Orthonormal Bases in the Orbit of Square-Integrable Representations of Nilpotent Lie Groups

TL;DR

This paper demonstrates the existence of orthonormal bases in $L^2( extbf{R}^d)$ constructed from square-integrable representations of nilpotent Lie groups, generalizing Gabor bases in time-frequency analysis.

Contribution

It establishes conditions under which discrete subsets of nilpotent groups generate orthonormal bases via group actions, extending known results to broader classes of Lie groups.

Findings

Existence of orthonormal bases from group representations.

Generalization of Gabor bases to nilpotent Lie groups.

Construction applicable to graded Lie groups with one-dimensional center.

Abstract

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$ there exist a discrete subset $\Gamma$ of $G/Z(G)$ and some (relatively) compact set $F \subseteq \mathbf{R}^d$ such that $$\bigl \{ |F|^{-1/2} \hspace{2pt} \pi(\gamma) 1_F \mid \gamma \in \Gamma \bigr\}$$ forms an orthonormal basis of $L^2(\mathbf{R}^d)$. This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis. The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set $\Gamma $ can be chosen to be a uniform subgroup of $G/Z$.

Equations

$$\displaystyle\bigl{\{}\left|F\right|^{-1/2}\hskip 2.0pt\pi(\gamma)1_{F}\mid\gamma\in\Gamma\bigr{\}}$$

$$({\pi }_{\lambda }(z,y,x)f)(t)=e^{2\pi i\lambda z}{e}^{2\pi i\lambda yt}f(t+x),$$

$$\displaystyle\left|\lambda\right|^{-d/2}\pi_{\lambda}(0,\vartheta,\gamma)1_{[0,\lambda^{-1})^{d}}=\Bigl{(}t\mapsto\left|\lambda\right|^{d/2}e^{2\pi i\lambda\vartheta t}\hskip 2.0pt1_{[0,\lambda^{-1})^{d}}(t+\eta)\Bigr{)},$$

$$\displaystyle\Bigl{\langle}\pi_{\lambda}(0,\vartheta,\eta)1_{[0,\lambda^{-1})^{d}},\pi_{\lambda}(0,\vartheta^{\prime},\eta^{\prime})$$

$$={\int }_{{\mathbb{R}}^d}{e}^{2\pi i\lambda \vartheta t}{1}_{[0,{\lambda }^{-1}{)}^d}(t+\eta )e^{-2\pi i\lambda {\vartheta }^{\prime }t}{1}_{[0,{\lambda }^{-1}{)}^d}(t+{\eta }^{\prime })dt$$

$$={\delta }_{\eta ,{\eta }^{\prime }}{\int }_{{\mathbb{R}}^d}{e}^{2\pi i\lambda (\vartheta -{\vartheta }^{\prime })t}{1}_{[0,{\lambda }^{-1}{)}^d}(t+\eta )dt$$

$$={\delta }_{\eta ,{\eta }^{\prime }}{\int }_{[0,{\lambda }^{-1}{)}^d}{e}^{2\pi i\lambda (\vartheta -{\vartheta }^{\prime })(t-\eta )}dt$$

$$={\delta }_{\eta ,{\eta }^{\prime }}{e}^{-2\pi i\lambda (\vartheta -{\vartheta }^{\prime })\eta }{\int }_{[0,{\lambda }^{-1}{)}^d}{e}^{2\pi i\lambda (\vartheta -{\vartheta }^{\prime })t}dt$$

$$={\delta }_{\vartheta ,{\vartheta }^{\prime }}{\delta }_{\eta ,{\eta }^{\prime }}{\left|\lambda \right|}^{-d}$$

$$(z,y,x)(z^{\prime },y^{\prime },x^{\prime })=(z+z^{\prime }+x{y}^{\prime },y+y^{\prime },x+x)$$

$$\varphi :{\mathbb{R}}^n\to G,$$

$$\varphi (t_1,\dots ,t_n):=\exp (t_1{X}_1)\dots \exp (t_n{X}_n)=\exp (t_1{X}_1\ast \dots \ast t_n{X}_n).$$

$$\varphi (t)\varphi (s)=\varphi (P_{X_1}(t,s),\dots ,P_{X_n}(t,s)).$$

$$\phi(t)=\exp\Bigl{(}\sum_{j=1}^{n}R_{X_{j}}(t)X_{j}\Bigr{)}$$

$$\Gamma =\exp (\mathbb{Z}{X}_1)\cdots \exp (\mathbb{Z}{X}_n).$$

$$\displaystyle\Gamma=\bigl{\{}\exp(k_{1}X_{1})\cdots\exp(k_{n}X_{n})\mid k_{1},\ldots,k_{n}\in\mathbb{Z}\bigr{\}}$$

$$\displaystyle\Sigma=\bigl{\{}\exp(t_{n}X_{n})\cdots\exp(t_{1}X_{1})\mid t_{1},\ldots,t_{n}\in[0,1)\bigr{\}}.$$

$$g=\exp (x_1{X}_1)\cdots \exp (x_n{X}_n)=\exp (t_n{X}_n)\cdots \exp (t_1{X}_1){\gamma }_1^{k_1}\cdots {\gamma }_n^{k_n}.$$

$$g=\exp (x_1{X}_1)\cdots \exp (x_{n+1}{X}_{n+1})=g_n\exp (x_{n+1}{X}_{n+1})$$

$$g$$

$$\displaystyle=\exp(t_{n+1}X_{n+1})\,\Bigl{(}\exp(-t_{n+1}X_{n+1})\,g_{n}\,\exp(t_{n+1}X_{n+1})\Bigr{)}\exp(k_{n+1}X_{n+1})$$

$$=\exp (t_{n+1}{X}_{n+1})g_n^{\prime }\exp (k_{n+1}{X}_{n+1})$$

$$g=\exp (t_{n+1}{X}_{n+1})\exp (t_n{X}_n)\cdots \exp (t_1{X}_1)\exp (k_1{X}_1)\cdots \exp (k_n{X}_n)\exp (k_{n+1}{X}_{n+1}).$$

$$\exp (t_n{X}_n)\cdots \exp (t_1{X}_1)$$

$$=\exp (t_n^{\prime }{X}_n)\cdots \exp (t_1^{\prime }{X}_1)\exp (k_1^{\prime }{X}_1)\cdots \exp (k_n^{\prime }{X}_n)$$

$$=\exp (t_n{X}_n)g_{n-1}\exp (k_n{X}_n)$$

$$=\exp (t_n^{\prime }{X}_n)g_{n-1}^{\prime }\exp (k_n^{\prime }{X}_n)$$

$$g_{n-1}$$

$$\displaystyle=\exp(y_{n}X_{n})\,g^{\prime}_{n-1}\,\exp(-y_{n}X_{n})\,\exp\bigl{(}(y_{n}+k^{\prime}_{n}-k_{n})X_{n}\bigr{)}$$

$$\displaystyle=g^{\prime\prime}_{n-1}\,\exp\bigl{(}(y_{n}+k^{\prime}_{n}-k_{n})X_{n}\bigr{)}$$

$$\displaystyle\mathop{\mathrm{pr}}(\Gamma)=\mathop{\mathrm{pr}}\bigl{(}\exp(k_{n^{\prime}+1}X_{n^{\prime}+1})\cdots\exp(k_{n}X_{n})\bigr{)}$$

$$\displaystyle\mathop{\mathrm{pr}}(\Sigma)=\bigl{\{}\exp(t_{n}X_{n})\cdots\exp(t_{n^{\prime}+1}X_{n^{\prime}+1})\mid t_{1},\ldots,t_{n}\in[0,1)\bigr{\}}.$$

$${\chi }_l(m)=e^{2\pi i⟨l,\log m⟩}\text{ for }m\in M=\exp (\mathfrak{m})$$

$$g=p(g)\sigma (q(g))=mh$$

$$h^{\prime }g=h^{\prime }m{h}^{\prime -1}{h}^{\prime }h=h^{\prime }m{h}^{\prime -1}p(h^{\prime }h)\sigma (q(h^{\prime }h)),$$

$$\displaystyle\Bigl{(}\mathop{\mathrm{ind}^{G}_{M}(\chi_{l})}(g)f\Bigr{)}(q(h^{\prime}))$$

$$\displaystyle=e^{2\pi i\left\langle l,\log\bigl{(}h^{\prime}mh^{\prime-1}\bigr{)}\right\rangle}\,e^{2\pi i\left\langle l,\log\bigl{(}p(h^{\prime}h)\bigr{)}\right\rangle}\,f\bigl{(}q(h^{\prime}\hskip 2.0pth)\bigr{)},$$

Full text

[TABLE]

forms an orthonormal basis of $L^{2}(\mathbb{R}^{d})$. This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis.

The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set $\Gamma$ can be chosen to be a uniform subgroup of $G/Z$.

K. G. and D. R. were supported in part by the project P26273 - N25 of the Austrian Science Fund (FWF)

1. Introduction

Haar, Gabor, wavelet — these names all refer to structured orthonormal bases for $L^{2}(\mathbb{R}^{d})$. So far these have been objects of applied harmonic analysis, because they are useful to distinguish and classify certain features of functions, such as smoothness [27] or phase space localization [11]. From an abstract point of view their common origin lies in the representation theory of certain groups. Each of these orthonormal bases is in the orbit of a unitary representation of a locally compact group. Specifically, the Gabor basis arises by applying phase space shifts to the characteristic function $1_{[0,1]}$ to yield the orthonormal basis $\{e^{2\pi ilx}1_{[0,1]}(x-k)\mid k,l\in\mathbb{Z}\}$ of $L^{2}(\mathbb{R})$; the Haar basis arises by applying shifts and dilations to a “wavelet”, e.g., the function $\psi(x)=1_{[0,1/2]}(x)-1_{[1/2,1]}(x)$ yields the orthonormal basis$\{2^{j/2}\psi(2^{j}x-k)\mid j,k\in\mathbb{Z}\}$ of $L^{2}(\mathbb{R})$.

The underlying question in representation theory is the following. Let $G$ be a locally compact group and $\pi$ be a unitary representation of $G$ on a Hilbert space $\mathcal{H}$. Under which conditions do there exist a template $f\in\mathcal{H}$ and a discrete subset $\Gamma\subseteq G$, such that $\{\pi(\lambda)f\mid\gamma\in\Gamma\}$ is an orthonormal basis for $\mathcal{H}$? In other words, do there exist orbits of the representation that contain an orthonormal basis?

To our knowledge the existence problem seems to be a new question in abstract harmonic analysis. Beyond the special cases that are studied for applications (wavelet bases, Gabor bases) no results seem to be known, let alone general results.

Wavelet theory deals with the construction of orthonormal bases  in the orbit of the quasi-regular representation of certain solvable products, usually a semidirect product of $\mathbb{R}^{d}$ with some matrix action [21, 24]. A special aspect of Gabor analysis is the existence and construction of orthonormal bases in the orbit of the Schrödinger representation of the Heisenberg group [14]. To our knowledge, certain decomposition results of representations of a semisimple Lie group restricted to a lattice subgroup yield the existence of an orthonormal basis in the orbit of discrete series representations. This was pointed out to us by Yurii Neretin [29] and is also implicit in Bekka’s work [5].

In this paper we study the existence of orthonormal bases in the orbit of irreducible unitary representations of a nilpotent Lie group. As it is not difficult to see that a necessary condition for the existence of an orthonormal basis (or a frame) in the orbit of a representation is the square-integrability modulo a suitable subgroup [1], we will restrict our attention to irreducible unitary representations of a connected, simply connected nilpotent Lie group that is square-integrable modulo its center. So our precise question is as follows:

Let $G$ be a connected, simply connected nilpotent Lie group and $(\pi,\mathcal{H})$ be an irreducible unitary representation of $G$ that is square-integrable modulo the center of $G$ (we write $\pi\in SI/Z(G)$). Does there exist a set $\Gamma\subseteq G/Z(G)$ and a vector $f\in\mathcal{H}$, such that $\{\pi(\gamma)f\mid\gamma\in\Gamma\}$ is an orthonormal basis of $\mathcal{H}$?

The answer is obviously yes for the Heisenberg group $\mathbf{H}_{d}$ and the (Schrödinger) representations $\pi_{\lambda}(x,y,z)f(t)=e^{2\pi i\lambda z}e^{2\pi\lambda yt}f(t+x)$ on $L^{2}(\mathbb{R}^{d})$, because $\{\pi_{\lambda}(k,\lambda^{-1}l,0)1_{[0,1]}(t)\mid k,l\in\mathbb{Z}^{d}\}$ is an orthonormal basis for $L^{2}(\mathbb{R})$. For low-dimensional nilpotent Lie groups the question has been studied in A. Höfler’s thesis [26]. She found that the answer is yes for all nilpotent Lie groups $G$ of dimension $\dim(G)\leq 6$ and all irreducible representations in $SI/Z(G)$. The analysis of explicit examples in low dimension justifies the conjecture that the answer to our question should be affirmative for all nilpotent Lie groups and representations in $SI/Z$, or at least for a significant, large class of nilpotent groups. However, as Höfler’s work is based on the case-by-case inspection of the explicit formulas for the irreducible representations in [30], her work gives little indication on what might happen for nilpotent groups in higher dimensions.

In this paper we prove a general result about orthonormal bases in the orbit of a square-integrable representation of a nilpotent group. Its spirit is captured in the following special case.

Theorem [math].

• Let $G$ be a graded Lie group with one-dimensional center and $(\pi,\mathcal{H})$ be a square-integrable irreducible unitary representation modulo center, $\pi\in SI/Z(G)$. Then there exist a discrete subset $\Gamma\subseteq G/Z(G)$ and an $f\in\mathcal{H}$ such that the set $\{\pi(\gamma)f\mid\gamma\in\Gamma\}$ is an orthonormal basis of $\mathcal{H}$. *

A more detailed formulation and the proof require the full details of Kirillov’s construction of irreducible representations via the coadjoint orbits and thus a non-negligeable amount of preparation. Other and more general formulations will be given in Theorems 4.1, 4.2, and 4.7.

Roughly, an irreducible unitary representation of $G$ is parametrized by a linear functional $l$ acting on the Lie algebra $\mathfrak{g}$ of $G$ and by a certain subgroup $M$ of $G$, a so-called polarization. The imposed conditions depend (a) on the structure of $G$, for instance, we assume that some polarization $M$ is a normal subgroup of $G$ and that we know some crucial facts about the coadjoint action, and (b) some rationality assumptions, for instance $\mathfrak{g}$ has rational structure coefficients and the functional $l$ also has rational coefficients with respect to the chosen basis.

(i) The set $\Gamma$ in Theorem [math] is not arbitrary. Under some rationality assumptions the construction yields a uniform subgroup $\Gamma$ in $G/Z(G)$ (Theorem 4.2 and 4.7). However, we will show that the existence of orthonormal bases is not tied to the rational structure of the group. Indeed, we will prove a version of Theorem [math] by using a quasi-lattice in $G$ in place of a lattice. Such structures exist in arbitrary nilpotent Lie groups [17].

(ii) In Theorem [math] we make heavy use of the description and construction of representations with Kirillov’s method of coadjoint orbits. The representation space $\mathcal{H}_{\pi}$ is then realized by $L^{2}(\mathbb{R}^{d})$; the representation is essentially of the form \bigl{(}\pi(x)f\bigr{)}(\zeta)=e^{2\pi iP(x,\zeta)}f(\zeta\cdot x), where $P$ is a polynomial of $x$ and $\zeta$, and $x$ acts on $\zeta\in\mathbb{R}^{d}$. Thus the main theorems prove the existence of a new class of orthonormal bases for $L^{2}(\mathbb{R}^{d})$. We believe that these bases could be useful in the non-Euclidean analysis of $\mathbb{R}^{d}$ (viewed as a homogeneous space).

(iii) Whereas in the low-dimensional examples this action $\zeta\to\zeta\cdot x$ is always Abelian and roughly a translation $\zeta\cdot x\simeq\zeta+x$ on $\mathbb{R}^{d}$, this action is generally non-Abelian in higher dimensions. We will demonstrate this new aspect with the example of the meta-Heisenberg group.

(iv) The construction of the orthonormal basis actually happens in $G/Z(G)$. Clearly, if $z_{\gamma}\in Z(G)$ is arbitrary and $\{\pi(\gamma)g\mid\gamma\in G\}$ is an orthonormal basis for $\mathcal{H}$, then so is $\{\pi(z_{\gamma}\gamma)g\mid\gamma\in G\}$. We could therefore investigate the problem for the projective representations of the nilpotent group $G/Z(G)$. However, since we use the Kirillov construction of irreducible representations, we prefer to work with the representations of $G$ and accept the ambiguity in the phase rather than passing to $G/Z(G)$.

(v) In $\mathcal{H}=L^{2}(\mathbb{R}^{d})$ the template $f$ generating the orthonormal orbit is a characteristic function $f=1_{F}$ of a suitable fundamental domain in $M\setminus G$. This raises the natural question whether there are smooth $f\in L^{2}(\mathbb{R}^{d})$ that generate orthonormal bases. This problem has been studied extensively for the Heisenberg group where the answer is a decisive “no”. Precisely, if $\{e^{2\pi ilt}g(t+k)\mid k,l\in\mathbb{Z}\}$ is an orthonormal basis for $L^{2}(\mathbb{R})$, then either $x\mapsto xf(x)\not\in L^{2}(\mathbb{R})$ or $f^{\prime}\not\in L^{2}(\mathbb{R})$. This is the famous Balian-Low theorem, which has inspired a whole direction of harmonic analysis [7, 6]. Since Kirillov’s lemma asserts that every nilpotent Lie group contains a subgroup isomorphic to the Heisenberg group, we expect that a Balian-Low type theorem also holds in the context of Theorem [math]. Indeed, it is possible to prove a version of the Balian-Low theorem for arbitrary homogeneous nilpotent groups. The proof is almost identical to the proof of the Balian-Low theorem via the theory of deformation of frames in [22] (with only technical modifications). We will exploit this direction in future work.

Our own motivation for the topic comes from studying the density of frames for representations of nilpotent groups. It has been known for a long time that frames always exist in the orbit of every square-integrable irreducible unitary representation of a locally compact group [10, 19]. In analogy to the sampling and interpolation theory for bandlimited functions one would like to understand how dense such a subset must necessarily be. Until recently, all proofs of density theorems required the existence of an orthonormal basis and the comparison of the frame to this orthonormal basis. See [4] for a very general density theory of frames. For frames in the orbits of nilpotent groups precise results have been derived by Höfler [26], but again under the hypothesis that there exists an orthonormal basis in the orbit of the representation. With Theorem [math] and its variations, Höfler’s results about the density of frames now apply to a general class of nilpotent groups. Let us mention that these density results have been generalized recently to arbitrary unimodular groups with square-integrable representations, but without requiring the existence of an orthonormal basis in an orbit [18].

Outlook: Our investigation raises several new questions in the theory of nilpotent Lie groups. It is certainly tempting to believe that every square-integrable irreducible unitary representation modulo the center possesses an orthonormal basis in some orbit. However, at this time we are far from a complete answer. We do not need a rational structure, as one might expect from examples and Bekka’s work [5], but we need that the representation is induced from a normal polarizing subgroup and we need special Malcev bases that are compatible with the parametrization of the coadjoint action (we call them Chevalley-Rosenlicht admissible). The normality of a polarizing subgroup may not be necessary, as is shown by an explicit construction of Führ [16] of an orthonormal basis for an $11$-dimensional nilpotent group with square-integrable representations that do not possess a normal polarizing subgroup. As for the existence of Chevalley-Rosenlicht admissible Malcev bases, we do not know for which groups they exist.

The paper is organized as follows: Section 2 is a warm-up and presents the example of the Heisenberg group and the Gabor basis. In Section 3 we give a quick review of Malcev bases of nilpotent groups and of graded nilpotent Lie algebras; we set the notation for induced representations, and review the standard construction of irreducible representations of nilpotent Lie groups. Section 4 contains several formulations of Theorem [math] and builds up the arguments towards its proof. In particular, we prove the existence of an orthonormal basis in the orbit of a square-integrable representation for all graded nilpotent Lie groups with one-dimensional center. Section 5 offers concrete examples and treats the meta-Heisenberg group in detail.

Acknowledgements. We would like to thank Hartmut Führ and Yuri Neretin for valuable discussions. K. G. would like to thank the Hausdorff Research Institute for Mathematics, Bonn, for the perfect conditions during the work on this paper.

2. Warm-Up

In this section we will review the example of the Gabor basis in $L^{2}(\mathbb{R}^{d})$ and the Heisenberg group, as the proof of our main result is modeled on the arguments of this example.

2.1. Gabor Bases and the Heisenberg Group

For $\lambda\neq 0$, let $\pi_{\lambda}$ be the operator

[TABLE]

acting on $f\in L^{2}(\mathbb{R}^{d})$.

The following statement is well-known in time-frequency analysis and yields an orthonormal basis for $L^{2}(\mathbb{R}^{d})$ in the orbit of a single function. (See [20, § 6.4] and [25, § 11.2].)

Example 2.1**.**

The family of time-frequency shifted functions

[TABLE]

for $(\vartheta,\eta)\in\mathbb{Z}^{d}\times\lambda^{-1}\mathbb{Z}^{d}$, forms an orthonormal basis for $L^{2}(\mathbb{R}^{d})$.

Proof.

We partition $\mathbb{R}^{d}$ into the cubes $\Omega_{\eta}:=[0,\lambda^{-1})^{d}+\eta$ with $\eta\in\lambda^{-1}\mathbb{Z}^{d}$. Then for arbitrary but fixed $\eta\in\lambda^{-1}\mathbb{Z}^{d}$ the set of functions $\{|\lambda|^{d/2}e^{2\pi i\lambda\vartheta t}\mid\vartheta\in\mathbb{Z}^{d}\}$ forms an orthonormal basis for $L^{2}([0,\lambda^{-1})^{d}+\eta)$ (the standard Fourier basis). Including shifts, we find that

[TABLE]

holds true for all $(\vartheta,\eta),(\vartheta^{\prime},\eta^{\prime})\in\mathbb{Z}^{d}\times\lambda^{-1}\mathbb{Z}^{d}$. Since the completeness is clear, the set $\{\left|\lambda\right|^{-d/2}\pi_{\lambda}(0,\vartheta,\eta)1_{[0,\lambda^{-1})^{d}}|(\vartheta,\eta)\in\mathbb{Z}^{d}\times\lambda^{-1}\mathbb{Z}^{d}\}$ forms an orthonormal basis for $L^{2}(\mathbb{R}^{d})$. ∎

A similar construction yields orthonormal bases  for $L^{2}(A)$ of an arbitary locally compact Abelian group $A$ [23].

In the language of abstract harmonic analysis, $\pi_{\lambda}$ is a unitary representation of the $2d+1$-dimensional Heisenberg group $\mathbf{H}_{d}=\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ with center $Z(\mathbf{H}_{d})=\mathbb{R}\times\{0\}\times\{0\}$ and multiplication

[TABLE]

for $z,z^{\prime}\in\mathbb{R},x,x^{\prime},y,y^{\prime}\in\mathbb{R}^{d}$. The representation $\pi_{\lambda}$ for $\lambda\neq 0$ is the Schrödinger representation and is square-integrable modulo center, in short $\pi_{\lambda}\in SI/Z(\mathbf{H}_{d})$.

We make the following observations:

(a) The variable $z$ in (1) (the center of $\mathbf{H}_{d}$) does not play any role in the definition of the orthonormal basis. In general, the construction of an orthonormal basis in the orbit of a representation depends only on the action of $G/Z(G)$ and not on the full representation.

(b) The set $\Gamma:=\lambda^{-1}\mathbb{Z}\times\lambda^{-1}\mathbb{Z}^{d}\times\mathbb{Z}^{d}\subset\mathbf{H}_{d}$ used in the example forms a uniform subgroup of $\mathbf{H}_{d}$, i.e., a discrete co-compact subgroup of $\mathbf{H}_{d}$. The quotient subgroup $\Gamma/Z(\mathbf{H}_{d})$ is isomorphic to the lattice $\mathbb{Z}^{d}\times\lambda^{-1}\mathbb{Z}^{d}$. Furthermore, the set $[0,\lambda^{-1})^{d}$ is a fundamental domain for the lattice $\lambda^{-1}\mathbb{Z}^{d}$ in $\mathbb{R}^{d}$.

(c) The Heisenberg group is a semidirect product $M\rtimes H$ of the normal subgroup $M=\{(z,y,0)\mid z\in\mathbb{R},y\in\mathbb{R}^{d}\}$ and the subgroup $H=\{(0,0,x)\mid x\in\mathbb{R}^{d}\}$ (in the jargon: of a polarizing normal subgroup $M$ and the subgroup $H\cong M\setminus\mathbf{H}_{d}$). The variables in $M$ act by modulation, whereas the variables in $H$ act by translation.

To generalize this elementary construction of orthonormal bases on $\mathbf{H}_{d}$ to other nilpotent groups, we need to find an appropriate coordinate system and representations of $G$ in which the variables split into some form of modulation and translation operators.

3. Some Tools from the Theory of Nilpotent Lie Groups

Throughout the paper we will make heavy use of the theory of nilpotent Lie groups and their irreducible unitary representations. Our main reference is the outstanding monograph of Corwin and Greenleaf [8]. We briefly recall some well-known facts about strong Malcev bases, rational structures, induced representations, square-integrable representations and graded groups. For more details confer Corwin and Greenleaf [8, Ch. 1] as well as Fischer and Ruzhansky [13, § 3.1.1]. To make the paper accessible also to applied harmonic analysts, we present these notions in more detail than is perhaps necessary for specialists.

In the following $G$ is always a connected, simply connected nilpotent Lie group with Lie algebra $\mathfrak{g}$. The vector space dual of $\mathfrak{g}$ is denoted by $\mathfrak{g}^{*}$, the Haar measure of $G$ is $\mu_{G}$.

3.1. Malcev Bases and Coordinates

On a technical level the choice of the appropriate coordinate system for $G$ will be essential. We will construct a particular set of strong Malcev coordinates [8, Thm. 1.1.13].

Lemma 3.1**.**

Let $\mathfrak{g}$ be a nilpotent Lie algebra of dimension $n$ and let $\mathfrak{g}_{1}\leq\ldots\leq\mathfrak{g}_{l}\leq\mathfrak{g}$ be ideals with $\dim(\mathfrak{g}_{j})=m_{j}$. Then there exists a basis $\{X_{1},\ldots,X_{n}\}$ such that

• (i)

for each $1\leq m\leq n$, $\mathfrak{h}_{m}:=\mathbb{R}\text{-}\mathrm{span}\{X_{1},\ldots,X_{m}\}$ is an ideal of $\mathfrak{g}$;

• (ii)

for $1\leq j\leq l$, $\mathfrak{h}_{m_{j}}=\mathfrak{g}_{j}$.

A basis satisfying (i) and (ii) is called a strong Malcev basis of $\mathfrak{g}$ passing through the ideals $\mathfrak{g}_{1},\ldots,\mathfrak{g}_{l}$.

If we do not list any particular ideals, we will simply refer to the basis as a strong Malcev basis.

Definition 3.2**.**

Given a strong Malcev basis $\{X_{1},\ldots,X_{n}\}$, the strong Malcev coordinates of $G$ are defined by the map

[TABLE]

The Haar measure $\mu_{G}$ of a set $E\subseteq G$ is given by $\mu_{G}(E)=m(\phi^{-1}(E))$, where $m$ is the Lebesgue measure on $\mathbb{R}^{n}$. In Section 4 we will fix a Malcev basis and the corresponding Haar measure. To determine the precise value of constants, we will therefore keep track of all basis changes.

The following two properties of strong Malcev coordinates [8, Prop. 1.2.7] will be used later.

Lemma 3.3**.**

Let $\{X_{1},\ldots,X_{n}\}$ be a strong Malcev basis for $G$. Then there exist polynomial functions $P_{X_{1}},\ldots,P_{X_{n}}:\mathbb{R}^{2n}\to\mathbb{R}$ of degree $\leq 2n$ such that for $t=(t_{1},\ldots,t_{n}),s=(s_{1},\ldots,s_{n})\in\mathbb{R}^{n}$ the group multiplication is given by

[TABLE]

The polynomials have the following properties:

• (i)

$P_{X_{n}}(t,s)=t_{n}+s_{n}$* and $P_{X_{n-1}}(t,s)=t_{n-1}+s_{n-1}$.*

• (ii)

If $1\leq j\leq n-2$, then $P_{X_{j}}(t,s)=t_{j}+s_{j}+\tilde{P}_{X_{j}}(t_{j+1},\ldots,t_{n},s_{j+1},\ldots,s_{n})$, where $2\leq\deg(\tilde{P}_{X_{j}})\leq(n-j)^{2}$.

• (iii)

$P_{X_{j}}(t,-t)=0$* for $j=1,\dots,n$.*

Given a strong Malcev basis, there exist polynomial functions $R_{X_{1}},\ldots,R_{X_{n}}:\mathbb{R}^{n}\to\mathbb{R}$ such that

[TABLE]

with similar properties as above:

• (i)

$R_{X_{n}}(t)=t_{n}$ and $R_{X_{n-1}}(t)=t_{n-1}$,

• (ii)

$R_{X_{j}}(t)=t_{j}+\tilde{R}_{X_{j}}(t_{i+1},\ldots,t_{n})$, where $2\leq\deg(\tilde{R}_{j})\leq n$.

The $R_{X_{j}}$ are the exponential coordinates of $g=\phi(t)$.

For detailed proofs see [8, Prop. 1.2.7].

3.2. Rational Structures, Uniform Subgroups, and Quasi-Lattices

Strong Malcev bases allow for the construction of discrete subsets of nilpotent groups which provide a convenient disjoint partition of $G$. Under certain conditions on the Lie bracket these discrete subsets are even subgroups. We recall the following definitions and statements from [8, §  5.1].

Definition 3.4**.**

Let $\mathfrak{g}$ be an $n$-dimensional Lie algebra. We say $\mathfrak{g}$ has a rational structure if there is an $\mathbb{R}$-basis $\{X_{1},\ldots,X_{n}\}$ for $\mathfrak{g}$ with rational structure constants. In particular, $\mathfrak{g}=\mathfrak{g}_{\mathbb{Q}}\otimes\mathbb{R}$ for $\mathfrak{g}_{\mathbb{Q}}:=\mathbb{Q}\text{-}\mathrm{span}\{X_{1},\ldots,X_{n}\}$.

Definition 3.5**.**

Let $G$ be a nilpotent Lie group and $\Gamma$ a discrete subgroup. We say that $\Gamma$ is a uniform subgroup if one of the following equivalent conditions holds:

• (i)

$G/\Gamma$* is compact. In this case there exists a relatively compact fundamental domain $\Sigma\subset G$ such that $\Sigma\Gamma=G$.*

• (ii)

There exists a strong Malcev basis $\{X_{1},\ldots,X_{n}\}$ of $\mathfrak{g}$ such that

[TABLE]

In this case we say $\{X_{1},\ldots,X_{n}\}$ is strongly based on $\Gamma$.

Theorem 3.6**.**

Let $G$ be nilpotent with Lie algebra $\mathfrak{g}$.

• (i)

If $G$ has a uniform subgroup $\Gamma$, then $\mathfrak{g}$ has a rational structure such that $\mathfrak{g}_{\mathbb{Q}}=\mathbb{Q}\text{-}\mathrm{span}\{\log(\Gamma)\}$.

• (ii)

Conversely, if $\mathfrak{g}$ has a rational structure from a $\mathbb{Q}$-algebra $\mathfrak{g}_{\mathbb{Q}}\leq\mathfrak{g}$, then $G$ has a uniform subgroup $\Gamma$ such that $\log(\Gamma)\subseteq\mathfrak{g}_{\mathbb{Q}}$.

Remark 3.7**.**

If $G$ possesses a uniform subgroup strongly based on the given Malcev basis $\{X_{1},\ldots,X_{n}\}$, then the polynomial functions from Lemma 3.3 have the additional property that

(iii) the $P_{X_{j}}$ and $R_{X_{j}}$ have rational coefficients.

See [8, Thm. 5.4.2].

A uniform $\Gamma$ as in (ii) can be obtained by rescaling any strong Malcev basis with $\mathfrak{g}_{\mathbb{Q}}:=\mathbb{Q}\text{-}\mathrm{span}\{X_{1},\ldots,X_{n}\}$ by a sufficiently large integer.

In the absence of a rational structure we can still work with quasi-lattices.

Definition 3.8**.**

A discrete subset $\Gamma$ of $G$ is called a quasi-lattice with relatively compact fundamental domain $\Sigma$ if $G=\bigcup_{\gamma\in\Gamma}\Sigma\gamma$ is a disjoint union of translates.

Quasi-lattices always exist [17], and an explicit construction can be obtained by adapting the proof of Theorem 5.3.1 in [8].

Lemma 3.9**.**

Let $\{X_{1},\ldots,X_{n}\}$ be a strong Malcev basis of $\mathfrak{g}$. Then

[TABLE]

is a quasi-lattice in $G$ with fundamental domain

[TABLE]

Proof.

Existence: Let us set $\gamma_{j}:=\exp(X_{j})$. We prove by induction on $\dim(G)=n$ that for every $g\in G$ there exist $k_{1},\ldots,k_{n}\in\mathbb{Z}$ and some $t_{1},\ldots,t_{n}\in[0,1)$ such that

[TABLE]

The base case $n=1$ is trivial.

Induction step $n\mapsto n+1$: For general $n=\dim(G)-1$ we denote by $\mathfrak{g}_{n}$ the ideal $\mathbb{R}\text{-}\mathrm{span}\{X_{1},\ldots,X_{n}\}$ and set $G_{n}:=\exp(\mathfrak{g}_{n})$. We know that for $g\in G$ we have

[TABLE]

with $g_{n}\in G_{n}$. We write $x_{n+1}=t_{n+1}+k_{n+1}$ for some uniquely determined $t_{n+1}\in[0,1)$ and $k_{n+1}\in\mathbb{Z}$. A short computation yields

[TABLE]

for some $g^{\prime}_{n}\in G_{n}\vartriangleleft G$. By applying the induction hypothesis to $g^{\prime}_{n}$, we then obtain

[TABLE]

Uniqueness: Assume that

[TABLE]

for some $g_{n-1},g^{\prime}_{n-1}\in G_{n-1}$. Then for $y_{n}:=t^{\prime}_{n}-t_{n}$ we have

[TABLE]

for some $g^{\prime\prime}_{n-1}\in G_{n-1}$. So ${g^{\prime\prime}_{n-1}}^{-1}\,g_{n-1}\in\exp(\mathbb{R}X_{n})$, but $G_{n-1}\bigcap\exp(\mathbb{R}X_{n})=\{e\}$. We conclude that $g^{\prime\prime}_{n-1}=g_{n-1}$ and $t^{\prime}_{n}-t_{n}+k^{\prime}_{n}-k_{n}=0$, hence $t^{\prime}_{n}=t_{n}$ and $k^{\prime}_{n}=k_{n}$. By repeating this argument for $n-1,n-2,\ldots,1$, we obtain the uniqueness. ∎

Corollary 3.10**.**

Let $\{X_{1},\ldots,X_{n}\}$ be a strong Malcev basis strongly based on $\Gamma$ passing through an ideal $\mathfrak{n}\vartriangleleft\mathfrak{g}$, and let $\Gamma$ be a quasi-lattice (or a uniform subgroup) of $G$ with fundamental domain $\Sigma$. Let $N=\exp(\mathfrak{n})$ and $\mathop{\mathrm{pr}}:G\to N\setminus G$ denote the natural map onto the quotient $N\setminus G$. Then $\mathop{\mathrm{pr}}(\Gamma)$ is a quasi-lattice (a uniform subgroup) of $N\setminus G$ with fundamental domain $\mathop{\mathrm{pr}}(\Sigma)$.

Proof.

Suppose $\dim(N)=n^{\prime}<n$, i.e., $\mathfrak{n}=\mathbb{R}\text{-}\mathrm{span}\{X_{1},\ldots,X_{n^{\prime}}\}$ (cf. Definition 3.1). If $\Gamma$ is a quasi-lattice, then it is clear from the proof of Lemma 3.9 that

[TABLE]

is a quasi-lattice of $N\setminus G$ with fundamental domain

[TABLE]

The same reasoning applies if $\Gamma$ is a uniform subgroup of $G$, since the proof of Lemma 3.9 is an adaption of the proof of [8, Thm. 5.3.1]. (Alternatively, it is mentioned in [8, Thm. 5.3.1] that $\Gamma\cap G_{n^{\prime}}$ is uniform in $G_{n^{\prime}}$; so by [8, Lem. 5.1.4 (a)], $\mathop{\mathrm{pr}}(\Gamma)$ is uniform in $G_{n^{\prime}}\setminus G$.) ∎

3.3. Kirillov Theory of Irreducible Representations

According to Kirillov, every irreducible unitary representation $\pi$ of $G$ is constructed by the following recipe: Choose $l\in\mathfrak{g}^{*}$ and choose a subalgebra $\mathfrak{m}$ of maximal dimension, a so-called polarization, such that $l([Y_{1},Y_{2}])=0$ for all $Y_{1},Y_{2}\in\mathfrak{m}$. Consequently, $\chi_{l}:M\to\mathbb{T}$ defined by

[TABLE]

is a character of $M$. The representation associated to $l$ is the representation $\pi_{l}=\mathop{\mathrm{ind}^{G}_{M}(\chi_{l})}$ induced from $\chi_{l}$ to $G$. It acts on the representation space $L^{2}(M\setminus G)\cong L^{2}(\mathbb{R}^{d})$. It is well known that $(a)$ different polarizations of $l$ yield equivalent representations, and $(b)$ different functionals $l$ and $l^{\prime}\in\mathfrak{g}^{*}$ in the same coadjoint orbit $\mathcal{O}_{l}=\mathop{\mathrm{Ad}^{*}}(G)l$ yield equivalent representations. Thus, every orbit corresponds to a unique equivalence class of irreducible unitary representations. In particular, we are free to choose the most convenient functional $l$ and polarization $\mathfrak{m}$ when we write a representation explicitly.

If a polarization $\mathfrak{m}$ is an ideal of $\mathfrak{g}$, $\mathfrak{m}\vartriangleleft\mathfrak{g}$, the induced representation $\pi_{l}=\mathop{\mathrm{ind}^{G}_{M}(\chi_{l})}$ possesses the following explicit description. Let $H=M\setminus G$ be the quotient group, $q:G\mapsto M\setminus G$ be the quotient map, and choose a Borel cross-section $\sigma:M\setminus G\to G$ such that $q\circ\sigma=\mathop{\mathrm{id}}$. Then every element $g\in G$ can be written in a unique way as

[TABLE]

with $p(g)=g\sigma(q(g))^{-1}\in M$. For $h^{\prime}\in\sigma(M\backslash G)$ we write

[TABLE]

and $h^{\prime}\hskip 2.0ptm\hskip 2.0pth^{\prime-1}\in M\vartriangleleft G$. Then the induced representation acting on $f\in L^{2}(M\setminus G)$ is given by

[TABLE]

where the second equality holds because $\chi_{l}$ is a character on $M$. Let us point out that

[TABLE]

This observation explains why a convenient description of the coadjoint orbit $\mathcal{O}_{l}$ helps to understand $\pi_{l}$. We will elaborate further in Subsection 4.1.

3.4. Square-Integrable Representations

An irreducible unitary representation $\pi$ of $G$ is square-integrable modulo the center of $G$ if there exists a vector $v\in\mathcal{H}_{\pi}$ such that

[TABLE]

We write $\pi\in SI/Z(G)$ when $\pi$ is square-integrable modulo the center of $G$. A fundamental theorem of Moore and Wolf [28] characterizes all irreducible representations in $SI/Z(G)$ of a nilpotent group by the flat-orbit condition.

Theorem 3.11** ([28]).**

A representation $\pi_{l}$ of a connected, simply connected nilpotent group $G$ is in $SI/Z(G)$ if and only if the coadjoint orbit $\mathcal{O}_{l}=\mathrm{Ad}(G)^{*}l$ is flat, if and only if $\mathcal{O}_{l}\oplus\mathfrak{z}(\mathfrak{g})^{*}=\mathfrak{g}^{*}$, if and only if $\mathcal{O}_{l}=l+\mathfrak{z}(\mathfrak{g})^{\perp}$, if and only if the symplectic form $B_{l}(\,.\,,\,.\,):=l([\,.\,,\,.\,])$ is non-degenerate on $\mathfrak{g}/\mathfrak{z}(\mathfrak{g})$.

For any such $\pi_{l}$ one can restrict to representatives $l\in\mathfrak{z}(\mathfrak{g})^{*}$. Furthermore, if $SI/Z(G)\neq\emptyset$, then up to a set of Plancherel measure zero all $\pi\in\widehat{G}$ are square-integrable modulo $Z$. For each $\pi\in SI/Z(G)$ there exists a number $d_{\pi}\in\mathbb{R}\setminus\{0\}$, called the formal degree or formal dimension of $\pi$, such that

[TABLE]

holds true for all $v_{1},v_{2},w_{1},w_{2}\in\mathcal{H}_{\pi}$.

The formal degree depends on the normalization of the Haar measure $\mu$ on $G$ and is intimately connected to the symplectic form $B_{l}$: Pick a strong Malcev basis $\{Z_{1},\ldots Z_{r},X_{1},\ldots,X_{2d}\}$ passing through $\mathfrak{z}(\mathfrak{g})$. Normalize the Lebesgue measure $\nu$ on $\mathfrak{g}$ such that

[TABLE]

and denote by $\mu$, $\mu_{G}$ or $dg$ the Haar measure on $G$ arising from $\nu$ via strong Malcev coordinates. Since $Z(G)$ is normal in $G$, we know that $dg=dz\hskip 2.0ptd\dot{g}$. We fix this Haar measure for once and all. Due to Theorem 3.11, a square-integrable representation $\pi\in SI/Z(G)$ and its orbit $\mathcal{O}_{\pi}$ depend only on the representative $l\in\mathfrak{z}(\mathfrak{g})^{*}$. So, the Pfaffian polynomial $\mathrm{Pf}$ defined by

[TABLE]

can be considered as a polynomial of $\mathfrak{z}(\mathfrak{g})^{*}$. By [28, Thm. 4], $d_{\pi}$ and $\mathrm{Pf}$ depend on the normalization of $\mu$ in the same way; in fact we have

[TABLE]

Example 3.12**.**

If $G$ is the Heisenberg group $\mathbf{H}_{d}$ and $\pi=\pi_{\lambda}$, the Schrödinger representation with parameter $\lambda\neq 0$, then $d_{\pi}=\left|\lambda\right|^{d}$. (Cf. Example 2.1.)

3.5. Graded Groups

Definition 3.13**.**

A Lie algebra $\mathfrak{g}$ is graded if it possesses a vector space decomposition $\mathfrak{g}=\bigoplus_{k=1}^{\infty}\mathfrak{g}_{k}$ such that $[\mathfrak{g}_{k},\mathfrak{g}_{k^{\prime}}]\subseteq\mathfrak{g}_{k+k^{\prime}}$ and $\mathfrak{g}_{k}\neq\{0\}$ for only finitely many $k$.

We will write $\mathfrak{g}_{N}$ for the summand of highest order, whence $\mathfrak{g}_{k}=\{0\}$ for all $k>N$. However, some $\mathfrak{g}_{k}$ with $k<N$ may be trivial, too.

Every graded Lie algebra is nilpotent and the corresponding connected, simply connected Lie group is called graded and is always nilpotent locally compact.

Remark 3.14**.**

Clearly $(\mathbb{R}^{n},+)$ and the Heisenberg group $\mathbf{H}_{d}$ are graded; $\mathbf{H}_{d}$ admits various gradations, one of which is given by

[TABLE]

Every nilpotent Lie algebra of $\dim(\mathfrak{g})\leq 6$ is graded. The lowest dimension for which there are non-graded nilpotent Lie algebras is $\dim(\mathfrak{g})=7$.

A graded Lie algebra is naturally equipped with a dilation.

Definition 3.15**.**

Let $\mathfrak{g}$ be graded Lie algebra with gradation $\mathfrak{g}=\bigoplus_{k=1}^{N}\mathfrak{g}_{k}$. We define the family of dilations $\{\delta_{s}\}_{s>0}$ on $\mathfrak{g}$ by

[TABLE]

Since we will deal with both gradations and rational structures at the same time, we need some compatibility of these structures.

Definition 3.16**.**

Let $\mathfrak{g}$ be a graded Lie algebra with gradation $\mathfrak{g}=\bigoplus_{k=1}^{N}\mathfrak{g}_{k}$. Furthermore, suppose there exists a rational structure $\mathfrak{g}_{\mathbb{Q}}\leq\mathfrak{g}$. We will say the rational structure is compatible with the gradation if there exists a strong Malcev basis $\{X_{1},\ldots,X_{n}\}$ of $\mathfrak{g}$ such that

• (i)

$\mathfrak{g}_{\mathbb{Q}}:=\mathbb{Q}\text{-}\mathrm{span}\{X_{1},\ldots,X_{n}\}$,

• (ii)

$\{X_{1},\ldots,X_{n}\}$* passes through the ideals $\bigoplus_{k=l}^{N}\mathfrak{g}_{k}\vartriangleleft\mathfrak{g}$, $l=N,\ldots,1$.*

4. Constructing Orthonormal Bases

In this section we construct orthonormal bases in the orbit of every square-integrable representation of a graded $SI/Z$-group. Our main theorems are the following.

Theorem 4.1**.**

[Quasi-Lattice] Let $G$ be a graded $SI/Z$-group of $\dim(G)=2d+1$ with $1$-dimensional center. Then there exists a normal subgroup $M\vartriangleleft G$ with the following properties:

• (i)

The ideal $\mathfrak{m}=\log(M)\vartriangleleft\mathfrak{g}$ is a polarization simultaneously for all $l\in\mathfrak{z}(\mathfrak{g})^{*}$. It thus induces all $\pi\in SI/Z(G)$ and permits a universal realization of all $\pi\in SI/Z(G)$ in $L^{2}(M\setminus G)$ related to the same set of strong Malcev coordinates for $G$.

• (ii)

For every $\pi\in SI/Z(G)$ with formal degree $d_{\pi}$ there exist a discrete subset $\Gamma\subseteq G/Z(G)$ and a relatively compact set $F\subseteq M\setminus G$, such that

[TABLE]

forms an orthonormal basis of $L^{2}(M\setminus G)$. Furthermore, $\left\|1_{F}\right\|_{L^{2}(M\setminus G)}=d_{\pi}^{1/2}$.

The construction shows that $\Gamma$ is derived from a quasi-lattice of $G/Z(G)$ by a small modification.

Theorem 4.2**.**

[Uniform Subgroup] Let $G$ be a graded $SI/Z$-group of $\dim(G)=2d+1$ with $1$-dimensional center and a rational structure $\mathfrak{g}_{\mathbb{Q}}$ compatible with the gradation. Then there exists a normal subgroup $M\vartriangleleft G$ with the following properties:

• (i)

The ideal $\mathfrak{m}=\log(M)\vartriangleleft\mathfrak{g}$ is a polarization simultaneously for all $l\in\mathfrak{z}(\mathfrak{g})^{*}$. It thus induces all $\pi\in SI/Z(G)$ and permits a universal realization of all $\pi\in SI/Z(G)$ in $L^{2}(M\setminus G)$ related to the same set of strong Malcev coordinates for $G$.

• (ii)

For every $\pi\in SI/Z(G)$ there exist a uniform subgroup $\Gamma\leq G/Z(G)$, a relatively compact set $F\subseteq M\setminus G$, such that

[TABLE]

forms an orthonormal basis of $L^{2}(M\setminus G)$. Furthermore $\left\|1_{F}\right\|_{L^{2}(M\setminus G)}=C^{-1}\hskip 2.0ptd_{\pi}^{1/2}$ for some integer $C$ depending on $\Gamma$.

We will prove a more general, but more technical statement (Theorem 4.7) and then derive Theorems 4.1 and 4.2 as special cases. We will structure the proofs as follows:

• (i)

Formulate an explicit parametrization of the coadjoint orbits by means of the theorem of Chevalley-Rosenlicht and construct a quasi-lattice or a uniform subgroup of $G$.

• (ii)

Prove the orthogonality.

• (iii)

Prove the completeness of the set in (10).

Throughout we make the following assumptions:

• –

$G$ is a connected, simply connected nilpotent $SI/Z$-group of dimension $n=r+2d$ with $r$-dimensional center $Z(G)$.

• –

$\pi$ is a square-integrable representation of $G$ modulo its center.

• –

The coadjoint orbit that determines $\pi$ has an element $l\in\mathfrak{z}(\mathfrak{g})^{*}$.

• –

There exists an ideal $\mathfrak{m}\vartriangleleft\mathfrak{g}$ which is a polarization for $l\in\mathfrak{z}(\mathfrak{g})^{*}$.

• –

The representation is realized on $L^{2}(M\setminus G)$, where $M:=\exp(\mathfrak{m})$.

We will denote by $\mathfrak{h}$ the complementary subspace such that $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$. Since $\mathfrak{m}$ is an ideal, $\mathfrak{h}$ is a nilpotent Lie algebra modulo $\mathfrak{m}$. The natural quotient maps will be denoted by $\mathop{\mathrm{pr}}:G\to G/Z(G)$, $q:G\to M\setminus G$. For $\mathop{\mathrm{pr}}(g)$ we also write $\dot{g}$. Given a cross-section $\sigma:G\setminus M\to G$, the projection $p:G\to M$ occurring in the induced representation is given by the map $p:G\to M$, $p(g)=g\sigma(q(g))^{-1}\in M$.

4.1. Explicit Parameterizations of Coadjoint Orbits

Our first goal is to write the induced representation $\pi=\pi_{l}$ associated to $l\in\mathfrak{z}(\mathfrak{g})$ as explicitly as possible. We recall that $\pi$ can be written as

[TABLE]

The occurrence of $\left\langle\mathop{\mathrm{Ad}^{*}}(h^{\prime-1})l,\log(m)\right\rangle$ suggests to describe the coadjoint orbit $\mathcal{O}_{l}$ in more detail.

A celebrated theorem by Chevalley and Rosenlicht provides useful parametrizations of the orbits of unipotent group actions. We will formulate a version of the theorem adapted to the coadjoint action. For a general version we refer to [8], Theorem 3.1.4. and Section 3.4, for related parametrizations in solvable groups to [2, 3].

Theorem 4.3** (Chevalley-Rosenlicht).**

Let $G$ be a connected, simply connected nilpotent $SI/Z$-group of dimension $n=r+2d$ with $(r+d)$-dimensional ideal $\mathfrak{m}\vartriangleleft\mathfrak{g}$. Denote by $\cdot:\mathfrak{m}^{*}\times G\to\mathfrak{m}^{*}$ the restriction of the coadjoint action to $\mathfrak{m}^{*}$, written as a right group action,

[TABLE]

Then $\mathfrak{m}^{*}\cdot G$ is a unipotent action.

If $\mathfrak{m}$ is a polarization for some $l\in\mathfrak{z}(\mathfrak{g})^{*}$, then the orbit $l\cdot G$ coincides with the $d$-dimensional affine subspace $l+\mathfrak{z}(\mathfrak{g})^{\perp}\subseteq\mathfrak{m}^{*}$ and has the following properties:

• (i)

For a fixed strong Malcev basis $\{Z_{1},\ldots,Z_{r},Y_{1},\ldots,Y_{d},X_{1},\ldots,X_{d}\}$ of $\mathfrak{g}$ passing through $\mathfrak{z}(\mathfrak{g})$ and $\mathfrak{m}$ there exists a basis $\{\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ of $\mathfrak{h}$ such that the map $\phi:\mathbb{R}^{d}\to l\cdot G$ defined by

[TABLE]

is a diffeomorphism. In particular, $l\cdot G=\mathrm{Ad}^{*}(G)l=l\cdot\exp(\mathbb{R}\tilde{X}_{d})\cdots\exp(\mathbb{R}\tilde{X}_{1})$.

• (ii)

The basis is chosen such that the infinitesimal action $\mathfrak{m}^{*}\cdot\mathfrak{g}$ satisfies

[TABLE]

In particular, $l\cdot G=l+l\cdot\mathfrak{g}=l+\mathbb{R}\text{-}\mathrm{span}\{Y^{*}_{1},\ldots,Y^{*}_{d}\}$.

• (iii)

The polynomials $Q_{1},\ldots,Q_{d}$ defined by

[TABLE]

are of the form

[TABLE]

for all $j=1,\ldots,d$.

Proof.

Since $l\to l\cdot g$ is the restriction of $\mathrm{Ad}^{*}(g^{-1})l$ to $\mathfrak{m}^{*}$ and $\mathfrak{m}$ is an ideal, $\mathfrak{m}^{*}$ is an invariant subspace of $\mathrm{Ad}^{*}$, and the restriction of a unipotent action remains unipotent.

To prove (i) — (iii), we apply [8, Thm. 3.1.4] to $\mathfrak{m}^{*}\cdot G$. Let $l\in\mathfrak{z}(\mathfrak{g})^{*}$ be such that $\mathfrak{m}$ is a polarization for $l$. Flatness of the orbit $l\cdot G$ follows from that fact that $l\cdot G$ is the image under the projection $\mathfrak{g}^{*}\to\mathfrak{m}^{*}=\mathfrak{g}^{*}/\mathfrak{m}^{\perp}$ of the coadjoint orbit $\mathop{\mathrm{Ad}^{*}}(G)l=l+\mathfrak{z}(\mathfrak{g})^{\perp}\subseteq\mathfrak{g}^{*}$.

Recall that for any strong Malcev basis $\{Z_{1},\ldots,X_{d}\}$ of $\mathfrak{g}$ the sequence of subspaces $\mathbb{R}\text{-}\mathrm{span}\{Z^{*}_{1},\ldots,X^{*}_{d}\}\leq\mathbb{R}\text{-}\mathrm{span}\{Z^{*}_{2},\ldots,X^{*}_{d}\}\leq\ldots\leq\mathfrak{g}^{*}$ forms a Jordan-Hölder flag for the coadjoint action of $G$. This property is then inherited by $\mathfrak{m}^{*}\cdot G$ if the Malcev basis passes through $\mathfrak{z}^{*}$ and $\mathfrak{m}^{*}$: If we quotient the vector space $\mathfrak{g}^{*}$ and, a fortiori, $G\to\mathop{\mathrm{Aut}}(\mathfrak{g}^{*}):g\mapsto\mathop{\mathrm{Ad}^{*}}(g^{-1})$ by the subspace $\mathfrak{h}^{*}$, then $\mathbb{R}\text{-}\mathrm{span}\{Z^{*}_{1},\ldots,Y^{*}_{d}\}\leq\mathbb{R}\text{-}\mathrm{span}\{Z^{*}_{2},\ldots,Y^{*}_{d}\}\leq\ldots\leq\mathfrak{m}^{*}$ forms a Jordan-Hölder flag for $\mathfrak{m}^{*}\cdot G$. So [8, Thm. 3.1.4] yields sufficiently many vectors $\tilde{X}_{j}\in\mathfrak{g}$ such that their combined action generates $l\cdot G$. The particular choice of $\{\tilde{X}_{j}\}_{j}$ in the proof of [8, Thm. 3.1.4] yields (i) — (iii) except for the fact that these vectors form a basis of $\mathfrak{h}$.

To show this, let $Y,v\in\mathfrak{m},X\in\mathfrak{h}$. Then $\mathop{\mathrm{Ad}}(\exp(X))v\in\mathfrak{m}\vartriangleleft\mathfrak{g}$ and

[TABLE]

since $l$ vanishes on $[\mathfrak{m},\mathfrak{m}]$. Thus, the contribution of $M$ to $l\cdot G$ is trivial and we can choose the $\tilde{X}_{j}$ to be in $\mathfrak{h}$. The linear independence of the $\tilde{X}_{j}$ becomes evident in view of the symplectic form $B_{l}$. The matrix representation of $B_{l}$ with respect to the basis $\{Z_{1},\ldots,X_{d}\}$ is of the form

[TABLE]

Since $B_{l}$ is non-degenerate (Theorem 3.11), the matrix $A$ is invertible; so the infinitesimal action restricted to $\mathfrak{h}$ has full rank and the vectors $\tilde{X}_{d},\ldots,\tilde{X}_{1}$ must form a basis of $\mathfrak{h}$. ∎

The following corollary is an immediate consequence of Theorem 4.3 (ii).

Corollary 4.4**.**

*The change of basis from $\{Z_{1},\ldots,Z_{r},Y_{1},\ldots,Y_{d},X_{1},\ldots,X_{d}\}$

to $\{Z_{1},\ldots,Z_{r},Y_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ in Theorem 4.3 (i) — (ii) yields the matrix representation of the symplectic form*

[TABLE]

∎

Note that the vectors $\tilde{X}_{d},\ldots,\tilde{X}_{1}$ are not uniquely determined. According to (12) one has the liberty to add contributions of higher index in each step: $\tilde{X}_{d}$ is fixed by the choice of $l$ and $Y_{d}$, whereas one can add some multiple of $\tilde{X}_{d}$ to $\tilde{X}_{d-1}$ without changing the assertions of the theorem, and so forth. The triangularization of $A$ can be interpreted as the first step of an algorithm which produces the coefficient polynomials $Q_{j},j=1,\ldots,d$, from Theorem 4.3 (iii).

In what follows we would like to use the basis $\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ to construct a quasi-lattice or a uniform subgroup in $G$. According to Lemma 3.9 we need a Malcev basis, not just a vector space basis, for this construction. This motivates the following definition.

Definition 4.5**.**

Let $G$, $\mathfrak{m}$, $\mathfrak{h}$ be as above and let $l\in\mathfrak{z}(\mathfrak{g})^{*}$. Let $\{Z_{1},\ldots,Y_{d}\}$ be a strong Malcev basis of $\mathfrak{m}$ passing through $\mathfrak{z}(\mathfrak{g})$ and let $\{\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ be a basis of $\mathfrak{h}$ satisfying the conclusions (i) and (ii) of Theorem 4.3 with respect to $l$. If the union $\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ forms a strong Malcev basis of $\mathfrak{g}$, it is called a Chevalley-Rosenlicht-admissible (or Ch-R-admissible) Malcev basis of $\mathfrak{g}$ subordinate to $l$.

In this paper Chevalley-Rosenlicht admissible Malcev bases are simply a technical tool. At this time we do not know whether and when a Ch-R-admissible basis exists in a nilpotent Lie algebra. In Subsection 4.3 we will see that every graded Lie group with one-dimensional center admits a Ch-R-admissible basis.

Since the basis of the complement $\mathfrak{h}$ in Theorem 4.3 is not unique, the first question is whether the definition of Ch-R-admissible bases depends on the choice of the basis. Fortunately this is not the case.

Lemma 4.6**.**

Assume that $\mathfrak{g}$ possesses a Ch-R-admissible Malcev basis subordinate to $l\in\mathfrak{z}(\mathfrak{g})^{*}$. Then every basis satisfying (12) is Ch-R-admissible.

Proof.

Let $\{\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ and $\{\tilde{\tilde{X}}_{d},\ldots,\tilde{\tilde{X}}_{1}\}$ be two bases of $\mathfrak{h}$ to parametrize the orbit $l\cdot G$ in $\mathfrak{m}^{*}\cdot G$. According to (32) this means that the $d\times d$-matrix $\tilde{A}$ with entries $\tilde{A}_{j,k}:=l([\tilde{X}_{j},Y_{k}])$ is an upper triangular matrix with ones on the diagonal. Likewise, $\tilde{\tilde{A}}$, with $\tilde{\tilde{A}}_{j,k}:=l([\tilde{\tilde{X}}_{j},Y_{k}])$, is upper triangular. Consequently, $C:=\tilde{\tilde{A}}\tilde{A}^{-1}$ is upper triangular. In fact, the matrix $C$ describes the basis change from $\{\tilde{X}_{j}\}_{j=1}^{d}$ to $\{\tilde{\tilde{X}}_{j}\}_{j=1}^{d}$, because for $\tilde{\tilde{X}}_{j}=\sum_{n\geq j}c_{j,n}\hskip 2.0pt\tilde{X}_{n}$ we have

[TABLE]

Now, assume that $\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ is a strong Malcev basis for $\mathfrak{g}$. Then for $j<k$ we have $[\tilde{X}_{j},\tilde{X}_{k}]\in\mathbb{R}\text{-}\mathrm{span}\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{k}\}:=\mathfrak{g}_{k}$, or $[\tilde{X}_{j},\tilde{X}_{k}]\in\mathfrak{g}_{\max(j,k)}$. Since $C$ is upper triangular, we find that $\tilde{\tilde{X}}_{k}=\sum_{m\geq k}c_{k,m}\hskip 2.0pt\tilde{X}_{m}\in\mathfrak{g}_{k}$ and consequently, $\mathfrak{g}_{k}=\mathbb{R}\text{-}\mathrm{span}\{Z_{1},\ldots,Y_{d},\tilde{\tilde{X}}_{d},\ldots,\tilde{\tilde{X}}_{k}\}$. We check the Lie bracket for the basis $\{\tilde{\tilde{X}}_{j}\}_{j=1}^{d}$. For $j<k$ we have

[TABLE]

and therefore $[\tilde{\tilde{X}}_{j},\tilde{\tilde{X}}_{k}]\in\mathfrak{g}_{k}$. This implies that $\{Z_{1},\ldots,Y_{d},\tilde{\tilde{X}}_{d},\ldots,\tilde{\tilde{X}}_{1}\}$ is a strong Malcev basis for $\mathfrak{g}$.

Clearly, the argument is symmetric. So, $\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ is Ch-R-admissible subordinate to $l$ if and only if $\{Z_{1},\ldots,Y_{d},\tilde{\tilde{X}}_{d},\ldots,\tilde{\tilde{X}}_{1}\}$ is so. ∎

Once we know that $\mathfrak{g}$ possesses a Ch-R-admissible basis subordinate to $l$, we have a certain freedom to choose. A distinguished Ch-R-admissible basis is obtained by choosing the basis transformation to be $C=\tilde{A}^{-1}$ in (33). Then $\tilde{\tilde{A}}=C\tilde{A}=\tilde{A}^{-1}\tilde{A}=I$ or, in other words,

[TABLE]

Thus, if the action $\mathfrak{m}^{*}\times G\to\mathfrak{m}^{*}$ possesses a Ch-R-admissible basis subordinate to $l$, then it possesses a distinguished Ch-R-admissible basis that diagonalizes the infinitesimal coadjoint action on $l$.

4.2. A Technical Version of the Main Theorem

We formulate and prove a general, but technical version for the existence of orthonormal bases associated to a representation $\pi\in SI/Z(G)$.

Theorem 4.7**.**

*Let $G$ be a connected, simply connected nilpotent $SI/Z$-group of dimension $n=r+2d$ with $(r+d)$-dimensional ideal $\mathfrak{m}\vartriangleleft\mathfrak{g}$. Fix a strong Malcev basis

$\{Z_{1},\ldots,Z_{r},Y_{1},\ldots,Y_{d},X_{1},\ldots,X_{d}\}$ of $\mathfrak{g}$ which passes through $\mathfrak{z}(\mathfrak{g})$ and $\mathfrak{m}$; use it to fix the Haar measures $\mu_{G}$ and $\mu_{Z(G)}$. Let $\pi\in SI/Z(G)$ with representative $l\in\mathfrak{z}(\mathfrak{g})^{*}$.*

(A) If

• (i)

$\mathfrak{m}$* is a polarization for $l$,*

• (ii)

$\mathfrak{g}$* has a Ch-R-admissible Malcev basis subordinate to $l$ of the form *

$\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$,

*then there exist a discrete subset $\Gamma\subseteq G/Z(G)$ and a relatively compact subset $F\subseteq M\setminus G$ such that the set *

[TABLE]

forms an orthonormal basis of $L^{2}(M\setminus G)$.

(B) If furthermore

• (iii)

$\mathfrak{g}_{\mathbb{Q}}:=\mathbb{Q}\text{-}\mathrm{span}\{Z_{1},\ldots,X_{d}\}$* forms a rational subalgebra of $\mathfrak{g}$,*

• (iv)

$l\in{\mathfrak{z}(\mathfrak{g}_{\mathbb{Q}})}^{*}$,

• (v)

$\mathfrak{g}_{\mathbb{Q}}$* possesses a Ch-R-admissible basis $\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ subordinate to $l$,*

then there exist a uniform subgroup $\Gamma\leq G/Z(G)$ of co-volume $\mu_{G/Z(G)}(\Sigma)=d_{\pi}^{-1}$ with fundamental domain $\Sigma$, such that for $F:=q(\Sigma)\subseteq M\setminus G$ the set

[TABLE]

forms an orthonormal basis of $L^{2}(M\setminus G)$. Furthermore, $\left\|1_{F}\right\|_{L^{2}(M\setminus G)}=C^{-1}\hskip 2.0ptd_{\pi}^{1/2}$ for some integer $C\in\mathbb{N}$.

The theorem will be proved in three steps: First we construct a quasi-lattice or a uniform subgroup in $G/Z(G)$, then we verify the orthonormality of the set (38), and finally we show its completeness.

Proposition 4.8**.**

[Step 1. Construction of Quasi-Lattice/Uniform Subgroup.]

Assume that $G$ satisfies the assumptions of Theorem 4.7.

• (i)

If $G$ satisfies conditions (i) and (ii), then there exists a quasi-lattice $\Gamma^{\prime}\subseteq G/Z(G)$ of co-volume $\mu_{G/Z(G)}(\Sigma)=d_{\pi}^{-1}$.

• (ii)

If $G$ satisfies all conditions (i) — (v), then there exists a uniform subgroup $\Gamma^{\prime}\leq G/Z(G)$ of co-volume $\mu_{G/Z(G)}(\Sigma)=d_{\pi}^{-1}$.

Proof.

(i) Let $\{Z_{1},\ldots,\tilde{X}_{d}\}$ be a Ch-R-admissible Malcev basis of $\mathfrak{g}$ subordinate to $l\in\mathfrak{z}(\mathfrak{g})^{*}$. By Lemma 3.9, the set

[TABLE]

forms a quasi-lattice of $G$ with fundamental domain

[TABLE]

and by Corollary 3.10 $\Gamma^{\prime}:=\mathop{\mathrm{pr}}(\Gamma^{\prime}_{G})$ is a quasi-lattice of $G/Z(G)$ with fundamental domain $\Sigma:=\mathop{\mathrm{pr}}(\Sigma_{G})$. Let $S_{l}$ denote the matrix of the basis change from $\{Z_{1},\ldots,Y_{d},X_{1},\ldots,X_{d}\}$ to $\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$, then the pullback under $S_{l}$ of the quotient Haar measure $\mu_{G}$ (fixed by (7)) is given by $\tilde{\mu}_{G}=\det(S_{l})^{-1}\hskip 2.0pt\mu_{G}$ [8, 1.2.11]. We now compute the formal degree $\widetilde{d_{\pi}}$ of $\pi$ with respect to $\widetilde{\mu}_{G/Z(G)}$ in two ways. On the one hand, $\widetilde{d}_{\pi}=\det(S_{l})\hskip 2.0ptd_{\pi}$ by (6). On the other hand, the relation (8) between the formal degree and the Pfaffian with respect to the basis $\{Z_{1},\ldots,\tilde{X}_{1}\}$ yields

[TABLE]

We conclude that

[TABLE]

Finally we compute the measure of the fundamental domain $\Sigma_{G}$ for $\Gamma_{G}^{\prime}$. Since $\Sigma_{G}$ is the image of the unit cube spanned by $\{Z_{1},\ldots,\tilde{X}_{1}\}$, the normalization (7) implies that $1=\widetilde{\mu}_{G/Z(G)}(\Sigma)=\det(S_{l})^{-1}\hskip 2.0pt\mu_{G/Z(G)}(\Sigma)$. Thus, we have

[TABLE]

(ii) To construct a uniform subgroup $\Gamma^{\prime}\leq G/Z(G)$, we choose a distinguished Ch-R-admissible basis $\{\tilde{X}_{j}\}_{j=1}^{d}$ which satisfies (37). By the proof of Lemma 4.6 this means that $C=A^{-1}$ and that

[TABLE]

Now assume that $\mathfrak{g}_{\mathbb{Q}}=\mathbb{Q}\text{-}\mathrm{span}\{Z_{1},\ldots,Y_{d},X_{1},\ldots,X_{d}\}$ and $l\in\mathfrak{z}(\mathfrak{\mathfrak{g}_{\mathbb{Q}}})^{*}$. Then $l=\sum_{n=1}^{r}l_{n}Z^{*}_{n}$ with coefficients $l_{n}\in\mathbb{Q}$ and

[TABLE]

again with coefficients $c^{j,k}_{m},d^{j,k}_{n}\in\mathbb{Q}$. Consequently, $l([X_{j},Y_{k}])=\sum_{n=1}^{r}l_{n}d_{n}^{j,k}$ is in $\mathbb{Q}$ and thus the matrix $A$ has only rational entries. By Kramer’s rule $A^{-1}$ has only rational entries, and (40) implies that $\tilde{X}_{j}\in\mathfrak{g}_{\mathbb{Q}}$. It follows that $\mathfrak{g}_{\mathbb{Q}}=\mathbb{Q}\text{-}\mathrm{span}\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$.

In order to construct a uniform subgroup $\Gamma^{\prime}$, we temporarily enumerate $\{Z_{1},\ldots,\tilde{X}_{1}\}$ by $\{V_{1},\ldots,V_{n}\}$. Let $P_{V_{1}},\ldots,P_{V_{n}}$ be the polynomials from (3) that describe the multiplication in $G$ in strong Malcev coordinates. Furthermore, let $K\in\mathbb{N}$ be large enough so that all denominators of all coefficients of all $P_{j}$ divide $K$. We then set

[TABLE]

The rescaled basis is

[TABLE]

If we denote by $P^{\prime}_{V_{j}}$ the polynomials from (3) for the strong Malcev basis $\{W_{1},\ldots,W_{n}\}$, then, by definition of the $q_{j}$’s,

[TABLE]

and $P^{\prime}_{V_{j}}$ has integer coefficients for all $j=1,\ldots,n$. Hence,

[TABLE]

forms a uniform subgroup of $G$ with fundamental domain

[TABLE]

By Corollary 3.10, $\Gamma^{\prime}:=\mathop{\mathrm{pr}}(\Gamma^{\prime}_{G})$ forms a uniform subgroup of $G/Z(G)$ with fundamental domain $\Sigma:=\mathop{\mathrm{pr}}(\Sigma_{G})$. Since the Jacobian determinant of the rescaling in $G/Z(G)$ equals $1$, the same reasoning as in (i) yields $\mu_{G/Z(G)}(\Sigma)=d_{\pi}^{-1}$ for case (ii). This completes the proof. ∎

Remark 4.9**.**

Multiplying the basis vectors by the factors $q_{j}$ does not change the assertion (ii) of the Chevalley-Rosenlicht theorem. The principal terms are linear in $t=(t_{d},\ldots,t_{1})$ because they correspond to the infinitesimal group action $\mathop{\mathrm{ad}^{*}}$. Indeed, our choice of factors in (41) yields

[TABLE]

Definition 4.10**.**

Let $\Gamma^{\prime}$ be a quasi-lattice or a uniform subgroup as constructed in Proposition 4.8. If $\Gamma^{\prime}$ is a quasi-lattice, set $K=1$; if $\Gamma^{\prime}$ is a uniform subgroup, let $K$ be the scaling factor in (41). Then we set

[TABLE]

Let $\Sigma$ be a fundamental domain for $\Gamma^{\prime}$ in $G$ and set $F=q(\Sigma)\subset M\setminus G$. In coordinates,

[TABLE]

Remark 4.11**.**

If $\Gamma^{\prime}$ is a uniform subgroup, then $\Gamma=\Gamma^{\prime}$. If $\Gamma^{\prime}$ is only a quasi-lattice, then $\Gamma$ is obviously related to $\Gamma^{\prime}$, but in general $\Gamma\neq\Gamma^{\prime}$.

Let $\vartheta\in\Gamma_{M}^{\prime}$ and $\eta\in\Gamma_{H}^{\prime}$, then every element of $\Gamma$ is of the form \gamma=\mathop{\mathrm{pr}}\bigr{(}\vartheta\hskip 2.0pt\eta^{-1}\bigl{)} for

[TABLE]

with $\vartheta_{1},\ldots,\eta_{1}\in\mathbb{Z}$.

We will need a more explicit description of the operators $\pi(\gamma)$, $\gamma\in\Gamma$, in terms of given strong Malcev coordinates. Of particular importance will be a specific type of products $g\cdot g^{\prime}\in G$, for which we give a description in the spirit of Lemma 3.3.

Lemma 4.12**.**

Let $\{X_{1},\ldots,X_{n}\}$ be a strong Malcev basis of $\mathfrak{g}$. Then there exist polynomial functions $S_{X_{j}},T_{X_{k}}:\mathbb{R}^{n}\to\mathbb{R}$ of degree $\leq 2n$ for all $j,k=1,\ldots,2n$ such that for $t=(t_{1},\ldots,t_{n}),s=(s_{1},\ldots,s_{n})\in\mathbb{R}^{n}$ we have

[TABLE]

with

• (i)

$S_{X_{j}}(t,s)=t_{j}+s_{j}+\tilde{S}_{X_{j}}(t_{j+1},\ldots,t_{n},s_{j+1},\ldots,s_{n})$,

• (ii)

$T_{X_{k}}(t,s)=t_{k}+s_{k}+\tilde{T}_{X_{k}}(t_{k+1},\ldots,t_{n},s_{j+1},\ldots,s_{n})$.

Proof.

The proof resembles the proof of [8, Proposition 1.2.7] and uses induction on $n$. For $\dim(\mathfrak{g})=n=1$ or $n=2$ the result is trivial since $\mathfrak{g}$ is Abelian.

Thus, suppose $n>2$ and that (i) holds true for all $j=2,\ldots,n$. Let $\mathfrak{g}\to\mathfrak{g}/\mathfrak{g}_{1}:X\mapsto\overline{X}$ be the canonical projection modulo the central ideal $\mathfrak{g}_{1}:=\mathbb{R}X_{1}$. By induction,

[TABLE]

with the $S_{\overline{X}_{j}}$ satisfying (i). For $j=2,\dots,n$ we set $S_{X_{j}}:=S_{\overline{X}_{j}}$. Since $\mathfrak{g}_{1}$ is central, there exists a polynomial function $\tilde{S}_{X_{1}}$ such that the multiplication in $G$ is

[TABLE]

and, consequently,

[TABLE]

This proves (i). The proof of (ii) works out analagously. ∎

Proposition 4.13**.**

[Step 2. Orthonormality]

Let $\Gamma$ be the discrete subset of $G/Z(G)$ from Definition 4.10 and let $C:=K^{2^{d}-1}\in\mathbb{N}$. Then the set

[TABLE]

forms an orthonormal system of $L^{2}(M\setminus G)$. Furthermore $\left\|1_{F}\right\|_{L^{2}(M\setminus G)}=C^{-1}\hskip 2.0ptd_{\pi}^{1/2}$.

Proof.

To prove orthogonality, we have to show that for $\gamma\neq\gamma^{\prime}\in\Gamma$

[TABLE]

Writing $\gamma=\mathop{\mathrm{pr}}(\vartheta\eta^{-1})$, the explicit formula for the representations (11) yields

[TABLE]

Hence, we need to show that

[TABLE]

Orthogonality when $\eta\neq\eta^{\prime}$: We focus on the characteristic functions in (43) and see that

[TABLE]

Since $\eta,\eta^{\prime}$ are in the quasi-lattice $\Gamma^{\prime}$ and $\Sigma$ is a fundamental domain for $\Gamma^{\prime}$, we conclude that either $1_{F}(q(h\eta^{-1}))1_{F}(q(h{\eta^{\prime}}^{-1}))=0$ or $\eta=\eta^{\prime}$. Consequently, $\langle\pi(\vartheta\eta^{-1})1_{F},\pi(\vartheta^{\prime}{\eta^{\prime}}^{-1})1_{F}\rangle=0$ for all $\vartheta,\vartheta^{\prime}\in\Gamma_{M}$ and $\eta\neq\eta^{\prime}\in\Gamma_{H}$.

Orthogonality when $\eta=\eta^{\prime}$ and $\vartheta\neq\vartheta^{\prime}$:

If $\eta=\eta^{\prime}$, then

[TABLE]

and the integral in (43) reduces to

[TABLE]

For further simplification, note that $\mathop{\mathrm{Ad}}(h)\log(\vartheta),\mathop{\mathrm{Ad}}(h)\log(\vartheta^{\prime})\in\mathfrak{m}$ for all $h\in H$ and $l$ vanishes on $[\mathfrak{m},\mathfrak{m}]$. Thus

[TABLE]

and after the change of variables $h\mapsto h\eta^{-1}=:h^{\prime}$ we need to compute the integral

[TABLE]

We now switch to the coordinates given by the parametrization of $l\cdot G$ in Theorem 4.3 and use the Haar measure $\mu^{\prime}_{M\setminus G}$ on $M\setminus G$ determined by the basis \bigl{\{}K^{2^{d}}\tilde{X}_{1},\ldots,K^{2}\tilde{X}_{d}\bigr{\}}. In this basis we have

[TABLE]

and the integral over $F$ becomes

[TABLE]

Note that for the verification of the orthogonality of the $\pi(\gamma)1_{F}$ the choice of the Haar measure on $M\setminus G$ is not relevant (because different Haar measures differ only by a multiplicative constant).

Let us first focus on $\mathop{\mathrm{Ad}^{*}}(\eta^{-1}{h^{\prime}}^{-1})l$. Due to Lemma 4.12 (i), there exist an $m_{h^{\prime}\eta}\in M$ and polynomials $\tilde{S}_{\tilde{X}_{j}}$ such that

[TABLE]

Since $M$ is normal in $G$, there furthermore exists some $m^{\prime}_{h^{\prime}\eta}\in M$ such that

[TABLE]

At this point we apply the parametrization of $l\cdot G$ of Theorem 4.3 and obtain

[TABLE]

For the $\log$-terms we employ Lemma 4.12 (ii) to rewrite

[TABLE]

for some $z_{\theta\theta^{\prime}}\in Z(G)$.

The conversion of strong Malcev coordinates to exponential coordinates (cf. Lemma 3.3) yields

[TABLE]

Now assume that $\vartheta\neq\vartheta^{\prime}$ and let $j$ be the largest index such that $\vartheta_{j}\neq\vartheta_{j}^{\prime}$ and thus $\vartheta_{d}=\vartheta_{d}^{\prime},\vartheta_{d-1}=\vartheta_{d-1}^{\prime},\vartheta_{j+1}=\vartheta_{j+1}^{\prime}$.

Since $\tilde{T}_{Y_{j}}(\vartheta_{j+1},\ldots,\vartheta_{d},-\vartheta_{j+1},\ldots,-\vartheta_{d})=0$ and thus $\zeta_{j+1}=\dots\zeta_{d}=0$, Lemma 3.3 implies that

[TABLE]

Combining (47) and (4.2), the action of $l\cdot\eta^{-1}{h^{\prime}}^{-1}$ on $\log(\vartheta{\vartheta^{\prime}}^{-1})$ yields

[TABLE]

So,

[TABLE]

Since $\int_{0}^{1}e^{2\pi it_{j}(\vartheta_{j}-\vartheta_{j}^{\prime})}dt_{j}=0$ when $\vartheta_{j}\neq\vartheta_{j}^{\prime}$, we have proved that the functions $\pi(\vartheta\eta^{-1})1_{F}$ and $\pi(\vartheta^{\prime}\eta^{-1})1_{F}$ are orthogonal.

Normalization: If $\gamma=\gamma^{\prime}$, then ${\left\langle\pi(\gamma)1_{F},\pi(\gamma^{\prime})1_{F}\right\rangle}_{L^{2}(M\setminus G)}=\|1_{F}\|^{2}_{L^{2}(M\setminus G)}=\mu_{M\setminus G}(F)$. The basis change $S_{l}$ from $\{Z_{1},\ldots,Y_{d},X_{1},\ldots,X_{d}\}$ to $\{Z_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ from the proof of Proposition 4.8 acts trivially on $\mathfrak{m}$. Since the according change of Haar measure obeys $\widetilde{\mu}_{G/Z(G)}=\det(S_{l})^{-1}\hskip 2.0pt\mu_{G/Z(G)}=d_{\pi}\hskip 2.0pt\mu_{G/Z(G)}$, we thus have

[TABLE]

Moreover, since $\tilde{\mu}_{G}(\Sigma_{G})=1$ for $\Sigma_{G}=\{\exp(\tilde{x}_{1}\tilde{X}_{1})\cdots\exp(z_{1}Z_{1})\mid z_{1},\ldots,\tilde{x}_{1}\in[0,1)\}$ and

[TABLE]

by Proposition 4.8 $(i)$ and $(ii)$, respectively, we have $\tilde{\mu}_{M\setminus G}(F)=\prod_{j=0}^{d-1}K^{2^{d-j}}=K^{2(2^{d}-1)}=C^{2}$ and, consequently,

[TABLE]

This yields the normalization of the system.

∎

Proposition 4.14**.**

[Step 3. Completeness.]

Let $\Gamma$ be the discrete subset of $G/Z(G)$ from Definition 4.10. Then the orthogonal system

[TABLE]

of $L^{2}(M\setminus G)$ from Proposition 4.13 is complete.

Proof.

By Corollary 3.10 $q(\Gamma^{\prime})$ is a quasi-lattice (a uniform subgroup) of $M\setminus G$ with fundamental domain $F=q(\Sigma)$, the sets $Fq(\eta)=q(\Sigma\eta),\eta\in\Gamma^{\prime}_{H}$, form a partition of $M\setminus G$. It suffices to show that for each $\eta\in\Gamma^{\prime}_{H}$ the orthogonal subsystem $\{\pi(\vartheta\eta^{-1})\,1_{F}\mid\vartheta\in\Gamma^{\prime}_{M}\}$ is complete in $L^{2}(q(\Sigma\eta))$. Thus, fix an arbitrary $\eta\in\Gamma^{\prime}_{H}$. Combining (42) and the change of variables $h\mapsto h\eta^{-1}=:h^{\prime}$ from the proof of Proposition 4.13, we have

[TABLE]

Since the map f(h)\mapsto e^{-2\pi i\left\langle l,\log\bigl{(}p(h^{\prime})\bigr{)}\right\rangle}f(h)=e^{-2\pi i\left\langle l,\log\bigl{(}p(h\eta^{-1})\bigr{)}\right\rangle}f(h) is unitary on $L^{2}(Fq(\eta))$, it suffices to show that the set

[TABLE]

is complete in $L^{2}(F)$. For $f\in L^{2}(F)$ this means that

[TABLE]

implies $f=0$ a.e. Note that by (47), we have

[TABLE]

for some polynomials $P_{j}$, and by (50) with $\theta^{\prime}=0$, we have

[TABLE]

Consequently,

[TABLE]

To verify (52), we therefore switch to the coordinates

[TABLE]

from $[0,1)^{d}$ to $F$. Substituting (53) into (52), it is equivalent to show that

[TABLE]

implies $f=0$ a.e.

Thus, suppose (54) holds true. First, we may omit the constant factors e^{2\pi i\eta_{j}\bigl{(}\vartheta_{j}+\tilde{R}_{Y_{j}}(\vartheta_{j+1},\ldots,\zeta_{d})\bigr{)}} of modulus one. Since $P_{1}=0$, we may rewrite (54) as

[TABLE]

As this holds true for all $\vartheta_{1}\in\mathbb{Z}$, it follows that

[TABLE]

for almost all $t_{1}\in[0,1)$. The unimodular factor outside the integral can be deleted and we apply the same argument to the remaining integrals. Repeating this procedure (induction), we arrive at

[TABLE]

for all $\vartheta_{k+1}\in\mathbb{Z}$ and almost all $(t_{1},\dots,t_{k})\in[0,1)^{k}$. Consequently,

[TABLE]

for almost all $(t_{1},\dots,t_{k+1})\in[0,1)^{k+1}$, which in turn implies the next step

[TABLE]

for all $\vartheta_{k+2}\in\mathbb{Z}$. For the last step $k=d-1$ we obtain from (55)

[TABLE]

for all $\vartheta_{d}\in\mathbb{Z}$ and almost all $(t_{1},\dots,t_{d-1})\in[0,1)^{d-1}$, from which we obtain $f(t_{1},\ldots,t_{d})=0$ a.e. This proves the completeness.

∎

In the case when $\Gamma$ is a quasi-lattice or a uniform subgroup, we can give a shorter and more elegant proof of completeness. In particular, the following observation yields a more structural proof of Theorem 4.7 under the assumptions (i) — (v).

Lemma 4.15**.**

Let $G$ be a nilpotent Lie group. Let $\Gamma$ be a quasi-lattice in $G/Z(G)$ with fundamental domain $\Sigma$, $\pi\in SI/Z(G)$ and $w\in\mathcal{H}_{\pi}$ with $\left\|w\right\|_{\mathcal{H}_{\pi}}=1$, such that the set $\{\pi(\gamma)w\mid\gamma\in\Gamma\}$ forms an orthonormal system in $\mathcal{H}_{\pi}$. Then the following are equivalent:

• (i)

$\{\pi(\gamma)w\mid\gamma\in\Gamma\}$* is complete in $\mathcal{H}_{\pi}$, and thus $\{\pi(\gamma)w\mid\gamma\in\Gamma\}$ is an orthonormal basis.*

• (ii)

$\mu_{G/Z(G)}(\Sigma)^{-1}=d_{\pi}$.

Proof.

$(i)\Rightarrow(ii)$ If $\{\pi(\gamma)w\mid\gamma\in\Gamma\}$ is an orthonormal basis, then for every $v\in\mathcal{H}_{\pi}$ and $\dot{g}\in G/Z(G)$ we have

[TABLE]

Hence, square-integrability yields

[TABLE]

$(ii)\Rightarrow(i)$ By Bessel’s inequality we have

[TABLE]

for all $\dot{g}\in\Sigma$. Since $\dot{g}\mapsto\sum_{\gamma\in\Gamma}\left|{\left\langle v,\pi(\dot{g}\gamma)w\right\rangle}_{\mathcal{H}_{\pi}}\right|^{2}$ is in $L^{2}(\Sigma,d\dot{g})$ and $d_{\pi}=\mu_{G/Z(G)}(\Sigma)^{-1}$, we can therefore estimate

[TABLE]

Since the integrand is non-negative, this implies that $\left\|v\right\|_{\mathcal{H}_{\pi}}^{2}-\sum_{\gamma\in\Gamma}\left|{\left\langle v,\pi(\dot{g}\gamma)w\right\rangle}_{\mathcal{H}_{\pi}}\right|^{2}=0$ for almost all $\dot{g}\in\Sigma$, thus $\{\pi(\gamma)w\mid\gamma\in\Gamma\}$ is complete. ∎

4.3. Graded $SI/Z$-Groups

In this subsection we show that the technical assumptions of Theorem 4.7 are satisfied for graded $SI/Z$-groups with one-dimensional center. We first prove several properties of graded groups.

In the following $G$ is a graded $SI/Z$-group of $\dim(G)=r+2d$ with $r$-dimensional center $Z(G)$. Let $\mathfrak{g}=\bigoplus_{k=1}^{N}\mathfrak{g}_{k}$ be a gradation of $\mathfrak{g}$ with the understanding that $\mathfrak{g}_{N}\neq\{0\}$ and $N$ is the smallest index such that $\mathfrak{g}_{N+k}=\{0\}$ for all $k\in\mathbb{N}$.

Lemma 4.16**.**

Let $\mathfrak{g}$ be a graded Lie algebra with gradation $\mathfrak{g}=\bigoplus_{k=1}^{N}\mathfrak{g}_{k}$.

(i) Then $\mathfrak{n}_{k_{0}}=\bigoplus_{k=k_{0}}^{N}\mathfrak{g}_{k}$ is an ideal of $\mathfrak{g}$ for each $k_{0}=1,\dots,N$.

(ii) $\mathfrak{g}_{N}\subseteq\mathfrak{z}(\mathfrak{g})$. In particular, if $\dim(\mathfrak{z}(\mathfrak{g}))=1$, then $\mathfrak{g}_{N}=\mathfrak{z}(\mathfrak{g})$.

(iii) Every basis of $\mathfrak{g}$ that is a union of bases of the $\mathfrak{g}_{k}$ is a strong Malcev basis for $\mathfrak{g}$.

Proof.

(i) follows directly from the definition of a gradation.

(ii) For every $X\in\mathfrak{g}_{k}$, $1\leq k\leq N$, and every $Y\in\mathfrak{g}_{N}$ we have $[X,Y]\in\mathfrak{g}_{N+k}=\{0\}$, hence $Y\in\mathfrak{z}(\mathfrak{g})$. If $\dim(\mathfrak{z}(\mathfrak{g}))=1$, then $1\leq\dim(\mathfrak{g}_{N})\leq\dim(\mathfrak{z}(\mathfrak{g}))=1$, so the subspaces must coincide.

(iii) Let $\{X_{k_{1}},\ldots,X_{k_{d}}\}$ be a basis of $\mathfrak{g}_{k}\neq\{0\}$ and $X\in\mathfrak{g}$ be arbitrary. Then $[X,X_{k_{j}}]\in\bigoplus_{k^{\prime}=1}^{N}\mathfrak{g}_{k+k^{\prime}}$. Consequently $\mathbb{R}\text{-}\mathrm{span}\{X_{k_{j}},\ldots X_{k_{d}}\}\bigoplus_{l=k+1}^{N}\mathfrak{g}_{k}$ is an ideal, which proves the strong Malcev property of the union of bases. ∎

Under the assumption $\dim(Z(G))=r=1$, we construct polarizing ideals $\mathfrak{m}\vartriangleleft\mathfrak{g}$ which are subordinate to all $l\in\mathfrak{z}(\mathfrak{g})^{*}$ simultaneously. This implies that all $\pi\in SI/Z(G)$ can be induced from the same polarization $\mathfrak{m}$. We write $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$ and we denote by $\mathop{\mathrm{pr}}_{\mathfrak{g}_{k}}$ the natural projection $\mathfrak{g}\to\mathfrak{g}_{k}$. We distinguish the cases of even and odd $N$.

Proposition 4.17** (N odd).**

Let $G$ be a graded $SI/Z$-group with one-dimensional center and gradation $\mathfrak{g}=\bigoplus_{k=1}^{N}\mathfrak{g}_{k}$ with $N=2N_{0}+1$, $N_{0}\in\mathbb{N}$.Then $\mathfrak{m}:=\bigoplus_{k=N_{0}+1}^{N}\mathfrak{g}_{k}$ is an ideal of $\mathfrak{g}$ that is polarizing for all $l\in\mathfrak{z}(\mathfrak{g})^{*}$. The normal subgroup $M=\exp(\mathfrak{m})$ therefore induces all $\pi\in SI/Z(G)$.

Proof.

Let $l=\lambda Z^{*}\in\mathfrak{z}(\mathfrak{g})^{*}$, $\lambda\neq 0$. By Lemma 4.16(i) the subspace $\bigoplus_{k=k_{0}}^{N}\mathfrak{g}_{k}\subseteq\mathfrak{g}$ is an ideal of $\mathfrak{g}$ for every $1\leq k_{0}\leq N$. Since $[\mathfrak{g}_{k},\mathfrak{g}_{k^{\prime}}]\subseteq\mathfrak{g}_{k+k^{\prime}}=\{0\}$ for all $k,k^{\prime}\geq N_{0}+1$, $\mathfrak{m}$ must be an Abelian ideal. Consequently, $\mathfrak{m}$ is subordinate to $l$.

We show that $\dim(\mathfrak{g}_{N-k})=\dim(\mathfrak{g}_{k})$. Choose a basis $\{V_{1},\ldots,V_{p}\}$ for $\mathfrak{g}_{N-k}$ and $\{W_{1},\ldots,W_{q}\}$ for $\mathfrak{g}_{k}$ and assume that $p\neq q$, $p<q$ say. Then the $p\times q$-matrix $C$ with entries $C_{jk}=l([V_{j},W_{k}])$ possesses a non-trivial kernel $c=(c_{1},\dots,c_{q})\in\mathbb{R}^{q}$. Set $Y=\sum_{m=1}^{q}c_{m}\hskip 2.0ptW_{m}$, then

[TABLE]

Consequently, $l([V,Y])=0$ for all $V\in\mathfrak{g}_{N-k}$ and, by the properties of the gradation, $B_{l}(X,Y)=l([X,Y])=0$ for all $X\in\mathfrak{g}$. This contradicts the assumption that $B_{l}$ restricted to $\mathfrak{g}/\mathfrak{z}$ is non-degenerate.

Finally, $\dim(\mathfrak{g})=1+\sum_{k=1}^{N}\dim(\mathfrak{g}_{k})=1+2\sum_{k=1}^{N_{0}}\dim(\mathfrak{g}_{k})$. It follows that $\dim(\mathfrak{m})=\sum_{k=N_{0}+1}^{N}\dim(\mathfrak{g}_{k})=d+1$, and thus $\mathfrak{m}$ is a polarization.

∎

Proposition 4.18** (N even).**

Let $G$ be a graded $SI/Z$-group with one-dimensional center and with gradation $\mathfrak{g}=\bigoplus_{k=1}^{N}\mathfrak{g}_{k}$ with $N=2N_{0}$, $N_{0}\in\mathbb{N}$. Then there exists an ideal $\mathfrak{m}\subseteq\mathfrak{n}_{N_{0}}=\bigoplus_{k=N_{0}}^{N}\mathfrak{g}_{k}\subseteq\mathfrak{g}$ that is polarizing for all $l\in\mathfrak{z}(\mathfrak{g})^{*}$. Again, the normal subgroup $M=\exp(\mathfrak{m})$ induces all $\pi\in SI/Z(G)$.

Proof.

Let $l=\lambda Z^{*}\in\mathfrak{z}(\mathfrak{g})^{*},\lambda\neq 0$. Again we obtain $\dim(\mathfrak{g}_{N-k})=\dim(\mathfrak{g}_{k})$ for $k=1,\dots,N_{0}-1$. Since $\dim(\mathfrak{z}(\mathfrak{g}))=\dim(\mathfrak{g}_{N})=1$ and $\dim(\mathfrak{g})$ is odd, the dimension $\dim(\mathfrak{g}_{N_{0}})$ must be even. If $\mathfrak{g}_{N_{0}}=\{0\}$, then nothing needs to be shown. If $\dim(\mathfrak{g}_{N_{0}})=2d_{N_{0}}\geq 2$, then B_{l}\Big{|}_{\mathfrak{g}_{N_{0}}\times\mathfrak{g}_{N_{0}}}, the restriction of $B_{l}$ to $\mathfrak{g}_{N_{0}}$, is a symplectic form. Therefore $\mathfrak{g}_{N_{0}}$ possesses a basis $\{Y_{1},\ldots,Y_{d_{N_{0}}},X_{1},\ldots,X_{d_{N_{0}}}\}$, such that $B_{l}([Y_{j},X_{k}])=\delta_{jk}$ and $B_{l}([Y_{j},Y_{k}])=B_{l}([X_{j},X_{k}])=0$ for $j,k=1,\dots d_{N_{0}}$.

Now set $\mathfrak{m}:=\mathbb{R}\text{-}\mathrm{span}\{Y_{1},\ldots,Y_{d_{N_{0}}}\}\bigoplus_{k=N_{0}+1}^{N}\mathfrak{g}_{k}$. Then $\mathfrak{m}$ is a $d+1$-dimensional Abelian ideal of $\mathfrak{g}$ and subordinate to $l$, as claimed. ∎

We next show that the technical assumptions of Theorem 4.7 are satisfied for graded groups with one-dimensional center. The main point is the existence of a Chevalley-Rosenlicht-admissible Malcev basis.

Proposition 4.19**.**

[Quasi-Lattice] Let $G$ be a graded $SI/Z$-group with one-dimensional center. Let $l\in\mathfrak{z}(\mathfrak{g})^{*}\setminus\{0\}$ and $\mathfrak{m}$ be the polarization constructed in Proposition 4.17 or 4.18. Then $\mathfrak{g}$ possesses a Ch-R-admissible basis $\{Z,Y_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ of $\mathfrak{g}$ which passes through $\mathfrak{m}$ and the gradation and is subordinate to $l$.

If, in addition, $\mathfrak{g}$ possesses a rational structure compatible with the gradation and $l=\lambda Z^{*}$ for $\lambda\in\mathbb{Q},\lambda\neq 0$, then this Ch-R-admissible basis generates the same rational structure $\mathfrak{g}_{\mathbb{Q}}=\mathbb{Q}\text{-}\mathrm{span}\{Z,Y_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$.

Proof.

For $k=1,\ldots,N_{0}$, choose a basis $\{V_{1},\ldots,V_{p}\}$ for $\mathfrak{g}_{N-k}$ and a basis $\{W_{1},\ldots,W_{q}\}$ for $\mathfrak{g}_{k}$. We have already seen that $p=q$. The same argument as in Proposition 4.17 shows that the $p\times p$-matrix $C$ with entries $C_{j,k}=l([W_{j},V_{k}])$ is invertible. Set $\tilde{W}_{j}=\sum_{k=1}^{p}(C^{-1})_{j,k}\hskip 2.0ptW_{k}$, then

[TABLE]

If, in addition, $N=2N_{0}$ is even, we partition the symplectic basis of $\mathfrak{g}_{N_{0}}$ as in the proof of Proposition 4.18.

Now take the union of bases $\{V_{j}\}$ of $\mathfrak{g}_{N-k}$, $k=0,\ldots,N_{0}$, and relabel it as $\{Z,\ldots,Y_{d}\}$, and take the union of bases $\{\tilde{W}_{j}\}_{j}$ of $\mathfrak{g}_{k}$, $k=1,\ldots,N_{0}$, and relabel it as $\{\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$. By construction, we have

[TABLE]

According to Theorem 4.3, $\{\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ is a basis that parametrizes the orbit $l\cdot G$. At the same time, $\{Z,\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$ is a union of bases of the subspaces $\mathfrak{g}_{k}$. By Lemma 4.16 (iii), this is a strong Malcev basis of $\mathfrak{g}$. This means that $\mathfrak{g}$ possesses a Ch-R-admissible basis subordinate to $l=\lambda Z^{*}$.

In the presence of a rational structure compatible with the gradation, the matrix $\langle Z^{*},[W_{j},V_{k}]\rangle_{j,k=1,\dots,p}$ has rational entries. Thus for $l=\lambda Z^{*},\lambda\in\mathbb{Q}$, the new basis elements $\tilde{W}_{j}$ defined by (56) are also in $\mathfrak{g}_{\mathbb{Q}}$ and thus $\mathfrak{g}_{\mathbb{Q}}=\mathbb{Q}\text{-}\mathrm{span}\{Z,Y_{1},\ldots,Y_{d},\tilde{X}_{d},\ldots,\tilde{X}_{1}\}$. ∎

Proof of Theorem 4.1.

By Propositions 4.17 and 4.18 every $l=\lambda Z^{*},\lambda\neq 0$ possesses the same polarization $\mathfrak{m}$, which by construction is an ideal of $\mathfrak{g}$. This is condition (i) of Theorem 4.7. By Proposition 4.19 $\mathfrak{g}$ possesses a Ch-R-admissible basis subordinate to $l$ passing through $\mathfrak{m}$ and $\mathfrak{z}$, which is condition (ii). In conclusion, Theorem 4.7 now yields the existence of an orthonormal basis in the orbit of $\pi_{l}$. ∎

Proof of Theorem 4.2.

If $l=\lambda Z^{*},\lambda\in\mathbb{Q},\lambda\neq 0$, then by Proposition 4.19 $\mathfrak{g}_{\mathbb{Q}}$ has a Ch-R-admissible basis that is compatible with the gradation. Therefore the additional conditions (iii) – (v) of Theorem 4.7 are satisfied. This implies the existence of an orthonormal basis with respect to a uniform subgroup. In particular, for $l=\pm Z^{*}$ with the associated representations $\pi_{\pm}$, there exist a uniform subgroup $\Gamma_{G}\subseteq G$ and a set $F\subseteq M\setminus G$, such that $\{\pi_{\pm}(\gamma)1_{F}\mid\gamma\in\mathrm{pr}(\Gamma_{G})\}$ is an orthogonal basis of $L^{2}(M\ \setminus G)$.

For $l=\lambda Z^{*}$ with irrational $\lambda\neq 0$ we use dilations. The dilation $\delta_{s}:\mathfrak{g}\to\mathfrak{g}$ in  (9) lifts to an automorphism $\delta_{s}:G\to G$ and leads to the one-parameter family of irreducible unitary representations $\pi_{s}=\pi_{\pm}\circ\delta_{s}$. These are irreducible and in $SI/Z(G)$. Now let $\Gamma_{s}:=\mathrm{pr}(\delta_{s^{-1}}(\Gamma_{G}))$. Then $\Gamma_{s}$ is a uniform subgroup of $G/Z(G)$ with fundamental domain \mathop{\mathrm{pr}}\bigl{(}\delta_{s^{-1}}(\Sigma)\bigr{)}. If $\tilde{\gamma}=\delta_{s^{-1}}(\gamma)$ for $\gamma\in\Gamma_{G}$, then $\pi_{s}(\tilde{\gamma})=\pi_{\pm}\circ\delta_{s}(\delta_{s^{-1}}\gamma)=\pi_{\pm}(\gamma)$, and for \tilde{h}\in\tilde{F}:=q\bigl{(}\delta_{s^{-1}}(\Sigma)\bigr{)} a short computation yields

[TABLE]

Consequently, the sets $\{\pi_{\pm}(\gamma)1_{F}\mid\gamma\in\mathrm{pr}(\Gamma_{G})\}$ and $\{\pi_{s}(\tilde{\gamma})1_{\tilde{F}}\mid\tilde{\gamma}\in\Gamma_{s}\}$ coincide, and we have found an orthonormal basis in the orbit of $\pi_{s}$.

Finally, we note that the $\pi_{s}$ exhaust all representations in $SI/Z(G)$: since

[TABLE]

the representation $\pi_{s}$ is associated to $l=\pm s^{N}Z^{*}$, and all representations in $SI/Z(G)$ are obtained in this way. Theorem 4.2 is thus proved completely.

∎

5. Examples

This section provides three examples that illustrate the main theorems.

Example 5.1**.**

Let $G$ be nilpotent with $\dim(G)\leq 6$. Then $G$ is graded with $\dim(Z(G)=1$ or $=2$. In the former case, the existence of an orthonormal basis in the orbit of a representation in $SI/Z(G)$ follows from Theorems 4.1 or 4.2. In the latter case, the existence follows from Theorem 4.7: The conditions (i) — (ii) are satisfied because groups of low dimension have a relatively simple structure: Most $SI/Z$-groups $G$ are a semidirect product of the form $G=M\rtimes\mathbb{R}^{d}$ of an Abelian polarization $M$ and another Abelian subgroup. The remaining ones have a normal Abelian and thus polarizing subgroup $M$ and the quotient group $M\setminus G$ is also Abelian.

This class of examples was obtained in Höfler’s thesis [26]. Höfler’s work is based on the explicit formulas for the irreducible representations of Nielsen [30] and could be done by hand without substantial input from the theory of nilpotent groups. Theorem 4.1 or 4.2 offer a systematic approach that carries over to a large class of nilpotent groups.

Example 5.2** (The Dynin-Folland Group $\mathbf{H}_{1,2}$).**

The first example of an $SI/Z$-group with a non-Abelian quotient group occurs in dimension $7$, this is the so-called Dynin-Folland group $\mathbf{H}_{1,2}$ as in [12]. As an illustration of the abstract construction of orthonormal bases we work out the details for this example.

The Dynin-Folland group was introduced by Dynin [9] to study pseudo-differential operators on the Heisenberg group and taken up again by Folland [15] for his detailed study of so-called meta-Heisenberg groups. Precisely, $\mathbf{H}_{1,2}$ is a semidirect product $\mathbf{H}_{1,2}=\mathbb{R}^{4}\rtimes\mathbf{H}_{1}$ of $\mathbb{R}^{4}$ and the $3$-dimensional Heisenberg group $\mathbf{H}_{1}$. Its $3$-step nilpotent Lie algebra $\mathfrak{h}_{2,1}$ is defined by a strong Malcev basis $\{Z,Y_{1},Y_{2},Y_{3},X_{1},X_{2},X_{3}\}$ with Lie brackets

[TABLE]

The center of $\mathfrak{h}_{2,1}$ is $\mathfrak{z}(\mathfrak{\mathfrak{h}_{2,1}})=\mathbb{R}Z$ and is one-dimensional, the Heisenberg Lie algebra $\mathfrak{h}_{1}=\{X_{1},X_{2},X_{3}\}$ is a subalgebra of $\mathfrak{h}_{2,1}$ and the complement $\mathfrak{m}:=\mathbb{R}\text{-}\mathrm{span}\{Z,Y_{1},Y_{2},Y_{3}\}$ is Abelian and an ideal of $\mathfrak{h}_{2,1}$.

For $l=\lambda Z^{*}\in\mathfrak{z}(\mathfrak{g})^{*}$ with $\lambda\neq 0$, the matrix representation of $B_{l}$ is given by

[TABLE]

Hence, $B_{l}$ is non-degenerate with $\mathrm{Pf}(l)=\left|\lambda\right|^{3}$ and the coadjoint orbit $\mathcal{O}_{l}$ is the $6$-dimensional affine subspace $\lambda Z^{*}+\mathbb{R}\text{-}\mathrm{span}\{Y_{1}^{*},\ldots,X_{3}^{*}\}$. This implies that, up to Plancherel measure zero, all coadjoint orbits are flat; these are precisely the orbits $\mathcal{O}_{\lambda Z^{*}},\lambda\in\mathbb{R}\setminus\{0\}$. Since $\mathfrak{m}$ is Abelian and of dimension $4$, it is a polarization for all $l\in\mathfrak{z}(\mathfrak{\mathfrak{h}_{2,1}})^{*}$; moreover, $M\setminus G\cong\mathbf{H}_{1}$. Consequently, all representations $\pi_{l}\in SI/Z(\mathbf{H}_{1,2})$ can be realized on $L^{2}(\mathbf{H}_{1})$ in terms of the strong Malcev coordinates $\{Z,\ldots,X_{3}\}$.

Comparing (72) and (32), we set $\tilde{X}_{3}:=\lambda^{-1}X_{1},\tilde{X}_{2}:=\lambda^{-1}X_{2},\tilde{X}_{1}:=\lambda^{-1}X_{3}$, then $\{\tilde{X}_{3},\tilde{X}_{2},\tilde{X}_{1}\}$ is a basis that parametrizes the orbit $l\cdot G$. Since we have only relabeled and rescaled the original basis, $\{Z,Y_{1},Y_{2},Y_{3},\lambda^{-1}X_{1},\lambda^{-1}X_{2},\lambda^{-1}X_{3}\}$ is a strong Malcev basis and thus also a Ch-R-admissible basis of $\mathfrak{g}$ subordinate to $l=\lambda Z^{*}$. The assumptions of Theorem 4.7 are satisfied and the representation $\pi_{l}$ possesses an orthonormal basis associated to the quasi-lattice

[TABLE]

If $\lambda$ is rational, then the additional conditions (iii) - (v) of Theorem 4.7 are satisfied, and there exists a uniform subgroup $\Gamma$ of $\mathbf{H}_{1,2}/\exp(\mathbb{R}Z)$, such that $\{C\left|\lambda\right|^{-3/2}\pi_{l}(\gamma)1_{F}\mid\gamma\in\Gamma\}$ is an orthonormal basis of $L^{2}(\mathbf{H}_{1})$.

In the following we perform the explicit computations for the case of a uniform subgroup. We begin with the group law in strong Malcev coordinates. Let $g:=\exp(zZ)\cdots\exp(\tilde{x}_{1}\tilde{X}_{1})$ and $g^{\prime}:=\exp(z^{\prime}Z)\cdots\exp(\tilde{x}^{\prime}_{1}\tilde{X}_{1})$ be two elements in $\mathbf{H}_{1,2}$. A straight-forward computation based on the pairwise commutation relations of their exponential factors yields

[TABLE]

The polynomial functions $P_{Z},\ldots,P_{\tilde{X}_{1}}$ from Lemma 3.3 are thus given by

[TABLE]

The coefficients of these polynomials are $1,\tfrac{1}{\lambda},\tfrac{1}{2\lambda},\tfrac{1}{2\lambda^{2}}$. For rational $\lambda=p/q\in\mathbb{Q}$ with $p,q$ relatively prime integers, these are the numbers $1,q/p,q/2p$ and $q^{2}/2p^{2}$. According to the construction of the uniform subgroup in Proposition 4.8, we must choose $K\in\mathbb{N}$ such that all denominators divide $K$; in this case $K=2p^{2}$ suffices. We then employ the change of basis from $\{Z,\ldots,\tilde{X}_{1}\}$ to $\{K^{-16}Z,K^{-8}Y_{1},K^{-4}Y_{2},K^{-2}Y_{3},K^{2}\tilde{X}_{3},K^{4}\tilde{X}_{2},K^{8}\tilde{X}_{1}\}$. In these coordinates the group multiplication is described by the polynomials $P^{\prime}_{Z},\ldots,P^{\prime}_{\tilde{X}_{1}}$ given by

[TABLE]

Moreover,

[TABLE]

forms a uniform subgroup of $\mathbf{H}_{1,2}$ with fundamental domain

[TABLE]

so we pick $\Gamma:=\mathop{\mathrm{pr}}(\Gamma_{G})$ with fundamental domain $\Sigma:=\mathop{\mathrm{pr}}(\Sigma_{G})\subseteq\mathbf{H}_{1,2}/\mathbb{R}Z$ to construct an orthonormal basis.

We now realize $\pi$ in terms of the new Malcev coordinates. Since $\mathbf{H}_{1}$ is a subgroup of $\mathbf{H}_{1,2}$, the natural splitting $g=p(g)s(g)=mh$ is satisfied by

[TABLE]

and

[TABLE]

Following (5), for a given $h^{\prime}:=\exp(t_{3}K^{2}\tilde{X}_{3})\exp(t_{2}K^{4}\tilde{X}_{2})\exp(t_{1}K^{8}\tilde{X}_{1})$ we compute

[TABLE]

Since \left\langle\lambda Z^{*},\log\big{(}h^{\prime}m{h^{\prime}}^{-1}\big{)}\right\rangle=\left\langle\lambda\mathrm{Ad}^{*}({h^{\prime}}^{-1})Z^{*},\log(m)\right\rangle, the coefficients of the $y_{j}$’s in the $Z$-component yield the polynomial functions describing the orbit $l\cdot G$ (cf. Theorem 4.3)

[TABLE]

We denote the coordinates of $m^{\prime\prime}:=p(h^{\prime}g)$ by $z^{\prime\prime},y^{\prime\prime}_{1},y^{\prime\prime}_{2},y^{\prime\prime}_{3}$. Since $\mathbf{H}_{1}$ is a subgroup, $q(h^{\prime}h)=h^{\prime}h$ and $p(h^{\prime}h)=e$, and the commutativity of $\mathfrak{m}$ implies

[TABLE]

Hence, the action of $\pi_{l}$ on a function $f\in L^{2}(\mathbf{H}_{1})$ is explicitly given by

[TABLE]

In this set of Malcev coordinates, the orthonormal basis \bigl{\{}K^{-7}\hskip 2.0pt\left|\lambda\right|^{-3/2}\hskip 2.0pt\pi(\gamma)1_{F}\mid\in\Gamma\bigr{\}} of $L^{2}(\mathbf{H}_{1})$ becomes the explicit orthonormal basis

[TABLE]

of $L^{2}(\mathbb{R}^{3})\cong L^{2}(\mathbf{H}_{1})$. Of course, once such an explicit formula is derived, it is not difficult to check directly that (74) is an orthonormal basis.

We discuss briefly another construction of an orthonormal basis of $L^{2}(\mathbb{R}^{3})$. This construction is based on a gradation of $\mathfrak{h}_{2,1}$ and Theorem 4.1. For contrast we treat the quasi-lattice case. We observe that $\mathfrak{h}_{2,1}$ admits the gradation $\mathfrak{h}_{2,1}=\mathfrak{g}_{3}\oplus\mathfrak{g}_{2}\oplus\mathfrak{g}_{1}$ with

[TABLE]

Setting $X^{\prime}_{1}:=Y_{3}$ and $Y^{\prime}_{3}:=X_{1}$, the table of the Lie algebra is

[TABLE]

We notice that $\mathfrak{m}:=\mathbb{R}\text{-}\mathrm{span}\{Z,Y_{1},Y_{2},Y^{\prime}_{3}\}=\mathfrak{g}_{3}\oplus\mathfrak{g}_{2}$ is Abelian and that $\mathfrak{g}/\mathfrak{m}$ is Abelian and thus $M\setminus G\cong(\mathbb{R}^{3},+)$, but the complement $\mathfrak{h}=\mathbb{R}\text{-}\mathrm{span}\{X^{\prime}_{1},X_{2},X_{3}\}$ is not a subalgebra. However, for $l=\lambda Z^{*}$, $\lambda\neq 0$, the polynomial functions $Q_{j}$ which parametrize the orbit $l\cdot G$ are much simpler, namely of order $1$, as we deduce from the lower left block in (82). Precisely, we choose $\tilde{X}^{\prime}_{3}:=-\lambda^{-1}X^{\prime}_{1},\tilde{X}_{2}:=\lambda^{-1}X_{2},\tilde{X}_{1}:=\lambda^{-1}X_{3}$ as a basis for the parametrization. Theorem 4.1 applies and asserts the existence of an orthonormal basis of $L^{2}(M\setminus\mathbf{H}_{1,2})$ arising from the quasi-lattice

[TABLE]

To be more specific, let $\pi_{l}^{\prime}$ be the unitary representation corresponding to $l=\lambda Z^{*}$, but now induced from the polarization $\mathfrak{m}=\mathfrak{g}_{3}\oplus\mathfrak{g}_{2}$. By the general theory $\pi_{l}^{\prime}$ is equivalent to $\pi_{l}$ in (73). This polarization leads to a simpler formula for the representation. Indeed, the first exponent of

[TABLE]

is a sum of first-order polynomial functions, and the translation $f(h^{\prime}h)$ is Abelian, because of $[\mathfrak{h},\mathfrak{h}]\subseteq\mathbb{R}\text{-}\mathrm{span}\{Y_{1},Y_{2},Y_{3}\}$; the $Z$-component of the non-vanishing factor

[TABLE]

enters the second exponent. All in all we have

[TABLE]

We therefore obtain the orthonormal basis $\{\left|\lambda\right|^{-3/2}\pi_{l}^{\prime}(\vartheta\eta^{-1})1_{F}\mid\vartheta\eta\in\Gamma\}$ or

[TABLE]

of $L^{2}(\mathbb{R}^{3})$. This construction resembles the Gabor orthonormal basis for $L^{2}(\mathbb{R}^{3})$ from Section 2.1. Again, once (83) is written down, the orthonormality and the completeness are easy to check. The point here is that the appropriate choice of coordinates and polarization are guided by a theory and are rather difficult to guess by hand.

Remark 5.3**.**

This example can be generalized to an infinite family of graded $3$-step nilpotent groups $\mathbf{H}_{j,2}$ of dimension $4j+3$. Technically, meta-Heisenberg groups $G$ such as $\mathbf{H}_{j,2}$ arise from the Lie algebra $\mathfrak{g}$ generated by a basis of left-invariant vector fields of a $2$-step nilpotent group $\bar{G}$ and multiplication by first-order polynomials on $\bar{\mathfrak{g}}\cong\mathbb{R}^{d}$ in the variables determined by the basis. In the case $G=\mathbf{H}_{j,2}$ we have $\bar{\mathfrak{g}}=\mathbf{H}_{j}\cong\mathbb{R}^{2j+1}=:\mathbb{R}^{d}$. See [12] for a detailed analysis of the Dynin-Folland groups.

Example 5.4**.**

The following $7$-dimensional Lie algebra is a slight variation of the Lie algebra in [13, Rem. 3.1.6]. It is an example for which only the quasi-lattice version of the technical Theorem 4.7 yields an orthonormal basis of $L^{2}(\mathbb{R}^{3})$, but not its special cases for graded groups.

Let $\{X_{1},\ldots,X_{7}\}$ be the Euclidean standard basis for $\mathfrak{g}:=\mathbb{R}^{7}$ endowed with the Lie bracket relations

[TABLE]

As in [13, Rem. 3.1.6] these Lie brackets define a $7$-dimensional Lie algebra with $1$-dimensional center that cannot be equipped with any gradation. The bracket $[X_{2},X_{3}]=\sqrt{2}X_{6}$ excludes the existence of a rational structure. Consequently we cannot apply Theorems 4.1 or 4.2.

Nevertheless, we can apply Theorem 4.7. Indeed, the basis $\{X_{7},\ldots,X_{1}\}$ is a strong Malcev basis of $\mathfrak{g}$ and $\mathfrak{m}:=\mathbb{R}\text{-}\mathrm{span}\{X_{7},X_{6},X_{5},X_{4}\}$ is a polarizing ideal for every $l:=\lambda X_{7}^{*}$, $\lambda\in\mathbb{R}\setminus\{0\}$. Since

[TABLE]

the basis $\{Z,Y_{1},\ldots,\tilde{X}_{1}\}:=\{X_{7},X_{6},X_{5},X_{4},\lambda^{-1}X_{3},-\lambda^{-1}X_{2},\lambda^{-1}X_{1}\}$ is Ch-R-admissible subordinate to $l$. Thus, conditions (i)—(ii) of Theorem 4.7 are satisfied and we obtain an orthonormal basis for $L^{2}(M\setminus G)\cong L^{2}(\mathbb{R}^{3})$ derived from a quasi-lattice.