MATH G9905 RESEARCH SEMINAR IN NUMBER THEORY
(FALL 2014)
LECTURE 2 (SEPTEMBER 16, 2014)
DORIAN GOLDFELD
MAASS–SELBERG RELATIONS FOR EISENSTEIN SERIES ON GL(n)
NOTES TAKEN BY PAK-HIN LEE
Abstract. I will introduce Arthur’s truncation operator and the inner product formula
for truncated Eisenstein series (Maass–Selberg relation) for GL(n) with n > 2.
Today we will generalize the Maass–Selberg relations to GL(n). Let me first talk about
the Eisenstein series for GL(n). Consider the basic set-up: G = GL(2n), Γ = GL(2n, Z),
K = O(2n, R). To keep things simple, we will look at the action of GL(2n, Z). If we
want to work adelically, we consider GL(2n, Ak ) and GL(2n, k) instead. We are focusing
on 2n because we will be applying thisto Rankin–Selberg. We are also
interested
in the
GL(n)
∗
In ∗
parabolic P = Pn,n =
, the unipotent radical N P =
, and the
0
GL(n)
0 In
GL(n)
0
P
Levi M =
. More generally, we have
0
GL(n)






GL(r1 ) 0
0
Ir1 ∗ ∗
GL(r1 ) ∗
∗



 P 

P
..
..
..
Pr1 ,··· ,rk =  0
.
. ∗ ,M =  0
.
0 .
∗ ,N =  0
0
0
GL(rk )
0
0
Irk
0
0
GL(rk )
Let F : N P → C be a smooth function such that F (ng) = F (g) for all n ∈ N P (Z). The
constant term along P of F is defined to be
Z
cP F (g) =
F (ng)dn.
N P (Z)\N P (R)
We are interested in the constant terms along a parabolic of Eisenstein series.
Definition 1. Let φ(pg) = φ(g) for all p ∈ P (Z). Then
X
E P (φ, g) :=
φ(γg).
γ∈P (Z)\G(Z)
Let me show how the constant term can be computed in general. The constant term can
be broken into pieces using the Bruhat decomposition. Let P, Q be two parabolics for G.
Then
[
[
G=
P wQ =
P wQ
w∈W
w∈(W ∩P )\W/(W ∩Q)
Last updated: September 30, 2014. Please send corrections and comments to [email protected].
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where W is the Weyl group.
Consider the Eisenstein series E Q (φ, g). The constant term along P is
Z
Q
cP E (φ, g) =
E Q (φ, ng)dn
NZP \NRP
Z
X
=
X
φ(γβng)dn
NZP \NRP γ∈Q \G /P β∈Q \Q γP
Z
Z
Z
Z
Z
Z
Z
X
=
w∈(W ∩Q)\W/(W ∩P )
Assume that QZ ∩ PZ

1
 ..
• If w = 
.
X
φ(wβng)dn.
NZP \NRP β∈((w−1 Q w)∩P )\P
Z
Z
Z
= PZ . There are two interesting cases.


 is the trivial Weyl element, then the integral is
1
Z
φ(ng)dn = Vol(NZP \NRP )φ(g).
NZP \NRP

1
−1
. 
 is the long element, then (w0 PZ w0 ) ∩ NZ is trivial, so
..


• If w0 = 
1
Z
NZP \NRP
X
Z
φ(w0 γng)dn =
φ(w0 ng)dn.
NRP
γ∈NZP
We willnow get to theMaass–Selberg relations. Again we are in the case G = GL(2n)
GL(n)
∗
. Fix a cusp form f on GL(n, R). For s ∈ C with Re(s) 1, we
and P =
0
GL(n)
define
X
EfP (g, s) =
φs (γg).
γ∈PZ \ SL(2n,Z)
To define φs , we need to introduce the height function
n
| det(m1 )|
P
h (g) :=
| det(m2 )|
m1 0
where g ∈ G is written as g = nmk with m =
in the Levi. Then
0 m2
φs (g) := hP (g)s f (m1 )f (m2 ).
If Re(s) is sufficiently large, EfP (g, s) converges because f is a cusp form.
m1 0
More generally, if we decompose the Levi as
where m1 ∈ GL(n1 ) and m2 ∈
0 m2
GL(n2 ), then
| det(m1 )|n2
.
hP (g) :=
| det(m2 )|n1
2


λ

This satisfies hP (zg) = hP (g) for central elements z = 
..
.

, λ 6= 0.
λ
The associated L-function to f is
−1
n YY
αp,i
L(s, f ) =
1− s
p
p i=1
for some αp,i ∈ C. The famous Ramanujan conjecture states that |αp,i | = 1. We also have
the Rankin–Selberg L-function, which is a double product
−1
n Y
n YY
αp,i αp,j
L(s, f × f ) =
.
1−
ps
p i=1 j=1
It is the Rankin–Selberg L-function that will appear in the constant term of Eisenstein series.
Langlands computed this and proposed to Shahidi (who was then a grad student) that this
can be used to prove the functional equation for the Eisenstein series.
L(s, f ) and L(s, f ×f ) have functional equations, due to Godement–Jacquet and Jacquet–
Piatetski-Shapiro–Shahidi respectively. Define
n
Y
s + αi
− ns
2
Λ(s, f ) = π
Γ
L(s, f )
2
i=1
where αi ∈ C are called the Langlands parameter. Conjecturally αi ∈ iR. This is now
understood to be the Ramanujan conjecture at infinity. The functional equation is
Λ(s, f ) = (f )Λ(1 − s, f˜)
where |(f )| = 1 and f˜(g) := f (w0 t g −1 w0 ) where w0 is the long element. In terms of
representation this is the contragredient. For the Rankin–Selberg product,
n
n
2 YY
s + αi + αj
− n2 s
Λ(s, f × f ) := π
L(s, f × f ),
Γ
2
i=1 j=1
which satisfies the functional equation
Λ(s, f × f ) = (f × f )Λ(1 − s, f˜ × f˜)
where |(f × f )| = 1.
Theorem 2 (Langlands). EfP (g, s) has meromorphic continuation with functional equation
EfP (g, s) = cs EfP˜ (g, 1 − s).
The functionalequation is more complicated if P is not self-associate, i.e. not of the form
GL(n)
∗
. The idea of the proof is to first show that the constant term cs has a
0
GL(n)
functional equation.
Theorem 3 (Langlands–Shahidi).
cs =
Λ(2ns − n, f × f )
Λ(1 + 2ns − n, f × f )
3
and
cP EfP (g, s) = φs (g) + cs φ1−s (g).
This is a rather long technical computation, and I will not give a proof. This is a general
phenomenon that holds for reductive groups.
As we saw last time, the Eisenstein series are not in L2 . We can truncate the constant
term and take an inner product. Selberg simply subtracted away the constant term. Arthur
introduced a truncation that remains automorphic.
Definition 4 (Arthur’s truncated constant term for T > 0).
(
cP F (g) if hP (g) ≥ T,
T
cP F (g) =
0
if 0 ≤ hP (g) < T.
m1 0
det m1 |n2
For g = nmk and m =
, h(g) = || det
is the height function.
m2 |n1
0 m2
Definition 5 (Truncation operator ΛT on Eisenstein series).
X
ΛT EfP (g, s) = EfP (g, s) −
cTP EfP (γg, s).
γ∈PZ \ SL(2n,Z)
If P is not a maximal parabolic, the truncation is more complicated.
Remark. ΛT EfP is automorphic for SL(2n, Z), i.e.
ΛT EfP (γg, s) = ΛT EfP (g, s)
for γ ∈ SL(2n, Z).
Now we want to talk about the Siegel sets. Recall the Iwasawa decomposition: if z ∈
G/(K · R)× , then g = y · x where




1
y1 · · · yn−1

 ..


..




.
.
x
ij
and
x
=
y=







1
y1
1
1
with yi > 0 and xij ∈ R.
Definition 6 (Siegel set). Fix δ > 0. The Siegel set is
1
1
Sδ = xy : − ≤ xij ≤ , yi > δ
2
2
Let F be the fundamental domain for Γ\G/(K · R× ), Γ = SL(2n, Z).
Theorem 7. F ⊂ S √3 .
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A proof can be found in my book Automorphic Forms and L-Functions for the Group
GL(n, R).
It is known that EfP (g, s) − cP EfP (g, s) is of rapid decay on Sδ (δ > 0).
Proposition 8. ΛT EfP is of rapid decay on any Sδ for T ≥ 1.
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This implies that ΛT EfP ∈ L2 (Γ\G/(K · R× )), and that hΛT EfP , ΛT EfP i < ∞. More
generally, we can take |hΛT EfP (∗, r), ΛT EfP (∗, s)i| < ∞.
Theorem 9 (Maass–Selberg relation). Assume r, s ∈ C with r 6= s and r + s − 1 6= 0. Then
r+s−1
T s−r
T 1−r−s
T r−s
T
T P
T P
+ cs
+ cr
+ cr cs
.
hΛ Ef (∗, r), Λ Ef (∗, s)i = hf, f i ·
r+s−1
r−s
s−r
1−r−s
GL(n)
∗
Again we are in the special situation G = GL(2n), P =
and f a cusp
GL(n)
form on GL(n). This was first proved by Langlands, but he used a different truncation. This
is the only way we know to prove the functional equation for Eisenstein series in the most
general case.
The original proof by Selberg uses Green’s theorem. Using Arthur’s truncation, we don’t
need any differential operators; the proof is just unfolding.
Proof. First we can show that
*
ΛT EfP (∗, r),
+
X
cTP EfP (γ · ∗, s)
= 0.
γ∈PZ \Γ
This implies
hΛT EfP (∗, r), ΛT EfP (∗, s)i
=hΛT EfP (∗, r), EfP (∗, s)i
Z
Z
=
−cr φ1−r (g)E(g, s)dg +
φr (g)E(g, s)dg.
PZ \G/(K·R× ),hP (g)<T
PZ \G/(K·R× ),hP (g)≥T
Unraveling the sum, we get a very simple integral.
Starting next week, Xiaoqing Li will give a series of lectures on the application of Maass–
Selberg relations on the lower bounds of Rankin–Selberg L-functions. For cusp forms f on
GL(n), we get
1
L(1 + it, f × f ) (log |t|)3+
for large t. We don’t know that f × f is automorphic. Since we are
with f × f ,
just working GL(n)
∗
we only need the Maass–Selberg relations on the parabolic P =
, which is
GL(n)
the case we considered today.
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