INVARIANT CR STRUCTURES ON COMPACT

HOMOGENEOUS MANIFOLDS

Dmitry V. Alekseevsky and Andrea F. Spiro

Abstract. An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds:

a) the standard CR homogeneous manifolds which are homogeneous S^{1}-bundles over a flag manifold F, with CR structure induced by an invariant complex structure on F;

b) the Morimoto-Nagano spaces, i.e. sphere bundles S(N) ⊂ T N of a compact rank one symmetric space N = G/H, with the CR structure induced by the natural complex structure of T N = G^{ℂ}/H^{ℂ};

c) the following manifolds: SU_{n}/T^{1}·SU_{n−2}, SU_{p}×SU_{q}/T^{1}·U_{p−2}·U_{q−2}, SU_{n}/T^{1}· SU_{2} · SU_{2} · SU_{n−4}, SO_{10}/T^{1} · SO_{6}, E_{6}/T^{1} · SO_{8}, E_{7}/T^{1} · SO_{10}; these manifolds admit canonical holomorphic fibrations over a flag manifold (F, J_{F}) with typical fiber S(S^{k}), where k = 2, 3, 5, 7, 9 or 11, respectively; the CR structure is determined by the invariant complex structure J_{F} on F and by an invariant CR structure on the typical fiber, depending on one complex parameter.

1. Introduction.

An almost CR structure on a manifold M is a pair (D, J), where D ⊂ T M is a distribution and J is a complex structure on D. The complexification D^{ℂ} can be decomposed as D^{ℂ} = D^{10} + D^{01} into sum of complex eigendistributions of J, with eigenvalues i and −i.

An almost CR structure is called integrable or, shortly, CR structure if the distribution D^{01} (and hence also D^{10}) is involutive, i.e. with space of sections closed under Lie bracket. This is equivalent to the following conditions:

J([JX, Y] + [X, JY]) ∈ D ,
[JX, JY] − [X, Y] − J([JX, Y] + [X, JY]) = 0 ,

for any two fields X, Y in D.

A map φ: (M, D, J) → (M′, D′, J′) between two CR manifolds is called holomorphic map if φ_{∗}(D) ⊂ D′ and φ_{∗}(JX) = J′φ_{∗}(X).

1991 Mathematics Subject Classification. Primary 32C16; Secondary 53C30 53C15. Key words and phrases. Homogeneous CR manifolds, Real Hypersurfaces, Contact Homoge-

neous manifolds.

Typeset by AMS-TEX

Two CR structures (D, J) and (D′, J′) are called equivalent if there exists a diffeomorphism such that ϕ_{∗}(D) = D′ and ϕ_{∗}J = J′.

The codimension of D is called codimension of the CR structure. Note that a CR structure of codimension zero is the same as a complex structure.

A codimension one CR structure (D, J) on a 2n + 1-dimensional manifold M is called Levi non-degenerate if D is a contact distribution. This means that any local (contact) 1-form θ, which defines the distribution (i.e. such that kerθ = D) is maximally non-degenerate, that is (dθ)^{n} ∧ θ ≠ 0.

Note that any real hypersurface M of a complex manifold N has a natural codimension one CR structure (D, J_{D}) induced by the complex structure J of N, where

D = { X ∈ T M , JX ∈ T M } , J_{D} = J|_{D} .

In the following, if the opposite is not stated, by CR structure we will mean integrable codimension one Levi non-degenerate CR structure. Sometimes, if the contact distribution D is given, we will identify a CR structure with the associated complex structure J.

A CR manifold, that is a manifold M with a CR structure (D, J), is called homogeneous if it admits a transitive Lie group of holomorphic transformations.

If the opposite is not stated, we will always assume that the homogeneous CR manifold (M, D, J) is simply connected.

The aim of this paper is to give a complete classification of simply connected homogeneous CR manifolds M = G/L of a compact Lie group G. This gives a classification of all simply connected homogeneous CR manifolds, since any compact homogeneous CR manifold admits a compact transitive Lie group of holomorphic transformations (see [12]).

The simplest example of compact homogeneous CR manifold is the standard sphere S^{2n−1} ⊂ ℂ^{n} with the induced CR structure.

More elaborated examples are provided by the following construction of A. Morimoto and T. Nagano ([9]). Let N = G/H be a compact rank one symmetric space (shortly ’CROSS’). The tangent space T N can be identified with the homogeneous space G^{ℂ}/H^{ℂ}. Hence, it admits a natural G^{ℂ}-invariant complex structure J. Any regular orbit G·p = S(N) ≃ G/L in T N = G^{ℂ}/H^{ℂ} is a sphere bundle; in particular it is a real hypersurface with (Levi non-degenerate) G-invariant CR structure.

Moreover, these examples together with the standard sphere S^{2n−1} ⊂ ℂ^{n} exhaust the class of CR structures induced on a codimension one orbit M = G · x ⊂ C of a compact Lie group G of holomorphic transformations of a Stein manifold C. We call the homogeneous CR manifolds which are equivalent to such orbits G·p = S(N) in the tangent space of a CROSS Morimoto-Nagano spaces.

In the fundamental paper [1], H. Azad, A. Huckleberry and W. Richthofer showed that these manifolds play a basic role in the description of compact homogeneous CR manifolds (see also [8] and [11]).

INVARIANT CR STRUCTURES ON COMPACT HOMOGENEOUS MANIFOLDS 3

More precisely, for any compact homogeneous CR manifold M = G/L they define a holomorphic map (called anticanonical map) ϕ: M = G/L → ℂP^{N}. This map is G-equivariant with respect to some explicitly defined projective action of G on ℂP^{N}. For any compact homogeneous CR manifold M only two possibilities may occur: the orbit ϕ(M) = G · p, p ∈ ϕ(M), is either a flag manifold with the complex structure induced by the complex structure J_{P} of ℂP^{N} and in this case ϕ: M → ϕ(M) is an S^{1}-fibering, or it is a CR manifold with CR structure induced by J_{P} and in this case ϕ: M → ϕ(M) is a finite covering.

This reduces the description of CR homogeneous manifolds of the second type to the description of compact orbits G · p ⊂ ℂP^{N} of a real subgroup G ⊂ Aut(ℂP^{N}) of projective transformations, on which J_{P} induces a CR structure.

A simple argument shows that an orbit G·p ⊂ ℂP^{N} of a connected Lie subgroup G ⊂ Aut(ℂP^{N}) carries a (possibly Levi degenerate) CR structure induced by ℂP^{N} if and only if G·p is a real hypersurface of G^{ℂ} ·p. Moreover, if the orbit is compact, one may assume that G is a compact semisimple Lie group.

The following important result in [1] describes the structure of such orbits.

Theorem. Let G^{ℂ} ⊂ Aut(ℂP^{N}) be a connected complex semisimple group of projective transformations and G its compact form. Assume that the orbit M = G·p = G/L carries a Levi non-degenerate CR structure induced by J_{P} and hence it is a real hypersurface in B = G^{ℂ}·p = G^{ℂ}/H. Denote by P a minimal parabolic subgroup of G^{ℂ} which properly contains H. Then the fiber C = P/H of the G^{ℂ}-equivariant fibration

π: B = G^{ℂ}/H → F = G^{ℂ}/P

over the flag manifold F = G^{ℂ}/P is a homogeneous Stein manifold biholomorphic to ℂ^{∗}, ℂ^{n} or to the tangent space of a CROSS.

This fibration is called Stein-rational fibration. Note that P not necessarily acts effectively on C.

The Stein-rational fibration induces a G-equivariant holomorphic fibration of the homogeneous CR manifold M = G/L over the flag manifold F

π′: M = G/L → F = G^{ℂ}/P

(it is a CRF fibration according to our definitions, see below). Moreover, in correspondence to a fiber of π, a fiber of π′ is either S^{1}, S^{2n−1} or a Morimoto-Nagano space.

This Theorem gives necessary conditions for the induced CR structure on M = G · p ⊂ ℂP^{N} being Levi non-degenerate. Our classification gives necessary and sufficient conditions. In particular, we show that only the sphere bundles S(S^{k}) with k = 2, 3, 5, 7, 9 and 11 occur as fibers of the fibration π′.

Now we describe the main results of this paper. Section §2 collects the basics facts on homogeneous CR manifolds.

Section §3 is devoted to the infinitesimal description of homogeneous contact manifolds M = G/L of a compact Lie group.

We prove that the center of G is at most one dimensional and we establish a natural one to one correspondence between simply connected homogeneous manifolds M = G/L with an invariant contact distribution D and an element Z ∈ 𝔤 = Lie(G) (defined up to scaling) such that:

a) the centralizer of Z has the following orthogonal decomposition

C_{𝔤}(Z) = 𝔩 ⊕ ℝZ , 𝔩 = Lie(L)

w.r.t. the Cartan-Killing form B;

b) the 1-parametric subgroup generated by Z is closed.

This element Z (called contact element) defines an orthogonal decomposition

𝔤 = 𝔩 + ℝZ + 𝔪 .

The subspace 𝔪 is Ad_{L}-invariant and defines the contact distribution D on M = G/L, while the Ad_{L}-invariant 1-form θ = B ∘ Z ∈ 𝔤^{∗} is extended to a G-invariant contact form θ on G/L.

We associate to Z a flag manifold F_{Z} , which is the adjoint orbit

F_{Z} = Ad_{G}(Z) = G/K ,

where K = C_{G}(Z) is the centralizer of Z. There is a natural principal S^{1}-fibration

π: M = G/L ⟶ F_{Z} = G/K .

In general, a homogeneous manifold G/L admits no more then one invariant contact structure. If it admits more then one then it is called special contact manifold.

The main examples of such manifolds can be described as follows.

Let G be a simple compact Lie group without center and let Q = G/Sp_{1} · H′ be the associated Wolf space, that is the homogeneous quaternionic Kähler manifold, where Sp_{1} · H′ is the normalizer in G of the 3-dimensional subalgebra 𝔰𝔭_{1}(μ) of 𝔤 associated with the maximal root μ. Then the associated 3-Sasakian homogeneous manifold M = G/H′ is a special contact manifold.

Any 0 ≠ Z ∈ 𝔰𝔭_{1}(μ) is a contact element. Furthermore, any two invariant contact structures on M are equivalent under a transformation, which commutes with G, defined by the right action of an element from Sp_{1}.

We prove the following theorem.

Theorem 1.1. Any special contact manifold M = G/L is either the 3-Sasakian homogeneous manifold G/H′ of a simple Lie group G, described above, or M = G_{2}/Sp_{1}, where Sp_{1} is the 3-dimensional subgroup of the exceptional Lie group G_{2}, with Lie algebra 𝔰𝔭_{1}(μ), where μ is the maximal root of G_{2}.

In section §4 we establish some general properties of compact homogeneous CR manifolds. Let (M = G/L, D) be a homogeneous contact manifold and

𝔤 = 𝔩 + ℝZ + 𝔪

the associated decomposition of 𝔤. Then any invariant (integrable) CR structure J is defined by the Ad_{L}-invariant decomposition

𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01}  (1.1)

of the complexified tangent space 𝔪^{ℂ} = T_{eL}^{ℂ}M into holomorphic and antiholomorphic subspaces; this decomposition is such that

𝔩^{ℂ} + 𝔪^{10} is a subalgebra of 𝔤^{ℂ} .  (1.2)

The subspace 𝔪 is naturally identified with the tangent space of the associated flag manifold F_{Z} = G/K, 𝔨 = 𝔩+ℝZ = Lie(K). It is known that any invariant complex structure on F_{Z} is defined by an Ad_{K}-invariant decomposition (1.1), where 𝔪^{10} is a subalgebra (in fact it is the nilradical of a parabolic subalgebra 𝔨^{ℂ} + 𝔪^{10}). Hence any invariant complex structure J_{F} on F_{Z} defines an invariant CR structure J_{M} on M = G/L. It is called standard CR structure induced by J_{F}.

The natural S^{1}-fibration π: M = G/L → F_{Z} = G/K is holomorphic with respect to the CR structure J_{M} and the complex structure J_{F}.

Since the description of all invariant complex structures on a flag manifold is known (see e.g. [10], [4], [5], [3]), it is sufficient to classify the non-standard CR structures.

The following notion is important for such classification.

A compact homogeneous CR manifold (M = G/L, D, J) is called non-primitive if it admits a holomorphic G-equivariant fibration π (called CRF-fibration)

π: M = G/L ⟶ F = G/Q ,

where F = G/Q is a flag manifold of positive dimension, equipped with an invariant complex structure J_{F}.

Note that a fiber of π is a homogeneous compact CR manifold Q/L and that any standard CR manifold is non-primitive.

The classification of primitive CR structures given in §5 and §6 is an important step for classification of non-standard CR structures.

A basic tool for studying the homogeneous CR manifolds is the anticanonical map ϕ defined in [1].

Let (M = G/L, D_{Z}, J) be a homogeneous CR manifold of a compact Lie group G and

𝔤^{ℂ} = 𝔩^{ℂ} + ℂZ + 𝔪^{10} + 𝔪^{01}

the corresponding decomposition of 𝔤^{ℂ}. Then the anticanonical map ϕ is the holomorphic map of M into the Grassmanian of k-planes, k = dim_{ℂ}(𝔩^{ℂ} + 𝔪^{01}), given by

ϕ: M = G/L ⟶ Gr_{k}(𝔤^{ℂ}) ⊂ ℂP^{N}
ϕ: gL ↦ Ad_{g}([𝔩^{ℂ} + 𝔪^{01}]) .

Note that ϕ is a G-equivariant map onto the orbit G · p of p = [𝔩^{ℂ} + 𝔪^{01}] ∈ Gr_{k}(𝔤^{ℂ}) under the natural adjoint action of G on Gr_{k}(𝔤^{ℂ}).

We obtain the following characterization of standard CR structures (see Theorems 4.9 and 4.11):

Theorem 1.2. Let (M = G/L, D_{Z}, J) be a homogeneous CR manifold.

(1) If it is standard, then the image ϕ(M) = G · p of the anticanonical map is the flag manifold F_{Z} = G/K, associated with the contact structure D_{Z}. Hence ϕ: M → ϕ(M) = F_{Z} is the natural S^{1}-fibration.

(2) If it is non-standard, then ϕ: M → ϕ(M) = G · p is a finite holomorphic covering, with respect to the CR structure of G · p ⊂ Gr_{k}(𝔤^{ℂ}) induced by the complex structure of Gr_{k}(𝔤^{ℂ}).

In section §5, we classify all invariant CR structures on special contact manifolds G/L. The result is the following:

Theorem 1.3. Let M = G/L be a special contact manifold with an invariant contact structure D_{Z}.

(1) if G ≠ SU_{n}, then there exists (up to sign of J) only one invariant CR structure (D_{Z}, J), which is standard;

(2) if G = SU_{2} and hence M = SU_{2}, then there exist (up to sign of J) one standard CR structure and one family of non-standard CR structures; any non-standard CR structure is primitive;

(3) if G = SU_{n}, n > 2, and hence M = SU_{n}/U_{n−2}, then there exist (up to sign of J) three standard CR structures, induced by the three invariant complex structures of the corresponding flag manifold F_{Z} = SU_{n}/T^{2} · SU_{n−2} (which is the twistor space of the Wolf space Q = SU_{n}/S(U_{2} · U_{n−2})), and two families of non-standard CR structures. Any non-standard CR structure is non-primitive and admits a CRF fibration

π: M = SU_{n}/U_{n−2} ⟶ Q = SU_{n}/S(U_{2} · U_{n−2})

with fiber SO_{3} over the Wolf space Q = SU_{n}/S(U_{2} · U_{n−2}) equipped with its (unique up to sign) complex structure.

The explicit description of all non-standard CR structures on SU_{2} and SU_{n}/U_{n−2} is given in Proposition 5.1.

In section §6, we obtain the classification of non-standard invariant CR structures on non-special homogeneous contact manifolds.

Together with above results it leads to the following classification of primitive CR structures.

Theorem 1.4. Let (M = G/L, D_{Z}, J) be a simply connected, primitive, homogeneous CR manifold and θ = B∘Z|_{𝔱} the dual form of the contact element Z restricted to a Cartan subalgebra 𝔱 of 𝔨 = C_{𝔤}(Z) = 𝔩 + ℝZ. Then G/L is isomorphic to a sphere bundle S(N) ⊂ T (N) of a CROSS N. The groups G, K = C_{G}(Z), the form ϑ = −iθ and the CROSS N are listed in the following table.

n^{o}  G  K = C_{G}(Z)  ϑ  N = G/H^{1  SU_{2}  T^{1}  ε_{1} − ε_{2}  S^{2} = \frac{SO_{3}}{SO_{2}}}^{2 SU_{2} × SU′_{2}  T^{1} × T^{1}′  (ε_{1} − ε_{2}) + (ε′_{1} − ε′_{2})  S^{3} = \frac{SO_{4}}{SO_{3}}}^{3  Spin_{7}  T^{1} · SU_{3}  ε_{1} + ε_{2} + ε_{3}  S^{7} = \frac{Spin_{7}}{SU_{3}}}^{4  F_{4}  T^{1} · SO_{7}  ε_{1}  𝕆P^{2} = \frac{F_{4}}{Spin_{9}}}^{5 SO_{2n+1}  T^{1} · SO_{2n−1}  ε_{1}  S^{2n} = \frac{SO_{2n+1}}{SO_{2n}}}^{6  SO_{2n}  T^{1} · SO_{2n−2}  ε_{1}  S^{2n−1} = \frac{SO_{2n}}{SO_{2n−1}}}^{7  Sp_{n}  T^{1} · Sp_{1} · Sp_{n−2}  ε_{1} + ε_{2}  ℍP^{n−1} = \frac{Sp_{n}}{Sp_{1}·Sp_{n−1}}}

In each of these cases, the set of all invariant CR structures (considered up to sign) on M = G/L is parameterized by the points of the unit disc D in ℝ^{2}. The center of D corresponds to the (unique) standard CR structure of M and all other points correspond to primitive CR structures.

The explicit description of all non-standard CR structures on these manifolds is given in Corollary 5.2 and Propositions 6.3 and 6.4.

For what concerns the non-primitive and non-standard CR structures, we have the following theorem (for an explicit description of the CR structures, see also Theorem 5.1 and Proposition 6.5).

Theorem 1.5. Let (M = G/L, D_{Z}, J) be a simply connected homogeneous CR manifold with a non-standard non-primitive CR structure. Then there exists a unique CRF fibration

π: M = G/L ⟶ F = G/Q

over a flag manifold F with an invariant complex structure J_{F}, such that the fiber C = Q/L is a primitive CR manifold. Moreover the groups G, L, the primitive fiber C = Q/L and the flag manifold F = G/Q are as in the following table (in n.2, the subgroups U_{p−2} and U′_{q−2} of L are subgroups of the factors SU_{p} and SU′_{q} of G, respectively):

n^{o}  G  L  C = Q/L  F = G/Q^{1 SU_{n} ^{n>2}  T^{1} · SU_{n−2}  SU_{2} = S(S^{2})  \frac{SU_{n}}{S(U_{2}·U_{n−2})}}^{2 SU_{p} × SU′_{q}_{p+q>4}  T^{1} · U_{p−2} · U′_{q−2}  \frac{SO_{4}}{SO_{2}} = S(S^{3}) \frac{SU_{p}}{S(U_{2}·U_{p−2})} × \frac{SU_{q}}{S(U_{2}·U_{q−2})}}^{3 SU_{n} ^{n>4} T^{1} · (SU_{2} × SU_{2}) · SU_{n−4} \frac{SO_{6}}{SO_{4}} = S(S^{5})  \frac{SU_{n}}{S(U_{4}×U_{n−4})}}^{4  SO_{10}  T^{1} · SO_{6}  \frac{SO_{8}}{SO_{6}} = S(S^{7})  \frac{SO_{10}}{T^{1}·SO_{8}}}^{5  E_{6}  T^{1} · SO_{8}  \frac{SO_{10}}{SO_{8}} = S(S^{9})  \frac{E_{6}}{T^{1}·SO_{10}}}^{6  E_{7}  T^{1} · SO_{10}  \frac{SO_{12}}{SO_{10}} = S(S^{11})  \frac{E_{7}}{T^{1}·SO_{12}}}

In particular, the primitive fiber C is always a sphere bundle S(S^{r}) ⊂ T S^{r} where r = 2, 3, 5, 7, 9 or 11.

Corollary 1.6. Let π: M = G/L → F = G/Q be the CRF fibration of a nonprimitive non-standard CR manifold (G/L, D, J_{0}) onto the flag manifold F = G/Q with a fixed invariant complex structure J_{F}. Then the set of all invariant CR structures (D, J) on G/L (up to sign of J), such that the fibering π: M = G/L → F = G/Q is holomorphic, is parameterized by the points of the unit disc D in ℝ^{2}. The center of D corresponds to the unique standard CR structure J_{s} of this family and all other points correspond to non-standard CR structures.

The unique standard CR structure J_{s} on M = G/L such that the fibration π: M = G/L → F = G/Q is holomorphic w.r.t. J_{s} and J_{F} is called the standard CR structure associated with the non-standard CR structure J_{0}.

Finally we give the description of all non-primitive CR manifolds G/L of a given compact Lie group G in terms of painted Dynkin graphs of 𝔤 = Lie(G), that is of Dynkin graphs of the Lie algebra 𝔤 with nodes painted in three colors: white (∘), black (•) and ’grey’ (⊗).

Recall that any flag manifold F = G/Q with an invariant complex structure J_{F} is defined (up to equivalence) by a black-white Dynkin graph, where the subalgebra 𝔮 = Lie(Q) is generated by the Cartan subalgebra and the root vectors associated with the white nodes. The complex structure J_{F} is determined by the decomposition

𝔤^{ℂ} = 𝔮^{ℂ} + 𝔪^{10} + 𝔪^{01}

where 𝔪^{10} is the nilpotent subalgebra generated by the root vectors associated to black nodes (see e.g. [3], [4]).

With a painted Dynkin graph Γ (equipped by simple roots in a standard way), we associate two flag manifolds F_{1}(Γ) = G/K and F_{2}(Γ) = G/Q and two invariant complex structure J_{1}(Γ) and J_{2}(Γ) on F_{1}(Γ) and F_{2}(Γ), respectively, as follows. The pairs (F_{1}(Γ) = G/K, J_{1}(Γ)) and (F_{2}(Γ) = G/Q, J_{2}(Γ)) are the flag manifolds with invariant complex structures defined by the black-white graphs obtained from Γ by considering the grey nodes as black and, respectively, white.

Note that Q contains K and that the natural fibration

ϖ: F_{1}(Γ) = G/K → F_{2}(Γ) = G/Q

is holomorphic and a fiber Q/K is a flag manifold with an induced invariant complex structure J′. Moreover, J_{1}(Γ) is canonically defined by J_{2}(Γ) and J′.

Conversely, if F_{1} = G/K and F_{2} = G/Q are two flag manifolds with invariant complex structures J_{1} and J_{2} such that Q ⊃ K and the equivariant fibration ϖ: F_{1} → F_{2} is holomorphic, then we may associate with F_{1} and F_{2} a painted Dynkin graph in an obvious way.

Definition 1.7. A CR-graph is a pair (Γ, ϑ(Γ)), formed by a painted Dynkin graph Γ and a linear combination ϑ(Γ) of simple roots, given in the following table:

type  𝔤  Γ  ϑ(Γ)^{I A_{n} ^{(n>1)}  ^{⊗} • ∘  ⋯  ∘  ε_{1} − ε_{2}}^{II A_{p} + A′_{q}_{(p+q>2)}  \begin{matrix}⊗ & • & ∘ & ⋯ & ∘ & ∘ \\ ⊗ & • & ∘ & ⋯ & ∘ & ∘ \end{matrix}  (ε_{1} − ε_{2}) − (ε′_{1} − ε′_{2})}^{III A_{n} ^{(n>3)} ∘ ^{⊗} ∘ • ∘  ⋯  ∘  ε_{1} + ε_{2} − ε_{3} − ε_{4}}^{IV  D_{5}  • ∘ ∘^{⧸^{⊗}}_{⧹∘}  ε_{n−3} + ε_{n−2} + ε_{n−1} + ε_{n}}^{V  E_{6}  ^{⊗} ∘ ∘ ∘ ∘ •  2ε_{1} + ε_{6} + ε}^{V I  E_{7}  ^{⊗} ∘ ∘ ∘ ∘ ∘ •  2ε_{1} + ε_{7} + ε_{8}}

The correspondence between nodes and simple roots is as in Table 4 of the Appendix.

The CR-graphs of type I are called special CR-graph. All the others are called non-special CR-graphs.

Let (Γ, ϑ(Γ)) be a CR-graph. We fix a Cartan subalgebra 𝔥 of the associated compact Lie algebra 𝔤 and define the element Z(Γ) = iB^{−1} ∘ ϑ(Γ) ∈ 𝔥. Then Z(Γ) is a contact element and we call the corresponding contact manifold (M(Γ) = G/L, D_{Z(Γ)}) the contact manifold associated with the CR-graph (Γ, ϑ(Γ)). Note that M(Γ) is special if and only if the CR-graph is special.

Using the concept of CR-graph, the results of our classification may be stated as follows.

Theorem 1.8. Let M = G/L be a homogeneous CR manifold with a non-primitive non-standard CR structure (D_{Z}, J). Denote by π: G/L → F_{Z} = G/K the natural (non-holomorphic) fibration associated with the contact structure D_{Z} and by π′: G/L → F_{2} = G/Q the unique CRF fibration with primitive fiber Q/L over a flag manifold F_{2} = G/Q with invariant complex structure J_{2}.

Then Q ⊃ K and the sequence of fibering

M = G/L ⟶ F_{Z} = G/K ⟶ F_{2} = G/Q

is holomorphic with respect to the standard CR structure (D, J_{s}) on M, associated to (D, J), the corresponding complex structure J_{s} on F_{Z} and the complex structure J_{2} on F_{2}.

Moreover, the painted Dynkin graph Γ associated to the flag manifolds F_{1} = F_{Z}, F_{2} with complex structures J_{1} = J_{s} and J_{2}, respectively, is a CR graph and (up to a transformation from the Weyl group) Z is proportional to Z(Γ).

Conversely, if Γ is a CR-graph, then there exists a unique homogeneous contact manifold (M = G/L, D_{Z}) such that Z = Z(Γ) and F_{Z} = F_{1}(Γ) = G/K. The complex structure J_{1}(Γ) defines the unique standard CR structure (D_{Z}, J_{1}(Γ)) on M such that the sequence of fibrations

M = G/L ⟶ F_{Z} = F_{1}(Γ) = G/K ⟶ F_{2}(Γ) = G/Q

is holomorphic w.r.t. (D_{Z}, J_{1}(Γ)), J_{1}(Γ) and J_{2}(Γ). The space of the invariant CR structures (D_{Z}, J) on M such that the projection π′: M → F_{2}(Γ) is holomorphic, is parameterized by the points of the unit disc D ∈ ℝ^{2}. The center of D corresponds to the CR structure (D_{Z}, J_{1}(Γ)) and the other points correspond to the non-standard CR structures. Moreover a CR structure is non-standard if and only if it induces a primitive CR structure on the fiber Q/L.

2. Basic facts about CR structures.

Definition 2.1.

(1) A CR structure on a manifold M is a pair (D, J), where D ⊂ T M is a distribution on M and J ∈ End D, J^{2} = −1, is a complex structure on D.

(2) A CR structure (D, J) is called integrable if J satisfies the following integrability condition:

J([JX, Y] + [X, JY]) ∈ D ,

[JX, JY] − [X, Y] − J([JX, Y] + [X, JY]) = 0  (2.1)

for any pair of vector fields X, Y in D.

In the sequel, by CR manifold we will understand a manifold M with integrable CR structure.

If (D, J) is a CR structure then the complexification D^{ℂ} ⊂ T^{ℂ}M of the distribution D is decomposed into a sum D^{ℂ} = D^{10} + D^{01} of two mutually conjugated (D^{10} = D̄^{01}) J-eigendistributions with eigenvalues i and −i. The integrability condition (2.1) means that these eigendistributions are involutive (i.e. closed under the Lie bracket).

The codimension of a CR structure (D, J) is defined as the codimension of the distribution D. Remark that a codimension zero CR structure is the same as a complex structure on a manifold. A codimension one CR structure (D, J) is also called a CR structure of hypersurface type, because such is the structure which is induced on a real hypersurface of a complex manifold. In this case the distribution D can be described locally as the kernel of a 1-form θ. The form θ defines an Hermitian symmetric bilinear form

L_{q}^{θ}: D_{q} × D_{q} → ℝ

given by

L^{θ}(v, w) = (dθ)(v, Jw)

for any v, w ∈ D. It is called the Levi form. Remark that the 1-form θ is defined up to multiplication by a function f everywhere different from zero and that L^{fθ} = fL^{θ}. In particular, the conformal class of a Levi form depends only on the CR structure.

A CR structure (D, J) of hypersurface type is called non-degenerate if it has non-degenerate Levi form or, in other words, if D is a contact distribution. In this case a 1-form θ with ker θ = D is called contact form.

A smooth map φ: M → M′ of one CR manifold (M, D, J) into another one (M′, D′, J′) is called holomorphic map if

a) φ_{∗}(D) ⊂ D′;

b) φ_{∗}(Jv) = J′φ_{∗}(v) for all v ∈ D.

In particular, we may speak about CR transformation of a CR manifold (M, D, J) as a transformation φ such that φ and φ^{−1} are CR maps. In general, the group of all CR transformations is not a Lie group, but it is a Lie group when (D, J) is of hypersurface type and it is Levi non-degenerate.

Definition 2.2. A CR manifold (M, D, J) is called homogeneous if it admits a transitive Lie group G of CR transformations.

Our aim is to classify compact homogeneous codimension one non-degenerate CR manifolds. The following result shows that we may identify any such manifold with a quotient space G/L of a compact Lie group G.

Theorem 2.3. [12] Let (M, D, J) be a compact non-degenerate CR manifold of hypersurface type. Assume that it is homogeneous, i.e. that there exists a transitive Lie group A of CR transformations. Then a maximal compact connected subgroup G of A acts on M transitively and one may identify M with the quotient space G/L where L is the stabilizer of a point p ∈ M.

Now we fix some notations. If the opposite is not stated, we will assume that a CR structure is of hypersurface type, integrable and Levi non-degenerate.

The Lie algebra of a Lie group is denoted by the corresponding gothic letter. For any subset A of a Lie group G or of its Lie algebra 𝔤, we denote by C_{G}(A)

and C_{𝔤}(A) its centralizer in G and 𝔤, respectively. Z(G) and Z(𝔤) denote the center of a Lie group G and a Lie algebra 𝔤. By homogeneous manifold M = G/L we mean a homogeneous manifold of a compact connected Lie group G with connected stability subgroup L and such that the action of G on M is effective.

3. Compact Homogeneous Contact Manifold.

3.1 Homogeneous contact manifolds of a compact Lie group G.

Let M = G/L be a homogeneous manifold of a compact Lie group G with connected stabilizer L.

A 1-form θ ∈ 𝔤^{∗} on the Lie algebra 𝔤 of G is called contact form if it is Ad_{𝔩}- invariant and vanishes on 𝔩 = Lie L. Such form defines a global invariant 1-form θ on the manifold M which is a contact form of the contact distribution D = ker θ. This establishes a 1-1 correspondence between invariant contact structures D on M and contact 1-forms θ ∈ 𝔤^{∗} up to a scaling (see e.g.[2]).

Fix now an Ad_{G}-invariant Euclidean metric B on 𝔤 and denote by 𝔩^{⊥} the orthogonal complement to 𝔩 in 𝔤.

The vector Z = B^{−1} ∘θ which corresponds to a contact form θ is called a contact element of the manifold M = G/L.

It is characterized by the properties that:

(1) Z ∈ 𝔩^{⊥} and

(2) the centralizer C_{𝔤}(Z) = 𝔩 ⊕ ℝZ. Hence, we have the following

Proposition 3.1. There exists a natural bijection between invariant contact structures on a homogeneous manifold M = G/L and contact elements Z defined up to a scaling.

We will denote by D_{Z} the contact structure on M defined by a contact element Z. A homogeneous manifold M = G/L with an invariant contact structure D is called homogeneous contact manifold.

Proposition 3.1 implies the following

Corollary 3.2. Let G/L be a homogeneous contact manifold of a compact Lie group G which acts effectively. Then the the center Z(G) of G has dimension 0 or

Moreover, if Z(G) is one dimensional, then any contact element Z has not zero orthogonal projections Z_{Z(𝔤)}, Z_{𝔤′} on Z(𝔤) and 𝔤′ = [𝔤, 𝔤], and the stability subalgebra 𝔩 can be written as

𝔩 = [C_{𝔤′}(Z_{𝔤′})]_{φ} =^{def} {X = Y + φ(Y) , Y ∈ C_{𝔤′}(Z_{𝔤′})}

where φ: C_{𝔤′}(Z_{𝔤′}) → Z(𝔤) ≈ ℝ is a non-trivial Lie algebra homomorphism.

Proof. Clearly C_{𝔤}(Z) ⊃ Z(𝔤). If dim Z(𝔤) ≥ 2 then 𝔩 ∩ Z(𝔤) ≠ {0} and this contradicts the fact that G acts effectively. The other claims follow immediately. □

Now we associate with a homogeneous contact manifold (M = G/L, D_{Z}) a flag manifold

F_{Z} =^{def} Ad_{G} Z = Ad_{G′}(Z_{𝔤′})

where K = C_{G}(Z) is the centralizer of the contact element Z. We will call F_{Z} the flag manifold associated to a contact element Z.

Note that the contact form θ = B ∘ Z is a connection (form) in the S^{1} bundle π : G/L → F_{Z} and that the corresponding contact structure D = ker θ is the horizontal distribution of this connection.

We describe now all homogeneous contact manifolds (G/L, D_{Z}) with given associated flag manifold F = G/K of a semisimple Lie group G.

Consider the orthogonal reductive decomposition

𝔤 = 𝔨 + 𝔪

associated with the flag manifold F = G/K.

We say that an element Z of the center Z(𝔨) is 𝔨-regular if it generates a closed 1-parametric subgroup of G and the centralizer C_{G}(Z) = K.

One can check that if Z is 𝔨-regular, then the subalgebra

𝔩_{Z} = 𝔨 ∩ (Z)^{⊥}

generates a closed subgroup, which we denote by L_{Z}. Therefore

Proposition 3.3. Let F = G/K be a flag manifold of a semisimple Lie group G. There is a natural 1-1 correspondence

Z ⟺ (G/L_{Z}, D_{Z})

between the 𝔨-regular elements Z ∈ 𝔤 (determined up to a scaling) and the homogeneous contact manifolds (G/L, D) associated flag manifold F = G/K.

Proof. The proof is straightforward. □

3.2 Invariant contact structures on a contact manifold M = G/L.

Now we describe all invariant contact structures on a given homogeneous manifold M = G/L. We will show that generically there is no more then one such structure.

Definition 3.4. A homogeneous manifold G/L is called homogeneous contact manifold of non-special type (respectively, of special type or, shortly, special) if it admits a unique (respectively, more then one) invariant contact structure.

3.2.1 Main examples of special homogeneous contact manifolds.

Let 𝔤 be a compact semisimple Lie algebra, 𝔥 a Cartan subalgebra of 𝔤 and R the root system of the pair (𝔤^{ℂ}, 𝔥^{ℂ}).

Recall that a root α ∈ R defines a 3-dimensional regular subalgebra 𝔤^{ℂ}(α) with standard basis given by the root vectors E_{α}, E_{−α} and

H_{α} = [E_{α}, E_{−α}] = \frac{2}{|α|^{2}} B^{−1} ∘ α  (3.1)

verifying the relation [H_{α}, E_{±α}] = ±2E_{±α}. Its intersection with 𝔤 is a 3-dimensional compact subalgebra denoted by 𝔤(α). We will call 𝔤(α) the subalgebra associated with the root α and denote by G(α) the 3-dimensional subgroup of the adjoint group G = Int(𝔤) = Aut(𝔤)^{0} generated by 𝔤(α).

Note that any two such subalgebras are conjugated by an inner automorphism of 𝔤 if and only if the corresponding roots have the same length.

Fix a system R^{+} of positive roots of R and put R^{−} = −R^{+}. The highest root μ of R^{+} defines the following gradation of the complex Lie algebra 𝔤^{ℂ}:

𝔤^{ℂ} = 𝔤_{−2} + 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} + 𝔤_{2} ,  (3.2)

where

𝔤_{−2} = ℂE_{−μ}  𝔤_{2} = ℂE_{μ}  𝔤_{0} = ℂH_{μ} + 𝔤′_{0}  𝔤′_{0} = C_{𝔤^{ℂ}}(𝔤(μ))  (3.3)
𝔤_{−1} = ∑_{β∈R^{−}\({−μ}∪R_{o})} ℂE_{β}  𝔤_{1} = ∑_{β∈R^{+}\({μ}∪R_{o})} ℂE_{β}

and R_{o} = {α ∈ R, α ⊥ μ} is the root system of the subalgebra 𝔤_{0} = C_{𝔤}(H_{μ}). (3.2) is called the gradation associated with the highest root. The explicit decomposition (3.2) for any simple complex Lie algebra is given in

Table 1 of the Appendix.

Denote by 𝔩 = C_{𝔤}(𝔤(μ)) = 𝔤′_{0} ∩ 𝔤 the centralizer of 𝔤(μ) in 𝔤 and by L the corresponding connected subgroup of G. It is easy to check that L = C_{G}(𝔤(μ)).

Lemma 3.5. Let G be a compact simple Lie group without center and let L = C_{G}(𝔤(μ)) be as defined above. Then any non zero vector Z ∈ 𝔤(μ) is a contact element of the manifold G/L. In particular, G/L is a homogeneous contact manifold of special type.

Proof. Observe that Z ∈ 𝔤(μ) is a contact element if and only if C_{𝔤}(Z) = 𝔩 + ℝZ and then g · Z is contact for any g ∈ G(μ). Since G(μ) acts transitively on the unit sphere of 𝔤(μ), the Lemma follows from the fact that

C_{𝔤}(iH_{μ}) = 𝔤_{0} ∩ 𝔤 = 𝔩 + ℝ(iH_{μ})

and hence that iH_{μ} is a contact element. □

Remark that the contact manifolds M = G/L = G/C_{G}(𝔤(μ)), with G simple, carry invariant 3-Sasakian structure and they exhaust all homogeneous 3-Sasakian manifolds (see [6]).

3.2.2 Classification of special homogeneous contact manifolds.

The previous examples almost exhaust the class of special homogeneous contact manifolds. In fact, we have the following classification theorem.

Theorem 3.6. Let M = G/L be a special homogeneous contact manifold of a compact Lie group G. Then the group G is simple and either L is the centralizer of the subalgebra 𝔤(μ) associated with the highest root and M is a homogeneous 3-Sasakian manifold or G = G_{2} and L is the centralizer of the subalgebra 𝔤(ν) associated with a short root ν.

Proof. We prove first that if G is not semisimple and, hence, dim Z(𝔤) = 1, then a contact element Z is unique up to a scaling and M is not special. Indeed, we have the decomposition

𝔨 = C_{𝔤}(Z) = 𝔩 ⊕ ℝZ = 𝔩 + Z(𝔤)

since Z(𝔤) ∩ 𝔩 = 0, by effectivity. The line ℝZ is determined uniquely as the orthogonal complement to 𝔩 in 𝔨 = 𝔩 + Z(𝔤).

Now we may assume that 𝔤 is semisimple. We need the following:

Lemma 3.7. Let 𝔤 be compact semisimple and let 𝔩 ⊂ 𝔤 be a closed subalgebra, which contains no ideal of 𝔤. If there exist two not proportional vectors Z, Z′ ∈ 𝔩^{⊥} such that

C_{𝔤}(Z) = 𝔩 + ℝZ, 𝔩 + ℝZ′ ⊆ C_{𝔤}(Z′) ,

then 𝔤 is simple and there exists a root α ∈ R such that:

(1) 𝔩 = C_{𝔤}(𝔤(α));

(2) Z, Z′ ∈ 𝔤(α) and C_{𝔤}(Z′) = C_{𝔤}(𝔤(α)) + ℝZ′;

(3) C_{𝔤}(𝔩) = Z(𝔩) + 𝔤(α);

(4) for any root β which is orthogonal to α, α ± β is not a root.

Proof. We put 𝔨 = C_{𝔤}(Z) and consider the orthogonal decomposition

𝔤 = 𝔨 + 𝔪 = (𝔩 + ℝZ) + 𝔪 .

Denote by R the root system of the complex Lie algebra 𝔤^{ℂ} with respect to a Cartan subalgebra 𝔥^{ℂ} which is the complexification of a Cartan subalgebra 𝔥 of 𝔨. Then the element Z′ can be written as

Z′ = cZ + ∑_{i=1}^{k} c_{i}E_{α_{i}}

for some root vectors E_{α_{i}} and constants c, c_{i}. The condition [𝔩, Z′] = 0 implies α_{i}(𝔥 ∩ 𝔩) = 0 if c_{i} ≠ 0. Since 𝔥 ∩ 𝔩 is of codimension one in 𝔥, there exist exactly two (proportional) roots with this property, say α and −α. This shows that 𝔩 ⊂ C_{𝔤}(𝔤(α)). Moreover, since Z ∈ 𝔥 ∩ 𝔩^{⊥}, we obtain also that Z is proportional to H_{α} = [E_{α}, E_{−α}] and (1) follows. In particular, 𝔤 must be simple and now (2) is clear. (3) follows from (2).

To prove (4), assume that there is a root β which is orthogonal to α and such that α + β is a root. Then the vector E_{β} + E_{−β} ∈ 𝔤^{ℂ} does not belong to 𝔩^{ℂ} = C_{𝔤^{ℂ}}(𝔤(α)), but it is orthogonal to Z (since Z is proportional to H_{α}) and belongs to the centralizer of Z: contradiction. □

Now we conclude the proof of Theorem 3.6. Let G be a compact semisimple Lie group and let Z, Z′ two non-proportional contact elements for G/L. By Lemma 3.7, G is simple and L = C_{G}(𝔤(α)). By direct inspection of the root systems of simple Lie groups, a root α verifies the condition (4) of Lemma 3.7 if and only if it is a long root or if it is a short root in the G_{2} type system. This concludes the proof. □

3.3 Isotropy representation of a homogeneous contact manifold.

Let M = G/L be a homogeneous contact manifold with invariant contact structure D associated to a contact element Z. Let 𝔤 = 𝔩 + ℝZ + 𝔪 be the corresponding orthogonal decomposition. Fix a Cartan subalgebra 𝔥 of 𝔤 which belongs to 𝔨 = 𝔩 + ℝZ = Z(𝔨) + 𝔨′ (where 𝔨′ = [𝔨, 𝔨] is the semisimple part of 𝔨). Then

𝔥 = Z(𝔨) + 𝔥′ = Z(𝔩) + ℝZ + 𝔥′ ,

where we denote by 𝔥′ a Cartan subalgebra of 𝔨′. Remark that 𝔥(𝔩) = Z(𝔩) + 𝔥′ is a Cartan subalgebra of 𝔩.

Denote by R (resp. R_{o}) the root system of 𝔤^{ℂ} (resp. 𝔨^{ℂ}) w.r.t. the Cartan subalgebra 𝔥^{ℂ} and let R′ = R \ R_{o}. We will denote by 𝔥(ℝ) the standard real form of 𝔥, spanned by R, that is

𝔥(ℝ) = 𝔥 ∩ B^{−1}(< R >) .

We put 𝔱 = 𝔷(𝔨) ∩ 𝔥(ℝ). Then Z ∈ i𝔱 and we may identify

ϑ = −iθ = −iB(Z, ·)

with the corresponding element in 𝔱^{∗} ⊂ 𝔥(ℝ)^{∗} = span_{ℝ}R.

Consider the decomposition of the 𝔨^{ℂ}-module 𝔪^{ℂ} into sum of irreducible 𝔨^{ℂ}- modules

𝔪^{ℂ} = ∑ 𝔪(γ) .  (3.4)

Here, 𝔪(γ) stands for the irreducible 𝔨^{ℂ}-module with highest weight γ ∈ R′.

The following Lemma states a well known property of flag manifolds (see e.g. [4] or [3]).

Lemma 3.8. The 𝔨^{ℂ}-modules 𝔪(γ) are pairwise not equivalent and, in particular, the decomposition (3.4) is unique. The moduli 𝔪(γ) are irreducible also as 𝔩^{ℂ}- modules.

Proof. We only need to check that a module 𝔪(γ) is irreducible also as an 𝔩^{ℂ}- module. But it is sufficient to observe that the semisimple parts of 𝔩^{ℂ} and of 𝔨^{ℂ} coincide and to recall that, whenever dim_{ℂ} 𝔪(γ) > 1, the semisimple part of 𝔨^{ℂ} acts non-trivially and irreducibly on 𝔪(γ). □

From Lemma 3.8 we derive the following technical proposition, which will be useful in the following sections.

Proposition 3.9. Let M = G/L be a homogeneous contact manifold and let Z be a contact element for M. Assume that G ≠ G_{2} or that G = G_{2} and ϑ = −iB ∘ Z is not proportional to a short root of R.

Then for any irreducible 𝔨^{ℂ}-module 𝔪(γ) there exists at most one distinct 𝔨^{ℂ}- module 𝔪(γ′) which is isomorphic to 𝔪(γ) as 𝔩^{ℂ}-module.

This is the case if and only if the highest weights γ and γ′ are ϑ-congruent, i.e. γ′ = γ + λϑ for some real number λ.

Corollary 3.10. Let M and Z as in the Proposition 3.9. Then:

a) if the modules 𝔪(γ), 𝔪(γ′) are equivalent as 𝔩^{ℂ}-modules, then for any weight α ∈ R′ of 𝔪(γ), there exists exactly one weight α′ ∈ R′ of 𝔪(γ′) which is ϑ-congruent to α;

b) for any root α ∈ R′ there exists at most one root α′ ∈ R′ which is ϑcongruent to α, i.e. such that α′ = α + λϑ for some real number λ ≠ 0.

Proof of Proposition 3.9. Observe that two irreducible 𝔩^{ℂ}-modules 𝔪(γ) and 𝔪(γ′) are isomorphic if and only if their highest weights γ|_{𝔥(𝔩)} and γ′|_{𝔥(𝔩)} coincide. This occurs if and only if γ′ = γ + λϑ for some λ ∈ ℝ.

Assume now that there exist three distinct isomorphic 𝔩^{ℂ}-modules 𝔪(γ), 𝔪(γ′) and 𝔪(γ′′). Then R̃ = span_{ℝ}(γ, γ′, γ′′) ∩ R is a 2-dimensional root system and γ, γ′ and γ′′ belong to the straight line γ +ℝϑ. Checking all 2-dimensional root systems, 2A_{1}, A_{2}, B_{2}, G_{2}, we conclude that this is possible only if R̃ is of type B_{2} or G_{2} and ϑ is proportional to a short root. We claim that both these cases cannot occur.

If R̃ has type G_{2}, then R̃ = R which contradicts to the assumptions.

If R̃ has type B_{2}, one of the roots γ, γ′, γ′′ is orthogonal to ϑ and this is impossible because

ϑ^{⊥} ∩ R = R_{o} = R \ R′

while γ, γ′, γ′′ ∈ R′. □

4. General Properties of Compact Homogeneous CR manifolds.

4.1 Infinitesimal description of invariant CR structures.

Let (M = G/L, D_{Z}) be a homogeneous contact manifold of a connected compact Lie group G with connected stabilizer L and let

𝔤 = 𝔩 + ℝZ + 𝔪  (4.1)

the associated orthogonal decomposition where 𝔨 = C_{𝔤}(Z) = 𝔩 + ℝZ.

Definition 4.1. A complex subspace 𝔪^{10} of 𝔪^{ℂ} is called holomorphic if

i) 𝔪^{10} ∩ 𝔪^{01} = \{0\}, where 𝔪^{01} = \overline{𝔪^{10}} and ’bar’ denotes the complex conjugation with respect to the real subspace 𝔤;

ii) 𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01}; iii) 𝔩^{ℂ} + 𝔪^{10} is a complex subalgebra of 𝔤^{ℂ}.

In the following we will referee to condition iii) as the integrability condition.

Note that if the integrability condition holds, also 𝔩^{ℂ} + 𝔪^{01} is a subalgebra. Furthermore, any holomorphic subspace 𝔪^{10} defines an ad_{𝔩}-invariant complex structure J on 𝔪, whose (+i)− and (−i)-eigenspaces are exactly 𝔪^{10} and 𝔪^{01}.

Proposition 4.2. Let (M = G/L, D_{Z}) be a compact homogeneous contact manifold and 𝔤 = 𝔩+ℝZ +𝔪 be the associated decomposition. Then there exists a natural one to one correspondence between the set of invariant CR structures (D_{Z}, J) on M and the set of holomorphic subspaces 𝔪^{10} of 𝔪^{ℂ}.

Proof. Recall that, under the natural identification of ℝZ + 𝔪 with the tangent space T_{eL}M, we have that 𝔪 = D_{Z}|_{eL}. Moreover, any invariant CR structure (D_{Z}, J) defines a decomposition D_{Z}^{ℂ} = D^{10} + D^{01} into two mutually conjugated invariant integrable distributions. Then one can easily check that the complex subspace 𝔪^{10} = D_{eL}^{10} ⊂ 𝔪^{ℂ} is a holomorphic subspace.

Conversely an holomorphic subspace 𝔪^{10} and its conjugate subspace 𝔪^{01} = \overline{𝔪^{10}} are ad_{𝔩}-invariant and also Ad_{L}-invariant since L is connected. Then they can be extended to two invariant integrable complex distributions D^{10} and D^{01} such that D^{ℂ} = D^{10} + D^{01} with D^{10} ∩ D^{01} = 0. Hence they may be considered as eigendistributions of an invariant CR structure (D_{Z}, J) on M. □

4.2 Standard CR structures.

We want to show how to construct an invariant CR structure (D_{Z}, J) on a homogeneous contact manifold (M = G/L, D_{Z}) starting from an invariant complex structure J on the associated flag manifold F_{Z}.

Let F = G/K be a flag manifold and let 𝔤 = 𝔨 + 𝔪 the associated reductive decomposition. Recall that an invariant complex structure J_{F} on F is associated with a decomposition 𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01} such that

a) 𝔪^{01} = \overline{𝔪^{10}} ; b) 𝔭 = 𝔨^{ℂ} + 𝔪^{10} is a subalgebra of 𝔤^{ℂ} .  (4.2)

We say that 𝔪^{10} is the holomorphic subspace associated with J_{F}.

It is known that 𝔭 is a parabolic subalgebra, with reductive part 𝔨^{ℂ} and nilradical 𝔪^{10}. Moreover, we can always choose a system of positive roots R^{+} for 𝔤^{ℂ}, such that 𝔪^{10} is generated by root vectors E_{α}, with α ∈ R^{+}. We say that such system R^{+} is compatible with the complex structure J_{F}.

Let (M = G/L, D_{Z}) be a homogeneous contact manifold, 𝔤 = (𝔩+ℝZ)+𝔪 = 𝔨+𝔪 the corresponding decomposition and F_{Z} = G/K the associated flag manifold. Any invariant complex structure J_{F} on F_{Z} induces an invariant CR structure (D_{Z}, J), which is the one corresponding to the same holomorphic subspace 𝔪^{10} ⊂ 𝔪^{ℂ} as J_{F}.

Definition 4.3. An invariant CR structure (D, J) on a homogeneous contact manifold (M = G/L, D), which is induced by an invariant complex structure J_{F} on the associated flag manifold F = G/K, is called standard CR structure.

Remark 4.4. Since any flag manifold admits at least one invariant complex structure, we may conclude that any homogeneous contact manifold (G/L, D), with G compact, admits an invariant CR structure (D, J).

The following Lemma gives an algebraic characterization of the standard CR structures.

Lemma 4.5. An invariant CR structure (D, J) on a homogeneous contact manifold (M = G/L, D) is standard if and only if the corresponding complex structure J on 𝔪 is Ad(K)-invariant.

Proof. The proof is straightforward. □

Since the description of all invariant complex structures on flag manifolds is well known (see [10], [4], [5], [3]), the problem of classification of the invariant CR structures on compact homogeneous spaces reduces to the description of nonstandard invariant CR structures.

The following proposition reduces the problem to the case of G semisimple.

Proposition 4.6. Let (M = G/L, D) be a contact manifold of a compact Lie group G with dim Z(G) = 1. Then any invariant CR structure with underlying distribution D is standard.

Proof. It follows immediately from the fact that any Ad(L)-invariant decomposition 𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01} is clearly also Ad(K)-invariant, since K = L · Z(G). □

4.3 Holomorphic fibering of homogeneous CR manifolds.

Let (M = G/L, D, J) be a homogeneous CR manifold with a standard CR structure J associated to a complex structure J_{F} on the associated flag manifold F = G/K. Then the natural projection

π: G/L ⟶ F = G/K

is a G-equivariant holomorphic fibration.

More generally we give the following definition.

Definition 4.7. Let M = G/L be a homogeneous manifold with invariant CR structure (D, J).

(1) Any G-equivariant holomorphic fibering

π: M = G/L ⟶ F = G/Q

of (M, D, J) over a flag manifold F = G/Q equipped with an invariant complex structure J_{F} is called CRF fibration;

(2) we say that a homogeneous CR manifold (M = G/L, D, J) is primitive if it doesn’t admit a non-trivial CRF fibration;

(3) a non-primitive homogeneous CR manifold (M = G/L, D, J), admitting a CRF fibration with typical fiber S^{1}, is called circular.

Remark that any standard CR structure is circular and that the typical fiber Q/L of a CRF fibration carries a natural invariant CR structure.

The following Lemma gives a characterization of primitive CR structures.

Lemma 4.8. A homogeneous CR manifold (G/L, D, J) admits a non-trivial CRF fibration if and only if there exists a proper parabolic subalgebra 𝔭 = 𝔯 + 𝔫 ⊊ 𝔤^{ℂ} (here 𝔯 is a reductive part and 𝔫 the nilpotent part) such that

a) 𝔯 = (𝔭 ∩ 𝔤)^{ℂ} ;  b) 𝔩^{ℂ} + 𝔪^{10} ⊂ 𝔭 ;  c) 𝔩^{ℂ} ⊊ 𝔯 .

In this case, G/L admits a CRF fibration with basis G/Q, where Q is the connected subgroup generated by 𝔮 = 𝔯 ∩ 𝔤.

Proof. Suppose that (M = G/L, D, J) is non-primitive and let π: G/L → G/Q be a CRF fibration over a flag manifold F = G/Q with invariant complex structure J_{F}. Consider the decompositions associated to J and J_{F}

𝔤 = 𝔩 + ℝZ + 𝔪 , 𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01} ,
𝔤 = 𝔮 + 𝔪′ , 𝔪′^{ℂ} = 𝔪′^{10} + 𝔪′^{01} .

Since π is holomorphic and non-trivial, the subalgebra 𝔩^{ℂ}+𝔪^{10} is properly contained in the parabolic subalgebra 𝔭 = 𝔮^{ℂ} + 𝔪′^{10}, with reductive part 𝔮^{ℂ} = (𝔤 ∩ 𝔭)^{ℂ}. Furthermore, since the fiber has positive dimension, 𝔩 ⊊ 𝔮.

Conversely, if 𝔭 = 𝔯 + 𝔫 ⊂ 𝔤^{ℂ} is a parabolic subalgebra with reductive subalgebra 𝔯 = 𝔮^{ℂ}, where 𝔮 = 𝔭 ∩ 𝔤, then we may consider the orthogonal decompositions

𝔤 = 𝔮 + 𝔪′ , 𝔤^{ℂ} = 𝔯 + 𝔪′^{ℂ} = 𝔯 + 𝔫 + 𝔫′ ,

where 𝔫′ = 𝔫^{⊥} ∩ 𝔪′^{ℂ}. By the remarks at the beginning of §4.2, there exists a unique invariant complex structure J_{F} with associated holomorphic space 𝔪′^{10} = 𝔫. Therefore if 𝔩^{ℂ} + 𝔪^{10} ⊂ 𝔭, 𝔩 ⊊ 𝔮 and Q is the reductive subgroup generated by 𝔮, it is clear that π: G/L → G/Q is a non-trivial CRF fibration. □

4.4 The anticanonical map of a homogeneous CR manifold.

Let (M = G/L, D_{Z}, J) be a homogeneous CR manifolds of a compact Lie group G and

𝔤 = 𝔩 + ℝZ + 𝔪 , 𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01}

the associated decompositions of 𝔤 and of 𝔪^{ℂ}.

To characterize the circular invariant CR structures we recall the definition of anticanonical map of a homogeneous CR manifold introduced for the first time in [1]. It is a G-equivariant holomorphic map

ϕ : M = G/L ⟶ Gr_{k}(𝔤^{ℂ})

into the Grassmanian of complex k-planes, k = dim_{ℂ}(𝔩^{ℂ} + 𝔪^{01}), of 𝔤^{ℂ} given by

ϕ: gL ↦ Ad_{g}(𝔩^{ℂ} + 𝔪^{01}) .

Due to the existence of standard holomorphic G-equivariant embedding

ı: Gr_{k}(𝔤^{ℂ}) ⟶ ℂP^{N} , N = ( \begin{matrix}dim 𝔤^{ℂ} \\ k \end{matrix} ) − 1 ,

V = span(e_{1}, …, e_{k}) ↦^{ı} [V] = ℂ(e_{1} ∧ ⋯ ∧ e_{k}) ,

we may consider ϕ as a G-equivariant map into ℂP^{N}. To prove that the map ϕ is holomorphic it is sufficient to check that the linear map

ϕ_{∗}: D_{0} = ker θ|_{T_{0}M} = 𝔪 ⟶ T_{[𝔩^{ℂ}+𝔪^{01}]}Gr_{k}(𝔤^{ℂ})

commutes with the complex structure.

Let v = X + X̄ ∈ 𝔪, where X ∈ 𝔪^{10}. Then

ϕ_{∗}(v) = ad_{(X+X̄)}([𝔩^{ℂ} + 𝔪^{01}]) = ad_{X}([𝔩^{ℂ} + 𝔪^{01}]) .

Therefore

ϕ_{∗}(Jv) = ϕ_{∗}(iX − iX̄) = ad_{iX}([𝔩^{ℂ} + 𝔪^{01}]) = i ad_{X}([𝔩^{ℂ} + 𝔪^{01}]) = iϕ_{∗}(v) .

This shows that the map ϕ is holomorphic.

Remark that the stabilizer Q of the point [𝔩^{ℂ} + 𝔪^{01}] in ϕ(M) = G/Q is the normalizer Q = N_{G}(𝔩^{ℂ} + 𝔪^{01}).

Now, the following theorem establishes some important properties of the anticanonical map.

Theorem 4.9. Let

ϕ : M = G/L ⟶ Gr_{k}(𝔤^{ℂ})

be the anticanonical map of a homogeneous CR manifold (M = G/L, D_{Z}, J).

(1) If the CR structure is circular, then the image ϕ(M) = G/Q is a flag

manifold and ϕ is a CRF fibration with fiber S^{1}. In this case the normalizer in 𝔤 of 𝔩^{ℂ} + 𝔪^{01} is

𝔮 = N_{𝔤}(𝔩^{ℂ} + 𝔪^{01}) = 𝔩 + ℝZ′

where Z′ ≠ 0 is an element from the centralizer of 𝔩 in 𝔤. Moreover, 𝔮 is the Lie algebra of the stabilizer of the point [𝔩^{ℂ} + 𝔪^{01}] ∈ ϕ(M) in G.

(2) If the CR structure is not circular then the image ϕ(M) = G/Q is a homogeneous CR manifold with CR structure induced by the complex structure of Gr_{k}(𝔤^{ℂ}) and ϕ: M → ϕ(M) is a finite covering.

Proof. We first need the following Lemma, which in fact was proved in [1].

Lemma 4.10. Let G/Q = ϕ(G/L) be the image of the anticanonical map. Then dim Q/L ≤ 1.

Proof. We need to prove that dim 𝔮/𝔩 ≤ 1, where 𝔮 = N_{𝔤}(𝔩^{ℂ} + 𝔪^{01}) is the stability subalgebra of the flag manifold G/Q. Since 𝔤 = 𝔩 + ℝZ + 𝔪, it is sufficient to check that 𝔮 ∩ 𝔪 = 0. Let v ∈ 𝔮 ∩ 𝔪. Then

B(Z, [v, 𝔩^{ℂ} + 𝔪^{01}]) ⊂ B(Z, 𝔩^{ℂ} + 𝔪^{01}) = {0}

and in particular

{0} = B(Z, [v, 𝔩 + 𝔪]) = −B([v, Z], 𝔩 + 𝔪) .

This means that v ∈ N_{𝔤}(Z) = 𝔨 = 𝔩 + ℝZ and hence that v ∈ 𝔨 ∩ 𝔪 = {0}. □

Let us prove (1). In case dim Z(G) = 1, the invariant CR structure is standard and the normalizer N_{G}(𝔩^{ℂ} + 𝔪^{01}) coincides with L · Z(G). Therefore the image ϕ(G/L) of the anticanonical map coincides with the flag manifold F = G/K = G/C_{G}(Z) associated to the contact structure. This proves (1) in this case.

Assume now that G is semisimple and consider a CRF a fibration with S^{1} fiber, i.e. a G-equivariant holomorphic map π: M = G/L → F = G/Q onto a flag manifold with invariant complex structure J_{F}. As usual, consider the associated decompositions

𝔤 = 𝔩 + ℝZ + 𝔪 , 𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01} ,
𝔤 = 𝔮 + 𝔪′ , 𝔪^{ℂ} = 𝔪′^{10} + 𝔪′^{01} ,

corresponding to the CR structure of M and to the complex structure J_{F} on F = G/Q. Clearly, the subalgebra 𝔩^{ℂ} + 𝔪^{01} is a subalgebra of the parabolic subalgebra 𝔭 = 𝔮^{ℂ} + 𝔪′^{01}.

Since the fiber is one dimensional, we may express 𝔮 as 𝔮 = 𝔩 + ℝZ′ for some element Z′ ∈ 𝔷(𝔮) and, from the previous observations,

[Z′, 𝔩^{ℂ} + 𝔪^{01}] ⊂ [Z′, 𝔪^{01}] ⊂ [𝔮^{ℂ}, 𝔮^{ℂ}] + 𝔪′^{01} ⊂ [𝔮^{ℂ}, 𝔮^{ℂ}] + 𝔪^{01} ,

where we used the fact that 𝔪^{01} ⊂ 𝔮^{ℂ} + 𝔪′^{01} and that 𝔪′^{01} ⊂ (𝔩^{ℂ} + ℂZ)^{⊥}. Since the semisimple parts of 𝔩^{ℂ} and 𝔮^{ℂ} coincide, it follows that

[Z′, 𝔩^{ℂ} + 𝔪^{01}] ⊂ 𝔩^{ℂ} + 𝔪^{01} .

In particular Z′ ∈ N_{𝔤}(𝔩^{ℂ}+𝔪^{01}). Lemma 4.10 implies that 𝔩+ℝZ′ = N_{𝔤}(𝔩^{ℂ}+𝔪^{01}) and that the anticanonical map is a CRF fibration onto the image ϕ(M) = G/N_{G}([𝔩^{ℂ} + 𝔪^{01}]) with fiber S^{1}. The other part of the claim is clear.

To prove (2) it is sufficient to observe that if the CR structure is not circular, the fiber of the anticanonical map cannot be 1-dimensional, because otherwise it would give a CRF fibration with S^{1} fiber. Lemma 4.10 shows that in this case φ : G/L → ϕ(G/L) is a finite covering. The other part of the claim follows immediately by the holomorphicity and the G-equivariance of ϕ. □

4.5 Any circular CR structure is standard.

Now we will prove that any circular CR structure is standard. Let (D, J) be a circular CR structure on G/L and let Z_{D} be a contact element associated to D. By Z_{J} = Z′ we denote the element given in Theorem 4.9 (1) such that the normalizer 𝔮 = N_{𝔤}(𝔩^{ℂ} + 𝔪^{01}) is of the form

𝔮 = N_{𝔤}(𝔩^{ℂ} + 𝔪^{01}) = 𝔩 + ℝZ_{J} .

¿From Theorem 4.9 and Lemma 4.5, the circular CR structure is standard if and only if ℝZ_{D} = ℝZ_{J}. Since Z_{D}, Z_{J} ∈ C_{𝔤}(𝔩) ∩ (𝔩)^{⊥}, if G/L is a contact manifold of non-special type, then dim C_{𝔤}(𝔩) ∩ (𝔩)^{⊥} = 1 and hence any circular CR structure is standard. We prove now that the same holds also for the special contact manifolds. In fact

Theorem 4.11. Let G/L be a homogeneous contact manifold of a compact Lie group G. An invariant CR structure (D, J) on G/L is circular if and only if it is standard.

Proof. By the previous remark we may assume that G/L is a special contact manifold. We have to prove that ℝZ_{D} = ℝZ_{J}.

Since C_{𝔤}(Z_{J}) ⊃ 𝔩 + ℝZ_{J} and C_{𝔤}(Z_{D}) = 𝔩 + ℝZ_{D}, by Lemma 3.7, we have that C_{𝔤}(Z_{J}) = 𝔩 + ℝZ_{J} and hence that Z_{J} is a contact element too.

It follows immediately from Theorem 3.6 that 𝔤 admits the following orthogonal decompositions

𝔤 = 𝔩 + 𝔞 + 𝔫 = 𝔩 + ℝZ_{D} + 𝔪 ,

where 𝔞 = 𝔤(α), for some root α of 𝔤^{ℂ}, and 𝔩 = C_{𝔤}(𝔞). Moreover, Z_{D}, Z_{J} ∈ 𝔞. Since by Lemma 3.7 (3) 𝔫^{ℂ} contains no trivial 𝔩^{ℂ}-module, we may decompose 𝔪^{ℂ} as follows

𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01} ,  𝔪^{10} = 𝔞^{10} + 𝔫^{10} ,  𝔪^{01} = 𝔞^{01} + 𝔫^{01} ,

where 𝔞^{10} = 𝔞^{ℂ} ∩ 𝔪^{10}, 𝔫^{10} = 𝔫^{ℂ} ∩ 𝔪^{10} and 𝔞^{01} = \overline{𝔞^{10}}, 𝔫^{01} = \overline{𝔫^{10}}. On the other hand, since Z_{J} ∈ N_{𝔤}(𝔩^{ℂ} + 𝔪^{10}) ∩ 𝔞, then

[Z_{J}, 𝔞^{10}] ⊂ 𝔞^{10} , [Z_{J}, 𝔞^{01}] ⊂ 𝔞^{01} ,

and hence Z_{J} is orthogonal to 𝔞^{10} + 𝔞^{01} = 𝔪^{ℂ} ∩ 𝔞^{ℂ}, because 𝔞^{ℂ} ≃ 𝔰𝔩_{2}(ℂ). From this it follows that Z_{J} and Z_{D} are proportional, because they are two elements of 𝔞, which are both orthogonal to the 2-plane 𝔞 ∩ 𝔪. □

5. Classification of CR structures on special contact manifolds.

We describe here all invariant CR structures (D_{Z}, J) on a special contact manifold G/L. Recall that in this case G is simple and L = C_{G}(𝔤(α)), by Theorem 3.6, where either α = μ is the highest root or G = G_{2} and α = ν is a short root.

We have the following orthogonal decomposition of 𝔤

𝔤 = 𝔩 + ℝZ + 𝔪 = 𝔩 + 𝔞 + 𝔫 ,  (5.1)

where 𝔞 = 𝔤(α) is the 3-dimensional subalgebra associated with the root α, Z = iH_{α} ∈ 𝔞 and 𝔩 = C_{𝔤}(𝔞) is its centralizer.

Let (D, J) be an invariant CR structure on G/L which is determined by the contact element Z = iH_{α} and by the decompositions

𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01} = 𝔞^{10} + 𝔫^{10} + 𝔞^{01} + 𝔫^{01} ,  (5.2)

where 𝔞^{10} = 𝔞^{ℂ} ∩ 𝔪^{10}, 𝔫^{10} = 𝔫^{ℂ} ∩ 𝔪^{10} and 𝔪^{01} = 𝔞^{01} + 𝔫^{01} = \overline{𝔪^{10}}.

Since 𝔞^{ℂ} ≃ 𝔰𝔩_{2}(ℂ) and 𝔞^{10} + 𝔞^{01} is the orthogonal complement to ℂZ in 𝔞^{ℂ}, we can write 𝔞^{10} = ℂZ′, for some Z′ ∈ 𝔪^{ℂ} ∩ 𝔞^{ℂ}.

Note that a regular element X of 𝔞^{ℂ} (up to rescaling) can be always identified with iH_{α}, where α is a root of 𝔤^{ℂ} with respect to some Cartan subalgebra 𝔥 of 𝔤^{ℂ} and such that 𝔞 = 𝔤(α). In particular, since any contact element Z of 𝔤 is a regular element for 𝔞^{ℂ}, it can always be identified with iH_{α}.

If α = μ is the highest root, the eigenspace decomposition of ad_{H_{α}} gives the gradation

𝔤^{ℂ} = 𝔤_{−2} + 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} + 𝔤_{2} ,  (5.3)

which is described in (3.3) and Table 1. Table 1 shows that for 𝔤^{ℂ} ≠ A_{ℓ}, the 𝔤_{0}-moduli 𝔤_{±1} are irreducible, their dimension is dim_{ℂ} 𝔤_{±1} = 1/2 dim_{ℂ} 𝔫^{ℂ} and

[𝔤_{±1}, 𝔤_{±1}] = 𝔤_{±2} .  (5.4)

If 𝔤^{ℂ} = A_{ℓ}, each 𝔤_{0}-module 𝔤_{±1} decomposes into two non-equivalent irreducible 𝔤_{0}-moduli: 𝔤_{±1} = 𝔤_{±1}^{(1)} + 𝔤_{±1}^{(2)}. Moreover, the following relations hold:

[𝔤_{1}^{(i)}, 𝔤_{1}^{(i)}] = {0} = [𝔤_{−1}^{(i)}, 𝔤_{−1}^{(i)}] ,  [𝔤_{1}^{(i)}, 𝔤_{1}^{(j)}] = 𝔤_{2} ,  [𝔤_{−1}^{(i)}, 𝔤_{−1}^{(j)}] = 𝔤_{−2} , (5.5)
[𝔤_{1}^{(i)}, 𝔤_{−2}] = 𝔤_{−1}^{(j)} ,  [𝔤_{−1}^{(i)}, 𝔤_{2}] = 𝔤_{1}^{(j)} ,  \overline{𝔤_{1}^{(i)}} = 𝔤_{−1}^{(i)}  (i ≠ j) .  (5.6)

The moduli 𝔤_{1}^{(i)} and 𝔤_{−1}^{(j)} (i ≠ j) are isomorphic as 𝔤′_{0}-moduli and, for both values of i, dim_{ℂ} 𝔤_{±1}^{(i)} = 1/4 dim_{ℂ} 𝔫^{ℂ}.

When 𝔤^{ℂ} = G_{2} and α = ν = ε_{1} is a short root, the eigenspace decomposition of operator ad_{H_{ν}} defines the following gradation of 𝔤^{ℂ}:

𝔤^{ℂ} = 𝔤_{−3} + 𝔤_{−2} + 𝔤_{−1} + 𝔤_{0} + 𝔤_{1} + 𝔤_{2} + 𝔤_{3} ,  (5.7)

where

𝔤_{0} = 𝔤′_{0} + ℂH_{ν} , 𝔤′_{0} = C_{𝔤^{ℂ}}(𝔤(ν)) =< E_{±(ε_{2}−ε_{3})}, H_{ε_{2}−ε_{3}} > ,
𝔤_{2} = ℂE_{ε_{1}} , 𝔤_{−2} = ℂE_{−ε_{1}} , 𝔤^{ℂ}(ν) = 𝔤_{2} + 𝔤_{−2} + ℂH_{ε_{1}} ,
𝔤_{1} =< E_{−ε_{3}}, E_{−ε_{2}} > , 𝔤_{3} =< E_{ε_{1}−ε_{3}}, E_{ε_{1}−ε_{2}} > ,
𝔤_{−i} = \overline{𝔤_{i}}  for i = 1, 3  (5.8)

(see Appendix for notation).

Note that all subspaces 𝔤_{i} are irreducible 𝔤′_{0}-moduli and that the moduli 𝔤_{j}, j = ±1, ±3, are equivalent 𝔤′_{0}-moduli. Furthermore, [𝔤_{±1}, 𝔤_{±1}] = 𝔤_{±2} and [𝔤_{±3}, 𝔤_{±3}] = {0}.

The following Theorem gives the complete classification of the invariant CR structures on special contact manifolds.

Theorem 5.1. Let (M = G/L, D_{Z}) be a special contact manifold. Then:

a) if G ≠ SU_{ℓ+1}, there exists (up to a sign) a unique invariant CR structure (D_{Z}, J), and it is the standard one.

b) if G = SU_{2} and hence M = SU_{2}, there exists a 1-1 correspondence between the invariant CR structures (determined up to a sign) and the points of the unit disc

D = {t ∈ ℂ , |t| < 1 } .  (5.9)

Under the identification Z = iH_{α}, a point t ∈ D corresponds to the CR structure (D, J_{t}) with the holomorphic subspace

𝔪^{10} = ℂ(E_{α} + tE_{−α}) .  (5.10)

The CR structure (D_{Z}, J_{t}) is standard if and only if t = 0.

c) if G = SU_{ℓ}, ℓ > 2, and hence M = SU_{ℓ}/U_{ℓ−2}, the set of all invariant CR structures (determined up to a sign) consists of:

c.1) the standard CR structure (D_{Z}, J^{(0)}), induced by the invariant complex structure J^{(0)} on F_{Z} = SU_{ℓ}/T^{2} · SU_{ℓ−2}, which is the natural complex structure of the twistor space of the Wolf space Gr_{2}(ℂ^{ℓ}) = SU_{ℓ}/S(U_{2} · U_{ℓ−2});

c.2) two families (D_{Z}, J_{t}) and (D_{Z}, J′_{t}) of invariant CR structures, parameterized by the points of the unit disc D. Under the identification Z = iH_{μ}, the CR structures (D_{Z}, J_{t}) and (D_{Z}, J′_{t}) have the following holomorphic subspaces

(for J_{t} ) 𝔪^{10} = ℂ(E_{μ} + tE_{−μ}) + 𝔤_{1}^{(1)} + 𝔤_{−1}^{(2)} ,  (5.11)
(for J′_{t} ) 𝔪′^{10} = ℂ(E_{μ} + tE_{−μ}) + 𝔤_{1}^{(2)} + 𝔤_{−1}^{(1)} ,  (5.12)

where

𝔤^{ℂ} = 𝔩^{ℂ} + ℂZ + 𝔪^{ℂ} = g′_{0} + ℂ(iH_{μ}) + (𝔤_{−2} + 𝔤_{−1} + 𝔤_{1} + 𝔤_{2}) .

A CR structure (D_{Z}, J_{t}) or (D_{Z}, J′_{t}) is standard if and only if t = 0.

Corollary 5.2. Let (M = G/L, D_{Z}) be a special contact manifold with G = SU_{ℓ}.

(1) if M = SU_{2}, then (M, D_{Z}) admits (up to a sign) only one standard CR structure and one family of non-standard CR structures parameterized by the punctured unit disc D \ {0} ⊂ ℂ; any non-standard CR structure is primitive;

(2) if M = SU_{ℓ}/U_{ℓ−2}, ℓ > 2, then (M, D_{Z}) admits (up to a sign) exactly three standard CR structures, namely (D_{Z}, J^{(0)}), (D_{Z}, J_{0}) and (D_{Z}, J′_{0}) that are induced by the three invariant complex structures of the corresponding flag manifold F_{Z} = SU_{ℓ}/T^{2} · SU_{ℓ−2}, plus two families of non-standard CR structures, parameterized by the punctured unit disc; any non-standard CR structure is non-primitive and admits a CRF fibration

π: M = SU_{ℓ}/U_{ℓ−2} ⟶ Gr_{2}(ℂ^{ℓ}) = SU_{ℓ}/S(U_{2} × U_{ℓ−2})

with a fiber SO_{3} over the Wolf space Gr_{2}(ℂ^{ℓ}) equipped with its (unique up to a sign) complex structure.

Remark 5.3. The complex structures J_{0} and J′_{0} on F_{Z} coincide on the fibers of the twistor fibration π : F_{Z} → Gr_{2}(ℂ^{ℓ}) but are projected into two opposite complex structures of Gr_{2}(ℂ^{ℓ}).

Proof. The proof of Theorem 5.1 reduces to classification of the decompositions (5.2), which correspond to an integrable CR structure, for each special contact manifold (G/L, D_{Z}). For any decomposition (5.2), the subspace 𝔞^{10} can be expressed as 𝔞^{10} = ℂZ′ for some suitable Z′ ∈ 𝔞^{ℂ}. Therefore we have to cases:

(1) Z′ is a regular element of 𝔞^{ℂ};

(2) Z′ is a non-regular (hence nilpotent) element of 𝔞^{ℂ} ≃ 𝔰𝔩_{2}(ℂ).

Case (1):

First of all we show that this case may occur only if 𝔤 = 𝔰𝔲_{ℓ}.

Consider first 𝔞 = 𝔤(μ), with μ long root of the simple group G. Since Z′ is regular, we may assume that Z′ = iH_{μ} and we may consider the corresponding graded decomposition (5.3). Recall that 𝔩^{ℂ} = C_{𝔤}(𝔤(μ)) = 𝔤′_{0}.

Hence the subalgebra 𝔟 = 𝔩^{ℂ} + 𝔪^{10} is contained in

𝔩^{ℂ} + 𝔪^{10} = 𝔤′_{0} + 𝔞^{10} + 𝔫^{10} = 𝔤′_{0} + ℂH_{μ} + 𝔫^{10} = 𝔤_{0} + 𝔫^{10} ⊂ 𝔤_{0} + 𝔤_{1} + 𝔤_{−1}

since 𝔫^{ℂ} ⊂ 𝔤_{1} + 𝔤_{−1}, being orthogonal to 𝔞^{ℂ} = ℂH_{μ} + 𝔤_{2} + 𝔤_{−2}. In case 𝔤^{ℂ} ≠ A_{ℓ}, 𝔤_{1} and 𝔤_{−1} are irreducible 𝔤_{0}-modules and hence either 𝔤_{1} or 𝔤_{−1} is included in 𝔫^{10}. However [𝔤_{1}, 𝔤_{1}] = 𝔤_{2} and [𝔤_{−1}, 𝔤_{−1}] = 𝔤_{−2}, and hence there is no subalgebra 𝔟 of 𝔤_{0} + 𝔤_{1} + 𝔤_{−1} which contains 𝔤_{0} properly. Hence 𝔤^{ℂ} = A_{ℓ}. In this case, each 𝔤_{±1} decomposes into two not equivalent irreducible 𝔤_{0}-moduli 𝔤_{±1}^{(i)}, i = 1, 2, of dimension equal to 1/4 dim_{ℂ} 𝔫^{ℂ}, which verify (5.5) and (5.6). Like before it is easy to check that the 𝔤′_{0}-moduli decomposition of 𝔫^{10} has the form 𝔫^{10} = 𝔤_{1}^{(i)} + 𝔤_{−1}^{(j)} for some choice of i and j. If i = j = 1, then 𝔫^{01} = \overline{𝔫^{10}} = 𝔤_{1}^{(1)} + 𝔤_{−1}^{(1)} = 𝔫^{10} and this contradicts the condition 𝔫^{10} ∩ 𝔫^{01} = {0}. A similar contradiction arises when i = j = 2.

In conclusion, if α = μ is a long root, then 𝔤^{ℂ} = A_{ℓ} and for any fixed 𝔞^{10} there exist at most two CR structures, i.e. those corresponding to the following two possibilities for 𝔫^{10}:

𝔫^{10} = 𝔤_{1}^{(1)} + 𝔤_{−1}^{(2)} ,  𝔫^{10} = 𝔤_{1}^{(2)} + 𝔤_{−1}^{(1)} .  (5.13)

It remains to consider the case in which G = G_{2} and 𝔞 = 𝔤(ν), with ν short root of 𝔤^{ℂ}. We assume that Z′ = iH_{ν} and we consider the corresponding graded decomposition (5.7).

Then 𝔩^{ℂ} + 𝔪^{10} is contained in

𝔩^{ℂ} + 𝔪^{10} = 𝔩^{ℂ} + 𝔞^{10} + 𝔫^{10} = 𝔤′_{0} + ℂH_{ν} + 𝔫^{10} ⊂ 𝔤_{0} + 𝔤_{1} + 𝔤_{−1} + 𝔤_{3} + 𝔤_{−3}

because 𝔫^{ℂ} is orthogonal to 𝔞^{ℂ} = ℂH_{ν} + 𝔤_{2} + 𝔤_{−2}. Since 𝔩^{ℂ} + 𝔪^{10} = 𝔤^{0} + 𝔫^{10} is a subalgebra and dim_{ℂ} 𝔤_{±1} = dim_{ℂ} 𝔤_{±3} = \frac{1}{2} dim_{ℂ} 𝔫^{10}, 𝔫^{10} contains two of the four irreducible 𝔤_{0}-moduli 𝔤_{±1} and 𝔤_{±3}. The only possibility for 𝔫^{10}, so that 𝔤_{0} + 𝔫^{10} is a subalgebra, is 𝔫^{10} = 𝔤_{−3} + 𝔤_{3}. This implies that 𝔫^{01} = \overline{𝔫^{10}} = 𝔤_{−3} + 𝔤_{3} = 𝔫^{10} and it contradicts the condition 𝔪^{10} ∩ \overline{𝔪^{10}} = \{0\}.

Now it remains to classify the invariant CR structures on (SU_{ℓ}/U_{ℓ−2}, D_{Z}).

For the following part of the proof, it is more convenient to identify the contact element Z (and no longer Z′) with iH_{μ}. We also consider the decomposition (5.3) determined by Z = iH_{μ}.

Since Z′ is a regular element which is orthogonal to Z = iH_{μ}, it is (up to a factor) of the form

Z′ = E_{μ} + tE_{−μ} ,  |t| ≠ 0 .  (5.14)

Exchanging 𝔞^{10} with 𝔞^{01} = \overline{𝔞^{10}} if necessary (which corresponds to changing sign to the complex structure), we may assume that 0 < |t| ≤ 1.

Since 𝔞^{10} ∩ 𝔞^{01} = {0} and hence E_{μ} + tE_{−μ} and t̄E_{μ} + E_{−μ} are linearly independent, t verifies the condition

det [ \begin{matrix}1 & t \\ t̄ & 1 \end{matrix} ] = 1 − |t|^{2} ≠ 0  (5.15)

and therefore t ∈ D \ {0} = {0 < |t| < 1}.

We claim that for any point t ∈ D \ {0} there exist exactly two invariant CR structures, whose associated subspace 𝔞^{10} is equal to ℂ(E_{μ}+tE_{−μ}). In fact, one can check immediately that the subspaces (5.11) and (5.12) are holomorphic subspaces of 𝔪^{ℂ}. Hence they define two invariant CR structures denoted by (D_{Z}, J_{t}) and (D_{Z}, J′_{t}).

If ℓ = 2 and hence M = SU_{2}, then 𝔫 = {0} and the two CR structures (D_{Z}, J_{t}) and (D_{Z}, J′_{t}) coincide for any t.

Since for any t ≠ 0 the holomorphic subspaces 𝔪^{10} and 𝔪′^{01} are not ad_{Z}- invariant, any CR structure (D_{Z}, J_{t}) or (D_{Z}, J′_{t}) (t ≠ 0) is non-standard by Lemma 4.5.

Case (2):

Since Z′ is not regular, it is a nilpotent element of 𝔞^{ℂ} = 𝔰𝔩_{2}(ℂ) = 𝔤^{ℂ}(α). Then we may always choose a Cartan subalgebra ℂH_{α} of 𝔞 so that Z′ ∈ ℂE_{α}. Furthermore, since the contact element Z is orthogonal to 𝔞^{10}+𝔞^{01} = ℂE_{α}+\overline{ℂE_{α}} = ℂE_{α}+ℂE_{−α}, we may assume (after rescaling) that Z = iH_{α}.

Consider first that α = μ is a long root of G and take the gradation (5.3) of 𝔤^{ℂ} determined with H_{μ}. Then 𝔤_{2} = ℂZ′ = 𝔞^{10} and hence

𝔩^{ℂ} + 𝔪^{10} = 𝔤′_{0} + 𝔤_{2} + 𝔫^{10} ⊂ 𝔤′_{0} + 𝔤_{2} + 𝔤_{1} + 𝔤_{−1} .

Assume that 𝔤^{ℂ} ≠ A_{ℓ}. Then the 𝔤′_{0}-moduli 𝔤_{±1} are irreducible and [𝔤_{±1}, 𝔤_{±1}] = 𝔤_{±2}. Hence the only subalgebra of 𝔤′_{0} + 𝔤_{2} + 𝔤_{1} + 𝔤_{−1}, which properly contains 𝔤′_{0} + 𝔤_{2}, is 𝔤′_{0} + 𝔤_{1} + 𝔤_{2}. Hence 𝔪^{10} = 𝔤_{1} + 𝔤_{2}.

Vice versa, 𝔪^{10} = 𝔤_{1} + 𝔤_{2} is a holomorphic subspace of 𝔪^{ℂ} = (𝔩^{ℂ} + ℂZ)^{⊥} = 𝔤_{0}^{⊥} and hence it corresponds to an invariant CR structure on (G/L, D_{Z}). Since Z = iH_{μ} ∈ N_{𝔤}(𝔤′_{0} + 𝔤_{−1} + 𝔤_{−2}) = N_{𝔤}(𝔩^{ℂ} + 𝔪^{01}), this CR structure is standard.

Assume now that 𝔤^{ℂ} = A_{ℓ} and again consider the decomposition (5.3) determined by Z = iH_{μ}. Since dim_{ℂ} 𝔤_{±1}^{(i)} = 1/4 dim_{ℂ} 𝔫^{ℂ}, the 𝔤′_{0}-module 𝔫^{10} can be written in one of the following five forms:

1) 𝔫^{10} = (𝔤_{1}^{(1)})_{φ} + (𝔤_{−1}^{(1)})_{ψ} ,  2) 𝔫^{10} = 𝔤_{1}^{(1)} + 𝔤_{−1}^{(2)} ,  3) 𝔫^{10} = 𝔤_{1}^{(2)} + 𝔤_{−1}^{(1)} ,

4) 𝔫^{10} = 𝔤_{1} , 5) 𝔫^{10} = 𝔤_{−1} ,

where φ : 𝔤_{1}^{(1)} → 𝔤_{−1}^{(2)} and ψ : 𝔤_{−1}^{(1)} → 𝔤_{1}^{(2)} are two 𝔤′_{0}-equivariant homomorphisms and where (𝔤_{1}^{(1)})_{φ} and (𝔤_{−1}^{(1)})_{ψ} denote the subspaces of the form

(𝔤_{1}^{(1)})_{φ} = {X + φ(X) : X ∈ 𝔤_{1}^{(1)}} , (𝔤_{−1}^{(1)})_{ψ} = {X + ψ(X) : X ∈ 𝔤_{−1}^{(1)}} .

Case 5) cannot occur because in that case [𝔫^{10}, 𝔫^{10}] = 𝔤_{−2} and this contradicts the fact that 𝔤′_{0} + 𝔫^{10} + 𝔤_{2} is a subalgebra.

Also case 1) may not occur. In fact, φ is either trivial or an isomorphism. In case φ is an isomorphism, for any 0 ≠ X ∈ 𝔤_{1}^{(1)}, it is possible to find an element Y ∈ 𝔤_{−1}^{(1)} so that [φ(X), Y] is non-trivial and belongs to 𝔤_{−2}. Hence,

[X + φ(X), Y + ψ(Y)] ≡_{mod 𝔤_{0}+𝔤_{1}+𝔤_{2}} [φ(X), Y] ∈ 𝔤_{−2} .

This contradicts the fact that 𝔩^{ℂ} + 𝔪^{10} is a subalgebra of 𝔤′_{0} + 𝔤_{1} + 𝔤_{2}. We conclude that, if case 1) occurred, 𝔫^{10} = 𝔤_{1}^{(1)} + (𝔤_{−1}^{(1)})_{ψ}. Now, for any X ∈ 𝔤_{1}^{(1)} we may consider an element Y + ψ(Y) ∈ (𝔤_{−1}^{(1)})_{ψ} so that [X, Y] = λH_{μ} for some λ ≠ 0. Hence

[X, Y + ψ(Y)] = λH_{μ} mod 𝔤′_{0} + 𝔤_{2}

This gives a contradiction with the fact that 𝔤′_{0} + 𝔫^{10} + 𝔤_{2} is a subalgebra and the claim is proved.

For the cases 2), 3) and 4), 𝔪^{10} equals one of the following three subspaces

𝔤_{1}^{(1)} + 𝔤_{−1}^{(2)} + 𝔤_{2} , 𝔤_{1}^{(2)} + 𝔤_{−1}^{(1)} + 𝔤_{2} , 𝔤_{1} + 𝔤_{2}  (5.16)

and one can check that any of them is a holomorphic subspace.

By Proposition 4.2, they determine three distinct CR structures denoted by (D, J), (D, J′) and (D, J^{(0)}), respectively. For any of the three subspaces (5.16), the normalizer N_{𝔤}(𝔩^{ℂ} + 𝔪^{01}) contains 𝔤_{0} ∩ 𝔤 = 𝔩 + ℝZ and hence the corresponding CR structures are standard.

Finally, observe that (D, J^{(0)}) is induced by the invariant complex structure J_{F} on the flag manifold F_{Z} = SU_{ℓ}/T^{2} · SU_{ℓ−2} which is associated to the following black-white Dynkin graph

• ∘ ∘  ⋯  ∘ ∘ •

and which is the invariant complex structure of the twistor space of the Wolf space Gr_{2}(ℂ^{ℓ}) = SU_{ℓ}/S(U_{2} · U_{ℓ−2}); the subspaces of J and J′ are the subspaces given in (5.11) and (5.12) for t = 0. The corresponding CR structures coincide if M = SU_{2}.

It remains to consider the case in which G = G_{2} and 𝔞 = 𝔤(ν), where ν is a short root. Consider the decomposition (5.7) determined by H_{ν} so that ℂZ′ = ℂE_{ν} = 𝔤_{2}.

As before, we identify Z with iH_{ν}. We have

𝔩^{ℂ} + 𝔪^{10} = 𝔤′_{0} + 𝔞^{10} + 𝔫^{10} ⊂ 𝔤′_{0} + 𝔤_{2} + 𝔤_{−1} + 𝔤_{1} + 𝔤_{−3} + 𝔤_{3}

because 𝔫^{ℂ} is orthogonal to 𝔞^{ℂ} = ℂH_{ν} + 𝔤_{−2} + 𝔤_{2}. We claim that 𝔤_{3} ⊂ 𝔫^{10}. In fact, for any element X ∈ 𝔫^{10} consider the decomposition

X = X_{−3} + X_{−1} + X_{1} + X_{3} , X_{i} ∈ 𝔤_{i} .

Then, one of the four vectors X, X′ = [E_{ν}, X], X′′ = [E_{ν}, [E_{ν}, X]], X′′′ = [E_{ν}, [E_{ν}, [E_{ν}, X]]] is a non-trivial element of 𝔤_{3} and it belongs to 𝔫^{10}. Since 𝔤_{3} is 𝔤′_{0}-irreducible, the claim follows.

Similarly, we claim that 𝔤_{1} ⊂ 𝔫^{10}. To prove this, take any element X ∈ 𝔫^{10} which has a decomposition of the form

X = X_{−3} + X_{−1} + X_{1} , X_{i} ∈ 𝔤_{i} .

Then, either X or X′ = [E_{ν}, X] or X′′ = [E_{ν}, [E_{ν}, X]] is a non-trivial element of 𝔤_{1} + 𝔤_{3}, with non-vanishing projection on 𝔤_{1}. This implies that 𝔤_{1} ∩ 𝔫^{10} ≠ {0} and hence that 𝔤_{1} ⊂ 𝔫^{10}. Since dim_{ℂ}(𝔤_{1}+𝔤_{3}) = dim_{ℂ} 𝔫^{10}, we conclude that 𝔫^{10} = 𝔤_{1}+𝔤_{3} and that 𝔪^{10} = 𝔤_{1} + 𝔤_{2} + 𝔤_{3}. Indeed, since 𝔩^{ℂ} + 𝔤_{1} + 𝔤_{2} + 𝔤_{3} is always a subalgebra, there exists an integrable CR structure whose associated holomorphic subspace is 𝔪^{10} = 𝔤_{1} + 𝔤_{2} + 𝔤_{3}. Furthermore, N_{𝔤}(𝔩^{ℂ} + 𝔪^{01}) contains Z = iH_{ν} and hence this CR structure is standard. □

Proof of Corollary 5.2. (1) By Theorem 5.1, we only need to prove that any nonstandard CR structure on M = SU_{2} is primitive. For this observe that the only flag manifold of the simple group SU_{2} is SU_{2}/T^{1}. This implies that any non-primitive CR structure on M = SU_{2} is circular and hence standard by Theorem 4.11.

(2) We need to prove that any non-standard CR structure on M = SU_{ℓ}/U_{ℓ−2} is non-primitive and it admits a CRF fibration onto Gr_{2}(ℂ^{ℓ}) = SU_{ℓ}/S(U_{2} · U_{ℓ−2}).

Indeed, note that, if we consider the decomposition (5.3) determined by the regular contact element Z = iH_{μ}, any CR structure (D_{Z}, J_{t}) or (D_{Z}, J′_{t}) corresponding to the holomorphic subspaces defined in (5.11) and (5.12) verify

𝔩^{ℂ} + 𝔪^{01} ⊂ 𝔭 = 𝔤_{0} + 𝔤_{−1}^{(1)} + 𝔤_{1}^{(2)} + 𝔤_{−2} + 𝔤_{2} ,  (5.17)
𝔩^{ℂ} + 𝔪′^{10} ⊂ 𝔭′ = 𝔤_{0} + 𝔤_{−1}^{(2)} + 𝔤_{1}^{(1)} + 𝔤_{−2} + 𝔤_{2} ,  (5.18)

respectively. A reductive part for both subalgebras 𝔭 and 𝔭′ is 𝔯 = 𝔯′ = (𝔩 + 𝔞)^{ℂ}. Therefore, by Lemma 4.8, the CR structures (D, J_{t}) and (D, J′_{t}) are non-primitive and they admit a CRF fibration over the Wolf space SU_{ℓ+1}/S(U_{2}·U_{ℓ−1}) with typical fiber S(U_{2} · U_{ℓ−1})/U_{ℓ−1} = SO_{3}. □

6. Classification of non-standard CR structures.

6.1 Notation.

In all this section,

- (G/L, D_{Z}) denotes a simply connected non-special homogeneous contact manifold of a compact Lie group G; - 𝔨 = C_{𝔤}(Z) = 𝔩 + ℝZ is the orthogonal decomposition of the centralizer 𝔨 of

Z and 𝔪 is the orthogonal complement to 𝔨 in 𝔤;

- 𝔥 ⊂ 𝔨 is a Cartan subalgebra of 𝔨 and hence of 𝔤;

- θ = B ∘ Z|_{𝔥} is the 1-form on 𝔥 dual to Z and ϑ = −iθ = −iB ∘ Z|_{𝔥}; we will refer to both of them as contact forms;

- R (resp. R_{o}) is the root system of (𝔤^{ℂ}, 𝔥^{ℂ}) (resp. of (𝔨^{ℂ}, 𝔥^{ℂ})) and R′ = R \ R_{o};

- E_{α} is the root vector with root α in the Chevalley normalization (see e.g. [7]);

- a subset S ⊂ R is called closed subsystem if (S + S) ∩ R ⊂ S;

- if S is a closed subsystem of roots, then 𝔤(S) ⊂ 𝔤^{ℂ} is the subalgebra generated by the root vectors E_{α}, α ∈ S;

- recall that the root vectors E_{α}, α ∈ R′, span 𝔪^{ℂ};

- 𝔪(α) denotes the irreducible 𝔨^{ℂ}-submodules of 𝔪^{ℂ}, with highest weight α ∈ R′;

- if 𝔪(α) and 𝔪(β) are equivalent as 𝔩^{ℂ}-moduli, we denote by 𝔪(α)+t𝔪(β) the irreducible 𝔩^{ℂ}-module with the highest weight vector E_{α} + tE_{β}, α, β ∈ R′, t ∈ ℂ; note that together with 𝔪(β), these moduli exhaust all the irreducible 𝔩^{ℂ}-submoduli of 𝔪(α) + 𝔪(β) (see Lemma 6.1);

- by Dynkin graph Γ we will understand the Dynkin graph associated with a root system R of a compact semisimple Lie algebra 𝔤; we associate with the nodes of Γ the simple roots of R as in [7] (see Table 4 in the Appendix).

6.2 Preliminaries.

By the results in §5, the classification of invariant CR structures reduces to the classification of non-standard CR structures on homogeneous contact manifolds of non-special type. This will be the contents of §6.3 and §6.4.

In this section we give two important lemmata that settle the main tools for the classification. The first Lemma is an immediate corollary of Proposition 3.9.

Lemma 6.1. Let (M = G/L, D_{Z}, J) be a homogeneous CR manifold associated with holomorphic subspace 𝔪^{10} ⊂ 𝔪^{ℂ} and J the associated complex structure on 𝔪. Assume also that G ≠ G_{2} or that G = G_{2} and that the contact form ϑ is not proportional to a short root of R.

Then a minimal J-invariant 𝔨^{ℂ}-submodule 𝔫 of 𝔪^{ℂ} is either 𝔨^{ℂ}-irreducible (and hence 𝔫 = 𝔪(α) for some α ∈ R′) or it is the sum 𝔪(α) + 𝔪(β) of two such 𝔨^{ℂ}- modules, where the roots α and β are ϑ-congruent (i.e. β = α + λϑ, for some λ ∈ ℝ).

Proof. Consider the decomposition 𝔪^{ℂ} = ∑ 𝔪(γ) into irreducible 𝔨-submodules as in §3.3. The claim follows immediately from the fact that any ad_{𝔩}-invariant complex structure J on 𝔪 preserves the 𝔩^{ℂ}-isotypic components (i.e. the sum of all mutually equivalent irreducible 𝔩^{ℂ}-modules) and that, under the hypotheses of Proposition 3.9, the multiplicity of any irreducible 𝔩-module 𝔪(γ) is less or equal to 2. □

Lemma 6.2. Let (G/L, D_{Z}, J) be a homogeneous CR manifold with non-standard CR structure. Then G is either simple or of the form G = G_{1} × G_{2}, where each G_{i} is simple.

Moreover, if G = G_{1} × G_{2} and R = R_{1} ∪ R_{2} is the corresponding decomposition of the root system, then there exist two roots μ_{1} ∈ R_{1}, μ_{2} ∈ R_{2}, such that the pairs of roots (μ_{1}, −μ_{2}) and (−μ_{1}, μ_{2}) are the only ones which are ϑ-congruent; in particular, ϑ = μ_{1} + μ_{2} is proportional to no root.

Proof. Since the CR structure (D_{Z}, J) is non-standard, the associated complex structure J on 𝔪 is not ad_{𝔨}-invariant; in particular there exists some minimal Jinvariant 𝔨^{ℂ}-module in 𝔪^{ℂ}, which is not 𝔨^{ℂ}-irreducible. By Lemma 6.1, there exist at least two roots α, β, which are ϑ-congruent. Without loss of generality, we may assume that ϑ = α − β.

If ϑ is proportional to some root γ, then this root belongs to some summand 𝔤_{i} of 𝔤, i = 1, …, r. Hence, 𝔨 = C_{𝔤}(Z) contains all other simple summands of 𝔤 and the same holds for 𝔩. By effectivity, this implies that 𝔤 = 𝔤_{1}.

If ϑ = α − β is not proportional to any root and α and β belong to the same summand 𝔤_{1}, then 𝔤 = 𝔤_{1} as before. Assume that they belong to two different summands 𝔤_{1} and 𝔤_{2}. The same arguments of before show that 𝔤 = 𝔤_{1} ⊕ 𝔤_{2} and that ±(α, β) are the only pairs of roots which are ϑ-congruent. □

We will perform the classification by considering separately two cases: when the contact form ϑ is proportional to a root and when it is not proportional to any root. Note that by Lemma 6.2, the first case may occur only when G is simple.

6.3 Case when the contact form is proportional to a root.

Recall that the Weyl group of a simple Lie group acts transitively on the set of roots of the same length. In particular any long root can be considered as a maximal root. Since we assume that the contact manifold (M = G/L, D_{Z}) is nonspecial and G is simple, we may suppose that ϑ is proportional to a short root (i.e. strictly shorter then a long root) and hence G equals either SO_{2n+1}, Sp_{n} or F_{4}. Note that if G = G_{2} then any contact manifold (G_{2}/L, D_{Z}), with contact form ϑ proportional to a short root, is special (see §3.2.2).

Proposition 6.3. Let (G/L, D_{Z}) be a homogeneous non-special contact manifold of a simple group G, such that the contact form ϑ is proportional to a root. Then:

(1) G/L is SO_{2n+1}/SO_{2n−1}, Sp_{n}/Sp_{1}×Sp_{n−2} or F_{4}/SO_{7} and ϑ is proportional to a short root of G;

(2) there exists a 1-1 correspondence between the invariant CR structures on (G/L, D_{Z}) (determined up to a sign) and the points of the unit disc D ⊂ ℂ;

(3) more precisely, any point t ∈ D corresponds to the CR structure (D_{Z}, J_{t}) whose holomorphic subspace 𝔪^{10} is listed in the following table (see §6.1 for notation):

G/L  ϑ  𝔪^{10}^{\frac{SO_{2n+1}}{SO_{2n−1}} = S(S^{2n})  ε_{1}  𝔪(ε_{1}+ε_{2})+t𝔪(−ε_{1}+ε_{2})}^{\frac{Sp_{n}}{Sp_{1}×Sp_{n−2}} = S(ℍP^{n−1}) ε_{1} + ε_{2}  (𝔪(2ε_{1})+t^{2}𝔪(−2ε_{2}))⊕(𝔪(ε_{1}+ε_{3})+t𝔪(−ε_{2}+ε_{3}))}^{\frac{F_{4}}{Spin_{7}} = S(𝕆P^{2})  ε_{1}  \begin{matrix}(𝔪(ε_{1}+ε_{2})+t^{2}𝔪(−ε_{1}+ε_{2}))⊕ \\ (𝔪(1/2(ε_{1}+ε_{2}+ε_{3}+ε_{4}))+t𝔪(1/2(−ε_{1}+ε_{2}+ε_{3}+ε_{4}))) \end{matrix}}

(4) a CR structure (D_{Z}, J_{t}) is standard if and only if t = 0; in all other cases it is primitive.

Proof. For each group G equal to SO_{2ℓ+1}, Sp_{ℓ} or F_{4} we may assume that ϑ is the short root ϑ = ε_{1}, ε_{1} + ε_{2} or ε_{1}, respectively. The associated decomposition 𝔤 = 𝔩 + ℝZ + 𝔪 is given in Table 2 of the Appendix. It is not difficult to determine the decomposition of 𝔪^{ℂ} into irreducible submoduli. The result is given in Table 2. Then one has to find all decompositions 𝔪^{ℂ} = 𝔪^{10} + 𝔪^{01} into two 𝔩^{ℂ}-modules which satisfy the following conditions: a) 𝔪^{01} = \overline{𝔪^{10}}; b) [𝔪^{10}, 𝔪^{10}] ⊂ 𝔪^{10} + 𝔩^{ℂ}. The moduli 𝔪^{10} which satisfy condition a) are of the following form:

G = SO_{2ℓ+1} : 𝔪^{10} = 𝔪_{t}^{10} = 𝔪(ε_{1} + ε_{2}) + t𝔪(−ε_{1} + ε_{2}) ;
G = Sp_{ℓ} :  𝔪^{10} = 𝔪_{t,s}^{10} = (𝔪(2ε_{1}) + s𝔪(−2ε_{2})) ⊕ (𝔪(ε_{1} + ε_{3}) + t𝔪(−ε_{2} + ε_{3})) ;
G = F_{4} :  𝔪^{10} = 𝔪_{t,s}^{10}′ =

= (𝔪(ε_{1}+ε_{2})+s𝔪(−ε_{1}+ε_{2}))⊕(𝔪(1/2(ε_{1}+ε_{2}+ε_{3}+ε_{4}))+t𝔪(1/2(−ε_{1}+ε_{2}+ε_{3}+ε_{4})))

for some s, t ≠ 0. One can easily check that 𝔪_{t}^{10} verifies condition b) for every t. The module 𝔪_{t,s}^{10} verifies condition b) if and only if s = t^{2}. To prove it one should observe that the only brackets between 𝔩^{ℂ}-weight vectors in [𝔪_{t,s}^{10}, 𝔪_{t,s}^{10}], which are non-trivial modulo 𝔩^{ℂ}, are

[E_{ε_{1}+ε_{i}} + tE_{−ε_{2}+ε_{i}},E_{ε_{1}−ε_{i}} + tE_{−ε_{2}−ε_{i}}] ≡
≡ N_{ε_{1}+ε_{i},ε_{1}−ε_{i}}E_{2ε_{1}} + t^{2}N_{−ε_{2}+ε_{i},−ε_{2}+ε_{i}}E_{−2ε_{2}}  mod 𝔩^{ℂ}
[E_{ε_{1}+ε_{i}} + tE_{−ε_{2}+ε_{i}},E_{ε_{2}−ε_{i}} + tE_{−ε_{1}−ε_{i}}] ≡
≡ N_{ε_{1}+ε_{i},ε_{2}−ε_{i}}E_{ε_{1}+ε_{2}} + t^{2}N_{−ε_{2}+ε_{i},−ε_{1}+ε_{i}}E_{−ε_{1}−ε_{2}} mod 𝔩^{ℂ}

By a straightforward computation, it follows that these vectors are in 𝔪_{t,s}^{10} if and only if s = t^{2}.

A similar argument shows that also 𝔪_{t,s}^{10}′ verifies condition b) if and only if s = t^{2}. Observe that up to an exchange between 𝔪^{10} and 𝔪^{01} (which corresponds to

changing the sign of complex structure J), we may always assume that |t| ≤ 1. It remains to check the condition 𝔪^{01}∩𝔪^{10} = \{0\}: in all cases, this implies det [ \begin{matrix}1 & t \\ t̄ & 1 \end{matrix} ] ≠ 0 and hence that |t| < 1.

To prove (4), note that, in all cases listed in the table above, N_{𝔤}(𝔩^{ℂ} + 𝔪^{01}) contains Z only if t = 0 and hence, by Theorems 4.9 and 4.11, this is the only case when the CR structure is standard. Moreover, in all cases, if t ≠ 0 there exists no proper parabolic subalgebra 𝔭 ⊃ 𝔩^{ℂ} which verifies the conditions of Lemma 4.8. □

6.3 Case when the contact form is not proportional to any root.

In this case we obtain the following classification.

Proposition 6.4. Let (M = G/L, D_{Z}) be a contact manifold with contact form ϑ not proportional to any root. If it admits a primitive invariant CR structure (D_{Z}, J), then it is one of the following.

If G is simple then

a) G/L = SO_{2n}/SO_{2n−2}, n > 2, and ϑ is either ε_{1} or, when n = 4, ε_{1} + ε_{2} + ε_{3} ± ε_{4}; moreover the holomorphic subspace of the CR structure (D_{Z}, J) is given by

𝔪^{10} = 𝔪(ε_{1} + ε_{2}) + t𝔪(β)  (6.1)

where β = −ε_{1} + ε_{2}, −ε_{3} − ε_{4} or −ε_{3} + ε_{4} (the last two cases occur only for n = 4) and t belongs to the punctured unit disc D \ {0} ⊂ ℂ;

b) G/L = Spin_{7}/SU_{3} = S(S^{7}) = S^{7}×S^{6}, ϑ = ε_{1}+ε_{2}+ε_{3} and the holomorphic subspace of (D_{Z}, J) is given by

𝔪^{10} = 𝔪(ε_{1} + ε_{2}) + t𝔪(−ε_{3}) + \overline{𝔪(ε_{1} + ε_{2})} + \frac{1}{t}\overline{𝔪(−ε_{3})}  (6.2)

for some t ∈ D \ {0}. If G is not simple then

c) G/L = SU_{2} × SU_{2}/T^{1} = S(S^{3}) = S^{3} × S^{2}, ϑ = (ε_{1} − ε_{2}) − (ε′_{1} − ε′_{2}) and the holomorphic subspace of (D_{Z}, J) is

𝔪^{10} = ℂ(E_{ε_{1}−ε_{2}} + tE_{ε′_{1}−ε′_{2}}) + ℂ(E_{−(ε_{1}−ε_{2})} + \frac{1}{t}E_{−(ε′_{1}−ε′_{2})}) .  (6.3)

In all cases we considered ϑ up to a factor and up to a transformation from the Weyl group W (R), and J up to a sign.

Proposition 6.5. A homogeneous contact manifold (G/L, D_{Z}) with contact form ϑ not proportional to any root, admits a non-standard non-primitive CR structure if and only if it is G-contact diffeomorphic to the contact manifold (M(Γ) = G/L, D_{Z(Γ)}) associated with a non-special CR-graph (Γ, ϑ(Γ)) (see Definition 1.7).

For any invariant CR structure (D_{Z(Γ)}, J) on M(Γ) = G/L the natural projection π : M(Γ) = G/L → F_{2}(Γ) = G/Q is holomorphic w.r.t. the complex structure J_{2}(Γ) or −J_{2}(Γ).

The CR structures for which π is holomorphic w.r.t. J_{2}(Γ) are in 1-1 correspondence with the invariant CR structures on the fiber C = Q/L subordinated to the induced contact structure D_{Z(Γ)} ∩ T C.

More precisely, if

𝔮^{ℂ} = 𝔩^{ℂ} + ℂZ + 𝔪_{C}^{10} + 𝔪_{C}^{01} , 𝔤^{ℂ} = 𝔮^{ℂ} + 𝔪_{J_{2}}^{10} + 𝔪_{J_{2}}^{01}

are the two decompositions of 𝔮^{ℂ} and 𝔤^{ℂ} associated with an invariant CR structure on the fiber C = Q/L and with the complex structure J_{2}(Γ) on F_{2}(Γ), then

𝔪^{10} = 𝔪_{C}^{10} + 𝔪_{J_{2}}^{10}  (6.4)

is the holomorphic subspace of the corresponding CR structure on M(Γ). Moreover, this CR structure is non-standard if and only if the CR structure on C is primitive.

The rest part of the paper is devoted to the proof of Propositions 6.4 and 6.5. We need some additional notations.

For a fixed CR structure (D_{Z}, J), we set

R_{J}^{±} = {α ∈ R′: J(E_{α}) = ±iE_{α}} ,  R_{J} = R_{J}^{+} ∪ R_{J}^{−} ,  R_{𝔢} =^{def} R′ \ R_{J} (6.5)

and we define the subspaces

𝔪_{J}^{±} = ∑_{β∈R_{J}^{±}} ℂE_{β} ,  𝔪_{J} = 𝔪_{J}^{+} + 𝔪_{J}^{−} ,  𝔢 =^{def} ∑_{β∈R_{𝔢}} ℂE_{β} ⊂ 𝔪^{ℂ} .  (6.6)

Note that J is standard if and only if R_{J} = R′. We define also the closed subsystem

R̃_{𝔢} = [R_{𝔢}] =^{def} R ∩ span_{ℝ}(R_{𝔢}) , R̃_{o} = R_{o} ∩ R̃_{𝔢} ,

and we set R′_{o} = R_{o} \ R̃_{o}.

The following Lemma collects some basic properties of these objects.

Lemma 6.6.

(1) R_{J} = −R_{J} and R_{𝔢} = −R_{𝔢};

(2) for any α ∈ R_{𝔢} there exists exactly one root β ∈ R_{𝔢} which is ϑ-congruent to α;

(3) for any pair α, β ∈ R_{𝔢} of ϑ-congruent roots, there exist two uniquely determined complex numbers λ, μ ≠ 0 such that

e_{α,β} = E_{α} + λE_{β} ∈ 𝔪^{10} ,  f_{α,β} = E_{α} + μE_{β} ∈ 𝔪^{01} .  (6.7)

(4) (R_{J}^{±} + R_{o}) ∩ R ⊂ R_{J}^{±} and (R_{𝔢} + R_{o}) ∩ R ⊂ R_{𝔢};

(5) (R_{J}^{±} + R_{𝔢}) ∩ R ⊂ R_{J}^{±} ∪ R_{𝔢} ∪ R_{o}.

Proof. (1) is clear. To see (2), (3) and (4), observe that α ∈ R_{J} if and only if E_{α} belongs to an irreducible 𝔨^{ℂ}-module which is also J-invariant; hence (2), (3) and (4) follow from Lemma 6.1 and Corollary 3.10.

The proof of (5) is the following. Let γ ∈ R_{J}^{+} and α, β ∈ R_{𝔢} a pair of two ϑ-congruent roots. If γ + α ∈ R_{J}^{−}, consider the element f_{−α,−β} ∈ 𝔪^{01} as defined in (6.7). Since E_{γ+α} ∈ 𝔪^{01}, by the integrability condition

[E_{γ+α}, f_{−α,−β}] = CE_{γ} + X ∈ 𝔪^{01} + 𝔩^{ℂ}

for some C ≠ 0 and X ∉ ℂE_{γ}. This implies that γ ∈ R_{J}^{−}: contradiction. □

For any α ∈ R, a root β ∈ R, which is ϑ-congruent to α, is said to be ϑ-dual to α and we say that (α, β) is a ϑ-dual pair. By Corollary 3.10 any root admits at most one ϑ-dual root; by Lemma 6.6 (3), any root in R_{𝔢} has exactly one ϑ-dual root.

Lemma 6.7. Let (α, α′) be a ϑ-dual pair in R_{𝔢}. Then the root subsystem R̃ = R ∩ span_{ℝ}{α, α′} is of type A_{1} + A_{1}. In particular α ⊥ α′ and α ± α′ ∉ R.

Proof. Assume that R̃ ≠ A_{1} + A_{1}. Then R̃ is a root system of type A_{2}, B_{2} or G_{2}. Since by assumptions ϑ = α − α′ is proportional to no root, looking at the corresponding root systems, we find that up to a transformation from the Weyl group there are the following possibilities:

R = A_{2} : α = ε_{0} − ε_{2} , α′ = ε_{2} − ε_{1} ;

R̃ = B_{2} : α = ε_{1} , α′ = −ε_{1} + ε_{2} ;
R̃ = G_{2} : α = −ε_{2} , α′ = −ε_{1} + ε_{2} .

Note that in each of these three cases, α + α′ = β ∈ R.

Case R̃ = A_{2}.

In this case ϑ = (ε_{0} − ε_{2}) − (ε_{2} − ε_{1}) = ε_{0} + ε_{1} − 2ε_{2} and β = α + α′ is orthogonal to ϑ and hence it belongs to R_{o}. Moreover 𝔩^{ℂ} = C_{𝔤^{ℂ}}(Z) contains the subalgebra

𝔩′ = ℂH_{ε_{0}−ε_{1}} + ℂE_{ε_{0}−ε_{1}} + ℂE_{ε_{1}−ε_{0}} .

At the same time, by Lemma 6.6 (3), 𝔪^{01} contains the element f_{ε_{0}−ε_{2},ε_{2}−ε_{1}} = E_{ε_{0}−ε_{2}} + μE_{ε_{2}−ε_{1}}, with some fixed μ ≠ 0. Since 𝔪^{01} is 𝔩^{ℂ}-invariant, 𝔪^{01} contains also the subspace

[E_{ε_{1}−ε_{0}}, ℂf_{ε_{0}−ε_{2},ε_{2}−ε_{1}}] = ℂ(E_{ε_{1}−ε_{2}} − μE_{ε_{2}−ε_{0}}) .

By integrability condition, this implies that

[E_{ε_{0}−ε_{2}} + μE_{ε_{2}−ε_{1}}, E_{ε_{1}−ε_{2}} − μE_{ε_{2}−ε_{0}}] = μ(−H_{ε_{0}−ε_{2}} + H_{ε_{2}−ε_{1}}) ∈ 𝔪^{01} + 𝔩^{ℂ}

and hence we conclude that −H_{ε_{0}−ε_{2}} + H_{ε_{2}−ε_{1}} ∈ 𝔩^{ℂ}. But this cannot be because −H_{ε_{0}−ε_{2}} + H_{ε_{2}−ε_{1}} is not orthogonal to Z = iB^{−1} ∘ ϑ.

Case R̃ = B_{2} or G_{2}.

Then β = α + α′ is not orthogonal to ϑ = α − α′ and, moreover,

(β + ℝϑ) ∩ R = ∅ .

These two facts show that β ∈ R \ (R_{𝔢} ∪ R_{o}) = R_{J}. Changing the sign of α and α′, if necessary, we may assume that β ∈ R_{J}^{+}.

Consider the vector f_{α,α′} = E_{α} + μE_{α′} ∈ 𝔪^{01} which is defined by (6.7). Then \overline{E_{α} + μE_{α′}} = E_{−α} + μ̄E_{−α′} ∈ 𝔪^{10} and by integrability condition its commutator with E_{β} is also in 𝔪^{10} + 𝔩^{ℂ}. Therefore

[E_{−α} + μ̄E_{−α′}, E_{β}] = N_{−α,β}E_{α′} + μ̄N_{−α′,β}E_{α} ∈ 𝔪^{10} .

Hence the coefficient λ of the vector e_{α,α′} defined by (6.7) is

λ = \frac{N_{−α,β}}{μ̄N_{−α′,β}} .  (6.8)

Since we use the Chevalley normalization (see §6.1), N_{−α,β} = ±(p + 1) for any two roots α, β, where p ≥ 0 is the maximal integer such that β + pα ∈ R̃ (see e.g. [7]). Using this formula, we obtain from (6.8) that if R̃ = B_{2}, λμ̄ = ±2, while if R̃ = G_{2}, λμ̄ = ±3.

On the other hand, by integrability condition

[e_{α,α′}, \overline{f_{α,α′}}] = [E_{α} + λE′_{α}, E_{−α} + μ̄E_{−α′}] = H_{α} + λμ̄H_{α′} ∈ 𝔩^{ℂ} .

This means that ϑ(H_{α} + λμ̄H_{α′}) = 0, i.e. that

< ϑ|α > +λμ̄ < ϑ|α′ >= 0 ,

where < ϑ|α >= 2(ϑ, α)/(α, α). Hence for ϑ = α − α′, we obtain

2 − < α′|α > +λμ̄[−2 + < α|α′ >] = 0 .

In case R̃ = B_{2}, < α′|α >= −2 and < α|α′ >= −1 so that λμ̄ = 4/3; in case R̃ = G_{2}, < α′|α >= −3 and < α|α′ >= −1 so that λμ̄ = 5/3. In both cases we get a contradiction with the previously determined values for λμ̄. □

Now we determine the possible types of the root subsystem R̃_{𝔢} = R∩span_{ℝ}(R_{𝔢}).

Lemma 6.8. If R̃_{𝔢} is not of the form A_{1} ∪ A_{1}, then R̃_{𝔢} and R are both indecomposable root systems.

Proof. By Lemma 6.7, we may assume that rank R̃_{𝔢} > 2. Suppose that R̃_{𝔢} is decomposable into two mutually orthogonal subsystems R_{1} and R_{2}. Let α ∈ R_{1} ∩ R_{𝔢}, α′ ∈ R_{2} ∩ R_{𝔢} and β, β′ the ϑ-dual roots of α and α′, respectively. Since ϑ cannot be in the span of R_{1}, it is clear that β ∈ R_{2} and that β′ ∈ R_{1}. Then the identity

ℝϑ = ℝ(α − β) = ℝ(α′ − β′)

implies that α + ρβ′ = ρα′ + β = 0 for some ρ ≠ 0. From this follows that β′ = −α, β = −α′ and that rankR̃_{𝔢} = 2: contradiction.

A similar contradiction arises if we replace R̃_{𝔢} by R. □

Note that by Lemma 6.8, if G = G_{1} × G_{2}, then the only possibility for R̃_{𝔢} is A_{1} ∪ A_{1}.

The following Lemma gives a more detailed description of the root subsystem R̃_{𝔢}.

Lemma 6.9. The root subsystem R̃_{𝔢} has type D_{ℓ}, ℓ > 1 or B_{3} and, up to a factor and a transformation from the Weyl group W = W (R), the contact form ϑ is one of the following:

(1) if R̃_{𝔢} = D_{2} = A_{1} + A′_{1} and α, α′ are roots of the summands A_{1} and A′_{1}, then ϑ = α − α′;

(2) if R̃_{𝔢} = D_{3} or D_{ℓ}, with ℓ > 4, then ϑ = 2ε_{1};

(3) if R̃_{𝔢} = D_{4} then ϑ = 2ε_{1} or ϑ = ε_{1} + ε_{2} + ε_{3} + ε_{4} or ϑ = ε_{1} + ε_{2} + ε_{3} − ε_{4};

(4) if R̃_{𝔢} = B_{3} then ϑ = ε_{1} + ε_{2} + ε_{3}.

Note that in case R̃_{𝔢} = D_{4}, all three contact forms ϑ in (3) are equivalent with respect to automorphisms of the root system.

Proof. From Lemma 6.8, it is sufficient to consider the case when rank R̃_{𝔢} > 2 and R̃_{𝔢} is indecomposable. For each indecomposable root system R̃_{𝔢} we describe, up to a transformation from the Weyl group, all pairs of roots (α, α′), which are orthogonal and such that α ± α′ ∉ R. By Lemma 6.7 such pairs are the only candidates for ϑ-dual pairs in R_{𝔢}. For each case, we consider the corresponding form ϑ = α − α′, and describe all ϑ-dual pairs in R̃_{𝔢} . Then, assuming that α, α′ ∈ R_{𝔢}, we check if the case is possible looking if the ϑ-dual pairs in R_{𝔢} may generate R̃_{𝔢}.

Case (A): R̃_{𝔢} = A_{ℓ}.

Up to a transformation from the Weyl group, the pair (α, α′) is equal to (ε_{1} − ε_{2}, ε_{3} − ε_{4}). Then ϑ = (ε_{1} − ε_{2}) − (ε_{3} − ε_{4}) and the ϑ-dual pairs are (up to sign)

(ε_{1} − ε_{2}, ε_{3} − ε_{4}) ; (ε_{1} − ε_{3}, ε_{2} − ε_{4}) .

Since β = ε_{2} − ε_{3} ∈ R_{o} = (ϑ)^{⊥} ∩ R, then ε_{1} − ε_{3} = α + β ∈ R_{𝔢} and hence also the second ϑ-dual pair is in R_{𝔢}. In particular rankR̃_{𝔢} = 3 and R̃_{𝔢} = A_{3} = D_{3}.

Case (B): R̃_{𝔢} = B_{ℓ}.

We have three possibilities for (α, α′) according to their lengths:

i) (α, α′) = (ε_{1} + ε_{2}, −(ε_{3} + ε_{4})); ii) (α, α′) = (ε_{1} + ε_{2}, −ε_{3}); iii) (α, α′) = (ε_{1}, −ε_{2}).

The last case is not possible, since we assume that ϑ = α − α′ is proportional to no root.

i) ϑ = ε_{1} + ε_{2} + ε_{3} + ε_{4} and the ϑ-dual pairs are (up to sign)

(ε_{1} + ε_{2}, −(ε_{3} + ε_{4})) ; (ε_{1} + ε_{3}, −(ε_{2} + ε_{4})) ; (ε_{1} + ε_{4}, −(ε_{2} + ε_{3})) . (6.9)

As in case (A), one can check that all these ϑ-dual pairs are in R_{𝔢} and that they span a space of dimension 4. Since the ϑ-dual pairs consist of long roots, they cannot generate the root system B_{ℓ} and hence this case is impossible.

ii) ϑ = ε_{1} + ε_{2} + ε_{3} and the ϑ-dual pairs are (up to sign)

(α = ε_{1} + ε_{2}, α′ = −ε_{3}) ; (β = ε_{2} + ε_{3}, β′ = −ε_{1}) ; (γ = ε_{3} + ε_{1}, γ′ = −ε_{2}) .

(6.10)

Again all pairs in (6.10) consist of roots in R_{𝔢}. This implies that rankR̃_{𝔢} = 3.

Case (C): R̃_{𝔢} = C_{ℓ}.

As in case (B), we have three possibilities.

i) (α, α′) = (ε_{1} + ε_{2}, −(ε_{3} + ε_{4})); ii) (α, α′) = (ε_{1} + ε_{2}, −2ε_{3}); iii) (α, α′) = (2ε_{1}, −2ε_{2}).

As in (B), the last case is not possible.

i) ϑ = ε_{1} + ε_{2} + ε_{3} + ε_{4} and the ϑ-dual pairs are given (up to sign) in (6.9). This implies that ±2ε_{i} ∈ R_{J}, i = 1, …, 4, because it has no ϑ-dual root and it is not orthogonal to ϑ. Note also that the roots ε_{i} − ε_{j}, i, j = 1, …, 4, belong to R_{o}, because they are orthogonal to ϑ. Therefore

R_{𝔢} ⊂ {±(ε_{i} + ε_{j}), , 1 ≤ i, j ≤ 4} ⊂ (R_{o} + {±2ε_{i}}) ∩ R ⊂ R_{J}

and this is a contradiction.

ii) ϑ = ε_{1} + ε_{2} + 2ε_{3}.

In this case, up to sign, there is only one ϑ-dual pair, that is (ε_{1} + ε_{2}, −2ε_{3}). On the other hand, ε_{1} − ε_{2} ∈ R_{o} and hence 2ε_{1} = (ε_{1} + ε_{2}) + (ε_{1} − ε_{2}) ∈ R_{𝔢}: contradiction.

Case (D): R̃_{𝔢} = D_{ℓ}.

Since D_{3} = A_{3}, we may assume that ℓ ≥ 4. Then we have three possibilities: i) (α, α′) = (ε_{1} + ε_{2}, −(ε_{3} + ε_{4})); ii) (α, α′) = (ε_{1} + ε_{2}, −(ε_{3} − ε_{4}); iii) (α, α′) = (ε_{1} + ε_{2}, −(ε_{1} − ε_{2})).

i) ϑ = ε_{1} + ε_{2} + ε_{3} + ε_{4} and the ϑ-dual pairs are given (up to sign) in (6.9) and they all belong to R_{𝔢}. Hence the rank of R̃_{𝔢} is 4.

A similar argument shows that rankR̃_{𝔢} = 4 in case ii), where ϑ = ε_{1}+ε_{2}+ε_{3}−ε_{4}.

iii) ϑ = 2ε_{1} and the ϑ-dual pairs are (ε_{1} + ε_{i}, ε_{1} − ε_{i}), with i = 2, … ℓ, they are all in R_{𝔢} and they span the whole system D_{ℓ}.

Case (E): R̃_{𝔢} = E_{6}, E_{7} or E_{8}.

Let α, α′ ∈ R_{𝔢} be a ϑ-dual pair. Since α and α′ are orthogonal, we may included them into a subsystem Π of simple roots. According to the type of R̃_{𝔢}, without loss of generality, we may assume that α′ is one of the following:

R̃_{𝔢} = E_{6} :  α′ = ε_{4} + ε_{5} + ε_{6} + ε ;
R̃_{𝔢} = E_{7} :  α′ = ε_{5} + ε_{6} + ε_{7} + ε_{8} ;
R̃_{𝔢} = E_{8} :  α′ = ε_{6} + ε_{7} + ε_{8} .

For each case, it follows that α = ε_{i} − ε_{i+1} for some i ≠ ℓ − 3 where ℓ = rankR̃_{𝔢}. It can be easily checked that, using permutations of the vectors ε_{i} which belong to the Weyl group of E_{ℓ} and which preserve α′, we may assume that either α = ε_{1} −ε_{2} or α = ε_{ℓ−1} − ε_{ℓ} = − ∑_{i=1}^{ℓ−2} ε_{i} − 2ε_{ℓ}. Therefore we have the following possibilities: if R̃_{𝔢} = E_{6}:

i) α = ε_{1} − ε_{2} and ϑ = α′ − α = −ε_{1} + ε_{2} + ε_{4} + ε_{5} + ε_{6} + ε;

ii) α = ε_{5} − ε_{6} and ϑ = ε_{4} + 2ε_{6} + ε = ε_{6} − ε_{1} − ε_{2} − ε_{3} − ε_{5} + ε;

if R̃_{𝔢} = E_{7}:

iii) α = ε_{1} − ε_{2} and ϑ = −ε_{1} + ε_{2} + ε_{5} + ε_{6} + ε_{7} + ε_{8}; iv) α = ε_{6} − ε_{7} and ϑ = ε_{5} + 2ε_{7} + ε_{8} = ε_{7} − ε_{1} − ε_{2} − ε_{3} − ε_{4} − ε_{6};

if R̃_{𝔢} = E_{8}:

v) α = ε_{1} − ε_{2} and ϑ = −ε_{1} + ε_{2} + ε_{6} + ε_{7} + ε_{8}; vi) α = ε_{7} − ε_{8} and ϑ = ε_{6} + 2ε_{8} = ε_{8} − ε_{1} − ε_{2} − ε_{3} − ε_{4} − ε_{5} − ε_{7}. We claim that all ϑ-dual pairs belong to R_{𝔢} and that the space they generate

has dimension 5 for the cases i), ii) and v); it has dimension 6 for the cases iii) and iv) and dimension 7 for the case vi). Since in all cases the dimension is strictly less then rankR̃_{𝔢} = ℓ, we conclude that the case R_{𝔢} = E_{ℓ} is impossible.

We prove the claim in the cases v) and vi) which occur when R̃_{𝔢} = E_{8}; in all other cases the proof is similar.

For case v), the ϑ-dual pairs are (up to sign) (−ε_{1} + ε_{i}, ϑ + ε_{1} − ε_{i}), where i = 2, 6, 7, 8 and they all belong to R_{𝔢}. These vectors generate a 5-dimensional vector space. In case vi) the ϑ-dual pairs are (−ε_{8}+ε_{i}, ϑ+ε_{8}−ε_{i}), where i = 1, …, 5 or 7, and again they are all in R_{𝔢}. These vectors generate a 7-dimensional vector space.

Case (F): R̃_{𝔢} = F_{4}.

We have the following possibilities:

i) (α, α′) = (ε_{1} + ε_{2}, −(ε_{3} + ε_{4}));

ii) (α, α′) = (ε_{1} + ε_{2}, −ε_{3});

iii) (α, α′) = (ε_{1} + ε_{2}, −(1/2(ε_{1} − ε_{2} + ε_{3} + ε_{4})); iv) (α, α′) = (ε_{1}, −ε_{2}).

Cases i) and iv) are impossible because ϑ = α − α′ should be proportional to no root. The admissible ϑ-dual pairs for case ii) are given by (6.10) and they all belong to R_{𝔢}. They generate a 3-dimensional subspace and this is impossible because rankR̃_{𝔢} = rankF_{4} = 4. A similar argument is applied for case iii). □

Corollary 6.10. If G is simple, then the only possibilities for the pair (R, R̃_{𝔢}) are

(A_{n}, A_{3}) , (A_{n}, B_{3}) , (B_{n}, A_{3}) , (B_{n}, B_{3}) ,
(B_{n}, D_{4}) , (D_{n}, D_{4}) , (D_{n}, D_{n}) , (E_{6}, D_{5}) ,
(E_{7}, D_{6}) , (E_{8}, D_{5}) , (E_{8}, D_{7}) , (F_{4}, A_{3}) , (F_{4}, B_{3}) .

Proof. If R is the root system of the simple Lie group G and (α, α′) is a ϑ-dual pair in R_{𝔢}, then the arguments used in the proof of Lemma 6.9 give the result. □

Lemma 6.11. Let R̃_{o} = R_{o} ∩ R̃_{𝔢}, R′_{o} = R_{o} \ R̃_{o} and α, α′ ∈ R_{𝔢} be a ϑ-dual pair. Then

a) R̃_{𝔢} = [({±α, ±α′} + R̃_{o}) ∩ R] ∪ R̃_{o};

b) R_{𝔢} = ({±α, ±α′} + R̃_{o}) ∩ R and R_{J} ∩ R̃_{𝔢} = ∅;

c) R_{Q} = R_{o} ∪ R_{𝔢} and R_{P} = R_{o} ∪ R_{𝔢} ∪ R_{J}^{+} are closed subsystem of R; R_{Q} is the maximal symmetric subset in R_{P} (i.e. the biggest subset such that −R_{Q} = R_{Q}), and R_{P} is parabolic (i.e. for any root α, either α or −α belongs to it);

d) (R_{Q}+R̃_{𝔢})∩R ⊂ R̃_{𝔢} and hence R_{Q} = R′_{o}∪R̃_{𝔢} is an orthogonal decomposition;

e) for any ϑ-dual pair (α, α′) let R_{o}(α) = (R_{o} + {α}) ∩ R and R_{o}(−α′) = (R_{o} + {−α′}) ∩ R); then the set of roots

S(α, α′) = R_{o} ∪ R_{o}(α) ∪ R_{o}(−α′) ∪ R_{J}^{+}  (6.11)

is a closed parabolic subsystem of R.

Proof. a) When rankR̃_{𝔢} = 2 the claim is trivial.

If R̃_{𝔢} = B_{3}, we may assume that α = ε_{1} + ε_{2}, α′ = −ε_{3} and ϑ = ε_{1} + ε_{2} + ε_{3}. Hence

R̃_{o} = R̃_{𝔢} ∩ (ϑ)^{⊥} = {ε_{i} − ε_{j} , i, j = 1, …, 3} .

By Lemma 6.6 (4),

({±α, ±α′} + R̃_{o}) ∩ R = {±(ε_{i} + ε_{j}), ±ε_{i} , i, j = 1, …, 3} ⊂ R_{𝔢} .

Since R̃_{𝔢} = R̃_{o} ∪ {±(ε_{i} + ε_{j}), ±ε_{i} , i, j = 1, …, 3}, the claim is proved for this case.

If R̃_{𝔢} = D_{ℓ}, the argument is similar. In particular, if ϑ = 2ε_{1}, one obtains that R̃_{o} = D_{ℓ−1} = {±ε_{i} ± ε_{j} , i, j > 1} and R_{𝔢} = {±ε_{1} ± ε_{i}}.

b) follows directly from a).

c) The closeness of R_{Q} and R_{P} follows from Lemma 6.6 (4) and (5) and from point b). The last statement is obvious.

d) The first claim follows from the facts that R̃_{𝔢} = span_{ℝ}(R_{𝔢}) ∩ R and (R_{o} + R_{𝔢}) ∩ R ⊂ R_{𝔢}. This implies that 𝔤(R̃_{𝔢}) is an ideal of the semisimple Lie algebra 𝔤(R_{Q}) and from this also the second claim follows.

e) By point b), R_{o}(α) ∪ R_{o}(−α′) ⊂ R_{𝔢} and hence R_{o} ∪ R_{o}(α) ∪ R_{o}(−α′) ⊂ R_{Q} and S(α, α′) ⊂ R_{P} = R_{Q} ∪ R_{J}^{+}. Since R_{Q} corresponds to a reductive part of the parabolic subalgebra 𝔤(R_{P}) and R_{J}^{+} corresponds to the nilradical, it follows that (S(α, α′) + R_{J}^{+}) ∩ R ⊂ R_{J}^{+}. By d), it remains to check that R_{o}(α) ∪ R_{o}(−α) is a closed subsystem.

In case R̃_{𝔢} = 2A_{1} = D_{2}, we have that R_{o}(α) ∪ R_{o}(−α) = {α, −α′} and hence the claim is trivial.

In case R̃_{𝔢} = D_{ℓ}, ℓ > 2, we may assume that ϑ = 2ε_{1}, α = ε_{1}+ε_{2}, α′ = −(ε_{1}−ε_{2}). Then R̃_{o} = {±ε_{i} ± ε_{j} , 1 < i, j} and

R_{o}(α) = {ε_{1} ± ε_{i} , 1 < i} = R_{o}(−α′)  (6.12)

and the conclusion follows. In case R̃_{𝔢} = B_{3}, then ϑ = ε_{1} + ε_{2} + ε_{3}, α = ε_{1} + ε_{2} and α′ = −ε_{3}. Then R̃_{o} = {±(ε_{i} − ε_{j}) } and

R_{o}(α) = {ε_{i} + ε_{j}} ,  R_{o}(−α′) = {ε_{i}}  (6.13)

and again the conclusion follows. □

Since 𝔤(R_{J}^{+}) is the nilradical of the parabolic subalgebra 𝔤(R_{P}), we may choose an ordering of the roots such that the positive root system R^{+} contains R_{J}^{+}. In the following α denotes the maximal root in R_{𝔢} w.r.t. this ordering and α′ is its associated ϑ-dual root.

Proposition 6.12. The orthogonal complement 𝔪^{ℂ} to 𝔨^{ℂ} in 𝔤^{ℂ} admits the following 𝔨^{ℂ}-invariant decomposition:

(1) if R̃_{𝔢} = B_{3} or D_{2} = A_{1} + A_{1}, then

𝔪^{ℂ} = 𝔢 + 𝔪_{J}^{+} + 𝔪_{J}^{−} = (𝔪(α) + 𝔪(α′) + \overline{𝔪(α)} + \overline{𝔪(α′)}) + 𝔪_{J}^{+} + 𝔪_{J}^{−} ,  (6.14)

(2) if R̃_{𝔢} = D_{ℓ}, then

𝔪^{ℂ} = 𝔢 + 𝔪_{J}^{+} + 𝔪_{J}^{−} = (𝔪(α) + 𝔪(α′)) + 𝔪_{J}^{+} + 𝔪_{J}^{−}  (6.15)

where 𝔪(α) and 𝔪(α′) are irreducible 𝔨^{ℂ}-moduli with highest weights α, α′, which are equivalent and irreducible as 𝔩^{ℂ}-moduli.

In terms of this decomposition, the holomorphic subspace 𝔪^{10} of the CR structure (D_{Z}, J) (up to sign) is of the form

(1) if R̃_{𝔢} = B_{3} or D_{2} = A_{1} + A′_{1}

𝔪^{10} = (𝔪(α) + t𝔪(α′)) + (\overline{𝔪(α)} + \frac{1}{t}\overline{𝔪(α′)}) + 𝔪_{J}^{+} ,  (6.16)

(2) if R̃_{𝔢} = D_{ℓ}

𝔪^{10} = (𝔪(α) + t𝔪(α′)) + 𝔪_{J}^{+}  (6.17)

for some t ∈ {x ∈ ℂ : 0 < |x| < 1} = D \ {0}.

Proof. From (R_{o} + α) ∩ R ⊂ R_{𝔢} and the definition of α, the root α is the maximal weight of the 𝔨^{ℂ}-module in 𝔪^{ℂ} which contains E_{α}. Moreover since α′ is ϑ-congruent to α, then also α′ is the maximal weight of an 𝔩^{ℂ}- and hence 𝔨^{ℂ}-module, and the 𝔩^{ℂ}-moduli 𝔪(α) and 𝔪(α′) are equivalent. By Lemma 6.11 b), it follows that the subspace 𝔢, spanned by the root vectors E_{γ}, γ ∈ R_{𝔢}, is given by

𝔢 = 𝔪(α) + 𝔪(α′) + \overline{𝔪(α)} + \overline{𝔪(α′)} .

Moreover if R̃_{𝔢} = D_{ℓ}, ℓ > 2, R_{o}(α) = R_{o}(−α′) (see (6.12)) and hence 𝔪(α) = \overline{𝔪(α′)} (see also Table 3 in the Appendix).

¿From Lemma 6.1 and the remark in the second to the last point of §6.1, we obtain that the holomorphic subspace 𝔪^{10} is of the form

𝔪^{10} = (𝔪(α) + t𝔪(α′)) + 𝔪_{J}^{+}

when R̃_{𝔢} = D_{ℓ}, ℓ > 2, and of the form

𝔪^{10} = (𝔪(α) + t𝔪(α′)) + (\overline{𝔪(α)} + s\overline{𝔪(α′)}) + 𝔪_{J}^{+}

when R̃_{𝔢} = B_{3} or D_{2} = A_{1} + A_{1}, for some t, s ≠ 0. By exchanging 𝔪^{10} with 𝔪^{01} (which corresponds to changing the sign of J) we may assume that |t| ≤ 1. Using the integrability condition and the assumption that ϑ = α − α′ ∉ R, we have

[E_{α} + tE_{α′}, E_{−α} + sE_{−α′}] = H_{α} + tsH_{α′} ∈ 𝔪^{10} + 𝔩^{ℂ}

and therefore H_{α} + tsH_{α′} ∈ 𝔩^{ℂ}. Using (3.1) we get

0 = ϑ(H_{α} + tsH_{α′}) =< ϑ|α > +ts < ϑ|α′ > .

So

s = −\frac{1}{t} \frac{< ϑ|α >}{< ϑ|α′ >} .

If R̃_{𝔢} = 2A_{1}, it is immediate to check that < ϑ|α >= 2 = − < ϑ|α′ >. In case R̃_{𝔢} = B_{3}, we may assume that ϑ = ε_{1} + ε_{2} + ε_{3}, α = ε_{1} + ε_{2} and α′ = −ε_{3}. Hence again < ϑ|α >= − < ϑ|α′ > and this shows that in both cases s = 1/t.

Finally, the condition 𝔪^{10} ∩ 𝔪^{01} = {0} implies that the vectors E_{α} + tE_{α′} and \overline{E_{−α} + \frac{1}{t}E_{−α′}} = E_{α} + \frac{1}{t̄}E_{α′} are linearly independent, and hence |t| ≠ 1. □

Lemma 6.13.

(1) Let 𝔮^{ℂ} = 𝔨^{ℂ} + 𝔢 and 𝔭 = 𝔮^{ℂ} + 𝔪_{J}^{+}. Then 𝔭 is a parabolic subalgebra of 𝔤^{ℂ}, with reductive part 𝔮^{ℂ} and nilradical 𝔪_{J}^{+}. Moreover, if Q is the connected subgroup of G with Lie algebra 𝔮 = 𝔮^{ℂ} ∩𝔤, then F_{2} = G/Q is a flag manifold and 𝔪_{J}^{+} is the holomorphic subspace of an invariant complex structure J_{2} on F_{2} = G/Q.

(2) The subspace 𝔪_{J_{1}}^{10} = 𝔪(α) + 𝔪(−α′) + 𝔪_{J}^{+} is the holomorphic subspace of an invariant complex structure J_{1} of F_{Z} = G/K.

(3) The natural G-equivariant projections

π : G/L ⟶ G/Q , π′ : G/K ⟶ G/Q

are holomorphic fibrations w.r.t. the CR structure (D_{Z}, J) on G/L, the complex structure J_{1} on F_{Z} = G/K and the complex structure J_{2} on F_{2} = G/Q, respectively. Moreover, the typical fiber C = Q/L of π is either Spin_{7}/SU_{3} = S^{7} × S^{6} or SO_{2ℓ}/SO_{2ℓ−2}, ℓ > 1 and the induced invariant CR structure is primitive.

(4) The typical fiber C = Q/L of π may be equal SO_{4}/SO_{2} = S^{3} × S^{2} only if G = G_{1} × G_{2}, with each G_{i} simple.

Proof. (1) The proof follows from Lemma 6.11 c) and the remark that 𝔭 = 𝔤(R_{P})+ 𝔥^{ℂ} and 𝔮^{ℂ} = 𝔤(R_{Q}) + 𝔥^{ℂ}.

(2) We have to check the conditions a) and b) of (4.2). Condition a) is obvious. Condition b) means that 𝔨^{ℂ} + 𝔪(α) + 𝔪(−α′) = 𝔤(S(α, α′)) + h^{ℂ} is a subalgebra. This follows from Lemma 6.11 e).

(3) The first claim follows from Lemma 4.8.

For the second claim, we recall that we have the following decompositions of the Lie algebras 𝔮^{ℂ} and 𝔩^{ℂ}:

𝔩^{ℂ} = 𝔤(R′_{o}) ⊕ (𝔤(R̃_{o}) + Z(𝔩^{ℂ})) ,

𝔮^{ℂ} = 𝔨^{ℂ} + 𝔢 = 𝔤(R′_{o}) ⊕ (𝔤(R̃_{𝔢}) + Z(𝔮^{ℂ})) .

Since the fiber Q/L has a non-standard CR structure, the group Q′ = Q/N, where N is its kernel of non-effectivity, is semisimple by Corollary 3.2 and Proposition 4.6. Therefore it has Lie algebra 𝔮′^{ℂ} = 𝔤(R̃_{𝔢}) = B_{3} or D_{ℓ}. The corresponding stability subalgebra 𝔩′^{ℂ} = 𝔩^{ℂ}/𝔫^{ℂ} has rank equal to rank(𝔮′^{ℂ}) − 1 and his semisimple part is 𝔤(R̃_{o}) = A_{2} or D_{ℓ−1}. Hence the fiber Q/L = Q′/L′, considered as homogeneous manifold of the effective group Q′, is either Spin_{7}/SU_{3} or SO_{2ℓ}/SO_{2ℓ−2} (note that SO_{7} does not contains SU_{3}). The manifold Spin_{7}/SU_{3} can be identified with the unit sphere bundle S(Spin_{7}/G_{2}) = S(S^{7}) = S^{7} × S^{6}.

The holomorphic subspace 𝔪^{10}(Q/L) of the CR structure of the fiber Q/L is of the form (𝔪(α) + t𝔪(α′)) + (\overline{𝔪(α)} + 1/t\overline{𝔪(α′)}) for some t ≠ 0 and the minimal 𝔨^{ℂ}-module generated by 𝔪^{10}(Q/L) is 𝔢. By Lemma 4.8, this implies that the CR structure on Q/L is primitive.

(4) It is sufficient to observe that if G is simple, the case R̃_{𝔢} = A_{1} ∪ A_{1} cannot occur by Corollary 6.10. □

Lemma 6.13 (3) and Proposition 6.12 directly imply Proposition 6.4.

Now it remains to prove Proposition 6.5. Let (M = G/L, D_{Z}, J) be a nonstandard non-primitive CR manifold with contact form ϑ not proportional to any root. We recall that in Lemma 6.13 (3) we defined a complex structure J_{1} on the flag manifold F_{Z} = G/K, associated with the decomposition 𝔤^{ℂ} = 𝔨^{ℂ} + 𝔪_{J_{1}}^{10} + 𝔪_{J_{1}}^{01}. We also defined another flag manifold F_{2} = G/Q, with 𝔮^{ℂ} = 𝔨^{ℂ} + 𝔢, with invariant complex structure J_{2} associated with the decomposition 𝔤^{ℂ} = 𝔮^{ℂ} + 𝔪_{J}^{+} + 𝔪_{J}^{−} and such that the projection π : (F_{Z} = G/K, J_{1}) → (F_{2} = G/Q, J_{2}) is holomorphic. Moreover the CR structure (D_{Z}, J) on G/L has the holomorphic subspace defined in (6.16) and (6.17).

The subalgebra 𝔨^{ℂ} corresponds to the root subsystem R_{o}, which has the orthogonal decomposition R_{o} = R′_{o} ∪ R̃_{o}, and 𝔮^{ℂ} corresponds to the root subsystem with the orthogonal decomposition R_{Q} = R′_{o} ∪ R̃_{𝔢} = R′_{o} ∪ (R̃_{o} ∪ R_{𝔢}) (see Lemma 6.11). Moreover there are only three possibilities for the pair of subsystems (R̃_{𝔢}, R̃_{o}) , namely (D_{2} = 2A_{1}, ∅,), (D_{ℓ}), D_{ℓ−1},, ℓ > 2, or (B_{3}, A_{2}). However, the following lemma shows that this last case cannot occur.

Lemma 6.14. If R_{J} ≠ ∅, then R̃_{𝔢} ≠ B_{3}.

In other words, the fiber C = Q/L of the CRF fibration π : G/L → G/Q described in Lemma 6.13 (3) cannot be Spin_{7}/SU_{3} if the base is not trivial.

Proof. Assume that R̃_{𝔢} = B_{3}. Then G is simple and R is indecomposable by Lemma 6.8. So R has type either B_{n} or F_{4}, because these are the only connected Dynkin graphs which contain a subgraph of type B_{3}.

If R = F_{4}, using the notation of the Appendix, we may assume that (α = ε_{2} +ε_{3}, α′ = −ε_{4}) is a ϑ-dual pair in R_{𝔢}. Since ϑ = ε_{2} +ε_{3} +ε_{4}, then −ε_{4} +ε_{1} ∈ R_{J}, because it is not orthogonal to ϑ nor has a ϑ-dual root; moreover −ε_{1} ∈ R_{o} = R ∩ (ϑ)^{⊥} and hence −ε_{4} = (−ε_{4} + ε_{1}) − ε_{1} ∈ R_{J}, by Lemma 6.6 (4): contradiction.

Assume now that R = B_{n}, n > 3. Then we may assume that (α, α′) = (ε_{1} + ε_{2}, −ε_{3}) is a ϑ-dual pair in R_{𝔢} and hence that ϑ = ε_{1} + ε_{2} + ε_{3}. Then, as before, we get that −ε_{3} + ε_{4} ∈ R_{J}, −ε_{4} ∈ R_{o} and hence that −ε_{3} = (−ε_{3} + ε_{4}) + (−ε_{4}) ∈ R_{J}: contradiction. □

Now we construct some special basis Π for R, which we will call good. For any basis Π let

Π_{o} = Π ∩ R_{o} , Π̃_{o} = Π_{o} ∩ R̃_{𝔢} , Π̃_{𝔢} = Π ∩ R̃_{𝔢} , Π_{𝔢} = Π ∩ R_{𝔢} .

Then

Π̃_{𝔢} = Π_{𝔢} ∪ Π̃_{o} .

A basis Π is called good if

R̃_{o} = [Π̃_{o}] , R̃_{𝔢} = [Π̃_{𝔢}] , R_{o} = [Π_{o}] ,

where for any subset A ⊂ Π we denote [A] = span(A) ∩ R.

A good basis exists because R_{o} ∪ R̃_{𝔢} = R′_{o} ∪ R̃_{𝔢} is a closed subset of roots, R′_{o} is orthogonal to R̃_{𝔢} and R_{o} = R′_{o} ∪ (R_{o} ∩ R̃_{𝔢}) = R′_{o} ∪ R̃_{o}. In fact, we may take a basis Π̃_{o} for R̃_{o}, extend it to a basis Π̃_{𝔢} for R̃_{𝔢}, add to it a basis for R′_{o} and finally extend everything to a basis Π for R.

By the remarks before Lemma 6.14, the pair (Π̃_{𝔢}, Π̃_{o}) is of type (D_{ℓ}, D_{ℓ−1}), ℓ > 2, or (2A_{1}, ∅) and it can be represented by the following two graphs

2^{⊗}  ^{2}∘  ⋯  ^{2}∘ ^{2}∘⧸^{∘ ^{1}} ⧹_{∘ _{1}}  (6.18)

{1 \atop ⊗}  {−1 \atop ⊗}  (6.19)

where the subdiagram of Π̃_{o} is obtained by deleting the grey nodes. Moreover, by Lemma 6.9, the contact form ϑ is the linear combination of the simple roots associated with the nodes of (6.18) and (6.19) with the indicated coefficients. For example, if (Π̃_{𝔢}, Π̃_{o}) = (D_{ℓ}, D_{ℓ−1}) and if we use the standard correspondence between nodes and roots, we get

ϑ = 2(ε_{1} − ε_{2}) + ⋯ + 2(ε_{ℓ−2} − ε_{ℓ−1}) + (ε_{ℓ−1} − ε_{ℓ}) + (ε_{ℓ−1} + ε_{ℓ}) = 2ε_{1} .

Note that if ℓ = 4, using two permutations of the simple roots corresponding to the end nodes, one gets the other two possible contact forms, namely ϑ = ε_{1}+ε_{2}+ε_{3}+ε_{4} and ϑ = ε_{1} + ε_{2} + ε_{3} − ε_{4}.

Remark that a good basis Π together with the subsets Π_{o} and Π_{𝔢} completely determines the homogeneous CR manifold M = G/L and the flag manifolds (F_{Z} = G/K, J_{1}) and (F_{2} = G/Q, J_{2}). In fact the root systems R_{o} = R(K) of K and R(Q) of Q are given by R_{o} = [Π_{o}] and R(Q) = [Π̃_{e} = Π_{o} ∪ Π_{𝔢}] and 𝔩 = 𝔨 ∩ (kerϑ), where ϑ is defined by (6.18)-(6.19).

Notice also that by definition of good basis

R ∩ (ϑ)^{⊥} = R_{o} = [Π_{o}]  (6.20)

and hence that

Π_{o} = Π ∩ (ϑ)^{⊥} .  (6.21)

Any good basis Π together with the subsets Π_{o} and Π_{𝔢} can be represented by a painted Dynkin graph Γ = Γ(Π) if we paint the nodes corresponding to the roots of Π_{𝔢} in grey, the nodes of Π_{o} in white and all others in black.

We call such graph Γ a painted Dynkin graph associated with the CR manifold (M = G/L, D_{Z}, J).

Any associated painted Dynkin graph has the following two properties.

(1) It contains a unique proper subgraph Γ_{𝔢} of type (6.18), if it is connected, or of type (6.19), if it is not connected; moreover in this second case, Γ = Γ_{1}∪Γ_{2} has two connected components and each of them contains exactly one grey node.

(2) The black nodes are exactly the nodes which are linked to Γ_{𝔢}. Indeed, (1) follows from definition of good basis, Lemma 6.8 and Lemma 6.2. (2) follows from (6.21).

A painted Dynkin graph which verifies (1) and (2) is called admissible graph.

Let Γ be an admissible graph and Γ_{𝔢} the corresponding subgraph of type (6.18) or (6.19). We denote by ϑ(Γ) the linear combination of roots associated with the nodes of Γ_{𝔢} as prescribed in (6.18)-(6.19).

An admissible graph Γ is called good if

[Π_{o}] = R ∩ (ϑ)^{⊥} ,  (6.22)

where Π_{o} is the set of simple roots associated with the white nodes of Γ. Remark that by (6.20) any graph associated with (M = G/L, D_{Z}, J) is a good graph. The converse of this statement is also true.

Lemma 6.15. Any good graph is a painted Dynkin graph associated to a homogeneous CR manifolds (G/L, D_{Z}, J), which have a contact form ϑ parallel to no roots and where (D_{Z}, J) is non-standard and non-primitive.

Proof. Let Γ be a good graph and ϑ(Γ) the corresponding contact form. As described in the Introduction, Γ defines two flag manifolds F_{1}(Γ) = G/K and F_{2}(Γ) = G/Q, with invariant complex structures J_{1}(Γ) and J_{2}(Γ), respectively. Denote by

𝔤^{ℂ} = 𝔨^{ℂ} + 𝔪_{J_{1}}^{10} + 𝔪_{J_{1}}^{01} , 𝔤^{ℂ} = 𝔮^{ℂ} + 𝔪_{J_{2}}^{10} + 𝔪_{J_{2}}^{01}

the corresponding associated decompositions. Consider also the element Z = iB^{−1}∘ ϑ(Γ). Since the 1-parametric subgroup generated by Z is closed, by Proposition 3.3 it defines a contact manifold (M = G/L, D_{Z}) with 𝔩 = 𝔨 ∩ (Z)^{⊥}. Moreover the fiber C = Q/L of the fibration π : G/L → G/Q, together with the contact structure induced on C by Z, is one of the contact manifolds described in Proposition 6.4 admitting a primitive CR structure.

If 𝔪_{C}^{10} is the holomorphic subspace of such CR structure, then 𝔪^{10} = 𝔪_{C}^{10} + 𝔪_{J_{2}}^{10} is the holomorphic subspace of a non-standard CR structure (D_{Z}, J) on G/L and the associated painted Dynkin graph is exactly (Γ, ϑ(Γ)). In fact, the conditions i) and ii) of Definition 4.1 are immediate. The integrability condition follows from the fact that 𝔪_{C}^{10} is a holomorphic subspace for a CR structure on C = Q/L (and hence that 𝔩^{ℂ} + 𝔪_{C}^{10} is a subalgebra), that 𝔪_{J_{2}}^{10} is the nilradical of the parabolic subalgebra 𝔮^{ℂ} + 𝔪_{J_{2}}^{10}, and that 𝔪_{C}^{10} ⊂ 𝔮^{ℂ}. □

Now the classification of homogeneous CR manifolds of the considered type reduces to the classification of good graphs Γ.

Case 1 . Γ is not connected.

In this case Γ_{𝔢} = A_{1} ∪ A_{1}, Γ = Γ_{1} ∪ Γ_{2}, where each Γ_{i} is a connected component which corresponds to a root system R_{i}, and R = R_{1} ∪ R_{2}. Moreover ϑ = α_{1} − α_{2}, where α_{i} ∈ R_{i}.

We prove that if Γ is good then R = A_{p} ∪ A_{q}, with p + q > 1 and that Γ is a CR-graph of type II.

First of all, one can easily check that if one of the connected components Γ_{i} is not of type A_{q}, then Γ is not good, that is that there exists a root β ∈ R ∩ (ϑ)^{⊥} which is not in [Π_{o}]. For example, if R_{1} = D_{q}, we may assume that ϑ = α_{1} − α_{2}, where α_{1} = ε_{1} − ε_{2}. Then β = ε_{1} + ε_{2} ∈ R ∩ (ϑ)^{⊥} but it is not in [Π_{o}].

Assume now that R = A_{p} ∪ A_{q}. Without loss of generality we may assume that α_{1} = ε_{k} − ε_{k+1}, α_{2} = ε′_{r} − ε′_{r+1} are the roots associated with the grey nodes of Γ_{1} and Γ_{2}, respectively. Then R ∩ (ϑ)^{⊥} = A_{p−2} ∪ A_{q−2} and it coincides with [Π_{o}] if and only if the nodes of the roots α_{i} are end nodes. This proves that Γ is good if and only if it is a CR-graph of type II ( see Definition 1.7).

Case 2 . Γ is connected.

In this case, Γ is a good graph only if the type of the pair (Γ, Γ_{𝔢}) is one of the following

(A_{n}, A_{3}) , (B_{n}, A_{3}) , (D_{n}, D_{4}) , (E_{6}, D_{5}) ,
(E_{7}, D_{6}) , (E_{8}, D_{5}) , (E_{8}, D_{7}) .

This follows from Corollary 6.10 and the fact that (A_{n}, A_{3}), (B_{n}, B_{3}), (B_{n}, D_{4}), (D_{n}, D_{n}), (F_{4}, A_{3}) and (F_{4}, B_{3}) do not correspond to any admissible graph.

We first prove that the cases (B_{n}, A_{3}), (E_{8}, D_{5}) and (E_{8}, D_{7}) are not possible. i) (Γ, Γ_{𝔢}) = (B_{n}, A_{3}). In this case Γ is of the form

∘  ⋯  ∘ • ^{α_{1}} ∘ ^{α_{2}}^{⊗} ^{α_{3}} ∘ •  ⋯  ∘ > ∘

where α_{1} = ε_{k} − ε_{k+1}, α_{2} = ε_{k+1} − ε_{k+2} and α_{3} = ε_{k+2} − ε_{k+3}. Then

ϑ(Γ) = α_{1} + 2α_{2} + α_{3} = ε_{k} + ε_{k+1} − ε_{k+2} − ε_{k+3} and
Π_{o} = {ε_{i} − ε_{i+1}, i = 1, …, k − 2; k; k + 2; k + 4, …, n − 1; ε_{n}} .

However the root β = ε_{k+1} + ε_{k+2} ∈ (ϑ(Γ))^{⊥} ∩ R but it does not belong to [Π_{o}]: contradiction.

ii) (Γ, Γ_{𝔢}) = (E_{8}, D_{5}). Then Γ and ϑ(Γ) are

∘ ∘ • ∘ ∘ ∘ ∘ ^{⊗} , ϑ(Γ) = ε_{4} + ε_{5} + ε_{6} + ε_{7} − ε_{8} .

One can easily check that the root β = ε_{1} + ε_{2} + ε_{4} is orthogonal to ϑ(Γ), but it doesn’t belong to the subsystem

[Π_{o}] = [{α_{2}, α_{3}, α_{4}, α_{5}, α_{6}; α_{8}}]

generated by white roots: cotradiction. iii) (Γ, Γ_{𝔢}) = (E_{8}, D_{7}).Then Γ and ϑ(Γ) are

⊗ ∘ ∘ ∘ ∘ ∘ ∘ • , ϑ(Γ) = 2ε_{1} + ε_{8} .

Also this case is not possible because ε_{7} − ε_{9} ∈ R ∩ (ϑ(Γ))^{⊥} but it is not in [Π_{o}].

It remains to describe the good graphs of the following types

1) (A_{n}, A_{3}) ,  2) (D_{n}, D_{4}) ,  3) (E_{6}, D_{5}) ,  4) (E_{7}, D_{6}) .

(1) (Γ, Γ_{𝔢}) = (A_{n}, A_{3}).

Assume that Γ_{𝔢} is not at an end of Γ, that is

∘  ⋯  ∘ • ^{α_{1}} ∘ ^{α_{2}}^{⊗} ^{α_{3}} ∘ • ∘  ⋯  ∘

Then we may assume that α_{i} = ε_{p+i} − ε_{p+i+1}. Then ϑ(Γ) = α_{1} + 2α_{2} + α_{3} = ε_{p+1} + ε_{p+2} − ε_{p+3} − ε_{p+4} and the root β = ε_{p} − ε_{p+5} ∈ R ∩ (ϑ(Γ))^{⊥} but it is not in the span of Π_{o}; hence the graph is not good. On the other hand one can easily check that the graph

∘ ^{⊗} ∘ • ∘  ⋯  ∘

is good.

(2) (Γ, Γ_{𝔢}) = (D_{n}, D_{4}).

In this case we have two admissible graphs:

∘  ⋯  ∘ • ^{α_{n−3}} ∘ ^{α_{n−2}} ∘^{⧸^{⊗ α_{n−1}}} ⧹_{∘ _{α_{n}}}  (6.23)

∘  ⋯  ∘ • ^{α_{n−3}}^{⊗} ^{α_{n−2}} ∘⧸^{∘ ^{α_{n−1}}} ⧹_{∘ _{α_{n}}}  (6.24)

Using the standard equipment, we have that if Γ is given by (6.23), then

ϑ(Γ) = α_{n−3} + 2α_{n−2} + α_{n−1} + 2α_{n} = ε_{n−3} + ε_{n−2} + ε_{n−1} + ε_{n}

and R ∩ (ϑ(Γ))^{⊥} = D_{n−4} ∪ A_{3}. If Γ is given by (6.24), then

ϑ(Γ) = 2α_{n−3} + 2α_{n−2} + α_{n−1} + α_{n} = 2ε_{n−3}

and R ∩ (ϑ(Γ))^{⊥} = D_{n−1}.

Since in both cases [Π_{o}] = A_{n−5} ∪ A_{3}, the graph (6.24) is not good, while the graph (6.23) is good only when n = 5.

(3) (Γ, Γ_{𝔢}) = (E_{6}, D_{5}).

Up to isomorphism, we have only one admissible graph

α_{1}^{⊗} ^{α_{2}} ∘ ^{α_{3}} ∘ ∘ α_{6}^{α_{4}} ∘ ^{α_{5}}•

Using the standard equipment, we get

ϑ(Γ) = 2(α_{1} + α_{2} + α_{3}) + α_{4} + α_{6} = 2ε_{1} + ε_{6} + ε .

Then

R ∩ (ϑ(Γ))^{⊥} = { ε_{a} − ε_{b} , ±(ε_{a} + ε_{b} + ε_{6} + ε) , a, b = 2, 3, 4, 5 } = [Π_{o}] = D_{4}

and hence the graph is good.

(4) (Γ, Γ_{𝔢}) = (E_{7}, D_{6}).

There exists only one admissible graph

α_{1}^{⊗} ^{α_{2}} ∘ ^{α_{3}} ∘ ^{α_{4}} ∘ ∘ α_{7}^{α_{5}} ∘ ^{α_{6}}•

Then

ϑ(Γ) = 2(α_{1} + α_{2} + α_{3} + α_{4}) + α_{5} + α_{7} = 2ε_{1} + ε_{7} + ε_{8} .

In this case

R ∩ (ϑ(Γ))^{⊥} = { ε_{a} − ε_{b} , ±(ε_{a} + ε_{b} + ε_{7} + ε_{8}) , a, b = 2, 3, 4, 5, 6 } = [Π_{o}] = D_{5}

and hence it is a good graph.

This concludes the classification of good graphs. We proved that the pairs (Γ, ϑ(γ)) given by all good graphs are exactly the non-special CR-graphs of Definition 1.7. By the remarks before Lemma 6.15 and the Lemmata 6.13 and 6.15, Proposition 6.5 follows.

APPENDIX

The notation used in the following Tables is the same of [7]. We recall that the weights of the groups B_{ℓ}, C_{ℓ}, D_{ℓ} and F_{4} are expressed in terms of an orthonormal basis (ε_{1}, …, ε_{ℓ}) of 𝔥(ℚ)^{∗}. The weights of the groups A_{ℓ}, E_{7}, E_{8} and G_{2} are expressed using vectors ε_{1}, …, ε_{ℓ+1} ∈ 𝔥(ℚ)^{∗} such that

∑ ε_{i} = 0 ,  (ε_{i}, ε_{j}) = \{ \begin{matrix}\frac{ℓ}{ℓ+1} & i = j \\ −\frac{1}{ℓ+1} & i ≠ j \end{matrix}  (A.1)

It is useful to recall that if ∑ a_{i} = 0, then (∑ a_{i}ε_{i}, ∑ b_{j}ε_{j}) = ∑ a_{i}b_{i}. For E_{6}, the weights are expressed by vectors ε_{1}, …, ε_{6}, which verify (A.1) with ℓ = 5, and by an auxiliary vector ε which is orthogonal to all ε_{i} and verifies (ε, ε) = 1/2.

In Table 1, for any simple complex Lie group 𝔤^{ℂ}, we give the corresponding root system R, the longest root μ (unique up to inner automorphisms), the subalgebra 𝔤′_{0} = C_{𝔤^{ℂ}}(𝔤(μ)), the subsystem of roots R_{o} corresponding to 𝔤′_{0}, the decomposition into irreducible submodules of the 𝔤_{0}-module 𝔤_{1} which appear in the decomposition (3.2), and the set of roots R_{1} = R^{+} \ (μ ∪ R_{o}).

For a set of simple roots of 𝔤′_{0}, we denote by {π_{1}, …, π_{ℓ}} the corresponding system of fundamental weights and, for any weight λ = ∑ a_{i}π_{i}, we denote by V (λ) the irreducible 𝔤′_{0}-module with highest weight λ.

In Table 2, we give the information needed to determine the holomorphic subspaces 𝔪^{10} when 𝔤^{ℂ} is a simple Lie algebra and the contact form ϑ = −iB ∘ Z|_{𝔥} is parallel to a short root.

In Table 3 we give the same information for the cases 𝔤^{ℂ} = B_{3} or D_{ℓ} and ϑ proportional to no root and associated with a primitive CR structure.

In both tables we give the root systems R, the contact form ϑ, the subalgebra 𝔩^{ℂ} = C_{𝔤^{ℂ}}(Z) ∩ (Z)^{⊥}, the root subsystem R_{o} of 𝔩^{ℂ} and the list of the highest weights for the irreducible 𝔨^{ℂ}-moduli in 𝔪^{ℂ} (𝔨^{ℂ} = C_{𝔤^{ℂ}}(Z)). We group the highest weights corresponding to equivalent 𝔩^{ℂ}-moduli with curly brackets.

In Table 4 we recall the Dynkin graphs associated with indecomposable root systems and the correspondence used in [7] between nodes and simple roots.

Table 1

𝔤  R  μ  𝔤′_{0}  R_{o}  𝔤_{1}  R_{1}^{A_{ℓ}  \begin{matrix}ε_{i}−ε_{j} \\ 1≤i,j≤ℓ+1 \end{matrix}  ε_{1}−ε_{ℓ−1}  A_{ℓ−2}+ℝ  \begin{matrix}ε_{a}−ε_{b} \\ 2≤a,b≤ℓ \end{matrix}  \begin{matrix}V (π_{1})+ \\ V (π_{ℓ−2}) \end{matrix} \begin{matrix}ε_{1}−ε_{a}, ε_{a}−ε_{ℓ+1} \\ 2≤a≤ℓ \end{matrix}}^{B_{ℓ}  \begin{matrix}±ε_{i}±ε_{j}, ±ε_{i} \\ 1≤i,j≤ℓ \end{matrix}  ε_{1}+ε_{2}  A_{1}+B_{ℓ−2}  \begin{matrix}±(ε_{1}−ε_{2}), ±ε_{a}±ε_{b} \\ ±ε_{a} \\ 3≤a,b≤ℓ \end{matrix} \begin{matrix}V (π_{1})⊗ \\ V (π′_{1}) \end{matrix}  \begin{matrix}ε_{1}, ε_{2} \\ ε_{1}±ε_{a}, ε_{2}±ε_{a} \\ 3≤a≤ℓ \end{matrix}}^{C_{ℓ} \begin{matrix}±ε_{i}±ε_{j} , ±2ε_{i} \\ 1≤i,j≤ℓ \end{matrix}  2ε_{1}  C_{ℓ−1}  \begin{matrix}±ε_{a}±ε_{b}, ±2ε_{a} \\ 2≤a,b≤ℓ \end{matrix}  V (π_{1})  \begin{matrix}ε_{1}±ε_{a} \\ 2≤a≤ℓ \end{matrix}}^{D_{ℓ}  \begin{matrix}±ε_{i}±ε_{j} \\ 1≤i,j≤ℓ \end{matrix}  ε_{1}+ε_{2}  A_{1}+D_{ℓ−2}  \begin{matrix}±(ε_{1}−ε_{2}), ±ε_{a}±ε_{b} \\ 3≤a,b≤ℓ \end{matrix} \begin{matrix}V (π_{1})⊗ \\ V (π′_{1}) \end{matrix}  \begin{matrix}ε_{1}±ε_{a}, ε_{2}±ε_{a} \\ 3≤a≤ℓ \end{matrix}}^{E_{6}  \begin{matrix}ε_{i}−ε_{j}, ±2ε \\ ε_{i}+ε_{j}+ε_{k}±ε \\ 1≤i,j,k≤6 \end{matrix}  2ε  A_{5}  ε_{i}−ε_{j}  V (π_{1})  ε_{i}+ε_{j}+ε_{k}±ε}^{E_{7}  \begin{matrix}ε_{i}−ε_{j} \\ ε_{i}+ε_{j}+ε_{k}+ε_{ℓ} \\ 1≤i,j,k,ℓ≤8 \end{matrix}  −ε_{7}+ε_{8}  D_{6}  \begin{matrix}ε_{a}−ε_{b} \\ ε_{7}+ε_{8}+ε_{a}+ε_{b} \\ ε_{a}+ε_{b}+ε_{c}+ε_{d} \\ 1≤a,b,c,d≤6 \end{matrix}  V (π_{1})  \begin{matrix}−ε_{7}+ε_{a}, ε_{8}−ε_{a} \\ ε_{8}+ε_{a}+ε_{b}+ε_{c} \\ 1≤a,b,c≤6 \end{matrix}}^{E_{8}  \begin{matrix}ε_{i}−ε_{j} \\ ±(ε_{i}+ε_{j}+ε_{k}) \\ 1≤i,j,k≤9 \end{matrix}  ε_{1}−ε_{9}  E_{7}  \begin{matrix}ε_{a}−ε_{b} \\ ±(ε_{1}+ε_{9}+ε_{a}) \\ ±(ε_{a}+ε_{b}+ε_{c}) \\ 2≤a,b,c≤8 \end{matrix}  V (π_{1})  \begin{matrix}ε_{1}−ε_{a}, −ε_{9}+ε_{a} \\ ε_{1}+ε_{a}+ε_{b} \\ 2≤a,b≤8 \end{matrix}}^{F_{4}  \frac{±ε_{1}±ε_{2}±ε_{3}±ε_{4}}{2} ; ±ε_{i}±ε_{j}, ±ε_{i} ; 1≤i,j≤4  ε_{1}+ε_{2}  C_{3}  \begin{matrix}±(ε_{1}−ε_{2}) \\ ± \frac{ε_{1}−ε_{2}±ε_{3}±ε_{4}}{2} \\ ±ε_{a}, ±ε_{a}±ε_{b} \\ 3≤a,b≤4 \end{matrix}  V (π_{1})  ε_{1}, ε_{2}_{\frac{ε_{1}+ε_{2}±ε_{3}±ε_{4}}{2}}}^{G_{2}  \begin{matrix}ε_{i}−ε_{j} , ±ε_{i} \\ 1≤i,j≤3 \end{matrix}  ε_{1}−ε_{2}  A_{1}  ±ε_{3}  V (π_{1})  \begin{matrix}ε_{1}−ε_{3} , ε_{1}−ε_{2} \\ ε_{3}−ε_{2} \end{matrix}}

Table 2

𝔤 R ϑ_{=iB∘Z} 𝔩^{ℂ}_{=C_{𝔤^{ℂ}} (Z)∩(Z)^{⊥}}

R_{o}

highest weights for 𝔪^{ℂ} grouped into sets of equivalent 𝔩^{ℂ}−moduli

B_{ℓ}  \begin{matrix}±ε_{i}±ε_{j}, ±ε_{i} \\ 1≤i,j≤ℓ \end{matrix}  ε_{1}  B_{ℓ−1}  \begin{matrix}±ε_{a}±ε_{b}, ±ε_{a} \\ 2≤a,b≤ℓ \end{matrix}  \{ε_{1}+ε_{2},−ε_{1}+ε_{2}\}
C_{ℓ}  \begin{matrix}±ε_{i}±ε_{j}, ±2ε_{i} \\ 1≤i,j≤ℓ \end{matrix} ε_{1}+ε_{2}  A_{1}+C_{ℓ−2}  \begin{matrix}±(ε_{1}−ε_{2}), ±2ε_{a} \\ ±ε_{a}±ε_{b} \\ 3≤a,b≤ℓ \end{matrix}  \begin{matrix}\{2ε_{1},−2ε_{2}\} \\ \{ε_{1}+ε_{3}, −ε_{2}+ε_{3}\} \end{matrix}
F_{4}  \frac{±ε_{1}±ε_{2}±ε_{3}±ε_{4}}{2} ; ±ε_{i}±ε_{j}, ±ε_{i} ; 1≤i,j≤4  ε_{1}  B_{3}  \begin{matrix}±ε_{a}±ε_{b}, ±ε_{a} \\ 2≤a,b≤4 \end{matrix} \begin{matrix}\{ε_{1}+ε_{2},−ε_{1}+ε_{2}\} \\ \{ \frac{ε_{1}+ε_{2}+ε_{3}+ε_{4}}{2} , \frac{−ε_{1}+ε_{2}+ε_{3}+ε_{4}}{2} \} \end{matrix}
G_{2}  \begin{matrix}ε_{i}−ε_{j} , ±ε_{i} \\ 1≤i,j≤3 \end{matrix}  ε_{1}  A_{1}  ±(ε_{2}−ε_{3})  \begin{matrix}\{ε_{1},−ε_{1}\} \\ \{ε_{3}−ε_{1}, ε_{3}, −ε_{2}, ε_{1}−ε_{2}\} \end{matrix}

Table 3

𝔤 R ϑ_{=iB∘Z} 𝔩^{ℂ}_{=C_{𝔤^{ℂ}} (Z)∩(Z)^{⊥}}

R_{o}

highest weights for 𝔪^{ℂ} grouped into sets of equivalent 𝔩^{ℂ}−moduli

B_{3} \begin{matrix}±ε_{i}±ε_{j}, ±ε_{i} \\ 1≤i,j≤3 \end{matrix} ε_{1}+ε_{2}+ε_{3}  A_{2}  \begin{matrix}±(ε_{a}−ε_{b}), \\ 1≤a,b≤3 \end{matrix} \{ε_{1}+ε_{2},−ε_{3}\} \{−ε_{2}−ε_{3}, ε_{1}\}
D_{ℓ}  \begin{matrix}±ε_{i}±ε_{j} \\ 1≤i,j≤ℓ \end{matrix}  ε_{1}  D_{ℓ−1}  \begin{matrix}±ε_{i}±ε_{j} \\ 2≤i,j≤ℓ \end{matrix}  \{ε_{1}+ε_{2},−ε_{1}+ε_{2}\}

Table 4

Type of G Dynkin graphs

Simple roots

A_{ℓ}  ^{1}∘ ^{2}∘  ⋯ ^{ℓ−1} ∘ ^{ℓ}∘  α_{i}=ε_{i}−ε_{i+1}

B_{ℓ}  ^{1}∘ ^{2}∘  ⋯ ^{ℓ−1} ∘ > ^{ℓ}∘  \begin{matrix}α_{i}=ε_{i}−ε_{i+1} (i<ℓ) , \\ α_{ℓ}=ε_{ℓ} \end{matrix}

C_{ℓ}  ^{1}∘ ^{2}∘  ⋯ ^{ℓ−1} ∘ < ^{ℓ}∘  \begin{matrix}α_{i}=ε_{i}−ε_{i+1} (i<ℓ) , \\ α_{ℓ}=2ε_{ℓ} \end{matrix}

D_{ℓ}  ^{1}∘ ^{2}∘  ⋯ ^{ℓ−2} ∘⧸^{∘ ^{ℓ−1}} ⧹_{∘ _{ℓ}}  \begin{matrix}α_{i}=ε_{i}−ε_{i+1} (i<ℓ) , \\ α_{ℓ}=ε_{ℓ−1}+ε_{ℓ} \end{matrix}

E_{6}  ^{1}∘ ^{2}∘ ^{3}∘ ∘ 6 ^{4}∘ ^{5}∘  \begin{matrix}α_{i}=ε_{i}−ε_{i+1} (i<6) , \\ α_{6}=ε_{4}+ε_{5}+ε_{6}+ε \end{matrix}

E_{7}  ^{1}∘ ^{2}∘ ^{3}∘ ^{4}∘ ∘ 7 ^{5}∘ ^{6}∘  \begin{matrix}α_{i}=ε_{i}−ε_{i+1} (i<7) , \\ α_{7}=ε_{5}+ε_{6}+ε_{7}+ε_{8} \end{matrix}

E_{8}  ^{1}∘ ^{2}∘ ^{3}∘ ^{4}∘ ^{5}∘ ∘ 8 ^{6}∘ ^{7}∘  \begin{matrix}α_{i}=ε_{i}−ε_{i+1} (i<8) , \\ α_{8}=ε_{6}+ε_{7}+ε_{8} \end{matrix}

F_{4}  ^{1}∘ ^{2}∘ < ^{3}∘ ^{4}∘  \begin{matrix}α_{1}=(ε_{1}−ε_{2}−ε_{3}−ε_{4})/2 , α_{2}=ε_{4} \\ α_{3}=ε_{3}−ε_{4} , α_{4}=ε_{2}−ε_{3} \end{matrix}

G_{2}  ^{1}∘ < ^{2}∘  α_{1}=−ε_{2} , α_{2}=ε_{2}−ε_{3}

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D. V. Alekseevsky Center Sophus Lie 117279 Moscow gen. Antonova 2-99 RUSSIA

   E-mail address: fort@slip.rsuh.ru

A. F. Spiro Dipartimento di Matematica Università di Ancona 60131 Ancona ITALY

   E-mail address: spiro@anvax1.unian.it