Mathematische Annalen

DOI 10.1007/s00208-012-0819-8

Mathematische Annalen

Moment maps of the spin action

and the Cartan–Münzner polynomials of degree four

Reiko Miyaoka

Received: 30 March 2011 / Revised: 28 March 2012 / Published online: 6 June 2012

© Springer-Verlag 2012

Abstract We express all the known Cartan–Münzner polynomials of degree four in terms of the moment map of certain group actions.

Mathematics Subject Classification 53C40·53D20·15A66 1 Introduction

It is known that a family of isoparametric hypersurfaces in Sⁿ (hypersurfaces with constant principal curvatures) is expressed by level sets of the so-called Cartan–

Münzner polynomial F : Rⁿ⁺¹ → Rrestricted to Sⁿ. Thus isoparametric hyper- surfaces inSⁿare all algebraic, but not necessarily homogeneous. The degree ofF(x) is equal to the numbergof distinct principal curvatures. The caseg =4 is the most interesting, because only in this case do there exist infinitely many non-homogeneous (and homogeneous, as well) examples. Except for two cases, they are constructed by using the representation of Clifford algebras, and in particular, there exists a spin action associated with this algebra. Such hypersurfaces are said to be of OT-FKM type, named after Ozeki and Takeuchi [22] and Ferus et al. [10].

The goal of this paper is to express the degree four Cartan–Münzner polynomials via the square norm of the moment mapµof a certain group action onTRⁿ⁺¹, especially of the spin action in the case of OT-FKM type. In order to do that, we prove the following rather general Theorem:

Partially supported by Grants-in-Aid for Scientific Research, 19204006, The Ministry of Education, Japan.

R. Miyaoka (

B

Mathematical Institute, Graduate School of Sciences, Tohoku University, Aoba-ku, Sendai 980-8578, Japan e-mail: [email protected]

Theorem 1.1 Let H be a compact Lie group acting onR^N as an isometry. Then a natural extension of the action to TR^Nbecomes a Hamiltonian action with respect to the standard symplectic structure of TR^N. The moment mapµ:TR^N →h^∗is given by

µ(x,Y)(ζ )= #ζY,x$ = −#ζx,Y$, ζ ∈h, (1) where#,$is the inner product ofR^N.

Now we state our main Theorems, using the notation given later. In the case of OT-FKM type, we seeN =2lfor somel, and we obtain:

Theorem 1.2 For a given Clifford system P₀, . . . ,Pm ∈ O(2l), we define a vector field(not necessarily continuous)Y :R^2l →TR^2l by

Y_x =





P₀x, if #P₀x,x$ =0

#P₁x,x$P₀x− #P₀x,x$P₁x

$#P₁x,x$²+ #P₀x,x$² , if #P₀x,x$ '=0. (2)

Then the degree four Cartan–Münzner polynomial F(x) on R^2l associated with P₀, . . . ,Pmis given by

F(x)= (x(⁴−2(µ(x,Y_x)(²= (µ₀(x,Y_x)(²−2(µ(x,Y_x)(², (3) whereµis the moment map of the Hamiltonian action of Spin(m+1)on TR^2l, and µ₀is that of U(1).

Although this spin action is small, and not transitive even in the homogeneous case in general, it is sufficient to describe F(x). An advantage of our argument is that it works in both homogeneous and non-homogeneous cases of OT-FKM type in a unified way.

There exist two homogeneous hypersurfaces not of OT-FKM type with g = 4.

These are given by isotropy orbits of rank two symmetric spacesG/K = SO(5)× SO(5)/SO(5)andSO(10)/U(5). In these cases, we extend the isotropy action of K onp∼=T₀(G/K)toTp, and use its moment map to describeF(x). In fact, when a= {ξ₁G₁₂+ξ₂G₃₄}is a maximal abelian subspace ofp, whereG_{i j}, 1≤i< j≤5, is the standard basis ofo(5), we obtain:

Theorem 1.3 When(m₁,m₂)=(2,2),(4,5), which are not of OT-FKM type, putting τ =G₂₅+G₄₅∈k, we define Y_H = [H, τ] ∈pfor H ∈a, and extend it to a vector field Y_xonpby the action of K . Then the Cartan–Münzner polynomial F(x)is given by

F(x)=3(x(⁴−4(µ(x,Yx)(²=3(µ₀(x,Yx)(²−4(µ(x,Yx)(², (4) whereµis the moment map of the Hamiltonian action of K on Tp, andµ₀is that of U(1).

In this way, all the known Cartan–Münzner polynomials of degree four are expressed in terms of the square norm of the moment map. This gives a new viewpoint in connection with the theory of integrable systems [9], in addition to the geometric characterizations obtained by Münzner. Also, fromC^2l ∼=TR^2l, the complexification of all this will expands the argument to the complex category, as is used in [3].

The paper is organized as follows: a brief introduction of isoparametric hypersur- faces and symplectic geometry is given in Sects.2and3. We prove Theorem1.1in Sect.4. Then the Clifford systems and OT-FKM type hypersurfaces are introduced, and the spin action is defined in Sect.5. A proof of Theorem1.2is given in Sect.6.

Next, we treat the homogeneous hypersurfaces in Sect.7, and give a proof of Theorem 1.3in Sect.8. Finally, the relation with the results in [11] is given in an Appendix.

2 Fundamental facts

Let M be a complete connected Riemannian manifold, and denote by∇ and-the Levi–Civita connection and the Laplacian ofM, respectively.

Definition(1) AC^∞function f :M →Ris called an isoparametric function, if it satisfies

(i) (∇f(x)(²=ϕ(f)

(ii)-f =ψ (f), (5)

for someC^∞functionsϕ, ψ : f(M)→R.

(2) A level set of a regular value of f is called an isoparametric hypersurface. A level set of a critical value of f is called a focal submanifold (shown to be a regular submanifold, [25]).

Remark 2.1 From (i), the level sets are parallel to each other. From (ii), each level set has constant mean curvature.

The origin of isoparametric hypersurfaces is in geometric optics, and the level hypersurfaces have physical meanings such as an isothermal levels, wave fronts, etc.

[9,23]. It is remarkable that the partial differential system (5) is solved by geometry.

Fact 1 (Cartan [2]) LetMtbe a family of parallel hypersurfaces inM =Rⁿ,Hⁿ,Sⁿ. Then the following are equivalent:

(a) EachM_t is an isoparametric hypersurface.

(b) EachM_t has constant mean curvature.

(c) A certainMt has constant principal curvatures.

Remark 2.2 (a) is a global condition, and (c) is a local condition. Hence it is non-trivial to deduce (a) from (c) (see Remark2.3).

Recall thatgis the number of distinct principal curvatures.

Fact 2 (Cartan [2])

(1) WhenM =Rⁿ orHⁿ, isoparametric hypersurfaces are either totally geodesic, totally umbilic (g=1), or a tube over a totally geodesic submanifold with constant radius (g=2).

(2) WhenM =Sⁿ, Cartan constructed examples withg=3,4.

Example All the homogeneous hypersurfaces inSⁿare isoparametric, since they have constant principal curvatures. They are given by isotropy orbits of rank two symmetric spaces [13], which are all classified. In particular,g∈ {1,2,3,4,6}[24].

Hereafter we consider the caseM =Sⁿ.

Fact 3 (Münzner [19,20]) Let M be an isoparametric hypersurface in Sⁿ with g principal curvatures. Then we have:

(a) g∈ {1,2,3,4,6}.

(b) For the principal curvatures λ₁ > λ₂ > · · · > λg, the multiplicities m₁,m₂, . . . ,mgsatisfy mi =mi+2.

(c) M is given by a level set of f = F|^Sn : Sⁿ → [−1,1]of a value in(−1,1), where F :Rⁿ⁺¹→Ris a homogeneous polynomial of degree g satisfying

(i) (D F(x)(²=g²(x(^2g−2 (ii)-F(x)=m₂−m₁

2 g²(x(^g−2. (6)

Here D and-are the Euclidean operators.

Remark 2.3 F(x)is called the Cartan–Münzner polynomial. Münzner defined the function f :Sⁿ→Rfirst as follows: LetM be an isoparametric hypersurface inSⁿ. Take the spherical distance functiond(x)=d(x,M)onSⁿ. Then f(x)=cos(gd(x)) is essentially the isoparametric function (Münzner takes a focal submanifold instead of M). In fact, althoughd(x)is not differentiable atx where d(x) =0 in general, cos(gd(x))is analytic, since costis an even power series oft. Moreover, the homoge- neous extensionFof f toRⁿ⁺¹becomes a degreeghomogeneous polynomial with respect to the standard coordinates ofRⁿ⁺¹, and satisfies (6).

Most of the isoparametric hypersurfaces inSⁿare classified:

g 1 2 3 4^∗ 6

M Sⁿ⁻¹ S^k×S^n−k−1 C_F Homogeneous or OT-FKM type N⁶,M¹²(homog.)

Here,C_F is the Cartan hypersurfaces, which are given by tubes over the standard embedding ofFP² (F = R,C,H, or Cayley numbers). N⁶ is a principal isotropy orbit ofG₂/SO(4), [8,17], andM¹²is that ofG₂×G₂/G₂, [16,18]. Wheng '=4, all isoparametric hypersurfaces are homogeneous. Wheng=4, the classification is done by [3,14] (see also [4]) except for(m₁,m₂)=(3,4), (4,5), (7,8), (6,9). Recently, Chi has settled the remaining case other than(m₁,m₂) =(7,8)[5–7]. The known isoparametric hypersurfaces withg =4 are as follows (all the symmetric spaces are compact and of rank two).

Isoparametric hypersurfaces inSⁿwithg=4:

Recently, Fujii [11], and Fujii and Tamaru [12] proved Fact4 below in the case associated with Hermitian symmetric spaces (* in the table). However, in other cases, their argument does not work.

3 Review of symplectic geometry

Definition (Q^2l, ω)is called a symplectic manifold whenQ^2lis aC^∞manifold and ωis a non-degenerate closed 2-form onQ.

For any f ∈C^∞(Q), there exists a vector fieldH_f such thatd f =ω(H_f, ). We callH_f the Hamiltonian vector field of f. Put Ham(Q)= {H_f | f ∈C^∞(Q)}.

When a compact Lie groupK acts onQ, an elementζ of the Lie algebrakof K determines the fundamental vector field onQby

X_ζ = d dt

%%

%_t=0(exptζ )x, x∈ Q.

Definition(1) The action ofKonQis symplectic ifk^∗ω=ωholds for anyk∈K. (2) The action ofK is Hamiltonian ifX_ζ ∈Ham(Q)for anyζ ∈k; i.e., there exists

µ_ζ ∈C^∞(Q)such thatdµ_ζ =ω(X_ζ, ).

(3) A mapµ:Q→k^∗is called the moment map if it satisfies

(i) µisK-equivariant with respect to the coadjoint action ofK onk^∗, (ii) dµ(ζ )=ω(X_ζ, )for eachζ ∈k.

When the moment mapµexists,µ_ζ ∈C^∞(Q)is determined byµ_ζ(x)=µ(x)(ζ ) forζ ∈k, and the action ofK is Hamiltonian. Obviously, the converse holds.

Example(1) Cⁿ is a symplectic manifold with the symplectic form ω(X, ) =

−#J X, $, where J is the standard Kähler structure and #,$ is the Hermitian metric. When there is a Hamiltonian action of a compact Lie groupKonCⁿ,

dµζ(Y)=ω(Xζ,Y)= −#J Xζ,Y$, ζ ∈k holds, anddµζ(Y)= #∇µζ,Y$impliesXζ = J∇µζ.

(2) In the Cartan decompositiong=k⊕pof a Hermitian symmetric spaceG/K,k has centerc, and for a fixed non-zero elementZ ∈c, the Kähler structureJonp is given by

J x =adZ(x)= −adx(Z), x∈p.

With respect to the symplectic structure onpgiven byJand the bi-invariant metric induced onp, the isotropy action ofK onpis Hamiltonian, and its moment map µ^H :p→k^∗is given by [21],

µ^H(x)= 1

2(adx)²Z. (7)

Fact 4 [11,12]When an isoparametric hypersurface with four principal curvatures is given by an isotropy orbit of a Hermitian symmetric space G/K,the Cartan–Münzner polynomial is expressed by using the moment mapµ^H =µ₀+µ₁decomposed into each component of k^∗=c^∗⊕k^∗₁,

F(x)=a(x(⁴−b(µ₁(x)(²=a(µ₀(x)(²−b(µ₁(x)(², where a,b are constants depending on m₁,m₂,

The idea of the proof is to find polynomials of degree four invariant under the isotropy action. However, in other cases in the table of Sect.2, that argument does not work, because of a lack of a suitable symplectic structure onpor onR^2l.

4 Proof of Theorem1.1

Before the proof of Theorem1.1, recall the symplectic structure onTR^N. First, the complex structureJ˜onTR^Nis given by

J˜(U,V)=(−V,U), (U,V)∈T(x,Y)(TR^N)∼=R^N⊕R^N. and thenTR^N has a symplectic form given by

ω(Z˜,W˜)= −# ˜JZ˜,W˜$, Z˜,W˜ ∈T_(x,Y)(TR^N),

where#,$is the inner product onR^N⊕R^N. In fact, sinceJ˜is parallel and#,$is non- degenerate,ωis a non-degenerate closed 2-form, and hence(TR^N, ω)is a symplectic manifold.

Expressing an isometricH-action onR^Nbyhxforh∈ Handx∈R^N, we extend it to an action onTR^Nby

h·(x,Y)=(hx,hY), (x,Y)∈TxR^N, (8) where we regardY as a vector inR^N. The fundamental vector field X˜_ζ onTR^N is then given by

˜

X_ζ =(x,Y;ζx, ζY)∈T_(x,Y)TR^N, ζ ∈h

which we abbreviate toX˜_ζ =(ζx, ζY). Then Theorem1.1follows from the following two lemmas.

Lemma 4.1 The H-action(8)on TR^Nis symplectic.

Proof Since

J˜(U,V)=(−V,U), (U,V)∈T(x,Y)TR^N∼=R^N⊕R^N, J˜is the right action ofη=

&

0−1 1 0

'

∈u(1)∼=o(2), i.e., J(U,˜ V)=(U,V)η, and

commutes with theHaction. 01

Lemma 4.2 The H-action(8)on TR^Nis Hamiltonian.

Proof We show that the moment mapµ:TR^N→h^∗is given by µ((x,Y))(ζ )= 1

2# ˜Xζ,J(x,˜ Y)$

= 1

2#(ζx, ζY), (−Y,x)$

= #ζY,x$ = −#ζx,Y$, ζ ∈h, (9) where we useh⊂o(N). In fact, this is equivariant with respect to the coadjoint action ofH onh^∗, because

µ(h·(x,Y))(ζ )= #ζ (hY),hx$ = #h(h⁻¹ζh)Y,hx$

= #(h⁻¹ζh)Y,x$ = #((Adh)⁻¹ζ )Y,x$

=µ(x,y)◦(Ad_h)⁻¹ζ

=(Ad^∗)_h(µ(x,Y))(ζ ).

Next, forZ˜ ∈T_(x,Y)TR^N, usingJζ˜ =ζJ˜, we obtain

dµ(ζ )(Z)˜ = ¹₂(#ζZ˜,J(x,˜ Y)$ + # ˜Xζ,J˜Z˜$)

= # ˜X_ζ,J˜Z˜$ = −# ˜JX˜_ζ,Z˜$

=ω(X˜_ζ,Z˜).

Thusµis the moment map. 01

Theorem 4.3 The U(1)action on TR^Ninduced byη∈u(1)is Hamiltonian, and the moment mapµ₀:TR^N →u(1)^∗is given by

µ₀(x,Y)=1

2# ˜X_η,J˜(x,Y)$η=1

2((x(²+ (Y(²)η∈u(1)∼=u(1)^∗. (10) Proof SinceU(1)is commutative,µ₀is equivariant. The fundamental vector field of ηis given by

˜

X_η= ˜J(x,Y)=(−Y,x),

and forZ˜ ∈T_(x,Y)TR^N,

2dµ₀(η)(Z)˜ = # ˜JZ˜,J˜(x,Y)$ + # ˜J(x,Y),J˜Z˜$

=2# ˜Xη,J˜Z˜$ =2ω(X˜η,Z˜$

holds. Thusµ₀is the moment map, and (10) follows. 01 5 Clifford systems onR^2l

In order to define isoparametric hypersurfaces of OT-FKM type, we introduce the Clifford system. In the following, we denote byO(n)the orthogonal group, and by o(n)its Lie algebra.

Definition(1) C^m−1 = {E₁, . . . ,Em−1},Ej ∈ o(l)is called a representation of a Clifford algebra when it satisfies

EiEj+EjEi = −2δi jid, 1≤i,j ≤m−1. (11) (2) P₀, . . .Pm∈ O(2l)is called a Clifford system when it satisfies

P_iP_j+P_jP_i =2δi jid, 0≤i,j ≤m. (12) This impliesP_i is symmetric.

Under a suitable identification of two Clifford representations or two Clifford sys- tems, we see [10, 3.3, pp. 482–483]:

Lemma 5.1 There exists a one-to-one correspondence betweenC^m−1and the Clifford system.

Proof FromC^m−1, putting(u, v)∈R^l⊕R^l, we obtain

P₀(u, v)=(u,−v), P₁(u, v)=(v,u), P_1+i(u, v)=(E_iv,−E_iu),

which satisfy (12). On the other hand, from a Clifford systemP₀, . . .P_m, we decom- poseR^2l into E₊(P₀)⊕E₋(P₀)where E_±(P₀)is the±1 eigenspace of P₀. From P₀Pj+PjP₀=0,Pj(j≥1) interchangesE_±(P₀). Now, it is easy to see thatP₁P₁₊i

(i ≥1) is skew and preserves eachE_±(P₀)∼=R^l. Thus we may put E_i =P₁P_1+i ∈o(l), i =1, . . .m−1,

which satisfy (11). 01

Remark 5.2(1) The possible pairs(m,l)are given in [1], [10, p. 483].

m 1 2 3 4 5 6 7 8 · · · m+8 · · ·

l=δ(m) 1 2 4 4 8 8 8 8 · · · 16δ(m) · · ·

(2) In the following, we use the inner product of linear operators onR^2l (symmetric or skew-symmetric) given by

#P,Q$ = 1

2lTr(P^tQ). (13)

With respect to this inner product, P₀, . . . ,P_m gives an orthonormal basis of the linear spaceV of symmetric orthogonal operators spanned by themselves. The unit hypersphere * of V is called the Clifford sphere. The orthogonal transformation O(m+1)ofV preserves*, and any orthonormal basis ofV satisfies the relation (12) because of^tPi =Pi.

Fact 5 [10]When a Clifford system P₀, . . . ,P_m is given, F(x)= #x,x$²−2

(m i=0

#P_ix,x$² (14)

is a Cartan–Münzner polynomial of degree four. When l−m−1 >0,it defines a family of isoparametric hypersurfaces in S^2l−1 with four principal curvatures with multiplicities m₁=m,m₂=l−m−1.

It is not difficult to see thatF(x)satisfies (6). Moreover,F(x)does not depend on the choice of P₀, . . . ,P_m. In fact, the second term of the right hand side of (14) is (x(²times the square norm of the component ofx∈R^2lprojected to the subspace of R^2l spanned byP₀x, . . . ,Pmx.

We call the hypersurfaces given in this way of OT-FKM type, because Ozeki–

Takeuchi first uses the representation of Clifford algebras to obtain new Cartan–

Münzner polynomials, and show that there exist infinitely many non-homogeneous examples among them [22]. Then Ferus et al. [10] extend the argument to be applica- ble to all the representations of the Clifford algebra. In some sense, isoparametric hypersurfaces of OT-FKM type give a geometric realization of Clifford algebras.

When a Clifford systemP₀, . . . ,Pm is given,PiPj, 0≤i < j ≤m, are skew by (12), and generate a Lie subalgebra ofo(2l)isomorphic too(m+1). Putζi j =PiPj ∈ o(2l), 0≤i < j ≤m.

Fact 6 [10, p. 496] Spin(m+1)acts onR^2l,and preserves the Cartan–Münzner polynomial.

Although this group is small in general, we show this is sufficient to describe the Cartan–Münzner polynomial. Since the metric ono(m+1)is given by (13), we have (A(²= 1

2lT r(−A²)for A∈o(m+1), and

(ζ_0i(²= (ζ_1i(²= (ζi j(²=1

follows. Sinceζi jare orthogonal to each other if the indices are different,ζi j(0≤i<

j≤m) is an orthonormal frame ofo(m+1). Now, we apply Theorem1.1to this spin action, and obtain:

Theorem 5.3 The Spin(m+1)action on TR^2lis Hamiltonian with the moment map µ:TR^2l →o^∗(m+1)given by

µ(x,Y)= − (

0≤i<j≤m

#ζ_{i j}x,Y$ζ_{i j} ∈o(m+1)∼=o^∗(m+1). (15)

It follows that

(µ(x,Y)(²= (

0≤i<j≤m

#ζi jx,Y$². (16)

6 Proof of Theorem1.2

Lemma 6.1 At each x∈R^2l, define

Xx = #P₁x,x$P₀x− #P₀x,x$P₁x∈TxR^2l. Then we have

(µ(x,Xx)(²=(#P₀x,x$²+ #P₁x,x$²) (m i=0

#Pix,x$². (17)

Proof From (15), we have

µ(x,P₀x)(ζ_{i j})= −#P_iP_jx,P₀x$. On the other hand, for 1≤i < j ≤m−1,

#P₀P₁x,P₀x$ = #P₁x,x$

#P₀P₁₊ix,P₀x$ = #P₁₊ix,x$

#P₁P_1+ix,P₀x$ = #(−E_iu,E_iv), (u,−v)$ =0

#P₁₊iP₁₊jx,P₀x$ = −#EiEju,u$ + #EiEjv, v$ =0 hold. Hence we obtain

µ(x,P₀x)= − (m

j=1

#Pjx,x$ζ_0j. (18)

Similarly, we have

µ(x,P₁x)= − (m j=0,j'=1

#P₀x,x$ζ_1j, ζ₁₀= −ζ₀₁. (19)

Thus we compute

µ(x,Xx)= #P₁x,x$µ(x,P₀x)− #P₀x,x$µ(x,P₁x)

= −#P₁x,x$ (m

j=1

#Pjx,x$ζ_0j + #P₀x,x$



 (m j=0,j'=1

#Pjx,x$ζ_1j





= −(#P₁x,x$²+ #P₀x,x$²)ζ₀₁ +

(m j=2

(−#P₁x,x$ζ_0j + #P₀x,x$ζ_1j)#P_jx,x$.

Therefore, it follows that

(µ(x,Xx)(²=(#P₁x,x$²+ #P₀x,x$²)² +

(m j=2

(#P₁x,x$²+ #P₀x,x$²)#P_jx,x$²

=(#P₁x,x$²+ #P₀x,x$²) (m i=0

#P_ix,x$². (20) 01 Proof of Theorem1.2 We obtain

(µ(x,Y_x)(²= (m

j=0

#P_ix,x$², (21)

from (18) if#P₀x,x$ =0, and from (20) if#P₀x,x$ '=0. Now, Theorem4.3implies (µ₀(x,Yx)(²= (x(⁴,

since(Yx(²= (x(², and we obtain our Theorem. 01 Remark 6.2 As a result, P₀,P₁can be replaced by any two orthogonal elements of the Clifford sphere*. Namely, whenm>1, there is no standard choice ofYx. For a background reason, there is no standard choice of a principal vector forλ₁ifm₁>1 (see [10] for details).

Corollary 6.3 (µ(x,Px)(² = -m

j=0#P_ix,x$² holds for any P ∈ * such that

#Px,x$ =0.

This follows immediately from (18) if we put P = P₀, where P depends onx.

Thus(µ(x,Px)(²takes the same value on theS^m−1bundle with fiber atxconsisting of Px such that#Px,x$ =0, along an isoparametric hypersurface. Whenxtends to a point of the focal submanifoldsM₊, this bundle tends to the normal sphere bundle of M₊, which is anS^m bundle overM₊. In fact, atx ∈ M₊,#Px,x$ =0 holds for allP ∈*∼=S^m, and henceF(x)=1 (see [10, p. 485, 4.2] for details).

7 Homogeneous case

There are two homogeneous hypersurfaces with g = 4 not of OT-FKM type. We review the general homogeneous case (not necessarilyg =4) here, and treat the case g =4 in the next section.

A homogeneous hypersurfaces in the sphere is given by an isotropy orbit of a rank two symmetric spaceG/K [13]. We denote the Cartan decomposition byg=k⊕p, and a maximal abelian subspace ofpbya. The isotropy action ofK onpis given by the adjoint action, which is an isometry with respect to the bi-invariant metric onp.

The fundamental vector field ofζ ∈kis given by X_ζ =adζ (x), x∈p.

We extend this action toTpas before, namely,

k·(x,Y)=(Adk(x),Adk(Y)), (x,Y)∈Tp,k∈K. Then the fundamental vector field onTpis given by

˜

X_ζ =(adζ (x),adζ (Y)), ζ ∈k. (22)

Now, applying Theorem1.1, we obtain

Theorem 7.1 Let G/K be a compact rank two symmetric space, and letg=k+p be the Cartan decomposition of the corresponding Lie algebra. Then the action of K on Tpis Hamiltonian, and the moment mapµ:Tp→k^∗is given by

µ(x,Y)= −adx(Y)∈k∼=k^∗, (x,Y)∈Tp. (23) Proof Theorem1.1implies

µ(x,Y)(ζ )= −#adζ (x),Y$ = #adx(ζ ),Y$ = −#adx(Y), ζ$.

01 Remark 7.2 This proposition holds for anyg∈ {1,2,3,4,6}.

Corollary 7.3 When G/K is a Hermitian symmetric space and J = adZ|p is the Kähler structure where Z∈c, we obtain

µ(x,J x)=2µ^H(x), (24)

whereµ^H is given in(7).

Proof With respect to the symplectic structure onpassociated withJ =adZ|p, the moment map of theK action is given by (7). If we putY = J x = −adx(Z)in (23), the right hand side becomes(adx)²(Z), and we obtain (24). 01

8 Case of non-OT-FKM type

Isoparametric hypersurfaces withg=4 which are not of OT-FKM type are given by the isotropy orbits of the symmetric spaceSO(5)×SO(5)/SO(5),(m₁,m₂)=(2,2), and of the Hermitian symmetric space SO(10)/U(5),(m₁,m₂) = (4,5). In these cases, we express F(x)in terms of the moment map of the isotropy action extended toTp.

PutGi j = Ei j −Eji ∈ o(5)⊂ u(5), 1 ≤ i < j ≤ 5, where, Ei j is the matrix with(i,j)component equal to one and all other components equal to 0. Recall that an isotropy orbit ofKinpalways passes througha.

For the proof of Theorem1.3, we use the notation in [22] and [15]. Let 1,i,j,kbe the standard units of the quaternion number fieldH. We identifyCwith a subalgebra of H by the natural map x +√

−1y 6→ x1+ yi. This identification induces an identificationgl(n,C)⊂gl(n,H). We consider a non-compact simple Lie algebra

g= {A∈gl(5,H)|^tAΨ¯ +ΨA=0, Ψ =√

−11₅} =k+p, where

.k =o(5) p= {√

−1Z | Z ∈M₅(R),^tZ = −Z}, .k =u(5)

p= {j Z| Z ∈M₅(C),^tZ = −Z},

which correspond to(m₁,m₂)=(2,2)and(4,5), respectively. Under the identifica- tion√

−1Z 6→Z, and j Z 6→Z, we use the inner product

#Z,W$ = −1

27Tr(Z W). (25)

As we use a classical Lie group here, we denote adζ (·)= [ζ,· ]. A maximal abelian subspace ofpis given by (under above identification)

a= {H(ξ₁, ξ₂)=ξ₁G₁₂+ξ₂G₃₄|ξ₁, ξ₂∈R}. Then a positive root space*_∗⁺is given by

*_∗⁺= {α₁= −ξ₁+ξ₂, α₂=ξ₁, α₃=ξ₁+ξ₂, α₄=ξ₂}. Letαbe a linear form onaand put

k_α = {X ∈k|(adH)²(X)= −α(H)²X,for allH∈a}, pα = {X ∈p|(adH)²(X)= −α(H)²X,for allH∈a}.

The corresponding root vectors are given in §11 of [15], X₁¹=z₁₃(E₁₃+E₂₄)− ¯z₁₃(E₃₁+E₄₂)∈k_α1, X₁²=z₁₄(E₁₄−E₂₃)− ¯z₁₄(E₄₁−E₃₂)∈kα₁, X₂¹=z₁₅E₁₅− ¯z₁₅E₅₁, X₂²=z₂₅E₂₅− ¯z₂₅E₅₂∈k_α2,

(X₂³=√

−1s(−E₁₁+E₂₂)∈k_α2), X₃¹=w₁₃(E₁₃−E₂₄)− ¯w₁₃(E₃₁−E₄₂)∈kα₃, X₃²=w₁₄(E₁₄+E₂₃)− ¯w₁₄(E₄₁+E₃₂)∈k_α3, X₄¹=z₃₅E₃₅− ¯z₃₅E₅₃, X₄²=z₄₅E₄₅− ¯z₄₅E₅₄∈k_α4,

(X₄³=√

−1t(E₃₃−E₄₄)∈kα₄),

wherez_{i j}, w_{i j} ∈RorCands,t ∈R, and in the real case,X³₂=X₄³=0. Moreover, we have

T₁¹=z₁₃(−E₁₄+E₂₃)− ¯z₁₃(−E₄₁+E₃₂)∈pα₁, T₁²=z₁₄(E₁₃+E₂₄)− ¯z₁₄(E₃₁+E₄₂)∈p_α1,

T₂¹=z₁₅E₂₅− ¯z₁₅E₅₂, T₂²= −z₂₅E₁₅+ ¯z₂₅E₅₁∈pα₂, (T₂³=√

−1s(E12+E₂₁)∈pα₂), T₃¹=w₁₃(E₁₄+E₂₃)− ¯w₁₃(E₄₁+E₃₂)∈p_α3, T₃²=w₁₄(−E₁₃+E₂₄)− ¯w₁₄(−E₃₁+E₄₂)∈pα₃, T₄¹=z₃₅E₄₅− ¯z₃₅E₅₄, T₄²=z₄₅E₃₅− ¯z₄₅E₅₃ ∈p_α4,

(T₄³=√

−1t(E₃₄+E₄₃)∈p_α4. These satisfy

adH(Xⁱ_j)=αj(H)(T_jⁱ), adH(Tⁱ_j)= −αj(H)(T_jⁱ).

Any AdKorbit in the unit sphereS^dinp(d =9 or 19) is obtained as an orbit through someH ∈S¹=S^d∩a. Thus if we considerH∈acorresponding to

H(ξ₁, ξ₂)=ξ₁G₁₂+ξ₂G₃₄, ξ₁, ξ₂∈R,

the value of the Cartan–Münzner polynomial is determined by (ξ₁, ξ₂). Here, the Cartan–Münzner polynomial is given by [22], p. 26–27,

F(z)= 3

4(Tr(zz))²−2Tr(zz)², z∈p, z∈M₅(R)orM₅(C),

and atz= H ∈ a, putting idi j = E_ii +E_{j j}, we haveHH¯ = −(ξ₁²id₁₂+ξ₂²id₃₄), and(HH)¯ ²=ξ₁⁴id₁₂+ξ₂⁴id₃₄. Thus we obtain

F(H)=3(ξ₁²+ξ₂²)²−4(ξ₁⁴+ξ₂⁴), (26) which is constant on the orbit throughH.

When(m₁,m₂)=(2,2),o(5)is generated byGi j, 1≤i < j ≤5. Since all the coefficients of the root vectors Xj = X¹_j,X_¯j = X²_j (1 ≤ j ≤ 4), are real, they are given by combinations ofG₁₂,G₃₄and

X₁=G₁₃+G₂₄, X_1¯ =G₁₄−G₂₃ ∈k_α1 X₂=G₂₅, X_2¯ =G₁₅∈k_α2

X₃=G₁₄+G₂₃, X_3¯ =G₁₃−G₂₄ ∈k_α3 X₄=G₄₅, X_4¯ =G₃₅ ∈k_α4.

Moreover,pis generated byaand Tj =√

−1X_¯j, T_¯j =√

−1Xj ∈pα_j, 1≤ j ≤4, and span{T_j,T_¯j}is the curvature distribution for each j ∈ {1,2,3,4}[24].

When(m₁,m₂)=(4,5),u(5)is generated byGi j, 1≤i < j ≤5, and symmetric matricesSkl = Ekl+Elk, 1 ≤ k ≤ l ≤ 5. The identification of these (as well as elements in the subspacep) with matrices inso(10)is given in [11], p. 207.

Proof of Theorem1.3: We extend the isotropy action ofK toTp. In the following, we use

X₂=G₂₅, X₄=G₄₅∈k, ([X₂,X₄] = −G₂₄).

Since the relation between roots and root vectors is partially common in the cases (m₁,m₂)=(2,2)and(4,5), and since the value ofF(z)is determined atH ∈a, the following proof works for both cases. Also, note that we use the identificaiton ofZ with√

−1Z or j Z given in the beginning of this section, which does not affect the final conclusions.

By Theorem7.1, the moment mapµ:Tp→k^∗is given by µ(H,Y)= −(adH)Y ∈k∼=k^∗. Now, fixing

τ =X₂+X₄=G₂₅+G₄₅∈k, (27)

putY_H =adH(τ )∈T_Hp. Then from(adH)²X₂= −ξ₁²X₂and(adH)²X₄= −ξ₂²X₄, we obtain

µ(H,YH)= −(adH)²τ =ξ₁²X₂+ξ₂²X₄. From (25),|X₂|²= |X₄|²=1 follows, and hence we obtain

(µ(H,Y_H)(²=ξ₁⁴+ξ₂⁴.

Restricting it to the adjoint orbits{(x,Y_x) =(Ads(H),Ads(YH)),s ∈ K}through (H,Y_H)inTp, we obtain, using the equivariancy ofµ,

(µ(x,Y_x)(²=ξ₁⁴+ξ₂⁴. (28) On the other hand, from(H(²=ξ₁²+ξ₂², and fromY_H =ξ₁G₁₅+ξ₂G₃₅,(Y_H(²= ξ₁²+ξ₂²follows, and by (10),

(µ₀(x,Y_x)(²= (µ₀(H,Y_H)(²=(ξ₁²+ξ₂²)² holds at(x,Y_x). Thus from (28), we obtain

F(x)=3(µ₀(x,Y_x)(²−4(µ(x,Y_x)(².

01 Therefore, the results (5.8) in [11] for(m₁,m₂)=(4,5)and here coincide up to constant multiple, through the formula next to (4.4) in p. 26 of [22]. Note that we give (4) by a unified argument for both(2,2)and(4,5).

9 Appendix

For hypersurfaces of OT-FKM type which are at the same time isotropy orbits of Hermitian symmetric spaces, we give a relation between the matrix expression in [11]

and the argument in Sect.6.

In the case of Hermitian symmetric spaceSO(2+n)/SO(2)×SO(n), we have k=

/&

A 0 0 A⁹

'

, A∈o(2), A⁹∈o(n) 0

,

p= /&

0 V

−^tV 0 '

,V ∈M_2,n(R) 0

,

and from dimp=2n,n=lholds in Sect.5. We may consider an elementH(ξ₁, ξ₂)∈ awhich corresponds to

V =

&

ξ₁ 0 . . .0 0 ξ₂. . .0 '

, ξ₁, ξ₂∈R.

Namely, with respect to an orthonormal basise₁, . . . ,el of R^l, take a double copy R^l⊕R^l consisting of±1 eigenspaces of P₀. We may considerH =(ξ₁e₁, ξ₂e₂)∈ a ⊂R^l⊕R^l. Since we can use P₀,P₁with respect to this decomposition (ark6.2), P₀H =(ξ₁e₁,−ξ₂e₂)andP₁H=(ξ₂e₂, ξ₁e₁)follow, and hence#P₀H,H$ =ξ₁²−ξ₂² and#P₁H,H$ =0 hold, which impliesY_H =P₁H. Becausem=1, it follows that

(H(²= (P₁H(²=ξ₁²+ξ₂², (µ(H,Y_H)(²= #P₀H,H$²=(ξ₁²−ξ₂²)²,

and from our result (3), we obtain

F(H)= (H(⁴−2(µ(H,YH)(²

=(ξ₁²+ξ₂²)²−2(ξ₁²−ξ₂²)²

=3(ξ₁²+ξ₂²)²−4(ξ₁⁴+ξ₂⁴).

This is nothing but (4.4) in [22], and (4.9) in [11] multiplied by_n⁸.

The case ofSU(2+n)/S(U(2)×U(n))is similarly discussed, where fromm=2, P₀P₁,P₁P₂,P₂P₀generatesu(2), and we may put P₂(u, v) = (√

−1v,−√

−1u), and#P₂(u, v), (u, v)$ = 0 follows. Thus we obtain the same F(x)as above, which coincides with (4.4) in [22].

In the case of E₆/U(1)×Spin(10), we know(m₁,m₂) = (9,6), and the spin action on p ∼= R³² in Sect. 6 coincides with the isotropy action of Spin(10). In fact, by the argument in 6.3 of [10], P_iP_jx,(i,j)= (0,1), (2,3), (1,2i), (1,2i+ 1), (0,2i), (0,2i+1)for 1≤i ≤4, and(2i,2j) (2i,2j+1)for 1≤i < j ≤4 span the 30 linearly independent Killing fields on the hypersurface, and hence the action is transitive. Up to now, [12] is not yet at hand, and so Theorem1.2gives the unique expression of this case in terms of the moment mapµ.

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