Generalized Maass Wave Forms

We initiate the study of generalized Maass wave forms, those Maass wave forms for which the multiplier system is not necessarily unitary. We then prove some basic theorems inherited from the classical theory of modular forms with a generalization of some examples from the classical theory of Maass forms.


Introduction
The space of generalized modular forms of integer weights arise naturally in rational conformal field theory or the theory of vertex operator algebras [8,26]. Those are meromorphic functions defined on the upper half plane that satisfy the transformation law under subgroups of finite index in the full modular group same as classical modular forms with a difference that the group multiplier appearing in the transformation law does not necessarily have absolute value one.
On the other hand, Maass wave forms are real analytic functions invariant under the action of subgroups of the full modular group, are eigenfunctions of the Laplacian operator and at most grow like polynomials at the cusps. Maass wave forms connects to several areas like L-series [19], representation theory [1,6] and other connections to Artin billiard and associated transfer operators [21].
In this paper, we initiate the study of generalized Maass wave forms, give basic properties and definitions and extend some theorems from the theory of classical modular forms. We shall show that we can construct Maass wave forms from generalized modular forms analogous to the construction of Maass wave forms from classical modular forms. We also construct Eisenstein series and Poincaré series associated to generalized Maass wave forms. We continue to study the analytic properties of the mentioned forms and in particular the Whittaker-Fourier expansion and the Maass operators.
In Lemma 2.5, we show that if we have a generalized Maass wave form on a subgroup of the full modular group and this same form is a classical Maass wave form of a smaller group contained in the subgroup, then the generalized Maass wave form in classical on the bigger group.
We then introduce vector valued Maass wave forms that might help in establishing the Eichler cohomology of generalized Maass wave forms in future work. Going to the vector valued case creates an easier tool to deal with integral transforms associated to the periods of Eichler integrals.

Generalized Maass Wave Forms
2.1. Preliminaries. Let SL(2, R) denote the group of 2 matrices with real entries and determinant 1. The subgroup SL(2, Z) ⊂ SL(2, R) denotes the full modular group, that is the subgroup matrices with integer entries. It is generated by where −1 ∈ SL(2, Z) and 1 denotes the identity matrix. The group SL(2, R) acts on the upper half-plane H = {z ∈ C | Im (z) > 0} and its boundary P R = R ∪ {∞} by fractional linear transformations Let Γ ⊂ SL(2, Z) be subgroup of the full modular group with finite index. It is known that the fundamental domain F = F Γ of Γ in H is a hyperbolic polygon containing finitely many inequivalent parabolic cusps q 1 , . . . q t , t ≥ 1. We denote the set of inequivalent cusps by C = C Γ := {q 1 , . . . q t }.
To each cusp q ∈ C Γ we denote the stabilizer of q by Γ q = γ q , −1 . There exists a scaling matrix g q ∈ SL(2, Z) such that where l q ∈ N is the width of the cusp q, see [11, (2.1), page 40], or [12, page 5].
A multiplier or multiplier system v compatible with (complex) weight k is a function allows non-zero solutions f . We call a multiplier system v weakly parabolic if as in [17,Equation (5)].
Remark 2.1. We use the convention with arg (z) ∈ (−π, π] for all z ∈ C =0 to determine the k th power in (2.6).
Remark 2.2. Condition (2.6) implies in particular that v satisfies the relation for all γ, δ ∈ Γ and z ∈ H. In particular (2.8) implies We also introduce the slash-action as notation. For γ = a b c d ∈ SL(2, R), k ∈ C and f be a function on H we define For example Equation (2.6) reads as f k γ = v(γ) f .

2.2.
Classical and Generalized Maass Wave Forms. We briefly recall Maass wave forms.
Definition 2.3. Let Γ ⊂ SL(2, Z) be subgroup with finite index and v : Γ → C =0 a unitary multiplier system compatible with the real weight k. A classical Maass wave form of weight k, multiplier v for the group Γ is a real-analytic function u : u is an eigenfunction of the Laplace operator ∆ k with eigenvalue λ ∈ R, i.e., ∆ k u = λ u with z = x + iy ∈ H and u satisfies the growth condition u(g q z) = O (y c ) at each cusp q ∈ C Γ as y → ∞ for some c ∈ R with g q beeing the associated scaling matrix in (2.4). We denote the space of classical Maass wave form by c M (Γ, k, v, λ).
u is called a classical Maass cusp form if u satisfies the stronger growth condition u(g q z) = O (y c ) at each cusp q ∈ C Γ as y → ∞ for all c ∈ R.
Maass has proved in [19,Theorem 28] that the space c M (Γ, k, v, λ) of Maass wave forms is finite dimensional. Generalized Maass wave forms still keep essentially the properties 1 and 2 of Definition 2.3. However we remove the condition that the multiplier system is unitary and we weaken the growth condition. This leads to the following Definition 2.4. Let Γ ⊂ SL(2, Z) be subgroup with finite index and v : Γ → C =0 a multiplier system compatible with the complex weight k. A generalized Maass wave form of weight k, multiplier v for the group Γ is a real-analytic function u : u is an eigenfunction of ∆ k with eigenvalue λ ∈ C, i.e., ∆ k u = λ u, (3) u satisfies the growth condition u(g q z) = O (e cy ) in each cusp q ∈ C Γ as y → ∞ for some c ∈ R with g q given in (2.4). We denote the space of generalized Maass wave form by gM (Γ, k, v, λ).

A Basic
Lemma. Similar to [12,Lemma 3] we have the following result.
Lemma 2.5. Suppose u : H → C is a classical Maass wave form form with respect to (Γ, k, v, λ), with Γ of finite index in SL(2, Z). (That means that k ∈ R and |v(γ)| = 1 for all γ ∈ Γ.) Suppose further that u is a generalized Maass wave form with respect to Then u is already a classical Maass form with respect to (Γ , k, v , λ). That is to say: k ∈ R and |v (γ)| = 1 for all γ ∈ Γ implies that |v (γ)| = 1 for all γ ∈ Γ .
We adapt the proof of [12, Lemma 3] to our situation.
Proof. From the consistency condition (2.8) and the fact that Γ has finite index in Γ , it follows easily that v assumes only finitely many distinct values on Γ . On the other hand, if there were γ ∈ Γ such that |v(γ )| = 1, then by (2.8) the set v (γ ) n ; n ∈ Z would contain infinite many distinct values for |v | on Γ .
We still have to check that the growth conditions of u are in fact as in Definition 2.3 (3). Since Γ ⊂ Γ the set of cusps satisfy C Γ ⊂ C Γ . The assumptions of the lemma imply that u satisfies the stronger growth condition in Definition 2.3 in each cusp of C Γ . The additional cusps in C Γ \ C Γ can be transformed into a cusp in C Γ by an element in Γ . (These additional cusps are Γ -equivalent to cusps in C Γ .) Hence the the stronger growth condition of Definition 2.3 is also valid for these cusps.
Hence u is a classical Maass wave form since it satisfies Definition 2.3.

Maass Operators.
We denote by E ± k the differential operators Remark 2.6. The Maass operators are named after Hans Maass. He studied operators K k and Λ k , see e.g. [19, page 177], which are essentially E ± k .
As shown for example in [4, §6.1.4], the operators E ± k∓2 E ∓ k are related to ∆ k by (2.14) Direct calculations show that the slash-action commutes with the Laplace operator and interacts as follows with the Maass operators for all k ∈ C and smooth u : H → C.
Lemma 2.7. E ± k map generalized Maass wave forms of weight k to generalized Maass forms of weight k ± 2: Proof. Using (2.14), we get the commutation relation As a result, the eigenfunctions of ∆ k are mapped to the eigenfunctions of ∆ k±2 by E ± k . (2.16) shows that the group action commute with E ± k and (2.12) shows that the growth condition of the generalized Maass waveform is compatible with E ± k .

Some Examples
3.1. Maass Wave Forms. We consider Maass wave forms as for example as introduced in [11]. These are real-analytic functions u : H → C which satisfy Maass wave forms are obviously also generalized Maass wave forms for weight 0 and trivial multiplier. Maass wave forms with real weight, as considered in [4] and [22], are also generalized Maass wave forms for real weight and unitary multiplier.
3.2. Generalized Modular Forms. Generalized modular forms are introduced a few years back. Following [12], a generalized modular form F is a holomorphic function F : H → C with a left finite Fourier-expansion at each cusp and it satisfies the transformation property for each γ ∈ Γ and using (2.14), and the property that u in (3.3) lies in the kernel of E − k : u satisfies the growth property in Definition 2.4 (3) since the generalized modular form F has a left finite Fourier-expansion. This generalizes the example of holomorphic modular forms in [22, page 6].

Eisenstein Series and Poincaré Series.
We use a method of constructing generalized Maass waveforms similar to the classical construction as in [5]. To twist the definition of the real analytic Eisenstein series and Poincaré series by introducing a non-unitary multiplier systetem inside the sum. However, this construction will definitely affect the convergence of the series.
Let v be a multiplier system for Γ which is compatible to weight k. For F : H → C an eigenfunction of ∆ k define formally the generalized Poincaré Series P (z) by the formal series runs to a complete set of coset representatives for Γ ∞ \Γ.
It is a straight forward calculation to show that P formally satisfies properties (1) and (2) of Definition 2.4, provided that the series converges absolutely. Here, |v| is not necessarily 1.
Assume for the moment that with h : H → C a bounded holomorphic function. Recall from [15, Lemma 6] that where K is a positive constant, α is another constant depending on the modulus of the multiplier system at the generators of Γ and µ(γ) = a 2 + b 2 + c 2 + d 2 where a, b, c, d are the entries of γ. Recall also that there exists a constant K 1 such that for all γ ∈ Γ ∞ \Γ. Moreover, we have from [15, Lemma 4] the following inequality Combining (3.7) and (3.8) explains the absolute convergence of the series in (3.4) for large k with k > 2α + 1: Similarly, taking  for smooth functions G : (0, ∞) → C and ν / ∈ − 1 2 N. According to [7, (13.14.2), (13.14. 3)], see also [20,Chapter 7], we have two solutions M k,ν (y) and W k,ν (y) with different behavior as y → ∞: The asymptotic behavior is valid for k − ν / ∈ 1 2 , 3 2 , 5 2 , . . . , see [7, (13.14.20), (13.14.21)]. These functions satisfy also the recurrence relations [7, (13.15 We consider a modified pair of solutions. The following lemma summarizes the action of the Maass operators E ± k onW k 2 ,ν : For n < 0 we have Lemma 4.4. Let k, ν ∈ C such that k ± ν / ∈ 1 2 + Z and λ = 1 4 − ν 2 . We have for Proof. Direct calculations using identities from [20, §7.2.1, page 302] to rewrite M k 2 ,ν and the functional equation xΓ(x) = Γ(x + 1) of the Gamma-function. 4.2. Whittaker-Fourier Expansions. We consider a generalized Maass wave form u ∈ gM (Γ, k, v, λ) with weakly parabolic multiplier v and assume that we have λ = 1 4 − ν 2 for some ν ∈ C \ − 1 2 N. Let q ∈ C Γ be a cusp of width l q and g q ∈ SL(2, R) be the associated scattering matrix, see (2.4). The action of the stabilizer γ q = g q T lq g −1 q implies that u q := u k g q is nearly periodic, i.e., u q k T lq = v(γ q ) u q . We expect an expansion of the form (4.7) u q (x + iy) = n≡κ mod 1 a n (y) e 2πinx lq at the cusp i∞, where κ ∈ R is given by v γ q = e 2πiκ . (κ is real since v is a weakly parabolic multiplier).
The coefficients a n (y) still depend on y. Since u q solves the partial differential equation we find by separation of variables that a n (y) = h 4πεn lq y , ε = sign (n) and n = 0, solves the ordinary differential equation which is the Whittaker differential equation. Solutions are theW-andM-Whittaker functions (4.10) t →W ε k 2 ,ν (t) and t →M ε k 2 ,ν (t). In the case n = 0 separation of variables shows that a 0 (y) solves the ordinary differential equation    for some M ∈ R (which corresponds to the constant c in Definition 2.4). The 0 th -term coefficients C + and C − vanish if κ / ∈ Z.
The calculation above shows the following Proposition 4.5. Let u ∈ gM (Γ, k, v, λ) and q be a cusp in C Γ . Then u(g q z) admits a Whittaker-Fourier expansion of the form (4.14). The 0 th -term term vanishes if v(γ q ) = 1.
Remark 4.6. SinceW 0,ν 1 2 y = y π K ν 1 2 y the expansion in (4.14) leads to the usual Fourier-Bessel expansion of classical Maass cusp forms in weight 0: Then . . , f (α µ ) is a linear isomorphism which transports χ H to the linear G-action on V µ given by . . , µ} is the unique index such that Hα j g = Hα kj . To see this, one simply calculates . In the case of the right regular representation the identification V G ∼ = C µ gives a matrix realization χ H (g) = χ(α i gα −1 j ) 1≤i,j≤µ whereχ(g) = χ(g) if g ∈ H andχ(g) = 0 otherwise.
Let's extend the matrix representation even more. for all z ∈ H and g, h ∈ SL(2, Z).

Example 5.2.
(1) The right regular representation w(h, z) = χ 0 (h) of the trivial character in (5.1) is a weight matrix of dimension µ.
(2) The scalar function w(h, z) = v(h)e ikarg(c h z+d h ) for v a multiplier with weight k for SL(2, Z) is a 1-dimensional weight matrix.
Proof. We have to verify (5.2). Indeed, using property (2.8) we find
To each u ∈ gM (Γ, k, v, λ) we associate the vector valued function Π(u) given by Indeed, take an u ∈ gM (Γ, k, v, λ) and an i ∈ {1, . . . , µ}. Obviously, Π(u) i = u k g i is real-analytic on H. As mentioned in (2.15), we have And the growth condition for Π(u) also follows directly from the growth condition for u. To check the transformation property, take a h ∈ SL(2, Z). There exists a h ∈ Γ and an unique j ∈ {1, . . . , µ} such that g i hg −1 j =: h ∈ Γ. Using the transformation property of generalized Maass wave forms in Definition 2.4 we find This shows the transformation property of vvgMFs. Hence Π(u) ∈ gM vv (Γ, µ, w k,v , λ).
On the other hand, consider the map where j ∈ {1, . . . , µ} satisfies g j ∈ Γ. The map is well defined since j is uniquely determined and the function u := π u is in gM (Γ, k, v, λ): u satisfies the transformation property u k h = v(h) u for all h ∈ Γ since = v(h) e ikarg(c h z+d h ) u(z) (using (2.8)).
Obviously, u is also an eigenfunction of ∆ k with eigenvalue λ.
To show that u satisfies the required growth condition in all cusps take a cuspidal point q ∈ C Γ of Γ and g q ∈ SL(2, Z) satisfying q = g q i∞, as in (2.4). Similar to the calculation above, we find u(γg q z) j = w k,v (g q , z) u(z) j = µ l=1 w j,l (g q , z) u(z) l (using (5.3)) = w j,q (g q , z) u(z) q (using (5.4)), since g j g q g −1 k / ∈ Γ except for k = q. Hence u(g q z) = w j,q (g q , z) u(z) q = v(g j ) e ikarg(cg j z+dg j ) u(z) q = O e M y as y → ∞ for some M ∈ R.

Conclusions and Outlook
In this paper, we introduced generalized Maass wave forms, which extend the generalized modular forms introduced in [12] and, simultaneously, Maass wave forms of real weight, as discussed in [4]. We also proved some related theorems and discussed the expansions of those forms which in turns extends from the classical theory of Maass forms. On the other hand, several examples were also introduced taking into account the bound for the multiplier system.
As a next step, we like to extend the concept of Eichler integrals leading to period polynomials [9] and period functions [18,22] attached to modular cusp forms and Maass cusp forms. That is we like to generalize objects of the form (g, γ) → γ z0 z0 g(z)(z − X) k−2 dz, where g : H → C is a modular cusp form of weight k and γ ∈ Γ, to the setting of generalized Maass wave forms. Hence we plan follow [16,17] in our setting and we plan as well to characterize the cohomology group associated to those forms. In the end, we aim at constructing an Eichler-Shimura-type map between the space of generalized Maass wave forms and the suitable group cohomology.
It is worth mentioning that the vector valued Maass wave forms are introduced in this paper for computational purposes in our future work similar to the use of vector valued Maass cusp forms in [23]. Also, to allow weight matrices instead of the scalar valued multiplier systems and weight factors seems to be an interesting generalization along [13]. This way, we can easily pull back relations on forms for Γ ⊂ SL(2, Z) to matrix valued relations on vector valued forms for SL(2, Z) as illustrated in [23].