The density theorem for discrete series representations restricted to lattices

This article considers the relation between the spanning properties of lattice orbits of discrete series representations and the associated lattice co-volume. The focus is on the density theorem, which provides a trichotomy characterizing the existence of cyclic vectors and separating vectors, and frames and Riesz sequences. We provide an elementary exposition of the density theorem, that is based solely on basic tools from harmonic analysis, representation theory, and frame theory, and put the results into context by means of examples.


Introduction
Let G be a second countable locally compact group and let (π, H π ) be an irreducible, square-integrable unitary representation of G, a so-called discrete series representation. For a lattice Γ ⊂ G, we consider the relation between certain spanning properties of lattice orbits of π under a vector g ∈ H π , π(Γ)g = π(γ)g : γ ∈ Γ , (1.1) and the lattice co-volume vol(G/Γ) of Γ, i.e., the volume of a fundamental domain of Γ.
The spanning properties that we consider are the existence of cyclic, separating, frame and Riesz vectors; see Section 3 for the precise definitions.
The notions of cyclic and separating vectors occur primarily in the theory of operator algebras, in particular, von Neumann algebras, and they provide (if they exist) a powerful tool in studying the structure of these algebras. The stronger notions of frames and Riesz sequences, on the other hand, form the core of Gabor and wavelet theory, and are important in applications as they guarantee unconditionally convergent and stable Hilbert space expansions.
The central theorem relating the spanning properties of systems (1.1) and the corresponding lattice co-volume is referred to as the density theorem. Under the assumption that the lattice Γ is an infinite conjugacy class (ICC) group, i.e., any conjugacy class {γγ 0 γ −1 | γ ∈ Γ} for γ 0 ∈ Γ \ {e} has infinite cardinality, the density theorem provides the following trichotomy: Theorem 1.1. Let (π, H π ) be a discrete series representation of a second countable unimodular group G of formal dimension d π > 0. Suppose Γ ⊂ G is an ICC lattice. Then the following assertions hold: (i) If vol(G/Γ)d π < 1, then π| Γ admits a Parseval frame, but neither a separating vector, nor a Riesz sequence; (ii) If vol(G/Γ)d π = 1, then π| Γ admits an orthonormal basis; (iii) If vol(G/Γ)d π > 1, then π| Γ admits an orthonormal system, but not a cyclic vector.
(While d π and vol(G/Γ) depend on the normalization of the Haar measure on G, their product vol(G/Γ)d π does not.) The density theorem characterizes the spanning properties of the lattice orbits (1.1) in terms of the lattice co-volume or its reciprocal, often called the density of the lattice. In the setting of a general unimodular group, the assumption that the lattice is ICC is essential and cannot be omitted -see Example 9.3 below-although a more general version of Theorem 1.1 for possibly non-ICC lattices was obtained by Bekka [10]. The existence claims in Theorem 1.1 are not accompanied by constructions of explicit vectors.
The criteria for the existence of cyclic and separating vectors in Theorem 1.1 are well-known to be consequences of the general theory underlying the so-called Atiyah-Schmid formula [3,4,32], and, for certain classes of representations, also a consequence of Rieffel's work [70,71]. The stronger statements on the existence of Parseval frames (part (i)) and orthonormal bases (part (ii)) can also be obtained by similar techniques as shown by Bekka [10,12]. The statement on orthonormal systems (part (iii)) does not seem to have explicitly occurred in the literature before.
While the interest in the density theorem is broad and manifold, as it encompasses operator algebras, representation theory, mathematical physics, and Gabor and wavelet analysis, the available proofs rely on advanced theory of von Neumann algebras, and may only be accessible to a smaller community of experts. This expository article provides an elementary and self-contained presentation of the density theorem, that is based solely on basic tools from harmonic analysis, representation theory, and frame theory, and should be accessible to an interested non-expert. While almost all methods employed exist in some antecedent form in the different specialized literatures, their particular combination here makes the basic structure underlying the density theorem transparent; see Section 1.3. The elementary arguments in this article fall, however, short of deriving the more general version of Theorem 1.1 by Bekka [10]. We hope that this article motivates the non-specialist to delve deeper into operator-algebraic methods. We also expect that the concrete exposition contributes to the study of quantitative aspects of Theorem 1.1, such as the relation between the distance between vol(G/Γ)d π and the critical value 1, and special qualities of the corresponding cyclic or separating vectors, such as smoothness in the case of Lie groups.
1.1. Context and related work. In the setting of Theorem 1.1, for any non-zero g ∈ H π , the system {π(x)g : x ∈ G} is overcomplete, i.e., it contains proper subsystems that are still complete. The fundamental question as to whether subsystems corresponding to lattices (1.1) remain complete was posed by Perelomov in his group-theoretical approach towards the construction of coherent states [63,65]. In fact, a criterion for the completeness of subsystems of coherent states similar to Theorem 1.1 was posed as a question in [65, p.226] 1 . These criteria have been considered for specific systems and vectors in, e.g., [8,36,47,56,60,64,66,68].
The related question as to whether a system (1.1) is a (discrete) frame is at the core of modern frame theory [20] and has, in particular, a long history in Gabor theory [39]. The existence of a frame vector is also studied in representation theory, in whose jargon such a vector is called admissible [26,27]. While the mere existence of a frame or Riesz vector for a given lattice is quite different from the validity of these properties for one specific vector, there is an interesting interplay between the two problems. In Section 1 Perelomov uses the term coherent state with a slightly different meaning, as systems are not the full orbit of a group representation, but parametrized by a homogeneous space to eliminate redundancies. See [57,58] for the relation between the two notions. 9 we discuss a selection of examples, including one where Theorem 1.1 yields seemingly unnoticed consequences.
1.2. Projective versions. The density theorem can also be formulated for projective unitary representations [10,31,37,74], and allows for applications to representations that are square-integrable only modulo a central subgroup (as in the case of nilpotent or reductive Lie groups). The proofs that we present work transparently for projective representations and we formulate the main results in that generality in Theorem 8.1. In the projective setting, the lattice is not assumed to be ICC, but is assumed to satisfy the weaker Kleppner condition [48], a compatibility condition between the lattice and the cocycle of the projective representation. The projective formulation greatly simplifies the treatment of concrete examples such as weighted Bergman spaces and Gabor systems in Section 9.
1.3. Technical comments. The common approach to the density theorem is through the coupling theory of von Neumann algebras, and a self-contained presentation in this spirit can be found in [12,32]. Although we make no explicit reference to the coupling theory, some of the arguments we give are simplifications of standard results, as we point out throughout the text. Most significantly, we circumvent certain technicalities associated with the so-called trace of a group von Neumann algebra. In finding elementary arguments, we benefited particularly from reading [2,18,27,51,72].
An important simplification in the proof of Theorem 1.1 occurs in the derivation of the necessity of the volume or density conditions for cyclicity and separateness, which also play an essential role in deriving the existence of frame and Riesz vectors. Our argument is inspired by Janssen's "classroom proof" of the density theorem for Gabor frames [42], and underscores the power of frame-theoretic methods. In this article such argument is pushed further to yield consequences for cyclicity and separateness. While the necessity of the density conditions for frames and Riesz sequences is an active field of research [6,28,55], most abstract results are not applicable to groups of non-polynomial growth. It is therefore remarkable that the particular lattice structure of the systems in question (1.1) leads to simple and conclusive results.

Preliminaries
Throughout the article, the locally compact group G is assumed to be second countable and unimodular. We fix a Haar measure µ G on G. Some of the notions below depend on this normalization, but the main results do not.
In this case, the map σ in (ii) is uniquely determined and it is a cocycle. A projective unitary representation with cocycle σ is called a σ-representation.
Common examples of a representation space H π are Hilbert spaces of real-variable or complex-variable functions; see Section 9 for a detailed discussion of some examples.

2.2.
Square-integrable σ-representations. Let (π, H π ) be a σ-representation of G. For f, g ∈ H π , the associated matrix coefficient is defined by C g f (x) = f, π(x)g for x ∈ G. The σ-representation (π, H π ) is called square-integrable if there exists a norm dense subspace D ⊂ H π such that C g f = f, π(·)g ∈ L 2 (G), f ∈ H π , g ∈ D. (2.1) The σ-representation (λ σ G , L 2 (G)) given by is called the σ-regular representation and satisfies the covariance property or intertwining property: 2) for all f ∈ H π , g ∈ D.
A σ-representation (π, H π ) is called irreducible if the only closed π(G)-invariant subspaces of H π are {0} and H π and is said to be a discrete series σ-representation if it is both square-integrable and irreducible.
Given a discrete series σ-representation (π, H π ), there exists a unique number d π > 0, called the formal dimension of π, such that the orthogonality relations hold for all f 1 , f 2 , g 1 , g 2 ∈ H π .
The formal dimension d π > 0 depends on the choice of Haar measure on G, and in certain concrete settings, such as real Lie groups, it can be explicitly computed. The book [17] treats nilpotent Lie groups while [49,59] treats semisimple Lie groups. Explicit expressions of d π for the simplest examples of such groups are also provided in Section 9.
2.3. Fundamental domains and lattices. Let Γ ⊆ G be a discrete subgroup. A left (resp. right) fundamental domain of Γ in G is a Borel set Ω ⊆ G satisfying G = Γ · Ω and γΩ ∩ γ ′ Ω = ∅ (resp. G = Ω · Γ and Ωγ ∩ Ωγ ′ = ∅) for all γ, γ ′ ∈ Γ with γ = γ ′ . If Ω is a left (resp. right) fundamental domain, then Ω −1 is a right (resp. left) fundamental domain. The discrete subgroup Γ ⊆ G is called a lattice if it admits a left (or right) fundamental domain of finite measure. Equivalently, a discrete subgroup Γ is a lattice if and only if the quotient G/Γ admits a finite G-invariant regular Borel measure. Any two fundamental domains have the same measure, and thus, we may define the covolume of Γ as vol(G/Γ) := µ G (Ω). This depends of course on the choice of the Haar measure for G.
Standard examples of lattices are Z d ⊆ R d and SL(2, Z) ⊆ SL(2, R). The lattice Z d is co-compact in R d , i.e., R d /Z d is compact, while SL(2, Z) is not co-compact in SL(2, R).
See [67] and [11,Appendix B] for more on lattices and fundamental domains.
2.6. Partial isometries and the polar decomposition. Let H and K be complex Hilbert spaces. A bounded linear operator U : K → H is called a partial isometry if U is an isometry when restricted to the orthogonal complement N (U) ⊥ of its null space N (U). The subspace N (U) ⊥ is called the initial space of U and the range R(U) of U is the final space of U, i.e., the image of N (U) ⊥ under the isometry U| N (U ) ⊥ A linear operator T : dom(T ) ⊂ H → K is densely defined if its domain dom(T ) is a norm dense subspace in H and is called closed if its graph G(T ) := {(f, T f ) | f ∈ H} is closed in H ⊕ K. For a closed, densely defined linear operator T : dom(T ) ⊂ H → K, its adjoint is denoted by T * and its modulus by |T | := (T * T ) 1/2 . The operator |T | is defined by Borel functional calculus and has domain dom(|T |) = dom(T ). The polar decomposition of T is uniquely given by where U T : H → K is a partial isometry with initial space N (T ) ⊥ = R(|T |) and final space R(T ). For more details and background, see, e.g., [24, VI, Section 13].
Intuitively, a vector g ∈ H π is cyclic if the corresponding orbit π(Γ)g is rich enough so as to provide approximations for every vector in H π . On the other hand, if g is separating for π(Γ) ′′ , then π(Γ) ′′ cannot be too rich, because π(Γ) ′′ ∋ T → T g ∈ H π is injective.
The central question of this article is the relation between the existence of cyclic and separating vectors on the one hand, and the co-volume of Γ within a larger group G. As a key tool, we consider certain strengthened notions of cyclicity and separation.

3.2.
Frames and Riesz sequences. A system π(Γ)g is called a frame for H π if there exist constants A, B > 0, called frame bounds, such that the following frame inequalities hold: A vector g is a frame vector if π(Γ)g is a frame. A system π(Γ)g forming a frame is complete by the first (lower) bound in (3.1). The second of the frame inequalities (upper bound), is known as a Bessel bound. A vector g satisfying (3.2) is a Bessel vector. Note that the definition concerns π(Γ)g as an indexed family. Two indexations of the same underlying set can have, for example, different frame bounds. The frame bounds of a given frame and indexation are of course not unique.
The Bessel condition (3.2) is equivalent to the frame operator S g,Γ : H π → H π , S g,Γ f = γ∈Γ f, π(γ)g π(γ)g being well-defined and bounded. The full two-sided frame inequality (3.1) is equivalent to the frame operator being a positive-definite (bounded, invertible) operator on H π . A frame π(Γ)g for which the frame bounds can be chosen as A = B = 1 is called a Parseval frame, because it gives the identity Equivalently, π(Γ)g is a Parseval frame for H π if and only if its frame operator S g,Γ is the identity on H π . Whenever well-defined and bounded, the frame operator S g,Γ commutes with π(γ) for all γ ∈ Γ.
Remark 3.1 (Turning a frame into a Parseval one). An arbitrary frame π(Γ)g can be turned into a Parseval frame by consideringg := S −1/2 g,Γ g. Indeed, if π(Γ)g is a frame, then S g,Γ is a positive operator, and, therefore,g is well-defined. Moreover, since S −1/2 g,Γ also commutes with each π(γ), for f ∈ H π , showing that π(Γ)g is a Parseval frame for H π .
A duality argument, shows that a Riesz sequence satisfies the Bessel bound (3.2). Moreover, a Riesz sequence is linearly independent and ω-independent, and hence cannot admit repetitions. A vector g yielding a Riesz sequence π(Γ)g is a Riesz vector.
A complete Riesz sequence π(Γ)g is called a Riesz basis for H π . Equivalently, a system π(Γ)g is a Riesz basis for H π if it is the image of an orthonormal basis under a bounded, invertible operator on H π . If π(Γ)g is a Riesz basis for H π , then π(Γ)g and π(Γ)S −1 It will be shown in Proposition 5.2 that, under Kleppner's condition, if π(Γ)g is a Riesz sequence, then g is separating for π(Γ) ′′ .

Bounded operators and Bessel vectors.
The coefficient operator and reconstruction operator associated with π(Γ)g are given respectively by and where c 00 (Γ) ⊆ C Γ denotes the space of finite sequences on Γ.
Recall that π(Γ)g is called a Bessel sequence if there exists B > 0 such that (3.2) holds. In this case, the coefficient operator is well-defined and bounded as a map from H π into ℓ 2 (Γ), and its adjoint D g,Γ is well-defined and bounded from ℓ 2 (Γ) into H π .
The space of Bessel vectors is denoted by B π . The assumption that (π, H π ) is squareintegrable in the sense of (2.1), together with the uniform boundedness principle, yields that the space B π is norm dense in H π .

Coefficient and reconstruction as unbounded operators.
In the sequel, we treat the coefficient mapping (3.3) and reconstruction mapping (3.4) as operators from domains and on images in which they do not necessarily act as bounded operators.
The reconstruction operator D g,Γ , with domain is given by D g,Γ c = f and well-defined from dom(D g,Γ ) into H π , where f ∈ H π is the vector occurring in the domain definition (3.5). Note that f is uniquely determined since B π is a dense subspace in H π .
For simplicity, we also sometimes write the series is however a formal expression for the vector f in (3.5).
The following result provides basic properties of the (possibly) unbounded coefficient and reconstruction operators. Proposition 3.3. Let (π, H π ) be a square-integrable σ-representation of a countable discrete group Γ. Let g ∈ H π be an arbitrary vector.
(ii) Note that the map D g,Γ is densely defined since the space of finite sequences Thus c ∈ dom(D g,Γ ) and D g,Γ c = f , which shows that D g,Γ is a closed operator.
Remark 3.4. For a general frame {f i } i∈I in an abstract Hilbert space H, the coefficient operator f → ( f, f i ) i∈I is always closed, but not necessarily densely defined, on its canonical domain. The reconstruction operator (c i ) i∈I → i∈I c i f i may fail to be closed on the domain c = (c i ) i∈I ∈ ℓ 2 (I) : i∈I c i f i converges in the norm of H , see [14]. Crucially, in (3.5) and part (ii) of Proposition 3.3, we define the series in a suitably weak form.

Uniqueness for the extended representation. Given
Note that π(c) = γ∈Γ c γ π(γ) is well-defined since the series representing π(c)g converges unconditionally in H π by the Bessel property.
In the notation of (3.6), conjugating the operator π(c) simply corresponds to (twisted) conjugation of the corresponding sequence c. Lemma 3.5. Let (π, H π ) be a square-integrable σ-representation of a countable discrete group Γ. Let c ∈ ℓ 2 (Γ). Then, for all γ ∈ Γ, holds for any γ ′ ∈ Γ. Therefore, where the second equality follows from the change of variable Under Kleppner's condition (see Section 2.4), we have the following important uniqueness result.
Proof. The proof is divided into four steps.
Step 2. (Minimal fixed point). Let c ∈ K be arbitrary and consider the norm-closed convex hull co(ϑ(Γ)c) in the Hilbert space K. Then there exists a unique d ∈ co(ϑ(Γ)c) of minimal norm. By uniqueness, the vector d must be ϑ(Γ)-invariant, that is, Therefore, |d| is constant on conjugacy classes.
Step 5. (Conclusion). The above shows that for an arbitrary c ∈ K, we have 0 ∈ co(ϑ(Γ)c). Since (ϑ(γ)c) e = c e for all γ ∈ Γ, it follows that c e = 0 e = 0. The This completes the proof.
Step 2 in the proof of Proposition 3.6 is an application of the minimal method for ergodic theorems [2, Section 10].
For the Heisenberg projective representation (π, L 2 (R d )) of a lattice Γ ≤ R 2d (see Section 9.2), an alternative proof for the uniqueness result of Proposition 3.6 can be given using the uniqueness of coefficients in Fourier series, see [34,Proposition 3.2]. In that setting, the statement of Proposition 3.6 is true even without Kleppner's condition, while in general it is not, e.g., for Γ = SL(2, Z) and a holomorphic discrete series representation π of SL(2, R), cf. Example 9.3.

Improving spanning properties
4.1. Mackey-type version of Schur's lemma. We will repeatedly use the following folklore result.
Mackey-type versions of Schur's lemma for representations of * -algebras can be found in [24].

4.2.
From cyclic vectors to Parseval frames. We show the existence of Parseval frames π(Γ)g whenever π admits a complete vector.
Proposition 4.2. Let (π, H π ) be a square-integrable σ-representation of a countable discrete group Γ. Let h ∈ H π be arbitrary. Then there exists g ∈ H π such that π(Γ)g is a Parseval frame for [π(Γ)h]. In particular, if π is cyclic, then there exists a Parseval frame π(Γ)g for H π .
Proof. We split the proof into two steps.

Expansions in the von Neumann algebra
5.1. Expansions. The following theorem provides, under Kleppner's condition, a Fouriertype series expansion for every operator in π(Γ) ′′ .

5.2.
Coherent Riesz sequences are generated by separating vectors. As a first application of Theorem 5.1, we show the following.

Doubly invariant subspaces.
As a second application of Theorem 5.1, we show that H π does not admit so-called doubly invariant subspaces.
Proof. Consider the orthogonal projection P K : H π → H π onto K. Since K is π(Γ) ′invariant, it follows that P K ∈ π(Γ) ′′ by the projection lemma. Theorem 5.1 then yields a unique sequence c ∈ ℓ 2 (Γ) such that as an operator on B π . Since K is also π(Γ)-invariant, it follows also that P K ∈ π(Γ) ′ . Therefore P K = π(γ)P K π(γ) * for all γ ∈ Γ. By Lemma 3.5, The uniqueness of the expansion (5.6) shows that for all γ, γ ′ ∈ Γ. Thus |c| is constant on conjugacy classes. We now use Kleppner's condition together with the fact that c ∈ ℓ 2 (Γ), as in Steps 3 and 4 of the proof of Proposition 3.6, to conclude that c γ = 0, for γ ∈ Γ \ {e}. This shows that either P K = 0 or P K = I Hπ , as claimed.

Existence of cyclic or separating vectors
In this section we investigate how to produce large cyclic subspaces for π(Γ). As a first step, we investigate when the sum of two orthogonal cyclic subspaces, [π(Γ)g 1 ] and [π(Γ)g 2 ] is again cyclic. The following key lemma shows that this is the case, provided that the corresponding cyclic subspaces generated by the commutant algebra π(Γ) ′ , i.e., [π(Γ) ′ g 1 ] and [π(Γ) ′ g 2 ], are also orthogonal. Lemma 6.1. Let (π, H π ) be a square-integrable σ-representation of a countable discrete group Γ. Suppose (g k ) k∈I is a countable family of unit-norm vectors g k ∈ H π satisfying the following simultaneous orthogonality conditions Let a ∈ ℓ 1 (I) with a k = 0 for all k ∈ I, and set g := k∈I a k g k . Then For the other inclusion, let k ∈ I, and note that the projection P K k onto K k = [π(Γ) ′ g k ] is in π(Γ) ′′ as [π(Γ) ′ g k ] is π(Γ) ′ -invariant. Therefore g k = a k −1 P K k g ∈ [π(Γ)g] for all k ∈ I. This gives (6.3). The identity (6.4) follows similarly, interchanging the roles of π(Γ) ′ and π(Γ) ′′ . Proposition 6.2. Let (π, H π ) be a square-integrable σ-representation of a countable discrete group Γ. Suppose that (Γ, σ) satisfies Kleppner's condition. Then π admits a cyclic vector or π(Γ) ′′ admits a separating vector (possibly both).
Proof. By Zorn's Lemma, we can select a family (g k ) k∈I of unit-norm vectors g k ∈ H π satisfying the simultaneous orthogonality conditions (6.1) and (6.2), and maximal with respect to that property. The set I is countable because H π is assumed to be separable.
Let g := k∈I a k g k be as in Lemma 6.1, so that (6.3) and (6.4) hold.

Discrete series representations restricted to lattices
Let G be a second countable unimodular group and let Γ ⊂ G be a lattice subgroup. Let (π, H π ) be a discrete series σ-representation of G, i.e., irreducible and squareintegrable. This section is devoted to orbits of the restriction π| Γ of (π, H π ) to Γ, i.e., π(Γ)g = π(γ)g : γ ∈ Γ for some g ∈ H π .
In order to apply the results obtained in the previous sections, it is essential that the restriction π| Γ be square-integrable in the sense of (2.1). The following observation guarantees this. Lemma 7.1. Let Γ ⊆ G be a lattice and let (π, H π ) be a discrete series σ-representation of G. The Bessel vectors B π of the restriction π| Γ are norm dense in H π .
Proof. Using the orthogonality relations (2.3), choose η ∈ H π such that the map C η : H π → L 2 (G) is an isometry. Let P K : L 2 (G) → L 2 (G) be the orthogonal projection onto the closed subspace K := C η (H π ), so that P K ∈ λ σ G (G) ′ . It suffices to show that the space of Bessel vectors of λ σ G | Γ is norm dense in K, since, if λ σ G (Γ)F is Bessel in K, then the unitary map C * η : K → H π produces a Bessel system in H π , namely π(Γ)C * η F = C * η λ σ G (Γ)F . To show that the space of Bessel vectors of λ σ G | Γ is norm dense in K, let Ω ⊆ G be a left fundamental domain for Γ ⊆ G and consider the collection (Here span denotes the set of finite C-linear combinations.) Since the sets {γΩ : γ ∈ Γ} have disjoint supports, any F ′ ∈ L 2 (G) can be written as F ′ = γ∈Γ F ′ · χ γΩ , where χ γΩ is the indicator function of γΩ and the series is norm convergent by orthogonality. Hence, S Ω is norm dense in L 2 (G). Therefore, the image space P K S Ω is dense in K, and it remains to show that P K S Ω consists of Bessel vectors for λ σ G | Γ . For this, note that if F ∈ S Ω is such that supp F ⊆ γΩ for some γ ∈ Γ, then the family λ σ G (Γ)F is orthogonal in L 2 (G), and thus λ σ G (Γ)P K F = P K λ σ G (Γ)F is a Bessel sequence in K. Taking finite linear combinations, it follows that any element of P K S Ω is a Bessel vector for λ σ G | Γ , with a finite Bessel constant depending on the coefficients. This completes the proof. 7.1. Frame bounds and density. The following proposition relates frame bounds (3.1), formal dimension and co-volume.
for f ∈ H π . This, together with the orthogonality relations (2.3), yields Thus, if π(Γ)g is Bessel with bound B, then d −1 π f 2 Hπ g 2 Hπ ≤ B Ω f 2 Hπ dµ G (x), which shows the upper bound in (7.1). The desired lower bound is proven similarly.
Remark 7.3. The proof of Proposition 7.2 also works for discrete subgroups Γ ⊂ G having possibly infinite co-volume. However, the lower bound in (7.1) shows that the restriction π| Γ admits a frame only if Γ ⊂ G has finite co-volume. The lattice assumption is in fact even necessary for π| Γ to admit a cyclic vector [10,Corollary 2].
The idea of periodizing the orthogonality relations by means of Weil's integral formula can also be found in [18,51]. Proposition 7.2 will be subsequently substantially sharpened by eliminating the frame bounds in the conclusion. 7.2. Necessary density conditions. The following result provides necessary density conditions for several spanning properties. Note that Kleppner's condition is not assumed in parts (i) and (ii).
Theorem 7.4. Let Γ ⊆ G be a lattice and let (π, H π ) be a discrete series σ-representation of G of formal dimension d π > 0.
The idea of relating the orthogonality relations and the frame inequalities for proving a density theorem as Theorem 7.4 was used in Janssen's "classroom proof" of the density theorem for Gabor frames [42]. The use of an auxiliary tight frame to deduce the density condition can be found in [15,Theorem 11.3.1]. A similar combination of these ideas have been used in [40]. In this article, these ideas are further refined, implying necessary conditions for completeness. The arguments for Riesz sequences seem to be new. 7.3. Critical density. This section is devoted to the spanning properties of π| Γ for lattices possessing the critical density vol(G/Γ)d π = 1. Lemma 7.5. Let Γ ⊆ G be a lattice and let (π, H π ) be a discrete series σ-representation of G of formal dimension d π > 0. Suppose g ∈ H π is a unit vector such that π(Γ)g is an orthonormal system in H π . Then the following are equivalent: (i) The system π(Γ)g is complete in H π .

Proof. That (i) implies (ii) follows from Proposition 7.2.
Conversely, suppose that vol(G/Γ)d π = 1. Let Ω ⊆ G be a right fundamental domain of Γ ⊆ G, and {f n : n ∈ N} a norm dense subset of H π .
Since n∈N E n has null measure, we can choose x 0 ∈ Ω \ n∈N E n . Therefore, holds for all f ∈ {f n : n ∈ N}, and extends by density to all f ∈ H π . Replacing f by π(x 0 )f gives γ∈Γ | f, π(γ)g | 2 = f 2 Hπ , for all f ∈ H π . This shows that π(Γ)g is complete.

Proof of the density theorem
We finally can prove the main result of the article.
(iii) Assume that vol(G/Γ)d π > 1. Then, by Theorem 7.4, π| Γ does not admit a cyclic vector. Combining this information with Proposition 6.2, it follows that π(Γ) ′′ admits a separating vector, and by Proposition 4.3, also an orthonormal sequence. A far reaching generalization of Theorem 1.1 without the ICC condition is due to Bekka [10]; see Section 9.
9. Examples and applications 9.1. The density theorem for semisimple Lie groups. For certain center-free semisimple Lie groups, a lattice is automatically ICC, and hence Kleppner's condition is satisfied. For reference purposes, we state Theorem 8.1 in this setting.
Theorem 9.1. Let G be a center-free connected semisimple real Lie group all of whose connected, normal, compact subgroups are trivial. 3 Let (π, H π ) be a discrete series σrepresentation of G of formal dimension d π > 0. Let Γ ⊆ G be a lattice. Then As we show in Example 9.3, the conclusion of Theorem 9.1 may fail when the center of the group is non-trivial. A more general version of Theorem 9.1, that does not require the ICC condition, was derived by Bekka [10], and applies to semisimple Lie groups with a possibly non-trivial center [10, Theorem 2], and to a class of algebraic groups over more general fields. Theorem 9.1 can also be phrased more generally for such algebraic groups, provided they have a trivial center.
We now illustrate an important instance of Theorem 9.1.
Example 9.2. The group G = PSL(2, R) = SL(2, R)/{−I, I} is a connected simple Lie group with trivial center [29,80], and acts on the upper half plane through Moebius transforms as The measure dµ(z) = (ℑ(z)) −2 dxdy, where z = x+iy and dxdy is the Lebesgue measure on C + , is G-invariant. Let PSO(2, R) := SO(2, R)/{−I, I} be the compact subgroup of rotations. We use the diffeomorphism, [m] → m · i, (9.2) to fix a Haar measure on G/PSO(2, R), and equip PSO(2, R) with a normalized Haar measure µ T of total measure 1. This fixes the Haar measure µ G on G as dµ G ≃ dµdµ T . With this normalization, for measurable E ⊆ C + , In the remainder of this article, the Haar measure on G = PSL(2, R) is always assumed to have this normalization.
Lattices Γ ⊆ G are known as Fuchsian groups. By the normalization (9.3), we have vol(G/Γ) = µ(D), where D ⊆ C + is a so-called Dirichlet fundamental domain for Γ, that provides the tessellation C + = γ∈Γ γD, up to sets of null measure.
According to Theorem 8.1, the existence of a function g ∈ A 2 α (C + ) such that π α (Γ)g is complete in (resp. frame for, resp. Parseval frame for) while the existence of a Riesz sequence π α (Γ)g (resp. orthonormal sequence, resp. g separating vector) in A 2 α (C + ) is equivalent to the condition For examples of Fuchsian groups, and formulae for their co-volume, see [9].
Consider the representation π α from Example 9.2. Since π ′ α (−I) = ±I, for any g ∈ A 2 α (C + ), We conclude that there exists g ∈ A 2 α (C + ) such that π ′ α (Γ)g is complete if and only if Second, note that there does not exist a Riesz sequence in A 2 α (C + ) of the form π ′ α (Γ)g, regardless of the value of vol(G/Γ), as the (indexed) system π ′ α (Γ)g is always linearly dependent: π ′ α (I)g = g = ±π ′ α (−I)g. Hence, also in that respect, the conclusion of Theorem 8.1 fails for G and Γ. 9.1.1. Perelomov's uniqueness problem. A set of points Λ ⊆ C + is called a set of uniqueness for the Bergman space A 2 α (C + ) if the only function f ∈ A 2 α (C + ) that vanishes identically on Λ is the zero function. Perelomov [66] studied this question when Λ is the orbit of a point w ∈ C + through a Fuchsian group Γ in G = PSL(2, R). 4 The link with lattice orbits of π α is provided by the special choice of vector k which has the reproducing property: where c α ∈ T is a unimodular constant and the notation of (9.5) is used. Hence, Λ = Γw is a set of uniqueness for A 2 α (C + ) if and only if π α (Γ)k (α) w is complete in A 2 α (C + ). Perelomov [66] showed that this is the case if where F w = {γ ∈ Γ : γ · w = w} is the stabilizer subgroup of w. 5 When #F w = 1, the sufficient condition for the completeness of π α (Γ)k (α) w in A 2 α (C + ) (9.11) almost matches (9.7), which is necessary for the completeness of any orbit π α (Γ)g. For #F w > 1, a necessary condition for the completeness of π α (Γ)k  [47], after observing that the physically-motivated restrictions the authors impose on α play no role in the argument.
Thus, while (π α , A 2 α (C + )) admits a cyclic vector g if and only if vol(G/Γ) ≤ 4π α−1 , in the smaller range vol(G/Γ) < 4π #Fw(α−1) the specific choice g = k (α) w is possible, and in the range 4π #Fw(α−1) < vol(G/Γ) ≤ 4π α−1 it is not. The completeness of π α (Γ)k (α) w when #F w vol(G/Γ) = 4π α−1 has recently been shown by Jones [43]. Perelomov's original work also contains a necessary condition for the completeness of π α (Γ)k (α) w in A 2 α (C + ), formulated in terms of the smallest weight m + 0 for which the space of parabolic Γ-modular forms on C + is at least two-dimensional [ where #P denotes the number of inequivalent cusps for Γ. Thus the necessity of (9.7) for cyclicity is stronger than Perelomov's automorphic weight bound for the cyclicity of one specific vector (9.13), but weaker than Kelly-Lyth's (9.12). Under the assumption that (9.13) fails, Perelomov uses certain Γ-modular forms to construct a non-zero function in A 2 α (C + ) that vanishes on Γw. Under the assumption that (9.12) fails, Kelly-Lyth also provides such function, by calculating the so-called upper Beurling-Seip density of Γw in terms of the co-volume of Γ, and by resorting to Seip's interpolation theorem [77]. While this article gives a very elementary argument for the necessity of (9.7) for the completeness of π α (Γ)g w , we do not have a similarly simple argument for (9.12). 9.1.2. Frames and Riesz sequences of reproducing kernels. By Theorem 9.1, under (9.7), there exists g ∈ A 2 α (C + ) such that the orbit π α (Γ)g is a (Parseval) frame for A 2 α (C + ). In light of Section 9.1.1, it is natural to ask whether the specific choice g = k (α) w also provides a frame. Here the answer depends on whether or not Γ is co-compact (that is, G/Γ is compact). Using (9.10), the frame property reads (9.14) for some constants A, B > 0. The stabilizer subgroup F w is finite because it is simultaneously contained in the discrete set Γ and in the compact subgroup m 0 PSO(2, R)m −1 0 , where m 0 i = w. Hence, we can rewrite (9.14) as a sampling inequality: Based on the characterization of sampling inequalities by Seip [77], Kelly-Lyth showed that if Γ is not co-compact, then Γw never satisfies (9.15), because its so-called lower Beurling-Seip density is zero [46, p.44]. Thus, in this case, π α (Γ)k (α) w fails to be a frame for A 2 α (C + ). On the other hand, if Γ is co-compact, the lower Beurling-Seip density of The coherent state subsystems associated with h (α) n can be more concretely described as follows [13,47]. The subgroup of affine transformations P := m x,y = √ y x/ √ y provides representatives for the quotient G/PSO(2, R), since G = P · PSO(2, R) and P ∩ PSO(2, R) = {I}. In particular, every m ∈ G can be written as m = m x,y r for unique m x,y ∈ P and r ∈ PSO(2, R). Recall that i ∈ C + is a fixed point of PSO(2, R), and, hence, (x, y) is x + iy = m x,y · i = m · i. Therefore, the coherent state associated with h (α) n can be realized as an affine system: Perelomov's problem concerns the completeness of A α (h (α) n , Γi) in A 2 α (C + ). While Theorem 9.1 shows that (9.7) is necessary for completeness, we are unaware of literature on corresponding sufficient conditions. We remark that, as G acts transitively on C + , the previous conclusions also apply to any other base point z ∈ C + in lieu of i. Indeed, if z = m · i with m ∈ G, then each element of is a unimodular multiple of an element of π α m)A α h The completeness problem can be alternatively reformulated on the real half-line. The connection is provided by the Paley-Wiener theorem for Bergman spaces [23,75]: the Fourier-Laplace transform is a multiple of an isometric isomorphism between the weighted L 2 -space and the Bergman space A 2 α (C + ). In addition, the special vectors h (α) n correspond under the isomorphism to multiples of (9.20) where L α−1 n is the Laguerre polynomial of degree n ∈ N and index α − 1; see [23]. The inverse Fourier-Laplace transform thus maps the affine system (9.17) into the system n , Γi) = d α n y −α/2+1 e −iπx · H (α) n (y·) : x + iy ∈ Γi , (9.21) in L 2 (R + , t −(α−1) dt) for a suitable d α n ∈ C. This yields another equivalent formulation of Perelomov's completeness problem. See also [16,Section 8.6].
Our proof of Theorem 7.4 is partially inspired by Janssen's "classroom proof" [42], which concerns frames and Riesz sequences. Instead of using the frame inequality, as in Proposition 7.2, Janssen uses the so-called canonical frame expansion f = γ∈Γ f, π(γ)S −1 g,Γ g π(γ)g associated with a frame π(Γ)g and frame operator S g,Γ . The coefficients f, π(γ)S −1 g,Γ g have minimal ℓ 2 norm among all sequences c such that f = γ∈Γ c γ π(γ)g and this property is leveraged to prove (9.25). In contrast, we prove Theorem 7.4 by resorting to the normalization procedure in Proposition 4.2, which applies also to complete systems π(Γ)g that may not be frames. Similarly, while Janssen treats Riesz sequences π(Γ)g by invoking properties of the corresponding biorthogonal element h characterized by π(γ)g, π(γ ′ )h = δ γ ′ ,γ , for γ, γ ′ ∈ Γ, and h ∈ [π(Γ)g], we use Proposition 4.3 to reduce the proof to orthonormal sequences, while also treating separating vectors.
As is the case with the necessity of the density conditions, the sufficiency of (9.25) and (9.26) for the existence of frames and Riesz vectors also holds without assuming Kleppner's condition. This deep fact, shown by Rieffel [70,71], and also a consequence of Bekka's work [10,Theorem 4], lies beyond the elementary approach presented in this article. Indeed, Rieffel's and Bekka's work require considering not only the operator algebras π(Γ) ′ and π(Γ) ′′ , but also certain so-called induced algebras, and in this way fully exploit the coupling theory of von Neumann algebras. We hope that our elementary introduction motivates the reader to delve deeper into operator-algebraic methods. For lattices of the form Γ = AZ d × BZ d , with A, B ∈ GL(d, R), Han and Wang gave a constructive proof of the sufficiency of (9.25) for the existence of frame vectors [38]. 9.2.1. Gaussians and Bargmann-Fock spaces. The question of choosing specific cyclic or frame vectors has been intensively studied for d = 1 and lattices in R 2 of the form Γ = αZ × βZ. In his work on foundations of quantum mechanics, von Neumann [82] claimed without proof that the Gabor system π(Γ)g generated by the Gaussian function g(t) = 2 −1/4 e −π|t| 2 , t ∈ R, (9.27) is complete in L 2 (R) if and only if (9.25) holds. Proofs of the claim were given by Perelomov [64], Bargmann [8], and Neretin [60]. For rational lattices (i.e., αβ ∈ Q), the same claim holds when the Gaussian function is multiplied by a rational function with no real poles [36].
The related question, under which conditions the Gabor system generated by the Gaussian (9.27) is a frame for L 2 (R) or a Riesz sequence was first considered by Daubechies and Grossmann [19], and fully answered independently by Lyubarskiȋ [52], and Seip and Wallstén [76,79]: vol(G/Γ) < 1, is necessary and sufficient for the frame property, while vol(G/Γ) > 1, is necessary and sufficient for the Riesz property.
The proofs of Lyubarskiȋ [52] and Seip-Wallstén [76] work with a σ-representation unitarily equivalent to (π, L 2 (R)) on the Bargmann-Fock space F 2 (C) of entire functions F : C → C having finite norm As in Example 9.2, the distinguished vector g corresponds under the new representation to the reproducing kernel, that is, the vector representing the evaluation functional F → F (0). A simple proof of the density results was derived by Janssen [41].
The characterization of the frame and Riesz property for other vectors g is a topic of intense study [35].