=B"
v
fv:, λ
.
Thus cκ=1.
Local applications
12. Local Bessel identities and functional equations—the general case In this section we obtain the local Bessel identities introduced in Section 3, i.e., we show that BI=bc-gen. More generally, we show that BIM=Mbc-gen for any Levi subgroup M. We also deduce functional equations for the local open periods. The Bessel identities generalize (and refine) the main local results of [LR00] and [Off07] which treat the case M=M0. New identities are also obtained for the closed periods. The proof is by global means using ideas originated in [LR00] and applying the Bessel identities obtained in Section7in the split case, Section8for unramified data and Section11 in the global setting. In order to use the global result we first give a standard globalization argument.
Many variants of this are known in the literature. For completeness we provide a proof.
In order to simplify notation, for this lemma only G=GLnover the number field F.
Lemma12.1. — Let S be a finite set of finite places of F and let u∈/ S be an auxiliary finite place. LetUS be a non-empty open subset of i
v∈Sa∗G andδS∈GusqrS . Suppose that there exist two distinct places u1,u2∈S such thatδui ∈Gcuspui , i=1,2. Then there existsπ ∈C which is unramified outside S∪ {u}andμ∈USsuch thatπSδS[μ].
Proof. — We first remark that we can assume without loss of generality thatUS= i
v∈Sa∗G. Indeed, granted the Lemma in that case, we can always find a unitary Hecke character χ :F∗\A∗F →C∗ which is unramified outside u, and such that χS =eμ,H(·) whereμlies in a prescribed open set of i
v∈Sa∗G(cf. [LR00, Corollary 2]).
Henceforth, assume thatUS=i
v∈Sa∗G. We apply Arthur’s trace formula for f =
vfv∈C∞c (GA) (or rather, its restriction to Ker|det|) where fv are chosen as follows.
For v∈S we take fv to be the product of the characteristic function of Ker|det|v by
a pseudo matrix-coefficient of δv (cf. [DKV84]). Thus Trσv(fv)=0 for any irreducible representationσvof Gvwhich is either an induced representation from a proper parabolic subgroup or a discrete series representation which is not an unramified twist ofδv. For v|∞we take fv=gv∗gv∨where gv∈C∞c (Gv)is any non-zero bi-Kv-invariant function. At u we take futo be the characteristic function of a sufficiently small compact open subgroup of Gu. At all other places we take fv=1Kv. By our choice of f and the assumption onδS, the spectral side of the trace formula applied to f is #
Tr(σ (f)) where σ ranges over the cuspidal representations of GA which are unramified outside S∪ {u}and such thatσS is an unramified twist ofδS. On the geometric side, note that if f(g−1γg)=0 for some g∈GAandγ ∈G then the coefficients of the characteristic polynomial ofγ are adelically bounded and u-adically close to those of(t−1)n(because of the choice of fu). Thusγ is unipotent. It follows from Arthur’s description of the geometric side of the trace formula [Art86] that only the unipotent conjugacy classes contribute. Suppose that γ is a non-trivial unipotent element of G. Then there exists a proper parabolic subgroup P such that the class ofγ intersects U in a Zariski open dense subset. The invariant unipotent orbital integral of fv atγ is given by%
Pv\Gv
%
Uvfv(g−1ug)du dg [LM09b, Lemma 5.3]. Therefore it vanishes for v=u1,u2 sinceδui is supercuspidal.10 Hence, by Arthur’s description of the unipotent contribution [Art85], the non-trivial unipotent contributions vanish as well and the geometric expansion reduces to vol(AGG\GA)f(e). It remains to recall that for allv∈S, fv(e)is the formal degree ofδv, and therefore f(e)=0.
12.1. The main local results. — We turn back to the local inert case. For the next result letδ∈sqrandδ=bc(δ). Recall thatδ∈sqr if and only ifbδ>1 if and only if δδ·η. In the Archimedean case this is further equivalent to the condition n=1.
Proposition12.2. — Letδ∈sqrandδ=bc(δ). Thenδ∈BI and the following condi-tions are equivalent for any x∈X.
(1) δis Gx-distinguished.
(2) αδx ≡0.
(3) Gx is quasi-split orδ∈sqr.
More generally, suppose thatδ∈Msqr and letδ=bc(δ)=δ1⊗ · · · ⊗δt. Thenδ∈BIMand the following conditions are equivalent for any x∈X.
(1) There existsλ∈a∗M,Csuch that JM(x, αδ, λ)≡0.
(2) There existsλ∈a∗M,Csuch thatJδ(x, αδ, λ)≡0.
(3) JM(x, αδ, λ)≡0 whenever holomorphic.
(4) There exists y∈M∩x•G such thatαδy ≡0.
(5) There exists y∈M∩x•G such thatδis My-distinguished.
10Actually, this vanishing holds for allv∈S.
(6) There exists y=diag(y1, . . . ,yt)∈M∩x•G such that Gyni
i is quasi-split except possibly (in the p-adic case) for a single index i0for whichδi0∈sqr.
(7) w(δ)˜ ≤w(x).
(8) In the p-adic case: Gxis quasi-split orbδ>1. In the Archimedean case: dimaM0 ≤w(x).
Proof. — We begin with the first set of equivalences. Suppose first thatbδ=1 and Gxis not quasi-split. Thenδ is not Gx-distinguished. In the p-adic case this follows from Theorem6.1. In the real case this is because Gxis conjugate to K andδis not unramified.
It further follows from the first part of Lemma3.8thatδ∈BIx(withαδx ≡0).
For the first part of the proposition, it remains to see that ifδ∈sqrand Gxis quasi-split or bδ >1 then αδx ≡0 (and δ∈BIx). We may assume without loss of generality that δ∈usqr. We use a global argument, starting with the p-adic case. Switching the notation, we consider now a quadratic extension of number fields E/F which is split at infinity and a placev1of F such that Ev1/Fv1 is our given local extension. Let y∈Xv1 and ifδδ·ηassume further that Gyv
1 is quasi-split (i.e., y∈X+v
1 since n is even). Letv2=v1 be an additional auxiliary finite inert place of F.Fixδv
2 ∈G
v2
usqrsuch thatδv
2 δv
2·η(for instance the Steinberg representation). Take yv2 ∈Xv2 such that ηv2(w0yv2)=ηv1(w0y).
Then there exists x∈X which is contained in the local orbits y•Gv1,yv2 •Gv2 and w0•Gvfor allv=v1, v2.By Lemma12.1, there existsπ∈Csuch thatπv
1 δ[s1]and πv
2 δv
2[s2]for some s1,s2∈iR. Note thatπ =bc(π)is cuspidal thanks to our choice ofδv
2 and that Gxvis quasi-split for allv=v1, v2. It follows from part(3)of Corollary10.3 thatπ is Gx-distinguished and thatπv ∈BIxandαπxv ≡0 for allv. By (3.11) and (3.12) we therefore get thatδ∈BIyandαδy≡0.
Consider now the case C/R. In this case n≤2. The case n=1 is trivial and we treat the case n=2 by using the quadratic extension E/F=Q[√
−1]/Q.Let y∈X∞. Note that δδ·η for any δ∈sqr and therefore we only need to consider the case y∈X+∞. Since n=2, we have X+∞=w0•G∞and X−∞=(±e)•G∞.Let D be the mul-tiplicative group of the quaternion algebra which ramifies at{3,∞}. Take any infinite-dimensional irreducible automorphic representationσ of DA which is trivial at 3 and up to an unramified twist transfers to δ via the Jacquet-Langlands correspondence at ∞. Then the Jacquet-Langlands transferπofσ satisfies the following conditions:
(1) π∈C,
(2) π∞ is an unramified twist ofδ, (3) π3 is the Steinberg representation.
As beforeπ =bc(π)is cuspidal (since 3 is inert). It follows from the last part of Corol-lary10.3thatπ is Gw0-distinguished,π∞ ∈BIw0 andαπw∞0 ≡0. As before, it follows that δ∈BIy andαδy ≡0. This completes the proof of the first statement.
As for the second statement, the first four conditions are clearly equivalent by part (4)of Proposition4.1. The other conditions are equivalent by virtue of the first part and
the structure of M∩x•G.
Corollary12.3. — Let π ∈τgen and x∈X be such that w(π )˜ ≤w(x). Thenπ is Gx -distinguished. Thus, Conjecture6.12 holds for generic representations. In particular,π ∈τgen is Gx -distinguished if Gxis quasi-split or ifπ ∈τ,an(and in particular, by Lemma3.3ifπ ∈unr).
Proof. — There existsδ∈Msqr,δ=bc(δ)such thatπ =I(δ). Sincew(π )˜ = ˜w(δ), the corollary follows from the second part of Proposition 12.2 together with
Corol-lary4.2.
We also remark that in the p-adic case, ifδ∈M,τsqr and is a set of representatives (of size 2t−1) for[B](δ)then
(12.1)
αδ|x•G∩M:δ∈
are linearly independent inEM
x•G∩M, δ∗ . This follows immediately from the non-vanishing of αδx and the relation (3.13) (for G=M).
We are now ready to state and prove the main local result. Recall the distribution B(σ˜ , λ)defined in (4.13).
Theorem12.4. — Let E/F be a quadratic extension of local fields. Then for all M we have Mbc-gen =BIM. In particular,bc-gen=BI. Moreover, for any n there exists a signυn=υn(E/F)=
±1 depending only on E/F withυ1=1 and with the following properties. Fix M andσ∈Mbc-gen . Then
(1) As an identity of meromorphic functions inλ∈a∗M,Cwe have
(12.2) α
σ, λ
=υκJσ
ασ, λ
and
(12.3) B˜
σ, λ
←→υκB σ, λ whereυκ=tυn
i=1υni. In particular,B(σ˜ , λ)is entire.
(2) Jσ(ασ,·)is holomorphic at any irreducibility point of I(σ, λ)and we have (12.4) αI(σ,λ)◦W(σ, λ)=υκJσ
ασ, λ . (3) If E/F is unramified thenυn=1 for all n.
(4) For anyw∈W(M)we have
(12.5) Jwσ
αwσ, wλ
◦N
w,W(σ ), λ
=Jσ
ασ, λ .
Proof. — First note that the relations (12.2) and (12.3) are equivalent from the definitions ofα(σ, λ),B(σ˜ , λ)andB(σ, λ).
We begin with the p-adic case. We first prove that Msqr ⊆BIM and that (12.2) holds forσ=δ=δ1 ⊗ · · · ⊗δt∈Msqr for some signυM. The relation (12.4) would then follow from Lemma 3.6. By Lemma3.7, we may assume without loss of generality that δ∈Musqr .
Let δ=bc(δ). By Proposition12.2we haveδ∈BIM. As in the proof of Propo-sition 12.2we switch to global notation. Thus, assume that δ∈Musqr(k) andδ=bc(δ) with respect to a quadratic extension k/k of p-adic fields. Choose a quadratic exten-sion of number fields E/˜ F which splits at infinity and such that at some place˜ v0 of F˜ we have E˜v0/F˜v0 k/k. We can choose a different quadratic extension F of F which˜ splits at v0 and such that Fv ˜Ev at all places v=v0 of F which are inert in˜ E and˜ are even or ramified at E. Let˜ v1, v2 be the places of F above v0. Then E= ˜EF is a quadratic extension of F which splits at infinity and such that Evi/Fvi k/k, i =1,2 and all other inert places of E/F are odd and unramified. Let u,u1,u2be three auxiliary finite places of F which split in E and let U ⊆(ia∗M)2 be an arbitrary non-empty open set. Denote by StMv2 =Stn1⊗ · · · ⊗Stnt the Steinberg representation of the group Mv2. It follows from Lemma12.1that there existsσ=σ1⊗ · · · ⊗σt∈CM unramified outside {v1, v2,u,u1,u2}such thatσv
1δ[μv1]andσv
2StMv2[μv2]for some(μv1, μv2)∈U. In particular,σi=σi·ηfor all i=1, . . . ,t since the same holds for the Steinberg represen-tation. Thereforeσ =bc(σ)is also cuspidal. Applying Proposition11.3and comparing the factorizations (11.3) and (11.12) we obtain that for every place v, σv ∈BIMv and there is a non-zero meromorphic functionυM,v(σv, λ)such that
α σv, λ
=υM,v σv, λ
Jσv
ασv, λ
and moreover
vυM,v(σv, λ)=1. Note that by (9.10), (9.12) and (9.8)υM,v(σv, λ)does not depend onψv. Taking into account (7.3) in the split case and (8.5) and Remark9.3 in the unramified case we infer that
υM,v1 σv
1, λ υM,v2
σv
2, λ
=1.
By Lemma3.7and (4.6) we haveδ∈BIMv1 and υM,v1
σv
1, λ
=υM,v1
δ, λ+μv1
.
Similarly StMv2 ∈BIMv2 and υM,v2
σv
2, λ
=υM,v2
StMv2, λ+μv2 . We infer that
υM,v1
δ, λ+μv1 υM,v2
StMv2, λ+μv2
=1.
Since(μv1, μv2)∈U andU was an arbitrary open set we infer that υM,v1
δ, λ+μ1 υM,v2
StMv2, λ+μ2
=1
for all μ1, μ2. Hence υM,v1 =υM,v1(δ, λ) is independent of both δ and λ. Moreover, takingδ=StMv2, we infer thatυM,v1 =υM,v2 is a sign depending only on M and the local quadratic extension. Switching back to the local notation, we simply denote it by υM. This shows (12.2) in the square-integrable case up to the determination of the sign.
Consider the general case ofσ∈Mbc-gen . Let Lbe a Levi subgroup of M,δ∈Lsqr and μ∈a∗L,C be such that σ IML(δ, μ). Applying the above argument for δ with respect to both Gand M and using Lemmas4.4and3.6we obtain
(12.6) υLM α σ, λ
=υLJσ
ασ, λ .
In particular, (12.2) holds whenever σ (resp. σ) is a principal series representa-tion of M (resp. M) with proportionality constant υM0/υMM
0. Using Lemma 12.1, and a global setting as before, this time for S = {v1,u1,u2}, we immediately obtain that υM=υM0/υMM
0. Thus υM= υn
iυni where we set υn=υMG
0. Hence υM=υκ, and it also follows thatυLυLM=υM. We obtain (12.2) from (12.6).
For E/F unramified it now follows from (8.5) thatυn=1.
We turn to the Archimedean case. Consider first the case where δ∈Musqr . Take E/F=Q(√
−1)/Q. Choose a globalization σ =σ1⊗ · · · ⊗σt of δ such that at the prime 2 (denoted by the place v2) we have (σi)v2 (σi)v2 ·ηv2 for all i and set σ = bc(σ). (For existence of suchσ we may argue as in the Archimedean part of the proof of Proposition 12.2.) It follows from Lemma 2.2 and strong multiplicity one that the distributions
p<∞rl(σp, λ) B(σp, λ), σ∈B(σ ) are linearly independent. It further follows from the result already established in the p-adic case that
"
p<∞
rl
σp, λB˜ σp, λ
←→υM,v2"
p<∞
rl σp, λ
B σp, λ
.
The factorizations (11.3) and (11.12) together with the global identity (11.13) imply that B˜
δ, λ
←→υM,v2B δ, λ
.
In particular, υM,v∞ =υM,v2 is independent of δ and λ. Back in a local Archimedean setting, arguing as before in the p-adic case, one now proves the identities (12.2) for any σ∈Mbc-gen withυn,v∞=υn,v2.
Finally, we show that (12.5) follows from (12.4). Indeed, since π :=I(σ, λ) I(wσ, wλ)andυwκ =υκ, applying (12.2) to(wσ, wMw−1)we get
απ◦W(wσ, wλ)=υκJwσ
αwσ, wλ .
Composing this with N(w,W(σ ), λ)and using the functional equation (1.10) we obtain απ◦W(σ, λ)=υκJwσ
αwσ, wλ
◦N
w,W(σ ), λ .
Using (12.4) once again we infer (12.5).
From (11.13) we now deduce
Corollary12.5. — Proposition 11.3holds for any quadratic extension of numbers fields E/F (not necessarily split at infinity). Moreover,
vυn(Ev/Fv)=1.
Remark12.6. — In order to determine the signsυnit is necessary to have more in-formation about the matching at the ramified places. For instance, a factorη(−1)cannot be discerned at the split or unramified places, and can be incorporated into the transfer factor at will, resulting in a possible sign change in theα’s.
12.2. Bessel identities for closed Bessel distributions. — Next, we prove a similar relation for the closed periods. Recall the distributionD˜L(, λ)defined in (5.16).
Proposition 12.7. — For any k≥0 and a local extension of quadratic fields E/F there exist signsςk with the following property. Let n=2k,κa composition of k, L=M(κ,←κ )− an even symmetric Levi and=σ⊗←σ−τ ∈L,wgen0. Then for anyλ=(μ,←μ )− ∈(a∗L)wC0 such that I(, λ)is irreducible we have
(12.7) απ◦W(, λ)=ςkλkψZ β, λ whereπ=I(ai(σ ), μ)(withμ∈a∗M
2κ,Cas in Corollary5.6) is the unique element ofB(I(, λ)) such thatπ·η=π. Equivalently,
(12.8) D(, λ)˜ ←→ςkBI(ai(σ ),μ).
Proof. — The equivalence of (12.7) and (12.8) follows from the definition (5.16) of D and the relation (2.9) applied to the equivalence˜
W(, λ),W(∨,−λ) :I
W(), λ
→W
I(, λ) .
By Corollary 5.10 and (2.14) we reduce (12.8) to the case where σ =δ1⊗ · · · ⊗δs∈ Musqrκ . In this case we argue exactly as in the proof of Theorem 12.4. First note that by (2.14), (9.5), (9.6) and (9.16) the identity (12.8) is independent ofψ. In the p-adic case, switching to a global (split at infinity) setting, we globalizeδitoi∈CGni,¬τ, i=1, . . . ,s (by using an auxiliary split place). By (11.17) we get (12.8) up to a constant. The argument in the proof of Theorem 12.4 using Proposition 7.5 and Corollary 8.4 shows that the proportionality constant is a sign which depends only on k. For the real case we argue once again as in Theorem12.4applying (11.17) toQ[√
−1]/Q.
Now let L=M(κ,m,←κ )− be odd symmetric. We will use the setup of Proposition5.4 and its notation. Namely, let w=w(m,k,k)(m+k,k), L˜ =w−1Lw=M(m,κ,←−κ ), M˜ =M(m,2k), w,∗= diag(1m, w0G2k).
Let =σ ⊗ δ ⊗ ←−δτ ∈ L,gen˜w,∗ with σ ∈ Ggenm,τ and δ ∈ Mgenκ. For any α ∈ EGm(Xm,W(σ )∗)andβ∈EM(κ,←κ )−(XGM2k
(κ,←−κ ),W(δ⊗←−δ τ)∗)defineα⊗β∈EL˜(XML˜˜,W()∗) by
(α⊗β)diag(x1,x2)=αx1⊗βx2, x1∈Xm, x2∈XGM2k
(κ,←−κ ).
We will view w(α⊗β) as an element of EL(XML(k,m,k),W(w)∗) by identifying wW() withW(w)(as in our convention for the intertwining operators N(w,W(), λ)).
Proposition12.8. — With the notation above, for anyλ=(z, μ,←μ )− ∈(a∗˜
L)wC˜ with z∈C, μ∈a∗M
κ,Csuch that I(, λ)∈gen and for anyσ∈B(σ )we have (12.9) αI(,0)◦W(w, wλ)= υm+2k
υmυ2kςkλ(1ψ−1−m)kZ w
ασ⊗βδ⊗←−δτ , wλ as elements ofEG(X,I(W(w), wλ)∗)where=σ[z] ⊗I(ai(δ), μ)∈Mgen˜.
Proof. — First note that by (1.10) and (1.6) we have W(w, wλ)=W(, λ)◦N
w,W(), λ−1
=WM˜
IML˜˜(, λ),0
◦IM˜
WML˜˜(, λ)
◦L,˜M˜
W(), λ
◦N
w,W(), λ−1
.
Recall that bc()=I(, λ)=IM˜(IML˜˜(, λ),0). Thus, by (12.4), the left-hand side of (12.9) is
υm+2k υmυ2kJ
α,0
◦IM˜
WML˜˜(, λ)
◦L,˜M˜
W(), λ
◦N
w,W(), λ−1
. Note thatα =ασ[z]⊗αI(ai(δ),μ). Therefore we infer from (12.7) that
J α,0
◦IM˜
WML˜˜(, λ)
=ςkλkψJ ZM˜
ασ⊗βδ⊗←−δτ, λ ,0
.
The proposition therefore follows from (5.14) sinceλψ−1=η(−1)λψ. The following Corollary strengthens Corollary12.3(taking into account the rela-tion (3.8)).
Corollary12.9. — Letπ∈bc-gen. Thenαπx ≡0 iffw˜(π)≤w(x). In particular, in the p-adic caseαπx≡0 unless Gxis not quasi-split andππ·η.
Proof. — The p-adic case follows from Lemma 3.8(and will also follow from the argument below).
By Lemma 3.2 we can write π =σ×ai() where ∈Ggenk, k = ˜w(π) and
˜
w(σ)=0. By (12.9) the non-vanishing of απx is equivalent to the non-vanishing of Z(x, w(ασ⊗β⊗←−τ),0). In view of Lemma 5.2 and Lemma 5.1 part (1), it suffices to prove the Corollary forσ. That is, we may assume thatw˜(π)=0.
Suppose therefore that w˜(π)=0. We have to show that απx ≡0 for all x. We reduce the statement further to the tempered case. Write π as π =I(σ, λ) where σ∈Mtemp and λ=(λ1, . . . , λt)∈(a∗M)+. By Theorem 12.4 the non-vanishing of απx is equivalent to that of Jσ(x, ασ, λ). Recall that CM(wM:σ,·;ψ ) is holomorphic and non-zero atλ(1.11) and therefore the same is true for CM(wM:σ, λ;ψ). Hence, the non-vanishing ofJσ(x, ασ, λ)is equivalent to that of J(x, ασ, λ). In turn, the latter con-dition is equivalent to the existence of y∈M∩x•G such that ασy ≡0. Therefore it suffices to prove the corollary forσ.
Hence, we can assume that π∈temp andw˜(π)=0. Thus π is induced from δ =δ1 ⊗ · · · ⊗δt ∈Musqr with δiδj·η for any i,j. Let δ=bc(δ)∈Musqr. It follows from Lemma 1.1 that nM(δ, λ) is holomorphic and non-zero at λ=0. By the same argument as before, the non-vanishing ofαπx follows from the analogous statement in the square-integrable case. The latter follows from Proposition12.2.
Remark12.10. — The Corollary gives an alternative proof of a result of Jacquet on the non-vanishing ofαπx in the quasi-split case [Jac10].
Combining Corollaries10.3and12.9we obtain Theorem0.1.
12.3. An example. — We finish this section with the following example which shows that in general, local distinction does not imply global distinction. Let E/F be˜ a quadratic extension of number fields with Galois involution τ˜ and corresponding quadratic character η˜ :F∗\A∗F→ {±1}. Let G˜n =ResE/F˜ GLn. Suppose thatσ ∈CG˜n,¬ ˜τ and let π =aiE/F˜ (σ )∈CG2n so that π =π· ˜η. Let E/F be another quadratic exten-sion such that π=π·η. (For instance, this is the case if Ev/Fv is ramified at a place where πv is unramified.) Thus,π =bc(π)∈CG2n. Let x∈X2n and let Sx be the (finite) set of places of F which are inert in E and such that x is not quasi-split with respect to Ev/Fv. Assume that for anyv∈Sxπvis unramified andE˜v/FvEv/Fv. Then by Corol-lary12.3π is locally Gx(Fv)-distinguished for allv. On the other hand, by Corollary10.3 π is not globally Gx-distinguished unless x is quasi-split with respect to E/F (i.e., unless Sx= ∅). For otherwise, if v∈Sx then πv =πv ·ηv since η˜v=ηv and hence απxv ≡0 by Corollary12.9. There is of course no difficulty in choosing x which is not quasi-split and satisfying the properties above sinceE˜v/FvEv/Fvfor infinitely many inert placesvof F.