Theta-functions on the Kodaira-Thurston manifold

The Kodaira--Thurston M manifold is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kaehler. We describe a construction of theta-functions associated to M which parallels the classical theory of theta-functions associated to the torus (from the point of view of representation theory and geometry), and yields pseudoperiodic complex-valued functions on R^4. There exists a three-step nilpotent Lie group G which acts transitively on the Kodaira--Thurston manifold M in a Hamiltonian fashion. The theta-functions discussed in this paper are intimately related to the representation theory of G in much the same way the classical theta-functions are related to the Heisenberg group. One aspect of our results which has not appeared in the classical theory is a connection between the representation theory of G and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in M.


Introduction
The classical theory of ϑ-functions is a rich and beautiful subject that weaves threads from a diverse set of mathematical disciplines. It is the purpose of this note to describe a generalization of this theory when viewed from a geometric/representation theoretic point of view. It is the authors' hope that this generalization will not only illustrate interesting new connections between ϑ-functions and symplectic geometry, but also clarify some aspects of the classical theory by comparison.
We will develop a theory of ϑ-functions associated to the Kodaira-Thurston manifold, a certain nontrivial 2-torus bundle over a 2-torus, realized here as a compact nilmanifold, which is symplectic but not Kähler. It seems that our constructions are not unique to this situation and could be adapted to other compact nilmanifolds. Just as in the classical theory, ϑ-functions associated to the Kodaira-Thurston M manifold arise when studying the decomposition of the L 2 -space of sections of certain line bundles over M . The construction we give is intimately related to the symplectic structure of M .
The main results of this paper are twofold. First, we give a construction of ϑ-functions on M which parallels the classical theory, where possible. The construction we present of ϑ-functions on M uses the representation theory of an associated nilpotent Lie group G, just as the classical ϑ-functions are intimately related to the Heisenberg group (in fact, G can be interpreted as "the Heisenberg group on Heis(3) × R").
The second main result of this paper is a connection between the algebraic structure of G and the symplectic structure of M . To make the construction of ϑ-functions explicit requires a choice: a subalgebra h of Lie( G) of a certain type (subordinate to a 4-dimensional integral coadjoint orbit, to be precise). It turns out that h is connected to the symplectic structure of M ; we will see that each subordinate subalgebra h corresponds to a Lagrangian foliation of M . If the subordinate subalgebra is an ideal, then the foliation is special Lagrangian. (Our proof of this fact is indirect; we enumerate all possible relevant subordinate subalgebras and observe that those which are ideals induce special Lagrangian fibrations.) The family of such subordinate subalgebras can be parameterized by R, and in a certain parametrization, those foliations associated to the subalgebras corresponding to 0 and ±∞ are torus fibrations.
In the remainder of this introduction, we will state our main results (though we leave some technical details for later). Next, we give an overview of the classical theory of ϑ-functions so that the analogy of our results with the classical theory is apparent. We then briefly review the tool which we use to generalize ϑ-functions to our situation: a generalization of geometric quantization to the symplectic category which is known as almost Kähler quantization. We conclude this section with a summary of the rest of the paper.

Main results
Let G = Heis(3) × R be the direct product of the three-dimensional Heisenberg group with the real line. Denote by Γ 0 the integer lattice in G. The Kodaira-Thurston manifold is the compact quotient M := Γ 0 \G. It can be equipped with a left G-invariant integral symplectic form ω and complex structure (see Kodaira's work [Kod64]), but, as Thurston was the first to observe [Thu76], it is not a Kähler manifold; that is, the metric defined by any choice of complex structure and symplectic structure is not positive definite. In this paper, we will primarily be interested in the symplectic structure of M .
Since we assume ω is integral, there is a Hermitian line bundle ℓ → M with compatible connection whose curvature is the symplectic form. As we will see, there exists a central extension such that G acts on M in a Hamiltonian fashion. This Hamiltonian action lifts to the line bundle ℓ, and induces the right (quasi)regular representation ρ of G on L 2 (M, ℓ), the space of L 2 -sections of ℓ, given by (ρ( g)s) (m) = g −1 s(m · g). This representation is unitary with respect to the Liouville measure on M . In Section 5, we will see that the quasiregular representation decomposes into a direct sum of unitary irreducible representations π k : G → End(V k ) as (Corollary 5.6) L 2 (M, ℓ ⊗k ) = 4k 2 V k . (1.1) Adapting a general construction due to Richardson [Ric71], we obtain: Theorem 1.1 Let k ∈ Z \ {0}. For each j = 1, ..., 4k 2 , there exists a map 3 θ j k : V k → L 2 (M, ℓ ⊗k ) such that 1. θ j k is unitary, up to a constant, 2. θ j k (V k ) is orthogonal to θ j′ k (V k ) whenever j = j ′ , and 3. θ j k intertwines the actions of G on L 2 (M, ℓ ⊗k ) and V k .
The maps θ j k are generalizations of maps introduced by Weil in [Wei64]. In [Bre70], Brezin considered in detail these maps in the case of Heisenberg groups. In [Bre70], Brezin also described an inductive procedure to obtain decompositions of the form (1.1) for a general nilmanifold, though his procedure is somewhat different from ours.
Each of the representation spaces V k is isomorphic to L 2 (H\ G), where H is any choice of a certain family of subgroups of G: those with Lie algebra subordinate to certain coadjoint orbits, described in Theorem 3.3. Both G and H are nilpotent, hence exponential, groups. The group G is diffeomorphic to R 5 while H\ G is diffeomorphic to R 2 , so that V k ≃ L 2 (R 2 ). An element of L 2 (H\ G) is already constant along H-cosets, and we will see in Section 5 that θ j k is essentially a sum over the remaining lattice directions. For this reason, we call the θ j k periodizing maps, even though they are not quite what one usually means by the term (the reason is, again, because we are really dealing with sections of a nontrivializable line bundle rather than functions).
Let h = Lie(H) and set h 0 = h ∩ Lie(G), where G ֒→ G as the zero section. Note that T 1 G ≃ T Γ0 M. The next theorem (a concatenation of Theorem 3.3, Lemma 3.4, and Theorems 4.3 and 4.4) exposes the symplectic structure of M in terms of the algebraic structure of G. (We recall the definitions related to Lagrangian subspaces in Section 2.2. The notion we use of special Lagrangian is due to Tomassini and Vezzoni [TV06].) Theorem 1.2 The left G-invariant distribution on M induced by the subspace h 0 ⊂ T Γ0 M is integrable and Lagrangian, hence defines a Lagrangian foliation of M . Moreover, the set of ideal subordinate subalgebras can be parameterized by e ∈ R ∪ {±∞}, and the foliation induced by h e , e ∈ R is special Lagrangian. Finally, the foliations induced by the subordinate subalgebras h e , e = 0, ±∞ are Lagrangian torus fibrations. 3 We will actually construct maps Θ j k : V k → L 2 k (P ), where L 2 k (P ) is a space of S 1 -equivariant functions on the circle bundle associated to ℓ; each such function can be identified with a section of ℓ ⊗k .
The universal cover of M is G (since G ≃ R 4 is contractible), and so ℓ → M lifts to a trivializable line bundleľ → G. Upon trivializingľ ≃ G × C, a section s ∈ Γ(M, ℓ) yields a function f s ∈ G → C. Such a function is necessarily pseudoperiodic, that is, it admits transformation rules associated to the lattice elements of the form f s (γ 0 g) = e(g, γ 0 )f s (g), for some multiplier e(g, γ 0 ) which is independent of f s . In particular, given φ ∈ L 2 (H\ G), the periodized image θ j k φ ∈ Γ(M, ℓ ⊗k ) lifts to a pseudoperiodic function ϑ j k φ : G → C. In Section 5, we prove the following pseudoperiodicity relations.
Theorem 5.7 Let γ 0 ∈ Γ 0 . Then where ψ(g 1 ,g 2 ) is defined by the group multiplication of G ≃ G ⋊ R : Our final result is a description of the almost Kähler quantization of M , one aspect of which yields a direct proof, in our case, of a general theorem of Guillemin and Uribe [GU88]. Choose a left-invariant metric on G. Associated to the resulting metric on M is a Laplacian ∆ (k) acting on Γ(ℓ ⊗k ). Since it is left-invariant, the Laplacian ∆ (k) induces a Laplacian ∆ k acting on V k . Theorem 1.4 There exist constants a, C > 0 such that for k sufficiently large, the lowest eigenvalue λ 0 of ∆ k − 4πk has multiplicity one (i.e. there is a unique ground state) and is contained in (−a, a). Moreover, the next largest eigenvalue λ 1 is bounded below by Ck. The spectrum of ∆ (k) − 4πk is identical to the spectrum of ∆ k − 4πk, except that each eigenvalue is repeated with multiplicity 4k 2 .
Throughout the rest of the paper, we will present specific computations exhibiting the above theorems (for specific choices of the relevant structures) in an effort to illustrate the similarities and differences with the classical theory of ϑ-functions. These computations will appear under the heading of Example, though they should be understood as instances of the main results and techniques we discuss.
Example In Section 4, we will see that there exists a subgroup H 0 < G such that the left-invariant Lagrangian foliation of M induced by h 0 = Lie(H 0 ) ∩ Lie(G) is a fibration of M by special Lagrangian tori. After choosing a matrix realization of G (listed in the Appendix) we can identify G ≃ R 4 and H\ G ≃ R 2 (equipped with the Lebesgue measure).
Associated to this data, for each k ∈ Z ≥0 there is a family of maps is an orthogonal decomposition of L 2 (M, ℓ ⊗k ) into irreducible G-spaces (Section 5.1). Identifying sections of ℓ ⊗k with sections of the pullback bundleľ ⊗k → G ≃ R 4 and hence with functions on R 4 , we obtain, in Section 5.2, for each square-integrable function f : R 2 → C and for each m, n = 0, 1, . . . , 2k − 1 a function ϑ m,n k f : R 4 → C given by These functions satisfy the pseudoperiodicity conditions (ϑ m,n k f )(x + 1, y, z, t) = (ϑ m,n k f )(x, y, z, t), (ϑ m,n k f )(x, y, z + 1, t) = e 4πikx (ϑ m,n k f )(x, y, z, t), and (ϑ m,n k f )(x, y, z, t + 1) = e 4πiky (ϑ m,n k f )(x, y, z, t).

1.2
The classical theory of ϑ-functions We give here a short description of the classical theory of ϑ-functions. Of course, we cannot hope do more than scratch the surface of this vast subject, so we will content ourselves here with recalling those pieces which suit our present interests (and even these points will be given a succinct treatment). There are many excellent references in the literature dealing with ϑ-functions; too many, in fact, for us to give any sort of inclusive list. Nevertheless, we would refer the interested reader to the Tata Lectures of Mumford [Mum83], [Mum84], for a treatment of ϑ-functions from both the algebraic and geometric point of view; in particular, the point of view taken in the third volume of the series [Mum91] (Mumford-Nori-Norman) is very much in the same vein as the approach taken in this paper. For connections with representation theory, and in particular the deep connections of the theory of ϑ-functions with the theory of nilpotent Lie groups, we recommend the work of Auslander and Tolimieri [AT75]. Let us emphasize that the following account of the classical theory of ϑ-functions consists entirely of well-known material that may be found in the references mentioned above.
In his Fundamenta Nova Theoriae Functionum Ellipticarum [Jac29], Jacobi gave the first treatment of what is now known as the ϑ-function, defined as the series where z ∈ C and τ ∈ H + := {z ∈ C : Im z > 0}. This series converges absolutely, and uniformly on compact sets. Hence, it defines an entire holomorphic function.
Because of these relations, ϑ(z, τ ) is said to be pseudoperiodic with respect to the lattice Z + τ Z ⊂ C. If ϑ(z, τ ) were periodic with respect to the lattice Z + τ Z, then it would descend to a function on the torus T 2 = C/(Z + τ Z). The geometric interpretation of ϑ(z, τ ) which we will generalize arises from the fact that because of the pseudoperiodicity conditions, ϑ(z, τ ) descends to a section of a (nontrivializable) line bundle over the torus, rather than a function.
We momentarily shift our point of view and recall some basic symplectic geometry. An action of a Lie group G on a symplectic manifold (M, ω) is said to be weakly Hamiltonian if each 1-parameter subgroup is infinitesimally generated by the symplectic gradient of some Hamiltonian function, that is, if for each ξ ∈ g := Lie(G) there exists a function φ ξ : M → R such that where X ξ is infinitesimal action of ξ on M. Such an action is Hamiltonian if the linear map ξ → φ ξ is a Poisson-Lie homomorphism, that is, if Consider R 2 with coordinates (x, y) equipped with the standard symplectic form ω = dx∧dy. The Abelian group R 2 acts on itself by translations which are infinitesimally generated by the vector fields ∂ x and ∂ y . Moreover, this action is weakly Hamiltonian; indeed ∂ x is the Hamiltonian flow of the function φ x := y and ∂ y is the Hamiltonian flow of φ y := x. A quick calculation, though, shows that {φ x , φ y } = 1, whereas [∂ x , ∂ y ] = 0 implies φ [∂x,∂y] = 0. Hence, the action of R 2 on itself by translations is not Hamiltonian.
Let us reflect on this situation for minute. On the one hand, [∂ x , ∂ y ] = 0 defines the Lie algebra structure of R 2 . On the other hand, we would like a Lie algebra structure which is reflected as a Poisson algebra satisfying {∂ x , ∂ y } = 1 (if we want a Hamiltonian action, that is). The resolution, it seems, is to take a central extension of R 2 whose Lie algebra structure is given by [∂ x , ∂ y ] = Z and assign the Hamiltonian function φ Z := 1. This means that Z acts trivially on R 2 , but the new group acts in a Hamiltonian fashion. This new group is, of course, the well-known Heisenberg group described by the short exact sequence The Heisenberg group can be realized in many equivalent ways. For what comes later, we will find it convenient to make the definition Heis(3) := a ∈ R 3 equipped with the group law Note that the first two components give the action of R 2 on itself by translations as claimed. That the Lie algebra of this Lie group satisfies the bracket relations [X, Y ] = Z, with X = ∂ R a 1 , Y = ∂ R a 2 , and Z = ∂ R a 3 , is an exercise left for the reader 4 .
Let Γ := {a ∈ Heis(3) : a ∈ Z 3 } denote the integer lattice in the Heisenberg group. The quotient is a compact manifold. In fact, the center {(0, 0, z)} ⊂ Heis(3) of the Heisenberg group acts (on the right) as S 1 on Q, and this action gives Q the structure of a principal S 1 -bundle over the torus T 2 whose Chern class is the class of the symplectic form (appropriately normalized).
The circle S 1 acts on C by multiplication, and this action induces a Hermitian line bundle ℓ → T 2 associated to Q. It turns out that this bundle has a unique (up to normalization) holomorphic section. Pulling back ℓ by the quotient map R 2 → T 2 yields a trivializable line bundle over R 2 , and up to factors arising from the choice of trivialization, ϑ(z) is this unique holomorphic section represented as a section of the pullback bundle. We will see this much more explicitly in a moment.
We can view this appearance of ϑ(z) as a section of a line bundle over T 2 through the lens of the representation theory of the Heisenberg group (the utility of this approach is that we can generalize it to the Kodaira-Thurston manifold).
The Heisenberg group acts on Q transitively on the right, and this action induces a unitary action on L 2 (Q) (with respect to the Lebesgue measure) via which is known as the right (quasi-)regular representation. Thus, L 2 (Q) can be decomposed into unitary irreducible representations of Heis(3).
The unitary irreducible representations of Heis(3) are well-known (and easily computed, see for example [Kir04, Sec. 2.3]). For our purposes, it is sufficient to know that for each λ ∈ R \ {0} there exists a unitary irreducible representation π λ of Heis(3) on V λ ≃ L 2 (R, dx) given by The decomposition of L 2 (Q) into unitary irreducible representations of Heis(3) is then where V 0 ≃ L 2 (T 2 ) and each invariant subspace |k|V k can be decomposed into |k| copies of the irreducible space V k (this result seems to be folklore in representation theory, but we refer the reader to [AB73, Theorem 1] for one proof).
Indeed, a very fruitful (at least in this paper) question to ask is: how is the decomposition (1.2) achieved? That is, given a function f ∈ L 2 (R, dx), how does one obtain a function in |k|V k ⊂ L 2 (P ), and moreover, is there some systematic way to achieve the decomposition of an invariant subspace of L 2 (P ) into |k| orthogonal copies of V k ?
The answer to both of these questions is achieved by the Weil-Brezin Θ-map [Bre70], [Wei64]. Let x, y and φ be coordinates on Q induced by the coordinates a 1 , a 2 and a 3 on Heis(3). For each k ∈ Z \ {0}, define a map Θ k : L 2 (R, dx) → L 2 (Q) by Each Θ k is unitary and intertwines the action of the Heisenberg group. Define a function 5 ϑ k f : The function ϑ k f is square-integrable on any fundamental domain of T 2 = R 2 /Z 2 , though not on R 2 itself.
Since it will make no difference for our purposes, we will henceforth take τ = i and write ϑ(z) := ϑ(z, i).
To see how ϑ(z) arises from the Weil-Brezin map requires one more piece of the puzzle. The basic fact of the matter is that ϑ(z) is, up to exponential factors, the image of the standard Gaussian under the Weil-Brezin map with k = 1, after ℓ → T 2 has been lifted to a trivial line bundleľ ≃ R 2 × C : is not as ad hoc as it seems at first sight: the factor e −πy 2 arises from the choice of trivialization ofľ. Why ϑ arises as the image of the standard Gaussian requires a bit more explanation. Sections of ℓ ⊗k can be identified with functions f : Q → C which satisfy the S 1 -equivariance condition f ((0, 0, c) · (x, y, φ)) = e −2πikc f ((x, y, φ)). (1.4) Consider the Hodge Laplacian 6 acting on sections of ℓ. It induces a second order elliptic differential operator ∆ (1) acting on L 2 1 (P ) which can be written in terms of the right quasi-regular representation as Since a function which transforms according to (1.4) satisfies the same S 1 -equivariance as V 1 , the Weil-Brezin map Θ 1 restricts to an S 1 -equivariant map Θ 1 : V 1 → L 2 1 (P ). The Hodge Laplacian ∆ (1) then yields a differential operator ∆ 1 acting on V 1 which is given by On ℓ → T 2 , the kernel of the Hodge Laplacian consists exactly of holomorphic sections (Hodge's theorem). On the other side, the kernel of ∆ 1 , acting on V 1 ≃ L 2 (R), is spanned by the Gaussian e −πt 2 . Hence, we see that (1.3) is simply an expression of the kernel of the Hodge Laplacian acting on S 1 -equivariant functions on P from two different points of view.

Quantization
Classical ϑ-functions, and also the ϑ-functions constructed in this paper, are related to a construction in mathematical physics known as geometric quantization. We will not go into detail about geometric quantization (the interested reader may refer to [Woo91] for comprehensive account), but since it will eventually provide the structure which we generalize, we now describe the relevant pieces.
Geometric quantization provides a systematic recipe which associates to each symplectic manifold (M, ω) a Hilbert space H M and a map Q from (some subalgebra of) C ∞ (M ) to the set of (usually unbounded) operators on H M . This association is rigged in such a way as to be nontrivial and approximately functorial. The construction works best when M is actually Kähler (for example, on the torus).
Suppose M is a compact Kähler manifold with integral symplectic form ω. Then there is a Hermitian line bundle ℓ → M with compatible connection with first Chern class the class of ω, called a prequantum line bundle. In this situation, for each k ∈ Z + one defines the quantum Hilbert space to be the L 2 -space of holomorphic sections of ℓ ⊗k : By Hodge's theorem, the quantum Hilbert space is precisely the kernel of the Hodge Laplacian ∆ k acting on ℓ ⊗k . Hence, we see that the geometric quantization of the torus consists exactly of ϑ-functions.
In order to generalize the construction of ϑ-functions, one should study the geometric quantization of other manifolds 7 . The description we have given, though, makes critical use of the assumption that M is Kähler (otherwise, we either have no notion of "holomorphic", if M is not complex, or there might be no holomorphic sections, if M is complex but the line bundle ℓ is not positive).
We will consider one possible generalization of the basic program of geometric quantization, known as almost Kähler quantization. Although the general theory has been around for some time, no nonKähler examples of this method have been worked out. (It was part of the original motivation of the current work to produce such an example.) Suppose that (M 2n , ω) is a compact symplectic manifold and that the class [ω/2π] is integral, whence there exists a prequantum line bundle ℓ → M . Choose a metric g on M , and construct the rescaled metric Laplacian ∆ (k) − 2πnk acting on sections of ℓ ⊗k . Denote the spectrum of ∆ (k) − 2πnk by If M is Kähler, and g is the Kähler metric, then ∆ (k) − 2πnk is equal to the Hodge Laplacian. So the geometric quantization of M consists of the kernel of ∆ (k) − 2πnk. In the nonKähler case, even though there is no Hodge Laplacian, it still makes sense to study ∆ (k) − 2πnk. The difficulty is that its kernel is generically empty.
The basic foundation on which almost Kähler quantization rests is that there is an approximate kernel of the rescaled metric Laplacian, described by the following theorem of Guillemin and Uribe [GU88]. Let the eigenvalues of ∆ (k) − 2πnk be denoted by λ 0 ≤ λ 1 ≤ λ 2 ≤ · · · → ∞. Theorem 1.5 There exist constants a ≥ 0 and C > 0 such that for k sufficiently large, The important point is that the constants a and C are independent of k. Thus, the span of the eigenfunctions of ∆ (k) − 2πnk with bound eigenvalues constitutes an approximate kernel. Indeed, if M is Kähler, then these bound eigenvalues are all exactly zero, and the span of the corresponding eigenfunctions is the kernel of the Hodge Laplacian.
Following this line of reasoning, Borthwick and Uribe [BU96] defined the almost Kähler quantization of (M, ω) to be the span of the bound eigenfunctions of the rescaled metric Laplacian: These bound eigenfunctions, in the case of the Kodaira-Thurston manifold, are the desired generalization of ϑ-functions.

Summary
In Section 2, we describe a nontrivial central extension 1 → R → G → G := Heis(3) × R → 1 which plays a central role in our analysis. The quotient of G by an integer lattice yields a principal circle bundle P over the Kodaira-Thurston manifold. Section 2 contains a description of the geometry of P , the complex line bundles ℓ ⊗k , k = 1, 2, . . . associated to P, and their lifts to (trivializable) line bundles over G, which are the source of ϑ-functions associated to M . Section 2 concludes with a review of the symplectic geometry we use later in the paper (Lagrangian and special Lagrangian foliations and fibrations).
Section 3 begins an analysis of the representation theory of G; we use Kirillov's orbit method to construct the unitary irreducible representations of G. After a brief review of the induction procedure (the basis of the orbit method), we discuss the set of subordinate subalgebras (the choice of which is the first step in orbit method).
The subordinate subalgebras provide the link with the symplectic geometry of M . In Section 4, we describe the correspondence between subordinate subalgebras and Lagrangian foliations. We then show that a certain subfamily of subordinate subalgebras, consisting exactly of the ideal subordinate subalgebras, correspond to special Lagrangian foliations. We also describe Lagrangian torus fibrations of M .
In Section 5, we return to the representation theory of G. In this section, we find a decomposition of L 2 (P ) into unitary irreducible representations of G. We also describe the periodizing maps Θ j k which realize this decomposition, and discuss the pseudoperiodicty of the images of Θ j k . In the final Section 6 we consider the harmonic analysis of P . After discussing the various Laplacians in the picture, we use semiclassical methods (in particular, the quantum Birkhoff canonical form) to analyze their spectra. Finally, we are able to define the ϑ-functions associated to M and hence the almost Kähler quantization of M .

Preliminaries
We begin by constructing a symplectic nonKähler 4-manifold (M, ω), known as the Kodaira-Thurston manifold. It is the product of S 1 and the quotient of the 3-dimensional Heisenberg group by a discrete uniform subgroup (that is, a discrete subgroup such that the quotient is compact). We will normalize ω so that [ω/2π] is an integral cohomology class.
Let G = Heis(3) × R be the product of the three dimensional Heisenberg group with R. Convenient faithful matrix representations of this group, as well as those defined below, are given in the Appendix. We will write g ∈ G as g = (a, r) := (a 1 , a 2 , a 3 , r), a ∈ Heis(3), r ∈ R.
The group law is given by The coordinates (a, r) on G may be expressed in terms of this basis: Such coordinates on G are called canonical coordinates.
Let Γ 0 ⊂ G be the integral lattice It is easy to check that Γ 0 is a subgroup (not normal) of G. The Kodaira-Thurston manifold is It is compact and symplectic, as we will see below, but not Kähler [Thu76]. A left invariant coframe on G is There is a right invariant frame which corresponds to the dual of the above frame at the identity: Recall that it is the right invariant frame that generates the left action of G on itself.
For easy reference and to fix sign conventions, recall that the Hamiltonian vector field X f associated to f ∈ C ∞ (G) is given by The corresponding Poisson brackets are {f, g} = X f (g) = ω(X f , X g ).

Lemma 2.1 The right invariant vector fields
The induced linear map g → C ∞ (G) is, however, not a Lie algebra homomorphism.
Proof. The first part is an easy computation. For the rest, observe that With the conventions thus far, writing 0 = (0, 0, 0, 0) ∈ G, we have (2.5) The fact that the Poisson brackets above do not close in g is an analogy of what happens in the case of translations on R 2 (see Section 1.2). Therefore, we are lead to consider the analogue of the Heisenberg group associated with G; namely, a specific central extensiong = span R {X 1 , X 2 , X 3 , T, U } of g subject to the relations The central extensiong is a three step nilpotent algebra whose center is spanned by U.
Observe that we are not using the bracket relations (2.4). This is because we want the restriction of the Lie algebra of G = exp(g) to the standard embedded copy of G to coincide with the algebra of left invariant vector fields along that embedded copy of G. Hence, due to (2.5), we need the change of signs.

Prequantum bundles
The Lie group G with Lie algebra g = span R {X 1 , X 2 , X 3 , T, U } subject to the relations (2.6) has the structure of a central extension An element (g, v) of G can be written in canonical coordinates as The group law in these coordinates, which can be worked out either with the Baker-Campbell-Hausdorff formula or the faithful matrix representation given in the Appendix, is In particular, A left G-invariant frame which corresponds to {X 1 , X 2 , X 3 , T, U } at the origin is given by The dual left G-invariant coframe is (2.9) Throughout the paper, if we need to choose a metric (for example in Section 6), we will use the left G-invariant At the origin in G, this metric yields a symmetric bilinear quadratic form, and orthogonal projection from g to g with respect to this form is given by (with the summation convention) (2.10) Moreover, the restriction of the metric g to G yields a metric (which we denote by the same symbol) The fundamental importance of G to our analysis is due to the following.
Lemma 2.2 The group G acts on (G, ω) in a Hamiltonian fashion, provided we associate to U the Hamil- The center of G is exp(R · U ) and can be identified with R if we set exp(U ) → 1. Denote by Z ⊂ G the subgroup of the center corresponding to the half-integers 8 , that is, The homomorphism F induces a homomorphism from K to G which we continue to denote by F : F ((a, r, [v])) = (a, r). 8 We are forced to consider half-integers because of the 1 2 that appears in ψ (2.8).

Let us denote by Γ
The projection π and the S 1 -action given by right multiplication by the center of K, i.e., give P the structure of a principal S 1 bundle. Equivalently, we can define an integral lattice in G and then identify P = Γ k \K = Γ\ G.
Lemma 2.3 P is a prequantum circle bundle over X, that is, a circle bundle with connection whose curvature 9 is ω.
Proof. By (2.2), we have dβ U L = da 1 ∧ da 3 + da 2 ∧ dr. The right hand side above is exactly 1/2π times the pullback tog of the symplectic form on g. Hence we can take 2πβ U for a connection 1-form. This means π : P → M is indeed a prequantum circle bundle.
Since the universal cover of M is G, the circle bundle P lifts to a circle bundle over G, and this circle bundle is just K.
We define the prequantum line bundle ℓ → M to be the Hermitian line bundle associated to P equipped with the connection induced by the connection 1-form 2πβ U . Recall that for a principal G-bundle P → M, if ρ : G → End(E) is a representation of G, then the vector bundle associated to P with fiber E is defined by where the equivalence is given by (p, v) ∼ (p · g, ρ(g −1 )v). Let ρ (k) (e 2πiθ ) = e 4πikθ . (2.11) Observe that this is not the standard action of S 1 on C; we have introduced an extra factor of 2 to compensate for the fact that the center of K is isomorphic to R/ 1 2 Z. The line bundles associated to P by this action are, for k ∈ Z >0 , ℓ ⊗k = P × ρ (k) C.
The line bundle ℓ ⊗k is equipped with a covariant derivative induced by the connection 1-form 2πβ U . The curvature of this connection is therefore 4πkω and so the Chern class of ℓ ⊗k is [2kω]; again, the factor of 2 arises because of the 1 2 that appears in (2.8). The lattice Γ 0 acts on K × ρ (k) C by Hence, there is a canonical isomorphism of line bundles The lift of ℓ ⊗k to G is therefore the line bundleľ ⊗k → G associated to K: The computations in this paper are greatly simplified by identifying sections of the prequantum line bundle ℓ (resp.ľ) with S 1 -equivariant functions on the total space of the associated prequantum circle bundle P (resp. K). The following lemma is standard, see for example [Ber04, Prop. 1.7]. 9 From the geometric point of view, it would be more natural to define U ′ = − √ −1U and then identify the center of K with S 1 via exp(2π √ −1U ′ ) → 1. The fiber of the S 1 -bundle P would then have tangent space 2π √ −1R. But since e G (and hence K) is a real Lie group, we omit the algebraically wieldy factors of 2π √ −1. This is responsible for the fact that P has curvature ω instead of the more standard − √ −1ω.

Lagrangians in
A submanifold p : L ֒→ M is special Lagrangian with respect to a generalized CY structure (ω, J, ε) if it is Lagrangian and p * (Im ε) = 0.
If J and ε are left G-invariant, then a CY structure (ω, J, ε) induces an algebraic structure on the Lie algebra g (denoted by the same symbols), and vice versa. We can therefore check that a left G-invariant Lagrangian foliation is special Lagrangian by checking the corresponding conditions in g.

3
Representation theory of G (Part I): subordinate subalgebras That the symplectic geometry of M is related to the algebraic structure of G becomes apparent after a careful analysis of the representation theory of G using Kirillov's orbit method, which is ideally suited to our situation since G is nilpotent (see [Kir04] for a thorough treatment of the orbit method). In fact, it is a seemingly innocuous choice, from a representation theoretic point of view, which provides the connection: the choice of subordinate subalgebra.
In this section, we begin the orbit method analysis and describe explicitly the relevant subordinate subalgebras. Their connection with the symplectic geometry of M will be taken up in the next section. The orbit method analysis will then be completed in Section 5, where we return to the idea of ϑ-functions on M .
The unitary dual of G is parameterized by the set of coadjoint orbits; among these, there is a family of 4-dimensional orbits Ω µ := Ad( G) * (µβ U ) parameterized by µ ∈ R \ {0}. The orbit method is (among other things) an explicit algorithm which constructs a unitary irreducible representation of G for each coadjoint orbit. The first step in the orbit method algorithm is to find the coadjoint orbits and associated subordinate subalgebras; we recall their definition. Before we get to the technicalities of the unitary dual of G, we make one final remark. Even from a representation theoretic point of view, the subordinate subalgebra plays a certain role which does not seem to have been observed: each choice of subalgebra subordinate to Ω µ=k , k ∈ 2Z leads to a different orthogonal decomposition L 2 k (P ) = 4k 2 L 2 (R 2 ). This fact will become clear after we study periodizing maps in Section 5.

Subordinate subalgebras
Equivalence classes of unitary irreducible representations of G are in one-to-one correspondence with the coadjoint orbits of G. Let h be a Ω-subordinate subalgebra for some coadjoint orbit Ω. A characterλ Ω of the connected analytic subgroup H of G with Lie algebra h is where λ ∈ Ω is any point in the coadjoint orbit, and ·, · denotes the canonical pairing of g * with g.
Since G is nilpotent, all of the unitary irreducible representations of G are induced from the characters of the analytic subgroups of G corresponding to the subordinate subalgebras; that is, given a subalgebra h subordinate to Ω and the corresponding Lie subgroup H, a unitary irreducible representation of G is defined on where h(x, g) is the solution to the so-called master equation for some choice of section s : H\ G → G (see [Kir04,Chap. 3] for details). It follows from (3.3) that h satisfies the cocycle condition h(x, g 1 g 2 ) = h(xg 1 , g 2 )h(x, g 1 ). (3.4) Assumption: We will always choose s : H\ G → G so that s(H) = 0. The task now is to enumerate the space of coadjoing orbits.
Topologically, this space is R with the origin removed and replaced by a copy of R 2 in which one axis is removed, each point of which is replaced by another copy of R 2 .
Observe that the center of G acts nontrivially only on the 4-dimensional orbits. Since it is the center of G which acts as S 1 on the fibers of the prequantum bundle P, we expect, and it is indeed the case, that these orbits will play a prominent role in the harmonic analysis of P .
To construct the unitary irreducible representation associated to a coadjoint orbit Ω we must find a corresponding Ω-subordinate subalgebra (Definition 3.1).
It is worth noting that any choice of subordinate subalgebra will do for the construction of a representation corresponding to Ω, but there are many such choices. Although they induce equivalent representations of G, different choices of subordinate subalgebra will induce different decompositions of L 2 (P ) into irreducible factors, and so we will take some care to enumerate here all such choices. Moreover, we will see in Section 4 that the different choices of subordinate subalgebra reflect the symplectic geometry of the Kodaira-Thurston manifold.
We have three types of orbits to consider. The choice of subordinate subalgebra will only be relevant for the four-dimensional orbits, and so it is only in that case that we enumerate all such choices.
That all of the subalgebras subordinate to Ad( G) * (0, 0, 0, 0, µ) are the ones given is a corollary of Theorem 4.2. One simply enumerates all of the Lagrangian subspaces of g and intersects with the set of subalgebras of g.

An important observation for what comes later (the proof is a straightforward computation using Theorem 3.3 and hence omitted):
Lemma 3.4 The family {h e , e ∈ R ∪ {±∞}} consists of ideals, and these are the only ideal subordinate subalgebras. Moreover, the subalgebras h e are commutative.
For these reasons, computations work especially nicely if we choose one of the h e subalgebras, and so throughout the remainder, if we need to do a model computation, we will use h 0 := h e=0 .

Lagrangian foliations
We turn our attention now to Lagrangian and special Lagrangian foliations and fibrations. First, we recall a generalization of the notion of special Lagrangian due to Tomassini and Vezzoni (see [TV06] for details). Then, we will show that the Lagrangian distributions associated to h e are in fact special Lagrangian foliations, and for certain values of e, these foliations are fibrations by tori.
The connection between the representation theory and symplectic geoemetry of our setup is a consequence of the following simple result.
Lemma 4.1 For X ∈g, let X 0 ∈ g be the g-orthogonal projection of X onto g (2.10). Then The Ω µ -subordinate subalgebras listed in Theorem 3.3 are 3-dimensional. It then follows from the general theory of the orbit method that all Ω µ -subordinate subalgebras are 3-dimensional (to avoid a circular argument, it is important that we do not assume here that Theorem 3.3 lists all of the Ω µ -subordinate subalgebras).
Theorem 4.2 A subalgebra h ⊆g is Ω µ -subordinate if and only if h = L⊕RU for some Lagrangian subspace L ⊂ g.
Proof. First, suppose h = L ⊕ RU is a subalgebra for some Lagrangian L. Then and h is of maximal dimension; hence h is subordinate.
In the other direction, suppose that h is subordinate. Then since [RU,g] = {0}, we must have RU ⊆ h. Let L ⊂ g be the projection of h onto span{X 1 , X 2 , X 3 , T }. Then so that L is Lagrangian as desired.
Be careful that it is necessary that h is a subalgebra in either direction; in fact, there is a 3-dimensional family of Lagrangian subspaces 10 L such that L ⊕ RU is not a subalgebra, and so the correspondence h → L is only injective.
Each subspace L ⊂ g defines a left-invariant distribution on M . If L is a subalgebra, then this distribution is integrable. If L is Langrangian, then so is the corresponding left-invariant distribution, and hence each Ω µ -subordinate subalgebra h induces an integrable Lagrangian distribution on M, that is, a Lagrangian foliation.  Remark The special Lagrangian torus defined by h e=0 was discovered by Tomassini and Vezzoni [TV06]. Given a point Ω ∈ H + , the corresponding ω-compatible complex structure is Hence, the complex structure J e on g corresponding to the point Define the (2, 0)-form (with respect to J e ) It is now routine (though somewhat tedious) to check that the foliation of M induced by h e is special Lagrangian; we leave the details to the reader (who may find it useful to recall that dβ 3 = −β 1 ∧ β 2 and dβ j = 0, j = 3).
At e = ±∞, the complex structure degenerates; this is a geometric feature of the foliation induced by h e=±∞ . Indeed, given an arbitrary Ω ∈ H + , one may write any left-invariant (2, 0)-form α (with respect to J Ω ) in terms of the components of Ω = a b b d : for some f ∈ C ∞ (M ), we obtain The condition that p * (Im ε) = 0 then implies the vanishing of the imaginary part of the coefficient of β 3 ∧ β 4 , that is, Im(det Ω) = 0. Hence, Ω lies on the boundary of H + so that the foliation is not special Lagrangian with respect to any CY structure. We return now to the study of the unitary dual of G, and in particular the decomposition of L 2 (P ) into unitary irreducible representations of G. We begin by showing that only those representations corresponding V k to certain integral 4-dimensional coadjoint orbits contribute nontrivially to the decomposition; in particular, we show that Next, we will compute the multiplicities appearing in the decomposition of L 2 (P ). Finally, we will construct periodizing maps-the analogues for the Kodaira-Thurston manifold of the Weil-Brezin map-which, for each choice of subordinate subalgebra, achieve an orthogonal decomposition of each invariant subspace of L 2 (P ) into irreducible factors. Finally, we will investigate the pseudoperiodicity of the periodizing maps. We are interested in the space of L 2 -sections of the k-th tensor power of the prequantum circle bundle P := Γ\ G over M, for k ≥ 1. Such a section is equivalent to a k-equivariant function f ∈ L 2 (P ), that is, one which is equivariant with respect to the circle action on the fibers of P (see the discussion following Lemma The isotypical subspace of L 2 (P ) consisting of k-equivariant functions is denoted by L 2 k (P ). Of course, the circle action on the fibers of P is just the action of the center of G (or, more precisely, K) on P .
Lemma 5.1 The representations π µ of G corresponding to the coadjoint orbits Ad( G) * (0, 0, 0, 0, µ), µ = 0, are the only unitary irreducible representations which are nontrivial on the center of G. The equivalence class of such unitary irreducible representation is uniquely determined by its value on the center of G, which is Proof. We first show that the represenations π µ have the desired properties. We will compute in the model case h e=0 = RX 2 ⊕ RX 3 ⊕ RU . By definition (3.2), Similar computations show that the representations associated to the other coadjoint orbits are trivial on the center; for example, the representation associated to Ad( G) * (0, 0, α 3 , ρ, 0), evaluated at the point (0, v), Hence, the isotypical subspace L 2 k (P ), k ∈ Z \ {0} is also isotypical with respect to the action of G, and decomposes as a direct sum of unitary irreducible representations corresponding to µ = −2k. There is no canonical way of choosing a canonical decomposition of the isotypical subspace L 2 k (P ) into irreducible representations. On the other hand, we may compute the multiplicity with which the representation π −2k appears in L 2 k (P ) unambiguously. Also, it will turn out that each choice of subalgebra subordinate to (0, 0, 0, 0, −2k) ∈ g * will induce a decomposition of L 2 k (P ). Each Ω µ -subordinate subalgebra h is 3-dimensional, so H := exp(h) is also 3-dimensional. The unitary irreducible representation induced by h is Ind e G H : G → End(V k = L 2 (H\ G)).
But H\ G ≃ R 2 , and since G is unimodular the measure on H\ G is identified with the Lebesgue measure on R 2 , so V k ≃ L 2 (R 2 , dx dt) [Kir04, Sec. V.2.2]. We compute Ind e G H in detail in the Example at the end of this section.
First, we consider the isotypical subspace more precisely. Let V k = L 2 (R 2 , dx dy) denote the representation space for π −2k : G → End(V k ), and consider the evaluation map where Hom e G (V k , L 2 k (P )) is the space of G-equivariant maps from V k to L 2 (P ). The image of this map is the isotypical subspace corresponding to π −2k . Since π −2k is uniquely determined by its value on the center of G (Lemma 5.1), which by (5.2) is exactly the k-equivariance condition (5.1), this image is precisely the isotypical subspace L 2 k (P ). The isotypical subspace L 2 k (P ) therefore decomposes into copies of V k , that is, where m(π k , L 2 k (P )) denotes the multiplicity with which (π k , V k ) appears in L 2 k (P ). As remarked in the Introduction, Brezin proved the existence of the decomposition (5.3)) in [Bre70], where he also gives a procedure for achieving the decomposition. Brezin's procedure is somewhat different from our approach, which is based on Richardson's periodizing maps [Ric71].
A multiplicity formula for the decomposition of the L 2 -space of a general nilmanifold was discovered by Moore [Moo73] and independently by Richardson [Ric71]. Richardson's proof of this formula will have important consequences later, so we recall the setup here.

The solution is
Again using the section s, we identify (H 0 ∩ Γ) ≃ Z 2 . The Haar measure on G descends to the Lebesgue measure on R 2 ≃ H 0 \ G.

Periodizing Maps
In this section, we describe the analogue Θ j k : L 2 (R 2 ) → L 2 k (P := Γ\ G) of the Weil-Brezin map (discussed in the Introduction) for the Kodaira-Thurston manifold; both maps are instances of a very general construction due to Richardson which we now describe.
Let (λ, H λ ) be an integral point for λ ∈ Ω = Ad( G) * (0, 0, 0, 0, µ), which is possible only if µ = −2k ∈ 2Z. To prove the multiplicity formula (Theorem 5.3), Richardson constructs a periodizing map 11 Θ (λ) k : L 2 (H λ \ G) → L 2 k (P ) from the induced representation space to the k-isotypical subspace of L 2 (P ). The image of Θ (λ) k is an irreducible subspace, and two integral points in the same G-orbit induce periodizing maps with the same image. Moreover, the images of two periodizing maps are orthogonal in L 2 k (P ) if the associated integral points lie in distinct G-orbits.
Since each function in L 2 k (P ) corresponds to a section of ℓ ⊗k , each map Θ (λ) k corresponds to a map θ (λ) The prequantum line bundle ℓ ⊗k lifts to a line bundleľ ⊗k → G ≃ R 4 . After trivializingľ, to each f ∈ L 2 (H λ \ G) there is associated a section θ (λ) k f and hence a function The function ϑ (λ) k f is square-integrable on any fundamental domain F D Γ0\G of Γ 0 \G; denote the set of such maps by The maps θ (λ) k were the maps referred to in the Introduction, but we will henceforth find it easier to work with Θ (λ) k and later with ϑ (λ) k . Although Richardson does not use the language of induced representations to do so, the periodizing maps Θ (λ) k can be described succinctly in terms of induced representations, where it becomes transparent that a periodizing map is essentially a sum over the remaining nonperiodic directions (i.e., over that portion of Γ which lies outside of Γ ∩ H).
Definition 5.5 Let (λ, H 0 ) be an integral point of a coadjoint orbit Ω = Ad( G) * (0, 0, 0, 0, −2k). The periodizing map Θ (λ) It is not hard to show that Ind e G H λ is constant on right (Γ ∩ H λ )-cosets, so that Θ (λ) k is well-defined. In [Ric71], Richardson also shows that Θ (λ) k is unitary up to a constant; specifically, that , and moreover that Θ (λ) k intertwines the right actions of G on L 2 (H 0 \ G) and L 2 k (P ). Combining the multiplicities given by Theorem 5.2 with the fact that the images of Θ Example

Transformation rules
The periodizing maps Θ j k are constructed so that the the resulting function is equivalent to a section of the nontrivializable line bundle ℓ ⊗k . Hence, when ℓ is lifted to a trivializable line bundleľ → G and then trivialized, the function which corresponds to Θ j k f is pseudoperiodic, that is, the functions ϑ j k f satisfy transformation rules associated to the integral lattice Γ 0 .
Remark In the classical theory, there is another aspect of the pseudoperiodicity of ϑ-functions: polarizations (complex structures); the classical ϑ-functions are holomorphic sections of a line bundle over the torus. Different trivializations of the lifted line bundle express the covariant notion of holomorphic differently. For example, in (1.3), the line bundleľ → R 2 was trivialized in such a way that a holomorphic section takes the form f (z)e −πy 2 . In the current situation, there is no relevant complex structure (polarization) with respect to which our ϑ-functions will be holomorphic. (this can also be easily checked by direct calculation). These are the pseudoperiodicity conditions given in the Introduction.

Harmonic analysis on P
We face the problem of computing the spectrum of the Laplacian on M acting on the k-th tensor power ℓ ⊗k of the prequantum line bundle associated to P . Although we do not obtain an exact description of the spectrum, a semiclassical analysis proves to be sufficient for our purposes. In this section, we will describe the Laplacian on M acting on sections of ℓ ⊗k and hence, with our usual identification, acting on k-equivariant functions on P . We will use the quantum Birkhoff canonical form of this Laplacian to deduce certain semiclassical properties and hence the structure of the almost Kähler quantization of M . The almost Kähler quantization of M is defined to be the approximate kernel of a rescaled metric Laplacian. In the classical case, this is the vector space of holomorphic sections of the prequantum bundle. Because of their holomorphicity, these sections are completely determined by their pseudoperiodicity. Here, there does not exist any complex structure with respect to which the sections in the almost Kähler quantization of M are holomorphic. Consequently, we cannot reconstruct them from their pseudoperiodicity alone; we are forced to try to solve for the approximate kernel directly.
As we have done throughout this paper, we identify a section s ∈ L 2 (M, ℓ ⊗k ) with a k-equivariant function (Lemma 2.4)s ∈ L 2 k (P ) in the standard way, i.e., for π(p) = x, wheres is k-equivariant ifs(p · e 2πiθ ) = e −4πikθs (p). We will find the computations are simpler when stated in terms of L 2 k (P ). As we will see in Section 6.1, the Laplacian on M acting on ℓ ⊗k can be written in terms of the standard Euclidean Laplacian ∆ E acting on P . Recall our left-invariant metric 14 on G and hence on P = Γ\ G: Since right translation is generated by the left-invariant vector fields, the Euclidean Laplacian on P is given by where ρ is the right regular representation of G on L 2 (P ), which is given by (ρ(g)f )(x) = f (xg).
The right action induces a representation of G on L 2 (P ) which commutes with ∆ E . Hence, ∆ E preserves G-invariant subspaces, and we can study the harmonic analysis of ∆ E by its pullback action on the representation spaces of G.

Laplacians
In the Kähler case, the Hodge Laplacian is equal to a rescaled metric Laplacian. Here, since the Kodaira-Thurston manifold does not admit any (positive) Kähler structure, any Hodge Laplacian will be badly behaved. But we can still write the metric (and rescaled metric) Laplacian (on M ) acting on the k-th tensor power of the prequantum bundle.
We have chosen a left-invariant metric on G defined by Since g is left-invariant, it descends to a metric, denoted also by g, on Γ 0 \G. The connection on P := Γ\ G defined by the connection 1-form 2πβ U induces a connection on ℓ ⊗k and hence a covariant derivative acting on sections of ℓ ⊗k . The corresponding covariant derivative on L 2 k (P ) is (see [Ber04,pp. 22], for example) The left-invariant frame {X L 1 , X L 2 , X L 3 , T L , U L } is given by Hence, the metric Laplacian acting on k-equivariant functions on the prequantum circle bundle is The rescaled metric Laplacian acting on k-equivariant functions on P (which, if M where Kähler, would be equal to the Hodge Laplacian) is then ∆ (k) Associated to the metric (6.1) is the Euclidean (i.e., standard) Laplacian acting on P : Using the fact that, when applied to a k-equivariant function, ∂ v = −4πik, we see that the three Laplacians are related by ∆ (k) Given a periodizing map Θ k : L 2 (H k \ G) → L 2 k (Γ\ G) associated to an integral point (λ, H) of an orbit Ad( G) * (0, 0, 0, 0, −2k), k ∈ Z \ {0}, we define the filtered Laplacian ∆ k ∈ O L 2 (H k \ G) by Since Θ k intertwines the G-action, we see that ∆ k = − ((π −2k ) * (X 1 )) 2 + ((π −2k ) * (X 2 )) 2 + ((π −2k ) * (X 3 )) 2 + ((π −2k ) * (T )) We will use the representation π 0,0 −2k of Section 5 and its associated periodizing map Θ 0,0 k to compute the filtered Laplacian. The result is The Laplacian −∂ xx − ∂ tt is a nonnegative operator, and therefore . Hence, the spectrum of ∆ k is nonnegative.
The metric Laplacian ∆ (k) commutes with the right action of G on P , and hence preserves any decomposition of L 2 (P ) into invariant subspaces. In particular, for each k ∈ Z \ {0} and each choice of representatives of the orbits (λ, H λ ) · G Z Γ, there exist periodizing maps Θ j k , j = 1, . . . , 4k 2 whose images are orthogonal irreducible subspaces of L 2 k (P ). Indeed, each Θ j k identifies an irreducible subspace with L 2 (H k \ G), and under this identification, the restriction of ∆ (k) to the irreducible subspace acts as ∆ k . We have therefore proved the following.
Theorem 6.1 For each k ∈ Z \ {0}, the spectrum of the metric Laplacian on M acting on sections of k-th tensor power ℓ ⊗k is equal to the spectrum of ∆ k , repeated with multiplicity 4k 2 .

Almost Kähler quantization of M
In order to study the spectrum of the family of operators ∆ k , we introduce a formal deformation parameter. In geometric quantization, the tensor power of the prequantum line bundle is interpreted as 1/4π , that is, The work of Charles and Vu Ngoc [CVN06] yields estimates on the spectrum of ∆ k from the quantum Birkhoff normal form of ∆ 1/ for small ; in particular, the estimates will hold for k sufficiently large (i.e. in the semiclassical limit). The main result is that the spectrum of ∆ k is an order 2 correction to the spectrum of the simple harmonic oscillator, that is, there are spectral bands around each eigenvalue of the simple harmonic oscillator whose widths are order 2 . The separation of the eigenvalues of the simple harmonic oscillator, on the other hand, is order . Hence, the separation between the lowest spectral bands of ∆ k is order -this is the simple verification of the expected spectral band gap.
In this section, we will find it useful to use a certain conjugation of our filtered Laplacian; let ε = √ and U : L 2 (R 2 ) → L 2 (R 2 ) be the unitary map U (f )(x) = 1/4 f ( √ x). Then we define 15 In this form, it is clear that ∆ k can be regarded as a perturbation of the simple harmonic oscillator. As is usually the case when dealing with the simple harmonic oscillator, computations are greatly simplified by the introduction of ladder operators. Let x 1 = x, x 2 = t, and define The standard commutators are then [a i , b j ] = δ ij and [a i , a j ] = [b i , b j ] = 0. We now recall the Birkhoff canonical form (see [CVN06] for details). Consider the graded algebra of differential operators D[[ε]] := ∞ j=2 ε j−2 D j where 16 15 This transformation is natural for semiclassical analysis; for example, one way to compute the semiclassical asymptotics of R e −x 2 / f (x)dx is to begin with the change of variables x → x/ √ . 16 We use standard multi-index notation.
17 Expanding and matching terms, one sees that we must choose A j , j = 3, 4, ... so that A 2 = 0, K 2 = H 2 , which is possible because of (6.3). Hence, we can find K j by computing K j = proj ker ad(H 2 )| D j (H j + ...).
Then, to find A j , compute A j = ad(H 2 ) −1 (H j + ... − K j ), which, since ad(H 2 ) is diagonal in our basis of ladder operators, is straightforward.
The utility of the quantum Birkhoff normal form for us is a result of Charles and Vu Ngoc in [CVN06] which says the spectrum of ∆ k is a perturbation of the spectrum of H 2 . In particular, around each eigenvalue of H 2 there is a spectral band of ∆ k whose width is O( ) (for large ε = √ , these spectral bands widen and eventually overlap, but we are mainly interested in the lowest band, centered at 1). Charles and Vu Ngoc prove the following theorem.
In our case, though, since K 3 = 0, the width of the spectral bands is O(ε 4 = 2 ) (that is, the Birkhoff canonical form of our operator is an O(ε 4 ) correction). Since the separation of the eigenvalues of the harmonic oscillator is O(ε 2 ), we see that as ε → 0, a spectral gap of width O(ε 2 ) appears between the ground state band (centered at 1) and the first excited band. This is the direct verification of the spectral band gap described in Theorem 1.4.
Remark Although it is not relevant to the almost Kähler quantization of the Kodaira-Thurston manifold, we note that the spectrum of the metric Laplacian on M acting on functions (i.e., the k = 0 case) can be computed exactly since the filtered Laplacians for the functional dimension-0 and -1 representations can be inverted explicitly.
The almost Kähler quantization of the Kodaira-Thurston manifold M is defined to be the C-span of the set of low-lying eigenstates of the rescaled metric Laplacian ∆ (k) • which acts on sections of the k-th tensor power ℓ ⊗k of the prequantum line bundle. The dimension of this space is, for k sufficiently large, the Riemann-Roch number of M twisted by ℓ ⊗k ; a routine computation shows that this Riemann-Roch number is 4k 2 . As we have seen in Section 6, the rescaled Laplacian ∆ (k) • decomposes as a direct sum of 4k 2 copies of the filtered Laplacian ∆ k acting on L 2 (R 2 ). We have therefore proved that: Corollary 6.5 The rescaled filtered Laplacian ∆ k − 4πk, for k sufficiently large, has a unique ground state which separates from the excited spectrum by a gap of order k.

Appendix: Faithful matrix representations
For computational convenience, we record here faithful matrix representations of the Lie groups and algebras studied in this paper. We begin with the product G = Heis(3)× R of the three-dimensional Heisenberg group with R, which we realize as the group of 5 × 5 matrices of the form These satisfy [X 1 , X 2 ] = X 3 . The canonical coordinates on G are then expressed in terms of the matrix exponential as [a 1 , a 2 , a 3 , r] = exp(a 1 X 1 ) exp(a 2 X 2 ) exp(a 3 X 3 ) ⊕ e r .