+m(A, B) =
+Ñ c(A)
+s(A)
+B
+s(A)
+c(A)
+é
+,
+c(A) = A + tA−1
+2
+,
+s(A) = A − tA−1
+2
+.
+Then (ξ, z) is a pure inner twist G∗ → G which restricts to a pure inner twist M ∗ → M such that z+ ∈ Z1(R, GL2)
+is trivial. We have Πτ1(GL2(R)) = {D1}, Πτ1(SO(3)) = {1}, and hence ΠφM (M) = {D1 ⊠ 1}, where 1 denotes the
+trivial representation of SO(3). Let us write π for the unique element in ΠφM (M).
+Proposition 4.27. Theorem 4.13 is valid for G, M, and φ.
+Proof. Put J =
+Å
+1
+−1
+ã
+, and J =
+Å14
+J
+ã
+. Let us define a mapping A �→ A(2) from the set of 3 × 3-matrices to
+those of 6 × 6-matrices by
+A =
+Ñ a1,1
+a1,2
+a1,3
+a2,1
+a2,2
+a2,3
+a3,1
+a3,2
+a3,3
+é
+�→ A(2) :=
+Ñ a1,112
+a1,212
+a1,312
+a2,112
+a2,212
+a2,312
+a3,112
+a3,212
+a3,312
+é
+.
+Since tτ1−1 = Jτ1J−1, the centralizer Sφ(G) is
+
+
+ Jh(2)J−1
+������
+h ∈ OC
+Ñ
+1
+1
+1
+é 
+
+
+∼= O(3, C).
+Thanks to this explicit structure, one can easily understand the diagram (4.6). The diagram has the form
+1
+1
+�
+�
+{ ±1 } × 1
+{ ±1 } × 1
+�
+�
+1 −−−−→ 1 × { ±1 } −−−−→ { ±1 } × { ±1 } −−−−→ { ±1 } × 1 −−−−→ 1
+���
+�
+�
+1 −−−−→ 1 × { ±1 } −−−−→
+1 × { ±1 }
+−−−−→
+1
+−−−−→ 1
+�
+�
+1
+1
+45
+
+where (−1, 1) and (1, −1) are represented by
+Ñ
+J−1
+−12
+J
+é
+∈ N(A �
+M, Sφ(G)◦),
+Ñ 12
+−12
+12
+é
+∈ Sφ(M),
+respectively. Note that (−1, 1) vanishes in Sφ(G). Let us write w for the image of (−1, 1) in Wφ(M, G) = Wφ(M, G)◦ ∼=
+W(M ∗, G∗) ∼= W(M, G). One can calculate the Langlands-Shelstad lift in “
+G of w as in [16]. It is
+Ñ
+J−1
+12
+J
+é
+,
+and thus the first sophisticated splitting s′ : Wψ(M, G) → Nψ(M, G) sends (−1, 1) to (−1, −1). Therefore, the other
+splitting s : Nψ(M, G) → Sψ(M) sends (−1, −1) to 1, and (−1, 1) to (1, −1).
+Let P ∗ be the standard parabolic subgroup of G∗ with Levi subgroup M ∗, and P = ξ(P ∗). Now in the same way
+as [18, pp.136-137], we know that the induced representation (IP (π), HP (π)) is irreducible and is the unique element
+of Πφ(G), and that the operator RP ((−1, 1), π, φ, ψR) is a scalar. Moreover, in order to prove the proposition, it is
+sufficient to show
+RP ((−1, 1), π, φ, ψR)f = f,
+for a nonzero element f in HP (π). Recall that
+RP ((−1, 1), π, φ, ψR) = IP (π((−1, 1))ξ,z) ◦ ℓP
+P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ).
+We focus first on IP (π((−1, 1))ξ,z) = IP (⟨(−1, 1), π⟩ξ,zπ( ˘w)ξ). Let �w be the Langlands-Shelstad lift of w in G∗(R).
+The calculation of �w has already been done in [16, §8]:
+�w =
+à
+J
+1
+1
+J
+1
+í
+.
+Thus we have
+˘w = ξ( �w) =
+Ñ
+J
+13
+J−1
+é
+∈ G(R).
+Since ˘w−1m(A, B) ˘w = m((det A)−1A, B) and the central character of D1 is trivial, the representations ˘wπ and π
+coincide. Thus π( ˘w)ξ is the identity map. On the other hand, we have ⟨(−1, 1), π⟩ξ,z = ⟨s(−1, 1), π⟩M = ⟨−1, π⟩M =
+−1. We obtain IP (π((−1, 1))ξ,z) = −1.
+Next we consider ℓP
+P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ). Let us define a connected compact subgroup K ⊂ G(R) as
+K =
+ß
+κ =
+Å κ2
+κ5
+ã ���� κm ∈ SO(m), (m = 2, 5)
+™
+.
+Although K is not maximal, the Iwasawa decomposition tells us that G(R) = P(R)K. Let N ∗ be the unipotent
+radical of P ∗, which is generated by {exp(Xα) | α = χ1 ± χ3, χ1, χ2 ± χ3, χ2, χ1 + χ2}. Here Xα denotes the Chevalley
+basis. The Haar measure on N ∗(R) is also given by the Chevalley basis. Put P = ξ(P ∗) and N = ξ(N ∗), which are
+46
+
+the standard parabolic subgroup with Levi subgroup M and its unipotent radical. Now we describe N(R) explicitly.
+Let ι1 and ι2 be embeddings of SO(1, 4) into G = SO(2, 5) given by
+h =
+Å a
+b
+β
+C
+ã
+�→ ι1(h) =
+Ü a
+b
+1
+β
+C
+1
+ê
+,
+h =
+Å A
+b
+β
+c
+ã
+�→ ι2(h) =
+Ü 1
+A
+b
+1
+β
+c
+ê
+,
+where A and C are 4-by-4 matrices. For x = (x1, x2, x3) ∈ C3 and u ∈ C, put n1(x) = ι1(n(x)) and n2(x) = ι2(n(x)),
+where n(x) is the element defined in the proof of Proposition 4.25, and put
+nc(u) =
+à
+1
+u
+2
+0
+− u
+2
+− u
+2
+1
+u
+2
+0
+13
+0
+u
+2
+1
+− u
+2
+− u
+2
+0
+u
+2
+1
+í
+,
+so that
+n1(x) = ξ(exp(
+√
+−1x1(Xχ1−χ3 + Xχ1+χ3) +
+√
+−1x2Xχ1 + x3(Xχ1−χ3 − Xχ1+χ3))),
+n2(x) = ξ(exp(
+√
+−1x1(Xχ2−χ3 + Xχ2+χ3) +
+√
+−1x2Xχ2 + x3(Xχ2−χ3 − Xχ2+χ3))),
+nc(u) = ξ(exp(uXχ1+χ2)).
+Let N1 (resp. N2, resp. Nc) be the unipotent subgroup of G consisting of n1(x) (resp. n2(x), resp. nc(u)). Then the
+Haar measure on N1(R) (resp. N2(R), resp. Nc(R)) is given by d(n1(x)) = 2dx1dx2dx3 (resp. d(n2(x)) = 2dx1dx2dx3,
+resp. d(nc(u)) = du). We have N(R) = N1(R)N2(R)Nc(R) = {n1(x)nc(u)n2(y) | x, y ∈ R3, u ∈ R} and the Haar
+measure is the product of those on N1(R), N2(R), and Nc(R).
+Let us realize the discrete series representation D1 of GL2(R) as a subrepresentation of a parabolically induced
+representation from a character | − |
+1
+2 ⊠ | − |− 1
+2 on the diagonal maximal torus of GL2(R), and let 0 ̸= v0 ∈ D1 be a
+lowest weight vector such that
+v0
+ÅÅ a
+b
+d
+ã Å
+cos θ
+sin θ
+− sin θ
+cos θ
+ãã
+=
+���a
+d
+��� e2√−1θ.
+For λ ∈ C, put φλ = τ(1,λ) ⊕ τ1 ⊕ τ ∨
+(1,λ) and πλ = (D1 ⊗ | · |λ) ⊠ 1, so that Πφλ(M) = {πλ} and Πφλ(G) = {IP (πλ)}.
+Since D1 is realized as a subrepresentation of a parabolically induced representation, we have a natural injection
+HP (πλ) ֒→ HP0(| − |
+1
+2 +λ ⊠ | − |− 1
+2 +λ ⊠ 1),
+f(·) �→ f(·)(1),
+where P0 is the minimal standard parabolic subgroup of G defined over R.
+Define a function f (λ) ∈ HP (πλ) ⊂
+HP0(| − |
+1
+2 +λ ⊠ | − |− 1
+2 +λ ⊠ 1) by
+f (λ)(m(A, B)nκ) = | det A|λ+2[τ1(A)v0](1)
+= |a|λ+3|d|λ+1e2√−1θ,
+A =
+Å a
+b
+d
+ã Å
+cos θ
+sin θ
+− sin θ
+cos θ
+ã
+,
+which is holomorphic in λ ∈ C. Assume that Re(λ) > 0. We have 0 ̸= f (0) ∈ HP (π) and
+î
+ℓP
+P(w, ξ, φ, ψR) ◦ RP w|P (ξ, φ)f (0)ó
+(g)
+= lim
+λ→+0
+î
+ℓP
+P (w, ξ, φλ, ψR) ◦ RP w|P (ξ, φλ)f (λ)ó
+(g)
+47
+
+= lim
+λ→+0 ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ)
+�
+N(R)
+f (λ)( ˘w−1ng)dn.
+A direct calculation implies that ρ∨
+P w|P ◦ φλ is isomorphic to
+τ(2,λ) ⊕ σωλ ⊕ ωλ ⊕ τ(2,2λ) ⊕ σω2λ.
+Therefore, we have
+ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ) = ΓC(λ + 2)
+ΓC(λ + 1)
+ΓR(λ + 2)
+ΓR(λ)
+ΓC(2λ + 2)
+ΓC(2λ + 1)
+ΓR(2λ + 2)
+ΓR(2λ + 1).
+We now turn to the integral. Let ιSL2 be an embedding of SL2 into G = SO(2, 5) given by ιSL2(A) = m(A, 13). Put
+w0 =
+Å1
+−14
+ã
+, which is the Langlands-Shelstad lift in SO(1, 4) given in the proof of proposition 4.25. Equations
+˘w = ι2(w0)ιSL2(J)ι2(w0),
+Ad(ιSL2(J)ι2(w0))−1(n1(x)) = n2(−x),
+Ad(ι2(w0))−1(nc(u)) = ιSL2(
+Å1
+u
+1
+ã
+),
+implies that
+�
+N(R)
+f (λ)( ˘w−1ng)dn = [M2 ◦ Mc ◦ M2(f (λ))](g),
+where we have put
+[M2f](g) =
+�
+N2(R)
+f(ι2(w0)−1n2g)dn2,
+[Mcf](g) =
+�
+Nc(R)
+f(ιSL2(J)−1ncg)dnc,
+for any function f on G(R).
+For α, β ∈ C, we define f (α,β) ∈ HP0(| − |α ⊠ | − |β ⊠ 1) by
+f (α,β)(m(A, B)nκ) = |a|α+ 5
+2 |d|β+ 3
+2 e2√−1θ,
+A =
+Å a
+b
+d
+ã Å
+cos θ
+sin θ
+− sin θ
+cos θ
+ã
+,
+so that f (λ) = f (λ+ 1
+2 ,λ− 1
+2 ). By Lemmas 4.28 and 4.29 below, we have
+M2 ◦ Mc ◦ M2(f (λ))
+= M2 ◦ Mc ◦ M2(f (λ+ 1
+2 ,λ− 1
+2 ))
+= 2− 1
+2 ΓC(λ − 1
+2)
+ΓC(λ + 1) M2 ◦ Mc(f (λ+ 1
+2 ,−λ+ 1
+2 ))
+= −2− 1
+2 ΓC(λ − 1
+2)
+ΓC(λ + 1)
+ΓR(2λ)ΓR(2λ + 1)
+ΓR(2λ + 3)ΓR(2λ − 1)M2(f (−λ+ 1
+2 ,λ+ 1
+2 ))
+= −2−1 ΓC(λ − 1
+2)
+ΓC(λ + 1)
+ΓR(2λ)ΓR(2λ + 1)
+ΓR(2λ + 3)ΓR(2λ − 1)
+ΓC(λ + 1
+2)
+ΓC(λ + 2) f (−λ).
+This leads to
+ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ)
+�
+N(R)
+f (λ)( ˘w−1ng)dn
+= ΓC(λ + 2)
+ΓC(λ + 1)
+ΓR(λ + 2)
+ΓR(λ)
+ΓC(2λ + 2)
+ΓC(2λ + 1)
+ΓR(2λ + 2)
+ΓR(2λ + 1) · −1
+2
+ΓC(λ − 1
+2)
+ΓC(λ + 1)
+ΓR(2λ)ΓR(2λ + 1)
+ΓR(2λ + 3)ΓR(2λ − 1)
+ΓC(λ + 1
+2)
+ΓC(λ + 2) f (−λ)(g),
+whose limit as λ approaching 0 from the right is −f (λ). Since f (0) ̸= 0, this completes the proof.
+48
+
+In order to finish the proof of Proposition 4.27, it remains to show the following two lemmas.
+Lemma 4.28. Suppose that Re(β) > 0. Then
+M2f (α,β) = 2− 1
+2
+ΓC(β)
+ΓC(β + 3
+2)f (α,−β).
+Proof. As in the proof of Lemma 4.26, one has
+M2f (α,β) = M2f (α,β)(1) · f (α,−β),
+if M2f (α,β)(1) converges absolutely. Since f (α,β) ◦ ι2 is equal to ϕ(β) defined in the previous subsection, we have
+[M2f (α,β)](1) =
+�
+x∈R3 f (α,β)(ι2(w0)−1ι2(n(x)))d(n(x))
+=
+�
+x∈R3 ϕ(β)(w−1
+0 n(x))d(n(x))
+= [MSO(1,4)ϕ(β)](1).
+Hence the assertion follows from Lemma 4.26.
+Lemma 4.29. Suppose that Re(α − β) > 0. Then
+Mcf (α,β) = −
+ΓR(α − β)ΓR(α − β + 1)
+ΓR(α − β + 3)ΓR(α − β − 1)f (β,α).
+Proof. As in the proof of Lemma 4.26, one has
+Mcf (α,β) = Mcf (α,β)(1) · f (β,α),
+if Mcf (α,β)(1) converges absolutely. Following the proof of [15, Lemma 1.4], for s ∈ C we define a function h(s) on
+SL2(R) by
+h(s)
+ÅÅ a
+b
+a−1
+ã Å
+cos θ
+sin θ
+− sin θ
+cos θ
+ãã
+=
+���a
+d
+���
+s+1
+e2√−1θ.
+Then f (α,β) ◦ ιSL2 is equal to h(α−β), and hence we have
+[Mcf (α,β)](1) =
+�
+u∈R
+f (α,β)
+Å
+ιSL2(J)−1ιSL2
+Å 1
+u
+1
+ãã
+du
+=
+�
+u∈R
+h(α−β)
+Å
+J−1
+Å 1
+u
+1
+ãã
+du.
+Hence the assertion follows from the second equation in the proof of [15, Lemma 1.4].
+5
+The decomposition into near equivalence classes and the standard
+model
+In this section we shall roughly recall from [18, 7] the stable multiplicity formula, which implies the decomposition
+of L2
+disc(G(F)\G(AF )) into near equivalence classes, the global intertwining relation, and its weaker identity.
+49
+
+5.1
+Stable multiplicity formula
+Let F be a number field, G∗ the split special orthogonal group SO2n+1 over F, and G an inner form of G∗.
+Although G does not have simply connected derived subgroup, every endoscopic data for G comes from an endoscopic
+triple. Hence the argument in [18, §§3.1-3.2] also holds for our G. Thus we have the decompositions
+L2
+disc(G(F)\G(AF )) =
+�
+c∈C(G)
+t≥0
+L2
+disc,t,c(G(F)\G(AF )),
+tr Rdisc(f) =
+�
+c∈C(G)
+t≥0
+Rdisc,t,c(f),
+for f ∈ H(G), where R♦ denotes the regular representation of G(AF ) on L2
+♦. Moreover, for t ∈ R≥0, c ∈ C(G), and
+e = (Ge, se, ηe) ∈ Eell(G), we have
+• the c-variant IG
+disc,t,c of the discrete part IG
+disc,t of the trace formula on the constituents of which the norm of the
+imaginary part of the infinitesimal character is t, where the central character datum is trivial;
+• the transfer mapping H(G) → S(Ge), f �→ f e = f Ge, where S(Ge) denotes a space of functions on conjugacy
+classes of semisimple elements defined in [7, p.53, p.132];
+• the stable linear form Se
+disc,t,c = SGe
+disc,t,c on H(Ge) and the associated linear form �Se
+disc,t,c on S(Ge) (see [7,
+(2.1.2)] for the notion of associated linear forms),
+and the stabilization
+IG
+disc,t,c(f) =
+�
+e∈Eell(G)
+ι(G, Ge)�Se
+disc,t,c(f e),
+f ∈ H(G).
+Here ι(G, Ge) are the global coefficients introduced by Kottwitz and Shelstad. See [7, (3.2.4)] for an explicit formula
+for them.
+Let ψ ∈ Ψ(G∗) and e = (Ge, se, ηe) ∈ Eell(G). To the former is associated t(ψ) ∈ R≥0 and c(ψ) ∈ C(G). As in [18,
+§3.3], put
+IG
+disc,ψ = IG
+disc,t(ψ),c(ψ),
+SGe
+disc,ψ = SGe
+disc,t(ψ),c(ψ),
+L2
+disc,ψ(G(F)\G(AF )) = L2
+disc,t(ψ),c(ψ)(G(F)\G(AF )),
+Rdisc,ψ = Rdisc,t(ψ),c(ψ).
+We also write RG
+disc,ψ if the group G is to be emphasized. Let us recall from [7, Theorem 4.1.2] the stable trace formula
+for the split odd special orthogonal group G∗ = SO2n+1:
+Proposition 5.1 (Stable trace formula). Let ψ ∈ ‹Ψ(N). Then we have
+SG∗
+disc,ψ(f) =
+�
+|Sψ|−1εG∗
+ψ (sψ)σ(S
+◦
+ψ)f G∗(ψ),
+if ψ ∈ Ψ(G∗),
+0,
+if ψ ∈ ‹Ψ(N) \ Ψ(G∗),
+for f ∈ H(G∗), where εG∗
+ψ
+is the character in the multiplicity formula, σ(S
+◦
+ψ) the constant given in [7, Proposition
+4.1.1], and f G∗(ψ) denotes the linear form defined by [7, (4.1.3)].
+As a consequence we have a decomposition
+L2
+disc(G(F)\G(AF )) =
+�
+ψ∈Ψ(G∗)
+L2
+disc,ψ(G(F)\G(AF )),
+tr Rdisc(f) =
+�
+ψ∈Ψ(G∗)
+Rdisc,ψ(f),
+f ∈ H(G),
+(5.1)
+of the discrete spectrum. Theorem 3.12 is now proven.
+50
+
+5.2
+Global intertwining operator
+Let ξ : G∗ → G be an inner twist. Let P ∗ ⊂ G∗ be a standard parabolic subgroup with a Levi decomposition
+P ∗ = M ∗N ∗ over F, and put P = ξ(P ∗), M = ξ(M ∗), and N = ξ(N ∗). We consider the case that P, M, and N are
+defined over F, and a restriction ξ|M∗ : M ∗ → M is an inner twist.
+Assume that M ̸= G. Let (π, Vπ) be an irreducible component of L2
+disc(A+
+M,∞G(F)\G(AF )), where A+
+M,∞ denotes
+the connected component of 1 in ResF/Q(M)(R). Let ψ ∈ Ψ2(M ∗) be the corresponding A-parameter (given by the
+induction hypothesis). We shall consider the induced representation (IP (π), HP (π)). It is a right regular representation
+on the Hilbert space HP (π) of measurable functions f : G(AF ) → Vπ such that f(nmg) = δ
+1
+2
+P (m)π(m)f(g) for any
+n ∈ N(AF ), m ∈ M(AF ), and g ∈ G(AF ) whose restriction to an open compact subgroup K ⊂ G(AF ) is square-
+integrable.
+Let P ′ ⊂ G be a parabolic subgroup over F with Levi component M. For λ ∈ a∗
+M,C, the intertwining operator
+JP ′|P : IP (πλ) → IP ′(πλ) is defined by
+[JP ′|P (πλ)(f)](g) =
+�
+N(AF )∩N ′(AF )\N ′(AF )
+f(n′g)dn′,
+where N ′ denotes the unipotent radical of P ′, and πλ = π ⊗ λ. As in [18], we take the Haar measure dn′ determined
+by the Haar measure on AF which assigns the quotient AF /F volume 1. It is known that the integral converges
+absolutely when the real part of λ lies in a certain open cone. As a function of λ, it has a meromorphic continuation
+and is nonzero and holomorphic at λ = 0. Then the operator JP ′|P (π) = JP ′|P (πλ)|λ=0 is defined. It is also known
+that JP ′|P (π) is unitary.
+We now consider the case P ′ = P w = w−1Pw for some w ∈ W(M, G)Γ. Let ˘w ∈ N(M, G)(F) be a representative
+of w. Then one has a twist ( ˘wπ, Vπ). We define two intertwining operators ℓ( ˘w) and C ˘
+w by
+ℓ( ˘w) : HP w(π) −→ HP ( ˘wπ),
+[ℓ( ˘w)f](g) = f( ˘w−1g),
+C ˘
+w : ( ˘wπ, Vπ) −→ (π, Vπ),
+[C ˘
+wϕ](m) = ϕ( ˘w−1m ˘w).
+The composition IP (C ˘
+w)◦ℓ( ˘w)◦JP w|P (π) is then a self-intertwining operator of the induced representation (IP (π), HP (π)),
+and independent of the choice of ˘w. We put
+MP (w, π) = IP (C ˘
+w) ◦ ℓ( ˘w) ◦ JP w|P (π).
+Let ρP ′|P be the adjoint representation of �
+M on �n ∩ �n′\�n′ ∼= �n ∩ “n′, where �n, “n′, and �n are the Lie algebras of “
+N,
+�
+N ′, and “
+N, respectively, where N denotes the unipotent radical of the opposite parabolic subgroup P of P containing
+M. Following [18], we define normalizing factors
+rP ′|P (ψ) =
+L(0, ψ, ρ∨
+P ′|P )
+L(1, ψ, ρ∨
+P ′|P )
+ǫ( 1
+2, ψ, ρ∨
+P ′|P )
+ǫ(0, ψ, ρ∨
+P ′|P ) ,
+rP (w, ψ) = rP w|P (ψ)ǫ(1
+2, ψ, ρ∨
+P w|P )−1,
+and normalized intertwining operators
+RP ′|P (π, ψ) = rP ′|P (ψ)−1JP ′|P (π),
+RP (w, π, ψ) = rP (w, ψ)−1MP (w, π),
+where L- and ǫ-factors are the automorphic ones.
+Lemma 5.2. We have
+RP ′|P (πλ, ψλ) =
+�
+v
+RP ′v|Pv(πλ,v, ψλ,v).
+Proof. The proof is similar to that of [18, Lemma 2.2.2].
+Lemma 5.3. Let P ′′ ⊂ G be a parabolic subgroup over F with Levi component M. We have
+RP ′′|P (πλ, ψλ) = RP ′′|P ′(πλ, ψλ) ◦ RP ′|P (πλ, ψλ).
+51
+
+Proof. The unnormalized intertwining operator JP ′|P (πλ) satisfies the multiplicativity property. It is known that at
+each place v the local factors of automorphic L- and ǫ-factors are equal to the Artin L- and ǫ-factors. Thus the
+normalizing factor has a decomposition
+rP ′|P (ψλ) =
+�
+v
+rP ′v|Pv(ξv, ψv, ψF,v).
+Since every local factor rP ′v|Pv(ξv, ψv, ψF,v) has the multiplicativity property, so is rP ′|P (ψλ). This completes the
+proof.
+5.3
+The global intertwining relation
+In this subsection, we define two global linear forms and state the global intertwining relation, which we shall call
+GIR for short. This subsection can be regarded as the global analogue of §4.6.
+Let ψM∗ ∈ Ψ2(M ∗), and ψ ∈ Ψ(G∗) be its image. Note the change of notation from the previous subsection that
+we write ψM∗ for a parameter for M ∗ rather than ψ. Let ψF : AF /F → C1 be a nontrivial additive character. For
+πM = �
+v πM,v ∈ ΠψM∗(M) and u♮ ∈ Nψ(M, G) we define a global intertwining operator
+RP (u♮, πM, ψM∗, ψF ) =
+�
+v
+RPv(u♮
+v, πM,v, ψM∗,v, ψF,v),
+where u♮
+v ∈ Nψv(Mv, Gv), ψM∗,v ∈ Ψ+(M ∗), and ψF,v are the localizations of u♮, ψM∗, and ψF respectively.
+Proposition 5.4. Assume that πM ∈ ΠψM∗(M) is automorphic, i.e., ⟨−, πM⟩ = εψM∗. Then for any u♮ ∈ Nψ(M, G)
+we have an equation
+�
+v
+πM,v(u♮
+v) = εψM∗(u♮)C ˘
+w,
+of isomorphisms ( ˘wπM, VπM ) → (πM, VπM ), where w is the image of u♮ in Wψ(M, G).
+Proof. The proof is similar to that of Proposition 3.5.3 of [18].
+Proposition 5.5. Let u♮ ∈ Nψ(M, G) and πM ∈ ΠψM∗(M). Then
+1. RP (yu♮, πM, ψM∗, ψF ) = ⟨y, πM⟩RP (u♮, πM, ψM∗, ψF ) for any y ∈ Sψ(M), where y is the image of y in Sψ(M).
+2. RP (wu, πM, ψM∗) = εψM∗(u♮)RP (u♮, πM, ψM∗, ψF ) if πM is automorphic.
+Proof. The first assertion follows from Lemma 4.10. The other one follows from Lemma 5.2 and Proposition 5.4.
+The first linear form H(G) ∋ f �→ fG(ψM∗, u♮) is defined by
+fG(ψM∗, u♮) =
+�
+v
+fv,Gv(ψM∗,v, u♮
+v)
+=
+�
+πM∈ΠψM∗ (M)
+tr �RP (u♮, πM, ψM∗, ψF ) ◦ IP (πM, f)� ,
+f =
+�
+v
+fv ∈ H(G).
+Note that the parameter ψM∗ is relevant since it is discrete and M = ξ(M ∗) is defined over F. Put fG(ψM∗, u♮) to be
+0 for M ∗ ⊂ G∗ which does not transfer to G. Proposition 5.5 implies the following lemma.
+Lemma 5.6. The linear form fG(ψM∗, u♮) depends only on the image of u♮ in Nψ(M, G).
+Thus we also write fG(ψM∗, u) for fG(ψM∗, u♮), where u ∈ Nψ(M, G) is the image of u♮.
+Next let us recall the definition of the second linear form. For a parameter ψ ∈ Ψ(G∗) and a semisimple element
+s ∈ Sψ, Lemma 3.9 attaches an endoscopic triple e = (Ge, se, ηe) ∈ E(G) and a parameter ψe ∈ Ψ(Ge), where ψe is
+not unique (determined up to OutG(Ge)-action). The second linear form H(G) ∋ f �→ f ′
+G(ψ, s) is defined by
+f ′
+G(ψ, s) = f e(ψe)
+=
+�
+v
+f e
+v(ψe
+v) =
+�
+v
+f ′
+v,Gv(ψv, sv),
+f =
+�
+v
+fv ∈ H(G),
+where sv ∈ Sψv denotes the localization of s. If ψ is the image of ψM∗ ∈ Ψ2(M ∗), we also write f ′
+G(ψM∗, s) for
+f ′
+G(ψ, s).
+52
+
+Lemma 5.7. The linear form f ′
+G(ψ, s) depends only on the image of s in Sψ, and hence on the image of e in E(G).
+Proof. By Lemma 4.12, we have f ′
+G(ψ, s) = f ′
+G(ψ, ss0) for any s0 ∈ S◦
+ψ. By Lemma 3.2 and the product formula (2.2),
+we have f ′
+G(ψ, s) = f ′
+G(ψ, sx) for any x ∈ Z(“
+G)Γ.
+The following theorem is called the global intertwining relation (GIR), which will follow from LIR.
+Theorem* 5.8 (Global intertwining relation). If u ∈ Nψ(M, G) and s ∈ Sψ,ss have the same image in Sψ(G), then
+we have
+f ′
+G(ψ, sψs−1) = fG(ψM∗, u).
+5.4
+Reduction of GIR to elliptic or exceptional (relative to G) parameters
+This subsection is the global version of §4.8. As in the local case, one can see that Ψ2
+ell(G∗) is the set of the
+equivalence classes of parameters ψ of the form
+ψ = 2ψ1 ⊞ · · · 2ψq ⊞ ψq+1 ⊞ · · · ⊞ ψr,
+(5.2)
+where ψ1, . . . , ψr are simple, symplectic, mutually distinct, and r ≥ q ≥ 1. Then we have
+Sψ ∼= O(2, C)q × O(1, C)r−q.
+Let now Ψexc1(G∗) and Ψexc2(G∗) be the subsets of Ψ(G∗) consisting of the parameters ψ of the form
+(exc1) ψ = 2ψ1 ⊞ψ2 ⊞· · ·⊞ψr, where ψ1 is simple and orthogonal, and ψ2, . . . , ψr are simple, symplectic, and mutually
+distinct,
+(exc2) ψ = 3ψ1 ⊞ ψ2 ⊞ · · · ⊞ ψr, where ψ1, . . . , ψr are simple, symplectic, and mutually distinct,
+respectively. They are disjoint. We then have
+(exc1) Sψ ∼= Sp(2, C) × O(1, C)r−1,
+(exc2) Sψ ∼= O(3, C) × O(1, C)r−1,
+respectively. We put Ψexc(G∗) = Ψexc1(G∗) ⊔ Ψexc2(G∗), and we shall say that ψ is exceptional (resp. of type (exc1),
+resp. of type (exc2)) if ψ ∈ Ψexc(G∗) (resp. ψ ∈ Ψexc1(G∗), resp. ψ ∈ Ψexc2(G∗)). One can see that Ψexc(G∗) and
+Ψell(G∗) are disjoint. Put Ψell,exc(G∗) = Ψ(G∗) \ (Ψell(G∗) ⊔ Ψexc(G∗)). Put also Φ♥(G∗) = Ψ♥(G∗) ∩ Φ(G∗) for
+♥ ∈ {exc, exc1, exc2}, and Φell,exc(G∗) = Ψell,exc(G∗) ∩ Φ(G∗).
+Let ψM∗ ∈ Ψ2(M ∗) and ψ ∈ Ψ(G∗) be parameters such that ψ is the image of ψM∗. For x ∈ Sψ, we define Tψ,
+Bψ, sx ∈ Sψ, and Tψ,x in the same way as in §4.8. Moreover, let T ψ, Bψ, sx, and T ψ,x be their images in Sψ. Then
+it can be seen that sx and T ψ,x are determined by the image of x in Sψ, and hence they are well-defined for x ∈ Sψ.
+Define subsets Sψ,ell ⊂ Sψ and Sψ,ell ⊂ Sψ as in the local case. The following four lemmas can be proved in the
+same way as Lemmas 4.19, 4.20, 4.21, and 4.23 were, respectively.
+Lemma 5.9. Suppose that ψ ∈ Ψell,exc(G∗). Then
+(1) every simple reflection w ∈ W ◦
+ψ(M ∗, G∗) centralizes a torus of positive dimension in T ψ and
+(2) dim T ψ,x ≥ 1 for all x ∈ Sψ.
+Lemma 5.10. If ψ ∈ Ψexc(G∗), then Sψ,ell = Sψ and Sψ,ell = Sψ.
+Let ξ : G∗ → G be an inner twist, and assume that M ∗ ⊊ G∗ is a proper standard Levi subgroup such that
+M = ξ(M ∗) ⊊ G is a proper Levi subgroup defined over F.
+Lemma 5.11. Let x ∈ Sψ(M, G). Assume that either
+(1) ψ is elliptic, or
+53
+
+(2) every simple reflection w ∈ W ◦
+ψ(M ∗, G∗) centralizes a torus of positive dimension in T ψ.
+Then fG(ψM∗, u) is the same for every u ∈ Nψ(M, G) mapping to x.
+Lemma 5.12. Let x ∈ Sψ. We have fG(ψM∗, u) = f ′
+G(ψ, sψs−1) whenever u ∈ Nψ(M, G) and s ∈ Sψ,ss map to x
+unless
+(1) ψ is elliptic and x ∈ Sψ,ell, or
+(2) ψ ∈ Ψexc(G∗).
+Lemmas 5.11 and 5.12 imply that fG(ψM∗, u) depends only on the image of u in Sψ(M, G) = Sψ(G), unless ψ is
+exceptional. Thus fG(ψM∗, x) is well-defined for x ∈ Sψ(M, G). On the other hand, when ψ is exceptional, we can
+define fG(ψM∗, x) for x ∈ Sψ(M, G) similarly to local setting in the end of §4.8. Now we have defined fG(ψM∗, x) for
+x ∈ Sψ(M, G).
+5.5
+The weaker identities
+In the previous subsection, GIR for ψ ∈ Ψell,exc(G∗) was deduced from the induction hypothesis (Lemma 5.12).
+Now we shall see two weaker identities. We omit Arthur’s procedure named the standard model for G, since it is very
+similar to that for other classical groups explained in [7, §4] and [18, §3.6]. We continue to be in the setup of the
+previous subsection, but do not assume that M = ξ(M ∗) is defined over F.
+Lemma 5.13. Suppose that ψ ∈ Ψexc(G∗). Let x ∈ Sψ and f ∈ H(G). If M ∗ does not transfer to G, then we have
+f ′
+G(ψ, sψx−1) = fG(ψ, x) = 0.
+If M ∗ transfers to G, then we have
+�
+x∈Sψ
+εG∗
+ψ (x) �f ′
+G(ψ, sψx−1) − fG(ψ, x)� = 0,
+and RP (w, πM, ψM∗) = 1 for w ∈ Wψ.
+Lemma 5.14. Suppose that ψ ∈ Ψ2
+ell(G∗) is of the form (5.2). Let f ∈ H(G). Then we have
+tr �RG
+disc,ψ(f)� = 2−q|Sψ|−1
+�
+x∈Sψ,ell
+εG∗
+ψ (x) �f ′
+G(ψ, sψx−1) − fG(ψ, x)� .
+Proof of Lemmas 5.13 and 5.14. The proofs of Lemmas 5.13 and 5.14 are similar to those of Lemmas 3.7.1 and 3.8.1
+of [18], respectively.
+6
+Globalizations and the proof of local classification
+In this section, we will finish the proof of LIR and ECR for generic parameters, and of LLC. The argument is
+similar to that of [18, §4]. The primary difference is the globalization of parameters, which is caused by the difference
+between unitary groups and orthogonal groups.
+6.1
+Globalizations of fields, groups, and representations
+First recall from [7, Lemma 6.2.1] the following lemma:
+Lemma 6.1. Let F be a local field other than C, and r0 a positive integer. Then we can find a totally real number
+field ˙F and a place u of ˙F such that ˙Fu is isomorphic to F and ˙F has at least r0 real places.
+As a corollary, we have the following.
+54
+
+Lemma 6.2. Let F be a local field other than C, and r0 a positive integer. Then we can find a totally real number
+field ˙F and places u1 and u2 of ˙F such that ˙Fu is isomorphic to F for u = u1, u2, and that ˙F has at least r0 real
+places.
+Proof. Let ˙F be as in Lemma 6.1. It can be easily seen that there exists an element α ∈ ˙F × that is not square in ˙F,
+totally positive, and square in ˙Fu = F. Then ˙F(√α) is a number field what we want.
+The classification of quadratic forms or the exact sequence (2.2) leads the following lemma of special orthogonal
+groups:
+Lemma 6.3. Let F be a local field, and G an inner form of G∗ = SO2n+1 over F. Let ˙F be a number field with a
+place u such that ˙Fu = F. Assume that ˙F is totally real unless F = C. Then for any place w of ˙F other than u, there
+exists an inner form ˙G of ˙G∗ = SO2n+1 over ˙F with following properties:
+1.
+˙Gu = G;
+2.
+˙Gv is split over ˙Fv for any place v of ˙F except u and w;
+3. if G is non-quasi-split and w is a finite (resp. real) place, then ˙Gw is the unique non-quasi-split inner form
+(resp. isomorphic to SO(n − 1, n + 2)).
+In place of the third property, we can also choose ˙G with the properties 1, 2, and
+3’ if ˙Fw is isomorphic to ˙Fu = F, then ˙Gw is isomorphic to ˙Gu = G.
+We have the following globalization theorems of discrete series representations.
+Lemma 6.4. Let ˙F be a totally real number field, and ˙G a simple twisted endoscopic group of GLN or an inner form
+of ˙G∗ = SO2n+1 over ˙F. Let V be a finite set of places of ˙F such that at least one real place v∞ is not contained
+in V . For all v ∈ V , let πv ∈ Π2,temp( ˙Gv), and let πv∞ be a discrete series representation with a sufficiently regular
+infinitesimal character. Then there exists a cuspidal automorphic representation ˙π of ˙G(A ˙F ) such that ˙πv = πv for all
+v ∈ V and ˙πv∞ = πv∞.
+Proof. The proof is the same as that of [18, Lemma 4.2.1].
+Lemma 6.5. Let ˙F be a totally real number field, and ˙G a simple twisted endoscopic group of GLN. Let V be a finite
+set of places of ˙F such that at least one real place v∞ is not contained in V . For all v ∈ V , let Mv ⊂ ˙Gv be a Levi
+subgroup such that Mv = ˙Gv if v is a real place. For each v ∈ V , let πMv ∈ Π2,temp(Mv). Then there exists a cuspidal
+automorphic representation ˙π of ˙G(A ˙F ) such that for all v ∈ V , if Mv = ˙Gv then ˙πv = πMv and if Mv ̸= ˙Gv then
+˙πv is an irreducible subquotient of the induced representation I
+˙Gv
+Pv (πMv ⊗ χv) for some unramified unitary character
+χv ∈ Ψ(Mv).
+Proof. The proof is the same as that of [18, Lemma 4.2.2].
+The following lemma is used in the proof of Lemma 4.3. Note that the proof of the following lemma is independent
+of §4.
+Lemma 6.6. Let F be a p-adic field, ξ : G∗ → G an inner twist of G∗ = SO2n+1 over F, M ∗ ⊂ G∗ a standard Levi
+subgroup such that M := ξ(M ∗) is defined over F, and π ∈ Πscusp(M) an irreducible supercuspidal representation.
+Let ˙F, u, v2, ˙G∗, and ˙G be as in Lemma 6.3. Then there exist an inner twist ˙ξ : ˙G∗ → ˙G over ˙F, a standard Levi
+subgroup
+˙M ∗ ⊂ ˙G∗, and an irreducible cuspidal automorphic representation ˙π of
+˙M(A ˙F ) with following properties:
+• ˙ξu = ξ;
+•
+˙M := ˙ξ( ˙M ∗) is defined over ˙F, and ˙ξ| ˙
+M∗ : ˙M ∗ → ˙M is an inner twist;
+•
+˙Mu = M, and
+˙Mv is split over ˙Fv for any place v of ˙F except u and v2;
+• ˙πu = π.
+Proof. The former three properties follow from Lemmas 6.1 and 6.3. The last property follows from Lemma 6.4 and
+[31, Proposition 5.1].
+55
+
+6.2
+Globalizations of parameters
+Now we shall globalize generic parameters. Let us first consider globalizations of simple parameters.
+Lemma 6.7. Let ˙F be a totally real number field, and ˙G∗ a simple twisted endoscopic group of GLN, i.e., the split
+symplectic group, the split odd special orthogonal group, or a quasi-split even special orthogonal group over ˙F. Let V
+be a finite set of places of ˙F in which at least one real place is not contained. For each v ∈ V , let φv ∈ �Φ2,bdd( ˙G∗
+v) be
+a square integrable parameter. Assume that for at least one v ∈ V , φv ∈ �Φsim( ˙G∗
+v). Then there exists a simple generic
+parameter ˙φ ∈ �Φsim( ˙G∗) such that ˙φv = φv for all v ∈ V .
+Proof. The proof is similar to that of [18, Lemma 4.3.1], except that we need the following one or two modifications.
+The first one is to use Lemma 6.4 of this paper in place of Lemma 4.2.1 of [18]. Then the same proof carries over word
+by word in the case when ˙G∗ is a symplectic or an odd special orthogonal group. The second one is, in the case of even
+special orthogonal groups, to replace Φsim by �Φsim and the ordinary equivalence classes of the representations by the
+ǫ-equivalence classes of the representations. Here, note that in the case of quasi-split even special orthogonal groups,
+the endoscopic classification (LLC, ECR, and AMF) is known up to ǫ-equivalence, by Arthur [7] and Atobe-Gan
+[8].
+Lemma 6.8. Let N ≥ 2. Let ˙F be a totally real number field, and ˙G∗ a simple twisted endoscopic group of GLN, i.e.,
+the split symplectic group, the split odd special orthogonal group, or a quasi-split even special orthogonal group over ˙F.
+Let V and {φv}v∈V be as in Lemma 6.7. Assume that for at least one v ∈ V , φv ∈ �Φsim( ˙G∗
+v). Let v2 /∈ V be a finite
+place. Then there exists a simple generic parameter ˙φ ∈ �Φsim( ˙G∗) such that ˙φv = φv for all v ∈ V and ˙φv2 is of the
+form
+˙φv2 = φ+ ⊕ φ∨
++ ⊕ φ−,
+where φ+ ∈ Φbdd(GL1) is a non-self-dual parameter and φ− ∈ �Φsim,bdd(N − 2) a simple self-dual parameter.
+Proof. The proof is similar to that of [18, Lemma 4.3.2], except that we need the modifications similar to that of
+Lemma 6.7.
+Next we shall consider globalizations of elliptic or exceptional parameters. Let F be a local field, and G an inner
+form of G∗ = SO2n+1 over F. Let M ∗ ⊂ G∗ be a Levi subgroup. Following [18, §4.4], we shall say that M ∗ is linear
+if M ∗ is isomorphic to a direct product of general linear groups, i.e., n0 = 0 in (3.4). As in the case of even unitary
+groups, any nontrivial inner form G of G∗ does not admit a globalization ˙G which localizes to G at one place and is
+split at all the other places. Hence the same complication is caused.
+Let φ ∈ Φbdd(G∗) be a bounded L-parameter for G, which is the image of a discrete parameter φM∗ ∈ Φ2(M ∗) for
+M ∗. Assume that φ is either elliptic or exceptional. Explicitly M ∗, φ, and φM∗ has the form
+M ∗ ≃ GLe1
+N1 × · · · × GLer
+Nr ×M ∗
+−,
+φ =
+r
+�
+i=1
+ℓiφi =
+r
+�
+i=1
+(ei(φi ⊕ φ∨
+i ) ⊕ δiφi) ,
+φM∗ = e1φ1 ⊕ · · · ⊕ erφr ⊕ φ−,
+where φi ∈ �Φsim,bdd(Ni) are mutually distinct self-dual simple bounded L-parameters for GLNi, φ− = ⊕iδiφi ∈
+Φ2(M ∗
+−), δi ∈ {0, 1}, Ni ∈ Z≥1, ei ∈ Z≥0, ℓi = 2ei + δi, M ∗
+− = SO2n0+1, and 2n0 = �
+i δiNi so that �
+i eiNi + n0 = n
+and ei = ⌊ℓi/2⌋.
+In the lemmas and propositions below, we will start from the following local data with the assumption given above:
+• a local field F;
+• G∗ = SO2n+1 and a Levi subgroup M ∗ ⊂ G∗ over F;
+• an inner form G of G∗;
+• an L-parameter φ ∈ Φbdd(G∗) that is the image of a parameter φM∗ ∈ Φ2(M ∗).
+First we treat the case when M ∗ = G∗.
+56
+
+Lemma 6.9. Assume that M ∗ = G∗, and hence φ ∈ Φ2(G∗). Then there exists a global data ( ˙F, ˙G∗, ˙φ, u, v1, v2),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙φ ∈ Φ( ˙G∗), and u, v1, v2 are places of ˙F, such that v1 is
+a finite place, v2 is a real place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗, and ˙φu = φ;
+2. ˙φ ∈ Φ2( ˙G∗);
+3. ˙φv ∈ Φ2,bdd( ˙G∗
+v) for v ∈ {v1, v2};
+4. the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.
+Proof. A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.1. Here we take ˙F to
+have more than two real places. Let v1 and v2 be finite and real places of ˙F, respectively. Put ˙G∗ = SO2n+1 over ˙F.
+For each i = 1, . . . , r, let G∗
+i ∈ �Esim(Ni) be a classical group over F such that φi ∈ �Φsim,bdd(G∗
+i ), which is given
+in the 1st seed theorem 3.5. Take ˙G∗
+i ∈ �Esim(Ni) to be a simple twisted endoscopic group over ˙F so that ˙G∗
+i,u = G∗
+i .
+Choose a collection φv1,i ∈ �Φsim,bdd( ˙G∗
+i,v1) (i = 1, . . . , r) of pairwise distinct parameters. Choose also a collection
+φv2,i ∈ �Φ2,bdd( ˙G∗
+i,v2) (i = 1, . . . , r) of parameters such that φv2,i and φv2,j do not have a common constituent for all
+i ̸= j. Then Lemma 6.7 gives us a collection of parameters ˙φi ∈ �Φsim( ˙G∗
+i ) such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and
+˙φi,v2 = φv2,i.
+By the assumption, we have ei = 0 and δi = 1 for all i. Put
+˙φ = ˙φ1 ⊞ · · · ⊞ ˙φr ∈ Φ( ˙G∗).
+Then by the construction the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}. Thus the fourth and second
+conditions are satisfied. The first condition is clearly satisfied. The third one follows immediately from the construction.
+Proposition 6.10. Assume that M ∗ = G∗, and φ ∈ Φ2(G∗) is relevant for G. Then there exists a global data
+( ˙F, ˙G∗, ˙G, ˙φ, u, v1, v2),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ2( ˙G∗) a parameter relevant
+for ˙G, and u, v1, v2 are places of ˙F, such that v1 is a finite place, v2 is a real place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗, ˙Gu = G, and ˙φu = φ;
+2.
+˙Gv is split unless v ∈ {u, v2};
+3. ˙φv ∈ Φ2,bdd( ˙G∗
+v) for v ∈ {v1, v2};
+4. the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.
+Proof. By Lemma 6.9, we obtain ˙F, ˙G∗, ˙φ, u, v1, and v2 satisfying the third and fourth conditions. If G is not split
+over F, then by Lemma 6.3, we obtain ˙G that satisfies the second condition for which ˙φ is relevant. The first condition
+is now clear. If G splits over F, then ˙G = SO2n+1 clearly satisfies the conditions.
+Now we treat the case that M ∗ is proper, which implies that φ /∈ Φ2(G∗). At first we consider the case that M ∗
+is not linear excluding the special cases.
+Lemma 6.11. Assume that M ∗ ⊊ G∗ is not linear.
+Assume also that (n, φ) is neither (2, of type (exc1)) nor
+(3, of type (exc2)).
+Then there exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙
+M∗, u, v1, v2), where
+˙F is a totally real number
+field, ˙G∗ = SO2n+1 over ˙F,
+˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗), and u, v1, v2 are places of
+˙F, such that v1 and v2 are finite places and
+1.
+˙Fu = F, ˙G∗
+u = G∗,
+˙M ∗
+u = M ∗, ˙φu = φ, and ˙φ ˙
+M∗,u = φM∗;
+2. if φ ∈ Φ2
+ell(G∗) (resp. Φexc1(G∗), resp. Φexc2(G∗)), then ˙φ ∈ Φ2
+ell( ˙G∗) (resp. Φexc1( ˙G∗), resp. Φexc2( ˙G∗));
+57
+
+3. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ Φ2,bdd( ˙M ∗
+v1);
+4. ˙φv2 ∈ Φell,exc
+bdd
+( ˙G∗
+v2) and it has a symplectic simple component with odd multiplicity;
+5. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u, v1}.
+Proof. A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.1. Let v1 and v2 be finite
+places of ˙F. Put ˙G∗ = SO2n+1 and
+˙M ∗
+− = SO2n0+1 over ˙F. Let
+˙M ∗ ⊂ ˙G∗ be a Levi subgroup over ˙F such that
+˙M ∗ ≃ GLe1
+N1 × · · · × GLer
+Nr × ˙M ∗
+−.
+For each i = 1, . . . , r, let G∗
+i ∈ �Esim(Ni) over F and ˙G∗
+i ∈ �Esim(Ni) over ˙F be as in the proof of Lemma 6.9. Choose
+a collection φv1,i ∈ �Φsim,bdd( ˙G∗
+i,v1), (i = 1, . . . , r) of pairwise distinct parameters.
+Since M ∗ is not linear, we have n0 ≥ 1, which implies that δi = 1 (i.e., ℓi is odd) for some i. For such i, φi must
+be symplectic, in particular Ni ≥ 2.
+Suppose that there exists 1 ≤ s ≤ r such that δs = 1, Ns = 2, and Nt = 1 for all t ̸= s. In this case ℓs is odd and
+ℓt is even for t ̸= s. We may assume s = 1. If φ is elliptic, it can be written as
+φ = φ1 ⊕ 2φq+1 ⊕ · · · ⊕ 2φr,
+where φi are all symplectic, in particular Ni ≥ 2 for all i. This leads to r = 1 and φ = φ1, which contradicts the
+assumption that M ∗ is proper. If φ is of type (exc1), it can be written as
+φ = φ1 ⊕ 2φ2,
+where φ1 is symplectic, and φ2 is orthogonal. Then the dimension of φ equals 2 + 2 × 1 = 4, which contradicts the
+assumption that (n, φ) is not (2, of type (exc1)). If φ is of type (exc2), it can be written as
+φ = 3φ1,
+where φ1 is symplectic. Then the dimension of φ equals 3 × 2 = 6, which contradicts the assumption that (n, φ) is not
+(3, of type (exc2)).
+Therefore, we have that either there exists 1 ≤ s ≤ r such that Ns ≥ 3, or there exists 1 ≤ s ≤ r such that Ns = 2
+and δt = 1 for some t ̸= s. In both cases, by Lemma 6.8, we have ˙φs ∈ �Φsim( ˙G∗
+s) such that ˙φs,u = φs, ˙φs,v1 = φv1,s,
+and ˙φs,v2 is of the form
+˙φs,v2 = φv2+ ⊕ ˙φ∨
+v2+ ⊕ φv2−,
+where φv2+ ∈ Φbdd(GL1) is a non-self-dual parameter, and φv2− ∈ �Φbdd(Ns − 2) a simple self-dual parameter. For
+i ̸= s, choose φv2,i ∈ �Φsim,bdd( ˙G∗
+i ) so that {φv2,i}i̸=s are mutually distinct and φv2,i ̸= φv2− for all i. Lemma 6.7 gives
+us ˙φi ∈ �Φsim( ˙G∗
+i ) such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and ˙φi,v2 = φv2,i.
+Put
+˙φ = ℓ1 ˙φ1 ⊞ · · · ⊞ ℓr ˙φr ∈ Φ( ˙G∗),
+˙φ ˙
+M∗ = e1 ˙φ1 ⊞ · · · ⊞ er ˙φr ⊞ ˙φ− ∈ Φ2( ˙M ∗),
+where ˙φ− = ⊞iδi ˙φi. The first and fifth conditions immediately follow from the construction, and hence so do the
+second and third ones.
+Let us consider the fourth condition. We have
+˙φv2 = ℓsφv2− ⊕ ℓs(φv2+ ⊕ φ∨
+v2+) ⊕
+�
+i̸=s
+ℓiφv2,i,
+which is neither elliptic nor exceptional, because φv2+ is not self-dual. If Ns ≥ 3, then ˙φv2 has a symplectic simple
+component with odd multiplicity because φ has. If Ns = 2 and δt = 1 for t ̸= s, then clearly φv2,t is a symplectic
+simple component with odd multiplicity. This completes the proof.
+58
+
+Proposition 6.12. Assume that M ∗ ⊊ G∗ is not linear. Assume also that (n, φ) is neither (2, of type (exc1)) nor
+(3, of type (exc2)). Then there exists a global data
+( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u, v1, v2),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,
+˙M ∗ ⊂ ˙G∗
+a Levi subgroup, ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u, v1, v2 are places of ˙F, such that v1
+and v2 are finite place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗
+u, ˙Gu = G, ˙φu = φ,
+˙M ∗
+u = M ∗, and ˙φ ˙
+M∗,u = φM∗;
+2.
+˙Gv is split over ˙Fv unless v ∈ {u, v2};
+3. if φ ∈ Φ2
+ell(G∗) (resp. Φexc1(G∗), resp. Φexc2(G∗)), then ˙φ ∈ Φ2
+ell( ˙G∗) (resp. Φexc1( ˙G∗), resp. Φexc2( ˙G∗));
+4. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ φ2,bdd( ˙M ∗
+v1);
+5. ˙φv2 ∈ Φell,exc
+bdd
+( ˙G∗
+v2) is relevant for ˙Gv2;
+6. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u, v1}.
+Proof. By Lemma 6.11, we obtain ˙F, ˙G∗, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u, v1, and v2 satisfying the third, fourth, and sixth conditions.
+By Lemma 6.3, we obtain ˙G satisfying the second condition. Since v2 is a finite place, the fifth condition follows from
+the fourth condition of Lemma 6.11, regardless of ˙G. The first condition is now clear.
+Next we shall consider the special cases.
+Lemma 6.13. Assume that M ∗ ⊊ G∗ is not linear. Assume also that n = 2 or 3. Let ♥ = (exc1) (resp. (exc2)) if
+n = 2 (resp. 3), and assume that φ is of type ♥. Then there exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙
+M∗, u, v1, v2), where
+˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F,
+˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗),
+and u, v1, v2 are places of ˙F, such that v1 is a finite place, v2 is a real place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗,
+˙M ∗
+u = M ∗, ˙φu = φ, and ˙φ ˙
+M∗,u = φM∗;
+2. ˙φ ∈ Φ♥( ˙G∗);
+3. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ Φ2,bdd( ˙M ∗
+v1);
+4. ˙φv2 = 2ω0 ⊕ τ1 or 3τ1;
+5. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u, v1}.
+Proof. A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.1. Let v1 be a finite place
+and v2 a real place of ˙F. Put ˙G∗ = SO2n+1 over ˙F.
+Consider first the case when n = 2 and ♥ = (exc1). By the assumption we have ˙G∗ = SO5, φ = 2φ1 ⊕ φ2,
+Sφ ≃ Sp(2, C) × O(1, C), and M ∗ ≃ GL1 × SO3, where φ1 is 1-dimensional orthogonal and φ2 is irreducible 2-
+dimensional symplectic. We also have N1 = 1 and N2 = 2. For i = 1, 2, choose ( ˙G∗
+i , si, ηi) ∈ �E(Ni) so that φi ∈ Φ( ˙G∗
+i )
+if we regard Φ( ˙G∗
+i ) as a subset of Φ(Ni) via ηi. Concretely, ˙G∗
+1 = Sp0 and ˙G∗
+2 = SO3. Let φv1,i be an element of
+Φsim,bdd( ˙G∗
+i ), for i = 1, 2. Put φv2,1 = ω0 and φv2,2 = τ1. Then Lemma 6.7 gives us a global parameter ˙φi ∈ Φsim( ˙G∗
+i )
+such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and ˙φi,v2 = φv2,i. Put
+˙φ = 2 ˙φ1 ⊞ ˙φ2,
+˙M ∗ = GL1 × SO3,
+˙φ ˙
+M∗ = ˙φ1 ⊞ ˙φ2.
+Consider next the case when n = 3 and ♥ = (exc2). By the assumption we have ˙G∗ = SO7, φ = 3φ1, Sφ ≃ O(3, C),
+and M ∗ ≃ GL2 × SO3, where φ1 is irreducible 2-dimensional symplectic. Hence we regard φ1 ∈ Φ(SO3 /F). Let φv1,1 be
+59
+
+an element of Φsim,bdd(SO3 / ˙Fv1), and put φv2,1 = τ1. Then Lemma 6.7 gives us a global parameter ˙φ1 ∈ Φsim(SO3 / ˙F)
+such that ˙φ1,u = φ1, ˙φ1,v1 = φv1,1, and ˙φ1,v2 = φv2,1. Put
+˙φ = 3 ˙φ1,
+˙M ∗ = GL2 × SO3,
+˙φ ˙
+M∗ = ˙φ1 ⊞ ˙φ1.
+In both cases, the conditions follow from the construction. This completes the proof.
+Proposition 6.14. Assume that M ∗ ⊊ G∗ is not linear. Assume also that n = 2 or 3. Let ♥ = (exc1) (resp. (exc2))
+if n = 2 (resp. 3). Assume that φ ∈ Φbdd,♥(G∗) and that G is not quasi-split. Then there exists a global data
+( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u, v1, v2),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,
+˙M ∗ ⊂ ˙G∗
+a Levi subgroup, ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u, v1, v2 are places of ˙F, such that v1
+is a finite place, v2 is a real place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗
+u, ˙Gu = G, ˙φu = φ,
+˙M ∗
+u = M ∗, and ˙φ ˙
+M∗,u = φM∗;
+2.
+˙Gv is split over ˙Fv unless v ∈ {u, v2};
+3. ˙φ ∈ Φ♥( ˙G∗);
+4. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ φ2,bdd( ˙M ∗
+v1);
+5. ˙φv2 ∈ Φbdd,♥( ˙G∗
+v2) is relevant for ˙Gv2 and satisfies Theorem 4.13 relative to
+˙M ∗
+v2;
+6. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u, v1}.
+Proof. By Lemma 6.13, we obtain ˙F, ˙G∗, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u, v1, and v2 satisfying the third, fourth, and sixth conditions.
+By Lemma 6.3, we obtain ˙G such that ˙Gu = G, ˙Gv2 ≃ SO(n − 1, n + 2), and ˙Gv is split if v /∈ {u, v2}. Hence the
+second condition is satisfied. The fifth condition follows from §4.9. The first condition is now clear.
+If M ∗ is linear and G is non-quasi-split, then M ∗ never transfer to G and φ is not relevant. The next proposition
+will be applied in such a case.
+Lemma 6.15. Assume that M ∗ ⊊ G∗ is proper. There exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙
+M∗, u1, u2, v1), where ˙F is
+a totally real number field, ˙G∗ = SO2n+1 over ˙F,
+˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗), and
+u1, u2, v1 are places of ˙F, such that v1 is a finite place and
+1.
+˙Fu = F, ˙G∗
+u = G∗,
+˙M ∗
+u = M ∗, ˙φu = φ, and ˙φ ˙
+M∗,u = φM∗, for u ∈ {u1, u2};
+2. if φ ∈ Φ2(G∗) (resp. Φ2
+ell(G∗), resp. Φexc1(G∗), resp. Φexc2(G∗)), then ˙φ ∈ Φ2( ˙G∗) (resp. Φ2
+ell( ˙G∗), resp.
+Φexc1( ˙G∗), resp. Φexc2( ˙G∗));
+3. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ Φ2,bdd( ˙M ∗
+v1);
+4. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u1, u2, v1}.
+Proof. A totally real field ˙F and places u1 and u2 such that ˙Fu1 = ˙Fu2 = F are given by Lemma 6.2. Let v1 be a
+finite place of ˙F. Put ˙G∗ = SO2n+1 and
+˙M ∗
+− = SO2n0+1 over ˙F. Let
+˙M ∗ ⊂ ˙G∗ be a Levi subgroup over ˙F such that
+˙M ∗ ≃ GLe1
+N1 × · · · × GLer
+Nr × ˙M ∗
+−.
+For each i = 1, . . . , r, let G∗
+i ∈ �Esim(Ni) be a classical group over F such that φi ∈ �Φsim,bdd(G∗
+i ), and take
+˙G∗
+i ∈ �Esim(Ni) to be a simple twisted endoscopic group over ˙F so that
+˙G∗
+i,u = G∗
+i for u ∈ {u1, u2}.
+Choose a
+60
+
+collection φv1,i ∈ �Φsim,bdd( ˙G∗
+i,v1) (i = 1, . . . , r) of pairwise distinct parameters. Then Lemma 6.7 gives us a collection
+of parameters ˙φi ∈ �Φsim( ˙G∗
+i ) such that ˙φi,v1 = φv1,i and ˙φi,u = φi for u ∈ {u1, u2}.
+Put
+˙φ = ℓ1 ˙φ1 ⊞ · · · ⊞ ℓr ˙φr ∈ Φ( ˙G∗),
+˙φ ˙
+M∗ = e1 ˙φ1 ⊞ · · · ⊞ er ˙φr ⊞ ˙φ− ∈ Φ2( ˙M ∗),
+where ˙φ− = ⊞iδi ˙φi. Then by the construction, the first and fourth conditions are satisfied. Hence the second and
+third ones follow.
+Proposition 6.16. Assume that M ∗ ⊊ G∗ is proper. Then there exists a global data
+( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u1, u2, v1),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,
+˙M ∗ ⊂ ˙G∗
+a Levi subgroup, ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u1, u2, v1 are places of ˙F, such that v1
+is a finite place and
+1.
+˙Fu = F, ˙G∗
+u = G∗
+u, ˙Gu = G, ˙φu = φ,
+˙M ∗
+u = M ∗, and ˙φ ˙
+M∗,u = φM∗, for u ∈ {u1, u2};
+2.
+˙Gv is split over ˙Fv unless v ∈ {u1, u2};
+3. if φ ∈ Φ2(G∗) (resp. Φ2
+ell(G∗), resp. Φexc1(G∗), resp. Φexc2(G∗)), then ˙φ ∈ Φ2( ˙G∗) (resp. Φ2
+ell( ˙G∗), resp.
+Φexc1( ˙G∗), resp. Φexc2( ˙G∗));
+4. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ Φ2,bdd( ˙M ∗
+v1);
+5. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u1, u2, v1}.
+Proof. By Lemma 6.15, we obtain ˙F, ˙G∗, ˙φ,
+˙M ∗, ˙φ ˙
+M∗ u1, u2, and v1 satisfying the first (except the assertion on ˙Gu),
+third, fourth, and fifth conditions. By Lemma 6.3, we obtain ˙G such that ˙Gv = G for v ∈ {u1, u2} and ˙Gv splits over
+˙Fv otherwise. This completes the proof.
+6.3
+On elliptic parameters
+The following lemma is proved in the same way as [18, Lemma 4.5.1].
+Lemma 6.17. Let ˙F be a number field, ˙G∗ = SO2n+1 over ˙F, ˙φ ∈ Φ2
+ell( ˙G∗) a parameter, and ˙G an inner form of ˙G∗.
+Let
+˙M ∗ ⊂ ˙G∗ be a Levi subgroup and ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗) a discrete parameter whose image in Φ( ˙G∗) is ˙φ. Assume that
+there is a place v1 of ˙F such that
+•
+˙Gv1 is split over ˙F;
+• ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗
+v1 ∈ φ2,bdd( ˙M ∗
+v1);
+• the canonical maps S ˙φ → S ˙φv1 and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v1 are isomorphic.
+Then we have
+tr R
+˙G
+disc, ˙φ( ˙f) =
+�
+x∈S ˙φ,ell
+Ä ˙f ′
+˙G( ˙φ, x) − ˙f ˙G( ˙φ, x)
+ä
+= 0,
+for all ˙f ∈ H( ˙G).
+61
+
+6.4
+Proof of LIR for L-parameters
+In this subsection we complete the proof of Theorem 4.13 for generic parameters (i.e., L-parameters). Let F be a
+local field, G∗ = SO2n+1 over F, (M ∗, P ∗) a standard parabolic pair of G∗, and ξ : G∗ → G an inner twist of G∗. If
+M ∗ transfers to G, then we take ξ so that M = ξ(M ∗) is defined over F. Let φM∗ ∈ Φbdd(M ∗) be a generic parameter
+for M, and φ ∈ Φbdd(G∗) its image. If G is split, the theorem is already proven by Arthur [7]. Therefore, we may
+assume that G is non-quasi-split, so in particular F ̸= C.
+Lemma 6.18. Assume that M ∗ ⊊ G∗ is proper, i.e., φ is not discrete. Assume also that φM∗ ∈ Φ2,bdd(M ∗) is
+discrete and that φ is elliptic or exceptional. Then for any x ∈ Sφ,ell, there exists a lift x ∈ Sφ of x such that
+f ′
+G(φ, x−1) = e(G)fG(φ, x),
+for any f ∈ H(G).
+Proof. The proof is similar to that of [18, Case N even of Lemma 4.6.1]. The difference is that we divide the case into
+the following cases:
+• M ∗ is not linear and (n, φ) is neither (2, of type (exc1)) nor (3, of type (exc2));
+• M ∗ is not linear and (n, φ) is either (2, of type (exc1)) or (3, of type (exc2));
+• M ∗ is linear,
+instead of the division into the cases
+• M ∗ is not linear and N ̸= 4;
+• M ∗ is not linear and N = 4;
+• M ∗ is linear,
+in loc. cit., and that we appeal to Lemmas 6.17, 5.13, 5.10, and Propositions 6.12, 6.14 6.16, instead of Lemmas 4.5.1,
+3.7.1, 3.5.10, and Propositions 4.4.4, 4.4.6, 4.4.7 of loc. cit., respectively.
+Lemma 6.19. The assertions 2 and 3 of Theorem 4.13 hold for generic parameters.
+Proof. Note that the assertion 3 implies the assertion 2. Thanks to §4.7 and §4.8, we may assume that φM∗ is discrete
+and that φ ∈ Φ2
+ell(G∗) ⊔ Φexc(G∗). By the consequence of §4.8, if φ is elliptic, then it remains to show that
+f ′
+G(φ, s−1) = e(G)fG(φ, u♮) for any u♮ ∈ Nφ and s ∈ Sφ,ss mapping to the same element in Sφ,ell,
+and if φ is exceptional, then it remains to show that
+f ′
+G(φ, s−1) = e(G)fG(φ, u♮) for any u♮ ∈ Nφ,reg and s ∈ Sφ,ss mapping to the same element in Sφ,ell.
+They can be proved similarly to [18, Lemma 4.6.2]. The difference is that we utilize Lemmas 4.20, 6.18, and 4.14
+instead of Lemmas 2.8.6, 4.6.1, and 2.6.5 of loc. cit. respectively. Note that in our case the equivalence classes of
+inner forms, inner twists, and pure inner twists are in bijection naturally.
+This completes the proof of the second and third assertions of Theorem 4.13 for generic parameters. The next two
+lemmas show the first assertion. The first lemma treats the case of φM∗ ∈ Φ2,bdd(M ∗):
+Lemma 6.20. Assume that φM∗ ∈ Φ2,bdd(M ∗). Then the assertion 1 of Theorem 4.13 holds for φM∗.
+Proof. By Lemmas 4.21 and 4.19, we may assume that φ ∈ Φexc(G∗). The proof is similar to that of [18, Case N
+even of Lemma 4.6.3]. The difference is that we appeal to Lemmas 6.16, 5.13, 6.20, and 6.19 instead of Lemmas 4.4.7,
+3.7.1, 4.6.3, and 4.6.2 of loc. cit. respectively.
+Recall that in §4.7 we reduce only part 2 and 3 of Theorem 4.13 to the case of discrete parameters. The next
+lemma is the reduction of part 1, and hence it completes the proof of part 1 for all generic parameters. As the lemmas
+above, in a similar way to the equation (4.6.3), Lemma 4.6.4, and Lemma 4.6.5 of loc. cit., one can obtain a surjection
+Rφ(M, G) = Wφ(M, G)/W ◦
+φ(M, G) ։ RπM (M, G) := WπM (M, G)/W ◦
+πM (M, G),
+(6.1)
+for πM ∈ ΠφM∗(M), and can show the following lemmas.
+62
+
+Lemma 6.21. Let φM∗ ∈ Φbdd(M ∗). Then the assertion 1 of Theorem 4.13 holds for φM∗.
+Lemma 6.22. The homomorphism (6.1) is bijective if φM∗ is relevant.
+Now Theorem 4.13 holds for any generic parameters. This completes the proof of LIR for generic parameters.
+6.5
+The construction of L-packets
+Theorem 3.11 for generic parameters (i.e., L-parameters) can be proven in the same way as Theorem 1.6.1 of [18].
+Let us roughly review the procedure. See [18, §§4.7-4.9] for detail.
+Let F be a local field, G∗ = SO2n+1 over F, and (ξ, z) : G∗ → G a pure inner twist of G∗ as in §4.4. If G is split,
+the theorem is already proven by Arthur [7]. Therefore, we assume that G is non-quasi-split, so in particular F ̸= C.
+In the archimedean case, the theorem is known by Langlands and Shelstad, so we assume that F is a p-adic field.
+First consider the non-discrete parameters. Since we now have Theorem 4.13 for generic parameters φ, the local
+packets Πφ(G) and the map Πφ(G) → Irr(Sφ, χG) for φ ∈ Φbdd(G∗) \ Φ2(G∗) have already constructed at the
+end of §4.6. The remaining assertions are that the map is bijective and that the packets are disjoint and exhaust
+Πtemp(G) \ Π2(G).
+The general classification [6] (cf. [7, §3.5], [18, §4.7]) of Πtemp(G) by harmonic analysis says that there is a bijective
+correspondence
+(M, σ, µ) �→ πµ ∈ Πtemp(G),
+from the G(F)-orbits of triples consisting of a Levi subgroup M ⊂ G over F, a discrete series representation σ
+of M(F), and an irreducible representation µ of the representation-theoretic R-group R(σ) = Rσ(M, G).
+Here,
+we do not need an extension �R(σ) of the R-group, as explained in [7, p.158] or [18, §4.7].
+The correspondence
+can be described explicitly as follows. For a triple (M, σ, µ), let P be a parabolic subgroup of G with Levi factor
+M. Choose a family {RP (r, σ)}r∈Rσ(M,G) ⊂ AutG(F )(IG
+P (σ)) of self-intertwining operators so that an assignment
+r �→ RP (r, σ) is a homomorphism from Rσ(M, G) to AutG(F )(IG
+P (σ)). Then we have a representation RP (r, σ)◦IG
+P (σ, g)
+of Rσ(M, G)×G(F) on HG
+P (σ), for which we shall write R. The representation πµ is characterized by a decomposition
+R =
+�
+µ∈Irr(Rσ(M,G))
+µ∨ ⊗ πµ.
+Note that Rσ(M, G) is abelian in our case.
+Then the bijectivity of the map Πφ(G) → Irr(Sφ, χG) and the disjointness and exhaustion of L-packets in Πtemp(G)
+follows from the same argument as in [18, §4.7], which we omit.
+Proposition 6.23. Theorem 3.11 holds for generic parameters φ ∈ Πbdd(G∗) \ Π2(G∗).
+Proof. The proof is similar to that of [18, Proposition 4.7.1]. Note that Πφ(G, Ξ), χΞ, S♮
+φ, N ♮
+φ, and S♮♮
+φ (M) in loc. cit.
+should be replaced by Πφ(G), χG, Sφ, Nφ, and Sφ(M) in our case, respectively.
+Next, before consider discrete parameters, we need some preparation. Put Tell(G) to be the set of G(F)-conjugacy
+classes of triples τ = (M, σ, r), where M ⊂ G is a Levi subgroup, σ a unitary discrete series representation of M(F),
+and r ∈ Rσ(M, G) a regular element. Here, r ∈ Rσ(M, G) is said to be regular if ar=1
+M
+:= {λ ∈ aM | rλ = λ} coincides
+with aG. The set Π2,temp(G) is naturally regarded as a subset of Tell(G) via π �→ (G, π, 1). The complement set will
+be denoted by T 2
+ell(G). In general, the trace Paley-Wiener theorem tells that ”orbital integrals” of cuspidal functions
+f ∈ Hcusp(G) are described by the functions on Tell(G) given by some intertwining operators.
+In order to utilize the local intertwining operator defined in §4.5 , we need to consider the set T ♮
+ell(G). Let T 2,♮
+ell (G) be
+the set of G(F)-conjugacy classes of triples τ ♮ = (M, σ, s), where M ⊊ G is a proper Levi subgroup, σ a unitary discrete
+series representation of M(F), and s ∈ Sφσ(G) an element whose image under the surjection Sφσ(G) ։ Rσ(M, G)
+is regular. Here, we write φσ for the L-parameter of σ, and we also write φσ for its image in Φbdd(G∗). Then put
+T ♮
+ell(G) := Π2,temp(G)⊔T 2,♮
+ell (G). The surjection Sφσ(G) ։ Rσ(M, G) induces the natural surjection T ♮
+ell(G) ։ Tell(G).
+For f ∈ H(G), put
+fG(τ ♮) =
+®
+tr �RP (u♮, σ, φσ, ψF ) ◦ IG
+P (σ, f)� ,
+for τ ♮ = (M, σ, s) ∈ T 2,♮
+ell (G),
+tr (π(f)) ,
+for τ ♮ = π ∈ Π2,temp(G),
+63
+
+where u♮ ∈ Nφσ(M, G) is a lift of s. It is independent of the choice of u♮. If τ ♮
+1 and τ ♮
+2 in T 2,♮
+ell (G) maps to a same
+element in Tell(G), then they are of the form τ ♮
+1 = (M, σ, s) and τ ♮
+2 = (M, σ, ys) for some y ∈ Sφσ(M). Then we shall
+write τ ♮
+2 = yτ ♮
+1. In this case, by Lemma 4.10 we have
+fG(τ ♮
+1) = ⟨y, σ⟩fG(τ ♮
+2).
+Let φ ∈ Φ2,bdd(G∗) be a discrete generic parameter, and x ∈ Sφ. Then by Lemma 3.8, one can construct an
+endoscopic triple e = (Ge, se, ηe) of G and a generic parameter φe of Ge, to obtain a linear form H(G) ∋ f �→
+f ′
+G(φ, x) := f e(φe). The trace Paley-Wiener theorem implies that there exists a family {cφ,x(τ ♮)}τ ♮∈T ♮
+ell(G) ⊂ C of
+complex numbers such that for any f �� Hcusp(G), we have
+f ′
+G(φ, x) = e(G)
+�
+τ∈Tell(G)
+cφ,x(τ ♮)fG(τ ♮),
+(6.2)
+and
+cφ,x(τ ♮) = ⟨y, τ⟩cφ,x(yτ ♮),
+where τ ♮ is a lift of τ, and ⟨y, τ⟩ = ⟨y, σ⟩ for τ = (M, σ, s) ∈ T 2
+ell(G). Note that the product cφ,x(τ ♮)fG(τ ♮) does not
+depend on the choice of τ ♮. Then one can show an orthogonality relation:
+Proposition 6.24. (a) Let φ ∈ Φ2,bdd(G∗) be a discrete generic parameter, and x ∈ Sφ. Then cφ,x(τ ♮) = 0 for all
+τ ♮ ∈ T 2,♮
+ell (G).
+(b) Let φ1, φ2 ∈ Φ2,bdd(G∗) be discrete generic parameters, and xi ∈ Sφi (i = 1, 2). Then
+�
+π∈Π2,temp(G)
+cφ1,x1(π)cφ2,x2(π) =
+®
+|Sφ1|,
+if φ1 = φ2 and x1 = x2,
+0,
+otherwise.
+Proof. The proof is similar to that of [18, Proposition 4.8.3] or [7, Lemma 6.5.3].
+Finally we shall consider discrete generic parameters, after introducing one lemma.
+Lemma 6.25. Let ξ : ˙G∗ → ˙G be an inner twist of ˙G∗ = SO2n+1 over a number field ˙F, and ˙φ ∈ Φ2( ˙G∗) a discrete
+global parameter. For any ˙f ∈ H( ˙G), we have
+�
+˙π
+n ˙φ( ˙π) ˙f ˙G( ˙π) =
+1
+|S ˙φ|
+�
+x∈S ˙φ
+˙f ′
+˙G( ˙φ, x),
+(6.3)
+where ˙π runs over irreducible representations of ˙G(A ˙F ), and n ˙φ( ˙π) it the multiplicity of ˙π in (R ˙G
+disc, ˙φ, L2
+disc, ˙φ( ˙G( ˙F)\ ˙G(A ˙F ))).
+Proof. The proof is similar to that of [18, Lemma 4.9.1] or [7, (6.6.6)].
+Let φ ∈ Φ2,bdd(G∗). Then Proposition 6.10 and Lemma 6.3 gives us a global data ( ˙F, ˙G∗, ˙G, ˙φ, u, v1, v2) such that
+˙F is totally real, and
+• v1 is a finite place, and v2 is a real place;
+•
+˙Fu = F, ˙G∗
+u = G∗, ˙Gu = G, and ˙φu = φ;
+•
+˙Gv is split for v /∈ {u, v2}, and χG = χ ˙Gv2;
+• ˙φv ∈ Φ2,bdd( ˙G∗
+v) for v ∈ {v1, v2};
+• the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.
+64
+
+Choose ˙f ∈ H( ˙G) so that ˙f is of the form ˙f = �
+v ˙fv, and that ˙fu ∈ Hcusp(G). Then we have the equation (6.3)
+due to Lemma 6.25. For v /∈ {u, v2}, the group ˙Gv is split and hence we have ECR (3.7). In the case v = v2, since
+˙Fv2 = R, ECR is known by the work of Shelstad. At the place u, we have the equation (6.2) since ˙fu is cuspidal.
+Substituting them, we obtain the following equation from (6.3):
+�
+˙π∈Πunit( ˙G(A ˙
+F ))
+n ˙φ( ˙π) ˙f ˙G( ˙π) =
+1
+|S ˙φ|
+�
+˙πu∈Π ˙φu
+�
+π∈Π2,temp(G)
+�
+x∈S ˙φ
+⟨ ˙xu, ˙πu⟩cφ, ˙xu(π) ˙f ˙G( ˙πu ⊗ π),
+where Πunit( ˙G(A ˙F )) denotes the set of isomorphism classes of irreducible unitary representations of ˙G(A ˙F ), ˙x ∈ S ˙φ
+is a lift of x ∈ S ˙φ, ˙xv ∈ S ˙φv is the image of ˙x, and
+Π ˙φu :=
+
+
+ ˙πu =
+�
+v̸=u
+˙πv
+������
+˙πv ∈ Π ˙φv, ⟨−, ˙πv⟩ = 1 for almost all v
+
+
+ ,
+˙xu := ( ˙xv)v̸=u ∈
+�
+v̸=u
+S ˙φv,
+⟨ ˙xu, ˙πu⟩ :=
+�
+v̸=u
+⟨ ˙xv, ˙πv⟩.
+For each place v other than u, v1 and v2, fix ˙πv ∈ Π ˙φv such that ⟨−, ˙πv⟩ is trivial. For a real place v2, fix ˙πv2 ∈ Π ˙φv2
+arbitrarily. They exist because ˙Gv splits if v /∈ {u, v1, v2} and ˙Fv2 = R.
+For any µ ∈ Irr(Sφ, χG), choose ˙πv1 ∈ Π ˙φv1 so that µ( ˙xu)⟨ ˙xu, ˙πu⟩ = 1 for all ˙x ∈ S ˙φ, where ˙πu = �
+v̸=u ˙πv. Such
+˙πv1 does exist because we have χG = χ ˙Gv2, v1 is a finite place, ˙Gv1 is split, and S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.
+Then for π ∈ Π2,temp(G), we set
+nφ(µ, π) := n ˙φ ( ˙πu ⊗ π) .
+Then by the same argument of the proof of [18, Proposition 4.9.2], we have
+nφ(µ, π) =
+1
+|Sφ|
+�
+x∈Sφ
+µ(x)−1cφ,x(π),
+(6.4)
+for any π ∈ Π2,temp(G), where x ∈ Sφ denotes for an arbitrary representative of x. Moreover, the orthogonality
+relation (Proposition 6.24) and the formula (6.4) implies the following formula:
+Proposition 6.26. Let φ, φ′ ∈ Φ2,bdd(G∗) be discrete generic parameters, and µ ∈ Irr(Sφ, χG) and µ′ ∈ Irr(Sφ′, χG).
+Then we have
+�
+π∈Π2,temp(G)
+nφ(µ, π)nφ′(µ′, π) =
+®
+1,
+if (φ, µ) = (φ′, µ′),
+0,
+otherwise.
+Proof. The proof is the similar to that of [18, Proposition 4.9.3].
+Since nφ(µ, π) is a non-negative integer, Proposition 6.26 tells us that for any φ ∈ Φ2,bdd(G∗) and µ ∈ Irr(Sφ, χG),
+there exists a unique π ∈ Π2,temp(G) such that nφ(µ, π) = 1. We shall write π(φ, µ) for this representation π. Then
+an assignment (φ, µ) �→ π(φ, µ) gives an injective map. The packet of φ ∈ Φ2,bdd(G∗) is defined by
+Πφ := { π(φ, µ) | µ ∈ Irr(S, χG) } ,
+and the associated character of the component group is defined by
+⟨−, π(φ, µ)⟩ := µ.
+One can now easily see that the packets are disjoint and the map πφ → Irr(Sφ, χG) is bijective.
+Let φ ∈ Φ2,bdd(G∗) and x ∈ Sφ. Combining the formula (6.4) with Proposition 6.26, we have
+cφ,x(π) =
+®µ(x) = ⟨x, π⟩,
+if π = π(φ, µ),
+0,
+otherwise.
+Then the formula (6.2) is none other than the endoscopic character relation for cuspidal function f ∈ Hcusp(G).
+65
+
+Proposition 6.27. For general f ∈ H(G), we have ECR:
+f ′(φ, x) = e(G)
+�
+π∈Πφ
+⟨x, π⟩fG(π).
+Proof. The proof is similar to that of [7, Corollary 6.7.4].
+Proposition 6.27 characterizes the packet Πφ and the bijection Πφ → Irr(Sφ, χG). Thus they does not depend on
+the choice of ˙φ or ˙πu, but only on φ. By the argument between Theorems 3.11 and 3.12, we now have the packet Πφ
+and the bijection Πφ → Irr(Sφ, χG) for all generic parameters φ ∈ Φ(G∗).
+It remains to show that the packets exhaust Π2,temp(G). Let π ∈ Π2,temp(G). Take a globalization ˙π of π as in
+Lemma 6.4. By the decomposition (5.1), we have the A-parameter ˙φ of ˙π. In particular, we have n ˙φ( ˙π) ̸= 0. Since
+˙πv′ has sufficiently regular infinitesimal character for some real place v′, the parameter ˙φ is generic. Thus Theorem
+4.13 and Hypothesis 7.1 hold for ˙φ, so does Theorem 7.2 for ˙φ. (Note that §7 is independent from the exhaustion.)
+Hence we have π ∈ Π ˙φu. If ˙φu is not bounded (resp. not discrete), then the packet does not contain a tempered
+representation π by definition in §3.5 (resp. the beginning of this subsection). Therefore, we have ˙φu ∈ Φ2,bdd(G∗).
+This completes the proof of the local classification theorem for generic parameters.
+7
+The proof of the global theorem
+Assuming the existence of local A-packets and LIR, Theorem 3.13 can be proven in the same way as Theorem
+1.7.1 of [18]. In particular, since the assumptions hold for generic parameters, generic part of the theorem holds true.
+In this section we record the statements. See [18, §5] for the proof.
+Let F be a number field, G∗ = SO2n+1 over F, and G an inner form of G∗. If G is split, the theorem is already
+proven by Arthur [7]. Therefore, we assume that G is non-quasi-split. Recall from §5.1 (5.1) the decomposition
+L2
+disc(G(F)\G(AF )) =
+�
+ψ∈Ψ(G∗)
+L2
+disc,ψ(G(F)\G(AF )),
+tr Rdisc(f) =
+�
+ψ∈Ψ(G∗)
+Rdisc,ψ(f),
+f ∈ H(G).
+Let us introduce a hypothesis on local A-packets:
+Hypothesis 7.1. Let ψ ∈ Ψ(G∗) be a global parameter for G. We have the local packet Πψv = Πψv(Gv) and the map
+Πψv → Irr(Sψv, χGv) satisfying ECR (3.7) for all places v of F.
+Theorem 7.2. Let ψ ∈ Ψ(G∗) be a global parameter for G. Assume that Theorem 4.13 and Hypothesis 7.1 hold for
+ψ.
+1. If ψ /∈ Ψ2(G∗), then
+L2
+disc,ψ(G(F)\G(AF )) = 0.
+2. If ψ ∈ Ψ2(G∗), then
+L2
+disc,ψ(G(F)\G(AF )) =
+�
+π∈Πψ(G,εψ)
+π.
+Proof. The proof is similar to that of [18, Theorem 5.0.5].
+Since Theorem 4.13 and Hypothesis 7.1 hold true if ψ = φ ∈ Φ(G∗) is generic, the global classification theorem is
+now established for the generic part.
+66
+
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