$w_{ij}^{\infty}(t)=\delta_{ij},$ $i\leq j,$ $w_{ij}^{0}(t)=0,$ $i>j$ $w_{ii}(t)\neq 0,$ $i=1,$ $\ldots,$ $n$ $W_{\infty}(t),$ $W_{0}(t)$ (14) $L(f)=W

Size: px

Start display at page:

Download "$w_{ij}^{\infty}(t)=\delta_{ij},$ $i\leq j,$ $w_{ij}^{0}(t)=0,$ $i>j$ $w_{ii}(t)\neq 0,$ $i=1,$ $\ldots,$ $n$ $W_{\infty}(t),$ $W_{0}(t)$ (14) $L(f)=W"

えいじろうさんきち
5 years ago
Views:

1 , 2000 pp72-87 $\overline{n}b_{+}/b_{+}$ scikumamoto-uacjp September 27, 2000 \S 1 Introduction $\#_{dt}^{1}d^{2}=\exp(q_{2}-q_{1})$ $arrow_{dt}^{d^{2}}2=\exp(q_{3}-q_{2})-\exp(q_{2}-q_{1})$ (11) $\{$ : $\frac{d^{2}q_{\tau\iota-1}}{\frac d^{2}q_{\iota}= dt^{2}dt^{2}},=\exp(q_{n}-q_{n-1})-\exp(q_{n-1}-\exp(q_{n}-q_{n-1})-q_{n-2})$ $p_{i}=\delta^{\underline{j}}ddt$ $i=1,$ $\ldots,$ $n$ $L(t)$ (12) $L_{ij}(t)=\{$ $0$ 1 $p_{i}$ $\exp(q_{i+1}-q_{i})$ if $j>i+1$ or $i>j+1$ if $j=i+1$ if $i=j$ if $i=j+1$ $L(t)$ (13) $\frac{dl(t)}{dt}=[l(t)_{+}, L(t)]$, (13) (12) 3 Bruhat Kostant 1 Theorem[K2] $W_{\infty}(t),$ $W_{0}(t)$ $TV_{\infty}(t)=(w_{ij}^{\infty}(t))_{1\leq i,j\leq n},$ $lf^{\gamma_{0}}(t)=(w_{ij}^{0}(t))_{t\leq i,\gamma\leq n}$, 72

2 $w_{ij}^{\infty}(t)=\delta_{ij},$ $i\leq j,$ $w_{ij}^{0}(t)=0,$ $i>j$ $w_{ii}(t)\neq 0,$ $i=1,$ $\ldots,$ $n$ $W_{\infty}(t),$ $W_{0}(t)$ (14) $L(f)=W_{\infty}(t)L(0)W_{\infty}^{-1}$ $(t)$ (14) $W_{\infty}^{-1}(t)W_{0}(t)=\exp(tL(O))$ $L(t)$ $L(t)$ $n\cross 7l$ A $(\Lambda)_{ij}=\delta_{i+1,j}$ $W_{\infty}(t)$ (15) $W_{\infty}(t)\Lambda TV_{\infty}^{-1}(t)=\Lambda+$ (14) $W_{\infty}(t)L(0)W_{\infty}^{-1}(t)=l\prime V_{0}(t)e^{-tL(0)}L(0)e^{tL(0)}lV_{0}^{-1}(t)$ $=W_{0}(t)L(0)W_{0}^{-1}(t)$ $W0(t)$ (16) $W_{0}(t)L(0)\ddagger\eta_{1}^{r_{0}-1}(t)=diag[a(s)]\Lambda^{-1}+$ $(\Lambda^{-1})_{ij}=\delta_{i-1,j}$ $diag[a(s)]$ (15) (16) $L(t)=$ $L(t)$ () (14) Riemann-Hilbert(R-H) $G=GL(n, C),$ $g=gl(n, C)$ $SL(n, C),$ $sl(n, C)$, $GL(n, C),$ $gl(n, C)$ $n,\overline{n}$ $g$, $h$ $g$ $N,\overline{N},$ $H$ $n,\overline{n},$ $h$ $G$ 73

3 (14) $W_{0}(t)$ $T(t)$ $\tilde{w}_{0}(t)=t^{-1}(t)\dagger V_{0}(t)$ (14) (17) $W_{\infty}^{-1}(t)T(t)\tilde{W}_{0}(t)=e^{tL(0)}$ $y\in g$ $G^{y}\subset G$ $G^{y}=\{g\in G gy=yg\}$ $e^{tl(0)}\in G^{L(0)}$ [K2] (17) $G_{*}^{L(0)}=G^{L(0)}\cap\overline{N}HN$ $W_{\infty}(t)L(0)i/t_{\infty}^{r-1}(t)$ Kostant [K2] Theorem 24 $b=n\oplus h$ $B_{+}$ $b$ $G$ $X$ $X=G/B_{+}$ $S_{n}$ $n$ $X$ $X=u_{\sigma\in S},,\overline{N}s(\sigma)B_{+}/B_{+}$, $s(\sigma)$ $\sigma$ $X$ $\overline{n}b_{+}/b_{+}\circ-$ Kostant type $7l_{\infty}^{\Gamma}(t)L(0)iV_{\infty}^{-1}(t)$ $X$ $\sigma\neq 1$ $X$ $\overline{n}s(\sigma)b_{+}/b_{+}$ R-H (18) $W_{\infty}^{-1}(t)s(\sigma)\nu V$ $(t)=e^{tl(0)}$ $W_{\infty}(t),$ $W_{0}(t)$ (18) (19) $IV_{\infty}^{-1}(t)(s(\sigma)W_{0}(t)s^{-1}(\sigma))=e^{tL(0)}s^{-1}(\sigma)$ $(18),(19)$ R-H (110) $W_{\infty}^{-1}(t)W_{0}(t)=e^{t\xi}s(\sigma)$ $\xi$ $L(t)=\Lambda+$ $L(t)=W_{\infty}(t)\xi M_{\infty}^{r-1}(t)$ $\frac{dl(t)}{dt}=[l(t)_{+}, L(t)]$ $L(t)$ full Kostant-Toda lattice [S] $L(t)$ (110) $\overline{n}s(\sigma)b_{+}/b_{+}$ 74

4 $01$ $\cdot$ $\backslash$ $\overline{n}b_{+}/b_{+}$ Flaschka Haine [FH] $\sigma$ $L(t)$ $t=t_{0}$ Weyl $\exp(t_{0}l(0))modb_{+}$ $G/B_{+}$ $\sigma$ $\sigma$, Weyl $\sigma$ \S 2 $V$ $0$ $V=\{$ $01$ $0$ $L_{n-1n-1}$ 1 $\cdot$ $ L_{ij}\in\not\subset\}$ $L_{nn-1}$ $L_{nn}/$ $V$ $C$ $C=C[\mathcal{L}_{ij}, i\geq j]$ $\mathcal{l}_{ij}$ $\mathcal{l}$ $L\in V$ $\mathcal{l}_{ij}(l)=l_{ij}$ $\mathcal{l}_{ij}$ $V$ $i,$ $j$ (21) $\mathcal{l}=(\mathcal{l}_{n1}\mathcal{l}_{n-11}\mathcal{l}_{21}\mathcal{l}_{11}$ $\mathcal{l}_{n2}\mathcal{l}_{n-12}\mathcal{l}_{22}1$ $\mathcal{l}_{nn-1}1^{\cdot}\mathcal{l}_{nn}00)$ $\mathcal{l}_{n-1n-1}$ [K1] $C$ $C=A\otimes C^{\overline{N}}$ (22), $A$ A, $C$ $C^{\overline{N}}$ $\overline{n}$ $C$ adjoint $\mathcal{w}_{\infty}\in A\otimes\overline{N},$ $\chi 0\in C^{\overline{N}}\otimes Mat(n, C)$ = $\mathcal{l}=\mathcal{w}_{\infty}\chi_{0}\mathcal{w}_{\infty}^{-1}$ $01$ $\chi_{0}=(\cdot\varphi_{n}\varphi_{2}\varphi_{1}$ $001$ $\cdot000)$ 75

5 $C^{\overline{N}}$ $\varphi_{1},$ $\ldots,$ $\varphi_{n}$ $C^{\overline{N}}$ $\mathcal{l}=$ $\varphi_{1}=\mathcal{l}_{11}+\mathcal{l}_{22}+\mathcal{l}_{33}+\mathcal{l}_{44}$, $\varphi_{2}=\mathcal{l}_{21}-\mathcal{l}_{11}\mathcal{l}_{22}+\mathcal{l}_{32}-\mathcal{l}_{11}\mathcal{l}_{33}-\mathcal{l}_{22}\mathcal{l}_{33}+\mathcal{l}_{43}-\mathcal{l}_{11}\mathcal{l}_{44}-\mathcal{l}_{22}\mathcal{l}_{44}-\mathcal{l}_{33}\mathcal{l}_{44}$, $\varphi_{3}=\mathcal{l}_{31}-\mathcal{l}_{11}\mathcal{l}_{32}-\mathcal{l}_{21}\mathcal{l}_{33}+\mathcal{l}_{11}\mathcal{l}_{22}\mathcal{l}_{33}+\mathcal{l}_{42}-\mathcal{l}_{11}\mathcal{l}_{43}-\mathcal{l}_{42}\mathcal{l}_{43}-\mathcal{l}_{21}\mathcal{l}_{44}$ $+\mathcal{l}_{11}\mathcal{l}_{22}\mathcal{l}_{44}-\mathcal{l}_{32}\mathcal{l}_{44}+\mathcal{l}_{11}\mathcal{l}_{33}\mathcal{l}_{44}+\mathcal{l}_{22}\mathcal{l}_{33}\mathcal{l}_{44}$, $\varphi_{4}=\mathcal{l}_{41}-\mathcal{l}_{11}\mathcal{l}_{42}-\mathcal{l}_{21}\mathcal{l}_{43}+\mathcal{l}_{11}\mathcal{l}_{22}\mathcal{l}_{43}-\mathcal{l}_{31}\mathcal{l}_{44}+\mathcal{l}_{11}\mathcal{l}_{32}\mathcal{l}_{44}$ $+\mathcal{l}_{21}\mathcal{l}_{33}\mathcal{l}_{44}-\mathcal{l}_{11}\mathcal{l}_{22}\mathcal{l}_{33}\mathcal{l}_{44}$ $w_{\infty}$ $(w_{ij}^{\infty})_{i,j=1}^{4}$ $w_{21}^{\infty}=-\mathcal{l}_{22}-\mathcal{l}_{33}-\mathcal{l}_{44}$, $w_{31}^{\infty}=-\mathcal{l}_{32}-\mathcal{l}_{43}+\mathcal{l}_{22}\mathcal{l}_{33}+\mathcal{l}_{22}\mathcal{l}_{44}+\mathcal{l}_{33}\mathcal{l}_{44}$, $w_{41}^{\infty}=-\mathcal{l}_{42}+\mathcal{l}_{32}\mathcal{l}_{44}+\mathcal{l}_{22}\mathcal{l}_{43}-\mathcal{l}_{22}\mathcal{l}_{22}\mathcal{l}_{33}\mathcal{l}_{44}$, $w_{32}^{\infty}=-\mathcal{l}_{33}-\mathcal{l}_{44}$, $w_{42}^{\infty}=-l_{43}+\mathcal{l}_{33}\mathcal{l}_{44}$, $w_{43}^{\infty}=-\mathcal{l}_{44}$ Proposition 21 $\varphi_{1},$ $\ldots,$ $\varphi_{n}$ $\mathcal{l}$ $\varphi_{i}=(-1)^{i+1}\sum_{i\subset\{1,\ldots,n\}, I =i}\det((\mathcal{l})_{\mu\nu})_{\mu,\nu\in I},$ $i=1,$ $\ldots,$ $n$ $nxn$ $x=(x_{\mu\nu})$ $\det(\lambda-x)=\lambda^{n}+\sum_{k=1}^{n}\lambda^{n-k}(-1)^{k}\sum_{i\subset\{1,\ldots,n\}, I =k}\det(x_{\mu\nu})_{\mu,\nu\in I}$ $\det=\lambda^{n}-\varphi_{1}\lambda^{n-1}-\cdots-\varphi_{n}$ 76

6 $\mathcal{w}_{\infty}$ Remark 1 $\mathcal{w}_{\infty}^{-1}\in $C$ Poisson A\otimes\overline{N}$ (23) $\{\mathcal{l}_{ij}, \mathcal{l}_{k\ell}\}=\delta_{j,k}\mathcal{l}_{il}-\delta_{\ell,j}\mathcal{l}_{kj}$ $C$ derivation $\mathcal{x}$ $f\in C$ (24) $\mathcal{x}f=\{\frac{1}{2}tr\mathcal{l}^{2}, f\}$ (25) $\mathcal{x}\mathcal{l}=[\mathcal{l}_{+}, \mathcal{l}]$ section $C$ (25) $C$ Hamiltonian $C$ orbit $\mathcal{l}(t)$ $f\in C$ $f( \mathcal{l}(t- -\Delta t))=f(\mathcal{l}(t))+\{\frac{1}{2}tr\mathcal{l}^{2}(t), f(\mathcal{l}(t))\}\triangle t+o(\triangle t)$ $\mathcal{l}(t)$ orbit $\mathcal{x}(t)$ $\mathcal{x}(t)=\{\frac{1}{2}tr\mathcal{l}^{2}(t), *\}$ Lemma 22 (26) $\frac{d\mathcal{l}(t)}{dt}=\mathcal{x}(t)\mathcal{l}(t)$ $(C\otimes C[[t]])\otimes Mat(n, C)$ $\mathcal{l}(t)=\mathcal{l}+tb^{(1)}+t^{2}b^{(2)}+\cdots$ $B^{(1)}= \frac{d}{dt} _{t=0}\mathcal{l}(t)=\{\frac{1}{2}tr\mathcal{l}^{2}, \mathcal{l}\}$ $\frac{d}{dt}tr\mathcal{l}^{2}(t)=\{tr\mathcal{l}^{2}, tr\mathcal{l}^{2}\}=0$ $d^{2}$ /9) 1, $- \backslash$, 1 $B^{(2)}= \frac{l}{2}\frac{tl}{db^{2}} _{t=0}\mathcal{l}(t)=\{\frac{1}{2}tr\mathcal{l}^{2}, \{tr\mathcal{l}^{2}, \mathcal{l}\}\}$ $B^{(i)},$ $i=3,4,$ $\ldots$ = $C(t)$ $C\otimes C[[t]]$ Lemma 22 $(26)arrow$ orbit $t=0$ $\mathcal{l}(t)$ $\mathcal{l}$ = $\mathcal{l}(t)$ (26) orbit 77

7 $\mathcal{i}$ $i-j\geq Lemma 23 orbit Poisson (27) $\{\mathcal{l}_{ij}(t), \mathcal{l}_{k}i(t)\}=\delta_{jk}\mathcal{l}_{i\ell}(t)-\delta_{\ell i}\mathcal{l}_{kj}(t)$ Lemma 22 $\frac{df(b)}{dt}=\mathcal{x}(t)f(t)$ $C(t)$ = $F(t)=\{\mathcal{L}_{ij}(t), \mathcal{l}_{k\ell}(t)\}$ $G(t)=$ $\delta_{jk}\mathcal{l}_{i\ell}(t)-\delta_{\ell i}\mathcal{l}_{kj}(t)$ $\frac{d}{db}f(t)=\{\frac{d\mathcal{l}_{ij}(t)}{dt}, \mathcal{l}_{kl}(t)\}+\{\mathcal{l}_{ij}(t), \frac{d\mathcal{l}_{k\ell}(l)}{dt}\}$ $=\{\mathcal{x}(t)\mathcal{l}_{ij}(t), \mathcal{l}_{k\ell}(t)\}+\{\mathcal{l}_{ij}(t), \mathcal{x}(t)\mathcal{l}_{k\ell}(t)\}$ $=\mathcal{x}(t)\{\mathcal{l}_{ij}(t), \mathcal{l}_{k\ell}(t)\}=\mathcal{x}(t)f(t)$ = $\frac{d}{dt}g(t)=\mathcal{x}(t)g(t)$ $F(0)=G(0)$ = $F(t)=G(t)$ Proposition 24 (26) (28) $\frac{d\mathcal{l}(t)}{dt}=[\mathcal{l}_{+}(t), \mathcal{l}(t)]$ Lemma 23 $\mathcal{x}(t)\mathcal{l}_{ij}(t)=\{\frac{1}{2}tr\mathcal{l}^{2}(t), \mathcal{l}(t)\}=([\backslash \mathcal{l}(t)_{+}, \mathcal{l}_{\backslash }^{(}t)_{j}^{\rceil})_{ij}$ $\mathcal{i}$ $\mathcal{l}_{ij},$ 2$ $C$ Poisson Poisson $\{C, \mathcal{i}\}\subset \mathcal{i}$ $C/\mathcal{I}$ quotient algebra Poisson $\rho$ : $Carrow C/\mathcal{I}$ $\rho$ Poisson $P=(P_{ij})$ $\rho(p)$ $\rho(p)=(\rho(p_{ij}))$ (28) $\mathcal{l}(t)\daggerh$ $\mathcal{l}(t)=\mathcal{l}+tb^{(1)}+t^{2}b^{(2)}+\cdots$ $\rho(\mathcal{l}(t))=\rho(\mathcal{l})+t\rho(b^{(1)})+t^{2}\rho(b^{(2)})+\cdots$ 78

8 $C/\mathcal{I}$ Proposition 25 $i-j\geq 2$ orbit $\mathcal{l}(t)$ $C/\mathcal{I}$ $\rho(\mathcal{l}_{ij}(t))=0$ $\mathcal{l}(t)$ $\mathcal{l}(t)=\mathcal{l}+tb^{(1)}+t^{2}b^{(2)}+\cdots$, $B^{(i)}\in C\otimes\Lambda/Iat(n, C)$ (210) $B^{(k)}= \frac{1}{k!}\frac{d^{k}\mathcal{l}(t)}{dt^{k}} _{t=0}$ lemma $\frac{\mathfrak{u}l(\iota\prime}{*k},$ Lemma 26,, $k=1,2,$ $\ldots\downarrowh$ $(^{\underline{9}}11)$ $[\Pi_{1}(t)_{+}, [\Pi_{2}(t)_{+}, [\cdots[\pi_{\mu}(t)_{+}, \mathcal{l}(t)]\cdots]]]$ $1\leq\mu\leq k$ $(t)\in C(t)\otimes A- Iat(n, C)$ Proposition 24 $\mathcal{l}(t)$ (28) lemma $k$ $k=1$ (28) $\frac{d^{k}\mathcal{l}(t)}{dt^{k}}$ (211) $\frac{d^{k+1}\mathcal{l}(t)}{dt^{k+1}}$ $\frac{d}{dt}[\pi_{1}(t)_{+}, [\Pi_{2}(t)_{+}, [\cdots, [\Pi_{\mu}(t)_{+}, \mathcal{l}(t)]\cdots]]]$ $=[( \frac{d\pi_{1}(t)}{dt})_{+}, [\Pi_{2}(t)_{+}, [\cdots, [\Pi_{\mu}(t)_{+}, \mathcal{l}(t)]\cdots]]]$ $+ \cdots+[\pi_{1}(t)_{+}, [\Pi_{2}(t)_{+}, [\cdots, [(\frac{d\pi_{\mu}(t)}{dt})_{+}, \mathcal{l}(t)]\cdots]]]$ $+[\Pi_{1}(t)_{+}, [\Pi_{2}(t)_{+}, [\cdots, [\Pi_{\mu}(t)_{+}, [\mathcal{l}(t)_{+}, \mathcal{l}(t)]]\cdots]]]$ (211) Lemma 26 Lemma 26 (210) $\mathcal{b}^{(k)}$ (212) $[\Pi_{1}(0)_{+}, [\Pi_{2}(0)_{+}, [\cdots, [\Pi_{\mu}(0)_{+}, \mathcal{l}]\cdots]]]$ $i-j\geq 2$ $B^{(k)}$ $\mathcal{i}$ $i,$ $j$ 1 $C\otimes Mat(n, C)$ $[\Pi 1+ \mathcal{l}]$ $i,$ $j$ $[ \Pi_{1+},\sum_{k-\ell\geq 2}\mathcal{L}_{k\ell}E_{k\ell}]$ 79

9 $i,j$ $E_{k\ell}$ $(k, \ell)$ $([\Pi_{1+}, \mathcal{l}])_{ij}\in \mathcal{i}$ 1,, $\Pi_{\mu}\in C\otimes Mat(n, C)$ $i-j\geq 2$ ([ $\Pi_{1+},$ [ $\Pi_{2+},$ $[\cdots, [\Pi_{\mu}+ \mathcal{l}]$ j $\in \mathcal{i}$ $\Pi_{\mu+1}\in C\otimes Mat(n, C)$ (213) $([\Pi_{1+}, [\Pi_{2+}, [\cdots, [\Pi_{\mu+1}\mathcal{L}]+ ]\cdots]]])_{ij}$ $=([ \Pi_{1+},\sum_{k-p\geq i-j}([\pi_{2+}, [\cdot\cdot, [\square _{\mu+1}\mathcal{l}]+ \cdots]])_{k\ell}e_{k\ell}])_{ij}$ $i-j\geq 2$ $k-\ell\geq 2$ $([\Pi_{2+}, [\cdots, [\Pi_{\mu+1}\mathcal{L}]+ \cdots]])_{k\ell}\in \mathcal{i}$ $\mathcal{i}$ (213) $\mathcal{l}_{ij}(t)=\mathcal{l}_{ij}+\sum_{k=1}^{\infty}t^{k}b_{ij}^{(k)}$ $i-j\geq 2$ $\mathcal{l}_{ij}(t)\in \mathcal{i}$ $\rho(\mathcal{l}_{ij}(t))=0$ Proposition27 $\mathcal{l}(t)$ $\rho(\mathcal{l}(t))$ $C$ (26) (214) $\frac{d\rho(\mathcal{l}(t))}{dt}=[\rho(\mathcal{l}(t))_{+}, \rho(\mathcal{l}(t))]$ (214) $\frac{1}{2}tr\rho(\mathcal{l}(t))^{2}$ $\rho$ $\rho(\frac{d\mathcal{l}(t)}{dt})=\frac{d\rho(\mathcal{l}(t))}{dt}$ $\rho$ $\frac{d\rho(\mathcal{l}(t))}{dt}=\rho(\frac{d\mathcal{l}(t)}{dt})=\rho([\mathcal{l}(t)_{+}, \mathcal{l}(t)])$ $=[\rho(\mathcal{l}(t))_{+,\rho}(\mathcal{l}(t))]$ $\rho$ Poisson $\frac{d\rho(\mathcal{l}(t))}{dt}=\rho(\frac{d\mathcal{l}(t)}{dt})=\rho(\{\frac{1}{2}tr\mathcal{l}^{2}(t), \mathcal{l}(t)\})$ $= \{\frac{1}{2}tr\rho(\mathcal{l}(t))^{2}, \rho(\mathcal{l}(t))\}$ \S 3 80

10 $\mathcal{l}$ $C^{\overline{N}}$ (21) \S 2 $\mathcal{l}=\mathcal{w}_{\infty}\chi $A\otimes\overline{N}$ $\chi_{0}$ 1 $\sigma\in S_{n}$ R-H 0\mathcal{W}_{\infty}^{-1}$ $\mathcal{w}_{\infty}\in$ $0$ (31) $\mathcal{w}_{\infty}^{-1}(t)\mathcal{w}_{0}(t)=\exp((t-t_{0})\chi_{0})_{-}^{-}-s(\sigma)$ $=\mathcal{w}_{\infty}^{-1}(0)$ $\mathcal{w}_{\infty}(t)$ $\overline{n}$ $\mathcal{w}_{0}(t)$ $0$ $\mathcal{l}(t)=\mathcal{w}_{\infty}(t)\chi 0\mathcal{W}_{\infty}^{-1}(t)$ $\mathcal{l}(t)=\lambda+$ $\frac{d\mathcal{l}(t)}{dt}=[\mathcal{l}(t)_{+}, \mathcal{l}(t)]$ $\mathcal{w}_{\infty}(t)=(w_{ij}^{\infty}(t))_{ij}$ $i\leq j$ $w_{ij}^{\infty}(t)=\delta_{ij}$ (31) (32) $(\mathcal{w}_{\infty}(t)\exp((t-t_{0})\chi_{0})_{-}^{\neg}-s(\sigma))_{-}=0$ $m(t)=\exp((t-to)\chi 0)_{-}^{-}-s(\sigma)$ (32) (33) $(w_{j1}^{\infty}(t), \ldots, w_{jj-1}^{\infty}(t))$ $=-(m_{j1}(t), \ldots, m_{jj-1}(t)),j=2,$ $\ldots,$ $m_{ij}(t)$ $m(t)$ $i,j$ $n$ (34) $\tau_{j}(t)=\det$ $\cdot m_{1j}(t)m_{jj}(t)),$ $j=1,$ $\ldots,$ $n-1$ $w_{jk}^{\infty}(t)= \frac{-\tau_{j-1,k}(t)}{\tau_{j-1}(t)},$ $k=1,$ $\ldots,j-1$ $\tau_{j-1,k}(t)=\det$ $\cdot m_{1j-1}(t)rn_{jj-1}(t)k- thm_{j-1j-1(t)}$ $row)$ 81

11 $\exp((t-t_{0})\chi_{0})=(e_{ij}(t))$ $\exp((t-t_{0})\chi_{0}) _{t=t}$ $=1$ (36) $e_{ii}(t)=1+(t-t_{0})g_{ii}(t),$ $i=1,$ $\ldots,$ $n$ (37) $e_{ij}(t)=(t-t_{0})g_{ij}(t),$ $i\neq j$ $gij(t)\in C[\varphi_{1}, \ldots, \varphi_{n}]\otimes C[[t]]$ $\sigma\in S_{n}$ $_{\sigma}$ $\Theta_{\sigma}=$ { $1\leq j\leq n-1 1\leq\exists i\leq j$ such that $\sigma(i)>j$ } Proposition 31 (34) $(t)$ t=t $0$ $j\in\theta_{\sigma}$ $(36),(37)$ $\exp((t-t_{0})\chi_{0})\equiv$ $mod (t-t_{0})$ $\tilde{a}$ $w_{ij}^{\infty}(0),$ $i>j$ $w_{ij}^{\infty}(0)$ w $\in\tilde{a}\otimes\overline{n}$ $01$ $\exp((t-t_{0})\chi_{0})_{-}^{-}-\equiv(\cdot001$ $\cdot$ $\cdot\cdot 001)mod (t-t_{0})$ $\exp((t-t_{0})\chi_{0})_{-}^{-}-s(\sigma)\equiv(e_{\sigma(1)}, \ldots, e_{\sigma(n)})mod (t - t_{0})$ $e_{j}$ $j$ $t---j=$ 82

12 $i-j\geq $\tilde{c}$ $i\geq : $\tau_{j}(t)=\det(t---j\exp((t-t_{0})\chi 0)_{-}^{-}-s(\sigma)_{-j}^{-}-)$ $\equiv\det(^{t-}--j(e_{\sigma(1)}, \ldots, e_{\sigma(j)}))1nod(t-t_{0})$ $\det(^{t-_{j}}-- (e_{\sigma(1)},, e_{\sigma(j)}))\neq 0$ $\sigma _{\{1,\ldots,j\}}\in S_{j}$ $\tau_{j}(t_{0})\neq 0\Leftrightarrow j\not\in\theta_{\sigma}$ $j\in\theta_{\sigma}$ $\tau_{j}(t)=(t-t_{0})^{\ell(\sigma,j)}(c+\cdots)$ $c$ $\ell(\sigma, j)=\#\{i 1\leq i\leq j, \sigma(i)>j\}$ $c=0$ $c\neq 0$ = $c=0$ $(t_{0})=0$ $k=j-1$ $\tau$ [UT] $w_{jj-1}^{\infty}(t)= \frac{-d\tau_{j-1}(t)}{dt}/\tau_{j-1}(t)=\frac{-d\log\tau_{j-1}(t)}{dt}$ $j-1\in_{\sigma}$ $c\neq 0$ $w_{jj-1}^{\infty}(t)= \frac{-\ell(\sigma,j-1)}{t-t_{0}}+$ regular part $\sigma\neq 1$ $_{\sigma}\neq\phi$ $\sigma\neq 0$ ( $c\neq 0$ $c=0$ ) $\mathcal{w}_{\infty}(t)$ $\tilde{\mathcal{l}}(t)=\mathcal{w}_{\infty}(t)\chi_{0}\mathcal{w}_{\infty}^{-1}(t)$ $\tilde{\mathcal{l}}(t)$ $\Lambda+$ $\tilde{\mathcal{l}}(0)$ $\tilde{\mathcal{l}}$ $\tilde{\mathcal{l}}_{ij},$ j$ $\mathcal{w}_{\infty}^{-1}(0)\tilde{\mathcal{l}}\mathcal{w}_{\infty}(0)=\chi 0$ $C[\varphi_{1}, \ldots, \varphi_{n}]$ $\tilde{c}\sigma$) $C[\varphi_{1}, \ldots, \varphi_{n}]=\tilde{c}^{\overline{n}}$ (39) $\tilde{c}=\tilde{a}\otimes\tilde{c}^{\overline{n}}$ (39) $w_{ij}^{\infty},$ $\tilde{\mathcal{i}}$ $\tilde{\mathcal{l}}_{ij},$ $i>j$ $\varphi_{i},$ $i=1,$ $n$ $\ldots,$ $\rho(w_{ij}^{\infty}),$ $\tilde{\mathcal{l}}_{ij},$ 2$ $\rho$ (t) $=\mathcal{w}_{\infty}(t)\chi 0\mathcal{W}_{\infty}^{-1}(t)$ $\mathcal{l}(t)$ $\tilde{\mathcal{l}}_{ij}(t)\in\tilde{c}\otimes C[[t]]$ $\mathcal{w}_{\infty}(t)$ $i\geq j$ $\tilde{c}arrow\tilde{c}/\tilde{\mathcal{i}}$ $i>j$ $\rho(\varphi_{i}),$ $i=1,$ $\ldots,$ $7\iota$ (32) $\mathcal{w}_{\infty}(t)$ $t=0$ $\tilde{\mathcal{l}}(t)=\mathcal{w}_{\infty}(t)\chi 0\mathcal{W}_{\infty}(t)^{-1}$ $t=0$ $t$ $\tilde{\mathcal{l}}(t)=\mathcal{e}_{0}+t\mathcal{e}_{1}+t^{2}\mathcal{e}_{2}+\cdots$ 83

13 $\tilde{\mathcal{l}}(0)=\mathcal{w}_{\infty}(0)\chi_{0}\mathcal{w}_{\infty}^{-1}(0)=\tilde{\mathcal{l}}$ $\mathcal{e}_{0}\in\tilde{c}\otimes Mat(n, C)$ $\tilde{\mathcal{l}}(t)$ Lemma 22 $\in\tilde{c}\otimes Mat(n, C),j=1,2,$ $\ldots$ $\mathcal{w}_{\infty}(t)\tilde{\mathcal{l}}(t)\mathcal{w}_{\infty}^{-1}(t)=\chi 0$ $w_{ij}^{\infty}(t)\in\tilde{c}(t)$ $\rho(w_{ij}^{\infty}(t))$ $\rho$ (310) $\rho(\tilde{\mathcal{l}}(t))=\rho(\mathcal{w}_{\infty}(t))\rho(\chi 0)\rho(\mathcal{W}_{\infty}(t))^{-1}$ (311) $\frac{d\rho(\tilde{\mathcal{l}}(t))}{dt}=[\rho(\tilde{\mathcal{l}}(t))_{+,\rho}(\tilde{\mathcal{l}}(t))]$ $\rho(\mathcal{w}_{\infty}(t))$ (312) $(\rho(\mathcal{w}_{\infty}(t))\exp((t-t_{0})\rho(\chi_{0}))\rho(_{-}^{-}-)s(\sigma))_{-}=0$ $\rho(\tilde{\mathcal{l}}(t))$, (310) $\rho(\mathcal{w}_{\infty}(t))\in\tilde{c}/\tilde{\mathcal{i}}(t)\otimes\overline{n}$ $\rho(\tilde{\mathcal{l}}(t))$ dressing operator, (312) (310) $\tau$ $\tilde{c}/\tilde{\mathcal{i}}(t)$ $\tilde{c}/\tilde{\mathcal{i}}(t)=\tilde{c}/\tilde{\mathcal{i}}\otimes C[[t]]$ \S 4 $V$ $V=\{$ $ L_{ij}\in C\}$ $Z$ $V$ $Z=$ $\{(\cdot0l_{21}l_{11} L_{22}1 0L_{nn-1}1 \cdotl_{nn}^{\cdot}\cdot) L_{ij}\in C\}$ $\gamma={}^{t}(\gamma_{1}, \ldots, \gamma_{n})\in C^{n}$ $Z(\gamma)\subset Z$ (41) $Z(\gamma)=\{L\in Z \varphi_{j}(l)=\gamma_{j}, j=1, \ldots, n\}$ section 3 $\mathcal{w}_{\infty}(t)$ (32), $\tilde{\mathcal{l}}(t)=$ $\mathcal{w}_{\infty}(t)\chi_{0}\mathcal{w}_{\infty}^{-1}(t)$, $\tilde{\mathcal{l}}_{ij}(t)\in\tilde{c}(t),$ $\mathcal{w}_{\infty}(t)\in\tilde{c}(t)\otimes\overline{n}$ 84

14 $0^{\cdot}L_{32}L_{22}1$ : $L\in V$ $\mu_{l}$ : $\mu_{l}(\tilde{l}_{ij}):=\tilde{\mathcal{l}}_{ij}(l)=l_{ij}$ $\tilde{c}arrow C$ $\mu_{l}$ $L$ (42) $Z(\gamma)=\{L\in Z \mu_{l}(\varphi_{i})=\gamma_{i}, i=1, \ldots, n\}$ $\tilde{\mathcal{l}}_{ij}(t)\in\tilde{c}\otimes C[[t]]$ $L\in V$ $\mu_{l}(\tilde{\mathcal{l}}_{ij}(t))\in C[[t]]$ $\mu_{l}(\tilde{\mathcal{l}}(t))=l(t)$ $\mu_{l}$ (28) $\mu_{l}(\frac{d\mathcal{l}(t)}{dt})=\mu_{l}([\tilde{\mathcal{l}}(t)_{+},\tilde{\mathcal{l}}(t)]$, $\mu_{l}$ $\mu_{l}$ $t$ (43) $\frac{dl(t)}{dt}=[l(t)_{+}, L(t)]$ $\tilde{c}/\tilde{\mathcal{i}}$ $\tilde{\mu}_{l}$ $\tilde{c}/\tilde{\mathcal{i}}arrow C$ $L\in V$ $0L_{33}1$ $\tilde{\mu}_{l}(\rho(\tilde{\mathcal{l}}))=(\cdot 00L_{21}L_{11}$ $\cdot000l_{nn})$ $L_{nn-1}$ $L\in Z$ (44) $\tilde{\mu}_{l}(\rho(\tilde{\mathcal{l}}))=l$ $\mathcal{w}_{\infty}(t)$ $\tilde{\mathcal{l}}(t)=\mathcal{w}_{\infty}(t)\chi_{0}\mathcal{w}_{\infty}^{-1}(t)$ (32) $\mathcal{w}_{\infty}(t)\in\tilde{c}(t)\otimes\overline{n},$ $i=1,$ $\varphi_{i}\in\tilde{c}^{\overline{n}},$ $n$ $\ldots,$ $\tau_{j}(t),$ $\tau_{j,k}(t)\in\overline{c}(t)$ $\tilde{\mu}_{l}(\rho(\mathcal{w}_{\infty}(t)),\tilde{\mu}_{l}(\rho(\tau_{j}(t))),\tilde{\mu}_{l}(\rho(\tau_{j,k}(t)))$ $L\in Z(\gamma)$ $L(t)$ $:=\mu_{l}(\rho(\tilde{\mathcal{l}}(t))),$ $W(t):=\tilde{\mu}_{L}(\rho(\mathcal{W}_{\infty}(t))),$ $\chi_{0}(\gamma):=\tilde{\mu}_{l}(\rho(\chi_{0}))$ $\chi 0(\gamma)$ $\chi 0(\gamma)=$ 85

15 $\tau$ section Theorem $L\in Z(\gamma)$ $L(t)=\tilde{\mu}_{L}(\rho(\tilde{\mathcal{L}}(t))),$ $W(t)=\tilde{\mu}_{L}(\rho(\mathcal{W}\infty(t)))$ $L(t)$ $Z(\gamma)$ orbit $L(t)\in Z(\gamma)$ $\frac{dl(t)}{dt}=[l(t)_{+}, L(t)]$ $W(t)$ $L(t)$ dressing operator $L(t)=W(t)\chi_{0}(\gamma)W(t)^{-1}$ $W(t)$ $t=t_{0}$ $G$ $\sigma\in S_{n}$ Bruhat $\overline{n}\sigma HN$ R-H $W(\theta)$ ( $j$, k)- $\tilde{\mu}_{l}(\rho(\tau_{j-1}(t)))\in C_{L}[[t]]$ $C_{L}$ $C$ $L\in Z(\gamma)$ [FH] Theorem $X=$ $G/B_{+}$ $\overline{n}s(\sigma)/b_{+}$ $W(t)$ R-H $W(t)^{-1}U(t)=\exp((t-t_{0})\chi_{0}(\gamma))_{-}^{-}-(L)s(\sigma)$ $U(t)=\tilde{\mu}_{L}(\rho(\mathcal{W}o(t)))$ $---(L)=\tilde{\mu}_{L}$ (\rho ()) $W(t)_{-}^{-}-(L)s(\sigma)U(t)^{-1}=W(t)\exp((t_{0}-t)\chi_{0}(\gamma))W(t)^{-1}$ $=\exp((t_{0}-t)l(t))$ $tv(t)_{-}^{-}-(l)s(\sigma)/b_{+}=\exp((t_{0}-t)l(t))/b_{+}$ $W(0)=_{-}--(L)^{-1}$ $s(\sigma)/b_{+}=\exp t_{0}l(0)/b_{+}$ [A] Audin,M, Spining tops, A course on integrable systems Cambridge University Press 1996 [FH] Flascka,H, Haine,L, Vari\ et\ es de drapeaux et r\ eseavx de Toda Math $Z,$ $208$, (1991) 86

16 [I] [K1] [K2] Ikeda,K, The solutions of the Toda lattice with poles for arbitrary cells of the fiag va $7\dot{v}eties$ preprint Kostant,B, On Whittaker vectors and representation theory InventMath 48, (1978) Kostant,B, The solution to a generalized Toda lattice and representation theory AdvMath 34, (1979) [OP] Ol shanetsky, MA, Perelmov, AM, Explicit solutions of the classical generalized Toda models InventMath 54, (1979) [S] [UT] Shipman,B, A symmetry of order two in the full Kostant- Toda lattice J of Algebra 215, (1999) Ueno,K, Takasaki,K Toda lattice hierarchy, Adv Studies in Pure Math 4, 1-95(1984) 87

可積分測地流を持つエルミート多様体のあるクラスについて (幾何学的力学系の新展開)

可積分測地流を持つエルミート多様体のあるクラスについて (幾何学的力学系の新展開) 1774 2012 63-77 63 Kazuyoshi Kiyoharal Department of Mathematics Okayama University 1 (Hermite-Liouville ) Hermite-Liouville (H-L) Liouville K\"ahler-Liouville (K-L $)$ Liouville Liouville ( FLiouville-St\"ackel