ON THE LIFTING OF HILBERT CUSP FORMS TO HILBERT-SIEGEL ...syamana.sub.jp/Lifting.pdf · 5....

ON THE LIFTING OF HILBERT CUSP FORMS TO

HILBERT-SIEGEL CUSP FORMS

TAMOTSU IKEDA AND SHUNSUKE YAMANA

Abstract. Starting from a Hilbert cusp form of weight 2κ, we willconstruct a Hilbert-Siegel cusp form of weight κ+ m

2and degree m and

its transfer to inner forms of symplectic groups. Applications includea relation between Fourier coefficients of Hilbert cusp forms of weightn + 1

2and a certain weighted sum of the representation numbers of a

quadratic form of rank 2n by a quadratic form of rank 4n.

Partant d’une forme modulaire parabolique de Hilbert de poids 2κ,nous construisons une forme modulaire parabolique de Hilbert-Siegel depoids κ + m

2et de degre m et son transfert aux formes interieures des

groupes symplectiques. Comme application, on obtient entre autres unerelation entre les coefficients de Fourier de formes modulaires paraboliquesde Hilbert de poids n+ 1

2et une certaine somme ponderee des nombres

de representation d’une forme quadratique de rang 2n par une formequadratique de rang 4n.

Contents

1. Introduction 22. Preliminaries 103. Degenerate Whittaker functions 134. Holomorphic cusp forms on quaternion unitary groups 175. Hilbert-Siegel cusp forms of half-integral weight 206. Main theorem 247. Fourier-Jacobi modules 268. Proofs of Theorems 1.1 and 1.2 339. Transfer to inner forms 3910. Translation to classical language 4411. The Duke-Imamoglu-Ikeda lifts and theta functions 4712. Hilbert-Siegel modular forms of degree 4 and weight 4 over

Q(√2) 51

References 58Index 61

Key words and phrases. Hilbert-Siegel cusp forms, Duke-Imamoglu-Ikeda lifts, degen-erate principal series, the Siegel-Weil formula, representation numbers, even unimodularlattices.

1

2 TAMOTSU IKEDA AND SHUNSUKE YAMANA

1. Introduction

The present investigation deals with the following problem: starting fromsimple automorphic data such as cusp forms on GL2, construct more compli-cated automorphic forms on groups of higher degree. Toward this problem,Ikeda [22] has constructed a lifting associating to an elliptic cusp form aSiegel cusp form of even genus. This paper generalizes it to Hilbert cuspforms with different methods. The resulting Hilbert-Siegel cusp forms areapplied to the theory of quadratic forms.

To illustrate our results, let F be a totally real number field of degree dwith adele ring A. . We write Af and A∞ for the finite part and the infinitepart of the adele ring. We denote the set of d real primes of F by S∞ andthe normalized absolute value by α =

∏v αv : A× → R×

+.Let Symm = z ∈ Mm | tz = z be the space of symmetric matrix of size

m and Wm a symplectic vector space of dimension 2m. We take matrixrepresentation

Spm =

g ∈ GL2m

∣∣∣∣ g( 0 −1m1m 0

)tg =

(0 −1m1m 0

)of the associated symplectic group Sp(Wm) by choosing a Witt basis ofWm.We define homomorphisms m : GLm → Spm and n : Symm → Spm by

m(a) =

(a 00 ta−1

), n(b) =

(1m b0 1m

).(1.1)

Let Mp(Wm)A ↠ Spm(A) be the metaplectic double cover. Denote theinverse image of Spm(A∞) (resp. Spm(Af )) by Mp(Wm)∞ (resp. Mp(Wm)f ).

Define the character e∞ : Cd → C× by e∞(Z) =∏v∈S∞

e2π√−1Zv .

Let ψ =∏v ψv be the additive character of A/F whose restriction to A∞

is e∞|Rd . We let p denote a finite prime of F and do not use p for an

archimedean place. For ξ ∈ Symm(F ) we define the character ψξf =∏

p ψξp :

Symm(Af ) → C× by ψξf (z) =∏

p ψp(tr(ξzp)). For t ∈ F×v there is an 8th

root of unity γ(ψtv) such that for all Schwartz functions ϕ on Fv∫Fv

ϕ(xv)ψv(tx2v) dxv = γ(ψtv)|2t|−1/2

v

∫Fv

Fϕ(xv)ψv(−x

2v

4t

)dxv,

where dxv is the self-dual Haar measure on Fv with respect to the Fouriertransform Fϕ(y) =

∫Fvϕ(xv)ψv(xvy) dxv. Set γψv(t) = γ(ψv)/γ(ψ

tv). We

denote the set of totally positive elements of F by F×+ , the set of totally

positive definite symmetric matrices of rank m over F by Sym+m and the set

of all complex symmetric matrices of size m with positive definite imaginarypart by Hm. For t ∈ F× we write χt =

∏v χ

tv for the quadratic character of

A×/F× associated to the extension F (√t)/F and denote its restriction to

the finite idele group A×f by χtf . For ℓ ∈ Rd we will set |t|ℓ =

∏v∈S∞

|t|ℓvv .The real metaplectic group Mp(Wm)v acts on Hm through Spm(Fv) for

v ∈ S∞. There is a unique factor of automorphy ȷ : Mp(Wm)v ×Hm → C×

3

satisfying ȷ(gv,Zv)2 = det(CvZv+Dv). We here write the projection of gv to

Spm(Fv) as

(∗ ∗Cv Dv

). Let ℓ be a tuple of d positive half integers such that

2ℓv ≡ 2ℓv′ (mod 2) for all v, v′ ∈ S∞. We set Jℓ(g,Z) =∏v∈S∞

ȷ(gv,Zv)2ℓvfor g ∈ Mp(Wm)∞ and Z ∈ Hd

m. If ℓ ∈ Zd, then Jℓ descends to thefunction on Spm(A∞) × Hd

m. Even when ℓ /∈ Zd, one can define it onsome congruence subgroup Γθm of Spm(F ). A Hilbert-Siegel modular formF of weight ℓ with respect to a congruence subgroup Γ of Spm(F ) is aholomorphic function on Hd

m which satisfies F(γZ) = F(Z)Jℓ(γ,Z) forevery γ ∈ Γ and also the additional condition at infinity ifm = 1 and F = Q.Let Mℓ(Γ) denote the vector space of such Hilbert-Siegel modular forms.The vector space Sℓ(Γ) of Hilbert-Siegel cusp forms consists of functions

F ∈ Mℓ(Γ) such that F(γZ)√J2ℓ(γ,Z)

−1has a Fourier expansion of the

form∑

ξ∈Sym+mA(ξ)e∞(tr(ξZ)) for all γ ∈ Spm(F ), where

√J2ℓ(γ,Z) means

any branch of the square root of J2ℓ(γ,Z). Let S(m)ℓ denote the union of

Sℓ(Γ) for congruence subgroups Γ ⊂ Γθm. The group Mp(Wm)f acts on the

space S(m)ℓ and it is important to know which representations appear in this

space. We shall explicitly construct a rather small irreducible submodule,which is neither tempered nor generic at any finite prime.

We define the space C2κ of Hilbert cusp forms on PGL2 of weight 2κ inDefinition 5.4. Let πf ≃ ⊗′

pπp be an irreducible admissible unitary genericrepresentation of PGL2(Af ). For some reason (see Remark 6.3) we supposethat none of πp is supercuspidal, i.e., there is a collection of continuous char-acters µp of the multiplicative groups of nonarchimedean local fields Fp suchthat πf is equivalent to the unique irreducible submodule of the principal se-ries representation ⊗′

pI(µp, µ−1p ), where µp is unramified for almost all p. Put

µf =∏

p µp. We form the restricted tensor product Iψfm (µf ) = ⊗′

pIψpm (µp),

where Iψpm (µp) is the representation of the local metaplectic group Mp(Wm)p

on the space of smooth functions hp on Mp(Wm)p transforming on the leftaccording to

hp((m(a)n(b), ζ)g) = ζmγψp(det a)mµp(det a)|det a|(m+1)/2p hp(g)

for all ζ ∈ ±1, a ∈ GLm(Fp), z ∈ Symm(Fp) and g ∈ Mp(Wm)p. This

representation has a unique irreducible submodule Aψfm (µf ), which is unitary.

Theorem 1.1. Notation being as above, Aψfm (µf ) appears in S

(m)(2κ+m)/2 if

and only if πf appears in C2κ and (−1)∑v∈S∞ κv

∏p µp(−1) = 1.

We here denote the tuple(κv+

m2

)v∈S∞

∈ 12Z

d simply by (2κ+m)/2. The

representation πf is the Shimura correspondence of Aψf1 (µf ) and Aψf

2 (µf )is the Saito-Kurokawa lifting of πf . Both are theta liftings. We discussthe connection of this result with Arthur’s endoscopic classification in §6.2.Though the trace formula will ultimately lead to another proof, our proof,


which relies heavily on the theory of the Shimura correspondence but not onthe Saito-Kurokawa lifting, is completely elementary. If κv < m for everyv ∈ S∞, then we obtain an irreducible cuspidal automorphic representationwhich is nontempered at all the places.

More importantly, our proof gives more precise information. We can

describe how the representation Aψfm (µf ) is embedded in S

(m)(2κ+m)/2 quite

explicitly. Fix a Haar measure db = ⊗pdbp on Symm(Af ). Then we canassociate to each ξ ∈ Sym+

m a basis vector wµfξ of the one-dimensional vector

space HomSymm(Af )(Iψfm (µf ) n, ψξf ) by

wµfξ (⊗php) =∏p

wµpξ (hp),

where wµpξ ∈ HomSymm(Fp)(I

ψpm (µp) n, ψξp) is defined by

wµpξ (hp) =

∫Symm(Fp)

hp

(((0 1m

−1m 0

)n(bp), 1

))ψξp(bp) dbp

×|det ξ|(m+1)/4

p

L(12 , µpχ

det ξp

) [(m+1)/2]∏j=1

L(2j−1, µ2p)×

1 if 2 ∤ m,

L(m+12 , µpχ

(−1)m/2

p

)if 2|m.

The integral diverges but makes sense as it stabilizes. One can check thatwµpξ (hp) = 1 for almost all p.

Theorem 1.2. If πf appears in C2κ and (−1)∑v∈S∞ κv

∏p µp(−1) = 1, then

Aψfm (µf ) appears in the decomposition of S

(m)(2κ+m)/2 with multiplicity one,

and there is a set ctt∈F×+

of complex numbers such that the Mp(Wm)f -

intertwining embeddings iηm : Aψfm (µf χ

ηf ) → S

(m)(2κ+m)/2 are given for all m

and η ∈ F×+ by means of the Fourier expansion

iηm(h)(Z) =∑

ξ∈Sym+m

| det ξ|(2κ+m)/4cη det ξe∞(tr(ξZ))wµf χ

ηf

ξ (h).

The constant ct is a mysterious part of the tth Fourier coefficient of aHilbert cusp form of weight κ + 1

2 . When F = Q, Kohnen and Zagier [34]

have given an exact relation between the square c2t and the central valueL(12 , π ⊗ χt

). We refer to Theorem 12.3 of [16] for its extension to Hilbert

cusp forms. The formula of Fourier coefficients looks like the classical Maassrelation.

Our proof is direct and simple. Section 7 is the technical heart of thispaper. Lemmas 7.4 and 7.7 play the important role in the proof of Theorem1.2. The series iηm(h) is a cusp form if and only if so are all its Fourier-Jacobicoefficients of degree 1, thanks to Lemma 7.7, which can apply to arbitraryHilbert-Siegel cusp forms. Taking into account the inductive structure de-scribed in Lemma 7.2, we will explicitly compute those Fourier-Jacobi series

5

in §8.3. From (8.5) we can choose the coefficients ct so that they are theShintani lifts of πf . We will prove Theorems 1.1 and 1.2 in §8.3 except for

the multiplicity of Aψfm (µf ), which is determined in §8.5. The proof in §§8.4-

8.5 is most technically difficult in this paper and may be skipped by readers,though a characterization of the lifting is given at the end of §8.5. Namely, ifan irreducible cuspidal automorphic representation of Mp(Wm) has degener-ate principal series as local components at nonarchimedean primes and haslowest weight representations of scalar K-type as archimedean components,then it is our lifting.

Theorem 6.1 constructs analogous liftings of πf for inner forms of sym-plectic groups of even rank, which are given by similar Fourier series withthe same coefficients ct. The series naturally extends to a cusp form on thesimilitude group for even m (see Remark 6.2(1)). In the proof of this casewe shall use transfers of the Saito-Kurokawa lifts instead of the Shimuracorrespondence due to the lack of Fourier-Jacobi coefficients of degree 1.

We will letm = 2n and construct Hilbert-Siegel cuspidal Hecke eigenformsof even degree by making Theorems 1.2 and 6.1 explicit with the test functionhp invariant under a maximal compact subgroup of Sp2n(Fp). Let I2n(µf ) =

Iψf2n

(µf χ

(−1)n

f

)be a representation of Sp2n(Fp). We denote the integer rings

of F and Fp by o and op, respectively, the different, trace, norm of F/Q byd, TrF/Q, NF/Q, and the cardinality of the residue field o/p by qp. Put

Rm = ξ ∈ Symm(F ) | tr(ξz) ∈ o for every z ∈ Symm(o).

Set R+m = Rm ∩ Sym+

m. With fractional ideals b, c of o such that bc ⊂ o, weput

(1.2) Γm[b, c] =

(α βγ δ

)∈ Spm(F )

∣∣∣∣ α, δ ∈ Mm(o)β ∈ Mm(b), γ ∈ Mm(c)

.

The norm and the order of a fractional ideal of o are defined by N(pi) = qipand ordp p

j = j. We denote the conductor of χη by dη and put

fηp =1

2(ordp η − ordp d

η) ∈ Z, fη =

√|NF/Q(η)|N(dη)

∈ Q×.

We put δp(η) = 1 if√η ∈ Fp, δp(η) = −1 if Fp(

√η) is an unramified

quadratic extension of Fp, and δp(η) = 0 if Fp(√η) is a ramified quadratic

extension of Fp. We define Ψp(η,X) ∈ C[X +X−1] by

Ψp(η,X) =

Xfηp +1−X−fηp −1

X−X−1 + q−1/2p δp(η)

Xfηp −X−fηp

X−X−1 if fηp ≥ 0,

0 if fηp < 0.

For ξ ∈ Sym+2n we set

D(ξ) = (−1)n det(2ξ), δp(ξ) = δp(D(ξ)), f ξp = fD(ξ)p , fξ = fD(ξ).


The Siegel series associated to ξ ∈ R+2n and p is defined by

bp(ξ, s) =∑

z∈Sym2n(Fp)/Sym2n(op)

ψ′p(−tr(ξz))ν[z]−s,

where ν[z] = [zo2np +o2np : o2np ] and ψ′p is an arbitrarily fixed additive character

on Fp of order zero. We define the polynomial γp(ξ,X) ∈ Z[X] by

γp(ξ,X) =1−X

1− δp(ξ)qnpX

n∏j=1

(1− q2jp X2).

There exists a monic polynomial Fp(ξ,X) ∈ Z[X] such that

bp(ξ, s) = Fp(ξ, q−sp )γp(ξ, q

−sp ), Fp(ξ,X) = q2n+1

p X2fξpFp(ξ, q−2n−1p X−1)

(see [29, 25]). We define Fp(ξ,X) ∈ C[X +X−1] by

Fp(ξ,X) = X−fξpFp(ξ, q−(2n+1)/2p X).

An explicit formula for Fp(ξ,X) is given by Ikeda and Katsurada (cf. [26]).We consider the parallel weight (k, . . . , k), which is denoted simply by

k, for the rest of this section. When d(k + n) is even, the Kohnen plus

subspace S+,n(2k+1)/2 of the space of Hilbert cusp forms of weight k + 1

2 with

respect to n and the congruence subgroup Γ1[d−1, 4d] is defined in [33] if

F = Q and in [16] in general. It should be remarked that our notationdiffers from that used in [16]. The superscript n indicates that one should

take η = (−1)n in Definition 13.2 of [16]. Let πf ≃ ⊗′pI(α

spp , α

−spp ) be an

irreducible summand of C2k. There is a Hilbert cusp form in S+,n(2k+1)/2 which

has a Fourier expansion of the form

hn(Z) =∑η∈F×

+

c(η)e∞(ηZ)f(2k−1)/2(−1)nη

∏p

Ψp((−1)nη, qspp )

and which generates ⊗′pIψp

1 (αspp χ

(−1)n

p ).The following result is a special case of Corollary 10.1 and the generaliza-

tion of the main result of [22].

Corollary 1.3. Notations and assumptions being as above, the series

Lift2n(π)(Z) =∑ξ∈R+

2n

c(det(2ξ))f(2k−1)/2ξ

∏p

Fp(ξ, qspp )e∞(tr(ξZ))

is a Hilbert-Siegel cusp form in Sk+n(Γ2n[d−1, d]) and generates ⊗′

pI2n(αspp ).

All the results and the proofs in this paper are applicable, with somemodifications, to holomorphic cusp forms on tube domains of other types,which generalizes the level one holomorphic cusp forms constructed in [23,59, 30] to higher level. Actually, analogous Hilbert-Hermitian cusp forms areconstructed for all the Hilbert cusp forms in a similar way. The assumptionon πf should not be essential, but if supercuspidal representations had been

7

included, then the construction would not have been so neat (cf. Remark6.3). Kim and Yamauchi [32] currently constructed higher level holomorphiccusp forms on the exceptional group of type E7, generalizing their previouswork [30].

Moreover, we shall construct Hilbert-Siegel cusp forms of Miyawaki typein Corollary 10.3, following the same technique as in [24]. Atobe [4] cur-rently used our construction of Hilbert-Siegel cusp forms to establish a the-ory of Miyawaki liftings. Miyawaki liftings were constructed in a classi-cal setting for quasisplit unitary groups in [5] and for GSpin(2, 10) in [31].Furthermore, our lifting combined with the theta lifting for the dual pairMp(Wm) × O(2m + 1), Spm × O(2m) or Sp(n, n) × O∗(4n) produces evenmore Hilbert-Siegel cusp forms and CAP representations of orthogonal orquaternion unitary groups.

Let n be a positive integer such that dn is even. Then there exists a 4n-dimensional totally positive definite quadratic space (Vn, qVn) over F whichis split over every nonarchimedean local field Fp. We write M(O(Vn)) forthe space of locally constant functions on O(Vn, F )\O(Vn,Af ). A function inM(O(Vn)) is called an algebraic modular form (of trivial weight) for O(Vn)(cf. [14]). Since O(Vn, Fp) is the split orthogonal group, it has a parabolicsubgroup P2n(Fp) whose Levi subgroup is isomorphic to GL2n(Fp). Let

I2n(µp) = IndO(Vn,Fp)P2n(Fp)

µp det be the normalized induced representation.

We here do not work in greatest possible generality, but rather considera reasonable special case of irreducible principal series, in which the mainideas of the theory become clear.

Corollary 1.4. Let π = ⊗′vπv be an irreducible cuspidal automorphic repre-

sentation of PGL2(A) such that πv is a discrete series with minimal weight±2n for v ∈ S∞ and such that πp ≃ I(µp, µ

−1p ) for every prime p. If dn is

even, then I2n(µf ) ≃ ⊗′pI2n(µp) occurs in M(O(Vn)) with multiplicity one.

This O(Vn,Af )-intertwining embedding is here denoted by

jn : I2n(µf ) → M(O(Vn)).

Theorem 1.1 applied to π ⊗ χ(−1)n implies that I2n(µf ) appears in S(2n)2n if

and only if ε(12 , π

)= 1. The Schrodinger model of the Weil representation

yields a representation ωψfVn

of Sp2n(Af )×O(Vn,Af ) on the space S(V 2nn (Af ))

of Schwartz functions on V 2nn (Af ). The theta function associated to ϕ ∈

S(V 2nn (Af )) is defined by

θ(Z, ϕ) =∑

x∈V 2nn (F )

ϕ(x)e∞

(1

2tr(qVn(x)Z)

),

where qVn(x) = (qVn(xi, xj)) ∈ Sym2n(F ) is the matrix of inner products ofthe components of x = (x1, . . . , x2n) ∈ V 2n

n (F ). We will explicitly constructa nonzero Sp2n(Af )-intertwining, O(Vn,Af )-invariant map

ϑψfn : ωψf

Vn⊗ I2n(µf ) → I2n(µf )


in §11.1. Such a map is unique up to scalar by the Howe principle.

Corollary 1.5. Notation being as in Corollary 1.4, if L(12 , π

)= 0, then there

is a nonzero constant c such that

(1.3) i12n(ϑψfn (ϕ⊗ f))(Z) = c

∫O(Vn,F )\O(Vn,Af )

jn(f)(g)θ(Z, ωψfVn(g)ϕ) dg

for all ϕ ∈ S(V 2nn (Af )) and f ∈ I2n(µf ).

This identity is analogous to the Siegel-Weil formula (cf. [51, 58, 38]). The

ξth Fourier coefficient of i12n(h) is built out of the local quantity wµpχ

(−1)n

p

ξ (hp)

and the det(2ξ)th Fourier coefficient of a Hilbert cusp form in Iψf1 (µf χ

(−1)n

f ).On the other hand, the theta function involves the representation numbers ofquadratic forms, essentially diophantine quantities. Our identity thus con-tains a link between local and global information. We will prove Corollaries1.4 and 1.5 in §11.2 and 11.3, respectively.

When L is a lattice in Vn(F ), it is one of the classical tasks of numbertheory to determine the number of representations of ξ ∈ Symm(F ) by L

N(L , ξ) = ♯(x1, . . . , xm) ∈ Lm | qVn(xa, xb) = 2ξab for 1 ≤ a, b ≤ m.

Let Ξn be the finite set of isometry classes of maximal lattices in Vn(F ). Tosolve this problem for L ∈ Ξn, it is sufficient to determine the sum

Rn(ξ, f) =∑L∈Ξn

f(L)N(L, ξ)

♯O(L)

for all Hecke eigenfunctions f : Ξn → C. It is important to note thatRn(ξ, f) appears in the ξth Fourier coefficient of the theta lift of f . Whenf : Ξn → C is associated to an irreducible everywhere unramified cuspidalautomorphic representation of PGL2(A) with parallel minimal weights ±2nin the sense of Corollary 1.4, we will give a product formula for Rn(ξ, f)with ξ ∈ R2n.

Corollary 1.6. Let πf ≃ ⊗′pI(α

spp , α

−spp ) be an irreducible summand of C2n.

Assume that dn is even. Let f : Ξn → C be a common eigenfunction of allHecke operators whose standard L-function is

∏2nj=1 L

(s+ n+ 1

2 − j, π).

(1) If L(12 , π

)= 0, then Rn(ξ, f) = 0 for every ξ ∈ R2n.

(2) If L(12 , π

)= 0, then Rn(ξ, f) = 0 unless ξ ∈ R+

2n, in which case

Rn(ξ, f) = c(det(2ξ))f(2n−1)/2ξ

∏p

Fp(ξ, qspp ).

The constant c(η) satisfies c(ηa2) = c(η)∏v∈S∞

sgnv(a)n for every

a ∈ F×, and the series∑η∈F×

+

c(η)e∞(ηZ)f(2n−1)/2(−1)nη

∏p

Ψp((−1)nη, qspp )

9

belongs to S+,n(2n+1)/2 and generates Iψf

1 (µf χ(−1)n

f ).

One can view this result, which is a consequence of Corollary 11.3, as ageneralization of the Siegel formula. The function f exists uniquely up toscalar by Corollary 1.4. Actually, one can obtain f as an eigenvector of theKneser p-neighbor matrix with eigenvalue

q(2n−1)/2p

q2np − 1

qp − 1(qspp + q

−spp )

thanks to Proposition 12.1 (cf. [13]).Since c(η) is determined uniquely up to scalar, one gets relations be-

tween the ratios of the variousRn(ξ, f) and those of c(det(2ξ)). In particular,

if L(12 , π ⊗ χD(ξ)

)= 0, then Rn(ξ, f) = 0. Whereas the generalized Kohnen-

Zagier formula mentioned above gives an explicit formula for |c(η)|2, ourformula involves c(η) itself.

When F = Q(√2), the genus of 8-dimensional totally positive definite

even unimodular lattices over Z[√2] has 6 classes E8, 2∆

′4,∆8, 2D4, 4∆2, ∅.

As an application we shall determine everywhere unramified representationsof O(V2,Af ) occurring in M(O(V2)) and their degrees and investigate thesubspace of M4(Sp4(Z[

√2])) spanned by the 6 associated theta series in

Section 12. This subspace is 6-dimensional and intersects the space of cuspforms in a one dimensional subspace, which is spanned by the lift of a Hilbertcusp form ϕ4 of weight 4. It corresponds to a Hilbert cusp form ϕ5/2(Z) =∑

η b(η)e∞(ηZ) in the Kohnen plus space of weight 52 (see §12.3 for their

constructions). We will give the following example of Corollary 1.6:

N(E8, ξ)

28 · 32 · 5 · 7+N(2∆′

4, ξ)

29 · 32−N(∆8, ξ)

28 · 3 · 7+N(2D4, ξ)

28 · 32−N(4∆2, ξ)

29 · 3+

N(∅, ξ)27 · 32 · 5

= b(ϖdξ)f3/2ξ

∏p

Fp(ξ, βp).

Here, ξ ∈ R+4 , ϖdξ is a totally positive generator of dξ, βp, β−1

p is thep-Satake parameter of ϕ4, and ϕ5/2 has been normalized so that b(1) = 1.

Theorem 1.2 is a generalization of the lifting constructed by Ikeda [22],

where he discussed the case in which F = Q, m is even and µpχ(−1)m/2

p isan unramified unitary character of Q×

p for all rational primes p. Ikeda’s firstproof in [22] is rather indirect and uses the Siegel Eisenstein series which alsohas a similar Fourier series. As it uses the algebraic independence of the p−s,it works only over rationals. This method cannot apply to nonsplit innerforms of symplectic groups even when F = Q. Subsequently, Ikeda inventeda more general approach and proved in his unpublished preprint that if

⊗′pI(µp, µ

−1p ) occurs in the space C2κ and its root number is 1, then Iψf

m (µf )

occurs in the space S(m)(2κ+m)/2 with multiplicity one. Later, Yamana refined

and generalized this new approach, giving the explicit Fourier expansionsand combining it with theta correspondence. The numerical examples in


Section 12 were added by Ikeda. The present article was written finally bycombining Yamana’s manuscript with Ikeda’s original preprint.

Acknowledgement. Ikeda is partially supported by the JSPS KAKENHIGrant Number 26610005. Yamana is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 18K03210. This paper was partly writtenwhile Yamana was visiting University of Rijeka, and he thanks Neven Gr-bac for his hospitality and encouragement. We thank Marcela Hanzer for

suggesting the unitarity of Aψfm (µf ) and Zhengyu Mao for stimulating dis-

cussions. We should also like to thank the anonymous referee for helpfulcomments. Especially, we have added the index according to his or hersuggestion. Last, but not least, Yamana thanks Max Planck Institute forMathematics for the excellent working environment.

2. Preliminaries

2.1. Notation. For an associative ring O with identity element we denoteby O× the group of all its invertible elements and by Mm

n (O) the O-moduleof all m × n matrices with entries in O. Put Om = Mm

1 (O), Mn(O) =Mnn(O) and GLn(O) = Mn(O)×. The zero element of Mm

n (O) is denotedby 0 and the identity element of the ring Mn(O) is denoted by 1n. Ifx1, . . . , xk are square matrices, then diag[x1, . . . , xk] denotes the matrix withx1, . . . , xk in the diagonal blocks and 0 in all other blocks. Assume that Ohas an involution a 7→ aι. For a matrix x over O, let tx be the transposeof x and x∗ = txι the conjugate transpose of x. Given ϵ ∈ ±1, we letSϵm = z ∈ Mm(O) | z∗ = ϵz be the space of ϵ-Hermitian matrices ofsize m. Set z[x] = x∗zx for matrices z ∈ Sϵm and x ∈ Mm

n (O). Given ϵ-Hermitian matrices B ∈ Sϵj and Ξ ∈ Sϵk, we sometimes write B ⊕Ξ instead

of diag[B,Ξ] ∈ Sϵj+k, particularly when we view them as ϵ-Hermitian forms.

We say that Ξ is represented by B if there is a matrix x ∈ Mjk(O) such that

B[x] = Ξ.We denote by N, Z, Q, R, C, R×

+, S and µk the set of strictly posi-tive rational integers, the ring of rational integers, the fields of rational,real, complex numbers, the groups of strictly positive real numbers, com-plex numbers of absolute value 1 and kth roots of unity. We define the signcharacter sgn : R× → µ2 by sgn(x) = x/|x|. When X is a totally discon-nected locally compact topological space or a smooth real manifold, we writeS(X) for the space of Schwartz-Bruhat functions on X.

2.2. Quaternionic unitary groups. Let F for the moment be an arbitraryfield and D a quaternion algebra over F , by which we understand a centralsimple algebra over F such that [D : F ] = 4. We denote by ι the maininvolution of D, by x∗ = txι the conjugate transpose of a matrix x ∈ Mn(D),by ν : GLn(D) → Gm the reduced norm and by τ : Mn(D) → Ga the reducedtrace, where Gm = GL1 and Ga = M1 are the multiplicative and additive

11

groups in one variable over F . If n = 1, then ν(x) = xxι and τ(x) = x+ xι

for x ∈ D. Put

Sn = B ∈ Mn(D) | B∗ = −B, Sndn = Sn ∩GLn(D).,

When n = 1, we simply write D− = S1 and Dnd− = Snd

1 . We identify Sn withthe space of D-valued skew Hermitian forms on the right D-module Dn, bywhich we understand an F -linear map B : Dn ×Dn → D such that

B(x, y)ι = −B(y, x), B(xa, yb) = aιB(x, y)b (a, b ∈ D; x, y ∈ Dn).

We frequently regard D as an algebraic variety over F and define theF -algebraic group Gn by

Gn = g ∈ GL2n(D) | gJng∗ = λn(g)Jn with λn(g) ∈ Gm,where

Jn =

(0 1n1n 0

)∈ GL2n(F ) ⊂ GL2n(D).

We call λn : Gn → Gm the similitude character. We are interested in itskernel Gn = g ∈ Gn | λn(g) = 1. For A ∈ GLn(D), z ∈ Sn and t ∈ Gm wedefine matrices in GL2n(D) by

m(A) =

(A 00 (A−1)∗

), n(z) =

(1n z0 1n

), d(t) =

(1n 00 t · 1n

).

Let Pn be the parabolic subgroup of Gn which has a Levi factor Mn =m(A) | A ∈ GLn(D) and the unipotent radical Nn = n(z) | z ∈ Sn.

2.3. The split case. We include the case in which D is the matrix algebraM2(F ) of degree 2 over F . Let us now take this case. We often identifyMm(D) with M2m(F ) by viewing an element (xij) of Mm(D) as a matrix ofsize 2m whose (i, j)-block of size 2 is xij . Put

J =

(0 1−1 0

)∈ GL2(F ), Bm = diag[J, . . . , J︸︷︷︸

m

] ∈ GL2m(F ).

Then we easily see that X∗ = B−1m

tXBm for X ∈ Mm(D), where tX denotesthe transpose of X as a matrix of size 2m. We are led to

σnGnσ−1n = Sp2n, BnSn = Sym2n,

where σn = diag[12n, Bn]. Since all automorphisms of Sp2n are inner, weconclude that Gn is an inner form of Sp2n for every D.

2.4. The case n = 1. When G is an algebraic group over a field F andZ is its center, we write PG for the adjoint group G/Z. It is importantto note that the group PG1 is isomorphic to a certain orthogonal group.To see this relation, we recall some well-known facts on Clifford algebras.The basic reference is [50]. For the time being, we will take V to be afinite dimensional vector space over a field F of characteristic different from2, and let qV : V → F be a nondegenerate quadratic form. The associatedsymmetric bilinear form is defined by qV (x, y) =

12(qV (x+y)−qV (x)−qV (y)).


A Clifford algebra of (V, qV ) is an F -algebra A with an F -linear mapp : V → A satisfying the following properties:

• A has an identity element, which we denote by 1A;• A as an F -algebra is generated by p(V ) and 1A;• p(x)2 = qV (x)1A for every x ∈ V ;• A has dimension 2ℓ over F , where ℓ = dimV .

It is known that such a pair (A, p) is unique up to isomorphism. Moreover,p is injective, and as such, V can be viewed as a subspace of A via p. Wedenote this algebra A by A(V ). The basic equalities are xy+yx = 2qV (x, y)for x, y ∈ V .

There is an automorphism β 7→ β′ of A(V ) such that v′ = −v for everyv ∈ V . Similarly, there is an anti-automorphism β 7→ βρ of A(V ) such thatvρ = v for every v ∈ V . The orthogonal group O(V ) consists of elementsg ∈ GL(V ) which satisfy qV (gx) = qV (x) for all x ∈ V . The Clifford groupG(V ) consists of elements β ∈ A(V )× such that βV β−1 = V . Let us put

A+(V ) = β ∈ A(V ) | β′ = β, A−(V ) = β ∈ A(V ) | β′ = −β,G+(V ) = G(V ) ∩A+(V ), G−(V ) = G(V ) ∩A−(V ), G·(V ) = G+(V ) ∪G−(V ).

It is known that G·(V ) is a subgroup of G(V ). Put µ1(β) = ββρ for β ∈A(V ). The map µ1 gives a homomorphism of G·(V ) to F×. For β ∈ G(V )we can define ϑ(β) ∈ GL(V ) by ϑ(β)v = βvβ−1 (v ∈ V ). In §11.2 we willuse the following result, which is proved in Theorem 3.6 of [50].

Lemma 2.1. (1) The map ϑ gives an isomorphism of G+(V )/F× ontothe special orthogonal group SO(V ) = SL(V ) ∩O(V ).

(2) If ℓ is even, then G·(V ) = G(V ) and ϑ is an isomorphism of G(V )/F×

onto O(V ).

By restricting the symmetric bilinear form on D2 given by (x, y) 7→12τ(xy), we obtain a three dimensional quadratic space VD = (D−, qD−)of discriminant 1. In what follows we take V = Fe ⊕ VD ⊕ Fe′ and definethe quadratic form qV by

qV (re+ x+ r′e′) = rr′ + qD−(x) = rr′ − ν(x) (r, r′ ∈ F ; x ∈ D−).

Lemma 2.2. Notation and assumption being as above, there is an F -linearring homomorphism Ψ : A(V ) → M2(D) such that

Ψ(βρ) =

(0 11 0

)Ψ(β)∗

(0 11 0

)for all β ∈ A(V ) and whose restriction gives isomorphisms

A+(V ) ≃ M2(D), G+(V ) ≃ G1.

Furthermore, for given t ∈ F×, a ∈ D× and z ∈ D−,

(ϑ Ψ−1)(d(t)m(A))(re+ x+ r′e′) = t−1ν(A)re+AxA−1 + tν(A)−1r′e′,

(ϑ Ψ−1)(n(z))(re+ x+ r′e′) = (r − τ(zx) + r′ν(z))e+ (x+ r′z) + r′e′,

13

(ϑ Ψ−1)(J1)(re+ x+ r′e′) = r′e− x+ re′.

Proof. This isomorphism is explained in §§A4.2 and A4.3 of [50], to whichwe refer for additional explanation. The last assertion can be verified by asimple calculation.

3. Degenerate Whittaker functions

The ground field F is a totally real number field or its completion. Ex-cluding the case of the real field, we let o be the maximal order of F andfix a maximal order O of D. We define the additive character of C by

e(z) = e2π√−1z for z ∈ C. In the real case we set ψ = e|R. When F

is an extension of Qp, we define the character ψ of F by ψ(x) = e(−y)with y ∈ Z[p−1] such that TrF/Qp(x) − y ∈ Zp. In the global case we putψ∞(x) =

∏v∈S∞

e(xv), ψf (x) =∏

p ψp(xp) and ψ(x) = ψ∞(x)ψf (x) forx ∈ A.

3.1. Degenerate principal series. In this and the next subsections F isa completion at a nonarchimedean prime. We denote by q the order of theresidue field of the valuation ring o, by α(t) = |t| the normalized absolutevalue of t ∈ F× and by χt the quadratic character of F× associated toF (

√t)/F via class field theory. For B ∈ Snd

n we set χB = χ(−1)nν(B). Wewrite Ω(F×) for the group of all continuous homomorphisms from F× toC×. Continuous homomorphisms of the form αs for some s ∈ C are calledunramified. When µ ∈ Ω(F×), we define ℜµ as the unique real number suchthat µα−ℜµ is unitary.

For µ ∈ Ω(F×) the normalized induced representation Jn(µ) is realizedon the space of smooth functions f : Gn → C satisfying

f(d(t)m(A)n(z)g) = µ(t−nν(A))|t−nν(A)|(2n+1)/2f(g)

for all t ∈ F×, A ∈ GLn(D), z ∈ Sn and g ∈ Gn. We denote its restrictionto Gn by In(µ). Since Gn = d(t) | t ∈ F×⋉Gn, these representations canalso be realized on the space of smooth functions f : Gn → C satisfying

f(m(A)n(z)g) = µ(ν(A))|ν(A)|(2n+1)/2f(g)

for all A ∈ GLn(D), z ∈ Sn and g ∈ Gn.Let B ∈ Sn. We define the character ψB : Sn → S by ψB(z) = ψ(τ(Bz))

for z ∈ Sn. For any smooth representation Π of Gn we put

WhB(Π) = HomSn(Π n, ψB).

Proposition 3.1 ([28, 39, 60]). (1) If −12 < ℜµ < 1

2 , then In(µ) is ir-reducible and unitary.

(2) dimWhB(In(µ)) = 1 for all µ ∈ Ω(F×) and B ∈ Sndn .

(3) Assume that µ2 = α. Then In(µ) has a unique irreducible subrep-resentation An(µ), which is unitary. Moreover, An(µ) is the uniqueirreducible subrepresentation of Jn(µ). Furthermore, WhB(An(µ))

is nonzero if and only if χB = µα−1/2.


Proof. The module structure of In(µ) is determined by Kudla-Rallis [39]in the symplectic case and by Yamana [60] in the quaternion case. Theunitarity follows from the general fact on irreducible subquotients of endsof complementary series explained in Section 3 of [55]. The second part isproved in [28]. We will prove (3). Since Jn(d(t), µ)An(µ) is an irreduciblesubmodule of In(µ), we know by its uniqueness that Jn(d(t), µ)An(µ) =An(µ) for all t ∈ F×. It is known that

In(µ)/An(µ) ≃∑

χB′=µα−1/2

Rψn(B′),

where B′ extends over all equivalence classes of nondegenerate skew Hermit-

ian matrices of size n with character µα−1/2 and Rψn(B′) is an irreduciblerepresentation arising via the Weil representation of Gn ×U(B′) associatedto B′, where U(B′) = g ∈ GLn(D) | B′[g] = B′. In the symplectic case

WhB(Rψn(B′)) is nonzero if and only if B is equivalent to B′ by Lemma 3.5

of [39]. One can see that this result is valid in the quaternion case by a basiccalculation based on Lemme on p. 73 of [43]. The claimed fact derives fromthe exactness of the Jacquet functor combined with these observations.

3.2. Jacquet integrals. For an ideal c of o we put

(3.1) KDn [c] =

(α βγ δ

)∈ Gn

∣∣∣∣ α, δ ∈ Mn(O)β ∈ c−1Mn(O), γ ∈ cMn(O)

.

For g ∈ Gn the quantity εc(g) is defined by writing g = pk with p =m(A)n(z) ∈ Pn, k ∈ KD

n [c], and setting εc(g) = |ν(A)|. For µ ∈ Ω(F×),

f ∈ In(µ) and s ∈ C we define f (s) ∈ In(µαs) by f (s)(g) = f(g)εo(g)

s. Put

Rn = Sn ∩Mn(O), RD2n = B ∈ Sn | τ(BRn) ⊂ o, RD,nd

2n = R2n ∩ Sndn .

Define a Haar measure dz on Sn so that the measure of Rn is 1.When µ is unramified, we put L(s, µ) = (1 − µ(ϖ)q−s)−1, where ϖ is

a generator of p. Set L(s, µ) = 1 if µ is ramified. When µ is the trivialcharacter, we write ζ(s) = L(s, µ). For B ∈ Snd

n the integral

wµαs

B (f (s)) =

∫Sn

f (s)(Jnn(z))ψB(z) dz

defines a formal Dirichlet series in the variable s, which is absolutely conver-gent for ℜs > n+ 1

2 −ℜµ. Actually, the integral stabilizes and consequently,

it is a polynomial of q−s, from which we can evaluate wµαs

B (f (s)) at s = 0so as to get a basis vector wµ

B ∈ WhB(In(µ)). From now on we assume that

ℜµ > −12 and set

wµB(f) = |ν(B)|(2n+1)/4wµB(f)

L(n+ 1

2 , µ)

L(12 , µχ

B) n∏j=1

L(2j − 1, µ2).

We write ϱ (resp. ℘) for the right regular action of Gn (resp. Gn) on thespace of smooth functions on Gn (resp. Gn).

15

Lemma 3.2. Let B ∈ Sndn and µ ∈ Ω(F×). Assume that ℜµ > −1

2 .

(1) 0 = wµB ∈ WhB(In(µ)).(2) When µ2 = α, the restriction of wµB to An(µ) is nonzero if and only

if χB = µα−1/2.(3) If t ∈ F× and A ∈ GLn(D), then

wµB ℘(d(t)m(A)) = µ(t−nν(A))−1wµt−1B[A]

.

Proof. The first part is clear. The second part is evident from Proposition3.1. The third part can be verified by obvious changes of variables.

We refer the reader to Lemma 3.3 of [62] for the following bound.

Lemma 3.3. Let f ∈ In(µ). For any compact subset C of Gn there area Schwartz function Φ on Sn and a positive constant M such that for all∆ ∈ C and B ∈ Snd

n

|wµB(ϱ(∆)f)| ≤ |ν(B)|−MΦ(B).

The Siegel series associated to B ∈ RD2n is defined by

b(B, s) =∑

z∈Sn/Rn

ψ′(−τ(Bz))ν[z]−s,

where ν[z] = [zOn + On : On]1/2 and ψ′ is an arbitrarily fixed additivecharacter on F of order zero. We define the function γ(B, s) by

γ(B, s) = ζ(s)−1L(s− n, χB)×

∏nj=1 ζ(2s− 2j)−1 if D ≃ M2(F ),∏[n/2]

j=1 ζ(2s− 4j)−1 otherwise.

Put F (B, q−s) = b(B, s)γ(B, s)−1. Then F (B,X) is a polynomial of degreefB with constant term 1 by [25, 62] (see Section 10 for the definition of fB).

We denote the different of F/Qp by d.

Lemma 3.4. If B ∈ RD,nd2n , then there is a nonzero constant c(s) indepen-

dent of B such that wαs

B (εs+(2n+1)/2d ) = c(s)|ν(B)|(2n+1)/4F (B, q−s−(2n+1)/2).

If d = o and D ≃ M2(F ), then c(s) = 1 and there is a positive constant M ,

depending only on n, such that |wαsB (εs+(2n+1)/2d )| ≤ |ν(B)|−M for B ∈ Snd

n

and ℜs > −12 .

Proof. We refer the reader to Lemma 4.5 of [62] except for the last statement.In the proof of Lemma 4.1 of [22] Ikeda shows that when ℜs > 0,

|F (B, q−s)| ≤ |ν(B)|−(13n2+13n+4)/2,

from which we obtain the desired estimate.


3.3. Degenerate Whittaker functions: the archimedean case. Wediscuss the case in which F = R and D = M2(R). Let Symm(R)+ be theset of positive definite symmetric matrices of rank m over R. Put

S+n = B ∈ Sn | BBn ∈ Sym2n(R)+, G+

n = g ∈ Gn | λn(g) > 0.

We can define the action of G+n on the space

Hn = Z ∈ M2n(C) | t(ZB−1n ) = ZB−1

n , ℑ(ZB−1n ) ∈ Sym2n(R)+

and the automorphy factor on G+n × Hn by

gZ = (αZ + β)(γZ + δ)−1, j(g, Z) = ν(g)−1/2ν(γZ + δ)

for Z ∈ Hn and g =

(α βγ δ

)∈ G+

n with matrices α, β, γ, δ of size n over

M2(R). There is a biholomorphic isomorphism from Hn onto the Siegelupper half space H2n (cf. §2.3). Define the origin of Hn and the standardmaximal compact subgroup of Gn by

i =√−1Bn ∈ Hn, Kn = g ∈ Gn | g(i) = i.

For ℓ ∈ N and B ∈ S+n we define a function W

(ℓ)B : G+

n → C by

W(ℓ)B (g) = ν(B)ℓ/2e(τ(Bg(i)))j(g, i)−ℓ.

Clearly,

(3.2) W(ℓ)B (n(z)d(t)m(A)gk) = e(τ(Bz))sgn(ν(A))ℓW

(ℓ)t−1B[A]

(g)j(k, i)−ℓ

for z ∈ Sn, A ∈ GL2n(R), t ∈ R×+, g ∈ G+

n and k ∈ Kn.

3.4. Degenerate Whittaker functions: the global case. Until the endof the next section D is a totally indefinite quaternion algebra over a totallyreal number field F . For each prime v of F and an algebraic group V definedover F , let Fv be the v-completion of F and put Vv = V(Fv) to make ourexposition simpler. The adele group, the finite part of the adele group, theinfinite part of the adele group and its connected component of the identityare denoted by V(A), V(Af ), V(A∞) and V(A∞)+, respectively. For an adelepoint x ∈ V(A) we denote its projections to V(Af ), V(A∞) and Vv by xf ,x∞ and xv, respectively.

We will denote the group of totally positive elements of F by F×+ . Put

S+n = B ∈ Sn(F ) | B ∈ Sn(Fv)

+ for all v ∈ S∞.

When n = 1, we write D+− = S+

1 . For B ∈ Sn(F ) we define the characters

ψB : Sn(A) → S by ψB(z) = ψ(τ(Bz)) whose restriction to Sn(Af ) isdenoted by ψBf . For any smooth representation Π of Gn(Af ) we put

WhB(Π) = HomSn(Af )(Π n, ψBf ).

17

For µ1, µ2 ∈ Ω(F×p ) let I(µ1, µ2) denote the representation of GL2(Fp) on

the space of all smooth functions f on GL2(Fp) satisfying

f

((a1 b0 a2

)g

)= µ1(a1)µ2(a2)αp(a1a

−12 )1/2f(g)

for all a1, a2 ∈ F×p ; b ∈ Fp and g ∈ GL2(Fp). This representation I(µ1, µ2)

is irreducible unless µ1µ−12 ∈ αp, α

−1p . If µ1µ

−12 = αp, then I(µ1, µ2) has a

unique irreducible submodule, which we denote by A(µ1, µ2).In what follows we fix, once and for all, an irreducible admissible uni-

tary generic representation πf of PGL2(Af ) whose local components are notsupercuspidal. Then πf is equivalent to the unique irreducible submoduleA(µf , µ

−1f ) of I(µf , µ

−1f ) = ⊗′

pI(µp, µ−1p ) for some character µf of the finite

idele group A×f whose restriction to F×

p is denoted by µp and which fulfillsthe following conditions:

• −12 < ℜµp ≤ 1

2 for all finite primes p;

• µ2p = αp whenever ℜµp = 12 ;

• µp is unramified and ℜµp < 12 for almost all finite primes p.

Let Sπf be the set of nonarchimedean primes p such that ℜµp = 12 .

Let In(µf ) and Jn(µf ) be the degenerate principal series representations ofGn(Af ) and Gn(Af ) induced from the character d(t)m(A) 7→ µf (t

−nν(A)).They have factorizations In(µf ) ≃ ⊗′

pIn(µp) and Jn(µf ) ≃ ⊗′pJn(µp). For

B ∈ S+n Lemma 3.4 defines a nonzero vector wµfB ∈ WhB(In(µf )) by

wµfB (f) =∏p

wµpB (fp),

provided that f = ⊗pfp is factorizable. Let us set

An(µf ) = (⊗p∈SπfAn(µp))⊗ (⊗′p/∈S∞∪Sπf

In(µp)).

We can view An(µf ) as the unique irreducible subrepresentation of bothIn(µf ) and Jn(µf ). We also regard An(µf ) as a subrepresentation of thespace of smooth functions on Gn(Af ) or Gn(Af ) on which Gn(Af ) or Gn(Af )acts by the right regular action ϱ or ℘. Put

Sπfn = B ∈ S+n | χBp = µpα

−1/2p for all p ∈ Sπf .

When n = 1, we will sometimes write Dπf− = Sπf1 . Proposition 3.1 and

Lemma 3.2 give the following result:

Lemma 3.5. Let B ∈ S+n . Then WhB(An(µf )) is nonzero if and only if the

restriction of wµfB to An(µf ) is nonzero if and only if B ∈ Sπfn .

4. Holomorphic cusp forms on quaternion unitary groups

When F is a smooth function on Nn(F )\Gn(A) and B ∈ Sn(F ), let

WB(g,F) =

∫Sn(F )\Sn(A)

F(n(z)g)ψB(z) dz


be the Bth Fourier coefficient of F . For ℓ ∈ Zd and B ∈ S+n we define a

function W(ℓ)B : Gn(A∞)+ → C by

W(ℓ)B (g) =

∏v∈S∞

W(ℓv)B (gv).

Definition 4.1. The symbol Gnℓ (resp. Tnℓ , C

nℓ ) denotes the space of all

smooth functions F on Gn(F )\Gn(A) (resp. Pn(F )\Gn(A), Gn(F )\Gn(A))that admit Fourier expansions of the form

F(g) =∑B∈S+

n

wB(gf ,F)W(ℓ)B (g∞)

which is absolutely and uniformly convergent on any compact neighborhoodof g = g∞gf ∈ Gn(A∞)+Gn(Af ) (resp. g = g∞gf ∈ Gn(A)).

The functions in Gnℓ (resp. Cnℓ ) are cuspidal automorphic forms on Gn(A)

(resp. Gn(A)) with scalar K-type k 7→∏v∈S∞

j(kv, i)−ℓ in the sense of

Langlands (cf. Proposition 8.9) and related to classical holomorphic cuspforms in the standard way (cf. Remark 5.3).

Remark 4.2. Put

Pn = d(t)p | t ∈ Gm, p ∈ Pn, P+n = d(t)p | t ∈ F×

+ , p ∈ Pn(F ).

Since Gn(A) = Pn(F )Gn(A∞)+Gn(Af ), smooth functions on Pn(F )\Gn(A)can naturally be identified with smooth functions on P+

n \Gn(A∞)+Gn(Af ).

Lemma 4.3. If i : An(µf ) → Tnℓ is a Gn(Af )-intertwining map, then thereare complex numbers CB such that

i(g, f) =∑B∈Sπfn

CBW(ℓ)B (g∞)wµfB (ϱ(gf )f)

for all f ∈ An(µf ), g = g∞gf , g∞ ∈ Gn(A∞) and gf ∈ Gn(Af ).

Proof. Notice that the coefficient wB(gf ,F) is given by

wB(gf ,F) = |ν(B)|−ℓ/2e∞(−τ(B(i, . . . , i)))WB(gf ,F).

Recall that |ν(B)|ℓ/2 =∏v∈S∞

|ν(B)|ℓv/2v . In particular, for z ∈ Sn(Af )

wB(n(z)gf ,F) = ψBf (z)wB(gf ,F).

Therefore the C-linear functional f 7→ wB(12n, i(f)) belongs to the spaceWhB(An(µf )). There is a complex number CB such that wB(12n, i(f)) =CBw

µfB (f) for all f ∈ An(µf ) in view of Lemma 3.5.

We associate to f ∈ An(µf ), ℓ = (ℓv)v∈S∞ ∈ Zd and complex numbersCB indexed by B ∈ Sπfn the Fourier series

Fℓ(g; f, CB) =∑B∈Sπfn

CBW(ℓ)B (g∞)wµfB (ϱ(gf )f), g = g∞gf ∈ Gn(A),

assuming that the series is absolutely convergent.

19

Lemma 4.4. Notation being as above, if for any lattice L in Sn(F ) thereare positive constants C and M such that |CB| ≤ CNF/Q(ν(B))M for all

B ∈ S+n ∩ L, then the series Fℓ(g; f, CB) is absolutely and uniformly

convergent on any compact subset of Gn(A) for every f ∈ An(µf ).

Proof. The proof goes along the same lines of the arguments in Section 4 of[22]. It suffices to show that the series∑

B∈Sπfn

CB|ν(B)|ℓ/2wµfB (f)e∞(τ(BZ))

is absolutely and uniformly convergent on any compact subset of Hdn. PutRn = Sn(F )∩Mn(O). Take a natural numberN such that wµfB (f) = 0 unless

B ∈ N−1Rn. Lemmas 3.3 and 3.4 say that wµfB (f) ≤ C ′NF/Q(ν(B))M′for

all B ∈ Sπfn with constants C ′ and M ′ depending only on f . Note thatNF/Q(ν(B)) ≤ (nd)−2nd(TrF/Qτ(B))2nd. The number of B ∈ N−1Rn ∩ S+

n

such that TrF/Qτ(B) ≤ T is O(T dn(2n+1)). From these estimates the seriesconverges absolutely and uniformly on

Z ∈ Hdn | ℑ(ZvB−1n )− ϵ1n ∈ Sym2n(Fv)

+ for all v ∈ S∞for any positive constant ϵ. Definition 4.5. Let Tnℓ (µf ) be the vector space which consists of setsCBB∈Sπfn of complex numbers such that the series Fℓ(g; f, CB) con-

verges absolutely and uniformly on any compact subset of Gn(A) for allf ∈ An(µf ) and such that for all B ∈ Sπfn and A ∈ GLn(D(F ))

CB[A] = CBµf (ν(A))−1

∏v∈S∞

sgnv(ν(A))ℓv .

Let Cnℓ (µf ) (resp. Gnℓ (µf )) stand for the space of coefficients CB ∈ Tnℓ (µf )

such Fℓ(f, CB) ∈ Cnℓ (resp. Gnℓ ) for all f ∈ An(µf ).

Lemma 4.6. (1) Fℓ(f, CB) ∈ Tnℓ for CB ∈ Tnℓ (µf ) and f ∈ An(µf ).(2) Let CB ∈ Cnℓ (µf ). Then CB ∈ Gnℓ (µf ) if and only if CtB =

CBµf (t)−n for all t ∈ F×

+ .

Proof. Fix f ∈ An(µf ) and put w(ℓ)B (g) = W

(ℓ)B (g∞)wµfB (ϱ(gf )f). Then

Lemma 3.2(3) and (3.2) say that

w(ℓ)B (n(z)d(t)m(A)g) =

ψB(z)

µf (t−nν(A))w

(ℓ)t−1B[A]

(g)∏v∈S∞

sgnv(ν(A))ℓv

for z ∈ Sn(A), A ∈ GLn(D(F )), t ∈ F×+ and g ∈ Gn(A∞)+Gn(Af ), from

which Fℓ(f, CB) is left invariant under Pn(F ). This proves (1).We can define the function F ′

ℓ(g; f, CB) : Gn(A∞)+Gn(Af ) → C by

F ′ℓ(g; f, CB) =

∑B∈Sπfn

CBW(ℓ)B (g∞)wµfB (℘(gf )f), f ∈ An(µf ).


Then F ′ℓ(f, CB) is left invariant under P+

n if and only if CtB = CBµf (t)−n

for all t ∈ F×+ , in which case F ′

ℓ(f, CB) naturally defines a cusp form inGnℓ by Remark 4.2.

5. Hilbert-Siegel cusp forms of half-integral weight

5.1. The metaplectic group. We first review the basic facts about themetaplectic double cover of Spm(A). This is mostly a well-known materialthat can be found in e.g. [21, 49, 9, 16, 43, 37]. Let Sp(Wm) = Spm bethe symplectic group of rank m acting on the 2m dimensional symplecticvector space Wm over a field F on the left. For a ∈ GLm and b ∈ Symm wedefine matrices of size 2m as in (1.1). These matrices generate the parabolicsubgroup Pm of Sp(Wm) with unipotent radical Um = n(b) | b ∈ Symm.Put Symnd

m = Symm ∩GLm.Let F be a local field of characteristic zero. We exclude the complex

case. The metaplectic group Mp(Wm) is the nontrivial central extension ofSp(Wm) by µ2. Let µ2 inject into the center of Mp(Wm). We call functionson Mp(Wm) or representations of Mp(Wm) genuine if −1 ∈ µ2 acts bymultiplication by −1. We choose the section s : Sp(Wm) → Mp(Wm) insuch a way that

ζs(g) · ζ ′s(g′) = ζζ ′c(g, g′)s(gg′) (ζ, ζ ′ ∈ µ2 ⊂ Mp(Wm); g, g′ ∈ Spm(F ))

where c(g, g′) is the Rao two cocycle on Spm(F ). The restriction of s to Um

is a group homomorphism, by which we view Um as a subgroup of Mp(Wm).We will write m = s m and n = s n. Note that

m(a)m(a′) = χdet a(det a′)m(aa′), m(a)n(b)m(a)−1 = n(ab ta)

for a, a′ ∈ GLm(F ) and b ∈ Symm(F ). For a subgroup H of Spm(F ) we

denote the inverse image of H in Mp(Wm) by H. As in Section 1 we definethe Weil index γψ : F×2\F× → µ4, which possesses the following properties:

γψ(tt′) = γψ(t)γψ(t′)χt(t′), γψt(t′) = γψ(t′)χt(t′).(5.1)

When F is of odd residual characteristic, there is a unique splittingSpm(o) → Mp(Wm), by which we regard Spm(o) as a subgroup of Mp(Wm).In other words there is a continuous map ζ : Spm(F ) → µ2 such thatc(k, k′) = ζ(k)ζ(k′)ζ(kk′) for k, k′ ∈ Spm(o). We shall set ζ(g) = 1 in thereal and dyadic cases to make our exposition uniform. We use a cocycle

c(g, g′) = c(g, g′)ζ(g)ζ(g′)ζ(gg′)

with global applications in view, i.e., we identity Mp(Wm) with the productSpm(F )×µ2 whose group law is given by (g, ζ)(g′, ζ ′) = (gg′, ζζ ′c(g, g′)). Itshould be remarked that the section s is now given by s(g) = (g, ζ(g)).

The real metaplectic group acts on the Siegel upper half-spaceHm throughSpm(R). There exists a unique factor of automorphy ȷ : Mp(Wm)×Hm →

C× satisfying ȷ(ζs(g),Z)2 = det(CZ +D) for ζ ∈ µ2 and g =

(∗ ∗C D

)∈

21

Mp(Wm). For each ℓ ∈ 12Z we put Jℓ(g,Z) = ȷ(g,Z)2ℓ. For each positive

definite ξ ∈ Symm(R) we define a function on the real metaplectic group by

W(ℓ)ξ (g) = (det ξ)ℓ/2e(tr(ξg(

√−11m)))Jℓ(g,

√−11m)

−1.

Put Km = g ∈ Spm(R) | g(√−11m) =

√−11m. Let Km be the preimage

of Km. If ζ ∈ µ2, a ∈ GLm(R), g ∈ Mp(Wm) and k ∈ Km, then

(5.2) W(ℓ)ξ (ζm(a)gk) = ζ2ℓγψ(det a)2ℓW

(ℓ)ξ[a](g)Jℓ(k,

√−11m)

−1.

5.2. Representations of the metaplectic group. Now we assume F tobe nonarchimedean. For ξ ∈ Symnd

m and a smooth representation Π ofMp(Wm) we set Whξ(Π) = HomSymm(F )(Π n, ψξ). When µ ∈ Ω(F×) and

ℜµ > −12 , we define the representation Iψm(µ) of Mp(Wm) and the nonzero

functional wµξ ∈ Whξ(Iψm(µ)) as in Section 1. We write ϱ for the right regular

action of Mp(Wm) on the space of smooth functions on Mp(Wm).

Proposition 5.1 ([57, Lemma 3], [54]). (1) If −12 < ℜµ < 1

2 , then Iψm(µ)

is irreducible and unitary.

(2) dimWhξ(Iψm(µ)) = 1 for all ξ ∈ Symnd

m and µ ∈ Ω(F×).

(3) Assume that µ2 = α. Then Iψm(µ) has a unique irreducible subrep-

resentation Aψm(µ), which is unitary. Furthermore, Whξ(Aψm(µ)) is

nonzero if and only if the restriction of wµξ to Aψm(µ) is nonzero if

and only if χdet ξ = µα−1/2.(4) If F is not dyadic, ψ is of order 0, µ is unramified, ξ ∈ Symnd

m ∩GLm(o) and h(k) = 1 for k ∈ Spm(o), then w

µξ (h) = 1.

(5) For all ξ ∈ Symndm and a ∈ GLm(F )

wµξ ϱ(ζm(a)) = ζmγψ(det a)mµ(det a)−1wµξ[a].

Proof. When m is even, all the results are included in Proposition 3.1 andLemma 3.2. The fourth part is Theorem 16.2 of [49]. The other assertionsare included in [54] or can be derived analogously. 5.3. Holomorphic cusp forms on Mp(Wm). Let F be a totally real num-ber field. For each place v of F we adopt the notation by adding a subscriptv for objects associated to Fv. We define the adelic cocycle

c : Spm(A)× Spm(A) → µ2, c(g, g′) =∏v

cv(gv, g′v)

for g, g′ ∈ Spm(A). Recall that cv(gv, g′v) = 1 for almost all v by the defi-

nition of the local cocycle cv. We write Mp(Wm)A for the central extensionof Spm(A) associated to c.

Let∏′vMp(Wm)v denote the restricted direct product with respect to the

subgroups Spm(op)p∤2. There exist a canonical embedding Mp(Wm)v →Mp(Wm)A and a canonical surjection

∏′vMp(Wm)v ↠ Mp(Wm)A. The

image of (gv) ∈∏′vMp(Wm)v is also denoted by (gv). If we are given a


collection of genuine admissible representations σv of Mp(Wm)v such thatthe space of Spm(op)-invariant vectors in σp is one-dimensional for almostall finite primes p, then we can form a genuine admissible representation ofMp(Wm)A by taking a restricted tensor product ⊗′

vσv.It is well-known that Mp(Wm)A splits over the subgroup of rational points

Spm(F ). An explicit splitting is given by

s : Spm(F ) → Mp(Wm)A, s(γ) = (γ,∏v

ζv(γ)),

where if γ ∈ Spm(F ), then ζv(γ) = 1 for almost all v. Though the expression∏v ζv(gv) does not make sense for all g ∈ Spm(A), we will denote the element

(g,∏v ζv(gv)) by s(g) whenever it makes sense. For example, s(m(a)n(b))

is defined for a ∈ GLm(A) and b ∈ Symm(A). We will regard Spm(F ) andUm(A) as subgroups of Mp(Wm)A via s. For ξ ∈ Symm(F ) and a smoothfunction F : Um(F )\Mp(Wm)A → C let

Wξ(g,F) =

∫Symm(F )\Symm(A)

F(s(n(b))g)ψξ(−b) db

denote the ξth Fourier coefficient of F .Let ℓ ∈ 1

2Zd be such that 2(ℓv − ℓv′) is even for every v, v′ ∈ S∞. For

g ∈ Mp(Wm)∞ and a function F on Hdm we define another function F|ℓg

on Hdm by F|ℓg(Z) = F(gZ)Jℓ(g,Z)−1. Define the origin of Hd

m and thestandard maximal compact subgroup of Mp(Wm)∞ by

im = (√−11m, . . . ,

√−11m) ∈ Hd

m, Km = g ∈ Mp(Wm)∞ | g(im) = im.

Definition 5.2. The symbol C(m)ℓ (resp. T

(m)ℓ ) denotes the space of all

smooth functions F on Mp(Wm)A which is left invariant under Spm(F )

(resp. Pm(F )) and transforms on the right by the character k 7→ Jℓ(k, im)−1

of Km and such that F∆ is a holomorphic function on Hdm having a Fourier

expansion of the form

F∆(Z) =∑

ξ∈Sym+m

| det ξ|ℓ/2wξ(∆,F)e∞(tr(ξZ))

for each ∆ ∈ Mp(Wm)f , where wξ(F) is a function on Mp(Wm)f and the

function F∆ : Hdm → C is defined by

F∆|ℓg(im) = F(g∆), g ∈ Mp(Wm)∞.

Remark 5.3. The group Mp(Wm)f acts on these spaces by right translation.

The linear map F 7→ Fem is a bijection of C(m)ℓ onto the space S

(m)ℓ of

Hilbert-Siegel cusp forms of weight ℓ defined in Section 1 by strong approx-

imation in Spm, where em is the identity element of Mp(Wm)f . If F ∈ C(m)ℓ

is right invariant under the open compact subgroup D of Mp(Wm)f , thenFem ∈ Sℓ(Γ), where Γ = Spm(F ) ∩ D.

23

For ξ ∈ Sym+m we define a function W

(ℓ)ξ on Mp(Wm)∞ by

W(ℓ)ξ (g) =

∏v∈S∞

W(ℓv)ξ (gv).

The group GL2(A∞)+ := (gv) ∈ GL2(A∞) | det gv > 0 for all v ∈ S∞acts componentwise on Hd

1. Given g ∈ GL2(A∞)+ and a function F on Hd1,

we define a function F|κg on Hd1 by

F|κg(Z) = F(gZ)Jκ(g,Z)−1, Jκ(g,Z) = | det g|−κ/2∏v∈S∞

(cvZv + dv)κv ,

where gv =

(∗ ∗cv dv

). Put Km = g ∈ Spm(A∞) | g(im) = im.

Definition 5.4. A Hilbert cusp form F on PGL2 of weight 2κ is a smoothfunction on GL2(A) satisfyingF(zγgk) = F(g)J2κ(k, i1)

−1 (z ∈ A×, γ ∈ GL2(F ), g ∈ GL2(A), k ∈ K1)

and having a Fourier expansion of the form

F∆(Z) =∑t∈F×

+

|t|κwt(∆,F)e∞(tZ)

for each ∆ ∈ GL2(Af ), where wt(F) is a function on GL2(Af ) and thefunction F∆ : Hd

1 → C is defined by F∆|2κg(i1) = F(g∆) for g ∈ GL2(A∞)+.We write C2κ for the space of Hilbert cusp forms on PGL2 of weight 2κ.

Notation being as in §3.4, we form the restricted tensor product

Aψfm (µf ) = (⊗p∈SπfA

ψpm (µp))⊗ (⊗′

p/∈S∞∪SπfIψpm (µp)).

For ξ ∈ Sym+m and a smooth representation Π of Mp(Wm)f we put

Whξ(Π) = HomSymm(Af )(Π s n, ψξf ).

We can define wµfξ ∈ Whξ(Aψfm (µf )) by setting wµfξ (h) =

∏pw

µpξ (hp) for

factorizable vectors h = ⊗php ∈ Aψfm (µf ). Put

Symπfm = ξ ∈ Sym+

m | χdet ξp = µpα

−1/2p for all p ∈ Sπf , F×

πf= Symπf

1 .

Proposition 5.1 tells us that Whξ(Aψfm (µf )) is nonzero if and only if the

restriction of wµfξ to Aψfm (µf ) is nonzero if and only if ξ ∈ Symπf

m .

Definition 5.5. Assume that 2ℓv − m is even for every v ∈ S∞. Let

T(m)ℓ (µf ) (resp. C

(m)ℓ (µf )) denote the vector space which consists of sets

Cξ of complex numbers indexed by ξ ∈ Symπfm such that the series

Fℓ(g;h, Cξ) =∑

ξ∈Symπfm

CξW(ℓ)ξ (g∞)wµfξ (ϱ(gf )h)

belongs to T(m)ℓ (resp. C

(m)ℓ ) for every h ∈ Aψf

m (µf ).


Lemma 5.6. (1) C(m)ℓ (µf ) = 0 if and only if there is a Mp(Wm)f

intertwining embedding Aψfm (µf ) → C

(m)ℓ .

(2) Assume that Fℓ(h, Cξ) converges for all h ∈ Aψfm (µf ). Then Cξ ∈

T(m)ℓ (µf ) if and only if for all a ∈ GLm(F ) and ξ ∈ Symπf

m ,

Cξ[a] = Cξµf (det a)−1

∏v∈S∞

sgnv(det a)(2ℓv−m)/2.

(3) If T(m)ℓ (µf ) = 0, then µf (−1)(−1)

∑v∈S∞ (2ℓv−m)/2 = 1.

(4) dim C(1)ℓ (µf ) ≤ 1.

(5) If ct ∈ C(1)ℓ (µf ), then cηt ∈ C

(1)ℓ (µf χ

ηf ) for all η ∈ F×

+ .

(6) We choose 0 = ct ∈ C(1)ℓ (µf ), assuming that C

(1)ℓ (µf ) = 0. Then

ct = 0 if and only if t ∈ F×πf

and L(12 , π ⊗ χt

)= 0.

Proof. The proof of (1) is similar to that of Lemma 4.3. Proposition 5.1(5)and (5.2) prove (2). One sees that the third statement is its simple con-sequence by taking a ∈ GLm(F ) so that ξ[a] = ξ and det ξ = −1. Theassertion (4) follows from the fact proved by Waldspurger [56] that everyirreducible representation of Mp(W1)A occurring in the decomposition of thespace of cusp forms on Mp(W1) appears with multiplicity one.

Now we prove (5). Let h ∈ Aψf1 (µf ). Put F = Fℓ(h, ct). Let us

set d(a) = diag[1, a] for a ∈ A×. We can realize Mp(W1)A as a normalsubgroup of a double cover of GL2(A) constructed by using the Kubotatwo cocycle. The conjugation action of d(a) | a ∈ F× on SL2(A) hasa lift to Mp(W1)A, which preserves the subgroup SL2(F ). We define acusp form F ′ : Mp(W1)A → C by F ′(g) = F(d(η)gd(η)−1γ−1), where γ =(m(

√η), 1) ∈ Mp(W1)∞. Then F ′ equals Fℓ(h′, cηtt) up to scalar, where

h′ is defined by h′(g) = h′(d(η)gd(η)−1). One can see that h′ ∈ Aψf1 (µf χ

ηf ).

The last assertion is Theorem 4.1 of [47].

6. Main theorem

6.1. Liftings to inner forms of Sp2n.

Theorem 6.1. Notations and assumptions being as in Theorem 1.2, theassignment

f 7→ ιηn(f)∆(Z) =∑B∈S+

n

|ν(B)|(κ+n)/2cην(B)e∞(τ(BZ))wµf χ

(−1)nηf

B (ϱ(∆)f)

defines an embedding ιηn : An(µf χ

(−1)nηf

)→ Cnκ+n for every n and η ∈ F×

+ ,

where ∆ ∈ Gn(Af ) and Z ∈ Hdn.

Remark 6.2. (1) In light of Lemma 4.6(2) this embedding naturally de-

fines an embedding An(µf χ(−1)nηf ) → Gn

κ+n.

25

(2) The multiplicity of A1(µf χ−ηf ) in G1

κ+1 is one by Corollary 7.7 of [9].However, we do not know if this result can imply the multiplicity ofA1(µf χ

−ηf ) in C1

κ+1.

6.2. Compatibility with Arthur’s endoscopic classification. We ex-plain how Theorem 6.1 can be viewed in the framework of Arthur’s conjec-ture. For details the reader should consult [1, 2]. The conjecture specializedto our current case is discussed in [9], Section 14 of [22] and [63].

Let L be the hypothetical Langlands group over F . Hypothetically,there is a bijective correspondence between the set of all equivalence classesof m-dimensional irreducible representations of L and the set of all irre-ducible cuspidal automorphic representations of GLm(A). If π is a cuspi-dal automorphic representation of PGL2(A), then π corresponds to a mapρ(π) : L → SL2(C). Let symm−1 be the irreducible m-dimensional rep-resentation of SL2(C). We may assume that sym2n−1(SL2(C)) ⊂ Spn(C).Thus ρ(π)⊠ sym2n−1 gives rise to a homomorphism L×SL2(C) → SO4n(C).Embedding SO4n(C) into SO4n+1(C) = Gn, we get a homomorphism L ×SL2(C) → Gn. One postulates that for each place v there is a naturalconjugacy class of embeddings Lv → L, where Lv is the Weil group of Fvif v ∈ S∞, and the Weil-Deligne group of Fv if v /∈ S∞. We obtain ahomomorphism ρ(πv)⊠ sym2n−1 : Lv × SL2(C) → Gn for each v.

The Arthur conjecture suggests that there exists a finite set Πn(πv) =Π+n (πv)⊔Π−

n (πv) of equivalence classes of unitary admissible representationsof Gn,v associated to ρ(πv) ⊠ sym2n−1. Moreover, it is required that ifGn,v ≃ Sp2n(Fv), then Π

+n (πv) contains the Langlands quotient I+n (πv) of

IndSp2n(Fv)P2,2,...,2(Fv)

(πv ⊗ α(2n−1)/2v )⊠ (πv ⊗ α(2n−3)/2

v )⊠ · · ·⊠ (πv ⊗ α1/2v ),

where P2,2,...,2 is the standard parabolic subgroup of Sp2n with Levi sub-group GL2 × · · · × GL2. Choose Πv ∈ Πϵv

n (πv) for each v. Then ⊗′vΠv

is an automorphic representation of Gn(A) generated by square-integrableautomorphic forms if and only if

∏v ϵv = ε

(12 , π

).

Let p be a finite prime such that Dp ≃ M2(Fp). Let µ ∈ Ω(F×p ). If

ℜµ > −12 , then we obtain an intertwining map

In(µ) → IndSp2n(Fp)P2,2,...,2(Fp)

(I(µ, µ−1)⊗ α−(2n−1)/2p )⊠ · · ·⊠ (I(µ, µ−1)⊗ α

−1/2p )

by applying Proposition 1 of [40] or Proposition 4.1 and Lemma 5.1 of [61]repeatedly. Therefore if −1

2 < ℜµ < 12 , then this map is nonzero by Lemma

8 of [40], so that In(µ) ≃ I+n (I(µ, µ−1)). If µ2 = αp and µ = α

1/2p , then

An(µ) ≃ I+n (A(µ, µ−1)) by Proposition 3.11(2) of [27]. On the other hand,

An(α1/2p ) is the Langlands quotient of

IndSp2n(Fp)Q2,2,...,2(Fp)

A(αnp , αn−1p )⊠ · · ·⊠A(α2

p, αp)⊠A1(α1/2p )

by Proposition 3.10(2) of [27], where Q2,2,...,2 is the standard parabolic sub-group of Sp2n with Levi subgroup GL2 × · · · × GL2 × Sp2. We guess that


An(α1/2p ) ∈ Π−

n (A(α1/2p , α

−1/2p )). We presume that the reasoning above is

correct even when Dp is division.Let π ≃ (⊗v∈S∞πv) ⊗ πf be an irreducible cuspidal automorphic repre-

sentation of PGL2(A) on which we impose the following conditions:

(i) πv is the discrete series with extremal weights ±2κv for v ∈ S∞;(ii) πf ≃ A(µf , µ

−1f );

(iii) µf (−1)(−1)∑v∈S∞ κv = 1.

Let v ∈ S∞ and assume that κv > n. Then W(κv+n)B generates a holomor-

phic discrete series representation of Sp2n(Fv). Fix η ∈ F×+ . Put

σ = π ⊗ χ(−1)nη, ℓ−σ = #p /∈ S∞ | σp ≃ A(α1/2p , α

−1/2p ).

The holomorphic discrete series representation with lowestK-type (det)κv+n

belongs to Π(−1)n

n (πv). Since

ε(1/2, σ) = (−1)ℓ−σ+

∑v∈S∞ κvµf (−1)χ

(−1)nηf (−1) = (−1)ℓ

−σ+nd,

the restriction (iii) is compatible with the Arthur conjecture.

Remark 6.3. If πp is supercuspidal, then we no longer have a description ofan element of Πn(πp) in terms of degenerate principle series, and we haveno simple expression of its degenerate Whittaker functionals. This is thereason why we assume that πp is not supercuspidal.

If m is odd, then we obtain a homomorphism ρ(σ) ⊠ symm−1 : L ×SL2(C) → Spm(C) = LSO2m+1, which should be the Arthur parameter ofthe lifting constructed in Theorems 1.1 and 1.2. Our result is compatiblewith an analogue of the Arthur conjecture for metaplectic groups formulatedby Wee Teck Gan [10]. We refer to [63] for a description of the associatedA-packet in the metaplectic case. The homomorphism ρ(σ) ⊠ sym2n−1 :L × SL2(C) → SO4n(C) = LSO4n should be the Arthur parameter of thelifting constructed in Corollary 1.4.

Let ρ(g) : L × SL2(C) → SO2r+1(C) = LSpr be the Arthur parameter forthe cuspidal automorphic representation generated by a Hecke eigenformg ∈ Sκ+n+r(Γr[d

−1, d]). The homomorphism ρ(g)× (ρ(σ)⊠ sym2n−1) : L ×SL2(C) → SO2r+4n+1(C) = LSp2n+r should be the Arthur parameter of theMiyawaki lifting constructed in Corollary 10.3.

7. Fourier-Jacobi modules

7.1. Jacobi groups. Fix 0 ≤ i ≤ n. Put n′ = n − i. For z ∈ Si andx, y ∈ Mi

n′(D) we use the notation

v(x, y; z) =

1i x0 1n′

z − yx∗ y−y∗ 0

01i 0−x∗ 1n′

, ηi =

1i

1n′

1i1n′

∈ Gn.

27

We define some subgroups of Gn by

Xi = v(x, 0; 0) | x ∈ Min′(D), Yi = v(0, y; 0) | y ∈ Mi

n′(D),Zi = v(0, 0; z) | z ∈ Si, Ni = v(x, y; z) | x, y ∈ Mi

n′(D), z ∈ Si.

We identity Xi and Yi with the space Min′(D). We view Gi and Gn′ as

subgroups of Gn via the embeddings

g1 7→

a1 b1

1n′

c1 d11n′

, g2 7→

1i

a2 b21i

c2 d2

,

where we write a typical element g1 ∈ Gi in the form

(a1 b1c1 d1

)with matrices

a1, b1, c1, d1 of size i over D, and similarly for g2 ∈ Gn′ . We also write

m′(A) =

(A 00 (A−1)∗

), n′(z) =

(1n′ z0 1n′

)for A ∈ GLn′(D) and z ∈ Sn′ . We will frequently specialize to the casei = n− 1 in our application to the proof of main theorems.

7.2. Weil representations of Jacobi groups. Let F be a local field.Fix S ∈ Snd

i . We regard S as a homomorphism Zi → Ga by z 7→ τ(Sz).Then Ni/KerS is a Heisenberg group with center Zi/KerS and a natural

symplectic structure Ni/Zi. The Schrodinger representation ωψS of Ni with

central character ψS is realized on the Schwartz space S(Xi) by

(7.1) (ωψS (v(x, y; z))ϕ)(u) = ϕ(u+ x)ψS(z)ψ(2τ(u∗Sy))

for ϕ ∈ S(Xi). By the Stone-von Neumann theorem, ωψS is the unique

irreducible representation of Ni on which Zi acts by ψS .

The embedding Gn′ → Gn and the conjugating action give an embeddingGn′ → Sp(Ni/Zi) and Kudla [37] gave an explicit local splitting Gn′ →Mp(Ni/Zi), where Mp(Ni/Zi) is the metaplectic extension of Sp(Ni/Zi).

The representation ωψS ofNi extends to theWeil representation of Mp(Ni/Zi)⋉Ni whose pullback to Gn′ is characterized by the following formulas:

(ωψS (m′(A))ϕ)(u) = χS(ν(A))|ν(A)|iϕ(uA),

(ωψS (n′(z))ϕ)(u) = ψS(uzu∗)ϕ(u),

(ωψS (Jn′)ϕ)(u) = γψ(S)(FSϕ)(u)(7.2)

for ϕ ∈ S(Xi), u ∈ Xi, A ∈ GLn′(D) and z ∈ Sn′ , where γψ(S) is a certain8th root of unity and FSϕ is the Fourier transform defined by

(FSϕ)(u) =∫Xi

ϕ(x)ψ(2τ(x∗Su)) dx.


7.3. The nonarchimedean case. Let F be a finite extension of Qp.

Lemma 7.1. Let f ∈ In(µ) and ϕ ∈ S(Xi). If ℜµ≫ 0, then the integral

BψS (g′; f ⊗ ϕ) =

∫Yi\Ni

f(ηivg′)(ωψS (vg

′)ϕ)(0) dv

is absolutely convergent. Moreover, it possesses analytic continuation to allµ ∈ Ω(F×) and gives an Ni-invariant and Gn′-intertwining map

BψS : In(µ)⊗ ωψS → In′(µχS).

Proof. The integral over Zi can be viewed as a Jacquet integral of the re-striction of f to Gi, which belongs to Ii(µα

n′). It is entire on the whole

of the complex manifold Ω(F×). Since (ωψS (xg′)ϕ)(0) = (ωψS (g

′)ϕ)(x) forx ∈ Xi, the integral over Xi is convergent for all µ. When D ≃ M2(F ),

Ikeda showed that BψS (f ⊗ ϕ) ∈ In′(µχS) in the proof of Theorem 3.2 of [21].The computation applies equally well to the quaternion case.

We modify the integral above in the following way:

βψS (g′; f ⊗ ϕ) =

L(n+ 1

2 , µ)

L(n′ + 1

2 , µχS)BψS (g′; f ⊗ ϕ)

i∏j=1

L(2n′ + 2j − 1, µ2).

The reason for this normalization is Corollary 7.3(1).

Lemma 7.2. Let S ∈ Sndi . Put n′ = n− i. There exists a nonzero constant

ES such that for all Ξ ∈ Sndn′ , f ∈ In(µ) and ϕ ∈ ωψS

wµχS

Ξ (βψS (f ⊗ ϕ)) = ES |ν(Ξ)|−i/2∫Xi

ϕ(x)wµS⊕Ξ(ϱ(x)f) dx.

Proof. Since ηi = Jn · Jn′ and Jn′v(0, y, z)Jn′ = v(y, 0, z), we observe that∫Nif(JnvJn′g′)(ωψS (vJn′g′)ϕ)(0) dv

=

∫Zi

∫Yi

∫Xi

f(JnxzyJn′g′)(ωψS (zyJn′g′)ϕ)(x) dxdydz

=

∫Si

∫Min′ (D)

∫Xi

f(Jnv(0, y; z)Jn′g′)(ωψS (Jn′v(y, 0; z)g′)ϕ)(x) dxdydz

=

∫Si

∫Min′ (D)

f(ηiv(y, 0; z)Jn′g′)(FSωψS (Jn′v(y, 0; z)g′)ϕ)(0) dydz

=γψ(S)−1

∫Zi

∫Xi

f(ηiv(y, 0; z)g′)(ωψS (v(y, 0; z)g

′)ϕ)(0) dydz

by (7.1), (7.2) and the Fourier inversion. Since

|ν(Ξ)|(2n′+1)/4L(n′ + 1

2 , µχS)

L(12 , µχ

SχΞ) n′∏

j=1

L(2j − 1, µ2)L(n+ 1

2 , µ)

L(n′ + 1

2 , µχS) i∏j=1

L(2n′ + 2j − 1, µ2)

29

= |ν(S)|−(2n+1)/4|ν(Ξ)|−i/2|ν(S ⊕Ξ)|(2n+1)/4 L(n+ 1

2 , µ)

L(12 , µχ

S⊕Ξ) n∏j=1

L(2j − 1, µ2),

it suffices to prove∫Sn′

∫Nif(Jnvn

′(u))(ωψS (vn′(u))ϕ)(0)ψΞ(−u) dvdu

=

∫Xi

∫Sn

f(Jnn(z)x)ϕ(x)ψS⊕Ξ(−z) dzdx

for ℜµ≫ 0. The left hand side is equal to∫Sn′

∫Nif(Jnn

′(u)v)(ωψS (n′(u)v)ϕ)(0)ψΞ(−u) dvdu

=

∫Sn′

∫Nif(Jnn

′(u)v)(ωψS (v)ϕ)(0)ψΞ(−u) dvdu

=

∫Sn′

∫Yi

∫Zi

∫Xi

f(Jnn′(u)yzx)ϕ(x)ψS(z)ψΞ(−u) dxdzdydu

=

∫Sn

∫Xi

f(Jnn(z)x)ϕ(x)ψS⊕Ξ(−z) dxdz.

Since this integral is absolutely convergent for ℜµ≫ 0, we can exchange theorder of integration. Corollary 7.3. (1) If p = 2, D ≃ M2(F ), ψ is of order 0, µ is unramified,

S ∈ Sndi ∩ GLi(O), ϕ is the characteristic function of Mi

n′(O) and

f ∈ In(µ) satisfies f(k) = 1 for all k ∈ KDn [o], then βψS (12n′ ; f⊗ϕ) =

1.

(2) If −12 < ℜµ < 1

2 , then βψS (In(µ)⊗ ωψS ) = In′(µχS).

(3) If µ2 = α, then βψS (An(µ)⊗ ωψS ) = An′(µχS).

Proof. Since ES = 1 in the unramified case, we can derive (1) from Lem-

mas 3.4 and 7.2. If χΞ = µχSα−1/2, then χS⊕Ξ = µα−1/2, and hence

wµχS

Ξ (βψS (f⊗ϕ)) = 0 for all f ∈ An(µ) and ϕ ∈ ωψS by Lemmas 3.2(2) and 7.2.

It follows from the proof of Proposition 3.1 that βψS (An(µ)⊗ωψS ) ⊂ An′(µχS).

Since the target spaces are irreducible, it is sufficient to show the nonvan-ishing of the intertwining maps. Lemma 3.2 enables us to take Ξ ∈ Snd

n′ andf so that wµS⊕Ξ(f) = 0. Using Lemma 7.2 and choosing ϕ to be supported

in a small neighborhood, one can show that wµχS

Ξ (βψS (f ⊗ ϕ)) = 0. 7.4. The metaplectic case. Fix 0 ≤ i ≤ m. Putm′ = m−i. For z ∈ Symi,x, y ∈ Mi

m′ , a ∈ GLm′ and b ∈ Symm′ we use the notation

u(x, y; z) =

1i x0 1m′

z − y tx yty 0

01i 0− tx 1m′

, ηi =

−1i

1m′

1i1m′

,


m′(a) =

(a 00 ta−1

), n′(b) =

(1m′ b0 1m′

).

We define the subgroups of Spm by Ji = Spm′ · Ni and

Xi = u(x, 0; 0) | x ∈ Mim′, Yi = u(0, y; 0) | y ∈ Mi

m′,Zi = u(0, 0; z) | z ∈ Symi, Ni = u(x, y; z) | x, y ∈ Mi

m′ , z ∈ Symi.

For the time being let F be a local field. Fix R ∈ Symndi . Recall the

parabolic subgroup Pm′ of Spm′ defined in §5.1. The pullback to Pm′ ⋉Ni

of the Weil representation ωψR of Mp(Wm′)⋉ Ni is given by

(ωψR(u(x, y; z))ϕ)(u) = ϕ(u+ x)ψR(z)ψ(2tr( tuRy)),

(ωψR(n′(b)m′(a))ϕ)(u) = ψR[u](b)γψ(det a)iχdetR(det a)| det a|i/2ϕ(ua)

for ϕ ∈ S(Xi), u ∈ Xi, a ∈ GLm′(F ) and b ∈ Symm′(F ).

Lemma 7.4. Let F be a finite extension of Qp. Assume that m′ is odd.

The following integral makes sense for all ℜµ > −m′

2 − 1 and gives an Ni-

invariant and Mp(Wm′)-intertwining map βψR : Iψm(µ)⊗ ωψR → Iψm′(µχdetR):

βψR(g′;h⊗ ϕ) =

[i/2]∏j=1

L(m′ + 2j, µ2)

∫Yi\Ni

h(s(ηiv)g′)(ωψR(vg

′)ϕ)(0) dv

×

1 if 2 ∤ m,

L(m+12 , µχ(−1)m/2

)if 2|m.

Moreover, for T ∈ Symndm′, h ∈ Iψm(µ) and ϕ ∈ ωψR

wµχdetR

T (βψR(h⊗ ϕ)) = ER|detT |−i/4∫

Xi

ϕ(x)wµR⊕T (ϱ(x)h) dx.

We omit the proof as it is the same as those of Lemmas 7.1 and 7.2. Wecan deduce the following corollary from Proposition 5.1 and Lemma 7.4 bythe same reasoning as in the proof of Corollary 7.3.

Corollary 7.5. (1) If p = 2, ψ is of order 0, µ is unramified, R ∈ Symndi ∩

GLi(o), ϕ is the characteristic function of Mim′(o) and h ∈ Iψm(µ)

satisfies h(k) = 1 for all k ∈ Spm(o), then βψR(12m′ ; f ⊗ ϕ) = 1.

(2) If −12 < ℜµ < 1

2 , then βψR(I

ψm(µ)⊗ ωψR) = Iψm′(µχdetR).

(3) If µ2 = α, then βψR(Aψm(µ)⊗ ωψR) = Aψm′(µχdetR).

7.5. The archimedean case. When m = i+m′, R ∈ Symndi , T ∈ Symnd

m′

and x ∈ Mim′(F ), we put

RT,x =

(R 00 T

)[(1i x0 1m′

)]=

(R RxtxR T +R[x]

).

31

Lemma 7.6. Suppose that F ≃ R, m = i + m′ and R ∈ Sym+i . Define

φR ∈ S(Xi) by φR(x) = e−2πtr(R[x]) for x ∈ Xi. Then there is a nonzeroconstant ER such that for T ∈ Sym+

m′ and g′ ∈ Mp(Wm′)∫Xi

W(ℓ)R⊕T (s(x)g

′)(ωψR(g′)φR)(x) dx = E−1

R (detT )i/4W(ℓ−i/2)T (g′).

Proof. Since the Gaussian is an eigenfunction for the action of Km′ witheigencharacter k′ 7→ Ji/2(k

′, i)−1 by [8], one can readily verify that

(7.3) (ωψR(g′)φR)(x) = Ji/2(g

′, im′)−1e(tr(R[x]g′(im′)))

for x ∈ Xi and g′ ∈ Mp(Wm′). Put Y = ℑg′(im′). Since

W(ℓ)R⊕T (s(x)g

′) =W(ℓ)R⊕T

(m

((1i x0 1m′

))g′)

=W(ℓ)RT,x

(g′)

by (5.2), the left hand side is equal to

det(R⊕ T )ℓ/2

Jℓ(g′, im′)

∫Xi

e

(tr

(RT,x

(ii

g′(im′)

)))(e(tr(R[x]g′(im′)))

Ji/2(g′, im′)

)dx

=W(ℓ−i/2)T (g′)(detR)ℓ/2(detT )i/4e−2πtrR(detY )i/2

∫Xi

e−4πtr(R[x]Y ) dx.

The factor (detY )i/2∫Xie−2πtr(R[x]Y ) dx is a constant independent of Y .

7.6. The global case. For the rest of this section D is a totally indefinitequaternion algebra over a totally real number field F . Fix S ∈ S+

i . Take

a maximal abelian subgroup A of Ni(A) to which the character ψS has an

extension ψSA . The Schrodinger representation is equivalent to IndNi(A)A ψSA .

Having chosen A = Yi(A)⊕ Zi(A), we obtain the Schrodinger model of the

Weil representation ωψS ≃ ⊗′vω

ψvS realized on S(Xi(A)). If we choose A =

Ni(F )Zi(A), then the space IndNi(A)A ψSA = C∞

ψS(Ni(F )\Ni(A)) consists of

smooth functions on Ni(F )\Ni(A) on which the center Zi(A) acts by ψS .The equivariant isomorphism S(Xi(A)) ≃ C∞

ψS(Ni(F )\Ni(A)) is given by

Θ(ωψS (v)φ) =∑

x∈Xi(F )

(ωψS (v)φ)(x).

Denote by ωψfS ≃ ⊗′

pωψp

S the finite part of the global Weil representation.For ϕ ∈ S(Xi(Af )) we define a Schwartz function ϕS on Xi(A) by

ϕS(x) = ϕ(xf )φ∞S (x∞), φ∞

S (x∞) = e−2π∑v∈S∞ τ(S[xv ]Bn)

for x = (xv) ∈ Xi(A). We write ρ for the right regular action of Gn(Af ) onTnℓ+n. For F ∈ Tnℓ+n we define the (S, ϕ)th Fourier-Jacobi coefficient of F by

FSϕ (g

′) =

∫Ni(F )\Ni(A)

F(vg′)Θ(ωψS (vg′)ϕS) dv.


Lemma 7.7. Let S ∈ S+i . Put n′ = n− i. Let

F(g) =∑B∈S+

n

wB(gf ,F)W(ℓ+n)B (g∞),

be the Fourier expansion of F ∈ Tnn+ℓ. Then FSϕ (g

′) is equal to∑Ξ∈S+

n′

NF/Q(ν(Ξ))i/2W(ℓ+n′)Ξ (g′∞)

∫Xi(Af )

wS⊕Ξ(xg′f ,F)(ωψf

S (g′f )ϕ)(x) dx

up to a nonzero constant multiple. Moreover, F ∈ Cnℓ+n if and only if

(ρ(∆)F)Sϕ ∈ Cn′ℓ+n′ for all ∆ ∈ Gn(Af ), S ∈ S+

i and ϕ ∈ S(Xi(Af )).

Remark 7.8. When D ≃ M2(F ), one can prove an analogous result for

R ∈ Sym+i and F ∈ T

(m)ℓ+m in the same way.

Proof. We abuse notation in writing w(ℓ+n)B (g,F) = wB(gf ,F)W

(ℓ+n)B (g∞).

The calculation in the proof of [19, Lemma 4.1] shows that

FSϕ (g

′) =∑Ξ∈S+

n′

∫Xi(A)

w(ℓ+n)S⊕Ξ (xg′,F)(ωψS (g

′)ϕS)(x) dx

for g′ ∈ Gn′(A). Employing Lemma 7.6, we arrive at the stated formula.The “only if” part is clear. The proof of the other direction is similar to

that of the Saito-Kurokawa lifting (cf. Proposition 1.3 of [21] and Section9 of [22]). Recall that we regard Gn′ as a subgroup of Gn as in §7.1. Notethat

F(g) =∑S∈S+

i

FS(g), FS(g) =∑Ξ∈S+

n′

∑u∈Xi(F )

w(ℓ+n)S⊕Ξ (ug,F).

Since the subgroups Pn(F ) and Gn′(F ) generate Gn(F ), if FS is left invari-ant under Gn′(F ) for all S ∈ S+

i , then F ∈ Cnℓ+n. Fix A ∈ GLi(D(A∞)). Put

A′ = m(diag[A,1n′ ]) and C = e−2π∑v τ(S[Av ]Bi)

∏v ν(S)

(ℓv+n)/2ν(Av)ℓv+n.

Then for v ∈ Ni(A∞) and g′ ∈ Gn′(A∞)

W(ℓ+n)S⊕Ξ (uvA′g′) = CNF/Q(ν(Ξ))i/2[ωψ∞

S (vg′)φ∞S ](u)W

(ℓ+n′)Ξ (g′)

by (7.1) and (7.3). Let C be a compact subgroup of Gn′(Af )⋉Ni(Af ) underwhich FS is right invariant. Take an orthonormal basis ϕ1, . . . , ϕk of the

finite dimensional space ϕ ∈ ωψfS | ωψf

S (c)ϕ = ϕ for c ∈ C. Then forA ∈ GLi(D(A∞)), v ∈ Ni(A) and g′ ∈ Gn′(A)

FS(vA′g′) = C

k∑l=1

Θ(ωψfS (vg′)ϕlS)FS

ϕl(g′).

Therefore, if (ρ(∆)F)Sϕ is left invariant under Gn′(F ) for all ∆ ∈ Gn(Af )

and ϕ ∈ S(Xi(Af )), then so is FS .

33

Lemma 7.9. Let CBB∈Sπfn ∈ Tnℓ+n(µf ). Then CB ∈ Cnℓ+n(µf ) if and

only if CS⊕ΞΞ∈D

πf⊗χS

f−

∈ C1ℓ+1(µf χ

Sf ) for all S ∈ S+

n−1.

Proof. Taking Corollary 7.3 into account, we define a surjection

βψfS = ⊗pβ

ψp

S : An(µf )⊗ ωψfS ↠ An′(µf χ

Sf ).

Lemma 4.6 says that Fℓ+n(f, CB) ∈ Tnℓ+n for all f ∈ An(µf ). In view of

Lemmas 7.2 and 7.7 the (S, ϕ)th Fourier-Jacobi coefficient of Fℓ+n(f, CB)equals Fℓ+1(β

ψfS (f ⊗ ϕ), CS⊕ΞΞ) up to a nonzero constant multiple. The

last statement of Lemma 7.7 finally proves the equivalence.

8. Proofs of Theorems 1.1 and 1.2

8.1. Theta lifts from Mp(Wm). We give a brief account of theta corre-spondence for the dual pair Mp(Wm) × O(V ). For a detailed treatmentone can consult [9, 43, 11, 12, 61]. Let (V, qV ) be a quadratic space of di-mension l. In the case of interest in this paper m = 1 and V = VD orV = Fe ⊕ VD ⊕ Fe′. In the former case G+(V ) ≃ D× and in the lattercase G+(V ) ≃ G1 by Lemma 2.2. We define the symplectic vector space(W,≪ , ≫) of dimension 2ml by W = V ⊗Wm and ≪ , ≫= ( , ) ⊗ ⟨ , ⟩.We have natural homomorphisms

Sp(Wm) → Sp(W), G+(V )ϑ↠ SO(V ) → O(V ) → Sp(W).

The groups O(V ) and Sp(Wm) form a dual pair inside Sp(W).

Fix η ∈ F×. We obtain the representation ωψη

V = ⊗′vω

ψηvVv

by pulling back

to O(V,A)×Mp(Wm)A (or G+(V,A)×Mp(Wm)A) the global Weil represen-tation of the metaplectic double cover Mp(W)A of Sp(W,A) associated to

ψη. Note that ωψη

V ≃ ωψηV , where ηV is the space V equipped with the qua-

dratic form ηqV . When m = l = 1, the local Weil representation ωψvη ≃ ωψηv

1is realized in S(Fv) and is the direct sum of two irreducible representations:

ωψvη = ωψηv

+ ⊕ ωψηv

− , where ωψηv

+ (resp. ωψηv

− ) consists of the even (resp. odd)functions in S(Fv). Given an irreducible admissible genuine representation

σv of Mp(Wm)v the maximal quotient of ωψvVv on which Mp(Wm)v acts as a

multiple of σv is of the form σv⊠ΘψvVv(σv), where Θ

ψvVv(σv) is a representation

of O(Vv). We say that ΘψvVv(σv) is zero if σv does not occur as a quotient

of ωψvVv . Let θψvVv (σv) be the maximal semisimple quotient of ΘψvVv(σv). Then

θψvVv (σv) is either zero or irreducible by the Howe conjecture, which wasproved by Wee Teck Gan and Takeda [12].

It turns out that there is a natural Sp2ml(F )-invariant map Θ : ωψV →C. Let σ be an irreducible genuine cuspidal automorphic representation of


Mp(Wm)A. For h ∈ σ and ϕ ∈ ωψV we set

θψV (g;h, ϕ) =

∫Spm(F )\Spm(A)

h(x)Θ(ωψV (x, g)ϕ) dx,

where x denotes a preimage of x in Mp(Wm)A. Note that since the integrandis a product of two genuine functions on the double cover of Spm(A), it is

defined on Spm(A). Then θψV (h, ϕ) is an automorphic form on O(V ). We

write θψV (σ) for the subspace of the space of automorphic forms on O(V )

spanned by θψV (h, ϕ) for all h ∈ σ and ϕ ∈ ωψV . It is a simple consequence

of the Howe conjecture that if θψV (σ) is nonzero and contained in the space

of square-integrable automorphic forms on O(V ), then θψV (σ) ≃ ⊗′vθψvVv(σ∨v ).

The correspondence Π 7→ θψWn(Π) in the opposite direction can be defined

similarly.Let A00 denote the space of genuine cusp forms on Mp(W1)A orthogonal

to elementary theta series of the Weil representation ωψη for any η ∈ F×.The following result can be deduced from the Rallis inner product formula.

Proposition 8.1 (Theorem 2.8 of [9]). Assume that m = 1 and l ≥ 5. Let σbe an irreducible genuine cuspidal automorphic representation in A00. Then

θψV (σ) is nonzero if and only if ΘψvVv(σv) is nonzero for all v.

8.2. The work of Waldspurger. The space A00 satisfies multiplicity onebut does not satisfy strong multiplicity one: there are nonequivalent cuspidalautomorphic representations σ and σ′ whose local components are equivalentfor almost all places. We say that such σ and σ′ are nearly equivalent.

Waldspurger has described the near equivalence classes of representationsin A00. In his papers [56, 57] he defined a surjective map Wdψv from the setof irreducible admissible genuine unitary representations of Mp(W1)v which

are not equivalent to ωψηv

+ for any η ∈ F×v to the set of irreducible infinite

dimensional unitary representations of PGL2(Fv). Let σv be such a repre-sentation of Mp(W1)v. Let V +

v (resp. V −v ) stand for a three dimensional

split (resp. anisotropic) quadratic space over Fv of discriminant 1. Then

precisely one of the representations θψvV +v(σv) and θψv

V −v(σv) is nonzero. Set

Wdψv(σv) = θψvV +v(σv) if it is nonzero. Otherwise Wdψv(σv) corresponds to

θψvV −v(σv) via the Jacquet-Langlands correspondence.

Given an irreducible infinite dimensional unitary representation πv ofPGL2(Fv), we put Πψv(πv) = Wd−1

ψv(πv). If πv is a discrete series, then

#Πψv(πv) = 2. Otherwise Πψv(πv) is a singleton. In the first case the set

Πψv(πv) has a distinguished element σψv+ (πv), which is characterized by the

fact that σψv+ (πv) ⊗ πv is a quotient of the Weil representation ωψvV +v. The

other element of Πψv(πv) is denoted by σψv− (πv): it is characterized by the

35

fact that σψv− (πv) ⊗ πJLv is a quotient of ωψvV −v, where πJLv is the Jacquet-

Langlands correspondence of πv. In the second case we shall let σψv+ (πv)

be the unique element in Πψv(πv), and set σψv− (πv) = 0. This partition ofrepresentations of Mp(W1)v into packets and their parametrization in termsof representations of PGL2(Fv) depend on the choice of ψv. But it is quiteexplicit. Denote the discrete series representation of PGL2(R) with extremalweights ±2ℓ by D2ℓ and the lowest weight representation of the real meta-plectic group of rank m with lowest K-type k 7→ J(2ℓ+1)/2(k,

√−11m)

−1 by

D(m)(2ℓ+1)/2.

Proposition 8.2 (Propositions 4, 5, 9 of [57]). (1) If πp is an irreducible

principal series I(µp, µ−1p ), then σ

ψp

+ (πp) ≃ Iψp

1 (µp).

(2) If πp ≃ A(α1/2p , α

−1/2p ), then σ

ψp

− (πp) ≃ Aψp

1 (α1/2p ).

(3) If µ2p = αp, µp = α1/2p and πp ≃ A(µp, µ

−1p ), then σ

ψp

+ (πp) ≃ Aψp

1 (µp).

(4) If v ∈ S∞, then σψv+ (D2κv) ≃ D(1)(2κv+1)/2.

Bear in mind the assumption that ψv = e|Fv for v ∈ S∞. Given anirreducible cuspidal automorphic representation π = ⊗′

vπv of PGL2(A), wedefine a set of irreducible unitary representations of Mp(W1)A by

Πψ(π) = ⊗′vσ

ψvϵv (πv) | ϵv ∈ ±, and for all most all v, ϵv = +.

For given σ = ⊗′vσ

ψvϵv (πv) ∈ Πψ(π), we set ϵ(σ) =

∏v ϵv. Corollaries 1 and

2 on p. 286 of [57] say that

(8.1) A00 ≃⊕π

⊕σ∈Πψ(π): ϵ(σ)=ε(1/2,π)

σ,

where the sum ranges over all irreducible cuspidal automorphic representa-tions π of PGL2(A) such that L

(12 , π ⊗ χt

)= 0 for some t ∈ F×.

This result includes the special case of Theorem 1.1 in which m = 1.

Lemma 8.3. Aψf1 (µf ) appears in C

(1)(2κ+1)/2 if and only if ⊗′

pA(µp, µ−1p ) ap-

pears in C2κ and µf (−1)(−1)∑v∈S∞ κv = 1.

Proof. Put

π = (⊗v∈S∞D2κv)⊗ (⊗′pA(µp, µ

−1p )), σ = (⊗v∈S∞D

(1)(2κv+1)/2)⊗Aψf

1 (µf ).

(8.2)

If σ is an irreducible cuspidal automorphic representation, then so is π =⊗′vWdψv(σv). Since πv is a discrete series for v ∈ S∞, Theorem A.2 of [47]

says that L(12 , π⊗ χt

)= 0 for some t. Let S−

π be the set of nonarchimedean

primes p of F such that µp = α1/2p . Put ℓ−π = #S−

π . Then

(8.3) ε(1/2, π) = µf (−1)(−1)ℓ−π+

∑v∈S∞ κv .


Note that

σ ≃ (⊗p∈S−πσψp

− (πp))⊗ (⊗′v/∈S−

πσψv+ (πv)) ∈ Πψ(π)

by Proposition 8.2. Thus ϵ(σ) = ϵ(12 , π

)is equivalent to the sign condition.

We have the desired conclusion by (8.1). Remark 8.4. Lemma 8.3 is consistent with Lemma 5.6(3).

8.3. Construction of an embedding Aψfm (µf ) → C

(m)(2m+κ)/2. We now

prove Theorem 1.2. We hereafter assume that m > 1. Fix R ∈ Sym+m−1.

Corollary 7.5 gives a Mp(W1)f -intertwining surjective homomorphism

(8.4) βψfR = ⊗pβ

ψp

R : Aψfm (µf )⊗ ωψf

R ↠ Aψf1 (µf χ

detRf ).

We associate to ϕ ∈ S(Xm−1(Af )) the function ϕR = ϕ ⊗ (⊗v∈S∞φR) ∈S(Xm−1(A)) and the theta function on Jm−1(F )\Jm−1(A) defined by

Θ(ωψR(vg′)ϕR) =

∑l∈Xm−1(F )

(ωψR(vg′)ϕR)(l) (v ∈ Nm−1(A), g′ ∈ Mp(W1)A).

The (R,ϕ)th Fourier-Jacobi coefficient of F ∈ T(m)ℓ is defined by

FRϕ (g

′) =

∫Nm−1(F )\Nm−1(A)

F(s(v)g′)Θ(ωψR(vg′)ϕR) dv.

Recall the function F(2κ+m)/2(h, Cξ) defined in Definition 5.5. Lemmas7.4 and 7.7 give a nonzero constant ER such that

(8.5) F(2κ+m)/2(h, Cξ)Rϕ = ERF(2κ+1)/2(βψfR (h⊗ ϕ), CR⊕tt)

for all Cξ ∈ T(m)(2κ+m)/2(µf ) and h ∈ Aψf

m (µf ) in view of Remark 7.8.

Lemma 8.5. If ct ∈ C(1)(2κ+1)/2(µf ), then cη det ξξ ∈ C

(m)(2κ+m)/2(µf χ

ηf ) for

all m and η ∈ F×+ .

Proof. The series iηm(h) = F(2κ+m)/2(h, cη det ξξ) is convergent for all h ∈Aψfm (µf χ

ηf ) by Lemma 4.4 and the estimate of ct given in Proposition A.6.4

of [49], and so by Lemma 5.6(2) iηm(Aψfm (µf χ

ηf )) ⊂ T

(m)(2κ+m)/2. Lemma 5.6(5),

(8.4) and (8.5) show that

iηm(h)Rϕ = ERi

η detR1 (βψf

R (h⊗ ϕ)) ∈ C(1)(2κ+1)/2

for all R ∈ Sym+m−1 and ϕ ∈ S(Xm−1(Af )). Lemma 7.7 eventually concludes

that iηm(Aψfm (µf χ

ηf )) ⊂ C

(m)(2κ+m)/2 (cf. Remark 7.8).

We have thus seen that the Fourier series given in Theorem 1.2 is a Hilbert-Siegel cusp form of weight κ+m

2 . In Section 10 we will show that it reduces tothe Duke-Imamoglu-Ikeda lift when F = Q, m = 2n and h is right invariantunder

∏p Sp2n(Zp).

37

8.4. Some lemma on quadratic forms. It remains to verify that the

multiplicity of Aψfm (µf ) is at most one. The proof relies on certain technical

results. When π is an irreducible cuspidal automorphic representation ofPGL2(A) of the form (8.2), we put

S πm = ξ ∈ Sym+

m | L(1/2, π ⊗ χdet ξ) = 0, χdet ξp = µpα

−1/2p for p ∈ Sπf .

Lemma 8.6. If C(1)(2κ+1)/2(µf ) = 0 and Ξ ∈ S π

3 represents t1, t2 ∈ F×+ ,

then there are a quadratic form S ∈ Sym+2 representing t1 and represented

by Ξ and a quadratic form T ∈ S π3 representing both S and t1 ⊕ t2.

Proof. Choose vectors x, y ∈ F 3 such that Ξ[x] = t1 and Ξ[y] = t2. IfΞ(x, y) = txΞy = 0, then we can take S = t1 ⊕ t2 and T = Ξ. Suppose thatΞ(x, y) = 0. Define functions S : F 3 → Sym2(F ) and T : F 3 → Sym3(F ) by

S(z) =

(Ξ(x, x) Ξ(x, z)Ξ(x, z) Ξ(z, z)

), T (z) =

Ξ(x, x) Ξ(x, z) 0Ξ(x, z) Ξ(z, z) Ξ(y, z)

0 Ξ(y, z) Ξ(y, y)

.

Clearly, S(z) and T (z) fulfill all the requirements besides the condition thatT (z) ∈ S π

3 . We define a quadratic form Q of three variables by

Q[z] = detT (z) = Ξ(x, x)Ξ(y, y)Ξ(z, z)− Ξ(y, y)Ξ(x, z)2 − Ξ(x, x)Ξ(y, z)2.

By a direct calculation detQ = − detΞ · Ξ(x, x)2Ξ(y, y)2Ξ(x, y)2 ∈ −F×+ .

Let SQ be the set of places of F at which Q is anisotropic. If z = 0 andΞ(x, z) = Ξ(y, z) = 0, then Q[z] = Ξ(x, x)Ξ(y, y)Ξ(z, z) ∈ F×

+ . Thus SQ

consists of finite primes. Since T (z) ≃ t1 ⊕ t2 ⊕ (t1t2)−1Q[z], if Q[z] ∈ F×

+ ,

then T (z) is totally positive definite. Then Q represents any element t ∈ F×+

such that t /∈ (det Ξ)F×2p for all p ∈ SQ.

Take η ∈ F×πf

such that η /∈ (det Ξ)F×2p for all p ∈ SQ. Since ε

(12 , π⊗χ

η)=

1 by Lemma 8.3 (cf. (8.3)), for any ϵ > 0 and a finite set S of primes of F ,Theorem 4 of [57] gives an element t ∈ F×

+ which satisfies |t − η|v < ϵ for

v ∈ S and such that L(12 , π ⊗ χt

)= 0.

Lemma 8.7. Suppose that C(1)(2κ+1)/2(µf ) = 0. For ξ0, ξ3 ∈ S π

m there are

ξ1, ξ2 ∈ S πm and R1, R2, R3 ∈ Sym+

m−1 such that Ri is represented by bothξi−1 and ξi for all i = 1, 2, 3.

Proof. If m ≥ 3, then ξ0 ⊕ (−ξ3) must have a totally isotropic subspace of

dimension m− 2 and hence there are ξ ∈ Sym+m−2 and ξ′0, ξ

′3 ∈ S π⊗χξ

2 suchthat ξ0 ≃ ξ ⊕ ξ′0 and ξ3 ≃ ξ ⊕ ξ′3. We may therefore assume that m = 2.

Set Ξ = 1 ⊕ ξ0 and Ξ′ = 1 ⊕ ξ3. Choose R2 ∈ F×+ represented by Ξ

and Ξ′. Applying Lemma 8.6 to Ξ, 1 and R2, we find a quadratic formS ∈ Sym+

2 representing 1 and represented by Ξ and find a quadratic formT ∈ S π

3 representing both S and 1⊕R2. Put R1 = detS. Then S ≃ 1⊕R1.There is ξ1 ∈ S π

2 such that T ≃ 1⊕ ξ1. Then ξ1 represents R2. Since bothΞ = 1 ⊕ ξ0 and T ≃ 1 ⊕ ξ1 represent S ≃ 1 ⊕ R1, both ξ0 and ξ1 represent


R1. Similarly, we can find a quadratic form ξ2 ∈ S π2 representing R2 and

find a totally positive element R3 represented by both ξ2 and ξ3.

8.5. Multiplicity of Aψfm (µf ). In light of Lemma 5.6(1), giving a Mp(Wm)f -

intertwining map from Aψfm (µf ) into C

(m)(2κ+m)/2(µf ) is equivalent to giving

complex numbers Cξξ∈Symπfm ∈ C(m)(2κ+m)/2. We now prove a stronger result.

Lemma 8.8. Suppose that there is a Mp(Wm)f -intertwining embedding i :

Aψfm (µf ) → C

(m)(2κ+m)/2. Then µf (−1)(−1)

∑v∈S∞ κv = 1, ⊗′

pA(µp, µ−1p ) occurs

in C2κ and there is ct ∈ C(1)(2κ+1)/2(µf ) such that i(h) = F(2κ+m)/2(h, cdet ξ)

for all h ∈ Aψfm (µf ). In particular, dim C

(m)(2κ+m)/2(µf ) = 1.

Proof. As mentioned above, Lemma 5.6(1) gives 0 = Cξ ∈ C(m)(2κ+m)/2(µf )

such that i(h) = F(2κ+m)/2(h, Cξ). Hence CR⊕t ∈ C(1)(2κ+1)/2(µf χ

detRf )

for all R ∈ Sym+m−1 by (8.4) and (8.5). This together with Lemma 8.3 proves

one implication of Theorem 1.1.

Fix a basis vector et of the one dimensional vector space C(1)(2κ+1)/2(µf ).

For each R ∈ Sym+m−1 Lemma 5.6(5) gives a complex number δR such that

CR⊕t = δRet detR for all t ∈ F×πf⊗χ

detRf

. Let ξ ∈ Symπfm . If there exists

a ∈ GLm(F ) such that ξ = (R⊕ t)[a], then

Cξ = δRet detRµf (det a)−1

∏v∈S∞

sgnv(det a)κv = δRet detR(det a)2 = δRedet ξ

by Lemma 5.6(2). Lemma 5.6(6) tells us that Cξ = 0 only if ξ ∈ S πm.

Lemma 8.7 now says that

Cξ0edet ξ0

= δR1 =Cξ1edet ξ1



for all ξ0, ξ3 ∈ S πm, which completes our proof.

Let Acusp(Mp(Wm)) be the space of cusp forms on Spm(F )\Mp(Wm)Aand g the complexified Lie algebra of Mp(Wm)∞.

Proposition 8.9. (1) C(m)ℓ ⊂ Acusp(Mp(Wm)).

(2) If e is a lowest weight vector of ⊗v∈S∞D(m)ℓv

and

ι : ⊗v∈S∞D(m)ℓv

→ Acusp(Mp(Wm))

is a (g, Km) intertwining map, then ι(e) ∈ C(m)ℓ .

Proof. Lemma 5 of [3] says that Hilbert-Siegel cusp forms are cusp forms inthe sense of Langlands, which just amounts to (1). By Lemma 7 of [3] ι(e)∆is a holomorphic function on Hd

m. We may assume thatm > 1 as ι(e) ∈ C(m)ℓ

is a part of the definition if m = 1. Then ι(e) is a Hilbert-Siegel modular

39

form by the Koecher principle. Since ι(e) ∈ Acusp(Mp(Wm)), Proposition

A4.5(2) of [48] says that ι(e) ∈ C(m)ℓ as expected.

We conclude this section by giving the following characterization.

Corollary 8.10. Let Π = ⊗′vΠv be an irreducible cuspidal automorphic rep-

resentation of Mp(Wm)A. Assume that there is a character µf =∏

p µp ∈Ω(A×

f ) with −12 < ℜµp ≤ 1

2 satisfying the following conditions:

• Πv ≃ D(m)(2κv+m)/2 with κv ∈ N for every v ∈ S∞;

• Πp is equivalent to a subrepresentation of Iψpm (µp) for every p.

Then the unique irreducible subrepresentation of ⊗′pI(µp, µ

−1p ) is a summand

of C2κ, (−1)∑v∈S∞ κvµf (−1) = 1 and Π is generated by i1m(A

ψfm (µf )).

This can be derived as a corollary from Lemma 8.8 and Proposition 8.9.

9. Transfer to inner forms

We retain the notation of §2.4. Thus V = Fe⊕VD⊕Fe′ and G+(V ) ≃ G1.In the first half of this section we switch to a local setting. Thus F = Fv isa local field of characteristic zero.

9.1. The Schrodinger model vs. the mixed model. The Weil repre-

sentation ωψV can be realized on S(V ) and has the following formulas:

(ωψV (ζm(t))Φ)(v) = ζγψ(t)|t|5/2Φ(tv),(9.1)

(ωψV (n(b))Φ)(v) = ψ(bqV (v))Φ(v),(9.2)

(ωψV (β)Φ)(v) = Φ(ϑ(β)−1v)(9.3)

for ζ ∈ µ2, t ∈ F×, b ∈ F , β ∈ G1 and v ∈ V . Since the map

(FΦ)(x;u, u′) =

∫FΦ(re+ x+ ue′)ψ(ru′) dr

is a C-linear isomorphism from S(V ) onto S(VD ⊕ F 2), we can define the

action ΩψV of Mp(W1)× G1 on S(VD ⊕ F 2) so that

ΩψV (x, g)FΦ = F (ωψV (x, g)Φ), (x, g) ∈ Mp(W1)× G1.

The following formulas are derived easily or read of from Lemma 46 of [57]:

(ΩψV (d(t)m(A))φ)(x;u, u′) =

∣∣∣∣ν(A)t∣∣∣∣φ(

A−1xA;ν(A)u

t,ν(A)u′

t

),(9.4)

(ΩψV (n(z))φ)(x;u, u′) = ψ(−(τ(zx) + uν(z))u′)φ(x− uz;u, u′),(9.5)

(ΩψV (m(t))φ)(x;u, u′) = γψ(t)|t|3/2φ(tx; tu, t−1u′),(9.6)

(ΩψV (n(b))φ)(x;u, u′) = ψ(−bν(x))φ(x;u, u′ + ub),(9.7)

(ΩψV (s(J))φ)(x;u, u′) = γψD

∫D−

φ(y;−u′, u)ψ(τ(xy)) dy(9.8)


for A ∈ D×; t ∈ F×; z, x ∈ D− and b, u, u′ ∈ F , where γψD is a certain 8th

root of the unitary and dy is the self-dual Haar measure on D− with respectto the Fourier transform defined by

(FDϕ)(x) =∫D−

ϕ(y)ψ(τ(xy)) dy, ϕ ∈ S(D−).

Remark 9.1. In the notation of [57] n2(z) = n(−z) (cf. Lemma 2.2).

9.2. Compatibility of Jacquet integrals. We first discuss the p-adiccase. Recall that χ−1 corresponds to F (

√−1)/F via class field theory.

Lemma 9.2. Let h ∈ Iψ1 (µ) and Φ ∈ S(V ). If ℜµ > −32 , then the integral

Γψ(g;h⊗ Φ) = L(3/2, µχ−1)−1

∫U1\SL2(F )

h(x)(ωψV (x, g)Φ)(e) dx

is absolutely convergent. It gives a Mp(W1)-invariant and G1-intertwining

map Γψ : Iψ1 (µ)⊗ ωψV → J1(µχ−1). If F is not dyadic, D ≃ M2(F ), ψ is of

order 0, µ is unramified, Φ is the characteristic function of oe ⊕ R1 ⊕ oe′

and h(k) = 1 for all k ∈ SL2(o), then Γψ(12;h⊗ Φ) = 1.

Proof. The integral defining Γψ(g;h⊗ Φ) makes sense by (9.2) and equals∫F×

∫SL2(o)

h(m(a)k)(ωψV (m(a)k, g)Φ)(e)|a|−2 dkda

=

∫F×

∫SL2(o)

χ−1(a)µ(a)|a|3/2h(k)(ωψV (k, g)Φ)(ae) dk da

by (9.1) and (5.1). It is absolutely convergent for ℜµ > −32 . It follows from

(9.1), (9.3) and Lemma 2.2 that for A ∈ D×, t ∈ F× and z ∈ D−

L(3/2, µχ−1)Γψ(d(t)m(A)n(z)g;h⊗ Φ)

=

∫U1\SL2(F )

h(x)(ωψV (x, g)Φ)(tν(A)−1e) dx

=(χ−1µα7/2)(t−1ν(A))

∫U1\SL2(F )

h(m(tν(A)−1)x)(ωψV (m(tν(A)−1)x, g)Φ)(e) dx

=(χ−1µα3/2)(t−1ν(A))L(3/2, µχ−1)Γψ(g;h⊗ Φ).

Therefore Γψ(h⊗ Φ) ∈ J1(µχ−1).

Recall that J1 =

(0 11 0

)∈ GL2(D) in the quaternion case.

Lemma 9.3. Let h ∈ Iψ1 (µ), Φ ∈ S(V ) and Ξ ∈ Dnd− . Put φ = FΦ ∈

S(VD ⊕ F 2). If ℜµ > −32 , then

Γψ(J1;h⊗ Φ) = L(3/2, µχ−1)−1

∫SL2(F )

h(x)(ΩψV (x)φ)(0; 1, 0) dx.

41

Proof. The product L(3/2, µχ−1)Γψ(J1;h⊗ Φ) equals∫U1\SL2(F )

h(x)(ωψV (x, J1)Φ)(e) dx =

∫U1\SL2(F )

h(x)(ωψV (x)Φ)(e′) dx

by (9.3) and Lemma 2.2. The Fourier inversion says that

(F−1φ)(re+ x+ r′e′) =

∫Fφ(x; r′, u)ψ(−ru) du.

We use this formula and (9.7) to see that the left hand side equals∫U1\SL2(F )

h(g)F−1(ΩψV (g)φ)(e′) dg =

∫U1\SL2(F )

h(g)

∫F(ΩψV (g)φ)(0; 1, u) dudg

=

∫U1\SL2(F )

h(g)

∫F(ΩψV (n(u)g)φ)(0; 1, 0) dudg.

We combine the integrals over U1 and U1\SL2(F ) into an integral overSL2(F ) to obtain the stated formula. The integral thus obtained equals∫

SL2(o)

∫F×

∫F|a|µ(a)h(k)|a|3/2χ−1(a)(ΩψV (k)φ)(0; a, ua

−1)|a|−2 dudadk

and converges absolutely for ℜµ > −32 , which justifies all the manipulations.

Lemma 9.4. Notation being as in Lemma 9.3, there is a constant C which

is independent of Ξ and such that wµχ−1

Ξ (Γψ(h⊗ Φ)) is equal to

C|ν(Ξ)|1/4∫

U1\SL2(F )wµν(Ξ)(ϱ(g)h)(Ω

ψV (g)φ)(−Ξ; 0, 1) dg.

Proof. By Lemma 9.3 we can write wµχ−1αs

Ξ (Γψ(h(s)⊗Φ)) as the product of

|ν(Ξ)|3/4 L(2s+ 1, µ2)

L(s+ 1

2 , µχ−1χΞ

) = |ν(Ξ)|1/4 · |ν(Ξ)|1/2L(2s+ 1, µ2)

L(s+ 1

2 , µχν(Ξ)

)and the integral∫

D−

∫SL2(F )

h(s)(Jg)(ΩψV (Jg,n(z))φ)(0; 1, 0)ψΞ(z) dgdz

=

∫D−

∫SL2(F )

h(s)(Jg)(ΩψV (Jg)φ)(−z; 1, 0)ψ(−τ(Ξz)) dgdz

by (9.5). Since the double integral is absolutely convergent for ℜs ≫ 0, wemay interchange the order of integration. Using (9.8) and (9.7), we get

1

γψD

∫SL2(F )

h(s)(Jg)(ΩψV (J2g)φ)(Ξ; 0,−1) dg

=γψ(−1)

γψD

∫SL2(F )

h(s)(Jg)(ΩψV (g)φ)(−Ξ; 0, 1) dg


=γψ(−1)

γψD

∫U1\SL2(F )

∫Fh(s)(J n(b)g)ψν(Ξ)(b) db (ΩψV (g)φ)(−Ξ; 0, 1) dg.

By (9.6) the outer integral converges absolutely for all s. The proof iscomplete by evaluating the equality at s = 0.

Corollary 9.5. (1) If −12 < ℜµ < 1

2 , then Γψ(Iψ1 (µ)⊗ ωψV ) = J1(µχ−1).

(2) If µ2 = α, then Γψ(Aψ1 (µ)⊗ ωψV ) = A1(µχ−1).

Proof. If χΞ = µχ−1α−1/2, then χν(Ξ) = µα−1/2, and hence

wµχ−1

Ξ (Γψ(h⊗ Φ)) = 0

for all h ∈ Aψ1 (µ) and Φ ∈ ωψV by Proposition 5.1(3) and Lemma 9.4. We can

therefore infer from Proposition 3.1(3) that Γψ(Aψ1 (µ) ⊗ ωψV ) ⊂ A1(µχ−1).

Employing Proposition 5.1 again, we can take Ξ ∈ Dnd− and a test vector h

for which wµν(Ξ)(h) = 0. If φ = φ′ ⊗ φ′′ with φ′ ∈ S(VD) and φ′′ ∈ S(F 2),

then we obtain

wµχ−1

Ξ (Γψ(h⊗ Φ)) = C|ν(Ξ)|1/4∫F

∫F\0

φ′′(c, a)

× wµν(Ξ)

(ϱ

((a−1 0c a

))h

)(ωψVD

((a−1 0c a

))φ′)(−Ξ) dadc

by rewriting the formula in Lemma 9.4, where da and dc are Haar measureson F . This integral can be made nonzero by choosing φ′′ to be supportedin a small neighborhood of (0, 1). Thus Γψ is nonzero, which verifies theclaimed results as the target spaces are irreducible.

Remark 9.6. The theta lift θψV (σ) of an irreducible admissible representation

σ of Mp(W1) is defined to be the unique irreducible quotient of σ∨⊗ωψV (see

§8.1). Since the map Γψ1 : Iψ1 (µ)⊗ωψV → J1(µχ−1) is Mp(W1)-invariant and

G1-equivariant by Lemma 9.2, we see by Corollary 9.5 that

θψV (Iψ1 (µ)

∨) ≃ J1(µχ−1), θψV (A

ψ1 (µ)

∨) ≃ A1(µχ−1).

This result is stated in Propositions 5.2 and 6.3 of [9].

In Lemma 7.6 of [20] Ichino explicitly constructed a Schwartz functionΛ ∈ S(VD ⊕ R2) with the following property (cf. Remark 9.1).

Lemma 9.7. Suppose that F = R and D ≃ M2(R). There is Λ ∈ S(VD⊕R2)such that for all Ξ ∈ D+

−

W(ℓ)Ξ (g) = 2ℓν(Ξ)1/4

∫U1\SL2(R)

W((2ℓ−1)/2)ν(Ξ) (x)(ΩψV (x, g)Λ)(−Ξ; 0, 1) dx

and for all Ξ ∈ −D+−∫

U1\SL2(R)W

((2ℓ−1)/2)ν(Ξ) (x)(ΩψV (x, g)Λ)(−Ξ; 0, 1) dx = 0.

43

9.3. Fourier coefficients for Saito-Kurokawa liftings. Let F be a to-tally real number field and D a totally indefinite quaternion algebra over

F . We denote by ωψV ≃ ⊗′vω

ψvVv

the Schrodinger model of the global Weil

representation and by ΩψV its mixed model. These models are related by the

intertwining map F : S(V (A)) → S(VD(A) ⊕ A2). We write ωψfV and Ωψf

Vfor their finite parts. For Φ ∈ S(V (Af )) we define a Schwartz function ΦΛ

on V (A) by

ΦΛ(x) = Φ(xf )∏v∈S∞

(F−1v Λ)(xv), x = (xv) ∈ V (A).

Taking Lemma 9.2 and Corollary 9.5 into account, we define a surjectivehomomorphism

(9.9) Γψf = ⊗pΓψp : Aψf

1 (µf )⊗ ωψfV ↠ A1(µf χ

−1f ).

The following result is nothing but Lemma 47 of [57] (cf. Remark 9.1).

Lemma 9.8. If F ∈ A00, φ ∈ S(VD(A)⊕ A2) and 0 = Ξ ∈ D−(F ), then

WΞ(θψV (F , φ)) =

∫U1(A)\SL2(A)

Wν(Ξ)(g,F)(ΩψV (g)φ)(−Ξ; 0, 1) dg.

Lemma 9.9. If 0 = ctt∈F×πf

∈ C(1)(2κ+1)/2(µf ), then

0 = cν(Ξ)Ξ∈D

πf⊗χ−1

f−

∈ C1κ+1(µf χ

−1f ).

Proof. By assumption F(2κ+1)/2(h, ct) ∈ C(1)(2κ+1)/2 for all h ∈ Aψf

1 (µf ). We

consider its Saito-Kurokawa lift

θψV (g;F(2κ+1)/2(h, ct),ΦΛ) =∑

Ξ∈D−(F )

WΞ(g, θψV (F(2κ+1)/2(h, ct),ΦΛ)).

Anti-holomorphic discrete series representations of Mp(W1)v do not occur

in the quotient of the Weil representation ωψvVDvfor v ∈ S∞. Consequently,

θψVD(C(1)(2κ+1)/2

)= 0, and so by the tower property, the space θψV

(C(1)(2κ+1)/2

)consists of cuspidal automorphic forms on G1. It follows that

W0(θψV (F(2κ+1)/2(h, ct),ΦΛ)) = 0.

Lemmas 9.7 and 9.8 show that

WΞ(θψV (F(2κ+1)/2(h, ct),ΦΛ)) = 0

unless Ξ ∈ Dπf⊗χ

−1f

− , in which case Lemma 9.4 gives a nonzero constant Cwhich is independent of Ξ and such that

WΞ(g∞, θψV (F(2κ+1)/2(h, ct),ΦΛ)) = Ccν(Ξ)W

(κ+1)Ξ (g∞)w

µf χ−1f

Ξ (Γψf (h,Φ)).

We conclude that

θψV (g;F(2κ+1)/2(h, ct),ΦΛ) = CFκ+1(g∞; Γψf (h, ωψfV (gf )Φ), cν(Ξ))


for all h ∈ Aψf1 (µf ) and Φ ∈ S(V (Af )). We therefore see by (9.9) that

Fκ+1(f, cν(Ξ)) ∈ G1κ+1 ⊂ C1

κ+1 for all f ∈ A1(µf χ−1f ). The map f 7→

Fκ+1(f, cν(Ξ)) is nonzero by Proposition 8.1 and Remark 9.6.

9.4. End of the proof of Theorem 6.1. Let ctt∈F×πf

∈ C(1)(2κ+1)/2(µf ).

Fix η ∈ F×+ . Put CB = cην(B) for B ∈ S

πf⊗χ(−1)nηf

n . Thanks to the esti-mate of Fourier coefficients given in Proposition A.6.4 of [49], we can in-voke Lemma 4.4 to guarantee convergence of the series Fκ+n(f, CB) for

f ∈ An(µf χ(−1)nηf ). Moreover, CB ∈ Tnκ+n(µf χ

(−1)nηf ) by Lemma 5.6(2),

(5). Note that cην(S)tt ∈ C(1)(2κ+1)/2(µf χ

ην(S)f ) by Lemma 5.6(5). Since

cην(S)ν(Ξ)Ξ∈D

πf⊗χ−ην(S)

f−

∈ C1κ+1(µf χ

−ην(S)f ) = C1

κ+1(µf χ(−1)nηf χSf )

for all S ∈ S+n−1 by Lemma 9.9, our proof is complete by Lemma 7.9.

10. Translation to classical language

10.1. Hilbert-Siegel cuspidal Hecke eigenforms with respect to ΓDn [d].We shall translate obtained results from adele language into classical ter-minology. If h is right invariant under an open compact subgroup D ofMp(Wm)f , then i

ηm(h)em ∈ S(2κ+m)/2(Γ) by Remark 5.3, where Γ = Spm(F )∩

D. When πf is ramified, it is a laborious task to find a suitable D, choose a

good test vector h and calculate wµf χ

ηf

ξ (h). Here we letm = 2n and explicate

ιηn(f) in Theorem 6.1 when πf is everywhere unramified and f is fixed bythe maximal compact subgroup

∏pK

Dn [dp] of Gn(Af ).

The norm and the order of a fractional ideal of o are defined byN(pk) = qkpand ordp p

k = k. We denote the different of F/Q by d, the product of allthe prime ideals of o ramified in D by eD and the Dedekind zeta function ofF by ζF (s) =

∏p(1− q−sp )−1. Put

ΓDn [c] =

(α βγ δ

)∈ Gn(F )

∣∣∣∣ α, δ ∈ Mn(O)β ∈ c−1Mn(O), γ ∈ cMn(O)

,

RD,+2n = B ∈ S+

n | τ(Bz) ∈ o for all z ∈ Sn(F ) ∩Mn(O)

for a fractional ideal c of o.We consider only the parallel weight case merely for expediency, so the

weight (k, . . . , k) is simply denoted by k. A Hilbert-Siegel cusp form F ofweight k with respect to an arithmetic subgroup Γ of Gn(F ) is a holomorphicfunction on Hdn which satisfies F|kγ = F for every γ ∈ Γ and such thatF|kγ has a Fourier expansion of the form

∑B∈S+

nc(B)e∞(τ(BZ)) for all

γ ∈ Gn(F ).

45

For t ∈ F× and B ∈ S+n we denote the conductors of χt and χB by dt and

dB, respectively, and define rational numbers ft, DB and fB by

ft =

√|NF/Q(t)|N(dt)

, DB = N(eD)2[(n+1)/2]NF/Q(ν(2B)), fB =

√DB

N(dB).

We write t ≡ (mod 4) if there is y ∈ o such that t ≡ y2 (mod 4). Fora finite prime p we define f tp ∈ Z by f tp = 1

2(ordp t − ordp dt). Recall the

polynomials Ψp(t,X) ∈ C[X+X−1] and Fp(B,X) ∈ Z[X] defined in Section1 and §3.2. Set

Fp(B,X) = X−fBp Fp(B, q−(2n+1)/2p X),

where

fBp = f(−1)nν(2B)p +

0 if p ∤ eD,[n+12

]otherwise.

Then Fp(B,X) = Fp(B,X−1) by [29, 25, 62].

Let k be a natural number such that d(k+n) is even. Let πf ≃ ⊗′pI(α

spp , α

−spp )

be an irreducible summand of C2k. Then πf is isomorphic as a Hecke module

to a certain subspace of the space C(1)(2k+1)/2 by the works of Shimura and

Waldspurger among others. Theorems 9.4, 10.1, 13.5 and Remark 9.1 of [16]give a Hecke eigenform hn in the Kohnen plus subspace of weight k+ 1

2 with

respect to Γ1[d−1, 4d] and η = (−1)n whose Fourier expansion is given by

hn(Z) =∑

t∈o∩F×+ , (−1)nt≡ (mod 4)

c(t)e∞(tZ)fk−1/2(−1)nt

∏p

Ψp((−1)nt, qspp ).

This result is a generalization of the work of Kohnen [33]. We extend c toa function on F×

+ /F×2+ when D ≃ M2(F ). One can obtain the following

explicit result from Theorem 6.1 and Lemma 3.4, which is a strengtheningof Theorems 3.2 and 3.3 of [22].

Corollary 10.1. Notations and assumptions being as above, we define a func-tion LiftD2n(π) : H

dn → C by

LiftD2n(π)(Z) =∑

B∈RD,+2n

c(ν(2B))e∞(τ(BZ))f(2k−1)/2B

∏p

Fp(B, qspp ).

Then LiftD2n(π) defines a cuspidal Hecke eigenform of weight k + n withrespect to ΓDn [d] whose standard (partial) L-function is equal to

ζeD

F (s)

2n∏i=1

LeD(s+ n− i+

1

2, π

),

where ζeD

F and LeD are defined with Euler factors for primes in eD removed.


Proof. Put βf =∏

p αspp . We will apply Theorem 6.1 with η = 1 and µf =

βf χ(−1)nηf . Proposition 4.5 of [16] gives an element f+K ∈ Iψf

1 (µf ) such that

|t|(2k+1)/4wµft (f+K) = |t|(2k−1)/4∏p

Ψp((−1)nt, qspp )

qspf

(−1)ntp

p

for every t ∈ F×+ (cf. Section 6 of [16]). Put

ct = N(d(−1)nt

)−(2k−1)/4c(t)

∏p

qspf

(−1)ntp

p .

Since |t|(2k−1)/4 = N(d(−1)nt

)(2k−1)/4f(2k−1)/2(−1)nt , we see that ct ∈ C

(1)(2k+1)/2

in view of the Fourier expansion of hn. Put eDπ =∏

p|e q[(n+1)/2]spp . Observe

that for B ∈ RD,+2n

c(ν(2B)) = eDπ N(dB)(2k−1)/4cν(2B)

∏p

q−spfBpp .

Let fp = εsp+(2n+1)/2dp ∈ In(α

spp ) be the normalized KD

n [dp]-invariant ele-

ment. Put f = ⊗pfp ∈ In(βf ). Lemma 3.4 shows that for B ∈ RD,+2n

|ν(B)|(k+n)/2wβfB (f) = cπN(dB)(2k−1)/4f(2k−1)/2B

∏p

Fp(B, q−spp )

qspf

Bp

p

,

where cπ is a constant independent of B. Thus LiftD2n(π) =eDπcπι1n(f)12n

Remark 10.2. (1) When n = 1 and D ≃ M2(F ), a weaker version of thisFourier expansion was proved in [46].

(2) When n = 1, F = Q and D is division, this lifting was explicitlycomputed by Oda [45] and Sugano [53].

(3) One can prove Conjectures 10.1 and 10.2 of [59] in a parallel way.

10.2. Miyawaki liftings. For simplicity, we let D ≃ M2(F ). Let κ bea tuple of d natural numbers such that

∑v∈S∞

κv ≡ d(n + r) (mod 2).

Let πf ≃ ⊗′pI(α

spp , α

−spp ) be an irreducible summand of C2κ. Theorem

1.2 gives a Hecke eigenform F ∈ Sκ+n+r(Γ2(n+r)[d−1, d]) which generates

⊗′pI2(n+r)(α

spp ) (cf. Corollary 10.1). Let g ∈ Sκ+n+r(Γr[d

−1, d]) be a Heckeeigenform. We consider the Miyawaki type integral

F (n)π,g (Z) =

∫Γr[d

−1,d]\Hdr

F((

W 00 Z

))g(−W)

∏v

(detℑWv)κv+n−1dW.

Then F(n)π,g ∈ Sκ+n+r(Γ2n+r[d

−1, d]). One can prove the following result bythe same type of reasoning as in [24].

47

Corollary 10.3. If F(n)π,g is not identically zero, then F

(n)π,g is a Hecke eigenform

whose standard L-function is equal to

L(s,F (n)π,g , st) = L(s, g, st)

2n∏i=1

L

(s+ n− i+

1

2, π

).

11. The Duke-Imamoglu-Ikeda lifts and theta functions

11.1. Theta corespondence between degenerate principal series.We define the split quadratic form on the vector space Hm of 2m-dimensional

column vectors by qHm(x) = 2 txHmx, where Hm = 12

(0 1m1m 0

). Let F

be a local field and µ ∈ Ω(F×) in this subsection. We write Pm for theparabolic subgroup of the orthogonal group O(Hm) stabilizing the maximalisotropic subspace t(x, 0, . . . , 0) ∈ F 2m | x ∈ Fm. Let Im(µ) denote thespace of smooth functions f on O(Hm) which satisfy

f

((a b ta−1

0 ta−1

)g

)= µ(det a)| det a|(m−1)/2f(g)

for all a ∈ GLm(F ), g ∈ O(Hm) and skew symmetric matrices b of size m.In the p-adic case we put Km = O(Hm)∩GL2m(o), define the open compactsubgroup Km[b, c] of Spm(F ) and the right Km[d

−1, d]-invariant function

εd : Spm(F ) → R×+ as in (3.1) and §3.2, respectively, and let f

(s)m be the

section of Im(αs) such that f(s)m (k) = 1 for every k ∈ Km. The normalized

induced representation Im(µ) is realized on the space of functions

h(m(a)n(b)g) = µ(det a)| det a|(m+1)/2h(g)

for all a ∈ GLm(F ), g ∈ Spm and b ∈ Symm.

TheWeil representation ωψHm is realized on the Schwartz space S(M2mm (F ))

as in §7.4. Let LGJ(s, µ−1 detGLm) =∏mj=1 L

(s+ m+1

2 − j, µ−1)denote the

Godement-Jacquet L-factor of the one-dimensional representation µ det ofGLm(F ). Following [41], we consider the following integral

Z(ϕ, s, µ) =

∫GL2n(F )

µ(det a)−1ϕ

(a02n

)| det a|s+(2n−1)/2da

for ϕ ∈ S(M4n2n(F )). This integral converges absolutely for ℜs > n+ℜµ− 1

2and is meromorphically continued to the whole C. We can take the limit

Z(ϕ, µ) = lims→0

LGJ(s, µ−1 detGL2n)−1Z(ϕ, s, µ).

A simple calculation gives

Z

(ωψH2n

((a1 b1

ta−11

0 ta−11

),

(a2 b2

ta−12

0 ta−12

))ϕ, µ

)= µ(det a1)| det a1|(2n+1)/2µ(det a2)

−1| det a2|(2n−1)/2Z(ϕ, µ).


We obtain a Sp2n-intertwining, O(H2n)-invariant map

ϑψn : ωψH2n⊗ I2n(µ) → I2n(µ)

by setting

ϑψn(h, ϕ⊗ f) =

∫P2n\O(H2n)

Z(ωψH2n(h, g)ϕ, µ)f(g) dg.

Proposition 11.1. If −12 < ℜµ < 1

2 , then the following statements hold.

(1) I2n(µ) is irreducible.

(2) θψH2n(I2n(µ)) ≃ I2n(µ).

Proof. The first part is included in Theorem 8.1 of [60]. Since the map

Z(µ) : ωψH2n→ I2n(µ)⊠ I2n(µ−1) is nonzero on account of the definition of

the Godement-Jacquet L-factor, the map ϑψn is nonzero, which implies that

θψH2n(I2n(µ)) ≃ I2n(µ)∨ ≃ I2n(µ−1) ≃ I2n(µ).

When ψ is trivial on c−1 but nontrivial on p−1c−1, we put ordψ = c.

Lemma 11.2. (1) If F is p-adic, ϕ0 ∈ S(M4n2n(F )) is the characteris-

tic function of M4n2n(o) and d = ordψ, then ϑψn(ϕ0 ⊗ f

(s)2n ) equals

εs+(2n+1)/2d up to a nonzero scalar multiple.

(2) If F = R and V is a positive definite quadratic space of dimension

m, then the O(V )-invariant subspace S(V l)O(V ) of ωψV is equivalent

to D(l)m/2.

Proof. It is the well-known fact that Z(ϕ0, αs) is a nonzero constant. Ob-

serve that ϕ0 is invariant under the action of the product K2n[d−1, d] ×

K2n of maximal compact subgroups. Thus ϑψn(ϕ0 ⊗ f(s)2n ) is nonzero, right

K2n[d−1, d]-invariant, and necessarily equal to a multiple of ε

s+(2n+1)/2d . It is

well-known that θψWl(1O(V )) ≃ D

(l)m/2, where 1O(V ) denotes the trivial repre-

sentation of O(V ). Since ΘψWl

(1O(V )) ≃ S(V l)O(V ) is irreducible by Section

3 of [17], the second part follows.

11.2. The embedding I2n(µf ) → M(O(Vn)). From non on F is a totallyreal number field of degree d and n is a positive integer such that dn iseven. Let Vn be a 4n-dimensional totally positive definite quadratic spaceover F such that Vn(Fp) ≃ H2n(Fp) for every prime p. Recall that the spaceM(O(Vn)) of algebraic modular forms for O(Vn) consists of locally constantfunctions on O(Vn,Af ) which are left invariant under O(Vn, F ). We nowprove Corollary 1.4.

Proof. We can choose η ∈ F× in such a way that L(12 , π ⊗ χη

)= 0 on

account of Theorem A.2 of [47]. Then Π ≃ D(2n)⊠d2n ⊗ I2n(µf χ

ηf ) occurs in

Acusp(Sp2n) with multiplicity one by Theorem 1.1 and Proposition 8.9.

49

The standard L-function of Π is ζF (s)∏2nj=1 L

(s+ n+ 1

2 − j, π ⊗ χη).

Since the archimedean L-factor of π is holomorphic for ℜs > 12 − n, the

partial function L(s, π ⊗ χη) has no zero for 12 − n < ℜs ≤ −1

2 . Therefore

the standard L-function has a pole at s = 1. Since D(2n)2n is a limit of dis-

crete series, its standard L-factor is holomorphic for ℜs > 0 by Lemma 7.2of [61]. Thus the complete standard L-function has a pole at s = 1 and isholomorphic for ℜs > 1, and so by the functional equation, it has a pole ats = 0 and is holomorphic for ℜs < 0.

We can now apply [61, Theorem 10.1] or [11, Theorem 11.6] to see that

θψVn(Π) is nonzero. The compact group O(Vn,A∞) acts trivially on θψVn(Π)

in view of Lemma 11.2(2). Thus θψVn(Π) occurs inM(O(Vn)). It is equivalent

to I2n(µf χηf ) by Proposition 11.1(2). Using the notation in §2.4, we define the

spinor norm spinv : O(Vn, Fv) → F×v /F

×2v as follows. Given gv ∈ O(Vn, Fv),

take an element βv ∈ G·(Vn(Fv)) so that ϑ(βv) = gv. We denote by spinv(gv)the coset represented by µ1(βv) in F

×v /F

×2v . Define a homomorphism spin :

O(Vn,A) → A×/A×2 by spin(g) = (spinv(gv)) for g = (gv) ∈ O(Vn,A). Thecomposition χη spin is an automorphic character of O(Vn,A). Let T be afinite set of places of F . When the cardinality of T is even, we can define anautomorphic character sgnT : O(Vn,A) → µ2 by sgnT (g) =

∏v∈T det gv.

We can choose T so that sgnT (g) = χη(spin(g)) for g ∈ O(Vn,A∞). Then

1O(Vn,A∞) ⊠ I2n(µf ) ≃ θψVn(Π)⊗ sgnT (χη spin).

by Lemma 4.9 of [50]. Therefore I2n(µf ) appears in M(O(Vn)).Let σ be an irreducible summand of M(O(Vn)) which is equivalent to

I2n(µf χηf ). Since its standard L-function is entire and has no zero at s = 1,

Theorem 2 and Lemma 10.2 of [61] say that θψW2n(σ) is nonzero and cuspidal.

Since θψW2n(σ) is equivalent to Π as an abstract representation, it is equal to

Π by Corollary 8.10. It follows that σ = θψVn(Π) by Lemma 12.2. As such,

the multiplicity of I2n(µf χηf ), which is equal to that of I2n(µf ), is one.

11.3. An identity of Hilbert-Siegel cusp forms. We have defined a

nonzero intertwining map ϑψfn = ⊗pϑ

ψpn : ωψf

Vn⊗ I2n(µf ) → I2n(µf ) in §11.1.

The space S(V ln(A∞))

O(Vn)2n consists of Schwartz functions φ on V l

n(A∞)

which satisfy ωψVn(k, g)φ = J2n(k, il)−1φ for every k ∈ Kl and g ∈ O(Vn,A∞).

Since the lowest K-type occurs in D(2n)2n with multiplicity one, this space is

one-dimensional by Lemma 11.2(2).We are now ready to prove Corollary 1.5.

Proof. As we have seen in the previous subsection, the standard L-functionof I2n(µf ) is entire and has no zero at s = 1 by assumption, and hence its

theta lift to Sp2n is nonzero, cuspidal and equivalent to D(2n)⊠d2n ⊗ I2n(µf ).


Fix a Gaussian 0 = φ ∈ S(V 2nn (A∞))

O(Vn)2n . Then

Θ(ωψVn(h, g)(φ⊗ ϕ)) = θ(h(i2n), ωψfVn(g)ϕ)/J2n(h, i2n)

is nonzero for some h ∈ Sp2n(A∞), g ∈ O(Vn,Af ) and ϕ ∈ S(V 2nn (Af )). The

right hand side of (1.3) is therefore equal to θψW2n(jn(f), φ⊗ϕ). We can define

a nonzero O(Vn,Af )-invariant, Sp2n(Af )-intertwining map ωψfVn

⊗ I2n(µf ) →S(2n)2n by ϕ⊗ f 7→ θψW2n

(jn(f), φ⊗ ϕ). The Howe principle forces it to factor

through the quotient map ϑψfn . The resulting map is proportional to i12n by

uniqueness.

We shall translate this result into a more classical language. Let L be alattice in Vn(F ). For g ∈ O(Vn,A), we write gL for the lattice defined by(gL)p = gpLp, where we denote the closure of L in Vn(Fp) by Lp. We meanby the genus (resp. class) of L the set of all lattices of the form gL withg ∈ O(Vn,A) (resp. g ∈ O(Vn, F )). We denote the genus of L by Υ(L) andthe class of L by [L]. Put

KL = g ∈ O(Vn,A) | gL = L, O(L) = O(Vn, F ) ∩ KL, E(L) = ♯O(L).

Then we can identify the set of classes in the genus of L with double cosets forO(Vn, F )\O(Vn,A)/KL via the map g 7→ [gL]. The KL-invariant subspaceof M(O(Vn)) can be considered to be the C-vector space V (Υ(L)) with basis[L] | L ∈ Υ(L) via the map f 7→

∑[L] f(L)[L].

Define the dual lattice of L by L∗ = x ∈ Vn(F ) | qVn(x, L) ⊂ o. Wecall L integral if L ⊂ L∗, even if qVn(L) ⊂ 2o, and unimodular if L = L∗.The set of even unimodular lattices in Vn forms a genus Υn. The thetaseries associated to L ∈ Υn is a Hilbert-Siegel modular form of degree mand weight 2n defined by

θ(m)L (Z) =

∑x∈Lm

e∞

(1

2tr(qVn(x)Z)

).

We define the theta lift of f ∈ V (Υn) by

(11.1) Θ(m)(Z, f) =∑[L]

f(L)

E(L)θ(m)L (Z).

Corollary 11.3. Let πf ≃ ⊗′pI(α

spp , α

−spp ) be an irreducible summand of C2n.

Assume that dn is even. Let f ∈ V (Υn) be a common eigenfunction of all

Hecke operators whose standard L-function is∏2nj=1 L

(s+ n+ 1

2 − j, π). If

L(12 , π

)= 0, then Θ(2n)(f) is a nonzero constant multiple of Lift2n(π). If

L(12 , π

)= 0, then Θ(2n)(f) is zero.

Proof. Let ϕ(2n)Lp

be the characteristic function of L 2np and f2n a nonzero

KL -invariant element of ⊗′pI2n(α

spp ). Then f equals jn(f2n) up to scaling by

51

C×. Put ϕ(2n)L = ⊗pϕ

(2n)Lp

. The theta lift∫O(Vn,F )\O(Vn,Af )

jn(f2n)(g)θ(Z, ωψVn(g)ϕ

(2n)L ) dg

=∑

g∈O(Vn,F )\O(Vn,A)/KL

jn(f2n)(g)θ(Z, ϕ(2n)gL )

∫O(gL )\KgL

dk

is equal to vol(KL )Θ(2n)(jn(f2n)). If L(12 , π

)= 0, then the standard L-

function of f has a zero at s = 1, and so by Theorem 2 of [61], the theta

lift is zero. Suppose that L(12 , π

)= 0. Then Θ(2n)(f) equals i12n(ϑ

ψfn (ϕ

(2n)L ⊗

f2n)) by Corollary 1.5. Since ϑψfn (ϕ

(2n)L ⊗ f2n) is a nonzero

∏pK2n[d

−1p , dp]-

invariant element of ⊗′pI2n(α

spp ) by Lemma 11.2(1), its image under i12n

coincides with Lift2n(π) by Corollary 10.1.

Remark 11.4. The linear subspace of M12(Sp12(Z)) spanned by the thetaseries associated to the 24 different Niemeier lattices has been extensivelystudied in [6, 44, 22, 24, 7]. It intersects the space of cusp forms in a onedimensional subspace whose basis vector is explicitly given in Theorem 4 of[6]. In Section 15 of [22] Ikeda verified that this sum of theta functions equals− 1

120Lift12(∆), where ∆ ∈ S12(SL2(Z)) is the Ramanujan delta function

and the corresponding cusp form h6 ∈ S+13/2(Γ0(4)) is normalized so that

c(1) = 1. This gives an explicit example of Corollary 1.6 for the Niemeierlattices.

12. Hilbert-Siegel modular forms of degree 4 and weight 4over Q(

√2)

12.1. Kneser neighbors and Hecke operators. For a prime p we denotethe residue field o/p by Fp. Let L ∈ Υn. The Fp-vector space L/pL comesequipped with a nondegenerate quadratic form x 7→ 1

2qVn(x) (mod p). We

say that K ∈ Υn is a pj-neighbor of L if L/(L∩K) ≃ K/(L∩K) ≃ Fjp. Thenumber of pj-neighbors of L in the class of K is denoted by N(L,K, pj). Itis important to note that the matrix N(L,K, pj) represents the action of aHecke operator on algebraic automorphic forms for O(Vn).

Fix L ∈ Υn. Put KLp = g ∈ O(Vn, Fp) | gLp = Lp. The spaceM(O(Vn)) comes equipped with the action of Hecke operators. Given f ∈V (Υn) and y ∈ O(Vn, Fp), decompose the double coset KLpyKLp into adisjoint union of right cosets

⊔j yjKLp and define f |KLpyKLp ∈ V (Υn) by

[f |KLpyKLp ](g) =∑j

f(gyj).

We can take an op-basis e1, . . . , e2n, f1, . . . , f2n of Lp such that

qVn(ei) = qVn(fj) = 0, qVn(ei, fj) = δij (1 ≤ i, j ≤ 2n),


where δij denotes Kronecker’s delta. In this basis we define a homomorphismm : GL2n(Fp) → O(Vn, Fp) by (1.1). Theorem 5.11 of [13] says that

[L]|KLpm(diag[ϖp, ϖp, . . . , ϖp︸︷︷︸j

, 1, 1, . . . , 1])KLp =∑[K]

N(K,L, pj)[K].

We consider the p-neighbor operator on V (Υn)

Kn(p) : [L] 7→∑[K]

N(K,L, p)[K].

Proposition 12.1. If f ∈ V (Υn) is an eigenvector of all Hecke operatorswith p-Satake parameter β±1

p,1 , . . . , β±1p,2n, then

Kn(p)f = q2n−1p f

2n∑i=1

(βp,i + β−1p,i ).

Proof. We have only to show that

f |KLpm(diag[ϖp, 1, 1, . . . , 1])KLp = q2n−1p f

2n∑i=1

(βp,i + β−1p,i ).

Let Tn be the maximal torus of O(Vn, Fp) consisting of diagonal matri-ces. The representation of O(Vn, Fp) generated by f is equivalent to theunique class one component of the unramified principal series representa-tion induced from the character of Tn defined by m(diag[t1, . . . , t2n]) 7→∏2ni=1 β

ordp tip,i . It is seen in Appendix A of [59] that∑

γ

ν[γ]−sf |KLpγKLp =f∏2nj=1(1− q1−2s−2j+4n−1

p )∏2nj=1(1− βp,jq

−s+2n−1p )(1− β−1

p,j q−s+2n−1p )

,

where γ runs over all the representatives for KLp\O(Vn, Fp)/KLp . Here, we

put ν[γ] = [γo4np + o4np : o4np ]. We obtain the declared result by looking at

the coefficient of q−sp . 12.2. Unramified theta correspondence.

Lemma 12.2 (Moeglin [42]). Let σ be an irreducible automorphic repre-

sentation of O(Vn,A) such that θψWm(σ) is nonzero and cuspidal. Then

θψ−1

Vn(θψWm

(σ)) = σ. If l > m, then θψWl(σ) is orthogonal to any cusp forms

on Spl(A).

We define the C-linear map Θ(m) : V (Υn) −→ M2n(Γm[d−1, d]) as in

(11.1). When f is a Hecke eigenfunction, we define its degree by

deg f = minm | Θ(m)(f) = 0.

Proposition 12.3. Let f ∈ V (Υn) be a nonzero Hecke eigenfunction. Putm = deg f .

(1) Θ(m)(f) is a nonzero Hecke eigenform in S2n(Γm[d−1, d]).

53

(2) If l > m, then Θ(l)(f) is nonzero and orthogonal to any cusp formsin S2n(Γl[d

−1, d]).

Proof. Let φ be a nonzero element of S(V mn (A∞))

O(Vn)2n and ϕ

(m)Lp

the charac-

teristic function of Lmp . Then Θ(m)(f) is a constant multiple of the theta lift

θψWm(f, φ⊗ (⊗pϕ

(m)Lp

)), as we have seen in the proof of Corollary 11.3. This

proves (1) as Θ(m)(f) is invariant under Km[d−1p , dp] for every p. Observe

that the

(0 00 ξ

)th Fourier coefficient of Θ(l)(f) is the ξth Fourier coefficient

of Θ(l−1)(f) for ξ ∈ Rl−1. Thus Θ(m)(f) is cuspidal in view of Proposition

A4.5(4) of [48]. Moreover, Θ(l)(f) is nonzero and has no cuspidal componentby Lemma 12.2. Lemma 12.4. Let F ∈ S2n(Γm[d

−1, d]) be a Hecke eigenform whose p-

Satake parameter is denoted by β±11,p , . . . , β

±1m,p. If F ∈ Θ(m)(V (Υn)) and

2n ≥ m, then there exists a Hecke eigenform f ∈ V (Υn) whose p-Satakeparameter is given by

β±11,p , . . . , β

±1m,p ∪ q±jp (0 ≤ j ≤ 2n−m− 1)

and such that F = Θ(m)(f).

Proof. By Proposition 12.3(1) there is a Hecke eigenform f ∈ V (Υn) such

that F = Θ(m)(f). We write Π for the automorphic representation gener-

ated by F . Then f ∈ θψVn(Π) by Lemma 12.2. Therefore θψVn(Π) is every-where unramified and so by Corollary 2.6 of [36] its p-Satake parameter isgiven as above.

12.3. Hilbert modular forms over Q(√2). Let F = Q(

√2). The narrow

class number of F is one. Denote a totally positive generator of an ideal aby ϖa. For example, the different d is generated by ϖd = 4− 2

√2.

For an integral ideal b of o we put σk(b) =∑

a|bN(a)k. We denote by µ

the Mobius function for ideals and by χη the ideal character associated to

χη. Put Fη =∏

p pfηp . We consider the Eisenstein series of weight 2n

G2n(Z) =ζF (1− 2n)

2d+

∑ξ∈o∩F×

+

σ2n−1((ξ))e∞(ξZ)

and the Cohen-Su Eisenstein series (cf. [52]) of weight n+ 12

G(2n+1)/2(Z) = ζF (1−2n)+∑

(−1)nη≡(4)

L(1−n, χ(−1)nη)Cn((−1)nη)e∞(ηZ),

where Cn(η) =∑

b|Fη µ(b)χη(b)N(b)n−1σ2n−1(F

ηb−1).

The ring⊕

k≥0M2k(SL2(Z[√2])) of Hilbert modular forms of even par-

allel weight is a polynomial ring generated by Eisenstein series of weight2, 4 and 6 (see [15]). This ring is isomorphic to

⊕k≥0M2k(Γ1[d

−1, d]).


In particular, the ring⊕

k≥0M2k(Γ1[d−1, d]) is generated by G2, G4 and

G6. We have dimS4(Γ1[d−1, d]) = 1 and dimS6(Γ1[d

−1, d]) = 2. Thespace S4(Γ1[d

−1, d]) is spanned by the normalized Hecke eigenform ϕ4(Z) =∑η∈o∩F×

+a(η)e∞(ηZ) given by

ϕ4(Z) = 44G2(Z)2− 5

6G4(Z) = e∞(Z)−2e∞((2−

√2)Z)−4e∞(2Z)+ · · · .

The Kohnen plus space of weight 52 with respect to Γ1[d

−1, 4d] is spannedby the cusp form ϕ5/2(Z) =

∑η≡ (mod 4) b(η)e∞(ηZ) defined by

ϕ5/2(Z) = 44G2(4Z)ϑ(Z)− 10G5/2(Z) = e∞(Z)− 4e∞(2Z) + · · · ,

where ϑ(Z) =∑

ξ∈o e∞(ξ2Z). Corollary 1.3 gives the Hilbert-Siegel cuspform

Lift4(ϕ4)(Z) =∑ξ∈R+

4

b(ϖdξ)e∞(tr(ξZ))f3/2ξ

∏p

Fp(ξ, βp) ∈ S4(Γ4[d−1, d]),

where βp satisfies βp + β−1p = q

−3/2p a(ϖp). The L-function of ϕ4 is given by

L(s, ϕ4) =∏p

(1− a(ϖp)q−sp + q3−2s

p )−1.

It has a functional equation

Λ(4− s, ϕ4) = Λ(s, ϕ4),

where Λ(s, ϕ4) = 8−sΓC(s)2L(s, ϕ4). By the generalized Kohnen-Zagier for-

mula

L(2, ϕ4) = 0

(see Theorem 12.3 of [16]). Let q = (√2) be the prime ideal above 2. Since

a(ϖq) = −2, the q-Satake parameter βq is given by

23/2(βq + β−1q ) = −2.

On the other hand, S6(Γ1[d−1, d]) is spanned by ϕ+6 , ϕ

−6 , where

ϕ±6 (Z) =−48240G2G4 + 2824320G3

2 − 7G6

1560±

√73(14160G2G4 − 470400G3

2 − 7G6)

1560

=e∞(Z) + (−1±√73)e∞((2−

√2)Z) + · · · .

The q-Satake parameter γ±q of ϕ±6 is given by

25/2(γ±q + (γ±q )−1) = −1±

√73.

55

12.4. 8-dimensional even unimodular lattices over Z[√2]. Totally pos-

itive definite even unimodular lattices of rank 8 over Z[√2] form a genus,

which we have denoted by Υ2. It is known that there are exactly 6 classesin this genus. They are labeled E8, 2∆

′4, ∆8, 2D4, 4∆2 and ∅. The orders

of the automorphism groups are as in the following table (see [18, p. 371]).

TABLE I.

L E8 2∆′4 ∆8 2D4 4∆2 ∅

E(L) 214 ·35 ·52 ·7 217 ·34 215 ·32 ·5·7 214 ·33 218 ·3 214 ·32 ·5·7

The numbers N(L,K, q) are given in the following table (see [18, p. 374]).

TABLE II.

L∖K E8 2∆′4 ∆8 2D4 4∆2 ∅

E8 0 0 135 0 0 02∆′

4 0 18 36 0 81 0∆8 2 35 28 70 0 02D4 0 0 3 96 36 04∆2 0 6 0 64 49 16

∅ 0 0 0 0 105 30

We denote by f1, . . . , f6 the eigenvectors of K2(q). The coefficients of fi(i = 1, . . . , 6) are as follows:

TABLE III.

E8 2∆′4 ∆8 2D4 4∆2 ∅

f1 1 1 1 1 1 1f2 135 36 −30 3 −8 14f3 −14175 −216 840 81 −304 840f4 −135 −36 −58 −3 8 30

f5 5775− 525√73 −88 + 104

√73 560 −81− 13

√73 16 + 16

√73 560

f6 5775 + 525√73 −88− 104

√73 560 −81 + 13

√73 16− 16

√73 560

The eigenvalues µi of K2(q) corresponding to fi are given by

TABLE IV.

f1 f2 f3 f4 f5 f6135 −30 −8 58 33 + 3

√73 33− 3

√73

It is important to note that the eigenvalues are mutually distinct.Given an even integral lattice L in Vn and a half-integral symmetric ma-

trix ξ of size m, we write N(L , ξ) for the number of elements (x1, . . . , xm) ∈Lm such that qVn(xa, xb) = 2ξab for 1 ≤ a, b ≤ m.

Lemma 12.5. If ξ ∈ R+4 satisfies det(2ξ) ∈ o×, then

N(E8, ξ) = N(∆8, ξ) = N(2D4, ξ) = N(4∆2, ξ) = N(∅, ξ) = 0, N(2∆′4, ξ) = 29 · 32.


Proof. It is known that Υ1 has only one class which we denote by∆′4. There-

fore if L is a totally positive definite even unimodular lattice of rank 8 suchthat N(L, ξ) = 0, then L ≃ 2∆′

4.Let ξ′4 denote the Gram matrix of ∆′

4. We may assume that 2ξ = ξ′4.Note that N(2∆′

4, ξ) is equal to the number of pairs of matrices A′, A′′ ∈M4(Z[

√2]) such that ξ′4[A

′] + ξ′4[A′′] = ξ′4. Let (A

′, A′′) be such a pair. Put

e1 =t(1, 0, 0, 0), e2 =

t(0, 1, 0, 0), e3 =t(0, 0, 1, 0), e4 =

t(0, 0, 0, 1),

a′i = A′ei, a′′i = A′′ei, T ′ = i | a′i = 0, T ′′ = i | a′′i = 0.

Since 2 = ξ′4[ei] = ξ′4[a′i] + ξ′4[a

′′i ], we have T ′ ∩ T ′′ = ∅. Therefore if i ∈ T ′

and j ∈ T ′′, then since a′′i = 0 and a′j = 0, we get

teiξ′4ej =

ta′iξ′4a

′j +

ta′′i ξ′4a

′′j = 0,

which is a contradiction. Thus T ′ = ∅ or T ′′ = ∅ and hence A′ = 0 or A′′ = 0.Since the mass of ∆′

4 is 128·32 (see p. 370 of [18]), we have N(2∆′

4, ξ) =

2E(∆′4) = 29 · 32.

Remark 12.6. (1) One can give an analytic proof by employing the Siegelformula (cf. [51]).

(2) The proof gives E(2∆′4) = 2E(∆′

4)2 = 217 · 34.

12.5. The degree of fi.

Proposition 12.7. We have

deg f1 = 0, deg f2 = 4, deg f4 = 1, deg f5 = deg f6 = 2.

Proof. Since Θ(m)(f1) equals the Siegel Eisenstein series for every m bythe Siegel-Weil formula [38], we understand that deg f1 = 0 (cf. Proposi-tion 12.3)). Indeed, the q-Satake parameter of the trivial representation is23, 22, 2, 1, 1, 12 ,

122, 123

and 23(1 +

∑3i=−3 2

i) = 135 is the eigenvalue of f1(see Table IV and Proposition 12.1).

We can appeal Corollary 11.3 to see that Lift4(ϕ4) ∈ Θ(4)(V (Υ2)). Since

23(23/2 + 21/2 + 2−1/2 + 2−3/2)(βq + β−1q ) = −30

agrees with the eigenvalue of f2, Proposition 12.1 and Lemma 12.4 implythat Lift4(ϕ4) and Θ(4)(f2) are equal up to scalar. Hence deg f2 = 4.

Theorem 1 of [35] shows that ϕ4 ∈ Θ(1)(V (Υ2)). Since

23(β2q + β−2q + 22 + 2 + 1 + 1 + 2−1 + 2−2) = 58

agrees with the eigenvalue of f4, Proposition 12.1 and Lemma 12.4 concludethat ϕ4 equals Θ(1)(f4) up to scalar. Hence deg f4 = 1.

We consider the Saito-Kurokawa lifting Lift2(ϕ±6 ) ∈ S4(Γ2[d

−1, d]). We

invoke Theorem 1 of [35] to see that Lift2(ϕ±6 ) ∈ Θ(2)(V (Υ2)). Since

23[(2−1/2 + 21/2)(γ±q + (γ±q )−1) + 2 + 1 + 1 + 2−1] =33± 3

√73

57

agree with the eigenvalues of f5 and f6, Proposition 12.1 and Lemma 12.4 im-ply that Lift2(ϕ

+6 ) (resp. Lift2(ϕ

−6 )) is a multiple of Θ(2)(f5) (resp. Θ

(2)(f6)).Hence deg f5 = deg f6 = 2.

To determine deg f3, we follow the method of Ikeda [24]. We define amultiplication x y and an inner product ( , ) on V (Υ2) by

[L] [K] = [L]δ[L],[K], ([L], [K]) = E(L)−1δ[L],[K].

Remark 12.8. The normalization is different from that of Ikeda [24]. Our [L]corresponds to [L]/E(L) in [24]. In terms of algebraic automorphic forms,these are just the product and inner product of algebraic automorphic forms.

Proposition 12.9. Put

ni = degfi, nj = degfj , Fi = Θ(ni)(fi), Fj = Θ(nj)(fj).

Then

⟨Θ(ni+nj)(fk)|H2ni

×H2nj,Fi ×Fj⟩ =

⟨Fi,Fi⟩⟨Fj ,Fj⟩(fi, fi)(fj , fj)

(fk, fi fj).

In particular, if (fk, fi fj) = 0, then deg fk ≤ degfi + degfj.

Proof. The proof is similar to Lemma 7.1 of [24] and omitted.

Proposition 12.10. deg f3 = 3.

Proof. One can easily verify (f3, f4 f5) = 0 and (f2, f3 f4) = 0. Wecombine Propositions 12.7 and 12.9 to have degf3 ≤ 3 and degf3 ≥ 3. (Seealso Nebe and Venkov [44, Proposition 2.3]).

Corollary 12.11. The Miyawaki lift F(1)ϕ4,ϕ4

is nonzero.

Proof. We have seen that

0 = ⟨Θ(4)(f2)|H21×H2

3, ϕ4 ×Θ(3)(f3)⟩ = ⟨F (1)

ϕ4,ϕ4,Θ(3)(f3)⟩

in the proof of Proposition 12.10.

Remark 12.12. One can show that F(1)ϕ4,ϕ4

is a multiple of Θ(3)(f3) by em-

ploying Theorem 11.6 of [11] and arguing as in the proof of Proposition12.7.

Corollary 12.13. The 6 theta series θ(4)E8

, θ(4)2∆′

4, θ

(4)∆8

, θ(4)2D4

, θ(4)4∆2

and θ(4)∅ are

linearly independent. Every degree 4 cusp form spanned by the 6 thetaseries is a constant multiple of

Lift4(ϕ4) =θ(4)E8

28 · 32 · 5 · 7+

θ(4)2∆′

4

29 · 32−

θ(4)∆8

28 · 3 · 7+

θ(4)2D4

28 · 32−θ(4)4∆2

29 · 3+

θ(4)∅

27 · 32 · 5.


Proof. Since Θ(4)(fi) (1 ≤ i ≤ 6) are nonzero and have different eigenvaluesby Propositions 12.3 and 12.7, they are linearly independent. By Proposi-tions 12.3, 12.7 and 12.10 Θ(4)(f2) is a cusp form but Θ(4)(fi) (i = 2) areorthogonal to cusp forms. We combine (11.1) with Tables I and III to obtain

Θ(4)(f2) =θ(4)E8

214 · 32 · 5 · 7+

θ(4)2∆′

4

215 · 32−

θ(4)∆8

214 · 3 · 7+

θ(4)2D4

214 · 32−θ(4)4∆2

215 · 3+

θ(4)∅

213 · 32 · 5.

Let ξ′4 be the Gram matrix of ∆′4. The

12ξ

′4th Fourier coefficient of Θ(4)(f2) is

N(2∆′4,∆

′4)

215·32 = 26 by Lemma 12.5. Since Lift4(ϕ4) is proportional to Θ(4)(f2)

by Corollary 11.3 and its 12ξ

′4th Fourier coefficient is 1, we conclude that

Lift4(ϕ4) = 26Θ(4)(f2).

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Index

deg f , 52S∞, 21O(V ), 48ordp, 5π, 35πf , 3, 6, 17, 45ℜµ, 13Ji, 30εc(g), 14i, 16fξ, 5fB , 45D, 10, 16D(ξ), 5d = [F : Q] = ♯S∞, 2DB , 45E(L) = ♯O(L), 50F×+ , 2

fξp , 5

fBp , 45

f tp = 12(ordp t− ordp d

t), 45Fv, 16Kn(p), 52L(s, µ), 14N(L , ξ), 8N(L,K, pj), 51qp = ♯o/p, 5R(ξ, f), 8Z(ϕ, µ), 47

adelesA, 2A∞, 2Af , 2

additive characterse of C, 13e∞ of A∞, 2ψ of F\A, 2ψξ of Symm(A), 2ψB of Sn, 13

algebraic modular formsM(O(Vn)), 7V (Υ(L)), 50

automorphy factorsj(g, Z), 16Jℓ(g,Z), 3Jκ(g,Z), 23

charactersα, 2χB , 13

χt, 13λn, 11spinT , 49spinv, 49µf , 3, 17Ω(F×), 13sgn, 10

compact subgroupsKL, 51Km, 21Km, 47KL, 50Kn, 16KDn [c], 14

compatible coefficients

C(m)ℓ (µf ), 23

T(m)ℓ (µf ), 23Cnℓ (µf ), 19Gnℓ (µf ), 19Tnℓ (µf ), 19

degenerate principal seriesIm(µ) of O(Hm), 47Im(µ) of Spm, 47I(µ1, µ2) of GL2, 17

Iψm(µ) of Mp(Wm), 3In(µ) of Gn, 13Jn(µ) of Gn, 13

degenerate Whittaker functionswµξ , 4

wµB , 14Wξ(F), 22

W(ℓ)ξ , 21, 23

WB(F), 17

W(ℓ)B , 16, 18

degenerate Whittaker spacesWhξ(Π), 21, 23WhB(Π), 13, 16

discrete subgroupsΓm[b, c], 5ΓDn [c], 44O(L), 50

embeddings

ιηn of An(µf χ

(−1)nηf

), 24

iηm of Aψfm (µf χ

ηf ), 4

jn of I2n(µf ), 7

Fourier seriesFℓ(h, Cξ), 23LiftD2n(π), 45

61


Lift2n(π), 6Fℓ(f, CB), 18

Fourier-Jacobi maps

βψR(h⊗ ϕ), 30

βψS (f ⊗ ϕ), 28

FRϕ , 36

FSϕ , 31

Γψ(h⊗ Φ), 40

Hilbert cusp formsC2κ, 23S+,n(2k+1)/2, 6

idealsd, 5, 15dB , 45dt, 45eD, 44p, 2q = (

√2), 54

involutions∗, 10ι, 10

irreducible subrepresentationsA(µ1, µ2), 17

Aψm(µ), 3, 23An(µ), 13

lattices[L], 50Υ(L), 50Υn, 50Ξn, 8gL, 50L∗, 50

lowest weight representations

D(m)

(2ℓ+1)/2, 35

D2ℓ, 35

mapss, 20m, 20n, 20ϑ, 12d, 11m, 11m′, 27n, 11n′, 27v, 26m, 2n, 2u, 29

matricesηi, 26, 29σn, 11Hm, 47Jn, 11

maximal ordersO, 13o, 5op, 5

modular formsF∆, 23F∆, 22Cnℓ , 18Gnℓ , 18Tnℓ , 18Acusp(Mp(Wm)), 38A00, 34

F (n)π,g , 46

C(m)ℓ , 22

T(m)ℓ , 22Mℓ(Γ), 3Sℓ(Γ), 3

S(m)ℓ , 3

normsN, 5ν, 10, 11NF/Q, 5

packetsΠ±n (πv), 25

Wdψv , 34parabolic subgroups

Pm, 47Pm, 20Pn, 18P+n , 18

Pn, 11

quadratic spacesHm, 47Vn, 7, 48

reductive groupsGn, 11Mp(Wm), 2G+n , 16G(V ), 12G+(V ), 12G·(V ), 12Gn, 11Spm = Sp(Wm), 2

right regular actions

63

ϱ, 14, 21℘, 14

Schwartz functionsΛ, 42ΦΛ, 43ϕR, 36ϕS , 31φ0, 48φR, 31

Siegel seriesΨp(η,X), 5

Fp(ξ,X), 6

Fp(B,X), 45bp(ξ, s), 6bp(B, s), 15Fp(ξ,X), 6Fp(B,X), 15

skew Hermitian matricesRn, 14

RD,nd2n , 14

RD2n, 14

RD,+2n , 44

Dπf− , 17

D+−, 16

Dnd− , 11

D−, 11Sπfn , 17Sn, 11S+n , 16Sndn , 11

symmetric domainsHm, 2Hn, 16

symmetric matricesSymπf

m , 23Rm, 5R+m, 5

S πm, 37

Sym+m, 2

Symndm , 20

theta functionsθ(ϕ), 7Θ(φ), 31

θ(m)L , 50

theta liftsσψ±, 34

ΘψV (σ), 33

θψV (σ), 33, 34

θψV (h, ϕ), 34

Θ(m)(f), 50

ϑψn(ϕ⊗ f), 48tracesτ , 10, 11TrF/Q, 5

unipotent radicalsUm, 20Ni, 27Ni, 30Nn, 11

unipotent subgroupsXi, 30Yi, 30Zi, 30Xi, 27Yi, 27Zi, 27

Weil constantγ(ψt), 2γψ(t), 2

Weil representationsωψ±, 33

ΩψV , 39

ωψHm, 47

ωψR, 30

ωψS , 27, 31

ωψV , 33, 39


Department of mathematics, Kyoto University, Kitashirakawa Oiwake-cho,Sakyo-ku, Kyoto, 606-8502, Japan

E-mail address: [email protected]

Hakubi Center, Yoshida-Ushinomiya-cho, Sakyo-ku, Kyoto, 606-8501, JapanE-mail address: [email protected]

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