On the degree five L-function for GSp(4)

0.33 MB·English

by  Daniel File

ON THE DEGREE FIVE L-FUNCTION FOR GSp(4)
DANIELFILE
Abstract. Igiveanewintegralrepresentationforthedegreefive(standard)
L-function for automorphic representations of GSp(4) that is a refinement
of integral representation of Piatetski-Shapiro and Rallis. The new integral
2
representation unfolds to produce the Bessel model for GSp(4) which is a
1
unique model. The local unramified calculation uses an explicit formula for
0
theBesselmodelanddifferscompletelyfromPiatetski-ShapiroandRallis.
2

n
a
J 1. Introduction

7
2 In 1978 Andrianov and Kalinin established an integral representation for the
degree 2n+1 standard L-function of a Siegel modular form of genus n [1]. Their

] integralinvolvesatheta functionandaSiegelEisensteinseries. Theintegralrepre-
T
sentation allowed them to prove the meromorphic continuation of the L-function,
N
and in the case when the Siegel modular form has level 1 they established a func-
.
h tional equation and determined the locations of possible poles.
t Piatetski-ShapiroandRallisbecameinterestedinthe constructionofAndrianov
a
m andKalininbecause itseems toproduce Eulerproducts withoutusing anyunique-
ness property. Previous examples of integral representations used either a unique
[
modelsuchastheWhittakermodel,ortheuniquenessoftheinvariantbilinearform

2
between an irreducible representation and its contragradient. It is known that an
v
automorphic representation of Sp (or GSp ) associated to a Siegel modular form
3 4 4
doesnothaveaWhittakermodel. Piatetski-ShapiroandRallisadaptedtheintegral
8
7 representation of Andrianov and Kalinin to the setting of automorphic represen-
2 tations and were able to obtain Euler products [23]; however, the factorization is
1. not the result of a unique model that would explain the local-global structure of
0 Andrianov and Kalinin. They considered the integral
2
1 φ(g)θ (g)E(s,g)dg
T
:
v Sp (F)Z\Sp (A)
2n 2n
i
X where E(s,g) is an Eisenstein series induced from a character of the Siegel para-
r bolic subgroup, φ is a cuspidal automorphic form, T is a n-by-n symmetric matrix
a
determining an n dimensional orthogonal space, and θ (g) is the theta kernel for
T
the dual reductive pair Sp ×O(V ).
2n T
Upon unfolding their integral produces the expansion of φ along the abelian
unipotent radical N of the Siegel parabolic subgroup. They refer to the terms in
this expansionas Fourier coefficients in analogywith the Siegelmodular case. The
Fourier coefficients are defined as
φ (g)= φ(ng)ψ (n)dn.
T T
N(F)Z\N(A)
Keywords and phrases. L-function,integralrepresentation, Besselmodels.
This work was done while I was a gruaduate student at Ohio State University as part of my
Ph.D. dissertation. I wish to thank my advisor Jim Cogdell for being a patient teacher and for
his helpful discussions about this work. I also thank Ramin Takloo-Bighash for many useful
conversations.
1
2 DANIELFILE
Here, T is associatedto a characterψ ofN(F)\N(A). These functions φ do not
T T
give a unique model for the automorphic representation to which φ belongs. The
correspondingstatementforafiniteplacev ofF isthatforacharacterψ ofN(F )
v v
the inequality
dimCHomN(Fv)(πv,ψv)≤1
does not hold for all irreducible admissible representation π of Sp (F ).
v 2n v
However, Piatetski-Shapiro and Rallis show that their local integral is indepen-
dentofthechoiceofFouriercoefficientwhenvisafiniteplaceandthelocalrepresen-
tation π is spherical. Specifically, they show that for any ℓ ∈ Hom (π ,ψ )
v T N(Fv) v v
the integral
g 2s+1
ℓ v |det(g)|s−1/2dg =d (s)L(π , )ℓ (v )
T tg−1 0 v v v 2 T 0
Z (cid:18)(cid:20) (cid:21) (cid:19)
Matn(Ov)∩GLn(Fv)
where v is the spherical vector for π , O is the ring of integers, and d (s) is a
0 v v v
product of local ζ-factors. At the remaining “bad” places the integral does not
factor, and there is no local integral to compute. However, they showed that the
integral over the remaining places is a meromorphic function of s.
InthispaperIpresentanewintegralrepresentationforthedegreefiveL-function
forGSp whichisarefinementofthe workofPiatetski-ShapiroandRallis. Instead
4
of working with the full theta kernel, the construction in this paper uses a theta
integral for GSp × GSO . This difference has the striking effect of producing
4 2
the Bessel model for GSp and the uniqueness that Piatetski-Shapior and Rallis
4
expected. Therefore, this integral factors as an Euler product over all places. I
compute the local unramified integral when the local representation is spherical
using the formula due to Sugano [32].
InsomeinstancesanintegralrepresentationofanL-functioncanbeusedtoprove
algebraicity of special values of that L-function (up to certain expected transcen-
dental factors). Harris [10], Sturm [33], Bocherer [3], and Panchishkin [21] applied
the integral representation of Andrianov and Kalinin to prove algebraicity of spe-
cialvaluesofthestandardL-functionofcertainSiegelmodularforms. Shimura[30]
alsoused anintegralrepresentationto provealgebraicityofthese specialvalues for
many Siegelmodular forms including forms for every congruencesubgroupofSp
2n
over a totally real number field.
Furusawa [7] gave an integral representation for the degree eight L-function for
GSp ×GL . Furusawa’s integral representation unfolds to give the Bessel model
4 2
for GSp times the Whittaker model for GL , and he uses Sugano’s formula for
4 2
the spherical Bessel model to compute the unramified local integral. Let Φ be a
genus 2 Siegel eigen cusp form of weight ℓ, and let π = ⊗ π be the automorphic
v v
representation for GSp associated to it. Let Ψ be an elliptic (genus 1) eigen cusp
4
form of weight ℓ, and let τ = ⊗ τ be the associated representation for GL . As
v v 2
anapplicationofhisintegralrepresentationFurusawaprovedanalgebraicityresult
for special values of the degree eight L-function L(s,Φ×Ψ) provided that for all
finite places v both π and τ are spherical. This conditionis satisfiedwhen Φ and
v v
Ψ are modular forms for the full modular groups Sp (Z) and SL (Z), respectively.
4 2
ArecentresultofSaha[28]includestheexplicitcomputationofBesselfunctions
oflocalrepresentationsthatareSteinberg. ThisallowedSahatoextendthespecial
value result of Furusawa to the case when π is Steinberg at some prime p. Pitale
p
and Schmidt [24] considered the local integral of Furusawa for a large class of rep-
resentations τ and as an application extended the algebraicity result of Furusawa
p
further.
In principle one could explicitly compute the local integral given in this paper
using the formula of Saha at a place where the local representation is Steinberg.
ON THE DEGREE FIVE L-FUNCTION FOR GSp(4) 3
Considering the algebraicity results of Harris [10], Sturm [33], Bocherer [3], and
Panchishkin[21]thatinvolvetheintegralofAndrianovandKalinin,andtheexplicit
computations for Besselmodels due to Sugano [32], Furusawa[7], and Saha [28], it
would be interesting to see if the integral representation of this paper can be used
to obtain any new algebraicity results. This is a question I intend to address in a
later work.
2. Summary of Results
Let π be an automorphic representation of GSp (A), φ∈V , ν an automorphic
4 π
characteronGSO (A)thesimilitudeorthogonalgroupthatpreservesthesymmetric
2
formdeterminedbythesymmetric matrixT, θ (ν−1)the thetalift ofν−1 toGSp
ϕ 4
with respect to a Schwartz-Bruhat function ϕ, and E(s,f,g) a Siegel Eisenstein
series for a section f(s,−)∈IndG(A)(δ1/3(s−1/2)). Consider the global integral
P(A) P
I(s;f,φ,T,ν,ϕ)=I(s):= E(s,f,g)φ(g)θ (ν−1)(g)dg.
ϕ
ZAGSp4(FZ)\GSp4(A)
Section 9 contains the proof that I(s) has an Euler product expansion
I(s)= f(s,g)φT,ν(g)ω(g,1)ϕ(1 )dg· I (s)
2 v
N(A∞)\ZG1(A∞) vY<∞
where integrals I (s) are defined to be
v
I (s)= f (s,g )φT,ν(g )ω (g ,1)ϕ (1 )dg .
v v v v v v v v 2 v
Z
N(Fv)\G1(Fv)
The function φT,ν belongs to the Bessel model of π .
v v
Section 11 includes the proof that under certain conditions that hold for all but
a finite number of places v, there is a normalization I∗(s) = ζ (s+1)ζ (2s)I (s)
v v v v
such that
I∗(s)=L(s,π ⊗χ )
v v T
where χ is a quadratic character associated to the matrix T. Section 14 deals
T
with the finite placesthatare notcoveredinSection11. For these placesthere is a
choice of data so that I (s)=1. Section 15 deals with the archimedeanplaces and
v
shows that there choice of data to control the analytic properties of I (s).
v
Combining these analyses give the following theorem.
Theorem 1. Let π be a cuspidal automorphic representation of GSp (A), and φ∈
4
V . Let T and ν be such that φT,ν 6= 0. There exists a choice of section f(s,−)∈
π
IndG(A)(δ1/3(s−1/2)), and some ϕ = ⊗ ϕ ∈ S(X(A)) such that the normalized
P(A) P v v
integral
I∗(s;f,φ,T,ν,ϕ)=d(s)·LS(s,π⊗χ )
T,v
where S is a finite set of bad places including all the archimedean places. Fur-
thermore, for any complex number s , there is a choice of data so that d(s) is
0
holomorphic at s , and d(s )6=0.
0 0
3. Notation
Let F be a number field, and let A =A be its ring of adeles. For a place v of
F
F denote by F the completion of F at v. For a non-archimedean place v let O
v v
be the ring of integers of F , and let p be its maximal ideal. Let q = [O : p ].
v v v v v
Let ̟ be a choice of uniformizer for p , and let |·| be the absolute value on F ,
v v v v
normalized so that |̟ | =q−1.
v v v
4 DANIELFILE
For a finite set of places S, let AS = ′F , and A = F . In particular,
v S v
v∈/S v∈S
A = F , and A = ′F . Q Q
∞ v fin v
v|∞ v<∞
DenoQtebyMat thevarQietyofn×nmatricesdefinedoverF. Sym isthevariety
n n
of symmetric n×n matrices defined over F.
Let G=GSp ={g ∈GL tgJg =λ (g)J} where
4 4 G
(cid:12) 1
(cid:12)
1
J = .
−1
 −1 
 
Fix a maximal compact subgroup K of G(A) such that K = K where K
v v v
is a maximalcompactsubgroupofG(F ), and atallbut finitely many finite places
v
Q
K = G(O ). According to [18, I.1.4] the subgroups K can be chosen so that for
v v v
everystandardparabolicsubgroupP,G(A)=P(A)K,andM(A)∩K isamaximal
compact subgroup of M(A).
4. Orthogonal Similitude Groups
AmatrixT ∈Sym (F)withdet(T)6=0determinesanon-degeneratesymmetric
2
bilinear form ( , ) on an V =F2:
T T
(v ,v ) :=tv Tv .
1 2 T 1 2
The orthogonal group associated to this form (and matrix T) is
O(V )={h∈GL thTh=T}.
T 2
Similarly, the similitude group GO(V ) =(cid:12) {h ∈ GL thTh = λ (h)T}, and
T (cid:12) n T
GSO(V ) is defined to be the Zariskiconnected component of GO(V ). Note that
T (cid:12) T
since dim(V )=2, and h∈GSO(V ), then λ(h)=det(h)(cid:12).
T T
Let χ be the quadratic character associated to V . If E/F is the discriminant
T T
field of V , i.e. E =F −det(T) , then
T
(cid:16)p χ (cid:17):F×\A× →C
T
isthe ideleclasscharacterassociatedtoE byclassfieldtheory. Ithasthe property
thatχ =⊗χ whereχ (a)=(a,−det(T)) ,and(, ) denotesthelocalHilbert
T T,v T,v v v
symbol[31, §0.3]. Consequently,eachχ ◦N ≡1whereN isthenorm
T.v Ev/Fv Ev/Fv
map [29, Chapter III, Proposition 1]. Note that N =det=λ .
Ev/Fv T
4.1. The Siegel Parabolic Subgroup. Let P = MN be the Siegel parabolic
subgroup of G, i.e. P stabilizes a maximal isotropic subspace X = span {e ,e }
F 1 2
where e is the ith standardbasis vector. Then P has Levi factor M ∼=GL ×GL
i 1 2
and unipotent radical N ∼=Sym ∼=G3. For g ∈GL , define
2 a 2
g
m(g)= ∈M.
tg−1
(cid:20) (cid:21)
For X ∈Sym , define
2
I X
n(X)= 2 ∈N.
I
2
(cid:20) (cid:21)
Let δ be the modular character of P.
P
g
For m= tg−1λ ∈M and n∈N, δP(mn)=|det(g)3·λ−3|A. It is possible
(cid:20) (cid:21)
to extend δ to all of G. For g = nmk where n ∈ N, m ∈ M, and k ∈ K, define
P
δ (g)=δ (m). This is well defined because δ (m)=1 for m∈M ∩K.
P P P
ON THE DEGREE FIVE L-FUNCTION FOR GSp(4) 5
5. Bessel Models and Coefficients
5.1. The Bessel Subgroup. Let ψ be an additive character of F\A. There is a
bijection between Sym (F) and the characters of N(F)\N(A). For T ∈ Sym (F)
2 2
define
ψ :N(F)\N(A)→C
T
ψ (n(X))=ψ(tr(TX)). (1)
T
Since M(F)acts onN(F)\N(A), it alsoacts onits characters. Define H to be
T
the connected component of the stabilizer of ψ in M. For g ∈GL , define
T 2
g
b(g)= .
det(g)· tg−1
(cid:20) (cid:21)
Then
H = b(g) tgTg=det(g)·T .
T
n (cid:12) o
Then HT is an algebraic group d(cid:12)efined over F isomorphic to GSO(VT) where
(cid:12)
V is defined as above.
T
The adjoint action of M(F) on the characters of N(F)\N(A) has two types of
orbits. They are represented by matrices
1 1
T = withρ∈/ F×,2, and T = .
ρ −ρ split 1
(cid:20) (cid:21) (cid:20) (cid:21)
The quadratic spaces corresponding to these matrices have similitude orthogonal
groups
x ρy
GSO(V )= x2−ρy2 6=0 .
Tρ y x
((cid:20) (cid:21)(cid:12)(cid:12) )
(cid:12)
and (cid:12)
(cid:12)
x
GSO(V )= xy 6=0 . (2)
Tsplit y
((cid:20) (cid:21)(cid:12)(cid:12) )
(cid:12)
For the rest of this article assume that ρ ∈/ F(cid:12) ×,2, and only consider T = Tρ.
(cid:12)
Define the Bessel subgroup R=R =H N. Consider a character
T T
ν :H (F)\H (A)→C.
T T
Then define
ν⊗ψ :R(F)\R(A)→C
T
ν⊗ψ (tn)=ν(t)ψ (n) t∈H (A), n∈N(A).
T T T
This is well defined since H normalizes ψ .
T T
Similarly, for a place v of F there are local characters
ν ⊗ψ :R(F )→C.
v T,v v
5.2. Non-Archimedean Local Bessel Models. Letv beafiniteplaceofF. Let
B be the space of locally constant functions φ:G(F )→C satisfying
v
φ(rg)=ν ⊗ψ (r)φ(g)
v T,v
for all r ∈R(F ) and all g ∈G(F ).
v v
Let π be an irreducible admissable representation of G(F ). Piatetski-Shapiro
v v
and Novodvorsky [20] showed that there is at most one subspace B(π ) ⊆ B such
v
that the right regular representation of G(F ) on B(π ) is equivalent to π . If the
v v v
subspace B(π ) exists, then it is called the ν ⊗ψ Bessel model of π .
v v T,v v
6 DANIELFILE
5.3. Archimedean Local Bessel Models. Now suppose v is an infinite place of
F. Let K be the maximalcompact subgroupof G(F ). Let B be the vector space
v v
of functions φ:G(F )→C with the following properties [24]:
v
(1) φ is smooth and K -finite.
v
(2) φ(rg)=ν ⊗ψ (r)φ(g) for all r ∈R(F ) and all g ∈G(F ).
v T,v v v
(3) φ is slowly increasing on Z(F )\G(F ).
v v
Let πv be a (gv,Kv)-module with space Vπv. Suppose that there is a subspace
B(π )⊂B, invariant under right translation by g and K , and is isomorphic as a
v v v
(g ,K )-moduletoπ ,thenB(π )iscalledtheν ⊗ψ Besselmodelofπ . Insome
v v v v v T,v v
instances the Bessel model at an archimedean place is known to be unique. For
example, when v is a realplace and π is a lowestor highest weight representation
v
of GSp (R) the Bessel model of π is unique [24]. It is also known to be unique
4 v
when the central character of π is trivial [20]. The results of this article do not
v
depend on the uniqueness of the Bessel model at any archimedean place; however,
if the model is not unique, then there is no local integral.
5.4. Bessel Coefficients. Let A (G) be the space of cuspidal automorphic forms
0
on G(A). Suppose that π is an irreducible cuspidal automorphic representation
of G(A) with space V ⊂ A (G). Let ω denote the central character of π. Let
π 0 π
φ∈V .
π
Suppose that ν is as above. Denote by ZA the center of G(A) so ZA ⊂ HT(A).
Suppose that ν =ω−1. Define the ν⊗ψ Bessel coefficient of φ to be
|ZA π T
φT,ν(g)= (ν⊗ψ )−1(r)φ(rg)dr. (3)
T
ZAR(FZ)\R(A)
6. Siegel Eisenstein Series
For more details about Siegel Eisenstein series of Sp see Kudla and Rallis [15]
2n
and Section 1.1 of Kudla, Rallis, and Soudry [16].
1(s−1)
Definition 6.1 (Induced Representation). The induced representation of δ3 2
P
to G(A) is defined to be
f :G(A)→C f is smooth, right K-finite, and for
IndG(A)(δ13(s−21))= .
P(A) P  (cid:12) p∈P(A), f(pg)=δ13(s+1)(p)f(g) 
 (cid:12)(cid:12) P 
(cid:12)
For convenience write Ind(s) = Ind(cid:12)(cid:12)G(A)(δ31(s−21)). Ind(s) is a representation of
P(A) P
(g ,K )×G(A ) under right translation. A standard section f(s,·) is one such
∞ ∞ fin
that its restriction to K is independent of s. Let f(s,·)∈Ind(s) be a holomorphic
standardsection. Thatisforallg ∈G(A)thefunctions7→f(s,g)isaholomorphic
function.
For a finite place v define f◦(s,·) to be the function so that f◦(s,k) = 1 for
v v
k ∈K .
v
There is an intertwining operator
M(s):Ind(s)→Ind(1−s).
For Re(s)>2, M(s) may be defined by means of the integral [14, 4.1]
M(s)f(s,g):= f(s,wng)dn
NZ(A)
ON THE DEGREE FIVE L-FUNCTION FOR GSp(4) 7
where
1
1
w = .
−1
 −1 
 
The induced representation factors as a restrictedtensor product with respect to
f◦(s,·):
v
′
Ind(s)= Ind (s),
v
v
O
and so does the intertwining operator
M(s)= M (s).
v
v
O
There is a normalization of M (s)
v
ζ (s+1)ζ (2s)
M∗(s)= v v M (s)
v ζ (s−1)ζ (2s−1) v
v v
where ζ (·) is the local zeta factor for F at v, so that
v
M∗(s)f◦(s,g)=f◦(1−s,g).
v v v
Define the Siegel Eisenstein series
E(s,f,g)= f(s,γg)
γ∈P(XF)\G(F)
which converges uniformly for Re(s)>2 and has meromorphic continuation to all
C [15]. Furthermore, the Eisenstein series satisfies the functional equation
E(s,f,g)=E(1−s,M(s)f,g)
[15,1.5]. Later,itwillbeusefultoworkwiththenormalizedEisensteinseries. Let
S be a finite set of places, including the archimedean places, such that for v ∈/ S
f =f◦. Define
v v
E∗(s,f,g)=ζS(s+1)ζS(2s)E(s,f,g). (4)
Kudla and Rallis completely determined the locations of possible poles of Siegel
Eisenstein series [15]. The normalized Eisenstein series E∗(s,f,g) has at most
simple poles at s =1,2 [15, Theorem 1.1].
0
7. The Weil Representation
7.1. TheSchro¨dingerModel. ConsidertheorthogonalspaceV withsymmetric
T
form(,) ,andthefourdimensionalsymplecticspaceW withsymplecticform<,>.
T
LetW=V ⊗W be the symplectic space with form≪,≫defined onpure tensors
T
by≪u⊗v,u′⊗v′ ≫=(u,u′) <v,v′ >andextendedtoallofWbylinearity. The
T
Weil representation ω = ωψ−1 is a representation of Sp(W). However, restricting
T
this representation to Sp(W)×O(V ) ֒→ Sp(W). Since the dimension of V is
T T
f
even, there is a splitting Sp(W)×O(V )֒→Sp(W)×O(V ) [26, Remark 2.1].
T T
Suppose that X is afmaximal isotropic sufbspace of W. Then X = X ⊗ V is
F T
a maximal isotropic subspace of W. The spface of the Schr¨odinger model, S(X),
is the space of Schwartz-Bruhat functions on X. Let v be a place of F. If v is a
finite place, then S(X(F )) is the space of locally constant functions with compact
v
support. If v is an infinite place, then S(X(F )) is the space of C∞ functions all
v
derivatives of which are rapidly decreasing.
Identify X with V2 =Mat .
T 2
8 DANIELFILE
The local Weil representation at a finite place v restricted to
Sp(W)(F )×O(V )(F )
v T v
acts in the following way on the Schr¨odinger model
ω (1,h)ϕ(x)=ϕ(h−1x),
v
ω (m(a),1)ϕ(x)=χ ◦det(a) |det(a)| ϕ(xa),
v T,v v
ω (n(X),1)ϕ(x)=ψ−1 (X)ϕ(x),
v txTx
ω (w,1)ϕ(x)=γ·ϕˆ(x).
v
whereγ isacertaineighthrootofunity,andϕˆistheFouriertransformofϕdefined
by
ϕˆ(x)= ϕ(x′)ψ((x,x′) )dx′.
1
Z
VT(Fv)2
Here ( , ) is defined as follows: for x,y ∈X=Mat define
1 2
(x,y) :=tr(x·y).
1
Note that matrices of the form m(a), n(X), and w generate Sp .
4
The space S(X(A)) is spanned by functions ϕ = ⊗ ϕ where ϕ = ϕ◦ is the
v v v v
normalized local spherical function for all but finitely many of the finite places v.
At an unramified place ϕ◦v = 1X(Ov). The global Weil representation, ω = ⊗v′ωv,
is the restricted tensor product with respect to the normalized spherical functions
ϕ◦.
v
SupposethatF =R. Assumethatψ =exp(2πix). LetK =Sp (R)∩O (R).
v T 1,v 4 4
Let V+ and V− be positive definite and negative definite, respectively, subspaces
T T
of V (F ) such that V (F )=V+⊕V−. For x∈V define
T v T v T T T
(x,x) if x∈V+
(x,x) = T
+ −(x,x) if x∈V−
(cid:26) T
For x∈V2 let (x,x)=((x ,x ) )∈V2. Define
T i j i,j T
ϕ◦(x)=exp(−πtr((x,x) )).
v +
Now, suppose F = C. Assume that ψ = exp(4πi(x+x¯). In this case K ∼=
v T 1,v
Sp(4), the compact real form of Sp(4,C). There is a choice of basis so
(x,x) =tx¯x,
+
and
ϕ◦(x)=exp(−2πtr((x,x) )).
v +
The subspace of K finite vectorsin the space of smooth vectors, S (X(F ))⊂
1,v 0 v
S(X(F )), consists of functions of the form p(x)ϕ◦ where p is a polynomial on
v v
V (F )2.
T v
7.2. ExtensiontoSimilitudeGroups. HarrisandKudladescribehowtoextend
the Weil representation to similitude groups [11, §3]. See also [12] and [27].
The Weil representation can be extended to the group
Y ={(g,h)∈GSp ×GSO(V ) | λ (g)=λ (h)}.
4 T G T
For (g,h)∈Y the action of ω is defined by
v
ω (g,h)ϕ(x)=|λ (h)|−1 ω (g ,1)ϕ(h−1x)
v T v v 1
where
I
g = 2 g.
1 λ (g)−1·I
G 2
(cid:20) (cid:21)
ON THE DEGREE FIVE L-FUNCTION FOR GSp(4) 9
Note that the natural projection to the first coordinate
p :Y →GSp(4)
1
(g,h)7→g
is generally not a surjective map. Indeed, g ∈ Im(p ) if and only if there is an
1
h∈GSO(V ) such that λ (g)=λ (h). Define
T G T
G+ :=p (Y).
1
7.3. Theta Lifts. Let H =GSO(V ), and H =SO(V ).
T 1 T
Definition 7.1. The theta lift of ν−1 to G+(A) is given by the integral
θ (ν−1)(g)= ω(g,h h )ϕ(x)ν−1(h h )dh .
ϕ g 1 g 1 1
H1(F)Z\H1(A) x∈XVT2(F)
Here,h ∈H(A)isanyelementsothatλ (h )=λ (g). NotethatDefinition7.1
g T g G
is independent of the choice h . Since H (F)\H (A) is compact the integral is
g 1 1
termwise absolutly convergent[34].
There is a natural inclusion
G(F)+\G(A)+ ֒→G(F)\G(A).
Consider θ (ν−1) as a function of G(F)\G(A) by extending it by 0 [8, §7.2].
ϕ
If ϕ is chosen to be a K-finite Schwartz-Bruhat function, then θ (ν−1) is a
ϕ
K-finite automorphic form on G(F)\G(A) [11].
8. The Degree Five L-function
Theconnectedcomponentofthe dualgroupofGSp isLG◦ =GSp (C)[4, I.2.2
4 4
(5)]. The degree five L-function of GSp corresponds to the map of L-groups [31,
4
page 88]
̺:GSp (C)→PGSp (C)∼=SO (C).
4 4 5
I describe the local L-factor explicitly when v is finite and π is equivalent to an
v
unramified principal series. Consider the maximal torus A of G and an element
0
t∈A :
0
a
1
a
t=diag(a1,a2,a0a−11,a0a−21):= 2 a a−1 . (5)
0 1
 a a−2
 0 2 
 
The character lattice of G is
X =Ze ⊕Ze ⊕Ze
0 1 2
where e (t)=a . The cocharacter lattice is
i i
X∨ =Zf ⊕Zf ⊕Zf
0 1 2
where
f (u)=diag(1,1,u,u), f (u)=diag(u,1,u−1,1),
0 1
f (u)=diag(1,u,1,u−1).
2
Suppose
π ∼=π (χ)=IndG(Fv)(χ)
v v B(Fv)
where
χ(t)=χ (a )χ (a )χ (a ), (6)
1 1 2 2 0 0
10 DANIELFILE
andtisgivenby(5). ThenLG◦ =Gˆ hascharacterlatticeX′ =X∨andcocharacter
lattice X′∨ =X. Let f′ =e ∈X′∨. Define
i i
3
tˆ= f′(χ (̟ ))∈LG◦. (7)
i i v
i=0
Y
Thentˆis the Satakeparameterfor π (χ) [2, Lemma2]. The LanglandsL-factor
v
is defined in [4, II.7.2 (1)] to be
L(s,π ,̺):=det(I −̺(tˆ)q−s)−1
v v
=(1−q−s)−1(1−χ (̟ )q−s)−1(1−χ (̟ )−1q−s)−1
v 1 v v 1 v v
(1−χ (̟ )q−s)−1(1−χ (̟)−1q−s)−1.
2 v v 2 v
Let S be a finite set of primes, including the archimedean primes, such that if
v ∈/ S, then π is unramified. Then the partial L-function is defined to be
v
LS(s,π)=LS(s,π,̺)= L(s,π ,̺).
v
vY∈/S
The product converges absolutely for Re(s)≫0 [17].
9. Global Integral Representation
The main result of this section is Theorem 9.4 which states that the integral
unfolds as an Euler product of local integrals.
As before G =GSp , G =Sp , P = MN is the Siegel parabolic subgoup of G,
4 1 4
and let P =M N =P ∩G where M =M ∩G .
1 1 1 1 1
The global integral is
I(s;f,φ,ν)=I(s):= E(s,f,g)φ(g)θ (ν−1)(g)dg (8)
ϕ
ZAG(FZ)\G(A)
= E(s,f,g)φ(g)θ (ν−1)(g)dg (9)
ϕ
ZAG(F)Z+\G(A)+
where equality holds because θ (ν−1) is supported on G(F)+\G(A)+. The central
ϕ
character of E(s,f,−) is trivial, and the central character of θ (ν−1) = ω−1, so
ϕ π
theintegrandisZA invariant. SinceE(s,f,−)andθϕ(ν−1)areautomorphicforms,
they are of moderate growth. Since φ is a cuspidal automorphic form, it is rapidly
decreasing on a Siegel domain [18, I.2.18]. Therefore, the integral (9) converges
everywhere that E(s,f,−) does not have a pole.
Define
A×,+ :=λ (H(A)), F×,+ :=F×∩A×,+ ⊆A×,+,
T
A×,2 :={a2|a∈A×}, C :=A×,2F×,+\A×,+.
There is an isomorphism
ZAG1(A)G(F)+\G(A)+ ∼=C. (10)
TheisomorphismisrealizedbyconsideringthemapfromG(A)+ −→C,g 7→λ (g).
G
It has kernel ZAG1(A)G(F)+. This fact is stated in [8].
Identify ZA with the subgroup of scalar linear transformations in H(A).
Proposition 9.1.
ZAH1(A)H(F)\H(A)∼=C. (11)