Higher order Maass forms

We determine the size of spaces of higher order Maass forms of even weight for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth at the cusps is allowed, the spaces of Maass forms of a given order are as large as algebraic constrictions allow. We show the analogous statement for the spaces of holomorphic forms. For functions on the universal covering group of PSL(2,R) we introduce the concept of generalized weight. For the resulting spaces of higher order Maass forms with even generalized weight we show that the size is maximal.


Introduction
In this work, the structure of the space of Maass forms of general order and integral weight as a linear vector space is determined. It is proved that, under suitable conditions, this space is as large as one would expect it to be.
There are mainly two objects and associated problems that suggest the study of specifically this type of higher-order form. The first is Eisenstein series modified with modular symbols defined by where Γ ∞ is the subgroup of translations of the congruence group Γ 0 (N), f a weight 2 newform and f, γ := −2πi γ ∞ ∞ f (w)dw. The study of this function has led to important results, such as the proof that the suitably normalised modular symbols follow the normal distribution ( [19]). The function E * (−, s) is not automorphic but transforms as a second-order automorphic form.
Here, the action | of Γ on functions on H is given by f |γ(z) := f (γz) .
and it is extended linearly to an action of the group ring C[Γ].
Clearly, several types of conditions on holomorphicity, growth etc. can be imposed on functions of general order. The function E * (−, s) in particular, is an eigenfunction of the Laplacian and therefore we view it as a Maass form of order 2.
The second object leading to functions that are Γ-invariant of second-order arises from considerations related to values of derivatives of L-functions of cusp forms: In [11] and [8] certain "period integrals" are associated to derivatives of L-functions of weight 2 cusp forms in a way analogous to the link between values of L-functions and modular integrals ( [17]). Specifically, let f be a newform of weight 2 for Γ 0 (N) and let L f (s) be its L-function. If L f (1) = 0, then, for each prime p, (p, N) = 1, L ′ f (1) can be written as a linear combination of integrals of the form γ ∈ Γ 0 (N) plus some "lower order terms". Here u(z) := log η(z) + log η(Nz), where η is the Dedekind η-function. The differential f (z)u(z) dz is not Γ 0 (N)-invariant. It does satisfy a transformation law which is reminiscent of (1.2), but is not quite Γ 0 (N)-invariant of order 2 in the narrow sense. If it were, the value of the derivative at 1 would be expressed as the value of the actual L-function of second-order Γ 0 (N) at 1. That could be advantageous for the study of L ′ f (1) in terms of the outstanding conjectures, especially since there is now evidence that a motivic structure underlies higher order forms (see [10] and [22]).
Here we show that it is indeed possible to obtain a second-order Γ 0 (N)-invariant function from u(z) provided we move to a different domain. This domain is the universal covering group which we will be defining in detail in §5.1.
As will become apparent in the sequel, it is natural, in higher orders, to unify the study of Maass forms and that of forms on universal covering groups. The full definition of the higher-order Maass forms with generalised weight on the universal covering group is discussed in §6. Theorem 6.4 then allows us to translate results on the universal covering group to the analogous results on the upper-half plane.
A fundamental question is how "large" this space is. In the case of holomorphic higher-order cusp forms, the corresponding spaces are finite-dimensional and the answer can be given by computing the dimensions ( [7] and [9]). In the present case, where the relevant space is not finite dimensional, a different characterisation of "size" is required. Such a characterisation is proposed in §3.
Although our results imply that there are "many" higher order Maass forms, the proofs are highly inductive and do not easily lead to explicit examples. In §4.3 and §6.4 we address this problem, by illustrating various methods that lead to explicit examples of higher order Maass. Surprisingly, these examples are derived very naturally from the theory which was developed in a completely different context in [2,3].
Finally, a particular aspect of the proof that deserves to be singled out because of its independent interest is the definition of genuinely higher-order Fourier expansions. Higher order automorphic forms need not be invariant under the group fixing a cusp, so there is no obvious Fourier expansion. To date, to address this problem one had to partially revert to the classical setting by imposing the somewhat unnatural extra condition of invariance under the parabolic elements of the group. In §7, appropriate higher-order Fourier terms are constructed, thus avoiding additional invariance conditions.

Structure of the paper
In §3 we first discuss higher-order invariants for general groups and modules. This allows a precise definition of the concept of "as large as possible" (maximally perturbable). A first maximal perturbability result for a general space of maps is also proved.
In §4, Maass forms on H (both general and holomorphic) are defined and the first two main theorems of the paper (4.2 and 4.3) are stated. The section includes an extended discussion of concrete examples of low-order forms on H.
In the next section the universal covering groupG is introduced and the basic facts aboutG are given. Maass forms on the universal covering group are defined in §6 and the counterparts of Theorems 4.2 and 4.3 for forms on the universal covering group are stated. The section concludes with concrete examples of low-order forms onG.
Section 7 is of independent interest. A theory of Fourier expansions for higher-order forms is developed. The proof of Theorems 4.2 and 4.3 is the content of §8. The proof involves the construction of two spaces with support conditions. To deduce their maximal perturbability we employ spectral techniques.

Higher order invariants
In this section, we discuss higher order invariants in general and then specialise their study to discrete cofinite subgroups Γ ⊂ PSL 2 (R). We introduce the concept of a "maximally perturbable" Γ-module to make precise the statement that there are as many higher order invariants of a given type as one can expect. A first maximal perturbability result in a general context is proved.
3.1. Higher order invariants on general groups. The concept of higher order invariant functions on the upper half plane is a special case of the concept "higher order invariants" for any group Γ and any Γ-module V. We work with right Γ-modules, an write the action as v → v|γ. It should be clear from the context when we refer to this general meaning of | and when to the more narrow meaning given in the Introduction. We define the higher order invariants inductively: We set V Γ,0 = {0}. Let now Γ be finitely generated and let I be the augmentation ideal in the group ring C[Γ], generated by γ − 1 with γ ∈ Γ. A fundamental role in the paper will be played by the map To define it we first quote from [5] (before Proposition 1.2): Next, we note that I q+1 \I q is generated by I q+1 + (γ 1 − 1) · · · (γ q − 1) , with γ i ∈ Γ. To each v ∈ V Γ,q+1 we associate the map on I q+1 \I q sending this element to v|(γ 1 −1) · · · (γ q −1). This map is well-defined because v|(γ 1 − 1) · · · (γ q+1 − 1) = 0. In this way, we obtain a map m q from V Γ,q+1 to hom C[Γ] (I q+1 \I q , V) hom C[Γ] (I q+1 \I q , V Γ ) (since the action induced on I q+1 \I q by the operation of Γ is trivial). It is easy to see that the kernel of m q is V Γ,q and thus we obtain the exact sequence The map m q may or may not be surjective and we will interpret the phrase "as large as possible" as surjectivity of m q for all q ∈ N.
Definition 3.1. Let Γ be a finitely generated group. We will call a Γ-module V maximally perturbable if the linear map m q : V Γ,q+1 → hom C[Γ] (I q+1 \I q , V Γ ) is surjective for all q ≥ 1.
A reformulation of this definition which is occasionally easier to use, uses the finite dimension (3.4) n(Γ, q) := dim C (I q+1 \I q ).
In [9] higher order cusps forms of weight k for a discrete group Γ are considered in the space of holomorphic functions on H with exponential decay at the cusps that moreover are invariant under the parabolic transformations. The dimensions of these spaces are computed and generally turn out to be strictly smaller than n(Γ, q). So the corresponding Γ-module is not maximally perturbable.
We call a perturbation commutative if µ f is invariant under all permutations of its arguments. If not, we call it non-commutative.

Canonical generators.
In this section we recall the "canonical generators" of cofinite discrete subgroups of PSL 2 (R), and use them to show that certain modules are maximally perturbable.
Let Γ ⊂ PSL 2 (R) be a cofinite discrete group of motions in the upper half-plane H. A system of canonical generators for Γ consists of • Parabolic generators P 1 , . . . , P n par , each conjugate in PSL 2 (R) to ± 1 0 1 1 . We shall assume that Γ has cusps: n par ≥ 1.
For the modular group we have n par = 1, In the sequel, we will need a basis for I q+1 \I q . Arguing as in Lemma 2.1 in [5] we can deduce that the elements By |i| we denote the length of the tuple i. We next set g i (γ) = g i ϕ 0 (γ) for γ ∈ Γ, where ϕ 0 is the homomorphism defined by ϕ 0 (E j ) = I, ϕ 0 (A j ) = A j . With the map m q in (3.3) and for for |i| = |j| we have on the basis elements in (3.8), where i ′ is the tuple (i(2), . . . , i(q)). Inductively we obtain Hence the g i with |i| = q form a dual system for the generators b(i). This implies that the image m q V Γ,q+1 has maximal dimension n(Γ, q).

Maass forms
We turn to spaces of functions on the upper half-plane that contain the classical holomorphic automorphic forms and the more general Maass forms. The first main results of this paper are stated in Theorems 4.2 and 4.3. In §4.3 we give some explicit examples of higher order Maass forms.

General Maass forms.
Let Γ be a cofinite discrete subgroup Γ of the group G = PSL 2 (R). For each cusp κ, we choose g κ ∈ PSL 2 (R) such that Here, Γ κ is the set of elements of Γ fixing κ. The elements g κ are determined up to right multiplication by elements ± a 0 b a −1 ∈ G. We choose the g κ for cusps in the same Γ-orbit so that g γκ ∈ γg κ Γ ∞ . We further consider a generalisation of the action | considered in the last section. For a fixed k and for a f : H → C we set We finally set With this notation we have uniformly for x in compact sets in R, for all cusps κ of Γ .
ii) E k (Γ, λ) denotes the space of smooth functions f such that L k f = λ f and for which there is some a ∈ R such that uniformly for x in compact sets in R, for all cusps κ of Γ.
iii) We denote the invariants in these spaces by We call the elements of E k (Γ, λ) (resp. M k (Γ, λ)) Maass forms of polynomial (resp. exponential) growth of weight k and eigenvalue λ ∈ C for Γ.

Remarks.
i) Since L k is elliptic, all its eigenfunctions are automatically real-analytic. (See, e.g., [14], §5 of App. A4, and the references therein.) If f is holomorphic, then it is an eigenfunction of L k with eigenvalue k 2 1 − k 2 . ii) The space M k (Γ, λ) is known to have finite dimension. The space E k (Γ, λ) has, for groups Γ with cusps, infinite dimension. The subspace of E k (Γ, λ) corresponding to a fixed value of a in the bound O(e ay ) has finite dimension. iii) In an alternative definition, suitable for functions not necessarily holomorphic, one replaces the Maass forms f as defined above by h(z) = y k/2 f (z). Then invariance under (4.2) becomes invariance under the action and the eigenproperty in the terms of the Laplacian The formulation of the growth conditions remains unchanged. Now antiholomorphic automorphic forms a(z) of weight k give Maass forms h(z) = y k/2 a(z) of weight −k. Our main result for general Maass forms on H is Theorem 4.2. Let Γ be a cofinite discrete group of motions in H with cusps. Then the Γ-module E k (Γ, λ) is maximally perturbable for each k ∈ 2Z and each λ ∈ C.
In the course of the proof in §8 we will see that even if we start with Maass forms with polynomial growth the construction of higher order invariants will lead us to functions that have exponential growth.

Holomorphic automorphic forms. For even
where the condition L k f = λ k f is replaced by the stronger condition that f is holomorphic. In the alternative definition, condition (4.8) is replaced by the condition that is the usual space of entire weight k automorphic forms for Γ, and E hol k (Γ, λ k ) Γ is the space of meromorphic automorphic forms with singularities only at cusps. Sometimes, e.g. in [1], the elements of E hol k (Γ, λ k ) Γ are called weakly holomorphic. There the elements of E k (Γ, λ k ) Γ are called harmonic weak Maass forms. We prefer to use the term harmonic for Maass forms in E k (Γ, 0) Γ . (Note that λ k 0 for k 0, 2.) Our main result for holomorphic automorphic forms on H is: For the modular group Γ mod = PSL 2 (Z) the space hom(Γ mod , C) is zero. Hence it does not accept higher order invariants. For the commutator subgroup Γ com = [Γ mod , Γ mod ] we will employ three different approaches to exhibit full sets of perturbations of 1 (as defined in Definition 3.2) of orders two and three. A reader only interested in the existence of higher order forms may prefer to skip this subsection. 4.3.1. Holomorphic perturbation of 1. In [15], Chap. XI, §3E, p. 362, one finds various facts concerning Γ com . It is freely generated by D = ± 2 1 1 1 and C = ± 2 −1 −1 1 . It has no elliptic elements, and one cuspidal orbit Γ com ∞ = P 1 Q . The group (Γ com ) ∞ fixing ∞ is generated by ± 1 0 6 1 . We have t(Γ com ) = 3. The space of holomorphic cusp forms of weight 2 has dimension g = 1. We use the basis element η 4 (power of the Dedekind eta-function). The map induces an embedding of Γ com \H into an elliptic curve, which can be described as C/Λ, with (4.10) .) The map H maps H onto C Λ, and satisfies for γ ∈ Γ com (4.11) where λ(C) = ρ̟ and λ(D) =ρ̟. So the lattice Λ is the image of λ : Γ com → C, and hom(Γ com , C) = Mult 1 (Γ com , C) has λ,λ as a basis. We note that the kernel ker(λ) is a subgroup with infinite index in Γ com ; it is in fact the commutator subgroup of Γ com . The element ± 1 0 6 1 generating the subgroup of Γ com fixing ∞ is in ker(λ). Since ker(λ) has no elliptic elements, composition with H gives a bijection from the holomorphic functions on C Λ to the holomorphic ker(λ)-invariant functions on H.
Clearly, H is a holomorphic second order perturbation of 1 with linear form λ. It is also a harmonic perturbation of 1, i.e., a perturbation which is harmonic as a function. By conjugation we obtain the antiholomorphic harmonic perturbation of 1 with linear formλ.
According to Theorem 4.3 there should also be a holomorphic second order perturbation of 1 with a linear form that is linearly independent of λ. Here we can use the Weierstrass zeta-function (4.12) ζ(u; See, e.g., [13], Chap. I, §6. It is holomorphic on C Λ and satisfies ζ(u + ω; Λ) = ζ(u; Λ) + h(ω) for all ω ∈ Λ, where h ∈ hom(Λ, C) is linearly independent of ω → ω. (The classical notation for h is η. We write h to avoid confusion with the Dedekind eta function.) Pulling back this zeta-function to H we get a second order holomorphic perturbation of 1 with the linear form γ → h λ(γ) . The Laurent expansion of the Weierstrass zeta-function at 0 starts with ζ(u; Λ) = u −1 + O(u 3 ). Hence W has a Fourier expansion at ∞ starting with (4.14) This shows that W has exponential growth at the cusps.
We may carry this out also for holomorphic forms of order three, to obtain the following commutative perturbations of 1 of order 3: We know that there also exist non-commutative holomorphic perturbations of order 3. To find an explicit example, we have to work on H, since the group Λ acting on C is abelian.
The closed holomorphic 1-forms on H transform as follows under Γ com : For an arbitrary base point z 0 ∈ H we put (4.17) This defines a holomorphic function on H that satisfies for γ ∈ Γ com : and hence for γ, δ ∈ Γ com : Thus, we have a holomorphic third order non-commutative perturbation K of 1 with non-symmetric multilinear form (h • λ) ⊗ λ. Since holomorphic forms are harmonic in weight zero these perturbations are also harmonic perturbations of 1.

Iterated integrals.
The construction of the third order form K in (4.17) is closely related to the iterated integrals used in [10] to prove maximal perturbability of spaces of smooth functions. The idea is that we have two closed Γ com -invariant differential forms on H, dH(z) = ω = −2πi η(z) 4 dz, and depends only on z 0 and z 1 , not on the actual path. For a fixed base point z 0 the holomorphic function z 1 → W(z 0 ) H(z 1 ) − H(z 0 ) is invariant of order two. So up to lower order terms the invariant K is given by an iterated integral, as in (3) of [10]; see also [4].

Differentiation of families.
We start by considering a general finitely generated group Γ acting on a space X. We will use the notation f |γ(x) = f (γx) for the action induced on functions defined on X. We consider a family of characters of Γ of the form χ r (γ) = e ir·α(γ) , where r · α(γ) = r 1 α 1 (γ) + · · · + r n α n (γ), α 1 , . . . , α n ∈ hom(Γ, R), r varying over an open set U in R n . In this way χ r is a family of unitary characters.
We consider a C ∞ family r → f r on a neighborhood U ⊂ R n of 0 of functions X → C that satisfy We assume that χ 0 is the trivial character and that f 0 is a Γ-invariant function f . We now set h(x) = ∂ r j f r (x) r=0 , for one of the coordinates of r. The transformation behaviour gives h(γx) = iα j (γ) f (x) + h(x), or, rewritten, The function h is a second order perturbation of f , with iα j as the corresponding element of hom(Γ, C). This can be generalised: We use the notations ∂ a r = ∂ a 1 r 1 · · · ∂ a n r n and |a| = a 1 + a 2 + · · · + a n .
Proof. We use induction on the length |a| of the multi-index. The case |a| = 1 has already been handled above. For |a| > 1 we have where b runs over the multi-indices with 0 ≤ b j ≤ a j , where b a = j a j b j , and where α(γ) c = j α j (γ) c j . Hence is a linear combination of higher order forms f (b) of orders 1, . . . , |a|. So f (a) is an invariant of order at most 1 + |a|. Furthermore .
For the commutativity of the perturbation we note by induction that, for all g 1 , . . . , g s ∈ Γ where the i j run through the set {1, . . . , s}. Application of (4.19) leads to Since α is a homomorphism, the factor α(γ i 1 γ i 2 · · · γ i l ) does not depend on the order of the γ i j . Hence we may rewrite the expression as follows.
where i in the sum i runs over the subsets of {1, . . . , |a|} with l elements. This is an expression that is invariant under permutations of the γ j , which shows that f (a) is a commutative perturbation.
Remark. Proposition 4.4 shows that commutative perturbations can arise as infinitesimal perturbations of a family of automorphic forms. That is our motivation to use the word perturbation in Definition 3.2.
Application to harmonic perturbations of 1. We use the method of differentiation of families to produce explicit harmonic higher order forms for Γ com of order 3. We employ families studied in [3].
Since Γ com is free on the generators C = ± 2 −1 −1 1 and D = ± 2 1 1 1 , the character group of Γ com is isomorphic to C * × C * . We can parametrise the characters by where (v, w) runs through C 2 , and where λ ∈ hom(Γ com , C) is as defined in (4.11). We are interested only in (v, w) in a neighborhood of 0 ∈ C 2 . In [3], §15.5 it is shown that there is a meromorphic Eisenstein family E(v, w, s) of automorphic forms for Γ com , with the character χ v,w and eigenvalue 1 4 − s 2 for ω 0 = −y 2 ∂ 2 x + ∂ 2 y . (In [3] the discussion of the family E is made in the context of families of automorphic forms of varying weight which are thus defined on the covering groupΓ com . However, in §15.5 the weight is zero, and the automorphic forms are, in effect, on the discrete group Γ com .) The restriction to s = 1 2 exists ( [3], §15.6) and forms a meromorphic , and L 0 f (v, w; z) = 0 for the dense set of (v, w) at which f is holomorphic. There is a meromorphic family . Chapter 15 of [3] gives a complicated but explicit construction (obtained with the help of D.Zagier) of such a family h with Jacobi theta-functions. Specifically, in §15.6.11 the function h is expressed as a sum where the function G µ (u, w), for µ Z and 0 < Im u < 1 2 ̟ √ 3 is given by with q = −e −π √ 3 , ξ = e 2πiu/̟ , and η = e −w̟ √ 3 We consider this for u, w, and µ near zero, but not equal to zero. Hence η ≈ 1 but η 1, and |q| < |ξ| < 1. The latter inequalities imply absolute convergence of the series. We shall derive the Taylor expansion ofh(v, w; u) := vw h(v, w; u) in terms of (v, w) near zero up to order two, from which we can obtain higher order forms by Proposition 4.4.
The term of G µ (u, w) with m = 0 has singularities at µ = 0, and, due to 1 η−1 , also at w = 0. This term has the following contribution to h(v, w; u) in (4.22).
We write the corresponding contribution toh(v, w; u) = vw h(v, w; u) as follows.
The last three quotients are holomorphic as a function of v + w in a neighborhood of 0. We replace them by their Taylor expansion up to the term (v + w) 2 and after that the Taylor expansion in both v and w up to order 2 is computed. This gives In the terms with m 0 in (4.23) we write ξ = e 2πiu/̟ ,ξ = e −2πiū/̟ , 3 , and µ = (v + w)̟/2π. We find the following contribution toh(v, w; u): This contribution is holomorphic near v = w = 0. Its expansion starts with the term vw. So for third order forms we need only the contribution to h(0, 0; u): Each of these terms gives a convergent series on the region 0 < Im u < 1 2 ̟ √ 3.
Commutative perturbations. In this expansion we find various higher order harmonic forms that we have seen above. Denoting f = −2π we find: The coefficient of vw gives a third order form By B 1,1 (z) = b 1,1 H(z) we denote the corresponding harmonic third order perturbation of 1 on H. The way B 1,1 has been derived, together with the proof of Proposition 4.4, ensures that it is a perturbation of 1 with a multilinear form that is a multiple of λ ⊗λ +λ ⊗ λ. However, b 1,1 (u) is represented by (4.29) only on the region 0 < Im u where F mod is the standard fundamental domain of the modular group) is shown to be the regular hexagon with centre 0 and one corner at − 1 3 (e πi/3 + 1)̟. Only the upper half of this hexagon is in the region where we have an expression for b 1,1 . We shall continue this function to the entire C.
We first note that the series in (4.29) defining S (u) converges absolutely for Im u > 0 yielding a holomorphic function in that region. To extend S (u) to other values we use the following identity, valid for Im u > 1 2 ̟ √ 3: Via this identity, we can define S (u) in the region Im u > − 1 2 ̟ √ 3. This extension of S is multivalued, since it depends on the way in which we extend the function u → log 1 − e 2πiu/̟ , which is given by the second series in (4.30) only for Im u > 0. However, the sum with logarithmic singularities at u = ̟n, n ∈ Z. Applying (4.31) repeatedly, we can extend S (u) + S (−ū) to all C to obtain a harmonic function with singularities at the points in Λ = ̟ Z[ρ] which have non-positive imaginary part.
Via (4.29), we then obtain the continuation of the function b 1,1 . It is harmonic on C Λ, with logarithmic singularities at all points of Λ.
Let us explicitly check the transformation behaviour: Since S is periodic with period ̟ (and, equiva- Let us denote by T ω the translation by ω ∈ Λ, and use the notations b 1, and a = 2π Since Λ is commutative we need not consider b 1,1 |(T ρ̟ − 1)(T ̟ − 1). We conclude that the pull-back •H is a harmonic commutative perturbation of 1 for the multilinear form µ determined by the following values at the generators C and CD of Γ com : We have used the values of λ given below (4.11).) With these values at the generators, µ coincides with λ ⊗λ +λ ⊗ λ as predicted above by the way B 1,1 was constructed.
Non-commutative perturbation. Proposition 4.4 shows that differentiation of families produces only commutative perturbations. However, by Theorem 4.2, there are non-commutative third order harmonic perturbations of 1. We can obtain such perturbations from B 1,1 upon decomposing it as B 1,1 = A + B for a holomorphic function A and an anti-holomorphic function B. Specifically, in view of (4.29), for those z ∈ H for which H(z) is in the upper half of the fundamental hexagon for C/Λ, we can set (4.34) gives an equality between a holomorphic and an antiholomorphic function, and therefore, there is ν : for all γ, δ ∈ Γ. This implies that A and B are third order invariants, and that ν ∈ Mult 2 (Γ, C).
To determine the bilinear form ν, we recall that λ(C) = ρ̟ and λ(D) =ρ̟ = (1 − ρ)̟. We consider the following four functions: The functions on the left are holomorphic, and those on the right are antiholomorphic. We consider the sum of the two functions on the first row, and denote u = H(z). With (4.32): Similarly the sum of the two functions on the second row gives The sums of the rows in (4.35) are zero, so the individual functions are constant. We do not try to determine these constants.
For A we have We conclude that − f −1 A is a non-commutative holomorphic third order holomorphic perturbation of 1 with multilinear formλ ⊗ λ. Then the multilinear form of the anticommutative third order perturbation of 1 given by

Universal covering group
5.1. Universal covering group of SL 2 (R). To define the universal covering group of SL 2 (R), which is also the universal covering group of G = PSL 2 (R), we first note that, as an analytic variety, SL 2 (R) is isomorphic to H × R/2πZ , by the Iwasawa decomposition expressing each element of SL 2 (R) uniquely as a product √ y 0 from H × R to itself, where we choose the argument such that −π < arg(cz + d) ≤ π. We note that the map g →g is injective.
Definition 5.1. The universal covering groupG of G is the group of operators H × R → H × R generated by the operatorsg in (5.2) for all g ∈ SL 2 (R).
A lengthy but routine calculation shows Suppose now thatg 1g2 · · ·g n is the identity as an operator on H×R. Then z → g 1 g 2 · · · g n z is the identity operator on H. So g 1 g 2 · · · g n ∈ {I, −I} ⊂ SL 2 (R). By Lemma 5.2, it is impossible that g 1 g 2 · · · g n = −I whileg 1g2 · · ·g n is the identity operator. So g 1 g 2 · · · g n = I. This implies that the mapg → g on the generators extends to a group homomorphism

Lemma 5.2. If the vertical maps in the diagram
The composition of pr 2 with the natural projection SL 2 (R) → PSL 2 (R) gives a map pr :G −→ PSL 2 (R) .
G H × R. Because of the last two facts, we can identifyG with H × R as analytic varieties. Furthermore, the group operations are analytic with respect to the structure of H × R as an analytic variety. SoG is a Lie group. The maps pr and pr 2 are covering maps. One can show that any covering of SL 2 (R) factors through G, henceG is the universal covering group of SL 2 (R).
Section g →g: This is a homeomorphism for g near the unit element of SL 2 (R), but it is discontinuous at a c b d ∈ SL 2 (R) with c = 0 and d < 0. This section is not a group homomorphism but instead there is a Z-valued 2-cocycle w on SL 2 (R) such thatgg 1 = gg 1 k 2πw(g, g 1 ) for all g, g 1 ∈ SL 2 (R). See Theorem 16 on p. 115 of [16] for an explicit description of this cocycle. Each element ofG has a unique decomposition asg k(2πn) with g ∈ SL 2 (R) and n ∈ Z. In this paper we will not use this description of the group structure ofG. We work with the interpretation as a group of operators in H × R, and occasionally use the "one-parameter subgroups" n, a and k.

5.2.
The Lie algebra of the universal covering group. The direction of the three one-parameter subgroups n, a and k at the origin determines elements of the (real) Lie algebra g R ofG. The groupsG, SL 2 (R) and PSL 2 (R) have the same Lie algebra, since they are locally isomorphic. The Lie algebra elements corresponding to n, a and k are, respectively, The Lie algebra acts on the functions onG by differentiation on the right: YF(g) = ∂ t F(g exp(tY))| t=0 for Y ∈ g R . This action can be extended to the complexified Lie algebra g = C ⊗ R g R , and to the universal enveloping algebra of g. All the resulting differential operators commute with the action ofG by left translation. With the identification ofG as H × R we have in the coordinates given by (x + iy, ϑ): The Casimir operator ω generates the centre of the enveloping algebra of g. The corresponding differential operator commutes with left and right translations inG.
As an example we consider the modular group Γ mod = PSL 2 (Z), with corresponding groupΓ mod ⊂G. It is known that PSL 2 (Z) is presented by the generators S = ± 0 1 with pr(s) = S and pr(t) = T . Then s 2 = k(−π) = ζ −1 ∈ Z, so s and t generateΓ mod . The relation S 2 = I is replaced by the centrality of s 2 . We have The conclusion is thatΓ mod has the presentation with generators s and t and relations s 2 t = ts 2 and tstst = s. This implies that the linear space hom(Γ mod , C) has dimension 1, and is generated by α : t → π 6 , α : s → −π 2 . For reasons that will become clear later, we take this basis element, and not an integral-valued one.
Among the canonical generators we single out the following elements: The α j together with the ε j generateΓ, with ε v j j = ζ and the centrality of ζ as the sole relations. For the modular groupΓ mod we have n par = 1, n ell = 2, g = 0, and t(Γ mod ) = 1. We may take By I we now denote the augmentation ideal of the group ring C[Γ]. In C[Γ] we have the elements We allow ourselves to use the same notation as in (3.8), since from now on we will useΓ. The centrality of ζ allows us to move (ζ − 1) through the product. So it suffices to consider only q-tuples i for which all i(l) = t(Γ) occur at the end. Such q-tuples we will callΓ-q-tuples.

Proposition 5.3. A C-basis of I q+1 \I q is induced by the elements
where i runs over theΓ-q-tuples.
(Lemma 1.1 in [5].) With the relation we can take the γ j in a system of generators, for instance α 1 , . . . , α t(Γ) , ε 1 , . . . , ε n ell . For the elliptic elements To see that the b(i) are linearly independent over C we proceed in rewriting terms ξ(α i(1) −1) · · · (α i(q) −1) by replacing ξ ∈ R := C[Γ] by n + η with n ∈ C and η ∈ I. In this way, we express each element of I q as a C-linear combination of products of q factors α j − 1 plus a term in I N , with N > q. To eliminate I N we consider the I-adic completionR of C[Γ], with closureÎ q of I q . Each element ofÎ ⊃ I is a countable sum of products of a complex number and finitely many factors α j − 1. SinceÎ q+1 \Î q and I q+1 \I q are isomorphic, it suffices to prove that the b(i) are linearly independent as elements ofÎ q+1 \Î q .
We suppose that there are x i ∈ C for all q-tuples i such that We can write this element ofÎ q+1 as j c j ξ j with c j ∈ C, and ξ j running over the countably many products We form the ring N = C Ξ 1 , . . . , Ξ t(Γ) of power series in the non-commuting, algebraically independent (over C) variables Ξ 1 , . . . , Ξ t , and the two-sided ideal Z in N generated by the commutators The quotient ring M := N/Z is non-commutative if t(Γ) ≥ 3. The relations between the generators imply that there is a group homomorphism ϕ :Γ → M * given by ϕ(α j ) = 1 + Ξ j for 1 ≤ j ≤ t(Γ), and This group homomorphism induces a ring homomorphismφ :R −→ M, for whicĥ where i runs over q-tuples, and j runs over countably many tuples with length strictly larger than q. Hence all x i (and c j ) vanish.
So forΓ with cusps the trivialΓ-module I q+1 \I q is always non-trivial. The dimension is equal to the number of allΓ-q-tuples. Thus we have We obtain for eachΓ-module V an exact sequence For the modular group, we have n par = 1, n ell = 2 and g = 0, hence t(Γ mod ) = 1, and n(Γ mod , q) = 1 for all q. So in contrast to Γ mod , forΓ mod we may hope for non-trivial higher order automorphic forms.
6. Maass forms with generalised weight on the universal covering group 6.1. The logarithm of the Dedekind eta function. In the introduction we mentioned that one of the motivating objects for the study of higher order forms on the universal covering group is the logarithm of the Dedekind eta function. Its branch is fixed by the second of the following expressions: where σ u (n) = d n d u . One can show that its behaviour under Γ mod is given by Except for the term 1 2 log z this looks like a second order holomorphic modular form of weight zero. In the next few sections we make this precise by generalizing the concept "weight" of Maass forms, and replacing the group Γ mod by the discrete subgroupΓ mod of the universal covering group of SL 2 (R), using the notation we introduced in the last section.
We first define the following function on H × R: With (6.2) we check easily that L γ(z, ϑ)) = L(z, ϑ) + iα(γ) for γ = t and γ = s, where α :Γ mod → π 6 Z is the group homomorphism at the end of §5. 3 The latter action corresponds to (4.7) when r ∈ Z. In general, this is an action ofG, not of SL 2 (R). The map f → f r defined above on the space of functions of strict weight is then equivariant in terms of these actions. Many important functions onG, such as L, are not eigenfunctions of the operator W, but they are annihilated by a power of W. This suggests the following definition.
Thus, L and all its powers have generalised weight 0. Next, holomorphy of F r = y −r/2 f r corresponds to the property E − f = 0. Definition 6.2. We call any differentiable function f onG holomorphic (resp. antiholomorphic) if E − f = 0, (resp. E + f = 0). We call any twice differentiable function f onG harmonic if it satisfies ω f = 0.
Note that, for functions of non-zero weight, this definition of harmonicity does not correspond to the use of the word harmonic in "harmonic weak Maass forms" in, e.g., [1].
With these definitions we set c) (Exponential growth) There exists a ∈ R such that for all compact sets X and Θ ⊂ R and for all cusps κ ofΓ we have as y → ∞ uniformly in x ∈ X and ϑ ∈ Θ.
The spaceẼ r (Γ, λ) is infinite dimensional. Further, since ω and W commute with left translations inG, the spaceẼ k (Γ, λ) is invariant under left translation by elements ofΓ.
If k ∈ 2Z, then the elements f ∈Ẽ k (Γ, λ) correspond bijectively to the Maass forms F ∈ E k (Γ, λ) by So the condition ofZ-invariance implies that the weight k is even, and that the weight is strict, i.e., condition b) holds with n = 1.
Proof of Theorem 6.4. Any smooth function f ∈ C ∞ (H × R) satisfying b) in Definition 6.3 can be written in the form f z, ϑ = n−1 j=0 ϕ j (z) e ikϑ ϑ j , with ϕ j ∈ C ∞ (H). If such a function is left-invariant underZ, then the action of k(πm) ∈Z ⊂Γ, implies for each m ∈ Z: With induction this gives k ∈ 2Z and ϕ j = 0 for j ≥ 1, hence f (z, ϑ) = ϕ 0 (z)e ikϑ . Moreover, the stronger condition f ∈Ẽ k (Γ, λ) =Ẽ k (Γ, λ)Γ can be checked to be equivalent to We have the following generalisation of Theorem 4.2.
In Section 8 we will prove this theorem. In this section we will show that it implies the corresponding result for E k (Γ, λ). We first give some facts that are of more general interest.
Suppose that w = v|(ζ − 1) 0. Take r ∈ [1, q] minimal such that w ∈ VΓ ,r . We will show that we can replace v by another element v 1 ∈ v + VΓ ,q with v 1 |(ζ − 1) ∈ VΓ ,r 1 and r 1 < r. Repeating this process brings us eventually to v j |(ζ − 1) = 0. For this v j we will have m q v j =f and v j |(ζ − 1) = 0 which, according to the remark of the last paragraph suffices to prove the proposition. From This shows that v 1 |(ζ − 1) has order less than r.
Proof of Theorem 4.2. From Theorem 6.5, V =Ẽ k (Γ, λ k ) is maximally perturbable. Therefore, by Proposition 6.6, the spaceẼ k (Γ, λ k )Z E k (Γ, λ k ) is maximally perturbable too. This proof illustrates the fact that, for groups with cusps, there are really more higher order forms with generalised weight than with strict weight: The basis in Proposition 5.3 is for all such discrete groups larger than the corresponding basis in §3.2.1.

6.3.
Holomorphic forms on the universal covering group.
This is aΓ-module for the action by left translation. We denote by H p k (Γ) (resp. H c k (Γ)) the space of f ∈ H k (Γ) satisfying f g κ (x + iy, ϑ) = O(y C ) for some C ∈ R (resp. f g κ (x + iy, ϑ) = O(e ay ) for some a < 0) instead of (6.4).
Proof of Theorem 4.3. As in the case of general Maass forms, we can show that, for k ∈ 2Z, E hol k (Γ, λ k ) H k (Γ)Z. Then, Proposition 6.6 implies Theorem 4.3.
Second order forms and derivatives of L-functions. With this definition, L is a second order invariant belonging to H 0 (Γ mod )Γ mod ,2 . (Incidentally, this example shows that, for generalised weight k, the space H k (Γ) need not be contained inẼ k (Γ, λ k ).) Based on L we can construct a second-order form which is related to derivatives of classical modular forms. Specifically, for positive integer N, denote by G N the group generated byg, g ∈< Γ 0 (N), Let now f be a newform in the space S 2 of cusp forms of weight 2 for Γ 0 (N) such that its L-function L f (s) vanishes at 1. Then, f (W N w)d(W N w) = f (w)dw and, for all ϑ ∈ R, where u(z) := log(η(z)) + log(η(Nz)). From this we see that, since, L ′ f (s) = 2π ∞ 0 f (iy) log(y)dy, we can retrieve, from a alternative perspective, the formula first derived in [11].
6.4. Examples of higher order forms for the full modular group. Theorems 6.5 and 6.8 show that there are perturbations of 1 for the full originalΓ mod of SL 2 (Z) in the universal covering group. Since t(Γ mod ) = 1 all these perturbations are commutative (see (5.12)).
2. Set χ r = e irα , r ∈ C, where α ∈ hom(Γ mod , C) is given by α n(1) = π 6 and α k(π/2) = π 2 . The family (6.9) r → e rL(z,ϑ) = y r/2 η(z) 2r e irϑ consists of elements of H r (Γ) that areΓ mod -invariant under the action given by 3. It is possible to obtain a more or less explicit description of a harmonic perturbation of 1 of order 3. We sketch how this can be done with the meromorphic continuation of the Eisenstein in weight and spectral parameter jointly. This family is studied in [2]. In that work, automorphic forms are described as functions on H transforming according to a multiplier system of Γ mod . These correspond to functions onG that transform according to a character ofΓ mod . Carrying out the reformulation, we can rephrase §2.18 in [2] as stating that there is a meromorphic family of Maass forms on U × C, where U is some neighborhood of (−12, 12) in C. We retrieve the exact family studied in [2] by considering z → E(r, s; z, 0). For each (r, s) ∈ U × C at which E is not singular it is an automorphic form of weight r for the character χ r = e irα of Γ mod with eigenvalue λ s = 1 4 − s 2 . It is a meromorphic family of automorphic forms onΓ mod with character χ r with a Fourier expansion of the form (6.10) E(r, s) = µ r (r/12, s) + C 0 (r, s) µ r (r/12, −s) + n 0 C n (r, s) ω r (n + r/12, s) , where the C n (r, s) are meromorphic functions, and where we use the following notations. This family and its Fourier coefficient C 0 satisfy the following functional equations. Further, the restriction of this family to the (complex) line r = 0 exists, and gives a meromorphic family of automorphic forms depending on one parameter s. This is a family of weight zero, so it does not depend on the parameter ϑ onG. The resulting family on H is the meromorphic continuation of the Eisenstein series for Γ mod in weight 0, with Fourier expansion (6.13) where µ 0 (0, s; z, ϑ) = y 1 2 +s , ω 0 (n, s; z, ϑ) = e 2πinx W 0,s (4π|n|y) = e 2πinx 2|n| 1/2 K s (2π|n|y) .
At (0, − 1 2 ) the family E is holomorphic in both variables r and s, with a constant as its value at (0, − 1 2 ). (This is a consequence of Proposition 6.5 ii) in [2].) So in principle, we obtain higher order harmonic perturbations of 1 by differentiating r → E(r, − 1 2 ). Here we encounter the problem that we have an explicit Fourier expansion (6.13) only for E(0, s) and thus we cannot describe the derivatives in the direction of r directly. To overcome this problem we use the fact that for r near 0 we have (6.14) The proof of the first equality is contained in 6.10 in [2]. The second one follows from the second functional equation in (6.12). Now we use the Taylor expansion of E of degree 2 at (r, s) = (0, − 1 2 ): By Proposition 4.4, the coefficients A 1,0 and A 2,0 are harmonic perturbations of 1 of order 2 and 3, respectively. From (6.14) we obtain the following results: (6.16) This confirms that Im L is a second order harmonic perturbation of 1. Differentiation in the direction of s preservesΓ mod -invariance. So A 0,1 = 2Re L and A 0,2 areΓ mod -invariant. However these functions are not in the kernel of ω.
Thanks to the identity A 2,0 + 1 4 A 0,2 = Re L 2 , to determine the third order harmonic perturbation A 2,0 it suffices to explicitly compute A 0,2 because Re L 2 is known in a fairly explicit way. The function A 0,2 can be obtained as the coefficient of 1 2 (s + 1 2 ) 2 in the Taylor expansion of E(0, s) at s = − 1 2 . As a by-product of this computation we will also obtain theΓ mod -invariant function A 0,1 as the coefficient of s + 1 2 in the same expansion. We shall examine each term of the expansion separately.
Set ξ := s + 1 2 . The first term of our expansion is (6.17) µ 0 (0, s; z, 0) = y 1 2 +s = 1 + ξ log y + ξ 2 1 2 (log y) 2 + · · · For the next term Λ(2s) Λ(2s+1) µ(0, −s; z, 0) = Λ(2−2ξ) Λ(1−2ξ) µ(0, −s; z, 0) with Λ(u) = π −u/2 Γ u 2 ζ(u) = Λ(1 − u), we define a 0 and b 1 by We get For the other terms we use with the incomplete gamma-function Γ(a, t) = The results in (6.17), (6.19) and (6.20) confirm that the constant term equals 1, and that with the notation q = e 2πiz . The term of order 2 leads to: which is a complicated, but explicit expression. A remarkable aspect of this computation that we have used an explicit computation of the derivatives of the Eisenstein series in weight zero to compute the second derivative in the r-direction of the more complicated Eisenstein family in two variables. The basic observation is (6.14), which shows that the Eisenstein family has easy derivatives in two directions. The Taylor expansion of E at 0, − 1 2 has three monomials in order 2. So it suffices to compute a second order derivative in one more direction to get hold of all terms. Higher order terms in the Taylor expansion have too many monomials for this method to work. We do not know how to compute all harmonic perturbations of 1 of higher order.

Higher order Fourier expansions
This section is needed for the constructions on which the proofs of Theorems 6.5 and 6.8 are based, but it is also of independent interest. It provides a higher-order analogue of the classical Fourier expansions.

Fourier expansion of Maass forms.
If f is inẼ r (Γ, λ), then for each cusp κ of Γ there is a Fourier expansion where ν runs through a class in C mod Z determined by χ and the cusp κ. The function F ν f satisfies F κ,ν f (z, ϑ) = e 2πiνx F κ,ν f (iy, 0) e irϑ and ωF κ,ν f = λ F κ,ν f . For each given ν, r and s set Because of the second relation in the definition, f ∈ W r (ν, s) can be thought of as a function of y. Therefore the space W r (ν, s) is isomorphic to the space of f : R → C satisfying We can choose a fixed s with Re s ≥ 0 corresponding to the eigenvalue λ = λ s under consideration. The spaces W r (ν, s) are two-dimensional. We will use the basis elements in §4.2 of [3].
• For Re ν 0 a basis of W r (ν, s) is formed by Here W µ,s (t) is the Whittaker function that decreases exponentially as t → ∞. We use the branch of W κ,s (z) that is holomorphic for − π 2 < arg z < 3π 2 . The asymptotic behaviour as y → ∞, by §4.2.1 in [21] is: (7.5) ω r (ν, s; z, ϑ) ∼ (4πνεy) rε/2 e 2πν(ix−εy)+irϑ , where ε denotes Sign(Re ν). The subspace of W r (ν, s) generated by ω r (ν, s) is denoted by W 0 r (ν, s). • For ν = 0, a basis is given by {y  Proof. The existence of such a Fourier expansion is a standard result. A detailed proof in a more general setting can be found in [3], §4.1-3. Fourier terms of automorphic forms inherit the growth behaviour of the automorphic form. So if f is in E k (Γ, λ s ), all Fourier terms satisfy F κ,ν f (z, ϑ) = O(e ay ) as y → ∞ for some a depending on f . Each Fourier term of non-zero order is a linear combination of ω k (n, s) andω k (n, s). From (7.6) we conclude that F κ,n f is a multiple of ω k (n, s) for all but finitely many n.
Conversely, suppose that for the cusp κ we have F κ,n f = c n ω k (n, s) for all n with |n| ≥ N. Then (7.5) and the convergence of the Fourier expansion at (z, ϑ) = (iy 0 , 0) with y 0 > 0 implies that c n = O y −k Sign(n)/2 0 e 2π|n|y 0 . This in turn shows that the sum over |n| ≥ N gives a bounded contribution in (7.7) for all y large enough. The terms with |n| < N cannot give a growth at the cusp κ larger than O y a e 2π(N−1)y for some a > 0.  .7) on the set y ≥ y 0 , and exponential growth of such a function is equivalent to the statement that all Fourier terms of sufficiently large order are in W 0 k (n, s). 7.2. Higher order Fourier terms. The higher order invariants of V k (n, s) that we will define now are the higher-order analogues of the classical Fourier terms. Definition 7.3. Let k ∈ 2Z, n ∈ Z, and s ∈ C. By V k (n, s) we denote the space of functions f onG that satisfy ω f = λ s f , have generalised weight k, and satisfy ∂ x − 2πin m f = 0 for some m ∈ N (which may depend on f ).
For n 0 we denote by V 0 k (n, s) the subspace of f ∈ V k (n, s) that satisfy f (z, ϑ) = O(y a e −2π|n|y as y → ∞ for some a ∈ R. The free commutative group∆ generated by τ = n(1) and ζ = k(π) acts on these spaces by left translation. Proposition 7.4. Let k, n, s be as above. The∆-modules V k (n, s) and V 0 k (n, s) are maximally perturbable. For each q ∈ N the elements f ∈ V k (n, s)∆ ,q satisfy, for each δ > 0, (7.8) f (z, ϑ) ≪ δ e (2π|n|+δ)y (y → ∞) uniformly for x and ϑ in compact sets. If n 0 then for each q ∈ N the elements f ∈ V 0 k (n, s)∆ ,q satisfy, for each δ > 0, uniformly for x and ϑ in compact sets.
Proof. To prove that V k (n, s) is maximally perturbable, we start with a characterisation of the space V k (n, s)∆. We first note that W k (n, s) ⊂ V k (n, s)∆. Conversely, if f ∈ V k (n, s)∆, then the reasoning in the proof of Theorem 6.4 shows that the weight of f is strict, and also that ∂ x f = 2πin f , hence f (z, ϑ) = e 2πinx f (iy, ϑ). So f ∈ W k (n, s). If, for n 0, the function f is also exponentially decreasing it has to be a multiple of ω k (n, s). Therefore, V 0 k (n, s)∆ = W 0 k (n, s). Let f be an arbitrary element of W k (n, s). Since each of the basis elements of W k (n, s) is a specialisation of a holomorphic family of elements of W r (ν, s), there is a holomorphic family of h(r, ν) ∈ W r (ν, s) such that h(k, n) = f . We have h r, ν; n(ξ)k(ℓπ)(z, ϑ) = e 2πiνξ+πirℓ h(r, ν; z, ϑ) for ξ ∈ R and ℓ ∈ Z.
Next consider the polynomials Q q ∈ Q[X] of degree q defined by (7.10) Then for each m = (m 1 , m 2 ), m j ≥ 0 set 2πi ∂ ν h(r, ν) ν=n, r=k . Upon applying the differential operator 1 2πi ∂ a ν on h(r, ν)|(τ − 1) = e 2πiν − 1 h(r, ν) we obtain Therefore, k (n, s). Therefore, for l 1 + l 2 = m 1 + m 2 (l 1 , l 2 ≥ 0), thus obtaining the maximal perturbability of V k (n, s). For convenience, we shall call perturbations statisfying the transformation law (7.14) perturbations of type m. Based on V 0 k (n, s)∆ = W 0 k (n, s), we deduce in an analogous way the maximal perturbability of V 0 k (n, s). To prove (7.8) and (7.9), we first note that the maximal perturbability we have just shown implies that the functions h m constructed from f 's ranging over a basis of W k (n, s) (resp. W 0 k (n, s)) induce a basis of the quotients V∆ ,q+1 /V∆ ,q . Therefore, it suffices to show (7.8) and (7.9) for h m only. In the case n 0, the family h may be taken to be ω r (ν, s) orω r (ν, s) in (7.4). For these functions the question reduces to the asymptotic behaviour of ∂ j t ∂ l κ W κ,s (t), since the factors e 2πiνx and e irϑ produce polynomials in x and ϑ, which yield constants when they vary through compact sets. The differentiation of 4π Sign(Re ν) ν y yields only a power of y, which can be absorbed by the factor e δy .
Differentiation of W κ,s (t) with respect to t does not change the exponential part of the asymptotic behaviour, since derivatives of W κ,s (t) are linear combinations of W κ,s (t) and W κ+1,s (t) with powers of t in the factors. See (2.4.24) in [21]. So we have to look only at differentiation with respect to κ.
For t ∈ R with t > 0, κ − 1 2 − s −1, −2, . . . , we shall use the integral representation (3.5.18) in [21]: where the contour comes from ∞ along a line slightly above the positive real axis, encircles 0 with radius δ < 1 and then goes back to ∞ on a line slightly below the positive real axis. By a routine computation we see that the part of the integral over the circular part is O(e δ|t| ). The integral over the remaining part of the contour is O(|t| A ) (A ∈ R). In all cases, the implied constants does not depend on t. Differentiation in terms of κ on W κ,s (t) leads to the appearance of additional factors log(−x) and log(1 + x/t) in the integrand. The arguments used in the last paragraph imply the same estimate. Thus we get the desired exponential decay of the perturbations of ω k (n, s).
The representation (7.15) is valid as long as −t = e −πit t is outside the path of integration. If we tilt the path of integration anti-clockwise by an angle φ we get a representation of W κ,s (t) for e −πi t outside the new path of integration, provided we keep ϕ ∈ (− π 2 , π 2 ) to have convergence. For 0 < ϕ < π 2 this gives a representation that can be used for arg e −πi t = 0 with |t| > δ, which leads to the desired growth of perturbations ofω k (n, s). If κ − 1 2 − s = −1, −2, . . . we take 0 < ϕ < π 2 and transform the integral representation (7.15) into Proceeding as before we obtained the same estimates. All these estimates taken together prove (7.8), (7.9) (when n 0). They further show that the derivatives of a family with exponential decay have exponential decay and thus V 0 k (n, s) is also maximally perturbable.
In the case n = 0 we might use the same method. However, many families of special functions have to be considered to cover all cases. Instead we argue directly that we can find functions h m k (0, s) in V k (0, s) of the form p m (x, y, ϑ) y Holomorphic Fourier terms onG are multiples of (7.17) η r (ν; z, ϑ) = y r/2 e 2πiνz e irϑ .
Thus we have the spectral parameter s = ± r−1 2 . For real values of ν and r we have with notations as in (7.4) and (6.11). The functions Q m 2 (z) η k (n; z, ϑ) satisfy (7.20) m m 1 +m 2 η m k : (ζ − 1) l 1 (τ − 1) l 2 → δ m 1 ,l 1 δ m 2 ,l 2 η k (n) for l 1 + l 2 = m 1 + m 2 , and as y → ∞ their growth is of order O(e (δ−2πn)y ). For the commutative group∆ and for a fixed m they yield a basis of the space of forms of order m 1 + m 2 + 1 modulo lower order forms.
As an example we note that the Fourier expansion (6.1) can be written in the following way: 8. Proofs of Theorems 6.5 and 6.8 The method of the proof is highly inductive. At each step we use the maximal perturbability of other spaces which has been proved in a previous step. The starting point for this process is the space Map(Γ, C) whose maximal perturbability is proved based on general algebraic principles in Proposition 8.1. This implies directly the maximal perturbability of theΓ-module Map(H × R, C). We proceed by imposing increasingly stringent regularity conditions on the functions H×R → C. We consider C ∞ (H×R) = C ∞ (G), the subspace C ∞ k (G) of functions in C ∞ (G) with generalised weight k and the subspace C k of C ∞ k (G) of functions that have compact support moduloΓ. In §7 we have considered higher order invariant functions for the group∆ generated by n(1) and k(π). These functions are related to the Fourier expansions of Maass forms. After proving that some more auxiliary subspaces of C ∞ k (H × R) are maximally perturbable, we finally prove in §8.5 the maximal perturbability ofẼ k (Γ, λ) and H k (Γ).
Now, the choice of the basis b(i) in (5.9) forΓ-q-tuples i shows that to prove that Map(Γ, C) is maximally perturbable it suffices to prove that for each i and for each function f onΓ\G a function h i ∈ Map(G, C) such that for allΓ-q-tuples j: To construct such functions we choose a strict fundamental domain FΓ ⊂G forΓ\G, i.e., a set meeting eachΓ-orbit exactly once. Such a fundamental domain can be constructed from a strict fundamental domain F H for Γ\H, by taking FΓ = (z, ϑ) : z ∈ F H , 0 ≤ ϑ ≤ π/n z , n z = min n ∈ N : there is γ ∈Γ fixing z in H conjugate to k(π/n) .
So n z = 1 for all z ∈ F H , except for the elliptic fixed points z 1 , . . . , z n ell in F H . These are conjugate to a fixed point of ε j and n z j = v j .
A choice for the sought function h i is then With the characteristic function ψ of FΓ, we can write this as

Higher order invariants in smooth functions onG.
We will use essentially the same construction as in the last section to prove that Proof. In order to show that C ∞ (G) is a maximally perturbableΓ-module, we need to have (8.7) with h i ∈ C ∞ (G) for each f ∈ C ∞ (Γ\G). Lemma A.1 in Appendix A shows that we can find functions ψ ∈ C ∞ (H×R) such that γ∈Γ ψ γ −1 (z, ϑ) = 1 for all (z, ϑ) ∈ H × R as a locally finite sum. If we define (8.9) with such a function ψ and f ∈ C ∞ (Γ\G), then the sum is locally finite, and the h i are smooth.

Higher order invariants and generalised weight. Set
, of generalised weight k} . Proposition 8.3. Let k ∈ 2Z. Then theΓ-module C ∞ k (G) is maximally perturbable. Proof. As with the previous proofs, our approach is to show that for everyΓ-q-tuple i = (i ′ , t(Γ), . . . , t(Γ)) with exactly m occurrences of t(Γ) at the end and for every f ∈ C ∞ k (Γ\G) there exists h i ∈ C ∞ k (G) satisfying equation (8.7) for allΓ-q-tuples j. We note that, by Theorem 6.4, theΓ-invariance of f implies that its weight k is strict, i.e., f (gk(ϑ)) = f (g)e ikϑ .
The support property of the partition of unity ψ ensures convergence; it is even a locally finite sum with a bounded number of non-zero terms. All factors depend smoothly on g. So h i ∈ C ∞ (G). We consider (W − ik)h i . Since Wψ = 0, we need only consider (8.12) Repeating this we obtain In a similar (but much simpler) way, one shows that, if Γ acts on C ∞ (H) via (4.2) and f ∈ C ∞ (H) Γ , then the function in In fact, since ψ 0 is bounded, if f has polynomial growth at all cusps, then so does h i thus proving that the submodule of C ∞ (H) of functions with polynomial growth is also maximally perturbable. 8.4. Higher order invariants with support conditions. We shall first discuss the motivation for the introduction of the invariants we will be dealing with. If Definition 6.3 of the spaceẼ k (Γ, λ) did not include a growth condition at the cusps, we could considerẼ k (Γ, λ) as the kernel K in the exact sequence With exponential growth, one might want to try to replace C ∞ k (G) by its subspace C ∞ l (Γ) eg of functions with exponential growth at the cusps ofΓ. This would lead to an exact sequence is exact. For this to be of use it seems that we need surjectivity of the map ω−λ : C ∞ k (Γ) eg Γ → C ∞ k (Γ) eg Γ , which we did not succeed in proving, and which may not hold. For this reason we will instead work with other better behaved subspaces of the spaces appearing in the exact sequence. We will therefore define subspaces C k , D k (λ) ⊂ C ∞ k (G) and E ′ k (λ) ⊂ E k (Γ, λ) related by an exact sequence 8.4.1. The spaces C k . For each cusp κ =g κ ∞, and each a > 0 we call There is a number A Γ such that for each a ≥ A Γ the D κ (a) are disjoint for different cusps. The sets satisfyΓG a =G a . This follows from the fact that the g κ have been chosen so that (8.17) γΓ κgκ =g γκΓ∞ for all cusps κ and for γ ∈Γ. HereΓ κ := pr −1 Γ κ = γ ∈Γ : γκ = κ .
So C k consists of the smooth functions with generalised weight k whose supports project to compact subsets of Γ\H. Clearly, the space C k isΓ-invariant. If we apply the construction of h i in the proof of Proposition 8.3 to functions f ∈ CΓ k ⊂ C ∞ k (Γ\G) then the support of each h i is contained in the same setG a that contains Supp( f ). This implies: Proposition 8.6. Let k ∈ 2Z. Then theΓ-module C k is maximally perturbable.

8.4.2.
The spaces D k (λ). The construction of D k (λ) and the proof of its maximal perturbability is much lengthier that those for C k . We will define D k (λ) essentially as the space of functions that accept higherorder analogues of Fourier expansions at the cusps. To make this formal we study spaces of functions defined on regions of the form (8.18) S (y 0 ) = (x + iy, ϑ) ∈ H × R : y > y 0 , with y 0 > 0.
Definition 8.7. Let k ∈ 2Z, λ ∈ C, and y 0 > 0. We denote by E k (y 0 , λ) the space of f ∈ C ∞ S (y 0 ) that satisfy ω f = λ f , (W − ik) n f = 0 for some n ∈ N, and have at most exponential growth as y → ∞, uniform for x and ϑ in compact sets. We denote by E hol k (y 0 ) the space of holomorphic functions on S (y 0 ) with generalised weight k and at most exponential growth as y → ∞ Proposition 8.8. Let k ∈ 2Z, s ∈ C and y 0 > 0. The spaces E k (y 0 , λ s ) and E hol k (y 0 ) are maximally perturbable∆-modules.
Let q ∈ N. Each f ∈ E k (y 0 , λ s )∆ ,q has an absolutely convergent expansion on S (y 0 ) with f n ∈ V k (n, s)∆ ,q for all n, and f n ∈ V 0 k (n, s)∆ ,q for almost all n. Each f ∈ E hol k (y 0 )∆ ,q has an absolutely convergent expansion on S (y 0 ) of the form where the inner sum ranges from some, possible negative, integer to infinity.
Proof. We start with the holomorphic case. Let f ∈ E hol k (y 0 )∆. Then the function z → y −k/2 f (z, 0) is holomorphic on {z ∈ H : y > y 0 } with period 1. So it has a finite to the left expansion of the form n a n e 2πinz converging absolutely on y > y 0 . For each y 1 > y 0 we have a n = O e 2πny 1 as n → ∞. Hence f (z, ϑ) = n a n η k (n; z, ϑ) converges absolutely on y > y 0 , and f m (z, ϑ) := n≥−N a n η m k (n; z, ϑ) converges absolutely on S (y 0 ), and the convergence is uniform on any set y ≥ y 1 with y 1 > y 0 , with x and ϑ in compact sets. These functions satisfy f m |(τ − 1) = f (m 1 ,m 2 −1) , f m |(ζ − 1) = f (m 1 −1,m 2 ) and f (0,0) = f , since all η m k have this property. Thus f m , with m such that m 1 + m 2 < q is a perturbation of type m and we deduce that E hol k (y 0 ) is maximally perturbable. An arbitrary element h ∈ E hol k (y 0 )∆ ,q can be written as a finite linear combination of such f m , which all have expansions of the type given in (8.20).
For f ∈ E k (y 0 , λ s )∆ we proceed similarly. By Theorem 7.1 in combination with Remark 7.2 and the integrality of k, there is an absolutely converging Fourier expansion on S (y 0 ) with f n ∈ W k (n, s). By the exponential growth, f n ∈ W 0 k (n, s) for |n| > N, for some N ∈ N. For |n| > N we have f n = a n ω k (n, s), and from (7.5) we conclude that a n = O e 2π|n|y 1 ) as |n| → ∞ for each y 1 > y 0 . So by (7.5) the series in E k (y 0 , λ s ). Thus we get (8.19) and the maximal perturbability of E k (y 0 , λ s ).
We are now ready to define D k (λ) and D hol k . Definition 8.9. Let k ∈ 2Z, and λ ∈ C. We define D k (λ) as the space of functions f ∈ C ∞ k (G) (hence with generalised weight k) for which there exist b ≥ A Γ , a ∈ R, and q ∈ N such that for each cusp κ ofΓ the function (z, ϑ) → f g κ (z, ϑ) is an element of E k (b, λ)∆ ,q , and satisfies a bound O(e ay ) as y → ∞.
We define D hol k similarly, with (z, ϑ) → f g κ (z, ϑ) in E hol k (b)∆ ,q , with bound O(e ay ). Remark 8.10. The numbers a, b and q may depend on the function f . In both cases we have exponential growth at each cusp. The definition requires that the order of this exponential growth stays bounded when we vary the cusp. Remark 8.12. In the definition we impose∆-invariance of bounded order near all cusps. This is a bit artificial, but serves our purpose.
The space C k is contained in D k (λ) and in D hol k . Indeed, for given f ∈ C k we can take b large so that κ D κ (b) is outside the support of f . Elements f ofẼ k (Γ, λ)Γ restricted to D κ (b) induce elements (z, ϑ) → f g κ (z, ϑ) in E k (b, λ)∆ for each cusp κ, and similarly in the holomorphic case. Hence Maximal perturbability of D k (λ) and D hol k . We first need a technical lemma in order to relate∆invariants toΓ-invariants.
We will write f = f cpt + κ f κ , with κ running over a set C of representatives of theΓ-orbits of cusps, where f cpt ∈ (C k )Γ, f κ ∈ D k (λ)Γ, and will produce perturbations for each of these components.
Since the supports of the f κ with κ ∈ C are disjoint, we can consider each of the f κ separately. Without loss of generality we can assume that ∞ is a cusp ofΓ withg κ = 1 and take ∞ ∈ C. Conjugation by the originalg κ then gives the same result for a general κ ∈ C.
The function v ∞ used in (8.27) is an element of E k (b, λ)∆. The proof of Proposition 8.8 shows that for each m ∈ N 2 0 there is a perturbation v m ∞ ∈ E k (b, λ)∆ ,m 1 +m 2 +1 of (z, ϑ) → f ∞ (z, ϑ) of type m. We define η i by η i = 0 onG b and on allΓD κ (b) for all κ ∈ C {∞}, and for y ≥ b and ρ in a system of representatives R ofΓ/∆. The functions a i l,m are as in (8.23). Since the sets ρD ∞ (b) are disjoint, this defines a smooth function, which can be checked to be an element of D k (λ).
Ignoring smoothness for a moment we have With our choice of fundamental domain, and using (8.23), we find for ρ ∈ R, δ ∈∆ and g = (x + iy, ϑ) with y ≥ b: OutsideΓD ∞ (b) the functions f ∞ , h i are zero. With Lem. 8.13 we conclude that the function induced by is in Map(G, C)Γ ,q . This implies that and behaves in the desired way under (α i ′ (1) − 1) · · · (α i ′ (q) − 1) for allΓ-q-tuples i ′ . Thus, we have proved that D k (λ) is maximally perturbable. Everywhere in this proof we can replace E k (b, λ) by E hol k (b), and D k (λ) by D hol k . In that way we also obtain the maximal perturbability of D hol k , thus completing the proof of Proposition 8.14. 8.4.3. Relations between the spaces C k and D k (λ). By Remark 8.11, for each f ∈ D k (λ) the support of (ω − λ) f is contained in some setG b , hence (ω − λ) f ∈ C k . So the differential operator ω − λ maps D k (λ) to C k . Since the operator ω commutes with the action ofΓ, we have (ω − λ)D k (λ)Γ ,q ⊂ CΓ Proof. Proposition 8.15 gives the case q = 1. The rows in the following commutative diagram are exact by Proposition 8.6 and 8.14. See (5.13) for m q .
The third column is exact by Proposition 8.15. With the exactness of the first column as induction hypothesis, we obtain the vanishing of coker(ω − λ) and thus the surjectivity of ω − λ : We work with the space of square integrable functions onΓ\G = Γ\G of strict weight k ∈ 2Z, where G = PSL 2 (R). We can view the elements of the Hilbert space H k = L 2 (Γ\G) k = L 2 (Γ\G) k as functions z → f (z, 0) on H, transforming according to weight k as indicated in (4.7). The inner product in H k is given by Here F can be any fundamental domain for Γ\H. We take it so that for each b > A Γ it has a decomposition (8.32) with C a system of representatives of the Γ-orbits of cusps, and x κ ∈ R depending on F and on the earlier choice of the g κ . The set F b has compact closure in H. The differential operator ω k = −y 2 ∂ 2 y − y 2 ∂ 2 x + iky∂ x in (4.8) determines a densely defined self-adjoint operator A k in H k . The spectral theory of automorphic forms gives the decomposition of this operator A k in terms of Maass forms. One may consult Chapters 4 and 7 in [12] for weight 0. For other weights the proofs are almost completely similar. (See [20].) There is a subspace H discr k with an at most countable orthonormal basis {ψ ℓ k } of Maass forms, indexed by some subset of Z. The ψ ℓ k are square integrable elements of the space of Maass forms E k (Γ, λ ℓ ) with λ ℓ ≥ k 2 (1 − k 2 ). We denote the eigenspace associated to λ (which is known to be finite-dimensional) by H k (λ). If k = 0 the eigenvalue 0 occurs with multiplicity one, corresponding to constant functions, and all other λ ℓ , if any, are positive. If k 0, then H discr k may be zero. If k ≥ 2 and the space S k (Γ) of holomorphic cusp forms of weight k is non-zero, then there are ψ ℓ k ∈ H discr k of the form ψ ℓ k (z, 0) = y k/2 h(z) with h ∈ S k (Γ). The corresponding eigenvalues are λ ℓ = k 2 1 − k 2 , which is negative if k ≥ 4. There may also be elements obtained by differentiation of holomorphic cusp forms of weights between 2 and k − 2. Similarly, for negative k there may be eigenfunctions corresponding to antiholomorphic cusp forms.
The orthogonal complement H cont k of H discr k in H k is isomorphic to a sum of n par copies of L 2 (0, ∞), dt , where n par is the number of Γ-orbits of cusps. The spectral decomposition gives the Parseval formula with κ running through a set of representatives of the cuspidal orbit. For each f ∈ H k we have a ℓ k ( f ) = f, ψ ℓ k . If f is sufficiently regular, then the functions e κ k ( f ; ·) are obtained by integration against the Eisenstein series E κ k (it) at the cusp κ. The space CΓ k is contained in H k . For f ∈ CΓ k the functions e κ k ( f ; ·) are given by for all s at which the meromorphic continuation of the Eisenstein series is holomorphic. In particular, e κ k ( f ; s) is holomorphic at points of the line iR. On the square integrable Maass forms and on the Eisenstein series the self-adjoint operator A k is given by ω k in (4.8). For f ∈ H k in the domain of A k , the self-adjointness of A k together with the eigenproperty . This implies that the spectral data of elements f ∈ H k such that A n k f is well defined for all n ∈ N, are quickly decreasing. The convergence in L 2 -sense of the Parseval formula in (8.33) is very fast for functions of this type, since the summands and integrands in the expansion are those of (A n k f, A n k f 1 ) divided by (λ ℓ ) n , respectively ( 1 4 + t 2 ) n for each n ∈ N. (If there is a term with λ ℓ = 0 we treat it separately; it does not influence the convergence.) The central point of the proof of Proposition 8.15 is that we transform the equation (A k − λ) f 1 = f with unknown f 1 ∈ H k for a given f ∈ CΓ k to the spectral decomposition. Application of A k − λ to f ∈ CΓ k amounts to multiplying a ℓ k ( f ) by λ ℓ − λ and multiplying e κ k ( f ; t) by 1 4 + t 2 − λ. This suggests the following Definition 8.17. Let λ ∈ C. We denote by C k (Γ, λ) the space of f ∈ CΓ k such that the following conditions are satisfied.
has a double zero at t = 0 if λ = 1 4 . Note that, for each λ, the conditions i), ii), iii) impose finitely many linear conditions, so C k (Γ, λ) has finite codimension in CΓ k . If λ is not in the spectrum of A k , then C k (Γ, λ) is equal to CΓ k .
Case I: f ∈ C k (Γ, λ). In this case, if we have the spectral decomposition then a solution of (A k − λ) f 1 = f is given by If λ is not in the spectrum of A k , then the convergence of this L 2 -expansion is better than that in (8.34), since λ ℓ → ∞ and hence the denominators improve the convergence. If λ is in the spectrum of A k , condition i) ensures that the a ℓ k ( f ) with λ ℓ = λ vanish, and that, by the other conditions, the simple or double zero of t → 1 4 + t 2 − λ at t = t λ is canceled by the zeros at s = it λ of the holomorphic functions s → e κ κ ( f ; s). The same reasoning shows that the obtained f 1 is in H k and, in fact, in the domain of A k . We Since ω k determines an elliptic differential operator on H, elliptic regularity implies that (ω − λ) f 1 = 0 holds as a relation for real-analytic functions on each D κ (b). Further, the square integrability implies that f 1 must have less than exponential growth at the cusps and hence it is an element of D k (λ)Γ. We have shown: Case II: f ∈ CΓ k C k (Γ, λ) for λ in the spectrum of A k . The following result enables us to pick representatives h of CΓ k /C k (Γ, λ) for which we can solve (ω − λ) f 1 = h directly. This procedure can be carried out by singling out one cusp κ, which we fix for the proof of Case II. Lemma 8.19. Let κ be the cusp that we keep fixed. Suppose that λ is in the spectrum of A k . Then there is a finite set X ⊂ Z such that, for each n ∈ X, there exist h n ∈ CΓ k of the form , such that {h n + C k (Γ, λ)} n spans CΓ k /C k (Γ, λ). If we can solve (A k − λ) f 1 = h n in another way for all n ∈ X, this lemma enables us to reduce the proof of Proposition 8.15(i) to Lemma 8.18.
Proof of Lemma 8.19. We shall examine each of the three cases for the eigenvalues of A k on H k separately: • λ = 1 4 − s 2 1 4 , ∞ . Assume s > 0. There are finitely many indices ℓ 1 , . . . , ℓ m such that λ ℓ j = λ. The ψ ℓ j k form a basis of ker (A k − λ). Each of these m linearly independent square integrable automorphic forms is given by its Fourier expansion at the fixed cusp κ. By Proposition 7.1, the Fourier terms of non-zero order are multiples of ω k (n, s). The Fourier term of order zero is a multiple of y 1 2 −s e ikϑ . We choose a set X of m elements in Z such that the m × m-matrix whose columns are the n-th Fourier coefficients of ψ ℓ j k (1 ≤ j ≤ m) with n ∈ X is invertible. We choose the χ n ∈ C ∞ c , n ∈ X, in the statement of the lemma, so that We have f 1 ∈ H k ⊖ H h k and E − f 1 = f . A reasoning as in the previous case shows that f 1 ∈ D hol k (λ)Γ. So we have solved the problem for a subspace of CΓ k−2 with finite codimension. A general element of CΓ k−2 will not be orthogonal to H a k−2 . We proceed as in the first case in the proof of Lemma 8.18. Instead of ψ ℓ j k we now use an orthogonal basis of H a k−2 , and form functions h n as in Lemma 8.18, corresponding to a set X of Fourier term orders such that elements of H a k−2 are determined by the Fourier coefficients in X. Solving E − k f 1 = h n leads to the differential equation (−2iy∂ x + 2y∂ y − k)e 2πinx ϕ(y) = χ(y) , ϕ(y 0 ) = ϕ ′ (y 0 ) = 0 , with which we proceed as in the previous case. This establishes the surjectivity of E − : (D hol k )Γ → CΓ k−2 in Proposition 8.15. 8.5. Higher order invariants and Maass forms. We now will derive the main results of this paper, Theorems 6.5 and 6.8, from the following result: The exactness of the columns follows from the definition ofẼ ′ k (λ), (3.2), the left-exactness of the functor hom C[Γ] (I q \C[Γ], -) and Corol. 8.16. Propositions 8.3 and 8.14 imply that the second and third row are exact. The Snake Lemma then implies that the first row is exact and that m q :Ẽ ′ k (λ)Γ ,q+1 → Ẽ ′ k (λ)Γ n(Γ,q) is surjective.
Replacing in this diagram the spaceẼ ′ l (λ) by H ′ k and the map ω − λ by E − , we obtain the maximal perturbability of H ′ k .
The proof of Theorem 6.8 is completely similar.
Appendix A. Partition of unity The following technical lemma gives partitions of unity that are adapted toΓ\G and Γ\H.
Proof. We fix a strict fundamental domain F H forΓ\H of the following form, based on the choice of a real number a > A Γ , as in §8.4.1. The set F H is bounded by finitely many geodesic segments and half-lines such that where C a is relatively compact in H, and is contained in the image ofG a under the projectionG → H. The disjoint union is over the set C of cusps κ in the closureF H of F H in H ∪ ∂H. We take F H such that C forms a system of representatives for theΓ-orbits of cusps. By taking the parameter a sufficiently large we arrange that all orbits of elliptic fixed points intersect F H in C a . These points are necessarily on the boundary of F H . We take a strict fundamental domain forΓ\G of the form F = (z, ϑ) : z ∈ F H , ϑ ∈ [0, π/v(z)) , where v(z) ∈ N is the order of the subgroup Γ z fixing z, or equivalentlyΓ z is conjugate inG to the group {k(nπ/v(z)) : n ∈ Z}. So v(z) is in general equal to 1, and only larger if z is an elliptic fixed point of Γ.
i. We first define a function onG satisfying a) and c), and a variant of b).
Let ω :G → {0, 1} be the characteristic function of F. It satisfies conditions a)-c) in part i) of the lemma, but is not smooth. To make it smooth we convolve it with a function ψ ∈ C ∞ c (G) with ψ ≥ 0 such that G ψ(g) dg = 1 for a choice dg of a Haar measure onG and such that Supp(ψ) is a compact neighborhood of the unit element inG.
Since ω is measurable, the integral ϕ 0 (g) = G ω(g 1 ) ψ(g −1 1 g) dg 1 = G ω(gg −1 1 ) ψ(g 1 ) dg 1 defines a smooth function ϕ 0 onG with values in [0, 1] and with support contained in the neighborhood F · Supp(ψ) (multiplication inG) of F. From the second form of the convolution integral we see that γ∈Γ ϕ 0 (γ −1 g) = 1 for all g ∈G. This smooth function ϕ 0 satisfies conditions a) and c) in part i) of the lemma. Condition b) is not satisfied, since although the support of ϕ 0 is contained in a neighborhood of F of the form F Supp(ψ), this neighborhood may meet near the cusps infinitely manyΓ-translates of F. We will construct two functions, one "away from the cusps" and another "close to the cusps" satisfying all conditions a), b), c) on overlapping regions. A suitable combination of these two functions will produce the sought function onG.
• The first function is simply the restriction of ϕ 0 toG b for any b ≥ a. We will show that this function satisfies condition b) (and thus all conditions). First we note that the projections p 1 :G → H and p 2 :G → R given by p 1 (z, ϑ) = z and p 2 (z, ϑ) = ϑ are continuous. Next we note that F Supp(ψ) ∩G b is contained in a compact set, and hence has compact image in H under p 1 . So Fix a g ∈G b . We will show that there is a finite number (independent of g) of γ ∈Γ with ϕ 0 (γ g) 0. Indeed, for each such γ we have γg ∈ F Supp(ψ) ∩G b , hence p 1 (γg) = pr(γ) p 1 (g) ∈ δ∈E δ F H . This leaves finitely many possibilities for the image pr(γ): for some δ 0 ∈ Γ. We conclude that γ = δδ −1 0 k(πm) with m ∈ Z. On the other hand, the image p 2 F Supp(ψ) ∩G b is contained in a compact set, hence it is contained in a set [−B, B] ⊂ R. For the γ = δδ −1 0 k(πm) with ϕ 0 (γg) 0 we conclude from (5.3) that p 2 δδ −1 0 k(πm)g = p 2 δδ −1 0 g + mπ. This leaves only finitely many possibilities for the integer m. This shows that condition b) is satisfied by the restriction of ϕ 0 toG b (b ≥ a).
• We now start the construction of another function ϕ 1 with the desired properties near the cusps. We take a compactly supported smooth partition β of unity for R/Z, i.e., β ∈ C ∞ c (R) with values in [0, 1] such that k∈Z β(x + k) = 1 for all x ∈ R. (For instance take a smooth function υ in C ∞ (R) with value 0 on a neighborhood of 0 and value 1 on a neighborhood of 1 2 . Then defines such a partition of unity.) We define a function ϕ 1 onG in the following way.
We choose a (bounded) function χ ∈ C ∞ (Γ\G) equal to 0 onG a and equal to 1 onG G a+1 . Put where ϕ 0 is as constructed above with b equal to a + 1. Since χ vanishes onG a the product χ · ϕ 1 is smooth onG. Similarly, (1 − χ) · ϕ 0 is smooth. So ψ ∈ C ∞ (G). Conditions a)-c) are easily checked to hold for ψ.
Appendix B. Index of commonly used notation