A generalized Cartan decomposition for connected compact Lie groups and its application (Topics in Combinatorial Representation Theory)

Size: px

Start display at page:

Download "A generalized Cartan decomposition for connected compact Lie groups and its application (Topics in Combinatorial Representation Theory)"

なおみょうだに
5 years ago
Views:

1 $\bullet$ $\bullet$ $\bullet$ $\prime \mathcal{h}$ A generalized Cartan decomposition for connected compact Lie groups and its application Graguate School of Mathematical Sciences, The University of Tokyo 1 $\sim$ 11. $G$ $H$ $G$ $\Leftrightarrow End_{G}(H)$ $\mathcal{h}$ $ \kappa$ $\mathcal{h}$ [ [ ( ) $U(n)r\vee S^{k}(\mathbb{C}^{n})$ $)$ $(GL_{k}-GL_{n}$ $U(k)\cross U(n)$ cpoly $[M(k, n;\mathbb{c})]$ $L^{2}$ ( ) $GL(n,\mathbb{R})$ $L^{2}(GL(n,\mathbb{R})/O(n))$

$$\sigma$ $\sigma$ $\sigma$ 118 3 ( ) ( ) 12. $G$ $D$ $D$ $S$ $G$. $D =G\cdot S$ $D$ id $\sigma _{S}=$ $s$ G-.$

2 $\sigma$ $\sigma$ $\sigma$ ( ) ( ) 12. $G$ $D$ $D$ $S$ $G$. $D =G\cdot S$ $D$ id $\sigma _{S}=$ $s$ G-. $D $ $S$ $D$ $G$ $S$ S 2 S $S$ ( ) 2 $D$ $\mathbb{h}$ $S$ $\sqrt{-1}\mathbb{r}+$ $\sigma(z)=-\overline{z},$ $z\in \mathbb{h}$ $\pm(\begin{array}{ll}l *0 1\end{array})\}$ $G=\{$ $G=SO(2)$ ([Ko4])

3 $\bullet$ $\bullet$ $G$ $D$ $\mathcal{v}arrow D$ $G$ $D$ $\mathcal{o}(d, V)$ $G$ 1( ) $G$ $D$ - $S$ 2( ) $+$ compatibility ( [Ko4]) $\ovalbox{\tt\small REJECT}$ $[Ko4]$?? $\sim?$? ( ) ( ) 1.4. $G$ $H,L$ $G/L$ $H$ $G/L$ 3 $G$ $\sigma$ 3 $(G,H, L)$ visible triple $\bullet$ $\sigma$ $H$ $L$ $\sigmag/larrow G/L$ $G=HG^{\sigma}L$ $G^{\sigma}$ $B$ $G=HBL$ $G$

4 $\sigma$ ( ). $G=GL(n,\mathbb{C})$ 3 $(G, H, L)=$ ( $G\cross G$, diag $(G)$, diag $(G)$ ) $G\sim(G\cross G)/diag(G)$ Program visible triple $(G, H, L)$ 2. visible triple $(G, H, L)$ $B$ $G=HBL$ (1) A $G$ $H,$ $L$ $G$ [KO5] (2) $(G, L),$ $(G, H)$ [Fl], $[Ho $, [Mal] visible action $(G, L)$ [Ko6] (1) $G$ $t$ $t$ $H,$ $L$ $t$ Weyl $H,$ $L$ $G$ $G/L,$ $G/H,$ $(G\cross G)/(L\cross H)$ $G^{\sigma}\cdot 0,$ $(G^{\sigma}\cross G^{\sigma})\cdot 0$ $G=HBL$ $($ $G=HG^{\sigma}L)$ 3 $H\sim G/L$, $L\sim G/H$, diag$(g)\sim(g\cross G)/(L\cross H)$ 3 $Ind_{L}^{G}\chi _{H}$, $Ind_{H}^{G}\chi _{L}$, $Ind_{L}^{G}\chi\otimes Ind_{H}^{G}\chi $ ( (triunity theorem for multiplicityheeness property) [Ko2] $\chi,$ $)$ $\chi $ $G=L_{\lambda}BL_{\mu}\Rightarrow m\lambda\otimes n\mu$ M.F. for $\forall m,$ $n\in N$

5 121 $\lambda$ ( $\mu$ $L_{\lambda},$ $L_{\mu}$ $f_{m.f.\rfloor}$ $m\lambda\otimes n\mu$ M.F. for $\forall $G$ (multiplicity-free) ) m,$ $n\in N\Rightarrow??G=L_{\lambda}BL_{\mu}$ $(\text{ ^{}\prime})$ $\sim?$? $\infty$ $(\text{ ^{}l})$? $U(n)$ J. R. Stembridge $GL(n, \mathbb{c})$ ([Stl]) $U(n)$ ( ) 1.7. ( ) [Ko2] A ( ) ( ) $U(n)$ Stembridge ([St2]) $U(n)$ ( ) 1.8. Programl6 ( ) ([Ko5]) $E_{7}$ ( 2.1) 2.1. $G$ $E_{7}$ Dynkin

6 $\alpha_{2}$ 122 $-\infty-\llcorner 0arrowarrow 0$ $\alpha_{1}$ $\alpha_{3}$ $\alpha_{4}$ $\alpha_{5}$ $\alpha_{6}$ $\alpha_{7}$ Type $E_{7}$ $H,$ $L$ $\{\alpha_{7}\},$ $\{\alpha_{1}\}$ $(G, H, L)$ visible triple $\tau.\tau $ Proof. Vogan (Figure 1,2) $G$ $\alpha_{2}$ $\alpha_{2}$ $ \llcorner_{-\bullet}$ $arrow-\llcorner 0\mapsto-\circ-0$ $\alpha_{1}$ $\alpha_{3}$ $\alpha_{4}$ $\alpha_{5}$ $\alpha_{6}$ $\alpha_{7}$ Figure 1 $\alpha_{1}$ $\alpha_{3}$ $\alpha_{4}$ $\alpha_{5}$ $\alpha_{6}$ $\alpha_{7}$ Figure 2 $\tau\tau^{l}$ $g^{\tau\tau }$ $\mathbb{r}\oplus \mathfrak{e}$ $=$ V $\mathfrak{g}^{\tau\tau }$ (6) $\mathfrak{g}^{\tau\tau }=\mathfrak{g}^{\tau,\tau }\oplus \mathfrak{g}^{-\tau,-\tau }$ $=(\mathbb{r}h_{1}\oplus(\sqrt{-1}\mathbb{r}\oplus\epsilon o(10)))\oplus \mathfrak{g}^{-\tau,-\tau }$ $=(\mathbb{r}h_{1}\oplus(\mathbb{r}h_{2}\oplus\epsilon o(10)))\oplus \mathfrak{g}^{-\tau,-\tau }$. $\mathbb{r}h_{1}$ $\mathfrak{g}^{\tau\tau }$ $\mathbb{r}h_{2}$ $\mathbb{r}\oplus$ T so(10) $\sigma$ $a$ $\mathfrak{g}^{-\tau.-\tau }$ Weyl -1 $\mathbb{r}\oplus $(e_{6}$ $\sqrt{}$ $)$, \mathfrak{s}$0(10) $H_{3}\in 5o(10)$ $\sqrt{-1}\mathbb{r}\oplus$ $\mathbb{r}(h_{2}+h_{3})\oplus\epsilon u(4)$ $\mathfrak{a}$ so(10) $Z_{\mathfrak{g}^{\tau.\tau }}(a)=\mathbb{r}h_{1}\oplus \mathbb{r}(h_{2}+h_{3})\oplus\epsilon u(4)$ $\mathfrak{g}^{\tau }=\epsilon u(2)\oplus$ so(12) $\mathfrak{g}^{\tau }=\mathfrak{g}^{\tau,\tau}\oplus \mathfrak{g}^{\tau,-\tau}$ $=(\mathfrak{s}u(2)^{\tau}\oplus so(12)^{\tau})\oplus \mathfrak{g}^{\tau,-\tau}$ $=(\mathbb{r}w_{1}\oplus(\sqrt{-1}\mathbb{r}\oplus \mathfrak{s}o(10)))\oplus \mathfrak{g}^{\tau,-\tau}$ $=$ $(\mathbb{r}w_{1}\oplus (\mathbb{r}w_{2}\oplus so(10)))\oplus \mathfrak{g}^{\tau,-\tau}$. $\mathbb{r}w_{1}=\epsilon u(2)^{\tau}$ so(10) $)$ $\mathbb{r}w_{2}$ $($ $)^{\tau}$ $(\sqrt{-1}\mathbb{r}\oplus$ so 12

7 $\tau^{l}$ $C^{\backslash }$ 123 $G^{l},$ $G $ $G$ $\epsilon u(2)$, so(12) $G^{\tau }$ $G^{\tau }=G G $ ( 22) $G=G^{\tau}\exp(a)G G $. (1) 2.2. (B.Hoogenboom, T.Matsuki) $G$ $\tau $ $\tau,$ $G$ $H,$ $L$ $\tau,$ $a$ $\mathfrak{g}^{-\tau,-\tau }$ $\tau\tau $ $\mathfrak{g}$ $G$ $G=H\exp(a)L$. \mathbb{r}w_{1})$ $(\mathfrak{g}^{l}, 22 $G =\exp(\mathbb{r}w_{1})\exp(a )\exp(\mathbb{r}w_{1})$ $a $ $\sigma$ $\mathfrak{g}^{l}=\epsilon u(2)$ 1 $\mathbb{r}w_{1}\oplus \mathbb{r}w_{2}=\mathbb{r}h_{1}\oplus \mathbb{r}h_{2}$ $a,$ $b\in \mathbb{r}$ $W_{1}=aH_{1}+bH_{2}$ $b\neq 0$ $(\exp(\mathbb{r}w_{1})\exp(a^{l})\exp(\mathbb{r}w_{1}))g $ $=(\exp(\mathbb{r}(ah_{1}+b(h_{2}+h_{3})))\exp(\mathfrak{a} )\exp(\mathbb{r}w_{1}))g $ (1) $G=G^{\tau}\exp(a)G G $ $=G^{\tau}\exp(a)(\exp(\mathbb{R}W_{1})\exp(a^{l})\exp(\mathbb{R}W_{1}))G^{l\prime}$ $=G$ $\exp(a)\exp(\mathbb{r}(ah_{1}+b(h_{2}+h_{3})))\exp(a^{l})\exp(\mathbb{r}w_{1})g $ $=G^{\tau}\exp(\mathbb{R}(aH_{1}+b(H_{2}+H_{3})))\exp(a)\exp(a )\exp(\mathbb{r}w_{1})g $ $=G^{\tau}\exp(a)\exp(a )\exp(\mathbb{r}w_{1})g $. $G^{\tau}=H$ $\exp(\mathbb{r}w_{1})g =L$ $M$ $G^{\tau }$ $c^{g}$ $L$ $H$

8 124 ( ) $(G, H, L)=$ $(SO(2n+1), U(n), U(n))$ $SO(2n+1)\supset SO(2n)\supset U(n)$ $(SO(2n+1), SO(2n))$ so $(2n+1)$ so $(2n)$-module $\epsilon o(2n+1)=\epsilon o(2n)\oplus q$ $(SO(2n), U(n))$ $= u(n)\oplus\bigcup_{g\in U(n)}Ad(g)(a)\oplus q$ $a$ $(SO(2n), U(n))$ Weyl ( ) $= u(n)\oplus\bigcup_{g\in U(n)}Ad(g)(a\oplus q_{0})$ qo $q$ ( ) Weyl $\dim(a)+\dim$(qo) $=n$ $G=H\exp(a+$ qo $)L$ quiver ([Ko5]) $G$ $(H, L)$ $G$ $\sigma$ $G$ Weyl $H\cross G^{\sigma}\cross Larrow G$ $g\in G$ $Hg\cap G^{\sigma}L=\emptyset$ ( ) $G$ $GL(N, \mathbb{c})$

9 $\bullet$ $\bullet$ 125 Weyl block diagonal $L$ $R\in$ $M(N, \mathbb{r})$ $G$ Ad $(G^{\sigma}L)R\subset M(N, \mathbb{r})$ $Ad(Hg)J\cap M(N, \mathbb{r})=\emptyset$ (2) $g\in G$ $H$ $G(n)=U(n),$ $SO(2n+1),$ $Sp(n)$ or $SO(2n)$ $n$ $n=n_{1}+\cdots+n_{k}$ $G(n)$ $U(n_{1})\cross\cdots\cross U(n_{k-1})\cross G(n_{k})$ $X$ $G(n)$ $H$ $X=(X_{ij})_{1\leq i,j\leq k}$ $G(n)$ $A$ $n=n_{1}+\cdots+n_{k}$ block $G(n)$ $B$ $2n+1=$ $n_{1}+\cdots+n_{k-1}+(2n_{k}+1)+n_{k-1}+\cdots+n_{1}$ $C,$ $D$ $2n=n_{1}+\cdots+n_{k-1}+2n_{k}+n_{k-1}+\cdots+n_{1}$ block $i_{*}\in\{1,2, \ldots, k\},$ $i_{*}\neq i_{*+1},$ $r\geq 2$ $i_{0}arrow i_{1}arrow\cdotsarrow i_{r}=i_{0}$, $A_{i_{O}\cdots i_{r}}(x)$ $=X_{i_{O}i_{1}}X_{i_{1}i_{2}}\cdots X_{i_{r-1}i_{r}}$ (3) ( 23 ) $H$ ( block diagonal ) $A_{i_{0}\cdots i_{r}}$ $(Ad$ $(h)x)=h_{i_{o}}a_{i_{0}\cdots i_{r}}(x)h_{i_{0}}^{-1}$ (4) block $h_{s}$ $h\in H=U(n_{1})\cross\cdots\cross U(n_{k-1})\cross G(n_{k})$ $s$ $H$ $A_{i_{\text{ }}\cdots i_{r}}(x)$ (quiver ) (2) $g\in G(n)$ $A_{i_{O}\cdots i_{r}}([x, R])$ $i_{0}arrow\cdotsarrow i_{r}$ $G(n)$ $X$ 2.3. (3)

10 126 A $u(n)=\{x\in \mathfrak{g}\mathfrak{l}(n, \mathbb{c})$ $r+x=o\}$ B $\mathfrak{s}o(2n+1)=\{x\in 5[(2n+1, \mathbb{c})$ ${}^{t}xj_{2n+1}+j_{2n+1}x=o,$ $\overline{x}+x=o\}$ C $s\mathfrak{p}(n)=\{x\in 5[(2n, \mathbb{c})$ ${}^{t}xj_{n}^{l}+j_{n}^{l}x=o,{}^{t}\overline{x}+x=o\}$ D $so$ $(2n)=\{X\in\epsilon 1(2n, \mathbb{c}){}^{t}xj_{2n}+j_{2n}x=o, \ulcorner X^{-}+X=O\}$ X A $\tilde{x}_{ij};=\{\begin{array}{ll}x_{ij} (i<j),x_{\overline{ji}} (i>j)\end{array}$ $B$ $\tilde{x}_{ij}=\{\begin{array}{ll}x_{ij} (i+j\leq 2k),J_{n_{1}}^{t}X_{2k-j,2k-i}J_{n_{j}} (i+j>2k, i,j\neq k),j_{2n_{k}+1^{t}}x_{2k-j.k}j_{n_{j}} (i=k,j>k),j_{n_{i}}{}^{t}x_{k,2k-i}j_{2n_{k}+1} (i>k,j=k)\end{array}$ $cg^{i\rfloor}$ $\tilde{x}_{ij}=\{\begin{array}{ll}x_{ij} (i+j\leq 2k),J_{n_{i}}{}^{t}X_{2k-j,2k-i}J_{n_{j}} (i+j>2k, i,j\neq k),j_{n_{k}} {}^{t}x_{2k-j,k}j_{n_{j}} (i=k,j>k),j_{n_{i}}{}^{t}x_{k,2k-i}j_{n} k (i>k,j=k)\end{array}$ D $\tilde{x}_{ij};=\{\begin{array}{ll}x_{ij} (i+j\leq 2k),J_{n_{l}}{}^{t}X_{2k-j,2k-i}J_{n_{j}} (i+j>2k, i,j\neq k),j_{2n_{k}}{}^{t}x_{2k-j,k}j_{n_{j}} (i=k,j>k),j_{n_{i}}{}^{t}x_{k,2k-i}j_{2n_{k}} (i>k,j=k)\end{array}$

11 $.\cdot\cdot$ $01$ $\Pi^{\prime l}$ 127 $(n_{2k-i}=n_{i},$ $1\leq i\leq k-1$ $)$ $A_{i_{0}\cdots i_{r}}(x)$ $=\tilde{x}_{i_{o}i_{1}}\tilde{x}_{i_{1}i_{2}}\cdots\tilde{x}_{i_{r-1}i_{r}}$ $O$ $J_{m};=[_{1}$ $1]\in GL(m, \mathbb{r})$, $\underline{m}$ $\underline{m}$ $J_{m} ;=$ $[_{-1}O.$ $-1$ 1 $\dot{o}$ $1]\in GL(2m, \mathbb{r})$ (4) 222 Stembridge 3 $G$ $t$ $G$ $\sigma$ $G$ Weyl $t$ $\Pi$ $\Pi,$ $\Pi$

12 l}$ 128 $L_{\Pi },$ $L_{\Pi }$ $\Pi,$ $\Pi $ $G$ $\Pi^{l}=\Pi$ $L_{\Pi }=G$ $(\Pi )^{c}=\pi\backslash \Pi $ $L_{\Pi }$ 1 ( ) $G=L_{\Pi }G^{\sigma}L_{\Pi }$ $(\Pi^{l}, \Pi^{ll})$ ( $G$ $\Pi $ $\Pi^{Jl}$ ) $(L_{\Pi }.L_{\Pi })$ $G$ [Ko6] ( polar ([Her]) polar $K\ddot{a}$hler visible polar ) $A_{n}$ 3.1 ([Ko5]) $\ovalbox{\tt\small REJECT}arrow\infty---\cdots\cdots\cdot$ $\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot$ $- \frac{-}{}c----$ $\alpha_{1}$ $\alpha_{2}$ $\alpha_{3}$ Type A $\alpha_{n-2}\alpha_{n-1}$ $\alpha_{n}$ I. $(\Pi^{l})^{c}=\{\alpha_{i}\}$, $(\Pi^{l\prime})^{c}=\{\alpha_{j}\}$. $1\leq i,j,$ $k\leq n$ I. $(\Pi^{l})^{c}=\{\alpha_{i}, \alpha_{j}\}$, $(\Pi )^{c}=\{\alpha_{k}\}$, $\min.\{p, n+1-p\}=1$, $p=i,g$ or $i=j\pm 1$. I. $(\Pi )^{c}=\{\alpha_{i}, \alpha_{j}\}$, $(\Pi )^{c}=\{\alpha_{k}\}$, $\min\{k, n+1-k\}=2$. anything, $i=1$ or. $n$ $m$. $(\Pi^{l})^{c}=\{\alpha_{i}\}$, $\Pi^{\prime $m$ $(\Pi^{l/})^{c}$ $B_{n}$ $-arrowarrow\cdots\cdot\cdots\cdot\cdot\cdot$ $\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot$$ =$ $\alpha_{1}$ $\alpha_{2}$ $\alpha_{3}$ Type $B$ $\alpha_{n-2}$ $\alpha_{n-1}$ $\alpha_{n}$ I., $(\Pi^{l})^{c}=\{\alpha_{1}\}$ $(\Pi^{l\prime})^{c}=\{\alpha_{1}\}$. I., $(\Pi^{l})^{c}=\{\alpha_{n}\}$ $(\Pi )^{c}=\{\alpha_{n}\}$. If. $(\Pi )^{c}=\{\alpha_{1}\}$, $(\Pi )^{c}=\{\alpha_{i}\}$, $2\leq i\leq n$.

13 $\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot$ $C_{n}$ $ arrow\cdots\cdot\cdot$ $\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot$ $-\mapsto=$ $\alpha_{1}$ $\alpha_{2}$ $\alpha_{3}$ Type $C$ $\alpha_{n-2}\alpha_{n-1}$ $\alpha_{n}$ I. $(\Pi )^{c}=\{\alpha_{n}\}$, $(\Pi^{lJ})^{c}=\{\alpha_{n}\}$. I. $(\Pi )^{c}=\{\alpha_{1}\}$, $(\Pi^{ll})^{c}=\{\alpha_{i}\}$, $1\leq i\leq n$. 3.4 $D_{n}$ $-arrow 0-\cdot\cdot\cdots\cdots\cdot$ $\cdot\cdot$ $ 0\nearrow\alpha_{n}$ $\alpha_{1}$ $\alpha_{2}$ $\alpha_{3}$ Type $D$ $\alpha_{n-3}\alpha_{n-}\backslash _{2}$ $\circ\alpha_{n-1}$ I. $(\Pi^{l})^{c}=\{\alpha_{i}\}$, $(\Pi )^{c}=\{\alpha j\}$, $i,j\in\{1, n-1, n\}$. I. $(\Pi^{l})^{c}=\{\alpha_{1}\}$, $(\Pi^{l\prime})^{c}=\{\alpha_{j}\}$, $j\neq 1,$ $n-1,$ $n$ IF. $(\Pi^{l})^{c}=\{\alpha_{i}\}$, $(\Pi^{ll})^{c}=\{\alpha_{j}\}$, $i\in\{n-1, n\},$ $j\in\{2,3\}$. $m$. $(\Pi )^{c}=\{\alpha_{i}\}$, $(\Pi )^{c}=\{\alpha_{j}, \alpha_{k}\}$, $i\in\{n-1, n\},$ $j,$ $k\in\{1, n-1, n\}$. IV. $(\Pi )^{c}=\{\alpha_{i}\}$, $(\Pi )^{c}=\{\alpha_{j}, \alpha_{k}\}$, $i\in\{n-1, n\},$ $j,$ $k\in\{1,2\}$. V. $(\Pi )^{c}=\{\alpha_{1}\}$, $(\Pi )^{c}=\{\alpha_{j}, \alpha_{k}\}$, $j\in\{n-1, n\}$ or $k\in\{n-1, n\}$. VI. $(\Pi^{l})^{c}=\{\alpha_{i}\}$, $(\Pi^{\prime l})^{c}=\{\alpha_{2}, \alpha_{j}\}$, $n=4,$ $(i,j)=(3,4)$ or (4, 3). 3.5 E6 $\alpha_{2}$ $-arrow\llcorner_{0}$ $\alpha_{1}$ $\alpha_{3}$ $\alpha_{4}$ $\alpha_{5}$ $\alpha_{6}$ Type $E_{6}$ I. $(\Pi )^{c}=\{\alpha_{i}\}$, I., $(\Pi^{l})^{c}=\{\alpha_{i}\}$ I. $(\Pi^{l})^{c}=\{\alpha_{i}\}$, $(\Pi^{l\prime})^{c}=\{\alpha_{j}\}$, $i,j\in\{1,6\}$. $(\Pi^{\prime l})^{c}=\{\alpha_{1}, \alpha_{6}\}$, $i=1$ or 6. $(\Pi^{l\prime})^{c}=\{\alpha_{j}\}$, $i=1$ or 6, $j\neq 1,4,6$.

14 E7 $\alpha_{2}$ $ \llcorner 0arrow-0$ $\alpha_{1}$ $\alpha_{3}$ $\alpha_{4}$ $\alpha_{5}$ $\alpha_{6}$ $\alpha_{7}$ Type $E_{7}$ $(\Pi )^{c}=\{\alpha_{7}\}$ I., $(\Pi^{l/})^{c}=\{\alpha_{7}\}$. I. $(\Pi )^{c}=\{\alpha_{7}\}$, $(\Pi )^{c}=\{\alpha_{i}\}$, $i=1$ or E8,F4,G2 LII $G=L_{\Pi }G^{\sigma}$ $G$ $(L_{\Pi }, L_{\Pi })$ 4 $\mu\otimes\nu$ $\omega_{*}$ (Dynkin ) $a,$ $b,c$ P. Littelmann [Li] J. R. Stembridge [St2] 4.1 $A_{n}$ ([Ko5]) $1\leq i,j,$ $k\leq n$ I. $\mu=a\omega_{i}$, $\nu=b\omega_{j}$. I. $\mu=a\omega_{i}+hv_{j}$, $\nu=c\omega_{k}$, $\min_{p=i,j}\{p, n+1-p\}=1$, or $i=j\pm 1$. I. $\mu=a\omega_{i}+b\omega_{j}$, $\nu=c\omega_{k}$, $\min\{k, n+1-k\}=2$. $m$. $\mu=a\omega_{i}$ $\nu$, anything, $i=1$ or. $n$ $m$ $\nu$ 1

15 $ $, $\mathfrak{a}$, $B_{n}$ I. $\mu=a\omega_{1}$, $\nu=b\omega_{1}$. I. $\mu=a\omega_{n}$, $\nu=b\omega_{n}$. I. $\mu=a\omega_{1}$, $\nu=b\omega_{i}$, $2\leq i\leq n$. 4.3 $C_{n}$ I. $\mu=a\omega_{n}$, $\nu=b\omega_{n}$. $l\ovalbox{\tt\small REJECT}=b\omega_{i}$ I. $\mu=a\omega_{1}$,, $1\leq i\leq n$. 4.4 $D_{n}$ I. $\mu=a\omega_{i}$, $\nu=b\omega_{j}$, $i,j\in\{1, n-1, n\}$. I. $\mu=a\omega_{1}$, $\nu=b\omega_{j}$, $j\neq 1,$ $n-1,$ $n$ I. $\mu=a\omega_{i}$, $\nu=b\omega_{j}$, $i\in\{n-1, n\},$ $j\in\{2,3\}$. $m$. $\mu=a\omega_{i}$, $\nu=b\omega_{j}+c\omega_{k}$, $i\in\{n-1, n\},$ $j,$ $k\in\{1, n-1, n\}$. IV. $\mu=a\omega_{i}$, $\nu=b\omega_{j}+c\omega_{k}$, $i\in\{n-1, n\},$ $j,$ $k\in\{1,2\}$. V. $\mu=a\omega_{1}$, $\nu=b\omega_{j}+c\omega_{k}$, $j\in\{n-1, n\}$ or $k\in\{n-1, n\}$. VI. $\mu=a\omega_{i}$, $\nu=b\omega_{2}+c\omega_{j}$, $n=4,$ $(i,j)=(3,4)$ or (4, 3). 4.5 E6 I. $\mu=a\omega_{i}$. I. $\mu=a\omega_{i-}$, $\nu=b\omega_{j}$, $\nu=b\omega_{1}+c\omega_{6}$, $i,j\in\{1,6\}$. $i=1$ or 6. I. $\mu=a\omega_{i}$. $\nu=b\omega_{j}$, $i=1$ or 6, $j\neq 1,4,6$. 4.6 E7 I. I.. $\mu=a\omega_{7}$, $\nu=b\omega_{7}$. $\mu=a\omega_{7}$, $\nu=b\omega_{i}$, $i=1$ or ( ). (1) [Ok]

16 132 A $i+j=n$ B I. C $D$ $i,j\in\{n-1, n\}$ - minor summation formula $([rw])$ (2) [Al] [ [Al] H. Alikawa, Multiplicity-free branching rules for outer automorphisms of simple Lie algebras. J. Math. Soc. Japan, 59, 2007, no. 1, [Fl] M. Flensted-Jensen, Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal., 30, 1978, no. 1, [Hel] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Pure and Appl. Math. New York, London Academic Press, [Her] R. Hermann, Variational completeness for compact symmetric spaces, Proc. Amer. Math. Soc., 11, 1960, [Ho] B. Hoogenboom, Intertwining functions on compact Lie groups, CWI Tract, 5. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, [IW] M. Ishikawa and M. Wakayama, Minor summation formulas of Pfaffians, Linear and Multilinear Algebra, 39, 1995, [Kn] A. W. Knapp, Lie groups beyond an introduction, 2nd ed., Progr. Math. 140, Birkh\"auser, Boston, [KO] [Kol] Multiplicity free theorem in branching problems of unitary highest weight modules, 1997 ), 1998, (

17 133 [Ko2] T. Kobayashi, Geometry of multiplicity-free representations of GL $(n)$, visible actions on flag varieties, and triunity, Acta Appl. Math., 81, 2004, [Ko3] T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci., 41, 2005, , special issue commemorating the fortieth anniversary of the founding of RIMS. [Ko4] T. Kobayashi, Propagation of multiplicity-freeness property for holomorphic vector bundles, Progr. Math., Birkh\"auser, 2012 (in press), math. RT/ [Ko5] T. Kobayashi, A generalized Cartan decomposition for the double coset space $(U(n_{1})\cross U(n_{2})\cross U(n_{3}))\backslash U(n)/(U(p)\cross U(q))$, J. Math. Soc. Japan, 59, 2007, [Ko6] T. Kobayashi, Visible actions on symmetric spaces, Thransform. Groups, 12, 2007, [Ko7] T. Kobayashi, Multiplicity-free theorems of the restriction of unitary highest weight modules with respect to reductive symmetric pairs, in Representation Theory and Automorphic Forms, Progr. Math., Birkh\"auser, Boston, 2007, , math. RT/ [KTl] K. Koike and I. Terada, Young diagrammatic methods for the representation theory of the classical groups of type $B_{n},$ $C_{n},$ $D_{n}$, J. Algebra, 107, 1987, [KT2] K. Koike and I. Terada, Young Diagrammatic Methods for the Restriction of Representations of Complex Classical Lie Groups to Reductive Subgroups of Maximal Rank, Adv. Math., 79, 1990, [Li] P. Littelmann, On spherical double cones, J. Algebra, 166, 1994, [Mal] 2 involution II, ( 1994), No. 895, 1995, [Ma2] T. Matsuki, Double coset decomposition of algebraic groups arising $hom$ two involutions. I, J. Algebra, 175, 1995, [Ma3] T. Matsuki, Double coset decompositions of reductive Lie groups arising from two involutions, J. Algebra, 197, 1997,

18 134 [Ok] S. Okada, Applications of minor summation formulas to rectangularshaped representations of classical groups, J. Algebra, , [Sal] A. Sasaki, A characterization of non-tube type Hermitian symmetric spaces by visible actions, Geom. Dedicata, 145, 2010, [Sa2] A. Sasaki, Visible action on irreducible multiplicity-free spaces, Int. Math. Res. Not. IMRN, 2009, no. 18, [Sa3] A. Sasaki, A generalized Cartan decomposition for the double coset space SU $(2n+1)\backslash$ SL $(2n+1,\mathbb{C})/$ Sp $(n,\mathbb{c})$, J. Math. Sci. Univ. Tokyo, 17, 2010, [Stl] J. R. Stembridge, Multiplicity-free products of Schur functions, Ann. Comb., 5, 2001, [St2] J. R. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory, 7, 2003, [Wo] J. A. Wolf, Harmonic Analysis on Commutative Spaces, Mathematical Surveys and Monographs, Amer. Math. Soc., yuichiro@ms.u-tokyo.ac.jp

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.

R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................