On homogeneous and symmetric CR manifolds
aa r X i v : . [ m a t h . DG ] O c t ON HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS
ANDREA ALTOMANI, COSTANTINO MEDORI, AND MAURO NACINOVICH
Abstract.
We consider canonical fibrations and algebraic geometric struc-tures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR struc-tures on semisimple Lie groups and of CR -symmetric structures on completeflag varieties. Introduction
In this paper we discuss some topics about the CR geometry of homogeneousmanifolds. Our main tool are CR algebras, introduced in [28] to parametrize hom-ogeneous partial complex structures. If M is a G -homogeneous CR -manifold, weassociate to each point p of M a pair ( g , q ), consisting of the Lie algebra g of G and of a complex Lie subalgebra of its complexification g . If p = g · p , with g ∈ G ,is another point of M , the CR algebra of M at p is ( g , Ad( g )( q )), so that q is de-termined by M modulo G -equivalence. Several questions about the CR geometryof M can be conveniently reduced to Lie-algebraic questions about the pair ( g , q ).This makes for a program that has already been started and carried on in severalpapers, see e.g. [1, 2, 23, 24], where the investigation focused on different specialclasses of homogeneous CR manifolds. In [20], W.Kaup and D. Zaitsev introducedthe notion of CR -symmetry, generalizing at the same time the Riemannian andHermitian cases, and showing that CR -symmetric manifolds are CR -homogeneous.In [23, 24] one of the Authors, in collaboration with A.Lotta, classified andinvestigated some classes of CR -symmetric manifolds. A key point was to representthe partial complex structure of M by an inner derivation J of g . The existence ofsuch a J was a crucial step in the classification of semisimple Levi-Tanaka algebrasin [25], and in establishing the structure of standard CR manifolds, which arehomogeneous CR manifolds with maximal CR -automorphisms groups (see [26, 28]).In this paper we will delve further into the relationship between the existence of J ,canonical CR fibrations, CR -symmetry.In the first section we survey the basic notions of CR and homogeneous CR manifolds, including the J -property, CR -symmetry, and explaining their relation-ship.In §
2, we discuss the existence of Levi-Malcev and Jordan-Chevalley fibrations,and the existence of suitable homogeneous CR structures on their total spaces,bases and fibers. These fibrations, interwoven with canonical decompositions of the CR algebras, were largely employed in [26, 27], and also in [1, 2] in the context ofparabolic CR manifolds. Here they are considered in full generality.In § G -homogeneous CR man-ifold starting from an abstractly given CR algebra ( g , q ). This is not alwayspossible, and the question arises to describe natural modifications of ( g , q ) leadingto new G -homogeneous CR manifolds. These are described in § § § Date : November 5, 2018.2000
Mathematics Subject Classification.
Primary: 32V05 Secondary: 14M15, 17B20, 57T20.
Key words and phrases.
Homogeneous CR manifold, CR algebra, J -property, CR -symmetry. The construction of § § CR manifolds in [1, 2]. Thus we consider CR manifoldsin an algebraic geometric context in §
4. We show that algebraic CR manifoldscanonically embed into the set of regular points of complex algebraic varieties. Animportant distinction arises between algebraic and weakly-algebraic CR manifolds,the latter admitting analytic, but not algebraic, embeddings.In the two final sections we deal with special applications. In § CR structures on semisimple real Lie groups. In §
6, we consider symmetric CR structures on full flags of complex Lie groups. They had been considered in [14]in a slightly different context. In our treatment we use the CR algebras approachand we are especially interested in the relationship between CR -symmetry and the J -property. All the CR -symmetric manifolds of [23, 24] also enjoyed the J -property.We are in the same situation when we consider the complete flags of the classicalgroups. On the complete flags of the exceptional groups we found examples of CR -symmetric structures which do not enjoy even the weaker version of the J -property, and also examples of CR -structures enjoying the weak- J -property, butnot the J -property.1. CR manifolds, CR algebras, J -property, CR -symmetry CR manifolds. Let M be a smooth real manifold. A CR structure on M isthe datum of an almost Lagrangian formally integrable smooth complex subbundle T , M of the complexified tangent bundle T C M . The subbundle T , M is requiredto satisfy: T , M ∩ T , M = 0 , (1.1) [Γ( M, T , M ) , Γ( M, T , M )] ⊂ Γ( M, T , M ) . (1.2)The rank n of T , M is the CR - dimension , and k = dim R M − n the CR - codimension of M . If n = 0, we say that M is totally real; if k = 0, M is acomplex manifold in view of the Newlander-Nirenberg theorem.When M is a real submanifold of a complex manifold X, for every p ∈ M wecan consider the C -vector space T , p M = T , p X ∩ T C p M of the anti-holomorphiccomplex tangent vectors on X which are tangent to M at p . If the dimensionof T , p M is independent of p ∈ M , then T , M = S p ∈ M T , p M is an almostLagrangian formally integrable complex subbundle of T C M , defining on M thestructure of a CR submanifold of X. If the complex dimension of X is the sum ofthe CR dimension and the CR codimension of M , the embedding M ֒ → X is called CR -generic .A smooth map f : M ′ → M is CR if M and M ′ are CR manifolds, and df ( T , M ′ ) ⊂ T , M . The notions of CR immersion, submersion, diffeomorphismand automorphism are defined in an obvious way. The set of all CR automorphismsof a CR manifold M is a group that we denote by Aut CR ( M ). Definition 1.1 (Characteristic bundle and Levi forms) . Let HM be the subbundleof T M consisting of the real parts of the elements of T , M . Its annihilator bundle H M ⊂ T ∗ M is called the characteristic bundle of M . We have(1.3) H p M = { ξ ∈ T ∗ p M | ξ ( z ) = 0 , ∀ z ∈ T , p M } , for all p ∈ M . If Z , Z are smooth sections of T , M , and Ξ a smooth section of H M , all definedon an open neighborhood of p in M , with Z i ( p ) = z i and Ξ( p ) = ξ , then we set(1.4) L ξ ( z , z ) = id Ξ( z , ¯ z ) = − iξ ([ Z , ¯ Z ]) . N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 3
In this way we define a Hermitian symmetric form L ξ on T , p M , which is calledthe scalar Levi form at ξ ∈ H M .If Z is a smooth section of T , M defined on a neighborhood of p ∈ M , with Z ( p ) = z , we define L p ( z ) = ̟ p ( i [ ¯ Z, Z ] p ) , (1.5)where ̟ p : T p M → T p M /H p M is the projection into the quotient . This map L p : T , p M → T p M /H p M is the vector valued Levi form of M at p .1.2. Homogeneous CR manifolds. Let M be a smooth CR manifold and G aLie group. Definition 1.2.
We say that M is a G -homogeneous CR manifold if G actstransitively on M by CR diffeomorphisms.Let M be a G -homogeneous CR manifold. Fix p ∈ M , let I = { g ∈ G | g · p = p } be the isotropy subgroup, and π : G → M ≃ G / I the associatedprincipal I -bundle. Denote by Z ( G ) the space of smooth sections of the pullback π ∗ T , M of T , M to G :(1.6) Z ( G ) = { Z ∈ X C ( G ) | π ∗ Z g ∈ T , π ( g ) M, ∀ g ∈ G } , where X C ( G ) is the space complex valued smooth vector fields on G . By (1.2),the complex system Z ( G ) is formally integrable, i.e.(1.7) [ Z ( G ) , Z ( G )] ⊂ Z ( G ) . Moreover, Z ( G ) is invariant by left translations, and therefore is generated, as aleft C ∞ ( G , C )-module, by its left invariant vector fields.Let g be the Lie algebra of G and g its complexification. By (1.7), the leftinvariant elements of Z ( G ) define a complex Lie subalgebra q of g , given by(1.8) q = π − ∗ ( T , p M ) ⊂ g ≃ T C e G . We can summarize these observations by
Proposition 1.3.
Let G be a Lie group, I a closed subgroup of G , g = Lie( G ) and i = Lie( I ) their Lie algebras. Then (1.8) establishes a one-to-one correspon-dence between G -homogeneous CR structures on M = G / I and complex Liesubalgebras q of g with q ∩ g = i . (cid:3) This lead us to introduce the notion of a CR algebra in [28]. Definition 1.4. A CR algebra is a pair ( g , q ), consisting of a real Lie algebra g and of a complex Lie subalgebra q of its complexification g , such that the quotient g / ( q ∩ g ) is finite dimensional. The real Lie subalgebra i = q ∩ g is called the isotropy subalgebra of ( g , q ).If M is a G -homogeneous CR manifold and q is defined by (1.8), we say thatthe CR algebra ( g , q ) is associated with M . Remark 1.5.
The CR -dimension and CR -codimension of M can be computed interms of its associated CR algebra ( g , q ). We have indeed CR -dim M = dim C q − dim C ( q ∩ ¯ q ) , (1.9) CR -codim M = dim C g − dim C ( q + ¯ q ) . (1.10)The CR algebra ( g , q ), is totally real when CR -dim M = 0, totally complex when CR -codim M = 0. A. ALTOMANI, C. MEDORI, AND M. NACINOVICH
The scalar and vector valued Levi forms of a G -homogeneous CR manifoldscan be computed in terms of the Lie product of g , by using G -left-invariant vectorfields. Indeed, for p ∈ M , ξ ∈ H p M , we have L ξ ( z , z ) = − iπ ∗ ( ξ )([ Z ∗ , ¯ Z ∗ ]) if Z , Z ∈ q , and π ∗ ( Z ∗ i ) p = z i , (1.11) L p ( z ) = ̟ p ( π ∗ ( i [ ¯ Z ∗ , Z ∗ ])) if Z ∈ q , and π ∗ ( Z ∗ ) p = z, (1.12) for z, z , z ∈ T , p M, ξ ∈ H p M ;here Z ∗ , Z ∗ , Z ∗ are the left invariant vector fields of Z, Z , Z ∈ q . The natural isomorphism between T p M/H p M and the quotient e = g / ( { q + ¯ q } ∩ g ) makes L p ( z ) correspond to the projection of i [ ¯ Z, Z ] into e . Definition 1.6.
Consider a CR algebra ( g , q ). Let Lie C ( g ) be the set of complexLie subalgebras of g . We recall that ( g , q ) is called: fundamental if q ′ ∈ Lie C ( g ) , q + ¯ q ⊂ q ′ = ⇒ q ′ = g weakly nondegenerate if q ′ ∈ Lie C ( g ) , q ⊂ q ′ ⊂ q + ¯ q = ⇒ q ′ = q Levi-nondegenerate if { Z ∈ q | ad( Z )(¯ q ) ⊂ q + ¯ q } = q ∩ ¯ q , effective if no nontrivial ideal of g is contained in i .If M is a G -homogeneous CR manifold with associated CR algebra ( g , q ), theabove properties are related to the CR geometry of M (see e.g. [1]) by:(1) ( g , q ) is fundamental if and only if M is of finite type in the sense of Bloomand Graham (see [6]).(2) ( g , q ) is Levi-nondegenerate if and only if the vector valued Levi form of M is nondegenerate. Levi-nondegeneracy implies weak nondegeneracy.(3) ( g , q ) is fundamental and weakly nondegenerate if and only if the groupof germs of CR diffeomorphisms at p ∈ M stabilizing p is a finite dimen-sional Lie group, i.e. if and only if M is holomorphically nondegenerate(see e.g. [5], [13]).(4) A fundamental ( g , q ) is weakly degenerate if and only if there exists a local G -equivariant CR fibration M → M ′ , with nontrivial complex fibers.(5) Effectiveness means that the normal subgroups of G contained in theisotropy I are discrete.Let g , g ′ be real Lie algebras and q , q ′ complex Lie subalgebras of their complexifi-cations g , g ′ . A Lie algebra homomorphism φ : g → g ′ is a CR algebras morphismfrom ( g , q ) to ( g ′ , q ′ ) if the complexification φ of φ transforms q into a subalgebraof q ′ . The pair ( φ , φ ) is a CR algebras immersion if φ − ( q ′ ∩ ¯ q ′ ) = q ∩ ¯ q , φ − ( q ′ ) = q , a CR algebras submersion if φ ( g )+ q ′ ∩ ¯ q ′ = g ′ , φ ( q )+ q ′ ∩ ¯ q ′ = q ′ , a CR algebras local isomorphism if it is at the same timea CR algebras immersion and submersion.The CR algebra ( g ′′ , q ′′ ) with g ′′ = φ − ( q ′ ∩ g ′ ) and q ′′ = q ∩ φ − ( q ′ ∩ ¯ q ′ ) is the fiber of ( φ , φ ) : ( g , q ) → ( g ′ , q ′ ).When g = g ′ , q ⊂ q ′ , and φ is the identity, the corresponding morphism( g , q ) → ( g , q ′ ) is said to be g -equivariant (see [28]).If M and M ′ are homogeneous CR manifolds with associated CR algebras ( g , q ),( g ′ , q ′ ), local CR maps that are local CR immersions, submersions or diffeomor-phisms, correspond to algebraic CR morphisms of their CR algebras that are CR algebras immersions, submersions, or local isomorphisms, respectively, and viceversa. N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 5
For later reference, it is convenient to restate [28, Lemma 5.1] in the followingform.
Proposition 1.7.
Let I ⊂ I ′ be closed subgroups of a Lie group G . Let i , i ′ , g be the Lie algebras of I , I ′ , G , and i , i ′ , g their complexifications, respectively. Let ( g , q ) be a CR algebra, defining a G -invariant CR structure on M = G / I .Then a necessary and sufficient condition for the existence of a G -invariant CR structure on M ′ = G / I ′ making the G -equivariant map π : M → M ′ a CR submersion is that: (1.13) q ′ = q + i ′ is a Lie algebra , and q ′ ∩ g = i ′ . When (1.13) holds, it defines the CR algebra ( g , q ′ ) at p ′ = [ I ′ ] which definesthe unique G -homogeneous CR structure on M ′ for which M π −→ M ′ is a CR submersion. The J -property. Let M be a CR manifold. Its partial complex structure isthe vector bundle isomorphism J : HM → HM that associates to X p ∈ H p M thevector JX p ∈ H p M for which X p + iJX p ∈ T , p M .Let M be G - CR -homogeneous, with CR algebra ( g , q ) at p ∈ M , and set V = { Re Z | Z ∈ q } . The partial complex structure of M yields a complexstructure on V / i , via its canonical identification with H p M . This is the partialcomplex structure of ( g , q ) . Definition 1.8.
We say that ( g , q ) has the J -property if J ∈ Der( g ) can bechosen in such a way that(1.14) J ( i ) ⊂ i , X + iJ ( X ) ∈ q , ∀ X ∈ V . We say that a CR algebra ( g , q ) has the weak- J -property if there is J ∈ Der( g )such that, for Υ = Ad(exp( πJ/ i ) = i , X + i Υ( X ) ∈ q , ∀ X ∈ V . If J ∈ Der( g ) satisfies (1.14), then Υ = Ad(exp( πJ/ Remark 1.9.
Conditions (1.14) and (1.15) can also be expressed in terms of thecomplexifications of J, Υ. Namely, denoting by the same letter also their complex-ifications, they are equivalent to(1.14) ′ J ( q ) ⊂ q , Z − iJ ( Z ) ∈ q ∩ ¯ q , ∀ Z ∈ q , (1.15) ′ Υ( q ) = q , Z − i Υ( Z ) ∈ q ∩ ¯ q , ∀ Z ∈ q . For a map A ∈ gl ( g ) , denote by A s and A n its semisimple and the nilpotentparts, respectively. If A ∈ Der( g ), then also A s and A n are derivations of g (seee.g [19, § Proposition 1.10.
Let ( g , q ) be a CR algebra, and J ∈ Der( g ) . If J satisfies (1.14) , then also J s satisfies (1.14) . If Υ = Ad(exp( πJ/ satisfies (1.15) , thenalso Υ s = Ad(exp( πJ s / satisfies (1.15) .Proof. Indeed, Ad( πJ s /
2) is the semisimple part of Ad( πJ/ J s and Υ s arepolynomials of J , Υ, respectively, (1.14) ′ for J implies (1.14) ′ for J s , and likewise(1.15) ′ for Υ implies (1.15) ′ for Υ s . (cid:3) As a consequence, we can always assume in Definition 1.8 that J be a semisimplederivation of g . A. ALTOMANI, C. MEDORI, AND M. NACINOVICH
Symmetric CR manifolds. Let M be a CR manifold, with partial complexstructure J . A Riemannian metric g on M is CR - compatible if g ( JX p , JX p ) = g ( X p , X p ) for all p ∈ M and X p ∈ H p M . Let Θ ( M ) be the Lie algebra of realvector fields generated by Γ( M, HM ) and Θ p M = { X p | X ∈ Θ ( M ) } . Note thatΘ p M = T p M when M is of finite type in the sense of Bloom and Graham. Denoteby Θ ⊥ p M the orthogonal of Θ p M in T p M for the Riemannian metric g . Definition 1.11 (see [20]) . Let M be a CR manifold, with a CR -compatibleRiemannian structure. We say that M is CR -symmetric if, for each p ∈ M , there isan isometry σ p : M → M that fixes p , is a CR map, and whose differential restrictsto − Id on H p M ⊕ Θ ⊥ p M .In [20, Proposition 3.6] the CR -isometries of a symmetric CR -manifold M areproved to form a transitive group G of transformations of M .Given a CR algebra ( g , q ), let q ♮ be the Lie subalgebra of g generated by q +¯ q .We recall that a subalgebra k of g is compact if the Killing form of g is negativedefinite on k . We say that a subalgebra i of g is almost compact if there existsa decomposition i = k ⊕ t with k compact in g and t contained in the kernelof the Killing form of g . Definition 1.12.
We say that ( g , q ) is CR -symmetric if i = q ∩ g is almostcompact in g , and there exists an involution λ of g with(1.16) λ ( g ) = g , ker(Id − λ ) ⊂ q ♮ , λ ( q ) = q ,Z + λ ( Z ) ∈ q ∩ ¯ q , ∀ Z ∈ q . Conditions (1.16) imply that(1.17) [ Z , Z ] ∈ q ∩ ¯ q , ∀ Z , Z ∈ q . The involution λ is equivalent to the datum of a Z -gradation g = g (0) ⊕ g (1) , [ g ( i ) , g ( j ) ] ⊂ g ( i + j ) , (1.18)where ( i ) denotes the congruence class of i ∈ Z in Z , compatible with ( g , q ).Compatibility means that:(1.19) g (0) ⊂ q ♮ , q ∩ g (0) ⊂ q ∩ ¯ q , q = ( q ∩ g (0) ) ⊕ ( q ∩ g (1) ) , g = ( g ∩ g (0) ) ⊕ ( g ∩ g (1) ) . The involution λ and the Z -gradation (1.18) are related by(1.20) g (0) = { Z ∈ g | λ ( Z ) = Z } , g (1) = { Z ∈ g | λ ( Z ) = − Z } , and (1.16), (1.19) are equivalent to define the CR -symmetry of ( g , q ). Proposition 1.13.
Let ( g , q ) be a fundamental CR algebra with i almost compact,and having the weak- J -property. If J ( i ) = 0 , then ( g , q ) is CR -symmetric.Proof. Indeed, by the assumptions, the automorphism λ = Ad(exp( πJ )) is aninvolution of g that satisfies (1.16). (cid:3) Proposition 1.14.
Let M be a CR -manifold. Assume that M is CR -symmetricfor a CR -compatible Riemannian structure. Let G be the transitive group of CR -isometries of M , and ( g , q ) the corresponding CR algebra of M at p ∈ M . Then ( g , q ) is CR -symmetric.Vice versa, if M is a G -homogeneous CR manifold, having at a point p ∈ M a CR algebra ( g , q ) which is CR -symmetric, and the analytic subgroup tangent to i is compact, then there is a compatible Riemannian metric on M for which M is CR -symmetric. (cid:3) N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 7 Levi-Malcev and Jordan-Chevalley fibrations A -fibrations. Let G be a Lie group, g its Lie algebra, a an ideal of g ,and A the corresponding analytic normal subgroup of G . Definition 2.1.
Let M = G / I be a homogeneous space of G . If the subgroup A I is closed in G , we call the G -equivariant fibration(2.1) M = G / I π −−−−→ M ′ = G / ( A I ) the A -fibration of M .Assuming that M admits an A -fibration (2.1), we will discuss the existence ofcompatible G -homogeneous CR structures on M = G / I and M ′ = G / ( A I ).Denote by a the complexification of a . Proposition 2.2.
Let ( g , q ) , with q ∩ g = i , be a CR algebra defining a G -homogeneous CR structure on M , and assume that the subgroup A I is closed.A necessary and sufficient condition for the existence of a G -homogeneous CR structure on M ′ = G / ( A I ) , making the A -fibration (2.1) a CR map is that: (2.2) q ∩ ¯ q + a = ( q + a ) ∩ (¯ q + a ) . Assume that (2.2) is satisfied and define the CR structure on M ′ by ( g , q ′ ) , with q ′ = q + a . Then: (1) (2.1) is a G -equivariant CR fibration. (2) Its typical fiber F is the A -homogeneous manifold A / ( A ∩ I ) , havingan A -homogeneous CR structure defined by the CR algebra ( a , q ∩ a ) .Proof. By Proposition 1.3, the G -homogeneous CR structures on M ′ are in one-to-one correspondence with the CR algebras ( g , q ′ ), with isotropy i ′ = q ′ ∩ g equalto ( i + a ). The map π : M → M ′ is CR if q ⊂ q ′ . Thus ( i + a ) ⊂ ( q ∩ ¯ q + a ) ⊂ q ′ ,and q + a ⊂ q ′ . By Proposition 1.7, the map π : M → M ′ is a CR submersionif and only if the last inclusion is an equality. Finally (1) and (2) follow by [28, § (cid:3) Proposition 2.3.
We keep the notation above. Assume that ( g , q ) has the weak- J -property and that a is Υ -invariant. Then: (1) Condition (2.2) is satisfied. (2)
The basis ( g , q + a ) and the fiber ( a , a ∩ q ) of the A -fibration enjoy theweak- J -property.If we assume that ( g , q ) has the J -property, then both the basis and the fiber of the A -fibration enjoy the J -property.Proof. Let J ∈ Der( g ) be such that Υ = Ad(exp( πJ/ q + a ) ∩ g ⊂ i + a . An element A of ( q + a ) ∩ g is a sum A = ( X + iY ) + ( U − iY ), with X, Y, U ∈ g , X + iY ∈ q and U, Y ∈ a . Since both Y − iX and Y + i Υ( Y ) belong to q , we obtain that X + Υ( Y ) ∈ q ∩ g = i . Moreover, Υ( Y ) ∈ a , because a is Υ-invariant. Hence A = ( X + Υ( Y )) + ( U − Υ( Y )) ∈ i + a .(2) The subalgebras q + a and ( q ∩ ¯ q ) + a are Υ-invariant. Thus Υ yields multi-plication by i on the quotient ( q + a ) / (( q ∩ ¯ q ) + a ). This proves the statement forthe base. The statement for the fiber is trivial.The last statement can be obtained by repeating with minor changes the argu-ments used above for the proof of (2). (cid:3) We will apply the results above to the cases where a is either the radical or thenilpotent radical of g . A. ALTOMANI, C. MEDORI, AND M. NACINOVICH
The Levi-Malcev fibration.
Let g be a real Lie algebra and r its radical.The Levi-Malcev decomposition of g has the form(2.3) g = r ⊕ s , where s is a semisimple Levi factor of g , i.e. a Lie subalgebra of g isomorphic tothe quotient g / r .Let G be a Lie group with Lie algebra g . Its radical R is its maximal con-nected solvable subgroup, and equals its analytic Lie subgroup with Lie algebra r . Definition 2.4.
Let M = G / I be a homogeneous space of G . If R I is aclosed subgroup of G , we call the G -equivariant fibration(2.4) M = G / I π −−−−→ M ′ = G / R I the Levi-Malcev fibration of M . Example 2.5.
Not all homogeneous spaces admit a Levi-Malcev fibration. Take,for instance, G = SU (3) × R + and I = { (exp( tX ) , e t ) | t ∈ R } for X = i diag( α, β, γ ), with α, β, γ ∈ R , α + β + γ = 0, and α, β linearly independent over Q .Let R be the radical of G . Then I is closed, but R I = { (exp( tX ) , e s ) | t, s ∈ R } is not closed in G . Example 2.6.
Let s be a semisimple real Lie algebra and V a nontrivial realirreducible s -module. Let g = s ⊕ V be the Abelian extension of s by V . ItsLie algebra structure is defined by ( [ X + v , X + v ] = [ X , X ] + X · v − X · v for X , X ∈ s , v , v ∈ V . Radical and nilradical of g are both equal to V ≃ ⊕ V , and s ≃ s ⊕ ⊂ g is a Levi subalgebra and a reductive component of g . Then g = s ⊕ V is aJordan-Chevalley and a Levi decomposition of g , at the same time.Fix a connected semisimple Lie group S with Lie algebra s , to which therepresentation of s on V lifts. The product( g , v ) · ( g , v ) = ( g g , v + g ( v )) , for g , g ∈ S , v , v ∈ V , defines on G = S × V the structure of a Lie group with Lie algebra g and radical R = { e S } × V .Let s and V be the complexifications of s and V , respectively, so that g = s ⊕ V is the complexification of g .Fix a closed subgroup A of S , with Lie algebra a . The stabilizer I of anyvector v ∈ V in A is a closed subgroup of S and hence of G . Let M = G / I ≃ ( S / I ) × V be endowed with the G -homogeneous CR structure defined by ( g , q ),for q = C · { X + iX · v | X ∈ a } ⊂ g . With a equal to the complexification of a , we have: i = q ∩ g = q ∩ s = { X ∈ a | X · v = 0 } , q + V = a ⊕ V, q + ¯ q = a ⊕ C · ( a · v ) . Hence the CR algebra ( g , q + V ), corresponding to the basis of the G -equivariantmap M → N = G / ( A × V ), is totally real and locally CR isomorphic to ( s , a ).The fiber F is an ( A ⋉ V )-homogeneous CR manifold, with CR algebra ( a ⊕ V , q ).Thus, if a = i , there is no G -homogeneous CR structure on M ′ = G / R I suchthat the fibration M → M ′ is an equivariant CR submersion. N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 9
The Levi subalgebras of g are parametrized by the elements of V : s ( v )0 = { X + X · v | X ∈ s } , for v ∈ V . We have s ( v )0 ∩ q = { X ∈ s | X ( v ) = 0 , X ( v ) = 0 } ⊂ i , s ( v ) ∩ q = { Z ∈ s | Z ( v ) = 0 , Z ( v ) = 0 } = (cid:0) s ( v )0 ∩ q (cid:1) C , so that, for every choice of v ∈ V , the CR algebra ( s ( v )0 , q ∩ s ( v ) ) is totally real. Thus,if a = i , there is no Levi factor s ( v )0 in g such that ( s ( v )0 , ( q ∩ s ( v ) )) ≃ ( g , q + V ).In [27] the homogeneous CR manifolds M with ( g , q ) of Levi-Tanaka type wereshown to admit a Levi-Malcev fibration; (2.4) is in this case a CR submersion withbasis and fiber having both CR structures of Levi-Tanaka type. Example 2.6 showsthat, even when it does exist, we cannot expect to find, on the basis M ′ of the Levi-Malcev fibration (2.4) of a G -homogeneous CR manifold M , a G -homogeneous CR structure that makes (2.4) a CR map. We have, by Proposition 2.2: Corollary 2.7.
Assume that the G -homogeneous CR structure of M = G / I is described by the CR algebra ( g , q ) and that M admits the Levi-Malcev fibra-tion (2.4) . Then a necessary and sufficient condition for the existence of a G -homogeneous CR structure on M ′ , making (2.4) a CR map, is that (2.5) q ∩ ¯ q + r = ( q + r ) ∩ (¯ q + r ) . Moreover we obtain:
Theorem 2.8.
Suppose that (2.5) is valid, and consider on the basis M ′ of theLevi-Malcev fibration (2.4) the G -homogeneous CR structure defined by ( g , q ′ ) ,with q ′ = q + r . Then: (1) M ′ is an S -homogeneous CR manifold M ′ ≃ S / ( S ∩ R I ) , with CR algebra ( s , s ∩ q ′ ) . (2) The fiber of (2.4) is the solvmanifold F ≃ R / ( R ∩ I ) , with R -homogeneous CR structure defined by ( r , r ∩ q ) .If ( g , q ) has the weak- J -property (resp. the J -property) then: (3) Condition (2.5) is satisfied. (4)
Both the basis M ′ and the fiber F of the Levi-Malcev fibration enjoy theweak- J -property (resp. the J -property).Proof. The result follows from Propositions 2.2 and 2.3, after noticing that r is acharacteristic ideal, hence J -invariant. (cid:3) The Jordan-Chevalley fibration.
An algebraic group G over a field k al-ways contains a maximal normal solvable subgroup R ∗ . The connected component R of the identity in R ∗ is the radical of G . The set N of unipotent elements of R is a connected normal subgroup of G , called the unipotent radical of G . Thealgebraic group G is reductive when its unipotent radical is trivial.If the field k is perfect , any algebraic group G over k admits a Jordan-Chevalleydecomposition (see e.g.[7, 12, 29]), i.e. there is a maximal reductive subgroup L of G such that(2.6) G = N ⋊ L . For the proof of the following Lemma, see e.g. [18, Ch.VII, Lemma 1.4]. This means that all algebraic extensions of k are separable. This is equivalent to the fact thateither k has characteristic 0, or, having positive characteristic p , every element of k admits a p -throot in k . Lemma 2.9.
Let g ⊂ gl ( n, k ) be a linear Lie algebra, n an ideal and a a subalgebraof g . If all the elements of n ∪ a are nilpotent, then all the elements of a + n arenilpotent. (cid:3) Proposition 2.10.
Let G be an algebraic group over a perfect field k and N itsunipotent radical. If I is an algebraic subgroup of G , then also N I is an algebraicsubgroup of G .Proof. Let n and i be the Lie algebras of N and I , respectively. Let U be theunipotent radical of I and u its Lie algebra. By Lemma 2.9, the sum n ′ = u + n isa nilpotent subalgebra of g . The set(2.7) G ′′ = { g ∈ G | Ad( g )( n ′ ) = n ′ } is an algebraic subgroup of G containing I . Let g ′′ be its Lie algebra and n ′′ , whichis contained in g ′′ , that of the unipotent radical N ′′ of G ′′ . The analytic subgroup N ′ corresponding to n ′ is a normal subgroup of G ′′ consisting of unipotent elements.Hence N ′ ⊂ N ′′ , and therefore N ′ is Zariski-closed and algebraic in both G ′′ and G .If I = L ′ ⋉ U is a Jordan-Chevalley decomposition of I , then we obtain a de-composition N I = L ′ ⋉ N ′ . Hence N I is algebraic, being a semidirect product ofalgebraic subgroups of G . (cid:3) Definition 2.11.
Let G be a real algebraic group, with unipotent radical N . If I is an algebraic subgroup of G , then by Proposition 2.10 also N I is algebraic.The G -equivariant fibration(2.8) M = G / I −−−−→ M ′ = G / N I is called the Jordan-Chevalley fibration of M .In this setting, Propositions 2.2 and 2.3 yield the following result. Theorem 2.12.
Let G be a real linear algebraic group and N its unipotentradical. We denote by g , n the Lie algebras of G , N , and by g , n their com-plexifications, respectively.Let M = G / I , for an algebraic subgroup I , have a G -invariant CR structuredefined by the CR algebra ( g , q ) .A necessary and sufficient condition in order that there exists a G -homogeneous CR structure on the basis M ′ = G / N I of the Jordan-Chevalley fibration, making (2.8) a CR map is that: (2.9) q ∩ ¯ q + n = ( q + n ) ∩ (¯ q + n ) . Assume that (2.9) is satisfied and consider on M ′ the CR structure defined by ( g , q ′ ) with q ′ = q + n . Then: (1) (2.8) is a CR fibration. (2) Its typical fiber F is the nilmanifold N / ( N ∩ I ) , with an N -homogeneous CR structure defined by the CR algebra ( n , q ∩ n ) . (3) The basis M ′ = G / N I is an L -homogeneous CR manifold, associatedwith the CR algebra ( l , l ∩ ( q + n )) , where L is a maximal reductive subgroupof G , l its Lie algebra, and l its complexification.If ( g , q ) has the weak- J -property (resp. the J -property) then: (4) Condition (2.9) is satisfied. (5)
Both the basis M ′ and the fiber F of (2.8) enjoy the weak- J -property (resp.the J -property).Proof. The first part of the statement is a consequence of Proposition 2.2. Finally,(1), (2), (3) follow by [28, § n is a characteristic ideal. (cid:3) N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 11 Attaching homogeneous CR manifolds to CR algebras CR manifolds associated to a CR algebra. Let us consider the questionof the existence of homogeneous CR manifolds associated with a given CR algebra( g , q ). Definition 3.1. A CR algebra ( g , q ) is factual if there exists a real Lie group G ,with Lie algebra g , and a closed subgroup I of G with Lie algebra i = q ∩ g .We have: Theorem 3.2.
Let ( g , q ) be a CR algebra, ˜ G a connected and simply connectedreal Lie group with Lie algebra g , and ˜ I its analytic subgroup with Lie algebra i = q ∩ g . Then ( g , q ) is factual if, and only if, ˜ I is closed in ˜ G .Proof. The statement is a special case of a general fact, only involving homogeneousspaces. If G is any connected Lie group with Lie algebra g , containing a closedsubgroup I with Lie algebra i , then the universal covering ˜ M of the homogeneousmanifold M = G / I is ˜ G -homogeneous, and of the form ˜ G / ˜ I , for the analyticLie subgroup ˜ I of ˜ G corresponding to the Lie subalgebra i . The subgroup ˜ I isthe connected component of the identity of the inverse image of I for the coveringmap ˜ G → G , hence closed when I is closed. (cid:3) Example 3.3.
Let G be a connected compact Lie group, with a simple Lie algebra g , of rank ℓ ≥
2. Fix a Cartan subalgebra h of g and let R be the correspondingroot system for the complexification g of g . Fix a lexicographic order “ ≺ ” of R and let α , . . . , α ℓ be the basis of positive simple roots, and H α , . . . , H α ℓ thecorresponding elements of i h . Fix c , . . . , c ℓ ∈ R , linearly independent over thefield Q of rational numbers. Set H = c H α + · · · + c ℓ H α ℓ , and take q = C · H + P α ≺ g α , where g α ⊂ g is the root space of α ∈ R . Then ( g , q ) is fundamentaland Levi-nondegenerate, but the analytic subgroup I = { exp ( itH ) | t ∈ R } of G ,corresponding to i = q ∩ g = i R · H , is not closed. Its closure coincides with theCartan subgroup H of G with Lie algebra h . We note that its G -closure (see § q G = q + C h . It is a Borel subalgebra of g , and ( g , q G ) is the totallycomplex CR algebra of a complex flag manifold.3.2. The G -closure of a CR algebra. We may canonically associate to every CR algebra a factual CR algebra. Proposition 3.4.
Let G be a Lie group with Lie algebra g , and ( g , q ) a CR algebra. Let I ⊂ G be the analytic subgroup of i = q ∩ g . Denote by ¯ I the closureof I in G , by i G ⊂ g the Lie algebra of ¯ I and by i G its complexification. Then: (1) q G = q + i G is a complex Lie subalgebra of the complexification g of g ,which contains q as an ideal. The quotient q G / q is Abelian. (2) If i ′ is any real linear subspace of g with i ⊂ i ′ ⊂ i G , and i ′ its com-plexification, then q ′ = q + i ′ is a complex Lie subalgebra of g and the g -equivariant map ( g , q ) → ( g , q ′ ) is an algebraic CR submersion, withLevi-flat fibers. (3) If ( g , q ) is fundamental, and q ′ is as in (2) , then also ( g , q ′ ) is fundamen-tal.Proof. Since Ad( g )( q ) = q for all g ∈ I , we also have Ad( g )( q ) = q for all g ∈ ¯ I .This implies that ad( X )( q ) ⊂ q for all X ∈ i G , hence q G = q + i G is a complex Liesubalgebra. Clearly q is an ideal in q G and q G / q is Abelian. Indeed the equality[ i G , i G ] = [ i , i ] ⊂ i (see e.g. [30, Chap.2, § q G , q G ] = q .Hence, if i ⊂ i ′ ⊂ i G , we obtain [ i ′ , i ′ ] ⊂ i . Therefore q ′ defined in (2) is acomplex Lie subalgebra of g C , and the g -equivariant map ( g , q ) → ( g , q ′ ) is an algebraic CR submersion, with Levi-flat fibers. Finally, (3) is obvious from theinclusion q ⊂ q ′ . (cid:3) Keeping the notation of Proposition 3.4, we give the following:
Definition 3.5.
Let ( g , q ) be a CR algebra and G a Lie group with Lie algebra g . The CR algebra ( g , q G ) is called the G -closure of ( g , q ) (cf. [30, pp.53-54],where i G is called the Malcev-closure of i ). Proposition 3.6.
Let ( g , q ) be a weakly nondegenerate CR algebra, i ′ any reallinear subspace of g with i ⊂ i ′ ⊂ i G , and denote by i ′ its complexification. Set q ′ = q + i ′ . Then the g -equivariant algebraic- CR submersion ( g , q ) → ( g , q ′ ) hastotally real fibers.Proof. Indeed, the intersection a = i ′ ∩ ( q + ¯ q ) is a real Lie subalgebra thatnormalizes q , hence q ′′ = q + a , where a is the complexification of a , is a complexLie subalgebra of g . Since q ⊂ q ′′ ⊂ q + ¯ q , the assumption that ( g , q ) is weaklynondegenerate implies that q ′′ = q . Then a ⊂ q and a ⊂ i , and the fiber of the g -equivariant CR map ( g , q ) → ( g , q ′ ) is ( i ′ , a ∩ q ) = ( i ′ , a ), thus totally real. (cid:3) Real analytic and algebraic CR manifolds CR submanifolds of analytic spaces. A G -homogeneous CR manifold isreal analytic, hence it can be realized as a generic CR submanifold of a complexmanifold (see e.g. [3, 31]). Let us consider, in general, the embedding of a realanalytic CR manifold into a complex space. Definition 4.1.
Let M be a real analytic CR manifold, N a complex space, and φ : M ֒ → N a real analytic map. The structure sheaf O N of germs of holomorphicfunctions on N pulls back to a subsheaf φ ∗ ( O N ) of the sheaf A M of germs ofcomplex valued real analytic functions on M . Let O M be the sheaf of germs of realanalytic CR functions on M . We say that φ is(1) a CR map if φ ∗ ( O N ) is contained in O M ,(2) a CR immersion if φ ∗ ( O N ) = O M ,(3) a generic CR immersion if moreover the composition φ − ( O N ) −−−−→ φ ∗ ( O N ) −−−−→ O M defines an isomorphism of the inverse image sheaf φ − ( O N ) onto O M .A CR immersion φ that is also a topological embedding will be called a CR embedding.If N is a smooth complex manifold, these notions coincide with the classic defi-nitions in § Lemma 4.2.
Let M be a real analytic CR manifold, generically embedded into acomplex space N . Then M is contained in the set N reg of non singular points of N .Proof. Indeed, the fact that M is real analytic implies that for each p ∈ M thelocal ring O M,p is regular. Hence O N,p , being isomorphic to a regular local ring, isalso regular. (cid:3)
Algebraic and weakly-algebraic CR manifolds. We consider now CR structures on real algebraic manifolds. Definition 4.3. An affine CR submanifold of C n is a smooth real algebraic sub-variety M of C n that is also a CR submanifold. N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 13 An affine CR manifold is a smooth real algebraic variety, endowed with a CR structure, and CR -isomorphic, by a smooth birational correspondence, with anaffine CR submanifold of some C n .An algebraic CR manifold is a smooth real algebraic variety, endowed with a CR structure, in which each point has a Zariski open neighborhood that is an affine CR manifold.An algebraic CR submanifold M of an algebraic complex variety N is a smoothreal algebraic subvariety of N , embedded as a CR submanifold in the set N reg ofits regular points.Likewise, we can define semialgebraic CR manifolds and submanifolds.A weakly-algebraic CR manifold M is a smooth real algebraic variety endowedwith an algebraic formally integrable partial complex structure. This is given by aformally integrable smooth complex valued algebraic distribution T , M ⊂ T C M ,with T , M ∩ T , M = 0 M .We observe that an irreducible real algebraic subvariety M ′ of an irreduciblecomplex algebraic variety N contains a maximal Zariski open subset M that is areal algebraic CR submanifold of a Zariski open subset of N .An algebraic (respectively, semialgebraic) CR submanifold of a complex algebraicvariety naturally is an algebraic (respectively, semialgebraic) CR manifold. Thevice versa may be false, as we shall see as a consequence of Theorem 4.7 below. Remark 4.4.
Since neither the complex nor the real Frobenius theorems are validin the algebraic category, weakly-algebraic CR manifolds may not be algebraic CR manifolds. For instance, consider the complex structure on R x,y defined by J x,y = (cid:16) x x − − x (cid:17) . This structure is weakly-algebraic, but not algebraic, becauseany rational function in C ( x, y ), holomorphic for this structure near a point of R ,is constant. Proposition 4.5.
Let M be an algebraic CR manifold. Then M has a real algebraicembedding into a complex variety N , that is also a generic CR -embedding into theset N reg of its regular points.Proof. We first consider the case where M is an affine CR manifold, of CR dimen-sion n , and CR codimension k , of a Euclidean complex space C ℓ . Denote by I theideal of polynomials P ∈ C [ z , . . . , z ℓ ] vanishing on M . We claim that N = V ( I )has the properties requested in the statement. To this aim, let us fix a point z ∈ M .We can assume that the restrictions to M of dz , . . . , dz n + k are linearly indepen-dent in a neighborhood of z in M , and that Re z , . . . , Re z n + k and Im z , . . . , Im z n define a system of coordinates in a neighborhood of z for the real analytic struc-ture of M . The restriction to M of the polynomials in P ∈ C [ z , . . . , z ℓ , ¯ z , . . . , ¯ z ℓ ]form a ring which is an algebraic extension of the ring of the restrictions to M ofpolynomials in C [ z , . . . , z n + k , ¯ z , . . . , ¯ z n ]. Let w ∈ C [ z , . . . , z ℓ ]. Then there is asmallest integer d ≥ w satisfies a monic equation: w d + a ( z ′ , ¯ z ′ ) w d − + · · · + a d ( z ′ , ¯ z ′ ) = 0 on M, where a j ( z ′ , ¯ z ′ ) are rational functions of z , . . . , z n + k , ¯ z , . . . , ¯ z n , for j = 1 , . . . , d .Eliminating denominators, we obtain an equation:(4.1) b ( z ′ , ¯ z ′ ) w d + b ( z ′ , ¯ z ′ ) w d − + · · · + b d ( z ′ , ¯ z ′ ) = 0 on M, with b j polynomials in C [ z , . . . , z n + k , ¯ z , . . . , ¯ z n ]. Moreover, we can assume that b ∈ C [ z , . . . , z n + k , ¯ z , . . . , ¯ z n ] has minimal total degree among the b ’s of the nonzero polynomial vectors ( b , . . . , b d ) that satisfy (4.1). For b ∈ C [ z , . . . , z n + k , ¯ z , . . . , ¯ z n ]the anti-holomorphic differential ¯ ∂b is a linear combination of the differentials d ¯ z , . . . , d ¯ z n and me may identify ¯ ∂ M b to the restriction of ¯ ∂b to M . The pull-backs to M of the differentials d ¯ z , . . . , d ¯ z n are linearly independent on a neighborhoodof z . Taking ¯ ∂ M of both sides of(4.1), we obtain: w d ¯ ∂b + w d − ¯ ∂b + · · · + w ¯ ∂b d − + ¯ ∂b d = 0 on M. By our choice of b , this implies that ¯ ∂b = 0 on M , hence that: w d − ¯ ∂b + · · · + ¯ ∂b d = 0 on M. This is a system of polynomial equations with polynomial coefficients on M , thus,by our choice of d , we obtain that all ¯ ∂b j ’s are zero on M , consequently zero becausethey only depend on z , . . . , z n + k , ¯ z , . . . , ¯ z n + k . Hence the b ′ j s are holomorphic, and a j = a j ( z ′ ) ∈ C ( z , . . . , z n + k ).Let A be the ring of the restrictions to M of the elements of C [ z , . . . , z ℓ ], and B the integral closure in A of the ring of the restrictions to M of the elements of C [ z , . . . , z n + k ]. We proved that A is contained in the integral closure of the fieldof fractions of B . By the theorem of the primitive element, we can find an element w ∈ C [ z , . . . , z ℓ ], a polynomial P ∈ C [ z , . . . , z n + k , w ], monic with respect to w ,such that, if ∆( z ′ ) is the discriminant of P with respect to w : P ( z ′ , w ) = w d + a ( z ′ ) w d − + · · · + a d ( z ′ ) ∈ I (4.2) ∀ j = n + k + 1 , . . . , ℓ there exists p j ∈ C [ z , . . . , z n + k , w ]such that ∆( z ′ ) z j − p j ( z ′ , w ) ∈ I . (4.3)This shows that N = V ( I ) is a complex algebraic subvariety of C ℓ , of pure dimen-sion n + k . The statement follows, because the points of M are contained in N reg by Lemma 4.2.The proof in the general case is obtained by patching together a finite atlas ofaffine charts of M by birational equivalence. (cid:3) Homogeneous algebraic CR manifolds. Let g be a finite dimensional realLie algebra and g its complexification. We recall that g (and g ) are algebraic Liealgebras , if any of the three equivalent conditions below is satisfied (see [10] for thedefinition of replica , [11], [16] for the equivalence of the conditions):(1) there exists a real linear algebraic group G with Lie algebra g ;(2) there exists an algebraic subgroup of Aut( g ) with Lie algebra ad( g );(3) for every X in g , the subalgebra ad( g ) of gl R ( g ) contains all replicas ofad( X ).Moreover, g is a real algebraic Lie algebra if and only if its complexification g isa complex algebraic Lie algebra, and the characterization of complex algebraic Liealgebras is given by conditions that are completely analogous to the ones listedabove.When g is an algebraic Lie subalgebra of some gl ( n, R ), the semisimple andnilpotent components X s and X n of an element X of g are replicas of X . Thus, inparticular, an algebraic Lie algebra g is ad-splittable: this means that, for every X ∈ g , also X s , X n ∈ g . Moreover, ad( X s ) = [ad( X )] s and ad( X n ) = [ad( X )] n are the semisimple and the nilpotent components of ad( X ), respectively. Lemma 4.6.
Let G be a real linear algebraic group. If M is a G -homogeneousreal algebraic manifold and a smooth G -homogeneous CR -manifold, then M is aweakly-algebraic CR manifold.Proof. Fix p ∈ M and let ( g , q ) be the CR algebra of M at p . The complexifi-cation T C G of the tangent space of G is algebraic and can be identified with theCartesian product G × g , the left action of G on G × g being defined by: h · ( g, Z ) = ( hg, Ad g ( h )( Z )) = ( h ◦ g, h ◦ Z ◦ h − ) . N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 15
The set T = { ( g, Z ) ∈ G × g | g − ◦ Z ◦ g ∈ q } is algebraic. The set T , M is the image of T by the differential of the map G ∋ g → g · p ∈ M , hence algebraic. This proves that the G -homogeneous CR structure of M defined by ( g , q ) is weakly algebraic. (cid:3) From Lemma 4.6 we obtain:
Theorem 4.7.
Let g be an algebraic Lie algebra. A necessary and sufficientcondition for the existence of a homogeneous weakly-algebraic CR manifold M with CR algebra ( g , q ) is that ∀ X ∈ i = q ∩ g all replicas of ad( X ) belong to ad( i ) . (4.4) Proof.
Let z be the center of g . By taking the quotient by the central ideal i ∩ z , we reduce to the case where z ∩ i = 0. Then we decompose i into thedirect sum of an ad g -reductive subalgebra l and an ideal n consisting of ad g -nilpotent elements. We can consider a maximal reductive subalgebra l ∗ of ad g ( g )containing ad g ( l ), and construct, as in [18, XVIII.1], an embedding of g as analgebraic subalgebra:(4.5) φ : g → l ∗ ⋊ n ⊂ gl ( n, R ) , where n is the maximum nilpotent ideal of g , for which the corresponding con-nected and simply connected Lie group has the structure of an algebraic group, con-sisting of unipotent matrices. Since, by [17], φ ( X ) is a nilpotent matrix in gl ( n, R )for all ad g -nilpotent X in g , we obtain that the semisimple parts in gl ( n, R ) ofthe elements of φ ( l ) belong to φ ( l ), and φ ( l ) is an algebraic subalgebra of φ ( g )by [11]. Finally, φ ( n ) is algebraic because it is a subalgebra of gl ( n, R ) consistingof nilpotent matrices. Hence φ ( g ) = φ ( l ) ⋊ φ ( n ) is an algebraic subalgebra of gl ( n, R ), and the statement follows from Lemma 4.6. (cid:3) Proposition 4.8.
Let g be an algebraic real Lie algebra and q an ideal of itscomplexification g , defining a complex structure on g . This means that: (4.6) g = q ⊕ q (direct sum of ideals) . Then we can find a complex algebraic group G with associated CR algebra ( g , q ) .Proof. We prove that q is an algebraic Lie subalgebra of g . To this aim we observethat [ q , ¯ q ] = 0. Hence the centralizer of ¯ q in g C is: z g (¯ q ) = q + z g ( g ) . This is an algebraic Lie subalgebra of g and therefore ad g ( q ) = ad g ( z g (¯ q )) is alge-braic in gl C ( g ). By the same argument of Theorem 4.7, we obtain that there existsa complex algebraic group G and a complex algebraic normal subgroup Q with Liealgebra q . Then G = G / Q is a complex algebraic group satisfying the conditionsof the statement. (cid:3) Example 4.9.
Let g = sl (2 m, R ), with m ≥
2. Consider a Borel subalgebra b of g ≃ sl (2 m, C ) such that h = b ∩ ¯ b is a Cartan subalgebra of g . Then h is the complexification of a maximally compact Cartan subalgebra h of g . Let n = [ b , b ] be the nilpotent ideal of b . Fix an element H ∈ h \ h , such thatexp( R H ) is not closed in SL (2 m, C ). We choose q = n ⊕ C H . Since q ∩ ¯ q = 0, thecomplex Lie subalgebra q defines a left homogeneous CR structure on SL (2 m, R ).However, in this case there is no semialgebraic G -equivariant CR embedding of G ≃ SL (2 m, R ) into an SL (2 m, C )-homogeneous complex manifold. Example 4.10.
Let g = sl (2 m − , C ), with m ≥
2. Define the conjugation: A = ( a i,j ) ≤ i,j ≤ m − → ¯ A = (¯ a m − i, m − j ) ≤ i,j ≤ m − . Consider the real form g = { A = ¯ A } ≃ sl (2 m − , R ) of g , and let G ≃ SL (2 m − , R ) be the analytic Lie subgroup of SL ( n, C ) with Lie algebra g . Define: q = { A = ( a i,j ) ∈ sl (2 m − , C ) | a i,j = 0 for i > j and for i = j > m } . Since q ∩ ¯ q = 0 and q + ¯ q = sl (2 m − , C ), the datum of the CR algebra ( g , q )yields on G a complex structure, which is only left G -invariant. Note that,being SL (2 m − , R ) a simple real Lie group corresponding to a connected Satakediagram, it can carry the structure of neither a complex Lie group, nor a complexalgebraic group. However, it is a quasi-projective smooth complex variety, open in SL (2 m − , C ) / Q , where Q is the algebraic subgroup of SL (2 m − , C ) correspondingto the solvable Lie subalgebra q . Example 4.11.
Let us take g and G as in Example 4.10, with m = 2, and define: q = A = a , a , a , a , a , a , ∈ sl (3 , C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a , + λ a , = 0 , for an irrational complex number λ , with | λ | 6 = 1. We have q ∩ ¯ q = 0 and q + ¯ q = sl (3 , C ). Since the subalgebra q is not algebraic, the complex structure on G defined by ( g , q ) is weakly algebraic, but not algebraic. Theorem 4.12.
Let G be a real linear algebraic group. Let M be a real algebraicmanifold and a smooth CR manifold, on which G acts as a transitive group ofalgebraic and CR transformations. Let ( g , q ) be the CR algebra of M at a point p .If G acts on M as a group of algebraic CR automorphisms, then: (1) q is an algebraic subalgebra of g . (2) There are a G -homogeneous complex algebraic manifold ˆ M and a G -equivariant CR generic algebraic embedding M ֒ → ˆ M .Vice versa, if q is algebraic, then M is a G -homogeneous algebraic CR manifold.Proof. First assume that M is affine. By Proposition 4.5, there is a generic CR -embedding M ֒ → N of M into the set of regular points of an affine complex variety N ֒ → C ℓ , in such a way that the ring C [ M ] of regular CR functions on M coincideswith the ring C [ N ] of regular holomorphic functions on N .Analogously, C [ G ] = C [ G ]. Let I ⊂ G be the isotropy subgroup at p ∈ M and π : G → G / I = M the natural projection. The subring π ∗ ( C [ M ]) of C [ G ] = C [ G ] is G -invariant, hence also G -invariant. Thus it defines a G -homogeneous complex algebraic variety ˆ M , and the isotropy subgroup is an al-gebraic subgroup Q with Lie algebra q . Indeed, in a G -equivariant way we have C [ M ] = C [ ˆ M ], and we also obtain a generic algebraic CR embedding M ֒ → ˆ M .Let us now turn to the general case. Let M ֒ → N reg be the embedding of Propo-sition 4.5, and let R M and R N be the sheaves of germs of regular CR functions on M and of regular holomorphic functions on N , respectively. Then R M and R N areisomorphic, as, for every open U ⊂ N , the restriction map R N ( U ) → R M ( U ∩ M )is an isomorphism. Then we apply the considerations of the affine case to thesubsheaf π − ( R M ) of R G ≃ R G .When q is algebraic, we consider the algebraic subgroup Q of G with Lie algebra q and the natural G -equivariant embedding M ֒ → G / Q . (cid:3) N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 17
Algebraic closure of a CR algebra. The considerations of § G . Indeed, if H is anysubgroup of G , we can define its algebraic closure H alg0 as the smallest algebraicsubgroup of G containing H . It coincides with the closure of H in the Zariskitopology of G . When H is the analytic subgroup of G corresponding to a Liesubalgebra h of its Lie algebra g , we denote by h alg0 the Lie algebra of H alg0 . Alsoin this case we have (see [33, Theorem 6.2])(4.7) [ h , h ] = [ h alg0 , h alg0 ] . As in § Proposition 4.13.
Let G be a real linear algebraic group, with Lie algebra g .Let ( g , q ) be a CR algebra, and denote by i alg0 the algebraic closure of the isotropysubalgebra i = q ∩ g . Set q alg = q + C ⊗ R i alg0 . Then q alg is a complex Lie subalgebraof g , contained in the normalizer of q in g . The quotient q alg / q is Abelian.Fix any real linear subspace i ′ of g with i ⊂ i ′ ⊂ i alg0 and define q ′ = q + C ⊗ R i ′ .Then: (1) q ′ is a complex Lie subalgebra of g and the g -equivariant map ( g , q ) → ( g , q ′ ) is an algebraic CR submersion, with Levi-flat fibers. (2) If ( g , q ) is fundamental, then also ( g , q ′ ) is fundamental. (3) If ( g , q ) is weakly nondegenerate, then the fiber of the g -equivariant map ( g , q ) → ( g , q ′ ) is totally real. (cid:3) Definition 4.14.
The CR algebra ( g , q alg ) is called the algebraic closure of the CR algebra ( g , q ).4.5. The G - and the g -anticanonical fibrations. In this section we describethe anticanonical fibration of [4] and [15] in terms of CR algebras.Let G be a Lie group, g its Lie algebra. Given a CR algebra ( g , q ), set: a = N g ( q ) = { X ∈ g | [ X, q ] ⊂ q } (4.8) q ′ = q + a , with a = C ⊗ R a , (4.9) A = N G ( q ) = { g ∈ G | Ad( g )( q ) = q } . (4.10) Proposition 4.15.
Keep the notation introduced above. Then: (1) q ′ is the complex Lie subalgebra of g characterized by the properties: (4.11) ( q ⊂ q ′ , q ′ ∩ g = a , ( g , q ) → ( g , q ′ ) is a g -equivariant CR -submersion. (2) q ′ is the smallest complex Lie subalgebra of g which satisfy q + a ⊂ q ′ ⊂ N g ( q ) . (3) The fiber ( a , a ∩ q ) of the g -equivariant CR fibration ( g , q ) → ( g , q ′ ) isLevi-flat. Indeed a ∩ q = q ∩ ¯ q ′ , and [ q ∩ ¯ q ′ , ¯ q ∩ q ′ ] ⊂ q ∩ ¯ q .Moreover: (4) If ( g , q ′ ) is totally real, then ( g , q ) is Levi-flat, and [ q , ¯ q ] ⊂ q ∩ ¯ q . (5) Conditions ( i ) , ( ii ) and ( iii ) below are equivalent and imply ( iv ) : a = g | {z } ( i ) ⇐⇒ q ′ = g | {z } ( ii ) ⇐⇒ q is an ideal of g | {z } ( iii ) = ⇒ a is an ideal of g | {z } ( iv ) . (6) A is a closed subgroup of G and hence ( g , q ′ ) is factual. (7) If g is an algebraic Lie algebra, then also a and a are algebraic. If G isa real linear algebraic group, then M ′ = G / A is a weakly algebraic CR manifold. If q is algebraic, then q ′ is algebraic too, and M ′ is an algebraic CR manifold. Proof.
Since i = q ∩ g ⊂ a , by Proposition 1.7 (4.9) and (4.11) are equivalent.This proves (1).Indeed, any complex Lie subalgebra containing a contains its complexification a . Hence (2) is obvious.By (4.8), we have [ a , q ] ⊂ q and [ a , ¯ q ] ⊂ ¯ q , hence [( a ∩ q , a ∩ q ] = [ a ∩ q , a ∩ ¯ q ] ⊂ q ∩ ¯ q yields (3).(4). When ( g , q ′ ) is totally real, ¯ q ⊂ q ′ , hence [ q , ¯ q ] ⊂ [ q , q ′ ] ⊂ q . By conjugation,we obtain [ q , ¯ q ] ⊂ q ∩ ¯ q .(5). We clearly have ( i ) ⇒ ( ii ) ⇒ ( iii ) ⇒ ( i ) and ( iii ) ⇒ ( iv ).(6) follows from (4.10), since a is the Lie algebra of A , and (7) is a consequenceof Theorems 4.7 and 4.12. (cid:3) Left invariant CR structures on semisimple Lie groups In this and the following section, we shall discuss special examples of homogen-eous CR structures. We begin by investigating left-invariant CR structures on realsemisimple Lie groups (see e.g. [32]). Note that a Lie group with a left and rightinvariant complex structure is in fact a complex Lie group.5.1. Existence of maximal CR structures.Theorem 5.1. Every semisimple real Lie group of even dimension admits a leftinvariant complex structure.Every semisimple real Lie group of odd dimension admits a left invariant CR structure of hypersurface type.Proof. Let G be a simple real Lie group, with Lie algebra g . Take a maximallycompact Cartan subalgebra h of g . The complexification g of g contains a Borelsubalgebra b with b ∩ ¯ b equal to the complexification h of h (see e.g. [21]). Let n = [ b , b ] be the nilpotent ideal of b .If the dimension of g is even, then the dimension of h is even too, and we canfind a complex structure J : h → h . Then V = { X + iJX | X ∈ h } is a complexsubspace of h , with V ∩ ¯ V = { } . Hence q = V ⊕ n is a complex subalgebra of g , with q ∩ ¯ q = 0 and g = q ⊕ ¯ q . The CR algebra ( g , q ) defines a left invariantcomplex structure on G .If g has odd dimension, then h has odd dimension too. Fix a hyperplane m of h , a complex structure J : m → m , set V = { X + iJX | X ∈ m } and take q = V ⊕ n . Since q ∩ ¯ q = 0 and q + ¯ q = n ⊕ ¯ n ⊕ m , the CR algebra ( g , q ) defines aleft invariant CR structure of hypersurface type on G . (cid:3) Example 5.2.
Let G = SL ( n, C ) and consider on its Lie algebra g = gl ( n, C ) theconjugation A → A ♯ , defined by A ♯ = (¯ a n +1 − i,n +1 − j ) ≤ i,j ≤ n for A = ( a i,j ) ≤ i,j ≤ n .Then g = { X ∈ g | X ♯ = X } ≃ sl ( n, R ) and G = { g ∈ G | g ♯ = g } ≃ SL ( n, R ).The diagonal matrices of g are a maximally compact Cartan subalgebra h of g .Let n = 2 m + 1 be odd. Fix p with 1 < p ≤ m and define q ′ as the complexLie subalgebra of g consisting of matrices ( a i,j ) ≤ i,j ≤ n with a i,j = 0 when either p < i ≤ n − p and j ≤ i , or i > n − p . Let a be a subspace of the Cartan subalgebra h of the diagonal matrices of g , with λ j = λ n for j > n − p , a ∩ ¯ a = 0 and a + ¯ a = h .By setting q = q ′ + a , we obtain a CR algebra ( g , q ), with a non solvable q , whichdefines a left invariant complex structure on G ≃ SL (2 m + 1 , R ).5.2. Classification of the regular maximal CR structures. We recall that acomplex Lie subalgebra q of a complex semisimple Lie algebra g is regular if itsnormalizer contains a Cartan subalgebra of g . N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 19
Definition 5.3.
We say that a CR algebra ( g , q ) is regular if q is normalized bya Cartan subalgebra of the real Lie algebra g .If G is a semisimple real Lie group with Lie algebra g , a G -invariant CR structure on a G -homogeneous CR manifold M is called regular if the associated CR algebra ( g , q ) is regular.Fix a semisimple real Lie algebra g . Let h be a Cartan subalgebra of g , and R the root system of its complexification h in g . For each α ∈ R we write g α forthe root space of α . The real form g defines a conjugation in g , which by dualitygives an involution α → ¯ α in R , with ¯ α ( H ) = α ( ¯ H ) for all H ∈ h . Lemma 5.4.
Assume that there is a closed system of roots
Q ⊂ R with (5.1)
Q ∩ ¯ Q = ∅ , Q ∪ ¯ Q = R . Then h is maximally compact.Set Q r = { α ∈ Q | − α ∈ Q} and Q n = { α ∈ Q | − α / ∈ Q} . Then: (1) Q r ∪ ¯ Q and Q r ∪ ¯ Q n are closed systems of roots; (2) the two systems of roots Q r and ¯ Q r are strongly orthogonal; (3) P = Q ∪ ¯ Q r is parabolic with P n := { α ∈ P | − α / ∈ P} = Q n ; (4) there is a system of simple positive roots α , . . . , α ℓ of R with the properties: (5.2) α , . . . , α ℓ ∈ P ,α , . . . , α p is a basis of Q r ,α p +1 , . . . , α ℓ − p ∈ Q n , ¯ α i ≺ ∀ i = 1 , . . . , ℓ, ¯ α i = − α ℓ +1 − i for i = 1 , . . . , p. Proof.
By (5.1), ¯ α = α for all α ∈ R , and this is equivalent to h being maximallycompact (see e.g. [21, Ch.VI, § R is partitioned into minimal disjoint subsets, invariant byaddition of roots of Q r . Since Q is a union of such Q r -invariant minimal subsets,its complement ¯ Q is Q r -invariant, too. Likewise, Q is ¯ Q r -invariant. This impliesthat Q r and ¯ Q r are strongly orthogonal. Indeed, assume by contradiction thatthere are α, β ∈ Q r such that α + ¯ β ∈ R . Then α + ¯ β ∈ Q ∩ ¯ Q would give acontradiction.Since ¯ Q r is Q r -invariant, then also ¯ Q n is Q r -invariant. This proves (1) and (2).From (5.1) we also deduce that ¯ Q n is equal to {− α | α ∈ Q n } , and this implies(3).To prove (4), we begin by fixing an element A ∈ h R that defines the parabolicset P : P = { α ∈ R | α ( A ) ≥ } . Next we note that, since R does not contain any real root, there is a regular element A in h R with ¯ A = − A , i.e. with iA ∈ h . Take ǫ > | α ( A ) | < ǫ − α ( A )for α ∈ Q n . Then A = A + ǫA is regular and we shall take B = { α , . . . , α ℓ } to be the simple roots of the system of positive roots R + = { α ∈ R | α ( A ) > } .Take { α , . . . , α p } = B ∩ Q r and { α p +1 , . . . , α r } = B ∩ Q n . By our choice of ǫ and A , the set { α , . . . , α p } is the set of the simple positive roots in { α ∈ Q r | α ( A ) > } . Likewise, the simple roots in { α ∈ ¯ Q r | α ( A ) > } are contained in { α ℓ − p +1 , . . . , α ℓ } ⊂ B . Hence r = ℓ − p .To conclude the proof of (4), it suffices to note that, since R + = R − = {− α | α ∈ R + } , the conjugate of each simple root is a simple negative root. Thus, bysuitably labelling the roots in B , since by (5.1) we have ¯ α = − α for α ∈ Q r , wealso obtain the last line of (5.2). The proof is complete. (cid:3) Proposition 5.5.
Let G be a real semisimple Lie group. Then any regular CR structure on G of maximal CR dimension is associated with a regular CR algebra ( g , q ) , with a q that is normalized by a maximally compact Cartan subalgebra h of g , and is of the form: (5.3) q = m ⊕ X α ∈Q g α for a closed system of roots Q ⊂ R satisfying (5.1) , and a complex subspace m ofthe complexification h of h , with the properties: (5.4) dim C m = (cid:2) ℓ (cid:3) , s ∩ h ⊂ m , m ∩ ¯ m = { } . Here ℓ is the rank of the complexification g of g and s is the Levi subalgebra of q associated with the root system Q r .Proof. We note that (5.3), with a choice of Q and m satisfying (5.1) and (5.4), has CR codimension equal to 0 or 1, according to whether g has even or odd rank,respectively. Thus it yields a CR structure of maximal CR dimension. Let ( g , q )be a regular CR algebra, of codimension 1 at the most, and set m = q ∩ h . Then m must satisfy (5.4) by the codimension constraint, and the set Q of the roots α with g α ⊂ q satisfies (5.1). (cid:3) Example 5.6.
Let g ≃ sl (3 , R ), consist of the matrices A = ( a i,j ) ∈ sl (3 , C ) thatsatisfy ¯ a i,j = a − i, − j . Set q = n(cid:16) z z z z z z − z (cid:17)(cid:12)(cid:12)(cid:12) z , z , z , z ∈ C o . The CR algebra ( g , q ) defines a left invariant complex structure on G ≃ SL (3 , R ),because q ∩ ¯ q = { } and q + ¯ q = sl (3 , C ). But ( g , q ) is not regular as a CR algebra,since q is self-normalizing in sl (3 , C ), hence, in particular, is not normalized by anyCartan subalgebra of g .In [9, 22] all complex structures on a compact semisimple Lie group of evendimension are shown to be regular. According to the example above, in the caseof non compact semisimple real Lie groups a complete classification of the leftinvariant maximal CR structure would require some extra consideration of nonregular structures.6. Symmetric CR structures on complete flags Symmetric maximal almost - CR structures (i.e. formally integrability is not re-quired) on complete flags were studied in [14]. Here we utilize CR algebras to studytheir CR -symmetric (formally integrable) structures, that are also of finite type.A complete flag is a homogeneous compact complex manifold, which is the quo-tient M ≃ G / B of a semisimple complex Lie group G by a Borel subgroup B . Amaximal compact subgroup U of G acts transitively on M , which is therefore alsoa quotient M ≃ U / T of U with respect to a maximal torus T .Let g , b , u , t be the Lie algebras of G , B , U , T , respectively. Then g is complex semisimple and is the complexification of its compact form u . Thecomplexification h of t is a Cartan subalgebra of g , contained in b .6.1. Homogeneous CR structures on complete flags. We shall consider M as a real compact manifold, and discuss its U -homogeneous CR structures. ByProposition 1.3, having fixed the point o = [ T ] of M , the U -homogeneous CR structures on M are in one-to-one correspondence with the complex Lie subalgebras q of g satisfying q ∩ u = t . In particular, any such q contains the Cartan subalgebra N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 21 h , hence is regular . Denote by R the root system of h in g , and let Q be the subsetof R consisting of the roots α for which q α ⊂ q . Then(6.1) q = h ⊕ n , where n = X α ∈Q g α . Conjugation with respect to the real form u yields on R the involution α → ¯ α = − α . Thus, the assumption that q ∩ ¯ q = h is equivalent to Q ∩ ( −Q ) = ∅ . Hence q is solvable (see e.g. [33, Proposition 1.2, p.183]), and(6.2) h ⊂ q ⊂ b . We may consider the ordering of R for which the roots α with g α ⊂ b are positive,so that Q can be regarded as a closed set of positive roots. Proposition 6.1.
Let M ≃ G / B ≃ U / T be a complete flag. We keep thenotation introduced above. (1) The U -homogeneous CR structures on M , modulo CR isomorphisms, arein one-to-one correspondence with the set of solvable complex Lie subal-gebras q of g satisfying (6.2) , modulo automorphisms of g which preserve b . (2) The maximally complex CR structure of M is its standard complex struc-ture, corresponding to the choice q = b , while q = h yields a totally real M . We conclude this subsection by considering CR structures that are related toparabolic subalgebras of g . Recall that a nilpotent subalgebra is horocyclic if it isthe nilradical of a parabolic subalgebra (cf.[34]). Proposition 6.2.
Consider on M the CR structure defined by a CR algebra ( u , q ) ,with q satisfying (6.2) . Assume that the nilpotent Lie algebra n in (6.1) is horocyclicand let q ′ be the normalizer of n in g . Then q ′ is parabolic and q ′ ∩ ¯ q ′ is a reductivecomplement of n in q ′ : (6.3) q ′ = f ⊕ n , with f = q ′ ∩ ¯ q ′ reductive , n nilpotent . The real Lie algebra f = f ∩ u is reductive, and f is its complexification. (1) ( u , q ′ ) is the CR algebra of a complex flag manifold N . (2) There is a natural U -equivariant CR fibration M π −→ N , with totally realfibers. For every p ∈ M , the restriction of dπ p defines a C -isomorphism of H p M with T π ( p ) N . (3) M is U -homogeneous CR -symmetric if and only if N is Hermitian sym-metric. Symmetric CR structures on complete flags. The natural complex struc-ture of the full flag M is not, in general, Hermitian symmetric. We will seek forconditions on the set Q in (6.1) for which M is U - CR -symmetric. We have Lemma 6.3.
Let q be defined by (6.1) , and assume that ( g , q ) defines on M = U / T a U -homogeneous CR -symmetric structure. Then: (1) there exists an involution λ of g with (6.4) λ ( u ) = u , λ | h = Id , λ | n = − Id;(2) n is Abelian and n + ¯ n generates g , or, equivalently, Q satisfies: α ∈ Q = ⇒ − α / ∈ Q , α, β ∈ Q = ⇒ α + β / ∈ R , (6.5) R ⊂ Z [ Q ] . (6.6) Proof.
By the assumption, there is an involution λ of u satisfying (1.16). Inparticular, λ transforms t into itself and equals minus the identity on (cid:0) ( q + ¯ q ) ∩ u (cid:1) / t . Its complexification, that we still denote by λ , is an involution of g leaving h and n invariant, hence equal to minus the identity on n . Since − [ Z , Z ] = λ ([ Z , Z ]) = [ λ ( Z ) , λ ( Z )] = [ − Z , − Z ] = [ Z , Z ] for Z , Z ∈ n , we get [ n , n ] = { } , which is equivalent to (6.5).The conditions that q + ¯ q generates g , that ( u , q ) is fundamental, and that M is of finite type are all equivalent (see § n + ¯ n generates g , then so does q + ¯ q . Vice versa, assume that q + ¯ q generates g , and let a be the subalgebra of g generated by n + ¯ n . From [ h , ¯ n ] = ¯ n , we obtainthat [ h , a ] = a . Hence a + h = g . Containing all root spaces, a contains also h andthus equals g .Condition (6.6) is obviously necessary for ( u , q ) to be fundamental. It is alsosufficient. Indeed, if β = P ℓi =1 ǫ i α i , with α , . . . , α ℓ ∈ Q and ǫ i = ±
1, then, uponreordering, we can assume that P hi =1 ǫ i α i is a root for all 1 ≤ h ≤ ℓ .Leaving h invariant, λ determines an involution λ ∗ on R , which is the identityon Q . Condition (6.6) implies that Q spans h ∗ , hence λ ∗ is the identity on R , andtherefore λ | h = Id. (cid:3) Remark 6.4.
When all roots in R have the same length, orthogonal roots arestrongly orthogonal and (6.5) is equivalent to(6.7) ( α | β ) ≥ , ∀ α, β ∈ Q . According to [14], a U - CR -symmetric structure on M is extrinsic symmetricif there is an isometric embedding of M into a Euclidean space V , and, for every x ∈ M , an isometry of V that restricts to a symmetry of M at x and to the identityon the normal bundle of M at x .The opposite of the Killing form defines a scalar product on u , which is invariantfor the adjoint action of U . The stabilizer of a regular element X of t in U isthe Cartan subalgebra T , so that the orbit Ad( U )( X ) is an embedding of M .The induced metric is U -invariant.The tangent space of M at X is identified, via the differential of the action atthe identity, to u / t ≃ P α u ∩ ( g α + g − α ). Under this identification, the subspace u ∩ ( g α + g − α ) is mapped onto itself, and t is its orthogonal complement in u .The involution λ of Lemma 6.3 is then an extrinsic symmetry at x . We haveproved: Proposition 6.5. If M = U / T , endowed with the CR structure defined by the CR algebra ( u , q ) , where q is given by (6.1) , is of finite type and U - CR -symmetric,then it is extrinsic CR -symmetric. (cid:3) CR symmetries, J -properties, and gradings. By Lemma 6.3 the invo-lution λ in (6.4) is inner . In fact, (6.4) implies that λ = Ad(exp( iπE )) for anelement E ∈ R ⋆ = { H ∈ h | α ( H ) ∈ Z , ∀ α ∈ R} such that(6.8) α ( E ) ≡ , ∀ α ∈ Q . The weak- J -property for ( u , q ) will then be equivalent to the possibility of choos-ing this E ∈ R ⋆ in such a way that(6.9) α ( E ) ≡ , ∀ α ∈ Q . Indded, the element J in Definition 1.8 will be equal to iE ∈ t .To discuss the symmetric CR structures on complete flags in terms of the setsof roots Q , it is convenient to introduce some notation. N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 23
Definition 6.6. If S ⊂ R , we shall indicate by Q ( S ) the set of all Q ⊂ S whichsatisfy (6.5) and (6.6). We set Q s ( S ) (resp. Q ( S ), Q Υ ( S )) for the subset of Q ( S ) consisting of those Q ⊂ S for which the ( u , q ) with q given by (6.1) is CR -symmetric (resp. has the J -property, has the weak- J -property). We will also saythat Q itself is CR -symmetric, or has the J or the weak- J -property when it holdsfor ( u , q ). Clearly Q ( S ) ⊂ Q Υ ( S ) ⊂ Q s ( S ) ⊂ Q ( S ).We indicate by Q ′ ( S ) the sets Q ⊂ S which satisfy (6.5).
Remark 6.7.
For E ∈ R ⋆ , Q ( { α | α ( E ) ≡ } ) ⊂ Q s ( R ) and Q ( { α | α ( E ) ≡ } ) ⊂ Q Υ ( R ).Given Q ⊂ R , let us define Q ∗ , = {± ( β − β ) ∈ R | β , β ∈ Q} , (6.10) Q ∗ h = nX k i β i ∈ R (cid:12)(cid:12)(cid:12) β i ∈ Q , k i ∈ Z , X k i = h o , h ∈ Z . (6.11)We have Q ⊂ Q ∗ , Q ∗ , ⊂ Q ∗ , and Q ∗− h = −Q ∗ h for all h ∈ Z . Moreover, Q ∗ is theroot system of the reductive complex Lie subalgebra of gq (0) = h ⊕ X α ∈Q ∗ g α , and, with(6.12) q ( h ) = X α ∈Q ∗ h g α for h ∈ Z \ { } , we have(6.13) [ q ( h ) , q ( k ) ] ⊂ q ( h + k ) . (6.14)We have g = P h ∈ Z q ( h ) if, and only if, R ⊂ Z [ Q ]. Lemma 6.8. If Q ∈ Q ( R ) and Q ∪ ( −Q ) = R , then Q ∗ , = ∅ . Moreover, h ∈ Z \ { } and Q ∗ , ∩ Q ∗ h = ∅ = ⇒ Q ∗ h ∩ Q ∗ h + h = ∅ , ∀ h ∈ Z . Proof.
Assume that
Q ∈ Q ( R ) and that Q ∪ ( −Q ) = R . Pick α ∈ R with ± α / ∈ Q .By (6.6), we get α = P ℓi =1 ǫ i β i , with ℓ ≥ ǫ i = ± β i ∈ Q for all 1 ≤ i ≤ ℓ ,and P i ≤ h ǫ i β i ∈ R for all 1 ≤ h ≤ ℓ . By (6.5) we have ǫ + ǫ = 0, and hence ǫ β + ǫ β ∈ Q ∗ , = ∅ .Assume that, for a pair of integers h = k , the intersection Q ∗ h ∩ Q ∗ k contains aroot α . Then we can find roots β i , γ j ∈ Q , and numbers ǫ i , η j = ±
1, for 1 ≤ i ≤ ℓ ,1 ≤ j ≤ ℓ , with P ǫ i = h , P η j = k , such that α = X ℓ i =1 ǫ i β i = X ℓ j =1 η j γ j , with (P ki =1 ǫ i β i ∈ R for k < ℓ , P kj =1 η j γ j ∈ R for k < ℓ . We can assume that h = ±
1. Then ℓ ≥ ǫ β + ǫ β ∈ Q ∗ , and, with h = h − k ,we obtain ǫ β + ǫ β = X ℓ j =1 η j γ j − X ℓ i =3 ǫ i β i ∈ Q ∗ , ∩ Q ∗ h . (cid:3) From (6.6), we obtain(6.15) q = X h ∈ Z g ( h ) , for Q ∈ Q ( R ) . Thus Lemma 6.8 and Remark 6.7 yield criteria for Q either to be symmetric or tohave the J or weak- J -property. Theorem 6.9.
Let M = U / T be a complete flag, with CR structure defined by ( u , q ) , for a q given by (6.1) , with Q ∈ Q ( R ) . Then (1) M is U - CR -symmetric if, and only if, (6.16) Q ∗ , ∩ Q ∗ h +1 = ∅ , for all h ∈ Z . (2) The CR algebra ( u , q ) has the weak- J -property if and only if (6.17) Q ∗ , ∩ Q ∗ h +1 = ∅ and Q ∗ , ∩ Q ∗ h +2 = ∅ , for all h ∈ Z . (3) The CR algebra ( u , q ) has the J -property if and only if (6.15) is a Z -gradation of g .Proof. By Lemma 6.8, conditions (6.16) (resp. (6.17)) implies that g admits a Z -gradation (resp. a Z -gradation) with h ⊂ g [0] and n ⊂ g [1] , where [ a ] means thecongruence class of a ∈ Z modulo 2 (resp. modulo 4). Since this gradation is inner,we obtain (1) (resp. (2)).Finally, if ( u , q ) has the J -property, and (1.15) is valid, then E = iJ ∈ R ⋆ and[ E, Z ] = Z for all Z ∈ n . By (6.6) we get g ( h ) = { Z ∈ g | [ E, Z ] = hZ } , and (6.15)is a direct sum decomposition, yielding a Z -gradation of g . Vice versa, if E ∈ R ⋆ defines a Z -gradation with α ( E ) = 1 for all α ∈ Q , we can take J = − iE to obtain(1.15). (cid:3) Complete flags of the classical groups.
In this section we classify thesymmetric CR structures on the complete flags of the classical groups.To fix notation, in the following we shall consider root systems R ⊂ R n , of thetypes A n − , B n , C n , D n explicitly described, according to [8], respectively, by:(A n − ) R = {± ( e i − e j ) | ≤ i Q ∈ Q ( R ) . We have: Proposition 6.10. Let R be an irreducible root system of one of the types A n − , B n , C n , D n .Then, modulo equivalence by the Weyl group W of R , the maximal Q ∈ Q ( R ) areequivalent to one of the following: (A n − ) Q p = { e i − e j | ≤ i ≤ p Q ∈ Q s ( R ) . Using the results of Proposition 6.10 we characterize,modulo equivalence, all maximal Q ∈ Q s ( R ), for R irreducible of one of the typesA , B , C , D. Theorem 6.11. If R is an irreducible root system of one of the classical types A n − , B n , C n , D n , then Q s ( R ) = Q ( R ) , i.e. all CR -symmetric ( u , q ) havethe J -property. Modulo equivalence w.r.t. the Weyl group W of R , the maximal Q ∈ Q s ( R ) are classified by: (A n − ) Q s ( R ) = Q ( R ) and all Q ∈ Q ( R ) are maximal. (B n ) Each maximal Q ∈ Q s ( R ) is equivalent, modulo W , to one of the followingsets: Q ′ i ,p,q ,...,q s = { e i } ∪ { e i + e j | ≤ i ≤ s, s +1 ≤ j ≤ p }∪ [ si =1 { e i ± e j | q i − Q ∈ Q s ( R ) is isomorphic, modulo W , to one of the followingsets: Q n = { e i + e j | ≤ i 1. If 1 ≤ i < j ≤ n and e i ± e j ∈ Q , then e i ( E ) ≡ , e j ( E ) ≡ e i + e j / ∈ Q for 1 ≤ i < j ≤ s .Hence we obtain that a maximal Q ∈ Q s ( R ) is equivalent to one of the sets listed above. We define the element J ∈ t by setting e h ( J ) = i for 1 ≤ h ≤ s , and e h ( J ) = 0 for s < j ≤ n .(C) With Q defined in Proposition 6.10, we define e h ( J ) = i/ ≤ h ≤ n .Then the corresponding ( u , q ) has the J -property.(D) We can repeat the argument of (B), to conclude that all maximal Q ∈ Q s ( R ) are described, modulo equivalence, by the list in (D n ) above. For Q = Q n ,we define J ∈ t by e h ( J ) = i/ ≤ h ≤ n . For Q = Q − n we set e i ( J ) = i/ ≤ i < n and e n ( J ) = − i/ 2. For Q = Q ′ p,q ,...,q s , we set e h ( J ) = i for 1 ≤ h ≤ s ,and e h ( J ) = 0 for s < h ≤ n . In this way we verify that all maximal Q ∈ Q s ( R ),hence all Q in Q s ( R ), have the J -property. (cid:3) Corollary 6.12. All CR symmetric Q contained in a root system of one of thetypes A , B , C , D have the J property. (cid:3) Complete flags of the exceptional groups. We turn finally to the com-plete flags of the exceptional groups.6.5.1. Type G . The root system is R = {± ( e i − e j ) | ≤ i < j ≤ } ∪ {± (2 e i − e j − e k ) | ( i, j, k ) ∈ S } . According to [33, Theorem3.11] there is, modulo automorphisms, a unique Z -grading of g , with S = {± ( e − e ) , ± ( e − e ) } ∪ {± (2 e − e − e ) , ± (2 e − e − e ) } , R = {± ( e − e ) } ∪ {± (2 e − e − e ) } . The sum of two short roots is always a root, while the sum of two long roots, ifit is a root, is long. Hence a Q ∈ Q ( R ) contains exactly one short root. Moduloisomorphisms, we can assume that ( e − e ) ∈ Q . Then Q ⊂ { e − e , e − e − e , ± (2 e − e − e ) , e + e − e } . The symmetry with respect to 2 e − e − e leaves e − e invariant and interchanges 2 e − e − e and e + e − e . Moreover, (2 e − e − e ) + (2 e − e − e ) ∈ R and ( e + e − e ) − (2 e − e − e ) ∈ R . Hence, moduloisomorphisms, there are two non equivalent maximal Q ∈ Q ( R ): Q = { e − e , e − e − e , e + e − e } , Q = { e − e , e − e − e , e + e − e } . Thus we obtain Proposition 6.13. Let R be simple of type G . Then: (1) Any maximal Q ∈ Q ( R ) is isomorphic either to Q or to Q . (2) Q ∈ Q s ( R ) , and Q / ∈ Q s ( R ) . (3) Q Υ ( R ) = Q ( R ) and all Q ∈ Q Υ ( R ) are isomorphic to Q = { e − e , e − e − e } . Type F . We split the root system of type F into two parts, by setting R = {± e i | ≤ i ≤ } ∪ {± e i ± e j | ≤ i The set R is a root system of type B . By Proposition 6.10, modulo equivalence,there are five maximal sets in Q ( R ), namely: Q (4)1 , = { e } ∪ { e i + e j | ≤ i Proposition 6.14. Modulo equivalence, there are five classes of non equivalentmaximal elements of Q ( R ) , corresponding to the terms of the following list: Q , = Q (4)1 , ∪ { β , β } , Q , , = Q (4)1 , , ∪ { β , β } , Q , , = Q (4)2 , , ∪ { β , β } , Q , , , = Q (4)1 , , , ∪ { β , β } , Q , , = Q (4)1 , , ∪ { β , β } , Proof. For any choice of three distinct roots in S , two of them sum to a root. Thena Q ∈ Q ( R ) contains at most two roots of S . The sets in the list are obtainedby adding a couple of roots of S to each maximal set in Q ( R ). Thus they aremaximal. The fact that they exhaust the list of maximal elements of Q ( R ) moduloequivalence is proved by considering the set of all roots in S that may be addedto a Q (4) ∗ without contradicting (6.5). (cid:3) Modulo equivalence, the maximal elements of Q s ( R ) are Q ′ , , = { e } ∪ { e ± e i | ≤ i ≤ } , Q ′ , , , = { e } ∪ { e ± e } ∪ { e ± e } . Thus we obtain Proposition 6.15. We have Q s ( R ) = Q Υ ( R ) = Q ( R ) , and the maximal ele-ments of Q s ( R ) are all equivalent to Q ′ , , , = { e , β , β , e ± e , e ± e } . Proof. In fact a maximal Q ∈ Q s ( R ) must contain two short roots. Hence, if E ∈ h has integral values on R and defines a Z -gradation with α ( E ) odd for α ∈ Q , thenthere are two even and two odd e i ( E )’s. This implies that all maximal elementsof Q s ( R ) are equivalent to Q ′ , , , = Q ′ , , , ∪ { β , β } . We observe that, with e ( E ) = 1, e ( E ) = 1, e ( E ) = 0, e ( E ) = 0 we obtain that α ( E ) = 1 for all α ∈ Q .Hence Q ′ , , , ∈ Q ( R ). (cid:3) Type E , E , E . We will write E ℓ for the root system of type E ℓ , and wewill use the explicit description of [8], with E ⊂ E ⊂ E ⊂ R .It is convenient to use the notation β ε = P i =1 ǫ i e i , where e , . . . , e is thecanonical basis of R , and ε = ( ǫ , . . . , ǫ ), with ǫ i = ± Q i =1 ǫ i = 1. We shallwrite some times v = e − e , v = e − e − e . We shall also employ, for the roots of S , the simplified notation: β = X i =1 e i , β i,j = β − ( e i + e j ) , β i,j,h,k = β i,j − ( e h + e k ) , for 1 ≤ i, j, h, k ≤ . Then E = {± e i ± e j | ≤ i Consider the set Q = { β , β , , β , , β , , β , } ∪ { β j,h , β h,k | j =4 , , ≤ h ≤ , h Q ∈ Q Υ ( E ) \ Q ( E ), but is not maximal in Q s ( E ). N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 29 Lemma 6.17. (1) Let ℓ ∈ { , , } . For every α ∈ Q ∈ Q ( E ℓ ) we can find β ∈ Q , β = α , with ( α | β ) > . (2) Let ( ℓ, i ) ∈ Ξ , and α , α , α ∈ S ℓi , then (6.20) ( α | α ) > , ( α | α ) > ⇒ α + α / ∈ R . (3) For ( ℓ, i ) ∈ Ξ \ { (6 , , (7 , } the group W ℓi is transitive on S ℓi . The set S is the union of the two orbits of W : Q , ,β , , − β , = { β , } ∪ {− β i, , β i,j, , | ≤ i ≤ , i 1. Hence ( α − α | α ) =2 yields α = α − α . Therefore there is no E ∈ E ∗ ℓ with α i ( E ) odd for i = 0 , , ℓ, i ) equal to either (6 , 1) or (7 , E = E ℓ,i ∈ h satisfies α ( E ) = 0 for all α ∈ R ℓi . Hence in this two cases S ℓi splits into { α ∈ S ℓi | α ( E ) = λ } ,for λ = ± 1. For ( ℓ, i ) ∈ Ξ \ { (6 , , (7 , } the transititivity of W ℓi on S i can beeasily checked by a case by case verification. (cid:3) Proposition 6.18. For ( ℓ, i ) ∈ Ξ \ { (6 , , (7 , } , and α ∈ S ℓi , the set (6.21) Q ℓ,i,α = { α ∈ S ℓi | ( α | α ) > } is a maximal element of Q s ( E ℓ ) and does not belong to Q Υ ( E ℓ ) .Proof. For each ( ℓ, i ) ∈ Ξ \ { (6 , , (7 , } , the Weyl group of R ℓi is transitive on S ℓi .Hence it suffices to consider Q ℓ,i,α when α is any specific element of S ℓi . We have: Q , ,β , , , = {− ( e + e ) , β , , , } ∪ { e i + e j , β i,r, , | ≤ i ≤ , i 1, ( β i,h | e i − e j ) = − β | − e i − e j ) = − 1, ( β i,j | β h,k,r,s ) = − i, j, h, k, r, s of distinct indiceswith 1 ≤ i, j, h, k, r, s ≤ Q , ,β is maximal in Q ( E ).Let us show that Q ,β / ∈ Q Υ ( R ). We argue by contradiction. From β ( E ) ≡ β ∈ Q ,β we obtain that e i ( E ) ± e j ( E ) ≡ ≤ i < j ≤ Then e i ( E ) = 2 k i is an even integer for all i = 1 , . . . , 8. But then β ( E ) ≡ β i,j ( E ) ≡ P i =1 k ≡ k i + k j ) ≡ k i ’s are odd, or all k i ’s are even. (cid:3) Example 6.19. Consider, for an integer p with 1 ≤ p ≤ 8, the set Q ′ p = { β } ∪ { e i + e r , β i,j , β r,s | ≤ i ≤ p, i 8. Each Q ∈ Q ( E ) which is maximal andis contained in { α ∈ E | ( β | α ) > } is equivalent, modulo W , to some Q ′ p . Oneeasily verifies that Q ′ p / ∈ Q s ( E ) for p odd and Q ′ p ∈ Q s ( E ) for p even.The definition in (6.21) can be generalized in the following way: Proposition 6.20. Let ( ℓ, i ) ∈ Ξ be fixed. Define, for α , . . . , α k ∈ S ℓi , with inf ≤ j 0. Then α ∈ Q jℓ,i,α ,...,α k . (cid:3) Lemma 6.21. Let ( ℓ, i ) ∈ Ξ and consider a sequence (6.22) , such that Q ℓ,i,α ,...,α k ∈ Q ( S i ) and is maximal. Then, for every integer p with ≤ p < k we have (6.23) Q ℓ,i,α ,...,α k = Q pℓ,i,α ,...,α k ∪ (cid:0) Q ℓ,i,α ,...,α k ∩ { α , . . . , α p } ⊥ (cid:1) . Proof. Let Q ℓ,i,α ,...,α k ∩{ α , . . . , α p } ⊥ = { γ , . . . , γ q } . Since we have Q ℓ,i,α ,...,α k ∩{ α | sup ≤ i ≤ p ( α | α i ) > } ⊂ Q pℓ,i,α ,...,α k , in (6.23) the left hand side is contained inthe right hand side. The right hand side of (6.23) is contained in Q α ,...,α p ,γ ,...,γ q .By the assumption that Q ℓ,i,α ,...,α k is maximal, it coincides with Q α ,...,α p ,γ ,...,γ q .This yields (6.23). (cid:3) Proposition 6.22. Let ℓ ∈ { , , } . Every maximal Q ∈ Q s ( E ℓ ) is equivalent,modulo W , to a Q ℓ,i,α ,...,α k with ( ℓ, i ) ∈ Ξ and (6.24) α , . . . , α k ∈ S ℓi , with ( α j | α h ) = 0 ∀ ≤ j < h ≤ k. Proof. First we observe that any Q ∈ Q s ( E ℓ ) is equivalent, modulo W , to a set Q ′ ∈ Q ( S ℓi ) for some i with ( ℓ, i ) ∈ Ξ. We can write Q ′ = Q ℓ,i,η ,...,η ℓ for asequence η , . . . , η ℓ of elements of S ℓi . Indeed, we can take for η , . . . , η ℓ the sequenceconsisting of the elements of Q ′ , listed in any order. Set α = η . If Q ′ ⊂ Q ℓ,i,α ,we have Q ′ = Q ℓ,i,α by maximality, and the thesis is verified. Otherwise, bythe argument of the proof of Lemma 6.21, Q ′ = Q α ,γ ,...,γ r for some γ , . . . , γ r ∈Q ′ ∩ α ⊥ . Set α = γ . Repeating the argument, either Q ′ = Q ℓ,i,α ,α , or, otherwise, Q ′ = Q α ,α ,µ ,...,µ s for a new sequence µ , . . . , µ s ∈ Q ′ ∩ { α , α } ⊥ . The generalargument is now clear, and the thesis follows by recurrence. (cid:3) Example 6.23. We have Q , ,β ,β , = Q , ,β ,β , , , ,β , , , ,β , , , . N HOMOGENEOUS AND SYMMETRIC CR MANIFOLDS 31 Remark 6.24. We have Q , ,β = Q , ,β , ,β , ,β , ,β , . In particular, the repre-sentation of the maximal symmetric Q given in Proposition 6.22 is not unique.We are interested in classifying modulo equivalence the maximal sets in Q ( S ℓi )for ( ℓ, i ) ∈ Ξ \ { (6 , , (7 , } . Using Proposition 6.22, they can be parametrized bysequences of orthogonal roots. Modulo the action of W ℓi , they can be taken to besubsets of the following:(A × A ) e + e , e + e , β , , , , β , , , , (D × A ) β , , β , , β , , , , β , , , , (A ) e + e , e + e , e + e , β , , , , β , , , , β , , , , β , , , , (D ) β , β , , , , β , , , , β , , , , β , , , , β , , , , β , , , , β , , , , (E × A ) β , , β , , β , , β , . The group W ℓi is transitive on the maximal systems of orthogonal roots of S ℓi when( ℓ, i ) ∈ { (6 , , (7 , , (8 , } , W is transitive on the pairs of orthogonal roots of S , and W on the triples of orthogonal roots of S . Thus we obtain Proposition 6.25. Every maximal element of Q s ( E ℓ ) is equivalent to an elementof Q ( S ℓi ) for some ( ℓ, i ) ∈ Ξ . The maximal elements of Q s ( S ℓi ) are conjugate tofor ( ℓ, i ) = (6 , , either Q , ,β , , − β , or its opposite Q , , − β , ,β , ;for ( ℓ, i ) = (7 , one of: Q , ,v ,e + e , Q , ,v ,e − e , Q , ,v ,e − e ,e + e , or theopposite of one of the above;for ( ℓ, i ) ∈ { (6 , , (7 , , (8 , } , one of the Q ℓ,i,α ,...,α k , and the equivalence classonly depends on the number k of orthogonal roots;some Q , ,α ,...,α k with α = e + e , α = e + e when k ≥ , and { α , . . . , α k } ⊂{ e + e , β , , , , β , , , , β , , , , β , , , } for ≤ k ≤ ,some Q , ,α ,...,α k with α = β , α = β , , , when k ≥ , α = β , , , when k ≥ , and { α , . . . , α k } ⊂ { β , , , , β , , , , β , , , , β , , , , β , , , } for ≤ k ≤ . Example 6.26. (1) We have Q , ,β ,β , , , = { β , β , , , } ∪ { β i,r , β i,j,h,r | ≤ i ≤ , i Q ∈ Q s ( E ) \ Q Υ ( E ), and is maximal.(4) Consider Q = Q , β ,β , , , ,β , , , ,β , , , ,β , , , We claim that Q ∈ Q ( E ). Indeed, ( β r,s | β i,j,h,k ) ≥ { r, s } ∩{ i, j, h, k } 6 = ∅ , hence { , , , }∩{ , , , }∩{ , , , }∩{ , , , } = ∅ = ⇒ β i, / ∈ Q , for i = 1 , . . . , . The conditions ( β r,s | β i,j,h,k ) ≥ 0, ( β a,b,c,d | β i,j,h,k ) ≥ { r, s } ∩{ i, j, h, k } 6 = ∅ , { a, b, c, d } ∩ { i, j, h, k } ≥ 2, respectively. Moreover, { β , , β , , β , , β , , β , , β , , β , , β , , β , , β , , β , } ⊂ Q . Then we can easily show that Q cannot contain any root of type β i,j,h, with1 ≤ i < j < h ≤ 7. Therefore α ( E , ) = 1 for all α ∈ Q β ,β , , , ,β , , , ,β , , , ,β , , , . (5) Consider Q = Q , ,β ,β , , , ,β , , , ,β , , , ,β , , , ,β , , , ,β , , , . We have Q = { β , β , , β , , β , , β , , β , , β , , , , β , , , , β , , , , β , , , , β , , , ,β , , , , β , , , , β , , , , β , , , , β , , , , β , , , , β , , , , β , , , } . Then Q is maximal in Q ( E ) and belongs to Q s ( E ) \ Q Υ ( E ).(6) Let Q = Q , ,β ,β , , , ,β , , , ,β , , , ,β , , , ,β , , , ,β , , , . 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Warner, Harmonic analysis on semi-simple Lie groups I , Springer-Verlag, New York,1972. A. Altomani: University of Luxembourg, 162a, avenue de la Fa¨ıencerie, L-2309 Lux-embourg E-mail address : [email protected] C. Medori: Dipartimento di Matematica, Universit`a di Parma, Viale G.P. Usberti,53/A, 43100 Parma (Italy) E-mail address : [email protected] M. Nacinovich: Dipartimento di Matematica, II Universit`a di Roma “Tor Vergata”,Via della Ricerca Scientifica, 00133 Roma (Italy) E-mail address ::