数理解析研究所講究録 1211 巻 2001 年 133-142

133

Explicit formulas for the reproducing kernels of thespace of harmonic polynomials in the case of real rank 1

Ryoko Wada
Kure University

Introduction.
be the space of homogeneous harmonic polynomials onof degree$(p\in \mathrm{N}, p\geq 2)$ . It is well known that the restriction mapping
is abijection
from Hn (Cp) onto $H_{n}(S^{p-1})$ , where $H_{n}(S^{p-1})$ is the space of spherical harmonics of degree
in dimension . This fact can be extended to the following form:Let$H_{n}(\mathrm{C}^{p})$$\mathrm{C}^{p}$$n$$farrow f|_{S^{\mathrm{p}-1}}$

$7l$

$p$

Theorem 0.1 (cf. [2], [5], , [11]). Let 0 $(\neq\{0\})$ be any SO(p)-orbit in
restric $tr,on$ mapping : $farrow f|_{0}$ is a bijection fromonto$[\overline{/}]$$H_{n}(\mathrm{C}^{p})$$r_{0}$$\mathrm{C}^{p}$$H_{n}(\mathrm{C}^{p})|_{0}$. Then, the.

In addition, we can express the inverse formula of this map
explicitly as an integral on
the orbit 0, by using tlle Legendre poynomials (for details, see [2], [5], [7], [11]).
On tlle other hand, according to the formulation in [4], classical harmonic polynomialson
can be canonically identified with the harmonic polynomials on the space , whereso(p, )
is the Cartan decomposition of the Lie algebra so(p, 1) and is thecomplexification of. Ill this situation, any SO(p)-orbit incorresponds to a
. Therefore, Theorem 0.1 can be reformulated in thisorbit in , where
Lie algebraic form, and we can expres5 the inverse formula
explicitly froni thisstandpoint (Theorem 1.2).
In this note we shall give explicit reproducing formulas of harmonic polynomials on each
andsingle-orbit0for two remaining classical real rank, 1casesand show that the similar results as in Theorem 0.1 hold for these cases. Thisis an extension of our previous note [10], where we expressed the inverse formula as anorbit in .
integral on every nilpotent$r_{\mathrm{O}}$$\mathrm{C}^{p}$$\mathfrak{p}$$1$$=\mathrm{g}_{\mathrm{R}}+\mathfrak{p}_{\mathrm{R}}$$\mathfrak{p}$$\mathrm{C}^{p}$$I\iota_{\mathrm{R}^{-}}^{r}$$\mathfrak{p}_{\mathrm{R}}$$\mathfrak{p}$$I\mathrm{i}_{\mathrm{R}}-=\exp \mathrm{a}\mathrm{d}\mathrm{P}_{\mathrm{R}}$$r_{\overline{\mathrm{o}}^{1}}(f)$$\mathfrak{g}_{\mathrm{R}}=\mathrm{s}\mathrm{u}(\mathrm{p}, 1)$$I\iota_{\mathrm{R}}^{r}$$\mathrm{s}\mathrm{p}(\mathrm{p}, 1)$$(p\geq 1)\backslash$$I\mathrm{f}_{\mathrm{R}}$$\mathfrak{p}$\S 1. Harmonic polynomials on .
$\mathfrak{p}$andIn this section we fix notations which we use in this note and recall thethe results on harmonic polynomials on .
be anoiicompact real form of , and let
Let 9be acomplex semisimple Lie algebra,. We put $G$
be the complexification of the Cartan decomposition$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}_{11}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$$\mathfrak{p}$$\mathfrak{g}_{\mathrm{R}}$$=\mathrm{g}$$\mathfrak{g}$$+\mathfrak{p}$$\mathfrak{g}$$\mathfrak{g}_{\mathrm{R}}=\mathrm{P}_{\mathrm{R}}+\mathfrak{p}_{\mathrm{R}}$134, where:andis defined by& $=1$ on and$\theta=-1$ on.
. Let If be the identity component of . Then we have
Now we define harmonic polynomials on . anddenote the spaces of polynomials on, $gf$
andpolynomials on of degree , respectively. For $f\in S$ andis defined by (gf)(.Y) $=f(g^{-1}X)(X\in \mathfrak{p})$ . denotes the ring of A-invariant polynomials.
on and we put $J_{+}=\{f\in J;f(0)=0\}$ . It is known that is also
According to the definition in [4], $f\in S$ is harmonic if and only if $(\partial P)f=0$ for any$P\in J_{+}$ .
denotes the space of homogeneous harmonic polynomials on of degree .
We put $\mathrm{Z}_{+}=\{0,1,2, \cdots\}$ . The following results are known:$I\mathrm{f}_{\theta}=\{g\in G’ ; fig=g\theta\}$$=\exp \mathrm{a}\mathrm{d}\mathfrak{g}$$\theta$$\mathrm{f}$$arrow \mathfrak{g}$$\mathfrak{g}$$I\mathrm{f}=\exp \mathrm{a}\mathrm{d}t$$I\mathrm{f}_{\theta}$$\mathfrak{p}$$S$$\mathfrak{p}$$1_{1}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}$$\mathfrak{p}$$S_{n}$$n$$\mathfrak{p}$$g\in I\iota_{\theta}^{-}$

$J$

$J$

$\mathfrak{p}$$I1_{\theta}^{-}\mathrm{A}\mathrm{V}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{t}$$n$$\mathfrak{X}_{n}$$\mathfrak{p}$Theorem 1.1 (cf. [1], [4]). (i) For any n$\in \mathrm{Z}_{+}we$$S_{n}=(J_{+}S)_{n}\oplus \mathrm{J}\mathrm{t}_{n}$have,where $(J_{+}S)_{n}=S_{n}\cap J_{+}S$ .
(ii) We put; $P(X)=0$ for any $P\in J_{+}$ } and{for$\{h(,
Z)^{n};
Z\in\mathrm{N}\}$.
. Then is generated by(iii) Let be a maximal dimensional -orbit in . Then the restriction mapping $farrow$.
onto
is a bijection from$\mathrm{N}$$X,\mathrm{Y}\in \mathfrak{p}$$=$$X\in \mathfrak{p}$$\mathrm{P}(\mathrm{X})\mathrm{Y})=\mathrm{T}\mathrm{r}$$(’ X\overline{1\vee})$$\mathrm{J}\{_{n}$$\Gamma$$I\dot{\mathrm{t}}_{\theta}$$\mathfrak{p}$$?\mathrm{t}_{n}$$f|_{\Gamma}$$\mathfrak{X}_{n}|_{\Gamma}$

For harmonic polynomials on for general semisimple Lie algebras , see [4].
,From now we consider the case whereis classical real rank 1, i.e.,, $p\geq 2$ , unless otherwise stated.
. In this note we assume thator(:
. Let $B(X, 1^{-})$
Letandbe the adjoint group of . We consider
be the Killing form of 9. Then, is generated by $B(X,X)(X\in \mathfrak{p})$ . We put$\Sigma=\{X\in \mathfrak{p} ;B(X,X)=1\}$ and. In this case we see thatconsists of(see [5], [8], [9]).
one-orbit and that$=so(p, 1)$ ,is generated by
Now we recall the results in the case of so(p, 1). When
. We put$\mathfrak{p}$$\mathfrak{g}$$\mathfrak{g}_{\mathrm{R}}=so(p, 1)$$\mathfrak{g}_{\mathrm{R}}$$\epsilon \mathrm{u}(p, 1)$$p\in \mathrm{N}$$\epsilon \mathfrak{p}(p, 1)$$I\mathrm{i}_{\mathrm{R}}$$I\dot{\mathrm{i}}_{\mathrm{R}}$$\mathrm{t}_{\mathrm{R}}$$I\dot{\mathrm{c}}$$\mathfrak{p}_{\mathrm{R}}\subset \mathfrak{p}$$J$$(X,\mathrm{Y}\in \mathfrak{g})$$\Sigma_{\mathrm{R}}=\Sigma\cap \mathfrak{p}_{\mathrm{R}}$$\Sigma_{\mathrm{R}}$$\Re_{n}\simeq g\{_{n}|\mathrm{z}_{\mathrm{R}}$$I\mathrm{t}_{\mathrm{R}}^{-}$$J$$\emptyset \mathrm{n}$$P(X)=\mathrm{T}\mathrm{r}$$(^{t}XX)(X\in \mathfrak{p})$$(X,\mathrm{Y}\in \mathfrak{p})$$Qn,p \{X,Y)=2^{-n}P_{n,p}(\frac{h(X,\mathrm{Y})}{(P(X)P(\overline{\mathrm{Y}}))^{1/2}})(P(X)P(\overline{\mathrm{Y}}))^{n/2}$

whereis the Legendre polynomial of degree
results hold:$P_{n,p}$$n$,

and dimension . Then the following$p$$=so(p, 1)$ $(p\geq 2)$ .
Theorem 1.2 (cf. [2], [5], [7], [11]). Assume that
: $farrow f|_{0}i,s$-orbit in . Then the restriction mapping(i) Let $0(\neq\{0\})$ be a.
onto
bijection from(ii) We have for$\mathrm{g}\mathrm{R}$$I\mathrm{f}_{\mathrm{R}}$$9\mathrm{t}_{n}$$r_{\mathrm{O}}$$\mathfrak{p}$$\mathrm{J}\{_{n}|_{0}$$f\in \mathrm{J}\mathrm{t}_{n}|_{0}$$r_{\overline{\mathrm{o}}^{1}}f(X)= \dim \mathfrak{X}_{n}\int_{0}f(Y)Q_{n,\mathrm{p}}(X, \mathrm{Y})d;r_{\mathrm{O}}(\mathrm{Y})$(X$\in \mathfrak{p})$

,

$a$

135
where

$d\mu_{\mathrm{C}\mathrm{J}}$is the normalized$I\iota_{\mathrm{R}}^{-}$-invarint measure on 0.

\S 2. Integral formulas of harmonic polynomials: The case of zu(p, 1).
In this section we give the reproducing kernel ofofandIn this case we have$\mathfrak{g}$$=\epsilon 1(p+1, \mathrm{C})$$\mathfrak{g}_{\mathrm{R}}=su(p, 1)(p\in \mathrm{N}, p\geq 2)$)$\mathrm{t}_{\mathrm{R}}=$($\}^{(}$$\mathfrak{p}_{\mathrm{R}}=$$\mathrm{f}$$\mathfrak{p}$

$=\{$

$=$

$\{$

$\mathfrak{X}_{n}$,on any$I1_{\mathrm{R}}^{-}$-orbit in$\mathfrak{p}$in the case.

$\alpha\in \mathrm{u}(1)$, Tr $A+\alpha=0\}$\prime\prime$t0 \frac{0}{x}A$$\alpha_{0}\mathrm{o}_{X)}$$.,jx\in \mathrm{C}^{p}\}A\in \mathrm{u}(p)$’,; $A\in M(p, \mathrm{C})$ , Tr $A+\alpha=0\}$ ,$(\begin{array}{ll}A 00 \alpha\end{array})$$(\begin{array}{ll}0 x{}^{t}y 0\end{array})$

;

$x$

,

,

$y\in \mathrm{C}^{p}\}$,gives agenerator of . We put $\mathrm{N}=\{X\in \mathfrak{p} ;P(X)=0\}$ ,.
andis thespace of homogeneous harm onic polynomials on of degree . For
weand$I\mathrm{t}_{\mathrm{R}}^{r}=\mathrm{A}\mathrm{d}$. For$S(U(p)\cross \mathrm{U}(1))=\{\mathrm{A}\mathrm{d}(\begin{array}{ll}A 00 1\end{array}) ; A\in U(p)\}$$\Sigma=\{X\in \mathfrak{p} j\mathrm{F}(\mathrm{X})=1\}$$\Sigma_{\mathrm{R}}=\Sigma\cap \mathfrak{p}_{\mathrm{R}}$$\mathfrak{X}_{n}=\{f\in S_{n} ; \sum_{j=1}^{p}\frac{\theta^{2}}{\partial x_{\mathrm{j}}\theta y\mathrm{j}}f=0\}$$n$$\mathfrak{p}$$\Psi$:$arrow \mathrm{C}^{2\mathrm{p}}$$\mathfrak{p}$and we have$f\circ\Psi^{-1}\in H_{n}(\mathrm{C}^{2\mathrm{p}})$(,$(^{\frac{0}{A}}y)$$Ax0$$\Psi(X)=\frac{1}{2}$ $(\begin{array}{l}x+yi(y-x)\end{array})$. Then$X=(\begin{array}{ll}0 x\prime y 0\end{array})$$H_{n}( \mathrm{C}^{2p})=\frac{2(n+p-1)(n+2p-3)’}{n!(2p-2)!}.$$\Sigma_{\mathrm{R}}$$g=\mathrm{A}\mathrm{d}(\begin{array}{ll}A 00 1\end{array})$$\sqrt{1-r^{2}}t0e_{2}$$E_{1}=(\begin{array}{ll}0 e_{\mathrm{l}}\iota_{e_{\mathrm{l}}} 0\end{array})$$\in$$\cdots$$(0\leq r\leq 1)$ ,$re_{1}0)\in \mathrm{N}$$\in\Sigma_{\mathrm{R}}$,$\overline{E}_{q}..,=(\begin{array}{ll}0 f’ e_{1}{}^{t}((1/\iota\cdot)e_{\mathrm{l}}+qe_{2}) 0\end{array})$

where $e_{1}=$ (10$\in\Sigma$$(r>0, q\geq 0)$0), and $e_{2}={}^{t}(01\cdots 0)$ . Then we have
$I\mathrm{i}_{\mathrm{R}}-E_{1}=\Sigma_{\mathrm{R}}$.

is bijective and $\overline{H}_{n}(X, 1’)=Q_{n,2\rho}(\Psi(X), \Psi(1’’))$(see [8] Proposition 2.1).
It $\mathrm{R}(A\in U(p))$ we have $gX=$). We put$\overline{E}_{r}=($$\in \mathfrak{p}$if all(l only if$f\in \mathrm{J}\{_{n}$$\Sigma_{\mathrm{R}}arrow S^{2p-1}$$\mathfrak{X}_{n}$$\in \mathfrak{p}$by$\dim \mathfrak{X}_{n}=\mathrm{d}\mathrm{i}\mathrm{m}$mark that the mapping :is tlle reproducing kernel ofon
For $X=$
and$\Psi$${\rm Re}$$(\begin{array}{ll}0 x{}^{t}y 0\end{array})$$\in \mathfrak{p}$$J$$P(X)= \frac{1}{2}\mathrm{T}\mathrm{r}(X^{2})={}^{t}yx$define tlle bijection$X=(\begin{array}{ll}0 x{}^{t}y 0\end{array})$and$=\mathrm{N}\mathrm{U}$$\mathfrak{p}$$\cup$$\lambda\in \mathrm{C}\backslash \{0\}$

Al.

,

136
Remark thatand N$\Sigma=\cup I\dot{\mathrm{i}}_{\mathrm{R}}\tilde{E}_{r,q}q\geq 0,r>0$

Proposition 2.1 (cf. [10]). (i) For any$f\in \mathrm{J}\mathrm{t}_{n,k}$

and$X\in \mathfrak{p}$we have.

(2.1)$\delta_{n,m}\delta_{k,l}f(X)=\dim \mathrm{J}\{_{n,k}\int_{I\mathrm{s}_{\acute{\mathrm{R}}}}f(gE_{0})\tilde{I\mathrm{f}}_{m,l}(X,gE_{0})dg$(ii) For any $f\in Jin,k$ and(2.2)$=\cup I\iota_{\mathrm{R}}^{-}(p\tilde{E}_{f})\rho\geq 0,0\leq r\leq 1$$h\in f\mathrm{f}_{m,l}$toe have$\int_{K_{\mathrm{R}}}f(gE_{1})\overline{h(gE_{1})}dg=\delta_{n,m}\delta_{k,l}$ $(\begin{array}{l}k+p-2k\end{array})(\begin{array}{ll}n +_{l)}-2 k\end{array})$

Now we define$\tilde{H}_{n,k}(X, Z)(X, Z\in \mathfrak{p})$.

by$\tilde{H}_{n,k}(X, Z)=\dim \mathfrak{X}_{n,k}$ $(\begin{array}{ll}n+p -2k \end{array})(\begin{array}{l}k+p-2k\end{array})$

Clearly we have$\int_{K_{\mathrm{R}}}f(gE_{0})\overline{/\iota(gE_{0})}dg$$\tilde{H}_{n,k}(, Z)\in \mathfrak{X}_{n}(Z\in \mathfrak{p})\backslash$$\int_{K_{\mathrm{R}}}\tilde{I\mathrm{f}}_{n,k}(X,gE_{0})\tilde{I\mathrm{i}}_{n,k}(gE_{0}, Z)dg$.

ind$\tilde{H}_{n,k}(X, Z)=\tilde{H}_{n,k}(Z, X)$,$\tilde{H}_{n,k}(gX,gZ)=\tilde{H}_{n,k}(X,Z)$$(g\in \mathrm{A}’ \mathrm{R})$.

We shall show that the reproducing kernel ofon each-orbit can |)e expressed in, Z)(Z(n,terms of. Our main theorem in this section is tlxe following$\mathfrak{X}_{n}$$\tilde{H}_{n,k}($$\in \mathfrak{p},$$I\mathrm{f}_{\mathrm{R}}$$k)\in\Lambda)$

Theorem 2.2. Letand(i) The restriction mapping ’.0 : $farrow f|_{0}$ is a bijection from(ii) Fortoe have$0=I\mathrm{i}_{\mathrm{R}}X_{0},$$-’\mathrm{Y}_{0}\in \mathfrak{p}$$\tilde{H}_{n,k}(X_{0},X_{0})\neq 0(^{\forall}(n, k)\in\Lambda)$$g\{_{n}$onto$\mathfrak{X}_{n}|_{0}$$f\in \mathfrak{X}_{n}|_{0}$

(2.3)

.

.

$r_{\overline{o}^{1}}(f)(X)= \sum_{k=0}^{n}\frac{\dim \mathrm{J}\mathrm{t}..,k}{\tilde{H}_{n,k}(X_{0},X_{0})}\int_{K_{\mathrm{R}}}f(gX_{0})\tilde{H}_{n,k}(X,gX_{0})dg$.

137
To prove Theorem 2.2 we need the following lemma.

Lemma 2.3. Let}. If
for any$g\in I1_{0}$$I\mathrm{t}_{0}^{-}$

be the isotropy group of, we have$E_{1}$

inand$IC_{\mathrm{R}}$$\mathfrak{X}_{n,k}’=\{f\in \mathrm{J}\mathrm{f}_{n,k}$$f\in \mathrm{J}\mathrm{t}_{n,k}’$(2.4)$f=f(E_{1})\tilde{H}_{n,k}(, E_{1})$.

Sketch of Proof. From [8] Lemmas 2.4 and 2.5 and the definition of(2.4).

Proof of Theorem 2.2. From (2.1)-(2.4) we have for any
(2.5)

dirn

because$\mathrm{J}\mathrm{t}_{n,k}$

we can prove
q.e.d.

$X_{0}\in \mathfrak{p}$.

$\mathfrak{X}_{n,k}\int_{K_{\mathrm{R}}}\tilde{H}_{m},\iota(gX_{0}, E_{1})\overline{H}\ldots k(X, gX_{0})dg=\delta_{n,m}\delta_{k,l}\tilde{H}_{n,k}(-\cdot \mathrm{Y}_{0},\lambda_{0})\overline{H}_{n,k}(X, E_{1})$(2.5) implies that for any(2.6); $gf=f$$f\in \mathfrak{X}_{m,l}$

and any$z\mathrm{Y}_{0}^{-}\in \mathfrak{p}$$\dim \mathrm{J}\{_{n,k}\int_{I\mathrm{s}_{\acute{\mathrm{R}}}}f(g_{A}\mathrm{Y}_{0})\overline{H}_{n,k}(X, g\lambda_{0}^{r})dg=\delta_{n,m}\delta_{k,l}\overline{H}_{n,k}(X_{0:}X_{0})f(X)$$\overline{H}_{n,k}($.

$E_{1})$

is agenerator of$\mathrm{J}\{_{n,k}$. (2.6) gives Theorem 2.2.

q.e.d.

Remark 2.4. From the definition it is valid that$\overline{H}_{n,k}(-\mathrm{Y}_{0}’, \lambda_{0}^{-})=0$

iff$\int_{I\mathrm{f}_{\mathrm{R}}}|I\mathrm{t}_{r\iota.k}^{-}(g\lambda_{0}^{-}, E_{0})|^{2}dq-.(,\mathrm{Y}_{0}^{-}\in \mathfrak{p})$.

Therefore the following two conditions (2.7) and (2.8) are equivalent.
(2.7)
(2.8)

Tliis implies that$\overline{H}_{n,k}(\lambda_{0}, \lambda_{0})\neq 0$$\tilde{H}_{n,k}(X_{0,\lrcorner}\mathrm{Y}_{0})=0$,$\mathfrak{X}_{n,k}|_{I\acute{\mathrm{z}}_{\mathrm{R}}X_{0}}=\{0\}$.

if and only if$-\mathrm{Y}_{0}\not\in\lambda I\mathrm{f}_{\mathrm{R}}\tilde{E}_{1}$and$X_{0}\not\in\lambda I\mathrm{e}_{\mathrm{R}}^{r}\tilde{E}_{0}$(A$\in \mathrm{C}$1).
\S 3. Integral formulas of harmonic polynomials: The case of
In this section we consider the case
. From now we put).

$\mathrm{s}\mathrm{p}(\mathrm{p},$$\epsilon \mathfrak{p}(p, 1)(p\in \mathrm{N}, p\geq 2)$$\epsilon \mathfrak{p}(p+1, \mathrm{C})$,$\mathfrak{g}_{\mathrm{R}}=g\mathfrak{p}(p,, 1)$$\mathrm{P}_{\mathrm{R}}=\{$$\mathfrak{p}_{\mathrm{R}}=\{$,$(\begin{array}{llll}A’ 0 B 00 a 0 b-\overline{B} 0 \overline{A}00 -\overline{b}0\overline{a}\end{array})$$(\begin{array}{llll}0 .\iota 0 yt_{\overline{X}}0 {}^{t}y 00\overline{y} 0 -\overline{..\iota\cdot}{}^{t}\overline{.y}0 -^{t}x 0\end{array})$,;$A\in \mathrm{u}(p)B\mathrm{i}\mathrm{s}pxp$

.
$|$

$x$

,

$a\in \mathrm{u}(1)b\in \mathrm{C}\mathrm{s}\mathrm{y}_{1}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\}$$y\in \mathrm{C}^{p}\}$.

,$\mathfrak{g}$

$=$

138
Then we have$\mathrm{f}$$\mathfrak{p}$$=\{$$(\begin{array}{llll}A 0 B 00 \alpha 0 \beta C 0 -^{t}A 00 \gamma 0 -\alpha\end{array})$$=\{$$(\begin{array}{llll}0 x 0 wt/t0 {}^{t}w 00z 0 -yt_{Z}0 -^{t}x 0\end{array})$;$\alpha,\beta,\gamma\in \mathrm{C}A,B,C\in hI(p {}^{t}B=B,{}^{t}C=C’ \mathrm{C})\}$; , , ,
$x$

$y$

$z$

$w\in \mathrm{C}^{p}\}$

,

,

and

$I\mathrm{f}_{\mathrm{R}}=\{\mathrm{A}\mathrm{d}$For $X=($$(\begin{array}{llll}A 0 B 00 \alpha 0 \beta-\overline{B} 0 \overline{A}00 -\overline{\beta} 0\overline{\alpha}\end{array})\in \mathrm{A}\mathrm{d}$,$\iota_{0},0yz0\epsilon 0\mathrm{z}$$-,yu00)\in \mathfrak{p}$$-\iota 0_{l}0w_{l}$$.\mathrm{Y}$$\in \mathfrak{p}$can see that

we define the mapping ?:$f\in \mathfrak{X}_{n}$if and only if.
$arrow \mathrm{C}^{4p}$$\mathfrak{p}$by$f\mathrm{o}\Psi^{-1}\in H_{n}(\mathrm{C}^{4p})$$\dim \mathrm{J}\mathrm{t}_{n}=\dim H_{n}(\mathrm{C}^{4p})=\frac{2\{n+2p-1)(n+4p-3)!}{n!(4p-2)!}$. We put$\mathrm{P}\{\mathrm{X}$)$= \frac{1}{2}$$(\begin{array}{l}x+yz+wi(y-r_{\prime})i(w-z)\end{array})$. We

and from this fact, we have$\mathrm{N}=\{X\in \mathfrak{p} ;P(X)=0\}$andandRemark that ?:$=Q_{n,4p}(\Psi(X), \Psi(\mathrm{Y}))$ is the reproducing kernel on(see [9] Theorem 2.2).
$\{X\in \mathfrak{p} ;P(X)=1\}$.

, $P(X)= \frac{1}{4}\mathrm{b}X^{2}={}^{t}xy+{}^{t}zw$ gives agenerator of.I and$J \mathrm{t}_{n}=\{f\in S_{n} ; \sum_{j=1}^{p}(.\frac{\theta^{2}}{\partial x_{\mathrm{j}}\theta y_{j}}+\frac{\theta^{2}}{\theta z_{\mathrm{j}}\theta u_{\mathrm{j}}\prime})f=0\}$For$U(2p+2);\alpha\overline{\alpha}+/f\overline{/\mathit{3}}=1tA{}^{\mathrm{t}}A_{\frac{\overline A}{B}}+{}^{t}B=I_{p}=^{t^{\frac{\overline B}{B}}}A,’\}$$\Sigma_{\mathrm{R}}\simeq S^{4p-1}$$\Sigma_{\mathrm{R}}=\sim\nabla\cap \mathfrak{p}_{\mathrm{R}}.\cdot$,$\Sigma=$$\overline{H}_{n}(X, 1’)$$\Sigma_{\mathrm{R}}$Let$g=\mathrm{A}\mathrm{d}$$(- \overline{B}A00-\overline{\beta}\alpha 00\frac{B0}{A,0},$$\frac{\beta 00}{\alpha})\in I\iota_{\mathrm{R}}^{-}$. If we put$\Phi(X)=(\begin{array}{l}xyzw\end{array})$$\in \mathrm{C}^{4p}$, we have.

$\Phi(gX)=(\begin{array}{l}A(\overline{\alpha}x+\overline{\beta}w)+B(\overline{\alpha}z-\overline{\beta}y)\overline{B}(-\beta x+\alpha w)+\overline{A}(\alpha y+\beta z)-\overline{B}(\overline{\alpha}x+\overline{\beta}w)+\overline{A}(\overline{\alpha}z-\overline{\beta}y)A(-\beta x+\alpha w)-B(\alpha y+\sqrt z)\end{array})$

We put E-r$=\Phi^{-1}$ $(\begin{array}{l}re_{1}0\sqrt{1-r^{2}}e_{2}0\end{array})\in er$$(0\leq r\leq 1),\tilde{E}_{r,q}=\Phi^{-1}$$(\begin{array}{l}re_{1}(\mathrm{l}/r)e_{1,0},+qe_{2}0\end{array})\in\Sigma$(’. $>0$ ,139
It is clear that

. Remark that$=\mathrm{N}\mathrm{t}\mathrm{J}$$\bigcup_{\lambda\in \mathrm{C}\backslash \{0\}}\lambda\Sigma$$\mathfrak{p}$and$\mathrm{N}=\cup I\mathrm{e}_{\mathrm{R}}^{-}(q\overline{E}_{r})q\geq 0,1/\sqrt{2}\leq r\leq 1$give the orbit decompositions of
we have$\mathrm{N}$and$\Sigma$$\Sigma=\cup I\mathrm{f}_{\mathrm{R}}\tilde{E}_{r,q}q\geq 0,\mathrm{r}>0$, respectively. We put([9] Lemma 2.1). For $X=\Phi^{-1}$$\Sigma_{\mathrm{R}}=It_{R}^{r}E_{l}$$(\begin{array}{l}lyzw\end{array})$,$E_{1}=\overline{E}_{1,0}\in\Sigma_{\mathrm{R}}$$\mathrm{Y}=\Phi^{-1}$$(\begin{array}{l}x’y’z’w\end{array})\in \mathfrak{p}$. Thenwe put,$(X, 1’ \rangle’=\frac{1}{2}\mathrm{T}\mathrm{r}(^{t}X\overline{\mathrm{Y}})=\Phi(X)\cdot\overline{\Phi(\mathrm{Y})}$$I\mathrm{i}_{2}(X,\mathrm{Y})=(x \cdot\overline{x’}+z\cdot\overline{z’})(\underline{y}\cdot\overline{y’}+w\cdot\overline{w’})+(x\cdot\overline{\iota v’}-z\cdot\overline{y’})(y\cdot\overline{z’}-w\cdot\overline{x’})$$\overline{I\mathrm{f}}_{m}(X,\mathrm{Y})=,.\frac{\Gamma(2p+m)}{m-\Gamma(2p)}\int_{K_{\mathrm{R}}}\langle gE_{1},\mathrm{Y}\rangle^{m}\langle X,g\overline{E}_{1}\rangle^{m}dg$$I^{-}\mathrm{i}_{n.k}^{-}.(X,1^{r})=I\acute{\iota}_{n-2k}(X, \mathrm{Y})\{I\mathrm{c}_{2}^{-}(X, \}^{-})\}^{k}$

In [10] we showed thatby$\tilde{I1}_{n,k}(, \mathrm{J}’)\in 9\mathrm{f}_{n}$if}’$g\{_{n}=\oplus_{q=0}^{[n/2]}mathrm{t}_{n,q}$

now we put$\Lambda=${$(\mathrm{n}$.

$\mathfrak{X}_{\mathfrak{n},k}$$\mathrm{i}\mathrm{K}_{n}$spanned$\mathfrak{X}_{n,k}=’\frac{(\mathrm{t}-2k+1)^{2}(2p+n-k-2)!(2p+k-3)!(2p+’ l-1)}{(n-k+1)!k!(2p-3)!(2p-1)!}$$I1_{\mathrm{R}}^{r}\mathrm{A}\mathrm{V}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{i}\mathrm{b}1\mathrm{e}$, As) ;.

denotes the subspace of. It is valid that dirn
gives thedecomposition of$\{I_{1_{n,k}^{r}}^{-}( , Z);Z\in \mathrm{N}\}$and,$(n\in \mathrm{Z}_{+}, 0\leq k\leq[n/2])$$\in \mathrm{N}$,$n\in \mathrm{Z}_{+}$$ff\{_{n}$, $0\leq k\leq[n/2]$ } and $E_{0}=\Phi^{-1}$(cf. [3], [10]). From$(\begin{array}{l}e_{1}00\mathrm{e}_{2}\end{array})\in \mathrm{N}$. Then wehave the following

Proposition 3.1 (cf. [10]). For any f(3.1)$\in \mathfrak{X}_{n,k}$

we have$\delta_{n,n\iota}\delta_{k,l}f(X)=\dim \mathrm{J}\mathrm{t}_{n,k}f_{K_{\mathrm{R}}}f(gE_{0})\tilde{I\mathrm{f}}_{m,l}(X, gE_{0})dg$

Now we define for $X$ ,. From (3.1)$\in \mathfrak{p})$.

$Y\in \mathfrak{p}$$\overline{H}_{7l,k}(X, 1^{r})=\dim \mathfrak{X}_{l,k},(n+2p-2k)$ $(\begin{array}{l}2p+k^{\wedge}-3k\end{array})$$((??, k)\in\Lambda)$(X$\overline{H}_{n,k}(, 1^{\Gamma})$$\overline{H}_{7l,k}(X, Y)=\overline{H}_{1lk}\underline,(\mathrm{Y}, X)$

belongs to$\mathfrak{X}_{n,k}$$\int_{K_{\mathrm{R}}}\overline{I\mathrm{f}}_{n,k}(X,gE_{0})\tilde{I\mathrm{t}}_{n,k}(gE_{0},\mathrm{Y})dg$

for any$\mathrm{Y}\in \mathfrak{p}$and we have,$\overline{H}_{\iota,k},(gX,gY)=H_{n,k}(X, 1^{-})$$(g\in I\iota_{\mathrm{R}})$,$\overline{H}_{1\iota,k}(X, 1’-)=(\begin{array}{ll}n +2p-2 k\end{array})(\begin{array}{ll}2p+k -3k \end{array})$$-1\overline{l\mathrm{i}^{-}}_{n,k}(X,1^{\sim}.)$($X\in \mathrm{N}$or$]’.\in \mathrm{N}$).

The purpose of this section is to show thatY) gives the reproducing kernel offor each-orbit. Our main theorem is the following$\tilde{H},.,k(,$$I\mathrm{t}_{\mathrm{R}}$,$\mathrm{J}\{_{\iota,k}$

140
Theorem 3.2. (i) For any
have
(3.2)

.and any$f\in \mathfrak{X}_{1,k}$such that$X_{0}\in \mathfrak{p}$$\tilde{H}_{n,k}(-\mathrm{Y}_{0}, \lambda_{0}’)\neq 0$$\delta_{n,m}\delta_{k,l}f(X)=\frac{\dim \mathrm{J}\{_{nk}}{\tilde{H}_{n,k}(_{J}\mathrm{Y}_{0},p\mathrm{Y}_{0}^{-})}\int_{K_{\mathrm{R}}}f(g_{d}\mathrm{Y}_{0}^{\cdot})\tilde{H}_{m,l}(X,gX_{0})d_{1}g$(ii) Assume
restriction mapping
we have$\tilde{H}_{n,k}(X_{0},X_{0})\neq 0(X_{0}\in \mathfrak{p}, \forall(n, k)\in\Lambda)$$r_{\mathrm{O}}$:$farrow f|_{0}$is a bijection from.

, and put$\mathrm{J}\mathrm{t}_{n}$

onto

.

$0=I\mathrm{i}_{\mathrm{R}}X0$$\mathrm{J}\mathrm{t},.|_{0}$we. Th.en the. And for$f\in \mathfrak{X}_{1},|_{0}$.

$r_{0}^{-1}(f)(X)= \sum_{k=0}^{n}\frac{\dim \mathfrak{X}_{n,k}}{\tilde{H}_{n.k}(X_{0},\lambda_{0}’)}\int_{I\mathrm{i}_{\mathrm{R}}}f\cdot(g.\lambda_{0}.)\tilde{H}_{n,k}(X,gd’\mathrm{Y}_{0}^{-})dg$

To prove this theorem we need some lemmas.

Lemma 3.3 (cf. [10]). For any f(3.3)$\tau vhe\mathit{7}e$

, h,

and X

$\in \mathfrak{X}_{m,l}$$\in \mathfrak{p}$we have$\int_{I\mathrm{f}_{\mathrm{R}}}f(gE_{1})\overline{h(gE_{1})}dg=C_{n.k}’\int_{I\mathrm{i}_{\mathrm{R}}}f(gE_{0})\overline{h(gE_{0})}dg$,$C_{n,k}=(k +2.p-3h)(\begin{array}{l},l+2p-2k\end{array})$

Lemma 3.4. Let$f$$\in f\{_{n,k}$

for any$g\in I\ddot{\backslash }0$$I\dot{\mathrm{t}}_{0}$}. Ifbe the isotropy group of, we have$E_{1}$

in$I\dot{\mathrm{C}}_{\mathrm{R}}$and let$ff\{_{n,k}’=\{f\in \mathfrak{X}_{\iota,kj},gf=$$f\in \mathrm{J}\{_{n,k}’$(3.4)$f=f(E_{1})\tilde{H}..,k(, E_{1})$.

Sketch of Proof From (3.1) and [9] Lemma 2.5 we can prove (3.4) with some calculations.
q.e.d.

Proof of Theorem 3.2. Using (3.1), (3,3) and (3.4), we can prove (i) and (ii) in the same

way as the proof of Theorem 2.2.

Remark 3.5.

$\tilde{H}_{n,k}(_{d}\mathrm{Y}_{0}, .\lambda_{\acute{0}})=0$$\tilde{H}_{n,k}(X_{0},X_{0})\neq 0$(A$\in \mathrm{C}$for any

q.e.d.

if and only if
if and only if$\mathfrak{X}_{n,k}|_{K_{\mathrm{R}}X_{0}}=\{0\}$$(n, k)\in \mathrm{A}$. Therefore we have$d\mathrm{Y}\mathrm{p}\not\in\lambda I\mathrm{f}_{\mathrm{R}}\tilde{E}_{1}$and$J\mathrm{Y}_{0}\not\in\lambda I\mathrm{t}_{\mathrm{R}}\tilde{E}_{0}$).

Appendix.
Combining Theorems 1.2, 2.2, and 3.2, we have the following

Theorem. Assume that 9n is a classical real simple Lie algebra with real
,
. Let
1)(p
1) or
p
1) (p

$\mathfrak{g}_{\mathrm{R}}=so(p,$$\in \mathrm{N},$$\geq 2)$$\epsilon \mathrm{u}(p,$$\mathrm{s}\mathrm{p}(\mathrm{p},$$\in \mathrm{N})$$\mathit{7}^{\cdot}a_{1}nk1$$\mathfrak{X}_{n}=\oplus_{k=0}^{N(n)}\mathfrak{X}_{n,k}$, i.e.,
be the141-irreducible decomposition of , where $N(n)$ is the number of irreducible components.
Then we have(i) If0is any-orbit in and, then the restiction mapping : $farrow$is bijection $hm$ クイー, onto ひ (n,k|O.
(ii) Letbe the reproducing kernel ofonand letbe thecomponent of. Assume,
satisfiesand put. Then the restriction mapping ’.0 : $farrow f|_{0}$ is a bijection fromonto
. And for
we have$g\{_{n}$$I\mathrm{t}_{\mathrm{R}}$$Ii_{\mathrm{R}}^{r}$$\mathfrak{p}$$f|_{0}$$a$$g\{_{n,k}|_{0}\neq\{0\}$$r_{\mathrm{O}}$$k$$\overline{H}_{\iota l}(X,\mathrm{Y})$$\mathrm{J}\{_{n}$$\frac{\dim \mathfrak{X}}{\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{J}\mathrm{t}_{n,\mathrm{k}}}\overline{\overline{H}}_{n}(X,\mathrm{Y})$$\lambda_{0}\in \mathfrak{p}$$\Sigma_{\mathrm{R}}$$\tilde{H}_{n,k}(X,\mathrm{Y})$$\overline{H}_{n,k}(X_{0},X_{0})\neq 0(^{\forall}(n, k)\in\Lambda)$$0=It_{\mathrm{R}}^{r}X_{0}$$\mathfrak{X}_{n}|_{0}$$\theta \mathrm{f}_{n,k^{-}}$$\mathfrak{X}_{n}$$f\in ff\{_{n}|_{\mathit{0}}$$r_{\mathit{0}}^{-1}‘(f)(X)= \sum_{k=0}^{n}\frac{\dim \mathfrak{X}_{l,k}}{\tilde{H}_{n,k}(-\mathrm{Y}_{0},X_{0})}.\int_{I\mathrm{i}_{\mathrm{R}}}f(garrow \mathrm{Y}_{0})\overline{H}_{n,k}(X, g_{J}\mathrm{Y}_{0})dg$.

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