Maximal dimension of groups of symmetries of homogeneous 2-nondegenerate CR structures of hypersurface type with a 1-dimensional Levi kernel
aa r X i v : . [ m a t h . C V ] F e b MAXIMAL DIMENSION OF GROUPS OF SYMMETRIES OFHOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES OFHYPERSURFACE TYPE WITH A 1-DIMENSIONAL LEVI KERNEL
DAVID SYKES AND IGOR ZELENKO
Abstract.
We prove that for every n ≥ the sharp upper bound for the dimension of the symme-try groups of homogeneous, 2-nondegenerate, (2 n + 1) -dimensional CR manifolds of hypersurfacetype with a -dimensional Levi kernel is equal to n + 7 . This supports Beloshapka’s conjecturestating that hypersurface models with a maximal finite dimensional group of symmetries for a givendimension of the underlying manifold are Levi nondegenerate. Introduction
A classical problem setting in differential geometry is to find homogeneous structures with thesymmetry group of maximal dimension among all geometric structure of a certain class. In Cauchy-Riemann (CR) geometry this problem is classically solved for the class of Levi nondegenerate CRstructures of hypersurface type of arbitrary dimension ([16, 2]). The present paper solves thisproblem for -nondegenerate CR structures of hypersurface type with a -dimensional Levi kernel.This class can be seen as the next one in a hierarchy of nondegeneracies to the class of Levinondegenerate CR structures of hypersurface type. Previously the answer to this problem wasgiven only in the -dimensional case [9, 11, 12], which is the case of the smallest possible dimensionin which -nondegenrate structures exist. We give the answer for arbitrary dimension (which a prioriis odd) greater than extending the previous result of [13] that worked under additional restrictionsof regularity of the CR symbol. This result supports Beloshapka’s conjecture [9, Conjecture 5.6]stating in that the hypersurface models with maximal finite dimensional group if symmetries for agiven dimension of the underlying manifold are Levi nondegenerate.In more detail, let M be a (2 n + 1) -dimensional homogeneous CR manifold with CR structure H of hypersurface type, meaning that H is an integrable, totally real, complex rank n distributioncontained in the complexified tangent bundle C T M of M , that is, [ H, H ] ⊂ H and H ∩ H = 0 where the overline in H denotes the natural complex conjugation in C T M . Homogeneity here refersto the property that for any two points x and x in M , there exists a symmetry of ( M, H ) sending x to x , that is, there exists a diffeomorphism ϕ : M → M such that ϕ ∗ ( H ) = H and ϕ ( x ) = x .Recall that the Levi form of the structure H is a field over M of Hermitian forms defined onfibers of H by the formula L ( X x , Y x ) := i (cid:2) X, Y (cid:3) x mod H x ⊕ H x ∀ X, Y ∈ Γ( H ) and x ∈ M. Here we are using the notation Γ( E ) to denote sections of a fiber bundle E . The kernel of the Leviform L is called the Levi kernel and will be denoted by K . CR-structures with K = 0 are calledLevi-nondegenerate. Mathematics Subject Classification.
Key words and phrases. -nondegenerate CR structures, homogeneous models, infinitesimal symmetry algebra,Tanaka prolongation, canonical forms in linear and multilinear algebra.I. Zelenko is supported by Simons Foundation Collaboration Grant for Mathematicians 524213. For a Levi-nondegenerate structure, if the Levi form has signature ( p, q ) with p + q = n then amaximally symmetric model can be obtained as a real hypersurface in the complex projective space CP n +1 , obtained by the complex projectivization of the cone of nonzero vectors in C n +2 that areisotropic with respect to a Hermitian form of signature ( p + 1 , q + 1) , and the algebra of infinitesimalsymmetries of this model is isomorphic to su ( p + 1 , q + 1) , having dimension ( n + 2) − .In the present paper we assume that the fiber K x of the Levi kernel is -dimensional at everypoint x ∈ M , that is, K is a rank distribution, and that the following nondegeneracy conditionholds: If for v ∈ K x and y ∈ H x /K x , we take V ∈ Γ( K ) and Y ∈ Γ( H ) such that V ( p ) = v and Y ( p ) ≡ y mod K , and define a linear mapad v : H x /K x → H x /K x y [ V, Y ] | x mod K x ⊕ H x , and similarly define a linear map ad v : H x /K x → H x /K x for v ∈ K x (or simply take complexconjugates), then there is no nonzero v ∈ K x (equivalently, no nonzero v ∈ K x ) such that ad v = 0 .A CR-structure is called -nondegenerate if this last condition holds.The term comes from the more general notion of k-nondegeneracy , see, forexample, [7] for the generalization of this definition to arbitrary k ≥ and arbitrary dimension ofLevi kernels via the Freeman sequence under analogous constant rank assumptions, [1, chapter XI]for more general definition without the assumption that K is a distribution, and [10, Appendix] forthe equivalence of the definitions in [7] and [1, chapter XI] under the constant rank assumptions.The focus of the present paper is on finding the sharp upper bound for the dimension of theLie group Aut ( M, H ) of symmetries of homogeneous 2-nondegenerate CR structures ( M, H ) ofhypersurface type with a 1-dimensional Levi kernel. As shown in [9, 11, 12] for the lowest dimensionalcase, i.e. when dim M = 5 , this sharp upper bound (even without homogeneity assumption) is equalto and for the maximally symmetric model the algebra if infinitesimal symmetries is isomorphicto is equal to so (3 , . The main result here, see Theorem 2.3 below, gives this sharp upper boundexpressed as a function of dim M ≥ (equivalently, n = (dim M − ≥ ), namely dim Aut ( M, H ) ≤
14 (dim M − + 7 = n + 7 . (1.1)We also show that symmetries of ( M, H ) are all determined by their third weighted jet. By the weighted jet we mean that the derivatives in various directions are calculated according to thefiltration ( K ⊕ K ) ∩ T M ⊂ ( H ⊕ H ) ∩ T M ⊂ T M of T M so that each derivative in a direction in ( K ⊕ K ) ∩ T M is assigned weight zero, each derivativein a direction in (cid:16) ( H ⊕ H ) \ ( K ⊕ K ) (cid:17) ∩ T M is assigned weight , and each derivative in a directionin T M \ H ⊕ H is assigned weight . These results (even without assumption of homogeneity)were previously obtained in [13] for the special class of CR structures whose symbols are known as regular , wherein it was shown by example that the upper bound in (1.1) is achieved.The essential technical bulk of this paper consists of showing that the dimension of Aut ( M, H ) for homogeneous structures with non-regular symbol is strictly less than the right side of (1.1) (infact it is shown in Theorem 3.8 below that it is strictly less than ( n − + 7 ) and that in thenon-regular case symmetries of ( M, H ) are all determined by their first weighted jet. The notion ofCR symbols and their regularity is explained in Section 2. Note that, for the considered case n ≥ ,the previously treated regular symbols constitute only a finite subset in the space of all CR symbolsfor each n , which itself depends on continuous parameters.In the proof of the bound (1.1) we use two main results from our previous papers [14] and [15]:the classification of CR symbols [14] and the description of the upper bound for the dimension ofsymmetry groups in terms of a Tanaka prolongation of the symbol or its reduced version [15]. In AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 3 the sequel, we calculate these prolongations and their dimensions for each reduced modified symbolcorresponding to a non-regular CR symbol. In particular, we show (Theorem 3.8) that the firstTanaka prolongation of each reduced modified symbol corresponding to a non-regular CR symbolis equal to zero and we find the upper bound for the dimension of its (entire) Tanaka prolongaiton.Analogous analysis for regular CR symbols was previously obtained in [13] with the help of thetheory of biagraded Tanaka prolongation. The result on the j th-jet determinacy follows from itsequivalence to the vanishing of the j th Tanaka prolongation. In Theorem 4.4 for each reducedmodified symbol corresponding to a non-regular CR symbol we give more precise upper bound forthe dimension of its (entire) Tanaka prolongaiton in terms of the parameters of this nonregularsymbol.Note that at this moment for structures with non-regular symbols (and therefore in the generalcase) we are not able to remove completely the homogeneity assumption in our results, as thisassumption implies that the modified (and reduced modified) symbols introduced in our previouswork [15] (see also Section 3 below for definition of these objects) are constant and therefore are Liealgebras, and we strongly use the latter fact. So, the assumption of homogeneity can be relaxed tothe assumption that the structures under consideration admit a constant reduced modified symbolin the sense of [15], but the question of whether or not there exist CR structures from the consideredclass with nontransitive symmetry group of dimension higher than the bound in (1.1) is still open,although the positive answer to this question is highly unlikely.In the very recent paper [3] it was shown that for dim M = 7 , without the homogeneoty as-sumption, the upper bound for the dimension of the group of symmetries of -nondegenerate CRstructures of hypersurface type with a 1-dimensional Levi kernel is . Our sharp bound (1.1)for homogeneous case is and an example of the structure from the considered class with 17-dimensional symmetry group is unknown. The result of the present paper (communicated in aprivate correspondence) was in fact used in [3] to reduce the bound from , obtained initially bythe methods of normal forms, to , see Proposition 16 there.In contrast to the case of dim M = 5 , in the case where M is of (odd) dimension greater than orequal to , the infinitesimal symmetry algebras of the maximally symmetric homogeneous modelsare not semisimple. These algebras were calculated in some form in [13, Subsection 5.3]. A morevisual description together with a hypersurface realizations of these models will feature in futurejoint work [5].In the case where dim M = 7 the infinitesimal symmetry algebra of the maximally symmetricmodels is isomorphic to one of the real forms of the following complex Lie algebra: Let s = C ⊕ sl (2 , C ) ⊕ sl (2 , C ) . The complexification of our algebra of interest is isomorphic to the naturalsemidirect sum of s and the -dimensional abelian Lie algebra C ∼ = C ⊗ C so that the first sl (2 , C ) component in s acts irreducibly on the first factor C in C ⊗ C , the second component sl (2 , C ) in s acts irreducibly on the second factor of C in C ⊗ C , and the component C in s acts just by rescaling. The desired real Lie algebra is the natural semidirect sum of the conformalLorenzian algebra co (3 , and the -dimensional real abelian Lie algebra R , where co (3 , actsirreducibly on R . This unique irreducible action is naturally induced from the standard actionof co (3 , on the Minkowski space, if one identifies R with the space of the traceless symmetricbilinear forms on the Minkowski space.Finally, for completeness, we offer without proof the (local) hypersurface realizations of themaximally symmetric homogeneous models in the considered class (the details will be given in [5]).If, as before, n = (dim M − , and the signature of the form obtained by the reduction of the Leviform at each point x to the space H x /K x is equal to ( p, q ) with p + q = n − , then in coordinates ( z , . . . , z n , w ) for C n +1 these are the hypersurfaces are given by the equation(1.2) Im( w + z ¯ z n ) = z ¯ z + ¯ z z + n − X i =3 ε i z i ¯ z i , DAVID SYKES AND IGOR ZELENKO where ε i ∈ {− , } and { ε i } n − i =3 consists of p − terms equal to and q − terms equal to − (notethat, for dim M = 7 , the last sum in the right side of (1.2) disappears).2. CR symbols and the main results
Our analysis branches depending on properties of the CR structure’s local invariants. A basiclocal invariant of a hypersurface-type CR structure called the
CR symbol is introduced in [13]. TheCR symbol of H (at a point x in M ) is a bigraded vector space(2.1) g := g − , ⊕ g − , − ⊕ g − , ⊕ g , − ⊕ g , ⊕ g , with involution ¯ whose bigraded components g i,j are defined as follows. For the structures consid-ered here, ultimately our definitions of g i,j will not depend on the point x because we are assumingthat ( M, H ) is homogeneous, but we still fix x to state the initial definitions. We let ℓ denotethe reduced Levi form , which is the field of nondegenerate Hermitian forms defined on fibers of thequotient bundle H/K by ℓ ( X x + K x ) := L ( X x ) . We define the coset spaces g − , := C T x M/H x , g − , − := H x /K x , and g − , := H x /K x . The space(2.2) g − := g − , ⊕ g − , − ⊕ g − , inherits a Heisenberg algebra structure with nontrivial Lie brackets defined in terms of the reducedLevi form by [ v, w ] := iℓ ( v, w ) ∀ v ∈ g − , , w ∈ g − , − . Note that ℓ formally takes values in g − , . By identifying g − , and C , we regard ℓ as a C -valuedHermitian form, but, since this identification is not naturally determined by the CR structure, inthe sequel we consider the real line R ℓ of C -valued Hermitian forms spanned by ℓ . While the one C -valued form ℓ is not an invariant of the CR structure, the line R ℓ is.To define g , , we consider special operators associated with vectors in K x . For a vector v in K x ,define the antilinear operator A v : g − , → g − , by A v ( x ) := ad v ( x ) . The dependence of A v on v is linear, that is, A λv = λA v ∀ λ ∈ C , so if the rank of K is equal to 1 then there exists an antilinear operator A such that { A v | v ∈ K x } = C A. The fact that H is -nondegenerate implies that A = 0 .The reduced Levi form ℓ naturally extends to define a symplectic form on the space g − := g − , − ⊕ g − , via a standard construction from the study of Heisenberg algebras. Hence g − inherets a symplectic structure from the CR structure with respect to which we obtain the conformalsymplectic algebra csp ( g − ) defined in the standard way. We define g , to be the subspace of csp ( g − ) given by the formula g , := ϕ : g − → g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ( v ) = 0 ∀ v ∈ g − , andthere exists λ ∈ C such that ϕ ( v ) = λA ( v ) ∀ v ∈ g − , − . The natural complex conjugation on C T x M induces an antilinear involution v v on g − , whichin turn induces an antilinear involution on csp ( g − ) by the rule(2.3) ϕ ( v ) := ϕ ( v ) . AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 5
Using this involution, we define g , − := { ϕ | ϕ ∈ g , } . Lastly, using the standard Lie brackets of csp ( g − ) we define(2.4) g , := (cid:8) v ∈ csp ( g − ) (cid:12)(cid:12) [ v, g ,i ] ⊂ g ,i ∀ i ∈ {− , } (cid:9) , which completes our definition of the CR symbol g of H (at the point x ). Note that by construction(2.5) [ g i ,j , g i ,j ] ⊂ g i + i ,j + j , i ≤ , { ( i , j ) , ( i , j ) } 6 = { (0 , , (0 , − } . Conversely a vector space g as in (2.1) with g − as in (2.2) being the Heisenberg algebra is called an abstract CR symbol for -nondegenerate, hypersurface-type CR structures if it satisfies (2.5), g , isthe maximal subalgebra of csp ( g − ) satisfying (2.4), and it is endowed with an antilinear involution ¯ satisfying (2.3). Remark 2.1.
The CR symbol g of a CR structure with a -dimensional kernel encodes and isencoded by the pair ( R ℓ, C A ) . Note that an abstract CR symbol g is not necessarily a Lie algebra, as the bigrading conditions in(2.5) are only applied for { ( i , j ) , ( i , j ) } 6 = { (0 , , (0 , − } , so that [ g , − , g , ] does not necessarilybelong to g , and therefore does not necessarily belong to g . Following the terminology of [13], wesay that a CR symbol is regular if it is a subalgebra of g − ⋊ csp ( g − ) and non-regular otherwise. Asshown in [13, Lemma 4.2], the symbol g of a CR structure with a -dimensional kernel correspondingto the pair ( R ℓ, C A ) is regular if and only if(2.6) A ∈ C A. To any abstract regular CR symbol g , we construct a corresponding special homogeneous CRstructure as follows. Denote by G and G , connected Lie groups with Lie algebras g and g , ,respectively, such that G , ⊂ G , and denote by ℜ G and ℜ G , the corresponding real parts withrespect to the involution on g , meaning that ℜ G and ℜ G , are the maximal subgroups of G and G , whose tangent spaces belong to the left translations of the fixed point set of the involution on g on G .Let M C = G /G , and M = ℜ G / ℜ G , . In both cases here we use left cosets. Let b D flat i,j be theleft-invariant distribution on G such that it is equal to g i,j at the identity. Since all g i,j are invariantunder the adjoint action of G , , the push-forward of each b D flat i,j to M C is a well defined distribution,which we denote by D flat i,j . Let D flat − be the distribution that is the sum of D flat i,j with i = − . Werestrict all of these distributions to M , considering them as subbundles of the complexified tangentbundle of M . The distribution H flat := D flat − , ⊕ D flat0 , defines a CR structure of hypersurface typeon M called the flat CR structure with constant CR symbol g .As a consequence of [13], see Theorems 3.2, 5.1, 5.3 and the last paragraph of section 5 there,one gets Theorem 2.2 (Porter and Zelenko [13]) . If ( M, H ) is a 2-nondegenerate CR structure of hypersur-face type with a 1-dimensional Levi kernel and constant regular symbol, then(1) the dimension of the algebra of infinitesimal symmetries of ( M, H ) is not greater than (dim M − + 7 ;(2) these symmetries are determined by their third weighted jet;(3) the dimension of the algebra of infinitesimal symmetries of ( M, H ) is equal to (dim M − + 7 if and only if ( M, H ) is locally equivalent to the flat structure with CR symbol suchthat the corresponding line of antilinear operators consists of nilpotent ones of rank 1. A natural question is whether or not the assumption of regularity of symbol can be removed in theprevious theorem. Addressing this question, the main result of the present paper is the following.
DAVID SYKES AND IGOR ZELENKO
Theorem 2.3. If ( M, H ) is a 2-nondegenerate homogeneous CR structure of hypersurface type witha 1-dimensional Levi kernel and constant symbol (not necessarily regular), then(1) statements (1) and (3) of Theorem 2.2 are valid;(2) if the symbol is non-regular then the (infinitesimal) symmetries of ( M, H ) are determinedby their first weighted jet. The proof of this theorem is given in Sections 3 through 5 and the appendix. Also, a generalizationof this theorem is described in Remark 3.9 below. In Section 3 we give the scheme of the proof of thistheorem, based on the constructions and results of our previous paper [15], namely the constructionof reduced modified symbols for sufficiently symmetric CR structures and the application of Tanakaprolongation of these reduced modified symbols to obtain an upper bound for the dimension oftheir infinitesimal symmetry algebras (see Theorem 3.7 below). In this way Theorem 2.3 will beessentially reduced to Theorem 3.8. The latter theorem is proved in Section 5 with the help of theappendix (Section 6). In this proof we also use the classification of symbols from our previous paper[14] and the system of matrix equations for the reduced modified symbols derived in [15, section 5].The latter two topics are briefly reviewed in Section 4 below.3.
Reduced modified symbol and the significance of its Tanaka prolongation
Now we will discuss the scheme of the proof of Theorem 2.3, based on the constructions and resultsof our previous paper [15]. In particular, there we introduced other local invariants of sufficientlysymmetric hypersurface-type CR structures encoded in objects called modified CR symbols and reduced modified CR symbols (see sections 4 and 6 of [15], respectively). Although modified andreduced modified CR symbols are defined in [15], we outline their definitions here for completenessbecause these objects (especially the latter one) are both nonstandard and fundamental for thepresent study. Some technical details that are not essential for understanding the principal conceptsare omitted here and we refer to [15] for those gaps.Let g be the CR symbol of H . Recall that g was actually defined in terms of a point x in M , solet us write g ( x ) to emphasize the point in M with respect to which this CR symbol was originallydefined. We also write g i,j ( x ) to denote the bigraded components of g ( x ) . Our homogeneityassumption on ( M, H ) implies that there exists a reference CR symbol that we will now refer to as g that is isomorphic g ( x ) for all x in M .There is a natural way to locally complexify M by working in local coordinates and replacingreal coordinates with complex ones, and, moreover, the CR structure H , as well as the distributions H , K , and K , naturally extend to this complexified manifold (see [15] for full details) yielding aso-called complexified CR manifold that we denote by C M (a detail omitted here is that, since theconstruction is local, this may only be well defined after replacing M with some neighborhood in M ). Note that dim R ( C M ) = 2 dim( M ) and there is a submanifold in C M that can be naturallyidentified with M . The distribution K + K on C M is involutive. We let N be the leaf space of thefoliation of C M generated by K + K , sometimes called the Levi leaf space , and let π : C M → N denote the natural projection. That is, points in N are maximal integral submanifolds of K + K in C M .From the resulting construction, g ( x ) remains well defined (in terms of H ) for all x in C M . Wedefine the fiber bundle pr : P → C M whose fiber pr − ( x ) over a point x in C M is comprised ofwhat we call adapted frames , that is, pr − ( x ) = ϕ : g − → g − ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ( g i,j ) = g i,j ( x ) ∀ ( i, j ) ∈ { ( − , ± , ( − , } , ϕ − ◦ g , ± ( x ) ◦ ϕ = g , ± , and ϕ ([ y , y ]) = [ ϕ ( y ) , ϕ ( y )] ∀ y , y ∈ g − . We also consider a second fiber bundle π ◦ pr : P → N , a bundle with total space P and basespace N . AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 7
For any ψ ∈ P and γ = π ◦ pr( ψ ) , the tangent space of the fiber ( P ) γ = ( π ◦ pr) − ( γ ) of thesecond bundle at ψ can be identified with a subspace of csp ( g − ) by the map θ : T ψ ( P ) γ → csp ( g − ) given by θ (cid:0) ψ ′ (0) (cid:1) := (cid:0) ψ (0) (cid:1) − ψ ′ (0) where ψ : ( − ǫ, ǫ ) → ( P ) γ denotes an arbitrary curve in ( P ) γ with ψ (0) = ψ . The notation θ isused here to match the notation in [15]. Let g mod0 ( ψ ) := θ ( T ψ ( P ) γ ) . Definition 3.1.
The space g , mod ( ψ ) := g − ⊕ g mod0 ( ψ ) is called the modified CR symbol of the CRstructure H at the point ψ ∈ P . Remark 3.2.
Modified CR symbols depend on points in the bundle P rather than points in theoriginal CR manifold. Accordingly, a modified CR symbol is not itself a local invariant of the CRstrucuture from which it arises, but rather, for x ∈ M , the set { g , mod ( ψ ) | pr( ψ ) = x } is a localinvariant at x . This invariant encodes more data than is encoded in the corresponding CR symbol. Remark 3.3.
Definition 3.1 can be made without assuming that ( M, H ) is homogeneous, and insteadassuming only that the CR symbols g ( x ) are constant on M . We consider the map ψ ϕ ( ψ ) sending each point in P to a subspace of csp ( g − ) . If, for somesubspace e g ⊂ csp ( g − ) , there is a maximal connected submanifold f P of P belonging to the levelset (cid:8) ψ ∈ P (cid:12)(cid:12) θ (cid:0) T ψ P (cid:1) = e g (cid:9) such that π ( f P ) = C M , then we call f P a reduction of P . After, replacing P and θ with f P and the restriction of θ to the tangent bundle of f P , we can repeat this reduction procedureby finding a maximal connected submanifold of P that is in the level set of the new mapping ψ θ ( T ψ f P ) also covering C M under the projection π , which we again call a reduction of P .In general this reduction procedure can be repeated many times, and eventually terminates in thesense that iterating the reduction procedure again will not yield new reductions. For a reduction P , red of P we label the corresponding space g red0 ( ψ ) := θ (cid:16) T ψ P , red (cid:17) ∀ ψ ∈ π − ( x ) ∩ θ (cid:16) T ψ P , red (cid:17) . (3.1) Definition 3.4. If P , red is a reduction of P then the space g , red ( ψ ) := g − ⊕ g red0 ( ψ ) , with g red0 ( ψ ) given by (3.1) , is called a reduced modified CR symbol of the CR structure H at ψ . We call g , red a constant reduced modified CR symbol of H if there exists a reduction P , red of P together with g red0 ( ψ ) given by (3.1) such that g , red = g , red ( ψ ) ∀ ψ ∈ P , red . Lemma 3.5. If ( M, H ) is homogeneous then it has a constant reduced modified symbol, that is,there exists a reduction P , red of P such that the map ψ g red0 ( ψ ) given by (3.1) is constant.Proof. Since ( M, H ) is homogeneous, so is P , and hence each reduction f P of P can be takenso that its fibers (cid:16) f P (cid:17) x := n ψ ∈ f P | π ( ψ ) = x o have the same image under the mapping ψ θ (cid:16) T ψ f P (cid:17) . Therefore, if ψ θ (cid:16) T ψ f P (cid:17) is not already constant on f P then we can repeat thereduction procedure to find a proper submanifold of f P that is also a reduction of P . Eventually,this iterated procedure ends with a reduction for which either the image of θ applied to its tangentspaces is constant, or a its fibers are 0-dimensional. But, in the latter case, using homogeneity, wecan take this final reduction P , red such that its fibers have the same image under the mapping DAVID SYKES AND IGOR ZELENKO ψ θ (cid:0) T ψ P , red (cid:1) . Accordingly, ψ θ (cid:0) T ψ P , red (cid:1) would be constant on P , red because it isconstant on fibers and the fibers are singletons. (cid:3) For the remainder of this paper, we let g , red denote a constant reduced modified CR symbolof H . Like the CR symbol of H , g , red is also a graded subspace of g − ⋊ csp ( g − ) . It has thedecomposition g , red = g − , ⊕ g − , − ⊕ g − , ⊕ g red0 where the components whose first weight isnegative coincide with those of the CR symbol. Here we state some of the properties of g red0 . Forthis we consider weighted components of csp ( g − ) defined by (cid:0) csp ( g − ) (cid:1) ,i = (cid:8) ϕ ∈ csp ( g − ) (cid:12)(cid:12) ϕ ( g − ,j ) ⊂ g − ,i + j ∀ j ∈ {− , } (cid:9) . The space g red0 is a subspace of csp ( g − ) with a decomposition g red0 = g red0 , ⊕ g red0 , − ⊕ g red0 , + (3.2)such that(1) g red0 , ⊂ g , ;(2) g red0 , + = g red0 , − ;(3) the natural projection of csp ( g − ) onto (cid:0) csp ( g − ) (cid:1) , defines an isomorphism between g red0 , + and g , ;(4) The subspace g red0 is invariant with respect to the involution on csp ( g − ) (5) The subspace g red0 is a subalgebra of csp ( g − ) .We stress that the decomposition g red0 = g red0 , ⊕ g red0 , − ⊕ g red0 , + satisfying these properties is not unique,and, furthermore, no such splitting is naturally determined by the CR structure. Remark 3.6.
The CR symbol g of ( M, H ) is determined by any of its modified CR symbols, whichin turn are all determined by the constant reduced modified CR symbol g , red . The underlying theory that we will apply to treat structures with non-regular CR symbols isdeveloped in [15], wherein it is shown that the upper bounds that we wish to compute can be foundby computing the universal Tanaka prolongation [17] of g , red , which is defined as follows. Startingwith k = 1 and setting g − = g − , , we recursively define the vector spaces g red k := ( ϕ ∈ − M i = − Hom( g i , g i + k ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ([ v , v ]) = [ ϕ ( v ) , v ] + [ v , ϕ ( v )] ∀ v , v ∈ g − ) ∀ k ≥ , The universal Tanaka prolongation of g , red is the vector space u ( g , red ) := g − ⊕ M k ≥ g red k . Theorem 3.7 (follows immediately from [15, Corollary 2.8 and Theorem 6.2]) . If ( M, H ) is a2-nondegenerate CR structure of hypersurface type with a 1-dimensional Levi kernel and constantreduced modified symbol g , red , then the dimension of the algebra of infinitesimal symmetries of ( M, H ) is not greater than dim u ( g , red ) . Hence, if we can explicitly calculate dim u ( g , red ) for non-regular CR symbols, then we can obtainan upper bound for the algebra of infinitesimal symmetries of ( M, H ) . This motivates the followingtheorem, proved in Section 5. Theorem 3.8.
If a reduced modified CR symbol g , red corresponds to a non-regular CR symbol thenthe following statements hold:(1) The first Tanaka prolongation g red1 of g , red vanishes or, equivalently, the universal Tanakaprolongation u ( g , red ) of g , red is equal to g , red . AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 9 (2) dim g , red and therefore the dimension of the algebra of infinitesimal symmetries of a ho-mogeneous, -nondegenerate, (2 n + 1) -dimensional, CR structure of hypersurface type withrank Levi kernel and non-regular CR symbol is strictly less than ( n − + 7 . Theorem 3.8 is proved in Section 5 with the help of the appendix (Section 6). Based on the well-known fact [17, Section 6] that an infinitesimal symmetry of a filtered structure is determined by the j th weighted jet, where j is the minimal nonnegative integer for which the j th Tanaka prolongationis equal to zero, this theorem immediately implies item (2) of Theorem 2.3. Item (1) will followfrom combining the last theorem with Theorem 3.7. In Theorem 4.4, for each reduced modifiedsymbol corresponding to a non-regular CR symbol, we give more precise upper bounds (than theones in item (2) of Theorem 3.8) for the dimension of its (entire) Tanaka prolongation in terms ofthe parameters of this non-regular symbol. Remark 3.9.
Finally note that the arguments of the previous paragraph imply that homogeneity as-sumption in Theorem 3.8 and our main Theorem 2.3 can be relaxed to the assumption that structuresunder consideration admit a constant reduced modified symbol. Matrix representations of CR and reduced modified CR symbols
Throughout this section, we work with a fixed CR symbol given by the pair ( R ℓ, C A ) , where ℓ is an Hermitian form and A is a self-adjoint antilinear operator on g − , . let us fix a basis of g − .This basis can be fixed such that the pair ( ℓ, A ) is represented with respect to it by matrices in acanonical form, which is shown in [14]. We recall one such canonical form below in Theorem 4.1(there are actually two canonical forms given in [14]).For λ ∈ C and a positive integer m , let J λ,m denote the m × m Jordan matrix with a singleeigenvalue λ and this eigenvalue has geometric multiplicity 1; let T m = J ,m , and let S m be the m × m matrix whose ( i, j ) entry is 1 if j + i = m + 1 and zero otherwise, that is(4.1) J λ,m := m columns z }| { λ · · · . . . . . . . . . ...... . . . . . . . . . ... . . . . . . · · · · · · λ m rows and S m = m columns z }| { · · · ... . . . . . . . . . . . . ... · · · m rows . In the sequel, given square matrices D , . . . D N we will denote by D ⊕ . . . ⊕ D N the block diagonalmatrix with diagonal blocks D , . . . , D N in the order from the top left to the bottom right and alloff-diagonal block equal to zero.For λ ∈ C , we define the k × k or k × k matrix M λ,k by M λ,k := J λ,k if λ ∈ R J λ ,k I ! otherwise , (4.2)where denotes a matrix of appropriate size with zero in all entries and I denotes the identitymatrix. We define corresponding matrices N λ,k by(4.3) N λ,k := ( S k if λ ∈ R S k otherwise . For the ℓ -self-adjoint antilinear operator A referred to in the following theorem, let us enumeratethe eigenvalues of A (counting them with multiplicity) that are contained in the upper-half plane { z ∈ C | ℜ ( Z ) ≥ } of C , labeling them as λ , . . . , λ γ . Furthermore, we take each λ i to be theprinciple square root of λ i . Theorem 4.1 (immediate consequence of the main result in [14]) . Given a nondegenerate Hermitianform ℓ on a vector space V and an ℓ -self-adjoing antilinear operator A , there exists a basis of V with respect to which ℓ and A are respectively represented by the matrices H ℓ and C given by H ℓ = γ M i =1 ǫ i N λ i ,m i and C = γ M i =1 M λ i ,m i , (4.4) for some sequence ǫ , . . . , ǫ γ satisfying ǫ i = ± and some sequence of positive integers m , . . . , m γ . Letting H ℓ and C be matrices representing ℓ and A respectively in some basis of g − , we considerthe Lie algebras of square matrices α satisfying αCH − ℓ + CH − ℓ α T = ηCH − ℓ for some η ∈ C and respectively α T H ℓ C + H ℓ Cα = ηH ℓ C for some η ∈ C , and define the algebra A to be their intersection, that is, A := (cid:26) α (cid:12)(cid:12)(cid:12)(cid:12) αCH − ℓ + CH − ℓ α T = ηCH − ℓ and α T H ℓ C + H ℓ Cα = η ′ H ℓ C for some η, η ′ ∈ C (cid:27) . (4.5)Let us fix a splitting of g red0 as given in (3.2). With respect to the basis of g − fixed above, thereexists some ( n − × ( n − matrix Ω such that g red0 , + and g red0 , − have the matrix representations(4.6) g red0 , + = span C (cid:26)(cid:18) Ω C − H − ℓ Ω T H ℓ (cid:19)(cid:27) and g red0 , − = span C (cid:26)(cid:18) − H ℓ − Ω ∗ H ℓ C Ω (cid:19)(cid:27) . In [15], we show that g red0 is a subalgebra of csp ( g − ) and establish the following lemma. Lemma 4.2 ([15, Proposition 5.4]) . There exists a subalgebra A of A invariant under the trans-formation α H ℓ − α ∗ H ℓ such that g red0 , = (cid:26) (cid:18) α − H − ℓ α T H ℓ (cid:19) + cI (cid:12)(cid:12)(cid:12)(cid:12) α ∈ A , and c ∈ C (cid:27) , (4.7) and there exist coefficients { η α } α ∈ A ⊂ C and µ ∈ C such that the system of relationsi) αCH − ℓ + CH − ℓ α T = η α CH − ℓ ii) [ α, Ω] − η α Ω ∈ A iii) Ω T H ℓ C + H ℓ C Ω = µH ℓ C iv) h H ℓ − Ω ∗ H ℓ , Ω i + CC − µ Ω − µH ℓ − Ω ∗ H ℓ ∈ A (4.8) holds for all α ∈ A . We have the following basic lemma.
Lemma 4.3 ([15, Proposition 3.6]) . The following are equivalent.(1) g is regular.(2) CCC is a scalar multiple of C .Moreover, if Ω is in A then g is regular. AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 11
Proof.
Equivalence of (1) and (2) follows from (2.6). The latter statement is also shown in [15],although we prove it more directly here because it is not given as a numbered result there. For this,let v + and v − be elements in g red0 , and g red0 , − respectively. Note that if Ω is in A then there existvectors w + , w − ∈ g , such that v ± + w ± belongs to g , ± . Accordingly, [ v + + w + , v − + w − ] = [ w − , w + ] + [ v + + w + , w − ] + [ w + , v − + w − ] + [ v + , v − ] . Since [ g , , g ] ⊂ g (4.9)by the definition of g , , the first three terms in the right side of this last equation belong to g . Since g red0 is closed under Lie brackets, [ v + , v − ] belongs to g red0 . Hence if Ω is in A then [ g , , g , − ] ⊂ g + g red0 . On the other hand if Ω is in A then g red0 ⊂ g . Therefore, if Ω is in A then [ g , , g , − ] ⊂ g . Noting (4.9), it follows that if Ω is in A then [ g , g ] ⊂ g , that is, g isregular. (cid:3) Now, for completeness, given a non-regular CR symbol g encoded by the pair ( ℓ, A ) , representedby the pair of matrices ( H ℓ , C ) in the canonical basis as in Theorem 4.1 we will give more precise(i.e., in terms of integers m , . . . m γ and numbers λ , . . . , λ γ ) upper bound for the dimension ofthe algebra of infinitesimal symmetries of a -nondegenerate (2 n + 1) -dimensional CR structure ofhypersurface type with -dimensional Levi kernel admitting a constant reduced modified symbolcorresponding to CR symbol g . For this, for every ≤ i, j ≤ γ , let d ( i, j ) = , ( λ i = λ j ) or ( i = j and λ i is not a nonpositive real number) min { m i , m j } ( i = j and λ i = λ j > or ( i = j and λ i < { m i , m j } i = j, λ i = λ j and ( λ i / ∈ R or λ i = 0)4 min { m i , m j } i = j, λ i = λ j and λ i < (cid:6) m i (cid:7) i = j and λ i = 0 where ⌈ m ⌉ denotes the ceiling function, i.e. the smallest integer not less than m i .Let d total := X i ≤ j d ( i, j ) . Then the following theorem is the direct consequence of item (1) of Theorem 3.8 and Lemmas6.2, 6.4, Corollary 6.5, and Lemma 6.8:
Theorem 4.4.
Given a nonregular CR symbol g encoded by the pair ( ℓ, A ) represented by thepair of matrices ( H ℓ , C ) in the canonical basis as in Theorem 4.1, the dimension of the algebra ofinfinitesimal symmetries of a -nondegenerate (2 n + 1) -dimensional CR structure of hypersurfacetype with 1-dimensional Levi kernel admitting a constant reduced modified symbol corresponding toCR symbol g is not greater than d total + 2 n + 3 , if the operator A (the matrix C ) is not nilpotent,and it is not greater than d total + 2 n + 4 , if the operator A (the matrix C) is nilpotent . Note that the mentioned Lemmas and Corollaries from the appendix (Section 6) together with(4.7) imply that dim g red , is either not greater than d total + 2 or d total + 3 depending whether ornot C is nilpotent. The estimate for u ( g , red ) = g , red in Theorem 4.4 follows from this and thefact that dim( g − + g , − + g , ) = 2 n + 1 .5. Proof of Theorem 3.8 §5.1.
Preparatory lemmas and notations.
Let σ : g , red → g , red denote the antilinear involu-tion induced by the natural complex conjugation of C T M . We introduce this σ notation to avoidconfusion because while working with matrix representations in coordinates we will use the overline notation to denote the standard complex conjugation of coordinates, which is a different involution.Let ( e , . . . , e n − ) be a basis of g − with respect to which we get the matrix representation of g red0 given by (4.6) and(4.7). Notice in particular that ( e , . . . , e n − ) spans g − , and σ ( e i ) = e n + i − ∀ ≤ i ≤ n − . Note that σ extends to an involution defined of g red1 by same formula (see (2.3)) that we used toextend the natural conjugation from g − to be defined on csp ( g − ) , that is σ ( ϕ )( v ) := σ ◦ ϕ ◦ σ ( v ) ∀ v ∈ g , red , ϕ ∈ g red1 (5.1)defines an involution of g red1 .An element ϕ in Hom ( g − , g − ) ⊕ Hom ( g − , g red0 ) belongs to g red1 if and only if(5.2) ϕ ([ e i , e j ]) = (cid:0) ϕ ( e i ) (cid:1) ( e j ) − (cid:0) ϕ ( e j ) (cid:1) ( e i ) ∀ i, j ∈ { , . . . , n − } . Note, here φ ( e i ) ∈ g red0 ⊂ csp ( g − ) .Given any element v ∈ g − let v − and v + be the canonical projections of v to g − , − and g − , ,respectively, with respect to the splitting g − = g − , − ⊕ g − , .As a direct consequence of (5.2) and (4.6), if n ≤ j ≤ n − and ≤ i ≤ n − , then (cid:0)(cid:0) ϕ ( e j ) (cid:1) e i ) + ∈ span { Ae j − n +1 } − (cid:0) ϕ ([ e i , e j ]) (cid:1) + ⊂ span { Ae j − n +1 , (cid:0) ϕ (1) (cid:1) + } , (5.3) (cid:0)(cid:0) ϕ ( e i ) (cid:1) e j ) − ∈ span (cid:8) σ (cid:0) Ae i (cid:1)(cid:9) − (cid:0) ϕ ([ e i , e j ]) (cid:1) − ⊂ span { Ae i , (cid:0) ϕ (1) (cid:1) − } In particular, the upper left ( n − × ( n − block in the matrix ϕ ( e j ) and the lower right ( n − × ( n − block in the matrix ϕ ( e i ) both have rank at most 2.Also from (5.2) and the fact that [ e i , e j ] = 0 for n ≤ i, j ≤ n − , we immediately have that(5.4) ϕ ( e i ) e j = ϕ ( e j ) e i , n ≤ i, j ≤ n − . Lemma 5.1.
If the antilinear operator A (or, equivalently the matrix C ) has rank greater than 1and i ≥ n then ϕ ( e i ) ∈ g red0 , ⊕ g red0 , − , or, equivalently, ϕ ( e i ) = (cid:18) α i cC − H − ℓ α Ti H ℓ (cid:19) for some c ∈ C and α i ∈ A + C ( H ℓ − Ω ∗ H ℓ ) . (5.5) Proof.
By (4.6), there exists c ∈ C such that for every n ≤ j ≤ n − (cid:0)(cid:0) ϕ ( e i ) (cid:1) e j (cid:1) + = cAe j − n +1 and (cid:0)(cid:0) ϕ ( e j ) (cid:1) e i (cid:1) + ∈ span { Ae i − n +1 } . By (5.4), for all n ≤ j ≤ n − , cAe j − n +1 ∈ span { Ae i − n +1 } . This implies that c = 0 , because otherwise rank A ≤ , contradicting our assumption. Therefore, (cid:0) ϕ ( e i ) v (cid:1) + = 0 for all v ∈ g − , − , which is equivalent to the statement of the Lemma. (cid:3) Similarly, we have the following Lemma.
Lemma 5.2.
If the antilinear operator A (or, equivalently the matrix C ) has rank greater than 1and i < n then ϕ ( e i ) ∈ g red0 , ⊕ g red0 , + or, equivalently, ϕ ( e i ) = (cid:18) α i cC − H − ℓ α Ti H ℓ (cid:19) for some c ∈ C and α i ∈ A + C Ω . (5.6) AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 13
Lemma 5.3. If C has rank greater than 1 and α i is the matrix defined by (5.5) and (5.6) then, for i < n , we have (cid:0) H ℓ Cα i (cid:1) T + H ℓ Cα i = ηH ℓ C for some η ∈ C (5.7) and, for n ≤ i , we have α i CH − ℓ + (cid:0) α i CH − ℓ (cid:1) T = ηCH − ℓ for some η ∈ C . (5.8) Proof. If α i is as in (5.6) then α i ∈ A + C Ω , so the definition of A and item (iii) of (4.8) imply(5.7). If, on the other hand, α i is as in (5.5) then α i ∈ A + C ( H ℓ − Ω ∗ H ℓ ) , so the definition of A and item (iii) of (4.8) imply (5.8). (cid:3) Corollary 5.4.
If the CR symbol is not regular and the matrix α i given in (5.5) or (5.6) is zero,then ϕ ( e i ) = 0 .Proof. Suppose α i = 0 . By (4.6), (4.7), and Lemmas 5.1 and 5.2, if ϕ ( e i ) = 0 then either Ω ∈ A or H ℓ − Ω ∗ H ℓ ∈ A . The conditions Ω ∈ A and H ℓ − Ω ∗ H ℓ ∈ A are, however, equivalent, so either ϕ ( e i ) = 0 or Ω ∈ A . If the CR symbol is not regular then, by Lemma 4.3, Ω A , and hence ϕ ( e i ) = 0 . (cid:3) Lemma 5.5.
If an element ϕ in g red1 satisfies ϕ (1) = 0 and ϕ ( e i ) = 0 ∀ i ≥ n (5.9) then ϕ ( e i ) = 0 ∀ i < n, (5.10) and so ϕ = 0 .Proof. Since ϕ (1) = 0 , the left side of (5.2) is zero for all i and j . Accordingly, for any i ∈{ , . . . , n − } and j ∈ { n, . . . , n − } , (5.2) and (5.9) imply that the j column of ϕ ( e i ) is zero.Hence, for all i ∈ { , . . . , n − } , the latter n − columns of ϕ ( e i ) are all zero. From this andLemma 5.2 (and specifically (5.6)), it follows that H − ℓ α Ti H ℓ = 0 . Hence α i = 0 and therefore by(5.6) again (5.10) holds. (cid:3) The general strategy of our proof of item (1) of Theorem 3.8 is, for a given arbitrary ϕ ∈ g red1 ,first to prove that ϕ (1) = 0 and then to prove (5.9).We will also need the following equations and notation. In the sequel every ( n − × ( n − matrix X will be also regarded as an operator having the matrix representation X with respect tothe basis ( e , . . . , e n − ) . Let { ϕ i } n − i =1 ⊂ C denote the coefficients satisfying ϕ (1) = n − X i =1 ϕ i e i . By (5.5), it follows that (cid:0) ϕ ( e i )) e j (cid:1) − = − (cid:0) H − ℓ α Ti H ℓ (cid:1) e j − n +1 , ∀ n ≤ i, j ≤ n − . This together with (5.4) yields(5.11) (cid:0) H − ℓ α Ti H ℓ (cid:1) e j − n +1 = (cid:0) H − ℓ α Tj H ℓ (cid:1) e i − n +1 , ∀ n ≤ i, j ≤ n − . Condition (5.11) is crucial in the subsequent analysis, namely in the proof of Lemmas 5.6 and5.11. Therefore, we need to describe the matrix H − ℓ α Tj H ℓ , which we begin by first describing thematrix α j . By (5.5), it follows that, for n ≤ j ≤ n − and ≤ i ≤ n − , (cid:0) ϕ ( e j ) e i (cid:1) + = α j e i . From this and (5.3), taking into account that the matrix C represents the antilinear operator A, wehave that there exists the unique tuple ( κ i ) n − i =1 such that(5.12) α j e i = κ i Ce j − n +1 − ( H ℓ ) i,j − n +1 (cid:0) ϕ (1) (cid:1) + for all ≤ i ≤ n − and n ≤ j ≤ n − . The uniqueness of ( κ i ) n − i =1 follows from the assumptionthat C = 0 and that κ i in (5.12) is independent of j .§5.2. The first special case:
In this subsection, §5.2, we consider the special case wherein, forsome integer m satisfying ≤ m ≤ n − , we have H ℓ = S m ⊕ H ′ ℓ (5.13)where H ′ ℓ is an arbitrary nondegenerate Hermitian matrix, and C = J λ,m ⊕ C ′ for some λ ≥ , (5.14)where C ′ is such that ( ℓ, A ) is represented by ( H ℓ , C ) . Moreover, we assume that ( H ℓ , C ) is in thecanonical form of Theorem 4.1. In particular,(5.15) Ce = λe , Ce i = λe i + e i − ∀ ≤ i ≤ m, and(5.16) H ℓ e i = e m +1 − i ∀ ≤ i ≤ m. Using (5.13) and (5.15) we obtain α n e i = κ i λe − δ i,m ( ϕ (1)) + ∀ i ∈ { , . . . , n − } , (5.17)and, for < p < m , α n + p e i = κ i e p + κ i λe p +1 − δ i,m − p ( ϕ (1)) + ∀ i ∈ { , . . . , n − } . (5.18)Now from (5.17), we get α Tn e = n − X j =1 κ j e j − ϕ e m and α Tn e i = − ϕ i e m ∀ ≤ i ≤ n − . Using this together with (5.16) we can get ( H − ℓ α Tn H ℓ ) e i = − ϕ m +1 − i e ∀ i ∈ { , . . . , m − } , (5.19) ( H − ℓ α Tn H ℓ ) e m ≡ − ϕ e + λ m X j =1 κ m +1 − j e j (mod span { e m +1 , e m +2 , . . . , e n − } ) , (5.20)and ( H − ℓ α Tn H ℓ ) e i = − n − X j = m +1 ( H ℓ ) j,i ϕ j e = − n − − m X j =1 ( H ′ ℓ ) j,i − m ϕ j + m e ∀ i > m, (5.21)where H ℓ is as in (5.13).Similarly, for < p < m , from (5.18) we have α Tn + p e i = − ϕ i e m − p , i ∈ { , . . . , n − } \ { p, p + 1 }− ϕ p e m − p + n − X j =1 κ j e j , i = p − ϕ p +1 e m − p + λ n − X j =1 κ j e j i = p + 1 , AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 15 ( H − ℓ α Tn + p H ℓ ) e i = − ϕ m +1 − i e p +1 ∀ i ∈ { , . . . , m } \ { m − p, m − p + 1 } , (5.22) ( H − ℓ α Tn + p H ℓ ) e m − p ≡ − ϕ p +1 e p +1 + λ m X j =1 κ m +1 − j e j (mod span { e m +1 , . . . , e n − } ) , (5.23)and ( H − ℓ α Tn + p H ℓ ) e m − p +1 ≡ − ϕ p e p +1 + m X j =1 κ m +1 − j e j (mod span { e m +1 , . . . , e n − } ) . For p ≥ m , ( H − ℓ α Tn + p H ℓ ) e i ∈ span { e m +1 , . . . , e n − } . (5.24) Lemma 5.6.
In the special case of §5.2 wherein (5.13) and (5.14) hold, if rank ( C ) > then (5.25) ϕ (1) = 0 . Proof.
We will begin by showing that(5.26) ( ϕ (1)) + = 0 . The proof consists of analysis of equation (5.11) in three cases: Equation (5.11) for i = n and j = n + p with ≤ p < m − . By (5.19) (cid:0) H − ℓ α Tn H ℓ (cid:1) e p +1 = ϕ m − p e ∀ ≤ p < m − , (5.27)and, by (5.22), (cid:0) H − ℓ α Tn + p H ℓ (cid:1) e = ϕ m e p +1 ∀ ≤ p < m − . (5.28)Applying (5.27) and (5.28) to (5.11) with i = n and j = n + p we get ϕ m − p e = ϕ m e p +1 ∀ ≤ p < m − . Therefore, using the last equation for ≤ p < m − (as for p=0 this equation is a tautology), weget ϕ = · · · = ϕ m − = 0 , and also that ϕ m = 0 for m > (we will give another way to prove the latter identity including thecase m = 2 in item 3 of the proof below). Equation (5.11) for i = n and j = n + p with p ≥ m . By (5.21) we get that (cid:0) H − ℓ α Tn H ℓ (cid:1) e p +1 = n − − m X j =1 ( H ′ ℓ ) j,p +1 − m ϕ j + m e . (5.29)Using (5.11), from (5.29) and (5.24) it follows that (cid:0) H − ℓ α Tn H ℓ (cid:1) e p +1 = 0 or, equivalently, n − − m X j =1 ( H ′ ℓ ) j,i ϕ j + m = 0 , ≤ i ≤ n − − m. Since the matrix H ′ ℓ is nonsigular, this yields ϕ m +1 = · · · = ϕ n − = 0 . Equation (5.11) for i = n and j = n + m − . If v = λ P mj =1 κ m +1 − j e j , then, by (5.20), ( H − ℓ α Tn H ℓ ) e m ≡ − ϕ e + v (mod span { e i } n − i = m +1 ) , (5.30)and, by (5.23), ( H − ℓ α Tn + m − H ℓ ) e ≡ − ϕ m e m + v (mod span { e i } n − i = m +1 ) . (5.31) Using (5.11) again and the fact that m ≥ , from (5.30) and (5.31) it follows that ϕ = 0 and ϕ m = 0 . This completes the proof of (5.26).Since (5.1) defines an involution of g red1 , σ ( ϕ ) also belongs to g red1 , so, since ϕ was an arbitraryelement in g red1 , the exact same arguments applied above show that ( σ ( ϕ )(1)) + = 0 . Since σ (1) = 1 , σ (cid:0) ( ϕ (1)) − (cid:1) = ( σ ◦ ϕ (1)) + = ( σ ( ϕ )(1)) + = 0 , and hence ( ϕ (1)) − = 0 , which, together with (5.26) implies (5.25). (cid:3) Lemma 5.7.
In the special case of §5.2 wherein (5.13) and (5.14) hold, if rank ( C ) > then ( κ , . . . , κ n ) C = 0 .Proof. Consider now the equation in (5.8) with i = n . The matrix on the right side of (5.8) is eitherzero or it has rank equal to rank ( C ) , which is at least 3 under this lemma’s hypothesis. On theother hand, applying (5.15), (5.16), (5.17) and Lemma 5.6, we get ( α n CH − ℓ ) e i ∈ span { e } ∀ i ∈ { , . . . , m − } , (5.32)and, applying (5.18) additionally, if λ = 0 then ( α n +1 CH − ℓ ) e i ∈ span { e } ∀ i ∈ { , . . . , m − } . (5.33)Hence, by (5.32),(5.34) rank (cid:0) α n CH − ℓ (cid:1) ≤ and rank (cid:16) α n CH − ℓ + (cid:0) α n CH − ℓ (cid:1) T (cid:17) ≤ because α n CH − ℓ has at most one nonzero row. Similarly,if λ = 0 then (5.33)(5.35) rank (cid:0) α n +1 CH − ℓ (cid:1) ≤ and rank (cid:16) α n +1 CH − ℓ + (cid:0) α n +1 CH − ℓ (cid:1) T (cid:17) ≤ . Since the matrix on the left side of (5.8) has rank atmost 2 whenever i = n or ( λ, i ) = (0 , n + 1) , the matrix on the right side of (5.8) is zero whenever i = n or ( λ, i ) = (0 , n + 1) . Thus by (5.8) the matrix α n CH − ℓ is skew symmetric, and the matrix α n +1 CH − ℓ is skew symmetric whenever λ = 0 . This together with (5.34) implies that α n CH − ℓ = 0 , (5.36)whereas applying (5.35) yields α n +1 CH − ℓ = 0 , (5.37)whenever λ = 0 . By (5.36) and (5.17) for λ = 0 , or by (5.37) and (5.18) for λ = 0 , we get that thevector ( κ , . . . , κ n ) CH − ℓ = 0 , which completes this proof. (cid:3) In the subsequent three Lemmas 5.8-5.10 we prove item (1) of Theorem 3.8 in three special specialcases that together cover all non-regular CR symbols not treated in subsequent sections.
Lemma 5.8.
In the special case of §5.2 wherein (5.13) and (5.14) hold, if rank ( C ) > and ( λ, m ) (0 , , (0 , } then g red1 = 0 .Proof. Let ϕ ∈ g red1 and let ( κ i ) n − i =1 be as in (5.12). It will suffice to show that κ i = 0 for every ≤ i ≤ n − . Indeed, first plugging this condition and the conclusion (5.25) of Lemma 5.6 intorelation (5.12) we obtain that α j = 0 for all n ≤ j ≤ n − . This and Corollary 5.4 imply (5.9).Thus, the conclusion of the present lemma will follow from (5.25) and Lemma 5.5.Notice that since ( κ , . . . , κ n ) C = 0 , we have that κ i = 0 for ≤ i ≤ m if λ = 0 , and κ i = 0 for ≤ i ≤ m − if λ = 0 . In particular, as m ≥ we have κ = κ = 0 always, and, since it isassumed that m > when λ = 0 , if λ = 0 then κ = 0 as well. AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 17
To produce a contradiction, assume that there exists an index r such that κ r = 0 and let r bethe minimal such index. By (5.17), α n e i = δ i,r κ i λe ∀ i ≤ r, (5.38)and, by (5.18), for < p < m , α n + p e i = δ i,r ( κ i e p + κ i λe p +1 ) ∀ i ≤ r. (5.39)Note that, by Lemma 5.1, span { α n , α n +1 } is a -dimensional subspace in A + C ( H − ℓ Ω ∗ H ℓ ) . Since A is a subspace in A + C ( H − ℓ Ω ∗ H ℓ ) of codimension at most , the subspaces span { α n , α n +1 } and A have a nontrivial intersection. That is, there exist b , b ∈ C such that ( b , b ) = (0 , and(5.40) b α n + b α n +1 ∈ A . By (5.38) and (5.39) again the first r − columns of the matrix b α n + b α n vanish and(5.41) ( b α n + b α n +1 ) e r = κ r (cid:16) ( λb + b ) e + λb e (cid:17) By applying formulas from the appendix (i.e., Section 6), we can derive a contradiction from theassumption λ = 0 as follows. Let b α n + b α n +1 be partitioned as a block matrix whose diagonalblocks have the same size as the diagonal blocks of C (referring to the block diagonal partition of C given in (4.4)).By (5.40), if λ > then each ( i, j ) block of b α n + b α n +1 is either characterized by Lemma 6.1or Corollary 6.5 and identically zero or it is characterized by Corollary 6.3 and more specificallycharacterized by (6.10). In particular, if the (1 , j ) block of b α n + b α n +1 is nonzero (and thereforecharacterized by (6.10)) and contains part of the r column of b α n + b α n +1 , then (6.10) implies thatthe ( j, block of b α n + b α n +1 is nonzero and contained in the first r − columns of b α n + b α n +1 ,which contradicts our definition of r . Accordingly, if λ > then the (1 , j ) block of b α n + b α n +1 containing part of the r column of b α n + b α n +1 is identically zero, which implies λb + b = 0 and λb = 0 by (5.41). So, if λ > , then we obtain the contradiction ( b , b ) = (0 , .On the other hand, if λ = 0 then, by Lemma 5.1, span { α n +2 , α n +3 } is a -dimensional subspacein A + C ( H − ℓ Ω ∗ H ℓ ) . Similarly to the previous case, A and span { α n +2 , α n +3 } have a nontrivialintersection, that is, there exist b , b ∈ C such that ( b , b ) = (0 , and b α n +2 + b α n +3 ∈ A . (5.42)Note that we are now redefining b and b because the previous definition is no longer needed, andthat the b i s in (5.42) are not related to the b i s in (5.40). By (5.38) and (5.39) the first r − columnsof the matrix b α n +2 + b α n +3 vanish and(5.43) ( b α n +2 + b α n +3 ) e r = κ r (cid:16) b e + b e (cid:17) . By applying formulas from the appendix again, we can derive a contradiction now from theassumption λ = 0 . For this, let b α n +2 + b α n +3 in (5.42) be partitioned as a block matrix whosediagonal blocks have the same size as the diagonal blocks of C . By (5.42), if λ = 0 then each ( i, j ) block of b α n + b α n +1 is either characterized by Lemma 6.1 and identically zero or it ischaracterized by Lemmas 6.4 and 6.8 and Corollary 6.5 and more specifically characterized by(6.15), (6.16), (6.17), and (6.23). In particular, if λ = 0 and the (1 , j ) block of b α n +2 + b α n +3 contains part of the r column of b α n +2 + b α n +3 , and, furthermore, we assume that the (1 , j ) blockis not identically zero, then this (1 , j ) block is either characterized by (6.17) and (6.23) or by (6.15)and (6.16).Considering the first possibility where the (1 , j ) block containing part of the r column of b α n +2 + b α n +3 is characterized by (6.17) and (6.23) (i.e., j = 1 ), by (5.43), the first m entries of b e + b e form the r column of the (1 , block of b α n +2 + b α n +3 . Since we are assuming that this (1 , block is a linear combination of matrices (6.17) and (6.23) with the latter being a diagonal matrix, noting that r > , it follows that the first entry in the r − column of this (1 , block is − b andthe second entry in the r − column of this (1 , block is − b . Yet the r − column of the (1 , block of b α n +2 + b α n +3 is zero by the definition of r , so we have obtained the contradiction that ( b , b ) = (0 , .Considering the remaining possibility, which is where the (1 , j ) block containing part of the r column of b α n +2 + b α n +3 is characterized by (6.15) or (6.16), if this (1 , j ) block is nonzero then(6.15) and (6.16) imply that the ( j, block is nonzero and contained in the first r − columns of b α n +2 + b α n +3 , which contradicts the definition of r .Hence, the (1 , j ) block containing part of the r column of b α n +2 + b α n +3 must be identicallyzero because all other possibilities yield contradictions, and yet, by (5.43), setting this (1 , j ) blockequal to zero again implies the contradiction ( b , b ) = (0 , . Therefore, there is no index r suchthat κ r = 0 . (cid:3) Lemma 5.9.
In the special case of §5.2 wherein (5.13) and (5.14) hold, if there is a basis withrespect to which A is represented by the matrix C = J , ⊕ J ,c ⊕ C ′′ for some c > (5.44) or C = J , ⊕ J ,c ⊕ J ,c ′ ⊕ C ′′ for some c, c ′ > . (5.45) then g red1 = 0 .Proof. Let ϕ ∈ g red1 and let ( κ i ) n − i =1 be as in (5.12). By the same arguments as in the beginning ofthe proof of Lemma 5.8 , it will suffice to show that κ i = 0 for every ≤ i ≤ n − . Note that, byLemma 5.6, in the considered cases ϕ (1) = 0 . It is more convenient to work with matrices e C = J c, ⊕ J , ⊕ C ′′ (5.46)or e C = J c, ⊕ J c ′ , ⊕ J , ⊕ C ′′ (5.47)instead of C in (5.44) and (5.45), respectively. This can be done by an obvious permutation of thebasis. Also, in the considered cases the rank assumptions of Lemma 5.7 with C replaced by e C holds.Therefore, using (5.12) with C replaced by e C we get(5.48) κ = κ = κ = 0 . Note that if we would not replace C by e C we could conclude that κ = κ = κ = 0 in the case of(5.44) and that κ = κ = κ = 0 in the case of (5.45), so that is why we make this permutation ofthe blocks.Assume for a proof by contradiction that there exists r such that κ r = 0 and moreover that this isthe minimal such index, that is, κ i = 0 for all i < r . By (5.48), r > . From (5.12) with C replacedby e C it follows that in both cases the first r − columns of the matrices α i with n ≤ i ≤ n + 3 vanish, α n e r = κ r ce , and α n +3 e r = κ r e . (5.49)Further, α n +2 e r = κ r e (5.50)if ˜ C satisfies (5.46) and α n +1 e r = κ r c ′ e (5.51)if ˜ C satisfies (5.47). Note that, by Lemma 5.1, each α i in these equations belongs to A + C (cid:16) H ℓ − Ω ∗ H ℓ (cid:17) . AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 19
Hence, using similar arguments as in the proof of Lemma 5.8 we get that the -dimensionalsubspace span { α n , α n +2 , α n +3 } in the first case and span { α n , α n +1 , α n +3 } in the second case has atleast a two dimensional intersection with A . Notice further that in either case, the r th columns ofmatrices in these intersections must have a two-dimensional span because the natural map from thespace span { α n , α n +2 , α n +3 } (or span { α n , α n +1 , α n +3 } ) to C n − sending a matrix to its r column inthis space is injective.Let us now first assume that ˜ C satisfies (5.46). Let B (1) and B (2) be matrices belonging to theintersection of span { α n , α n +2 , α n +3 } and A such that the r column of B (1) is linearly independentfrom the r column of B (2) . For an ( n − × ( n − matrix B , let ( B ( i,j ) ) be a partition of B intoa block matrix whose diagonal blocks have the same size as the diagonal blocks of C . Let j be theindex such that B (1 ,j ) contains part of the r column of B . By Lemma 6.1, since c = 0 there exists i ∈ { , } such that B ( i,j ) = 0 for all B ∈ A , because otherwise Lemma 6.1 implies that the (1 , and (2 , blocks of CC have the same eigenvalues. In particular, at most one of the (1 , j ) and (2 , j ) blocks of any linear combination of B (1) and B (2) is nonzero. It follows that, for each k ∈ { , } , B ( k )(1 ,j ) = 0 and B ( k )(2 ,j ) = 0 because otherwise the r column of each B ( k ) belongs to span { e } , whichcontradicts our choice of B (1) and B (2) . Moreover, by (5.49) and (5.50), the first nonzero columnof each block B ( k )(2 ,j ) has zero in all but its first two entries.Each B ( k )(2 ,j ) is either characterized by Lemma 6.1 and is identically zero or characterized by Lemma6.4 and Corollary 6.5 and more specifically characterized by (6.15), (6.16), or (6.17) (with λ i = 0 ).If B ( k )(2 ,j ) is characterized by (6.17) then j = 2 and, by (6.17), the second entry of the first nonzerocolumn of B ( k )(2 , is zero. If, on the other hand, B ( k )(2 ,j ) is characterized by (6.15) (or (6.16)) andthe second entry of the first nonzero column of B ( k )(2 ,j ) is nonzero, then, by (6.16) (or respectively(6.15)), the B ( k )( j, block of B ( k ) is nonzero and contained in the first r − columns of B ( k ) , whichcontradicts our choice of r . Therefore if B ( k )(2 ,j ) is nonzero then the second entry of the first nonzerocolumn of B ( k )(2 ,j ) is zero. Yet this contradicts our choice of B (1) and B (2) because it means that theonly nonzero entry in the r column of B (1) and B (2) is the second entry.Let us now address the remaining case, that is, assume that ˜ C satisfies (5.47). Again, let j be theindex such that B (1 ,j ) contains part of the r column a given ( n − × ( n − matrix B . Let B (1) and B (2) be matrices belonging to the intersection of span { α n , α n +1 , α n +3 } and A such that the r column of B (1) is linearly independent from the r column of B (2) . From this independence conditionand the fact that nonzero entries of these respective r th columns of B (1) and of B (2) appear withintheir first three entries (the latter is a consequence of (5.49) and (5.51)), it follows that there existsa matrix B in span { B (1) , B (2) } such that there exists i ∈ { , } with B ( i,j ) = 0 (because otherwise,the third entry is the only nonzero entry of r th columns of B (1) and B (2) , which contradicts theindependence of these columns). Since r > it follows that j > . Thus, it follows from Lemma6.1 and Corollary 6.3 that this nonzero B ( i,j ) with i ∈ { , } is characterized by (6.10). Yet (6.10)implies that the B ( j,i ) is a nonzero block contained in the first r − rows of B , which contradictsour choice of r . (cid:3) Lemma 5.10.
In the special case of §5.2 wherein (5.13) and (5.14) hold, if C = k copies z }| { J , ⊕ · · · ⊕ J , ⊕ J c, ⊕ J , ⊕ · · · ⊕ J , , for some integer k and some c > then g red1 = 0 . Proof.
Let ϕ ∈ g red1 and let ( κ i ) n − i =1 be as in (5.12). By the same arguments as in the beginning ofthe proof of Lemma 5.8 , it will suffice to show that κ i = 0 for every ≤ i ≤ n − . We work with ( H ℓ , C ) in the canonical form of Theorem 4.1, so H ℓ is as in (4.4), that is H ℓ = ǫ N , ⊕ · · · ⊕ ǫ k N , ⊕ ǫ k +1 N c, ⊕ · · · ⊕ ǫ γ N , for some coefficients ǫ i = ± .For a matrix B in A , let ( B ( i,j ) ) be a partition of B into a block matrix whose diagonal blockshave the same size as the diagonal blocks of C . By Lemma 6.4 and Corollary 6.5 (in the appendixbelow), we have B ( i,j ) = ǫ i (cid:18) b c d (cid:19) and B ( j,i ) = − ǫ j (cid:18) b e d (cid:19) ∀ i, j ≤ k and B ( i,j ) = (cid:18) a (cid:19) and B ( j,i ) = (cid:0) b (cid:1) ∀ i ≤ k < j for some b, c, d, e ∈ C that depend on ( i, j ) . By Corollary 6.5 and Lemma 6.8 (in the appendixbelow), B , = B , = · · · = B k +1 , k +1 , (5.52)where here B i,j denotes the ( i, j ) entry of B rather than the ( i, j ) block B ( i,j ) . By Lemma 6.1and Corollary 6.5 (in the appendix below), B ( i,k +1) = 0 and B ( k +1 ,i ) = 0 ∀ i = k. (5.53)Since, by Lemma 5.7, ( κ , . . . , κ n − ) C = 0 , we have κ i = 0 whenever i is odd and i ≤ k + 1 . (5.54)From (5.12) and Lemma 5.6 it follows that, for ≤ p ≤ n − , the i column of the matrix α n + p isequal to κ i times the p + 1 column of C . In particular, the ( i, j ) entry of α n +2 k is ( α n +2 k ) i,j = κ j cδ i, k +1 . (5.55)Since, by Lemma 5.1, each α n + p belongs to A + C (cid:16) H ℓ − Ω ∗ H ℓ (cid:17) and α n +2 k does not belong to A \ { } , which can be seen by contrasting (5.53) and (5.55), it follows thateither α n +2 k = 0 or H ℓ − Ω ∗ H ℓ ∈ A + span C { α n +2 k } . But α n +2 k = 0 if and only if κ = · · · = κ n − = 0 , which is equivalent to what we want to show, solet us proceed assuming H ℓ − Ω ∗ H ℓ ∈ A + span C { α n +2 k } in order to produce a contradiction. Accordingly, let Ω ∈ A and s ∈ C be such that H ℓ − Ω ∗ H ℓ = H ℓ − Ω ∗ H ℓ + sα n +2 k , (5.56)or, equivalently, Ω = Ω + sH ℓ − α ∗ n +2 k H ℓ . (5.57)Here we will apply another result from the appendix (below), namely Corollary 6.9, which statesthat for B ∈ A , since C is not nilpotent, if (cid:0) H ℓ CB (cid:1) T + H ℓ CB = µH ℓ C then BCH − ℓ + CH − ℓ B T = µCH − ℓ . Noting that, by (5.54) and (5.55), CH ℓ − α ∗ n +2 k H ℓ = 0 , item (iii) in (4.8) and (5.57) implythat (cid:0) H ℓ C Ω (cid:1) T + H ℓ C Ω = µH ℓ C, (5.58) AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 21 and hence Corollary 6.9 implies that η Ω = µ, where this notation η Ω refers to the coefficient with that label in items (i) and (ii) or (4.8).Since the matrix equation (cid:0) H ℓ CX (cid:1) T + H ℓ CX = µH ℓ C is equivalent to (cid:16) H − ℓ X ∗ H ℓ (cid:17) CH − ℓ + CH − ℓ (cid:16) H − ℓ X ∗ H ℓ (cid:17) T = µCH − ℓ , (5.58) implies η H ℓ − Ω ∗ H ℓ = µ. (5.59)By (5.59), items (i) and (ii) in (4.8) imply h Ω , H ℓ − Ω ∗ H ℓ i + µ Ω ∈ A , (5.60)and applying the transformation X H ℓ − X ∗ H ℓ to the matrix in (5.59) yields h H ℓ − Ω ∗ H ℓ , Ω i − µH ℓ − Ω ∗ H ℓ ∈ A . (5.61)Now we analyze item (iv) of (4.8). Using (5.56), (5.57), and lastly (5.60), we have h H ℓ − Ω ∗ H ℓ , Ω i = h H ℓ − Ω ∗ H ℓ , Ω i + [ sα n +2 k , Ω ] + | s | h α n +2 k , H ℓ − α ∗ n +2 k H ℓ i ≡ µ Ω + [ sα n +2 k , Ω ] + | s | h α n +2 k , H ℓ − α ∗ n +2 k H ℓ i (mod A ) . Substituting the last equation into item (iv) of (4.8) we get, after the obvious cancellation, that [ sα n +2 k , Ω ] + | s | h α n +2 k , H ℓ − α ∗ n +2 k H ℓ i + CC − µH ℓ − Ω ∗ H ℓ ∈ A . (5.62)Similarly, (5.56), (5.57), and then (5.61) yields h H ℓ − Ω ∗ H ℓ , Ω i = h H ℓ − Ω ∗ H ℓ , Ω i + h H ℓ − Ω ∗ H ℓ , sH ℓ − α ∗ n +2 k H ℓ i + | s | h α n +2 k , H ℓ − α ∗ n +2 k H ℓ i ≡ µH ℓ − Ω ∗ H ℓ + h H ℓ − Ω ∗ H ℓ , sH ℓ − α ∗ n +2 k H ℓ i + | s | h α n +2 k , H ℓ − α ∗ n +2 k H ℓ i , where the equivalence is modulo A . Substituting the last equation into item (iv) of (4.8) we get h H ℓ − Ω ∗ H ℓ , sH ℓ − α ∗ n +2 k H ℓ i + | s | h α n +2 k , H ℓ − α ∗ n +2 k H ℓ i + CC − µ Ω ∈ A . (5.63)On the other hand, again from (5.56) , (5.57), and using that h H ℓ − Ω ∗ H ℓ , Ω i ∈ A , we can write h H ℓ − Ω ∗ H ℓ , Ω i ≡ [ sα n +2 k , Ω ] + h H ℓ − Ω ∗ H ℓ , sH ℓ − α ∗ n +2 k H ℓ i + | s | h α n +2 k , H ℓ − α ∗ n +2 k H ℓ i , where here again the equivalence is modulo A . By subtracting the matrix in item (iv) of (4.8) fromthe sum of the matrices in (5.62) and (5.63) and using the last relation, we get CC + | s | h α n +2 k , H ℓ − α ∗ n +2 k H ℓ i ∈ A , or, equivalently, (cid:16) CC + | s | α n +2 k H ℓ − α ∗ n +2 k H ℓ (cid:17) − | s | H ℓ − α ∗ n +2 k H ℓ α n +2 k ∈ A . (5.64)Notice that the first two terms in (5.64), grouped together by parentheses, are matrices whose onlypotentially nonzero entry is the (2 k + 1 , k + 1) entry, whereas the other term has the same value inthe first k + 1 entries of its main diagonal. By (5.52), each matrix in A also has the same values in the first k + 1 entries of its main diagonal. Moreover, the (2 k + 1 , k + 1) entry of CC is nonzero.Therefore, by (5.64), CC = −| s | α n +2 k H ℓ − α ∗ n +2 k H ℓ . (5.65)Defining α := | s | H ℓ − α ∗ n +2 k H ℓ α n +2 k , (5.64) and (5.65) imply that α is in A .It is straightforward to check that, with this definition for α , η α = 0 in the notation of item (i)of (4.8) (by calculating, for example, the (1 , entries of the terms in item (i)), and hence items (i)and (ii) of (4.8) yield [Ω , α ] ∈ A . Or, equivalently, by (5.57), noting that [Ω , α ] ∈ A , s h H ℓ − α ∗ n +2 k H ℓ , α i ∈ A . (5.66)Notice that H ℓ − α ∗ n +2 k H ℓ α = 0 because (cid:16) H ℓ − α ∗ n +2 k H ℓ (cid:17) = 0 , and hence (5.66) implies s | s | H ℓ − α ∗ n +2 k H ℓ (cid:16) α n +2 k H ℓ − α ∗ n +2 k H ℓ (cid:17) ∈ A . (5.67)Applying (5.65), we get − s | s | | c | H ℓ − α ∗ n +2 k H ℓ (cid:16) α n +2 k H ℓ − α ∗ n +2 k H ℓ (cid:17) = s | c | H ℓ − α ∗ n +2 k H ℓ (cid:0) CC (cid:1) (5.68) = sH ℓ − α ∗ n +2 k H ℓ , where this last equality follows easily from (5.55).By (5.57), (5.67), and (5.68), we get that Ω is in A , but this contradicts Lemma 4.3. Therefore,the assumption that α n +2 k = 0 must be false, which in turn implies that a = · · · = a n − = 0 ,completing this proof. (cid:3) §5.3. The second special case:
In this subsection, §5.3, we consider the special case where wehave some integer ≤ m ≤ n − such that H ℓ = (cid:18) S m H ′ ℓ (cid:19) , (5.69)where H ′ ℓ is an arbitrary nondegenerate Hermitian matrix, and C = m columns z }| { J m,λ I C ′ (cid:27) m rows , for some λ ∈ C \ { x ∈ R | x ≥ } ,(5.70)where C ′ is a matrix such that ( ℓ, A ) is represented by ( H, C ) . The analysis in §5.3 is similar tothat of §5.2, but some formulas differ.By Lemma 5.3 there exist coefficients κ , . . . , κ n − , given in (5.12), such that, first, α n + m e i = − δ i,m − (cid:0) ϕ (1) (cid:1) + + λκ i e , second, for any nonnegative integer p < m , α n + p e i = − δ i, m − p (cid:0) ϕ (1) (cid:1) + + λκ i e m + p , and, third, if < p < m then α n + m + p e i = − δ i, m − p (cid:0) ϕ (1) (cid:1) + + κ i e p + λκ i e p +1 , which we use to obtain the following formulas. For ≤ p < m , we have ( α n + p CH − ℓ ) e i = ( κ m − i λ − κ m +1 − i λ ) e m + p +1 ∀ i ∈ { , . . . , m − } , (5.71) AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 23 ( α n + p CH − ℓ ) e m = κ λ e m + p +1 , (5.72) ( α n + p CH − ℓ ) e i = κ m +1 − i λe m + p +1 − δ i,m + p +1 (cid:0) ϕ (1) (cid:1) + ∀ i ∈ { m + 1 , . . . , m } , (5.73)and ( α n + p CH − ℓ ) e i ∈ span { e m + p +1 } ∀ i > m. (5.74)For any nonnegative integer p < m and ≤ i ≤ m (cid:0) H − ℓ α Tn + p H ℓ (cid:1) e i ≡ − ϕ m +1 − i e p +1 + δ i,m − p m X j =1 κ m +1 − j λe j (mod span { e k } n − k =2 m +1 ) (5.75)and, moreover, this equivalence modulo span { e k } n − k =2 m +1 can be replaced with ordinary strict equiv-alence whenever δ i,m − p = 0 . Also, for ≤ i ≤ m , (cid:0) H − ℓ α Tn + m H ℓ (cid:1) e i ≡ − ϕ m +1 − i e m +1 + δ i, m m X j =1 κ m +1 − j λe j (mod span { e k } n − k =2 m +1 ) , (5.76)where equivalence modulo span { e k } n − k =2 m +1 can be replaced with ordinary strict equivalence when-ever δ i, m − = 0 . For any < p < m and < i < m + 1 , (cid:0) H − ℓ α Tn + m + p H ℓ (cid:1) e i = − ϕ m +1 − i e m + p +1 + δ i, m − p m X j =1 κ m +1 − j λe j + n − X k =2 m +1 κ k λe k (5.77) + δ i, m − p +1 m X j =1 κ m +1 − j e j + n − X k =2 m +1 κ k e k , and for any < p < m and m < i < n (cid:0) H − ℓ α Tn + m + p H ℓ (cid:1) e i = − ϕ i e m + p +1 . Lastly, for all i ≥ m ( H − ℓ α Tn + p H ℓ ) e i = − n − X j =2 m +1 ( H ℓ ) j,i ϕ j e p +1 ∀ ≤ p < m (5.78)and (cid:0) H − ℓ α Tn + p H ℓ (cid:1) e i ⊂ span { e m +1 , . . . , e n − } ∀ m ≤ p. (5.79) Lemma 5.11.
In the special case of §5.3 wherein (5.69) and (5.70) hold ϕ (1) = 0 . Proof.
By the same argument applied at the end of the proof of Lemma 5.6, it will suffice to showthat (cid:0) ϕ (1) (cid:1) + = 0 . Similar to the proof of Lemma 5.6, this proof consists of analysis of equation(5.11) in four cases: Equation (5.11) for i = n and j = n + p with ≤ p < m and m = 1 . By (5.75) replacing p with and replacing i with p + 1 , (cid:0) H − ℓ α Tn H ℓ (cid:1) e p +1 = − ϕ m − p e ∀ ≤ p < m − , (5.80)and, by (5.75) with i = 1 , (cid:0) H − ℓ α Tn + p H ℓ (cid:1) e = − ϕ m e p +1 ∀ ≤ p < m − . (5.81) Applying (5.11), (5.80), and (5.81) we get ϕ m +2 = ϕ m +3 = · · · = ϕ m = 0 . Furthermore, by (5.75)with p = 0 and i = m , (cid:0) H − ℓ α Tn H ℓ (cid:1) e m ≡ − ϕ m +1 e + m X j =1 κ m +1 − j λe j (mod span { e k } n − k =2 m +1 ) (5.82)whereas, by (5.75) with p = m − and i = 1 , (cid:0) H − ℓ α Tn + m − H ℓ (cid:1) e ≡ − ϕ m e m + m X j =1 κ m +1 − j λe j (mod span { e k } n − k =2 m +1 ) . (5.83)Applying (5.11), (5.83), and (5.82) yields ϕ m +1 = 0 so, altogether, we have shown ϕ m +1 = · · · = ϕ m = 0 . (5.84) Equation (5.11) for i = n + m and j = n + m + p with ≤ p < m and m = 1 . By (5.77)replacing p with and replacing i with m + p + 1 , (cid:0) H − ℓ α Tn + m H ℓ (cid:1) e m + p +1 = − ϕ m − p e m +1 ∀ < p < m − , (5.85)and, by (5.77) with i = m + 1 , (cid:0) H − ℓ α Tn + m + p H ℓ (cid:1) e m +1 = − ϕ m e m + p +1 ∀ < p < m − . (5.86)Applying (5.11), (5.85), and (5.86) we get ϕ = ϕ = · · · = ϕ m = 0 . Furthermore, by (5.77) with p = 0 and i = 2 m , (cid:0) H − ℓ α Tn + m H ℓ (cid:1) e m = − ϕ e m +1 + m X j =1 κ m +1 − j λe j + n − X k =2 m +1 κ k λe k (5.87)and, by (5.75) with p = m − and i = m + 1 , (cid:0) H − ℓ α Tn +2 m − H ℓ (cid:1) e m +1 = − ϕ m e m + m X j =1 κ m +1 − j λe j + n − X k =2 m +1 κ k λe k . (5.88)Applying (5.11), (5.87), and (5.88) yields ϕ = ϕ m = 0 so, altogether, noting (5.84), we have shown ϕ = · · · = ϕ m = 0 if m > . (5.89) Equation (5.11) for i = n and j = n + m . By (5.75) and (5.76) (cid:0) H − ℓ α Tn H ℓ (cid:1) e m +1 = − ϕ m e and (cid:0) H − ℓ α Tn + m H ℓ (cid:1) e = − ϕ m e m +1 . (5.90)By (5.11), (cid:0) H − ℓ α Tn H ℓ (cid:1) e m +1 = (cid:0) H − ℓ α Tn + m H ℓ (cid:1) e , and hence (5.90) implies ϕ m = ϕ m . This is truein particular when m = 1 , which together with (5.89) yields the general result ϕ = · · · = ϕ m = 0 . (5.91) Equation (5.11) for i = n and j = n + p with p ≥ m . By (5.78) we get that (cid:0) H − ℓ α Tn H ℓ (cid:1) e p +1 = n − X j =2 m +1 ( H ′ ℓ ) j,i ϕ j e . (5.92)Using (5.11) again, from (5.79) and (5.92) it follows that (cid:0) H − ℓ α Tn H ℓ (cid:1) e p +1 = 0 or, equivalently, n − − m X j =1 ( H ′ ℓ ) j,i ϕ j = 0 , ∀ ≤ i ≤ n − − m. (5.93) AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 25
Since the matrix H ′ ℓ is nonsigular, (5.93) implies ϕ m +1 = · · · = ϕ n − = 0 , which together with(5.91) yields ϕ = · · · = ϕ n − = 0 , that is, (cid:0) ϕ (1) (cid:1) + = 0 . (cid:3) Lemma 5.12.
In the special case of §5.3 wherein (5.69) and (5.70) hold, ( κ , . . . , κ n − ) C = 0 .Proof. First we want to show that α n CH − ℓ is skew symmetric, and we do so by considering twoseparate cases.First, consider the case where m = 1 . By (5.72), the (1 , entry of α n CH − ℓ is zero. But the (1 , entry of CH − ℓ is nonzero, so (5.8) implies that α n CH − ℓ is skew symmetric.Now let us consider the second case, which is where m > . The right side of (5.8) is either zeroor its right side has rank equal to rank ( C ) (which is at least 4 because m > ). On the other hand,using formulas (5.71), (5.72), (5.73), and (5.74) for the matrix α n CH − ℓ together with Lemma 5.11,we can see that the matrix α n CH − ℓ has rank at most . Therefore the matrix on the left side of(5.8) (when setting i = n ) has rank at most , and hence the matrix α n CH − ℓ appearing in (5.71)must be skew symmetric if m > .So, for all values of m , we have shown that α n CH − ℓ is skew symmetric and of rank at most .Thus it is identically zero, which implies that the rows of α n are in the left kernel of CH − ℓ . Inparticular, ( κ , . . . , κ n ) CH − ℓ = 0 , which completes this proof because H ′ ℓ is nonsingular. (cid:3) Lemma 5.13.
In the special case of §5.3 wherein (5.69) and (5.70) hold, if C corresponds to anon-regular CR structure then g red1 = 0 .Proof. Let ϕ ∈ g red1 and let ( κ i ) n − i =1 be as in (5.12). By the same arguments as in the beginning ofthe proof of Lemma 5.8 , it will suffice to show that κ i = 0 for every ≤ i ≤ n − .To produce a contradiction, let us assume there exists an index i such that κ i = 0 , and let r bethe smallest such index. Since, by Lemma 5.12, ( κ , . . . , κ n ) C = 0 , we have κ = κ = 0 , and hence < r . Also, α n + m e i = δ i,r κ r λe and α n + m +1 e i = δ i,r ( κ r e + κ r λe ) ∀ i ≤ r. (5.94)By Lemma 5.1, span { α n + m , α n + m +1 } is a -dimensional subspace in A + C ( H − ℓ Ω ∗ H ℓ ) . Since A is a subspace in A + C ( H − ℓ Ω ∗ H ℓ ) of codimension at most it has a nontrivial intersection with span { α n + m , α n + m +1 } , and hence there exist b , b ∈ C such that ( b , b ) = (0 , and(5.95) b α n + m + b α n + m +1 ∈ A . By (5.94) the first r − columns of the matrix b α n + b α n vanish and(5.96) ( b α n + b α n +1 ) e r = κ r (cid:16) ( λb + b ) e + λb e (cid:17) . Using results from the appendix (Section 6 below), we can now derive a contradiction as follows.Let b α n + b α n +1 be partitioned as a block matrix whose diagonal blocks have the same size asthe diagonal blocks of C . By (5.95), each ( i, j ) block of b α n + b α n +1 is either characterized byLemma 6.1 and identically zero or it is characterized by Corollaries 6.3 and 6.5 and more specificallycharacterized by (6.11), (6.12), (6.13), (6.14), and (6.17). Notice that if this (1 , j ) block of b α n + b α n +1 is characterized by (6.17) then j = 1 , and clearly no matrix of the form in (6.17) can havenonzero values in either of the first two entries of its first nonzero column, which shows that this (1 , j ) block of b α n + b α n +1 containing part of the r column of b α n + b α n +1 must be zero if it ischaracterized by (6.17).If, on the other hand, the (1 , j ) block of b α n + b α n +1 is characterized by (6.11) or (6.12)(respectively (6.13) or (6.14)), is nonzero, and contains part of the r column of b α n + b α n +1 , then(6.11) and (6.12) (respectively (6.13) and (6.14)) imply that the ( j, block of b α n + b α n +1 isnonzero and contained in the first r − columns of b α n + b α n +1 , which contradicts our definitionof r . Therefore, the (1 , j ) block of b α n + b α n +1 containing part of the r column of b α n + b α n +1 is identically zero, which, by (5.96), implies that λb + b = 0 and λb = 0 . Yet this yields thecontradiction ( b , b ) = (0 , . (cid:3) §5.4. The third special case:
In this subsection, §5.4, we consider the special case where ( H ℓ , C ) corresponds to a non-regular CR structure and C is diagonal. Working in the normal form ofTheorem 4.1, H ℓ is diagonal too. Since C corresponds to a non-regular CR structure, the matrix CC has at least two distinct nonzero eigenvalues, so we can assume without loss of generality thatthere are numbers λ , . . . , λ n − , ∈ C and ǫ , . . . , ǫ n − ∈ { , − } such that | λ | 6 = | λ | , λ = 0 , λ = 0 , and C = diag ( λ , . . . , λ n − ) and H ℓ = diag ( ǫ , . . . , ǫ n − ) . Accordingly, by (5.12), α n + p e i = κ i λ p +1 e p +1 − δ i,p +1 ϕ (1) ∀ ≤ p < n, (5.97) α n CH − ℓ e i = λ i ε i κ i (cid:0) λ e − δ i, ϕ (1) (cid:1) , (5.98) H − α Tn + p He = ± ϕ e p +1 ∀ ≤ p < n, (5.99)and H − α Tn He p +1 = ± ϕ p +1 e ∀ < p < n. (5.100)By (5.11), we can equate H − α Tn He p +1 , and hence (5.99) and (5.100) yields ϕ = ϕ = · · · = ϕ n − = 0 . (5.101)Formula in (5.98) now simplifies giving that α n CH − ℓ is a matrix with at most 1 nonzero row,and hence the left side of (5.8) (when setting i = n ) cannot be a diagonal matrix of rank greaterthan one. Yet the right side of (5.8) is a diagonal matrix that is either zero or of rank greater than1, so the right side of (5.8) must be zero for the equation to hold. Since the left side of (5.8) is zero,(5.98) and (5.101) imply that λ κ = λ κ = · · · = λ n − κ n − = 0 because λ = 0 . In particular, κ = κ = 0 (5.102)because λ and λ are both nonzero. Lemma 5.14. If ( H ℓ , C ) corresponds to a non-regular CR structure and C is diagonal then g red1 = 0 .Proof. Let ϕ ∈ g red1 and let ( κ i ) n − i =1 be as in (5.12). Recall that (cid:0) ϕ (1) (cid:1) + = 0 implies ϕ (1) = 0 , bythe same argument applied at the end of the proof of Lemma 5.6, and hence ϕ (1) = 0 by (5.101).Accordingly, by the same arguments as in the beginning of the proof of Lemma 5.8, it will sufficeto show that κ i = 0 for every ≤ i ≤ n − .Assume that there exists r such that κ r = 0 and r is the minimal index with this property. By(5.102) we have that r > . Noting (5.97), by Lemma 5.1, κ r = 0 implies span { α n , α n +1 } is a -dimensional subspace in A + C ( H − ℓ Ω ∗ H ℓ ) . Accordingly, κ r = 0 yields that span { α n , α n +1 } and A have at least a -dimensional intersection. By (5.102) and(5.97), nonzero entries in the matricesin span { α n , α n +1 } can only appear in their first two rows and moreover they do not appear intheir first two columns. Yet, in the appendix (Section 6 below), we describe the matrices in A explicitly. In particular, given that H ℓ and C are diagonal, the description of A in the appendiximplies that every matrix in A with nonzero entries in its first two rows also has nonzero entries inits first two columns, which implies that span { α n , α n +1 } and A have a trivial intersection, a clearcontradiction. (cid:3) AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 27
By combining the results of Lemmas 5.8, 5.9, 5.10, 5.13 , and 5.14, we finish the proof of item(1) of Theorem 3.8, because these Lemmas account for all non-regular symbols.To prove item (2) of Theorem 3.8 note that by (4.7) and Lemma 6.10 for the reduced modifiedCR symbol corresponding to a non-regular symbol dim g red0 , = dim A + 1 < n − n + 7 Therefore, from item (1) of the theorem under consideration and the fact that dim g red0 = dim g red0 , +2 and dim g − = 2 n − , it follows that dim u ( g , red ) = dim g , red < (2 n −
1) + ( n − n + 7) + 2 = ( n − + 7 , which together with Theorem 3.7 completes the proof of item (2) of Theorem 3.8.6. Appendix: Matrix representations of the algebra A In this appendix we give a general formula for matrices in the algebra A defined in (4.5) togetherwith an outline for how the formula can be verified. Naturally, it is easier to verify the formula thanto derive it, and, since the formula is ancillary to this paper’s topic, we omit the analysis used toderive it. The formula depends on the matrices H ℓ and C representing the pair ( ℓ, A ) .In the sequel we assume that H ℓ and C are in the canonical form prescribed by Theorem 4.1,namely as given in (4.4). We will also use the notation of Section 4, and, in particular, we let λ , . . . , λ γ , m , . . . , m γ , ǫ , . . . , ǫ γ , M λ i ,m i and N λ i ,m i as in Theorem 4.1. Recall that, in particular,this means the real and imaginary parts of each λ i are both nonnegative.Define the bi-orthogonal subalgebra of A to be A o := { B ∈ A | BCH − ℓ + CH − ℓ B T = B T H ℓ C + H ℓ CB = 0 } , where this name is reflecting the observation that A o is analogous to an intersection of two orthog-onal algebras. In this appendix, we first obtain a formula describing the elements in A o and thenobtain a formula for a subspace A s ⊂ A complimentary to A o , that is, such that A = A o ⊕ A s . (6.1)Such a space A s is spanned by elements that we call conformal scaling elements of A , referringto the observation that these are analogous to non-orthogonal elements in an intersection of twoconformally orthogonal algebras.To begin, let B be an ( n − × ( n − matrix in A o and partition B into blocks { B ( i,j ) } γi,j =1 where the number of rows in B ( i,j ) is the same as in the matrix M λ i ,m i and the number of columnsin B ( i,j ) is the same as in the matrix M λ j ,m j . Similarly, we partition H ℓ CB and BCH − ℓ into blocks { ( H ℓ CB ) ( i,j ) } γi,j =1 and { ( BCH − ℓ ) ( i,j ) } γi,j =1 whose sizes are the same as in the partition of B .Let us now derive a relationship between the blocks B ( i,j ) and B ( j,i ) . To simplify formulas, weassume ǫ i = ǫ j . To treat the more general case where possibly ǫ i = ǫ j , one can simply replace N λ i ,m i (or N λ j ,m j ) with ǫ i N λ i ,m i (or ǫ j N λ j ,m j ) in all of the subsequent formulas.We have ( BCH − ℓ ) ( i,j ) = B ( i,j ) M λ j ,m j N λ j ,m j and ( H ℓ CB ) ( i,j ) = N λ i ,m i M λ i ,m i B ( i,j ) , so, since B ∈ A , ( M λ i ,m i N λ i ,m i ) T B T ( j,i ) = − B ( i,j ) M λ j ,m j N λ j ,m j and B T ( j,i ) (cid:0) N λ j ,m j M λ j ,m j (cid:1) T = − N λ i ,m i M λ i ,m i B ( i,j ) . Since A is ℓ -self-adjoint, each matrix N λ k ,m k M λ k ,m k and M λ k ,m k N λ k ,m k is symmetric (one can alsoverify this by directly using the canonical form), and hence M λ i ,m i N λ i ,m i B T ( j,i ) = − B ( i,j ) M λ j ,m j N λ j ,m j , (6.2) and B T ( j,i ) N λ j ,m j M λ j ,m j = − N λ i ,m i M λ i ,m i B ( i,j ) . (6.3)Multiplying both sides of (6.3) by M λ j ,m j N λ j ,m j from the right and then applying (6.2) yields B T ( j,i ) N λ j ,m j M λ j ,m j M λ j ,m j N λ j ,m j = − N λ i ,m i M λ i ,m i B ( i,j ) M λ j ,m j N λ j ,m j (6.4) = N λ i ,m i M λ i ,m i M λ i ,m i N λ i ,m i B T ( j,i ) . Multiplying (6.4) by N λ i ,m i from the left and by N λ j ,m i from the right yields (cid:16) N λ i ,m i B T ( i,j ) N λ j ,m j (cid:17) M λ j ,m j M λ j ,m j = M λ i ,m i M λ i ,m i (cid:16) N λ i ,m i B T ( i,j ) N λ j ,m j (cid:17) . (6.5)Notice that (6.2) is also equivalent to N λ i ,m i M λ i ,m i (cid:0) N λ j ,m j B ( j,i ) N λ i ,m i (cid:1) T = − (cid:0) N λ i ,m i B ( i,j ) N λ j ,m j (cid:1) N λ j ,m j M λ j ,m j . (6.6)Equation (6.5) gives us all restrictions on the general form of B ( i,j ) that are not coming fromthe relationship between B ( i,j ) and other blocks in the matrix B . Equation (6.6), on the otherhand, gives us the restrictions on the general form of B ( i,j ) coming from its relationship with B ( j,i ) .Moreover, if (6.5) and (6.6) are satisfied for i and j then B is in A o because (6.2) and (6.3) hold.In other words, our present goal is to solve the system of matrix equations in (6.5) and (6.6), andwhenever ( λ i , λ j ) = (0 , , this exercise is equivalent to first solving the matrix equation XM λ j ,m j M λ j ,m j = M λ i ,m i M λ i ,m i X, (6.7)and then, for the case where i = j , solving the system of equations consisting of (6.7) and N λ i ,m i M λ i ,m i X T = − XN λ i ,m i M λ i ,m i . The case where λ i = λ j = 0 requires special treatment because, in this case, contrary to the casewhere ( λ i , λ j ) = (0 , , even if i = j solutions for B ( i,j ) in (6.5) need not satisfy (6.6) for any matrix B ( j,i ) .Equation (6.7) is of the form analyzed in [8, Chapter 8]. In fact, an explicit solution to (6.7)is given in [8, Chapter 8], but the solution is expressed in terms of a basis with respect to which M λ i ,m i M λ i ,m i and M λ j ,m j M λ j ,m j have their Jordan normal forms. On the other hand, the transitionmatrix from the initially considered basis to a basis of the Jordan normal form is block-diagonalwith the blocks corresponding to the Jordan blocks. Hence, the following lemma can be obtainedfrom the solution in [8, Chapter 8]. Lemma 6.1. If λ i = λ j then B ( i,j ) = 0 .Proof. Since the real and imaginary parts of λ i and λ j are all nonnegative, if λ i = λ j then theeigenvalues of M λ i ,m i M λ i ,m i all differ from the eigenvalues of M λ j ,m j M λ j ,m j . Accordingly, by [8,Chapter 8, Theorem 1 and Equation (11)], the matrix X in (6.7) is zero. (cid:3) Given Lemma 6.1, all that remains is to find the general formula for B ( i,j ) when λ i = λ j . We willsay that a Toeplitz p × q matrix is an upper-triangular Toeplitz matrix , if the only nonzero entriesappear on or above the main diagonal in their right-most p × p block if p ≤ q , and the top-most q × q block if p ≥ q (in the terminology of [8, Chapter 8] they are called regular upper-triangular,but we avoid this terminology because the term “regular" is already assigned in the present paperto another concept). Lemma 6.2.
Suppose λ i = λ j and m i ≤ m j . The dimension of the space of solutions of (6.7) isequal to(1) m i if λ i > ;(2) m i if λ i R ; AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 29 (3) m i if λ i < .Proof. We use [8, Chapter 8, Theorem 1] again for each of the cases.Suppose first that λ i > . If λ > then M λ,m M λ,m is similar to the Jordan matrix J λ ,m . Let U i and U j be invertible matrices such that U j M λ j ,m j M λ j ,m j U − j = J λ j ,m j and U i M λ i ,m i M λ i ,m i U − i = J λ i ,m i . For a matrix X satisfying (6.7), set e X = U − j XU i so that, by (6.7), e XJ λ j ,m j = J λ i ,m i e X. (6.8)It is shown in [8, Chapter 8, Theorem 1] that the space of solutions of (6.8) consists of upper-triangular Toeplitz matrices. Therefore, the space of solutions of (6.8) has dimension m i , whichshows item (1) because X U − j XU i gives an isomorphism between the space of solutions of (6.8)and the space of solutions of (6.7).Let us now suppose λ i R or λ i < . If λ R or λ < then M λ,m M λ,m = J λ ,m ⊕ J λ ,m . (6.9)For a matrix X satisfying (6.7), consider the × block matrix partition ( X ( r,s ) ) r,s ∈{ , } of X whose blocks are all m i × m j matrices. It is shown in [8, Chapter 8, Theorem 1] that the spaceof solutions of (6.7) with M λ i ,m i M λ i ,m i and M λ j ,m j M λ j ,m j of the form in (6.9) consists of matrices ( X ( r,s ) ) r,s ∈{ , } for which each X ( r,s ) is an upper-triangular Toeplitz matrix, where, moreover, if λ i = λ i then X (1 , = X (2 , = 0 . Accordingly, if λ i R (respectively λ i < ) then solutions to(6.7) are determined by two (respectively four) upper-triangular Toeplitz m i × m i matrices. Items(2) and (3) follow because each upper-triangular Toeplitz m i × m i is determined by m i variables. (cid:3) Corollary 6.3. If m i ≤ m j , λ i = λ j = λ and λ = 0 then the matrices B ( i,j ) and B ( j,i ) are describedby one of three formulas, where the correct formula depends on λ . In the formulas below , as before, T m denote the m × m nilpotent Jordan block J ,m .(1) If λ > then B ( i,j ) and B ( j,i ) respectively equal (6.10) m j − m i columns z }| { · · · ... ...... ... · · · m i − P k =0 b k T km i , and − ǫ i ǫ j m i − P k =0 b k T km i · · · ... ... · · · m j − m i rows, for some coefficients { b k } .(2) If λ R then B ( i,j ) = m j − m i columns z }| { · · · ... ...... ... · · · m i − P k =0 a k T m i m j − m i columns z }| { · · · ... ...... ... · · · m i P k =0 b k T m i , (6.11) and B ( j,i ) = − ǫ i ǫ j m i P k =0 a k T m i · · · · · · · · · · · · · · · ... ... · · · · · · · · · · · · · · · m i P k =0 b k T m i · · · · · · · · · · · · · · · ... ... m j − m i rows ) m j − m i rows, (6.12) for some coefficients { a k , b k } .(3) If λ < then B ( i,j ) = m j − m i columns z }| { · · · ... ...... ... · · · m i − P k =0 c k T km i m i − P k =0 (cid:16)P kr =0 e r (cid:17) T km i m j − m i columns z }| { · · · ... ...... ... · · · m i P k =0 d k T km i m i P k =0 f k T km i , (6.13) and B ( j,i ) = ǫ i ǫ j − m i P k =0 c k T km i m i P k =0 e k T km i · · · · · · · · · · · · ... ... · · · · · · · · · · · · m i P k =0 (cid:16)P kr =0 d r (cid:17) T km i − m i P k =0 f k T km i · · · · · · · · · · · · ... ... m j − m i rows ) m j − m i rows, (6.14) for some coefficients { a k , b k , c k , d k , e k , f k } .Proof. Using the formula for B ( i,j ) given in (6.10), (6.11), and (6.13), it is straightforward to checkthat (6.7) holds with X = B ( i,j ) . Moreover, this formula for B ( i,j ) is the most general formula withthis property because, by Lemma 6.2, it has the maximum number of parameters possible. Lastly,the formula for B ( j,i ) given in (6.10), (6.12), and (6.14) is obtained through another straightforwardcalculation by applying (6.6) directly to the formula for B ( i,j ) . (cid:3) To simplify notation in the following lemma, for an integer q , we let [ q ] denote the residue of q modulo 2, that is, [ q ] = 0 if q is even and [ q ] = 1 if q is odd. AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 31
Lemma 6.4. If m i ≤ m j and λ i = λ j = 0 then B ( i,j ) = m j − m i columns z }| { · · · ... ...... ...... ... · · · c c · · · · · · c m i c c · · · · · · c m i − c c · · · c m i − ... . . . c · · · c m i − ... . . . . . . ... · · · · · · c [ m i ] , (6.15) and B ( j,i ) = − ǫ i ǫ j c [ m i +1] c [ m i +2] · · · · · · c [2 m i ] m i c [ m i +2] c [ m i +3] · · · · · · c [2 m i ] m i − c [ m i +3] c [ m i +4] · · · c [2 m i ] m i − ... . . . c [ m i +4] · · · c [2 m i ] m i − ... . . . . . . ... · · · · · · · · · c [2 m i ] · · · · · · · · · · · · · · · ... ... · · · · · · · · · · · · · · · m j − m i rows (6.16) for some coefficients { a k , b k , c k , c k } .Proof. Let us refer to the main diagonal of the upper right m i × m i block in each matrix B ( i,j ) and B ( j,i ) as that matrix’s reference diagonal.Notice that equations (6.2) and (6.3) hold in the present context with λ i = λ j = 0 and ǫ i = ǫ j .Let us assume ǫ i = ǫ j , noting that for the other case, where ǫ i = ǫ j , we would first change the signof the right side of (6.2) and (6.3) and then proceed with exactly the same calculations.Applying (6.2), we find that the last row of B ( i,j ) contains only zeros below the reference diagonal,and, applying (6.3), we find that the first column of B ( i,j ) contains only zeros to the left of thereference diagonal. Similarly, by (6.2) and (6.3), the first column and last row of B ( j,i ) contain zerosin their entries that are below or to the left of the reference diagonal. After substituting 0 in forthose entries, applying (6.2) again, we now find that the second to last row of B ( i,j ) (or of B ( j,i ) )contains only zeros below (or to the left of) the reference diagonal, whereas, by applying (6.3) again,we find that the second column of B ( i,j ) (or of B ( j,i ) ) contains only zeros to the left of (or below) thereference diagonal. Repeating this analysis, we eventually find that all entries in B ( i,j ) and B ( j,i ) that are below or to the left of the reference diagonal are zero.Let us now calculate the restrictions that (6.2) and (6.3) impose on the remaining nonzero entriesin B ( i,j ) and B ( j,i ) . For the next observations, we use the term secondary transpose to refer to thetransformation of square matrices described by reflecting their entries over the secondary diagonal,that is, sending the ( i, j ) entry of an m × m matrix to the ( m + 1 − j, m + 1 − i ) entry. Applying (6.2),we see that upper left ( m i − × ( m i − block of the upper right m i × m i block of B ( i,j ) is equal to − (or − ǫ i ǫ j in the general case) times the secondary transpose of the upper left ( m i − × ( m i − block of B ( j,i ) . Similarly, applying (6.3), we see that lower right ( m i − × ( m i − block of theupper right m i × m i block of B ( i,j ) is equal to − (or − ǫ i ǫ j in the general case) times the secondarytranspose of the lower right ( m i − × ( m i − block of B ( j,i ) . These last two observations, takentogether, complete this proof. (cid:3) Corollary 6.5.
For all i ∈ { , . . . , γ } , B ( i,i ) = (cid:16)P ⌈ m i / ⌉ k =1 a k T m i − k +1 m i (cid:17) I alt ,m i if λ i = 0 m i − P k =0 a k T km i m i − P k =0 (cid:16)P kr =0 a r (cid:17) T km i if λ i < otherwise, (6.17) where I alt ,m denotes the m × m diagonal matrix with a 1 in its odd columns and a -1 in its evencolumns.Proof. This follows immediately from the formulas in Corollary 6.3 and Lemma 6.4 with i = j . (cid:3) The previous results provide a general formula for matrices in A o . We now focus on obtaining ageneral formula of a subspace A s satisfying (6.1). Lemma 6.6.
Either dim( A ) − dim( A o ) = 1 or dim( A ) − dim( A o ) = 2 , and the latter case occursif and only if there exists a matrix X in A satisfying XCH − ℓ + CH − ℓ X T = 2 CH − ℓ ⇔ ( X − I ) T H ℓ C − + H ℓ C − ( X − I ) = 0 , (6.18) X T H ℓ C + H ℓ CX = 0 . Proof.
Define A o := (cid:8) X (cid:12)(cid:12) XCH − ℓ + CH − ℓ X T = 0 (cid:9) and A o := (cid:8) X (cid:12)(cid:12) X T H ℓ C + H ℓ CX = 0 (cid:9) . Since A o = A o ∩ A o , dim( A o ) + dim( A o + A o ) = dim( A o ) + dim( A o ) , (6.19)and, letting C I denote span { I } , since A = ( A o + C I ) ∩ ( A o + C I ) , dim( A ) + dim ( A o + A o + C I ) = dim ( A o + C I ) + dim ( A o + C I )= dim( A o ) + dim( A o ) + 2= dim( A o ) + dim( A o + A o ) + 2 , where this last equation holds by (6.19). Therefore, dim( A ) − dim( A o ) = dim( A o + A o ) − dim ( A o + A o + C I ) + 2 , and hence dim( A ) − dim( A o ) = ( if I A o + A o if I ∈ A o + A o . In particular, dim( A ) − dim( A o ) = 2 if and only if there exists X ∈ A o such that ( I − X ) ∈ A o ,which is equivalent to (6.18). (cid:3) Lemma 6.7. If C = M m,λ and λ = 0 then dim( A ) − dim( A o ) = 1 .Proof. We assume that ( H ℓ , C ) is in the canonical form of Theorem 4.1, so H ℓ = S m , where S m isdefined in (4.1). Fix a subspace A s of A satisfying (6.1). To produce a contradiction, let us assumethat dim( A ) − dim( A o ) = 1 . By Lemma 6.6, we can assume that there exists a matrix X in A s satisfying (6.18). Since H ℓ C − and H ℓ C are symmetric, condition (6.18) is fundamentally relatedto the two symmetric forms Q and Q defined by Q ( v, w ) := w T H ℓ C − v and Q ( v, w ) := w T H ℓ Cv.
AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 33
Note that Q ( v, w ) = Q (cid:0) CCv, w (cid:1) = Q (cid:0) A v, w (cid:1) , where A is, again, the antilinear operator represented by C .Let us now work instead with respect to a basis that is orthonormal with respect to Q , that is,letting L denote the matrix representing the linear operator A in this basis, we have Q ( v, w ) = w T v and Q = w T Lv in this new basis. By [8, Chapter 11.3, Corollary 2], we can assume without loss of generality that L = ( ( I + iS m ) J λ,m ( I − iS m ) if λ > ( I + iS m ) J λ,m ( I − iS m ) ⊕ ( I + iS m ) J λ,m ( I − iS m ) otherwise.The second equation in (6.18) implies that X is in the Lie algebra of the transformation groupthat preserves Q , whereas the first equation of (6.18) implies that X − I is in the Lie algebra of thetransformation group that preserves Q . That is, with respect to the new basis, ( X − I ) = − ( X − I ) T and X T L + LX = 0 ,which is equivalent to ( X − I ) = − ( X − I ) T and [ X, L ] = 0 . (6.20)Defining the pair of matrices ( S, J ) by ( S, J ) = ( ( I + iS m , J λ,m ) if λ > I + iS m ) ⊕ ( I + iS m ) , J λ,m ⊕ J λ,m ) otherwise,the condition [ X, L ] = 0 is equivalent to (cid:2) S − XS, J (cid:3) = 0 . (6.21)Solving for the matrix X in [ X, L ] = 0 is a classical problem of Frobenious whose general solutionis given in [8, Chapter 8]. In [8, Chapter 8], a formula is given for matrices that commute witha Jordan matrix such as J , so we have rewritten [ X, L ] = 0 as in (6.21), in order to apply thesolution of [8, Chapter 8] directly. The formula in [8, Chapter 8] gives that, after partitioning thematrix S − XS into size m × m blocks, each block of S − XS in this partition is an upper-triangularToeplitz matrix. If X is a Toeplitz matrix then ( I + iS m ) X ( I − iS m ) is symmetric because S m X and XS m are both symmetric whereas X T = S m XS m . Accordingly, letting X ′ denote the upperleft m × m block of X , since ( I − iS m )( X ′ − I )( I + iS m ) is Toeplitz, X ′ − I = 14 ( I + iS m ) (cid:2) ( I − iS m )( X ′ − I )( I + iS m ) (cid:3) ( I − iS m ) (6.22) = (cid:18)
14 ( I + iS m ) (cid:2) ( I − iS m )( X ′ − I )( I + iS m ) (cid:3) ( I − iS m ) (cid:19) T = ( X ′ − I ) T . By (6.20) and (6.22), X ′ = I , which contradicts the upper left m × m block of the second matrixequation in (6.18). (cid:3) With Lemmas 6.6 and 6.7 established we now give a general formula for a subspace A s of A satisfying (6.1). Lemma 6.8.
For a subspace A s of A satisfying (6.1) , dim( A s ) = 2 if and only if C is nilpotent.In particular, if C = J ,m ⊕ . . . ⊕ J ,m γ then, to satisfy (6.1) , we can take the subspace A s of A spanned by the identity matrix and thematrix γ M i =1 D m i , where, for an integer m , D m denotes the m × m diagonal matrix defined by (6.23) D m := Diag (cid:16) m , m − , . . . , m − m + 1 (cid:17) . Proof.
Suppose that ( H ℓ , C ) is in the canonical form of Theorem 4.1, specifically such that C = J λ ,m ⊕ · · · ⊕ J λ γ ,m γ , and suppose that dim( A s ) = 2 . As is shown in the proof of Lemma 6.6, we can assume withoutloss of generality that there exists a matrix X in A s satisfying (6.18). In particular, partitioning X into a block matrix whose diagonal blocks X ( i,i ) are size m i × m i , the blocks X ( i,i ) satisfy X ( i,i ) M m i ,λ i N m i ,λ i + M m i ,λ i N m i ,λ i X T ( i,i ) = 2 M m i ,λ i N m i ,λ i (6.24)and X T ( i,i ) N m i ,λ i M m i ,λ i + N m i ,λ i M m i ,λ i X ( i,i ) = 0 . (6.25)Lemma 6.7 implies that (6.24) and (6.24) are consistent if and only if λ i = 0 , and hence if A s = 2 then C is nilpotent.Conversely, if C is nilpotent then λ = · · · = λ γ = 0 . Hence, by (4.2) and (4.3) the relations(6.24) and (6.25) can be rewritten as(6.26) X ( i,i ) J ,m i S m i + J ,m i S m i X T ( i,i ) = 2 J ,m i S m i and X T ( i,i ) S m i J ,m i + S m i J ,m i X ( i,i ) = 0 for each i individually. Assuming that B ( i,i ) = Diag (cid:0) x i , . . . x im i (cid:1) , by comparing the entries of (6.26)with the help of the expressions for matrices J ,m and S m i from (4.1), one gets that (6.26) isequivalent to x ij + x im i − j = 2 ∀ ≤ j ≤ m − , (6.27) x ij + x im − j +2 = 0 ∀ ≤ j ≤ m. Finally, it is clear that taking X ( i,i ) = D m i , where D m i is as in (6.23), satisfies (6.27) which completesthe proof. (cid:3) As a direct consequence of the previous Lemma, since for nilpotent C we have A = A o + C I ,one gets immediately the following Corollary 6.9. If C is not nilpotent then in (4.5) one can take η ′ = η . Now we prove one more result.
Lemma 6.10. If H ℓ and C are in the canonical form prescribed by Theorem 4.1 and C = 0 then dim( A ) ≤ n − n + 6 . (6.28) Moreover, this bound is attained if and only if ( ℓ, A ) can be represented by the pair ( H ℓ , C ) in thecanonical form of Theorem 4.1 with C = J , ⊕ n − copies z }| { J , ⊕ · · · ⊕ J , . (6.29) AXIMAL DIMENSION OF SYMMETRIES OF HOMOGENEOUS 2-NONDEGENERATE CR STRUCTURES 35
Proof.
Assume that dim( A ) ≥ n − n + 6 , (6.30)and that ( H ℓ , C ) are in the canonical form of Theorem 4.1. We will still use the notation of (4.4),in particular referring to the sequence ( λ , . . . , λ γ ) .Suppose that the λ i s are not all the same. Without loss of generality, we can assume that ( λ , . . . , λ γ ) is enumerated so that there exists an integer k such that λ = . . . = λ k and λ j = λ ∀ j > k. (6.31)Define s = k X i =1 [ number of rows in M λ i ,m i ] where k is as in (6.31). By Lemma 6.1, for every matrix B in dim( A o + span { I } ) , the upper right ( s ) × ( n − − s ) block and the lower left ( n − − s ) × ( s ) block of B is zero. Moreover, since the λ i s are not all zero, there is at least one index i such that B ( i,i ) has zeros on its main diagonal.Accordingly, if the λ i s are not all the same, then dim( A o ) + 1 = dim( A o + span { I } ) ≤ ( n − − s ( n − − s ) . Since n − ≤ j ( n − − j ) ∀ ≤ j < n − , it follows that dim( A ) = dim( A o ) + 1 ≤ ( n − − s ( n − − s ) ≤ ( n − − n + 4 = n − n + 5 , where the identity dim( A ) = dim( A o ) + 1 follows from Lemma 6.8 and the assumption that the λ i s are not all the same. Clearly, this contradicts (6.30), so if (6.30) holds then there exists a value λ ∈ C such that λ = λ i ∀ i. (6.32)If (6.32) holds with λ = 0 then Corollaries 6.3 and 6.5 imply that each matrix B in A o is fullydetermined by its entries above the main diagonal, and hence, applying Lemma 6.8, dim( A ) ≤ ( n − n − < n − n + 6 , ∀ n ≥ Therefore, if (6.32) holds with λ = 0 then our assumption (6.30) fails.In other words, – assuming for a moment that (6.30) can be satisfied, which we will prove below bygiving an explicit example – if dim( A ) is maximized then we can assume without loss of generalitythat C = J ,m ⊕ · · · ⊕ J ,m γ with m ≥ · · · ≥ m γ . (6.33)For B in A o , let us partition B as is done in Lemma 6.4. By Lemma 6.4, for i < j the B ( i,j ) and B ( j,i ) blocks are together determined by m j parameters, whereas, by Corollary 6.5, the B ( i,i ) blockis determined by ⌈ m i ⌉ parameters, where ⌈ m i ⌉ denotes the ceiling function, i.e. the smallest integernot less than m i . Hence, by counting the number of parameters determining B , Lemma 6.4 andCorollary 6.5 imply that if (6.33) holds then dim( A o ) = γ X k =1 (cid:16)l m k m + 2( k − m k (cid:17) . (6.34)Let r ∈ { , . . . , γ } be an integer such that m i = 1 ∀ i > r, and to compare with C , let us also consider the matrix C ′ = J ,m ⊕ · · · ⊕ J ,m r − ⊕ J , ⊕ · · · ⊕ J , . In other words, C ′ is obtained from C by replacing the last nonzero block on the diagonal of C with zeros. We will compute the dimension of A o corresponding to the case where C = C ′ , but,since are going to compare this to the sum in (6.34), for clarity let A ′ denote the algebra that wewould otherwise denote by A o corresponding to this case where C = C ′ , and let A o still denotethe algebra refered to in (6.34).Notice that the k th summand in (6.34) counts the number of parameters determining the blocks B ( i,j ) of a matrix B in A o for which max { i, j } = k . If we compare the general formula for a matrix B in A o to that of a matrix B ′ in A ′ , the only difference appears in the blocks B ( i,j ) of B for which max { i, j } = r , and hence a formula for dim( A ′ ) should match the formula in (6.34), except thatthe r th summand will change. Using Lemma 6.4 and Corollary 6.5, it is however straightforwardto work out exactly how this r th summand of (6.34).Specifically, in replacing the formula for B with the formula for B ′ , the B ( r,r ) block is replaced withthe m r × m r matrix having m r independent parameters, whereas, for all i < r , B ( i,r ) (respectively B ( r,i ) ) is replaced with a matrix having m r independent parameters in its first row (respectivelycolumn) and zeros elsewhere. Accordingly, dim( A ′ ) = dim( A o ) − (cid:16)l m r m + 2( r − m r (cid:17) + m r + 2( r − m r ≥ dim( A o ) . (6.35)Since equality holds in (6.35) if and only if m r = 1 , the dimension of A o is maximized with C asin (6.33) if and only if C = J ,m ⊕ n − − m copies z }| { J , ⊕ · · · ⊕ J , , (6.36)in which case, by (6.34), dim A o = l m m + n − m X k =2 (2 k −
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Email address : [email protected] URL : Igor Zelenko, Department of Mathematics Texas A&M University College Station Texas, 77843USA
Email address : [email protected] URL ::