Notes on L-functions for GLn
J.W. Cogdell
∗
Oklahoma State University, Stillwater, USA
Lectures given at the
School on Automorphic Forms on GL(n)
Trieste, 31 July - 18 August 2000
LNS0821003
∗
cogdell@math.okstate.edu
Abstract
The theory of L-functions of automorphic forms (or modular forms)
via integral representations has its origin in the paper of Riemann on
the ζ-function. However the theory was really developed in the classical
context of L-functions of modular forms for congruence subgroups of
SL2 (Z) by Hecke and his school. Much of our current theory is a direct
outgrowth of Hecke’s. L-functions of automorphic representations were
first developed by Jacquet and Langlands for GL2 . Their approach followed Hecke combined with the local-global techniques of Tate’s thesis.
The theory for GLn was then developed along the same lines in a long
series of papers by various combinations of Jacquet, Piatetski-Shapiro,
and Shalika. In addition to associating an L-function to an automorphic form, Hecke also gave a criterion for a Dirichlet series to come
from a modular form, the so-called Converse Theorem of Hecke. In the
context of automorphic representations, the Converse Theorem for GL2
was developed by Jacquet and Langlands, extended and significantly
strengthened to GL3 by Jacquet, Piatetski-Shapiro, and Shalika, and
then extended to GLn .
What we have attempted to present here is a synopsis of this work
and in doing so present the paradigm for the analysis of automorphic
L-functions via integral representations. We begin with the Fourier
expansion of a cusp form and results on Whittaker models since these
are essential for defining Eulerian integrals. We then develop integral representations for L-functions for GLn × GLm which have nice
analytic properties (meromorphic continuation, finite order of growth,
functional equations) and have Eulerian factorization into products of
local integrals. We next turn to the local theory of L-functions for
GLn , in both the archimedean and non-archimedean local contexts,
which comes out of the Euler factors of the global integrals. We finally
combine the global Eulerian integrals with the definition and analysis
of the local L-functions to define the global L-function of an automorphic representation and derive their major analytic properties. We next
turn to the various Converse Theorems for GLn and their applications
to Langlands liftings.
Contents
Introduction
79
1 Fourier Expansions and Multiplicity One Theorems
80
1.1 Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . 81
1.2 Whittaker Models and the Multiplicity One Theorem . . . . . 86
1.3 Kirillov Models and the Strong Multiplicity One Theorem . . 89
2 Eulerian Integrals for GLn
2.1 Eulerian Integrals for GL2 . . . . . . .
2.2 Eulerian Integrals for GLn × GLm with
2.2.1 The projection operator . . . .
2.2.2 The global integrals . . . . . .
2.3 Eulerian Integrals for GLn × GLn . . .
2.3.1 The mirabolic Eisenstein series
2.3.2 The global integrals . . . . . .
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m<n
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91
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3 Local L-functions
3.1 The Non-archimedean Local Factors . . . . . . .
3.1.1 The local L-function . . . . . . . . . . . .
3.1.2 The local functional equation . . . . . . .
3.1.3 The unramified calculation . . . . . . . .
3.1.4 The supercuspidal calculation . . . . . . .
3.1.5 Remarks on the general calculation . . . .
3.1.6 Multiplicativity and stability of γ–factors
3.2 The Archimedean Local Factors . . . . . . . . . .
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106
106
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111
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119
4 Global L-functions
4.1 The Basic Analytic Properties . . . . . . . . . .
4.2 Poles of L-functions . . . . . . . . . . . . . . .
4.3 Strong Multiplicity One . . . . . . . . . . . . .
4.4 Non-vanishing Results . . . . . . . . . . . . . .
4.5 The Generalized Ramanujan Conjecture (GRC)
4.6 The Generalized Riemann Hypothesis (GRH) .
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124
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5 Converse Theorems
134
5.1 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 Inverting the Integral Representation . . . . . . . . . . . . . . 138
5.3
5.4
5.5
Remarks on the Proofs . . . . . .
5.3.1 Theorem 5.1 . . . . . . .
5.3.2 Theorem 5.2 . . . . . . .
5.3.3 Theorem 5.3 . . . . . . .
5.3.4 Theorem 5.4 . . . . . . .
Converse Theorems and Liftings
Some Liftings . . . . . . . . . . .
References
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141
141
142
145
146
148
151
153
L-functions for GLn
79
Introduction
The purpose of these notes is to develop the analytic theory of L-functions
for cuspidal automorphic representations of GLn over a global field. There
are two approaches to L-functions of GLn : via integral representations or
through analysis of Fourier coefficients of Eisenstein series. In these notes
we develop the theory via integral representations.
The theory of L-functions of automorphic forms (or modular forms) via
integral representations has its origin in the paper of Riemann on the ζfunction [53]. However the theory was really developed in the classical context of L-functions of modular forms for congruence subgroups of SL2 (Z) by
Hecke and his school [25]. Much of our current theory is a direct outgrowth
of Hecke’s. L-functions of automorphic representations were first developed
by Jacquet and Langlands for GL2 [21, 28, 30]. Their approach followed
Hecke combined with the local-global techniques of Tate’s thesis [64]. The
theory for GLn was then developed along the same lines in a long series
of papers by various combinations of Jacquet, Piatetski-Shapiro, and Shalika [31–38, 47, 48, 62]. In addition to associating an L-function to an automorphic form, Hecke also gave a criterion for a Dirichlet series to come
from a modular form, the so-called Converse Theorem of Hecke [26]. In
the context of automorphic representations, the Converse Theorem for GL2
was developed by Jacquet and Langlands [30], extended and significantly
strengthened to GL3 by Jacquet, Piatetski-Shapiro, and Shalika [31], and
then extended to GLn [7, 9].
What we have attempted to present here is a synopsis of this work and in
doing so present the paradigm for the analysis of automorphic L-functions
via integral representations. Section 1 deals with the Fourier expansion of
automorphic forms on GLn and the related Multiplicity One and Strong
Multiplicity One theorems. Section 2 then develops the theory of Eulerian
integrals for GLn . In Section 3 we turn to the local theory of L-functions for
GLn , in both the archimedean and non-archimedean local contexts, which
comes out of the Euler factors of the global integrals. In Section 4 we finally
combine the global Eulerian integrals with the definition and analysis of the
local L-functions to define the global L-function of an automorphic representation and derive their major analytic properties. In Section 5 we turn
to the various Converse Theorems for GLn .
We have tried to keep the tone of the notes informal for the most part.
We have tried to provide complete proofs where feasible, at least sketches of
80
J.W. Cogdell
most major results, and references for technical facts.
There is another body of work on integral representations of L-functions
for GLn which developed out of the classical work on zeta functions of algebras. This is the theory of principal L-functions for GLn as developed by
Godement and Jacquet [22,28]. This approach is related to the one pursued
here, but we have not attempted to present it here.
The other approach to these L-functions is via the Fourier coefficients of
Eisenstein series. This approach also has a classical history. In the context
of automorphic representations, and in a broader context than GLn , this
approach was originally laid out by Langlands [43] but then most fruitfully
pursued by Shahidi. Some of the major papers of Shahidi on this subject
are [55–61]. In particular, in [58] he shows that the two approaches give the
same L-functions for GLn . We will not pursue this approach in these notes.
For a balanced presentation of all three methods we recommend the book
of Gelbart and Shahidi [16]. They treat not only L-functions for GLn but
L-functions of automorphic representations of other groups as well.
We have not discussed the arithmetic theory of automorphic representations and L-functions. For the connections with motives, we recommend the
surveys of Clozel [5] and Ramakrishnan [50].
1
Fourier Expansions and Multiplicity One Theorems
In this section we let k denote a global field, A, its ring of adeles, and ψ will
denote a continuous additive character of A which is trivial on k. For the
basics on adeles, characters, etc. we refer the reader to Weil [68] or the book
of Gelfand, Graev, and Piatetski-Shapiro [18].
We begin with a cuspidal automorphic representation (π, Vπ ) of GLn (A).
For us, automorphic forms are assumed to be smooth (of uniform moderate
growth) but not necessarily K∞ –finite at the archimedean places. This is
most suitable for the analytic theory. For simplicity, we assume the central
character ωπ of π is unitary. Then Vπ is the space of smooth vectors in an
irreducible unitary representation of GLn (A). We will always use cuspidal in
this sense: the smooth vectors in an irreducible unitary cuspidal automorphic
representation. (Any other smooth cuspidal representation π of GLn (A) is
necessarily of the form π = π ◦ ⊗ | det |t with π ◦ unitary and t real, so there
is really no loss of generality in the unitarity assumption. It merely provides
L-functions for GLn
81
us with a convenient normalization.) By a cusp form on GLn (A) we will
mean a function lying in a cuspidal representation. By a cuspidal function
we
R will simply mean a smooth function ϕ on GLn (k)\ GLn (A) satisfying
U(k)\ U(A) ϕ(ug) du ≡ 0 for every unipotent radical U of standard parabolic
subgroups of GLn .
The basic references for this section are the papers of Piatetski-Shapiro
[47, 48] and Shalika [62].
1.1
Fourier Expansions
Let ϕ(g) ∈ Vπ be a cusp form in the space of π. For arithmetic applications,
and particularly for the theory of L-functions, we will need the Fourier
expansion of ϕ(g).
If f (τ ) is a holomorphic cusp form on the upper half plane H, say with
respect to SL2 (Z), then f is invariant under integral translations, f (τ + 1) =
f (τ ) and thus has a Fourier expansion of the form
f (τ ) =
∞
X
an e2πinτ .
n=1
If ϕ(g) is a smooth cusp form on GL2 (A)
correspond
thenthe translations
1 x
to the maximal unipotent subgroup N2 = n =
and ϕ(ng) = ϕ(g)
0 1
for n ∈ N2 (k). So, if ψ is any continuous character of k\A we can define the
ψ-Fourier coefficient or ψ-Whittaker function by
Z
1 x
ϕ
g ψ −1 (x) dx.
Wϕ,ψ (g) =
0 1
k\A
We have the corresponding Fourier expansion
X
Wϕ,ψ (g).
ϕ(g) =
ψ
(Actually from abelian Fourier theory, one has
X
1 x
ϕ
g =
Wϕ,ψ (g)ψ(x)
0 1
ψ
as a periodic function of x ∈ A. Now set x = 0.)
82
J.W. Cogdell
If we fix a single non-trivial character ψ of k\A, then by standard duality
theory [18,68] the additive characters of the compact group k\A are isomorphic to k via the map γ ∈ k 7→ ψγ where ψγ is the character ψγ (x) = ψ(γx).
γ
Now, an elementary calculation shows that Wϕ,ψγ (g) = Wϕ.ψ
g
1
if γ 6= 0. If we set Wϕ = Wϕ,ψ for our fixed ψ, then the Fourier expansion
of ϕ becomes
X
γ
ϕ(g) = Wϕ,ψ0 (g) +
Wϕ
g .
1
×
γ∈k
Since ϕ is cuspidal
Wϕ,ψ0 (g) =
Z
k\A
ϕ
1 x
g dx ≡ 0
0 1
and the Fourier expansion for a cusp form ϕ becomes simply
X
γ
ϕ(g) =
Wϕ
g .
1
×
γ∈k
We will need a similar expansion for cusp forms ϕ on GLn (A). The
translations still correspond to the maximal unipotent subgroup



1 x1,2
∗









.

.




.
1





.
.
Nn = n = 
.. ..
 ,








1 xn−1,n 






0
1
but now this is non-abelian. This difficulty was solved independently by
Piatetski-Shapiro [47] and Shalika [62]. We fix our non-trivial continuous
character ψ of k\A as above. Extend it to a character of Nn by setting
ψ(n) = ψ(x1,2 + · · · + xn−1,n ) and define the associated Fourier coefficient or
Whittaker function by
Z
ϕ(ng)ψ −1 (n) dn.
Wϕ (g) = Wϕ,ψ (g) =
Nn (k)\ Nn (A)
Since ϕ is continuous and the integration is over a compact set this integral
is absolutely convergent, uniformly on compact sets. The Fourier expansion
takes the following form.
L-functions for GLn
83
Theorem 1.1 Let ϕ ∈ Vπ be a cusp form on GLn (A) and Wϕ its associated
ψ-Whittaker function. Then
X
γ
ϕ(g) =
Wϕ
g
1
γ∈Nn−1 (k)\GLn−1 (k)
with convergence absolute and uniform on compact subsets.
The proof of this fact is an induction. It utilizes the mirabolic subgroup
Pn of GLn which seems to be ubiquitous in the study of automorphic forms
on GLn . Abstractly, a mirabolic subgroup of GLn is simply the stabilizer
of a non-zero vector in (either) standard representation of GLn on kn . We
denote by Pn the stabilizer of the row vector en = (0, . . . , 0, 1) ∈ kn . So
h y n−1
h ∈ GLn−1 , y ∈ k
Pn = p =
≃ GLn−1 ⋉ Yn
1
where
Yn =
I
y y = n−1
y ∈ kn−1 ≃ kn−1 .
1
Simply by restriction of functions, a cusp form on GLn (A) restricts to a
smooth cuspidal function on Pn (A) which remains left invariant under Pn (k).
(A smooth Rfunction ϕ on Pn (A) which is left invariant under Pn (k) is called
cuspidal if U(k)\ U(A) ϕ(up) du ≡ 0 for every standard unipotent subgroup
U ⊂ Pn .) Since Pn ⊃ Nn we may define a Whittaker function attached to a
cuspidal function ϕ on Pn (A) by the same integral as on GLn (A), namely
Z
ϕ(np)ψ −1 (n) dn.
Wϕ (p) =
Nn (k)\ Nn (A)
We will prove by induction that for a cuspidal function ϕ on Pn (A) we
have
X
γ 0
ϕ(p) =
Wϕ
p
0 1
γ∈Nn−1 (k)\ GLn−1 (k)
with convergence absolute and uniform on compact subsets.
The function on Yn (A) given by y 7→ ϕ(yp) is invariant under Yn (k)
since Y n (k) ⊂ Pn (k) and ϕ is automorphic on Pn (A). Hence by standard
abelian Fourier analysis for Yn ≃ kn−1 we have as before
X
ϕ(p) =
ϕλ (p)
\
λ∈(k n−1
\An−1 )
84
J.W. Cogdell
where
ϕλ (p) =
Z
ϕ(yp)λ−1 (y) dy.
Yn (k)\ Y n (A)
\
Now, by duality theory [68], (kn−1
\An−1 ) ≃ kn−1 . In fact, if we let h , i
P
denote the pairing kn−1 × kn−1 → k by hx, yi = xi yi we have
X
ϕx (p)
ϕ(p) =
x∈k n−1
where now we write
ϕx (p) =
Z
ϕ(yp)ψ −1 (hx, yi) dy.
k n−1 \An−1
GLn−1 (k) acts on kn−1 with two orbits: {0} and kn−1 −{0} = GLn−1 (k)·t
en−1 where en−1 = (0, . . . , 0, 1). The stabilizer of ten−1 in GLn−1 (k) is t Pn−1 .
Therefore, we may write
X
ϕ(p) = ϕ0 (p) +
ϕγ·ten−1 (p).
γ∈GLn−1 (k)/t Pn−1 (k)
Since ϕ(p) is cuspidal and Yn is a standard unipotent subgroup of GLn ,
we see that
Z
ϕ(yp) dy ≡ 0.
ϕ0 (p) =
Yn (k)\ Y n (A)
On the other hand an elementary calculation as before gives
t
γ 0
ϕγ·ten−1 (p) = ϕten−1
p .
0 1
Hence we have
ϕ(p) =
X
γ∈Pn−1 (k)\ GLn−1 (k)
ϕten−1
γ 0
p
0 1
and the convergence is still absolute and uniform on compact subsets.
Note that if n = 2 this is exactly the fact we used previously for GL2 .
This then begins our induction.
Next, let us write the above in a form more suitable for induction.
Structurally, we have Pn = GLn−1 ⋉ Yn and Nn = Nn−1 ⋉ Y n . Therefore,
e n−1 = Pn−1 ⋉ Y n ⊂ Pn ,
Nn−1 \ GLn−1 ≃ Nn \ Pn . Furthermore, if we let P
L-functions for GLn
85
e n−1 \ Pn . Next, note that the function ϕte (p) satthen Pn−1 \ GLn−1 ≃ P
n−1
isfies, for y ∈ Y n (A) ≃ An−1 ,
ϕten−1 (yp) = ψ(yn−1 )ϕten−1 (p).
Since ψ is trivial on k we see that ϕten−1 (p) is left invariant under Yn (k).
Hence
ϕ(p) =
X
ϕten−1
γ∈Pn−1 (k)\ GLn−1 (k)
γ 0
p =
0 1
X
ϕten−1 (δp).
e n−1 (k)\ Pn (k)
δ∈P
To proceed with the induction, fix p ∈ Pn (A) and consider the function
= ϕ′p (p′ ) on Pn−1 (A) given by
ϕ′ (p′ )
ϕ′ (p′ ) = ϕten−1
p′ 0
p .
0 1
ϕ′ is a smooth function on Pn−1 (A) since ϕ was smooth. One checks that ϕ′
is left invariant by Pn−1 (k) and cuspidal on Pn−1 (A). Then we may apply
our inductive assumption to conclude that
′
′
ϕ (p ) =
X
W ϕ′
γ ′ ∈Nn−2 \ GLn−2
=
X
γ′ 0 ′
p
0 1
Wϕ′ (δ′ p′ ).
δ′ ∈Nn−1 \ Pn−1
If we substitute this into the expansion for ϕ(p) we see
ϕ(p) =
=
=
X
e n−1 (k)\ Pn (k)
δ∈P
X
e n−1 (k)\ Pn (k)
δ∈P
X
ϕten−1 (δp)
ϕ′δp (1)
X
e n−1 (k)\ Pn (k) δ′ ∈Nn−1 \ Pn−1
δ∈P
Wϕ′δp (δ′ ).
86
J.W. Cogdell
e n−1 and Nn ≃ Nn−1 ⋉ Y n−1 . Thus
Now, as before, Nn−1 \ Pn−1 ≃ Nn \P
Z
′
ϕ′δp (n′ δ′ )ψ −1 (n′ ) dn′
Wϕ′δp (δ ) =
Nn−1 (k)\ Nn−1 (A)
Z
Z
ϕ(yn′ δ′ δp)ψ −1 (yn−1 )ψ −1 (n′ ) dy dn′
=
N
(k)\ Nn−1 (A) Y n (k)\ Yn (A)
Z n−1
ϕ(nδ′ δp)ψ −1 (n) dn
=
=
Nn (k)\ Nn (A)
Wϕ (δ′ δp)
and so
ϕ(p) =
=
X
X
e n−1 (k)\ Pn (k) δ′ ∈Nn \P
e n−1
δ∈P
X
Wϕ (δp)
δ∈Nn (k)\ Pn (k)
=
Wϕ (δ′ δp)
X
Wϕ
γ∈Nn−1 (k)\ GLn−1 (k)
γ 0
p
0 1
which was what we wanted.
To obtain the Fourier expansion on GLn from this, if ϕ is a cusp form
on GLn (A), then for g ∈ Ω a compact subset the functions ϕg (p) = ϕ(pg)
form a compact family of cuspidal functions on Pn (A). So we have
X
γ 0
ϕg (1) =
W ϕg
0 1
γ∈Nn−1 (k)\ GLn−1 (k)
with convergence absolute and uniform. Hence
X
γ 0
ϕ(g) =
Wϕ
g
0 1
γ∈Nn−1 (k)\ GLn−1 (k)
again with absolute convergence, uniform for g ∈ Ω.
1.2
Whittaker Models and the Multiplicity One Theorem
Consider now the functions Wϕ appearing in the Fourier expansion of a
cusp form ϕ. These are all smooth functions W (g) on GLn (A) which satisfy
L-functions for GLn
87
W (ng) = ψ(n)W (g) for n ∈ Nn (A). If we let W(π, ψ) = {Wϕ | ϕ ∈ Vπ }
then GLn (A) acts on this space by right translation and the map ϕ 7→ Wϕ
intertwines Vπ with W(π, ψ). W(π, ψ) is called the Whittaker model of π.
The notion of a Whittaker model of a representation makes perfect sense
over a local field or even a finite field. Much insight can be gained by pursuing
these ideas over a finite field [20,49], but that would take us too far afield. So
let kv be a local field (a completion of k for example [18,68]) and let (πv , Vπv )
be an irreducible admissible smooth representation of GLn (kv ). Fix a nontrivial continuous additive character ψv of kv . Let W(ψv ) be the space of
all smooth functions W (g) on GLn (kv ) satisfying W (ng) = ψ(n)W (g) for
all n ∈ Nk (kv ), that is, the space of all smooth Whittaker functions on
GLn (kv ) with respect to ψv . This is also the space of the smooth induced
n
representation IndGL
Nv (ψv ). GLn (kv ) acts on this by right translation. If we
have a non-trivial continuous intertwining Vπv → W(ψv ) we will denote its
image by W(πv , ψv ) and call it a Whittaker model of πv .
Whittaker models for a representation (πv , Vπv ) are equivalent to continuous Whittaker functionals on Vπv , that is, continuous functionals Λv satisfying Λv (πv (n)ξv ) = ψv (n)Λv (ξv ) for all n ∈ Nn (kv ). To obtain a Whittaker
functional from a model, set Λv (ξv ) = Wξv (e), and to obtain a model from
a functional, set Wξv (g) = Λv (πv (g)ξv ). This is a form of Frobenius reciprocity, which in this context is the isomorphism between HomNn (Vπv , Cψv )
n
and HomGLn (Vπv , IndGL
Nn (ψv )) constructed above.
The fundamental theorem on the existence and uniqueness of Whittaker
functionals and models is the following.
Theorem 1.2 Let (πv , Vπv ) be a smooth irreducible admissible representation of GLn (kv ). Let ψv be a non-trivial continuous additive character of kv .
Then the space of continuous ψv –Whittaker functionals on Vπv is at most
one dimensional. That is, Whittaker models, if they exist, are unique.
This was first proven for non-archimedean fields by Gelfand and Kazhdan [19] and their results were later extended to archimedean local fields by
Shalika [62]. The method of proof is roughly the following. It is enough
n
to show that W(πv ) = IndGL
Nn (ψv ) is multiplicity free, i.e., any irreducible
representation of GLn (kv ) occurs in W(ψv ) with multiplicity at most one.
This in turn is a consequence of the commutativity of the endomorphism
algebra End(Ind(ψv )). Any intertwining map from Ind(ψv ) to itself is given
by convolution with a so-called Bessel distribution, that is, a distribution B
on GLn (kv ) satisfying B(n1 gn2 ) = ψv (n1 )B(g)ψv (n2 ) for n1 , n2 ∈ Nn (kv ).
88
J.W. Cogdell
Such quasi-invariant distributions are analyzed via Bruhat theory. By the
Bruhat decomposition for GLn , the double cosets Nn \ GLn / Nn are parameterized by the monomial matrices. The only double cosets that can support
are associated to permutation matrices of the form

 Bessel distributions
Irk
.
 and the resulting distributions are then stable under the

..
Ir1


1
. 
involution g 7→ gσ = wn tg wn with wn =  . .
the long Weyl
1
element of GLn . Thus for the convolution of Bessel distributions we have
B1 ∗B2 = (B1 ∗B2 )σ = B2σ ∗B1σ = B2 ∗B1 . Hence the algebra of intertwining
Bessel distributions is commutative and hence W(ψv ) is multiplicity free.
A smooth irreducible admissible representation (πv , Vπv ) of GLn (kv ) which
possesses a Whittaker model is called generic or non-degenerate. Gelfand
and Kazhdan in addition show that πv is generic iff its contragredient π
ev is
generic, in fact that π
e ≃ π ι where ι is the outer automorphism gι = tg−1 , and
in this case the Whittaker model for π
ev can be obtained as W(e
πv , ψv−1 ) =
t
−1
f
{W (g) = W (wn g ) | W ∈ W(π, ψv )}.
As a consequence of the local uniqueness of the Whittaker model we can
conclude a global uniqueness. If (π, Vπ ) is an irreducible smooth admissible
representation of GLn (A) then π factors as a restricted tensor product of
local representations π ≃ ⊗′ πv taken over all places v of k [14, 18]. Consequently we have a continuous embedding Vπv ֒→ Vπ for each local component.
Hence any Whittaker functional Λ on Vπ determines a family of local Whittaker functionals Λv on each Vπv and conversely such that Λ = ⊗′ Λv . Hence
global uniqueness follows from the local uniqueness. Moreover, once we fix
the isomorphism of Vπ with ⊗′ Vπv and define global and local Whittaker
functions via Λ and the corresponding family Λv we have a factorization of
global Whittaker functions
Y
Wξv (gv )
Wξ (g) =
v
for ξ ∈ Vπ which are factorizable in the sense that ξ = ⊗′ ξv corresponds to a
pure tensor. As we will see, this factorization, which is a direct consequence
of the uniqueness of the Whittaker model, plays a most important role in
the development of Eulerian integrals for GLn .
Now let us see what this means for our cuspidal representations (π, Vπ )
of GLn (A). We have seen that for any smooth cusp form ϕ ∈ Vπ we have
L-functions for GLn
89
the Fourier expansion
ϕ(g) =
X
γ∈Nn−1 (k)\GLn−1 (k)
Wϕ
γ
1
g .
We can thus conclude that W(π, ψ) 6= 0 and that π is (globally) generic with
Whittaker functional
Z
Λ(ϕ) = Wϕ (e) = ϕ(ng)ψ −1 (n) dn.
Thus ϕ is completely determined by its associated Whittaker function Wϕ .
From the uniqueness of the global Whittaker model we can derive the Multiplicity One Theorem of Piatetski-Shapiro [48] and Shalika [62].
Theorem (Multiplicity One) Let (π, Vπ ) be an irreducible smooth admissible representation of GLn (A). Then the multiplicity of π in the space
of cusp forms on GLn (A) is at most one.
Proof: Suppose that π has two realizations (π1 , Vπ1 ) and (π2 , Vπ2 ) in the space
of cusp forms on GLn (A). Let ϕi ∈ Vπi be the two cusp forms associated
to the vector ξ ∈ Vπ . Then we have two nonzero Whittaker functionals on
Vπ , namely Λi (ξ) = Wϕi (e). By the uniqueness of Whittaker models, there
is a nonzero constant c such that Λ1 = cΛ2 . But then we have Wϕ1 (g) =
Λ1 (π(g)ξ) = cΛ2 (π(g)ξ) = cWϕ2 (g) for all g ∈ GLn (A). Thus
X
γ
ϕ1 (g) =
W ϕ1
g
1
γ∈Nn−1 (k)\GLn−1 (k)
X
γ
=c
W ϕ2
g = cϕ2 (g).
1
γ∈Nn−1 (k)\GLn−1 (k)
But then Vπ1 and Vπ2 have a non-trivial intersection. Since they are irreducible representations, they must then coincide.
2
1.3
Kirillov Models and the Strong Multiplicity One Theorem
The Multiplicity One Theorem can be significantly strengthened. The Strong
Multiplicity One Theorem is the following.
90
J.W. Cogdell
Theorem (Strong Multiplicity One) Let (π1 , Vπ1 ) and (π2 , Vπ2 ) be two
cuspidal representations of GLn (A). Suppose there is a finite set of places S
such that for all v ∈
/ S we have π1,v ≃ π2,v . Then π1 = π2 .
There are two proofs of this theorem. One is based on the theory of Lfunctions and we will come to it in Section 4. The original proof of PiatetskiShapiro [48] is based on the Kirillov model of a local generic representation.
Let kv be a local field and let (πv , Vπv ) be an irreducible admissible
smooth generic representation of GLn (kv ), such as a local component of a
cuspidal representation π. Since πv is generic it has its Whittaker model
W(πv , ψv ). Each Whittaker function W ∈ W(πv , ψv ), since it is a function
on GLn (kv ), can be restricted to the mirabolic subgroup Pn (kv ). A fundamental result of Bernstein and Zelevinsky in the non-archimedean case [1]
and Jacquet and Shalika in the archimedean case [36] says that the map
ξv 7→ Wξv |Pn (kv ) is injective. Hence the representation has a realization on a
space of functions on Pn (kv ). This is the Kirillov model
K(πv , ψv ) = {W (p)|W ∈ W(πv , ψv )}.
Pn (kv ) acts naturally by right translation on K(πv , ψv ) and the action of all
of GLn (kv ) can be obtained by transport of structure. But for now, it is the
structure of K(πv , ψv ) as a representation of Pn (kv ) which is of interest.
For kv a non-archimedean field, let (τv , Vτv ) be the compactly induced
P (k )
representation τv = indNnn (kvv ) (ψv ). Then Bernstein and Zelevinsky have
analyzed the representations of Pn (kv ) and shown that whenever πv is an
irreducible admissible generic representation of GLn (kv ) then K(πv , ψv ) contains Vτv as a Pn (kv ) sub-representation and if πv is supercuspidal then
K(πv , ψv ) = Vτv [1].
For kv archimedean, we then let (τv , Vτv ) be the smooth vectors in the
P (k )
irreducible smooth unitarily induced representation IndNnn (kvv ) (ψv ). Then
Jacquet and Shalika have shown that as long as πv is an irreducible admissible smooth unitary representation of GLn (kv ) then in fact K(πv , ψv ) = Vτv
as representations of Pn (kv ) [36, Remark 3.15].
Therefore, for a given place v the local Kirillov models of any two irreducible admissible generic smooth unitary representations have a certain
Pn (kv )-submodule in common, namely Vτv .
Let us now return to Piatetski-Shapiro’s proof of the Strong Multiplicity
One Theorem [48].
L-functions for GLn
91
Proof: We begin with our cuspidal representations π1 and π2 . Since π1
and π2 are irreducible, it suffices to find a cusp form ϕ ∈ Vπ1 ∩ Vπ2 . If
we let Bn denote the Borel subgroup of upper triangular matrices in GLn ,
then Bn (k)\ Bn (A) is dense in GLn (k)\ GLn (A) and so it suffices to find
two cusp forms ϕi ∈ Vπi which agree on Bn (A). But Bn ⊂ Pn Zn with Zn
the center. If we let ωi be the central character of πi then by assumption
ω1,v = ω2,v for v ∈
/ S and the weak approximation theorem then implies
ω1 = ω2 . So it suffices to find two ϕi which agree on Pn (A). But as in the
proof of the Multiplicity One Theorem, via the Fourier expansion, to show
that ϕ1 (p) = ϕ2 (p) for p ∈ Pn (A) it suffices to show that Wϕ1 (p) = Wϕ2 (p).
Q
Since we can take each Wϕi to be of the form v Wϕi,v this then reduces
to a question about the local Kirillov models. For v ∈
/ S we have by assumption that K(π1,v , ψv ) = K(π2,v , ψv ) and for v ∈ S we have seen that
Vτv ⊂ K(π1,v , ψv ) ∩ K(π2,v , ψv ). So we can construct a common Whittaker function in the restriction of W(πi , ψ) to Pn (A). This completes the
proof.
2
2
Eulerian Integrals for GLn
Let f (τ ) again be a holomorphic cusp form of weight k on H for the full
modular group with Fourier expansion
f (τ ) =
X
an e2πinτ .
Then Hecke [25] associated to f an L-function
L(s, f ) =
X
an n−s
and analyzed its analytic properties, namely continuation, order of growth,
and functional equation, by writing it as the Mellin transform of f
Z ∞
−s
f (iy)y s d× y.
Λ(s, f ) = (2π) Γ(s)L(s, f ) =
0
An application of the modular transformation law for f (τ ) under the transformation τ 7→ −1/τ gives the functional equation
Λ(s, f ) = (−1)k/2 Λ(k − s, f ).
92
J.W. Cogdell
Moreover, if f was an eigenfunction of all Hecke operators then L(s, f ) had
an Euler product expansion
Y
L(s, f ) =
(1 − ap p−s + pk−1−2s )−1 .
p
We will present a similar theory for cuspidal automorphic representations
(π, Vπ ) of GLn (A). For applications to functoriality via the Converse Theorem (see Lecture 5) we will need not only the standard L-functions L(s, π)
but the twisted L-functions L(s, π × π ′ ) for (π ′ , Vπ′ ) a cuspidal automorphic
representation of GLm (A) for m < n as well. One point to notice from the
outset is that we want to associate a single L-function to an infinite dimensional representation (or pair of representations). The approach we will take
will be that of integral representations, but it will be broadened in the sense
of Tate’s thesis [64].
The basic references for this section are Jacquet-Langlands [30], Jacquet,
Piatetski-Shapiro, and Shalika [31], and Jacquet and Shalika [36].
2.1
Eulerian Integrals for GL2
Let us first consider the L-functions for cuspidal automorphic representations
(π, Vπ ) of GL2 (A) with twists by an idele class character χ, or what is the
same, a (cuspidal) automorphic representation of GL1 (A), as in JacquetLanglands [30].
Following Jacquet and Langlands, who were following Hecke, for each
ϕ ∈ Vπ we consider the integral
Z
a
χ(a)|a|s−1/2 d× a.
ϕ
I(s; ϕ, χ) =
1
×
×
k \A
Since a cusp form on GL2 (A) is rapidly decreasing upon restriction to A×
as in the integral, it follows that the integral is absolutely convergent for all
s, uniformly for Re(s) in an interval. Thus I(s; ϕ, χ) is an entire function of
s, bounded in any vertical strip a ≤ Re(s) ≤ b. Moreover, if we let ϕ(g)
e
=
t
−1
t
−1
ϕ( g ) = ϕ(wn g ) then ϕ
e ∈ Vπe and the simple change of variables a 7→
a−1 in the integral shows that each integral satisfies a functional equation of
the form
I(s; ϕ, χ) = I(1 − s; ϕ,
e χ−1 ).
So these integrals individually enjoy rather nice analytic properties.
L-functions for GLn
93
If we replace ϕ by its Fourier expansion from Lecture 1 and unfold, we
find
I(s; ϕ, χ) =
=
Z
Z
k × \A×
X
Wϕ
γ∈k ×
Wϕ
A×
a
1
γa
1
χ(a)|a|s−1/2 d× a
χ(a)|a|s−1/2 d× a
where we have used the fact that the function χ(a)|a|s−1/2 is invariant under
k× . By standard gauge estimates on Whittaker functions [31] this converges
for Re(s) >> 0 after the unfolding. As we have seen in Lecture 1, if Wϕ ∈
W(π, ψ) corresponds to a decomposable vector ϕ ∈ Vπ ≃ ⊗′ Vπv then the
Whittaker function factors into a product of local Whittaker functions
Wϕ (g) =
Y
Wϕv (gv ).
v
Since the character χ and the adelic absolute value factor into local components and the domain of integration A× also factors we find that our global
integral naturally factors into a product of local integrals
Z
A×
Wϕ
a
1
s−1/2
×
χ(a)|a|
d a=
YZ
v
kv×
W ϕv
av
1
χv (av )|av |s−1/2 d× av ,
with the infinite product still convergent for Re(s) >> 0, or
I(s; ϕ, χ) =
Y
Ψv (s; Wϕv , χv )
v
with the obvious definition of the local integrals
Ψv (s; Wϕv , χv ) =
Z
kv×
W ϕv
av
1
χv (av )|av |s−1/2 d× av .
Thus each of our global integrals is Eulerian.
In this way, to π and χ we have associated a family of global Eulerian
integrals with nice analytic properties as well as for each place v a family of
local integrals convergent for Re(s) >> 0.
94
2.2
J.W. Cogdell
Eulerian Integrals for GLn × GLm with m < n
Now let (π, Vπ ) be a cuspidal representation of GLn (A) and (π ′ , Vπ′ ) a cuspidal representation of GLm (A) with m < n. Take ϕ ∈ Vπ and ϕ′ ∈ Vπ′ . At
first blush, a natural analogue of the integrals we considered for GL2 with
GL1 twists would be
Z
h
ϕ
ϕ′ (h)| det(h)|s−(n−m)/2 dh.
I
n−m
GLm (k)\ GLm (A)
This family of integrals would have all the nice analytic properties as before
(entire functions of finite order satisfying a functional equation), but they
would not be Eulerian except in the case m = n − 1, which proceeds exactly
as in the GL2 case.
The problem is that the restriction of the form ϕ to GLm is too brutal to
allow a nice unfolding when the Fourier expansion of ϕ is inserted. Instead we
will introduce projection operators from cusp forms on GLn (A) to cuspidal
functions on on Pm+1 (A) which are given by part of the unipotent integration
through which the Whittaker function is defined.
2.2.1
The projection operator
In GLn , let Y n,m be the unipotent radical of the standard parabolic subgroup
attached to the partition (m + 1, 1, . . . , 1). If ψ is our standard additive
character of k\A, then ψ defines a character of Yn,m (A) trivial on Yn,m (k)
since Yn,m ⊂ Nn . The group Yn,m is normalized by GLm+1 ⊂ GLn and
the mirabolic subgroup Pm+1 ⊂ GLm+1 is the stabilizer in GLm+1 of the
character ψ.
Definition If ϕ(g) is a cusp form on GLn (A) define the projection operator
Pnm from cusp forms on GLn (A) to cuspidal functions on Pm+1 (A) by
“
”Z
n−m−1
p
−
2
Pnm ϕ(p) = | det(p)|
ϕ y
ψ −1 (y) dy
I
n−m−1
Yn,m (k)\ Yn,m (A)
for p ∈ Pm+1 (A).
As the integration is over a compact domain, the integral is absolutely
convergent. We first analyze the behavior on Pm+1 (A).
Lemma The function Pnm ϕ(p) is a cuspidal function on Pm+1 (A).
L-functions for GLn
95
Proof: Let us let ϕ′ (p) denote the non-normalized projection, i.e., for p ∈
Pm+1 (A) set
“
”
ϕ′ (p) = | det(p)|
n−m−1
2
Pnm ϕ(p).
It suffices to show this function is cuspidal. Since ϕ(g) was a smooth function
on GLn (A), ϕ′ (p) will remain smooth on Pm+1 (A). To see that ϕ′ (p) is
automorphic, let γ ∈ Pm+1 (k). Then
Z
γ 0
p 0
′
ϕ y
ϕ (γp) =
ψ −1 (y) dy.
0 1
0 1
Yn,m (k)\ Y n,m (A)
Since γ ∈ Pm+1 (k) and Pm+1 normalizes Yn,m and stabilizes ψ we may make
−1
γ 0
γ 0
the change of variable y 7→
y
in this integral to obtain
0 1
0 1
Z
γ 0
p 0
′
ϕ
y
ϕ (γp) =
ψ −1 (y) dy.
0 1
0 1
Yn,m (k)\ Y n,m (A)
Since ϕ(g) is automorphic on GLn (A) it is left invariant under GLn (k) and
we find that ϕ′ (γp) = ϕ′ (p) so that ϕ′ is indeed automorphic on Pm+1 (A).
We next need to see that ϕ′ is cuspidal on Pm+1 (A). To this end, let
U ⊂ Pm+1 be the standard unipotent subgroup associated to the partition
(n1 , . . . , nr ) of m + 1. Then we must compute the integral
Z
ϕ′ (up) du.
U(k)\ U(A)
Inserting the definition of ϕ′ we find
Z
ϕ′ (up) du
U(k)\ U(A)
=
Z
U(k)\ U(A)
Z
u 0 p 0
ϕ y
ψ −1 (y) dy du.
0
1
0
1
Yn,m (k)\ Yn,m (A)
The group U′ = U ⋉ Y n,m is the standard unipotent subgroup of GLn associated to the partition (n1 , . . . , nr , 1, . . . , 1) of n. We may decompose this
group in a second manner. If we let U′′ be the standard unipotent subgroup
e n−m−1
of GLn associated to the partition (n1 , . . . , nr , n−m−1) of n and let N
be the subgroup of GLn obtained by embedding Nn−m−1 into GLn by
I
0
n 7→ m+1
0
n
96
J.W. Cogdell
e n−m−1 ⋉ U′′ . If we extend the character ψ of Ym,n to U′ by
then U′ = N
e n−m−1 ⋉ U′′ , ψ
making it trivial on U, then in the decomposition U′ = N
e n−m−1 component and there it is the standard
is dependent only on the N
character ψ on Nn−m−1 . Hence we may rearrange the integration to give
Z
ϕ′ (up) du
U(k)\ U(A)
=
Z
Nn−m−1 (k)\ Nn−m−1 (A)
Z
′′
ϕ u
U′′ (k)\ U′′ (A)
1 0
0 n
p 0
du′′ ψ −1 (n) dy.
0 1
But since ϕ is cuspidal on GLn and U′′ is a standard unipotent subgroup of
GLn then
Z
p 0
′′ 1 0
du′′ ≡ 0
ϕ u
0 n
0 1
U′′ (k)\ U′′ (A)
from which it follows that
Z
ϕ′ (up) du ≡ 0
U(k)\ U(A)
so that ϕ′ is a cuspidal function on Pm+1 (A).
2
From Lecture 1, we know that cuspidal functions on Pm+1 (A) have a
Fourier expansion summed over Nm (k)\ GLm (A). Applying this expansion
to our projected cusp form on GLn (A) we are led to the following result.
h
n
Lemma Let ϕ be a cusp form on GLn (A). Then for h ∈ GLm (A), Pm ϕ
1
has the Fourier expansion
“
”
n−m−1
X
h
γ
0
h
−
2
Pnm ϕ
= | det(h)|
Wϕ
1
0 In−m
In−m
γ∈Nm (k)\ GLm (k)
with convergence absolute and uniform on compact subsets.
Proof: Once again let
′
“
ϕ (p) = | det(p)|
n−m−1
2
”
Pnm ϕ(p)
with p ∈ Pm+1 (A). Since we have verified that ϕ′ (p) is a cuspidal function
on Pm+1 (A) we know that it has a Fourier expansion of the form
X
γ 0
′
ϕ (p) =
W ϕ′
p
0 1
γ∈Nm (k)\ GLm (k)
L-functions for GLn
where
Wϕ′ (p) =
Z
ϕ′ (np)ψ −1 (n) dn.
Nm+1 (k)\ Nm+1 (A)
To obtain our expansion for Pnm ϕ we need
terms of ϕ rather than ϕ′ .
We have
Wϕ′ (p) =
=
Z
Z
97
to express the right-hand side in
ϕ′ (n′ p)ψ −1 (n′ ) dn′
Nm+1 (k)\ Nm+1 (A)
Nm+1 (k)\ Nm+1 (A)
′
np 0
ϕ y
g ·
0 1
Y n,m (k)\ Y n,m (A)
Z
· ψ −1 (y) dy ψ −1 (n′ ) dn′ .
It is elementary to see that the maximal unipotent subgroup Nn of GLn can
be factored as Nn = Nm+1 ⋉ Y n,m and if we write n = n′ y with n′ ∈ Nm+1
and y ∈ Yn,m then ψ(n) = ψ(n′ )ψ(y). Hence the above integral may be
written as
Z
p
0
p
0
−1
′
ϕ n
ψ (n) dn = Wϕ
Wϕ (p) =
.
0 In−m−1
0 In−m−1
Nn (k)\ Nn (A)
Substituting this expression into the above we find that
h
n
Pm ϕ
1
”
“
n−m−1
X
γ
0
h
−
2
Wϕ
= | det(h)|
0 In−m
In−m
γ∈Nm (k)\ GLm (k)
and the convergence is absolute and uniform for h in compact subsets of
GLm (A).
2
2.2.2
The global integrals
We now have the prerequisites for writing down a family of Eulerian integrals
for cusp forms ϕ on GLn twisted by automorphic forms on GLm for m <
n. Let ϕ ∈ Vπ be a cusp form on GLn (A) and ϕ′ ∈ Vπ′ a cusp form on
GLm (A). (Actually, we could take ϕ′ to be an arbitrary automorphic form
on GLm (A).) Consider the integrals
Z
h 0
n
′
Pm ϕ
I(s; ϕ, ϕ ) =
ϕ′ (h)| det(h)|s−1/2 dh.
0
1
GLm (k)\ GLm (A)
98
J.W. Cogdell
The integral I(s; ϕ, ϕ′ ) is absolutely convergent for all values of the complex
parameter s, uniformly in compact subsets, since the cusp forms are rapidly
decreasing. Hence it is entire and bounded in any vertical strip as before.
Let us now investigate the Eulerian properties of these integrals. We first
replace Pnm ϕ by its Fourier expansion.
Z
h
0
n
′
ϕ′ (h)| det(h)|s−1/2 dh
Pm ϕ
I(s; ϕ, ϕ ) =
0 In−m
GLm (k)\ GLm (A)
Z
X
γ
0
h
0
Wϕ
·
=
0 In−m
0 In−m
GLm (k)\ GLm (A)
γ∈Nm (k)\ GLm (k)
· ϕ′ (h)| det(h)|s−(n−m)/2 dh.
Since ϕ′ (h) is automorphic on GLm (A) and | det(γ)| = 1 for γ ∈ GLm (k)
we may interchange the order of summation and integration for Re(s) >> 0
and then recombine to obtain
Z
h
0
′
Wϕ
I(s; ϕ, ϕ ) =
ϕ′ (h)| det(h)|s−(n−m)/2 dh.
0
I
n−m
Nm (k)\ GLm (A)
This integral is absolutely convergent for Re(s) >> 0 by the gauge estimates
of [31, Section 13] and this justifies the interchange.
Let us now integrate first over Nm (k)\ Nm (A). Recall that for n ∈
Nm (A) ⊂ Nn (A) we have Wϕ (ng) = ψ(n)Wϕ (g). Hence we have
I(s; ϕ, ϕ′ ) =
Z
Nm (A)\ GLm (A)
=
=
=
Z
Z
Z
Wϕ
Nm (k)\ Nm (A)
Wϕ
Nm (A)\ GLm (A)
Wϕ
Nm (A)\ GLm (A)
Ψ(s; Wϕ , Wϕ′ ′ )
n
0
0 In−m
h
0
0 In−m
·
· ϕ′ (nh) dn | det(h)|s−(n−m)/2 dh
Z
0
ψ(n)ϕ′ (nh) dn
h
0 In−m
h
0 In−m
Nm (k)\ Nm (A)
· | det(h)|s−(n−m)/2 dh
0
Wϕ′ ′ (h)| det(h)|s−(n−m)/2 dh
where Wϕ′ ′ (h) is the ψ −1 -Whittaker function on GLm (A) associated to ϕ′ ,
i.e.,
Z
Wϕ′ ′ (h) =
ϕ′ (nh)ψ(n) dn,
Nm (k)\ Nm (A)
L-functions for GLn
99
and we retain absolute convergence for Re(s) >> 0.
From this point, the fact that the integrals are Eulerian is a consequence of the uniqueness of the Whittaker model for GLn . Take ϕ a smooth
cusp form in a cuspidal representation π of GLn (A). Assume in addition
that ϕ is factorizable, i.e., in the decomposition π = ⊗′ πv of π into a restricted tensor product of local representations, ϕ = ⊗ϕv is a pure tensor.
Then as we have seen there is a choice of local Whittaker models so that
Q
Wϕ (g) = Wϕv (gv ). Similarly for decomposable ϕ′ we have the factorizaQ
tion Wϕ′ ′ (h) = Wϕ′ ′v (hv ).
If we substitute these factorizations into our integral expression, then
Q
since the domain of integration factors Nm (A)\ GLm (A) = Nm (kv )\ GLm (kv )
we see that our integral factors into a product of local integrals
YZ
hv
0
′
W ϕv
Wϕ′ ′v (hv ) ·
Ψ(s; Wϕ , Wϕ′ ) =
0
I
n−m
Nm (kv )\ GLm (kv )
v
·| det(hv )|vs−(n−m)/2 dhv .
If we denote the local integrals by
Z
′
Ψv (s; Wϕv , Wϕ′v ) =
W ϕv
Nm (kv )\ GLm (kv )
hv
0
Wϕ′ ′v (hv ) ·
0 In−m
·| det(hv )|vs−(n−m)/2 dhv ,
which converges for Re(s) >> 0 by the gauge estimate of [31, Prop. 2.3.6],
we see that we now have a family of Eulerian integrals.
Now let us return to the question of a functional equation. As in the case
of GL2 , the functional equation is essentially a consequence of the existence
of the outer automorphism g 7→ ι(g) = gι = tg−1 of GLn . If we define
the action of this automorphism on automorphic forms by setting ϕ(g)
e
=
e n = ι ◦ Pn ◦ ι then our integrals naturally satisfy
ϕ(gι ) = ϕ(wn gι ) and let P
m
m
the functional equation
where
e ϕ, ϕ′ ) =
I(s;
Z
e − s; ϕ,
I(s; ϕ, ϕ′ ) = I(1
e ϕ
e′ )
GLm (k)\ GLm (A)
h
n
e
Pm ϕ
We have established the following result.
1
ϕ′ (h)| det(h)|s−1/2 dh.
100
J.W. Cogdell
Theorem 2.1 Let ϕ ∈ Vπ be a cusp form on GLn (A) and ϕ′ ∈ Vπ′ a cusp
form on GLm (A) with m < n. Then the family of integrals I(s; ϕ, ϕ′ ) define
entire functions of s, bounded in vertical strips, and satisfy the functional
equation
e − s; ϕ,
I(s; ϕ, ϕ′ ) = I(1
e ϕ
e′ ).
Moreover the integrals are Eulerian and if ϕ and ϕ′ are factorizable, we have
Y
I(s; ϕ, ϕ′ ) =
Ψv (s; Wϕv , Wϕ′ ′v )
v
with convergence absolute and uniform for Re(s) >> 0.
The integrals occurring on the right-hand side of our functional equation
are again Eulerian. One can unfold the definitions to find first that
e − s; ϕ,
e − s; ρ(wn,m )W
fϕ , W
f′ ′ )
I(1
e ϕ
e′ ) = Ψ(1
ϕ
where the unfolded global integral is

Z Z
h
′
e

Ψ(s; W, W ) =
W x In−m−1

 dx W ′ (h)| det(h)|s−(n−m)/2 dh
1
with the h integral over Nm (A)\ GLm (A) and the x integral over Mn−m−1,m (A),
the space of (n − m − 1) × m matrices, ρ denoting right translation,
and


1
.
Im

with wn−m =  . .
wn,m the Weyl element wn,m =
wn−m
1
the standard long Weyl element in GLn−m . Also, for W ∈ W(π, ψ) we set
f (g) = W (wn gι ) ∈ W(e
W
π , ψ −1 ). The extra unipotent integration is the reme W, W ′ ) is absolutely convergent for Re(s) >> 0.
en . As before, Ψ(s;
nant of P
m
′
e Wϕ , W ′ ′ ) will factor
For ϕ and ϕ factorizable as before, these integrals Ψ(s;
ϕ
as well. Hence we have
Y
e Wϕ , W ′ ′ ) =
e v (s; Wϕv , W ′ ′ )
Ψ(s;
Ψ
ϕ
ϕv
v
where
e v (s; Wv , Wv′ ) =
Ψ
Z Z

hv

Wv xv In−m−1

 dxv Wv′ (hv )| det(hv )|s−(n−m)/2 dhv
1
L-functions for GLn
101
where now the hv integral is over Nm (kv )\ GLm (kv ) and the xv integral is
over the matrix space Mn−m−1,m (kv ). Thus, coming back to our functional
equation, we find that the right-hand side is Eulerian and factors as
Y
e v (1−s; ρ(wn,m )W
fϕv , W
f ′ ′ ).
e
e
fϕ , W
f′ ′ ) =
Ψ
I(1−s;
ϕ,
e ϕ
e′ ) = Ψ(1−s;
ρ(wn,m )W
ϕv
ϕ
v
2.3
Eulerian Integrals for GLn × GLn
The paradigm for integral representations of L-functions for GLn × GLn is
not Hecke but rather the classical papers of Rankin [52] and Selberg [54].
These were first interpreted in the framework of automorphic representations
by Jacquet for GL2 × GL2 [28] and then Jacquet and Shalika in general [36].
Let (π, Vπ ) and (π ′ , Vπ′ ) be two cuspidal representations of GLn (A). Let
ϕ ∈ Vπ and ϕ′ ∈ Vπ′ be two cusp forms. The analogue of the construction
above would be simply
Z
ϕ(g)ϕ′ (g)| det(g)|s dg.
GLn (k)\ GLn (A)
This integral is essentially the L2 -inner product of ϕ and ϕ′ and is not
suitable for defining an L-function, although it will occur as a residue of our
integral at a pole. Instead, following Rankin and Selberg, we use an integral
representation that involves a third function: an Eisenstein series on GLn (A).
This family of Eisenstein series is constructed using the mirabolic subgroup
once again.
2.3.1
The mirabolic Eisenstein series
To construct our Eisenstein series we return to the observation that Pn \ GLn ≃
kn − {0}. If we let S(An ) denote the Schwartz–Bruhat functions on An , then
each Φ ∈ S defines a smooth function on GLn (A), left invariant by Pn (A),
by g 7→ Φ((0, . . . , 0, 1)g) = Φ(en g). Let η be a unitary idele class character.
(For our application η will be determined by the central characters of π and
π ′ .) Consider the function
Z
Φ(aen g)|a|ns η(a) d× a.
F (g, Φ; s, η) = | det(g)|s
A×
If we let P′n = Zn Pn be the parabolicof GL
n associated to the partition
h y
(n − 1, 1) then one checks that for p′ =
∈ P′n (A) with h ∈ GLn−1 (A)
0 d
102
J.W. Cogdell
and d ∈ A× we have,
F (p′ g, Φ; s, η) = | det(h)|s |d|−(n−1)s η(d)−1 F (g, Φ; s, η)
= δPs ′ (p′ )η −1 (d)F (g, Φ; s, η),
n
with the integral absolutely convergent for Re(s) > 1/n, so that if we extend
η to a character of P′n by η(p′ ) = η(d) in the above notation we have that
F (g, Φ; s, η) is a smooth section of the normalized induced representation
GLn (A) s−1/2
s−1/2
IndP′ (A)
(δP′ η). Since the inducing character δP′ η of P′n (A) is invarin
n
n
ant under P′n (k) we may form Eisenstein series from this family of sections
by
X
E(g, Φ; s, η) =
F (γg, Φ; s, η).
γ∈P′n (k)\ GLn (k)
If we replace F in this sum by its definition we can rewrite this Eisenstein
series as
Z
X
s
Φ(aξg)|a|ns η(a) d× a
E(g, Φ; s, η) = | det(g)|
k × \A× ξ∈k n −{0}
= | det(g)|s
Z
k × \A×
Θ′Φ (a, g)|a|ns η(a) d× a
and this first expression is convergent absolutely for Re(s) > 1 [36].
The second expression essentially gives the Eisenstein series as the Mellin
transform of the Theta series
X
ΘΦ (a, g) =
Φ(aξg),
ξ∈k n
where in the above we have written
X
Θ′Φ (a, g) =
Φ(aξg) = ΘΦ (a, g) − Φ(0).
ξ∈k n −{0}
This allows us to obtain the analytic properties of the Eisenstein series from
the Poisson summation formula for ΘΦ , namely
X
X
ΘΦ (a, g) =
Φ(aξg) =
Φa,g (ξ)
ξ∈k n
=
X
ξ∈k n
d
Φ
a,g (ξ) =
ξ∈k n
−n
= |a|
X
b −1 ξ tg−1 )
|a|−n | det(g)|−1 Φ(a
ξ∈k n
| det(g)|−1 ΘΦ̂ (a−1 ,t g−1 )
L-functions for GLn
103
where the Fourier transform Φ̂ on S(An ) is defined by
Z
Φ(y)ψ(y tx) dy.
Φ̂(x) =
A×
This allows us to write the Eisenstein series as
Z
s
E(g, Φ, s, η) = | det(g)|
Θ′Φ (a, g)|a|ns η(a) d× a
|a|≥1
Z
s−1
+ | det(g)|
Θ′Φ̂ (a,t g−1 )|a|n(1−s) η −1 (a) d× a + δ(s)
|a|≥1
where
δ(s) =
(
0
if η is ramified
s
−cΦ(0) | det(g)|
s+iσ
+
s−1
cΦ̂(0) | det(g)|
s−1+iσ
if η(a) = |a|inσ with σ ∈ R
with c a non-zero constant. From this we easily derive the basic properties
of our Eisenstein series [36, Section 4].
Proposition 2.1 The Eisenstein series E(g, Φ; s, η) has a meromorphic continuation to all of C with at most simple poles at s = −iσ, 1 − iσ when η
is unramified of the form η(a) = |a|inσ . As a function of g it is smooth of
moderate growth and as a function of s it is bounded in vertical strips (away
from the possible poles), uniformly for g in compact sets. Moreover, we have
the functional equation
E(g, Φ; s, η) = E(gι , Φ̂; 1 − s, η −1 )
where gι = tg−1 .
Note that under the center the Eisenstein series transforms by the central
character η −1 .
2.3.2
The global integrals
Now let us return to our Eulerian integrals. Let π and π ′ be our irreducible
cuspidal representations. Let their central characters be ω and ω ′ . Set
η = ωω ′ . Then for each pair of cusp forms ϕ ∈ Vπ and ϕ′ ∈ Vπ′ and each
Schwartz-Bruhat function Φ ∈ S(An ) set
Z
′
ϕ(g)ϕ′ (g)E(g, Φ; s, η) dg.
I(s; ϕ, ϕ , Φ) =
Zn (A) GLn (k)\ GLn (A)
104
J.W. Cogdell
Since the two cusp forms are rapidly decreasing on Zn (A) GLn (k)\ GLn (A)
and the Eisenstein series is only of moderate growth, we see that the integral
converges absolutely for all s away from the poles of the Eisenstein series
and is hence meromorphic. It will be bounded in vertical strips away from
the poles and satisfies the functional equation
I(s; ϕ, ϕ′ , Φ) = I(1 − s; ϕ,
e ϕ
e′ , Φ̂),
coming from the functional equation of the Eisenstein series, where we still
have ϕ(g)
e
= ϕ(gι ) = ϕ(wn gι ) ∈ Vπe and similarly for ϕ
e′ .
These integrals will be entire unless we have η(a) = ω(a)ω ′ (a) = |a|inσ
is unramified. In that case, the residue at s = −iσ will be
Z
′
ϕ(g)ϕ′ (g)| det(g)|−iσ dg
Res I(s; ϕ, ϕ , Φ) = −cΦ(0)
s=−iσ
Zn (A) GLn (A)\ GLn (A)
and at s = 1 − iσ we can write the residue as
Z
′
Res I(s; ϕ, ϕ , Φ) = cΦ̂(0)
s=1−iσ
Zn (A) GLn (k)\ GLn (A)
ϕ(g)
e ϕ
e′ (g)| det(g)|iσ dg.
Therefore these residues define GLn (A) invariant pairings between π and
π ′ ⊗ | det |−iσ or equivalently between π
e and π
e′ ⊗ | det |iσ . Hence a residues
′
iσ
can be non-zero only if π ≃ π
e ⊗ | det | and in this case we can find ϕ, ϕ′ ,
and Φ such that indeed the residue does not vanish.
We have yet to check that our integrals are Eulerian. To this end we
take the integral, replace the Eisenstein series by its definition, and unfold:
Z
′
ϕ(g)ϕ′ (g)E(g, Φ; s, η) dg
I(s; ϕ, ϕ , Φ) =
Zn (A) GLn (k)\ GLn (A)
Z
ϕ(g)ϕ′ (g)F (g, Φ; s, η) dg
=
Zn (A) P′n (k)\ GLn (A)
Z
Z
ϕ(g)ϕ′ (g)| det(g)|s
Φ(aen g)|a|ns η(a) da dg
=
×
Z (A) Pn (k)\ GLn (A)
A
Z n
ϕ(g)ϕ′ (g)Φ(en g)| det(g)|s dg.
=
Pn (k)\ GLn (A)
We next replace ϕ by its Fourier expansion in the form
X
ϕ(g) =
Wϕ (γg)
γ∈Nn (k)\ Pn (k)
L-functions for GLn
105
and unfold to find
Z
′
Wϕ (g)ϕ′ (g)Φ(en g)| det(g)|s dg
I(s; ϕ, ϕ , Φ) =
Nn (k)\ GLn (A)
Z
Z
ϕ′ (ng)ψ(n) dn Φ(en g)| det(g)|s dg
Wϕ (g)
=
Nn (k)\ Nn (A)
N (A)\ GLn (A)
Z n
Wϕ (g)Wϕ′ ′ (g)Φ(en g)| det(g)|s dg
=
=
Nn (A)\ GLn (A)
Ψ(s; Wϕ , Wϕ′ ′ , Φ).
This expression converges for Re(s) >> 0 by the gauge estimates as before.
To continue, we assume that ϕ, ϕ′ and Φ are decomposable tensors under
the isomorphisms π ≃ ⊗′ πv , π ′ ≃ ⊗′ πv′ , and S(An ) ≃ ⊗′ S(kvn ) so that we
Q
Q
Q
have Wϕ (g) = v Wϕv (gv ), Wϕ′ ′ (g) = v Wϕ′ ′v (gv ) and Φ(g) = v Φv (gv ).
Then, since the domain of integration also naturally factors we can decompose this last integral into an Euler product and now write
Y
Ψ(s; Wϕ , Wϕ′ ′ , Φ) =
Ψv (s; Wϕv , Wϕ′ ′v , Φv ),
v
where
Ψv (s; Wϕv , Wϕ′ ′v , Φv )
=
Z
Nn (kv )\ GLn (kv )
Wϕv (gv )Wϕ′ ′v (gv )Φv (en gv )| det(gv )|s dgv ,
still with convergence for Re(s) >> 0 by the local gauge estimates. Once
again we see that the Euler factorization is a direct consequence of the
uniqueness of the Whittaker models.
Theorem 2.2 Let ϕ ∈ Vπ and ϕ′ ∈ Vπ′ cusp forms on GLn (A) and let
Φ ∈ S(An ). Then the family of integrals I(s; ϕ, ϕ′ , Φ) define meromorphic
functions of s, bounded in vertical strips away from the poles. The only
possible poles are simple and occur iff π ≃ π
e′ ⊗ | det |iσ with σ real and are
then at s = −iσ and s = 1 − iσ with residues as above. They satisfy the
functional equation
fϕ , W
f ′ ′ , Φ̂).
I(s; ϕ, ϕ′ , Φ) = I(1 − s; W
ϕ
Moreover, for ϕ, ϕ′ , and Φ factorizable we have that the integrals are Eulerian and we have
Y
I(s; ϕ, ϕ′ , Φ) =
Ψv (s; Wϕv , Wϕ′ ′v , Φv )
v
with convergence absolute and uniform for Re(s) >> 0.
106
J.W. Cogdell
We remark in passing that the right-hand side of the functional equation
also unfolds as
Z
fϕ (g)W
f ′ ′ (g)Φ̂(en g)| det(g)|1−s dg
W
I(1 − s; ϕ,
e ϕ
e′ , Φ̂) =
ϕ
Nn (A)\ GLn (A)
=
Y
v
fϕ , W
f ′ ′ , Φ̂)
Ψv (1 − s; W
ϕ
with convergence for Re(s) << 0.
We note again that if these integrals are not entire, then the residues give
us invariant pairings between the cuspidal representations and hence tell us
structural facts about the relation between these representations.
3
Local L-functions
If (π, Vπ ) is a cuspidal representation of GLn (A) and (π ′ , Vπ′ ) is a cuspidal
representation of GLm (A) we have associated to the pair (π, π ′ ) a family of
Eulerian integrals {I(s; ϕ, ϕ′ )} (or {I(s; ϕ, ϕ′ , Φ)} if m = n) and through the
Euler factorization we have for each place v of k a family of local integrals
{Ψv (s; Wv , Wv′ )} (or {Ψv (s; Wv , Wv′ , Φv )}) attached to the pair of local components (πv , πv′ ). In this lecture we would like to attach a local L-function
(or local Euler factor) L(s, πv × πv′ ) to such a pair of local representations
through the family of local integrals and analyze its basic properties, including the local functional equation. The paradigm for such an analysis of local
L-functions is Tate’s thesis [64]. The mechanics of the archimedean and nonarchimedean theories are slightly different so we will treat them separately,
beginning with the non-archimedean theory.
3.1
The Non-archimedean Local Factors
For this section we will let k denote a non-archimedean local field. We
will let o denote the ring of integers of k and p the unique prime ideal
of o. Fix a generator ̟ of p. We let q be the residue degree of k, so
q = |o/p| = |̟|−1 . We fix a non-trivial continuous additive character ψ of k.
(π, Vπ ) and (π ′ , Vπ′ ) will now be the smooth vectors in irreducible admissible
unitary generic representations of GLn (k) and GLm (k) respectively, as is
true for local components of cuspidal representations. We will let ω and ω ′
denote their central characters.
The basic reference for this section is the paper of Jacquet, PiatetskiShapiro, and Shalika [33].
L-functions for GLn
3.1.1
107
The local L-function
For each pair of Whittaker functions W ∈ W(π, ψ) and W ′ ∈ W(π ′ , ψ −1 )
and in the case n = m each Schwartz-Bruhat function Φ ∈ S(kn ) we have
defined local integrals
′
Ψ(s; W, W ) =
e W, W ′ ) =
Ψ(s;
Z
Z
W
Nm (k)\ GLm (k)
Nm (k)\ GLm (k)
Z
h
In−m
Mn−m−1,m (k)
W ′ (h)| det(h)|s−(n−m)/2 dh

h

W x In−m−1
1

 dx
W ′ (h)| det(h)|s−(n−m)/2 dh
in the case m < n and
′
Ψ(s; W, W , Φ) =
Z
W (g)W ′ (g)Φ(en g)| det(g)|s dg
Nn (k)\ GLn (k)
in the case n = m, both convergent for Re(s) >> 0. To make the notation
more convenient for what follows, in the case m < n for any 0 ≤ j ≤ n−m−1
let us set


Z
Z
h
 dx
W x Ij
Ψj (s : W, W ′ ) =
Nm (k)\ GLm (k) Mj,m (k)
In−m−j
W ′ (h)| det(h)|s−(n−m)/2 dh,
e W, W ′ ) = Ψn−m−1 (s; W, W ′ ),
so that Ψ(s; W, W ′ ) = Ψ0 (s; W, W ′ ) and Ψ(s;
which is still absolutely convergent for Re(s) >> 0.
We need to understand what type of functions of s these local integrals
are. To this end, we need to understand the local Whittaker functions. So
let W ∈ W(π, ψ). Since W is smooth, there is a compact open subgroup
K, of finite index in the maximal compact subgroup Kn = GLn (o), so that
W (gk) = W (g) for all k ∈ K. If we let {ki } be a set of coset representatives
of GLn (o)/K, using that W transforms on the left under Nn (k) via ψ and
the Iwasawa decomposition on GLn (k) we see that W (g) is completely determined by the values of W (aki ) = Wi (a) for a ∈ An (k), the maximal split
(diagonal) torus of GLn (k). So it suffices to understand a general Whittaker
function on the torus. Let αi , i = 1, . . . , n − 1, denote the standard simple
108
J.W. Cogdell

a1

roots of GLn , so that if a = 
..
.


 ∈ An (k) then αi (a) = ai /ai+1 .
an
By a finite function on An (k) we mean a continuous function whose translates span a finite dimensional vector space [30, 31, Section 2.2]. (For the
field k× itself the finite functions are spanned by products of characters and
powers of the valuation map.) The fundamental result on the asymptotics
of Whittaker functions is then the following [31, Prop. 2.2].
Proposition 3.1 Let π be a generic representation of GLn (k). Then there
is a finite set of finite functions X(π) = {χi } on An (k), depending only
on π, so that for every W ∈ W(π, ψ) there are Schwartz–Bruhat functions
φi ∈ S(kn−1 ) such that for all a ∈ An (k) with an = 1 we have
X
W (a) =
χi (a)φi (α1 (a), . . . , αn−1 (a)).
X(π)
The finite set of finite functions X(π) which occur in the asymptotics
near 0 of the Whittaker functions come from analyzing the Jacquet module W(π, ψ)/hπ(n)W − W |n ∈ Nn i which is naturally an An (k)–module.
Note that due to the Schwartz-Bruhat functions, the Whittaker functions
vanish whenever any simple root αi (a) becomes large. The gauge estimates
alluded to in Section 2 are a consequence of this expansion and the one in
Proposition 3.6.
Several nice consequences follow from inserting these formulas for W and
W ′ into the local integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) [31, 33].
Proposition 3.2 The local integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) satisfy
the following properties.
1. Each integral converges for Re(s) >> 0. For π and π ′ unitary, as we
have assumed, they converge absolutely for Re(s) ≥ 1. For π and π ′
tempered, we have absolute convergence for Re(s) > 0.
2. Each integral defines a rational function in q −s and hence meromorphically extends to all of C.
3. Each such rational function can be written with a common denominator
which depends only on the finite functions X(π) and X(π ′ ) and hence
only on π and π ′ .
L-functions for GLn
109
In deriving these when m < n − 1 note that one has that


h
 6= 0
W x Ij
In−m−j−1
implies that x lies in a compact set independent of h ∈ GLm (k) [33].
Let Ij (π, π ′ ) denote the complex linear span of the local integrals Ψj (s; W ,
′
W ) if m < n and I(π, π ′ ) the complex linear span of the Ψ(s; W, W ′ , Φ) if
m = n. These are then all subspaces of C(q −s ) which have “bounded denominators” in the sense of (3). In fact, these subspaces have more structure
– they are modules for C[q s , q −s ] ⊂ C(q −s ). To see this, note that for any
h ∈ GLm (k) we have
h
′
′
W, π (h)W = | det(h)|−s−j+(n−m)/2 Ψj (s; W, W ′ )
Ψj s; π
In−m
and
Ψ(s; π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ψ(s; W, W ′ , Φ).
So by varying h and multiplying by scalars, we see that each Ij (π, π ′ ) and
I(π, π ′ ) is closed under multiplication by C[q s , q −s ]. Since we have bounded
denominators, we can conclude:
Proposition 3.3 Each Ij (π, π ′ ) and I(π, π ′ ) is a fractional C[q s , q −s ]–ideal
of C(q −s ).
Note that C[q s , q −s ] is a principal ideal domain, so that each fractional ideal
Ij (π, π ′ ) has a single generator, which we call Qj,π,π′ (q −s ), as does I(π, π ′ ),
which we call Qπ,π′ (q −s ). However, we can say more. In the case m < n
recall that from what we have said about the Kirillov model that when we
restrict Whittaker functions in W(π, ψ) to the embedded GLm (k) ⊂ Pn (k)
we get all functions of compact support on GLm (k) transforming by ψ. Using
this freedom for our choice of W ∈ W(π, ψ) one can show that in fact the
constant function 1 lies in Ij (π, π ′ ). In the case m = n one can reduce
to a sum of integrals over Pn (k) and then use the freedom one has in the
Kirillov model, plus the complete freedom in the choice of Φ to show that
once again 1 ∈ I(π, π ′ ). The consequence of this is that our generator
can be taken to be of the form Qj,π,π′ (q −s ) = Pj,π,π′ (q s , q −s )−1 for m < n
or Qπ,π′ (q −s ) = Pπ,π′ (q s , q −s )−1 for appropriate polynomials in C[q s , q −s ].
110
J.W. Cogdell
Moreover, since q s and q −s are units in C[q s , q −s ] we can always normalize
our generator to be of the form Pj,π,π′ (q −s )−1 or Pπ,π′ (q −s )−1 where the
polynomial P (X) satisfies P (0) = 1.
Finally, in the case m < n one can show by a rather elementary although somewhat involved manipulation of the integrals that all of the ideals
Ij (π, π ′ ) are the same [33, Section 2.7]. We will write this ideal as I(π, π ′ )
and its generator as Pπ,π′ (q −s )−1 .
This gives us the definition of our local L-function.
Definition Let π and π ′ be as above. Then L(s, π × π ′ ) = Pπ,π′ (q −s )−1 is
the normalized generator of the fractional ideal I(π, π ′ ) formed by the family
of local integrals. If π ′ = 1 is the trivial representation of GL1 (k) then we
write L(s, π) = L(s, π × 1).
One can easily show that the ideal I(π, π ′ ) is independent of the character
ψ used in defining the Whittaker models, so that L(s, π × π ′ ) is independent
of the choice of ψ. So it is not included in the notation. Also, note that
for π ′ = χ an automorphic representation (character) of GL1 (A) we have
the identity L(s, π × χ) = L(s, π ⊗ χ) where π ⊗ χ is the representation of
GLn (A) on Vπ given by π ⊗ χ(g)ξ = χ(det(g))π(g)ξ.
We summarize the above in the following Theorem.
Theorem 3.1 Let π and π ′ be as above. The family of local integrals form
a C[q s , q −s ]–fractional ideal I(π, π ′ ) in C(q −s ) with generator the local Lfunction L(s, π × π ′ ).
Another useful way of thinking of the local L-function is the following. L(s, π × π ′ ) is the minimal (in terms of degree) function of the form
P (q −s )−1 , with P (X) a polynomial satisfying P (0) = 1, such that the raΨ(s; W, W ′ )
Ψ(s; W, W ′ , Φ)
tios
(resp.
) are entire for all W ∈ W(π, ψ) and
L(s, π × π ′ )
L(s, π × π ′ )
W ′ ∈ W(π ′ , ψ −1 ), and if necessary Φ ∈ S(kn ). That is, L(s, π × π ′ ) is the
standard Euler factor determined by the poles of the functions in I(π, π ′ ).
One should note that since the L-factor is a generator of the ideal I(π, π ′ ),
then in particular it lies in I(π, π ′ ). Since this ideal is spanned by our local
integrals, we have the following useful Corollary.
Corollary There is a finite collection of Wi ∈ W(π, ψ), Wi′ ∈ W(π ′ , ψ −1 ),
and if necessary Φi ∈ S(kn ) such that
X
X
L(s, π × π ′ ) =
Ψ(s; Wi , Wi′ ) or L(s, π × π ′ ) =
Ψ(s; Wi , Wi′ , Φi ).
i
i
L-functions for GLn
111
For future reference, let us set
e(s; W, W ′ ) =
ẽ(s; W, W ′ ) =
Ψ(s; W, W ′ )
,
L(s, π × π ′ )
e W, W ′ )
Ψ(s;
,
L(s, π × π ′ )
and
e(s; W, W ′ , Φ) =
ej (s; W, W ′ ) =
Ψj (s; W, W ′ )
,
L(s, π × π ′ )
Ψ(s; W, W ′ , Φ)
.
L(s, π × π ′ )
Then all of these functions are Laurent polynomials in q ±s , i.e., elements of
C[q s , q −s ]. As such they are entire and bounded in vertical strips. As above,
P
there are choices of Wi , Wi′ , and if necessary Φi such that e(s; Wi , Wi′ ) ≡ 1
P
or
e(s; Wi , Wi′ , Φi ) ≡ 1. In particular we have the following result.
Corollary The functions e(s; W, W ′ ) and e(s; W, W ′ , Φ) are entire functions, bounded in vertical strips, and for each s0 ∈ C there is a choice of W ,
W ′ , and if necessary Φ such that e(s0 ; W, W ′ ) 6= 0 or e(s0 ; W, W ′ , Φ) 6= 0.
3.1.2
The local functional equation
Either by analogy with Tate’s thesis or from the corresponding global statement, we would expect our local integrals to satisfy a local functional equation. From the functional equations for our global integrals, we would expect
e − s; ρ(wn,m )W
f, W
f ′ ) when
these to relate the integrals Ψ(s; W, W ′ ) and Ψ(1
f, W
f ′ , Φ̂) when m = n. This will
m < n and Ψ(s; W, W ′ , Φ) and Ψ(1 − s; W
indeed be the case. These functional equations will come from interpreting the local integrals as families (in s) of quasi-invariant bilinear forms on
W(π, ψ) × W(π ′ , ψ −1 ) or trilinear forms on W(π, ψ) × W(π ′ , ψ −1 ) × S(kn )
depending on the case.
First, consider the case when m < n. In this case we have seen that
h
Ψ s; π
In−m
W, π ′ (h)W ′
= | det(h)|−s+(n−m)/2 Ψ(s; W, W ′ )
f, W
f ′ ) has the same quasi-invariance
and one checks that Ψ(1 − s; ρ(wn,m )W
as a bilinear form on W(π, ψ) × W(π ′ , ψ −1 ). In addition, if we let Yn,m
112
J.W. Cogdell
denote the unipotent radical of the standard parabolic subgroup associated
to the partition (m + 1, 1, . . . , 1) as before then we have the quasi-invariance
Ψ(s; π(y)W, W ′ ) = ψ(y)Ψ(s; W, W ′ )
e − s; ρ(wn,m )W
f, W
f ′ ) satisfies the
for all y ∈ Yn,m . One again checks that Ψ(1
same quasi-invariance as a bilinear form on W(π, ψ) × W(π ′ , ψ −1 ).
For n = m we have seen that
Ψ(s; π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ψ(s; W, W ′ , Φ)
f, W
f ′ , Φ̂) satisfies the same quasiand it is elementary to check that Ψ(1−s; W
invariance as a trilinear form on W(π, ψ) × W(π ′ , ψ −1 ) × S(kn ). Our local
functional equations will now follow from the following result [33, Propositions 2.10 and 2.11].
Proposition 3.4 (i) If m < n, then except for a finite number of exceptional
values of q −s there is a unique bilinear form Bs on W(π, ψ) × W(π ′ , ψ −1 )
satisfying
h
′
′
Bs π
W, π (h)W = | det(h)|−s+(n−m)/2 Bs (W, W ′ )
In−m
and
Bs (π(y)W, W ′ ) = ψ(y)Bs (W, W ′ )
for all h ∈ GLm (k) and y ∈ Yn,m (k).
(ii) If n = m, then except for a finite number of exceptional values of
−s
q there is a unique trilinear form Ts on W(π, ψ) × W(π ′ , ψ −1 ) × S(kn )
satisfying
Ts (π(h)W, π ′ (h)W ′ , ρ(h)Φ) = | det(h)|−s Ts (W, W ′ , Φ)
for all h ∈ GLn (k).
Let us say a few words about the proof of this proposition, because it
is another application of the analysis of the restriction of representations
of GLn to the mirabolic subgroup Pn [33, Sections 2.10 and 2.11]. In the
case where m < n the local integrals involve the restriction of the Whittaker
functions in W(π, ψ) to GLm (k) ⊂ Pn , that is, the Kirillov model K(π, ψ)
of π. In the case m = n one notes that S0 (kn ) = {Φ ∈ S(kn ) | Φ(0) = 0},
which has co-dimension one in S(kn ), is isomorphic to the compactly induced
L-functions for GLn
GL (k)
113
−1/2
n
(δPn ) so that by Frobenius reciprocity a GLn (k)
representation indPn (k)
quasi-invariant trilinear form on W(π, ψ) × W(π ′ , ψ −1 ) × S0 (kn ) reduces to
a Pn (k)-quasi-invariant bilinear form on K(π, ψ) × K(π ′ , ψ −1 ). So in both
cases we are naturally working in the restriction to Pn (k). The restrictions
of irreducible representations of GLn (k) to Pn (k) are no longer irreducible,
but do have composition series of finite length. One of the tools for analyzing
the restrictions of representations of GLn to Pn , or analyzing the irreducible
representations of Pn , are the derivatives of Bernstein and Zelevinsky [2,11].
These derivatives π (n−r) are naturally representations of GLr (k) for r ≤
n. π (0) = π and since π is generic the highest derivative π (n) corresponds
to the irreducible common submodule (τ, Vτ ) of all Kirillov models, and is
hence the non-zero irreducible representation of GL0 (k). The poles of our
local integrals can be interpreted as giving quasi-invariant pairings between
derivatives of π and π ′ [11]. The s for which such pairings exist for all
but the highest derivatives are the exceptional s of the proposition. There is
always a unique pairing between the highest derivatives π (n) and π ′(m) , which
are necessarily non-zero since they correspond to the common irreducible
subspace (τ, Vτ ) of any Kirillov model, and this is the unique Bs or Ts of the
proposition.
As a consequence of this Proposition, we can define the local γ-factor
which gives the local functional equation for our integrals.
Theorem 3.2 There is a rational function γ(s, π × π ′ , ψ) ∈ C(q −s ) such
that we have
e − s; ρ(wn,m )W
f, W
f ′ ) = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ )
Ψ(1
if m < n
or
f, W
f ′ , Φ̂) = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ , Φ)
Ψ(1 − s; W
if m = n
for all W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ −1 ), and if necessary all Φ ∈ S(kn ).
Again, if π ′ = 1 is the trivial representation of GL1 (k) we write γ(s, π, ψ) =
γ(s, π × 1, ψ). The fact that γ(s, π × π ′ , ψ) is rational follows from the fact
that it is a ratio of local integrals.
An equally important local factor, which occurs in the current formulations of the local Langlands correspondence [23, 26], is the local ε-factor.
114
J.W. Cogdell
Definition The local factor ε(s, π × π ′ , ψ) is defined as the ratio
ε(s, π × π ′ , ψ) =
γ(s, π × π ′ , ψ)L(s, π × π ′ )
.
L(1 − s, π
e×π
e′ )
With the local ε-factor the local functional equation can be written in the
form
e − s; ρ(wn,m )W
f, W
f′)
Ψ(1
Ψ(s; W, W ′ )
′
n−1
′
=
ω
(−1)
ε(s,
π
×π
,
ψ)
L(1 − s, π
e×π
e′ )
L(s, π × π ′ )
if m < n
or
f, W
f ′ , Φ̂)
Ψ(s; W, W ′ , Φ)
Ψ(1 − s; W
′
n−1
′
=
ω
(−1)
ε(s,
π
×
π
,
ψ)
L(1 − s, π
e×π
e′ )
L(s, π × π ′ )
if m = n .
This can also be expressed in terms of the e(s; W, W ′ ), etc.. In fact, since
we know we can choose a finite set of Wi , Wi′ , and if necessary Φi so that
X Ψ(s, ; Wi , W ′ )
=
X Ψ(s; Wi , W ′ , Φi )
=
i
i
or
L(s, π × π ′ )
i
i
L(s, π × π ′ )
X
i
we see that we can write either
ε(s, π × π ′ , ψ) = ω ′ (−1)n−1
X
e(s; Wi , Wi′ , Φi ) = 1
i
X
i
or
ε(s, π × π ′ , ψ) = ω ′ (−1)n−1
fi , W
f′)
ẽ(1 − s; ρ(wn,m )W
i
X
i
π ′ , ψ)
C[q s , q −s ].
and hence ε(s, π ×
∈
functional equation twice we get
e(s; Wi , Wi′ ) = 1
fi , W
f ′ , Φ̂i )
e(1 − s; W
i
On the other hand, applying the
ε(s, π × π ′ , ψ)ε(1 − s, π
e×π
e′ , ψ −1 ) = 1
so that ε(s, π × π ′ , ψ) is a unit in C[q s , q −s ]. This can be restated as:
Proposition 3.5 ε(s, π × π ′ , ψ) is a monomial function of the form cq −f s .
L-functions for GLn
115
Let us make a few remarks on the meaning of the number f occurring
in the ε–factor in the case of a single representation. Assume that ψ is
unramified. In this case write ε(s, π, ψ) = ε(0, π, ψ)q −f (π)s . In [34] it is
shown that f (π) is a non-negative integer, f (π) = 0 iff π is unramified, that
in general the space of vectors in Vπ which is fixed by the compact open
subgroup








f (π)
∗
K1 (p
) = g ∈ GLn (o)g ≡ 





0 ···

∗
.. 
.

∗
0 1





f (π)
(mod p
)




has dimension exactly 1, and that if t < f (π) then the dimension of the
space of fixed vectors for K1 (pt ) is 0. Depending on the context, either the
integer f (π) or the ideal f(π) = pf (π) is called the conductor of π. Note that
the analytically defined ε-factor carries structural information about π.
3.1.3
The unramified calculation
Let us now turn to the calculation of the local L-functions. The first case
to consider is that where both π and π ′ are unramified. Since they are
assumed generic, they are both full induced representations from unramified
n
characters of the Borel subgroup [69]. So let us write π ≃ IndGL
Bn (µ1 ⊗
′
′
′
m
· · · ⊗ µn ) and π ′ ≃ IndGL
Bm (µ1 ⊗ · · · ⊗ µm ) with the µi and µj unramified
characters of k× . The Satake parameterization of unramified representations
associates to each of these representations the semi-simple conjugacy classes
[Aπ ] ∈ GLn (C) and [Aπ′ ] ∈ GLm (C) given by


Aπ = 

µ1 (̟)
..
.
µn (̟)




Aπ′ = 
µ′1 (̟)
..
.


.
′
µm (̟)
(Recall that ̟ is a uniformizing parameter for k, that is, a generator of p.)
In the Whittaker models there will be unique normalized K = GL(o)–
fixed Whittaker functions, W◦ ∈ W(π, ψ) and W◦′ ∈ W(π ′ , ψ −1 ), normalized
by W◦ (e) = W◦′ (e) = 1. Let us concentrate on W◦ for the moment. Since
this function is right Kn –invariant and transforms on the left by ψ under Nn
we have that its values are completely determined by its values on diagonal
116
J.W. Cogdell
matrices of the form


̟J = 
̟ j1
..
.
̟ jn



for J = (j1 , . . . , jn ) ∈ Zn . There is an explicit formula for W◦ (̟ J ) in terms
of the Satake parameter Aπ due to Shintani [63] for GLn and generalized to
arbitrary reductive groups by Casselman and Shalika [4].
Let T + (n) be the set of n–tuples J = (j1 , . . . , jn ) ∈ Zn with j1 ≥ · · · ≥ jn .
Let ρJ be the rational representation of GLn (C) with dominant weight ΛJ
defined by


t1


j
..
ΛJ 
 = t11 · · · tjnn .
.
tn
Then the formula of Shintani says that
(
0
W◦ (̟ J ) =
1/2
δBn (̟ J ) tr(ρJ (Aπ ))
if J ∈
/ T + (n)
if J ∈ T + (n)
under the assumption that ψ is unramified. This is proved by analyzing the
recursion relations coming from the action of the unramified Hecke algebra
on W◦ .
We have a similar formula for W◦′ (̟ J ) for J ∈ Zm .
If we use these formulas in our local integrals, we find [36, I, Prop. 2.3]
J
X
̟
′
Ψ(s; W◦ , W◦ ) =
W◦
W◦′ (̟ J )
I
n−m
+
J∈T (m), jm ≥0
−1
(̟ J )
· | det(̟ J )|s−(n−m)/2 δB
m
X
=
tr(ρ(J,0) (Aπ )) tr(ρJ (Aπ′ ))q −|J|s
J∈T + (m), jm ≥0
=
X
tr(ρ(J,0) (Aπ ) ⊗ ρJ (Aπ′ ))q −|J|s
J∈T + (m), jm ≥0
where we let |J| = j1 +· · ·+jm and we embed Zm ֒→ Zn by J = (j1 , · · · , jm ) 7→
(J, 0) = (j1 , · · · , jm , 0, · · · , 0). We now use the fact from invariant theory
that
X
tr(ρ(J,0) (Aπ ) ⊗ ρJ (Aπ′ )) = tr(S r (Aπ ⊗ Aπ′ )),
J∈T + (m), jm ≥0, |J|=r
L-functions for GLn
117
where S r (A) is the r th -symmetric power of the matrix A, and
∞
X
tr(S r (A))z r = det(I − Az)−1
r=0
for any matrix A. Then we quickly arrive at
Ψ(s; W◦ , W◦′ ) = det(I − q −s Aπ ⊗ Aπ′ )−1 =
Y
(1 − µi (̟)µ′j (̟)q −s )−1
i,j
a standard Euler factor of degree mn. Since the L-function cancels all poles
of the local integrals, we know at least that det(I − q −s Aπ ⊗ Aπ′ ) divides
L(s, π × π ′ )−1 . Either of the methods discussed below for the general calculation of local factors then shows that in fact these are equal.
There is a similar calculation when n = m and Φ = Φ◦ is the characteristic function of the lattice on ⊂ kn . Also, since π unramified implies that
f◦ as its normalized unramified
its contragredient π
e is also unramified, with W
Whittaker function, then from the functional equation we can conclude that
in this situation we have ε(s, π × π ′ , ψ) ≡ 1.
Theorem 3.3 If π, π ′ , and ψ are all unramified, then
(
Ψ(s; W◦ , W◦′ )
−1
′
−s
L(s, π × π ) = det(I − q Aπ ⊗ Aπ′ ) =
Ψ(s; W◦ , W◦′ , Φ◦ )
m<n
m=n
and ε(s, π × π ′ , ψ) ≡ 1.
For future use, let us recall a consequence of this calculation due to
Jacquet and Shalika [36].
Corollary Suppose π is irreducible unitary generic admissible (our usual
assumptions on π) and unramified. Then the eigenvalues µi (̟) of Aπ all
satisfy q −1/2 < |µi (̟)| < q 1/2 .
To see this, we apply the above calculation to the case where π ′ = π̄ the
complex conjugate representation. Then Aπ′ = Aπ , the complex conjugate
matrix, and we have from the above
det(I − q −s Aπ ⊗ Aπ )Ψ(s; W◦ , W◦ , Φ◦ ) = 1.
The local integral in this case is absolutely convergent for Re(s) ≥ 1 and so
the factor det(I − q −s Aπ ⊗ Aπ ) cannot vanish for Re(s) ≥ 1. If µi (̟) is an
eigenvalue of Aπ then we have 1−q −σ |µi (̟)|2 6= 0 for σ ≥ 1. Hence |µi (̟)| <
q 1/2 . Note that if we apply this to the contragredient representation π
e as
−1/2
1/2
well we conclude that q
< |µi (̟)| < q .
118
3.1.4
J.W. Cogdell
The supercuspidal calculation
The other basic case is when both π and π ′ are supercuspidal. In this case
the restriction of W to Pn or W ′ to Pm lies in the Kirillov model and is
hence compactly supported mod N . In the case of m < n we find that
in our integral we have W evaluated along GLm (k) ⊂ Pn (k). Since W is
smooth, and hence stabilized by some compact open subgroup, we find that
the local integral always reduces to a finite sum and hence lies in C[q s , q −s ].
In particular it is always entire. Thus in this case L(s, π × π ′ ) ≡ 1. In the
case n = m the calculation is a bit more involved and can be found in [11,15].
In essence, in the family of integrals Ψ(s; W, W ′ , Φ), if Φ(0) = 0 then the
integral will again reduce to a finite sum and hence be entire. If Φ(0) 6= 0
and if s0 is a pole of Ψ(s; W, W ′ , Φ) then the residue of the pole at s = s0
will be of the form
Z
W (g)W ′ (g)| det(g)|s0 dg
cΦ(0)
Zn (k) Nn (k)\ GLn (k)
which is the Whittaker form of an invariant pairing between π and π ′ ⊗
| det |s0 . Thus we must have s0 is pure imaginary and π
e ≃ π ′ ⊗ | det |s0 for
the residue to be nonzero. This condition is also sufficient.
Theorem 3.4 If π and π ′ are both (unitary) supercuspidal, then L(s, π ×
π ′ ) ≡ 1 if m < n and if m = n we have
Y
L(s, π × π ′ ) =
(1 − αq −s )−1
with the product over all α = q s0 with π
e ≃ π ′ ⊗ | det |s0 .
3.1.5
Remarks on the general calculation
In the other cases, we must rely on the Bernstein–Zelevinsky classification
of generic representations of GLn (k) [69]. All generic representations can be
realized as prescribed constituents of representations parabolically induced
from supercuspidals. One can proceed by analyzing the Whittaker functions
of induced representations in terms of Whittaker functions of the inducing
data as in [33] or by analyzing the poles of the local integrals in terms of quasi
invariant pairings of derivatives of π and π ′ as in [11] to compute L(s, π × π ′ )
in terms of L-functions of pairs of supercuspidal representations. We refer
you to those papers or [42] for the explicit formulas.
L-functions for GLn
3.1.6
119
Multiplicativity and stability of γ–factors
To conclude this section, let us mention two results on the γ-factors. One
is used in the computations of L-factors in the general case. This is the
multiplicativity of γ-factors [33]. The second is the stability of γ-factors [37].
Both of these results are necessary in applications of the Converse Theorem
to liftings, which we discuss in Section 5.
Proposition (Multiplicativity of γ-factors) If π = Ind(π1 ⊗ π2 ), with
πi and irreducible admissible representation of GLri (k), then
γ(s, π × π ′ , ψ) = γ(s, π1 × π ′ , ψ)γ(s, π2 × π ′ , ψ)
and similarly for π ′ . Moreover L(s, π × π ′ )−1 divides [L(s, π1 × π ′ )L(s, π2 ×
π ′ )]−1 .
Proposition (Stability of γ-factors) If π1 and π2 are two irreducible
admissible generic representations of GLn (k), having the same central character, then for every sufficiently highly ramified character η of GL1 (k) we
have
γ(s, π1 × η, ψ) = γ(s, π2 × η, ψ)
and
L(s, π1 × η) = L(s, π2 × η) ≡ 1.
More generally, if in addition π ′ is an irreducible generic representation of
GLm (k) then for all sufficiently highly ramified characters η of GL1 (k) we
have
γ(s, (π1 ⊗ η) × π ′ , ψ) = γ(s, (π2 ⊗ η) × π ′ , ψ)
and
L(s, (π1 ⊗ η) × π ′ ) = L(s, (π2 ⊗ η) × π ′ ) ≡ 1.
3.2
The Archimedean Local Factors
We now take k to be an archimedean local field, i.e., k = R or C. We
take (π, Vπ ) to be the space of smooth vectors in an irreducible admissible
unitary generic representation of GLn (k) and similarly for the representation
120
J.W. Cogdell
(π ′ , Vπ′ ) of GLm (k). We take ψ a non-trivial continuous additive character
of k.
The treatment of the archimedean local factors parallels that of the nonarchimedean in many ways, but there are some significant differences. The
major work on these factors is that of Jacquet and Shalika in [38], which we
follow for the most part without further reference, and in the archimedean
parts of [36].
One significant difference in the development of the archimedean theory
is that the local Langlands correspondence was already in place when the
theory was developed [45]. The correspondence is very explicit in terms of
the usual Langlands classification. Thus to π is associated an n dimensional
semi-simple representation τ = τ (π) of the Weil group Wk of k and to π ′ an
m-dimensional semi-simple representation τ ′ = τ (π ′ ) of Wk . Then τ (π) ⊗
τ (π ′ ) is an nm dimensional representation of Wk and to this representation
of the Weil group is attached Artin-Weil L– and ε–factors [65], denoted
L(s, τ ⊗ τ ′ ) and ε(s, τ ⊗ τ ′ , ψ). In essence, Jacquet and Shalika define
L(s, π×π ′ ) = L(s, τ (π)⊗τ (π ′ ))
and
ε(s, π×π ′ , ψ) = ε(s, τ (π)⊗τ (π ′ ), ψ)
and then set
γ(s, π × π ′ , ψ) =
ε(s, π × π ′ , ψ)L(1 − s, π
e×π
e′ )
.
L(s, π × π ′ )
For example, if π is unramified, and hence of the form π ≃ Ind(µ1 ⊗ · · · ⊗
µn ) with unramified characters of the form µi (x) = |x|ri then
L(s, π) = L(s, τ (π)) =
n
Y
Γv (s + ri )
i=1
is a standard archimedean Euler factor of degree n, where
(
π −s/2 Γ( 2s )
if kv = R
Γv (s) =
.
2(2π)−s Γ(s) if kv = C
They then proceed to show that these functions have the expected relation to the local integrals. Their methods of analyzing the local integrals
Ψj (s; W, W ′ ) and Ψ(s; W, W ′ , Φ), defined as in the non-archimedean case
for W ∈ W(π, ψ), W ′ ∈ W(π ′ , ψ −1 ), and Φ ∈ S(kn ), are direct analogues
of those used in [33] for the non-archimedean case. Once again, a most
important fact is [38, Proposition 2.2]
L-functions for GLn
121
Proposition 3.6 Let π be a generic representation of GLn (k). Then there
is a finite set of finite functions X(π) = {χi } on An (k), depending only
on π, so that for every W ∈ W(π, ψ) there are Schwartz functions φi ∈
S(kn−1 × Kn ) such that for all a ∈ An (k) with an = 1 we have
X
W (nak) = ψ(n)
χi (a)φi (α1 (a), . . . , αn−1 (a), k).
X(π)
Now the finite functions are related to the exponents of the representation
π and through the Langlands classification to the representation τ (π) of
Wk . We retain the same convergence statements as in the non-archimedean
case [36, I, Proposition 3.17; II, Proposition 2.6], [38, Proposition 5.3].
Proposition 3.7 The integrals Ψj (s; W, W ′ ) and Ψ(s; W, W ′ , Φ) converge
absolutely in the half plane Re(s) ≥ 1 under the unitarity assumption and
for Re(s) > 0 if π and π ′ are tempered.
The meromorphic continuation and “bounded denominator” statement
in the case of a non-archimedean local field is now replaced by the following.
Define M(π × π ′ ) to be the space of all meromorphic functions φ(s) with
the property that if P (s) is a polynomial function such that P (s)L(s, π × π ′ )
is holomorphic in a vertical strip S[a, b] = {s a ≤ Re(s) ≤ b} then P (s)φ(s)
is bounded in S[a, b]. Note in particular that if φ ∈ M(π × π ′ ) then the
quotient φ(s)L(s, π × π ′ )−1 is entire.
Theorem 3.5 The integrals Ψj (s; W, W ′ ) or Ψ(s; W, W ′ , Φ) extend to meromorphic functions of s which lie in M(π × π ′ ). In particular, the ratios
ej (s; W, W ′ ) =
Ψj (s; W, W ′ )
L(s, π × π ′ )
or
e(s; W, W ′ , Φ) =
Ψ(s; W, W ′ , Φ)
L(s, π × π ′ )
are entire and in fact are bounded in vertical strips.
This statement has more content than just the continuation and “bounded
denominator” statements in the non-archimedean case. Since it prescribes
the “denominator” to be the L factor L(s, π × π ′ )−1 it is bound up with
the actual computation of the poles of the local integrals. In fact, a significant part of the paper of Jacquet and Shalika [38] is taken up with the
simultaneous proof of this and the local functional equations:
122
J.W. Cogdell
Theorem 3.6 We have the local functional equations
f, W
f ′ ) = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψj (s; W, W ′ )
Ψn−m−j−1 (1 − s; ρ(wn,m )W
or
f, W
f ′ , Φ̂) = ω ′ (−1)n−1 γ(s, π × π ′ , ψ)Ψ(s; W, W ′ , Φ).
Ψ(1 − s; W
The one fact that we are missing is the statement of “minimality” of
the L-factor. That is, we know that L(s, π × π ′ ) is a standard archimedean
Euler factor (i.e., a product of Γ-functions of the standard type) and has the
property that the poles of all the local integrals are contained in the poles
of the L-factor, even with multiplicity. But we have not established that
the L-factor cannot have extraneous poles. In particular, we do know that
we can achieve the local L-function as a finite linear combination of local
integrals.
Towards this end, Jacquet and Shalika enlarge the allowable space of local
integrals. Let Λ and Λ′ be the Whittaker functionals on Vπ and Vπ′ associated
with the Whittaker models W(π, ψ) and W(π ′ , ψ −1 ). Then Λ̂ = Λ ⊗ Λ′
defines a continuous linear functional on the algebraic tensor product Vπ ⊗
Vπ′ which extends continuously to the topological tensor product Vπ⊗π′ =
ˆ π′ , viewed as representations of GLn (k) × GLm (k).
Vπ ⊗V
Before proceeding, let us make a few remarks on smooth representations.
If (π, Vπ ) is the space of smooth vectors in an irreducible admissible unitary
representation, then the underlying Harish-Chandra module is the space of
Kn -finite vectors Vπ,K . Vπ then corresponds to the (Casselman-Wallach)
canonical completion of Vπ,K [66]. The category of Harish-Chandra modules
is appropriate for the algebraic theory of representations, but it is useful to
work in the category of smooth admissible representations for automorphic
forms. If in our context we take the underlying Harish-Chandra modules
Vπ,K and Vπ′ ,K then their algebraic tensor product is an admissible HarishChandra module for GLn (k) × GLm (k). The associated smooth admissible
representation is the canonical completion of this tensor product, which is in
fact Vπ⊗π′ , the topological tensor product of the smooth representations π
and π ′ . It is also the space of smooth vectors in the unitary tensor product
of the unitary representations associated to π and π ′ . So this completion is a
natural place to work in the category of smooth admissible representations.
Now let
W(π ⊗ π ′ , ψ) = {W (g, h) = Λ̂(π(g) ⊗ π ′ (h)ξ)|ξ ∈ Vπ⊗π′ }.
L-functions for GLn
123
Then W(π⊗π ′ , ψ) contains the algebraic tensor product W(π, ψ)⊗W(π ′ , ψ −1 )
and is again equal to the topological tensor product. Now we can extend all
our local integrals to the space W(π ⊗ π ′ , ψ) by setting

 
Z Z
h
 , h dx | det(h)|s−(n−m)/2 dh
Ψj (s; W ) =
W x Ij
In−m−j
and
Ψ(s; W, Φ) =
Z
W (g, g)Φ(en g)| det(g)|s dh
for W ∈ W(π ⊗ π ′ , ψ). Since the local integrals are continuous with respect
to the topology on the topological tensor product, all of the above facts
remain true, in particular the convergence statements, the local functional
equations, and the fact that these integrals extend to functions in M(π ×π ′ ).
At this point, let Ij (π, π ′ ) = {Ψj (s; W )|W ∈ W(π ⊗ π ′ )} and let I(π, π ′ )
be the span of the local integrals {Ψ(s; W, Φ)|W ∈ W(π ⊗π ′ , ψ), φ ∈ S(kn )}.
Once again, in the case m < n we have that the space Ij (π, π ′ ) is independent
of j and we denote it also by I(π, π ′ ). These are still independent of ψ. So
we know from above that I(π, π ′ ) ⊂ M(π × π ′ ). The remainder of [38] is
then devoted to showing the following.
Theorem 3.7 I(π, π ′ ) = M(π × π ′ ).
As a consequence of this, we draw the following useful Corollary.
Corollary There is a Whittaker function W in W(π ⊗ π ′ , ψ) such that
L(s, π × π ′ ) = Ψ(s; W ) if m < n or finite collectionof functions Wi ∈ W(π ⊗
P
π ′ , ψ) and Φi ∈ S(kn ) such that L(s, π × π ′ ) = i Ψ(s; Wi , Φi ) if m = n.
In the cases of m = n − 1 or m = n, Jacquet and Shalika can indeed get
the local L-function as a finite linear combination of integrals involving only
K-finite functions in W(π, ψ) and W(π ′ , ψ −1 ), that is, without going to the
completion of W(π, ψ) ⊗ W(π ′ , ψ −1 ), but this has not been published.
As a final result, let us note that in [12] it is established that the linear
functionals e(s; W ) = Ψ(s; W )L(s, π × π ′ )−1 and e(s; W, Φ) = Ψ(s; W, Φ)
L(s, π × π ′ )−1 are continuous on W(π ⊗ π ′ , ψ), uniformly for s in compact
sets. Since there is a choice of W ∈ W(π ⊗ π ′ , ψ) such that e(s; W ) ≡ 1
P
or Wi ∈ W(π ⊗ π ′ , ψ) and Φi ∈ S(kn ) such that
e(s; Wi , Φi ) ≡ 1, as
a result of this continuity and the fact that the algebraic tensor product
W(π, ψ) ⊗ W(π ′ , ψ −1 ) is dense in W(π ⊗ π ′ , ψ) we have the following result.
124
J.W. Cogdell
Proposition 3.8 For any s0 ∈ C there are choices of W ∈ W(π, ψ), W ′ ∈
W(π ′ , ψ −1 ) and if necessary Φ such that e(s0 ; W, W ′ ) 6= 0 or e(s0 ; W, W ′ , Φ) 6=
0.
4
Global L-functions
Once again, we let k be a global field, A its ring of adeles, and fix a non-trivial
continuous additive character ψ = ⊗ψv of A trivial on k.
Let (π, Vπ ) be a cuspidal representation of GLn (A) (see Section 1 for
all the implied assumptions in this terminology) and (π ′ , Vπ′ ) a cuspidal
representation of GLm (A). Since they are irreducible we have restricted
tensor product decompositions π ≃ ⊗′ πv and π ′ ≃ ⊗′ πv′ with (πv , Vπv ) and
(πv′ , Vπv′ ) irreducible admissible smooth generic unitary representations of
GLn (kv ) and GLm (kv ) [14,18]. Let ω = ⊗′ ωv and ω ′ = ⊗′ ωv′ be their central
characters. These are both continuous characters of k× \A× .
Let S be the finite set of places of k, containing the archimedean places
S∞ , such that for all v ∈
/ S we have that πv , πv′ , and ψv are unramified.
For each place v of k we have defined the local factors L(s, πv × πv′ ) and
ε(s, πv × πv′ , ψv ). Then we can at least formally define
Y
Y
L(s, π × π ′ ) =
L(s, πv × πv′ ) and ε(s, π × π ′ ) =
ε(s, πv × πv′ , ψv ).
v
v
We need to discuss convergence of these products. Let us first consider
the convergence of L(s, π × π ′ ). For those v ∈
/ S, so πv , πv′ , and ψv are
′
unramified, we know that L(s, πv × πv ) = det(I − qv−s Aπv ⊗ Aπv′ )−1 and that
1/2
the eigenvalues of Aπv and Aπv′ are all of absolute value less than qv . Thus
the partial (or incomplete) L-function
Y
Y
LS (s, π × π ′ ) =
L(s, πv × πv′ ) =
det(I − q −s Aπv ⊗ Aπv′ )−1
v∈S
/
v∈S
/
is absolutely convergent for Re(s) >> 0. Thus the same is true for L(s, π ×
π ′ ).
For the ε–factor, we have seen that ε(s, πv × πv′ , ψv ) ≡ 1 for v ∈
/ S so
that the product is in fact a finite product and there is no problem with
convergence. The fact that ε(s, π × π ′ ) is independent of ψ can either be
checked by analyzing how the local ε–factors vary as you vary ψ, as is done in
[7, Lemma 2.1], or it will follow from the global functional equation presented
below.
L-functions for GLn
4.1
125
The Basic Analytic Properties
Our first goal is to show that these L-functions have nice analytic properties.
Theorem 4.1 The global L–functions L(s, π × π ′ ) are nice in the sense that
1. L(s, π × π ′ ) has a meromorphic continuation to all of C,
2. the extended function is bounded in vertical strips (away from its poles),
3. they satisfy the functional equation
L(s, π × π ′ ) = ε(s, π × π ′ )L(1 − s, π
e×π
e′ ).
To do so, we relate the L-functions to the global integrals.
Let us begin with continuation. In the case m < n for every ϕ ∈ Vπ and
′
ϕ ∈ Vπ′ we know the integral I(s; ϕ, ϕ′ ) converges absolutely for all s. From
the unfolding in Section 2 and the local calculation of Section 3 we know
that for Re(s) >> 0 and for appropriate choices of ϕ and ϕ′ we have
Y
I(s; ϕ, ϕ′ ) =
Ψv (s; Wϕv , Wϕ′v )
v
=
Y
v∈S
=
=
!
Ψv (s; Wϕv , Wϕ′v ) LS (s, π × π ′ )
!
Y Ψv (s; Wϕv , Wϕ′ )
v
L(s, π × π ′ )
L(s, πv × πv′ )
v∈S
!
Y
ev (s; Wϕv , Wϕ′v ) L(s, π × π ′ )
v∈S
We know that each ev (s; Wv , Wv′ ) is entire. Hence L(s, π × π ′ ) has a meromorphic continuation. If m = n then for appropriate ϕ ∈ Vπ , ϕ′ ∈ Vπ′ , and
Φ ∈ S(An ) we again have
!
Y
′
′
ev (s; Wϕv , Wϕ′v , Φv ) L(s, π × π ′ ).
I(s; ϕ, ϕ , Φ) =
v∈S
Once again, since each ev (s; Wv , Wv′ , Φv ) is entire, L(s, π × π ′ ) has a meromorphic continuation.
126
J.W. Cogdell
Let us next turn to the functional equation. This will follow from the
functional equation for the global integrals and the local functional equations. We will consider only the case where m < n since the other case is
entirely analogous. The functional equation for the global integrals is simply
˜ − s; ϕ,
I(s; ϕ, ϕ′ ) = I(1
e ϕ
e′ ).
Once again we have for appropriate ϕ and ϕ′
′
I(s; ϕ, ϕ ) =
Y
!
ev (s; Wϕv , Wϕ′ ′v )
v∈S
L(s, π × π ′ )
while on the other side
˜ − s; ϕ,
I(1
e ϕ
e′ ) =
Y
v∈S
!
fϕv , W
f ′ ′ ) L(1 − s, π
ẽv (1 − s; ρ(wn,m )W
e×π
e′ ).
ϕv
However, by the local functional equations, for each v ∈ S we have
e
f f′
fv , W
fv′ ) = Ψ(1 − s; ρ(wn,m )Wv , Wv )
ẽv (1 − s; ρ(wn,m )W
L(1 − s, π
e×π
e′ )
Ψ(s; Wv , Wv′ )
L(s, π × π ′ )
′
n−1
′
= ωv (−1)
ε(s, πv × πv , ψv )ev (s, Wv , Wv′ )
= ωv′ (−1)n−1 ε(s, πv × πv′ , ψv )
Combining these, we have
′
L(s, π × π ) =
Y
ωv′ (−1)n−1 ε(s, πv
×
!
πv′ , ψv )
v∈S
L(1 − s, π
e×π
e′ ).
Now, for v ∈
/ S we know that πv′ is unramified, so ωv′ (−1) = 1, and also that
′
ε(s, πv × πv , ψv ) ≡ 1. Therefore
Y
Y
ωv′ (−1)n−1 ε(s, πv × πv′ , ψv ) =
ωv′ (−1)n−1 ε(s, πv × πv′ , ψv )
v
′
v∈S
= ω (−1)n−1 ε(s, π × π ′ )
= ε(s, π × π ′ )
and we indeed have
L(s, π × π ′ ) = ε(s, π × π ′ )L(1 − s, π
e×π
e′ ).
L-functions for GLn
127
Note that this implies that ε(s, π × π ′ ) is independent of ψ as well.
Let us now turn to the boundedness in vertical strips. For the global
integrals I(s; ϕ, ϕ′ ) or I(s; ϕ, ϕ, Φ) this simply follows from the absolute convergence. For the L-function itself, the paradigm is the following. For every
′ , and Φ
finite place v ∈ S we know that there is a choice of Wv,i , Wv,i
v,i if
necessary such that
L(s, πv × πv′ ) =
L(s, πv × πv′ ) =
X
X
Ψ(s; Wv,i , Wv′ ′ i )
or
Ψ(s; Wv,i , Wv′ ′ i , Φv,i ).
If m = n − 1 or m = n then by the unpublished work of Jacquet and
Shalika mentioned toward the end of Section 3 we know that we have similar
statements for v ∈ S∞ . Hence if m = n − 1 or m = n there are global choices
ϕi , ϕ′i , and if necessary Φi such that
L(s, π × π ′ ) =
X
I(s; ϕi , ϕ′i )
or
L(s, π × π ′ ) =
X
I(s; ϕi , ϕ′i , Φi ).
Then the boundedness in vertical strips for the L-functions follows from that
of the global integrals.
However, if m < n − 1 then all we know at those v ∈ S∞ is that there
′
−1
ˆ
is a function Wv ∈ W(πv ⊗ πv′ , ψv ) = W(πv , ψv )⊗W(π
v , ψv ) or a finite
collection of such functions Wv,i and of Φv,i such that
L(s, πv × πv′ ) = I(s; Wv )
or
L(s, πv × πv′ ) =
X
I(s; Wv,i , Φv,i ).
To make the above paradigm work for m < n − 1 we would need to rework
ˆ π′ . This is
the theory of global Eulerian integrals for cusp forms in Vπ ⊗V
naturally the space of smooth vectors in an irreducible unitary cuspidal
representation of GLn (A) × GLm (A). So we would need extend the global
theory of integrals parallel to Jacquet and Shalika’s extension of the local
integrals in the archimedean theory. There seems to be no obstruction to
carrying this out, and then we obtain boundedness in vertical strips for
L(s, π × π ′ ) in general.
We should point out that if one approaches these L-functions by the
method of constant terms and Fourier coefficients of Eisenstein series, then
Gelbart and Shahidi have shown a wide class of automorphic L-functions,
including ours, to be bounded in vertical strips [17].
128
J.W. Cogdell
4.2
Poles of L-functions
Let us determine where the global L-functions can have poles. The poles of
the L-functions will be related to the poles of the global integrals. Recall
from Section 2 that in the case of m < n we have that the global integrals
I(s; ϕ, ϕ′ ) are entire and that when m = n then I(s; ϕ, ϕ′ , Φ) can have at
most simple poles and they occur at s = −iσ and s = 1 − iσ for σ real when
π≃π
e′ ⊗ | det |iσ . As we have noted above, the global integrals and global
L-functions are related, for appropriate ϕ, ϕ′ , and Φ, by
!
Y
ev (s; Wϕv , Wϕ′ ′v ) L(s, π × π ′ )
I(s; ϕ, ϕ′ ) =
v∈S
or
I(s; ϕ, ϕ′ , Φ) =
Y
!
ev (s; Wϕv , Wϕ′ ′v , Φv ) L(s, π × π ′ ).
v∈S
On the other hand, we have seen that for any s0 ∈ C and any v there is a
choice of local Wv , Wv′ , and Φv such that the local factors ev (s0 ; Wv , Wv′ ) 6= 0
or ev (s0 ; Wv , Wv′ , Φv ) 6= 0. So as we vary ϕ, ϕ′ and Φ at the places v ∈ S
we see that division by these factors can introduce no extraneous poles in
L(s, π × π ′ ), that is, in keeping with the local characterization of the Lfactor in terms of poles of local integrals, globally the poles of L(s, π ×
π ′ ) are precisely the poles of the family of global integrals {I(s; ϕ, ϕ′ )} or
{I(s; ϕ, ϕ′ , Φ)}. Hence from Theorems 2.1 and 2.2 we have.
Theorem 4.2 If m < n then L(s, π × π ′ ) is entire. If m = n, then L(s, π ×
π ′ ) has at most simple poles and they occur iff π ≃ π
e′ ⊗ | det |iσ with σ real
and are then at s = −iσ and s = 1 − iσ.
If we apply this with π ′ = π
e we obtain the following useful corollary.
Corollary L(s, π × π
e) has simple poles at s = 0 and s = 1.
4.3
Strong Multiplicity One
Let us return to the Strong Multiplicity One Theorem for cuspidal representations. First, recall the statement:
Theorem (Strong Multiplicity One) Let (π, Vπ ) and (π ′ , Vπ′ ) be two
cuspidal representations of GLn (A). Suppose there is a finite set of places S
such that for all v ∈
/ S we have πv ≃ πv′ . Then π = π ′ .
L-functions for GLn
129
We will now present Jacquet and Shalika’s proof of this statement via Lfunctions [36]. First note the following observation, which follows from our
analysis of the location of the poles of the L-functions.
Observation For π and π ′ cuspidal representations of GLn (A), L(s, π×e
π′ )
has a pole at s = 1 iff π ≃ π ′ .
Thus the L-function gives us an analytic method of testing when two cuspidal
representations are isomorphic, and so by the Multiplicity One Theorem, the
same.
Proof: If we take π and π ′ as in the statement of Strong Multiplicity One,
we have that πv ≃ πv′ for v ∈
/ S and hence
Y
Y
e′ )
LS (s, π × π
e) =
L(s, πv × π
ev ) =
L(s, πv × π
ev′ ) = LS (s, π × π
v∈S
/
v∈S
/
Since the local L-factors never vanish and for unitary representations they
have no poles in Re(s) ≥ 1 (since the local integrals have no poles in this
region) we see that for s = 1 that L(s, π × π
e′ ) has a pole at s = 1 iff
S
′
L (s, π × π
e ) does. Hence we have that since L(s, π × π
e) has a pole at
S
S
S
s = 1, so does L (s, π × π
e). But L (s, π × π
e) = L (s, π × π
e′ ), so that
S
′
′
both L (s, π × π
e ) and then L(s, π × π
e ) have poles at s = 1. But then
the L-function criterion above gives that π ≃ π ′ . Now apply Multiplicity
One.
2
In fact, Jacquet and Shalika push this method much further. If π is
an irreducible automorphic representation of GLn (A), but not necessarily
cuspidal, then it is a theorem of Langlands [44] that there are cuspidal
representations, say τ1 , . . . , τr of GLn1 , . . . , GLnr with n = n1 + · · ·+ nr , such
that π is a constituent of Ind(τ1 ⊗ · · · ⊗ τr ). Similarly, π ′ is a constituent of
Ind(τ1′ ⊗ · · · ⊗ τr′ ′ ). Then the generalized version of the Strong Multiplicity
One theorem that Jacquet and Shalika establish in [36] is the following.
Theorem (Generalized Strong Multiplicity One) Given π and π ′ irreducible automorphic representations of GLn (A) as above, suppose that
/ S. Then
there is a finite set of places S such that πv ≃ πv′ for all v ∈
r = r ′ and there is a permutation σ of the set {1, . . . , r} such that ni = n′σ(i)
′ .
and τi = τσ(i)
Note, the cuspidal representations τi and τi′ need not be unitary in this
statement.
130
4.4
J.W. Cogdell
Non-vanishing Results
Of interest for questions from analytic number theory, for example questions
of equidistribution, are results on the non-vanishing of L-functions. The
fundamental non-vanishing result for GLn is the following theorem of Jacquet
and Shalika [35] and Shahidi [56, 57].
Theorem 4.3 Let π and π ′ be cuspidal representations of GLn (A) and
GLm (A). Then the L-function L(s, π × π ′ ) is non-vanishing for Re(s) ≥ 1.
The proof of non-vanishing for Re(s) > 1 is in keeping with the spirit of
these notes [36, I, Theorem 5.3]. Since the local L-functions are never zero,
to establish the non-vanishing of the Euler product for Re(s) > 1 it suffices
to show that the Euler product is absolutely convergent for Re(s) > 1, and
for this it is sufficient to work with the incomplete L-function LS (s, π × π ′ )
where S is as at the beginning of this Section. Then we can write
Y
Y
LS (s, π × π ′ ) =
L(s, πv × πv′ ) =
det(I − qv−s Aπv ⊗ Aπv′ )−1
v∈S
/
v∈S
/
with absolute convergence for Re(s) >> 0.
Q
Recall that an infinite product (1 + an ) is absolutely convergent iff the
P
associated series
log(1 + an ) is absolutely convergent.
Let us write

 ′


µv,1
µv,1




..
..
and Aπv′ = 
Aπv = 

.
.
.
µ′v,m
µv,n
1/2
1/2
and |µ′v,j | < qv . Then
X
log L(s, πv × πv′ ) = −
log(1 − µv,i µ′v,j qv−s )
We have seen that |µv,i | < qv
i,j
=
∞
XX
(µv,i µ′v,j )d
i,j d=1
dqvds
=
∞
X
tr(Adπv ) tr(Adπ′ )
v
d=1
dqvds
with the sum absolutely convergent for Re(s) >> 0. Then, still for Re(s) >>
0,
∞ tr(Ad ) tr(Ad )
XX
πv
πv′
.
log(LS (s, π × π ′ )) =
ds
dqv
v∈S
/ d=1
L-functions for GLn
131
e we find
If we apply this to π ′ = π = π
log(LS (s, π × π)) =
∞
XX
| tr(Adπv )|2
dqvds
v∈S
/ d=1
which is a Dirichlet series with non-negative coefficients. By Landau’s Lemma
this will be absolutely convergent up to the its first pole, which we know is
at s = 1. Hence this series, and the associated Euler product L(s, π × π
e), is
absolutely convergent for Re(s) > 1.
An application of the Cauchy–Schwatrz inequality then implies that the
series
∞
XX
tr(Adπv ) tr(Adπv′ )
log(LS (s, π × π ′ )) =
dqvds
v∈S
/ d=1
is also absolutely convergent for Re(s) > 1. Thus L(s, π × π ′ ) is absolutely
convergent and hence non-vanishing for Re(s) > 1.
To obtain the non-vanishing on the line Re(s) = 1 requires the technique of analyzing L-functions via their occurrence in the constant terms
and Fourier coefficients of Eisenstein series, which we have not discussed.
They can be found in the references [35] and [56, 57] mentioned above.
4.5
The Generalized Ramanujan Conjecture (GRC)
The current version of the GRC is a conjecture about the structure of cuspidal representations.
Conjecture (GRC) Let π be a (unitary) cuspidal representation of GLn (A)
with decomposition π ≃ ⊗′ πv . Then the local components πv are tempered
representations.
However, it has an interesting interpretation in terms of L-functions
which is more in keeping with the origins of the conjecture. If π is cuspidal, then at every finite place v where π
v is unramified we
 have associated
µv,1


..
a semisimple conjugacy class, say Aπv = 
 so that
.
µv,n
L(s, πv ) = det(I −
qv−s Aπv )−1
n
Y
(1 − µv,i qv−s )−1 .
=
i=1
132
J.W. Cogdell
If v is an archimedean place where πv is unramified, then we can similarly
write
n
Y
Γv (s + µv,i )
L(s, π) =
i=1
where
(
π −s/2 Γ( 2s )
Γv (s) =
2(2π)−s Γ(s)
if kv ≃ R
.
if kv ≃ C
Then the statement of the GRC in these terms is
Conjecture (GRC for L-functions) If π is a cuspidal representation of
GLn (A) and if v is a place where πv is unramified, then |µv,i | = 1 for v
non-archimedean and Re(µv,i ) = 0 for v archimedean.
Note that by including the archimedean places, this conjecture encompasses not only the classical Ramanujan conjectures but also the various
versions of the Selberg eigenvalue conjecture [27].
−1/2
Recall that by the Corollary to Theorem 3.3 we have the bounds qv
<
1/2
|µv,i | < qv for v non-archimedean, and a similar local analysis for v archimedean would give | Re(µv,i )| < 21 . The best bound for general GLn is due to
Luo, Rudnick, and Sarnak [46]. They are the uniform bounds
−( 21 −
qv
1
)
n2 +1
1
− 1
2 n2 +1
≤ |µv,i | ≤ qv
if v is non-archimedean
and
1
1
−
for v archimedean.
2 n2 + 1
Their techniques are global and rely on the theory of Rankin–Selberg Lfunctions as presented here, a technique of persistence of zeros in families of
L-functions, and a positivity argument.
| Re(µv,i )| ≤
For GL2 there has been much recent progress. The best general estimates
I am aware of at present are due to Kim and Shahidi [41], who use the
holomorphy of the symmetric ninth power L-function for Re(s) > 1 to obtain
− 91
qv
1
< |µv,i | < qv9
for i = 1, 2, and v non-archimedean,
and Kim and Sarnak, who obtain the analogous estimate for v archimedean
(with possible equality) in the appendix to [39].
For some applications, the notion of weakly Ramanujan [8] can replace
knowing the full GRC.
L-functions for GLn
133
Definition A cuspidal representation π of GLn (A) is called weakly Ramanujan if for every ǫ > 0 there is a constant cǫ > 0 and an infinite sequence
of places {vm } with the property that each πvm is unramified and the Satake
parameters µvm ,i satisfy
−ǫ
ǫ
c−1
ǫ qvm < |µvm ,i | < cǫ qvm .
For example, if we knew that all cuspidal representations on GLn (A)
were weakly Ramanujan, then we would know that under Langlands liftings
between general linear groups, the property of occurrence in the spectral
decomposition is preserved [8].
For n = 2, 3 our techniques let us show the following.
Proposition 4.1 For n = 2 or n = 3 all cuspidal representations are weakly
Ramanujan.
Proof: First, let π be a cuspidal representation or GLn (A). Recall that
from the absolute convergence of the Euler product for L(s, π × π) we know
X X | tr(Adπ )|2
v
is absolutely convergent for Re(s) > 1, so
that the series
dqvds
d
v∈S
/
X | tr(Aπ )|2
v
that in particular we have that
is absolutely convergent for
qvs
v∈S
/
Re(s) > 1. Thus, for a set of places of positive density, we have the estimate
| tr(Aπv )|2 < qvǫ for each ǫ. Since Aπv = A−1
πv for components of cuspidal
representations, we have the same estimate for | tr(A−1
πv )|.
In the case of n = 2 and n = 3, these estimates and the fact that
| det Aπv | = |ωv (̟v )| = 1 give us estimates on the coefficients of the characteristic polynomial for Aπv . For example, if n = 3 and the characteristic
ǫ/2
polynomial of Aπv is X 3 + aX 2 + bX + c then we know |a| = | tr(Aπv )| < qv ,
ǫ/2
|b| = | tr(A−1
πv ) det(Aπv )| < qv , and |c| = | det(Aπv )| = 1. Then an application of Rouche’s theorem gives that the roots of this polynomial all lie in the
circle of radius qvǫ as long as qv > 3. Applying this to both Aπv and A−1
πv we
find that for our set primes of positive density above we have the estimate
qv−ǫ < |µvm ,i | < qvǫ . Thus we find that for n = 2, 3 cuspidal representations
of GLn are weakly Ramanujan.
2
134
4.6
J.W. Cogdell
The Generalized Riemann Hypothesis (GRH)
This is one of the most important conjectures in the analytic theory of Lfunctions. Simply stated, it is
Conjecture (GRH) For any cuspidal representation π, all the zeros of
the L-function L(s, π) lie on the line Re(s) = 12 .
Even in the simplest case of n = 1 and π = 1 the trivial representation this
reduces to the Riemann hypothesis for the Riemann zeta function!
For an interesting survey on these and other conjectures on L-functions
and their relation to number theoretic problems, we refer the reader to the
survey of Iwaniec and Sarnak [27].
5
Converse Theorems
Let us return first to Hecke. Recall that to a modular form
f (τ ) =
∞
X
an e2πinτ
n−1
for, say, SL2 (Z) Hecke attached an L function L(s, f ) and they were related
via the Mellin transform
Z ∞
f (iy)y s d× y
Λ(s, f ) = (2π)−s Γ(s)L(s, f ) =
0
and derived the functional equation for L(s, f ) from the modular transformation law for f (τ ) under the modular transformation law for the transformation τ 7→ −1/τ . In his fundamental paper [24] he inverted this process
by taking a Dirichlet series
D(s) =
∞
X
an
n=1
ns
and assuming that it converged in a half plane, had an entire continuation to
a function of finite order, and satisfied the same functional equation as the
L-function of a modular form of weight k, then he could actually reconstruct
a modular form from D(s) by Mellin inversion
Z 2+i∞
X
1
−2πny
f (iy) =
an e
=
(2π)−s Γ(s)D(s)y s ds
2πi 2−i∞
i
L-functions for GLn
135
and obtain the modular transformation law for f (τ ) under τ 7→ −1/τ from
the functional equation for D(s) under s 7→ k − s. This is Hecke’s Converse
Theorem.
In this Section we will present some analogues of Hecke’s theorem in
the context of L-functions for GLn . Surprisingly, the technique is exactly
the same as Hecke’s, i.e., inverting the integral representation. This was
first done in the context of automorphic representation for GL2 by Jacquet
and Langlands [30] and then extended and significantly strengthened for
GL3 by Jacquet, Piatetski-Shapiro, and Shalika [31]. For a more extensive
bibliography and history, see [10].
This section is taken mainly from our survey [10]. Further details can be
found in [7, 9].
5.1
The Results
Once again, let k be a global field, A its adele ring, and ψ a fixed non-trivial
continuous additive character of A which is trivial on k. We will take n ≥ 3
to be an integer.
To state these Converse Theorems, we begin with an irreducible admissible representation Π of GLn (A). In keeping with the conventions of these
notes, we will assume that Π is unitary and generic, but this is not necessary. It has a decomposition Π = ⊗′ Πv , where Πv is an irreducible admissible
generic representation of GLn (kv ). By the local theory of Section 3, to each
Πv is associated a local L-function L(s, Πv ) and a local ε-factor ε(s, Πv , ψv ).
Hence formally we can form
Y
Y
L(s, Π) =
L(s, Πv )
and
ε(s, Π, ψ) =
ε(s, Πv , ψv ).
v
v
We will always assume the following two things about Π:
1. L(s, Π) converges in some half plane Re(s) >> 0,
2. the central character ωΠ of Π is automorphic, that is, invariant under
k× .
Under these assumptions, ε(s, Π, ψ) = ε(s, Π) is independent of our choice
of ψ [7].
Our Converse Theorems will involve twists by cuspidal automorphic representations of GLm (A) for certain m. For convenience, let us set A(m)
136
J.W. Cogdell
to be the set of automorphic representations of GLm (A), A0 (m) the set of
m
a
cuspidal representations of GLm (A), and T (m) =
A0 (d).
d=1
Let π ′ = ⊗′ π ′ v be a cuspidal representation of GLm (A) with m < n.
Then again we can formally define
Y
Y
L(s, Π×π ′ ) =
L(s, Πv ×π ′ v )
and
ε(s, Π×π ′ ) =
ε(s, Πv ×π ′ v , ψv )
v
v
since again the local factors make sense whether Π is automorphic or not.
A consequence of (1) and (2) above and the cuspidality of π ′ is that both
e × πe′ ) converge absolutely for Re(s) >> 0, where
L(s, Π × π ′ ) and L(s, Π
e and πe′ are the contragredient representations, and that ε(s, Π × π ′ ) is
Π
independent of the choice of ψ.
We say that L(s, Π × π ′ ) is nice if it satisfies the same analytic properties
it would if Π were cuspidal, i.e.,
e × πe′ ) have analytic continuations to entire
1. L(s, Π × π ′ ) and L(s, Π
functions of s,
2. these entire continuations are bounded in vertical strips of finite width,
3. they satisfy the standard functional equation
e × πe′ ).
L(s, Π × π ′ ) = ε(s, Π × π ′ )L(1 − s, Π
The basic Converse Theorem for GLn is the following.
Theorem 5.1 Let Π be an irreducible admissible representation of GLn (A)
as above. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 1). Then Π is
a cuspidal automorphic representation.
In this theorem we twist by the maximal amount and obtain the strongest
possible conclusion about Π. The proof of this theorem essentially follows
that of Hecke [24] and Weil [67] and Jacquet–Langlands [30]. It is of course
valid for n = 2 as well.
For applications, it is desirable to twist by as little as possible. There are
essentially two ways to restrict the twisting. One is to restrict the rank of
the groups that the twisting representations live on. The other is to restrict
ramification.
When we restrict the rank of our twists, we can obtain the following
result.
L-functions for GLn
137
Theorem 5.2 Let Π be an irreducible admissible representation of GLn (A)
as above. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 2). Then Π is
a cuspidal automorphic representation.
This result is stronger than Theorem 5.1, but its proof is a bit more
delicate.
The theorem along these lines that is most useful for applications is one
in which we also restrict the ramification at a finite number of places. Let
us fix a finite set of S of finite places and let T S (m) denote the subset of
T (m) consisting of representations that are unramified at all places v ∈ S.
Theorem 5.3 Let Π be an irreducible admissible representation of GLn (A)
as above. Let S be a finite set of finite places. Suppose that L(s, Π × π ′ ) is
nice for all π ′ ∈ T S (n − 2). Then Π is quasi-automorphic in the sense that
there is an automorphic representation Π′ such that Πv ≃ Π′v for all v ∈
/ S.
Note that as soon as we restrict the ramification of our twisting representations we lose information about Π at those places. In applications we
usually choose S to contain the set of finite places v where Πv is ramified.
The second way to restrict our twists is to restrict the ramification at
all but a finite number of places. Now fix a non-empty finite set of places S
which in the case of a number field contains the set S∞ of all archimedean
places. Let TS (m) denote the subset consisting of all representations π ′ in
T (m) which are unramified for all v ∈
/ S. Note that we are placing a grave
restriction on the ramification of these representations.
Theorem 5.4 Let Π be an irreducible admissible representation of GLn (A)
as above. Let S be a non-empty finite set of places, containing S∞ , such that
the class number of the ring oS of S-integers is one. Suppose that L(s, Π×π ′ )
is nice for all π ′ ∈ TS (n − 1). Then Π is quasi-automorphic in the sense that
there is an automorphic representation Π′ such that Πv ≃ Π′v for all v ∈ S
and all v ∈
/ S such that both Πv and Π′v are unramified.
There are several things to note here. First, there is a class number
restriction. However, if k = Q then we may take S = S∞ and we have a
Converse Theorem with “level 1” twists. As a practical consideration, if we
let SΠ be the set of finite places v where Πv is ramified, then for applications
we usually take S and SΠ to be disjoint. Once again, we are losing all
information at those places v ∈
/ S where we have restricted the ramification
unless Πv was already unramified there.
138
J.W. Cogdell
The proof of Theorem 5.1 essentially follows the lead of Hecke, Weil,
and Jacquet–Langlands. It is based on the integral representations of Lfunctions, Fourier expansions, Mellin inversion, and finally a use of the weak
form of Langlands spectral theory. For Theorems 5.2, 5.3, and 5.4, where
we have restricted our twists, we must impose certain local conditions to
compensate for our limited twists. For Theorem 5.2 and 5.3 there are a finite
number of local conditions and for Theorem 5.4 an infinite number of local
conditions. We must then work around these by using results on generation
of congruence subgroups and either weak or strong approximation.
5.2
Inverting the Integral Representation
Let Π be as above and let ξ ∈ VΠ be a decomposable vector in the space VΠ
of Π. Since Π is generic, then fixing local Whittaker models W(Πv , ψv ) at
all places, compatibly normalized at the unramified places, we can associate
Q
to ξ a non-zero function Wξ (g) = Wξv (gv ) on GLn (A) which transforms
by the global character ψ under left translation by Nn (A), i.e., Wξ (ng) =
ψ(n)Wξ (g). Since ψ is trivial on rational points, we see that Wξ (g) is left
invariant under Nn (k). We would like to use Wξ to construct an embedding of
VΠ into the space of (smooth) automorphic forms on GLn (A). The simplest
idea is to average Wξ over Nn (k)\ GLn (k), but this will not be convergent.
However, if we average over the rational points of the mirabolic P = Pn then
the sum
X
Uξ (g) =
Wξ (pg)
Nn (k)\ P(k)
is absolutely convergent. For the relevant growth properties of Uξ see [7].
Since Π is assumed to have automorphic central character, we see that Uξ (g)
is left invariant under both P(k) and the center Zn (k).
Suppose now that we know that L(s, Π × π ′ ) is nice for all π ′ ∈ T (m).
Then we will hope to obtain the remaining invariance of Uξ from the GLn ×
GLm functional equation by inverting the integral representation for L(s, Π×
π ′ ). With this in mind, let Q = Qm be the mirabolic subgroup of GLn which
stabilizes the standard unit vector tem+1 , that is, the column vector all of
whose entries are 0 except the (m + 1)th , which is 1. Note that if m = n − 1
then Q is nothing more than the opposite mirabolic P =t P−1 to P. If we let
L-functions for GLn
139
αm be the permutation matrix in GLn (k) given by


1

αm = Im
In−m−1
−1
then Qm = α−1
m αn−1 Pαn−1 αm is a conjugate of P and for any m we have
that P(k) and Q(k) generate all of GLn (k). So now set
X
Vξ (g) =
Wξ (αm qg)
N′ (k)\ Q(k)
where N′ = α−1
m Nn αm ⊂ Q. This sum is again absolutely convergent and
is invariant on the left by Q(k) and Z(k). Thus, to embed Π into the space
of automorphic forms it suffices to show Uξ = Vξ , for the we get invariance
of Uξ under all of GLn (k). It is this that we will attempt to do using the
integral representations.
Now let (π ′ , Vπ′ ) be an irreducible subrepresentation of the space of automorphic forms on GLm (A) and assume ϕ′ ∈ Vπ′ is also factorizable. Let
Z
h
n
′
Pm Uξ
ϕ′ (h)| det(h)|s−1/2 dh.
I(s; Uξ , ϕ ) =
1
GLm (k)\ GLm (A)
This integral is always absolutely convergent for Re(s) >> 0, and for all s
if π ′ is cuspidal. As with the usual integral representation we have that this
unfolds into the Euler product
Z
h
0
′
Wξ
Wϕ′ ′ (h)| det(h)|s−(n−m)/2 dh
I(s; Uξ , ϕ ) =
0
I
n−m
Nm (A)\ GLm (A)
YZ
hv
0
Wξv
=
Wϕ′ ′ v (hv )| det(hv )|vs−(n−m)/2 dhv
0
I
n−m
Nm (kv )\ GLm (kv )
v
Y
=
Ψv (s; Wξv , Wϕ′ ′ v ).
v
Note that unless π ′ is generic, this integral vanishes.
Assume first that π ′ is cuspidal. Then from the local theory of Lfunctions from Section 3, for almost all finite places we have Ψv (s; Wξv , Wϕ′ ′ )
v
= L(s, Πv ×π ′ v ) and for the other places Ψv (s; Wξv , Wϕ′ ′ v ) = ev (s; Wξv , Wϕ′ ′ v )
L(s, Πv ×π ′ v ) with the ev (s; Wξv , Wϕ′ ′ v ) entire and bounded in vertical strips.
So in this case we have I(s; Uξ , ϕ′ ) = e(s)L(s, Π × π ′ ) with e(s) entire and
140
J.W. Cogdell
bounded in vertical strips. Since L(s; Π × π ′ ) is assumed to be nice we may
conclude that I(s; Uξ , ϕ′ ) has an analytic continuation to an entire function
which is bounded in vertical strips. When π ′ is not cuspidal, it is a subrepresentation of a representation that is induced from (possibly non-unitary)
P
cuspidal representations σi of GLri (A) for ri < m with
ri = m and is
in fact, if our integral doesn’t vanish, the unique generic constituent of this
induced representation. Then we can make a similar argument using this
induced representation and the fact that the L(s, Π × σi ) are nice to again
Q
conclude that for all π ′ , I(s; Uξ , ϕ′ ) = e(s)L(s, Π × π ′ ) = e′ (s) L(s, Π × σi )
is entire and bounded in vertical strips. (See [7] for more details on this
point.)
Similarly, consider I(s; Vξ , ϕ′ ) for ϕ′ ∈ Vπ′ with π ′ an irreducible subrepresentation of the space of automorphic forms on GLm (A), still with
Z
h
n
′
Pm Vξ
ϕ′ (h)| det(h)|s−1/2 dh.
I(s; Vξ , ϕ ) =
1
GLm (k)\ GLm (A)
Now this integral converges for Re(s) << 0. However, when we unfold, we
find
Y
f ′ ′ ) = ẽ(1 − s)L(1 − s, Π
e × πe′ )
e v (1 − s; ρ(wn,m )W
fξ , W
I(s; Vξ , ϕ′ ) =
Ψ
v
ϕv
as above. Thus I(s; Vξ , ϕ′ ) also has an analytic continuation to an entire
function of s which is bounded in vertical strips.
Now, utilizing the assumed global functional equation for L(s, Π × π ′ ) in
the case where π ′ is cuspidal, or for the L(s, Π × σi ) in the case π ′ is not
cuspidal, as well as the local functional equations at v ∈ S∞ ∪ SΠ ∪ Sπ′ ∪ Sψ
as in Section 3 one finds
e × πe′ ) = I(s; Vξ , ϕ′ )
I(s; Uξ , ϕ′ ) = e(s)L(s, Π × π ′ ) = ẽ(1 − s)L(1 − s, Π
for all ϕ′ in all irreducible subrepresentations π ′ of GLm (A), in the sense of
analytic continuation. This concludes our use of the L-function.
We now rewrite our integrals I(s; Uξ , ϕ′ ) and I(s; Vξ , ϕ′ ) as follows. We
first stratify GLm (A). For each a ∈ A× let GLam (A) = {g ∈ GLm (A)|
a
det(g) = a}. We can, and will, always take GLam (A) = SLm (A) ·
.
Im−1
Let
Z
h
n
n
′
Pm Uξ
hPm Uξ , ϕ ia =
ϕ′ (h) dh
1
a
SLm (k)\ GLm (A)
L-functions for GLn
141
and similarly for hPnm Vξ , ϕ′ ia . These are both absolutely convergent for
all a and define continuous functions of a on k× \A× . We now have that
I(s; Uξ , ϕ′ ) is the Mellin transform of hPnm Uξ , ϕ′ ia ,
Z
′
hPnm Uξ , ϕ′ ia |a|s−1/2 d× a,
I(s; Uξ , ϕ ) =
k × \A×
similarly for I(s; Vξ , ϕ′ ), and that these two Mellin transforms are equal in
the sense of analytic continuation. By Mellin inversion as in Lemma 11.3.1
of Jacquet-Langlands [30], we have that hPnm Uξ , ϕ′ ia = hPnm Vξ , ϕ′ ia for all
a, and in particular for a = 1. Since this is true for all ϕ′ in all irreducible
subrepresentations of automorphic forms on GLm (A), then by the weak form
of Langlands’ spectral theory for SLm we may conclude that Pnm Uξ = Pnm Vξ
as functions on Pm+1 (A). More specifically, we have the following result.
Proposition 5.1 Let Π be an irreducible admissible representation of GLn (A)
as above. Suppose that L(s, Π × π ′ ) is nice for all π ′ ∈ T (m). Then for each
ξ ∈ VΠ we have Pnm Uξ (Im+1 ) = Pnm Vξ (Im+1 ).
This proposition is the key common ingredient for all our Converse Theorems.
5.3
Remarks on the Proofs
All of our Converse Theorems take Proposition 5.1 as their starting point.
Theorem 5.1 follows almost immediately. In Theorems 5.2, 5.3, and 5.4 we
must add local conditions to compensate for the fact that we do not have
the full family of twists from Theorem 5.1 and then work around them. We
will sketch these arguments here. Details for Theorems 5.1 and 5.4 can be
found in [7] and for Theorems 5.2 and 5.3 can be found in [9].
5.3.1
Theorem 5.1
Let us first look at the proof of Theorem 5.1. So we now assume that Π
is as above and that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 1). Then we
have that for all ξ ∈ VΠ , Pnn−1 Uξ (In ) = Pnn−1 Vξ (In ). But for m = n − 1 the
projection operator Pnn−1 is nothing more than restriction to Pn . Hence we
have Uξ (In ) = Vξ (In ) for all ξ ∈ VΠ . Then for each g ∈ GLn (A), we have
Uξ (g) = UΠ(g)ξ (In ) = VΠ(g)ξ (In ) = Vξ (g). So the map ξ 7→ Uξ (g) gives our
embedding of Π into the space of automorphic forms on GLn (A), since now
142
J.W. Cogdell
Uξ is left invariant under P(k), Q(k), and hence all of GLn (k). Since we still
have
X
Uξ (g) =
Wξ (pg)
Nn (k)\ P(k)
we can compute that Uξ is cuspidal along any parabolic subgroup of GLn .
Hence Π embeds in the space of cusp forms on GLn (A) as desired.
5.3.2
Theorem 5.2
Next consider Theorem 5.2, so now suppose that n ≥ 3, and that L(s, Π×π ′ )
is nice for all π ′ ∈ T (n−2). Then from Proposition 5.1 we may conclude that
Pnn−2 Uξ (In−1 ) = Pnn−2 Vξ (In−1 ) for all ξ ∈ VΠ . Since the projection operator
Pnn−2 now involves a non-trivial integration over kn−1 \An−1 we can no longer
argue as in the proof of Theorem 5.1. To get to that point we will have to
impose a local condition on the vector ξ at one place.
Before we place our local condition, let us write Fξ = Uξ − Vξ . Then Fξ is
rapidly decreasing as a function on Pn−1 . We have Pnn−2 Fξ (In−1 ) = 0 and we
would like to have simply that Fξ (In ) = 0. Let u = (u1 , . . . , un−1 ) ∈ An−1
and consider the function
tu
I
fξ (u) = Fξ n−1
.
1
Now fξ (u) is a function on kn−1 \An−1 and as such has a Fourier expansion
X
fξ (u) =
fˆξ (α)ψα (u)
α∈k n−1
where ψα (u) = ψ(α ·t u) and
fˆξ (α) =
Z
k n−1 \An−1
fξ (u)ψ−α (u) du.
In this language, the statement Pnn−2 Fξ (In−1 ) = 0 becomes fˆξ (en−1 ) = 0,
where as always, ek is the standard unit vector with 0’s in all places except
the kth where there is a 1.
Note that Fξ (g) = Uξ (g) − Vξ (g) is left invariant under (Z(k) P(k)) ∩
(Z(k) Q(k)) where Q = Qn−2 . This contains the subgroup




In−2


R(k) = r =  α′ αn−1 αn  α′ ∈ kn−2 , αn−1 6= 0 .


1
L-functions for GLn
143
Using this invariance of Fξ , it is now elementary to compute that, with
this notation, fˆΠ(r)ξ (en−1 ) = fˆξ (α) where α = (α′ , αn−1 ) ∈ kn−1 . Since
fˆξ (en−1 ) = 0 for all ξ, and in particular for Π(r)ξ, we see that for every ξ
we have fˆξ (α) = 0 whenever αn−1 6= 0. Thus
fξ (u) =
X
fˆξ (α)ψα (u) =
α∈k n−1
X
fˆξ (α′ , 0)ψ(α′ ,0) (u).
α′ ∈k n−2
P
ˆ ′
Hence fξ (0, . . . , 0, un−1 ) =
α′ ∈k n−2 fξ (α , 0) is constant as a function of
un−1 . Moreover, this constant is fξ (en−1 ) = Fξ (In ), which we want to be 0.
This is what our local condition will guarantee.
If v is a finite place of k, let ov denote the ring of integers of kv , and let
pv denote the prime ideal of ov . We may assume that we have chosen v so
that the local additive character ψv is normalized, i.e., that ψv is trivial on
ov and non-trivial on p−1
v . Given an integer nv ≥ 1 we consider the open
compact group
K00,v (pnv v ) = {g = (gi,j ) ∈ GLn (ov ) |(i) gi,n−1 ∈ pnv v for 1 ≤ i ≤ n − 2;
(ii) gn,j ∈ pnv v for 1 ≤ j ≤ n − 2;
v
(iii) gn,n−1 ∈ p2n
v }.
(As usual, gi,j represents the entry of g in the i-th row and j-th column.)
Lemma Let v be a finite place of k as above and let (Πv , VΠv ) be an irreducible admissible generic representation of GLn (kv ). Then there is a vector
ξv′ ∈ VΠv and a non-negative integer nv such that
1. for any g ∈ K00,v (pnv v ) we have Πv (g)ξv′ = ωΠv (gn,n )ξv′


I
n−2
R
2.
Πv 
1 u ξv′ du = 0.
−1
pv
1
The proof of this Lemma is simply an exercise in the Kirillov model of
Πv and can be found in [9].
If we now fix such a place v0 and assume that our vector ξ is chosen so
144
J.W. Cogdell
that ξv0 = ξv′ 0 , then we have
Fξ (In ) = fξ (en−1 ) =
−1
Vol(p−1
v0 )
−1
= Vol(p−1
v0 )
Z
Z
p−1
v0
p−1
v0
fξ (0, . . . , 0, uv0 ) duv0

Fξ 
In−2

1 uv0  duv0 = 0
1
for such ξ.
Hence we now have Uξ (In ) = Vξ (In ) for all ξ ∈ VΠ such that ξv0 = ξv′ 0
nv
at our fixed place. If we let G′ = K00,v0 (pv0 0 ) Gv0 , where we set Gv0 =
Q′
v6=v0 GLn (kv ), then we have that this group preserves the local component
ξv′ 0 up to a constant factor so that for g ∈ G′ we have Uξ (g) = UΠ(g)ξ (In ) =
VΠ(g)ξ (In ) = Vξ (g).
We now use a fact about generation of congruence type subgroups. Let
Γ1 = (P(k) Z(k)) ∩ G′ , Γ2 = (Q(k) Z(k)) ∩ G′ , and Γ = GLn (k) ∩ G′ . Then
Uξ (g) is left invariant under Γ1 and Vξ (g) is left invariant under Γ2 . It is
essentially a matrix calculation that together Γ1 and Γ2 generate Γ. So,
as a function on G′ , Uξ (g) = Vξ (g) is left invariant under Γ. So if we let
Πv0 = ⊗′v6=v0 Πv then the map ξ v0 7→ Uξv′ o ⊗ξ v0 (g) embeds VΠv0 into A(Γ\ G′ ),
the space of automorphic forms on G′ relative to Γ. Now, by weak approximation, GLn (A) = GLn (k) · G′ and Γ = GLn (k) ∩ G′ , so we can extend
Πv0 to an automorphic representation of GLn (A). Let Π0 be an irreducible
component of the extended representation. Then Π0 is automorphic and
coincides with Π at all places except possible v0 .
One now repeats the entire argument using a second place v1 6= v0 .
Then we have two automorphic representations Π1 and Π0 of GLn (A) which
agree at all places except possibly v0 and v1 . By the generalized Strong
Multiplicity One for GLn we know that Π0 and Π1 are both constituents of
the same induced representation Ξ = Ind(σ1 ⊗ · · · ⊗ σr ) where each σi is a
P
mi = n. We
cuspidal representation of some GLmi (A), each mi ≥ 1 and
can write each σi = σi◦ ⊗ | det |ti with σi◦ unitary cuspidal and ti ∈ R and
assume t1 ≥ · · · ≥ tr . If r > 1, then either m1 ≤ n − 2 or mr ≤ n − 2 (or
both). For simplicity assume mr ≤ n − 2. Let S be a finite set of places
containing all archimedean places, v0 , v1 , SΠ , and Sσi for each i. Taking
L-functions for GLn
145
π′ = σ
er ∈ T (n − 2), we have the equality of partial L-functions
LS (s, Π × π ′ ) = LS (s, Π0 × π ′ ) = LS (s, Π1 × π ′ )
Y
Y
=
LS (s, σi × π ′ ) =
LS (s + ti − tr , σi◦ × σ
er◦ ).
i
i
Now LS (s, σr × σ
er ) has a pole at s = 1 and all other terms are non-vanishing
at s = 1. Hence L(s, Π × π ′ ) has a pole at s = 1 contradicting the fact that
L(s, Π × π ′ ) is nice. If m1 ≤ 2, then we can make a similar argument using
e × σ1 ). So in fact we must have r = 1 and Π0 = Π1 = Ξ is cuspidal.
L(s, Π
Since Π0 agrees with Π at v1 and Π1 agrees with Π at v0 we see that in fact
Π = Π0 = Π1 and Π is indeed cuspidal automorphic.
5.3.3
Theorem 5.3
Now consider Theorem 5.3. Since we have restricted our ramification, we no
longer know that L(s, Π × π ′ ) is nice for all π ′ ∈ T (n − 2) and so Proposition
5.1 above is not immediately applicable. In this case, for each place v ∈ S
we fix a vector ξv′ ∈ VΠv as in the above Lemma. (So we must assume we
Q
have chosen ψ so it is unramified at the places in S.) Let ξS′ = v∈S ξv′ ∈
ΠS . Consider now only vectors ξ of the form ξ S ⊗ ξS′ with ξ S arbitrary
h
′
n
and
in VΠS and ξS fixed. For these vectors, the functions Pn−2 Uξ
1
h
Pnn−2 Vξ
are unramified at the places v ∈ S, so that the integrals
1
I(s; Uξ , ϕ′ ) and I(s; Vξ , ϕ′ ) vanish unless ϕ′ (h) is also unramified at those
places in S. In particular, if π ′ ∈ T (n − 2) but π ′ ∈
/ T S (n − 2) these integrals
will vanish for all ϕ′ ∈ Vπ′ . So now, for this fixed class of ξ we actually have
I(s; Uξ , ϕ′ ) = I(s; Vξ , ϕ′ ) for all ϕ′ ∈ Vπ′ for all π ′ ∈ T (n − 2). Hence, as
before, Pnn−2 Uξ (In−1 ) = Pnn−2 Vξ (In−1 ) for all such ξ.
Now we proceed as before. Our Fourier expansion argument is a bit more
subtle since we have to work around our local conditions, which now have
been imposed before this step, but we do obtain that Uξ (g) = Vξ (g) for all
Q
g ∈ G′ = ( v∈S K00,v (pnv v )) GS . The generation of congruence subgroups
goes as before. We then use weak approximation as above, but then take for
Π′ any constituent of the extension of ΠS to an automorphic representation
of GLn (A). There is no use of strong multiplicity one nor any further use of
the L-function in this case. More details can be found in [9].
146
5.3.4
J.W. Cogdell
Theorem 5.4
Let us now sketch the proof of Theorem 5.4. We fix a non-empty finite set
of places S, containing all archimedean places, such that the ring oS of Sinteger has class number one. Recall that we are now twisting by all cuspidal
representations π ′ ∈ TS (n − 1), that is, π ′ which are unramified at all places
v∈
/ S. Since we have not twisted by all of T (n − 1) we are not in a position
to apply Proposition 5.1. To be able to apply that, we will have to place
local conditions at all v ∈
/ S.
We begin by recalling the definition of the conductor of a representation
Πv of GLn (kv ) and the conductor (or level) of Π itself. Let Kv = GLn (ov )
be the standard maximal compact subgroup of GLn (kv ). Let pv ⊂ ov be the
unique prime ideal of ov and for each integer mv ≥ 0 set




∗









.. 


mv
mv
.
∗
K0,v (pv ) = g ∈ GLn (ov )g ≡ 
 (mod p )



∗






0 ··· 0 ∗
mv
mv
v
and K1,v (pm
v ) = {g ∈ K0,v (pv ) | gn,n ≡ 1 (mod pv ))}. Note that for
mv = 0 we have K1,v (p0v ) = K0,v (p0v ) = Kv . Then for each local component
v
Πv of Π there is a unique integer mv ≥ 0 such that the space of K1,v (pm
v )–
fixed vectors in Πv is exactly one. For almost all v, mv = 0. We take the ideal
Q mv
v
pm
v = f(Πv ) as the conductor of Πv . Then the ideal n = f(Π) =
v pv ⊂ o
is called the conductor of Π. For each place v we fix a non-zero vector ξv◦ ∈ Πv
v
which is fixed by K1,v (pm
v ), which at the unramified places is taken to be
the vector with respect to which the restricted tensor product Π = ⊗′ Πv is
◦
◦
v
taken. Note that for g ∈ K0,v (pm
v ) we have Πv (g)ξv = ωΠv (gn,n )ξv .
Now fix a non-empty finite set of places S, containing the archimedean
Q
places if there are any. As is standard, we will let GS = v∈S GLn (kv ),
Q
(k ), Π = ⊗ Π , ΠS = ⊗′v∈S
GS = v∈S
/ GL
/ Πv , etc. Then the compact
Qn v mv S S v∈S v
S
⊂
k
or
the
ideal
it
determines
nS = k ∩ kS nS ⊂ oS
subring n = v∈S
p
v
/
Q
mv
is called the S–conductor of Π. Let KS1 (n) = v∈S
/ K1,v (pv ) and similarly
S
◦
◦
S
for K0 (n). Let ξ = ⊗v∈S
by KS1 (n) and
/ ξv ∈ Π . Then this vector is fixed
Q
transforms by a character under KS0 (n). In particular, since v∈S
/ GLn−1 (ov )
h
embeds in KS1 (n) via h 7→
we see that when we restrict ΠS to GLn−1
1
the vector ξ ◦ is unramified.
Now let us return to the proof of Theorem 5.4 and in particular the
version of Proposition 5.1 we can salvage. For every vector ξS ∈ ΠS consider
L-functions for GLn
147
the functions UξS ⊗ξ ◦ and VξS ⊗ξ ◦ . When we restrict these functions to GLn−1
they become unramified for all places v ∈
/ S. Hence we see that the integrals
I(s; UξS ⊗ξ ◦ , ϕ′ ) and I(s; VξS ⊗ξ ◦ , ϕ′ ) vanish identically if the function ϕ′ ∈ Vπ′
is not unramified for v ∈
/ S, and in particular if ϕ′ ∈ Vπ′ for π ′ ∈ T (n−1) but
′
π ∈
/ TS (n − 1). Hence, for vectors of the form ξ = ξS ⊗ ξ ◦ we do indeed have
that I(s; UξS ⊗ξ ◦ , ϕ′ ) = I(s; VξS ⊗ξ ◦ , ϕ′ ) for all ϕ′ ∈ Vπ′ and all π ′ ∈ T (n − 1).
Hence, as in Proposition 5.1 we may conclude that UξS ⊗ξ ◦ (In ) = VξS ⊗ξ ◦ (In )
for all ξS ∈ VΠS . Moreover, since ξS was arbitrary in VΠS and the fixed vector
ξ ◦ transforms by a character of KS0 (n) we may conclude that UξS ⊗ξ ◦ (g) =
VξS ⊗ξ ◦ (g) for all ξS ∈ VΠS and all g ∈ GS KS0 (n).
What invariance properties of the function UξS ⊗ξ ◦ have we gained from
our equality with VξS ⊗ξ ◦ . Let us let Γi (nS ) = GLn (k) ∩ GS KSi (n) which
we may view naturally as congruence subgroups of GLn (oS ). Now, as a
function on GS KS0 (n), UξS ⊗ξ ◦ (g) is naturally left invariant under Γ0,P (nS )
= Z(k) P(k) ∩ GS KS0 (n) while VξS ⊗ξ ◦ (g) is naturally left invariant under
Γ0,Q (nS ) = Z(k) Q(k) ∩ GS KS0 (n) where Q = Qn−1 . Similarly we set
Γ1,P (nS ) = Z(k) P(k) ∩ GS KS1 (n) and Γ1,Q (nS ) = Z(k) Q(k) ∩ GS KS1 (n).
The crucial observation for this Theorem is the following result.
Proposition The congruence subgroup Γi (nS ) is generated by Γi,P (nS ) and
Γi,Q (nS ) for i = 0, 1.
This proposition is a consequence of results in the stable algebra of GLn
due to Bass which were crucial to the solution of the congruence subgroup
problem for SLn by Bass, Milnor, and Serre. This is the reason for the
restriction to n ≥ 3 in the statement of Theorem 5.4.
From this we get not an embedding of Π into a space of automorphic
forms on GLn (A), but rather an embedding of ΠS into a space of classical
automorphic forms on GS . To this end, for each ξS ∈ VΠS let us set
ΦξS (gS ) = UξS ⊗ξ ◦ ((gS , 1S )) = VξS ⊗ξ ◦ ((gS , 1S ))
for gS ∈ GS . Then ΦξS will be left invariant under Γ1 (nS ) and transform
by a Nebentypus character χS under Γ0 (nS ) determined by the central character ωΠS of ΠS . Furthermore, it will transform by a character ωS = ωΠS
under the center Z(kS ) of GS . The requisite growth properties are satisfied
and hence the map ξS 7→ ΦξS defines an embedding of ΠS into the space
A(Γ0 (nS )\ GS ; ωS , χS ) of classical automorphic forms on GS relative to the
congruence subgroup Γ0 (nS ) with Nebentypus χS and central character ωS .
148
J.W. Cogdell
We now need to lift our classical automorphic representation back to an
adelic one and hopefully recover the rest of Π. By strong approximation for
GLn and our class number assumption we have the isomorphism between the
space of classical automorphic forms A(Γ0 (nS )\ GS ; ωS , χS ) and the KS1 (n)
invariants in A(GLn (k)\ GLn (A); ω) where ω is the central character of Π.
Hence ΠS will generate an automorphic subrepresentation of the space of
automorphic forms A(GLn (k)\ GLn (A); ω). To compare this to our original
Π, we must check that, in the space of classical forms, the ΦξS ⊗ξ ◦ are Hecke
eigenforms for a classical Hecke algebra and that their Hecke eigenvalues
agree with those from Π. We do this only for those v ∈
/ S which are unramified, where it is a rather standard calculation. As we have not talked about
Hecke algebras, we refer the reader to [7] for details.
Now if we let Π′ be any irreducible subrepresentation of the representation generated by the image of ΠS in A(GLn (k)\ GLn (A); ω), then Π′
is automorphic and we have Π′v ≃ Πv for all v ∈ S by construction and
Π′v ≃ Πv for all v ∈
/ S ′ by the Hecke algebra calculation. Thus we have
proven Theorem 5.4.
5.4
Converse Theorems and Liftings
In this section we would like to make some general remarks on how to apply
these Converse Theorems to the problem of functorial liftings [3].
In order to apply these theorems, you must be able to control the global
properties of the L-function. However, for the most part, the way we have of
controlling global L-functions is to associate them to automorphic forms or
representations. A minute’s thought will then lead one to the conclusion that
the primary application of these results will be to the lifting of automorphic
representations from some group H to GLn .
Suppose that H is a split classical group, π an automorphic representation of H, and ρ a representation of the L-group of H. Then we should
be able to associate an L-function L(s, π, ρ) to this situation [3]. Let us
assume that ρ :L H → GLn (C) so that to π should be associated an automorphic representation Π of GLn (A). What should Π be and why should it
be automorphic.
We can see what Πv should be at almost all places. Since we have the
(arithmetic) Langlands (or Langlands-Satake) parameterization of representations for all archimedean places and those finite places where the representations are unramified [3], we can use these to associate to πv and the map
L-functions for GLn
149
ρv :L Hv → GLn (C) a representation Πv of GLn (kv ). If H happens to be GLm
then we in principle know how to associate the representation Πv at all places
now that the local Langlands conjecture has been solved for GLm [23, 26],
but in practice this is still not feasible. For other situations, we do not know
what Πv should be at the ramified places. We will return to this difficulty
momentarily. But for now, let’s assume we can finesse this local problem
and arrive at a representation Π = ⊗′ Πv such that L(s, π, ρ) = L(s, Π). Π
should then be the Langlands lifting of π to GLn associated to ρ.
For simplicity of exposition, let us now assume that ρ is simply the
standard embedding of L H into GLn (C) and write L(s, π, ρ) = L(s, π) =
L(s, Π). We have our candidate Π for the lift of π to GLn , but how to
tell whether Π is automorphic. This is what the Converse Theorem lets
us do. But to apply them we must first be able to define and control the
twisted L-functions L(s, π × π ′ ) for π ′ ∈ T with an appropriate twisting set
T from one of our Converse Theorems. This is one reason why it is always
crucial to define not only the standard L-functions for H, but also the twisted
versions. If we know, from the theory of L-functions of H twisted by GLm
for appropriate π ′ , that L(s, π × π ′ ) is nice and L(s, π × π ′ ) = L(s, Π × π ′ )
for twists, then we can use Theorem 5.1 or 5.2 to conclude that Π is cuspidal
automorphic or Theorem 5.3 or 5.4 to conclude that Π is quasi-automorphic
and at least obtain a weak automorphic lifting Π′ which is verifiably the
correct representation at almost all places. At this point this relies on the
state of our knowledge of the theory of twisted L-functions for H.
Let us return now to the (local) problem of not knowing the appropriate
local lifting πv 7→ Πv at the ramified places. We can circumvent this by
a combination of global and local means. The global tool is simply the
following observation.
Observation Let Π be as in Theorem 5.3 or 5.4. Suppose that η is a fixed
(highly ramified) character of k× \A× . Suppose that L(s, Π × π ′ ) is nice for
all π ′ ∈ T ⊗ η, where T is either of the twisting sets of Theorem 5.3 or 5.4.
Then Π is quasi-automorphic as in those theorems.
The only thing to observe, say by looking at the local or global integrals,
is that if π ′ ∈ T then L(s, Π×(π ′ ⊗η)) = L(s, (Π⊗η)×π ′ ) so that applying the
Converse Theorem for Π with twisting set T ⊗η is equivalent to applying the
Converse Theorem for Π ⊗ η with the twisting set T . So, by either Theorem
5.3 or 5.4, whichever is appropriate, Π ⊗ η is quasi-automorphic and hence
Π is as well.
150
J.W. Cogdell
Now, if we begin with π automorphic on H(A), we will take T to be the
set of finite places where πv is ramified. For applying Theorem 5.3 we want
S = T and for Theorem 5.4 we want S ∩ T = ∅. We will now take η to be
highly ramified at all places v ∈ T . So at v ∈ T our twisting representations
are all locally of the form (unramified principal series)⊗(highly ramified
character).
We now need to know the following two local facts about the local theory
of L-functions for H.
1. Multiplicativity of γ-factors: If π ′ v = Ind(π ′ 1,v ⊗ π ′ 2,v ), with π ′ i,v and
irreducible admissible representation of GLri (kv ), then
γ(s, πv × π ′ v , ψv ) = γ(s, πv × π ′ 1,v , ψv )γ(s, πv × π ′ 2,v , ψv )
and L(s, πv × π ′ v )−1 should divide [L(s, πv × π ′ 1,v )L(s, π × π ′ 2,v )]−1 .
If πv = Ind(σv ⊗ πv′ ) with σv an irreducible admissible representation
of GLr (kv ) and πv′ an irreducible admissible representation of H′ (kv )
with H′ ⊂ H such that GLr × H′ is the Levi of a parabolic subgroup of
H, then
ev × π ′ v , ψv ).
γ(s, πv × π ′ v , ψv ) = γ(s, σv × π ′ v , ψv )γ(s, πv′ × π ′ v , ψv )γ(s, σ
2. Stability of γ-factors: If π1,v and π2,v are two irreducible admissible
representations of H(kv ), then for every sufficiently highly ramified
character ηv of GL1 (kv ) we have
γ(s, π1,v × ηv , ψv ) = γ(s, π2,v × ηv , ψv )
and
L(s, π1,v × ηv ) = L(s, π2,v × ηv ) ≡ 1.
Once again, for these applications it is crucial that the local theory of
L-functions is sufficiently developed to establish these results on the local
γ-factors. As we have seen in Section 3, both of these facts are known for
GLn .
To utilize these local results, what one now does is the following. At the
places where πv is ramified, choose Πv to be arbitrary, except that it should
have the same central character as πv . This is both to guarantee that the
central character of Π is the same as that of π and hence automorphic and
L-functions for GLn
151
to guarantee that the stable forms of the γ–factors for πv and Πv agree. Now
form Π = ⊗′ Πv . We choose our character η so that at the places v ∈ T we
have that the L– and γ–factors for both πv ⊗ ηv and Πv ⊗ ηv are in their
stable form and agree. We then twist by T ⊗ η for this fixed character η. If
π ′ ∈ T ⊗ η, then for v ∈ T , π ′ v is of the form π ′ v = Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv
with each µv,i an unramified character of kv× . So at the places v ∈ T we have
γ(s, πv × π ′ v ) = γ(s, πv × (Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv ))
Y
=
γ(s, πv ⊗ (µv,i ηv )) (by multiplicativity)
Y
=
γ(s, Πv ⊗ (µv,i ηv )) (by stability)
= γ(s, Πv × (Ind(µv,1 ⊗ · · · ⊗ µv,m ) ⊗ ηv )) (by multiplicativity)
= γ(s, Πv × π ′ v )
and similarly for the L-factors. From this it follows that globally we will have
L(s, π×π ′ ) = L(s, Π×π ′ ) for all π ′ ∈ T ⊗η and the global functional equation
for L(s, π × π ′ ) will yield the global functional equation for L(s, Π × π ′ ). So
L(s, Π × π ′ ) is nice and we may proceed as before. We have, in essence,
twisted away all information about π and Π at those v ∈ T . The price we
pay is that we also lose this information in our conclusion since we only
know that Π is quasi-automorphic. In essence, the Converse Theorem fills
in a correct set of data at those places in T to make the resulting global
representation automorphic.
5.5
Some Liftings
To conclude, let us make a list of some of the liftings that have been accomplished using these Converse Theorems. Some have used the above trick
of multiplicativity and stability of γ–factors to handle the ramified places.
Others, principally those that involve GL2 , have adopted a technique of Ramakrishnan [51] involving a sequence of base changes and descents to get a
more complete handle on the ramified places.
1. The symmetric square lifting from GL2 to GL3 by Gelbart and Jacquet
[15].
2. Non-normal cubic base change for GL2 by Jacquet, Piatetski-Shapiro,
and Shalika [32].
152
J.W. Cogdell
3. The tensor product lifting from GL2 × GL2 to GL4 by Ramakrishnan
[51].
4. The lifting of generic cusp forms from SO2n+1 to GL2n , with Kim,
Piatetski-Shapiro, and Shahidi [6].
5. The tensor product lifting from GL2 × GL3 to GL6 and the symmetric
cube lifting from GL2 to GL4 by Kim and Shahidi [40].
6. The exterior square lifting from GL4 to GL6 and the symmetric fourth
power lift from GL2 to GL5 by Kim [39].
For the most part, it was Theorem 5.3 that was used in each case, with
the exception of (4), where a simpler variant was used requiring twists by
T S (n − 1). For the non-normal cubic base change both Theorem 5.3 with
n = 3 and Theorem 5.1 with n = 2 were used.
Acknowledgments
Most of what I know about L-functions for GLn I have learned through my
years of work with Piatetski-Shapiro. I owe him a great debt of gratitude
for all that he has taught me.
For several years Piatetski-Shapiro and I have envisioned writing a book
on L-functions for GLn [13]. The contents of these notes essentially follows
our outline for that book. In particular, the exposition in Sections 1, 2, and
parts of 3 and 4 is drawn from drafts for this project. The exposition in
Section 5 is drawn from the survey of our work on Converse Theorems [10].
I would also like to thank Piatetski-Shapiro for graciously allowing me to
present part of our joint efforts in these notes.
I would also like to thank Jacquet for enlightening conversations over the
years on his work on L-functions for GLn .
L-functions for GLn
153
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