A characterization of CR quadrics with a symmetry property
AA CHARACTERIZATION OF CR QUADRICSWITH A SYMMETRY PROPERTY
ANDREA ALTOMANI AND COSTANTINO MEDORI
Abstract.
We study CR quadrics satisfying a symmetry property ( ˜ S )which is slightly weaker than the symmetry property ( S ), recently in-troduced by W. Kaup, which requires the existence of an automorphismreversing the gradation of the Lie algebra of infinitesimal automorphismsof the quadric.We characterize quadrics satisfying the ( ˜ S ) property in terms of theirLevi-Tanaka algebras. In many cases the ( ˜ S ) property implies the ( S )property; this holds in particular for compact quadrics.We also give a new example of a quadric such that the dimensionof the algebra of positive-degree infinitesimal automorphisms is largerthan the dimension of the quadric. Introduction
The affine quadrics Q = { ( z, w ) ∈ C nz × C kw | Im w = H ( z, z ) } , for anondegenerate hermitian form H : C n × C n → C k , provide the simplestnontrivial examples of CR manifolds. Any such quadric Q has a canonicalcompletion as a real submanifold ˆ Q , not necessarily closed when k > CP N . Both Q and its completion ˆ Q are CR-homogeneous and any local CR automorphism of Q or ˆ Q extends to a globalprojective automorphism of CP N transforming ˆ Q into itself (see [7]).The choice of a point of Q , e.g. the point 0 ∈ C n × C k , yields a naturalgradation g = g − ⊕ g − ⊕ g ⊕ g ⊕ g of the Lie algebra g of the infinitesimal CR automorphisms of Q . The groupAut CR ( ˆ Q ) of CR automorphisms of ˆ Q has g as its Lie algebra and acts on g via the adjoint action.In [8] W. Kaup defined a quadric Q to have the symmetry property ( S ) ifthere is an involutive CR automorphism γ of ˆ Q such that Ad( γ )( g j ) = g − j for j = 0 , ± , ± S ) to an ( ˜ S ), that requires theexistence of a degree reversing CR automorphism γ of ˆ Q of finite order.Quadrics enjoying property ( ˜ S ) are characterized in terms of a propertyof the Levi-Tanaka algebra g . We prove that ( ˜ S ) holds true if and only ifthe gradation of g is inner and defined by a semisimple element of a Levi Date : October 29, 2018.2010
Mathematics Subject Classification.
Primary: 32V05; Secondary: 16W10, 17B66,32V20, 53C30, 57S20.
Key words and phrases.
CR quadric, homogeneous CR manifold, Levi-Tanaka algebra,involutive automorphism. a r X i v : . [ m a t h . C V ] D ec A. ALTOMANI AND C. MEDORI factor of g . The order of γ can be required to be either 2, or 4, and weshow that in several instances, including the case where ˆ Q is compact, weget actually property ( S ). We have no example to show that ( ˜ S ) and ( S )are not equivalent.We also provide a simple example of a CR quadric Q for which dim g > dim g − and dim g > dim g − , giving a new counterexample to a questionthat was formulated by V.Ezhov and G.Schmalz in [6].The authors wish to thank I. Kossovskiy for pointing out the result ofUtkin [17]. 2. CR manifolds and Levi-Tanaka algebras We first recall the definition of a CR manifold.
Definition 2.1.
A CR manifold of type ( n, k ) is the datum (
M, T , M ) of areal smooth manifold M of dimension 2 n + k and a smooth complex vectorsubbundle T , M , with constant complex rank n , of the complexification T C M of the tangent bundle of M , satisfying the following conditions:(1) T , M ∩ T , M = 0,(2) [ C ∞ ( M, T , M ) , C ∞ ( M, T , M )] ⊂ C ∞ ( M, T , M ).The integers n and k are called the CR dimension and CR codimension of M . We also set T , M = T , M and HM = T M ∩ ( T , M + T , M ).Let J : T , M + T , M → T , M + T , M be the linear semisimple iso-morphism with eigenvalues i on T , M and − i on T , M . Then J preserves HM and is called a partial complex structure .A CR map between two CR manifolds M and N is a smooth map f : M → N such that df C ( T , M ) ⊂ T , N . The notions of CR isomorphism andautomorphism are defined in the natural way. Definition 2.2.
Let M be a real submanifold of a complex manifold X .Define T , M = T C M ∩ T , X . If T , M has constant rank, then ( M, T , M )is a CR manifold, called a CR submanifold of X .We introduce two further definitions. Definition 2.3.
A CR manifold (
M, T , M ) is said to be:(1) Levi nondegenerate at a point x ∈ M if for every vector field Z ∈ C ∞ ( M, T , M ) there exists a vector field ¯ W ∈ C ∞ ( M, T , M ) suchthat [ Z, ¯ W ] x / ∈ T , x M + T , x M ;(2) of finite type at a point x ∈ M if the Lie algebra generated by allvector fields in C ∞ ( M, T , M + T , M ) spans T C x M .2.1. Levi-Tanaka algebras and standard CR manifolds. Let M be aCR manifold, and x ∈ M a point where M is Levi nondegenerate and offinite type. We associate to x a graded Lie algebra, called the Levi-Tanakaalgebra of M at x . We refer to [9] for a more detailed discussion of Levi-Tanaka algebras. Define: D = 0 , D − = C ∞ ( M, HM ) , and inductively, for p ≥ D − p = D − p +1 + [ D − p +1 , D − ] . R QUADRICS WITH A SYMMETRY PROPERTY 3
Then we set, for p ≥ m − p = D − p ( x ) / D − p +1 ( x ) . The vector field bracket induces a graded Lie algebra structure on m − = (cid:80) p ≤− m p . Note that m − is canonically isomorphic to H x M . Then it isnaturally defined a complex structure J on m − and [ J X, J Y ] = [
X, Y ] forevery
X, Y ∈ m − .Let m = { D ∈ Der ( m − ) | [ D | m − , J ] = 0 } be the set of zero-degree derivations on m − commuting with J on m − . Then m + m − is a graded Lie algebra. Definition 2.4.
The
Levi-Tanaka algebra associated to M at a point x ∈ M where M is Levi nondegenerate and of finite type is the (unique) graded Liealgebra g = (cid:80) p ∈ Z g p with the following properties:(1) g p = m p for p ≤ X ∈ g p , with p ≥
0, the action ad g ( X ) | g − is nonzero,(3) g is maximal with those properties. Definition 2.5.
In general, we can start with any graded Lie algebra m − = (cid:80) p ≤− m p , such that m − generates m − and with a complex structure J on m − such that [ X, Y ] = [
J X, J Y ] ∀ X, Y ∈ m − , and perform the same prolongation procedure as in Definition 2.4.The resulting algebra g = (cid:80) p ∈ Z g p (with complex structure J on g − ) isa Levi-Tanaka algebra .We fix the following notation: for a graded Lie algebra g = (cid:80) p ∈ Z g p , weset g − = (cid:88) p< g p , p = (cid:88) p ≥ g p , g + = (cid:88) p> g p , p opp = (cid:88) p ≤ g p . A Levi-Tanaka algebra has trivial center and contains a unique element E ∈ g , called characteristic element , such that ad g ( E ) | g j = j Id g j for all j ∈ Z . Definition 2.6.
For a Levi-Tanaka algebra g = (cid:80) p ∈ Z g p , with complexstructure J g on g − , it is possible to construct a CR manifold such that theassociated Levi-Tanaka algebra at every point is isomorphic to g .Let ˜ G be the connected and simply connected group with Lie algebra g ,and P the analytic subgroup with Lie algebra p . Then P is closed, and welet S = S ( g ) = ˜ G / P .There is a unique ˜ G -homogeneous CR structure on S such that at thebase point o = e P the partial complex structure is given by H o S = g − and J o = J g , where we identified T o S and g − , in the natural way.The CR manifold S = S ( g ) is the standard CR manifold associated to g (see [9]). A. ALTOMANI AND C. MEDORI
The standard CR manifold S = S ( g ) is simply connected, and g is iso-morphic both to the Lie algebra of its infinitesimal automorphisms and tothe Lie algebra of the group of (global) CR automorphisms.We recall that S is compact if and only if g is semisimple (see [11, Corol-lary 5.3]). The group of CR automorphisms of S is in general not connected,but we can give a description of its connected component of the identity. Proposition 2.7.
Let g be a Levi-Tanaka algebra, and S the associatedstandard CR manifold. Then the connected component of the identity of thegroup Aut CR ( S ) of CR automorphisms of S is the group G = Int( g ) ofinner automorphisms of g .Proof. Let P = { g ∈ G | Ad( g )( p ) = p } . Then the Lie algebra of P is p and the manifold M = G / P has a natural CR structure. The naturalquotient ˜ G → G induces a covering map π : S → M .The manifold M is maximally homogeneous (the dimension of its auto-morphism group is equal to the dimension of the group of automorphismsof the standard manifold S ), then M is isomorphic to S and, in particular,simply connected (see [12]). It follows that π is a diffeomorphism. (cid:3) The standard CR manifold S can then be identified to the set of innerconjugates of p in g .This observation provides also another construction of standard CR man-ifolds. The group G acts, via the complexification of the adjoint action, onall the complex grassmannians of subspaces of g C . Let q = g C + g C + g C + { X + i J X | X ∈ g − } . The G -orbit through the point o = q in the complex grassmannian Gr dim q ( g C ),with the CR structure given by the embedding, is CR-isomorphic to thestandard CR manifold associated to g .Although we will not use it, we give a characterization of the full auto-morphism group of a standard CR manifold. Proposition 2.8.
Let g be a Levi-Tanaka algebra, and S the associatedstandard CR manifold. Then the group Aut CR ( S ) of CR automorphisms of S is the group G = { g ∈ Aut( g ) | g · q is Int( g ) -conjugate to q } . Proof. (Aut CR ( S ) ⊂ G ). Let φ ∈ Aut CR ( S ) be a CR automorphism of S and X ∈ g . Denote by X † the vector field on S generated by X . Then( φ · X ) † = dφ ( X † ) defines an action of φ on g , which is an automorphism.Let g ∈ Int( g ) be an element such that φ ◦ g ( o ) = o . Then φ ◦ g · p = p , and φ ◦ g · q = q because φ ◦ g is a CR map.(Aut CR ( S ) ⊃ G ). Let g be an element of G , and h ∈ Int( g ) an elementwith g · q = h · q . Define an action of g on S as follows: for k ∈ Int( g ), let g · ( k · o ) = ( gkhg − ) · o . (cid:3) CR quadrics
Let H : C n × C n → C k be a vector valued hermitian form, linear in thefirst variable and anti- C -linear in the second one. R QUADRICS WITH A SYMMETRY PROPERTY 5
Definition 3.1.
The vector valued hermitian form H : C n × C n → C k issaid to be: nondegenerate: if for all z ∈ C n \ { } there exists z (cid:48) ∈ C n such that H ( z, z (cid:48) ) (cid:54) = 0; fundamental: if the set { H ( z, z ) | z ∈ C n } ⊂ R k spans R k .To a vector valued hermitian form it is naturally associated a CR sub-manifold of C n + k in the following way. Definition 3.2.
The affine CR quadric associated to a vector valued her-mitian form H : C n × C n → C k is the CR-submanifold of C n ⊕ C k givenby: Q = Q H = { ( z, w ) ∈ C n ⊕ C k | (cid:61) w = H ( z, z ) } . It is straightforward to see that Q H is a CR manifold of CR-dimension n and CR-codimension k , it is finitely nondegenerate (in fact Levi nondegen-erate) if and only if H is nondegenerate, and it is of finite type (indeed oftype 2) if and only if H is fundamental. Remark 3.3.
Any affine quadric Q can be written as a product Q = Q (cid:48) × C m × R h , where Q (cid:48) is a Levi nondegenerate affine quadric of finite type, m isthe dimension of the null space of H , and h is the codimension of the imageof H in C k . We assume, from now on, that H is nondegenerate and fundamental. The Lie algebra of infinitesimal automorphisms of Q is finite-dimensionaland possesses a natural grading g = (cid:76) i = − g i . It is canonically isomorphicto the Levi-Tanaka algebra associated to Q (see [15] and [5]).Let ( g = ⊕ i ∈ Z g i , J ) be the Levi-Tanaka algebra associated to Q . Then g i = 0 for i < − i >
2, and dim R g − = 2 n , dim R g − = k . TheLie algebra structure on g − = g − ⊕ g − is given by H in the followingway. Identify g − , endowed with its complex structure J , to C n , and g − to R k ⊂ C k . Then: [ X, Y ] = (cid:61) ( H ( X, Y )) , ∀ X, Y ∈ g − . Definition 3.4.
The quadric ˆ Q = ˆ Q H associated to H is the standard CRmanifold S ( g ) associated to g . This definition agrees with the definitionin [8] (see also [7]).The affine quadric Q is CR diffeomorphic to the G − orbit through o , andit is open and dense in ˆ Q . The complement ˆ Q \ Q is the intersection of ˆ Q and a complex-algebraic subvariety of Gr dim q ( g C ).In [8] W. Kaup introduced a symmetry property, called property ( S ), forthe quadric ˆ Q . Here we consider the following generalization. Definition 3.5.
The quadric ˆ Q is said to have: • the ( S ) property if there exists an involutive automorphism γ ∈ G such that Ad g ( γ )( E ) = − E ; • the ( ˜ S ) property if there exists an automorphism γ ∈ G of finiteorder such that Ad g ( γ )( E ) = − E . A. ALTOMANI AND C. MEDORI
We use the same notation for the Levi-Tanaka algebra associated to thequadric.Our aim is to characterize quadrics with the ( ˜ S ) property, and show thatin many cases the ( S ) and ( ˜ S ) properties are equivalent.4. Levi-Malcev decomposition
We recall that Levi Tanaka algebras have a pseudocomplex graded
Levi-Malcev decomposition, i.e. compatible with the grading and the complexstructure [13]. More precisely, given a Levi-Tanaka algebras g , with radical r , there exist a semisimple subalgebra s such that:(1) g = s ⊕ r ;(2) s and r are graded;(3) s − and r − are J -invariant. Lemma 4.1.
Let g = (cid:80) p ∈ Z g p be a finite dimensional Levi-Tanaka algebra,and r = (cid:80) r p the radical of g . If there is an automorphism Γ ∈ Aut( g ) with Γ( E ) = − E , then r − := r ∩ g − (cid:54) = g − . In particular, g is not solvable.Proof. Assume r ∩ g − = g − . Then we have r ∩ g p = g p for every p ≤ − n = (cid:80) n p be the nilradical of g and consider the descending centralsequence n = n , n k +1 = [ n k , n ] , for k ≥ . Note that (cid:80) p (cid:54) =0 r p ⊂ n . Let d be the minimal integer such that n d (cid:54) = 0 and n d +1 = 0. Then n d is a characteristic ideal of g and [ n , n d ] = 0.Consider X ∈ n d := n d ∩ g . We have(4.1) [ X, g p ] = [ X, r p ] = [ X, n p ] = 0 , ∀ p ≤ − , hence X = 0 (see [9, Theorem 3.1]). Then n d = 0 and in general n dp := n d ∩ g p = 0, for any p > g p and g − p , and the ideal n d is characteristic, wehave also n dp = 0 for p (cid:54) = 0, therefore n d ⊂ g .Finally, [ n d , g − ] ⊂ n d − = { } , hence n d = { } . Then we have n d = { } , obtaining a contradiction. (cid:3) We fix now a pseudocomplex graded Levi-Malcev decomposition g = s ⊕ r of the Levi-Tanaka algebra g associated to a quadric ˆ Q having the ( ˜ S ) prop-erty. From Lemma 4.1 it follows that s (cid:54) = 0 and s − (cid:54) = 0.Let E ∈ g be the characteristic element. Then E = E s + E r with E r ∈ r ,and E s ∈ s is the characteristic element of s . Proposition 4.2.
If a quadric ˆ Q admits an automorphism γ with Ad( γ )( E ) = − E , then E = E s .Proof. Assume that ˆ Q admits such a γ . The isotropy Lie algebra at thepoint γ · o ∈ ˆ Q is g ⊕ g − . It follows that there exists an element X + ∈ g + with exp( X + ) γ · o ∈ Q . Since exp( g − ) acts transitively on Q , we also havean element X − ∈ g − such that exp( X + ) γ · o = exp( X − ) · o or in other words: R QUADRICS WITH A SYMMETRY PROPERTY 7 γ = exp( − X + ) exp( X − ) h , where h is an element of the isotropy at o . Sincethe isotropy at o is exactly G G + , we finally obtain, for a g ∈ G and an X (cid:48) + ∈ G + : γ = exp( − X + ) exp( X − ) exp( X (cid:48) + ) g = exp( Y ) exp( Y ) exp( Y − ) exp( Y − ) exp( Y ) exp( Y ) g (here the subscripts indicate the degrees of the homogeneous elements Y ij ).From Ad( γ )( E ) = − E we obtain2 E = 2[ Y − , Y ] + 12 [ Y − , Y + Y ] . Let n be the nilradical of g . Note that it is graded, and r p = n p for all p (cid:54) = 0.Decompose each element Y ij into its s and n component. It follows2 E r ∈ ([ s , n ] + [ r , r ]) ∩ g ⊂ n and E r is ad-nilpotent.Since ad( E ) preserves s , and r is an ideal, we have ad( E ) | s = ad( E s ) | s ,and E s is a ad-semisimple element of s . Then E = E s + E r is a Wedderburndecomposition of E , and since E is semisimple element, it follows that E = E s . (cid:3) We also have the following
Lemma 4.3.
If the quadric ˆ Q has property ( ˜ S ) , then p opp is conjugate to p by an inner automorphism of g .Proof. Assume that ˆ Q has property ( ˜ S ). Since Int( g ) acts transitively on ˆ Q and Ad( γ )( p ) = p opp is the isotropy Lie algebra at the point γ · o ∈ ˆ Q , thecondition is necessary. (cid:3) The semisimple case
We assume now that g is semisimple. For a standard CR manifold (andin particular for quadrics ˆ Q H ) this is equivalent to compactness (see [11,Corollary 5.3]). First we recall the description of semisimple Levi-Tanakaalgebras (see [10] for a more detailed treatment of the topic).Let g be a semisimple Levi-Tanaka algebra. Since every semisimple Levi-Tanaka algebra is a direct sum of simple Levi-Tanaka algebras, we can as-sume that g is simple. Choose a maximally noncompact Cartan subalgebra h of g , let g C and h C be the complexifications of g and h , and R = R ( g C , h C )be the corresponding root system.The conjugation of g C with respect to the real form g leaves h C invariant,and then induces a conjugation σ : R → R . Let R • = { α ∈ R | σα = − α } be the set of compact roots. There exists a choice of a set of positive roots R + such that σ ( R + \ R • ) ⊂ R + . Let B be the system of simple positiveroots for R + , that we identify to the nodes of the associated Dynkin diagram∆, and B • = B ∩ R • .The action of the conjugation σ on simple positive roots can be describedas follows: there exists an involution of the Dynkin diagram (cid:15) : B → B suchthat σα − (cid:15)α ∈ (cid:104)B • (cid:105) Z . The datum of (∆ , B • , (cid:15) ) completely determines g andis known as Satake diagram of g . A. ALTOMANI AND C. MEDORI
Fix a subset Φ of B with the following properties:(1) Φ ∩ B • = ∅ ,(2) Φ ∩ (cid:15) Φ = ∅ (in particular (cid:15) is nontrivial),(3) every connected component of ∆ intersects both Φ and (cid:15) Φ,(4) every path in ∆ connecting two elements of Φ contains elements of (cid:15)
Φ.Let
E, J be the elements of h such that: (cid:40) α ( E ) = 1 for α ∈ Φ ∪ (cid:15) (Φ), α ( E ) = 0 for α / ∈ Φ ∪ (cid:15) (Φ), α ( J ) = − i for α ∈ Φ, α ( J ) = i for α ∈ (cid:15) (Φ), α ( J ) = 0 for α / ∈ Φ ∪ (cid:15) (Φ).Then E defines a gradation on g , and J defines a complex structure on g − . The largest p ∈ N such that g p (cid:54) = { } is called the kind of g . Itcoincides with the degree of a maximal positive root. Conversely, everysimple Levi-Tanaka algebra is isomorphic to one obtained in this way.It is straightforward then to classify the simple Levi-Tanaka algebras ofkind 2. The names of the simple Lie algebras of real type are those of thecorresponding symmetric spaces in Cartan’s classification, the order of theroots { α , . . . , α (cid:96) } = B follows Bourbaki (see the table in the appendix of[1] or [2]). For simple algebras of the complex type, the simple roots aredenoted { α , . . . , α (cid:96) , α (cid:48) , . . . , α (cid:48) (cid:96) } with (cid:15)α j = α (cid:48) j . Proposition 5.1.
The simple Levi-Tanaka algebras of kind are directsums of simple factor of the following types: (1) Type A (cid:96) III / IV , Φ = { α i } , with ≤ i ≤ p and i (cid:54) = ( (cid:96) + 1) / or q ≤ i ≤ (cid:96) and i (cid:54) = ( (cid:96) + 1) / Type D (cid:96) Ib / IIIb , Φ = { α (cid:96) } ( or Φ = { α (cid:96) − } );(3) Type
E II / III , Φ = { α } ( or Φ = { α } );(4) Type A C (cid:96) , Φ = { α i , α (cid:48) j } with i (cid:54) = j ;(5) Type D C (cid:96) , Φ = { α , α (cid:48) (cid:96) − } or Φ = { α (cid:96) − , α (cid:48) (cid:96) } ( or Φ = { α , α (cid:48) (cid:96) } or Φ = { α (cid:96) − , α (cid:48) } , Φ = { α (cid:96) , α (cid:48) (cid:96) − } or Φ = { α (cid:96) , α (cid:48) } );(6) Type E C with Φ = { α , α (cid:48) } ( or Φ = { α , α (cid:48) } ) . (cid:3) We fix then a compact quadric ˆ Q , the corresponding semisimple Levi-Tanaka algebra g , a maximally noncompact Cartan subalgebra h of g con-tained in g , a system B of positive simple roots of the root system R = R ( g C , h C ).Of course in this case E = E s . We recall that G = Int( g ) = Aut CR ( ˆ Q ) is the adjoint group, and ˆ Q can be identified to the set of Ad( G )-conjugatesof p in g or of q in g C . We also recall that the analytic Weyl group W ( G , h )is the quotient of the normalizer in G of h by the centralizer in G of h .First we prove that in the semisimple case the converse of Lemma 4.3holds true. Lemma 5.2.
A compact quadric ˆ Q has property ( ˜ S ) if and only if p opp isconjugate to p by an inner automorphism of g . R QUADRICS WITH A SYMMETRY PROPERTY 9
Proof. If p opp is conjugate by an inner automorphism to p , we can choose aWeyl group element w with w · p = p opp . A representative γ of finite orderin G , which exists thanks to [16], satisfies the ( ˜ S ) property. (cid:3) Simple factors of the real type.
We show that, for simple Lie al-gebras of the real type, the ( ˜ S ) property always holds true. We recall thatthe analytic Weyl group of a real connected semisimple Lie group G withrespect to a real Cartan subalgebra h is the group: W ( G , h ) = N G ( h C ) / Z G ( h C ) . It is a subgroup of the usual Weyl group W ( g C , h C ). Lemma 5.3. If g is a simple algebra of the real types A III / IV , D Ib , D IIIb , E II / III and h is a maximally split Cartan subalgebra, then the longestelement w of the Weyl group W ( g C , h C ) is in the analytic Weyl group W ( G , h ) .Proof. First of all, we recall that, for Lie algebras of type D (cid:96) and (cid:96) even,the longest element w of the Weyl group is minus the identity, while in theother cases of the lemma, w is equal to minus the identity composed withthe root involution associated to the symmetry of the Dynkin diagram (see[4]).If g is of type A III / IV (cid:96) , then the roots β j = e j − e (cid:96) +2 − j , 1 ≤ j ≤ ( (cid:96) +1) / s β j are in theanalytic Weyl group. The longest element is w = Π j s β j .If g is of type D Ib (cid:96) with (cid:96) = 2 k + 1 odd, or of type D IIIb n , the rootse i − ± e i , for 1 ≤ i ≤ k , are either real or compact, hence the associatedsymmetries s e i − ± e i are in the analytic Weyl group, and their product isthe longest element w .If g is of type D Ib (cid:96) with (cid:96) = 2 k even, the roots e i − ± e i , for 1 ≤ i ≤ k ,are real, hence the associated symmetries s e i − ± e i are in the analytic Weylgroup, and furthermore also the symmetry s e k − − e k ◦ s e k − +e k belongs toit. Their product is the longest element w .If g is of type E II / III, the roots α + α + α + α + α and α + 2 α +2 α + 3 α + 2 α + α are real, and the roots α and α + α + α are eitherreal or compact, hence the associated symmetries are in the analytic Weylgroup, and their product is the longest element w . (cid:3) Proposition 5.4. If g is a simple algebra of the real types A III / IV , D Ib , D IIIb , E II / III , then there exists an element of finite order γ ∈ G suchthat Ad( γ )( E ) = − E .Proof. The longest element w of the Weyl group acts on h either by − Idor by − Id ◦ (cid:15) , where (cid:15) is the map induced by the nontrivial automorphismof the diagram. Since E is (cid:15) -invariant, w · E = − E .Finally, according to [16], there exists a representative γ of w in G , oforder 2 or 4. (cid:3) Simple factors of the complex type.
We consider now the casewhere g is a simple algebra of the complex types A C , D C , or E C . Lemma 5.5.
If a quadric ˆ Q admits an automorphism of finite order γ with Ad( γ )( E ) = − E , then there exists a maximally split Cartan subalgebra,containing E , self-conjugate, contained in p , and Ad( γ ) -invariant.Proof. Let Γ ⊂ Aut( g C ) be the group generated by Ad( γ ) and complexconjugation. It is a finite group, and it is the direct product of a cyclicgroup and Z / Z . The subalgebra g C is Γ-invariant. By [3] there exists aΓ-invariant Cartan subalgebra h C of g C . It contains E , because E is in thecenter. Since g C contains a Cartan subalgebra of g C , also h C is a Cartansubalgebra of g C . Finally, h = h C ∩ g is maximally split because there existsonly one conjugacy class of Cartan subalgebras. (cid:3) Fix an S -adapted Weyl chamber and system of simple positive roots.In this case the analytic Weyl group W ( G , h ) is exactly the Weyl group W ( g , h ), where g and h are considered as complex Lie algebras. Thus theconclusion of Lemma 5.3 is trivially true.We recall a result about conjugacy of parabolic subalgebras. Lemma 5.6.
Let g be a complex semisimple Lie algebra, b a Borel subal-gebra, and q , q (cid:48) ⊃ b two parabolic subalgebras. If q (cid:48) is Int( g ) -conjugate to q then q = q (cid:48) .Proof. Assume that w ∈ Int( g ) transforms q into q (cid:48) . Let b (cid:48) = w · b . Both b and b (cid:48) are Borel subalgebras of g contained in q (cid:48) . In particular they areBorel subalgebras of q (cid:48) , hence conjugated by an element u ∈ Int( q (cid:48) ), whichwe can lift to an element u (cid:48) ∈ Int( g ) that preserves q (cid:48) . Then u (cid:48) w · q = q (cid:48) and u (cid:48) w · b = b , and it follows u (cid:48) w · q = q . (cid:3) Conclusion.
From the results above it follows:
Theorem 5.7.
A quadric ˆ Q with a semisimple associated Levi-Tanaka al-gebra g has the ( ˜ S ) property if and only if the simple factors of g are all ofthe following real types: (1) A (cid:96) III / IV , Φ = { α i } , with ≤ i ≤ p and i (cid:54) = ( (cid:96) + 1) / or q ≤ i ≤ (cid:96) and i (cid:54) = ( (cid:96) + 1) / (cid:96) Ib / IIIb , Φ = { α (cid:96) } ( or Φ = { α (cid:96) − } );(3) E II / III , Φ = { α } ( or Φ = { α } ); or of the following complex types: (1 (cid:48) ) A C (cid:96) with Φ = { α j , α (cid:48) (cid:96) +1 − j } ;(2 (cid:48) ) D C (cid:96) with (cid:96) even and Φ = { α , α (cid:48) (cid:96) − } or Φ = { α (cid:96) − , α (cid:48) (cid:96) } ( or Φ = { α , α (cid:48) (cid:96) } or Φ = { α (cid:96) − , α (cid:48) } , Φ = { α (cid:96) , α (cid:48) (cid:96) − } or Φ = { α (cid:96) , α (cid:48) } );(3 (cid:48) ) D C (cid:96) with (cid:96) odd and Φ = { α (cid:96) , α (cid:48) (cid:96) − } ( or Φ = { α (cid:96) − , α (cid:48) (cid:96) } );(4 (cid:48) ) E C with Φ = { α , α (cid:48) } ( or Φ = { α , α (cid:48) } ) .Proof. For simple factors of the real type the statement is a consequence ofPropostion 5.4.For simple factores of the complex type, the types listed are exactly thosefor which p opp is conjugate to p for the action of w . By Lemma 5.2 theseare the algebras with the ( ˜ S ) property.Assume now that p opp is conjugate to p for the action of some element w ofthe analytic Weyl group. Then ( ww · p ) ∩ p contains some Borel subalgebra.From Lemma 5.6 it follows that ww · p = p that is p opp = w · p . (cid:3) R QUADRICS WITH A SYMMETRY PROPERTY 11
We will see later that for a semisimple g , the ( S ) property and the ( ˜ S )property are equivalent. 6. The general case
We drop now the hypothesis that the Levi-Tanaka algebra associated toa quadric ˆ Q is semisimple. Lemma 6.1.
If a quadric ˆ Q has property ( ˜ S ) , then there exists a Ad( γ ) -invariant graded Levi factor s of g , as described in § .Proof. Taft [14] proves that if Γ is a finite group of automorphisms of a realLie algebra g , and a ⊂ g is a Γ-invariant semisimple subalgebra, then thereexists a Γ-invariant Levi factor s and a Γ-fixed element X in the nilradical of g such that Ad(exp X )( a ) ⊂ s . Actually his proof is valid for any Γ-invariantsubalgebra a contained in some (non necessarily invariant) Levi factor. Itfollows that if a is a Γ-invariant subalgebra contained in some Levi factor,then there exists a Γ-invariant Levi factor s containing a .Let Γ be the group generated by Ad( γ ), and let a = C · E . By Proposi-tion 4.2, there exists a Levi factor containing a . It follows that there existsa Γ-invariant Levi factor s containing a . It is graded, because it contains E ,and it has a compatible complex structure on s − again by [13]. (cid:3) We fix then a Levi-Malcev decomposition g = s ⊕ r as in §
4. Since thegroup G is semi-algebraic, we also have a corresponding Levi decomposition G = SR (note that S ∩ R is discrete, but not necessarily trivial). Lemma 6.2. If p opp is Int( g ) -conjugate to p , then p opp ∩ s is Int( s ) -conjugateto p ∩ s .Proof. Decompose an element γ ∈ Int( g ) = G such that Ad( γ )( p ) = p opp as γ = γ S γ R . Then Ad( γ S )( p ∩ s ) = p opp ∩ s . (cid:3) The simple ideals of the Levi factor s belong to three families. Those ofkind 2 are Levi-Tanaka algebras and, by Lemma 4.1, there is at least one ofthem. Those of kind 1 are of the complex type and correspond to compacthermitian symmetric spaces. We can ignore for the moment those of kind 0. Theorem 6.3.
The quadric ˆ Q has property ( ˜ S ) if and only if E = E s andthe simple ideals of kind of a Levi factor are of the types described inTheorem , and the simple ideals of kind of a Levi factor are of thefollowing types: (1) A C (cid:96) with (cid:96) odd and Φ = { α ( (cid:96) +1) / } ; (2) D C (cid:96) with (cid:96) even and Φ = { α } or Φ = { α (cid:96) − } or Φ = { α (cid:96) } ; (3) D C (cid:96) with (cid:96) odd and Φ = { α } . (cid:3) Proof.
Indeed the same proof as in Theorem 5.7 applies to the Levi factor.The types listed are exactly those for which p opp is conjugate to p for theaction of w . The resulting element γ is still of finite order in G , because S is a finite covering of Int( s ). (cid:3) Recovering an involution
So far we have proved only the existence of a finite order inner automor-phism reversing the degree. Now we investigate the existence of an involutiveautomorphism with this property.We keep the notation of the previous section. Moreover, let ˆ S be theuniversal connected linear group with Lie algebra s , i.e. the set of realpoints of the simply connected group with Lie algebra s C . There is a naturalprojection π : ˆ S → S which is a finite covering map.We proceed in two steps. For simple Levi factors of kind 2, we look forelements γ, γ (cid:48) ∈ ˆ S , with the properties that: (i) Ad g ( γ )( E ) = Ad g ( γ (cid:48) )( E ) = − E , (ii) γ (cid:48) = e , (iii) γ ∈ Z (ˆ S ), and γ | V λ = ( − λ ( E ) for every irreduciblerepresentation V and weight λ . For simple Levi factors of kind 1, we providea general construction for such an element γ . In many cases the image of γ or γ (cid:48) in S , and hence in G , is the identity, and thus we obtain the ( S )property.We remark that in the following discussion the algebraic structure of theradical does not play any role, and we only consider it as a s -module.We introduce the following notation. If α is a root of s , then let s ( α ) bethe (complex) Lie subalgebra isomorphic to sl (2 , C ) containing g α and g − α ,and S ( α ) the corresponding analytic subgroup in ˆ S . Let ˜ s α be the image of (cid:0) (cid:1) in S ( α ). We have that Ad(˜ s α ) = s α , ˜ s α = 1, and ˜ s α | V λ = ( − ( α ∨ ,λ ) for any representation V and weight λ .7.1. Simple ideals of kind . Case A (cid:96) . In this case λ ( E ) ∈ Z for all weights λ . Denote by A k the k × k matrix with entries equal to 1 on the antidiagonal, and 0 elsewhere. Let γ ∈ S be the block matrix: γ = A [ (cid:96) ] BA [ (cid:96) ] with B = (1) , ( − , I , A depending on the class of (cid:96) modulo 4, in such away that det γ = 1. Then γ satisfies our hypotheses. Case D (cid:96) with (cid:96) = 2 k + 1 odd. In this case λ ( E ) ∈ Z for all weights λ . Let˜ w = Π ki =1 ˜ s e i − + e i ˜ s e i − − e i . Then, if { ω j } are the fundamental weights,˜ w | V ωj = (cid:40) Id if 1 ≤ j ≤ k − , ( − k Id if j = 2 k, k + 1 . If k is even, i.e. (cid:96) ≡ γ = γ (cid:48) = ˜ w satisfies γ = 1.If k is odd, i.e. (cid:96) ≡ {± α (cid:96) − , ± α (cid:96) − , ± α (cid:96) } to su (1 ,
3) or su (2 ,
2) or sl (4 , C ) and let h be the image of iId in the corresponding subgroup. Then ˜ w and h commute,and γ = γ (cid:48) = ˜ w h is the sought after element. R QUADRICS WITH A SYMMETRY PROPERTY 13
Case D (cid:96) Ib or D C (cid:96) with (cid:96) = 2 k even, Φ = { α (cid:96) − } or Φ = { α (cid:96) − , α (cid:48) (cid:96) } . Inthis case ω j ( E ) = j ∈ Z for the fundamental weights ω , . . . , ω (cid:96) − , and ω (cid:96) − ( E ) = ω (cid:96) ( E ) = ( (cid:96) − /
2. Let ˜ w = Π ki =1 ˜ s e i − + e i ˜ s e i − − e i . Then, if { ω j } are the fundamental weights,˜ w | V ωj = (cid:40) Id if 1 ≤ j ≤ k − , ( − k Id if j = 2 k − , k. If k is odd, i.e. (cid:96) ≡ γ = ˜ w satisfies γ | V λ = ( − λ ( E ) .Let I ∈ Spin ( (cid:96) − , (cid:96) + 1) be an element covering − Id ∈ SO ( (cid:96) − , (cid:96) + 1).Then γ (cid:48) = ( I · ˜ w ) satisfies γ (cid:48) = IdIf k is even, i.e. (cid:96) ≡ γ (cid:48) = ˜ w satisfies γ (cid:48) = Id. In generalhowever it is not possible to find an element γ with the required properties. Case D C (cid:96) with (cid:96) = 2 k even, Φ = { α , α (cid:48) (cid:96) − } . In this case ω i ( E ) ∈ Z for allfundamental weights ω i , and ω i ( E ) ∈ Z exactly for ω , ω , . . . , ω k − and for ω (cid:96) − (resp. ω (cid:96) ) if (cid:96) ≡ (cid:96) ≡ γ (cid:48) with γ (cid:48) = e satisfyingall conditions. In general however it is not possible to find an element γ with the required properties. Case E . In this case λ ( E ) ∈ Z for all weights λ . Let γ = γ (cid:48) = ˜ w = ˜ s α + α + α + α + α ˜ s α +2 α +2 α +3 α +2 α + α ˜ s α ˜ s α + α + α . Then γ = e Summarizing, we found an element γ (cid:48) for all simple factors of kind 2, andan element γ for all simple factors of kind 2 excepts some those of kind D (cid:96) with (cid:96) even.7.2. Simple ideals of kind . First we consider the existence of a suitableelement γ . The longest element w of the Weyl group can be written as aproduct of reflections w = Π s β i where { β i } is a maximal set of positive strongly orthogonal roots. Let { α j } ⊂{ β i } be the subset of roots of degree 1 (i.e. α j ( E ) = 1), and w = Π s α j , γ = Π˜ s α j .Since w ( E ) = − E , we have E = (cid:88) j α j ( E ) α j ( α j , α j ) = 12 (cid:88) j α j ( E ) α ∨ j = 12 (cid:88) j α ∨ j . Then γ | V λ = Π j ( − ( α ∨ j ,λ ) = ( − ( (cid:80) j α ∨ j ,λ ) = ( − λ ( E ) . We turn now to the problem of the existence of γ (cid:48) with γ (cid:48) = 1. For simpleideals of type D C (cid:96) or E C the element γ (cid:48) found in the previous subsection isa representative of the longest element of the Weyl group, thus satisfies allrequirements. For simple ideals of type A C (cid:96) with (cid:96) ≡ γ (cid:48) . For simple ideals of type A C (cid:96) with (cid:96) ≡ γ (cid:48) . Proposition 7.1.
Let ˆ Q be a quadric with the ( ˜ S ) property, and g theassociated Levi-Tanaka algebra, with Levi-Malcev decomposition g = s ⊕ r .If any of the following conditions is satisfied, then ˆ Q has the ( S ) property: (1) g is semisimple; (2) s does not contain any simple factor of kind and type A C (cid:96) with (cid:96) ≡ , (3) s does not contain any simple factor of kind and type D (cid:96) with (cid:96) even.Proof. In case (1) all the ideals of s are of kind 2, so case (1) is a subcase ofcase (2).For each simple ideal s i of s , let γ i , γ (cid:48) i be the images in S i of the elementsdescribed in the previous sections, if defined. For Levi factors of kind 0 welet γ i = γ (cid:48) i = e .In case (2) the elements γ i are defined for every simple factor s i , and welet γ = Π i γ i . In case (3) the elements γ (cid:48) i are defined for every simple factor s i , and we let γ = Π i γ (cid:48) i . In both cases the element γ ∈ G has order 2. (cid:3) Since compact quadrics have a semisimple group of automorphisms, wehave the following.
Corollary 7.2.
Every compact CR quadric has propoerty ( S ) . (cid:3) Remark 7.3.
If a quadric has property ( ˜ S ) the above construction showsthat it is anyway possible to find an appropriate automorphism γ with order2 or 4. Remark 7.4.
As the next example shows, the conditions in Proposition 7.1are not necessary. In fact we have no example of quadrics with the ( ˜ S )property but without the ( S ) property. Example 7.5.
Let s = o (8 , C ) ⊕ sl (2 , C ), with the grading and the CRstructure defined respectively by: E = diag(1 , , , , , − , − , − ⊕ diag(1 / , − / ,J = diag(0 , , , i , − i , , , ⊕ diag(i / , − i / . Let C and C denote the standard representations of o (8 , C ) and sl (2 , C ),respectively, and let V = C ⊗ C . The same elements E and J definea grading and CR structure on the semidirect product s ⊕ V . We finallydefine g = s ⊕ V ⊕ C T , where T is an element commuting with s andsuch that ad( T ) | V = Id. Then g is a Levi-Tanaka algebra associated to aquadric with the ( ˜ S ) property. It has the ( S ) property too, with the element γ = γ (cid:48) o (8 , C ) γ sl (2 , C ) exp(i πT / γ in S .8. An example
The CR dimension of a quadric ˆ Q is n = dim R g − /
2, while the CR-codimension is k = dim R g − , and hence dim R g − = 2 n + k . For quadricswith the ( S ) or ( ˜ S ) property, the dimension of g + = Ad( γ )( g − ) is 2 n + k . Itwas an open question whether also in the general case the dimension of g + can be estimated by 2 n + k (see, for example, [6, p.445]). A first negative R QUADRICS WITH A SYMMETRY PROPERTY 15 answer was given by P.B. Utkin in 2002 (see [17]). Here we give a newexample.
Example 8.1.
For n = 7 and k = 8, we consider the quadric ˆ Q = ˆ Q H associated to the hermitian form H parametrized by α, β, γ, δ ∈ C (cid:39) R : α γ δα β γ δ
00 ¯ β γ γ δ δ . Let s = sl (3 , C ) = (cid:76) i = − s i be endowed with the unique Levi-Tanaka struc-ture, given by the elements E s = diag(1 , , − , J s = diag( − i / , / , − i / . Let V = C be the space of the standard representation ρ of s and U , U two copies the adjoint representation of s . We assume on V the grading V = V − + V − + V given by the eigenspace decomposition of ( ρ ( E s ) − Id)and on V − the complex structure given by multiplication by the imaginaryunit J = iId.On U k , k = 1 ,
2, we put the grading U k = ⊕ i = − U ki induced by Ad( E s )and the complex structure induced by Ad( J s ).On h = s ⊕ V ⊕ U ⊕ U we have a natural Lie algebra structure, with V ⊕ U ⊕ U an abelian ideal and s acting through the standard or adjointrepresentation. Then h is a graded Lie algebra, with a complex structure on h − and it is fundamental, nondegenerate and transitive.Let W , W be two copies of the dual space V ∗ of V . The algebra s acts onthem via the contragradient representation − t ρ . We assume on W k , k = 1 ,
2a grading W k = W k + W k + W k given by the eigenspace decomposition of( − t ρ ( E s ) + Id). We define a product of elements of V and W k [ v, w ] := v ⊗ w with values in V ⊗ W k , which we identify with U k ⊕ C (cid:39) gl ( n, C ).Assuming W + W + U + U + C abelian, we obtain a graded Liealgebra a = s + V + W + W + U + U + C which is nondegenerate andfundamental. It is also pseudocomplex and transitive (see [9]). Its maximalpseudocomplex prolongation g = ⊕ − g i is finite dimensional with dim g ≥ dim a = 16 >
14 = dim g − and dim g ≥ dim a = 10 > g − . References [1] A. Altomani, C. Medori and M. Nacinovich,
The CR structure of minimal orbits incomplex flag manifolds , J. Lie Theory (2006), 483–530.[2] , On the topology of minimal orbits in complex flag manifolds , Tohoku Math.J. (2) (2008), 403–422.[3] A. Borel and G. D. Mostow, On semi-simple automorphisms of Lie algebras , Ann. ofMath. (1955), 389–405.[4] N. Bourbaki, ´El´ements de math´ematique. Fasc. XXXIV. Groupes et alg`ebres de Lie.Chapitre IV: Groupes de Coxeter et syst`emes de Tits. Chapitre V: Groupes engendr´es par des r´eflexions. Chapitre VI: syst`emes de racines , Actualit´es Scientifiques et In-dustrielles, No. 1337, Hermann (Paris), 1968.[5] V. V. Ezhov and A. V. Isaev, Canonical isomorphism of two Lie algebras arising inCR-geometry , Publ. Res. Inst. Math. Sci. (1999), 249–261.[6] V. V. Ezhov and G. Schmalz, Automorphisms of nondegenerate CR quadrics andSiegel domains. Explicit description , J. Geom. Anal. (2001), 441–467.[7] A. Isaev and W. Kaup, Regularization of Local CR-Automorphisms of Real-AnalyticCR-Manifolds , J. Geom. Anal., DOI 10.1007/s12220-010-9181-9.[8] W. Kaup,
CR-quadrics with a symmetry property , Manuscripta Math. (2010),505–517.[9] C. Medori and M. Nacinovich,
Levi-Tanaka algebras and homogeneous CR manifolds ,Compositio Math. (1997), 195–250.[10] ,
Classification of semisimple Levi-Tanaka algebras , Ann. Mat. Pura Appl. (4) (1998), 285–349.[11] ,
Complete nondegenerate locally standard CR manifolds , Math. Ann. (2000), 509–526.[12] ,
Maximally homogeneous nondegenerate CR manifolds , Adv. Geom. (2001),89–95.[13] , The Levi-Malcev theorem for graded CR Lie algebras , Recent advances in Lietheory (Vigo, 2000), Res. Exp. Math., vol. 25, Heldermann, Lemgo, 2002, pp. 341–346.[14] E. J. Taft,
Orthogonal conjugacies in associative and Lie algebras , Trans. Amer. Math.Soc. (1964), 18–29.[15] N. Tanaka,
On generalized graded Lie algebras and geometric structures. I , J. Math.Soc. Japan (1967), 215–254.[16] J. Tits, Normalisateurs de tores I. Groupes de Coxeter ´etendus , J. Algebra (1966),96–116.[17] P. B. Utkin, The dimension conjecture for quadrics of codimension three and higher.(Russian)
Mat. Zametki (2002), 152–156; translation in Math. Notes (2002),138–141. A. Altomani: Research Unit in Mathematics, University of Luxembourg, 6,rue Coudenhove-Kalergi, L-1359 Luxembourg
E-mail address : [email protected] C. Medori: Dipartimento di Matematica, Universit`a di Parma, Parco Areadelle Scienze 53/A, 43124 Parma (Italy)
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