Classification of stationary compact

homogeneous special pseudo-Kähler manifolds of semisimple groups

D.V. Alekseevsky^{⋆}^{1}, V. Cortés^{⋆⋆}^{2}

⋆ Sophus Lie Center, Gen. Antonova 2 - 99, 117279 Moscow, Russia ⋆⋆ Mathematisches Institut der Universität Bonn, Beringstr. 1, 53115 Bonn, Germany

Abstract

The variation of the Hodge structure of a Calabi-Yau 3-fold induces a canonical Kähler metric on its Kuranishi moduli space, known as the Weil-Petersson metric. Similarly, special pseudo-Kähler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classification of homogeneous special pseudo-Kähler manifolds of semisimple group with compact stabilizer.

Key words: Homogeneous Kähler manifolds, Special Geometry, Calabi-Yau 3-folds/Mirror Symmetry

MSC: 53C30

Introduction

The Kuranishi moduli space of a Calabi-Yau threefold carries a canonical negative definite Kähler metric, known as the Weil-Petersson metric. More generally, any variation of principally polarized Hodge structure of weight 3 with h^{3,0} = 1 gives rise to a canonical negative definite Kähler metric g on the complex manifold M parametrizing the variation. The pair (M, g) is considered as formal moduli space. Generalizing the construction to the case of arbitrary polarizations one obtains pseudo-Kähler manifolds (M, g) of arbitrary signature. Such manifolds are called special pseudo-Kähler manifolds. The notion of special (pseudo-) Kähler geometry was introduced by physicists, see e.g. [10]. In physical terminology, special Kähler geometry is the geometry allowed for the coupling of N = 2 supergravity to vector multiplets in D = 4 spacetime dimensions. Homogeneous special Kähler manifolds were first considered by Cecotti [6], who related them to homogeneous quaternionic Kähler manifolds, cf. [1], [11], [12], [7], [3], [8] and [9]. The particularly interesting homogeneous special Kähler manifolds associated to homogeneous real affine cubic hypersurfaces were completely classified in [11] and [8]. In general, none of the two factors in the Levi-Malcev decomposition of the full isometry group of these homogeneous Kähler manifolds is trivial, see [12] and [3]. The reader who has learned about special Kähler manifolds from [13] may be surprised to read there exist homogeneous special Kähler manifolds. In fact it follows from [18] that any special Kähler manifold in the sense of [13] which has a transitive isometry group is necessarily flat. However this result does not apply to special Kähler manifolds in our sense, which in the terminology of [13] are not special Kähler manifolds but projective special Kähler manifolds.

1Supported by Erwin Schrödinger Institut (Vienna). e-mail: daleksee@esi.ac.at 2Supported by SFB 256 (Bonn University). e-mail: vicente@math.uni-bonn.de

In the following text, we give the classification of homogeneous stationary compact special pseudo-Kähler manifolds M of a real semisimple group G^{0}. (A homogeneous manifold M ≅ G^{0}/H^{0} is called stationary compact if its stationary or isotropy group H^{0} is compact.) This includes the classification of all homogeneous special Kähler manifolds of a real semisimple group. The precise definitions of special pseudo-Kähler manifolds, formal moduli spaces etc. are given in the first section. In the second section, we prove that if M is a stationary compact homogeneous special Kähler manifold of a real semisimple group G^{0} then there exists a symplectic module V of the complexified group G = (G^{0})ℂ ⊂ GL(V) such that the highest weight vector orbit C ⊂ V is a Lagrangian cone and M is embedded into its projectivization P (C) ⊂ P (V) as an open G^{0}-orbit. Then, in section 3, we classify all Lagrangian highest weight vector orbits of complex semisimple groups G. In the last section, we study the real forms of the symplectic G-modules V occurring in section 3, obtaining the solution to our classification problem.

In particular, we establish a natural 1-1 correspondence between simple complex Lie algebras 𝔩 different from 𝔠_{n} = 𝔰𝔭_{n}(ℂ) and homogeneous special Kähler manifolds G^{0}/H^{0} of a semisimple Lie group G^{0} which are formal moduli spaces. All such formal moduli spaces are Hermitian symmetric manifolds of nonpositive curvature and belong to the following list :

A) 𝔩 = 𝔰𝔩_{n+2}(ℂ), M = ℂH^{n−1} = SU_{1,n−1}/S(U_{1} · U_{n−1})

BD) 𝔩 = 𝔰𝔬_{n+4}(ℂ), M = (SL_{2}(ℝ)/SO_{2}) × (SO_{2,n−2}/SO_{2} · SO_{n−2})

G) 𝔩 = 𝔤_{2}(ℂ), M = ℂH^{1} = SL_{2}(ℝ)/SO_{2}

F) 𝔩 = 𝔣_{4}(ℂ), M = Sp_{3}(ℝ)/U_{3}

E6) 𝔩 = 𝔢_{6}(ℂ), M = SU_{3,3}/S(U_{3} · U_{3})

E7) 𝔩 = 𝔢_{7}(ℂ), M = SO_{12}^{∗}/U_{6}

E8) 𝔩 = 𝔢_{8}(ℂ), M = E_{7}^{(−25)}/E_{6} · SO_{2}.

In order to explain the correspondence we recall Wolf’s construction of the symmetric quaternionic Kähler manifolds of compact type [25]. Let 𝔩 be a simple complex Lie algebra, L^{0} the corresponding compact Lie group with trivial centre, Sp(1) ⊂ L^{0} the 3-dimensional regular subgroup associated to the highest root and K = Z_{L^{0}}(Sp(1)) its centralizer. The homogeneous manifold W (𝔩) := L^{0}/Sp(1)K is called the Wolf space associated to 𝔩. The Wolf spaces are precisely the symmetric quaternionic Kähler manifolds of positive scalar curvature. Next we consider the isotropy representation Sp(1)K → GL(V), V = T_{[e]}W, of the Wolf space W = W (𝔩) and denote by ρ = ρ_{𝔩} : K → GL(V) its restriction to the subgroup K. The Kmodule V carries a K-invariant structure of a complex symplectic vector space. In other words, to any complex simple Lie algebra 𝔩 we have associated a compact Lie group K and a complex representation ρ_{𝔩} : K → Spℂ(V) of symplectic type. If 𝔩 ≠ 𝔰𝔭_{n+1}(ℂ) then V admits a Lagrangian highest weight vector orbit C = ρ_{𝔩}(K)ℂv = ℂ^{∗}ρ_{𝔩}(K)v = ℝ^{∗}ρ_{𝔩}(K)v ⊂ V. (In the case 𝔩 = 𝔰𝔭_{n+1}(ℂ) we have that V = ℍ^{n} = ℂ^{2n} is the standard module of K = Sp(n), which has an open highest weight vector orbit C = ρ_{𝔩}(K)ℂv = V − {0}.) The projectivization P (C) of the complex cone C is a compact Hermitian symmetric manifold and the dual noncompact Hermitian symmetric manifold is a formal moduli space. Conversely, any homogeneous formal moduli space can be obtained by this construction.

It is natural to conjecture that there exist Calabi-Yau 3-folds whose Kuranishi moduli spaces are the formal moduli spaces from the above list. The recent remarkable results of Looijenga and Lunts [17] concerning the action of a semisimple Lie group on the cohomology of a Kähler manifold support this conjecture (due to Mirror Symmetry).

Finally, we point out a remarkable coincidence with the irreducible holonomy groups of torsion free connections of symplectic type, see [19] and [23]. Let 𝔩 ≠ 𝔰𝔩_{n}(ℂ) be a complex simple Lie algebra. Then the complex symplectic representation ρ : K → Spℂ(V) constructed above is irreducible and there exists a complex torsionfree connection with holonomy group ρ_{𝔩}(K)ℂ ⊂ Spℂ(V) ≅ Sp_{m}(ℂ) = Sp(ℂ^{2m}). Conversely, if Hol ⊂ Sp_{m}(ℂ), m > 2, is the holonomy group of a complex torsionfree connection and if Hol is irreducible then Hol = ρ_{𝔩}(K)ℂ for some complex simple Lie algebra 𝔩 ≠ 𝔰𝔩_{n}(ℂ). Notice that the generic holonomy group Hol = Sp_{m}(ℂ) corresponds to the simple Lie algebra l = 𝔰𝔭_{m+1}(ℂ).

We thank Simon Salamon for his useful comments on our paper.

1 Definitions

Let (V, ω) be a complex symplectic vector space and τ : V → V a compatible real structure on V, i.e. a ℂ-antilinear involution such that ω(τx, τy) = \overline{ω(x, y)} for all x, y ∈ V. In other words, (V, ω) is the complexification of the real symplectic vector space (V^{τ}, ω|V^{τ}); where V^{τ} = {x ∈ V |τx = x} is the fixed point set of τ. Choosing a real symplectic basis for V we can identify V = T^{∗}ℂ^{n}, ω = ∑_{i=1}^{n} dq^{i} ∧dp_{i} in canonical coordinates (q^{i}, p_{i}) and V^{τ} = T^{∗}ℝ^{n}. Note that the automorphism group of the triple (V, ω, τ) is

Aut(V, ω, τ) = {φ ∈ GL(V)|φ^{∗}ω = ω , φ^{∗}τ = τ} ≅ Sp_{n}(ℝ) ⊂ GL_{2n}(ℝ) .

Combining ω and τ we define a Hermitian form γ on V by:

γ(x, y) := \sqrt{−1}ω(x, τy) , x, y ∈ V .

In canonical coordinates it has the following expression:

γ = \sqrt{−1} ∑_{i=1}^{n}(dq^{i} ⊗ dp̄_{i} − dp_{i} ⊗ dq̄^{i}) = ∑_{i=1}^{n}(e_{−}^{i} ⊗ ē_{−}^{i} − e_{+}^{i} ⊗ ē_{+}^{i}) ,

where e_{±}^{i} := \frac{1}{\sqrt{2}}(dq^{i} ± \sqrt{−1}dp_{i}). In particular, γ has signature (n, n).

Definition 1 Let (V, ω, τ) be as above and ι : C ↪ V a holomorphic immersion of a connected complex manifold C. The pair (C, ι) is called

i) conic if λ · ι(C) ⊂ ι(C) for all λ ∈ ℂ^{∗},

ii) totally isotropic if ι^{∗}ω = 0,

iii) Lagrangian if (C, ι) is totally isotropic and 2 dim C = dim V,

iv) pseudo-Kähler if ι^{∗}γ is nondegenerate,

v) a special cone if conditions i)-iv) hold and γ(u, u) ≠ 0 for all u ∈ ι(C).

A connected complex submanifold C ⊂ V is called conic (resp. totally isotropic, Lagrangian etc.) if (C, ι) is conic (resp. totally isotropic, Lagrangian etc.), where ι is the canonical inclusion C ↪ V.

Note that n = dim V/2 is the maximal possible dimension of a totally isotropic complex submanifold C ⊂ V. To any conic complex submanifold C ⊂ V we associate its projectivization P (C) ⊂ P (V), which is a complex submanifold of the projective space P (V). If C is a special cone, a canonical pseudo-Kähler metric g on M = P (C) is defined as follows:

g_{πu}(dπ v, dπ v) := \frac{γ(v, v)}{γ(u, u)} − |\frac{γ(v, u)}{γ(u, u)}|^{2} , u ∈ C , v ∈ T_{u}C ,

where π : C → P (C) is the natural projection.

Definition 2 The pseudo-Kähler manifold (M = P (C), g) canonically associated to a special cone C ⊂ V is called a special pseudo-Kähler manifold. (M, g) is called special Kähler manifold if g or −g is positive definite. If g is negative definite, we call (M, g) a formal moduli space (of Calabi-Yau 3-folds).

Remark 1: Let X be a (general) Calabi-Yau 3-fold, i.e. a compact Kähler manifold with holonomy group SU_{3}, M_{X} its Kuranishi moduli space and g_{W P} the Weil-Petersson metric on M_{X}. Then (M_{X}, g_{W P}) is a formal moduli space in the sense of Def. 2, cf. [9].

Definition 3 A special pseudo-Kähler manifold (M, g) ⊂ P (V) is called homogeneous if there exists a closed subgroup G ⊂ Aut(V, ω, τ) ≅ Sp_{n}(ℝ) ⊂ GL_{2n}(ℝ) such that M = Gp is the G-orbit of some point p ∈ P (V).

Note that, under the assumptions of Def. 3, G acts transitively on the pseudo-Kähler manifold (M, g) by holomorphic isometries and we can identify M with the coset space G/G_{p}, G_{p} = {φ ∈ G|φp = p}.

Proposition 1 Let C ⊂ V be a special cone which is not contained in a linear Lagrangian subspace of V and G ⊂ Aut(V, ω, τ) a subgroup preserving C. Then P G = G/G ∩ {±id_{V} } acts effectively on M = P (C) and hence we have an inclusion

P G ⊂ Aut(M) = {φ : M → M|φ is a holomorphic isometry}.

Proof: We prove that if φ ∈ Aut(V, ω, τ) preserves C and acts as identity on M = P (C) then φ = ±id_{V}. First we show that φ is completely determined by its value at a single point u ∈ C. Since φ acts as identity on M we know that φ(u) = λu for some λ ∈ ℂ^{∗}. For the same reason the differential dφ_{u} : T_{u}C → T_{λu}C = T_{u}C induces the identity map on T_{p}M, p = ℂ^{∗}u, and hence dφ_{u}(v) = λv for all v ∈ T_{u}C ⊂ V. This shows that φ(v) = dφ_{u}(v) = λv for all v ∈ T_{u}C. The nondegeneracy of γ|T_{u}C is equivalent to V = T_{u}C ⊕ τT_{u}C. Now using φ^{∗}τ = τ we can conclude that φ ∈ Aut(V, ω, τ) is completely determined by its value φ(u) = λu at the point u ∈ C. In fact, φ preserves the decomposition V = T_{u}C ⊕τT_{u}C and acts as scalar multiplication by λ and λ̄ on the Lagrangian subspaces T_{u}C and τT_{u}C respectively. From the fact that C consists of eigenvectors of φ it follows that λ = λ̄ or C ⊂ T_{u}C or C ⊂ τT_{u}C. By assumption C is not contained in a Lagragian subspace. So λ = λ̄ and from φ^{∗}γ = γ it follows that λ = ±1. This shows that φ = ±id_{V} proving the proposition.

Corollary 1 Let M = P (C) be a homogeneous special Kähler manifold of the group G such that C is not contained in a linear Lagrangian subspace. Then P G ⊂ Aut(M) is a closed subgroup with compact stabilizer P G_{p}, p ∈ M.

Proof: Recall that for any Hermitian manifold M the subgroup Aut(M) ⊂ Isom(M) is closed with respect to the compact open topology. Since G acts transitively on M, we can express the closure of P G in Aut(M) as \overline{P G} = P G\overline{P G}_{p}. This shows that it is sufficient to prove the compactness of G_{p}. Consider the compact group

K_{p} = {φ ∈ Aut(V, ω, τ)|φT_{u}C ⊂ T_{u}C, φ(u) ∈ ℂu} ≅ U_{n−1} × U_{1} ,

where p = πu, π : C → M = P (C) the natural projection. The claim follows from the fact that G_{p} is a closed subgroup of K_{p} ⊂ Aut(V, ω, τ). This proves the corollary.

Our aim is to classify all homogeneous special pseudo-Kähler manifolds (M, g) of (real) semisimple group G with compact stabilizer G_{p}. By Corollary 1 this includes the classification of homogeneous special Kähler manifolds of semisimple group.

2 Compactification of homogeneous pseudo-Käh-ler manifolds

Let (M, g) be a homogeneous special pseudo-Kähler manifold of the real semisimple group G with compact stabilizer G_{p}, p ∈ M. We consider the complexified group Gℂ ⊂ GL(V), i.e. the (algebraic) linear group with Lie algebra 𝔤ℂ = 𝔤 ⊗ ℂ ⊂ 𝔤𝔩(V), where 𝔤 = Lie G ⊂ 𝔤𝔩(V^{τ}) ≅ 𝔤𝔩_{2n}(ℝ). The orbit \overline{M} = Gℂp of any point p ∈ M is a complex submanifold \overline{M} ⊂ P (V) containing M as an open subset. We will prove that \overline{M} is precisely the Zariski closure of M and hence a projective homogeneous algebraic variety.

More generally, let G^{0} be a connected real semisimple group acting effectively and transitively on a pseudo-Kähler manifold (M, g, J), J the complex structure, by holomorphic isometries. We assume that the stabilizer H^{0} of a point p ∈ M is compact.

Theorem 1 Under the conditions above any choice of base point p ∈ M defines a canonical open holomorphic embedding M ↪ G/P into a flag manifold of the connected complex semisimple group G = Aut_{e}(𝔤), 𝔤 = (𝔤^{0}) ⊗ ℂ, 𝔤^{0} = Lie G^{0}. Moreover, the stabilizer H^{0} = Z_{G^{0}}(h) is the centralizer of some element h ∈ 𝔥^{0} = Lie H^{0} and is connected; M ≅ G^{0}/H^{0} is simply connected.

Proof: The (pseudo-) Kähler form on M is a G^{0}-invariant symplectic structure. It follows from Kirillov, Kostant and Souriau’s description of homogeneous symplectic manifolds that the stabilizer H^{0} = Z_{G^{0}}(h) for some h ∈ 𝔥^{0} = Lie H^{0}. The maximal compact subgroup K^{0} of G^{0} is connected (see [14] Ch. VI, Theorem 2.1) and hence H^{0} = Z_{K^{0}}(h) is the centralizer of a torus in a compact connected Lie group. This shows that H^{0} is connected and M simply connected (see [14] Ch. VII Corollary 2.7). Moreover, since G^{0} acts effectively, we can identify G^{0} with its adjoint form Ad G^{0}.

Any parabolic subalgebra 𝔭 ⊂ 𝔤 = 𝔤^{0} ⊗ ℂ defines a flag manifold G/P, where G = Aut_{e}(𝔤) is the adjoint form of 𝔤 and P is its parabolic subgroup with Lie P = 𝔭. If P ∩ G^{0} = H^{0} then we have an open embedding M = G^{0}/H^{0} ↪ G/P.

The effective action of G^{0} on M gives rise to a monomorphism ρ^{0} of 𝔤^{0} into the Lie algebra 𝔛_{p}(M) of germs of vector fields at p ∈ M. Note that 𝔥^{0} = {x ∈ 𝔤^{0}|ρ^{0}(x)(p) = 0}. The infinitesimal action ρ^{0} : 𝔤^{0} → 𝔛_{p}(M) by (germs of) holomorphic Killing vector fields extends naturally to an infinitesimal action ρ : 𝔤 → 𝔛_{p}(M) by holomorphic vector fields: ρ(x + iy) = ρ^{0}(x) + Jρ^{0}(y), x, y ∈ 𝔤^{0}.

Lemma 1 ρ : 𝔤 → 𝔛_{p}(M) is injective.

Proof: ker ρ is an ideal and hence a semisimple direct summand of 𝔤. It follows that ker ρ is the complexification of the ideal 𝔤^{0} ∩ ker ρ ⊂ 𝔤^{0}. Now the lemma follows from 𝔤^{0} ∩ ker ρ = ker ρ^{0} = 0.

The next proposition finishes the proof of Theorem 1.

Proposition 2 The infinitesimal stabilizer 𝔤_{p} = {x ∈ 𝔤|ρ(x)(p) = 0} is a parabolic subalgebra 𝔤_{p} = 𝔭 ⊂ 𝔤 with maximally reductive subalgebra 𝔥 = 𝔷_{𝔤}(h). The corresponding parabolic subgroup P ⊂ G satisfies P ∩ G^{0} = H^{0}.

Proof: The Lie group H^{0} = Z_{G^{0}}(h) contains a compact Cartan subgroup A^{0} of G^{0} such that h ∈ 𝔞^{0} = Lie A^{0} ⊂ 𝔥^{0}. Let R denote the root system of 𝔤 with respect to the Cartan subalgebra 𝔞 = 𝔞^{0} + i𝔞^{0} ⊂ 𝔤. Note that 𝔞^{0} = {x ∈ 𝔞|α(x) ∈ iℝ for all α ∈ R}. Since 𝔞 ⊂ 𝔤_{p}, we have

𝔤_{p} = 𝔞 + ∑_{α∈R_{p}} 𝔤_{α}

for some root subsystem R_{p} ⊂ R. More precisely, since 𝔤_{p} ∩ 𝔤^{0} = 𝔥_{0}, we have

𝔤_{p} = 𝔥 + 𝔪 ,

𝔥 = 𝔥^{0} ⊗ ℂ = 𝔷_{𝔤}(h) = 𝔞 + ∑_{α∈R_{0}} 𝔤_{α} , 𝔪 = ∑_{α∈R_{m}} 𝔤_{α} ,

where R_{0} = {α ∈ R|α(h) = 0} ⊂ R_{p} and R_{m} = R_{p} − R_{0}.

Lemma 2 R_{p} = R_{0} ∪^{·} R_{m} is the decomposition of the root system R_{p} into its maximally symmetric root subsystem R_{0} and the asymmetric root subsystem R_{m}; 𝔤_{p} = 𝔥 + 𝔪 is the decomposition of the Lie algebra 𝔤_{p} into the maximally reductive subalgebra 𝔥 and the nilpotent ideal 𝔪.

Proof: Let τ denote the ℂ-antilinear involutive automorphism of 𝔤 with fixed point set 𝔤^{τ} = 𝔤^{0}. The Cartan subalgebra 𝔞 = 𝔞^{0} + i𝔞^{0}, 𝔞^{0} = 𝔞^{τ}, is τ-invariant and a straightforward computation shows that τ𝔤_{α} = 𝔤_{α^{τ}}, α ∈ R, where α^{τ}(a) = \overline{α(τ(a))}, a ∈ 𝔞. Using the fact that α is purely imaginary on 𝔞^{0} we conclude that α^{τ} = −α and hence τ𝔤_{α} = 𝔤_{−α}. Let α ∈ R_{p} and assume that also −α ∈ R_{p}, i.e. 𝔤_{α} + 𝔤_{−α} ⊂ 𝔭. If 𝔤_{α} = ℂe_{α}, then it follows that

0 ≠ ℝ(e_{α} + τe_{α}) = (𝔤_{α} + 𝔤_{−α})^{τ} ⊂ 𝔤_{p} ∩ 𝔤^{0} = 𝔥^{0}

and hence 𝔤_{α} + 𝔤_{−α} ⊂ 𝔥. This implies that α ∈ R_{0}. We have proved that R_{0} ⊂ R_{p} is the (unique) maximally symmetric root subsystem and that the complement R_{m} = R_{p} − R_{0} is asymmetric. From the Levi-Malcev decomposition it follows that there exists a nilpotent ideal 𝔫 ⊂ 𝔤_{p} complementary to 𝔥. Since 𝔥 contains the Cartan subalgebra 𝔞, the complement 𝔪 = ∑_{α∈R_{m}} 𝔤_{α} of 𝔥 in 𝔤_{p} is the only 𝔥-invariant complement and hence 𝔪 = 𝔫. This shows, in particular, that 𝔪 is a subalgebra and hence R_{m} is a root system finishing the proof of the lemma.

We continue the proof of Proposition 2. The solvable Lie algebra 𝔞 + 𝔪 ⊂ 𝔤 is contained in some Borel subalgebra and so R_{m} ⊂ R^{+} for some system of positive roots R^{+} ⊂ R. Put R_{0}^{+} := R_{0} ∩ R^{+}. Then R_{0}^{+} is a system of positive roots for the root system R_{0}.

Lemma 3 R^{+} = R_{0}^{+} ∪^{·} R_{m}.

Proof: Since R_{0}^{+} ∪^{·} R_{m} ⊂ R^{+} it is sufficient to check that the cardinalities coincide, i.e. |R| − |R_{0}| = 2|R_{m}|. We have the natural isomorphisms

T_{p}M ≅ 𝔤^{0}/𝔥^{0} ≅ 𝔤/𝔤_{p} , (T_{p}M) ⊗ ℂ ≅ 𝔤/𝔥

and hence

|R| − |R_{0}| = dimℂ 𝔤/𝔥 = dimℝ 𝔤/𝔤_{p} = 2(|R| − |R_{0}|) − 2|R_{m}| .

This proves Lemma 3 and shows that 𝔤_{p} = 𝔞 + ∑_{α∈R_{0}∪R_{m}} 𝔤_{α} ⊂ 𝔤 is a parabolic subalgebra. Let P ⊂ G be the parabolic subgroup with Lie algebra 𝔭 = 𝔤_{p}. The intersection P ∩G^{0} is a Lie group with Lie algebra 𝔭∩𝔤^{0} = 𝔥^{0}. Therefore, H^{0} ⊂ P ∩G^{0} is the identity component of P ∩ G^{0} and

P ∩ G^{0} = N_{P} (H^{0}) ∩ G^{0} = N_{P} (Z_{G}(h)) ∩ G^{0} = Z_{G}(h) ∩ G^{0} = H^{0} .

We have used that P = Z_{G}(h)⋉N is the semidirect product of the maximally reductive subgroup Z_{G}(h) with the nilpotent radical N and hence Z_{G}(h) equals its own normalizer in P. This completes the proof of Proposition 2.

Now we specialize Theorem 1:

Theorem 2 Let (M, g) ⊂ P (V) be a homogeneous special pseudo-Kähler manifold of the connected real semisimple group G^{0} with compact stabilizer H^{0} = G_{p}^{0}, p ∈ M. Then the stabilizer H^{0} = Z_{G^{0}}(h) is the centralizer of some element h ∈ 𝔥^{0} = Lie H^{0} and is connected; M = G^{0}/H^{0} is simply connected. The orbit \overline{M} = Gp = GM under the complexified linear group G = (G^{0})ℂ ⊂ GL(V) is a flag manifold and the stabilizer P = G_{p} ⊂ G is a parabolic subgroup with maximally reductive subgroup H = (H^{0})ℂ. Finally, \overline{M} is precisely the Zariski closure of M in P (V). In particular, we have the canonical open inclusion M ⊂ \overline{M}.

Proof: We denote by J the G^{0}-invariant complex structure on M induced by the inclusion M ⊂ P (V). The first part of Theorem 2 follows immediately from Theorem 1. Moreover, since G^{0} acts effectively on M, the Lie group G^{0} is isomorphic to its adjoint group Ad G^{0}. We will write G = (G^{0})ℂ. Now Proposition 2 shows that the stabilizer P = G_{p} is a parabolic subgroup (with maximally reductive subgroup H = (H^{0})ℂ) and therefore Gp ≅ G/P a flag manifold. In particular, \overline{M} = Gp ⊂ P (V) is a compact complex submanifold and hence Zariski closed (by Chow’s Theorem). \overline{M} being connected and dim M = dim \overline{M} this implies that M ⊂ \overline{M} is open (in the standard Hausdorff topology) and \overline{M} its Zariski closure. This proves Theorem 2.

3 Classification of Lagrangian cones over flag manifolds

If M ⊂ P (V) is a homogeneous special pseudo-Kähler manifold as in Theorem 2 then, by virtue of this theorem, its Zariski closure \overline{M} ⊂ P (V) is an orbit of the complex semisimple group G ⊂ Sp(V). The stabilizer of any point p ∈ \overline{M} is a parabolic subgroup of G and hence p is a highest weight direction of the G-module V, i.e. p = ℂ^{∗}v for some highest weight vector v ∈ V (with respect to some choice of Cartan subalgebra 𝔞 ⊂ 𝔤 and system of positive roots R^{+}). The cone C\overline{M} = \{v ∈ V |πv ∈ \overline{M}\} (π : V → P (V) the canonical projection) over the Zariski closure \overline{M} is Lagrangian, since the cone C_{M} over M is Lagrangian. Moreover, G acts transitively on C\overline{M}. In fact, C\overline{M} consists of highest weight vectors and hence for all v ∈ C\overline{M} we can find an algebraic torus in G acting transitively on the direction ℂ^{∗}v spanned by v. The claim follows now from the fact that G acts transitively on the set of directions \overline{M}. In this section we will classify all symplectic modules V of complex semisimple groups G which contain a highest weight vector v whose orbit Gv ⊂ V is Lagrangian. Propositions 3 and 5 reduce this problem to the case of irreducible modules of simple groups.

Proposition 3 Let (V, ω) be a symplectic G-module of a connected complex semisimple group G and C = Gv a Lagrangian G-orbit of a highest weight vector v ∈ V. Then either

i) V = U ⊕ U^{∗} is the direct sum of an irreducible G-module U and its dual U^{∗} and we have U = C ∪ {0} or

ii) V is irreducible.

In the first case 𝔤 = Lie G = 𝔰𝔩_{n}(ℂ) or 𝔰𝔭_{n}(ℂ) and U is the standard 𝔤-module, i.e. U = ℂ^{n} if 𝔤 = 𝔰𝔩_{n}(ℂ) and U = ℂ^{2n} if 𝔤 = 𝔰𝔭_{n}(ℂ).

Proof: The cone C is contained in a unique irreducible submodule U ⊂ V. By Schur’s Lemma the restriction ω|U of the symplectic form ω to the irreducible submodule U ⊂ V is either zero or nondegenerate. In the first case U is totally isotropic and contains the Lagrangian cone C as an open orbit. On the other hand, it is known, see [26], that the orbit Gp of the highest weight direction p = ℂ^{∗}v is closed in P (V). Thus U = C ∪ {0} is a Lagrangian subspace . Let U′ be a G-invariant complement to U in V. The symplectic structure ω defines an isomorphism U′ ≅ U^{∗} of G-modules. From the fact that G is a complex semisimple Lie group acting transitively on the projective space P (U) by projective transformations it follows that 𝔤 = 𝔰𝔩_{n}(ℂ) or 𝔤 = 𝔰𝔭_{n}(ℂ) and U is the standard 𝔤-module (up to an automorphism of 𝔤), see [20]. This completes the discussion of the first case. In the second case C is Lagrangian in (U, ω|U) and hence V = U. In particular, V is irreducible. This proves the proposition.

Recall that any connected and simply connected complex semisimple Lie group G is the direct product G = G_{1} × ⋯ × G_{k} of simple groups G_{i}. Any irreducible G-module V is the tensor product V = V_{1} ⊗ ⋯ ⊗ V_{k} of irreducible G_{i}-modules V_{i}.

Lemma 4 The irreducible G-module V = V_{1} ⊗ ⋯ ⊗ V_{k} is symplectic if and only if each of the G_{i}-modules V_{i} is self-dual and the number of symplectic factors V_{i} is odd.

Proof: We choose a Cartan subalgebra 𝔞_{i} ⊂ 𝔤_{i} = Lie G_{i} and a system of positive 𝔞_{i}- roots R_{i}^{+} for each of the simple summands 𝔤_{i} of 𝔤 = Lie G. Let Λ^{i} (respectively M^{i}) be the highest (respectively lowest) weight of the G_{i}-module V_{i} with respect to R_{i}^{+}. Then 𝔞 = ⊕_{i=1}^{k}𝔞_{i} ⊂ 𝔤 is a Cartan subalgebra and R^{+} = ∪_{i}R_{i}^{+} a system of positive 𝔞-roots for 𝔤, where we have identified 𝔞_{i}^{∗} = {α ∈ 𝔞^{∗}|α(a) = 0 for all a ∈ 𝔞_{j}, j ≠ i}. The G-module V has highest weight Λ = ∑_{i=1}^{k} Λ^{i} and lowest weight M = ∑_{i=1}^{k} M^{i} with respect to R^{+}. V is self-dual if and only if Λ + M = 0, see [21]. Restricting Λ + M to 𝔞_{i} we see that V is self-dual if and only if Λ^{i} + M^{i} = 0 for all V_{i}, i.e. if and only if all V_{i} are self-dual. Let b_{i} ≠ 0 be a G_{i}-invariant symmetric or skew symmetric bilinear form on V_{i}. Then b = b_{1} ⊗ ⋯ ⊗ b_{k} is a G-invariant bilinear form on V. It is skew symmetric if and only if the number of skew symmetric factors b_{i} is odd. This proves the lemma.

The next proposition shows, in particular, that the highest weight vector orbit C of an irreducible symplectic module V of a simple group is automatically Lagrangian provided that dim V = 2 dim C.

Proposition 4 Let (V, ω) be an irreducible symplectic module of a complex semisimple group G ⊂ Sp(V) and C ⊂ V its highest weight vector orbit. Then C is a totally isotropic cone, unless V = V_{1} ⊗ ⋯ ⊗ V_{k} is the tensor product of an odd number k of symplectic vector spaces (V_{i}, ω_{i}), ω = ω_{1} ⊗ ⋯ ⊗ ω_{k}, and G = Sp(V_{1}) ⊗ ⋯ ⊗ Sp(V_{k}) is the image of the group Sp(V_{1}) × ⋯ × Sp(V_{k}) under the tensor product representation on V. If (V, g) is an orthogonal module of a complex semisimple group G then any highest weight vector orbit is contained in the null cone of (V, g).

Proof: The stabilizer of a point p ∈ P (C) being parabolic, there exists a Cartan subalgebra 𝔞 ⊂ 𝔤 = Lie G and a system of positive roots R^{+} (of 𝔤 with respect to 𝔞) such that ∑_{α∈R^{+}} 𝔤_{α}v = 0 and av = Λ(a)v for all a ∈ 𝔞, where Λ is the highest weight of the irreducible 𝔤-module V. In particular, we have

T_{v}C = 𝔤v = ℂv + ∑_{α∈R^{−}} 𝔤_{α}v , R^{−} = −R^{+} .

It is sufficient to show that the tangent space T_{v}C is totally isotropic, unless 𝔤 = sp(V_{1}) ⊕ ⋯ ⊕ sp(V_{k}). For a ∈ 𝔞, 𝔤_{α} = ℂe_{α}, α ∈ R^{−}, using the G-invariance of ω we compute

Λ(a)ω(v, e_{α}v) = ω(av, e_{α}v) = −ω(v, [a, e_{α}]v + e_{α}av) = −(α(a) + Λ(a))ω(v, e_{α}v)

and hence ω(v, e_{α}v) = 0, unless −α = 2Λ.

Assume for the moment that −α ≠ 2Λ. By our computation, this implies that ω(v, 𝔤v) = 0. In order to prove that T_{v}C = 𝔤v is totally isotropic it only remains to show that ω(e_{α}v, e_{β}v) = 0 for all α, β ∈ R^{−}. Using the fact that ω(v, 𝔤v) = 0, we compute

ω(e_{α}v, e_{β}v) = −ω(v, e_{α}e_{β}v) = −ω(v, [e_{α}, e_{β}]v) − ω(v, e_{β}e_{α}v)
= 0 + ω(e_{β}v, e_{α}v) = −ω(e_{α}v, e_{β}v) ,

which proves that ω(e_{α}v, e_{β}v) = 0.

To discuss the case −α = 2Λ let 𝔤 = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{k} be the decomposition of 𝔤 into simple ideals and 𝔞 = 𝔞_{i} ⊕ ⋯ ⊕ 𝔞_{k} the corresponding decomposition of the Cartan subalgebra 𝔞 ⊂ 𝔤 into Cartan subalgebras 𝔞_{i} ⊂ 𝔤_{i}. Since G acts faithfully on V we have Λ|𝔞_{i} ≠ 0. If −α = 2Λ then −α|𝔞_{i} ≠ 0 is a dominant root of the simple Lie algebra 𝔤_{i}, i.e. ⟨−α|𝔞_{i}, β|𝔞_{i}⟩ ≥ 0 for all β ∈ R^{+}. This implies that −α|𝔞_{i} = δ(𝔤_{i}) is the highest root of 𝔤_{i}. Moreover, δ(𝔤_{i})/2 = Λ|𝔞_{i} is an element of the weight lattice, i.e. is an integral linear combination of fundamental weights. This does only occur if 𝔤_{i} is the symplectic Lie algebra and Λ|𝔞_{i} is the first fundamental weight, which is the highest weight of the standard representation. This proves the proposition for the symplectic case. In the orthogonal case it is sufficient to show that the highest weight vector v is a null vector. This follows from the computation:

Λ(a)g(v, v) = g(av, v) = −g(v, av) = −Λ(a)g(v, v) , a ∈ 𝔞 .

This finishes the proof of Proposition 4.

Proposition 5 Let (V, ω) be an irreducible symplectic module of a connected complex semisimple group G ⊂ Sp(V) and C ⊂ V its highest weight vector orbit. Assume that C is Lagrangian. Then either G is simple or V = ℂ^{2} ⊗ ℂ^{n} and G = SL_{2}(ℂ) ⊗ G′, where ℂ^{2} is the standard module of SL_{2}(ℂ) and ℂ^{n} is the standard module of SO_{n}(ℂ) considered as module of a simple subgroup G′ ⊂ SO_{n}(ℂ) acting transitively on the cone C′ ⊂ ℂ^{n} of nonzero null vectors. More precisely, G′ = SO_{n}(ℂ) or G′ = G_{2}(ℂ) ⊂ SO_{7}(ℂ). Moreover, P (C) = P (ℂ^{2}) × Q ⊂ P (ℂ^{2} ⊗ ℂ^{n}) = P (V) is the product of the projective line P (ℂ^{2}) with the quadric Q = P (C′) ⊂ P (ℂ^{n}).

Proof: If G is not simple, by Lemma 4, we can write V = V_{1}⊗V′ where (V_{1}, ω_{1}) is an irreducible symplectic module of a simple group G_{1}, (V′, g′) an irreducible orthogonal module of a semisimple group G′ and G = G_{1} ⊗ G′. Let v ∈ C, then v = v_{1} ⊗ v′, where v_{1} (respectively v′) is a highest weight vector of the G_{1}-module V_{1} (respectively of the G′-module V′). The tangent space

T_{v}C = 𝔤v = 𝔤_{1}v_{1} ⊗ v′ + v_{1} ⊗ 𝔤′v′

(𝔤 = Lie G, 𝔤_{1} = Lie G_{1}, 𝔤′ = Lie G′) is Lagrangian. By Proposition 4, v′ ∈ V′ is a null vector for the metric g′ and hence 𝔤′v′ is tangent to the null cone C′ ∋ v′. This implies that g(v′, 𝔤′v′) = 0. Now a straightforward calculation shows that V_{1} ⊗ v′ + v_{1} ⊗ T_{v′}C′ is totally isotropic for the symplectic form ω = ω_{1} ⊗ g′. The subspace T_{v}C ⊂ V_{1} ⊗ v′ + v_{1} ⊗ T_{v′}C′ being Lagrangian, it follows that 𝔤_{1}v_{1} = V_{1}, 𝔤′v′ = T_{v′}C′ and 2(dim V_{1} + dim V′ − 2) = dim V_{1} · dim V′. Since G is semisimple, this is only possible if dim V_{1} = 2 and hence G_{1} = Sp_{1}(ℂ) = SL_{2}(ℂ). The remaining statements follow from the classification of transitive projective actions of complex semisimple groups on the quadric Q = P (C′) ⊂ P (ℂ^{n}), see [20]. This establishes the proposition.

Now we associate with a complex simple Lie algebra 𝔩 a symplectic G-module V such that the orbit of the highest weight vector is a Lagrangian cone.

Let μ be the highest root of 𝔩 (with respect to some Cartan subalgebra and choice of positive root system). Denote by

𝔤(μ) = span{h_{μ} = [e_{μ}, e_{−μ}], e_{μ}, e_{−μ}} ≅ 𝔰𝔩_{2}(ℂ)

the corresponding 3-dimensional subalgebra, where the root vectors e_{±μ} are normalized by the condition

⟨e_{μ}, e_{−μ}⟩ = \frac{2}{⟨μ, μ⟩},

(and ⟨·, ·⟩ is the Killing form). One can check easily that the adjoint operator ad(h_{μ}) has eigenvalues 0, ±1, ±2 and the corresponding eigenspace decomposition

𝔩 = 𝔩_{−2} + 𝔩_{−1} + 𝔩_{0} + 𝔩_{1} + 𝔩_{2}

defines a grading on the Lie algebra 𝔩. Moreover, 𝔩_{±2} = ℂe_{±μ} and 𝔩_{0} = ℂh_{μ} + 𝔩′_{0} where 𝔩′_{0} is the centralizer of 𝔤(μ). Denote by L^{τ} the compact Lie group without centre associated with 𝔩 and by

L_{0}^{′τ},  L_{0}^{τ} = T^{1} · L_{0}^{′τ},  L_{ev}^{τ} = Sp_{1} · L_{0}^{′τ}

its compact subgroups associated with the subalgebras

𝔩′_{0}, 𝔩_{0} and 𝔩_{ev} = 𝔩_{−2} + 𝔩_{0} + 𝔩_{2} respectively.

The following result is known.

Theorem 3 1. M = L^{τ}/L_{ev}^{τ} is a compact symmetric quaternionic Kähler manifold (called Wolf space). Moreover, the Wolf spaces exhaust all homogeneous quaternionic Kähler manifolds with positive scalar curvature.

2.

Z = L^{τ}/L_{0}^{τ} ⟶ M = L^{τ}/L_{ev}^{τ}

is the twistor fibration of the quaternionic Kähler manifold M. In particular, the manifold Z has the structure of a complex contact homogeneous manifold and admits an invariant Kähler-Einstein metric, which is unique up to scaling. Moreover, such manifolds Z exhaust all compact homogeneous complex contact manifolds.

Denote by 𝔤 = [𝔩_{0}, 𝔩_{0}] ⊂ 𝔩′_{0} the Levi subalgebra of 𝔩_{0} and by G the corresponding complex Lie subgroup of Aut(𝔩). Note that 𝔤 = 𝔩′_{0}, unless 𝔩 = 𝔰𝔩_{n+2}(ℂ); 𝔤 = 𝔰𝔩_{n}(ℂ) ⊂ 𝔩′_{0} = 𝔤𝔩_{n}(ℂ) if 𝔩 = 𝔰𝔩_{n+2}(ℂ). The adjoint representation of G in the space V = 𝔩_{−1} is symplectic, with the symplectic form defined by the Lie bracket [·, ·] : ∧^{2}𝔩_{−1} → 𝔩_{−2}. We will call V the standard symplectic G-module associated with the simple Lie algebra 𝔩.

In the following table we indicate for any complex simple Lie algebra 𝔩 the associated Wolf space N = L^{τ}/L_{ev}^{τ} , complex Lie group G , symplectic G-module V = 𝔩_{−1} and the semisimple part H′ of the stabilizer of a highest weight vector of V. We denote by (V^{(d)} = V (Λ), 𝔤) the irreducible 𝔤-module with highest weight Λ = ∑ Λ_{i}π_{i}, π_{i} the fundamental weights, and the superindex d indicates the dimension.

Table 1

\begin{matrix}𝔩 & N & G & V & H′ \\ 𝔰𝔩_{n+2}(ℂ) & SU_{n+2}/S(U_{2} · U_{n}) & SL_{n}(ℂ) & V^{(2n)} = ℂ^{n} ⊕ ℂ^{n∗} & SL_{n−1}(ℂ) \\ 𝔰𝔬_{n+4}(ℂ) & SO_{n+4}/SO_{4} · SO_{n} & SL_{2}(ℂ) · SO_{n}(ℂ) & V^{(2n)} = ℂ^{2} ⊗ ℂ^{n} & SO_{n−2}(ℂ) \\ 𝔰𝔭_{n+1}(ℂ) & Sp_{n+1}/Sp_{1} · Sp_{n} & Sp_{n}(ℂ) & V^{(2n)} = ℂ^{2n} & Sp_{n−1}(ℂ) \\ 𝔤_{2}(ℂ) & G_{2}/SU_{2} · SU_{2} & SL_{2}(ℂ) & V^{(4)} = V (3π_{1}) & \{e\} \\ 𝔣_{4}(ℂ) & F_{4}/Sp_{1} · Sp_{3} & Sp_{3}(ℂ) & V^{(14)} = V (π_{3}) & SL_{3}(ℂ) \\ 𝔢_{6}(ℂ) & E_{6}/Sp_{1} · SU_{6} & SL_{6}(ℂ) & V^{(20)} = V (π_{3}) = ∧^{3}ℂ^{6} & SL_{3}(ℂ) · SL_{3}(ℂ) \\ 𝔢_{7}(ℂ) & E_{7}/Sp_{1} · Spin_{12} & Spin_{12}(ℂ) & V^{(32)} = V (π_{5}) & SL_{6}(ℂ) \\ 𝔢_{8}(ℂ) & E_{8}/Sp_{1} · E_{7} & E_{7}(ℂ) & V^{(56)} = V (π_{1}) & E_{6}(ℂ) \end{matrix}

The following theorem shows that with a few number of exceptions the standard symplectic modules exhaust all G-modules with Lagrangian orbit of the highest weight vector v.

Theorem 4 1. Let V be the standard G-module associated with a simple Lie algebra 𝔩 ≠ 𝔰𝔭_{𝔫}(ℂ). Then the orbit C = Gv of the highest weight vector v is a Lagrangian cone.

2. Let V be a symplectic G-module of a semisimple Lie group G such that the orbit of a highest weight vector v is a Lagrangian cone. Assume that V is not a standard symplectic module. Then the structure of G-module on V can be extended to the structure of a standard symplectic G̃-module V, where G̃ is a semisimple Lie group which contains G , such that the G-orbit and the G̃-orbit of the highest weight vector v coincide. More precisely, this happens only in the following three cases:

i) G = Sp_{n}(ℂ) ⊂ G̃ = SL_{2n}(ℂ),  V = ℂ^{2n} ⊕ ℂ^{2n}^{∗},  H′ = Sp_{n−1}(ℂ)

ii) G = Spin_{11}(ℂ) ⊂ G̃ = Spin_{12}(ℂ), V = V^{(32)} = V (π_{5}) (the semispinor module of Spin_{12}(ℂ), which is the spinor module of Spin_{11}(ℂ)), H′ = SL_{5}(ℂ)

iii) G = SL_{2}(ℂ) × G_{2}(ℂ) ⊂ G̃ = SL_{2}(ℂ) × SO_{7}(ℂ), V = V^{(14)} = V (π_{1}) ⊗ V (π_{1}) = ℂ^{2} ⊗ ℂ^{7}, H′ = SL_{2}(ℂ).

(Here H′ is the semisimple part of the stabilizer of v in G).

Proof: First we prove 1). In the case 𝔩 = 𝔰𝔩_{n+2}(ℂ) the highest weight vector orbits ℂ^{n} − {0} and (ℂ^{n})^{∗} − {0} ⊂ ℂ^{n} ⊕ (ℂ^{n})^{∗} = V are Lagrangian orbits of the group G = SL_{n}(ℂ). In the remaining cases V is irreducible and by Proposition 4 and Lemma 5 the G-orbit C of the highest weight vector v is Lagrangian if and only if

2 dim C = dim V.

The dimension of C can be calculated by the formula

dim C = \frac{1}{2}(dim G − rk G − dim H′ + rk H′) + 1 ,

where H′ is the semisimple part of the stabilizer of v. Indeed, dim P (C) = |R^{+}|−|R_{0}^{+}|, where R^{+} (respectively R_{0}^{+}) is a system of positive roots for 𝔤 (respectively 𝔥′). Since |R^{+}| = (dim G−rk G)/2 and |R_{0}^{+}| = (dim H′ −rk H′)/2 we get this formula. Using it and the table, one can check immediately that for each standard symplectic G-module the orbit of the highest weight vector is Lagrangian.

Now we prove 2). By Proposition 3 and Proposition 5 it is sufficient to consider the case when V is an irreducible symplectic module of a simple Lie group G. The proof is based on the following lemma and proposition.

Lemma 5 Let V be a symplectic module of a complex semisimple group G such that the highest weight vector orbit C ⊂ V is Lagrangian. Then

dim V ≤ dim G − rk G + 2 .

Proof: Since P (C) ⊂ P (V) is a flag manifold of the group G, we have dim P (C) ≤ (dim G − rk G)/2 and hence

dim V = 2 dim C = 2(dim P (C) + 1) ≤ dim G − rk G + 2 ,

proving the lemma.

Proposition 6 The irreducible symplectic G-modules of complex simple (connected) Lie group G which satisfy the condition

dim V ≤ dim G − rkG + 2

are listed below. If the module is standard symplectic, we indicate also the corresponding simple Lie algebra 𝔩.

A) G = SL_{2}(ℂ), V^{(4)} = V (3π_{1}) = ∨^{3}ℂ^{2}, 𝔩 = 𝔤_{2}(ℂ), G = SL_{6}(ℂ), V^{(20)} = V (π_{3}) = ∧^{3}ℂ^{6}, 𝔩 = 𝔢_{6}(ℂ),

B) G = Spin_{11}(ℂ), V^{(32)} = V (π_{5}) = spinor module, G = Spin_{13}(ℂ), V^{(64)} = V (π_{6}) = spinor module,

C) G = Sp_{l}(ℂ), V^{(2l)} = V (π_{1}) = ℂ^{2l}, 𝔩 = 𝔰𝔭_{l+1}(ℂ), G = Sp_{3}(ℂ), V^{(14)} = V (π_{3}), 𝔩 = 𝔣_{4}(ℂ),

D) G = Spin_{12}(ℂ), V^{(32)} = V (π_{5} or π_{6}) = semi spinor modules, 𝔩 = 𝔢_{7}(ℂ),

E) G = E_{7}(ℂ), V^{(56)} = V (π_{1}), 𝔩 = 𝔢_{8}(ℂ).

Proof: The proof is a straightforward (but long) exercise; the needed representation theory and useful tables can be found in [21] and [22].

Now to finish the proof of the theorem, it only remains to investigate the two modules in the case B, which are the only nonstandard symplectic modules from the list.

For the spinor module V^{(64)} of Spin_{13}(ℂ) the semisimple part of the stabilizer of the highest weight vector v is SL_{6}(ℂ) . This can be easily derived from the fact that the Dynkin diagram of the root system R_{0} of 𝔥′ = Lie H′ is obtained from the Dynkin diagram of the Lie algebra 𝔤 of G by deleting all simple roots α_{i} such that ⟨α_{i}, Λ⟩ ≠ 0. Hence,

dim C = \frac{1}{2}(dim G − rk G − dim H′ + rk H′) + 1 =

= \frac{1}{2}(6 · 13 − 6 − 35 + 5) + 1 = 22 < \frac{1}{2} dim V = 32

and the orbit C is not Lagrangian in this case.

The restriction of the semispinor module of Spin_{12}(ℂ) to a module of Spin_{11}(ℂ) ⊂ Spin_{12}(ℂ) gives the spinor module of Spin_{11}(ℂ). This implies that the highest weight vector orbit of the group Spin_{11}(ℂ) is contained in the highest weight vector orbit of Spin_{12}(ℂ). A short dimension count as above shows that the two orbits coincide and, hence, are Lagrangian cones. This establishes the proposition. Remark 2: For any of the complex contact manifolds Z listed in Theorem 3 and Table 1 with the exception of Z = SU_{n+2}/U_{1} · U_{n} = P (T^{∗}ℂP^{n+1}) the second Betti number is 1. Hence, using the exact sequence 0 → ℤ → O → O^{∗} → 0, one can easily show that the Picard group P ic(Z) = H^{1}(Z, O^{∗}) ≅ ℤ. Let L be the ample generator of P ic(Z). The degree k of a rational curve ℂP^{1} ↪ Z is defined by

L|ℂP^{1} ≅ O(k)

and we can consider the cone C_{min} ⊂ T_{o}Z of nonzero tangent vectors to rational curves of minimal degree k = 1 through the point o ∈ Z. If Z = ℂP^{2n+1} then C_{min} = T_{o}Z −{0}. In the remaining cases C_{min} is contained in the contact hyperplane W ⊂ T_{o}Z and is precisely the highest weight vector orbit of the G-module W, where G is the complexification of the Levi subgroup of the isotropy group L_{0}^{τ} of Z = L^{τ}/L_{0}^{τ}, cf. [16].

4 Real forms

By Theorem 2 any homogeneous special pseudo-Kähler manifold of a real semisimple group G^{0} with compact stabilizer is an open orbit of G^{0} on a flag manifold P (C), where C ⊂ V is a Lagrangian highest weight vector orbit in a symplectic G-module (V, ω), Lie G = (Lie G^{0}) ⊗ ℂ. Moreover, V carries the G^{0}-invariant real structure τ and Hermitian form γ = \sqrt{−1}ω(·, τ·). In Theorem 4 we have enumerated all symplectic modules V of complex semisimple group G for which a highest weight vector orbit is Lagrangian. Now we list all real forms of those modules.

Proposition 7 For any symplectic module (V, ω) of a complex semisimple Lie algebra 𝔤 with Lagrangian highest weight vector orbit the list of real forms 𝔤^{0} of 𝔤 for which there exists a 𝔤^{0}-invariant real structure on V is given below (up to an automorphism of 𝔤). For the simple Lie algebras 𝔤 we have:

A) ∨^{3}ℂ^{2}: 𝔰𝔩_{2}(ℝ).

∧^{3}ℂ^{6}: 𝔰𝔩_{6}(ℝ), 𝔰𝔲_{1,5} and 𝔰𝔲_{3,3}.

ℂ^{n} ⊕ (ℂ^{n})^{∗}: 𝔰𝔩_{n}(ℝ) and 𝔰𝔲_{k,l}, k + l = n.

B) Spinor module: 𝔰𝔬_{1,10} and 𝔰𝔬_{2,9}.

C) V (π_{3}): 𝔰𝔭_{3}(ℝ).

ℂ^{2n} ⊕ (ℂ^{2n})^{∗} ≅ 2V (π_{1}): 𝔰𝔭_{n}(ℝ) and 𝔰𝔭(k, l), k + l = n.

D) V (π_{5}): 𝔰𝔬_{2,10} and 𝔰𝔬_{6,6}. V (π_{6}): 𝔰𝔬_{2,10}, 𝔰𝔬_{6,6} and 𝔰𝔬_{12}^{∗}.

E) V (π_{1}): 𝔢_{7}^{(i)}, i = −25, −5 or 7.

The remaining cases are: ℂ^{2} ⊗ ℂ^{n}: 𝔰𝔩_{2}(ℝ) ⊕ 𝔰𝔬_{k,l} and, if n is even, also 𝔰𝔲_{2} ⊕ 𝔰𝔬_{n}^{∗}. ℂ^{2} ⊗ ℂ^{7}: 𝔰𝔩_{2}(ℝ) ⊕ 𝔤_{2}^{(2)}.

Proof: The proof is straightforward using the tables [24].

Let (V, ω) be a symplectic module of a complex semisimple group G which admits a Lagrangian highest weight vector orbit C and G^{0} a connected real form. There exists a G^{0}-invariant real structure τ on V if and only if 𝔤^{0} = Lie G^{0} occurs in the list of Proposition 7. Consider first the case where V is irreducible. Up to multiplying ω with λ ∈ ℂ^{∗} we can assume that ω and τ are compatible, i.e. ω|V^{τ} is a real symplectic structure. The nondegenerate G^{0}-invariant Hermitian form γ = \sqrt{−1}ω(·, τ·) is unique up to scaling by λ ∈ ℝ^{∗}.

Consider now the reducible case 𝔤 = 𝔰𝔩_{n}(ℂ) or 𝔤 = 𝔰𝔭_{n}(ℂ) and V = U ⊕ U^{∗}, U the standard 𝔤-module. Let C ⊂ V be a Lagrangian highest weight vector orbit. Then L = C ∪ {0} ⊂ V is a G-invariant linear subspace. Let τ be a G^{0}-invariant real structure on V compatible with ω (such pair (ω, τ) exists). The corresponding G^{0}- invariant Hermitian form γ|L is unique up to scaling by λ ∈ ℝ. It is nondegenerate if and only if V = L ⊕ τL and zero otherwise, i.e. if τL = L. In the first (respectively second) case V is the sum of two irreducible complex or quaternionic (respectively real) G^{0}-modules. This implies that if γ|L is nondegenerate then 𝔤^{0} = 𝔰𝔲_{k,l} if 𝔤 = 𝔰𝔩_{n}(ℂ) and 𝔤^{0} = 𝔰𝔭(k, l) if 𝔤 = 𝔰𝔭_{n}(ℂ).

Now we return to the general case, i.e. V, G, G^{0}, ω, τ, γ and C are as above and the G-module V may be reducible or irreducible. Assume that G^{0} has an open orbit M = G^{0}p ⊂ P (C) with compact stabilizer H^{0}. Then by Theorem 2 the corresponding complex Lie group H = (H^{0})ℂ ⊂ G is precisely the maximally reductive subgroup of the parabolic stabilizer G_{p} = P. Its Lie algebra 𝔥 = 𝔥′ ⊕ 𝔱 is the direct sum of the semisimple Lie algebra 𝔥′ (specified in Theorem 4 and Table 1) and the centre 𝔱 which has dimension

dim 𝔱 = rk 𝔤 − rk 𝔥′ = \{ \begin{matrix}1 & if 𝔤 is simple \\ 2 & else. \end{matrix}

The real form 𝔤^{0} of 𝔤 contains the compact real form 𝔥^{0} of 𝔥. This is only the case for the real forms included in the next proposition.

Proposition 8 Let V be a G-module as in Proposition 7 and 𝔥 the maximally reductive subalgebra of 𝔭 = Lie P, where P = G_{p} is the stabilizer of a highest weight direction p ∈ P (V). Any real form 𝔤^{0} listed in Proposition 7 which contains a subalgebra isomorphic to the compact form 𝔥^{0} of 𝔥 is (up to automorphism of 𝔤) listed in one of the triples (V, 𝔤^{0}, 𝔥^{0}) below. For the simple Lie algebras 𝔤 we have:

A) (∨^{3}ℂ^{2}, 𝔰𝔩_{2}(ℝ), 𝔰𝔬_{2}), (∧^{3}ℂ^{6}, 𝔰𝔲_{3,3}, 𝔰(𝔲_{3} ⊕ 𝔲_{3})) and (ℂ^{n} ⊕ (ℂ^{n})^{∗}, 𝔰𝔲_{n} or 𝔰𝔲_{1,n−1}, 𝔰(𝔲_{1} ⊕ 𝔲_{n−1})).

B) (Spinor module, 𝔰𝔬_{1,10}, 𝔲_{5}).

C) (V (π_{3}), 𝔰𝔭_{3}(ℝ), 𝔲_{3}) and (ℂ^{2n} ⊕ (ℂ^{2n})^{∗} ≅ 2V (π_{1}), 𝔰𝔭(n) or 𝔰𝔭(1, n − 1), 𝔰𝔭(n − 1) ⊕ 𝔲_{1}).

D) (V (π_{6}), 𝔰𝔬_{12}^{∗}, 𝔲_{6}).

E) (V (π_{1}), 𝔢_{7}^{(−25)}, 𝔢_{6} ⊕ 𝔰𝔬_{2}).

The remaining cases are: (ℂ^{2} ⊗ ℂ^{n}, 𝔰𝔩_{2}(ℝ) ⊕ 𝔰𝔬_{2,n−2}, 𝔰𝔬_{2} ⊕ 𝔰𝔬_{2} ⊕ 𝔰𝔬_{n−2}) and (ℂ^{2} ⊗ ℂ^{7}, 𝔰𝔩_{2}(ℝ) ⊕ 𝔤_{2}^{(2)}, 𝔰𝔬_{2} ⊕ 𝔲_{2}).

We will show now that each of these triples really corresponds to a special pseudo-Kähler homogeneous manifold M = G^{0}/H^{0} with compact stabilizer. More precisely, we prove the following main theorem.

Theorem 5 Let (V, 𝔤^{0}, 𝔥^{0}) be any of the triples listed in Proposition 8, G, G^{0} and H^{0} the connected linear Lie groups with Lie algebras 𝔤 = 𝔤^{0} ⊗ ℂ, 𝔤^{0} and 𝔥^{0} respectively. Then V has a Lagrangian highest weight vector orbit C ⊂ V such that there exists an open orbit of G^{0} on P (C) ⊂ P (V) with compact stabilizer H^{0} and any such orbit is a homogeneous special pseudo-Kähler manifold of the group G^{0}.

Conversely, any homogeneous special pseudo-Kähler manifold of a real semisimple group with compact stabilizer arises in this way. Moreover, it is homothetic to one of the following pseudo-Hermitian locally symmetric spaces.

The special pseudo-Kähler metric is positively defined in the cases i) and ii):

i) ℂP^{n} = SU_{n+1}/S(U_{1} · U_{n}) : (ℂ^{n+1} ⊕ (ℂ^{n+1})^{∗}, 𝔰𝔲_{n+1}, 𝔰(𝔲_{1} ⊕ 𝔲_{n})).

ii) ℂP^{2n+1} = Sp_{n+1}/U_{1} · Sp_{n} : (ℍ^{n+1} ⊕ (ℍ^{n+1})^{∗}, 𝔰𝔭(n + 1), 𝔰𝔭(n) ⊕ 𝔲_{1}).

The cases iii)-ix) correspond to formal moduli spaces, i.e. the special pseudo-Kähler metric is negative definite:

iii) ℂH^{n} = SU_{1,n}/S(U_{1} · U_{n}) : (ℂ^{n+1} ⊕ (ℂ^{n+1})^{∗}, 𝔰𝔲_{1,n}, 𝔰(𝔲_{1} ⊕ 𝔲_{n})).

iv) ℂH^{1} = SL_{2}(ℝ)/SO_{2} : (∨^{3}ℂ^{2}, 𝔰𝔩_{2}(ℝ), 𝔰𝔬_{2}).

v) SU_{3,3}/S(U_{3} · U_{3}) : (∧^{3}ℂ^{6}, 𝔰𝔲_{3,3}, 𝔰(𝔲_{3} ⊕ 𝔲_{3})).

vi) Sp_{3}(ℝ)/U_{3} : (V (π_{3}), 𝔰𝔭_{3}(ℝ), 𝔲_{3}). vii) SO_{12}^{∗}/U_{6} : (V (π_{6}), 𝔰𝔬_{12}^{∗}, 𝔲_{6}).

viii) E_{7}^{(−25)}/E_{6} · SO_{2} : (V (π_{1}), 𝔢_{7}^{(−25)}, 𝔢_{6} ⊕ 𝔰𝔬_{2}).

ix) ℂH^{1} × (SO_{2,n}/SO_{2} · SO_{n}) : (ℂ^{2} ⊗ ℂ^{n+2}, 𝔰𝔩_{2}(ℝ) ⊕ 𝔰𝔬_{2,n}, 𝔰𝔬_{2} ⊕ 𝔰𝔬_{2} ⊕ 𝔰𝔬_{n}).

In the remaining cases the metric is indefinite of (complex) signature (10,5), (1,2n) and (1,5) in the cases x), xi) and xii) respectively.

x) Spin_{1,10}/U_{5} ⊂ SO_{2,10}/U_{1,5} : (spinor module, 𝔰𝔬_{1,10}, 𝔲_{5}).

xi) Sp(1, n)/U_{1} ·Sp(n) ⊂ SU_{2,2n}/S(U_{1} ·U_{1,2n}) : (ℍ^{n+1} ⊕(ℍ^{n+1})^{∗}, 𝔰𝔭(1, n), 𝔰𝔭(n)⊕ 𝔲_{1}).

xii) (SL_{2}(ℝ)/SO_{2}) × (G_{2}^{(2)}/SO_{2} · U_{2}) ⊂ (SL_{2}(ℝ)/SO_{2}) × (SO_{3,4}/SO_{2} · SO_{1,4}) : (ℂ^{2} ⊗ ℂ^{7}, 𝔰𝔩_{2}(ℝ) ⊕ 𝔤_{2}^{(2)}, 𝔰𝔬_{2} ⊕ 𝔲_{2}).

Proof: First we treat the case of reducible modules: 𝔤 = 𝔤^{0} ⊗ ℂ = 𝔰𝔩_{n+1}(ℂ) or 𝔤 = 𝔰𝔭_{n+1}(ℂ) and the symplectic 𝔤-module (V, ω) admits the 𝔤-invariant Lagrangian splitting V = L ⊕ L′, where L is isomorphic to the standard 𝔤-module and L′ is isomorphic to its dual. Let τ be a real structure on V invariant under the real form 𝔤^{0} = 𝔰𝔲_{k,l} (respectively 𝔰𝔭_{k,l}) of 𝔤 = 𝔰𝔩_{n+1}(ℂ) (respectively 𝔤 = 𝔰𝔭_{n+1}(ℂ)), k + l = n + 1. As discussed above, we can choose the data (ω, L, L′, τ) such that τ is compatible with ω and interchanges L and L′, i.e. L′ = τL. The corresponding 𝔤^{0}- invariant nondegenerate Hermitian structure γ|L, γ = \sqrt{−1}ω(·, τ·), is unique up to scaling by λ ∈ ℝ^{∗}. Up to multiplying τ by −1 we can assume that γ|L has signature (k, l) (respectively (2k, 2l)) if 𝔤^{0} = 𝔰𝔲_{k,l} (respectively 𝔰𝔭(k, l)).

Let G^{0} be the linear group with Lie algebra 𝔤^{0}. We consider its action on the projectivization P (C) of the highest weight vector orbit C = L − {0}. The open G^{0}-orbits M = G^{0}p on P (C) are precisely the orbits of non-null directions p = ℂ^{∗}v, i.e. γ(v, v) ≠ 0. The cone C_{M} ⊂ V over any open orbit M is a special cone, see Def. 1. The pseudo-Kähler metric γ|L is invariant under the group SU_{k,l} if the signature of γ|L is (k, l). This shows that, up to a scaling of the metric by a positive factor, the special pseudo-Kähler manifold (M = G^{0}p, g) is isometric to the pseudo-Hermitian symmetric space SU_{k,l}/S(U_{1} · U_{k−1,l}) (respectively SU_{k,l}/S(U_{k,l−1} · U_{1})) of signature (k − 1, l) (respectively (l − 1, k)) if γ(v, v) > 0 (respectively γ(v, v) < 0), p = ℂ^{∗}v.

It follows that if G^{0} = SU_{k,l} and H^{0} is the stabilizer of p in G^{0} then H^{0} is compact if and only if the special pseudo-Kähler metric g on M = G^{0}p ⊂ P (C) is of definite signature. Moreover, (M, g) is homothetic to complex projective space ℂP^{n} (respectively complex hyperbolic space ℂH^{n}) if g > 0 (respectively g < 0). Up to an automorphism of 𝔤 = 𝔰𝔩_{n+1}(ℂ) we obtain the cases i) and iii) of the theorem.

If G^{0} = Sp(k, l) then H^{0} is compact if and only if either G^{0} is compact, the special pseudo-Kähler metric g on M = G^{0}p ≅ Sp(n + 1)/Sp(n) · U_{1} is positive definite and (M, g) is homothetic to ℂP^{2n+1} = SU_{2n+2}/S(U_{2n+1} · U_{1}) or G^{0} ≅ Sp(1, n), the special pseudo-Kähler metric on M = G^{0}p ≅ Sp(1, n)/Sp(n) · U_{1} is negative definite and (M, g) is homothetic to an open Sp(1, n)-orbit on the pseudo-Hermitian symmetric space SU_{2,2n}/S(U_{1} · U_{1,2n}). This covers the cases ii) and xi) and completes the discussion of the reducible modules.

Next we study the irreducible modules. So let (V, 𝔤^{0}, 𝔥^{0}) be a triple listed in Proposition 8 such that 𝔤^{0} is a noncompact real form of the complex semisimple Lie algebra 𝔤 and V is an irreducible symplectic 𝔤-module. The highest weight vector orbit C = Gv ⊂ V of the complex linear group G with Lie G = 𝔤 is a Lagrangian cone. The stabilizer G_{p} = P of p = ℂ^{∗}v ∈ P (V) in G is a parabolic subgroup with maximally reductive subgroup H. The maximal rank subalgebra 𝔥^{0} ⊂ 𝔤^{0} is the compact real form of 𝔥 = Lie H. Any Cartan subalgebra 𝔞^{0} ⊂ 𝔥^{0} is a Cartan subalgebra of 𝔤^{0} and its complexification 𝔞 = 𝔞^{0} ⊗ ℂ ⊂ 𝔥 ⊂ 𝔤 is a Cartan subalgebra of 𝔤. Let R^{+} be a system of positive roots of 𝔤 with respect to 𝔞 and p the highest weight direction of the irreducible 𝔤-module V with respect to R^{+}. We can write

𝔥 = 𝔞 + ∑_{α∈R_{0}} 𝔤_{α}

for some root subsystem R_{0} ⊂ R = R^{+} ∪ R^{−} and

𝔭 = Lie P = 𝔥 + ∑_{α∈R^{+}−R_{0}} 𝔤_{α} , R_{0}^{+} = R_{0} ∩ R^{+} .

We denote by σ the complex antilinear involution of 𝔤 with fixed point set 𝔤^{σ} = 𝔤^{0}. Since 𝔞^{0} ⊂ 𝔤^{0} is a compact Cartan subalgebra, we have that

𝔞ℝ := {α ∈ R|α(𝔞) ⊂ ℝ} = i𝔞^{0} , σ𝔤_{α} = 𝔤_{−α}

and hence 𝔭 ∩ σ𝔭 = 𝔥 and 𝔭 ∩ 𝔤^{0} = 𝔥^{0}. This shows that M = G^{0}p ≅ G^{0}/H^{0} is open in \overline{M} = Gp = P (C). Moreover, any G^{0}-orbit on P (C) with stabilizer H^{0} is obtained in this way. In fact, let 𝔭 be the Lie algebra of the stabilizer in G of a point p ∈ P (C) and assume that 𝔭 ∩ 𝔤^{0} = 𝔥^{0}. The point p is a highest weight direction for some choice of Cartan subalgebra 𝔞 ⊂ 𝔥 and system of positive roots R^{+} of 𝔤. Since any two Cartan subalgebras of 𝔥 are conjugated by an element of H, we can assume that 𝔞 = 𝔞^{0} + i𝔞^{0}, 𝔞^{0} = 𝔞 ∩ 𝔥^{0}.

The next lemma shows that, under our assumptions, the cone C_{M} ⊂ V over M is special and the signature of the Hermitian metric γ on C_{M} is easily computed.

Lemma 6 Let (V, ω) be an irreducible symplectic G-module of a complex semisimple Lie group G with Lagrangian highest weight vector orbit C ⊂ V and G^{0} a connected real form of G for which there exists a G^{0}-invariant real structure τ on V. Assume that the stabilizer H^{0} of p ∈ P (C) in G^{0} is a compact real form of the maximally reductive subgroup H of the stabilizer G_{p} = P of p in G. Then the orbit M = G^{0}p ⊂ P (C) is open and the cone C_{M} ⊂ V over M is special in the sense of Def. 1. The Hermitian form γ = \sqrt{−1}ω(·, τ·) has signature (k,l) or (l,k) on C_{M}, where k (respectively l) is the number of compact (respectively noncompact) roots in R^{+} −R_{0}^{+}. Here R^{+} (respectively R_{0}^{+} ⊂ R^{+}) is a system of positive roots of 𝔤 (respectively 𝔥) with respect to the Cartan subalgebra 𝔞 = 𝔞^{0} ⊗ ℂ, 𝔞^{0} ⊂ 𝔥^{0} a Cartan subalgebra of 𝔤^{0}. If γ has signature (k, l) on C_{M}, then the special pseudo-Kähler metric g of M = G^{0}p, p = ℂ^{∗}v, has signature (k − 1, l) if γ(v, v) > 0 and signature (l − 1, k) if γ(v, v) < 0.

Proof: As above, we denote by σ the ℂ-antilinear involution of 𝔤 with fixed point set 𝔤^{σ} = 𝔤^{0} and recall that σ𝔤_{α} = 𝔤_{−α} for all α ∈ R, where R is the system of roots of 𝔤 with respect to 𝔞. We choose a system of positive roots R^{+} ⊂ R such that our base point v ∈ M is a highest weight vector with respect to R^{+} and denote by Λ the highest weight. Then 𝔥 = 𝔥^{0} ⊗ ℂ = 𝔞 + ∑_{α∈R_{0}} 𝔤_{α} for some root subsystem R_{0} = R_{0}^{+} ∪ −R_{0}^{+} ⊂ R, R_{0}^{+} = R^{+} ∩ R_{0}. Then the tangent space T_{v}C_{M} is precisely

T_{v}C_{M} = 𝔤v = ℂv ⊕ ⨁_{α∈R^{+}−R_{0}^{+}} 𝔤_{−α}v = V_{Λ} ⊕ ⨁_{α∈R^{+}−R_{0}^{+}} V_{Λ−α} .

The 𝔤^{0}-invariance of the real structure τ on V is equivalent to τ(xu) = σ(x)τ(u) for all x ∈ 𝔤, u ∈ V and hence we have that:

τT_{v}C_{M} = ℂτv ⊕ ⨁_{α∈R^{+}−R_{0}^{+}} 𝔤_{α}τv = V_{−Λ} ⊕ ⨁_{α∈R^{+}−R_{0}^{+}} V_{α−Λ} .

This shows that V = T_{v}C_{M} ⊕ τT_{v}C_{M} and therefore γ is nondegenerate on L = T_{v}C_{M}. Moreover, any G^{0}-invariant Hermitian form on L is proportional to γ|L over the reals. A straightforward calculation shows that γ|L has the orthogonal basis (v, e_{α}v, α ∈ R^{−} − R_{0}^{−}), R^{−} = −R^{+}, R_{0}^{−} = −R_{0}^{−}, which satisfies:

γ(e_{α}v, e_{α}v) = Λ([e_{α}, σe_{α}])γ(v, v) = ⟨α, Λ⟩⟨e_{α}, σe_{α}⟩ \{ \begin{matrix}> 0 & if α is compact \\ < 0 & if α is noncompact. \end{matrix}

Here we have used that the Killing form ⟨·, ·⟩ is negative (respectively positive) definite on (𝔤^{α} + 𝔤^{−α})^{σ} if α is a compact (respectively noncompact) root and the fact that R^{−} − R_{0}^{−} = {α ∈ R|⟨α, Λ⟩ < 0}. This proves the lemma.

Now we complete the proof of the theorem. Thanks to Lemma 6 it only remains to show that the special pseudo-Kähler metric on M = G^{0}p ≅ G^{0}/H^{0} is pseudo-Hermitian locally symmetric. This is immediate if the isotropy representation is irreducible. In the remaining two cases we have the following inclusions (𝔤^{0}, 𝔥^{0}) ⊂ (𝔤̃^{0}, 𝔥̃^{0}) of pairs:

(𝔰𝔬_{1,10}, 𝔲_{5}) ⊂ (𝔰𝔬_{2,10}, 𝔲_{1,5}) and
(𝔰𝔩_{2}(ℝ) ⊕ 𝔤_{2}^{(2)}, 𝔰𝔬_{2} ⊕ 𝔲_{2}) ⊂ (𝔰𝔩_{2}(ℝ) ⊕ 𝔰𝔬_{3,4}, 𝔰𝔬_{2} ⊕ 𝔰𝔬_{2} ⊕ 𝔰𝔬_{1,4}) .

The irreducible symplectic 𝔤-module V (V = spinor module in the case 𝔤 = 𝔰𝔬_{11}(ℂ) and V = ℂ^{2} ⊗ ℂ^{7} for 𝔤 = 𝔰𝔩_{2}(ℂ) ⊕ 𝔤_{2}(ℂ)) carries the structure of irreducible symplectic 𝔤̃-module, 𝔤̃ = 𝔤̃^{0} ⊗ ℂ (semispinor module if 𝔤̃ = 𝔰𝔬_{12}(ℂ), tensor product of standard modules if 𝔤̃ = 𝔰𝔩_{2}(ℂ) ⊕ 𝔰𝔬_{7}(ℂ)), and a 𝔤̃^{0}-invariant real structure, as follows from Proposition 7. This shows that the special pseudo-Kähler metric of M = G^{0}p ≅ G^{0}/H^{0} ↪ G̃^{0}/H̃^{0} is canonically extended to a G̃^{0}-invariant special pseudo-Kähler metric on M̃ = G̃^{0}/H̃^{0}. Here G̃^{0} and H̃^{0} are the connected linear groups with Lie G^{0} = 𝔤^{0} and Lie H^{0} = 𝔥^{0}. On the other hand, M̃ = G̃^{0}/H̃^{0} carries the structure of an irreducible pseudo-Hermitian symmetric space. Hence the special pseudo-Kähler metric of M̃ is pseudo-Hermitian symmetric. Since the special pseudo-Kähler manifold M is isometrically embedded into M̃ as an open G^{0}-orbit it follows that M is pseudo-Hermitian locally symmetric finishing the proof of Theorem 5.

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