The local Langlands conjecture for GSp(4) III: Stability and twisted endoscopy

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The local Langlands conjecture for GSp(4) III: Stability and twisted endoscopy Ping-Shun Chan a , Wee Teck Gan b,∗ a Department of Mathematics, Room 220 Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong b Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076, Singapore

a r t i c l e

i n f o

Article history: Received 11 June 2013 Received in revised form 22 July 2013 Accepted 26 July 2013 Available online xxxx Communicated by James Cogdell, Herve Jacquet, Dihua Jiang, and Steve Kudla

a b s t r a c t In this paper, we complete the analysis of the local Langlands correspondence for GSp(4) and its inner form over a nonarchimedean local field of characteristic 0, which was shown in two earlier papers of the second author with S. Takeda and W. Tantono. We show that the L-packets satisfy the local character identities postulated in the theory of (twisted) endoscopy. © 2013 Elsevier Inc. All rights reserved.

In fond memory of Steve Rallis Keywords: Local Langlands conjecture Stability Endoscopy Character identities

1. Introduction In [GT] and [GTW], the local Langlands correspondence for GSp(4) and its inner form GU2 (D) over a p-adic field F was established. More precisely, suppose that H = GSp(4) and H  = GU2 (D) and let Π(H) (respectively Π(H  )) denote the set of equivalence classes of irreducible smooth representations of H(F ) (respectively H  (F )). On the other * Corresponding author. E-mail addresses: [email protected] (P.-S. Chan), [email protected] (W.T. Gan). 0022-314X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jnt.2013.07.009

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hand, let Φ(H) denote the set of L-parameters of H, consisting of admissible homomorphisms  φ : WD F → H  = GSp (C) is the complex dual group where WD F is the Weil–Deligne group of F and H 4 of H and H  . Then there is a natural surjective map   L : Π(H)  Π H  → Φ(H)

with finite fibers (which are the so-called Vogan L-packets). For each φ ∈ Φ(H), the associated Vogan L-packet   Πφ = Πφ (H)  Πφ H 

with Πφ (H) ⊂ Π(H) (resp. Πφ (H  ) ⊂ Π(H  )) is in natural bijection with the set of irreducible characters of the component group   Bφ = π0 ZH sc (φ) ,  sc = Sp (C) is the derived group of H.  In addition, the map L was shown to where H 4 satisfy certain expected properties. For example, it preserves certain natural invariants (L-functions, -factors and Plancherel measures of pairs) which one can attach to both sides. Furthermore, it turns out that L is characterized by these properties. However, the papers [GT] and [GTW] did not address a very important aspect of the local Langlands correspondence, namely the relation of the map L with the theory of (twisted) endoscopy. Indeed, the partition of Π(H) (or Π(H  )) into L-packets is supposed to be governed by the character theory of H(F ) and the fact that L-packets may not be singletons is a reflection of the difference between the relations of conjugacy and stable conjugacy in H(F ). More precisely, the theory of (twisted) endoscopy dictates that the sum of the character distributions of the elements in a given L-packet Πφ (H) is supposed to be a stable distribution on H(F ) and there should be character identities relating an L-packet of H to associated L-packets on endoscopic groups of H and on G = GL(4) × GL(1) (which possesses H as a twisted endoscopic group). Unfortunately, the approach used in [GT] and [GTW] to construct the map L relies on the method of theta correspondence, which does not seem to be well-suited for handling such questions. The purpose of this paper is to complete the analysis of the local Langlands correspondence for H = GSp(4) and its inner form H  by addressing the issues of stability of L-packets and (twisted) endoscopic transfer. To state the precise result, we need to introduce some notations and briefly recall the theory of (twisted) endoscopy. The group GSp(4) has a unique proper elliptic endoscopic group      C = GSO(2, 2) ∼ = GL(2) × GL(2) / z, z −1 : t ∈ Gm

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with dual group  = GSpin (C) ⊂ H  = GSpin (C) ∼ C = GSp4 (C). 4 5  As a 4-dimensional An L-parameter φ ∈ Φ(H) is endoscopic if it factors through C. representation of the Weil–Deligne group WD F , an endoscopic φ has the form

φ = φ1 ⊕ φ2 where φ1 and φ2 are 2-dimensional and det φ1 = det φ2 . The associated L-packet Πφ (H) has size 2 if and only if φ1 and φ2 are both irreducible (but may be equivalent), in which case one has   Πφ (H) = πφ+ , πφ− ,

with πφ+ generic and πφ− non-generic.

When at least one of the φi ’s is reducible, the endoscopic L-packet Πφ (H) has size 1, but we shall still denote it as above, with the understanding that πφ− = 0. Similarly, when φ is not endoscopic, then its associated L-packet Πφ (H) is a singleton and we denote it by   Πφ (H) = πφ+ .  it determines an Since an endoscopic L-parameter φ = φ1 ⊕ φ2 factors through C,  L-packet on C (up to the outer automorphism action of C which switches the two copies of GL(2)). Indeed, if τ1 and τ2 are the irreducible representations of GL2 (F ) associated to φ1 and φ2 , then the L-packet of C determined by φ consists simply of the representation τ1  τ2 of C(F ). Now the theory of endoscopy gives rise to a map (called the transfer)   TransH C : stably invariant distributions on C(F )   → invariant distributions on H(F ) .

Since every invariant distribution on C(F ) is stable, we may consider the transfer of (the character of) any admissible representation of C(F ), and in particular of τ1  τ2 . On the other hand, H = GSp(4) is itself a twisted endoscopic group of G = GL(4) × GL(1) with respect to the following involution σ on G:   σ : (g, e) → J t g −1 J −1 , e det(g) ,

for all (g, e) ∈ G,

where ⎛

1

⎜ J := ⎜ ⎝

1 −1 −1

⎞ ⎟ ⎟. ⎠

(1.1)

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In particular, there is a natural embedding  = GSp (C) → G  = GL4 (C) × GL1 (C) H 4

given by   g → g, sim(g) ,  as the subgroup of G  fixed by: which realizes H   σ ˆ : (g, e) → eJ t g −1 J −1 , e ,

ˆ ∀(g, e) ∈ G.

The theory of twisted endoscopy furnishes a map   TransG H : stably invariant distributions on H(F )   → σ-invariant distributions on G(F ) .

Given any L-parameter φ of H, we may regard the L-parameter φ as an L-parameter (φ, sim φ) for G, via the natural embedding of dual groups above. We write φG for the corresponding L-parameter for G, which corresponds to a representation Π  μ of GL4 × GL1 . This representation turns out to be σ-invariant and so can be extended to the semi-direct product G(F )  σ (in two possible ways, to be made precise later). The character distribution of the extended representation, which we will still denote by Π  μ, is a σ-invariant distribution when restricted to the non-identity coset G(F ) · σ. With this preparation, we can now state our main theorem for H = GSp(4). Main Theorem. (i) Let φ = φ1 ⊕φ2 be a generic endoscopic L-parameter of H = GSp(4) with associated L-packet Πφ (H) = {πφ+ , πφ− }. Let τ1  τ2 be the representation of the endoscopic group C(F ) determined by φ. Then + − TransH C (τ1  τ2 ) = πφ − πφ .

(ii) In the setting of (i), πφ+ + πφ− is a stable distribution on H(F ) and  +  − TransG = Π  μ. H π +π

(iii) Let φ be a non-endoscopic generic L-parameter of H with associated L-packet Πφ (H) = {πφ+ }. Then πφ+ is a stable distribution on H(F ) and  + TransG H πφ = Π  μ.

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Here, by a generic L-parameter φ, we mean one such that the adjoint L-function L(s, Ad ◦ φ) is holomorphic at s = 1; by [GT, Main Theorem (vii)], this is equivalent to saying that the L-packet Πφ (H) contains a generic representation. There is an analogous theorem for the inner form H  = GU2 (D), whose statement is slightly more subtle because of issues with normalization of the transfer factor. Indeed, figuring how the precise character identities for the inner form case is easily the more interesting part of this paper. We refer the reader to Section 11 for the result. Not surprisingly, the proof of our main theorem relies on the stable trace formula of Arthur for GSp(4) and a simple stable twisted trace formula for (GL(4) × GL(1), σ) due to Kottwitz–Shelstad [KoS]. These stable trace formulas, by themselves, are designed to lead to a description of the automorphic discrete spectrum of GSp(4), obtaining in the process the partition of Π(GSp(4)) into L-packets with the desired stability and transfer properties (as announced by Arthur in [A3], and partially shown by Flicker [F2] and Weissauer [We]). For the classical groups (symplectic, orthogonal and unitary groups), this program has recently been carried out by Arthur [A7] and Mok [M] assuming the full stabilization of the twisted trace formula. As the local Langlands correspondence was established in [GT] and [GTW] via the method of theta correspondence, our task is essentially to relate the two approaches. Note however that we do not assume that the analog of [A7] (as announced in [A3]) has been carried out for GSp(4), nor do we assume the full stabilization of the twisted trace formula. Since our goals are local, we can make do with the simple stable twisted trace formula of [KoS], and as such our results are unconditional. Indeed, the fact that the theta correspondence is capable of constructing the local L-packets and also large parts of the automorphic discrete spectrum leads to much simplification in the application of the trace formula. For example, part (i) of the main theorem was obtained years ago by Weissauer [We], using the trace formula, but though our treatment here is along the lines of [We], it is more streamlined because we make use of much more input from the theory of theta correspondence. Indeed, as we recall in Section 3, the endoscopic part of the automorphic discrete spectrum of H and its inner forms, including the Arthur multiplicity formula, can be completely obtained by theta correspondence. 2. The local Langlands correspondence In this section, we briefly recall the local Langlands correspondence for H = GSp(4) and its inner form H  = GU2 (D) over a p-adic field F . Here, D is the unique quaternion division algebra over F . 2.1. The case of GSp(4) In the case of H = GSp(4), the construction of the map L : Π(H) → Φ(H) is based on the local theta correspondence associated to the following three dual pairs

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GSO(3, 3) GSp(4) GSO(4, 0)

GSO(2, 2)

and exploits the accidental isomorphisms:  −1   ⎧ × ∼ ⎪ ⎨ GSO(2, 2) = (GL2 × GL2 )/ z, z  : z ∈ F  , GSO(4, 0) ∼ = D× × D× / z, z −1 : z ∈ F × , ⎪    ⎩ GSO(3, 3) ∼ = (GL4 × GL1 )/ z, z −2 : z ∈ F × .

In particular, the local Langlands correspondence is known for each of these 3 groups. The proof in [GT] then consists of: (a) showing that each element of Π(GSp4 ) participates in theta correspondence with exactly one of GSO(4, 0) or GSO(3, 3). Moreover, those representations which participate in theta correspondence with GSO(3, 3) can further be divided into two disjoint subsets, depending on whether they participate in theta correspondence with GSO(2, 2) or not; (b) characterizing the representations Π  μ of GL4 × GL1 with ωΠ = μ2 (so that Π  μ is a representation of GSO(3, 3)) which participates in theta correspondence with GSp4 as those such that the L-parameter φΠ : WD F → GL4 (C) of Π factors through GSp4 (C) with similitude character μ. From the above, one can describe the L-packets Πφ (H) concretely. Recall that the unique proper elliptic endoscopic group of H is      C = GSO(2, 2) ∼ = GL(2) × GL(2) / t, t−1 : t ∈ Gm

with dual group ι  = GSpin (C) −→  = GSpin (C) ∼ ι:C H = GSp4 (C). 4 5

An L-parameter φ ∈ Φ(H) is endoscopic if it factors through ι, in which case it is of the form φ = φ1 ⊕ φ2

with det φ1 = det φ2 .

The L-packet Πφ can then be described as follows. If τi is the irreducible representation of GL2 with L-parameter φi , then τ1  τ2 is an irreducible representation of C and if

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both τi are discrete series representations (i.e. when both φi are irreducible), then one may consider its Jacquet–Langlands lift τ1D  τ2D on CD = GSO(4, 0). Setting 

πφ+ = θ(τ1  τ2 ) = θ(τ2  τ1 ) ∈ Π(H),     πφ− = θ τ1D  τ2D = θ τ2D  τ1D ∈ Π(H) (if it exists)

we have   Πφ (H) = πφ+ , πφ− ,

where π − is interpreted as zero if some φi is reducible. This describes the endoscopic L-packets of H. On the other hand, H itself is a twisted endoscopic group of G = GL(4) × GL(1) with respect to the involution σ on G described in (1.1). One has an embedding of dual groups  = GSp (C) → G  = GL4 (C) × GL1 (C) H 4

  via g → g, sim(g)

 as the subgroup fixed by the induced action of σ on G.  Given φ ∈ Φ(H), which realizes H we may regard (φ, sim φ) as an L-parameter of G via the above inclusion. Let Π  μ be the corresponding irreducible representation of G(F ) and set     Πφ (H) = πφ+ = θ(Π  μ)

if φ is not endoscopic. 2.2. Inner forms For the case of H  = GU2 (D) = GSp(1, 1), the definition of L is obtained by considering the inner form versions of the above theta correspondences, namely theta correspondences associated to quaternionic hermitian groups. Indeed, given a Hermitian space V over D and a skew-Hermitian space W over D, one has a theta correspondence between GU(V ) = GSp(m, n) (an inner form of a symplectic similitude group) and GU(W ) = GO∗ (p, q) (an inner form of an orthogonal similitude group). More precisely, we shall consider the theta correspondence associated to the two dual pairs: GO∗ (3, 0)

GSp(1, 1)

GO∗ (1, 1)

and exploit the accidental isomorphisms: 

     GO∗ (1, 1) ∼ = D× × GL2 / z, z −1 : z ∈ F × ,      GO∗ (3, 0) ∼ = D4× × GL1 / z, z −2 : z ∈ F × ,

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where D4× denotes the multiplicative group of a degree 4 division algebra over F . There are two such division algebras, but they are opposite of each other and hence have canonically isomorphic multiplicative groups. The proof in [GTW] then consists of showing the analogs of statements (a) and (b) in the split case: (a ) each element π of Π(H  ) participates in theta correspondence with exactly one of GO∗ (1, 1) or GO∗ (3, 0); (b ) a representation Π  μ of D4× × F × participates in theta correspondence with GU2 (D) if and only if the L-parameter of Π factors through GSp4 (C) with similitude character μ. Given these, we may give the description of the resulting L-packets as follows. Suppose  that φ ∈ Φ(H  ) is an endoscopic L-parameter, so that φ = φ1 ⊕ φ2 factors through C.  Because φ is relevant for H [B], we must have φ1 = φ2 and at least one of them is irreducible. Let τi be the representation of GL2 with L-parameter φi and let τiD be its Jacquet–Langlands lift to D× (if it exists). Then we have the representations τ1D  τ2 and τ2D  τ1 of D× × GL2 (or rather of GO∗ (1, 1)). If we set 

    πφ+− = θ τ2D  τ1 ∈ Π H  ,     πφ−+ = θ τ1D  τ2 ∈ Π H 

(if they exist; at least one of them does), then we have     Πφ H  = πφ+− , πφ−+ .

This describes the endoscopic L-packets of H  . On the other hand, if φ is not an endoscopic L-parameter, then as in the split case, we may regard φ as an L-parameter of D4× × GL1 . If Π  μ denotes the corresponding representation of D4× × GL1 then we set       Πφ H  = πφ− = θ(Π  μ) .

2.3. Vogan packets For each L-parameter φ, we may consider the Vogan packet Πφ (H)  Πφ (H  ), and the above parametrization can be subsumed in a single statement. Namely, if we set   Bφ = π0 ZSp4 (C) (φ)

(a finite group),

φ denote its set of irreducible characters, then the description of the map L and let B gives a bijection   φ . Πφ (H)  Πφ H  ↔ B

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For example, if φ = φ1 ⊕ φ2 is endoscopic with φi irreducible, then Bφ = Z/2Z × Z/2Z, and if we denote the two irreducible characters of Z/2Z by ±, then the 4 representations in the Vogan packet are πφ++ = πφ+ ,

πφ−− = πφ− ,

πφ+− ,

and πφ−+ .

2.4. Archimedean case Since we will be using global arguments later on, it is necessary to have a good understanding of the archimedean places, especially the case when F = R. In this case, it is necessary to work with the notion of “K-groups” (see [Sh3] and [W5]). What this means concretely in our case is the following. While H is still GSp(4), the inner form H  should be taken as the “disjoint union” of the two inner forms GSp(1, 1) and GSp(2, 0) (where the latter has compact derived group), in the sense that       Π H  = Π GSp(1, 1)  Π GSp(2, 0) .

Thus, in this case, for an L-parameter φ, the L-packet Πφ (H  ) consists of irreducible representations of both GSp(1, 1) and GSp(2, 0). Such an interpretation is in fact necessary in order to have the same description of Vogan packets as in the non-archimedean case. We illustrate this for a discrete series L-parameter. When F = R, an L-parameter φ of H is necessarily endoscopic, so that φ = φ1 ⊕φ2 . We suppose that φi corresponds to a discrete series representation τi of GL2 (R) with extremal weights ±ki , in which case we shall say that τi has weight ki . For this L-parameter to be relevant for H  , we need k1 = k2 . Then Bφ ∼ = Z/2Z × Z/2Z as before, and the Vogan packet should have 4 irreducible representations π ±,± . These can be described as follows. We have the two representations of H(R) = GSp4 (R) given by:   π ++ = θ(τ1  τ2 ) and π −− = θ τ1D  τ2D .

On the other hand, if we consider the representations τ1D  τ2 and τ2D  τ1 of the inner form D× × GL2 (R), one finds that under local theta correspondence, these representations has nonzero theta lift to exactly one of GSp(1, 1) or GSp(2, 0), and moreover, they have nonzero theta lift to different groups. Thus, if we consider the nonzero representations     π −+ = θ τ1D  τ2 and π +− = θ τ2D  τ1 ,

then one of these is a representation of GSp(1, 1) and the other is a representation of GSp(2, 0). Indeed, if k1 > k2 , then π −+ is a representation of GSp(1, 1). These assertions can be checked from the results of [LPTZ].

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3. Global endoscopic packets via theta correspondence In this section, let k be a number field with adèle ring A. Let B be a quaternion k-algebra (possibly split). Let SB denote the set of places of k where B is ramified. Let V be a 2-dimensional Hermitian B-module, so that its associated similitude group HV = GU(V ) is an inner form of the split group H = GSp(4). If Z denotes the center of HV , fix a unitary character χ of Z(k)\Z(A) and consider the right regular representation L2χ (HV (k)\HV (A)) of HV (A). We would like to describe the generic part of the endoscopic discrete spectrum L2HV ,χ,endo of L2χ,disc (HV (k)\HV (A)) using theta correspondence. 3.1. Endoscopic discrete spectrum We define precisely what we mean by the generic part of the endoscopic discrete spectrum L2HV ,χ,endo . If τ1  τ2 is a cuspidal representation of the endoscopic group C(A) of HV (A), then outside a finite set S of places of k, τ1,v  τ2,v is unramified and thus one obtains a collection of semisimple conjugacy classes  v∈ {sv ∈ C: / S}

which are the Satake parameters of the unramified local components. The inclusion  → H  V gives a collection {ι(sv ): v ∈ V / S} of semisimple conjugacy classes in H ι : C and this data specifies a near equivalence class in L2HV ,χ,disc . This near equivalence class  L2HV ,τ1 τ2 consists of all irreducible summands π = v πv of L2HV ,χ,disc such that for almost all v, πv has Satake parameter ι(sv ). The generic part of the endoscopic discrete spectrum L2HV ,χ,endo is by definition 

L2HV ,χ,endo =

L2HV ,τ1 τ2

τ1 τ2

where the direct sum runs over all cuspidal representations τ1  τ2 of C(A) with τ1 = τ2 and ωτi = χ. It will be convenient to treat all the inner forms HV of H simultaneously, so we set L2χ,endo =



L2HV ,χ,endo

V

and L2τ1 τ2 =



L2HV ,τ1 τ2 ,

V

where the sum runs over all 2-dimensional Hermitian B-modules, as B varies over all quaternion k-algebra.

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Let us fix the cuspidal representation τ1  τ2 of C(A) and describe the construction and structure of L2HV ,τ1 τ2 . For each place v, the representation τ1,v  τ2,v determines an endoscopic local L-parameter φv = φ1,v ⊕ φ2,v of Hv and an associated local Vogan  V,v , then packet Πφv . When τ1,v  τ2,v is unramified with Satake parameter sv ∈ H Πφv is a singleton set containing the unramified representation of HV (kv ) with Satake parameter ι(sv ). 3.2. Skew-Hermitian spaces Let W be a 2-dimensional skew-Hermitian B-module with trivial discriminant and let CW = GU (W ) denote its similitude group. Then there are two quaternion k-algebras B1 and B2 (possibly split) such that SB = (SB ∩ SB1 )  (S ∩ SB2 ) and SB1  SB = SB2  SB and      CW ∼ = B1× × B2× / z, z −1 : z ∈ Gm .

In particular, CW is an inner form of C, and the cuspidal representation τ1  τ2 of C(A) determines a cuspidal representation (possibly zero) τ1B1  τ2B2 of CW (A) via the Jacquet–Langlands transfer. 3.3. Global theta correspondence Now one may consider the global theta correspondence for CW × HV . Assuming that τ1B1  τ2B2 = 0, let us set   πV,W = θV,W τ1B1  τ2B2 to be the global theta lift of τ1B1  τ2B2 to HV . Then it follows by [GI] and [Y] that πV,W is either zero or is a nonzero irreducible cuspidal representation of HV (A) with πV,W ⊂ B B L2HV ,τ1 τ2 . Moreover, πV,W is nonzero if and only if all local theta lifts θV,W (τ1,v1,v τ2,v2,v ) are nonzero for every place v. This question of local theta correspondence is completely understood, as we discussed in the previous section. Conversely, suppose that σ is a cuspidal representation in L2HV ,τ1 τ2 , with V a quaternionic Hermitian space defined over B. Then its partial degree 5 standard L-function LS (s, σ, std) factors as: LS (s, σ, std) = ζ S (s) · LS (s, τ1 × τ2 ) and thus has a pole at s = 1. By [KR1] and [Y], this implies that σ has nonzero theta lift to CW for some 2-dimensional skew-Hermitian B-module W with trivial discriminant. Together with the above results, one sees that

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L2HV ,τ1 τ2 =



πV,W ,

W

as W ranges over all 2-dimensional skew-Hermitian B-module with trivial discriminant and   πV,W = θV,W τ1B1  τ2B2 (possibly zero). We summarize the above discussion in the following theorem: Theorem 3.1. The generic part of the endoscopic discrete spectrum of H and its inner forms is given by:  L2χ,endo = L2τ1 τ2 τ1 τ2

where the direct sum runs over all cuspidal representations of C(A) (up to the action of the outer automorphism group of C) with τ1 = τ2 and ωτi = χ. For a fixed cuspidal representation τ1  τ2 of C(A) as above, L2τ1 τ2 can be described as follows. Set    v  A(τ1  τ2 ) = π = πφv : v ∈ Bφv with v trivial for almost all v . v

Then 

L2τ1 τ2 =

m(π) · π,

π∈A(τ1 τ2 )

with



m(π) =

Δ

∗



  v , 1

,

Bφ

v

where Δ : Bφ →



Bφv

v

is the natural diagonal map and Bφ = Z/2Z × Z/2Z is the global component group. This describes the generic part of the endoscopic discrete spectrum of H and its inner forms in the framework of Arthur’s conjecture. 4. Transfers In this section, we discuss the precise definition of the transfer maps in the theory of (twisted) endoscopy, as given in [LS] and [KoS]. See [Sh1,Sh4] and [W5] for a more recent account.

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4.1. Local transfer factors We return to the setting of the local field F . Let G be a connected reductive group over F equipped with an automorphism σ (possibly trivial). When F = R, we shall work with the notion of “K-groups” (cf. [Sh3] and [W5]), though we will not explicitly say so in order to simplify the exposition. Let H be an elliptic σ-twisted endoscopic group of G (so H is quasi-split by definition). There is a norm correspondence on the set of stable conjugacy classes of H(F )reg × G(F )σ-reg , where the superscript reg denotes the subset of strongly regular elements. Then [LS] and [KoS] defines a function ΔG,H : H(F )reg × G(F )σ-reg → C which is called a transfer factor. It satisfies: • ΔG,H is a function of stable conjugacy classes of H(F )reg and of σ-conjugacy classes of G(F )reg ; • ΔG,H (s, t) = 0 unless s ∈ H(F ) is a norm of t ∈ G(F ); • If ΔG,H (s, t) = 0, then |ΔG,H (s, t)| = 1. Note that the transfer factor used here does not include the term typically denoted by ΔIV which is the ratio of the square roots of the Weyl discriminant of G and H. The function ΔG,H is however only well-defined up to scaling by S 1 and we need to discuss a precise normalization for our purposes. The following cases will be relevant for us. 4.2. Quasi-split case When G is quasi-split, [KoS] has defined the notion of a σ-stable Whittaker datum (U, ψ), where • U is the unipotent radical of an F -rational σ-stable Borel subgroup B = T · U of G; • ψ is a generic character of U such that ψ σ = ψ (here ψ σ (u) = ψ(σ(u))). Such σ-stable Whittaker datum always exists for a quasi-split group G, and given such a Whittaker datum (U, ψ), [KoS] gives a normalization of ΔG,H depending on the orbit of ψ under the action of T. We shall call such a transfer factor Whittakernormalized. Let us specialize to the cases of interest in this paper. When (G, H) = (H, C), the group H is split and all generic characters of the unipotent radical of a Borel subgroup are in the same torus orbit. Thus, we have a Whittaker-normalized transfer factor ΔH,C which does not depend on any choices. Similarly, when (G, H) = (G, H), one has a Whittaker-normalized transfer factor ΔG,H which does not depend on any choices.

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4.3. Inner forms case When G is not quasi-split, one may normalize the transfer factor ΔG,H in certain favorable cases as follows. To every σ-stable Levi subgroup M of G, there is a corresponding Levi subgroup MH of H such that MH is an elliptic σ-twisted endoscopic group of M. Thus, one may speak of the transfer factor ΔM,MH , which is again well-defined up to scaling. Now there is a notion of the two transfer factors ΔG,H and ΔM,MH being compatibly normalized [Sh1, Section 14 and Lemma 14.1], namely ΔM,MH (s, t) = ΔG,H (s, t) for (s, t) ∈ MH (F )reg × M(F )σ-reg with s a norm of t. Thus, to fix a normalization of ΔG,H , it suffices to pick a normalization of ΔM,MH , and then insist that the two are compatibly normalized. A favorable case arises when it turns out that MH is simply the quasi-split inner form of M. In this case, the transfer factor is a constant function where it is nonzero, and the canonical normalization of this transfer factor is to set ΔM,MH (s, t) = e(M) when s is a norm of t, where e(M) is the Kottwitz sign [Ko]. As another example, in the quasi-split cases discussed in the previous subsection, the choice of a Whittaker datum on G induces one on M, and if we use the Whittaker-normalized transfer factors for (G, H) and (M, MH ), then the two transfer factors are compatibly normalized. Specializing to the case when (G, H) = (H  , H), we see that if M  ∼ = D× × GL1 is  the unique (up to conjugation) proper Levi subgroup of H , then the corresponding Levi subgroup M of H is simply its split inner form GL2 × GL1 . Thus, we are in the favorable situation mentioned above, and we normalize ΔH  ,H so that ΔH  ,H (s, t) = −1

if s ∈ M (F ) is a norm of t ∈ M  (F ).

On the other hand, when (G, H) = (H  , C), there are two Levi subgroups M1 = GL2 × GL1 and M2 = GL1 × GL2 of C corresponding to M  = D× × GL1 . The two Levi subgroups M1 and M2 are conjugate under the outer automorphism σC of C which switches the two factors of GL2 ’s in C. Since Mi is the split inner form of M  , we may again normalize ΔM  ,Mi to be −1 where it is nonzero. However, as we explain below, the two possible normalizations here are not the same.

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4.4. Outer automorphism There is a notion of the outer automorphism group OutG (H) of an endoscopic datum. When G is quasi-split, the transfer factor ΔG,H is invariant under OutG (H). However, when G is not quasi-split, it was noted by Arthur [A5] that the transfer factor may transform by a nontrivial character χG,H of OutG (H). In the case of interest, where (G, H) = (H, C), the group OutG (H) has order 2 and the nontrivial element is the outer automorphism of C which switches the two factors of GL(2). Thus, the transfer factor ΔH,C satisfies:     ΔH,C (s1 , s2 ), t = ΔH,C (s2 , s1 ), t .

On the other hand, when (G, H) = (H  , C), one has     ΔH  ,C (s1 , s2 ), t = −ΔH  ,C (s2 , s1 ), t .

Thus, the outer automorphism group exchanges the two normalizations which are compatible with ΔM  ,M1 and ΔM  ,M2 respectively. Hence, we have only normalized ΔH  ,C up to ±. This ambiguity makes the determination of the precise local character identities in the inner form case somewhat delicate. 4.5. Global transfer factor If k is a number field with ring of adeles Ak , and (G, H) is a (twisted) endoscopic pair over k, there is a canonical global transfer factor ΔG,H . For the cases of interest in this paper, we note: • if G is quasi-split, we may fix a σ-stable global Whittaker datum (U, ψ), so that ψ is a generic automorphic character of U(Ak ) (trivial on U(k)). Then we inherit a σ-stable Whittaker datum (Uv , ψv ) for each place v of k, and thus have the local Whittaker-normalized transfer factor ΔGv ,Hv . It then follows that ΔG,H (s, t) =



ΔGv ,Hv (sv , tv )

(4.1)

v

for s ∈ H(Ak ) and t ∈ G(Ak ); • when G is an inner form of GSp(4) and H = GSp(4), the product formula (4.1) holds because the Kottwitz sign satisfies an analogous product formula [Ko]; • when G is an inner form of GSp(4) and H = C, our local transfer factors have an ambiguity of ± at those places v where Gv is non-split. However, since the number of such places v is even, the ambiguity disappears in the product on the RHS of (4.1), so that (4.1) continues to hold.

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4.6. Transfer of class functions Going back to the local situation (over F ), the local transfer factor Δ = ΔG,H serves as a kernel function for the transfer of stable-conjugacy-invariant functions on H(F ) to σ-conjugacy-invariant functions on G(F ) and vice versa. More precisely, let Γσ-reg (G) denote the space of σ-regular σ-conjugacy classes of G and let ΓG-reg (H) denote the space of strongly G-regular conjugacy classes of H. Given any function Φ on Γσ-reg (G), let ΦH = TransH G Φ denote the function on ΓG-reg (H) defined by: ΦH (h) =



Δ(h, g)Φ(g),

∀h ∈ ΓG-reg (H).

g∈Γσ-reg (G)

Note that we view Δ here as a function on ΓG-reg (H) × Γσ-reg (G). We call ΦH the transfer of Φ to H. We shall be particularly interested in those Φ which arise as orbital integrals of functions in the Hecke algebra Cc∞ (G(F ), χ). 4.7. Orbital integrals For a test function f ∈ Cc∞ (G(F ), χ), we may consider its (normalized, twisted) orbital integral. For t ∈ G(F )reg , let Gtσ denote its σ-twisted centralizer in G. Then we set    Oσ (t, f ) = ΔG (tσ) · f g −1 tg σ dg, Gtσ (F )\G(F )

where   1/2  ΔG (tσ) = det 1 − Ad(tσ) Lie(G)/ Lie(Gtσ ) 

is the square root of the Weyl discriminant. Thus, Oσ (−, f ) is a σ-class function on G(F ). As in [A2], we set    I(G) = Oσ (−, f ): f ∈ Cc∞ G(F ), χ .

We shall frequently suppress σ from the notation Oσ , especially when σ is the trivial automorphism. We define the σ-twisted stable orbital integral of f to be SO(t, f ) =



O(ti , f )

i

where {ti } is a set of representatives of the σ-conjugacy classes in the stable σ-conjugacy classes of t. Thus, SO(−, f ) is a σ-stable class function on G(F ). As in [A2], we set    SI(G) = SO(−, f ): f ∈ Cc∞ G(F ), χ .

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4.8. Elliptic inner product One may define an inner product −, −G,σ on I(G). We will only do so on the subspace Ie (G) of elements supported on the elliptic set (this subspace is denoted by Icusp (G) in [A2]). For a torus T in G with the property that σT = T, let:   T1−σ := tσ(t)−1 : t ∈ T .

Let W σ (T(F )) = (NG(F ) (T(F ))/T(F ))σ denote the group of σ-fixed elements in the Weyl group of T(F ) in G(F ). For two functions Φ, Ψ ∈ Ie (G), set

Φ, Ψ G,σ,e :=

 T



1 σ |W (T(F ))|

Φ(t)Ψ (t) dt, ZG

T1−σ \T

where the sum is over representatives of the σ-conjugacy classes of the σ-elliptic maximal tori in G. We call Φ, Ψ G,σ,e the σ-elliptic inner product of Φ and Ψ . If σ is the identity automorphism, we put Φ, Ψ G,e := Φ, Ψ G,σ,e . Indeed, one may define an extension of this inner product to the whole space I(G); we refer to [A2] for the details. Similarly, we may define an elliptic inner product on the subspace SI e (G) of SI(G) consisting of elements supported on the elliptic set, as follows. For Φ, Ψ ∈ SI e (G), set

Φ, Ψ G,σ,e =

 T

1 σ |W (T)(F )|

 ZG T1−σ \T

1 · Φ(t)Ψ (t) dt, nσ (t)

where the sum runs over stable σ-conjugacy classes of σ-elliptic maximal tori T,  σ W σ (T)(F ) = NG (T)/T (F ),

and nσ (t) is the number of σ-conjugacy classes in the stable σ-conjugacy class of t. As before, this inner product can be extended to the whole space SI(G). The two inner products are related in the following way. If f ∈ Cc∞ (G(F ), χ) is such that its orbital integral Oσ (f ) is already stable, then for any f  ∈ Cc∞ (G(F ), χ), one has: 

  Oσ (f ), Oσ f 

G,σ,e

   = SO σ (f ), SO σ f 

G,σ,e

.

(4.2)

Though we have used the same notation for the inner products on two different spaces, it should be clear from the context which one is meant. 4.9. Transfer of orbital integrals Given f ∈ Cc∞ (G(F ), χ), we define a function on the σ-twisted elliptic endoscopic group H(F ) by:

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OG,H (s, f ) :=



ΔG,H (s, t) · Oσ (t, f )

t

with the sum running over a set of representatives of the σ-conjugacy classes in G(F ). Then OG,H (−, f ) is a so-called κ-orbital integral of f and it defines a stable class function on H(F ). The following big theorem is the culmination of the fundamental work of Ngo [N], Waldspurger [W1,W2,W3,W5] and Arthur [A2,A5] (among many others). Theorem 4.3. (i) (Transfer) Given f ∈ Cc∞ (G(F ), χ), there exists fH ∈ Cc∞ (H, χ) such that SO(−, fH ) = OG,H (−, f ) as functions on H(F )G-reg . (ii) (Fundamental Lemma) If G is an unramified group and f = f0 is the characteristic function of a σ-stable hyperspecial maximal compact subgroup of G(F ), then one may take fH = fH,0 to be the characteristic function of a hyperspecial maximal compact subgroup of H(F ). More generally, there is a natural algebra homomorphism η : HG → HH of spherical Hecke algebras such that for any f ∈ HG , one may take fH = η(f ) ∈ HH . (iii) (Isomorphism) Let {Hi } be the set of σ-twisted elliptic endoscopic groups of G. The ! map O(−, f ) → i SO G,Hi (−, fHi ) defines a topological isomorphism Ie (G) →



SI e (Hi )χG,Hi ,

i

where χG,Hi is a character of OutG (Hi ) noted in Section 4.4, and SI e (Hi )χG,Hi is the χG,Hi -eigenspace of SI e (Hi ). (iv) (Isometry) The isomorphism in (iii) is an isometry where Ie (G) is given the inner ! χG,Hi product introduced in Section 4.8, and the inner product on is a i SI e (Hi ) weighted sum of those introduced in Section 4.8. For the cases of interest in this paper, it is given by:

Φ, Ψ G,σ,e =

 i

for Φ = (Φi ) and Ψ = (Ψi ) ∈

! i

1 · Φi , Ψi Hi ,e , |OutG (Hi )|

SI e (Hi )χG,Hi

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4.10. Transfer of distributions In view of the above theorem, we have a transfer map   TransG H : stably invariant distributions on H(F )   → σ-invariant distributions on G(F )

defined by: TransG H (D)(f ) = D(fH ). Here we shall frequently regard a σ-invariant distribution on G(F ) as a linear form on I(G) and a stable distribution as a linear form on SI(G). 5. Character distributions We shall be interested in the twisted character distribution associated to σ-invariant representations of G(F ). In this section, we introduce some basic results about these. We shall assume henceforth that σ has order 2. 5.1. Extensions Given any representation π of G(F ), we let π σ denote the representation of G(F ) realized on the same space as π but with the twisted action   π σ (g) = π σ(g) .

If π is a σ-invariant irreducible representation of G(F ), then HomG(F ) (π, π σ ) is a 1-dimensional space. For any A ∈ HomG(F ) (π, π σ ), A2 defines an element of EndG(F ) (π) = C. Hence we may normalize A so that A2 = 1. This determines A up ˜ = G  σ, to ±1 and defines an extension of the representation π to extended group G so that π(σ) = A. For our purposes, we shall need to normalize the choice of A precisely, and we shall see how to achieve this in some favorable situations. 5.2. Whittaker datum Given a σ-stable Whittaker datum (U, ψ), suppose that π is generic with respect to (U, ψ). For any  ∈ HomU(F ) (π, ψ), we have:  ◦ A ∈ HomU(F ) (π, ψ), so that we may normalize A by requiring that  ◦ A = . This is called the Whittaker normalization of A and we have a corresponding Whittaker-normalized extension of π ˜ to G.

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5.3. Spherical representations Suppose that G is an unramified group and K ⊂ G(F ) is a hyperspecial maximal compact subgroup such that σ(K) = K. If π is an irreducible σ-invariant representation of G(F ) such that π K = 0 (so dim π K = 1), then (π σ )K = π K = 0. For any nonzero v ∈ π K , we can normalize A ∈ HomG (π, π σ ) by requiring that A(v) = v. We shall call this the spherical normalization of A. Observe that if π is both K-spherical and (U, ψ)-generic, then a nonzero Whittaker functional l on π does not vanish on a nonzero K-spherical vector v. This implies that the Whittaker normalization and spherical normalization of π are the same. 5.4. Parabolically induced representations Suppose now that P = M · N is a parabolic subgroup of G such that σ(P) = P. Then σ acts naturally on the Levi factor M. Suppose further that ρ is an irreducible σ-invariant representation of M(F ) and we have fixed an Aρ ∈ HomM (ρ, ρσ ), thus obtaining an extension of ρ to M(F )  σ. Let π be the normalized induced representation:   1/2 IPG (ρ) := φ ∈ C ∞ (G, ρ): φ(mng) = δN (m)φ(g), m ∈ M, n ∈ N, g ∈ G ,

where (π(g)φ)(h) := φ(hg), and:    δN (m) := det Ad(m)|Lie N .

Then π is σ-invariant and inherits a natural extension to G  σ: this natural extension Gσ is simply IndPσ ρ. More concretely, the operator Aπ on π is given by:    Aπ (f )(g) = Aρ f σ −1 (g) .

We now observe: • a σ-stable Whittaker datum (U, ψ) on G induces a σ-stable (UM , ψM ) on M. If ρ is generic with respect to (UM , ψM ), then π is generic with respect to (U, ψ). If we choose Aρ to be Whittaker-normalized, then the operator Aπ defined above is easily checked to be Whittaker-normalized with respect to (U, ψ); • suppose that G is unramified with σ-stable hyperspecial maximal compact subgroup K such that K ∩ M(F ) is a hyperspecial maximal compact subgroup of M(F ). If we choose Aρ to be spherical-normalized, then it is easily checked that Aπ as defined above is also spherical-normalized; • suppose that ρ is a 1-dimensional character of M(F ) which is σ-invariant, then ρ has a canonical extension Aρ given by Aρ (σ) = 1. In this case, we shall always use this canonical extension for Aρ . In particular, if P is a Borel subgroup, so that M = T,

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then the resulting normalization on π is Whittaker-normalized with respect to any σ-stable Whittaker datum. 5.5. Global extension Assume now that k is a number field with ring of adeles Ak , and the pair (G, σ) is defined over k. Then one obtains an action of σ on the space of automorphic forms of G by:   σ(f )(g) = f σ −1 (g) . This action preserves the subspace of cusp forms. Suppose that Π is a cuspidal representation of G(A). Then the map f → σ(f ) intertwines Π σ with the right translation action of G(A) on the space {σ(f ): f ∈ Π}. In particular, if Π is σ-invariant and occurs with multiplicity one in the cuspidal spectrum of G, then the action of σ preserves Π, and the map f → σ(f ) defines an extension of the G(A)-action on Π to G(A)  σ. In this paper, we shall consider only the case when G = GL4 × GL1 (or rather a quotient of this by a split torus), so that the above multiplicity one condition is satisfied. Suppose further that (U, ψ) is a σ-invariant global Whittaker datum and Π is globally generic with respect to (U, ψ), so that the Whittaker functional lψ : Π → C associated to the global Whittaker–Fourier coefficient is nonzero. One can easily see that   lψ σ(f ) = lψ (f ).

Thus, if we denote by σv the action of σ on Πv which is Whittaker-normalized with respect to (Uv , ψv ), then we have the local-global compatibility  σ(f ) = σv (fv ) v

if f =

 v

fv .

5.6. Twisted characters We now return to the local setting, with F a local field. If π is an admissible representation of G(F ) with central character ω = ωπ which has an extension to G  σ, then one has the linear form on Cc∞ (G, ωπ−1 ) defined by   Trσ (π) : f → Tr π(σ)π(f ) . Then this is a σ-conjugacy-invariant distribution on G(F ). One knows that this distribution can be represented by a locally integrable function on G(F ) which is σ-conjugacyinvariant and which is smooth on G(F )σ-reg . We set

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Θπ,σ (t) = ΔG (tσ) · Trσ (π)(t)

for t ∈ G(F )reg ,

and call Θπ,σ the twisted character of π. Thus, in the favorable situations discussed above, one has this twisted character for any σ-invariant π, since we have a natural normalization of the extension of π to G(F )  σ. 5.7. The Weyl integration formula We recall here a basic tool in invariant harmonic analysis. There is a decomposition of Gσ-reg as follows: " " σ-reg " Gσ-reg = σ(g)−1 tg , (5.1) T t∈T/T1−σ g∈Tσ \G

where the first union is over representatives of the σ-conjugacy classes of the σ-invariant maximal tori T of G. For f ∈ Cc∞ (G, 1), the following identity is the σ-twisted version of the Weyl integration formula: #  $     −1  1 2 f (g) dg = Δ (tσ) f g tσ(g) dg dt, G |W σ (T(F ))| ZG \G

T

Tσ \G

ZG T1−σ \T

where the sum is over representatives of the σ-conjugacy classes of the σ-invariant maximal tori of G. Using the Weyl integration formula, we see that Θπ,σ (f ) = Tr π(σ)π(f ) =

 T



1 σ |W (T(F ))|

Θπ,σ (t) · Oσ (f, t) dt. ZG

(5.2)

T1−σ \T

5.8. Example of principal series We illustrate the above discussion with the example of principal series representations. Thus we shall work in the setting of Section 5.4. Let K be a maximal compact subgroup of G(F ) such that we have the Iwasawa decomposition G(F ) = N(F ) · M(F ) · K. For h ∈ G(F ), f ∈ Cc (G, ω −1 ) and φ ∈ π, we have:    %  & Aπ π(f )φ (h) = Aρ f (g)π(g)φ σ −1 (h) dg ZG \G



= ZG \G

%  & f (g)Aρ φ σ −1 (h)g dg

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      f σ −1 h−1 g Aρ φ(g) dg

= ZG \G



23





=

    f σ −1 h−1 nmk · δN (m)1/2

N ZG \M K

   · Aρ ρ(m) φ(k) · δN−1 (m) dk dm dn.

Hence, Aπ ◦ π(f ) may be viewed as an integral operator on a space of functions on K. Its trace is obtained by integrating over K the trace of its kernel:  



Θπ,σ (f ) =

      f σ −1 k−1 nmk δN (m)−1/2 · tr Aρ ρ(m) dm dn dk.

K N ZG \M

Applying the change of variable n → n , such that nm = σ −1 (n −1 )mn , we have: 





Θπ,σ (f ) =

      f σ −1 k−1 n −1 mn k · tr Aρ ρ(m) · ΔM\G (mσ) dk dm dn

K ZG \M N





=



     tr Aρ ρ(m) · ΔM\G (mσ) · f σ −1 g −1 mg dm dg,

M\G ZG \M

where: −1/2

ΔM\G (mσ) = δN

    ΔG (m) · det 1 − Ad(mσ) Lie N  = (mσ). ΔM

Using the σ-twisted Weyl integration formula for M, we obtain: Θπ,σ (f ) =

 TM

1 σ |W (M(F ), TM (F ))|



·  TM

1 |W σ (M(F ), TM (F ))|

#  · ΔG (tσ) ·



f g

−1

=

TM

1−σ ZG TM \TM



  ΔM (tσ) tr ρ(σ)ρ(t)

1−σ ZG TM \TM



$

tσ(g) dg dt

σ \G TM



  ΔM\G (tσ) tr ρ(σ)ρ(t) · Δ2M (tσ)

  f g −1 tσ(g) dg dt

σ \G TM

=



1 |W σ (M(F ), TM (F ))|



Θρ,σ (t) · Oσ (t, f ) dt, 1−σ ZG TM \TM

(5.3)

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where the sum is over representatives of the σ-conjugacy classes of the σ-invariant maximal tori of M, and W σ (M(F ), TM (F )) is the subgroup of σ-fixed elements in the Weyl group of TM in M. We shall make extensive use of this identity in the next section. 6. Transfer of parabolically induced representations After the preparation of the previous sections, we can now begin our investigation of the transfer and the resulting local character identities. In this section, we investigate the transfer of stable characters associated to parabolically induced representations. We shall begin by investigating the twisted endoscopic transfer of the various families of parabolically induced representations of H = GSp(4) to G = GL(4) × GL(1) and then conclude with the endoscopic transfer from C = GSO(4) to H = GSp(4). 6.1. The case of GL(4) × GL(1) and GSp(4) We now let G = GL(4) × GL(1), H = GSp(4), and let σ be the automorphism of G defined in the introduction:   σ : (g, e) → J t g −1 J −1 , e det(g) ,

for all (g, e) ∈ G,

where ⎛

1

⎜ J := ⎜ ⎝

1 −1

⎞ ⎟ ⎟. ⎠

−1 Then H is an elliptic σ-twisted endoscopic group of G corresponding to the identity eleˆ = GL(4, C) × C× . In this case, the norm correspondence ment s = 1 in the dual group G is a bijection between the set of stable σ-conjugacy classes of G(F ) with the set of stable conjugacy classes of H(F ) [F1]. We are going to make an observation about the transfer factor of Kottwitz–Shelstad in this particular case. Since both G and H have unique orbits of Whittaker data, we shall use the Whittaker normalization of the transfer factor. Since H is a twisted endoscopic group of G corresponding to the identity element s = 1, the functions ΔI , ΔIII on H(F )reg ×G(F )σ-reg which appear in the definition of the (Whittaker-normalized) transfer factor are identically 1. Moreover, each root of G is “from” H in the sense of [KoS, Section 4.3], hence we also have ΔII,v = 1. Hence, the following lemma holds: Lemma 6.1. For each pair of elements (tH , t) ∈ H(F )reg × G(F )σ-reg such that tH is a norm of t, we have: Δ(tH , t) = 1.

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What this lemma says is that the transfer of orbital integrals from G(F ) to H(F ) identifies the space of σ-twisted stable orbital integrals on G(F ) with the space of stable orbital integrals on H(F ). 6.2. Borel subgroup Given characters α, β, χ of F × , let μH := α ⊗ β ⊗ χ denote the character: diag(a, b, λ/b, λ/a) → α(a)β(b)χ(λ) of the diagonal maximal torus T0,H of H and let H π = α × β  χ := IB (α ⊗ β ⊗ λ) H

be the associated principal series representation. The character distribution of this representation is a stable distribution on H(F ), since normalized induction preserves stability. In particular, we may consider its twisted endoscopic transfer TransG H (Θπ ) to G(F ). On the other hand, for characters μ1 , μ2 , μ3 , μ4 of F × , let μ := μ1 ⊗ μ2 ⊗ μ3 ⊗ μ4 be the character diag(t1 , t2 , t3 , t4 ) →

4 

μi (ti )

i=1

of the diagonal maximal torus A of GL4 (F ), with associated principal series representation GL(4,F )

Π = μ1  μ2  μ3  μ4 := IB

(μ).

Consider the particular representation Π := (αβχ  αχ  βχ  χ) ⊗ αβχ2 . It is easy to check that the inducing data for Π is σ-invariant and thus Π is σ-invariant. We have a canonical (Whittaker-normalized) extension of Π to G(F )  σ as discussed in Section 5.4 with associated twisted character ΘΠ,σ . Now we have: Proposition 6.2. The G-module Π := (αβχ  αχ  βχ  χ) ⊗ αβχ2 and the H-module π := α × β  χ satisfy the following character relation for matching test functions f ∈ Cc∞ (G, ωπ−2 ⊗ ωπ−1 ) and fH ∈ Cc∞ (H, ωπ−1 ):   ΘΠ,σ (f ) = Θπ f H .

In particular:

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ΘΠ,σ (t) = Θπ (tH ) for any pair of regular elements (t, tH ) which are in norm correspondence with each other. Proof. Let T0 = A × k× be the diagonal maximal torus of G, and T0,H that of H. The norms in H of each σ-regular element t = (diag(x, y, z, u), w) ∈ T0 are conjugate to tH = diag(xyw, xzw, uyw, uzw) ∈ T0,H (see [F1]). Observe that:   (αβχ ⊗ αχ ⊗ βχ ⊗ χ) ⊗ αβχ2 (t) = (α ⊗ β ⊗ χ)(tH ).

By Lemma 6.1, the matching condition on f and fH implies that: Oσ (f, t) = O(fH , tH ). The lemma now follows from the identity (5.3) and the observation that the two relevant Weyl groups have the same size. 2 6.3. Heisenberg parabolic Now we consider the principal series representations associated to the Heisenberg parabolic Q of H. The parabolic subgroup Q = LU has Levi factor L consisting of elements of the form   blockdiag a, m, (det m)/a , with a ∈ GL(1) and m ∈ GL(2). Let μ be a character of F × and let τ be an irreducible admissible representation of GL(2, F ) with central character ωτ . Let μ ⊗ τ denote Q-module defined by: %  & (μ ⊗ τ ) blockdiag a, m, (det m)/a = μ(a) · τ (m) and put H π = μ  τ := IQ (μ ⊗ τ ).

On the other hand, let Q = LU be the maximal parabolic subgroup of GL(4) × GL(1) with Levi factor L consisting of elements of the form:   blockdiag(g1 , g2 ), e , g1 , g2 ∈ GL(2, F ), e ∈ F × . The action of automorphism σ preserves L. Indeed, for m = (blockdiag(m1 , m2 ), e) ∈ L(F ), we have     σ(m) = blockdiag σ(m2 ), σ(m1 ) , e · det(m1 ) · det(m2 ) ,

where

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σ(g) := w t g −1 w,

w=

1

27



1

,

for all g ∈ GL(2, F ). Since: w t g −1 w = (det g)−1 ·



1 −1

  1 g



−1

,

we have τ σ ∼ = ωτ−1 τ , with the isomorphism given by the intertwining operator:     A := τ diag(1, −1) ∈ HomGL(2) ωτ−1 τ, τ σ .

The representation ρ = (μτ ⊗ τ ) ⊗ μωτ of L(F ) is therefore σ-invariant, with nonzero intertwining operator ρ(σ) ∈ HomL (ρ, ρσ ) defined by: ρ(σ)(v ⊗ w) = Aw ⊗ Av,

for all v, w ∈ τ.

(6.3)

We note that ρ(σ) is Whittaker-normalized if ρ is generic, and it is normalized as in Section 5.4 if ρ is 1-dimensional. Put Π := (μτ  τ ) ⊗ μωτ := IPG (ρ) so that Π inherits an extension to G  σ and thus has a twisted character ΘΠ,σ . Proposition 6.4. The following character identity holds for matching functions f and fH :   ΘΠ,σ (f ) = Θμτ f H .

Proof. Applying the identity (5.3), we have: ΘΠ,σ (f ) =

 TL

1 |W σ (L(F ), TL (F ))|



Θρ,σ (t) · Oσ (f, t) dt. ZG TL1−σ \TL

Let L1 be the subgroup of L consisting of elements of the form: z(blockdiag(g1 , 1), e1 ), with g1 ∈ GL(2, F ), z ∈ ZG . Similarly, let L2 be the subgroup consisting of elements of the form: z(blockdiag(1, g2 ), e2 ), with g2 ∈ GL(2, F ). For each element m ∈ L, we have: m = m−1 2 m1 σ(m2 ) for a unique m1 ∈ L1 , and an m2 ∈ L2 which is unique up to ZG . In other words, each element in L is σ-conjugate to an element in L1 . Hence, writing t = t−1 2 t1 σ(t2 ) for t ∈ TL , we deduce that   1 ΘΠ,σ (f ) = Θρ,σ (t1 ) · Oσ (f, t1 ) dt. (6.5) |W σ (L(F ), TL (F ))| TL

ZG TL1−σ \TL

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On the side of H = GSp(4), with L the Levi component of Q, one sees by the same argument that   H    1 Θμτ f = Θμ⊗τ (tH ) · O f H , tH dtH . (6.6) |W (L(F ), TL (F ))| TL

ZH \TL

To relate the above two displayed equations, we note that for each σ-invariant maximal torus TL of L, there is a (unique up to conjugation) maximal torus TL of L such that: ZH \TL ∼ = ZG TL1−σ \TL , and vice versa. Moreover, the two relevant Weyl groups have the same size. Since f and fH are matching, we see that to relate (6.5) and (6.6), it remains to verify that Θρ,σ (t1 ) = Θμ⊗τ (tH ) if tH is in norm correspondence with t = t−1 2 t1 σ(t2 ). Recall that ρ = (μτ ⊗ τ ) ⊗ μωτ , and A := τ (diag(1, −1)) ∈ HomGL(2,F ) (ωτ−1 τ, τ σ ). Let {vi } be a basis of the vector space Vτ of τ , so that {vi ⊗ vj } is a basis of ρ. Let M be an operator of finite rank on Vτ , and let M be the operator M ⊗ 1 on Vτ ⊗ Vτ . By the definition of ρ(σ), for each vi ⊗ vj , we have: ρ(σ)M (vi ⊗ vj ) = Avj ⊗ AMvi , and: tr ρ(σ)M =

   Avj ⊗ AMvi , vi∗ ⊗ vj∗ = Avj , vi∗ AMvi , vj∗ i,j

 = AAMvi , vi∗ = tr M,

i,j

(6.7)

i

since A2 = 1. Hence: 

      tr ρ(σ)ρ a−1 g, I2 , a = μωτ (a)(tr μτ ) a−1 g = μ (det g)/a (tr τ )(g),

which is precisely the value at blockdiag((det g)/a, g, a) of the character of the L-module μ ⊗ τ . This proves the proposition. 2 6.4. Siegel parabolic Let P = M N be the upper-triangular Siegel parabolic subgroup of H. Its Levi component M consists of elements of the form:       1 t −1 1 blockdiag m, λm∗ , λ ∈ GL(1), m ∈ GL(2), m∗ := m . 1 1 For τ being an irreducible admissible GL(2, F )-module, and μ a character of F × , let τ ⊗ μ denote the M -module on the space of τ defined as follows:

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   (τ ⊗ μ) blockdiag m, λm∗ = μ(λ) · τ (m)

and let π = τ  μ = IPH (τ ⊗ μ) denote the associated parabolically induced representation. Its character is a stable distribution on H(F ). On the other hand, let P = MN ⊂ G denote the parabolic subgroup of type (1, 2, 1), so that its Levi factor is of the form   M = GL(1) × GL(2) × GL(1) × GL(1).

Let ρ = (μ ⊗ μτ ⊗ ωτ μ) ⊗ μ2 ωτ be the M -module on the vector space of τ defined as follows:     ρ(m) := μ(a) · (ωτ μ)(d) · μτ (g) · μ2 ωτ (e),

  m = blockdiag(a, g, d), e ∈ M.

Observe that ρσ (m) = ρ(m). Thus, the identity map Aρ = idρ is a nonzero element of HomM (ρ, ρσ ). Put Π := (μ  μτ  μωτ ) ⊗ μ2 ωτ := IPG (ρ). It is a σ-invariant representation of G, with Whittaker-normalized intertwining operator Π(σ) ∈ HomG (Π, Π σ ) defined as in Section 5.4. Proposition 6.8. The following character identity holds for matching local test functions f and fH :   ΘΠ,σ (f ) = Θπ f H .

Proof. By (5.3), and the fact that ρ(σ) is the identity automorphism, we have: ΘΠ,σ (f ) =

 TM

1 |W σ (M(F ), TM (F ))|



Θρ (t) · Oσ (f, t) dt 1−σ ZG TM \TM

where the sum is over representatives of the σ-conjugacy classes of the σ-invariant maximal tori of M. On the other hand,    Θπ f H = TM

1 |W (M (F ), TM (F ))|

 ZH \TM

  Θτ ⊗μ (tH ) · O f H , tH dtH .

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For g ∈ GL(2, F ) and a, d, e ∈ F × , the conjugacy class of blockdiag(deg, aeg) in H is the norm of the conjugacy class of (blockdiag(a, g, d), e) in G. The lemma now follows from the matching condition on the test functions and the observation that Θρ (t) = Θτ ⊗μ (tH ) if t and tH correspond, as well as the fact that the two Weyl groups in question have the same size. 2 6.5. The case of GSp(4) and GSO(2, 2) We now consider the endoscopic transfer of parabolically induced representations from C = GSO(2, 2) to H = GSp(4). For two characters α, β of F × , let αβ denote the associated principal series representation of GL(2, F ). For two GL(2, F )-modules (τ1 , V1 ), (τ2 , V2 ) with common central character ω, we let (τ1  τ2 , V1 ⊗ V2 ) denote the representation of C(F ) defined as follows: (τ1  τ2 )(g1 , g2 ) = τ1 (g1 ) ⊗ τ2 (g2 ),

(g1 , g2 ) ∈ C(F ).

Now we have: Proposition 6.9. Let α, β be characters of F × , and ρ an admissible representation of GL(2, F ) with central character αβ. The following character identity holds for matching test functions f H and f C :     Θ(αβ)ρ f C = Θβ −1 ρβ f H . (6.10) Proof. This is similar to the proofs of Propositions 6.4 and 6.8; we omit the details.

2

6.6. Endoscopic character identities Using the results of this section, we can now verify the desired endoscopic character identities for all L-packets of H(F ) associated to non-discrete series generic L-parameters. These include all tempered L-parameters. Suppose that φ is a non-discrete series generic L-parameter for H. Then the associated L-packet Πφ (H) is described explicitly in [GT2, Proposition 13.1]. In particular, we shall consider the 3 families of non-discrete series generic L-packets as given in [GT2, Proposition 5.3] and their twisted endoscopic transfer to G. (a) (Heisenberg case) suppose that φ = φτ ⊕ φτ ⊗ χ and

sim(φ) = χ · ωτ ,

with τ an irreducible discrete series representation of GL2 (F ) and χ a character of F × such that |χ| = |−|−s with s  0. Then Πφ (H) consists of the irreducible

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submodules of the induced representation χ  τ . In fact, under the hypothesis that φ is a generic L-parameter, this induced representation is irreducible (so that #Πφ (H) = 1) unless χ = 1, in which case 1  τ is a semisimple representation of length 2. Now when χ  τ is irreducible or semisimple, then the sum of the characters of elements in Πφ (H) is simply the character of χ  τ which is stable. Denoting this sum by Θφ , we see by Proposition 6.4 that TransG H (Θφ ) = ΘΠ,σ where Π = (χτ  τ ) ⊗ χωτ . But the representation Π is irreducible and is precisely the representation of G = GL(4) × GL(1) with L-parameter (φ, sim(φ)). This gives the desired local character identity for the transfer of Πφ (H) to G.  if and only if χ = 1. In this The non-discrete-series parameter φ factors through C case, the corresponding representation of C(F ) is the discrete series representation τ  τ of GL2 (F ) × GL2 (F ). We shall only be able to determine its endoscopic lifting later on (in Section 8). (b) (Siegel case) Now suppose that φ = χ ⊕ χφτ ⊕ χωτ

and

sim(φ) = χ2 ωτ

where χ is a character of F × and τ an irreducible essentially discrete series representation of GL2 (F ) such that |ωτ | = |−|−s with s  0. Then the associated L-packet Πφ (H) is a singleton consisting of the irreducible representation π = τ  χ. Thus Proposition 6.8 gives the desired local character identity TransG H (Θφ ) = ΘΠ,σ where Π = (χ  χτ  χωτ ) ⊗ χ2 ωτ has L-parameter (φ, sim(φ)).  and determines the irreducible representation The parameter φ factors through C, δ = χτ  (χ  χωτ ) of C(F ) when τ  χ is irreducible. Then Proposition 6.9 shows that TransH C (Θδ ) = Θπ .

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(c) (Borel case) Finally, suppose that φ = χ1 χ2 χ ⊕ χ1 χ ⊕ χ2 χ ⊕ χ and

sim(φ) = χ1 χ2 χ2 ,

with |χi | = |−|−si and s1  s2  0. Then the L-packet Πφ (H) is a singleton consisting of the irreducible representation χ1 × χ2  χ. Proposition 6.2 then gives the desired local character identity.  Proposition 6.9 gives the desired character Similarly, since φ factors through C, identity for the transfer from C to H; we omit the details. 7. The trace formula To deal with the character identities for the remaining L-packets (mainly discrete series ones), we shall need to resort to the use of global arguments, using the (stable) Arthur–Selberg trace formula. In this section, we recall the precise trace formula we need in the various situations. The results are largely due to Arthur [A4,A6], Clozel–Labesse– Langlands [CLL], Kottwitz–Shelstad [KoS] and Labesse–Waldspurger [LW]. Let k be a number field with ring of adeles Ak . Recall that H = GSp(4) has endoscopic group C = GL(2) × GL(2)/Δ GL(1). 7.1. Stable trace formula for H We first take note of the stable trace formula for H established in general by Arthur [A4,A6]. In particular, we are interested in the discrete part of the stable trace formula, which is an identity: IH,disc (f ) = SH (f ) +

  1 · SC f C 4

(7.1)

for a test function f . Here, IH,disc (f ) =

  W (G, M )−1 · M



 det(s − 1)a

M /aG

−1 

s∈W (G,M )reg



· tr MP (s, 0) · IP,disc (0, f )



(7.2)

where the first sum runs over association classes of standard Levi subgroups of H; we refer the reader to [A4] for the precise definitions of other terms in the formula above. For our purpose, we simply note that the term corresponding to M = H is 

m(π) · Tr π(f ),

π⊂L2H,disc

which is the character distribution of the automorphic discrete spectrum of H (with fixed central character) and is the term of primary interest. On the other hand, the

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distribution SH is a stable distribution on H(A), and the term SC is the analogous stable distribution on C(A). However, since C has no proper elliptic endoscopic group, one has     SC f C = IC,disc f C =



  Tr(τ1  τ2 ) f C + (other terms).

(7.3)

τ1 τ2 ⊂L2C,disc

7.2. Stable trace formula for H  We also state the discrete part of the stable trace formula for the inner form H  of H. We have:   1   IH  ,disc (f ) = SH f H + · SC f C 4

(7.4)

for matching test functions. Here the distribution IH  ,disc is analogously defined as in (7.2) and the stable distributions SH and SC are the ones appearing in (7.1). In particular, with the given function f H on H(A), we may find a test function (f H )C on C(A) which matches f H . Then applying the stable trace formula for H, one has   1  C  1   IH  ,disc (f ) = IH,disc f H − · SC f H + · SC f C . 4 4

Note that the functions (f H )C and f C are different in general, since they are defined by matching conditions with respect to H  and H respectively. Indeed, O(−, (f H )C ) is invariant under the outer automorphism of C(A), whereas O(−, f C ) is anti-symmetric under the outer automorphism. 7.3. Elliptic twisted endoscopic groups for (G, σ) Before we describe the twisted trace formula for G = GL(4) × GL(1) with the automorphism σ, we enumerate the elliptic twisted endoscopic groups. The elliptic σ-twisted endoscopic groups (see [KoS] and [F2]) of G are as follows: • H = GSp(4), • CE = RE/F GL(2) := {(g1 , g2 ) ∈ RE/F GL(2): det g1 = det g2 }, E • C+ = (GL(2) × RE/F GL(1))/ GL(1), E . where E ranges over the étale quadratic F -algebras, and E is not split in the case of C+ E In the case of C+ , the group GL(1) embeds in GL(2) × RE/F GL(1) via: z → (diag(z, z), (z −1 , z −1 )), for all z ∈ GL(1).

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7.4. Simple stable twisted trace formula In [KoS], Kottwitz and Shelstad have stabilized the elliptic part of the twisted trace  formula. More precisely, suppose that the global test function f = v fv has the property that for at least 3 finite places v, the twisted orbital integrals of fv are supported on the regular elliptic set, then we have an identity: IG,σ,disc (f ) =

  1    1   E 1 · SH f H + · SCE f CE + · SC+E f C+ 2 4 8 2 E

(7.5)

E =F

where ' • the sum E is over the quadratic extensions E of F ; E • the functions f, f H , f CE , f C+ are global matching test functions; • the stable distributions SH , SCE and SC+E are those appearing in the discrete part E respectively; of the stable trace formula of H, CE and C+ • IG,σ,disc is an invariant distribution which is the twisted analog of (7.2) given by [LW, Theorem 14.3.1 and Proposition 14.3.2] (see also [A7, pp. 125–128]):

IG,σ,disc (f ) =

  W (G, M )−1 · M



  det(1 − s · σ)−1 s·σ a /a M

s∈W (G,M )σ-reg

  · Tr M (s, 0)IP,disc (0, f )IP,disc (σ) ,

G

(7.6)

where the first sum runs over the set of association classes of all standard Levi subgroups M of G. 8. Endoscopic character identities In this section, we shall use the stable trace formula of H to deduce the desired endoscopic character identities for the discrete series L-packets of H(F ). 8.1. Archimedean case We first recall the results of Shelstad. Suppose F = R or C. Then endoscopic character identities for reductive groups have been established in the archimedean case (see [Sh1, Sh2,Sh3]). We consider the case of interest here. An irreducible, tempered, admissible C(F )-module τ1  τ2 lifts to a tempered packet on H(F ) which consists of: (i) two distinct discrete series H(R)-modules π + , π − , if τ1 , τ2 are distinct discrete series representations; (ii) two distinct tempered H(R)-modules π + , π − , if τ1 , τ2 are equivalent discrete series representations;

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(iii) a single tempered representation π + , if at least one of τ1 , τ2 is a principal series representation. Moreover, the following character identity holds for matching test functions:       Θπ+ f H − Θπ− f H = Θτ1 τ2 f C ,

(8.1)

where π − := 0 if {π} is a singleton packet as in case (iii). Equivalently, we have the equality of distribution on H(F ): Θπ+ − Θπ− = TransH C (Θτ1 τ2 ). More generally, suppose that τ1  τ2 is any nontempered generic representation of C(F ). Then we have seen in Sections 5 and 6 that the same character identity holds (with π − = 0 in this case). This more general version is necessary for global applications where we may not know the Ramanujan conjecture on temperedness of cusp forms. Finally, as we have noted in Section 2.4, we have: π + = θ(τ1  τ2 )

  and π − = θ τ1D  τ2D .

8.2. Endoscopic packets of GSp(4) To deduce the analogous character identity for an L-packet Πφ of H(F ) corresponding to a discrete series L-packet of C(F ), we shall use global arguments. Let τ1 and τ2 be irreducible, square-integrable, admissible representations of GL(2, F ), with a common central character. As we recall in Section 2, the associated endoscopic L-packet on H(F ) consists of the representations π + := θ(τ1  τ2 )

  and π − := θ τ1D  τ2D

of H(F ), where τiD denotes the Jacquet–Langlands lift of τi to D× . Moreover: (i) If τ1 ∼ = τ2 then π + (resp. π − ) is the unique generic (resp. nongeneric) constituent of the length two parabolically induced representation 1  τ of H. (ii) If τ1  τ2 , then π + is square-integrable, π − is nongeneric supercuspidal; and π + is supercuspidal if and only if both τ1 and τ2 are supercuspidal. We want to prove: Proposition 8.2. Let F be a non-archimedean local field. Let τ1  τ2 be an irreducible, square-integrable representation of C(F ). Let π + = θ(τ1  τ2 ), π − = θ(τ1D  τ2D ). The following character identity holds for matching test functions:       Θπ+ f H − Θπ− f H = Θτ1 τ2 f C .

(8.3)

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8.3. Globalization Before we begin the proof of the proposition, we first establish the following lemma: Lemma 8.4. Let k = Q be a number field with an archimedean place w∞ , and a finite place w, such that kw∞ = R and kw = F . Let τ∞ be an irreducible discrete series representation of GL2 (R) and τF an irreducible discrete series representation of GL(2, F ). Suppose that Ω is a unitary Hecke character of k such that Ωw and Ωw∞ are equal to the central characters of τF and τ∞ , and such that Ωv is unramified at all finite places v = w. Then there exists an irreducible automorphic representation Ξ of GL(2, Ak ) with central character Ω such that: (i) Ξw ∼ = τF ; (ii) Ξv is unramified for all finite places v = w; (iii) Ξw∞ = τ∞ . Proof. Let B be a quaternion algebra over k which is ramified at precisely w and w∞ . Now apply the trace formula for the anisotropic (modulo center) k-group B × and the  given central character Ω, using as test function f = v fv given by • fw = a matrix coefficient of τFD ; D ; • fw∞ = a matrix coefficient of τ∞ • fv = the characteristic function of GL2 (Ov ) for every finite place v = w. Since k = Q, there is at least one other archimedean place of k, and we may shrink the   support of the test function fw∞ at all other archimedean places w∞ so that the only contribution from the geometric side of the trace formula is that of the trivial element of B × (F ). In this way, we construct an irreducible automorphic representation Ξ B of B D B ∼ B × (Ak ) such that Ξw = τ∞ , Ξw = τF , and ΞvB is unramified for all finite places v = w ∞ of k. The global Jacquet–Langlands lift of Ξ B to GL(2, Ak ) satisfies the conditions of the lemma. 2 8.4. Stable trace formula We can now begin the proof of Proposition 8.2. Let k be a number field as in Lemma 8.4 and let V∞ denote the set of archimedean places of k; in particular, V∞ contains at least two elements, including the distinguished real place w∞ . By Lemma 8.4, we construct an irreducible cuspidal representation Ξ1 Ξ2 of C(Ak ), such that Ξ1 and Ξ2 have the same central characters and: (i) Ξi,w ∼ = τi (i = 1, 2); (ii) the local representations Ξi,v are unramified for all finite places v = w of k; (iii) Ξ1,w∞ , Ξ2,w∞ are inequivalent discrete series representations of GL(2, R).

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By [GRS], one knows that the global theta lift Θ(Ξ1  Ξ2 ) of Ξ1  Ξ2 to H(Ak ) is nonzero and cuspidal. Let ξC : L C → L H be a fixed L-embedding. For each place v ∈ / S, let t(Ξ1,v Ξ2,v ) be the Satake parameter of the unramified representation Ξ1,v Ξ2,v and let tH (Ξ1,v Ξ2,v ) denote the conjugacy class of the image of t(Ξ1,v  Ξ2,v ) under ξC .   Now we let fH = v fH,v and fC = v fC,v be matching global test functions on H(Ak ) and C(Ak ) respectively, such that fH,v and fC,v are in the relevant spherical Hecke algebra for all finite places v of k outside S := V∞ ∪{w}. Using these test functions in the stable trace formula, and using the independence of characters of the spherical Hecke algebra, we obtain a semi-local identity:

SΞ1 Ξ2 (fH,S ) =

 Π

m(Π) Tr(Π)(fH,S ) −

  1  Tr Ξ1  Ξ2 (fC,S ) 4  

(8.5)

Ξ1 Ξ2

where ( • SΞ1 Ξ2 is a stable distribution on HS := v∈S H(kv ); • the first sum on the RHS is over representatives of the equivalence classes of irreducible H(Ak )-modules in the discrete spectrum whose local component at every place v ∈ / S is unramified and parameterized by tH (Ξ1,v  Ξ2,v ). Thus, Π belongs to the endoscopic part of the discrete spectrum indexed by Ξ1  Ξ2 , as described in Section 3; • the coefficient m(Π) is the multiplicity of Π in the discrete spectrum of H(Ak ); • the second sum on the RHS is over representatives of the equivalence classes of irreducible automorphic C(Ak )-modules Ξ1  Ξ2 such that, for every place v ∈ / S,     the representation Ξ1,v  Ξ2,v is unramified, and tH (Ξ1,v  Ξ2,v ) is equal to tH (Ξ1,v  Ξ2,v ).

By construction, the representation Θ(Ξ1  Ξ2 ) contributes to the first sum on the RHS of (8.5), and Ξ1  Ξ2 contributes to the second sum. 8.5. Contributions from C We now need to determine what other representations Ξ1  Ξ2 of C(Ak ) con  tribute to the second sum. Observe that if t(Ξ1,v  Ξ2,v ) is represented by (t1,v , t2,v ) ∈ GL(2, C) × GL(2, C), then the conjugacy class of the matrix blockdiag(t1,v , t2,v ) in   GL(4, C) parameterizes the GL(4, kv )-module Ξ1,v  Ξ2,v induced from the type (2, 2)   parabolic subgroup. Consequently, if Ξ1  Ξ2 contributes to the second sum in (8.5), then the following equivalence of GL(4, kv )-modules holds for all places v ∈ / S:   ∼ Ξ1,v  Ξ2,v = Ξ1,v  Ξ2,v .

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It now follows from the strong multiplicity one theorem for GL(4) that Ξ1  Ξ2 contributes to the second sum in (8.5) if and only if it is equivalent to either Ξ1  Ξ2 or Ξ2  Ξ1 . So we have: SΞ1 Ξ2 (fH,S ) =



m(Π) Tr(Π)(fH,S ) −

Π

−

1 · Tr(Ξ1  Ξ2 )(fC,S ) 4

1 · Tr(Ξ2  Ξ1 )(fC,S ). 4

(8.6)

Since fC matches a test function on H(Ak ), we obtain by the matching condition that for every place v,     O (t1 , t2 ), fC,v = O (t2 , t1 ), fC,v ,

for all (t1 , t2 ) ∈ C reg (kv ).

Hence, Tr(Ξ1  Ξ2 )(fC,S ) = Tr(Ξ2  Ξ1 )(fC,S ), and Eq. (8.5) becomes: SΞ1 Ξ2 (fH,S ) =



m(Π) Tr(Π)(fH,S ) −

Π

1 · Tr(Ξ1  Ξ2 )(fC,S ). 2

(8.7)

8.6. Choosing fH,w∞ We shall now specify the test function fw∞ at the place w∞ . By (8.1), we see that at any archimedean place v, we have Θπv+ (fH,v ) − Θπv− (fH,v ) = ΘΞ1,v Ξ2,v (fC,v ),

(8.8)

+ − where πv− may be zero. At the place w∞ , the representations πw , πw of Hw∞ form ∞ ∞ + a discrete series L-packet. Let f+ (resp. f− ) be a pseudo-coefficient [A1,K] of πw ∞ − (resp. πw ). Since the tempered local L-packets of Hw∞ are mutually disjoint, we deduce ∞ that



π, f+ − f−  = 0

π∈Πφ

for any tempered L-packet Πφ of Hw∞ . By [Sh1, Theorem 4.1], the stable orbital integral of f+ − f− vanishes on all regular elements, and thus any stable distribution on Hw∞ vanishes at f+ − f− . In particular, we choose the global test function fH such that fH,w∞ = f+ −f− . Since SΞ1 Ξ2 is a stable distribution on HS , we have SΞ1 Ξ2 (fH,S ) = 0, and the identity (8.7) becomes:  Π

m(Π) Tr(Π)(fH,S ) =

1 Tr(Ξ1  Ξ2 )(fC,S ). 2

(8.9)

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8.7. Contributions from H Now we examine the sum of the LHS of (8.9). Recall that the sum runs over all Π in / S. In Section 3, the endoscopic submodule L2Ξ1 Ξ2 such that Πv is unramified for all v ∈ we have completely described the submodule L2Ξ1 Ξ2 using theta correspondence. In  particular, for any Π = v Πv ⊂ L2ΞΞ2 , we have Πv = πv+ := θ(Ξ1,v  Ξ2,v ) or

 D  D πv− = θ Ξ1,v  Ξ2,v .

Moreover, for any such Π, the multiplicity of Π in the discrete spectrum is given by the formula: m(Π) =

n(Π) 1 1 + (−1) , 2

(8.10)

where n(Π) is the number of local components of Π which are of the form πv− . In particular, this multiplicity is nonzero if and only if n(Π) is even. 8.8. The final comparison Hence, in Eq. (8.9), if we use the linear independence of characters of the group ( HV∞ −{w∞ } := v∈V∞ −{w∞ } H(kv ) to extract the term on both sides corresponding to ( v∈V∞ −{w∞ } Θπv+ (fH,v ), we obtain the identity: Θπ+ (fH,w ) · Θπw+ (fH,w∞ ) + Θπ− (fH,w ) · Θπw− (fH,w∞ ) ∞

=

∞

1 · Θτ1 τ2 (fC,w ) · ΘΞ1,w∞ Ξ2,w∞ (fC,w∞ ). 2

Since fH,w∞ is equal to f+ − f− , we have: Θπw+ (fH,w∞ ) = 1 and Θπw− (fH,w∞ ) = −1 ∞

∞

so that, by (8.8), ΘΞ1,w∞ Ξ2,w∞ (fC,w∞ ) = 2 for every test function fC,w∞ on Cw∞ matching f+ − f− . It follows that: Θπ+ (fH,w ) − Θπ− (fH,w ) = Θτ1 τ2 (fC,w ) as desired. This completes the proof of the proposition.

(8.11)

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8.9. Some corollaries We note some corollaries of the proposition. Corollary 8.12. Suppose that {π ± } is an endoscopic L-packet of size 2. Let f+ be a pseudo-coefficient of π + , and f− that of π − . The stable orbital integral of f+ − f− is zero on regular elements. Proof. Since f± are pseudo-coefficients, their orbital integrals vanish on non-elliptic elements. Let t be a regular elliptic element in H(F ) such that the orbital integral O(t, f+ − f− ) is nonzero. Then we have: O(t, f+ − f− ) = Θπ+ (t) − Θπ− (t). The stable conjugacy class of t contains another conjugacy class, represented by t [F1] and the transfer factor ΔH,C satisfies Δ(tC , t ) = −Δ(tC , t) for tC the norm of {t, t }. By Proposition 8.2 and the formula (5.2), we have: )τ τ (sC ) Θπ+ (s) − Θπ− (s) = Δ(sC , s) · Θ 1 2

(8.13)

for all s ∈ H reg (F ) with norm sC ∈ C H -reg (F ), and where )τ τ := Θτ τ + Θτ τ . Θ 1 2 1 2 2 1

By the identity (8.13), one has:   SO(t, f+ − f− ) = O(t, f+ − f− ) + O t , f+ − f− .     = Θπ+ (t) − Θπ− (t) + Θπ+ t − Θπ− t   )τ τ (tC ) + Δ tC , t · Θ )τ τ (tC ) = Δ(tC , t) · Θ 1 2 1 2

= 0, since Δ(tC , t ) = −Δ(tC , t). This proves the corollary. 2 Lemma 8.14. Suppose that {π + , π − } is a discrete series L-packet of H(F ) where π − may be 0. Then the class function χ := Θπ+ + Θπ− is stable on H reg (F ). Proof. The non-elliptic conjugacy classes of H(F ) are stable. Thus, to prove the lemma, it suffices to consider the regular elliptic conjugacy classes. On the regular elliptic set, χ = O(−, f+ ) + O(−, f− ) where f± is a pseudo-coefficient of π ± . Thus, we need to show that the transfer f C of f = f+ + f− to C is zero on elliptic conjugacy classes of C(F ). By Theorem 1 in the Appendix of [K], this is equivalent to showing that:   Θτ1 τ2 f C = 0 (8.15) is zero for all tempered representations τ1  τ2 of C(F ).

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But by Proposition 8.2, we have:   Θτ1 τ2 f C = Θπ+ (f ) − Θπ− (f ).

Since f = f+ + f− is a sum of pseudo-coefficients, the RHS of the above identity is zero ± unless possibly when π  = π ± . But in that case, the packet {π ± } is endoscopic and the RHS is equal to 1 − 1 = 0. The lemma is proved. 2 The following corollary is called the stable multiplicity formula in [A7, p. 169]. Corollary 8.16. Let k be any number field and let Ξ1  Ξ2 be a cuspidal representation of C(Ak ). For any finite set of places S of k, let SΞ1 Ξ2 be the stable distribution of ( HS = v∈S H(kv ) defined by (8.5). Then

SΞ1 Ξ2 =

1  · (ΘΠv+ + ΘΠv− ) 2 v∈S

D D with Πv+ = θ(Ξ1,v  Ξ2,v ) and Πv− = θ(Ξ1,v  Ξ2,v ).

9. Twisted endoscopic character identities I Now we shall consider the twisted endoscopic character identities for H = GSp(4) and G = GL(4) × GL(1). For any discrete series L-packet Πφ , we have seen by Lemma 8.14 ' that Θφ := π∈Πφ Θπ is a stable distribution. Thus, we may consider the transfer of Θφ to G. If we regard (φ, sim φ) as an L-parameter for G, the associated L-packet is a singleton consisting of an essentially tempered representation Π. We would like to relate the transfer of Θφ to the twisted character of Π. More precisely, we will show: Proposition 9.1. If φ is a discrete series L-packet of H(F ) with associated L-packet Πφ = {π ± }, then   ΘΠ,σ (f ) = (Θπ+ + Θπ− ) f H

for matching functions f and f H , where Π is the representation of G = GL(4) × GL(1) with L-parameter (φ, sim φ) and such that the extension of Π to G  σ is Whittakernormalized. In this section, we consider the case of endoscopic L-packets of H(F ).

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9.1. Endoscopic case Suppose that φ is associated to a discrete series representation τ1 τ2 of C(F ), where τ1 and τ2 are inequivalent discrete series representations of GL2 (F ) with common central character ω. The L-packet Πφ consists of the representations π + = θ(τ1  τ2 ), and π − = θ(τ1D  τ2D ). Let L be the Levi subgroup of G = GL(4) × GL(1) which is of the form: 

  blockdiag(g1 , g2 ), e : g1 , g2 ∈ GL(2), e ∈ GL(1) .

Let P be the upper-triangular proper parabolic subgroup of GL(4) containing L. Let (τ1 ⊗ τ2 ) ⊗ ω denote the L-module defined by:   (τ1 ⊗ τ2 ) ⊗ ω : blockdiag(g1 , g2 ), e → ω(e)τ1 (g1 ) ⊗ τ2 (g2 ),

and let Π = (τ1 τ2 )⊗ω denote the parabolically induced representation IPG ((τ1 ⊗τ2 )⊗ω) of G. Then the generic representation Π is the representation of G with L-parameter (φ, sim φ). Moreover, the Levi subgroup M is σ-stable and we have: 

(τ1 ⊗ τ2 ) ⊗ ω

σ

∼ = (τ2 ⊗ τ1 ) ⊗ ω.

Consequently, the representation Π is σ-invariant and we have fixed an element A ∈ HomG (Π, Π σ ) which is Whittaker-normalized, thus obtaining an extension of Π to G  σ. The rest of this section is devoted to the proof of Proposition 9.1 in this case. 9.2. Globalization Let k be a totally complex number field with 4 places v0 , v1 , v2 , v3 such that kvj = F . Let S be the set consisting of v1 , v2 , v3 and the archimedean places of k. For j = 1, 2, 3, fix two inequivalent supercuspidal, irreducible, admissible representations τ1,vj and τ2,vj of GL(2, kvj ) with common central character independent of vj . Let {πv+j , πv−j } be the local endoscopic packet on H(kvj ) which is the lift of τ1,vj  τ2,vj . As in Lemma 8.4, one can find cuspidal representations Ξ1 , Ξ2 of GL(2, Ak ), with the same central character Ω, such that Ξi,vj ∼ = τi,vj (i = 1, 2; j = 1, 2, 3), and Ξi,v is unramified for all v ∈ / S. The cuspidal automorphic representation Ξ1  Ξ2 of C(Ak ) gives rise to a family of Satake parameters {t(Ξ1,v  Ξ2,v )}v∈S / , which gives by virtue of the maps ξ : LC → LH → LG a family of Satake parameters {tH (Ξ1,v  Ξ2,v )}v∈S / for H and {tG (Ξ1,v  Ξ2,v )}v ∈S / for G. This gives a near equivalence class L2H,Ξ1 Ξ2 in the automorphic discrete

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spectrum of H, and by the strong multiplicity one property, an automorphic representation ˜ := (Ξ1  Ξ2 ) ⊗ Ω Π of G(Ak ), where Ξ1  Ξ2 := IPG (Ξ1 ⊗ Ξ2 ). 9.3. Picking test functions To apply the simple stable twisted trace formula, we need to put some restrictions on the test functions used. Since πv±j is square-integrable, we may consider their pseudo-coefficients fπH± . These vj

pseudo-coefficients are supported on the elliptic set of H(kvj ) and their orbital integrals are equal to Θπv± on the regular elliptic set. In particular, the orbital integral of fπH+ +fπH− vj

j

vj

is not identically zero on the regular elliptic set and is stable there by Lemma 8.14. By Theorem 4.3(iii), we can find a test function fv0j on G(kvj ) which matches fπH+ + vj

fπH− on H(kvj ), and whose transfer to all other (elliptic) twisted endoscopic groups of vj

(G, σ) is 0. We choose the matching global test functions f and f H on G(Ak ) and H(Ak ) such that: • their local components are spherical and matching at all places outside of S; • fvj = fv0j for j = 1, 2, 3; • fvHj = fπH+ + fπH− for j = 1, 2, 3. vj

vj

9.4. Applying the trace formula We may now apply the simple stable twisted trace formula given in (7.5), taking f and f H as above and taking the test function on other elliptic twisted endoscopic group of (G, σ) to be zero. Thus we have an identity IG,σ,disc (f ) =

  1 · SH f H , 2

where we regard both sides as linear functionals on Cc∞ (G(k∞ )) × H(G(AS ), K S ), where H(G(AS ), K S ) is the spherical Hecke algebra of G(AS ). Then using the linear independence of characters of spherical Hecke algebras, one may extract the tG (Ξ1  Ξ2 ) eigenspaces on both sides to obtain IG,σ,disc,Ξ1 Ξ2 (f ) =

  1 · SH,Ξ1 Ξ2 f H . 2

Now the RHS of the above identity is given by Corollary 8.16. On the LHS, only the ˜ will contribute. More precisely, if A0 denotes the maximal diagonal representation Π

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˜ to the LHS torus of G, then by [CLL] and [LW, Theorem 14.3.1], the contribution of Π is: 2·

 −1 1 ˜ )I(σ), · det(1 − s × σ)|aL /as×σ tr M (s, 0)Π(f G |W (G, L)|

where: (i) (ii) (iii) (iv)

aL := X∗ (AL ) ⊗ R, where AL ∼ = GL(1)3 is the center of L. s is the Weyl action diag(z1 , z1 , z2 , z2 ; z0 ) → diag(z2 , z2 , z1 , z1 ; z0 ) on aL . is the space of (s × σ)-fixed points in aG . as×σ G M (s, 0) is the normalized intertwining operator in:     HomG(Ak ) (Ξ1  Ξ2 ) ⊗ Ω , s (Ξ1  Ξ2 ) ⊗ Ω .

Since Ξ1  Ξ2 is irreducible, M (s, 0) is the identity map by [KeS]. (v) I(σ) is the intertwining operator in HomG(Ak ) (Π, Π σ ) defined by:     I(σ)ϕ (g) = ϕ σ(g) ,

for all ϕ ∈ (Ξ1  Ξ2 ) ⊗ Ω and g ∈ G(Ak ).

Here, the 2 which occurs on the LHS accounts for the fact that the two cuspidal repre˜ under parabolic induction: sentations Ξ1  Ξ2 and Ξ2  Ξ1 of L(Ak ) give rise to Π ˜ Ξ1  Ξ2 ∼ = Ξ2  Ξ1 ∼ = Π. Since   W (G, L) = 2

and

det(1 − s × σ)|aL /as×σ = 4, G

˜ to we conclude that the total contribution of representations which are equivalent to Π the trace formula is: 2·

1 1 1 · · ΘΠ,σ ˜ (f ) = ΘΠ,σ ˜ (f ). 2 4 4

Thus we have:   1 1 · ΘΠ,σ · SH,Ξ1 Ξ2 f H , ˜ (f ) = IG,σ,disc (f ) = 4 2

so that  H ΘΠ,σ . ˜ (f ) = 2 · SH,Ξ1 Ξ2 f

(9.2)

Since k is totally complex, the archimedean components of the automorphic representations in (9.2) are irreducible representations induced from the Borel subgroups.

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By Proposition 6.2 and the linear independence of local characters, these parabolically induced archimedean components cancel one another. Hence (9.2) gives the following identity: 3 

ΘΠ˜ v

 ,σ j

3     fv0j = (Θπv+ + Θπv− ) fvHj = 23 = 8.

j=1

j=1

j

(9.3)

j

9.5. Final step Now we may repeat the above consideration, but starting with an irreducible cuspidal automorphic representation Ξ1  Ξ2 of C(Ak ) such that for i = 1, 2, we have: • Ξi,v0 ∼ = τi ; • Ξi,vj ∼ = τi,vj (j = 1, 2, 3); • Ξi,v is unramified for all non-archimedean places different from v0 , v1 , v2 , v3 . Using global test functions f and f H as we did above, but with the local test function at the place v0 to be free, the same argument which leads to the identity (9.3) gives:

ΘΠ˜ v

0 ,σ

(fv0 ) ·

3 



ΘΠ,σ fv0j ˜



= (Θπ+

3  H    + Θπ− ) fv0 · (Θπv+ + Θπv− ) fvHj .

j=1

j=1

j

(9.4)

j

It now follows from (9.3) that on cancelling 8 from both sides of (9.4), we have the desired identity: ΘΠ˜ v

0 ,σ

  (fv0 ) = (Θπ+ + Θπ− ) fvH0 .

Proposition 9.1 is thus proved in the endoscopic case. 10. Twisted endoscopic character identities II In this section, we shall prove Proposition 9.1 when φ is a stable discrete series parameter, so that the associated L-packet Πφ = {π} is a singleton consisting of a discrete series representation π of H(F ) whose character Θπ we have shown to be a stable distribution on H(F ). If (φ, sim φ) is the associated L-parameter for G with corresponding σ-invariant discrete series representation Π of G(F ), then Π = θ(π). We restate Proposition 9.1 here: Proposition 10.1. The following identity holds for matching test functions:   ΘΠ,σ (f ) = Θπ f H .

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The rest of the section is devoted to proving this proposition. We first treat the case of twisted Steinberg representations directly, and then deal with the rest using a global argument. 10.1. Twisted Steinberg We first consider the case when φ is the L-parameter of a twisted Steinberg representation StH ⊗ μ of H(F ), with μ a character of F × . The character μ defines characters μG := (μ ◦ det) ⊗ μ2 on G = GL(4, F ) × F × , and μH := μ ◦ λ on H = GSp(4, F ), where λ is the similitude character of H(F ). The one-dimensional representation μG of G is σ-invariant, and according to our convention, we take the fixed intertwining operator μG (σ) ∈ HomG (μG , μσG ) to be the identity map. If h ∈ H is a norm of (g, e) ∈ G, then λ(h) is equal to (det g)e2 . By Lemma 6.1, for tH ∈ H reg which is a norm of t ∈ Gσ-reg , the transfer factor Δ(tH , t) is simply equal to 1. Hence, the following lemma follows from the formula (5.2): Lemma 10.2. The following character identity holds for matching local test functions: ΘμG , σ(f ) = ΘμH (fH ). Let ν denote the normalized absolute value character of F × . Let StGL(4,F ) denote the Steinberg representation of GL(4, F ) and let μStG denote the G-module μStGL(4,F ) ⊗ μ2 . It is the unique square-integrable subrepresentation of the parabolically induced representation:   I := μ ν −3/2  ν −1/2  ν 1/2  ν 3/2 ⊗ μ2 .

The inducing data of this principal series is σ-invariant and hence so is I and μStG . In Section 5, we have already fixed an element I(σ) ∈ HomG (I, I σ ) which is Whittaker-normalized. Its restriction to the submodule μStG is thus the element of HomG (μStG , μStσG ) which is Whittaker-normalized. The twisted Steinberg representation μH StH is the unique square-integrable subrepresentation of the parabolically induced H-module ν 2 × ν  μν −3/2 . In particular, ΘμH StH is a stable conjugation-invariant function on H(F )reg . Lemma 10.3. For matching test functions f ∈ Cc∞ (G, μ−4 ⊗ μ−2 ) and fH ∈ Cc∞ (H, μ−2 ), we have: ΘμStGL(4) ⊗μ2 ,σ (f ) = ΘμH StH (fH ). In particular, for any pair (t, tH ) with t ∈ G(F )reg and tH ∈ H(F )reg which are in norm correspondence, we have: ΘμStGL(4) ⊗μ2 ,σ (t) = ΘμH StH (tH ).

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Proof. The H-modules ν 3/2 1GL(2)  ν −3/2 μ and ν 2  ν −1 μStGL(2) are reducible of length two. Their semisimplifications are as follows:   ν 3/2 1GL(2)  ν −3/2 μ = μH + L ν 2 , ν −1 μStGL(2) ,   ν 2  ν −1 μStGL(2) = μH StH + L ν 2 , ν −1 μStGL(2) .

Hence, we have:   μH StH = ν 2  ν −1 μStGL(2) − ν 3/2 1GL(2)  ν −3/2 μ − μH .

(10.4)

Now let π1 denote the σ-invariant G-module μ(ν −1 StGL(2)  νStGL(2) ) ⊗ μ2 , and let π2 denote the σ-invariant G-module μ(ν −3/2  1GL(2)  ν 3/2 ) ⊗ μ2 . Let πi (σ) ∈ HomG (πi , πiσ ) (i = 1, 2) be the intertwining operators which we have fixed in Section 5. By Proposition 6.4, Proposition 6.8 and Lemma 10.2, we have the following character identities for matching local test functions: Θπ1 ,σ (f ) = Θν 2 ν −1 μStGL(2) (fH ), Θπ2 ,σ (f ) = Θν 3/2 1GL(2) ν −3/2 μ (fH ), ΘμG ,σ (f ) = ΘμH (fH ). Hence,   ΘμH StH (fH ) = Θπ1 ,σ (f ) − Θπ2 ,σ (f ) − ΘμG ,σ (f )

(10.5)

by (10.4). On the group G, we have: π1 = μStG + L(π1 ), π2 = μG + L(π2 ),

(10.6)

where L(π1 ) and L(π2 ) are equivalent irreducible nontempered subquotients of π1 and π2 respectively. Since μStG and μG are σ-invariant, the operator πi (σ) (for i = 1 or 2) induces an intertwining operator L(σ)i in HomG (L(πi ), L(πi )σ ). Since πi (σ)2 is the identity, one has ΘL(π1 ),σ =  · ΘL(π2 ),σ for some sign  = ±1. By the identities (10.5) and (10.6), we have: ΘμH StH (fH ) = ΘμStG ,σ (f ) + ΘL(π1 ),σ (f ) −  · ΘL(π1 ),σ (f ). Hence, the lemma is proved once we show that  = 1.

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Suppose  = −1. Then, ΘμH StH = ΘμStG ,σ + 2 · ΘL(π1 ),σ .

(10.7)

To see that this leads to a contradiction, we consider the central exponents of the representations on both sides. Here, a central exponent of a representation π is the central character of an irreducible subquotient of the Jacquet module of π with respect to a parabolic subgroup P = M N . By a well-known result of Casselman [C], the asymptotics of the character of π is controlled by its central exponents. Now, since μH StH and μStG are square-integrable, their central exponents decay by Casselman’s square-integrability criterion. On the other hand, the central exponents of the nontempered representation L(π1 ) are unbounded. Thus the character functions on both sides of (10.7) will have different growth properties (see, for example, [FK, Section 21] or [R, Lemma 12.7.2] for a similar argument). With this contradiction, we conclude that  = 1. 2 10.2. Twisted pseudo-coefficients We say that an irreducible σ-invariant tempered G-module π is σ-discrete if it is either square-integrable, or it is not parabolically induced from any σ-invariant representation of a proper Levi subgroup of G. For a σ-discrete generic representation π of G, with Whittaker-normalized intertwining operator π(σ) ∈ HomG (π, π σ ), let fπ denote the corresponding σ-twisted pseudo-coefficient of π [W4], so that fπ is supported on σ-elliptic elements. Moreover, for any σ-invariant, irreducible tempered representation π  of G, we have: Θπ ,σ (fπ ) = δπ,π , and Θπ,σ (t) = Oσ (t, fπ ) for all t ∈ Gσ-reg,e . The following corollary follows from Lemma 10.3: Corollary 10.8. The μStG (σ)-twisted pseudo-coefficient fμStG of μStG and the pseudocoefficient fμH StH of μH StH are matching functions. 10.3. Globalization Let k be a totally complex number field such that at 4 finite places v0 , v1 , v2 and v3 , we have kvi = F . Now we would like to globalize Π; more precisely, we would like to ˜ of G(Ak ) such that: find a cuspidal representation Π ˜ v = Π; • Π 0 ˜ v = μi StG for some character μi ; • for i = 1, 2, 3, Π i

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˜ v is unramified for all other finite places v of k; • Π ˜ • Π has nonzero global theta lift to H(Ak ). ˜ especially to ensure the last condition, we shall To show the existence of such a Π, need to first globalize on an inner form of G and then use the Jacquet–Langlands correspondence. Thus, we fix a quaternion division algebra B over k ramified precisely at vi , for i = 0, 1, 2, 3. As in the construction of the local Langlands correspondence for the inner forms of H, we consider the split rank 2 quaternionic Hermitian space VB over B and a rank 3 quaternionic skew-Hermitian space WB over B. The similitude groups HB and GB of these quaternionic spaces are then inner forms of the split groups H and G. In particular, there is a degree 4 division algebra B4 over k which is ramified precisely at vi , for i = 0, 1, 2, 3, such that    GB ∼ = B4× × GL1 / t, t−2 : t ∈ k× as linear algebraic groups over k. ˜ B of GB such that: Now we would like to find a cuspidal representation Π • • • •

˜ B,v = ΠD (the Jacquet–Langlands lift of Π to GD ); Π 0 ˜ v = μi ; for i = 1, 2, 3, Π i ˜ ΠB,v is unramified for all other finite places v of k; ˜ B has nonzero global theta lift to HB (Ak ). Π

˜ B , we shall appeal to [PSP, Theorem 4.1] and argue as in [GTW, To construct such a Π proof of Proposition 8.1]. More precisely, one first globalizes the central character of ΠD to a unitary Hecke character Ω of k which is unramified at all finite places v = v0 . Choose an unramified character μj of kvj (for j = 1, 2, 3) such that μ2j = Ωvj . At the places vj , the local rep˜ B,v = ΠD or μj of GB (kv ) are all supercuspidal and have nonzero local resentations Π j j ˜ B,v ) are infinite dimentheta lift to HB (kvj ) = H  (F ). Now these local theta lifts θ(Π j sional and thus will have some nonzero local Bessel period, i.e. a nonzero nondegenerate local Fourier coefficient along the Siegel parabolic subgroup of H  (F ) (see [PT]). These nondegenerate local Fourier coefficients are indexed by D-rank 1 quaternionic skew˜ B,v ) has nonzero local Fourier coefficient with respect Hermitian spaces L over D. If θ(Π j to Lvj , then Lvj is a subspace of WB,vj and the isometry group of its orthogonal complement L⊥ vj is a subgroup CLvj of GD . Now by a standard computation of the Bessel model of the Weil representation of HB (kvj ) × GB (kvj ) (see [PT, Section 6]), one sees ˜ B,v ˜ B,v ) has nonzero local Fourier coefficient associated to Lv if and only if Π that θ(Π j

j

j

has nonzero local period over CL (kvj ). Choose a B-rank 1 subspace L ⊂ WB , such that Lvj is chosen as above for j = ˜ B of GB (Ak ) 0, 1, 2, 3. By [PSP, Theorem 4.1], one may find a cuspidal representation Π ˜ which satisfies the first 3 desired conditions and such that ΠB has nonzero global period ˜ B has nonzero global theta lift to HB (Ak ). integral over CL (Ak ). This then implies that Π

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˜ B (à la By [Y], we then deduce that the (complete) standard degree 6 L-function of Π Piatetski–Shapiro and Rallis) has a pole at s = 1. ˜ B at hand, we consider its Jacquet–Langlands transfer Π ˜ to G(Ak ). Then Π ˜ With Π ˜ has nonzero satisfies the first 3 desired conditions above, and we need to justify that Π global theta lift to H(Ak ). But we know (by [Y] or [GTW, Proposition 6.2 and its proof]) ˜ is the same as that of Π ˜ B , and hence it has that the standard degree 6 L-function of Π a pole at s = 1. Moreover, we know that at all places v of k, the local representation ˜ has nonzero ˜ v has nonzero local theta lift to HB (kv ). Hence, it follows by [Y] that Π Π global theta lift to H(Ak ), as desired. We remark that the need to move to the inner form in order to apply [PSP, Theorem 4.1] is because [PSP, Theorem 4.1] requires the local representations which one is interested in globalizing to be supercuspidal. In our case, we have 3 local components where the representations involved are the Steinberg representation. ˜ to H(Ak ). Then we note: Let π ˜ denote the global theta lift of Π • • • •

π ˜ is an irreducible cuspidal representation of H(Ak ); π ˜v is unramified at all finite places v = vi ; π ˜v0 = π; for i = 1, 2, 3, π ˜vi = μi StHvi .

Moreover, we see that π ˜ does not belong to the endoscopic part of the automorphic discrete spectrum of H(Ak ) because the twisted Steinberg representation μi StH does not intervene in the endoscopic part. 10.4. Picking test functions Let V∞ be the set of archimedean places of k. Let S = V∞ ∪ {v0 , v1 , v2 , v3 }. To apply the simple stable twisted trace formula, we shall specify the test functions we  will be using. We consider f = v fv ∈ Cc∞ (G(Ak )) satisfying: • for finite v = vi , fv belongs to the spherical Hecke algebra; • for i = 1, 2, 3, fvi is the twisted pseudo-coefficient of the twisted Steinberg representation μi StGvi ; • for v = v0 or v ∈ V∞ , fv is free.  Since there are 3 places where the test function f = v fv is supported only on the σ-elliptic set, we may apply the simple stable twisted trace formula to such an f .

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10.5. Applying the twisted trace formula Applying the simple stable twisted trace formula, we have: IG,σ,disc (f ) =

  1    1   E 1 · SH f H + · SCE f CE + · SC+E f C+ . 2 4 8 2 E

E =F

Now, for i = 1, 2, 3, the twisted orbital integral fvi = fStGv is a stable function, and i hence the κ-orbital integral of fvi is zero for κ = 1. It follows that the transfers of fvi to all elliptic twisted endoscopic groups of (G, σ) vanish, except possibly for H. On the other hand, by Corollary 10.8, we may take its transfer to H to be the pseudo-coefficient of StHvi . Thus, in the twisted trace formula above, all terms on the RHS vanish except  for the term indexed by H. Moreover, we may take the test function f H = v fvH to satisfy: • for i = 1, 2, 3, fvHi = fμi StHv is the pseudo-coefficient of the twisted Steinberg reprei sentation μi StHvi of H(kvi ); • for all other finite places v, fvH lies in the spherical Hecke algebra. Then for the matching test functions f and f H as above, we have: IG,σ,disc (f ) =

  1 · SH f H . 2

We may now use the stable trace formula for H to explicate the RHS. Namely, we have:     1  C  SH f H = IH,disc f H − · Idisc,C f H 4

for matching test functions. However, since for i = 1, 2, 3, the orbital integral of fvHi = fμi StHv is stable, we see that its transfer to the endoscopic group C is zero. Thus, for i the test function f H at hand, we have:     SH f H = IH,disc f H .

In particular, we have the identity IG,σ,disc (f ) =

  1 · IH,disc f H 2

which relates spectral information on G to that on H. Now we shall consider the part of the LHS corresponding to the cuspidal representation ˜ which we have constructed in Section 10.3. By the linear independence of characters Π of the spherical Hecke algebra outside S, and using the strong multiplicity one property for G, we deduce the semi-local identity:

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ΘΠ˜ S ,σ (fS ) =



  m(Ξ)ΘΞS fSH

(10.9)

Ξ

where the sum on the RHS is over inequivalent irreducible automorphic representations Ξ of H(Ak ) with the following properties: (i) Ξ has nonzero contribution to the discrete part IH,disc of the trace formula of H(Ak ); (ii) the coefficient m(Ξ) is the coefficient associated with the trace of Ξ in IH,disc ; (iii) at every place v ∈ / S, the local representation Ξv is unramified and its Satake ˜v; parameter lifts to that of Π (iv) for i = 1, 2, 3, Ξvi is a representation which does not kill the pseudo-coefficient of the twisted Steinberg representation μi StHvi . 10.6. Simplifying RHS We can now simplify the RHS further by making the following observations: • Note that non-discrete spectrum representations of H(Ak ) which intervenes in IH,disc are parabolically induced from the discrete spectrum of proper Levi subgroups, and by the results of Section 5.8, parabolically induced representations of H(Ak ) lift to ˜ is cuspidal, it follows by the parabolically induced representations of G(Ak ). Since Π strong multiplicity one theorem for G that the representations Ξ which appear on the right-hand side of Eq. (10.9) occur in the discrete spectrum of H(Ak ), in which case the coefficients m(Ξ) are simply the multiplicity of Ξ in the discrete spectrum. In particular, m(Ξ) is a positive integer. • Since the local components at v1 , v2 , v3 of the global test functions are (σ-twisted) pseudo-coefficients of twisted Steinberg representations, for any Ξ which has nonzero contribution to the right-hand side of (10.9), the local component Ξvi (i = 1, 2, 3) is equivalent to a constituent of μi · (ν 2 × ν  ν −3/2 ). (Here, ν denotes the normalized absolute value character of F × .) • Since the local components of Ξ are unitarizable, we deduce that, if ΘΞvi (fvi ) = 0 for i = 1, 2, 3, then Ξvi is either the twisted Steinberg representation μi StH or the character μi , since these are the only two unitarizable representations occurring in ν 2 × ν  ν −3/2 (see [ST, Lemma 3.5, Theorem 4.4]). By the strong approximation theorem, an irreducible automorphic representation has a one-dimensional local component if and only if it is a one-dimensional representation. Since one-dimensional automorphic representations of H(Ak ) do not weakly lift to cuspidal automorphic representations of G(Ak ), the automorphic representations Ξ which contribute to the right-hand side of (10.9) all have twisted Steinberg local components at the places v1 , v2 , v3 . Consequently, ΘΞvi (fH,vi ) = 1 for i = 1, 2, 3. • Since k is totally imaginary, the archimedean local components of the representations Ξ in (10.9) are induced from the Borel subgroup. Hence, their characters cancel those

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˜ at the archimedean places, due to Proposition 6.2 and the linear independence of Π of local characters. ˜v ∼ Noting that Π 0 = Π, we thus obtain: ΘΠ,σ (fv0 ) =



  m(Ξ) · ΘΞv0 fvH0

(10.10)

Ξ

for all matching local test functions fv0 and fvH0 . Here, the sum on the RHS is over a set of automorphic representations occurring in the discrete spectrum of H and satisfying the various conditions listed above; moreover, m(Ξ) is its discrete multiplicity. Note that one of the Ξ’s on the RHS is the cuspidal representation π ˜ which is the global theta lift ˜ of Π. Moreover, though the sum on the RHS of (10.10) is not necessarily finite a priori, it is a locally finite sum, in the sense that for any test function fvH0 , only finitely many terms in the sum will survive. It remains to argue that in fact, there is only one term on the RHS of (10.10), namely the term given by π ˜. 10.7. Final step We first argue that all representations Ξv0 appearing on the RHS of (10.10) are squareintegrable. Since Π is square-integrable, this follows by the consideration of central exponents, applying Casselman’s theorem relating character distributions and Jacquet modules and the fact that all m(Ξ) are positive integers, as we have done at the end of the proof of Lemma 10.3 (see also [FK, Section 21] and [R, Lemma 12.7.2] for a similar argument). There is thus no loss in restricting both sides of (10.10) to the regular elliptic set. Suppose there is some other representation Ξ2,v0 appearing on the RHS of (10.10) besides Ξ1,v0 = π. Let fi be the pseudo-coefficient of Ξi,v0 . If Ξ2,v0 belongs to an endoscopic L-packet, let f2 be the pseudo-coefficient of the other member of the L-packet; otherwise, set f2 to be 0. Then consider the test function fvH0 = f1 + f2 + f2 , and pick a test function fv0 on G(F ) matching fvH0 . Observe that the orbital integral of fvH0 is stable, and the σ-twisted orbital integral of fv0 is supported on the σ-elliptic set. Now (10.10) reads: ΘΠ,σ (fv0 )  m(Ξ1 ) + m(Ξ2 )  2. On the other hand, by the Cauchy–Schwarz inequality, we have:  ΘΠ,σ (fv0 ) = ΘΠ,σ , Oσ (fv0 )

G,σ,e

 1/2  ΘΠ,σ , ΘΠ,σ G,σ,e · Oσ (fv0 ), Oσ (fv0 )

By elliptic orthonormality [W4], we have:

ΘΠ,σ , ΘΠ,σ G,σ,e = 1,

1/2 . G,σ,e

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and by Theorem 4.3(iv) and (4.2), we have:  Oσ (fv0 ), Oσ (fv0 )

G,σ,e

     = SO fvH0 , SO fvH0

H,e

     = O fvH0 , O fvH0

H,e

 3.

Thus we deduce that √

32

which is a contradiction. Hence, only π ˜ contributes to the right-hand side of (10.10), with m(˜ π ) = 1. This completes the proof of Proposition 10.1. We record a global consequence which will be used in the next section, where we deal with the inner form case: ˜ Corollary 10.11. Let π ˜ be the cuspidal representation constructed in Section 10.3. Then π occurs with multiplicity one in the automorphic discrete spectrum of H(Ak ). Moreover, if π ˜  is an automorphic representation occurring in the automorphic discrete spectrum of H such that π ˜v ∼ ˜ ∼ ˜v for all v = v0 , then π ˜. =π =π 11. Inner form of GSp(4) In this section, we deal with the character identities for the L-packets of the unique inner form H  = GU2 (D) of GSp(4); here D denotes the unique quaternion division F -algebra. Recall that we have discussed the notion of the local transfer factors ΔH  ,H and ΔH  ,C in the inner form case in Section 4.3. In particular, ΔH  ,H = −1, whereas ΔH  ,C is only well-defined up to ±1, depending on the choice of a conjugacy class of maximal parabolic subgroups of C. There are two such conjugacy classes which are exchanged by the outer automorphism of C. To fix matters, we shall fix an isomorphism    C∼ = GL(2) × GL(2)/ t, t−1 : t ∈ GL(1) ,

and use the local transfer factor which is compatibly chosen with respect to the parabolic subgroup B1 := B × GL(2) of GL(2) × GL(2). 11.1. Character identities Let φ be an L-parameter of H  , so that φ is also an L-parameter for H, with associated L-packet Πφ (H  ) and Πφ (H). Then Πφ (H  ) is a singleton, unless φ is an endoscopic L-parameter. If φ is an endoscopic L-parameter, then φ is associated to the representation τ1  τ2 of C(F ) and we have        Πφ H  = π +− = θ τ1  τ2D , π −+ = θ τ2  τ1D .

We shall show:

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Proposition 11.1. 

(i) For matching test functions f H and f H , one has 



  Θπ  f H =

π  ∈Πφ (H  )

  Θπ f H .

π∈Πφ (H)

In particular, as functions on the regular set, 

Θπ (tH  ) = −

π  ∈Πφ (H  )



Θπ (tH )

π∈Πφ (H)

if tH  corresponds to tH under the norm correspondence. (ii) Fix the local transfer factor to be compatible with the parabolic subgroup B1 of C. If φ is an endoscopic L-parameter, associated to the representation τ1  τ2 of C, then         Θπ+− f H − Θπ−+ f H = Θτ1 τ2 f C = −Θτ2 τ1 f C 

for matching test functions f H and f C . The rest of this section is devoted to the proof of the proposition. As some of the arguments are analogous to those used in the split case, we will be somewhat more sketchy in this section. 11.2. Lifting of principal series As in the split case, we begin with treating the case of principal series representations. This uses the character formula for induced representation given in (5.3). We consider each case in turn: • (Transfer from H to H  .) If τ is an irreducible discrete series representation of GL2 (F ) and μ a character of F × , then τ μ is a representation of M (F ), where P = M N is the Siegel parabolic subgroup of H(F ), and we have the principal series representation π + = τ  μ of H(F ). On the other hand, if τ D is the Jacquet–Langlands lift of τ to D× , then τ D  μ is a representation of M  (F ) = D× × GL1 (F ), where P  = M  N  is the Siegel parabolic subgroup of H  (F ), and we have the induced representation π − = τ D  μ of H  (F ). Then it follows readily from (5.3) that     Θπ− f H = Θπ+ f H 

for matching test functions f H and f H . • (Transfer from C to H  .) If τ1 = α  β is a principal series representation of GL2 (F ) and τ2 = ρ is a discrete series representation of GL2 (F ), with ωτ2 = αβ, then τ1  τ2 and τ2 τ1 are two representations of C(F ). On the other hand, let π +− = β −1 ρD β

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be the principal series representation of H  (F ) induced from the representation β −1 ρD  β of M  (F ). Using (5.3), and the compatibility of the local transfer factor with the parabolic B1 of C, we deduce readily that       Θπ+− f H = Θτ1 τ2 f C = −Θτ2 τ1 f C .

This identity will be crucially used in the global argument below. In particular, the above identities establish Proposition 11.1 for principal series representations of H  (F ). 11.3. Endoscopic packets Now we consider the discrete series L-parameters φ, and we begin with the case when φ is endoscopic. Let τ1 , τ2 be two inequivalent discrete series representations of GL(2, F ), with common central character ω. For i = 1, 2, let τiD denote the Jacquet–Langlands lift of τi to D× . Then we have:     π +− := θ τ1  τ2D and π −+ := θ τ1D  τ2 .

To establish Proposition 11.1 for Πφ (H  ), we shall use a global argument. 11.4. Globalization Let k be a totally real number field, and B a quaternion algebra over k, such that: (i) kw0 = kw1 = kw2 = F for three finite places w0 , w1 and w2 of k; (ii) B is ramified precisely at w1 and w2 . As in Section 10.3, we can consider a split rank 2 quaternionic Hermitian space VB over B whose associated similitude group HB is an inner form of H, so that HB,w1 ∼ =  . H HB,w2 ∼ = Fix an irreducible supercuspidal representation τw0 = τ1,w0  τ2,w0 of C(kw0 ) = C(F ), such that τ1,w0 and τ2,w0 are inequivalent; this determines an endoscopic L-packet + − {πw , πw } of HB (kw0 ) = H(F ). As in Section 8.3, we may construct an irreducible 0 0 cuspidal representation Ξ = Ξ1  Ξ2 of C(Ak ) such that: (i) (ii) (iii) (iv) (v)

Ξ1 and Ξ2 have the same central character; Ξw0 ∼ = τ1,w0  τ2,w0 ; Ξw1 ∼ = τ1  τ 2 ; Ξ1,w2 is unramified and Ξ2,w2 is supercuspidal; Ξv is unramified for all non-archimedean places v = w0 , w1 , w2 .

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11.5. Applying the stable trace formula We are going to apply the stable trace formula for HB . For this, we need to specify  the test functions f HB = v fvHB . For the purpose of proving Proposition 11.1(i), we consider f HB such that: • • • •

fwH0B fwH1B fvHB fvHB

+ − = fw+0 + fw−0 where fw+0 (resp. fw−0 ) is a pseudo-coefficient of πw (resp. πw ); 0 0 HB and fw2 are free; belongs to the spherical Hecke algebra for all finite places v = w0 , w1 , w2 ; is free if v is archimedean.

We pick the test function f H =

 v

fvH on H(A) such that:

• fvH = fvHB for all places v where Bv is split; • fvH matches fvHB if Bv is ramified. Then observe that the orbital integral of fwH0B = fwH0 is stable by Lemma 8.14. Thus, the transfer of both fwH0B and fwH0 to C(kw0 ) is zero. Thus, applying the stable trace formula of HB , we obtain:     IHB ,disc f HB = SH,disc f H .

Let S = V∞ ∪ {w0 , w1 , w2 }, where V∞ is the set of archimedean places of k. By the linear independence of characters of the spherical Hecke algebra outside S, we may extract the near equivalence classes on both sides corresponding to Ξ. Now by Section 3, we know precisely what are the representations which contribute to both sides. By linear independence of characters at archimedean places, we may further extract the contribution of both sides associated to a particular representation of H(F∞ ), for example the representation corresponding to the trivial character of the archimedean component group. Taking into account Corollary 8.16, we obtain:  Π

      ΘΠw 0 fwH0B · ΘΠw 1 fwH1B · ΘΠw 2 fwH2B

=

      1 · (Θπw+ + Θπw− ) fwH0 · (Θπw+ + Θπw− ) fwH1 · Θπw++ fwH2 0 1 2 0 1 2

(11.2)

where the sum on the LHS runs over all cuspidal Π  in the endoscopic part of the automorphic discrete spectrum of HB indexed by Ξ with • Πv unramified for all finite places v = w0 , w1 , w2 ;  • Π∞ corresponds to the trivial representation of the archimedean component group; +− . • Πw 2 = πw 2

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Thus, only the local components of Π  at w0 and w1 are not completely specified. By our knowledge of the endoscopic part of the automorphic discrete spectrum, we see that there are 2 Π  which contribute to the LHS, namely with ++ and Πw 1 = π +− , or • Πw 0 = πw 0  −− • Πw0 = πw0 and Πw 1 = π −+ .

In either case, we have:   ΘΠw 0 fwH0B = 1

  and (Θπw+ + Θπw− ) fwH0 = 2. 0

0

In addition, by the case of principal series considered in Section 11.2, we have     Θπw++ fwH2 = Θπw+− fwH2B . 2

2

Thus we conclude that:         Θπ+− fwH1B + Θπ−+ fwH1B = Θπ++ fwH1 + Θπ−− fwH1 .

This proves Proposition 11.1(i). To prove the more interesting part (ii), where the normalization of the local transfer factor plays a crucial role, we pick test functions f HB and f H as above, except that at the place w0 , we have fwH0B = fwH0 = f + − f − . In this case, fwH0 is killed by any stable distribution and hence the stable trace formula for HB reads:   1   IHB ,disc f HB = · SC f C . 4

As above, by linear independence of characters outside of {w0 , w1 , w2 }, we extract the part on both sides corresponding to the near equivalence class associated to Ξ, with the archimedean representations corresponding to the trivial character of the archimedean component group. The only representations which contribute to the RHS are Ξ1  Ξ2 and Ξ2  Ξ1 . Thus, we have:  Π

      ΘΠw 0 fwH0B · ΘΠw 1 fwH1B · ΘΠw 2 fwH2B

=

      1 · Θτ1,w0 τ2,w0 fwC0 · Θτ1 τ2 fwC1 · ΘΞ1,w2 Ξ2,w2 fwC2 4       1 + · Θτ2,w0 τ1,w0 fwC0 · Θτ2 τ1 fwC1 · ΘΞ2,w2 Ξ1,w2 fwC2 , 4

(11.3)

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where the sum on the LHS runs over the two cuspidal Π  which contribute to the LHS of (11.2). Thus the LHS of (11.3) is:             Θπw++ fwH0B · Θπ+− fwH1B · Θπw+− fwH2B + Θπw−− fwH0B · Θπ−+ fwH1B · Θπw+− fwH2B . 0

2

0

2

Now we note: • At the place w0 , with fwH0B = f + − f − , we have:     Θπw++ fwH0B = 1 = −Θπw−− fwH0B 0

0

on the LHS of (11.3), and         Θτ1,w0 τ2,w0 fwC0 = Θτ2,w0 τ1,w0 fwC0 = Θπw++ fwH0B − Θπw−− fwH0B = 2 0

0

on the RHS of (11.3). • At the place w1 , we have     Θτ1 τ2 fwC1 = −Θτ2 τ1 fwC1

on the RHS of (11.3). • At the place w2 , we have         ΘΞ1,w2 Ξ2,w2 fwC2 = −ΘΞ2,w2 Ξ1,w2 fwC2 = ΘΠw 2 fwH2B = Θπw+− fwH2B 2

by the result for principal series representations shown in Section 11.2. Putting these observations together, we obtain the desired identity       Θπ+− fwH1B − Θπ−+ fwH1B = Θτ1 τ2 fwC1 .

This proves Proposition 11.1(ii). 11.6. Twisted Steinberg We are left with proving Proposition 11.1(i) for the non-endoscopic discrete series L-packets of H  (F ). We begin by treating the case of twisted Steinberg representations; the treatment is analogous to that in Section 10.1. Let StGL(2) denote the Steinberg representation of GL(2, F ). Let 1GL(2) (resp. 1D× ) denote the trivial representation of GL(2, F ) (resp. D× ). Let StH (resp. StH  ) denote the Steinberg representation of H (resp. H  ). For a character χ of F × , we have the twisted Steinberg representations StH ⊗ χ and StH  ⊗ χ, with associated pseudo-coefficients fStH ⊗χ and fStH  ⊗χ respectively. We shall show:

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Lemma 11.4. The following identity holds for all t ∈ H  (F )reg with norm t ∈ H(F ):   ΘStH  ⊗χ t = −ΘStH ⊗χ (t).

In particular, ΘStH  ⊗χ is a stable function, and fStH ⊗χ and fStH  ⊗χ are matching test functions. Proof. Let M (resp. M  ) denote the Levi subgroup of the Siegel parabolic subgroup of H (resp. H  ). We let ν denote the absolute value character of D× or F × , depending on the context. We have the following semi-simplifications of parabolically induced representations of H and H  (see [ST,GTW]):   ν 3/2 StGL(2)  ν −3/2 χ = StH ⊗ χ + L ν 3/2 StGL(2) , ν −3/2 χ ,

ν 3/2 1D×  ν −3/2 χ = StH  ⊗ χ + χH  .

(11.5)

For all regular g  ∈ D× with norm g ∈ GL(2, F ), we have:   Θ1D× g  = −ΘStGL(2) (g).

By the transfer of principal series representations, we obtain:   Θ(ν 3/2 1D× ν −3/2 χ) t = −Θ(ν 3/2 StGL(2) ν −3/2 χ) (t),

for all t ∈ H  (F )reg with norm t ∈ H(F ). Hence, by the identities (11.5), we have:     ΘStH  ⊗χ t + ΘχH  t = −ΘStH ⊗χ (t) − ΘL(ν 3/2 StGL(2) ,ν −3/2 χ) (t).

(11.6)

Observing that χH (t) = χH  (t ), we see that to prove the lemma, it remains to show that the character of χH + L(ν 3/2 StGL(2) , ν −3/2 χ) vanish on all elements of H(F ) which are in the image of the norm map from H  (F ); these are precisely the elliptic elements and the elements which are elliptic in the Siegel Levi subgroup M (F ). To see this, we note that   ν 2  ν −1 χ1GL(2) = L ν 3/2 StGL(2) , ν −3/2 χ + χH ,

where the representation on the LHS is induced from the Heisenberg parabolic subgroup Q. Since the elements of H(F ) which are in the image of the norm map from H  (F ) do not lie in the Levi subgroup of Q, we have the desired vanishing. The lemma is proved. 2

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11.7. Applying the stable trace formula again We can now deal with the remaining non-endoscopic discrete series L-packets by using a global argument. Given such an L-parameter φ with associated representation π  of H  (F ), we have the L-parameter (φ, sim φ) of G(F ) with associated representation Π. We shall make use of the globalization which we carried out in Section 9.2. As in the context of Section 9.2, we work over a totally complex number field k with 4 places v0 , v1 , v2 and v3 such that kvi = F , and let B be a quaternion k-algebra ramified at precisely the places vi . Let VB (resp. WB ) be the rank 2 (resp. rank 3) quaternionic Hermitian (resp. skew-Hermitian) space over B with similitude group HB (resp. GB ), which is an inner form of H (resp. G). In Section 9.2, we constructed a ˜ B to HB is nonzero. ˜ B of GB such that the global theta lift of Π cuspidal representation Π We consider the cuspidal representation ˜B) π ˜B = Θ(Π on HB (Ak ), so that: • π ˜B,v0 = π  ; • for i = 1, 2, 3, π ˜B,vi = μi StH  is a twisted Steinberg representation; • π ˜B,v is unramified for all finite places v = vi . ˜ be the Jacquet–Langlands lift of Π ˜ B to G(Ak ). In Section 9.2, we showed that Π ˜ Let Π has nonzero theta lift π ˜ to H(Ak ).  Now we apply the stable trace formula of HB with test functions f HB = v fvHB and  f H = v fvH satisfying: • for i = 1, 2, 3, fvHi B is the pseudo-coefficient of the twisted Steinberg representation μi StH  ; • for finite places v = vi , fvHB belongs to the spherical Hecke algebra; • at v0 and archimedean places, fvHB is free; and • for i = 1, 2, 3, fvHi is the pseudo-coefficient of the twisted Steinberg representation μi StH ; • at v0 , fvH0 matches fvH0B ; • at all other places, fvHB = fvH . Then it follows by Lemma 11.4 that f HB and f H are matching and the transfer of f HB and f H to C is zero. Thus, the stable trace formula for HB reads:       IHB ,disc f HB = SH f H = IH,disc f H .

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Now using linear independence of characters of spherical Hecke algebras and at the archimedean places, we extract the part on both sides corresponding to the near equivalence class of π ˜B . This gives:  ρ

3        m ρ · Θρv0 fvH0B · Θρv fvHi B i

i=1

=



3      m(ρ) · Θρv0 fvH0 · Θρvi fvHi

ρ

(11.7)

i=1

where the sum runs over cuspidal representations ρ or ρ such that • at all archimedean or finite places v = vi , ρv = ρv = π ˜v ; • for i = 1, 2, 3, ρv and ρv are unitary representations which are not killed by the pseudo-coefficient of the relevant twisted Steinberg representations. Moreover, m(ρ) (reps. m(ρ )) is the multiplicity of ρ (resp. ρ ) in the automorphic discrete spectrum of H(Ak ) (resp. HB (Ak )) and so is a positive integer. In particular, we see that ˜ contributes to the RHS. π ˜B contributes to the LHS whereas π We have seen in Section 10.6 that for i = 1, 2, 3, ρvi is necessarily equal to the twisted Steinberg representation μi StH . Using the same argument, and (11.5), we see that ρvi is equal to the twisted Steinberg representation μi StH  as well. Thus, we have 3 

3      Θρv fvHi B = Θρvi fvHi = 1. i

i=1

i=1

Moreover, by Corollary 10.11, we conclude that the only representation which contributes to the RHS of (11.7) is the representation π ˜ , with m(˜ π ) = 1. Hence, (11.7) simplifies to:  ρ

      m ρ · Θρv0 fvH0B = Θπ fvH0 .

11.8. Final step We shall see that the only representation which contributes to the LHS is ρ = π ˜B , with m(˜ πB ) = 1, using a similar argument as in Section 10.7. More precisely, by consideration of central exponents, an application of Casselman’s theorem relating central exponents to asymptotics of character distributions and application of Casselman’s square-integrability criterion, we first deduce that all representations ρv0 on the LHS are discrete series representations. ˜ , we let fi be the Now if there is another representation ρ2 intervening besides ρ1 = π   pseudo-coefficients of ρi and set fvH0 = f1 + f2 , so that the orbital integral of fvH0 is supported on the elliptic set. Then we have

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           2  m ρ1 + m ρ2 = m ρ · Θρv0 fvH0B = Θπ fvH0 . ρ

Now if fπ is a pseudo-coefficient of π, then the orbital integral O(fπ ) is stable since we know that π is stable. Hence we have:      Θπ fvH0 = Θπ , O fvH0

H,e

   = O(fπ ), O fvH0

H,e

   = SO(fπ ), SO fvH0

H,e

where the last equality is (4.2). By Cauchy–Schwarz inequality, (4.2) and elliptic orthonormality of irreducible discrete series characters, we thus see that        1/2 1/2  Θπ fvH0  SO(fπ ), SO(fπ ), H,e · SO fvH0 , SO fvH0 H,e      1/2 1/2  = O(fπ ), O(fπ ), H,e · SO fvH0 , SO fvH0 H,e      1/2 = SO fvH0 , SO fvH0 H,e .

By Theorem 4.3(iv), we deduce that (since OutH (H  ) is trivial),   H   SO fv0 , SO fvH0

1/2 H,e

      O fvH0 , O fvH0

1/2 H  ,e

=

√

2.

√ Thus we obtain the desired contradiction: 2  2. We have thus completed the proof of Proposition 11.1. Acknowledgments This project was started and largely completed in the academic year 2008–2009, when both authors were at the University of California, San Diego. Due to the indolence of the second author, the paper has languished in an electronic folder for the past 4 years, and would not have seen the light of day if not for the persistence of Chung Pang Mok. We would like to thank him for his encouragement to complete the project and for his many comments on an earlier draft of the paper. We also thank the referee of our paper for many useful comments and corrections to numerous inaccuracies and typos. In the intervening period, several progress in the theory of the twisted trace formula and its stabilization as well as in the theory of theta correspondence have occurred, including the recent papers of Shelstad [Sh1,Sh2,Sh3,Sh4], Labesse–Waldspurger [LW], Waldspurger [W4] and Yamana [Y]. As a result of these advances, parts of our paper have been streamlined and we have incorporated these improvements into the current version. During the course of this work, the second author was partially supported by NSF grant DMS-0801071 and is currently partially supported by an MOE-AcRF Tier One grant R-146-000-155-112. Finally, the work of Professor Rallis has had a huge impact on the work of the second author. His mathematical insight and enthusiasm will be dearly missed.

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