MATH G9905 RESEARCH SEMINAR IN NUMBER THEORY
(FALL 2014)
LECTURE 10 (NOVEMBER 18, 2014)
YOUNGJU CHOIE
SCHUBERT EISENSTEIN SERIES
NOTES TAKEN BY PAK-HIN LEE
Abstract. We define Schubert Eisenstein series as sums like usual Eisenstein series but
with the summation restricted to elements of a particular Schubert cell, indexed by an
element of the Weyl group. They are generally not fully automorphic. We will develop
some results and methods for GL3 that may be suggestive about the general case. The six
Schubert Eisenstein series are shown to have meromorphic continuation and some functional
equations. This is a joint work with D. Bump.
Today I will introduce Schubert Eisenstein series, which are not automorphic but have
nice arithmetic properties.
1. Schubert Variety
Let G be a split reductive algebraic group defined over a global field F, and B be a Borel
subgroup. We know B\G is a flag variety. Since G is a reductive group, we can decompose
[
G=
BwB
w∈W
where W = NG (T )/T is the Weyl group, B = T U , and T is a maximal torus.
Let Yw be the image of BwB in B\G. Take Xw to be the Zariski closure of Yw , i.e.,
[
Xw := Yw =
Yu ,
u≤w
where ≤ is the Bruhat order. Xw is called the Schubert variety.
Example 1. If G = GL3 , then W = {id, s1 , s2 , s1 s2 , s2 s1 , s1 s2 s1 }, where for example


1

s 1 = 1
1
satisfies s21 = 1 and corresponds to the simple root α1 = (1, −1, 0).
For example, Xs1 s2 = B\B ∪ B\Bs1 s2 B.
Last updated: February 28, 2015. Please send corrections and comments to [email protected].
1
2. Bott–Samelson Variety
Let w ∈ W , which we write as si1 si2 · · · sik where sij are simple reflections. Denote by
αij the simple root corresponding to sij . Let w = (si1 , · · · , sik ). Let Pij be the parabolic
subgroup spanned by B and sij , i.e., Pij = B ∪ Bsij B. We have B\Pij ' P1 .
Consider the left action of B k on Pi1 × Pi2 × · · · Pik given by
−1
(b1 , · · · , bk )(pi1 , pi2 , · · · , pik ) := (b1 pi1 b−1
2 , b2 pi2 b3 , · · · , bk pik ).
The quotient Zw := B k \Pi1 × · · · × Pik is called the Bott–Samelson variety. We have the
following facts.
(1) There exists a morphism ϕw : Zw → Xw induced from the “multiplication’ map
(pi1 , · · · , pik ) 7→ pi1 pi2 · · · pik .
(2) ϕw is a surjective birational map.
(3) Zw is always nonsingular while Xw may be singular. Therefore, this gives a resolution
of singularities of Xw .
(4) ϕw may not be an isomorphism.
(5) We have:
Lemma 2. When ϕw : Zw → Xw is an isomorphism, every element in Xw can be represented
uniquely as a product of ιαi1 (γ1 )ιαi2 (γ2 ) · · · ιαik (γk ), where ιαij : SL2 ,→ G is an embedding
such that the image is in the Levi subgroup of Pij (Chevalley embedding), and γj ∈ BSL2 \ SL2 .
γ1
1
Example 3. Let G = GL3 . Then Xs1 s2 3 A = ια1 (γ1 )ια2 (γ2 ) =
.
1
γ2
Remark. When ϕw is not an isomorphism, then every element of Xw can still be written in
this form, but the representation will not necessarily be unique.
3. Schubert Eisenstein Series
Let G be defined over F, and B = T U where T is a maximal torus and U is the unipotent
subgroup. Write A as the ring of adeles of F. For every place v of F, Gv := G(Fv ).
Let χ : T (A)/T (F) → C× be a quasi-character. Let (Πv (χv ), Vv (χv )) be a principal series
representation, where
n
o
1
Vv (χv ) = fv : Gv → C : fv (bg) = (δ 2 χv )(b)fv (g), fv is Kv -finite ,
Q
δ is the modular quasi-character,
N and K = v Kv is a maximal compact subgroup.
For simplicity, we let χ = v χv , where χv is unramified at every nonarchimedean place,
i.e., Vv (χv ) has a nonzero Kv -fixed vector fv0 , N
normalized such that fv0 (1) = 1. Let V (χ) be
the space of finite linear combinations of f = v fv , where fv = fv0 for almost all v.
For each w ∈ W and f ∈ V (χ), we define
X
Ew (g) :=
f (γg).
γ∈Xw
This is called the “Schubert Eisenstein series”.
If w = w0 is the longest element, then Ew0 (g) = E(g) is the usual Eisenstein series.
2
4. GL3 examples
For simplicity, we take χ to be unramified at every place v. Take


y1 y2 ∗ ∗
1
y1 ∗ = |y1 |2ν1 +ν2 |y2 |ν1 +2ν2
(δ 2 χ) 
1
for ν1 , ν2 ∈ C, so that, for any kv ∈ Kv = SL(3, Ov ), we have



y1 y2 ∗ ∗
y1 ∗ kv ; ν1 , ν2  = |y1 |2ν1 +ν2 |y2 |ν1 +2ν2 .
fv0 
1
We define
Ew (g; ν1 , ν2 ) =
X
f (γg; ν1 , ν2 ).
γ∈Xw (F)
Example 4.
(1) If w = id, then
Eid (g; ν1 , ν2 ) = f (g; ν1 , ν2 ).
(2) If w = s1 , then
Es1 (g; ν1 .ν2 ) =
X
f (γg; ν1 , ν2 ).
γ∈Xs1 (F)
But we know Z(s1 ) ' Xs1 , where every element is of the form γ = ια1 (γ1 ) with
γ1 ∈ BSL2 \ SL2 , so the sum can be written as
X
γ1
f
g; ν1 , ν2 .
1
γ1 ∈BSL2 \ SL2
This is essentially the GL2 -Eisenstein series.
(3) If w = s1 s2 (the case w = s2 s1 is similar), then
X
Es1 s2 (g; ν1 , ν2 ) =
f (γg; ν1 , ν2 ).
γ∈Xs1 s2 (F)
Since Z(s1 s2 ) ' Xs1 s2 3 γ = ια1 (γ1 )ια2 (γ2 ) with γ1 , γ2 ∈ BSL2 \ SL2 , the sum can be
written as
X
X
f (ια1 (γ1 )ια2 (γ2 )g; ν1 , ν2 ) =
Es1 (ια2 (γ2 )g; ν1 , ν2 ).
γ1 ,γ2 ∈BSL2 \ SL2
γ2 ∈BSL2 \ SL2
We normalize
Ew∗ (g; ν1 , ν2 ) = ζ ∗ (3ν1 )ζ ∗ (3ν2 )ζ ∗ (3ν1 + 3ν2 + 1)Ew (g; ν1 , ν2 )
Q
where ζ ∗ (s) = v ζv (s), with
(
Γv (s)
if v is archimedean,
ζv (s) =
(1 − qv−s )−1 if v is nonarchimedean,
and qv = |Ov /pv | where pv is the maximal ideal of Ov .
We have the following facts for w = w0 .
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(1) Ew0 (g; ν1 , ν2 ) is analytic except poles at ν1 , ν2 , 1 − ν1 − ν2 ∈ 0, 23 .
(2) Ew∗ 0 (g; ν1 , ν2 ) = Ew∗ 0 (g; w(ν1 , ν2 )) for all w ∈ W , where
(
2
1
if w = s1 ,
−
ν
,
ν
+
ν
−
1
1
2
3
3
w(ν1 , ν2 ) =
1 2
ν1 + ν2 − 3 , 3 − ν2 if w = s2 .
Proposition 5.
(1) Es∗1 (g; ν1 , ν2 ) has a meromorphic continuation for all ν1 , ν2 ∈ C, and similarly for
Es∗2 .
(2) Es∗1 (g; ν1 , ν2 ) = Es∗1 (g; s2 (ν1 , ν2 )).
(3) Es∗1 s2 (g; ν1 , ν2 ) has a meromorphic continuation for all ν1 , ν2 ∈ C.
(4) Es∗1 s2 (g; ν1 , ν2 ) = Es∗1 s2 (g; s2 (ν1 , ν2 )).
5. Fourier Expansion of Eisenstein Series
We write E(g) = E(g; ν1 , ν2 ) = Ew0 (g; ν1 , ν2 ) following Bump’s notations. It is known
that
X
X
E(g) = E00 (g) +
E0,1 (ια1 (γ1 )g) +
W (ια1 (γ1 )g)
γ1 ∈BSL2 \ SL2
γ1 ∈USL2 \ SL2
where:
• for c, d,

Ecd (g) =
Z
(A/F)2
E 
1
 
x3
1 x1  g  ψ(cx3 + dx1 ) dx3 dx1
1
and

 
1 x2 x3
1 x1  g  ψ(cx2 + dx1 ) dx3 dx2 dx1
Ec,d (g) =
E 
3
(A/F)
1
Z
where ψ is an additive character of A/F.
•

 
1 x2 x3
1 x1  g  ψ(x2 + x1 ) dx1 dx2 dx3
W (g) =
E 
3
(A/F)
1
Z
and we denote W ∗ (g) to be the same integral of E ∗ .
Theorem 6.
(1) E ∗ (g; ν1 , ν2 ) = H ∗ (g; ν1 , ν2 ) +
X
Es∗1 s2 (g; w(ν1 , ν2 )) − Es∗1 (g; w(ν1 , ν2 )) .
w2 =1
w∈{id,s1 ,s2 }
(2) H ∗ (g) is entire in ν1 , ν2 .
4
6. Applications
• It is known that the GL3 -Eisenstein series E(g) has a pole at ν1 = ν2 = 0.
• Let κ(g) be the coefficient of ν1−1 in the Taylor expansion of E at ν1 = ν2 = 0.
• (Bump–Goldfeld) If F/Q is a cubic field, and a is an ideal class in F , one may
associate with a a compact torus of GL3 . If La is the period of κ(g) on this torus,
then the Taylor expansion of L-function is
ρ
L(s, a) = + La + O(s).
s
Therefore,
X
L(s; θ) =
θ(a)L(s, a)
a ideal class
for θ a character on ideal classes.
• (Connection with Schubert Eisenstein series) Write
ρ
ζ ∗ (s) = + δ + O(s).
s
Then
ρ
Es∗∗1 (g; ν1 , ν2 ) = ζ ∗ (3ν1 )Es1 (g; ν1 , ν2 ) =
+ φs1 (g; ν2 ) + O(ν1 ).
3ν1
Theorem 7.
ρ ∗ b ∗∗
∗∗
κ(g) = ζ (2) Es2 s1 (g; 0, 0) + Es1 (g; 1, 0) + c0
s
where
bs∗∗s (g; ν1 , ν2 ) = Es∗∗s (g; ν1 , ν2 ) − Es∗∗ (g; ν1 , ν2 ) − Es∗∗ (g; s1 (ν1 , ν2 ))
E
2 1
2 2
2
2
and
ρ
c0 = (δζ ∗ (−1) + ρ(ζ ∗ )0 (−1)).
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5