Regular Representations of Time-Frequency Groups
aa r X i v : . [ m a t h . R T ] S e p REGULAR REPRESENTATIONS OF TIME-FREQUENCY GROUPS
AZITA MAYELI AND VIGNON OUSSA
Abstract.
In this paper, we study the Plancherel measure of a class of non-connectednilpotent groups which is of special interest in Gabor theory. Let G be a time-frequencygroup. More precisely, that is G = (cid:10) T k , M l : k ∈ Z d , l ∈ B Z d (cid:11) , T k , M l are translations andmodulations operators acting in L ( R d ) , and B is a non-singular matrix. We compute thePlancherel measure of the left regular representation of G which is denoted by L. The actionof G on L ( R d ) induces a representation which we call a Gabor representation. Motivatedby the admissibility of this representation, we compute the decomposition of L into directintegral of irreducible representations by providing a precise description of the unitary dualand its Plancherel measure. As a result, we generalize Hartmut F¨uhr’s results which areonly obtained for the restricted case where d = 1, B = 1 /L, L ∈ Z and L > . Even inthe case where G is not type I, we are able to obtain a decomposition of the left regularrepresentation of G into a direct integral decomposition of irreducible representations when d = 1. Some interesting applications to Gabor theory are given as well. For example, when B is an integral matrix, we are able to obtain a direct integral decomposition of the Gaborrepresentation of G. Introduction
Let G be a locally compact group with type I left regular representation. The Planchereltheorem guarantees the existence of a measure µ on the unitary dual of G such that once aHaar measure is fixed on the group G , µ is uniquely determined. Although the existence ofthe Plancherel measure is given; it is generally a hard but interesting problem to computeit. Let b G be the unitary dual of the group G. The group Fourier transform f b f maps L ( G ) ∩ L ( G ) into R ⊕ b G H π ⊗ H π dµ ( π ) . Also, the group Fourier transform extends to aunitary map defined on L ( G ). This extension is known as the Plancherel transform P . Forarbitrary functions f and g ∈ L ( G ) , the following holds true: h f, g i L ( G ) = Z b G trace h b f ( π ) b g ( π ) ∗ i dµ ( π ) . Moreover, if L is the left regular representation of G acting in L ( G ) then we obtain a unique direct integral decomposition of L into unitary irreducible representations of G suchthat P◦ L ◦ P − = Z ⊕ b G π ⊗ H π dµ ( π ) . Let B be an invertible matrix of order d. For k ∈ Z d , and l ∈ B Z d , we define the followingunitary operators T k , M l : L (cid:0) R d (cid:1) → L (cid:0) R d (cid:1) such that T k f ( x ) = f ( x − k ) and M l f ( x ) = e πi h l,x i f ( x ) . Let G be the group generated by T k , and M l . T k is a time-shift operator, and M l is a modulation operator (or frequency shift operator). We write G = (cid:10) T k , M l | k ∈ Z d , l ∈ B Z d (cid:11) . (1) Mathematics Subject Classification.
Characteristics of the closure of the linear span of orbits of the type G ( S ) where S ⊂ L ( R d ) have been studied in [2] when B only has rational entries. Also a thorough presen-tation of the theory of time-frequency analysis is given in [10]. In this paper, we are mainlyinterested in the following questions. Question 1.1. If G is type I, can we provide a description of the unitary dual of G, and aprecise formula for the Plancherel measure of G ? Question 1.2. If G is not type I, can we obtain a decomposition of the left regular repre-sentation into unitary irreducible representations of G ? Is it possible to provide a centraldecomposition of its left regular representation? For obvious reasons, we term the group G a time-frequency group . There are threecases to consider. First, it is easy to see that G is a commutative discrete group if and onlyif B is an integral matrix. In that case, all irreducible representations of G are characters,and thanks to the Pontrjagin duality, the Plancherel measure is well-understood. In fact if B is a matrix with integral entries, the Plancherel measure of G is supported on a measurablefundamental domain of the lattice Z d × ( B − ) tr Z d ; namely the set [0 , d × ( B − ) tr [0 , d . Interestingly, it can be shown that the Gabor representation Z d × B Z d ∋ ( k, l ) T k ◦ M l is unitarily equivalent with a subrepresentation of the left regular representation if and onlyif | det B | ≤ . Otherwise, the Gabor representation is equivalent to a direct sum of regularrepresentations. Although the previous statement is not technically new, the proof givenhere is based on the representation theory of the time-frequency group. Secondly, if B hassome rational entries, some of them non-integer, then G is a non-commutative discrete typeI group. Using well-known techniques developed by Mackey, and later on by Kleppner andLipsman in [14], precise descriptions of the unitary dual of G and its Plancherel measureare obtained. Two main ingredients are required to compute the Plancherel measure of G .Namely, a closed normal subgroup of N whose left regular representation is type I, and afamily of subgroups of G/N known as the ‘little groups’. We will show that the Plancherelmeasure in the case where B has some non integral rational entries but no irrational entry,is a fiber measure supported on a fiber space with base space: the unitary dual of thecommutator subgroup of G . That is, the base space is \ [ G, G ] . Using some procedure givenin [9], we can construct a non-singular matrix A, such that A Z d = Z d ∩ ( B − ) tr Z d and weshow that each fiber can be identified with some compact setΛ × E σ × { χ σ } ⊂ R d × R d × { χ σ } where χ σ ∈ \ [ G, G ] , Λ is a measurable fundamental domain of ( A − ) tr Z d and E σ is the cross-section of the action of a little group (which is a finite group here) in R d / ( B − ) tr Z d . Wealso show that all irreducible representations of G are monomial representations modelled asacting in some finite-dimensional Hilbert spaces with dimensions bounded above by | det A | .It is worth noticing that Hartmut F¨uhr has already computed the Plancherel measure of thesimplest example (Section 5 . d = 1 and B Z d = 1 /L Z where L is some positive integer greater than one. Forthe more general case in which we are interested, we obtain a parametrization of the unitarydual of G, and we derive a precise formula for the Plancherel measure. In the case where G is not type I ( B has some irrational entries); unfortunately the ‘Mackey machine’ fails. Inthe particular case where d = 1 ,G = h T k , M l | k ∈ Z , l ∈ α Z i , α ∈ R − Q , LANCHEREL MEASURE AND ADMISSIBILITY 3 we are able to obtain a central decomposition of the left regular representation of L as wellas a direct integral decomposition of the left regular representation of G ∼ = Γ = m l k : ( m, k, l ) ∈ Z into its irreducible components. In fact, we derive that the left regular representation of G is unitarily equivalent to Z ⊕ [ − , Z ⊕ [0 , | λ | ) Ind Γ K (cid:0) χ ( | λ | ,t ) (cid:1) | λ | dtdλ (2)acting in R ⊕ [ − , R ⊕ [0 , | λ | ) l ( Z ) | λ | dtdλ where K = l k : ( k, l ) ∈ Z and χ ( | λ | ,t ) l k = exp (2 πi ( | λ | l + tk )) . To the best of our knowledge, the decomposition given in (2) and the central decompositionobtained in Theorem 6.6 are both appearing for the first time in the literature.
Acknowledgements
Sincerest thanks go to B. Currey, for providing support and suggestions during the prepa-ration of this work. We also thank the anonymous referees. Their suggestions and correctionswere crucial to the improvement of our work.2.
Preliminaries
Let us start by setting up some notation. Given a matrix A of order d, A tr stands for thetranspose of A, and A − tr = ( A − ) tr stands for the inverse transpose of A. Let F be a fieldor a ring. It is standard to use GL ( d, F ) to denote the set of invertible matrices of order d with entries in F. Let G be a locally compact group. The unitary dual, which is the setof all irreducible unitary representations of G is denoted by b G. Given x ∈ R d , we define acharacter which is a one-dimensional unitary representation of R d into the one-dimensionaltorus as χ x : R d → T where χ x ( y ) = e πi h x,y i . Let H be a subgroup of G. The index of H in G is denoted by ( G : H ) = | G/H | . We will use to denote the identity operator acting insome Hilbert space. Given two isomorphic groups G, H we write G ∼ = H. The reader who is not familiar with the theory of direct integrals is invited to refer to [5].
Definition 2.1.
Given a countable sequence { f i } i ∈ I of vectors in a Hilbert space H , we say { f i } i ∈ I forms a frame if and only if there exist strictly positive real numbers A, B such thatfor any vector f ∈ H A k f k ≤ X i ∈ I |h f, f i i| ≤ B k f k . In the case where A = B , the sequence of vectors { f i } i ∈ I is called a tight frame , and if A = B = 1 , { f i } i ∈ I is called a Parseval frame . LANCHEREL MEASURE AND ADMISSIBILITY 4
Remark 2.2. If { f i } i ∈ I is a Parseval frame such that for all i ∈ I, k f i k = 1 , then { f i } i ∈ I is an orthonormal basis for H . Definition 2.3.
A lattice Λ in R d is a discrete additive subgroup of R d . In other words,every point in Λ is isolated and Λ = A Z d for some matrix A . We say Λ is a full rank latticeif A is nonsingular, and we denote the dual of Λ by ( A − ) tr Λ . A fundamental domain(or transveral) D for a lattice in R d is a measurable set such that the following hold: ( D + l ) ∩ ( D + l ′ ) = ∅ for distinct l, l ′ in Λ , and R d = S l ∈ Λ ( D + l ) . Definition 2.4.
Let
Λ = A Z d × B Z d be a full rank lattice in R d and g ∈ L (cid:0) R d (cid:1) . Thefamily of functions in L (cid:0) R d (cid:1) , G (cid:0) g, A Z d × B Z d (cid:1) = (cid:8) e πi h k,x i g ( x − n ) : k ∈ B Z d , n ∈ A Z d (cid:9) (3) is called a Gabor system . Definition 2.5.
Let m be the Lebesgue measure on R d , and consider a full rank lattice Λ = A Z d inside R d . (a) The volume of Λ is defined as vol (Λ) = m (cid:0) R d / Λ (cid:1) = | det A | . (b) The density of Λ is defined as d (Λ) = 1 | det A | . Lemma 2.6.
Given a separable full rank lattice
Λ = A Z d × B Z d in R d . The followingstatements are equivalent (a) There exits f ∈ L ( R d ) such that G (cid:0) f, A Z d × B Z d (cid:1) is a Parseval frame in L (cid:0) R d (cid:1) . (b) vol (Λ) = | det A det B | ≤ . (c) There exists f ∈ L (cid:0) R d (cid:1) such that G (cid:0) f, A Z d × B Z d (cid:1) is complete in L (cid:0) R d (cid:1) . For a proof of the above lemma, we refer the reader to Theorem 3 . Lemma 2.7.
Let Λ be a full-rank lattice in R . There exists a function f ∈ L (cid:0) R d (cid:1) suchthat G ( f, Λ) is an orthonormal basis if and only if vol (Λ) = 1 . Also, if G ( f, Λ) is a Parsevalframe for L ( R d ) then k f k = vol (Λ) . A proof of the Lemma 2.7 is given in [11]. Now, let F = (cid:10) T x , M y | x ∈ R d , y ∈ R d (cid:11) and let f be a square-integrable function over R d . It is easy to see that T x M y T − x M − y f = e − πi h y,x i f, (4)and e − πi h y,x i is a central element of the group F . Thus F is a non-commutative, connected, non-simply connected two-step nilpotent Lie group. In fact F is isomorphic to the reduced d + 1 dimensional Heisenberg group. The 2 d + 1-dimensional Heisenberg group has Liealgebra h = R -span { X , · · · , X d, Y , · · · , Y d , Z } with non-trivial Lie brackets [ X i , Y i ] = Z. Let H = exp ( h ) (5)Clearly, H is a simply connected, connected 2 n + 1-dimensional Heisenberg group, andexp ( Z Z ) is a discrete central subgroup of H . Moreover, H is the universal covering group of F . That is, F is isomorphic to the group H / exp ( Z Z ) . LANCHEREL MEASURE AND ADMISSIBILITY 5
We will now provide a light introduction to the notion of admissibility of unitary repre-sentations. A more thorough exposure to the theory is given in [7]. However, we will discusspart of the material which is necessary to fully understand the results obtained in our work.Let π be a unitary representation of a locally compact group X, acting in some Hilbert space H. We say that π is admissible , if and only if there exists some vector φ ∈ H such that theoperator W φ W φ : H → L ( X ) where W φ ψ ( x ) = h ψ, π ( x ) φ i defines an isometry from H to L ( X ). Let ( ρ, K ) be an arbitrary type I representation of thegroup G = (cid:10) T k , M l : k ∈ Z d , l ∈ B Z d (cid:11) . Moreover, let us suppose that X is a type I unimod-ular group. Let us also suppose that we are able to obtain a direct integral decompositionof ρ as follows ρ ∼ = Z ⊕ b X ( ⊕ n π k =1 π ) dµ ( π )where dµ is a measure defined on the unitary dual of X. According to well-known theoremsdeveloped in [7]; ( ρ, K ) is admissible if and only if it is unitarily equivalent with a subrepre-sentation of the left regular representation, and the multiplicity function is integrable overthe spectrum ρ . That is: µ is absolutely continuous with respect to the Plancherel measure µ supported on b X and R b X n π dµ ( π ) < ∞ . If a representation ρ is admissible , in theory it isknown (see [7]) how to construct all admissible vectors . Let us describe such process ingeneral terms. First, we must construct a unitary operator U : K → Z ⊕ b X ( ⊕ n π k =1 H π ) dµ ( π )intertwining the representation ρ with R ⊕ b X n π π dµ ( π ) . Next, we define a measurable field( F λ ) λ ∈ b X of operators in R ⊕ b X ⊕ n π k =1 H π dµ ( π ) such that each operator F λ is an isometry. Alladmissible vectors are of the type U − (cid:0) ( F λ ) λ ∈ b X (cid:1) . In the remainder of the paper, we will focus on time-frequency groups. We will computethe Plancherel measure of the group whenever G is type I, and we will obtain a directintegral decomposition of the left regular representation if G is not type I and d = 1. Someapplication to Gabor theory will be discussed throughout the paper as well.3. Normal Subgroups of G In this section, we will study the structure of normal subgroups of the time frequencygroup. The reason why this section is important, is because part of what the Mackeymachine [15] needs is an explicit description of the unitary dual of type I normal subgroupsin order to compute the unitary dual of the whole group. In this paper, unless we stateotherwise, G stands for the following group: (cid:10) T k , M l | k ∈ Z d , l ∈ B Z d (cid:11) . We recall that the subgroup generated by operators of the type T k ◦ M l ◦ T − k ◦ M − l is calledthe commutator subgroup of G and is denoted [ G, G ]. From now on, to simplify the notation,we will simply omit the symbol ◦ whenever we are composing operators. Lemma 3.1. [ G, G ] is isomorphic to a subgroup of the torus. (a) If B ∈ GL ( d, Z ) then G is commutative and isomorphic to Z d × B Z d . LANCHEREL MEASURE AND ADMISSIBILITY 6 (b) If B is in GL ( d, Q ) − GL ( d, Z ) then [ G, G ] is a central subgroup of G , and is a cyclicgroup . G is not commutative but it is a type I discrete unimodular group. (c) If B ∈ GL ( d, R ) − GL ( d, Q ) then, G is a non-commutative two-step nilpotent group,and its commutator subgroup is a infinite subgroup of the circle group.Proof. To show part (a), let l ∈ B Z d , k ∈ Z d where B is a non-invertible matrix. Let f ∈ L (cid:0) R d (cid:1) . We have T k M l T − k M − l f = e − πi h l,k i f = χ l ( k ) f. If B ∈ GL ( d, Z ) then the commutator subgroup of G is trivial, and G is an abelian groupisomorphic to Z d × B Z d . For part (b), let us suppose that B ∈ GL ( d, Q ) − GL ( d, Z ) . So, B = [ p ij /q ij ] ≤ i,j ≤ d where p ij , q ij are integral values, gcd ( p ij , q ij ) = 1 and q ij = 0 for1 ≤ i, j ≤ d. Let m = lcm ( q ij ) ≤ i,j ≤ d . Clearly χ ml ( k ) = 1 for all l ∈ B Z d , and k ∈ Z d . Thus, [
G, G ] is a finite abelian proper closed subgroup of the circle group. As a result [
G, G ]is cyclic. For part (c), if B ∈ GL ( d, R ) − GL ( d, Q ) then the commutator subgroup of G isnot isomorphic to a finite subgroup of the torus. That is, there exist l ∈ B Z d , k ∈ Z d suchthat the set { χ ml ( k ) : m ∈ Z } is not closed and is dense in the torus T . (cid:3) Example 3.2.
Let G be group generated T k , M l such that k ∈ Z , l ∈ B Z B = (cid:20) / / / − / (cid:21) then, [ G, G ] is isomorphic to Z . Example 3.3.
Let G be group generated T k , M l such that k ∈ Z , l ∈ B Z such that B = (cid:20) √ − (cid:21) then [ G, G ] is isomorphic to an infinite subgroup of the circle. Assuming that B is in GL ( d, Q ) , we will construct an abelian normal subgroup of G. Forthat purpose, we will need to define the groups N = (cid:10) T k , M l | k ∈ B − tr Z d , l ∈ B Z d (cid:11) , and N = (cid:10) T k , M l | k ∈ B − tr Z d ∩ Z d , l ∈ B Z d (cid:11) . Notice that in general N is not a subgroup of G because the lattice Z d is not invariantunder the action of B − tr if B − tr has non integral rational entries. However, the group N will be important in constructing the unitary dual of G, and we will need to study some ofits characteristics. Lemma 3.4. If B is an element of GL ( d, Q ) − GL ( d, Z ) then N = (cid:10) T k , M l | k ∈ B − tr Z d , l ∈ B Z d (cid:11) is an abelian group. LANCHEREL MEASURE AND ADMISSIBILITY 7
Proof.
Given k ∈ B − tr Z d and l ∈ B Z d , we recall that T k M l T − k M − l f ( x ) = e − πi h l,k i f. Since k ∈ B − tr Z d , and l ∈ B Z d then there exist k ′ , l ′ ∈ Z d such that T k M l T − k M − l f = e − πi h Bl ′ ,B − tr k ′ i f = e − πi h l ′ ,k ′ i f ( x ) = f. Thus, for any given k ∈ B − tr Z d , and l ∈ B Z d , T k M l T − k M − l is equal to the identity operator.It follows that the commutator subgroup of N is trivial. (cid:3) We recall the following lemmas from [9].
Lemma 3.5.
Given two lattices Γ , Γ , Γ ∩ Γ is a lattice in R d if and only if there existsa lattice Γ that contains both Γ and Γ . Definition 3.6.
Let Γ be a full-rank lattice in R d with generators v ... v d , v ... v d , · · · , v d ... v dd ∈ R d . The matrix of order d below v · · · v d ... ... ... v d · · · v dd is called a basis for Γ . Lemma 3.7.
Let Γ , Γ be two distinct lattices with bases J, K respectively . Γ + Γ , Γ ∩ Γ are two lattices in R d if and only if J K − is a rational matrix. Moreover dim (Γ + Γ ) + dim (Γ ∩ Γ ) = dim Γ + dim Γ Remark 3.8. If Γ ∩ Γ is a lattice, there is a well-known technique given in [9] (see page ) used to compute the basis of the lattice Γ ∩ Γ . We describe the procedure here. Let J, K be bases for lattices Γ , and Γ respectively. First we compute the d × d matrix [ J | K ] . Secondly, we evaluate the
Hermite lower triangular form of [ J | K ] . This form has thestructure [ L | , and is obtained as [ J | K ] E = [ L | , where E is matrix of order d obtainedby applying elementary row operations to [ J | K ] . In fact, E is a block matrix of the type E = (cid:20) R SC D (cid:21) , and R, S, C, and D are matrices of order d. Finally, a basis for the lattice Γ ∩ Γ is thengiven by KD.
That is Γ ∩ Γ = ( KD ) Z d . Corollary 3.9. If B is in GL ( d, Q ) − GL ( d, Z ) then B − tr Z d ∩ Z d is a full-rank latticesubgroup of R d . Proof.
The fact that B − tr Z d ∩ Z d is a full-rank lattice follows directly from Lemma 3.7. (cid:3) We assume that B is an element of GL ( d, Q ) − GL ( d, Z ) . We will prove that N is anormal subgroup of G . However it is not a maximal normal subgroup of G since it does notcontain the center of the group. Thus, N needs to be extended. For that purpose, we definethe group N = (cid:10) T k , M l , τ | k ∈ B − tr Z d ∩ Z d , l ∈ B Z d , τ ∈ [ G, G ] (cid:11) ⊂ G. Proposition 3.10. If B is in GL ( d, Q ) then N is an abelian normal subgroup of G. LANCHEREL MEASURE AND ADMISSIBILITY 8
Proof.
From Lemma 3.4, we have already seen that T k commutes with M l for arbitrary k ∈ B − tr Z d ∩ Z d , l ∈ B Z d . Since [
G, G ] commutes with T k and M l for k ∈ B − tr Z d ∩ Z d , l ∈ B Z d ,then N is abelian. For the second part of the proof, let k ∈ B − tr Z d ∩ Z d , l ∈ B Z d and s ∈ Z d . First, we compute the conjugation action of the translation operator on an arbitraryelement of N. Let s ∈ Z d . Then T s ( T k M l τ ) T − s = τ e − πi h l,s i T k M l . Next, we compute theconjugation action of the modulation operator on an arbitrary element of N as follows: M s ( T k M l τ ) M − s = τ T k M l . Clearly,
GN G − ⊂ N. Thus, N is a normal subgroup of G. (cid:3) Lemma 3.11.
Assuming that B is in GL ( d, Q ) − GL ( d, Z ) , then the following holds. (a) The quotient group Z d B − tr Z d ∩ Z d is isomorphic to a finite abelian group. (b) The group
G/N is a finite group isomorphic to Z d B − tr Z d ∩ Z d . Proof.
For the first part of the proof, since B − tr Z d ∩ Z d is a full-rank lattice, there exists anon-singular matrix A such that B − tr Z d ∩ Z d = A Z d . Thus, referring to the discussion [9](page 95), the index of B − tr Z d ∩ Z d in Z d is (cid:0) Z d : B − tr Z d ∩ Z d (cid:1) = | det A | . As a result, Z d B − tr Z d ∩ Z d is a finite abelian group. For the second part of lemma, there exist k , k , · · · , k m ∈ Z d suchthat Z d B − tr Z d ∩ Z d = (cid:8)(cid:0) B − tr Z d ∩ Z d (cid:1) , k + (cid:0) B − tr Z d ∩ Z d (cid:1) , · · · , k m + (cid:0) B − tr Z d ∩ Z d (cid:1)(cid:9) ∼ = { , T k · · · , T k m } . Also,
G/N ∼ = { , T k · · · , T k m }∼ = Z d B − tr Z d ∩ Z d and this completes the proof. (cid:3) The Plancherel Measure and Application: The Integer Case If B ∈ GL ( d, Z ) then G is commutative and isomorphic to Z d × B Z d . In this case,the unitary dual, and the Plancherel measure of G = (cid:10) T k , M l : k ∈ Z d , l ∈ B Z d (cid:11) are well-understood through the Pontrjagin duality. Let L be the left regular representation of G. The unitary dual of G is isomorphic to c R d Z d × c R d B − tr Z d and the Plancherel measure is up to multiplication by a constant equal to dxdy | det ( B − tr ) | which is supported on a measurable set Λ ⊂ R d parametrizing the group c R d Z d × c R d B − tr Z d . Moreprecisely, the collection of sets (cid:8)
Λ + j : j ∈ Z d × B − tr Z d (cid:9) LANCHEREL MEASURE AND ADMISSIBILITY 9 forms a measurable partition for R d and Λ is called the spectrum of the left regularrepresentation of π. It is worth noticing that there is no canonical way to choose Λ . More-over, via the Plancherel transform, the left regular representation of Z d × B Z d is decom-posed into a direct integral decomposition of characters R ⊕ Λ χ ( x,y ) | det B | dxdy acting in R ⊕ Λ C | det B | dxdy ∼ = L (Λ) . Now, we will discuss some application of the Plancherel theory of G to Gabor theory. Let B ∈ GL ( d, Z ) , and let e G = Z d × B Z d . There is a representation of e G where π : e G → U (cid:0) L (cid:0) R d (cid:1)(cid:1) , such that G is the image of e G via the representation π. The representation π is called a Gabor representation of e G. Since G is abelian, once a choice for a transversalof c R d Z d × c R d B − tr Z d is made, there is a decomposition of π into irreducible representations of thegroup e G Z ⊕ Λ χ ς ⊗ C n ( ς ) dµ ( ς ) (6)acting in R ⊕ Λ C ⊗ C n ( ς ) dµ ( ς ) . The function n : Λ → N ∪ { } is the multiplicity function of the irreducible representations appearing in the decomposition of π. Let us define W f : L (cid:0) R d (cid:1) → l (cid:16) e G (cid:17) where W f h ( k, l ) = h h, T k M l f i . We recall that π is admissible if and only if W f defines an isometry on L (cid:0) R d (cid:1) . Lemma 4.1.
Let B ∈ GL ( d, Z ) . π is admissible if and only if X T k M l ∈ G |h h, T k M l f i| = k h k for all h ∈ L (cid:0) R d (cid:1) . That is f is a Gabor Parseval frame . We recall that the
Zak transform is a unitary operator Z : L (cid:0) R d (cid:1) → L (cid:16) [0 , d × [0 , d (cid:17) ∼ = Z ⊕ [0 , d × [0 , d C dxdy where Zf ( x, y ) = X m ∈ Z d f ( x + m ) exp (2 πi h m, y i ) . It is easy to see that if B ∈ GL ( d, Z ) then Z ( T k M l f ) ( x, y ) = exp ( − πi h k, y i ) exp (2 πi h l, x i ) Zf ( x, y ) . Thus, the Zak transform intertwines the Gabor representation with the representation Z ⊕ [0 , d × [0 , d χ ( x,y ) dxdy. (7)Now, we would like to compare the representations given in (6) and (7). Proposition 4.2. If B ∈ GL ( d, Z ) and | det B | 6 = 1 then the Gabor representation is notadmissible. That is, there is no Parseval frame of the type π (cid:16) e G (cid:17) f. Moreover, the Gaborrepresentation is unitarily equivalent to the direct integral Z ⊕ Λ χ ς ⊗ C | det B | dµ ( ς ) . LANCHEREL MEASURE AND ADMISSIBILITY 10
Proof.
First, let us notice that if B ∈ GL ( d, Z ) then | det B − tr | ≤ . As a result, there isa measurable set Λ ⊂ R d tiling R d by the lattice Z d × B − tr Z d such that Λ is containedin [0 , d × [0 , d (see [11]). Thus if | det B | 6 = 1 , the representation R ⊕ [0 , d R ⊕ [0 , d χ ( x,y ) dxdy cannot be contained in R ⊕ Λ χ ( x,y ) | det ( B tr ) | dxdy. Picking Λ = [0 , d × E ⊂ R d such that E ⊆ [0 , d , we obtain π ∼ = Z ⊕ Λ χ ς ⊗ C n ( ς ) dµ ( ς ) (8)We now claim that the multiplicity function is given by n ( ς ) = (cid:16)n j ∈ B − tr Z d : { ς + j } ∩ [0 , d = ∅ o(cid:17) = | det B | . To show that the above holds, we partition [0 , d × [0 , d into | det B | many subsets Λ k suchthat each set Λ k is a fundamental domain for Z d × B − tr Z d . Writing[0 , d × [0 , d = | det B | [ k =1 Λ k , we obtain π ∼ = Z ⊕ [0 , d × [0 , d χ ( x,y ) dxdy ∼ = Z ⊕ Λ | det B | M k =1 χ ( x,y ) dxdy ∼ = Z ⊕ Λ χ ς ⊗ C | det B | dµ ( ς ) . (cid:3) Example 4.3.
Let G = h T k , M l | k ∈ Z , l ∈ Z i . The spectrum of the left regular represen-tation of G is given by Λ = [0 , × [0 , / and π ∼ = R ⊕ Λ χ ς ⊗ C dµ ( ς ) . Thus, π is not anadmissible representation since π ∼ = L ⊕ L ⊕ L. Example 4.4.
Let G = h T k , M l | k ∈ Z , l ∈ B Z i where B = (cid:20) (cid:21) . The spectrum of the left regular representation is [0 , × (cid:20) − −
15 35 (cid:21) [0 , . In the graph below we illustrate the idea that there exists a collection of sets (cid:8) Λ k (cid:9) k =1 suchthat each set Λ k is a fundamental domain of [0 , × B − tr [0 , and the spectrum of π isgiven as follows [0 , × [0 , = [ k =1 Λ k . LANCHEREL MEASURE AND ADMISSIBILITY 11
Thus, π ∼ = L ⊕ L ⊕ L ⊕ L ⊕ L. As a result, the Gabor representation π is not admissible. A projection of the spectrum of π in R . Remark 4.5.
Let K be a π -invariant closed subspace of L (cid:0) R d (cid:1) ( K is a shift-modulationinvariant space [2] ). There exists a unitary operator J : K → Z ⊕ [0 , d × B − tr [0 , d C ⊗ C n ( ς ) dµ ( ς ) intertwining the representations π |K with R ⊕ Λ χ ς ⊗ C n ( ς ) dµ ( ς ) and n ( ς ) ≤ | det B | a.e. As aresult, we have a general characterization of shift-modulation invariant spaces in the specificcase where B has integral entries.Proof. The proof follows from the fact that ( π |K , K ) is a subrepresentation of the Gaborrepresentation of e G. (cid:3) Definition 4.6.
Two unitary representations ( π , H ) , ( π , H ) of a group X are quasi-equivalent if there exist unitarily-equivalent representations ρ , ρ such that ρ k is a multipleof π k for k = 1 , . Remark 4.7. If B ∈ GL ( d, Z ) then the Gabor representation π is quasi-equivalent to theleft regular representation of e G. The proof of Remark 4.7 is a direct application of Proposition 4.2.
Remark 4.8.
In the case where B is the identity matrix, π is an admissible representation.In fact, a well-known admissible vector is the indicator function of the cube [0 , d . The Plancherel Measure and Application: The Rational Case
In this section, we assume that the given matrix B has at least one rational non-integralentry. We recall from Lemma 3.1 that G = (cid:10) T k , M l : k ∈ Z d , l ∈ B Z d (cid:11) LANCHEREL MEASURE AND ADMISSIBILITY 12 is not commutative but is a discrete type I group. Thus, its unitary dual exists, and itsPlancherel measure is computable. Using
Mackey’s Machine and results developed byRonald Lispman and Adam Kleppner in [14], we will describe the unitary dual of the group G , and a formula for the Plancherel measure. We recall that if B is in GL ( d, Q ) − GL ( d, Z )then G contains a normal subgroup N = (cid:10) T k , M l , τ | k ∈ B − tr Z d ∩ Z d , l ∈ B Z d , τ ∈ [ G, G ] (cid:11) which is isomorphic with a direct product of abelian groups. Since N is isomorphic to (cid:0) B − tr Z d ∩ Z d (cid:1) × B Z d × Z m , its unitary dual is a group of characters. The underlying set for the group G is Z d × B Z d × Z m ,and we define the representation π of the group G as follows: π ( k, l, j ) = T k M l e πijm (9) Lemma 5.1.
Let B ∈ GL ( d, Q ) . If π is admissible then there exists a function φ ∈ L (cid:0) R d (cid:1) such that given h ∈ L (cid:0) R d (cid:1) , X T k M l ∈ G |h h, T k M l φ i| = k h k . That is π ( G ) φ is Parseval Gabor frame .Proof.
Let h ∈ L (cid:0) R d (cid:1) . If π is admissible, and if f is an admissible vector, X T k M l e πiθ ∈ G (cid:12)(cid:12)(cid:10) h, T k M l e πiθ f (cid:11)(cid:12)(cid:12) = X T k M l X e πiθ ∈ [ G,G ] |h h, T k M l f i| = X T k M l G, G ]) |h h, T k M l f i| = X T k M l (cid:12)(cid:12)(cid:12)D h, T k M l G, G ]) / f E(cid:12)(cid:12)(cid:12) = k h k . Thus, the statement of the lemma holds by replacing
G, G ]) / f with φ. (cid:3) Using the procedure provided in Remark 3.8, we construct an invertible matrix A withintegral entries such that B − tr Z d ∩ Z d = A Z d . Thus, the unitary dual of N is isomorphic tothe commutative group c R d A − tr Z d × c R d B − tr Z d × c Z m . Letting Λ ⊂ c R d be a measurable fundamental domain for A − tr Z d and Λ ⊂ c R d a measurablefundamental domain for B − tr Z d , the unitary dual of N is parameterized by the set n ( γ , γ , σ ) : γ ∈ Λ , γ ∈ Λ , σ ∈ c Z m o . Next, the action of a fixed character of N is computed as follows. χ ( γ ,γ ,σ ) (cid:0) T k M l e πiθ (cid:1) = exp [2 πi h γ , k i ] exp [2 πi h γ , l i ] exp [2 πiσθ ] . We recall one important result due to Mackey which is also presented in [15]. We willneed this proposition to compute the unitary dual of the group G. LANCHEREL MEASURE AND ADMISSIBILITY 13
Proposition 5.2.
Let N be a normal subgroup of G. Assume that N is type I and is regularlyembedded. Let π be an arbitrary element of the unitary dual of N. G acts on the unitarydual of N as follows: x · π ( y ) = π (cid:0) x − yx (cid:1) , x ∈ G, y ∈ N. Let G π be the stabilizer group of the G -action on π. b G = [ π ∈ b N/G n Ind GG π ( υ ) : υ ∈ c G π and υ | N is a multiple of π o . Now, we will apply Proposition 5.2 to G = (cid:10) T k , M l : k ∈ Z d , l ∈ B Z d (cid:11) . First, we recall from Lemma 3.11, that
G/N is a finite group, and thus G is a compactextension of an abelian group. Referring to [14] I Chapter 4, G is type I and N is regularlyembedded. Let P ∈ G , k ∈ B − tr Z d ∩ Z d , l ∈ B Z d and let χ ( γ ,γ ,σ ) be a character of N. Wedefine the action of G on the unitary dual of N multiplicatively such that for P ∈ G , P · χ ( γ ,γ ,σ ) (cid:0) T k M l e πiθ (cid:1) = χ ( γ ,γ ,σ ) (cid:0) P − (cid:0) T k M l e πiθ (cid:1) P (cid:1) . (10) Definition 5.3.
We define a measurable map ρ : c R d → Λ such that ρ ( x ) = y x if and only y x is the unique element in Λ such that x = y x + l, where l ∈ B − tr Z d . Since the collection of sets (cid:8) Λ + j : j ∈ B − tr Z d (cid:9) is a measurable partition of c R d then the map ρ makes sense. Lemma 5.4.
For s ∈ Z d , l ∈ B Z d and e πiθ ∈ [ G, G ] we have (a) T s · χ ( γ ,γ ,σ ) = χ ( γ ,ρ ( γ + σs ) ,σ ) (b) M l · χ ( γ ,γ ,σ ) = χ ( γ ,γ ,σ ) (c) e πiθ · χ ( γ ,γ ,σ ) = χ ( γ ,γ ,σ ) Proof.
Since e πiθ is a central element of G then clearly part (c) is correct. Now, for part(a), let T k , M l , such that k ∈ B − tr Z d ∩ Z d , l ∈ B Z d and s ∈ Z d T s · χ ( γ ,γ ,σ ) (cid:0) T k M l e πiθ (cid:1) = χ ( γ ,γ ,σ ) (cid:0) T − s (cid:0) T k M l e πiθ (cid:1) T s (cid:1) = χ ( γ ,γ ,σ ) (cid:0) T k M l e πi ( θ + h l,s i ) (cid:1) = χ ( γ ,γ + sσ,σ ) (cid:0) T k M l e πiθ (cid:1) . Thus, T s · χ ( γ ,γ ,σ ) = χ ( γ ,ρ ( γ + σs ) ,σ ) . For part (b), since k ∈ B − tr Z d ∩ Z d , l ∈ B Z d χ ( γ ,γ ,σ ) (cid:0) M − l (cid:0) T k M l e πiθ (cid:1) M l (cid:1) = χ ( γ ,γ ,σ ) (cid:0) M − l M l T k M l e πiθ e πi ( h l,k i ) (cid:1) = χ ( γ ,γ ,σ ) (cid:0) T k M l e πiθ (cid:1) . (cid:3) Lemma 5.5.
The stabilizer group of a fixed character χ ( γ ,γ ,σ ) in the unitary dual of N isgiven by G ( γ ,γ ,σ ) = (cid:28) T k M l e πiθ ∈ G | σk = B − tr j for some j ∈ Z d l ∈ B Z d , e πiθ ∈ [ G, G ] (cid:29) (11) LANCHEREL MEASURE AND ADMISSIBILITY 14
Thanks to (11), we may write G ( γ ,γ ,σ ) = A ( σ ) Z d × B Z d × [ G, G ] where A ( σ ) is a full-rankmatrix of order d and A ( σ ) Z d ≥ A Z d . For a fixed element ( γ , γ , σ ) in the unitary dual of N, we define the set U σ = Λ ( σ ) × Λ × \ [ G, G ] (12)where Λ ( σ ) is a fundamental domain for c R d A ( σ ) − tr Z d such that Λ is contained in Λ ( σ ) . Sincewe need Proposition 5.2 to compute the unitary dual of G, we would like to be able to assertthat if χ ( γ ,γ ,σ ) is a character of the group N, it is possible to extend χ ( γ ,γ ,σ ) to a characterof the stabilizer group G ( γ ,γ ,σ ) . However, we will need a few lemmas first.
Lemma 5.6.
Let λ = ( η, γ , σ ) ∈ U σ such that η = γ + A − tr j for some j ∈ Z d . If χ ( γ + A − tr j,γ ,σ ) is a character of G ( γ ,γ ,σ ) , then χ ( γ + A − tr j,γ ,σ ) (cid:0) T k M l e πiθ (cid:1) = χ ( γ ,γ ,σ ) (cid:0) T k M l e πiθ (cid:1) Proof.
Let T k M l e πiθ ∈ N. Since k ∈ A Z d , there exists k ′ ∈ Z d such that k = Ak ′ χ λ (cid:0) T k M l e πiθ (cid:1) = exp (2 πi h γ , k i ) exp (2 πi h j, k ′ i ) exp (2 πi h γ , l i ) e πiσθ = exp (2 πi h γ , k i ) exp (2 πi h γ , l i ) e πiσθ = χ ( γ ,γ ,σ ) (cid:0) T k M l e πiθ (cid:1) . (cid:3) Lemma 5.7.
For a fixed ( γ , γ , σ ) ∈ Λ × Λ × c Z m in the unitary dual of N, if λ = ( η, γ , σ ) ∈U σ , η = γ + A − tr j for some j ∈ Z d then χ λ (cid:2) G ( γ ,γ ,σ ) , G ( γ ,γ ,σ ) (cid:3) = 1 . Proof.
Let T k M l τ ∈ G ( γ ,γ ,σ ) . It suffices to check that χ λ (cid:0) T k M l τ T − k M − l τ − (cid:1) = 1where τ ∈ [ G, G ] . First, we observe that χ λ (cid:0) T k M l τ T − k M − l τ − (cid:1) = χ λ (cid:0) T k M l T − k M − l (cid:1) = exp [ − πi h σk, l i ] . Applying Lemma 5.5, since T k ∈ G ( γ ,γ ,σ ) there exists some p ∈ Z d such that χ λ (cid:0) T k M l τ T − k M − l τ − (cid:1) = exp (cid:2) πiσ (cid:10) B − tr p, l (cid:11)(cid:3) . Secondly, using the fact that l ∈ B Z d there exists l ′ ∈ Z d such that χ λ (cid:0) T k M l τ T − k M − l τ − (cid:1) = exp (cid:2) πiσ (cid:10) B − tr p, Bl ′ (cid:11)(cid:3) = 1 . (cid:3) The following lemma allows us to establish the extension of characters from the normalsubgroup N to the stabilizer groups. Lemma 5.8.
Fix ( γ , γ , σ ) in the unitary dual of N. Let λ = ( η, γ , σ ) ∈ U σ . Then χ λ defines a character on G ( γ ,γ ,σ ) .Proof. Given T k M l e πiθ , and T k M l e πiθ ∈ G ( γ ,γ ,σ ) , it is easy to see that (cid:0) T k M l e πiθ (cid:1) (cid:0) T k M l e πiθ (cid:1) = T k + k M l + l e πi ( θ + θ ) e πi h l ,k i LANCHEREL MEASURE AND ADMISSIBILITY 15 where e πi h l ,k i ∈ (cid:2) G ( γ ,γ ,σ ) , G ( γ ,γ ,σ ) (cid:3) . We want to show that χ λ defines a homomorphismfrom G ( γ ,γ ,σ ) into the circle group. Since χ λ (cid:2) G ( γ ,γ ,σ ) , G ( γ ,γ ,σ ) (cid:3) = 1 then χ λ (cid:0)(cid:0) T k M l e πiθ (cid:1) (cid:0) T k M l e πiθ (cid:1)(cid:1) = χ λ (cid:0) T k T k M l M l e πiθ e πiθ e πi h l ,k i (cid:1) = χ λ (cid:0) T k + k M l + l e πi ( θ + θ + h l ,k i ) (cid:1) = exp (2 πi h η, k + k i ) exp (2 πi h γ , l + l i ) e πiσ ( θ + θ ) e πi ( σ h l ,k i ) From Lemma 5.7, e πi ( σ h l ,k i ) = 1 and χ λ (cid:0)(cid:0) T k M l e πiθ (cid:1) (cid:0) T k M l e πiθ (cid:1)(cid:1) = exp (2 πi h η, k + k i ) exp (2 πi h γ , l + l i ) e πiσ ( θ + θ ) Now, using Lemma 5.6 χ λ (cid:0)(cid:0) T k M l e πiθ (cid:1) (cid:0) T k M l e πiθ (cid:1)(cid:1) = χ λ (cid:0) T k M l e πiθ (cid:1) χ λ (cid:0) T k M l e πiθ (cid:1) . Thus χ λ defines a character on G ( γ ,γ ,σ ) . (cid:3) Remark 5.9.
Fix ( γ , γ , σ ) in the unitary dual of N. Let η = γ + A − tr j ∈ U σ where j ∈ Z d . The character χ ( η,γ ,σ ) is called an extension of χ ( γ ,γ ,σ ) from N to the stabilizergroup G ( γ ,γ ,σ ) . Proposition 5.10.
The unitary dual of G is parameterized by the set Σ = [ λ ∈ Ω c G λ where Ω = [ σ ∈ \ [ G,G ] (Λ × E σ × { σ } ) , and E σ is a measurable subset of Λ satisfying the condition [ s ∈ Z d ρ ( E σ + σs ) = Λ . Proof.
Fixing σ ∈ \ [ G, G ] , recall that G · ( γ , γ , σ ) = (cid:8) ( γ , ρ ( γ + σs ) , σ ) : s ∈ Z d (cid:9) . For a fixed σ ∈ \ [ G, G ] , we pick a measurable set E σ ⊂ Λ such that [ s ∈ Z d ρ ( E σ + σs ) = Λ . (13)The set Ω = [ σ ∈ \ [ G,G ] (Λ × E σ × { σ } )parameterizes the orbit space b N /G.
Thus, following Mackey’s result (see Proposition 5.2),the unitary dual of G is parametrized by the setΣ = [ ( γ ,γ ,σ ) ∈ Ω \ G ( γ ,γ ,σ ) . (cid:3) LANCHEREL MEASURE AND ADMISSIBILITY 16
Following the description given in [14], Section 4, let χ ( γ ,γ ,σ ) ∈ b N and let χ j ( γ ,γ ,σ ) = χ ( γ + A − tr j,γ ,σ ) be its extended representation from N to G ( γ ,γ ,σ ) . Let ζ be an irreduciblerepresentation of G ( γ ,γ ,σ ) /N and define its lift to G ( γ ,γ ,σ ) which we denote by ζ . For a fixed( γ , γ , σ, ζ ) we define the representation ρ ( γ ,γ ,σ,ζ ) = Ind GG ( γ ,γ ,σ ) (cid:16) χ j ( γ ,γ ,σ ) ⊗ ζ (cid:17) acting in the Hilbert space H ( γ ,γ ,σ,ζ ) = u : G → C : u ( T k P ) = h(cid:16) χ j ( γ ,γ ,σ ) ⊗ ζ (cid:17) ( P ) i − u ( T k ) , where P ∈ G ( γ ,γ ,σ ) (14)which is naturally identified with l (cid:0) G/G ( γ ,γ ,σ ) (cid:1) ∼ = C card ( G/G ( γ ,γ ,σ ) ) . The inner product on H ( γ ,γ ,σ,ζ ) is given by h u , v i H ( γ ,γ ,σ,ζ ) = X P G ( γ ,γ ,σ ) ∈ G/G ( γ ,γ ,σ ) u ( P ) v ( P )Notice that the number of elements in G/G ( γ ,γ ,σ ) is bounded above by the order of thefinite group G/N ∼ = Z d A Z d which has precisely | det A | elements. Remark 5.11.
Every irreducible representation of G is monomial. That is, every irreduciblerepresentation of G is induced by a one-dimensional representation of some subgroup of G. Given u ∈ H ( γ ,γ ,σ,ζ ) ,ρ ( γ ,γ ,σ,ζ ) (cid:0) T k M l e πiθ (cid:1) u ( T s ) = u (cid:16)(cid:0) T k M l e πiθ (cid:1) − T s (cid:17) and u (cid:16)(cid:0) T k M l e πiθ (cid:1) − T s (cid:17) is computed by following the rules defined in (14) where T s ∈ n T k , · · · , T k | det A | o and (cid:8) k + A Z d , · · · , k | det A | + A Z d (cid:9) is a set of representative elements of the quotient group Z d A Z d . The lemma above is a standard computation of an induced representation. The interestedreader is referred to [5]Next, we define the setΣ σ = ( Ind GG ( γ ,γ ,σ ) (cid:16) χ j ( γ ,γ ,σ ) ⊗ ζ (cid:17) : ζ ∈ \ G ( γ ,γ ,σ ) /N ( γ , γ , σ ) ∈ Λ × E σ × { σ } ) . LANCHEREL MEASURE AND ADMISSIBILITY 17
A more convenient description of the unitary dual of G which will be helpful when wecompute the Plancherel measure is Σ = [ σ ∈ \ [ G,G ] Σ σ . (15)Now that we have a complete description of the unitary dual of the group G , we willprovide a computation of its Plancherel measure.Theorem 5.12.
The Plancherel measure is a fiber measure which is given by dµ (cid:0) ρ ( γ ,γ ,σ,ζ ) (cid:1) = dm ( γ ) dm ( γ ) dm ( σ ) dm ( ζ ) | det A | − m ( E σ ) . The measures dm , dm are the canonical Lebesgue measures defined on Λ and Λ respec-tively. The measure dm is the counting measure on the unitary dual of the commutator [ G, G ] and dm is the counting measure on the dual of the little group \ G ( γ ,γ ,σ ) /N withweight (cid:0) G ( γ ,γ ,σ ) : N (cid:1) . Moreover if L is the left regular representation of G, the direct inte-gral decompositon of L into irreducible representations of G is Z ⊕ Σ ρ ( γ ,γ ,σ,ζ ) ⊗ C n ( γ ,γ ,σ,ζ ) dµ (cid:0) ρ ( γ ,γ ,σ,ζ ) (cid:1) acting in Z ⊕ Σ C n ( γ ,γ ,σ,ζ ) ⊗ C n ( γ ,γ ,σ,ζ ) dµ (cid:0) ρ ( γ ,γ ,σ,ζ ) (cid:1) and n ( γ , γ , σ, ζ ) = card (cid:0) G/G ( γ ,γ ,σ ) (cid:1) . Proof.
The theorem above is an application of the abstract case given in [14] II (Theorem2.3), and the precise weight of the Plancherel measure is obtained by some normalization. (cid:3)
Let us suppose that B is in GL ( d, Q ) − GL ( d, Z ) . Let ϕ be any type I representation of G. Then, there is a unique measure, µ defined on the spectral set Σ such that ϕ is unitarilyequivalent to Z ⊕ Σ ρ ( γ ,γ ,σ,ζ ) ⊗ C m ( γ ,γ ,σ,ζ ) dµ (cid:0) ρ ( γ ,γ ,σ,ζ ) (cid:1) (16)acting in Z ⊕ Σ C n ( γ ,γ ,σ,ζ ) ⊗ C m ( γ ,γ ,σ,ζ ) dµ (cid:0) ρ ( γ ,γ ,σ,ζ ) (cid:1) where m is the multiplicity function of the irreducible representations occurring in the de-composition of ϕ. Proposition 5.13.
The representation ϕ is admissible if and only if (a) dµ (cid:0) ρ ( γ ,γ ,σ,ζ ) (cid:1) is absolutely continuous with respect to the Plancherel measure of G. (b) m ( γ , γ , σ, ζ ) ≤ card (cid:0) G/G ( γ ,γ ,σ ) (cid:1) . Proof.
See [7] page 126 (cid:3)
Remark 5.14. If B is in GL ( d, Q ) − GL ( d, Z ) and | det B | ≤ and the Gabor representa-tion π (9) is unitarily equivalent to (16) then π is admissible and the Proposition above isapplicable.Proof. This remark is just an application of the density condition given in Lemma 2.6. (cid:3)
LANCHEREL MEASURE AND ADMISSIBILITY 18 Non-Type I Groups
In general if B has at least one non-rational entry, then the commutator subgroup of G isa infinite abelian group. In this section, we will consider the case where d = 1 , and G = h T k , M l | k ∈ Z , l ∈ α Z i where α is irrational positive number. Unfortunately, the Mackey machine will not beapplicable here, and we will have to rely on different techniques to obtain a decompositionof the left regular representation in this case. Let H be the three-dimensional connected,simply connected Heisenberg group. We defineΓ = P l,m,k = m l k : ( m, k, l ) ∈ Z . Lemma 6.1.
There is a faithful representation Θ α of Γ such that Θ α ( P l, , ) = χ α ( l ) , Θ α ( P ,m, ) = T m , and Θ α ( P , ,k ) = M kα . Proof. Θ α is the restriction of an irreducible infinite-dimensional representation of the Heisen-berg group [3] to the lattice Γ . Sinceker Θ α = the representation Θ α is clearly faithful. (cid:3) Thus, G ∼ = Γ via Θ α and for our purpose, it is more convenient to work with Γ . We defineΓ = (cid:8) P ,m,k : ( m, k ) ∈ Z (cid:9) . Let L H be the left regular representation of the simply connected, connected Heisenberggroup H = x z y : ( z, y, x ) ∈ R . In fact, it is not too hard to show that Γ is a lattice subgroup of H . Let P be the Planchereltransform of the Heisenberg group. We recall that P : L ( H ) → Z ⊕ R ∗ L ( R ) ⊗ L ( R ) | λ | dλ where the Fourier transform is defined on L ( H ) ∩ L ( H ) by P ( f ) ( λ ) = Z R ∗ π λ ( n ) f ( n ) dn where π λ ( n ) f ( t ) = π λ ( P z,x,y ) f ( t ) = e πizλ e − πiλyt f ( t − x ) , and the Plancherel transform is the extension of the Fourier transform to L ( H ) inducingthe equality k f k L ( H ) = Z R ∗ kP ( f ) ( λ ) k HS | λ | dλ. LANCHEREL MEASURE AND ADMISSIBILITY 19
In fact, || · || HS denotes the Hilbert Schmidt norm on L ( R ) ⊗ L ( R ). Let u ⊗ v and w ⊗ y be rank-one operators in L ( R ) ⊗ L ( R ) . We have h u ⊗ v, w ⊗ y i HS = h u, w i L ( R ) h v, y i L ( R ) . It is well-known that L H ∼ = P◦ L H ◦ P − = Z ⊕ R ∗ π λ ⊗ L ( R ) | λ | dλ, where L ( R ) is the identity operator on L ( R ) , and for a.e. λ ∈ R ∗ , P ( L H ( x ) φ )( λ ) = π λ ( x ) ◦ P φ ( λ ) . Let ( u λ ) λ ∈ [ − , ∪ (0 , be a measurable field of unit vectors in L ( R ) . We define two left-invariant multiplicity-free subspaces of L ( H ) such that H + = P − (cid:18)Z ⊕ (0 , L ( R ) ⊗ u λ | λ | dλ (cid:19) , H − = P − (cid:18)Z ⊕ [ − , L ( R ) ⊗ u λ | λ | dλ (cid:19) and H + , H − are mutually orthogonal. The following lemma has also been proved in moregeneral terms in [16]. However the proof is short enough to be presented again in this section. Lemma 6.2.
The representation ( L H | Γ , H + ) is cyclic and H + admits a Parseval frame ofthe type L H (Γ) f with k f k L ( H ) = . Proof.
Let f, φ ∈ L ( H ). X γ ∈ Γ (cid:12)(cid:12)(cid:12) h φ, L H ( γ ) f i L ( H ) (cid:12)(cid:12)(cid:12) = X γ ∈ Γ (cid:12)(cid:12)(cid:12)(cid:12)Z (0 , hP φ ( λ ) , π λ ( γ ) P f ( λ ) i HS | λ | dλ (cid:12)(cid:12)(cid:12)(cid:12) = X γ ∈ Γ X j ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)Z (0 , e πijλ hP φ ( λ ) , π λ ( γ ) P f ( λ ) i HS | λ | dλ (cid:12)(cid:12)(cid:12)(cid:12) . Since (cid:8) e πijλ : j ∈ Z (cid:9) forms a Parseval frame for L ((0 , , we have X γ ∈ Γ (cid:12)(cid:12)(cid:12) h φ, L H ( γ ) f i L ( H ) (cid:12)(cid:12)(cid:12) = X γ ∈ Γ Z (0 , (cid:12)(cid:12)(cid:12)D P φ ( λ ) , π λ ( γ ) P ( f ) ( λ ) | λ | / E HS (cid:12)(cid:12)(cid:12) | λ | dλ. Let P ( f ) ( λ ) | λ | / = | λ | / w λ ⊗ u λ ∈ L ( R ) ⊗ u λ a.e. Then X γ ∈ Γ (cid:12)(cid:12)(cid:12) h φ, L H ( γ ) f i L ( H ) (cid:12)(cid:12)(cid:12) = Z (0 , X γ ∈ Γ (cid:12)(cid:12)(cid:12)(cid:12)D s ( λ ) , π λ ( γ ) (cid:16) | λ | / w λ (cid:17)E L ( R ) (cid:12)(cid:12)(cid:12)(cid:12) | λ | dλ (17) LANCHEREL MEASURE AND ADMISSIBILITY 20 where P φ ( λ ) = s λ ⊗ u λ . Notice that by definition π λ ( γ ) f = exp (2 πiλjt ) f ( t − k ) where( λj, k ) ∈ λ Z × Z with λ ∈ (0 , . By the density condition given in Lemma 2.6, it is possibleto find v λ such that the system { π λ ( γ ) ( v λ ) : γ ∈ Γ } is a Parseval frame in L ( R ) for a.e. λ ∈ (0 , . So let us suppose that we pick f ∈ H + sothat P ( f ) ( λ ) = | λ | − / v λ ⊗ u λ . We will later see that f is indeed square-integrable. Coming back to (17), X γ ∈ Γ (cid:12)(cid:12)(cid:12) h φ, L H ( γ ) f i L ( H ) (cid:12)(cid:12)(cid:12) = Z (0 , X γ ∈ Γ (cid:12)(cid:12)(cid:12) h s λ , π λ ( γ ) ( v λ ) i L ( R ) (cid:12)(cid:12)(cid:12) | λ | dλ = Z (0 , kP φ ( λ ) k HS | λ | dλ = k φ k L ( H ) . To prove that k f k L ( H ) = , we appeal to Lemma 2.7 and we obtain (cid:13)(cid:13)(cid:13) | λ | − / v λ (cid:13)(cid:13)(cid:13) L ( R ) = | λ | − k v λ k L ( R ) = | λ | − vol ( λ Z × Z )= 1 . The above holds for almost every λ ∈ (0 , kP f ( λ ) k HS = 1 we obtain k f k L ( H ) = Z | λ | dλ = 12 . (cid:3) Similarly, we also have the following lemma
Lemma 6.3.
The representation ( L H | Γ , H − ) is cyclic. Moreover H − admits a Parsevalframe of the type L H (Γ) h and k h k L ( H ) = 12 . Lemma 6.4.
Let H = H + ⊕ H − . Then there exists an orthonormal basis of the type L H (Γ) f for H . We remark that in general the direct sum of two Parseval frames in H, and K is not aneven a Parseval frame for the space H ⊕ K , unless the ranges of the coefficients operatorsare orthogonal to each other. A proof of Lemma 6.4 is given by Currey and Mayeli in [4],where they show how to put together f and h in order to obtain an orthonormal basis for H .Now, we are in a good position to obtain a decomposition of the left regular representationof the time-frequency group. First, let us define K = l k : ( k, l ) ∈ Z . LANCHEREL MEASURE AND ADMISSIBILITY 21
It is easy to see that K is an abelian subgroup of Γ . Let L be the left regular representationof Γ . Theorem 6.5.
A direct integral decomposition of L is obtained as follows Z ⊕ [ − , π λ | Γ | λ | dλ ∼ = Z ⊕ [ − , Z ⊕ [0 , | λ | ) Ind Γ K (cid:0) χ ( | λ | ,t ) (cid:1) | λ | dtdλ (18) acting in the Hilbert space Z ⊕ [ − , Z ⊕ [0 , | λ | ) l ( Z ) | λ | dtdλ Proof.
First, let us define R f : H → l (Γ) such that R f g ( γ ) = h g, L ( γ ) f i . Using Lemma6.4 (see [4] also), we construct a vector f ∈ H such that R f is an unitary. As a result, theoperator R f ◦ P − is unitary and (cid:0) R f ◦ P − (cid:1) ◦ (cid:18)Z ⊕ [ − , π λ | Γ | λ | dλ (cid:19) ( · ) ◦ (cid:0) P◦ R − f (cid:1) = L ( · ) (19)where π λ m l k f ( t ) = e πilλ e πikλt f ( t − m ) . Thus, Z ⊕ [ − , π λ | Γ | λ | dλ ∼ = L. Notice that ( π λ | Γ , L ( R )) is not an irreducible representation. However, we may use Baggett’sdecomposition given in [1]. For each λ ∈ [ − , , the representation π λ is decomposed intoits irreducible components as follows: π λ | Γ ∼ = Z ⊕ [0 , | λ | ) Ind Γ K (cid:0) χ ( | λ | ,t ) (cid:1) dt (20)where χ ( | λ | ,t ) l k = exp (2 πi ( | λ | l + tk )) . Combining (19) with (20), we obtain the decomposition given in (18). (cid:3)
It is worth noticing that in the case where α ∈ R − Q , that the group G is a non-typeI group. Moreover, it is well-known that the left regular representation of G is a non-typeI representation. In fact, (see[13]) the left regular representation of this group is type II.In this case, in order to obtain a useful decomposition of the left regular representation, itis better to consider a new type of dual. Let us recall the following well-known facts (seeSection 3 . . G be a locally compact group. Let ˘ G be the collection of all quasi-equivalence classes of primary representations of G, and let π be a unitary representationof G acting in a Hilbert space H π . Essentially, there exists a unique way to decompose therepresentation π into primary representations such that the center of the commuting algebraof the representation is decomposed as well. This decomposition is known as the centraldecomposition . More precisely, there exist LANCHEREL MEASURE AND ADMISSIBILITY 22 (a) A standard measure ν π on the quasi-dual of G : ˘ G (b) A ν π -measurable field of representations ( ρ t ) t ∈ ˘ G (c) A unitary operator R : H π → Z ⊕ ˘ G H π,t dν π ( t )such that R π ( · ) R − = Z ⊕ ˘ G ρ t ( · ) dν π ( t ) . Moreover, letting Z ( C ( π )) be the center of the commuting algebra of the represen-tation π, then R ( Z ( C ( π ))) R − = Z ⊕ ˘ G C · H π,t dν π ( t ) . The importance of the central decomposition is illustrated through the following facts.Let π, θ be representations of a locally compact group and let ν π and ν θ be the measuresunderlying the respective central decompositions. Then π is quasi-equivalent to a subrepre-sentation of θ if and only if ν π is absolutely continuous with respect to ν θ . In particular, π is quasi-equivalent to θ if and only if the measures ν π and ν θ are equivalent. Furthermore,the representations ν π and ν θ are disjoint if and only if the measures ν π and ν θ are disjointmeasures. Since the central decomposition provides information concerning the commutingalgebra of the left regular representation L , and because such information is crucial in theclassification of admissible representations of Γ then it is important to mention the following. Theorem 6.6. L ∼ = R ⊕ (0 , π λ | Γ ⊗ C | λ | dλ and this decomposition is the central decompositionof the left regular representation of G. Proof.
We have already seen that L ∼ = Z ⊕ [ − , π λ | Γ | λ | dλ and π λ | Γ ∼ = Z ⊕ [0 , | λ | ) Ind Γ K (cid:0) χ ( | λ | ,t ) (cid:1) dt. Therefore given ℓ , ℓ ∈ [ − , − { } , if | ℓ | = | ℓ | then π ℓ | Γ ∼ = π ℓ | Γ . As a result, the leftregular representation of G is equivalent to Z ⊕ (0 , π λ | Γ ⊗ C | λ | dλ (21)Also, it is well-known that the representation π λ | Γ is a primary or a factor representationof G whenever λ is irrational (see Page 127 of [5]). So, the decomposition given in (21) isindeed a central decomposition of L . This completes the proof. (cid:3) References [1] L. Baggett, Processing a radar signal and representations of the discrete Heisenberg group, Colloq.Math. 60/61 (1990) 195–203.[2] M. Bownik, The structure of shift-modulation invariant spaces: the rational case, J. Funct. Anal. 244(2007), 172-219.
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