The Jacquet Langlands correspondence via twisted descent

0.28 MB·English

by  Dihua Jiang

THE JACQUET-LANGLANDS CORRESPONDENCE VIA TWISTED DESCENT
DIHUAJIANG,BAIYINGLIU,BINXU,ANDLEIZHANG
Abstract. Theexistenceofthewell-knownJacquet-Langlandscorrespondencewasestab-
lishedbyJacquetandLanglandsviathetraceformulamethodin1970([13]). Anexplicit
construction of such a correspondence was obtained by Shimizu via theta series in 1972
6
([30]).Inthispaper,weextendtheautomorphicdescentmethodofGinzburg-Rallis-Soudry
1
0 ([10])toanewsetting. Asaconsequence,werecovertheclassicalJacquet-Langlandscor-
2 respondenceforPGL(2)viaanewexplicitconstruction.

n
a
J

7

] 1. Introduction
T
N The classical Jacquet-Langlands correspondence between automorphic forms on GL(2)
. and D , with a quaternion division algebra D, is one of the first established instances of
h ×
t Langlands functorial transfers. The existenceof such a correspondence was established by
a
m JacquetandLanglandsviathetraceformulamethodin1970([13]). Anexplicitconstruction
oftheJacquet-Langlandscorrespondencewas obtainedbyShimizuviathetaseriesin 1972
[

([30]). Shimizu’s construction was extended in the general framework of theta correspon-
2
v dencesforreductivedualpairsinthesenseofHowe([12]),andhasbeenimportanttomany
6 arithmeticapplications, including the famous Waldspurger formula for the central value of
0
the standard L-function of GL(2) ([33]), and the work of Harris and Kudla on Jacquet’s
5
0 conjectureforthecentral valueofthetripleproduct L-function ([11]), forinstance.
0
TheautomorphicdescentmethoddevelopedbyGinzburg,RallisandSoudryin[10]con-
.
1 structs a map which is backward to the Langlands functorial transfer from quasi-split clas-
0
sicalgroupsto thegeneral lineargroup. Thestartingpointoftheirmethodis thesymmetry
5
1 of the irreducible cuspidal automorphic representation on GL(n). However, their theory is
:
v not able to cover the classical groups which are not quasi-split. In this paper we extend
i their method by considering more invariance properties of irreducible cuspidal automor-
X
phicrepresentationsofGL(n),sothatthisextendeddescentisabletoreachcertainclassical
r
a
Date:January8,2016.
2000MathematicsSubjectClassification. Primary11F70,22E55;Secondary11F30.
Key words and phrases. Jacquet-Langlandscorrespondence, Automorphic Forms, Langlands Functorial
Transfers,ClassicalGroups.
The work of the first named author is supported in part by the NSF Grant DMS–1301567, that of the
second named author is supported in part by NSF Grant DMS-1302122 and by a postdoc research fund
from Department of Mathematics, University of Utah, that of the third named author is supported in part
byNationalNaturalScienceFoundationofChina (No.11501382),andbythe NationalKey Basic Research
Program of China (No. 2013CB834202), and that of the fourth named author is supported in part by the
NationalUniversityofSingaporesstartupgrant.
1
2 DIHUAJIANG,BAIYINGLIU,BINXU,ANDLEIZHANG
groupswhicharenotquasi-split. Duetothenatureofthenewlyintroducedinvarianceprop-
erty and the new setting of the construction, we call the method introduced below twisted
automorphicdescent.
Let τ be an irreducible unitary cuspidal automorphic representation of GL (A), where
2n
A is the ring of adeles of a number field F. Assume that the partial exterior square L-
functionLS(s,τ, 2)hasasimplepoleat s = 1. ItiswellknownnowthatτistheLanglands
∧
functorialtransferfromanirreduciblegenericcuspidalautomorphicrepresentationπ ofthe
0
F-splitoddspecialorthogonalgroupG (A) = SO (A). Theautomorphicdescentmethod
0 2n+1
of Ginzburg, Rallis and Soudry ([10]) and the irreducibilityof thedescent ([19]) showthat
this π is unique and can be explicitly constructed by means of a certain Bessel-Fourier
0
coefficients oftheresidual representationofSO (A)withcuspidaldatum(GL ,τ).
4n 2n
The objective of this paper is to extend the descent method of [10] to construct more
generalcuspidalrepresentationsandmoregeneralgroupswhicharepureinnerformsofthe
F-split odd special orthogonal group G (A). To this end, we take σ to be an irreducible
0
cuspidal automorphic representation of an F-anisotropic SOδ associated to a non-square
2
class δ of F , and assume that the central value L(1,τ σ) is nonzero. The main idea is
× 2 ×
to make the conditions such as L(1,τ σ) , 0 into play in the construction of more gen-
2 ×
eral cuspidal automorphic representations of classical groups using the irreducible unitary
cuspidal automorphicrepresentation τ of GL and the irreducible cuspidal representation σ
of SOδ. We refer to [22] for a more general framework of such constructions, which are
2
technicallymuch moreinvolvedthanthecurrent case in thispaper. Hence weleaveto [22]
the detailed discussionsfor general construction. We givebelow more detailed description
oftheconstructionandthemainresultsofthepaper.
1.1. A residual representation of SOδ (A). Let V be a quadratic space of dimension
4n+2
4n+2 defined over F with a non-degeneratequadratic form , . We assume that the Witt
h· ·i
indexofV is2n withapolardecomposition
V = V+ V V ,
0 −
⊕ ⊕
where V+ is a maximal isotropic subspace of V, and V is an anisotropic subspace of di-
0
1 0
mension 2. We may take the quadratic form on V to be associated to J = , where
0 δ 0 δ!
δ < F 2, and thequadraticform ofV istaken tobeassociated to
×
−
w
2n
J ,
 δ 
where w is the (r r)-matrix with 1w’s2non its anti-diagonal and zero elsewhere. Denote
r
×
by Hδ = SOδ the corresponding F-quasisplit special even orthogonal group. We fix a
4n+2
maximalflag
: 0 V+ V+ V+ = V+
F ⊂ 1 ⊂ 2 ⊂ ··· ⊂ 2n
in V+, and choose a basis e , ,e of V+ over F such that V+ = Span e , ,e . We
{ 1 ··· 2n} i { 1 ··· i}
alsolet e , ,e beabasisforV , whichis dualto e , ,e in thesensethat
1 2n − 1 2n
{ − ··· − } { ··· }
e,e = δ for1 i, j 2n,
i j i,j
h − i ≤ ≤
JACQUET-LANGLANDSAND TWISTEDDESCENT 3
andlet V = Span e ,...,e .
Let P bℓ−e the par{ab−1olic sub−gℓ}roup fixing V+. Then P has a Levi decomposition P = MU
such that M GL SOδ, where SOδ is the F-quasisplit special orthogonal group of
(V ,J ). Let τ≃be an2nir×reduc2ible unitary2cuspidal automorphic representation of GL (A),
0 δ 2n
and σ an irreducible unitary (cuspidal) representation of SOδ(A). Then τ σ is an irre-
ducibleunitarycuspidalrepresentationof M(A). For s Can2d an automorp⊗hicfunction
∈
φ (M(F)U(A) Hδ(A)) ,
τ σ τ σ
⊗ ∈ A \ ⊗
following[26, II.1], one defines λ φ to be (λ m )φ , whereλ XHδ C (see[26,
I.1] for the de§finition of XHδ and stheτ⊗σmap m ),sa◦nd Pdefiτn⊗eσs the corress∈ponMdin≃g Eisenstein
§ M P
seriesby
E(h,s,φ ) = λ φ (γh),
τ σ s τ σ
⊗ X ⊗
γ P(F) Hδ(F)
∈ \
which converges absolutelyfor Re(s) 0 and has meromorphiccontinuationto thewhole
≫
complex plane ([26, IV]). We note here that under the the normalization of Shahidi, we
§
cantakeλ = sα(see[28]),whereαistheuniquereducedrootofthemaximalF-splittorus
s
of Hδ in U.
As in[18], deefine
β(s) = L(s+1,τ σ) L(2s+1,τ, 2),
× · ∧
where the L-functions are defined through the Langlands-Shahidi method ([29]). Then
definethenormalizedEisensteinseries
E (h,s,φ ) = β(s)E(h,s,φ ),
∗ τ σ τ σ
⊗ ⊗
whichhas thefollowingproperties.
Proposition 1.1 ([18], Proposition 4.1). Let τ and σ be as above. Then the normalized
EisensteinseriesE (h,s,φ )hasasimplepoleat s = 1 ifandonlyifL(s,τ, 2)hasapole
at s = 1and L(1,τ∗ σ) ,τ0⊗.σ 2 ∧
2 ×
Let denote the automorphic representation of Hδ(A) generated by the residues at
τ σ
s = 1 Eof⊗E(h,s,φ ) for all φ (M(F)U(A) Hδ(A)) . From now on, we assume
that L2(s,τ, 2) hasτ⊗aσpole at s τ=⊗σ1∈anAd L(1,τ σ)\, 0. Inτ⊗tσhis case, τ has trivial central
∧ 2 ×
character ([15]). By Proposition 1.1, the residual representation is nonzero. By the
τ σ
L2-criterionin[26],[18,Theorem6.1]showsthattheresidualrepreEse⊗ntation issquare-
τ σ
E ⊗
integrable. Moreover, theresidual representation is irreducible, followingTheorem A
τ σ
E ⊗
ofMoeglinin [24], for instance. Notethat theglobalArthurparameter (see[2]) for is
τ σ
(τ,2)⊞ψ ([18, 6]). E ⊗
σ
§
1.2. Fouriercoefficientsattachedtopartition[(2ℓ+1)14n 2ℓ+1]. Following[16],onede-
−
fines Fourier coefficients of automorphic forms of classical groups attached to nilpotent
orbits and hence to a partition. For Hδ = SOδ , we consider here the Fourier coefficients
4n+2
attachedtothepartition[(2ℓ+1)14n 2ℓ+1]of4n+2with1 ℓ 2n,whichisalsoknownas
−
≤ ≤
Bessel-Fourier coefficients. More precisely, for 1 ℓ 2n, consider the following partial
≤ ≤
flag
: 0 V+ V+ V+,
Fℓ ⊂ 1 ⊂ 2 ⊂ ··· ⊂ ℓ
4 DIHUAJIANG,BAIYINGLIU,BINXU,ANDLEIZHANG
whichdefines thestandardparabolicsubgroup P of Hδ,withtheLevidecomposition P =
ℓ ℓ
M N . TheLevipart M (GL )ℓ SO(W ), with
ℓ ℓ ℓ 1 ℓ
· ≃ ×
W = (V+ V ) .
ℓ ℓ ⊕ ℓ− ⊥
Following [35], the F-rational nilpotent orbits in the F-stable nilpotent orbits in the Lie
algebra of SOδ associated to the partition [(2ℓ + 1)14n 2ℓ+1] are parameterized by F-
4n+2 −
rationalorbitsintheF-anisotropicvectorsinW undertheactionofGL (F) SO(W ). The
ℓ 1 ℓ
×
actionofGL (F) SO(W )onW isinducednaturallyfromtheadjointactionof M onthe
1 ℓ ℓ ℓ
×
unipotentradical N .
ℓ
Takeananisotropicvectorw W with w ,w inagivensquareclassofF ,anddefine
0 ℓ 0 0 ×
ahomomorphismχ : N G∈ by h i
ℓ,w0 ℓ −→ a
ℓ
χ (u) = u e,e + u w ,e .
ℓ,w0 Xi=2h · i −(i−1)i h · 0 −ℓi
Recall thathere , denotes thequadraticform onV. Define alsoacharacter
h· ·i
ψ = ψ χ : N (A) C ,
ℓ,w0 ◦ ℓ,w0 ℓ −→ ×
whereψ : F A C isa fixed non-trivialadditivecharacter. Hencethecharacter ψ is
\ −→ × ℓ,w0
trivialon N (F). Nowtheadjointactionof M on N inducesanactionofSO(W )ontheset
ℓ ℓ ℓ ℓ
ofallsuch characters ψ . Thestabilizer L ofχ inSO(W )equalsto SO(w W ).
ℓ,w0 ℓ,w0 ℓ,w0 ℓ ⊥0 ∩ ℓ
Let Π bean automorphicrepresentation of Hδ(A) (see [3, Section 4.6]), occurring in the
discrete spectrum. For f V and h Hδ(A), we define the ψ -Fourier coefficients of f
∈ Π ∈ ℓ,w0
by
(1.1) fψℓ,w0(h) = Z f(vh)ψ−ℓ,1w0(v) dv,
[Nℓ]
here [N ] denotes the quotient N (F) N (A). This is one of the Fourier coefficients of f
ℓ ℓ ℓ
\
associatedtothepartition[(2ℓ+1)14n−2ℓ+1]. Itisclearthat fψℓ,w0(h)isleftLℓ,w0(F)-invariant,
with L = SO(w W ). Following[10, Section 3.1],wedefine thespace
ℓ,w0 ⊥0 ∩ ℓ
σψℓ,w0(Π) = Lℓ,w0(A)−Spanfψℓ,w0(cid:12)(cid:12)(cid:12)Lℓ,w0(A) (cid:12)(cid:12)(cid:12) f ∈ VΠ,
which is a representation of Lℓ,w0(A), with right tran(cid:12)(cid:12)slation(cid:12)(cid:12)action.The situation that is
the main concern of the paper is the case of the so-called first occurrence, as described in
Theorem 1.2. In such a situation, the function fψℓ,w0(h) will be cuspidal for all f VΠ, in
∈
particular, square-integrable. Hence we may identify the space σ (Π) as a subspace of
the discrete spectrum of the space of square-integrable automorphψicℓ,wf0unctions on L (A),
ℓ,w0
andσ (Π)becomesan automorphic L (A)-moduleinthesenseof[3, Section 4.6].
ψℓ,w0 ℓ,w0
To makeaprecisechoiceoftheanisotropicvectorw forthecases 1 ℓ < 2n, wetake
0
≤
α
w = y = e e
0 α 2n 2n
− 2 −
for α F , and hence we have w ,w = α. For such an α, we consider the three-
× 0 0
∈ h i −
1
dimensionalquadraticform given by J = δ , which can be splitor non-splitover
δ,α  
 α
JACQUET-LANGLANDSAND TWISTEDDESCENT 5
F, depending on the Hilbert symbol (δ,α). For the cases ℓ = 2n, we take any nonzero
w V . Hence for 1 ℓ < 2n and w = y , we have L = SOδ,α , which is a special
0 ∈ 0 ≤ 0 α ℓ,α 4n 2ℓ+1
oddorthogonalgroupdefined by theform −
w
2n ℓ 1
J = J − − .
 δ,α 
Wedenoteψ by ψ , and σ (Πw)2bny−ℓσ−1 (Π). 
ℓ,yα ℓ,α ψℓ,yα ψℓ,α
1.3. The twisted automorphic descent. Now we apply the ψ -Fourier coefficients to
ℓ,w0
the residual representation of SOδ (A) and investigate more carefully its properties
Eτ⊗σ 4n+2
dependingontheintegerℓ. Oneofthemainresultsin thispaperisthefollowingtheorem.
Theorem1.2. Assumethatanirreducibleunitarycuspidalautomorphicrepresentationτof
GL (A)hasthepropertythat L(s,τ, 2)hasapoleat s = 1andthereexistsan irreducible
2n
unitary(cuspidal)representationσof∧SOδ(A)suchthatL(1,τ σ) , 0. Thenthefollowing
2 2 ×
hold.
(1) Therepresentationσ ( )of SOδ,α (A) iszeroforalln < ℓ 2n.
(2) Foranysquareclassαψℓ,iαnEFτ⊗σ,therepr4ens−e2ℓn+t1ationσ ( )ofSOδ,α≤(A)iscuspidal
× ψn,α Eτ⊗σ 2n+1
automorphic.
(3) There exists a square class α in F such that the representation σ ( ) of
SOδ,α (A)is nonzero,andin thiscas×e ψn,α Eτ⊗σ
2n+1
σ ( ) = π π π ,
ψn,α Eτ⊗σ 1 ⊕ 2 ⊕···⊕ r ⊕···
whereπ areirreduciblecuspidalautomorphicrepresentationsofSOδ,α (A),which
i 2n+1
are nearly equivalent, but are not globally equivalent, i.e. the decomposition is
multiplicityfree.
(4) Whenσ ( )isnonzero,anydirectsummandπ ofσ ( )hasaweakLang-
ψn,α Eτ⊗σ i ψn,α Eτ⊗σ
lands functorial transfer to τ in the sense that the Satake parameter of the local
unramified component τ of τ is the local functorial transfer of that of the local
v
unramifiedcomponentπ ofπforalmostallunramifiedlocalplaces vof F.
v
(5) When σ ( ) isnonzero,every irreducibledirect summandof σ ( )has a
nonzeroψFn,oαuErτie⊗rσcoefficient attachedto thepartition[(2n 1)12]. ψn,α Eτ⊗σ
−
(6) The residual representation has a nonzero Fourier coefficient attached to the
τ σ
partition[(2n+1)(2n 1)12E]. ⊗
−
For these square classes α in F such that σ ( ) is nonzero, we call σ ( ) the
twistedautomorphicdescentof τ×to SOδ,α (A)ψ.n,αItEisτ⊗cσlear from the notation aψnnd,α mEτo⊗rσe im-
2n+1
portantlytheexplicitconstructionthatthetwistedautomorphicdescentσ ( )depends
ψn,α Eτ⊗σ
onthedata(τ;δ,σ;α). TheGL (F) SO(W )(F)-orbitsofanisotropicvectorsinW labeled
1 ℓ ℓ
×
by the pair (δ,α) determine more refined properties of the twisted automorphic descent
σ ( ),whichwillbebrieflydiscussedattheendofSection3inconnectiontoParts(5)
ψn,α Eτ⊗σ
and (6) of Theorem 1.2 and to the structure of the relevant global Vogan packets. We refer
to [22] for more discussion on the general framework of the twisted automorphic descent
constructions for classical groups with connections to explicit structures of global Arthur
packets and global Vogan packets. In order to show the potential of the theory of twisted
6 DIHUAJIANG,BAIYINGLIU,BINXU,ANDLEIZHANG
automorphic descents, we provide in Section 5 a complete description of the method and
resultforthecaseofn = 1,whichrecoverstheclassicalJacquet-Langlandscorrespondence.
Hence the twisted automorphic descent discussed in this paper provides a new method to
establishtheJacquet-LanglandscorrespondenceforPGL(2).
Parts (1) and (2) of the theorem are usually called the towerpropertyand cuspidality.
They will be proved in Section 2 after establishing the vanishing property of the corre-
sponding local Jacquet module at one unramified local place, following the idea of [10].
In order to establish thefirstoccurrenceat ℓ = n, which is the first assertion in Part (3) of
the theorem, we have to understand the structure of the Fourier coefficients of the residual
representation followingthe general framework of [16]. We first provethat the resid-
τ σ
ual representatiEon⊗ has a nonzero Fourier coefficient attached to the partition [(2n)212]
τ σ
E ⊗
in Section 3 (Proposition 3.1), which is based on the cuspidal support of . After that,
τ σ
E ⊗
weusetheargumentsimilartothatin[6, Section5], butwithfulldetail,toprovethatthere
existsatleastonesquareclassαinF suchthatthetwistedautomorphicdescentσ ( )
× ψn,α Eτ⊗σ
is nonzero (Proposition 3.3), which yields the first assertion in Part (3) of the theorem. It
is clear that the second assertion in Part (3) of the theorem follows from Part (4) for the
assertionofnearly equivalence,whilethemultiplicityfree decompositionfollowsfrom the
local uniqueness of Bessel models ([1], [20] and [5]). If we assume that the global Gan-
Gross-Prasadconjectureholdsforthiscase,thenmaydeducethatσ ( )isirreducible.
ψn,α Eτ⊗σ
Wewillnotdiscussthisissueinsuchageneralityinthispaper,butwewilldiscussthecase
of n = 1 in Section 5. Next, we continue in Section 3 to obtain Part (5) of the theorem
by showing that the irreducible summands in the twisted automorphic descent σ ( )
have nonzero Fourier coefficients attached to the partition [(2n 1)12]. Note thatψinf,αnE=τ⊗σ1,
−
this partition is trivial, and the corresponding Fourier coefficient is the identity map. It is
clear that if the group SOδ,α is F-anisotropic, then its only nilpotent orbit is the trivial one,
3
corresponding to the trivial partition. However, if the group SOδ,α is F-split, because of
3
Part (4), we expect that the twisted automorphic descent has a nonzero Whittaker-Fourier
coefficient. There will be a more detailed discussion for this case in Section 5. In Section
4, we are going to prove that the Satake parameter of the unramified local component of
any of the direct summand π of σ ( ) transfers canonically to the Satake parameter
i ψn,α Eτ⊗σ
of the unramified local component of τ, and hence the direct summands of σ ( ) are
ψn,α Eτ⊗σ
nearly equivalent. This proves Parts (3) and (4) of the theorem. The last part (Part (6) of
the theorem) follows from Part (5) as consequence of the Fourier coefficients associated to
a composite of two partitions as discussed in [9] and [17], which will be briefly discussed
beforetheendofSection 3.
Acknowledgement. Thismaterial isbased uponworksupportedby theNationalScience
Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions
or recommendations expressed in this material are those of the authors and do not neces-
sarily reflect the views of the National Science Foundation. We also would like to thank
thereferee forveryhelpfulcommentsonthepreviousversionofthepaper,whichmakethe
papermorereadable.
JACQUET-LANGLANDSAND TWISTEDDESCENT 7
2. OnVanishing and Cuspidality
We are going to show that the ψ -Fourier coefficients of the residual representation
ℓ,w0
ofSOδ (A)vanishesforalln < ℓ 2n. Followingthetowerpropertyprovedin[10],
Eτ⊗σ 4n+2 ≤
the twisted automorphic descent σ ( ) is cuspidal, which might also vanish. Such
vanishing property of the family oψfn,αthEeτF⊗σourier coefficients ψℓ,w0 should be determined
Eτ σ
by the global Arthur parameter of the square-integrable residua⊗l representation and
τ σ
the structure of its Fourier coefficients as conjectured in [16]. Through the locEal-⊗global
relation, one may take the local approach to prove such vanishing property as shown in
[10]. However, a purely global argument should be interesting and expected ([8]). We will
comeback to thisissueinfuture.
In the following, we follow [10] to take the local approach, which is a calculation of
twistedJacquet moduleat oneunramifiedlocal place. ThisprovesPart(1)ofTheorem1.2.
Following the argument of the tower property in [10], we deduce in the second subsection
thecuspidalityofthetwistedautomorphicdescent, whichis Part (2)ofTheorem1.2.
2.1. Calculation of certain Jacquet modules. Most of the results are deduced from the
general resultsof[10,Chapter 5].
Let k be a non-Archimedean local field and V be a non-degenerate quadratic space of
dimension4n+2 overk. We haveapolardecomposition
V = V+ V V ,
0 −
⊕ ⊕
where V+ is a maximal isotropic subspace of V and V is anisotropic. There are two cases
0
tobeconsidered:
(1) dimV = 0, ifSO(V)issplit;
0
(2) dimV = 2, ifSO(V)isquasi-split.
0
We denote the Witt index of V by m, which can equal to 2n + 1 or 2n in case (1) or (2),
respectively.
Let H = SO(V) be the special ortehogonal group on V over k. Fix a basis e , ,e of
1 2n
{ ··· }
V+,andifV , 0,wetakeabasis e(1),e(2) forV suchthat e(1),e(1) = 1and e(2),e(2) = δ
0 { 0 0 } 0 h 0 0 i h 0 0 i
for some δ k . Otherwise we fix a basis e , ,e for V+. As in the global case
× 1 2n+1
∈ { ··· }
( 2.2),for0 < ℓ 2n,let P betheparabolicsubgroupof H preserving thepartial flag
ℓ
§ ≤
: 0 V+ V+ V+.
Fℓ ⊂ 1 ⊂ 2 ⊂ ··· ⊂ ℓ
Then P = M N with M Gℓ SO(W ), where W is the same as in 1.2. For α k ,
ℓ ℓ · ℓ ℓ ≃ m × ℓ ℓ § ∈ ×
wetakean anisotropicvectorw V as follows:
0
∈
y = e αe , ifℓ < 2n;
α 2n − 2 −2n
(2.1) w = 
0 αe(1) orαe(2) inV , ifℓ = 2nand V , 0.
0 0 0 0
Remark2.1. By achangeofbasisif necessary,we seethateach anisotropicvector in V is
oftheformαe +βe orαe(j) (thisoccursonlyifV , 0)forsomeα,β k . Thentheabove
i −i 0 0 ∈ ×
choicesofw essentiallyrepresentanyanisotropicvector in V.
0
8 DIHUAJIANG,BAIYINGLIU,BINXU,ANDLEIZHANG
Similar to the global case ( 2.2), one defines a character ψ on N (k) by ψ = ψ χ ,
§ ℓ ℓ ℓ ◦ ℓ,w0
where ψ is a non-trivial additivecharacter on k. Let L = SO(W w ) be the stabilizer
ℓ,w0 ℓ ∩ ⊥0
ofχ inSO(W ).
ℓ,w0 ℓ
Foranysmoothrepresentation(Π,V )of H(k), wedefinethe(twisted)Jacquetmodule
Π
J (V ) = V /Span Π(u)ξ ψ (u)ξ u N (k), ξ V ,
ψℓ,w0 Π Π { − ℓ,w0 | ∈ ℓ ∈ Π}
whichisanL -module. Tosimplifythenotation,wewilluseχ ,L and J ifw = y .
ℓ,w0 ℓ,α ℓ,α ψℓ,α 0 α
Note that with the notation of the previous section, for 0 ℓ < 2n, the form of the odd
≤
orthogonalgroup L is
ℓ,α
w
2n ℓ 1
J = J − − ,
 δ,α 
andhence wehavethefollowingcaswes2n:−ℓ−1 
(i) if J isnon-splitoverk,then L is non-split,and Witt(k y +V ) = 0;
δ,α ℓ,α α 0
(ii) otherwise,thegroup L isk-split. In thiscase, Witt(k y · +−V ) = 1ifV , 0(i.e.
ℓ,α α 0 0
δ < k 2), and Witt(k y +V ) = 0 ifV = 0. · −
× α 0 0
− · −
Moreover,ifℓ = 2n,thiswillbeadegeneratecase, and L is atrivialgroup.
2n,w0
For any 0 < j 2n, let V+ = Span e , ,e , and let Q be the standard parabolic
≤ j { 1 ··· j} j
subgroup of H which preserves V+. The group Q has a Levi decomposition Q = D U
j j j j · j
withD GL (k) SO(W ). For0 t < j,letτ(t) isthet-thBernstein-Zelevinskyderivative
j j j
≃ × ≤
ofτalongthesubgroup
I y
Z = s GL (k) z Z(k)
t′ ( 0 z! ∈ j ∈ t )
(cid:12)
(cid:12)
andcorrespondingto thecharacter (cid:12)
I y
ψ s = ψ 1(z + +z ).
′t 0 z!! − 1,2 ··· t−1,t
Then τ(t) is the representation of GL (k) with s = j t, acting on the Jacquet module
s
−
J (V ) viatheembedding
Z ,ψ τ
t′ ′t
d diag(d,I) GL (k).
t j
7−→ ∈
Fora ∈ k×, wealso considerthecharacter ψ′t′,a onZt′ defined by
I y
ψ′t′,a( 0s z!) = ψ−1(z1,2 +···+zt−1,t +ays,1).
Denote the corresponding Jacquet module J (V ) by τ , which is a representation of
Zt′,ψ′t,′a τ (t),a
the mirabolic subgroup P1 of GL (k). Since τ τ for any a,a k , according to
s 1 s (t),a ≃ (t),a′ ′ ∈ ×
[10, Lemma5.2],wesome−timesdenotebyτ any ofsuchrepresentationsτ .
(t) (t),a
Let τ and σ be smooth representations of GL (k) and SO(W ) respectively. The Jacquet
j j
module J (IndH τ σ)has been studiedin[10]. Westateithereforcompleteness.
ψℓ,α Qj ⊗
Proposition2.2 (Theorem 5.1of[10]). Set := det(). Thefollowinghold.
|·| | · |
JACQUET-LANGLANDSAND TWISTEDDESCENT 9
(1) Assumethat0 ℓ < m and1 j < m, then
≤ ≤
Jψℓ,α(IndHQjτ⊗σ) ≡ℓ+j mM<t ℓe, 0 t jindLQℓ′j,−αt|·|e12−tτ(t) ⊗ Jψ′ℓ−t,α(σωtb)
− ≤ ≤≤
eindLℓ,α −2ℓτ(ℓ) σωℓb, ifℓ < j,
Q′ |·| ⊗
j ℓ,ℓ
⊕0, − otherwise;
⊕δα ·0in,dLQℓ′α,α|·|1+2n2−ℓ−jτ(ℓ+j−me) ⊗ Jψ′m−j,vα(σωℓb+j−2n), oifth0e<rw2inse−. ℓ ≤ j,
 e
(2) Assumethat0 ℓ m and j = m, then
≤ ≤
Jψℓ,α(IndHQjτ⊗σ) ≡ inedLQℓ′m,αℓ|·|−2ℓτe(ℓ) ⊗σωℓb ⊕δα ·indLQℓ′α,α|·|1−2ℓτ(ℓ) ⊗ Jψ′0,vα(σωℓb).
−
(3) Assumethatℓ = m andwe V , then
0 0
∈
e Jψℓ,w0(IndHQjτ⊗σ) ≡ dτ · Jψ′ℓ−j,w0(σωbj),
hered = dimτ(j).
τ
Here Q = L η 1Q(w)η with w being the representative of Q H/Q corresponding to
′s ℓ,α ∩ −t ℓ,j t j\ ℓ
thepair(0,s)asin [10, Chapter4], and witht = j s,
−
ǫ
I
Q(w) = Q w 1Q w, η = I , andǫ = ℓ t .
Also Q = Lℓ,j η 1ℓQ∩(w)η− wjitht betingthes4an+m2−e2aℓs aǫb∗ove,and It −!
′s,t ℓ,α ∩ −s,t ℓ,j s,t
0 I
s
ǫ I 0
 2n ℓ s 
ηs,t =  γs ǫ∗, γs =  − − IV0 I0s I2n0−ℓ−s.
1 0 1 0
Set ω = diag I ,ω0,I , where ω0 = if m = 2n, and ω0 = if m = 2n+1.
b m b m b 0 1! b 0 1!
(cid:16) (cid:17) − −
Moreover, δ =e0 unleess Witt(k y + V ) = 1, and in such cases, set v V such that
α α 0 e α e0
v ,v = α, · − ∈
α α
h i −
I
2n ℓ 1
− −
1
 
γ = diag(I ,γ ,I ), anγdαη=  is of th−eα2vsαamIevV′α0for1m aI2sn−ηℓ−1w,here γ is replaced by γ .
s 1 α s 1 γα,t s,t s α
Finally, Q −= L −η 1 Q(w)η , and ψ (or ψ ) denotes the corresponding character
′α ℓ,α ∩ −γα,t ℓ,j γα,t ′l,α ′l.vα
likeψ buton thegroupsofsmallerrank. Notethat“ind”denotesthecompact induction,
ℓ,α
and“ ” denotesisomorphismofrepresentations,upto semi-simplification.
≡
10 DIHUAJIANG,BAIYINGLIU,BINXU,ANDLEIZHANG
Remark2.3. Denote
V = Span e , ,e W .
ℓ±,s k{ ±(ℓ+1) ··· ±(ℓ+s)} ⊂ ℓ
When w W or H(k) is split, Q is the maximal parabolic subgroup of L which
0 ∈ ℓ+s ′s ℓ,w0
preserves the isotropic subspace ωtV+ w . Otherwise, it is a proper subgroup of the
b ℓ,s ∩ ⊥0
parabolic subgroup. Moreover, Q is a subgroup of the maximal parabolic subgroup of
′s,t
L , which preserves theisotropicsubspaceη 1V+ y .
ℓ,α −s,t ℓ,s ∩ ⊥α
With abovepreparation,we turnto thecalculationofthetwistedJacquetmodulefor any
unramifiedlocal componentoftheresidualrepresentation .
τ σ
Let τ be an irreducible unramified representation of GLE(⊗k) with central character ω =
2n τ
1. Since τ is generic and self-dual, we may write τ as the full induced representation from
theBorel subgroupas follows:
τ = µ µ µ 1 µ 1,
1 ×···× n × −n ×···× −1
where µ’s are unramified characters on k . Also, let σ be an irreducible unramified rep-
i ×
resentation of SO(W ). When H = SO(V) is k-split, the representation σ is given by a
2n
unramifiedcharacter µ ofk ; and when H is k-quasisplit,SO(W )is acompact torus,and
0 × 2n
henceσis thetrivialrepresentation.
Letπ betheunramifiedconstituentofIndH τ det 1 σ. WewillcalculatetheJacquet
τ⊗σ Q2n ·| |2⊗
module J (π ),usingProposition2.2,andleadtothefollowingproposition,concerning
ψℓ,α τ σ
⊗
thevanishingof J (π ).
ψℓ,α τ σ
⊗
Proposition 2.4. Let τ be an irreducible unramified representation of GL (k) with cen-
2n
tral character ω = 1, and σ an irreducible unramified representation of SO(W ). The
τ 2n
followinghold.
(i) Assumethat J is non-splitover k.
δ
(1) If w = y forα k suchthat J issplit,then J (π ) = 0 forℓ n+1.
(2) If w0 Vα, then J∈ × (π ) = 0.δ,α ψℓ,α τ⊗σ ≥
0 ∈ 0 ψ2n,w0 τ⊗σ
(ii) Assumethat J splitsover k. Foranychoiceofα k , J (π ) = 0 forℓ n+1.
δ ∈ × ψℓ,α τ⊗σ ≥
Proof. Wesupposefirst that H is non-split,i.e. J isnon-splitoverk.
δ
In this case, π is the unramified constituent of the representation of H induced from
τ σ
thecharacter (here⊗ σ = 1)
(2.2) µ1|·|12 ⊗···⊗µn|·|21 ⊗µ−n1|·|21 ⊗···⊗µ−11|·|12 ⊗1.
Moreover,onecan find aWeyl elementofSO(V)whichconjugatesabovecharacter to
1 1 1 1
µ1 2 µ1 −2 µn 2 µn −2 1.
|·| ⊗ |·| ⊗···⊗ |·| ⊗ |·| ⊗
Then, induction by stages, one sees that π is the unramified constituent of IndH τ σ,
τ⊗σ Q2n ′ ⊗
where
τ = IndGL2n(k)µ (det ) µ (det ).
′ P2, ,2 1 GL2 ⊗···⊗ n GL2
···
Then since the twisted Jacquet functor corresponding to the descent is exact, we get (i) (1)
by Proposition 2.2 (2) if we take w = y as in (2.1) (here j = m = 2n, and δ = 1 by our
0 α α
assumptiononα k ). Notethatwehaveusedthefact thatτ = 0.
∈ × ′(n)
e