Articulo 9.dvi CUBO A Mathematical Journal Vol.12, No¯ 02, (127–143). June 2010 Examples of a complex hyperpolar action without singular orbit Naoyuki Koike Department of Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka Shinjuku-ku, Tokyo 162-8601, Japan email: koike@ma.kagu.tus.ac.jp ABSTRACT The notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as a counterpart to the hyperpolar action on a symmetric space of compact type. As examples of a complex hyperpolar action, we have Hermann type actions, which admit a totally geodesic singular orbit (or a fixed point) except for one example. All principal orbits of Hermann type actions are curvature-adapted and proper complex equifocal. In this paper, we give some examples of a complex hyperpolar action without singular orbit as solvable group free actions and find complex hyperpolar actions all of whose orbits are non-curvature-adapted or non-proper complex equifocal among the examples. Also, we show that some of the examples possess the only minimal orbit. RESUMEN La noción de una acción hiperpolar compleja sobre un espacio simétrico de tipo no com- pacto fue recientemente introducida como el análogo de la acción hiperpolar sobre un espacio simétrico de tipo compacto. Como ejemplos de una acción hiperpolar complejas, nosotros tenemos acciones de tipo Hermann, las cuales admiten una orbita (o un punto fijo) singular totalmente geodesica excepto para un ejemplo. Todas las orbitas principales de acciones de tipo Hermann son curvatura-adaptadas y unifocales complejas propias. En 128 Naoyuki Koike CUBO 12, 2 (2010) este art́ıculo, nosotros damos algunos ejemplos de una acción hiperpolar compleja sin or- bitas singulares como grupo soluble de acciones libres y encontramos acciones complejas hiperpolares cuyas orbitas son no curvatura-adaptadas o no propias unifocales complejas. También, mostramos que algunos de los ejemplos poseen solamente orbitas minimales. Key words and phrases: symmetric space, complex hyperpolar action, complex equifocal submani- fold. AMS (MOS) Subj. Class.: 53C35; 53C40 1 Introduction In symmetric spaces, the notion of an equifocal submanifold was introduced in [30]. This notion is defined as a compact submanifold with globally flat and abelian normal bundle such that the focal radius functions for each parallel normal vector field are constant. However, this conditions of the equifocality is rather weak in the case where the symmetric spaces are of non-compact type and the submanifold is non-compact. So we [13, 14] have recently introduced the notion of a complex equifocal submanifold in a symmetric space G/K of non-compact type. This notion is defined by imposing the constancy of the complex focal radius functions instead of focal radius functions. Here we note that the complex focal radii are the quantities indicating the positions of the focal points of the extrinsic complexification of the submanifold, where the submanifold needs to be assumed to be complete and of class Cω (i.e., real analytic). On the other hand, Heintze-Liu-Olmos [10] has recently defined the notion of an isoparametric submanifold with flat section in a general Riemannian manifold as a submanifold such that the normal holonomy group is trivial, its sufficiently close parallel submanifolds are of constant mean curvature with respect to the radial direction and that the image of the normal space at each point by the normal exponential map is flat and totally geodesic. We [14] showed the following fact: All isoparametric submanifolds with flat section in a symmetric space G/K of non-compact type are complex equifocal and that conversely, all curvature-adapted and complex equifocal submanifolds are isoparametric ones with flat section. Here the curvature-adaptedness means that, for each normal vector v of the submanifold, the Jacobi operator R(·, v)v preserves the tangent space of the submanifold invariantly and the restriction of R(·, v)v to the tangent space commutes with the shape operator Av, where R is the curvature tensor of G/K. Furthermore, as a subclass of the class of complex equifocal submanifolds, we [15] defined that of the proper complex equifocal submanifolds in G/K as a complex equifocal submanifold whose lifted submanifold to H0([0, 1], g) (g := Lie G) through some pseudo-Riemannian submersion of H0([0, 1], g) onto G/K is proper complex isoparametric in the sense of [13], where we note that H0([0, 1], g) is a pseudo-Hilbert space consisting of certain kind of paths in the Lie algebra g of G. Let G/K be a symmetric space of non-compact type and H be a closed subgroup of G which admits an embedded complete flat submanifold meeting all H-orbits orthogonally. Then the H-action on G/K is called a complex hyperpolar action. This action was named thus because this action has not necessarily a singular orbit (which should be called a pole of this action) but the complexified action CUBO 12, 2 (2010) Examples of a complex hyperpolar action without singular orbit 129 has a singular orbit. Note that all cohomogeneity one actions are complex hyperpolar. We [14] showed that principal orbits of a complex hyperpolar actions are isoparametric submanifolds with flat section and hence they are complex equifocal. Conversely we [17] have recently showed that all homogeneous complex equifocal submanifolds occurs as principal orbits of complex hyperpolar actions. Let H′ be a symmetric subgroup of G (i.e., there exists an involution σ of G with (Fix σ)0 ⊂ H′ ⊂ Fix σ), where Fix σ is the fixed point group of σ and (Fix σ)0 is the identity component of Fix σ. Then the H′-action on G/K is called a Hermann type action. A Hermann type action admits a totally geodesic orbit or a fixed point. Except for one example, the totally geodsic orbit is singular (see Theorem E of [17]). We [15] showed that principal orbits of a Hermann type action are proper complex equifocal and curvature-adapted. We [17] have recently showed that all complex hyperpolar actions of cohomogeneity greater than one on G/K admitting a totally geodesic orbit and all complex hyperpolar actions of cohomogeneity one on G/K admitting reflective orbit are orbit equivalent to Hermann type actions (see Theorems B, C and Remark 1.1 in [17]). Let G/K be a symmetric space of non-compact type, g = f + p (f := Lie K) be the Cartan decomposition associated with (G, K), a be the maximal abelian subspace of p, ã be the Cartan subalgebra of g containing a and g = f + a + n be the Iwasawa’s decomposition. Let A, à and N be the connected Lie subgroups of G having a, ã and n as their Lie algebras, respectively. Let π : G → G/K be the natural projection. The symmetric space G/K is identified with the solvable group AN with a left-invariant metric through π|AN . In this paper, we first prove the following fact for a complex hyperpolar action without singular orbit. Theorem A. Any complex hyperpolar action on G/K(= AN ) without singular orbit is orbit equiv- alent to the free action of some solvable group contained in ÃN . Next we give some examples of a complex hyperpolar action without singular orbit as the free actions of solvable groups contained in AN (see Examples 1 and 2 of Section 3), which contain examples of cohomogeneity one actions without singular orbit constructed by J. Berndt and H. Tamaru [3] as special cases (see also [1]). Among these examples, we find complex hyperpolar actions all of whose orbits are non-proper complex equifocal or non-curvature-adapted. As its result, we have the following facts. Theorem B. (i) For any symmetric space G/K of non-compact type and any positive integer r with r ≤ rank(G/K), there exists a complex hyperpolar action without singular orbit such that the cohomogeneity is equal to r and that any of the orbits is not proper complex equifocal. (ii) Let G/K be one of SU (p, q)/S(U (p) × U (q)) (p < q), Sp(p, q)/Sp(p) × Sp(q) (p < q), SO∗(2n)/U (n) (n : odd), E−146 /Spin(10) · U (1) or F −20 4 /Spin(9). Then, for any positive integer r with r ≤ rank(G/K), there exists a complex hyperpolar action without singular orbit such that the cohomogeneity is equal to r and that any of the orbits is not curvature-adapted. Also, among those examples, we find complex hyperpolar actions possessing the only minimal orbit. As its result, we have the following fact. 130 Naoyuki Koike CUBO 12, 2 (2010) Theorem C. For any irreducible symmetric space G/K of non-compact type and any positive integer r ≤ [ 1 2 (rank(G/K) + 1)], there exists a complex hyperpolar action without singular orbit such that the cohomogeneity is equal to r and that the only orbit is minimal. 2 Complex equifocal submanifolds In this section, we recall the notions of a complex equifocal submanifold and a proper complex equifocal submanifold. We first recall the notion of a complex equifocal submanifold. Let M be an immersed submanifold with abelian normal bundle in a symmetric space N = G/K of non-compact type. Denote by A the shape tensor of M . Let v ∈ T ⊥x M and X ∈ TxM (x = gK). Denote by γv the geodesic in N with γ̇v(0) = v. The strongly M -Jacobi field Y along γv with Y (0) = X (hence Y ′(0) = −AvX) is given by Y (s) = (Pγv|[0,s] ◦ (D co sv − sDsisv ◦ Av))(X), where Y ′(0) = ∇̃vY, Pγv|[0,s] is the parallel translation along γv|[0,s] and Dcosv (resp. Dsisv) is given by Dcosv = g∗ ◦ cos( √ −1ad(sg−1∗ v)) ◦ g−1∗( resp. Dsisv = g∗ ◦ sin( √ −1ad(sg−1∗ v))√ −1ad(sg−1∗ v) ◦ g−1∗ ) . Here ad is the adjoint representation of the Lie algebra g of G. All focal radii of M along γv are obtained as real numbers s0 with Ker(D co s0v −s0Dsis0v ◦Av) 6= {0}. So, we call a complex number z0 with Ker(Dcoz0v−z0D si z0v ◦Acv) 6= {0} a complex focal radius of M along γv and call dim Ker(Dcoz0v−z0D si z0v ◦Acv) the multiplicity of the complex focal radius z0, where A c v is the complexification of Av and D co z0v (resp. Dsiz0v) is a C-linear transformation of (TxN ) c defined by Dcoz0v = g c ∗ ◦ cos( √ −1adc(z0g−1∗ v)) ◦ (gc∗)−1( resp. Dsisv = g c ∗ ◦ sin( √ −1adc(z0g−1∗ v))√ −1adc(z0g−1∗ v) ◦ (gc∗)−1 ) , where gc∗ (resp. ad c) is the complexification of g∗ (resp. ad). Here we note that, in the case where M is of class Cω, complex focal radii along γv indicate the positions of focal points of the extrinsic complexification M c(→֒ Gc/Kc) of M along the complexified geodesic γcι∗v, where G c/Kc is the anti-Kaehlerian symmetric space associated with G/K and ι is the natural immersion of G/K into Gc/Kc. See Section 4 of [14] about the definitions of Gc/Kc, M c(→֒ Gc/Kc) and γcι∗v. Also, for a complex focal radius z0 of M along γv, we call z0v (∈ (T ⊥x M )c) a complex focal normal vector of M at x. Furthermore, assume that M has globally flat normal bundle, that is, the normal holonomy group of M is trivial. Let ṽ be a parallel unit normal vector field of M . Assume that the number (which may be 0 and ∞) of distinct complex focal radii along γṽx is independent of the choice of x ∈ M . Furthermore assume that the number is not equal to 0. Let {ri,x | i = 1, 2, · · ·} be the set of all complex focal radii along γṽx , where |ri,x| < |ri+1,x| or ”|ri,x| = |ri+1,x| & Re ri,x > Re ri+1,x” or ”|ri,x| = |ri+1,x| & Re ri,x = Re ri+1,x & Im ri,x = −Im ri+1,x < 0”. Let ri (i = 1, 2, · · · ) be complex valued functions on M defined by assigning ri,x to each x ∈ M . We call these functions ri (i = 1, 2, · · · ) complex focal radius functions for ṽ. We call riṽ a complex focal normal vector field for ṽ. If, for each parallel unit normal vector field ṽ of M , the number of distinct complex focal radii CUBO 12, 2 (2010) Examples of a complex hyperpolar action without singular orbit 131 along γṽx is independent of the choice of x ∈ M , each complex focal radius function for ṽ is constant on M and it has constant multiplicity, then we call M a complex equifocal submanifold. Let N = G/K be a symmetric space of non-compact type and π be the natural projection of G onto G/K. Let (g, θ) be the orthogonal symmetric Lie algebra of G/K, f = {X ∈ g | θ(X) = X} and p = {X ∈ g | θ(X) = −X}, which is identified with the tangent space TeK N . Let 〈 , 〉 be the Ad(G)-invariant non-degenerate symmetric bilinear form of g inducing the Riemannian metric of N . Note that 〈 , 〉|f×f (resp. 〈 , 〉|p×p) is negative (resp. positive) definite. Denote by the same symbol 〈 , 〉 the bi-invariant pseudo-Riemannian metric of G induced from 〈 , 〉 and the Riemannian metric of N . Set g+ := p, g− := f and 〈 , 〉g± := −π∗g−〈 , 〉 + π ∗ g+ 〈 , 〉, where πg− (resp. πg+ ) is the projection of g onto g− (resp. g+). Let H 0([0, 1], g) be the space of all L2-integrable paths u : [0, 1] → g (with respect to 〈 , 〉g± ). Let H0([0, 1], g−) (resp. H0([0, 1], g+)) be the space of all L2-integrable paths u : [0, 1] → g− (resp. u : [0, 1] → g+) with respect to −〈 , 〉|g−×g− (resp. 〈 , 〉|g+×g+ ). It is clear that H0([0, 1], g) = H0([0, 1], g−) ⊕ H0([0, 1], g+). Define a non-degenerate symmetric bilinear form 〈 , 〉0 of H0([0, 1], g) by 〈u, v〉0 := ∫ 1 0 〈u(t), v(t)〉dt. It is easy to show that the decomposition H0([0, 1], g) = H0([0, 1], g−)⊕H0([0, 1], g+) is an orthogonal time-space decomposition with respect to 〈 , 〉0. For simplicity, set H0± := H0([0, 1], g±) and 〈 , 〉0,H0 ± := −π∗ H0 − 〈 , 〉0+π∗H0 + 〈 , 〉0, where πH0 − (resp. πH0 + ) is the projection of H0([0, 1], g) onto H0− (resp. H 0 +). It is clear that 〈u, v〉0,H0 ± = ∫ 1 0 〈u(t), v(t)〉g± dt (u, v ∈ H0([0, 1], g)). Hence (H0([0, 1], g), 〈 , 〉0,H0 ± ) is a Hilbert space, that is, (H0([0, 1], g), 〈 , 〉0) is a pseudo-Hilbert space. Let H1([0, 1], G) be the Hilbert Lie group of all absolutely continuous paths g : [0, 1] → G such that the weak derivative g′ of g is squared integrable (with respect to 〈 , 〉g± ), that is, g−1∗ g′ ∈ H0([0, 1], g). Define a map φ : H0([0, 1], g) → G by φ(u) = gu(1) (u ∈ H0([0, 1], g)), where gu is the element of H1([0, 1], G) satisfying gu(0) = e and g−1u∗ g ′ u = u. We call this map the parallel transport map (from 0 to 1). This submersion φ is a pseudo-Riemannian submersion of (H0([0, 1], g), 〈 , 〉0) onto (G, 〈 , 〉). Let π : G → G/K be the natural projection. It follows from Theorem A of [13] (resp. Theorem 1 of [14]) that, in the case where M is curvature adapted (resp. of class Cω ), M is complex equifocal if and only if each component of (π ◦ φ)−1(M ) is complex isoparametric. See [13] about the definition of a complex isoparametric submanifold in a pseudo-Hilbert space. In particular, if each component of (π ◦ φ)−1(M ) are proper complex isoparametric, that is, for each normal vector v, there exists a pseudo-orthonormal base of the complexified tangent sapce consisting of the eigenvectors of the complexified shape operator for v, then we call M a proper complex equifocal submanifold. Next we recall the notion of an infinite dimensional anti-Kaehlerian isoparametric submanifold. Let M be an anti-Kaehlerian Fredholm submanifold in an infinite dimensional anti-Kaehlerian space V and A be the shape tensor of M . See [14] about the definitions of an infinite dimensional anti- Kaehlerian space and anti-Kaehlerian Fredholm submanifold in the space. Denote by the same symbol J the complex structures of M and V . Fix a unit normal vector v of M . If there exists X(6= 0) ∈ T M with AvX = aX +bJX, then we call the complex number a+b √ −1 a J-eigenvalue of Av (or a complex principal curvature of direction v) and call X a J-eigenvector for a + b √ −1. Also, we call the space of all J-eigenvectors for a + b √ −1 a J-eigenspace for a + b √ −1. The J-eigenspaces are orthogonal to one another and each J-eigenspace is J-invariant. We call the set of all J-eigenvalues of Av the J-spectrum of Av and denote it by SpecJ Av. The set SpecJ Av \ {0} is described as follows: SpecJ Av \ {0} = {λi | i = 1, 2, · · · } 132 Naoyuki Koike CUBO 12, 2 (2010) ( |λi| > |λi+1| or ”|λi| = |λi+1| & Re λi > Re λi+1” or ”|λi| = |λi+1| & Re λi = Re λi+1 & Im λi = −Im λi+1 > 0” ) . Also, the J-eigenspace for each J-eigenvalue of Av other than 0 is of finite dimension. We call the J-eigenvalue λi the i-th complex principal curvature of direction v. Assume that M has globally flat normal bundle. Fix a parallel normal vector field ṽ of M . Assume that the number (which may be ∞) of distinct complex principal curvatures of direction ṽx is independent of the choice of x ∈ M . Then we can define functions λ̃i (i = 1, 2, · · · ) on M by assigning the i-th complex principal curvature of direction ṽx to each x ∈ M . We call this function λ̃i the i-th complex principal curvature function of direction ṽ. If M satisfies the following condition (AKI), then we call M an anti-Kaehlerian isoparametric submanifold: (AKI) For each parallel normal vector field ṽ, the number of distinct complex principal curvatures of direction ṽx is independent of the choice of x ∈ M , each complex principal curvature function of direction ṽ is constant on M and it has constant multiplicity. Let {ei}∞i=1 be an orthonormal system of TxM . If {ei}∞i=1 ∪ {Jei}∞i=1 is an orthonormal base of TxM , then we call {ei}∞i=1 a J-orthonormal base. If there exists a J-orthonormal base consisting of J-eigenvectors of Av, then Av is said to be diagonalized with respect to the J-orthonormal base. If M is anti-Kaehlerian isoparametric and, for each v ∈ T ⊥M , the shape operator Av is diagonalized with respect to a J-orthonormal base, then we call M a proper anti-Kaehlerian isoparametric submanifold. For arbitrary two unit normal vector v1 and v2 of a proper anti-Kaehlerian isoparametric submanifold, the shape operators Av1 and Av2 are simultaneously diagonalized with respect to a J-orthonormal base. Let M be a proper anti-Kaehlerian isoparametric submanifold in an infinite dimensional anti- Kaehlerian space V . Let {Ei | i ∈ I} be the family of distributions on M such that, for each x ∈ M , {Ei(x) | i ∈ I} is the set of all common J-eigenspaces of Av’s (v ∈ T ⊥x M ). The relation TxM = ⊕ i∈I Ei holds. Let λi (i ∈ I) be the section of (T ⊥M )∗ ⊗ C such that Av = Reλi(v)id + Imλi(v)J on Ei(π(v)) for each v ∈ T ⊥M , where π is the bundle projection of T ⊥M . We call λi (i ∈ I) complex principal curvatures of M and call distributions Ei (i ∈ I) complex curvature distributions of M . In the case where M is a real analytic submanifold in a symmetric space G/K of non-compact type, it is shown that M is complex equifocal if and only if (πc ◦ φc)−1(M c) is anti-Kaehlerian isoparametric, where πc is the natural projection of Gc onto Gc/Kc and φc is the parallel transport map for Gc (which is defined in similar to the above φ). Also, it is shown that M is proper complex equifocal if and only if (πc ◦ φc)−1(M c) is proper anti-Kaehlerian isoparametric. 3 Proof of Theorems A and B In this section, we first prove Theorem A. Proof of Theorem A. Let H be a complex hyperpolar action on G/K(= AN ) without singular orbit, H = LR (L : semi-simple, R : solvable) be the Levi decomposition of H and L = KLALNL (KL : compact, AL : abelian, NL : nilpotent) be the Iwasawa decomposition of L. Since KL is compact, it has a fixed point p0 by the Cartan’s fixed point theorem. Suppose that KL · p 6⊂ ALNLR · p for some CUBO 12, 2 (2010) Examples of a complex hyperpolar action without singular orbit 133 p ∈ G/K. Then we have dim H · p0 < dim H · p, which implies that H · p0 is a singular orbit. This contradicts the fact that the H-action has no singular orbit. Hence it follows that KL · p ⊂ ALNLR · p for any p ∈ G/K. Therefore we can show that the ALNLR-action has the same orbits as the H-action. The group ALNLR is decomposed into the product of some compact subgroup T ′ and some solvable normal subgroup S′ admitting a maximal compact normal subgroup S′K contained in the center of S′ such that S′/S′K is simply connected (see Theorem 6 of [19]). Since T ′ is compact, it is shown by the same argument as above that the S′-action has the same orbit as the ALNLR-action (hence the H-action). Take any p ∈ G/K and any g ∈ S′ with g 6= e. Since S′ acts on G/K effectively, there exists p1 ∈ G/K with g(p1) 6= p1. The section Σp1 through p1 is mapped into the section Σg(p1) through g(p1) by g. Since the S ′-action has no singular orbit, we have Σp1 ∩ Σg(p1) = ∅. Let q be the intersection of H·p with Σp1 . Then g(q) is the intersection of H·p with Σg(p1). Hence we have g(q) 6= q. Therefore S′ acts on each H-orbit effectively. Since the isotropy group S′p of S ′ at any p ∈ G/K is compact, it is contained in a conjugate of S′K (see Theorem 4 of [19]). Hence S ′ p is contained in the center of S′. Therefore, since the S′p-action has a fixed point p and it is effective, it is trivial. Thus the S′-action is free. Let s′ := Lie S′ (the Lie algebra of S′), s̃′ be a maximal solvable subalgebra of g containing s′ and S̃′ be the connected subgroup of G with Lie S̃′ = s̃′. Since g is a real semi-simple Lie algebra and s̃′ is a maximal solvable subalgebra of g, s̃′ contains a Cartan subalgebra ã′ of g. Let t′ (resp. a′) be the toroidal part (resp. the vector part) of ã′. There exists a Cartan decomposition g = f′ + p′ of g with t′ ⊂ f′ and a′ ⊂ p′. Let g = g′0 + ∑ λ∈△′ g′λ be the root space decomposition with respect to a′ (i.e., g′0 is the centralizer of a ′ in g and g′λ = {X ∈ g | ad(a)(X) = λ(a)X for all a ∈ a′} and △′ = {λ ∈ (a′)∗ \ {0} | g′λ 6= {0}}). Let n′ := ∑ λ∈△′ + g′λ, where △′+ is the positive root system with respect to some lexicographic ordering of a′. The algebra ã′ + n′ is a maximal solvable subalgebra of g. According to a result of [21], we may assume that s̃′ = ã′ + n′ by retaking ã′ if necessary. By imitating the proof of Lemma 5.1 of [3], it is shown that a′ is a maximal abelian subspace of p′ because the S′-action has flat section. There exists g ∈ G satisfying Ad(g)(f′) = f, Ad(g)(p′) = p, Ad(g)(a′) = a and Ad(g)(ã′) = ã, where Ad is the adjoint representation of G, a and ã are as in Introduction. Let s := Ad(g)(s′) and S be the connected subgroup of G with Lie S = s. Since the S-action is conjugate to the S′-action and S ⊂ ÃN , we obtain the statement of Theorem A. q.e.d. Let a be a maximal abelian subspace of p. Fix a lexicographic ordering of a. Let g = g0 + ∑ λ∈△ gλ, p = a + ∑ λ∈△+ pλ and f = f0 + ∑ λ∈△+ fλ be the root space decompositions of g, p and f with respect to a, where we note that gλ = {X ∈ g | ad(a)X = λ(a)X for all a ∈ a} (λ ∈ △), pλ = {X ∈ p | ad(a)2X = λ(a)2X for all a ∈ a} (λ ∈ △+), fλ = {X ∈ f | ad(a)2X = λ(a)2X for all a ∈ a} (λ ∈ △+ ∪ {0}). Also, let g = f+a+n be the Iwasawa decomposition of g and G = KAN be the corresponding Iwasawa decomposition of G, where we note that n = ∑ λ∈△+ gλ. Now we shall give examples of a solvable group contained in AN whose action on G/K(= AN ) is complex hyperpolar. Denote by π the natural projection of G onto G/K. Since G/K is of non-compact type, π gives a diffeomorphism of AN onto G/K. Denote by 〈 , 〉 the left-invariant metric of AN induced from that of G/K by π|AN . Also, 134 Naoyuki Koike CUBO 12, 2 (2010) denote by 〈 , 〉G the bi-invariant metric of G inducing that of G/K. Note that 〈 , 〉 6= ι∗〈 , 〉G, where ι is the inclusion map of AN into G. Let l be a r-dimensional subspace of a + n and set s := (a + n)⊖l , where (a + n) ⊖ l denotes the orthogonal complement of l in a + n with respect to 〈 , 〉e, where e is the identity element of G. If s is a subalgebra of a + n and lp := prp(l ) (prp : the orthogonal projection of g onto p) is abelian, then the S-action (S := expG(s)) is a complex hyperpolar action without singular orbit. We shall give examples of such a subalgebra s of a + n and investigate the structure of the S-orbit. Example 1. Let b be a r(≥ 1)-dimensional subspace of a and sb := (a + n) ⊖ b. It is clear that bp(= b) is abelian and that sb is a subalgebra of a + n. Hence the Sb-action (Sb := expG(sb)) on G/K is a complex hyperpolar action without singular orbit. Example 2. Let {λ1, · · · , λk} be a subset of a simple root system Π of △ such that Hλ1 , · · · , Hλk are mutually orthogonal, b be a subspace of a ⊖ Span{Hλ1 , · · · , Hλk } (where b may be {0}) and li (i = 1, · · · , k) be a one-dimensional subspace of RHλi + gλi with li 6= RHλi , where Hλi is the element of a defined by 〈Hλi , ·〉λi(·) and RHλi is the subspace of a spanned by Hλi . Set l := b + k∑ i=1 li. Then, it follows from Lemma 3.1 (see the below) that lp is abelian and that sb,l1,··· ,lk := (a + n) ⊖ l is a subalgebra of a + n. Hence the Sb,l1,··· ,lk -action (Sb,l1,··· ,lk := expG(sb,l1,··· ,lk )) on G/K is a complex hyperpolar action without singular orbit. Lemma 3.1. Let l and sb,l1,··· ,lk be as in Example 2. Then lp is abelian and sb,l1,··· ,lk is a subalgebra of a + n. Proof. Let H ∈ b and Xi ∈ li (i = 1, · · · , k). Since λi(H) = 0 and (Xi)p ∈ RHλi ⊕ pλi , we have [H, (Xi)p] = 0. Fix i, j ∈ {1, · · · , k} (i 6= j). Since λi and λj are simple roots and 〈Hλi , Hλj 〉 = 0, we have [(Xi)p, (Xj )p] = 0. Thus lp is abelian. Let V, W ∈ sb,l1,··· ,lk . Since sb,l1,··· ,lk = (a ⊖ (b + k∑ i=1 RHλi )) ⊕ ( ∑ λ∈△+\{λ1,··· ,λk} gλ) ⊕ ( k∑ i=1 ((RHλi + gλi ) ⊖ li)), V and W are described as V = V0 + ∑ λ∈△+\{λ1,··· ,λk} Vλ + k∑ i=1 Vi and W = W0 + ∑ λ∈△+\{λ1,··· ,λk} Wλ + k∑ i=1 Wi, respectively, where V0, W0 ∈ a ⊖ (b + k∑ i=1 RHλi ), Vλ, Wλ ∈ gλ and Vi, Wi ∈ (RHλi + gλi ) ⊖ li. Easily we have [V, W ] ≡ ∑ λ,µ∈△+\{λ1,··· ,λk} [Vλ, Wµ] + ∑ λ∈△+\{λ1,··· ,λk} k∑ i=1 ([Vλ, Wi] + [Vi, Wλ]) + k∑ i=1 k∑ j=1 [Vi, Wj ] (mod sb,l1,···lk ). Since λ1, · · · , λk are simple roots, [Vλ, Wµ], [Vλ, Wi], [Vi, Wλ] and [Vi, Wj ] (λ, µ ∈ △+\{λ1, · · · , λk}, 1 ≤ i, j ≤ k) belong to sb,l1,··· ,lk . Therefore we have [V, W ] ∈ sb,l1,··· ,lk . Thus sb,l1,···lk is a subalgebra of a+n. q.e.d. CUBO 12, 2 (2010) Examples of a complex hyperpolar action without singular orbit 135 For the orbit Sb,l1,··· ,lk · e, we have the following facts. Lemma 3.2. Let sb,l1,··· ,lk be as in Example 2, ξ0 ∈ b, ξiti := 1 cosh(|λi|ti) ξi − 1 |λi| tanh(|λi|ti)Hλi be a unit vector of li (i = 1, · · · , k), where ξi is a unit vector of gλi . Denote by A the shape tensor of the orbit Sb,l1,··· ,lk · e (⊂ AN ). Then, for Aξ0 and Aξiti , the following statements (i) ∼ (vii) hold: (i) For X ∈ a ⊖ (b + k∑ i=1 RHλi ), we have Aξ0 X = Aξiti X = 0 (i = 1, · · · , k). (ii) For X ∈ Ker(ad(ξi)|gλi ) ⊖ Rξ i, we have Aξ0 X = 0 and Aξiti X = −|λi| tanh(|λi|ti)X. (iii) Assume that 2λi ∈ △+. For X ∈ g2λi , we have Aξ0 ([θξi, X]) = 0 and Aξiti X = −2|λi| tanh(|λi|ti)X − 1 2 cosh(|λi|ti) [θξi, X], Aξiti ([θξi, X]) = − |λi| 2 cosh(|λi|ti) X − |λi| tanh(|λi|ti)[θξi, X], where θ is the Cartan involution of g with Fix θ = f. (iv) For X ∈ (Rξi + RHλi ) ⊖ li, we have Aξ0 X = 0 and Aξiti X = −|λi| tanh(|λi|ti)X. (v) For X ∈ (gλj ⊖ Rξj ) + ((Rξj + RHλj ) ⊖ lj ) + g2λj (j 6= i), we have Aξ0 X = Aξiti X = 0. (vi) For X ∈ gµ (µ ∈ △+ \ {λ1, · · · , λk}), we have Aξ0 X = µ(ξ0)X. (vii) Let ki := exp ( π√ 2|λi| (ξi + θξi) ) , where exp is the exponential map of G. Then Ad(ki) ◦ Aξiti = −Aξiti ◦ Ad(ki) holds over n ⊖ k∑ i=1 (gλi + g2λi ), where Ad is the adjoint representation of G. Proof. Let pr1a+n (resp. pr 2 a+n) be the projection of g onto a + n with respect to the decomposition g = f + (a + n) (resp. g = (f0 + ∑ λ∈△+ pλ) + (a + n)), prf (resp. prp) be the projection of g onto f (resp. p) with respect to the decomposition g = f + p and prf0 be the projection of g onto f0 with respect to the decomposition g = f0 + (a + ∑ λ∈△ gλ). Then we have (3.1) prp ◦ pr1a+n = prp and prf ◦ pr2a+n = prf − prf0 . Let H ∈ a, N1, N2 ∈ n and E ∈ gλ (λ ∈ △+). Denote by ad(H)∗ (resp. ad(E)∗) the adjoint operator of ad(H) (resp. ad(E)) : a + n → a + n with respect to 〈 , 〉e. Easily we can show (3.2) ad(H)∗ = ad(H). For simplicity, we denote prf(·) (resp. prp(·)) by (·)f (resp. (·)p). From (3.1) and the skew- symmetricness of ad(·) with respect to 〈 , 〉Ge , we have 〈ad(E)N1, N2〉e = 〈ad(Ef)((N1)p) + ad(Ep)(((N1)f), (N2)p〉Ge = −〈(N1)p, ad(Ef)((N2)p)〉Ge − 〈(N1)f, ad(Ep)((N2)p)〉Ge = −〈(N1)p, (pr1a+n(ad(Ef)N2))p〉Ge −〈(N1)f, (pr2a+n(ad(Ep)N2))f + prf0 (ad(Ep)N2)〉 G e = −〈N1, pr1a+n(ad(Ef)N2)〉e + 〈N1, pr2a+n(ad(Ep)N2)〉e 136 Naoyuki Koike CUBO 12, 2 (2010) and hence prn(ad(E) ∗N2) = prn(−pr1a+n(ad(Ef)N2) + pr2a+n(ad(Ep)N2)), where prn is the projection of a + n onto n. Also, we have 〈ad(E)H, N2〉e = −λ(H)〈E, N2〉e = −〈H, 〈E, N2〉eHλ〉e and hence pra(ad(E) ∗N2) = −〈E, N2〉eHλ, where pra is the projection of a + n onto a. Also, we can show ad(E)∗H = 0. Therefore, we have (3.3) ad(E)∗ =    0 on a −〈E, ·〉e ⊗ Hλ − prn ◦ pr1a+n ◦ ad(Ef) +prn ◦ pr2a+n ◦ ad(Ep) on n On the other hand, according to the Koszul’s formula, we have 〈AξX, Y 〉e = 1 2 (〈[X, Y ], ξ〉e − 〈[Y, ξ], X〉e + 〈[ξ, X], Y 〉e) = 1 2 〈(ad(ξ) + ad(ξ)∗)X, Y 〉e for any X, Y ∈ Te(Sb,l1,··· ,lk · e)sb,l1,··· ,lk and any ξ ∈ T ⊥e (Sb,l1,··· ,lk · e) = b + k∑ i=1 li. That is, we have (3.4) Aξ = 1 2 prT ◦ (ad(ξ) + ad(ξ)∗), where prT is the orthogonal projection of a + n onto sb,l1,··· ,lk . From (3.2) and (3.4), we have Aξ0 X =    0 (X ∈ sb,l1,··· ,lk ⊖ ∑ µ∈△+\{λ1,··· ,λk} gλ) µ(ξ0)X (X ∈ gµ), where µ ∈ △+ \ {λ1, · · · , λk}. From (3.3) and (3.4), we have Aξiti X = 0 (X ∈ a ⊖ (b + k∑ i=1 RHλi )). Set gKλj := Ker(ad(ξ j )|gλj ) and g I λj := Im(ad(θξj )|g2λj ) (j = 1, · · · , k). Then we have gλj = g K λj ⊕ gIλj . By simple calculations, it is shown that this decomposition is orthogonal with respect to 〈 , 〉e. If X ∈ gKλj ⊖ Rξ j , then it follows from (3.2), (3.3), (3.4), λi, λj ∈ Π and 〈Hλi , Hλj 〉 = 0 (when i 6= j) that Aξiti X = { −|λi| tanh(|λi|ti)X (i = j) 0 (i 6= j). If X ∈ g2λj , then it follows from (3.2), (3.3), (3.4), λi, λj ∈ Π and 〈Hλi , Hλj 〉 = 0 (when i 6= j) that Aξiti X =    −2|λi| tanh(|λi|ti)X − 1 2 cosh(|λi|ti) [θξi, X] (i = j) 0 (i 6= j). Also, we have Aξiti ([θξi, X]) = − |λi| 2 cosh(|λi|ti) X − |λi| tanh(|λi|ti)[θξi, X]. CUBO 12, 2 (2010) Examples of a complex hyperpolar action without singular orbit 137 Let X := tanh(|λj|tj )ξj + 1|λj | cosh(|λj |tj ) Hλj , which is a unit vector of (Rξ j + RHλj ) ⊖ lj . From (3.2), (3.3), (3.4), λi, λj ∈ Π and 〈Hλi , Hλj 〉 = 0 (when i 6= j), we have Aξiti X = − 1 2 |λi| tanh(|λi|ti)X + 1 2 cosh(|λi|ti) prT (ad(ξ i)∗X) − 1 2|λi| tanh(|λi|ti)prT (ad(Hλ)∗X) = { −|λi| tanh(|λi|ti)X (i = j) 0 (i 6= j). This completes the proof of (i) ∼ (vi). Finally we shall show the statement (vii). Let X ∈ n ⊖ k∑ i=1 (gλi + g2λi ) and ki be as in the statement (vii). From (3.2), (3.3), (3.4), λj ∈ Π (j = 1, · · · , k) and 〈Hλi , Hλj 〉 = 0 (when i 6= j), we have Aξiti X = 1 cosh(|λi|ti) [ξip, X] − 1 |λi| tanh(|λi|ti)[Hλi , X]. By operating Ad(ki) to both sides of this relation, we have Ad(ki)(Aξiti X) = −Aξiti (Ad(ki)X), where we use Ad(ki)(ξ i p) = −ξip and Ad(ki)(Hλi ) = −Hλi . Thus the statement (vii) is shown. q.e.d. Also, we have the following fact. Lemma 3.3. Let sb,l1,··· ,lk be as in Example 2 and l̄i be the orthogonal projection of li onto gλi . Set sb,̄l1,··· ,̄lk := (a + n) ⊖ (b + k∑ i=1 l̄i) and Sb,̄l1,··· ,̄lk := expG(sb,̄l1,··· ,̄lk ). Then the Sb,̄l1,··· ,̄lk -action is conjugate to the Sb,l1,··· ,lk -action. Proof. Denote by ∇ the Levi-Civita connection of the left-invariant metric of AN . Let H be a vector of b, ξi be a unit vector of l̄i (i = 1, · · · , k) and γξi be the geodesic in AN with γ̇ξi (0) = ξi. Let ti be a real number with 1 cosh(|λi|ti) ξi − tanh(|λi|ti)Hλi ∈ li (i = 1, · · · , k). Denote by the same symbols H, ξi and Hλi the left-invariant vector fields arising from H, ξ i and Hλi , respectively. By using the relation (5.4) of Section 5 of [20] (arising the Koszul formula for the left-invariant vector fields), we can show ∇ξ1 ξ1 = |λ1|Hλ1 , ∇ξ1 Hλ1 = −|λ1|ξ1 ∇ξ1 ξi = ∇ξ1 H = ∇Hλ1 ξ 1 = ∇Hλ1 ξ i = ∇Hλ1 Hλ1 = ∇Hλ1 H = 0, where i = 2, · · · , k. From ∇ξ1 ξ1 = |λ1|Hλ1 , ∇ξ1 Hλ1 = −|λ1|ξ1, ∇Hλ1 ξ 1 = ∇Hλ1 Hλ1 = 0, it follows that exp R{ξ1, Hλ1} is a totally geodesic subgroup of AN . Hence γ̇ξ1 (t) is expressed as γ̇ξ1 (t) = a(t)(Hλ1 )γξ1 (t) + b(t)(ξ 1)γ ξ1 (t). Furthermore, we have ∇γ̇ξ1 γ̇ξ1 = (a ′ + |λ1|b2)Hλ1 + (b′ −|λ1|ab)ξ1 = 0, that is, a′ = −|λ1|b2 and b′ = |λ1|ab. By solving this differential equation under the initial conditions a(0) = 0 and b(0) = 1, we have a(t) = − tanh(|λ1|t) and b(t) = 1cosh(|λ1|t) . Hence we obtain 138 Naoyuki Koike CUBO 12, 2 (2010) γ̇ξ1 (t) = 1 cosh(|λ1|t) (ξ1)γ ξ1 (t) − tanh(|λ1|t)(Hλ1 )γξ1 (t). From ∇ξ1 ξ i = ∇ξ1 H = ∇Hλ1 ξ i = ∇Hλ1 H = 0 (i = 2, · · · , k), it follows that ξi (i = 2, · · · , k) and H are parallel along γξ1 (with respect to ∇). Denote by Pγ ξ1 |[0,t] the parallel translation along γξ1|[0,t] (with respect to ∇) and Lγξ1 (t) the left translation by γξ1 (t). From the above facts, we have T ⊥γ ξ1 (t1) (Sb,̄l1,··· ,̄lk ) = Pγξ1 |[0,t1] (b + k∑ i=1 l̄i) = (Lγ ξ1 (t1))∗(b + k∑ i=2 l̄i + l1) = (Lγ ξ1 (t1))∗(T ⊥ e Sb,l1,̄l2,··· ,̄lk ), which implies γξ1 (t1) −1Sb,̄l1,··· ,̄lk γξ1 (t1) = Sb,l1,̄l2,··· ,̄lk . By repeating the same discussion, we obtain (γξ1 (t1) · · · γξk (tk))−1Sb,̄l1,··· ,̄lk (γξ1 (t1) · · · γξk (tk)) = Sb,l1,··· ,lk . Thus the Sb,̄l1,··· ,̄lk -action is conjugate to the Sb,l1,··· ,lk -action. q.e.d. For parallel submanifolds of a proper complex equifocal submanifold and a curvature-adapted complex equifocal submanifold, we have the following facts. Lemma 3.4. (i) All parallel submanifolds of a proper complex equifocal submanifold are proper complex equifocal. (ii) All parallel submanifolds of a curvature-adapted complex equifocal submanifold are curvature- adapted and complex equifocal. Proof. First we shall show the statement (i). Let M be a proper complex equifocal submanifold in a symmetric space G/K of non-compact type and ṽ be the parallel normal vector field of M which is not a focal normal vector field. Denote by η ev the end-point map for ṽ and Mev := ηev(M ), which is a parallel submanifold of M . The vector field ṽ is regarded as a parallel normal vector field of the complexification M c along M . Let ṽL be the horizontal lift of ṽ to H0([0, 1], gc) by the anti- Kaehlerian submersion πc ◦ φc : H0([0, 1], gc) → Gc/Kc, which is a parallel normal vector field of M̃ c(:= (πc ◦ φc)−1(M c)). Set M̃ c evL := ηevL (M̃ c), where η evL is the end-point map for ṽ L. Note that M̃ c evL = (π c ◦ φc)−1((M ev) c). Denote by à and Ãev L the shape tensors of M̃ c and M̃ c evL , respectively. Let {λi | i ∈ I} be the set of all complex principal curvatures of M̃ c and Ei be the complex curvature distribution for λi. Then, according to Lemma 3.2 of [16], we have (3.5) Ãev L w |(Ei)u = (λi)u(w) 1 − (λi)u(ṽLu ) id (i ∈ I, u ∈ M̃ c evL ), where we note that Tη evL (u)M̃ c evL = TuM̃ c(= ⊕ i∈I (Ei)u). This implies that M̃ c evL is proper anti- Kaehlerian isoparametric, that is, M ev is proper complex equifocal. Thus the statement (i) is shown. Next we shall show the statement (ii). Let M be a curvature-adapted complex equifocal submanifold in G/K and ṽ be the parallel normal vector field of M . Set M ev := ηev(M ). Denote by A and A ev the shape tensors of M and M ev, repsectively. Let w ∈ T ⊥x M . Without loss of generality, we may assume that x = eK. Let a be a maximal abelian subspace of p := TeK (G/K) containing T ⊥ eK M and p = a + ∑ α∈△+ pα be the root space decomposition with respect to a. Let X ∈ Ker(Av − λ id) ∩ Ker(Aw − µ id) ∩ pα CUBO 12, 2 (2010) Examples of a complex hyperpolar action without singular orbit 139 (λ ∈ Spec Av, µ ∈ Spec Aw, α ∈ △+). Let w̃ be the parallel tangent vector field on the (flat) section Σ of M through eK with w̃eK = w. Since M ev is regarded as a partial tube over M , it follows from (ii) of Corollary 3.2 in [15] that (3.6) (Aev) ewη ev (eK) ((η ev )∗X) = 1 α(v) − λ tanh α(v) {−α(v)α(w) tanh α(v) +λ ( 1 − tanh α(v) α(v) ) α(w) + µ tanh α(v)}(η ev )∗X. Let Z be the element of p with expG(Z)K = ηev(eK). For simplicity, set g := expG(Z). Since g∗ : p → Tη ev(eK)(G/K) is the parallel translation along the normal geodesic γZ (⇔ def γZ (t) := expG(tZ)K), it follows from (3.1) of [15] that (η ev)∗X = g∗(D co v (X) − Dsiv (AvX)) = ( cosh α(v) − λ sinh α(v) α(v) ) g∗X ∈ g∗pα. Also, we have g−1∗ (T ⊥ η ev (eK) M ev) = T ⊥ eK M ⊂ a. Hence we have R((ηev)∗X, w̃η ev (eK))w̃ηev (eK) = −α(w)2(η ev)∗X, which together with (3.6) implies [(Aev) ewη ev (eK) , R(·, w̃η ev (eK))w̃ηev (eK)]((ηev)∗X) = 0. Therefore, it follows from the arbitrariness of X that [(Aev) ewη ev (eK) , R(·, w̃η ev (eK))w̃ηev (eK)] vanishes over (η ev)∗(Ker(Av − λ id) ∩ Ker(Aw − µ id) ∩ pα). Since M is curvature-adapted, we have ⊕ λ∈Spec Av ⊕ µ∈Spec Aw ⊕ α∈△+ (η ev)∗(Ker(Av − λ id) ∩ Ker(Aw − µ id) ∩ pα) = Tη ev(eK)Mev. Hence we have [(Aev) ewη ev (eK) , R(·, w̃η ev (eK))w̃ηev (eK)] = 0. Therefore, it follows from the arbitrariness of w that M ev is curvature-adapted. It is clear that Mev is complex equifocal. Thus the statement (ii) is shown. q.e.d. For the Sb-action and the Sb,l1,··· ,lk -action, we have the following facts. Proposition 3.5. (i) All orbits of the Sb-action are curvature-adapted but they are not proper complex equifocal. (ii) Let λ1, · · · , λk (∈ △+) be as in Example 2. If the root system △ of G/K is non-reduced and 2λi0 ∈ △+ for some i0 ∈ {1, · · · , k}, then all orbits of the Sb,l1,··· ,lk -action are not curvature-adapted. Also, if b 6= {0}, then they are not proper complex equifocal. Proof. First we shall show the statement (i). The group Sb acts isometrically on (AN, 〈 , 〉). Denote by A the shape tensor of the orbit Sb · e in AN . Since 〈 , 〉 is left-invariant, it follows from the Koszul formula that 〈AvX, Y 〉 = 〈ad(v)X, Y 〉 for any v ∈ l = T ⊥e (Sb · e) and X, Y ∈ s = Te(Sb · e). Hence we have Av|a⊖l = 0 and Av|gλ = λ(v)id (λ ∈ △+), where v ∈ T ⊥e (Sb · e) = l (⊂ p). Therefore, the orbit Sb · e is curvature-adapted but it is not proper complex equifocal by (ii) of Theorem 1 of [14]. Hence so are all orbits of the Sb-action by Lemma 3.3. 140 Naoyuki Koike CUBO 12, 2 (2010) Next we shall show the statement (ii). Assume that the root system △ of G/K is non-reduced. Denote by A the shape tensor of the orbit Sb,l1,··· ,lk ·e (⊂ AN ). Also, let ξ0 ∈ b and ξiti : 1 cosh(|λi|ti) ξi − 1 |λi| tanh(|λi|ti)Hλi (ξi ∈ gλi ) be a unit (tangent) vector of li. Then, according to Lemma 3.2, we see that (3.7) Aξ0|sb,l1,···lk ∩(a+ P k i=1 gλi ) = 0, Aξ0|gµ = µ(ξ0)id (µ ∈ △+ \ k ∪ i=1 {λi}), Aξiti |a⊖(b+Pk j=1 RHλj ) = 0, Aξiti |Ker(ad(ξi)|gλi )⊖Rξi = −|λi| tanh(|λi|ti)id Aξiti |(Rξi+RHλi )⊖li = −|λi| tanh(|λi|ti)id and that, in case of 2λi ∈ △+, Aξiti |Im(ad(θξi)|g2λi )+g2λi has two eigenvalues µ+i := − 3 2 |λi| tanh(|λi|ti) + 1 2 |λi| √ 2 − tanh2(|λi|ti) and µ−i := − 3 2 |λi| tanh(|λi|ti) − 1 2 |λi| √ 2 − tanh2(|λi|ti) with the same multiplicity. Note that gλi = Ker(ad(ξ i)|gλi ) ⊕ Im(ad(θξ i)|g2λi ). The eigenspace for µ+i (resp. µ − i ) is spanned by Z+ ξi,Y := [θξi, Y ] + |λi| ( sinh(|λi|ti) − √ sinh2(|λi|ti) + 2 ) Y ′s (Y ∈ g2λi ) (resp. Z− ξi,Y := [θξi, Y ] + |λi| ( sinh(|λi|ti) + √ sinh2(|λi|ti) + 2 ) Y ′s (Y ∈ g2λi ))). Denote by R the curvature tensor of 〈 , 〉. Also, denote by Xf (resp. Xp) the f-component (resp. the p-component) of X ∈ g. Then we have (3.8) ( R(Z± ξi,Y , ξiti )ξ i ti ) p = −a[[(Z± ξi,Y )p, (ξ i ti )p], (ξ i ti )p] = a(−[[Z± ξi,Y , ξiti ], ξ i ti ]p + [[(Z ± ξi,Y )f, (ξ i ti )f], (ξ i ti )p] +[[(Z± ξi,Y )f, (ξ i ti )p], (ξ i ti )f] + [[(Z ± ξi,Y )p, (ξ i ti )f], (ξ i ti )f]) for some non-zero constant a, where we note that a = 1 if the metric of G/K is induced from the restriction of the Killing form of g to p. Also we have (3.9) [[(Z± ξi,Y )p, (ξ i ti )f], (ξ i ti )f] = 0, (3.10) [[(Z± ξi,Y )f, (ξ i ti )f], (ξ i ti )p] = − tanh(|λi|ti) |λi| cosh(|λi|ti) [[[θξi, Y ]f, ξ i f ], Hλi ] and (3.11) [[(Z± ξi,Y )f, (ξ i ti )p], (ξ i ti )f] = |λi| tanh(|λi|ti) cosh(|λi|ti) [[θξi, Y ]p, ξ i f ]. Let η (resp. η̄) be the element of a + n with ηf = [[θξ i, Y ]f, ξ i f ] (resp. η̄p = [[θξ i, Y ]p, ξ i f ]). Then it follows from (3.8) ∼ (3.11) that (R(Z± ξi,Y , ξiti )ξ i ti )p = −a[[Z±ξi,Y , ξ i ti ], ξiti ]p + a|λi| tanh(|λi|ti) cosh(|λi|ti) (2ηp + η̄p), CUBO 12, 2 (2010) Examples of a complex hyperpolar action without singular orbit 141 that is, (3.12) R(Z± ξi,Y , ξiti )ξ i ti = −a[[Z± ξi,Y , ξiti ], ξ i ti ] + a|λi| tanh(|λi|ti) cosh(|λi|ti) (2η + η̄). We have [ξi, θξi] = bHλi for some non-zero constant b. By simple calculation, we have (3.13) [[Z± ξi,Y , ξiti ], ξ i ti ] = 2|λi| tanh2(|λi|ti) ( − 3b|λi| 2 sinh(|λi|ti) + sinh(|λi|ti) ∓ √ sinh2(|λi|ti) + 2 ) Y + tanh2(|λi|ti)[θξi, Y ]. From (3.12) and (3.13), it follows that R(Z± ξi,Y , ξiti )ξ i ti belongs to Im ad(θξi) ⊕ g2λi . Hence R(·, ξiti )ξ i ti preserves Im ad(θξi) ⊕ g2λi invariantly. It is clear that so is also Aξiti . From (3.12) and (3.13), we have [R(·, ξiti )ξ i ti , Aξiti ]|Im ad(θξi)⊕g2λi 6= 0, under a suitable choice of ti. Therefore, Sb,l1,··· ,lk · e is not curvature-adapted under suitable choices of l1, · · · , lk. Then, so are all orbits of the Sb,l1,··· ,lk -action by Lemma 3.4. Furthermore, it follows from Lemma 3.3 that all orbits of the Sb,l1,··· ,lk -action are not curvature-adapted under arbitrary choices of l1, · · · , lk. Also, it follows from the second relation of (3.7) that Sb,l1,··· ,lk · e (hence all orbits of the Sb,l1,··· ,lk -action) is not proper complex equifocal in case of b 6= {0}. q.e.d. From this proposition, we obtain the statements of Theorem B. Also, we have the following fact. Proposition 3.6. If b = {0}, then the Sb,l1,··· ,lk -action possesses the only minimal orbit. Proof. According to Lemma 3.3, the Sb,l1,··· ,lk -action is conjugate to Sb,̄l1,··· ,̄lk -action, where l̄i is the orthogonal projection of li onto gλi . Hence they are orbit equivalent to each other. Hence we suffice to show that the statement of this proposition holds for the Sb,̄l1,··· ,̄lk - action. Let ξ i be a unit vector of l̄i. Take p ∈ AN . We can express as p = γξ1 (t1) · · · γξk (tk) for some t1, · · · , tk ∈ R, where γξi is the geodesic with γ̇ξi (0) = ξ i. Set l̂i := R{ 1cosh(|λi|ti) ξ i − 1 |λi| tanh(|λi|ti)Hλi } (i = 1, · · · , k). For simplicity, set ξiti := 1 cosh(|λi|ti) ξi − 1 |λi| tanh(|λi|ti)Hλi . According to the proof of Lemma 3.3, we have (γξ1 (t1) · · · γξk (tk))−1Sb,̄l1,··· ,̄lk (γξ1 (t1) · · · γξk (tk)) = Sb,̂l1,··· ,̂lk . Hence the orbit Sb,̄l1,··· ,̄lk · p is congruent to the orbit Sb,̂l1,··· ,̂lk · e. Denote by A the shape tensor of S b,̂l1,··· ,̂lk · e. According to Lemma 3.2, we have Tr Aξiti = −|λi| tanh(|λi|ti) × (dim gλı + 2dim g2λi ) (i = 1, · · · , k). Hence the orbit S b,̂l1,··· ,̂lk · e is minimal if and only if t1 = · · · = tk = 0, where we note that T ⊥e (Sb,̂l1,··· ,̂lk · e) = R{ξ 1 t1 , · · · , ξktk } because of b = {0}. That is, the orbit Sb,̄l1,··· ,̄lk · p is mini- mal if and only if p = e. Thus the orbit Sb,̄l1,··· ,̄lk -action posseses the only minimal orbit Sb,̄l1,··· ,̄lk · e. This completes the proof. q.e.d. 142 Naoyuki Koike CUBO 12, 2 (2010) From this proposition, we obtain the statement of Theorem C. At the end of this paper, we propose the following question. Question. Is any complex hyperpolar action without singular orbit on a symmetric space of non- compact type orbit equivalent to either the Sb-action (b ⊂ a) as in Example 1 or the Sb,l1,··· ,lk -action (li : a one dimensional subspace of gλi (i = 1, · · · , k), b ⊂ a ⊖ Span{Hλi | i = 1, · · · , k}) as in Example 2 ? Received: September 2008. 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