Metamorphism as a covariant transform
for the SSR group

Let $G$ be a group and $H$ be a subgroup of $G$. The space $X=G/H$ of the left cosets $gH$ of the subgroup $H$ is given by the equivalence relation: $g\sim g^{\prime}$ if there exists $h\in H$ such that $g=g^{\prime}h$. We define the natural projection $\mathbf{p}:G\rightarrow X$ such that $\mathbf{p}(g)=gH$.

Let us fix a section $\mathbf{s}:X\rightarrow G$ such that $\mathbf{p}\circ\mathbf{s}=I$, where $I$ is the identity map on $X$. An associated map $\mathbf{r}:G\rightarrow H$ by

provides the unique decomposition of the form [Kirillov76]*§13.2

Thus, $X$ is a left homogeneous space with the $G$ action as follows:

where $*$ is the multiplication of $G$ and $\cdot$ is the action of $G$ on $X$ from the left.

Suppose $\chi:H\rightarrow\mathbb{T}{}$ be a character of the subgroup $H$. Let $\mathsf{L}_{2}^{\chi}{}(G)$ be a Hilbert space of functions on $G$ with a $G$-invariant inner product and the $H$-covariance property [Kisil17a],

The space $\mathsf{L}_{2}^{\chi}{}(G)$ is invariant under the left regular representation by $G$-shifts

The restriction of $\Lambda$ to the space $\mathsf{L}_{2}^{\chi}{}(G)$ is called the induced representation from the character $\chi$.

An equivalent form of the induced representation can be constructed as follows [Kirillov76, Kisil17a]. We define a lifting $\mathcal{L}^{\chi}:\mathsf{L}_{2}{}(X)\rightarrow\mathsf{L}_{2}^{\chi}{}(G)$ as the map

The pulling $\mathcal{P}:\mathsf{L}_{2}{}^{\chi}(G)\rightarrow\mathsf{L}_{2}{}(X)$ given by

Clearly $\mathcal{P}\circ\mathcal{L}^{\chi}=I$ on $\mathsf{L}_{2}{}(X)$. From (10), (11), the induced representation $\rho:\mathsf{L}_{2}{}(X)\rightarrow\mathsf{L}_{2}{}(X)$ is defined by the formula:

where $\Lambda(g)$ is the left regular representation (9). The representation $\rho_{\chi}$ explicitly is

where $g\in G$ and $x\in X$ and $g^{-1}\cdot x$ is defined by (7). For a $G$-invariant measure $\mu$ on $X$ the representation (12) is unitary on the space $\mathsf{L}_{2}{}(X,\mu)$

In this subsection $G$ is a Lie group with the corresponding Lie algebra $\mathfrak{g}$. Let $\rho$ be a representation of $G$ in a Hilbert space $\mathcal{H}$, the derived representation of $X\in\mathfrak{g}$ denoted as $\mathrm{d}\rho^{X}$ is given by

where the vector $\phi\in\mathcal{H}$ is such that the vector-function $g\rightarrow\rho(g)\phi$ is infinitely-differentiable for any $g\in G$. These vectors are called smooth and constitute a linear subspace, denoted $\mathcal{D}^{\infty}$, of $\mathcal{H}$ which is dense in $\mathcal{H}$. It is easy to show that $\mathcal{D}^{\infty}$ is invariant under $\rho(g)$ [Lang85]*§6.1. If $\mathcal{H}$ is $\mathsf{L}_{2}{}(\mathbb{R}^{n}{})$ then the space $D^{\infty}$ contains the Schwartz space, which is a dense subspace of $\mathsf{L}_{2}{}(\mathbb{R}^{n}{})$.

Also, we define the Lie derivative $\mathcal{L}^{X}$ for $X\in\mathfrak{g}$ as the derived right regular representation [Lang85]*§6.1, that is

for any differentiable function $F$ on $G$.

The left and right invariant Haar measures of the group $\mathbb{G}{}$ are given by

Thus, the group $\mathbb{G}{}$ is non-unimodular with the modular function $\Delta(s,x,y,b,r)=\textstyle\frac{1}{r^{2}}$.

We extend the action (9) of $\mathbb{G}{}$ on itself by left shifts to the left regular unitary representation on the linear space of functions $\mathsf{L}_{2}{}(\mathbb{G}{},d_{l})$:

where $(s,x,y,b,r)$, $(s^{\prime},x^{\prime},y^{\prime},b^{\prime},r^{\prime})\in\mathbb{G}{}$.

This representation is reducible, i.e. there are $\Lambda$-invariant proper subspaces in $\mathsf{L}_{2}{}(\mathbb{G}{},d_{l})$. In particular, many types of induced representations of $\mathbb{G}{}$ are realised as restrictions of the left regular representations (15) to some subspaces with a covariance property (8). We describe here two of them—called the quasi-regular type representation and the Schrödinger type representation—together with equivalent forms on the respective homogeneous spaces.

Let

be the centre of the group $\mathbb{G}{}$. The space of left cosets $X=\mathbb{G}{}/Z$ can be parametrised by

Consider the natural projection and the section maps

We calculate the respective map $\mathbf{r}$ (6) as follows

Let $\chi_{\hbar}:Z\rightarrow\mathbb{T}{}$ be an unitary character of $Z$:

defined by a parameter $\hbar\in\mathbb{R}{}$. In quantum mechanical framework $\hbar$ is naturally associated to the Planck constant [Folland89, Kisil02e, Kisil09e, Kisil17a]. The corresponding induced representation $\tilde{\rho}:\mathsf{L}_{2}{}(\mathbb{R}^{4}_{+}{})\rightarrow\mathsf{L}_{2}{}(\mathbb{R}^{4}_{+}{})$ is [Kisil21b]

It is called the quasi-regular type representation on $\mathsf{L}_{2}{}(\mathbb{R}^{4}_{+}{})$. One can check that $\tilde{\rho}$ is unitary and we will discuss its reducibility below.

Let

be a subgroup of $\mathbb{G}{}$, which is a semidirect product of a maximal abelian subgroup of $\mathbb{H}{}$ and the affine group $\mathbb{A}{}$. The space of the left cosets $\mathbb{G}{}/H_{1}$ is parameterized by $\mathbb{R}{}$. We define the natural projection $\mathbf{p}:\mathbb{G}{}\rightarrow\mathbb{R}{}$ and a section map $\mathbf{s}:\mathbb{R}{}\rightarrow\mathbb{G}{}$ by

The respective map $\mathbf{r}$ (6) is

Let ${\chi}_{\hbar\lambda}:H_{1}\rightarrow\mathbb{T}{}$ be a character $H_{1}$

where $\hbar\in\mathbb{R}{}$, $\lambda\in i{\mathbb{R}{}}$. For simplicity, we will consider here the case of $\lambda=0$ only. The induced representation on $\mathsf{L}_{2}{}(\mathbb{R}{})$ is [Kisil21b]

In view of the mentioned connections, we call representation (18) as Schrödinger type representation. It is irreducible since its restriction to the Heisenberg group coincides with the irreducible Schrödinger representation [Folland89, Kisil17a].

The derived representation (13) of the Schrödinger type representation (18) is

It is easy to check that the above sets of operators (19) represents commutators (5) of the Lie algebra $\mathfrak{g}$ of the group $\mathbb{G}{}$.

Let $G$ be a group and let $\rho$ be a unitary irreducible representation of the group $G$ in a Hilbert space $\mathcal{H}$. For a fixed unit vector $\phi\in\mathcal{H}$, called here a fiducial vector (aka vacuum vector, ground state, mother wavelet, etc.), the covariant transform $\mathcal{W}_{\phi}:\mathcal{H}\rightarrow\mathsf{L}{}(G)$ is \citelist[AliAntGaz14a]*§8.1 [Berezin86] [Perelomov86]

Here $\mathsf{L}{}(G)$ is a certain linear space of functions on $G$ usually linked to some additional conditions. The common focus is on $\mathsf{L}{}(G)=\mathsf{L}_{2}{}(G,\mathrm{d}\mu)$—the square integrable functions with respect to a Haar measure $\mathrm{d}\mu$, for this we need a square-integrable representation $\rho$ and an admissible fiducial vector $\phi$ [AliAntGaz14a, Ch. 8]. However, many topics in analysis prompt to study other situations, e.g. related to Hardy-type invariant functionals [Kisil12d, Kisil13a], Gelfand triples [FeichtingerGrochenig88a], Banach spaces [Kisil98a], etc.

The main property of (20) is that ${\mathcal{W}_{\phi}}$ intertwines the representation $\rho$ on $\mathcal{H}$ and the left regular action $\Lambda$ (9) on $G$:

A representation $\rho$ is square-integrable if for some $\phi\in\mathcal{H}$, the map $\mathcal{W}_{\phi}:\mathcal{H}\rightarrow\mathsf{L}_{2}{}(G,dg)$ is unitary for a left Haar measure $dg$ on $G$. Some representations are not square-integrable, but can still be treated by the following modification of covariant transform by Perelomov [Perelomov86]. Let $H$ be a closed subgroup of the group $G$ and the corresponding homogeneous space is $X=G/H$. Let for some character $\chi$ of $H$ a fiducial vector $\phi\in\mathcal{H}$ is a joint eigenvector

Then, the respective covariant transform satisfies the covariant property, cf. (8):

Thus, the image space of $\mathcal{W}_{\phi}$ belongs to the induced representation by the character $\chi$ of the subgroup $H$. This prompts to adopt the covariant transform to the space of function on the homogeneous space $X=G/H$. To this end, let us fix a section $\mathbf{s}:X\rightarrow G$ and a fiducial vector $\phi\in\mathcal{H}$ satisfying (22). The induced covariant transform from the Hilbert space $\mathcal{H}$ to a space of functions $\mathsf{L}_{\phi}{}(X)$ is

Then, the induced covariant transform intertwines $\rho$ and $\tilde{\rho}$—an induced representation from the character $\chi$ of the subgroup $H$, cf. (21):

In particular, the image space $\mathsf{L}_{\phi}{}(G/H)$ of the induced covariant transform is invariant under $\tilde{\rho}$. Induced covariant transforms for the Heisenberg group [Kisil09e] and the affine group [Kisil12d] are the most familiar examples.

On the same way as above, we can calculate the induced covariant transform of $\mathbb{G}{}$. Consider the subgroup $Z$ of $\mathbb{G}{}$, which is $Z=\{(s,0,0,0,1),s\in\mathbb{R}{}\}$. For the Schrödinger type representation (18), any function $\phi\in\mathsf{L}_{2}{}(\mathbb{R}{})$ satisfies the eigenvector condition $\rho(s,0,0,0,1)\phi=\mathrm{e}^{2\pi\mathrm{i}\hbar s}\phi$ with the character $\chi(s,0,0,0,1)=\mathrm{e}^{-2\pi\mathrm{i}\hbar s}$, cf. (22). Thus, the respective homogeneous space is $\mathbb{G}{}/Z\simeq\mathbb{R}^{4}_{+}{}$ and we take the above section $\mathbf{s}:\mathbb{G}{}/Z\rightarrow\mathbb{G}{}:\ \mathbf{s}(x,y,b,r)=(0,x,y,b,r)$ (16). Then, the induced covariant transform is

From (23), $\mathcal{W}_{\phi}$ intertwines the Schrödinger type representation (18) with quasi-regular (17).

The last integral in (24) is a composition of five unitary operators $\mathsf{L}_{2}{}(\mathbb{R}^{2}{})\rightarrow\mathsf{L}_{2}{}(\mathbb{R}^{2}{})$ applied to a function $F(y,u)=f(y)\overline{\phi}(u)$ in the space $\mathsf{L}_{2}{}(\mathbb{R}{})\otimes\mathsf{L}_{2}{}(\mathbb{R}{})\simeq\mathsf{L}_{2}{}(\mathbb{R}^{2}{})$:

1. (1)

   The unitary operator $R:\mathsf{L}_{2}{}(\mathbb{R}^{2}{})\rightarrow\mathsf{L}_{2}{}(\mathbb{R}^{2}{})$ based on the dilation

   $$R:F(y,u)\rightarrow\sqrt{r}\,F(y,ru),\qquad\text{ where }r>0.$$

2. (2)

   The change of variables $T:\mathsf{L}_{2}{}(\mathbb{R}^{2}{})\rightarrow\mathsf{L}_{2}{}(\mathbb{R}^{2}{})$

   $$T:F(y,u)\rightarrow F(u,u-y).$$

3. (3)

   The operator of multiplication by an unimodular function $\psi_{b}(x,y)=\mathrm{e}^{\pi\mathrm{i}\hbar b(u-y)^{2}}$,

   $$M_{b}:F(y,u)\rightarrow\mathrm{e}^{\pi\mathrm{i}\hbar b(u-y)^{2}}\,F(y,u),\qquad\text{ where }b\in\mathbb{R}{}.$$

4. (4)

   The partial Fourier transform $u\rightarrow x$ in the second variable

   $$[\mathcal{F}_{2}F](y,x)=\int_{\mathbb{R}{}}{F(y,u)\,\mathrm{e}^{-2\pi\mathrm{i}\hbar xu}\,\mathrm{d}u}.$$

5. (5)

   The multiplication $M$ by the unimodular function $\mathrm{e}^{2\pi\mathrm{i}\hbar xy}$.

Thus, we can write $\mathcal{W}_{\phi}$ as

and obtain

Also, the induced covariant transform preserves the Schwartz space, that is, if $f,\phi\in\mathcal{S}(\mathbb{R}{})$ then $\mathcal{W}_{\phi}f(\cdot,\cdot,b_{0},r_{0})\in\mathcal{S}(\mathbb{R}^{2}{})$. This is because the $\mathcal{S}(\mathbb{R}^{2}{})$ is invariant under the all five above components of $\mathcal{W}_{\phi}f$ in (25).

Note, that the induced covariant transform (24) does not define a square-integrable function on $\mathbb{G}{}/Z\sim\mathbb{R}^{4}_{+}{}$. To discuss unitarity we need to introduce a suitable inner product. In general we can start from a probability measure $\mu$ on $\mathbb{R}^{2}_{+}{}$, that is $\int_{\mathbb{R}^{2}_{+}{}}\,\mathrm{d}\mu=1$. Then we define the inner product

for $f,g\in\mathsf{L}_{\phi}{}(\mathbb{R}^{4}_{+}{})$. The factor $\hbar$ in the measure $\textstyle\frac{\hbar\,\mathrm{d}x\,\mathrm{d}y}{\sqrt{2r_{0}}}$ makes it dimensionless, see discussion of this in [Kisil02e, AlmalkiKisil18a]. Important particular cases of probability measures parametrised by $(b_{0},r_{0})\in\mathbb{R}^{2}_{+}{}$ are

where $\delta(t)$ is the Dirac delta. The respective inner products becomes:

From now on we consider $\mathsf{L}_{\phi}{}(\mathbb{R}^{4}_{+}{})$ as a Hilbert space with the inner product (26) or specifically (28). The respective norms are denoted by $\|\cdot\|_{\mu}$ and $\|\cdot\|_{(b_{0},r_{0})}$.

Using the above inner product, we can derive from Proposition 4.1 the following orthogonality relation: