Harmonic Analysis on the Affine Group of the Plane

Abstract

For any natural number n, the group $G_n$ of all invertible affine transformations of n-dimensional Euclidean space has, up to equivalence, just one square-integrable representation and the left regular representation of $G_n$ is a multiple of this square-integrable representation. We provide a concrete realization $\sigma$ of this distinguished representation in the two-dimensional case. We explicitly decompose the Hilbert space $L^2(G_2)$ as a direct sum of left-invariant closed subspaces on each of which the left regular representation acts as a representation equivalent to $\sigma$.

1 Introduction

The purpose of this paper is to provide concrete tools, analogous to those provided by the Peter–Weyl Theorem for compact groups, for the analysis of square-integrable functions on the group $G_2$ of all invertible affine transformations of the Euclidean plane, equipped with its left Haar measure. This group is isomorphic, as a topological group, to the semidirect product of the vector group ${\mathbb {R}}^2$ with ${{{\textrm{GL}}}}_2({\mathbb {R}})$, the group of invertible $2\times 2$ real matrices, acting on ${\mathbb {R}}^2$ in a natural manner. Although $G_2$ is not compact, it shares an important feature with compact groups: Its left regular representation decomposes as a direct sum of irreducible representations. Indeed, for any positive integer n, if $G_n$ denotes the group of invertible affine transformations of ${\mathbb {R}}^n$, then the left regular representation of $G_n$ is an infinite multiple of a single distinguished irreducible representation. It is not so difficult to prove this fact about $G_n$ using general results from Mackey’s theory of induced representations as introduced in [15]. However, since we are not aware of an explicit statement of this feature of $G_n$ in the literature, we provide it in Theorem 4.8 along with a proof. For the affine group of the line, $G_1$, it has been known for a long time that its left regular representation is a multiple of a single square-integrable representation (see Example 11.3 of [14], for example) and it is relatively routine to work out various equivalent descriptions of the distinguished representation. Most of the original work in this current paper consists of finding an explicit description of the one irreducible representation of $G_2$ that occurs in its left regular representation and to give an explicit way to write the left regular representation of $G_2$ as an infinite multiple of this one irreducible representation.

For the rest of this introduction, please refer to Sect. 2 for unfamiliar or ambiguous notational conventions. Also, in Sect. 2, we review known results on square-integrable representations; in particular, we recall the Duflo–Moore Theorem and the generalization of the Peter-Weyl Theorem that holds for any group whose regular representation is a direct sum of irreducible representations.

Section 3 is devoted to induced representations. We recall one of the equivalent ways of defining ${{\textrm{ind}}}_H^G\pi$, where H is a closed subgroup of a locally compact group G and $\pi$ is a unitary representation of H. In the case that there exists a closed subgroup K of G that is complementary to H (see Sect. 3 for the exact definition), we show how $\textrm{ind}_H^G\pi$ is equivalent to a representation realized on the Hilbert space $L^2(K,{\mathcal {H}}_\pi )$ (see Proposition 3.1). In that process, we correct an error in Example 2.29 of [13].

In Sect. 4, we consider $G_n$, for general n. If N denotes the closed abelian normal subgroup of pure translations and if $\chi _{{\underline{\omega }}}$ is any nontrivial character of N, we construct a representation $\pi ^{{\underline{\omega }}}$, which acts on the Hilbert space $L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ and is equivalent to $\textrm{ind}_N^{G_n}\chi _{{\underline{\omega }}}$. In fact, the $\pi ^{{\underline{\omega }}}$ are all mutually equivalent, for ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n} {\setminus }\{{\underline{0}}\}$. To make notation as simple as possible, let ${\underline{\omega }}_0=(1,0\cdots ,0)$. In Theorem 4.3, we give an explicit way of writing $L^2(G_n)$ as a direct sum of left-invariant closed subspaces so that the restriction of the left regular representation of $G_n$ to each of these subspaces is equivalent to $\pi ^{{\underline{\omega }}_0}$. When $n=1$, ${\underline{\omega }}_0=1$. The representation $\pi ^1$ turns out to be the unique infinite-dimensional irreducible representation of $G_1$. We show how $\pi ^1$ is equivalent to the well-known natural representation of $G_1$ on $L^2({\mathbb {R}})$. We note that the fact that the natural representation of $G_1$ on $L^2({\mathbb {R}})$ is irreducible, even square-integrable, and is closely tied to the reproducing property of the continuous wavelet transform on ${\mathbb {R}}$ (see [9]). However, for $n>1$, $\pi ^{{\underline{\omega }}_0}$ is reducible. Using mathematical induction and the Induction in Stages Theorem, we show that $\pi ^{{\underline{\omega }}_0}$ can be further decomposed when $n>1$ and prove that there exists an irreducible representation $\sigma _n$ of $G_n$ such that the left regular representation of $G_n$ is equivalent to a countably infinite multiple of $\sigma _n$ (Theorem 4.8). In particular, $\sigma _n$ is square-integrable. Moreover, up to equivalence, there must be only one square-integrable representation of $G_n$, for each n. This is useful to know theoretically, but the real nature of $\sigma _n$, for $n>1$, is obscured by nested operations of inducing, direct sums, and taking tensor products. For general n, it is a challenging problem to find a realization of this abstract $\sigma _n$ that acts on a concrete Hilbert space. We succeed in Sect. 5 for the case of $n=2$. This provides a bridge between the soft methods of abstract harmonic analysis and the tools of classical harmonic analysis for study of functions on the affine group of the plane. We will use $\sigma$ instead of $\sigma _2$ to denote the concrete realization of the unique square-integrable representation of $G_2$.

One key to the success when $n=2$ is a factorization of ${{{\textrm{GL}}}}_2({\mathbb {R}})$ as a product of two closed subgroups. They are

$$\begin{aligned} K_0=\left\{ \begin{pmatrix} s &{} -t\\ t &{} s \end{pmatrix}: s,t\in {\mathbb {R}}, s^2+t^2>0\right\} \text { and } H_{(1,0)}=\left\{ \begin{pmatrix} 1 &{} 0\\ u &{} v \end{pmatrix}: u,v\in {\mathbb {R}}, v\ne 0\right\} . \end{aligned}$$

Direct calculations show that ${{{\textrm{GL}}}}_2({\mathbb {R}})=K_0H_{(1,0)}$, $K_0\cap H_{(1,0)}=\{{{\textrm{id}}}\}$, where id denotes the identity matrix, and $(M,C)\rightarrow MC$ is a homeomorphism of $K_0\times H_{(1,0)}$ with ${{{\textrm{GL}}}}_2({\mathbb {R}})$. This immediately provides a similar decomposition of $G_2$. Let $K=\{[{\underline{0}},M]:M\in K_0\}$ and $H=\{[{\underline{x}},C]:{\underline{x}}\in {\mathbb {R}}^2,C\in H_{(1,0)}\}$. Then $K\cap H=\{[{\underline{0}},{{\textrm{id}}}]\}$ and $G_2=KH$. Note that ${\mathcal {O}}=\{(1,0)A:A\in {{{\textrm{GL}}}}_2({\mathbb {R}})\}=\widehat{{\mathbb {R}}^2}{\setminus } \{(0,0)\}$ is the nontrivial orbit in $\widehat{{\mathbb {R}}^2}$ under the action of $G_2$ and H is the stability subgroup of ${\underline{\omega }}_0=(1,0)$ under that action. Because K is complementary to H, we can identify K with both the quotient space $G_2/H$ and the orbit ${\mathcal {O}}$. To make our formula for $\sigma$ readable, we define a homeomorphism $\gamma :{\mathcal {O}}\rightarrow K_0$ by $\gamma ({\underline{\omega }})=\frac{1}{\Vert {\underline{\omega }}\Vert ^2} \begin{pmatrix} \omega _1 &{} -\omega _2\\ \omega _2 &{} \omega _1 \end{pmatrix}$, for ${\underline{\omega }}\in {\mathcal {O}}$, where $\Vert {\underline{\omega }}\Vert$ is the Euclidean norm of ${\underline{\omega }}$. We also define two rational functions of six real variables, expressed as functions of ${\underline{\omega }}\in {\mathcal {O}}$ and $A\in {{{\textrm{GL}}}}_2({\mathbb {R}})$. They are denoted $u_{{\underline{\omega }},A}$ and $v_{{\underline{\omega }},A}$. The precise expressions for $u_{{\underline{\omega }},A}$ and $v_{{\underline{\omega }},A}$ are given in Proposition 5.3. We can now present a detailed expression for $\sigma$ realized on a concrete Hilbert space.

The Hilbert space of $\sigma$ is $L^2\big (K, L^2({\mathbb {R}}^*)\big )$, where ${\mathbb {R}}^*={\mathbb {R}}\setminus \{0\}$ viewed as a group under multiplication of real numbers. Both K and ${\mathbb {R}}^*$ are equipped with their respective Haar measures. We note that $K=\{[{\underline{0}},\gamma ({\underline{\omega }})]:{\underline{\omega }}\in {\mathcal {O}}\}$. For $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$, $[{\underline{x}},A]\in G_2$, and a.e. ${\underline{\omega }}\in {\mathcal {O}}$, $\sigma [{\underline{x}},A]F[{\underline{0}},\gamma ({\underline{\omega }})]$ is the element of $L^2({\mathbb {R}}^*)$ given by

$$\begin{aligned} \big (\sigma [{\underline{x}},A]F[{\underline{0}},\gamma ({\underline{\omega }})]\big )(t) = \textstyle \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}\Vert }{\Vert {\underline{\omega }}A\Vert } e^{2\pi i({\underline{\omega }}\,{\underline{x}}+t^{-1}u_{{\underline{\omega }},A})} \big (F[{\underline{0}},\gamma ({\underline{\omega }}A)]\big ) (v_{{\underline{\omega }},A}^{-1}t),\nonumber \\ \end{aligned}$$

(1)

for a.e. $t\in {\mathbb {R}}^*$ (Eq. (23) in Sect. 5). We note that Mackey theory (see [5, 15], or [13]) tells us that

$$\begin{aligned}{} & {} \widehat{G_2}=\{[\sigma ]\}\cup \left\{ \tilde{[\pi ]}: [\pi ]\in {\widehat{{{{\textrm{GL}}}}_2({\mathbb {R}})}}\right\} \cup \{[\textrm{ind}_{H}^{G_2}(X_{(1,0)}\otimes \psi )]:\psi \mathrm{is\, a\, character\, of} H_{(1,0)}\}. \end{aligned}$$

Here $[\tau ]$ denotes the unitary equivalence class of a representation $\tau$ and, for a representation $\pi$ of ${{{\textrm{GL}}}}_2({\mathbb {R}})$, the lift of $\pi$ to $G_2$ is denoted ${\tilde{\pi }}$. A parametrization of ${\widehat{{{{\textrm{GL}}}}_2({\mathbb {R}})}}$ is complex (see, for example, [19]), but only $\sigma$ is needed for the analysis of square-integrable functions on $G_2$.

We show that $\sigma$ is square-integrable by a multi-step process that concludes with the following core result (Theorem 6.5). Let $g\in L^2({\mathbb {R}}^*)$ satisfy $\int _{{\mathbb {R}}^*}\big | |\nu |^{1/2}g(\nu )\big |^2d\mu _{{\mathbb {R}}^*}(\nu )=1$ and let $\eta \in L^2(\widehat{{\mathbb {R}}^2})$ satisfy $\Vert \eta \Vert _{_{L^2(\widehat{{\mathbb {R}}^2})}}=1$. Let $E\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$ be defined as

$$\begin{aligned} E[{\underline{0}},L](\nu )= \frac{{\overline{\eta }}((1,0)L^{-1})}{|\det (L)|}\,g(\nu ), \end{aligned}$$

for a.e. $[{\underline{0}},L]\in K$ and a.e. $\nu \in {\mathbb {R}}^*$. Define $V_EF[{\underline{x}},A]= \langle F, \sigma [{\underline{x}},A]E\rangle _{_{L^2(K,L^2({\mathbb {R}}^*))}}$, for $[{\underline{x}},A]\in G_2$ and $F \in L^2\big (K,L^2({\mathbb {R}}^*)\big )$. Then $V_E$ is a linear isometry of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ into $L^2(G_2)$ that intertwines $\sigma$ with the left regular representation of $G_2$. In particular, $\sigma$ is a square-integrable representation of $G_2$.

In Theorem 6.7, we present a doubly indexed family $\{{\mathcal {M}}_{i,j}:(i,j)\in I\times J\}$ of closed, left-invariant, subspaces of $L^2(G_2)$ such that $L^2(G_2)=\sum ^{\oplus }_{(i,j)\in I\times J}{\mathcal {M}}_{i,j}$ and the restriction of the left regular representation of $G_2$ to each ${\mathcal {M}}_{i,j}$ is unitarily equivalent to $\sigma$, showing that the regular representation is supported by the singleton $\{[\sigma ]\}$ in $\widehat{G_2}$. This also enables us to easily formulate what the abstract Plancherel Theorem looks like for the group $G_2$ and we do this in Proposition 7.1. We end with a section of concluding remarks.

2 Notation and Background

In this section, we introduce our notational conventions as well as those properties of square-integrable representations that we need. Let $n\in {\mathbb {N}}$.

• ${\mathbb {R}}$ denotes the field of real numbers. Under addition as operation, ${\mathbb {R}}$ is a group.

• ${\mathbb {R}}^*={\mathbb {R}}\setminus \{0\}$, which is a group under multiplication.

• ${\mathbb {R}}^+$ is the subgroup of ${\mathbb {R}}^*$ consisting of positive real numbers.

• ${\mathbb {R}}^n=\left\{ {\underline{x}}=\begin{pmatrix} x_1\\ \vdots \\ x_n \end{pmatrix}: x_1, \cdots , x_n\in {\mathbb {R}}\right\}$, $\widehat{{\mathbb {R}}^n}=\{{\underline{\omega }}=(\omega _1,\cdots ,\omega _n): \omega _1,\cdots ,\omega _n\in {\mathbb {R}}\}$.

• The symbol ${\underline{0}}$ will be used for either a column vector or row vector of zeroes; whichever fits in the context.

• For $f\in L^1({\mathbb {R}}^n)$, the Fourier transform of f is ${\widehat{f}}({\underline{\omega }})= \int _{{\mathbb {R}}^n}f({\underline{x}})e^{2\pi i{\underline{\omega }}\,{\underline{x}}}d{\underline{x}}$, for all ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n}$.

• ${\mathcal {F}}:L^2({\mathbb {R}}^n)\rightarrow L^2\big (\widehat{{\mathbb {R}}^n}\big )$ is the unitary such that ${\mathcal {F}}f={\widehat{f}}$, for all $f\in L^2({\mathbb {R}}^n)\cap L^1({\mathbb {R}}^n)$.

• ${{{\textrm{GL}}}}_n({\mathbb {R}})$ denotes the general linear group of invertible $n\times n$ real matrices.

• For ${\underline{x}}\in {\mathbb {R}}^n$ and $A\in {{{\textrm{GL}}}}_n({\mathbb {R}})$, $[{\underline{x}},A]$ is the affine transformation ${\underline{z}}\rightarrow {\underline{x}}+A{\underline{z}}$ of ${\mathbb {R}}^n$.

• $G_n=\{[{\underline{x}},A]:{\underline{x}}\in {\mathbb {R}}^n,A\in {{{\textrm{GL}}}}_n({\mathbb {R}})\}$.

For $[{\underline{x}},A],[{\underline{y}},B]\in G_n$, the composition, denoted $[{\underline{x}},A][{\underline{y}},B]$, is the affine transformation $[{\underline{x}}+A{\underline{y}},AB]$. In fact, $G_n$ forms a group when equipped with composition as a product. The identity in $G_n$ is $[{\underline{0}},{{\textrm{id}}}]$, where ${{\textrm{id}}}$ denotes the identity $n\times n$ matrix. If ${{{\textrm{GL}}}}_n({\mathbb {R}})$ is considered as an open subset of ${\mathbb {R}}^{n^2}$, then ${\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}})$ is a locally compact Hausdorff space with the product topology and we move this topology to $G_n$ with the bijection $({\underline{x}},A)\rightarrow [{\underline{x}},A]$. With this topology, $G_n$ is a locally compact group which can be viewed as the semidirect product ${\mathbb {R}}^n\rtimes {{{\textrm{GL}}}}_n({\mathbb {R}})$ (see the discussion following Example 1.5 in [13] for basic information on semidirect products). For any locally compact group G, we use the following notation:

• $\mu _G$ denotes left Haar measure on G. It is a nonzero Radon measure with the property that $\mu _G(xE)=\mu _G(E)$, for any Borel $E\subseteq G$ and $x\in G$. Left Haar measure is unique up to multiplication by a positive constant.

• Integrals with respect to $\mu _G$ are denoted $\int _Gf\,d\mu _G$ or $\int _Gf(x)\,d\mu _G(x)$, for any function f on G for which the integral has meaning. Thus, $\int _Gf(yx)\,d\mu _G(x)= \int _Gf(x)\,d\mu _G(x)$, for any $y\in G$.

• The modular function on G is the continuous homorphism $\Delta _G:G\rightarrow {\mathbb {R}}^+$ that satisfies $\Delta _G(y)\int _Gf(xy)\,d\mu _G(x)= \int _Gf(x)\,d\mu _G(x)$, for any $y\in G$ and any f where the integral makes sense. We call G unimodular if $\Delta _G$ is identically 1 on G; otherwise G is nonunimodular.

• For $1\le p\le \infty$, $L^p(G)$ denotes the usual Lebesgue space $L^p(G,{\mathcal {B}}_G,\mu _G)$, where ${\mathcal {B}}_G$ denotes the $\sigma$-algebra of Borel subsets of G.

• $L^2(G)$ is a Hilbert space with inner product given by $\langle f,g\rangle _{L^2(G)}= \int _Gf(x)\overline{g(x)} d\mu _G(x)$, for all $f,g\in L^2(G)$.

• $C_b(G)$ is the space of bounded continuous ${\mathbb {C}}$-valued functions on G, a Banach space when equipped with the uniform norm.

• $C_c(G)$ denotes the space of continuous ${\mathbb {C}}$-valued functions of compact support on G.

For a particular locally compact group G, we often identify $\mu _G$ by giving a formula for $\int _Gf\,d\mu _G$ for any $f\in C_c(G)$. This uniquely identifies the left Haar measure, and the normalization we are using, by the Riesz Representation Theorem. For example, the left Haar measure on ${{{\textrm{GL}}}}_n({\mathbb {R}})$ is described as follows (see [11]). For $A\in {{{\textrm{GL}}}}_n({\mathbb {R}})$, let $a_{ij}$ denote the (i, j) entry of A. Then,

$$\begin{aligned} \int _{{{{\textrm{GL}}}}_n({\mathbb {R}})}f\,d\mu _{{{{\textrm{GL}}}}_n({\mathbb {R}})}= \int _{-\infty }^\infty \cdots \int _{-\infty }^\infty f\begin{pmatrix} a_{11} &{} \cdots &{} a_{1n}\\ \vdots &{} \ddots &{} \vdots \\ a_{n1} &{} \cdots &{} a_{nn} \end{pmatrix} \frac{da_{nn}\cdots da_{11}}{|\det (A)|^n}, \end{aligned}$$

(2)

for any $f\in C_c({{{\textrm{GL}}}}_n({\mathbb {R}}))$. In particular,

$$\begin{aligned} \int _{{{{\textrm{GL}}}}_2({\mathbb {R}})}f\,d\mu _{{{{\textrm{GL}}}}_2({\mathbb {R}})}= \int _{-\infty }^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty \int _{-\infty }^\infty f\begin{pmatrix} w &{} x\\ y &{} z \end{pmatrix}\frac{dz\,dy\,dx\,dw}{(wz-xy)^2}, \end{aligned}$$

for any $f\in C_c({{{\textrm{GL}}}}_2({\mathbb {R}}))$. Note that $\mu _{{{{\textrm{GL}}}}_n({\mathbb {R}})}$ is also right invariant, so ${{{\textrm{GL}}}}_n({\mathbb {R}})$ is a unimodular group.

For a semidirect product group such as $G_n$, the left Haar measure can be computed as described on page 9 of [13]. This gives

$$\begin{aligned} \int _{G_n}f\,d\mu _{G_n}=\int _{{{{\textrm{GL}}}}_n({\mathbb {R}})}\int _{{\mathbb {R}}^n} f[{\underline{x}},A]\,\frac{d{\underline{x}}\,d\mu _{{{{\textrm{GL}}}}_n({\mathbb {R}})}(A)}{|\det (A)|}, \end{aligned}$$

(3)

for all $f\in C_c(G_n)$. The modular function of $G_n$ is given by $\Delta _{G_n}[{\underline{x}},A]=|\det (A)|^{-1}$, for $[{\underline{x}},A]\in G_n$.

For the theory of Hilbert spaces and operators on Hilbert spaces, a good reference is [12]. Let ${\mathcal {H}}$ be a Hilbert space.

• The norm and inner product on ${\mathcal {H}}$ are denoted by $\Vert \cdot \Vert _{{\mathcal {H}}}$ and $\langle \cdot ,\cdot \rangle _{{\mathcal {H}}}$, respectively.

• The Banach $*$-algebra of bounded linear operators on ${\mathcal {H}}$ is denoted ${\mathcal {B}}({\mathcal {H}})$.

• The group of all unitary operators on ${\mathcal {H}}$ is ${\mathcal {U}}({\mathcal {H}})$.

• $A\in {\mathcal {B}}({\mathcal {H}})$ is called a Hilbert–Schmidt operator on ${\mathcal {H}}$ if $\sum _{j\in J}\Vert A\xi _j\Vert _{{\mathcal {H}}}^{\,\,2}<\infty$, for any orthonormal basis $\{\xi _j:j\in J\}$ of ${\mathcal {H}}$. The sum $\sum _{j\in J}\Vert A\xi _j\Vert _{{\mathcal {H}}}^{\,\,2}$ is independent of the basis.

• Let ${\mathcal {B}}_2({\mathcal {H}})$ denote the set of all Hilbert–Schmidt operators on ${\mathcal {H}}$. This is a Hilbert space with norm given by $\Vert A\Vert _{{{\textrm{HS}}}}=\left( \sum _{j\in J}\Vert A\xi _j\Vert _{{\mathcal {H}}}^{\,\,2}\right) ^{1/2}$, for any orthonormal basis $\{\xi _j:j\in J\}$ of ${\mathcal {H}}$.

• If $(X,\mu )$ is a $\sigma$-finite measure space and $g:X\rightarrow {\mathbb {C}}$ is measurable, for any $f\in L^2(X,\mu )$, let $M_gf=gf$. If g is bounded, then $M_g\in {\mathcal {B}}\big (L^2(X,\mu )\big )$. In general, $M_g$ is a not necessarily bounded operator on $L^2(X,\mu )$ (see Example 2.7.2 of [12]).

Let G be a locally compact group. Let ${\mathcal {H}}_\pi$ be a Hilbert space and let $\pi :G\rightarrow {\mathcal {U}}({\mathcal {H}}_\pi )$ be a group homomorphism. We call $\pi$, more formally the pair $({\mathcal {H}}_\pi ,\pi )$, a unitary representation of G if, for every $\xi ,\eta \in {\mathcal {H}}_\pi$, the map $x\rightarrow \langle \xi ,\pi (x)\eta \rangle _{{\mathcal {H}}_\pi }$ is a continuous function on G. When no confusion can arise, we will simply write representation of G to mean a unitary representation of G. The left regular representation of G is denoted $\lambda _G$. The Hilbert space of $\lambda _G$ is $L^2(G)$ and, for each $x\in G$, $\lambda _G(x)f(y)= f(x^{-1}y)$, for $\mu _G$-almost every $y\in G$ and any $f\in L^2(G)$. For a given representation $\pi$ of G, we have the following notation and concepts:

• For any $f\in L^1(G)$, there exists $\pi (f)\in {\mathcal {B}}({\mathcal {H}}_\pi )$ that satisfies, for all $\xi ,\eta \in {\mathcal {H}}_\pi$.

  $$\begin{aligned} \langle \pi (f)\xi ,\eta \rangle = \int _G f(x)\langle \pi (x)\xi ,\eta \rangle d\mu _G(x), \end{aligned}$$

  Then $\pi :L^1(G)\rightarrow {\mathcal {B}}({\mathcal {H}}_\pi )$ is a non-degenerate $*$-homomorphism.

• For the left regular representation, $\lambda _G(f)h=f*h$, for $h\in L^2(G)$ and $f\in L^1(G)$.

• Suppose $\sigma$ is another representation of G and $A:{\mathcal {H}}_\sigma \rightarrow {\mathcal {H}}_\pi$ is a bounded linear operator. We say that A intertwines $\sigma$ and $\pi$ if $A\sigma (x)=\pi (x)A$, for all $x\in G$.

• If $\sigma$ is another representation of G, we say $\sigma$ is equivalent to $\pi$ if there exists a unitary map $U:{\mathcal {H}}_\sigma \rightarrow {\mathcal {H}}_\pi$ that intertwines $\sigma$ and $\pi$. Then $U^{-1}$ intertwines $\pi$ and $\sigma$. We write $\sigma \sim \pi$ when $\sigma$ and $\pi$ are equivalent. Then $\sim$ is an equivalence relation on any set of representations of G.

• A closed subspace ${\mathcal {K}}$ of ${\mathcal {H}}_\pi$ is $\pi$-invariant if $\xi \in {\mathcal {K}}$ implies $\pi (x)\xi \in {\mathcal {K}}$, for all $x\in G$.

• If ${\mathcal {K}}$ is a $\pi$-invariant closed subspace of ${\mathcal {H}}_\pi$, let $\pi _{{\mathcal {K}}}$ denote the representation whose Hilbert space is ${\mathcal {K}}$ and satisfies $\pi _{{\mathcal {K}}}(x)=\pi (x)|_{{\mathcal {K}}}$, for all $x\in G$. We call $\pi _{{\mathcal {K}}}$ a subrepresentation of $\pi$.

• If J is an index set and $\pi _j$ is a representation of G on ${\mathcal {H}}_j$, for each $j\in J$, then $\oplus _{j\in J}\pi _j$ denotes the representation of G acting on ${\underline{\xi }}=(\xi _j)_{j\in J}\in \oplus _{j\in J}{\mathcal {H}}_j$ by, for $x\in G$,

  $$\begin{aligned} \big (\oplus _{j\in J}\pi _j\big )(x){\underline{\xi }}= \big (\pi _j(x)\xi _j\big )_{j\in J}. \end{aligned}$$

• If J is an index set with cardinality m, suppose $\pi _j=\pi$ and ${\mathcal {H}}_j={\mathcal {H}}_\pi$, for all $j\in J$, then $\oplus _{j\in J}\pi _j$ may be denoted $m\cdot \pi$. If $\sigma$ is a representation of G such that $\sigma \sim m\cdot \pi$, then $\sigma$ may be called a multiple of $\pi$.

• If $\{0\}$ and ${\mathcal {H}}_\pi$ are the only $\pi$-invariant closed subspaces of ${\mathcal {H}}_\pi$, then $\pi$ is called an irreducible representation of G.

• For each $\eta \in {\mathcal {H}}_\pi$, define the linear map $V_\eta :{\mathcal {H}}_\pi \rightarrow C_b(G)$ by, for each $\xi \in {\mathcal {H}}_\pi$,

  $$\begin{aligned} V_\eta \xi (x)=\langle \xi ,\pi (x)\eta \rangle _{{\mathcal {H}}_\pi }, \text { for all }x\in G. \end{aligned}$$

• $\pi$ is called a square-integrable representation if $\pi$ is irreducible and the exists a nonzero $\eta \in {\mathcal {H}}_\pi$ such that $V_\eta \eta \in L^2(G)$.

Examples of square-integrable representations arise in an elementary manner for many groups of affine transformations. Let K be a closed subgroup of ${{{\textrm{GL}}}}_n({\mathbb {R}})$ and let $G={\mathbb {R}}^n\rtimes K=\{[{\underline{x}},A]:{\underline{x}}\in {\mathbb {R}}^n, A\in K\}$, a closed subgroup of $G_n$. For ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n}$, The K-orbit of ${\underline{\omega }}$ is ${\underline{\omega }}K=\{{\underline{\omega }}A:A\in K\}$ and the stabilizer is $K_{{\underline{\omega }}}= \{A\in K:{\underline{\omega }}A={\underline{\omega }}\}$. The natural representation $\rho$ of G on $L^2({\mathbb {R}}^n)$ is given by, for $[{\underline{x}},A]\in G$ and $f\in L^2({\mathbb {R}}^n)$,

$$\begin{aligned} \rho [{\underline{x}},A]f({\underline{y}})=|\det (A)|^{-1/2}f\left( A^{-1}({\underline{y}}-{\underline{x}})\right) , \end{aligned}$$

(4)

for a.e. ${\underline{y}}\in {\mathbb {R}}^n$. If ${\mathcal {O}}$ is an open subset of $\widehat{{\mathbb {R}}^n}$, let ${\mathcal {H}}^2_{{\mathcal {O}}}$ denote the closed subspace of $L^2({\mathbb {R}}^n)$ consisting of functions whose Fourier transform is supported on ${\mathcal {O}}$. If ${\mathcal {O}}K=\{{\underline{\omega }}A:{\underline{\omega }}\in {\mathcal {O}}, A\in K\}={\mathcal {O}}$, then ${\mathcal {H}}^2_{{\mathcal {O}}}$ is a $\rho$-invariant subspace of $L^2({\mathbb {R}}^n)$. Let $\rho _{{\mathcal {O}}}$ denote the corresponding subrepresentation of $\rho$. If ${\mathcal {O}}$ is a single K-orbit that is free, in the sense that $K_{{\underline{\omega }}}=\{{{\textrm{id}}}\}$, for any ${\underline{\omega }}\in {\mathcal {O}}$, then $\rho _{{\mathcal {O}}}$ was shown to be square-integrable in [2]. The following more general proposition was established for open K-orbits with compact stability subgroups in [6].

Proposition 2.1

Let K be a closed subgroup of ${{{\textrm{GL}}}}_n({\mathbb {R}})$ and let $G={\mathbb {R}}^n\rtimes K$. Suppose there exists an ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n}$ such that ${\mathcal {O}}={\underline{\omega }}K$ is open. Then $\rho _{{\mathcal {O}}}$ is a square-integrable representation of G if and only if $K_{{\underline{\omega }}}$ is compact.

Our primary reference for the properties of square-integrable representations is [4]. The following theorem is formulated from Theorem 3 of [4] and the preliminary results in [4]. Also see Theorem 2.25 of [7].

Theorem 2.2

(Duflo-Moore) Let $\pi$ be a square-integrable representation of a locally compact group. Let ${\mathcal {D}}_\pi =\{\eta : V_\eta \eta \in L^2(G)\}$. Then

• (a) ${\mathcal {D}}_\pi$ is a dense linear subspace of ${\mathcal {H}}_\pi$,

• (b) if $\eta \in {\mathcal {D}}_\pi$, then $V_\eta \xi \in L^2(G)$, for all $\xi \in {\mathcal {H}}_\pi$,

• (c) there exists a nonzero positive selfadjoint operator $C_\pi$ in ${\mathcal {H}}_\pi$ with domain ${\mathcal {D}}_\pi$ satisfying

  $$\begin{aligned} \pi (x)C_\pi \pi (x)^*=\Delta _G(x)^{1/2}C_\pi , \text { for all } x\in G, \end{aligned}$$

• (d) for any $\xi _1,\xi _2\in {\mathcal {H}}_\pi$ and $\eta _1,\eta _2\in {\mathcal {D}}_\pi$,

  $$\begin{aligned} \langle V_{\eta _1}\xi _1, V_{\eta _2}\xi _2\rangle _{L^2(G)}= \langle \xi _1,\xi _2\rangle _{{\mathcal {H}}_\pi } \langle C_\pi \eta _2,C_\pi \eta _1\rangle _{{\mathcal {H}}_\pi } \end{aligned}$$

  Moreover, if T is any densely defined nonzero positive selfadjoint operator in ${\mathcal {H}}_\pi$ that satisfies the identity $\pi (x)T\pi (x)^*=\Delta _G(x)^{1/2}T$, for all $x\in G$, then the domain of T is ${\mathcal {D}}_\pi$ and there exists a constant $r> 0$ such that $T=rC_\pi$.

Definition 2.3

Let $\pi$ be a square-integrable representation of a locally compact group G. The operator $C_\pi$ named in Theorem 2.2 is called the Duflo-Moore operator of $\pi$.

Let $\pi$ be a square-integrable representation of a locally compact group G. For any $\eta \in {\mathcal {D}}_\pi$, by (b) and (d) of Theorem 2.2, $\Vert V_\eta \xi \Vert _{L^2(G)}=\Vert C_\pi \eta \Vert _{{\mathcal {H}}_\pi } \Vert \xi \Vert _{{\mathcal {H}}_\pi }$, for all $\xi \in {\mathcal {H}}_\pi$. Thus, ${\mathcal {K}}^\pi _\eta =\{V_\eta \xi :\xi \in {\mathcal {H}}_\pi \}$ is a closed subspace of $L^2(G)$ and $V_\eta :{\mathcal {H}}_\pi \rightarrow {\mathcal {K}}^\pi _\eta \subseteq L^2(G)$ is a bounded linear map. Moreover, if $\Vert C_\pi \eta \Vert _{{\mathcal {H}}_\pi }=1$, then $V_\eta$ is an isometry into $L^2(G)$. In fact, (d) of Theorem 2.2 shows that $V_\eta$ is a unitary map when considered as a map of ${\mathcal {H}}_\pi$ onto ${\mathcal {K}}^\pi _\eta$. Note that, for $x\in G$ and $\xi \in {\mathcal {H}}_\pi$,

$$\begin{aligned} \lambda _G(x)V_\eta \xi (y){} & {} =V_\eta \xi (x^{-1}y)= \langle \xi ,\pi (x^{-1}y)\eta \rangle _{{\mathcal {H}}_\pi }= \langle \pi (x)\xi ,\pi (y)\eta \rangle _{{\mathcal {H}}_\pi }\nonumber \\{} & {} = V_\eta \pi (x)\xi (y), \end{aligned}$$

(5)

for all $y\in G$. Thus, ${\mathcal {K}}^\pi _\eta$ is a $\lambda _G$-invariant subspace of $L^2(G)$ and $V_\eta$ intertwines $\pi$ with $\lambda _G$. This means $\pi$ is equivalent to the restriction of $\lambda _G$ to ${\mathcal {K}}^\pi _\eta$. If $\eta _1,\eta _2 \in {\mathcal {D}}_\pi$ are such that $C_\pi \eta _1\perp C_\pi \eta _2$, then ${\mathcal {K}}^\pi _{\eta _1}\perp {\mathcal {K}}^\pi _{\eta _2}$ by Theorem 2.2(d). Following [7], let $L^2_\pi (G)$ denote the smallest closed subspace of $L^2(G)$ containing $\cup _{\eta \in {\mathcal {D}}_\pi }{\mathcal {K}}^\pi _\eta$. A Gram–Schmidt process shows that, when G is separable, there exists a countable set $\{\eta _j:j\in J\}$ in ${\mathcal {D}}_\pi$ such that $\{C_\pi \eta _j:j\in J\}$ is an orthonormal basis of ${\mathcal {H}}_\pi$. Then $L^2_\pi (G)=\oplus _{j\in J} {\mathcal {K}}^\pi _{\eta _j}$ (see Theorem 2.33(c) of [7]). Note that the cardinality of J is $\dim ({\mathcal {H}}_\pi )$.

Suppose $\pi$ and $\pi '$ are both square-integrable representations of G and $\pi '$ is not equivalent to $\pi$. A simple argument (see, for example, Theorem 2.33(d) of [7]) shows that, for $\eta \in {\mathcal {D}}_\pi$ and $\eta '\in {\mathcal {D}}_{\pi '}$, ${\mathcal {K}}^\pi _\eta \perp {\mathcal {K}}^{\pi '}_{\eta '}$. Thus, $L^2_\pi (G)\perp L^2_{\pi '}(G)$.

Definition 2.4

For a locally compact group G, let ${\widehat{G}}$ denote the set of equivalence classes of irreducible representations of G. Let ${\widehat{G}}^r$ denote the subset of ${\widehat{G}}$ consisting of equivalence classes of square-integrable representations.

We will sometimes abuse notation and use the same symbol for an irreducible representation and its equivalence class. A locally compact group G is called an [AR]-group when its left regular representation is a direct sum of irreducible representations. See [18] for some implications of this poperty. When G is an [AR]-group, we have

$$\begin{aligned} L^2(G)=\oplus _{\pi \in {\widehat{G}}^r}L^2_\pi (G). \end{aligned}$$

From Theorem 2.33 of [7] and Theorem 5.2 of [8], we can formulate a generalization of the Peter-Weyl Theorem for compact groups to [AR]-groups.

Theorem 2.5

Let G be a separable [AR]-group. For each $\pi \in {\widehat{G}}^r$, let $\{\eta ^\pi _j:j\in J_\pi \}$ be a set in ${\mathcal {D}}_\pi$ such that $\{C_\pi \eta ^\pi _j:j\in J_\pi \}$ is an orthonormal basis of ${\mathcal {H}}_\pi$ and let $\{\xi ^\pi _k:k\in J_\pi \}$ be any orthonormal basis of ${\mathcal {H}}_\pi$. Then

$$\begin{aligned} \cup _{\pi \in {\widehat{G}}^r} \left\{ V_{\eta ^\pi _j}\xi ^\pi _k: (j,k)\in J_\pi \times J_\pi \right\} \end{aligned}$$

is an orthonormal basis of $L^2(G)$.

Note that, if G is compact, then ${\widehat{G}}^r={\widehat{G}}$. Moreover, each $\pi \in {\widehat{G}}$ is finite-dimensional and $C_\pi =\dim (\pi )^{-1/2}I_{{\mathcal {H}}_\pi }$, where $\dim (\pi )=\dim ({\mathcal {H}}_\pi )$ and $I_{{\mathcal {H}}_\pi }$ is the identity operator on ${\mathcal {H}}_\pi$.

In order for the kind of basis that is guaranteed by Theorem 2.5 to be useful when working with a particular [AR]-group, one usually needs explicit descriptions of the square-integrable representations on concrete Hilbert spaces and a precise identification of the operator $C_\pi$, for each such $\pi$. As a result of Proposition 2.1, this works out well for groups of the form ${\mathbb {R}}^n\rtimes K$, where K is a closed subgroup of ${{{\textrm{GL}}}}_n({\mathbb {R}})$ and there exist open K-orbits ${\mathcal {O}}_1,\cdots ,{\mathcal {O}}_m$ in $\widehat{{\mathbb {R}}^n}$, each with compact stability subgroups, such that $\cup _{j=1}^m{\mathcal {O}}_j$ is co-null in $\widehat{{\mathbb {R}}^n}$. However, there exist [AR]-groups of the form ${\mathbb {R}}^n\rtimes K$ where there are open K-orbits, but the associated stability subgroups are not compact. In fact, it was shown in [1], example (iii), that ${\mathbb {R}}^2\rtimes {{\textrm{GL}}}_2^+({\mathbb {R}})$, where ${{\textrm{GL}}}_2^+({\mathbb {R}})=\{A\in {{{\textrm{GL}}}}_2({\mathbb {R}}):\det (A)>0\}$, is an [AR]-group. The same is true of the full affine group in two-dimensions $G_2$. It is worthwhile formulating a proposition collecting known results.

Proposition 2.6

Let K be a closed subgroup of ${{{\textrm{GL}}}}_n({\mathbb {R}})$ such that there exist open K-orbits ${\mathcal {O}}_1,\cdots ,{\mathcal {O}}_m$ in $\widehat{{\mathbb {R}}^n}$ with $\cup _{j=1}^m{\mathcal {O}}_j$ co-null in $\widehat{{\mathbb {R}}^n}$. For $1\le j\le m$, select ${\underline{\omega }}_j\in {\mathcal {O}}_j$. Then ${\mathbb {R}}^n\rtimes K$ is an [AR]-group if and only if $K_{{\underline{\omega }}_j}$ is an [AR]-group for $1\le j\le m$.

Proof

This follows from the calculations on page 595 of [1] or Corollary 11.1 of [14]. $\square$

For example, in the group $G_2$, $K={{{\textrm{GL}}}}_2({\mathbb {R}})$ and there are just two K-orbits in $\widehat{{\mathbb {R}}^2}$. One orbit is the trivial orbit $\{{\underline{0}}\}$ and the other is ${\mathcal {O}}=\widehat{{\mathbb {R}}^2}\setminus \{{\underline{0}}\}$. Pick ${\underline{\omega }}_0=(1,0)\in {\mathcal {O}}$. Then, the stability subgroup is $K_{(1,0)}=\{A\in {{{\textrm{GL}}}}_2({\mathbb {R}}):(1,0)A=(1,0)\}$. If $A=\begin{pmatrix} s &{} t\\ u &{} v \end{pmatrix}$, then $(1,0)A=(s,t)$. So $A\in K_{(1,0)}$ exactly when $s=1$ and $t=0$. Thus $K_{(1,0)}=\left\{ \begin{pmatrix} 1 &{} 0\\ u &{} v \end{pmatrix}:u\in {\mathbb {R}},v\in {\mathbb {R}}^*\right\}$. Note that $[u,v]\rightarrow \begin{pmatrix} 1 &{} 0\\ u &{} v \end{pmatrix}$ is an isomorphism of $G_1$ with $K_{(1,0)}$ and it is well-known that $G_1$ has a regular representation that is an infinite multiple of just one square-integrable representation (see Example 1, Section 10, of [14], for example). This is essentially the way ${\mathbb {R}}^2\rtimes {{\textrm{GL}}}_2^+({\mathbb {R}})$ is shown to be an [AR]-group in [1].

3 Induced Representations

The main purpose of this study is to make the various ingredients that appear in Theorem 2.5 explicit for $G_2$, the group of invertible affine transformations of the plane. We make extensive use of induced representations.

The theory of induced representations for locally compact groups in general was initially developed by Mackey in [15]. The particular results we need are found in [5] or [13] and we briefly summarize them here. We introduce a representation, equivalent to the induced representation we need, for a particular situation that holds in all the cases that arise in the study of $G_2$.

If $\pi$ is a unitary representation of a closed subgroup H of a locally compact group G, there are a variety of ways (all resulting in mutually equivalent representations) of defining the induced representation ${{\textrm{ind}}}_H^G\pi$, which is a representation of G. We will use the construction presented in Section 6.1 of [5] and following Proposition 2.28 of [13]. With G and H fixed, a rho-function is a Borel map $\rho :G\rightarrow [0,\infty )$ that is locally integable and satisfies

$$\begin{aligned} \rho (xh)=\frac{\Delta _H(h)}{\Delta _G(h)}\rho (x), \text { for }x\in G \text { and } h\in H. \end{aligned}$$

(6)

Let $p:G/H\rightarrow G$ be a cross-section of the H-cosets. That is, $p(yH)\in yH$, for any $yH\in G/H$. There always exists a continuous rho-function $\rho$ such that $\rho (x)>0$, for all $x\in G$, and an associated regular Borel measure $\mu _\rho$ on G/H such that Weil’s integration formula holds. That is, for any $f\in C_c(G)$

$$\begin{aligned} \int _Gf(x)\rho (x)\,d\mu _G(x)=\int _{G/H}\int _H f\big (p(\omega )h\big )\,d\mu _H(h)\,d\mu _\rho (\omega ). \end{aligned}$$

(7)

See Theorem 1.18, Lemma 1.20, and Corollary 1.21 of [13]. If $\rho$ is a continuous and strictly positive rho-function on G, (6) implies that, for $x,y\in G$, $\frac{\rho (x^{-1}yh)}{\rho (yh)}= \frac{\rho (x^{-1}y)}{\rho (y)}$, for all $h\in H$. We need this term $\frac{\rho (x^{-1}y)}{\rho (y)}$ in the definition of ${{\textrm{ind}}}_H^G\pi$. Fix a continuous, strictly positive, rho-function $\rho$ and let $\mu _\rho$ be the associated measure on G/H so that (7) holds. The Hilbert space of ${{\textrm{ind}}}_H^G\pi$ is denoted ${\mathcal {H}}_{{{\textrm{ind}}}\pi }$ and consists of all equivalence classes of weakly measurable functions $\xi :G\rightarrow {\mathcal {H}}_\pi$ such that $\xi (xh)=\pi (h^{-1})\xi (x)$, for all $h\in H$ and $\mu _G$-almost all $x\in G$, and $\int _{G/H}\Vert \xi \big (p(\omega )\big )\Vert _{{\mathcal {H}}_\pi }^{\,\,2}d\mu _\rho (\omega )<\infty$. Note that the latter condition does not depend on the choice of the cross-section p. For $\xi _1,\xi _2\in {\mathcal {H}}_{{{\textrm{ind}}}\pi }$, the inner product is given by

$$\begin{aligned} \langle \xi _1,\xi _2\rangle _{{\mathcal {H}}_{{{\textrm{ind}}}\pi }}= \int _{G/H}\langle \xi _1\big (p(\omega )\big ), \xi _2\big (p(\omega )\big )\rangle _{{\mathcal {H}}_\pi } d\mu _\rho (\omega ). \end{aligned}$$

Then ${\mathcal {H}}_{{{\textrm{ind}}}\pi }$ is a Hilbert space. For $x\in G$ and $\xi \in {\mathcal {H}}_{{{\textrm{ind}}}\pi }$, let

$$\begin{aligned} {{\textrm{ind}}}_H^G\pi (x)\xi (y)=\textstyle \left[ \frac{\rho (x^{-1}y)}{\rho (y)}\right] ^{1/2} \xi (x^{-1}y), \end{aligned}$$

(8)

for $\mu _G-{{\mathrm{a.e.}}} \,y\in G$. This defines a unitary operator ${{\textrm{ind}}}_H^G\pi (x)$ on ${\mathcal {H}}_{{{\textrm{ind}}}\pi }$, for each $x\in G$, and a representation ${{\textrm{ind}}}_H^G\pi$ of G. See [13] or [5] for more details. Using a different rho-function in the definition results in an equivalent representation. In some situations, one can show that ${{\textrm{ind}}}_H^G\pi$ is equivalent to a representation acting on a more natural Hilbert space of functions on a space that has a natural G-action. One situation that is useful to us is treated in Example 2.29 of [13]. However, there is an error in the definition of the rho-function in the last line of page 74 of [13], so we will point out the details necessary to correct the formula given there.

Suppose H is a closed subgroup of G that is complemented in the sense that there exists a closed subgroup K of G such that $K\cap H= \{e\}$, where e is the identity element of G, and the map $(k,h)\rightarrow kh$ is a homeomorphism of $K\times H$ onto G. When this holds, the restriction $q|_K$ of the quotient map q to K is a homeomorphism of K with G/H. For each $x\in G$, there exist unique $k_x\in K$ and $h_x \in H$ such that $x=k_xh_x$. We take the cross-section $p:G/H\rightarrow G$ to be given by $p(xH)=k_x$, for any $xH\in G/H$. Then p is a homeomorphism of G/H with its image K. The left action of G on G/H is transferred to a left action of G on K by this homeomorphism. That is, for $x\in G$ and $k\in K$, $x\cdot k=k_{xk}$. Define $\rho :G\rightarrow {\mathbb {R}}^+$ by $\rho (x)=\frac{\Delta _H(h_x)}{\Delta _G(h_x)}$, for all $x\in G$. For $x\in G$ and $h\in H$, $h_{xh}=h_xh$, so $\rho (xh)= \frac{\Delta _H(h)}{\Delta _G(h)}\rho (x)$. Thus, $\rho$ is a rho-function on G that is obviously continuous and positive. Since $h_{kx}=h_x$, we have $\rho (kx)=\rho (x)$, for $k\in K$ and $x\in G$.

The measure $\mu _\rho$ that is associated to $\rho$ so that (7) holds can be moved to K by the homeomorphism $q|_K$. That is, let ${\tilde{\mu }}_\rho (E)= \mu _\rho \big (q(E)\big )$, for any Borel $E\subseteq K$, and $\int _K\psi (k)\,d{\tilde{\mu }}_\rho (k)= \int _{G/H}\psi \big (p(\omega )\big )\,d\mu _\rho (\omega )$, for any $\psi \in C_c(K)$. It turns out that ${\tilde{\mu }}_\rho$ is left Haar measure on K. To see this, let $\varphi \in C_c(K)$ and $\ell \in K$. Let $\varphi '\in C_c(G/H)$ be defined by $\varphi '(kH)=\varphi (k)$, for all $k\in K$. By Proposition 1.9 of [13], there exists an $f\in C_c(G)$ so that $\varphi (k)=\varphi '(kH)=\int _Hf(kh)\,d\mu _H(h)$, for all $k\in K$. Now, using (7), we have

$$\begin{aligned} \int _K\varphi (\ell k)\,d{\tilde{\mu }}_\rho (k)= \int _{G/H}\int _Hf\big (\ell p(\omega )h\big )\,d\mu _H(h)\,d\mu _\rho (\omega ) =\int _Gf(\ell x)\rho (x)\,d\mu _G(x). \end{aligned}$$

But $\rho (\ell x)=\rho (x)$, for every $x\in G$, and left-invariance of $\mu _G$ yields $\int _K\varphi (\ell k)\,d{\tilde{\mu }}_\rho (k)= \int _Gf(x)\rho (x)\,d\mu _G(x)= \int _K\varphi (k)\,d{\tilde{\mu }}_\rho (k)$. Since $\varphi \in C_c(K)$ and $\ell \in K$ were arbitrary, ${\tilde{\mu }}_\rho$ must be left Haar measure of K. This normalizes the choices of left Haar measures on G, H, and K so that, for all $f\in C_c(G)$,

$$\begin{aligned} \textstyle \int _K\int _Hf(kh)\,d\mu _H(h)\,d\mu _K(k)= \int _Gf(x)\frac{\Delta _H(h_x)}{\Delta _G(h_x)}\, d\mu _G(x). \end{aligned}$$

(9)

Equivalently,

$$\begin{aligned} \textstyle \int _Gf\,d\mu _G=\int _K\int _Hf(kh)\frac{\Delta _G(h)}{\Delta _H(h)}\,d\mu _H(h)\,d\mu _K(k). \end{aligned}$$

(10)

Let $L^2(K,{\mathcal {H}}_\pi )$ denote the space of ${\mathcal {H}}_\pi$-valued weakly measurable functions F on K such that $\int _K\Vert F(k)\Vert _{{\mathcal {H}}_\pi }^{\,\,2} d\mu _K(k)<\infty$. As usual, identify functions that agree $\mu _K$-almost everywhere and equip $L^2(K,{\mathcal {H}}_\pi )$ with the natural inner product. Then $L^2(K,{\mathcal {H}}_\pi )$ is a Hilbert space that is Hilbert space isomorphic to ${\mathcal {H}}_{{{\textrm{ind}}}\pi }$. To define the unitary map $W:{\mathcal {H}}_{{{\textrm{ind}}}\pi }\rightarrow L^2(K,{\mathcal {H}}_\pi )$, let ${\mathcal {X}}$ consist of all elements of ${\mathcal {H}}_{{{\textrm{ind}}}\pi }$ that are continuous (more precisely, have a continuous member of the equivalence class) and let ${\mathcal {Y}}$ denote the continuous members of $L^2(K,{\mathcal {H}}_\pi )$. Then ${\mathcal {X}}$ is a dense subspace of ${\mathcal {H}}_{{{\textrm{ind}}}\pi }$ and ${\mathcal {Y}}$ is a dense subspace of $L^2(K,{\mathcal {H}}_\pi )$. For $\xi \in {\mathcal {X}}$, let $W\xi$ denote the restriction of $\xi$ to K. Then $W\xi \in {\mathcal {Y}}$. For any $F\in {\mathcal {Y}}$, define $\xi :G\rightarrow {\mathcal {H}}_\pi$ by $\xi (x)=\pi (h_x^{-1})F(k_x)$, for all $x\in G$. Then $\xi$ is continuous, $\xi (xh)=\pi (h^{-1})\xi (x)$, for all $x\in G, h\in H$, and $\int _{G/H} \Vert \xi \big (p(\omega )\big )\Vert _{{\mathcal {H}}_\pi }^{\,\,2} d\mu _\rho (\omega )=\int _K \Vert \xi (k)\Vert _{{\mathcal {H}}_\pi }^{\,\,2} d\mu _K(k)=\int _K \Vert F(k)\Vert _{{\mathcal {H}}_{\pi }}^{\,\,2} d\mu _K(k)<\infty$. Thus $\xi \in {\mathcal {X}}$ and, clearly, $W\xi =F$. So W is an isometry of ${\mathcal {X}}$ onto ${\mathcal {Y}}$. Therefore, W extends to a unitary map, also denoted W, of ${\mathcal {H}}_{{{\textrm{ind}}}\pi }$ onto $L^2(K,{\mathcal {H}}_\pi )$.

We can now formulate a proposition that corrects and clarifies the expression given in Example 2.29 of [13] for a representation equivalent to ${{\textrm{ind}}}_H^G\pi$ in the special situation we are considering.

Proposition 3.1

Let H and K be closed subgroups of a locally compact group G that satisfy $K\cap H= \{e\}$ and $(k,h)\rightarrow kh$ is a homeomorphism of $K\times H$ onto G. Let $\pi$ be a representation of H. Then ${{\textrm{ind}}}_H^G\pi$ is equivalent to $\sigma ^\pi$ acting on $L^2(K,{\mathcal {H}}_\pi )$ by

$$\begin{aligned} \sigma ^\pi (x)F(k)=\textstyle \left[ \frac{\Delta _H(h_{x^{-1}k})}{\Delta _G(h_{x^{-1}k})} \right] ^{1/2}\pi \left( h_{x^{-1}k}^{\,\,\,\,-1}\right) F(x^{-1}\cdot k), \end{aligned}$$

(11)

for all $k\in K$, for all $F\in L^2(K,{\mathcal {H}}_\pi )$, and for every $x\in G$.

Proof

For each $x\in G$, let $\sigma ^\pi (x)=W{{\textrm{ind}}}_H^G(x)W^{-1}$. Then $\sigma ^\pi$ is a representation of G whose Hilbert space is $L^2(K,{\mathcal {H}}_\pi )$. To show that $\sigma ^\pi$ satisfies (11), it suffices to verify (11) for $F\in {\mathcal {Y}}$, where ${\mathcal {Y}}$ is the dense subspace defined above. Let $F\in {\mathcal {Y}}$ and let $\xi =W^{-1}F\in {\mathcal {X}}$. For each $x\in G$,

$$\begin{aligned} \begin{aligned} \sigma ^\pi (x)F(k)&= W\big ({{\textrm{ind}}}_H^G\pi (x)\xi \big )(k)= {{\textrm{ind}}}_H^G\pi (x)\xi (k)=\textstyle \left[ \frac{\rho (x^{-1}k)}{\rho (k)}\right] ^{1/2}\xi (x^{-1}k)\\&= \textstyle \left[ \frac{\rho (x^{-1}k)}{\rho (k)}\right] ^{1/2}W^{-1}F(x^{-1}k)= \left[ \frac{\rho (x^{-1}k)}{\rho (k)}\right] ^{1/2}\pi \left( h_{x^{-1}k }^{\,\,\,\,-1}\right) F(x^{-1}\cdot k), \end{aligned} \end{aligned}$$

for all $k\in K$. But $\rho (z)=\frac{\Delta _H(h_z)}{\Delta _G(h_z)}$, for all $z\in G$. Thus $\rho (x^{-1}k)= \frac{\Delta _H(h_{x^{-1}k})}{\Delta _G(h_{x^{-1}k})}$, while $\rho (k)=1$, for any $k\in K$. Therefore, (11) holds. $\square$

This formula simplifies further when H is an abelian normal subgroup and $\pi$ is a one-dimensional representation of H. When this simplified form is used below, H is actually a vector subgroup. Suppose $n\in {\mathbb {N}}$ and $K_0$ is a closed subgroup of ${{{\textrm{GL}}}}_n({\mathbb {R}})$. Let

$$\begin{aligned} G={\mathbb {R}}^n\rtimes K_0=\{[{\underline{x}},A]:{\underline{x}}\in {\mathbb {R}}^n, A\in K_0\}. \end{aligned}$$

Let $H=\{[{\underline{x}},{{\textrm{id}}}]:{\underline{x}}\in {\mathbb {R}}^n\}$. Then H is an abelian normal closed subgroup of G. Let $K=\{[{\underline{0}},A]:A\in K_0\}$, a closed subgroup of G. We have $K\cap H=\{[{\underline{0}},{{\textrm{id}}}]\}$ and $\big ([{\underline{0}},A],[{\underline{x}},{{\textrm{id}}}]\big )\rightarrow [{\underline{0}},A][{\underline{x}},{{\textrm{id}}}]=[A{\underline{x}},A]$ is a homeomorphism of $K\times H$ with G. Note that, for $[{\underline{x}},A]\in G$,

$$\begin{aligned} k_{[{\underline{x}},A]}=[{\underline{0}},A]\quad \text {and}\quad h_{[{\underline{x}},A]}=[A^{-1}{\underline{x}},{{\textrm{id}}}]. \end{aligned}$$

(12)

The modular function of G is given by $\Delta _G[{\underline{x}},A]=\frac{\Delta _{K_0}(A)}{|\det (A)|}$, for all $[{\underline{x}},A]\in G$. Note that $\Delta _G\equiv 1$ on H and H, itself, is unimodular. So $\left[ \frac{\Delta _H[{\underline{x}},{\underline{0}}]}{\Delta _G[{\underline{x}},{\underline{0}}]}\right] ^{1/2}=1$, for all $[{\underline{x}},{\underline{0}}]\in H$. The irreducible representations of H are all of the form $\chi _{{\underline{\omega }}}$, for ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n}$, where

$$\begin{aligned} \chi _{{\underline{\omega }}}[{\underline{x}},{{\textrm{id}}}]=e^{2\pi i{\underline{\omega }}\,{\underline{x}}}, \text { for all } [{\underline{x}},{{\textrm{id}}}]\in H. \end{aligned}$$

Corollary 3.2

Let $G={\mathbb {R}}^n\rtimes K_0$, where $K_0$ is a closed subgroup of ${{{\textrm{GL}}}}_n({\mathbb {R}})$. Let $H=\{[{\underline{x}},{{\textrm{id}}}]: {\underline{x}}\in {\mathbb {R}}^n\}$ and let ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n}$. Then ${{\textrm{ind}}}_H^G\chi _{{\underline{\omega }}}$ is unitarily equivalent to $\sigma ^{{\underline{\omega }}}$, which acts on $L^2(K_0)$ as follows: For $[{\underline{x}},A]\in G$, $f\in L^2(K_0)$, and for all $B\in K_0$,

$$\begin{aligned} \sigma ^{{\underline{\omega }}}[{\underline{x}},A]f(B)= e^{2\pi i{\underline{\omega }}B^{-1}{\underline{x}}}f(A^{-1}B). \end{aligned}$$

Proof

For $[{\underline{x}},A]\in G$ and $[{\underline{0}},B]\in K$, $[{\underline{x}},A]^{-1}[{\underline{0}},B]=[-A^{-1}{\underline{x}},A^{-1}B]$, so

$$\begin{aligned} {[}{\underline{x}},A]^{-1}\cdot [{\underline{0}},B]= k_{[{\underline{x}},A]^{-1}[{\underline{0}},B]}=[{\underline{0}},A^{-1}B]\quad \text {and}\quad h_{[{\underline{x}},A]^{-1}[{\underline{0}},B]}= [-B^{-1}{\underline{x}},{{\textrm{id}}}]. \end{aligned}$$

By Proposition 3.1, ${{\textrm{ind}}}_H^G\chi _{{\underline{\omega }}}$ is unitarily equivalent to $\sigma ^{\chi _{{\underline{\omega }}}}$ acting on $L^2(K)$ by, for $[{\underline{x}},A]\in G$ and $f\in L^2(K)$,

$$\begin{aligned} \sigma ^{\chi _{{\underline{\omega }}}}[{\underline{x}},A]f[{\underline{0}},B]= \chi _{{\underline{\omega }}}\left( h_{[{\underline{x}},A]^{-1}[{\underline{0}},B]}^{\,\,\,-1}\right) f[{\underline{0}},A^{-1}B]= e^{2\pi i{\underline{\omega }}B^{-1}{\underline{x}}}f[{\underline{0}},A^{-1}B], \end{aligned}$$

for all $[{\underline{0}},B]\in K$. Let $U:L^2(K)\rightarrow L^2(K_0)$ be the obvious unitary map $Uf(B)=f[{\underline{0}},B]$, for all $B\in K_0$ and $f\in L^2(K_0)$. Then define $\sigma ^{{\underline{\omega }}}$ acting on $L^2(K_0)$ by $\sigma ^{{\underline{\omega }}}[{\underline{x}},A]= U\sigma ^{\chi _{{\underline{\omega }}}}[{\underline{x}},A]U^{-1}$, for all $[{\underline{x}},A]\in G$. Thus, $\sigma ^{{\underline{\omega }}}[{\underline{x}},A]f(B)= e^{2\pi i{\underline{\omega }}B^{-1}{\underline{x}}}f(A^{-1}B)$, for all $B\in K_0$ and ${{\textrm{ind}}}_H^G\chi _{{\underline{\omega }}}$ is unitarily equivalent to both $\sigma ^{\chi _{{\underline{\omega }}}}$ and $\sigma ^{{\underline{\omega }}}$. $\square$

4 The Affine Group on ${\mathbb {R}}^n$

In this section, we decompose the left regular representation of $G_n$ as an infinite multiple of a representation induced from a character of the normal vector subgroup we now call N, where $N=\{[{\underline{y}},{{\textrm{id}}}]:{\underline{y}}\in {\mathbb {R}}^n\}$. Note that, for $[{\underline{x}},A]\in G_n$ and $[{\underline{y}},{{\textrm{id}}}]\in N$,

$$\begin{aligned} {[}{\underline{x}},A]^{-1}[{\underline{y}},{{\textrm{id}}}][{\underline{x}},A]= \left[ -A^{-1}{\underline{x}},A^{-1}\right] [{\underline{y}}+{\underline{x}},A]= \left[ A^{-1}{\underline{y}},{{\textrm{id}}}\right] . \end{aligned}$$

For ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n}$, we continue to denote by $\chi _{{\underline{\omega }}}$ the character of N given by $\chi _{{\underline{\omega }}}[{\underline{y}},{{\textrm{id}}}]=e^{2\pi i{\underline{\omega }}\,{\underline{y}}}$, for all $[{\underline{y}},{{\textrm{id}}}] \in N$. Then ${\widehat{N}}=\{\chi _{{\underline{\omega }}}:{\underline{\omega }}\in \widehat{{\mathbb {R}}^n}\}$. For $[{\underline{x}},A]\in G_n$ and $\chi \in {\widehat{N}}$, $[{\underline{x}},A]\cdot \chi \in {\widehat{N}}$ is defined by

$$\begin{aligned} \big ([{\underline{x}},A]\cdot \chi \big )[{\underline{y}},{{\textrm{id}}}]= \chi \big ([{\underline{x}},A]^{-1}[{\underline{y}},{{\textrm{id}}}][{\underline{x}},A] \big )=\chi \left[ A^{-1}{\underline{y}},{{\textrm{id}}}\right] . \end{aligned}$$

Therefore, the action of $G_n$ on ${\widehat{N}}$ reduces to, for $[{\underline{x}},A]\in G_n$ and ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n}$, $[{\underline{x}},A]\cdot \chi _{{\underline{\omega }}}=\chi _{{\underline{\omega }}A^{-1}}$. All information about this action is obtained from the action of ${{{\textrm{GL}}}}_n({\mathbb {R}})$ on $\widehat{{\mathbb {R}}^n}$ given by $(A,{\underline{\omega }})\rightarrow {\underline{\omega }}A^{-1}$. There are just two orbits, $\{{\underline{0}}\}$ and ${\mathcal {O}}_n =\widehat{{\mathbb {R}}^n}{\setminus }\{{\underline{0}}\}$. Let ${\underline{\omega }}_0=(1,0,\cdots ,0)$. Then ${\mathcal {O}}_n =\{{\underline{\omega }}_0A^{-1}:A\in {{{\textrm{GL}}}}_n({\mathbb {R}})\}= \{{\underline{\omega }}_0A:A\in {{{\textrm{GL}}}}_n({\mathbb {R}})\}$. It is not difficult to construct a measurable map $\gamma :{\mathcal {O}}_n \rightarrow {{{\textrm{GL}}}}_n({\mathbb {R}})$ that satisfies, for all ${\underline{\omega }}\in {\mathcal {O}}_n$,

$$\begin{aligned} {\underline{\omega }}_0\gamma ({\underline{\omega }})^{-1}={\underline{\omega }}\,\,\text {(equivalently}\,\, {\underline{\omega }}\gamma ({\underline{\omega }})={\underline{\omega }}_0). \end{aligned}$$

(13)

Fix such a map $\gamma$. For ${\underline{\omega }}\in \widehat{{\mathbb {R}}^n}$, by Corollary 3.2, the induced representation ${{\textrm{ind}}}_N^{G_n}\chi _{{\underline{\omega }}}$ is unitarily equivalent to a representation we denote $\pi ^{{\underline{\omega }}}$ which acts on $L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ as follows: For $[{\underline{x}},A]\in G_n$,

$$\begin{aligned} \pi ^{{\underline{\omega }}}[{\underline{x}},A]f(B)= e^{2\pi i{\underline{\omega }}B^{-1}{\underline{x}}}f\left( A^{-1}B\right) , \quad \text {for all}\,\,B\in {{{\textrm{GL}}}}_n({\mathbb {R}}), f\in L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big ).\nonumber \\ \end{aligned}$$

(14)

It is a basic result of Mackey theory that inducing two representations from the same orbit results in equivalent representations (see Proposition 2.39 of [13], for example). Thus, for each ${\underline{\omega }}\in {\mathcal {O}}_n$, we should have $\pi ^{{\underline{\omega }}}\sim \pi ^{{\underline{\omega }}_0}$. There is some value in identifying the unitary map that institutes this equivalence. Define $V_{{\underline{\omega }}}:L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )\rightarrow L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ by, for $f\in L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$,

$$\begin{aligned} V_{{\underline{\omega }}}f(B)=f\big (B\gamma ({\underline{\omega }})^{-1} \big ),\quad \text {for all}\,\,B\in {{{\textrm{GL}}}}_n({\mathbb {R}}). \end{aligned}$$

Since ${{{\textrm{GL}}}}_n({\mathbb {R}})$ is unimodular, $\Vert V_{{\underline{\omega }}}f\Vert _2=\Vert f\Vert _2$, for all $f\in L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$. It is also clear that $V_{{\underline{\omega }}}$ is linear, one-to-one, and onto. So $V_{{\underline{\omega }}}$ is a unitary map of $L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ with itself. For $[{\underline{x}},A]\in G_n$, $f\in L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$, and $B\in {{{\textrm{GL}}}}_n({\mathbb {R}})$,

$$\begin{aligned} V_{{\underline{\omega }}}\pi ^{{\underline{\omega }}}[{\underline{x}},A] V_{{\underline{\omega }}}^{-1}f(B)= & {} \pi ^{{\underline{\omega }}}[{\underline{x}},A] V_{{\underline{\omega }}}^{-1}f\big (B\gamma ({\underline{\omega }})^{-1}\big )\\= & {} e^{2\pi i{\underline{\omega }}\gamma ({\underline{\omega }})B^{-1}{\underline{x}}} V_{{\underline{\omega }}}^{-1}f\big (A^{-1}B\gamma ({\underline{\omega }})^{-1}\big )\\= & {} e^{2\pi i{\underline{\omega }}\gamma ({\underline{\omega }})B^{-1}{\underline{x}}} f\big (A^{-1}B\big )\\= & {} e^{2\pi i{\underline{\omega }}_0 B^{-1}{\underline{x}}} f\big (A^{-1}B\big )=\pi ^{{\underline{\omega }}_0}[{\underline{x}},A] f(B). \end{aligned}$$

This shows that $\pi ^{{\underline{\omega }}}\sim \pi ^{{\underline{\omega }}_0}$.

Our goal now is to establish an explicit unitary equivalence of the left regular representation, $\lambda _{G_n}$, of $G_n$ with an infinite multiple of $\pi ^{{\underline{\omega }}_0}$, as expressed in Theorem 4.3. The key part of the process is to transform $L^2(G_n)$ into $L^2({\mathcal {O}}_n\times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ with somewhat natural unitary maps and tracking how $\lambda _{G_n}$ changes. The steps are summarized in the following diagram.

The unitary maps $U:L^2(G_n)\rightarrow L^2({\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}}))$, ${\mathcal {F}}_1:L^2({\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}}))\rightarrow L^2\big (\widehat{{\mathbb {R}}^n}\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ and $W:L^2\big (\widehat{{\mathbb {R}}^n}\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big ) \rightarrow L^2({\mathcal {O}}\times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ are defined below. As seen in the diagram, we compose ${\mathcal {F}}_1$ with U to reduce some of the notational clutter. In the diagram, the displayed unitary intertwines the representation above with the one below.

For any $f\in L^2(G_n)$, define Uf on ${\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}})$ by $Uf({\underline{y}},B)=f[B{\underline{y}},B]$, for all $({\underline{y}},B)\in {\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}})$. It is straightforward to verify that U is a unitary map of $L^2(G_n)$ onto $L^2({\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ when ${\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}})$ is equipped with the product of Lebesgue measure on ${\mathbb {R}}^n$ with Haar measure on ${{{\textrm{GL}}}}_n({\mathbb {R}})$. Moreover, $U^{-1}: L^2\big ({\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )\rightarrow L^2(G_n)$ is given by

$$\begin{aligned} U^{-1}f[{\underline{y}},B]=f(B^{-1}{\underline{y}},B),\quad \text {for}\,\,[{\underline{y}},B]\in G_n, f\in L^2\big ({\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big ). \end{aligned}$$

Let ${\mathcal {F}}_1: L^2\big ({\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )\rightarrow L^2\big (\widehat{{\mathbb {R}}^n}\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ be the unitary map such that, for $f\in C_c\big ({\mathbb {R}}^n\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ and any $({\underline{\omega }},A)\in \widehat{{\mathbb {R}}^n}\times {{{\textrm{GL}}}}_n({\mathbb {R}})$,

$$\begin{aligned} {\mathcal {F}}_1f({\underline{\omega }},B)=\int _{{\mathbb {R}}^n}f({\underline{y}},B) e^{2\pi i{\underline{\omega }}{\underline{y}}}d{\underline{y}}. \end{aligned}$$

The left regular representation $\lambda _{G_n}$ of $G_n$ is unitarily equivalent, via ${\mathcal {F}}_1\circ U$ to a unitary representation $\widetilde{\lambda _{G_n}}$ acting on $L^2\big (\widehat{{\mathbb {R}}^n}\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$. That is, for all $[{\underline{x}},A]\in G_n$,

$$\begin{aligned} \widetilde{\lambda _{G_n}}[{\underline{x}},A]=({\mathcal {F}}_1\circ U) \lambda _{G_n}[{\underline{x}},A]({\mathcal {F}}_1\circ U)^{-1}. \end{aligned}$$

A short computation shows that, for $f\in L^2\big (\widehat{{\mathbb {R}}^n}\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$, $[{\underline{x}},A]\in G_n$, and almost every $({\underline{\omega }},B)\in \widehat{{\mathbb {R}}^n}\times {{{\textrm{GL}}}}_n({\mathbb {R}})$, we have

$$\begin{aligned} \widetilde{\lambda _{G_n}}[{\underline{x}},A]f({\underline{\omega }},B) =e^{2\pi i{\underline{\omega }}B^{-1}{\underline{x}}} f\left( {\underline{\omega }}, A^{-1}B\right) . \end{aligned}$$

Notice the similarity with the $\pi ^{{\underline{\omega }}}$. This shows how we could write the left regular representation as a direct integral of the $\pi ^{{\underline{\omega }}}$. But $\pi ^{{\underline{\omega }}}$ is equivalent to $\pi ^{{\underline{\omega }}_0}$, for each ${\underline{\omega }}\in {\mathcal {O}}_n$. Thus, $\lambda _{G_n}$ must be equivalent to a multiple of $\pi ^{{\underline{\omega }}_0}$. We will make this equivalence explicit.

Note that ${\mathcal {O}}_n$ is a co-null open subset of $\widehat{{\mathbb {R}}^n}$. We will consider Lebesgue measure on ${\mathcal {O}}_n$ as its standard measure, so $L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ is the same Hilbert space as $L^2\big (\widehat{{\mathbb {R}}^n}\times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$. We define W on $L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ as follows: For $f\in L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}}))$,

$$\begin{aligned} (Wf)({\underline{\omega }},B)= f\big ({\underline{\omega }},B\gamma ({\underline{\omega }})^{-1}\big ), \end{aligned}$$

for a.e. $({\underline{\omega }},B)\in {\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})$. Then Wf is measurable and, using Fubini’s Theorem and that ${{{\textrm{GL}}}}_n({\mathbb {R}})$ is unimodular,

$$\begin{aligned} \begin{aligned}&\int _{{{{\textrm{GL}}}}_n({\mathbb {R}})}\int _{{\mathcal {O}}_n }|(Wf)({\underline{\omega }},B)|^2 d{\underline{\omega }}\,d\mu _{{{{\textrm{GL}}}}_n({\mathbb {R}})}(B)\\&\quad = \int _{{\mathcal {O}}_n }\int _{{{{\textrm{GL}}}}_n({\mathbb {R}})} |f\big ({\underline{\omega }},B\gamma ({\underline{\omega }})^{-1}\big )|^2 d\mu _{{{{\textrm{GL}}}}_n({\mathbb {R}})}(B)\,d{\underline{\omega }}\\&\quad = \int _{{{{\textrm{GL}}}}_n({\mathbb {R}})}\int _{{\mathcal {O}}_n } |f\big ({\underline{\omega }},B\big )|^2 d{\underline{\omega }}\,d\mu _{{{{\textrm{GL}}}}_n({\mathbb {R}})}(B)=\Vert f\Vert _2^{\,\,2}. \end{aligned} \end{aligned}$$

Thus $Wf\in L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ and W is a linear isometry on $L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}}))$. Clearly, W is onto and $W^{-1}$ is given by $\big (W^{-1}g\big )({\underline{\omega }},B)= g\big ({\underline{\omega }},B\gamma ({\underline{\omega }})\big )$, for all $({\underline{\omega }},B)\in {\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})$. So W is a unitary.

Define the unitary representation $\lambda _{G_n}^0$ of $G_n$ on $L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ by, for $[{\underline{x}},A]\in G_n$,

$$\begin{aligned} \lambda _{G_n}^0[{\underline{x}},A]= W\widetilde{\lambda _{G_n}}[{\underline{x}},A]W^{-1}. \end{aligned}$$

So, for $f\in L^2\big ({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ and a.e. $({\underline{\omega }},B)\in {\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})$,

$$\begin{aligned} \lambda _{G_n}^0[{\underline{x}},A]f({\underline{\omega }},B)= & {} W\widetilde{\lambda _{G_n}}[{\underline{x}},A]W^{-1} f({\underline{\omega }},B)= \widetilde{\lambda _{G_n}}[{\underline{x}},A]W^{-1} f\big ({\underline{\omega }},B\gamma ({\underline{\omega }})^{-1}\big )\\= & {} e^{2\pi i{\underline{\omega }}\gamma ({\underline{\omega }})B^{-1}{\underline{x}}} \left( W^{-1}f\right) \big ({\underline{\omega }},A^{-1}B \gamma ({\underline{\omega }})^{-1}\big )\\= & {} e^{2\pi i{\underline{\omega }}_0B^{-1}{\underline{x}}} f\big ({\underline{\omega }},A^{-1}B\big ). \end{aligned}$$

Therefore, we have $\lambda _{G_n}^0[{\underline{x}},A]f({\underline{\omega }},B)= e^{2\pi i{\underline{\omega }}_0B^{-1}{\underline{x}}} f\big ({\underline{\omega }},A^{-1}B\big )$, for a.e. $({\underline{\omega }},B)\in {\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})$, for all $f\in L^2\big ({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$, and all $[{\underline{x}},A]\in G_n$. The representation $\lambda _{G_n}^0$ is equivalent to $\lambda _{G_n}$.

Let $\eta \in L^2({\mathcal {O}}_n)$ be such that $\Vert \eta \Vert _{L^2({\mathcal {O}}_n)}=1$. Let ${\mathcal {H}}_\eta =\{\eta \otimes g:g\in L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )\}$, where $\eta \otimes g$ is defined by $(\eta \otimes g)({\underline{\omega }},B)=\eta ({\underline{\omega }})g(B)$, for a.e. $({\underline{\omega }},B)\in {\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})$. Then ${\mathcal {H}}_\eta$ is a closed subspace of $L^2\big ({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$. Note that

$$\begin{aligned} \lambda _{G_n}^0[{\underline{x}},A](\eta \otimes g)({\underline{\omega }},B)= e^{2\pi i{\underline{\omega }}_0B^{-1}{\underline{x}}} \eta ({\underline{\omega }})g\big (A^{-1}B\big )= \eta ({\underline{\omega }})\pi ^{{\underline{\omega }}_0}[{\underline{x}},A]g(B),\nonumber \\ \end{aligned}$$

(15)

for a.e. $({\underline{\omega }},B)\in {\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})$ and all $g\in L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$.

Define $W_\eta :L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big ) \rightarrow L^2\big ({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})\big )$ by $W_\eta f=\eta \otimes f$, for $f\in L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$. Clearly, $W_\eta$ is a linear isometry with $W_\eta L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )={\mathcal {H}}_\eta$, for any unit vector $\eta$ in $L^2({\mathcal {O}}_n)$. Moreover, (15) shows that ${\mathcal {H}}_\eta$ is $\lambda _{G_n}^0$-invariant and $W_\eta$ intertwines $\pi ^{{\underline{\omega }}_0}$ with $\lambda _{G_n}^0$ restricted to ${\mathcal {H}}_\eta$. We can now exhibit $\lambda _{G_n}^0$ as a multiple of $\pi ^{{\underline{\omega }}_0}$.

Proposition 4.1

Let $\{\eta _j:j\in J\}$ be an orthonormal basis of $L^2({\mathcal {O}}_n )$. For each $j\in J$, ${\mathcal {H}}_{\eta _j}$ is a closed $\lambda _{G_n}^0$-invariant subspace of $L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ and the map $W_{\eta _j}$ intertwines $\pi ^{{\underline{\omega }}_0}$ with the restriction of $\lambda _{G_n}^0$ to ${\mathcal {H}}_{\eta _j}$. Moreover, $L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})) =\textstyle \sum ^{\oplus }_{j\in J}{\mathcal {H}}_{\eta _j}.$

Returning to a single $\eta \in L^2({\mathcal {O}}_n )$ with $\Vert \eta \Vert _{L^2({\mathcal {O}}_n)}=1$, let us see where $\eta \otimes f$ goes as we map it with the above unitaries back into $L^2(G_n)$. First, we have $W^{-1}(\eta \otimes f)({\underline{\omega }},B)= \eta ({\underline{\omega }})f\big (B\gamma ({\underline{\omega }})\big )$, for $({\underline{\omega }},B)\in {\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}})$. Next,

$$\begin{aligned} {\mathcal {F}}_1^{-1}W^{-1}(\eta \otimes f)({\underline{y}},B)= \int _{\widehat{{\mathbb {R}}^n}} \eta ({\underline{\omega }})f\big (B\gamma ({\underline{\omega }})\big ) e^{-2\pi i{\underline{\omega }}{\underline{y}}}d{\underline{\omega }}. \end{aligned}$$

Finally,

$$\begin{aligned} \begin{aligned} U^{-1}{\mathcal {F}}_1^{-1}W^{-1}(\eta \otimes f)[{\underline{y}},B]&= {\mathcal {F}}_1^{-1}W^{-1}(\eta \otimes f)(B^{-1}{\underline{y}},B)\\&= \int _{\widehat{{\mathbb {R}}^n}} \eta ({\underline{\omega }})f\big (B\gamma ({\underline{\omega }})\big ) e^{-2\pi i{\underline{\omega }}B^{-1}{\underline{y}}}d{\underline{\omega }}. \end{aligned} \end{aligned}$$

We can define $U_\eta :L^2({{{\textrm{GL}}}}_n({\mathbb {R}}))\rightarrow L^2(G_n)$ by

$$\begin{aligned} U_\eta f[{\underline{y}},B]=\int _{\widehat{{\mathbb {R}}^n}} \eta ({\underline{\omega }})f\big (B\gamma ({\underline{\omega }})\big ) e^{-2\pi i{\underline{\omega }}B^{-1}{\underline{y}}}d{\underline{\omega }}, \end{aligned}$$

(16)

for a.e. $[{\underline{y}},B]\in G_n, f\in L^2({{{\textrm{GL}}}}_n({\mathbb {R}}))$. Thus, we obtain the following result.

Proposition 4.2

Let $\eta \in L^2({\mathcal {O}}_n )$ satisfy $\Vert \eta \Vert _{L^2({\mathcal {O}}_n)}=1$. Then $U_\eta$ is an isometric linear map of $L^2({{{\textrm{GL}}}}_n({\mathbb {R}}))$ into $L^2(G_n)$ that intertwines $\pi ^{{\underline{\omega }}_0}$ with $\lambda _{G_n}$.

Fix an orthonormal basis $\{\eta _j:j\in J\}$ of $L^2({\mathcal {O}}_n )$. Let ${\mathcal {L}}_{\eta _j}=U^{-1}{\mathcal {F}}_1^{-1}W^{-1}{\mathcal {H}}_{\eta _j}$, for each $j\in J$. Since $U^{-1}{\mathcal {F}}_1^{-1}W^{-1}$ is a unitary map of $L^2({\mathcal {O}}_n \times {{{\textrm{GL}}}}_n({\mathbb {R}}))$ onto $L^2(G_n)$, we see that $L^2(G_n)$ is the direct sum of the ${\mathcal {L}}_{\eta _j}$ Thus, we have a decomposition of the left regular representation of $G_n$. Note that $L^2({\mathcal {O}}_n )$ is identified with $L^2(\widehat{{\mathbb {R}}^n})$.

Theorem 4.3

Let ${\mathcal {O}}_n =\widehat{{\mathbb {R}}^n}{\setminus }\{{\underline{0}}\}$, let ${\underline{\omega }}_0= (1,0,\cdots ,0)$, and define the representation $\pi ^{{\underline{\omega }}_0}$ of $G_n$ by $\pi ^{{\underline{\omega }}_0}[{\underline{x}},A]f(B)= e^{2\pi i{\underline{\omega }}_0 B^{-1}{\underline{x}}}f\left( A^{-1}B\right)$, for $B\in {{{\textrm{GL}}}}_n({\mathbb {R}})$, $f\in L^2\big ({{{\textrm{GL}}}}_n({\mathbb {R}})\big )$, and $[{\underline{x}},A]\in G_n$. Fix a measurable map $\gamma :{\mathcal {O}}_n \rightarrow {{{\textrm{GL}}}}_n({\mathbb {R}})$ such that ${\underline{\omega }}_0\gamma ({\underline{\omega }})^{-1}={\underline{\omega }}$, for all ${\underline{\omega }}\in {\mathcal {O}}_n$. Let $\{\eta _j:j\in J\}$ be an orthonormal basis of $L^2(\widehat{{\mathbb {R}}^n})$. For each $j\in J$, define $U_{\eta _j}:L^2({{{\textrm{GL}}}}_n({\mathbb {R}}))\rightarrow L^2(G_n)$ by

$$\begin{aligned} U_{\eta _j} f[{\underline{y}},B]=\int _{\widehat{{\mathbb {R}}^n}} \eta _j({\underline{\omega }})f\big (B\gamma ({\underline{\omega }})\big ) e^{-2\pi i{\underline{\omega }}B^{-1}{\underline{y}}}d{\underline{\omega }}, \end{aligned}$$

for $[{\underline{y}},B]\in G_n$, and $f\in L^2({{{\textrm{GL}}}}_n({\mathbb {R}}))$. Let ${\mathcal {L}}_{\eta _j}=U_{\eta _j}L^2({{{\textrm{GL}}}}_n({\mathbb {R}}))$. Then ${\mathcal {L}}_{\eta _j}$ is a closed $\lambda _{G_n}$-invariant subspace of $L^2(G_n)$ and $U_{\eta _j}$ intertwines $\pi ^{{\underline{\omega }}_0}$ with the restriction of $\lambda _{G_n}$ to ${\mathcal {L}}_{\eta _j}$. Moreover, $L^2(G_n)=\sum _{j\in J}^{\oplus }{\mathcal {L}}_{\eta _j}$.

When $n=1$, ${{\textrm{GL}}}_1({\mathbb {R}})$ can be identified with ${\mathbb {R}}^*$ and $G_1$ identified with ${\mathbb {R}}\rtimes {\mathbb {R}}^*$. We recall that $\int _{{\mathbb {R}}^*}f\,d\mu _{{\mathbb {R}}^*}=\int _{{\mathbb {R}}}f(b) \frac{db}{|b|}$, where the integral on the right-hand side is the Lebesgue integral on ${\mathbb {R}}$, and $\int _{G_1}f\,d\mu _{G_1}=\int _{{\mathbb {R}}}\int _{{\mathbb {R}}} f[y,b]\frac{dy\,db}{b^2}$. We continue to write $N=\{[y,1]:y\in {\mathbb {R}}\}$ and ${\widehat{N}}=\{\chi _\omega :\omega \in {\mathbb {R}}\}$, where $\chi _\omega [y,1]=e^{2\pi i\omega y}$, for $[y,1]\in N$. Select ${\underline{\omega }}_0=\omega _0=1$. Then, from (14), $\pi ^{{\underline{\omega }}_0}=\pi ^1$ acts on $L^2({\mathbb {R}}^*)$ via, for $[x,a]\in G_1$, $f\in L^2({\mathbb {R}}^*)$, and a.e. $b\in {\mathbb {R}}^*$,

$$\begin{aligned} \pi ^1[x,a]f(b)= e^{2\pi ib^{-1}x}f\big (a^{-1}b\big ). \end{aligned}$$

(17)

Now, the ${\mathbb {R}}^*$-orbit ${\mathcal {O}}_1={\widehat{{\mathbb {R}}}}\setminus \{0\}$ is a free orbit. Condition (13) forces $\gamma :{\mathcal {O}}_1 \rightarrow {\mathbb {R}}^*$ to actually be the homeomorphism given by $\gamma (\omega )=\omega ^{-1}$. We can use $\gamma$ to move $\pi ^1$ to a representation acting on $L^2({\widehat{{\mathbb {R}}}})$. Note that $L^2({\widehat{{\mathbb {R}}}})=L^2({\mathcal {O}}_1)$ when ${\mathcal {O}}_1$ is equipped with Lebesgue measure. For $f\in L^2({\mathbb {R}}^*)$, define Zf as a Borel function on ${\widehat{{\mathbb {R}}}}$ such that

$$\begin{aligned} (Zf)(\omega )=|\omega |^{-1/2}f\big (\gamma (\omega )\big ), \text { for a.e. }\omega \in {\mathcal {O}}_1. \end{aligned}$$

Then $\int _{{\widehat{{\mathbb {R}}}}}|(Zf)(\omega )|^2d\omega = \int _{{\widehat{{\mathbb {R}}}}}|f(\omega ^{-1})|^2|\omega |^{-1}\, d\omega =\int _{{\mathbb {R}}}|f(u)|^2\frac{du}{|u|}= \Vert f\Vert _{L^2({\mathbb {R}}^*)}^{\,\,\,2}$. Thus $Zf\in L^2({\widehat{{\mathbb {R}}}})$. It is clear that Z is a unitary map of $L^2({\mathbb {R}}^*)$ onto $L^2({\widehat{{\mathbb {R}}}})$. Let $\pi [x,a]=Z\pi ^1[x,a]Z^{-1}$, for all $[x,a]\in G_1$. For any $\xi \in L^2({\widehat{{\mathbb {R}}}})$, we have (writing $f=Z^{-1}\xi$), for a.e. $\omega \in {\widehat{{\mathbb {R}}}}$,

$$\begin{aligned} \pi [x,a]\xi (\omega )=Z\pi ^1[x,a]f(\omega )= & {} |\omega |^{-1/2}\pi ^1[x,a]f\big (\omega ^{-1}\big ) =|\omega |^{-1/2}e^{2\pi i\omega x} f\big (a^{-1}\omega ^{-1}\big )\\= & {} |a|^{1/2} e^{2\pi i\omega x}\big |\omega a\big |^{-1/2} f\big ((\omega a)^{-1}\big ) \nonumber \\= & {} |a|^{1/2} e^{2\pi i\omega x}\xi (\omega a). \end{aligned}$$

Thus, $\pi [x,a]\xi (\omega )=|a|^{1/2} e^{2\pi i\omega x}\xi (\omega a)$, for a.e. $\omega \in {\widehat{{\mathbb {R}}}},\xi \in L^2({\widehat{{\mathbb {R}}}})$, and $[x,a]\in G_1$. Finally, we use the inverse Fourier transform to move $\pi$ to a representation on $L^2({\mathbb {R}})$. Let $\rho [x,a]={\mathcal {F}}^{-1}\pi [x,a]{\mathcal {F}}$, for all $[x,a]\in G_1$. Then, for $f\in L^2({\mathbb {R}})$, $\xi ={\mathcal {F}}f$, and a.e. $t\in {\mathbb {R}}$,

$$\begin{aligned} \begin{aligned} \rho [x,a]f(t)&= {\mathcal {F}}^{-1}\pi [x,a]\xi (t)= \int _{{\widehat{{\mathbb {R}}}}}\pi [x,a]\xi (\omega ) e^{-2\pi i\omega t}d\omega \\&=\int _{{\widehat{{\mathbb {R}}}}}|a|^{1/2} e^{2\pi i\omega x}\xi (\omega a) e^{-2\pi i\omega t}d\omega =\int _{{\widehat{{\mathbb {R}}}}}|a|^{1/2} \xi (\omega a) e^{-2\pi i\omega (t-x)}d\omega \\&=\int _{{\widehat{{\mathbb {R}}}}}|a|^{-1/2} \xi (\nu ) e^{-2\pi i\nu a^{-1}(t-x)}d\nu =|a|^{-1/2}f\big (a^{-1}(t-x)\big ). \end{aligned} \end{aligned}$$

Thus, $\rho$ is just the natural representation of $G_1$ on $L^2({\mathbb {R}})$, compare with (4), which is square-integrable by Proposition 2.1. We can now state a well-known result as a corollary of Theorem 4.3.

Corollary 4.4

The left regular representation $\lambda _{G_1}$ is equivalent to $\aleph _0\cdot \rho$, where $\rho$ is the natural representation of $G_1$ on $L^2({\mathbb {R}})$ and $\rho$ is square-integrable.

Since $\pi ^1$ is equivalent to $\rho$, we have that $\pi ^1$ is square-integrable. The next two results are well-known in connection with the continuous wavelet transform on ${\mathbb {R}}$. We present them in a manner convenient for later use. We need to identify which $g\in L^2({\mathbb {R}}^*)$ is such that $V_g$, defined via the representation $\pi ^1$, is an isometry of $L^2({\mathbb {R}}^*)$ into $L^2(G_2)$.

Proposition 4.5

The Duflo-Moore operator for $\pi ^1$ is given by $C_{\pi ^1}g(t)=|t|^{1/2}g(t)$, a.e. $t\in {\mathbb {R}}^*$, for $g\in {\mathcal {D}}_{\pi ^1}$, where ${\mathcal {D}}_{\pi ^1}=\left\{ h\in L^2({\mathbb {R}}^*): \int _{{\mathbb {R}}^*}\big ||t|^{1/2}h(t)\big |^2d\mu _{{\mathbb {R}}^*}(t) < \infty \right\}$.

Proof

Calculations modeled on the standard proof, such as used in Theorem 3.3.5 of [10] or Example 2.28 of [7], show that, for any $f,g\in L^2({\mathbb {R}}^*)$,

$$\begin{aligned} \int _{G_1}|V_gf[x,a]|^2 d\mu _{G_1}\big ([x,a]\big )= \Vert f\Vert _2^{\,\,2} \int _{{\mathbb {R}}^*}\big ||t|^{1/2} g(t)\big |^2d\mu _{{\mathbb {R}}^*}(t). \end{aligned}$$

(18)

If $f\in L^2({\mathbb {R}}^*)$ is nonzero, then the right-hand side of (18) is finite if and only if

$$\begin{aligned} \int _{{\mathbb {R}}^*}\big ||t|^{1/2} g(t)\big |^2d\mu _{{\mathbb {R}}^*}(t)<\infty . \end{aligned}$$

An appeal to Theorem 2.2 completes the proof. $\square$

Corollary 4.6

If $g\in L^2({\mathbb {R}}^*)$ satisfies $\int _{{\mathbb {R}}^*}\big ||t|^{1/2} g(t)\big |^2d\mu _{{\mathbb {R}}^*}(t)=1$, then $V_g$ is an isometry of $L^2({\mathbb {R}}^*)$ into $L^2(G_1)$ that intertwines $\pi ^1$ with $\lambda _{G_1}$.

Remark 4.7

Note that a measurable function g on ${\mathbb {R}}$ can be considered as in $L^2({\mathbb {R}}^*)$ exactly when $\int _{-\infty }^\infty |g(x)|^2\frac{dx}{|x|}<\infty$. So

$$\begin{aligned} {\mathcal {D}}_{\pi ^1}{} & {} =\left\{ h:{\mathbb {R}}\rightarrow {\mathbb {C}}\big | h \text { is measurable, } \int _{-\infty }^\infty |h(x)|^2\frac{dx}{|x|}<\infty \right. \\{} & {} \quad \left. \text { and } \int _{-\infty }^\infty |h(x)|^2 dx<\infty \right\} . \end{aligned}$$

Now, we can use basic facts of the theory of induced representations to show that $\lambda _{G_n}$ is a countably infinite multiple of a single square-integrable representation, for any $n\in {\mathbb {N}}$. The proof of the following theorem follows the ideas in [1].

Theorem 4.8

Let $n\in {\mathbb {N}}$. Then there exists a square-integrable representation $\sigma _n$ of $G_n$ such that $\lambda _{G_n}$ is equivalent to $\aleph _0\cdot \sigma _n$.

Proof

Use mathematical induction. The case of $n=1$ is simply Corollary 4.4. Suppose that $n\in {\mathbb {N}}$ and there exists a square-integrable representation $\sigma _n$ of $G_n$ such that $\lambda _{G_n}$ is equivalent to $\aleph _0\cdot \sigma _n$. Now consider the action of ${{\textrm{GL}}}_{n+1}({\mathbb {R}})$ on $\widehat{{\mathbb {R}}^{n+1}}$. Let ${\underline{\omega }}_0=(1,0,\cdots ,0)\in \widehat{{\mathbb {R}}^{n+1}}$. The stability subgroup of ${\underline{\omega }}_0$ is

$$\begin{aligned} H_{{\underline{\omega }}_0}=\left\{ \begin{pmatrix} 1 &{} {\underline{0}}\\ {\underline{x}}&{} A \end{pmatrix}:{\underline{x}}\in {\mathbb {R}}^n, A\in {{{\textrm{GL}}}}_n({\mathbb {R}})\right\} , \end{aligned}$$

where ${\underline{0}}$ here denotes a row of n zeros. Note that $[{\underline{x}},A]\rightarrow \begin{pmatrix} 1 &{} {\underline{0}}\\ {\underline{x}}&{} A \end{pmatrix}$ is a topological group isomorphism of $G_n$ with $H_{{\underline{\omega }}_0}$. Let $\sigma _n'\begin{pmatrix} 1 &{} {\underline{0}}\\ {\underline{x}}&{} A \end{pmatrix}=\sigma _n[{\underline{x}},A]$, for each $\begin{pmatrix} 1 &{} {\underline{0}}\\ {\underline{x}}&{} A \end{pmatrix}\in H_{{\underline{\omega }}_0}$. Then $\sigma _n'$ is a square-integrable representation of $H_{{\underline{\omega }}_0}$ and $\lambda _{H_{{\underline{\omega }}_0}}$ is equivalent to $\aleph _0\cdot \sigma _n'$. Use N to denote $\{[{\underline{z}},{{\textrm{id}}}]:{\underline{z}}\in {\mathbb {R}}^{n+1}\}$, where ${{\textrm{id}}}$ is the identity in ${{\textrm{GL}}}_{n+1}({\mathbb {R}})$.

By Theorem 4.3, $\lambda _{G_{n+1}}$ is equivalent to $\aleph _0\cdot \pi ^{{\underline{\omega }}_0}$. Moreover, $\pi ^{{\underline{\omega }}_0}$ is equivalent to ${{\textrm{ind}}}_N^{G_{n+1}}\chi _{{\underline{\omega }}_0}$. Let $H={\mathbb {R}}^{n+1}\rtimes H_{{\underline{\omega }}_0}=\{[{\underline{z}},B]:{\underline{z}}\in {\mathbb {R}}^{n+1},B\in H_{{\underline{\omega }}_0}\}$. By the induction in stages theorem (Theorem 2.47 in [13]) ${{\textrm{ind}}}_N^{G_{n+1}}\chi _{{\underline{\omega }}_0}$ is equivalent to ${{\textrm{ind}}}_H^{G_{n+1}}\big ({{\textrm{ind}}}_N^{H}\chi _{{\underline{\omega }}_0}\big )$. Since ${\underline{\omega }}_0 B^{-1} ={\underline{\omega }}_0$, for any $B\in H_{{\underline{\omega }}_0}$, Corollary 3.2 implies ${{\textrm{ind}}}_N^{H}\chi _{{\underline{\omega }}_0}$ is equivalent to the representation $\sigma ^{{\underline{\omega }}_0}$ acting on $L^2(H_{{\underline{\omega }}_0})$ by, for $[{\underline{z}},B]\in H$, $\sigma ^{{\underline{\omega }}_0} [{\underline{z}},B]f(C)= e^{2\pi i{\underline{\omega }}_0{\underline{z}}} f(B^{-1}C)$, for $C\in H_{{\underline{\omega }}_0}$ and $f\in L^2(H_{{\underline{\omega }}_0})$. That is, $\sigma ^{{\underline{\omega }}_0}=\chi _{{\underline{\omega }}_0}\otimes (\lambda _{H_{{\underline{\omega }}_0}}\circ q)$, where $q:H\rightarrow H_{{\underline{\omega }}_0}$ is the homomorphism given by $q[{\underline{z}},B]=B$, for all $[{\underline{z}},B]\in H$ (see Remark 1.45 of [13]). By the inductive hypothesis, $\sigma ^{{\underline{\omega }}_0}$, and hence also ${{\textrm{ind}}}_N^{H}\chi _{{\underline{\omega }}_0}$, is equivalent to $\chi _{{\underline{\omega }}_0}\otimes \big (\aleph _0\cdot (\sigma _n'\circ q)\big )=\aleph _0\cdot \big ( \chi _{{\underline{\omega }}_o}\otimes (\sigma _n'\circ q)\big )$. The process of inducing commutes with taking direct sums (see Proposition 2.42 of [13], for example). Thus,

$$\begin{aligned} \pi ^{{\underline{\omega }}_0}\sim {{\textrm{ind}}}_N^{G_{n+1}}\chi _{{\underline{\omega }}_0}\sim {{\textrm{ind}}}_H^{G_{n+1}}\big ({{\textrm{ind}}}_N^{H}\chi _{{\underline{\omega }}_0}\big )\sim \aleph _0\cdot {{\textrm{ind}}}_H^{G_{n+1}}\big ( \chi _{{\underline{\omega }}_o}\otimes (\sigma _n'\circ q)\big ). \end{aligned}$$

Let $\sigma _{n+1}= {{\textrm{ind}}}_H^{G_{n+1}}\big ( \chi _{{\underline{\omega }}_o}\otimes (\sigma _n'\circ q)\big )$. By Theorem 4.24 of [13] or Theorem 6.42 in [5], $\sigma _{n+1}$ is an irreducible representation of $G_{n+1}$. We have $\pi ^{{\underline{\omega }}_0}$ is equivalent to $\aleph _0\cdot \sigma _{n+1}$ and $\lambda _{G_{n+1}}$ is equivalent to $\aleph _0\cdot \pi ^{{\underline{\omega }}_0}$. Therefore, $\lambda _{G_{n+1}}$ is equivalent to $\aleph _0\cdot \sigma _{n+1}$. Finally, since $\sigma _{n+1}$ is irreducible and equivalent to a subrepresentation of $\lambda _{G_{n+1}}$, we have that $\sigma _{n+1}$ is a square-integrable representation of $G_{n+1}$ (see Theorem 2.25 of [7], for example). $\square$

Remark 4.9

Theorem 4.8 says that $G_n$ is an [AR]-group and $\widehat{G_n^r}$ is a singleton. We have detailed knowledge in the case of $n=1$ and $\sigma _1$ could be taken to be either $\pi ^1$ as in (17) or the natural representation $\rho$ acting on $L^2({\mathbb {R}})$, or any other equivalent realization. When $n=2$, we will reserve the notation $\sigma _2$ for one concrete realization of the unique member of $\widehat{G_2^r}$.

5 Factoring ${{{\textrm{GL}}}}_2({\mathbb {R}})$ and $G_2$

For the rest of this paper, the focus is on $G_2$. The first step in this section is to establish decompositions of ${{{\textrm{GL}}}}_2({\mathbb {R}})$ and $G_2$ so that Proposition 3.1 can be applied, which we do at the end of the section. Moreover, we introduce a particular choice of the map $\gamma$ from the open orbit into ${{{\textrm{GL}}}}_2({\mathbb {R}})$ that is a cross-section of the group action. Properties of this $\gamma$ turn out to be effective in simplifying computations later.

As usual N denotes the abelian normal subgroup $\{[{\underline{y}},{{\textrm{id}}}]:{\underline{y}}\in {\mathbb {R}}^2\}$. The nontrivial orbit in $\widehat{{\mathbb {R}}^2}$ will be denoted ${\mathcal {O}}$, rather than ${\mathcal {O}}_2$. So ${\mathcal {O}}=\widehat{{\mathbb {R}}^2}\setminus \{{\underline{0}}\}$. The point ${\underline{\omega }}_0=(1,0)$ will serve as a representative point in ${\mathcal {O}}$. The stability subgroup inside ${{{\textrm{GL}}}}_2({\mathbb {R}})$ for this point is now denoted $H_{(1,0)}$. So

$$\begin{aligned} H_{(1,0)}=\big \{A\in {{{\textrm{GL}}}}_2({\mathbb {R}}):(1,0)A=(1,0)\big \}= \left\{ \begin{pmatrix} 1 &{} 0\\ u &{} v \end{pmatrix}:u,v\in {\mathbb {R}}, v\ne 0 \right\} . \end{aligned}$$

Let $K_0=\left\{ \begin{pmatrix} s &{} -t\\ t &{} s \end{pmatrix}:s,t\in {\mathbb {R}},s^2+t^2>0\right\}$. Then $K_0$ is a closed subgroup of ${{{\textrm{GL}}}}_2({\mathbb {R}})$ such that $K_0\cap H_{(1,0)}=\{{{\textrm{id}}}\}$. The fact that ${{{\textrm{GL}}}}_2({\mathbb {R}})=K_0H_{(1,0)}$ is established in the following proposition which is proven by direct computation (see [16]).

Proposition 5.1

If $A=\begin{pmatrix} a &{} b\\ c &{} d \end{pmatrix}\in {{{\textrm{GL}}}}_2({\mathbb {R}})$, then A can be uniquely decomposed as the product $A=M_AC_A$ with $M_A\in K_0$ and $C_A\in H_{(1,0)}$. In fact

$$\begin{aligned} M_A=\begin{pmatrix} s &{} -t\\ t &{} s \end{pmatrix}, \text { with } s=\frac{d(ad-bc)}{b^{2}+d^{2}}, t=\frac{-b(ad-bc)}{b^{2}+d^{2}}, \end{aligned}$$

and

$$\begin{aligned} C_A=\begin{pmatrix} 1 &{} 0\\ u &{} v \end{pmatrix}, \text { with } u=\frac{cd+ab}{(ad-bc)}, v=\frac{b^{2}+d^{2}}{(ad-bc)}. \end{aligned}$$

This factorization leads to a parallel factorization of $G_2$. Let

$$\begin{aligned} H={\mathbb {R}}^2\rtimes H_{(1,0)}=\{[{\underline{x}},C]:C\in H_{(1,0)}\} \text { and } K=\{[{\underline{0}},M]:M\in K_0\}. \end{aligned}$$

Then $K\cap H=\{[{\underline{0}},{{\textrm{id}}}]\}$ and $G_2=KH$. For any $[{\underline{x}},A]\in G_2$,

$$\begin{aligned} {[}{\underline{x}},A]=[{\underline{0}},M_A][M_A^{-1}{\underline{x}},C_A], \end{aligned}$$

(19)

where $M_A$ and $C_A$ are as defined in Proposition 5.1. Note that the map $\big ([{\underline{0}},M],[{\underline{x}},C]\big )\rightarrow [M{\underline{x}},MC]$ is a homeomorphism of $K\times H$ with $G_2$, so the conditions necessary for Proposition 3.1 are satisfied when we wish to induce a representation of H to $G_2$.

For each ${\underline{\omega }}= (\omega _1,\omega _2)\in {\mathcal {O}}$, there is a unique element $\gamma ({\underline{\omega }})\in K_0$ such that $\gamma ({\underline{\omega }})\cdot (1,0)={\underline{\omega }}$. Since ${\underline{\omega }}=\gamma ({\underline{\omega }})\cdot (1,0)= (1,0)\gamma ({\underline{\omega }})^{-1}$, the top row of $\gamma ({\underline{\omega }})^{-1}$ must be $(\omega _1 \quad \omega _2)$. Thus

$$\begin{aligned} \gamma ({\underline{\omega }})^{-1}=\begin{pmatrix} \omega _1 &{} \omega _2\\ -\omega _2 &{} \omega _1 \end{pmatrix}\quad \text {and}\quad \gamma ({\underline{\omega }})=\frac{1}{\Vert {\underline{\omega }}\Vert ^2} \begin{pmatrix} \omega _1 &{} -\omega _2\\ \omega _2 &{} \omega _1 \end{pmatrix}. \end{aligned}$$

We will frequently use that $(1,0)\gamma ({\underline{\omega }})^{-1} ={\underline{\omega }}$ and ${\underline{\omega }}\gamma ({\underline{\omega }})=(1,0)$, for any ${\underline{\omega }}\in {\mathcal {O}}$. We also will need the observation that Haar integration on K is given by, for $f\in C_c(K)$,

$$\begin{aligned} \int _K f\,d\mu _K=\int _{\widehat{{\mathbb {R}}^2}} f[{\underline{0}},\gamma ({\underline{\omega }})]\, \frac{d{\underline{\omega }}}{\Vert {\underline{\omega }}\Vert ^2}= \int _{\widehat{{\mathbb {R}}^2}} f[{\underline{0}},\gamma ({\underline{\omega }})^{-1}]\, \frac{d{\underline{\omega }}}{\Vert {\underline{\omega }}\Vert ^2}. \end{aligned}$$

(20)

In our calculations later, various matrices related to $A\in {{{\textrm{GL}}}}_2({\mathbb {R}})$ and ${\underline{\omega }}\in {\mathcal {O}}$ arise and there are a number of identities involving these matrices that are useful. We also use the entries of the matrix $C_{A^{-1}\gamma ({\underline{\omega }})}^{\,\,-1}$ and need a notation for these entries. Let $u_{{\underline{\omega }},A}= (0,1)C_{A^{-1}\gamma ({\underline{\omega }})}^{\,\,-1}\begin{pmatrix} 1\\ 0 \end{pmatrix}$ and $v_{{\underline{\omega }},A}= (0,1)C_{A^{-1}\gamma ({\underline{\omega }})}^{\,\,-1}\begin{pmatrix} 0\\ 1 \end{pmatrix}$. Then $C_{A^{-1}\gamma ({\underline{\omega }})}^{\,\,-1}=\begin{pmatrix} 1 &{} 0\\ u_{{\underline{\omega }},A} &{} v_{{\underline{\omega }},A} \end{pmatrix}$. The identities we need are collected into a proposition.

Proposition 5.2

Let $A,B\in {{{\textrm{GL}}}}_2({\mathbb {R}})$ and ${\underline{\omega }}\in {\mathcal {O}}$. Then

• (a) $M_A=\gamma \big ((1,0)A^{-1}\big )$ and $C_A=\gamma \big ((1,0)A^{-1}\big )^{-1}A$,

• (b) $M_{A\gamma ({\underline{\omega }})}= \gamma \big ({\underline{\omega }}A^{-1}\big )$ and $C_{A\gamma ({\underline{\omega }})}= \gamma \big ({\underline{\omega }}A^{-1}\big )^{-1}A \gamma ({\underline{\omega }})$,

• (c) $M_{A^{-1}\gamma ({\underline{\omega }})}= \gamma \big ({\underline{\omega }}A\big )$ and $C_{A^{-1}\gamma ({\underline{\omega }})}= \gamma \big ({\underline{\omega }}A\big )^{-1}A^{-1}\gamma ({\underline{\omega }})$,

• (d) $u_{{\underline{\omega }},AB}= u_{{\underline{\omega }},A}+v_{{\underline{\omega }},A}u_{{\underline{\omega }}A,B}$ and $v_{{\underline{\omega }},AB}=v_{{\underline{\omega }},A}v_{{\underline{\omega }}A,B}$,

• (e) $\det \left( \gamma ({\underline{\omega }})^{-1}\right) =\Vert {\underline{\omega }}\Vert ^2$ and $\det \left( \gamma ({\underline{\omega }})\right) =\Vert {\underline{\omega }}\Vert ^{-2}$,

• (f) $\det \left( C_{A^{-1}\gamma ({\underline{\omega }})}\right) = \frac{\Vert {\underline{\omega }}A\Vert ^2}{\det (A)\Vert {\underline{\omega }}\Vert ^2}$, and

• (g) $v_{{\underline{\omega }},A}= \det \left( C_{A^{-1}\gamma ({\underline{\omega }})}^{\,\,-1}\right) = \frac{\det (A)\Vert {\underline{\omega }}\Vert ^2}{\Vert {\underline{\omega }}A\Vert ^2}$.

Proof

(a) and (c) follow from (b), since $\gamma \big ((1,0)\big )$ is the identity matrix. For any ${\underline{\omega }}\in {\mathcal {O}}$, $\gamma \big ({\underline{\omega }}A^{-1}\big )\in K_0$ by definition of $\gamma$. On the other hand,

$$\begin{aligned} \begin{aligned} (1,0)\left( \gamma \big ({\underline{\omega }}A^{-1}\big )^{-1} A\gamma ({\underline{\omega }})\right)&= \left( (1,0)\gamma \big ({\underline{\omega }}A^{-1}\big )^{-1} \right) A\gamma ({\underline{\omega }})\\&= {\underline{\omega }}A^{-1}A\gamma ({\underline{\omega }})={\underline{\omega }}\gamma ({\underline{\omega }}) =(1,0). \end{aligned} \end{aligned}$$

Thus $C_{A\gamma ({\underline{\omega }})}= \gamma \big ({\underline{\omega }}A^{-1}\big )^{-1}A \gamma ({\underline{\omega }})$ and $M_{A\gamma ({\underline{\omega }})}= \gamma \big ({\underline{\omega }}A^{-1}\big )$, by uniqueness. Clearly (e) is true while (f) and (g) follow from (c) and (e). It remains to verify (d). By (c), $C_{A^{-1}\gamma ({\underline{\omega }})}^{\,\,-1}= \gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)$. Thus

$$\begin{aligned} \begin{aligned} C_{(AB)^{-1}\gamma ({\underline{\omega }})}^{\,\,-1}&= \gamma ({\underline{\omega }})^{-1}(AB)\gamma ({\underline{\omega }}AB)= \gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A) \gamma ({\underline{\omega }}A)^{-1}B\gamma ({\underline{\omega }}AB)\\&= C_{A^{-1}\gamma ({\underline{\omega }})}^{\,\,-1} C_{B^{-1}\gamma ({\underline{\omega }}A)}^{\,\,-1}. \end{aligned} \end{aligned}$$

That is,

$$\begin{aligned} \begin{aligned} \begin{pmatrix} 1 &{} 0\\ u_{{\underline{\omega }},AB} &{} v_{{\underline{\omega }},AB} \end{pmatrix}&= \begin{pmatrix} 1 &{} 0\\ u_{{\underline{\omega }},A} &{} v_{{\underline{\omega }},A} \end{pmatrix}\begin{pmatrix} 1 &{} 0\\ u_{{\underline{\omega }}A,B} &{} v_{{\underline{\omega }}A,B} \end{pmatrix}\\&= \begin{pmatrix} 1 &{} 0\\ u_{{\underline{\omega }},A}+v_{{\underline{\omega }},A}u_{{\underline{\omega }}A,B} &{} v_{{\underline{\omega }},A}v_{{\underline{\omega }}A,B} \end{pmatrix}, \end{aligned} \end{aligned}$$

which establishes (d). $\square$

The detailed values of $u_{{\underline{\omega }},A}$ and $v_{{\underline{\omega }},A}$ are usually not needed, but may sometimes be useful.

Proposition 5.3

Let $A=\begin{pmatrix} a &{} b\\ c&{} d \end{pmatrix}\in {{{\textrm{GL}}}}_2({\mathbb {R}})$ and ${\underline{\omega }}=(\omega _1,\omega _2)\in {\mathcal {O}}$. Then

$$\begin{aligned} u_{{\underline{\omega }},A}=\frac{(ac+bd)(\omega _1^2-\omega _2^2)-(a^2+b^2-c^2-d^2) \omega _1\omega _2}{(a\omega _1+c\omega _2)^2+(b\omega _1+d\omega _2)^2} \end{aligned}$$

and

$$\begin{aligned} v_{{\underline{\omega }},A}=\frac{(ad-bc)(\omega _1^2+\omega _2^2)}{(a\omega _1+c\omega _2)^2+(b\omega _1+d\omega _2)^2}. \end{aligned}$$

Proof

These are both obtained by straightforward calculation from the definitions of $u_{{\underline{\omega }},A}$ and $v_{{\underline{\omega }},A}$. $\square$

The map $[u,v]\rightarrow \begin{pmatrix} 1 &{} 0\\ u &{} v \end{pmatrix}$ is an isomorphism of the group $G_1={\mathbb {R}}\rtimes {\mathbb {R}}^*$ with $H_{(1,0)}$. We saw that $G_1$ has, up to equivalence, one square-integrable representation. One realization is $\pi ^1$, which acts on $L^2({\mathbb {R}}^*)$ as in (17). We will simplify notation by also considering $\pi ^1$ as an irreducible representation of $H_{(1,0)}$. That is, if $C=\begin{pmatrix} 1 &{} 0\\ u &{} v \end{pmatrix}\in H_{(1,0)}$, then we let $\pi ^1(C)=\pi ^1[u,v]$. Lift $\pi ^1$ to H. That is, let ${\tilde{\pi }}^1[{\underline{x}},C]=\pi ^1(C)$, for every $[{\underline{x}},C]\in H$. The representation $\chi _{(1,0)}\otimes {\tilde{\pi }}^1$ of H, given by

$$\begin{aligned} \left( \chi _{(1,0)}\otimes {\tilde{\pi }}^1\right) [{\underline{x}},C]= \chi _{(1,0)}({\underline{x}})\pi ^1(C),\quad \text {for } [{\underline{x}},C]\in H, \end{aligned}$$

is an irreducible representation of H on $L^2({\mathbb {R}}^*)$. Then ${{\textrm{ind}}}_H^{G_2}\left( \chi _{(1,0)}\otimes {\tilde{\pi }}^1\right)$ is an irreducible representation of $G_2$ by Theorem 4.24 of [13]. By Proposition 3.1, ${{\textrm{ind}}}_H^G\left( \chi _{(1,0)}\otimes {\tilde{\pi }}^1\right)$ is equivalent to a representation $\sigma$ acting on $L^2\big (K,L^2({\mathbb {R}}^*)\big )$. In preparation for defining $\sigma$, take $[{\underline{x}},A]\in G_2$, and $[{\underline{0}},L]\in K$ and compute $[{\underline{x}},A]^{-1}[{\underline{0}},L]= [-A^{-1}{\underline{x}},A^{-1}L]$. Now factor

$$\begin{aligned} {[}-A^{-1}{\underline{x}},A^{-1}L]=[{\underline{0}},M_{A^{-1}L}] [-M_{A^{-1}L}^{\,\,-1}A^{-1}{\underline{x}},C_{A^{-1}L}], \end{aligned}$$

with the elements $[{\underline{0}},M_{A^{-1}L}]\in K$ and $[-M_{A^{-1}L}^{\,\,-1}A^{-1}{\underline{x}},C_{A^{-1}L}]\in H$. Observe that

$$\begin{aligned} C_{_{A^{-1}L}}^{\,\,\,\,-1}M_{A^{-1}L}^{\,\,-1}A^{-1}= \big (M_{A^{-1}L}C_{_{A^{-1}L}}\big )^{-1}A^{-1}=L^{-1}. \end{aligned}$$

Therefore,

$$\begin{aligned} {[}-M_{A^{-1}L}^{\,\,-1}A^{-1}{\underline{x}},C_{A^{-1}L}]^{-1}= [L^{-1}{\underline{x}}, C_{_{A^{-1}L}}^{\,\,\,\,-1}]. \end{aligned}$$

Thus, Proposition 3.1 gives, for $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$, $[{\underline{x}},A]\in G$, and $[{\underline{0}},L]\in K$,

$$\begin{aligned} \begin{aligned} \sigma [{\underline{x}},A]F[{\underline{0}},L]&= \left| \det \left( C_{_{A^{-1}L}}\right) \right| ^{-1/2}\big (\chi _{(1,0)}\otimes \pi ^1\big )\left[ L^{-1}{\underline{x}}, C_{_{A^{-1}L}}^{\,\,\,\,-1}\right] F[{\underline{0}},M_{A^{-1}L}]\\&=\left| \det \left( C_{_{A^{-1}L}}\right) \right| ^{-1/2}e^{2\pi i(1,0)L^{-1}{\underline{x}}}\, \pi ^1\!\left( C_{_{A^{-1}L}}^{\,\,\,\,-1}\right) F[{\underline{0}},M_{A^{-1}L}]. \end{aligned} \end{aligned}$$

(21)

Use of the homeomorphism $\gamma :{\mathcal {O}}\rightarrow K_0$ and the identities collected in Proposition 5.2 help make the expression given in (21) easier to read. For $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$, $[{\underline{x}},A]\in G_2$, and a.e. ${\underline{\omega }}\in {\mathcal {O}}$,

$$\begin{aligned} \sigma [{\underline{x}},A]F[{\underline{0}},\gamma ({\underline{\omega }})] = \textstyle \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}\Vert }{\Vert {\underline{\omega }}A\Vert } e^{2\pi i{\underline{\omega }}\,{\underline{x}}}\, \pi ^1\big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)\big ) F[{\underline{0}},\gamma ({\underline{\omega }}A)].\nonumber \\ \end{aligned}$$

(22)

In (22), $\sigma [{\underline{x}},A]F[{\underline{0}},\gamma ({\underline{\omega }})]\in L^2({\mathbb {R}}^*)$. Before evaluating it, we note that

$$\begin{aligned} \gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A) =C_{A^{-1}\gamma ({\underline{\omega }})}^{\,\,-1}= \begin{pmatrix} 1 &{} 0\\ u_{{\underline{\omega }},A} &{} v_{{\underline{\omega }},A}, \end{pmatrix} \end{aligned}$$

so $\pi ^1\big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)\big )=\pi ^1[u_{{\underline{\omega }},A}, v_{{\underline{\omega }},A}]$, when we consider $\pi ^1$ as a representation of $G_1$. Using (17) and (22), we have, for a.e. $t\in {\mathbb {R}}^*$,

$$\begin{aligned} \big (\sigma [{\underline{x}},A]F[{\underline{0}},\gamma ({\underline{\omega }})]\big )(t) = \textstyle \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}\Vert }{\Vert {\underline{\omega }}A\Vert } e^{2\pi i({\underline{\omega }}\,{\underline{x}}+t^{-1}u_{{\underline{\omega }},A})} \big (F[{\underline{0}},\gamma ({\underline{\omega }}A)]\big ) (v_{{\underline{\omega }},A}^{-1}t),\nonumber \\ \end{aligned}$$

(23)

for a.e. ${\underline{\omega }}\in {\mathcal {O}}$, with $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$ and $[{\underline{x}},A]\in G_2$. By construction, $\sigma$ is equivalent to ${{\textrm{ind}}}_H^G\left( \chi _{(1,0)}\otimes {\tilde{\pi }}^1\right)$, which is irreducible.

Proposition 5.4

The representation $\sigma$ of $G_2$ given by (23) is irreducible.

6 Decomposing the Regular Representation of $G_2$

This section contains the detailed calculations that lead to an explicit decomposition of $\lambda _{G_2}$ into an infinite sum of representations, each of which is equivalent to $\sigma$, as given in (23). Our first set of tasks is to construct a family of isometries from $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ into $L^2(G_2)$ that intertwine $\sigma$ with the left regular representation of $G_2$. This is done in three steps: We map $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ into $L^2\big (K,L^2(H_{(1,0)})\big )$ by essentially using the continuous wavelet transform in dimension one as expressed by Corollary 4.6. Then we use the fact that $K\times H_{(1,0)}$ maps to ${{{\textrm{GL}}}}_2({\mathbb {R}})$ in a manner that connects the Haar measures to move from $L^2\big (K,L^2(H_{(1,0)})\big )$ to $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$. The composition of these first two isometries will intertwine $\sigma$ with $\pi ^{(1,0)}$. Then, we can use Proposition 4.2 to get isometries that map $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$ into $L^2(G_2)$ intertwining $\pi ^{(1,0)}$ with $\lambda _{G_2}$. Let us proceed with the details.

Corollary 4.6 says that, if $g\in L^2({\mathbb {R}}^*)$ satisfies $\int _{{\mathbb {R}}^*}\big ||\nu |^{1/2}g(\nu )\big |^2 d\mu _{{\mathbb {R}}^*}(\nu )=1$, then $V_g:L^2({\mathbb {R}}^*)\rightarrow L^2(H_{(1,0)})$ is an isometry that intertwines $\pi ^1$ with $\lambda _{H_{(1,0)}}$. Recall that

$$\begin{aligned} V_gf(D)=\langle f,\pi ^1(D)g\rangle _{_{L^2({\mathbb {R}}^*)}}, \end{aligned}$$

for all $D\in H_{(1,0)}, f\in L^2({\mathbb {R}}^*)$. Let ${\mathcal {K}}_g=V_g L^2({\mathbb {R}}^*)$, a closed $\lambda _{H_{(1,0)}}$-invariant subspace of $L^2(H_{(1,0)})$.

Let $V_g':L^2\big (K,L^2({\mathbb {R}}^*)\big )\rightarrow L^2\big (K,L^2(H_{(1,0)})\big )$ be given by $\big (V_g'F\big )[{\underline{0}},L]=V_g\big (F[{\underline{0}},L]\big )$, for all $[{\underline{0}},L]\in K$ and $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$. Since $V_g$ is an isometry, so is $V_g'$ and the range of $V_g'$ is $L^2(K,{\mathcal {K}}_g)$. For $[{\underline{x}},A]\in G_2$, $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$, and ${\underline{\omega }}\in {\mathcal {O}}$,

$$\begin{aligned} (V_g'\sigma [{\underline{x}},A]F)[{\underline{0}},\gamma ({\underline{\omega }})]&= \textstyle \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}\Vert }{\Vert {\underline{\omega }}A\Vert } e^{2\pi i{\underline{\omega }}\,{\underline{x}}}\,V_g\big ( \pi ^1\big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)\big ) F[{\underline{0}},\gamma ({\underline{\omega }}A)]\big )\\&= \textstyle \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}\Vert }{\Vert {\underline{\omega }}A\Vert } e^{2\pi i{\underline{\omega }}\,\int _{-\infty }^\infty |h(x)|^2\frac{dx}{|x|}<\infty {\underline{x}}}\, \lambda _{H_{(1,0)}}\\&\quad \big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)\big ) V_g\big ( F[{\underline{0}},\gamma ({\underline{\omega }}A)]\big ). \end{aligned}$$

Thus, $\sigma$ is equivalent to a representation ${\tilde{\sigma }}$ acting on $L^2(K,{\mathcal {K}}_g)$ as follows: For $[{\underline{x}},A]\in G_2$, $\varphi \in L^2(K,{\mathcal {K}}_g)$, and ${\underline{\omega }}\in {\mathcal {O}}$,

$$\begin{aligned} \big ({\tilde{\sigma }}[{\underline{x}},A]\varphi \big )[{\underline{0}},\gamma ({\underline{\omega }})]= \textstyle \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}\Vert }{\Vert {\underline{\omega }}A\Vert } e^{2\pi i{\underline{\omega }}\,{\underline{x}}}\, \lambda _{H_{(1,0)}}\big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)\big ) \varphi [{\underline{0}},\gamma ({\underline{\omega }}A)].\nonumber \\ \end{aligned}$$

(24)

The next step is to map $L^2(K,{\mathcal {K}}_g)$ isometrically into $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$.

Let $W_1:L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )\rightarrow L^2\big (K,L^2(H_{(1,0)})\big )$ be given by

$$\begin{aligned} \big (W_1f[{\underline{0}},M]\big )(C)=|\det (C)|^{1/2}f(MC), \end{aligned}$$

for all $C\in H_{(1,0)}$, $[{\underline{0}},M]\in K$.

Proposition 6.1

The map $W_1$ is a unitary map onto $L^2\big (K,L^2(H_{(1,0)})\big )$ and its inverse is given by, for $F\in L^2\big (K,L^2(H_{(1,0)})\big )$ and $B\in {{{\textrm{GL}}}}_2({\mathbb {R}})$,

$$\begin{aligned} \big (W_1^{-1}F\big )(B) =|\det (C_B)|^{-1/2} (F[{\underline{0}},M_B])(C_B). \end{aligned}$$

Proof

Using (10) and the facts that $\Delta _{H_{(1,0)}}(C)=|\det (C)|^{-1}$ and ${{{\textrm{GL}}}}_2({\mathbb {R}})$ is unimodular, the Haar integral on ${{{\textrm{GL}}}}_2({\mathbb {R}})$ can be expressed as

$$\begin{aligned} \int _{{{{\textrm{GL}}}}_2({\mathbb {R}})}h\,d\mu _{{{{\textrm{GL}}}}_2({\mathbb {R}})}=\int _{K_0}\int _{H_{(1,0)}} h(MC)\,|\det (C)|\,d\mu _{H_{(1,0)}}(C)\,d\mu _{K_0}(M), \end{aligned}$$

for any integrable h on ${{{\textrm{GL}}}}_2({\mathbb {R}})$. Thus, for any $f\in L^2({{{\textrm{GL}}}}_2({\mathbb {R}}))$,

$$\begin{aligned} \begin{aligned}&\int _{K_0}\Vert W_1f[{\underline{0}},M]\Vert _{_{L^2(H_{(1,0)})}}^{\,\,\,2} d\mu _{K_0}(M) \\&\quad = \int _{K_0}\int _{H_{(1,0)}} |(W_1f[{\underline{0}},M])(C)|^2d\mu _{H_{(1,0)}}d\mu _{K_0}(M)\\&\quad = \int _{K_0}\int _{H_{(1,0)}} |f(MC)|^2|\det (C)|\, d\mu _{H_{(1,0)}}\,d\mu _{K_0}(M)\\&\quad =\int _{{{{\textrm{GL}}}}_2({\mathbb {R}})}|f|^2\,d\mu _{{{{\textrm{GL}}}}_2({\mathbb {R}})}<\infty . \end{aligned} \end{aligned}$$

Hence, $W_1f\in L^2\big (K,L^2(H_{(1,0)})\big )$ and $W_1$ is an isometry. It is clear $W_1$ is linear. Therefore, the range of $W_1$ is a closed subspace of $L^2\big (K,L^2(H_{(1,0)})\big )$. We need to show that the range of $W_1$ is all of $L^2\big (K,L^2(H_{(1,0)})\big )$. For any $k\in C_c(K_0)$ and $h\in C_c(H_{(1,0)})$, define $F_{k,h}\in L^2\big (K,L^2(H_{(1,0)})\big )$ by $\big (F_{k,h}[{\underline{0}},M]\big )(C)=k(M)h(C)$. The linear span of $\{F_{k,h}:k\in C_c(K_0), h\in C_c(H_{(1,0)})\}$ is dense in $L^2\big (K,L^2(H_{(1,0)})\big )$. So we just need to show each $F_{k,h}$ is in $W_1L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$. For $k\in C_c(K_0)$ and $h\in C_c(H_{(1,0)})$, let $f_{k,h}(B)=|\det (C_B)|^{-1/2}k(M_B)h(C_B)$, for all $B\in {{{\textrm{GL}}}}_2({\mathbb {R}})$. Since $B\rightarrow (M_B,C_B)$ is a homeomorphism of ${{{\textrm{GL}}}}_2({\mathbb {R}})$ with $K_0\times H_{(1,0)}$ and $B\rightarrow |\det (C_B)|^{-1/2}$ is continuous, $f_{k,h}\in C_c\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )\subseteq L^{2}\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$. Moreover, since $M_{MC}=M$ and $C_{MC}=C$,

$$\begin{aligned} \begin{aligned} \big (W_1f_{k,h}[{\underline{0}},M]\big )(C)&=|\det (C)|^{1/2} f_{k,h}(MC)=|\det (C)|^{1/2}|\det (C)|^{-1/2}k(M)h(C)\\&= \big (F_{k,h}[{\underline{0}},M]\big )(C), \end{aligned} \end{aligned}$$

for any $C\in H_{(1,0)}$ and $[{\underline{0}},M]\in K$. Thus, $F_{k,h}\in W_1L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$, for any $k\in C_c(K_0)$ and $h\in C_c(H_{(1,0)})$. This implies $W_1$ is a unitary map onto $L^2\big (K,L^2(H_{(1,0)})\big )$. Also, $W_1^{-1}F_{k,h}=f_{k,h}$ so, for any $B\in {{{\textrm{GL}}}}_2({\mathbb {R}})$,

$$\begin{aligned} \begin{aligned} W_1^{-1}F_{k,h}(B)&=f_{k,h}(B)=|\det (C_B)|^{-1/2}k(M_B)h(C_B)\\&=|\det (C_B)|^{-1/2} (F_{k,h}[{\underline{0}},M_B])(C_B). \end{aligned} \end{aligned}$$

Since $\{F_{k,h}:k\in C_c(K_0),h\in C_c(H_{(1,0)}\}$ is total in $L^2\big (K,L^2(H_{(1,0)})\big )$,

$$\begin{aligned} \big (W_1^{-1}F\big )(B)=|\det (C_B)|^{-1/2} (F[{\underline{0}},M_B])(C_B), \end{aligned}$$

for all $B\in {{{\textrm{GL}}}}_2({\mathbb {R}})$ and $F\in L^2\big (K,L^2(H_{(1,0)})\big )$. $\square$

Continuing with a fixed $g\in L^2({\mathbb {R}}^*)$ satisfying $\int _{{\mathbb {R}}^*}\big ||\nu |^{1/2}g(\nu )\big |^2 d\mu _{{\mathbb {R}}^*}(\nu )=1$, let

$$\begin{aligned} {\mathcal {H}}_g=W_1^{-1}L^2(K,{\mathcal {K}}_g)\subseteq L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big ). \end{aligned}$$

Then ${\mathcal {H}}_g$ is a closed subspace of $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}}) \big )$ and $W_1:{\mathcal {H}}_g\rightarrow L^2(K,{\mathcal {K}}_g)$ is a unitary map. Note that we use the same notation for $W_1$ and its restriction to ${\mathcal {H}}_g$. Recall from (14) the representation $\pi ^{(1,0)}$ of $G_2$ acting on the Hilbert space $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$. For $[{\underline{x}},A]\in G_2$, $f\in L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$ and a.e. $B\in {{{\textrm{GL}}}}_2({\mathbb {R}})$,

$$\begin{aligned} \pi ^{(1,0)}[{\underline{x}},A]f(B)=e^{2\pi i(1,0) B^{-1}{\underline{x}}} f(A^{-1}B). \end{aligned}$$

Proposition 6.2

Let $g\in L^2({\mathbb {R}}^*)$ satisfy $\int _{{\mathbb {R}}^*}\big ||\nu |^{1/2}g(\nu )\big |^2 d\mu _{{\mathbb {R}}^*}(\nu )=1$. The subspace ${\mathcal {H}}_g$ of $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$ is $\pi ^{(1,0)}$-invariant and the restriction of $\pi ^{(1,0)}$ to ${\mathcal {H}}_g$ is equivalent to ${\tilde{\sigma }}$ via the unitary map $W_1:{\mathcal {H}}_g\rightarrow L^2(K,{\mathcal {K}}_g)$.

Proof

Let $[{\underline{x}},A]\in G_2$. For $f\in {\mathcal {H}}_g$, let $F=W_1f \in L^2(K,{\mathcal {K}}_g)$. Since $L^2(K,{\mathcal {K}}_g)$ is ${\tilde{\sigma }}$-invariant, ${\tilde{\sigma }}[{\underline{x}},A]F\in L^2(K,{\mathcal {K}}_g)$ as well. Thus $W_1^{-1}{\tilde{\sigma }}[{\underline{x}},A]F= W_1^{-1}{\tilde{\sigma }}[{\underline{x}},A]W_1f\in {\mathcal {H}}_g$.

For any $B\in {{{\textrm{GL}}}}_2({\mathbb {R}})$, let ${\underline{\omega }}=(1,0)B^{-1}$. By Proposition 5.2 (a) $M_B=\gamma ({\underline{\omega }})$ and $C_B=\gamma ({\underline{\omega }})^{-1}B$. Then, using (24) and $|\det \big (\gamma ({\underline{\omega }})\big )|^{1/2}= \Vert {\underline{\omega }}\Vert ^{-1}$,

$$\begin{aligned} \begin{aligned} W_1^{-1}{\tilde{\sigma }}[{\underline{x}},A]F(B)&=|\det (C_B)|^{-1/2} \big ({\tilde{\sigma }}[{\underline{x}},A]F[{\underline{0}},M_B]\big )(C_B)\\&= \textstyle \frac{|\det (\gamma ({\underline{\omega }}))|^{1/2}}{|\det (B)|^{1/2}} \big ({\tilde{\sigma }}[{\underline{x}},A] F[{\underline{0}},\gamma ({\underline{\omega }})]\big )(\gamma ({\underline{\omega }})^{-1}B)\\&=\textstyle \frac{|\det (A)|^{1/2}}{|\det (B)|^{1/2}\Vert {\underline{\omega }}A\Vert } e^{2\pi i{\underline{\omega }}\,{\underline{x}}}\, \lambda _{H_{(1,0)}}\big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)\big ) \big (F[{\underline{0}},\gamma ({\underline{\omega }}A)]\big ) \\&\quad \times (\gamma ({\underline{\omega }})^{-1}B)\\&= \textstyle \frac{|\det (A)|^{1/2}}{|\det (B)|^{1/2}\Vert {\underline{\omega }}A\Vert } e^{2\pi i{\underline{\omega }}\,{\underline{x}}}\, \big (F[{\underline{0}},\gamma ({\underline{\omega }}A)]\big ) (\gamma ({\underline{\omega }}A)^{-1}A^{-1}B)\\&= \textstyle \frac{|\det (A)|^{1/2}}{|\det (B)|^{1/2}\Vert {\underline{\omega }}A\Vert } e^{2\pi i{\underline{\omega }}\,{\underline{x}}}\, \big (W_1f[{\underline{0}},\gamma ({\underline{\omega }}A)]\big ) \big (\gamma ({\underline{\omega }}A)^{-1}A^{-1}B\big ). \end{aligned} \end{aligned}$$

Before applying $W_1$, note that $|\det \big (\gamma ({\underline{\omega }}A)^{-1}A^{-1}B\big )|^{1/2} =\frac{|\det (B)|^{1/2}\Vert {\underline{\omega }}A\Vert }{|\det (A)|^{1/2}}$, which will cancel the first factor in the previous expression. Therefore, recalling that ${\underline{\omega }}=(1,0)B^{-1}$,

$$\begin{aligned} W_1^{-1}{\tilde{\sigma }}[{\underline{x}},A]W_1f(B){} & {} = e^{2\pi i{\underline{\omega }}\,{\underline{x}}}f(A^{-1}B) =e^{2\pi i(1,0)B^{-1}{\underline{x}}}f(A^{-1}B) \\{} & {} =\pi ^{(1,0)}[{\underline{x}},A]f(B). \end{aligned}$$

This implies that ${\mathcal {H}}_g$ is $\pi ^{(1,0)}$-invariant and the restriction of $\pi ^{(1,0)}$ to ${\mathcal {H}}_g$ is equivalent to ${\tilde{\sigma }}$. $\square$

Recall Theorem 4.3. The nontrivial orbit in the one-dimensional case is ${\mathcal {O}}_1={\widehat{{\mathbb {R}}}}\setminus \{0\}$, which is naturally identified with ${\mathbb {R}}^*$. Let ${\mathcal {D}}=L^2({\mathcal {O}}_1)\cap L^2({\mathbb {R}}^*)$. Note that

$$\begin{aligned} {\mathcal {D}}=\left\{ h:{\widehat{{\mathbb {R}}}}\setminus \{0\}\rightarrow {\mathbb {C}}\,\big |\, h \text { measurable },\int _{-\infty }^\infty |h(t)|^2dt<\infty \text { and } \int _{-\infty }^\infty |h(t)|^2\frac{dt}{|t|}<\infty \right\} , \end{aligned}$$

which can be considered as a dense subspace of either $L^2({\mathcal {O}}_1)$ or $L^2({\mathbb {R}}^*)$. Note that, if $g\in L^2({\mathbb {R}}^*)$ satisfies $\int _{{\mathbb {R}}^*}\big ||\nu |^{1/2}g(\nu )\big |^2 d\mu _{{\mathbb {R}}^*}(\nu )=1$ as in Proposition 6.2, then

$$\begin{aligned} \int _{-\infty }^\infty |g(\nu )|^2d\nu =\int _{-\infty }^\infty \big ||\nu |^{1/2}g(\nu )\big |^2\frac{d\nu }{|\nu |}= \int _{{\mathbb {R}}^*}\big ||\nu |^{1/2}g(\nu )\big |^2 d\mu _{{\mathbb {R}}^*}(\nu )=1. \end{aligned}$$

So such a g is a unit vector in $L^2({\mathcal {O}}_1)$ and, thus, lies in ${\mathcal {D}}$. Fix $\{g_j:j\in J\} \subseteq {\mathcal {D}}$ such that $\{g_j:j\in J\}$ is an orthonormal basis in $L^2({\mathcal {O}}_1)$. Let $w_2(\nu )=|\nu |^{1/2}$, for $\nu \in {\widehat{{\mathbb {R}}}} \setminus \{0\}$. We state the following elementary lemma for later use.

Lemma 6.3

The map $h\rightarrow w_2h$ is a unitary map of $L^2({\mathcal {O}}_1)$ onto $L^2({\mathbb {R}}^*)$. In particular, $\{w_2g_j:j\in J\}$ is an orthonormal basis of $L^2({\mathbb {R}}^*)$.

Identifying $H_{(1,0)}$ with $G_1$, Theorem 4.3, with $n=1$, says that $L^2(H_{(1,0)})=\sum _{j\in J}^{\oplus }{\mathcal {K}}_{g_j}$. Therefore, $L^2\big (K,L^2(H_{(1,0)})\big )=\textstyle \sum _{j\in J}^{\oplus }L^2(K,{\mathcal {K}}_{g_j})$. Applying $W_1^{-1}$, now considered as a unitary map of $L^2\big (K,L^2(H_{(1,0)})\big )$ onto $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$, we get a decomposition of $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$.

Proposition 6.4

Let $\{g_j:j\in J\} \subseteq {\mathcal {D}}$ be an orthonormal basis in $L^2({\mathcal {O}}_1)$. Then each ${\mathcal {H}}_{g_j}$ is a closed $\pi ^{(1,0)}$-invariant subspace of $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$ and the restriction of $\pi ^{(1,0)}$ to ${\mathcal {H}}_{g_j}$ is equivalent to $\sigma$. Moreover, $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )=\sum _{j\in J}^{\oplus } {\mathcal {H}}_{g_j}$.

Recall from Proposition 4.2, if $\eta \in L^2({\mathcal {O}})$ satisfies $\Vert \eta \Vert _{_{L^2({\mathcal {O}})}} =1$, then there is an isometric linear map $U_\eta :L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )\rightarrow L^2(G_2)$ that intertwines $\pi ^{(1,0)}$ with $\lambda _{G_2}$. The map $U_\eta$ is defined, for all $[{\underline{y}},B]\in G_2$ and $f\in L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$, by

$$\begin{aligned} U_\eta f[{\underline{y}},B]=\int _{\widehat{{\mathbb {R}}^2}} \eta ({\underline{\omega }})f\big (B\gamma ({\underline{\omega }})\big ) e^{-2\pi i{\underline{\omega }}B^{-1}{\underline{y}}}\,d{\underline{\omega }}. \end{aligned}$$

The steps we have taken to move from $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ to $L^2(G_2)$ are summarized in the following diagram.

The vertical maps are linear isometries from the upper Hilbert space into the lower Hilbert space intertwining the corresponding unitary representations.

Thus, if $g\in {\mathcal {D}}$ satisfies $\Vert g\Vert _{L^2({\mathcal {O}}_1)}=1$ and $\eta \in L^2({\mathcal {O}})$ satisfies $\Vert \eta \Vert _{L^2({\mathcal {O}})}=1$, then $\Phi _{\eta ,g}=U_\eta \circ W_1^{-1} \circ V_g'$ is a linear isometry of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ into $L^2(G_2)$ that intertwines $\sigma$ with $\lambda _{G_2}$. Let $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$. For $[{\underline{x}},A]\in G_2$,

$$\begin{aligned} \begin{aligned} \Phi _{\eta ,g}F[{\underline{x}},A]&= \int _{\widehat{{\mathbb {R}}^2}} \eta ({\underline{\omega }})\big (W_1^{-1}V_g'F\big )\big )\big (A\gamma ({\underline{\omega }})\big ) e^{-2\pi i{\underline{\omega }}A^{-1}{\underline{x}}}d{\underline{\omega }}\\&= |\det (A)|\int _{\widehat{{\mathbb {R}}^2}} \eta ({\underline{\omega }}A)\big (W_1^{-1}V_g'F\big )\big )\big (A\gamma ({\underline{\omega }}A)\big ) e^{-2\pi i{\underline{\omega }}\,{\underline{x}}}d{\underline{\omega }}. \end{aligned} \end{aligned}$$

(25)

Observe that $M_{A\gamma ({\underline{\omega }}A)}= \gamma ({\underline{\omega }})$ and $C_{A\gamma ({\underline{\omega }}A)}= \gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)$. So

$$\begin{aligned} \det (C_{A\gamma ({\underline{\omega }}A)})= \det \big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)\big ) =\frac{\Vert {\underline{\omega }}\Vert ^2\det (A)}{\Vert {\underline{\omega }}A\Vert ^2} \end{aligned}$$

Thus,

$$\begin{aligned}{} & {} \big (W_1^{-1}V_g'F\big )\big )\big (A\gamma ({\underline{\omega }}A)\big )\\{} & {} \quad = \textstyle \frac{\Vert {\underline{\omega }}A\Vert }{\Vert {\underline{\omega }}\Vert \cdot |\det (A)|^{1/2}}V_g(F[{\underline{0}},\gamma ({\underline{\omega }})]) \big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A)\big )\\{} & {} \quad =\textstyle \frac{\Vert {\underline{\omega }}A\Vert }{\Vert {\underline{\omega }}\Vert \cdot |\det (A)|^{1/2}}\langle F[{\underline{0}},\gamma ({\underline{\omega }})], \pi ^1\big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A) \big )g\rangle _{_{L^2({\mathbb {R}}^*)}}\\{} & {} \quad = {\textstyle \frac{\Vert {\underline{\omega }}A\Vert }{\Vert {\underline{\omega }}\Vert \cdot |\det (A)|^{1/2}}}\int _{{\mathbb {R}}^*} (F[{\underline{0}},\gamma ({\underline{\omega }})])(\nu )\overline{\pi ^1\big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A) \big )g(\nu )}\,d\mu _{{\mathbb {R}}^*}(\nu ). \end{aligned}$$

Inserting this into (25) gives

$$\begin{aligned} \Phi _{\eta ,g} F[{\underline{x}},A]{} & {} =\int _{\widehat{{\mathbb {R}}^2}}\int _{{\mathbb {R}}^*} {\textstyle \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}A\Vert }{\Vert {\underline{\omega }}\Vert }} \eta ({\underline{\omega }}A) e^{-2\pi i{\underline{\omega }}\,{\underline{x}}}\\{} & {} \quad \times (F[{\underline{0}},\gamma ({\underline{\omega }})])(\nu )\overline{ \pi ^1\big (\gamma ({\underline{\omega }})^{-1}A\gamma ({\underline{\omega }}A) \big )g(\nu )}\,d\mu _{{\mathbb {R}}^*}(\nu )\,d{\underline{\omega }}. \end{aligned}$$

We will compare the expression for $\Phi _{\eta ,g}F$ with a coefficient function of the irreducible representation $\sigma$.

If $E\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$ is fixed, then, for any $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$, $V_EF$ is the continuous function on $G_2$ defined by $V_EF[{\underline{x}},A]= \langle F,\sigma [{\underline{x}},A]E\rangle _{_{L^2(K,L^2({\mathbb {R}}^*))}}$, for all $[{\underline{x}},A]\in G_2$. Recall that the Haar integral over K can be expressed using the parametrization ${\underline{\omega }}\rightarrow \gamma ({\underline{\omega }})$ by ${\mathcal {O}}$, which is co-null in $\widehat{{\mathbb {R}}^2}$. Then, for $[{\underline{x}},A] \in G_2$,

$$\begin{aligned} \begin{aligned} V_EF[{\underline{x}},A]&= \langle F,\sigma [{\underline{x}},A]E\rangle _{_{L^2(K,L^2({\mathbb {R}}^*))}}\\&=\int _{\widehat{{\mathbb {R}}^2}}\int _{{\mathbb {R}}^*} (F[{\underline{0}},\gamma ({\underline{\omega }})])(\nu )\overline{\sigma [{\underline{x}},A] (E[{\underline{0}},\gamma ({\underline{\omega }})])(\nu )}\,d\mu _{{\mathbb {R}}^*}(\nu ) \frac{d{\underline{\omega }}}{\Vert {\underline{\omega }}\Vert ^2}. \end{aligned}\nonumber \\ \end{aligned}$$

(26)

Note that

$$\begin{aligned} \overline{\sigma [{\underline{x}},A] (E[{\underline{0}},\gamma ({\underline{\omega }})])(\nu )}{} & {} = \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}\Vert }{\Vert {\underline{\omega }}A\Vert }\\{} & {} \quad \times e^{-2\pi i{\underline{\omega }}\,{\underline{x}}}\, \overline{\pi ^1\big (\gamma ({\underline{\omega }})^{-1} A\gamma ({\underline{\omega }}A)\big )E[{\underline{0}},\gamma ({\underline{\omega }}A)](\nu ) }. \end{aligned}$$

If we select E as a function built from $\eta$ and g, then we can make the expression for $V_EF$ coincide with that for $\Phi _{\eta ,g}$. Define E as follows: For $[{\underline{0}},L]\in K$ and $\nu \in {\mathbb {R}}^*$,

$$\begin{aligned} E[{\underline{0}},L](\nu )=\textstyle \frac{{\overline{\eta }} ((1,0)L^{-1})}{|\det (L)|}\,g(\nu ). \end{aligned}$$

If $L=\gamma ({\underline{\omega }}A)$, then $E[{\underline{0}},\gamma ({\underline{\omega }}A)](\nu )= \frac{{\overline{\eta }}({\underline{\omega }}A)}{|\det (\gamma ({\underline{\omega }}A))|}\,g(\nu )=\Vert {\underline{\omega }}A\Vert ^2 {\overline{\eta }}({\underline{\omega }}A)\,g(\nu )$. Thus

$$\begin{aligned} \overline{\sigma [{\underline{x}},A] (E[{\underline{0}},\gamma ({\underline{\omega }})])(\nu )}\frac{1}{\Vert {\underline{\omega }}\Vert ^2}{} & {} = \frac{|\det (A)|^{1/2}\Vert {\underline{\omega }}A\Vert }{\Vert {\underline{\omega }}\Vert } \eta ({\underline{\omega }}A) \\{} & {} \quad \times e^{-2\pi i{\underline{\omega }}\,{\underline{x}}}\, \overline{\pi ^1\big (\gamma ({\underline{\omega }})^{-1} A\gamma ({\underline{\omega }}A)\big )g(\nu ). } \end{aligned}$$

Substituting into (26), there is a perfect match with the expression for $\Phi _{\eta ,g}F$. Thus, we have the following theorem.

Theorem 6.5

Let $g\in L^2({\mathbb {R}}^*)$ satisfy $\int _{{\mathbb {R}}^*}\big | |\nu |^{1/2}g(\nu )\big |^2d\mu _{{\mathbb {R}}^*}(\nu )=1$ and let $\eta \in L^2(\widehat{{\mathbb {R}}^2})$ satisfy $\Vert \eta \Vert _{_{L^2(\widehat{{\mathbb {R}}^2})}}=1$. Let $E\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$ be defined as

$$\begin{aligned} E[{\underline{0}},L](\nu )= \frac{{\overline{\eta }}((1,0)L^{-1})}{|\det (L)|}\,g(\nu ), \text { for each } [{\underline{0}},L]\in K \text { and } \nu \in {\mathbb {R}}^*. \end{aligned}$$

Define $V_EF[{\underline{x}},A]= \langle F, \sigma [{\underline{x}},A]E\rangle _{_{L^2(K,L^2({\mathbb {R}}^*))}}$, for $[{\underline{x}},A]\in G_2$ and $F \in L^2\big (K,L^2({\mathbb {R}}^*)\big )$. Then $V_E$ is a linear isometry of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ into $L^2(G_2)$ that intertwines $\sigma$ with $\lambda _{G_2}$. In particular, $\sigma$ is a square-integrable representation of $G_2$.

Combining Theorem 4.3 with Proposition 6.4 will now provide a decomposition of the regular representation, $\lambda _{G_2}$ on $L^2(G_2)$, into infinitely many copies of $\sigma$. Fix an orthonormal basis $\{g_j:j\in J\}$ of $L^2({\mathcal {O}}_1)$ consisting of functions in ${\mathcal {D}}$ as in Proposition 6.4. For each $j\in J$, ${\mathcal {K}}_{g_j}=V_{g_j}L^2({\mathbb {R}}^*)$ and ${\mathcal {H}}_{g_j}=W_1^{-1}L^2(K,{\mathcal {K}}_{g_j})$. Then ${\mathcal {H}}_{g_j}$ is a closed $\pi ^{(1,0)}$-invariant subspace of $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$ and the restriction of $\pi ^{(1,0)}$ to ${\mathcal {H}}_{g_j}$ is equivalent to $\sigma$ via $W_1^{-1}\circ V_g'$. Moreover, Proposition 6.4 says $L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )=\sum _{j\in J}^{\oplus } {\mathcal {H}}_{g_j}$.

On the other hand, ${\mathcal {O}}=\widehat{{\mathbb {R}}^2}\setminus \{(0,0)\}$ is the orbit of (1, 0) in $\widehat{{\mathbb {R}}^2}$ under the action of ${{{\textrm{GL}}}}_2({\mathbb {R}})$. It is equipped with the Lebesgue measure of $\widehat{{\mathbb {R}}^2}$. Fix an orthonormal basis $\{\eta _i:i\in I\}$ of $L^2(\widehat{{\mathbb {R}}^2})=L^2({\mathcal {O}})$. At this point, we pause to formulate an analog of Lemma 6.3. There is a useful conjugate linear map $W_\gamma$ from $L^2({\mathcal {O}})$ to $L^2(K)$ provided by the map $\gamma$. We observe that $\gamma ^{-1}: K_0\rightarrow {\mathcal {O}}$ is such that $\gamma ^{-1}(L)=(1,0)L^{-1}$, for all $L\in K_0$. For $\xi \in L^2({\mathcal {O}})$, let $W_\gamma \xi [{\underline{0}},L]= |\det (L)|^{-1/2}{\overline{\xi }}\big (\gamma ^{-1}(L)\big )$, for a.e. $L\in K$.

Lemma 6.6

The map $\xi \rightarrow W_\gamma \xi$ is a conjugate linear isometry of $L^2({\mathcal {O}})$ onto $L^2(K)$. In particular, $\{W_\gamma \eta _i:i\in I\}$ is an orthonormal basis of $L^2(K)$.

Proof

We use the expression for Haar integration on K given in (20). Then, for $\xi \in L^2({\mathcal {O}})$,

$$\begin{aligned} \begin{aligned} \int _K|W_\gamma \xi |^2d\mu _K&=\int _{\widehat{{\mathbb {R}}^2}} \left| W_\gamma \xi [{\underline{0}},\gamma ({\underline{\omega }})]\right| ^2 \frac{d{\underline{\omega }}}{\Vert {\underline{\omega }}\Vert ^2}\\&=\int _{\widehat{{\mathbb {R}}^2}} \left| |\det \big (\gamma ({\underline{\omega }})\big )|^{-1/2} {\overline{\xi }}\big ((1,0)\gamma ({\underline{\omega }})^{-1}\big )\right| ^2 \frac{d{\underline{\omega }}}{\Vert {\underline{\omega }}\Vert ^2}\\&= \int _{\widehat{{\mathbb {R}}^2}} \Vert {\underline{\omega }}\Vert ^2\left| {\overline{\xi }}\big ((1,0)\gamma ({\underline{\omega }})^{-1}\big )\right| ^2 \frac{d{\underline{\omega }}}{\Vert {\underline{\omega }}\Vert ^2}\\&= \int _{\widehat{{\mathbb {R}}^2}}|\xi ({\underline{\omega }})|^2d{\underline{\omega }}= \Vert \xi \Vert _{L^2({\mathcal {O}})}^{\,\,2}. \end{aligned} \end{aligned}$$

Thus, $W_\gamma \xi \in L^2(K)$ and $\Vert W_\gamma \xi \Vert _{L^2(K)}^{\,\,2}= \Vert \xi \Vert _{L^2({\mathcal {O}})}^{\,\,2}$. Clearly, $W_\gamma$ is additive and $W_\gamma \alpha \xi = {\overline{\alpha }}W_\gamma \xi$ for all $\alpha \in {\mathbb {C}}$ and $\xi \in L^2({\mathcal {O}})$. Moreover, a calculation similar to that above shows that $W_\gamma$ maps $L^2({\mathcal {O}})$ onto $L^2(K)$ and $W_\gamma ^{-1}$ is given by $W_\gamma ^{-1}f({\underline{\omega }})= \Vert {\underline{\omega }}\Vert ^{-1}{\overline{f}}[{\underline{0}},\gamma ({\underline{\omega }})]$, for a.e. ${\underline{\omega }}\in {\mathcal {O}}$ and $f\in L^2(K)$. $\square$

For each $i\in I$, $U_{\eta _i}:L^2({{{\textrm{GL}}}}_2({\mathbb {R}}))\rightarrow L^2(G_2)$ is given by

$$\begin{aligned} U_{\eta _i}f[{\underline{y}},B]=\int _{\widehat{{\mathbb {R}}^2}} \eta _i({\underline{\omega }})f\big (B\gamma ({\underline{\omega }})\big ) e^{-2\pi i{\underline{\omega }}B^{-1}{\underline{y}}}\,d{\underline{\omega }}, \end{aligned}$$

for $[{\underline{y}},B]\in G_2$ and $f\in L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$. By Theorem 4.3, each $U_{\eta _i} L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$ is a $\lambda _{G_2}$-invariant closed subspace of $L^2(G_2)$ and $U_{\eta _i}$ intertwines $\pi ^{(1,0)}$ with the restriction of $\lambda _{G_2}$ to $U_{\eta _i} L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$ and $L^2(G_2)=\sum _{i\in I}^{\oplus } U_{\eta _i}L^2\big ({{{\textrm{GL}}}}_2({\mathbb {R}})\big )$.

For each $(i,j)\in I\times J$, form $E_{i,j} \in L^2\big (K,L^2({\mathbb {R}}^*)\big )$ by

$$\begin{aligned} E_{i,j}[{\underline{0}},L](\nu )= \textstyle \frac{\overline{\eta _i}((1,0)L^{-1})}{|\det (L)|}\,g_j(\nu ), \end{aligned}$$

(27)

for $\nu \in {\mathbb {R}}^*$ and $[{\underline{0}},L]\in K$. The orthogonal decompositions just recalled imply the following theorem.

Theorem 6.7

Let $\sigma$ be the representation of $G_2$ on the Hilbert space $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ given in (23). Let $\{\eta _i:i\in I\}$ be an orthonormal basis of $L^2({\mathcal {O}})$ and let $\{g_j:j\in J\}$ be an orthonormal basis of $L^2({\mathcal {O}}_1)$ consisting of functions in $L^2({\mathbb {R}}^*)$. For each $(i,j) \in I\times J$, form $E_{i,j} \in L^2\big (K,L^2({\mathbb {R}}^*)\big )$ as in (27) and let ${\mathcal {M}}_{i,j}= V_{E_{i,j}}L^2\big (K,L^2({\mathbb {R}}^*)\big )$. Then each ${\mathcal {M}}_{i,j}$ is a closed $\lambda _{G_2}$-invariant subspace of $L^2(G_2)$ and $V_{E_{i,j}}$ is an isometry that intertwines $\sigma$ with the restriction of $\lambda _{G_2}$ to ${\mathcal {M}}_{i,j}$. Moreover $L^2(G_2)=\sum _{(i,j)\in I\times J}^{\oplus }{\mathcal {M}}_{i,j}.$

7 The Plancherel Formula for $G_2$

For certain classes of locally compact groups, an abstract Plancherel Theorem has been established. For example, Theorem 18.8.2 of [3] implies that, if G is a Type I, separable, unimodular group, then there exists a unique measure $\mu _{{\widehat{G}}}$ on ${\widehat{G}}$ such that, for any $f\in L^1(G)\cap L^2(G)$, $\pi (f)$ is a Hilbert–Schmidt operator on ${\mathcal {H}}_\pi$, for $\mu _{{\widehat{G}}}$-a.e. $\pi \in {\widehat{G}}$, and

$$\begin{aligned} \Vert f\Vert _{L^2(G)}^{\,\,\,2}= \int _{{\widehat{G}}}\Vert \pi (f)\Vert _{{{\textrm{HS}}}}^{\,\,2} \,d\mu _{{\widehat{G}}}(\pi ). \end{aligned}$$

(28)

There are generalizations of (28) to appropriate classes of nonunimodular groups in [14] and [4] (see [7] for an organized treatment and some extensions). The group $G_2$ satisfies the hypotheses for the results in both [14] and [4]. Using Theorem 6.7, the Plancherel formula for $G_2$ is quite concrete. The method below is a special case of Theorem 2.34 in [7].

We fix $\{\eta _i:i\in I\}$, an orthonormal basis of $L^2({\mathcal {O}})$, and $\{g_j:j\in J\}$, an orthonormal basis of $L^2({\mathcal {O}}_1)$ consisting of functions in $L^2({\mathbb {R}}^*)$ as in Theorem 6.7. By Lemma 6.3, $\{w_2g_j:j\in J\}$ is an orthonormal basis of $L^2({\mathbb {R}}^*)$. Likewise, by Lemma 6.6, $\{W_\gamma \eta _i:i \in I\}$ is an orthonormal basis of $L^2(K)$. Since we can view $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ as $L^2(K\times {\mathbb {R}}^*)$, where $K\times {\mathbb {R}}^*$ is given the product of the Haar measures, and $L^2(K\times {\mathbb {R}}^*)$ can be identified with $L^2(K)\otimes L^2({\mathbb {R}}^*)$ (see Example 2.6.11 of [12]), we can construct an orthonormal basis of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ from the orthonormal basis $\{(W_\gamma \eta _i)\otimes (w_2g_j):(i,j)\in I\times J\}$ of $L^2(K)\otimes L^2({\mathbb {R}}^*)$. For each $(i,j)\in I\times J$, define $F_{i,j}\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$ by $\big (F_{i,j}[{\underline{0}},L]\big )(\nu )= \big (W_\gamma \eta _i[{\underline{0}},L]\big )\big ((w_2g_j)(\nu )\big )$, for a.e. $\nu \in {\mathbb {R}}^*$ and $[{\underline{0}},L]\in K$. That is

$$\begin{aligned} \big (F_{i,j}[{\underline{0}},L]\big )(\nu )= \frac{\overline{\eta _i}\big ((1,0)L^{-1}\big )}{|\det (L)|^{-1/2}}|\nu |^{1/2}g_j(\nu ), \end{aligned}$$

for a.e. $[{\underline{0}},L]$ and $\nu$. Then $\{F_{i,j}:(i,j)\in I\times J\}$ is an orthonormal basis of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$. Note the close relationship with the $E_{i,j}$ as defined in (27). Define the positive unbounded operator T on $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ by, for $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$,

$$\begin{aligned} \big ((TF)[{\underline{0}},L]\big )(\nu )=|\det (L)\nu |^{-1/2} \big (F[{\underline{0}},L]\big )(\nu ), \end{aligned}$$

for a.e. $[{\underline{0}},L]$ and $\nu$. Then $F_{i,j}$ is in the domain of T and $E_{i,j}=TF_{i,j}$, for all $(i,j)\in I\times J$.

Since $\{F_{i,j}:(i,j)\in I\times J\}$ is an orthonormal basis of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$, Theorem 6.7 implies that $\left\{ V_{E_{i,j}}F_{i',j'}:(i,j),(i',j')\in I\times J \right\}$ is an orthonormal basis of $L^2(G_2)$. Therefore, for any $f\in L^2(G_2)$,

$$\begin{aligned} \Vert f\Vert _{L^2(G_2)}^{\,\,2}=\sum _{(i,j)\in I\times J}\,\, \sum _{(i',j')\in I\times J}\left| \langle f, V_{E_{i,j}}F_{i',j'}\rangle _{L^2(G_2)}\right| ^2. \end{aligned}$$

(29)

But, if $f\in L^1(G_2)\cap L^2(G_2)$, then we can further analyze each of the inner products in the sum in (29). For $(i,j),(i',j')\in I\times J$,

$$\begin{aligned} \langle f, V_{E_{i,j}}F_{i',j'}\rangle _{L^2(G_2)}&= \int _{G_2}f[{\underline{x}},A]\overline{\langle F_{i',j'}, \sigma [{\underline{x}},A]E_{i,j}\rangle _{L^2(K,L^2({\mathbb {R}}^*))}} \,d\mu _{G_2}[{\underline{x}},A]\\&= \int _{G_2}f[{\underline{x}},A]\langle \sigma [{\underline{x}},A]E_{i,j},F_{i',j'}\rangle _{L^2(K,L^2({\mathbb {R}}^*))} \,d\mu _{G_2}[{\underline{x}},A]\\&= \langle \sigma (f)E_{i,j},F_{i',j'}\rangle _{L^2(K,L^2({\mathbb {R}}^*))}= \langle \sigma (f)TF_{i,j},F_{i',j'}\rangle _{L^2(K,L^2({\mathbb {R}}^*))}. \end{aligned}$$

So, for $(i,j)\in I\times J$ fixed,

$$\begin{aligned} \sum _{(i',j')\in I\times J}\left| \langle f, V_{E_{i,j}}F_{i',j'}\rangle _{L^2(G_2)}\right| ^2= \sum _{(i',j')\in I\times J}\left| \sigma (f)TF_{i,j},F_{i',j'}\rangle _{L^2(K,L^2({\mathbb {R}}^*))} \right| ^2. \end{aligned}$$

Since $\{F_{i',j'}:(i',j')\in I\times J\}$ is an orthonormal basis of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$,

$$\begin{aligned} \sum _{(i',j')\in I\times J}\left| \langle f, V_{E_{i,j}}F_{i',j'}\rangle _{L^2(G_2)}\right| ^2= \Vert \sigma (f)TF_{i,j}\Vert _{L^2(K,L^2({\mathbb {R}}^*))}^{\,\,2}. \end{aligned}$$

(30)

Inserting (30) into (29) gives $\Vert f\Vert _{L^2(G_2)}^{\,\,2}= \sum _{(i,j)\in I\times J} \Vert \sigma (f)TF_{i,j}\Vert _{L^2(K,L^2({\mathbb {R}}^*))}^{\,\,2}= \Vert \sigma (f)T\Vert _{{{\textrm{HS}}}}^{\,2}$, when $f\in L^1(G_2)\cap L^2(G_2)$. This establishes a concrete version of the abstract Plancherel Theorem for the non-unimodular group $G_2$ and serves as an example of the considerably more general Theorem 2.34 of [7].

Proposition 7.1

Define the positive operator T on $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ by,

$$\begin{aligned} \big ((TF)[{\underline{0}},L]\big )(\nu )=|\det (L)\nu |^{-1/2} \big (F[{\underline{0}},L]\big )(\nu ) \text { for. a.e. } ([{\underline{0}},L],\nu )\in K\times {\mathbb {R}}^*, \end{aligned}$$

for $F\in L^2\big (K,L^2({\mathbb {R}}^*)\big )$. Let $\sigma$ be the irreducible representation of $G_2$ given by (23). If $f\in L^1(G_2)\cap L^2(G_2)$, then $\sigma (f)T$ extends to a Hilbert-Schmidt operator on $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ and $\Vert f\Vert _{L^2(G_2)}= \Vert \sigma (f)T\Vert _{{{\textrm{HS}}}}$.

8 Concluding Remarks

The bases $\{\eta _i:i\in I\}$ of $L^2({\mathcal {O}})$ and $\{g_j:j\in J\}$ of $L^2({\mathcal {O}}_1)$ can be selected from functions in $C_c({\mathcal {O}})$ and $C_c({\mathcal {O}}_1)$, respectively, and as smooth as one may need. The subspaces $\{{\mathcal {M}}_{i,j}:(i,j)\in I\times J\}$ are then constructed from these bases. The decomposition in Theorem 6.7 should be useful for analysis on $G_2$ in a manner similar to how the Peter–Weyl Theorem plays a role in analysis on a compact group.

The representation $\sigma$ can be presented in several equivalent forms using natural isomorphisms of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ with related function spaces. For example, $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ is identified with $L^2\big (K_0,L^2({\mathbb {R}}^*)\big )$ with a simple change of notation. Also, $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ can be viewed as $L^2(K_0\times {\mathbb {R}}^*)$ by mapping $F\rightarrow {\tilde{F}}$, where ${\tilde{F}}(L,\nu )= \big (F[{\underline{0}},L]\big )(\nu )$, for a.e. $(L,\nu )\in K_0\times {\mathbb {R}}^*$. The expressions for $\sigma$ can be easily modified. A more substantial change of Hilbert space is carried out in [17], a companion paper to this one. There, we define a unitary map U of $L^2\big (K,L^2({\mathbb {R}}^*)\big )$ onto $L^2\big (\widehat{{\mathbb {R}}^2}\times {\widehat{{\mathbb {R}}}}\big )$ and let $\sigma _2[{\underline{x}},A]=U\sigma [{\underline{x}},A]U^{-1}$, for all $[{\underline{x}},A]\in G_2$. We apply the conclusions of the Duflo–Moore Theorem to $\sigma _2$ and obtain an analog of the continuous wavelet transform for functions of three variables, with one of the variables treated in a manner distinct from the other two.