数理解析研究所講究録
第 822 巻 1993 年 62-69

62

Extension problems for spinors on$S^{4}$

Tosiaki KORI
$S^{4}$

上のスピノールに対する延長問題
郡 敏昭

1. The space of spinors on(早大・理工)$S^{3}$

Here we shall explain the complex analytic point of view of Dirac operatoronand discuss the eigenvalues of Hamiltonian acting on spinors on the$equator\simeq S^{3}$ . These were obtained in [K].
and a smooth. Let us consider two copies of complex planes
and. We patch
bijection :given by$S^{4}$$a$$\hat{C}_{w}^{2}$$C_{z}^{2}$$v$$C_{z}^{2}\backslash \{0\}arrow\hat{C}_{w}^{2}\backslash \{0\}$$w=v(z)=- \frac{\overline{z}}{|z|^{2}}$by to obtain a differentiable manifold $M=c^{2}u{}_{v}\hat{C}^{2}$ , which isandhomeomorphic to .
We endow $M$ with a riemannian metric defined by$C_{z}^{2}$$C_{w}^{2}$$v$$S^{4}$$g=\{(1(1:|_{w|)^{-}\sum_{i}^{i=_{=^{1_{1}}}}dw_{i}^{i}\otimes r_{i}^{i}}^{z|^{2_{2}})^{-2_{2}}\sum^{2_{2}}dz\otimes d\overline{z}_{w}}onC_{z}^{2}on\hat{C}_{w}^{2}$.
The Levi-Civita connection on $M$ is given by gauge potentials$\Gamma(z)=\frac{|z|^{2}}{1+|z|^{2}}\sigma(z)^{-1}\cdot(d\sigma)_{z}$$\hat{\Gamma}(w)=\frac{|w|^{2}}{1+|w|^{2}}\sigma(w)^{-1}\cdot(d\sigma)_{w}$for$z\in C_{z}^{2}$for$w\in\hat{C}_{w}^{2}$,

Typed by $A_{M}S$.TEX

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iswhere $\sigma(z)=|z|^{2}(v_{*})_{z}$ ,being the differential of , anda one-form valued in $\mathcal{G}=\{X\in gl(4, C) : {}^{t}XK+KX=0\}\simeq o(4, C),$ $K=$$(\begin{array}{ll}0 I_{2}I_{2} 0\end{array})$$\sigma(z)^{-1}(d\sigma)_{z}$$v$$v_{*}$.

On $M$ there are a unique spin-structure Spin $(M)$ and the associated spinor
bundle $S=Spin(M)\cross s_{pin(4)}\triangle\cdot\triangle$ is a basic representation space of Spin $(4)$which is the direct sum of two irreducible representations ofeach
andof dimension 2. Letandbe the corresponding bundles whose crosssections are spinors of positive (respectively negative) chirality. We shallchoose a frame ofand denote the spinors in matrix form$\triangle^{+}$$S^{+}$$\triangle^{-}$$S^{-}$$S^{\pm}$$(\begin{array}{l}\phi\psi\end{array})\in\Gamma(S)$,$\phi=(\begin{array}{l}\phi_{1}\phi_{2}\end{array})\in\Gamma(S^{+})$,$\psi=(\begin{array}{l}\psi_{1}\psi_{2}\end{array})\in\Gamma(S^{-})$,

signifies the sections of a bundle. The inner product of two spinors
is defined by $<\phi(z),$.
The Dirac operator acting on the spinors is defined as the composition
where
is the covariant derivative induced by the Levi-Civita
connection and is Clifford multiplication. The Dirac operator switches

where$\phi,$$\Gamma$$\varphi\in\Gamma(S^{\pm})$$\varphi(z)>=\phi_{1}(z)\overline{\varphi}_{1}(z)+\phi_{2}(z)\overline{\varphi}_{2}(z)$$b$$\nabla$$\mathcal{D}=\mu\cdot\nabla$$S^{+}$

$\mu$

and

$S^{-}$and is of the form

$D=(\begin{array}{ll}0 D^{\uparrow}D 0\end{array})$

where $D:\Gamma(S^{+})arrow\Gamma(S^{-})$ .

We gave in [K] the following matrix representation of the Dirac operator.

$D=(\begin{array}{ll}(1+|z|^{2})\frac{\partial}{\partial z_{1}}-\frac{3}{2}\overline{z}_{l} -(1+|z|^{2})\frac{\partial}{\partial\overline{z}_{2}}+\frac{3}{2}z_{2}(1+|z|^{2})\frac{\partial}{\partial z_{2}}-\frac{3}{2}\overline{z}_{2} (1+|z|^{2})\frac{\partial}{\partial\overline{z}_{1}}-\frac{3}{2}z_{l}\end{array})$$D^{\uparrow}=(\begin{array}{ll}(l+|z|^{2})\frac{\partial}{\partial\overline{z}_{1}}-\frac{3}{2}z_{l} (1+|z|^{2})\frac{\partial}{\partial\overline{z}_{2}}-\frac{3}{2}z_{2}-(1+|z|^{2})\frac{\partial}{\partial z_{2}}+\frac{3}{2}\overline{z}_{2} (1+|z|^{2})\frac{\partial}{\partial z_{1}}-\frac{3}{2}\overline{z}_{1}\end{array})$

We have a decomposition ofparts;$D$and$D=\gamma_{0}(n-\mathcal{P})$$D^{\uparrow}$

to their longitudinal parts and radial$D^{\uparrow}=(n+\mathcal{P})\gamma_{0}$.

Here. We shallsignifies Clifford multiplication of the radial vector
explainsoon after. First we introduce an orthonormal frame on $M$ , buthere we shall write down it only on the local coordinate, the formulas$n$$\gamma_{0}$$\mathcal{P}$$C^{2}\subset M$

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are easily obtained by the transition relation. This frame isonimportant not only as it gives a neat expression of Dirac operators on $M$ andon the $equator\simeq S^{3}$ but also as is associated to the Lie group structure of$S^{3}\simeq SU(2)$ (see c). Let$\hat{C}^{2}\subset M$$\nu=\frac{1+|z|^{2}}{|z|}(z_{1}\frac{\partial}{\partial z_{1}}+z_{2}\frac{\partial}{\partial z_{2}})$$\epsilon=\frac{1+|z|^{2}}{|z|}(-\overline{z}_{2}\frac{\partial}{\partial z_{1}}+\overline{z}_{1}\frac{\partial}{\partial z_{2}})$

The radial vector field is given by$n=\frac{1}{2}(\nu+\overline{\nu})$.

Put$\theta_{1}=\frac{1}{2}(\epsilon+\overline{\epsilon})$$\theta_{0}=\frac{1}{2\sqrt{-1}}(\nu-\overline{\nu})$$\theta_{2}=\frac{1}{2\sqrt{-1}}(\epsilon-\overline{\epsilon})$.

form an orthonormal frame on $M$ andThen
are tangent to the constant altitude $\{|z|=const\}$ .
$P$ : $S^{+}arrow S^{+}$ is given by
withcoming from
Clifford multiplication of .
The matrix representation of is written as$\sqrt{2}n,$$\sqrt{2}\theta_{0},$$\sqrt{2}\theta_{2}$$\sqrt{2}\theta_{1},$$\theta_{0},$$\theta_{1},$$\mathcal{P}=-(\gamma_{0}|S^{-})\sum_{i=0}^{2}\theta_{i}\nabla_{\theta;}$$\gamma_{0}$$n$$\mathcal{P}$$\mathcal{P}=(^{-\sqrt{-1}\theta_{0}}$$\sqrt{-1}^{\overline{\epsilon}}\theta_{0}$$-\epsilon$)$+ \frac{3}{2}|z|(\begin{array}{ll}1 00 1\end{array})$,

Let $E=\{|z|=1\}$ be the equator of $M;E\simeq S^{3}$ . $E$ is endowed with theriemannian metric $g|E$ . Since Spin (3) has the spinor representation onis the spinor bundle corresponding to thethe restrictions on $E$ of bundle. The Diracspin structure Spin $(E)$ .
gives the isomorphism between
operator on $E$ acting on spinors of positive chirality is given by $-\gamma_{0}P|E$ .
The restriction on $E$ is called Hamiltonian on $E$ .
$\triangle^{\pm}$$S^{\pm}$$S^{\pm}$$\gamma_{0}$$\mathcal{P}$

Here we shall discuss alittle about infinitesimal representations of $SU(2)$given by the vector fields$i=0,1,2$ . First we note the commutation
relations same as those of $sl(2)$ ;$c$$\sqrt{-1}\theta_{i},$$[\sqrt{-1}\theta_{0}, \epsilon]=-2\epsilon$,,$[\sqrt{-1}\theta_{0},\overline{\epsilon}]=2\overline{\epsilon}$$[\epsilon,\overline{\epsilon}]=4\sqrt{-1}\theta_{0}$.

We now follow the isomorphism $B\simeq S^{3}\simeq SU(2)$ and look the point $z\in B$
as$\ddot{z}=(\begin{array}{ll}z_{1} -\overline{z}_{2}z_{2} \overline{z}_{1}\end{array})\in SU(2)$. We shall then define the right action on$E$$\theta_{2}|$65. Put $R_{g}f(z)=f(z\cdot g)$
of $g\in SU(2)$ by $z\cdot g=$ the first column of$E$for a continupus function on . Then the differentials become $dR(e_{k})=$, $k=0,1,2$ , where$\ddot{z}\cdot g$$f$$-\theta_{k}$$e_{0}=(^{\sqrt{-1}}$$-\sqrt{-1}0$$0$)$e_{2}=(_{\sqrt{-1}^{0}}$$e_{1}=(\begin{array}{ll}0 1-l 0\end{array})$$\sqrt{-1}0)$

are the basis of $su(2)$ .
A polynomials that satisfies$P(az_{1}, bz_{2}, b\overline{z}_{1}.a\overline{z}_{2})=a^{k}b^{l}P(z_{1}, z_{2},\overline{z}_{1},\overline{z}_{2})$.
is called of class $(k, l)$ . The set of polynomials of class $(k, l)$ is denoted by. We
Letbe the set of harmonic polynomials on
and put, hence $\dim \mathcal{H}_{k,l}=k+l+1$ . It foollws also
have
$E$
.
that , on , every polynomial is a sum of harmonic polynomials in$E$
This ensures the fact that our family of eigenspinors of Hamiltonian on
obtained later is a complete system.
Put, for $r\geq 0$ and $0\leq k,$ $q\leq r$ ,$S_{k,l}$$C^{2}$$\mathcal{H}$$\mathcal{H}_{k.l}=\mathcal{H}\cap S_{k}\iota$$S_{k,l}=\mathcal{H}_{k,l}\oplus|z|^{2}S_{k-1,l-1}$$\mathcal{H}_{k,l}’ s$$h_{k,r-k}^{q}(z)=\epsilon^{q}(z_{1}^{k}z_{2}^{r-k})$pair$\mathcal{H}_{k,r-k}Foreach$$p_{\Gamma O}$$r$.

and $k\leq r$ the set $\{h_{k,r-k}^{q};q=0, \cdots ’ r\}$ forms a basis of

暇科謡 ion.
$t$(1) ’] 乙\mbox{\boldmath $\theta$}0 $h_{k,r-k}^{q}=(r-2q)h_{k,r-k}^{q}$(2)(3) $\overline{\epsilon}h_{k,r-k}^{q}=-4q(r-q+1)h_{k,r-k}^{q-1}$$\epsilon h_{k,r-k}^{q}=h_{k,r-k}^{q+1}$

Hence the space of harmonic polynomials(restricted on $B$ ) is decom. Each inposed by the right action $R$ of $SU(2)$ into
duced representation $R_{k,r-k}=(R, \mathcal{H}_{k,r-k})$ is an irrreducible representation
with the highest weight$\mathcal{H}$$\mathcal{H}=\sum_{r}\sum_{k=0}^{r}\mathcal{H}_{k,r-k}$$\frac{r}{2}$$d$Put, for $r\leq 0,0\leq k\leq r$ , and $0\leq q\leq r+1$ ,$\phi_{k,r-k}^{q}=(\begin{array}{l}q2^{-q+1}h_{k,r-k}^{q-1}-2^{-q}h_{k,r-k}^{q}\end{array})$.

66
Then we have from the matrix representation of the Hamiltonian and the
Proposition in ;$c$

$P \phi_{k,r-k}^{q}=(r+\frac{3}{2})\phi_{k,r-k}^{q}$.

Thus the positive eigenvalues and eigenfunctions of are obtained. In particular the multiplicity of the eigenvalue is $(r+1)(r+2)$ .
The investigation of negative eigenspinors is related to the left action of$SU(2)$ on the harmonic polynomials. The left action of a $g\in SU(2)$ on $E$
is defined by $g\cdot z=$ the first column of. Let $L_{g}f(z)=f(g^{-1}\cdot z)$ for acontinuous function on $E$ .
We introduce the following vector fields on $M-\{0,\hat{0}\}$ , that have the following local expressions on $C^{2}-\{0\}$ :$\mathcal{P}$$r$$g\cdot\ddot{z}$$\mu=\frac{1+|z|^{2}}{|z|}(z_{2}\frac{\partial}{\partial z_{2}}+\overline{z}_{1}$ $f_{f\overline{z}_{1}}^{\partial})$$\tau_{0}=\frac{1}{2\sqrt{-1}}(\mu-\overline{\mu})$,

We have $dL(e_{i})=-\tau_{i}|E$ ;Let,$\delta=\frac{1+|z|^{2}}{|z|}(\overline{z}_{2}$$\tau_{1}=\frac{1}{2}(\delta+\overline{\delta})$,$f_{f\overline{z}_{1}}^{\partial}-z_{1}\frac{\partial}{\partial z_{2}})$$\tau_{2}=\frac{1}{2\sqrt{-1}}(\delta-\overline{\delta})$.

.

$i=0,1,2$ .
$\hat{h}_{q}^{r-k,k}(z)=\delta^{q}(\overline{z}_{1}^{k}z_{2}^{r-k})$.

: the space of harmonic polynomials
give a basis of. Put,
that satisfy the condition
for $r\geq 0,0\leq k\leq r$ , and $0\leq q\leq r+1$ ,$\hat{\mathcal{H}}^{l,k}$$\{\hat{h}_{q}^{l,k}; q=0, \cdots , r\}$$P(az_{1}, az_{2}, b\overline{z}_{1}.b\overline{z}_{2})=a^{l}b^{k}P(z_{1}, z_{2},\overline{z}_{1},\overline{z}_{2})$$\pi_{q}^{r-k,k}=(\begin{array}{l}2^{-q^{\wedge}}h_{q}^{r-k+l,k}2^{-q^{\wedge}}h_{q}^{r-k,k+1}\end{array})$.

By an easy calculus we have
$P \pi_{q}^{r-k,k}=-(r+\frac{3}{2})\pi_{q}^{r-k,k}$Thus we have.

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$Th$ 伽科 $rHY1$ ]. The eigenvalues

with; $r=0,1,2,$
multiplicity $(r+1)(r+2)$ , in ar ticular, there is no zero mode spinor ofand the $sp$ ectrum are symmetric rela tive to .
$[K ]$$of \mathcal{P}axe\pm(\frac{3}{2}+r)$$\cdot$$\mathcal{P}$

$p$

$0$

Here we note corresponding subjects on the other coordinate neigh. The transition function to describe the bundle Spin $(M)$ isborhoodandand a spinor on $M$ is a pair of. The matrix
that are patched by
has the same form as those
representations of the Dirac operator onofin (1-5) but the first and the second are changed since a section on“). This is(resp.
the bundle(resp.
) is valued inontheorem. The counterpart of is defined as.
, we have. For aand$e$$\hat{C}_{w}^{2}$-$\varphi(z)\in\Gamma(C_{z}^{2}\cross\Delta)$${}^{t}(\gamma_{0})=-\overline{\gamma}_{0}$$\varphi\wedge(w)\in\Gamma(\hat{C}^{2}\cross\triangle)$$\varphi\wedge(v(z))=-(\gamma_{0}\varphi)(z))$$\hat{C}^{2}\subset M$$\hat{C}^{2}$$S^{+}$$S^{-}$$\triangle^{-}$$\mathcal{P}$$\hat{\varphi}\in\Gamma(\hat{C}_{w}^{2}\cross\triangle^{-})$$\prime cPT$$\triangle^{+}$$\mathcal{P}=(\gamma_{0}|S^{+}\underline{)\sum}\theta_{1}\nabla_{\theta\underline{:}}acting$$D\varphi\wedge=D\varphi$$\varphi\in\Gamma(C_{z}^{2}\cross\triangle^{+})$$\mathcal{P}\varphi=\mathcal{P}\varphi\wedge$

2 Extension of spinors from the equatora Let $H$ be the space of square integrable spinors of positive chirality on$E$ .
Let $H\pm be$ the closed subspace of $H$ spanned by the eigenvectors
corresponding to the positive (resp. negative) eigenvalues of .
$\phi_{\lambda}$$\lambda$

Puteigenspinors of$c(r, q, k)=( \frac{q!k!(r-k)!}{(r+1-q)!})^{-\frac{1}{2}}$$\mathcal{P}$$\mathcal{P}$. Then a complete orthonormal system ofis given by$\{c(r, q, k)\phi_{k,r-k}^{q},$Take an eigenspinor$c(r, q, k)\pi_{q}^{r-k,k},$ $r\geq 0,0\leq k\leq r,$$\varphi_{\lambda}$

and extend it by$0\leq q\leq r+1\}$$\Phi_{\lambda}(z)=r_{\lambda}(|z|)\varphi_{\lambda}(\frac{z}{|z|})$. Thenis a zero-mode spinor of $D$ on
proved by the following calculus:$r_{\lambda}(t)=t^{\lambda-\frac{3}{2}}( \frac{1+t^{2}}{2})^{\frac{3}{2}}$$\Phi_{\lambda}(z)$.

to $C^{2},where$$C^{2}$. This is$D\Phi_{\lambda}(z)=\gamma_{0}(n-\mathcal{P})(\Phi_{\lambda}(z))$.

$= \gamma_{0}((1+|z|^{2})r_{\lambda}’(|z|)-(\lambda-\frac{3}{2})\frac{1+|z|^{2}}{|z|}r_{\lambda}(|z|)-3|z|r_{\lambda}(|z|))\varphi_{\lambda}(\frac{z}{|z|})$But$r_{\lambda}(t)$satisfies the equation$(1+t^{2})r_{\lambda}’(t)-( \lambda-\frac{3}{2})\frac{1+t^{2}}{t}r_{\lambda}(t)-3tr_{\lambda}(t)=0$.
ThereforeLet(resp.
$D$
operator (resp.

.

$D\Phi_{\lambda}=0$$\mathcal{N}(U)$$D\dagger$$\mathcal{N}^{\uparrow}(U)$

) on

$U$

) be the space of zero-mode spinors of Diracthat have-boundary values.
$L^{2}$

68$T$

heor em 2 [K]. Let $R=\{z\in c^{2};|z|<1\}$ and $\hat{R}=\{w\in\hat{c}^{2};, |w|<1\}$ .
,is isomorphic to(1),(2) H-is isomorphic to(3) Every spinor in $H$ is equal to the difference of the restrictions of zeromode spinors on $R$ and on .
$\mathcal{N}(R)$$H_{+}$$\mathcal{N}(\hat{R})$$\hat{R}$$Pr$

oof: Let $\varphi\in H+and$ expand it in;$\Phi(z)=\sum_{\lambda>0}a_{\lambda}\Phi_{\lambda}(z)$$\Phi_{m}^{n}=\sum_{\lambda=m+\frac{3}{2}}^{n+\frac{3}{2}}a_{\lambda}\Phi_{\lambda}$. The spinor on $R$
is well defined. In fact, consider the finite sum;. Then$<\Phi_{m}^{n},$$\varphi=\sum_{\lambda>0}a_{\lambda}\phi_{\lambda}$

is subharmonic on $R$ and is$\Phi_{m}^{n}>(z)$-norm on $E$ , hence converges
dominated by some constant multiple of its
there to compact uniformly as $m,$ tend to infinity. If we note the fact that-boundary value.
each component ofis harmonic we see that it has$E$and let be its restriction to . We can show that
Conversely let
the eigenfunction expansion of by
can not contain the term with $\lambda<0$, $t\geq 0$ ,andAs for (2) consider the function
where $- \mu=-r-\frac{3}{2}r=0,1,$
and do the same argument as in (1).
Relations in transform the result to that onLet
be the space of square integrable spinors of negative chirality$(\gamma_{0}|S^{+})H=H^{*}$ , $(\gamma_{0}|S^{-})H^{*}=H$ . We shallon E.
switches $H$ anddefine $H_{+}^{*}=(\gamma_{0}|S^{+})H+andH_{-}^{*}=(\gamma_{0}|S^{+})H_{-}$ .
Let
and suppose thatis an eigenspinor belonging
to a negative eigenvalue $\lambda=-(r+\frac{3}{2})$ . Let, where$L^{2}$$0$$n$$L^{2}$$\Phi$$\Phi\in \mathcal{N}(R)$$\varphi$$\{\phi_{\lambda}\}$$\varphi$$r_{-\mu}(t)=t^{\mu-\frac{3}{2}}( \frac{1+t^{2}}{2})^{\frac{3}{2}}$$\varphi\in H+\cdot$$\cdots$

バ

$e$

$b$

$H^{*}$$H^{*};$$\gamma_{0}$$\psi*\in H_{-}^{*}$$\psi=(\gamma_{0}|S^{-})\psi^{*}$$\Psi(z)=s_{\lambda}(|z|)\psi(\frac{z}{|z|})$$s_{\lambda}(t)=t^{-(\lambda-\frac{3}{2})}( \frac{2}{1+t^{2}})^{\frac{3}{2}}$$C^{2},$. Then as before we can verify that$\Psi(z)$extend$\psi$

to$\Psi(0)=0$ and $D^{\uparrow}\psi^{*}=(n+P)\gamma_{0}\psi^{*}=(n+P)\psi=0$ .

Thus in the same manner as in Theorem 2 we have the following;$Th$ 伽科 $rem3$.

(1) $H$ 二 is isomorphic to,
is isomorphic to(2),(3) very spinor inis equal to the difFerence of the restrictions of zeromode spinors on $R$ and on$\{\phi\in \mathcal{N}^{\uparrow}(R);\phi(0)=0\}$$\{\psi\in \mathcal{N}^{t}(\hat{R});\psi(\hat{0})=0\}$$H_{+}^{*}$$E$$H^{*}$$\wedge$

From the definition $<\phi,$ $\psi>=0$ for all $\phi\in H$ and.
Let and be spinors on $R=\{|z|\leq 1\}$ , Stokes’ theorem statts;$\psi\in H^{*}$$c$$\phi$$\psi$$\int_{R}\frac{1}{(1+|z|^{2})^{4}}(<D\phi, \psi>+<\phi, D^{\uparrow}\psi>)dV=\frac{1}{8}\int_{E}<\phi,$ $\gamma_{0}\psi>d\sigma$.

69
Theorems 2,3 and Stokes’ theorem yield immediately that$\int_{E}<\phi,\gamma_{0}\psi>d\sigma=0$

for $\phi\in H_{+},$$\psi\in H_{-}^{*}$.

$\int_{E}<\phi,\gamma_{0}\psi>d\sigma=0$

for $\phi\in H_{-},$$\psi\in H_{+}^{*}$.

Similarly

The coupling betweeen
and $H\pm does$ not vanish and is important toconstruct the geometric model of conformal field theory on
which will betreated in the next paper.
Actually eigenspinors; $\lambda>0$ are extended toand those for$\lambda<0$ are extended to. We list here a table of expansion formula forforandfor $\lambda<0$ .
$H_{\pm}^{*}$$S^{4}$$d$$\mathcal{N}(C^{2})$$\phi_{\lambda}$$\mathcal{N}(\hat{C}^{2})$$\phi_{\lambda},$$\lambda>0$$\phi_{\lambda}^{*}=\gamma_{0}\phi_{\lambda}$(1)$\pi_{\lambda}\pi_{\lambda}^{*}=\gamma_{0}\phi_{\lambda}$$\Phi_{\lambda}(z)=|z|^{\lambda-\frac{3}{2}}(\frac{1+|z|^{2}}{2})^{\frac{3}{2}}\phi_{\lambda}(\frac{z}{|z|})\in \mathcal{N}(C^{2}),$$\lambda>0$

and$\Phi_{\lambda}(z)=\phi_{\lambda}(z)$for $|z|=1$ .

(2)$\overline{\Phi_{\lambda}^{*}}(w)=|w|^{\lambda+\frac{3}{2}}(\frac{2}{1+|w|^{2}})^{\frac{3}{2}}\overline{\phi_{\lambda}^{*}}(\frac{w}{|w|})\in N^{\uparrow}(\hat{C}^{2})_{0},$$-\overline{\gamma_{0}\phi_{\lambda}^{*}}(z)$

(3)
(4)

and

$\overline{\Phi_{\lambda}^{*}}(-\overline{z})=$$\lambda<0$and$\hat{\Pi}_{\lambda}(-\overline{z})=$for $|z|=1$ .

$\hat{\Pi}_{\lambda}(w)=|w|^{-\lambda-\frac{3}{2}}(\frac{1+|w|^{2}}{2})^{\frac{3}{2}\wedge}\pi_{\lambda}(\frac{w}{|w|})\in \mathcal{N}(\hat{C}^{2}),$$-\overline{\gamma_{0}\pi_{\lambda}}(z)$$\lambda>0$

for $|z|=1$ .

$\Pi_{\lambda}^{*}(z)=|z|^{-\lambda+\frac{3}{2}}(\frac{2}{1+|z|^{2}})^{\frac{3}{2}}\pi_{\lambda}^{*}(\frac{z}{|z|})\in \mathcal{N}^{\uparrow}(C^{2})_{0},$$\lambda<0$

and$\Pi_{\lambda}^{*}(z)=\pi_{\lambda}^{*}(z)$for $|z|=1$ .

Ref ff 伽寡 伽歌$C$[K] Kori, T., Dirac$S^{3}$.

$Univff\dot{s}ty$$\sqrt at\alpha s\alpha\rceil S^{4}$

and $mS^{3}$ .

Ct Wada, Slinjuku-ku, $T*W$$ln$nite dimen 白\alpha lal Gr

mani$a\mathfrak{n}$on