A SPANNING SET FOR THE SPACE OF SUPER CUSP FORMS

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A Spanning Set for the Space of Super Cusp Forms

Roland Knevel vol. 10, iss. 1, art. 2, 2009

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A SPANNING SET FOR THE SPACE OF SUPER CUSP FORMS

ROLAND KNEVEL

Unité de Recherche en Mathématiques Luxembourg Campus Limpertsberg, 162 a, avenue de la Faiencérie L - 1511 Luxembourg

EMail:roland.knevel@uni.lu URL:http://math.uni.lu/˜knevel/

Received: 18 July, 2008

Accepted: 09 February, 2009

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: Primary: 11F55; Secondary: 32C11.

Key words: Automorphic and cusp forms, super symmetry, semisimple LIEgroups, partially hyperbolic flows, unbounded realization of a complex bounded symmetric domain.

Abstract: The aim of this article is the construction of a spanning set for the spacesSk(Γ) of super cusp forms on a complex bounded symmetric super domainBof rank 1with respect to a latticeΓ. The main ingredients are a generalization of the ANOSOVclosing lemma for partially hyperbolic diffeomorphisms and an unbounded realizationHofB, in particular FOURIERdecomposition at the cusps of the quotientΓ\B mapped to∞via a partial CAYLEYtransformation. The elements of the spanning set are in finite-to-one correspondence with closed geodesics of the bodyΓ\BofΓ\B, the number of elements corresponding to a geodesic growing linearly with its length.

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1. Introduction

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 LIEgroups of Hermitian type, and they have applications to number theory, especially in the simplest case whereB is the unit disc inC, biholomorphic to the upper half plane H via a CAYLEYtransform, G=SL(2,R)acting onH via MÖBIUStransformations andΓ@SL(2,Z)of finite index. The aim of the present paper is to generalize an approach used by Tatyana FOTHand Svetlana KATOKin [4] and [8] for the construction of spanning sets for the space of cusp forms on a complex bounded symmetric domainB of rank1, which then by classification is (biholomorphic to) the unit ball of someCⁿ, n ∈ N, and a latticeΓ @ G = Aut₁(B)for sufficiently high weightk. This is done in Theorem 4.3, which is the main theorem of this article, again for sufficiently large weightk.

The new idea in [4] and [8] is to use the concept of a hyperbolic (or ANOSOV) diffeomorphism resp. flow on a Riemannian manifold and an appropriate version of the ANOSOV closing lemma. This concept originally comes from the theory of dynamical systems, see for example in [7]. Roughly speaking a flow (ϕ_t)_t∈

R on a Riemannian manifold M is called hyperbolic if there exists an orthogonal and (ϕt)_t∈_R-stable splittingT M =T⁺⊕T⁻⊕T⁰of the tangent bundleT M such that the differential of the flow(ϕ_t)_t∈

Ris uniformly expanding onT⁺, uniformly contracting onT⁻ and isometric on T⁰, and finallyT⁰ is one-dimensional, generated by ∂_tϕ_t. In this situation the ANOSOVclosing lemma says that given an ’almost’ closed orbit of the flow (ϕ_t)_t∈

R there exists a closed orbit ’nearby’. Indeed given a complex bounded symmetric domainB of rank1, G = Aut₁(B)is a semisimple LIE group of real rank1, and the root space decomposition of its LIEalgebragwith respect to a CARTANsubalgebraa@ gshows that the geodesic flow(ϕ_t)_t∈

Ron the unit tangent bundleS(B), which is at the same time the left-invariant flow onS(B)generated by

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a'R, is hyperbolic. The final result in this direction is Theorem5.5(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. BEREZINas 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 domainB is an object which has a super dimension(n, r) ∈ N² and the characteristics:

(i) it has a bodyB =B^#being an ordinary domain inCⁿ,

(ii) the complex unital graded commutative algebra O(B) of holomorphic super functions on Bis (isomorphic to) O(B)⊗V

(C^r), where V

(C^r) denotes the exterior algebra of C^r. FurthermoreO(B) naturally embeds into the first two factors of the complex unital graded commutative algebraD(B)' C^∞(B)^C⊗ V(C^r) V

(C^r) ' C^∞(B)^C ⊗ V

(C^2r) of ’smooth’ super functions on B, whereC^∞(B)^C =C^∞(B,C)denotes the algebra of ordinary smooth functions with values inC, which is at the same time the complexification ofC^∞(B), and

’’ denotes the graded tensor product.

We see that for each pair(B, r)whereB ⊂Cⁿis an ordinary domain andr ∈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, . . . , ζn ∈ C^r denote the standard basis vectors of C^r. Then they are the standard generators of V(C^r), and so we get the standard even (commuting) holomorphic coordinate functionsz₁, . . . , z_n∈ O(B),→ O B^|r

and odd (anticommuting) coordinate functions ζ1, . . . , ζr ∈ V

(C^r) ,→ O B^|r

. So omitting the tensor products, as there is no

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danger of confusion, we can decompose everyf ∈ O B^|r

uniquely as f = X

I∈℘(r)

f_Iζ^I,

where ℘(r) denotes the power set of {1, . . . , r}, all f_I ∈ O(B), I ∈ ℘(r), and ζ^I :=ζ_i₁· · ·ζ_i_s for allI ={i₁, . . . , i_s} ∈℘(r),i₁ <· · ·< i_s.

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 andf h=hf for allf, h∈ D B^|r ,

{ii} isC-antilinear, and restricted toC^∞(B)it is just the identity, {iii} ζi is thei-th standard generator ofV

(C^r) ,→ D B^|r

embedded as the third factor, whereζ_i denotes thei-th odd holomorphic standard coordinate onB^|r, which is the i-th standard generator of V

(C^r) ,→ 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

I,J∈℘(r)

f_IJζ^Iζ^J,

wheref_IJ ∈ C^∞(B)^C,I, J ∈ ℘(r), andζ^J := ζ_i₁. . . ζ_i_s for allJ ={j₁, . . . , j_s} ∈

℘(r),j₁ <· · ·< j_s.

For a discussion of super automorphic and super cusp forms we restrict ourselves to the case of the LIEgroupG:=sS(U(n,1)×U(r)),n ∈N\ {0},r ∈N, acting

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on the complex(n, r)-dimensional super unit ballB^|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 Gis the body of the super LIE groupSU(n,1|r)studied in [2] acting on B^|r. The fact that an ordinary discrete subgroup (which means a sub super LIEgroup of super dimension(0,0)) of a super LIE group is just an ordinary discrete subgroup of the body justifies our restriction to an ordinary LIEgroup acting onB^|rsince 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 domainB^|r already in G since it is an almost direct product of the semisimple LIE groupSU(n,1)acting on the bodyB andU(r)acting onV

(C^r). Observe that ifr >0the full automorphism group ofB^|r, without any isometry condition, is never a super LIE group since one can show that otherwise its super LIE algebra would be the super LIE algebra of integrable super vector fields onB^|r, which has unfortunately infinite dimension.

Let us remark on two striking facts:

(i) the construction of our spanning set uses FOURIERdecomposition exactly three times, which is not really surprising, since this corresponds to the three factors in the IWASAWAdecompositionG=KAN.

(ii) super automorphic resp. cusp forms introduced this way are equivalent (but not one-to-one) 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 UPMEIER for mentoring during my PhD but also Martin SCHLICHENMAIERand Martin OLBRICH

for their helpful comments.

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2. The Space of Super Cusp Forms

Letn ∈N\ {0},r ∈Nand

G:=sS(U(n,1)×U(r)) := g⁰ 0

0 E

∈U(n,1)×U(r)

detg⁰ = detE

,

which is a real((n+ 1)²+r²−1)-dimensional LIEgroup. LetB:=B^|r, where B :={z∈Cⁿ|z^∗z<1} ⊂Cⁿ

denotes the usual unit ball, with even coordinate functionsz₁, . . . , z_n and odd coordinate functionsζ₁, . . . , ζ_r. Then we have a holomorphic action ofGonBgiven by super fractional linear (MÖBIUS) transformations

g z

ζ

:=

(Az+b) (cz+d)⁻¹ Eζ(cz+d)⁻¹

, where we split

g :=





A b c d 0

0 E



 }n

←n+ 1 }r

. The stabilizer of0,→ Bis

K :=sS((U(n)×U(1))×U(r))

=











A 0

0 d 0

0 E



∈U(n)×U(1)×U(r)

ddetA= detE





 .

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OnG×B we define the cocyclej ∈ C^∞(G)^C⊗O(B)ˆ asj(g,z) := (cz+d)⁻¹ for allg ∈Gandz∈B. Observe thatj(w) :=j(w,z)∈U(1)is independent ofz∈B for allw∈Kand therefore defines a character on the groupK.

Letk∈Zbe fixed. Then we have a right-representation ofG

|_g :D(B)→ D(B), f 7→f|_g :=f

g z

ζ

j(g,z)^k, for allg ∈G, which fixesO(B). Finally letΓbe a discrete subgroup ofG.

Definition 2.1 (Super Automorphic Forms). Let f ∈ O(B). Then f is called a super automorphic form forΓof weightk if and only iff|_γ = f for allγ ∈ Γ. We denote the space of super automorphic forms forΓof weightkbysMk(Γ).

Let us define a lift:

e:D(B)→ C^∞(G)^C⊗ D C^0|r

' C^∞(G)^C⊗^

(C^r)^ (C^r), f 7→f ,e

where

fe(g) :=f|_g 0

η

=f

g 0

η

j(g,0)^k

for allf ∈ D(B)andg ∈ Gand we use the odd coordinate functionsη₁, . . . , η_r on C^0|r. Letf ∈ O(B). Then clearlyfe∈ C^∞(G)^C⊗ O C^0|r

andf ∈sM_k(Γ)⇔fe∈

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C^∞(Γ\G)^C⊗ O C^0|r

since for allg ∈G C^∞(G)^C⊗ D C^0|r ^lg

−→ C^∞(G)^C⊗ D C^0|r

↑_e ↑_e

D(B) −→

|g

D(B)

commutes, where l_g : C^∞(G) → C^∞(G)denotes the left translation with g ∈ G, l_g(f)(x) := f(gx) for all x ∈ G. Let h , i be the canonical scalar product on D C^0|r

'V

(C^2r)(semilinear in the second entry). Then for alla ∈ D C^0|r we write|a|:=p

ha, ai, andh , iinduces a ’scalar product’

(f, h)_Γ :=

Z

Γ\G

D eh,feE for allf, h∈ D(B)such thatD

eh,feE

∈L¹(Γ\G), and for alls∈]0,∞]a ’norm’

||f||^(k)_s,Γ :=

fe

_s,Γ\G for allf ∈ D(B)such that

fe

∈ C^∞(Γ\G). OnGwe always use the (left and right) HAARmeasure. Let us define

L^s_k(Γ\B) :=

f ∈ D(B)

fe∈ C^∞(Γ\G)^C⊗ D C^0|r

,||f||^(k)_s,Γ <∞

. Definition 2.2 (Super Cusp Forms). Let f ∈ sM_k(Γ). f is called a super cusp form forΓof weightkif and only iff ∈L²_k(Γ\B). TheC- vector space of all super cusp forms forΓof weightkis denoted bysS_k(Γ). It is a HILBERTspace with inner product( , )_Γ.

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Observe that|_grespects the splitting O(B) =

r

M

ρ=0

O^(ρ)(B) for allg ∈G, whereO^(ρ)(B)is the space of allf =P

I∈℘(r),|I|=ρf_I, allf_I ∈ O(B), I ∈ ℘(r), |I| = ρ, ρ = 0, . . . , r, and emaps the space O^(ρ)(B) into C^∞(G)^C ⊗ O^(ρ) C^0|r

. Therefore we have splittings sM_k(Γ) =

r

M

ρ=0

sM_k^(ρ)(Γ) and sS_k(Γ) =

r

M

ρ=0

sS_k^(ρ)(Γ),

wheresM_k^(ρ)(Γ) :=sM_k(Γ)∩O^(ρ)(B),sS_k^(ρ)(Γ) :=sS_k(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 SATAKE’s theorem in the super case:

Theorem 2.3. Letρ ∈ {0, . . . , r}. AssumeΓ\Gis compact orn ≥ 2andΓ @ G is a lattice (discrete such that vol Γ\G < ∞, Γ\G not necessarily compact). If k ≥2n−ρthen

sS_k^(ρ)(Γ) =sM_k^(ρ)(Γ)∩L^s_k(Γ\B) for alls∈[1,∞].

As in the classical case this theorem implies that if Γ\G is compact or n ≥ 2, Γ @ G is a lattice and k ≥ 2n −ρ, then the HILBERT space sS_k^(ρ)(Γ) is finite dimensional.

We will use the JORDANtriple determinant∆ :Cⁿ×Cⁿ→Cgiven by

∆ (z,w) := 1−w^∗z

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for allz,w∈Cⁿ. Let us recall the basic properties:

(i) |j(g,0)|= ∆ (g0, g0)¹² for allg ∈G,

(ii) ∆ (gz, gw) = ∆ (z,w)j(g,z)j(g,w)for allg ∈Gandz,w∈B, and (iii) R

B∆ (z,z)^λdV_Leb<∞if and only ifλ >−1.

We have theG-invariant volume element∆(z,z)⁻⁽ⁿ⁺¹⁾dV_LebonB.

For allI ∈℘(r),h∈ O(B),z∈Band g =

∗ 0 0 E

∈G we have

hζ^I

g(z) =h(gz) (Eη)^Ij(g,z)^k+|I|, whereE ∈U(r). So for all s ∈]0,∞], f = P

I∈℘(r)f_Iζ^I andh = P

I∈℘(r)h_Iζ^I ∈ O(B)we have

||f||^(k)_s,Γ ≡

s X

I∈℘(r)

f_I²∆ (z,z)^k+|I|

s,Γ\B,∆(z,z)⁻⁽ⁿ⁺¹⁾dVLeb

iffe∈ C^∞(G)⊗ O C^0|r and (f, h)Γ ≡ X

I∈℘(r)

Z

Γ\B

fIhI∆ (z,z)k+|I|−(n+1)

dVLeb

ifD eh,feE

∈L¹(Γ\G), where ’≡’ means equality up to a constant6= 0depending on Γ.

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For the explicit computation of the elements of our spanning set in Theorem4.3 we need the following lemmas:

Lemma 2.4 (Convergence of relative POINCARÉseries). LetΓ₀ @Γbe a subgroup and

f ∈sM_k(Γ₀)∩L¹_k(Γ₀\B). Then

Φ := X

γ∈Γ₀\Γ

f|_γandΦ⁰ := X

γ∈Γ₀\Γ

fe(γ♦) converge absolutely and uniformly on compact subsets ofBresp. G,

Φ∈sM_k(Γ)∩L¹_k(Γ\B), Φ = Φe ⁰, and for allϕ ∈sM_k(Γ)∩L^∞_k (Γ\B)we have

(Φ, ϕ)_Γ = (f, ϕ)_Γ

0.

The symbol ’♦’ here and also later simply stands for the argument of the function.

Sofe(γ♦)∈ C^∞(G)^C⊗V

(C^r)is a short notation for the smooth map G→^

(C^r), g 7→fe(γg).

Proof. Standard, on using the mean value property of holomorphic functions for all k ∈Zwithout any further assumption onk.

Lemma 2.5. LetI ∈℘(r)andk ≥2n+ 1− |I|. Then for allw∈B

∆ (♦,w)^−k−|I|ζ^I ∈ O^|I|(B)∩L¹_k(B),

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and for allf =P

J∈℘(r)f_Jζ^J ∈ O(B)∩L^∞_k (B)we have

∆ (♦,w)^−k−|I|ζ^I, f

≡f_I(w), where( , ) := ( , )_{1}.

Since the proof is also standard, we will omit it here. It can be found in [11].

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3. The Structure of the Group G

We have a canonical embedding

G⁰ :=SU(p, q),→G, g⁰ 7→

g⁰ 0 0 1

, and the canonical projection

G→U(r), g :=

g⁰ 0 0 E

7→E_g :=E induces a group isomorphism

G/G⁰ 'U(r).

Obviously K⁰ = K ∩ G⁰ = S(U(n)×U(1)) is the stabilizer of 0 in G⁰. Let A denote the common standard maximal split abelian subgroup ofGandG⁰ given by the image of the LIEgroup embedding

R,→G⁰, t7→a_t :=





cosht 0

0 1

sinht₁ 0 sinht 0 cosht



. Then the centralizerM ofAinKis the group of all







ε 0 0 u 0

0 ε

0

0 E





 ,

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where ε ∈ U(1), u ∈ U(p−1) and E ∈ U(r) such that ε²detu = detE. Let M⁰ =K⁰ ∩M = G⁰ ∩M be the centralizer ofAinK⁰. The centralizer ofG⁰ inG is precisely

Z_G(G⁰) := ε1 0 0 E

ε∈U(1), E ∈U(r), ε^p+1 = detE

@M, andG⁰ ∩Z_G(G⁰) = Z(G⁰). An easy calculation shows that G = G⁰Z_G(G⁰). So K =K⁰Z_G(G⁰)andM =M⁰Z(G⁰). Therefore if we decompose the adjoint representation ofAas

g=M

α∈Φ

g^α, where for allα∈R

g^α :=

ξ∈g

Ad_a_t(ξ) = e^αt is the corresponding root space and

Φ :={α ∈R|g^α 6= 0}

is the root system, then we see thatΦis 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 g^α @ g⁰ is at the same time the corresponding root space of g⁰, and finally g⁰ =a⊕m=a⊕m⁰ ⊕z_g(g⁰).

Lemma 3.1.

N(A) =AN_K(A) =N(AM)@N(M).

Proof. Simple calculation.

In particular we have the WEYLgroup

W :=M\NK(A) 'M⁰\NK⁰(A) ' {±1}

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acting onA ' Rvia sign change. For the main result, Theorem4.3, of this article the following definition is crucial:

Definition 3.2. Letg₀ ∈G.

(i) g₀ is called loxodromic if and only if there existsg ∈Gsuch thatg₀ ∈gAM g⁻¹. (ii) Ifg₀is loxodromic, it is called regular if and only ifg₀ =ga_twg⁻¹witht∈R\{0}

andw∈M.

(iii) If γ ∈ Γ is regular loxodromic then it is called primitive in Γ if and only if γ =γ^0ν impliesν∈ {±1}for all loxodromicγ⁰ ∈Γandν∈Z.

Clearly for all γ ∈ Γ regular loxodromic there existsγ⁰ ∈ Γ primitive regular loxodromic andν ∈N\ {0}such thatγ =γ^0ν.

Lemma 3.3. Letg₀ ∈ Gbe regular loxodromic,g ∈ G, w ∈ M andt ∈ R\ {0}

such thatg₀ = ga_twg⁻¹. Theng is uniquely determined up to right translation by elements ofAN_K(A), andtis uniquely determined up to sign.

Proof. By straight forward computation or using the following strategy: Letg⁰ ∈G, w⁰ ∈Mandt⁰ ∈Rsuch thatg₀ =g⁰a_t⁰w⁰g⁰⁻¹also. Thena_tw= (g⁻¹g⁰)a_t⁰w⁰(g⁻¹g⁰)⁻¹. Sincet ∈ R\ {0}and because of the root space decomposition,a+mmust be the largest subspace of g on which Adatw is orthogonal with respect to an appropiate scalar product. SoAd_g⁻¹_g⁰ mapsa+minto itself. This impliesg⁻¹g⁰ ∈N(AM) = AN_K(A)by Lemma3.1.

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4. The Main Result

Letρ∈ {0, . . . , r}. AssumeΓ\Gcompact orn ≥2,vol Γ\G <∞andk ≥2n−ρ.

Let C > 0 be given. Let us consider a regular loxodromic γ₀ ∈ Γ. Let g ∈ G, w₀ ∈M andt₀ >0such thatγ₀ =ga_t₀w₀g⁻¹.

There exists a torus T := hγ0i\gAM belonging to γ0. From Lemma 3.3 it follows that T is independent of g up to right translation with an element of the WEYLgroupW =M\N_K(A).

Letf ∈sS_k(Γ). Thenfe∈ C^∞(Γ\G)^C⊗ O C^0|r

. Defineh∈ C^∞(R×M)^C⊗ O C^0|r

as

h(t, w) := fe(ga_tw)

for all(t, w) ∈ R×M ’screening’ the values offeonT. Then clearly h(t, w) = h(t,1, Ewηj(w))j(w)^k, and soh(t, w) = h(t,1, Ewη)j(w)^k+ρiff ∈ sS_k^(ρ)(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 ∈ R^r×r diagonal such thatexp(2πiD) = E₀ andχ ∈ Rsuch that j(w₀) = e^2πiχ. D andχ are uniquely determined byw₀ up toZ. If

D=





d₁ 0

. ..

0 d_r





withd₁, . . . , d_r ∈RandI ∈℘(r), then we definetr_ID:=P

j∈Id_j.

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Theorem 4.1 (FOURIERexpansion ofh).

(i) h(t+t₀, w) =h t, w₀⁻¹w

for all(t, w)∈R×M, and there exist uniqueb_I,m ∈ C,I ∈℘(r),m∈ _t¹

0 (Z−(k+|I|)χ−tr_ID), such that h(t, w) = X

I∈℘(r)

j(w)^k+|I| X

m∈_t¹

0(Z−(k+|I|)χ−tr_ID)

b_I,me^2πimt(E_wη)^I

for all(t, w)∈R×M, where the sum converges uniformly in all derivatives.

(ii) If f ∈ sS_k^(ρ)(Γ),bI,m = 0for allI ∈ ℘(r), |I| = ρ, andm ∈ _t¹

0(Z−(k+ρ)χ

−tr_ID)∩]−C, C[then there exists H ∈ C^∞(R×M)^C⊗V

(C^r)uniformly LIPS-

CHITZcontinuous with a LIPSCHITZconstantC₂ ≥0independent ofγ₀ such that h=∂_tH,

H(t, w) =j(w)^kH(t,1, Ewηj(w)) and

H(t+t₀, w) =H t, w₀⁻¹w for all(t, w)∈R×M.

Proof. (i) Lett ∈Randw∈M. Then

h(t+t₀, w) = fe(ga_t₀a_tw) =f γe ₀gw⁻¹₀ a_tw

=f gae _tw⁻¹₀ w

=h t, w⁻¹₀ w ,

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and so

h(t+t₀,1) =h t, w⁻¹₀

=j(w₀)^−kh t,1, E₀⁻¹ηj(w₀)⁻¹

=j(w₀)^−k X

I∈℘(r)

h(t,1)e^−2πitr^I^Dη^Ij(w₀)^−|I|

= X

I∈℘(r)

e−2πi((k+|I|)χ+trID)h_I(t,1)η^I.

Thereforeh_I(t+t₀,1) = e−2πi((k+|I|)χ+tr_ID)h_I(t,1)for all I ∈ ℘(r), and the rest follows by a standard FOURIERexpansion.

To prove (ii) we need the following lemma:

Lemma 4.2 (Generalization of the reverse BERNSTEINinequality). Lett0 ∈ R\ {0},ν ∈RandC >0. LetSbe the space of all convergent FOURIERseries

s= X

m∈_t¹

0(Z−ν),|m|≥C

sle^2πim♦ ∈ C^∞(R)^C,

for allsm ∈C. Then b:S → S, s= X

m∈_t¹

0(Z−ν),|m|≥C

s_me^2πim♦ 7→bs:= X

m∈_t¹

0(Z−ν),|m|≥C

s_m

2πime^2πim♦

is a well-defined linear map, and||bs||_∞≤ _πC⁶ ||s||_∞for alls∈ S.

Proof. This can be deduced from the ordinary reverse BERNSTEIN inequality, see for example Theorem 8.4 in Chapter I of [9].

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Now we prove Theorem4.1(ii). Fix someI ∈℘(r)such that|I|=ρandb_I,m = 0for allm ∈ _t¹

0 (Z−(k+ρ)χ−tr_ID)∩]−C, C[. Then if we defineν := (k+ρ)χ+ tr_ID∈Rwe have

h_I(♦,1) = X

m∈¹

t0(Z−ν),|m|≥C

b_I,me^2πim♦,

and so we can apply the generalized reverse BERNSTEINinequality, Lemma4.2, to h_I. Therefore we can define

H_I⁰ :=h_I\(♦,1) = X

m∈¹

t0(Z−ν),|m|≥C

b_I,m

2πime^2πim♦ ∈ C^∞(R)^C.

fe

∈ L^∞(G) by SATAKE’s theorem, Theorem 2.3, and so there exists a constant C⁰ >0independent ofγ₀andIsuch that||h_I||_∞< C⁰, and now Lemma4.2tells us that

||H_I⁰||_∞≤ 6

πC ||h(♦,1)||_∞≤ 6C⁰ πC. ClearlyhI(♦,1) =∂tH_I⁰.

Since j is smooth on the compact setM, j^k+ρ(E_wη)^I is uniformly LIPSCHITZ

continuous onM with a common LIPSCHITZ constantC⁰⁰independent of γ₀andI.

So we see thatH ∈ C^∞(R, M)^C⊗V

(C^r)defined as H(t, w) := X

I∈℘(r)

j(w)^k+ρH_I⁰(t) (E_wη)^I

for all(t, w)∈R×M is uniformly LIPSCHITZcontinuous with LIPSCHITZconstant C₂ := ^6C_πC⁰⁰ + 1

C⁰independent ofγ₀, and the rest is trivial.

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LetI ∈℘(r)andm ∈ _t¹

0 (Z−(k+|I|)χ−tr_ID). SincesS_k(Γ)is a HILBERT

space and sS_k(Γ) → C, f 7→ b_I,m is linear and continuous, there exists exactly one ϕ_γ₀_,I,m ∈ sS_k(Γ) such that b_I,m = (ϕ_γ₀_,I,m, f) for all f ∈ sS_k(Γ). Clearly ϕ_γ₀_,I,m ∈sS_k^(|I|)(Γ).

For the remainder of the article for simplicity we writem ∈]−C, C[instead of m ∈ _t¹

0 (Z−(k+|I|)χ−trID)∩]− C, C[. In the last section we will compute ϕγ0,I,mas a relative POINCARÉseries. One can check that the family

{ϕ_γ₀_,I,m}I∈℘(r),|I|=ρ,m∈]−C,C[

is independent of the choice ofg,Dandχup to multiplication with a unitary matrix with entries inCand invariant under conjugatingγ₀ with elements ofΓ.

Now we can state our main theorem: LetΩbe a fundamental set for all primitive regular loxodromicγ0 ∈Γmodulo conjugation by elements ofΓand

Ze:=n

m∈Z_G(G⁰)

∃g⁰ ∈G⁰ :mg⁰ ∈Γo

@Z_G(G⁰). Then clearlyΓ@G⁰Ze. Recall that we still assume

• Γ\Gcompact or

• n ≥2,vol Γ\G < ∞andk ≥2n−ρ.

Theorem 4.3 (Spanning set forsS_k(Γ)). Assume that the right translation ofAon Γ\G⁰Zeis topologically transitive. Then

{ϕ_γ₀_,I,m|γ₀ ∈Ω, I ∈℘(r),|I|=ρ, m∈]−C, C[}

is a spanning set forsS_k^(ρ)(Γ).

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For proving this result we need an ANOSOV type theorem for G and the unbounded realization ofB, which we will discuss in the following two sections.

Remark 1.

(i) If there is some subgroupMf@Z_G(G⁰)such thatΓ@G⁰Mfand the right translation ofAonΓ\G⁰Mfis topologically transitive then necessarilyM Z(Gf ⁰) =Ze and there existsg₀ ∈G⁰ such thatG⁰Ze= Γg₀A. The latter statement is a trivial consequence of the fact thatZe@M.

(ii) In the case whereΓ∩G⁰ @ Γis of finite index or equivalentlyZeis finite then we know that the right translation of A on Γ\G⁰Ze is topologically transitive because of MOORE’s ergodicity theorem, see [13] Theorem 2.2.6, and since thenΓ∩G⁰ @G⁰ is a lattice.

(iii) There is a finite-to-one correspondence betweenΩand the set of closed geodesics ofΓ\Bassigning to each primitive loxodromic element

γ₀ = ga_t₀w₀g⁻¹ ∈ Γ, g ∈ G, t₀ > 0 andw₀ ∈ M, the image of the unique geodesicgA0ofBnormalized byγ₀under the canonical projectionB →Γ\B.

It is of lengtht₀ if there is no irregular point ofΓ\B ongA0.

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5. An A

NOSOV

Type Result for the Group G

On the LIE groupGwe have a smooth flow(ϕ_t)_t∈

Rgiven by the right translation by elements ofA:

ϕ_t :G→G, g 7→ga_t.

This turns out to be partially hyperbolic, and so we can apply a partial ANOSOVclos- ing lemma. Let me mention that the flow(ϕt)_t∈_Rdescends to the ordinary geodesic flow on the unit tangent bundleSB ' G/M. 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). LetC > 1.

(i) Let ϕ be aC^∞-diffeomorphism of W. Then ϕis called partially hyperbolic with constant C if and only if there exists an orthogonal Dϕ (and therefore Dϕ⁻¹ ) - invariantC^∞-splitting

(5.1) T W =T⁰ ⊕T⁺⊕T⁻

of the tangent bundleT W such thatT⁰⊕T⁺,T⁰⊕T⁻,T⁰,T⁺ andT⁻are closed under the commutator,Dϕ|_T⁰ is an isometry,||Dϕ|_T⁻|| ≤ _C¹ and||Dϕ⁻¹|_T⁺|| ≤ _C¹. (ii) Let(ϕ_t)_t∈

Rbe aC^∞-flow onW. Then(ϕ_t)_t∈

Ris called partially hyperbolic with constantCif and only if allϕ_t,t >0are partially hyperbolic diffeomorphisms with a common splitting (5.1) and constantse^Ct resp. andT⁰ contains the generator of the flow.

A partially hyperbolic diffeomorphismϕgives rise toC^∞-foliations onW corresponding to the splittingT W =T⁰⊕T⁺⊕T⁻. Let us denote the distances along theT⁰⊕T⁺-,T⁰-,T⁺- respectivelyT⁻-leaves byd^0,+,d⁰,d⁺andd⁻.

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Definition 5.2. Let T W = T⁰ ⊕T⁺⊕ T⁻ be an orthogonal C^∞-splitting of the tangent bundleT W ofW such thatT⁰⊕T⁺,T⁰,T⁺andT⁻are closed under the commutator, C⁰ ≥ 1 andU ⊂ W. U is called C⁰-rectangular (with respect to the splittingT W =T⁰⊕T⁺⊕T⁻) if and only if for ally, z ∈U

{i} there exists a unique intersection pointa ∈ U of theT⁰⊕T⁺-leaf containing y and theT⁻-leaf containing z and a unique intersection point b ∈ U of the T⁰⊕T⁺-leaf containingzand theT⁻-leaf containingy,

d^0,+(y, a), d⁻(y, b), d⁻(z, a), d^0,+(z, b)≤C⁰d(y, z), and

1

C⁰d^0,+(z, b)≤d^0,+(y, a)≤C⁰d^0,+(z, b), 1

C⁰d⁻(z, a)≤d⁻(y, b)≤C⁰d⁻(z, a).

{ii} ify andz belong to the sameT⁰ ⊕T⁺-leaf there exists a unique intersection point c ∈ U of the T⁰-leaf containingy and theT⁺-leaf containingz and a unique intersection point d ∈ U of theT⁰-leaf containing z and theT⁺-leaf containingy,

d⁰(y, c), d⁺(y, d), d⁺(z, c), d⁰(z, d)≤C⁰d^0,+(y, z), and

1

C⁰d⁰(z, d)≤d⁰(y, c)≤C⁰d⁰(z, d), 1

C⁰d⁺(z, c)≤d⁺(y, d)≤C⁰d⁺(z, c).

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Since the splittingT W = T⁰⊕T⁺⊕T⁻ is orthogonal and smooth we see that for allx∈W andC⁰ >1there exists aC⁰-rectangular neighbourhood ofx.

Theorem 5.3 (Partial ANOSOV closing lemma). Let ϕ be a partially hyperbolic diffeomorphism with constantC, letx ∈ W,C⁰ ∈]1, C[andδ > 0such thatU_δ(x) is contained in aC⁰-rectangular subsetU ⊂W.

Ifd(x, ϕ(x))≤δ¹⁻

C0 C

C⁰²+1 then there existy, z ∈U such that (i) xandybelong to the sameT⁻-leaf and

d⁻(x, y)≤ C⁰

1− ^C_C⁰d(x, ϕ(x)),

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(ii) yandϕ(y)belong to the sameT⁰⊕T⁺-leaf and d^0,+(y, ϕ(y))≤C⁰²d(x, ϕ(x)), (iii) yandz belong to the sameT⁺-leaf and

d⁺(ϕ(y), ϕ(z))≤ C⁰³

1−^C_C⁰d(x, ϕ(x)), (iv) z andϕ(z)belong to the sameT⁰-leaf and

d⁰(z, ϕ(z))≤ C⁰⁴d(x, ϕ(x)).

The proof, which will not be given here, uses a standard argument obtaining the pointsyandϕ(z)as limits of certain CAUCHYsequences. The interested reader will find it in [11].

Now let us return to the flow(ϕt)_t∈_RonGand choose a left invariant metric onG such thatg^α,α∈Φ\{0},aandmare pairwise orthogonal and the isomorphismR' A ⊂ Gis isometric. Then since the flow(ϕ_t)_t∈

Rcommutes with left translations it is indeed partially hyperbolic with constant1and the unique left invariant splitting ofT Ggiven by

T₁G=g= a⊕m

| {z }

T₁⁰:=

⊕ M

α∈Φ,α>0

g^α

| {z }

T₁⁻:=

⊕ M

α∈Φ,α<0

g^α

| {z }

T₁⁺:=

.

For allL⊂Gcompact,T, ε > 0define ML,T :=

gatg⁻¹

g ∈L, t∈[−T, T]

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and

N_L,T,ε :={g ∈G|dist (g, M_L,T)≤ε}.

Lemma 5.4. For allL⊂Gcompact there existT₀, ε₀ >0such thatΓ∩N_L,T₀_,ε₀ = {1}.

Proof. Let L ⊂ G be compact and T > 0. Then M_L,T is compact, and so there existsε > 0 such that N_L,T,ε is again compact. Since Γ is discrete, Γ∩N_L,T,ε is finite. Clearly for allT, T⁰, εandε⁰ >0ifT ≤T⁰ andε≤ε⁰ thenN_L,T,ε ⊂N_L,T⁰_,ε⁰, and finally

\

T,ε>0

N_T,ε={1}.

Here now is the quintessence of this section:

Theorem 5.5.

(i) For allT₁ > 0there existC₁ ≥1andε₁ >0such that for allx ∈G, γ ∈ Γand T ≥T₁if

ε:=d(γx, xa_T)≤ε₁

then there existz ∈G,w∈M andt₀ >0such thatγz =za_t₀w(and soγis regular loxodromic),d((t₀, w),(T,1))≤C₁εand for allτ ∈[0, T]

d(xa_τ, za_τ)≤C₁ε e^−τ +e^−(T^−τ) .

(ii) For allL ⊂ G compact there existsε₂ > 0such that for all x ∈ L, γ ∈ Γ and T ∈[0, T₀],T₀ >0given by Lemma5.4, if

ε:=d(γx, xa_T)≤ε₂ thenγ = 1andT ≤ε.

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Proof. (i) LetT₁ >0and define

C₁ := max e³²^T¹ 1−e⁻^T²¹

, e^2T¹

!

≥1.

DefineC⁰ :=e^T²¹, letU be aC⁰-rectangular neighbourhood of1∈ Gand letδ > 0 such thatU_δ(1) ⊂U. Then by the left invariance of the splitting and the metric onG we see thatgU is aC⁰-rectangular neighbourhood ofg andU_δ(g) = gU_δ(1) ⊂ gU for allg ∈G. Define

ε₁ := min δ1−e⁻^T²¹ e^T¹ + 1 , T₁

C₁

!

>0.

Now assumeγ ∈ΓandT ≥T₁such that

ε :=d(γx, xa_T_v)≤ε₁.

Then ϕ : G → G, g 7→ γ⁻¹gaT is a partially hyperbolic diffeomorphism with constante^T¹ >1and the corresponding splittingT G=T⁰⊕T⁺⊕T⁻. Then since

ε≤δ1−e⁻^T²¹

e^T¹ + 1 =δ1−C⁰e^−T¹ C⁰²+ 1

the partial ANOSOV closing lemma, Theorem 5.3, tells us that there existy, z ∈ G such that

(i) xandybelong to the sameT⁻-leaf and d⁻(x, y)≤ε C⁰

1− ^C_C⁰,

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(iii) yandz belong to the sameT⁺-leaf and

d⁺(yaTv, zaTv)≤ε C⁰³ 1− ^C_C⁰, (iv) γz andza_T_v belong to the sameT⁰-leaf and

d⁰(γz, za_T_v)≤εC⁰⁴.

In (iii) and (iv) we already used that the metric and the flow are left invariant. So by (iv) and since theT⁰-leaf containingza_T iszAM, there existw∈M andt₀ ∈R such thatγz =za_t₀w. So

d⁰(a_t₀−Tw,1)≤εC⁰⁴, and so, sinceAM 'R×M isometrically, we see that

d((t₀, w),(T,1)) ≤εC⁰⁴ =εe^2T¹ ≤εC₁. In particular,|t₀−T| ≤T₁, and sot₀ >0.

Now let τ ∈ [0, T]. Then sincexandy belong to the sameT⁻-leaf, the same is true forxa_τ andya_τ, and

d⁻(xa_τ, ya_τ)≤d⁻(x, y)e^−τ ≤ε C⁰

1−^C_C⁰e^−τ ≤εC₁e^−τ.

Sinceyandz belong to the sameT⁺-leaf, the same is true forya_τ andza_τ, and d⁺(ya_τ, za_τ)≤d⁺(ya_T, za_T)e^−(T^−τ)

≤ε C⁰³

1− ^C_C⁰e^−(T^−τ) ≤εC₁e^−(T^−τ).