11institutetext: Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia-CSIC, Institutos de Investigación de Paterna, Aptdo.
22085, 46071 Valencia, Spain

# The $T_{c\bar{s}}(2900)$ as a threshold effect from the interaction of the
$D^{*}K^{*}$, $D^{*}_{s}\rho$ channels

Raquel Molina 11<EMAIL_ADDRESS>Eulogio Oset 11
<EMAIL_ADDRESS>
###### Abstract

We investigate the $D^{*}K^{*}$ and $D^{*}_{s}\rho$ interaction in coupled
channels within the hidden gauge formalism. A structure is developed around
their thresholds, short of producing a bound state, which leads to a peak in
the $D_{s}^{+}\pi^{-}$ mass distribution in the
$B^{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ decay compatible with the experimental
data. We conclude that the interaction between the $D^{*}K^{*}$ and
$D^{*}_{s}\rho$ is essential to produce the cusp structure that we associate
to the recently seen $T_{c\bar{s}}(2900)$, and that its experimental width is
mainly due to the decay width of the $\rho$ meson. The peak obtained together
with a smooth background reproduces fairly well the experimental mass
distribution observed in the $B_{0}\to\bar{D}^{0}D_{s}^{+}\pi^{-}$ decay.

##  1 Introduction

The $D^{*}K^{*}$ system was investigated in branz  and three states were found
corresponding to $I=0;J^{P}=0^{+},1^{+}$ and $2^{+}$. The $2^{+}$ state was
identified with the $D^{*}_{s2}(2573)$ state, and served to set the scale for
the regularization of the loops, allowing predictions in the other sectors.
There, the $I=1$ interaction of the $D^{*}K^{*}$ and $D^{*}_{s}\rho$ channels
was also studied and, a cusp was found for $J=0$ and $J=1$ around the
$D^{*}_{s}\rho$ threshold.

Recently, the LHCb Collaboration has observed an state in the
$D^{+}_{s}\pi^{-}$, $D^{+}_{s}\pi^{+}$ mass distributions in the
$B^{0}\to\bar{D}^{0}D^{+}_{s}\pi^{-}$ and $B^{+}\to D^{-}D_{s}^{+}\pi^{+}$
decays, respectively, at $2900$ MeV lhcb1 ; lhcb2 . Indeed, the state branded
as $T_{c\bar{s}}(2900)$ with $J^{P}=0^{+}$, as seen in $D^{+}_{s}\pi^{-}$ and
$D^{+}_{s}\pi^{+}$, exhibits an $I=1$ character and it has also been
associated with $J^{P}=0^{+}$. On the other hand, $2900$ MeV is just the
threshold of the $D^{*}K^{*}$ channel. Thus, one is finding a $I=1$
$J^{P}=0^{+}$ state in the threshold of $D^{*}K^{*}$ (the $D^{*}_{s}\rho$ is
only $14$ MeV below neglecting the $\rho$ width), which could correspond to
the cusp found in branz .

In the present work we look again at the interaction of $D^{*}K^{*}$ and
$D^{*}_{s}\rho$ channels, taking into account the $K^{*}$ and $\rho$ widths
and also the decay of the states found into the $D_{s}\pi$ channel where it
has been observed, comparing our results with the recent experimental
findings.

##  2 Formalism

In Ref. lhcb1  a peak is found in the $D_{s}\pi$ invariant mass in the
$B^{0}\to\bar{D}^{0}D^{+}_{s}\pi^{-}$ and $B^{+}\to D^{-}D^{+}_{s}\pi^{+}$
decays. In order to have a $b$ quark rather than a $\bar{b}$ quark, we look at
the reaction $\bar{B}^{0}\to D^{0}D_{s}^{*-}\rho^{+}$. We produce this state
with the external emission Cabibbo favored decay shown in Fig. 1 (left). In
Fig. 1 (right) we depict the direct decay $\bar{B}^{0}\to
D_{s}^{-}D^{0}\pi^{+}$ considered as background.

|
---|---
Figure 1: Left: $\bar{B}^{0}$ decay to $D^{*-}_{s}c\bar{d}$ with hadronization
of the $c\bar{d}$ pair to produce $D^{*-}_{s}D^{0}\rho^{+}$. Right:
$\bar{B}^{0}$ decay into $D_{s}^{-}D^{0}\pi^{+}$ (contribution to the
background).

We evaluate the scattering matrix using the Bethe-Salpeter equation in the
$D^{*}K^{*}$ and $D^{*}_{s}\rho$ channels,

$T=[1-VG]^{-1}V\ ,$ (1)

with $G$ the diagonal loop function for the intermediate mesons and $V$ the
transition potential. However, the state is observed in $D_{s}\pi$, hence, the
mechanism by means of which the reaction proceeds is given in Fig. 2.

Figure 2: Mechanism by means of which the resonance is produced and decays
into $\pi^{+}D_{s}^{-}$.

The amplitude for the process of Fig. 2 is given by,

$t=aG_{\rho D^{*}_{s}}(M_{\mathrm{inv}})t_{\rho
D^{*}_{s},K^{*}D^{*}}(M_{\mathrm{inv}})\tilde{V}(\pi D_{s},M_{\mathrm{inv}})$
(2)

where $a$ is a normalization constant that we do not evaluate, unnecessary to
show the shape of the $\pi D_{s}$ mass distribution in the $\bar{B}^{0}$
decay, and $M_{\mathrm{inv}}$ is the invariant mass distribution of the
$D_{s}\pi$ final state. The vertex function $\tilde{V}$ corresponding to the
triangle loop of Fig. 3 can be easily evaluated.

Figure 3: Triangle diagram accounting for the $R\to\pi\bar{D}_{s}$ decay of
the $R$ resonance of $I=1$ generated with the $\rho\bar{D}_{s}$,
$\bar{D}^{*}\bar{K}^{*}$ coupled channels.

We assume the resonance to be in $J=0$, and that the vectors have small
momenta with respect to their masses. Then, the structure of the triangle
diagram of Fig. 3 is given by

$\displaystyle\tilde{V}=-i\int\frac{d^{4}q}{(2\pi)^{4}}\epsilon_{\bar{K}^{*}}^{l}\epsilon_{\bar{D}^{*}}^{l}\epsilon_{\bar{K}^{*}}^{i}\epsilon_{\bar{D}^{*}}^{j}\frac{(2k+q)^{i}(2k+q)^{j}}{(P-q-k)^{2}-m^{2}_{K}+i\epsilon}\frac{\theta(q_{\mathrm{max}}-q)}{(P-q)^{2}-m_{K^{*}}^{2}+i\epsilon}\frac{1}{q^{2}-m^{2}_{D^{*}}+i\epsilon}\
.$ (3)

The loop function $\tilde{V}$ is naturally regularized with a cutoff
$q_{\mathrm{max}}$, the same one used to regularize the $D^{*}K^{*}$ and
$D^{*}_{s}\rho$ loops when studying their interactions. The equivalent
$q_{\mathrm{max}}$ used in branz  was $1100$ MeV. We find,

$\displaystyle\tilde{V}=-\int\frac{d^{3}q}{(2\pi)^{3}}\frac{(2\vec{k}+\vec{q})^{2}}{8\omega_{K^{*}}(q)\omega_{D^{*}}(q)\omega_{K}(\vec{q}+\vec{k})}\frac{1}{P^{0}-\omega_{D^{*}_{s}}(q)-\omega_{K^{*}}(q)+i\epsilon}$
$\displaystyle\times\left\\{\frac{1}{P^{0}-k^{0}-\omega_{D^{*}}(q)-\omega_{K}(\vec{q}+\vec{k})+i\epsilon)}\right.\left.+\frac{1}{k^{0}-\omega_{K^{*}}(q)-\omega_{K}(\vec{q}+\vec{k})+i\epsilon}\right\\}\
,$ (4)

which shows the different cuts of the loop diagram when pairs of the internal
particles of the loop are placed on-shell.

Then, we consider that the transition amplitude for $\bar{B}^{0}\to
D^{0}D^{-}_{s}\pi^{+}$ is given by a constant background (considering the
dominance of s-wave in the coupling of the bottom meson to the pseudoscalars),
see Fig. 1 (right), together with the scattering amplitude of the diagram in
Fig. 2, which accounts for the interaction of the $VV$ coupled channels. It
reads as

$t^{\prime}=aG_{\rho D^{*}_{s}}(M_{\mathrm{inv}})t_{\rho
D^{*}_{s},K^{*}D^{*}}(M_{\mathrm{inv}})\tilde{V}(\pi
D_{s},M_{\mathrm{inv}})+b$ (5)

Therefore, the mass distribution of $\pi D_{s}^{-}$ in the $\bar{B}^{0}$ decay
is given by,

$\frac{d\Gamma}{dM_{\mathrm{inv}}}=\frac{1}{(2\pi)^{3}}\frac{1}{4M^{2}_{B}}p_{D^{0}}\tilde{p}_{\pi}|t^{\prime}|^{2}\
,$ (6)

where
$p_{D^{0}}=\frac{\lambda^{1/2}(M^{2}_{B},m^{2}_{D^{0}},M^{2}_{\mathrm{inv}})}{2M_{B}}$
and
$\tilde{p}=\frac{\lambda^{1/2}(M^{2}_{\mathrm{inv}},m^{2}_{D_{s}},m^{2}_{\pi})}{2M_{\mathrm{inv}}}$.

###  2.1 Results

We take into account the decay widths of the vector mesons $K^{*}$ and $\rho$
by means the convolution of the $G$ function in Eq. (1) tcsbar . The result
for the $T$ matrix in $I=1;J=0$ is shown in Fig. 4. The cusp obtained for
$J=0$ has become wider. The position of the cusp is similar, it shows up
slightly above the $D^{*}K^{*}$ threshold and around $2920$ MeV, with a width
coming basically from the decay of the $\rho$ into $\pi\pi$. We have also
obtained visible peaks in the scattering amplitudes for $J=1$ and $2$ tcsbar .

Figure 4: $|T|^{2}$ for $C=1;S=1;I=1;J=0$ with $\alpha=-1.474$.

.

Finally, we show the result of the invariant mass distribution of the decay
$\bar{B}^{0}\to D^{-}_{s}D^{0}\pi^{+}$, Eq. (6), in comparison with the LHCb
experimental data lhcb1 ; lhcb2  in Fig. 5 (left). In Eq. (6), we adjusted the
constants $a$ and $b$ to reproduce well the experimental data around the
$T_{c\bar{s}}(2900)$ resonance, and we obtain $a=2.1\times 10^{3}$ and
$b=-1.45\times 10^{3}$. As can be seen, our model describes well the
experimental data. A peak is obtained around the threshold of the $D^{*}K^{*}$
channel. Since these results were obtained fixing the subtraction constant to
obtain the $T_{cs}(2900)$, this also supports the molecular picture of this
state as $D^{*}\bar{K}^{*}$ of raquel . Thus, our model strongly supports the
$T_{c\bar{s}}(2900)$ as a cusp structure originated by the non-diagonal
interaction $D^{*}K^{*}\to D^{*}_{s}\rho$, with a width mainly due to the
decay of the $\rho$ meson into $\pi\pi$

|
---|---
Figure 5: Left: invariant mass distribution for $D_{s}\pi$ from the decay
$B\to\bar{D}D_{s}\pi$ compared to the experimental data from Ref. lhcb1 ;
lhcb2 . Right: the same but with the error band obtained by changing the
parameter for the background ($b$) 5% up and down.

Finally, it is interesting to give a band of errors by changing the
background, we do this to show the sensitivity of the results to this
background. We have done this, keeping the value of $a$, needed to get the
strength of the peak of the distribution, by varying the parameter $b$ of the
background by $5\%$ (up and down). This is shown in Fig. 5 (right). The band
obtained overlaps with the errors of the data.

## References

  * (1) R. Molina, T. Branz, E. Oset, Phys. Rev. D 82, 014010 (2010), 1005.0335
  * (2) R. Aaij et al. (LHCb), Phys. Rev. Lett. 131, 041902 (2023), 2212.02716
  * (3) R. Aaij et al. (LHCb), Phys. Rev. D 108, 012017 (2023), 2212.02717
  * (4) R. Molina, E. Oset, Phys. Rev. D 107, 056015 (2023), 2211.01302
  * (5) R. Molina, E. Oset, Phys. Lett. B 811, 135870 (2020), 2008.11171
  * (6) C. Chen (CERN), Particle Zoo 2.0: New tetra- and pentaquarks at LHCb p. July 5 (2022)


# Erlangen Program at Large: Outline

Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds, LS2 9JT,
UK<EMAIL_ADDRESS>http://www.maths.leeds.ac.uk/~kisilv/

###### Abstract.

This is an outline of _Erlangen Program 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 provide an extensive description 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.

###### Contents

  1. 1 Introduction
    1. 1.1 Make a Guess in Three Attempts
    2. 1.2 Erlangen program at large
  2. 2 Geometry
    1. 2.1 Cycles as Invariant Objects
    2. 2.2 Invariance of FSCc
    3. 2.3 Invariants: algebraic and geometric
    4. 2.4 Joint invariants: orthogonality
    5. 2.5 Higher order joint invariants: s-orthogonality
    6. 2.6 Distance, length and perpendicularity
  3. 3 Analytic Functions
    1. 3.1 Wavelet Transform and Cauchy Kernel
    2. 3.2 The Dirac (Cauchy-Riemann) and Laplace Operators
    3. 3.3 The Taylor expansion
  4. 4 Functional Calculus
    1. 4.1 Another Approach to Analytic Functional Calculus
    2. 4.2 Representations in Banach Algebras
    3. 4.3 Jet Bundles and Prolongations
    4. 4.4 Spectrum and the Jordan Normal Form of a Matrix
    5. 4.5 Spectral Mapping Theorem
  5. 5 Open Problems
    1. 5.1 Geometry
    2. 5.2 Analytic Functions
    3. 5.3 Functional Calculus
    4. 5.4 Quantum Mechanics

##  1\. Introduction

The simplest objects with non-commutative multiplication may be $2\times 2$
matrices with real entries. Such matrices _of determinant one_ form a closed
set under multiplication (since $\det(AB)=\det A\cdot\det B$), the identity
matrix is among them 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) $g:x\mapsto g\cdot x=\frac{ax+b}{cx+d},\text{ where
}g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix},x\in\mathbb{R}{}.$

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

The Erlangen program 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 $z=x+\mathrm{i}y$, $\mathrm{i}^{2}=-1$.
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.

However there is no need to be restricted to the traditional route of complex
numbers only. Less-known _dual_ and _double_ numbers [Yaglom79]*Suppl. C have
also the form $z=x+\mathrm{i}y$ but different assumptions on the imaginary
unit $\mathrm{i}$: $\mathrm{i}^{2}=0$ or $\mathrm{i}^{2}=1$ correspondingly.
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 $\sigma:=\mathrm{i}^{2}$ 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 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, ellipticity of quadratic forms, metrics and operators. On the other
hand there are still a lot of white spots and obscure gaps between some
subjects as well.

To understand the action (1.1) in all EPH cases we use the Iwasawa
decomposition [Lang85] 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. $z\mapsto z+\nu$.

Figure 1. Action of the $K$ subgroup. The corresponding $K$-orbits are thick
circles, parabolas and hyperbolas. Thin traversal lines are images of the
vertical axis for certain values of the parameter $\phi$.

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.

###  1.2. Erlangen program at large

As we already mentioned the division of mathematics into areas is only
apparent. Therefore it is unnatural to limit Erlangen program 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):

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

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].

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.

In the spirit of Erlangen program 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 program provides an
important input on this fashionable topic [Kisil02c].

Of course, there is no reasons to limit Erlangen program 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

###  2.1. Cycles as Invariant Objects

######  Definition 2.1.

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 2.2 ([Kisil06a]).

The family of all cycles from Defn. 2.1 is invariant under the action (1.1).

According to Erlangen ideology we shall study invariant properties of cycles.

###  2.2. Invariance of FSCc

Fig. 2 suggests that we may get a unified treatment of cycles in all EPH 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

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

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
the same equation (2.1). 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

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

with a new imaginary 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 (2.2) is the cornerstone of (extended)
Fillmore–Springer–Cnops construction (FSCc) [Cnops02a] and closely related to
technique recently used by A.A. Kirillov to study the Apollonian gasket
[Kirillov06].

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

######  Theorem 2.3.

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 (2.2):

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

In other words FSCc (2.2) _intertwines_ Möbius action (1.1) on cycles with
linear map (2.3).

There are several ways to prove (2.3): 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 section 2.3.

The important observation here is that FSCc (2.2) uses an imaginary unit
$\mathrm{\breve{\i}}$ which is not related to $\mathrm{i}$ 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
$c_{e,p,h}=(\frac{l}{k},-\sigma\frac{n}{k})$ are their respective
e/p/h-centres. They are related to each other through several identities:

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

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:

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

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. Thms. 2.5 and 2.7, 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 (2.3) (and thus under Möbius action
(1.1)): the _trace_ and the _determinant_. The latter was already used in
(2.5) 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.

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 number 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 $(\frac{l}{k},\frac{n}{k})$:

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

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

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

In general for every notion there is 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 on 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 [Kisil05a]*Rem. 3.4. 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. 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:

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

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 2.4.

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

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 Defn. 2.4 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 Defn. 2.4, 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 [Kisil05a].

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_ $\chi(\sigma)$:

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

######  Theorem 2.5.

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)

Since $\left\langle
C_{\breve{\sigma}}^{s},{C}_{\breve{\sigma}}^{s}\right\rangle=\det{C}_{\breve{\sigma}}^{s}$
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 $Z^{s}_{\breve{\sigma}}$.

###  2.5. Higher order joint invariants: s-orthogonality

With appetite already wet one may wish to build more joint invariants. Indeed
for any homogeneous 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 (2.3)
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 (2.2) 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 2.6.

A cycle ${C^{s}_{\breve{\sigma}}}$ is s-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 Defn. 2.4) to the real line. Analytically this is defined by:

(2.8)
$\mathop{tr}({C^{s}_{\breve{\sigma}}}{\tilde{C}^{s}_{\breve{\sigma}}}{C^{s}_{\breve{\sigma}}}R^{s}_{\breve{\sigma}})=0.$

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

Figure 6. Orthogonality of the second kind 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 s-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. Thm. 2.5.

######  Theorem 2.7.

A cycle is s-orthogonal to a cycle $C^{s}_{\breve{\sigma}}$ if its orthogonal
in the traditional sense to its _s-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_
(2.7). 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 s-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
s-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)  (c)

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.
(c) Perpendicular as the shortest route to a line.

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 _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 2.8.

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 distance
between endpoints of a vector $z=u+\mathrm{i}v$ as follows:

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

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 (2.4) or three foci (2.5).

######  Definition 2.9.

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 $B$.

These 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 program due to its
$SL_{2}{}(\mathbb{R}{})$-_conformal invariance_ :

######  Theorem 2.10 ([Kisil05a]).

Let $l$ denote either the EPH distances (2.9) or any length from Defn. 2.9.
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
$y^{\prime}$.

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.
7(c).

######  Definition 2.11.

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 (Defn. 2.9) coincides with already defined in Section 2.5
s-orthogonality as follows from Thm. 2.7(i). It is also possible [Kisil08a] to
make $SL_{2}{}(\mathbb{R}{})$ action isometric in all three cases.

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

##  3\. Analytic Functions

We saw in the previous 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].

###  3.1. Wavelet Transform and Cauchy Kernel

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

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

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 only compact subgroup of $SL_{2}{}(\mathbb{R}{})$:

(3.7) $\displaystyle{\left(\\!\\!\begin{array}[]{cc}\alpha&\bar{\beta}\\\
\beta&\bar{\alpha}\end{array}\\!\\!\right)}$ $\displaystyle=$
$\displaystyle\left|\alpha\right|{\left(\\!\\!\begin{array}[]{cc}1&\bar{\beta}\bar{\alpha}^{-1}\\\
{\beta}{\alpha}^{-1}&1\end{array}\\!\\!\right)}{\left(\\!\\!\begin{array}[]{cc}\frac{{\alpha}}{\left|\alpha\right|}&0\\\
0&\frac{\bar{\alpha}}{\left|\alpha\right|}\end{array}\\!\\!\right)}$ (3.12)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-\left|u\right|^{2}}}{\left(\\!\\!\begin{array}[]{cc}1&u\\\
\bar{u}&1\end{array}\\!\\!\right)},{\left(\\!\\!\begin{array}[]{cc}e^{i\omega}&0\\\
0&e^{-i\omega}\end{array}\\!\\!\right)}$

where

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

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:

(3.13)
$g^{-1}:z\mapsto\frac{\bar{\alpha}z-\bar{\beta}}{\alpha-{\beta}z},\qquad\textrm{
where }\quad
g^{-1}={\left(\\!\\!\begin{array}[]{cc}\bar{\alpha}&-\bar{\beta}\\\
-\beta&\alpha\end{array}\\!\\!\right)}.$

In the decomposition (3.7) the first matrix on the right hand side acts by
transformation (1.1) 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 standard linearisation procedure [Kirillov76, § 7.1] leads from Möbius
transformations (1.1) to the unitary representation $\rho_{1}$ irreducible on
the _Hardy space_ :

(3.14)
$\rho_{1}(g):f(z)\mapsto\frac{1}{\alpha-{\beta}{z}}\,f\left(\frac{\bar{\alpha}z-\bar{\beta}}{\alpha-{\beta}z}\right)\qquad\textrm{
where }\quad
g^{-1}={\left(\\!\\!\begin{array}[]{cc}\bar{\alpha}&-\bar{\beta}\\\
-\beta&\alpha\end{array}\\!\\!\right)}.$

Möbius transformations provide a natural family of intertwining operators for
$\rho_{1}$ coming from inner automorphisms of $SL_{2}{}(\mathbb{R}{})$ (will
be used later).

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:

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

We start from the following observation reflected in the almost any textbook
on complex analysis:

######  Proposition 3.1.

_Analytic function theory_ in the unit disk $\mathbb{D}{}$ is a manifestation
of the mock discrete series representation $\rho_{1}$ of
$SL_{2}{}(\mathbb{R}{})$:

(3.16)
$\rho_{1}(g):f(z)\mapsto\frac{1}{\alpha-{\beta}{z}}\,f\left(\frac{\bar{\alpha}z-\bar{\beta}}{\alpha-{\beta}z}\right),\quad\textup{
where }{\left(\\!\\!\begin{array}[]{cc}\bar{\alpha}&-\bar{\beta}\\\
-\beta&\alpha\end{array}\\!\\!\right)}\in SL_{2}{}(\mathbb{R}{}).$

Other classical objects of complex analysis (the Cauchy-Riemann equation, the
Taylor series, the Bergman space, etc.) can be also obtained [Kisil97c,
Kisil01a] from representation $\rho_{1}$ as shown below.

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

Consideration of Lie groups is hardly possible without consideration of their
Lie algebras, which are naturally represented by left and right invariant
vectors fields on groups. On a homogeneous space $\Omega=G/H$ we have also
defined a left action of $G$ and can be interested in left invariant vector
fields (first order differential operators). Due to the irreducibility of
$F_{2}{}(\Omega)$ under left action of $G$ every such vector field $D$
restricted to $F_{2}{}(\Omega)$ is a scalar multiplier of identity
$D|_{F_{2}{}(\Omega)}=cI$. We are in particular interested in the case $c=0$.

######  Definition 3.2.

[AtiyahSchmid80, KnappWallach76] A $G$-invariant first order differential
operator

$D_{\tau}:C_{\infty}{}(\Omega,\mathcal{S}\otimes V_{\tau})\rightarrow
C_{\infty}{}(\Omega,\mathcal{S}\otimes V_{\tau})$

such that $\mathcal{W}(F_{2}{}(X))\subset\mathrm{ker}\,D_{\tau}$ is called
_(Cauchy-Riemann-)Dirac operator_ on $\Omega=G/H$ associated with an
irreducible representation $\tau$ of $H$ in a space $V_{\tau}$ and a spinor
bundle $\mathcal{S}$.

The Dirac operator is explicitly defined by the formula [KnappWallach76,
(3.1)]:

(3.17) $D_{\tau}=\sum_{j=1}^{n}\rho(Y_{j})\otimes c(Y_{j})\otimes 1,$

where $Y_{j}$ is an orthonormal basis of
$\mathfrak{p}=\mathfrak{h}^{\perp}$—the orthogonal completion of the Lie
algebra $\mathfrak{h}$ of the subgroup $H$ in the Lie algebra $\mathfrak{g}$
of $G$; $\rho(Y_{j})$ is the infinitesimal generator of the right action of
$G$ on $\Omega$; $c(Y_{j})$ is Clifford multiplication by
$Y_{i}\in\mathfrak{p}$ on the Clifford module $\mathcal{S}$. We also define an
invariant Laplacian by the formula

(3.18) $\Delta_{\tau}=\sum_{j=1}^{n}\rho(Y_{j})^{2}\otimes\epsilon_{j}\otimes
1,$

where $\epsilon_{j}=c(Y_{j})^{2}$ is $+1$ or $-1$.

######  Proposition 3.3.

Let all commutators of vectors of $\mathfrak{h}^{\perp}$ belong to
$\mathfrak{h}$, i.e.
$[\mathfrak{h}^{\perp},\mathfrak{h}^{\perp}]\subset\mathfrak{h}$. Let also
$f_{0}$ be an eigenfunction for all vectors of $\mathfrak{h}$ with eigenvalue
$0$ and let also $\mathcal{W}f_{0}$ be a null solution to the Dirac operator
$D$. Then $\Delta f(x)=0$ for all $f(x)\in F_{2}{}(\Omega)$.

###### Proof.

Because $\Delta$ is a linear operator and $F_{2}{}(\Omega)$ is generated by
$\pi_{0}(s(a))\mathcal{W}f_{0}$ it is enough to check that
$\Delta\pi_{0}(s(a))\mathcal{W}f_{0}=0$. Because $\Delta$ and $\pi_{0}$
commute it is enough to check that $\Delta\mathcal{W}f_{0}=0$. Now we observe
that

$\Delta=D^{2}-\sum_{i,j}\rho([Y_{i},Y_{j}])\otimes c(Y_{i})c(Y_{j})\otimes 1.$

Thus the desired assertion is follows from two identities
$\rho([Y_{i},Y_{j}])\mathcal{W}f_{0}=0$ for $[Y_{i},Y_{j}]\in H$ and
$D\mathcal{W}f_{0}=0$. ∎

######  Example 3.4.

Let $G=SL_{2}{}(\mathbb{R}{})$ and $H$ be its one-dimensional compact subgroup
$K$ generated by an element $Z\in\mathfrak{sl}(2,\mathbb{R}{})$. Then
$\mathfrak{h}^{\perp}$ is spanned by two vectors $Y_{1}=A$ and $Y_{2}=B$. In
such a situation we can use $\mathbb{C}{}$ instead of the Clifford algebra.
Then formula (3.17) takes a simple form $D=r(A+iB)$. Infinitesimal action of
this operator in the upper-half plane follows from calculation in [Lang85,
VI.5(8), IX.5(3)], it is $[D_{\mathbb{H}{}}f](z)=-2iy\frac{\partial
f(z)}{\partial\bar{z}}$, $z=x+iy$. Making the Caley transform we can find its
action in the unit disk $D_{\mathbb{D}{}}$: again the Cauchy-Riemann operator
$\frac{\partial}{\partial\bar{z}}$ is its principal component. We calculate
$D_{\mathbb{H}{}}$ explicitly now to stress the similarity with
$\mathbb{R}^{1,1}{}$ case.

For the upper half plane $\mathbb{H}{}$ we have following formulas:

$\displaystyle s$ $\displaystyle:$ $\displaystyle\mathbb{H}{}\rightarrow
SL_{2}{}(\mathbb{R}{}):z=x+iy\mapsto
g={\left(\\!\\!\begin{array}[]{cc}y^{1/2}&xy^{-1/2}\\\
0&y^{-1/2}\end{array}\\!\\!\right)};$ $\displaystyle s^{-1}$ $\displaystyle:$
$\displaystyle
SL_{2}{}(\mathbb{R}{})\rightarrow\mathbb{H}{}:{\left(\\!\\!\begin{array}[]{cc}a&b\\\
c&d\end{array}\\!\\!\right)}\mapsto z=\frac{ai+b}{ci+d};$
$\displaystyle\rho(g)$ $\displaystyle:$
$\displaystyle\mathbb{H}{}\rightarrow\mathbb{H}{}:z\mapsto s^{-1}(s(z)*g)$
$\displaystyle\qquad\qquad\qquad=s^{-1}{\left(\\!\\!\begin{array}[]{cc}ay^{-1/2}+cxy^{-1/2}&by^{1/2}+dxy^{-1/2}\\\
cy^{-1/2}&dy^{-1/2}\end{array}\\!\\!\right)}$
$\displaystyle\qquad\qquad\qquad=\frac{(yb+xd)+i(ay+cx)}{ci+d}$

Thus the right action of $SL_{2}{}(\mathbb{R}{})$ on $\mathbb{H}{}$ is given
by the formula

$\rho(g)z=\frac{(yb+xd)+i(ay+cx)}{ci+d}=x+y\frac{bd+ac}{c^{2}+d^{2}}+iy\frac{1}{c^{2}+d^{2}}.$

For $A$ and $B$ in $\mathfrak{sl}(2,\mathbb{R}{})$ we have:

$\rho(e^{At})z=x+iye^{2t},\qquad\rho(e^{Bt})z=x+y\frac{e^{2t}-e^{-2t}}{e^{2t}+e^{-2t}}+iy\frac{4}{e^{2t}+e^{-2t}}.$

Thus

$\displaystyle[\rho(A)f](z)$ $\displaystyle=$ $\displaystyle\frac{\partial
f(\rho(e^{At})z)}{\partial t}|_{t=0}=2y\partial_{2}f(z),$
$\displaystyle{}[\rho(B)f](z)$ $\displaystyle=$ $\displaystyle\frac{\partial
f(\rho(e^{Bt})z)}{\partial t}|_{t=0}=2y\partial_{1}f(z),$

where $\partial_{1}$ and $\partial_{2}$ are derivatives of $f(z)$ with respect
to real and imaginary party of $z$ respectively. Thus we get

$D_{\mathbb{H}{}}=i\rho(A)+\rho(B)=2yi\partial_{2}+2y\partial_{1}=2y\frac{\partial}{\partial\bar{z}}$

as was expected.

###  3.3. The Taylor expansion

For any decomposition $f_{a}(x)=\sum_{\alpha}\psi_{\alpha}(x)V_{\alpha}(a)$ of
the coherent states $f_{a}(x)$ by means of functions $V_{\alpha}(a)$ (where
the sum can become eventually an integral) we have the _Taylor expansion_

(3.22) $\displaystyle\widehat{f}(a)$ $\displaystyle=$
$\displaystyle\int_{X}f(x)\bar{f}_{a}(x)\,dx=\int_{X}f(x)\sum_{\alpha}\bar{\psi}_{\alpha}(x)\bar{V}_{\alpha}(a)\,dx$
$\displaystyle=$
$\displaystyle\sum_{\alpha}\int_{X}f(x)\bar{\psi}_{\alpha}(x)\,dx\bar{V}_{\alpha}(a)$
$\displaystyle=$
$\displaystyle\sum_{\alpha}^{\infty}\bar{V}_{\alpha}(a)f_{\alpha},$

where $f_{\alpha}=\int_{X}f(x)\bar{\psi}_{\alpha}(x)\,dx$. However to be
useful within the presented scheme such a decomposition should be connected
with the structures of $G$, $H$, and the representation $\pi_{0}$. We will use
a decomposition of $f_{a}(x)$ by the eigenfunctions of the operators
$\pi_{0}(h)$, $h\in\mathfrak{h}$.

######  Definition 3.5.

Let $F_{2}{}=\int_{A}H_{\alpha}{}\,d\alpha$ be a spectral decomposition with
respect to the operators $\pi_{0}(h)$, $h\in\mathfrak{h}$. Then the
decomposition

(3.23) $f_{a}(x)=\int_{A}V_{\alpha}(a)f_{\alpha}(x)\,d\alpha,$

where $f_{\alpha}(x)\in H_{\alpha}{}$ and
$V_{\alpha}(a):H_{\alpha}{}\rightarrow H_{\alpha}{}$ is called the Taylor
decomposition of the Cauchy kernel $f_{a}(x)$.

Note that the Dirac operator $D$ is defined in the terms of left invariant
shifts and therefor commutes with all $\pi_{0}(h)$. Thus it also has a
spectral decomposition over spectral subspaces of $\pi_{0}(h)$:

(3.24) $D=\int_{A}D_{\delta}\,d\delta.$

We have obvious property

######  Proposition 3.6.

If spectral measures $d\alpha$ and $d\delta$ from (3.23) and (3.24) have
disjoint supports then the image of the Cauchy integral belongs to the kernel
of the Dirac operator.

For discrete series representation functions $f_{\alpha}(x)$ can be found in
$F_{2}{}$ (as in Example 3.7), for the principal series representation this is
not the case. To overcome confusion one can think about the Fourier transform
on the real line. It can be regarded as a continuous decomposition of a
function $f(x)\in L_{2}{}(\mathbb{R}{})$ over a set of harmonics $e^{i\xi x}$
neither of those belongs to $L_{2}{}(\mathbb{R}{})$. This has a lot of common
with the Example 3.10(b) in [Kisil97c].

######  Example 3.7.

Let $G=SL_{2}{}(\mathbb{R}{})$ and $H=K$ be its maximal compact subgroup and
$\pi_{1}$ defined in (3.14). $H$ acts on $\mathbb{T}{}$ by rotations. It is
one dimensional and eigenfunctions of its generator $Z$ are parametrized by
integers (due to compactness of $K$). Moreover, on the irreducible Hardy space
these are positive integers $n=1,2,3\ldots$ and corresponding eigenfunctions
are $f_{n}(\phi)=e^{i(n-1)\phi}$. Negative integers span the space of anti-
holomorphic function and the splitting reflects the existence of analytic
structure given by the Cauchy-Riemann equation. The decomposition of coherent
states $f_{a}(\phi)$ by means of this functions is well known:

$f_{a}(\phi)=\frac{\sqrt[]{1-\left|a\right|^{2}}}{\bar{a}e^{i\phi}-1}=\sum_{n=1}^{\infty}\sqrt[]{1-\left|a\right|^{2}}\bar{a}^{n-1}e^{i(n-1)\phi}=\sum_{n=1}^{\infty}V_{n}(a)f_{n}(\phi),$

where $V_{n}(a)=\sqrt[]{1-\left|a\right|^{2}}\bar{a}^{n-1}$. This is the
classical Taylor expansion up to multipliers coming from the invariant
measure.

##  4\. Functional Calculus

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. All traditional definitions of functional
calculus are covered by the following rigid template based on _algebra
homomorphism_ property:

######  Definition 4.1.

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. $v_{0}(z)=z$.

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

######  Definition 4.2.

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 4.3 (Spectral Mapping).

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

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

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 [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 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 (4.1) or
some its variation.

Thus the entire scheme depends from the notion of the functional calculus and
our ability to escape limitations of Definition 4.1. 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] in terms of _intertwining operators_ for group
representations. It was initially targeted for several non-commuting operators
because no non-trivial algebra homomorphism with a commutative algebra of
function is possible 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 present note 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 matrix. Other more difficult questions are
the subject of ongoing research.

###  4.1. Another Approach to Analytic Functional Calculus

Anything called “ _functional_ calculus” uses properties of _functions_ to
model properties of _operators_. Thus changing our viewpoint on functions, as
was done in Section 3, we could get another approach to operators.

The representation (3.16) is unitary irreducible when acts on the Hardy space
$H_{2}{}$. Consequently we have one more reason to abolish the template
definition 4.1: $H_{2}{}$ is _not_ an algebra. Instead we replace the
homomorphism property by a symmetric covariance:

######  Definition 4.4.

An _analytic functional 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 $\rho_{1}$
(3.16) and $\rho_{a}$ defined below in (4.2).

  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 form 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 4.4 it
will means the set of non-vanishing intertwining operators with primary
subrepresentations.

######  Definition 4.5.

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 _prime representations_ [Kirillov76, § 8.3].

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

###  4.2. Representations of $SL_{2}{}(\mathbb{R}{})$ in Banach Algebras

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

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

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

(4.3)
$R(g_{1},\mathsf{a})R(g_{2},g_{1}^{-1}\mathsf{a})=R(g_{1}g_{2},\mathsf{a}).$

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 (4.2) (cf. (3.14)) 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 again 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. (3.15)):

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

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 4.4 easily follows from our construction.

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

Spectrum was defined in 4.5 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 4.6.

[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:

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

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

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

The graph $\Gamma^{(n)}_{f}$ 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 $J_{n}:f\mapsto\Gamma^{(n)}_{f}$ of a holomorphic $f$ to the graph
$\Gamma^{(n)}_{f}$ of its $n$-jet $\mathrm{j}_{n}f(z)$ (4.6).

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

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

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 4.7.

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 $\rho_{1}^{k}$.

###  4.4. Spectrum and the Jordan Normal Form of a Matrix

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 prime representations (we could not expect
generally that these prime 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 4.7 to the complete characterisation of $\rho_{a}$ for matrices.

######  Proposition 4.8.

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 8. Classical spectrum of the matrix from the Ex. 4.9 is shown at (a).
Covariant spectrum of the same matrix in the jet space is drawn at (b). The
image of the covariant spectrum under the map from Ex. 4.11 is presented (c).

######  Example 4.9.

Let $J_{k}(\lambda)$ denote the Jordan block of the length $k$ for the
eigenvalue $\lambda$. On the Fig. 8 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 the Fig. 8(b)
completely characterise $a$ up to a similarity. Note that each point of
$\mathbf{sp}\,a$ on Fig. 8(b) corresponds to a particular root vector, which
spans a primary subrepresentation.

###  4.5. Spectral Mapping Theorem

As was mentioned in the Introduction 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 4.3). 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 (4.7) 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:

(4.8)
$\phi_{*}^{(n)}(\lambda,k)=\left(\phi(\lambda),\left[\frac{k}{\deg_{\lambda}\phi}\right]\right),$

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 4.10 (Spectral mapping).

Let $\phi$ be a holomorphic mapping $\phi:\mathbb{D}{}\rightarrow\mathbb{D}{}$
and its prolonged action $\phi_{*}^{(n)}$ defined by (4.8), then

$\mathbf{sp}\,\phi(a)=\phi_{*}^{(n)}\mathbf{sp}\,a.$

The explicit expression of (4.8) for $\phi_{*}^{(n)}$, which involves
derivatives of $\phi$ upto $n$th order, is known, see for example
[HornJohnson94, Thm. 6.2.25], but was not recognised before as form of
spectral mapping.

######  Example 4.11.

Let us continue with Example 4.9. Let $\phi$ map all four eigenvalues
$\lambda_{1}$, …, $\lambda_{4}$ of the matrix $a$ into themselves. Then Fig.
8(a) will represent the classical spectrum of $\phi(a)$ as well as $a$.

However Fig. 8(c) shows mapping of the new spectrum for the case $\phi$ has
orders of zeros at these points as follows: the order $1$ at $\lambda_{1}$,
exactly the order $3$ at $\lambda_{2}$, an order at least $2$ at
$\lambda_{3}$, and finally any order at $\lambda_{4}$.

##  5\. Open Problems

In this section we indicate several directions for further work which go
through three main areas described in the paper..

###  5.1. Geometry

Geometry is most elaborated area so far, yet many directions are waiting for
further exploration.

  1. (i)

Möbius transformations (1.1) with three types of imaginary units appear from
the action of the group $SL_{2}{}(\mathbb{R}{})$ on the homogeneous space
$SL_{2}{}(\mathbb{R}{})/H$ [Kisil09c], where $H$ is any subgroup $A$, $N$, $K$
from the Iwasawa decomposition (1.3). Which other actions and hypercomplex
numbers can be obtained from semisimple Lie groups and their subgroups?

  2. (ii)

Lobachevsky geometry of the upper half-plane is extremely beautiful and well-
developed subject [Beardon05a] [CoxeterGreitzer]. However the traditional
study is limited to one subtype out of nine possible: with the complex numbers
for Möbius transformation and the complex imaginary unit used in FSCc (2.2).
The remaining eight cases shall be explored in various directions, notably in
the context of discrete subgroups [Beardon95].

  3. (iii)

The Filmore-Springer-Cnops construction, see subsection 2.2, is closely
related to the orbit method [Kirillov99] applied to $SL_{2}{}(\mathbb{R}{})$.
An extension of the orbit method from the Lie algebra dual to matrices
representing cycles may be fruitful for semisimple Lie groups.

###  5.2. Analytic Functions

It is known that in several dimensions there are different notions of
analyticity, e.g. several complex variables and Clifford analysis. However,
analytic functions of a complex variable are usually thought to be the only
options in a plane domain. The following seems to be promising:

  1. (i)

Development of the basic components of analytic function theory (the Cauchy
integral, the Taylor expansion, the Cauchy-Riemann and Laplace equations,
etc.) from the same construction and principles in the elliptic, parabolic and
hyperbolic cases and subcases.

  2. (ii)

Identification of Hilbert spaces of analytic functions of Hardy and Bergman
types, investigation of their properties. Consideration of the corresponding
Töplitz operators and algebras generated by them.

  3. (iii)

Application of analytic methods to elliptic, parabolic and hyperbolic
equations and corresponding boundary and initial values problems.

  4. (iv)

Generalisation of the results obtained to higher dimensional spaces. Detailed
investigation of physically significant cases of three and four dimensions.

###  5.3. Functional Calculus

The functional calculus of a finite dimensional operator considered in Section
4 is elementary but provides a coherent and comprehensive treatment. It shall
be extended to further cases where other approaches seems to be rather
limited.

  1. (i)

Nilpotent and quasinilpotent operators have the most trivial spectrum possible
(the single point $\\{0\\}$) while their structure can be highly non-trivial.
Thus the standard spectrum is insufficient for this class of operators. In
contract, the covariant calculus and the spectrum give complete description of
nilpotent operators—the basic prototypes of quasinilpotent ones. For
quasinilpotent operators the construction will be more complicated and shall
use analytic functions mentioned in 5.2.i.

  2. (ii)

The version of covariant calculus described above is based on the _discrete
series_ representations of $SL_{2}{}(\mathbb{R}{})$ group and is particularly
suitable for the description of the _discrete spectrum_ (note the remarkable
coincidence in the names).

It is interesting to develop similar covariant calculi based on the two other
representation series of $SL_{2}{}(\mathbb{R}{})$: _principal_ and
_complementary_ [Lang85]. The corresponding versions of analytic function
theories for principal [Kisil97c] and complementary series [Kisil05a] were
initiated within a unifying framework. The classification of analytic function
theories into elliptic, parabolic, hyperbolic [Kisil05a, Kisil06a] hints the
following associative chains:

Representations of $SL_{2}{}(\mathbb{R}{})$ — |  Function Theory — | Type of Spectrum
---|---|---
discrete series — | elliptic — | discrete spectrum
principal series — | hyperbolic — | continuous spectrum
complementary series — | parabolic — | residual spectrum

  3. (iii)

Let $a$ be an operator with $\mathbf{sp}\,a\in\bar{\mathbb{D}{}}$ and
$\left\|a^{k}\right\|<Ck^{p}$. It is typical to consider instead of $a$ the
_power bounded_ operator $ra$, where $0<r<1$, and consequently develop its
$H_{\infty}{}$ calculus. However such a regularisation is very rough and hides
the nature of extreme points of $\mathbf{sp}\,{a}$. To restore full
information a subsequent limit transition $r\rightarrow 1$ of the
regularisation parameter $r$ is required. This make the entire technique
rather cumbersome and many results have an indirect nature.

The regularisation $a^{k}\rightarrow a^{k}/k^{p}$ is more natural and accurate
for polynomially bounded operators. However it cannot be achieved within the
homomorphic calculus Defn. 4.1 because it is not compatible with any algebra
homomorphism. Albeit this may be achieved within the covariant calculus Defn.
4.4 and Bergman type space from 5.2.ii.

  4. (iv)

Several non-commuting operators are especially difficult to treat with
functional calculus Defn. 4.1 or a joint spectrum. For example, deep insights
on joint spectrum of commuting tuples [JTaylor72] refused to be generalised to
non-commuting case so far. The covariant calculus was initiated [Kisil95i] as
a new approach to this hard problem and was later found useful elsewhere as
well. Multidimensional covariant calculus [Kisil04d] shall use analytic
functions described in 5.2.iv.

###  5.4. Quantum Mechanics

Due to the space restrictions we did not mentioned connections with quantum
mechanics [Kisil96a] [Kisil02e] [Kisil05c] [Kisil04a] [Kisil09a] [Kisil10a].
In general Erlangen approach is much more popular among physicists rather than
mathematicians. Nevertheless its potential is not exhausted even there.

  1. (i)

There is a possibility to build representation of the Heisenberg group using
characters of its centre with values in dual and double numbers rather than in
complex ones. This will naturally unifies classical mechanics, traditional QM
and hyperbolic QM [Khrennikov08a].

  2. (ii)

Representations of nilpotent Lie groups with multidimensional centres in
Clifford algebras as a framework for consistent quantum filed theories based
on De Donder–Weyl formalism [Kisil04a].

######  Remark 5.1.

This work is performed within the “Erlangen programme at large” framework
[Kisil06a, Kisil05a], thus it would be suitable to explain the numbering of
various papers. Since the logical order may be different from chronological
one the following numbering scheme is used:

Prefix |  Branch description
---|---
“0” or no prefix |  Mainly geometrical works, within the classical field of Erlangen programme by F. Klein, see [Kisil05a] [Kisil09c]
“1” |  Papers on analytical functions theories and wavelets, e.g. [Kisil97c]
“2” |  Papers on operator theory, functional calculi and spectra, e.g. [Kisil02a]
“3” |  Papers on mathematical physics, e.g. [Kisil10a]

For example, this is the first paper in the mathematical physics area.

## References