On the Local Converse Theorem for p-adic GLn

0.23 MB·

by  Herve Jacquet

ON THE LOCAL CONVERSE THEOREM FOR p-ADIC
GL
n
6
1 HERVE´ JACQUET AND BAIYING LIU
0
2
Abstract. Inthispaper,wecompletelyproveastandardconjec-
n
a ture on the local converse theorem for generic representations of
J GLn(F), where F is a non-archimedean local field.

8
1

]
T 1. Introduction
R
Let F be a non-archimedean local field. Let G := GL (F) and let
. n n
h
π be an irreducible generic representation of G . The family of local
t n
a gamma factors γ(s,π ×τ,ψ), for τ any irreducible generic representa-
m
tion of G , ψ an additive character of F and s ∈ C, can be defined
r
[
using Rankin–Selberg convolution [JPSS83] or the Langlands–Shahidi

2 method[S84]. Thefollowingisastandardconjectureonpreciselywhich
v
family of gamma factors determine π.
6
5
6 Conjecture 1.1. Let π1,π2 be irreducible generic representations of
3 G . Suppose that they have the same central character. If
n
0
. γ(s,π ×τ,ψ) = γ(s,π ×τ,ψ),
1 1 2
0
as functions of the complex variable s, for all irreducible generic rep-
6
1 resentations τ of G with 1 ≤ r ≤ [n], then π ∼= π .
r 2 1 2
:
v
Theconjectureandtheglobalversionoftheconjecture([CPS99, Sec-
i
X tion8,Conjecture1])emergedfromearlydiscussionsbetweenPiatetski-
r Shapiro, Shalika and the first mentioned author. In particular, they
a
proved 1.1 in the case n = 3 ([JPSS79]).
The fact that the representations have the same central character
implies that if, for a given r, the above equality is true for one choice
of ψ, then it is true for all choices of ψ. Moreover, if the above equality
is true for r = 1 and one choice of ψ, then the representations have the
same central character ([JNS15, Corollary 2.7]).
Date: January 19, 2016.
2000 Mathematics Subject Classification. Primary 11S70, 22E50; Secondary
11F85, 22E55.
Key words and phrases. Local Converse Theorem, Generic Representations.
ThesecondmentionedauthorissupportedinpartbyNSFGrantsDMS-1302122.
1
2 HERVE´ JACQUET ANDBAIYINGLIU
One can propose a more general family of conjectures as follows (see
[ALSX15]). We say that π and π satisfy hypothesis H if they have
1 2 0
the same central character. For m ∈ Z , we say that they satisfy
≥1
hypothesis H if they satisfy hypothesis H and satisfy
m 0
γ(s,π ×τ,ψ) = γ(s,π ×τ,ψ)
1 2
as functions of the complex variable s, for all irreducible generic repre-
sentations τ of G . For r ∈ Z , we say that π ,π satisfy hypothesis
m ≥0 1 2
H if they satisfy hypothesis H , for 0 ≤ m ≤ r.
≤r m
Conjecture J(n,r). If π ,π are irreducible generic representations
1 2
of G which satisfy hypothesis H , then π ≃ π .
n ≤r 1 2
Conjecture1.1isexactlyConjectureJ(n,[n]). HenniartprovedCon-
2
jecture J(n,n−1) in [H93]. Conjecture J(n,n−2) (for n ≥ 3) is a
Theorem due to Chen [Ch96, Ch06], to Cogdell and Piatetski-Shapiro
[CPS99], and to Hakim and Offen [HO15]. As we mentioned above,
Conjecture J(3,1) is first due to the first mentioned author, Piatetski-
Shapiro and Shalika [JPSS79].
In [JNS15, Section 2.4], Conjecture 1.1 is shown to be equivalent to
the same conjecture with the adjective “generic” replaced by “unita-
rizable supercuspidal” as follows:
Conjecture 1.2. Let π ,π be irreducible unitarizable supercuspidal
1 2
representations of G . Suppose that they have the same central char-
n
acter. If
γ(s,π ×τ,ψ) = γ(s,π ×τ,ψ),
1 2
as functions of the complex variable s, for all irreducible supercuspidal
representations τ of G with 1 ≤ r ≤ [n], then π ∼= π .
r 2 1 2
In [JNS15], Jiang, Nien and Stevens introduced the notion of a spe-
cial pair of Whittaker functions for a pair of irreducible unitarizable
supercuspidal representations π , π of G . They proved that if there
1 2 n
is such a pair, and π1, π2 satisfy hypothesis H≤[n], then π1 ∼= π2. They
2
alsofoundspecial pairsofWhittaker functions inmany cases, inpartic-
ular the case of depth zero representations. In [ALSX15], Adrian, the
second mentioned author, Stevens and Xu proved part of the case left
open in [JNS15]. In particular, the results in [JNS15] and [ALSX15]
together imply that Conjecture 1.2 is true for G , n prime. We remark
n
thatboth[JNS15] and[ALSX15]make use ofthe construction ofsuper-
cuspidal representations of G in [BK93] and properties of Whittaker
n
functions of supercuspidal representations constructed in [PS08].
In this paper we prove Conjecture 1.1, hence conjecture 1.2. We use
analytic methods. We do not resort to the construction of special pairs
LOCAL CONVERSE THEOREM 3
of Whittaker functions for supercuspidal representations. The idea is
inspired by the proof of Conjecture J(n,n − 2) in [Ch06]. We state
the main result of the paper as the following theorem.
Theorem 1.3. Conjecture 1.1 is true.
We were recently informed that Chai has an independent and differ-
ent proof of Conjecture 1.1 ([Ch15]).
One straightforward application of Theorem 1.3 is that it reduces
the amount of necessary GL-twisted local factors, in order to obtain
the uniqueness of local Langlands correspondence (proved by Henniart
in [H02]), and it also gives a corresponding local converse theorem for
local Langlands parameters via the local Langlands correspondence.
By the argument of [CPS99, Section 7, Theorem], one can see that
Conjecture 1.2 is a consequence of the global version of Conjecture
1.1 ([CPS99, Section 8, Conjecture 1]). Hence, Theorem 1.3 provides
evidence for the global version of Conjecture 1.1 on the global converse
theorem.
It is easy to find pairs of generic representations showing that in
Conjecture 1.1, [n] is sharp for the generic dual of G . In Conjecture
2 n
1.2, sharpness of [n] for the supercuspidal dual of G is being consid-
2 n
ered in [ALST15], in particular, it has been shown in [ALST15] that
[n] is sharp for the supercuspidal dual of G , n prime. However, it is
2 n
expected that for certain families of supercuspidal representations, [n]
2
may not be sharp, for example, for simple supercuspidal representa-
tions (of depth 1), the upper bound may be lowered to 1 (see [BH14,
n
Proposition 2.2] and [AL15, Remark 3.18] in general, and [X13] in the
tame case).
Nien in [N14] proved thefinite field analogueof Conjecture 1.1, using
special properties of normalized Bessel functions. We remark that the
idea in this paper also applies to the finite field case, and could give a
new proof for the result in [N14]. Moss in [M15] proved an analogue of
Conjecture J(n,n−1) for ℓ-adic families of smooth representations of
GL (F), where F is a finite extension of Q and ℓ is different from p.
n p
The local converse problem has been studied for irreducible generic
representations ofgroupsotherthanGL : U(2,1)andGSp(4)(Baruch,
n
[B95] and [B97]); SO(2n+1) (Jiang and Soudry, [JS03]); U(1,1) and
U(2,2) (Zhang, [Z15a] and [Z15b]). We remark that since the local
converse theorem for SO in [JS03] is eventually reduced to the
2n+1
localconverse theorem forGL , following exactly thesame proofgiven
2n
in [JS03], Theorem 1.3 implies that twisting up to irreducible generic
representations of GL is enough in the local converse theorem for
n
SO in [JS03].
2n+1
4 HERVE´ JACQUET ANDBAIYINGLIU
Section 2 will be preparation on properties of irreducible generic
representations of GL (F) and Rankin-Selberg convolution. Theorem
n
1.3willbeprovedinSection3. Section4willbetheproofofProposition
3.6.
Finally, we would like to thank J. Cogdell, D. Jiang and F. Shahidi
for their interest in the problems discussed in this paper and for their
encouragements, and S. Stevens for a helpful suggestion which makes
the paper more readable.
Acknowledgements. This material is based upon work supported by
the National Science Foundation under agreement No. DMS-1128155.
Any opinions, findings and conclusions or recommendations expressed
in this material are those of the authors and do not necessarily reflect
the views of the National Science Foundation.
2. Generic representations and Rankin-Selberg
convolution
In this section, we review basic results on generic representations
and the Rankin-Selberg convolution, which will be used in the proof of
Theorem 1.3 in Section 3.
Let F be a non-archimedean local field, and let q be the cardinality
of the residue field of F. Let G := GL (F). All representations of G
n n n
considered in this paper are irreducible smooth and complex.
2.1. Whittaker models. Let B = T U be the standard Borel sub-
n n n
group of G consisting of upper triangular matrices, with unipotent
n
radical U := U and diagonal group T . Fix a nontrivial additive char-
n n
acter ψ of F. Define a non-degenerate character ψ of U also denoted
Un n
by ψ as follows:
U
n−1
ψ (u) := ψ u , u ∈ U .
Un i,i+1 n
!
i=1
X
An irreducible representation (π,V) of G is generic if
n
Hom (V,IndGnψ ) 6= 0.
Gn U U
It is known that if π is generic, then the above Hom-space is of di-
mension 1. Let π be an irreducible generic representation of G , fix
n
a nonzero functional ℓ in the above Hom-space, then the image of V
under ℓ is called the Whittaker model of π, denoted by W(π,ψ). It is
known that W(π,ψ) is independent of the choice of ℓ. For each v ∈ V,
let W = ℓ(v). Then for u ∈ U, g ∈ G ,
v n
W (ug) = ψ (u)W (g),
v U v
LOCAL CONVERSE THEOREM 5
W (g) = ℓ(v)(g) = ℓ(π(g)v)(I ) = W (I ).
v n π(g)v n
For W ∈ W(π,ψ), let
W(g) = W(ω tg−1),
n
where
f 0 1
ω = 1, ω = .
1 n ω 0
n−1
(cid:18) (cid:19)
It is well known that W ∈ W(π,ψ), where π is the representation
contragradient to π.
Let P be the maximafl paraboliec subgroup ofeGn with Levi subgroup
G ×G . Let Z be the center of G . Given two irreducible generic
n−1 1 n
representations π and π of G with the same central character, to
1 2 n
show that π ∼= π , it suffices to show that their Whittaker models
1 2
W(π ,ψ) and W(π ,ψ) have a nonzero intersection. The following two
1 2
propositions allow us to study Whittaker functions by restricting them
to P.
Proposition 2.1 ([GK75]). Let π be an irreducible generic represen-
tation of G with central character ω . Then the restriction W(π,ψ)|
n π P
has a Jordan-Ho¨lder series of finite length which contains the compact
induction indP ω ψ as an irreducible subrepresentation.
ZU π U
The following proposition is proven in [JPSS79] for n = 3, and the
same argument works for general n. The proof can also be found in
[BZ77, Theorem 4.9].
Proposition 2.2. Let (π,V) be an irreducible generic representation
of G . Then
n
v 7→ W |
v P
is an injective map from V to the space of smooth functions on P.
Let π , π be two irreducible generic representations of G with the
1 2 n
same central character ω. Let V = indP ωψ . For p ∈ P, let ρ(p) be
0 ZU U
the operator of right translation on complex functions v on P:
ρ(p)v(x) = v(xp).
By Propositions 2.1 and 2.2, for any v ∈ V there is a unique element
0
Wi in theWhittaker model ofπ such that, forall p ∈ P, Wi(p) = v(p).
v i v
Thus, we have
W1(p) = W2(p), ∀p ∈ P , ∀v ∈ V .
v v 0
Note that for p ∈ P, we have
Wi(gp) = Wi (g), ∀g ∈ G , ∀v ∈ V .
v ρ(p)v n 0
6 HERVE´ JACQUET ANDBAIYINGLIU
2.2. Rankin-Selberg convolution. Let n,t ∈ Z , and let π and τ
≥1
be irreducible generic representations of G and G , with Whittaker
n t
models W(π,ψ) and W(τ,ψ), respectively. Let W ∈ W(π,ψ) and
W′ ∈ W(τ,ψ). Assume that n > t, which is the case of interest to us
in this paper.
For any integer j with 0 ≤ j ≤ n−t−1, let k = n−t−1−j, define
a local zeta integral as follows:
(2.1)
g 0 0
Ψ(s,W,W′;j) := W X Ij 0 W′(g)|detg|s−n2−tdxdg,
 
Z Z 0 0 Ik+1
 
with integration being over g ∈ U \G and X ∈ M (F).
t t j×t
For g ∈ G , let ρ(g) be the operator of right translation on complex
n
functions f on G :
n
ρ(g)f(x) = f(xg).
Let
I 0
ω = t .
n,t 0 ω
n−t
(cid:18) (cid:19)
The following result is about functional equations for a pair of ir-
reducible generic representations, proved by the first named author,
Piatetski-Shapiro and Shalika in [JPSS83]. It plays an important role
in proving the main result of this paper.
Theorem 2.3 ([JPSS83], Section 2.7). With notation as above, the
followings hold.
(1) Each integral Ψ(s,W,W′;j) is absolutely convergent for Re(s)
large and is a rational function of q−s. More precisely, for any
fixed j, the integrals Ψ(s,W,W′;j) span a fractional ideal (in-
dependent of j) of C[qs,q−s]:
C[qs,q−s]L(s,π ×τ),
where the local factor L(s,π × τ) has the form P(qs)−1, with
P ∈ C[qs] and P(0) = 1.
(2) For any 0 ≤ j ≤ n − t − 1, there is a factor ǫ(s,π × τ,ψ),
independent of j, such that
Ψ(1−s,ρ(ω )W,W′;k) Ψ(s,W,W′;j)
n,t = ω (−1)n−1ǫ(s,π ×τ,ψ) ,
τ
L(1−s,π ×τ) L(s,π ×τ)
f f
where k = n−t−1−j and ω is the central character of τ.
τ
e e
LOCAL CONVERSE THEOREM 7
The local gamma factor attached to a pair (π,τ) is defined to be
L(1−s,π ×τ)
γ(s,π×τ,ψ) = ǫ(s,π ×τ,ψ) .
L(s,π ×τ)
e e
Then the functional equation in Part (ii) of Theorem 2.3 canbewritten
as
(2.2)
Ψ(1−s,ρ(ω )W,W′;k) = ω (−1)n−1γ(s,π×τ,ψ)Ψ(s,W,W′;j).
n,t τ
At the end of this section, we introduce the following important
f f
lemma.
Lemma 2.4. Let π and π be two irreducible generic representations
1 2
of G . Let t ≤ n−2 and j with 0 ≤ j ≤ t. Suppose that W1 and W2 are
n
elements in the Whittaker models of π and π respectively. Suppose
1 2
further that for all irreducible generic representations τ of G we
n−t−1
have
Ψ(s,W1,W′;j) = Ψ(s,W2,W′;j)
for all W′ ∈ W(τ,ψ) and for Re(s) ≫ 0. Then
I 0 0 I 0 0
n−t−1 n−t−1
W1 X I 0 dX = W2 X I 0 dX,
j j
   
Z 0 0 It+1−j Z 0 0 It+1−j
   
where the integrals are over X ∈ M (F).
j×(n−t−1)
Proof. For j = 0, the assumption is that
W1 g 0 W′(g)|detg|s+t+21dg
0 I
(2.3) ZUn−t−1\Gn−t−1 (cid:18) t+1(cid:19)
g 0
= W2 W′(g)|detg|s+t+21dg,
0 I
ZUn−t−1\Gn−t−1 (cid:18) t+1(cid:19)
for all W′. The conclusion is that W1(I ) = W2(I ). Indeed, recall
n n
that given C > 0 the relations
g 0
|detg| = C, Wi 6= 0
0 I
t+1
(cid:18) (cid:19)
imply that g is in a set compact modulo U . Both sides of the
n−t−1
identity (2.3) converge for Re(s) ≫ 0. Thus they can be interpreted as
formal Laurent series in q−s. We conclude that for any C > 0
g 0 g 0
W1 W′(g)dg = W2 W′(g)dg.
0 I 0 I
Z|detg|=C (cid:18) t+1 (cid:19) Z|detg|=C (cid:18) t+1 (cid:19)
8 HERVE´ JACQUET ANDBAIYINGLIU
One then applies the spectral theory of the space L2(U \G0 )
n−t−1 n−t−1
where G0 = {g ∈ G : |detg| = 1}. For more details, see [H93,
n−t−1 n−t−1
Section 3] and [Ch06, Section 2].
For 0 < j ≤ t, one observes that there is a compact subset Ω of
M (F) such that for all g ∈ G and i = 1,2,
j×(n−t−1) n−t−1
g 0 0
Wi X I 0 6= 0
j
 
0 0 I
t+1−j
 
implies that X ∈ Ω. Thus, for i = 1,2, there is an element Wi ∈
0
W(π ,ψ) such that for all g ∈ G
i n−t−1
g 0 0 g 0 0
Wi X I 0 dX = Wi 0 I 0 .
 j  0 j 
ZMj×(n−t−1)(F) 0 0 It+1−j 0 0 It+1−j
We are therefore reduced to the casej = 0.   (cid:3)
3. Proof of Theorem 1.3
In this section, we prove Theorem 1.3. Let π and π be irreducible
1 2
generic representations of G with the same central character ω. We
n
recall from Section 2.1 that P is the maximal parabolic subgroup of G
n
with Levi subgroup G ×G , Z is the center of G , V = indP ωψ ,
n−1 1 n 0 ZU U
and we have
(3.1) W1(p) = W2(p), ∀p ∈ P , ∀v ∈ V ,
v v 0
(3.2) Wi(gp) = Wi (g), ∀g ∈ G , ∀p ∈ P , ∀v ∈ V , i = 1,2.
v ρ(p)v n 0
We recall the decomposition of G into double cosets of U and P as
n
in [Ch06]:
˙ n−1
G = UαiP ,
n
i=0
where [
0 I
α = n−1 .
1 0
(cid:18) (cid:19)
0 I
Note that αi = n−i , in particular, α0 = αn = I .
I 0 n
i
(cid:18) (cid:19)
Definition 3.1. For each double coset UαiP, 0 ≤ i ≤ n−1, we call i
the height of the double coset. We say that π and π agree at height i
1 2
if
W1(g) = W2(g), ∀g ∈ UαiP , ∀v ∈ V .
v v 0
LOCAL CONVERSE THEOREM 9
By (3.1), π and π agree at height 0. The following lemma gives a
1 2
characterization of π and π agreeing at height i.
1 2
Lemma 3.2 ([Ch06], Lemma 3.1). π and π agree at height i if and
1 2
only if
W1(αi) = W2(αi), ∀v ∈ V .
v v 0
The following proposition is one of the main ingredients for this
paper.
Lemma 3.3 ([Ch06], Proposition 3.1). Let t with 1 ≤ t ≤ n−1. If π
1
and π satisfy hypothesis H , then they agree at height t.
2 t
To proceed, we give a characterization of the matrices in the double
coset PαsU, 0 ≤ s ≤ n−1.
Lemma 3.4. Suppose 0 ≤ s ≤ n−1, Then g ∈ PαsU if and only if
the last row of g has the form
(0,...,0,a ,a ,...,a ), a 6= 0.
s s+1 n s
Proof. Recall that
0 I
αs = n−s .
I 0
s
(cid:18) (cid:19)
It is clear that the last row of any matrix in Pαs has the form
(0,...,0,a ,0,...,0),
s
wherea 6= 0occursinthes-thcolumnofthematrix. Aftermultiplying
s
by matrices in U fromthe right, one can see that last row of any matrix
in PαsU has the form (0,...,0,a ,a ,...,a ), with a 6= 0. (cid:3)
s s+1 n s
In fact, this lemma gives at once the decomposition in the disjoint
double cosets
˙ n−1 ˙ n−1
G = PαiU = UαiP .
n
i=0 i=0
The next lemma is a g[eneralization o[f [Ch06, Lemma 3.2].
Lemma 3.5. Let t with [n] ≤ t ≤ n − 2. Suppose that for any s
2
with 0 ≤ s ≤ t the representations π and π agree at height s. Then
1 2
the following equality holds for all X ∈ M (F), all g ∈
(n−t−1)×(2t+2−n)
G , and all v ∈ V :
n−t−1 0
I 0 0 I 0 0
n−t−1 n−t−1
W1 0 I 0 = W2 0 I 0 .
v  2t+2−n  v  2t+2−n 
0 X g 0 X g
   
10 HERVE´ JACQUET ANDBAIYINGLIU
Proof. First note that the hypothesis [n] ≤ t ≤ n − 2 implies that
2
n−t−1 ≥ 1 and 2t+2−n ≥ 1.
I 0 0
n−t−1
Let A = 0 I 0 , where X ∈ M (F),
2t+2−n (n−t−1)×(2t+2−n)
 
0 X g
 I 0 0
n−t−1
and g ∈ G . Then A−1 = 0 I 0 . By Lemma
n−t−1 2t+2−n
 
0 −g−1X g−1
3.4, A−1 ∈ PαiU, where i ≥ n−t, hence, A ∈ Uαn−iPwith n−i ≤ t.
Since π and π agree at heights 0,1,2,...,t, W1 and W2 agree on A,
1 2 v v
for any v ∈ V .
0
(cid:3)
This completes the proof of the lemma.
ThefollowingpropositionallowsustoproveTheorem1.3inductively.
Proposition 3.6. Assume that π1 and π2 satisfy hypothesis H≤[n]. Let
2
t with [n] ≤ t ≤ n − 2. Suppose that for any s with 0 ≤ s ≤ t, the
2
representations π and π agree at height s. Then they agree at height
1 2
t+1.
Before proving the proposition, we apply it to the proof of our main
result as follows.
Proof of Theorem 1.3. Assume that π and π satisfy hypothesis
1 2
H≤[n2]. By Lemma 3.3, π1 and π2 agree at heights 1,2,...,[n2]. Note
thatby(3.1), π andπ alreadyagreeatheight0. ApplyingProposition
1 2
3.6 repeatedly for t from [n] to n − 2, we obtain that π and π also
2 1 2
agree at heights [n] + 1,...,n−1. Hence, π and π agree at all the
2 1 2
heights 0,1,...,n−1, that is, W1(g) = W2(g), for all g ∈ G and for
v v n
all v ∈ V . Therefore, π ∼= π . This concludes the proof of Theorem
0 1 2
(cid:3)
1.3.
Therefore, we only need to prove Proposition 3.6, which will be done
in Section 4.
4. Proof of Proposition 3.6
In this section, we prove Proposition 3.6.
Proof of Proposition 3.6. By Lemma 3.5,
I 0 0 I 0 0
n−t−1 n−t−1
W1 0 I 0 = W2 0 I 0
v  2t+2−n  v  2t+2−n 
0 X g 0 X g
   