The cubic Dirac operator on compact quotients of the oscillator group

We determine the spectrum of Kostant's cubic Dirac operator $D^{1/3}$ on locally symmetric Lorentzian manifolds of the form $\Gamma\backslash {\rm Osc}_1$, where ${\rm Osc}_1$ is the four-dimensional oscillator group and $\Gamma\subset {\rm Osc}_1$ is a (cocompact) lattice. Moreover, we give an explicit decomposition of the regular representation of ${\rm Osc}_1$ on $L^2$-sections of the spinor bundle into irreducible subrepresentations and we determine the eigenspaces of $D^{1/3}$.


Introduction
This paper is a contribution to the spectral theory of the cubic Dirac operator on compact locally symmetric Lorentzian manifolds.The cubic Dirac operator has been introduced by Kostant [Ko] on naturally reductive spaces as a purely algebraic object.However, it can be also considered as a geometric Dirac operator D 1/3 that belongs to a family of Dirac operators D t .These operators are induced by non-standard connections [A, Go].The square of the cubic Dirac operator satisfies a simple formula, actually it equals minus the Casimir operator up to terms of order zero [Ko].The spectrum of geometric Dirac operators on pseudo-Riemannian manifolds has been calculated for some specific examples.For the Dirac operator D = D 1/2 associated with the Levi-Civita connection, the point spectrum of the pseudo-Riemannian torus T p,q has been computed, see [Ba] for T 1,2 and [Ln] for the general case.Kunstmann [Ku] studied the spectrum for pseudo-Riemannian spheres.In the case of even dimension of the manifold or even index of the metric, he computed the point spectrum and proved that the imaginary axis belongs to the continuous spectrum and that the residual spectrum is empty.Reincke [Rei] explicitly computed the full spectrum of D on R p,q , the flat torus T p,q and products of the form T 1,1 × F , where F is an arbitrary compact, even-dimensional Riemannian spin manifold.Here we will consider the Dirac operator on four-dimensional compact homogeneous spaces G/Γ, where G is a solvable Lie group endowed with a bi-invariant Lorentzian metric and Γ is a (cocompact) lattice in G.More exactly, G will be the four-dimensional oscillator group.Homogeneous spaces of the form G/Γ for solvable G are not only examples of compact homogeneous Lorentzian manifolds but play a central role in their classification.Indeed, Baues and Globke [BG] proved the following result.Let M = G/H be a compact homogeneous pseudo-Riemannian manifold, and let G be connected and solvable.Then H is a lattice in G and the pseudo-Riemannian metric on M pulls back to a bi-invariant metric on G.In the Lorentzian case such Lie groups G admitting a biinvariant metric were classified by Medina [Me].They are products of an abelian group by a so-called oscillator group, which is a certain semi-direct product of a Heisenberg group by the real line.Combining these results of Baues, Globke and Medina, Revoy, one obtains a classification of all solvable Lie groups G for which there exist compact Lorentzian G-homogeneous spaces.This classification in the Lorentzian case can already be found in [Z].If we restrict ourselves to four-dimensional manifolds, we now see that the group G either is isomorphic to the abelian group R 4 or to the four-dimensional oscillator group Osc 1 .Thus M is a flat Lorentzian torus or a quotient of Osc 1 by a lattice and the metric on M is induced by the bi-invariant metric on Osc 1 .Let us explain the four-dimensional oscillator group Osc 1 in more detail.This group is a semi-direct product of the 3-dimensional Heisenberg group H by the real line R, where R acts trivially on the centre Z(H) of H and by rotation on H/Z(H).In particular, it is solvable.As mentioned above, it admits a bi-invariant Lorentzian metric.This metric is a particular case of a plane wave metric.The Lie algebra osc 1 of Osc 1 is 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.Actually, 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 .The oscillator group contains lattices.Each lattice L in Osc 1 gives rise to a compact locally-symmetric Lorentzian manifold L\Osc 1 .Notice that for a lattice, we take the quotient of the left action and therefore write the subgroup on the left side.The problem of classifying lattices in Osc 1 was first considered by Medina and Revoy [MeRe].Note, however, that the result in [MeRe] is 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 Fischer [Fi] and up to inner automorphisms of Osc 1 by Fischer and Kath [FiK].Here we will not consider arbitrary lattices but we will concentrate on basic lattices, see Section 4 for a justification of this assumption.Let X be a quotient of Osc 1 by a basic lattice L. On X, we fix a spin structure and consider the spinor bundle Σ.We use the canonical indefinite inner product on smooth sections of Σ(X) to define a Krein space L 2 (Σ(X)).Then the cubic Dirac operator is defined on L 2 (Σ(X)) and iD 1/3 : L 2 (Σ(X)) → L 2 (Σ(X)) is essentially self-adjoint (as an operator on a Krein space).Our aim is to determine the spectrum of this operator.The scalar curvature of X = L\Osc 1 is zero.This implies that the zero order terms vanish in the formula for the square of D 1/3 .Thus (D 1/3 ) 2 = −Ω, where Ω denotes the Casimir operator on L 2 (Σ(X)).In order to determine the point spectrum of D 1/3 , we will consider L 2 (Σ(X)) as a representation of Osc 1 .We provide an explicit decomposition of L 2 (Σ(X)) into a (discrete) direct sum of irreducible subrepresentations with finite multiplicities, see Theorem 5.2.The cubic Dirac operator preserves each summand and its square −Ω acts by scalar multiplication on it.So we determine first the eigenvalues of −Ω.To do so, we use results of [FiK].We consider a finite cover X of X such that the pull back of the spin structure of X to X becomes trivial.Then we decompose L 2 (Σ( X)) = L 2 ( X)⊗∆ according to [FiK] and afterwards we determine the space of sections that are invariant under all decktransformations of the cover X → X.This indeed allows to determine the point spectrum of −Ω, see Theorem 6.1.In order to determine the point spectrum of D 1/3 , we finally show that all square roots of eigenvalues of −Ω are eigenvalues of D 1/3 .Using the decomposition of L 2 (Σ(X)), we can also prove that the whole spectrum of D 1/3 is all of C. From general properties of Dirac operators, it follows, moreover, that the residual spectrum is empty.In summary, we obtain: Theorem 1.1 Let L ⊂ Osc 1 be a basic lattice and X := L\Osc 1 be the quotient space.Let D 1/3 : L 2 (Σ(X)) → L 2 (Σ(X)) be the cubic Dirac operator for a fixed spin structure on X.The spectrum of D 1/3 is equal to C. The point spectrum of D 1/3 depends on the spin structure.It consists of 0 if the kernel of Ω is non-trivial and the two roots of each of the non-zero eigenvalues of −Ω (see Theorem 6.1 for an explicit description of the point spectrum of −Ω depending on the spin structure).The residual spectrum is empty.
In particular, it turns out that the point spectrum for a basic lattice is always discrete.This is no longer true for arbitrary lattices.In Section 6.4, we give examples of shifted lattices for which the point spectrum of D 1/3 on the quotient has accumulation points.We not only determine the eigenvalues of D 1/3 , but we also describe the corresponding eigenspaces explicitly, see Section 6.2.In Section 6.5, we use these results to determine the spectrum also for all other D t .We describe the point spectrum of D t in terms of the eigenvalues of D 1/3 and show that the full spectrum of D t is also equal to C and that the residual spectrum of D t is empty.

Invariant connections on Lie groups
In this short subsection, we want to recall some basic facts on connections on Lie groups.Let G be a simply-connected Lie group endowed with a bi-invariant semi-Riemannian metric • , • G .Let g denote the Lie algebra of G.The metric • , • G on G corresponds to an ad(g)-invariant scalar product • , • on g.As a Lie group, G is endowed with a canonical left-invariant connection ∇ 0 , for which all left-invariant vector fields are parallel.We use ∇ 0 to define a one-parameter family of metric connections ∇ t by for left-invariant vector fields X, Y .For t = 1/2, we obtain the Levi-Civita connection of • , • G .All these connections define the same divergence operator on vector fields since Let us explain the relation between the connection ∇ t of G and the canonical connections of G understood as a reductive homogeneous space K/H for K := G × G and Let k and h denote the Lie algebras of K and H, respectively.Then

Spin structures on quotients of Lie groups
Here we gather some facts on spin structures on quotients of Lie groups by discrete subgroups.Let G be a simply-connected Lie group endowed with a bi-invariant semi-Riemannian metric • , • G , an orientation and a time orientation.As above, let g denote the Lie algebra of G and • , • the induced scalar product on g.
Let Γ ⊂ G be a discrete subgroup and consider the quotient Γ\G.The metric, the orientation and the time orientation on G induce a metric, an orientation and a time orientation on the quotient.The tangent bundle of Γ\G is trivial.Let π : G → Γ\G denote the projection, then is an isomorphism of vector bundles.Consequently, the bundle of oriented and timeoriented orthonormal frames on Γ\G equals where SO + (g) denotes the identity component of SO(g) := SO(g, • , • ).The spin structures on Γ\G are classified by H 1 (Γ\G, Z 2 ), see [Ba], Satz 2.6.or [Fr].Since we obtain a one-to-one correspondence between isomorphism classes of spin structures on Γ\G and homomorphisms ε : Γ → Z 2 = Z/2Z, see [Fr,Section 2.2] for the more general case of a covering map.The spin structure corresponding to ε equals where the action of Γ on G × Spin + (g) is given by see [Ba,Folg. 2.3] or [Gi,Prop. 1.4.2].Let ∆ g denote the spinor representation of Spin + (g).We obtain for the spinor bundle associated with P ε Spin + (Γ\G).We identify vector fields on Γ\G with functions X : G → g that are invariant under left translation by Γ and we identify smooth sections of Σ(Γ\G) with smooth functions ψ : G → ∆ g that are invariant under the action of Γ, that is for all γ ∈ Γ.

Krein spaces
Since the natural scalar product on the spinor bundle of a Lorentzian spin manifold is indefinite, sections of the spinor bundle do not constitute a Hilbert space in a natural way.Therefore we will work in Krein spaces as it is done in [Ba].For a general theory of such spaces see [Bo, Lr].
Let K be a complex vector space and • , • a possibly indefinite inner product on K.
We define symmetric operators and selfadjoint operators on K in the same way as in the definite case.
Definition 2.1 A Krein space (K, • , • ) consists of a complex vector space K and an indefinite inner product • , • on K such that there exists a selfadjoint linear map J : K → K with the following properties: • is a positive definite inner product that makes K a Hilbert space, 2. J 2 = id.
A linear map J that satisfies this condition is called a fundamental symmetry.On K, we consider the strong topology.It is defined to be the norm topology of the Hilbert space (K, (• , •)), where (• , •) = • , J• for any fundamental symmetry J. Although, in general, the linear map J is not uniquely determined by (K, • , • ), the strong topology is well defined, i.e., independent of J.
If A is a closed linear operator with a dense domain, then spec(A) denotes the spectrum of A and spec p (A), spec c (A), and spec r (A) denote the discrete, continuous, and residual spectra, respectively.
Fact 2.2 [Lr] If A is a closed selfadjoint operator on a Krein space, then the complex conjugate of spec(A) satisfies 1. spec r (A) ⊂ spec p (A),

The space of spinors
In this section, we recall some basic facts on inner products on the space of spinor fields on a pseudo-Riemannian manifold.Since later on we will be interested in Dirac operators on Lorentzian manifolds, we restrict the explanations to the case where the metric of the manifold has Lorentzian signature.Moreover, we will concentrate on the case where the manifold is a quotient of a Lie group G by a discrete subgroup Γ although most of the results could be stated as well for general Lorentzian manifolds.Then, on the spinor module ∆ g , there exists a scalar product for all X ∈ g, where '•' denotes the Clifford multiplication.This scalar product is unique up to multiplication by a real number different from zero.It defines a scalar product on the bundle Σ(Γ\G), which we also denote by • , • ∆ .We choose a time-oriented left-invariant vector field ξ on G with ξ, ξ G = −1.This vector field defines a vector field on the quotient Γ\G, which we also denote by ξ.We use ξ to define a map The stabiliser of a timelike vector is a maximal compact subgroup of the Lorentz group.Therefore the vector field ξ defines a reduction of the frame bundle of Γ\G to a maximal compact subgroup of the Lorentz group.The scalar product (• , •) ∆,ξ is invariant under this subgroup.
The volume form of the metric • , • G induces a measure µ on Γ\G, which is invariant under G.We define inner products on the space of compactly supported smooth sections of Σ(Γ\G) by The first one is indefinite, the second one is positive definite.We can identify spinors with smooth functions with values in ∆ g satisfying ( 1).If we identify, in addition, (∆ g , (• , •) ∆,ξ ) with the standard unitary space by choosing an orthonormal basis, then the scalar product (• , •) ξ on smooth sections of Σ(Γ\G) becomes the standard L 2 -product on functions (with several components).We define L 2 ξ (Σ(Γ\G)) as the completion of the space of compactly supported smooth sections in Σ(Γ\G) with respect to the norm induced by (• , •) ξ .
We want to compare the spaces L 2 ξ (Σ(Γ\G)) for different choices of ξ.Let r ξ be the leftinvariant Riemannian metric on G defined by reversing the sign of ( for X, Y in the orthogonal complement of ξ with respect to • , • G .Let ξ 1 and ξ 2 be time-oriented left-invariant vector fields with ξ 1 , ξ 1 = ξ 2 , ξ 2 = −1.Since r ξ 1 and r ξ 2 are left-invariant, they are quasi-isometric, i.e., there exists a constant C > 0 such that Of course, also the metrics induced by r ξ 1 and r ξ 2 on Γ\G are quasiisometric.This implies that the spaces L 2 ξ 1 (Σ(Γ\G)) and L 2 ξ 2 (Σ(Γ\G)) are the same in the following sense.They coincide as vector spaces (whose elements are equivalence classes of Cauchy series in the space of compactly supported smooth sections of Σ(Γ\G)), and the identity ) is a bounded isomorphism with bounded inverse [Rei,Theorem 3.8].Let us fix ξ as above and put K := L 2 (Σ(Γ\G)) := L 2 ξ (Σ(Γ\G)) as a vector space.The map J ξ can be extended to K, and we can define an indefinite inner product [Ba,Satz 3.16].Its definition is independent of ξ.For any time-oriented left-invariant vector field ξ ′ the map J ξ ′ is a fundamental symmetry.

The cubic Dirac operator
Every connection ∇ t on G induces a connection on Σ(Γ\G), which we will also denote by ∇ t .Consider a smooth section ψ : G → ∆ g of Σ(Γ\G), see (1).Let X ∈ g be a left-invariant vector field on G. Then X can also be considered as a vector field on Γ\G.Then ∇ Mei,p. 152].Here and in the following, {e a | a = 1, . . ., n} denotes a basis of g and {e a | a = 1, . . ., n} its dual basis with respect to • , • .These elements of g can also be understood as vector fields on Γ\G or, equivalently, as constant maps G → g, g → e a and G → g, g → e a .Then the Dirac operator corresponding to ∇ t is equal to If we apply this to a smooth section ψ of the spinor bundle, we obtain where e a e b e c is understood as an element of the Clifford algebra [A, Eq. ( 5)].
For t = 1/3 we obtain the cubic Dirac operator D 1/3 .The square of this operator is related to the Casimir operator Ω = e a e a ∈ U (g) with respect to • , • by Here Scal denotes the scalar curvature of • , • G on Γ\G.
For the following remark, let us again concentrate on the Lorentzian case in order to simplify the exposition.
is essentially selfadjoint in the Krein space (L 2 (Σ(Γ\G)), • , • ).This can be seen as follows.We have noticed that ∇ t defines the same divergence operator as the Levi-Civita connection ∇ 1/2 .Furthermore, the Riemannian metric r ξ on Γ\G that is obtained by reversing the sign in direction of a time-oriented left-invariant vector field ξ is complete.Indeed, r ξ is left-invariant on G, hence (G, r ξ ) is a homogeneous Riemannian manifold and therefore complete.Hence (Γ\G, r ξ ) is also complete.Now the assertion follows from [Ba,Satz 3.19].

The right regular representation
Let (G, • , • G ) be as above and let Γ be a cocompact discrete subgroup of G.The right regular representation ρ of G on L 2 (Γ\G) is the unitary representation given by It is a classical result that (ρ, L 2 (Γ\G)) is a discrete direct sum of irreducible unitary representations of G with finite multiplicities, see e.g.[Wo].
Let F be an automorphism of G.For a representation (σ, V ) of G we define a representation is an equivalence of representations.
Recall that a smooth section of the spinor bundle Σ(Γ\G) is identified with a smooth Γ-invariant function ϕ : G → ∆ g .In this way we can also define an action of G on L 2 (Σ(Γ\G)) by ( 4).
3 The oscillator group and its Lie algebra

The 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 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 If we identify Osc 1 ∼ = R 4 as sets, then the Lebesgue measure is left-and right-invariant with respect to multiplication in Osc 1 .
Let us consider the automorphisms of this group.For η ∈ C, let C η : Osc 1 → Osc 1 be the conjugation by (η, 0, 0).Then Furthermore, we define an automorphism T u of Osc 1 for u ∈ R by Finally, consider an R-linear isomorphism S of C such that S(iξ) = ǫiS(ξ) for an element ǫ ∈ {1, −1} and for all ξ ∈ C. Then ǫ = 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 [Fi].Besides C η also F S is an inner automorphism if S ∈ SO(2, R).
In some of our computations we will use a slightly different multiplication rule for the oscillator group.It looks more complicated than the usual one but it will make the 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) parametrized by x, y, z ∈ R with group multiplication We define an action l of R on H(1) by 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 ( 9) is denoted by (t).It is easy to check that is an isomorphism.Also here we can identify Osc M 1 ∼ = R 4 .Then φ preserves the Lebesgue measure.

The oscillator algebra
The Lie algebra osc 1 of the four-dimensional oscillator group is spanned by elements Z, X, Y, T , whose non-vanishing commutators are The following result about the centre of the universal enveloping algebra of g is known, see [MüRi].We give a short self-contained proof.
Proposition 3.1 The centre Z(U (g)) of the universal enveloping algebra U (g) of g is generated by Proof.Obviously, Ω 0 belongs to Z(U (g)).By the Poincaré-Birkhoff-Witt Theorem, the symmetrisation map sym : S(g induces an isomorphism S(g) g ∼ = gr(Z(U (g))).Hence it suffices to show that S(g) g is generated by the centre of g and the preimage of Ω 0 under sym.Obviously, it suffices to show this for any S k (g) g .
Instead of g, we consider its complexification g C .The vectors Z, T , N + := X + iY , N − := X − iY constitute a basis of g C .Their non-vanishing Lie brackets are In this new basis, we have Ω 0 = N + N − +2ZT +iZ.Hence Ω S := N + N − +2ZT ∈ S(g) g is a preimage of Ω 0 under sym.Let ω be in S(g) g and assume that ω is homogeneous. Then The symmetrisation map ( 11) is not a homomorphism.For the sake of completeness let us determine Duflo's factor for g although we will not use it in the present paper.See [Mei] for a general introduction to this subject.For ξ = zZ with respect to the basis Z, N + , N − , T .This gives J(ξ) = det(j(ad(ξ))) = j(−it)j(it), where Hence Duflo's factor equals

The biinvariant metric and the cubic Dirac operator
On Osc 1 , there exists a 2-parameter family of bi-invariant metrics.The metrics are defined by the ad-invariant scalar products on osc 1 given by span{X, Y } ⊥ span{Z, T }, and for r > 0 and s ∈ R. It is well known that there is only one bi-invariant Lorentzian metric on Osc 1 up to isometric Lie group isomorphisms [MeRe].The above defined family of metrics arises as the orbit of such a metric under the action of the automorphism group of Osc 1 .The Casimir operator corresponding to the metric with parameters r > 0 and s ∈ R is equal to 1 r (Ω 0 − sZ 2 ), where Ω 0 ∈ Z(U (g)) is as defined in Proposition 3.1.In the present paper, we consider the metric for r = 1 and s = 0, i.e., The Casimir operator Ω of this metric equals Let ∆ = C 4 denote the spinor module of the metric Lie algebra osc 1 .We can choose a basis u 1 , . . ., u 4 of ∆ such that the Clifford multiplication by Z, X, Y and T is given with respect to this basis by In particular, According to section 2.4, there is an indefinite scalar product • , • ∆ on ∆ satisfying (2) and this scalar product is uniquely defined up to a constant.We fix it by u 1 , u 2 ∆ = u 3 , u 4 ∆ = 1, and u i , u j ∆ = 0 for all other indices.We choose the timelike left-invariant vector field ξ = 1 √ 2 (Z − T ) in order to define a definite spin(3)-invariant scalar product: (u, v) ∆ := u, ξ • v ∆ .The vectors u 1 , . . ., u 4 constitute an orthonormal basis with respect to (• , •) ∆ .

Unitary representations of the oscillator group
The irreducible unitary representations of osc 1 can be determined by applying a generalised version of Kirillov's orbit method.An explicit description of these representations can be found in [Ki,§4.3],where the oscillator Lie algebra is called diamond Lie algebra.Let us recall this description.Note that the case c < 0 in item (iii) does not appear in [Ki].The infinite-dimensional representations will be given only on the Lie algebra level.Every irreducible unitary representation of the oscillator group is equivalent to one of the following representations, see also [FiK]: ) where σ is given by ).The orthonormal system φ n := e int , n ∈ Z satisfies (iii) For c > 0, d ∈ R, we consider the Hilbert space for ϕ 1 , ϕ 2 ∈ F c (C).Then the representation σ := σ c,d on F c (C) is given by The functions ψ n := ( √ πcξ) n √ n! , n ≥ 0, constitute a complete orthonormal system of F c (C) and we have Furthermore, for c < 0, d ∈ R, we consider with scalar product given by ( 13) with c replaced by −c, now for ϕ 1 , ϕ 2 ∈ F c (C).The representation σ := σ c,d on F c (C) is given by Here, the functions , n ≥ 0, constitute a complete orthonormal system and we have Now, Equations ( 14) and ( 15) hold for A + := σ * (X − iY ) and A − := σ * (X + iY ).We will use the notation F c,d := (σ c,d , F c (C)) for all c = 0 and d ∈ R.
Let F be an automorphism of Osc 1 .In (5), we defined the pullback of a representation of G by F .The following table shows (the equivalence class of) F * (σ, V ) for the case that V is one of the irreducible unitary representations of osc 1 and F is one of the (outer) automorphisms T u or F S introduced in Section 3.1 by ( 7) and ( 8).Note that C η as an inner automorphism does not change V .

Straight and basic lattices
In Section 6 we will study the spectrum of the cubic Dirac operator on compact quotients of the oscillator group.More exactly, we consider quotients of Osc 1 by discrete uniform subgroups of Osc 1 .We will call such subgroups lattices.This is justified by the fact that the group Osc 1 is solvable and therefore a quotient by a discrete subgroup is of finite measure (for the measure inherited from Haar measure on Osc 1 ) if and only if it is compact.The lattices of the oscillator group are known.They were classified up to automorphisms of Osc 1 by Fischer [Fi].Since here we are interested in the spectrum of the quotient and therefore in the right regular representation, we need a classification up to inner automorphisms, which can be found in [FiK].
To avoid too much technical effort, we will concentrate on straight lattices, where a lattice in Osc 1 is called straight if it is generated by a lattice in H and an element δ of the centre of Osc 1 .It can be shown that each lattice in Osc 1 contains a sublattice of finite index which is a straight, see [Fi,Section 8].In other words, each lattice in Osc 1 is virtually straight.Moreover, we will assume that the lattice is unshifted and normalised in the sense of [FiK].
has covolume one with respect to the standard metric of R 2 .A normalised straight lattice is called unshifted if δ can be chosen in the R-factor of Osc 1 = H ⋊ R, i.e., δ = (0, 0, 2πκ).This leads us to the following definition.
Definition 4.1 A lattice of Osc 1 is called a basic lattice if it is normalised and generated by a lattice in the Heisenberg group and an element (0, 0, 2πκ) ∈ R ⊂ H ⋊ R.
The additional assumptions to be normalised and unshifted are justified by the fact that each straight lattice can be normalised and shifted by (outer) automorphisms of Osc 1 .More exactly, the following holds.Let M strt denote the set of all isomorphism classes of straight lattices of Osc 1 with respect to inner automorphisms of Osc 1 and let B ⊂ M strt be the set of isomorphism classes of basic lattices.For a basic lattice L we define numbers I is a bijection.Here I 2 denotes the identity on R 2 .
Proof.The assertion follows from Thm. 4.12 in [FiK].Indeed, the property to be straight is invariant under automorphisms.Therefore we can restrict the bijections in [FiK,Thm. 4.12 ] to straight lattices.In the notation of [FiK], we thus obtain bijections from It remains to check that the composition of these bijections has the form asserted in the proposition.Let L be a basic lattice.Then we have q = 1 and x δ = y δ = 0 in item 1 in [FiK,Def. 4.8], which implies v = w = 0 and therefore s 0 = 1.The assertion follows.✷ The computation of the spectrum relies on the decomposition of the right regular representation into irreducible subrepresentations.Once this decomposition is known for basic lattices, the decomposition for arbitrary lattices can be derived using Proposition 4.2.Indeed, according to ( 16) the decomposition of L 2 (F S (T u (L))\Osc 1 ) can be computed from that of L 2 (L\Osc 1 ).That is why we focus on basic lattices here.
We will denote this lattice by L r (κ, µ, ν).In [FiK], it is denoted by L r (2πκ, µ, ν, 0, 0), but here we do not need the last two parameters since we only consider straight lattices.
As an abstract group, the lattice L is isomorphic to the direct product of a discrete Heisenberg group We fix a spin structure on X = L\Osc 1 .As explained in Section 2, it is determined by a homomorphism ε : L → (Z 2 , +).We will use the notation and write ε = (ε 1 , . . ., ε 4 ).Note that a map ε : L → Z 2 is a homomorphism if and only if rε 3 = 0. Let again ∆ = C 4 denote the spinor module of the metric Lie algebra osc 1 .
5 The right regular representation for basic lattices

Strategy
Let L be a basic lattice of Osc 1 .We consider X := L\Osc 1 .The aim of this subsection is to decompose the representation L 2 (Σ(X)) of Osc 1 into irreducible components.We want to apply the results of [FiK] for the decomposition of the right regular representation on L 2 -functions.In [FiK], the decomposition of L 2 (L\Osc 1 ) is determined for arbitrary lattices, where first the computation is reduced to the case of unshifted and normalised lattices and then explicit formulas are given in this case.In particular, [FiK,Prop. 7.2] describes the decomposition for basic lattices.In order to apply these results, we consider a finite covering of X such that the lifted spin structure becomes trivial.More exactly, we consider the covering X = L ′ \Osc 1 of X, where L ′ is the subgroup of L generated by l 2 1 , . . ., l 2 4 .Since L is a basic lattice, this subgroup is normal.We obtain that X = I\ X, where I ⊂ Iso( X) is the finite group generated by the actions of l 1 , . . ., l 4 on X.The spin structure on X lifts to a spin structure on X, which is now the trivial one since 2ε j = 0, j = 1, . . ., 4. Therefore the associated spinor bundle of X equals X × ∆ and sections in this bundle can be identified with functions from Osc 1 to ∆ that are invariant under left translation by elements of the lattice L ′ .To recover the sections in the spinor bundle of X from these sections we have to find those sections in X × ∆ that are invariant under the action of the group I of decktransformations, where this action is defined as follows where l * denotes the left translation by l ∈ L. Thus we can identify Consequently, we can obtain a decomposition of L 2 (Σ(X)) into irreducible subspaces in the following way.First we decompose L 2 ( X) according to [FiK].Then, for each isotypic component, we determine the subspace of sections that are invariant under the action of l 1 , . . ., l 4 by ( 17).Finally, tensoring by ∆ gives the result.As said above, the explicit formulas in [FiK] for the decomposition work under the assumption that the lattice is normalised.However, note, that our new lattice L ′ generated by l 2 1 := (2T −1 µ,ν e 1 , 0, 0), l 2 2 := (2T −1 µ,ν e 2 , 0, 0), l 2 3 := 0, 2 r , 0 , l 2 4 := 0, 0, 4πκ , is not normalised.Indeed, the projection of L ′ ∩ H to H/Z(H) ∼ = R 2 is generated by the projections of l 2 1 and l 2 2 .Thus it has covolume 4. Therefore we apply the automorphism 2r and κ ′ = 2κ.Thus the formulas in [FiK] apply to F S (L ′ ).

5.2
The decomposition of L 2 (Σ(X)) As explained in Subsection 3.3 we have to describe the decomposition of L 2 (Σ(X)) into irreducible subrepresentations up to equivalence.We will use the representations introduced in Subsection 3.4.To formulate the result we need the following notations.Let Theorem 5.2 On X = L r (κ, µ, ν)\Osc 1 we consider the spin structure given by ε = (ε 1 , . . ., ε 4 ), where we assume (ε 1 , ε 2 ) = (0, 1) if ε 1 = ε 2 .Then we have L 2 (Σ(X)) = 4H 0 ⊕ 4H 1 with where Proof.We proceed according to the strategy outlined in Subsection 5.1.By Proposition 5.1, we have to determine the decomposition of F * S (L 2 (F S (L ′ )\Osc 1 ) F S (I) ), where F S (ξ, z, t) = ( 1 2 ξ, 1 4 z, t) and L ′ is the lattice generated by l 2 1 , . . ., l 2 4 .It turns out that calculations on Osc M 1 are easier than on Osc 1 .Therefore we transform the lattice F S (L) by the isomorphism φ : Osc 1 → Osc M 1 defined in (10).We obtain , where r ′ = 2r and κ ′ = 2κ.We denote the lattices (φ • F S )(L) and (φ • F S )(L ′ ) by Γ and Γ ′ , respectively.The push-forward of a representation (σ, In the following, we will identify these representations with each other and omit (φ −1 ) * in the notation.In particular, we identify the representation It is natural to use the push-forwards of the irreducible representations of Osc 1 as models for the irreducible representations of Osc M 1 .Then the irreducible unitary representations of Osc M 1 are (φ −1 ) * C d , (φ −1 ) * S τ a and (φ −1 ) * F c,d .According to the above remark we simply write C d , S τ a and F c,d instead of (φ −1 ) * C d , (φ −1 ) * S τ a and (φ −1 ) * F c,d .Our task now is to describe explicitly the irreducible subrepresentations of the right regular representation L 2 (Γ ′ \Osc M 1 ) of Osc M 1 .Here we can use the results of [FiK].The representation , where the subrepresentations H ′ 0 and H ′ 1 are given as follows: The subspaces of H ′ 0 and H ′ 1 corresponding to the irreducible subrepresentations of L 2 (Γ ′ \Osc M 1 ) in the above formulas are explicitly given as follows.For a function f : Osc M 1 → C, we will denote by f (x, y, z, t) the image of M (x, y, z)(t) under f .Then the representation H ′ 0 is the direct sum of subspaces Next we want to describe the decomposition of H ′ 1 .For m ∈ Z =0 and n ∈ Z, we consider the subspace We denote by A + the ladder operator X + sgn(m)iY .Then H ′ 1 decomposes into the direct sum of subspaces In order to obtain L 2 (Γ ′ \Osc M 1 ) Γ , we determine the elements in the isotypic components of that are invariant under γ 1 , . . ., γ 4 .We compute where we used that κ ′ is even.Thus, the action defined by ( 17) is now given by Furthermore, Indeed, the formula for γ 2 .θm,n,k follows from the following observation.For k ∈ Z r ′ |m| , we have , where the subrepresentations (H ′ 0 ) Γ and (H ′ 1 ) Γ are given by where m C (n), m S (a, K) and m F (m, n) are given by ( 21), ( 22) and ( 23), respectively.Indeed, the formula for m C (n) follows from ( 24) and the one for m S (a, K) from ( 25).
Finally, m F (m, n) is obtained from ( 26), where we used that ε 3 = 0 if r is odd.
In order to obtain L 2 (Σ(X)), we have to pull back (H ′ 0 ) Γ ⊕ (H ′ 1 ) Γ by F S and to tensor the result by ∆, i.e., to multiply by 4. By ( 16) we have .
Finally, we replace r ′ by 2r and κ ′ by 2κ and obtain the assertion.✷ Remark 5.3 Actually, the proof shows more than we claimed in the theorem.It gives an explicit decomposition of the representation and not only an equivalence.
Example 5.5 In the case, where r is even and ε 3 = 1, we have m C (n) = 0 for all n ∈ Z, m S (a, K) = 0 for all a and K, and 6 The spectrum of the cubic Dirac operator In this section, we compute the spectrum of (the closure of) the cubic Dirac operator D 1/3 on X = L\Osc 1 for any basic lattice L of Osc 1 .We obtain that spec(D 1/3 ) consists only of the point spectrum and the continuous spectrum, which we will compute in Section 6.1 and Section 6.3, respectively.It will turn out that spec(D 1/3 ) = C.This will prove Theorem 1.1.Finally we determine the spectrum of the Dirac operators D t for all other t.

The spectrum of −Ω
The operator (D 1/3 ) 2 acts as −Ω in the first factor of the tensor product in ( 18) and trivially on the second one.On each irreducible representation, Ω acts by a scalar.Hence, Theorem 5.2 allows us to compute the spectrum of −Ω on quotients of the oscillator group by basic lattices.If L r (κ, µ, ν) is a basic lattice, the volume of X = L r (κ, µ, ν)\Osc 1 only depends on the quotient of κ by r.For given r, κ ∈ N >0 we have .
The following theorem describes the spectrum of −Ω in dependence of the spin structure on X. Recall that we consider only the case (ε 1 , ε 2 ) = (0, 1) if ε 1 = ε 2 , see Remark 4.5.
6.4 A remark on the point spectrum of quotients by shifted lattices In subsections 6.1 and 6.2 we proved that for basic lattices in Osc 1 the point spectrum of the cubic Dirac operator on the quotient space is discrete.Here we will show that there also exist (non-basic) lattices such that the point spectrum is not discrete.A similar statement for the wave operator has been proven in [FiK].
Example 6.7 Let L = L r (κ, µ, ν) be a basic lattice.We choose a real number u such that ũ := 2πκru is irrational and consider L ′ := T u (L), where T u is the automorphism defined in (7).We consider a spin structure for which ε 3 = 0 and ε 4 = κ.By ( 6  The spectrum of D t is equal to C and the residual spectrum is empty.