, 2000 pp.19-32 1950 Mackey Dixmier = 50 Pukanzsky, Duflo, Pedersen $\text{}$ 60 Kirillov Dixmier = Mackey 62 Kirillov Auslander-Kostant 1 Pukanzsky 1 70 $ $) $-$ = = Corwin, Pedersen - $-$ 19
\S 1. $g$ $n$ $\{0\}=$ $\subset\alpha\subset\cdots\subset r_{n-1}\subset$ $=r,$ $\dim_{c}r_{j}=j(0\leq j\leq n)$ $A_{j+1}/A_{j}(0\leq j\leq n-1)$ $\lambda_{j}$ 1. $\lambda_{j}(0\leq j\leq n-1)$ $ $\Phi$ $g$ $-$ 1. : $g=\langle T,P,Q\rangle_{R};[T,P$ $=-Q,$ $[T,Q]=P$ 2. Diamond algebra : $\infty=\langle T,P,Q,Z\rangle_{R}$ ; $[T,P]=-Q,$ $[T,Q]=P,$ $[P,Q]=Z$ 2. $G$ () $g$ $exp:oarrow G$ $G$ Ll. (Dixmier [13], Saito [42]) $G$ $\Phi$ $-$ (1) $\sim(6)$ (1) $G$ {2) $exp:\iotaarrow G$ (3) $exp:aarrow G$ (4) (5) $r$ $\lambda$ A $X\ovalbox{\ttREJECT}\mapsto\lambda(XX1+i\alpha)$ $(\alpha\in \mathbb{r})$ $i=\sqrt{-1}$ (6) $p$ & $-$ $G$ $H$ $G$ $\Delta_{G}$ $H$ $\Delta_{H,G}=\Delta_{H}/\Delta_{G}$ $G$ $\psi:garrow \mathbb{c}$ $\psi(gh)=\delta_{h,g}(h)\psi(g)(g\in G, h\in H)$ 20
\mathfrak{n}^{\ell}$ $mdh$ $G-$ $v_{g,h}$ = $v_{g.h}(\psi)=\oint_{g}l^{h}\psi(g)dv_{g,h}(g)$ $H$ $P$ $P$ $G$ $F$ $F(gh)=\Delta_{HG}^{1}p,(h)p(h)^{-1}(F(g))(g\in G, h\in H)$ $mod H$ $ P =( v_{g,h}( F ^{2}))^{1/2}=(\oint_{G H} F(g) \zeta dv_{g,h}(g))^{\psi 2}$ $G$ $V_{G,H}$ $G$ $G$ $P$ $ind_{g}^{hp}$ 1 \S 2. Kirillov [27] t) - $g$ $G=\exp \mathfrak{g}$ $G$ $G$ $\hat{g}$ $G$ $g^{*}$ = $g$ $G$ $\mathfrak{g}^{*}$ 1)- () $g\in G,$ $P\in \mathfrak{g}^{*}$ $\hat{g}$ $l\in \mathfrak{g}^{*}$ $(g\cdot l)(x)=l(g^{-1}\cdot X)(\forall X\in \mathfrak{g})$ $G$ $\Phi^{*}/G$ 4 = $B_{\ell}$ $g$ $\mathfrak{n}$ $\mathfrak{n}\subset $\mathfrak{n}$ $\Leftrightarrow \mathfrak{n}=\mathfrak{n}^{\ell}\leftrightarrow u\subset \mathfrak{n}^{\ell}$ $B_{\ell}(X,Y)=\ell([X.Y])(X,Y\in o)$ $\mathfrak{n}^{1}=\{x\in p:b_{\ell}(x,y)=0, \forall Y\in \mathfrak{n}\}$ $P$ $\mathfrak{n}$ $2\dim \mathfrak{n}=\dim A^{+\dim}A^{l}$ $\hat{g}$ $\ell$ 3. $\mathfrak{g}$ $\mathfrak{g}$ )- $0$ $\ell\in g^{*}$ $0$ polarization $\mathfrak{g}$ $S(\ell,\mathfrak{g})$ $M(\ell,\mathfrak{g})$ $l\in J^{s}$ $p$ A polarization $ff\in S(\ell,\mathfrak{g})$ $\chi_{\ell}(\exp X)=e^{r(X\rangle}(X\in 0)$ $G$ $H=\exp 0$ $G$ \ell,o, $G$ ) $=ind_{h}^{g}\chi_{\ell}$ $p(p,0,g)$ $O\in S(\ell,\mathfrak{g})$ $I(P,\mathfrak{g})$ $I(P,\mathfrak{g})$ 21 Pukanszky
2.1. (Pukanszky [40]) $H=\exp 0$ $\ell\in A^{s},$ $0\in S(\ell,\mathfrak{g})$ (1) $\sim(3)$ (1) $H$.\ell =\ell +01 $0^{\perp}=\{\lambda\in \mathfrak{g}^{*} : \lambda 1_{0}=0\}_{\circ}$ (2) $\phi\in M(\ell,\mathfrak{g})$ $\ell+0^{\perp}\subset G\cdot\ell$ (3) $\lambda\in 0^{\perp}$ $O\in M(\ell+\lambda,g)$ 1. $G^{-}\text{_{}}$ $M(P,\mathfrak{g})$ Pukanszky 2.1. (Bemmat [6], Pukanszky [40], Takenouchi [43]) $l\in n^{*},$ $0\in S(\ell,g)$ (1 ) $I(\ell,g)\neq\otimes$ (2) $O\in 1(\ell,g)\Leftrightarrow 0$ Pukanszky (3) $0_{1},0_{z}\in I(P,\mathfrak{g})$ \ell,y,, $G$ ) $\cong$ (4) (1) (3) $\hat{p}=\hat{p}_{g}$ (5) $\ell_{1},$ ) $\Theta_{G}$ : $\ell_{2}\in g^{*}$ $\text{}\ell_{1}$ : $g^{*}/garrow\hat{g}$ $\mathfrak{g}^{*}arrow\hat{g}$ \ell,y2 $G$ ) () $\hat{p}$ ) ) $=\hat{\mu}p_{2}$ $\Leftrightarrow G\cdot P_{1}=G\cdot\ell_{2}$ (Kirillov $0^{*}/G$ Fell $\hat{g}$ 2.2. (Ludwig [28]) $e_{g}$ : $\mathfrak{g}^{*}/garrow\hat{g}$ \S 3. Frobenius $G=\exp\Phi$ $H=\exp 0$ $\chi$ $\tau=ind_{h}^{g}\chi$ $f\in $O\in S(f,g)$ $\chi=\chi_{f}$ \mathfrak{g}^{*}$ Pukanszky $\mathfrak{g}^{*}$ G $H-$ $\Gamma_{\tau}$ $\tilde{\mu}$ $0^{*}$ $\Gamma_{\tau}=f+O^{\perp}$ Kirillov $\mu=(\hat{p}_{g})_{*}(\tilde{\mu})$ $\pi\in\hat{g}$ $\Omega(\pi)$ $\Omega(\pi)=(\hat{p}_{G})^{-1}(\pi)_{\text{}}\Gamma_{\tau}\cap\Omega(\pi)$ $H$ $m(\pi)$ $\hat{g}$ $\mu$ $m(\pi)$ 22
$\hat{\mu}$ 3.1. (cf. [18]) $\tau\cong\int_{\hat{g}}^{\oplus}m(\pi)\pi d\mu(\pi)$ $K=\exp f$ $G$ $\pi\in\hat{g}$ $G-$ $\Omega(\pi)$ $\tilde{\gamma}=\tilde{\gamma}_{\pi}$ $\Omega(\pi)$ $\mathfrak{g}^{*}$ $p:g^{*}arrow f^{*}$ $\gamma=(\hat{p}_{k}\circ p)_{*}(\tilde{\gamma})$ $\sigma\in\hat{k}$ $0$) $(\sigma)=(\hat{p}_{k})^{-1}(\sigma)\subset 9^{*}$ $\Omega(\pi)\cap p^{-1}(ox\sigma))$ $K-$ $n_{\pi}(\sigma)$ $\hat{k}$ $\gamma$ $n_{\pi}(\sigma)$ $\pi\in\hat{g}$ $K$ $\pi _{K}$ 3.2. (cf. [19]) $\pi 1_{K}$ $\cong\int_{\hat{k}}^{\oplus}n_{\pi}(\sigma)d\gamma(\sigma)$ $\pi_{j}(j=1,2)$ $G$ 2 - $\pi_{1}\cross\pi_{2}$ $G\cross G$ $(\Omega(\pi_{1}), = $\pi_{1}\cross\pi_{2}$ $\pi_{1}\otimes\pi_{2}$ \Omega(\pi_{2}))\subset \mathfrak{g}^{*}\oplus \mathfrak{g}^{*}$ $G$ $G\cross G$ $\pi_{1}$ $\pi_{2}$ 1. $p:\mathfrak{g}^{*}\oplus \mathfrak{g}^{*}arrow g^{*}$ $\gamma=(\hat{p}_{g}\circ p)_{*}(\tilde{\gamma}_{\pi_{1}\cross\pi_{2}})$ $\pi\in\hat{g}$ $(\Omega(\pi_{1}),\Omega(\pi_{2}))\cap p^{-1}(\omega(\pi))$ $G$ $m(\pi)$ $\pi_{1}\otimes\pi_{2}\cong\int_{\hat{g}}^{\oplus}m(\pi)\pi d\gamma(\pi)$ $\sigma\in\hat{k}$ 3.1 $ind_{k}^{g}\sigma$ $g^{*}$ $p^{-1}(ox\sigma))$ $0$) $(\sigma)$ $K$ $\hat{\mu}$ $\hat{g}$ $\mu$ ind $GH\sigma$ $g^{*}$ $\tilde{\mu}$ $\mu=(\hat{p}_{g})_{*}(\tilde{\mu})$ 3.2 $\hat{g}\ni\pi$ $\vdash\rightarrow n_{\pi}(\sigma)$ 3.3. (cf. [19]) $ind_{h}^{g}\sigma\equiv\int_{\hat{g}}^{\oplus}n_{\pi}(\sigma)\pi d\mu(\pi)$ 32 33 2. Frobenius $G=\exp \mathfrak{g}$ 3.1 $\tau=ind_{h}^{g}\chi_{f}$ 23
$\ovalbox{\ttreject}_{\pi}^{arrow}$ \S 4. $G$ $\pi$ () $\ovalbox{\ttreject}_{\pi}$ $G\ni g\mapsto\pi(g)v\in$ $C^{\infty}$. $v\in$ $\pi \text{}c^{\infty}-$ $g$ $d\pi$ : $d \pi(x)v=\frac{d}{dt}\pi(\exp(tx))v[_{=0}(x\in A, v\in\ovalbox{\ttreject}_{\pi}^{\infty})$ $\pi$ $d\pi$ A $\mathscr{u}(p)$ $\{X_{j}\}_{1\leq j\leq n}$ $g$ $\mathfrak{p}_{\pi}(x^{a})\mathcal{v} = \oint\pi(x)^{\alpha_{1}}\cdots d4x_{n})^{\alpha_{*}}v t\alpha=(\alpha_{1},\ldots,\alpha_{n})\in \mathbb{z}_{+}^{n},$ $\mathbb{z}_{+}=\{0\}\cup N$ $\pi$ = $\ovalbox{\ttreject}_{\pi}^{arrow}$ $\langle a,b\rangle$ $a\in\ovalbox{\ttreject}_{n}^{b$ $b\in\ovalbox{\ttreject}_{\pi}^{\mp\infty}$ $G$ A $G$ $C^{\infty}$ (G) $\phi\in$ G) \mbox{\boldmath $\pi$}(\mbox{\boldmath $\phi$}) $\ovalbox{\ttreject}_{n}^{arrow}\subset$ $-j$ I/ $dg$ $\pi(\phi)$ $G$ $\pi(\phi)=\int_{g}\phi(g)\pi(g)dg$ $[7]_{\text{}}[8]_{\text{}}$ [38] $G=\exp$ $ $\pi$ 2.1 $\ell\in g^{*}$ $r$ polarization $\mathfrak{n}$ $\pi$ $\mu\ell,h,g$) $h=f_{0}\subset f_{1}\subset\cdots\subset f_{d}=\mathfrak{g}\dim f_{j+1}/f_{j}=1(0\leq j\leq d-1)$ $\tilde{x}_{j}\in f_{j+1}\backslash f_{j}(0\leq j\leq d-1)$ $\Phi:\mathbb{R}^{d}\cross B\ni(t,b)I\Rightarrow(\prod_{j=d}^{1}\exp(t_{j}X_{j}))b\in G,$ $C=(t_{1},\ldots,t_{d}),$ $B=\exp b$ $\Phi$ $\Phi$ $G/H$ = $\pi$ $L^{2}(\mathbb{R}^{d})$ $i$) $-$ 4.1. $d\pi(\mathscr{u}(a))$ (cf. [11]) (Rd) = $\pi$ = $G$ 24
$a_{\pi}^{k}$ : j $K$ $\chi:karrow \mathbb{c}^{*}$ $(\ovalbox{\ttreject}_{\pi}^{arrow})^{\kappa_{x} =\{a\in\ovalbox{\ttreject}_{\pi}^{arrow} : \pi(k)a=\chi(k)a, \forall k\in K\}$ $\tau=ind_{h}^{g}\chi_{f}$ $G$ $e$ $6_{\tau}\in(\ovalbox{\ttREJECT}_{\tau}^{\infty})^{H,\chi_{f}$ $6_{\tau}$ $\ovalbox{\ttreject}_{\pi}^{\infty}\ni\psiarrow\overline{\psi(e)}\in \mathbb{c}$ : ( 3.1) $m(\pi)$ Penney [37] = $6_{\tau}$ : $6_{\tau}= \int_{\hat{g}}^{\oplus}\sum_{k=1}^{m(\pi)}a_{\pi}^{k}d\mu(\pi)_{\text{}}$ (41) $a_{\pi}^{k}\in(\ovalbox{\ttreject}_{\pi}^{r)^{h,\chi_{f}}(1\leq k\leq m(\pi))$ $H$ $1\backslash$ $\phi\in$ G) $dh$ $\phi_{h}^{f}\in $\phi_{h}^{f}(g)=\int_{h}\phi(gh)\chi_{f}(h)dh(g\in G)$ \text{_{}\tau}^{\infty}$ $\tau(\phi)$ $G$ $J\backslash$-$js$ $dg_{\text{}}$ $v_{g,h}$ $v_{g,h}=dg/dh$ $\phi\in \mathfrak{u}(g)$ $\phi_{h}^{f}(e)=\langle\tau(\phi)6_{\tau},6_{\tau}\rangle$ (4-1) $\phi_{h}^{f}(e)=\int_{\hat{g}}\sum_{k=1}^{m(\pi)}\langle\pi(\phi)a_{\pi}^{k},a_{\pi}^{k}\rangle d\mu(\pi)$ (4-2) = $a_{\pi}^{k}$ (4-1) (4-2) $\pi\in\hat{g}$ $l\in\omega(\pi)$ polarisation $u\in M(\ell,\mathfrak{g})$ \ell, $u$, $G$ ) 3.1 $\Gamma_{\tau}\cap\Omega(\pi)$ $g_{k}\in G$ $H$ $C_{1},$ $\ldots,$ $C_{m(\pi)}$ $1\leq k\leq m(\pi)$ $g_{k}\cdot\ell\in C_{k}$ 4.2. (cf. [17]) $H/H\cap g_{k}bg_{k}^{-1}$ $(\ovalbox{\ttreject}_{\pi}^{arrow})^{h,\chi_{f}$ $d_{k}h$ $\ovalbox{\ttreject}_{\pi}^{\infty}\ni\psiarrow\int_{h/h\cap g,bg_{\overline{t}^{1}}\overline{\psi(hg_{k})\chi_{f}(h)}d_{k}\dot{h}(1\leq k\leq m(\pi))$ $m(\pi)$ == (4-2) 2. ( 3.1) $m(\pi)$ = (4-1) (cf. [20]. [33]) (cf. [31) Lipsman = = 25
(cf. $[12]_{\text{}}[\mathfrak{B}]_{\text{}}$ [30]) \S 5. $H$ $\chi_{f}$ $G/H$. $--$. $--$ $G-$ $D_{\tau}(G/H)$ $0$ $\{Y_{j}\}_{1\leq j\leq d}(d=\dim 0)$ (g) $= \sum_{j1}^{\text{}}\mathbb{c}(y_{j}+if(y_{j}))$ (g) $\mathscr{u}(p)\mathfrak{n}_{\tau}$ $\mathscr{u}(g,\tau)=$ { $A\in\%(\mathfrak{g}):[A,Y]\in u(a)\mathfrak{n}_{\tau}$ fy $\in 0$ } $\mathscr{u}($ $G$ D\tau (G/ $\mathscr{u}(g,\tau)/\mathscr{u}(a)0_{\tau}$ D\tau (G/ Corwin-Greenleaf [10] 5.1. (cf. [101) ( 3.1) D\tau (G/ 1. () \Leftrightarrow D\tau (G/ 2. () D\tau (G/ 1 $H-$ $C[\Gamma_{\tau}]^{H}$ 3. 1 Duflo [16] Corwin-Greenleaf [10] 2 $\phi\in$ $G$ ) (4-2) $\phi_{h}^{f}(g)=\int_{\hat{g}}\sum_{k=1}^{m(\pi)}\langle\pi(\phi_{g})a_{\pi}^{k},a_{\pi}^{k}\rangle d\mu(\pi)(\forall g\in G)_{\text{}}$ $\phi_{g}$ $\phi_{g}(x)=\phi(gx)(\forall x\in G)$ $\Phi\langle G$) $X\in \mathscr{u}(\mathfrak{g})$ $(X \phi_{h}^{f})(g)=\int_{\hat{g}}\sum_{k=1}^{m(\pi)}\langle\pi(\phi)a_{\pi}^{k},d\pi(\overline{x})a_{\pi}^{k}\rangle d\mu(\pi)_{\text{}}$ (5-1) 5.2. (cf. [22]) $\mu$ $\hat{g}$ $a_{\pi}^{k}(1\leq k\leq m(\pi))$ (g,\tau ) 26
$X\in \mathscr{u}(\mathfrak{g},\tau)$ $\Gamma_{\tau}$ $P_{X}$ $\ell\in\gamma_{\tau}$ : $H-$ k $\Omega(\pi)$ $d\pi(\overline{x})a_{\pi}^{k}=\overline{p_{x}(\ell}\nu_{\pi}^{k}$ $P_{X}$ $H$ $\Gamma_{\tau}$ X\in l(g) $P_{X}\equiv 0$ (5-1) $D_{\tau}(G/H)\cong \mathscr{u}(g,\tau)/\mathscr{u}(\mathfrak{g})\mathfrak{n}_{\tau}\ni[x]\vdash\rightarrow P_{X}$ 2 (1) $P_{X}$ $\Gamma_{\tau}$ ; (2) (1) (g,\tau )/?X(9) $\mathfrak{n}_{\tau}\ni[x]$ $\vdash\rightarrow P_{X}\in C[\Gamma_{r},]^{H}$ ; 2 2 (cf. $[4]_{\text{}}[21]$ ) = $P_{X}$? 2 \S 6. 1 1 Baklouti $Lion_{\text{}}Magneron_{\text{}}$ Mehdi $\Phi$ $\{0\}=\iota_{0}\subset \mathfrak{g}_{1}\subset\cdots\subset$ -1\subset \sim $=\mathfrak{g}$ $O\cap g_{i}\neq 0\cap \mathfrak{g}_{i-1}$ $i(1\leq i\leq n)$ $\mathcal{j}=\{\dot{q}<\dot{a}<\cdots<i_{d}\}$ $(d=\dim 0)$ $\ovalbox{\ttreject}=\{1,2, \ldots, n\}\backslash \ovalbox{\ttreject}=\{j_{1<j_{2}<\cdots<j_{p}\(p=\dim$ $/O} $f_{0}=\mathfrak{h},$ $f_{r}=\mathfrak{h}+\mathfrak{g}_{j_{r}}(1\leq r\leq p)$ $g$ $0=l_{0}\subset f_{1}\subset\cdots\subset f_{p-1}\subset e_{p}=\mathfrak{g}(\dim t_{r}/f_{r-1}=1)$ $0_{0}=\{0\},$ $0_{S}=0\cap \mathfrak{g}_{i_{s}}(1\leq s\leq d)$ $0$ $\{0\}=0_{0}\subset 0_{1}\subset\cdots\subset 0_{d-1}\subset\phi_{d}=1(\dim 0_{s}=s)$ $\overline{y}_{s}\in 0_{s}\backslash 0_{s-1}(1\leq s\leq d)$ $X_{r}\in f_{r}\backslash f_{r- 1}(1\leq r\leq p)$ $0$ $\mathfrak{g}$ $K_{j}=\exp f_{j}$ $\tau_{j}=ind_{h}^{k_{j}}\chi_{f}(0\leq j\leq p)$ $\mathbb{c}=d_{\tau_{\text{}}}(k_{0}/h)\subset D_{\tau_{1}}(K_{1}/H)\subset\cdots\subset D_{\tau_{p-1}}(K_{p-1}/H)\subset D_{\tau_{p}}(K_{P}/H)=D_{\tau}(G/H)$ = $D_{\tau}(G/H)$ $\tau_{j_{0}- 1}$ = $\tau_{j_{0}}$ = $i_{0}(1\leq j_{0}\leq p)$ 6.1. (cf. Greenleaf [26]. Fujiwan-Lion-Mehdi [24]) 27
$D_{\tau_{j_{0^{-1}}}}(K_{j\text{}-1}/H) \neq D_{\tau_{h}}(K_{j_{0}}\int H)$ $D_{\tau_{j\text{}}}(K_{j_{0}}/H)$ = $V$ $K$ $V$ $0\neq x\in V$ $K-$ $K$ $\Omega\subset V$ 2 = $\Omega+\mathbb{R}x\subset\Omega$ $v\in\omega$ (v+&)\cap \Omega $=\{v\}$ $\Omega$ $x-$ $\tau_{j}(1\leq j\leq p)$ $H-$ $r_{j_{0}}=\{p\in f_{j_{0}}^{*} : \ell 1_{1}=f1_{0}\}$ $K_{j_{0}}$ $H-$ - $f_{j_{0^{-}}1^{-}}$. $\Gamma_{j}$ $H$ 1- ) ( $D_{\tau_{j}}(K_{j}/H$ $=D_{\tau_{j-1}}(K_{j-1}/H)$ = Corwin-Greenleaf [10] ) $n-1$ $G$ $j=n$ $d=\dim\phi=1$ $(\alpha 0)$ $0$ $m$ $ad0-$ $\beta:s(g)arrow \mathscr{u}(p)$ $s(m)$ = $\beta$ (B) $\mathfrak{n}_{\tau}$ = $\dim 0$ $0 =0_{d-1},$ $g =f_{p-1}$, $H=\exp 0,$ $G =\exp p,$ $\tau =ind_{h }^{G}\chi_{f}$, $=ind_{h }^{G }\chi_{f}$ $\Gamma_{\tau}$ $D_{\tau}(G/H)\neq D_{\tau_{p- 1}}(G /H)$ $H-$ $(\mathfrak{g} )^{\perp}-$ $H$ $(\mathfrak{g} )^{\perp}$ $\Gamma_{r }$, $H$ $W\in \mathscr{u}(n,\tau )$ $\mathscr{u}(g )+\mathscr{u}(a)\mathfrak{n}_{\tau }$ $W$ $W\not\in\ (9 )+0\ (9)\text{}$ $ady_{d}$ (g,\tau ) $ady_{d}$ $W$ 6.1. (cf. Baklouti-Fujiwara [2]) $\mathscr{u}(g )+\mathscr{u}(\mathfrak{g})\text{}$ $\mathscr{u}(a,\tau )\not\subset \mathscr{u}(\mathfrak{g},\tau_{p-,)}$ $\mathscr{u}(g,\tau)$ 28
6.1 $\mathscr{u}(9,\tau )\not\subset \mathscr{u}(p,\tau_{p-1})$ = $l\in\gamma_{\tau}$ $B_{\ell}$ $Y_{d}\in\phi$ $\mathscr{u}(\mathfrak{g},\tau )=\mathscr{u}(\phi^{j},\tau_{p-1})$ $0_{d-1}\cap 0^{\ell}\neq 0_{d}\cap$ $\ell\in\gamma_{\tau}$ = $\tau_{p-1}$ == $[Y_{d},W]\in \mathscr{u}(\mathfrak{g})\mathfrak{n}_{\tau}$ $T(\Gamma_{\tau})=$ $\{\text{} <m_{2}<\cdots<m_{q}\}$ = $\ell\in\gamma_{\pi}$ $A_{k-1}\cap g^{\ell}\neq$ $\cap$ $k(1\leq k\leq n)$ Corwin-Greenleaf [10] $\Gamma_{\tau}-$ $\{\sigma_{1}, \sigma_{2}, \ldots, \sigma_{q}\}$ $\sigma_{j}(1\leq j\leq q)$ $m_{j}$ $\sim \text{},$ ) $\backslash 1(\text{_{}-1})$ = $\ell\in\gamma_{\tau}$ $d\hat{p}(u\sigma_{j})$ $=m_{k}$ $k(1\leq k\leq q)$ $\mathscr{u}(g)$ $\gamma$ ; 6.2. (Fujiwara-Lion-Magneron-Mehdi) $\mathscr{u}\mathscr{u}(g$,-1 $Y_{d}$ $\gamma(\sigma_{1})$, $)$ $\cdot$.., $\gamma(o_{k})$ $[V,\gamma(\sigma_{j})]\in \mathscr{u}(\mathfrak{g})u_{\tau}(1\leq j\leq q)$ $\mathscr{u}(\mathfrak{g})$ $V$ 62 $[W,Y_{d}]\in \mathscr{u}(n)\mathfrak{n}_{\tau}$ $W\in \mathscr{u}(g,\tau)$ 6.2. 3. [1] L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. math. 14 (1971), 255-354. [2] A. Baklouti et H. Fujiwara, Op\ erateurs diff\ erentiels associ\ es \ a certaines repr\ esentations unitaires d un groupe de Lie r\ esoluble exponentiel, Pr\ epublication. [3] A. Baklouti and J. Ludwig, The Penney-Fujiwara Plancherel formula for nilpotent Lie groups, J. Maffi. Kyoto Univ. 40 $(20(\mathfrak{P}),$ $1-11$. [4] A. Baklouti and J. Ludwig, Invariant differential operators on certain nilpotent homogeneous 29
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