Higher order Levi forms on homogeneous CR manifolds

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous $CR$ manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal $3$ and the compact ones less or equal~$2$. Finally we construct by Lee extensions weakly nondegenerate $CR$ vector bundles with arbitrary orders of nondegeneracy.


Introduction
The Levi form is a basic invariant of CR geometry (see e.g. [9]). It is a hermitian symmetric form on the space of tangent holomorphic vector fields, which, when the CR codimension is larger than one, is vector valued. Its nondegeneracy was shown in [26] to be a sufficient condition to apply Cartan's method to investigate equivalence and automorphisms of CR structures and is an obvious obstruction for locally representing the manifold as a product of a CR manifold of smaller dimension and of a nontrivial complex manifold. Sufficient more general conditions preventing a CR manifold M from being foliated by complex leaves of positive dimension or from having an infinite dimensional group of local CR automorphisms can be expressed by the nondegeneracy of higher order Levi forms (see e.g. [13]). In the case of homogeneous CR manifolds these properties can be rephrased in terms of their associated CR algebras and lead to the notions of weak nondegeneracy and ideal nondegeneracy in [22]. The last one was renamed contact nondegeneracy and proved sufficient for the finite dimensionality of the group of CR automorphisms in [19].
Iterations of the Levi forms can be described by building descending chains of algebras of vector fields, whose lengths can be taken as a measure of nondegeneracy (see §1.1). One of these numbers, that we call here Levi order, and relates to weak nondegeneracy, is the main topic of this paper. The real submanifolds M of a complex flag manifold F of a semisimple complex group S that are orbits of its real form S R form an interesting class of homogeneous CR manifolds, that has been studied e.g. in [1,3]. In [10] G. Fels showed that when the isotropy Q of F is a maximal parabolic subgroup, and M is weakly nondegenerate, then its Levi order is at most 3 and found an example where it is in fact equal to 3. In §2 we prove that this bound is valid for general weakly nondegenerate real orbits, dropping the maximality assumption on Q and give further examples of weakly nondegenerate real orbits having Levi order 3. Moreover we show that the minimal orbit (the single one which is compact, cf. [28]) cannot have a finite Levi order larger than 2 and that the same result is valid for a larger class or orbits, that we name of the minimal type. Orbits which are not of the minimal type may have any finite order 1, 2, 3. Our methods are illustrated by several examples. We point out that, together with the new results obtained here, those in [1], where descriptions in terms of cross-marked Satake diagrams are emphasised, would allow to list the minimal orbits of Levi orders 1 and 2.
In [10] G. Fels posed the question of the existence of weakly nondegenerate homogeneous CR manifolds with Levi order larger than 3. In §3 we exhibit, by constructing some CR vector bundles over CP 1 , weakly nondegenerate homogeneous CR manifolds having Levi order q, for every positive integer q.

Nondegeneracy conditions
1.1. Abstract CR manifolds. In this subsection we discuss some notions of nondegeneracy for general smooth abstract CR manifolds of type (n, k). We will eventually be interested in the locally homogeneous case and therefore, in the rest of this section, in their reformulation in the framework of Lie algebras theory.
We recall that an abstract CR manifold of type (n, k) is defined by the datum, on a smooth manifold M of real dimension 2n+k, of a rank n smooth complex linear subbundle T 0,1 M of its complexified tangent bundle T C M, satisfying The map J M squares to −I H and is the partial complex structure of M. An equivalent definition of the CR structure can be given by assigning first an even dimensional distribution HM and then a smooth partial complex structure J M on HM in such a way that the complex distribution (1.4) satisfies (1.2).
Let us denote by H (resp. T , H C , T 0,1 , T 1,0 ) the sheaf of germs of smooth sections of HM (resp. TM, H C M, T 0,1 M, T 1,0 M). We define recursively a nested sequence of sheaves of germs of smooth complex valued vector fields on M By conjugation we obtain another nested sequence of sheaves These sequences, considered by Freeman in [12,Thm.3.1], correspond to the chain (1.11) that we construct in the locally homogeneous case.
In the same paper (cf. [12,Remarks 4.5]) also another sequence, introduced before in [15,16], was considered, which will correspond to (1.15), namely We say that M is at its point x • Levi (resp. contact) nondegenerate if it has Levi (resp. contact) order 1; • weakly (resp. contact) nondegenerate if it has finite Levi (resp. contact) order p≥1; • holomorphically (resp. contact) degenerate if it is not weakly (resp. contact) nondegenerate.
The Levi order at x is the smallest q for which, given any nonzero germ Z ∈T 0,1 x , we can find a p≤q and Z 1 , . . . , Z p ∈ T 1,0 x of M and let π : G R ∋ g → g·x ∈ M be the natural projection. The differential at x defines a map π * : g R → T x M of the Lie algebra g R of G R onto the tangent space to M at x. By the formal integrability of the partial complex structure of M, the pullback q = (π C * ) −1 (T 0,1 x M) of the space of tangent vectors of type (0, 1) at x by the complexification of the differential is a complex Lie subalgebra q of the complexification g = C ⊗ R g R of g R . Vice versa, the assignment of a complex Lie subalgebra q of g yields a formally integrable G R -equivariant partial complex structure on a locally homogeneous space M of G R by the requirement that T 0,1 x M = π C * (q) (see e.g. [1,22]). These considerations led to the following definition. Definition 1.1. A CR algebra is a pair (g R , q), consisting of a real Lie algebra g R and a complex Lie subalgebra q of its complexification g= C ⊗ R g R , such that the quotient g R /(g R ∩ q) is a finite dimensional real vector space.
We call the intersection q ∩ g R its isotropy subalgebra and say that (g R , q) is effective when q ∩ g R does not contain any nontrivial ideal of g R .
If G R is the real form of a complex Lie algebra G and q the Lie algebra of its closed subgroup Q, then M is locally CR diffeomorphic to the orbit of G R in the complex homogeneous space G /Q and its CR structure is induced by the complex structure of G /Q. These considerations can be generalized to locally homogeneous CR manifolds (see e.g. [1]).
The CR-dimension and codimension of M are expressed in terms of its associated CR algebra (g R , q) by Definition 1.3. We call fundamental a CR algebra (g R , q) such that q+q generates g as a Lie algebra and we say that it is • of complex type if q+q = g, • of contact type if q+q g.
A corresponding CR manifold M is in the first case a complex manifold by Newlander-Nirenberg theorem (cf. [4,23]), while contact type is equivalent to the fact that its CR distribution is strongly non-integrable.
1.3. Levi-order of weak nondegeneracy. The Levi form is a basic invariant of CR geometry. When M is locally homogeneous, it can be computed by using its associated CR algebra (g R , q) (for definitions and basic properties, cf. e.g. [9]). Nondegeneracy of the Levi form can be stated by This is equivalent to When this condition is not satisfied, we say that (g R , q) is Levi-degenerate. To measure the degeneracy of the Levi form, one can consider its iterations: in the homogeneous case this means, given a Z ∈ q\(q∩q), to seek whether it is possible to find L 1 , . . . , L p ∈q such that [L 1 , . . . , L p , Z] q+q. To this aim, it is convenient to consider the descending chain (see e.g. [10,13,19,22]) Note that q∩q ⊆ q (p) for all integers p≥0. Since by assumption q/(q∩q) is finite dimensional, there is a smallest nonnegative integer q such that q (p) = q (q) for all p≥q. Definition 1.4. We call (1.11) the descending Levi chain of (g R , q).
Let q be a positive integer. The CR algebra (g R , q) is said to be • weakly nondegenerate of Levi order q if q (q−1) q (q) = q∩q.
• strictly nondegenerate if it is weakly nondegenerate of Levi order 1.
If q (q) q∩q for all integers q>0, we say that (g R , q) is holomorphically degenerate.
Proposition 1.2. The terms q (p) of (1.11) are Lie subalgebras of q.
Proof. By definition, ⊆ q+q by the definition of q (1) . This shows that q (1) is a Lie subalgebra of q.
Next we argue by recurrence. Let p≥1 and assume that q (p) is a Lie subalgebra of q. If Z 1 , Z 2 ∈ q (p+1) , then [Z 1 , Z 2 ] ∈ q (p) by the inductive assumption that q (p) is a Lie subalgebra and showing that also [Z 1 , Z 2 ] ∈ q (p+1) . This completes the proof. Let us introduce the notation The weak nondegeneracy defined here is equivalent to the notion of [22], consisting in the requirement that, for a complex Lie subalgebra f of g, Indeed, it easily follows from [22, Lemma 6.1] that (1.14) is the largest complex Lie subalgebra f of g with q ⊆ f ⊆ H .

Contact nondegeneracy.
A less restrictive nondegeneracy condition in terms of iterations of the Levi form can be expressed by requiring that, given Z ∈ q\(q∩q) there are L 1 , . . . , L p ∈ H such that [L 1 , . . . , L p , Z] H . For a CR algebra of the contact type this is equivalent to the fact that any ideal a of g R that is contained in H R is contained in q ∩ g R . Thus this property was called ideal nondegeneracy in [22]. When (g R , q) is the CR algebra at x of a (locally) homogeneous CR manifold M, the subspace H R is the pullback to g R of the real contact distribution associated to the CR structure of M. Thus this notion was renamed contact nondegeneracy in [19]. It was shown in [22, Lemma 7.2] that, for the complexification a of the largest ideal of g R contained in H R , the sum a +q ∩q is the limit of the descending chain Since q ∩q ⊆ q [p] for all integers p≥0, the chain (1.15) stabilizes.
Remark 1.3. Note that, for p≥1, This is true in fact for p=1 and for p>2 follows from Definition 1.5. We call (1.15) the descending contact chain. Let q be a positive integer. The CR algebra (g R , q) is said to have If q [q] q∩q for all integers q>0, we say that (g R , q) is contact degenerate.
We note that we can equivalently use the descending chain . This follows from Lemma 1.4. With the notation introduced above, we have: R is a Lie algebra and, for p>1 an ideal of a (1) R ; (2) Let a (p) be the complexification of a Proof. The first statement is trivial. We can check the second one by recurrence. This is in fact true for p=0, since q+q is the complexification of H .
We already considered two descending chains whose length defines the order of contact nondegeneracy. It is in fact convenient to consider a third one, which is easier to deal with (see §2 below), namely: The equivalence is a consequence of Proposition 1.5. With the notation above: Since we obviously have the inclusion we obtain Proposition 1.6. If the CR algebra (g R , q) has Levi order q<∞, then (g R , q) has contact order q ′ ≤q. A contact degenerate (g R , q) is also holomorphically degenerate.

Orbits of real forms in complex flag manifolds
2.1. Complex flag manifolds. A complex flag manifold F is a smooth compact algebraic variety that can be described as the quotient of a complex semisimple Lie group S by a parabolic subgroup Q; according to Wolf [28], a real form S R of S has finitely many orbits in F. Only one of them, having minimal dimension, is compact. With the partial complex structures induced by F, these orbits make a class of homogeneous CR manifolds that were studied by many authors (see e.g. [1,2,3,8,10,11,14,18,20]). Cross-marked Dynkin diagrams. Being connected and simply connected, a complex flag manifold F = S/Q is completely described by the Lie pair (s, q) consisting of the Lie algebras of S and of Q and vice versa to any Lie pair (s, q) of a complex semisimple Lie algebra and its parabolic subalgebra q corresponds a unique flag manifold F. Therefore the classification of complex flag manifolds reduces to that of parabolic subalgebras of semisimple complex Lie algebras. Parabolic subalgebras q of s are classified, modulo automorphisms, by a finite set of parameters. In fact, after fixing any Cartan subalgebra h of s, their equivalence classes are in one to one correspondence with the subsets of a basis B of simple roots of the root system R of (s, h) (see e.g. [7, Ch.VIII, §3.4]). We recall that the Dynkin diagram ∆ B is a graph with no loops, whose nodes are the roots in B and in which two nodes may be joined by at most 3 edges. Each root β in R can be witten in a unique way as a notrivial linear combination with integral coefficients k β,α which are either all ≥ 0, or all ≤ 0 and we set The parabolic subalgebras q are paremetrized, modulo isomorphisms, by subsets Φ of B: to a Φ ⊆ B we associate The set Q Φ is a parabolic set of roots, i.e.
To specify the q Φ of (2.3) we can cross the nodes corresponding to the roots in Φ. In this way each cross-marked Dynkin diagram encodes a specific complex flag manifold F Φ .
Notation 2.1. Let ξ Φ be the linear functional on the linear span of R which equals one on the roots in Φ and zero on those in B\Φ. Then and we get partitions 2.2. Real forms. Let us take, as we can, S connected and simply connected. Then real automorphisms of its Lie algebra s lift to automorphisms of the Lie group S, so that real forms S R of S are in one-to-one correspondence with the anti-C-linear involutions σ of s . We will denote by s σ the real Lie subalgebra consisting of the fixed points of σ: it is the Lie algebra of the real form S σ of fixed points of the liftσ of σ to S. Its orbits are CR submanifolds M Φ,σ of F Φ whose CR algebra at the base point Q is the pair (s σ , q Φ ).
Definition 2.1 (cf. [1, §5]). A parabolic CR algebra is a pair (s σ , q Φ ) consisting of a real semisimple Lie algebra s σ and a parabolic complex Lie subalgebra q Φ of its complexification s . We say that ( When s σ is not simple, the corresponding orbits M Φ,σ are CR diffeomorphic to a cartesian product of orbits of simple real Lie groups (see e.g. [3]).If A simple s σ is of the real type if also s is simple; otherwise, s is the direct sum of two complex simple Lie algebras s ′ , s ′′ , which are R-isomorphic to s σ , and we say in this case that s σ is of the complex type.
To list all the orbits of a real form, one can use the fact that the isotropy subalgebra s σ ∩ q contains a Cartan subalgebra h R of s σ (see e.g. [3]). By choosing h equal to its complexification, we obtain on R a conjugation which is compatible with the one defined on s by its real form s σ (and which, for simplicity, we still denote by σ). Vice versa, an orthogonal involution σ of R lifts, although in general not in a unique way, to a conjugation of s. The conjugation on s depends indeed also on the description of which roots This is determined by the choice of a Cartan involution θ on s, with θ(h) = h and σ•θ = θ•σ, which induces a map, that we will denote by the same symbol, We will write for simplicityᾱ instead of σ(α) and R • for R σ • when this will not cause confusion. We recall that κ σ = {X ∈ s σ | θ(X)=X} is a maximal compact Lie subalgebra of s σ and that we have the Cartan decomposition When θ(α) =α, then either s α is contained in the complexification κ of κ σ (compact root) or in the complexification p of p σ (hermitian root).
The subalgebras q Φ ∩q Φ , q Φ andq Φ turn out to be direct sums of h and root subspaces s α ; in particular q Φ ∩q Φ is the direct sum of h and the root subspaces Having fixed a base B of simple roots of the root system R associated to (s, h), the orbit of the real form is determined by the data of: • a subset Φ of B specifying the parabolic subalgebra q Φ ; • a conjugation σ of R ; • into a first set consisting of the compact and a second of the hermitian roots. We point out that different choices of σ may yield the same CR submanifold M Φ,σ . In particular, we can conjugate σ by any element of the subgroup of the Weyl group generated by reflections with respect to roots in B\Φ.

2.3.
Contact nondegeneracy for parabolic CR algebras. Since by definition simple Lie algebra have no proper nontrivial ideals, we obtain Proposition 2.1. A real orbit M Φ,σ which is fundamental and does not have a totally complex factor is contact nondegenerate.
Proof. We can indeed reduce to the case where (s σ , q Φ ) is effective and s σ is simple, in which the proof is straigthforward.
We have the following criterion Let (s σ , q Φ ) be a parabolic CR algebra and set To show the vice versa, we note that any Lie subalgebra of s containing q Φ is of the form q Ψ for some Ψ ⊆ Φ. If it contains q Φ +q Φ , then Ψ ⊆ Φ σ • . This proves the first item. It suffices to prove the second item in the case where Φ = Φ σ • . Then condition (2.8) is equivalent to the fact that each α ∈ B belongs either to Q Φ or toQ Φ and is therefore clearly sufficient for (s σ , q Φ ) being fundamental. Vice versa, when this condition is not satisfied, we can pick α ∈Q c Φ ∩Φ. Then q {α} is a proper parabolic subalgebra of s containing both q Φ andq Φ . Therefore q Φ +q Φ generates a proper Lie subalgebra of s and hence (s σ , q Φ ) is not fundamental. This completes the proof.

2.4.
Conditions for weak nondegeneracy. To discuss weak nondegeneracy, we observe that the terms of the chain (1.11) for (s σ , q Φ ) can be described by the combinatorics of the root system. We recall that the chain is Φ in the chain is the direct sum of h and root spaces s α . Let us set With the notation of §2.1, we have Q 0 This yields a characterization of weak nondegeneracy in terms of roots: Proposition 2.4. A necessary and sufficient condition for (s σ , q Φ ) being weakly nondegenerate of Levi order q is that Q Remark 2.5. The necessary and sufficient condition for (s σ , q Φ ) being weakly nondegenerate is that (cf. [1, Lemma 12.1]) For any root β ∈ Q Φ \Q Φ we denote by q σ Φ (β) and call its Levi order the smallest number k for which (2.11) is valid. We put q σ Φ (β) = +∞ when (2.11) is not valid for any positive integer k.
Lemma 2.6. Assume that β ∈ Q Φ \Q Φ has finite Levi order q σ Φ (β)=q and (2.11) is satisfied for a sequence α 1 , . . . , α q . Then Proof. Let us first prove (ii). With the notation in (2.11), we observe that γ h Q Φ for h<q, because, otherwise, γ q ∈Q Φ . Next we prove (iii). Let {Z α } α∈R ∪ {H i ∈ h | 1≤i≤ℓ} be a Chevalley basis for (s, h). Then (2.11) is equivalent to the fact that The item (iii) follows because and, by the minimality assumption, the right hand side belongs to q Φ +q Φ . Let us prove (i) by contradiction. If α i ∈ Q Φ ∩Q Φ for some 1≤i≤q, then we could assume by (iii) that it was α q . Then yields the contradiction. Also (iv) is an easy consequence of (iii), because if α i +α j (1≤i, j≤q) is a root, than it would belong toQ Φ ∩ Q c Φ and, by substituting to the two roots α i , α j the single root α i +α j we would obtain a sequence satisfying (2.11) and containing q−1 terms.
The proof is complete.
Proof. By Lemma 2.6 the condition is necessary. To prove that it is also sufficient, we can argue by contradiction: if we could find β ∈ Q ∩Q c Φ with q σ Φ (β) = +∞, then by (2.13) we could construct an infinite sequence ( Since ξ Φ (γ h ) ≥ ξ Φ (β)+h and ξ Φ is bounded, we get a contradiction.
More precisely we obtain: , if β belongs to a root system of type A 2 ; or is a long root of a root system of type B 2 ; q(β)=2, if β belongs to a root system of type A ≥3 , C, or is a short root of a system of type B 2 , G; q(β)=3, if β is a short root of a root system of type B ≥3 , F; q(β)=4, if β belongs to a root system of type D, E, or is a long root of a root system of type B ≥3 , F, G.
Proof. For short we will call admissible a sequence (α i ) for which (2.14) is valid. Let us set We consider the different cases using for root systems the notation of [6]. Type A. We have R = {±(e i − e j ) | 1≤i< j≤n} where e 1 , . . . , e n is an orthonormal basis of R n . We can take β = e 2 −e 1 . Then An admissible sequence (α i ) can contain at most one element from each of the two sets in the right hand side of ( * A). If n = 3, then R add (β) = {e 3 −e 2 , e 1 −e 3 } contains two elements, whose sum is still a root and therefore q(β) = 1.
If n>3, then the only possible choice is that of a couple of roots e i −e 2 , e 1 −e j with 3 ≤i j≤n and hence q(β) = 2. Type B. We have R = {±e i ±e j | 1≤i< j≤n}∪{±e i | 1≤i≤n}, for an orthonormal basis e 1 , . . . , e n of R n (n≥2).
If β is a short root, we can take β = −e 1 . Then An admissible sequence contains at most one root from the first and two from the second set in the right hand side of ( * B). Thus q(−e 1 )≤3. The sequence e 1 −e 2 , e 1 +e 2 satisfies (2.14) and therefore q(−e 1 )≥2. We have equality if n=2, because in this case R add (−e 1 ) ={±e 2 , e 1 ±e 2 } and the maximal admissible sequences are then (e 2 ), (−e 2 ), (e 1 +e 2 , e 1 −e 2 ).
An admissible sequence contains at most two equal terms from the first and two from each of the second and third on the right hand side of ( * * B). Moreover, if one term is taken from the first, we can take at most one from each one of the other two. This implies that q(−e 1 −e 2 )≤4 and in fact q(−e 1 −e 2 )=4, with maximal sequences isomorphic to one of e 1 + e 3 , e 1 − e 3 , e 2 + e 4 , e 2 − e 4 , e 1 , e 1 , e 1 −e 3 , e 1 +e 3 , which, summed up to (−e 1 −e 2 ), gives e 1 +e 2 .
Type C. We can take R = {±e i ±e j | 1≤i< j≤n}∪{±2e i | 1≤i≤n}, for an orthonormal basis e 1 , . . . , e n of R n (n≥3). If β is a short root, we can assume that β=(−e 1 −e 2 ). Then An admissible sequence may contain both roots of the first, but at most one root from each the second and third sets on the right hand side of ( * C). Moreover, a term in one of the last two forbids the corresponding term in the first one. This yields q(−e 1 −e 2 )=2, with maximal sequences isomorphic to (the third one should be omitted if n=3) (2e 1 , 2e 2 ), (2e 1 , e 2 +e 3 ), (e 1 +e 3 , e 2 +e 4 ) If β is a long root, we can assume that β = −2e 1 . Then We note that q(−2e 1 )≤4. We cannot take in an admissible sequence both the element e 1 +e i and e 1 −e i , because they add up to the root 2e i . Hence in fact q(−2e 1 )=2, with maximal sequence isomorphic to e 1 +e 2 , e 1 +e 3 .
Type D. We can take R = {±e i ±e j | 1≤i< j≤n}, where e 1 , . . . , e n is an orthonormal basis of R n (n≥4). We can assume that β = −e 1 −e 2 . We have An admissible sequence contains at most two elements from each set in the right hand side of ( * D). Therefore q(−e 1 −e 2 )≤4 and in fact we have equality with maximal admissibe sequences isomorphic to e 1 +e 3 , e 1 −e 3 , e 2 +e 4 , e 2 −e 4 , which, summed up to (−e 1 −e 2 ), give e 1 +e 2 .
Type E. Since the root systems E 6 and E 7 can be considered as subsystems of E 8 , we will restrain to this case. We consider, for an orthonormal basis e 1 , . . . , e 8 of R 8 , We can take β = (−e 1 −e 2 ). Then An admissible sequence may contain at most two roots from each set on the right hand side of ( * E) and no more than four terms. Clearly we can take the maximal sequence showing that q(−e 1 −e 2 ) = 4. Moreover, any admissible sequence containing four terms sums up to (−e 1 −e 2 ) to yield e 1 +e 2 .
Type F. For an orthonormal basis e 1 , e 2 , e 3 , e 4 of R 4 we take If β is a short root, we can take β=−e 1 . Then To build an an admissible sequence we can take at most one element from the first, two from the second and from the third set in the right hand side of ( * F). Indeed two roots of the form 1 2 (±e 1 ±e 2 ±e 3 ±e 4 ) do not add up to a root if and only if they differ by only one sign. Moreover, no root can be taken from the first if one is taken from the last set. These considerations imply that q(−e 1 )≤3 and in fact equality holds, as (−e 1 ) is contained in a subsystem of type B 3 .
If β is a long root, we can assume β=(−e 1 −e 2 ). We have We note that the sum of four terms of R add (−e 1 −e 2 ) is a linear combination β + k 1 e 1 +k 2 e 2 +k 3 e 3 +k 4 e 4 with k 1 +k 2 ≥2 and therefore, if they form an admissible sequence, is equal to e 1 +e 2 . Since R contains subsystems of type B 3 , there are indeed admissible sequences with four elements.
As an easy consequence we obtain: Theorem 2.11. Let M Φ,σ be a real orbit which is fundamental and weakly nondegenerate. Then its Levi order is less or equal to 3.
Proof. This is a consequence of Prop.2.10 and the fact that, if β does not belong toQ Φ , then −β ∈Q Φ becauseQ Φ is a parabolic set of roots.
for an orthonormal basis e 1 , . . . , e n of R n . Fix k with 1<k<n. The crossmarked Dynkin diagram × represents the grassmannian of totally b-isotropic k-planes in C 2n+1 . Here α i =e i −e i+1 for 1≤i<n and α n = e n . We have Φ = {α k } and Real forms are obtained by fixing a conjugation σ on C 2n+1 . Then is hermitian symmetric and nondegenerate, of signature (p, q) for a pair of nonnegative integers with p+q=2n+1. The Lie algebra of the group of Clinear transformations which keep fixed both b and b σ is a real form s σ of s ≃o 2n+1 (C), which is isomorphic to the real simple Lie algebra o(p, q).
We define a conjugation σ on R by According to the number of compact roots between e 2 , . . . , e n−1 , this conjugation corresponds to any of the Lie algebras o(p, 2n+1 − p) with 1≤p≤2n. The orbit M Φ,σ consists of b σ -isotropic k-spaces ℓ k with dim(ℓ k ∩σ(ℓ k ))=k−1. By (1.13), since q Φ is maximal, if the parabolic CR algebra (s σ , q Φ ) is fundamental and not totally complex, then it is also weakly nondegenerate. To compute its Levi order we observe that The fact that Q c Φ ∩Q c Φ ∅ shows that (s σ , q Φ ) is not totally complex. Since e n ∈ Q Φ ∩Q c Φ and e 1 ∈Q Φ ∩Q c Φ add up to e 1 +e n , the CR algebra (s σ , q Φ ) is fundamental and therefore, as we noticed above, weakly nondegenerate.

We have
This shows that (s σ , q Φ ) is not totally complex and therefore, since q Φ is maximal parabolic, this CR algebra is weakly nondegenerate iff it is fundamental. We have Note that e 3 +e 4 ∈ Q Φ ∩Q c and e 1 −e 3 ∈Q ∩ Q c sum up to e 1 +e 4 . This shows that (s σ , q Φ ) is not totally complex and fundamental. Since q Φ is maximal parabolic, this implies that (s σ , q Φ ) is weakly nonedegenerate.
Example 2.14. Consider a semisimple complex Lie algebra s of type G 2 . Having fixed a Cartan subalgebra, we can write its root system in the form for an orthonormal basis e 1 , e 2 , e 3 of R 3 . We consider the complex flag manifold F Φ corresponding to the cross-marked Dynkin diagram It corresponds to the grading functional ξ Φ with ξ Φ (e 1 ) = 1, ξ Φ (e 2 ) = 1, ξ Φ (e 3 ) = 0.
We consider the conjugation defined by σ(e 1 ) = e 3 , σ(e 2 ) = e 2 , σ(e 3 ) = e 1 . Then then is sufficient to check that (s σ , q Φ ) is weakly nondegenerate to find that it is also fundamental.
The root β = 2e 3 −e 1 −e 2 belongs to Since ξ Φ equals one on every root of Q c Φ ∩Q Φ , a sequence satisfying (2.11) has at least three roots. We find indeed that e 2 − e 3 , e 2 − e 3 , e 2 − e 3 is a sequence with the desired properties, proving that (s σ , q Φ ) is fundamental and has Levi order three.
2.6. Levi order of orbits of the minimal type. Weak nondegeneracy for minimal orbits was characterized in [1, Thm.11.5] by using their description in terms of cross-marked Satake diagrams (see e.g. [5,25]).
Let h R be a maximally vectorial Cartan subalgebra of s σ , h its complexification and R the root system of (s, h). Then all roots in R • are compact. We can select a basis B such that the conjugate of any positive noncompact root stays positive. This condition defines an involution ǫ : B→B, which keeps fixed the elements of B • = B ∩ R • and such that, for nonnegative n α,β ∈ Z, The Satake diagram Σ B is obtained from ∆ B by painting black the roots in B • and joining by an arch the pairs of distinct simple roots α 1 , α 2 with ǫ(α 1 ) = α 2 .
Minimal orbits correspond to cross-marked Satake diagrams: they are associated to parabolic q Φ for which all roots in Q c Φ ∩Q n Φ are compact. Let us drop the assumption that h R is maximally vectorial. The map θ : α → −ᾱ induced on R by the Cartan involution (see §2.2) acts on Q c Φ ∩Q n Φ , which is therefore the union of its fixed points, which are roots in R • , and of pairs (α, −ᾱ) of distinct roots.
Definition 2.4. We say that the CR algebra (s σ , q Φ ) and the corresponding CR manifold M Φ,σ are of the minimal type if the roots in Q c Φ ∩Q n Φ are fixed by the Cartan involution, i.e. if 15. For a parabolic CR algebra (s σ , q Φ ) the following are equivalent to the fact that it is of the minimal type: Consider the conjugation σ(e 1 ) = e 2 , σ(e 2 ) = e 1 , σ(e 3 ) = −e 3 .
This (s σ , q Φ ) is of the minimal type, because Lemma 2.18. The parabolic CR algebra (s σ , q Φ ) of a minimal orbit is of the minimal type.
Proof. Suppose that Φ is the set of crossed roots in a cross-marked Satake diagram. Since all roots β in Q c Φ are positive, by (2.17), if β ∈ Q c Φ \R • , then its conjugateβ is positive, and hence has ξ Φ (β)≥0. This shows that (2.18) is valid, i.e. that M Φ,σ is of the minimal type.
We obtain Since Q c Φ ∩Q c Φ is nonempty, e 1 −e 4 = (e 3 −e 4 )+(e 1 −e 3 ) and q Φ is maximal, we obtain that (s σ , q Φ ) is fundamental and weakly nondegenerate. Since ξ Φ (e 3 −e 2 )=−1 and ξ Φ is 1 on all the elements ofQ Φ ∩ Q c Φ , the Levi order is at least, and thus equal, by Thm.2.17, to 2. We have in fact Example 2.21. Consider the CR algebra described by the cross-marked Satake diagram It corresponds to the CR algebra (s σ , q Φ ), with s σ ≃ sl 3 (H), of the grassmannian M Φ,σ of 3-planes of C 6 ≃ H 3 containing a quaternionic line. We have , with s σ ≃ sl 7 (C), is fundamental and weakly nondegenerate. This can be proved e.g. by applying the criteria in [1]. Since R • =∅, its Levi order is one.
Remark 2. 24. It was observed in [3] that a parabolic CR algebra (s σ , q Φ ) can always be described by using a base B associated to an S -chamber: this means one withᾱ ≻ 0 for all α ∈B\(Φ∪B • ). The condition of being of the minimal type translates for this choice of B into the fact thatᾱ ≻ 0 also for the elements in Φ\B • . The real dimension of M Φ,σ is the difference dim C (s)− dim C (q Φ ∩q Φ ), i.e. #R − #(Q Φ ∩Q Φ ). Thus, in case Φ contains a root α B • withᾱ ≺ 0, the symmetry with respect to α yields a new basis B ′ that, with the crosses in the same positions, describes a new real orbit whose dimension is smaller by one unit. Then, parametrizing the real orbits that we can describe, after having made a fixed choice of h R , by using the Weyl chambers of R , those of the minimal type are a sort of local minima with respect to dimension. One has to be cautious because, unless h R is maximally vectorial, there can be several inequivalent choices of B such thatᾱ ≻ 0 for all α ∈ B\B • that we can look at as yielding different local minima for the dimension of a class of real orbits. Example 2.25. Consider sl 3 (C) as a simple real Lie algebra. Its complexification is the direct sum of two copies of sl 3 (C). Its root system can be described, after fixing orthogonal basis e 1 , e 2 , e 3 , e 4 and e ′ 1 , e ′ 2 , e ′ 3 , e ′ 4 of two copies of R 4 , by j≤4}. Let us consider the cross-marked Dynkin diagram  Example 2.27. Consider a root system of type C 3 and the cross-marked Dynkin diagram Consider the conjugation σ(e 1 ) = e 3 , σ(e 2 ) = −e 2 , σ(e 3 ) = e 1 .

Weakly nondegenerate CR manifolds with larger Levi orders
Fix any integer q≥1. In this last section we discuss in detail the example of a homogeneous CR manifold M of CR dimension q+1 and CR codimension 1 which is fundamental and weakly nondegenerate of Levi order q.
The compact group SU(2) acts transitively on the complex projective line CP 1 . The homogeneous complex structure of CP 1 can be defined by the totally complex CR algebra (su(2), b), where su(2) is the real Lie algebra of anti-Hermitian 2×2 matrices and b a Borel subalgebra of its complexification sl 2 (C). This CR algebra corresponds to the simple cross-marked Satake diagram α × The root system of the complexification sl 2 (C) is R = {±(e 1 −e 2 )}. and we take α = (e 1 −e 2 ), with fundamental weight ω = α/2.
The irreducible finite dimensional complex linear representations of sl 2 (C) are indexed by the nonnegative integral multiples k·ω of ω and the corresponding irreducible sl 2 (C)-module V k·ω can be identified with the space of complex homogeneous polynomials of degree k in two indeterminates where, for a diagonal H in the canonical Cartan subalgebra of sl 2 (C), are the one-dimensional weight spaces contained in V kω . Sinceω=−ω, we have V kω = V kω . The anti-C-linear automorphism θ kω of V kω defined by the conjugation σ comes from (z, w) → (−w,z) and therefore Then θ 2 kω equals id V kω for k even and −id V kω for k odd. Accordingly, for k even V kω is the complexification of an irreducible (k+1)-dimensional representation of the real type, that we will denote by V R kω ; for k odd is isomorphic to a 2(k+1)-dimensional irreducible representation of the quaternionic type of su 2 (see e.g. [7, Ch.IX, App.II, Prop.2]).
Remark 3.1. Studying irreducible representation of su 2 turns out to be of some interest in quantum physics, as they arise when considering rotations on fermionic and bosonic systems (for more details see [27,Ch.5 , §5]).

The subspace
is a b-submodule of V kω and we can consider the semidirect sum b ⊕ V − kω as a subalgebra of the abelian extension sl 2 (C) ⊕ V kω (cf. e.g. [24, Ch.VII, §3]). We may consider the map SL 2 (C) → CP 1 associated to our choice of a Borel subalgebra B as a principal bundle with structure group B. Then the Lie pair (sl 2 (C) ⊕ V kω , b ⊕ V − kω ) defines a complex holomorphic vector bundle E k with base CP 1 and typical fiber V kω /V − kω ≃ 2h≤k V (k−2h)ω kω (this is an example of Mostow fibration, see [21] for more details).
Proposition 3.2. Let q be any positive integer. Then is the CR algebra of a CR manifold E 2q , of CR dimension q+1 and CR codimension 1, which is fundamental and weakly nondegnenerate of Levi order q.
Proof. We havē H is the canonical basis of sl 2 (C) and w a nonzero vector of V −2qω 2qω , then the images of X −α , w , X α w , . . . , X q−1 we obtain that E 2q is fundamental and weakly nondegenerate. With the notation of the previous section, we have Q c Φ ∩Q Φ ={α}, with ξ Φ (α)=1 and ξ Φ (−2 jω)=− j. Since g/(q ′ Φ +q ′ Φ ) is generated by the image of V 0 2qω , by the above considerations the Levi order of an element of V −2 jω 2qω equals j. This shows that the Levi order of (g R , q ′ Φ ) is q. In an analogous way we can also prove Proposition 3.3. For each positive integer q, the homogeneous CR manifold E 2q is contact nondegenerate of order q.
Remark 3.4. Representations V kω with an odd k are canonically associated with complex holomorphic vector bundles E k , of rank (k+1), with base CP 1 .