# Erlangen Programme at Large: An Overview

Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds, LS2 9JT,
UK<EMAIL_ADDRESS>http://www.maths.leeds.ac.uk/~kisilv/ Dedicated
to Prof. Hans G. Feichtinger on the occasion of his 60th birthday  School of
Mathematics, University of Leeds, Leeds LS2 9JT, England
E-mail<EMAIL_ADDRESS>
###### Abstract.

This is an overview of _Erlangen Programme at Large_. Study of objects and
properties, which are invariant under a group action, is very fruitful far
beyond the traditional geometry. In this paper we demonstrate this on the
example of the group $SL_{2}{}(\mathbb{R}{})$. Starting from the conformal
geometry we develop analytic functions and apply these to functional calculus.
Finally we link this to quantum mechanics and conclude by a list of open
problems.

###### Key words and phrases:

Special linear group, Hardy space, Clifford algebra, elliptic, parabolic,
hyperbolic, complex numbers, dual numbers, double numbers, split-complex
numbers, Cauchy-Riemann-Dirac operator, Möbius transformations, functional
calculus, spectrum, quantum mechanics, non-commutative geometry.

###### 2000 Mathematics Subject Classification:

Primary 30G35; Secondary 22E46, 30F45, 32F45, 43A85, 30G30, 42C40, 46H30,
47A13, 81R30, 81R60.

On leave from the Odessa University.

††copyright: ©:

###### Contents

  1. Erlangen Programme at Large: An Overview
    1. 1 Introduction
      1. 1.1 Make a Guess in Three Attempts
      2. 1.2 Erlangen Programme at Large
    2. 2 Geometry
      1. 2.1 Hypercomplex Numbers
      2. 2.2 Cycles as Invariant Families
      3. 2.3 Invariants: Algebraic and Geometric
      4. 2.4 Joint Invariants: Orthogonality
      5. 2.5 Higher Order Joint Invariants: f-Orthogonality
      6. 2.6 Distance, Length and Perpendicularity
    3. 3 Linear Representations
      1. 3.1 Hypercomplex Characters
      2. 3.2 Induced Representations
      3. 3.3 Similarity and Correspondence: Ladder Operators
        1. 3.3.1 Elliptic Ladder Operators
        2. 3.3.2 Hyperbolic Ladder Operators
        3. 3.3.3 Parabolic Ladder Operators
    4. 4 Covariant Transform
      1. 4.1 Extending Wavelet Transform
      2. 4.2 Examples of Covariant Transform
      3. 4.3 Symbolic Calculi
      4. 4.4 Inverse Covariant Transform
    5. 5 Analytic Functions
      1. 5.1 Induced Covariant Transform
      2. 5.2 Induced Wavelet Transform and Cauchy Integral
      3. 5.3 The Cauchy-Riemann (Dirac) and Laplace Operators
      4. 5.4 The Taylor Expansion
      5. 5.5 Wavelet Transform in the Unit Disk and Other Domains
    6. 6 Covariant and Contravariant Calculi
      1. 6.1 Intertwining Group Actions on Functions and Operators
      2. 6.2 Jet Bundles and Prolongations
      3. 6.3 Spectrum and Spectral Mapping Theorem
      4. 6.4 Functional Model and Spectral Distance
      5. 6.5 Covariant Pencils of Operators
    7. 7 Quantum Mechanics
      1. 7.1 The Heisenberg Group and Its Automorphisms
        1. 7.1.1 The Heisenberg group and induced representations
        2. 7.1.2 Symplectic Automorphisms of the Heisenberg Group
      2. 7.2 p-Mechanic Formalism
        1. 7.2.1 Convolutions (Observables) on $\mathbb{H}^{n}{}$ and Commutator
        2. 7.2.2 States and Probability
      3. 7.3 Elliptic characters and Quantum Dynamics
        1. 7.3.1 Fock–Segal–Bargmann and Schrödinger Representations
        2. 7.3.2 Commutator and the Heisenberg Equation
        3. 7.3.3 Quantum Probabilities
      4. 7.4 Ladder Operators and Harmonic Oscillator
        1. 7.4.1 Ladder Operators from the Heisenberg Group
        2. 7.4.2 Symplectic Ladder Operators
      5. 7.5 Hyperbolic Quantum Mechanics
        1. 7.5.1 Hyperbolic Representations of the Heisenberg Group
        2. 7.5.2 Hyperbolic Dynamics
        3. 7.5.3 Hyperbolic Probabilities
        4. 7.5.4 Ladder Operators for the Hyperbolic Subgroup
        5. 7.5.5 Double Ladder Operators
      6. 7.6 Parabolic (Classical) Representations on the Phase Space
        1. 7.6.1 Classical Non-Commutative Representations
        2. 7.6.2 Hamilton Equation
        3. 7.6.3 Classical probabilities
        4. 7.6.4 Ladder Operator for the Nilpotent Subgroup
        5. 7.6.5 Similarity and Correspondence
    8. 8 Open Problems
      1. 8.1 Geometry
      2. 8.2 Analytic Functions
      3. 8.3 Functional Calculus
      4. 8.4 Quantum Mechanics

##  Chapter Erlangen Programme at Large:
An Overview

Vladimir V. Kisil

_Dedicated to Prof. Hans G. Feichtinger on the occasion of his 60th birthday_

Abstract. This is an overview of _Erlangen Programme at Large_. Study of
objects and properties, which are invariant under a group action, is very
fruitful far beyond the traditional geometry. In this paper we demonstrate
this on the example of the group $SL_{2}{}(\mathbb{R}{})$. Starting from the
conformal geometry we develop analytic functions and apply these to functional
calculus. Finally we link this to quantum mechanics and conclude by a list of
open problems.

2010 Mathematics Subject Classification: Primary 30G35; Secondary 22E46,
30F45, 32F45, 43A85, 30G30, 42C40, 46H30, 47A13, 81R30, 81R60.

Key words and phrases: Special linear group, Hardy space, Clifford algebra,
elliptic, parabolic, hyperbolic, complex numbers, dual numbers, double
numbers, split-complex numbers, Cauchy-Riemann-Dirac operator, Möbius
transformations, functional calculus, spectrum, quantum mechanics, non-
commutative geometry.

A mathematical idea should not be petrified in a formalised axiomatic
settings, but should be considered instead as flowing as a river.

Sylvester (1878)

###  1\. Introduction

The simplest objects with non-commutative (but still associative)
multiplication may be $2\times 2$ matrices with real entries. The subset of
matrices _of determinant one_ has the following properties:

  * •

form a closed set under multiplication since $\det(AB)=\det A\cdot\det B$;

  * •

the identity matrix is the set; and

  * •

any such matrix has an inverse (since $\det A\neq 0$).

In other words those matrices form a _group_ , the $SL_{2}{}(\mathbb{R}{})$
group [Lang85]—one of the two most important Lie groups in analysis. The other
group is the Heisenberg group [Howe80a]. By contrast the $ax+b$ group, which
is often used to build wavelets, is only a subgroup of
$SL_{2}{}(\mathbb{R}{})$, see the numerator in (1.1).

The simplest non-linear transforms of the real line— _linear-fractional_ or
_Möbius maps_ —may also be associated with $2\times 2$ matrices
[Beardon05a]*Ch. 13:

(1.1) $g:x\mapsto g\cdot x=\frac{ax+b}{cx+d},\quad\text{ where
}g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix},\ x\in\mathbb{R}{}.$

An enjoyable calculation shows that the composition of two transforms (1.1)
with different matrices $g_{1}$ and $g_{2}$ is again a Möbius transform with
matrix the product $g_{1}g_{2}$. In other words (1.1) it is a (left) action of
$SL_{2}{}(\mathbb{R}{})$.

According to F. Klein’s _Erlangen programme_ (which was influenced by S. Lie)
any geometry is dealing with invariant properties under a certain transitive
group action. For example, we may ask: _What kinds of geometry are related to
the $SL_{2}{}(\mathbb{R}{})$ action (1.1)_?

The Erlangen programme has probably the highest rate of
$\frac{\text{praised}}{\text{actually used}}$ among mathematical theories not
only due to the big numerator but also due to undeserving small denominator.
As we shall see below Klein’s approach provides some surprising conclusions
even for such over-studied objects as circles.

####  1.1. Make a Guess in Three Attempts

It is easy to see that the $SL_{2}{}(\mathbb{R}{})$ action (1.1) makes sense
also as a map of complex numbers $z=x+\mathrm{i}y$, $\mathrm{i}^{2}=-1$
assuming the denominator is non-zero. Moreover, if $y>0$ then $g\cdot z$ has a
positive imaginary part as well, i.e. (1.1) defines a map from the upper half-
plane to itself. Those transformations are isometries of the Lobachevsky half-
plane.

However there is no need to be restricted to the traditional route of complex
numbers only. Moreover in Subsection 2.1 we will naturally come to a necessity
to work with all three kinds of hypercomplex numbers. Less-known _double_ and
_dual_ numbers, see [Yaglom79]*Suppl. C, have also the form $z=x+\iota y$ but
different assumptions on the hypercomplex unit $\iota$: $\iota^{2}=0$ or
$\iota^{2}=1$ correspondingly. We will write $\varepsilon$ and $\mathrm{j}$
instead of $\iota$ within dual and double numbers respectively. Although the
arithmetic of dual and double numbers is different from the complex ones, e.g.
they have divisors of zero, we are still able to define their transforms by
(1.1) in most cases.

Three possible values $-1$, $0$ and $1$ of $\sigma:=\iota^{2}$ will be
refereed to here as _elliptic_ , _parabolic_ and _hyperbolic_ cases
respectively. We repeatedly meet such a division of various mathematical
objects into three classes. They are named by the historically first
example—the classification of conic sections—however the pattern persistently
reproduces itself in many different areas: equations, quadratic forms,
metrics, manifolds, operators, etc. We will abbreviate this separation as
_EPH-classification_. The common origin of this fundamental division of any
family with one-parameter can be seen from the simple picture of a coordinate
line split by zero into negative and positive half-axes:

(1.2)

Connections between different objects admitting EPH-classification are not
limited to this common source. There are many deep results linking, for
example, the ellipticity of quadratic forms, metrics and operators, e.g. the
Atiyah-Singer index theorem. On the other hand there are still a lot of white
spots, empty cells, obscure gaps and missing connections between some subjects
as well.

To understand the action (1.1) in all EPH cases we use the Iwasawa
decomposition [Lang85]*§ III.1 of $SL_{2}{}(\mathbb{R}{})=ANK$ into three one-
dimensional subgroups $A$, $N$ and $K$:

(1.3) $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}={\begin{pmatrix}\alpha&0\\\
0&\alpha^{-1}\end{pmatrix}}{\begin{pmatrix}1&\nu\\\
0&1\end{pmatrix}}{\begin{pmatrix}\cos\phi&-\sin\phi\\\
\sin\phi&\cos\phi\end{pmatrix}}.$

Subgroups $A$ and $N$ act in (1.1) irrespectively to value of $\sigma$: $A$
makes a dilation by $\alpha^{2}$, i.e. $z\mapsto\alpha^{2}z$, and $N$ shifts
points to left by $\nu$, i.e. $z\mapsto z+\nu$.

The corresponding orbits are circles, parabolas and hyperbolas shown by thick
lines. Transverse thin lines are images of the vertical axis under the action
of the subgroup $K$. Grey arrows show the associated derived action.

Figure 1. Action of the subgroup $K$.

By contrast, the action of the third matrix from the subgroup $K$ sharply
depends on $\sigma$, see Fig. 1. In elliptic, parabolic and hyperbolic cases
$K$-orbits are circles, parabolas and (equilateral) hyperbolas
correspondingly. Thin traversal lines in Fig. 1 join points of orbits for the
same values of $\phi$ and grey arrows represent “local velocities”—vector
fields of derived representations. We will describe some highlights of this
geometry in Section 2.

####  1.2. Erlangen Programme at Large

As we already mentioned the division of mathematics into areas is only
apparent. Therefore it is unnatural to limit Erlangen programme only to
“geometry”. We may continue to look for $SL_{2}{}(\mathbb{R}{})$ invariant
objects in other related fields. For example, transform (1.1) generates
unitary representations on certain $L_{2}{}$ spaces, cf. (1.1) and Section 3:

(1.4) $g:f(x)\mapsto\frac{1}{(cx+d)^{m}}f\left(\frac{ax+b}{cx+d}\right).$

For $m=1$, $2$, … the invariant subspaces of $L_{2}{}$ are Hardy and
(weighted) Bergman spaces of complex analytic functions. All main objects of
_complex analysis_ (Cauchy and Bergman integrals, Cauchy-Riemann and Laplace
equations, Taylor series etc.) may be obtaining in terms of invariants of the
_discrete series_ representations of $SL_{2}{}(\mathbb{R}{})$ [Kisil02c]*§ 3.
Moreover two other series (_principal_ and _complimentary_ [Lang85]) play the
similar rôles for hyperbolic and parabolic cases [Kisil02c, Kisil05a]. This
will be discussed in Sections 4 and 5.

Moving further we may observe that transform (1.1) is defined also for an
element $x$ in any algebra $\mathfrak{A}$ with a unit $\mathbf{1}$ as soon as
$(cx+d\mathbf{1})\in\mathfrak{A}$ has an inverse. If $\mathfrak{A}$ is
equipped with a topology, e.g. is a Banach algebra, then we may study a
_functional calculus_ for element $x$ [Kisil02a] in this way. It is defined as
an _intertwining operator_ between the representation (1.4) in a space of
analytic functions and a similar representation in a left
$\mathfrak{A}$-module. We will consider the Section 6.

In the spirit of Erlangen programme such functional calculus is still a
geometry, since it is dealing with invariant properties under a group action.
However even for a simplest non-normal operator, e.g. a Jordan block of the
length $k$, the obtained space is not like a space of point but is rather a
space of $k$-th _jets_ [Kisil02a]. Such non-point behaviour is oftenly
attributed to _non-commutative geometry_ and Erlangen programme provides an
important input on this fashionable topic [Kisil02c].

It is noteworthy that ideas of F. Klein ans S. Lie are spread more in physics
than in mathematics: it is a common viewpoint that laws of nature shall be
invariant under certain transformations. Yet systematic use of Erlangen
approach can bring new results even in this domain as we demonstrate in
Section 7. There are still many directions to extend the present work thus we
will conclude by a list of some open problems in Section 8.

Of course, there is no reasons to limit Erlangen programme to
$SL_{2}{}(\mathbb{R}{})$ group only, other groups may be more suitable in
different situations. However $SL_{2}{}(\mathbb{R}{})$ still possesses a big
unexplored potential and is a good object to start with.

###  2\. Geometry

We start from the natural domain of the Erlangen Programme—geometry.
Systematic use of this ideology allows to obtain new results even for very
classical objects like circles.

####  2.1. Hypercomplex Numbers

Firstly we wish to demonstrate that hypercomplex numbers appear very naturally
from a study of $SL_{2}{}(\mathbb{R}{})$ action on the homogeneous spaces
[Kisil09c]. We begin from the standard definitions.

Let $H$ be a subgroup of a group $G$. Let $X=G/H$ be the corresponding
homogeneous space and $s:X\rightarrow G$ be a smooth section [Kirillov76]*§
13.2, which is a left inverse to the natural projection $p:G\rightarrow X$.
The choice of $s$ is inessential in the sense that by a smooth map
$X\rightarrow X$ we can always reduce one to another. We define a map
$r:G\rightarrow H$ associated to $p$ and $s$ from the identities:

(2.1) $r(g)={(s(x))}^{-1}g,\qquad\text{where }x=p(g)\in X.$

Note that $X$ is a left homogeneous space with the $G$-action defined in terms
of $p$ and $s$ as follows:

(2.2) $g:x\mapsto g\cdot x=p(g*s(x)),$

######  Example 2.1 ([Kisil09c]).

For $G=SL_{2}{}(\mathbb{R}{})$, as well as for other semisimple groups, it is
common to consider only the case of $H$ being the maximal compact subgroup
$K$. However in this paper we admit $H$ to be any one-dimensional subgroup.
Then $X$ is a two-dimensional manifold and for any choice of $H$ we define
[Kisil97c]*Ex. 3.7(a):

(2.3) $s:(u,v)\mapsto\frac{1}{\sqrt{v}}\begin{pmatrix}v&u\\\
0&1\end{pmatrix},\qquad(u,v)\in\mathbb{R}^{2}{},\ v>0.$

Any continuous one-dimensional subgroup $H\in SL_{2}{}(\mathbb{R}{})$ is
conjugated to one of the following:

(2.4) $\displaystyle K$ $\displaystyle=$
$\displaystyle\left\\{{\begin{pmatrix}\cos t&\sin t\\\ -\sin t&\cos
t\end{pmatrix}=\exp\begin{pmatrix}0&t\\\ -t&0\end{pmatrix}},\
t\in(-\pi,\pi]\right\\},$ (2.5) $\displaystyle N^{\prime}$ $\displaystyle=$
$\displaystyle\left\\{{\begin{pmatrix}1&0\\\
t&1\end{pmatrix}=\exp\begin{pmatrix}0&0\\\ t&0\end{pmatrix},}\
t\in\mathbb{R}{}\right\\},$ (2.6) $\displaystyle A\\!^{\prime}$
$\displaystyle=$ $\displaystyle\left\\{\begin{pmatrix}\cosh t&\sinh t\\\ \sinh
t&\cosh t\end{pmatrix}=\exp\begin{pmatrix}0&t\\\ t&0\end{pmatrix},\
t\in\mathbb{R}{}\right\\}.$

Then [Kisil09c] the action (2.2) of $SL_{2}{}(\mathbb{R}{})$ on
$X=SL_{2}{}(\mathbb{R}{})/H$ coincides with Möbius transformations (1.1) on
complex, dual and double numbers respectively.

####  2.2. Cycles as Invariant Families

We wish to consider all three hypercomplex systems at the same time, the
following definition is very helpful for this.

######  Definition 2.2.

The common name _cycle_ [Yaglom79] is used to denote circles, parabolas and
hyperbolas (as well as straight lines as their limits) in the respective EPH
case.

(a) (b)

Figure 2. $K$-orbits as conic sections: circles are sections by the plane
$EE^{\prime}$; parabolas are sections by $PP^{\prime}$; hyperbolas are
sections by $HH^{\prime}$. Points on the same generator of the cone correspond
to the same value of $\phi$.

It is well known that any cycle is a _conic sections_ and an interesting
observation is that corresponding $K$-orbits are in fact sections of the same
two-sided right-angle cone, see Fig. 2. Moreover, each straight line
generating the cone, see Fig. 2(b), is crossing corresponding EPH $K$-orbits
at points with the same value of parameter $\phi$ from (1.3). In other words,
all three types of orbits are generated by the rotations of this generator
along the cone.

$K$-orbits are $K$-invariant in a trivial way. Moreover since actions of both
$A$ and $N$ for any $\sigma$ are extremely “shape-preserving” we find natural
invariant objects of the Möbius map:

######  Theorem 2.3.

The family of all cycles from Definition 2.2 is invariant under the action
(1.1).

According to Erlangen ideology we should now study invariant properties of
cycles.

Fig. 2 suggests that we may get a unified treatment of cycles in all EPH cases
by consideration of a higher dimension spaces. The standard mathematical
method is to declare objects under investigations (cycles in our case,
functions in functional analysis, etc.) to be simply points of some bigger
space. This space should be equipped with an appropriate structure to hold
externally information which were previously inner properties of our objects.

A generic cycle is the set of points $(u,v)\in\mathbb{R}^{2}{}$ defined for
all values of $\sigma$ by the equation

(2.7) $k(u^{2}-\sigma v^{2})-2lu-2nv+m=0.$

This equation (and the corresponding cycle) is defined by a point $(k,l,n,m)$
from a _projective space_ $\mathbb{P}^{3}{}$, since for a scaling factor
$\lambda\neq 0$ the point $(\lambda k,\lambda l,\lambda n,\lambda m)$ defines
an equation equivalent to (2.7). We call $\mathbb{P}^{3}{}$ the _cycle space_
and refer to the initial $\mathbb{R}^{2}{}$ as the _point space_.

In order to get a connection with Möbius action (1.1) we arrange numbers
$(k,l,n,m)$ into the matrix

(2.8) $C_{\breve{\sigma}}^{s}=\begin{pmatrix}l+\mathrm{\breve{\i}}sn&-m\\\
k&-l+\mathrm{\breve{\i}}sn\end{pmatrix},$

with a new hypercomplex unit $\mathrm{\breve{\i}}$ and an additional parameter
$s$ usually equal to $\pm 1$. The values of
$\breve{\sigma}:=\mathrm{\breve{\i}}^{2}$ is $-1$, $0$ or $1$ independently
from the value of $\sigma$. The matrix (2.8) is the cornerstone of an extended
_Fillmore–Springer–Cnops construction_ (FSCc) [Cnops02a].

The significance of FSCc in Erlangen framework is provided by the following
result:

######  Theorem 2.4.

The image $\tilde{C}_{\breve{\sigma}}^{s}$ of a cycle $C_{\breve{\sigma}}^{s}$
under transformation (1.1) with $g\in SL_{2}{}(\mathbb{R}{})$ is given by
similarity of the matrix (2.8):

(2.9) $\tilde{C}_{\breve{\sigma}}^{s}=gC_{\breve{\sigma}}^{s}g^{-1}.$

In other words FSCc (2.8) _intertwines_ Möbius action (1.1) on cycles with
linear map (2.9).

There are several ways to prove (2.9): either by a brute force calculation
(fortunately performed by a CAS) [Kisil05a] or through the related
orthogonality of cycles [Cnops02a], see the end of the next Subsection 2.3.

The important observation here is that our extended version of FSCc (2.8) uses
an hypercomplex unit $\mathrm{\breve{\i}}$, which is not related to $\iota$
defining the appearance of cycles on plane. In other words any EPH type of
geometry in the cycle space $\mathbb{P}^{3}{}$ admits drawing of cycles in the
point space $\mathbb{R}^{2}{}$ as circles, parabolas or hyperbolas. We may
think on points of $\mathbb{P}^{3}{}$ as ideal cycles while their depictions
on $\mathbb{R}^{2}{}$ are only their shadows on the wall of Plato’s cave.

(a) (b)

Figure 3.  (a) Different EPH implementations of the same cycles defined by
quadruples of numbers.
(b) Centres and foci of two parabolas with the same focal length.

Fig. 3(a) shows the same cycles drawn in different EPH styles. Points
$c_{e,p,h}=(\frac{l}{k},-\breve{\sigma}\frac{n}{k})$ are their respective
e/p/h-centres. They are related to each other through several identities:

(2.10) $c_{e}=\bar{c}_{h},\quad c_{p}=\frac{1}{2}(c_{e}+c_{h}).$

Fig. 3(b) presents two cycles drawn as parabolas, they have the same focal
length $\frac{n}{2k}$ and thus their e-centres are on the same level. In other
words _concentric_ parabolas are obtained by a vertical shift, not scaling as
an analogy with circles or hyperbolas may suggest.

Fig. 3(b) also presents points, called e/p/h-foci:

(2.11) $f_{e,p,h}=\left(\frac{l}{k},-\frac{\det
C_{\breve{\sigma}}^{s}}{2nk}\right),$

which are independent of the sign of $s$. If a cycle is depicted as a parabola
then h-focus, p-focus, e-focus are correspondingly geometrical focus of the
parabola, its vertex, and the point on the directrix nearest to the vertex.

As we will see, cf. Theorems 2.6 and 2.8, all three centres and three foci are
useful attributes of a cycle even if it is drawn as a circle.

####  2.3. Invariants: Algebraic and Geometric

We use known algebraic invariants of matrices to build appropriate geometric
invariants of cycles. It is yet another demonstration that any division of
mathematics into subjects is only illusive.

For $2\times 2$ matrices (and thus cycles) there are only two essentially
different invariants under similarity (2.9) (and thus under Möbius action
(1.1)): the _trace_ and the _determinant_. The latter was already used in
(2.11) to define cycle’s foci. However due to projective nature of the cycle
space $\mathbb{P}^{3}{}$ the absolute values of trace or determinant are
irrelevant, unless they are zero.

Alternatively we may have a special arrangement for normalisation of
quadruples $(k,l,n,m)$. For example, if $k\neq 0$ we may normalise the
quadruple to $(1,\frac{l}{k},\frac{n}{k},\frac{m}{k})$ with highlighted
cycle’s centre. Moreover in this case $\det{C^{s}_{\breve{\sigma}}}$ is equal
to the square of cycle’s radius, cf. Section 2.6. Another normalisation
$\det{C^{s}_{\breve{\sigma}}}=1$ is used in [Kirillov06] to get a nice
condition for touching circles. Moreover, the Kirillov normalisation is
preserved by the conjugation (2.9).

We still get important characterisation even with non-normalised cycles, e.g.,
invariant classes (for different $\breve{\sigma}$) of cycles are defined by
the condition $\det C_{\breve{\sigma}}^{s}=0$. Such a class is parametrises
only by two real numbers and as such is easily attached to certain point of
$\mathbb{R}^{2}{}$. For example, the cycle $C_{\breve{\sigma}}^{s}$ with $\det
C_{\breve{\sigma}}^{s}=0$, $\breve{\sigma}=-1$ drawn elliptically represent
just a point $(\frac{l}{k},\frac{n}{k})$, i.e. (elliptic) zero-radius circle.
The same condition with $\breve{\sigma}=1$ in hyperbolic drawing produces a
null-cone originated at point $(\frac{l}{k},\frac{n}{k})$:

$\textstyle(u-\frac{l}{k})^{2}-(v-\frac{n}{k})^{2}=0,$

i.e. a zero-radius cycle in hyperbolic metric.

Figure 4. Different $\sigma$-implementations of the same
$\breve{\sigma}$-zero-radius cycles and corresponding foci.

In general for every notion there are (at least) nine possibilities: three EPH
cases in the cycle space times three EPH realisations in the point space. Such
nine cases for “zero radius” cycles is shown in Fig. 4. For example, p-zero-
radius cycles in any implementation touch the real axis.

This “touching” property is a manifestation of the _boundary effect_ in the
upper-half plane geometry. The famous question on hearing drum’s shape has a
sister:

> _Can we see/feel the boundary from inside a domain?_

Both orthogonality relations described below are “boundary aware” as well. It
is not surprising after all since $SL_{2}{}(\mathbb{R}{})$ action on the
upper-half plane was obtained as an extension of its action (1.1) on the
boundary.

According to the categorical viewpoint internal properties of objects are of
minor importance in comparison to their relations with other objects from the
same class. As an illustration we may put the proof of Theorem 2.4 sketched at
the end of of the next section. Thus from now on we will look for invariant
relations between two or more cycles.

####  2.4. Joint Invariants: Orthogonality

The most expected relation between cycles is based on the following Möbius
invariant “inner product” build from a trace of product of two cycles as
matrices:

(2.12) $\left\langle
C_{\breve{\sigma}}^{s},\tilde{C}_{\breve{\sigma}}^{s}\right\rangle=\mathop{tr}(C_{\breve{\sigma}}^{s}\tilde{C}_{\breve{\sigma}}^{s})$

By the way, an inner product of this type is used, for example, in GNS
construction to make a Hilbert space out of $C^{*}$-algebra. The next standard
move is given by the following definition.

######  Definition 2.5.

Two cycles are called _$\breve{\sigma}$ -orthogonal_ if $\left\langle
C_{\breve{\sigma}}^{s},\tilde{C}_{\breve{\sigma}}^{s}\right\rangle=0$.

The orthogonality relation is preserved under the Möbius transformations, thus
this is an example of a _joint invariant_ of two cycles. For the case of
$\breve{\sigma}\sigma=1$, i.e. when geometries of the cycle and point spaces
are both either elliptic or hyperbolic, such an orthogonality is the standard
one, defined in terms of angles between tangent lines in the intersection
points of two cycles. However in the remaining seven ($=9-2$) cases the
innocent-looking Definition 2.5 brings unexpected relations.

Figure 5. Orthogonality of the first kind in the elliptic point space.
Each picture presents two groups (green and blue) of cycles which are
orthogonal to the red cycle $C^{s}_{\breve{\sigma}}$. Point $b$ belongs to
$C^{s}_{\breve{\sigma}}$ and the family of blue cycles passing through $b$ is
orthogonal to $C^{s}_{\breve{\sigma}}$. They all also intersect in the point
$d$ which is the inverse of $b$ in $C^{s}_{\breve{\sigma}}$. Any orthogonality
is reduced to the usual orthogonality with a new (“ghost”) cycle (shown by the
dashed line), which may or may not coincide with $C^{s}_{\breve{\sigma}}$. For
any point $a$ on the “ghost” cycle the orthogonality is reduced to the local
notion in the terms of tangent lines at the intersection point. Consequently
such a point $a$ is always the inverse of itself.

Elliptic (in the point space) realisations of Definition 2.5, i.e. $\sigma=-1$
is shown in Fig. 5. The left picture corresponds to the elliptic cycle space,
e.g. $\breve{\sigma}=-1$. The orthogonality between the red circle and any
circle from the blue or green families is given in the usual Euclidean sense.
The central (parabolic in the cycle space) and the right (hyperbolic) pictures
show non-local nature of the orthogonality. There are analogues pictures in
parabolic and hyperbolic point spaces as well, see [Kisil05a, Kisil12a].

This orthogonality may still be expressed in the traditional sense if we will
associate to the red circle the corresponding “ghost” circle, which shown by
the dashed line in Fig. 5. To describe ghost cycle we need the _Heaviside
function_ $\chi(\sigma)$:

(2.13) $\chi(t)=\left\\{\begin{array}[]{ll}1,&t\geq 0;\\\
-1,&t<0.\end{array}\right.$

######  Theorem 2.6.

A cycle is $\breve{\sigma}$-orthogonal to cycle $C_{\breve{\sigma}}^{s}$ if it
is orthogonal in the usual sense to the $\sigma$-realisation of “ghost” cycle
$\hat{C}_{\breve{\sigma}}^{s}$, which is defined by the following two
conditions:

  1. (i)

$\chi(\sigma)$-centre of $\hat{C}_{\breve{\sigma}}^{s}$ coincides with
$\breve{\sigma}$-centre of $C_{\breve{\sigma}}^{s}$.

  2. (ii)

Cycles $\hat{C}_{\breve{\sigma}}^{s}$ and $C^{s}_{\breve{\sigma}}$ have the
same roots, moreover $\det\hat{C}_{\sigma}^{1}=\det
C^{\chi(\breve{\sigma})}_{\sigma}$.

The above connection between various centres of cycles illustrates their
meaningfulness within our approach.

One can easy check the following orthogonality properties of the zero-radius
cycles defined in the previous section:

  1. (i)

Due to the identity $\left\langle
C_{\breve{\sigma}}^{s},{C}_{\breve{\sigma}}^{s}\right\rangle=\det{C}_{\breve{\sigma}}^{s}$
zero-radius cycles are self-orthogonal (isotropic) ones.

  2. (ii)

A cycle ${C^{s}_{\breve{\sigma}}}$ is $\sigma$-orthogonal to a zero-radius
cycle $Z^{s}_{\breve{\sigma}}$ if and only if ${C^{s}_{\breve{\sigma}}}$
passes through the $\sigma$-centre of $Z^{s}_{\breve{\sigma}}$.

As we will see, in parabolic case there is a more suitable notion of an
infinitesimal cycle which can be used instead of zero-radius ones.

####  2.5. Higher Order Joint Invariants: f-Orthogonality

With appetite already wet one may wish to build more joint invariants. Indeed
for any polynomial $p(x_{1},x_{2},\ldots,x_{n})$ of several non-commuting
variables one may define an invariant joint disposition of $n$ cycles
${}^{j}\\!{C^{s}_{\breve{\sigma}}}$ by the condition:

$\mathop{tr}p({}^{1}\\!{C^{s}_{\breve{\sigma}}},{}^{2}\\!{C^{s}_{\breve{\sigma}}},\ldots,{}^{n}\\!{C^{s}_{\breve{\sigma}}})=0.$

However it is preferable to keep some geometrical meaning of constructed
notions.

An interesting observation is that in the matrix similarity of cycles (2.9)
one may replace element $g\in SL_{2}{}(\mathbb{R}{})$ by an arbitrary matrix
corresponding to another cycle. More precisely the product
${C^{s}_{\breve{\sigma}}}{\tilde{C}^{s}_{\breve{\sigma}}}{C^{s}_{\breve{\sigma}}}$
is again the matrix of the form (2.8) and thus may be associated to a cycle.
This cycle may be considered as the reflection of
${\tilde{C}^{s}_{\breve{\sigma}}}$ in ${C^{s}_{\breve{\sigma}}}$.

######  Definition 2.7.

A cycle ${C^{s}_{\breve{\sigma}}}$ is f-orthogonal (focal orthogonal) _to_ a
cycle ${\tilde{C}^{s}_{\breve{\sigma}}}$ if the reflection of
${\tilde{C}^{s}_{\breve{\sigma}}}$ in ${C^{s}_{\breve{\sigma}}}$ is orthogonal
(in the sense of Definition 2.5) to the real line. Analytically this is
defined by:

(2.14)
$\mathop{tr}({C^{s}_{\breve{\sigma}}}{\tilde{C}^{s}_{\breve{\sigma}}}{C^{s}_{\breve{\sigma}}}R^{s}_{\breve{\sigma}})=0.$

Due to invariance of all components in the above definition f-orthogonality is
a Möbius invariant condition. Clearly this is not a symmetric relation: if
${C^{s}_{\breve{\sigma}}}$ is f-orthogonal to
${\tilde{C}^{s}_{\breve{\sigma}}}$ then ${\tilde{C}^{s}_{\breve{\sigma}}}$ is
not necessarily f-orthogonal to ${C^{s}_{\breve{\sigma}}}$.

Figure 6. Focal orthogonality for circles. To highlight both similarities and
distinctions with the ordinary orthogonality we use the same notations as that
in Fig. 5.

Fig. 6 illustrates f-orthogonality in the elliptic point space. By contrast
with Fig. 5 it is not a local notion at the intersection points of cycles for
all $\breve{\sigma}$. However it may be again clarified in terms of the
appropriate s-ghost cycle, cf. Theorem 2.6.

######  Theorem 2.8.

A cycle is f-orthogonal to a cycle $C^{s}_{\breve{\sigma}}$ if its orthogonal
in the traditional sense to its _f-ghost cycle_
${\tilde{C}^{\breve{\sigma}}_{\breve{\sigma}}}={C^{\chi(\sigma)}_{\breve{\sigma}}}\mathbb{R}^{\breve{\sigma}}_{\breve{\sigma}}{}{C^{\chi(\sigma)}_{\breve{\sigma}}}$,
which is the reflection of the real line in
${C^{\chi(\sigma)}_{\breve{\sigma}}}$ and $\chi$ is the _Heaviside function_
(2.13). Moreover

  1. (i)

$\chi(\sigma)$-Centre of ${\tilde{C}^{\breve{\sigma}}_{\breve{\sigma}}}$
coincides with the $\breve{\sigma}$-focus of ${C^{s}_{\breve{\sigma}}}$,
consequently all lines f-orthogonal to ${C^{s}_{\breve{\sigma}}}$ are passing
the respective focus.

  2. (ii)

Cycles ${C^{s}_{\breve{\sigma}}}$ and
${\tilde{C}^{\breve{\sigma}}_{\breve{\sigma}}}$ have the same roots.

Note the above intriguing interplay between cycle’s centres and foci. Although
f-orthogonality may look exotic it will naturally appear in the end of next
Section again.

Of course, it is possible to define another interesting higher order joint
invariants of two or even more cycles.

####  2.6. Distance, Length and Perpendicularity

Geo _metry_ in the plain meaning of this word deals with _distances_ and
_lengths_. Can we obtain them from cycles?

(a)  (b)

Figure 7. (a) The square of the parabolic diameter is the square of the
distance between roots if they are real ($z_{1}$ and $z_{2}$), otherwise the
negative square of the distance between the adjoint roots ($z_{3}$ and
$z_{4}$).
(b) Distance as extremum of diameters in elliptic ($z_{1}$ and $z_{2}$) and
parabolic ($z_{3}$ and $z_{4}$) cases.

We mentioned already that for circles normalised by the condition $k=1$ the
value
$\det{C^{s}_{\breve{\sigma}}}=\left\langle{C^{s}_{\breve{\sigma}}},{C^{s}_{\breve{\sigma}}}\right\rangle$
produces the square of the traditional circle radius. Thus we may keep it as
the definition of the $\breve{\sigma}$-_radius_ for any cycle. But then we
need to accept that in the parabolic case the radius is the (Euclidean)
distance between (real) roots of the parabola, see Fig. 7(a).

Having radii of circles already defined we may use them for other measurements
in several different ways. For example, the following variational definition
may be used:

######  Definition 2.9.

The _distance_ between two points is the extremum of diameters of all cycles
passing through both points, see Fig. 7(b).

If $\breve{\sigma}=\sigma$ this definition gives in all EPH cases the
following expression for a distance $d_{e,p,h}(u,v)$ between endpoints of any
vector $w=u+\mathrm{i}v$:

(2.15) $d_{e,p,h}(u,v)^{2}=(u+\mathrm{i}v)(u-\mathrm{i}v)=u^{2}-\sigma v^{2}.$

The parabolic distance $d_{p}^{2}=u^{2}$, see Fig. 7(b), algebraically sits
between $d_{e}$ and $d_{h}$ according to the general principle (1.2) and is
widely accepted [Yaglom79]. However one may be unsatisfied by its degeneracy.

An alternative measurement is motivated by the fact that a circle is the set
of equidistant points from its centre. However the choice of “centre” is now
rich: it may be either point from three centres (2.10) or three foci (2.11).

######  Definition 2.10.

The _length_ of a directed interval $\overrightarrow{AB}$ is the radius of the
cycle with its _centre_ (denoted by $l_{c}(\overrightarrow{AB})$) or _focus_
(denoted by $l_{f}(\overrightarrow{AB})$) at the point $A$ which passes
through $B$.

This definition is less common and have some unusual properties like non-
symmetry: $l_{f}(\overrightarrow{AB})\neq l_{f}(\overrightarrow{BA})$. However
it comfortably fits the Erlangen programme due to its
$SL_{2}{}(\mathbb{R}{})$-_conformal invariance_ :

######  Theorem 2.11 ([Kisil05a]).

Let $l$ denote either the EPH distances (2.15) or any length from Definition
2.10. Then for fixed $y$, $y^{\prime}\in\mathbb{R}^{\sigma}{}$ the limit:

$\lim_{t\rightarrow 0}\frac{l(g\cdot
y,g\cdot(y+ty^{\prime}))}{l(y,y+ty^{\prime})},\qquad\text{ where }g\in
SL_{2}{}(\mathbb{R}{}),$

exists and its value depends only from $y$ and $g$ and is independent from
$y^{\prime}$.

Figure 8. Perpendicular as the shortest route to a line.

We may return from distances to angles recalling that in the Euclidean space a
perpendicular provides the shortest root from a point to a line, see Fig. 8.

######  Definition 2.12.

Let $l$ be a length or distance. We say that a vector $\overrightarrow{AB}$ is
_$l$ -perpendicular_ to a vector $\overrightarrow{CD}$ if function
$l(\overrightarrow{AB}+\varepsilon\overrightarrow{CD})$ of a variable
$\varepsilon$ has a local extremum at $\varepsilon=0$.

A pleasant surprise is that $l_{f}$-perpendicularity obtained thought the
length from focus (Definition 2.10) coincides with already defined in Section
2.5 f-orthogonality as follows from Thm. 2.8(i). It is also possible
[Kisil08a] to make $SL_{2}{}(\mathbb{R}{})$ action isometric in all three
cases.

Further details of the refreshing geometry of Möbius transformation can be
found in the paper [Kisil05a]and the book [Kisil12a].

All these study are waiting to be generalised to high dimensions, quaternions
and Clifford algebras provide a suitable language for this [Kisil05a,
JParker07a].

###  3\. Linear Representations

A consideration of the symmetries in analysis is natural to start from _linear
representations_. The previous geometrical actions (1.1) can be naturally
extended to such representations by induction [Kirillov76]*§ 13.2 [Kisil97c]*§
3.1 from a representation of a subgroup $H$. If $H$ is one-dimensional then
its irreducible representation is a character, which is always supposed to be
a complex valued. However hypercomplex number naturally appeared in the
$SL_{2}{}(\mathbb{R}{})$ action (1.1), see Subsection 2.1 and [Kisil09c], why
shall we admit only $\mathrm{i}^{2}=-1$ to deliver a character then?

####  3.1. Hypercomplex Characters

As we already mentioned the typical discussion of induced representations of
$SL_{2}{}(\mathbb{R}{})$ is centred around the case $H=K$ and a complex valued
character of $K$. A linear transformation defined by a matrix (2.4) in $K$ is
a rotation of $\mathbb{R}^{2}{}$ by the angle $t$. After identification
$\mathbb{R}^{2}{}=\mathbb{C}{}$ this action is given by the multiplication
$e^{\mathrm{i}t}$, with $\mathrm{i}^{2}=-1$. The rotation preserve the
(elliptic) metric given by:

(3.1) $x^{2}+y^{2}=(x+\mathrm{i}y)(x-\mathrm{i}y).$

Therefore the orbits of rotations are circles, any line passing the origin (a
“spoke”) is rotated by the angle $t$, see Fig. 9.

Dual and double numbers produces the most straightforward adaptation of this
result.

Figure 9. Rotations of algebraic wheels, i.e. the multiplication by $e^{\iota
t}$: elliptic ($E$), trivial parabolic ($P_{0}$) and hyperbolic ($H$). All
blue orbits are defined by the identity $x^{2}-\iota^{2}y^{2}=r^{2}$. Thin
“spokes” (straight lines from the origin to a point on the orbit) are
“rotated” from the real axis. This is symplectic linear transformations of the
classical phase space as well.

######  Proposition 3.1.

The following table show correspondences between three types of algebraic
characters:

Elliptic | Parabolic | Hyperbolic
---|---|---
$\mathrm{i}^{2}=-1$ | $\varepsilon^{2}=0$ | $\mathrm{j}^{2}=1$
$w=x+\mathrm{i}y$ | $w=x+\varepsilon y$ | $w=x+\mathrm{j}y$
$\bar{w}=x-\mathrm{i}y$ | $\bar{w}=x-\varepsilon y$ | $\bar{w}=x-\mathrm{j}y$
$e^{\mathrm{i}t}=\cos t+\mathrm{i}\sin t$ | $e^{\varepsilon t}=1+\varepsilon t$ | $e^{\mathrm{j}t}=\cosh t+\mathrm{j}\sinh t$
$\left|w\right|_{e}^{2}=w\bar{w}=x^{2}+y^{2}$ | $\left|w\right|_{p}^{2}=w\bar{w}=x^{2}$ | $\left|w\right|_{h}^{2}=w\bar{w}=x^{2}-y^{2}$
$\arg w=\tan^{-1}\frac{y}{x}\frac{}{}$ | $\arg w=\frac{y}{x}$ | $\arg w=\tanh^{-1}\frac{y}{x}$
unit circle $\left|w\right|_{e}^{2}=1$ |  “unit” strip $x=\pm 1$ |  unit hyperbola $\left|w\right|_{h}^{2}=1$

Geometrical action of multiplication by $e^{\iota t}$ is drawn in Fig. 9 for
all three cases.

Explicitly parabolic rotations associated with $\mathrm{e}^{\varepsilon t}$
acts on dual numbers as follows:

(3.2) $\mathrm{e}^{\varepsilon x}:a+\varepsilon b\mapsto a+\varepsilon(ax+b).$

This links the parabolic case with the Galilean group [Yaglom79] of symmetries
of the classic mechanics, with the absolute time disconnected from space.

The obvious algebraic similarity and the connection to classical kinematic is
a wide spread justification for the following viewpoint on the parabolic case,
cf. [HerranzOrtegaSantander99a, Yaglom79]:

  * •

the parabolic trigonometric functions are trivial:

(3.3) $\mathop{\operator@font cosp}\nolimits t=\pm
1,\qquad\mathop{\operator@font sinp}\nolimits t=t;$

  * •

the parabolic distance is independent from $y$ if $x\neq 0$:

(3.4) $x^{2}=(x+\varepsilon y)(x-\varepsilon y);$

  * •

the polar decomposition of a dual number is defined by [Yaglom79]*App. C(30’):

(3.5) $u+\varepsilon v=u(1+\varepsilon\frac{v}{u}),\quad\text{ thus
}\quad\left|u+\varepsilon v\right|=u,\quad\arg(u+\varepsilon v)=\frac{v}{u};$

  * •

the parabolic wheel looks rectangular, see Fig. 9.

Those algebraic analogies are quite explicit and widely accepted as an
ultimate source for parabolic trigonometry [LavrentShabat77,
HerranzOrtegaSantander99a, Yaglom79]. Moreover, those three rotations are all
non-isomorphic symplectic linear transformations of the phase space, which
makes them useful in the context of classical and quantum mechanics [Kisil10a,
Kisil11a], see Section 7. There exist also alternative characters [Kisil09a]
based on Möbius transformations with geometric motivation and connections to
equations of mathematical physics.

####  3.2. Induced Representations

Let $G$ be a group, $H$ be its closed subgroup with the corresponding
homogeneous space $X=G/H$ with an invariant measure. We are using notations
and definitions of maps $p:G\rightarrow X$, $s:X\rightarrow G$ and
$r:G\rightarrow H$ from Subsection 2.1. Let $\chi$ be an irreducible
representation of $H$ in a vector space $V$, then it induces a representation
of $G$ in the sense of Mackey [Kirillov76]*§ 13.2. This representation has the
realisation ${\rho_{\chi}}$ in the space $L_{2}{}(X)$ of $V$-valued functions
by the formula [Kirillov76]*§ 13.2.(7)–(9):

(3.6) $[{\rho_{\chi}}(g)f](x)=\chi(r(g^{-1}*s(x)))f(g^{-1}\cdot x),.$

where $g\in G$, $x\in X$, $h\in H$ and $r:G\rightarrow H$, $s:X\rightarrow G$
are maps defined above; $*$ denotes multiplication on $G$ and $\cdot$ denotes
the action (2.2) of $G$ on $X$.

Consider this scheme for representations of $SL_{2}{}(\mathbb{R}{})$ induced
from characters of its one-dimensional subgroups. We can notice that only the
subgroup $K$ requires a complex valued character due to the fact of its
compactness. For subgroups $N^{\prime}$ and $A\\!^{\prime}$ we can consider
characters of all three types—elliptic, parabolic and hyperbolic. Therefore we
have seven essentially different induced representations. We will write
explicitly only three of them here.

######  Example 3.2.

Consider the subgroup $H=K$, due to its compactness we are limited to complex
valued characters of $K$ only. All of them are of the form $\chi_{k}$:

(3.7) $\chi_{k}\begin{pmatrix}\cos t&\sin t\\\ -\sin t&\cos
t\end{pmatrix}=e^{-\mathrm{i}kt},\qquad\text{ where }k\in\mathbb{Z}{}.$

Using the explicit form (2.3) of the map $s$ we find the map $r$ given in
(2.1) as follows:

$r\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\frac{1}{\sqrt{c^{2}+d^{2}}}\begin{pmatrix}d&-c\\\
c&d\end{pmatrix}\in K.$

Therefore:

$r(g^{-1}*s(u,v))=\frac{1}{\sqrt{(cu+d)^{2}+(cv)^{2}}}\begin{pmatrix}cu+d&-cv\\\
cv&cu+d\end{pmatrix},\quad\text{where }g^{-1}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}.$

Substituting this into (3.7) and combining with the Möbius transformation of
the domain (1.1) we get the explicit realisation ${\rho_{k}}{}$ of the induced
representation (3.6):

(3.8)
${\rho_{k}}{}(g)f(w)=\frac{\left|cw+d\right|^{k}}{(cw+d)^{k}}f\left(\frac{aw+b}{cw+d}\right),\quad\text{
where }g^{-1}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix},\ w=u+\mathrm{i}v.$

This representation acts on complex valued functions in the upper half-plane
$\mathbb{R}^{2}_{+}{}=SL_{2}{}(\mathbb{R}{})/K$ and belongs to the discrete
series [Lang85]*§ IX.2. It is common to get rid of the factor
$\left|cw+d\right|^{k}$ from that expression in order to keep analyticity and
we will follow this practise for a convenience as well.

######  Example 3.3.

In the case of the subgroup $N$ there is a wider choice of possible
characters.

  1. (i)

Traditionally only complex valued characters of the subgroup $N$ are
considered, they are:

(3.9) $\chi_{\tau}^{\mathbb{C}{}}\begin{pmatrix}1&0\\\
t&1\end{pmatrix}=e^{\mathrm{i}\tau t},\qquad\text{ where
}\tau\in\mathbb{R}{}.$

A direct calculation shows that:

$r\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}1&0\\\
\frac{c}{d}&1\end{pmatrix}\in N^{\prime}.$

Thus:

(3.10) $r(g^{-1}*s(u,v))=\begin{pmatrix}1&0\\\
\frac{cv}{d+cu}&1\end{pmatrix},\quad\text{ where }g^{-1}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}.$

A substitution of this value into the character (3.9) together with the Möbius
transformation (1.1) we obtain the next realisation of (3.6):

${\rho^{\mathbb{C}{}}_{\tau}}(g)f(w)=\exp\left(\mathrm{i}\frac{\tau
cv}{cu+d}\right)f\left(\frac{aw+b}{cw+d}\right),\quad\text{where
}w=u+\varepsilon v,\ g^{-1}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

The representation acts on the space of _complex_ valued functions on the
upper half-plane $\mathbb{R}^{2}_{+}{}$, which is a subset of _dual_ numbers
as a homogeneous space $SL_{2}{}(\mathbb{R}{})/N^{\prime}$. The mixture of
complex and dual numbers in the same expression is confusing.

  2. (ii)

The parabolic character $\chi_{\tau}$ with the algebraic flavour is provided
by multiplication (3.2) with the dual number:

$\chi_{\tau}\begin{pmatrix}1&0\\\ t&1\end{pmatrix}=e^{\varepsilon\tau
t}=1+\varepsilon\tau t,\qquad\text{ where }\tau\in\mathbb{R}{}.$

If we substitute the value (3.10) into this character, then we receive the
representation:

${\rho_{\tau}}{}(g)f(w)=\left(1+\varepsilon\frac{\tau
cv}{cu+d}\right)f\left(\frac{aw+b}{cw+d}\right),$

where $w$, $\tau$ and $g$ are as above. The representation is defined on the
space of dual numbers valued functions on the upper half-plane of dual
numbers. Thus expression contains only dual numbers with their usual algebraic
operations. Thus it is linear with respect to them.

All characters in the previous Example are unitary. Then the general scheme of
induced representations [Kirillov76]*§ 13.2 implies their unitarity in proper
senses.

######  Theorem 3.4 ([Kisil09c]).

Both representations of $SL_{2}{}(\mathbb{R}{})$ from Example 3.3 are unitary
on the space of function on the upper half-plane $\mathbb{R}^{2}_{+}{}$ of
dual numbers with the inner product:

(3.11) $\left\langle
f_{1},f_{2}\right\rangle=\int_{\mathbb{R}^{2}_{+}{}}f_{1}(w)\bar{f}_{2}(w)\,\frac{du\,dv}{v^{2}},\qquad\text{
where }w=u+\varepsilon v,$

and we use the conjugation and multiplication of functions’ values in algebras
of complex and dual numbers for representations ${\rho^{\mathbb{C}{}}_{\tau}}$
and ${\rho_{\tau}}$ respectively.

The inner product (3.11) is positive defined for the representation
${\rho^{\mathbb{C}{}}_{\tau}}$ but is not for the other. The respective spaces
are parabolic cousins of the _Krein spaces_ [ArovDym08], which are hyperbolic
in our sense.

####  3.3. Similarity and Correspondence: Ladder Operators

From the above observation we can deduce the following empirical principle,
which has a heuristic value.

######  Principle 3.5 (Similarity and correspondence).

  1. (i)

Subgroups $K$, $N^{\prime}$ and $A\\!^{\prime}$ play a similar rôle in the
structure of the group $SL_{2}{}(\mathbb{R}{})$ and its representations.

  2. (ii)

The subgroups shall be swapped simultaneously with the respective replacement
of hypercomplex unit $\iota$.

The first part of the Principle (similarity) does not look sound alone. It is
enough to mention that the subgroup $K$ is compact (and thus its spectrum is
discrete) while two other subgroups are not. However in a conjunction with the
second part (correspondence) the Principle have received the following
confirmations so far, see [Kisil09c] for details:

  * •

The action of $SL_{2}{}(\mathbb{R}{})$ on the homogeneous space
$SL_{2}{}(\mathbb{R}{})/H$ for $H=K$, $N^{\prime}$ or $A\\!^{\prime}$ is given
by linear-fractional transformations of complex, dual or double numbers
respectively.

  * •

Subgroups $K$, $N^{\prime}$ or $A\\!^{\prime}$ are isomorphic to the groups of
unitary rotations of respective unit cycles in complex, dual or double
numbers.

  * •

Representations induced from subgroups $K$, $N^{\prime}$ or $A\\!^{\prime}$
are unitary if the inner product spaces of functions with values in complex,
dual or double numbers.

######  Remark 3.6.

The principle of similarity and correspondence resembles _supersymmetry_
between bosons and fermions in particle physics, but we have similarity
between three different types of entities in our case.

Let us give another illustration to the Principle. Consider the Lie algebra
$\mathfrak{sl}_{2}$ of the group $SL_{2}{}(\mathbb{R}{})$. Pick up the
following basis in $\mathfrak{sl}_{2}$ [MTaylor86]*§ 8.1:

(3.12) $A=\frac{1}{2}\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix},\quad
B=\frac{1}{2}\ \begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\quad
Z=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$

The commutation relations between the elements are:

(3.13) $[Z,A]=2B,\qquad[Z,B]=-2A,\qquad[A,B]=-\frac{1}{2}Z.$

Let ${\rho}$ be a representation of the group $SL_{2}{}(\mathbb{R}{})$ in a
space $V$. Consider the derived representation $d{\rho}$ of the Lie algebra
$\mathfrak{sl}_{2}$ [Lang85]*§ VI.1 and denote $\tilde{X}=d{\rho}(X)$ for
$X\in\mathfrak{sl}_{2}$. To see the structure of the representation ${\rho}$
we can decompose the space $V$ into eigenspaces of the operator $\tilde{X}$
for some $X\in\mathfrak{sl}_{2}$, cf. the Taylor series in Section 5.4.

#####  3.3.1. Elliptic Ladder Operators

It would not be surprising that we are going to consider three cases: Let
$X=Z$ be a generator of the subgroup $K$ (2.4). Since this is a compact
subgroup the corresponding eigenspaces $\tilde{Z}v_{k}=\mathrm{i}kv_{k}$ are
parametrised by an integer $k\in\mathbb{Z}{}$. The _raising/lowering_ or
_ladder operators_ $L^{\\!\pm}$ [Lang85]*§ VI.2 [MTaylor86]*§ 8.2 are defined
by the following commutation relations:

(3.14) $[\tilde{Z},L^{\\!\pm}]=\lambda_{\pm}L^{\\!\pm}.$

In other words $L^{\\!\pm}$ are eigenvectors for operators
$\mathop{\operator@font ad}\nolimits Z$ of adjoint representation of
$\mathfrak{sl}_{2}$ [Lang85]*§ VI.2.

######  Remark 3.7.

The existence of such ladder operators follows from the general properties of
Lie algebras if the element $X\in\mathfrak{sl}_{2}$ belongs to a _Cartan
subalgebra_. This is the case for vectors $Z$ and $B$, which are the only two
non-isomorphic types of Cartan subalgebras in $\mathfrak{sl}_{2}$. However the
third case considered in this paper, the parabolic vector $B+Z/2$, does not
belong to a Cartan subalgebra, yet a sort of ladder operators is still
possible with dual number coefficients. Moreover, for the hyperbolic vector
$B$, besides the standard ladder operators an additional pair with double
number coefficients will also be described.

From the commutators (3.14) we deduce that $L^{\\!+}v_{k}$ are eigenvectors of
$\tilde{Z}$ as well:

(3.15) $\displaystyle\tilde{Z}(L^{\\!+}v_{k})$ $\displaystyle=$
$\displaystyle(L^{\\!+}\tilde{Z}+\lambda_{+}L^{\\!+})v_{k}=L^{\\!+}(\tilde{Z}v_{k})+\lambda_{+}L^{\\!+}v_{k}=\mathrm{i}kL^{\\!+}v_{k}+\lambda_{+}L^{\\!+}v_{k}$
$\displaystyle=$ $\displaystyle(\mathrm{i}k+\lambda_{+})L^{\\!+}v_{k}.$

Thus action of ladder operators on respective eigenspaces can be visualised by
the diagram:

(3.16)
$\textstyle{\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!+}}$$\textstyle{\,V_{\mathrm{i}k-\lambda}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!-}}$$\scriptstyle{L^{\\!+}}$$\textstyle{\,V_{\mathrm{i}k}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!-}}$$\scriptstyle{L^{\\!+}}$$\textstyle{\,V_{\mathrm{i}k+\lambda}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!-}}$$\scriptstyle{L^{\\!+}}$$\textstyle{\,\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L^{\\!-}}$

Assuming $L^{\\!+}=a\tilde{A}+b\tilde{B}+c\tilde{Z}$ from the relations (3.13)
and defining condition (3.14) we obtain linear equations with unknown $a$, $b$
and $c$:

$c=0,\qquad 2a=\lambda_{+}b,\qquad-2b=\lambda_{+}a.$

The equations have a solution if and only if $\lambda_{+}^{2}+4=0$, and the
raising/lowering operators are $L^{\\!\pm}=\pm\mathrm{i}\tilde{A}+\tilde{B}$.

#####  3.3.2. Hyperbolic Ladder Operators

Consider the case $X=2B$ of a generator of the subgroup $A\\!^{\prime}$ (2.6).
The subgroup is not compact and eigenvalues of the operator $\tilde{B}$ can be
arbitrary, however raising/lowering operators are still important
[HoweTan92]*§ II.1 [Mazorchuk09a]*§ 1.1. We again seek a solution in the form
$L_{h}^{\\!+}=a\tilde{A}+b\tilde{B}+c\tilde{Z}$ for the commutator
$[2\tilde{B},L_{h}^{\\!+}]=\lambda L_{h}^{\\!+}$. We will get the system:

$4c=\lambda a,\qquad b=0,\qquad{a}=\lambda c.$

A solution exists if and only if $\lambda^{2}=4$. There are obvious values
$\lambda=\pm 2$ with the ladder operators $L_{h}^{\\!\pm}=\pm
2\tilde{A}+\tilde{Z}$, see [HoweTan92]*§ II.1 [Mazorchuk09a]*§ 1.1. Each
indecomposable $\mathfrak{sl}_{2}$-module is formed by a one-dimensional chain
of eigenvalues with a transitive action of ladder operators.

Admitting double numbers we have an extra possibility to satisfy
$\lambda^{2}=4$ with values $\lambda=\pm 2\mathrm{j}$. Then there is an
additional pair of hyperbolic ladder operators $L_{\mathrm{j}}^{\\!\pm}=\pm
2\mathrm{j}\tilde{A}+\tilde{Z}$, which shift eigenvectors in the “orthogonal”
direction to the standard operators $L_{h}^{\\!\pm}$. Therefore an
indecomposable $\mathfrak{sl}_{2}$-module can be parametrised by a two-
dimensional lattice of eigenvalues on the double number plane, see Fig. 10

$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!+}}$$\textstyle{\,V_{(n-2)+\mathrm{j}(k-2)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{n+\mathrm{j}(k-2)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{(n+2)+\mathrm{j}(k-2)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\textstyle{\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!+}}$$\textstyle{\,V_{(n-2)+\mathrm{j}k}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{n+\mathrm{j}k}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{(n+2)+\mathrm{j}k}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\textstyle{\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!+}}$$\textstyle{\,V_{(n-2)+\mathrm{j}(k+2)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{n+\mathrm{j}(k+2)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,V_{(n+2)+\mathrm{j}(k+2)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\scriptstyle{L_{h}^{\\!+}}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\scriptstyle{L_{\mathrm{j}}^{\\!+}}$$\textstyle{\,\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{h}^{\\!-}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$$\textstyle{\,\ldots\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L_{\mathrm{j}}^{\\!-}}$

Figure 10. The action of hyperbolic ladder operators on a 2D lattice of
eigenspaces. Operators $L_{h}^{\\!\pm}$ move the eigenvalues by $2$, making
shifts in the horizontal direction. Operators $L_{\mathrm{j}}^{\\!\pm}$ change
the eigenvalues by $2\mathrm{j}$, shown as vertical shifts.

#####  3.3.3. Parabolic Ladder Operators

Finally consider the case of a generator $X=-B+Z/2$ of the subgroup
$N^{\prime}$ (2.5). According to the above procedure we get the equations:

$b+2c=\lambda a,\qquad-a=\lambda b,\qquad\frac{a}{2}=\lambda c,$

which can be resolved if and only if $\lambda^{2}=0$. If we restrict ourselves
with the only real (complex) root $\lambda=0$, then the corresponding
operators $L_{p}^{\\!\pm}=-\tilde{B}+\tilde{Z}/2$ will not affect eigenvalues
and thus are useless in the above context. However the dual number roots
$\lambda=\pm\varepsilon t$, $t\in\mathbb{R}{}$ lead to the operators
$L_{\varepsilon}^{\\!\pm}=\pm\varepsilon t\tilde{A}-\tilde{B}+\tilde{Z}/2$.
These operators are suitable to build an $\mathfrak{sl}_{2}$-modules with a
one-dimensional chain of eigenvalues.

######  Remark 3.8.

The following rôles of hypercomplex numbers are noteworthy:

  * •

the introduction of complex numbers is a necessity for the _existence_ of
ladder operators in the elliptic case;

  * •

in the parabolic case we need dual numbers to make ladder operators _useful_ ;

  * •

in the hyperbolic case double numbers are not required neither for the
existence or for the usability of ladder operators, but they do provide an
enhancement.

We summarise the above consideration with a focus on the Principle of
similarity and correspondence:

######  Proposition 3.9.

Let a vector $X\in\mathfrak{sl}_{2}$ generates the subgroup $K$, $N^{\prime}$
or $A\\!^{\prime}$, that is $X=Z$, $B-Z/2$, or $B$ respectively. Let $\iota$
be the respective hypercomplex unit.

Then raising/lowering operators $L^{\\!\pm}$ satisfying to the commutation
relation:

$[X,L^{\\!\pm}]=\pm\iota L^{\\!\pm},\qquad[L^{\\!-},L^{\\!+}]=2\iota X.$

are:

$L^{\\!\pm}=\pm\iota\tilde{A}+\tilde{Y}.$

Here $Y\in\mathfrak{sl}_{2}$ is a linear combination of $B$ and $Z$ with the
properties:

  * •

$Y=[A,X]$.

  * •

$X=[A,Y]$.

  * •

Killings form $K(X,Y)$ [Kirillov76]*§ 6.2 vanishes.

Any of the above properties defines the vector $Y\in\mathop{\operator@font
span}\nolimits\\{B,Z\\}$ up to a real constant factor.

The usability of the Principle of similarity and correspondence will be
illustrated by more examples below.

###  4\. Covariant Transform

A general group-theoretical construction [Perelomov86, FeichGroech89a,
Kisil98a, AliAntGaz00, Fuhr05a, ChristensenOlafsson09a, KlaSkag85] of
_wavelets_ (or _coherent state_) starts from an irreducible square integrable
representation—in the proper sense or modulo a subgroup. Then a mother wavelet
is chosen to be _admissible_. This leads to a wavelet transform which is an
isometry to $L_{2}{}$ space with respect to the Haar measure on the group or
(quasi)invariant measure on a homogeneous space.

The importance of the above situation shall not be diminished, however an
exclusive restriction to such a setup is not necessary, in fact. Here is a
classical example from complex analysis: the Hardy space
$H_{2}{}(\mathbb{T}{})$ on the unit circle and Bergman spaces
$B_{2}^{n}{}(\mathbb{D}{})$, $n\geq 2$ in the unit disk produce wavelets
associated with representations $\rho_{1}$ and $\rho_{n}$ of the group
$SL_{2}{}(\mathbb{R}{})$ respectively [Kisil97c]. While representations
$\rho_{n}$, $n\geq 2$ are from square integrable discrete series, the mock
discrete series representation $\rho_{1}$ is not square integrable [Lang85]*§
VI.5 [MTaylor86]*§ 8.4. However it would be natural to treat the Hardy space
in the same framework as Bergman ones. Some more examples will be presented
below.

####  4.1. Extending Wavelet Transform

To make a sharp but still natural generalisation of wavelets we give the
following definition.

######  Definition 4.1.

[Kisil09d] Let ${\rho}$ be a representation of a group $G$ in a space $V$ and
$F$ be an operator from $V$ to a space $U$. We define a _covariant transform_
$\mathcal{W}$ from $V$ to the space $L{}(G,U)$ of $U$-valued functions on $G$
by the formula:

(4.1) $\mathcal{W}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v),\qquad v\in V,\ g\in
G.$

Operator $F$ will be called _fiducial operator_ in this context.

We borrow the name for operator $F$ from fiducial vectors of Klauder and
Skagerstam [KlaSkag85].

######  Remark 4.2.

We do not require that fiducial operator $F$ shall be linear. Sometimes the
positive homogeneity, i.e. $F(tv)=tF(v)$ for $t>0$, alone can be already
sufficient, see Example 4.14.

######  Remark 4.3.

Usefulness of the covariant transform is in the reverse proportion to the
dimensionality of the space $U$. The covariant transform encodes properties of
$v$ in a function $\mathcal{W}v$ on $G$. For a low dimensional $U$ this
function can be ultimately investigated by means of harmonic analysis. Thus
$\dim U=1$ (scalar-valued functions) is the ideal case, however, it is
unattainable sometimes, see Example 4.11 below. We may have to use a higher
dimensions of $U$ if the given group $G$ is not rich enough.

As we will see below covariant transform is a close relative of wavelet
transform. The name is chosen due to the following common property of both
transformations.

######  Theorem 4.4.

The covariant transform (4.1) intertwines ${\rho}$ and the left regular
representation $\Lambda$ on $L{}(G,U)$:

$\mathcal{W}{\rho}(g)=\Lambda(g)\mathcal{W}.$

Here $\Lambda$ is defined as usual by:

(4.2) $\Lambda(g):f(h)\mapsto f(g^{-1}h).$

###### Proof.

We have a calculation similar to wavelet transform [Kisil98a]*Prop. 2.6. Take
$u={\rho}(g)v$ and calculate its covariant transform:

$\displaystyle{}[\mathcal{W}({\rho}(g)v)](h)$ $\displaystyle=$
$\displaystyle[\mathcal{W}({\rho}(g)v)](h)=F({\rho}(h^{-1}){\rho}(g)v)$
$\displaystyle=$ $\displaystyle F({\rho}((g^{-1}h)^{-1})v)$ $\displaystyle=$
$\displaystyle[\mathcal{W}v](g^{-1}h)$ $\displaystyle=$
$\displaystyle\Lambda(g)[\mathcal{W}v](h).$

∎

The next result follows immediately:

######  Corollary 4.5.

The image space $\mathcal{W}(V)$ is invariant under the left shifts on $G$.

######  Remark 4.6.

A further generalisation of the covariant transform can be obtained if we
relax the group structure. Consider, for example, a _cancellative semigroup_
$\mathbb{Z}_{+}{}$ of non-negative integers. It has a linear presentation on
the space of polynomials in a variable $t$ defined by the action
$m:t^{n}\mapsto t^{m+n}$ on the monomials. Application of a linear functional
$l$, e.g. defined by an integration over a measure on the real line, produces
_umbral calculus_ $l(t^{n})=c_{n}$, which has a magic efficiency in many
areas, notably in combinatorics [Rota95, Kisil97b]. In this direction we also
find fruitful to expand the notion of an intertwining operator to a _token_
[Kisil01b].

####  4.2. Examples of Covariant Transform

In this Subsection we will provide several examples of covariant transforms.
Some of them will be expanded in subsequent sections, however a detailed study
of all aspects will not fit into the present work. We start from the classical
example of the group-theoretical wavelet transform:

######  Example 4.7.

Let $V$ be a Hilbert space with an inner product
$\left\langle\cdot,\cdot\right\rangle$ and ${\rho}$ be a unitary
representation of a group $G$ in the space $V$. Let
$F:V\rightarrow\mathbb{C}{}$ be a functional $v\mapsto\left\langle
v,v_{0}\right\rangle$ defined by a vector $v_{0}\in V$. The vector $v_{0}$ is
oftenly called the _mother wavelet_ in areas related to signal processing or
the _vacuum state_ in quantum framework.

Then the transformation (4.1) is the well-known expression for a _wavelet
transform_ [AliAntGaz00]*(7.48) (or _representation coefficients_):

(4.3)
$\mathcal{W}:v\mapsto\hat{v}(g)=\left\langle{\rho}(g^{-1})v,v_{0}\right\rangle=\left\langle
v,{\rho}(g)v_{0}\right\rangle,\qquad v\in V,\ g\in G.$

The family of vectors $v_{g}={\rho}(g)v_{0}$ is called _wavelets_ or _coherent
states_. In this case we obtain scalar valued functions on $G$, thus the
fundamental rôle of this example is explained in Rem. 4.3.

This scheme is typically carried out for a square integrable representation
${\rho}$ and $v_{0}$ being an admissible vector [Perelomov86, FeichGroech89a,
AliAntGaz00, Fuhr05a, ChristensenOlafsson09a]. In this case the wavelet
(covariant) transform is a map into the square integrable functions
[DufloMoore] with respect to the left Haar measure. The map becomes an
isometry if $v_{0}$ is properly scaled.

However square integrable representations and admissible vectors does not
cover all interesting cases.

######  Example 4.8.

Let $G=\mathrm{Aff}$ be the “$ax+b$” (or _affine_) group [AliAntGaz00]*§ 8.2:
the set of points $(a,b)$, $a\in\mathbb{R}_{+}{}$, $b\in\mathbb{R}{}$ in the
upper half-plane with the group law:

(4.4) $(a,b)*(a^{\prime},b^{\prime})=(aa^{\prime},ab^{\prime}+b)$

and left invariant measure $a^{-2}\,da\,db$. Its isometric representation on
$V=L_{p}{}(\mathbb{R}{})$ is given by the formula:

(4.5)
$[{\rho_{p}}(g)\,f](x)=a^{\frac{1}{p}}f\left(ax+b\right),\qquad\text{where
}g^{-1}=(a,b).$

We consider the operators
$F_{\pm}:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ defined by:

(4.6) $F_{\pm}(f)=\frac{1}{2\pi
i}\int_{\mathbb{R}{}}\frac{f(t)\,dt}{x\mp\mathrm{i}}.$

Then the covariant transform (4.1) is the Cauchy integral from
$L_{p}{}(\mathbb{R}{})$ to the space of functions $\hat{f}(a,b)$ such that
$a^{-\frac{1}{p}}\hat{f}(a,b)$ is in the Hardy space in the upper/lower half-
plane $H_{p}{}(\mathbb{R}^{2}_{\pm}{})$. Although the representation (4.5) is
square integrable for $p=2$, the function $\frac{1}{x\pm\mathrm{i}}$ used in
(4.6) is not an admissible vacuum vector. Thus the complex analysis become
decoupled from the traditional wavelet theory. As a result the application of
wavelet theory shall relay on an extraneous mother wavelets [Hutnik09a].

Many important objects in complex analysis are generated by inadmissible
mother wavelets like (4.6). For example, if
$F:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ is defined by $F:f\mapsto
F_{+}f+F_{-}f$ then the covariant transform (4.1) reduces to the _Poisson
integral_. If $F:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}^{2}{}$ is defined
by $F:f\mapsto(F_{+}f,F_{-}f)$ then the covariant transform (4.1) represents a
function $f$ on the real line as a jump:

(4.7) $f(z)=f_{+}(z)-f_{-}(z),\qquad f_{\pm}(z)\in
H_{p}{}(\mathbb{R}^{2}_{\pm}{})$

between functions analytic in the upper and the lower half-planes. This makes
a decomposition of $L_{2}{}(\mathbb{R}{})$ into irreducible components of the
representation (4.5). Another interesting but non-admissible vector is the
_Gaussian_ $e^{-x^{2}}$.

######  Example 4.9.

For the group $G=SL_{2}{}(\mathbb{R}{})$ [Lang85] let us consider the unitary
representation ${\rho}$ on the space of square integrable function
$L_{2}{}(\mathbb{R}^{2}_{+}{})$ on the upper half-plane through the Möbius
transformations (1.1):

(4.8)
${\rho}(g):f(z)\mapsto\frac{1}{(cz+d)^{2}}\,f\left(\frac{az+b}{cz+d}\right),\qquad
g^{-1}=\ \begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

This is a representation from the discrete series and $L_{2}{}(\mathbb{D}{})$
and irreducible invariant subspaces are parametrised by integers. Let $F_{k}$
be the functional $L_{2}{}(\mathbb{R}^{2}_{+}{})\rightarrow\mathbb{C}{}$ of
pairing with the lowest/highest $k$-weight vector in the corresponding
irreducible component (Bergman space) $B_{k}{}(\mathbb{R}^{2}_{\pm}{})$,
$k\geq 2$ of the discrete series [Lang85]*Ch. VI. Then we can build an
operator $F$ from various $F_{k}$ similarly to the previous Example. In
particular, the jump representation (4.7) on the real line generalises to the
representation of a square integrable function $f$ on the upper half-plane as
a sum

$f(z)=\sum_{k}a_{k}f_{k}(z),\qquad f_{k}\in B_{n}{}(\mathbb{R}^{2}_{\pm}{})$

for prescribed coefficients $a_{k}$ and analytic functions $f_{k}$ in question
from different irreducible subspaces.

Covariant transform is also meaningful for principal and complementary series
of representations of the group $SL_{2}{}(\mathbb{R}{})$, which are not square
integrable [Kisil97c].

######  Example 4.10.

Let $G=\mathrm{SU}(2)\times\mathrm{Aff}$ be the Cartesian product of the
groups $\mathrm{SU}(2)$ of unitary rotations of $\mathbb{C}^{2}{}$ and the
$ax+b$ group $\mathrm{Aff}$. This group has a unitary linear representation on
the space $L_{2}{}(\mathbb{R}{},\mathbb{C}^{2}{})$ of square-integrable
(vector) $\mathbb{C}^{2}{}$-valued functions by the formula:

${\rho}{}(g)\begin{pmatrix}f_{1}(t)\\\
f_{2}(t)\end{pmatrix}=\begin{pmatrix}\alpha f_{1}(at+b)+\beta f_{2}(at+b)\\\
\mathsf{a}mmaf_{1}(at+b)+\delta f_{2}(at+b)\end{pmatrix},$

where $g=\begin{pmatrix}\alpha&\beta\\\
\mathsf{a}mma&\delta\end{pmatrix}\times(a,b)\in\mathrm{SU}(2)\times\mathrm{Aff}$.
It is obvious that the vector Hardy space, that is functions with both
components being analytic, is invariant under such action of $G$.

As a fiducial operator
$F:L_{2}{}(\mathbb{R}{},\mathbb{C}^{2}{})\rightarrow\mathbb{C}{}$ we can take,
cf. (4.6):

(4.9) $F\begin{pmatrix}f_{1}(t)\\\ f_{2}(t)\end{pmatrix}=\frac{1}{2\pi
i}\int_{\mathbb{R}{}}\frac{f_{1}(t)\,dt}{x-\mathrm{i}}.$

Thus the image of the associated covariant transform is a subspace of scalar
valued bounded functions on $G$. In this way we can transform (without a loss
of information) vector-valued problems, e.g. matrix _Wiener–Hopf
factorisation_ [BoetcherKarlovichSpitkovsky02a], to scalar question of
harmonic analysis on the group $G$.

######  Example 4.11.

A straightforward generalisation of Ex. 4.7 is obtained if $V$ is a Banach
space and $F:V\rightarrow\mathbb{C}{}$ is an element of $V^{*}$. Then the
covariant transform coincides with the construction of wavelets in Banach
spaces [Kisil98a].

######  Example 4.12.

The next stage of generalisation is achieved if $V$ is a Banach space and
$F:V\rightarrow\mathbb{C}^{n}{}$ is a linear operator. Then the corresponding
covariant transform is a map $\mathcal{W}:V\rightarrow
L{}(G,\mathbb{C}^{n}{})$. This is closely related to M.G. Krein’s works on
_directing functionals_ [Krein48a], see also _multiresolution wavelet
analysis_ [BratJorg97a], Clifford-valued Fock–Segal–Bargmann spaces
[CnopsKisil97a] and [AliAntGaz00]*Thm. 7.3.1.

######  Example 4.13.

Let $F$ be a projector $L_{p}{}(\mathbb{R}{})\rightarrow
L_{p}{}(\mathbb{R}{})$ defined by the relation $(Ff)\hat{\
}(\lambda)=\chi(\lambda)\hat{f}(\lambda)$, where the hat denotes the Fourier
transform and $\chi(\lambda)$ is the characteristic function of the set
$[-2,-1]\cup[1,2]$. Then the covariant transform
$L_{p}{}(\mathbb{R}{})\rightarrow C{}(\mathrm{Aff},L_{p}{}(\mathbb{R}{}))$
generated by the representation (4.5) of the affine group from $F$ contains
all information provided by the _Littlewood–Paley operator_ [Grafakos08]*§
5.1.1.

######  Example 4.14.

A step in a different direction is a consideration of non-linear operators.
Take again the “$ax+b$” group and its representation (4.5). We define $F$ to
be a homogeneous but non-linear functional $V\rightarrow\mathbb{R}_{+}{}$:

$F(f)=\frac{1}{2}\int\limits_{-1}^{1}\left|f(x)\right|\,dx.$

The covariant transform (4.1) becomes:

(4.10)
$[\mathcal{W}_{p}f](a,b)=F({\rho_{p}}(a,b)f)=\frac{1}{2}\int\limits_{-1}^{1}\left|a^{\frac{1}{p}}f\left(ax+b\right)\right|\,dx=a^{\frac{1}{p}}\frac{1}{2a}\int\limits^{b+a}_{b-a}\left|f\left(x\right)\right|\,dx.$

Obviously $M_{f}(b)=\max_{a}[\mathcal{W}_{\infty}f](a,b)$ coincides with the
Hardy _maximal function_ , which contains important information on the
original function $f$. From the Cor. 4.5 we deduce that the operator
$M:f\mapsto M_{f}$ intertwines ${\rho_{p}}$ with itself
${\rho_{p}}M=M{\rho_{p}}$.

Of course, the full covariant transform (4.10) is even more detailed than $M$.
For example, $\left\|f\right\|=\max_{b}[\mathcal{W}_{\infty}f](\frac{1}{2},b)$
is the shift invariant norm [Johansson08a].

######  Example 4.15.

Let $V=L_{c}{}(\mathbb{R}^{2}{})$ be the space of compactly supported bounded
functions on the plane. We take $F$ be the linear operator
$V\rightarrow\mathbb{C}{}$ of integration over the real line:

$F:f(x,y)\mapsto F(f)=\int_{\mathbb{R}{}}f(x,0)\,dx.$

Let $G$ be the group of Euclidean motions of the plane represented by ${\rho}$
on $V$ by a change of variables. Then the wavelet transform $F({\rho}(g)f)$ is
the _Radon transform_ [Helgason11a].

####  4.3. Symbolic Calculi

There is a very important class of the covariant transforms which maps
operators to functions. Among numerous sources we wish to single out works of
Berezin [Berezin72, Berezin86]. We start from the Berezin _covariant symbol_.

######  Example 4.16.

Let a representation ${\rho}$ of a group $G$ act on a space $X$. Then there is
an associated representation ${\rho_{B}}$ of $G$ on a space $V=B{}(X,Y)$ of
linear operators $X\rightarrow Y$ defined by the identity [Berezin86,
Kisil98a]:

(4.11) $({\rho_{B}}(g)A)x=A({\rho}(g^{-1})x),\qquad x\in X,\ g\in G,\ A\in
B{}(X,Y).$

Following the Remark 4.3 we take $F$ to be a functional
$V\rightarrow\mathbb{C}{}$, for example $F$ can be defined from a pair $x\in
X$, $l\in Y^{*}$ by the expression $F:A\mapsto\left\langle Ax,l\right\rangle$.
Then the covariant transform is:

$\mathcal{W}:A\mapsto\hat{A}(g)=F({\rho_{B}}(g)A).$

This is an example of _covariant calculus_ [Kisil98a, Berezin72].

There are several variants of the last Example which are of a separate
interest.

######  Example 4.17.

A modification of the previous construction is obtained if we have two groups
$G_{1}$ and $G_{2}$ represented by ${\rho_{1}}$ and ${\rho_{2}}$ on $X$ and
$Y^{*}$ respectively. Then we have a covariant transform $B{}(X,Y)\rightarrow
L{}(G_{1}\times G_{2},\mathbb{C}{})$ defined by the formula:

$\mathcal{W}:A\mapsto\hat{A}(g_{1},g_{2})=\left\langle
A{\rho_{1}}(g_{1})x,{\rho_{2}}(g_{2})l\right\rangle.$

This generalises the above _Berezin covariant calculi_ [Kisil98a].

######  Example 4.18.

Let us restrict the previous example to the case when $X=Y$ is a Hilbert
space, ${\rho_{1}}{}={\rho_{2}}{}={\rho}$ and $x=l$ with $\left\|x\right\|=1$.
Than the range of the covariant transform:

$\mathcal{W}:A\mapsto\hat{A}(g)=\left\langle
A{\rho}(g)x,{\rho}(g)x\right\rangle$

is a subset of the _numerical range_ of the operator $A$. As a function on a
group $\hat{A}(g)$ provides a better description of $A$ than the set of its
values—numerical range.

######  Example 4.19.

The group $\mathrm{SU}(1,1)\simeq SL_{2}{}(\mathbb{R}{})$ consists of $2\times
2$ matrices of the form $\begin{pmatrix}\alpha&\beta\\\
\bar{\beta}&\bar{\alpha}\end{pmatrix}$ with the unit determinant [Lang85]*§
IX.1. Let $T$ be an operator with the spectral radius less than $1$. Then the
associated Möbius transformation

(4.12) $g:T\mapsto g\cdot T=\frac{\alpha T+\beta
I}{\bar{\beta}T+\bar{\alpha}I},\qquad\text{where}\quad
g=\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}\in
SL_{2}{}(\mathbb{R}{}),\ $

produces a well-defined operator with the spectral radius less than $1$ as
well. Thus we have a representation of $SU(1,1)$.

Let us introduce the _defect operators_ $D_{T}=(I-T^{*}T)^{1/2}$ and
$D_{T^{*}}=(I-TT^{*})^{1/2}$. For the fiducial operator $F=D_{T^{*}}$ the
covariant transform is, cf. [NagyFoias70]*§ VI.1, (1.2):

${}[\mathcal{W}T](g)=F(g\cdot
T)=-e^{\mathrm{i}\phi}\,\Theta_{T}(z)\,D_{T},\qquad\text{for
}g=\begin{pmatrix}e^{\mathrm{i}\phi/2}&0\\\
0&e^{-\mathrm{i}\phi/2}\end{pmatrix}\begin{pmatrix}1&-z\\\
-\bar{z}&1\end{pmatrix},$

where the _characteristic function_ $\Theta_{T}(z)$ [NagyFoias70]*§ VI.1,
(1.1) is:

$\Theta_{T}(z)=-T+D_{T^{*}}\,(I-zT^{*})^{-1}\,z\,D_{T}.$

Thus we approached the _functional model_ of operators from the covariant
transform. In accordance with Remark 4.3 the model is most fruitful for the
case of operator $F=D_{T^{*}}$ being one-dimensional.

The intertwining property in the previous examples was obtained as a
consequence of the general Prop. 4.4 about the covariant transform. However it
may be worth to select it as a separate definition:

######  Definition 4.20.

A _covariant calculus_ , also known as _symbolic calculus_ , is a map from
operators to functions, which intertwines two representations of the same
group in the respective spaces.

There is a dual class of covariant transforms acting in the opposite
direction: from functions to operators. The prominent examples are the Berezin
_contravariant symbol_ [Berezin72, Kisil98a] and symbols of a
_pseudodifferential operators_ (PDO) [Howe80b, Kisil98a].

######  Example 4.21.

The classical _Riesz–Dunford functional calculus_ [DunfordSchwartzI]*§ VII.3
[Nikolskii86]*§ IV.2 maps analytical functions on the unit disk to the linear
operators, it is defined through the Cauchy-type formula with the resolvent.
The calculus is an intertwining operator [Kisil02a] between the Möbius
transformations of the unit disk, cf. (5.33), and the actions (4.12) on
operators from the Example 4.19. This topic will be developed in Subsection
6.1.

In line with the Defn. 4.20 we can directly define the corresponding calculus
through the intertwining property [Kisil95i, Kisil02a]:

######  Definition 4.22.

A _contravariant calculus_ , also know as _functional calculus_ , is a map
from functions to operators, which intertwines two representations of the same
group in the respective spaces.

The duality between co- and contravariant calculi is the particular case of
the duality between covariant transform and the inverse covariant transform
defined in the next Subsection. In many cases a proper choice of spaces makes
covariant and/or contravariant calculus a bijection between functions and
operators. Subsequently only one form of calculus, either co- or
contravariant, is defined explicitly, although both of them are there in fact.

####  4.4. Inverse Covariant Transform

An object invariant under the left action $\Lambda$ (4.2) is called _left
invariant_. For example, let $L$ and $L^{\prime}$ be two left invariant spaces
of functions on $G$. We say that a pairing
$\left\langle\cdot,\cdot\right\rangle:L\times
L^{\prime}\rightarrow\mathbb{C}{}$ is _left invariant_ if

(4.13) $\left\langle\Lambda(g)f,\Lambda(g)f^{\prime}\right\rangle=\left\langle
f,f^{\prime}\right\rangle,\quad\textrm{ for all }\quad f\in L,\ f^{\prime}\in
L^{\prime}.$

######  Remark 4.23.

  1. (i)

We do not require the pairing to be linear in general.

  2. (ii)

If the pairing is invariant on space $L\times L^{\prime}$ it is not
necessarily invariant (or even defined) on the whole $C{}(G)\times C{}(G)$.

  3. (iii)

In a more general setting we shall study an invariant pairing on a homogeneous
spaces instead of the group. However due to length constraints we cannot
consider it here beyond the Example 4.26.

  4. (iv)

An invariant pairing on $G$ can be obtained from an _invariant functional_ $l$
by the formula $\left\langle f_{1},f_{2}\right\rangle=l(f_{1}\bar{f}_{2})$.

For a representation ${\rho}$ of $G$ in $V$ and $v_{0}\in V$ we fix a function
$w(g)={\rho}(g)v_{0}$. We assume that the pairing can be extended in its
second component to this $V$-valued functions, say, in the weak sense.

######  Definition 4.24.

Let $\left\langle\cdot,\cdot\right\rangle$ be a left invariant pairing on
$L\times L^{\prime}$ as above, let ${\rho}$ be a representation of $G$ in a
space $V$, we define the function $w(g)={\rho}(g)v_{0}$ for $v_{0}\in V$. The
_inverse covariant transform_ $\mathcal{M}$ is a map $L\rightarrow V$ defined
by the pairing:

(4.14) $\mathcal{M}:f\mapsto\left\langle f,w\right\rangle,\qquad\text{ where
}f\in L.$

######  Example 4.25.

Let $G$ be a group with a unitary square integrable representation $\rho$. An
invariant pairing of two square integrable functions is obviously done by the
integration over the Haar measure:

$\left\langle f_{1},f_{2}\right\rangle=\int_{G}f_{1}(g)\bar{f}_{2}(g)\,dg.$

For an admissible vector $v_{0}$ [DufloMoore] [AliAntGaz00]*Chap. 8 the
inverse covariant transform is known in this setup as a _reconstruction
formula_.

######  Example 4.26.

Let $\rho$ be a square integrable representation of $G$ modulo a subgroup
$H\subset G$ and let $X=G/H$ be the corresponding homogeneous space with a
quasi-invariant measure $dx$. Then integration over $dx$ with an appropriate
weight produces an invariant pairing. The inverse covariant transform is a
more general version [AliAntGaz00]*(7.52) of the _reconstruction formula_
mentioned in the previous example.

Let $\rho$ be not a square integrable representation (even modulo a subgroup)
or let $v_{0}$ be inadmissible vector of a square integrable representation
$\rho$. An invariant pairing in this case is not associated with an
integration over any non singular invariant measure on $G$. In this case we
have a _Hardy pairing_. The following example explains the name.

######  Example 4.27.

Let $G$ be the “$ax+b$” group and its representation ${\rho}$ (4.5) from Ex.
4.8. An invariant pairing on $G$, which is not generated by the Haar measure
$a^{-2}da\,db$, is:

(4.15) $\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b)\,\bar{f}_{2}(a,b)\,db.$

For this pairing we can consider functions $\frac{1}{2\pi i(x+i)}$ or
$e^{-x^{2}}$, which are not admissible vectors in the sense of square
integrable representations. Then the inverse covariant transform provides an
_integral resolutions_ of the identity.

Similar pairings can be defined for other semi-direct products of two groups.
We can also extend a Hardy pairing to a group, which has a subgroup with such
a pairing.

######  Example 4.28.

Let $G$ be the group $SL_{2}{}(\mathbb{R}{})$ from the Ex. 4.9. Then the
“$ax+b$” group is a subgroup of $SL_{2}{}(\mathbb{R}{})$, moreover we can
parametrise $SL_{2}{}(\mathbb{R}{})$ by triples $(a,b,\theta)$,
$\theta\in(-\pi,\pi]$ with the respective Haar measure [Lang85]*III.1(3). Then
the Hardy pairing

(4.16) $\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b,\theta)\,\bar{f}_{2}(a,b,\theta)\,db\,d\theta.$

is invariant on $SL_{2}{}(\mathbb{R}{})$ as well. The corresponding inverse
covariant transform provides even a finer resolution of the identity which is
invariant under conformal mappings of the Lobachevsky half-plane.

###  5\. Analytic Functions

We saw in the first section that an inspiring geometry of cycles can be
recovered from the properties of $SL_{2}{}(\mathbb{R}{})$. In this section we
consider a realisation of the function theory within Erlangen approach
[Kisil97c, Kisil97a, Kisil01a, Kisil02c]. The covariant transform will be our
principal tool in this construction.

####  5.1. Induced Covariant Transform

The choice of a mother wavelet or fiducial operator $F$ from Section 4.1 can
significantly influence the behaviour of the covariant transform. Let $G$ be a
group and ${H}$ be its closed subgroup with the corresponding homogeneous
space $X=G/{H}$. Let ${\rho}$ be a representation of $G$ by operators on a
space $V$, we denote by ${\rho_{H}}$ the restriction of ${\rho}$ to the
subgroup $H$.

######  Definition 5.1.

Let $\chi$ be a representation of the subgroup ${H}$ in a space $U$ and
$F:V\rightarrow U$ be an intertwining operator between $\chi$ and the
representation ${\rho_{H}}$:

(5.1) $F({\rho}(h)v)=F(v)\chi(h),\qquad\text{ for all }h\in{H},\ v\in V.$

Then the covariant transform (4.1) generated by $F$ is called the _induced
covariant transform_

The following is the main motivating example.

######  Example 5.2.

Consider the traditional wavelet transform as outlined in Ex. 4.7. Chose a
vacuum vector $v_{0}$ to be a joint eigenvector for all operators ${\rho}(h)$,
$h\in H$, that is ${\rho}(h)v_{0}=\chi(h)v_{0}$, where $\chi(h)$ is a complex
number depending of $h$. Then $\chi$ is obviously a character of $H$.

The image of wavelet transform (4.3) with such a mother wavelet will have a
property:

$\hat{v}(gh)=\left\langle v,{\rho}(gh)v_{0}\right\rangle=\left\langle
v,{\rho}(g)\chi(h)v_{0}\right\rangle=\chi(h)\hat{v}(g).$

Thus the wavelet transform is uniquely defined by cosets on the homogeneous
space $G/H$. In this case we previously spoke about the _reduced wavelet
transform_ [Kisil97a]. A representation ${\rho_{0}}$ is called _square
integrable_ $\mod H$ if the induced wavelet transform $[\mathcal{W}f_{0}](w)$
of the vacuum vector $f_{0}(x)$ is square integrable on $X$.

The image of induced covariant transform have the similar property:

(5.2)
$\hat{v}(gh)=F({\rho}((gh)^{-1})v)=F({\rho}(h^{-1}){\rho}(g^{-1})v)=F({\rho}(g^{-1})v)\chi{}{}(h^{-1}).$

Thus it is enough to know the value of the covariant transform only at a
single element in every coset $G/H$ in order to reconstruct it for the entire
group $G$ by the representation $\chi$. Since coherent states (wavelets) are
now parametrised by points homogeneous space $G/H$ they are referred sometimes
as coherent states which are not connected to a group [Klauder96a], however
this is true only in a very narrow sense as explained above.

######  Example 5.3.

To make it more specific we can consider the representation of
$SL_{2}{}(\mathbb{R}{})$ defined on $L_{2}{}(\mathbb{R}{})$ by the formula,
cf. (3.8):

${\rho}(g):f(z)\mapsto\frac{1}{(cx+d)}\,f\left(\frac{ax+b}{cx+d}\right),\qquad
g^{-1}=\ \begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

Let $K\subset SL_{2}{}(\mathbb{R}{})$ be the compact subgroup of matrices
$h_{t}=\begin{pmatrix}\cos t&\sin t\\\ -\sin t&\cos t\end{pmatrix}$. Then for
the fiducial operator $F_{\pm}$ (4.6) we have
$F_{\pm}\circ{\rho}(h_{t})=e^{\mp\mathrm{i}t}F_{\pm}$. Thus we can consider
the covariant transform only for points in the homogeneous space
$SL_{2}{}(\mathbb{R}{})/K$, moreover this set can be naturally identified with
the $ax+b$ group. Thus we do not obtain any advantage of extending the group
in Ex. 4.8 from $ax+b$ to $SL_{2}{}(\mathbb{R}{})$ if we will be still using
the fiducial operator $F_{\pm}$ (4.6).

Functions on the group $G$, which have the property
$\hat{v}(gh)=\hat{v}(g)\chi(h)$ (5.2), provide a space for the representation
of $G$ induced by the representation $\chi$ of the subgroup $H$. This explains
the choice of the name for induced covariant transform.

######  Remark 5.4.

Induced covariant transform uses the fiducial operator $F$ which passes
through the action of the subgroup ${H}$. This reduces information which we
obtained from this transform in some cases.

There is also a simple connection between a covariant transform and right
shifts:

######  Proposition 5.5.

Let $G$ be a Lie group and ${\rho}$ be a representation of $G$ in a space $V$.
Let $[\mathcal{W}f](g)=F({\rho}(g^{-1})f)$ be a covariant transform defined by
the fiducial operator $F:V\rightarrow U$. Then the right shift
$[\mathcal{W}f](gg^{\prime})$ by $g^{\prime}$ is the covariant transform
$[\mathcal{W^{\prime}}f](g)=F^{\prime}({\rho}(g^{-1})f)]$ defined by the
fiducial operator $F^{\prime}=F\circ{\rho}(g^{-1})$.

In other words the covariant transform intertwines right shifts on the group
$G$ with the associated action ${\rho_{B}}$ (4.11) on fiducial operators.

Although the above result is obvious, its infinitesimal version has
interesting consequences.

######  Corollary 5.6 ([Kisil10c]).

Let $G$ be a Lie group with a Lie algebra $\mathfrak{g}$ and ${\rho}$ be a
smooth representation of $G$. We denote by $d{\rho_{B}}$ the derived
representation of the associated representation ${\rho_{B}}$ (4.11) on
fiducial operators.

Let a fiducial operator $F$ be a null-solution, i.e. $AF=0$, for the operator
$A=\sum_{J}a_{j}d{\rho^{X_{j}}_{B}}$, where $X_{j}\in\mathfrak{g}$ and $a_{j}$
are constants. Then the covariant transform
$[\mathcal{W}f](g)=F({\rho}(g^{-1})f)$ for any $f$ satisfies:

$DF(g)=0,\qquad\text{where}\quad D=\sum_{j}\bar{a}_{j}\mathfrak{L}^{X_{j}}.$

Here $\mathfrak{L}^{X_{j}}$ are the left invariant fields (Lie derivatives) on
$G$ corresponding to $X_{j}$.

######  Example 5.7.

Consider the representation ${\rho}$ (4.5) of the $ax+b$ group with the $p=1$.
Let $A$ and $N$ be the basis of the corresponding Lie algebra generating one-
parameter subgroups $(e^{t},0)$ and $(0,t)$. Then the derived representations
are:

$[d{\rho^{A}}f](x)=f(x)+xf^{\prime}(x),\qquad[d{\rho^{N}}f](x)=f^{\prime}(x).$

The corresponding left invariant vector fields on $ax+b$ group are:

$\mathfrak{L}^{A}=a\partial_{a},\qquad\mathfrak{L}^{N}=a\partial_{b}.$

The mother wavelet $\frac{1}{x+\mathrm{i}}$ is a null solution of the operator
$d{\rho^{A}}+\mathrm{i}d{\rho^{N}}=I+(x+\mathrm{i})\frac{d}{dx}$. Therefore
the covariant transform with the fiducial operator $F_{+}$ (4.6) will consist
with the null solutions to the operator
$\mathfrak{L}^{A}-\mathrm{i}\mathfrak{L}^{N}=-\mathrm{i}a(\partial_{b}+\mathrm{i}\partial_{a})$,
that is in the essence the Cauchy-Riemann operator in the upper half-plane.

There is a statement which extends the previous Corollary from differential
operators to integro-differential ones. We will formulate it for the wavelets
setting.

######  Corollary 5.8.

Let $G$ be a group and ${\rho}$ be a unitary representation of $G$, which can
be extended to a vector space $V$ of functions or distributions on $G$. Let a
mother wavelet $w\in V^{\prime}$ satisfy the equation

$\int_{G}a(g)\,{\rho}(g)w\,dg=0,$

for a fixed distribution $a(g)\in V$ and a (not necessarily invariant) measure
$dg$. Then any wavelet transform $F(g)=\mathcal{W}f(g)=\left\langle
f,{\rho}(g)w_{0}\right\rangle$ obeys the condition:

$DF=0,\qquad\text{where}\quad D=\int_{G}\bar{a}(g)\,R(g)\,dg,$

with $R$ being the right regular representation of $G$.

Clearly, the Corollary 5.6 is a particular case of the Corollary 5.8 with a
distribution $a$, which is a combination of derivatives of Dirac’s delta
functions. The last Corollary will be illustrated at the end of Section 6.1.

######  Remark 5.9.

We note that Corollaries 5.6 and 5.8 are true whenever we have an intertwining
property between ${\rho}$ with the right regular representation of $G$.

####  5.2. Induced Wavelet Transform and Cauchy Integral

We again use the general scheme from Subsection 3.2. The $ax+b$ group is
isomorphic to a subgroups of $SL_{2}{}(\mathbb{R}{})$ consisting of the lower-
triangular matrices:

$F=\left\\{\frac{1}{\sqrt{a}}\begin{pmatrix}a&0\\\ b&1\end{pmatrix},\
a>0\right\\}.$

The corresponding homogeneous space $X=SL_{2}{}(\mathbb{R}{})/F$ is one-
dimensional and can be parametrised by a real number. The natural projection
$p:SL_{2}{}(\mathbb{R}{})\rightarrow\mathbb{R}{}$ and its left inverse
$s:\mathbb{R}{}\rightarrow SL_{2}{}(\mathbb{R}{})$ can be defined as follows:

(5.3) $p:\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto\frac{b}{d},\qquad
s:u\mapsto\begin{pmatrix}1&u\\\ 0&1\end{pmatrix}.$

Thus we calculate the corresponding map $r:SL_{2}{}(\mathbb{R}{})\rightarrow
F$, see Subsection 2.1:

(5.4) $r:\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}d^{-1}&0\\\ c&d\end{pmatrix}.$

Therefore the action of $SL_{2}{}(\mathbb{R}{})$ on the real line is exactly
the Möbius map (1.1):

$g:u\mapsto p(g^{-1}*s(u))=\frac{au+b}{cu+d},\quad\text{ where
}g^{-1}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

We also calculate that

$r(g^{-1}*s(u))=\begin{pmatrix}(cu+d)^{-1}&0\\\ c&cu+d\end{pmatrix}.$

To build an induced representation we need a character of the affine group. A
generic character of $F$ is a power of its diagonal element:

${\rho_{\kappa}}\begin{pmatrix}a&0\\\ c&a^{-1}\end{pmatrix}=a^{\kappa}.$

Thus the corresponding realisation of induced representation (3.6) is:

(5.5)
${\rho_{\kappa}}(g):f(u)\mapsto\frac{1}{(cu+d)^{\kappa}}\,f\left(\frac{au+b}{cu+d}\right)\qquad\text{
where }g^{-1}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

The only freedom remaining by the scheme is in a choice of a value of _number_
$\kappa$ and the corresponding functional space where our representation acts.
At this point we have a wider choice of $\kappa$ than it is usually assumed:
it can belong to different hypercomplex systems.

One of the important properties which would be nice to have is the unitarity
of the representation (5.5) with respect to the standard inner product:

$\left\langle
f_{1},f_{2}\right\rangle=\int_{\mathbb{R}^{2}{}}f_{1}(u)\bar{f}_{2}(u)\,du.$

A change of variables $x=\frac{au+b}{cu+d}$ in the integral suggests the
following property is necessary and sufficient for that:

(5.6) $\kappa+\bar{\kappa}=2.$

A mother wavelet for an induced wavelet transform shall be an eigenvector for
the action of a subgroup $\tilde{H}$ of $SL_{2}{}(\mathbb{R}{})$, see (5.1).
Let us consider the most common case of $\tilde{H}=K$ and take the
infinitesimal condition with the derived representation:
$d{\rho^{Z}_{n}}w_{0}=\lambda w_{0}$, since $Z$ (3.12) is the generator of the
subgroup $K$. In other word the restriction of $w_{0}$ to a $K$-orbit should
be given by $e^{\lambda t}$ in the exponential coordinate $t$ along the
$K$-orbit. However we usually need its expression in other “more natural”
coordinates. For example [Kisil11b], an eigenvector of the derived
representation of $d{\rho^{Z}_{n}}$ should satisfy the differential equation
in the ordinary parameter $x\in\mathbb{R}{}$:

(5.7) $-\kappa xf(x)-f^{\prime}(x)(1+x^{2})=\lambda f(x).$

The equation does not have singular points, the general solution is globally
defined (up to a constant factor) by:

(5.8)
$w_{\lambda,\kappa}(x)=\frac{1}{(1+x^{2})^{\kappa/2}}\left(\frac{x-\mathrm{i}}{x+\mathrm{i}}\right)^{\mathrm{i}\lambda/2}=\frac{(x-\mathrm{i})^{(\mathrm{i}\lambda-\kappa)/2}}{(x+\mathrm{i})^{(\mathrm{i}\lambda+\kappa)/2}}.$

To avoid multivalent functions we need $2\pi$-periodicity along the
exponential coordinate on $K$. This implies that the parameter
$m=-\mathrm{i}\lambda$ is an integer. Therefore the solution becomes:

(5.9)
$w_{m,\kappa}(x)=\frac{(x+\mathrm{i})^{(m-\kappa)/2}}{(x-\mathrm{i})^{(m+\kappa)/2}}.$

The corresponding wavelets resemble the Cauchy kernel normalised to the
invariant metric in the Lobachevsky half-plane:

$\displaystyle w_{m,\kappa}(u,v;x)$ $\displaystyle=$
$\displaystyle{\rho^{F}_{\kappa}}(s(u,v))w_{m,\kappa}(x)=v^{\kappa/2}\frac{\left(x-u+\mathrm{i}v\right)^{(m-\kappa)/2}}{\left(x-u-\mathrm{i}v\right)^{(m+\kappa)/2}}$

Therefore the wavelet transform (4.3) from function on the real line to
functions on the upper half-plane is:

$\displaystyle\hat{f}(u,v)$ $\displaystyle=$ $\displaystyle\left\langle
f,{\rho^{F}_{\kappa}}(u,v)w_{m,\kappa}\right\rangle=v^{\bar{\kappa}/2}\int_{\mathbb{R}{}}f(x)\,\frac{(x-(u+\mathrm{i}v))^{(m-\kappa)/2}}{(x-(u-\mathrm{i}v))^{(m+\kappa)/2}}\,dx.$

Introduction of a complex variable $z=u+\mathrm{i}v$ allows to write it as:

(5.10) $\hat{f}(z)=(\Im
z)^{\bar{\kappa}/2}\int_{\mathbb{R}{}}f(x)\frac{(x-{z})^{(m-\kappa)/2}}{(x-\bar{z})^{(m+\kappa)/2}}\,dx.$

According to the general theory this wavelet transform intertwines
representations ${\rho^{F}_{\kappa}}$ (5.5) on the real line (induced by the
character $a^{\kappa}$ of the subgroup $F$) and ${\rho^{K}_{m}}$ (3.8) on the
upper half-plane (induced by the character $e^{\mathrm{i}mt}$ of the subgroup
$K$).

####  5.3. The Cauchy-Riemann (Dirac) and Laplace Operators

Ladder operators $L^{\\!\pm}=\pm\mathrm{i}A+B$ act by raising/lowering indexes
of the $K$-eigenfunctions $w_{m,\kappa}$ (5.8), see Subsection 3.3. More
explicitly [Kisil11b]:

(5.11)
$d{\rho^{L^{\\!\pm}}_{\kappa}}:w_{m,\kappa}\mapsto-\frac{\mathrm{i}}{2}(m\pm\kappa)w_{m\pm
2,\kappa}.$

There are two possibilities here: $m\pm\kappa$ is zero for some $m$ or not. In
the first case the chain (5.11) of eigenfunction $w_{m,\kappa}$ terminates on
one side under the transitive action (3.16) of the ladder operators; otherwise
the chain is infinite in both directions. That is, the values $m=\mp\kappa$
and only those correspond to the maximal (minimal) weight function
$w_{\mp\kappa,\kappa}(x)=\frac{1}{(x\pm\mathrm{i})^{\kappa}}\in
L_{2}{}(\mathbb{R}{})$, which are annihilated by $L^{\\!\pm}$:

(5.12) $\displaystyle
d{\rho^{L^{\\!\pm}}_{\kappa}}w_{\mp\kappa,\kappa}=(\pm\mathrm{i}d{\rho^{A}_{\kappa}}+d{\rho^{B}_{\kappa}})\,w_{\mp\kappa,\kappa}=0.$

By the Cor. 5.6 for the mother wavelets $w_{\mp\kappa,\kappa}$, which are
annihilated by (5.12), the images of the respective wavelet transforms are
null solutions to the left-invariant differential operator
$D_{\pm}=\overline{\mathfrak{L}^{{L^{\\!\pm}}}}$:

(5.13)
$D_{\pm}=\mp\mathrm{i}\mathfrak{L}^{A}+\mathfrak{L}^{B}=\textstyle-\frac{\mathrm{i}\kappa}{2}+v(\partial_{u}\pm\mathrm{i}\partial_{v}).$

This is a conformal version of the Cauchy–Riemann equation. The second order
conformal Laplace-type operators
$\Delta_{+}=\overline{\mathfrak{L}^{L^{\\!-}}\mathfrak{L}^{L^{\\!+}}}$ and
$\Delta_{-}=\overline{\mathfrak{L}^{L^{\\!+}}\mathfrak{L}^{L^{\\!-}}}$ are:

(5.14)
$\Delta_{\pm}=\textstyle(v\partial_{u}-\frac{\mathrm{i}\kappa}{2})^{2}+v^{2}\partial_{v}^{2}\pm\frac{\kappa}{2}.$

For the mother wavelets $w_{m,\kappa}$ in (5.12) such that $m=\mp\kappa$ the
unitarity condition $\kappa+\bar{\kappa}=2$, see (5.6), together with
$m\in\mathbb{Z}{}$ implies $\kappa=\mp m=1$. In such a case the wavelet
transforms (5.10) are:

(5.15) $\hat{f}^{+}(z)=(\Im
z)^{\frac{1}{2}}\int_{\mathbb{R}{}}\frac{f(x)\,dx}{x-z}\quad\text{and}\quad\hat{f}^{-}(z)=(\Im
z)^{\frac{1}{2}}\int_{\mathbb{R}{}}\frac{f(x)\,dx}{x-\bar{z}},$

for $w_{-1,1}$ and $w_{1,1}$ respectively. The first one is the Cauchy
integral formula up to the factor $2\pi\mathrm{i}\sqrt{\Im z}$. Clearly, one
integral is the complex conjugation of another. Moreover, the minimal/maximal
weight cases can be intertwined by the following automorphism of the Lie
algebra $\mathfrak{sl}_{2}$:

$A\rightarrow B,\quad B\rightarrow A,\quad Z\rightarrow-Z.$

As explained before $\hat{f}^{\pm}(w)$ are null solutions to the operators
$D_{\pm}$ (5.13) and $\Delta_{\pm}$ (5.14). These transformations intertwine
unitary equivalent representations on the real line and on the upper half-
plane, thus they can be made unitary for proper spaces. This is the source of
two faces of the Hardy spaces: they can be defined either as square-integrable
on the real line with an analytic extension to the half-plane, or analytic on
the half-plane with square-integrability on an infinitesimal displacement of
the real line.

For the third possibility, $m\pm\kappa\neq 0$, there is no an operator spanned
by the derived representation of the Lie algebra $\mathfrak{sl}_{2}$ which
kills the mother wavelet $w_{m,\kappa}$. However the remarkable _Casimir
operator_ $C=Z^{2}-2(L^{\\!-}L^{\\!+}+L^{\\!+}L^{\\!-})$, which spans the
centre of the universal enveloping algebra of $\mathfrak{sl}_{2}$
[MTaylor86]*§ 8.1 [Lang85]*§ X.1, produces a second order operator which does
the job. Indeed from the identities (5.11) we get:

(5.16) $d{\rho^{C}_{\kappa}}w_{m,\kappa}=(2\kappa-\kappa^{2})w_{m,\kappa}.$

Thus we get $d{\rho^{C}_{\kappa}}w_{m,2}=0$ for $\kappa=2$ or $0$. The mother
wavelet $w_{0,2}$ turns to be the _Poisson kernel_ [Grafakos08]*Ex. 1.2.17.
The associated wavelet transform

(5.17) $\hat{f}(w)=\Im
z\int_{\mathbb{R}{}}\frac{f(x)\,dz}{\left|x-z\right|^{2}}$

consists of null solutions of the left-invariant second-order Laplacian, image
of the Casimir operator, cf. (5.14):

$\Delta(:=\mathfrak{L}^{C})=v^{2}\partial^{2}_{u}+v^{2}\partial_{v}^{2}.$

Another integral formula producing solutions to this equation delivered by the
mother wavelet $w_{m,0}$ with the value $\kappa=0$ in (5.16):

(5.18)
$\hat{f}(z)=\int_{\mathbb{R}{}}f(x)\left(\frac{x-{z}}{x-\bar{z}}\right)^{m/2}\,dx.$

Furthermore, we can introduce higher order differential operators. The
functions $w_{\mp 2m+1,1}$ are annihilated by $n$-th power of operator
$d{\rho^{L^{\\!\pm}}_{\kappa}}$ with $1\leq m\leq n$. By the Cor. 5.6 the the
image of wavelet transform (5.10) from a mother wavelet
$\sum_{1}^{n}a_{m}w_{\mp 2m,1}$ will consist of null-solutions of the $n$-th
power $D^{n}_{\pm}$ of the conformal Cauchy–Riemann operator (5.13). They are
a conformal flavour of _polyanalytic_ functions [Balk97a].

We can similarly look for mother wavelets which are eigenvectors for other
types of one dimensional subgroups. Our consideration of subgroup $K$ is
simplified by several facts:

  * •

The parameter $\kappa$ takes only complex values.

  * •

The derived representation does not have singular points on the real line.

For both subgroups $A\\!^{\prime}$ and $N^{\prime}$ this will not be true. The
further consideration will be given in [Kisil11b].

####  5.4. The Taylor Expansion

Consider an induced wavelet transform generated by a Lie group $G$, its
representation ${\rho}$ and a mother wavelet $w$ which is an eigenvector of a
one-dimensional subgroup $\tilde{H}\subset G$. Then by Prop. 5.5 the wavelet
transform intertwines ${\rho}$ with a representation ${\rho^{\tilde{H}}}$
induced by a character of $\tilde{H}$.

If the mother wavelet is itself in the domain of the induced wavelet transform
then the chain (3.16) of $\tilde{H}$-eigenvectors $w_{m}$ will be mapped to
the similar chain of their images $\hat{w}_{m}$. The corresponding derived
induced representation $d{\rho^{\tilde{H}}}$ produces ladder operators with
the transitive action of the ladder operators on the chain of $\hat{w}_{m}$.
Then the vector space of “formal power series”:

(5.19) $\hat{f}(z)=\sum_{m\in\mathbb{Z}{}}a_{m}\hat{w}_{m}(z)$

is a module for the Lie algebra of the group $G$.

Coming back to the case of the group $G=SL_{2}{}(\mathbb{R}{})$ and subgroup
$\tilde{H}=K$. Images $\hat{w}_{m,1}$ of the eigenfunctions (5.9) under the
Cauchy integral transform (5.15) are:

$\hat{w}_{m,1}(z)=(\Im
z)^{1/2}\frac{(z+\mathrm{i})^{(m-1)/2}}{(z-\mathrm{i})^{(m+1)/2}}.$

They are eigenfunctions of the derived representation on the upper half-plane
and the action of ladder operators is given by the same expressions (5.11). In
particular, the $\mathfrak{sl}_{2}$-module generated by $\hat{w}_{1,1}$ will
be one-sided since this vector is annihilated by the lowering operator. Since
the Cauchy integral produces an unitary intertwining operator between two
representations we get the following variant of Taylor series:

$\hat{f}(z)=\sum_{m=0}^{\infty}c_{m}\hat{w}_{m,1}(z),\qquad\text{ where }\quad
c_{m}=\left\langle f,w_{m,1}\right\rangle.$

For two other types of subgroups, representations and mother wavelets this
scheme shall be suitably adapted and detailed study will be presented
elsewhere [Kisil11b].

####  5.5. Wavelet Transform in the Unit Disk and Other Domains

We can similarly construct an analytic function theories in unit disks,
including parabolic and hyperbolic ones [Kisil05a]. This can be done simply by
an application of the _Cayley transform_ to the function theories in the upper
half-plane. Alternatively we can apply the full procedure for properly chosen
groups and subgroups. We will briefly outline such a possibility here, see
also [Kisil97c].

Elements of $SL_{2}{}(\mathbb{R}{})$ could be also represented by $2\times
2$-matrices with complex entries such that, cf. Example 4.21:

$g={\left(\\!\\!\begin{array}[]{cc}\alpha&\bar{\beta}\\\
\beta&\bar{\alpha}\end{array}\\!\\!\right)},\qquad
g^{-1}={\left(\\!\\!\begin{array}[]{cc}\bar{\alpha}&-\bar{\beta}\\\
-\beta&\alpha\end{array}\\!\\!\right)},\qquad\left|\alpha\right|^{2}-\left|\beta\right|^{2}=1.$

This realisations of $SL_{2}{}(\mathbb{R}{})$ (or rather $SU(2,\mathbb{C}{})$)
is more suitable for function theory in the unit disk. It is obtained from the
form, which we used before for the upper half-plane, by means of the Cayley
transform [Kisil05a]*§ 8.1.

We may identify the _unit disk_ $\mathbb{D}{}$ with the homogeneous space
$SL_{2}{}(\mathbb{R}{})/\mathbb{T}{}$ for the _unit circle_ $\mathbb{T}{}$
through the important decomposition
$SL_{2}{}(\mathbb{R}{})\sim\mathbb{D}{}\times\mathbb{T}{}$ with
$K=\mathbb{T}{}$—the compact subgroup of $SL_{2}{}(\mathbb{R}{})$:

(5.26) $\displaystyle{\left(\\!\\!\begin{array}[]{cc}\alpha&\bar{\beta}\\\
\beta&\bar{\alpha}\end{array}\\!\\!\right)}$ $\displaystyle=$
$\displaystyle\left|\alpha\right|{\left(\\!\\!\begin{array}[]{cc}1&\bar{\beta}\bar{\alpha}^{-1}\\\
{\beta}{\alpha}^{-1}&1\end{array}\\!\\!\right)}{\left(\\!\\!\begin{array}[]{cc}\frac{{\alpha}}{\left|\alpha\right|}&0\\\
0&\frac{\bar{\alpha}}{\left|\alpha\right|}\end{array}\\!\\!\right)}$ (5.31)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-\left|u\right|^{2}}}{\left(\\!\\!\begin{array}[]{cc}1&u\\\
\bar{u}&1\end{array}\\!\\!\right)},{\left(\\!\\!\begin{array}[]{cc}e^{ix}&0\\\
0&e^{-ix}\end{array}\\!\\!\right)}$

where

$x=\arg\alpha,\qquad u=\bar{\beta}\bar{\alpha}^{-1},\qquad\left|u\right|<1.$

Each element $g\in SL_{2}{}(\mathbb{R}{})$ acts by the linear-fractional
transformation (the Möbius map) on $\mathbb{D}{}$ and $\mathbb{T}{}$
$H_{2}{}(\mathbb{T}{})$ as follows:

(5.32) $g:z\mapsto\frac{\alpha
z+\beta}{{\bar{\beta}}z+\bar{\alpha}},\qquad\textrm{ where }\quad
g={\left(\\!\\!\begin{array}[]{cc}\alpha&\beta\\\
\bar{\beta}&\bar{\alpha}\end{array}\\!\\!\right)}.$

In the decomposition (5.26) the first matrix on the right hand side acts by
transformation (5.32) as an orthogonal rotation of $\mathbb{T}{}$ or
$\mathbb{D}{}$; and the second one—by transitive family of maps of the unit
disk onto itself.

The representation induced by a complex-valued character $\chi_{k}(z)=z^{-k}$
of $\mathbb{T}{}$ according to the Section 3.2 is:

(5.33)
$\rho_{k}(g):f(z)\mapsto\frac{1}{(\alpha-{\beta}{z})^{k}}\,f\left(\frac{\bar{\alpha}z-\bar{\beta}}{\alpha-{\beta}z}\right)\qquad\textrm{
where }\quad g={\left(\\!\\!\begin{array}[]{cc}\alpha&\beta\\\
\bar{\beta}&\bar{\alpha}\end{array}\\!\\!\right)}.$

The representation ${\rho_{1}}$ is unitary on square-integrable functions and
irreducible on the _Hardy space_ on the unit circle.

We choose [Kisil98a, Kisil01a] $K$-invariant function $v_{0}(z)\equiv 1$ to be
a _vacuum vector_. Thus the associated _coherent states_

$v(g,z)=\rho_{1}(g)v_{0}(z)=(u-z)^{-1}$

are completely determined by the point on the unit disk
$u=\bar{\beta}\bar{\alpha}^{-1}$. The family of coherent states considered as
a function of both $u$ and $z$ is obviously the _Cauchy kernel_ [Kisil97c].
The _wavelet transform_ [Kisil97c, Kisil98a]
$\mathcal{W}:L_{2}{}(\mathbb{T}{})\rightarrow
H_{2}{}(\mathbb{D}{}):f(z)\mapsto\mathcal{W}f(g)=\left\langle
f,v_{g}\right\rangle$ is the _Cauchy integral_ :

(5.34) $\mathcal{W}f(u)=\frac{1}{2\pi
i}\int_{\mathbb{T}{}}f(z)\frac{1}{u-z}\,dz.$

This approach can be extended to arbitrary connected simply-connected domain.
Indeed, it is known that Möbius maps is the whole group of biholomorphic
automorphisms of the unit disk or upper half-plane. Thus we can state the
following corollary from the _Riemann mapping theorem_ :

######  Corollary 5.10.

The group of biholomorphic automorphisms of a connected simply connected
domain with at least two points on its boundary is isomorphic to
$SL_{2}{}(\mathbb{R}{})$.

If a domain is non-simply connected, then the group of its biholomorphic
mapping can be trivial [MityushevRogosin00a, Beardon07a]. However we may look
for a rich group acting on function spaces rather than on geometric sets. Let
a connected non-simply connected domain $D$ be bounded by a finite collection
of non-intersecting contours $\mathsf{a}mma_{i}$, $i=1,\ldots,n$. For each
$\mathsf{a}mma_{i}$ consider the isomorphic image $G_{i}$ of the
$SL_{2}{}(\mathbb{R}{})$ group which is defined by the Corollary 5.10. Then
define the group $G=G_{1}\times G_{2}\times\ldots\times G_{n}$ and its action
on $L_{2}{}(\partial D)=L_{2}{}(\mathsf{a}mma_{1})\oplus
L_{2}{}(\mathsf{a}mma_{2})\oplus\ldots\oplus L_{2}{}(\mathsf{a}mma_{n})$
through the Moebius action of $G_{i}$ on $L_{2}{}(\mathsf{a}mma_{i})$.

######  Example 5.11.

Consider an _annulus_ defined by $r<\left|z\right|<R$. It is bounded by two
circles: $\mathsf{a}mma_{1}=\\{z:\left|z\right|=r\\}$ and
$\mathsf{a}mma_{2}=\\{z:\left|z\right|=R\\}$. For $\mathsf{a}mma_{1}$ the
Möbius action of $SL_{2}{}(\mathbb{R}{})$ is

$\begin{pmatrix}\alpha&\bar{\beta}\\\
\beta&\bar{\alpha}\end{pmatrix}:z\mapsto\frac{\alpha z+\bar{\beta}/r}{\beta
z/r+\bar{\alpha}},\qquad\text{where}\quad\left|\alpha\right|^{2}-\left|\beta\right|^{2}=1,$

with the respective action on $\mathsf{a}mma_{2}$. Those action can be
linearised in the spaces $L_{2}{}(\mathsf{a}mma_{1})$ and
$L_{2}{}(\mathsf{a}mma_{2})$. If we consider a subrepresentation reduced to
analytic function on the annulus, then one copy of $SL_{2}{}(\mathbb{R}{})$
will act on the part of functions analytic outside of $\mathsf{a}mma_{1}$ and
another copy—on the part of functions analytic inside of $\mathsf{a}mma_{2}$.

Thus all classical objects of complex analysis (the Cauchy-Riemann equation,
the Taylor series, the Bergman space, etc.) for a rather generic domain $D$
can be also obtained from suitable representations similarly to the case of
the upper half-plane [Kisil97c, Kisil01a].

###  6\. Covariant and Contravariant Calculi

United in the trinity functional calculus, spectrum, and spectral mapping
theorem play the exceptional rôle in functional analysis and could not be
substituted by anything else. Many traditional definitions of functional
calculus are covered by the following rigid template based on the _algebra
homomorphism_ property:

######  Definition 6.1.

An _functional calculus_ for an element $a\in\mathfrak{A}$ is a continuous
linear mapping $\Phi:\mathcal{A}\rightarrow\mathfrak{A}$ such that

  1. (i)

$\Phi$ is a unital _algebra homomorphism_

$\Phi(f\cdot g)=\Phi(f)\cdot\Phi(g).$

  2. (ii)

There is an initialisation condition: $\Phi[v_{0}]=a$ for for a fixed function
$v_{0}$, e.g. $v_{0}(z)=z$.

The most typical definition of the spectrum is seemingly independent and uses
the important notion of resolvent:

######  Definition 6.2.

A _resolvent_ of element $a\in\mathfrak{A}$ is the function
$R(\lambda)=(a-\lambda e)^{-1}$, which is the image under $\Phi$ of the Cauchy
kernel $(z-\lambda)^{-1}$.

A _spectrum_ of $a\in\mathfrak{A}$ is the set $\mathbf{sp}\,a$ of singular
points of its resolvent $R(\lambda)$.

Then the following important theorem links spectrum and functional calculus
together.

######  Theorem 6.3 (Spectral Mapping).

For a function $f$ suitable for the functional calculus:

(6.1) $f(\mathbf{sp}\,a)=\mathbf{sp}\,f(a).$

However the power of the classic spectral theory rapidly decreases if we move
beyond the study of one normal operator (e.g. for quasinilpotent ones) and is
virtually nil if we consider several non-commuting ones. Sometimes these
severe limitations are seen to be irresistible and alternative constructions,
i.e. model theory cf. Example 4.19 and [Nikolskii86], were developed.

Yet the spectral theory can be revived from a fresh start. While three
components—functional calculus, spectrum, and spectral mapping theorem—are
highly interdependent in various ways we will nevertheless arrange them as
follows:

  1. (i)

Functional calculus is an _original_ notion defined in some independent terms;

  2. (ii)

Spectrum (or more specifically _contravariant spectrum_) (or spectral
decomposition) is derived from previously defined functional calculus as its
_support_ (in some appropriate sense);

  3. (iii)

Spectral mapping theorem then should drop out naturally in the form (6.1) or
some its variation.

Thus the entire scheme depends from the notion of the functional calculus and
our ability to escape limitations of Definition 6.1. The first known to the
present author definition of functional calculus not linked to algebra
homomorphism property was the Weyl functional calculus defined by an integral
formula [Anderson69]. Then its intertwining property with affine
transformations of Euclidean space was proved as a theorem. However it seems
to be the only “non-homomorphism” calculus for decades.

The different approach to whole range of calculi was given in [Kisil95i] and
developed in [Kisil98a, Kisil02a, Kisil04d, Kisil10c] in terms of
_intertwining operators_ for group representations. It was initially targeted
for several non-commuting operators because no non-trivial algebra
homomorphism is possible with a commutative algebra of function in this case.
However it emerged later that the new definition is a useful replacement for
classical one across all range of problems.

In the following Subsections we will support the last claim by consideration
of the simple known problem: characterisation a $n\times n$ matrix up to
similarity. Even that “freshman” question could be only sorted out by the
classical spectral theory for a small set of diagonalisable matrices. Our
solution in terms of new spectrum will be full and thus unavoidably coincides
with one given by the Jordan normal form of matrices. Other more difficult
questions are the subject of ongoing research.

####  6.1. Intertwining Group Actions on Functions and Operators

Any functional calculus uses properties of _functions_ to model properties of
_operators_. Thus changing our viewpoint on functions, as was done in Section
5, we could get another approach to operators. The two main possibilities are
encoded in Definitions 4.20 and 4.22: we can assign a certain function to the
given operator or wise verse. Here we consider the second possibility and
treat the first in the Subsection 6.4.

The representation ${\rho_{1}}$ (5.33) is unitary irreducible when acts on the
Hardy space $H_{2}{}$. Consequently we have one more reason to abolish the
template definition 6.1: $H_{2}{}$ is _not_ an algebra. Instead we replace the
_homomorphism property_ by a _symmetric covariance_ :

######  Definition 6.4 ([Kisil95i]).

An _contravariant analytic calculus_ for an element $a\in\mathfrak{A}$ and an
$\mathfrak{A}$-module $M$ is a _continuous linear_ mapping
$\Phi:A{}(\mathbb{D}{})\rightarrow A{}(\mathbb{D}{},M)$ such that

  1. (i)

$\Phi$ is an _intertwining operator_

$\Phi\rho_{1}=\rho_{a}\Phi$

between two representations of the $SL_{2}{}(\mathbb{R}{})$ group $\rho_{1}$
(5.33) and $\rho_{a}$ defined below in (6.1).

  2. (ii)

There is an initialisation condition: $\Phi[v_{0}]=m$ for $v_{0}(z)\equiv 1$
and $m\in M$, where $M$ is a left $\mathfrak{A}$-module.

Note that our functional calculus released from the homomorphism condition can
take value in any left $\mathfrak{A}$-module $M$, which however could be
$\mathfrak{A}$ itself if suitable. This add much flexibility to our
construction.

The earliest functional calculus, which is _not_ an algebraic homomorphism,
was the Weyl functional calculus and was defined just by an integral formula
as an operator valued distribution [Anderson69]. In that paper (joint)
spectrum was defined as support of the Weyl calculus, i.e. as the set of point
where this operator valued distribution does not vanish. We also define the
spectrum as a support of functional calculus, but due to our Definition 6.4 it
will means the set of non-vanishing intertwining operators with primary
subrepresentations.

######  Definition 6.5.

A corresponding _spectrum_ of $a\in\mathfrak{A}$ is the _support_ of the
functional calculus $\Phi$, i.e. the collection of intertwining operators of
$\rho_{a}$ with _primary representations_ [Kirillov76]*§ 8.3.

More variations of contravariant functional calculi are obtained from other
groups and their representations [Kisil95i, Kisil98a, Kisil02a, Kisil04d,
Kisil10c].

A simple but important observation is that the Möbius transformations (1.1)
can be easily extended to any Banach algebra. Let $\mathfrak{A}$ be a Banach
algebra with the unit $e$, an element $a\in\mathfrak{A}$ with
$\left\|a\right\|<1$ be fixed, then

(6.2) $g:a\mapsto g\cdot a=(\bar{\alpha}a-\bar{\beta}e)(\alpha e-\beta
a)^{-1},\qquad g\in SL_{2}{}(\mathbb{R}{})$

is a well defined $SL_{2}{}(\mathbb{R}{})$ action on a subset
$\mathbb{A}{}=\\{g\cdot a\,\mid\,g\in
SL_{2}{}(\mathbb{R}{})\\}\subset\mathfrak{A}$, i.e. $\mathbb{A}{}$ is a
$SL_{2}{}(\mathbb{R}{})$-homogeneous space. Let us define the _resolvent_
function $R(g,a):\mathbb{A}{}\rightarrow\mathfrak{A}$:

$R(g,a)=(\alpha e-\beta a)^{-1}\quad$

then

(6.3)
$R(g_{1},\mathsf{a})R(g_{2},g_{1}^{-1}\mathsf{a})=R(g_{1}g_{2},\mathsf{a}).$

The last identity is well known in representation theory [Kirillov76]*§
13.2(10) and is a key ingredient of _induced representations_. Thus we can
again linearise (6.2), cf. (5.33), in the space of continuous functions
$C{}(\mathbb{A}{},M)$ with values in a left $\mathfrak{A}$-module $M$, e.g.
$M=\mathfrak{A}$:

$\displaystyle\rho_{a}(g_{1}):f(g^{-1}\cdot a)$ $\displaystyle\mapsto$
$\displaystyle R(g_{1}^{-1}g^{-1},a)f(g_{1}^{-1}g^{-1}\cdot a)$
$\displaystyle\quad=(\alpha^{\prime}e-\beta^{\prime}a)^{-1}\,f\left(\frac{\bar{\alpha}^{\prime}\cdot
a-\bar{\beta}^{\prime}e}{\alpha^{\prime}e-\beta^{\prime}a}\right).$

For any $m\in M$ we can define a $K$-invariant _vacuum vector_ as
$v_{m}(g^{-1}\cdot a)=m\otimes v_{0}(g^{-1}\cdot a)\in C{}(\mathbb{A}{},M)$.
It generates the associated with $v_{m}$ family of _coherent states_
$v_{m}(u,a)=(ue-a)^{-1}m$, where $u\in\mathbb{D}{}$.

The _wavelet transform_ defined by the same common formula based on coherent
states (cf. (5.34)):

(6.5) $\mathcal{W}_{m}f(g)=\left\langle f,\rho_{a}(g)v_{m}\right\rangle,$

is a version of Cauchy integral, which maps $L_{2}{}(\mathbb{A}{})$ to
$C{}(SL_{2}{}(\mathbb{R}{}),M)$. It is closely related (but not identical!) to
the Riesz-Dunford functional calculus: the traditional functional calculus is
given by the case:

$\Phi:f\mapsto\mathcal{W}_{m}f(0)\qquad\textrm{ for }M=\mathfrak{A}\textrm{
and }m=e.$

The both conditions—the intertwining property and initial value—required by
Definition 6.4 easily follows from our construction. Finally, we wish to
provide an example of application of the Corollary 5.8.

######  Example 6.6.

Let $a$ be an operator and $\phi$ be a function which annihilates it, i.e.
$\phi(a)=0$. For example, if $a$ is a matrix $\phi$ can be its minimal
polynomial. From the integral representation of the contravariant calculus on
$G=SL_{2}{}(\mathbb{R}{})$ we can rewrite the annihilation property like this:

$\int_{G}\phi(g)R(g,a)\,dg=0.$

Then the vector-valued function $[\mathcal{W}_{m}f](g)$ defined by (6.5) shall
satisfy to the following condition:

$\int_{G}\phi(g^{\prime})\,[\mathcal{W}_{m}f](gg^{\prime})\,dg^{\prime}=0$

due to the Corollary 5.8.

####  6.2. Jet Bundles and Prolongations of $\rho_{1}$

Spectrum was defined in 6.5 as the _support_ of our functional calculus. To
elaborate its meaning we need the notion of a _prolongation_ of
representations introduced by S. Lie, see [Olver93, Olver95] for a detailed
exposition.

######  Definition 6.7.

[Olver95]*Chap. 4 Two holomorphic functions have $n$th _order contact_ in a
point if their value and their first $n$ derivatives agree at that point, in
other words their Taylor expansions are the same in first $n+1$ terms.

A point $(z,u^{(n)})=(z,u,u_{1},\ldots,u_{n})$ of the _jet space_
$\mathbb{J}^{n}{}\sim\mathbb{D}{}\times\mathbb{C}^{n}{}$ is the equivalence
class of holomorphic functions having $n$th contact at the point $z$ with the
polynomial:

(6.6) $p_{n}(w)=u_{n}\frac{(w-z)^{n}}{n!}+\cdots+u_{1}\frac{(w-z)}{1!}+u.$

For a fixed $n$ each holomorphic function
$f:\mathbb{D}{}\rightarrow\mathbb{C}{}$ has $n$th _prolongation_ (or _$n$
-jet_) $\mathrm{j}_{n}f:\mathbb{D}{}\rightarrow\mathbb{C}^{n+1}{}$:

(6.7) $\mathrm{j}_{n}f(z)=(f(z),f^{\prime}(z),\ldots,f^{(n)}(z)).$

The graph $\mathsf{a}mma^{(n)}_{f}$ of $\mathrm{j}_{n}f$ is a submanifold of
$\mathbb{J}^{n}{}$ which is section of the _jet bundle_ over $\mathbb{D}{}$
with a fibre $\mathbb{C}^{n+1}{}$. We also introduce a notation $J_{n}$ for
the map $J_{n}:f\mapsto\mathsf{a}mma^{(n)}_{f}$ of a holomorphic $f$ to the
graph $\mathsf{a}mma^{(n)}_{f}$ of its $n$-jet $\mathrm{j}_{n}f(z)$ (6.7).

One can prolong any map of functions $\psi:f(z)\mapsto[\psi f](z)$ to a map
$\psi^{(n)}$ of $n$-jets by the formula

(6.8) $\psi^{(n)}(J_{n}f)=J_{n}(\psi f).$

For example such a prolongation $\rho_{1}^{(n)}$ of the representation
$\rho_{1}$ of the group $SL_{2}{}(\mathbb{R}{})$ in $H_{2}{}(\mathbb{D}{})$
(as any other representation of a Lie group [Olver95]) will be again a
representation of $SL_{2}{}(\mathbb{R}{})$. Equivalently we can say that
$J_{n}$ _intertwines_ $\rho_{1}$ and $\rho^{(n)}_{1}$:

$J_{n}\rho_{1}(g)=\rho_{1}^{(n)}(g)J_{n}\quad\textrm{ for all }g\in
SL_{2}{}(\mathbb{R}{}).$

Of course, the representation $\rho^{(n)}_{1}$ is not irreducible: any jet
subspace $\mathbb{J}^{k}{}$, $0\leq k\leq n$ is $\rho^{(n)}_{1}$-invariant
subspace of $\mathbb{J}^{n}{}$. However the representations $\rho^{(n)}_{1}$
are _primary_ [Kirillov76]*§ 8.3 in the sense that they are not sums of two
subrepresentations.

The following statement explains why jet spaces appeared in our study of
functional calculus.

######  Proposition 6.8.

Let matrix $a$ be a Jordan block of a length $k$ with the eigenvalue
$\lambda=0$, and $m$ be its root vector of order $k$, i.e. $a^{k-1}m\neq
a^{k}m=0$. Then the restriction of $\rho_{a}$ on the subspace generated by
$v_{m}$ is equivalent to the representation $\rho_{1}^{k}$.

####  6.3. Spectrum and Spectral Mapping Theorem

Now we are prepared to describe a spectrum of a matrix. Since the functional
calculus is an intertwining operator its support is a decomposition into
intertwining operators with primary representations (we could not expect
generally that these primary subrepresentations are irreducible).

Recall the transitive on $\mathbb{D}{}$ group of inner automorphisms of
$SL_{2}{}(\mathbb{R}{})$, which can send any $\lambda\in\mathbb{D}{}$ to $0$
and are actually parametrised by such a $\lambda$. This group extends
Proposition 6.8 to the complete characterisation of $\rho_{a}$ for matrices.

######  Proposition 6.9.

Representation $\rho_{a}$ is equivalent to a direct sum of the prolongations
$\rho_{1}^{(k)}$ of $\rho_{1}$ in the $k$th jet space $\mathbb{J}^{k}{}$
intertwined with inner automorphisms. Consequently the _spectrum_ of $a$
(defined via the functional calculus $\Phi=\mathcal{W}_{m}$) labelled exactly
by $n$ pairs of numbers $(\lambda_{i},k_{i})$, $\lambda_{i}\in\mathbb{D}{}$,
$k_{i}\in\mathbb{Z}_{+}{}$, $1\leq i\leq n$ some of whom could coincide.

Obviously this spectral theory is a fancy restatement of the _Jordan normal
form_ of matrices.

(a)  (b) (c)

Figure 11. Classical spectrum of the matrix from the Ex. 6.10 is shown at (a).
Contravariant spectrum of the same matrix in the jet space is drawn at (b).
The image of the contravariant spectrum under the map from Ex. 6.12 is
presented at (c).

######  Example 6.10.

Let $J_{k}(\lambda)$ denote the Jordan block of the length $k$ for the
eigenvalue $\lambda$. In Fig. 11 there are two pictures of the spectrum for
the matrix

$a=J_{3}\left(\lambda_{1}\right)\oplus J_{4}\left(\lambda_{2}\right)\oplus
J_{1}\left(\lambda_{3}\right)\oplus J_{2}\left(\lambda_{4}\right),$

where

$\lambda_{1}=\frac{3}{4}e^{i\pi/4},\quad\lambda_{2}=\frac{2}{3}e^{i5\pi/6},\quad\lambda_{3}=\frac{2}{5}e^{-i3\pi/4},\quad\lambda_{4}=\frac{3}{5}e^{-i\pi/3}.$

Part (a) represents the conventional two-dimensional image of the spectrum,
i.e. eigenvalues of $a$, and (b) describes spectrum $\mathbf{sp}\,{}a$ arising
from the wavelet construction. The first image did not allow to distinguish
$a$ from many other essentially different matrices, e.g. the diagonal matrix

$\mathop{\operator@font
diag}\nolimits\left(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\right),$

which even have a different dimensionality. At the same time Fig. 11(b)
completely characterise $a$ up to a similarity. Note that each point of
$\mathbf{sp}\,a$ in Fig. 11(b) corresponds to a particular root vector, which
spans a primary subrepresentation.

As was mentioned in the beginning of this section a resonable spectrum should
be linked to the corresponding functional calculus by an appropriate spectral
mapping theorem. The new version of spectrum is based on prolongation of
$\rho_{1}$ into jet spaces (see Section 6.2). Naturally a correct version of
spectral mapping theorem should also operate in jet spaces.

Let $\phi:\mathbb{D}{}\rightarrow\mathbb{D}{}$ be a holomorphic map, let us
define its action on functions $[\phi_{*}f](z)=f(\phi(z))$. According to the
general formula (6.8) we can define the prolongation $\phi_{*}^{(n)}$ onto the
jet space $\mathbb{J}^{n}{}$. Its associated action
$\rho_{1}^{k}\phi_{*}^{(n)}=\phi_{*}^{(n)}\rho_{1}^{n}$ on the pairs
$(\lambda,k)$ is given by the formula:

(6.9)
$\phi_{*}^{(n)}(\lambda,k)=\left(\phi(\lambda),\left[\frac{k}{\deg_{\lambda}\phi}\right]\right),$

where $\deg_{\lambda}\phi$ denotes the degree of zero of the function
$\phi(z)-\phi(\lambda)$ at the point $z=\lambda$ and $[x]$ denotes the integer
part of $x$.

######  Theorem 6.11 (Spectral mapping).

Let $\phi$ be a holomorphic mapping $\phi:\mathbb{D}{}\rightarrow\mathbb{D}{}$
and its prolonged action $\phi_{*}^{(n)}$ defined by (6.9), then