with a capital sans serif letter with a superscript preceding it to indicate
its rank, for example, ${}^{2}\mathsf{T}$ is a dyadic. The set of all
polyadics is a real linear space of dimension $d^{r}$. Indeed, if $r=0$,
then $(\mathbb{R}^{d})^{r}=\mathbb{R}^{1}$, $d^{r}=1$, and a linear
functional $f\colon \mathbb{R}^{1}\rightarrow \mathbb{R}$ has the form $%
f(x_{1})=y_{1}x_{1}$ with $y_{1}\in \mathbb{R}^{1}$. Otherwise, if $r\geq 1$
, we define a special polyadic as follows.

A polyadpolyad (monad,monad dyad,dyad triad,triad …) is the
polyadic $\mathbf{x}_{1}\mathbf{x}_{2}\cdots \mathbf{x}_{r}$ given by

\[
\mathbf{x}_{1}\mathbf{x_{2}}\cdots \mathbf{x}_{r}(\mathbf{y}_{1},\mathbf{%
y_{2}},\dots ,\mathbf{y}_{r})=(\mathbf{x}_{1}\cdot \mathbf{y}_{1})(\mathbf{x}%
_{2}\cdot \mathbf{y}_{2})\cdots (\mathbf{x}_{r}\cdot \mathbf{y}_{r})
\]

for all $(\mathbf{y}_{1},\mathbf{y}_{2},\dots ,\mathbf{y}_{r})\in (\mathbb{R}%
^{d})^{r}$, where $\mathbf{x}_{1}$, $\mathbf{x_{2}}$, …, $\mathbf{x}%
_{r} $ are arbitrary vectors in $\mathbb{R}^{d}$.

The $d^{r}$ polyads $\mathbf{e}_{i_{1}}\mathbf{e}_{i_{2}}\cdots \mathbf{e}%
_{i_{r}}$ with $1\leq i_{j}\leq d$ for all $j$ with $1\leq j\leq r$, form a
basis in the linear space of all polyadics. With the $r$-dot product
the above basis becomes orthonormal. In particular, if $r=1$, then the $1$
-dot product of two monads $\mathbf{x}$ and $\mathbf{y}$ coincides with the
scalar product $\mathbf{x\cdot y}$, the $2$-dot product of two dyads is $%
\mathbf{x}_{1}\mathbf{x_{2}}\colon \mathbf{y}_{1}\mathbf{y_{2}}=(\mathbf{x}%
_{1}\cdot \mathbf{y}_{1})(\mathbf{x}_{2}\cdot \mathbf{y}_{2})$, and the $3$
-dot product of two triads by $\mathbf{x}_{1}\mathbf{x}_{2}\mathbf{x}%
_{3}\,\vdots \,\mathbf{y}_{1}\mathbf{y}_{2}\mathbf{y}_{3}=(\mathbf{x}%
_{1}\cdot \mathbf{y}_{1})(\mathbf{x}_{2}\cdot \mathbf{y}_{2})(\mathbf{x}%
_{3}\cdot \mathbf{y}_{3})$. A polyad $\mathbf{x}_{1}\mathbf{x_{2}}\cdots
\mathbf{x}_{r}$ is identified with the tensor product $\mathbf{x}_{1}\otimes
\mathbf{x_{2}}\otimes \cdots \otimes \mathbf{x}_{r}$ and the linear space of
all polyadics with the tensor product $(\mathbb{R}^{d})^{\otimes r}$ of $r$
copies of the space $\mathbb{R}^{d}$. We can prove that, for $r\geq 1$, any
$r$-adic can be represented as a sum of not more than $d^{r-1}$ $r$-ads.

If, under a shift map $\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$, $\mathbf{x}%
\mapsto \mathbf{x}+\mathbf{y}$, the one-point correlation
polyadic $\langle {}^{r}\mathsf{T}(\mathbf{x})\rangle $ and the
two-point covariance polyadic

\[
\langle {}^{r}\mathsf{T}(\mathbf{x}),{}^{r}\mathsf{T}(\mathbf{y})\rangle
=\left\langle ({}^{r}\mathsf{T}(\mathbf{x})-\langle {}^{r}\mathsf{T}(\mathbf{%
x})\rangle ),({}^{r}\mathsf{T}(\mathbf{y})-\langle {}^{r}\mathsf{T}(\mathbf{y}%
)\rangle )\right\rangle  \label{correlation polyadics}
\]

do not change, a second-order TRF is WSS. It is WSSI if its correlation
polyadics satisfy

\[
\langle {}^{r}\mathsf{T}(g\mathbf{x})\rangle =g\cdot \langle {}^{r}\mathsf{T}%
(\mathbf{x})\rangle ,\qquad \langle ^{r}\mathsf{T}(g\mathbf{x}),{}^{r}%
\mathsf{T}(g\mathbf{y})\rangle =(\rho \otimes \rho )(g)\langle {}^{r}\mathsf{%
T}(g\mathbf{x}),{}^{r}\mathsf{T}(g\mathbf{y})\rangle ,
\label{WSSI correlation polyadics}
\]

where $g$ $\mapsto \mathsf{S}^{2}(g)$ is an orthogonal representation of the
group $\mathrm{O}(3)$. Some open issues are:

While it is clear how to construct the polyadic TRFs from scalar random
fields, the resulting statistics and correlation structures are open
research topics.

§ RANDOM CROSS-SECTIONS OF HOMOGENEOUS VECTOR BUNDLES

This section is motivated by the following observation: the Cosmic Microwave Background can be described mathematically as a random cross-section of a homogeneous vector bundle over the sky sphere, see [7, 34, 62, 74, 113].

A random cross-sectioncross-section!random of a homogeneous vector bundle $(E,B,\pi)$ is a collection of random vectors $\{\,s(x,\omega)\colon x\in B\,\}$ such that $s(x,\omega)$ is a random vector in $\pi^{-1}(x)$. In what follows, we use the terms “random cross-section”  and “random field” for $s(x)$ interchangeably.

A random cross-section $s(x)$ of a homogeneous vector bundle $(E,B,\pi)$ is called strictly isotropiccross-section!random!strictly isotropic if for any positive integer $n$, for any $n$ distinct points $x_1$, …, $x_n$ in $B$, and for arbitrary $g\in G$, the finite-dimensional distributions $(s(x_1),\dots,s(x_n))$ and $(s(gx_1),\dots,s(gx_n))$ are identical.

To simplify the subsequent notation, introduce the following agreements. Let $tL=L$ and $t\mathbf{l}=\mathbf{l}$ whenever $\mathbb{K}\in\{\mathbb{R},\mathbb{H}\}$. For a complex Hilbert space $V$ with inner product $(\cdot,\cdot)$, and for $\mathbf{v}\in V$, let $t\mathbf{v}$ be the element of the space $tV$ satisfying $(t\mathbf{v},\mathbf{v}_1)=(\mathbf{v}_1,\mathbf{v})$ for each $\mathbf{v}_1\in V$.

Assume that the random cross-section $s(x)$ of a homogeneous vector bundle $(E,B,\pi)$ with fibre $L_0$ is second-order,cross-section!random!second-order that is, $\mathsf{E}[\|s(x)\|^2_{L_0}]<\infty$, $x\in B$. If, in addition, such a field is strictly isotropic, then it is wide-sense isotropic.

A second-order random cross-section $s(x)$ is called wide-sense isotropiccross-section!random!wide-sense isotropic if and only if its one-point correlation tensorcorrelation tensor!one-point $\langle s(x)\rangle=\mathsf{E}[s(x)]$ and the two-point correlation tensorcorrelation tensor!two-point
$\langle s(x),s(y)\rangle=\mathsf{E}[t(s(x)-\langle s(x)\rangle)\otimes(s(y)-\langle s(y)\rangle)]$ are $G$-invariant, that is, for any $g\in G$ we have

\begin{equation}\label{eq:41}
\langle s(gx)\rangle=\langle s(x)\rangle,\qquad
\langle s(gx),s(gy)\rangle=\langle s(x),s(y)\rangle.
\end{equation}

Note that the one-point correlation tensor of a  random field $s(x)$ is $\mathbb{K}$-valued, while the two-point one is $\mathbb{K}'$-valued. See Section <ref> for explanation of the above symbols.

In what follows, we consider only wide-sense isotropic random fields in homogeneous vector bundles and call them just isotropic.

The next step is to define a mean-square continuous random field in a homogeneous vector bundle. The standard definition, Equation (<ref>), does not work, because if $x$, $y\in B$ and $x\neq y$, then the random vectors $s(x)$ and $s(y)$ belong to different linear spaces and cannot be subtracted one from another. See a discussion of this problem in [62, 63]. Here we propose an alternative definition.

Let $L^2(\Omega,L_0)$ be the Hilbert space of $L_0$-valued random vectors $\mathbf{X}$ with $\mathsf{E}[\|\mathbf{X}\|^2_{L_0}]<\infty$ and inner product

\begin{equation}\label{eq:34}
\end{equation}

Let $H$ be the closed linear span of the set of random vectors $\{\,s(x)\colon x\in B\,\}$ in $L^2(\Omega,L_0)$. For each $g\in G$, there is a unique bounded linear operator $A(g)$ in $H$ with $A(g)T(x)=s(g^{-1}x)$. Moreover, the correspondence $g\mapsto A(g)$ is a representation of $G$ in $H$.

A random cross-section $s(x)$ is called mean-square continuous,cross-section!random!mean-square continuous if the above representation is continuous.

Moreover, it is not difficult to prove that a random field $s(x)$ with constant one-point correlation tensor is isotropic if and only if the inner product (<ref>) is $G$-invariant.

How to find a spectral expansion of an isotropic random cross-section? First, exactly as in the case of ordinary random fields, we conclude that the one-point correlation tensor $\langle s(x)\rangle$ is an arbitrary element of the isotypic subspace of the space $L_0$ that corresponds to the trivial representation of the group $H$.

Let ${}_{L^n,p}Y_{ijkLm}(x)$ be the cross-sections of the orthonormal basis constructed in Subsection <ref>. In this notation, the index $L^n$ runs over the set of all irreducible representations of the group $H$, for which $\Hom_{\mathbb{K}H}(L^n,L_0)\neq\{\mathbf{0}\}$, the index $p$ enumerates the copies of the representation $L^n$ inside $L_0$, the index $L$ runs over the set of all irreducible representations of the group $G$, for which $\Hom_{\mathbb{K}H}(\res^G_HL,L^n)\neq\{\mathbf{0}\}$, the index $i$ enumerates the copies of the representation $L^n$ inside the representation $\Hom_{\mathbb{K}H}(\res^G_HL,L^n)$, the index $j$ enumerates identical blocks inside a certain matrix, the index $k$ runs from $1$ to the dimension of a certain division algebra over $\mathbb{K}'$, and the index $m$ enumerates the basis vectors in $L$.

Let $d_L$ be the multiplicity of the representation $L$ in the induced representation $L^2(E)$, and let $\{\,{}_{L^n,p}\mathbf{a}_{Lm}\colon L\in\hat{G}_{\mathbb{K}},1\leq m\leq\dim L\,\}$ be a sequence of centred uncorrelated $\mathbb{K}^{d_L}$-valued random vectors such that the covariance matrix of the vector ${}_{L^n,p}\mathbf{a}_{Lm}$ does not depend on $m$ and for fixed $L^n$ and $p$ we have

\[
\sum_{L\in\hat{G}_{\mathbb{K}}}\dim L\mathsf{E}[\|{}_{L^n,p}\mathbf{a}_{L1}\|^2]<\infty.
\]

A random field $s(x)$ is isotropic if and only if it has the form

\[
\]

Let $(E,\pi,B\times B)$ be the homogeneous vector bundle where the representation of the group $G\times G$ induced by the representation $tL_0\otimes L_0$ of the subgroup $H\times H$ acts. Observe that the two-point correlation tensor
$\langle s(x),s(y)\rangle$ of the random field $s(x)$ is a continuous cross-section of the above bundle. The cross-sections $t{}_{L^n,p}Y_{ijkLm}(x)\otimes{}_{L'^n,p'}Y_{i'j'k'L'm'}(y)$
form an orthonormal basis in the Hilbert space $L^2(E)$.

We note that $t{}_{L^n,p}Y_{ijkLm}(x)={}_{L^n,p}Y_{ijk(tL)m}(x)$, and, for a fixed $L\in\hat{G}_{\mathbb{K}}$, the functions ${}_{L^n,p}Y_{ijkLm}(x)$ form an orthonormal basis in the isotypic subspace of the space $L^2(E)$ that correspond to $L$, call it $\mathcal{H}_L$. Moreover, by the construction of this basis, for fixed $L^n$, $p$, $i$, $j$, and $k$, we have

\[
{}_{L^n,p}Y_{ijkLm}(x)=\sum_{q=1}^{\dim L}\theta^L_{mq}(g){}_{L^n,p}Y_{ijkLq}(x).
\]

Let $C_{LL'}$ be the matrix of the Fourier coefficients of the cross-section $\langle s(x),s(y)\rangle$ in the above basis. The second equation in condition (<ref>) becomes

\[
\]

that is, $C_{LL'}\in\Hom_{\mathbb{K}G}(\mathcal{H}_L,\mathcal{H}_{L'})$. Schur's Lemma immediately shows that $C_{LL'}$ is the zero matrix whenever $tL\neq L'$. Otherwise, the matrix $C^{tLL}$ is symmetric when $\mathbb{K}'=\mathbb{R}$, Hermitian when $\mathbb{K}'=\mathbb{C}$, consists of square sub-matrices with $\dim V$ rows, and each sub-matrix is block-diagonal, with blocks described in Subsection <ref>. Because the blocks of sizes $2\times 2$ and $4\times 4$ are skew-symmetric, they must be diagonal and multiple to the identity matrix. In the space $\mathcal{H}_L$, the Fourier coefficients form a square matrix with $d^2_L$ blocks, where each block is a multiple of the $\dim L\times \dim L$ identity matrix. The elements of each block which are located at the intersection of the $m$th row and the $m$th column, form the covariance matrix of the random vector ${}_{L^n,p}\mathbf{a}_{Lm}$. The rest is obvious.

§ APPLICATIONS TO COSMOLOGY

We make a simplification in Example <ref>. Suppose that the spacetime $M$ is the Minkowski one, that is, the nonzero components of the bilinear form $g(x)$ are $g_{00}(x)=-c^2$, $g_{11}(x)=g_{22}(x)=g_{33}(x)=1$, where $c$ is the speed of light. Physically, this means that we neglect possible secondary effects of the CMB (the effects that appear after the last scattering of the CMB photons).

Consider an observer at the event $\mathbf{0}\in M$. Light rays through her eye correspond to null straight lines through $\mathbf{0}$. Their past directions with $t<0$ constitute her field of vision. She can imagine herself at the centre of her sphere of vision, a unit sphere $S^2$, the celestial sphere, see Example <ref>. It can be regarded in different ways.

[The $(\Theta,\mathcal{Q}\pm\mathrm{i}\mathcal{U},\mathcal{V})$ model]
Regard the celestial sphere as the complex projective line $\mathbb{C}P^1$. Let $T_{\zeta}\mathbb{C}P^1$ be the tangent space at a point $\zeta\in\mathbb{C}P^1$. According to Example <ref>, the electric field of the CMB at the point $\zeta$ belongs to the real linear space  $rT_{\zeta}\mathbb{C}P^1$. The space $crT_{\zeta}\mathbb{C}P^1$ is the complexified tangent spacetangent space!complexified at $\zeta\in\mathbb{C}P^1$, see [82]. We have $crT_{\zeta}\mathbb{C}P^1=T^{1,0}_{\zeta}\mathbb{C}P^1\oplus T^{0,1}_{\zeta}\mathbb{C}P^1$, where the first (resp. the second) term is the holomorphictangent space!holomorphic (resp. anti-holomorphic)tangent space!anti-holomorphic tangent space at $\zeta\in\mathbb{C}P^1$. The projection $\pi(crT_{\zeta}\mathbb{C}P^1)=\zeta$ determines a vector bundle, which is the vector bundle $(\Sigma\mathbb{C}P^1,\mathbb{C}P^1,\pi)$ with fibre $V_1\oplus tV_1$ of Example <ref>. That is, the electric field of the CMB is a spinor field.

Identify a point $\zeta\in\mathbb{C}P^1$ with the point $\mathbf{x}\in S^2\subset\mathbb{R}^3$ with coordinates given by <cit.>:

\[
x=\frac{\zeta+\zeta^*}{\zeta\zeta^*+1},\qquad y=\frac{\zeta-\zeta^*}{\mathrm{i}(\zeta\zeta^*+1)},\qquad z=\frac{\zeta\zeta^*-1}{\zeta\zeta^*+1},
\]

where the point at infinity $\zeta=\infty$ is identified with the north pole of the sphere. If we observe an electromagnetic wave of the CMB at a point $\mathbf{x}\in S^2$, it propagates in the direction $\mathbf{n}=-\mathbf{x}$.
Choose two vectors $\bm{\varepsilon}^{(1)}$ and $\bm{\varepsilon}^{(2)}$ in the space $crT_{\zeta}\mathbb{C}P^1$ in such a way that the vectors $(\bm{\varepsilon}^{(1)},\bm{\varepsilon}^{(2)},\mathbf{n})$ form a right-handed orthonormal basis. The pair $(\bm{\varepsilon}^{(1)},\bm{\varepsilon}^{(2)})$ is called the linear polarisation basis.polarisation basis!linear It is customary to represent the electric field of the wave in the complex form

\[
\mathbf{E}(t)=E_1(t)\bm{\varepsilon}^{(1)}+E_2(t)\bm{\varepsilon}^{(2)},\qquad E_1(t),E_2(t)\in\mathbb{C}.
\]

According to <cit.>:

…only the real part of the fields has a physical meaning. We make use of the complex notation because many formulas are thereby simplified…

A CMB detector observes the time-averaged electric field $\mathbf{E}=E_1\bm{\varepsilon}^{(1)}+E_2\bm{\varepsilon}^{(2)}$. Following [23], introduce the Hermitian $2\times 2$ matrix $\tilde{\mathcal{P}}=\left(
\begin{smallmatrix}
|E_1|^2 & E_1^*E_2 \\
E_2^*E_1 & |E_2|^2
\end{smallmatrix}
\right)$. The standard orthonormal basis in the real $4$-dimensional linear space of Hermitian $2\times 2$ matrices with inner product $(\tilde{\mathcal{P}}_1,\tilde{\mathcal{P}}_2)=\tr(\tilde{\mathcal{P}}_1\tilde{\mathcal{P}}_2)$ consists of the matrices $\frac{1}{\sqrt{2}}\sigma^{(i)}$, $0\leq i\leq 3$, where $\sigma^{(i)}$ are the Pauli matricesPauli matrices $\sigma^{(0)}=\left(
\begin{smallmatrix}
1 & 0 \\
0 & 1
\end{smallmatrix}
\right)$, $\sigma^{(1)}=\left(
\begin{smallmatrix}
0 & 1 \\
1 & 0
\end{smallmatrix}
\right)$, $\sigma^{(2)}=\left(
\begin{smallmatrix}
0 & -\mathrm{i} \\
\mathrm{i} & 0
\end{smallmatrix}
\right)$, $\sigma^{(3)}=\left(
\begin{smallmatrix}
1 & 0 \\
0 & -1
\end{smallmatrix}
\right)$. The expansion coefficients are called the Stokes parametersStokes parameters and defined as follows:

\begin{equation}\label{eq:45}
\begin{aligned}
I(\zeta)&=\tr(\tilde{\mathcal{P}}(\zeta)\sigma^{(0)}), & U(\zeta)&=\tr(\tilde{\mathcal{P}}(\zeta)\sigma^{(1)}),\\ V(\zeta)&=\tr(\tilde{\mathcal{P}}(\zeta)\sigma^{(2)}), & Q(\zeta)&=\tr(\tilde{\mathcal{P}}(\zeta)\sigma^{(3)}).
\end{aligned}
\end{equation}

The matrix $\tilde{\mathcal{P}}$ takes the form

\[
\tilde{\mathcal{P}}(\zeta)=\frac{1}{2}(I(\zeta)\sigma^{(0)}
\]

The matrix $\tilde{\mathcal{P}}$ lies in the space

\begin{equation}\label{eq:47}
L_0=t(V_1\oplus tV_1)\otimes(V_1\oplus tV_1)=(tV_1\oplus V_1)\otimes(V_1\oplus tV_1)=2cU_0\oplus V_2\oplus tV_2,
\end{equation}

and the group $H$, the subgroup of diagonal matrices in $\mathrm{SU}(2)$, acts on $\tilde{\mathcal{P}}$ by $h\cdot\tilde{\mathcal{P}}=h\tilde{\mathcal{P}}h^{-1}$.

The theory of irreducible unitary representations of the group $\mathrm{SU}(2)$ is developed in numerous sources, we mention [8, 30, 33, 61, 84] among others. The set $\hat{G}_{\mathbb{C}}$ contains the unitary irreducible representations of real typerepresentation!irreducible complex of $\mathrm{SU}(2)$!$cU_{\ell}$ $\{\,cU_{\ell}\colon\ell\geq 0\,\}$ and of quaternionic typerepresentation!irreducible complex of $\mathrm{SU}(2)$!$c'W_{\ell}$ $\{\,c'W_{\ell}\colon\ell=\frac{1}{2},\frac{3}{2},\dots\,\}$. To determine the structure of the restrictions $\res^G_HcU_{\ell}$ and $\res^G_Hc'W_{\ell}$, we use the characters.

By definition, the charactercharacter of a complex finite-dimensional representation $V$ is a function $G\to\mathbb{C}$, given by $\chi_V(g)=\tr\theta^V(g)$. The character of a real finite-dimensional representation $U$ (resp. quaternionic finite-dimensional representation $W$) is the character of the complex representation $cU$ (resp. $c'W$). We have $\chi_{V_1\oplus V_2}(g)=\chi_{V_1}(g)+\chi_{V_2}(g)$, and similarly for real and quaternionic representations.

According to [121], the restriction of the character of an irreducible representations with index $\ell$ to the subgroup $H$ is

\[
\chi_{cU_{\ell}}\left(\left(
\begin{smallmatrix}
\alpha & 0 \\
0 & \alpha^*
\end{smallmatrix}
\right)\right)=\chi_{c'W_{\ell}}\left(\left(
\begin{smallmatrix}
\alpha & 0 \\
0 & \alpha^*
\end{smallmatrix}
\right)\right)=\alpha^{2\ell}+\alpha^{-2\ell-2}
\]

In other words, the restriction $\res^G_HcU_{\ell}$ has the form

\begin{equation}\label{eq:35}
\res^G_HcU_{\ell}=tV_{2\ell}\oplus tV_{2\ell-2}\oplus\cdots\oplus tV_2\oplus cU_0\oplus V_2\oplus\cdots\oplus V_{2\ell},
\end{equation}

while the restriction $\res^G_Hc'W_{\ell}$ has the form

\begin{equation}\label{eq:36}
\res^G_Hc'W_{\ell}=tV_{2\ell}\oplus tV_{2\ell-2}\oplus\cdots\oplus tV_1\oplus V_1\oplus\cdots\oplus V_{2\ell}.
\end{equation}

The first copy of the irreducible representation $cU_0$ of the group $H$ acts in the complex linear space $L_{01}$ generated by the matrix $\frac{1}{\sqrt{2}}\sigma^{(0)}$. Schur's Lemma gives $D=\Hom_{\mathbb{C}H}(cU_0,cU_0)=\mathbb{C}$, and we have $\dim_{\mathbb{C}}D=1$. By Equations (<ref>) and (<ref>), $\dim_{\mathbb{C}}\Hom_{\mathbb{C}H}(\res^G_HcU_{\ell},cU_0)=1$, while $\dim_{\mathbb{C}}\Hom_{\mathbb{C}H}(\res^G_Hc'W_{\ell},cU_0)=0$. The space $\Hom_{\mathbb{C}H}(\res^G_HcU_{\ell},cU_0)$ contains $n_{cU_{\ell}}=1$ copy of the representation $cU_0$, while the space $\Hom_{\mathbb{C}H}(\res^G_Hc'W_{\ell},cU_0)$ contains no copies of the above representation. The matrix $f^{111}$ has $1$ row and $2\ell+1$ columns. We have $f^{111}_{1\,\ell+1}=1$, the remaining entries are zeroes.

Choose a model for $cU_{\ell}$,representation!irreducible complex of $\mathrm{SU}(2)$!$cU_{\ell}$ described in [121] and in many other sources. The linear space $cU_{\ell}$ is the space of all polynomials in a real variable $x$ of degree $2\ell$. The group $G$ acts by

\[
\left(
\begin{smallmatrix}
\alpha & \beta \\
-\beta^* & \alpha^*
\end{smallmatrix}
\right)\cdot p(x)=(\beta x+\alpha^*)^{2\ell}p\left(\frac{\alpha x-\beta^*}{\beta x+\alpha^*}\right),\qquad p\in cU_{\ell}.
\]

The orthonormal basis in $cU_{\ell}$ is formed by the polynomials

\[
p_m(x)=\frac{x^{\ell-m}}{\sqrt{(\ell-m)!(\ell+m)!}},\qquad-\ell\leq m\leq\ell.
\]

The matrix $\left(
\begin{smallmatrix}
\alpha & 0 \\
0 & \alpha^*
\end{smallmatrix}
\right)\in H$ acts on $p_m(x)$ by multiplication by $\alpha^{-2m}$, that is, by one of the representations in the right hand side of (<ref>). The Frobenius reciprocity isomorphism maps the matrix $f^{111}$ to the function  $F^{111}\colon cU_{\ell}\to C(G,L_{01})$. This function maps an element $g\in G$ to the zeroth component of the vector $g^{-1}\cdot p\in cU_{\ell}$ for $p\in U_{\ell}$. In particular, $F^{111}(p_m)$ is the $L_{01}$-valued function on $G$ given by $D^{*(\ell)}_{m0}(g)\frac{1}{\sqrt{2}}\sigma^{(0)}$, where $D^{(\ell)}_{m0}(g)$ are the Wigner $D$-functions,Wigner $D$-functions the matrix entries of the representations $cU_{\ell}$. After normalisation, the cross-sections ${}_{L_{01}}Y_{ijkLm}(gH)$ become

\[
\]

and we recognise the spherical harmonicsspherical harmonics!complex-valued $Y_{\ell,m}(\zeta)$, see <cit.>. Theorem <ref> gives

\begin{equation}\label{eq:42}
I(\zeta)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a_{\ell m}Y_{\ell,m}(\zeta),
\end{equation}

where $a_{\ell m}$ are complex-valued uncorrelated random variables with $\mathsf{E}[a_{\ell m}]=0$ for $\ell\geq 1$, $\mathsf{E}[|a_{\ell m}|^2]=C_{\ell}$, and

\[
\sum_{\ell=0}^{\infty}(2\ell+1)C_{\ell}<\infty.
\]

Note that we absorbed the constant cross-section $s$ into the expected value of $a_{00}$. The expansion (<ref>) was proved in [87] long before the discovery of the relic radiation. The spherical harmonics $Y_{\ell,m}(\zeta)$ have the form

\[
\begin{aligned}
\end{aligned}
\]

It follows from this equation that $Y_{\ell,-m}(\zeta)=(-1)^mY_{\ell,m}^*(\zeta)$. We see that the random field $I(\zeta)$ is real-valued if and only if $a_{\ell\,-m}=(-1)^ma^*_{\ell m}$.

The Stokes parameters, as defined above, have units of intensity $I$. It is conventional to express the CMB fluctuations in terms of the brightness temperature,brightness temperature $T$. Denote the Stokes parameters after re-scaling by $T$, $U_T$, $V_T$, and $Q_T$. Denote $T_0=\mathsf{E}[a_{00}]$, the averaged temperature of the CMB. Finally, the dimensionless Stokes parametersStokes parameters!dimensionless are

\[
\Theta=\frac{T-T_0}{T_0},\qquad\mathcal{U}=\frac{U_T}{T_0},\qquad\mathcal{V}
\]

Note that our definition of $\mathcal{U}$ and $\mathcal{Q}$ differs from <cit.> by a factor of $\frac{1}{4}$.

The number $\ell$ is called the multipole numbermultipole number (the monopolemonopole if $\ell=0$, the dipoledipole if $\ell=1$). The number $m$ is called the azimuthal number,azimuthal number the number $C_{\ell}$ the multipole moment.multipole moment The multipole expansionmultipole expansion of the dimensionless temperature deviation $\Theta(\zeta)$ generally starts at $\ell=2$:

\[
\Theta(\zeta)=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\Theta}_{\ell m}Y_{\ell,m}(\zeta).
\]

This is because with our definition of $\Theta$, the monopole term vanishes, while the dipole term is affected by our own motion across space and dominates over the intrinsic cosmological dipole term.

The analysis for $\mathcal{V}$ is similar and gives

\[
\mathcal{V}(\zeta)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{V}}_{\ell m}Y_{\ell,m}(\zeta).
\]

\[
\mathcal{P}(\zeta)=\frac{1}{2}(\mathcal{U}(\zeta)\sigma^{(1)}+\mathcal{Q}(\zeta)\sigma^{(3)})
\begin{pmatrix}
\mathcal{Q}(\zeta) & \mathcal{U}(\zeta) \\
\mathcal{U}(\zeta) & -\mathcal{Q}(\zeta)
\end{pmatrix}
\]

Rotate the basis $\bm{\varepsilon}^{(1)}$ and $\bm{\varepsilon}^{(2)}$ of the space $crT_{\zeta}\mathbb{C}P^1$ by an angle $\psi$ around the direction $\mathbf{n}$. The new basis is

\[
\bm{\varepsilon}^{(1)'}=\cos\psi\bm{\varepsilon}^{(1)}+\sin\psi\bm{\varepsilon}^{(2)},
\qquad\bm{\varepsilon}^{(2)'}=-\sin\psi\bm{\varepsilon}^{(1)}+\cos\psi\bm{\varepsilon}^{(2)}.
\]

In the rotated basis, the coefficients of the electric field become

\[
E_1=E_1\cos+E_2\sin\psi,\qquad E_2'=-E_1\sin\psi+E_2\cos\psi.
\]

Equation (<ref>) gives

\[
I=|E_1|^2+|E_2|^2,\quad Q=|E_1|^2-|E_2|^2,\quad U=2\RE(E^*_1E_2),\quad V=2\IM(E^*_1E_2),
\]

and we see that under the above rotation

\[
I'=I,\quad V'=V,\quad Q'=Q\cos(2\psi)-U\sin(2\psi),\quad U'=U\cos(2\psi)+Q\sin(2\psi).
\]

It follows that $Q'\pm\mathrm{i}U'=\mathrm{e}^{\pm 2\mathrm{i}\psi}(Q+\mathrm{i}U)$. In other words, $O+\mathrm{i}U\in V_2$, $Q-\mathrm{i}U\in tV_2$.

The irreducible representation $V_2$ of the group $H$ acts in the complex linear space $L_2$ generated by the matrix $\frac{1}{2}(\sigma^{(3)}+\mathrm{i}\sigma^{(1)})$. Schur's Lemma gives $D=\Hom_{\mathbb{C}H}(V_2,V_2)=\mathbb{C}$, and we have $\dim_{\mathbb{C}}D=1$. By Equations (<ref>) and (<ref>), $\dim_{\mathbb{C}}\Hom_{\mathbb{C}H}(\res^G_HcU_{\ell},V_2)=1$ for $\ell\geq 2$, while $\dim_{\mathbb{C}}\Hom_{\mathbb{C}H}(\res^G_Hc'W_{\ell},V_2)=0$. The space $\Hom_{\mathbb{C}H}(\res^G_HcU_{\ell},V_2)$ contains $n_{cU_{\ell}}=1$ copy of the representation $V_2$, while the space $\Hom_{\mathbb{C}H}(\res^G_Hc'W_{\ell},cU_0)$ contains no copies of the above representation. The matrix $f^{111}$ has $1$ row and $2\ell+1$ columns. We have $f^{111}_{1\,\ell+1}=1$, the remaining entries are zeroes.

This time, the Frobenius reciprocity isomorphism maps the matrix $f^{111}$ to the function $F^{111}\colon V_2\to C(G,L_2)$. This function maps an element $g\in G$ to the zeroth component of the vector $g^{-1}\cdot p\in V_2$ for $p\in U_{\ell}$. In particular, $F^{111}(p_m)$ is the $L_2$-valued function on $G$ given by $D^{*(\ell)}_{m2}(g)\frac{1}{2}(\sigma^{(3)}+\mathrm{i}\sigma^{(1)})$. After normalisation, the cross-sections ${}_{L_2}Y_{ijkLm}(gH)$ become

\[
\frac{1}{2}(\sigma^{(3)}+\mathrm{i}\sigma^{(1)}).
\]

with the help of <cit.>, we recognise the spin 2 spherical harmonicsspherical harmonics!spin 2!complex-valued ${}_2Y_{\ell,m}(\zeta)$. Theorem <ref> gives

\[
(\mathcal{Q}+\mathrm{i}\mathcal{U})(\zeta)=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a_{\ell m}^{(2)}\,{}_2Y_{\ell,m}(\zeta),
\]

where $a_{\ell m}^{(2)}$ are centred complex-valued uncorrelated random variables with

\[
\mathsf{E}[|a_{\ell m}^{(2)}|^2]=C^+_{\ell},\qquad\sum_{\ell=0}^{\infty}(2\ell+1)C^+_{\ell}<\infty.
\]

We say that the random fields $(\mathcal{Q}\pm\mathrm{i}\mathcal{U})(\zeta)$ have spin $\pm 2$ in contrast to the fields $\Theta(\zeta)$ and $\mathcal{V}(\zeta)$ which have spin $0$.

Similar analysis for the representation $tV_2$ gives

\[
(\mathcal{Q}\pm\mathrm{i}\mathcal{U})(\zeta)=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a_{\ell m}^{(\pm 2)}\,{}_{\pm 2}Y_{\ell,m}(\zeta).
\]

This expansion was proved in [141]. The spin $s$ spherical harmonics have the form

\[
\begin{aligned}
\end{aligned}
\]

It follows from this equation that ${}_{-s}Y_{\ell,-m}(\zeta)=(-1)^{m+s}{}_sY_{\ell,m}^*(\zeta)$. We see that

\begin{equation}\label{eq:52}
a^{(-2)*}_{\ell m}=a^{(2)}_{\ell\,-m}.
\end{equation}

Combining everything together, we obtain

\begin{equation}\label{eq:48}
\begin{aligned}
\tilde{\mathcal{P}}(\zeta)&=\frac{1}{2}\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\Theta}_{\ell m}\sigma^{(0)}Y_{\ell,m}(\zeta)+\frac{1}{2}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{V}}_{\ell m}\sigma^{(2)}Y_{\ell,m}(\zeta)\\
&\quad+\frac{1}{2}\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}\sigma^{(3)}\RE(a_{\ell m}^{(2)}\,{}_2Y_{\ell,m}(\zeta))+\frac{1}{2}\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}\sigma^{(1)}\IM(a_{\ell m}^{(2)}\,{}_2Y_{\ell,m}(\zeta)).
\end{aligned}
\end{equation}

[The $(\Theta,\mathcal{Q},\mathcal{U},\mathcal{V})$ model]
Introduce the circular polarisationpolarisation basis!circular or helicity basishelicity basis $\bm{\varepsilon}^{(\pm)}=\frac{1}{\sqrt{2}}(\bm{\varepsilon}^{(1)}\pm\mathrm{i}\bm{\varepsilon}^{(2)})$ and the symmetric trace free tensor spherical harmonicsspherical harmonics!symmetric trace free tensor

\begin{equation}\label{eq:46}
\begin{aligned}
Y^E_{\ell m}(\zeta)&=-\frac{1}{\sqrt{2}}[{}_{-2}Y_{\ell,m}(\zeta)\bm{\varepsilon}^{(+)}
\otimes\bm{\varepsilon}^{(+)}+{}_2Y_{\ell,m}(\zeta)\bm{\varepsilon}^{(-)}\otimes\bm{\varepsilon}^{(-)}],\\
\bm{\varepsilon}^{(+)}\otimes\bm{\varepsilon}^{(+)}-{}_2Y_{\ell,m}(\zeta)
\bm{\varepsilon}^{(-)}\otimes\bm{\varepsilon}^{(-)}].
\end{aligned}
\end{equation}

Then we obtain

\[
\mathcal{P}(\zeta)=\frac{1}{2}\sum_{\ell=2}^{\infty}[e_{\ell m}Y^E_{\ell,m}(\zeta)+b_{\ell m}Y^B_{\ell,m}(\zeta)],
\]

\[
e_{\ell m}=-\frac{1}{\sqrt{2}}(a^{(2)}_{\ell m}+a^{(-2)}_{\ell m}),\qquad b_{\ell m}=\frac{\mathrm{i}}{\sqrt{2}}(a^{(2)}_{\ell m}-a^{(-2)}_{\ell m}).
\]

This expansion was proved in [50], but we need to prove that their definition of symmetric trace free tensor spherical harmonics is equivalent to our definition.

To perform this, we use the results of [19]. In our notation, a part of their Equation (92) takes the form

\[
\begin{aligned}
\mathsf{T}^{2\,\ell-2,\ell m}(\zeta)\\
\mathsf{T}^{2\,\ell,\ell m}(\zeta)\\
\mathsf{T}^{2\,\ell+2,\ell m}(\zeta),\\
\mathsf{T}^{2\,\ell-1,\ell m}(\zeta)+\mathrm{i}\left(\frac{\ell-2}{2\ell+1}\right)^{\frac{1}{2}}
\mathsf{T}^{2\,\ell+1,\ell m}(\zeta),
\end{aligned}
\]

where $\mathsf{T}^{2\,\ell',\ell m}(\zeta)$ are the pure-orbital tensor spherical harmonics,spherical harmonics!pure-orbital tensor see [76, 116]. Compare these equations with <cit.>. We obtain

\[
Y^E_{\ell,m}(\zeta)=-\mathsf{T}^{E2,\ell m}(\zeta),\qquad Y^B_{\ell,m}(\zeta)=-\mathsf{T}^{B2,\ell m}(\zeta),
\]

where $\mathsf{T}^{E2,\ell m}(\zeta)$ and $\mathsf{T}^{E2,\ell m}(\zeta)$ are the pure-spin tensor spherical harmonics,spherical harmonics!pure-spin tensor see [116, 143]. <cit.> become equivalent to our Equation (<ref>).

The complete expansion takes the form

\begin{equation}\label{eq:49}
\begin{aligned}
\tilde{\mathcal{P}}(\zeta)&=\frac{1}{2}\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\Theta}_{\ell m}\sigma^{(0)}Y_{\ell,m}(\zeta)+\frac{1}{2}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{V}}_{\ell m}\sigma^{(2)}Y_{\ell,m}(\zeta)\\
&\quad+\frac{1}{2}\sum_{\ell=2}^{\infty}[e_{\ell m}Y^E_{\ell,m}(\zeta)+b_{\ell m}Y^B_{\ell,m}(\zeta)].
\end{aligned}
\end{equation}

[The $(\Theta,\mathcal{E},\mathcal{B},\mathcal{V})$ model]
Is is well-known that

\[
\eth^2({}_{-2}Y_{\ell,m}(\zeta))=\overline{\eth}^2({}_2Y_{\ell,m}(\zeta))
\]

If we assume that

\[
\sum_{\ell=2}^{\infty}\frac{(2\ell+1)\mathsf{E}[|a^{(2)}_{\ell 0}|^2](\ell+2)!}{(\ell-2)!}<\infty,
\]

then we may differentiate the random fields $(\mathcal{Q}\pm\mathrm{i}\mathcal{U})(\zeta)$ term by term:

\[
\begin{aligned}
\overline{\eth}^2((\mathcal{Q}+\mathrm{i}\mathcal{U})(\zeta))&=\sum_{\ell=2}^{\infty}
\sum_{m=-\ell}^{\ell}a^{(2)}_{\ell m}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}Y_{\ell,m}(\zeta),\\
\eth^2((\mathcal{Q}-\mathrm{i}\mathcal{U})(\zeta))&=\sum_{\ell=2}^{\infty}
\sum_{m=-\ell}^{\ell}a^{(-2)}_{\ell m}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}Y_{\ell,m}(\zeta).
\end{aligned}
\]

Introduce the random fields

\[
\begin{aligned}
\mathcal{E}(\zeta)&=\frac{1}{2}[\overline{\eth}^2((\mathcal{Q}+\mathrm{i}\mathcal{U})(\zeta))
\mathcal{B}(\zeta)&=-\frac{\mathrm{i}}{2}[\overline{\eth}^2((\mathcal{Q}+\mathrm{i}\mathcal{U})(\zeta))
\end{aligned}
\]

Their spectral expansion takes the form

\[
\begin{aligned}
\mathcal{E}(\zeta)&=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{E}}_{\ell m}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}Y_{\ell,m}(\zeta),\\
\mathcal{B}(\zeta)&=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{B}}_{\ell m}\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}Y_{\ell,m}(\zeta),
\end{aligned}
\]

\begin{equation}\label{eq:53}
a^{\mathcal{E}}_{\ell m}=\frac{1}{2}(a^{(2)}_{\ell m}+a^{(-2)}_{\ell m}),\qquad a^{\mathcal{B}}_{\ell m}=-\frac{\mathrm{i}}{2}(a^{(2)}_{\ell m}-a^{(-2)}_{\ell m}).
\end{equation}

All random fields $\Theta(\zeta)$, $\mathcal{E}(\zeta)$, $\mathcal{B}(\zeta)$, and $\mathcal{V}(\zeta)$ are spin $0$ fields. The complete expansion takes the form

\begin{equation}\label{eq:50}
\begin{aligned}
&=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\Theta}_{\ell m}Y_{\ell,m}(\zeta)\mathbf{e}_{\Theta}+\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{E}}_{\ell m}Y_{\ell,m}(\zeta)\mathbf{e}_{\mathcal{E}}\\
&\quad+\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{B}}_{\ell m}Y_{\ell,m}(\zeta)\mathbf{e}_{\mathcal{B}}+\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{V}}_{\ell m}Y_{\ell,m}(\zeta)\mathbf{e}_{\mathcal{V}},
\end{aligned}
\end{equation}

where $\mathbf{e}_{\Theta}=(1,0,0,0)^{\top}$, and so on.

Regard the celestial sphere as the base space of the principal bundle $(\mathrm{SO}(3),\mathrm{SO}(2),S^2,\tilde{\pi})$ with $\tilde{\pi}(g)=gH$. Let the group $\mathrm{SO}(2)$ acts in the space $L_0$ given by Equation (<ref>) by $h\cdot\tilde{\mathcal{P}}=h\mathcal{P}h^{-1}$. Let $(\theta,\varphi)$ be the spherical coordinates on $S^2$. The polarisation tensor of the CMB can be described as a random cross-section of a vector bundle associated to the principal bundle $(\mathrm{SO}(3),\mathrm{SO}(2),S^2,\tilde{\pi})$ by the above representation. The spectral expansions of these random sections have the form of Equations (<ref>), (<ref>), and (<ref>), where $\zeta$ is replaced with the spherical coordinates $(\theta,\varphi)$ on the sphere $S^2$.

However, in this model the electric field of the CMB is not a spinor field, because $\mathrm{SO}(2)\neq\mathrm{Spin}(2)$. The group $\mathrm{Spin}(2)=\mathrm{U}(2)$ covers the group $\mathrm{SO}(2)$ by the “square” map (<ref>). Nevertheless, the term “spin 0 field” is traditionally used for $\Theta(\theta,\varphi)$ and similar random fields, while the term “spin 2 field” is used for $\mathcal{Q}(\theta,\varphi)$ and similar random fields.

The transformation $\mathbf{n}\mapsto -\mathbf{n}$ is called the parity transformation.parity transformation Under parity, the spherical harmonics $Y_{\ell,m}$ transform according to <cit.>: $Y_{\ell,m}(-\mathbf{n})=(-1)^{\ell}Y_{\ell,m}(\mathbf{n})$. The naming convention for harmonic types is called the even-odd convention,even-odd convention see <cit.>. We say that the spherical harmonics $Y_{\ell,m}$ have “electric-type” parity.parity!electric-type It follows that the random fields $\Theta(\theta,\varphi)$ and $\mathcal{V}(\theta,\varphi)$ have electric type. Using Equations (<ref>) and (<ref>), we see that $a^{\mathcal{E}}_{\ell m}\mapsto(-1)^{\ell}a^{\mathcal{E}}_{\ell\,-m}$ (electric type), but $a^{\mathcal{B}}_{\ell m}\mapsto(-1)^{\ell+1}a^{\mathcal{B}}_{\ell\,-m}$. This is called “magnetic-type” parity.parity!magnetic-type

In Equation (<ref>), consider the matrix of cross-correlations $C^{ij}_{\ell}=\mathsf{E}[a^{(i)*}_{\ell m}a^{(j)}_{\ell m}]$ with $i$, $j\in\{\Theta,\mathcal{E},\mathcal{B},\mathcal{V}\}$. The current standard cosmological model assumes that the random field that generates the initial fluctuations, is invariant under parity. In this case, the cross-correlations between the components with different parity vanish. That is, $C^{\Theta\mathcal{B}}_{\ell}=C^{\mathcal{E}\mathcal{B}}_{\ell}=C^{\mathcal{B}\mathcal{V}}_{\ell}=0$.

Again, according to the standard cosmological model, in the early universe, photons and baryons are coupled by the physical process called Thomson scattering.Thomson scattering This process does introduce linear polarisation, but no circular one. Only a small amount of circular polarisation may be introduced by other physical processes, see [47]. By this reason, we may put $a^{\mathcal{V}}_{\ell m}=0$. The expansion takes the form

\[
\begin{aligned}
&=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\Theta}_{\ell m}Y_{\ell,m}(\theta,\varphi)\mathbf{e}_{\Theta}
+\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{E}}_{\ell m}Y_{\ell,m}(\theta,\varphi)\mathbf{e}_{\mathcal{E}}\\
&\quad+\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{B}}_{\ell m}Y_{\ell,m}(\theta,\varphi)\mathbf{e}_{\mathcal{B}}
\end{aligned}
\]

\[
\mathsf{E}[(a^{\Theta*}_{\ell m},a^{\mathcal{E}*}_{\ell m},a^{\mathcal{B}*}_{\ell m})^{\top}(a^{\Theta}_{\ell' m'},a^{\mathcal{E}}_{\ell'm'},a^{\mathcal{B}}_{\ell'm'})]
\begin{smallmatrix}
C^{\Theta\Theta}_{\ell} & C^{\Theta\mathcal{E}}_{\ell} & 0 \\
C^{\mathcal{E}\Theta}_{\ell} & C^{\mathcal{E}\mathcal{E}}_{\ell} & 0 \\
0 & 0 & C^{\mathcal{B}\mathcal{B}}_{\ell}
\end{smallmatrix}
\right).
\]

Finally, the standard cosmological model predicts arising primordial $\mathcal{B}$-polarisation due to a background of gravitational waves created either just after the Big Bang during a physical process called inflationinflation or between the end of inflation and Big Bang nucleosynthesis. Currently, this mode is not detected, the best current upper estimates for its amplitude may be found in [17].

Recall that the spinor representation $V_1\oplus tV_1$ of the group $H=\mathrm{U}(1)$ has a real structure $j(z_1,z_2)=(z^*_2,z^*_1)$, see Example <ref>. Consider the corresponding real spinor representation $rV_1$: $\mathrm{e}^{\mathrm{i}\theta}\mapsto\left(
\begin{smallmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{smallmatrix}
\right)$. Its tensor square has the form $rV_1\otimes rV_1=2U_0\oplus rV_2$, which easily follows from the corresponding relation for characters: $4\cos^2\theta=2+2\cos(2\theta)$.

Introduce the matrices $\sigma^{(i)}_{\mathbb{R}}$ by $\sigma^{(i)}_{\mathbb{R}}=\sigma^{(i)}$ for $i=0$, $1$, $3$, and $\sigma^{(2)}_{\mathbb{R}}=-\mathrm{i}\sigma^{(2)}$. The first (resp. second) copy of $U_0$ acts in the one-dimensional real space generated by  $\sigma^{(i)}_{\mathbb{R}}$ (resp.  $\sigma^{(2)}_{\mathbb{R}}$). The representation $rV_2$ acts in the two-dimensional space generated by  $\sigma^{(1)}_{\mathbb{R}}$ and  $\sigma^{(3)}_{\mathbb{R}}$.

On the other hand, the real irreducible representations of the group $G=\mathrm{SU}(2)$ are $\{\,U_{\ell}\colon\ell=0,1,\dots\,\}$representation!irreducible real of $\mathrm{SU}(2)$!$U_{\ell}$ and $\{\,rc'W_{\ell}\colon\ell=\frac{1}{2},\frac{3}{2},\dots\,\}$.representation!irreducible real of $\mathrm{SU}(2)$!$rc'W_{\ell}$ The character of $U_{\ell}$ is equal to that of $cU_{\ell}$, while the character of $rV_m$, $m>0$, is

\[
\chi_{rV_m}\left(\left(
\begin{smallmatrix}
\alpha & 0 \\
0 & \alpha^*
\end{smallmatrix}
\right)\right)=\chi_{crV_m}\left(\left(
\begin{smallmatrix}
\alpha & 0 \\
0 & \alpha^*
\end{smallmatrix}
\right)\right)=\chi_{V_m\oplus tV_m}\left(\left(
\begin{smallmatrix}
\alpha & 0 \\
0 & \alpha^*
\end{smallmatrix}
\right)\right)=\alpha^{2m}+\alpha^{-2m},
\]

and $\chi_{U_0}\left(\left(
\begin{smallmatrix}
\alpha & 0 \\
0 & \alpha^*
\end{smallmatrix}
\right)\right)=1$, where we used the relation $cr=1+t$ (<cit.>). It follows that

\[
\res^G_HU_{\ell}=U_0\oplus rV_1\oplus\cdots\oplus rV_{\ell}.
\]

For the Hilbert space $L^2(E_0)$ of square-integrable section of the line bundle, where the real representation of $\mathrm{SU}(2)$ induced by the representation $U_0$ of the subgroup $\mathrm{U}(1)$ acts, the Frobenius reciprocity gives that $L^2(E_0)$ contains

\[
\frac{\dim_{\mathbb{R}}\Hom_{\mathbb{R}H}(\res^G_HU_{\ell},U_0)}
\]

copy of the representation $U_{\ell}$ for $\ell\geq 0$. The Hilbert space $L^2(E_2)$ contains

\[
\frac{\dim_{\mathbb{R}}\Hom_{\mathbb{R}H}(\res^G_HU_{\ell},rV_2)}
\]

copies of the representation $U_{\ell}$ for $\ell\geq 2$.

Similarly, the character of $rc'W_{\ell}$ is $\chi_{rc'W_{\ell}}=\chi_{crc'W_{\ell}}=2\chi_{c'W_{\ell}}$, and we obtain

\[
\res^G_Hrc'W_{\ell}=2rV_1\oplus 2rV_3\oplus\cdots\oplus 2rV_{2\ell}.
\]

Neither $L^2(E_0)$ nor $L^2(E_2)$ contain copies of the representations $rc'W_{\ell}$.

Instead of constructing the bases of the above real Hilbert spaces from scratch, we use the existing bases of the corresponding complex spaces. Indeed, the basis in $cL^2(E_0)$ is given by the complex-valued spherical harmonics $Y_{\ell,m}(\zeta)$. The complex conjugation is a real structure in $cL^2(E_0)$. Therefore, the real-valued spherical harmonicsspherical harmonics!real-valued

\[
\begin{cases}
\frac{1}{\sqrt{2}}[Y_{\ell,m}(\zeta)+(-1)^mY_{\ell,-m}(\zeta)], & \mbox{if } m>0, \\
Y_{\ell,0}(\zeta), & \mbox{if } m=0, \\
\frac{1}{\sqrt{2}\mathrm{i}}[Y_{\ell,m}(\zeta)-(-1)^mY_{\ell,-m}(\zeta)], & \mbox{if } m<0
\end{cases}
\]

form the basis in $L^2(E_0)$. Likewise, the basis in $cL^2(E_2)\oplus cL^2(E_{-2})$ is given by the $\mathbb{C}^2$-valued spin 2 real-valued spherical harmonicsspherical harmonics!spin 2!real-valued

\[
\left(
\begin{smallmatrix}
{}_2Y_{\ell,m}(\zeta) \\
\end{smallmatrix}
\right),\quad\left(
\begin{smallmatrix}
0 \\
\end{smallmatrix}
\right),\qquad \ell=2, 3,\dots,\quad -\ell\leq m\leq\ell.
\]

The real structure of Example <ref> maps $\left(
\begin{smallmatrix}
{}_2Y_{\ell,m}(\zeta) \\
\end{smallmatrix}
\right)$ to $\left(
\begin{smallmatrix}
0 \\
\end{smallmatrix}
\right)$ and $\left(
\begin{smallmatrix}
0 \\
\end{smallmatrix}
\right)$ to $\left(
\begin{smallmatrix}
(-1)^m{}_2Y_{\ell,m}(\zeta) \\
\end{smallmatrix}
\right)$. Therefore, the $\mathbb{R}^2$-valued spin-weighted spherical harmonics take the form

\[
\begin{aligned}
\begin{smallmatrix}
{}_2Y_{\ell,m}(\zeta)+(-1)^m{}_{-2}Y_{\ell,-m}(\zeta) \\
\end{smallmatrix}
\right),\\
\begin{smallmatrix}
\mathrm{i}\,{}_2Y_{\ell,m}(\zeta)-\mathrm{i}(-1)^m\,{}_{-2}Y_{\ell,-m}(\zeta) \\
\end{smallmatrix}
\right).
\end{aligned}
\]

It is easy to check that the real-valued spherical harmonics have “electric-type” parity. We obtain

\[
\begin{aligned}
\Theta(\zeta)&=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\Theta}_{\ell m}Y^m_{\ell}(\zeta),\\
\left(\begin{smallmatrix}
\mathcal{Q} \\
\mathcal{U}
\end{smallmatrix}
\right)(\zeta)&=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}\left({}_2a^{\mathcal{Q},\mathcal{U}}_{\ell m}{}_2\mathbf{Y}^m_{\ell}(\zeta)+{}_{-2}a^{\mathcal{Q},\mathcal{U}}_{\ell m}{}_{-2}\mathbf{Y}^m_{\ell}(\zeta)\right),\\
\mathcal{V}(\zeta)&=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}a^{\mathcal{V}}_{\ell m}Y^m_{\ell}(\zeta),
\end{aligned}
\]

where the coefficients are centred random variables with cross-correlations given by

\[
\mathsf{E}[(a^{\Theta}_{\ell m},{}_2a^{\mathcal{Q},\mathcal{U}}_{\ell m},{}_{-2}a^{\mathcal{Q},\mathcal{U}}_{\ell m},a^{\mathcal{V}}_{\ell m})(a^{\Theta}_{\ell'm'},{}_2a^{\mathcal{Q},\mathcal{U}}_{\ell' m'},{}_{-2}a^{\mathcal{Q},\mathcal{U}}_{\ell'm'},a^{\mathcal{V}}_{\ell' m'})^{\top}]=\delta_{\ell\ell'}\delta_{mm'}C_{\ell},
\]

and the symmetric positive-definite matrices $C_{\ell}$ satisfy

\[
\sum_{\ell=2}^{\infty}(2\ell+1)\tr C_{\ell}<\infty.
\]

Observe that the spinor representation $V_1\oplus tV_1$ of the group $H=\mathrm{U}(1)$ has a quaternionic structure $j(z_1,z_2)=(-z^*_2,z^*_1)$. The corresponding quaternionic irreducible representation $qV_1$ of the group $\mathrm{U}(1)$ acts in $\mathbb{H}^1$ by $\mathrm{e}^{\mathrm{i}\theta}\mapsto\mathrm{e}^{\mathrm{i}\theta}$. By <cit.>, its character is the character of $c'qV_1=V_1\oplus tV_1$. Again by [1], the tensor square of $c'qV_1$ is a real representation, and its character is the same as that of the representation $2U_0\oplus rV_2$ of Example <ref>. No new expansions appears here.

§ CONCLUSIONS

The theory of random sections of various fibre bundles is now a mature part of Probability with various interesting links to Special functions, Invariant Theory, Group Representations, Convex Analysis, etc. The case of trivial tensor bundles over Euclidean spaces found interesting and important applications in Continuum Physics, while that of nontrivial spinor bundles over the two-dimensional sphere $S^2$ in Cosmology.

This survey reflects the scientific interests of the authors and is concentrated on spectral expansions of random functions in question and their applications to Physics. On the one hand, many research questions are missing; for example, tensor-valued random fields as solutions of boundary value problems for stochastic partial differential equations similar to Equation (<ref>), statistics of random sections of spinor bundles, properties of sample paths of random sections, and many others. On the other hand, we formulated several unsolved research questions above.

The authors hope that this survey can be useful both for mathematicians who are working in different physical applications and for applied specialists who need to use advanced mathematical tools.

§ ACKNOWLEDGEMENTS

We are grateful to Victor Abramov for useful discussion on principal bundles.

§ THE MATHEMATICAL LANGUAGE OF THE THEORY

We give a short description of mathematical terms used in the paper. For a more detailed account, see <cit.>, <cit.>, and the literature cited there.

§.§ Tensors

The symbol $\mathbb{R}$ (resp. $\mathbb{C}$, resp. $\mathbb{H}$) denotes the field of real numbers (resp. the field of complex numbers, resp. the skew field of quaternions). The symbol $U$ (resp. $V$, resp. $W$) denotes a finite-dimensional linear space over $\mathbb{R}$ (resp. over $\mathbb{C}$, resp. a right finite-dimensional linear space over $\mathbb{H}$). The round brackets $(\cdot,\cdot)$ denote an inner product in either $U$, $V$, or $W$. It is $\mathbb{K}$-linear in the second argument, has the property $\langle \mathbf{x}_1,\mathbf{x}_2\rangle=\langle \mathbf{x}_2,\mathbf{x}_1\rangle^*$ for all $\mathbf{x}_1$, $\mathbf{x}_2\in U$ (resp. in $V$, resp. in $W$), and is positive-definite: $\langle\mathbf{x},\mathbf{x}\rangle\geq 0$ with equality only if $\mathbf{x}=\mathbf{0}$, where a $\mathbb{R}$-linear map $\mathbb{K}\to\mathbb{K}$, $x\mapsto x^*$ is given by $a^*=a$ for $a\in\mathbb{R}$, $(a+b\mathrm{i})^*=a-b\mathrm{i}$ for $(a+b\mathrm{i})\in\mathbb{C}$, and

\[
\]

for $(a+b\mathrm{i}+c\mathrm{j}+d\mathrm{k})\in\mathbb{H}$.
By choosing a basis $\{\,\mathbf{e}^i\colon 1\leq i\leq d\,\}$ which is orthonormal with respect to the introduced inner product, the linear space $U$ (resp. $V$, resp. $W$) can be identified with the coordinate space $\mathbb{R}^d$ (resp. $\mathbb{C}^d$, resp. $\mathbb{H}^d$).

The symbol $U^*$ denotes the set of all linear forms on $U$. A real rank $r$ tensortensor is an element of a tensor product of $r\geq 0$ finite-dimensional real linear spaces. By definition, the tensor producttensor product!of real linear spaces of an empty family of real linear spaces ($r=0$) is the coordinate space $\mathbb{R}^1$. The tensor product of the family $\{U_1,\dots,U_r\}$ with $r\geq 1$ is the set of all $r$-linear forms defined on the Cartesian product $U^*_1\times\cdots\times U^*_r$. The inner product on each space $U_i$, $1\leq i\leq r$, enables to identify $U_i$ with $U^*_i$ with the help of the map $U_i\to U^*_i$, $\mathbf{x}\mapsto(\mathbf{x},\cdot)$ and do not distinguish between covariant and contravariant tensors. The tensor producttensor product!of vectors of the vectors $\mathbf{x}_1\in U_1$, …, $\mathbf{x}_r\in U_r$ is the $r$-linear form

\[
\mathbf{x}_1\otimes\cdots\otimes\mathbf{x}_r(\mathbf{y}_1,\dots,\mathbf{y}_r)
\]

Similarly, the inner product on each complex linear space $V_i$, $1\leq i\leq r$, enables to identify $V_i$ with $V^*_i$. Moreover, let the space $tV$ has the same underlying set as $V$, but let $z\in\mathbb{C}$ acts on $tV$ as $z^*$ used to act on $V$. We can identify $V^*$ and $tV$ with the help of the map $tV\to V^*$, $\mathbf{x}\mapsto(\mathbf{x},\cdot)$ and do not distinguish between four types of complex tensors, described, e.g., in <cit.>.

The inner product on each quaternionic linear space $W_i$, $1\leq i\leq r$, enables to identify $W_i$ with $W^*_i$. However, assume that $W_1$ and $W_2$ are two right quaternionic linear spaces, and $q_1$, $q_2$ are two quaternions. An $\mathbb{H}$-bilinear form $\mu\colon W_1\times W_2\to\mathbb{H}^1$ must satisfy $\mu(\mathbf{x}_1q_1,\mathbf{x}_2q_2)=\mu(\mathbf{x}_1,\mathbf{x}_2q_2)q_1=\mu(\mathbf{x}_1,\mathbf{x}_2)q_1q_2$ and $\mu(\mathbf{x}_1q_1,\mathbf{x}_2q_2)=\mu(\mathbf{x}_1q_1,\mathbf{x}_2)q_2=\mu(\mathbf{x}_1,\mathbf{x}_2)q_2q_1$. In general, $q_1q_2\neq q_2q_1$, and this does not work. On the other hand, a bilinear form is well-defined if $W_2$ is a left quaternionic space. The set of such forms is not a right quaternionic linear space. Indeed, the form $(\mathbf{x}_1,\mathbf{x}_2)\mapsto\mu(\mathbf{x}_1,\mathbf{x}_2)q$ is not bilinear: $\mu(\mathbf{x}_1q_1,\mathbf{x}_2)q$ must be equal to $\mu(\mathbf{x}_1,\mathbf{x}_2)qq_1$, but it is equal to $\mu(\mathbf{x}_1,\mathbf{x}_2)q_1q$. By similar reasons, it is not a left quaternionic linear space. However, the set $W_1\otimes W_2$ is a real linear space.

By this reasons, we put $\mathbb{R}'=\mathbb{H}'=\mathbb{R}$ and $\mathbb{C}'=\mathbb{C}$. In this notation, the tensor product of two $\mathbb{K}$-linear spaces is a $\mathbb{K}'$-linear space.

[The Levi–Civita permutation tensor in three dimensions]
Let $U_1$, $U_2$, and $U_3$ be three copies of a real $3$-dimensional linear space with inner product. define the Levi–Civita permutation tensortensor!Levi–Civita in a coordinate-free form by

\[
\varepsilon(\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3)=(\mathbf{x}_1,\mathbf{x}_2\times\mathbf{x}_3),
\]

that is, the scalar triple product of vectors in a right-handed coordinate system.

In coordinates, let $\{\mathbf{e}^i\colon 1\leq i\leq 3\}$ be the standard basis of the space $\mathbb{R}^3$. Denote

\[
\varepsilon_{ijk}=\varepsilon(\mathbf{e}^i,\mathbf{e}^j,\mathbf{e}^k).
\]

It is easy to check that the introduced symbol is totally antisymmetric. That is, when any two indices are interchanged, the symbol is negated, if any two indices are equal, the symbol is zero, and when all indices are unequal, the symbol $\varepsilon_{ijk}$ is equal to the signature of the permutation $(i,j,k)$.

§.§ Group representations

A left actiongroup action!left of a group $G$ with identity element $e$ on a non-empty set $L$ is such a map $G\times L \to L$, $(g,x)\mapsto g\cdot x$, that $e\cdot x=x$ and $g\cdot(h\cdot x)=(gh)\cdot x$ for all $g$, $h\in G$ and for all $x\in L$. Similarly, a right actiongroup action!right of a group $G$ with identity element $e$ on a non-empty set $L$ is such a map $L\times G \to L$, $(x,g)\mapsto x\cdot g$, that $x\cdot e=x$ and $(x\cdot h)\cdot g=x\cdot(hg)$ for all $g$, $h\in G$ and for all $x\in L$.

As an example of a left action, think about the matrix-vector multiplication. A more interesting example is as follows. Consider a real finite-dimensional space $U$ as a group with respect to the addition. Assume that the group $U$ acts on a set $E$ transitively,group action!transitive that is, for each pair $A$, $B\in E$ there exists a $x\in U$ such that $x\cdot A=B$, and faithfully,group action!faithful that is, for any $x\in U\setminus\{0\}$ there is an $A\in E$ with $x\cdot A\neq A$. We change notation and denote the result of the action of a vector $x\in U$ on a point $A\in E$ by $A+x$. Denote by $B-A$ such a vector in $U$ that $A+(B-A)=B$. The inner product on $V$ enables to define the distance in $E$:

\[
\rho(A,B)=\sqrt{(A-B,A-B)}.
\]

An affine space with this distance is called a Euclidean space.Euclidean space If the dimension of $U$ is equal to $d$, then we denote the pair $(E,\rho)$ by $E^d$. According to [2]:

Formerly, the universe was provided not with an affine, but with a linear structure (the geocentric system of the universe).

We omit the dot in the notation $g\cdot x$ in the case when $G$ is a topological group, $L$ is a Hilbert space over $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, $g\cdot x$ is a $\mathbb{K}$-linear function of $x$ and a continuous function of $g$ and $x$ (in the strong operator topology, when $X$ is infinite-dimensional). Such an action is called a representationrepresentation of $G$. Recall that a topological group is a set $G$ which is a group and a topological space such that the map

\begin{equation}\label{eq:1}
G\times G\to G,\qquad (g,h)\mapsto g^{-1}h
\end{equation}

is continuous.

The translationtranslation $\theta^L(g)\colon x\to gx$ is a bounded linear operator in $L$. For two Hilbert spaces $L_1$ and $L_2$ over the same (skew) field $\mathbb{K}$, a bounded linear operator $f\colon X_1\to X_2$ is called intertwiningintertwining operator if $f(gx)=g(fx)$, $g\in G$, $x\in L_1$. Two representations are equivalent if the $\mathbb{K}'$-linear space $\Hom_G(L_1,L_2)$ of intertwining operators contains an invertible operator with bounded inverse.

If $G$ is a compact topological group, that we can and will give $L$ an inner product which is invariant under $G$. We then speak of an orthogonalrepresentation!orthogonal (resp. unitary,representation!unitary resp. symplectic)representation!symplectic representation in the case of $\mathbb{K}=\mathbb{R}$ (resp. $\mathbb{K}=\mathbb{C}$, resp. $\mathbb{K}=\mathbb{H}$). In what follows the symbol $G$ always denotes a compact group.

A non-zero linear space $L$ is reduciblerepresentation!reducible if some proper non-zero closed subspace $M$ of $X$ is invariant: $gy\in M$ for all $y\in M$, otherwise $L$ is irreducible.representation!irreducible For a compact group, every representation $L$ is a Hilbert direct sum of irreducible components. Moreover, let $L_i$ runs over the inequivalent irreducible representations, as $i$ runs over some set $I$. For a finite-dimensional space $L$, there exists a unique set $\{\,m_i\colon i\in I\,\}$ of nonnegative integers such that all of them except finitely many are zeroes and the space $L$ is uniquely decomposed into the Hilbert direct sum of isotypical subspacesisotypic subspace in which the direct sum of $m_i>0$ copies of $L_i$ acts.

Let $U_1$ and $U_2$ be two real finite-dimensional representations of a topological group $G$ with the translations $\theta^{U_1}$ and $\theta^{U_2}$. A map $f\colon U_2\to U_1$ is called form-invariantform-invariant map for the pair $(U_1,U_2)$ if

\[
f(\theta^{U_2}(g)x)=\theta^{U_1}(g)f(x),\qquad g\in G,\quad x\in U_2.
\]

If $U_1=\mathbb{R}^1$ and $\theta^{U_1}(g)=1$, then the function $f\colon X_2\to\mathbb{R}^1$ is called just invariant of the representation $U_2$, hence the name Invariant Theory.

We formulate an important result of Invariant Theory. Let $U$, $U_1$, …, $U_N$ be finitely many finite-dimensional orthogonal representations of a closed subgroup $G$ of the group $\mathrm{O}(3)$ of $3\times 3$ orthogonal matrices. It turns out that there exists finitely many, say $K$, polynomials $\{\,I_k(x_1,\dots,x_N)\colon 1\leq k\leq K\,\}$ with $x_n\in U_n$ for $1\leq n\leq N$ that constitute an integrity basisintegrity basis for polynomial invariants of the representation $U_1\oplus\cdots\oplus U_N$. That is, every polynomial invariant of the above representation is a polynomial in $I_1$, …, $I_K$. Similarly, there exists an integrity basis $\{\,T_l\colon 1\leq l\leq L\,\}$ for form-invariant polynomials of the pair $(U,U_1\oplus\cdots\oplus U_N)$. Following [134], call $T_l$ the basis form-invariant tensors.tensor!basis form-invariant

A function $T\colon U_1\oplus\cdots\oplus U_N\to U$ is a measurable form-invariant map of the pair $(U,U_1\oplus\cdots\oplus U_N)$ if and only if it has the form

\[
\]

where $\varphi_l$ are real-valued measurable functions.

§.§ The Adams construction

In this subsection, we introduce a convenient notation system for irreducible representations of compact topological groups due to Adams [1].

Let $V$ be a finite-dimensional complex representation of a compact topological group $G$. Let the representation $tV$ has the same underlying set as $V$ and the same action of $G$, but let $z\in\mathbb{C}$ acts on $tV$ as $z^*$ used to act on $V$. For example, let $G=\mathrm{U}(1)$, the set of all complex numbers $z$ with $|z|=1$, and let an element $g=\mathrm{e}^{\mathrm{i}\varphi}\in\mathrm{U}(1)$ acts on $V_n=\mathbb{C}^1$representation!irreducible complex of $\mathrm{U}(1)$!$V_n$ by $g\cdot z=\mathrm{e}^{\mathrm{i}n\varphi}z$ for $n\geq 1$ and $z\in\mathbb{C}^1$. We have $tV_n=\mathbb{C}^1$representation!irreducible complex of $\mathrm{U}(1)$!$tV_n$ with the action $g\cdot z=\mathrm{e}^{-\mathrm{i}n\varphi}z$, and $tV_n$ is not equivalent to $V_n$. An irreducible complex representation with this property is called a complex representation of complex type.representation!complex!of complex type

Let the representation $rV_n$representation!irreducible real of $\mathrm{U}(1)$!$rV_n$ has the same underlying set as $V_n$ and the same action of $G$, but regard it as a $\mathbb{R}$-linear space. The matrix of the representation $rV_n$ is $\left(
\begin{smallmatrix}
\cos(n\varphi) & -\sin(n\varphi) \\
\sin(n\varphi) & \cos(n\varphi)
\end{smallmatrix}
\right)$. An irreducible real representation of the form $rV$, where $V$ is a complex representation of complex type, is called a real representation of complex type.representation!real!of complex type The representation $rtV$ is equivalent to $rV$ by <cit.>. In particular, when $n=1$, we conclude, that the group $\mathrm{U}(1)$ is isomorphic to the group $\mathrm{SO}(2)$ of orthogonal $2\times 2$ matrices with unit determinant.

Define $qV_n=V_n\otimes_{\mathbb{C}}\mathbb{H}$ and regard it as a right quaternionic linear space with the vector-scalar multiplication $(\mathbf{v}\otimes_{\mathbb{C}}q')q=\mathbf{v}\otimes_{\mathbb{C}}q'q$ and a quaternionic representation of $G$ with the action $g\cdot(\mathbf{v}\otimes_{\mathbb{C}}q)=(g\cdot\mathbf{v})\otimes_{\mathbb{C}}q$. The quaternionic representation $qV_n$representation!irreducible quaternionic of $\mathrm{U}(1)$!$qV_n$ is irreducible and is called a quaternionic representation of complex type.representation!quaternionic!of complex type

Assume that an irreducible complex representation $V$ is not of complex type. This may happen by one of two mutually exclusive reasons.

* There exists a conjugate-linear map $j\colon V\to V$ that commutes with the representation and satisfies the condition $j^2=\id$.

* There exists a conjugate-linear map $j\colon V\to V$ that commutes with the representation and satisfies the condition $j^2=-\id$.

As an example, consider the trivial representation $V_0=\mathbb{C}^1$ of the group $\mathrm{U}(1)$ with the action $g\cdot z=z$. This representation clearly commutes with the map $j(z)=z^*$. For such a representation $V$, there always exists a real irreducible representation $U$ such that $cU=V$. Here $cU=U\otimes_{\mathbb{R}}\mathbb{C}$ with the vector-scalar multiplication $(\mathbf{u}\otimes_{\mathbb{R}}z')z=\mathbf{u}\otimes_{\mathbb{R}}z'z$ and the action of $G$ by $g\cdot(\mathbf{u}\otimes_{\mathbb{R}}z)=(g\cdot\mathbf{u})\otimes_{\mathbb{R}}z$. In this case the representation $V$ (resp. $U$) is called a complexrepresentation!complex!of real type (resp. real)representation!real!of real type representation of real type, while the irreducible representation $W=qcU$ is called a quaternionic representation of real type.representation!quaternionic!of real type

As an another example, consider the group $G=\mathrm{SU}(2)$ of unitary $2\times 2$ matrices with unit determinant and elements $g=\left(
\begin{smallmatrix}
\alpha & \beta \\
-\beta^* & \alpha^*
\end{smallmatrix}
\right)$. Put $V_{1/2}=\mathbb{C}^2$, where the group $G$ acts by matrix-vector multiplication, and $j(z_1,z_2)=(z^*_2,-z^*_1)$. Then we have

\[
\begin{aligned}
\begin{pmatrix}
\alpha & \beta \\
-\beta^* & \alpha^*
\end{pmatrix}
\begin{pmatrix}
z^*_2 \\
\end{pmatrix}
\begin{pmatrix}
\alpha z^*_2-\beta z^*_1 \\
\end{pmatrix}
\begin{pmatrix}
\alpha & \beta \\
-\beta^* & \alpha^*
\end{pmatrix}
\begin{pmatrix}
z_1 \\
\end{pmatrix}
\begin{pmatrix}
\alpha z_1+\beta z_2 \\
\end{pmatrix}
\begin{pmatrix}
-\beta z^*_1+\alpha z^*_2 \\
\end{pmatrix}
\end{aligned}
\]

see also <cit.>. In this case, there exists an irreducible quaternionic representation, call it $W_{1/2}$, such that $c'W_{1/2}=V_{1/2}$, where the representation $c'W_{1/2}$ has the same underlying set as $W_{1/2}$ and the same action of $G$, but regard it as a $\mathbb{C}$-linear space. Any irreducible complex representation $V$ with the above property is called a complex representation of quaternionic type,representation!complex!of quaternionic type the corresponding representation $W$ a quaternionic representation of quaternionic type,representation!quaternionic!of quaternionic type and the real irreducible representation $U=rV=rc'W$ a real representation of quaternionic type.representation!real!of quaternionic type It is easy to see that $W_{1/2}=\mathbb{H}^1$, where a matrix $g$ acts by $g\cdot q=(\alpha+\beta\mathrm{j})q$. The representation $W_{1/2}$ establishes an isomorphism between the group $\mathrm{SU}(2)$ and the group $\mathrm{Sp}(1)$ of all $q\in\mathbb{H}$ with $qq^*=1$. The irreducible real representation $U_{1/2}=rV_{1/2}=rc'W_{1/2}$ acts in $\mathbb{R}^4$ by

\[
\begin{pmatrix}
\RE\alpha & -\IM\alpha & \RE\beta & -\IM\beta \\
\IM\alpha & \RE\alpha & \IM\beta & \RE\beta \\
-\RE\beta & -\IM\beta & \RE\alpha & \IM\alpha \\
\IM\beta & -\RE\beta & -\IM\alpha & \RE\alpha
\end{pmatrix}
\mathbf{u},\qquad\mathbf{u}\in\mathbb{R}^4,
\]

see <cit.>.

It turns out that for any compact topological group $G$ and for any skew field $\mathbb{K}$, the partition of the set $\hat{G}_{\mathbb{K}}$ of equivalence classes of irreducible representations of $G$ over $\mathbb{K}$ into representations of real, complex, and quaternionic types is exhaustive, see <cit.> and <cit.>.

§.§ Manifolds

A chartchart in a set $M$ is a pair $(\mathcal{U},\varphi)$, where $\mathcal{U}$ is a subset of $M$, and $\varphi$ is a one-to-one map from $\mathcal{U}$ to an open subset $\varphi(\mathcal{U})$ of the coordinate space $\mathbb{R}^d$ or $\mathbb{C}^d$.

The overlap mapoverlap map is the map

\[
\varphi_{\beta}\circ\varphi^{-1}_{\alpha}\colon\varphi_{\alpha}
(\mathcal{U}_{\alpha}\cap \mathcal{U}_{\beta})
\]

provided that the intersection $\mathcal{U}_{\alpha}\cap\mathcal{U}_{\beta}$ of the domains of the charts $(\mathcal{U}_{\alpha},\varphi_{\alpha})$ and $(\mathcal{U}_{\beta},\varphi_{\beta})$ is not empty.

A family $\mathcal{A}=\{(\mathcal{U}_{\alpha},\varphi_{\alpha})\colon\alpha\in A\}$ of charts on $M$ is called an differentiableatlas!differentiable (resp. real-analytic,atlas!real-analytic resp. holomorphic)atlas!holomorphic atlas if the sets $\mathcal{U}_{\alpha}$ cover all of $M$, and for any two charts $(\mathcal{U}_{\alpha},\varphi_{\alpha})$ and $(\mathcal{U}_{\beta},\varphi_{\beta})$ with $\mathcal{U}_{\alpha}\cap\mathcal{U}_{\beta}\neq\varnothing$, both the domain and range of the corresponding overlap map are open subsets of $\mathbb{R}^d$ (resp. $\mathbb{R}^d$, resp. $\mathbb{C}^d$), and the overlap map is infinitely differentiable (resp. real-analytic, resp. holomorphic).

Two atlases are equivalent if their union is also an atlas. A differentiable (resp. real-analytic, resp. holomorphic) structurestructure!differentiablestructure!real-analyticstructure!holomorphic on $M$ is an equivalence class of differentiable (resp. real-analytic, resp. holomorphic) atlases. A set $M$ together with a differentiable (resp. real-analytic, resp. holomorphic) structure is called a differentiable (resp. real-analytic, resp. holomorphic) manifold.manifold!differentiablemanifold!real-analyticmanifold!holomorphic  A chart is called admissiblechart!admissible if it belongs to an atlas from the structure. The domains of the admissible charts form a base for a topology on $M$. To avoid pathologies, we assume that this topology is Hausdorff and second-countable.

Let $M_1$ and $M_2$ be two manifolds of the same type, and let $\Phi\colon M_1\to M_2$ be a continuous map. Let $x\in M$ and let $(\mathcal{V},\psi)$ be a chart for $M_2$ with $\Phi(x)\in\mathcal{V}$. It is possible to choose a chart $(\mathcal{U},\varphi)$ for $M_1$ with $\Phi(\mathcal{U})\subset\mathcal{V}$. The map

\[
\Phi_{\varphi\psi}\colon\varphi(\mathcal{U})\to\psi(\mathcal{V}),\qquad x\mapsto\psi\circ\Phi\circ\varphi^{-1}(x)
\]

is called the the local representativelocal representative of $\Phi$ with respect to the charts $(\mathcal{U},\varphi)$ and $(\mathcal{V},\psi)$. A map $\Phi$ is called differentiablemap!differentiable (resp. real-analytic,map!real-analytic resp. holomorphic)map!holomorphic if for every $x\in M$ and every chart $(\mathcal{V},\psi)$ for $M_2$ with $\Phi(x)\in\mathcal{V}$ there exists a chart $(\mathcal{U},\varphi)$ for $M_1$ such that $\Phi(\mathcal{U})\subset\mathcal{V}$ and the local representative $\Phi(\mathcal{U})\subset\mathcal{V}$ is differentiable (resp. real-analytic, resp. holomorphic). If $\Phi$ is invertible and both $\Phi$ and $\Phi^{-1}$ are differentiable (resp. real-analytic, resp. holomorphic), then $\Phi$ is called a diffeomorphism.diffeomorphism

A group $G$ is called a Lie groupLie group if it has a structure of either real-analytic or holomorphic manifold, and the map $G\times G\to G$, $(g_1,g_2)\mapsto g^{-1}_1g_2$ is either real-analytic or holomorphic.

Let $S^d\subset\mathbb{R}^{d+1}$ be the unit sphere.sphere Define two chart domains $\mathcal{U}_{\pm}\subset S^d$ by
$\mathcal{U}_{\pm}=S^d\setminus\{(0,\dots,0,\pm 1)^{\top}\}$ and tho chart maps $\varphi_{\pm}\colon\mathcal{U}_{\pm}\to\mathbb{R}^d$ by

\[
\varphi_{\pm}(x_1,\dots,x_{d+1})=\frac{1}{1\mp x_{d+1}}(x_1,\dots,x_d)^{\top}.
\]

It is easy to check that $\varphi_{\pm}(\mathcal{U}_{\pm})=\mathbb{R}^d\setminus\{\mathbf{0}\}$ and

\[
\varphi^{-1}_+(\mathbf{x})
\]

The overlap map $(\varphi_-\circ\varphi^{-1}_+)(\mathbf{x})=\|\mathbf{x}\|^{-2}\mathbf{x}$ for $\mathbf{x}\in\mathbb{R}^d\setminus\{\mathbf{0}\}$ is real-analytic and makes $S^d$ into a real-analytic manifold.

In the case of $d=2$, we will also use an admissible chart

\[
\left(
\begin{smallmatrix}
\cos\varphi\sin\theta \\
\sin\varphi\sin\theta \\
\cos\theta
\end{smallmatrix}
\right)\mapsto(\theta,\varphi)\in(0,\pi)\times(0,2\pi)\subset\mathbb{R}^2,
\]

the angular part of the spherical coordinates. The domain of this chart is dense in $S^2$.

In the case of $d=3$, we will also use an admissible chart

\[
\left(
\begin{smallmatrix}
\cos(\theta/2)\cos((\psi+\varphi)/2) \\
-\cos(\theta/2)\sin((\psi+\varphi)/2) \\
-\sin(\theta/2)\cos((\psi-\varphi)/2) \\
\sin(\theta/2)\sin((\psi-\varphi)/2)
\end{smallmatrix}
\right)\mapsto(\theta,\varphi,\psi)\in(0,\pi)\times(0,2\pi)\times(0,4\pi)
\subset\mathbb{R}^3
\]

with dense domain.

When $d=2$, replace the maps $\varphi_{\pm}$ with the map $\tilde{\varphi}_{\pm}\colon\mathcal{U}_{\pm}\to\mathbb{C}^1$ given by

\[
\tilde{\varphi}_+(x_1,x_2,x_3)=\frac{x_1+\mathrm{i}x_2}{1-x_3},\qquad
\tilde{\varphi}_-(x_1,x_2,x_3)=\frac{x_1-\mathrm{i}x_2}{1+x_3}.
\]

For this choice, we obtain $(\tilde{\varphi}_-\circ\tilde{\varphi}^{-1}_+)(z)=z^{-1}$. By verifying the Cauchy–Riemann equations, we see that this map is a holomorphic on $\mathbb{C}^1\setminus\{0\}$. We made the real-analytic structure into the holomorphic one.

Let $\mathbb{C}P^1$ be the complex projective line,complex projective line the set of $1$-dimensional subspaces of $\mathbb{C}^2$. If $(z_0,z_1)\in\mathbb{C}^2\setminus\{\mathbf{0}\}$, denote by $\pcoor{z_0:z_1}$ the line through this point. For $j\in\{0,1\}$, define the charts $(\mathcal{U}_j,\varphi_j)$ as follows:

\[
\mathcal{U}_j=\{\,\pcoor{z_0:z_1}\in\mathbb{C}P^1\colon z_j\neq 0\,\},\qquad\varphi_0(\pcoor{z_0:z_1})=\frac{z_1}{z_0},\quad
\varphi_1(\pcoor{z_0:z_1})=\frac{z_0}{z_1}.
\]

The holomorphic overlap map is $(\varphi_0\circ\varphi^{-1}_1)(z)=z^{-1}$. The maps $\Phi_{\pm}\colon\mathcal{U}_{\pm}\to\mathbb{C}P^1$ given by

\[
\Phi_+(x,y,z)=\pcoor{\tilde{\varphi}_+(x_1,x_2,x_3):1},\qquad
\Phi_-(x,y,z)=\pcoor{1:\tilde{\varphi}_-(x_1,x_2,x_3)}
\]

coincide on the sphere without the poles. Together, they give a holomorphic diffeomorphism between $S^2$ and $\mathbb{C}P^1$.

For the group $\mathrm{U}(1)=\{\,z\in\mathbb{C}\colon|z|=1\,\}$, the map $\Phi\colon S^1\to\mathrm{U}(1)$, $(x_1,x_2)^{\top}\to x_1+\mathrm{i}x_2$ is one-to-one. For the group $\mathrm{SO}(2)$ of orthogonal $2\times 2$ matrices with unit determinant, the map $\Psi\colon\mathrm{SO}(2)\to S^1$, $\left(
\begin{smallmatrix}
a & b \\
-b & a \\
\end{smallmatrix}
\right)\mapsto(a,b)^{\top}$, is one-to-one. As $(\mathcal{U},\psi)$ runs over the set of all admissible atlases for the real-analytic structure on $S^1$, the charts $(\Phi(\mathcal{U}),\psi\circ\Phi^{-1})$ constitute the structure of a Lie group on $\mathrm{U}(1)$.Lie group!$\mathrm{U}(1)$ We will use an admissible chart $\mathrm{e}^{\mathrm{i}\varphi}\mapsto\varphi\in(0,2\pi)\subset\mathbb{R}^1$ with dense domain.

The charts $(\Psi^{-1}(\mathcal{U}),\psi\circ\Psi^)$ constitute the structure of a Lie group on $\mathrm{SO}(2)$.Lie group!$\mathrm{SO}(2)$ We will use an admissible chart $\left(
\begin{smallmatrix}
\cos\varphi & -\sin\varphi \\
\sin\varphi & \cos\varphi \\
\end{smallmatrix}
\right)\mapsto\varphi\in(0,2\pi)\subset\mathbb{R}^1$ with dense domain.

For the group $\mathrm{SU}(2)$ of unitary $2\times 2$ matrices with unit determinant, the map $\Phi\colon\mathrm{SU}(2)\to S^3$, $\left(
\begin{smallmatrix}
a & b \\
-b^* & a^* \\
\end{smallmatrix}
\right)\mapsto(\RE a,\IM a,-\RE b,\IM b)^{\top}$, is one-to-one. The charts $(\Phi(\mathcal{U}),\psi\circ\Phi^{-1})$ constitute the structure of a Lie group on $\mathrm{SU}(2)$.Lie group!$\mathrm{SU}(2)$ We will use an admissible chart $\left(
\begin{smallmatrix}
\cos(\beta/2)\mathrm{e}^{-\mathrm{i}(\gamma+\alpha)/2} & \sin(\beta/2)\mathrm{e}^{\mathrm{i}(\gamma-\alpha)/2} \\
-\sin(\beta/2)\mathrm{e}^{-\mathrm{i}(\gamma-\alpha)/2} & \cos(\beta/2)\mathrm{e}^{\mathrm{i}(\gamma+\alpha)/2}
\end{smallmatrix}
\right)\mapsto(\alpha,\beta,\gamma)\in(0,2\pi)\times(0,\pi)\times(0,4\pi)\subset\mathbb{R}^3$ with dense domain and call it the Euler angles.Euler angles

§.§ Bundles

A quadruple $\lambda=(P,H,B,\tilde{\pi})$ is called a principal bundlebundle!principal if and only if

* the total spacetotal space!of a principal bundle $P$ and the base spacebase space!of a principal bundle $B$ are either real-analytic of holomorphic manifolds, the fibrefibre!of a principal bundle $H$ is a Lie group;

* $H$ acts on $P$ from the right, and the map $P\times H\to P$, $(p,h)\mapsto p\cdot h$ is either real analytic or holomorphic;

* the bundle projectionbundle projection!of a principal bundle $\tilde{\pi}\colon P\to B$ is either a real-analytic or holomorphic map that satisfies to the following local triviality condition:local triviality condition for every $b\in B$, there exists a neighbourhood $\mathcal{U}$ of $b$, and either a real-analytic or holomorphic one-to-one map $f\colon\mathcal{U}\times H\to\tilde{\pi}^{-1}(\mathcal{U})$ such that for all $u\in\mathcal{U}$ and for all $h$, $h'\in H$ we have

\[
\tilde{\pi}(f(u,h))=u,\qquad f(u,hh')=f(u,h)\cdot h'.
\]

[A cylinder and a Möbius bundle]
Let $n$ be an integer. Put

\[
\]

The bundle projection $\tilde{\pi}(p_1,p_2,p_3,p_4)=(p_1,p_2)$ maps $P_n$ onto the circle $B=S^1\subset\mathbb{R}^2$. The group $H$ is $\mathbb{R}^1$ acting by scalar-vector multiplication. The principal bundle $\lambda_n=(P_n,H,B,\tilde{\pi})$ is a cylindercylinder for $n=0$ and a Möbius bundleMöbius bundle if $n=1$.

[Spin groups]
Let $n$ be a positive integer. There is a Lie group called the spin groupspin group and denoted by $\mathrm{Spin}(n)$, and a map $\tilde{\pi}_n\colon\mathrm{Spin}(n)\to\mathrm{SO}(n)$ such that the quadruple $\lambda_n=(\mathrm{Spin}(n),\mathrm{O}(1),\mathrm{SO}(n),\tilde{\pi}_n)$ is a principal bundle, see <cit.>. In particular, $\mathrm{Spin}(1)=\mathrm{O}(1)$ with $\tilde{\pi}_1(t)=t^2$, and $\mathrm{Spin}(2)=\mathrm{U}(1)$ with

\begin{equation}\label{eq:51}
\tilde{\pi}_2(\mathrm{e}^{\mathrm{i}\theta})=\left(
\begin{smallmatrix}
\cos(2\theta) & -\sin(2\theta) \\
\sin(2\theta) & \cos(2\theta)
\end{smallmatrix}
\right).
\end{equation}

Under the identification of $\mathrm{SO}(2)$ with $\mathrm{U}(1)$, the map $\tilde{\pi}_2$ is again a “square” map $z\mapsto z^2$.

The map $\tilde{\pi}_3\colon\mathrm{Spin}(3)=\mathrm{SU}(2)\to\mathrm{SO}(3)$ has the form

\[
\tilde{\pi}_3\left(
\begin{smallmatrix}
a & b \\
-b^* & a^*
\end{smallmatrix}
\right)=\left(
\begin{smallmatrix}
\RE(a^2-b^2) & \IM(a^2+b^2) & -2\RE(ab) \\
-\IM(a^2-b^2) & \RE(a^2+b^2) & 2\IM(ab)  \\
2\RE(ab^*) & 2\IM(ab^*) & |a|^2-|b|^2
\end{smallmatrix}
\right).
\]

[The Hopf bundle]
Let $P=\mathrm{SU}(2)$, $H$ be the subgroup of diagonal matrices in $P$, isomorphic to $\mathrm{U}(1)$, $B=\mathbb{C}P^1$, and let $\tilde{\pi}\colon P\to B$ acts by

\[
\tilde{\pi}\left(
\begin{smallmatrix}
a & b \\
-b^* & a^*
\end{smallmatrix}
\right)=\pcoor{a:b}.
\]

The quadruple $(\mathrm{SU}(2),\mathrm{U}(1),\mathbb{C}P^1,\tilde{\pi})$ is a principal bundle over $\mathbb{C}P^1$ called the Hopf bundle.Hopf bundle The fibre $\tilde{\pi}^{-1}(\pcoor{\alpha:\beta})$ of this bundle over a point $\pcoor{a:b}\in\mathbb{C}P^1$ is a copy of the torsortorsor (the underlying set) of the group $\mathrm{U}(1)$, embedded into $\mathrm{SU}(2)$ as the subset $\left(
\begin{smallmatrix}
\mathrm{e}^{\mathrm{i}\theta}a & \mathrm{e}^{\mathrm{i}\theta}b \\
-\mathrm{e}^{-\mathrm{i}\theta}b^* & \mathrm{e}^{-\mathrm{i}\theta}a^*
\end{smallmatrix}
\right)$ with $\theta\in\mathbb{R}$.

[Principal bundles over a sphere]
Let $H$ be the group of orthogonal $3\times 3$ matrices $h$ with $\det h=h_{33}=1$. In fact, $H$ is isomorphic to the group $\mathrm{SO}(2)$. The quadruple $(\mathrm{SO}(3),\mathrm{SO}(2),S^2,\tilde{\pi})$, where $\tilde{\pi}(g)=(0,0,1)g$, is a principal bundle over $S^2$. The fibre of this bundle over a point

\begin{equation}\label{eq:44}
b=(2\RE(\alpha\beta^*),2\IM(\alpha\beta^*),|\alpha|^2-|\beta|^2)^{\top}\in S^2\subset\mathbb{R}^3
\end{equation}

is a copy of $\mathrm{SO}(2)$ embedded into $\mathrm{SO}(3)$ as the subset

\begin{equation}\label{eq:43}
\left(
\begin{smallmatrix}
\RE(\mathrm{e}^{2\mathrm{i}\theta}(\alpha^2-\beta^2)) & \IM(\mathrm{e}^{2\mathrm{i}\theta}(\alpha^2+\beta^2)) & -2\RE(\mathrm{e}^{2\mathrm{i}\theta}\alpha\beta) \\
-\IM(\mathrm{e}^{2\mathrm{i}\theta}(\alpha^2-\beta^2)) & \RE(\mathrm{e}^{2\mathrm{i}\theta}(\alpha^2+\beta^2)) & 2\IM(\mathrm{e}^{2\mathrm{i}\theta}\alpha\beta) \\
2\RE(\alpha\beta^*) & 2\IM(\alpha\beta^*) & |\alpha|^2-|\beta|^2
\end{smallmatrix}
\right)
\end{equation}

with $\theta\in\mathbb{R}$.

The quadruple $(\mathrm{O}(3),\mathrm{O}(2),S^2,\tilde{\pi})$ is another principal bundle over $S^2$.

Let $\lambda=(P,H,B,\tilde{\pi})$ be a principal bundle. Assume that $H$ acts on either a real-analytic or a holomorphic manifold $F$ from the left, and the map $H\times F\to F$, $(h,f)\mapsto h\cdot f$ is either real-analytic or holomorphic. The group $H$ acts on the right on the Cartesian product $P\times F$ by $(p,f)\cdot h=(p\cdot h,h^{-1}\cdot f)$. Denote by $E$ the set of orbits for this action. Denote the orbit of a point $(p,f)\in P\times F$ by $[p,f]$. Define the map $\pi\colon E\to B$ by $\pi([p,f])=\tilde{\pi}(p)$. The map $\pi$ is continuous with respect to the quotient topology on $E$.

The triple $(E,B,\pi)$ is called the fibre bundlebundle!fibre associated with the principal bundle $\lambda=(P,H,B,\tilde{\pi})$ by the action $(h,f)\mapsto h\cdot f$. The manifold $E$ is the total spacetotal space!of a fibre bundle of the bundle, the manifold $B$ is its base space,base space!of a fibre bundle the map $\pi$ is the bundle projection,bundle projection!of a fibre bundle while $F$ is the fibre.fibre!of a fibre bundle

The triple $(E,B,\pi)$ satisfies the local triviality condition: for each $b\in B$ there is an open set $\mathcal{U}$ containing $b$ and a homeomorphism $\Psi\colon\pi^{-1}(\mathcal{U})\to\mathcal{U}\times F$ such that for all $b\in\mathcal{U}$ and $f\in F$ we have $\pi(\Psi^{-1}(b,f))=b$. If $H$ were not a Lie group, then this statement would be wrong, see a counterexample in <cit.>.

In what follows, we consider a particular case, when the fibre $F=L_0$ is a finite-dimensional linear space over a (skew) field $\mathbb{K}$, and $H$ is a Lie group that acts on $L_0$ by a representation $\theta$. The fibre bundle $(E,B,\pi)$ is called a vector bundlebundle!vector (resp. a tensor bundle,bundle!tensor resp. a line bundle)bundle!line if the elements of $L_0$ are vectors (resp. tensors, resp. elements of $\mathbb{K}^1$).

A cross-sectioncross-section of a fibre bundle $(E,B,\pi)$ is a map $f\colon B\to E$ satisfying $\pi(f(b))=b$ for all $b\in B$.

Conversely, consider a vector bundle $(E,B,\pi)$, that is, for any $b\in B$, the set $\pi^{-1}(b)$ is an $n$-dimensional linear space over a (skew) field $\mathbb{F}$, and the local triviality condition is satisfied. In addition, assume that the set $\pi^{-1}(b)$ carries an inner product $(\cdot,\cdot)$ which is either a real-analytic or holomorphic function of $b$. Let $P$ be the set of points $\mathbf{p}=(p_1,\dots,p_n)\in E^n$ such that $\pi(p_1)=\pi(p_2)=\cdots=\pi(p_n)=b\in B$, and the vectors $p_1$, …, $p_n$ constitute an orthonormal basis in $\pi^{-1}(b)$.

The quadruple $(P,H,B,\tilde{\pi})$, where $\tilde{\pi}(\mathbf{p})=\pi(p_1)$ and $H$ acts on $P$ by a representation $(p_1,\dots,p_n)h=(p_1\cdot h,\dots,p_n\cdot h)$, is called the principal bundle of orthonormal framesbundle!principal!of orthonormal frames on the vector bundle $(E,B,\pi)$.

Consider the principal fibre bundle $(\mathrm{SU}(2),\mathrm{U}(1),\mathbb{C}P^1,\tilde{\pi})$ of Example <ref>. For any integer or half-integer $s$, consider the representation $z\mapsto z^{2s}$ of the group $\mathrm{U}(1)$ in the linear space $L_0=\mathbb{C}^1$. The cross-sections of the line bundle $(E,\mathbb{C}P^1,\pi_{2s})$ associated to the principal fibre bundle by the above representation, are called functions of spin weight $s$,functions of spin weight $s$ see [18, 42, 85].

The line bundle $(E,\mathbb{C}P^1,\pi_2)$ is the tangent bundle $T\mathbb{C}P^1$, see proof in <cit.>. Let $(\mathrm{Spin}(T\mathbb{C}P^1),\mathrm{U}(1),\mathbb{C}P^1,\tilde{\pi})$ be the principal bundle of orthonormal frames corresponding to $T\mathbb{C}P^1$. The fibres of this bundle are copies of $\mathrm{U}(1)$.

On the other hand, consider the principal fibre bundle $(\mathrm{SO}(3),\mathrm{SO}(2),S^2,\tilde{\pi}')$ of Example <ref>. The representation $\left(
\begin{smallmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{smallmatrix}
\right)\mapsto\mathrm{e}^{2\mathrm{i}\theta}$ determines a line bundle over $S^2$. In fact, it is the tangent bundle $TS^2$, in which the tangent space at a point $\mathbf{u}\in S^2\subset\mathbb{R}^3$ is first identified with the two-dimensional real linear space $T_{\mathbf{u}}S^2=\{\,\mathbf{v}\in\mathbb{R}^3\colon(\mathbf{v},\mathbf{u})=0\,\}$, and then $T_{\mathbf{u}}S^2$ is converted into a complex line by defining $z\mathbf{v}=\RE z\mathbf{v}+\IM z(\mathbf{u}\times\mathbf{v})$. Denote by $(\mathrm{SO}(TS^2),\mathrm{SO}(2),S^2,\pi')$ the principal bundle of orthonormal frames corresponding to $TS^2$. The fibres of this bundle are copies of $\mathrm{SO}(2)$.

The map $S^2\to\mathbb{C}P^1$, $\mathbf{u}\mapsto\pcoor{u_1+u_2\mathrm{i}:1-u_3}$ is a homeomorphism. Consider the map $\Theta\colon\mathrm{Spin}(T\mathbb{C}P^1)\to\mathrm{SO}(TS^2)$ given by $\Theta(\pcoor{u_1+u_2\mathrm{i},1-u_3},z)=(\mathbf{u},z^2)$. By construction, this map is fibre-preserving. Moreover, for all $(\pcoor{u_1+u_2\mathrm{i}:1-u_3},z)\in\mathrm{Spin}(T\mathbb{C}P^1)$ and for all $w\in\mathrm{U}(1)=\mathrm{Spin}(2)$ we have

\[
\Theta((\pcoor{u_1+u_2\mathrm{i}:1-u_3},z)w)
\]

By <cit.>, the map $\Theta$ defines a spin structurespin structure on the manifold $S^2$.

Recall that the complex spin representationrepresentation!complex spin of the group $\mathrm{Spin}(2)=\mathrm{U}(1)$ is the complex representation $\rho_2(\mathrm{e}^{\mathrm{i}\theta})=\left(
\begin{smallmatrix}
\mathrm{e}^{\mathrm{i}\theta} & 0 \\
0 & \mathrm{e}^{-\mathrm{i}\theta}
\end{smallmatrix}
\right)$. Its irreducible components $\rho_2^+(\mathrm{e}^{\mathrm{i}\theta})=\mathrm{e}^{\mathrm{i}\theta}$ and $\rho_2^-(\mathrm{e}^{\mathrm{i}\theta})=\mathrm{e}^{-\mathrm{i}\theta}$ are called complex spin half-representations.half-representation!complex spin The space of the representation $\rho_2$ is equal to the direct sum $V_1\oplus tV_1$ of linear spaces where the above components act. Note that these representations do not descend to the group $\mathrm{SO}(2)$ since $\rho_2(-1)=-\id$ and $\rho_2^{\pm}(-1)=-1$.

The complex vector bundle $(\Sigma\mathbb{C}P^1,\mathbb{C}P^1,\pi)$ with fibre $V_1\oplus tV_1$ is the bundle associated with the principal fibre bundle $(\mathrm{SU}(2),\mathrm{U}(1),\mathbb{C}P^1,\tilde{\pi})$ by the complex spin representation. We see that this bundle is the direct sum of the two bundles $(\Sigma^+\mathbb{C}P^1,\mathbb{C}P^1,\pi_1)$ and $(\Sigma^-\mathbb{C}P^1,\mathbb{C}P^1,\pi_{-1})$ with fibres $V_1$ and $tV_1$ that correspond to the cases of $s=\pm 1$ above. The cross-sections of the bundle $(\Sigma\mathbb{C}P^1,\mathbb{C}P^1,\pi)$ are called spinor fields.spinor field

Note that the map $j\colon V_1\oplus tV_1\to V_1\oplus tV_1$, $j(z_1,z_2)=(z^*_2,z^*_1)$ is a real structure on the space $V_1\oplus tV_1$ that commutes with the complex spin representation. The corresponding real spin representationrepresentation!real spin $\mathrm{e}^{\mathrm{i}\theta}\mapsto\left(
\begin{smallmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{smallmatrix}
\right)$ is $rV_1$. Similarly, the map $j:V_1\oplus tV_1\to V_1\oplus tV_1$, $j(z_1,z_2)=(z^*_2,-z^*_1)$ is a quaternionic structure on the space $V_1\oplus tV_1$ that commutes with the complex spin representation. The corresponding quaternionic spin representationrepresentation!quaternionic spin $\mathrm{e}^{\mathrm{i}\theta}\mapsto\mathrm{e}^{\mathrm{i}\theta}\in\mathrm{Sp}(1)$
is $qV_1$.

It turns out that not all manifolds may carry a spin structure, there exist topological obstructions for that, see <cit.> and <cit.>.

Finally, note that the maps $\tilde{\pi}_1(t)=t^2$ and $\tilde{\pi}_2(z)=z^2$ of Example <ref> lead to the idea of a “square root” between vectors and spinors, see [117].

[The edth operator]
There exists a beautiful coordinate-free description of the operator $\eth$ in [24], where it is associated with the $\partial$-operator and the Dolbeault resolution of complex analysis, see also <cit.>. We give a coordinate description instead. Note that $\eth$ is the phonetic symbol for voiced “th”, see <cit.>.

Introduce a chart $\mathbb{C}P^1\setminus\{\pcoor{1:0}\}\to\mathbb{C}$, $\pcoor{z_1:z_2}\mapsto\zeta=\frac{z_1}{z_2}$. This is a standard stereographic correspondence between the Argand planeArgand plane of $\zeta$ and the complex projective line without the point at infinity. To include the above point, we regard $\zeta=\infty$ as a “point” added to the Argand plane, and map it to the point $\pcoor{1:0}\in\mathbb{C}P^1$. The union $\mathbb{C}\cup\{\infty\}$ is the Riemann sphere.Riemann sphere

Let $f$ be a smooth cross-section of the vector bundle $(\mathrm{SU}(2),\mathbb{C}P^1,\pi_s)$ of Example <ref>. We define

\[
\begin{aligned}
(\eth f)(\zeta)&=(1+\zeta\zeta^*)^{1-s}\frac{\partial}{\partial\zeta}
(\overline{\eth} f)(\zeta)&=(1+\zeta\zeta^*)^{1+s}\frac{\partial}{\partial\zeta^*}
\end{aligned}
\]

where we adapted definition <cit.> in order to be consistent with cosmological literature. Then, the spin rising operatorspin rising operator $\eth$ maps smooth cross-sections of the bundle $(\mathrm{SU}(2),\mathbb{C}P^1,\pi_s)$ to smooth cross-sections of the bundle $(\mathrm{SU}(2),\mathbb{C}P^1,\pi_{s+1})$. Similarly, the spin lowering operatorspin lowering operator $\overline{\eth}$ maps smooth cross-sections of the bundle $(\mathrm{SU}(2),\mathbb{C}P^1,\pi_s)$ to smooth cross-sections of the bundle $(\mathrm{SU}(2),\mathbb{C}P^1,\pi_{s-1})$.

§.§ Induced representations

Let $H$ be a closed subgroup of a compact Lie group $G$, and let $B=G/H$. Consider the principal fibre bundle $(G,H,B,\tilde{\pi})$, where $\tilde{\pi}(g)=gH$. Let $(E,B,\pi)$ be the vector bundle associated to the above principal fibre bundle by a representation of $H$ ia a finite-dimensional linear space $L_0$ over a (skew) field $\mathbb{K}$. Define a left action of $G$ on $G\times L_0$ by $g_0\cdot(g,\mathbf{l})=(g_0g,\mathbf{l})$. The above action induces an action of $G$ on $E$. It is easy to check that with this action, the vector bundle $(E,B,\pi)$ becomes homogeneous.

A vector bundle  $(E,G/H,\pi)$ is called homogeneousbundle!vector!homogeneous if $G$ acts on $E$ such that $g\cdot\pi^{-1}(x)=\pi^{-1}(g\cdot x)$ and the action of $G$ on $\pi^{-1}(x)$ is linear for all $x\in G/H$ and $g\in G$.

The construction given above describes all homogeneous vector bundles, see [130].

Let $C(G,L_0)$ be the linear space of all continuous functions $f\colon G\to L_0$ satisfying the condition $f(gh)=h^{-1}\cdot f(g)$, $h\in H$, $g\in G$. Introduce the notation

\begin{equation}\label{eq:37}
\langle f_1,f_2\rangle=\int_G(f_1(g),f_2(g))\,\mathrm{d}g,
\end{equation}

where $f_1$, $f_2\in C(G,L_0)$, and $\mathrm{d}g$ is the probabilistic $G$-invariant measure on the Borel $\sigma$-field of subsets of $G$. We observe that the map $C(G,L_0)\times C(G,L_0)\to\mathbb{K}$, $(f_1,f_2)\mapsto\langle f_1,f_2\rangle$, is an inner product. The map $(\tilde{\theta}(g_0)f)(g)=f(g^{-1}_0g)$ extends to a representation of $G$ in the Hilbert space $L^2(G,L_0)$, which is the completion of the linear space $C(G,L_0)$ with respect to the norm induced by the inner product (<ref>). Moreover, the above inner product is $G$-invariant. This fact is proved in <cit.> for the case of $\mathbb{K}=\mathbb{C}$, and can be proved literally in the same way for the remaining (skew) fields. In what follows, in similar cases we refer to proofs of our statements for the complex case.

Denote by $\Gamma E$ the $\mathbb{K}$-linear space of continuous cross-sections of the homogeneous vector bundle $(E,B,\pi)$. Let $G$ act on $\Gamma E$ by $(\theta(g)s)(x)=g\cdot s(g^{-1}x)$ for $g\in G$, $s\in\Gamma E$, and $x\in B$. The representation $(\Gamma E,\theta)$ extends to a representation of $G$ in the Hilbert space $L^2(E)$ of the square-integrable cross-sections of the vector bundle $(E,B,\pi)$ with the $G$-invariant inner product

\[
\langle s_1,s_2\rangle=\int_B(s_1(x),s_2(x))_{L_0}\,\mathrm{d}x,
\]

where $\mathrm{d}x$ is the probabilistic $G$-invariant measure on $B$. The complex case is proved in <cit.>. We call $\theta$ the representation of $G$ inducedrepresentation!induced by the representation $L_0$ of the subgroup $H$.

Define the map $\Gamma E\to C(G,L_0)$, $s\mapsto\tilde{s}$ by $\tilde{s}(g)=g^{-1}\cdot s(gH)$ for $g\in G$. This map extends to an equivalence between the representations $(L^2(E),\theta)$ and $(L^2(G,L_0),\tilde{\theta})$, see proof for the complex case in <cit.>. Act by $g$ to both hand sides of this equality. We obtain, that the inverse map has the form

\begin{equation}\label{eq:40}
\end{equation}

§.§ The structure of the induced representation

The Hilbert space $L^2(E)$ is uniquely decomposed into the Hilbert direct sum of isotypical subspaces. Let $L$ be an irreducible representation of $G$, and let $\res^G_HL$ be the restriction of this representation to the group $H$. Let $\Hom_{\mathbb{K}}(\res^G_HL,L_0)$ be the $\mathbb{K}'$-linear space of $\mathbb{K}$-linear maps from $L$ to $L_0$. The group $H$ acts on this space by $(h\cdot f)\mathbf{l}=h\cdot(f(h^{-1}\cdot\mathbf{l}))$ for $h\in H$, $f\in\Hom_{\mathbb{K}}(\res^G_HL,L_0)$, and $\mathbf{l}\in L$. Let $\Hom_{\mathbb{K}H}(\res^G_HL,L_0)$ be the isotypical subspace of the space $\Hom_{\mathbb{K}}(\res^G_HL,L_0)$ that corresponds to the trivial representation of $H$. In other words, $\Hom_{\mathbb{K}H}(\res^G_HL,L_0)$ is the subspace of elements in $\Hom_{\mathbb{K}}(\res^G_HL,L_0)$ which are invariant under $H$, or the linear space of intertwining operators between $\res^G_HL$ and $L_0$.

Similarly, construct the $\mathbb{K}'$-linear space $\Hom_{\mathbb{K}G}(L,L^2(E))$ of intertwining operators between $L$ and $L^2(E)$. The celebrated Frobenius reciprocityFrobenius reciprocity states that the two above constructed spaces are isomorphic, see <cit.> and <cit.>. In particular, $\dim_{\mathbb{K}'}\Hom_{\mathbb{K}H}(\res^G_HL,L_0)=\dim_{\mathbb{K}'}\Hom_{\mathbb{K}G}(L,L^2(E))$ and the representation $(L^2(E),\theta)$ contains

\[
\]

copies of the irreducible representation $L$.

The particular case of the above statement, when $\mathbb{K}=\mathbb{R}$ and $H$ contains only the identity element, is called the Fine Structure Theorem,Fine Structure Theorem see [43]. We used this result in Example <ref>.

The next step is construct an orthonormal basis in the Hilbert space $L^2(E)$ in such a way that different kinds of “spherical harmonics” are particular cases of our general construction. See also alternative approaches to construction of various spherical harmonics in [21, 116].

§.§ The construction of a basis in the space of an induced representation

§.§.§ Schur's Lemma

This result describes the structure of the space $\Hom_{\mathbb{K}G}(L_1,L_2)$ for $L_1$, $L_2\in\hat{G}_{\mathbb{K}}$.

Schur's Lemma
Let $L$, $L_1$, $L_2\in\hat{G}_{\mathbb{K}}$.

* Any element of the space $\Hom_{\mathbb{K}G}(L_1,L_2)$ is either zero or an isomorphism. Moreover, $\Hom_{\mathbb{K}G}(L_1,L_2)=\{0\}$ if and only if $L_1$ is not equivalent to $L_2$.

* As a linear space (resp. as a division algebra), the set $\Hom_{\mathbb{R}G}(U,U)$ is isomorphic to $\mathbb{R}^1$ (resp. to $\mathbb{R}$) if and only if $U$ is of real type; to $\mathbb{R}^2$ (resp. to $\mathbb{C}$) if and only if $U$ is of complex type; to $\mathbb{R}^4$ (resp. to $\mathbb{H}$) if and only if $U$ is of quaternionic type.

* As a linear space (resp. as a division algebra), the set $\Hom_{\mathbb{C}G}(V,V)$ is isomorphic to $\mathbb{C}^1$ (resp. to $\mathbb{C}$).

* As a linear space (resp. as a division algebra), the set $\Hom_{\mathbb{H}G}(W,W)$ is isomorphic to $\mathbb{R}^4$ (resp. to $\mathbb{H}$) if and only if $W$ is of real type; to $\mathbb{R}^2$ (resp. to $\mathbb{C}$) if and only if $W$ is of complex type; to $\mathbb{R}^1$ (resp. to $\mathbb{R}$) if and only if $W$ is of quaternionic type.

All items except the last one are proved in <cit.> and <cit.>.

To prove the last item, note that $c\Hom_{\mathbb{H}G}(W,W)=\Hom_{\mathbb{C}G}(c'W,c'W)$ by <cit.>. If $W$ is of real type, then, by <cit.>, there exists an irreducible complex representation $V$ of real type such that $c'W=V\oplus V$. By <cit.>, $\Hom$ is bilinear over the direct sum $\oplus$, therefore

\[
\Hom_{\mathbb{C}G}(c'W,c'W)=\Hom_{\mathbb{C}G}(V\oplus V,V\oplus V)=\mathbb{C}^4.
\]

If $W$ is of complex type, then, by <cit.>, there exists an irreducible complex representation $V$ of complex type such that $c'W=V\oplus tV$. Therefore

\[
\Hom_{\mathbb{C}G}(c'W,c'W)=\Hom_{\mathbb{C}G}(V\oplus tV,V\oplus tV)=\mathbb{C}^2.
\]

If $W$ is of quaternionic type, then, by <cit.>, there exists an irreducible complex representation $V$ of quaternionic type such that $c'W=V$. Therefore

\[
\Hom_{\mathbb{C}G}(c'W,c'W)=\Hom_{\mathbb{C}G}(V,V)=\mathbb{C}^1.
\]

The statement about the isomorphism of linear spaces follows. The Frobenius classification of finite-dimensional real division algebras [29] states that every such an algebra is isomorphic either to $\mathbb{R}$ or to $\mathbb{C}$ or to $\mathbb{H}$ and implies the statement about the isomorphism of division algebras.

§.§.§ The construction of a basis

Consider the $\mathbb{K}'$-linear space $\Hom_{\mathbb{K}H}(\res^G_HL,L_0)$ and assume that $L_0$ is irreducible. Denote by $D$ the division algebra $\Hom_{\mathbb{K}H}(L_0,L_0)$. The above space contains

\[
\]

copies of the representation $L_0$. Choose and fix the $\mathbb{K}$-linear subspaces $L_0^1$, …, $L^{n_L}_0$ of the linear space $\res^G_HL$ where the above copies act. This choice is not unique. However, we have

\[
\Hom_{\mathbb{K}H}(\res^G_HL,L_0)=\Hom_{\mathbb{K}H}(L^1_0,L_0)\oplus\cdots
\oplus\Hom_{\mathbb{K}H}(L^{n_L}_0,L_0).
\]

The isomorphism of division algebras between $D$ and $\Hom_{\mathbb{K}H}(L^i_0,L_0)$, $1\leq i\leq n_L$, described in Schur's Lemma, determines a representation of $D$ in $\Hom_{\mathbb{K}H}(L^i_0,L_0)$.

It is well-known that any finite-dimensional complex representation of the algebra $\mathbb{C}$ is a direct sum of finitely many copies of the unique irreducible representation $\mathbb{C}\to\Hom_{\mathbb{C}}(\mathbb{C}^1,\mathbb{C}^1)$. We choose it in the form $z\mapsto z$. A real representation of the algebra $\mathbb{R}$ (resp. $\mathbb{C}$, resp. $\mathbb{H}$) is a direct sum of finitely many copies of the unique irreducible representation $\mathbb{R}\to\Hom_{\mathbb{R}}(\mathbb{R}^1,\mathbb{R}^1)$ (resp. $\mathbb{C}\to\Hom_{\mathbb{R}}(\mathbb{R}^2,\mathbb{R}^2)$, resp. $\mathbb{H}\to\Hom_{\mathbb{R}}(\mathbb{R}^4,\mathbb{R}^4)$). We choose it in the form $a\mapsto a$ (resp.

\begin{equation}\label{eq:38}
\begin{smallmatrix}
a & -b \\
b & a
\end{smallmatrix}
\right)\in\Hom_{\mathbb{R}}(\mathbb{R}^2,\mathbb{R}^2),
\end{equation}

\begin{equation}\label{eq:39}
\begin{smallmatrix}
a & -b & c & -d \\
b & a & d & c  \\
-c & -d & a & b \\
d & -c & -b & a
\end{smallmatrix}
\right)\in\Hom_{\mathbb{R}}(\mathbb{R}^4,\mathbb{R}^4)).
\end{equation}

For proofs, see [16] and [118].

We proved the existence of a basis, in which the elements of the linear space $\Hom_{\mathbb{K}H}(\res^G_HL,L_0)$ are matrices over $\mathbb{K}'$ with $\dim_{\mathbb{K}'}L_0$ rows and $\dim_{\mathbb{K}'}L$ columns, in which only $n_L$ components are potentially non-zero. Each component is a square block-diagonal matrix and contains $\frac{\dim_{\mathbb{K}'}L_0}{\dim_{\mathbb{K}'}D}$ identical blocks. Each block is a square matrix with $\dim_{\mathbb{K}'}D$ rows. If $\mathbb{K}'=\mathbb{R}$ and $D=\mathbb{C}$, then each block has the form of the matrix in Equation (<ref>), if $\mathbb{K}'=\mathbb{R}$ and $D=\mathbb{H}$, then each block has the form of the matrix in Equation (<ref>).

Define the matrices $\{\,f^{ijk}\colon 1\leq i\leq n_L,1\leq j\leq\frac{\dim_{\mathbb{K}'}L_0}{\dim_{\mathbb{K}'}D},1\leq k\leq\dim_{\mathbb{K}'}D\,\}$ as follows. All the components of the matrix $f^{ijk}$ but the $i$th one are zero matrices. If $\dim_{\mathbb{K}'}D=1$, then each block contains the number $\frac{1}{\sqrt{\dim_{\mathbb{K}'}L_0}}$. If $\dim_{\mathbb{K}'}D=2$, then each block of the matrix $f^{ij1}$ has the form $\frac{1}{\sqrt{\dim_{\mathbb{K}'}L_0}}\left(
\begin{smallmatrix}
1 & 0 \\
0 & 1
\end{smallmatrix}
\right)$, and each block of the matrix $f^{ij2}$ the form $\frac{1}{\sqrt{\dim_{\mathbb{K}'}L_0}}\left(
\begin{smallmatrix}
0 & -1 \\
1 & 0
\end{smallmatrix}
\right)$. If $\dim_{\mathbb{K}'}D=4$, then each block of the matrix $f^{ij1}$ has the form $\frac{1}{\sqrt{\dim_{\mathbb{K}'}L_0}}\left(
\begin{smallmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0  \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{smallmatrix}
\right)$, each block of the matrix $f^{ij2}$ the form $\frac{1}{\sqrt{\dim_{\mathbb{K}'}L_0}}\left(
\begin{smallmatrix}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0  \\
0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0
\end{smallmatrix}
\right)$, each block of the matrix $f^{ij3}$ the form $\frac{1}{\sqrt{\dim_{\mathbb{K}'}L_0}}\left(
\begin{smallmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1  \\
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0
\end{smallmatrix}
\right)$, and each block of the matrix $f^{ij4}$ the form $\frac{1}{\sqrt{\dim_{\mathbb{K}'}L_0}}\left(
\begin{smallmatrix}
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0  \\
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{smallmatrix}
\right)$. An orthonormal basis of the space $\Hom_{\mathbb{K}H}(\res^G_HL,L_0)$ is constructed.

Apply the Frobenius reciprocity isomorphism to the matrices $f^{ijk}$. According to the explicit construction of this isomorphism given in <cit.>, the matrix $f^{ijk}$ becomes the map $F^{ijk}\colon L\to C(G,L_0)$ that acts by $F^{ijk}(\mathbf{l})(g)=f^{ijk}(g^{-1}\cdot\mathbf{l})$ for $\mathbf{l}\in L$ and $g\in G$. The composition of the map (<ref>) and $F^{ijk}$ acts from $L$ to $L^2(E)$ and maps a vector $\mathbf{l}\in L$ to the cross-section $s^{ijk}_{\mathbf{l}}(gH)=g\cdot f^{ijk}(g^{-1}\cdot\mathbf{l})$ of the vector bundle $(E,G/H,\pi)$. In particular, for the vectors $\mathbf{e}_1$, …, $\mathbf{e}_{\dim L}$ of an orthonormal basis in $L$, their images

\[
{}_{L_0}Y_{ijkLm}(gH)=g\cdot f^{ijk}(g^{-1}\cdot\mathbf{e}_m)
\]

form an orthogonal basis in the isotypic subspace of the irreducible representation $L$ of the space $L^2(E)$. It is not necessarily orthonormal. By abuse of notation, we denote an orthonormal basis by the same symbol. The union of the above images over all representations $L\in\hat{G}_{\mathbb{K}}$ for which $\Hom_{\mathbb{K}G}(L,L^2(E))\neq\{0\}$, constitutes an orthonormal basis in the space $L^2(E)$.

If $L_0$ is reducible, then we decompose it into irreducible components and use <cit.>, according to which the space $L^2(E)$ is the Hilbert direct sum of subspaces that correspond to each component, and construct an orthonormal basis in each subspace as above.

An alternative construction of the basis for the case of $\mathbb{K}=\mathbb{C}$, $G=\mathrm{SU}(2)$, and $H=\mathrm{U}(1)$ using the Euler angles, is given in [21].

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