1. Preliminaries
Let F be a number field. For each place
v of F , let Fv be the completion of F at
v. For each finite v, let Ov be the ring of
integers of Fv and denote by Pv its maximal
ideal. Let $v be a generator of Pv . Let
qv = [Ov : Pv ] and fix an absolute value | |v
such that |$v |v = qv−1 .
Let G be a split connected reductive algebraic group over F . This simply means
a Zariski closed subgroup of GLN (F ) for
some N , where F is the algebraic closure
of F , whose radical (maximal normal connected solvable subgroup) consists of only
1
2
semisimple elements. The radical is then
equal to the connected component of the
center of G. Being split simply means that
G has a maximal abelian subgroup consisting entirely of diagonalizable elements, a
maximal torus, which is isomorphic over F
∗
to some power of F .
Let B be a Borel subgroup of G (over F ),
i.e., a maximal connected solvable subgroup
of G. Let T be a maximally split torus of G
contained in B. Then B = TU, where U is
the unipotent radical of B. The unipotent
subgroup U determines a set of simple roots
∆ and positive roots R+ for T, upon acting
on the Lie algebra g of G.
3
Let P be a parabolic subgroup of G, i.e.,
a conjugate of a closed subgroup of G containing B. We will assume P is standard
by P ⊃ B. We let N be the unipotent radical (maximal connected normal unipotent
subgroup) of P. Then P = MN, where M
is a reductive subgroup, called a Levi subgroup. We will fix M by assuming T ⊂ M.
Let A be the split component of M, i.e.,
the connected component of the center of
M (the maximally split subtorus of the center of M, if the group is not necessarily
split over F ). The parabolic subgroup P
is maximal if the dimension of A/A ∩ ZG
is one, where ZG is the center of G. Then
4
the adjoint action of A on the Lie algebra
of N has a unique reduced eigenfunction α,
the simple root of A in N. There exists
a unique simple root of T whose restriction to A is α. We will denote this root
of T also by α and always identify them
with each other. Other roots of A in N
are simply multiples (considered additively)
of α. Throughout these lectures P is always assumed to be maximal. We refer to
[B2,Sat,Sp] as our main references for algebraic groups and their structure theory.
Let H to be any connected reductive algebraic group defined over F . Considering
H as a group over Fv , for each v, we let
5
Hv = H(Fv ). If AF is the ring of adeles of F , we set H = H(AF ). It may be
considered as a restricted product of groups
Hv with respect to H(Ov ) for all v, where
H splits over an unramified extension [B1].
There will be no restriction if H splits over
F . Moreover, in this case each Kv = H(Ov )
is a maximal compact subgroup of Gv . We
Q
let K = v Kv , where each Kv is a good
maximal compact subgroup of Hv and Kv =
H(Ov ) for almost all v(cf. [B1,Sh1]). Then
G = P K.
For every algebraic group H over F , let
X(H)F denote the group of F rational characters of H. We let X(H) = X(H)F . Note
6
that if T is a split torus over F , then
X(T)F = X(T). We set
a = Hom(X(M)F , R) = Hom(X(A)F , R).
Then a∗ = X(M)F ⊗Z R = X(A)F ⊗Z R and
a∗C = a∗ ⊗R C is the complex dual of a, via
hλ, χ ⊗ zi = λ(χ)z, λ ∈ a, χ ∈ X(M)F
and z ∈ C. For each v, the embedding
X(M)F ,→ X(M)Fv induces a map from
av = Hom(X(M)Fv , R) to a. There exists
a homomorphism HM : M −→ a defined by
Y
|χ(mv )|v
exphχ, HM (m)i =
v
for every χ ∈ X(M)F and m = (mv ). Extend HM to HP on G by making it trivial
on N and K.
7
If ρP is half the sum of roots in N, we set
α̃ = hρP , αi−1 ρP ∈ a∗ . It is a fundamental
weight for T (cf. [Sh1,Sh2]).
Finally having fixed M with M ⊃ T, let
θ ⊂ ∆ denote the subset of simple roots,
generating M. We sometimes write Mθ for
M. Let W be the Weyl group of T in G.
We use WM to denote its Weyl group in
M. There exists a unique element w̃0 ∈ W
such that w̃0 (θ) ⊂ ∆, while w̃0 (α) < 0. We
will always choose a representative w0 for
w̃0 in G(F ) and use w0 to denote each of
its components. We will be more specific
about the choice of w0 later. Finally, let
M0 be the Levi subgroup of G generated
8
by w̃0 (θ). There exists a parabolic subgroup
P0 ⊃ B which has M0 as a Levi factor, in
fact the unique one containing T. Let N0 be
the unipotent radical of P0 (cf. [La1,Sh3]).
2. L–Groups, L–Functions
and Generic Representations
Denote by X ∗ (T) = X(T) the character
group of T which is the same as X(T)F .
Let X∗ (T) be the group of cocharacters of
∗
T, i.e., homomorphisms from Gm = F into
T = T(F ). Let ∆∨ = ∆∨ (T) be the set
of simple coroots of T, i.e., α∨ : Gm −→ T
satisfying α(α∨ (t)) = t2 . Let
(2.1)
ψ0 (G) = (X ∗ (T), ∆(T), X∗ (T), ∆∨ (T))
9
denote the based root datum. By
Chevalley–Grothendieck theorem [Sp],
there exist a complex connected reductive
group G∨ with a maximal torus T ∨ such
that
ψ0 (G)∨ = (X∗ (T), ∆∨ (T), X ∗ (T), ∆(T))
= ψ0 (G∨ )
= (X ∗ (T ∨ ), ∆(T ∨ ), X∗ (T ∨ ), ∆∨ (T ∨ )).
The L–group
L
G of G is
L
G = G∨ × ΓF ,
where ΓF = Gal(F /F ). In general, one
carries the action of ΓF on roots and coroots dually to G∨ and let L G = G∨ o ΓF .
(Observe that G∨ = L G0 .) In fact, we have
an exact sequence
10
(2.2)
→ 0,
0−
→ Int(G) −
→ Aut(G) −
→ Autψ0 (G) −
where Int(G) is the subgroup of inner automorphisms. One can show that if {Xβ }β∈∆0 ,
∆0 being the set of simple roots of T, is a set
of simple root vectors invariant under ΓF ,
then
Aut ψ0 (G) = Aut(G, B, T, {Xβ }β∈∆0 ).
The set {Xβ }β∈∆0 is called a splitting, as
it splits (2.2). The map
ΓF −→ Aut(G, B, T, {Xβ }β )
defines a map
ΓF −−−−→ Autψ0 (G)∨ = Aut ψ0 (G∨ )
11
which is a subset of Aut(G∨ ).
This de-
fines the action of ΓF on G∨ and defines
L
G (cf. [B1, Sat]).
Given a connected reductive algebraic
group H over F , let L H be its L–group.
Considering H as a group over Fv , we then
denote by L Hv its L–group over Fv . If G is
split over F and if we decide to only consider
L
G0 = G∨ , then we may assume that the L–
groups are all the same, no matter the place
v. Finally, the natural map ΓFv → ΓF leads
to a map ηv : L Mv → L M for all v.
In our setting L M is a Levi subgroup of
L
G and one can define a unipotent group
L
N (cf. [B1]) so that L M L N is a parabolic
12
subgroup of L G with unipotent radical L N .
The L–group L M acts on the (complex) Lie
algebra L n of
L
N . Let r be this represenm
L
tation. Decompose r =
ri to its irrei=1
ducible subrepresentations, indexed according to the values hα̃, βi = i as β ranges
among the positive roots of T. More precisely, Xβ ∨ ∈ L n lies in the space Vi of ri if
and only if hα̃, βi = i. Here Xβ ∨ is a root
vector attached to the coroots β ∨ , considered as a root of the L–group. Clearly the
integer m is equal to the nilpotence class of
L
n. We let ri,v = ri · ηv for each i. Again if
G is split over F , we may assume ri,v = ri
for all v and i (cf. [La,Sh1,Sh2]).
13
Let π = ⊗v πv be a cuspidal representation of M = M(AF ). Then for almost all v,
πv is an unramified representation of Mv =
M(Fv ). This means that πv has a vector
which remains invariant under M(Ov ). In
this case, the class of πv is determined by a
semisimple LM –conjugacy class Av ⊂ LMv =
L
M . Given a complex analytic (finite dimen-
sional) representation ρ of LM , we define the
local Langlands L–function attached to πv ,
ρ and a complex number s by
(2.3) L(s, πv , ρ) = det(I − ρ(Av )qv−s )−1 .
When M is quasisplit over Fv , to split over
an unramified extension Lw /Fv , and τv is
14
the unique Frobenius conjugacy class in Gal
(Lw /Fv ), then Av must be replaced by tv o
τv , tv ∈
L
T 0 . We may moreover assume
that tv is fixed by τv ([B1, La1, Sh2]).
In these lectures, we will mainly be concerned with the case where ρ = ri,v , i =
1, . . . , m.
We shall now discuss the notion of generic
representations. We will first assume that F
is a local field. Fix a F –splitting {Xβ }β∈∆0
as before. This then determines a map
Y
(2.4)
φ: U →
Ga −→ Ga ,
where the product runs over all β ∈ ∆0 ,
sending exp(xβ Xβ ) to Σxβ .Let ψF be a non–
15
trivial character of F . We shall then define
a generic character χ of U = U(F ) by
(2.5)
χ(u) = ψF (Σβ xβ ),
where φ(u) = (xβ )β and the β–component
of u is exp(xβ Xβ ). Conversely, given a
generic character χ of U , i.e., one which
is non–trivial on every simple root group,
there exists an F –splitting {Xβ }β such that
χ is defined by (2.4) and (2.5).
If π is an irreducible (admissible) unitary representation of M = M(F ), then
π is called generic, or more precisely χ–
generic for a generic character χ of U 0 =
(U ∩ M)(F ), if there exists a functional λ
16
on the space of π, called a Whittaker functional, which is continuous with respect to
the seminorm topology defined by the Hilbert
space norm on the space H(π) of π if F
is archimedean (cf. [S,Sh4,Sh5]) (continuous with respect to the trivial locally convex
topology on H(π) for which every seminorm
is continuous if F is non–archimedean) and
satisfies
(2.6)
hπ(u)x, λi = χ(u)hx, λi,
u ∈ U 0 , x ∈ H(π)∞ , the subspace of C ∞ –
vectors. By a theorem of Shalika [S], the
space of all the Whittaker functionals on
H(π) is at most one–dimensional. Changing
17
the splitting, we may assume χ is defined by
(2.4) and (2.5).
Now, assume F is global. Let ψ = ⊗v ψv
be a non–trivial character of F \AF . Then
each ψv is non–trivial. Moreover, for almost all v, ψv is unramified, i.e., Ov is the
largest ideal on which ψv is trivial. The map
(2.4) is F –rational and therefore extends to
a map from U = U(AF ) into Ga (AF ), sending U(F ) into Ga (F ). We then define a
character χ of U(F )\U by (2.5). Consequently, if χv (uv ) is defined by (2.5) and ψv
Q
for each v, then χ(u) = v χv (uv ), where
u = (uv )v ∈ U .
Now, let π = ⊗v πv is a cuspidal represen-
18
tation of M = M(AF ). Choose a function
ϕ in the space of π and set
Z
(2.7) Wϕ (m) =
ϕ(um)χ(u)du.
U0 (F )\U 0
We shall say π is (globally) χ–generic if Wϕ 6=
0 for some ϕ. Then each πv is χv –generic.
If ϕ = ⊗v ϕv , ϕv ∈ H(πv ), then
Wϕ (m) =
Y
hπv (mv )ϕv , λv i
v
(2.8)
=
Y
Wϕv (mv )
v
for some χv –Whittaker functional λv on H(πv ).
We finally point out that every generic
character χ of U(F )(U(F )\U(AF ) if F is
19
global) points to a F –splitting and conversely,
each F –splitting defines a generic character,
both by means of (2.4) and (2.5).
Now assume F is local and πv is an irreducible admissible representation of Mv .
Let s ∈ C and denote by I(s, πv ) = I(sα̃, πv ),
the induced representation
(2.9)
hsα̃,HMv ( )i
I(s, πv ) = IndMv Nv ↑Gv πv ⊗qv
⊗1.
This is the right regular action of Gv on
the space of smooth functions f from Gv to
H(πv ), satisfying:
(2.10)
hsα̃+ρP ,HMv (m)i
f (mng) = πv (m)qv
f (g).
If v = ∞, smooth would mean that f is
20
infinitely many times differentiable. On the
other hand for p < ∞, it simply means that
f is locally constant, i.e., f (gk) = f (g) for k
in some open compact subgroup depending
on f (cf. [Ca1,Car]).
One can now define a Whittaker functional λχv (s, πv ) for I(s, πv ) as follows:
(2.11)
λχv (s, πv )(f ) =
Z
Nv0
hf (w0−1 n0 ), λiχ(n0 )dn0 ,
where λ is a fixed Whittaker functional on
H(πv ). The integral converges as a principal value integral and stabilizes as Nv0 is approached by an increasing sequence of open
compact subgroups (cf. [CS,Sh3]), i.e., we
21
may replace Nv0 with a sufficiently large open
compact subgroup depending on f .
3. Eisenstein Series and Intertwining
Operators; The Constant Term
Let π = ⊗v πv be a cusp form on M .
Given a KM –finite function ϕ in the space
of π, we extend ϕ to a function ϕ̃ on G as
follows. The representation π is a subrepre0
0
sentation of L20 (ZM
M(F )\M, ρ), where ZM
is the AF –points of the connected component Z0M of the center of M and ρ is a
0
. The function ϕ
character of Z0M (F )\ZM
is then in this L2 –space and being KM –
finite, its right translations by elements in
22
KM = Πv KM,v , KM,v = Kv ∩ Mv , generate
a finite dimensional representation τ of KM
(cf. [Car]). We may assume ϕ is so chosen
that τ is irreducible and write τ = ⊗v τv ,
where for almost all v, τv is trivial. Next we
will choose irreducible (finite dimensional)
representations τ̃v of each Kv , containing
τv .
Moreover, we assume τ̃v is the triv-
ial representation for almost all Kv . Set
τ̃ = ⊗v τ̃v (cf. [Sh6]).
Let Pτ be the projection on the space of
τ and fix measures dkv on each KM,v whose
total mass is 1. Let dk be the product measure on KM . Set
23
Z
(3.1)
ϕ̃(m) =
KM
ϕ(mk)τ̃ (k)dk · Pτ .
Observe that ϕ̃(mk −1 ) = τ̃ (k)ϕ̃(m). This is
a τ̃ –function on M in Harish–Chandra’s terminology [HC]. We extend τ̃ to all of G by
ϕ̃(nmk) = τ̃ (k −1 )ϕ̃(m). It is easily checked
to be a well–defined operator valued function on G ([Sh6]).
Next, set
(3.2) Φ̃s (g) = ϕ̃(g) exphsα̃ + ρP , HP (g)i
and let Φs be a matrix coefficient of this
operator valued function. (See (3.7) below.)
The corresponding Eisenstein series is then
24
defined by
(3.3) E(s, Φs , g, P ) =
X
Φs (γg).
γ∈P(F )\G(F )
We will also define the operator valued
Eisenstein series by
(3.4) Ẽ(s, Φ̃s , g, P ) =
X
Φ̃s (γg).
γ∈P(F )\G(F )
They both converge for Re(s) >> 0 and
have a finite number of simple poles for Re(s)
≥ 0, none with Re(s) = 0 (cf. [HC,La2,MW1]).
Let I(s, π) = ⊗v I(s, πv ) be the representation of G = G(AF ) induced from
(3.5)
π ⊗ exphsα̃ + ρP , HM ( )i ⊗ 1.
25
Let f ∈ V (s, π) = ⊗v V (s, πv ) be defined
by
(3.6)
f (n0 m0 k0 ) = exphsα̃ + ρP , HP (m0 )i·
Z
hτ̃ (k0−1 )τ (k)x, x̂iπ(m0 k)ϕdk,
KM
with m0 ∈ M, n0 ∈ N and k0 ∈ K. Here
x ∈ H(τ ) and x̂ ∈ H(τ̂ ), where τ̂ is the
contragredient of τ . Moreover
(3.7)
f (n0 m0 k0 )(e) = hΦ̃s (g)x, x̂i
= Φs (g),
where the left hand side is the value of the
cusp form f (n0 m0 k0 ) at identity and g =
n0 m0 k0 (cf. [Sh6]).
26
Observe that
(3.8)
E(s, Φs , g, P ) = hẼ(s, Φ̃s , g, P )x, x̂i.
Given f ∈ V (s, π) and Re(s) >> 0, define the global intertwining operator A(s, π)
by
(3.9)
Z
A(s, π)f (g) =
N0
f (w0−1 n0 g)dn0 .
Finally, if at each v we define a local intertwining operator by
(3.10)
Z
A(s, πv , w0 )fv (g) =
Nv0
fv (w0−1 n0 g)dn0 ,
then
(3.11)
A(s, π) = ⊗v A(s, πv , w0 ).
27
Observe that
(3.12)
A(s, π): I(s, π) → I(−s, w0 (π))
and
(3.13)
A(s, πv , w0 ): I(s, πv ) → I(−s, w0 (πv )).
Using (3.7), we now define
(3.14) (M (s, π)Φs )(g) = A(s, π)f (g)(e),
where by the left hand side we understand
the value of the cusp form A(s, π)f (g) at e.
This is basically the Langlands’ M (s, π) introduced in [La2], or as denoted by Harish–
Chandra in [HC], his function c(s, π).
28
Constant Term Theorem. The constant
term
(3.15)
EP0 (s, Φs , g, P) =
Z
N0 (F )\N 0
E(s, Φs , n0 g, P)dn0
is equal to
(3.16)
EP0 (s, Φs , g, P) = δM,M0 Φs (g)+(M (s, π)Φs )(g).
Here δM,M0 is the Kronecker δ–function.
Its analytic properties and therefore those
of M (s, π) are exactly the same as E(s, Φs , −, P).
In the split case the proof of the first
part is in [La1], as in elsewhere. But the
rest is among the main properties of Eisen-
29
stein series and included in several places
[La2,HC,MW1].
To prove the first part, one just substitutes (3.3) in (3.15) and expands and uses
Bruhat decomposition and cuspidality of ϕ.
We finally express the functional equation of Eisenstein series by (cf. [HC,La2]),
(3.17)
E(−s, M (π, s)Φs , g, P 0 ) = E(s, Φs , g, P ).
4. Constant Term
and Automorphic L–Functions
It follows immediately from the Constant
Term Theorem that ⊗v A(s, πv , w0 ) is a meromorphic function of s with a finite num-
30
ber of simple poles for Re(s) > 0, none on
Re(s) = 0.
Assume v is an unramified place for π,
i.e., that πv is spherical. Take fv0 ∈ V (s, πv )
such that fv0 (k) is a fixed vector invariant
under M(Ov ) for all k ∈ G(Ov ). With nom
L
tation as in Section 2, let r =
ri be the
i=1
adjoint action of L M on L n. We have
Lemma 4.1 [La1]. Assume πv is unramified. Then
A(s, πv , w0 )fv0 (ev )
=
m
Y
i=1
L(is, πv , r̃i )/L(1 + is, πv , r̃i )fv0 (ev ).
31
With a clever induction [La1,La2,Sh3] the
problem reduces to that of SL2 , which we
will now explain.
∗
Here G = SL2 and M = T = Gm = F v .
We need to calculate
Z
1 x
0 1
0
dx
fv
0
1
−1
0
Fv
(4.1.1)
Z
1 0
=
dx,
fv0
x 1
Fv
0 −1
since Rw fv0 = fv0 , where w =
.
1
0
We then have Z
Z
1 0
0
dx =
fv
dx
x 1
Fv
|x|v ≤1
(4.1.2)
Z
1 0
dx.
fv0
+
x 1
|x|v >1
32
We now write
(4.1.3)
1
x
0
1
=
−1
x
0
1
x
0
1
−1
x−1
,
and therefore for |x|v > 1, which implies
x−1 ∈ Ov ,
fv0
1
x
0
1
=
fv0
−1
x
0
1
x
= ηv−1 (x)|x|−1−s
,
v
where ηv is the character of F ∗ (with cusp
form η = ⊗v ηv on A∗F ) defining the induced
representation. Then (4.1.2) equals
33
1+
∞
X
ηv ($vn )qv−n−ns
n=1
∞
X
=1+
n=1
∞
X
=1+
Z
($v−n )−($v−n+1 )
ηv ($vn )qv−n−ns |$v−n |v
Z
|x|v d∗ x
d∗ x
Ov∗
ηv ($vn )(1 − qv−1 )qv−ns .
n=1
Write ηv (x) = |x|µv v to get
(4.1.4)
1+
∞
X
qv−nµv −ns (1 − qv−1 )
n=1
= 1 + (1 −
=
1−
qv−1 )
1
qvµv +1+s
1−
1
qvµv +s
.
·
qv−µv −s
·
1
1−
1
qvµv +s
34
Now if α∨ is the standard coroot of SL2 ,
then
qvµv = α∨ (Av ),
following our HT identification. (See the
remark below). Here Av ∈ P GL2 (C) represents the semisimple conjugacy class parametrizing πv and α∨ is the root of P GL2 (C), or
the coroot of SL2 . Clearly α∨ (Av ) is the
eigenvalue for the adjoint action of
L
L
T on
n, evaluated at Av . Therefore
(4.1.5)
qv−µv = α∨ (Av )−1 = r̃(Av )
We thus get that (4.1.1) equals
(4.1.6)
(1−α∨ (Av )−1 qv−s )−1 /(1−α∨ (Av )−1 qv−s−1 )−1
35
which equals
(4.1.7)
L(s, ηv , r̃1 )/L(1 + s, ηv , r̃1 )
since m = 1, i.e., r = r1 .
SL2 (R). It is instructive to also compute
the case of SL2 (R)(GL2 (R), respectively).
We again need to calculate
Z
1 0
0
dx.
fv
(4.1.12)
x 1
R
Here K = SO2 (R)(O2 (R), respectively) and
we can write
1 0
a
=
x 1
0
where
k(θ) =
cos θ
− sin θ
y
b
k(θ),
sin θ
cos θ
.
36
We can then take tan θ = −x, b = a−1 =
√
√
2
x + 1 and y = x/ x2 + 1. We need to
calculate
Z
∞
−∞
1
√
x2 + 1
where η
Z
a
0
∞
2
s1 p
( x2 + 1)s2 (x2 +1)−1/2 dx,
0
b
= |a|s1 |b|s2 , or
(x2 + 1)−(s1 −s2 +1)/2 dx.
0
Let s = s1 − s2 and set x = tan θ, we need
to calculate
Z
π/2
2
0
(cos θ)s−1 dθ.
37
Using the standard formula
Z
p+1
q
p
cos θ sin θdθ = Γ
2
0
p+q
q+1
/2Γ
·Γ
+1 ,
2
2
π/2
where
Z
∞
Γ(s) =
ts e−t d∗ t,
0
(4.1.12) then equals
(4.1.13)
Γ(1/2)Γ(s/2)/Γ((s + 1)/2).
√
Using Γ(1/2) = π, (4.1.13) equals
π −s/2 Γ(s/2)/π −(s+1/2) Γ(s + 1/2)
38
which is again L(s)/L(s + 1), where L(s) is
the archimedean Hecke–Tate L–function attached to the character |x|s , the R–component
of our cusp form on T .
The main result of Langlands in [La1] can
be stated as follows. Let S be a finite set of
places with the property that if v 6∈ S, πv
is an unramified representation. Every f ∈
V (s, π) is of the form
f ∈ ⊗v∈S V (s, πv )
O
⊗v6∈S {fv0 },
where fv0 is Kv –spherical for some S. To
be precise, the decomposition π = ⊗v πv depends on a choice of M(Ov )–invariant vectors {xv } for all the unramified places which
39
one fixes once for all (cf. [S]). The functions
fv0 must then satisfy fv0 (kv ) = xv for all
kv ∈ Kv and all v 6∈ S. Assume further
that f = ⊗v fv , with fv = fv0 for all v 6∈ S.
For each i, let
Y
S
(4.1)
L (s, π, ri ) =
L(s, πv , ri ).
v6∈S
Then by Lemma 4.1
A(s, π)f (e) = (
m
Y
LS (is, π, r̃i )/LS (1 + is, π, r̃i ))
i=1
(4.2)
⊗v6∈S fv0 (ev )
O
⊗v∈S A(s, πv , w0 )fv (ev ).
It now follows from the properties of the
constant term A(s, π) that
40
Theorem 4.2 (Langlands) [La1].
The product quotient
m
Y
LS (is, π, r̃i )/LS (1 + is, π, r̃i )
i=1
is meromorphic on all of C.
Clearly one needs an induction to get this
to lead to meromorphic continuation of each
L–function in the product. We will soon
discuss this induction.
5. Examples
We shall now give a number of important
examples of L–functions which appear in
constant terms for appropriate pairs (G, M).
We refer to [La1,Sh2] for the complete list.
41
5.1.
Let G = GLn+t , M = GLn ×
GLt , π = ⊗v πv cusp form on GLn (AF ) and
π 0 = ⊗v πv0 one on GLt (AF ). Then m = 1
and we get L(s, π × π̃ 0 ), the Rankin–Selberg
product L–function for the pair (π, π̃ 0 )
(cf. [JPSS1] and [Sh10]). It will be discussed
by Cogdell in more length.
5.2. Let G be a classical group, split
over F and let M = GLn × G0 , where G0 is
a classical group of the same type, but lower
rank. Let π and π 0 be cuspidal representations of GLn (AF ) and G0 . Then m = 2.
One gets L(s, π × π̃ 0 ) as its first L–function.
For i = 2, we get L(s, π, ρ), where ρ = Λ2
if L G is orthogonal and ρ = Sym2 if L G is
42
symplectic (cf. [CKPSS1,2].
5.3. Let G = GSpinn+t , M = GLn ×
GSpint , π and π 0 cusp forms on GLn (AF )
and GSpint (AF ), respectively. Then m =
2. Again we get L(s, π × π̃ 0 ) as our first
L–function. The second L–function is then
an appropriate twist of either L(s, π, Λ2 ) or
L(s, π, Sym2 ).
5.3.a. Let G = GSpin5+2n , M = GLn ×
Gpin5 = GLn × GSp4 , and let (π, π 0 ) be a
cusp form on GLn (AF ) × GSp4 (AF ). Again
we get L(s, π × π̃ 0 ) as our first L–function.
This is very important.
5.3.b. G = GSpin6+2n , M = GLn ×
43
GSpin6 . We have 0 → {±1} → GL4 (C) →
GSO6 (C) → 0. Suppose π is on GLn (AF )
and π 0 on GL4 (AF ). We then get L(s, π ⊗
π̃ 0 , ρn ⊗Λ2 ρ4 ) as our first L–function (cf. [K4]).
5.4. Let G be a simply connected group
of either type E6 or E7 . Choose M such
that MD , the derived group of M, is either SL3 × SL2 × SL3 or SL3 × SL2 × SL4 ,
respectively. There exist F –rational injections from M into GL3 × GL2 × GLt , t = 3
or 4, which are identity on SL3 ×SL2 ×SLt .
Let π1 ⊗ π2 ⊗ σ be a cuspidal representation
of GL3 (AF ) × GL2 (AF ) × GLt (AF ). Then
m = 3 or 4 according as if G = E6 or E7 .
The first L–function is then L(s, π1 ×π2 × σ̃)
44
(cf. [KS2]).
All these L–functions will be revisited
later in connection with functoriality.
6. Local Coefficients, Non–constant
Term and the Crude Functional Equation
Changing the splitting in U, we may assume each χv is defined by means of ψv
through equation (2.5). At each v, let λχv
(s, πv ) be the Whittaker functional defined
by equation (2.11). Next, let A(s, πv , w0 )
be the local intertwining operator defined
by (3.10). Finally, let λχv (−s, w0 (πv )) be
the corresponding functional defined for
I(−s, w0 (πv )) by means of (2.11). Using our
45
assumption on χv , Rodier’s theorem points
to the existence of a complex function Cχv (s, πv )
of s, depending on πv , χv and w0 such that
(6.1)
λχv (s, πv ) = Cχv (s, πv )
λχv (−s, w0 (πv )) · A(s, πv , w0 ).
This is what we call the “Local Coefficient” attached to s, πv , χv and w0
(cf. [Sh2,Sh3]). The choice of w0 is now
specified by our fixed splitting as in [Sh4].
Now, let
(6.2)
Z
Eχ (s, Φs , g, P )
=
U(F )\U
E(s, Φs , ug, P )χ(u)du,
the χ–nonconstant term of E(s, Φs , g, P )
(cf. [Sh2,Sh3,Sh6]), where χ = ⊗v χv .
46
If we substitute for E(s, Φs , ug, P ) its definition in (3.3) and do some telescoping, we
get, using orthogonality of χ, that
(6.3) Eχ (s, Φs , e, f ) =
Y
λχv (s, πv )(fv ),
v
where fv is the local component of f defined
by (3.6) in which ϕ = ⊗v ϕv is identified
with Wϕ = ⊗v Wv , where Wϕ is defined by
(2.7). As explained in [S], Wv (ev ) = 1 for
almost all v. We now appeal to the following
formula of Casselman–Shalika [CS]:
Theorem 6.1(Casselman-Shalika [CS]).
Assume πv and ψv are both unramified and
if fv (ev ) defines a Whittaker function Wv in
47
the Whittaker model of πv , assume Wv (ev ) =
1. Observe that this is the case for almost
all v. Then
(6.4)
λχv (s, πv )(fv ) =
m
Y
L(1 + is, πv , r̃i )−1 .
i=1
In fact, if Wfv (gv ) = λχv (s, πv )(Iv (gv )fv )
is the Whittaker function attached to fv ,
then (6.3) can be written as
Theorem 6.2. One has
Eχ (s, Φs , e, f ) =
(6.5)
Y
Wfv (ev )
v∈S
m
Y
i=1
LS (1 + is, π, r̃i )−1 .
48
Remark. As opposed to intertwining operators, Whittaker functions are by no means
multiplicative and therefore proof of Theorem 6.1 cannot be reduced to rank one calculations by means multiplicativity (cocycle
relations). It should be pointed out that in
the case of SL2 , Theorem 6.1 is an easy exercise.
Corollary 6.3 [Sh3]. The product
m
Y
LS (1 + is, π, ri ) 6= 0
i=1
for Re(s) = 0. In particular, if π and π 0
are cusp forms on GLn (AF ) and GLt (AF ),
then
L(1, π × π 0 ) 6= 0,
49
where local L–functions at every place are
corresponding Artin ones ([HT], [He1]).
Proof. Modulo non–vanishing of Wfv (e) for
Re(s) = 1, which is highly non–trivial if
v = ∞ (cf. Casselman–Wallach [Ca2,W]),
this follows from unitarity (and therefore
holomorphy) of M (s, π) for Re(s) = 0, and
Theorem 6.2. Observe that the integration
in (6.2) is over a compact set.
Now, computing the non–constant terms
from the two sides of the functional equation (3.17), Lemma 4.1 and Theorems 6.1
and 6.2, together with Definition (6.1) implies:
50
Theorem 6.4 (Crude Functional Equation) [Sh3,Sh6]. We have
m
Y
(6.6)
S
L (is, π, ri ) =
i=1
Y
Cχv (s, π̃v )
v∈S
m
Y
LS (1 − is, π, r̃i ).
i=1
We just point out that by Lemma 4.1,
Theorem 6.1 and Definition (6.1)
(6.7)
Cχv (s, π̃v ) =
m
Y
L(1−is, πv , r̃i )/L(is, πv , ri ),
i=1
whenever πv is unramified.
51
7. The Main Induction, Functional
Equations and Multiplicativity
To prove the individual functional equations, i.e., for each L(s, π, ri ) with precise
root numbers and L–functions , we appeal
to the following induction statement
(cf. [Sh1,Sh2]). It is crucial in all the results
that we prove from now on.
Proposition 7.1 [Sh1]. Given 1 < i ≤
m, there exists a split group Gi over F , a
maximal F –parabolic subgroup Pi = Mi Ni
and a cuspidal automorphic representation
π 0 of Mi = Mi (AF ), unramified for every
v 6∈ S, such that if the adjoint action of
52
0
L
Mi on L ni decomposes as r0 =
m
L
j=1
rj0 , then
LS (s, π, ri ) = LS (s, π 0 , r10 ).
Moreover m0 < m.
It was observed by Arthur [A], that each
Mi can be taken equal to M and π 0 = π.
More precisely:
Proposition 7.2 [A]. Given i, 1 < i ≤ m,
there exists a split connected reductive F –
group Gi with M as a Levi subgroup and
m0 < m for which r10 = ri . Each Gi can be
taken to be an endoscopic group for G. (Its
L–group is the centralizer of a semisimple
element in L G.)
53
Next, we need the following variant of a
result of Henniart and Vigneras. In it, we
assume that the defining additive character
ψ for χ is local component of a global one.
Here we shift the ramification to archimedean
places. Consequently, we need to use a result of Dixmier–Malliavin on convolution algebras for real semisimple groups.
Proposition 7.3 [Sh1]. Let σ be an irreducible χ–generic supercuspidal representation of G = G(F ), where F is a non–
archimedean local field and G is defined over
F . Let B = TU be the Borel subgroup of
G defining χ. Then there exists a number field K with a ring integers O, a split
54
group H over K, a non–degenerate character χ̃ = ⊗v χ̃v of UH (K)\ UH (AK ), and a
globally χ̃–generic cusp form π = ⊗v πv on
H = H(AK ) such that:
a) Kv0 = F for some place v0 of K,
b) χ̃v0 = χ,
c) as a group over F, H = G,
d) πv0 = σ, and finally
e) for every other finite place v of K, v 6=
v0 , πv is of class one with respect to a
special maximal compact subgroup Qv
of H(Kv ). Here UH is the unipotent
radical of a Borel subgroup of H for
which UH as a group over F equals
U.
55
Proposition 7.3 [Sh1,Sh4]. Assume either Fv is archimedean or πv has a vector
fixed by an Iwahori subgroup. Let ϕv : WF0 v →
L
Mv be the homorphism of the Deligne–Weil
group parametrizing πv . For each i, let L(s, ri ·
ϕv ) and ε(s, ri ·ϕv , ψv ) be the Artin L–function
and root number attached to ri · ϕv . Then
(7.1)
m
Y
L(1 − is, ri · ϕv )
.
Cχv(s, πv ) = ε(is, r̃i ·ϕv , ψv )
L(is, r̃i · ϕv )
i=1
Remark. If G is quasisplit, but not split,
then a product of Langlands λ–functions
(Hilbert symbols) will also appear on the
right hand side of (7.1).
56
Applying these propositions inductively
and using the Crude Functional Equations (6.6) then implies:
Theorem 7.4 [Sh1]. Assume G is a split
reductive algebraic group over a local field F
of characteristic zero. Let P = MN, P ⊃
B, be a maximal parabolic subgroup as before. Let χ be a generic character defined
by the splitting and ψF ∈ F̂ . Given an irreducible admissible χ–generic representation
σ of M = M(F ), these exist m complex
functions γ(s, σ, ri , ψF ), 1 ≤ i ≤ m, such
that:
57
1) If F and σ satisfy the conditions of Proposition 7.3, then
(7.2)
γ(s, σ, ri , ψF ) = ε(s, ri · ϕ, ψF )
L(1 − s, r̃i · ϕ)/L(s, ri · ϕ).
2) Equation (7.1) holds (in the form
m
Q
γ(is, σ, r̃i , ψ F )).
i=1
3) γ(s, σ, ri , ψF ) is multiplicative under induction (to be discussed below).
4) Whenever σ becomes a local component
of a globally generic cusp from, then γ’s
become the local factors needed in their
functional equations. Moreover 1), 3) and
4) determine the γ–functions uniquely.
58
What is multiplicativity?
We will
discuss this only in examples since the general formulation is complicated. It simply
says that the γ–functions are multiplicative
under parabolic induction and is a consequence of multiplicativity of intertwining operators (3.10) under that (cf. [Sh3]). This is
very deep from the point of view of Rankin–
Selberg method and usually quite hard to
prove. Here are some examples:
Example 1 (cf. [Sh7]). Suppose
G = Sp(2n + 2t) and M = GLn × Sp(2t),
where n and t are positive integers. Write
σ = σ1 ⊗ τ . Suppose M0 = GLn1 × . . . ×
GLnk × GLt1 × . . . × GLt` × Sp(2a), where
59
n1 +. . .+nk = n and t1 +. . .+t` +a = t. By
case Cn of [Sh2], r1 is equal to the tensor
product of the standard representation of
GLn (C) and SO2t+1 (C).
0
If σ =
k
N
j=1
σj0
⊗
Ǹ
b=1
00
σb ⊗ τ 0 , then multi-
plicativity simply means that, if
σ⊂
0
σ
⊗ 1,
Ind
0
0
M N ↑G
then
(7.3)
γ(s, σ1 × τ, ψF ) =
k Ỳ
Y
γ(s, σj0
00
× σb , ψ F )
j=1 b=1
γ(s, σj0
00
× σ̃b , ψF )
k
Y
j=1
γ(s, σj0 × τ 0 , ψF ).
60
If ρn is the standard representation of
GLn (C), then r2 = Λ2 ρn . With σ1 and σj0
as before, multiplicativity for r2 means
γ(s, σ1 , Λ2 ρn , ψF ) =
k
Y
γ(s, σj0 , Λ2 ρnj , ψF )
j=1
Y
γ(s, σi0 × σj0 , ψF ).
1≤i<j≤k
No more ri beyond r2 shows up and this is
the case for all the classical groups. Equality of the dimension on both sides of (7.4)
simply means the following trivial identity:
k
X
!2
ni −
i=1
k
X
i=1
!
ni
k
X
X
2
ni nj .
= (ni −ni )+2
i=1
1≤i<j≤k
61
L–function and root number
L–functions are now defined using
γ–functions. When σ is tempered, we define
L(s, σ, ri ) as the inverse of the normalized
polynomial P (q −s ) in q −s satisfying P (0) =
1 and
(7.5)
γ(s, σ, r1 , ψF ) = ε(s, σ, ri , ψF )
L(1 − s, σ, r̃i )/L(s, σ, ri ).
The L–function L(s, σ, r̃i ) and the root
number ε(s, σ, ri , ψF ) are also uniquely defined by (7.5). To proceed we need the following theorem.
62
Theorem 7.5. Suppose σ is tempered. Then
L(s, σ, ri ) are all holomorphic for Re(s) > 0.
With this theorem in hand, L(s, σ, ri ) are
now also multiplicative if σ is tempered.
(See below.)
To define L(s, σ, ri ) for any irreducible χ–
generic representation, we appeal to Langlands classification [La3,Si]. We embed σ ⊂
IndM 0 (N 0 ∩M )↑M σν0 ⊗ 1, where σν0 is quasi–
tempered with a negative Langlands parameter ν. Then σ00 is tempered. By multiplicativity, we then write γ(s, σ, ri , ψF ) as a
product of appropriate γ–functions
0
γ(s, σν,j
, rij , ψF ), where j runs over a finite
63
index set determined by M0 and M, i.e.,
(7.6)
γ(s, σ, ri , ψF ) =
Y
0
γ(s, σν,j
, rij , ψF ).
j
More precisely, for each j, there exist
Levi subgroups (not necessarily maximal)
M0j and M̃j of G with T ⊂ M0j ⊂ M̃j as
a maximal Levi subgroup. The representa0
tion σν,j
is a quasi–tempered representation
0
is tempered. The repreof Mj0 for which σ0,j
sentation rij of L Mj0 is an irreducible consti-
tutent of the action of L Mj0 on the Lie alge-
bra of the L–group of M̃j ∩N0j , where N0j ⊂
U is the unipotent radical of P0j = M0j N0j .
0
Thus the γ–function γ(s, σν,j
, rij , ψF ) is a
64
γ–function attached to the pair (M̃j , M0j ).
When ν = 0, by Conjecture 7.5, the prod0
uct of the numerators of γ(s, σ0,j
, rij , ψF )
equals to the numerator of the product and
0
, rij )−1 denotes the normalized
if L(s, σ0,j
0
numerator of γ(s, σ0,j
, rij , ψF ), i.e., the re-
ciprocal of the tempered L–function attached
0
and rij by means of the pair (M̃j , M0j ),
to σ0,j
0
, rij ) to denote its anwe then use L(s, σν,j
alytic continuation to ν. We now set
Y
0
L(s, σν,j
, rij ).
(7.7)
L(s, σ, ri ) =
j
This agrees with the way Artin L–functions
are defined [La3,KSh,Sh1,T]. Details are given
in [Sh3]; also see the discussion in pages
65
862 and 863 of [KS2]. The root number is
then defined uniquely to satisfy (7.5). We
should point out that in Definition (7.7) we
do not need to assume the validity of Conjecture 7.5. But if valid, it implies that L–
functions are also multiplicative, if the representations are tempered.
Having defined our L–functions and root
numbers everywhere, we set
Y
L(s, π, ri ) =
L(s, πv , ri )
v
and
ε(s, π, ri ) =
Y
v
We then have:
ε(s, πv , ri , ψv ).
66
Theorem 7.6(Functional Equation [Sh1]).
For each i, 1 ≤ i ≤ m,
(7.8) L(s, π, ri ) = ε(s, π, ri )L(1 − s, π, r̃i ).
Exercise 1. Use the pair (G, M), G = GL3
and M = GL2 × GL1 , to get the standard L–function for GL2 . Determine L–
functions using our method and show that
they are equal to those of Jacquet–Langlands.
Exercise 2. Let G = E6sc and M be such
that MD = SL3 × SL2 × SL3 .
Fact 1. There exists a F –rational map (injection) f : M → GL3 × GL2 × GL3 whose
restriction to MD is identity.
67
Fact 2. m = 3 and if π2 ⊗π1 ⊗σ is an unramified representation of GL3 (F ) × GL2 (F ) ×
GL3 (F ), where F is a local field, then
L(s, π2 ×π1 × σ̃) = L(s, (π2 ⊗π1 ⊗σ)·f, r1 ).
Define γ(s, π2 × π1 × σ̃) to be γ(s, (π2 ⊗
π1 ⊗ σ) · f, r1 ), using our method for arbitrary local representations π2 ⊗ π1 ⊗ σ. Assume σ̃ =
(µ1 ⊗ µ2 ⊗ µ3 ) ⊗ 1.
Ind
(F ∗ )3 ×U ↑GL3 (F )
Show that multiplicativity implies:
γ(s, π2 × π1 × σ̃) =
3
Y
γ(s, π2 × (π1 ⊗ µj )),
j=1
where the γ–function on the right are those
of Rankin–Selberg for GL3 (F ) × GL2 (F ).
68
(This is crucial to Kim–Shahidi’s proof of
functoriality [KS2] of the inclusion GL2 (C)⊗
GL3 (C) ,→ GL6 (C), to be discussed later.)
Exercise 3. Let G = SO(2m + 2n + 1) and
M = GLm × SO(2n + 1). Let σ ⊗ π be an
irreducible admissible χ–generic representation of GLm (F ) × SO2n+1 (F ), where F is a
m
N
µj .
local field. Assume σ ,→ ∗ m Ind
(F ) ×U ↑GLm (F ) j=1
Show that multiplicativity implies:
γ(s, σ × π) =
m
Y
γ(s, π ⊗ µj ).
j=1
(This is crucial to Cogdell–Kim–Piatetski–
Shapiro–Shahidi’s proof [CKPSS1] of func-
69
torial transfer from generic cusp forms on
classical groups to GL2n (AF ).)
8. Twists by Highly Ramified Characters
Holomorphy and Boundedness
Since our aim is to establish those analytic properties of L–functions from our
method which are crucial in proving the striking new cases of functoriality, we will limit
our discussion on the issue of holomorphy of
L–functions only to twists by highly ramified characters. In fact, as we explained in
earlier sections, the functional equations for
L–functions within our method are proved
quite generally and multiplicativity and the
70
related machinary necessary for applying converse theorems to our L–functions are in
perfect shape.
Nothing that general can be said about
the holomorphy and possible poles of these
L–functions. On the other hand, there has
recently been some remarkable new progress
on the question of holomorphy of these L–
functions, mainly due to Kim [K2,K3,KS1].
They rely on reducing the existence of the
poles to that of existence of certain unitary
automorphic forms, which in turn point to
the existence of certain local unitary representations.
One then disposes of these
representations, and therefore the pole, by
71
checking the corresponding unitary dual of
the local group. In view of the functional
equation, this needs to be checked only for
Re(s) ≥ 1/2, if a certain local assumption
on normalized local intertwining operators
is valid. To explain, let A(s, πv , w0 ) be the
local intertwining operator attached by equation (3.10) to our inducing representation
πv . We recall that we are dealing with a
pair (G, M) and a χ–generic cuspidal representation π = ⊗v πv of M = M(AF ). Let,
for each i, 1 ≤ i ≤ m, L(s, πv , ri ) and
ε(s, πv , ri , ψv ) be the corresponding L–function
and root number specified earlier. We de-
72
fine a normalized operator N (s, πv , w0 ) by
(8.1)
N (s, πv , w0 ) = r(s, πv , ψv )A(s, πv , w0 ),
where the normalizing factor is defined as
[Sh1]
(8.2)
r(s, πv , ψv ) =
m
Y
ε(is, πv , r̃i )
i=1
L(1 + is, πv , r̃i )/L(is, πv , r̃i ).
Theorem 8.1. The operator N (s, πv , w0 )
is holomorphic and non–zero for Re(s) ≥
1/2.
It should be mentioned that it is a result
of Yuanli Zhang [Z] that, if Theorem 7.5
73
is valid for the pair (G, M), then the non–
vanishing of N (s, πv , w0 ) for Re(s) ≥ 1/2
follows from its holomorphy over the same
range.
Arguments given in [CKPSS1,K4,KS2],
then proceed under the validity of Theorem
8.1 for (G, M) as well as for all other related
lower rank pairs (that come into the multiplicativity), which consequently are verified
in each of the cases in [CKPSS1,K4,KS2].
The main issue with this argument is that
one cannot always get such a contradiction
and rule out the pole. In fact, there are
many unitary duals whose complementary
series extend all the way to Re(s) = 1, mak-
74
ing the results far from general.
On the other hand, if one considers a
highly ramified twist πη (see below) of π,
then it can be shown quite generally that
every L(s, πη , ri ) is entire (cf. [Sh8] for its
local analogue). In fact, if η is highly ramified, then by checking central characters,
w0 (πη ) 6' πη , whose negation is a necessary
condition for M (s, πη ) to have a pole, a basic fact from Langlands spectral theory of
Eisenstein series (Lemma 7.5 of [La2]). This
lemma was used by Kim in [K2] and in view
of the present powerful converse theorems of
Cogdell and Piatetski–Shapiro [CPS1,2,3],
that is all one needs to prove our cases of
75
functoriality [CKPSS1,K4,KS2]. We formalize this by quoting the following (Proposition 2.1) from [KS2].
Theorem 8.2. There exists a rational character ξ ∈ X(M)F = X(M), with the following property. Let S be a non–empty finite
set of finite places of F . For every globally
generic cuspidal representation π of M =
M(AF ), there exist non–negative integers
fv , v ∈ S, such that for every grössencharacter
η = ⊗v ηv of F for which the conductor of
ηv , v ∈ S, is larger than or equal to fv ,
every L–function L(s, πη , ri ), i = 1, . . . , m,
is entire, where πη = π ⊗ (η · ξ). The rational character ξ can be simply taken to be
76
ξ(m) = det(Ad(m)|n), m ∈ M, where n is
the Lie algebra of N.
The last ingredient in applying converse
theorems is that of boundedness of each
L(s, π, ri ) in every vertical strip of finite width,
away from its finite number of poles. The
finiteness of poles is again a consequence
of the finiteness of the poles of M (s, π) for
Re(s) ≥ 0 and the functional equation satisfied by each L(s, π, ri ), but under the validity of Assumption 8.1 (cf. [GS1]). In this full
generality, the boundedness in finite vertical
strips, away from their poles, were proved
by Gelbart–Shahidi in [GS1], again using
our method. The main difficulty in proving
77
this result is having to deal with reciprocals
of each L(s, π, ri ), 2 ≤ i ≤ m, near and on
the line Re(s) = 1, the edge of critical strip,
whenever m ≥ 2, which is unfortunately the
case for each of the L–functions appearing
in our cases of functoriality. We handle this
by appealing to equation (6.5) (Theorem
6.2) and estimating the non–constant term
(6.2) by means of Langlands [HC,La2] and
Müller [Mu], and a non–trivial result from
complex function theory (Matsaev’s theorem). Here is the statement of the main result of [GS1] as formulated for πη to avoid
the issue of poles.
78
Theorem 8.3. Let ξ and η be as in Theorem 8.2. Assume η is sufficiently ramified so that each L(s, πη , ri ) is entire. Then,
given a finite real interval I, each L(s, πη , ri )
remains bounded for all s with Re(s) ∈ I.
Remark. Another proof of Theorem 8.3 is
due to Gelbart and Lapid, following some
ideas of Sarnak.
9. Examples of Functoriality with Applications
Consider the embedding
i: GL2 (C) ⊗ GL3 (C) ,→ GL6 (C).
This is a homomorphism from L GL2 ×L GL3
into L GL6 . Accordingly functoriality pre-
79
dicts a map
Aut (GL2 (AF )) × Aut (GL3 (AF ))
→ Aut (GL6 (AF )).
More precisely, let π1 = ⊗v π1v and π2 =
⊗v π2v be cusp forms on GL2 (AF ) and GL3 (AF ),
respectively, with π1v given by t1v ∈ GL2 (C)
and π2v by t2v ∈ GL3 (C) for almost all v.
Q
Let v be the irreducible admissible spherical representation of GL6 (Fv ) defined by
{t1v ⊗ t2v } ⊂ GL6 (C). Then we can state
the functoriality in this case as:
Functoriality. There exists an automorQ0
Q0
= ⊗v v of GL6 (AF )
phic representation
Q0
Q
such that v = v for almost all v.
80
More precisely, let ρiv : WF0 v → GLi+1 (C),
i = 1, 2, parametrize πiv (Harris–Taylor [HT],
Henniart [He1]) for all v. Let ρ1v ⊗ ρ2v be
the six dimensional representation of WF0 v ,
i.e., the homomorphism
ρ1v ⊗ ρ2v : WF0 v → GL6 (C).
Denote by π1v π2v the irreducible admissible representation of GL6 (Fv ) attached to
ρ1v ⊗ ρ2v . Let π1 π2 = ⊗v (π1v π2v ).
Theorem 9.1 (Kim–Shahidi [KS2]). The
irreducible admissible representation π1 π2
of GL6 (AF ) is automorphic. Thus GL2 (C)⊗
GL3 (C) ,→ GL6 (C) is functorial.
81
For the proof, one applies an appropriate
version of the converse theorem [CPS2] to
the following cases of our method. In each
case G and MD , the derived group of M,
are given as follows.
1) G = SL(5)(orGL(5)),
MD = SL2 ×
SL3
2) G = Spin(10),
MD = SL3 × SL2 ×
SL2
3) G = E6sc ,
MD = SL3 × SL2 × SL3
4) G = E7sc ,
MD = SL3 × SL2 × SL4
We then get the necessary analytic properties of the highly ramified twisted L–functions
L(s, π1 × π2 × (σ ⊗ η)), σ ⊗ η = σ ⊗ η · det,
82
η a highly ramified grössencharacter, where
σ’s are appropriate cusp forms on GLj (AF ),
j = 1, 2, 3, 4, respectively. Observe that
L(s, π1v × π2v × σv ) = L(s, (π1v π2v ) × σv )
for almost all v. Similarly for root numbers.
We in fact prove these equalities for all v.
This is immediate from the fact that the
local components of the weak transfer established through the converse theorem (Theorem 3.8 of [KS2]) is in fact π1v π2v for
each v (Theorem 5.1 of [KS2]). Proof of
this is quite delicate and beside the techniques already discussed (functional equations, multiplicativity,...), relies on several
83
other techniques such as base change, both
normal [AC] and non–normal [JPSS2], as
well as certain results from the theory of
types [BH].
Next, let π = ⊗v πv be a cusp form on
GL2 (AF ). Let Ad(π) be its Gelbart–Jacquet
transfer [GJ]. Then
π Ad(π) = Sym3 (π) ⊗ ωπ−1 π
implies that Sym3 (π) is an automorphic representation of GL4 (AF ).
More precisely, for every v let ρv : WF0 v →
GL2 (C) be the two dimensional representation of the Deligne–Weil group WF0 v attached to πv (cf. [Ku]).
84
Consider:
Sym3 · ρv = Sym3 ρv : WF0 v → GL4 (C).
Let Sym3 πv be the irreducible admissible
representation of GL4 (Fv ) attached to Sym3 ρv .
Set
Sym3 π = ⊗v Sym3 πv .
Theorem 9.2 (Kim–Shahidi [KS2]).
Sym3 π is an automorphic representation of
GL4 (AF ). It is cuspidal unless π is of dihedral or tetrahedral type.
Next, let
Q
= ⊗v
Q
v
be a cuspidal repre-
sentation of GL4 (AF ). Let Λ2 : GL4 (C) →
GL6 (C) be the exterior square map. Let
85
ϕv : WF0 v
→ GL4 (C) parametrize
Q
v
for all
v [HT,He1]. Then Λ2 ϕv : WF0 v → GL6 (C)
Q
2
parametrizes Λ
v , an irreducible admissible representation of GL6 (Fv ).
Theorem 9.3 (Kim [K4]). There exists
Q0
Q0
an automorphic representation
= ⊗v v
Q0
Q
2
of GL6 (AF ) such that v = Λ
v.
Corollary 9.4 [K4]. The representation
Sym4 (π) is automorphic, where
Sym4 (π) = ⊗v Sym4 (πv )
and π = ⊗v πv is a cusp form on GL2 (AF ).
Proof. Apply Λ2 to Sym3 π;
Λ2 (Sym3 π) = Sym4 π ⊗ ωπ ωπ3
86
Proposition 9.5 (Kim-Shahidi [KS3]).
Sym4 (π) is cuspidal unless π is of dihedral,
tetrahedral or octahedral type.
Theorem 9.3 is proved by using
G = Spin2k+8 ,MD = SLk+1 ×SL4 , k = 0, 1, 2, 3.
We get L(s, π ⊗ σ, Λ2 ρ4 ⊗ ρk+1 ) for each
cusp form σ on GLk+1 (AF ), k = 0, 1, 2, 3.
If we use the isogeny SL4 → SO6 →
0, we note that functoriality established in
Theorem 9.3 is a special case of functoriality
for the embedding
GSO2n (C) ,→ GL2n (C).
87
The result is the transfer
Aut(GSpin2n (AF ))
Aut(GL2n (AF )).
This is now established by Asgari–Shahidi
in Duke J. in the weak form. The full transfer is now being written. It needs generalization of the descent of Ginzburg–Ralis–
Soundry to GSpin groups. This is half–
done by Hundley–Sayag. We have now completed the rest of it and expect to prove
results as strong as those of [CKPSS] for
GSpinm , m = odd or even. We need some
results on LS (s, π, Λ2 ⊗χ) and LS (s, π, Sym2 ⊗
χ) when π is a cusp form on GLn (AF ).
(We need holomorphy for both for Re(s) >
88
1. This implies non–vanishing for both for
Re(s) > 1. Much stronger results now seem
to be available through the work of D. Belt
for Λ2 ⊗ χ and S. Takeda for Sym2 ⊗ χ, using Jacquet–Shalika and Bump–Ginzburg,
respectively.)
In the cases of Asgari–Shahidi we have
G = GSpin2(m+n) , M = GLm × GSpin2n
for 1 ≤ m ≤ 2n − 1. The L–functions are
product L–functions. Apply the converse
theorem.
Applications:
Theorem 9.6 (Kim–Shahidi [KS3]). Let
F be an arbitrary number field. Let π =
89
⊗v πv be a cusp form on GL2 (AF ). Assume
πv is parametrized by
αv 0
tv =
∈ GL2 (C).
0 βv
Then
qv−1/9 < |αv | and |βv | < qv1/9 .
Similar inequality holds at archimedean places
(λ > 0.23765432...).
This is proved using the techniques of
[Sh2] (which led to 1/5 using Sym2 π and
groups of either type F4 or E6 , by applying a general theorem from [Sh2] which implies L(s, πv , Sym5 (ρ2 )) is holomorphic for
90
Re(s) ≥ 1 for all such v). When this general
theorem is applied to a simply connected
group of type E8 with a Levi M for which
MD = SL5 ×SL4 , together with a representation related to Sym4 π⊗ Sym3 π leading to
L(s, πv , Sym9 (ρ2 )), one gets 1/9. We refer
to [CPSS] for an important application of
Theorem 9.6. Next, we have
Theorem 9.7 (Kim–Sarnak [KSa]).
Suppose F = Q, then
p−7/64 ≤ |αp | and |βp | ≤ p7/64 .
At the archimedean places we get the esti975
= 0.2380371 for the first
mate λ ≥ 4096
positive eigenvalue of ∆.
91
Proof. This is proved by means of analytic
methods of Duke–Iwaniec [DI] applied to
L(s, Sym4 π, Sym2 ), (cf. [BG]) along the lines
of Bump–Duke–Hoffstein–Iwaniec [BDHI]
which led to 5/28 + ε over Q, when applied
to L(s, Sym2 π, Sym2 ).
92
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Department of Mathematics
Purdue University
West Lafayette, IN 47907 USA
[email protected]