Volume 10 (2009), Issue 1, Article 2, 33 pp.
A SPANNING SET FOR THE SPACE OF SUPER CUSP FORMS
ROLAND KNEVEL
U NITÉ DE R ECHERCHE EN M ATHÉMATIQUES L UXEMBOURG
C AMPUS L IMPERTSBERG
162 A , AVENUE DE LA FAIENCÉRIE
L - 1511 L UXEMBOURG
roland.knevel@uni.lu
URL: http://math.uni.lu/ knevel/
Received 18 July, 2008; accepted 09 February, 2009
Communicated by S.S. Dragomir
A BSTRACT. The aim of this article is the construction of a spanning set for the space sSk (Γ)
of super cusp forms on a complex bounded symmetric super domain B of rank 1 with respect
to a lattice Γ. The main ingredients are a generalization of the A NOSOV closing lemma for partially hyperbolic diffeomorphisms and an unbounded realization H of B, in particular F OURIER
decomposition at the cusps of the quotient Γ\B mapped to ∞ via a partial C AYLEY transformation. The elements of the spanning set are in finite-to-one correspondence with closed geodesics
of the body Γ\B of Γ\B, the number of elements corresponding to a geodesic growing linearly
with its length.
Key words and phrases: Automorphic and cusp forms, super symmetry, semisimple L IE groups, partially hyperbolic flows,
unbounded realization of a complex bounded symmetric domain.
2000 Mathematics Subject Classification. Primary: 11F55; Secondary: 32C11.
1. I NTRODUCTION
Automorphic and cusp forms on a complex bounded symmetric domain B are already a well
established field of research in mathematics. They play a fundamental role in representation
theory of semisimple L IE groups of Hermitian type, and they have applications to number
theory, especially in the simplest case where B is the unit disc in C, biholomorphic to the upper
half plane H via a C AYLEY transform, G = SL(2, R) acting on H via M ÖBIUS transformations
and Γ @ SL(2, Z) of finite index. The aim of the present paper is to generalize an approach
used by Tatyana F OTH and Svetlana K ATOK in [4] and [8] for the construction of spanning
sets for the space of cusp forms on a complex bounded symmetric domain B of rank 1, which
then by classification is (biholomorphic to) the unit ball of some Cn , n ∈ N, and a lattice
Γ @ G = Aut1 (B) for sufficiently high weight k. This is done in Theorem 4.3, which is the
main theorem of this article, again for sufficiently large weight k.
The new idea in [4] and [8] is to use the concept of a hyperbolic (or A NOSOV) diffeomorphism resp. flow on a Riemannian manifold and an appropriate version of the A NOSOV closing
lemma. This concept originally comes from the theory of dynamical systems, see for example
204-08
2
ROLAND K NEVEL
in [7]. Roughly speaking a flow (ϕt )t∈R on a Riemannian manifold M is called hyperbolic if
there exists an orthogonal and (ϕt )t∈R -stable splitting T M = T + ⊕ T − ⊕ T 0 of the tangent
bundle T M such that the differential of the flow (ϕt )t∈R is uniformly expanding on T + , uniformly contracting on T − and isometric on T 0 , and finally T 0 is one-dimensional, generated by
∂t ϕt . In this situation the A NOSOV closing lemma says that given an ’almost’ closed orbit of
the flow (ϕt )t∈R there exists a closed orbit ’nearby’. Indeed given a complex bounded symmetric domain B of rank 1, G = Aut1 (B) is a semisimple L IE group of real rank 1, and the root
space decomposition of its L IE algebra g with respect to a C ARTAN subalgebra a @ g shows
that the geodesic flow (ϕt )t∈R on the unit tangent bundle S(B), which is at the same time the
left-invariant flow on S(B) generated by a ' R, is hyperbolic. The final result in this direction
is Theorem 5.3 (i).
For the super case, first it is necessary to develop the theory of super automorphic resp.
cusp forms, while the general theory of (Z2 -) graded structures and super manifolds is already
well established, see for example [3]. It was first developed by F. A. B EREZIN as a mathematical method for describing super symmetry in physics of elementary particles. However,
even for mathematicians the elegance within the theory of super manifolds is really amazing
and satisfying. Here I deal with a simple case of super manifolds, namely complex super domains. Roughly speaking a complex super domain B is an object which has a super dimension
(n, r) ∈ N2 and the characteristics:
(i) it has a body B = B # being an ordinary domain in Cn ,
(ii) the complex unital graded commutative
algebra O(B)
V
V of holomorphic super functions
on B is (isomorphic to) O(B) ⊗ (Cr ), where (Cr ) denotes the exterior algebra
of Cr . Furthermore O(B) naturally embeds into the first
factors
V two
V rof the∞complex
∞
C
r
unital
graded
commutative
algebra
D(B)
'
C
(B)
⊗
(C
)
(C ) ' C (B)C ⊗
V 2r
(C ) of ’smooth’ super functions on B, where C ∞ (B)C = C ∞ (B, C) denotes the
algebra of ordinary smooth functions with values in C, which is at the same time the
complexification of C ∞ (B), and ’’ denotes the graded tensor product.
We see that for each pair (B, r) where B ⊂ Cn is an ordinary domain and r ∈ N there exists
exactly one (n, r)-dimensional complex super domain B of super dimension (n, r) with body
B, and we denote it by B |r . Now let ζ1 , V
. . . , ζn ∈ Cr denote the standard basis vectors of Cr .
Then they are the standard generators of (Cr ), and so we get the standard even (commuting)
holomorphic coordinate functions z1 , . . . , zn ∈ O(B)
,→ O B |r and odd (anticommuting)
V r
coordinate functions ζ1 , . . . , ζr ∈ (C ) ,→ O B |r . So omitting
the tensor products, as there
|r
is no danger of confusion, we can decompose every f ∈ O B uniquely as
f=
X
fI ζ I ,
I∈℘(r)
where ℘(r) denotes the power set of {1, . . . , r}, all fI ∈ O(B), I ∈ ℘(r), and ζ I := ζi1 · · · ζis
for all I = {i1 , . . . , is } ∈ ℘(r), i1 < · · · < is .
D B |r is a graded ∗ -algebra, and the graded involution
: D B |r → D B |r
is uniquely defined by the rules
{i} f = f and f h = hf for all f, h ∈ D B |r ,
{ii} is C-antilinear, and restricted to C ∞ (B) it is just the identity,
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
3
V
{iii} ζi is the i-th standard generator of (Cr ) ,→ D B |r embedded as the third factor,
where ζi denotes the i-th
holomorphic
standard coordinate on B |r , which is the i-th
V odd
standard generator of (Cr ) ,→ D B |r embedded as the second factor, i = 1, . . . , r.
With the help of this graded involution we are able to decompose every f ∈ D B |r uniquely
as
f=
X
J
fIJ ζ I ζ ,
I,J∈℘(r)
J
where fIJ ∈ C ∞ (B)C , I, J ∈ ℘(r), and ζ := ζi1 . . . ζis for all J = {j1 , . . . , js } ∈ ℘(r),
j1 < · · · < j s .
For a discussion of super automorphic and super cusp forms we restrict ourselves to the case
of the L IE group G := sS (U (n, 1) × U (r)), n ∈ N \ {0}, r ∈ N, acting on the complex (n, r)dimensional super unit ball B |r . So far there seems to be no classification of super complex
bounded symmetric domains although we know the basic examples, see for example Chapter
IV of [2], which we follow here. The group G is the body of the super L IE group SU (n, 1|r)
studied in [2] acting on B |r . The fact that an ordinary discrete subgroup (which means a sub
super L IE group of super dimension (0, 0)) of a super L IE group is just an ordinary discrete
subgroup of the body justifies our restriction to an ordinary L IE group acting on B |r since
purpose of this article is to study automorphic and cusp forms with respect to a lattice. In any
case one can see the odd directions of the complex super domain B |r already in G since it
is an almost direct
V product of the semisimple L IE group SU (n, 1) acting on the body B and
U (r) acting on (Cr ). Observe that if r > 0 the full automorphism group of B |r , without
any isometry condition, is never a super L IE group since one can show that otherwise its super
L IE algebra would be the super L IE algebra of integrable super vector fields on B |r , which has
unfortunately infinite dimension.
Let us remark on two striking facts:
(i) the construction of our spanning set uses F OURIER decomposition exactly three times,
which is not really surprising, since this corresponds to the three factors in the I WASAWA
decomposition G = KAN .
(ii) super automorphic resp. cusp forms introduced this way are equivalent (but not one-toone) to the notion of ’twisted’ vector-valued automorphic resp. cusp forms.
Acknowledgement: Since the research presented in this article is partially based on my PhD
thesis I would like to thank my doctoral advisor Harald U PMEIER for mentoring during my PhD
but also Martin S CHLICHENMAIER and Martin O LBRICH for their helpful comments.
2. T HE S PACE OF S UPER C USP F ORMS
Let n ∈ N \ {0}, r ∈ N and
G := sS (U (n, 1) × U (r))
0
g 0
0
:=
∈ U (n, 1) × U (r) det g = det E ,
0 E
which is a real ((n + 1)2 + r2 − 1)-dimensional L IE group. Let B := B |r , where
B := {z ∈ Cn | z∗ z < 1} ⊂ Cn
denotes the usual unit ball, with even coordinate functions z1 , . . . , zn and odd coordinate functions ζ1 , . . . , ζr . Then we have a holomorphic action of G on B given by super fractional linear
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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4
ROLAND K NEVEL
(M ÖBIUS) transformations
g
z
ζ
:=
(Az + b) (cz + d)−1
Eζ (cz + d)−1
,
where we split

A b

c d
g :=
0

}n
0 
←n+1 .
}r
E
The stabilizer of 0 ,→ B is
K := sS ((U (n) × U (1)) × U (r))



A 0


0 
=  0 d
∈ U (n) × U (1) × U (r) d det A = det E .


0
E
ˆ
On G × B we define the cocycle j ∈ C ∞ (G)C ⊗O(B)
as j(g, z) := (cz + d)−1 for all g ∈ G
and z ∈ B. Observe that j(w) := j(w, z) ∈ U (1) is independent of z ∈ B for all w ∈ K and
therefore defines a character on the group K.
Let k ∈ Z be fixed. Then we have a right-representation of G
z
|g : D(B) → D(B), f 7→ f |g := f g
j(g, z)k ,
ζ
for all g ∈ G, which fixes O(B). Finally let Γ be a discrete subgroup of G.
Definition 2.1 (Super Automorphic Forms). Let f ∈ O(B). Then f is called a super automorphic form for Γ of weight k if and only if f |γ = f for all γ ∈ Γ. We denote the space of super
automorphic forms for Γ of weight k by sMk (Γ).
Let us define a lift:
^
^
e : D(B) → C ∞ (G)C ⊗ D C0|r ' C ∞ (G)C ⊗ (Cr ) (Cr ) ,
f 7→ fe,
where
0
e
f (g) := f |g
η
0
=f g
j (g, 0)k
η
for all f ∈ D(B) and g ∈ G and we use the odd coordinate functions η1 , . . . , ηr on C0|r . Let f ∈
O(B). Then clearly fe ∈ C ∞ (G)C ⊗ O C0|r and f ∈ sMk (Γ) ⇔ fe ∈ C ∞ (Γ\G)C ⊗ O C0|r
since for all g ∈ G
lg
C ∞ (G)C ⊗ D C0|r −→ C ∞ (G)C ⊗ D C0|r
↑e
↑e
D(B)
D(B)
−→
|g
commutes, where lg : C ∞ (G) → C ∞ (G) denotes the left translation with
g∈
V G, 2rlg (f )(x) :=
0|r
f (gx) for all x ∈ G. Let h , i be the canonical scalar product on D C
' (C ) (semilinp
0|r
ear in the second entry). Then for all a ∈ D C
we write |a| := ha, ai, and h , i induces
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
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a ’scalar product’
Z
D
E
e
e
h, f
(f, h)Γ :=
Γ\G
D
E
for all f, h ∈ D(B) such that e
h, fe ∈ L1 (Γ\G), and for all s ∈]0, ∞] a ’norm’
(k)
||f ||s,Γ := fe s,Γ\G
for all f ∈ D(B) such that fe ∈ C ∞ (Γ\G). On G we always use the (left and right) H AAR
measure. Let us define
(k)
s
∞
C
0|r
e
Lk (Γ\B) := f ∈ D(B) f ∈ C (Γ\G) ⊗ D C
, ||f ||s,Γ < ∞ .
Definition 2.2 (Super Cusp Forms). Let f ∈ sMk (Γ). f is called a super cusp form for Γ of
weight k if and only if f ∈ L2k (Γ\B). The C- vector space of all super cusp forms for Γ of
weight k is denoted by sSk (Γ). It is a H ILBERT space with inner product ( , )Γ .
Observe that |g respects the splitting
O(B) =
r
M
O(ρ) (B)
ρ=0
P
for all g ∈ G, where O(ρ) (B) is the space of all f = I∈℘(r),|I|=ρ fI , all fI ∈ O(B), I ∈ ℘(r),
|I| = ρ, ρ = 0, . . . , r, and e maps the space O(ρ) (B) into C ∞ (G)C ⊗ O(ρ) C0|r . Therefore we
have splittings
sMk (Γ) =
r
M
(ρ)
sMk (Γ) and sSk (Γ) =
r
M
(ρ)
sSk (Γ),
ρ=0
ρ=0
(ρ)
(ρ)
where sMk (Γ) := sMk (Γ) ∩ O(ρ) (B), sSk (Γ) := sSk (Γ) ∩ O(ρ) (B), ρ = 0, . . . , r, and the
last sum is orthogonal.
As shown in [10] and in Section 3.2 of [11] there is an analogon to S ATAKE’s theorem in the
super case:
Theorem 2.1. Let ρ ∈ {0, . . . , r}. Assume Γ\G is compact or n ≥ 2 and Γ @ G is a lattice
(discrete such that vol Γ\G < ∞, Γ\G not necessarily compact). If k ≥ 2n − ρ then
(ρ)
(ρ)
sSk (Γ) = sMk (Γ) ∩ Lsk (Γ\B)
for all s ∈ [1, ∞].
As in the classical case this theorem implies that if Γ\G is compact or n ≥ 2, Γ @ G is a
(ρ)
lattice and k ≥ 2n − ρ, then the H ILBERT space sSk (Γ) is finite dimensional.
We will use the J ORDAN triple determinant ∆ : Cn × Cn → C given by
∆ (z, w) := 1 − w∗ z
for all z, w ∈ Cn . Let us recall the basic properties:
1
(i) |j (g, 0)| = ∆ (g0, g0) 2 for all g ∈ G,
(ii) ∆
R (gz, gw)λ= ∆ (z, w) j (g, z) j (g, w) for all g ∈ G and z, w ∈ B, and
(iii) B ∆ (z, z) dVLeb < ∞ if and only if λ > −1.
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6
ROLAND K NEVEL
We have the G-invariant volume element ∆(z, z)−(n+1) dVLeb on B.
For all I ∈ ℘(r), h ∈ O(B), z ∈ B and
∗ 0
g=
∈G
0 E
we have
hζ I g (z) = h (gz) (Eη)I j (g, z)k+|I| ,
P
P
where E ∈ U (r). So for all s ∈]0, ∞], f = I∈℘(r) fI ζ I and h = I∈℘(r) hI ζ I ∈ O(B) we
have
s X
(k)
k+|I|
||f ||s,Γ ≡ fI2 ∆ (z, z)
I∈℘(r)
s,Γ\B,∆(z,z)−(n+1) dVLeb
if fe ∈ C ∞ (G) ⊗ O C0|r and
X Z
(f, h)Γ ≡
fI hI ∆ (z, z)k+|I|−(n+1) dVLeb
I∈℘(r)
Γ\B
D
E
e
e
if h, f ∈ L1 (Γ\G), where ’≡’ means equality up to a constant 6= 0 depending on Γ.
For the explicit computation of the elements of our spanning set in Theorem 4.3 we need the
following lemmas:
Lemma 2.2 (Convergence of relative P OINCARÉ series). Let Γ0 @ Γ be a subgroup and
f ∈ sMk (Γ0 ) ∩ L1k (Γ0 \B) .
Then
Φ :=
X
X
f |γ and Φ0 :=
γ∈Γ0 \Γ
fe(γ♦)
γ∈Γ0 \Γ
converge absolutely and uniformly on compact subsets of B resp. G,
Φ ∈ sMk (Γ) ∩ L1k (Γ\B) ,
e = Φ0 , and for all ϕ ∈ sMk (Γ) ∩ L∞ (Γ\B) we have
Φ
k
(Φ, ϕ)Γ = (f, ϕ)Γ0 .
The symbol ’♦’ here and also later simply stands for the argument of the function. So
V
e
f (γ♦) ∈ C ∞ (G)C ⊗ (Cr ) is a short notation for the smooth map
^
G→
(Cr ) , g 7→ fe(γg).
Proof. Standard, on using the mean value property of holomorphic functions for all k ∈ Z
without any further assumption on k.
Lemma 2.3. Let I ∈ ℘(r) and k ≥ 2n + 1 − |I|. Then for all w ∈ B
∆ (♦, w)−k−|I| ζ I ∈ O|I| (B) ∩ L1k (B),
and for all f =
P
J∈℘(r)
fJ ζ J ∈ O(B) ∩ L∞
k (B) we have
∆ (♦, w)−k−|I| ζ I , f ≡ fI (w) ,
where ( , ) := ( , ){1} .
Since the proof is also standard, we will omit it here. It can be found in [11].
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
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3. T HE S TRUCTURE OF THE G ROUP G
We have a canonical embedding
0
0
G := SU (p, q) ,→ G, g 7→
g0 0
0 1
,
and the canonical projection
G → U (r), g :=
g0 0
0 E
7→ Eg := E
induces a group isomorphism
G /G0 ' U (r).
Obviously K 0 = K ∩ G0 = S(U (n) × U (1)) is the stabilizer of 0 in G0 . Let A denote the
common standard maximal split abelian subgroup of G and G0 given by the image of the L IE
group embedding


sinh t1
cosh t 0
.
0
1
0
R ,→ G0 , t 7→ at := 
sinh t 0
cosh t
Then the centralizer M of A in K is the group of all


ε 0
 0 u 0 0 
,



0
ε
0
E
where ε ∈ U (1), u ∈ U (p − 1) and E ∈ U (r) such that ε2 det u = det E. Let M 0 = K 0 ∩M =
G0 ∩ M be the centralizer of A in K 0 . The centralizer of G0 in G is precisely
ε1 0 0
p+1
ZG (G ) :=
ε ∈ U (1), E ∈ U (r), ε
= det E @ M,
0 E and G0 ∩ ZG (G0 ) = Z (G0 ). An easy calculation shows that G = G0 ZG (G0 ). So K =
K 0 ZG (G0 ) and M = M 0 Z (G0 ). Therefore if we decompose the adjoint representation of A
as
M
g=
gα ,
α∈Φ
where for all α ∈ R
gα := ξ ∈ g Adat (ξ) = eαt
is the corresponding root space and
Φ := {α ∈ R| g α 6= 0}
is the root system, then we see that Φ is at the same time the root system of G0 , so Φ = {0, ±2}
if n = 1 and Φ = {0, ±1, ±2} if n ≥ 2. Furthermore, if α 6= 0 then gα @ g0 is at the same time
the corresponding root space of g0 , and finally g0 = a ⊕ m = a ⊕ m0 ⊕ zg (g0 ).
Lemma 3.1.
N (A) = ANK (A) = N (AM ) @ N (M ).
Proof. Simple calculation.
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ROLAND K NEVEL
In particular we have the W EYL group
W := M \NK (A) ' M 0 \NK 0 (A) ' {±1}
acting on A ' R via sign change. For the main result, Theorem 4.3, of this article the following
definition is crucial:
Definition 3.1. Let g0 ∈ G.
(i) g0 is called loxodromic if and only if there exists g ∈ G such that g0 ∈ gAM g −1 .
(ii) If g0 is loxodromic, it is called regular if and only if g0 = gat wg −1 with t ∈ R \ {0} and
w ∈ M.
(iii) If γ ∈ Γ is regular loxodromic then it is called primitive in Γ if and only if γ = γ 0ν implies
ν ∈ {±1} for all loxodromic γ 0 ∈ Γ and ν ∈ Z.
Clearly for all γ ∈ Γ regular loxodromic there exists γ 0 ∈ Γ primitive regular loxodromic
and ν ∈ N \ {0} such that γ = γ 0ν .
Lemma 3.2. Let g0 ∈ G be regular loxodromic, g ∈ G, w ∈ M and t ∈ R \ {0} such that
g0 = gat wg −1 . Then g is uniquely determined up to right translation by elements of ANK (A),
and t is uniquely determined up to sign.
Proof. By straight forward computation or using the following strategy: Let g 0 ∈ G, w0 ∈ M
−1
and t0 ∈ R such that g0 = g 0 at0 w0 g 0−1 also. Then at w = (g −1 g 0 ) at0 w0 (g −1 g 0 ) . Since t ∈
R \ {0} and because of the root space decomposition, a + m must be the largest subspace of
g on which Adat w is orthogonal with respect to an appropiate scalar product. So Adg−1 g0 maps
a + m into itself. This implies g −1 g 0 ∈ N (AM ) = ANK (A) by Lemma 3.1.
4. T HE M AIN R ESULT
Let ρ ∈ {0, . . . , r}. Assume Γ\G compact or n ≥ 2, vol Γ\G < ∞ and k ≥ 2n − ρ. Let
C > 0 be given. Let us consider a regular loxodromic γ0 ∈ Γ. Let g ∈ G, w0 ∈ M and t0 > 0
such that γ0 = gat0 w0 g −1 .
There exists a torus T := hγ0 i\ gAM belonging to γ0 . From Lemma 3.2 it follows that T is
independent of g up to right translation with an element of the W EYL group W = M \NK (A).
Let f ∈ sSk (Γ). Then fe ∈ C ∞ (Γ\G)C ⊗ O C0|r . Define h ∈ C ∞ (R × M )C ⊗ O C0|r as
h (t, w) := fe(gat w)
for all (t, w) ∈ R × M ’screening’ the values of fe on T. Then clearly h (t, w) = h(t, 1,
(ρ)
Ew ηj(w))j(w)k , and so h(t, w) = h(t, 1, Ew η)j(w)k+ρ if f ∈ sSk (Γ), for all (t, w) ∈ R×M .
Clearly E0 := Ew0 ∈ U (r). So we can choose g ∈ G such that E0 is diagonal without changing
T. Choose D ∈ Rr×r diagonal such that exp(2πiD) = E0 and χ ∈ R such that j (w0 ) = e2πiχ .
D and χ are uniquely determined by w0 up to Z. If


d1
0
...

D=
0
dr
with d1 , . . . , dr ∈ R and I ∈ ℘(r), then we define trI D :=
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P
j∈I
dj .
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
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Theorem 4.1 (F OURIER expansion
of h ).
−1
(i) h (t + t0 , w) = h t, w0 w for all (t, w) ∈ R × M , and there exist unique bI,m ∈ C,
I ∈ ℘(r), m ∈ t10 (Z − (k + |I|) χ − trI D), such that
X
X
h (t, w) =
j(w)k+|I|
bI,m e2πimt (Ew η)I
m∈ t1 (Z−(k+|I|)χ−trI D)
I∈℘(r)
0
for all (t, w) ∈ R × M , where the sum converges uniformly in all derivatives.
(ρ)
(ii) If f ∈ sSk (Γ), bI,m = 0 for all I ∈ ℘(r), |I| = ρ, and m ∈ t10 (Z − (k + ρ) χ −trI D)∩] −
V
C, C[ then there exists H ∈ C ∞ (R × M )C ⊗ (Cr ) uniformly L IPSCHITZ continuous with a
L IPSCHITZ constant C2 ≥ 0 independent of γ0 such that
h = ∂t H,
H (t, w) = j(w)k H (t, 1, Ew ηj(w))
and
H (t + t0 , w) = H t, w0−1 w
for all (t, w) ∈ R × M .
Proof. (i) Let t ∈ R and w ∈ M . Then
h (t + t0 , w) = fe(gat0 at w) = fe γ0 gw0−1 at w = fe gat w0−1 w
= h t, w0−1 w ,
and so
h (t + t0 , 1) = h t, w0−1
= j (w0 )−k h t, 1, E0−1 ηj (w0 )−1
X
= j (w0 )−k
h (t, 1) e−2πitrI D η I j (w0 )−|I|
I∈℘(r)
X
=
e−2πi((k+|I|)χ+trI D) hI (t, 1)η I .
I∈℘(r)
Therefore hI (t + t0 , 1) = e−2πi((k+|I|)χ+trI D) hI (t, 1) for all I ∈ ℘(r), and the rest follows by a
standard F OURIER expansion.
To prove (ii) we need the following lemma:
Lemma 4.2 (Generalization of the reverse B ERNSTEIN inequality). Let t0 ∈ R \ {0}, ν ∈ R
and C > 0. Let S be the space of all convergent F OURIER series
X
s=
sl e2πim♦ ∈ C ∞ (R)C ,
m∈ t1 (Z−ν),|m|≥C
0
for all sm ∈ C. Then
X
b : S → S, s =
X
sm e2πim♦ 7→ sb :=
m∈ t1 (Z−ν),|m|≥C
m∈ t1 (Z−ν),|m|≥C
0
is a well-defined linear map, and ||b
s||∞ ≤
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
sm 2πim♦
e
2πim
0
6
πC
||s||∞ for all s ∈ S.
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10
ROLAND K NEVEL
Proof. This can be deduced from the ordinary reverse B ERNSTEIN inequality, see for example
Theorem 8.4 in Chapter I of [9].
Now we prove Theorem 4.1 (ii). Fix some I ∈ ℘(r) such that |I| = ρ and bI,m = 0 for all
m ∈ t10 (Z − (k + ρ) χ − trI D) ∩] − C, C[. Then if we define ν := (k + ρ) χ + trI D ∈ R we
have
X
hI (♦, 1) =
bI,m e2πim♦ ,
m∈ t1 (Z−ν),|m|≥C
0
and so we can apply the generalized reverse B ERNSTEIN inequality, Lemma 4.2, to hI . Therefore we can define
X
bI,m 2πim♦
e
∈ C ∞ (R)C .
HI0 := hI\
(♦, 1) =
2πim
1
m∈ t (Z−ν),|m|≥C
0
e
f ∈ L∞ (G) by S ATAKE’s theorem, Theorem 2.1, and so there exists a constant C 0 > 0
independent of γ0 and I such that ||hI ||∞ < C 0 , and now Lemma 4.2 tells us that
||HI0 ||∞ ≤
6
6C 0
||h (♦, 1)||∞ ≤
.
πC
πC
Clearly hI (♦, 1) = ∂t HI0 .
Since j is smooth on the compact set M , j k+ρ (Ew η)I is uniformly L IPSCHITZ continuous
on M with a common
L IPSCHITZ constant C 00 independent of γ0 and I. So we see that H ∈
V
C ∞ (R, M )C ⊗ (Cr ) defined as
X
H(t, w) :=
j(w)k+ρ HI0 (t) (Ew η)I
I∈℘(r)
for all (t,w) ∈ R × M is uniformly L IPSCHITZ continuous with L IPSCHITZ constant C2 :=
6C 00
+ 1 C 0 independent of γ0 , and the rest is trivial.
πC
Let I ∈ ℘(r) and m ∈ t10 (Z − (k + |I|) χ − trI D). Since sSk (Γ) is a H ILBERT space and
sSk (Γ) → C, f 7→ bI,m is linear and continuous, there exists exactly one ϕγ0 ,I,m ∈ sSk (Γ) such
(|I|)
that bI,m = (ϕγ0 ,I,m , f ) for all f ∈ sSk (Γ). Clearly ϕγ0 ,I,m ∈ sSk (Γ).
For the remainder of the article for simplicity we write m ∈] − C, C[ instead of m ∈
1
(Z
− (k + |I|) χ − trI D) ∩] − C, C[. In the last section we will compute ϕγ0 ,I,m as a relt0
ative P OINCARÉ series. One can check that the family
{ϕγ0 ,I,m }I∈℘(r),|I|=ρ,m∈]−C,C[
is independent of the choice of g, D and χ up to multiplication with a unitary matrix with entries
in C and invariant under conjugating γ0 with elements of Γ.
Now we can state our main theorem: Let Ω be a fundamental set for all primitive regular
loxodromic γ0 ∈ Γ modulo conjugation by elements of Γ and
n
o
0
0
0
0
e
Z := m ∈ ZG (G ) ∃g ∈ G : mg ∈ Γ @ ZG (G0 ) .
e Recall that we still assume
Then clearly Γ @ G0 Z.
• Γ\G compact or
• n ≥ 2, vol Γ\G < ∞ and k ≥ 2n − ρ.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
11
Theorem 4.3 (Spanning set for sSk (Γ) ). Assume that the right translation of A on Γ\G0 Ze is
topologically transitive. Then
{ ϕγ0 ,I,m | γ0 ∈ Ω, I ∈ ℘(r), |I| = ρ, m ∈] − C, C[}
(ρ)
is a spanning set for sSk (Γ).
For proving this result we need an A NOSOV type theorem for G and the unbounded realization of B, which we will discuss in the following two sections.
Remark 1.
f @ ZG (G0 ) such that Γ @ G0 M
f and the right translation of
(i) If there is some subgroup M
0f
fZ(G0 ) = Ze and there exists
A on Γ\G M is topologically transitive then necessarily M
g0 ∈ G0 such that G0 Ze = Γg0 A. The latter statement is a trivial consequence of the fact
that Ze @ M .
(ii) In the case where Γ ∩ G0 @ Γ is of finite index or equivalently Ze is finite then we know
that the right translation of A on Γ\G0 Ze is topologically transitive because of M OORE’s
ergodicity theorem, see [13] Theorem 2.2.6, and since then Γ ∩ G0 @ G0 is a lattice.
(iii) There is a finite-to-one correspondence between Ω and the set of closed geodesics of
Γ\B assigning to each primitive loxodromic element
γ0 = gat0 w0 g −1 ∈ Γ, g ∈ G, t0 > 0 and w0 ∈ M , the image of the unique geodesic
gA0 of B normalized by γ0 under the canonical projection B → Γ\B. It is of length t0
if there is no irregular point of Γ\B on gA0.
5. A N A NOSOV T YPE R ESULT FOR THE G ROUP G
On the L IE group G we have a smooth flow (ϕt )t∈R given by the right translation by elements
of A:
ϕt : G → G, g 7→ gat .
This turns out to be partially hyperbolic, and so we can apply a partial A NOSOV closing lemma.
Let me mention that the flow (ϕt )t∈R descends to the ordinary geodesic flow on the unit tangent
bundle SB ' M \G. Let us first have a look at the general theory of partial hyperbolicity: Let
W be, for the moment, a smooth Riemannian manifold.
Definition 5.1 (Partially Hyperbolic Diffeomorphism and Flow). Let C > 1.
(i) Let ϕ be a C ∞ -diffeomorphism of W . Then ϕ is called partially hyperbolic with constant C
if and only if there exists an orthogonal Dϕ (and therefore Dϕ−1 ) -invariant C ∞ -splitting
(5.1)
TW = T0 ⊕ T+ ⊕ T−
of the tangent bundle T W such that T 0 ⊕ T + , T 0 ⊕ T − , T 0 , T + and T − are closed under the
commutator, Dϕ|T 0 is an isometry, ||Dϕ|T − || ≤ C1 and ||Dϕ−1 |T + || ≤ C1 .
(ii) Let (ϕt )t∈R be a C ∞ -flow on W . Then (ϕt )t∈R is called partially hyperbolic with constant C
if and only if all ϕt , t > 0 are partially hyperbolic diffeomorphisms with a common splitting
(5.1) and constants eCt resp. and T 0 contains the generator of the flow.
A partially hyperbolic diffeomorphism ϕ gives rise to C ∞ -foliations on W corresponding to
the splitting T W = T 0 ⊕ T + ⊕ T − . Let us denote the distances along the T 0 ⊕ T + -, T 0 -, T + respectively T − -leaves by d0,+ , d0 , d+ and d− .
Definition 5.2. Let T W = T 0 ⊕ T + ⊕ T − be an orthogonal C ∞ -splitting of the tangent bundle
T W of W such that T 0 ⊕ T + , T 0 , T + and T − are closed under the commutator, C 0 ≥ 1 and
U ⊂ W . U is called C 0 -rectangular (with respect to the splitting T W = T 0 ⊕ T + ⊕ T − ) if and
only if for all y, z ∈ U
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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12
ROLAND K NEVEL
{i} there exists a unique intersection point a ∈ U of the T 0 ⊕ T + -leaf containing y and
the T − -leaf containing z and a unique intersection point b ∈ U of the T 0 ⊕ T + -leaf
containing z and the T − -leaf containing y,
d0,+ (y, a) , d− (y, b) , d− (z, a) , d0,+ (z, b) ≤ C 0 d (y, z) ,
and
1 0,+
d (z, b) ≤ d0,+ (y, a) ≤ C 0 d0,+ (z, b) ,
C0
1 −
d (z, a) ≤ d− (y, b) ≤ C 0 d− (z, a) .
C0
{ii} if y and z belong to the same T 0 ⊕T + -leaf there exists a unique intersection point c ∈ U
of the T 0 -leaf containing y and the T + -leaf containing z and a unique intersection point
d ∈ U of the T 0 -leaf containing z and the T + -leaf containing y,
d0 (y, c) , d+ (y, d) , d+ (z, c) , d0 (z, d) ≤ C 0 d0,+ (y, z) ,
and
1 0
d (z, d) ≤ d0 (y, c) ≤ C 0 d0 (z, d) ,
C0
1 +
d (z, c) ≤ d+ (y, d) ≤ C 0 d+ (z, c) .
C0
Figure 5.1: Intersection points in {i}.
Since the splitting T W = T 0 ⊕ T + ⊕ T − is orthogonal and smooth we see that for all x ∈ W
and C 0 > 1 there exists a C 0 -rectangular neighbourhood of x.
Theorem 5.1 (Partial A NOSOV closing lemma). Let ϕ be a partially hyperbolic diffeomorphism
with constant C, let x ∈ W , C 0 ∈]1, C[ and δ > 0 such that Uδ (x) is contained in a C 0 rectangular subset U ⊂ W .
1− C
0
C
then there exist y, z ∈ U such that
If d (x, ϕ(x)) ≤ δ C 02 +1
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
13
(i) x and y belong to the same T − -leaf and
d− (x, y) ≤
C0
0 d (x, ϕ(x)) ,
1 − CC
(ii) y and ϕ(y) belong to the same T 0 ⊕ T + -leaf and
d0,+ (y, ϕ(y)) ≤ C 02 d (x, ϕ(x)) ,
(iii) y and z belong to the same T + -leaf and
C 03
d (ϕ(y), ϕ(z)) ≤
0 d (x, ϕ(x)) ,
1 − CC
+
(iv) z and ϕ(z) belong to the same T 0 -leaf and
d0 (z, ϕ(z)) ≤ C 04 d (x, ϕ(x)) .
The proof, which will not be given here, uses a standard argument obtaining the points y and
ϕ(z) as limits of certain C AUCHY sequences. The interested reader will find it in [11].
Now let us return to the flow (ϕt )t∈R on G and choose a left invariant metric on G such
that gα , α ∈ Φ \ {0}, a and m are pairwise orthogonal and the isomorphism R ' A ⊂ G
is isometric. Then since the flow (ϕt )t∈R commutes with left translations it is indeed partially
hyperbolic with constant 1 and the unique left invariant splitting of T G given by
M
M
T1 G = g = a ⊕ m ⊕
gα ⊕
gα .
α∈Φ,α<0
| {z } α∈Φ,α>0
| {z } | {z }
T10 :=
T1− :=
T1+ :=
For all L ⊂ G compact, T, ε > 0 define
ML,T := gat g −1 g ∈ L, t ∈ [−T, T ]
and
NL,T,ε := {g ∈ G |dist (g, ML,T ) ≤ ε } .
Lemma 5.2. For all L ⊂ G compact there exist T0 , ε0 > 0 such that Γ ∩ NL,T0 ,ε0 = {1}.
Proof. Let L ⊂ G be compact and T > 0. Then ML,T is compact, and so there exists ε > 0
such that NL,T,ε is again compact. Since Γ is discrete, Γ ∩ NL,T,ε is finite. Clearly for all T, T 0 , ε
and ε0 > 0 if T ≤ T 0 and ε ≤ ε0 then NL,T,ε ⊂ NL,T 0 ,ε0 , and finally
\
NT,ε = {1}.
T,ε>0
Here now is the quintessence of this section:
Theorem 5.3.
(i) For all T1 > 0 there exist C1 ≥ 1 and ε1 > 0 such that for all x ∈ G, γ ∈ Γ and T ≥ T1 if
ε := d (γx, xaT ) ≤ ε1
then there exist z ∈ G, w ∈ M and t0 > 0 such that γz = zat0 w (and so γ is regular
loxodromic), d ((t0 , w), (T, 1)) ≤ C1 ε and for all τ ∈ [0, T ]
d (xaτ , zaτ ) ≤ C1 ε e−τ + e−(T −τ ) .
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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14
ROLAND K NEVEL
(ii) For all L ⊂ G compact there exists ε2 > 0 such that for all x ∈ L, γ ∈ Γ and T ∈ [0, T0 ],
T0 > 0 given by Lemma 5.2, if
ε := d (γx, xaT ) ≤ ε2
then γ = 1 and T ≤ ε.
Proof. (i) Let T1 > 0 and define
!
3
C1 := max
e 2 T1
−
1−e
T1
2
, e2T1
≥ 1.
T1
Define C 0 := e 2 , let U be a C 0 -rectangular neighbourhood of 1 ∈ G and let δ > 0 such that
Uδ (1) ⊂ U . Then by the left invariance of the splitting and the metric on G we see that gU is a
C 0 -rectangular neighbourhood of g and Uδ (g) = gUδ (1) ⊂ gU for all g ∈ G. Define
!
T1
1 − e− 2 T1
ε1 := min δ T1
,
> 0.
e + 1 C1
Now assume γ ∈ Γ and T ≥ T1 such that
ε := d (γx, xaT v ) ≤ ε1 .
Then ϕ : G → G, g 7→ γ −1 gaT is a partially hyperbolic diffeomorphism with constant eT1 > 1
and the corresponding splitting T G = T 0 ⊕ T + ⊕ T − . Then since
T1
1 − e− 2
1 − C 0 e−T1
ε ≤ δ T1
=δ
e +1
C 02 + 1
the partial A NOSOV closing lemma, Theorem 5.1, tells us that there exist y, z ∈ G such that
(i) x and y belong to the same T − -leaf and
C0
d (x, y) ≤ ε
0 ,
1 − CC
−
(iii) y and z belong to the same T + -leaf and
d+ (yaT v , zaT v ) ≤ ε
C 03
0 ,
1 − CC
(iv) γz and zaT v belong to the same T 0 -leaf and
d0 (γz, zaT v ) ≤ εC 04 .
In (iii) and (iv) we already used that the metric and the flow are left invariant. So by (iv) and
since the T 0 -leaf containing zaT is zAM , there exist w ∈ M and t0 ∈ R such that γz = zat0 w.
So
d0 (at0 −T w, 1) ≤ εC 04 ,
and so, since AM ' R × M isometrically, we see that
d ((t0 , w) , (T, 1)) ≤ εC 04 = εe2T1 ≤ εC1 .
In particular, |t0 − T | ≤ T1 , and so t0 > 0.
Now let τ ∈ [0, T ]. Then since x and y belong to the same T − -leaf, the same is true for xaτ
and yaτ , and
C 0 −τ
d− (xaτ , yaτ ) ≤ d− (x, y) e−τ ≤ ε
e ≤ εC1 e−τ .
C0
1− C
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
15
Since y and z belong to the same T + -leaf, the same is true for yaτ and zaτ , and
d+ (yaτ , zaτ ) ≤ d+ (yaT , zaT ) e−(T −τ )
≤ε
C 03 −(T −τ )
≤ εC1 e−(T −τ ) .
0 e
1 − CC
Combining these two inequalities we obtain
d (xaτ , zaτ ) ≤ εC1 e−τ + e−(T −τ ) .
≤ c and therefore
(ii) Let L ⊂ G be compact and let c ≥ 1 be given such that ||Adg || , Ad−1
g
1
d(ag, bg) ≤ d(a, b) ≤ cd(ag, bg)
c
for all g ∈ L and a, b ∈ G. Let ε0 > 0 be given by Lemma 5.2 and define
ε0
> 0.
ε2 :=
c
Let x ∈ L, γ ∈ Γ and T ∈ [0, T0 ] such that
ε := d (γx, xaT ) ≤ ε2 .
Then since x ∈ L, we get
d γ, xaT x−1 ≤ cε ≤ ε0
and so γ ∈ Γ ∩ NL,T0 ,ε0 . This implies γ = 1 and so d (1, aT ) = ε and therefore T ≤ ε.
6. T HE U NBOUNDED R EALIZATION
Let n @ g0 be the standard maximal nilpotent sub L IE algebra, which is at the same time the
direct sum of all root spaces of g0 of positive roots with respect to a. Let N := exp n. Then we
have an I WASAWA decomposition
G = N AK,
N is 2-step nilpotent, and so N 0 := [N, N ] is at the same time the center of N .
Now we transform the whole problem to the unbounded realization via the partial C AYLEY
transformation
 1

√
0 √12
←1
2
1 0  }n − 1 ∈ G0C = SL(n + 1, C)
R :=  0
←n+1
− √12 0 √12
mapping B biholomorphically onto the unbounded domain
1 ∗
w1
←1
n
∈ C Re w1 > w2 w2 .
H := w =
}n − 1
w2
2
We see that
RG0 R−1 @ G0C = SL(n + 1, C) ,→ GL(n + 1, C) × GL(r, C)
acts holomorphically and transitively on H via
calculations show that
 t
e
0
−1

0
at := Rat R =
0
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
fractional linear transformations, and explicit

0 0
←1
1 0  }n − 1
←n+1
0 e−t
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16
ROLAND K NEVEL
for all t ∈ R, and RN R−1 is the image of

1 u∗ iλ + 12 u∗ u
,
u
:=  0 1
0 0
1

R × Cn−1 → RG0 R−1 , (λ, u) 7→ n0λ,u
which is a C ∞ -diffeomorphism onto its image, with the multiplication rule
n0λ,u n0µ,v = n0λ+µ+Im (u∗ v),u+v
for all λ, µ ∈ R and u, v ∈ Cn−1 , so N is exactly the H EISENBERG group Hn acting on H as
pseudo translations
w1 + u∗ w2 + iλ + 21 u∗ u
w 7→
.
w2 + u
√
Define j (R, z) =
2
1−z1
−1
√
∈ O(B), j (R−1 , w) := j (R, R−1 w) = 1+w2 1 ∈ O(H), and for all


A b
0 
g ∈ RGR−1 =  c d
∈ RGR−1
0
E
define
1
.
cw + d
Let H := H |r with even coordinate functions w1 , . . . , wn and odd coordinate functions ϑ1 , . . . , ϑr .
R commutes with all g ∈ ZG (G0 ), and we have a right-representation of the group RGR−1 on
D(H) given by
♦
|g : D(H) → D(H), f 7→ f g
j (g, ♦)k
ϑ
j (g, w) = j R, R−1 gw j R−1 gR, R−1 w j R−1 , w =
for all g ∈ RGR−1 . If we define
♦
j (R, ♦)k
|R : D(H) → D(B), f →
7 f R
ζ
and
|R−1 : D(B) → D(H), f 7→ f
R
−1
♦
ϑ
k
j R−1 , ♦ ,
then we see that we get a commuting diagram
D(H)
|R ↓
D(B)
|RgR−1
−→
−→
|g
D(H)
↓ |R .
D(B)
Now define the sesqui polynomial ∆0 on H ×H, holomorphic in the first and antiholomorphic
in the second variable, as
−1
−1
∆0 (z, w) := ∆ R−1 z, R−1 w j R−1 , z
j (R−1 , w) = z1 + w1 − w2∗ z2
for all z, w ∈ H. Clearly det (z 7→ Rz)0 = |j (R, z)|n+1 for all z ∈ B. So
det (w 7→ gw)0 = |j (g, w)|n+1 ,
1
|j (g, e1 )| = ∆0 (ge1 , ge1 ) 2
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
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for all gP∈ RGR−1 and ∆0 (w, w)−(n+1) dVLeb is the RGR−1 -invariant volume element on H.
If f = I∈℘(r) fI ζ I ∈ O(B), all fI ∈ O(B)C , I ∈ ℘(r), then
X
k+|I| I
f |R−1 =
fI R−1 ♦ j R−1 , ♦
ϑ ∈ O(H),
I∈℘(r)
and if f =
P
I∈℘(r)
fI ϑI ∈ O(H), all fI ∈ C ∞ (H)C , I ∈ ℘(r), and g ∈ RGR−1 , then
X
f |g =
fI (g♦) j (g, ♦)k+|I| (Eg ϑ)I ∈ O(H).
I∈℘(r)
Let ∂H = w ∈ Cn Re w1 = 21 w2∗ w be the boundary of H in Cn . Then ∆0 and ∂H are
RN R−1 -invariant, and RN R−1 acts transitively on ∂H and on each
w ∈ H ∆0 (w, w) = e2t = RN at 0,
t ∈ R.
Figure 6.1: The geometry of H.
All geodesics in H can be written in the form
R → H, t 7→ wt := Rgat 0 = RgR−1 a0t e1
with some g ∈ G, and conversely all these curves are geodesics in H. We have to distinguish
two cases: Either the geodesic connects ∞ with a point in ∂H, or it connects two points in ∂H.
In the second case we have
lim ∆0 (wt , wt ) = 0,
t→±∞
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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18
ROLAND K NEVEL
so we may assume without loss of generality that ∆0 (wt , wt ) is maximal for t = 0, otherwise
we have to reparametrize the geodesic using gaT , T ∈ R appropriately chosen, instead of g.
Lemma 6.1.
(i) Let
R → H, t 7→ wt := Rgat 0 = RgR−1 a0t e1
be a geodesic in H such that limt→∞ wt = ∞ and limt→−∞ wt ∈ ∂H with respect to the
euclidian metric on Cp . Then for all t ∈ R
∆0 (wt , wt ) = e2t ∆0 (w0 , w0 ) ,
and if instead limt→−∞ wt = ∞ and limt→∞ wt ∈ ∂H, then
∆0 (wt , wt ) = e−2t ∆0 (w0 , w0 ) .
(ii) Let
R → H, t 7→ wt := Rgat 0 = RgR−1 a0t e1
be a geodesic in H connecting two points in ∂H such that ∆0 (wt , wt ) is maximal for t = 0.
Then
R → R>0 , t 7→ ∆0 (wt , wt )
is strictly increasing on R≤0 and strictly decreasing on R≥0 , and for all t ∈ R
∆0 (w−t , w−t ) = ∆0 (wt , wt )
and
e−2|t| ∆0 (w0 , w0 ) ≤ ∆0 (wt , wt ) ≤ 4e−2|t| ∆0 (w0 , w0 ) .
Proof. (i) Since RN R−1 acts transitively on ∂H and ∆0 is RN R−1 -invariant we can assume
without loss of generality that the geodesic connects 0 and ∞. But in H a geodesic is uniquely
determined up to reparametrization by its endpoints. So we see that in the first case
wt = a0t xe1 = e2t xe1
and in the second case
wt = a0−t xe1 = e−2t xe1
both with an appropriately chosen x > 0.
(ii) Let u, y ∈ R and s ∈ Cp−1 such that y 2 + s∗ s = 1. Then
u
eu
e √1 − y 2 tanh2 t + 2iy tanh t
(u,y,s)
R → H, t 7→ wt
:=
2 tanh t (1 + iy tanh t) s
1 + y 2 tanh2 t
is a geodesic through e2u e1 in H since it is the image of the standard geodesic
tanh t
R → B, t 7→ at 0 =
0
in B under the transformation

iy −s∗ 0
R  s −iy 0  .
0
0 1
|
{z
}

a0u
| {z }
∈RAR−1 @RG0 R−1
So we see that
(u,y,s) ∂t wt
=
t=0
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
∈K 0 @G0
2ie2u y
√
2eu s
∈ Te2u e1 H
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
19
is a unit vector with respect to the RGR−1 -invariant metric on H coming from B via R. Now
since RN R−1 acts transitively on each
w ∈ H ∆0 (w, w) = e2t = RN at 0,
t ∈ R, and ∆0 is invariant under RN R−1 we may assume without loss of generality that w0 =
e2u e1 with an appropriate u ∈ R. Since ∆0 (wt , wt ) is maximal for t = 0 we know that ∂t wt |t=0
is a unit vector in iR ⊕ Cp−1 @ Te1 H, and therefore there exist y ∈ R and s ∈ Cp−1 such that
y 2 + s∗ s = 1 and
2ie2u y
√
∂t wt |t=0 =
.
2eu s
(u,y,s)
Since the geodesic is uniquely determined by w0 and ∂t wt |t=0 we see that wt = wt
t ∈ R, and so a straight forward calculation shows that
for all
1 − tanh2 t
1 + y 2 tanh2 t
8e2u
.
=
(1 + y 2 ) (e2t + e−2t ) + 2s∗ s
∆0 (wt , wt ) = 2e2u
The rest is an easy exercise using y 2 + s∗ s = 1.
For all t ∈ R define A>t := { aτ | τ > t} ⊂ A.
Theorem 6.2 (A ’fundamental domain’ for Γ\G ). There exist η ⊂ N open and relatively
compact, t0 ∈ R and Ξ ⊂ G0 finite such that if we define
[
Ω :=
gηA>t0 K
g∈Ξ
then
(i) g −1 Γg ∩ N ZG (G0 ) @ N ZG (G0 ) and g −1 Γg ∩ N 0 ZG (G0 ) @ N 0 ZG (G0 ) are lattices, and
N ZG (G0 ) = g −1 Γg ∩ N ZG (G0 ) ηZG (G0 )
for all g ∈ Ξ ,
(ii) G = ΓΩ,
(iii) the set {γ ∈ Γ|γΩ ∩ Ω 6= ∅} is finite.
Proof. The theorem is a direct consequence of Theorem 0.6 (i) - (iii), Theorem 0.7, Lemma
3.16 and Lemma 3.18 of [5]. For a detailed derivation see [10] or Section 3.2 of [11].
Now clearly the set of cusps of Γ\B in Γ\∂B is contained in the set
lim Γgat 0|g ∈ Ξ ,
t→+∞
and is therefore finite as expected, where the limits are taken with respect to the Euclidian metric
on B.
Corollary 6.3. Let t0 ∈ R, η ⊂ N and Ξ ⊂ G be given by Theorem 6.2. Let h ∈ C (Γ\G)C and
s ∈]0, ∞]. Then h ∈ Ls (Γ\G) if and only if h (g♦) ∈ Ls (ηA>t0 K) for all g ∈ Ξ.
P
Let f ∈ sMk (Γ) and g ∈ Ξ. Then we can decompose f |g |R−1 = I∈℘(r) qI ϑI ∈ O(H), all
qI ∈ O(H), I ∈ ℘(r), and by Theorem 6.2 (i) we know that g −1 Γg ∩ N 0 ZG (G0 ) 6@ ZG (G0 ).
So let n ∈ g −1 Γg ∩ N 0 ZG (G0 ) \ ZG (G0 ),
ε1 0
−1
0
RnR = nλ0 ,0
,
0 E0
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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20
ROLAND K NEVEL
λ0 ∈ R \ {0}, ε ∈ U (1), E0 ∈ U (r), εn+1 = det E.
j (RnR−1 ) := j (RnR−1 , w) = ε−1 ∈ U (1) is independent of w ∈ H. So there exists χ ∈ R
such that j (RnR−1 ) = e2πiχ . Without loss of generality we can assume that E0 is diagonal,
otherwise conjugate n with an appropriate element of ZG (G0 ). So there exists D ∈ Rr×r
diagonal such that E0 = exp (2πiD).
Theorem 6.4 (F OURIER expansion of f |g |R−1 ).
(i) There exist unique cI,m ∈ O (Cn−1 ), I ∈ ℘(r), m ∈
X
qI (w) =
1
λ0
(Z − trI D − (k + |I|) χ), such that
cI,m (w2 ) e2πmw1
m∈ λ1 (Z−trI D−(k+|I|)χ)
0
for all w ∈ H and I ∈ ℘(r), and so
X
f |g |R−1 (w) =
I∈℘(r)
X
cI,m (w2 ) e2πmw1 ϑI
m∈ λ1 (Z−trI D−(k+|I|)χ)
0
w1
←1
∈ H, where the convergence is absolute and compact.
}n − 1
w2
(ii) cI,m = 0 for all I ∈ ℘(r) and m > 0 (this is a super analogon for KOECHER’s principle,
see for example Section 11.5 of [1] ), and if trI D + (k + |I|) χ ∈ Z, then cI,0 is a constant.
(iii) Let I ∈ ℘(r) and s ∈ [1, ∞] . If trI D + (k + |I|) χ 6∈ Z, then
for all w =
qI ∆0 (w, w)
k+|I|
2
∈ Ls (RηA>t0 0)
with respect to the RGR−1 -invariant measure ∆0 (w, w)−(n+1) dVLeb on H. If trI D+(k + |I|) χ ∈
Z and k ≥ 2n − |I| then
qI ∆0 (w, w)
k+|I|
2
∈ Ls (RηA>t0 0)
with respect to the RGR−1 -invariant measure on H if and only if cI,0 = 0.
A proof can be found in [10] or [11] Section 3.2.
7. P ROOF OF THE M AIN R ESULT
We have a L IE algebra embedding
ρ : sl(2, C) ,→ g0C = sl(n + 1, C),
a b
c −a

a 0

0 0
7→
c 0

b
0 .
−a
Obviously the preimage of g0 under ρ is su(1, 1), the preimage of k0 under ρ is s (u(1) ⊕ u(1)) '
u(1) and ρ lifts to a L IE group homomorphism


a 0
b
a b
0 
ρe : SL(2, C) → G0C = SL(n + 1, C),
7→  0 0
c d
c 0
d
such that ρe (SU (1, 1)) @ G0 .
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
21
Let us now identify the elements of g with the corresponding left invariant differential operators. They are defined on a dense subset of L2 (Γ\G), and define
0 1
0 i
0
D := ρ
∈ a , D := ρ
∈ g0 and
1 0
−i 0
i 0
φ := ρ
∈ k0 .
0 −i
The R-linear span of D, D0 and φ is the 3-dimensional sub L IE algebra ρ (su(1, 1)) of g0 @ g,
and D generates the flow ϕt . φ generates a subgroup of K 0 , being the image of the L IE group
embedding
it
e
0
R/2πZ ,→ K, t 7→ exp (tφ) = ρe
.
0 e−it
Now define
1
1
(D − iD0 ) , D− := (D + iD0 ) and Ψ := −iφ
2
2
as left invariant differential operators on G. Then we get the commutation relations
Ψ, D+ = 2D+ , Ψ, D− = −2D− and D+ , D− = Ψ,
D+ :=
and since G is unimodular
D+
∗
= −D− , D−
∗
= −D+ and Ψ∗ = Ψ.
So by standard F OURIER analysis
L2 (Γ\G) =
M
d
Hν
ν∈Z
as an orthogonal sum, where
Hν := F ∈ L2 (Γ\G) ∩ domain Ψ ΨF = νF
for all ν ∈ Z. By a simple calculation we obtain
D+ Hν ∩ domain D+ ⊂ Hν+2 and D− Hν ∩ domain D− ⊂ Hν−2
for all ν ∈ Z.
Lemma 7.1. D−e
h = 0 for all h ∈ O(B).
fg . So
Proof. Let g ∈ G. Then again h|g ∈ O(B), and e
h (g♦) = h|
D −e
h(g) = D− e
h (g♦) (1) = ∂ 1 h|g = 0.
(ρ)
Lemma 7.2. Let f ∈ sSk (Γ). Then fe is uniformly L IPSCHITZ continuous.
Proof. Since on G we use a left invariant metric it suffices to show that there exists a constant
c ≥ 0 such that for all g ∈ G and ξ ∈ g with ||ξ||2 ≤ 1
e ξ f (g) ≤ c.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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22
ROLAND K NEVEL
Then c is a L IPSCHITZ constant for fe. So choose an orthonormal basis (ξ1 , . . . , ξN ) of g and a
compact neighbourhood L of 0 in B. Then by C AUCHY’s integral formula there exist C 0 , C 00 ≥
0 such that for all h ∈ O(B) ∩ L∞
k (B) and n ∈ {1, . . . , N }
Z
e
0
0
00
|h| ≤ C vol L ||h||∞,L ≤ C vol L e
h ,
ξn h (1) ≤ C
∞
L
and since g → C, ξ 7→ ξe
h (1) is linear we obtain
e
e 00
ξ
h
(1)
≤
N
C
vol
L
h
∞
for general ξ ∈ g with ||ξ||2 ≤ 1. Now let g ∈ G. Then again f |g ∈ O(B), fe(g♦) = ff
|g , and
∞
by S ATAKE’s theorem, Theorem 2.1, f and so f |g ∈ Lk (B). So
e e
e
e
00
00
ξ
f
(g)
=
ξ
f
(g♦)
(1)
≤
N
C
vol
L
f
(g♦)
≤
N
C
vol
L
f ,
∞
∞
and we can define c := N C 00 vol L fe .
∞
(ρ)
Now let f ∈ sSk (Γ) such that (ϕγ0 ,I,m , f )Γ = 0 for all ϕγ0 ,I,m , γ0 ∈ Γ primitive loxodromic,
I ∈ ℘(r), |I| = ρ, m ∈] − C, C[. We will show that f = 0 in several steps.
V
Lemma 7.3. There exists F ∈ C (Γ\G)C ⊗ (Cr ) uniformly L IPSCHITZ continuous on compact
sets and differentiable along the flow ϕt such that
f = ∂τ F (♦aτ )|τ =0 = DF.
Proof. Here we use that the right translation with A on Γ\G0 Ze is topologically transitive. So
V
let g0 ∈ G0 such that Γg0 A = G0 Ze and define s ∈ C ∞ (R)C ⊗ (Cr ) by
Z t
s(t) :=
fe(g0 aτ ) dτ
0
for all t ∈ R.
Step I. Show that for all L ⊂ G0 Ze compact there exist constants C3 ≥ 0 and ε3 > 0 such
that for all t ∈ R, T ≥ 0 and γ ∈ Γ if g0 at ∈ L and
ε := d (γg0 at , g0 at+T ) ≤ ε3
then |s(t) − s(t + T )| ≤ C3 ε.
Let L ⊂ G0 Ze be compact, T0 > 0 be given by Lemma 5.2 and C
1≥ 1 and ε1 be given by
e
Theorem 5.3 (i) with T1 := T0 . Define C3 := max C1 (C2 + 2c) , f ≥ 0, where C2 ≥ 0
∞
e
is the L IPSCHITZ constant from
Theorem 4.1 (ii) and c ≥ 0 is the L IPSCHITZ constant of f .
T0
Define ε3 := min ε1 , ε2 , 2C
> 0, where ε2 > 0 is given by Theorem 5.3 (ii).
1
Let t ∈ R, T ≥ 0 and γ ∈ Γ such that g0 at ∈ L and ε := d (γg0 at , g0 at+T ) ≤ ε3 .
First assume T ≥ T0 . Then by Theorem 5.3 (i) since ε ≤ ε1 there exist g ∈ G, w0 ∈ M and
t0 > 0 such that γg = gat0 w0 , d ((t0 , w0 ) , (T, 1)) ≤ C1 ε, and for all τ ∈ [0, T ]
d (g0 at+τ , gaτ ) ≤ C1 ε e−τ + e−(T −τ ) .
We get
Z
s(t + T ) − s(t) =
|0
T
Z T
e
e
e
f (gaτ ) dτ +
f (g0 at+τ ) − f (gaτ ) dτ
0
{z
} |
{z
}
I1 :=
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
I2 :=
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
23
and
T
Z
e
f (g0 at+τ ) − fe(gaτ ) dτ
|I2 | ≤
0
Z
T
≤c
d (g0 at+τ , gaτ ) dτ
Z T
≤ cC1 ε
e−τ + e−(T −τ ) dτ
0
0
≤ 2cC1 ε.
Since γ ∈ Γ is regular loxodromic, there exists γ0 ∈ Γ primitive loxodromic and ν ∈ N \ {0}
such that γ = γ0ν . γ0 ∈ gAW g −1 since Lemma 3.2 tells us that g ∈ G is already determined
by γ up to right translation with elements of ANK (A). Choose w0 ∈ NK (M ), t00 > 0 and
w00 ∈ M such that Ew00 is diagonal and γ = gw0 at00 w00 (gw0 )−1 , and let g 0 := gw0 . We define
V
h ∈ C ∞ (R × M )C ⊗ (Cr ) as
h(τ, w) := fe(g 0 aτ w) = fe(gaτ w0 w)
for all τ ∈ R and w ∈ M . Then
Z
T
h τ, w0−1 dτ.
I1 =
0
We can apply Theorem 4.1 (i) and, since f is perpendicular to all ϕγ0 ,I,m , I ∈ ℘(r), m ∈
] − C, C[, also Theorem 4.1 (ii) with g 0 := gw0 instead of g, and so
|I1 | = H T, w0−1 − H 0, w0−1 = H T, w0−1 − H t0 , w0−1 w0 ≤ C2 d ((T, 1) , (t0 , w0 ))
≤ C1 C2 ε,
where we used that H (0, w0−1 ) = H (t00 , w00 w0−1 ), choosing the left invariant metric on M , and
the claim follows.
Now assume T ≤ T0 . Then by Theorem 5.3 (ii), since ε ≤ ε0 we get T ≤ ε and so
Z T
e
|s(t + T ) − s(t)| = f (g0 at+τ ) dτ ≤ ε fe .
∞
0
C V
Step II. Show that there exists a unique F1 ∈ C Γ\G0 Ze ⊗ (Cr ) uniformly L IPSCHITZ
continuous on compact sets such that for all t ∈ R
s(t) = F1 (g0 at ) .
C
By Step I for all L ⊂ Γ\G0 Ze compact with L◦ ⊂ L there exists a unique FL ∈ C Γ\G0 Ze
dense
uniformly L IPSCHITZ continuous such that for all t ∈ R if Γg0 at ∈ L then s(t) = FL (Γg0 at ).
C V
0e
So we see that there exists a unique F1 ∈ C Γ\G Z ⊗ (Cr ) such that F1 |L = FL for all
L ⊂ Γ\G0 Ze compact with L◦ ⊂ L.
dense
Step III. Show that F1 is differentiable along the flow and that for all g ∈ G0 Ze
∂τ F1 (gaτ ) |τ =0 = fe(g).
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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24
ROLAND K NEVEL
e It suffices to show that for all T ∈ R
Let g ∈ G0 Z.
Z T
fe(gaτ ) dτ = F1 (gaT ) − F1 (g).
0
e since
If g = g0 at for some t ∈ R then it is clear by construction. For general g ∈ G0 Z,
N
0e
Γg0 A = G Z there exists (γn , tn )n∈N ∈ (Γ × R) such that
lim γn g0 atn = g,
n→∞
and so
lim γn g0 aτ +tn = gaτ
n→∞
compact in τ ∈ R. Finally fe is uniformly L IPSCHITZ continuous. Therefore we can interchange
integration and taking the limit n
∞:
Z T
Z T
e
f (gaτ ) dτ = lim
fe(γn g0 aτ +tn ) dτ
n→∞
0
0
= lim (F1 (γn g0 aT +tn ) − F1 (γn g0 atn ))
n→∞
= F1 (gaT ) − F1 (g).
Step IV. Conclusion. V
Define F ∈ C(G)C ⊗ (Cr ) as
Z
F (gw) :=
F1 gu−1 , Euw η j(uw)k+ρ du
e
Z
for all g ∈ G0 Ze and w ∈ ZG (G0 ), where we normalize the H AAR measure on the compact L IE
group Ze such that vol Ze = 1. Then we see that F is well defined and fulfills all the desired
properties.
Lemma 7.4.
(i) For all L ⊂ G compact there exists ε4 > 0 such that for all g, h ∈ L if g and h belong to the
same T − -leaf and d− (g, h) ≤ ε4 then
lim (F (gat ) − F (hat )) = 0,
t→∞
and if g and h belong to the same T + -leaf and d+ (g, h) ≤ ε4 then
lim (F (gat ) − F (hat )) = 0.
t→−∞
(ii) F is continuously differentiable along T − - and T + -leafs, more precisely if ρ : I → G is a
continuously differentiable curve in a T − -leaf, then
Z ∞
∂t (F ◦ ρ) (t) = −
∂t fe(ρ(t)aτ ) dτ,
0
and if ρ : I → G is a continuously differentiable curve in a T + -leaf then
Z 0
∂t (F ◦ ρ) (t) =
∂t fe(ρ(t)aτ ) dτ.
−∞
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
25
Proof. (i) Let L ⊂ G be compact, and let L0 ⊂ G be a compact neighbourhood of L. Let T0 > 0
be given by Lemma 5.2 and ε2 > 0 by Theorem 5.3 (ii) both with respect to L0 . Define
1
T0
ε4 := min ε1 , ε2 ,
> 0,
3
2C1
where ε1 > 0 and C1 ≥ 1 are given by Theorem 5.3 (i) with T1 := T0 . Let δ0 > 0 such that
Uδ0 (L) ⊂ L0 and let
δ ∈ ]0, min (δ0 , ε4 )[ .
−
Let g, h ∈ L be in the same T -leaf such that ε := d− (g, h) ≤ ε4 . Since the splitting of T G
is left invariant and T1− (G) @ g0 we see that there exist g 0 , h0 ∈ G0 and u ∈ ZG (G0 ) such that
g = g 0 u and h = h0 u. Fix some T 0 > 0. Again by assumption there exists g0 ∈ G0 such that
e and so g, h ∈ Γg0 uA. So there exist γg , γh ∈ Γ and tg , th ∈ R such that
Γg0 A = G0 Z,
d gat , γg g0 uatg +t , d (hat , γh g0 uath +t ) ≤ δ
for all t ∈ [0, T 0 ], and so in particular γg g0 uatg , γh g0 uath ∈ L0 . We will show that for all
t ∈ [0, T 0 ]
F γg g0 uatg +t − F (γh g0 uat +t ) ≤ C30 εe−t + 2δ
h
with the same constant C30 ≥ 0 as in Step I of the proof of Lemma 7.3 with respect to L0 .
Without loss of generality we may assume T := th − tg ≥ 0. Define γ :=
γg γh−1 ∈ Γ. Then for all t ∈ [0, T 0 ]
d γγh g0 uatg +t , γh g0 uatg +t+T ≤ εe−t + 2δ.
First assume T ≥ T0 andfix t ∈ [0, T 0 ]. Then by Theorem 5.3 (i), since
T0
εe−t + 2δ ≤ ε + 2δ ≤ min ε1 , 2C
, there exist z ∈ G, t0 ∈ R and w ∈ M
1
such that γz = zat0 w,
d ((t0 , w) , (T, 1)) ≤ C1 2δ + εe−t ,
and for all τ ∈ [0, T ]
d γg g0 uatg +t+τ , zaτ ≤ C1 εe−t + 2δ e−τ + e−(T −τ ) .
And so by the same calculations as in the proof of Lemma 7.3 we obtain the
estimate
F γg g0 uatg +t − F (γh g0 uat +t ) ≤ C30 εe−t + 2δ .
h
Now assume T ≤ T0 . Then by Theorem 5.3 (ii) since γg g0 matg ∈ L0 and
ε + 2δ ≤ ε2 we obtain γ = 1 and so by the left invariance of the metric on G
0
d (1, aT ) ≤ εe−T + 2δ,
0
therefore T ≤ εe−T + 2δ. So as in the proof of Lemma 7.3,
−T 0
F γg g0 uatg +t − F (γh g0 uat +t ) ≤ fe
εe
+ 2δ
h
∞
≤ C30 εe−t + 2δ .
Now let us take the limit δ
continuous
0. Then γg g0 uatg
g and γh g0 uath
h, so since F is
|F (gat ) − F (hat )| ≤ C30 εe−t
for all t ∈ [0, T 0 ], and since T 0 > 0 has been arbitrary, we obtain this estimate for all t ≥ 0 and
so limt→∞ F (gat )−F (hat ) = 0. By similar calculations we can prove that limt→−∞ F (gat )−
F (hat ) = 0 if g and h belong to the same T + -leaf and d+ (g, h) ≤ ε4 .
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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26
ROLAND K NEVEL
(ii) Let ρ : I → G be a continuously differentiable curve in a T − -leaf, and let t0 , t1 ∈ I, t1 > t0 .
It suffices to show that
Z t1 Z ∞
F (ρ (t1 )) − F (ρ (t0 )) = −
∂t fe(ρ(t)aτ ) dτ dt.
t0
0
0
0
Let C ≥ 0 such that ||∂t ρ(t)|| ≤ C for all t ∈ [t0 , t1 ]. Then since ρ lies in a T − -leaf we have
||∂t (ρ(t)aτ )|| ≤ C 0 e−τ and so
e
∂t f (ρ(t)aτ ) ≤ cC 0 e−τ
for all τ ≥ 0 and t ∈ [t0 , t1 ] where c ≥ 0 is the L IPSCHITZ constant of fe. So the double integral
on the right side is absolutely convergent and so we can interchange the order of integration:
Z t1 Z ∞
Z ∞ Z t1
e
∂t f (ρ(t)aτ ) dτ dt =
∂t fe(ρ(t)aτ ) dtdτ
t0
0
Z0 ∞ t0
=
fe(ρ (t1 ) aτ ) − fe(ρ (t0 ) aτ ) dτ
0
= lim (F (ρ (t1 ) aT ) − F (ρ (t0 ) aT ))
T →∞
− F (ρ (t1 )) + F (ρ (t0 )) .
Now let L ⊂ G be compact such that ρ([t1 , t2 ]) ⊂ L and let ε4 > 0 as in (i). Without loss of
generality we may assume that d− (ρ (t0 ) , ρ (t1 )) ≤ ε4 . Then
lim (F (ρ (t1 ) aT ) − F (ρ (t0 ) aT )) = 0
T →∞
by (i). By similar calculations one can also prove
Z 0
∂t (F ◦ ρ) (t) =
∂t fe(ρ(t)aτ ) dτ
−∞
in the case when ρ : I → G is a continuously differentiable curve in a T + -leaf.
Lemma 7.5.
V r
(i) F ∈ L2 (Γ\G) ⊗ V
(C ),
2
(ii) ξF ∈ L (Γ\G) ⊗ (Cr ) for all ξ ∈ RD ⊕ g ∩ (T + ⊕ T − ).
Proof. (i) If Γ\G is compact then the assertion is trivial. So assume that Γ\G is not compact,
then we use the unbounded realization H of B introduced in Section 6. Since vol (Γ\G) < ∞, it
suffices to prove that F is bounded, and by Corollary 6.3 it is even enough to show that F (g♦)
is bounded on N A>t0 K for all g ∈ Ξ, where t0 ∈ R and Ξ ⊂ G0 are given by Theorem 6.2. So
let g ∈ Ξ.
Step I. Show that F (g♦) is bounded on N at0 K.
Let η ⊂ N also be given by Theorem 6.2. Then F (g♦) is clearly bounded on the compact
set ηat0 K. On the other hand F (g♦) is left-g −1 Γg -invariant, so it is also bounded on
N at0 K = gΓg −1 ∩ N ZG (G0 ) ηat0 K
by Theorem 6.2 (i).
Step II. Show that there exists C 0 ≥ 0 such that for all g 0 ∈ N A>t0 K
C0
e 0 .
f (gg ) ≤ 0
∆ (Rg 0 0, Rg 0 0)
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
27
P
As in Section 6, let qI ∈ O(H) such that f |g |R−1 = I∈℘(r) qI ϑI . Then since fe(g♦) ∈
V
L2 (ηA>t0 K) ⊗ (Cr ), by Theorem 6.4 we have F OURIER expansions
X
(7.1)
qI (w) =
cI,m (w2 ) e2πmw1
m∈ λ1 (Z−trI D−(k+|I|)χ)∩R<0
0
w1
←1
∈ H, where cI,m ∈ O (Cn−1 ), I ∈ ℘(r),
for all I ∈ ℘(I) and w =
}n − 1
w2
m ∈ λ10 (z − trI D − (k + |I|) χ) ∩ R<0 . Define
[ 1
M0 := max
(Z − trI D − (k + |I|) χ) ∩ R<0 < 0.
λ0
I∈℘(r)
Rηat0 0 ⊂ H is compact, and so since the convergence of the F OURIER series (7.1) is absolute
and compact we can define
2t0
C 00 := e−2πM0 e
× max
I∈℘(r)
X
cI,m (w2 ) e2πmw1 m∈ λ1 (Z−trI D−(k+|I|)χ)∩R<0
0
Then we have
∞,Rηat0 0
< ∞.
0
|qI (w)| ≤ C 00 eπM0 ∆ (w,w)
for all I ∈ ℘(r) and w ∈ RηA>t0 0. Now let
∗ 0
0
g =
∈ ηA>0 K,
0 E0
E 0 ∈ U (r). Then
fe(gg 0 ) = f |g |R−1 RgR−1 (e1 )
k
e1
0 −1
j Rg 0 R−1 , e1
= f |g |R−1 Rg R
η
k
Rg 0 0
j Rg 0 R−1 , e1
= f |g |R−1
0 −1
Eηj (Rg R )
X
k+|I|
=
qI (Rg 0 0) (Eη)I j Rg 0 R−1 , e1
.
I∈℘(r)
p
Therefore since |j (Rg 0 R−1 , e1 )| = ∆0 (Rg 0 0, Rg 0 0) we get
0
0
0
e 0 f (gg ) ≤ 2r C 00 eπM0 ∆ (Rg 0,Rg 0)
k+r
k
× ∆0 (Rg 0 0, Rg 0 0) 2 + ∆0 (Rg 0 0, Rg 0 0) 2 .
So we see that there exists C 0 > 0 such that
e 0 f (gg ) ≤
C0
∆0 (Rg 0 0, Rg 0 0)
for all g 0 ∈ ηA>t0 K. However, on the one hand fe(g♦) is left- g −1 Γg -invariant, and on the
other hand ∆0 is RN ZG (G0 ) R−1 -invariant. Therefore the estimate is correct even for all
g 0 ∈ N A>t0 K = gΓg −1 ∩ N ZG (G0 ) ηA>t0 K
by Theorem 6.2 (i).
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
http://jipam.vu.edu.au/
28
ROLAND K NEVEL
Step III. Conclusion: Prove that
+ 2C 0 e−2t0
|F (g♦)| ≤ ||F (g♦)||∞,N at
0K
on N A>t0 K.
Let g 0 ∈ G be arbitrary. We will show the estimate on g 0 A ∩ N A>t0 K.
R → H, t 7→ wt := Rg 0 at 0
is a geodesic in H, and for all t ∈ R we have g 0 at ∈ N A>t0 K if and only if ∆0 (wt , wt ) > 2e2t0 .
Now we have to distinguish two cases.
In the first case the geodesic connects ∞ with a point in ∂H. First assume that limt→∞ wt =
∞ and limt→−∞ wt ∈ ∂H. Then limt→∞ ∆0 (wt , wt ) = ∞ and limt→−∞ ∆0 (wt , wt ) = 0. So
we may assume without loss of generality that ∆0 (w0 , w0 ) = 2e2t0 , and therefore g 0 = g 0 a0 ∈
N at0 K and g 0 at ∈ N A>t0 K if and only if t > 0. So let t > 0. Then
0
Z
0
t
fe(gg 0 aτ ) dτ,
F (gg at ) = F (gg ) +
0
and so
0
|F (gg at )| ≤ ||F (g♦)||∞,N at
0K
By Step II and Lemma 6.1 (i),
Z
Z t
e 0 0
f (gg aτ ) dτ ≤ C
Z t
e 0 +
f
(gg
a
)
τ dτ.
0
t
dτ
0
0 ∆ (wτ , wτ )
Z t
C0
= 0
e−2τ dτ
∆ (w0 , w0 ) 0
≤ C 0 e−2t0 .
0
The case where limt→−∞ = ∞ and limt→∞ ∈ ∂H is done similarly.
In the second case the geodesic connects two points in ∂H. Then without loss of generality
we may assume that ∆0 (Rwt , Rwt ) is maximal for t = 0. So if ∆0 (w0 , w0 ) < 2e2t0 , we have
g 0 A ∩ N A>t0 K = ∅. Otherwise by Lemma 6.1 (ii) there exists T ≥ 0 such that ∆0 (wT , wT ) =
4
0
∆0 (w−T , w−T ) = 2e2t0 , and since ∆0 (wT , wT ) ≤ e2|T
| ∆ (w0 , w0 ), we see that
T ≤
1
log (2∆0 (wT , wT )) − t0 .
2
So g 0 aT , g 0 a−T ∈ N at0 K and g 0 at ∈ N A>t0 K if and only if t ∈] − T, T [. Let t ∈] − T, T [ and
assume t ≥ 0 first. Then
Z T
0
0
F (gg at ) = F (gg aT ) −
fe(gg 0 aτ ) dτ,
t
and so
Z
0
|F (gg at )| ≤ ||F (g♦)||∞,N at
0K
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
+
0
T
e 0 f (gg aτ ) dτ.
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
By Step II and Lemma 6.1 (ii), now
Z T
Z
e 0 0
f (gg aτ ) dτ ≤ C
29
T
dτ
(wτ , wτ )
0
Z T
0
C
e2τ dτ
≤ 0
∆ (w0 , w0 ) 0
C0
≤
e2T
0
2∆ (w0 , w0 )
≤ 2C 0 e−2t0 .
0
∆0
The case t ≤ 0 is done similarly.
V
(ii) Since on one hand ∂τ F (♦aτ ) |τ =0 = fe ∈ L2 (Γ\G) ⊗ (Cr ) and on the other hand
vol (Γ\G) < ∞, it suffices to show that ξF is bounded for all α ∈ Φ \ {0} and ξ ∈ gα . So let
α ∈ Φ \ {0} and ξ ∈ gα . First assume α > 0, which clearly implies that α ≥ 1 and ξ ∈ T − . So
there exists a continuously differentiable curve ρ : I → G contained in the T − -leaf containing
1 such that 0 ∈ I, ρ(0) = 1 and ∂t ρ(t)|t=0 = ξ. Let g ∈ G. Then by Theorem 7.4 (ii), we have
(ξF ) (g) = ∂t F (gρ(t))|t=0
Z ∞
=−
∂t fe(gρ(t)aτ ) dτ
t=0
Z0 ∞
=−
∂t fe(gaτ a−τ ρ(t)aτ ) dτ
t=0
Z0 ∞ =−
Ada−τ (ξ) fe (gaτ ) dτ
Z0 ∞
=−
e−ατ ξ fe (gaτ ) dτ,
0
so
|(ξF ) (g)| ≤ c ||ξ||2 < ∞,
where c is the L IPSCHITZ constant of fe. The case α < 0 is done similarly.
Therefore by the F OURIER decomposition described above we have
X
X
FIν η I ,
F =
I∈℘(r),|I|=ρ
ν∈Z
where FIν ∈ Hν for all I ∈ ℘(r), |I| = ρ, and ν ∈ Z. D = D+ + D− , and a simple calculation
V
shows that D+ and D− ∈ RD ⊕ g ∩ (T + ⊕ T − ), and so D+ F, D− F ∈ L2 (Γ\G) ⊗ (Cr ) by
Lemma 7.5 (ii). So we get the F OURIER decomposition of fe as
X
X
fe = DF =
D+ FI,ν−2 + D− FI,ν+2 η I
I∈℘(r),|I|=ρ
ν∈Z
with D FI,ν−2 + D FI,ν+2 ∈ Hν for all ν ∈ Z. But since f ∈ sSkρ (Γ) the F OURIER decomposition of fe is exactly
X
fe =
qI η I
−
+
I∈℘(r),|I|=ρ
∞
with qI ∈ C (G) ∩ Hk+ρ , and so for all I ∈ ℘(r), |I| = ρ, and ν ∈ Z
qI if ν = k + ρ
+
−
D FI,ν−2 + D FI,ν+2 =
.
0 otherwise
C
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
http://jipam.vu.edu.au/
30
ROLAND K NEVEL
Lemma 7.6. FI,ν = 0 for I ∈ ℘(r), |I| = ρ, and ν ≥ k + ρ.
Proof. Similar to the argument of G UILLEMIN and K AZHDAN in [6]. Let I ∈ ℘(r) such that
|I| = ρ. Then by the commutation relations of D+ and D− we get for all n ∈ Z
+
D FI,n 2 = D− FI,n 2 + ν ||FI,n ||2 ,
(7.2)
2
2
2
and for all n ≥ k + ρ + 1 we have D+ FI,n−2 + D− FI,n+2 = 0 and so
−
D FI,n+2 = D+ FI,n−2 .
2
2
Now let ν ≥ k + ρ. We will prove that
+
D FI,ν+4l ≥ ||FI,ν ||
2
2
for all l ∈ N by induction on l:
If l = 0 then the inequality is clear by (7.2). So let us assume that the inequality
is true for some l ∈ N. Then again by (7.2) we have
+
D FI,ν+4l+4 2 ≥ D− FI,ν+4l+4 2 = D+ FI,ν+4l 2 ≥ ||FI,ν ||2 .
2
2
2
2
On the other hand, D+ FI ∈ L2 (Γ\G) by Lemma 7.5 and so ||D+ FI,n ||2
0 for n
∞.
This implies Fν = 0.
So for all I ∈ ℘(r), |I| = ρ, we obtain D+ FI,k+ρ−2 = qI and finally D− qI = 0 by Lemma
7.1, since f ∈ O(B), so
||qI ||22 = qI , D+ FI,k+ρ−2 = − D− qI , FI,k+ρ−2 = 0,
and so fe = 0, which completes the proof of our main theorem.
8. C OMPUTATION OF THE ϕγ0 ,I,m
Fix a regular loxodromic γ0 ∈ Γ, g ∈ G, t0 > 0 and w0 ∈ M such that E0 := Ew0 is diagonal
and γ0 = gat0 w0 g −1 ∈ gAM g −1 . Let D ∈ Rr×r be diagonal such that exp(2πiD) = E0 and
χ ∈ R such that j(w0 ) = e2πiχ . Now we will compute ϕγ0 ,I,m ∈ sSk (Γ), I ∈ ℘(r), m ∈
1
(Z − (k + |I|) χ − trI D), as a relative P OINCARÉ series with respect to Γ0 := hγ0 i @ Γ.
t0
Hereby again ’≡’ means equality up to a constant 6= 0 not necessarily independent of γ0 , I and
m.
Theorem 8.1. Let I ∈ ℘(r) and k ≥ 2n+1−|I|. Then for all m ∈
(i)
X
(|I|)
ϕγ0 ,I,m ≡
q|γ ∈ sSk (Γ),
1
t0
(Z − (k + |I|) χ − trI D)
γ∈Γ0 \Γ
where
Z
∞
q :=
k+|I|
e2πimt ∆ (♦, gat 0)−k−|I| j (gat , 0)
dt Eg−1 ζ
I
−∞
(|I|)
∈ sMk
(Γ0 ) ∩ L1k (Γ0 \B) .
(ii) For all z ∈ B we have
+
q (z) ≡ ∆ z, X
−
∆ z, X
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
− k+|I|
2
1 + v1
1 − v1
πim
Eg−1 ζ
I
,
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
31
where

1
 0 

X+ := g 
 ... 


−1
 0 

X− := g 
 ... 

and
0
0
are the two fixpoints of γ0 in ∂B, and
v := g −1 z ∈ B ⊂ Cp .
Proof. Let ρ := |I|.
P
V
(ρ)
(i) Let f ∈ sSk (Γ), and define h = J∈℘(r),|J|=ρ hJ η J ∈ C ∞ (R × M )C ⊗ (Cr ), all hJ ∈
C ∞ (R × M )C , and bI,m ∈ C, m ∈ t10 (Z − (k + |I|) χ − trI D), as in Theorem 4.1. Then by
standard F OURIER theory and Lemma 2.3 we have
Z t0
bI,m ≡
e−2πimt hI (t, 1)dt
0
Z t0
I ≡
e−2πimt ∆ (♦, gat 0)−k−ρ Eg−1 ζ , f j (gat , 0)k+ρ dt
0
Z D Z t0
I ∼ E
−2πimt
e
fe, ∆ (♦, gat 0)−k−ρ Eg−1 ζ
j (gat , 0)k+ρ dt.
=
0
G
V
Since by S ATAKE’s theorem, Theorem 2.1, fe ∈ L∞ (G) ⊗ (Cr ), and
Z t0 Z I ∼
k+ρ −k−ρ
−1
j
(ga
,
0)
∆
(♦,
ga
0)
E
ζ
dt
t
t
g
0
G
Z t0 Z ∼
−1 −k−ρ I
=
(gat ) ♦ dt
ζ
∆ (♦, 0)
0
G
Z eI ≡
ζ ZG k+ρ =
j (♦, 0) ZG
k+ρ
∆ (Z, Z) 2 −(p+1) dVLeb < ∞,
≡
B
by T ONELLI’s and F UBINI’s theorem we can interchange the order of integration:
Z Z t0
I ∼
k+ρ
−k−ρ
2πimt
−1
j (gat , 0) dt
bI,m ≡
fe,
e
∆ (♦, gat 0)
Eg ζ
G
0
Z t0
I
k+ρ
−k−ρ
−1
2πimt
j (gat , 0) dt Eg ζ , f
=
e
∆ (♦, gat 0)
0
= (q, f )Γ0 ,
where
Z
t0
2πimt
e
−k−ρ
∆ (♦, gat 0)
k+ρ
j (gat , 0)
dt
I
Eg−1 ζ
∼
∈ L1 (G) ⊗
^
(Cr ) ,
0
Z
t0
k+ρ
e2πimt ∆ (♦, gat 0)−k−ρ j (gat , 0)
dt Eg−1 ζ
I
∈ O(B)
0
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
http://jipam.vu.edu.au/
32
ROLAND K NEVEL
since ∆ (♦, w) ∈ O(B) for all w ∈ B and the convergence of the integral is compact, and so
by Lemma 2.2,
X Z t0
I k+ρ
−k−ρ
−1
2πimt
q :=
e
∆ (♦, gat 0)
j (gat , 0) dt Eg ζ γ 0 ∈Γ0
γ0
0
∈ sMk (Γ0 ) ∩ L1k (Γ0 \B) .
Clearly
I Eg−1 ζ −k−ρ
∆ (♦, gat 0)
−k−ρ
= ∆ (γ0 ♦, gat 0)
γ0
I
j (γ0 , ♦)k+ρ
I
k+ρ
,
E0 Eg−1 ζ j γ0−1 , gat 0
E0 Eg−1 ζ
−k−ρ
= ∆ ♦, γ0−1 gat 0
so for all z ∈ B we can compute q (z) as
q (z) =
XZ
XZ
t0
t0
XZ
t0
k+ρ
j (gat , 0)
dt (z)
γ0ν
k+ρ
−k−ρ ν −1 I
e2πimt ∆ z, γ0−ν gat 0
E0 Eg ζ × j γ0−ν gat , 0
dt
e2πimt ∆ (z, gat−νt0 0)−k−ρ Eg−1 ζ
I
k+ρ 2πiν(k+ρ)χ
e2πiνtrI D j (gat−νt0 , 0)
e
k+ρ
e2πim(t−νt0 ) ∆ (z, gat−νt0 0)−k−ρ j (gat−νt0 , 0)
ν∈Z 0
∞
2πimt
Z
=
∆ (♦, gat 0)
I
Eg−1 ζ
e
dt
0
ν∈Z
=
e
−k−ρ
0
ν∈Z
=
2πimt
0
ν∈Z
=
t0
XZ
k+ρ
∆ (z, gat 0)−k−ρ j (gat , 0)
dt Eg−1 ζ
I
dt Eg−1 ζ
I
.
−∞
Again by Lemma 2.2 we see that
(ρ)
P
γ∈Γ0 \Γ
q|γ ∈ sMk (Γ) ∩ L1k (Γ\B), and so by S ATAKE’s
(ρ)
theorem, Theorem 2.1, it is even an element of sSk (Γ), such that


X
bI,m ≡ 
q|γ , f  ,
γ∈Γ0 \Γ
and so we conclude that ϕγ0 ,I,m ≡
P
γ∈Γ0 \Γ
Γ
q|γ .
(ii)
Z
∞
k+ρ
e2πimt ∆ (z, gat 0)−k−ρ j (gat , 0) dt
−∞
Z
k+ρ ∞ 2πimt
−k−ρ
k+ρ
−1
= j g ,z
e
∆ g −1 z, at 0
j (at , 0) dt
Z−∞
∞
1
k+ρ
= j g −1 , z
e2πimt (1 − v1 tanh t)−k−ρ
dt
(cosh t)k+ρ
−∞
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 2, 33 pp.
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A S PANNING S ET FOR THE S PACE OF S UPER C USP F ORMS
−1
= j g ,z
k+ρ
Z
∞
33
e2πimt
dt
(cosh t − v1 sinh t)k+ρ
πim
k+ρ
1
1 + v1
−1
≡ j g ,z
k+ρ
1 − v1
(1 − v12 ) 2
πim
k+ρ
1
+
v
1
− k+ρ
−1
= j g ,z
((1 − v1 ) (1 + v1 )) 2
1 − v1
πim
− k+ρ
1 + v1
2
≡ ∆ z, X+ ∆ z, X−
.
1 − v1
−∞
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