Stability of the local gamma factor in the unitary case

Journal of Number Theory 128 (2008) 1358–1375 www.elsevier.com/locate/jnt

Stability of the local gamma factor in the unitary case Eliot Brenner ∗ Center for the Advanced Study of Mathematics at Ben Gurion University, Be’er Sheva 84105, Israel Received 3 August 2006; revised 12 February 2007 Available online 4 March 2008 Communicated by S. Rallis

Abstract In [Stephen Rallis, David Soudry, Stability of the local gamma factor arising from the doubling method, Math. Ann. 333 (2) (2005) 291–313, MR2195117 (2006m:22026)], Rallis and Soudry prove the stability under twists by highly ramified characters of the local gamma factor arising from the doubling method, in the case of a symplectic group or orthogonal group G over a local non-archimedean field F of characteristic zero, and a representation π of G, which is not necessarily generic. This paper extends their arguments to show the stability in the case when G is a unitary group over a quadratic extension E of F , thereby completing the proof of the stability for classical groups. This stability property is important in Cogdell, Kim, Piatetski-Shapiro, and Shahidi’s use of the converse theorem to prove the existence of a weak lift from automorphic, cuspidal, generic representations of G(A) to automorphic representations of GLn (A) for appropriate n, to which references are given in [Stephen Rallis, David Soudry, Stability of the local gamma factor arising from the doubling method, Math. Ann. 333 (2) (2005) 291–313, MR2195117 (2006m:22026)]. © 2008 Elsevier Inc. All rights reserved. MSC: 22E50 Keywords: Doubling method; Gamma factor; L-functions; Unitary groups

1. Introduction Let G be either a symplectic or orthogonal group over a local non-archimedean field F of characteristic zero, or a unitary group over a quadratic extension E of F . We consider the local * Current address: University of Minnesota School of Mathematics, Minneapolis, MN 55455, USA.

E-mail address: [email protected]. URL: http://www.math.umn.edu/~brenn240/. 0022-314X/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2007.10.015

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gamma factor, associated to an irreducible admissible representation π of G, by the doubling method of Piatetski-Shapiro and Rallis [2,4]. Denote the local gamma factor by γ (π, χ, s, ψ), where χ is a character of F ∗ , and ψ is a fixed non-trivial character F . In this paper we treat the last open case of the following result (cf. Theorem 1 in [5]). Theorem 1. The local gamma factor γ (π, χ, s, ψ) is stable, for χ sufficiently ramified. This means that for two given irreducible admissible representations π1 , π2 of G, there exists an integer N > 0, such that γ (π1 , χ, s, ψ) = γ (π2 , χ, s, ψ), for all characters χ , with conductor having an exponent larger than N . Rallis and Soudry, in [5], prove Theorem 1 in the symplectic and orthogonal cases, and this paper completes the proof of Theorem 1 by extending their arguments to the unitary case. The stability property of the local gamma factor, under highly ramified twists, is well known for GLn × GLm . It was proved by Jacquet and Shalika in [3]. For generic representation of split classical groups, it is known thanks to the works of Cogdell, Piatetski-Shapiro and Shahidi. The stability property is a key ingredient in the proof, by the converse theorem, of the existence of a weak lift from automorphic, cuspidal, generic representations of G(A) (G a split classical group) to automorphic representations of GLn (A) (appropriate n), where A is the adele ring of a given number field. See [5] for precise references to the literature. In Theorem 1, π is any irreducible representation of G; even when G is quasi-split, π is not necessarily generic. The proof of the stability in this paper follows the argument of [5] closely. In this paper, we freely refer to [5], instead of repeating arguments that are unchanged in the unitary case, restricting ourselves mainly to pointing out the points of difference between the unitary and the other classical cases. The reader who is familiar with [5] is advised to turn to Proposition 15 first, since this elementary, but apparently new, observation is the heart of the matter concerning the extension of the arguments of [5] to the unitary case. Recall that in the local theory of the doubling method, we consider the integrals  Z(v1 , vˆ2 , fχ,s ) =



   π(g)v1 , vˆ2 fχ,s i(g, 1) dg.

(1)

G

π (affording the Here v1 lies in Vπ —a space for π , and vˆ2 lies in the smooth dual of Vπ , V contragredient representation πˆ ). Thus, g → π(g)v1 , vˆ2  is a matrix coefficient of π ; fχ,s is a holomorphic section in an induced representation of the split “doubled” group H —induced from the Siegel parabolic subgroup P of H , and a character, which is of the form χ(det ·)| det ·|s−1/2 . Finally, there is an embedding i : G × G → H , such that P · i(G × G) = P · i(G × 1) is an open and dense subset in H . The integrals (1) converge absolutely in a right-half plane and continue meromorphically to the whole plane, being rational functions in q −s , where q is the number of elements in the residue field of F . The functions Z(v1 , vˆ2 , fχ,s ) satisfy a functional equation   Γ (π, χ, s)Z(v1 , vˆ2 , fχ,s ) = Z v1 , vˆ2 , M(χ, s)fχ,s ,

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where M(χ, s) is the intertwining operator associated to the element w = i(1, −1). The proportionality factor Γ (π, χ, s) is a rational function of q −s which depends only on π and χ . Note that Γ (π, χ, s) is independent of ψ . The local gamma factor γ (π, χ, s, ψ) is obtained from ωπ (−1)Γ (π, χ, s) (where ωπ is the central character of π ) by multiplication by a factor which depends on χ , ψ (and G) and not on π . See [5, pp. 292–293], [4, pp. 329, 337] for the details. Therefore, Theorem 1 will follow from Theorem 2. Let π be an irreducible admissible representation of G. Then ωπ (−1)Γ (π, χ, s) is stable, for sufficiently ramified χ . More precisely, there is a positive integer N , such that for all ramified characters χ of F ∗ , with conductor having exponent larger than N , we have   ωπ (−1)Γ (π, χ, s) = M(χ, s)fχ,s i(−1, 1) for certain choice of fχ,s . A more precise form of this stability is given in Theorem 3. 2. Notation and preliminaries As far as possible, we keep the notation consistent with [5]. Let E be any local nonarchimedean field, of characteristic zero. We denote by OE its ring of integers, and by PE the prime ideal of OE . We assume that the residue field OE /PE has qE elements. We denote by | · |E the absolute value E, such that |E | = qE−1 , for any generator E of PE . Now let E be a local non-archimedean field, of characteristic zero with an involution θ . In certain situations it will be more convenient to denote θ by conjugation, so, as a matter of notation, we set e¯ = θ (e),

for all e ∈ E.

Let F be the fixed field of θ . Since F is again a local non-archimedean field of characteristic zero, all of the above notation again applies to F . Further, we may write E = F ⊕ F ω, as a vector space, where ω ∈ E − F, ω2 = a ∈ F − {0},

(2)

θ (ω) := ω¯ = −ω.

(3)

and where,

Since ω ∈ E − F , by assumption, the relation (3) completely determines the involution θ of E. Let V be a pair (V , b) consisting of an m-dimensional vector space V over E and a sesquilinear form b on V such that   θ b(v, u) = b(u, v) for all u, v ∈ V . We will also assume that b is non-degenerate. Unless otherwise mentioned we will always denote by G the group of isometries Isom(V) of V, considered as an algebraic group over F . A concrete way of doing this is via the “restriction of scalars” construction. That is we consider V to be a 2m-dimensional vector space over F ,

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and then considering G to be the F -linear transformations of V satisfying an additional set of conditions corresponding to E-linearity and (E, T )-unitarity with respect to θ . This point of view will be developed in greater detail when we need it, in the proof of Lemma 6 below. It will be convenient to fix a basis B of V as follows. We fix an orthogonal E-basis of V, B = {v1 , . . . , vm }, such that     b(v1 , v1 ) = · · · = b(vk , vk ) = q,     b(vk+1 , vk+1 ) = · · · = b(vm , vm ) = 1.

(4)

The choice of an orthogonal E-basis for V satisfying (4) is possible by Théorème IX.6.1.1 of Bourbaki Algèbre [1]. For x ∈ Matn (E), let x ∗ = t θ (x), where the superscripted t on the left indicates the usual transpose of the matrix, and θ (x) denotes the “conjugation” operation θ applied entry-wise to x. We set   T = diag b(v1 , v1 ), . . . , b(vm , vm ) .

(5)

Note that T =T∗

∗  and T −1 = T −1 ,

(6)

since T is diagonal with entries in F . Using the basis B, we write G = Um (F ) as a matrix group. We have an isomorphism of G with the group 

Um (T ) = g ∈ GLm (E)  g ∗ T g = T ≡ G.

(7)

We will write the Lie algebra g of G in the matrix form 

g∼ = um (F ) := x ∈ Mm (E)  x ∗ T + T x = 0 .

(8)

All representations π of G, considered here, are assumed to be admissible. We denote by Vπ a vector space realization of π , and, if it has a central character, we denote it by ωπ . Note that the center Z(G) of G is isomorphic to U1 (F ), the elements of norm 1 in E, and Z(G)’s isomorphic image in Um (T ) is U1 (F )Im . 3. The doubling method Consider V × V, consisting of the doubled space V × V equipped with the bilinear form b∗ = b ⊕ (−b). Denote by H the isometry group of (V × V , b∗ ). Since the subspace 

V = (v, v)  v ∈ V

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is an m-dimensional isotropic subspace of V × V , hence a maximal totally isotropic subspace V × V , the group H is quasi-split. The elements (g1 , g2 ) of G × G act on V × V by   (g1 , g2 )(v1 , v2 ) = g1 (v1 ), g2 (v2 ) , and they clearly preserve b∗ . Thus we get a natural embedding i : G × G → H . Consider the maximal parabolic subgroup PV of H which preserves V . This is a Siegel type parabolic subgroup of H . Its Levi part is isomorphic to GL(V ) ∼ = GL(V ). Denote the unipotent radical of PV by UV . We have the “transversality.”   (9) i(G × G) ∩ PV = i G , where 

G = (g, g)  g ∈ G . Recall that PV \ H / i(G × G) is finite and contains only one open orbit, which is PV · i(G × G) = PV · (G × 1). This equality follows from (9), as in [2, p. 8]. Denote by det(·) the algebraic character of PV given by P → det(P |V ). Let χ be a (unitary) character of E ∗ . Consider, for s ∈ C, s−1/2 . ρχ,s = IndH P (χ ◦ det ·)| det ·| V

The induction is normalized as in §3 of [5]. Let π be an irreducible representation of G, acting in a space Vπ . Consider the contragredient π , the smooth dual of Vπ . Denote by ·,· the canonical G-invariant representation πˆ acting in V  π , and let fχ,s ∈ Vρχ,s be a holomorphic bilinear form on Vπ × Vπ . Let v1 ∈ Vπ and vˆ2 ∈ V section. The local zeta integrals attached to π by the doubling method are      (10) Z(v1 , vˆ2 , fχ,s ) = π(g)v1 , vˆ2 fχ,s i(g, 1) dg. G

By Theorem 3 in [4], the integral in (10) converges absolutely in a right-half plane and continues to a meromorphic function in the whole plane. This function is rational in q −s . We keep denoting the analytic continuation by Z(v1 , vˆ2 , fχ,s ). Consider the intertwining operator M(χ, s) = M(s) : ρχ,s → ρθ(χ)−1 ,1−s , defined, first for Re(s) 0, as an absolutely convergent integral  fχ,s (wuh) du, M(χ, s)fχ,s (h) =

(11)

UV

and then, by meromorphic continuation, to the whole plane. Here, we take, as in [4], w = i(1, −1) ∈ i(G × G). Note that 

  w V = V − = (v, −v)  v ∈ V .

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The subspace V − is a maximal isotropic subspace of V × V which is transversal to V , i.e., V ∩ V − = {0}. By Theorem 3 in [4], we have a functional equation (as an identity of meromorphic functions in the whole plane)   Γ (π, χ, s)Z(v1 , vˆ2 , fχ,s ) = Z v1 , vˆ2 , M(χ, s)fχ,s

(12)

π , fχ,s ∈ Vρ,s (holomorphic section). The function Γ (π, χ, s) depends on for all v1 ∈ Vπ , vˆ2 ∈ V the choice of measure du made in the definition of M(χ, s). Section 9 of [4] explains how to obtain the local gamma factor γ (π, χ, s, ψ) from Γ (π, χ, s). In the hermitian case, the relation between the two is given in (25), [4] as γ (π, χ, s, ψ) =

ξG (χ, A) ωπ (−1)Γ (π, χ, s), CH (χ, s, A, ψ)

(13)

where CH (χ, s, A, ψ) is a certain rational function of q −s , which depends only on χ , ψ , a certain matrix A, and H —see §5 of [4] for the exact definition—and where ξG (χ, A) = χ −1 (det A)| det A|s−1/2 . It is shown in [4, §§8–9], that γ (π, χ, s, ψ) is independent of the choice of A. Since by (13), γ (π, χ, s, ψ) is obtained from ωπ (−1)Γ (π, χ, s) by multiplication by a factor which depends only on χ , ψ (and G), and not on π , Theorem 1 will follow from the explicit formula for ωπ (−1)Γ (π, χ, s) in Theorem 3, valid for χ sufficiently ramified, which evidently does not depend on π . Theorem 3. Let π be an irreducible representation of G. Then there exists a positive integer N , N such that for any ramified character χ of E × , having conductor 1 + PE χ with Nχ > N , we have 

2 −m (−1) × ωπ (−1)Γ (π, χ, s) = |2|m F χ

  χ −1 det(Im − v) dμ(v),

nχ

g(PE )

where Nχ + 1 nχ := . 2 

The measure dμ(v) is to be specified below, in (26). We end this section with an explicit description of i(g, 1), g ∈ G, as a matrix, following the decomposition V × V = V ⊕ V −

(14)

and a choice of a standard basis of V × V whose Gram matrix, with respect to b∗ is w2m . Here, we are using the notation

1

wn =

... 1

 ,

for n ∈ N,

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as in [5]. Let T be the diagonal matrix representing b, as defined in (5). In order to obtain a standard basis of V × V , in the sense of a basis consistent with the decomposition (14) whose Gram matrix (with respect to b∗ ) is w2m , we proceed as follows. We let ui = b(vi1,vi ) vi for i = 1, . . . , m. Then {u1 , . . . , um } is a basis of V , dual to B (with respect to b). Then

B = (v1 , v1 ), . . . , (vm , vm ), 12 (um , −um ), . . . 12 (u1 , −u1 )

(15)

is a standard basis of V × V . Writing the elements of G as matrices, with respect to B, and the  it is now easy to verify Lemma 4. elements of H as matrices, with respect to B, Lemma 4. We have, for all g ∈ G,  i(g, 1) =

1 2 (g

+ Im )

1 4 (g

wm T (g − Im )

− Im )T −1 wm

1 2 wm T (g



+ Im )T −1 wm

(16)

and  w = i(1, −1) =

2wm T

 1 −1 wm 2T .

(17)

4. Proof of Theorem 3 Choose fχ,s so that it is supported in the open orbit PV · i(G × G) = PV · i(G × 1), and so that the restriction fχ,s |i(G×1) , thought of as a function of G, is the characteristic function φU of a small compact open subgroup U of G. We assume that U is small enough, so that VπU = 0. π , such that v1 , vˆ2  = 1. Then the integral (10) converges Let 0 = v1 ∈ VπU , and choose vˆ2 ∈ V for all s, and is easily seen to be    Z(v1 , vˆ2 , fχ,s ) = π(u)v1 , vˆ2 du = m(U )v1 , vˆ2  = m(U ), (18) U

where m(U ) is the measure of U , and so, from the functional equation (12),      1 Γ (π, χ, s) = π(g)v1 , vˆ2 M(χ, s)fχ,s i(g, 1) dg, m(U )

(19)

G

for Re(s)  0. Our next task is to compute M(χ, s)fχ,s (i(g, 1)) for our choice of fχ,s . From now till after the completion of the proof of Lemma 9, we assume that Re(s) 0, so that the expression for M(χ, s)fχ,s given in (11) is valid. Lemma 5. For the above choice of fχ,s , we have, for Re(s) 0,   m(1−2s) −m M(χ, s)fχ,s i(g, 1) = |2|F χ (−2)  s−m/2−1/2   × χ det(Im + h) det(Im + h)E φU (−hg) dh. G

(20)

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Proof. Let Re(s) 0, so that the integral in (11) converges absolutely. We choose dx to be the standard measure of matrices which are “skew-hermitian with respect to the second diagonal,” i.e., dx =



dxij

i+j
m 

(ω)

dxi,m+1−i

(21)

i=1 (ω)

where dxij is the Haar measure of E which assigns the measure 1 to OE , and dxi,m+1−i is the F -invariant measure of F ω which assigns the measure 1 to OF ω . From this point the proof is the same as the proof of Lemma 4.1 in [5], except in place of Lemma 4.2 in [5], we use Lemma 29. Instead of repeating all the arguments, we confine ourselves to pointing out the minor differences compared with the proof of Lemma 4.1 in [5] and in the proof of Lemma 29 compared with the proof of Lemma 4.2 in [5]. Consider the Cayley transform c : glm (E) → GLm (E), given by c(y) =

Im + y , Im − y

where 

glm (E) = y ∈ g  det(Im + y)(Im − y) = 0 , 

GLm (E) = t ∈ G  det(t + Im ) = 0 .

(22)

Now observe that c is a bijection from glm (E) to its image GLm (E), with inverse c−1 (t) =

t − Im . t + Im

The restriction of c from glm (E) to g := um (T ) ∩ glm (E) is a bijection onto the image Um (T ) := Um (T ) ∩ GLm (E) . Note that in the case of the restriction to the Lie algebra of the unitary group, we can drop explicit mention of the requirement that det(Im + y) = 0, since det(Im + y) is merely the conjugate det(Im − y), and the requirement that the latter is nonzero implies that the former is as well. As in (4.8) of [5], we make the change of variables, h = −c(2xwm T ).

(23)

It is easy to show that (wm x)∗ = −(wm x)

if and only if 2xwm T ∈ g.

Here the Lie algebra g = um (F ) is written in matrix form as in (8). As in (4.13) of [5], we get for Re(s) 0,

(24)

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  2 m M(χ, s)fχ,s i(g, 1) = |2|−m F χ(−1)  ×

  χ −1 det(Im − xwm T )

(wm x)∗ =−(wm x)∗ det(Im −xwm T )=0

−s−m/2+1/2    φU −c(xwm T )g dx. × det(Im − xwm T )E

(25)

Letting y = xwm T , we see by (24) that the domain of integration in (25) in the variable y is g . Denote by dμ(y) the measure dμ(y) = dx,

where dx is as in (21).

(26)

Then   2 m M(χ, s)fχ,s i(g, 1) = |2|−m F χ(−1)  −s−m/2+1/2   × χ −1 det(Im − y) det(Im − y)E g

  × φU −c(y)g dμ(y).

(27)

Now, in (27), we want to make the change of variable c(y) = h.

(28)

We do this using the analogue of Lemma 4.2 from [5]. Lemma 6. Let f be a function in Cc∞ (G) such that the function   −m  y → f c(y) det(Im − y)E , defined on g = u , and extended by 0 to g, is integrable. Then there is a choice of Haar measure on G such that 

−m   f c(y) det(Im − y)E dμ(y) =

g



 m/2 f (h)det(h)E dh

G



=

f (h) dh.

(29)

G

Proof. Since the Jacobian of (28) is given by a rational function, defined over F , it is enough to compute it over an algebraic closure F of F containing E. Thus, we may assume, for this proof, that F is algebraically closed. Over the algebraically closed field F , G becomes the split group GLm (F ). In order to make the statement more comprehensible, we break down the passage from F to F into two steps, as follows:

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• Step 1. Replace F with the quadratic extension E, over which G becomes GLm (E). • Step 2. Replace E with F = E, over which GLm (E) becomes GLm (E) = GLm (F ), hence split. It is well known that a split group over an algebraically closed field has a conjugacy class of Cartan subgroups, so this argument also validates the application of the “simple” form of the Weyl integration formulas, stated in (4.17) and (4.21) of [5]. Of the two steps above, Step 2 amounts to a straightforward tensoring operation on the Lie algebra level. Alternatively, considering the abstract Chevalley group GLm as a functor from fields (Field) to matrix groups, we may consider this step as a simple substitution of objects belonging to the domain category Field. Therefore, only Step 1 needs any further explanation, for which we introduce the following concepts. Let F ⊂ E be a quadratic extensions of fields of characteristic different from 2. By Section 2, this implies that as a vector space, E is of the form F ⊕ ωF , with ω2 = a ∈ F . For V an m-dimensional vector space over E, let V (F ) be the 2m-dimensional F -vector space which equals V as a set. If B = {v1 , . . . , vm } is a basis for V , then call B F := {v1 , ωv1 , . . . , vm , ωvm }, the corresponding basis for V (F ). Fix a B as above so as to consider gl(V (F )) as the matrixalgebra gl2m (F ). For an element X ∈ gl(V (F )), denote by X (ij ) the ij th two-by-two block counting from the upper-left of the matrix X, for 1  i, j  m. Thus X (ij ) ∈ gl2 (F ), and its (ij ) entries will be denoted by Xkl for 1  k, l  2. Definition 7. The operation transp on gl(V (F )) is defined by the relation  (ij ) transp(X) = X (j i) ,

for 1  i, j  m,

in other words, the transpose operation operating a matrix of 2-by-2 blocks instead of individual entries. Define the operation conj on gl(V (F )) by the relations  (ij ) (ij ) conj(X) kl = (−1)k−l (X)kl ,

for 1  i, j  m, 1  k, l  2,

i.e. conj negates “off-diagonal” entries of each two-by-two block and leaves the diagonal entries alone. For T as in (5), with T = diag(t1 , . . . , tm ), set t = diag(t1 I2 , . . . , tm I2 ). Let g be the subalgebra of gl(V (F )) satisfying the conditions:

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• E-linearity. Each X (ij ) is of the form 

f1 f2

af2 f1

 .

• (E, T )-unitarity. We have conj ◦ transp(X) = −tXt −1 . Proposition 8. With g defined as above, we have g ∼ = um (T ) as a Lie algebra over F . Proof. Consider the F -isomorphism of V with V (F ) induced by the identification of the underlying sets. This isomorphism induces a natural embedding of glm (E) into gl2m (F ). The first condition, E-linearity, is equivalent to X belonging to the image of this embedding, that is to X’s actually from a “restriction of scalars” from an element of glm (E). Given that X satisfies the condition of E-linearity, it is clear that the inverse image of X belongs to um (F ) if and only if X satisfies the second condition, of (E, T )-unitarity. 2 Therefore, in order to justify Step 1, we can replace the original task of showing that um (F ) ⊗ E∼ = glm (E) over E, with showing that g ⊗ E ∼ = glm (E) over E. The isomorphism is simply the isomorphism induced on the level of vector space endomorphisms by the canonical vector-space isomorphism V (F ) ⊗ E ∼ = V . This can be made obvious by showing how a basis of the former maps to a basis of the latter. In order to define the basis, we adopt the notation E (ij )

 a b  cd

∈ Mat2m (F )

with

 a b cd

as the ij th 2-by-2 block and zeros elsewhere.

Also, we use eij to denote the (usual) m-by-m elementary matrix with 1 in the ij th position and zeros elsewhere. Then it is easily calculated that the E-basis of g ⊗ E

 

 

 E (ii) 01 0a 1im ∪ E (ij ) + ti tj−1 E (j i) 01 0a 1i
∪ E (ij ) − ti tj−1 E (j i) (I2 ) 1i
maps to the basis of glm (E), 

 {ωeii }ij m ∪ ω eij + ti tj−1 ej i 1i
(11)

af2

(12)

f1

(12)

f2 (12)

af2

(12)

−f1

(22)

f2

(12) ⎞

af2

⎫ ⎪ ⎪ ⎪ ⎬

⎟ ⎟  (ij )  fk ∈ F for i, j, k ∈ {1, 2} . (22) ⎟ ⎪ af2 ⎠ ⎪ ⎪ ⎭ (12)

f1

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The  0 a  can now verify by inspection the above claims concerning the bases (ii)reader ∪ · · · and {eii }1im ∪ · · · in this case of m = 2 and see how to extend E 1im 10 the arguments to general m. Throughout this proof we will use | · | to denote | · |F . We may replace the first equality of (29), to be proved, with the new formula   m −2m      f c(y) det(Im − y) dμ(y) = f (h)det(h) dh. (30) glm (F )

GLm (F )

We compare the two sides of (30) by computing both using the appropriate forms of Weyl’s Integration Formula. Because, over the field F , which by the above argument may be assumed to be algebraically closed, there is only one conjugacy class of Cartan subgroups of GLm (F ). The proof of the first equality of Lemma 6 from this point is essentially the same as the proof of Lemma 4.2 in [5]. For the purposes of comparison, we give the polynomials D(x) and d(t) in the case when G is GL(n, F ), namely, 

  D(x) =

|xi − xj |2 ,

(31)

1i
and   d(t) =

 1i
|ti − tj |2

m 

|ti |−m+1 .

(32)

i=1

See pp. 302–303 of [5] for the meaning of the above notation. Finally, it follows easily from the definition (7) that | det(h)| = 1 for h ∈ G. 2 We continue with the proof of Lemma 5. We make the change of variable (28) in (27) (Re(s) is still large enough), and by (29), we get that there is a choice of Haar measure dh on G such that   M(χ, s)fχ,s i(g, 1) = |2|Fm(1−2s) χ −m (−2)  s−m/2−1/2   φU (−hg) dh. × χ det(Im + h) det(Im + h)E G

This proves Lemma 5.

2

Let N be a positive even integer such that 4 q N > |8|−1 E q ,

(33)

  N−2  c g PE 2 ⊂ U.

(34)

and such that

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E. Brenner / Journal of Number Theory 128 (2008) 1358–1375 N

From now on, we assume that the conductor 1 + PE χ of χ is such that Nχ > N . We now return to the integral in (20), evaluated at −g −1 in place of g. In order for a point h in the domain of integration to make a nonzero contribution to the integral (i.e., in order for the integrand to be nonzero at h), we must have hg −1 = u, where u ∈ U . That is, we must have h = ug for some u ∈ U . Denote, for a matrix x ∈ Matm (E), x = max |xij |E ,

(35)

1i,j m

as in [5]. This is a norm on Matm (E). Then, according to (20), we have   m(1−2s) −m M(χ, s)fχ,s i −g −1 , 1 = |2|F χ (−2)  ∞ × L=−∞

  χ det(Im + ug)

c−1 (ug)=q L u∈U

 s−m/2−1/2 × det(Im + ug)E du,

(36)

for Re(s) 0. Denote, for g ∈ G, Re(s) 0, and L ∈ Z,  IL (χ, s; g) :=

s−m/2−1/2   χ det(Im + ug) det(Im + ug)E du.

(37)

c−1 (ug)=q L u∈U

Set 

Nχ + 1 nχ = . 2

(38)

Our main aim from this point is to prove an analogue of Lemmas 4.4–4.6 from [5], namely that Lemma 9. We have IL (χ, s; g) = 0, for all L  nχ . The first main step towards proving Lemma 9 will be making a change of variable in the integral IL (χ, s; g) that will allow us to replace the single integral of IL (χ, s; g) with a double integral. The following easy consequence of basic facts in theory p-adic fields will be key to the computation of the inner integral. Proposition 10. Let n, N satisfy n  N  2n,

(39)

E. Brenner / Journal of Number Theory 128 (2008) 1358–1375

1371

and let ψ0 a fixed character of F whose conductor is OF . Let χ be a character of F × such that: The conductor of χ is no greater than N.

(40)

Then there exists a fixed a ∈ F ∗ with −ν(a) equal to the conductor of χ

(41)

such that     χ det(1 − v) = ψ0 a tr(v) ,

for all v ∈ Matm (F ) with v  q −n .

(42)

In order to state Lemma 12, it is convenient to introduce the following piece of notation. Definition 11. Let + (·) be the “non-negativity function” from the reals to the non-negative reals. That is, let + (·) be defined piecewise by ! r if r  0, + r= 0 if r < 0. Let − (·) be the “non-positivity function” from the reals to the non-negative reals defined analogously so that for all r ∈ R,

r = +r − −r

and |r| = + r + − r.

(43)

Henceforth, we abbreviate g(PEn ) as gn . Lemma 12. Assume that L satisfies L > −nχ ,

equivalently

−

L < nχ .

(44)

Then, there exist a, b ∈ E × satisfying the properties |a|E = |b|E = q Nχ

(45)

|a + b|E  q nχ ,

(46)

and

such that IL =

1 μ(g+ L+nχ )  ×



s−m/2−1/2   χ det(Im + ug) det(Im + ug)

c−1 (ug)=q L u∈U

   ψ0 b tr v + a tr vc−1 (ug) dμ(v) du,

v∈g+ L+nχ

where ψ0 is a fixed character of E whose conductor is OE .

(47)

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E. Brenner / Journal of Number Theory 128 (2008) 1358–1375

Proof. Until the analogue of (4.36) in [5], the proof follows the proofs of Lemmas 4.4–4.6 in [5] closely. We make the change of variable u → c(v)u for v ∈ g+ L+nχ . Unlike in the proofs of [5], we do not have χ −1 (det(Im − v)) ≡ 1. Instead, we apply Proposition 10, below, to rewrite each of the factors χ −1 (det(Im − v)) and χ(det(Im + vc−1 (ug))). In the first case n = + L + nχ , while in the second n = nχ , and in both cases N = 2n. Since Nχ  2n, the hypothesis (40) is verified. The corollary now gives a, b ∈ E × satisfying (45), with         χ −1 det(Im − v) χ det Im + vc−1 (ug) = ψ0 (b tr v)ψ0 a tr vc−1 (ug)    = ψ0 b tr v + a tr vc−1 (ug)

(48)

n

for all v such that v ∈ g+ L+nχ , vc−1 (ug) ∈ PEχ . Together with the calculation of (4.36) in [5], this proves the lemma with the exception of the claim (46). In order to verify (46), note that, by construction, a, b are elements of E × such that and χ −1 (1 + x) = ψ0 (bx),

x(1 + x) = ψ0 (ax)

n

for all x ∈ PEχ .

Consequently, we have   n q 0 = 1 = ψ0 (ax + bx) = ψ (a + b)x for all x ∈ PEχ , implying that n

(a + b)PEχ ⊆ OE , This immediately yields (46).

so that |a + b|E q −nχ  1.

2

Rewriting the single integral IL (χ, s; g) as the iterated integral of (47) was the first main step in showing that IL (χ, s; g) = 0. From this point, the strategy of the proof consists in changing the order of integration and rewriting the inner integrand in such a way that this inner integrand can be seen to be zero, for all u in the range of integration of the outer integrand. As a result of form of the inner integrand in (47) being different from (4.42) and its analogs in Lemmas 4.5, 4.6 in [5], we are faced with the most significant complications inherent in the treatment of the unitary case, namely, the replacement of the simple arguments on pp. 306–308 of [5] with Lemmas 13 and 14 and Proposition 15, below. Lemma 13. Let L ∈ Z and assume that L = 0. Let ψ0 be a fixed character of E with conductor OE , a, b ∈ E × such that nχ +1 |a|E = |b|E > |2|−1 . E q

(49)

X = q L ,

(50)

Then for X ∈ glm (E) such that

E. Brenner / Journal of Number Theory 128 (2008) 1358–1375

1373

we have 

  ψ0 b tr v + a tr(vX) dμ(v) = 0.

(51)

g+ L+nχ

Proof. Note that the expression b tr v + a tr(vX) is equal to   tr v(bIm + aX) . As in the proof of Lemma 4.4 of [5], at (4.42) we see that in order for the integral (51) to be nonzero, we must have bIm + aX  |2|−1 E q

+ L+n +1 χ

.

(52)

Since L = 0, by assumption, q L = 1, so that (50) and the equality of (49) together imply that bIm  = aX. Therefore, (4.27) from [5] implies that   bIm + aX = max bIm , aX .

(53)

By the strict inequality of (49), (50), and the definition of + L before (43)   + L+n +1 χ max bIm , aX > |2|−1 , E q so that by (53) bIm + aX > |2|−1 E q

+ L+n +1 χ

.

With (52) this gives a contradiction. Therefore, by the above comments, the integral of (51) equals zero. 2 Lemma 14. Let ψ0 be a fixed character of E with conductor OE , a, b ∈ E × satisfying (45) and (46). Then for X ∈ g,

such that X = 1,

(54)

we have 

  ψ0 b tr v + a tr(vX) dμ(v) = 0.

(55)

gnχ

Proof. In order for the integral of (55) not to vanish, we must have nχ +1 bIm + aX  |2|−1 , E q

(56)

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E. Brenner / Journal of Number Theory 128 (2008) 1358–1375

paralleling (52). The hypothesis (46) is equivalent to assuming that a + b is of the form  −nχ o, for  the generator of PE and some o ∈ OE . So in order for the integral of (55) to be nonzero, we must have " " −n " χ oIm + a(X − Im )"  |2|−1 q nχ +1 . E By (4.27) of [5], because  −nχ o  q nχ , this implies that " " "a(X − Im )"  |2|−1 q nχ +1 , E so by (45), nχ −Nχ +1 , X − Im   |2|−1 E q

meaning that  N −n −1−νE (2)  . X − Im ∈ glm PE χ χ Thus X ∈ Im + glm (P Nχ −n−1−νE (2) ). Because of (33), this, combined with Proposition 15, implies that X ∈ / g. Thus, we obtain a contradiction with (54). 2 In the proof of Lemma 14, we needed to use the fact that the elements of g cannot get “too close” to the identity Im , in a precise quantitative sense. The purpose of the following proposition is simply to prove this claim. Proposition 15. For T a form matrix as in (4)–(5), let g be the realization of um given by 

g = xwm T  x ∈ glm (E) and (xwm )∗ = −xwm . Let n > vE (2). Then there is no X ∈ g such that X − Im ∈ glm (PEn ). Proof. Suppose otherwise, so that we have   xwm T − Im ∈ glm PEn .

(57)

By (4) and (5), because T −1 is a diagonal matrix with entries of norm 1 or q −1 , and it follows that these entries are in OE . Therefore, multiplying an element of glm (PEn ) on the right by T −1 multiplies each column by an integer. Multiply (57) on the right by T −1 to obtain   (58) xwm − T −1 ∈ glm PEn . ¯ E , we have Then apply (·)∗ to (57). Since for any x ∈ E, |x|E = |x| ∗    (xwm )∗ − T −1 ∈ glm PEn ,

E. Brenner / Journal of Number Theory 128 (2008) 1358–1375

1375

so that by the description of g in the hypotheses and (6)     −xwm − T −1 ∈ glm PEn .

(59)

Add (58) and (59) to get   2T −1 ∈ glm PEn . By (4) and (5) this means that 2b(vi , vi ) ∈ PEn , implying |2|E q or |2|E × 1  q −n , i.e., |2|E < q −n−1

or

|2|E  q −n .

Since n > νE (2), either of these conditions will produce a contradiction.

2

Completion of proof of Lemma 9. By applying Lemma 13 or 14 to Lemma 12, depending on whether L = 0 or L = 0, we deduce the vanishing of IL for all L in the required range of L  −nχ . The reason the hypotheses of Lemma 13 are satisfied is that according to Lemma 12 we have (45) and then (33) implies (49). The reason the hypotheses of Lemma 14 are satisfied in the case L = 0 is that X = c−1 (ug) satisfies (54) for all u in the domain of integration of the outer integral for IL . 2 The completion of the proof of Theorem 3 from Lemmas 9, (37) and (38) now proceeds the same practically word-for-word as the completion of the proof of Theorem 3.1 on p. 308 of [5]. Acknowledgments The author thanks David Soudry for suggesting this problem and Soudry, Nadya Gurevich, and Omer Offen for helpful conversations. He also thanks Mr. Tony Petrello for additional financial support during the writing of this paper. References [1] N. Bourbaki, Éléments de mathématique. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind., vol. 1272, Hermann, Paris, 1959, MR0107661 (21 #6384). [2] Stephen Gelbart, Ilya Piatetski-Shapiro, Stephen Rallis, Explicit Constructions of Automorphic L-Functions, Lecture Notes in Math., vol. 1254, Springer-Verlag, Berlin, 1987, MR892097 (89k:11038). [3] Herve Jacquet, Joseph Shalika, A lemma on highly ramified -factors, Math. Ann. 271 (3) (1985) 319–332, MR0787183 (87i:22048). [4] Erez M. Lapid, Stephen Rallis, On the local factors of representations of classical groups, in: Automorphic Representations, L-Functions and Applications: Progress and Prospects, in: Ohio State Univ. Math. Res. Inst. Publ., vol. 11, 2005, pp. 309–359, MR2192828 (2006j:11071). [5] Stephen Rallis, David Soudry, Stability of the local gamma factor arising from the doubling method, Math. Ann. 333 (2) (2005) 291–313, MR2195117 (2006m:22026).