Download Notes on Automorphic Representations

Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
NAVA CHITRIK
Referenced heavily (and often directly) from Goldfeld and Hundley (2011), Automorphic Representations and L-Functions for the General Linear Group,
Cambridge University Press and from Daniel
Bump (1991), Automorphic Representations
Contents
Introduction
Maass Forms for Γ0 (N )
Automorphic forms for GL(2, AQ )
Automorphic Representations
The tensor Product Theorem
The Whittaker Models and the Multiplicity One Theorem
Poisson Summation
The Global Zeta Integral
1
1
2
3
6
7
10
11
Introduction
These notes are meant to get one as quickly as possible to the denitions are results.
To be honest, they are not comprehensive and they are not very interesting.
Maass Forms for Γ0 (N )
Denition 1. (The slash operator)
f|k γ(z) :=
cz + d
|cz + d|
−k
f (γz)
Denition 2. Fix a character χ mod N . A Maass form for Γ0 (N ) of weight k is
a function f (z) on the upper half plane satisfying
• f|k γ(z) = χ(d)f (z) where γ =
a b
c d
∈Γ0 (N )
∂
• ∆k f = ν(1 − ν)f where ∆k = −y 2 ( ∂x
2 +
2
1
∂2
∂y 2 )
∂
+ iky ∂x
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
2
• Moderate Growth + L2
Denition 3. A Maass form is called cuspidal if in addition, the constant term
in its Fourier-Whittaker expansion is 0. Equivalently, it vanishes at each cusp of
Γ0 (N )
Denition 4. Hecke Operators
d
X
1 X
az + b
Tn f (z) := √
χ(a)
f
d
n
ad=n
b=1
Fact 5. The Hecke operators commute with each other and with ∆k
Automorphic forms for GL(2, AQ )
Let G = GL(2, AQ ) from here on.
Denition 6. (Moderate growth). Let
g =
a b
c d
=
Q
av bv
cv dv
= {gv }v ∈
GL(2, AQ ). We dene a norm ||g|| by
||g|| =
Y
||g||v
v
where
||g||v = max |av |, |bv |, |cv |, |dv |, | det gv−1 |
We say that a function φ(g) has moderate growth if there exists C and N such that
|φ(g)| < C||g||N
∀g ∈ G
Denition 7. An automorphic form for G with central character ω : F × → C is a
function φ : GL(2, Q)\GL(2, AQ ) → C that satises
• φ(zg) = ω(a)φ(g) ∀g ∈ G and z = ( a a )
• φ is K nite (The vector space spanned by φ(gk) for xed g and k ∈ K is
nite dimensional)
• φ is Z(U (g)) nite (The vector space spanned by D.φ(g) for all D ∈ Z(U (g))
is nite dimensional)
• φ has moderate growth
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
Denition 8. If in addition
3
ˆ
φ (( 1 x1 ) g) dx = 0 ∀g
Q\A
then φ is called a cuspidal automorphic form.
Denition 9. Lift of a weight zero Maass cusp form. Weight zero just means
f (γz) = f (z). We use the following decomposition, if g ∈ G we can write g
uniquely as
g = γ · {( y∞
x∞
1
) ( r∞
r∞
Where γ ∈ GL(2, Q) and k ∈ K = O(2, R) ·
) , I2 , ..., I2 , ...} · k
Q
GL(2, Zp ).
Now we dene fadelic (g) = f ( y∞ x1∞ )
Another way to describe the lift is as follows. A Maass form is a function on
GL(2, R)/K for some compact group. Consider the quotient space
0
GL(2, Q)\GL(2, AQ ) ∼
= GL(2, Z)\GL(2, R) ×
Y
GL(2, Zp )
If we quotient further by some compact set K we get something like GL(2, R)/K
0
0
so we can lift the Maass form to GL(2, AQ ) by letting
φlif t (g) = φ(ḡ)
where
ḡ = g(mod GL(2, Q) × K )
0
Fact 10. Such a lift will be a cusp form
Automorphic Representations
Let Aω (GL(2, AQ ) be the vector space of adelic automorphic forms for G with
central character ω . We consider the following actions
• Right translation by the nite adeles
π(af in ) · φ(g) = φ(gaf in )
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
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• Right translation on the innite part by O(2, R)
π(k)φ({g∞ , g2 , ...}) = φ({g∞ k, g2 , ...}) with k ∈ O(2, R)
• Action by U (g)= dierential operators on the innite place
π(D).φ(g) = D(φ(g)) D acts on the variables in g∞ ∈ GL(2, R)
Remark 11. The last two actions do not commute (but the other pairs do, obvi-
ously). instead we have that
−1
π(Dα ).π(k)φ(g) = π(k).π(Dk∞
αk∞ )φ(g)
If you trace through the denitions and recall
∂
tα Dα φ(g) := φ(g · e )
∂t
t=0
Denition 12. Let
g = gl(2, C), K∞ = O(2, R). A (g, K∞ ) module is a vector
space V with actions
• πg : U (g) → End(V )
• π∞ : K∞ → GL(V )
• πg (Dα )π∞ (k) = π∞ (k)πg (Dk−1 αk )
Denition 13. A (g, K∞ ) × GL(2, Af in ) module is a vector space V with actions
by Kf in ⊂ GL(2, Af in ) that commute with the actions above.
Denition 14. A smooth
(g, K∞ ) × GL(2, Af in ) module is one such that for
every vector v there's an open subgroup U ⊂ GL(2, Af in ) such that U.v = v
Lemma 15. The space of adelic automorphic forms is a smooth (g, K∞ )×GL(2, Af in )
module.
Proof. This follows from the fact that automorphic forms are smooth and K - nite.
Denition 16. Intertwiners among (g, K∞ ) × GL(2, Af in ) modules must respect
all these actions.
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
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Here's the big denition:
Denition 17. By an Automorphic Representation with central character
×
×
Q \A → C
×
ω :
we mean a smooth (g, K∞ ) × GL(2, Af in ) module which is isomor-
phic to a sub-quotient of Aω (GL2 (AQ )).
Lemma 18. The 3 actions denoted π preserve the subspace of cusp forms.
Denition. If an automorphic representation is a sub quotient of Aω (GL2 (AQ ))cusp
then it is called a cuspidal representation. This can also be stated as the condition
that
ˆ
π ( 1 u1 ) .vdu = 0 ∀v ∈ V
Q\A
Denition 19. An admissible (g, K∞ ) × GL(2, Af in ) module is one that is smooth
and satises
VnK is nite dimensional
0
where
Vn =
v ∈ V π∞
cosθ sinθ
−sinθ cosθ
.v = einθ v
and K 0 ⊆ GL2 (Af in ) any compact open subset.
Remark 20. Aω (GL2 (AQ ) is not admissible! For example, if ω = 1, any weight 0
level 1 Maass form is xed by Kf in . And there are innitely many of these.
Theorem 21. Irreducible automorphic cuspidal representations are admissible.
Example 22. Suppose we start with a Maass form, a newform f of weight k level
N and character χ (mod N ), and type v . When we lift the Maass form, we will get
an automorphic form fadelic (g) with central character χidelic . We can dene the
vector space
(
Vf =
N
X
)
ci R
mi
fadelic (g · hi )
i=1
where hi ∈ Af in and R is some dierential operator (raising or lowering). The
other ones act by scalars or zero on fadelic .
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
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Then, obviously Vf is preserved under the three actions
(1) πK∞ : K∞ → GL(Vf )
(2) πAf in : Af in → GL(Vf )
(3) πU (g) : U (g) → End(Vf )
Furthermore, it is a cuspidal representation since the function fidelic is a cusp form.
The tensor Product Theorem
Denition 23. (Restricted Tensor Product) Let {Vv }v≤∞ be a family of representations and let ξv◦ ∈ Vv be a vector specied for all v except for possibly some
nite set S . We dene the restricted tensor product of the representations Vv with
respect to ξv◦ to be the set of vectors
ξ=
O
ξv such that ξv = ξv◦ at all but nitely many places
v≤∞
Denition 24. (Spherical Hecke Algebra)
K
HK = {f (k1 gk2 ) = f (g), k ∈ GL(2, Zp )}
with a convolution
ˆ
f1 (gh−1 )f2 (h)d× h
f1 ∗ f2 (g) =
•
K
HK is commutative and associative
• Inherits a representation
ˆ
f (g)π(g).vd× g
π(f ).v =
G
• Maps V → V
K
• Therefore the dimension of K -xed vectors is always at most 1
• Therefore when the representation is unramied there exists a spherical
Hecke character
Theorem 25. Let
(π, V ) be an irreducible, admissible (g, K∞ ) × GL(2, Af in ) -
module. Then (π, V )ramies at only nitely many places and
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
π∼
=
O
7
πv
v
with respect to the distinguished vector at each unramied place.
Theorem 26. The factors are unique
Theorem 27. Products of irreducibles remain so.
The Whittaker Models and the Multiplicity One Theorem
The main result here is the following (very powerful) theorem
Theorem 28. (Multiplicity One) Let
N
N 0
0
V ∼
πv and V ∼
πv be irreducible,
=
=
admissible automorphic sub-representations of A0 (GL(2, Q)\GL(2, AQ ) such that
0
0
πv ∼
= πv for all but nitely many v . Then V = V .
Remark. Sometimes the theorems that are used to prove the above theorem are
also called multiplicity one.
Denition 29. A Whittaker model of a representation (π, V ) is a space of functions
W∼
=G V such that
W = W : G → CW (( 1 x1 ) g) = ψ(x)W (g)
and with the right regular action by G. (We may need K− niteness in the global
case and moderate growth)
An alternative way to state this is that if we consider ψ to be a representation of
N = {( 1 x1 )} then the Whittaker model is a subrepresentation in the representation
ψ induced to G.
Theorem 30. Let ψ be a non-trivial additive character on a non-archimedian local
eld F and let (π, V ) be an irreducible admissible representation of GL(2, F ). There
is at most one local Whittaker functional
Λ:V →C
Λ (π ( 1 x1 ) .v) = ψ(x).v
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
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There is a bijection between spaces of Whittaker functions (whittaker models) and
Whittaker functionals. Specifying one is the same as specifying the other, as follows.
Let Λ be a whittaker functional. Then we can form a Whittaker model using the
isomorphism
v 7→ Λ(π(g).v) ∈ W
Conversely, if we have a Whittaker model, then we dene the functional
Λ(W (g)) = W (1)
then
Λ (π ( 1 x1 ) .W (g)) = (ψ(x).W (g)) = ψ(x)Λ(W )
So a standard procedure for showing that there's at most one Whittaker model is
to show that there's at most one Whittaker functional. We can also look at this
fact as Frobenius reciprocity
∼
HomG (V, IndG
N ψ) = HomN (VN , ψ)
The rst one is the number of whittaker models and the second is the number of
whittaker functionals.
The next thing to show is that Whittaker models exist. Well, they sometimes do.
Theorem 31. Let (π, V ) ⊂ A0,ω (GL(2, Q)\GL(2, AQ )) be an irreducible, admissible cuspidal representation. Then there exists a Whittaker model for (π, V ), given
by
ˆ
φ (( 1 x1 ) g) ψ(−x)dx
Wφ (g) =
Q\A
we also have a fourier-like expansion
φ(g) =
X
Wφ (( α 1 ) g)
α∈Q×
Proof. Suppose that the expansion held. Then consider the map φ 7→ Wφ . Suppose
that Wφ = 0 then the expansion gives us that φ(g) = 0 so that the map is injective.
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
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To get the expansion, notice that φ(γg) = φ(g) because φ is an automorphic form.
Therefore, for a xed g, φ (( 1 x1 ) g) = φ(g) is a function on Q\A which is compact
so that we can expand
φ (( 1 x1 ) g) =
X
C(α)ψ(αx)
α∈Q
where, if α 6= 0
ˆ
1y
1
φ
C(α) =
g ψ(−αy)dy
Q\A
ˆ
φ (α 1)
=
1y
1
g ψ(−αy)dy
Q\A
(by automorphicity of φ)
ˆ
α
=
φ 1 αy
( 1 ) g ψ(−αy)dy
1
Q\A
now take y 7→ α−1 y
ˆ
=
1y
1
φ
( α 1 ) g ψ(−y)dy
Q\A
= Wφ (( α 1 ) g)
If α = 0 then
ˆ
C(0) =
φ
1y
1
g dy = 0
Q\A
since φ is a cuspidal automorphic form.
What goes wrong when we try to make a model out of a non-cuspidal automorphic
representation?
Well, we can still write a function as a fourier series,
φ(g) =
X
α∈Q
φ̂α (g)
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
where
10
ˆ
φ (( 1 x1 ) g) ψ(−αx)dx
φ̂α (g) =
Q\A
If we try to make a whittaker function out of this:
ˆ
φ (( 1 x1 ) g) ψ(−x)dx
Wφ (g) =
Q\A
ˆ
X
=
Q\A
φ̂α (( 1 x1 ) g) ψ(−x)dx
α∈Q
ˆ
ˆ
X
φ̂0 (( 1 x1 ) g) ψ(−x)dx +
=
Q\A
ˆ
Q\A
ˆ
φ
=
φ̂α (( 1 x1 ) g) ψ(−x)dx
α∈Q×
0
0
1x
g ψ(−x)ψ(−x )dx + ...
1
Q\AQ\A
And the rst piece is equal to zero by standard tricks. In other words, when
we construct Wφ it doesn't see the 0th fourier coecient. So this construction
clearly won't inject V into a Whittaker model unless we started with φ's which are
cuspidal.
Poisson Summation
Denition 32.
M at(2, AQ ) = M at(2, R)·
Y0
M at(2, Qp ) with respect to M at(2, Zp )
p
Denition 33. A Bruhat-Schwartz function on
functions
Q
M at(2, AQ ) is a linear sum of
φv (av ) where each φv is locally constant, compactly supported at the
nite places and all but nitely many are 1M at(2,Zp ) . The real part is Schwartz in
all four variables.
Denition 34. (Fourier Transform) Let Φ be a Bruhat Schwartz function. Then
we dene the fourier transform of Φ by
Φ̂
αβ
γ δ
ˆ ˆ ˆ ˆ
=
Φ
A4Q
a b
c d
e(−aα − bγ − cβ − dδ dadbdcdd
{z
}
|
−T r
αβ
γ δ
a b
c d
Some Basic Ideas in the Theory of
Automorphic Representations of GL(2,AQ )
11
Theorem 35. (Poisson Summation) Let Φ be a Bruhat-Schwartz function then
X
X
Φ(ξ) =
ξ∈M at(2,Q)
Φ̂(ξ)
ξ∈M at(2,Q)
Proof. This follows by doing poisson summation on each
The Global Zeta Integral
Let (π, V ) be a cuspidal automorphic representation. A matrix coecient is usually of the form < π(g).v, ṽ > but if we're in the unitary scenario (and cuspidal
representations are), then usuing the inner product we get an association V → Ṽ
and so we can write any matrix coecient as
ˆ
f1 (hg)f2 (h)d× h
β(g) =
GL2 (Q)·Z\G
Denition 36. Let
<(s) 0 and Φ be a Bruhat-Schwartz function. We dene
the global-zeta integral to be
ˆ
1
Φ(g)β(g)|det(g)|s+ 2 d× g
Z(s, Φ, β) :=
G
Theorem 37. With the zeta integral dened as above,
Z(s, Φ, β) = Z(1 − s, Φ̂, β̌)
Proof. Similar to GL(1). Use Poisson summation
Proposition 38.
β factors into
Q
βv . If Φ is factorizable then
Z(s, Φ, β) =
Y
Zv (s, Φv , βv )
v≤∞
Proof. Follows from the tensor product theorem.