数理解析研究所講究録
第 1868 巻 2013 年 9-23

9

SPECIAL LAGRANGIAN SUBMANIFOLDSINVARIANT UNDER THE ISOTROPY ACTION OF
SYMMETRIC SPACES OF RANK TWO
大阪市立大学数学研究所橋本要 (Kaname Hashimoto)
Osaka City University Advanced Mathematical Institute

1. INTRODUCTION
Special Lagrangian submanifold in Calabi-Yau manifolds play an important role in the explanation of Mirror symmetry. They are examples of calibrated submanifolds, appearing in Harvey and Lawson ([1]),which generalizes the concept of volume minimizing property of complex submanifolds of K\"aher manifolds. Let $M$ be a Calabi-Yau manifold with a complex volume form . Then naturally
is a calibra$M$
tion on , and a calibrated submanifold is called a special Lagmngiansubmanifold. For examples, Joyce constructed many interesting examples of special Lagrangian submanifolds in, using various methods.
In particular, cohomogeneity one special Lagrangian submanifolds areconstructed using moment map techniques.
The cohomogeneity one actions on spheres have been classified by
Hsiang and Lawson ([2]). Every cohomogeneity one action on
isorbit equivalent to the isotropy representation of a Riemannian symmetric space of rank 2.
A compact hypersurface $N$ in the unit standard sphereis homogeneous if it is obtained as an orbit of a compact connected subgroupof $SO(n+1)$ . It is well known that any homogeneous hypersurface
incan be obtained as a principal orbit of the isotropy representation of a Riemannian symmetric space of rank 2 ([2]).
homogeneous$N$hypersurface
inis a hypersurface with constant principal curvatures, which is called([13]). Then the number ofdistinct principal curvatures must be 1, 2, 3, 4 or 6 ([13], [8] and see[9], [10] for general isoparametric hypersurfaces). Denote by $(m_{1}, m_{2})$${\rm Re}\Omega$$\Omega$$\mathbb{C}^{n}$$S^{n}$$S^{n}$$S^{n}$$A$$S^{n}$$i_{\mathcal{S}}opammetr’ic$

2010 Mathematics Subject Classification. Primary:$53C42.$$g$$53C40$ ; Secondary:$53D12,$

10
the multiplicities of its principal curvatures. The isotropy representation of a Riemannian symmetric space $G/K$ of rank 2 induces a groupin a natural way. In the cases ofand thus
action of $K$ on$g=1,2$ such group actions of $SO(p)\cross SO(n+1-p)(1\leq p\leq n)$ are. We classified cohomogeneity one special Lagrangian
induced onsubmanifolds inunder the group actions ([5]).
In this paper we shall discuss the construction of cohomogeneity
one special Lagrangian submanifolds in the case when $g=1,2,3,$
4 and $G/K$ is of classical type. We refer Halgason) textbook ([7])$T^{*}S^{n}$$S^{n}$$T^{*}S^{n}$$T^{*}S^{n}$$s$

TABLE 1. Homogeneous hypersurfaces in spheres

for the general theory of Riemannian symmetric space. Let $M$ bea simply connected semisimple Riemannian symmetric space. If
is the identity component of the full group of isometrics of $M$ , thenacts transitively on $M$ and we can write $M=G/K$ , where $K$ is
the isotropy subgroup of
at a point $p\in M$ . Since
is simply$K$
connected and is connected, is also connected. If
is thecanonical decomposition of associated to the symmetric pair $(G, K)$ ,then the isotropy representation of $K$ on $T_{p}M$ is equivalent to theadjoint representation of $K$ on . The isotropy representation of $G/K$
$G$

$G$

$G$

$S$

$G$

$g=P\oplus \mathfrak{p}$$\mathfrak{g}$$\mathfrak{p}$11:. So a $K$ -orbit
at is a Lie group homomorphism
throughis denoted by
Let $G/K$ be an-dimensional rank 2 symmetric space of compact type. Define an-invariant inner product of from theKilling from of . Then the vector space can be identified with
with respect to the inner product. Let be a maximal abelian subspace of . Since for each
there is an element $k\in K$ such that, every orbit in under $K$ meets . The unit hyperspherein is denoted by. Since the action of $K$ on is an orthogonal representation, an orbit
is a submanifold of the hyperspherein . For a regular element, we obtain a homogeneous
hypersurface
Conversely, every
homogeneous hypersurfaces in a sphere is obtained in this way ([2]).
$p$$Karrow SO(\mathfrak{p})$$Ad_{\mathfrak{p}}$$Ad_{\mathfrak{p}}(K)X.$$X\in \mathfrak{p}$$(\prime n+1)$$Ad_{\mathfrak{p}}(K)$$\mathfrak{p}$$\mathbb{R}^{n+1}$$\mathfrak{p}$$\mathfrak{g}$$a$$X\in \mathfrak{p}$$\mathfrak{p}$$Ad_{\mathfrak{p}}(K)X\in \mathfrak{a}$$\mathfrak{p}$$\mathfrak{p}$$S^{n}$$\mathfrak{p}$$\mathfrak{p}$$S^{n}$$Ad_{\mathfrak{p}}(K)X$$H\in \mathfrak{a}\cap S^{n}$$\mathfrak{p}$$N=Ad_{\mathfrak{p}}(K)H\subset S^{n}\subset \mathfrak{p}\cong \mathbb{R}^{n+1}$

2. CALABI-YAU MANIFOLDS AND SPECIAL LAGRANGIANSUBMANIFOLDSWe shall review some definitions and basic notions of Calabi-Yau
manifolds and special Lagrangian submanifolds. See [4] for details.
There are several different definitions of Calabi-Yau manifolds. In
this paper, we use the following definition.
Definition 2.1. Let. An almost Calabi-Yau’ -fold is a quadruple
such that $(M, J, \omega)$ is a K\"ahler manifold of complex
dimension with a complex structure and a K\"ahler form , andis a nonvanishing holomorphic $(n, 0)$ -form on $M$ . In addition, if andsatisfy$\prime n\geq 2$$(M, J, \omega, \Omega)$

$J$

$n$

$\Omega$$\omega$$\omega$$\Omega$(1)$\frac{\omega^{n}}{n!}=(-1)^{n(n-1)/2}(\frac{\sqrt{-1}}{2})^{n_{\Omega\wedge\overline{\Omega}}})$

then we call$(M, J, \omega, \Omega)$a Calabi-Yau -fold.
$n$

If and satisfy (1), then the K\"ahler metric of $(M, J, \omega)$ is Ricciflat. Its holonomy group Hol $(g)$ is a subgroup of $SU(n)$ , and this is
another definition of a Calabi-Yau manifold.
A closed -form on a Riemannian manifold $(M, g)$ is called a calibration if
for any oriented -plane $V\subset T_{x}M$ for all $x\in M.$$Ap$-dimensional submanifold $N$ of $M$ is said to be calibrated by a calibration if
for all $x\in N.$$\Omega$$\omega$$g$$p$$\varphi$$\varphi|_{V}\leq vo1_{V}$$\varphi$$\varphi|_{T_{x}N}=vo1_{T_{x}N}$

$p$

12
Remark 2.2. The constant factor in (1) is chosen so thatis a calibration for any${\rm Re}(e^{\sqrt{-1}\theta}\Omega)$$\theta\in \mathbb{R}.$Definition 2.3. Let
be a Calabi-Yau -fold and be areal -dimensional submanifold of $M$ . Then, foris calleda special Lagmngian ubmanifold of phase if it is calibrated by the.
calibration$(M, J,\omega, \Omega)$

$L$

$n$

$n$

$\theta\in \mathbb{R},$$L$$\theta$$\mathcal{S}$${\rm Re}(e^{\sqrt{-1}\theta}\Omega)$

Harvey and Lawson gave the following alternative characterizationof special Lagrangina submanifolds.
Proposition 2.4 ([1]). Let
be a Calabi-Yau -fold andbe a real -dimensional submanifold of M. Then is a special Lagmngian submanifold of phase if and only if
and$(M, J, \omega, \Omega)$

$L$

$n$

$L$

$n$

$\omega|_{L}\equiv 0$$\theta$${\rm Im}(e^{\sqrt{-1}\theta}\Omega)|_{L}$$\equiv 0.$

3. STENZEL METRIC AND MOMENTA MAPS

. We denote the cotanWe briefly recall the Stenzel metric ongent bundle of the -sphere $S^{n}\cong SO(n+1)/SO(n)$ by $T^{*}S^{n}=$. We identify the tangent
by the Riemannian metric onbundle and the cotangent bundle ofSince any unit cotangent vector of can be translated to anotherone, the Lie group $SO(n+1)$ acts onwith cohomogeneity one by$g\cdot(x,\xi)=(gx, g\xi)$ for $g\in SO(n+1)$ . Let
be a complex quadric in
defined by$T^{*}S^{n}$$n$$\{(x, \xi)\in \mathbb{R}^{n+1}\cross \mathbb{R}^{n+1}|\Vert x\Vert=1, \langle x, \xi\rangle=0\}$$S^{n}$$S^{n}$$S^{n}$$T^{*}S^{n}$$Q^{n}$$\mathbb{C}^{n+1}$$Q^{n}= \{z=(z_{1}, \ldots, z_{n+1})\in \mathbb{C}^{n+1} \sum_{i=1}^{n+1}z_{i}^{2}=1\}.$transitively, hence $Q^{n}\cong SO(n+$The Lie group $SO(n+1, \mathbb{C})$ acts on. According to Sz\"oke ([12]), we can identify
withthrough the following diffeomorphism:$Q^{n}$$T^{*}S^{n}$$1,$ $\mathbb{C})/SO(n, \mathbb{C})$$Q^{n}$$\Phi$:$\prime 1^{\prime*}S^{n}\ni(x, \xi)\mapsto x\cosh(\Vert\xi\Vert)+\sqrt{-1}\frac{\xi}{\Vert\xi\Vert}\sinh(\Vert\xi\Vert)\in Q^{n}$

The diffeomorphism is equivariant under the action of $SO(n+1)$ .
. Then consider a holomorThus we frequently identify
withphic -form
given by$\Phi$$T^{*}S^{n}$$n$$Q^{n}$$\Omega_{Stz}$$\frac{1}{2}d(z_{1}^{2}+z_{2}^{2}+\cdots+z_{n+1}^{2}-1)\wedge\Omega_{Stz}=dz_{1}\wedge\ldots\wedge dz_{n+1}.$

13
The Stenzel metric is a complete Ricci-flat K\"ahler metric on
definedby, whereand is asmooth real-valued function satisfying the differential equation$Q^{n}$$\omega_{Stz};=\sqrt{-1}\partial\overline{\partial}u(r^{2}-)$$r^{2}= \Vert z\Vert^{2}=\sum_{i=0}^{n+1}z_{i}\overline{z}_{i}$$u$$\frac{d}{dt}(U’(t))^{n}=cn(\sinh t)^{n-1} (c>0)$where $U(t)=u(\cosh t)$ . The K\"ahler form
is exact, that is,.
where
Let $K$ be a compact connected Lie subgroup of $SO(n+1)$ with Liealgebra . Then the group action of $K$ on
is Hamiltonian withrespect toand its moment map :is given by$\omega_{Stz}=$$\omega_{Stz}$$d\alpha_{Stz}$$\alpha_{Stz}$$:=-{\rm Im}(\overline{\partial}u(r^{2}))$$Q^{n}$$\mathfrak{k}$$\omega_{Stz}$(2)$\langle\mu(z),$$\mu$$Q^{n}arrow \mathfrak{k}^{*}$$X\rangle=\alpha_{Stz}(Xz)=u’(\Vert z\Vert^{2})\langle Jz,$ $Xz\rangle$$(z\in Q^{n}, X\in \mathfrak{k})$.

Choose a subsetofsuch that every $K$ -orbit in
meets
$K$
. In general assume that has the Hamiltonian group action on a
symplectic manifold $M$ . We define the center of to be$\mathfrak{k}^{*}|Ad^{*}(k)X=X(\forall k\in K)\}$ . It is easy to see that the inverse imageofis invariant under the group action of $K$ if and only if$\Sigma$$T^{*}S^{n}$$T^{*}S^{n}$$\Sigma$$Z(\mathfrak{k}^{*})=\{X\in$$\mathfrak{k}^{*}$$\mu^{-1}(c)$$c\in \mathfrak{k}^{*}$$c\in Z(\mathfrak{k}^{*})$.

Proposition 3.1. Let
be a connected isotropic submanifold, i.e.,, of $M$ invariant under the action of K. Then $L\subset\mu^{-1}(c)$ for$L$$\omega|_{L}\equiv 0$

some$c\in Z(\mathfrak{k}^{*})$.

Proposition 3.2. Let
be a connected submanifold of $M$ invariant
under the action $ofK$ . Suppose that the action of $K$ on is of cohomoone (possibly tmnsitive). Then is an isotropic submanifold,$i.e,$, if and only if.
for some
$L$

$L$

$L$

$genei\cdot ty$$\omega|_{L}\equiv 0$$L\subset\mu^{-1}(c)$$c\in Z(\mathfrak{k}^{*})$For the group action of $K$ induced by the isotropy representation of$G/K$ , the moment map formula (2) becomes(3)$\mu(Z)=-u’(\Vert Z\Vert^{2})\sqrt{-1}[Z, \overline{Z}]=-2u’(\Vert Z\Vert^{2})[X, Y]\in \mathfrak{k}\cong \mathfrak{k}^{*}$

for eachwith $X,$
Now we consider only the case where the inverse image
In the same way as [2], the orbit space of $K$-action on
be explicitly parametrized by a complex coordinate$Z=X+\sqrt{-1}Y\in Q^{n}\subset \mathfrak{p}^{\mathbb{C}}\cong \mathbb{C}^{n+1}$$Y\in \mathfrak{p}\cong \mathbb{R}^{n+1}$$\mu^{-1}(0)$$\mu^{-1}(0)$$0\in \mathfrak{k}^{*}$

of

can

$\tau=t+\sqrt{-1}\xi_{1}\in \mathbb{C}.$4. MAIN RESULTS

In [5] we studied in detail and classified cohomogeneity one special
Lagrangian submanifolds inunder the group action of $SO(p)\cross$$T^{*}S^{n}$

14$SO(n+1-p)(1\leq p\leq n)$ . In these cases, we give explicit descriptionsof the special Lagrangian submanifolds in terms of ordinary differential equations and determine the diffeomorphism type of the principal
orbits. These special Lagrangian submanifolds are generically smooth,but in some degenerate cases they are singular and we explicitly describe the form of the singularities. We observe the asymptotic behavior of the ends and singularities of special Lagrangian submanifolds in$\tau*s^{n}.$In this section, by generalizing the arguments of [5], we provide aconstruction of cohomogeneity one special Lagrangian submanifolds inunder the group action induced by the isotropy representation ofa Riemannian symmetric space $G/K$ of rank 2,$T^{*}S^{n}$4.1. Case $g=1.$
4.1.1. $(S^{1}\cross SO(n+1), SO(n))$ .
Theorem 4.1. Letain$cu7^{\cdot}ve\sigma$$\tau$

be a regular curve in the complex planeby$\mathbb{C}$. Define$\mu^{-1}(0)\cap\Phi(\Sigma)$$\sigma(s)=(\cos\tau(s), \sin\tau(s), 0, \ldots, 0)$.

Then the $K$ -orbit $L=K\cdot\sigma$ through is a Lagmngian submanifold in. Moreover, the smooth part $ofL$ is a special Lagmngian submanifoldof phase if and only if satisfies$\sigma$$Q^{n}$$\theta$$\tau$(4)${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(\sin\tau(s))^{n-1})=0.$

4.2. Case $g=2.$

4.2.1. $(SO(p+1)\cross SO(n+1-p), SO(p)\cross SO(n+1-p))$ .
Theorem 4.2. Letain$cur^{v}ue\sigma$$\tau$

be a regular curve in the complex planeby$\mathbb{C}$. Define$\mu^{-1}(0)\cap\Phi(\Sigma)$$\sigma(s)=(\cos\tau(s), 0, \ldots, 0, \sin\tau(s), 0, \ldots, 0)$.

Then the $K$ -orbit $L=K\cdot\sigma$ through is a Lagmngian submanifold in. Moreover, the smooth part of is a special Lagmngian submanifoldof phase if and only if satisfies$\sigma$$L$$Q^{n}$$\theta$$\tau$(5)${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(\cos\tau(s))^{p-1}(\sin\tau(s))^{q-1})=0.$

4.3. Case $g=3.$

15
4.3.1. $(G, K)=(SU(3), SO(3))$ . We consider the case of $(G, K)=$$(SU(3), SO(3))$ .

We denote by and the Lie algebras ofrespectively. The canonical decomposition of is given by
where
and

and $K$$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p},$$g$$\mathfrak{k}=\mathfrak{s}0(3)$$G$$\mathfrak{k}$$\mathfrak{g}$$\mathfrak{p}=\{\sqrt{-1}X\in M_{3}(\mathbb{R})|tX=X, TrX=0\}.$

Then the isotropy representation of $K$ is defined by $Ad_{\mathfrak{p}}(k)X=kX^{t}k$for $k\in SO(3)$ and. We define an inner product on by$Y\rangle=-Tr(XY)$ for $X,$
. Let
$X\in \mathfrak{p}$$\langle X,$$\mathfrak{p}$$Y\in \mathfrak{p}$$\mathfrak{a}=\{\sqrt{-1}(\alpha_{1} a_{2} a_{3}) (\iota_{1},\alpha_{2},\alpha_{3}\in \mathbb{R}a_{1}+a_{2}+a_{3}=0, \}$Then is a maximal abelian subspace of . The group action of $K=$
$SO$ (3) is naturally induced on the complex quadricin$a$$\mathfrak{p}$$Q^{4}$$\mathfrak{p}^{\mathbb{C}}=\{Z\in$$M_{3}(\mathbb{C})|tZ=Z,$ $TrZ=0\}.$

Theorem 4.3. Let\‘a$cu7^{\cdot}ve\sigma$

in$\tau$

be a regular curve in the complex planeby$\mathbb{C}$. Define$\mu^{-1}(0)\cap\Phi(\Sigma)$$\sigma(s)=$$\frac{1}{\sqrt{6}}(2cos\tau(s) -cos\tau(s)+\sqrt{3}sin\tau(s) -cos\tau(s)- \sqrt{3}sin\tau(s)) \in \mathfrak{a}^{\mathbb{C}}$

Then $the_{J}K$ -orbit $L=K\cdot\sigma$ through a cume
is a cohomogeneity
one Lagmngian submanifold under the gmup action of $K$ in
Conversely, such a cohomogeneity one Lagrangian submanifold in
isobtained in this way. Moreover, is a special Lagmngian $\mathcal{S}ubmanifold$of phase if and only if satisfies$\sigma$$Q^{4}$$Q^{4}$$L$$\theta$$\tau$(6)${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(3\cos^{2}\tau(s)-\sin^{2}\tau(s))\sin\tau(s))=0.$

4.3.2. $(G, K)=(SU(3)\cross SU(3), SU(3))$ . We consider the case of$(G, K)=(SU(3)\cross SU(3), SU(3))$ . The canonical decomposition ofis given by, where$\mathfrak{g}$$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$$\mathfrak{k}=\{(X, X)|X\in \mathfrak{s}u(3)\}$and$\mathfrak{p}=\{(X, -X)|X\in Bt\downarrow(3)\}\cong Bu(3)$.

16, we identify them. Then theis linearly isomorphic tolinearly isotropy representation of $K$ is defined by $Ad_{\mathfrak{p}}(k)X=kX^{t}k$. We define an inner product on by
for $k\in SU(3)$ and$Y\rangle=-Tr(XY)$ for $X,$
. Let

Since

$\epsilon u(3)$$\mathfrak{p}$$X\in \mathfrak{p}$$\mathfrak{p}$$Y\in \mathfrak{p}=\epsilon u(3)$$\langle X,$$\alpha=\{\sqrt{-1}((\iota_{1} a_{2} a_{3}) a_{1},(\iota_{2},a_{3}\in \mathbb{R}a_{1}+a_{2}+a_{3}=0, \}$is a maximal abelian subspace of . The group action of $K=$
$SU(3)$ is naturally induced on
in
.

Then

$\mathfrak{a}$$\mathfrak{p}$$Q^{7}$

Theorem 4.4. Letain$cu7^{\cdot}ve\sigma$$\tau$$\mathfrak{p}^{\mathbb{C}}=\mathfrak{s}\mathfrak{l}(3, \mathbb{C})$be a regular curve in the complex planeby$\mathbb{C}$. Define$\mu^{-1}(0)\cap\Phi(\Sigma)$$\sigma(s)=$$\frac{1}{\sqrt{6}}(2cos\tau(s) -cos\tau(s)+\sqrt{3}sin\tau(s) -cos\tau(s)- \sqrt{3}sin\tau(s)) \in \mathfrak{a}^{\mathbb{C}}$

Then the $K$ -orbit $L=K\cdot\sigma$ through a curue is a cohomogeneity
Conone Lagmngian submanifold under the gmup action of $K$ inversely, such a cohomogeneity one Lagmngian submanifold in
isobtained in this way. Moreover, is a special Lagmngian submanifoldof phase if and only if satisfies$\sigma$$Q^{7}$$Q^{7}$$L$$\theta$(7)$\tau$${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(3\cos^{2}\tau(s)-\sin^{2}\tau(s))^{2}(\sin\tau(s))^{2})=0.$

4.3.3. $(G, K)=(SU(6), Sp(3))$ . We consider the case of $(G, K)=$$(SU(6), Sp(3))$ . The canonical decomposition of is given by
where$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p},$$\mathfrak{g}$$\mathfrak{k}=\epsilon u(6)$and$\mathfrak{p}=\{ (\frac{X}{Y} -\overline{X}Y)|X\in su(n), Y\in \mathfrak{o}(n, \mathbb{C})\}.$

Then$\mathfrak{p}^{\mathbb{C}}=\{(\begin{array}{ll}V_{11} V_{12}V_{21} V_{22}\end{array})$$V_{11},$$V_{22}\in\epsilon \mathfrak{l}(n, \mathbb{C}),$$V_{12},$$V_{21}\in o(n, \mathbb{C})\}.$$Y\rangle=-$ Tr $(XY)$ for $X,$
We define an inner product on by
Then the isotropy representation of $K$ is defined by $Ad_{\mathfrak{p}}(k)X=kX^{t}k$$\mathfrak{p}$$\langle X,$$Y\in \mathfrak{p}.$17
for $k\in Sp(3)$ and$X\in \mathfrak{p}$. Let$\mathfrak{a}=\{\sqrt{-1}(\begin{array}{ll}H OO H\end{array})|H=(a_{1} a_{2} a_{3}),$$\}$$a_{1},\alpha_{2},a_{3}\in \mathbb{R}a_{1}+a_{2}+a_{3}=0,$

is a maximal abelian subspace of . The group action of $K=$
$Sp(3)$ is naturally induced on
in

Then

$\mathfrak{a}$$\mathfrak{p}$$Q^{14}$

Theorem 4.5. Leta curve$\sigma$in$\tau$$\mathfrak{p}^{\mathbb{C}}.$be a regular curve in the complex plane$\mu^{-1}(0)\cap\Phi(\Sigma)$by$\sigma(s)=\sqrt{-1}(\begin{array}{ll}H OO H\end{array})\in \mathfrak{a}^{\mathbb{C}}$$\mathbb{C}$. Define, where$H= \frac{1}{\sqrt{6}}(2cos\tau(s) -cos\tau(s)+\sqrt{3}sin\tau(s) -cos\tau(s)- \sqrt{3}sin\tau(s))$

Then the $K$ -orbit $L=K.$ through a cume is a cohomogeneity
one Lagmngian submanifold under the group action of $K$ in
Conversely, such a cohomogeneity one Lagrangian submanifold in
isobtained in this way. Moreover, is a special Lagmngian submanifoldof phase if and only if satisfies$\sigma$$\sigma$$Q^{14}$$Q^{14}$$L$$\theta$$\tau$(8)${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(3\cos^{2}\tau(s)-\sin^{2}\tau(s))^{4}(\sin\tau(s))^{4})=0.$

4.4. Case $g=4.$
4.4.1. $(G, K)=(SO(m+2), SO(2)xSO(m))$ We consider the caseof $(G, K)=(SO(m+2), SO$ (2) $\cross SO(m))$ . We denote by and the
Lie algebras of and $K$ respectively. The canonical decomposition ofis given by, where,$\mathfrak{k}$$\mathfrak{g}$$G$$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$$\mathfrak{g}$$\mathfrak{k}=\{(\begin{array}{ll}A OO B\end{array}) A\in o(2), B\in 0(\prime rr\iota)\}$and$\mathfrak{p}=\{(\begin{array}{ll}O X-tX O\end{array}) X\in M_{2,m}(\mathbb{R})\}.$

Since is linearly isomorphic to, we identify them. We define$(X^{t}Y)$an inner product byTr
for $X,$. Thenthe isotropy representation of $K$ is defined byfor$M_{2,m}(\mathbb{R})$$\mathfrak{p}$$\langle X,$$Y\rangle=$$Y\in M_{2,m}(\mathbb{R})$$Ad_{\mathfrak{p}}(k)X=k_{1}Xk_{2}^{-1}$$k=(\begin{array}{ll}k_{1} OO k_{2}\end{array})\in SO(2)\cross SO(m+1)$

and$X\in \mathfrak{p}$. We take a maximal

18abelian subspace ofas$\mathfrak{p}$$H=(\begin{array}{lllll}a_{1} 0 0 \cdots 00 a_{2} 0 \cdots 0\end{array})\in M_{2,m}(\mathbb{R})\}.$$\mathfrak{a}=\{(\begin{array}{ll}O H-tH O\end{array})$

The group action of $K=SO(2)\cross SO(\gamma\gamma)$ is naturally induced on$Q^{2m-1}$ in.
$\mathfrak{p}^{\mathbb{C}}\cong M_{2,m}(\mathbb{C})$

Theorem 4.6. Leta curve$\sigma$in

be a regular curve in the complex plane$\tau$$\mu^{-1}(0)\cap\Phi(\Sigma)$by$\sigma(s)=(\begin{array}{ll}O H-tH O\end{array})\in \mathfrak{a}^{\mathbb{C}}$$\mathbb{C}$. Define, where$H=(\begin{array}{lllll}cos\tau(s) 0 0 \cdots 00 sin\tau(s) 0 \cdots 0\end{array})$

Then the $K$ -orbit $L=K\cdot\sigma$ thmugh is a cohomogeneity one Lagmngian submanifold under the group action of $K$ in $Q^{2m-1}$ Conversely, such a cohomogeneity one Lagmngian submanifold in $Q^{2m-1}$ isobtained in this way. Moreover, is a special Lagmngian submanifoldof phase if and only if satisfies$\sigma$$L$$\theta$$\tau$(9)${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)\cos 2\tau(s)(\sin 2\tau(s))^{m-2})=0.$

4.4.2. $(G, K)=(SU(m+2), S(U(2)\cross U(m)))$ . We consider the case of$(G, K)=(SU(m+2), S(U(2)\cross U(m)))$ . The canonical decompositionof is given by, where$\mathfrak{g}=P\oplus \mathfrak{p}$$\mathfrak{g}$$\mathfrak{k}=\{(\begin{array}{ll}A OO B\end{array}) B\in u(rr\iota)A\in u(,2), \}$and$\mathfrak{p}=\{(\begin{array}{ll}O X-t\overline{X}O \end{array}) X\in M_{2,m}(\mathbb{C})\}.$

Then$\mathfrak{p}^{\mathbb{C}}=\{(\begin{array}{ll}O VtW O\end{array}) V, W\in M_{\Delta,m}(\mathbb{C})\}.$$Y\rangle=-$ Tr $(XY)$ for $X,$
We define an inner product by
the isotropy representation of $K$ is defined by$\langle X,$$Y\in \mathfrak{p}$. Then$Ad_{\mathfrak{p}}(k)X=k_{1}X^{t}\overline{k_{2}}$$k=(\begin{array}{ll}k_{1} OO k_{2}\end{array})\in S(U(2)\cross U(m))$

abelian subspace of$\mathfrak{p}$and$X\in \mathfrak{p}$. We take a maximalas$\mathfrak{a}=\{(\begin{array}{ll}O H-tH O\end{array})$for$H=(\begin{array}{lllll}a_{l} 0 0 \cdots 00 o_{2} 0 \cdots 0\end{array})\in M_{2,m}(\mathbb{R})\}.$

19
The group action of $K=S(U(2)\cross U(m))$ is naturally induced on$Q^{4m-1}$ in$\mathfrak{p}^{\mathbb{C}}.$

Theorem 4.7. Leta curve$\sigma$in$\tau$

be a regular curve in the complex plane$\mu^{-1}(0)\cap\Phi(\Sigma)$by$\mathbb{C}$. Define, where$\sigma(\mathcal{S})=(\begin{array}{ll}O H-tH O\end{array})\in \mathfrak{a}^{\mathbb{C}}$$H=(\begin{array}{lllll}cos\tau(s) 0 0 \cdots 00 sin\tau(\mathcal{S}) 0 \cdots 0\end{array})$

Then the $K$ -orbit $L=K.$ through a cume is a cohomogeneity
one Lagrangian submanifold under the gmup action of $K$ in $Q^{4m-1}$
Moreover, is a special Lagmngian submanifold of phase if and only$\sigma$$\sigma$$L$$\theta$

if satisfies$\tau$(10)${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(\cos 2\tau(s))^{2}(\sin 2\tau(s))^{2m-3})=0.$

4.4.3. $(G, K)=(Sp(m+2)_{)}Sp(2)\cross Sp(m))$ . We consider the case of$(G, K)=(Sp(m+2), Sp(2)\cross Sp(m))$ . The canonical decompositionof$\mathfrak{g}$is given by$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$, where$\mathfrak{k}=\{(\begin{array}{llll}A_{1l} O B_{l1} OO A_{22} O B_{22}-\overline{B}_{11} O \overline{A}_{11} OO -\overline{B}_{22} O \overline{A}_{22}\end{array})$$B_{22}=M_{m}(\mathbb{C}),tB_{22}=B_{2\Delta}B_{11}\in M_{2}(\mathbb{C}),tB_{11}=B_{11)}A_{11}\in u(2),A_{22}\in u(\prime rnn),\}$

and$\mathfrak{p}=\{$$(-t^{\frac{O}{X}}1’2-t^{\frac{O}{Y}}12$$-12X_{2} \frac{o^{1}}{OY}$$-tX_{12}-tY_{12}OO$$\overline{x^{O}}_{12}Y_{12}O)$$X_{12},$ $Y_{12}\in M_{2,m}(\mathbb{C})\}$

Then$\mathfrak{p}^{\mathbb{C}}=\{(\begin{array}{llll}O V_{12} O V_{14}-tW_{12} O tV_{14} OO -W_{14} O W_{12}-tW_{14} O -tV_{12} O\end{array})$$V_{12},$$V_{14},$$W_{12},$ $W_{14}\in M_{2,m}(\mathbb{C})\}$$Y\rangle=-$ TY $(XY)$ for $X,$
We define an inner product by
the isotropy representation of $K$ is defined by$\langle X,$$Y\in \mathfrak{p}$. Thenfor$Ad_{\mathfrak{p}}(k)X=k_{1}X^{t}\overline{k_{2}}$20$k=(\begin{array}{ll}k_{1} OO k_{\Delta}\end{array})\in Sp(2)\cross Sp(m)$and$\mathfrak{a}=\{(\begin{array}{llll}O H O O-tH O O OO O O HO O -tH O\end{array})|$$X\in \mathfrak{p}$. Then$a_{2}\in \mathbb{R}\}$$H=(\begin{array}{lllll}a_{1} 0 0 \cdots 00 a_{2} 0 \cdots 0\end{array})a_{1},$is a maximal abelian subspace of . The group action of $K=Sp(2)\cross$is naturally induced on $Q^{8m-1}$ in$\mathfrak{p}$$Sp(\prime rn)$$\mathfrak{p}^{\mathbb{C}}.$

Theorem 4.8. Let
a $cu7’ue\sigma$ in$\tau$

be a regular curve in the complex planeby$\mathbb{C}$. Define$\mu^{-1}(0)\cap\Phi(\Sigma)$$\sigma(s)=(\begin{array}{llll}O H O O-tH O O OO O O HO O -tH O\end{array})\in \mathfrak{a}^{\mathbb{C}},$

where$H=(\begin{array}{lllll}cos\tau(s) 0 0 \cdots 00 sin\tau(s) 0 \cdots 0\end{array})$

Then the $K$ -orbit $L=K\cdot\sigma$ thmugh a curve is a cohomogeneity one
Lagmngian submanifold under the gmup action of $K$ in $Q^{8m-1}$ Convers $ely$ , such a cohomogeneity one Lagmngian submanifold in $Q^{8m-1}$ isobtained in this way. Moreover, is a special Lagmngian submanifoldof phase if and only if satisfies$\sigma$$L$$\theta$(11)$\tau$${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(\cos 2\tau(s))^{4}(\sin 2\tau(s))^{4m-5})=0.$

4.4.4. $(G, K)=(SO(5)xSO(5), SO(5))$ . We consider the case of$(G, K)=(SO(5)\cross SO(5), SO(5))$ . The canonical decomposition of, whereis given by$\mathfrak{g}$$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$$\mathfrak{k}=\{(X, X)|X\in o(5)\}\cong \mathfrak{o}(5)$and$\mathfrak{p}=\{(X, -X)|X\in \mathfrak{o}(5)\}.$

Then
for $X,$$Y\in \mathfrak{p}$. We use the inner product by

Tr $(XY)$. Then the isotropy representation of $K$ is defined by$\mathfrak{p}^{\mathbb{C}}=\mathfrak{o}(5, \mathbb{C})$$\langle X,$$Y \rangle=-\frac{1}{2}$21$Ad_{\mathfrak{p}}(k)X=kXk^{-1}$

for $k\in SO(5)$ and$X\in \mathfrak{p}$. Let$\mathfrak{a}=\{(H, -H)|H=(\begin{array}{lllll}0 a_{1} 0 a_{2} -a_{1} 0 -a_{2} 0 0\end{array}),$$a_{1},$$a_{2}\in \mathbb{R}\}$is a maximal abelian subalgebra in . The group action of$K=SO(5)$ is naturally induced on
in

Then

$\mathfrak{a}$$\mathfrak{p}$$Q^{9}$

Theorem 4.9. Letain$cu7^{\cdot}ve\sigma$$\tau$$\mathfrak{p}^{\mathbb{C}}.$be a regular curve in the complex planeby, where$\mu^{-1}(0)\cap\Phi(\Sigma)$$\mathbb{C}$. Define$\sigma(s)=(H, -H)\in \mathfrak{a}^{\mathbb{C}}$$H=(\begin{array}{lllll}0 cos\tau(s) 0 sin\tau(s) -COb^{1}\mathcal{T}(S) 0 -sin\tau(s) 0 0\end{array})$

Then the $K$ -orbit $L=K.$ thmugh a curvea cohomogeneity
one Lagmngian submanifold under the group action of $K$ in
Conversely, such a cohomogeneity one Lagmngian submanifold in
isobtained in the way. Moreover, is a special Lagrangian submanifold
so that satisfies
of phase if and only if there exists a constant$\sigma i\mathcal{S}$$\sigma$$Q^{9}$$Q^{9}$$L$$\theta$(12)$c\in \mathbb{R}$$\tau$${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(\cos 2\tau(s))^{2}(\sin 2\tau(s))^{2})=0.$

4.4.5. $(G, K)=(SO(10), U(5))$ . We consider the case of $(G, K)=$$(SO(10), U(5))$ . The canonical decomposition of is given by
where
$g$

$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p},$$\mathfrak{k}=\{(\begin{array}{ll}A B-B A\end{array}) tA=-A,tB=B\}\cong u(5)$and$\mathfrak{p}=\{(\begin{array}{ll}X YY -X\end{array}) X, Y\in\epsilon \mathfrak{o}(5)\}.$

Then$\mathfrak{p}^{\mathbb{C}}=\{(\begin{array}{ll}V WW -V\end{array})\in \mathfrak{o}(10, \mathbb{C}) V, W\in \mathfrak{o}(5, \mathbb{C})\}.$We define an inner product by
Tr $(XY)$ for $X,$
Then the isotropy representation of $K$ is defined by$\langle X,$$Y \rangle=-\frac{1}{2}$$Y\in \mathfrak{p}.$$Ad_{\mathfrak{p}}(k)X=kX^{t}\overline{k}$22for $k\in U(5)$ and$X\in \mathfrak{p}$. Then$\mathfrak{a}=\{(\begin{array}{ll}H OO -H\end{array})|H=(\begin{array}{lllll}0 a_{1} 0 a_{2} -a_{1} 0 -a_{2} 0 0\end{array}),$ $a_{1},$ $a_{2}\in \mathbb{R}\}$is a maximal abelian subspace of . The group action of $K=U(5)$ isin
naturally induced on$\mathfrak{p}$$Q^{19}$Theorem 4.10. Let

fine a curve$\sigma$in$\tau$$\mathfrak{p}^{\mathbb{C}}.$be a regular curve in the complex planeby$\mu^{-1}(0)\cap\Phi(\Sigma)$$\sigma(s)=(\begin{array}{ll}H OO -H\end{array})\in \mathfrak{a}^{\mathbb{C}}$$\mathbb{C}$. De-, where$H=(\begin{array}{lllll}0 cos\tau(s) 0 sin\tau(s) -cos\tau(s) 0 -sin\tau(s) 0 0\end{array})$

Then the $K$ -orbit $L=K\cdot\sigma$ thmugh a curve is a cohomogeneity
Conone Lagmngian submanifold under the gmup action of $K$ inisversely, such a cohomogeneity one Lagmngian submanifold in
obtained in this way. Moreover, is a special Lagmngian submanifoldof phase if and only if satisfies$\sigma$$Q^{19}$$Q^{19}$$L$$\theta$$\tau$(13)${\rm Im}(e^{\sqrt{-1}\theta}\tau’(s)(\cos 2\tau(s))^{4}(\sin 2\tau(s))^{5})=0.$

In the forthcoming paper we will study the remaining cases when
$G/K$ are of exceptional type.

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