Regular Sampling on Metabelian Nilpotent Lie Groups: The...

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Regular Sampling on Metabelian Nilpotent Lie Groups: The Multiplicity-Free Case Vignon S Oussa August 6, 2016 Abstract Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra n having rational structure constants. We assume that N = P o M, M is commutative, and for all λ n * in general posi- tion the subalgebra p = log(P ) is a polarization ideal subordinated to λ (p is a maximal ideal satisfying [p, p] ker λ for all λ in general position and p is necessarily commutative.) Under these assumptions, we prove that there exists a discrete uniform subgroup Γ N such that L 2 (N ) admits band- limited spaces with respect to the group Fourier/Plancherel transform which are sampling spaces with respect to Γ. We also provide explicit sufficient conditions which are easily checked for the existence of sampling spaces. Suf- ficient conditions for sampling spaces which enjoy the interpolation property are also given. Our result bears a striking resemblance with the well-known Whittaker-Kotel’nikov-Shannon sampling theorem. 1 Introduction It is a well-established fact that a function on the real line with its Fourier transform vanishing outside of an interval - 1 2 , 1 2 can be reconstructed by the Whittaker-Kotel’nikov-Shannon sampling series from its values at points in the lattice Z (see [12]). This series expansion takes the form f (t)= X kZ f (k) sin (π (t - k)) π (t - k) with convergence in L 2 (R) as well as convergence in L (R). A relatively novel problem in harmonic analysis has been to find analogues of Whittaker- Kotel’nikov-Shannon sampling series for non-commutative groups. Since R is a commutative nilpotent Lie group, it is natural to investigate if it is possible to extend Whittaker-Kotel’nikov-Shannon’s theorem to nilpotent Lie groups which are not commutative. 1

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Regular Sampling on Metabelian Nilpotent LieGroups: The Multiplicity-Free Case

Vignon S Oussa

August 6, 2016

Abstract

Let N be a simply connected, connected non-commutative nilpotent Liegroup with Lie algebra n having rational structure constants. We assumethat N = P o M, M is commutative, and for all λ ∈ n∗ in general posi-tion the subalgebra p = log(P ) is a polarization ideal subordinated to λ (pis a maximal ideal satisfying [p, p] ⊆ kerλ for all λ in general position and pis necessarily commutative.) Under these assumptions, we prove that thereexists a discrete uniform subgroup Γ ⊂ N such that L2(N) admits band-limited spaces with respect to the group Fourier/Plancherel transform whichare sampling spaces with respect to Γ. We also provide explicit sufficientconditions which are easily checked for the existence of sampling spaces. Suf-ficient conditions for sampling spaces which enjoy the interpolation propertyare also given. Our result bears a striking resemblance with the well-knownWhittaker-Kotel’nikov-Shannon sampling theorem.

1 Introduction

It is a well-established fact that a function on the real line with its Fouriertransform vanishing outside of an interval

[−1

2 ,12

]can be reconstructed by

the Whittaker-Kotel’nikov-Shannon sampling series from its values at pointsin the lattice Z (see [12]). This series expansion takes the form

f (t) =∑k∈Z

f (k)

(sin (π (t− k))

π (t− k)

)with convergence in L2 (R) as well as convergence in L∞ (R). A relativelynovel problem in harmonic analysis has been to find analogues of Whittaker-Kotel’nikov-Shannon sampling series for non-commutative groups. Since R isa commutative nilpotent Lie group, it is natural to investigate if it is possibleto extend Whittaker-Kotel’nikov-Shannon’s theorem to nilpotent Lie groupswhich are not commutative.

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Let G be a locally compact group and Γ a discrete subset of G. Let H bea left-invariant closed subspace of L2 (G) consisting of continuous functions.We say that H is a sampling space with respect to the set Γ [8] if thefollowing conditions are satisfied. Firstly, the restriction map f 7→ f |Γ definesa constant multiple of an isometry of H into the Hilbert space of square-summable sequences defined over Γ. In other words, there exists a positiveconstant cH such that ∑

γ∈Γ

|f (γ)|2 = cH ‖f‖22 (1)

for all f in H. Secondly, there exists a vector s in H such that an arbitraryelement f in the given Hilbert space has the expansion

f (x) =∑γ∈Γ

f (γ) s(γ−1x

)(2)

with convergence in the norm of L2 (G) . If Γ is a discrete subgroup of G, wesay that H is a regular sampling space with respect to Γ. Also, if H is asampling space with respect to Γ and if the restriction mapping f 7→ f |Γ ∈l2 (Γ) is surjective then we say that H has the interpolation propertywith respect to Γ. This notion of sampling space is taken from [8] and isanalogous to Whittaker-Kotel’nikov-Shannon’s theorem. In [9], the authorsused a less restrictive definition. They defined a sampling space to be a left-invariant closed subspace of L2 (G) consisting of continuous functions withthe additional requirement that the restriction map f 7→ f |Γ is a topologicalembedding of H into l2 (Γ) in the sense that there exist positive real numbersa ≤ b such that

a ‖f‖22 ≤∑γ∈Γ

|f (γ)|2 ≤ b ‖f‖22

for all f ∈ H. The positive number b/a is called the tightness of the samplingset. Notice that in (1) the tightness of the sampling is required to be equal toone. Using oscillation estimates, the authors in [9] provide general but preciseresults on the existence of sampling spaces on locally compact groups. Theband-limited vectors in [9] are functions that belong to the range of a spectralprojection of a self-adjoint positive definite operator on L2(G) called the sub-Laplacian. This notion of band-limitation is essentially due to Pesenson [18]and does not rely on the group Fourier transform.

We shall employ in this work a different concept of band-limitation whichin our opinion is consistent with the classical one (the Whittaker-Kotel’nikov-Shannon band-limitation), and the main objective of the present work isto prove that under reasonable assumptions (see Condition 3) Whittaker-Kotel’nikov-Shannon Theorem naturally extends to a large class of non com-mutative nilpotent Lie groups.

Let G be a simply connected, connected nilpotent Lie group with Liealgebra g. A subspace H of L2(G) is said to be a band-limited space with

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respect to the group Fourier transform if there exists a bounded subset E ofthe unitary dual of the group G such that E has positive Plancherel measure,and H consists of vectors whose group Fourier transforms are supported onthe bounded set E. In this work, we address the following.

Problem 1 Let N be a simply connected and connected nilpotent Lie groupwith Lie algebra n with rational structure constants. Are there conditions onthe Lie algebra n under which there exists a uniform discrete subgroup

Γ ⊂ N = exp n

such that L2 (N) admits a band-limited (in terms of the group Fourier/Planchereltransform) sampling subspace with respect to Γ?

Here are some partial answers to Problem 1.

• If N = Rd then Γ can be taken to be an integer lattice, and the Hilbertspace of functions vanishing outside the cube[

−1

2,1

2

]dis a sampling space which enjoys the interpolation property with respectto Zd.

• Put

N =

1 x z

0 1 y0 0 1

: x, y, z ∈ R

and

Γ =

1 m k

0 1 l0 0 1

: k, l,m ∈ Z

.

Then N is the three-dimensional Heisenberg Lie group and Γ is a discreteuniform subgroup of N . The unitary dual of N is up to a null setparametrized by the punctured line R\ 0, and the Plancherel measureis the weighted Lebesgue measure |λ| dλ. It is shown in [8, 5] that thereexist band-limited subspaces with respect to the group Fourier transformof L2 (N) which are sampling subspaces with respect to Γ.

• Let N be a step-two nilpotent Lie group with Lie algebra n of dimen-sion n with center z such that n = z ⊕b ⊕ a where [a, b] ⊆ z, a, b arecommutative Lie algebras,

a =

d∑k=1

RXk, b =

d∑k=1

RYk, z =

n−2d∑k=1

RZk (d ≥ 1, n > 2d)

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and

det

[X1, Y1] · · · [X1, Yd]...

. . ....

[Xd, Y1] · · · [Xd, Yd]

(3)

is a non-vanishing polynomial in the variables Z1, · · · , Zn−2d. The map

n = z⊕ b⊕ a Z + Y +X 7→ exp (Z + Y ) exp (X) ∈ N

defines an analytic diffeormorphism between n and N which is used toendow N with a coordinate system. Moreover, given Z,Z ′ ∈ z, Y, Y ′ ∈ band X,X ′ ∈ a, the multiplication law for N is given by

(exp (Z + Y ) exp (X))(exp

(Z ′ + Y ′

)exp

(X ′))

= exp(Z + Z ′ + Y + eadX

′Y ′)

exp(X +X ′

)where the linear operator adX is defined as follows: adX (Y ) = [X,Y ] .The unitary dual of N is up to a null set parametrized by

λ ∈ z∗ : det B (λ) 6= 0

where

B (λ) =

〈λ, [X1, Y1]〉 · · · 〈λ, [X1, Yd]〉...

. . ....

〈λ, [Xd, Y1]〉 · · · 〈λ, [Xd, Yd]〉

and the corresponding Plancherel measure is the weighted Lebesgue mea-sure |det B (λ)| dλ. It is proved in [15] that L2 (N) admits band-limitedsampling subspaces with respect to the discrete subset

Γ = exp

(n−2d∑k=1

ZZk +d∑

k=1

ZYk

)exp

(d∑

k=1

ZXk

)⊂ N.

In the case where the structure constants of the Lie algebra are integersthen Γ is a discrete uniform subgroup of N.

The main objective of the present work is to exploit well-known represen-tation theoretic tools for nilpotent Lie groups to establish in a unified mannerthe existence of regular sampling spaces on a large class of nilpotent Lie groups(see Condition 3). It is worth pointing out that the class of groups consideredhere contains nilpotent Lie groups of arbitrary step.

1.1 Overview of the Paper

Let us start by fixing notation and by recalling some relevant concepts.

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• Let Q be a linear operator acting on an n-dimensional real vector spaceV. The norm of the matrix Q induced by the max-norm of the vectorspace V is given by

‖Q‖∞ = sup ‖Qv‖max : v ∈ V and ‖v‖max = 1

and the max-norm of an arbitrary vector is given by

‖v‖max = max |vk| : 1 ≤ k ≤ n .

Next, letting [Q] be the matrix representation of Q with respect to afixed basis, the transpose of this matrix is denoted [Q]T . Additionally,the adjoint of a linear operator Q is denoted Q∗.

• Given a countable sequence (fi)i∈I of vectors in a Hilbert space H, wesay that(fi)i∈I forms a frame [1, 11, 21] if and only if there exist strictlypositive real numbers a, b such that for any vector f ∈ H,

a ‖f‖2 ≤∑i∈I|〈f, fi〉|2 ≤ b ‖f‖2 .

In the case where a = b, the sequence (fi)i∈I is called a tight frame. Ifa = b = 1, (fi)i∈I is called a Parseval frame.

• Let π be a unitary representation of a locally compact group G actingon a Hilbert space Hπ. We say that the representation π is admissible[8] if there exists a vector h in Hπ such that the linear map V π

h given by

V πh f (x) = 〈f, π (x)h〉 (4)

defines an isometry of the Hilbert space Hπ into L2 (G) . In this case,the vector h is called an admissible vector for the representation π.

• Let (A,M) be a measurable space. A family (Ha)a∈A of Hilbert spacesindexed by the set A is called a field of Hilbert spaces over A [6]. Anelement f of Πa∈AHa is a vector-valued function a 7→ f (a) ∈Ha definedon the set A. Such a map is called a vector field on A. A measurablefield of Hilbert spaces defined on a measurable set A is a field of Hilbertspaces together with a countable set (ej)j∈J of vector fields such thatthe functions

a 7→ 〈ej (a) , ek (a)〉Ha

are measurable for all j, k ∈ J, and the linear span of ej (a)j∈J is densein Ha for each a. A vector field f is called a measurable vector fieldif

a 7→ 〈f (a) , ej (a)〉Ha

is a measurable function for each index j.

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• Let n be a nilpotent Lie algebra of dimension n, and let n∗ be the dualvector space of n. A polarizing subalgebra p (λ) subordinated to alinear functional λ ∈ n∗ (see [2, 14]) is a maximal algebra satisfying

[p (λ) , p (λ)] = Span- [X,Y ] ∈ n : X,Y ∈ p (λ) ⊆ ker (λ) .

• The coadjoint action on the dual of n is the dual of the adjoint actionof N = exp n on n. In other words, for X ∈ n, and a linear functionalλ ∈ n∗, the coadjoint action is defined as follows:

(expX · λ) (Y ) =⟨(e−ad(X)

)∗λ, Y

⟩=[(e−ad(X)

)∗λ]

(Y ) . (5)

The following is a concept which is central to our results.

Definition 2 Let p be a subalgebra (or ideal) of n. We say that p is a con-stant polarization subalgebra (or ideal) of n if there exists a Zariskiopen set Ω ⊂ n∗ which is invariant under the coadjoint action of N and p isa polarization subalgebra subordinated to every linear functional in Ω.

In other words, p is a constant polarization subalgebra of n if p is a polar-ization algebra for all linear functionals in general position, and it can thenbe shown (see Proposition 8) that p is necessarily commutative.

1.1.1 Summary of Main Results

Let us suppose that

N = P oM = exp (p) o exp (m)

is a simply connected, connected non-commutative nilpotent Lie group withLie algebra n = p + m such that

Condition 3

1. p is a constant polarization ideal of n (thus commutative) m is commu-tative as well, p = dim p, m = dimm and p−m > 0.

2. There exists a strong Malcev basis Z1, · · · , Zp, A1, · · · , Am for n suchthat Z1, · · · , Zp is a basis for p, A1, · · · , Am is a basis for m and

Γ = exp (ZZ1 + · · ·+ ZZp) exp (ZA1 + · · ·+ ZAm)

is a discrete uniform subgroup of N. This is equivalent to the fact thatn has rational structure constants (see Chapter 5, [2]).

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First of all, we observe that the mapping

p + m Z +A 7→ exp (Z) exp (A) ∈ N

defines an analytic diffeomorphism between n and N which provides a coor-dinate system on the Lie group N. This diffeomorphism takes the canonicalLebesgue measure on the algebra n to a Haar measure on the Lie group Nwhich is both left and right invariant. The multiplication law on N is givenby

(exp (Z) exp (A))(exp

(Z ′)

exp(A′))

= exp(Z + ead(A)Z ′

)exp

(A+A′

)where Z,Z ′ ∈ p and A,A′ ∈ m.

In order to properly introduce the concept of band-limitation with respectto the group Fourier transform, we appeal to Kirillov’s theory [2] which statesthat the unitary irreducible representations of N are parametrized by orbitsof the coadjoint action of N on the dual of its Lie algebra and can be modeledas acting in the Hilbert space L2(Rm). Let Σ be a parameterizing set for theunitary dual of N. In other words, Σ is a cross-section for the coadjoint orbitsin an N -invariant Zariski open set Ω ⊂ n∗. If the ideal p is a constant polariza-tion subalgebra for n then the orbits in general position are 2m-dimensionalsubmanifolds of n∗ and we shall (this is a slight abuse of notation) regard Σas a Zariski open subset of Rp−m = Rn−2m. Next, let L be the left regularrepresentation of N acting on L2 (N) by left translations. Let

P : L2 (N) −−−−→ L2(Σ, L2 (Rm)⊗ L2 (Rm) , dµ (λ)

)be the Plancherel transform which defines a unitary map on L2 (N) (see Sub-section 2.2.2). The Plancherel transform intertwines the left regular represen-tation with a direct integral of irreducible representations of N. The measureused in the decomposition is the so-called Plancherel measure: dµ; which is aweighted Lebesgue measure on Σ. More precisely dµ (λ) is equal to |P (λ)| dλ.P (λ) is a polynomial defined over Σ and dλ is the Lebesgue measure on Σ(see Lemma 23.) Given a µ-measurable bounded set A ⊂ Σ, and a fixedmeasurable field of unit vectors (u (λ))λ∈A in L2 (Rm) , we define the Hilbertspace HA as follows. HA consists of vectors f ∈ L2(N) such that

Pf (λ) =

v (λ)⊗ u (λ) if λ ∈ A

0⊗ 0 if λ /∈ A

and (v (λ)⊗ u (λ))λ∈A is a measurable field of rank-one operators. Con-sequently, HA is a left-invariant multiplicity-free band-limited subspace ofL2 (N) which is naturally identified with the function space L2(A×Rm) (seeSubsection 2.3.) Conjugating the operators L(x) by the Plancherel transform,we obtain that

[P L(·) P−1](v (λ)⊗ u (λ))λ∈Σ = ([σλ(·)v(λ)]⊗ u(λ))λ∈Σ

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where σλ is the unitary irreducible representation corresponding to the linearfunctional λ ∈ Σ, and the action of σλ on L2(Rm) is given by (see Lemma 22)σλ

exp

m∑j=1

ajAj

f

(x) = f (x− a)

andσλexp

m∑j=1

zjZj

f

(x) = exp

2πi

⟨λ, e−ad(x1A1+···+xmAm)

m∑j=1

zjZj

⟩ f (x) .

Let LHAbe the representation induced by the action of the left regular repre-

sentation on the Hilbert space HA. It can be shown that if the spectral set Asatisfies precise conditions specified in Theorem 4 then the restriction of LHA

to the discrete group Γ is unitarily equivalent with a subrepresentation of theleft regular representation of Γ acting on l2 (Γ) . More precisely if A satisfiesthe conditions described in Theorem 4, it can be proved that there exists avector η ∈ HA such that

HA F 7→(⟨F,L

(γ−1

)η⟩)γ∈Γ

defines an isometric embedding of HA into l2 (Γ) which intertwines the rep-resentation LHA

with a subrepresentation of the right regular representationof Γ. Let

projp∗ : n∗ → p∗

be the restriction mapping given by

projp∗

p∑j=1

(λjZ

∗j

)+

m∑j=1

(ξjA

∗j

) =n∑j=1

λjZ∗j (6)

whereZ∗1 , · · · , Z∗p , A∗1, · · · , A∗m

is a dual basis for Z1, · · · , Zp, A1, · · · , Am .

Next define the map β : Σ× Rm → Rp by

β (λ, t) = projp∗

exp

m∑j=1

tjAj

· λ = projp∗

((e−ad

∑mj=1 tjAj

)∗λ)

(7)

where · denotes the coadjoint action of N on the dual of its Lie algebra andt = (t1, · · · , tm). Under the assumptions listed in Condition 3, we prove inLemma 20 that β is a diffeomorphism with rational inverse, and the followingholds true.

Theorem 4 LetN = P oM = exp (p) o exp (m)

be a simply connected, connected nilpotent Lie group with Lie algebra n satis-fying Condition 3. Let A be a µ-measurable bounded subset of Σ.

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1. If β (A× [0, 1)m) has positive Lebesgue measure in Rp and is containedin a fundamental domain of Zp then there exists a vector η ∈ HA suchthat V L

η (HA) is a left-invariant subspace of L2 (N) which is a samplingspace with respect to Γ.

2. If β (A× [0, 1)m) is equal to a fundamental domain of Zp then thereexists a vector η ∈ HA such that V L

η (HA) is a left-invariant subspaceof L2 (N) which is a sampling space with the interpolation property withrespect to Γ.

Lets = (s1, s2, · · · , sm)

be an element of Rm and define A (s) to be the restriction of the linear map

ad(−∑m

j=1 sjAj

)to the ideal p ⊂ n. Let [A (s)] be the matrix representation

of the linear map A (s) with respect to the basis Z1, · · · , Zp. Let e[A(s)] bethe matrix obtained by exponentiating [A (s)] . Since

s 7→∥∥∥e[A(s)]T

∥∥∥∞

is a continuous function of s, it is bounded over any compact set and inparticular over the cube [0, 1]m. As such, letting ε be a positive real numbersatisfying

ε ≤ δ =1

2

(sup

∥∥∥e[A(s)]T∥∥∥∞

: s ∈ [0, 1)m)−1

<∞, (8)

we shall prove that under the assumptions provided in Condition 3, the set

β(

(−ε, ε)n−2m × [0, 1)m)

has positive Lebesgue measure and is contained in a fundamental domainof Zp. Appealing to Theorem 4, we are then able to establish the followingresult which provides us with a concrete formula for the bandwidth of varioussampling spaces.

Corollary 5 LetN = PM = exp (p) exp (m)

be a simply connected, connected nilpotent Lie group with Lie algebra n satis-fying Condition 3. For any positive number ε satisfying (8) there exists a band-

limited vector η = ηε in the Hilbert space H(−ε,ε)n−2m such that V Lη

(H(−ε,ε)n−2m

)is a left-invariant subspace of L2 (N) which is a sampling space with respectto Γ.

Next, we exhibit several examples to illustrate that the class of groupsunder consideration is fairly large.

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Example 6

1. Let N be a simply connected, connected nilpotent Lie group with Lie al-gebra n of dimension four or less. Then, there is a basis for the Liealgebra n with respect to which there exists a uniform discrete subgroupΓ ⊂ N such that L2 (N) admits a band-limited sampling subspace withrespect to Γ. Additionally, the Heisenberg Lie group admits a samplingspace which has the interpolation property with respect to a uniform dis-crete subgroup (this result for the Heisenberg group was originally provedby Currey and Mayeli in [5]).

2. (Nilpotent Lie groups of the type Rp o R) Let N be a simply connected,connected nilpotent Lie group with Lie algebra spanned by Z1, Z2, · · · , Zp, A1,the vector space generated by Z1, Z2, · · · , Zp is a commutative ideal,[adA1]|p = A is a nonzero rational upper triangular nilpotent matrix

of order p, and eAZp ⊆ Zp. Then L2 (N) admits a band-limited samplingsubspace with respect to the discrete uniform subgroup

exp (ZZ1 + · · ·+ ZZp) exp (ZA1) .

3. Let N be a simply connected, connected nilpotent Lie group with Liealgebra spanned by the vectors Z1, Z2, · · · ,Zp, A1, · · · , Am where p = m+1, the vector space generated by Z1, Z2, · · · , Zp is a commutative ideal,the vector space generated by A1, · · ·Am is commutative and the matrixrepresentation of ad (

∑mk=1 tkAk) restricted to p is given by

A (t) =

[ad

m∑k=1

tkAk

]∣∣∣∣∣ p = m!

0 t1 t2 · · · tm−1 tm

0 t1 t2. . . tm−1

0 t1. . .

...

0. . . t2. . . t1

0

. (9)

Then L2 (N) admits a band-limited sampling subspace with respect to thediscrete uniform subgroup

exp (ZZ1 + · · ·+ ZZp) exp (ZA1 + · · ·+ ZAm) .

The work is organized as follows. In Section 2, we present general well-known results of harmonic analysis on nilpotent Lie groups. Section 3 containsintermediate results leading to the proofs of Theorem 4, Corollary 5 andExample 6 which are given in Section 4.

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2 Harmonic Analysis on Nilpotent Lie Groups

2.1 Parametrization of Coadjoint Orbits

Let n be a finite-dimensional nilpotent Lie algebra of dimension n. We saythat n has a rational structure [2] if there is a real basis Z1, · · · , Zn for theLie algebra n having rational structure constants. Moreover, the rational spannQ Z1, · · · , Zn provides a rational structure such that n is isomorphic to thevector space nQ ⊗ R. Let B = Z1, · · · , Zn be a basis for the Lie algebra nsuch that for any Zi, Zj ∈ B, we have:

[Zi, Zj ] =n∑k=1

cijkZk

and cijk ∈ Q. We say that B is a strong Malcev basis (see Page 10, [2])if and only if for each 1 ≤ j ≤ n the real span of Z1, Z2, · · · , Zj is an idealof n. Now, let N be a connected, simply connected nilpotent Lie group withLie algebra n having a rational structure. The following result is taken fromCorollary 5.1.10, [2]. Let Z1, · · · , Zn be a strong Malcev basis for the Liealgebra n. There exists a suitable integer q such that

Γq = exp (qZZ1) · · · exp (qZZn)

is a discrete uniform subgroup of N (there is a compact set K ⊂ G such thatΓK = N). Setting Xk = qZk for 1 ≤ k ≤ n, from now on, we fix X1, · · · , Xnas a strong Malcev basis for the Lie algebra n such that

Γ = exp (ZX1) · · · exp (ZXn)

is a discrete uniform subgroup of N .We shall next discuss the Plancherel theory for N. This theory is well

exposed in [2] for nilpotent Lie groups. Let s be a subset of n = log(N). Foreach linear functional λ ∈ n∗, we define the corresponding set

s (λ) = Z ∈ n : 〈λ, [Z,X]〉 = 0 for every X ∈ s .

Next, we consider a fixed strong Malcev basis B′ = X1, · · · , Xn and weconstruct a sequence of ideals

n1 ⊆ n2 ⊆ · · · ⊆ nn−1 ⊆ n

where each ideal nk is spanned by X1, · · · , Xk . It is easy to see that thedifferential of the coadjoint action on λ at the identity is given by the matrix

[〈λ, [Xj , Xk]〉]1≤j,k≤n = [λ ([Xj , Xk])]1≤j,k≤n .

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Defining the skew-symmetric matrix-valued function

λ 7→M (λ) =

λ [X1, X1] · · · λ [X1, Xn]...

. . ....

λ [Xn, X1] · · · λ [Xn, Xn]

(10)

on n∗, it is worth noting that n (λ) is equal to the null-space of M (λ), if M (λ)is regarded as a matrix with respect to the ordered basis B′ acting on n [14].According to the orbit method [2], the unitary dual of N is in one-to-onecorrespondence with the set of coadjoint orbits in the dual of the Lie algebra.For each λ ∈ n∗ we define

e (λ) = 1 ≤ k ≤ n : nk * nk−1 + n (λ) . (11)

The set e (λ) collects all basis elements Xi : i ∈ e (λ) ⊂ B′ such that if

e (λ) = e1 (λ) < · · · < e2m (λ)

then the dimension of the manifold

exp(RXe1(λ)

)· · · exp

(RXe2m(λ)

)· λ

is equal to the dimension of the N -orbit of λ. Each element of the set e (λ)is called a jump index and clearly the cardinality of the set of jump indicese (λ) must be equal to the dimension of the coadjoint orbit of λ. Referring toTheorem 3.1.6, [2], the following holds true

Proposition 7 For each subset e ⊆ 1, 2, · · · , n , the set

Ωe = λ ∈ n∗ : e (λ) = e

is algebraic and N -invariant [4]. Moreover, there exists a set of jump indicese such that Ωe = Ω is a Zariski open set in n∗ which is invariant under theaction of N.

Put Ω = Ωe. We recall that a polarization subalgebra subordinated to thelinear functional λ is a maximal subalgebra p(λ) of n∗ satisfying the condition

[p(λ), p(λ)] ⊆ kerλ.

Notice that if p(λ) is a polarization subalgebra associated with the linearfunctional λ then χλ(expX) = e2πiλ(X) defines a character on exp(p(λ). It isalso well-known that dim (n(λ)) = n−2m and dim (n/p(λ)) = m. Finally, thealgebra defined by the formula

p(λ) =n∑k=1

nk (λ|nk)

(see Page 30, [2] and [14]) is called a Vergne polarization.

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Proposition 8 If p is a constant polarization subalgebra for n then p is com-mutative.

Proof. Let Ω be a Zariski open and N -invariant subset of n∗ such thatp is a subalgebra subordinated to every linear functional λ ∈ Ω. Next, letW1, · · · ,Wr be a basis for [p, p] . Define [p, p]∗ to be the vector subspace ofn∗ spanned by the elements W ∗1 , · · · ,W ∗r . Here W ∗k is the unique elementin n∗ satisfying the property 〈W ∗k ,Wj〉 = δk,j . Observe that Ω∩ [p, p]∗ is openin [p, p]∗. Next, for arbitrary ` ∈ Ω ∩ [p, p]∗, by assumption [p, p] is containedin the kernel of `. Thus, [p, p] must be a trivial vector space and it followsthat p is commutative.

The following result is established in Theorem 3.1.9, [2]

Proposition 9 A cross-section for the coadjoint orbits in Ω is given by

Σ = λ ∈ Ω : λ (Zk) = 0 for all k ∈ e . (12)

2.2 Unitary Dual and Plancherel Theory

The setting in which we are studying sampling spaces requires the followingingredients:

1. An explicit description of the irreducible representations occurring inthe decomposition of the left regular representation of N.

2. The Plancherel measure and a formula for the group Fourier/Planchereltransform.

3. A description of left-invariant multiplicity-free spaces.

2.2.1 A Realization of the Irreducible Representations of N

The following discussion is taken from Chapter 6, [6]. Let G be a locallycompact group, and let K be a closed subgroup of G. Let us define q : G →G/K to be the canonical quotient map and let ϕ be a unitary representationof the group K acting in some Hilbert space which we call H. Next, let K1

be the set of continuous H-valued functions f defined over G satisfying thefollowing properties:

• The image of the support of f under the quotient map q is compact.

• f (gk) =[ϕ (k)−1 f

](g) for g ∈ G and k ∈ K.

Clearly, G acts on the set K1 by left translation. Now, to simplify thepresentation, let us suppose that G/K admits a G-invariant measure (thisassumption is not always true.) However, since we are mainly dealing with

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unimodular groups, the assumption holds. First, we endow K1 with thefollowing inner product:⟨

f, f ′⟩

=

∫G/K

⟨f (g) , f ′ (g)

⟩Hd (gK)

for f, f ′ ∈ K1. Second, let K be the Hilbert completion of the space K1 withrespect to this inner product. The translation operators extend to unitaryoperators on K inducing the unitary representation IndGK (ϕ) which acts onK as follows: [

IndGK (ϕ) (x) f]

(g) = f(x−1g

)for f ∈ K. We notice that if ϕ is a character, then the Hilbert space K canbe naturally identified with L2 (G/K) . The reader who is not familiar withthese notions is invited to refer to Chapter 6 of the book of Folland [6] for athorough presentation.

For each linear functional in the set Σ (see (12)), there is a correspondingunitary irreducible representation of N which is realized as acting in L2 (Rm)as follows. Define a character χλ on the normal subgroup exp (p(λ)) such that

χλ (expX) = e2πiλ(X)

for X ∈ p(λ). In order to realize an irreducible representation correspondingto the linear functional λ, we induce the character χλ as follows:

σλ = IndNPλ (χλ) , where Pλ = exp (p(λ)) . (13)

The induced representation σλ acts by left translations on the Hilbert space

Hλ =f : N −−−−→ C : f (xy) = χλ (y)−1 f (x) for y ∈ Pλ

and∫N/Pλ

|f (x)|2 d (xPλ) <∞,

(14)

which is endowed with the following inner product:⟨f, f ′

⟩=

∫N/Pλ

f (n) f ′ (n)d (nPλ)

. Picking a cross-section in N (which we may identify with RdimN ) for N/Pλ,since χλ is a character there is an obvious identification between Hλ and theHilbert space

L2 (N/Pλ) = L2 (Rm) .

2.2.2 The Plancherel Measure and the Plancherel Transform

For a linear functional λ ∈ Ω, put

e = e1 < e2 < · · · < e2m

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and define

B (λ) =[⟨λ,[Xei , Xej

]⟩]1≤i,j≤2m

=[λ[Xei , Xej

]]1≤i,j≤2m

. (15)

Then B (λ) is a skew-symmetric invertible matrix of rank 2m. Let dλ be theLebesgue measure on Σ which is parametrized by a Zariski subset of Rn−2m.Put

dµ (λ) = |detB (λ)|1/2 dλ.

It is proved in Section 4.3, [2] that up to multiplication by a constant, themeasure dµ (λ) is the Plancherel measure for N, which can be understoodas follows. The group Fourier transform F is an operator-valued boundedoperator which is weakly defined on L2(N) ∩ L1(N) as follows:

σλ (f) = F (f) (λ) =

∫Nf (n)σλ

(n−1

)dn where f ∈ L2(N) ∩ L1(N). (16)

σλ (f) is weakly defined as follows. Given u,v ∈ L2 (Rm) ,

〈σλ (f) u,v〉 =

∫Nf (n)

⟨σλ(n−1

)u,v

⟩dn.

Next, the Plancherel transform is a unitary operator

P : L2(N)→ L2(Σ, L2 (Rm)⊗ L2 (Rm) , dµ (λ)

)which is obtained by extending the Fourier transform to L2(N). This exten-sion induces the equality

‖f‖2L2(N) =

∫Σ

∥∥∥f (σλ)∥∥∥2

HSdµ (λ)

wheref (σλ) = Pf (λ)

and ‖·‖HS stands for the Hilbert-Schmidt norm. Let L be the left regularrepresentation of the nilpotent group N. It is easy to check that for almostevery λ ∈ Σ (with respect to the Plancherel measure)(

PL (n)P−1A)

(λ) = σλ (n) A (λ) .

In other words, the Plancherel transform intertwines the regular representa-tion with a direct integral of irreducible representations of N. The irreduciblerepresentations occurring in the decomposition are parametrized up to a nullset by the manifold Σ and each irreducible representation occurs with infinitemultiplicities in the decomposition.

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2.3 Bandlimited Multiplicity-Free Spaces

Given a Lebesgue measurable set A ⊆ Σ, it is easily checked that the Hilbertspace

P−1(L2(A, L2 (Rm)⊗ L2 (Rm)

), dµ (λ)

)is a left-invariant subspace of L2 (N) . Let us suppose that A is a boundedsubset of Σ of positive Plancherel measure. Letting |A| be the Lebesguemeasure of the set A

µ (A) =

∫A|detB (λ)|1/2 dλ ≤ |A| × sup

|detB (λ)|1/2 : λ ∈ A

. (17)

Next, since λ 7→ | detB(λ)|12 is a continuous function then µ(A) is finite.

Fixing a measurable field (u (λ))λ∈A of unit vectors in L2 (Rm), we define

HA =

f ∈ L2 (N) : Pf (λ) =

v (λ)⊗ u (λ) if λ ∈ A

0⊗ 0 if λ /∈ Aand

(v (λ)⊗ u (λ))λ∈A is a measurablefield of rank-one operators

.

(18)

Then HA is a left-invariant, band-limited and multiplicity-free subspace ofL2 (N) . Let h ∈ HA such that the Plancherel transform of h is a measurablefield of projections. More precisely, let us assume that

Ph (σλ) = h (σλ) =

u (λ)⊗ u (λ) if λ ∈ A

0⊗ 0 if λ /∈ A(19)

and [V Lh (f)

](exp (X)) = 〈f, L (exp (X))h〉 = f ∗ h∗ (exp (X))

where h∗ (x) = h (x−1) and ∗ is the convolution operator given by

f ∗ g (n) =

∫Nf (m) g

(m−1n

)dm.

Proposition 10 If A is a bounded subset of Σ of positive Plancherel measureand if h is as given in (19) then h is an admissible vector for the representation(L,HA) .

Proof. To check that h is an element of the Hilbert space HA, it is enoughto verify that

‖h‖2L2(N) =

∫A‖u (λ)⊗ u (λ)‖2HS dµ (λ) = µ (A)

is finite. Next, for any vector f ∈ HA, the square of the norm of the imageof f under the map V L

h is computed as follows:∥∥V Lh (f)

∥∥2

L2(N)=

∫A

∥∥∥f (σλ) (u (λ)⊗ u (λ))∥∥∥2

HSdµ (λ)

=

∫A

⟨f (σλ) u (λ) , f (σλ) u (λ)

⟩L2(Rm)

dµ (λ) .

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Letting f (σλ) = v (λ)⊗ u (λ) where v (λ) is in L2 (Rm) , it follows that∥∥V Lh (f)

∥∥2

L2(N)=

∫A〈v (λ) ,v (λ)〉L2(Rm) dµ (λ) = ‖f‖2L2(N) .

In other words, the map V Lh defines an isometry from HA into L2 (N) and

the representation (L,HA) which is a subrepresentation of the left regularrepresentation of N is admissible. Thus, the vector h is an admissible vector.

Remark 11 It is worth noting that h is convolution idempotent in the sensethat h = h ∗ h∗ = h∗. Next, V L

h (HA) is a left-invariant vector subspace ofL2 (N) consisting of continuous functions. Moreover, the projection onto theHilbert space V L

h (HA) is given by right convolution in the sense that

V Lh (HA) = L2 (N) ∗ h.

In order to simplify our presentation, we shall naturally identify the Hilbertspace P (HA) with the function space

L2 (A× Rm, dµ (λ) dt) .

This identification is given by the map

(v (λ)⊗ u (λ))λ∈A 7→ [v (λ)] (t) := v (λ, t)

for any measurable field of rank-one operators (v (λ)⊗ u (λ))λ∈A .

Lemma 12 Let π be a unitary representation of a group N acting in a Hilbertspace Hπ. Assume that π is admissible, and let h be an admissible vector forπ. Furthermore, suppose that π (Γ)h is a tight frame with frame bound Ch.Then the vector space Vh (Hπ) is a left-invariant closed subspace of L2 (N)consisting of continuous functions and Vh (Hπ) is a sampling space with sinc-type function

1

ChVh (h) .

Lemma 12 is proved in Proposition 2.54, [8]. This result establishes aconnection between admissibility and sampling theories. This connection willplay a central role in the proof of our main results. The following result is aslight extension of Proposition 2.61 [8], and the proof given here is essentiallyinspired by the one given in the Monograph [8].

Lemma 13 Let Γ be a discrete subgroup of N with positive co-volume. Let πbe a unitary representation of N acting in a Hilbert space Hπ. If the restrictionof π to the discrete subgroup Γ is unitarily equivalent to a subrepresentationof the left regular representation of Γ then there exists a subspace of L2 (N)which is a sampling space with respect to Γ. Moreover, if π is equivalent to theleft regular representation of Γ then there exists a subspace of L2 (N) whichis a sampling space with the interpolation property with respect to Γ.

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Proof. LetT : Hπ → H ⊂ l2 (Γ)

be a unitary map which is intertwining the restricted representation of πto Γ with a representation which is a subrepresentation of the left regularrepresentation of the lattice Γ. Since Γ is a discrete group, the left regularrepresentation of Γ is admissible. To see this, let κ be the sequence which isequal to one at the identity of Γ and zero everywhere else. By shifting thesequence κ by elements in Γ, we generate an orthonormal basis for the Hilbertspace l2 (Γ) . Now, let

P : l2 (Γ)→ H

be an orthogonal projection. Next, we claim that the vector η = T−1 (P (κ))is an admissible vector for π|Γ as well. We recall that

V πη (f) = 〈f, π (·) η〉 .

Let N = AΓ where A is a set of finite measure with respect to the Haarmeasure of N. Without loss of generality, let us assume that a Haar measurefor N is fixed so that |A| = 1. Then∥∥V π

η (f)∥∥2

L2(N)=

∫N|〈f, π (x) η〉|2 dx =

∫A

∑γ∈Γ

∣∣⟨π (x−1)f, π (γ) η

⟩∣∣2 dx.Next, since π (Γ) η is a Parseval frame in Hπ,∑

γ∈Γ

∣∣⟨π (x−1)f, π (γ) η

⟩∣∣2 =∥∥π (x−1

)f∥∥2

and it follows that∥∥V πη (f)

∥∥2

L2(N)=

∫A

∥∥π (x−1)f∥∥2

Hπdx = ‖f‖2Hπ

|A| = ‖f‖2Hπ.

Thus η is a continuous wavelet for the representation π, π (Γ) (η) is a Parsevalframe, and the Hilbert space V π

η (Hπ) is a sampling space of L2 (N) withrespect to the lattice Γ. Now, for the second part, if we assume that π isequivalent to the left regular representation, then the operator P describedabove is just the identity map. Next,

π (Γ) η = π (Γ)(T−1 (κ)

)is an orthonormal basis of Hπ and Vη (η) is a sinc-type function. It followsfrom Theorem 2.56 [8] that Vη (Hπ) is a sampling space of L2 (N) which hasthe interpolation property with respect to the lattice Γ.

Remark 14 Let HA be the Hilbert space of band-limited functions as de-scribed in (18). We recall that

Γ = exp (ZX1) · · · exp (ZXn)

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is a discrete uniform subgroup of N. Since HA is left-invariant, the regularrepresentation of N admits a subrepresentation obtained by restricting the ac-tion of the left regular representation to the Hilbert space HA. Let us denotesuch a representation by LHA

. Furthermore, let LHA,Γ be the restriction ofLHA

to Γ. If the representation LHA,Γ is unitarily equivalent to a subrepresen-tation of the left regular representation of the discrete group Γ then accordingto arguments used in the proof of Lemma 13, there exists a vector η such thatV Lη (HA) is a sampling space of L2 (N) with respect to the discrete uniform

group Γ. In the present work, we are aiming to find conditions on the spectralset A which guarantees that LHA,Γ is unitarily equivalent to a subrepresenta-tion of the left regular representation of the discrete group Γ. We shall alsoprove that under the assumptions given in Condition 3, it is possible to findA such that V L

η (HA) is a sampling subspace of L2 (N) with respect to thediscrete uniform group Γ.

3 Intermediate Results

Let us now fix assumptions and specialize the theory of harmonic analysis ofnilpotent Lie groups to the class of groups being considered here (see Condi-tion 3)

Let N be a simply connected, connected non-commutative nilpotent Lie groupwith Lie algebra n with rational structure constants such that

N = P oM = exp (p) exp (m)

where p and m are commutative Lie algebras, and p is an ideal of n. We fix astrong Malcev basis

Z1, · · · , Zp, A1, · · · , Am (20)

for n such that Z1, · · · , Zp is a basis for p and A1, · · · , Am is a basis form. Moreover it is assumed that

Γ = exp

(p∑

k=1

ZZk

)exp

(m∑k=1

ZAk

)

is a discrete uniform subgroup of N. Indeed, in order to ensure that Γ is adiscrete uniform group, it is enough to pick A1, · · · , Am such that the matrixrepresentation of eadAk |p with respect to the basis Z1, · · · , Zp has entries inZ. Let M (λ) be the skew-symmetric matrix defined in (10). Regarding M (λ)as the matrix representation of a linear operator acting on n, we recall thatthe null-space of M (λ) corresponds to n (λ) . Now, let Mk (λ) be the matrix

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obtained by retaining the first k columns of M (λ) (see illustration below)

M (λ) =

0...0

λ [A1, Z1]...

λ [Am, Z1]︸ ︷︷ ︸M1(λ)

· · · 0. . .

...· · · 0· · · λ [A1, Zp]. . .

...· · · λ [Am, Zp]

︸ ︷︷ ︸Mp(λ)

λ [Z1, A1]...

λ [Zp, A1]0...0

︸ ︷︷ ︸Mp+1(λ)

· · · λ [Z1, Am]. . .

...· · · λ [Zp, Am]· · · 0. . .

...· · · 0

︸ ︷︷ ︸M(λ)=Mn(λ)

.

For convenience of notation, we use the convention rank (M0 (λ)) = 0. Put

X1 = Z1, · · · , Xp = Zp, Xp+1 = A1, · · · , Xn−1 = Am−1, Xn = Am.

Lemma 15 Given λ ∈ n∗, the following holds true.

1 ≤ k ≤ n : rank (Mk (λ)) > rank (Mk−1 (λ))= 1 ≤ k ≤ n : nk * nk−1 + n (λ)= e (λ) .

Proof. First, assume that the rank of Mi(λ) is greater than the rank ofMi−1(λ). Then it is clear that Xi cannot be in the null-space of the matrixM(λ). Thus,

ni = ni−1 + RXi * ni−1 + n(λ).

Next, if ni * ni−1 + n(λ) and ni = ni−1 + RXi since the basis element Xi

cannot be in n + n(λ). Thus the rank of Mi(λ) is greater than the rank ofMi−1(λ) and the stated result is established.

It is proved in Theorem 3.1.9, [2] that there exist a Zariski open subset Ωof n∗ and a fixed set e ⊂ 1, 2, · · · , n such that the map

λ 7→ 1 ≤ k ≤ n : rank (Mk (λ)) > rank (Mk−1 (λ)) = e

is constant, Ω is invariant under the coadjoint action of N and

Σ = λ ∈ Ω : λ (Xk) = 0 for all k ∈ e

is an algebraic set which is a cross-section for the coadjoint orbits of N in Ω,as well as a parameterizing set for the unitary dual of N.

Lemma 16 If p is a constant polarization for n then the set

p+ 1, p+ 2, · · · , p+m = n

is contained in e and card (e) = 2m.

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Proof. Let λ ∈ Σ. Let us suppose that there exists one index k such that

k ∈ p+ 1, p+ 2, · · · , p+m = n

and k is not an element of the set e (λ) = e. Then the dimension of thecoadjoint orbit of the linear functional λ is equal to 2(m − 1). However, thedimension of any polarization algebra subordinated to λ must be equal to

n− (m− 1) = n−m+ 1.

This contradicts the fact that the ideal p is a polarization algebra subordinatedto λ and dim p = n−m. The second part of the lemma is true because M(λ)is a skew symmetric rank of positive rank.

Remark 17 From now on, we shall assume that p is a constant polarizationideal for n. Since P = exp p is normal in N, it is clear that the dual of the Liealgebra p is invariant under the coadjoint action of the commutative group M .

We recall thatβ : Σ× Rm → Rp

and

β (λ, t) = projp∗

exp

m∑j=1

tjAj

· λ = projp∗

((e−ad

∑mj=1 tjAj

)∗λ).

(21)

Remark 18 In vector-form, β (λ, t1, · · · , tm) is easily computed as follows.Let P (A(t)) be the transpose of the matrix representation of

e−ad(∑mk=1 tkAk)|p

with respect to the ordered basis Zk : 1 ≤ k ≤ p . We write

P (A(t)) =[e−ad(

∑mk=1 tkAk)|p

]Tand β (λ, t1, · · · , tm) is identified with

P (A(t))

f1...fp

where λ =

p∑k=1

fkZ∗k

Z∗k : 1 ≤ k ≤ p is a dual basis to Zk : 1 ≤ k ≤ p . We shall generally makeno distinction between linear functionals and their representations as eitherrow or column vectors. Secondly for any linear functional

λ =

p∑k=1

fkZ∗k ∈ Σ

since P (A(t)) is a unipotent matrix, the components of β (λ, t1, · · · , tm) arepolynomials in the variables fk where k /∈ e and t1, · · · tm.

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The following result is proved in [2], Proposition 1.3.6

Lemma 19 Let λ ∈ Σ. p is a polarizing ideal subordinated to the linear func-tional λ if and only if for any given X ∈ n, p is also a polarizing ideal subor-dinated to the linear functional exp (X) · λ.

Lemma 20 If for each λ ∈ Σ, p = R-span Z1, · · · , Zp is a commutativepolarizing ideal which is subordinated to the linear functional λ then β definesa diffeomorphism between Σ× Rm and its range. Moreover, the inverse of βis a rational map.

Proof. In order to prove this result, it is enough to show that β is a bijectivesmooth map with constant full rank (see Proposition 6.5, [13]). First, we needa formula for the coadjoint action. Let us define βe : Σ×R2m → n∗ such that

βe (λ, te1 , · · · , te2m) = exp (te1Xe1) · · · exp (te2mXe2m) · λ

ande1 < e2 < · · · < e2m = e.

For a fixed linear functional λ in the cross-section Σ, the map

(te1 , · · · , te2m) 7→ exp (te1Xe1) · · · exp (te2mXe2m) · λ

defines a diffeomorphism between R2m and the N -orbit of λ which is a closedsubmanifold of the dual of the Lie algebra n. Next, since all orbits in Ω are2m-dimensional manifolds, the map βe is a bijection with constant full-rank.Thus, βe defines a diffeomorphism between Σ×R2m and Ω . Next, there existindices i1, · · · , im ≤ p such that for

g = exp (te1Xe1) · · · exp (te2mXe2m)

we have

g · λ = exp

(m∑k=1

tikZik

)exp

m∑j=1

sjAj

· λ.In order to compute the coadjoint action of N on the linear functional λ, itis quite convenient to identify n∗ with p∗ ×m∗ via the map

ι : n∗ = p∗ + m∗ → p∗ ×m∗

given by

ι (f1 + f2) =

[f1

f2

]where f1 ∈ p∗ and f2 ∈ m∗. (22)

Thus, for any linear functional λ ∈ Σ,

ι (λ) =

[f0

]22

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for some f ∈ p. Put

A (s) =

m∑j=1

(sjAj) ∈ m, Z (t) =

m∑k=1

(tikZik) ∈ p

wheres = (s1, · · · , sm) , t = (ti1 , · · · , tim) .

Clearly, with the fixed choice of the Malcev basis described in (20), it is easyto check that (see Remark 18)

ι (exp (A (s)) · λ) =

[P (A(s)) f

0

]and ι (exp (Z (t)) · λ) =

[f

σ (t, f)

](23)

where σ (t, f) is an m× 1 vector which depends on t and f. Next, (23) gives

exp (Z (t)) exp (A (s)) · λ =

[P (A(s)) f

σ (t,P (A(s)) f)

].

Now, we consider the map

(f, s, t) 7→[

P (A(s)) fσ (t,P (A(s)) f)

]. (24)

The Jacobian of (24) takes the form

T (f, s, t) =

[A (f, s)

C (f, s, t)0

D (f, s, t)

]where A (f, s) is a matrix of order p obtained by taking the partial derivativesof the components of P (A(s)) f with respect to the coordinates correspondingto (f, s) ∈ Rp. C (f, s, t) is an m × p matrix obtained by computing thepartial derivatives of the components of σ (t,P (A(s)) f) with respect to (f, s) .Finally D (f, s, t) is a matrix of order m which is given by computing thepartial derivatives of the components of σ (t,P (A(s)) f) with respect to thecoordinates of s ∈ Rm. Observe that

det T (f, s, t) = det (A (f, s)) det (D (f, s, t)) 6= 0.

Since the Jacobian of β is the submatrix A (f, s) , clearly

det (A (f, s)) 6= 0.

It follows that the Jacobian of the map β has constant full rank as well. Since

ι (β (λ, s)) = P (A(s)) f

and (24) is a bijection then it is clear that β is a smooth bijection with constantfull-rank. Thus, β is a diffeomorphism. For the second part we write

β (λ, t) = (β1 (λ, t) , · · · , βp (λ, t))

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where the range of each βk is a subset of R. Since N is a nilpotent Lie group,for each index k the function βk is a polynomial function in the variables λ andt. Next we consider the system of equations sk = βk (λ, t) where the variablesare given by the coordinates of (λ, t). Since each βk (λ, t) is a polynomial,the solutions for the equations above are given by some rational functionsR1 (s) , · · · , Rp (s) in the variables s and

β−1 (s) = (R1 (s) , · · · , Rp (s)) .

Remark 21 For λ ∈ Σ there exist real numbers fk such that

λ =∑k/∈e

fkZ∗k = fk1Z

∗k1 + · · ·+ fkn−2mZ

∗kn−2m

.

Thus, defining

j : fk1Z∗k1 + · · ·+ fkn−2mZ

∗kn−2m

7→(fk1 , · · · , fkn−2m

)it is clear that the unitary dual of the Lie group N is parametrized by j (Σ)which is a Zariski open subset of Rdim Σ. Although this is an abuse of notation,we shall make no distinction between j (Σ) and Σ.

Lemma 22 Given a linear functional λ ∈ Σ and a vector h ∈ L2 (Rm) , theirreducible representation σλ of N acts on L2 (Rm) as follows

[σλ (exp (Z (t)) exp (A (a)))h] (x) (25)

= e2πi〈λ,e−ad(x1A1+···+xmAm)Z(t)〉h (x1 − a1, · · · , xm − am) . (26)

Proof. For each λ ∈ Σ, the corresponding unitary irreducible representationσλ of N (see (13)) is obtained by inducing the character χλ of the normalsubgroup P which is defined as follows: χ

λ (exp (t1Z1 + · · ·+ tpZp)) = e2πi〈λ,t1Z1+···+tpZp〉

= e2πiλ(t1Z1+···+tpZp).

Puta = (a1, · · · , am) , x = (x1, · · · , xm) , t = (t1, · · · , tp) .

DefineA (a) = a1A1 + · · ·+ amAm,

andZ (t) = t1Z1 + · · ·+ tpZp.

We observe that

exp (A (a))−1 expA (x) = exp (A (x− a))

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and(expZ (t))−1 expA (x) = expA (x) exp

(eadA(−x)Z (−t)

).

Sinceexp (RA1 + · · ·+ RAm)

is a cross-section for N/P in N, we shall realize the unitary representationσλ as acting on the Hilbert space L2(N/P ) which we identify with L2 (Rm) .Following the discussion given in Subsection 2.2.1 , if h ∈ L2 (Rm) , then forevery linear functional λ ∈ Σ

[σλ (exp (Z (t)) exp (A (a)))h] (x) = e2πi〈λ,e−adA(x)Z(t)〉h (x1 − a1, · · · , xm − am) .(27)

We remark that although the Plancherel measure for an arbitrary nilpotentLie group has already been computed in general form in the book of Corwinand Greenleaf [2], in order to prove the main results stated in the introduction,we will need to establish a connection between the Plancherel measure of Nand the determinant of the Jacobian of the map β. To make this connectionas transparent as possible, we shall need the following lemma.

Lemma 23 Let Jβ (λ, s1, · · · , sm) be the Jacobian of the smooth map β de-fined in (20). The Plancherel measure of N is up to multiplication by aconstant equal to

dµ (λ) = |det Jβ (λ, 0)| dλ (28)

where dλ is the Lebesgue measure on Rdim Σ.

Proof. Let N = P oM as defined above. Since the set of smooth functionsof compact support is dense in L2 (N) , it suffices to show that for any smoothfunction F of compact support on the group N ,∫

Σ

∥∥∥F (σλ)∥∥∥2

HS|det Jβ (λ, 0)| dλ = ‖F‖2L2(N) .

In order to simplify our presentation, we shall identify the set N = PM withRp × Rm via the map

exp (t1Z1 + · · ·+ tpZp) exp (a1A1 + · · ·+ amAm)

7→ (t1, · · · , tp, a1, · · · , am) = (t, a) .

As such the Haar measure on the group N is taken to be the Lebesgue measureon Rp×Rm. For any smooth function F of compact support on the group N,the operator F (σλ) (see (16)) is defined as acting on L2 (Rm) as follows. For

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φ ∈ L2 (Rm) we have[F (σλ)φ

](x) =

∫Rp

∫Rm

F (t, a) [(σλ (t, a))φ] (x) dadt (29)

=

∫Rp

∫Rm

F (t, a) e2πi〈λ,e−adA(x)Z(t)〉φ (x− a) dadt (30)

=

∫Rp

∫Rm

F (t, a) e2πi〈exp(A(x))·λ,Z(t)〉φ (x− a) dadt. (31)

Next, we recall that

P (A(x)) f =[e−adA(x)|p

]Tf

and

ι (exp (A (x)) · λ) =

[P (A(x)) f

0

]where ι (λ) =

[f0

].

Next,[F (σλ)φ

](x) =

∫Rp

∫Rm

F (t, a) e2πi〈exp(A(x))·λ,Z(t)〉φ (x− a) dadt

=

∫Rp

∫Rm

F (t, a) e2πi〈β(λ,x),Z(t)〉φ (x− a) dadt

=

∫Rp

∫Rm

F (t, x− a) e2πi〈β(λ,x),Z(t)〉φ (a) da dt

=

∫Rp

(∫Rm

F (t, x− a) e2πi〈β(λ,x),Z(t)〉 dt

)φ (a) da.

Thus, F (σλ) is an integral operator on L2 (Rm) with kernel Kλ,F given by

Kλ,F (x, a) =

∫Rp

F (t, x− a) e2πi〈β(λ,x),Z(t)〉dt. (32)

Now, let F1 be the partial Euclidean Fourier transform in the direction of t.It is clear that

Kλ,F (x, a) = [F1F] (β (λ, x) , x− a) .

Additionally, the square of the Hilbert-Schmidt norm of the operator F (σλ)is given by ∥∥∥F (σλ)

∥∥∥2

HS=

∫Rm

∫Rm|Kλ,F (x, a)|2 dx da

=

∫Rm

∫Rm|F1F (β (λ, x) , x− a)|2 dx da

=

∫Rm

∫Rm|F1F (β (λ, x) , a)|2 dx da.

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Observing thatι (β (λ, x+ t)) = P (A(x))P (A(t)) f,

the components of β (λ, x+ t) may be computed by multiplying a unipotentmatrix by the matrix representation of β (λ, t), the determinant of the Jaco-bian of the map β at (λ, x) is then given by

det Jβ ((λ, x+ t)) = det([e(−adx1A1−···−adxmAm)|p

])det Jβ (λ, t)

= 1× det Jβ (λ, t) .

It follows thatdet Jβ (λ, x) = detJβ (λ, 0) = P (λ)

where P (λ) is a polynomial in the coordinates of λ. Next,∫Σ

∥∥∥F (σλ)∥∥∥2

HS|det Jβ (λ, 0)| dλ

=

∫Σ

(∫Rm

∫Rm|(F1F) (β (λ, x) , a)|2 da dx |det Jβ (λ, 0)|

)dλ

=

∫Σ

∫Rm

∫Rm|(F1F) (β (λ, x) , a)|2 da dx |det Jβ (λ, 0)| dλ

=

∫Σ×Rm

∫Rm|(F1F) (β (λ, x) , a)|2 da |det Jβ (λ, 0)| d (λ, x)

=

∫Σ×Rm

∫Rm|(F1F) (β (λ, x) , a)|2 da d (β (λ, x))

=

∫Ω

∫Rm|(F1F) (z, a)|2 da dz.

Finally, appealing to the Plancherel Theorem for Rm∫Σ

∥∥∥F (σλ)∥∥∥2

HS|det Jβ (λ, 0)| dλ =

∫Rp

∫Rm|F (z, a)|2 dadz = ‖F‖2L2(N) .

We are now interested in finding conditions on the spectral set A whichwill allow us to isometrically embed the Hilbert space L2 (A× Rm, dµ (λ) dt)into the sequence space l2 (Γ) . Moreover, this map should also intertwine theoperators PL (γ)P−1 with the right shift operators acting on l2 (Γ) ( γ ∈ Γ).Let A be a Lebesgue measurable subset of Rn−2m. Define (in a formal way)the map

JA : L2 (A× Rm, dµ (λ) dt)→ Set of sequences over Γ

such that for l = (l1, · · · , lm) ∈ Zm we have

[JAF ]

exp

p∑j=1

kjZj

exp

m∑j=1

ljAj

(33)

=

∫A

∫[0,1)m

F (λ, t− l) e2πi〈β(λ,t),∑pj=1 kjZj〉 |det Jβ (λ, 0)| dt dλ (34)

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where

exp

p∑j=1

kjZj

exp

m∑j=1

ljAj

∈ Γ.

Next, we define Z (k) =∑p

j=1 kjZj and A (l) =∑m

j=1 ljAj . Let 1X denotesthe indicator function for a given set X.

Lemma 24 For any γ ∈ Γ, and F ∈ L2 (A× Rm, dµ (λ) dt) ,

|[JAF ] (γ)| ≤ ‖F‖ ×∫A

∫[0,1)m

|det Jβ (λ, 0)| dt dλ.

Proof. For γ ∈ Γ such that

γ = exp (Z (k)) exp (A (l))

we obtain:

[JAF ] (γ)

=

∫A

∫[0,1)m

F (λ, t− l) e2πi〈β(λ,t),∑pj=1 kjZj〉 |det Jβ (λ, 0)| dt dλ

=

∫A

∫Rm

[F (λ, t− l) e2πi〈β(λ,t),

∑pj=1 kjZj〉

]×(1A×[0,1)m (λ, t)

)dt︸ ︷︷ ︸

=〈σλ(γ)F (λ,·),1A×[0,1)m (λ,·)〉L2(Rm)

|det Jβ (λ, 0)| dλ

=

∫A

⟨σλ (γ)F (λ, ·) , 1A×[0,1)m (λ, ·)

⟩L2(Rm)

|det Jβ (λ, 0)| dλ

=⟨L (γ)P−1

((F (λ, ·)⊗ u (λ))λ∈A

),P−1

((1A×[0,1)m (λ, ·)⊗ u (λ)

)λ∈A

)⟩L2(N)

.

Applying Cauchy–Schwarz inequality

|[JAF ] (γ)| ≤ ‖F‖ ·∥∥∥P−1

((1A×[0,1)m (λ, ·)⊗ u (λ)

)λ∈A

)∥∥∥= ‖F‖ ×

∫A

∫[0,1)m

|det Jβ (λ, 0)| dt dλ.

Recall that Σ is identified with a Zariski open subset of Rn−2m.

Proposition 25 Let A be a dµ-measurable bounded subset of Σ. If

H ∈ L2 (A× Rm, dµ (λ) dt)

is a smooth function of compact support, then JAH ∈ l2 (Γ) .

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Proof. First, observe that

[JAH] (exp (Z (k)) exp (A (l)))

=

∫A

∫[0,1)m

H (λ, t− l) e2πi〈β(λ,t),Z(k)〉dt∣∣det Jβ(λ,0)

∣∣ dλand for fixed Z (k) , the sequence ([JAH] (expZ (k) expA (l)))l∈Zm has boundedsupport. Making the change of variable s = t− l, we obtain that

[JAH] (expZ (k) expA (l))

=

∫A

∫[0,1)m−l

H (λ, s) e2πi〈β(λ,s+l),Z(k)〉ds∣∣det Jβ(λ,0)

∣∣ dλ.Next, since

β (λ, s+ l) = expA (s) expA (l) · λ

it follows that

[JAH] (expZ (k) expA (l))

=

∫A×([0,1)m−l)

H (λ, s) e2πi〈β(λ,s+l),Z(k)〉 ∣∣det Jβ(λ,0)

∣∣ d (λ, s)

=

∫A×([0,1)m−l)

H (λ, s) e2πi〈exp(A(s)) exp(A(l))·λ,Z(k)〉 ∣∣det Jβ(λ,0)

∣∣ d (λ, s)

=

∫A×([0,1)m−l)

H (λ, s) e2πi〈exp(A(s))·λ,e−adA(l)Z(k)〉 ∣∣det Jβ(λ,0)

∣∣ d (λ, s)

=

∫A×([0,1)m−l)

H (λ, s) e2πi〈β(λ,s),e−adA(l)Z(k)〉 ∣∣det Jβ(λ,0)

∣∣ d (λ, s) .

Next, the change of variable r = β (λ, s) yields

[JAH] (exp (Z (k)) exp (A (l)))

=

∫β(A×([0,1)m−l))

(H β−1

)(r) e2πi〈r,e−adA(l)Z(k)〉 dr

=

∫β(A×([0,1)m−l)) ∩ support(Hβ−1)

(H β−1

)(r) e2πi〈r,e−adA(l)Z(k)〉 dr

where dr is the Lebesgue measure on Rp. Next for each fixed l ∈ Zm, we write

E = β (A× ([0, 1)m − l)) ∩(support

(H β−1

))as a finite disjoint union of subsets of Rp; each contained in a fundamentaldomain of Zp as follows:

E =·⋃

j∈J(H,l)

(KA,l + j) where J(H, l) is a finite subset of Zp.

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In the disjoint union given above, each KA,l is a measurable subset of Rpcontained in some fundamental domain of Zp. Letting 1KA,l+j be the indicatorfunction of the set KA,l + j, we obtain

[JAH] (exp (Z (k)) exp (A (l))) =∑

j∈J(H,l)

∫KA,l+j

(H β−1

)(r) e2πi〈r,e−adA(l)Z(k)〉 dr.

Since β−1 is a diffeomorphism and since H is smooth and has compact supportthen H β−1 is smooth and has compact support as well. Put

ΓZp =

p∑k=1

ZZk ' Zp.

Since det[e−adA(l)

∣∣ p] = 1 and the entries of the matrix[e−adA(l)

∣∣ p] are allintegers, it is clear that

e−adA(l) (ΓZp) = ΓZp .

Let Φ ⊂ Rp be a fundamental domain of Zp. We define the Fourier transformon L2 (Φ) as follows:

(FΓZpF) (Z (k)) =

∫Φ

F (r) e2πi〈r,Z(k)〉dr where F ∈ L2 (Φ) .

For a fixed l ∈ Zm, we have

[JAH] (expZ (k) expA (l)) =∑

j∈J(H,l)

[FΓZp

((H β−1

)× 1KA,l+j

)] (e−adA(l)Z (k)

).

In order to avoid cluster of notation, we set e−adA(l)Z (k) = al (k) and γk,l =exp (Z (k)) exp (A (l)) . Then

‖[JAH]‖2l2(Γ) =∑k∈Zp

∑l∈Zm

|JAH (γk,l)|2

=∑k∈Zp

∑l∈F

∣∣∣∣∣∣∑

j∈J(H,l)

[FΓZp

((H β−1

)× 1KA,l+j

)](al (k))

∣∣∣∣∣∣2

where F is a finite subset of Zm (because H has compact support) and

‖[JAH]‖2l2(Γ) =∑l∈F

∑k∈Zp

∣∣∣∣∣∣∑

j∈J(H,l)

[FΓZp

((H β−1

)× 1KA,l+j

)](al (k))

∣∣∣∣∣∣2

.

LettingΘH,l,j =

(H β−1

)× 1KA,l+j ∈ L

2 (KA,l + j)

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and applying the Triangle Inequality, since J (H, l) is a finite set, it followsthat

‖[JAH]‖2l2(Γ) ≤∑l∈F

∑k∈Zp

∑j∈J(H,l)

|[FΓZpΘH,l,j ] (al (k))|

2

.

and

‖[JAH]‖2l2(Γ)

≤∑l∈F

∑k∈Zp

∑j∈J(H,l)

∑j′∈J(H,l)

|[FΓZpΘH,l,j ] (al (k))|∣∣[FΓZpΘH,l,j′

](al (k))

∣∣=∑l∈F

∑j∈J(H,l)

∑j′∈J(H,l)

(∑k∈Zp|[FΓZp (ΘH,l,j)] (al (k))| ×

∣∣[FΓZp

(ΘH,l,j′

)](al (k))

∣∣)

=∑l∈F

∑j∈J(H,l)

∑j′∈J(H,l)

⟨(|[FΓZp (ΘH,l,j)] (al (k))|)k∈Zp ,

(∣∣[FΓZp

(ΘH,l,j′

)](al (k))

∣∣)k∈Zp

⟩l2(Zp)

.

By Cauchy–Schwarz inequality on l2 (Zp) ,⟨(|[FΓZp (ΘH,l,j)] (al (k))|)k∈Zp ,

(∣∣[FΓZp

(ΘH,l,j′

)](al (k))

∣∣)k∈Zp

⟩l2(Zp)

is less than or equal to∥∥(|[FΓZp (ΘH,l,j)] (al (k))|)k∈Zp∥∥l2(Zp)

×∥∥∥(∣∣[FΓZp

(ΘH,l,j′

)](al (k))

∣∣)k∈Zp

∥∥∥l2(Zp)

and it follows that

‖[JAH]‖2l2(Γ)

≤∑l∈F

∑j∈J(H,l)

∑j′∈J(H,l)

∥∥(|[FΓZp (ΘH,l,j)] (al (k))|)k∈Zp∥∥l2(Zp)

×∥∥∥(∣∣[FΓZp

(ΘH,l,j′

)](al (k))

∣∣)k∈Zp

∥∥∥l2(Zp)

=∑l∈F

∑j∈J(H,l)

∑j′∈J(H,l)

‖ΘH,l,j‖L2(KA,l+j) ×∥∥ΘH,l,j′

∥∥L2(KA,l+j′)

.

The last equality above is due to Parseval equality. Next, appealing to thefact that each

ΘH,l,j =(H β−1

)× 1KA,l+j

is square-integrable over KA,l + j, we obtain

‖[JAH]‖2l2(Γ) ≤∑l∈F

∑j∈J(H,l)

∑j′∈J(H,l)

∥∥(H β−1)× 1KA,l+j

∥∥L2(KA,l+j)

×∥∥(H β−1

)× 1KA,l+j

∥∥L2(KA,l+j′)

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Thus,‖[JAH]‖2l2(Γ)

is a finite sum of finite quantities. Therefore, ‖[JAH]‖l2(Γ) is finite and thiscompletes the proof.

Let τ be the unitary representation of Γ which acts on the Hilbert space

SA = L2 (A× Rm, dµ (λ) dt)

as follows:

[τ (γ)F ] (λ, t) = σλ (γ)F (λ, t) = e2πi〈β(λ,t),∑pj=1 kjZj〉F (λ, t− l)

where γ = expZ (k) expA (l) ∈ Γ. We shall prove the following three impor-tant facts.

• JA intertwines τ with the right regular representation of the discreteuniform group Γ (Lemma 26)

• If |β (A× [0, 1)m)| > 0 and if β (A× [0, 1)m) is contained in a funda-mental domain of Zp then JA defines an isometry on a dense subset ofSA into l2 (Γ) which extends uniquely to an isometry of SA into l2 (Γ)(Lemma 27)

• If |β (A× [0, 1)m)| > 0 and if β (A× [0, 1)m) is up to a null set equalto a fundamental domain of Zp then JA defines a unitary map (Lemma28)

Lemma 26 The map JA intertwines τ with R (the right regular representa-tion of Γ).

Proof. Let F ∈ SA and γ ∈ Γ. Then clearly,

[JAF ] (γ) =⟨τ (γ)F, 1A×[0,1)m

⟩.

Next, given α ∈ Γ and letting R denotes the right regular representation ofΓ, we obtain

[JAτ (α)F ] (γ) =⟨τ (γα)F, 1A×[0,1)m

⟩= [JAF ] (γα)

= R (α) [JAF ] (γ) .

Lemma 27 If β (A× [0, 1)m) has positive Lebesgue measure in Rp and iscontained in a fundamental domain of Zp then JA defines an isometry on adense subset of SA into l2 (Γ) which extends uniquely to an isometry of SAinto l2 (Γ) .

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Proof. Let F ∈ SA. Furthermore, let us assume that F is smooth withcompact support in A× Rm. Computing the norm of F, we obtain

‖F‖2L2(A×Rm,dµ(λ)) =

∫A

∫Rm|F (λ, t)|2 dt dµ (λ)

=

∫A

∑l∈Zm

∫[0,1)m

|F (λ, t− l)|2 dt dµ (λ) . (35)

LettingGl (λ, t) = F (λ, t− l) ,

making the change of variable s = β (λ, t) and using the fact that β is adiffeomorphism (see Lemma 20) we obtain∫

A

∑l∈Zm

∫[0,1)m

|F (λ, t− l)|2 dt dµ (λ)

=∑l∈Zm

∫[0,1)m

∫A|Gl (λ, t)|2 dµ (λ) dt

=∑l∈Zm

∫A×[0,1)m

|Gl (λ, t)|2 |det Jβ (λ, 0)| d (λ, t)

=∑l∈Zm

∫β(A×[0,1)m)

∣∣Glβ−1 (s)∣∣2 |det Jβ (λ, 0)| d

(β−1 (s)

)=∑l∈Zm

∫β(A×[0,1)m)

∣∣Glβ−1 (s)∣∣2 ds.

Since β (A× [0, 1)m) is contained in a fundamental domain of Zp then∫A

∑l∈Zm

∫[0,1)m

|F (λ, t− l)|2 dt dµ (λ)

=∑l∈Zm

∥∥s 7→ Glβ−1 (s)

∥∥2

=∑l∈Zm

∥∥FΓZp

(Glβ

−1)∥∥2

=∑l∈Zm

∑k∈Zp

∣∣FΓZp

(Glβ

−1)

(k)∣∣2

=∑l∈Zm

∑k∈Zp

∣∣∣∣∣∫β(A×[0,1)m)

(Glβ

−1)

(s) exp (2πi 〈s, k〉) ds

∣∣∣∣∣2

.

Thus

‖F‖2L2(A×Rm,dµ(λ)) =∑l∈Zm

∑k∈Zp

∣∣∣∣∣∫β(A×[0,1)m)

Glβ−1 (s) exp (2πi 〈s, k〉) ds

∣∣∣∣∣2

.

(36)

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Next, since β (A× [0, 1)m) is contained in a fundamental domain of Zp, thetrigonometric system

χβ(A×[0,1)m) (s)× exp (2πi 〈s, k〉) : k ∈ Zp

forms a Parseval frame in L2 (β (A× [0, 1)m) , ds) . Clearly this is true becausethe orthogonal projection of an orthonormal basis is always a Parseval frame.Letting

Glβ−1 = FL2(β(A×[0,1)m))

(s 7→ Gl

(β−1 (s)

))be the Fourier transform of the function

s 7→ Gl(β−1 (s)

)∈ L2 (β (A× [0, 1)m) , ds)

it follows that∑l∈Zm

∑k∈Zp

∣∣∣∣∣∫β(A×[0,1)m)

Gl(β−1 (s)

)exp (2πi 〈s, k〉) ds

∣∣∣∣∣2

=∑l∈Zm

∑k∈Zp

∣∣∣Glβ−1 (k)∣∣∣2 .

Next, ∑l∈Zm

∑k∈Zp

∣∣∣Glβ−1 (k)∣∣∣2 =

∑l∈Zm

∥∥∥Glβ−1∥∥∥2

l2(Zp)

=∑l∈Zm

∫β(A×[0,1)m)

∣∣Gl (β−1 (s))∣∣2 ds.

Now substituting (λ, t) for β−1 (s),

∑l∈Zm

∑k∈Zp

∣∣∣∣∣∫β(A×[0,1)m)

Gl(β−1 (s)

)exp (2πi 〈s, k〉) ds

∣∣∣∣∣2

=∑l∈Zm

∫A

∫[0,1)m

|Gl (λ, t)|2 |det Jβ (λ, 0)| dt dλ (37)

=∑l∈Zm

∫A

∫[0,1)m

|F (λ, t− l)|2 dt dµ (λ) . (38)

Equation (35) together with (38) gives

∑l∈Zm

∑k∈Zp

∣∣∣∣∣∫β(A×[0,1)m)

Gl(β−1 (s)

)exp (2πi 〈s, k〉) ds

∣∣∣∣∣2

= ‖F‖2L2(A×Rm,dµ(λ)) .

Finally, we obtain

‖F‖L2(A×Rm,dµ(λ)dt) = ‖JAF‖l2(Γ) .

Now, since the set of smooth functions of compact support is dense in SA andsince JA defines an isometry on a dense set, then J extends uniquely to anisometry on SA.

The proof given for Lemma 27 can be easily modified to establish thefollowing result

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Lemma 28 If β (A× [0, 1)m) has positive Lebesgue measure in Rp and isequal to a fundamental domain of Zp then JA defines an isometry on a densesubset of SA into l2 (Γ) which extends uniquely to a unitary map of SA intol2 (Γ) .

Remark 29 Suppose that β (A× [0, 1)m) has positive Lebesgue measure inRp and is contained in a fundamental domain of Zp. We have shown thatJA is an isometry. Moreover, the image of SA under the map JA is stableunder the action of the right regular representation of Γ. Now, let Φ be theorthogonal projection of l2 (Γ) onto the Hilbert space JA (SA) and let κ bethe indicator sequence of the singleton containing the identity element in Γ.Identifying SA with P (HA) , it is clear that

P−1 (J∗A (Φκ)) ∈ HA ⊂ L2 (N)

and L (Γ)(P−1 (J∗A (Φκ))

)is a Parseval frame for the band-limited Hilbert

space HA. We remark that the vector κ could be replaced by any other vectorwhich generates an orthonormal basis or a Parseval frame under the actionof the right regular representation of Γ. Additionally, we observe that

[JAF ] (γ) =⟨L (γ)P−1

((F (λ, ·)⊗ u (λ))λ∈A

),P−1

((1A×[0,1)m (λ, ·)⊗ u (λ)

)λ∈A

)⟩L2(N)

.

Letting

F = P−1((F (λ, ·)⊗ u (λ))λ∈A

)and SA = P−1

((1A×[0,1)m (λ, ·)⊗ u (λ)

)λ∈A

)then

[JAF ] (γ) =⟨F, L

(γ−1

)SA

⟩= F ∗ S∗A

(γ−1

).

Thus, if β (A× [0, 1)m) has positive Lebesgue measure in Rp and is containedin a fundamental domain of Zp then L (Γ) SA is a Parseval frame for HA ⊂L2 (N) . Next since ‖SA‖ ≤ 1 it follows that

‖SA‖2 =

∫A

∫[0,1)m

|det Jβ (λ, 0)| dt dλ =

∫A|det Jβ (λ, 0)| dλ ≤ 1

and ∫A|det Jβ (λ, 0)| dλ ≤ 1.

Thus if ∫A|det Jβ (λ, 0)| dλ > 1

then JA cannot be an isometry.

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4 Proof of Main Results

4.1 Proof of Theorem 4

First, we observe that the right regular and left regular representations of Γare unitarily equivalent ([6], Page 69). To prove Part 1, we appeal to Lemma27, and Lemma 26. Assuming that β (A× [0, 1)m) has positive Lebesgue mea-sure in Rp and is contained in a fundamental domain of Zp, the restrictionof the representation (L,HA) to the discrete group Γ is equivalent to a sub-representation of the left regular representation of Γ. Appealing to Lemma13, there exists a vector η such that V L

η (HA) is a sampling space with re-spect to Γ. For Part 2, Lemma 27, Lemma 26 together with the assumptionthat β (A× [0, 1)m) is equal to a fundamental domain of Zp imply that therestriction of the representation (L,HA) to the discrete group Γ is equivalentto the left regular representation of Γ. Finally, there exists a vector η ∈ HA

such that V Lη (HA) is a left-invariant subspace of L2 (N) which is a sampling

space with the interpolation property with respect to Γ.

4.2 Proof of Corollary 5

For s = (s1, · · · , sm) ∈ Rm, let

A (s) = s1A1 + · · ·+ smAm ∈ m.

Since the linear operators adA1, · · · , adAm are pairwise commutative andnilpotent, and e−adA(s)|p is unipotent, there is a unit vector which is an eigen-vector for e−adA(s)|p with corresponding eigenvalue 1. So, it is clear that∥∥∥∥[e−adA(s)|p

]T∥∥∥∥∞≥ 1

and

sup

∥∥∥∥[e−adA(s)|p]T∥∥∥∥

∞: s ∈ E

≥ 1

for any nonempty E ⊆ Rm. We recall again that

P (A(s)) =[ead(−

∑mj=1 sjAj)|p

]T. (39)

Lemma 30 Let E be an open bounded subset of Rm. If ε is a positive numbersatisfying

ε ≤ δ = (2 sup ‖P (A(s))‖∞ : s ∈ E)−1

thenβ((

(−ε, ε)dim Σ ∩ Σ)×E

)is open in Rp and is contained in a fundamental domain of Zp.

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Proof. Since the map β is a diffeomorphism (see Lemma 20) and since theset (

(−ε, ε)dim Σ ∩ Σ)×E

is an open set in Σ × Rm, it is clear that its image under the map β is alsoopen in Rp. Next, it remains to show that it is possible to find a positive realnumber δ such that if 0 < ε ≤ δ then

β((

(−ε, ε)dim Σ ∩ Σ)×E

)is an open set contained in a fundamental domain of Zp. Let λ ∈ Σ. Thenthere exists a linear functional f in the dual of the ideal p such that

ι (λ) =

[f0

]and

ι

exp

m∑j=1

sjAj

· λ =

[P (A(s)) f

0

]. (40)

Moreover, it is worth noting that∥∥∥∥∥∥exp

m∑j=1

sjAj

· λ∥∥∥∥∥∥

max

= ‖P (A(s)) f‖max .

Let δ be a positive real number defined as follows:

δ = (2 sup ‖P (A(s))‖∞ : s ∈ E)−1 . (41)

Iff ∈ (−ε, ε)dim Σ ⊆ (−δ, δ)dim Σ

and if s ∈ E then

‖B(A(s))f‖∞ ≤ ‖f‖max × sup ‖P (A(s))‖∞ : s ∈ E =1

2×‖f‖max

δ.

Now since ‖f‖max < δ, it follows that ‖P (A(s)) f‖max <12 . As a result,

β((

(−ε, ε)dim Σ ∩ Σ)×E

)⊆(−1

2,1

2

)pand clearly

(−1

2 ,12

)pis contained in a fundamental domain of Zp.

Appealing to Lemma 27, and Lemma 30 the following is immediate

Proposition 31 If

0 < ε ≤ δ = (2 sup ‖P (A(s))‖∞ : s ∈ [0, 1)m)−1

then J(−ε,ε)n−2m∩Σ defines an isometry between

L2((

(−ε, ε)n−2m ∩ Σ)× Rm, dµ (λ)

)and l2 (Γ).

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4.2.1 Proof of Corollary 5

Let δ be a positive number defined by

δ = (2 sup ‖P (A(s))‖∞ : s ∈ [0, 1)m)−1

. We want to show that for ε ∈ (0, δ] there exists a band-limited vector

η = ηε ∈ H(−ε,ε)n−2m such that the Hilbert space V Lη

(H(−ε,ε)n−2m

)is a

left-invariant subspace of L2 (N) which is a sampling space with respect to Γ.According to Lemma 30 the set

β((

(−ε, ε)dim Σ ∩ Σ)× [0, 1)m

)has positive Lebesgue measure and is contained in a fundamental domain ofZp. The desired result follows immediately from Theorem 4.

4.3 Proof of Example 6 Part 1

The case of a commutative simply connected, and connected nilpotent Liegroup is already known to be true. Thus, to prove this result, it remains tofocus on the non commutative algebras. According to the classification offour-dimensional nilpotent Lie algebras [10] there are three distinct cases toconsider. Indeed if n is a non-commutative nilpotent Lie algebra of dimensionthree, then n must be isomorphic with the three-dimensional Heisenberg Liealgebra. If n is four-dimensional then up to isomorphism either n is the directsum of the Heisenberg Lie algebra with a one-dimensional algebra, or there isa basis Z1, Z2, Z3, A1 for n with the following non-trivial Lie brackets

[A1, Z2] = 2Z1, [A1, Z3] = 2Z2.

Case 1 (The Heisenberg Lie algebra) Let N be the simply connected, con-nected Heisenberg Lie group with Lie algebra n which is spanned by Z1, Z2, A1

with non-trivial Lie brackets [A1, Z2] = Z1.We check that N = PM whereP = exp (RZ1 + RZ2) and M = exp (RA1) . Put

Γ = exp (ZZ1) exp (ZZ2) exp (ZA1) .

It is easily checked that

M (λ) =

0 0 00 0 −λ (Z1)0 λ (Z1) 0

.Next, since

e (λ) =

∅ if λ (Z1) = 0

2, 3 if λ (Z1) 6= 0

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we obtain that e = 2, 3 . It follows that

Ωe = λ ∈ n∗ : λ (Z1) 6= 0 .

Next, the unitary dual of N is parametrized by

Σ = λ ∈ Ωe : λ (Z2) = λ (A1) = 0

which we identify with the punctured line: R∗. It is not hard to check that

δ−1 = 2 sup

∥∥∥∥[ 1 0−s 1

]∥∥∥∥∞

: s ∈ [0, 1)

= 4.

So, there exists a band-limited vector η ∈ H(− 14, 14) such that V L

η

(H(− 1

4, 14)

)is a sampling space with respect to Γ.

To prove that the Heisenberg group admits sampling spaces with the inter-polation property with respect to Γ, we claim that the set

B(1) = β ((−1, 1)× [0, 1)) =

[f−sf

]: f ∈ (−1, 1) , s ∈ [0, 1)

is up to a null set equal to a fundamental domain of Z2 (see illustration ofB(1) below)

To prove this we write

β ((−1, 1)× [0, 1)) = β ((0, 1)× [0, 1)) ∪ β ((−1, 0)× [0, 1)) .

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Next, it is easy to check that(β ((0, 1)× [0, 1)) +

[10

])∪(β ((−1, 0)× [0, 1)) +

[01

])is up to a null set equal to the unit square [0, 1)2 . Thus the set β ((−1, 1)× [0, 1))is up to a null set equal to a fundamental domain of Z2. Appealing to Theo-rem 4, the following result confirms the work proved in [5][15]. There existsa band-limited vector η ∈ H(−1,1) such that V L

η

(H(−1,1)

)is a sampling space

with respect to Γ which also enjoys the interpolation property.

Case 2 (Four-dimensional and step two) Assume that n is the direct sumof the Heisenberg Lie algebra with R. Let us suppose that the Lie algebra nis spanned by Z1, Z2, Z3, A1 with non-trivial Lie brackets [A1, Z2] = Z1. Wecheck that

M (λ) =

0 0 0 00 0 0 − (Z1)0 0 0 00 λ (Z1) 0 0

and

e (λ) =

∅ if λ (Z1) = 0

2, 4 if λ (Z1) 6= 0.

Fix e = 2, 4 such that

Ωe = λ ∈ n∗ : λ (Z1) 6= 0 .

The unitary dual of N is parametrized by

Σ = λ ∈ Ωe : λ (Z2) = λ (A1) = 0 .

For any linear functional λ ∈ Σ, the ideal spanned by Z1, Z2, Z3 is a polariza-tion algebra subordinated to λ and

δ =

2 sup

∥∥∥∥∥∥ 1 0 0−s 1 00 0 1

∥∥∥∥∥∥∞

: s ∈ [0, 1)

−1

=1

4.

So, there exists a band-limited vector η ∈ H(− 14, 14) such that V L

η

(H(− 1

4, 14)

)is a sampling space with respect to

Γ = exp (ZZ1 + ZZ2 + ZZ3) exp (ZA1) .

Case 3 (Four-dimensional and three step) Assume that n is a four-dimensionalZ1, Z2, Z3, A1 such that

[A1, Z2] = 2Z1, [A1, Z3] = 2Z2.

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With respect to the ordered basis Z1, Z2, Z3, we have

[adA1]| p =

0 2 00 0 20 0 0

and exp [adA1]| p =

1 2 20 1 20 0 1

.Next, we check that

δ =

2 sup

∥∥∥∥∥∥ 1 0 0−2s 1 02s2 −2s 1

∥∥∥∥∥∥∞

: s ∈ [0, 1)

−1

=1

2

(max

1, 1 + 2 |s| , 1 + 2 |s|+ 2 |s|2 : s ∈ [0, 1)

)−1=

1

10.

Indeed, the set

β

((− 1

10,

1

10

)2

× [0, 1)

)⊂(−1

2,1

2

)3

is contained in a fundamental domain of Z3. Thus, there exists a band-limited

vector η ∈ H(− 110, 110) such that V L

η

(H(− 1

10, 110)

)is a sampling space with

respect toΓ = exp (ZZ1 + ZZ2 + ZZ3) exp (ZA1) .

4.4 Proof of Example 6 Part 2

Let N be a simply connected, connected nilpotent Lie group with Lie algebraspanned by

Z1, Z2, · · · , Zp, A1

such that [adA1]|p = A is a nonzero rational upper triangular nilpotent matrix

of order p such that eAZp ⊆ Zp and the algebra generated by Z1, Z2, · · · , Zpis commutative. Then N is isomorphic to a semi-direct product group RpoRwith multiplication law given by

(x, t)(x′, t′

)=(x+ etAx′, t+ t′

).

Clearly since A is not the zero matrix then

max rank (M (λ)) : λ ∈ n∗ = 2

and the unitary dual of N is parametrized by a Zariski open subset of Rp−1.Finally, let

δ =

(2× sup

∥∥∥∥∥m−1∑k=0

(−sAT

)kk!

∥∥∥∥∥∞

: s ∈ [0, 1)

)−1

> 0.

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For ε ∈ (0, δ] there exists a band-limited vector

η = ηε ∈ H(−ε,ε)p−1

such that the Hilbert space V Lη

(H(−ε,ε)p−1

)is a left-invariant subspace of

L2 (N) which is a sampling space with respect to Γ.

4.5 Proof of Example 6 Part 3

Let A (t) be as given in (9). Then

exp (A (t)) =

m+1∑k=0

A (t)k

k!

and N is a nilpotent Lie group of step p = m+ 1. Moreover, the unitary dualof N is parametrized by the manifold:

Σ = λ ∈ n∗ : λ (Z1) 6= 0and λ (Zk+1) = λ (Ak) = 0 for 1 ≤ k ≤ m ' R∗

and the Plancherel measure is up to multiplication by a constant given by|λ|m dλ. The positive number δ described in Corollary 5 is equal to

δ =

sup

2

∥∥∥∥∥∥∥m+1∑k=0

(−A (t)T

)kk!

∥∥∥∥∥∥∥∞

: t ∈ [0, 1)m

−1

.

Thus, for ε ∈ (0, δ] there exists a band-limited vector

η = ηε ∈ H(−ε,ε)

such that the Hilbert space V Lη

(H(−ε,ε)

)is a left-invariant subspace of L2 (N)

which is a sampling space with respect to Γ.

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