From Conformal Group to Symmetries of Hypergeometric Type Equations

From Conformal Group to Symmetries
of Hypergeometric Type Equations

Jan DEREZIŃSKI † and Przemysław MAJEWSKI †‡

J. Dereziński and P. Majewski

† Department of Mathematical Methods in Physics, Faculty of Physics,
University of Warsaw,
† Pasteura 5, 02-093 Warszawa, Poland<EMAIL_ADDRESS>http://fuw.edu.pl/~derezins/

‡ Bureau of Air Defence and Anti-missile Defence Systems, PIT-RADWAR S.A.,
‡ Poligonowa 30, 04-025 Warszawa, Poland<EMAIL_ADDRESS>http://fuw.edu.pl/~pmaj/

Received February 24, 2016, in final form October 20, 2016; Published online
November 05, 2016

We show that properties of hypergeometric type equations become transparent if
they are derived from appropriate 2nd order partial differential equations
with constant coefficients. In particular, we deduce the symmetries of the
hypergeometric and Gegenbauer equation from conformal symmetries of the 4- and
3-dimensional Laplace equation. We also derive the symmetries of the confluent
and Hermite equation from the so-called Schrödinger symmetries of the heat
equation in 2 and 1 dimension. Finally, we also describe how properties of the
${}_{0}F_{1}$ equation follow from the Helmholtz equation in 2 dimensions.

Laplace equation; hypergeometric equation; confluent equation; Kummer’s table;
Lie algebra; conformal group

35J05; 33Cxx; 35B06

###### Contents

  1. 1 Introduction
    1. 1.1 Hypergeometric type operators
    2. 1.2 Group-theoretical derivation
    3. 1.3 Organization of the paper
    4. 1.4 Comparison with literature
  2. 2 Hypergeometric type operators and their symmetries
    1. 2.1 Hypergeometric operator
    2. 2.2 Gegenbauer operator
    3. 2.3 Confluent operator
    4. 2.4 Hermite operator
    5. 2.5 ${}_{0}F_{1}$ operator
  3. 3 Basic complex geometry
    1. 3.1 Vector fields
    2. 3.2 Point transformations
    3. 3.3 Local cocycles
    4. 3.4 Half-integer powers of a cocycle
    5. 3.5 Generalized symmetries
  4. 4 Line bundles
    1. 4.1 Scaling
    2. 4.2 Vector fields on a line bundle
    3. 4.3 Point transformations of a line bundle
    4. 4.4 Homogeneous functions of integer degree
    5. 4.5 Homogeneous functions of non-integer degree
  5. 5 Complex Euclidean spaces
    1. 5.1 Linear transformations
    2. 5.2 Bilinear scalar product
    3. 5.3 Split coordinates
    4. 5.4 Weyl symmetries
  6. 6 Conformal invariance
    1. 6.1 Conformal invariance of Riemannian manifolds
    2. 6.2 Null quadric
    3. 6.3 Conformal invariance of projective quadric
    4. 6.4 Conformal invariance of complex Euclidean space
    5. 6.5 Laplacian on bundle of homogeneous functions
    6. 6.6 Conformal invariance of Laplacian for a general section
    7. 6.7 Conformal invariance of Laplacian on $\mathbb{C}^{n}$
    8. 6.8 Dimension $n=1$
    9. 6.9 Dimension $n=2$
  7. 7 $\mathrm{so}(\mathbb{C}^{6})$ and the hypergeometric equation
    1. 7.1 Extended space $\mathbb{C}^{6}$
    2. 7.2 Spherical section
    3. 7.3 Balanced hypergeometric operator
    4. 7.4 Standard hypergeometric operator
    5. 7.5 Factorizations
    6. 7.6 Conformal symmetries of $\Delta_{\mathbb{C}^{4}}$
    7. 7.7 Deriving balanced hypergeometric operator from $\Delta_{\mathbb{C}^{4}}$
    8. 7.8 Deriving standard hypergeometric operator from $\Delta_{\mathbb{C}^{4}}$
  8. 8 $\mathrm{so}(\mathbb{C}^{5})$ and the Gegenbauer equation
    1. 8.1 Extended space $\mathbb{C}^{5}$
    2. 8.2 Spherical section
    3. 8.3 Balanced Gegenbauer operator
    4. 8.4 Standard Gegenbauer operator
    5. 8.5 Factorizations
    6. 8.6 Conformal symmetries of $\Delta_{\mathbb{C}^{3}}$
    7. 8.7 Deriving balanced Gegenbauer operator from $\Delta_{\mathbb{C}^{3}}$
    8. 8.8 Deriving standard Gegenbauer operator from $\Delta_{\mathbb{C}^{3}}$
  9. 9 Symmetries of the heat equation – the Schrödinger algebra
    1. 9.1 The Schrödinger Lie algebra and group on $\mathbb{C}^{n+2}$
    2. 9.2 Structure of $\mathrm{sch}(\mathbb{C}^{n-2})$
    3. 9.3 The Schrödinger Lie algebra and group on $\mathbb{C}^{n}$
    4. 9.4 The Schrödinger Lie algebra and group on $\mathbb{C}^{n-2}\oplus\mathbb{C}$
    5. 9.5 Hermite operator
    6. 9.6 Schrödinger symmetries in coordinates
  10. 10 $\mathrm{sch}(\mathbb{C}^{2})$ and the confluent equation
    1. 10.1 $\mathbb{C}^{6}$
    2. 10.2 $\mathbb{C}^{4}$
    3. 10.3 $\mathbb{C}^{2}\oplus\mathbb{C}$
    4. 10.4 Coordinates $u$, $w$, $s$
    5. 10.5 Sandwiching with an exponential
    6. 10.6 Balanced confluent operator
    7. 10.7 Standard confluent operator
    8. 10.8 Factorizations
  11. 11 $\mathrm{sch}(\mathbb{C}^{1})$ and the Hermite equation
    1. 11.1 $\mathbb{C}^{5}$
    2. 11.2 $\mathbb{C}^{3}$
    3. 11.3 $\mathbb{C}\oplus\mathbb{C}$
    4. 11.4 Coordinates $w$, $s$
    5. 11.5 Sandwiching with a Gaussian
    6. 11.6 Balanced Hermite operator
    7. 11.7 Standard Hermite operator
    8. 11.8 Factorizations
  12. 12 $\mathbb{C}^{2}\rtimes\mathrm{so}(\mathbb{C}^{2})$ and the ${}_{0}F_{1}$ equation
    1. 12.1 Space $\mathbb{C}^{5}$
    2. 12.2 Space $\mathbb{C}^{3}$
    3. 12.3 Space $\mathbb{C}^{2}$ and the Helmholtz equation
    4. 12.4 Balanced ${}_{0}F_{1}$ operator
    5. 12.5 Standard ${}_{0}F_{1}$ operator
    6. 12.6 Factorizations
    7. Acknowledgements

##  1 Introduction

The paper is devoted to the properties of the hypergeometric equation

$\displaystyle\big{(}z(1-z)\partial_{z}^{2}+(c-(a+b+1)z)\partial_{z}-ab\big{)}F(z)=0,$
(1.1)

the Gegenbauer equation

$\displaystyle\big{(}\big{(}1-z^{2}\big{)}\partial_{z}^{2}-({a}+{b}+1)z\partial_{z}-{a}{b}\big{)}F(z)=0,$
(1.2)

the confluent equation

$\displaystyle\big{(}z\partial_{z}^{2}+(c-z)\partial_{z}-a\big{)}F(z)=0,$
(1.3)

the Hermite equation

$\displaystyle\big{(}\partial_{z}^{2}-2z\partial_{z}-2{a}\big{)}F(z)=0,$ (1.4)

and the ${}_{0}F_{1}$ equation (closely related to the Bessel equation, see,
e.g., [6])

$\displaystyle\big{(}z\partial_{z}^{2}+c\partial_{z}-1\big{)}F(z)=0.$ (1.5)

Here, $z$ is a complex variable, $\partial_{z}$ is the differentiation with
respect to $z$, and $a$, $b$, $c$ are arbitrary complex parameters.

Special functions that solve these equations are typical representatives of
hypergeometric type functions [34]. They often appear in applications [11,
32]. In old times they were considered one of the central topics of
mathematics, see, e.g., [48]. In our opinion, they indeed belong to the most
natural objects in mathematics.

Properties of hypergeometric type functions look quite complicated. According
to our observations, when these properties are discussed, most people react
with boredom and/or irritation. We would like to convince the reader that in
reality identities related to hypergeometric type equations are beautiful and
can be derived in an elegant and transparent way.

We will show that in order to understand hypergeometric type equations it is
helpful to start from certain 2nd order PDE’s in several variables with
constant coefficients. If we start from rather obvious properties of these
PDE’s, reduce the number of variables and change the coordinates, we can
observe how these properties become more complicated. At the end one obtains
relatively complicated sets of identities for hypergeometric type equations.

###  1.1 Hypergeometric type operators

Equations (1.1)–(1.5) are determined by an operator of the form

$\displaystyle\mathcal{C}=\mathcal{C}(z,\partial_{z}):=\sigma(z)\partial_{z}^{2}+\tau(z)\partial_{z}+\eta.$

In our paper we will concentrate on the study of the operator $\mathcal{C}$
itself, rather than on individual solutions $F$ of the equation

$\displaystyle\mathcal{C}F=0.$

Note, however, that properties of $F$’s can be to a large extent inferred from
the properties of $\mathcal{C}$ itself.

According to the terminology used in [34], and then in [6, 7], equations
(1.1)–(1.5) belong to the class of hypergeometric type equations. This class
is defined by demanding that $\sigma$ is a polynomial of at most 2nd order,
$\tau$ is a polynomial of at most 1st order and $\eta$ is a number. More
precisely, (1.1)–(1.5) constitute standard forms of all nontrivial classes of
hypergeometric type equations, as explained, e.g., in [6].

Equations (1.1)–(1.5) depend on a number of (complex) parameters. For
instance, in the case of the hypergeometric equation these parameters are $a$,
$b$, $c$. We will prefer to use different sets of parameters introduced in a
systematic way in [6], which are more convenient to express the symmetries of
these equations. In [6] these new parameters were called the Lie-algebraic
parameters. Indeed, as we will describe, they are eigenvalues of the “Cartan
algebra” of appropriate Lie algebras. For instance, for the hypergeometric
equation the Lie-algebraic parameters, denoted $\alpha$, $\beta$, $\mu$, are
the differences of the indices at the three singular points.

We will prefer not to use the operators $\mathcal{C}$ directly. As explained
in [6], we can write

$\displaystyle\mathcal{C}(z,\partial_{z})=\rho^{-1}(z)\partial_{z}\rho(z)\partial_{z}+\eta,$

which defines (up to a multiplicative factor) a certain function $\rho(z)$
called the weight. Following [6], the operator

$\displaystyle\mathcal{C}^{\mathrm{bal}}(z,\partial_{z}):=\rho(z)^{\frac{1}{2}}\mathcal{C}(z,\partial_{z})\rho(z)^{-\frac{1}{2}},$

will be called the balanced form of $\mathcal{C}$. The study of the balanced
form is obviously equivalent to that of $\mathcal{C}$, since both are related
by a simple similarity transformation. The original operator $\mathcal{C}$
will be sometimes called the standard form of $\mathcal{C}$.

We will consider 3 classes of identities:

  1. (1)

discrete symmetries,

  2. (2)

transmutation relations,

  3. (3)

factorizations.

Discrete symmetries involve a transformation of the independent variable,
together with a change of the parameters. The family of the discrete
symmetries of the hypergeometric equation is especially famous. In the
literature it is sometimes known under the name of the Kummer’s table [1, 21,
22, 25].

The balanced form is especially convenient for a presentation of discrete
symmetries, because some of them simply reduce to the change of sign of
parameters.

Transmutation relations say that $\mathcal{C}$ multiplied from the right by an
appropriate 1st order operator equals $\mathcal{C}$ for shifted parameters
multiplied from the left by a similar 1st order operator. In quantum physics
the corresponding 1st order operators are often called creation/annihilation
operators.

Factorizations say that $\mathcal{C}$ can be written as a product of two 1st
order operators, up to an additional term that does not contain the
independent variable. It is easy to see that factorizations imply
transmutation relations, as described, e.g., in [6]. Factorizations play an
important role in quantum mechanics. They are often interpreted as the
manifestation of supersymmetry [5]. In quantum mechanics discussion of these
factorizations has a long history going back at least to [14].

Discrete symmetries, transmutation relations and factorizations are far from
being trivial. Nevertheless, in our opinion they belong to the most elementary
properties of hypergeometric type equations and functions. There exist many
other properties, notably addition formulas and integral representations,
which we view as more advanced. We do not consider them in our paper.

###  1.2 Group-theoretical derivation

We will see that all hypergeometric type equations can be obtained by
separating the variables of 2nd order PDE’s with constant coefficients. We
will always use the complex variable, to avoid discussing various signatures
of these PDE’s.

Every such a PDE has a Lie algebra and a Lie group of generalized symmetries.
In this Lie algebra we can fix a certain maximal commutative algebra, which we
will call the “Cartan algebra”. Operators whose adjoint action is diagonal in
the “Cartan algebra” will be called “root operators”. In the Lie group of
generalized symmetries we will distinguish a discrete group, which we will
call the group of “Weyl symmetries”. This group will implement automorphisms
of the Lie algebra leaving invariant the “Cartan algebra”.

Note that in some cases the Lie algebra of generalized symmetries is
semisimple, and then the names Cartan algebra, root operators and Weyl
symmetries correspond to the standard names. In other cases the Lie algebra is
non-semisimple, and then the names are less standard – this is the reason for
the quotation marks that we use.

Parameters of hypergeometric type equation can be interpreted as the
eigenvalues of elements of the “Cartan algebra”. In particular, the number of
parameters of a given class of equations equals the dimension of the
corresponding “Cartan algebra”. Each transmutation relation is related to a
“root operator”. Finally, each discrete symmetry of a hypergeometric type
operator corresponds to a “Weyl symmetry” of the Lie algebra.

We can distinguish 3 kinds of PDE’s of the complex variable with constant
coefficients:

  1. (1)

The Laplace equation on $\mathbb{C}^{n}$

$\displaystyle\Delta_{\mathbb{C}^{n}}f=0,$

whose Lie algebra of generalized symmetries is
$\mathrm{so}(\mathbb{C}^{n+2})$.

  2. (2)

The heat equation on $\mathbb{C}^{n-2}\oplus\mathbb{C}$,

$\displaystyle\big{(}\Delta_{\mathbb{C}^{n-2}}+\partial_{t}\big{)}f=0,$

whose Lie algebra of generalized symmetries is
$\mathrm{sch}(\mathbb{C}^{n-2})$, the so-called $($complex$)$ Schrödinger Lie
algebra.

  3. (3)

The Helmholtz equation on $\mathbb{C}^{n-1}$,

$\displaystyle\big{(}\Delta_{\mathbb{C}^{n-1}}-1\big{)}f=0,$

whose Lie algebra of symmetries is
$\mathbb{C}^{n-1}\rtimes\mathrm{so}(\mathbb{C}^{n-1})$.

Separating the variables in these equations usually leads to differential
equations with many variables. Only in a few cases it leads to ordinary
differential equations, which turn out to be of hypergeometric type. All these
cases are described in the following table:

Table 1:  PDE | $\begin{array}[]{@{}c@{}}\hbox{Lie}\\\ \hbox{algebra}\end{array}$ | $\begin{array}[]{@{}c@{}}\hbox{dimension of}\\\ \hbox{Cartan algebra}\end{array}$ | $\begin{array}[]{@{}c@{}}\hbox{discrete}\\\ \hbox{symmetries}\end{array}$ | equation
---|---|---|---|---
$\Delta_{\mathbb{C}^{4}}$ | $\mathrm{so}(\mathbb{C}^{6})$ | 3 | cube | hypergeometric 2pt
$\Delta_{\mathbb{C}^{3}}$ | $\mathrm{so}(\mathbb{C}^{5})$ | 2 | square | Gegenbauer
$\Delta_{\mathbb{C}^{2}}+\partial_{t}$ | $\mathrm{sch}(\mathbb{C}^{2})$ | 2 | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ | confluent
$\Delta_{\mathbb{C}}+\partial_{t}$ | $\mathrm{sch}(\mathbb{C}^{1})$ | 1 | $\mathbb{Z}_{4}$ | Hermite
$\Delta_{\mathbb{C}^{2}}-1$ | $\mathbb{C}^{2}\rtimes\mathrm{so}(\mathbb{C}^{2})$ | 1 | $\mathbb{Z}_{2}$ | ${}_{0}F_{1}$

The Laplace equation on $\mathbb{C}^{n}$, the heat equation on
$\mathbb{C}^{n-2}$ and the Helmholtz equation on $\mathbb{C}^{n-1}$ together
with their generalized symmetries can be elegantly derived by an appropriate
reduction from the Laplace equation in $n+2$ dimensions

$\displaystyle\Delta_{\mathbb{C}^{n+2}}K=0.$

Thus, as follows from Table 1, to derive symmetries of the hypergeometric and
confluent equations one should start from

$\displaystyle\Delta_{\mathbb{C}^{6}}K=0.$ (1.6)

To derive the Gegenbauer, Hermite and ${}_{0}F_{1}$ equation together with all
its symmetries it is enough to start with

$\displaystyle\Delta_{\mathbb{C}^{5}}K=0.$ (1.7)

It is easy to reduce the Laplace equation from 6 to 5 dimensions. Thus the
Laplace equation in 6 dimensions is the “mother” of all hypergeometric type
equations.

Let us describe these derivations in more detail.

  * •

We start from (1.6), where the symmetries $\mathrm{so}(\mathbb{C}^{6})$ are
obvious. By what we call the conformal reduction, we can reduce
$\Delta_{\mathbb{C}^{6}}$ to $\Delta_{\mathbb{C}^{4}}$, and then further to
the hypergeometric operator. Alternatively, one can reduce
$\Delta_{\mathbb{C}^{6}}$ to an appropriate Laplace–Beltrami operator, and
then we obtain (1.1) more directly.

  * •

We can repeat an analogous procedure one dimension lower. We start from (1.7),
and at the end we obtain the Gegenbauer operator.

  * •

One can reduce (1.6) to $\Delta_{\mathbb{C}^{2}}+\partial_{t}$, the heat
operator in 2 dimensions, and then further to the confluent operator. Note
that $\mathrm{sch}(\mathbb{C}^{2})$ is contained in
$\mathrm{so}(\mathbb{C}^{6})$.

  * •

One can repeat the above steps one dimension lower, reducing (1.7) to
$\Delta_{\mathbb{C}}+\partial_{t}$, the heat operator in 1 dimension, and then
further to the Hermite operator. Note that $\mathrm{sch}(\mathbb{C})$ is
contained in $\mathrm{so}(\mathbb{C}^{5})$.

  * •

To obtain the ${}_{0}F_{1}$ operator one needs to separate variables in the
Helmholtz operator $\Delta_{\mathbb{C}^{2}}-1$. Its symmetries
$\mathbb{C}^{2}\rtimes\mathrm{so}(\mathbb{C}^{2})$ are contained in
$\mathrm{so}(\mathbb{C}^{5})$ and one can start the derivation from (1.7).

One can ask whether Table 1 can be enlarged, e.g., by considering
$\Delta_{\mathbb{C}^{n}}f=0$ with its conformal symmetries
$\mathrm{so}(\mathbb{C}^{n+2})$ for $n\geq 5$. One can argue that the answer
is negative and Table 1 is complete. Indeed, the Cartan algebra of
$\mathrm{so}(n+2)$ has dimension $[n/2]$, and $n-[n/2]>1$ for $n\geq 5$.
Therefore, separation of variables in the Laplace equation in dimension $n\geq
5$ leads to a differential equation in more than one variable.

###  1.3 Organization of the paper

The paper can be considered as a sequel to [6]. Nevertheless, it is to a large
degree self-contained and independent of [6].

In Section 2 we list the identities that we would like to derive/explain in
our article. As described in the introduction, these identities involve 5
classes of differential operators (1.1)–(1.5). All these operators are first
transformed to the balanced form.

The versions of these identities for the standard form of equations
(1.1)–(1.5) can be found in [6]. In order to reduce the length of the paper,
in this paper we concentrate on the balanced form, which is more symmetric.

Sections 3, 4 and 5 provide basic definitions and concepts, mostly related to
(complex) differential geometry, Lie groups and Lie algebras. This material is
very well known, especially in the real context. Unfortunately, the use of
complex manifolds, natural in our context, has some disadvantages due to the
rigidity of holomorphic functions and their multivaluedness. This is the
reason for some annoying minor complications in these sections, such as local
representations of groups.

In Section 6 we describe the action of the conformal group/Lie algebra in $n$
dimensions. We do this first for a general $n$. As a simple, but instructive
exercise we consider the cases $n=1,2$.

In Section 7 we consider the case $n=4$, which yields the hypergeometric
operator. In Section 8 we consider $n=3$, which leads to the Gegenbauer
operator. These two sections are very parallel to one another. Both are direct
applications of the formalism of Section 6.

In Section 9 we consider the Schrödinger group
$\mathrm{Sch}(\mathbb{C}^{n-2})$ and its Lie algebra
$\mathrm{sch}(\mathbb{C}^{n-2})$. They describe generalized symmetries of the
heat equation in $n-2$ dimensions. We first do this for a general dimension.

In Section 10 we consider the case $n=4$, which yields the confluent operator.
In Section 11 we consider $n=3$, which leads to the Hermite operator. Again,
these two sections are quite parallel. They are applications of the formalism
of Section 9.

In the final Section 12 we consider the Helmholtz equation in 2 dimensions
together with the affine Euclidean symmetries
$\mathbb{C}^{2}\rtimes\mathrm{O}(\mathbb{C}^{2})$ and
$\mathbb{C}^{2}\rtimes\mathrm{so}(\mathbb{C}^{2})$. This leads to the
${}_{0}F_{1}$ equation (or, equivalently, to the Bessel equation). We included
this section for completeness, however its material is well-known and well
documented in the literature.

Note that Sections 3, 4, 5, 6 and 9 are quite general and abstract. On the
other hand, Sections 7, 8, 10, 11 and 12 are more concrete and present
applications of the general theory to the classes of hypergeometric type
equations (1.1), (1.2), (1.3), (1.4) and (1.5), respectively. To a large
extent they can be read independently of the “general” part of the paper.

###  1.4 Comparison with literature

Properties of functions of hypergeometric type are described in numerous
books, such as [1, 8, 9, 10, 13, 23, 26, 35, 38, 48]. In particular, the
properties presented in Section 2 (transmutation relations, discrete
symmetries and factorizations) are known in one form or another. A similar
presentation can be found in [6]. (Unlike in this paper, the presentation of
[6] involves the standard form of hypergeometric type equations and not their
balanced form).

Lie algebras associated with the Bessel and Hermite functions can be found in
papers by Weisner [46, 47].

The idea of studying hypergeometric type equations with help of Lie algebras
was developed further by Miller. His early book [27] considers mostly small
Lie algebras/Lie groups, typically
$\mathrm{sl}(2,\mathbb{C})$/$\mathrm{SL}(2,\mathbb{C})$ and its contractions,
and applies them to obtain various identities about hypergeometric type
functions. These Lie algebras have 1-dimensional “Cartan algebras” and a
single pair of roots. This kind of analysis is able to explain only a single
pair of transmutation relations, whereas to explain bigger families of
transmutation relations one needs larger Lie algebras.

A Lie algebra strictly larger than $\mathrm{sl}(2,\mathbb{C})$ is
$\mathrm{so}(4,\mathbb{C})$. There exists a large literature on the relation
of the hypergeometric equation with $\mathrm{so}(4,\mathbb{C})$ and its real
forms, see, e.g., [16]. This Lie algebra is however still too small to account
for all symmetries of the hypergeometric equation – its Cartan algebra is only
2-dimensional, whereas the equation has three parameters.

An explanation of symmetries of the Gegenbauer equation in terms of
$\mathrm{so}(5)$ and of the hypergeometric equation in terms of
$\mathrm{so}(6)\simeq\mathrm{sl}(4)$ was first given by Miller, see [30], and
especially [29].

Miller and Kalnins wrote a series of papers where they studied the symmetry
approach to separation of variables for various 2nd order partial differential
equations, such as the Laplace and wave equation, see, e.g., [15]. A large
part of this research is summed up in the book by Miller [31]. As an important
consequence of this study, one obtains detailed information about symmetries
of hypergeometric type equations.

The main tool that we use to describe properties of hypergeometric type
functions is the theory of generalized symmetries of 2nd order linear PDE’s.
This theory is described in another book by Miller [28], and further developed
in [31].

The fact that conformal transformations of the Euclidean space are generalized
symmetries of the Laplace equation was apparently known already to Kelvin. Its
explanation in terms of the null quadric first appeared in [3]. Null quadric
as a tool to study conformal symmetries of the Laplace equation is the basic
tool of [15, 17].

The conformal invariance of the Laplace equation generalizes to arbitrary
pseudo-Riemannian manifolds. In fact, the Laplace–Beltrami operator plus an
appropriate multiple of the scalar curvature, sometimes called the Yamabe
Laplacian, is invariant in a generalized sense with respect to conformal maps.
This can be found for instance in [37, 41].

The group of generalized symmetries of the heat equation was known already to
Lie [24]. It was rediscovered (in the essentially equivalent context of the
free Schrödinger equation) by Schrödinger [40]. It was then studied, e.g., in
[12, 33]. Elementary notions from differential geometry used in our paper are
well known. One of standard references in this subject is [18, 19].

A topic that is extensively treated in the literature on the relation of
special functions to group theory, such as [27, 36, 43, 44, 45], is derivation
of various addition formulas. Addition formulas say that a certain special
function can be written as a sum, often infinite, of some related functions.
These identities can be typically interpreted in terms of a certain
representation of an appropriate Lie group. These identities are very
interesting, however we do not discuss them. The only elements of Lie groups
that we consider are very special – they are the “Weyl symmetries”. They yield
discrete symmetries of hypergeometric type equations, such as Kummer’s table.
We leave out addition formulas, because their theory is considerably more
complicated than what we consider in our paper.

The relationship of Kummer’s table with the group of symmetries of a cube
(which is the Weyl group of $\mathrm{so}(\mathbb{C}^{6})$) was discussed in
[25]. A recent paper, where symmetries of the hypergeometric equation play an
important role is [20]. (We learned the term “transmutation relations” from
this paper).

The use of transmutation relations as a tool to derive recurrence relations
for hypergeometric type functions is well known and can be found, e.g., in the
book by Nikiforov–Uvarov [34], in the books by Miller [27] or in older works
such as [42, 46, 47].

There exists various generalizations of hypergeometric type functions. Let us
mention the class of $\mathcal{A}$-hypergeometric functions, which provides a
natural generalization of the usual hypergeometric function to many-variable
situations [2, 4]. Saito [39] considers generalized symmetries in the
framework of $\mathcal{A}$-hypergeometric functions. Note, however, that the
results of Saito are incomplete in the case of the classic hypergeometric
equation. He admits this: “When $p=2$, the symmetry Lie algebra is much larger
than ${\mathfrak{g}}_{2}$”, and he quotes the paper by Miller [29]. Similarly,
the (surprisingly large) Lie algebras of symmetries of the Gegenbauer and
Hermite equations cannot be easily seen from a (seemingly very general)
analysis of Saito.

There are a number of topics related to the hypergeometric type equation that
we do not touch. Let us mention the question whether hypergeometric functions
can be expressed in terms of algebraic functions. This topic, in the context
of $\mathcal{A}$-hypergeometric functions was considered, e.g., in the
interesting papers [2, 4].

In our paper we stick to a rather limited class of equations. We do not have
the ambition to go for generalizations. This limited class has a surprisingly
rich structure, which seems to be lost when we consider their generalizations.

Many, perhaps most identities and ideas described in our paper can be found in
one form or another in the literature, especially in the works by Miller, also
by Miller and Kalnins, as we discussed above. Nevertheless, we believe that
our work raises important points that are not explicitly described in the
literature. We argue that symmetry properties of all hypergeometric type
equations become almost obvious if we add a certain number of variables
obtaining the Laplace equation. We describe this idea in a unified framework,
identifying the relationship of theory of hypergeometric type equations with
such elements of group theory as roots, Cartan algebras and Weyl groups. These
ideas are summed in Table 1, which to our knowledge has not appeared in the
literature, except for the paper [6] written by one of us.

We use various (minor but helpful) ideas to make our presentation as short and
transparent as possible: e.g., the balanced form of hypergeometric type
equations, Lie algebraic parameters and split coordinates in $\mathbb{C}^{n}$.
In our derivations the symmetries are completely obvious at the starting
point, then at each step they become more and more complicated.

The derivation of generalized symmetries of the Laplacian, given after Theorem
6.4, is probably partly original. It leads to an interesting geometric object,
which we call $\Delta^{\diamond}$. It satisfies identities (6.18) and (6.19),
which seem quite important in the context of conformal invariance of the
Laplace equation. These identities are elementary and quite simple, however we
have never seen them in the literature. They can be used to derive
factorizations of hypergeometric type equations, relating them to Casimir
operators of certain distinguished subalgebras, another point that is probably
original.

##  2 Hypergeometric type operators and their symmetries

In this section we describe the families of identities that we would like to
interpret in a group-theoretical fashion in this article. As mentioned above,
all of them involve the balanced form of the operators (1.1)–(1.5).

###  2.1 Hypergeometric operator

In the hypergeometric equation (1.1) we prefer to replace the parameters $a$,
$b$, $c$ with

$\displaystyle\alpha:=c-1,\qquad\beta:=a+b-c,\qquad\mu:=b-a.$

We obtain the $($standard$)$ hypergeometric operator

$\displaystyle\mathcal{F}_{\alpha,\beta,\mu}(w,\partial_{w})=w(1-w)\partial_{w}^{2}+\big{(}(1+\alpha)(1-w)-(1+\beta)w\big{)}\partial_{w}$
$\displaystyle\hphantom{\mathcal{F}_{\alpha,\beta,\mu}(w,\partial_{w})=}{}+\tfrac{1}{4}\mu^{2}-\tfrac{1}{4}(\alpha+\beta+1)^{2}.$
(2.1)

Instead of (2.1) we prefer to consider the balanced hypergeometric operator

$\displaystyle\mathcal{F}_{\alpha,\beta,\mu}^{\mathrm{bal}}(w,\partial_{w}):=w^{\frac{\alpha}{2}}(1-w)^{\frac{\beta}{2}}\mathcal{F}_{\alpha,\beta,\mu}(w,\partial_{w}){(1-w)^{-\frac{\beta}{2}}w^{-\frac{\alpha}{2}}}$
$\displaystyle\hphantom{\mathcal{F}_{\alpha,\beta,\mu}^{\mathrm{bal}}(w,\partial_{w})}{}=\partial_{w}w(1-w)\partial_{w}-\frac{\alpha^{2}}{4w}-\frac{\beta^{2}}{4(1-w)}+\frac{\mu^{2}-1}{4}.$
(2.2)

_Discrete symmetries._
$\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$ does not
change if we flip the signs of $\alpha$, $\beta$, $\mu$. Besides, the
following operators coincide with
$\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$:

$\displaystyle w=z\colon\qquad$
$\displaystyle\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(z,\partial_{z}),$
$\displaystyle w=1-z\colon\qquad$
$\displaystyle\mathcal{F}_{\beta,\alpha,\mu}^{{\mathrm{bal}}}(z,\partial_{z}),$
$\displaystyle w=\frac{1}{z}\colon\qquad$ $\displaystyle
z^{\frac{1}{2}}(-z)\mathcal{F}_{\mu,\beta,\alpha}^{{\mathrm{bal}}}(z,\partial_{z})z^{-\frac{1}{2}},$
$\displaystyle w=1-\frac{1}{z}\colon\qquad$ $\displaystyle
z^{\frac{1}{2}}(-z)\mathcal{F}_{\mu,\alpha,\beta}^{{\mathrm{bal}}}(z,\partial_{z})z^{-\frac{1}{2}},$
$\displaystyle w=\frac{1}{1-z}\colon\qquad$
$\displaystyle(1-z)^{\frac{1}{2}}(z-1)\mathcal{F}_{\beta,\mu,\alpha}^{{\mathrm{bal}}}(z,\partial_{z})(1-z)^{-\frac{1}{2}},$
$\displaystyle w=\frac{z}{z-1}\colon\qquad$
$\displaystyle(1-z)^{\frac{1}{2}}(z-1)\mathcal{F}_{\alpha,\mu,\beta}^{{\mathrm{bal}}}(z,\partial_{z})(1-z)^{-\frac{1}{2}}.$

Transmutation relations:

$\displaystyle\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha}{2w}+\frac{\beta}{2(1-w)}\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha+1,\beta+1,\mu}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha}{2w}+\frac{\beta}{2(1-w)}\right),$
$\displaystyle\sqrt{w(1-w)}\left(\partial_{w}+\frac{\alpha}{2w}-\frac{\beta}{2(1-w)}\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha-1,\beta-1,\mu}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{w(1-w)}\left(\partial_{w}+\frac{\alpha}{2w}-\frac{\beta}{2(1-w)}\right),$
$\displaystyle\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha}{2w}-\frac{\beta}{2(1-w)}\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha+1,\beta-1,\mu}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha}{2w}-\frac{\beta}{2(1-w)}\right),$
$\displaystyle\sqrt{w(1-w)}\left(\partial_{w}+\frac{\alpha}{2w}+\frac{\beta}{2(1-w)}\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha-1,\beta+1,\mu}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{w(1-w)}\left(\partial_{w}+\frac{\alpha}{2w}+\frac{\beta}{2(1-w)}\right),$
$\displaystyle\sqrt{w}\left(2(1-w)\partial_{w}-\frac{\alpha}{\vphantom{(}w\vphantom{)}}-\mu-3\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha+1,\beta,\mu+1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{w}\left(2(1-w)\partial_{w}-\frac{\alpha}{\vphantom{(}w\vphantom{)}}-\mu-1\right),$
$\displaystyle\sqrt{w}\left(2(1-w)\partial_{w}+\frac{\alpha}{\vphantom{(}w\vphantom{)}}+\mu-3\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha-1,\beta,\mu-1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{w}\left(2(1-w)\partial_{w}+\frac{\alpha}{\vphantom{(}w\vphantom{)}}+\mu-1\right),$
$\displaystyle\sqrt{w}\left(2(1-w)\partial_{w}-\frac{\alpha}{\vphantom{(}w\vphantom{)}}+\mu-3\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha+1,\beta,\mu-1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{w}\left(2(1-w)\partial_{w}-\frac{\alpha}{\vphantom{(}w\vphantom{)}}+\mu-1\right),$
$\displaystyle\sqrt{w}\left(2(1-w)\partial_{w}+\frac{\alpha}{\vphantom{(}w\vphantom{)}}-\mu-3\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha-1,\beta,\mu+1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{w}\left(2(1-w)\partial_{w}+\frac{\alpha}{\vphantom{(}w\vphantom{)}}-\mu-1\right),$
$\displaystyle\sqrt{1-w}\left(-2w\partial_{w}-\frac{\beta}{1-w}-\mu-3\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha,\beta+1,\mu+1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w}\left(-2w\partial_{w}-\frac{\beta}{1-w}-\mu-1\right),$
$\displaystyle\sqrt{1-w}\left(-2w\partial_{w}+\frac{\beta}{1-w}+\mu-3\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha,\beta-1,\mu-1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w}\left(-2w\partial_{w}+\frac{\beta}{1-w}+\mu-1\right),$
$\displaystyle\sqrt{1-w}\left(-2w\partial_{w}-\frac{\beta}{1-w}+\mu-3\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha,\beta+1,\mu-1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w}\left(-2w\partial_{w}-\frac{\beta}{1-w}+\mu-1\right),$
$\displaystyle\sqrt{1-w}\left(-2w\partial_{w}+\frac{\beta}{1-w}-\mu-3\right)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{F}_{\alpha,\beta-1,\mu+1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w}\left(-2w\partial_{w}+\frac{\beta}{1-w}-\mu-1\right).$

Factorizations:

$\displaystyle\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle=\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha-1}{2w}+\frac{\beta-1}{2(1-w)}\right)\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha}{2w}+\frac{\beta}{2(1-w)}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\beta+\alpha\right)\left(\beta+\alpha-2\right)+\mu^{2}-1\right)$
$\displaystyle=\sqrt{w(1-w)}\left(\partial_{w}+\frac{\alpha+1}{2w}+\frac{\beta-1}{2(1-w)}\right)\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha}{2w}-\frac{\beta}{2(1-w)}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\beta-\alpha\right)\left(\beta-\alpha-2\right)+\mu^{2}-1\right)$
$\displaystyle=\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha-1}{2w}-\frac{\beta+1}{2(1-w)}\right)\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha}{2w}-\frac{\beta}{2(1-w)}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\beta-\alpha\right)\left(\beta-\alpha+2\right)+\mu^{2}-1\right)$
$\displaystyle=\sqrt{w(1-w)}\left(\partial_{w}+\frac{\alpha+1}{2w}-\frac{\beta+1}{2(1-w)}\right)\sqrt{w(1-w)}\left(\partial_{w}-\frac{\alpha}{2w}+\frac{\beta}{2(1-w)}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\beta+\alpha\right)\left(\beta+\alpha+2\right)+\mu^{2}-1\right),$
$\displaystyle-(1-w)\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle=-\sqrt{w}\left(-\frac{1}{2}+(1-w)\partial_{w}-\frac{\alpha-1}{2w}-\frac{\mu-1}{2}\right)\sqrt{w}\left(-\frac{1}{2}+(1-w)\partial_{w}+\frac{\alpha}{2w}+\frac{\mu}{2}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\mu+\alpha\right)\left(\mu+\alpha-2\right)+\beta^{2}-1\right)$
$\displaystyle=-\sqrt{w}\left(-\frac{1}{2}+(1-w)\partial_{w}+\frac{\alpha+1}{2w}-\frac{\mu-1}{2}\right)\sqrt{w}\left(-\frac{1}{2}+(1-w)\partial_{w}-\frac{\alpha}{2w}+\frac{\mu}{2}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\mu-\alpha\right)\left(\mu-\alpha-2\right)+\beta^{2}-1\right)$
$\displaystyle=-\sqrt{w}\left(-\frac{1}{2}+(1-w)\partial_{w}-\frac{\alpha-1}{2w}+\frac{\mu+1}{2}\right)\sqrt{w}\left(-\frac{1}{2}+(1-w)\partial_{w}+\frac{\alpha}{2w}-\frac{\mu}{2}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\mu-\alpha\right)\left(\mu-\alpha+2\right)+\beta^{2}-1\right)$
$\displaystyle=-\sqrt{w}\left(-\frac{1}{2}+(1-w)\partial_{w}+\frac{\alpha+1}{2w}+\frac{\mu+1}{2}\right)\sqrt{w}\left(-\frac{1}{2}+(1-w)\partial_{w}-\frac{\alpha}{2w}-\frac{\mu}{2}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\mu+\alpha\right)\left(\mu+\alpha+2\right)+\beta^{2}-1\right),$
$\displaystyle-w\mathcal{F}_{\alpha,\beta,\mu}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle=-\sqrt{1-w}\left(-\frac{1}{2}-w\partial_{w}-\frac{\beta-1}{2(1-w)}-\frac{\mu-1}{2}\right)\sqrt{1-w}\left(-\frac{1}{2}-w\partial_{w}+\frac{\beta}{2(1-w)}+\frac{\mu}{2}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\mu+\beta\right)\left(\mu+\beta-2\right)+\alpha^{2}-1\right)$
$\displaystyle=-\sqrt{1-w}\left(-\frac{1}{2}-w\partial_{w}+\frac{\beta+1}{2(1-w)}-\frac{\mu-1}{2}\right)\sqrt{1-w}\left(-\frac{1}{2}-w\partial_{w}-\frac{\beta}{2(1-w)}+\frac{\mu}{2}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\mu-\beta\right)\left(\mu-\beta-2\right)+\alpha^{2}-1\right)$
$\displaystyle=-\sqrt{1-w}\left(-\frac{1}{2}-w\partial_{w}-\frac{\beta-1}{2(1-w)}+\frac{\mu+1}{2}\right)\sqrt{1-w}\left(-\frac{1}{2}-w\partial_{w}+\frac{\beta}{2(1-w)}-\frac{\mu}{2}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\mu-\beta\right)\left(\mu-\beta+2\right)+\alpha^{2}-1\right)$
$\displaystyle=-\sqrt{1-w}\left(-\frac{1}{2}-w\partial_{w}+\frac{\beta+1}{2(1-w)}+\frac{\mu+1}{2}\right)\sqrt{1-w}\left(-\frac{1}{2}-w\partial_{w}-\frac{\beta}{2(1-w)}-\frac{\mu}{2}\right)$
$\displaystyle\hphantom{=}{}-\frac{1}{4}\left(\left(\mu+\beta\right)\left(\mu+\beta+2\right)+\alpha^{2}-1\right).$

It is striking how symmetric the above formulas look like. The main goal of
our paper is to explain why this is so.

###  2.2 Gegenbauer operator

In the hypergeometric equation (1.1) let us move the singular points to $-1$,
$1$ and demand that it is reflection invariant. Then we can eliminate one of
the parameters, say $c$. We obtain the Gegenbauer equation (1.2). We introduce
new parameters

$\displaystyle\alpha:=\frac{{a}+{b}-1}{2},\qquad\lambda:=\frac{{b}-{a}}{2}.$

We obtain the $($standard$)$ Gegenbauer operator

$\displaystyle\mathcal{S}_{\alpha,\lambda}(w,\partial_{w}):=\big{(}1-w^{2}\big{)}\partial_{w}^{2}-2(1+\alpha)w\partial_{w}+\lambda^{2}-\left(\alpha+\frac{1}{2}\right)^{2}.$
(2.3)

The balanced Gegenbauer operator is

$\displaystyle\mathcal{S}_{\alpha,\lambda}^{\mathrm{bal}}(w,\partial_{w}):=\big{(}w^{2}-1\big{)}^{\frac{\alpha}{2}}\mathcal{S}_{\alpha,\lambda}(w,\partial_{w})\big{(}w^{2}-1\big{)}^{-\frac{\alpha}{2}}$
$\displaystyle\hphantom{\mathcal{S}_{\alpha,\lambda}^{\mathrm{bal}}(w,\partial_{w})}{}=\partial_{w}\big{(}1-w^{2}\big{)}\partial_{w}-\frac{\alpha^{2}}{1-w^{2}}+\lambda^{2}-\frac{1}{4}.$
(2.4)

_Discrete symmetries._
$\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$ does not
change if we flip the signs of $\alpha$, $\lambda$. Besides, the following
operators coincide with
$\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$:

$\displaystyle w=z\colon\qquad$
$\displaystyle\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(z,\partial_{z}),$
$\displaystyle w=\frac{z}{\sqrt{z^{2}-1}}\colon\qquad$
$\displaystyle\big{(}z^{2}-1\big{)}^{\frac{1}{4}}\big{(}z^{2}-1\big{)}\mathcal{S}_{\lambda,\alpha}^{{\mathrm{bal}}}(z,\partial_{z})\big{(}z^{2}-1\big{)}^{-\frac{1}{4}}.$

_Transmutation relations:_

$\displaystyle\sqrt{1-w^{2}}\left(-\frac{5}{2}-w\partial_{w}-\frac{\alpha}{1-w^{2}}-\lambda\right)\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{S}_{\alpha+1,\lambda+1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w^{2}}\left(-\frac{1}{2}-w\partial_{w}-\frac{\alpha}{1-w^{2}}-\lambda\right),$
$\displaystyle\sqrt{1-w^{2}}\left(-\frac{5}{2}-w\partial_{w}+\frac{\alpha}{1-w^{2}}+\lambda\right)\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{S}_{\alpha-1,\lambda-1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w^{2}}\left(-\frac{1}{2}-w\partial_{w}+\frac{\alpha}{1-w^{2}}+\lambda\right),$
$\displaystyle\sqrt{1-w^{2}}\left(-\frac{5}{2}-w\partial_{w}-\frac{\alpha}{1-w^{2}}+\lambda\right)\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{S}_{\alpha+1,\lambda-1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w^{2}}\left(-\frac{1}{2}-w\partial_{w}-\frac{\alpha}{1-w^{2}}+\lambda\right),$
$\displaystyle\sqrt{1-w^{2}}\left(-\frac{5}{2}-w\partial_{w}+\frac{\alpha}{1-w^{2}}-\lambda\right)\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\mathcal{S}_{\alpha-1,\lambda+1}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w^{2}}\left(-\frac{1}{2}-w\partial_{w}+\frac{\alpha}{1-w^{2}}-\lambda\right),$
$\displaystyle
w\left(-\frac{5}{2}+\frac{1-w^{2}}{w}\partial_{w}-\lambda\right)\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{S}_{\alpha,\lambda+1}^{{\mathrm{bal}}}(w,\partial_{w})w\left(-\frac{1}{2}+\frac{1-w^{2}}{w}\partial_{w}-\lambda\right),$
$\displaystyle
w\left(-\frac{5}{2}+\frac{1-w^{2}}{w}\partial_{w}+\lambda\right)\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{S}_{\alpha,\lambda-1}^{{\mathrm{bal}}}(w,\partial_{w})w\left(-\frac{1}{2}+\frac{1-w^{2}}{w}\partial_{w}+\lambda\right),$
$\displaystyle\sqrt{1-w^{2}}\left(\partial_{w}+\frac{w}{1-w^{2}}\alpha\right)\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{S}_{\alpha+1,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w^{2}}\left(\partial_{w}+\frac{w}{1-w^{2}}\alpha\right),$
$\displaystyle\sqrt{1-w^{2}}\left(\partial_{w}-\frac{w}{1-w^{2}}\alpha\right)\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{S}_{\alpha+1,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})\sqrt{1-w^{2}}\left(\partial_{w}-\frac{w}{1-w^{2}}\alpha\right).$

_Factorizations:_

$\displaystyle-\big{(}1-w^{2}\big{)}\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=-w\left(\frac{1}{2}+\frac{1-w^{2}}{w}\partial_{w}-\lambda\right)w\left(-\frac{1}{2}+\frac{1-w^{2}}{w}\partial_{w}+\lambda\right)-\lambda^{2}+\lambda+\alpha^{2}-\frac{1}{4}$
$\displaystyle\qquad{}=-w\left(\frac{1}{2}+\frac{1-w^{2}}{w}\partial_{w}+\lambda\right)w\left(-\frac{1}{2}+\frac{1-w^{2}}{w}\partial_{w}-\lambda\right)-\lambda^{2}-\lambda+\alpha^{2}-\frac{1}{4},$
$\displaystyle\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=\sqrt{1-w^{2}}\left(\partial_{w}+\frac{w}{1-w^{2}}\left(\alpha-1\right)\right)\sqrt{1-w^{2}}\left(\partial_{w}-\frac{w}{1-w^{2}}\alpha\right)$
$\displaystyle\qquad\quad{}-\alpha^{2}+\alpha+\lambda^{2}-\frac{1}{4}$
$\displaystyle\qquad{}=\sqrt{1-w^{2}}\left(\partial_{w}-\frac{w}{1-w^{2}}\left(\alpha+1\right)\right)\sqrt{1-w^{2}}\left(\partial_{w}+\frac{w}{1-w^{2}}\alpha\right)-\alpha^{2}-\alpha+\lambda^{2}-\frac{1}{4},$
$\displaystyle-w^{2}\mathcal{S}_{\alpha,\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$
$\displaystyle\qquad{}=-\sqrt{1-w^{2}}\left(\frac{1}{2}-w\partial_{w}-\frac{\alpha-1}{1-w^{2}}-\lambda\right)\sqrt{1-w^{2}}\left(-\frac{1}{2}-w\partial_{w}+\frac{\alpha}{1-w^{2}}+\lambda\right)$
$\displaystyle\qquad\quad{}-\left(\lambda+\alpha\right)\left(\lambda+\alpha-2\right)-\frac{3}{4}$
$\displaystyle\qquad{}=-\sqrt{1-w^{2}}\left(\frac{1}{2}-w\partial_{w}+\frac{\alpha+1}{1-w^{2}}-\lambda\right)\sqrt{1-w^{2}}\left(-\frac{1}{2}-w\partial_{w}-\frac{\alpha}{1-w^{2}}+\lambda\right)$
$\displaystyle\qquad\quad{}-\left(\lambda-\alpha\right)\left(\lambda-\alpha-2\right)-\frac{3}{4}$
$\displaystyle\qquad{}=-\sqrt{1-w^{2}}\left(\frac{1}{2}-w\partial_{w}+\frac{\alpha+1}{1-w^{2}}+\lambda\right)\sqrt{1-w^{2}}\left(-\frac{1}{2}-w\partial_{w}-\frac{\alpha}{1-w^{2}}-\lambda\right)$
$\displaystyle\qquad\quad{}-\left(\lambda+\alpha\right)\left(\lambda+\alpha+2\right)-\frac{3}{4}$
$\displaystyle\qquad{}=-\sqrt{1-w^{2}}\left(\frac{1}{2}-w\partial_{w}-\frac{\alpha-1}{1-w^{2}}+\lambda\right)\sqrt{1-w^{2}}\left(-\frac{1}{2}-w\partial_{w}+\frac{\alpha}{1-w^{2}}-\lambda\right)$
$\displaystyle\qquad\quad{}-\left(\lambda-\alpha\right)\left(\lambda-\alpha+2\right)-\frac{3}{4}.$

###  2.3 Confluent operator

In the confluent equation (1.3) we introduce new parameters

$\displaystyle\alpha:=c-1,\qquad\theta:=2a-c.$

The $($standard$)$ confluent operator is

$\displaystyle\mathcal{F}_{\theta,\alpha}(w,\partial_{w}):=w\partial_{w}^{2}+(1+\alpha-w)\partial_{w}-\frac{1}{2}(1+\theta+\alpha).$
(2.5)

The balanced confluent operator is

$\displaystyle\mathcal{F}_{\theta,\alpha}^{\mathrm{bal}}(w,\partial_{w}):=w^{\frac{\alpha}{2}}\mathrm{e}^{-\frac{w}{2}}\mathcal{F}_{\theta,\alpha}(w,\partial_{w}){\mathrm{e}^{\frac{w}{2}}w^{-\frac{\alpha}{2}}}=\partial_{w}w\partial_{w}-\frac{w}{4}-\frac{\theta}{2}-\frac{\alpha^{2}}{4w}.$
(2.6)

_Discrete symmetries._
$\mathcal{F}_{\theta,\alpha}^{\mathrm{bal}}(w,\partial_{w})$ does not change
if we flip the sign of $\alpha$. Besides, the following operators coincide
with $\mathcal{F}_{\theta,\alpha}^{\mathrm{bal}}(w,\partial_{w})$:

$\displaystyle
w=z\colon\quad\mathcal{F}_{\theta,\alpha}^{\mathrm{bal}}(z,\partial_{z}),\qquad
w=-z\colon\quad\mathcal{F}_{-\theta,\alpha}^{\mathrm{bal}}(z,\partial_{z}).$

_Transmutation relations:_

$\displaystyle\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}+\frac{w}{2}\right)\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{F}_{\theta+1,\alpha-1}^{{\mathrm{bal}}}(w,\partial_{w})\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}+\frac{w}{2}\right),$
$\displaystyle\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}+\frac{w}{2}\right)\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{F}_{\theta+1,\alpha+1}^{{\mathrm{bal}}}(w,\partial_{w})\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}+\frac{w}{2}\right),$
$\displaystyle\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}-\frac{w}{2}\right)\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{F}_{\theta-1,\alpha-1}^{{\mathrm{bal}}}(w,\partial_{w})\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}-\frac{w}{2}\right),$
$\displaystyle\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}-\frac{w}{2}\right)\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{F}_{\theta-1,\alpha+1}^{{\mathrm{bal}}}(w,\partial_{w})\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}-\frac{w}{2}\right),$
$\displaystyle\left(-w\partial_{w}-\frac{\theta}{2}-\frac{w}{2}-\frac{3}{2}\right)\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{F}_{\theta+2,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})\left(-w\partial_{w}-\frac{\theta}{2}-\frac{w}{2}-\frac{1}{2}\right),$
$\displaystyle\left(w\partial_{w}-\frac{\theta}{2}-\frac{w}{2}+\frac{3}{2}\right)\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{F}_{\theta-2,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})\left(w\partial_{w}-\frac{\theta}{2}-\frac{w}{2}+\frac{1}{2}\right).$

_Factorizations:_

$\displaystyle-w\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})=\left(w\partial_{w}-\frac{\theta+1}{2}-\frac{w}{2}\right)\left(-w\partial_{w}-\frac{\theta+1}{2}-\frac{w}{2}\right)-\frac{1}{4}\left(\theta+1\right)^{2}+\frac{1}{4}\alpha^{2}$
$\displaystyle\hphantom{-w\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})}{}=\left(-w\partial_{w}-\frac{\theta-1}{2}-\frac{w}{2}\right)\left(w\partial_{w}-\frac{\theta-1}{2}-\frac{w}{2}\right)-\frac{1}{4}\left(\theta-1\right)^{2}+\frac{1}{4}\alpha^{2},$
$\displaystyle\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})=\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha-1}{2}-\frac{w}{2}\right)\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}+\frac{w}{2}\right)-\frac{1}{2}\left(\theta-\alpha+1\right)$
$\displaystyle\hphantom{\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})}{}=\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha+1}{2}+\frac{w}{2}\right)\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}-\frac{w}{2}\right)-\frac{1}{2}\left(\theta-\alpha-1\right)$
$\displaystyle\hphantom{\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})}{}=\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha+1}{2}-\frac{w}{2}\right)\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}+\frac{w}{2}\right)-\frac{1}{2}\left(\theta+\alpha+1\right)$
$\displaystyle\hphantom{\mathcal{F}_{\theta,\alpha}^{{\mathrm{bal}}}(w,\partial_{w})}{}=\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha-1}{2}+\frac{w}{2}\right)\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}-\frac{w}{2}\right)-\frac{1}{2}\left(\theta+\alpha-1\right).$

###  2.4 Hermite operator

In the Hermite equation (1.4) we prefer to use the parameter

$\displaystyle\lambda={a}-\tfrac{1}{2}.$

The $($standard$)$ Hermite operator is

$\displaystyle\mathcal{S}_{\lambda}(w,\partial_{w}):=\partial_{w}^{2}-2w\partial_{w}-2\lambda-1.$
(2.7)

The balanced Hermite operator is

$\displaystyle\mathcal{S}_{\lambda}^{\mathrm{bal}}(w,\partial_{w}):=\mathrm{e}^{-\frac{w^{2}}{2}}\mathcal{S}_{\lambda}(w,\partial_{w})\mathrm{e}^{\frac{w^{2}}{2}}=\partial_{w}^{2}-w^{2}-2\lambda.$
(2.8)

_Discrete symmetries._ The following operators coincide with
$\mathcal{S}_{\lambda}^{{\mathrm{bal}}}(w,\partial_{w})$:

$\displaystyle w=z\colon\quad$
$\displaystyle\mathcal{S}_{\lambda}^{\mathrm{bal}}(z,\partial_{z}),\quad\qquad$
$\displaystyle w=\mathrm{i}z\colon\quad$
$\displaystyle-\mathcal{S}_{-\lambda}^{\mathrm{bal}}(z,\partial_{z}),$
$\displaystyle w=-z\colon\quad$
$\displaystyle\mathcal{S}_{\lambda}^{\mathrm{bal}}(z,\partial_{z}),\quad\qquad$
$\displaystyle w=-\mathrm{i}z\colon\quad$
$\displaystyle-\mathcal{S}_{-\lambda}^{\mathrm{bal}}(z,\partial_{z}).$

_Transmutation relations:_

$\displaystyle(\partial_{w}+w)\mathcal{S}_{\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{S}_{\lambda+1}^{{\mathrm{bal}}}(w,\partial_{w})(\partial_{w}+w),$
$\displaystyle(\partial_{w}-w)\mathcal{S}_{\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{S}_{\lambda-1}^{{\mathrm{bal}}}(w,\partial_{w})(\partial_{w}-w),$
$\displaystyle\big{(}{-}w\partial_{w}-\lambda-w^{2}-\tfrac{5}{2}\big{)}\mathcal{S}_{\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{S}_{\lambda+2}^{{\mathrm{bal}}}(w,\partial_{w})\big{(}{-}w\partial_{w}-\lambda-w^{2}-\tfrac{1}{2}\big{)},$
$\displaystyle\big{(}w\partial_{w}-\lambda-w^{2}+\tfrac{5}{2}\big{)}\mathcal{S}_{\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\mathcal{S}_{\lambda-2}^{{\mathrm{bal}}}(w,\partial_{w})\big{(}w\partial_{w}-\lambda-w^{2}+\tfrac{1}{2}\big{)}.$

_Factorizations:_

$\displaystyle-w^{2}\mathcal{S}_{\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=\big{(}w\partial_{w}-\lambda-\tfrac{3}{2}-w^{2}\big{)}\big{(}{-}w\partial_{w}-\lambda-\tfrac{1}{2}-w^{2}\big{)}-(\lambda+1)^{2}+\tfrac{1}{4}$
$\displaystyle\hphantom{-w^{2}\mathcal{S}_{\lambda}^{{\mathrm{bal}}}(w,\partial_{w})}{}=\big{(}{-}w\partial_{w}-\lambda+\tfrac{3}{2}-w^{2}\big{)}\big{(}w\partial_{w}-\lambda+\tfrac{1}{2}-w^{2}\big{)}-(\lambda-1)^{2}+\tfrac{1}{4},$
$\displaystyle\mathcal{S}_{\lambda}^{{\mathrm{bal}}}(w,\partial_{w})=(\partial_{w}-w)(\partial_{w}+w)-2\lambda-1=(\partial_{w}+w)(\partial_{w}-w)-2\lambda+1.$

###  2.5 $\boldsymbol{{}_{0}F_{1}}$ operator

In the ${}_{0}F_{1}$ equation (1.5) we prefer to use the parameter

$\displaystyle\alpha:=c-1.$

The $($standard$)$ ${}_{0}F_{1}$ operator is

$\displaystyle\mathcal{F}_{\alpha}(w,\partial_{w}):=w\partial_{w}^{2}+(\alpha+1)\partial_{w}-1.$

The balanced ${}_{0}F_{1}$ operator is

$\displaystyle\mathcal{F}_{\alpha}^{\mathrm{bal}}(w,\partial_{w}):=w^{\frac{\alpha}{2}}\mathcal{F}_{\alpha}(w,\partial_{w})w^{-\frac{\alpha}{2}}=\partial_{w}w\partial_{w}-1-\frac{\alpha^{2}}{4w}.$

_Discrete symmetries._ $\mathcal{F}_{\alpha}(w,\partial_{w})$ does not change
if we flip the sign of $\alpha$.

_Transmutation relations:_

$\displaystyle\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}\right)\mathcal{F}_{\alpha}^{\mathrm{bal}}(w,\partial_{w})=\mathcal{F}_{\alpha+1}^{\mathrm{bal}}(w,\partial_{w})\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}\right),$
$\displaystyle\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}\right)\mathcal{F}_{\alpha}^{\mathrm{bal}}(w,\partial_{w})=\mathcal{F}_{\alpha-1}^{\mathrm{bal}}(w,\partial_{w})\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}\right).$

_Factorizations:_

$\displaystyle\mathcal{F}_{\alpha}^{\mathrm{bal}}(w,\partial_{w})=\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha-1}{2}\right)\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha}{2}\right)-1$
$\displaystyle\hphantom{\mathcal{F}_{\alpha}^{\mathrm{bal}}(w,\partial_{w})}{}=\frac{1}{\sqrt{w}}\left(w\partial_{w}+\frac{\alpha+1}{2}\right)\frac{1}{\sqrt{w}}\left(w\partial_{w}-\frac{\alpha}{2}\right)-1.$

##  3 Basic complex geometry

In this section we describe basic notation for complex geometry. Throughout
the section, $\Omega$, $\Omega_{1}$, $\Omega_{2}$ are open subsets of
$\mathbb{C}^{n}$ or, more generally, complex manifolds. We will write
$\mathbb{C}^{\times}$ for the multiplicative group
$\mathbb{C}\backslash\\{0\\}$.

We will write $\mathcal{A}(\Omega)$ for the set of holomorphic functions on
$\Omega$. $y=(y^{1},\dots,y^{n})$ will denote generic coordinates on $\Omega$.
We will write $\mathcal{A}^{\times}(\Omega)$ for the set of nowhere vanishing
holomorphic functions on $\Omega$.

###  3.1 Vector fields

Let $\mathrm{hol}(\Omega)$ denote the Lie algebra of holomorphic vector fields
on $\Omega$. Every $A\in\mathrm{hol}(\Omega)$ can be identified with the
differential operator

$\displaystyle Af(y)=\sum_{i}A^{i}(y)\partial_{y^{i}}f(y),\qquad
f\in\mathcal{A}(\Omega),$

where $A^{i}\in\mathcal{A}(\Omega)$, $i=1,\dots,n$.

We will denote by $\mathcal{A}\rtimes\mathrm{hol}(\Omega)$ the Lie algebra of
1st order differential operators on $\Omega$ with holomorphic coefficents.
Such operators can be written as

$\displaystyle(A+M)f(y):=\sum_{i}A^{i}(y)\partial_{y^{i}}f(y)+M(y)f(y),$

where $A\in\mathrm{hol}(\Omega)$ and $M\in\mathcal{A}(\Omega)$.

Let ${\mathfrak{g}}$ be a Lie subalgebra of $\mathrm{hol}(\Omega)$. A linear
function ${\mathfrak{g}}\ni A\mapsto M_{A}\in\mathcal{A}(\Omega)$ satisfying

$\displaystyle A_{1}M_{A_{2}}-A_{2}M_{A_{1}}=M_{[A_{1},A_{2}]}$

will be called a cocycle for ${\mathfrak{g}}$. Every cocycle together with
$\eta\in\mathbb{C}$ determines a homomorphism

$\displaystyle{\mathfrak{g}}\ni A\mapsto A+\eta
M_{A}\in\mathcal{A}\rtimes\mathrm{hol}(\Omega).$

###  3.2 Point transformations

The set of biholomorphic maps $\Omega_{1}\to\Omega_{2}$ will be denoted
$\mathrm{Hol}(\Omega_{1},\Omega_{2})$. We set
$\mathrm{Hol}(\Omega):=\mathrm{Hol}(\Omega,\Omega)$.

Let ${\alpha}\in\mathrm{Hol}(\Omega_{1},\Omega_{2})$. The transport of
functions, vector fields, etc. by the map $\alpha$ will be also denoted by
$\alpha$. More precisely, for $f\in\mathcal{A}(\Omega_{1})$ we define
${\alpha}f\in\mathcal{A}(\Omega_{2})$ by

$\displaystyle({\alpha}f)(y):=f\big{(}{\alpha}^{-1}(y)\big{)}.$

For $A\in\mathrm{hol}(\Omega_{1})$, ${\alpha}(A)\in\mathrm{hol}(\Omega_{2})$
is defined as

$\displaystyle{\alpha}(A):=\alpha A\alpha^{-1}.$

If $m\in\mathcal{A}^{\times}(\Omega_{2})$, then we have a map
$m{\alpha}\colon\mathcal{A}(\Omega_{1})\to\mathcal{A}(\Omega_{2})$ given by

$\displaystyle(m{\alpha}f)(y):=m(y)f\big{(}{\alpha}^{-1}(y)\big{)}.$

$\mathcal{A}^{\times}\rtimes\mathrm{Hol}(\Omega_{1},\Omega_{2})$ will denote
the set of transformations $\mathcal{A}(\Omega_{1})\to\mathcal{A}(\Omega_{2})$
of this form. Clearly, $\mathcal{A}^{\times}\rtimes\mathrm{Hol}(\Omega)$ is a
group.

Suppose that $G$ is a subgroup of $\mathrm{Hol}(\Omega)$. A family
$G\ni{\alpha}\mapsto m_{\alpha}\in\mathcal{A}^{\times}(\Omega)$ satisfying

$\displaystyle
m_{{\alpha}_{2}}(y)m_{{\alpha}_{1}}\big{(}{\alpha}_{2}^{-1}(y)\big{)}=m_{{\alpha}_{2}{\alpha}_{1}}(y),\qquad{\alpha}_{1},{\alpha}_{2}\in
G,\qquad y\in\Omega,$

will be called a cocycle for $G$. Every cocycle together with
$\eta\in\mathbb{Z}$ determines a homomorphism

$\displaystyle G\ni{\alpha}\mapsto
m_{\alpha}^{\eta}{\alpha}\in\mathcal{A}^{\times}\rtimes\mathrm{Hol}(\Omega).$
(3.1)

###  3.3 Local cocycles

Unfortunately, the above definition of a cocycle on a group is too rigid for
our purposes. Below we introduce a weaker version of this concept, which we
will be better adapted to our goals.

As before, we assume that $G$ is a subgroup of $\mathrm{Hol}(\Omega)$.
Besides, we fix $\Omega_{0}$ open in $\Omega$. For $\alpha\in G$ we will write

$\displaystyle\Omega_{0}^{\alpha}:=\Omega_{0}\cap\alpha(\Omega_{0}).$ (3.2)

Furthermore, we suppose that to every $\alpha\in G$ we associate
$m_{\alpha}\in\mathcal{A}^{\times}\big{(}\Omega_{0}^{\alpha}\big{)}$
satisfying

$\displaystyle
m_{\alpha_{2}}(y)m_{\alpha_{1}}\big{(}\alpha_{2}^{-1}(y)\big{)}=m_{\alpha_{2}\alpha_{1}}(y),\qquad\alpha_{1},\alpha_{2}\in
G,\qquad
y\in\Omega_{0}\cap\alpha_{2}(\Omega_{0})\cap\alpha_{2}\circ\alpha_{1}(\Omega_{0}).$

Then $G\mapsto m_{\alpha}$ will be called a local cocycle for $G$ based on
$\Omega_{0}$.

Let $p\in\mathcal{A}^{\times}(\Omega_{0})$. Then

$\displaystyle
m_{\alpha}(y):=\frac{p(y)}{p\big{(}\alpha^{-1}(y)\big{)}},\qquad
y\in\Omega_{0}^{\alpha}$ (3.3)

is a (trivial) example of a local cocycle based on $\Omega_{0}$. Note that if
$p$ cannot be extended to a holomorphic function on the whole $\Omega$, then
(3.3) cannot be extended to a true cocycle.

Let $\eta\in\mathbb{Z}$. For any $\alpha\in G$ we can define the map

$\displaystyle
m_{\alpha}^{\eta}\alpha\in\mathcal{A}^{\times}\rtimes\mathrm{Hol}\big{(}\Omega_{0}^{\alpha^{-1}},\Omega_{0}^{\alpha}\big{)}.$
(3.4)

For $\alpha_{1},\alpha_{2}\in G$ and $\eta\in\mathbb{Z}$ we have the following
weak form of the chain rule:

$\displaystyle\big{(}m_{\alpha_{2}}^{\eta}\alpha_{2}\circ
m_{\alpha_{1}}^{\eta}\alpha_{1}\big{)}(y)=m_{\alpha_{2}{\circ}\alpha_{1}}^{\eta}\alpha_{2}{\circ}\alpha_{1}(y),\qquad
y\in\Omega_{0}\cap\alpha_{2}(\Omega_{0})\cap\alpha_{2}{\circ}\alpha_{1}(\Omega_{0}).$

It will be convenient have a special notation for such a collection of maps
(3.4): We will write that

$\displaystyle G\ni\alpha\mapsto
m_{\alpha}^{\eta}{\alpha}\mathop{\in}_{\mathrm{loc}}\mathcal{A}^{\times}\rtimes\mathrm{Hol}(\Omega_{0})$

is a local representation of $G$.

###  3.4 Half-integer powers of a cocycle

For non-integer exponents the power function is unfortunately multivalued.
Because of that, strictly speaking, $\eta\not\in\mathbb{Z}$ should not be
allowed in (3.1). However, we will be forced to consider situations when
$\eta$ is a half integer. This can be handled by the following formalism.

The non-identity element of the group $\mathbb{Z}_{2}$ acts on
$\mathbb{C}^{\times}$ by $\mathbb{C}^{\times}\ni
a\mapsto-a\in\mathbb{C}^{\times}$. This defines
$\mathbb{C}^{\times}/\mathbb{Z}_{2}$, which is the space of pairs of non-zero
complex numbers differing by a sign.

Let $\eta\in\frac{1}{2}+\mathbb{Z}$. Then for any $a\in\mathbb{C}^{\times}$,
the power $a^{\eta}$ can be interpreted as an element in
$\mathbb{C}^{\times}/\mathbb{Z}_{2}$.

Let us restrict our attention to $\Omega$ that are simply connected. We then
define

$\displaystyle\mathcal{A}^{\times}(\Omega)/\mathbb{Z}_{2}:=\big{\\{}(f,-f)\colon
f\in\mathcal{A}^{\times}(\Omega)\big{\\}},$ (3.5)

If $f\in\mathcal{A}^{\times}(\Omega)$, then $f^{\eta}$ is well defined as an
element of $\mathcal{A}^{\times}(\Omega)/\mathbb{Z}_{2}$.

######  Remark 3.1.

If $\Omega$ is not simply connected, then on the left hand side of (3.5)
instead of $\Omega$ we need to put the double cover of $\Omega$. Then
$f^{\eta}$ is still well defined. However we will not use this construction.

Let us go back to the setup of Section 3.2. We can then define
$m_{\alpha}^{\eta}\in\mathcal{A}^{\times}(\Omega)/\mathbb{Z}_{2}$. Therefore,
(3.1) can be interpreted as a group of transformations of
$\mathcal{A}^{\times}(\Omega)/\mathbb{Z}_{2}$.

A similar remark applies to Section 3.3.

###  3.5 Generalized symmetries

Let $\mathcal{C}$ be a linear differential operator on a complex manifold
$\Omega$. Let $\alpha\in\mathrm{Hol}(\Omega)$. We say that it is a symmetry of
$\mathcal{C}$ iff

$\displaystyle\alpha\mathcal{C}=\mathcal{C}\alpha.$

Let $m^{\sharp},m^{\flat}\in\mathcal{A}^{\times}(\Omega)$. Define a pair of
transformations in $\mathcal{A}^{\times}\rtimes\mathrm{Hol}(\Omega)$:

$\displaystyle\alpha^{\sharp}:=m^{\sharp}\alpha,\qquad\alpha^{\flat}:=m^{\flat}\alpha.$

We say that a pair $(\alpha^{\sharp},\alpha^{\flat})$ is a generalized
symmetry of $\mathcal{C}$ if

$\displaystyle\alpha^{\flat}\mathcal{C}=\mathcal{C}\alpha^{\sharp}.$

Clearly, the kernel of $\mathcal{C}$ is invariant wrt the action of
$\alpha^{\sharp}$:

$\displaystyle\mathcal{C}f=0\qquad\text{implies}\quad\mathcal{C}\alpha^{\sharp}f=0.$

Generalized symmetries of $\mathcal{C}$ form a group.

Let $A\in\mathrm{hol}(\Omega)$. We say that it is an infinitesimal symmetry of
$\mathcal{C}$ iff

$\displaystyle A\mathcal{C}=\mathcal{C}A.$

Let $M^{\sharp},M^{\flat}\in\mathcal{A}(\Omega)$. One can also consider a pair
of operators in $\mathcal{A}\rtimes\mathrm{hol}(\Omega)$

$\displaystyle A^{\sharp}:=A+M^{\sharp},\qquad A^{\flat}:=A+M^{\flat}.$

We say that a pair $(A^{\sharp},A^{\flat})$ is a generalized infinitesimal
symmetry of $\mathcal{C}$ if

$\displaystyle A^{\flat}\mathcal{C}=\mathcal{C}A^{\sharp}.$

Clearly, the kernel of $\mathcal{C}$ is invariant wrt the action of
$A^{\sharp}$:

$\displaystyle\mathcal{C}f=0\qquad\text{implies}\quad\mathcal{C}A^{\sharp}f=0.$

Infinitesimal generalized symmetries of $\mathcal{C}$ form a Lie algebra.

##  4 Line bundles

###  4.1 Scaling

A holomorphic bundle $\pi\colon\mathcal{V}\to\mathcal{Y}$ is called a line
bundle if its fibers are modelled on $\mathbb{C}^{\times}$. $\mathcal{V}$ is
equipped with scaling, a homomorphism $\mathbb{C}^{\times}\ni
s\mapsto\lambda_{s}\in\mathrm{Hol}(\mathcal{V})$ preserving the fibers, that
is, satisfying $\pi\lambda_{s}=\pi$. The vector field obtained by
differentiating $\lambda_{s}$ is called the vertical vector field and denoted
$V$:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\lambda_{s}\Big{|}_{s=1}=:V.$

For $v\in\mathcal{V}$, we will often simply write $sv$ instead of
$\lambda_{s}v$. We will also write $s=\frac{\lambda_{s}(v)}{v}$.

Let $\mathcal{Y}_{0}\subset\mathcal{Y}$ be open. A section based on
$\mathcal{Y}_{0}$ is a holomorphic map
$\gamma\colon\mathcal{Y}_{0}\to\mathcal{V}$ such that
$\pi\circ\gamma=\mathrm{id}$. Every section based on $\mathcal{Y}_{0}$
determines a trivialization of $\pi^{-1}(\mathcal{Y}_{0})$

$\displaystyle\mathcal{Y}_{0}\times\mathbb{C}^{\times}\ni(y,s)\mapsto
s\gamma(y)\in\pi^{-1}(\mathcal{Y}_{0}).$

###  4.2 Vector fields on a line bundle

Let $\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V})$ denote the Lie algebra
of scaling invariant vector fields, that is,

$\displaystyle\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V}):=\big{\\{}B\in\mathrm{hol}(\mathcal{V})\colon\lambda_{s}B=B\lambda_{s},\;s\in\mathbb{C}^{\times}\big{\\}}.$

Let $B\in\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V})$. Then $B$ determines
a unique element of $\mathrm{hol}(\mathcal{Y})$, which will be denoted
$B^{\diamond}$.

Let $\gamma$ be a section based on $\mathcal{Y}_{0}$. $B^{\diamond}$ can be
transported by $\gamma$ onto $\gamma(\mathcal{Y}_{0})$. Thus we obtain two
vector fields on $\gamma(\mathcal{Y}_{0})$:
$B\big{|}_{\gamma(\mathcal{Y}_{0})}$ and $\gamma\big{(}B^{\diamond}\big{)}$.

######  Proposition 4.1.

For any $v\in\gamma(\mathcal{Y}_{0})$, $B(v)-\gamma(B^{\diamond})(v)$ is
parallel to $V(v)$. Therefore, there exists
$M_{B}^{\gamma}\in\mathcal{A}(\mathcal{Y}_{0})$ such that

$\displaystyle
B\big{(}\gamma(y)\big{)}=M_{B}^{\gamma}(y)V\big{(}\gamma(y)\big{)}+\gamma\big{(}B^{\diamond}\big{)}(y),\qquad
y\in\mathcal{Y}_{0}.$ (4.1)

Moreover,

$\displaystyle\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V})\ni B\mapsto
M_{B}^{\gamma}$

is a cocycle. Hence, for any $\eta\in\mathbb{C}$,

$\displaystyle\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V})\ni B\mapsto
B^{\gamma,\eta}:=B^{\diamond}+\eta
M_{B}^{\gamma}\in\mathcal{A}\rtimes\mathrm{hol}(\mathcal{Y}_{0})$ (4.2)

is a representation of the Lie algebra of scaling invariant vector fields.

###### Proof.

Every $B\in\mathrm{hol}(\mathcal{V}_{0})$ can be written uniquely as

$\displaystyle B=\tilde{M}_{B}^{\gamma}V+B^{\gamma},$ (4.3)

where $\tilde{M}_{B}^{\gamma}\in\mathcal{A}(\mathcal{V}_{0})$ and for any
$s\in\mathbb{C}^{\times}$ the vector field $\mathcal{Y}_{0}\ni y\mapsto
B^{\gamma}(s\gamma(y))$ is tangent to the section $s\gamma(\mathcal{Y}_{0})$.
Assume now that $B\in\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V}_{0})$.
This means $[V,B]=0$, which is equivalent to

$\displaystyle\big{(}V\tilde{M}_{B}^{\gamma}\big{)}V+[V,B^{\gamma}]=0.$ (4.4)

We also have

$\displaystyle\gamma(B^{\diamond})(y)=B^{\gamma}\big{(}\gamma(y)\big{)},$
(4.5)

and we set

$\displaystyle
M_{B}^{\gamma}(y):=\tilde{M}_{B}^{\gamma}\big{(}\gamma(y)\big{)},\qquad
y\in\mathcal{Y}_{0}.$ (4.6)

Restricting (4.3) to $\gamma(\mathcal{Y}_{0})$, using (4.5) and setting (4.6),
we obtain (4.1).

Now let $B_{1},B_{2}\in\mathrm{hol}(\mathcal{V}_{0})$. Replacing $B$ in (4.3)
with $[B_{1},B_{2}]$, we can write

$\displaystyle[B_{1},B_{2}]=\tilde{M}_{[B_{1},B_{2}]}^{\gamma}V+[B_{1},B_{2}]^{\gamma}.$
(4.7)

If in addition
$B_{1},B_{2}\in\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V}_{0})$, then
using (4.4) with $B$ replaced with $B_{1}$ and $B_{2}$, we obtain

$\displaystyle[B_{1},B_{2}]=\big{(}B_{1}^{\gamma}\tilde{M}_{B_{2}}^{\gamma}-B_{2}^{\gamma}\tilde{M}_{B_{1}}^{\gamma}\big{)}V+[B_{1}^{\gamma},B_{2}^{\gamma}].$
(4.8)

Comparing (4.8) with (4.7), we obtain

$\displaystyle
B_{1}^{\gamma}\tilde{M}_{B_{2}}^{\gamma}-B_{2}^{\gamma}\tilde{M}_{B_{1}}^{\gamma}=\tilde{M}_{[B_{1},B_{2}]}^{\gamma}.$
(4.9)

Restricting (4.9) to the section $\gamma(\mathcal{Y}_{0})$, using (4.5) and
setting (4.6), we obtain the cocycle relation

$\displaystyle
B_{1}^{\diamond}M_{B_{2}}^{\gamma}-B_{2}^{\diamond}M_{B_{1}}^{\gamma}=M_{[B_{1},B_{2}]}^{\gamma}.$
∎

###  4.3 Point transformations of a line bundle

Let $\mathrm{Hol}^{\mathbb{C}^{\times}}(\mathcal{V})$ denote the group of
scaling invariant biholomorphic maps of $\mathcal{V}$, that is

$\displaystyle\mathrm{Hol}^{\mathbb{C}^{\times}}(\mathcal{V}):=\big{\\{}\alpha\in\mathrm{Hol}(\mathcal{V})\colon\alpha\lambda_{s}=\lambda_{s}\alpha,\;s\in\mathbb{C}^{\times}\big{\\}}.$

Let $\alpha\in\mathrm{Hol}^{\mathbb{C}^{\times}}(\mathcal{V})$. Then $\alpha$
determines a unique element of $\mathrm{Hol}(\mathcal{Y})$, which will be
denoted by ${\alpha}^{\diamond}$.

Let $\gamma$ be a section over $\mathcal{Y}_{0}$. As in (3.2), we set
$\mathcal{Y}_{0}^{\alpha^{\diamond}}:={\alpha^{\diamond}}(\mathcal{Y}_{0})\cap\mathcal{Y}_{0}$.
We define
$m_{\alpha}^{\gamma}\in\mathcal{A}\big{(}\mathcal{Y}_{0}^{\alpha^{\diamond}}\big{)}$
by

$\displaystyle
m_{\alpha}^{\gamma}(y):=\frac{\gamma(y)}{\alpha\circ\gamma\circ(\alpha^{\diamond})^{-1}(y)},\qquad
y\in\mathcal{Y}_{0}^{\alpha^{\diamond}}.$

######  Proposition 4.2.

$\displaystyle\mathrm{Hol}^{\mathbb{C}^{\times}}(\mathcal{V})\ni\alpha\mapsto
m_{\alpha}^{\gamma}$

is a cocycle. Hence for any $\eta\in\mathbb{Z}$

$\displaystyle\mathrm{Hol}^{\mathbb{C}^{\times}}(\mathcal{V})\ni\alpha\mapsto\alpha^{\gamma,\eta}:=(m_{\alpha}^{\gamma})^{\eta}\alpha^{\diamond}\mathop{\in}_{\mathrm{loc}}\mathcal{A}^{\times}\rtimes\mathrm{Hol}(\mathcal{Y}_{0})$
(4.10)

is a local representation.

###  4.4 Homogeneous functions of integer degree

As before, $\mathcal{Y}_{0}\subset\mathcal{Y}$ is open. We set
$\mathcal{V}_{0}:=\pi^{-1}(\mathcal{Y}_{0})$. For $\eta\in\mathbb{Z}$, let
$\Lambda^{\eta}\big{(}\mathcal{V}_{0}\big{)}$ denote the space of holomorphic
functions on $\mathcal{V}_{0}$ homogeneous of degree $\eta$, that is,
functions $k\in\mathcal{A}\big{(}\mathcal{V}_{0})$ satisfying

$\displaystyle k(sv)=s^{\eta}k(v),\qquad v\in\mathcal{V}_{0},\qquad
s\in\mathbb{C}^{\times}.$ (4.11)

Clearly, (4.11) implies

$\displaystyle Vk=\eta k.$ (4.12)

Let $\gamma$ be a section based on $\mathcal{Y}_{0}$. We then have an obvious
map
$\psi^{\gamma,\eta}\colon\Lambda^{\eta}(\mathcal{V}_{0})\to\mathcal{A}(\mathcal{Y}_{0})$:
for $k\in\Lambda^{\eta}(\mathcal{V}_{0})$ we set

$\displaystyle\big{(}\psi^{\gamma,\eta}k\big{)}(y):=k\big{(}\gamma(y)\big{)},\qquad
y\in\mathcal{Y}_{0}.$ (4.13)

$\psi^{\gamma,\eta}$ is bijective and we can introduce its inverse, denoted
$\phi^{\gamma,\eta}$, defined for any $f\in\mathcal{A}(\mathcal{Y}_{0})$ by

$\displaystyle\big{(}\phi^{\gamma,\eta}f\big{)}\big{(}s\gamma(y)\big{)}=s^{\eta}f(y),\qquad
s\in\mathbb{C}^{\times},\qquad y\in\mathcal{Y}_{0}.$ (4.14)

######  Proposition 4.3.

With the notation of (4.2) and (4.10),

$\displaystyle
B^{\gamma,\eta}:=\psi^{\gamma,\eta}B\phi^{\gamma,\eta}\in\mathcal{A}\rtimes\mathrm{hol}(\mathcal{Y}_{0}),$
$\displaystyle{\alpha}^{\gamma,\eta}:=\psi^{\gamma,\eta}{\alpha}\phi^{\gamma,\eta}\in\mathcal{A}^{\times}\rtimes\mathrm{Hol}\big{(}\mathcal{Y}_{1},\mathcal{Y}_{2}\big{)}.$
(4.15)

###  4.5 Homogeneous functions of non-integer degree

One can try to generalize the above construction to
$\eta\in\mathbb{C}\backslash\mathbb{Z}$. In this case, there is a problem with
the definition of functions homogeneous of degree $\eta$, because the power
function is multivalued on $\mathbb{C}^{\times}$. Therefore, we cannot use
$\mathcal{V}_{0}:=\pi^{-1}(\mathcal{Y}_{0})$. Instead, let us we assume that
$\mathcal{V}_{0}\subset\mathcal{V}$ is open, connected,
$\pi(\mathcal{V}_{0})=\mathcal{Y}_{0}$ and $\pi^{-1}(y)\cap\mathcal{V}_{0}$ is
simply connected for any $y\in\mathcal{Y}_{0}$. We say that
$k\in\Lambda^{\eta}(\mathcal{V}_{0})$ if
$k\in\mathcal{A}\big{(}\mathcal{V}_{0})$ and

$\displaystyle k(sv)=s^{\eta}k(v),\qquad v,sv\in\mathcal{V}_{0},\qquad
s\in\mathbb{C}^{\times}.$ (4.16)

Note that (4.16) is unambiguous, because, for any $y\in\mathcal{Y}_{0}$, on
$\pi^{-1}(y)\cap\mathcal{V}_{0}$ we have a unique continuation of holomorphic
functions. (4.12) still holds.

Let $\gamma$ be a section based on $\mathcal{Y}_{0}$ whose image is contained
in $\mathcal{V}_{0}$. $\psi^{\gamma,\eta}$ is still bijective and we can
introduce its inverse, denoted $\phi^{\gamma,\eta}$, defined for any
$f\in\mathcal{A}(\mathcal{Y})$ by

$\displaystyle\big{(}\phi^{\gamma,\eta}f\big{)}\big{(}s\gamma(y)\big{)}=s^{\eta}f(y),\qquad
s\in\mathbb{C}^{\times},\qquad y\in\mathcal{Y}_{0},\qquad
s\gamma(y)\in\mathcal{V}_{0}.$

With this definition, (4.15) is still true.

##  5 Complex Euclidean spaces

###  5.1 Linear transformations

Let us first consider the vector space $\mathbb{C}^{n}$ without the Euclidean
structure. The affine general linear Lie algebra
$\mathbb{C}^{n}\rtimes\mathrm{gl}(\mathbb{C}^{n})$ can be identified with the
subalgebra of $\mathrm{hol}(\mathbb{C}^{n})$ spanned by

$\displaystyle\partial_{y^{j}},\quad j=1,\dots,n,\qquad
y^{i}\partial_{y^{j}},\quad i,j=1,\dots,n.$

Similarly, the affine general linear group
$\mathbb{C}^{n}\rtimes\mathrm{GL}(\mathbb{C}^{n})$ is a subgroup of
$\mathrm{Hol}(\mathbb{C}^{n})$.

We will have a special notation for the generator of dilations

$\displaystyle D_{\mathbb{C}^{n}}:=\sum_{i=1}^{n}y^{i}\partial_{y^{i}}.$

Obviously,

$\displaystyle D_{\mathbb{C}^{n}}B=BD_{\mathbb{C}^{n}},\qquad$ $\displaystyle
B\in\mathrm{gl}(\mathbb{C}^{n}),$ $\displaystyle
D_{\mathbb{C}^{n}}\alpha=\alpha D_{\mathbb{C}^{n}},\qquad$
$\displaystyle\alpha\in\mathrm{GL}(\mathbb{C}^{n}).$

Let $\sigma=(\sigma_{1},\dots,\sigma_{n})$ be a permutation of
$\\{1,\dots,n\\}$. Then

$\displaystyle\sigma\big{(}y^{1},\dots,y^{n}\big{)}:=\big{(}y^{\sigma_{1}^{-1}},\dots,y^{\sigma_{n}^{-1}}\big{)}$

defines an element of $\mathrm{GL}(\mathbb{C}^{n})$. On the level of point
transformations it acts as

$\displaystyle(\sigma
f)\big{(}y^{1},\dots,y^{n}\big{)}:=f\big{(}y^{\sigma_{1}},\dots,y^{\sigma_{n}}\big{)}.$

###  5.2 Bilinear scalar product

Suppose that

$\displaystyle\mathbb{C}^{n}\ni y,x\mapsto\langle
y|x\rangle=\sum_{i,j}g_{i,j}y^{i}x^{j}$

is a nondegenerate symmetric bilinear form on $\mathbb{C}^{n}$ called the
scalar product. Clearly, if we know the square of each vector

$\displaystyle\langle y|y\rangle=\sum_{i,j}g_{i,j}y^{i}y^{j},$

we have the complete information about the scalar product.

$[g^{ij}]$ will denote the inverse of $[g_{ij}]$. The orthogonal Lie algebra
of $\mathbb{C}^{n}$, understood as a Lie subalgebra of
$\mathrm{hol}(\mathbb{C}^{n})$, is defined as

$\displaystyle\mathrm{so}(\mathbb{C}^{n}):=\big{\\{}B\in\mathrm{gl}(\mathbb{C}^{n})\colon
B\langle y|y\rangle=0\big{\\}}.$

For $i,j=1,\dots,n$, define

$\displaystyle
B_{i,j}:=\sum_{k}(g_{j,k}y_{i}\partial_{y_{k}}-g_{i,k}y_{k}\partial_{y_{j}}).$

$\\{B_{i,j}\colon i<j\\}$ is a basis of $\mathrm{so}(\mathbb{C}^{n})$.
Clearly, $B_{i,j}=-B_{j,i}$, in particular $B_{i,i}=0$.

Likewise, recall that the orthogonal and the special orthogonal group of
$\mathbb{C}^{n}$ is defined as

$\displaystyle\mathrm{O}(\mathbb{C}^{n}):=\big{\\{}{\alpha}\in\mathrm{GL}(\mathbb{C}^{n})\colon\langle{\alpha}y|{\alpha}x\rangle=\langle
y|x\rangle,\;y,x\in\mathbb{C}^{n}\big{\\}},$
$\displaystyle\mathrm{SO}(\mathbb{C}^{n}):=\big{\\{}\alpha\in\mathrm{O}(\mathbb{C}^{n})\colon\det\alpha=1\big{\\}}.$

We define

$\displaystyle\text{the
Laplacian}\quad\Delta_{\mathbb{C}^{n}}:=\sum_{i,j=1}^{n}g^{i,j}\partial_{y^{i}}\partial_{y^{j}},\qquad\text{and}$
$\displaystyle\text{the Casimir
operator}\quad\mathcal{C}_{\mathbb{C}^{n}}:=\frac{1}{2}\sum_{i,j=1}^{n}g^{i,k}g^{j,l}B_{i,j}B_{k,l}.$

Clearly,

$\displaystyle\Delta_{\mathbb{C}^{n}}B=B\Delta_{\mathbb{C}^{n}},\quad$
$\displaystyle
B\in\mathbb{C}^{n}\rtimes\mathrm{so}(\mathbb{C}^{n}),\quad\qquad$
$\displaystyle\mathcal{C}_{\mathbb{C}^{n}}B=B\mathcal{C}_{\mathbb{C}^{n}},\quad$
$\displaystyle B\in\mathrm{so}(\mathbb{C}^{n}),$
$\displaystyle\Delta_{\mathbb{C}^{n}}\alpha=\alpha\Delta_{\mathbb{C}^{n}},\quad$
$\displaystyle\alpha\in\mathbb{C}^{n}\rtimes{\rm
O}(\mathbb{C}^{n}),\quad\qquad$
$\displaystyle\mathcal{C}_{\mathbb{C}^{n}}\alpha=\alpha\mathcal{C}_{\mathbb{C}^{n}},\quad$
$\displaystyle\alpha\in{\rm O}(\mathbb{C}^{n}).$

Note the identity

$\displaystyle\Delta_{\mathbb{C}^{n}}=\frac{1}{\langle
y|y\rangle}\big{(}D_{\mathbb{C}^{n}}^{2}+(n-2)D_{\mathbb{C}^{n}}+\mathcal{C}_{\mathbb{C}^{n}}\big{)}.$
(5.1)

We will denote by $\mathcal{S}^{n-1}(R)$ the $($complex$)$ sphere in
$\mathbb{C}^{n}$ of squared radius $R$, that is

$\displaystyle\mathcal{S}^{n-1}(R):=\big{\\{}y\in\mathbb{C}^{n}\colon\langle
y|y\rangle=R\big{\\}}.$

We also introduce the null quadric

$\displaystyle\mathcal{V}^{n-1}:=\mathcal{S}^{n-1}(0)\backslash\\{0\\}.$ (5.2)

###  5.3 Split coordinates

The coordinates that we describe in this subsection are particularly
convenient for the analysis of $\mathrm{so}(\mathbb{C}^{n})$ and
$\mathrm{O}(\mathbb{C}^{n})$. Let $n=2m$ if $n$ is even and $n=2m+1$ if $n$ is
odd. Set

$\displaystyle I_{n}:=\begin{cases}\\{-1,1,\dots,-m,m\\},&\text{for even}\
n,\\\ \\{0,-1,1,\dots,-m,m\\},&\text{for odd}\ n.\end{cases}$

The coordinates in $\mathbb{C}^{n}$ will be labelled by $I_{n}$, so that the
square of $y=[y^{i}]_{i\in I_{n}}$ is given by

$\displaystyle\langle y|y\rangle=\sum\limits_{i\in
I_{n}}y_{i}y_{-i}=\begin{cases}\displaystyle\sum\limits_{i=1}^{m}2y_{-i}y_{i},&\text{for
even}\ n,\\\ \displaystyle
y_{0}^{2}+\sum\limits_{i=1}^{m}2y_{-i}y_{i}&\text{for odd}\ n.\end{cases}$

Clearly, $g_{i,j}=g^{i,j}=\delta_{i,-j}$.

For $n=2m$, $\mathrm{so}(\mathbb{C}^{n})$ has a basis consisting of

$\displaystyle
N_{i}:=B_{i-i}=y_{-i}\partial_{y_{-i}}-y_{i}\partial_{y_{i}},\qquad$
$\displaystyle j=1,\dots,m,$ (5.3) $\displaystyle
B_{ij}:=y_{-i}\partial_{y_{j}}-y_{-j}\partial_{y_{i}},\qquad$ $\displaystyle
1\leq|i|<|j|\leq m.$ (5.4)

For $n=2m+1$ we have to add

$\displaystyle
B_{0j}=y_{0}\partial_{y_{j}}-y_{-j}\partial_{y_{0}},\qquad|j|=1,\dots,m.$
(5.5)

The subalgebra of $\mathrm{so}(\mathbb{C}^{n})$ spanned by (5.3) is its Cartan
algebra. (5.4), and in the odd case also (5.5), are its root operators:

$\displaystyle[N_{k},B_{i,j}]=(\mathrm{sgn}(i)\delta_{k,|i|}+\mathrm{sgn}(j)\delta_{k,|j|})B_{i,j},\qquad[N_{k},B_{0,j}]=\mathrm{sgn}(j)\delta_{k,|j|}B_{i,j}.$

We have

$\displaystyle\Delta_{\mathbb{C}^{n}}=\sum_{i\in
I_{n}}\partial_{y_{i}}\partial_{y_{-i}},\qquad\mathcal{C}_{\mathbb{C}^{n}}=\frac{1}{2}\sum_{i,j\in
I_{n}}B_{i,j}B_{{-i,-j}}.$

###  5.4 Weyl symmetries

In our applications of the group invariance we will restrict ourselves only to
the so-called “Weyl symmetries”. It will be convenient to consider “Weyl
symmetries” contained not only in $\mathrm{SO}(\mathbb{C}^{n})$, but in the
whole $\mathrm{O}(\mathbb{C}^{n})$.

Permutations of $I_{2m}$ that preserve its decomposition into pairs
$\\{-1,1\\},\dots,\\{-m,m\\}$ correspond to a subgroup of $\mathrm{O}(2m)$
that will be denoted $\mathrm{W}(\mathbb{C}^{2m})$. It is isomorphic to
${\mathbb{Z}}_{2}^{m}\rtimes S_{m}$. It is generated by two kinds of
transformations: $\tau_{j}$, $j=1,\dots,m$, which swap the elements of the
$j$th pair, and permutations from $S_{m}$, which permute the pairs. If
$\sigma=(\sigma_{1},\dots,\sigma_{m})\in S_{m}$, then

$\displaystyle\sigma
f(y_{-1},y_{1},\dots,y_{-m},y_{m}):=f(y_{-\sigma_{1}},y_{\sigma_{1}},\dots,y_{-\sigma_{m}},y_{\sigma_{m}}).$

For $j=1,\dots,n$,

$\displaystyle\tau_{j}f(y_{-1},y_{1},\dots,y_{-j},y_{j},\dots):=f(y_{-1},y_{1},\dots,y_{j},y_{-j},\dots).$

We have

$\displaystyle\sigma B_{i,j}\sigma^{-1}=B_{\sigma_{i},\sigma_{j}},\qquad$
$\displaystyle\tau_{k}B_{i,j}\tau_{k}^{-1}=(-1)^{\delta_{|i|,k}+\delta_{|j|,k}}B_{i,j},$
$\displaystyle\sigma N_{j}\sigma^{-1}=N_{\sigma_{j}},\qquad$
$\displaystyle\tau_{k}N_{j}\tau_{k}^{-1}=(-1)^{\delta_{j,k}}N_{j}.$

Using $\mathbb{C}^{2m+1}=\mathbb{C}\oplus\mathbb{C}^{2m}$, we embed
$\mathrm{W}(\mathbb{C}^{2m})$ in $\mathrm{O}(\mathbb{C}^{2m+1})$. We also
introduce a transformation $\tau_{0}\in\mathrm{O}(\mathbb{C}^{2m+1})$ given by

$\displaystyle\tau_{0}f(y_{0},y_{-1},y_{1},\dots,y_{-m},y_{m}):=f(-y_{0},y_{-1},y_{1},\dots,y_{-m},y_{m}).$

Clearly, $\tau_{0}$ commutes with $\mathrm{W}(\mathbb{C}^{2m})$. The group
$\mathrm{W}(\mathbb{C}^{2m+1})$ is defined as the group generated by
$\mathrm{W}(\mathbb{C}^{2m})$ and $\tau_{0}$, and is isomorphic to
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{m}\rtimes S_{m}$. We have for
$i,j=1,\dots,m$

$\displaystyle\tau_{0}B_{0,j}\tau_{0}^{-1}=-B_{0,j},\qquad\tau_{0}B_{i,j}\tau_{0}^{-1}=B_{i,j},\qquad\tau_{0}N_{j}\tau_{0}^{-1}=N_{j}.$

In both even and odd cases $\mathrm{W}(\mathbb{C}^{n})$ acts as a group of
automorphisms of $\mathrm{so}(\mathbb{C}^{n})$ leaving invariant the Cartan
algebra. To compute the determinant of elements of
$\mathrm{W}(\mathbb{C}^{n})$ it suffices to remember that $\det\sigma=1$ for
$\sigma\in S_{m}$ and $\det\tau_{j}=-1$.

##  6 Conformal invariance

The main subject of this section is the description of generalized
(infinitesimal) symmetries of the Laplace equation

$\displaystyle\Delta_{\mathbb{C}^{n}}f=0.$ (6.1)

We will see in particular that the Lie algebra of generalized symmetries is
$\mathrm{so}(\mathbb{C}^{n+2})$. We will see that it is convenient to start
the description of these symmetries from the space $\mathbb{C}^{n+2}$, which
we will call the extended space. The space $\mathbb{C}^{n}$ will be embedded
inside $\mathbb{C}^{n+2}$ as a section of the null quadric. We will see how
the Laplacian $\Delta_{\mathbb{C}^{n+2}}$ reduces to the Laplacian
$\Delta_{\mathbb{C}^{n}}$.

###  6.1 Conformal invariance of Riemannian manifolds

Suppose that a (complex) manifold $\Omega$ is equipped with a nondegenerate
holomorphic covariant 2-tensor field $g$, called the $($complex$)$ metric
tensor. We will say that $(\Omega,g)$ is a $($complex$)$ Riemannian space.

Thus if $A,B\in\mathrm{hol}(\Omega)$, then we have a holomorphic function

$\displaystyle\Omega\ni y\mapsto g(A,B)(y)=g_{i,j}(y)A^{i}(y)B^{j}(y)$

called the scalar product.

Let ${\alpha}\in\mathrm{Hol}(\Omega)$. We can transport $g$ by ${\alpha}$:

$\displaystyle{\alpha}(g)(A,B):=g\big{(}{\alpha}^{-1}(A),{\alpha}^{-1}(B)\big{)}.$

We say that ${\alpha}$ is conformal if there exists
$m_{\alpha}\in\mathcal{A}^{\times}(\Omega)$ such that

$\displaystyle{\alpha}(g)=m_{\alpha}g.$

Let $\mathrm{Cf}(\Omega)$ denote the group of conformal maps on $(\Omega,g)$.
One can check that

$\displaystyle\mathrm{Cf}(\Omega)\times\Omega\ni({\alpha},y)\mapsto
m_{\alpha}(y)\in\mathcal{A}^{\times}(\Omega)$

is a cocycle.

Let $C\in\mathrm{hol}(\Omega)$. The Lie derivative of $g$ in the direction of
$C$ is denoted $Cg$ and defined by

$\displaystyle(Cg)(A,B):=C(g(A,B))-g([C,A],B)-g(A,[C,B]).$

We say that $C$ is infinitesimally conformal if there exists
$M_{C}\in\mathcal{A}(\Omega)$ such that

$\displaystyle Cg=M_{C}g.$

Let $\mathrm{cf}(\Omega)$ denote the Lie algebra of infinitesimally conformal
fields. One can check that

$\displaystyle\mathrm{cf}(\Omega)\times\Omega\ni(C,y)\mapsto
M_{C}(y)\in\mathcal{A}(\Omega)$

is a cocycle.

We say that a manifold $\Omega$ has a conformal structure, if it is covered by
a family of open sets $\Omega_{i}$ equipped with bilinear scalar products
$g_{i}$ such that on $\Omega_{i}\cap\Omega_{j}$ we have

$\displaystyle g_{i}=\rho_{i,j}g_{j}$

for some $\rho_{i,j}\in\mathcal{A}^{\times}(\Omega_{i}\cap\Omega_{j})$.
Clearly, a Riemannian structure on $\Omega$ is not necessary to define
$\mathrm{Cf}(\Omega)$ and $\mathrm{cf}(\Omega)$ – we need only a conformal
structure on $\Omega$.

###  6.2 Null quadric

Consider the extended space, that is, the complex Euclidean space
$\mathbb{C}^{n+2}$. The central role will be played by the representations

$\displaystyle\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)}\ni B\mapsto
B\in\mathrm{hol}\big{(}\mathbb{C}^{n+2}\big{)},$ (6.2)
$\displaystyle\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}\ni{\alpha}\mapsto{\alpha}\in\mathrm{Hol}\big{(}\mathbb{C}^{n+2}\big{)},$
(6.3)

and the symmetry

$\displaystyle B\Delta_{\mathbb{C}^{n+2}}=\Delta_{\mathbb{C}^{n+2}}B,\qquad$
$\displaystyle B\in\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)},$ (6.4)
$\displaystyle{\alpha}\Delta_{\mathbb{C}^{n+2}}=\Delta_{\mathbb{C}^{n+2}}{\alpha},\qquad$
$\displaystyle{\alpha}\in\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}.$ (6.5)

As in (5.2), we introduce

$\displaystyle\mathcal{V}:=\big{\\{}z\in\mathbb{C}^{n+2}\colon z\neq
0,\;\langle z|z\rangle=0\big{\\}}$

called the null quadric.

Multiplication by $s\in\mathbb{C}^{\times}$ preserves $\mathcal{V}$.
Therefore, we can define the projective quadric

$\displaystyle\mathcal{Y}:=\mathcal{V}/\mathbb{C}^{\times}=\big{\\{}\mathbb{C}^{\times}z\colon
z\in\mathcal{V}\big{\\}}.$

It is an $n$-dimensional complex manifold. Let
$\pi\colon\mathcal{V}\to\mathcal{Y}$ denote the natural projection. Clearly,
$\mathcal{V}$ is a complex line bundle over $\mathcal{Y}$. As in Section 4.1,
the multiplication by $s$ will be often denoted by $\mathbb{C}^{\times}\ni
s\mapsto\lambda_{s}\in\mathrm{Hol}(\mathcal{V})$ and the corresponding
vertical vector field by $V\in\mathrm{hol}(\mathcal{V})$.

We can restrict (6.2) and (6.3) to $\mathcal{V}$ and note that they are
scaling invariant. Thus, we have natural embeddings

$\displaystyle\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)}\ni B\mapsto
B\in\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V}),$ (6.6)
$\displaystyle\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}\ni{\alpha}\mapsto{\alpha}\in\mathrm{Hol}^{\mathbb{C}^{\times}}(\mathcal{V}).$
(6.7)

(Recall that $\mathrm{hol}^{\mathbb{C}^{\times}}(\mathcal{V})$, resp.
$\mathrm{Hol}^{\mathbb{C}^{\times}}(\mathcal{V})$ denote the scaling invariant
holomorphic vector fields, resp. bijections). Therefore, (6.6) and (6.7)
induce their actions on $\mathcal{Y}$:

$\displaystyle\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)}\ni B\mapsto
B^{\diamond}\in\mathrm{hol}(\mathcal{Y}),$ (6.8)
$\displaystyle\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}\ni{\alpha}\mapsto{\alpha}^{\diamond}\in\mathrm{Hol}(\mathcal{Y}).$
(6.9)

###  6.3 Conformal invariance of projective quadric

Let $g$ denote the restriction of the metric tensor on $\mathbb{C}^{n+2}$ to
$\mathcal{V}$. Note that the null space of $g$ is 1-dimensional and is spanned
by the vertical field $V$. In particular,

$\displaystyle g(V,A)=0,\qquad A\in\mathrm{hol}(\mathcal{V}).$ (6.10)

Moreover, the scaling scales the metric tensor:

$\displaystyle\lambda_{s}(g)=s^{2}g,\qquad s\in\mathbb{C}^{\times},$ (6.11)
$\displaystyle Vg=2g.$ (6.12)

Using (6.10), we can extend (6.11) to multiplication by nonconstant functions:

######  Proposition 6.1.

Let $\mathcal{U}$ be open in $\mathcal{Y}$,
$m\in\mathcal{A}^{\times}(\mathcal{U})$. Define
$\lambda_{m}\in\mathrm{Hol}\big{(}\pi^{-1}(\mathcal{U})\big{)}$ by

$\displaystyle\lambda_{m}(z):=m(\pi(z))z,\qquad z\in\pi^{-1}(\mathcal{U}).$

Then $\lambda_{m}g=m^{2}g$. In particular, for any section $\gamma$, the
restriction $\lambda_{m}\colon\gamma(\mathcal{U})\to
m\circ\pi\gamma(\mathcal{U})$ is conformal.

Let $\mathcal{U}$ be open in $\mathcal{Y}$. Let $\gamma$ be a section over
$\mathcal{U}$. The tensor $g$ restricted to $\gamma(\mathcal{U})$ is
nondegenerate. We can transport it by $\gamma^{-1}$ onto $\mathcal{U}$. This
way we endow $\mathcal{U}$ with a metric tensor.

For $i=1,2$, let $\mathcal{U}_{i}$ be two open subsets of $\mathcal{Y}$
equipped with sections $\gamma_{i}$. Let $g_{i}$ be the corresponding complex
Riemannian tensors. Obviously, there exists
$\rho_{2,1}\in\mathcal{A}^{\times}(\mathcal{U}_{1}\cap\mathcal{U}_{2})$ such
that

$\displaystyle\gamma_{2}(y)=\rho_{2,1}(y)\gamma_{1}(y),\qquad
y\in\mathcal{U}_{1}\cap\mathcal{U}_{2}.$

Therefore, by Proposition 6.1,

$\displaystyle
g_{2}=\big{(}\gamma_{2}^{-1}\gamma_{1}\big{)}g_{1}=\rho_{2,1}^{2}g_{1}.$

Cover $\mathcal{Y}$ with open subsets $\mathcal{U}_{i}$, $i=1,\dots,N$,
equipped with sections $\gamma_{i}$. Let $g_{i}$ be the corresponding
Riemannian tensors on $\mathcal{U}_{i}$. Then on
$\mathcal{U}_{i}\cap\mathcal{U}_{j}$ we have

$\displaystyle
g_{j}=\rho_{j,i}^{2}g_{i},\qquad\rho_{j,i}\in\mathcal{A}^{\times}(\mathcal{U}_{i}\cap\mathcal{U}_{j}).$

This way we endow $\mathcal{Y}$ with a conformal structure. It is easy to see
that it does not depend on the choice of the covering and sections.

######  Proposition 6.2.

(6.8) is infinitesimally conformal and (6.9) is conformal.

###### Proof.

Let $\gamma$ be a section over $\mathcal{U}\subset\mathcal{Y}$.

Let $B\in\mathrm{so}(\mathbb{C}^{n+2})$. By Proposition 4.1, there exists
$M_{B}^{\gamma}\in\mathcal{A}(\mathcal{U})$ such that

$\displaystyle\gamma(B^{\diamond})=\big{(}B-M_{B}^{\gamma}V\big{)}\big{|}_{\gamma(\mathcal{U})}.$

Now $Bg=0$ and $Vg=2g$. Hence

$\displaystyle\gamma(B^{\diamond})g=-2M_{B}^{\gamma}g.$

Therefore, $\gamma(B^{\diamond})$ is infinitesimally conformal on
$\gamma(\mathcal{U})$. Hence $B^{\diamond}$ is infinitesimally conformal on
$\mathcal{U}$.

Let ${\alpha}\in\mathrm{O}(\mathbb{C}^{n+2})$. Clearly,
$\gamma_{\alpha}:={\alpha}\circ\gamma\circ({\alpha}^{\diamond})^{-1}$ is a
section based on ${\alpha}^{\diamond}(\mathcal{U})$. Therefore, there exists
$m\in\mathcal{A}^{\times}\big{(}{\alpha}^{\diamond}(\mathcal{U})\cap\mathcal{U}\big{)}$
such that

$\displaystyle\gamma(y)=m(y)\gamma_{\alpha}(y),\qquad
y\in{\alpha}^{\diamond}(\mathcal{U})\cap\mathcal{U}.$ (6.13)

Define now
$\beta\colon\pi^{-1}\big{(}\mathcal{U}\cap({\alpha}^{\diamond})^{-1}(\mathcal{U})\big{)}\to\pi^{-1}\big{(}{\alpha}^{\diamond}(\mathcal{U})\cap\mathcal{U}\big{)}$

$\displaystyle\beta(z):=m\big{(}\pi(z)\big{)}\alpha(z).$

Substituting $z=\gamma(y)$ and using (6.13) we obtain

$\displaystyle\beta(\gamma(y))=m(y)\alpha\big{(}\gamma(y)\big{)}=\gamma(\alpha^{\diamond}(y)).$

Thus $\beta$ maps
$\gamma\big{(}\mathcal{U}\cap({\alpha}^{\diamond})^{-1}(\mathcal{U})\big{)}$
onto $\gamma\big{(}{\alpha}^{\diamond}(\mathcal{U})\cap\mathcal{U}\big{)}$.
Let $\beta^{\gamma}$ denote this restriction of $\beta$. By Proposition 6.1,
$\beta^{\gamma}$ is conformal. Clearly, it satisfies the identity

$\displaystyle\beta^{\gamma}=\gamma\alpha^{\diamond}\gamma^{-1}.$

Hence $\alpha^{\diamond}$ is conformal. ∎

###  6.4 Conformal invariance of complex Euclidean space

Fix a vector $e\in\mathcal{V}$, and set

$\displaystyle\mathcal{V}_{e}:=\\{z\in\mathcal{V}\colon\langle z|e\rangle\neq
0\\}.$

Clearly, $\mathcal{V}_{e}$ is invariant with respect to the action of
$\mathbb{C}^{\times}$ and
$\mathcal{Y}_{e}:=\mathcal{V}_{e}/\mathbb{C}^{\times}$ is an open dense subset
of $\mathcal{Y}$. We have a natural section of the line bundle
$\mathcal{V}\to\mathcal{Y}$ based on $\mathcal{Y}_{e}$:

$\displaystyle\mathcal{Y}_{e}\ni\mathbb{C}^{\times}z\mapsto\frac{z}{\langle
z|e\rangle}\in\mathcal{V}_{e}.$ (6.14)

Next choose a vector $d\in\mathcal{V}$ such that $\langle e|d\rangle=1$.
Clearly, $\\{e,d\\}^{\perp}$ is $n$-dimensional. It will be convenient further
on to choose coordinates $(z_{i})_{i\in I_{n}}$ in $\\{e,d\\}^{\perp}$. Each
$z\in\mathbb{C}^{n+2}$ can be written as

$\displaystyle z=\left[\begin{matrix}(z_{i})_{i\in I_{n}}\\\ z_{-m-1}\\\
z_{m+1}\end{matrix}\right]=(z_{i})_{i\in I_{n}}+z_{-m-1}e+z_{m+1}d,$

where $(z_{i})_{i\in I_{n}}\in\\{e,d\\}^{\perp}\simeq\mathbb{C}^{n}$,
$(z_{-m-1},z_{m+1})\in\mathbb{C}^{2}$.

Using $y=(y_{i})_{i\in I_{n}}$ as generic variables in $\mathbb{C}^{n}$, and
noting that $z_{m+1}=\langle z|e\rangle$, we see that $\mathcal{Y}_{e}$ can be
identified with $\mathbb{C}^{n}$ through the map

$\displaystyle\mathbb{C}^{n}\ni
y\mapsto\mathbb{C}^{\times}\left[\begin{matrix}y\\\ -\frac{\langle
y|y\rangle}{2}\\\ 1\end{matrix}\right]\in\mathcal{Y}_{e}.$

With this identification, the section (6.14) can be written as

$\displaystyle\mathbb{C}^{n}\ni y\mapsto\left[\begin{matrix}y\\\
-\frac{\langle y|y\rangle}{2}\\\ 1\end{matrix}\right]\in\mathcal{V}_{e}.$
(6.15)

######  Remark 6.3.

The above discussion shows that $\mathcal{Y}_{e}$ has a natural structure of
the affine $n$-dimensional Euclidean space. The choice of
$d\in\mathcal{V}_{e}$ (which does not influence the definition of
$\mathcal{Y}_{e}$) determines the origin of coordinates in $\mathcal{Y}_{e}$.

The stabilizer of $\left[\begin{matrix}0\\\ 1\\\
0\end{matrix}\right]=e\in\mathcal{V}$ inside $\mathrm{O}(\mathbb{C}^{n+2})$ is
isomorphic to $\mathbb{C}^{n}\rtimes\mathrm{O}(\mathbb{C}^{n})$, and is given
by

$\displaystyle\left[\begin{matrix}{\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}&0&w\\\
-w^{\\#}&1&-\frac{1}{2}\langle w|w\rangle\\\
0&0&1\end{matrix}\right]\left[\begin{matrix}\beta&0&0\\\ 0&1&0\\\
0&0&1\end{matrix}\right],\qquad\beta\in\mathrm{O}\big{(}\mathbb{C}^{n}\big{)},\qquad
w\in\mathbb{C}^{n}.$

The stabilizer of $\mathbb{C}^{\times}e\in\mathcal{Y}$ inside
$\mathrm{O}(\mathbb{C}^{n+2})$ is isomorphic to
$\mathbb{C}^{n}\rtimes\big{(}\mathrm{O}(\mathbb{C}^{n})\times\mathrm{O}(\mathbb{C}^{2})\big{)}$
and is given by

$\displaystyle\left[\begin{matrix}{\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}&0&w\\\
-w^{\\#}&1&-\frac{1}{2}\langle w|w\rangle\\\
0&0&1\end{matrix}\right]\left[\begin{matrix}\beta&0&0\\\ 0&s&0\\\
0&0&s^{-1}\end{matrix}\right],\qquad\beta\in\mathrm{O}\big{(}\mathbb{C}^{n}\big{)},\qquad
w\in\mathbb{C}^{n},\qquad s\in\mathbb{C}^{\times}.$

###  6.5 Laplacian on bundle of homogeneous functions

Let $\mathcal{V}_{0}$ be an open subset of $\mathcal{V}$ and
$\eta\in\mathbb{C}$. We define $\Lambda^{\eta}(\mathcal{V}_{0})$ to be the set
of holomorphic functions on $\mathcal{V}_{0}$ homogeneous of degree $\eta$.
(See Sections 4.4 and 4.5 for a discussion.)

Clearly, $B\in\mathrm{so}(\mathbb{C}^{n+2})$ preserves
$\Lambda^{\eta}(\mathcal{V}_{0})$. We will denote by $B^{\diamond,\eta}$ the
restriction of $B\in\mathrm{hol}(\mathcal{V})$ to
$\Lambda^{\eta}(\mathcal{V}_{0})$.

Clearly, ${\alpha}\in\mathrm{O}(\mathbb{C}^{n+2})$ maps
$\Lambda^{\eta}(\mathcal{V}_{0})$ onto
$\Lambda^{\eta}\big{(}{\alpha}(\mathcal{V}_{0})\big{)}$. We will denote by
${\alpha}^{\diamond,\eta}$ the restriction of ${\alpha}$ to
$\Lambda^{\eta}(\mathcal{V}_{0})$. Thus we have representations

$\displaystyle\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)}\ni B\mapsto
B^{\diamond,\eta},\qquad\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}\ni{\alpha}\mapsto{\alpha}^{\diamond,\eta}.$

We find the following theorem curious because it allows in some situations to
restrict a second order differential operator to a submanifold.

######  Theorem 6.4.

Let $\Omega\subset\mathbb{C}^{n+2}$ be open. Let $K\in\mathcal{A}(\Omega)$ be
homogeneous of degree $\frac{2-n}{2}$ such that

$\displaystyle K\big{|}_{cV\cap\Omega}=0.$

Then

$\displaystyle\Delta_{\mathbb{C}^{n+2}}K\big{|}_{\mathcal{V}\cap\Omega}=0.$

###### Proof.

We will give two proofs. Each of the proofs will provide us with a formula,
which will be useful later on.

Method I. We use the decomposition
$\mathbb{C}^{n+2}=\mathbb{C}^{n}\oplus\mathbb{C}^{2}$. As usual, we will
denote by $\langle z|z\rangle$ the square of $z\in\mathbb{C}^{n+2}$, by
$D_{\mathbb{C}^{n+2}}$ the generator of dilations, by
$\mathcal{C}_{\mathbb{C}^{n+2}}$ the Casimir operator and by
$\Delta_{\mathbb{C}^{n+2}}$ the Laplacian on $\mathbb{C}^{n+2}$. We will also
need the corresponding objects on $\mathbb{C}^{n}$: $\langle
z|z\rangle_{\mathbb{C}^{n}}$, $D_{\mathbb{C}^{n}}$,
$\mathcal{C}_{\mathbb{C}^{n}}$, and $\Delta_{\mathbb{C}^{n}}$. We will write

$\displaystyle\langle z|z\rangle_{m+1}:=2z_{-m-1}z_{m+1},\qquad$
$\displaystyle\Delta_{m+1}:=2\partial_{z_{-m-1}}\partial_{z_{m+1}},$
$\displaystyle
D_{m+1}:=z_{-m-1}\partial_{z_{-m-1}}+z_{m+1}\partial_{z_{m+1}},\qquad$
$\displaystyle
N_{m+1}:=z_{-m-1}\partial_{z_{-m-1}}-z_{m+1}\partial_{z_{m+1}}.$

Thus we have

$\displaystyle\langle z|z\rangle=\langle z|z\rangle_{\mathbb{C}^{n}}+\langle
z|z\rangle_{m+1},\qquad\Delta_{\mathbb{C}^{n+2}}=\Delta_{\mathbb{C}^{n}}+\Delta_{m+1},\qquad
D_{\mathbb{C}^{n+2}}=D_{\mathbb{C}^{n}}+D_{m+1}.$

The following identity is a consequence of (5.1):

$\displaystyle\langle
z|z\rangle_{\mathbb{C}^{n}}\Delta_{\mathbb{C}^{n+2}}=\langle
z|z\rangle_{\mathbb{C}^{n}}\Delta_{\mathbb{C}^{n}}+\big{(}\langle
z|z\rangle-\langle z|z\rangle_{m+1}\big{)}\Delta_{m+1}$
$\displaystyle\hphantom{\langle
z|z\rangle_{\mathbb{C}^{n}}\Delta_{\mathbb{C}^{n+2}}}{}=\mathcal{C}_{\mathbb{C}^{n}}+\Bigl{(}D_{\mathbb{C}^{n}}-\frac{2-n}{2}\Bigr{)}^{2}-\left(\frac{2-n}{2}\right)^{2}+\langle
z|z\rangle\Delta_{m+1}-D_{m+1}^{2}+N_{m+1}^{2}$
$\displaystyle\hphantom{\langle
z|z\rangle_{\mathbb{C}^{n}}\Delta_{\mathbb{C}^{n+2}}}{}=\langle
z|z\rangle\Delta_{m+1}+\left(D_{\mathbb{C}^{n}}-\frac{2-n}{2}-D_{m+1}\right)\left(D_{\mathbb{C}^{n+2}}-\frac{2-n}{2}\right)$
$\displaystyle\hphantom{\langle
z|z\rangle_{\mathbb{C}^{n}}\Delta_{\mathbb{C}^{n+2}}=}{}-\left(\frac{2-n}{2}\right)^{2}+\mathcal{C}_{\mathbb{C}^{n}}+N_{m+1}^{2}.$
(6.16)

$\bigl{(}\frac{2-n}{2}\bigr{)}^{2}$ is a scalar.
$\mathcal{C}_{\mathbb{C}^{n}}$ and $N_{m+1}^{2}$ are polynomials in elements
of $\mathrm{so}(\mathbb{C}^{n+2})$. $\mathcal{V}$ is tangent to
$\mathrm{so}(\mathbb{C}^{n+2})$. Therefore, all operators in the last line of
(6.16) can be restricted to $\mathcal{V}$. The operator
$D_{\mathbb{C}^{n+2}}-\frac{2-n}{2}$ vanishes on functions in
$\Lambda^{\frac{2-n}{2}}(\Omega)$. The operator $\langle
z|z\rangle\Delta_{m+1}$ is zero when restricted to $\mathcal{V}$.

Method II. We write $\mathbb{C}^{n+2}=\mathbb{C}^{n+1}\oplus\mathbb{C}$ with
the distinguished variable denoted by $t$. We assume that the square of
$z\in\mathbb{C}^{n+2}$ is given by

$\displaystyle\langle z|z\rangle=\langle z|z\rangle_{\mathbb{C}^{n+1}}+t^{2}.$

We will use various operators on $\mathbb{C}^{n+1}$: $D_{\mathbb{C}^{n+1}}$,
$\mathcal{C}_{\mathbb{C}^{n+1}}$, and $\Delta_{\mathbb{C}^{n+1}}$. We have

$\displaystyle
D_{\mathbb{C}^{n+2}}=D_{\mathbb{C}^{n+1}}+t\partial_{t},\qquad\Delta_{\mathbb{C}^{n+2}}=\Delta_{\mathbb{C}^{n+1}}+\partial_{t}^{2}.$

We have the following identity

$\displaystyle\langle
z|z\rangle_{\mathbb{C}^{n+1}}\Delta_{\mathbb{C}^{n+2}}=\langle
z|z\rangle_{\mathbb{C}^{n+1}}\Delta_{\mathbb{C}^{n+1}}+\langle
z|z\rangle\partial_{t}^{2}-t^{2}\partial_{t}^{2}$
$\displaystyle\hphantom{\langle
z|z\rangle_{\mathbb{C}^{n+1}}\Delta_{\mathbb{C}^{n+2}}}{}=\mathcal{C}_{\mathbb{C}^{n+1}}+\left(D_{\mathbb{C}^{n+1}}+\frac{n-1}{2}\right)^{2}-\left(\frac{n-1}{2}\right)^{2}+\langle
z|z\rangle\partial_{t}^{2}$ $\displaystyle\hphantom{\langle
z|z\rangle_{\mathbb{C}^{n+1}}\Delta_{\mathbb{C}^{n+2}}=}{}-\left(t\partial_{t}-\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}$
$\displaystyle\hphantom{\langle
z|z\rangle_{\mathbb{C}^{n+1}}\Delta_{\mathbb{C}^{n+2}}}{}=\langle
z|z\rangle\partial_{t}^{2}+\left(D_{\mathbb{C}^{n+1}}+\frac{n}{2}-t\partial_{t}\right)\left(D_{\mathbb{C}^{n+2}}-\frac{2-n}{2}\right)$
$\displaystyle\hphantom{\langle
z|z\rangle_{\mathbb{C}^{n+1}}\Delta_{\mathbb{C}^{n+2}}=}{}-\left(\frac{n-2}{2}\right)\frac{n}{2}+\mathcal{C}_{\mathbb{C}^{n+1}}.$
(6.17)

Then we argue similarly as in Method I. ∎

Using Theorem 6.4 we can restrict the Laplacian to functions in
$\Lambda^{\eta}(\mathcal{V}_{0})$ for $\eta=\frac{2-n}{2}$. More precisely, we
introduce the following definition.

Let $k\in\Lambda^{\frac{2-n}{2}}(\mathcal{V}_{0})$. Let $\Omega$ be any open
subset of $\mathbb{C}^{n+2}$ such that
$\mathcal{V}_{0}=\Omega\cap\mathcal{V}$, let $K\in\mathcal{A}(\Omega)$ be
homogeneous of degree $\frac{2-n}{2}$ and

$\displaystyle k=K\big{|}_{\mathcal{V}_{0}}.$

(We can always find such $\Omega$ and $K$.) Note that
$\Delta_{\mathbb{C}^{n+2}}K$ is homogeneous of degree $\frac{-2-n}{2}$. We set

$\displaystyle\Delta_{\mathbb{C}^{n+2}}^{\diamond}k:=\Delta_{\mathbb{C}^{n+2}}K\big{|}_{\mathcal{V}_{0}}.$

By Theorem 6.4, the above definition does not depend on the choice of $\Omega$
and $K$ and defines a map

$\displaystyle\Delta_{\mathbb{C}^{n+2}}^{\diamond}\colon\
\Lambda^{\frac{2-n}{2}}(\mathcal{V}_{0})\to\Lambda^{\frac{-2-n}{2}}(\mathcal{V}_{0}).$

######  Remark 6.5.

Let us explain the notation $\Delta_{\mathbb{C}^{n+2}}^{\diamond}$ for the
reduced Laplacian. We do not put the degree of homogeneity
$\eta=\frac{2-n}{2}$ as a superscript, because it is fixed by Theorem 6.4,
unlike in the case of the representations of $\mathrm{so}(\mathbb{C}^{n+2})$
and ${\rm O}(\mathbb{C}^{n+2})$. The subscript $\mathbb{C}^{n+2}$ is a little
confusing, because $\Delta_{\mathbb{C}^{n+2}}^{\diamond}$ acts on functions of
only $n+1$ variables, and after fixing a section on functions of $n$
variables. However, the initial operator is clearly
$\Delta_{\mathbb{C}^{n+2}}$. Finally, the diamond $\diamond$ is a symbol that
we have already used in the context of line bundles.

Restricting (6.4) and (6.5) to $\Lambda^{\frac{2-n}{2}}(\mathcal{V}_{0})$ we
obtain

$\displaystyle
B^{\diamond,\frac{-2-n}{2}}\Delta_{\mathbb{C}^{n+2}}^{\diamond}=\Delta_{\mathbb{C}^{n+2}}^{\diamond}B^{\diamond,\frac{2-n}{2}},\qquad$
$\displaystyle B\in\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)},$
$\displaystyle{\alpha}^{\diamond,\frac{-2-n}{2}}\Delta_{\mathbb{C}^{n+2}}^{\diamond}=\Delta_{\mathbb{C}^{n+2}}^{\diamond}{\alpha}^{\diamond,\frac{2-n}{2}},\qquad$
$\displaystyle{\alpha}\in\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}.$

The following proposition is the consequence of the proof of Theorem 6.4.

######  Proposition 6.6.

  1. $(1)$

In the notation of Method I of the proof of Theorem 6.4, we have

$\displaystyle\langle
z|z\rangle_{\mathbb{C}^{n}}\Delta_{\mathbb{C}^{n+2}}^{\diamond}=\left(\frac{n-2}{2}\right)^{2}+\mathcal{C}_{\mathbb{C}^{n}}+N_{m+1}^{2}.$
(6.18)

  2. $(2)$

In the notation of Method II of the proof of Theorem 6.4, we have

$\displaystyle\langle
z|z\rangle_{\mathbb{C}^{n+1}}\Delta_{\mathbb{C}^{n+2}}^{\diamond}=-\left(\frac{n-2}{2}\right)\frac{n}{2}+\mathcal{C}_{\mathbb{C}^{n+1}}.$
(6.19)

###### Proof.

(6.18) follows from (6.16). (6.19) follows from (6.17). ∎

###  6.6 Conformal invariance of Laplacian for a general section

The operator $\Delta_{\mathbb{C}^{n+2}}^{\diamond}$ is quite abstract. In this
subsection we describe how to make it more explicit.

Consider an open set $\mathcal{Y}_{0}$ in $\mathcal{Y}$. Let
$\mathcal{V}_{0}:=\pi^{-1}(\mathcal{Y}_{0})$ and
$\eta\in\frac{1}{2}\mathbb{Z}$. Choose a section $\gamma$ of the line bundle
$\mathcal{V}\to\mathcal{Y}$ based on $\mathcal{Y}_{0}$. As in (4.14) and
(4.13), we can introduce
$\psi^{\gamma,\eta}\colon\Lambda^{\eta}(\mathcal{V}_{0})\to\mathcal{A}(\mathcal{Y}_{0})$
and its left inverse $\phi^{\gamma,\eta}$. We set

$\displaystyle
B^{\gamma,\eta}:=\psi^{\gamma,\eta}B^{\diamond,\eta}\phi^{\gamma,\eta}=\psi^{\gamma,\eta}B\phi^{\gamma,\eta},\qquad$
$\displaystyle B\in\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)},$
$\displaystyle\alpha^{\gamma,\eta}:=\psi^{\gamma,\eta}\alpha^{\diamond,\eta}\phi^{\gamma,\eta}=\psi^{\gamma,\eta}\alpha\phi^{\gamma,\eta},\qquad$
$\displaystyle\alpha\in\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}.$

As explained in Propositions 4.1, 4.2 and 4.3, we obtain a representation and
a local representation

$\displaystyle\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)}\ni B\mapsto
B^{\gamma,\eta}\in\mathcal{A}\rtimes\mathrm{hol}(\mathcal{Y}_{0}),$ (6.20)
$\displaystyle\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}\ni{\alpha}\mapsto{\alpha}^{\gamma,\eta}\underset{\mathrm{loc}}{\in}\mathcal{A}^{\times}\rtimes\mathrm{Hol}(\mathcal{Y}_{0}).$
(6.21)

We also define

$\displaystyle\Delta_{\mathbb{C}^{n+2}}^{\gamma}:=\psi^{\gamma,\frac{-2-n}{2}}\Delta_{\mathbb{C}^{n+2}}^{\diamond}\phi^{\gamma,\frac{2-n}{2}}=\psi^{\gamma,\frac{-2-n}{2}}\Delta_{\mathbb{C}^{n+2}}\phi^{\gamma,\frac{2-n}{2}}.$

We have the identities

$\displaystyle
B^{\gamma,\frac{-2-n}{2}}\Delta_{\mathbb{C}^{n+2}}^{\gamma}=\Delta_{\mathbb{C}^{n+2}}^{\gamma}B^{\gamma,\frac{2-n}{2}},\qquad$
$\displaystyle B\in\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)},$ (6.22)
$\displaystyle{\alpha}^{\gamma,\frac{-2-n}{2}}\Delta_{\mathbb{C}^{n+2}}^{\gamma}=\Delta_{\mathbb{C}^{n+2}}^{\gamma}{\alpha}^{\gamma,\frac{2-n}{2}},\qquad$
$\displaystyle{\alpha}\in\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}.$ (6.23)

Thus we have shown that (infinitesimal) conformal transformations of the
$n$-dimensional manifold $\mathcal{Y}_{0}$ lead to generalized (infinitesimal)
symmetries of $\Delta_{\mathbb{C}^{n+2}}^{\gamma}$. Even if (in a somewhat
different form) this is a known fact, it seems that our derivation is new and
of interest. In particular, it shows that a large class of second order
$n$-dimensional operators together with their generalized symmetries directly
come from the $(n+2)$-dimensional Laplacian with its true symmetries.

###  6.7 Conformal invariance of Laplacian on $\boldsymbol{\mathbb{C}^{n}}$

Let us describe more closely the above construction in the case of the section
(6.14). In this case, instead of $\gamma$ we will write “$\mathrm{f\/l}$”, for
flat. We identify of $\mathcal{Y}_{e}$ with $\mathbb{C}^{n}$. We can restrict
(6.8) to an action of $\mathrm{so}(\mathbb{C}^{n+2})$ on $\mathcal{Y}_{e}$,
and (6.9) to a local action of $\mathrm{O}(\mathbb{C}^{n+2})$ on
$\mathcal{Y}_{e}$. Using (4.2) and (4.10), we obtain

$\displaystyle\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)}\ni B\mapsto
B^{\mathrm{f\/l}}\in\mathrm{hol}\big{(}\mathbb{C}^{n}\big{)},\qquad\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}\ni{\alpha}\mapsto{\alpha}^{\mathrm{f\/l}}\underset{\mathrm{loc}}{\in}\mathrm{Hol}\big{(}\mathbb{C}^{n}\big{)}.$

We introduce
$\psi^{\mathrm{f\/l},\eta}\colon\Lambda^{\eta}(\mathcal{V}_{e})\to\mathcal{A}(\mathbb{C}^{n})$
and its left inverse $\phi^{\mathrm{f\/l},\eta}$. (6.20) and (6.21) can be
rewritten as

$\displaystyle\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)}\ni B\mapsto
B^{\mathrm{f\/l},\eta}\in\mathcal{A}\rtimes\mathrm{hol}\big{(}\mathbb{C}^{n}\big{)},\qquad\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}\ni{\alpha}\mapsto{\alpha}^{\mathrm{f\/l},\eta}\underset{\mathrm{loc}}{\in}\mathcal{A}^{\times}\rtimes\mathrm{Hol}\big{(}\mathbb{C}^{n}\big{)}.$

The $(n+2)$-dimensional Laplacian reduced to the flat section is just the
usual $n$-dimensional Laplacian:

$\displaystyle\Delta_{\mathbb{C}^{n+2}}^{\mathrm{f\/l}}=\Delta_{\mathbb{C}^{n}}.$

The symmetries (6.22) and (6.23) become the generalized symmetries of the
usual Laplacian:

$\displaystyle
B^{\mathrm{f\/l},\frac{-2-n}{2}}\Delta_{\mathbb{C}^{n}}=\Delta_{\mathbb{C}^{n}}B^{\mathrm{f\/l},\frac{2-n}{2}},\qquad$
$\displaystyle B\in\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)},$
$\displaystyle{\alpha}^{\mathrm{f\/l},\frac{-2-n}{2}}\Delta_{\mathbb{C}^{n}}=\Delta_{\mathbb{C}^{n}}{\alpha}^{\mathrm{f\/l},\frac{2-n}{2}},\qquad$
$\displaystyle{\alpha}\in\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}.$

Thus $\mathbb{C}^{n+2}$ serves to describe in a simple way conformal
symmetries of $\mathbb{C}^{n}$. When used in this fashion, the space
$\mathbb{C}^{n+2}$ will be sometimes called the extended space.

Below we sum up information about conformal symmetries on the level of the
extended space $\mathbb{C}^{n+2}$ and the space $\mathbb{C}^{n}$. We will use
the split coordinates, that is, $z\in\mathbb{C}^{n+2}$ and
$y\in\mathbb{C}^{n}$ have the square

$\displaystyle\langle z|z\rangle=\sum_{j\in I_{n+2}}z_{-j}z_{j},\qquad\langle
y|y\rangle=\sum_{j\in I_{n}}y_{-j}y_{j}.$

As a rule, if a given operator does not depend on $\eta$, we will omit $\eta$.

Cartan algebra of $\mathrm{so}(\mathbb{C}^{n+2})$. Cartan operators of
$\mathrm{so}(\mathbb{C}^{n})$, $i=1,\dots,m$:

$\displaystyle N_{i}=z_{-i}\partial_{z_{-i}}-z_{i}\partial_{z_{i}},\qquad
N_{i}^{\mathrm{f\/l}}=y_{-i}\partial_{y_{-i}}-y_{i}\partial_{y_{i}}.$

Generator of dilations:

$\displaystyle
N_{m+1}=z_{-m-1}\partial_{z_{-m-1}}-z_{m+1}\partial_{z_{m+1}},\qquad
N_{m+1}^{\mathrm{f\/l},\eta}=\sum\limits_{i\in
I_{n}}y_{i}\partial_{y_{i}}-\eta=D_{\mathbb{C}^{n}}-\eta.$

Root operators. Roots of $\mathrm{so}(\mathbb{C}^{n})$, $|i|<|j|$, $i,j\in
I_{n}$:

$\displaystyle B_{i,j}=z_{-i}\partial_{z_{j}}-z_{-j}\partial_{z_{i}},\qquad
B_{i,j}^{\mathrm{f\/l}}=y_{-i}\partial_{y_{j}}-y_{-j}\partial_{y_{i}}.$

Generators of translations, $j\in I_{n}$:

$\displaystyle
B_{-m-1,j}=z_{m+1}\partial_{z_{j}}-z_{-j}\partial_{z_{-m-1}},\qquad
B_{-m-1,j}^{\mathrm{f\/l}}=\partial_{y_{j}}.$

Generators of special conformal transformations, $j\in I_{n}$:

$\displaystyle
B_{m+1,j}=z_{-m-1}\partial_{z_{j}}-z_{-j}\partial_{z_{m+1}},\qquad
B_{m+1,j}^{\mathrm{f\/l},\eta}=-\frac{1}{2}\langle
y|y\rangle\partial_{y_{j}}+y_{-j}\sum\limits_{i\in
I_{n}}y_{i}\partial_{y_{i}}-\eta y_{-j}.$

Weyl symmetries. We will write $K$ for a function on $\mathbb{C}^{n+2}$ and
$f$ for a function on $\mathbb{C}^{n}$. Reflection:

$\displaystyle\tau_{0}K(z_{0},\dots)=K(-z_{0},\dots),$
$\displaystyle\tau_{0}^{\mathrm{f\/l}}f(y_{0},\dots)=f(-y_{0},\dots).$

Flips, $j=1,\dots,m$:

$\displaystyle\tau_{j}K(\dots,z_{-j},z_{j},\dots,z_{-m-1},z_{m+1})=K(\dots,z_{j},z_{-j},\dots,z_{-m-1},z_{m+1}),$
$\displaystyle\tau_{j}^{\mathrm{f\/l}}f(\dots,y_{-j},y_{j},\dots)=f(\dots,y_{j},y_{-j},\dots).$

Inversion:

$\displaystyle\tau_{m+1}K(\dots,z_{-m-1},z_{m+1})=K(\dots,z_{m+1},z_{-m-1}),$
$\displaystyle\tau_{m+1}^{\mathrm{f\/l},\eta}f(y)=\left(-\frac{\langle
y|y\rangle}{2}\right)^{\eta}f\left(-\frac{2y}{\langle y|y\rangle}\right).$

Permutations, $\sigma\in S_{m}$:

$\displaystyle\sigma
K(\dots,z_{-j},z_{j},\dots,z_{-m-1},z_{m+1})=K(\dots,z_{-\sigma_{j}},z_{\sigma_{j}},\dots,z_{-m-1},z_{m+1}),$
$\displaystyle\sigma^{\mathrm{f\/l}}f(\dots,y_{-j},y_{j},\dots)=f(\dots,y_{-\sigma_{j}},y_{\sigma_{j}},\dots).$

Special conformal transformations, $j=1,\dots,m$:

$\displaystyle\sigma_{(j,m+1)}K(z_{-1},z_{1},\dots,z_{-j},z_{j},\dots,z_{-m-1},z_{m+1})$
$\displaystyle\qquad{}=K(z_{-1},z_{1},\dots,z_{-m-1},z_{m+1},\dots,z_{-j},z_{j}),$
$\displaystyle\sigma_{(j,m+1)}^{\mathrm{f\/l},\eta}f(y_{-1},y_{1},\dots,y_{-j},y_{j},\dots)=y_{j}^{\eta}f\left(\frac{y_{-1}}{y_{j}},\frac{y_{1}}{y_{j}},\dots,-\frac{\langle
y|y\rangle}{2y_{j}},\frac{1}{y_{j}},\dots\right).$

Laplacian:

$\displaystyle\Delta_{\mathbb{C}^{n{+}2}}=\sum\limits_{i\in
I_{n+2}}\partial_{z_{i}}\partial_{z_{-i}},\qquad\Delta_{\mathbb{C}^{n+2}}^{\mathrm{f\/l}}=\sum\limits_{i\in
I_{n}}\partial_{y_{i}}\partial_{y_{-i}}=\Delta_{\mathbb{C}^{n}}.$

Computations. Let us describe how to derive these formulas in an easy way.
Consider $\mathbb{C}^{n+1}\times\mathbb{C}^{\times}$ (defined by $z_{m+1}\neq
0$), which is an open dense subset of $\mathbb{C}^{n+2}$. Clearly,
$\mathcal{V}_{e}$ is contained in $\mathbb{C}^{n+1}\times\mathbb{C}^{\times}$.

We will write $\Lambda^{\eta}(\mathbb{C}^{n+1}\times\mathbb{C}^{\times})$ for
the space of functions homogeneous of degree $\eta$ on
$\mathbb{C}^{n+1}\times\mathbb{C}^{\times}$.

Instead of using the maps $\phi^{\mathrm{f\/l},\eta}$ and
$\psi^{\mathrm{f\/l},\eta}$, as in (4.14) and (4.13), we will prefer
$\Phi^{\mathrm{f\/l},\eta}\colon\mathcal{A}(\mathbb{C}^{n})\to\Lambda^{\eta}(\mathbb{C}^{n+1}\times\mathbb{C}^{\times})$
and
$\Psi^{\mathrm{f\/l},\eta}\colon\Lambda^{\eta}(\mathbb{C}^{n+1}\times\mathbb{C}^{\times})\to\mathcal{A}(\mathbb{C}^{n})$
defined below.

For
$K\in\Lambda^{\eta}\bigl{(}\mathbb{C}^{n+1}\times\mathbb{C}^{\times}\bigr{)}$,
we define $\Psi^{\mathrm{f\/l},\eta}K\in\mathcal{A}(\mathbb{C}^{n})$ by

$\displaystyle\big{(}\Psi^{\mathrm{f\/l},\eta}K\big{)}(y)=K\left(y,-\frac{\langle
y|y\rangle}{2},1\right),\qquad y\in\mathbb{C}^{n}.$

Let $f\in\mathcal{A}(\mathbb{C}^{n})$. Then there exists a unique function in
$\Lambda^{\eta}\bigl{(}\mathbb{C}^{n+1}\times\mathbb{C}^{\times}\bigr{)}$ that
extends $f$ and does not depend on $z_{-m-1}$. It is given by

$\displaystyle\big{(}\Phi^{\mathrm{f\/l},\eta}f\big{)}(\dots,z_{m},z_{-m-1},z_{m+1}):=z_{m+1}^{\eta}f\left(\dots,\frac{z_{m}}{z_{m+1}}\right).$

$\Psi^{\mathrm{f\/l},\eta}$ is a left inverse of $\Phi^{\mathrm{f\/l},\eta}$:

$\displaystyle\Psi^{\mathrm{f\/l},\eta}\Phi^{\mathrm{f\/l},\eta}=\mathrm{id}.$

Clearly,

$\displaystyle\Phi^{\mathrm{f\/l},\eta}f\big{|}_{\mathcal{V}_{e}}=\phi^{\mathrm{f\/l},\eta}f,\qquad\Psi^{\mathrm{f\/l},\eta}K=\psi^{\mathrm{f\/l},\eta}\big{(}K\big{|}_{\mathcal{V}_{e}}\big{)}.$

Moreover, functions in
$\Lambda^{\eta}(\mathbb{C}^{n+1}\times\mathbb{C}^{\times})$ restricted to
$\mathcal{V}_{e}$ are in $\Lambda^{\eta}(\mathcal{V}_{e})$. Therefore,

$\displaystyle
B^{\mathrm{f\/l},\eta}=\Psi^{\mathrm{f\/l},\eta}B\Phi^{\mathrm{f\/l},\eta},\qquad$
$\displaystyle B\in\mathrm{so}\big{(}\mathbb{C}^{n+2}\big{)},$
$\displaystyle{\alpha}^{\mathrm{f\/l},\eta}=\Psi^{\mathrm{f\/l},\eta}{\alpha}\Phi^{\mathrm{f\/l},\eta},\qquad$
$\displaystyle{\alpha}\in\mathrm{O}\big{(}\mathbb{C}^{n+2}\big{)}.$

Note also that

$\displaystyle\Delta_{\mathbb{C}^{n+2}}^{\mathrm{f\/l}}=\Psi^{\mathrm{f\/l},\eta}\Delta_{\mathbb{C}^{n+2}}\Phi^{\mathrm{f\/l},\eta}=\Delta_{\mathbb{C}^{n}}.$

In practice, the above idea can be implemented by the following change of
coordinates on $\mathbb{C}^{n+2}$:

$\displaystyle y_{i}:=\frac{z_{i}}{z_{m+1}},\quad i\in I_{n},\qquad
R:=\sum\limits_{i\in I_{n+2}}z_{i}z_{-i},\qquad p:=z_{m+1}.$

The inverse transformation is

$\displaystyle z_{i}=py_{i},\quad i\in I_{n},\qquad
z_{-m-1}=\frac{1}{2}\left(\frac{R}{p}-p\sum\limits_{i\in
I_{n}}y_{i}y_{-i}\right),\qquad z_{m+1}=p.$

The derivatives are equal to

$\displaystyle\partial_{z_{i}}=z_{m+1}^{-1}\partial_{y_{i}}+2z_{-i}\partial_{R},\quad
i\in I_{n},\qquad\partial_{z_{-m-1}}=2z_{m+1}\partial_{R},$
$\displaystyle\partial_{z_{m+1}}=\partial_{p}-z_{m+1}^{-2}\sum\limits_{i\in
I_{n}}z_{i}\partial_{y_{i}}+2z_{-m-1}\partial_{R}.$

Note that these coordinates are defined on
$\mathbb{C}^{n+1}\times\mathbb{C}^{\times}$. $\mathcal{V}_{e}$ is given by the
condition $R=0$. The section (6.14) (see also (6.15)) is given by $p=1$.

For a function $y\mapsto f(y)$ we have

$\displaystyle\big{(}\Phi^{\mathrm{f\/l},\eta}f\big{)}(y,R,p)=p^{\eta}f(y).$

For a function $(y,R,p)\mapsto K(y,R,p)$ we have

$\displaystyle\big{(}\Psi^{\mathrm{f\/l},\eta}K\big{)}(y)=K(y,0,1).$

Note also that on $\Lambda^{\eta}(\mathbb{C}^{n+1}\times\mathbb{C}^{\times})$
we have

$\displaystyle p\partial_{p}=\eta.$

###  6.8 Dimension $\boldsymbol{n=1}$

Let us illustrate the constructions of this section by describing the
projective quadric in the lowest dimensions, where everything is very
explicit. We start with dimension $n=1$.

The 1-dimensional projective quadric is isomorphic to the Riemann sphere or,
what is the same, the 1-dimensional projective complex space:

$\displaystyle\mathcal{Y}^{1}\simeq\mathbb{C}\cup\\{\infty\\}=P^{1}\mathbb{C}.$

Indeed, consider $\mathbb{C}^{3}$ with the scalar product

$\displaystyle\langle z|z\rangle=z_{0}^{2}+2z_{-1}z_{+1}.$

We can cover $\mathcal{Y}^{1}$ with two maps:

$\displaystyle\mathbb{C}\ni
s\mapsto\phi_{+}(s)=\left(s,1,-\frac{1}{2}s^{2}\right)\mathbb{C}^{\times}\in\mathcal{Y}^{1},\qquad\mathbb{C}\ni
s\mapsto\phi_{-}(s)=\left(s,-\frac{1}{2}s^{2},1\right)\mathbb{C}^{\times}\in\mathcal{Y}^{1}.$

The transition map is

$\displaystyle\phi_{+}^{-1}\phi_{-}(s)=-\frac{2}{s}.$

The Lie algebra $\mathrm{so}(\mathbb{C}^{3})$ is spanned by

$\displaystyle B_{0,1},\quad B_{0,-1},\quad N_{1},$

with the commutation relations

$\displaystyle[B_{0,1},B_{0,-1}]=N_{1},\qquad[N_{1},B_{0,1}]=B_{0,1},\qquad[N_{1},B_{0,-1}]=-B_{0,-1}.$

The Casimir operator is

$\displaystyle\mathcal{C}=2B_{0,1}B_{0,-1}-N_{1}^{2}-N_{1}=2B_{0,-1}B_{0,1}-N_{1}^{2}+N_{1}.$
(6.24)

###  6.9 Dimension $\boldsymbol{n=2}$

The 2-dimensional projective quadric is isomorphic to the product of two
Riemann spheres:

$\displaystyle\mathcal{Y}^{2}\simeq P^{1}\mathbb{C}\times P^{1}\mathbb{C}.$

Indeed, consider $\mathbb{C}^{4}$ with the scalar product

$\displaystyle\langle z|z\rangle=2z_{-1}z_{+1}+2z_{-2}z_{+2}.$

We can cover $\mathcal{Y}^{2}$ with four maps:

$\displaystyle\mathbb{C}\times\mathbb{C}\ni(t,s)\mapsto\phi_{+1}(t,s)=(-ts,1,t,s)\mathbb{C}^{\times}\in\mathcal{Y}^{2},$
$\displaystyle\mathbb{C}\times\mathbb{C}\ni(t,s)\mapsto\phi_{-1}(t,s)=(1,-ts,s,t)\mathbb{C}^{\times}\in\mathcal{Y}^{2},$
$\displaystyle\mathbb{C}\times\mathbb{C}\ni(t,s)\mapsto\phi_{+2}(t,s)=(-s,-t,1,ts)\mathbb{C}^{\times}\in\mathcal{Y}^{2},$
$\displaystyle\mathbb{C}\times\mathbb{C}\ni(t,s)\mapsto\phi_{-2}(t,s)=(-t,-s,-ts,1)\mathbb{C}^{\times}\in\mathcal{Y}^{2}.$

Here are the transition maps:

$\displaystyle\begin{split}&\phi_{-1}^{-1}\phi_{+1}(t,s)=\phi_{-2}^{-1}\phi_{+2}(t,s)=\big{(}{-}t^{-1},-s^{-1}\big{)},\\\
&\phi_{-2}^{-1}\phi_{-1}(t,s)=\phi_{+2}^{-1}\phi_{+1}(t,s)=\big{(}{-}t^{-1},s\big{)},\\\
&\phi_{-2}^{-1}\phi_{+1}(t,s)=\phi_{+2}^{-1}\phi_{-1}(t,s)=\big{(}t,-s^{-1}\big{)}.\end{split}$

The Lie algebra $\mathrm{so}(\mathbb{C}^{4})$ is spanned by

$\displaystyle N_{1},\quad N_{2},\quad B_{1,2},\quad B_{1,-2},\quad
B_{-1,2},\quad B_{-1,-2}.$

Its Casimir operator is

$\displaystyle\mathcal{C}=2B_{1,2}B_{-1,-2}+2B_{1,-2}B_{-1,2}-N_{1}^{2}-N_{2}^{2}+2N_{1}.$

As is well known, $\mathrm{so}(\mathbb{C}^{4})$ decomposes into the direct sum
$\mathrm{so}_{+}(\mathbb{C}^{3})\oplus\mathrm{so}_{-}(\mathbb{C}^{3})$ of two
commuting Lie algebras isomorphic to $\mathrm{so}(\mathbb{C}^{3})$ spanned by

$\displaystyle B_{1,2},\quad B_{-1,-2},\quad N_{1}+N_{2}\qquad\text{and}\qquad
B_{1,-2},\quad B_{-1,2},\quad N_{1}-N_{2}$

with the commutation relations

$\displaystyle[B_{1,2},B_{-1,-2}]=N_{1}+N_{2},\qquad$
$\displaystyle[B_{1,-2},B_{-1,2}]=N_{1}-N_{2},$
$\displaystyle[N_{1}+N_{2},B_{1,2}]=2B_{1,2},\qquad$
$\displaystyle[N_{1}-N_{2},B_{1,-2}]=2B_{1,-2},$
$\displaystyle[N_{1}+N_{2},B_{-1,-2}]=-2B_{-1,-2},\qquad$
$\displaystyle[N_{1}-N_{2},B_{-1,2}]=-2B_{-1,2}.$

The corresponding Casimir operators are

$\displaystyle\mathcal{C}_{+}=2B_{1,2}B_{-1,-2}-\frac{1}{2}(N_{1}+N_{2})^{2}-N_{1}-N_{2}=2B_{-1,-2}B_{1,2}-\frac{1}{2}(N_{1}+N_{2})^{2}+N_{1}+N_{2},$
$\displaystyle\mathcal{C}_{-}=2B_{1,-2}B_{-1,2}-\frac{1}{2}(N_{1}-N_{2})^{2}-N_{1}+N_{2}=2B_{-1,2}B_{1,-2}-\frac{1}{2}(N_{1}-N_{2})^{2}+N_{1}-N_{2}.$

Thus

$\displaystyle\mathcal{C}=\mathcal{C}_{+}+\mathcal{C}_{-}.$

In the enveloping algebra of $\mathrm{so}(\mathbb{C}^{4})$ the operators
$\mathcal{C}_{+}$ and $\mathcal{C}_{-}$ are distinct. They satisfy
$\alpha(\mathcal{C}_{-})=\mathcal{C}_{+}$ for
$\alpha\in\mathrm{O}(\mathbb{C}^{4})\backslash\mathrm{SO}(\mathbb{C}^{4})$,
for instance for $\tau_{i}$, $i=1,2$.

However, inside the associative algebra of differential operators on
$\mathbb{C}^{4}$ we have the identity

$\displaystyle B_{1,2}B_{-1,-2}-B_{1,-2}B_{-1,2}=N_{1}N_{2}+N_{2},$

which implies

$\displaystyle\mathcal{C}_{+}=\mathcal{C}_{-}$

inside this algebra. Therefore, represented in the algebra of differential
operators we have

$\displaystyle\mathcal{C}=4B_{1,2}B_{-1,-2}-(N_{1}+N_{2})^{2}-2N_{1}-2N_{2}$
$\displaystyle\hphantom{\mathcal{C}}{}=4B_{-1,-2}B_{1,2}-(N_{1}+N_{2})^{2}+2N_{1}+2N_{2}$
$\displaystyle\hphantom{\mathcal{C}}{}=4B_{1,-2}B_{-1,2}-(N_{1}-N_{2})^{2}-2N_{1}+2N_{2}$
$\displaystyle\hphantom{\mathcal{C}}{}=4B_{-1,2}B_{1,-2}-(N_{1}-N_{2})^{2}+2N_{1}-2N_{2}.$
(6.25)

##  7 $\boldsymbol{\mathrm{so}(\mathbb{C}^{6})}$ and the hypergeometric
equation

In this section we derive the hypergeometric operator and its
$\mathrm{so}(\mathbb{C}^{6})$ symmetries. We will consider the following
levels:

  1. (1)

extended space $\mathbb{C}^{6}$ and the Laplacian $\Delta_{\mathbb{C}^{6}}$,

  2. (2)

reduction to the so-called spherical section and the corresponding
Laplace–Beltrami operator,

  3. (3)

depending on the choice of coordinates, separation of variables leads to the
balanced or standard hypergeometric operator.

Alternatively, one can use a different derivation:

  1. $(2)^{\prime}$

reduction to $\mathbb{C}^{4}$ and $\Delta_{\mathbb{C}^{4}}$ with help of the
flat section,

  2. $(3)^{\prime}$

with appropriate coordinates, separation of variables leads to the balanced or
standard hypergeometric operator.

A separate subsection is devoted to factorizations of the hypergeometric
operator. We will see that they are closely related to
$\mathrm{so}(\mathbb{C}^{4})$ subalgebras of $\mathrm{so}(\mathbb{C}^{6})$ and
their Casimir operators.

###  7.1 Extended space $\boldsymbol{\mathbb{C}^{6}}$

We consider $\mathbb{C}^{6}$ with the coordinates

$\displaystyle z_{-1},\quad z_{1},\quad z_{-2},\quad z_{2},\quad z_{-3},\quad
z_{3}$ (7.1)

and the scalar product given by

$\displaystyle\langle z|z\rangle=2z_{-1}z_{1}+2z_{-2}z_{2}+2z_{-3}z_{3}.$
(7.2)

Lie algebra $\mathrm{so}(\mathbb{C}^{6})$. Cartan algebra:

$\displaystyle N_{i}=z_{-i}\partial_{z_{-i}}-z_{i}\partial_{z_{i}},\qquad
i=1,2,3.$

Root operators:

$\displaystyle B_{i,j}=z_{-i}\partial_{{j}}-z_{-j}\partial_{{i}},\qquad
1\leq|i|<|j|\leq 3.$

Weyl symmetries. Transpositions:

$\displaystyle\sigma_{(12)}K(z_{-1},z_{1},z_{-2},z_{2},z_{-3},z_{3})=K(z_{-2},z_{2},z_{-1},z_{1},z_{-3},z_{3}),$
$\displaystyle\sigma_{(13)}K(z_{-1},z_{1},z_{-2},z_{2},z_{-3},z_{3})=K(z_{-3},z_{3},z_{-2},z_{2},z_{-1},z_{1}),$
$\displaystyle\sigma_{(23)}K(z_{-1},z_{1},z_{-2},z_{2},z_{-3},z_{3})=K(z_{-1},z_{1},z_{-3},z_{3},z_{-2},z_{2}).$

Cycles:

$\displaystyle\sigma_{(123)}K(z_{-1},z_{1},z_{-2},z_{2},z_{-3},z_{3})=K(z_{-3},z_{3},z_{-1},z_{1},z_{-2},z_{2}),$
$\displaystyle\sigma_{(132)}K(z_{-1},z_{1},z_{-2},z_{2},z_{-3},z_{3})=K(z_{-2},z_{2},z_{-3},z_{3},z_{-1},z_{1}).$

Flips:

$\displaystyle\tau_{1}K(z_{-1},z_{1},z_{-2},z_{2},z_{-3},z_{3})=K(z_{1},z_{-1},z_{-2},z_{2},z_{-3},z_{3}),$
$\displaystyle\tau_{2}K(z_{-1},z_{1},z_{-2},z_{2},z_{-3},z_{3})=K(z_{-1},z_{1},z_{2},z_{-2},z_{-3},z_{3}),$
$\displaystyle\tau_{3}K(z_{-1},z_{1},z_{-2},z_{2},z_{-3},z_{3})=K(z_{-1},z_{1},z_{-2},z_{2},z_{3},z_{-3}).$

Laplacian:

$\displaystyle\Delta_{\mathbb{C}^{6}}=2\partial_{z_{-1}}\partial_{z_{1}}+2\partial_{z_{-2}}\partial_{z_{2}}+2\partial_{z_{-3}}\partial_{z_{3}}.$
(7.3)

Symmetries:

$\displaystyle
N_{i}\Delta_{\mathbb{C}^{6}}=\Delta_{\mathbb{C}^{6}}N_{i},\qquad$
$\displaystyle 1\leq i\leq 3,$ (7.4a) $\displaystyle
B_{i,j}\Delta_{\mathbb{C}^{6}}=\Delta_{\mathbb{C}^{6}}B_{i,j},\qquad$
$\displaystyle 1\leq|i|<|j|\leq 3,$ (7.4b)
$\displaystyle\sigma\Delta_{\mathbb{C}^{6}}=\Delta_{\mathbb{C}^{6}}\sigma,\qquad$
$\displaystyle\sigma\in S_{3},$ (7.4c)
$\displaystyle\tau_{j}\Delta_{\mathbb{C}^{6}}=\Delta_{\mathbb{C}^{6}}\tau_{j},\qquad$
$\displaystyle 1\leq j\leq 3.$ (7.4d)

###  7.2 Spherical section

In this subsection we consider the section of the quadric

$\displaystyle\mathcal{V}^{5}:=\big{\\{}z\in\mathbb{C}^{6}\colon
2z_{-1}z_{1}+2z_{-2}z_{2}+2z_{-3}z_{3}=0\big{\\}}$

given by equations

$\displaystyle 4=2\left(z_{-1}z_{1}+z_{-2}z_{2}\right)=-2z_{3}z_{-3}.$

We will call it the spherical section, because it coincides with
$\mathcal{S}^{3}(4)\times\mathcal{S}^{1}(-4)$. The superscript used for this
section will be “$\mathrm{sph}$” for spherical.

We will see that this section is well suited to obtain the hypergeometric
equation, both in the balanced and standard form, because its conformal factor
is trivially equal to $1$.

As a preparation for a discussion of this section, let us choose the
coordinates

$\displaystyle r=\sqrt{2\left(z_{-1}z_{1}+z_{-2}z_{2}\right)},\qquad$
$\displaystyle p=\sqrt{2z_{3}z_{-3}},\qquad$ $\displaystyle
w=\frac{z_{-1}z_{1}}{z_{-1}z_{1}+z_{-2}z_{2}},$ $\displaystyle
u_{1}=\sqrt{\frac{z_{-1}}{z_{1}}},\qquad$ $\displaystyle
u_{2}=\sqrt{\frac{z_{-2}}{z_{2}}},\qquad$ $\displaystyle
u_{3}=\sqrt{\frac{z_{-3}}{z_{3}}}.$

The null quadric in these coordinates is given by $r^{2}+p^{2}=0$. The
generator of dilations is

$\displaystyle D_{\mathbb{C}^{6}}=r\partial_{r}+p\partial_{p}.$

The spherical section is given by the condition $r^{2}=4$.

Let us now describe in detail various objects in the spherical section.

Lie algebra $\mathrm{so}(\mathbb{C}^{6})$. Cartan operators:

$\displaystyle N_{1}^{\mathrm{sph}}=u_{1}\partial_{u_{1}},\qquad
N_{2}^{\mathrm{sph}}=u_{2}\partial_{u_{2}},\qquad
N_{3}^{\mathrm{sph}}=u_{3}\partial_{u_{3}}.$

Roots:

$\displaystyle
B_{2,1}^{\mathrm{sph}}=u_{1}u_{2}\sqrt{w(1-w)}\left(\partial_{w}-\frac{N^{\mathrm{sph}}_{1}}{2w}+\frac{N^{\mathrm{sph}}_{2}}{2(1-w)}\right),$
(7.5a) $\displaystyle
B_{-2,-1}^{\mathrm{sph}}=\frac{1}{u_{1}u_{2}}\sqrt{w(1-w)}\left(\partial_{w}+\frac{N^{\mathrm{sph}}_{1}}{2w}-\frac{N^{\mathrm{sph}}_{2}}{2(1-w)}\right),$
(7.5b) $\displaystyle
B_{2,-1}^{\mathrm{sph}}=\frac{u_{2}}{u_{1}}\sqrt{w(1-w)}\left(\partial_{w}+\frac{N^{\mathrm{sph}}_{1}}{2w}+\frac{N^{\mathrm{sph}}_{2}}{2(1-w)}\right),$
(7.5c) $\displaystyle
B_{-2,1}^{\mathrm{sph}}=\frac{u_{1}}{u_{2}}\sqrt{w(1-w)}\left(\partial_{w}-\frac{N^{\mathrm{sph}}_{1}}{2w}-\frac{N^{\mathrm{sph}}_{2}}{2(1-w)}\right),$
(7.5d) $\displaystyle
B_{3,1}^{\mathrm{sph},{\eta}}=\frac{\mathrm{i}}{2}u_{1}u_{3}\sqrt{w}\left(\eta+2(1-w)\partial_{w}-\frac{N^{\mathrm{sph}}_{1}}{w}-N^{\mathrm{sph}}_{3}\right),$
(7.5e) $\displaystyle
B_{-3,-1}^{\mathrm{sph},{\eta}}=\frac{\mathrm{i}}{2}\frac{1}{u_{1}u_{3}}\sqrt{w}\left(\eta+2(1-w)\partial_{w}+\frac{N^{\mathrm{sph}}_{1}}{w}+N^{\mathrm{sph}}_{3}\right),$
(7.5f) $\displaystyle
B_{3,-1}^{\mathrm{sph},{\eta}}=\frac{\mathrm{i}}{2}\frac{u_{3}}{u_{1}}\sqrt{w}\left(\eta+2(1-w)\partial_{w}+\frac{N^{\mathrm{sph}}_{1}}{w}-N^{\mathrm{sph}}_{3}\right),$
(7.5g) $\displaystyle
B_{-3,1}^{\mathrm{sph},{\eta}}=\frac{\mathrm{i}}{2}\frac{u_{1}}{u_{3}}\sqrt{w}\left(\eta+2(1-w)\partial_{w}-\frac{N^{\mathrm{sph}}_{1}}{w}+N^{\mathrm{sph}}_{3}\right),$
(7.5h) $\displaystyle
B_{3,2}^{\mathrm{sph},{\eta}}=\frac{\mathrm{i}}{2}u_{2}u_{3}\sqrt{1-w}\left(\eta-2w\partial_{w}-\frac{N^{\mathrm{sph}}_{2}}{1-w}-N^{\mathrm{sph}}_{3}\right),$
(7.5i) $\displaystyle
B_{-3,-2}^{\mathrm{sph},{\eta}}=\frac{\mathrm{i}}{2}\frac{1}{u_{2}u_{3}}\sqrt{1-w}\left(\eta-2w\partial_{w}+\frac{N^{\mathrm{sph}}_{2}}{1-w}+N^{\mathrm{sph}}_{3}\right),$
(7.5j) $\displaystyle
B_{3,-2}^{\mathrm{sph},{\eta}}=\frac{\mathrm{i}}{2}\frac{u_{3}}{u_{2}}\sqrt{1-w}\left(\eta-2w\partial_{w}+\frac{N^{\mathrm{sph}}_{2}}{1-w}-N^{\mathrm{sph}}_{3}\right),$
(7.5k) $\displaystyle
B_{-3,2}^{\mathrm{sph},{\eta}}=\frac{\mathrm{i}}{2}\frac{u_{2}}{u_{3}}\sqrt{1-w}\left(\eta-2w\partial_{w}-\frac{N^{\mathrm{sph}}_{2}}{1-w}+N^{\mathrm{sph}}_{3}\right).$
(7.5l)

Weyl symmetries. Transpositions:

$\displaystyle\sigma_{12}^{\mathrm{sph},{\eta}}f(w,u_{1},u_{2},u_{3})=f(1-w,u_{2},u_{1},u_{3}),$
$\displaystyle\sigma_{13}^{\mathrm{sph},{\eta}}f(w,u_{1},u_{2},u_{3})=\left(\mathrm{i}\sqrt{w}\right)^{\eta}f\left(\frac{1}{w},u_{3},u_{2},u_{1}\right),$
$\displaystyle\sigma_{23}^{\mathrm{sph},{\eta}}f(w,u_{1},u_{2},u_{3})=\left(\mathrm{i}\sqrt{1-w}\right)^{\eta}f\left(\frac{w}{w-1},u_{1},u_{3},u_{2}\right).$

Cycles:

$\displaystyle\sigma_{312}^{\mathrm{sph},{\eta}}f(w,u_{1},u_{2},u_{3})=\left(\mathrm{i}\sqrt{1-w}\right)^{\eta}f\left(\frac{1}{1-w},u_{3},u_{1},u_{2}\right),$
$\displaystyle\sigma_{231}^{\mathrm{sph},{\eta}}f(w,u_{1},u_{2},u_{3})=\left(\mathrm{i}\sqrt{w}\right)^{\eta}f\left(1-\frac{1}{w},u_{2},u_{3},u_{1}\right).$

Flips:

$\displaystyle\tau_{1}^{\mathrm{sph},{\eta}}f(w,u_{1},u_{2},u_{3})=f\left(w,\frac{1}{u_{1}},u_{2},u_{3}\right),$
$\displaystyle\tau_{2}^{\mathrm{sph},{\eta}}f(w,u_{1},u_{2},u_{3})=f\left(w,u_{1},\frac{1}{u_{2}},u_{3}\right),$
$\displaystyle\tau_{3}^{\mathrm{sph},{\eta}}f(w,u_{1},u_{2},u_{3})=f\left(w,u_{1},u_{2},\frac{1}{u_{3}}\right).$

The Laplacian in coordinates (7.5) is

$\displaystyle\Delta_{\mathbb{C}^{6}}=\frac{4}{r^{2}}\biggl{(}\frac{1}{4}\left((r\partial_{r})^{2}+2(r\partial_{r})+\frac{r^{2}}{p^{2}}(p\partial_{p})^{2}\right)$
$\displaystyle\hphantom{\Delta_{\mathbb{C}^{6}}=\frac{4}{r^{2}}\biggl{(}}{}+\partial_{w}w(1-w)\partial_{w}-\frac{(u_{1}\partial_{u_{1}})^{2}}{4w}-\frac{(u_{1}\partial_{u_{2}})^{2}}{4(1-w)}-\frac{r^{2}}{p^{2}}\frac{(u_{3}\partial_{u_{3}})^{2}}{4}\biggr{)}.$

Using

$\displaystyle(r\partial_{r})^{2}+2r\partial_{r}+\tfrac{r^{2}}{p^{2}}(p\partial_{p})^{2}=\left(r\partial_{r}+\frac{r^{2}}{p^{2}}(p\partial_{p})+1\right)(r\partial_{r}+p\partial_{p}+1)-1$
$\displaystyle\hphantom{(r\partial_{r})^{2}+2r\partial_{r}+\tfrac{r^{2}}{p^{2}}(p\partial_{p})^{2}=}{}-\left(\frac{r^{2}}{p^{2}}+1\right)(p\partial_{p})(r\partial_{r}+1),$

$r^{2}+p^{2}=0$ and $r\partial_{r}+p\partial_{p}=-1$, we obtain

$\displaystyle\Delta_{\mathbb{C}^{6}}^{\diamond}=\frac{4}{r^{2}}\left(\partial_{w}w(1-w)\partial_{w}-\frac{(u_{1}\partial_{u_{1}})^{2}}{4w}-\frac{(u_{2}\partial_{u_{2}})^{2}}{4(1-w)}+\frac{(u_{3}\partial_{u_{3}})^{2}}{4}-\frac{1}{4}\right).$

To convert $\Delta_{\mathbb{C}^{6}}^{\diamond}$ into the reduced Laplacian
$\Delta_{\mathbb{C}^{6}}^{\mathrm{sph}}$, we simply remove the prefactor
$\frac{4}{r^{2}}$, obtaining the Laplace–Beltrami operator on
$\mathcal{S}^{3}(4)\times\mathcal{S}^{1}(-4)$:

$\displaystyle\Delta_{\mathbb{C}^{6}}^{\mathrm{sph}}=\partial_{w}w(1-w)\partial_{w}-\frac{\big{(}N_{1}^{\mathrm{sph}}\big{)}^{2}}{4w}-\frac{\big{(}N_{2}^{\mathrm{sph}}\big{)}^{2}}{4(1-w)}+\frac{\big{(}N_{3}^{\mathrm{sph}}\big{)}^{2}}{4}-\frac{1}{4}.$
(7.6)

Generalized symmetries:

$\displaystyle
N_{i}^{\mathrm{sph}}\Delta_{\mathbb{C}^{6}}^{\mathrm{sph}}=\Delta_{\mathbb{C}^{6}}^{\mathrm{sph}}N_{i}^{\mathrm{sph}},\qquad$
$\displaystyle 1\leq i\leq 3,$ (7.7) $\displaystyle