Spectra of Compact Quotients of the Oscillator Group

This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group Osc1, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of Osc1 up to inner automorphisms of Osc1. For every lattice L in Osc1, we compute the decomposition of the right regular representation of Osc1 on L (L\Osc1) into irreducible unitary representations. This decomposition allows the explicit computation of the spectrum of the wave operator on the compact locally-symmetric Lorentzian manifold L\Osc1.


Introduction
Let G be a Lie group and L be a cocompact discrete subgroup in G. Then we can consider the right regular representation of G on L 2 (L\G), which is unitary. This representation can be decomposed into irreducible unitary representations. It is a classical result that this decomposition is a discrete sum and each summand appears with finite multiplicity. We will call the set of irreducible representations that appear in this decomposition together with their multiplicities the spectrum of L\G. It allows to compute the spectrum of differential operators on L\G that come from bi-invariant operators on G.
One of the basic questions in harmonic analysis is to determine the spectrum of L\G. For semisimple G, this question is very deep and has been studied for decades. For arithmetic L, it is related to the Langlands program. In any case, only for very few irreducible representations, their multiplicities can be explicitly determined. For nilpotent G, the situation is quite different. Here explicit computations are possible and there are closed formulas for the multiplicity of every irreducible subrepresentation [4,8,17]. Compared to nilmanifolds, solvmanifolds are less systematically studied, even in low dimensions. The first comprehensive texts on the subject were a book by Brezin [3] and a long article by Howe [9], both published in the 70s. To date, there has been surprisingly little further research except for exponential solvable Lie groups [6]. We think that solvable Lie groups deserve much more interest. They play an important role in pseudo-Riemannian geometry. There are large families of solvable Lie groups that admit a bi-invariant pseudo-Riemannian metric. In particular, quotients of these groups by discrete subgroups are locally-symmetric spaces.
Indeed, our motivation to write this paper comes from Lorentzian geometry. We are interested in the spectrum of the wave operator on certain compact Lorentzian locally-symmetric spaces. These spaces arise as quotients of an oscillator group by a discrete subgroup. Oscillator groups are solvable and admit a bi-invariant Lorentzian metric. The wave operator can be considered as minus the quadratic Casimir operator, which is bi-invariant on the group. Hence its spectrum can be easily computed if the decomposition of the right regular representation is known, see Section 8.4. We will see that there exist lattices such that the spectrum is discrete as well as lattices for which the quotient has a non-discrete spectrum.
For solvable Lie groups, we prefer to speak of lattices instead of discrete cocompact subgroups. By definition, a lattice in a Lie group G is a discrete subgroup L with the property that the quotient space L\G has finite invariant measure. Lattices in a solvable Lie group are uniform, that is, they are the same as discrete cocompact subgroups.
The above mentioned book by Brezin contains an informal discussion of ideas how to attack the problem of decomposing the regular representation in the solvable case. There are also some explicit calculations for a series of four-dimensional examples. These examples are semidirect products of the Heisenberg group H by the real line in which selected examples of lattices are considered. Inexplicably, the book has only very few citations and Brezin's ideas were not picked up and developed except in [10]. Here we will continue the study of the mentioned fourdimensional groups. We will concentrate on the case where R acts on H/Z(H) by rotations, which gives exactly the four-dimensional oscillator group. We will treat this example more systematically than it was done in [3]. In particular, we consider quotients of this group by arbitrary lattices and not only by "suitable" ones as done in [3], see Remark 8.13 for more information. We will use a somewhat different approach than proposed by Brezin and are able to completely determine the spectrum for all compact quotients.
Let us introduce the four-dimensional oscillator group and its Lie algebra in more detail. The four-dimensional oscillator algebra osc 1 is a Lie algebra spanned by a basis X, Y , Z, T , where Z spans the centre and the remaining basis elements satisfy the relations [X, Y ] = Z, [T, X] = Y , [T, Y ] = −X. This Lie algebra is strongly related to the one-dimensional quantum harmonic oscillator. Indeed, the Lie algebra spanned by the differential operators P := d/dx, Q := x, H = P 2 + Q 2 /2 and the identity I is isomorphic to osc 1 . In differential geometry, the oscillator algebra osc 1 is also called warped Heisenberg algebra. It appears in the classification of (connected) isometry groups of compact Lorentzian manifolds as the Lie algebra of one of such groups [21].
The oscillator group is the simply-connected Lie group Osc 1 that is associated with the Lie algebra osc 1 . It is a semi-direct product of the 3-dimensional Heisenberg group H by the real line R, where R acts by rotation on the quotient of H by its centre. In particular, it is solvable. As mentioned above, it admits a bi-invariant Lorentzian metric. Moreover, this group contains lattices. So, each lattice L in Osc 1 gives rise to a compact Lorentzian manifold L\Osc 1 .
The group Osc 1 is not exponential. However, it is of type I. Hence, irreducible unitary representations can be determined by the orbit method, which was originally developed by Kirillov for nilpotent Lie groups and generalised to type I solvable Lie groups by Auslander and Kostant. The polynomials z and x 2 + y 2 + 2zt (in the obvious notation) generate the algebra of invariant polynomials on osc * 1 . The level sets z = c, x 2 + y 2 + 2zt = cd ∼ = R 2 with fixed c = 0 and d ∈ R are the generic coadjoint orbits. The set {z = 0} is stratified into two-dimensional cylindrical orbits x 2 + y 2 = a, z = 0 , a > 0, and orbits consisting of one point {x = y = z = 0, t = d}. The latter orbits correspond to one-dimensional representations C d . Each cylindric orbit corresponds to a set S τ a , τ ∈ S 1 , of infinite-dimensional representations and each generic orbit corresponds to a an infinite-dimensional representation F c,d , cf. [12].
The study of lattices in oscillator groups was initiated by Medina and Revoy [13]. Note, however, that the classification results in [13] are not correct due to a wrong description of the automorphism group of an oscillator group. Lattices of Osc 1 (as subgroups) were classified up to automorphisms of Osc 1 by the first author [5]. Since outer automorphisms can change the spectrum of the quotient, here we need a classification only up to inner automorphisms of Osc 1 . We derive such a classification in Section 4. First we classify the abstract groups (up to isomorphism) that appear as lattices of Osc 1 . We will call these groups discrete oscillator groups. These groups are semi-direct products of a discrete Heisenberg group by Z. A discrete Heisenberg group is generated by a central element γ and elements α, β satisfying the relation [α, β] = γ r for some r ∈ N >0 . It is characterised by r and will be denoted by H r 1 (Z). If L = H r 1 (Z) Z is a discrete oscillator group and if δ is a generator of the Z-factor, then the action of δ on the quotient of H r 1 (Z) by its centre has order q ∈ {1, 2, 3, 4, 6}. For q = 1, L is almost determined by r and q: for fixed r, q, there are only one or two non-isomorphic discrete oscillator groups. If q = 1, then the situation is slightly more complicated, see Section 4.1. Each discrete oscillator group has different realisations as a lattice in Osc 1 . More exactly, given a discrete oscillator group L we consider the set M L of lattices of Osc 1 isomorphic to L, where we identify subgroups that differ only by an inner automorphisms of Osc 1 . We want to parametrise M L . First we observe, that every lattice arises from a normalised and unshifted one by outer automorphisms of Osc 1 that can be viewed as rescaling and shifting. Here, "normalised" means that the covolume of the lattice obtained by projection of L ∩ H to H/Z(H) ∼ = R 2 equals one. The explanation of the notion "unshifted" is more involved, see Section 4.2. Basically, it means that the lattice is adapted to the fixed splitting of Osc 1 into the normal subgroup H and a complement R. Hence, we get a description of M L by the subset M L,0 ⊂ M L of normalised unshifted lattices modulo inner automorphisms and by two continuous parameters corresponding to rescaling and shifting. For q ∈ {3, 4, 6}, the set M L,0 is discrete and can be parametrised by the generator λ ∈ 2π/q + 2πZ of the projection of L to the R-factor of Osc 1 = H R. For q ∈ {1, 2}, M L,0 is parametrised by a continuous parameter (µ, ν) in a fundamental domain F of the SL(2, R)-action on the upper halfplane and some further discrete parameters besides λ. The final classification result for lattices in Osc 1 up to inner automorphisms is formulated in Theorems 4.12 and 4.15. In particular, we determine a set of representatives of unshifted normalised lattices, which we will call standard lattices.
For the computation of the spectrum of a quotient L\Osc 1 , we will concentrate on normalised and unshifted lattices. The spectrum for arbitrary lattices can easily be derived from these. The first step is to consider only lattices that are generated by a lattice in the Heisenberg group H and an element of the centre of Osc 1 . Such lattices will be called straight. For a straight lattice L, we derive a Fourier decomposition for functions in L 2 (L\Osc 1 ). For the calculations, we use another model of the oscillator group, i.e., a group G that is isomorphic to Osc 1 . Lattices in G will usually be denoted by Γ. If Γ ⊂ G corresponds to a straight lattice in Osc 1 under this isomorphism, then a (continuous) function in L 2 (Γ\G) is periodic in three of the four variables and it is not hard to describe its Fourier decomposition explicitly. From this decomposition, we obtain the spectrum. Having solved the problem for straight lattices, we can turn to standard lattices. We show that each standard lattice L is generated by a straight lattice L ⊂ L and an additional element l ∈ L. We can identify L 2 (L\Osc 1 ) with the space of functions in L 2 (L \Osc 1 ) that are invariant under l. The isotypic components of L 2 (L \Osc 1 ) are invariant by l. Hence, it remains to determine the invariants of the action of l on each of the isotypic components. This is done in Section 8, where in practice we again use G instead of Osc 1 and pass from the lattice L ⊂ Osc 1 to the corresponding lattice Γ ⊂ G. Since, up to an inner automorphism of Osc 1 , every normalised and unshifted lattice L is equal to a standard lattice, we obtain the spectrum of L\Osc 1 from our computations for the standard lattices. The representation C d appears in L 2 (L\Osc 1 ) for d = n/λ, n ∈ Z. The discrete parameter τ for which the representation S τ a appears depends only on λ if q ∈ {2, 3, 4, 6}. For q = 1, it depends also on the above mentioned additional discrete parameters of L. The parameter a depends on λ and, for q ∈ {1, 2}, in addition on (µ, ν) ∈ F. Finally, the representation F c,d appears for (c, d) = rm, n/(qλ) , m ∈ Z =0 , n ∈ Z, if q ∈ {2, 3, 4, 6}. For q = 1, the parameter d of the representation depends besides on λ on one of the additional discrete parameters of L. For each irreducible unitary representation of type C d or F c,d , we compute the multiplicity with which it appears in L 2 (L\Osc 1 ). Furthermore, for each summand of type S τ a we describe a and τ explicitly in terms of the parameters of the lattice. However, the exact determination of the multiplicity of S τ a for a given real number a ∈ R remains a number theoretic problem, see Remark 9.2.
Finally, given an arbitrary lattice L , then we can determine an (outer) automorphism F ∈ Aut(Osc 1 ) that transforms L into a normalised and unshifted lattice L. This allows the computation of the spectrum of L from that of L since we can control the action of F on the irreducible representations of Osc 1 . We summarise the results in Section 9 at the end of the paper.

Notation
. . , q − 1} remainder after dividing n ∈ Z by q ∈ N >0 ; sgn sign function on R; span closed span in a Hilbert space.

The four-dimensional oscillator group
The 4-dimensional oscillator group is a semi-direct product of the 3-dimensional Heisenberg group H by the real line. Usually, the Heisenberg group H is defined as the set H = C × R with multiplication given by where ω(ξ 1 , ξ 2 ) := Im ξ 1 ξ 2 . Hence in explicit terms, the oscillator group is understood as the set Osc 1 = H × R with multiplication defined by where ω(ξ 1 , ξ 2 ) := Im ξ 1 ξ 2 .
In the following we want to describe the automorphisms of Osc 1 , see [5] for a proof. For η ∈ C, let F η : Osc 1 → Osc 1 be the conjugation by (η, 0, 0). Furthermore, we define an automorphism F u of Osc 1 for u ∈ R by (2.1) Finally, consider an R-linear isomorphism S of C such that S(iξ) = µiS(ξ) for an element µ ∈ {1, −1} and for all ξ ∈ C. Then µ = sgn(det S) and also is an automorphism of Osc 1 . Each automorphism F of Osc 1 is of the form for suitable u ∈ R, η ∈ C and S ∈ GL(2, R) as considered above. Besides F η also F S is an inner automorphism if S ∈ SO(2, R).

Irreducible unitary representations of Osc 1
The irreducible unitary representations of the oscillator group were determined for the first time by Streater [19]. Actually, in [19], not the simply-connected group Osc 1 was studied but a quotient of this group which is a semidirect product of the Heisenberg group and the circle and can be considered as a matrix group. The group Osc 1 is not exponential, i.e., the exponential map exp : osc 1 → Osc 1 is not a diffeomorphism. More exactly, exp is not surjective. However, Osc 1 has the following weaker property which ensures that one can describe its irreducible unitary representations. A locally compact group G is said to be of type I if every factor representation of G is isotypic. It is known that the universal cover of any connected solvable algebraic group is of type I, see [15, p. 440]. In particular, Osc 1 is of type I. The irreducible unitary representations of solvable Lie groups of type I can be determined by applying a generalised version of Kirillov's orbit method, see [1,11,12] for a description of this method. An explicit computation of the irreducible unitary representations of osc 1 can be found in [12,Section 4.3], where the oscillator Lie algebra is called diamond Lie algebra. The difference between the representations of the non-simply-connected group considered in [19] and the representations of Osc 1 lies in an additional parameter τ ∈ R/Z for the irreducible representations corresponding to cylindrical orbits, see (ii) below.
Let us recall the results of [12,Section 4.3]. Note that the case c < 0 in item (iii) below is not discussed in [12]. We consider coadjoint orbits of Osc 1 in osc * 1 = C ⊕ R ⊕ R, which are represented by The orbit of (0, 0, d) is a point, the orbit of (a, 0, 0) is diffeomorphic to S 1 × R, and the orbit of (0, c, d) for c = 0 is diffeomorphic to R 2 . We will describe the representations corresponding to (ii) and (iii) only on the Lie algebra level. Let X, Y , Z, T be the standard basis of osc 1 = C ⊕ R ⊕ R, The non-vanishing brackets of basis elements are [X, Y ] = Z, [T, X] = Y and [T, Y ] = −X.
We will use the notation Every irreducible unitary representation of the oscillator group is isomorphic to one of these representations.
Let S be an R-linear isomorphism of C such that S(iξ) = µiS(ξ) for µ ∈ {1, −1} and for all ξ ∈ C. Recall that this implies µ = sgn(det S). Let F S be defined as in (2.2).

Lattices in Osc 1
Oscillator groups can be defined in every even dimension as a generalisation of the classical fourdimensional one. The study of lattices in such groups was initiated by Medina and Revoy [13]. However, their classification of lattices in Osc 1 is not correct, see Remark 4.21. Finally, lattices of Osc 1 were classified up to automorphisms of Osc 1 by the first author [5]. Since outer automorphisms can change the spectrum of the quotient, here we need a classification up to inner automorphisms of Osc 1 . Such a classification is achieved in this section. In order to formulate it, we introduce the new concept of normalised and unshifted lattices. This concept will also play an important role in Section 8, where we compute the spectrum of quotients by standard lattices.

Classification of discrete oscillator groups
Consider the discrete Heisenberg group where r ∈ N >0 . It is well known that H r 1 (Z) and H r 1 (Z) are isomorphic if and only if r = r . Let S be a homomorphism from the infinite cyclic group Zδ generated by δ into the automorphism group of H r 1 (Z). A discrete oscillator group is a semidirect product H r 1 (Z) S Zδ, where the map S(δ) induced by S(δ) on H r 1 (Z)/Z(H r 1 (Z)) is conjugate to a rotation in GL(2, R). In the following we just write S instead of S(δ) andS instead of S(δ). Using the projections of α and β to H r 1 (Z)/Z(H r 1 (Z)) as a basis, we identify this quotient with Z 2 andS with an element of SL(2, Z). Since the mapS ∈ SL(2, Z) is conjugate over GL(2, R) to a rotation, it is of finite order. This implies Lemma 4.1. Given a discrete oscillator group H r 1 (Z) S Zδ, we find (new) generators α, β, γ of H r 1 (Z) such that the relations in (4.1) are satisfied andS equals one of the matrices In particular, each discrete oscillator group is isomorphic to a group H r 1 (Z) S Zδ for whichS is one of the maps in (4.2).
Proof . It is well known that each matrix of finite order in SL(2, Z) is conjugate over GL(2, Z) to one of the matrices S 1 , S 2 , S 3 , S 4 , S 6 , see, e.g., [14]. The map T ∈ GL(2, Z) that is used for this conjugation can be extended to an isomorphism of discrete oscillator groups which preserves δ.
Proof . First take q ∈ {2, 3, 4, 6}. We may assume thatS is one of the maps S 2 , S 3 , S 4 , S 6 and it is easy to check that not onlyS q = I 2 but also S q = id, which proves the claim. Now considerS = I 2 and suppose that l := α a β b δ d , d = 0, is in Z(L). Then thus br = dk = n 1 lcm(k, r) and dl = −ar = n 2 lcm(l, r) for suitable n 1 , n 2 ∈ Z. This implies d = n 1 lcm(k, r) k = n 2 lcm(l, r) l = n 1 r gcd(k, r) = n 2 r gcd(l, r) .
Hence d is a multiple of lcm r gcd(k,r) , r gcd(l,r) = r/r 0 , from which the assertion easily follows.
For each r ∈ N >0 , we define discrete oscillator groups L i r , i ∈ {1, 2, 3, 4, 6} and L j,+ r , j ∈ {2, 3, 4} by the data in the following table: Note that, compared to [5], we use a slightly different system of representatives to assure that L 2 r , L 2,+ r and L 3 r are subgroups of L 4 r , L 4,+ r and L 6 r , respectively. Clearly, if r is odd, then L 2,+ r ∼ = L 2 r , L 4,+ r ∼ = L 4 r and if r ≡ 0 mod 3, then L 3,+ r ∼ = L 3 r . We recall the classification of discrete oscillator groups from [5]. We will present also the main ideas of a direct proof since the proof in [5] already uses the classification of lattices in Osc 1 . for i ∈ {1, 2, 3, 4, 6}, j ∈ {2, 3, 4} and r ∈ N >0 are pairwise non-isomorphic discrete oscillator groups. Conversely, each discrete oscillator group is isomorphic to one of these groups.
More exactly, let L := H r 1 (Z) S Zδ be a discrete oscillator group with S(α) =S(α)γ k , S(β) =S(β)γ l , where we already assume thatS is one of the maps listed in (4.2): (i) ifS = I 2 , then L is isomorphic to L 1 r 0 for r 0 = gcd(r, k, l), if r is odd or if k and l are even, Proof . Assume first thatS = I 2 , i.e., S(α) = αγ k and S(β) = βγ l . We put again a = −l/r 0 , b = k/r 0 , d = r/r 0 . Then gcd(a, b, d) = 1. Hence we can find elements (m 1 , m 2 , m), (n 1 , n 2 , n) We define a map from the set of generators {α, β, γ, δ} of L 1 r 0 into L by This map extends to a homomorphism from L 1 r 0 to L since [α m 1 β m 2 δ m , α n 1 β n 2 δ n ] = γ N with Obviously, this homomorphism is bijective. Now supposeS = S 2 . Let (k, l) and (k , l ) be pairs of integers and let S and S be automorphisms of H 1 r (Z) defined by S(α) = S 2 (α)γ k , S(β) = S 2 (β)γ l and S (α) = S 2 (α)γ k , S(β) = S 2 (β)γ l , respectively. We define maps T 1 , . . . , T 4 from the set of generators of H 1 r (Z) S δZ into H 1 r (Z) S δZ by Then If r is odd, the latter two groups are isomorphic and L ∼ = L 2 r . If r is even, then We have ord S = 3 if and only if L ab ∼ = Z 3r × Z or L ab ∼ = Z 3 × Z r × Z. If 3 r, then these groups are isomorphic and L ∼ = L 3 r . If 3 | r, then Finally, ord S = 4 holds if and only if L ab ∼ = Z 2r × Z or L ab ∼ = Z 2 × Z r × Z. As above, these groups are isomorphic for odd r and L ∼ = L 4 r . If r is even, then

Classification of lattices up to inner automorphisms of Osc 1
Let L be a lattice in Osc 1 . The intersection of L with the Heisenberg group is a lattice in the Heisenberg group (see [16,Corollary 3.5]). In particular, it is a discrete Heisenberg group H r 1 (Z) for some unique r ∈ N >0 . Hence L considered as an abstract group is isomorphic to exactly one of the discrete oscillator groups listed in Proposition 4.3. The projection of L ⊂ Osc 1 = H × R to the second factor is generated by a unique real number λ, where The order of λ coincides with the order ofS in the abstract discrete oscillator group L. In particular, L is isomorphic to L q r or L q,+ r as an abstract group. We will say that L is of type L q r or L q,+ r , respectively. Furthermore, we define For fixed r and q, there are different lattices in Osc 1 (up to automorphisms of Osc 1 ) of type L q r and L q,+ r . The aim of this subsection is to give a classification of all lattices in Osc 1 up to inner automorphisms of Osc 1 . The problem will be reduced to the classification of lattices with the additional property to be normalised and unshifted, which we will explain in the following.
The lattice L is called normalised ifL 0 is a normalised lattice in R 2 , i.e., ifL 0 ⊂ R 2 has covolume one with respect to the standard metric of R 2 .
(a) If L is of type L q,+ r then we set where z 0 is the real number uniquely defined by Remark 4.9.
(i) The number s L is well-defined, see Proposition 4.11.
which proves the claim.
(iii) One easily computes that z 0 = 0 for q = 2 and for L = L q,+ r , q ∈ {3, 4, 6}, and (iv) If L is a normalised lattice, then s Fu(L) = s L + u · λ(L) mod 1 s 0 r Z, where s 0 is defined as in Definition 4.8.
Proposition 4.11. For a given lattice L, the number s L is independent of the chosen adapted generators. It is invariant under inner automorphisms of Osc 1 , i.e., s F (L) = s L for every inner automorphism F of Osc 1 .
We also denote the remaining data associated with α , . . . , δ by z α , z β , v , w , z 0 . Then and an easy calculation shows that s Fη(L) = s L . Now, we want to show that s L does not depend on the choice of the generators α, β and δ. We will do that only for q = 1 and q = 4, the proof of the remaining cases follows the same strategy. We may assume z α = z β = 0 since we already proved invariance under inner automorphisms F η . Thus α = (ᾱ, 0, 0), β = (β, 0, 0), γ = 0, 1 r , 0 and δ = v rᾱ + w rβ , z δ , λ . We will change the set of generators and denote the new data by z α , z β , z δ , v , w , z 0 and s L . In each case we will show that s L = s L holds.
where λ = λ(L), r = r(L) and s 0 is associated with L according to Definition 4.8.
It remains to determine M 0 , which we want to do before proving Theorem 4.12.
Definition 4.14. We denote the lattice obtained in Lemma 4.13 by The group SL(2, Z) acts on the Poincaré half plane H. The set F = F + ∪ F − for is a fundamental domain of this action. For M ∈ SL(2, Z), we denote by M the subgroup of SL(2, Z) that is generated by M .
Let us first consider the case q = 1. Let L be unshifted with generators as in (4.16). In particular, T := (ᾱ,β) = T −1 µ,ν . Define v, w as in (4.6). Then v = rx δ ∈ Z and w = ry δ ∈ Z by Remark 4.9(ii). Hence L = L r λ, µ, ν, v r + i w r , z δ . Inner automorphisms of Osc 1 do not change the number r. Furthermore, under our assumption that µ + iν belongs to F, also µ and λ are uniquely determined. Therefore, we consider the set L := L = L r λ, µ, ν, ι 1 r + i ι 2 r , z ι 1 , ι 2 ∈ Z; z ∈ R such that s L = 0 for fixed r > 0, λ ∈ 2πN >0 and µ + iν ∈ F. Lattices in L that differ only in the parameter z are considered to be equivalent. Then the set L/ ∼ of equivalence classes is in bijection with Z r × Z r . We denote by [ι 1 , ι 2 ] the equivalence class of the lattice L r (λ, µ, ν, ι 1 r + i ι 2 r , z) ∈ L. Equivalent lattices in L are isomorphic via an inner automorphism of Osc 1 , see Lemma 4.18. Thus it remains to check which equivalence classes can be represented by lattices that are isomorphic via an inner automorphism. Let F be an inner automorphism of Osc 1 . If there exist lattices L 1 , L 2 ∈ L such that F (L 1 ) = L 2 , then an easy calculation shows that F belongs to the set I consisting of automorphisms F η F S that satisfy the conditions Conversely, each element of I maps equivalence classes of L to equivalence classes of L. More exactly, , which proves the claim for q = 1. Let L be an unshifted normalised lattice of type L 2 r . According to the above considerations, modulo an appropriate inner automorphism, L is generated by elements given as in (4.16), where λ ∈ π + 2π N. Let v, w be defined as in (4.6). Note that v and w are even for r ∈ 2N >0 by Proposition 4.3 and equation (4.7). Then Thus, L is generated by α, β, γ and We define η = η 1ᾱ + η 2β by Since v and w are even for even r, F η preserves the subgroup of L ∩ H = α, β, γ . Furthermore, F η (δ ) = (0, z δ , λ) for an appropriate z δ . Since with L also F η (L) is unshifted and since F η (L) arises by a shift F u from the unshifted lattice L 2 r (λ, µ, ν) = α, β, γ, (0, 0, λ) , there exists an inner automorphism that maps F η (L) to L 2 r (λ, µ, ν), see Lemma 4.18. If L is unshifted, normalised and of type L 2,+ r , then we again may assume that L is generated by elements of the form (4.16), where λ ∈ π/2 + 2π · Z. In this case v is odd or w is odd by Proposition 4.3 and equation (4.7). Denoteṽ = rem 2 (v) andw = rem 2 (w). Here we define η = η 1ᾱ + η 2β by Since v −ṽ, w −w and r are even, F η preserves the subgroup of L ∩ H = α, β, γ . Furthermore, F η (δ) = (ṽᾱ +wβ, z δ , λ) for an appropriate z δ . Now we again use Lemma 4.18 to see that there exists an inner automorphism that maps F η (L) to the unshifted lattice L r (λ, µ, ν,ṽ/r + iw/r, 0). For µ+iν = i, we have F S 4 L 2,+ r (λ, µ, ν, 1, 0) = L r (λ, µ, ν, i/r, 0). Furthermore, for µ+iν = e iπ/3 , we have F S (L r (λ, µ, ν, 1/r, 0)) = L 2,+ r (λ, µ, ν, 1, 1) and F S L 2,+ r (λ, µ, ν, 1, 1) = L r (λ, µ, ν, i/r, 0), where S = T −1 µ,ν S 6 T µ,ν for T µ,ν as defined in (4.15). This shows that L is isomorphic to some standard lattice of type L 2,+ r given in the list of the theorem. Conversely, it is not hard to see that these standard lattices are not isomorphic to each other, which proves the assertion for type L 2,+ r . Let L be an unshifted normalised lattice of type L 4 r . As above we may assume that L is generated by elements given as in (4.16), where λ ∈ π/2 + 2π · Z. Let v, w be defined as in (4.6).
Since v + w is odd, F η preserves the subgroup of L ∩ H = α, β, γ . Furthermore, F η (δ) = 1/r, z δ , λ for an appropriate z δ . Now we can argue as above that there exists an inner automorphism that maps L to L 4,+ r (λ), The remaining cases q = 3 and q = 6 follow a similar strategy.

Classification of lattices up to all automorphisms of Osc 1
To complete this subsection, we recall from [5] the classification of lattices in Osc 1 up to all automorphisms of Osc 1 , i.e., we consider lattices L 1 , L 2 ⊂ Osc 1 as isomorphic if and only if there exists an automorphism F of Osc 1 such that F (L 1 ) = L 2 .
Clearly, every lattice of Osc 1 is isomorphic to some normalised unshifted lattice in the list of Theorem 4.15. On the other hand, there are standard lattices of type L 1 r 0 , L 2 r and L 2,+ r which are isomorphic to each other with respect to all automorphisms. Which ones are isomorphic was shown in [5].
Before formulating the classification result, let us recall some notions concerning the group GL(2, Z). We can extend the action of SL(2, Z) on the Poincaré half plane H to GL(2, Z) by S · z := −z for S := diag(1, −1) ∈ GL(2, Z). The set F + defined in (4.13) is a fundamental domain of this action. Furthermore, for M 1 , . . . , M k ∈ GL(2, Z), we denote by M 1 , . . . , M k the subgroup of GL(2, Z) that is generated by M 1 , . . . , M k . In particular, we will consider subgroups of GL(2, Z) that are isomorphic to the dihedral groups D 2 , D 4 and D 6 .

Remark 4.21.
There is the following generalisation of the classical four-dimensional oscillator group. Let H n be the (2n + 1)-dimensional Heisenberg group, which we identify with R × C n (as a set). Let us fix an element λ ∈ R n . Then we can define the (2n + 2)-dimensional oscillator group Osc n (λ) := H n R, where t ∈ R acts on H n by t.(z, ξ) = (z, exp(diag(itλ))(ξ)). Medina and Revoy [13] proved a criterion for the existence of lattices in Osc n (λ) in terms of λ. In [13, p. 94], they tried to classify lattices of Osc 1 (up to automorphisms of Osc 1 ). For every r ∈ N >0 , they found only a finite number of (isomorphism classes of) lattices L such that r(L) = r, where r(L) was defined in (4.4), which is obviously wrong. Note that the map given on p. 92 of [13] is not an automorphism of Osc n (λ). Therefore, in the proof of Theorem III, one cannot assume that (0, 0, t) belongs to L without changing L ∩ H n . Thus, Theorem III is not correct, which is one reason for the wrong classification.

The model G of the oscillator group
In the following, we want to use a slightly different multiplication rule for the oscillator group, which will make our computations easier. We use the well known fact that the Heisenberg group H is isomorphic to the set H(1) of elements M (x, y, z) parametrised by x, y, z ∈ R with group multiplication M (x, y, z)M (x , y , z ) = M (x + x , y + y , z + z + xy ).
We define an action l of R on H(1) by l(t)(M (x, y, z)) = M x cos t− y sin t, x sin t+ y cos t, z+ xy 2 (cos(2t)− 1)+ x 2 − y 2 4 sin(2t) and consider the semi-direct product The image of an element t ∈ R under the identification of R with the second factor of G in (5.1) is denoted by (t). It is easy to check that is an isomorphism. The isomorphism φ maps L r (λ, µ, ν, ξ 0 , z) to the lattice Γ r (λ, µ, ν, ξ 0 , z) generated by where ξ 0 = x 0 + iy 0 .
They are called standard lattices in G.
Definition 5.1. A lattice L ⊂ Osc 1 is called straight if it is generated by a lattice in the Heisenberg group and an element of the centre of Osc 1 . A lattice in G is called straight if its preimage in Osc 1 is straight. Similarly, a lattice in G is called unshifted or normalised if its preimage in Osc 1 has this property.

Remark 5.3. An unshifted normalised lattice Γ in G is straight if and only if it is isomorphic
under inner automorphisms of G to a lattice Γ r (λ, µ, ν, 0, 0) for λ ∈ 2π N >0 . Indeed, consider the preimage L of Γ in Osc 1 . After applying an inner automorphism we have (4.16) and δ is an element of the centre of Osc 1 , thus x δ = y δ = 0. Since L is unshifted, z δ ∈ 1 s 0 r Z. Now we apply Lemma 4.18.

The right regular representation
Let L be a lattice in Osc 1 . Then L acts on the left of Osc 1 and we can consider the quotient L\Osc 1 . Furthermore, L acts by left translation on functions ϕ : Osc 1 → C. For γ ∈ L this action is defined by Let L 2 (L\Osc 1 ) denote the Banach space completion of the normed space of all left Linvariant continuous functions ϕ : Osc 1 → C that are compactly supported mod L with norm where F is a fundamental domain for the action of L on Osc 1 . For integration we use the Lebesgue measure, which is left-and right-invariant with respect to multiplication in Osc 1 .
It is a classical result that ρ, L 2 (L\Osc 1 ) is a discrete direct sum of irreducible unitary representations of Osc 1 with finite multiplicities, see, e.g., [20]. We already described the irreducible unitary representations of Osc 1 in Section 3. Our aim is to determine how often they occur in ρ, L 2 (L\Osc 1 ) for a given lattice L ⊂ Osc 1 . In the previous section we identified Osc 1 with the group G using the isomorphism φ : Osc 1 → G defined by (5.2). Let L be a lattice in Osc 1 and let Γ = φ(L) denote the corresponding lattice in G. We identify the right regular representation ρ, L 2 (L\Osc 1 ) of Osc 1 with the right regular representation ρ G , L 2 (Γ\G) of G via the isomorphism φ * : L 2 (Γ\G) → L 2 (L\Osc 1 ), which satisfies ρ(g) for all g ∈ Osc 1 . Then our problem now consists in the decomposition of ρ G , L 2 (Γ\G) into irreducible subrepresentations of G. It would be natural to use the push-forwards of the irreducible representations of Osc as models for the irreducible representations of G. Let us denote by However, in practice we will work with the slightly different representations C dG , S τ a G and F c,d G , which are defined by the same formulas as C d , S τ a and F c,d of Osc 1 , but where now X, Y , Z, T is the basis of the Lie algebra g of G that satisfies These representations are equivalent to φ −1 * C d , φ −1 * S τ a and φ −1 * F c,d , respectively. In the following we simply write ρ instead of ρ G and C d , S τ a and F c,d instead of C dG , S τ a G and F c,d G . In the following sections we will mainly concentrate on the spectrum of quotients by standard lattices, only the final result will be formulated for arbitrary ones. This is justified by the following observation. An arbitrary lattice L can be transformed into an unshifted and normalised one by an automorphism F of Osc 1 . Furthermore, the normalised and unshifted lattice L we get is isomorphic to a standard latticeL under an inner isomorphism of Osc 1 . Let ρ andρ denote the corresponding right regular representation on L 2 (L \Osc 1 ) and L 2 L \Osc 1 , respectively. Then we haveρ ∼ = ρ and F * ρ ∼ = ρ and we can apply (3.4) to obtain the spectrum of L\Osc 1 from that ofL\Osc 1 .
We begin the study of ρ by calculating the action of the basis elements X, Y , Z, T of g. In order to simplify the notation, we often write ϕ(x, y, z, t) instead of ϕ(M (x, y, z) · (t)) for ϕ ∈ L 2 (Γ\G).
Denote by H 0 the sum of all irreducible subrepresentations of L 2 (Γ\G) for which Z acts trivially and let H 1 be the orthogonal complement of H 0 in L 2 (Γ\G), thus By (6.1), H 0 consists of those functions in L 2 (Γ\G) that do not depend on z.
In particular, ϕ is periodic in y, z and t with periodicity 1 √ ν , 1 r and 2πκ. Hence ϕ has a Fourier expansion of the form (7.1). Since ϕ is Γ-invariant, we have which gives (7.2). Then (7.3) follows by induction. The set is a fundamental domain of the Γ-action on G. Thus Recall that we put Proposition 7.2. We have L 2 (Γ\G) = H 0 ⊕ H 1 . The first summand is equivalent to for a(l, k) = νk 2 + 1 ν (−µk + l) 2 1 2 = T −1 µ,ν · (l, k) and τ (K) = K/κ. The second one is equivalent to a(l,k) , which appear in equation (7.7) are not pairwise non-isomorphic since a(l, k) = a l ,k does not imply (l, k) = l ,k . For instance, Proof of Proposition 7.2. Assume ϕ ∈ H 0 . Using equations (7.1) and (7.2), we obtain where ϕ k,n : R → R satisfies The latter equation implies that ϕ k,n σ −1 k is periodic with periodicity √ ν for Consequently, √ νky e i n κ t | k, l, n ∈ Z .
Using Lemma 6.1, we compute Hence, for each n ∈ Z, φ 0 0,n spans a representation of type C n/2πκ . Fix k, l, where now at least one of these numbers does not vanish. Furthermore, fix K ∈ N, 0 ≤ K < κ and put φ j := φ k l,jκ+K .

Lemma 6.1 implies
Consequently, the representation of G on span{φ j | j ∈ Z} is equivalent to S τ (K) a(k,l) , which proves (7.7).
Let us turn to H 1 . For fixed m ∈ Z and k ∈ Z, we define Obviously, V k,m = V k+lmr,m holds.
Proof . Obviously, L 2 m (R) ⊗ L 2 S 1 = n∈Z X n m as a vector space. Furthermore, (7.11) and (7.14) imply Zψ n m,q = 2πirmψ n m,q , T ψ n m,q = i n κ ψ n m,q . Using (7.15) and (7.16), we obtain These equations together with (7.12) and (7.13) give for m > 0 and for m < 0. This shows that, for fixed n ∈ Z, the space X n m := span ψ n−qκ m,q | q ∈ N is invariant under G and equivalent to F rm, n/λ as a G-representation.

Strategy
We will see that each standard lattice Γ contains an unshifted normalised straight lattice Γ , which is generated by γ 1 , γ 2 , γ 3 and a power of γ 4 . In particular, λ := λ(Γ ) = 2πκ . We can identify L 2 (Γ\G) with the space of functions in L 2 (Γ \G) that are invariant under γ 4 . By Proposition 7.2, the representation L 2 (Γ \G) decomposes as a direct sum H 0 ⊕ H 1 , where H 0 and H 1 are defined as in (7.7) and (7.8) but with λ and κ instead of λ and κ. The subspaces H 0 and H 1 are invariant under γ 4 . Moreover, each isotypic component of H 0 and H 1 is invariant by γ 4 . Hence we have to determine the invariants of the action of γ 4 on each of these isotypic components.
How do these isotypic components look like? Of course, the subrepresentations C n/λ of H 0 are pairwise non-equivalent. We have C n/λ ∼ = span φ 0 0,n = e i n κ t .
The situation for the second part of H 0 is more complicated. As already mentioned, the summands S K/κ a(l,k) and S K/κ a(l ,k ) can be equal even if (l, k) = (l , k ). For instance, it will turn out that a(l, k) = a(l , k ) if (l, k) and (l , k ) belong to the same orbit of the Z q -action on Z 2 defined by the matrix S q , where q = ord(λ(Γ)). Thus the corresponding representations are equal. Let O denote the Z q -orbit of a non-zero element of Z 2 . Then (l,k)∈O S K/κ a(l,k) ∼ = span φ k l,n = e 2πi(x,y)T −1 µ,ν (l,k) e i n κ t | (l, k) ∈ O, n ∈ K + κ Z for K ∈ N, 0 ≤ K < κ . Let us now turn to the isotypic components of H 1 . These are the summands |m|r · F rm, n/λ for m = 0 and n ∈ Z. By (3.2), each F rm, n/λ is spanned by a ground state ψ 0 and by A j + ψ 0 for j ∈ N >0 . Let m = 0 and n ∈ Z be fixed and consider the subrepresentation of H 1 that is equivalent to |m|r · F rm, n/λ . By Lemma 7.5, the space W m,n of ground states in this subrepresentation is spanned by θ k,n := (2πκ ) √ νy e 2πirmz e int/κ for k = 0, . . . , r|m| − 1. Thus as a vector space. Since Φ k+lmr = Φ k , we may write The action of γ 4 commutes with the regular representation, hence the subspace of γ 4 -invariants in |m|r · F rm, n/λ is isomorphic to W 0 ⊕ A + (W 0 ) ⊕ A 2 + (W 0 ) ⊕ · · · , where W 0 is the space of γ 4invariant elements in W m,n .

The subrepresentation H 0
Recall that we defined integer valued matrices S q by (4.2). Since (S q ) q = I 2 holds, S q defines a left action of Z q on Z 2 by (l, k) → S q · (l, k) . Each orbit of this action contains exactly q elements except that one of (0, 0) ∈ Z 2 . We denote the orbit space by Z 2 /Z q .
We will use again the matrix T µ,ν , see (7.6). Let Γ be a standard lattice and let λ, µ, ν be the corresponding parameters. We put q = ord(λ). Then In particular, the function (l, k) → T −1 µ,ν · (l, k) is constant on orbits of the Z q -action.
In particular, the subrepresentation of γ 4 -invariant elements in H O is equivalent to S K/κ a(l,k) .
Recall from the proof of Proposition 8.2 that and where κ = s 0 κ, n j = s 0 ι j /r for j = 1, 2 and a suitable n 3 ∈ Z. In particular, Γ := γ 1 , γ 2 , γ 3 , γ s 0 4 is a straight lattice. Let us first assume that m > 0. The action of γ 4 on functions is given by In particular, this yields Consequently, γ 4 -invariant ground states can only exist if a + b ∈ 2Z for some k ∈ Z. By the definition of b, this condition is equivalent to the existence of an integer l such that Note that a is an integer. Moreover, as + 2mz 0 for k ∈ K. The summands on the right hand side satisfy L * γ 4 j θ k,n ∈ span{θ k−jmι 2 ,n }. It follows that they are linearly independent since the order of mι 2 in Z rm equals rm/ gcd(rm, mι 2 ) = r/s 2 .
Hence the dimension of the space of γ 4 -invariant ground states equals #K · s 2 r = mr 0 . This proves the assertion for m > 0.
Now, the dimension of the space of γ 4 -invariant functions in W m,n equals the multiplicity of the eigenvalue (−1) n of S ι 1 ,ι 2 . Moreover, the multiplicity m ± of the eigenvalue ±1 of S ι 1 ,ι 2 equals Thus, it remains to compute if ι 2 m is odd and rm is even, 1, if ι 2 m is even and rm is odd, e πimι 1 , if ι 2 m is odd and rm is odd, 1 + e πimι 1 , if ι 2 m is even and rm is even.
Lemma 8.6. For m > 0, the action of ( π 2 ) on W m,n is given by k ∈Zrm e −2πi kk rm θk ,n .
Recall that we assume m > 0. We define Then γ 4 = (2π). We already proved that π 2 acts on W m,n . Thus γ acts on W m,n if a = 0. This is also true for a = 1 since then γ 1 , γ 2 , γ 3 , (2πκ ) is also a sublattice of γ 1 , γ 2 , γ 3 , γ . This action can be computed from the one in the case a = 0 using m r e 2πi k r θ k,n .
We obtain Hence for a = 0 we get if rm = 2 mod 4, −i, if rm = 3 mod 4, and for r even and a = 1 we get Moreover, one easily checks S 2 a (θ k,n ) = e −aπi m r e 2aπi k r θ am−k,n .
Thus for a = 0 we get tr S 2 0 = 1, if rm is odd, 2, if rm is even, and for r even and a = 1 we have Lemma 8.7. Let D be a diagonalisable linear automorphism of order 4 of a vector space of dimension d. Let t, t 1 , t 2 ∈ R be defined by Then the multiplicities m 1 , m i , m −1 and m −i of the eigenvalues 1, i, −1 and −i of D are equal to Using Lemma 8.7 for D = S 0 , we obtain Similary, we obtain for even r and D = S 1 , if m is odd and r ≡ 0 mod 4, Recall that a = 0 if Γ = Γ 4 r (λ) and a = 1 if Γ = Γ 4,+ r (λ). In both cases, we have γ 4 = γ ·(2πκ). Thus L * γ 4 θ k,n = L * (2πκ) L * γ θ k,n = e in 2πκ κ e in π 2κ S a (θ k,n ) = e i sgn(λ)n π 2 S a (θ k,n ).
In particular, the dimension of the space of γ 4 -invariant functions in W m,n equals the multiplicity of the eigenvalue e −in π 2 of S a for λ > 0, and the multiplicity of the eigenvalue e in π 2 of S a for λ < 0. In particular, we have that m(m, n) for λ equals m(m, −n) for −λ. Now, the previous computations for S a give the assertion for m > 0. Furthermore, using m(m, n) = m(−m, −n), we obtain the assertion also for m < 0. r (λ) with λ ∈ π 3 + 2πZ, then the representation H 1 is equivalent to where m(m, n) = 1 6 r|m| − rem 6 (sgn(λm) · n) + 1, if rm + n is even, 1 6 r|m| − rem 6 (sgn(λm) · n) + 3 , if rm + n is odd.
Remark 8.13. In the introduction, we mentioned Brezin's book [3] on harmonic analysis on solvmanifolds, which contains an informal discussion of approaches to the decomposition of the right regular representation. In Chapter 1, Brezin studies a series of four-dimensional examples. These examples are general semi-direct products S σ = H σ R of the Heisenberg group H by the real line defined by a one-parameter subgroup σ of SL(2, R) Aut(H), where σ(1) ∈ SL(2, Z). He fixes a lattice Γ in H which is generated by the set Z 2 in the complement R 2 of the centre Z(H). Then he considers lattices Λ σ that are generated by Γ and the subset Z of the real line. This requires an additional condition for σ(1), which is called suitability by Brezin. Let us now restrict these considerations to the case where S σ is isomorphic to the oscillator group and let Λ σ ⊂ S σ be a lattice in such a group. Then it is easy to see that Λ σ must be of type L 1 1 , L 2 1 or L 4 1 . Moreover, it satisfies a further condition corresponding to ι 1 = ι 2 = 0 in our classification of standard lattices. Compared to the general case, this special choice of lattices simplifies the decomposition of the right regular representation.
Brezin uses a decomposition of the space of L 2 -functions on the quotient into two subspaces Y (0) and n =0 Y (n), which correspond to our spaces H 0 and H 1 . However, his strategy to treat these subspaces is somewhat different from ours. He prefers an inductive approach. He identifies functions in Y (0) with functions on the three-dimensional quotient of R 2 σ R by a lattice generated by Z 2 ⊂ R 2 and Z ⊂ R. For functions in n =0 Y (n), he considers their restriction to the Heisenberg group H. This yields a representation of H σ Z. This representation is decomposed into irreducible ones. Then all extensions of these irreducible representations of H σ Z to representations of S σ are determined and those that actually occur in the right regular representation are identified.
In Brezin's book, the emphasis is more on a general discussion of approaches than on explicit results. However, the case of lattices of type L 4 1 is studied in more detail and some computations of multiplicities are done. Unfortunately, no final result is formulated, but we think that, basically, these computations coincide with ours.

The spectrum of the wave operator
The (Lorentzian) Laplace-Beltrami operator, i.e., the wave operator, equals minus the Casimir operator of the right regular representation. On each irreducible subrepresentation σ it acts as multiplication by a scalar x σ . This scalar is known for all irreducible unitary representations of Osc 1 , see Section 3. Consequently, once the decomposition of ρ, L 2 (L\Osc 1 ) into irreducible summands is known, the spectrum of the Laplace-Beltrami operator can be computed explicitly. More exactly, the spectrum is equal to the closure of the set of all numbers x σ ∈ R, where σ is an irreducible subrepresentation of ρ. Let us do this, for example, for straight lattices.
Then the spectrum of the wave operator of L\Osc 1 equals 4π r κ Z ∪ A(µ, ν) if κ is even and 2π r κ Z ∪ A(µ, ν) if κ is odd. In particular, the spectrum is discrete.
Proof . The assertion follows from Proposition 7.2 and the formulas for ∆ C d , ∆ S τ a , and ∆ F c,d in Section 3.
Corollary 8.15. On each quotient of Osc 1 by a standard lattice, the spectrum of the wave operator is discrete.
Proof . As we have seen, every quotient by a standard lattice is finitely covered by a quotient by a straight lattice.
There exist lattices in Osc 1 such that the spectrum of the wave operator on the quotient space is not discrete.

Generalised quadratic Gauss sums
We used several times the following reciprocity formula for generalised quadratic Gauss sums [18], see also [2]. Let a, b, c be integers with ac = 0 such that ac + b is even. Then

Summary
Suppose we are given a lattice L in Osc 1 = H R and we want to determine the spectrum of L\Osc 1 . First, we reduce the problem to the situation where L is normalised and unshifted. Let L be arbitrary. Then L ∩ H is isomorphic to a discrete Heisenberg group H r 1 (Z) for a uniquely determined natural number r. We put r(L) := r. The projection of L ⊂ Osc 1 = H R to the second factor is generated by a uniquely determined element λ of πN >0 ∪ {λ 0 + 2πZ | λ 0 = π/3, π/2, 2π/3}. We set λ(L) := λ. We choose adapted generators α, β, γ, δ (see Definition 4.7) and determine the type of L, i.e., the isomorphism class of L considered as a discrete oscillator group, using Proposition 4.3. The commutator group [L ∩ H, L ∩ H] is generated by an element (0, h 2 , 0) for some h ∈ R >0 . We put S := h −1 · I 2 . Then L := F S (L) is a normalised lattice. Now we compute the number s L for L as in Definition 4.8, where for suitable z α , z β , z δ ∈ R, which is isomorphic to Γ r (λ, µ, ν, ξ 0 , z 0 ) under inner automorphisms take φF η φ −1 with η = −z β 1 √ ν + z α − µ √ ν + i √ ν and Lemma 4.18 . By Proposition 8.2, we obtain that H 0 for Γ is equivalent to a representation given by formula (9.2) but with a(l, k) = T −1 µ,ν (l, k) . Since T −1 µ,ν (l, k) = T (l, k) , the assertion for q = 1 follows. Now, assume that q is in {2, 3, 4, 6}. As before there is an inner automorphism ϕ such that ϕ(Γ) is a standard lattice. Let µ and ν be the corresponding parameters for ϕ(Γ). Hence, by Proposition 8.1, H 0 is equivalent to where a (l , k ) = T −1 µ,ν · (l , k ) . It remains to show that this representation is indeed equal to the one in (9.1). We know that there exist matrices M ∈ SL(2, Z) and S ∈ SO(2, R) such that ST M = T −1 µ,ν . Since S q = T µ,ν e −iλ T −1 µ,ν holds, the map