On sufficient density conditions for lattice orbits of relative discrete series

This note provides new criteria on a unimodular group G and a discrete series representation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\pi , \mathcal {H}_{\pi })$$\end{document}(π,Hπ) of formal degree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\pi } > 0$$\end{document}dπ>0 under which any lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \le G$$\end{document}Γ≤G with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{vol}\,}}(G/\Gamma ) d_{\pi } \le 1$$\end{document}vol(G/Γ)dπ≤1 (resp. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{vol}\,}}(G/\Gamma ) d_{\pi } \ge 1$$\end{document}vol(G/Γ)dπ≥1) admits \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \in \mathcal {H}_{\pi }$$\end{document}g∈Hπ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (\Gamma ) g$$\end{document}π(Γ)g is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.

If π(Γ)g satisfies the upper bound in (1.2), then it is a Bessel sequence. The lower bound in (1.2) implies, in particular, that a Riesz sequence is linearly independent.
This note is concerned with the existence of vectors g ∈ H π such that its orbit π(Γ)g is a frame or Riesz sequence. A simple necessary condition is that if π(Γ)g is a frame or Riesz sequence, so that it admits a Bessel constant B > 0, then g ∈ H π \ {0} satisfies G | g, π(x)g | 2 dμ G (x) = G/Γ γ∈Γ | π(γ) * g, π(x)g | 2 dμ G/Γ (xΓ) ≤ vol(G/Γ)B g 2 Hπ < ∞. An irreducible π with a non-zero L 2 -integrable matrix coefficient is called a (projective) discrete series; see Section 2.3 for several basic properties. Since nilpotent and (unimodular) exponential Lie groups do not admit genuine representations that are square-integrable in the strict sense, the use of projective representations is particularly convenient; see Section 3.
Let C φ be the σ-twisted convolution operator on 2 (Γ) defined by Then the following assertions hold: (i) There exists g ∈ H π such that π(Γ)g is a frame if and only if C φ ≤ I 2 .
(ii) There exists g ∈ H π such that π(Γ)g is a Riesz sequence if and only if C φ ≥ I 2 .
(ii) in Theorem 1.1 are consequences of the underlying theory of von Neumann algebras. The paper [2] provides the statements of Theorem For a simple proof of Corollary 1.2 based on frame and representation theory, see [16].
The density conditions provided by Corollary 1.2 are generally not sharp, in the sense that they are not sufficient for the existence of frames and Riesz sequences of the form π(Γ)g. For example, this might fail for discrete series of semi-simple Lie groups with a non-trivial center, cf. [2, Example 1]. However, for semi-simple Lie groups with a trivial center, the convolution kernel φ of the operator C φ in Theorem 1.1 is simply given by (1. 4) In general, if the identity (1.4) holds, then the density conditions provided by Corollary 1.2 are also sufficient for the existence of frames and Riesz sequences. In particular, this holds for lattices in which every non-trivial σ-regular conjugacy class has infinite cardinality; such pairs (Γ, σ) are sometimes said to satisfy "Kleppner's condition" [11]. It is the aim of this note to provide new criteria under which the convolution kernel φ in Theorem 1.1 takes the simple form (1.4). In particular, this will imply the optimality of the density conditions provided by Corollary 1.2.
In order to state the key result of this note, let B(G) be the set of all elements with pre-compact conjugacy classes in G. Then B(G) is a normal subgroup of G containing the center Z(G) and was studied for classes of locally compact groups in, e.g., [8,13,18,21]. In particular, it was shown that exponential solvable Lie groups and reductive algebraic groups (with no simple factors) have the property B(G) = Z(G), cf. Example 2.1 for references.
The following result provides criteria for a unimodular G with B(G) = Z(G) under which the convolution operator C φ of Theorem 1.1 is a scalar multiple of the identity operator. Theorem 1.3. Let G be such that B(G) = Z(G) and let Γ ≤ G be a lattice. Suppose that either G is locally connected or Γ is co-compact. Suppose (π, H π ) is a discrete series of formal degree d π > 0 such that the projective kernel P π := {x ∈ G : π(x) ∈ C · I Hπ } is trivial. Then the twisted convolution operator C φ on 2 (Γ) defined in (1.3) is given by (1.5) Consequently, the following assertions hold: (i) If vol(G/Γ)d π ≤ 1, then there exists a frame π(Γ)g for H π .
(ii) If vol(G/Γ)d π ≥ 1, then there exists a Riesz sequence π(Γ)g in H π . Theorem 1.3 is applicable to all cases in which the necessary density conditions of Corollary 1.2 are known to be sharp, namely for Abelian groups [6,Corollary 4.6], linear algebraic semi-simple groups [2,Theorem 2], and square-integrable representations modulo the center of nilpotent Lie groups [2,Theorem 3]. In addition, it is applicable to exponential Lie groups and reductive algebraic groups and it allows to treat representations that are only square-integrable modulo their projective kernel since any such representation is naturally treated as a projective discrete series of the quotient (cf. Section 3).
The assumption in Theorem 1.3 that the projective kernel P π is trivial is essential for its validity. For example, both conclusions (i) and (ii) fail for a holomorphic discrete series π of SL(2, On the other hand, for the existence of Riesz sequences π(Γ)g in general, it is necessary that π| Γ acts projectively faithful.
A particular motivation for obtaining Theorem 1.3 was to investigate the optimality of the density conditions in Corollary 1.2 for the existence of frames and Riesz sequences for general exponential Lie groups, i.e., Lie groups for which the exponential map is a diffeomorphism. For a description of the projective discrete series of an exponential Lie group in terms of the Kirillov correspondence, see [12,20]; in particular, cf. [20,Proposition 4]. A non-nilpotent example to which Theorem 1.4 is applicable is given in Section 4.

2.
Restrictions of discrete series to lattices. Throughout, unless stated otherwise, G denotes a second-countable unimodular locally compact group. A fixed Haar measure on G will be denoted by μ G . If H ≤ G is a closed subgroup with Haar measure μ H , then there exists a unique G-invariant Radon measure μ G/H on the space G/H of left cosets of H such that Weil's formula holds: (2022) On sufficient density conditions 283 The measure μ G/H will always be assumed to be normalized such that (2.1) holds. If H is discrete, then μ H will be assumed to be the counting measure.

Bounded conjugacy classes. For a subset
Locally compact groups G for which B(G) = Z(G) will play a key role in this note. The following example lists (classes of) groups for which this condition is satisfied.

Lattices.
A discrete subgroup Γ ≤ G is said to be a lattice if the unique invariant Radon measure on G/Γ provided by (2.1) is finite. A lattice Γ is called uniform if G/Γ is compact. For classes of amenable groups, including connected solvable Lie groups, any lattice is automatically uniform, see [1,14].

Lemma 2.2.
Let G be such that B(G) = Z(G) and let Γ ≤ G be a lattice. Suppose that either G is locally connected or that Γ ≤ G is uniform. Then the following assertions hold: (i) For every γ ∈ Γ, the conjugacy class C Γ (γ) in Γ is either trivial or infinite.
Secondly, if Γ is a uniform lattice, then there exists a compact set Ω ⊆ G such that G = Ω · Γ. The conjugacy class C G (γ) is therefore given by . Suppose first that G is locally connected. The finite G-invariant measure on G/Γ can be pushed forward to a finite G-invariant measure on G/Z G (x), which implies that G/Z G (x) is compact by [19,Theorem]. Hence, Lastly, if Γ is a uniform lattice, then the continuous surjective map G/Γ → G/Z G (x) yields that G/Z G (x) is compact. Therefore, the continuous bijection Lemma 2.2 applies, in particular, to arbitrary lattices in Lie groups. For this setting, there are alternative proofs of the used [19,Theorem], see [9, Theorem 1] and [7, Theorem 2]. It is not known whether Lemma 2.2 holds for non-uniform lattices in general unimodular groups.
A projective unitary representation with 2-cocycle σ will simply be referred to as a σ-representation. For σ ≡ 1, it will simply be said that π is a representation.
A σ-representation (π, H π ) is irreducible if the only closed π(G)-invariant subspaces are {0} and H π . It is called square-integrable if there exist nonzero f, g ∈ H π such that An irreducible, square-integrable σ-representation is called a discrete series σ-representation, or a projective discrete series if the associated cocycle is irrelevant.

Vol. 119 (2022)
On sufficient density conditions 285 The significance of a discrete series π is the existence of a unique d π > 0, called its formal degree, such that the orthogonality relations hold for all f, f , g, g ∈ H π .

The projective kernel.
The projective kernel of a σ-representation π is defined by The σ-representation π is projectively faithful if P π = {e}. Throughout, χ π : P π → T denotes the measurable function satisfying π(x) = χ π (x)I Hπ for all x ∈ P π . Then, for f, g ∈ H π , x ∈ G, and y ∈ P π , so that xP π → | f, π(x)g | is a well-defined function on the coset space G/P π .

Lemma 2.3.
If π is a σ-representation of G, then P π is a closed normal subgroup. If, in addition, π is square-integrable, then P π is compact.
Proof. Let P(H π ) := U(H π )/T · I Hπ be the projective unitary group of H π , equipped with the quotient topology relative to the strong operator topology on U(H π ). Let p : U(H π ) → P(H π ) be the canonical projection. By [22,Theorem 7.5], the map π := p•π : G → P(H π ) is a continuous homomorphism, and hence P π = ker(π ) is a closed normal subgroup. Suppose π is square-integrable. Letting f, g ∈ H π \ {0}, we apply (2.1) and and thus the Haar measure of P π is finite, so that P π must be compact.

Lemma 2.4.
Let G be such that B(G) = Z(G) and let (π, H π ) be a discrete series σ-representation of G. Then the projective kernel coincides with the σregular elements of the center of G. In particular, the following are equivalent: (i) π is projectively faithful.
(ii) The only σ-regular element of G with precompact conjugacy class is the identity.
Since B(G) = Z(G), it follows that the projective kernel coincides with σregular elements with pre-compact conjugacy classes. In particular, P π = {e} if and only if the only σ-regular element with pre-compact conjugacy class is the identity.
A combination of the previous lemmata allows a proof of Theorem 1.3: Proof of Theorem 1.3. Suppose γ ∈ Γ is σ-regular in Γ and C Γ (γ) is finite.
The G-invariance of the measure on G/Γ gives that which means that Vol. 119 (2022) On sufficient density conditions 287 Note that ω γ ≡ 1 if and only if σ(γ, y) = σ(y, γ) for all y ∈ G, i.e., if and only if γ is σ-regular in G. Since γ ∈ Z(G), γ is σ-regular in G if and only if γ = e by Lemma 2.4 and the assumption that π is projectively faithful. Hence, as required.
Proof of Theorem 1.4. Since G is exponential, we have Z(G) = B(G) by [8,Theorem 9.4]. By Lemma 2.3, the projective kernel P π of a discrete series π must be compact, hence trivial, since G does not contain nontrivial compact subgroups, see, e.g., [10,Theorem 14.3.12]. The conclusion follows therefore directly from Theorem 1.3.

Discrete series modulo the projective kernel.
This section considers projective representations obtained from genuine representations that are squareintegrable modulo their projective kernel. Such projective representations are projectively faithful, and they form an important class to which Theorem 1.3 applies. Let (ρ, H ρ ) be an irreducible representation of a second countable unimodular group H. It is called a relative discrete series (modulo P ρ ) if there exist non-zero f, g ∈ H ρ such that whereẋ = xP ρ and μ H/Pρ denotes the Haar measure on H/P ρ .
A relatively discrete series (ρ, H ρ ) of H can be treated as a (projective) discrete series of G := H/P ρ . For this, choose a Borel section s : H/P ρ → H of the canonical quotient map, and set π := ρ • s. Then a direct calculation shows that π(ẋ)π(ẏ) = σ(ẋ,ẏ)π(ẋẏ),ẋ,ẏ ∈ G = H/P ρ , where the 2-cocycle σ is given by A different choice of the section s yields a 2-cocycle cohomologous to σ and a representation unitarily equivalent to π.
The following proposition is a special case of Theorem 1.3.
Proposition 3.1. Let (ρ, H ρ ) be a relative discrete series (modulo P ρ ) of a unimodular group H. Suppose that G = H/P ρ is unimodular and denote by π a σ-representation of G associated to ρ. Let Γ ≤ G be a lattice. If B(G) = Z(G) and either G is locally connected or Γ is uniform, then C φ = vol(G/Γ)d π · I 2 .
Proof. Denote by π a σ-representation of G = H/P ρ obtained from ρ via a Borel section s. If xP ρ ∈ P π ≤ H/P ρ , then ρ(s(xP ρ )) = π(xP ρ ) ∈ T · I Hπ , so that s(xP ρ ) ∈ P ρ . This implies that xP ρ = s(xP ρ )P ρ = P ρ , and thus Then g is completely solvable, i.e., it admits a sequence of ideals In particular, this shows that g is an exponential solvable Lie algebra. Its nilradical is given by n = span R {X 1 , X 2 , X 3 }⊕RX 4 , so that g is non-nilpotent. The center of g is z(g) = RX 1 . Let G be the connected, simply connected Lie group with Lie algebra g. Denote by N and T the connected Lie subgroups with Lie algebras n and RX 5 , respectively. Then G is a semi-direct product G = NT with group multiplication ⎛ x + x − tw − 1 2 (e t zy − e −t yz ) y + e −t y z + e t z w + w t + t The center of G is given by Z(G) = {(x, 0, 0, 0, 0) : x ∈ R}.