Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration

We discuss a fine tuning of the co- and contra-variant transforms through construction of specific fiducial and reconstructing vectors. The technique is illustrated on three different forms of induced representations of the Heisenberg group. The covariant transform provides intertwining operators between pairs of representations. In particular, we obtain the Zak transform as an induced covariant transform intertwining the Schr\"odinger representation on $\mathsf{L}_2(\mathbb{R})$ and the lattice (nilmanifold) representation on $\mathsf{L}_2\big(\mathbb{T}^2\big)$. Induced covariant transforms in other pairs are Fock-Segal-Bargmann and theta transforms. Furthermore, we describe peelings which map the group-theoretical induced representations to convenient representation spaces of analytic functions. Finally, we provide a condition which can be imposed on the reconstructing vector in order to obtain an intertwining operator from the induced contravariant transform.


Introduction
The purpose of this paper is to present an advanced use of the induced co-and contra-variant transform, which is created by the Gilmore-Perelomov coherent states (see [40] and [3, Section 7.1]). The transform is an intertwining operator to an induced representation, which explains our choice of the name for it. The approach is illustrated here by the crucial example of the Heisenberg group H 1 , however the technique is not limited to this case, cf. [5,33,36]. The topics of coherent states and covariant transform (also known under many other names) are extensively covered in the existing literature, e.g., [28,Section 13], [30,Appendix V.2], and [9,18,19,22,34,37,40], and we refer to authoritative surveys [3,15,20] for further references. Our purpose is to present some additional aspects which are commonly shadowed or missing in the existing sources. If these properties are explicitly stated then many known important results immediately follow as their direct corollaries.
For example, the standard induction (see [16,Chapter 6], [30,Appendix V.2] and [7,20,27]) from a character of the Heisenberg group centre gives the representation (2.5) below, which is different from the commonly used celebrated Fock-Segal-Bargmann (FSB) representation (4.4) in the space of analytic function [15,Section 1.6]. Those two representations are linked by the peeling map which transforms the annihilator of the representation space to the Cauchy-Riemann operator, see Section 4. The origin of the annihilator operator is revealed as the Lie arXiv:2105.13811v4 [math-ph] 1 Sep 2022 derivative with a special relation to the chosen fiducial vector, see Section 3. Section 5 provides the respective consideration of the reconstructing vector for the contravariant transform.
In this paper we present a machinery which allows to design fiducial and reconstructing vectors to ensure specific properties within the induced co-and contra-variant transforms. This technique does not rely on ad hoc knowledge and allows one to get new insights even within the much-studied framework of the Heisenberg group [7,21]. As an basic illustration, we apply this technique to find contra-and co-variant transforms, which intertwine the coordinate and momentum representations of H 1 , and predictably obtain the Fourier transform and its inverse, see Example 3.2. Less elementary example is the interpretation of the Zak transform as an induced covariant transform in Theorem 3.6, which emerges as follows.
There are three forms of induction of representations of the Heisenberg group [30, Section 2.2]: the left quasi-regular representation, the Schrödinger representation and the lattice representation (see Section 2.2) for details. We systematically and uniformly use the covariant transform for them. In particular, the Zak transform and its inverse are expressed as the covariant transform between the Schrödinger and lattice representations, with the Jacobi theta function appearing as a vacuum state of the latter, see Theorem 3.6. Similarly, expressing the pre-theta transform and its inverse throughout the same technique is also new, see Theorems 3.10 and 5.6. The pre-theta transform and its inverse intertwine the (pre-)FSB and the lattice representations. The (pre-) Fock-Segal-Bargamann (FSB) transform and its inverse (see [39,Section 4.2] and [15]) intertwines the Schrödinger representation on L 2 (R) and left quasi-regular representation on L 2 R 2 . We name it as the FSB transform from quantum mechanics (see [39,Section 4.2] and [15]), it is also known as the Gabor or time-frequency transform or windowed Fourier transform of a signal [17,38].
The classical Zak transform [44], also known as Weil-Brezin transform [15,Section 1.10], can be traced back to the works of Gelfand in 1950, see also [40,Section 1.5], [17,Chapter 8], [39,Section 8.1] and [20,21,22] for further applications and historical notes. This transform is an isometric isomorphism from L 2 (R) onto L 2 T 2 given, for f ∈ L 2 (R), by Weil [43] defined the abstract Zak transform on arbitrary locally compact abelian (LCA) groups with respect to arbitrary closed subgroups. Subsequently, the Zak transform was reviewed and generalised by many authors, see for example [6,8,20,22,26]. Various connections between the Zak transform and (left) shift-invariant spaces were studied by several authors, cf. [7,21,24,25], further references may be found in the recent survey paper [20]. Yet, an explicit interpretation of the Zak transform and its inverse as a co-and contravariant transforms respectively, cf. Theorems 3.6 and 5.6, appears to be new in this paper.
2 Preliminaries on the Heisenberg group and its induced representations

The Heisenberg group and its Lie algebra
The polarised Heisenberg group H n p is the set of triples (s, x, y), where s ∈ R and x, y ∈ R n , with the group law given as follows [15,Section 1.2] (s, x, y) · (s , x , y ) = (s + s + xy , x + x , y + y ).
For the sake of simplicity in this paper, we will work with the one-dimensional case of H 1 p and call it the Heisenberg group. It is a non-commutative group and its centre is a one-dimensional subgroup Z = (s, 0, 0) ∈ H 1 p : s ∈ R . (2.1) The left action of H 1 p on itself is given bỹ Λ(g) : g → g −1 g .
We extend this action to a linear representation Here, we introduce which forms the basis of the Lie algebra h 1 of H 1 p . The commutator of X and Y is given by the celebrated Heisenberg commutation relation It is common for a representation ρ of a Lie group to pass to the derived representations dρ of the respective Lie algebra (cf. [30,Chapter 2]). Consider the derived representations of h 1 spanned by If ρ is irreducible, dρ S is a multiple of the identity operator I; that is, dρ S = −i I (cf. [ Definition 2.1. Let κ > 0 be some fixed number and ρ be a representation of H p such that ρ(s, 0, 0) = e 2πi s I. The ladder operators are defined as follows The operators a + and a − are known as the creation and annihilation operators, respectively.
In this paper, we fix the parameter κ > 0 of the ladder operators a ± [4] and indicate the dependence of ladder operators upon it. The Heisenberg commutator relation (2.3) implies Also, for a unitary representation ρ, we have a −1 * = a + . Definition 2.2 ([1, Section 5.3] and [30, Section 2.6]). In the above notations, a vector φ 0 ∈ H is called a vacuum vector if it is a null solution of the annihilation operator: We will need the following properties of the vacuum (see [30,Section 2.6]): Lemma 2.3. If φ 0 is a vacuum of an irreducible representation ρ, then 1. φ 0 is unique up to a scalar multiple.

2.
For an intertwining operator W between ρ and another representation ρ 1 , the image Wφ 0 is a vacuum for ρ 1 .

Induced representations of the Heisenberg group
We are interested in three representations of H 1 p , which are induced by characters of certain subgroups of H 1 p . Here, we briefly set the notations for this particular case of a general theory of representations induced in the sense of Mackey (the reader is referred to [16,27,28] for detailed presentations).
Let H be a subgroup of H 1 p and χ be a (complex unitary) character of H. Let X = H 1 p /H be the corresponding left H 1 p -homogeneous space with the measure dx, which factorises the Haar measure dg = dx dh for a Haar measure dh of H. We write L χ 2 H 1 p for the space of functions F (g) on H 1 p having these properties: 2) because the left and right shifts commute. An alternative realisation of the same representation is obtained from a given section s : H 1 p /H → H 1 p which is a right inverse for the quotient map p : For such a section s and the character χ, we can define the lifting L χ : where r = (s(p(g))) −1 g.
Then, the lifting L χ intertwines a representation ρ χ on L 2 H 1 p /H and the left regular representation Λ (2.2) restricted to L χ 2 H 1 p : Combining the previous identities, we obtain an explicit expression for the induced representation The representation Λ , which is induced from χ on the space of square-integrable functions L 2 R 2 , is given by This is called the left quasi-regular representation. It is unitary reducible, and it can be decomposed into unitary irreducible components in many different ways (see [1,Section 6.4]). For physical (aka mathematical) reasons, the most popular irreducible module is the pre-Fock-Segal-Bargmann space (pre-FSB space): where the Lie derivative L A = dR A is the derivation of the right quasi-regular representation [1, Section 5.4.2] on L 2 R 2 : We will explain the origin of pre-FSB space in Section 3.2.
2. For a two-dimensional maximal Abelian continuous subgroup of H 1 p : the homogeneous space H 1 p /H x can be identified with R. The respective maps are The character χ (s, 0, y) = e 2πi s of H x induces the representation ρ on a space of squareintegrable functions, L 2 (R), is given by This is the celebrated Schrödinger representation, which is a unitary irreducible representation on L 2 (R) [15,Chapter 1].
There is another two-dimensional maximal Abelian continuous subgroup of H 1 p : Subgroups H x and H y are conjugated by the automorphism of i : There exist small but important differences between the respective maps cf. (2.8): The character χ (s, x, 0) = e 2πi s of H y induced an alternative form of the Schrödinger representation In quantum mechanics, (2.9) and (2.11) serve as coordinate and momentum representations of H 1 p . 3. For a non-commutative discontinuous subgroup H d of H 1 p : we consider a homogeneous space We have two possibilities to treat f : (a) f is a square-integrable and double quasi-periodic function on R 2 (periodic in u and quasi-periodic in v), that is, for all (n, k) ∈ Z 2 , we have If f is considered as a double quasi-periodic function on R 2 , the representation ρ m induced from the character χ m of H d on L 2 T 2 is given by 14) The representation ρ m is unitary irreducible on L 2 T 2 and is called the lattice representation [12]. The respective maps are is the fractional part. Using them we express on T 2 as follows For a unitary irreducible representation ρ of H 1 p , its restriction to the centre Z is a character ρ(s, 0, 0) = e 2πi s , where the real parameter is known as the Planck constant in quantum mechanical contexts (see [

An induced covariant transform
In this work we need an extended version of the covariant transform which covers the Banach space situation (see [31,Section 2] and [35]). All representations in this paper are assumed to be strongly continuous.
Definition 3.1. Let ρ be a representation of a group in a space V , and F be an operator from V to a space U . We define a covariant transform W F from V to the space L(G, U ) of a U -valued function on G by the formula The fundamental property of the covariant transform W F (3.1) is that W F intertwines the representation ρ and the left regular action Λ of G: For the Gilmore-Perelomov coherent states (see [40] and [3, Section 7.1.2]), it is enough to have the covariant transform values on a homogeneous space rather than the entire group. We name it the induced covariant transform [35] due to its connections with induced representations (cf. (3.5)). More specifically, for the Heisenberg group it is defined as follows. Let ρ be an irreducible unitary representation of H 1 p on a Hilbert space H, and H be a closed subgroup of H 1 p . Let X = H 1 p /H be a homogeneous space. Let φ 0 ∈ H be a fiducial vector, that is, for some character χ of H. The induced covariant transform W ρ φ 0 is a map from the Hilbert space H to a space W (X) of functions on X = H 1 p /H given as follows where s : X → H 1 p is a Borel section (the right inverse of the natural projection p : H 1 p → H 1 p /H). Note that, in the definition (3.4) we use a linear functional φ 0 ∈ V as a special case of the operator F : V → C in (3.1). Then, an adjusted notation of the covariant transform W F will be W ρ φ 0 . The main algebraic property of the induced covariant transform (3.4) is that it intertwines ρ on H with a representation ρ χ on W (X) induced by the character χ of the subgroup H. That is, Alternatively, this can be observed by the fact that any function of the image of the induced covariant transform (3.4) has the H-covariance propertỹ The main analytic property of the induced covariant transform is formulated in terms of the matrix coefficient: Example 3.2 (the inverse Fourier transform). To find the covariant transform which intertwines two forms of the Schrödinger representations (2.9) and (2.11), we shall take the fiducial vector which would be the eigenfunction for all representations ρ (s, x, 0) (2.9). That is, Of course the only solution is the constant function f (t) ≡ c. Then the corresponding induced covariant transform based on the maps (2.10) is Of course, this transformation is the (inverse) Fourier transform which intertwines the representations (2.9) and (2.11). The significance of this intertwining property for the harmonic analysis is revealed in the work of Howe [23].
Although many functions can be taken as fiducial vectors, some of them turn out to be much more preferable. The origin of their advantages is revealed by the following observation [33,Section 5]. Let G be a Lie group and ρ be its representation in a Hilbert space H. Let [W φ f ](g) = f, ρ(g)φ be the covariant transform defined by a fiducial vector φ ∈ H. Then, the covariant transform intertwines right shifts R(g) : f (g ) → f (g g) on the group G with the associated action ρ on fiducial vectors There are many interesting applications of this simple observation [4,5,33,36,37], in particular, 33,36]). Let G be a Lie group with a Lie algebra g and ρ be a representation of G on a Hilbert space L 2 (R n ). We denote the derived representation of ρ by dρ X . Let φ be a fiducial vector in the Schwartz space S(R n ) such that n j=1 a j dρ X j φ = 0, for some a j ∈ C. Then, the image of the covariant transform consists of functions f such that: where L X denotes the Lie derivative -the derivation of the right regular representation R of G: An induced covariant transform from a representation ρ on a Banach space B can be defined in a similar fashion. Let H be a closed subgroup of H 1 p and X = H 1 p /H be a homogeneous space. Consider a continuous section s : Both applications will be illustrated below.

The (pre-)Fock-Segal-Bargmann transform
We look for an induced covariant transform W ρ φ : L 2 (R) → L 2 R 2 , which intertwines the Schrödinger representation (2.9) and the left quasi-regular representation restricted to an irreducible component of L 2 R 2 . In fact, for the character χ (s, 0, 0) = e 2πi s of the centre Z = (s, 0, 0) ∈ H 1 p : s ∈ R , any vector φ ∈ L 2 (R) satisfies the following specialisation of (3.3) Let s : H 1 p /Z → H 1 p : (x, y) → (0, x, y) be a continuous section [1, Section 5.1.2]. Thus, for all f ∈ L 2 (R), the induced covariant transform W ρ φ for any fiducial vector φ ∈ L 2 (R) is The main properties of W ρ φ follow from the general properties of the covariant transform.
Remark 3.4. Let φ ∈ L 2 (R) be a fiducial vector such that φ = 1. The covariant transform W ρ φ : L 2 (R) → L 2 R 2 is a unitary intertwining operator between the Schrödinger representation ρ on L 2 (R) and the left quasi-regular representation Λ restricted on the image space In particular, Λ is an irreducible representation on F φ R 2 .
So far all fiducial vectors seem to be equally suitable, yet its is common to give the strong preference to a vacuum vector of the Schrödinger representation -the Gaussian This preference is explained by Proposition 3.3: since the Gaussian is a null-solution to the annihilation operator the image space of the induced covariant transform W ρ φ κ (3.4) is annihilated by the Lie derivative L κX+iY (the derived representation from the right regular action R ). This allows to give an intrinsic characterisation of the image space.
Explicitly, W ρ φ κ is given bỹ where the measure is renormalised by the factor ( κ ) 1/2 . For reasons explained here, we call it the pre-FSB transform (see [39,Section 4.2] and [15, Section 1.6]) from L 2 (R) into the pre-FSB space F φ κ R 2 (2.6). The image space F φ κ R 2 is a subspace of square-integrable functions on R 2 . The left quasi-regular representation Λ restricted on the pre-FSB space F φ κ R 2 is called the pre-FSB representation. The prefix "pre-" is removed by a unitary operator -the peeling, which will produce the FSB space of analytic functions on C in Section 4.1.
Of course, the last two conditions imply that supp(l 0 ) = Z. The simplest non-zero vector l 0 satisfying (3.13) would be the Dirac comb distribution, that is, which is a periodic distribution constructed from the Dirac delta δ(t).
Remark 3.5. Let K be a compact subset of R, and C(K) be the space of continuous functions on R supported in K, which is a Banach space equipped with the uniform norm. Let C c (R) be the union of these Banach spaces, where C c (R) inherits a natural inductive limit topology [16,41].
We denote by C * c (R) the space of all continuous functionals (pseudomeasures) on C c (R), which is the intersection of all duals of C(K). As the Dirac comb l 0 (t) (3.14) is a finite measure in any compact set K, thus l 0 ∈ C * c (R) is a pseudomeasure. Therefore, we consider the Schrödinger representation ρ of H 1 p on C c (R) ⊂ L 2 (R) to be restricted to the Banach space C(K), for any compact subset K of R. ). Let C c (K) be the space of smooth functions that are compact support in K. For f ∈ C c (K) ⊂ L 2 (R), we calculate the induced covariant transform as follows: Remark 3.7. Since the co-Zak transform is defined on C c (R), which is dense on L 2 (R), the co-Zak transform can be extended to be defined on the entire L 2 (R).
The general properties of the covariant transform W ρ l 0 yield corresponding properties of the Zak transform.  where ω = m(v + iu/κ) ∈ C and Θ mκ is the Jacobi theta function. We will use the notation Φ mκ (ω,ω) = Φ mκ (u, v) as well. By Lemma 2.3(2) Φ mκ , is a vacuum of the lattice representation.

The (pre-)theta transform
In the present subsection, we look for an intertwining operator between the lattice representation and the left quasi-regular representation restricted to an irreducible component of L 2 R 2 . Although the formula of the left quasi-regular representation (2.5) looks very similar to the lattice representation's formula (2.14): they act on different spaces L 2 R 2 and L 2 T 2 , respectively. Let χ (s, 0, 0) = e 2πi s be the character of the centre Z ⊂ H 1 p . As was already mentioned (Section 3.2), any vector Φ ∈ L 2 T 2 satisfies version ρ m (s, 0, 0)Φ = χ (s, 0, 0)Φ of (3.3), for all (s, 0, 0) ∈ Z and = m. Thus, the respective covariant transform W Φ intertwines the lattice and quasi-regular representations.
We may be more specific and request that W Φ map: L 2 T 2 to the pre-FSB space F 2 R 2 . As in the case of the pre-FSB transform, the vacuum Φ mκ (3.16) shall be taken as the fiducial vector. Indeed, by Proposition 3.3, the image space of W ρ m Φmκ is annihilated by the right ladder iκL X + L Y (2.7).
We call W

Peeling representations of H 1 p and analyticity
Induced representations are a common tool to construct representations of groups. However, a representation prepared using a generic methodology may not be particularly suited for a special situation. It often needs to be tuned to be enriched with useful features. In this section, we demonstrate such a simple tool which produces the required enhancements needed for various situations.
For example: since the annihilation operator a − provides a useful characterisation of an irreducible component of a representation, we are interested in expressing a − in the most transparent form. A map, called here peeling, then simplifies the corresponding annihilation operator into a linear combination of first-order derivatives only. Therefore, the structure of the eigenvectors φ n (cf. [1,Section 5.4] and [30, Section 2.6]) forms an orthonormal basis of the initial irreducible space H becomes more transparent.
On the other hand, if a representation is reducible, its irreducible component can be characterised as the space of null-solutions for the Lie derivative (cf. (2.6)). In such cases, we seek to peel the irreducible component to a space of analytic functions. This allows one to use the power of complex variable theory to study the induced representations of H 1 p .
Definition 4.1. A peeling ε d is an invertible operator of multiplication defined by a function d(x) on X: The operator ε d is unitary for suitably related measures We use such peeling operators to improve some properties of covariant transform related to specific representations. In this paper, all considered peelings use smooth d(x) on a domain X in a Euclidean space. We will discuss the choice of d(x) for the pre-FSB, Schrödinger and lattice representations in Sections 4.1, 4.2 and 4.3, respectively.
Consider the variables z,z ∈ C, where z = h 2κ (x+iκy) and h = 2π > 0. In this subsection, we peel the representation Λ into the corresponding oneΛ that acts on the FSB space of analytic functions. To perform this, we look for a peeling operator satisfying the following conditions: 1. The peeling defined by e d(z,z) shall intertwine the right annihilation operator L κX+iY = 2π x + (κ∂ x + i∂ y ) and the Cauchy-Riemann operators ∂z = κ∂ x + i∂ y : A simple differential equation for (4.1) implies that whereψ is an arbitrary smooth function of z alone.
2. Let a − Λ = dΛ κX−iY = 2πi κy − (κ∂ x − i∂ y ) be the left annihilation operator of Λ (2.5). The same peeling shall intertwine a − Λ with (a multiple of) the complex derivative ∂ z = (κ∂ x −i∂ y ). This fixesψ(z) = − 1 2 z 2 −c, c ∈ C in (4.2). Thus, the peeling operator becomes Let c 0 ∈ C and c 0 = 0. There is a special vacuum of the representation Λ annihilated by both a − Λ and L κX+iY , given as follows The consequence of the conditions (1) and (2) is that the peeling maps the vacuum φ 0 , which is killed by both the left and right annihilation operators to the function identically equal to c 0 , which is killed by both ∂ z and ∂z. The peeled representationΛ is which is called the FSB representation. The composition of the peeling (4.3) with the covariant transform (3.12) is where F (z) is an analytic function of z = h 2κ (x + iκy). Indeed, by Proposition 3.3 and the intertwining property (4.1), the function F (z) (4.5) satisfies ∂zF (z) = (κ∂ x + i∂ y )F (x, y) = 0, which is essentially the Cauchy-Riemann equation. The integral (4.5) is known as the FSB transform. The image F 2 of the FSB transforms is called the FSB space. It is a closed subspace of 2κ (x 2 +κ 2 y 2 )+2c dx dy = L 2 C, e −|z| 2 +2c dz dz consisting of the analytic functions. Note that often, only the values = 1, κ = 1 and c = 0 are used [42]. Similar to the pre-FSB transform, we calculate the composition of the pre-theta transformf (3.17) and the peeling (4.3), h = 2πm, as follows where z = h 2κ (x + iκy) ∈ C as before. We callF (z) the theta transform of f (u, v). As with the FSB transform, the intertwining property (4.1) implies that the image of the theta transform consists of analytic functions, which can be found in many references [10,15,39].

Peeling the Schrödinger representation
In this subsection, we are performing less common peeling the Schrödinger representation ρ (2.9), so that the corresponding annihilation operator will only be the derivative ∂ t . For a representation space realised as a function on a set, we have the following simple result. Recall the annihilation operator a − ρ = dρ κX−iY = −2π t − κ∂ t (3.11) for the Schrödinger representation. By Lemma 4.2, a function ε d which intertwines a − ρ with the plain derivative κ∂ t shall satisfy the vacuum condition: a − ρ ε −1 d = 0. Thus, the peeling defined by ε d (t) = ce π t 2 κ , c ∈ C is unitary: The composition of the Schrödinger representation with the peeling acting on L 2 R, e −2 π κ t 2 dt is Consequently, the corresponding derived representations of the Lie algebra h 1 are and the annihilation operator a − ρ is a − ρ = dρ κX−iY = −∂ t , which annihilates the vacuumφ(t) = c. A notable consequence is that ε d transforms the Hermite functions H n (t)e − π κ t 2 ∈ L 2 (R) to the corresponding Hermite polynomials H n (t) ∈ L 2 R, e −2 π κ t 2 dt .

Peeling the lattice representation
The purpose of peeling the lattice representation ρ m (2.14) seems to be difficult to formulate while staying within the framework of the Heisenberg group itself. However, the lattice representation is better understood if it is extended to the Schrödinger group (aka the Jacobi group [10, Chapter 9]) -the semi-direct product of H 1 p with the group SL 2 (R) acting on H 1 p through symplectic automorphism [15,Section 1.2]. The representation of the Schrödinger group can then be peeled in a way that its vacuum will be a null-solution of the heat equation.
Since an accurate treatment of the Schrödinger group is beyond the scoop of the present paper, we simply provide the form of the required peeling, cf. (3.16): where ω = m κ (κv + iu).
Then, ε d is a unitary operator: Therefore, the corresponding annihilation operator is simply which annihilates the theta functionΦ =: cΘ mκ ω, im κ . Therefore, the peeling maps the vacuum vector Φ mκ from Section 3.4 to the theta function: The goal of this section is to introduce the contravariant transform M ψ , which is the adjoint of the covariant transform W φ (see [3,Section 8.1] and [31, Section 2]).

A Contravariant transform for induced representations
Let H be a closed subgroup of H 1 p and X = H 1 p /H be the respective homogeneous space, which is a subset of Euclidean space with the respective Lebesgue measure. Let ρ be a representation of the Heisenberg group H 1 p on the vector space V.
The right-hand side of (5.1) is If the intertwining property holds for every functionν, then the following condition for the reconstructing vector ψ is required: χ(r(g))ρ (s(p(g))) ψ = ρ(g)ψ, for all g ∈ G. (5. 2) The abstract framework of the contravariant transform is well known for a unitary irreducible representation ρ in a Hilbert space V (see for example [ [2,31,35]. We use W * : B * → W * (X) and M * : W * (X) → B * to denote the adjoint operators to W and M, respectively. This results in the following identity: The contravariant transform construction is particularly simple for maps acting from the pre-FSB space as the next two examples show.
Example 5.2 (the inverse of the (pre-)FSB transform). For the Schrödinger representation ρ (2.9), the intertwining condition (5.2) is trivially satisfied by an arbitrary reconstructing vector ψ. From the sesqui-unitary property (3.8), it follows that M ψ •W φ = I for vectors ψ and φ such that φ, ψ = 1. In particular, for the Gaussian ψ κ (t) = 2 1/4 e − π κ t 2 as a reconstruction vector, the contravariant transform is This is known as the inverse of the pre-FSB transform [39, Section 4.2].
Thus, we obtain the inverse operator of W Example 5.4 (the Fourier transform). We can look for a contravariant transform which intertwines the two forms (2.9) and (2.11) and will be an inverse of the covariant transform (the Fourier transform) from Example 3.2. Using maps (2.10) and representation (2.9), we obtain the form of the compatibility condition (5.2): e 2πi s e −2πi ty ψ(t) = e 2πi (s−ty) ψ(t − x), which again delivers the solution ψ(λ) ≡ 1. The respective contravariant transform is, as expected, the Fourier transform: A bit more care is required in the next case.

The inverse of the Zak transform
In Section 3.3, we derived the co-Zak transform Z : L 2 (R) → L 2 T 2 (3.15) through the induced covariant transform W ρ φ 0 . Now, we calculate its inverse using the contravariant transform. The intertwining property (  Since g(t, v) is contained in the space L 2 T 2 of square-integrable functions that are periodic in t and quasi-periodic in v, multiplying g(t, v) by e −2πimtv produces a function that has the same double quasi-periodicity property of g(t, v) but in the opposite way. In other words, g(t, v) = g(t, v) · e −2πimtv ∈L 2 T 2 is quasi-periodic in t and periodic in v and square-integrable. Moreover, since t ∈ R ≈ [0, 1] × Z, then t = x + n, for some x ∈ [0, 1] and n ∈ Z. Therefore, for t = x + n, (5.5) becomes The computations of this subsection allows us to presentZ −1 as a contravariant transform.
Theorem 5.6. Letg(x, v) = g(x, v) · e −2πimxv such that g ∈ L 2 T 2 . The contravariant transform (5.6), is the inverse of the Zak transform. For g ∈ L 2 T 2 , one can write the inverse of the co-Zak transform as Z −1 g =Z −1 e −2πimuv g.

Conclusion
Our work in this paper illustrates the technique which allows to obtain co-and contra-variant transforms with desired properties. For the covariant transform, the fiducial vector needs to agree with Proposition 3.3. An induced contravariant transform will be an intertwining operator with an induced representation if the reconstructing vector satisfies (5.2). This approach is illustrated on various representations of the Heisenberg group, and produces an interpretation of the Zak transform and its inverse as induced co-and contra-variant transforms. Furthermore, we used peeling operators to obtain the familiar representation spaces of analytic function space of analytic functions. Of course, this approach is not limited to the illustrative example of the Heisenberg group and can be fruitfully applied in many other situations.