An Integral Representation of Standard Automorphic L Functions for Unitary Groups

Let F be a number field, G the general linear group of degree n defined over F. Let π be an irreducible cuspidal automorphic representation of G(A). In [1–3], a Rankin-Selbergtype integral is constructed to represent the L function of π. That the integrals of Jacquet, Piatetski-Shapiro, and Shalika are Eulerian follows from the uniqueness of Whittaker models and the fact that cuspidal representations of GLn are always generic. For other reductive group whose cuspidal representations are not always generic, in [4], PiatetskiShapiro and Rallis construct a Rankin-Selberg integral for symplectic group G= Sp2n to represent the partial L function of a cuspidal representation π of G(A). In this paper, we apply similar method to the quasi-split unitary group of rank n. Let F be a number field, E a quadratic field extension of F. Let V be a 2n-dimensional vector space over E with an anti-Hermitian form


Introduction
Let F be a number field, G the general linear group of degree n defined over F. Let π be an irreducible cuspidal automorphic representation of G(A).In [1][2][3], a Rankin-Selbergtype integral is constructed to represent the L function of π.That the integrals of Jacquet, Piatetski-Shapiro, and Shalika are Eulerian follows from the uniqueness of Whittaker models and the fact that cuspidal representations of GL n are always generic.For other reductive group whose cuspidal representations are not always generic, in [4], Piatetski-Shapiro and Rallis construct a Rankin-Selberg integral for symplectic group G = Sp 2n to represent the partial L function of a cuspidal representation π of G(A).In this paper, we apply similar method to the quasi-split unitary group of rank n.
Let F be a number field, E a quadratic field extension of F. Let V be a 2n-dimensional vector space over E with an anti-Hermitian form on it.Let G = U(η 2n ) be the unitary group of η 2n .Let π be an irreducible cuspidal automorphic representation of G(A), f a cusp form belonging to the isotypic space of π.The Rankin-Selberg-type integral is defined by f (g)E(g,s)θ(g)dg, ( where E(g,s) is an Eisenstein series associated with a degenerate principle series, θ is a theta series defined by the Weil representation of Sp(V ⊗ W), where W is a nondegenerate Hermitian space of dimension n.We show in Theorem 6.3 that (1.2) represent the standard partial L function L S (s,π,σ) of π.
In [4], after showing the Rankin-Selberg integral has a Euler product decomposition, Piatetski-Shapiro and Rallis continued to show that if n/2 + 1 is a pole of partial L function, then theta lifting is nonvanishing [4, Proposition on page 120].There should be a parallel application of our paper, that is, relate the largest possible pole with nonvanishing of period integral.

Notations and conventions
Let F be a field of characteristic 0, E a commutative F-algebra with rank two.Let ρ be an F-linear automorphism of E. We are interested in (E,ρ) of the following two types: (1) E is a quadratic field extension of F, ρ is the nontrivial element of Gal(E/F); (2) E = F ⊕ F, (x, y) ρ = (y,x).Let tr be the trace of E over F, that is, it is defined by tr(z) = z + z ρ , z ∈ E. ( Let V be a left E-module, ϕ : V × V → E a nonsingular ε-Hermitian form on V , here ε = ±1.The unitary group of ϕ is U(ϕ) = α ∈ GL(V ,E) | ϕ(xα, yα) = ϕ(x, y), ∀x, y ∈ V . (2.2) Let ε = − so that εε = −1.Let (W,ϕ ) be a nonsingular ε -Hermitian space.Put Then W is a nonsingular symplectic space over F with symplectic form Let G = U(ϕ), G = U(ϕ ) be the unitary groups corresponding to ϕ and ϕ , respectively.It is well known that G × G embeds as a dual pair in Sp(φ).
We often express various objects by matrices.For a matrix x with entries in E, put assuming x to be square and invertible if necessary.Assume that V ∼ = E for some nonzero positive integer .Let ϕ 0 be an × matrix satisfying ϕ * 0 = εϕ 0 .We can define an ε-Hermitian form ϕ on V by requiring ϕ(x, y) = xϕ 0 y * . (2.6) Yujun Qin 3 Then the unitary group U(ϕ) is isomorphic to the subgroup of GL (E) consisting elements g satisfying gϕ 0 g * = ϕ 0 . (2.7) In the following we let ε = −1.Then ϕ is a nonsingular skew-Hermitian form, hence = 2n for some positive integer n.Let e 1 ,...,e 2n be a basis of V such that ϕ is represented by Then X, Y are maximal isotropic spaces of V .Let P be the maximal parabolic subgroup of G preserving Y .Then Here is the set of Hermitian matrices of degree n.Let N be the unipotent radical of P. Then N(F) consists of elements of the following type: Then M is a Levi subgroup of P. The F-rational points M(F) of M consists of elements of the following form: (2.16) We will say "the action of M(F) on S(F)" if no confusion is caused.
Let O be the unique open orbit of M(F)\S(F), then (2.17) For β ∈ O, let M β be the stabilizer of β.Since β is a nonsingular Hermitian matrix, is the unitary group of β.

Localization of various objects
Let F be a number field, E a quadratic field extension of F. Let v be the set of all places of F, a, f be the sets of Archimedean and non-Archimedean places, respectively.Then A E be the rings of adeles of F and E, respectively.Let ρ be the generator of Gal(E/F).For belongs to one of the following two cases.
(1) Case NS: For an algebraic group H defined over F, we let H(F v ) be the set of F v -points of H. Put Yujun Qin 5 where the prime indicates restricted product with respect to H(ᏻ v ).Then Let G = U(η n ) be the quasi-split even unitary group of rank n defined over F. We have defined the standard Siegel parabolic subgroup P = MN of G in Section 2. Keep notations of last section.For v ∈ f, the localization of these algebraic groups are as follows.
(1) Case NS: v remains prime in E. In this case, (3.4) (2) Case S: v splits in E. In this case, For g ∈ G(F v ), we have Iwasawa decomposition for some k ∈ K 0,v , n(X)m(a) or n(X)m(A,B) belong to P(F v ).

Local computation
Our result relies heavily on the L function of unitary group in [5] derived by Li.So in this section, we review the doubling method of Gelbart et al. [6] briefly and the main theorem of [5].
Let F be non-Archimedean local field with characteristic 0, ᏻ the valuation ring of F with uniformizer .Let | • | be the normalized absolute value of F. Let (E,ρ) be a couple as in Section 1.If E is a field extension of F, let ᏻ E be the ring of integer of E with uniformizer E , | • | E the normalized absolute value of E.
Let V be 2n-dimensional space over E with skew-Hermitian form ϕ = η 2n , G = U(V ).Then Let −V be the space V with Hermitian form −ϕ. Define Then ϕ ⊕ (−ϕ) is a nonsingular skew-Hermitian form on V. Let H = U(V) be the unitary group of V. Then K = H(ᏻ) is a maximal open compact subgroup of H(F).We embed G × G into H as a closed subgroup.Define two maximal isotropic subspaces of V as follows: Then V = X ⊕ Y .Let Q be the maximal parabolic subgroup of H preserving Y .Following [5], we define a rational character x of Q by Choose a basis of V compatible with the decomposition (4.3), we can write p as a matrix: H(F) acts on I(s,γ) by right multiplication.Let I(s,γ) K be the subspace of K-invariant elements of I(s,γ).Since γ is unramified, by Frobenius reciprocity, Let Φ K,s be the unique K-invariant function in I(s,γ) such that One important property of Φ K,s is the following.
Let r be the natural action of GL 2n (C) on C 2n , σ the induced representation where q is the cardinality of residue field of F.
The relation between the functions Φ K,s , ω π , and L(s,π,σ) is as follows.
Theorem 4.2 (see [5,Theorem 3.1]).Notations as above.For s ∈ C, .17) We will derive a formula from (4.17) which is applicable for our computation later.For this purpose, for g ∈ G(F), let such that g ∈ K 0 m(δ(g))K 0 (Case NS) or g ∈ K 0 δ(g)K 0 (Case S).Define a function Δ(g) on G(F) by Furthermore, reasoning as in [5, page 197], one can show that Hence Theorem 4.2 is equivalent to the following.

.23)
Here d H (s) is the meromorphic functions in Theorem 4.2.
Yujun Qin 9 Before we end this section, we record a formula for the value on Δ(g) for some special elements in G(F).For β ∈ M n×n (F), let L(β) be the set of all minors of β.Lemma 4.4 (see [8,Proposition 3.9]).( 1) (Case NS) Let (4.27)

Fourier coefficients
In this section, we will compute Fourier coefficients of Δ(g).Our method is similar to that of [4].Notations are as in the last section.Let ψ be a nontrivial additive character of F. Let (π,V 0 ) be an unramified irreducible admissible representation of G(F), T a square matrix such that T ∈ S(F)(Case NS) or T ∈ M n×n (F)(Case S).Let l T be a linear functional on V 0 satisfying Example 5.1.Let F be a number field, π an irreducible cuspidal automorphic representation of G(A) for a moment [9].Then where the integral is taken on , so l Tv is a linear functional on π v satisfying (5.1).
Back to the assumption that F is non-Archimedean local field, (π,V 0 ) is an unramified irreducible representation of G(F).Define a subset M(ᏻ) of M 2n (E)(Case NS) or of M 2n (F)(Case S) as follows: ( (5.4) Lemma 5.2.Let ψ be an unramified additive character of F. Let T be a square matrix such that

.5)
Proof.As in [3], the convergence of left-hand side of the equation when Re s is sufficiently large comes from the vanishing of l T (π(a) f 0 ) when a is sufficiently large, here a belongs to the maximal F-torus consisting of diagonal elements in G(F).
Since both sides are meromorphic functions of s, we only need to show the equation for Res sufficiently large.We first claim that K0 l T π(kg) f 0 dk = l T f 0 ω π (g), g ∈ G(F). (5.6) In fact, the left-hand side is a bi-K 0 -invariant matrix coefficient of π, so there is some (5.7) Let g = 1, then λ = l T ( f 0 ).Back to the proof of the lemma.If Re s is sufficiently large, the left-hand side of (5.5) converges absolutely.Hence L.H.S of (5.5 Yujun Qin 11 we have computed the inside integral in (5.6), so , by Theorem 4.3. (5.9) Apply Iwasawa decomposition (3.6) g = n(X)m(a)k in the integrand of (5.5).When Re s is sufficiently large, −1 dn(X)dm(a)dk. (5.10) Here δ P (m(a)) is the modular function of P(F), hence δ P (m(a) −1 dn(X)dm(a). ( If we let Δ −(s+n) n(X)m(a) ψ tr(XT) dn(X), (5.12) for m(a) ∈ M(F), then (5.13) Properties of J T (s,a), such as convergent when s sufficiently large, having meromorphic continuation to C, is discussed by Shimura [10], for example, Proposition 3.3 there.
Lemma 5.3.Let ψ be an unramified character of F. Let T be a square matrix such that (5.15) Proof.Both sides of (5.14) are meromorphic functions for a given m(a) ∈ M(F).We only need to prove this lemma for Res sufficiently large.
(Case NS).Let a ∈ GL n (E).By the principle of elementary divisors, and (5.17) Let S(ᏻ) be the set of elements in S(F) with entries in ᏻ E .Let Ᏽ be a set of representative of S(F)/S(ᏻ).Decompose the integral in (5.17) as a sum of integrals indexed by Ᏽ: (5.17 for all X ∈ S(ᏻ), since max (5.21) Hence for all ξ ∈ S(F), X ∈ S(ᏻ), (5.22) Apply (5.22) to (5.18), we then get ψ tr Xw −ρ T t w −1 dX.
The proof for Case S is similar, and we omit it here.
Theorem 5.4.Let ψ be an unramified character of F, (π,V 0 ) an unramified irreducible admissible representation of G(F).Let T be a square matrix such that T ∈ GL n (ᏻ E ) ∩ S(F)(Case NS) or T ∈ GL n (ᏻ)(Case S).Let l T be a linear functional on V 0 satisfying (5.1).Then for where d H (s) and j T (s) are given in Theorem 4.2 and Lemma 5.3.
Proof.Lemma 5.2 and the paragraph after Lemma 5.2 have shown that (5.28)By Lemma 5.3, J T (s,a) vanishes when a ∈ M(ᏻ).Substitute the formula of J T (s,a) for a ∈ M(ᏻ) and δ −1 P , the conclusion follows.

Global computation
Let F be a number field, E a quadratic field extension of F. As usual, let v be the set of all places of F, a,f the set of archimedean and non-archimedean places of F respectively.Let F v be the localization of F at the place v of v, Let V be a 2n-dimensional vector space over E with an anti-Hermitian form η 2n on it.Let W be an n-dimensional vector space over E with a nonsingular Hermitian form T. Let G = U(η 2n ), G = U(T) be the corresponding unitary groups.Then G × G is a dual pair in Sp(W), where W = V ⊗ W is symplectic space with symplectic form tr E/F (η 2n ⊗ T).
Let P = MN be the maximal parabolic subgroup of G defined in Section 2. For v ∈ v, let K v be a maximal compact subgroup of G(F v ) such that for almost all v ∈ v,

is a left Haar measure on P(A). Since P(A) = M(A)N(A), d l p
, where d × a, dX are Haar measure on GL n (A E ), S(A), respectively.We then let dg = d l p dk be an Haar measure on G(A).
Let s ∈ C, let γ be a Hecke character of E. Denote by I(s,γ) the set of smooth functions G(A) acts on I(s,γ) by right multiplication.Let Φ(g,s) be a smooth function in I(s,γ) holomorphic at s.The Eisenstein series associated to Φ(g,s) is given by Φ(ξg,s). (6.1) In [9], it has been shown that (6.1) is convergent when Res > n/2 and has a meromorphic continuation to the whole complex plane.

Yujun Qin 15
Let π be a cusp automorphic representation of G(A) (cf.[9]).Let f be cusp form in the isotypic space of π.Let β ∈ S(F).The βth Fourier coefficient of f is Let χ be a Hecke character of E satisfying χ| , where E/F is the quadratic character of A × /F × by global class field theory.Associate with ψ a Weil representation ω ψ of G(A) acting on (Y(A)), the set of Schwartz-Bruhat functions on Y(A).In fact, ω ψ is the restriction of Weil representation (associated with ψ) of Sp(W)(A) to G(A) (see Section 2 for the definition of Y, W).We will omit the subscript ψ when ψ is clear from the context.The explicit formula of ω is given in [11], we cite here the formula on P(A). Here The theta series θ φ for φ ∈ (Y(A)) is a smooth function on G(A) of moderate growth ω(g)φ(ξ), g ∈ G(A).(6.5)

Vanishing lemma.
Let π be a cuspidal automorphic representation of G(A).We make the following assumption: There is some cusp form f in the isotypic space of π such that f n(X)g ψ tr(XT) = 0. ( In [4], Piatetski-Shapiro and Rallis do not propose this assumption, because Li has shown in [12] that every cusp forms supports some nonsingular symmetric matrix. For φ ∈ (Y(A)), Φ(g,s) ∈ I(s,γ), f ∈ A(G(F) \ G(A)) π the isotypic space of π in the space of automorphic forms on G(A), define Although θ φ is slowly increasing function on G(A), E(g,s,Φ) is of moderate growth, but f is rapidly decreasing on G(A), (6.7) is convergent at s where the Eisenstein series is holomorphic.We will show that when we choose appropriate φ, Φ, f , I(s,φ,Φ, f ) is product of meromorphic function with partial L function of π.
Substitute Eisenstein series (6.1), theta series (6.5) into (6.7),then (6.7 By the assumption that Φ(g,s Apply the formula of Weil representation (6.4) to (6.8), then Recall that in Section 2, we let Ꮿ ⊂ S(F) be the image of moment map, which is invariant under the action of M(F).Let be a set of representatives of orbits Ꮿ/M(F) such that T ∈ .We then write (6.9) as a sum of integrals indexed by : (6.9) Here f β is βth Fourier coefficient of f , M β is the stabilizer of β under the action of M (cf.Section 2).For β ∈ , let where T is a nondegenerate r × r Hermitian matrix.So without loss of generality, we assume that β = diag(0 n−r ,T ).Then (6.18) Then M β = M 1 • L. We use this decomposition to compute the inner integral over M β (F)\ M β (A) of (6.15), (Here because Φ(k,s) is independent of m 1 so we remove it from the integral over M β (F)\ M(A).)The above integral equals to f n(X) m 1 mk ψ tr(Xβ) Let U be the subgroup of N consisting of elements of the following form: Then LU is the unipotent radical of the maximal parabolic group P preserving the flag 0 ⊂ ⊗ n−r i=1 Ee n+i ⊂ Y (see Section 2 for the choice of basis of V ).On the other hand, let Δ + be the set of positive roots of G with respect to the Borel subgroup of G consisting of element of following form:

A B
A with A be upper triangular matrix.( For α ∈ Δ + , let N α be the 1-parameter unipotent subgroup of G corresponding to α. Set Γ = {α ∈ Δ + | N α ⊂ N}.Let α 0 be the simple root corresponding to P , w = s α0 be the simple reflection of α 0 .Then U = β∈Γ,wβ∈Γ N β .If we put U 1 = β∈Γ,wβ∈−Γ N β , then N = U • U 1 .Hence we have decomposition Corresponding to the decomposition of N, we have a decomposition of S(F): Then the isomorphism n : S(F) → N send S U and S U1 onto U and U 1 , respectively.Substitute the decomposition of S(F) into (6.20),then (6.25) Direct computation shows that L centralizes U 1 .We can change the order of the above integration, then Let a 1 ,...,a n be the column vectors of ξ.Recall that the right lower corner of ξ is an r × r nonsingular matrix T , the space generated by a n−r+1 ,...,a n is of rank r.Hence there is a ∈ M β (depends on ξ, but it does not affect our computation) such that for some nonsingular r × r matrix u.
(6.32)By (6.28), (6.29), and (6.31), (6.32 which is 0, since LU is the unipotent radical of P .This finishes the proof of the lemma. By Lemma 6.1, I β (s) = 0 if β is singular.Recall that we choose T to be the representative of the open orbit of Ꮿ/M.The stabilizer M T is isomorphic to G = U(T) the unitary group of W. Then (6.12) reduces to Let f be a cusp form in the isotypic space of a cuspidal automorphic representation of G(A).Let S be a finite subset of v containing all archimedean places such that if Let Ω be a finite subset of v containing S. Put Proof.We will apply results in Section 5, F v will be F there, (6.39) Φ v is the standard section, then Φ v (k v ,s) = 1 for all k v ∈ K v .Moreover, .14) Define an action of GL n (E) on S(F) by (a,b) −→ aba * , with a ∈ GL n (E), b ∈ S(F).(2.15)It is equivalent to the adjoint action of M on N, since m(a)n(b)m(a) −1 = n(aba * ).

( 4 .
18)ξ(s) is the zeta function of F, E/F is the character of order 2 associated to the extension E/F by local class field theory, L(s,χ) is the local Hecke L function for a character χ of F × .

( 6 .
40)Asφ v = char(Y(ᏻ v )), M v ∩ Y(ᏻ) = M(ᏻ v ) (cf.Section 5), M(Fv) f T m a v m a Ω k Ω φ a v γ s 0 a v (γχ) deta v d × a v = M(ᏻv) f T m a v m a Ω k Ω γ s 0 a v (γχ) deta v d × a v = L s + 1/2,π v ,γ v χ v ,σ j Tv (s)d Hv (s) f T m a Ω k Ω ,by Theorem 5.4.(6.41)Here we are viewing f T (m(a v )m(a Ω )k Ω ) as a functional l Tv on π v by Example 5.1 in Section 5. Hence the ring of integers of F v .If v remains prime in E, then E v is a quadratic field extension of F v , let ᏻ Ev be the ring of integer of E v .The ring of adeles of F (resp., E) is denoted by A (resp., A E ).Denote by | • | (resp., | • | E ) the normalized absolute value of A × (resp.,A× E ).Let ψ be a nontrivial continuous character of A trivial on F.
which is K v fixed element for the Weil representation, hence ω(k v )φ v = φ v , (6.39) = π v ,γ v χ v ,σ j Tv (s)d Hv (s) Theorem 6.3.Choose f ,φ,Φ and S ⊂ v as in Section 6.1.Then for all s ∈ C, = I S (s) is a meromorphic function of s.Proof.Argue as [6, Theorem 6.1], the partial L function is a meromorphic function.Also by the analytic property of Eisenstein series, I(s,φ,Φ, f ) itself is a meromorphic function, hence R(s) = I S (s) is a meromorphic function of s.Remark 6.4.We remark here that following[4, pages 118-119], under our assumption one can show that by choosing appropriate φ, Φ, f , we can let that R(s) = 0.
v , γ v χ v ,σ .(6.44) Since I(s) = lim Ω I Ω (s), by Theorem 6.2, let Ω be a finite set of v approaching to v by adding one place each time, then the following holds.