tube realizations of hyperquadrics

arX

iv:0

906.

5549

v2 [

mat

h.C

V]

8 Ju

l 201

0

Classification of commutative algebras andtube realizations of hyperquadrics

By Gregor FelsandWilhelm Kaup

ABSTRACT: In this paper we classify up to affine equivalence all local tube realizationsof real hyperquadrics inCn. We show that this problem can be reduced to the classifica-tion, up to isomorphism, of commutative nilpotent real and complex algebras. We alsodevelop some structure theory for commutative nilpotent algebras over arbitrary fieldsof characteristic zero.

1. Introduction

It is a well-known fact that every real-analytic manifoldM together with an invo-lutive CR-structure(HM,J) admits at least locally a generic embedding into someCn,such that the CR-structure induced from the ambient spaceCn coincides with the originalone. A particularly important class of CR-submanifolds ofCn are the so-called CR-tubes,i.e., product manifoldsM = iF + IRn ⊂ iIRn ⊕ IRn = Cn together with the inheritedCR-structure, whereF ⊂ IRn is a submanifold. One important point here is that the CR-structure ofiF + IRn is closely related to real-geometric properties of the baseF , whichare often easier to deal with, see e.g. [9]. In general, a CR-manifold will not admit a localrealization inCn as a CR-tube. On the other hand, as shown by the example of the sphereS = {z ∈ Cn : |z1|2 + · · · + |zn|2 = 1}, it is not immediate, thatS does admit severalaffinely inequivalent local tube realizations, see [5].

It is quite obvious that the existence of a CR-tube realization for a given CR-manifold(M,HM,J) is related to the presence of certain abelian subalgebrasv in hol (M), theLie algebra of infinitesimal CR-transformations, which areinduced by all real translationsz 7→ z + x, x ∈ IRn. It is perhaps a little bit more subtle to give sufficient and neces-sary conditions for abelian Lie subalgebras ofhol (M) to give local CR-tube realization ofM . This has been worked out in [10]. For short, let us call everysuch abelian subalgebraa ‘qualifying’ subalgebra ofhol (M). Curiously, the notion of locally affine equivalenceamong various (germs of) tube realizations for a givenM proved to be less appropriate forthe study of CR-manifolds as it is too fine for many applications: Even a homogeneous CR-manifold may admit an affinely non-homogeneous tube realization, and in such a case theaforementioned equivalence relation will give rise to uncountable many equivalence classesof tube realizations. A coarser equivalence relation has been introduced in [10] which seemsto be most natural in the context of CR-tubes. Moreover, it isquite surprising that under cer-tain assumptions the pure geometric question of globally affine equivalence can be reducedto the purely algebraic problem of classifying conjugacy classes of certain maximal abeliansubalgebras ofhol (M) with respect to a well-chosen groupG.

The purpose of this paper is to give a full classification of all local CR-tube realiza-tions of every hyperquadric

Sp,q ={[z] ∈ IP(Cm) : |z1|2 + · · ·+ |zp|2 = |zp+1|2 + · · ·+ |zm|2

}

in the complex projective spaceIP(Cm), wherem := p + q andp, q ≥ 1, applying thegeneral methods from [10]. The (compact) hyperquadricSp,q is the unique closed orbit of

2000 Mathematics Subject Classification: 32V30, 13C05.

http://arxiv.org/abs/0906.5549v2

2 Classification of commutative algebras

SU(p, q) ⊂ SL(m,C) acting by biholomorphic transformations onIP(Cm). In this situa-tion hol (Sp,q) = hol (Sp,q, a) ∼= su (p, q) holds for everya ∈ Sp,q. It is well known andeasy to see thatSp,q is locally CR-equivalent to the affine real quadric inCr, r := m− 1,

{z ∈ Cr : Im(zr) =

∑

1≤k<p

|zk|2 −∑

p≤k<r

|zk|2}

with non-degenerate Levi form of type(p−1, q−1). Therefore the classification problemfor local tube realizations for both classes is the same, compare also [5], [10], [11], [12],[13], [14], [20] for partial results in this context.

We have shown in [10] that every abelian subalgebrav ⊂ hol (M), which yieldsa tube realization, determines an involutionτv : hol (M) → hol (M). For a given hy-perquadricSp,q however, it turns out that all arising involutionsτv are conjugate ing :=hol (Sp,q) ∼= su(p, q). We therefore fix an involutionτ : g → g (with fixed point setg τ ∼= so (p, q)) once and for all and reduce the classification of tube realizations to thealgebraic classification of all maximal abelian subalgebras v of g contained in the(−1)-eigenspace of the non-riemannian symmetric pair(g , τ) up to conjugation bySU(p, q) (infact, up to conjugation by the normalizerG of SU(p, q) in SL(m,C), but these two groupsdiffer only if p = q, and in this case the classification with respect to one groupcan easilybe derived from the classification up to conjugation with respect to the other). In contraryto the special case of toral maximal subalgebras (i.e., Cartan subalgebras)t ⊂ g only lit-tle is known about the general case of arbitrary abelian maximal subalgebrasv ⊂ g . Thekey point here is that after some reduction procedures the conjugacy class of a maximalabelian subalgebrav ⊂ g−τ is completely determined by itsD-invariant (which is a firstrough invariant ofv , determined by its toral part, see (4.14) for more details) and a finite setof maximal abelian subalgebrasn, consisting of ad-nilpotent elements only insu(p, q)andsl (m,C). Hence, the classification task reduces essentially to the classification of ad-nilpotent abelian subalgebrasn j up to conjugation inSU(p, q), resp.SL(m,C). By ourconstructions, to every suchn there is associated a finite-dimensional commutative asso-ciative nilpotent algebraN over IF = IR, resp.IF = C. Our main algebraic result is thenthe following

1.1 Theorem. Let G = SU(p, q) or G = SL(m,C) and τ : G → G an involutiveautomorphism withGτ ∼= SO(p, q), resp.Gτ ∼= SO(m,C). For any two maximal abelianad-nilpotent subalgebrasn1, n 2 ⊂ g , contained in the(−1)-eigenspace ofτ , the followingconditions are equivalent:

(i) n1 andn2 are conjugate by an element inG,(ii) n1 andn2 are conjugate by an element inGτ ,

(iii) the associated algebrasN1 andN2 are isomorphic as abstractIF-algebras.

The commutative nilpotent algebrasN occurring in the above theorem all have a 1-di-mensional annihilator. On the other hand, given any nilpotent commutativeIF-algebraNwith 1-dimensional annihilatorA, we construct an invariant ofN which is a certain non-degenerate symmetric 2-formbπ : N/A × N/A → A. Depending on the type ofbπ, thealgebraN gives rise to a maximal abelian subalgebra ing−τ (g andτ as in the preceedingtheorem) and in turn to a tube realization of a hyperquadric.

Summarizing, the classification of all local tube realizations of hyperquadrics is es-sentially equivalent to the classification of finite-dimensional nilpotent commutativeIF-algebras up to isomorphism. In this paper we give an explicitclassification for low valuesof p or q, i.e., we carry out all those cases(p, q) where there are only finitely many isomor-phy classes. The classification in terms of explicit lists seems to be hopeless in the generalcase. For big values ofp andq there are always uncountable many inequivalent nilpotentcommutativeIF-algebras.

and tube realizations of hyperquadrics 3

As the algebraic results developed in this paper might be of broader interest, we col-lect in the Appendix all relevant results concerning the finestructure of nilpotent commu-tative algebras. These are formulated in a more general setup (e.g. over arbitrary fields ofcharacteristic zero).

For certain applications one would like to have explicit defining equations for thevarious tube realizations, determined by qualifying subalgebrasv ⊂ su (p, q). One of ourmain geometric results is a procedure which produces for every qualifying v an explicitdefining equation which describes the tube realizationiFv ⊕ Vv of Sp,q. In that way weobtain quite transparent formulae, reflecting the algebraic structure ofv .

The paper is organized as follows: In Section 2 we relate our results to existing resultsin the literature, in particular to those in [5], [11], [12].In Section 3 we recall the necessarytools from [10] and give a short outline of the classificationprocedure. In particular weintroduce certain abelian subalgebrasv ⊂ su(p, q) asqualifying MASAs – these are thealgebraic objects to be classified. In Section 4 we split every MASA v into its toralv red

and its nilpotent partvnil and classify the centralizers ofv red. Crucial for the classificationis the decomposition given by Lemma 4.6 that leads to a combinatorial invariantD(v) thatwe call theD-invariant ofv . For fixedp, q the setDp,q of all D-invariants of MASAs insu(p, q) is finite, but still, in general there are infinitely many equivalence classes of MASAsin su (p, q) with a fixedD-invariant. In Sections 5 and 6 we study MANSAs (maximalcommutative associative nilpotent subalgebras) insu(pj , qj) andsl(mj,C) as these are thebuilding blocks for general MASAs insu (p, q). In Section 7 we demonstrate briefly howfor every MANSAv ⊂ su(p, q) with corresponding tube realizationiF + IRn ⊂ Cn ofSp,q the baseF ⊂ IRn can be written in terms of a canonical equation. In section 9 wegive two examples of MANSAs and in the Appendix 10 we collect several algebraic toolsneeded in the paper that might also be of independent interest.

2. Preliminaries

In the following we characterize algebraically the local tube realizations of the hy-perquadricS = Sp,q ⊂ IPr := IP(Cr+1) with p, q ≥ 1 andr := p + q − 1, compare (3.3)in [10]. SinceSp,q andSq,p only differ by a biholomorphic automorphism of the projectivespaceIPr it would be enough to discuss the casep ≥ q.

The local tube realizations ofS up to affine equivalence in the casesq = 1, 2 wereobtained in the papers [5], [11] respectively by solving certain systems of partial differentialequations coming from the Chern-Moser theory [3]. A classification of the caseq = 3 hasbeen announced in [12], proofs are intended to appear in the forthcoming book [14].

In this paper we give a classification for arbitraryp, q. It turns out that this, after sev-eral reducing steps, essentially boils down to the classification of abstract abelian nilpotentreal and complex algebrasN of dimensionr := p + q − 1 with 1-dimensional annihilator.For small values ofq these can be determined explicitly while for largep, q this appearsto be hopeless. On the other hand, we associate to every localtube realization ofSp,q acombinatorial invariantD out of a finite setDp,q in such a way that for any two local tuberealizations the equations for the corresponding tube bases F, F ⊂ IRr are essentially ofthe same type (up to some polynomial terms in the coordinatesof IRr coming from theaforementioned different abelian nilpotent algebras arising naturally in thiscontext).

To compare this with the known results in caseq ≤ 3 let us introduce the numbern := r− 1 = p+ q− 2, so that every local tube realizationT of S = Sp,q is a hypersurfacein Cn+1 with CR-dimensionn, andr is the rank of the Lie algebrahol (S). Let furthermorecp,q be the cardinality of all affine equivalence classes of closed tube submanifolds inCr

that are locally CR-equivalent toS.


In caseq = 1, that is the case of the standard sphere inCp, [5] impliescp,1 = p+2 = n+3.In casep ≥ q = 2 we haven = p and the explicit list of tube realizations in [11] impliesthe estimatecp,2 ≤ p(p+ 9)/2. Our considerations will give

(2.1) cp,2 = 5p + k(p− k)− δp,2 with k := ⌈p/2⌉ ,

where for everyt ∈ IR the ceiling ⌈t⌉ is the smallest integer≥ t andδ is the Kroneckerdelta. Therefore, the list in the Theorem of [11] p. 442 must contain repetitions. Indeed, intype 7) for everys the parameterst and t := n − 2 + s − t give affinely equivalent tuberealizations. The same holds in casep = 2 for s = 1 in type 1) ands = 0 in type 2).In casep ≥ q = 3 it has been announced in [12] thatcp,3 is finite if and only ifp ≤ 5.

In casep, q ≥ 4 we show thatcp,q always is infinite. Except forc4,4, this has alreadybeen announced in [12].

3. The algebraic setup

For fixedp, q ≥ 1 with m := p + q ≥ 3 let E ∼= Cm be a complex vector space andh : E×E → C a hermitian form of type(p, q) (p positive andq negative eigenvalues). Sinceany two hermitian forms of the same type onE are equivalent (up to a positive multiplicativeconstant) with respect to the groupL := SL(E) ∼= SL(m,C) it does not matter whichh hasbeen chosen above. More important for computational purposes is to choose a convenientvector basis ofE in such a way that the corresponding matrix representation of h is optimallyadapted.

The complex Lie groupL acts in a canonical way transitively on the complex projec-tive spaceZ := IP(E) with finite kernel of ineffectivity (the center ofL). The subgroup

(3.1) G := {g ∈ L : h(gz, gz) = ±h(z, z) for all z ∈ E}

is a real Lie group with(1 + δp,q) connected components acting transitively on the hyper-surface

S = Sp,q := {[z] ∈ IP(E) : h(z, z) = 0} .We write for the corresponding Lie algebras

l = sl(E) and g := su (E, h) = {ξ ∈ l : Reh(ξz, z) = 0 for all z ∈ E} .

As a matter of fact,l coincides with the complex Lie algebrahol (Z) of holomorphic vectorfields onZ. Further, for everya ∈ S the canonical inclusionsg → hol (S) → hol (S, a)turn out to be isomorphisms, and therefore we identifyhol (S) with g . With σ : l → l wedenote the antilinear involutive Lie automorphism withFix(σ) = g .

The hyperquadricS = Sp,q satisfies the assumptions of Theorem 7.1 in [10], and thevarious tube realizations are, up to the global affine equivalence as defined in [10] Definition6.1, in a 1-1-correspondence toGlob(S, a)-conjugacy classes of certain abelian subalgebrasin hol (S, a) (see [10] for the definition ofGlob(S, a) and its basic properties). In the caseunder considerationGlob(S, a) = Ad(G) ⊂ Aut(hol (S, a)) for everya ∈ S; our task thenwill be to classify up to the action ofAd(G) = Ad(NL(g)) on l all σ-invariant abeliansubalgebrase ⊂ l which have an open orbit inZ. Every suche automatically has complexdimensionr := m−1 and is maximal abelian inl by Lemma 2.1 in [10].

Every involutionτv : (S, a) → (S, a) extends to a global involutionτv : Z → Z.Moreover, any two such involutions are conjugate by an element ofG (even of the connectedidentity componentSU(E, h) ofG), compare [10]. The search can therefore be restricted by


fixing once and for all an involutionτ of S whose fixed point setSτ = Fix(τ) is not emptyand has dimensionr− 1. Such aτ has a unique extension to an antiholomorphic involutionof IP(E) that comes from a conjugationE → E, z 7→ z, that is,τ [z] = [z] for all [z] ∈ S.By our results it is enough to classify up to conjugation byG all abelian Lie subalgebrasv ⊂ g−τ with εa(v) = T−τ

a S for a given pointa ∈ S. Thesev are automatically maximalabelian ing ∼= su(p, q) and have dimensionr = m−1 = rank(g).

3.2 Setup For the rest of the paper we fix the following notation: Forp, q andm =p + q as above,E is a complex vector space of dimensionm with (positive definite) innerproduct(z|w) (complex linear in the first and antilinear in the second variable). Further-more,τ : E → E, z 7→ z, is an (antilinear) conjugation onE with (z|w) = (w|z) for allz, w ∈ E. With the same symbolτ we also denote the induced antiholomorphic involutionof Z = IP(E) as well as of the complex Lie algebral := sl (E) = hol (IP(E)). In addition,(ej)1≤j≤m is an orthonormal basis ofE with ej = ej for all j, and the hermitian formh = hp,q onE is given by

(3.3) h(ej , ek) = ϑp,jδj,k with ϑp,j :={

1 if p ≥ j−1 otherwise .

The involutionτ on Z leaves the hyperquadricS = Sp,q invariant. Therefore alsog =su(p, q) = hol (S) is invariant under the involutionτ of l . As before,σ is the involution ofl defining the real formg of l . Clearly, the involutionsσ, τ commute onl .

SU(E, h) = SU(p, q) is the connected identity componentG0 of the groupG definedin (3.1). Only in casep = q the groupG is disconnected and then

(0 11−110

)∈ SL(2p,C) is

contained in the second connected component ofG.With End(E) we denote the endomorphism algebra ofE, a unital complex associative

algebra with involutiong 7→ g∗ (the adjoint with respect to the inner product). With respectto the Lie bracket[f, g] = fg − gh it becomes a reductive complex Lie algebra that isdenoted bygl(E) and containssl (E) as semisimple part. For everyz, w ∈ E we denote byz ⊗ w∗ ∈ gl(E) the endomorphismx 7→ (x|w)z. Then(z ⊗ w∗)∗ = w ⊗ z∗ is obvious.We also consider adjoints with respect toh and writeg⋆ for the endomorphism satisfyingh(gw, z) = h(w, g⋆z) for all w, z ∈ E.

3.4 The task Let G, g , l = sl(E), σ be as before and let a compatible conjugationτ :l → l be fixed once and for all, induced byz 7→ z onE. In order to classify all local tuberealizations ofS = Sp,q up to globally affine equivalence (compare Section 6 in [10]) wehave to classify all abelian subalgebrasv ⊂ g = hol (Sp,q) = su(p, q) up to conjugationwith respect toAd(G) which have the following property:

(A) The complexificationvC has an open orbit inZ = IP(E), that is,εa(vC) = TaZ forsomea ∈ Z (and hence even for somea ∈ S).

This condition, justified by Proposition 4.2 in [10], is of geometric nature but implies thefollowing purely algebraic properties:

(B) v is maximal abelian ing – we call every such subalgebra a MASA ing .(C) dim v = rank g = dimZ (= r := m− 1).(D) Ad(g)(v ) ⊂ g−τ for someg ∈ G.

Instead of classifying allG-conjugation classes ofv with property (A) we classify moregenerally the classes ofv satisfying (B) and (D), let us call themqualifying MASAs ing for the following. It will turn out a posteriori that thesev automatically satisfy (A) andhence also (C).

3.5 A short outline of the classification procedureWe proceed by analyzing the alge-braic structure of maximal abelian subalgebrasv ⊂ g . We will need some well known factsfrom the structure theory of semisimple Lie algebras (we refer to [17] and [21] as general


references). Write

Na(b) : = {x ∈ a : [x, b ] ⊂ b} for the normalizer and

Ca(b) : = {x ∈ a : [x, b ] = 0} for the centralizer

of any subalgebrab in a Lie algebraa . Also letZ(a) := Ca(a) be the center ofa . Theclassification idea is based on the observation that each maximal abelianv ⊂ g ⊂ End(E)has a unique decomposition into toral and nilpotent part, i.e.,v = v red⊕ vnil , wherev red

consists of semisimple andvnil of nilpotent elements inEnd(E). Each toral subalgebra, inparticularv red of a qualifying MASAv , gives rise to the real reductive subalgebraCg(v

red).On the other hand, the maximality ofv implies thatv red = Z(Cg(v

red)). Hence, there is anatural bijection between [theG-conjugacy classes of] toral partsv red of qualifying MASAsv and [theG-conjugacy classes of] certain reductive subalgebrasCg(v

red). A particularclass of qualifying MASAs is formed by thosev -s for which vnil = 0, i.e., v is a realCartan subalgebra ofg . It turns out that each of themin{p, q}+1 conjugacy classes (withrespect toG0 - or equivalently toG) of real CSAs has qualifying representatives.

It is well-known that general centralizers of tori incomplexsemisimple Lie algebrascan be characterized by subsets of simple roots. In our case however we have to classifyrealcentralizers. An additional complication is that not all (conjugacy classes of) centralizers oftori are of the formCg(v

red) with a qualifying MASAv . In the first part of our classifica-tion we provide a combinatorial tool giving an explicit characterization of these conjugacyclasses of centralizes which are related to qualifying MASAs.

The nilpotent partvnil of a qualifying MASA is contained in the semisimple part ofCg(v

red), more precisely, we have the following diagram:

(3.6)

v = v red ⊕ vnil

∩ ‖ ∩Cg(v

red)−τ = Z(Cg(vred))⊕ Css

g (vred)−τ .

Moreover,vnil is a maximal abelian and nilpotent subalgebra ofCssg (v

red) – we call suchsubalgebras MANSAs. An important observation is that the simple factors occurring inCssg (v

red) are not arbitrary real forms ofCl(vred): A simple factor inCss

g (vred) is isomorphic

either tosu(p′, q′) or slm(C). Consequently,vnil is a product of qualifying MANSAs insu(p′, q′) andslm(C). The classification of the last mentioned Lie subalgebras turns outto be equivalent to the classification of arbitrary real or complex associativecommutativenilpotent algebrasN with 1-dimensional annihilator.

We will analyze the consequences of condition (C) later on; one outcome is thatdim vnil = rank(Css

g (vred)), hencedim v = rank g and each suchvC has an open or-

bit in Z. Summarizing, our task then is reduced to the solution of thefollowing algebraicproblems:

[R] Classify up to conjugation all reductive subalgebrasr ⊂ g which are centralizers ofthe reductive part of a qualifying MASAv ⊂ g (compare 3.4).

[N] Given aτ -stable reductive subalgebrar = Z(r) ⊕ rss of the above type, classify upto conjugation all maximal abelian nilpotent subalgebrasn of r ss with n ⊂ (r ss)−τ .

4. Classification of the centralizersCg ( v red )

4.1 Qualifying Cartan subalgebras. Particular examples of MASAsv ⊂ g are maximaltoral subalgebras, i.e., Cartan subalgebras. This is precisely the case whenv = v red andvnil = 0. It is well-known thatsu (p, q) (with q ≤ p) hasq + 1 G0–conjugacy classesof real Cartan subalgebras. We need to know that each such conjugacy class contains aqualifying MASA of g :


4.2 Lemma. (Maximal toral subalgebras)Let g , τ andS = Sp,q with p ≥ q be as before.Then:

(i) Every complex Cartan subalgebra ofsl (E) has an open orbit inIP(E).(ii) EveryG0-conjugacy class of real Cartan subalgebras ing has a representative con-

tained ing−τ .(iii) To every maximal abelian subalgebrav ⊂ g−τ there exists a real Cartan subalgebra

h of g and ag ∈ G with Ad(g)(v red) ⊂ h ⊂ g−τ .

Proof. (i) is an easy consequence of the fact that there is only one conjugacy class ofcomplex CSAs insl (E) and that the subspace of all diagonal matrices insl (E) is one ofthem. For the proof of (ii) fix an arbitrary real CSAh of g . ThenhC is a complex CSA ofsl (E) and hence has an open orbit inIP(E). Therefore, by Propositions 4.2 and 3.2 in [10],there is a pointa ∈ S and an involutionθ of (S, a) with h ⊂ g−θ. Also, θ satisfies (3.1) in[10] and extends to an antiholomorphic involution ofIP(E). Thereforeτ = gθg−1 for someg ∈ G0, that is,Ad(g)(h ) ⊂ g−τ . Below, we also give an alternative, algebraic proof of(ii) without referring to results from [10]. Assertion (iii) follows from (ii) since there existsa CSAh of g with v red ⊂ h .

Every toral subalgebrat ⊂ g ⊂ End(E) has a unique decompositiont = t+⊕ t−into its compact and its vector part, that is, all elements int+ (t− respectively) have imag-inary (real respectively) spectrum as operators onE. Clearly the dimensions of these partsare invariants of theG-conjugacy class oft in g , and in the case of a semisimple Lie algebrag of Hermitian type, (as for instancesu(p, q)) dim t+ determines uniquely its conjugacyclass. For later use we construct explicitly for everyℓ = 0, 1, . . . , q a CSAℓh of g withℓh ⊂ g−τ andℓ = dim(ℓh−) .

4.3 Diagonal bases.Consider on the integer interval{1, 2, . . . ,m} the reflection definedby j 7→ j• := m+1−j and recall the choice of the orthonormal basis(ej)1≤j≤m and ofϑp,j as in (3.3). Fix an integerℓ with 0 ≤ ℓ ≤ q, a complex numberω with 2ω2 = i anddefine a new orthonormal basis(ℓfj)1≤j≤m of E by

ℓfj :=

{ej if ℓ < j < ℓ•

ωej + ωej• otherwise, for which ej =

{ ℓfj if ℓ < j < ℓ•

ω ℓfj + ω ℓfj• otherwise

is easily verified. Then for all1 ≤ j ≤ k ≤ m we have

(4.4) h(ℓfj ,ℓfk) =

{iδj,k• if j ≤ ℓϑp,jδj,k if ℓ < j < ℓ• .

Let ℓh ⊂ g = su(E, h) be the abelian subalgebra of all endomorphisms that are diagonalwith respect to(ℓfj). From

ℓfj ⊗ ℓf∗j + ℓfj• ⊗ ℓf∗

j• = ej ⊗ e∗j + ej• ⊗ e∗j•ℓfj ⊗ ℓf∗

j − ℓfj• ⊗ ℓf∗j• = iej ⊗ e∗j• − iej• ⊗ e∗j .

for all j ≤ ℓ we derive that the decompositionℓh =ℓ h+⊕ ℓh− into compact and vectorparts is given by

(4.5)

ℓh− =⊕

j≤ℓ

iIR(ej ⊗ e∗j• − ej• ⊗ e∗j

)

ℓh+ =(⊕

j≤ℓ

iIR(ej ⊗ e∗j + ej• ⊗ e∗j•

)⊕

⊕

ℓ<j<ℓ•

iIR(ej ⊗ e∗j ))

tr=0.

As a consequence,ℓh = Cg(ℓh ) is a CSA ofg with dim(ℓ h−) = ℓ andℓh ⊂ g−τ . This

gives a constructive proof for (ii) in Lemma 4.2.


4.6 The general casev red ⊂ v . We proceed to the general case whereCg(vred) may

containv red properly, that isvnil 6= 0. Givenv red, or equivalentlyCg(vred), after conjugat-

ing with an element ofSU(E, h) we may assume thatv red ⊂ ℓ h ⊂ g−τ for someℓ ≤ q asabove. In the complex situation, (i.e., for the centralizerin gC , or equivalently ingl(E)) itis well known that there is a unique direct sum decompositionwith summandsE 6= 0

(4.7) E =⊕

∈J

E such that Cgl(E)(vred) =

⊕

∈J

gl (E) .

The subspacesEj correspond to joint eigenspaces of the toral abelian subalgebrav red ⊂sl (E) with respect to certain functionalsγ ∈ (v red)∗. Sincev red, and in turnCl(v

red) isinvariant under the conjugationτ , we conclude that there is an involution 7→ of the indexsetJ with τE = E for all ∈ J . One key point here is that for every ∈ J the restrictionh of h toE +E is non-degenerate while in case 6= the spacesE andE are totallyh–isotropic and have zero intersection. (A priori, the non-degeneracy of the restrictionsh doesnot follow from the mereτ–invariance of the decompositionE =

⊕E. However, since

v red ⊂ ℓh for someℓ, one can show using root theory that every subspaceE ⊂ E occurringin the above decomposition is invariant under every orthogonal projectionℓfk⊗ ℓf∗

k , k ∈ J .With 4.4 then the non-degeneracy ofh follows).

4.8 Restrictions ofσ and τ to the simple factors of the centralizer. Choose a subsetL ⊂ J such thatJ = K ∪ L ∪ L is a disjoint union forK := { ∈ J : = }. For every ∈ K the subalgebrasl(E) ⊂ sl(E) is invariant underτ as well asσ, that is

sl (E)σ = su (E, h) sl (E)

τ = sl (Eτ )

(with su(E, h) = 0 = sl (E) in casedimE = 1). Also, for every ∈ L we haveE = τ(E), σ(sl (E)) = τ(sl (E)) = sl (E) andσ, τ ∈ AutIR(sl (E) ⊕ sl (E)) aregiven by

τ(x, y) = (τyτ , τxτ) and σ(x, y) = (−y⋆,−x⋆) ,wherex⋆, y⋆ are the adjoints ofx, y with respect to the hermitian formh. The symmetriccomplex bilinear formβ : E × E → C defined byβ(x, y) = h(x, τy) is non-degenerateand

(4.9)(sl (E)⊕ sl(E)

)σ ∼= RCIR

(sl (E)

) (sl (E)⊕ sl(E)

)στ ∼= RCIR

(so(Em

,C)),

whereRCIR is the forgetful functor restricting scalars fromC to IR. With these ingredients

we can state:4.10 Lemma. For the decomposition(4.7)we have

v red = Z(Cg(vred)) =

( ⊕

K∪L

iIR idE+E

)tr=0

⊕⊕

L

IR(idE− idE

)

Cssg (v

red) =⊕

K

su (E, h)⊕⊕

L

(sl (E)⊕ sl (E)

)σ

∼=⊕

K

su (p, q)⊕⊕

L

sl (m,C) ,

wherem = dim(E) and (p, q) for ∈ K is the type of the restrictionh on E. Ifv red ⊂ ℓh for a Cartan subalgebra as in(4.5) then eachE is spanned by some of thevectors in the basis(ℓfj). For each fixedg ∼= su (p, q) there are only finitely manyG0-conjugacy classes of centralizersCg(v

red) asv ⊂ g varies through all qualifying MASAsin g .


For the sake of clarity let us mention that in general there are infinitely many con-jugacy classes of qualifying MASAsv while the above Lemma asserts that there are onlyfinitely many conjugacy classes of the corresponding toral parts v red. The point here isthat for a fixedv red there may be infinitely many non-conjugate qualifying MANSAs inCg(v

red)ss.

The above lemma describes the structure ofv red and its centralizerCg(vred) in an

elementary-geometric way and shows that both determine each other uniquely. For the de-scription ofv = v red⊕ vnil it is therefore enough to determine all possiblevnil . These splitinto a direct sum

(4.11)

vnil =⊕

K∪L

n with

n : = vnil ∩ s j for s :=

{su(E, h) ∈ K

(sl (E)⊕ sl (E)

)σ ∈ L .

Every n is an abelianad-nilpotent subalgebra ofs . In case ∈ K the algebran hasdimensionp + q − 1. As a consequence,p = 0 is possible only ifq = 1 (since in caseq > 1 the formh is definite and everyad-nilpotent element ins is zero). In the same wayq = 0 impliesp = 1. In case ∈ L the algebran has dimension2m − 2.

Before we turn to the corresponding classification result weneed to extract someinvariants from the equations in 4.10. For every setA denote byF(A) thefree commutativemonoid overA. We write the elements ofF(A) in the form

∑α∈A nα·α with nα ∈ IN

and∑

A nα < ∞. Here we use the free monoids over the following sets, whereIN ={0, 1, 2, . . . .}:

(4.12)K : = {(s, t) ∈ IN2 : (st = 0) ⇒ (s+ t = 1)} ,L : = IN\{0} and J := K ∪L .

Then D := F(J) = F(K) + F(L) ,

and the permutation ofJ defined by(s, t) 7→ (t, s) on K and the identity onL inducesan involutionD 7→ Dopp of D. As an example, theoppositeof D = 4·(3, 5) + 2·7 isDopp = 4·(5, 3) + 2·7. Notice that2·7 and 7·2 are different elements inF(L) ⊂ D.The setJ can be considered in a canonical way as subset ofF(J) by identifying j ∈ J

with 1·j ∈ D, but for better distinction we writej instead of1·j only if no confusion islikely. Also, for better distinction we write the natural numbers inIN\{0} in boldface if weconsider them as element ofL.

4.13 Definition. For everyp, q ∈ IN we denote byDp,q ⊂ D the subset of all

D =∑

j∈J

nj ·j ∈ F(J) satisfying

p =∑

j=(s,t)∈K

njs+∑

j∈L

njj and q =∑

j=(s,t)∈K

njt+∑

j∈L

njj ,

wherej is the natural number underlyingj. ThenDoppp,q = Dq,p andDp,q + Dp′,q′ ⊂

Dp+p′,q+q′ are obvious.

To every qualifying MASAv ⊂ su(p, q) we associate an elementD(v) of D thatonly depends on theSU(p, q)-conjugation class ofv and is called theD-invariant ofv :


Suppose thatv (after a suitable conjugation) gives rise to the equations (4.7) and (4.10).Then just put

(4.14) D(v) :=∑

∈K

1·(p, q) +∑

∈L

1·m .

For instance, the CSAs ing = su (p, q) are precisely the qualifying MASAsv ⊂ g withD(v) ∈ F(A) with A := {(1, 0), (0, 1),1}. Indeed, in the notation of (4.5) we haveD( ℓh) = (p− ℓ)·(1, 0) + (q − ℓ)·(0, 1) + ℓ·1.

The relevance of theD-invariants for our classification problem is demonstratedbythe following two results. Recall thatG = NSL(p+q,C)(su (E, hp,q)) andG0 = SU(E, hp,q)with G 6= G0 only if p = q.

4.15 Proposition. D ∈ D is theD-invariant of a qualifying MASAv in su (p, q) if andonly if D ∈ Dp,q.

4.16 Proposition. Let v1, v2 be two qualifying MASAs ing = su(p, q). Then the toralpartsv red

1 , v red2 (and hence also the corresponding centralizers) areG0-conjugate ing if and

only if D(v1) = D(v2). In casep = q the toral partsv red1 , v red

2 areG-conjugate if and onlyif D(v1) = D(v2) orD(v1) = D(v2)

opp (in this caseSU(E, hp,q) has index two inG).

Our classification problem now reduces to the following task: For everyp, q ≥ 1 withp+q ≥ 3 and everyD in the finite setDp,q determine allG-conjugacy classes of qualifyingMASAs v ⊂ su(p, q) with D(v) = D.

4.17 Explicit classification for small values ofq: ForDp,q in the casesp ≥ q = 1, 2 wehave the following explicit lists (without repetitions).Dp,1 consists of all invariants

(i) (p − s)·(1, 0) + (s, 1) for s = 1, 2, . . . , p.(ii) (p − 1)·(1, 0) + 1 and p·(1, 0) + (0, 1).

Dp,2 consists of all invariants(iii) 1+Dp−1,1,(iv) (p − s− t)·(1, 0) + (s, 1) + (t, 1) for all 1 ≤ s ≤ t with s+ t ≤ p ,(v) (p − s)·(1, 0) + (s, 2) , (p− s)·(1, 0) + (0, 1) + (s, 1) for 1 ≤ s ≤ p ,(vi) (p − 2)·(1, 0) + 2 , p·(1, 0) + 2·(0, 1) .

Notice that inD2,2 there exists an invariant that is not self-opposite, e.g.(1, 0) + (1, 2).As a consequence, in caseg ∼= su(2, 2) there are elevenSU(2, 2)-conjugation classes ofcentralizers in contrast to the only tenG-conjugation classes in this case.

4.18 Nilpotent parts of qualifying MASAs. So far we have given a description of con-jugacy classes of the toral partsv red of qualifying MASAs ing . For the description ofv =v red⊕ vnil it is therefore sufficient to determine all possible nilpotent partsvnil. Accordingto 3.5, each suchvnil is a maximal abelian nilpotent subalgebra (MANSA) ofCss

g (vred),

compare (3.6). On the other hand, given any qualifying MASAv ⊂ g , each MANSAnof Css

g (vred) in the−1-eigenspace ofτ gives rise to a qualifying MASAvn := v red⊕ n

in g . Given one of the finitely may conjugacy classes of centralizersCD with D ∈ Dp,q,our task is therefore to classify the MANSAs in the semisimple partCss

D of CD. As alreadyexplained,Css

D decomposes uniquely into simple idealsg , each of them being isomorphiceither tosu(p, q) or to slm

(C). Consequently each qualifying MANSAn ⊂ CssD has the

unique decompositionn =⊕

J1n with J1 := { ∈ K ∪ L : dimE > 1} ⊂ J , compare


4.10. Each of the factorsn is a qualifying MANSA ing , more precisely:

(4.19)

Cg(vred) = CD = v red ⊕

⊕

J1

g

∪ ‖ ∪vn = v red

n ⊕⊕

J1

n

g ∼= su(p, q)

or

g ∼= RC

IR(sl (m,C)) .

Summarizing the results of the present subsection, our nexttask is to determine max-imal abelianad-nilpotent subalgebrasn ⊂ g with eitherg = su (E, h) ∼= su(p, q)

or g = (sl (E) ⊕ sl (E))σ ∼= RC

IR(slm(C)). Here we can restrict to subalgebras that

are contained in the(−1)-eigenspace of an involutionτ coming from a conjugation on thevector spacesE andE ⊕ E respectively. In the following we discuss the casessl(E)σ =su(E, h) ∼= su(p, q) and

(sl (E)⊕ sl(E)

)σ ∼= sl(m,C) separately.

5. MANSAs in su (p, q)−τ

For notational simplicity let us drop the subscript for the rest of this section andwrite E = Ej as well as(p, q) = (p, q) for the type of the restriction ofh to E andalso g = su (E, h). As before we denote byσ the conjugation on the complexificationgC = sl (E) with Fix(σ) = g . For everyz ∈ End(E) we denote byz⋆ ∈ End(E) theadjoint with respect toh.

Without loss of generality we assumepq 6= 0, since otherwisep + q = 1 and thussu(E, h) = 0, see (4.12). As before,m = p+ q andr = m− 1.

In order to classify the maximal nilpotent Lie subalgebrasn = vnil ⊂ su (E, h) werelate them to nilpotent commutative andassociativeIR-algebras, see the Appendix for theterminology.

5.1 Proposition. Let n ⊂ g = su (E, h) be a Lie subalgebra. Then the following conditionsare equivalent:

(i) n is maximal among all abelian Lie subalgebrasa of g such that every element ofais a nilpotent endomorphism ofE.

(ii) n is maximal among all abelian subalgebras ofg that are ad-nilpotent ing .(iii) The complexificationnC is σ-stable and is maximal among all abelian and nilpotent

associative subalgebras ofEnd(E).If these conditions are satisfied thennC⊕ C idE is a maximal abelian subalgebra of thecomplexassociativealgebraEnd(E). Further, in the notation of9.3ff. for everyn ⊂ g

satisfying one (and hence all) of the conditions(i) - (iii) the following holds:(iv) The subspacesKnC andBnC areh-orthogonal inE, implying d1 = d3.(v) There exists annC-adapted decompositionE1⊕E2⊕E3 of E such thatE1 andE3 = K

areh-isotropic andE2 ⊥h (E1 ⊕ E3). If, in addition,n is τ -stable then the adapteddecomposition can be chosen to beτ -stable, too. The restriction ofh toE2 is of type(p − d1, q − d1).

Now the conjugationτ of E comes into play. ForV := Eτ ∼= IRm we haveE = V ⊕ iV,

and we identifyEnd(V) in the obvious way with the real subalgebraEnd(E)τ of End(E).A crucial observation is the following refinement of Proposition 9.7.

5.2 Proposition. Assume thatn ⊂ su(E, h)−τ is maximal among abelian and ad-nilpotentsubalgebras ofsu (E, h). As before, letN := i n ⊂ End(E). Then, with the notation ofProposition9.7, the following holds:


(i) dimAnn(N ) = 1. In particular,dimV1 = dimV3 = 1 for anyN-adapted decom-position of V.

(ii) Fix generatorsv1 ∈ V1 and v3 ∈ V3 with h(v1, v3) = 1. This yields canonicalidentificationsV1 = IR = V3, Ann(N ) = IR, N21 = Hom(IR,V2) = V2 andN32 = Hom(V2, IR) = V

∗2 (the dual ofV2). The mapJ : V2 → V

∗2 is given

by J(y)(x) = h(x, y) for all x, y ∈ V2. With all these identifications the matrixpresentation in Proposition9.7 reads

N =

0 0 0y N(y) 0t J(y) 0

: y ∈ V2, t ∈ IR

⊂ S(V, h) ⊂ End(E) ,

whereS(V, h) ⊂ End(V) is the linear subspace of allh-selfadjoint operators onV.(iii) The restriction ofh to V2 has type(p − 1, q − 1) andN(y) ∈ S(V2, h) for every

y ∈ V2.

Proof. (ii): From 5.1 and 9.7.(ii) follows that for a maximal abelian and ad-nilpotent sub-algebran ⊂ su (E, h) we haveAnn(i n ) = {x ∈ Ann(nC) = Hom(E1,E3) : x = x⋆}.Since at the same timen is contained in the(−1)-eigenspace ofτ the first part of the lemmatogether with 9.7.(ii) implyAnn(i n ) = Hom(V1,V3) = Hom(E1,E3)

τ . This is only pos-sible if dimE1 = dimE3 = 1.

The next proposition shows that the classification of maximal nilpotent subalgebrasn ⊂ su (E, h), contained also insl (E)−τ , reduces to the classification of abstract associativenilpotent subalgebras (ofEnd(E)) with 1-dimensional annihilator. Crucial for the followingtheorem is the construction of a non-degenerate 2-formb = bπ depending on a suitableprojectionπ, see 9.12. Keeping also in mind Proposition 9.10 and Lemma 9.14 we have:

5.3 Theorem. LetN be an arbitrary commutative associative and nilpotentIR-algebra withdimAnn(N ) = 1 and letV := N 0 be its unital extension. Fix an identificationAnn(N ) =IR and a projectionπ onV with rangeAnn(N ) = IR satisfyingπ(11) = 0. Then for the leftregular representationL of V = N 0 and the symmetric real 2-formb : V × V → IR wehave:

(i) L(N ) is a maximal nilpotent and abelian subalgebra ofEnd(V) contained inS(V, b).(ii) Let E := V ⊕ iV be the complexification ofV and τ the conjugation onE with

Eτ = V. Furthermore, denote the unique hermitian extension ofb to E × E by the

same symbolb. Thenn := iL(N ) is a subset ofsu(E, b)−τ and is a maximal abelianand ad-nilpotent Lie subalgebra ofsu(E, b). Finally, exp nC ⊂ SL(E) has an openorbit in IP(E).

(iii) Every maximal abelian and ad-nilpotent subalgebra ofsu(E, h) which is also con-tained insl(E)−τ for τ as in3.2 is equivalent to someiL(N ) as above.

Suppose that fork = 1, 2 there are given two abelian nilpotent associativeIR-algebrasNk each with 1-dimensional annihilatorAk and assume that the corresponding 2-formsbk on the corresponding unital extensions have types(pk, qk) with respect to the linearisomorphismsλk : Ak

∼= IR. Then forI := {(x, y) ∈ A1 ⊕ A2 : λ1(x) = λ2(y)}the quotient algebraN := (N1 ⊕N2)/I is an abelian nilpotent associative algebra with1-dimensional annihilatorA := (A1 ⊕A2)/I, and the 2-formb onN induced byb1 × b2onN1 ×N2 has type(p1 + p2 − 1, q1 + q2 − 1).

5.4 MANSAs insu (p, q) for low values ofq. By our above considerations, to determineall MANSAs n ⊂ su (p, q) up to SU(p, q)-conjugacy it is equivalent to determine up to


isomorphism all real nilpotent abelian associative algebrasN with annihilatorA such thatfor some linear isomorphismA ∼= IR the formb has type(p, q) onN 0. For low values ofqthis can be done:

q = 1: There is precisely 1 equivalence class of MANSAs insu(p, 1)−τ for everyp ≥ 1.Indeed, for everyN with annihilatorA ∼= IR the factor algebraN/A must be a zero productalgebra.

q = 2: There are preciselymin(p, 3) equivalence classes of MANSAs insu(p, 2)−τ forevery p ≥ 1. Representing algebrasN are obtained as follows: For everyn ≥ 1 withn ≤ min(p, 3) let N1 be the cyclic abelian algebra of dimensionn, compare Example9.40, and identifyt ∈ IR with tξn in the annihilatorA1 of N1. Then the correspondingform b1 has type(1, 1), (2, 1), (2, 2) for n = 1, 2, 3 respectively. Next choose an abeliannilpotent algebraN2 with 1-dimensional annihilatorA2 such that the constructionN :=N1 ⊕ N2/I as above leads to a nilpotent algebra with 1-dimensional annihilator A suchthat the corresponding 2-formb onN 0 has type(p, 2). This is always possible sinceN2/A2

must be a zero product algebra.

6. MANSAs in sl (m,C)In this section we deal with the simple factorsg

∼= sl (m,C) in (4.10). We retain ourconvention from the last section and drop the index from our notation, that is, we considerabelian nilpotent subalgebras ofEnd(E) that are contained ing = sl(m,C), where thelatter space is considered as a real Lie algebra. Recall thatin this case the restriction of theinvolution τ to g is given by the mapx 7→ −x′, wherex′ is the adjoint with respect to thecomplex bilinear non-degenerate symmetric 2-formβ given byβ(v,w) := h(v, τw) for allv,w ∈ E, compare (4.9). Note that the complexificationgC is isomorphic to the productsl (m,C)× sl(m,C).

6.1 Proposition. Let g = RCIR(sl (E)) and letn ⊂ g be a Lie subalgebra. Then the follow-

ing conditions(i) – (iii) are equivalent:(i) n is maximal among ad-nilpotent and abelianIR-subalgebras ofsl(E).(ii) n is maximal among all abelianC-subalgebras ofsl(E) that are ad-nilpotent insl (E).

(iii) n is maximal among abelian and nilpotent subalgebras of the associative complexalgebraEnd(E).

If these conditions are satisfied, thennC ⊕ C· idE is a maximal abelian subalgebra of thecomplex associative algebraEnd(E). Furthermore we have:

(iv) Let n be maximal among ad-nilpotent and abelianC-subalgebras ofsl (E) containedin sl (E)−τ . ThendimAnn(n) = 1, i.e.,1 = dimE1 = dimE3 for anyn -adapteddecomposition ofE (dimensions overC).

(v) Identifying C = E1 = E3 = Ann(n), the matrix presentation in Proposition9.7reads

N =

0 0 0y N(y) 0t J(y) 0

: y ∈ E2, t ∈ C

⊂ S(E, β) ⊂ End(E) ,

whereS(E, β) ⊂ End(E) is the linear subspace of allβ-selfadjoint operators onE.

Proof. (iv): Proposition 9.7 implies thatAnn(n ) = Hom(E1,E3) for a givenn -adapteddecomposition ofE. On the other hand,n is also contained insl (E)−τ , i.e.,x = x′ for allx ∈ sl (E). Hence, we can choose ann -adapted decompositionE = E1⊕E2⊕E3 such thatE1 andE3 areβ-isotropic and(E1⊕E3), E2 areβ-orthogonal. In the matrix presentation ofn as in Proposition 9.7 the two above conditions implyAnn(n) = Hom(E1,E3) = {x ∈Hom(E1,E3) : x = x′}. This is only possible ifdimE1 = dimE3 = 1 = dimAnn(n ).

Similar to Theorem 5.3 we have


6.2 Theorem. LetN be an arbitrary associativeC-algebra which is commutative nilpotentand has annihilatorAnn(N ) of dimension 1. Let furthermoreπ be an arbitrary (complexlinear) projection on its unital extensionE := N 0 with rangeAnn(N ) andπ(11) = 0, andfix an identificationAnn(N ) = C. ThenL(N ) is maximal in the class of all nilpotent andabelian subalgebrasA ⊂ End(E) which are contained inS(E, b)tr=0 = sl (E)−τ . Hereb = bπ is as in9.12, x′ is the adjoint with respect to the complex bilinear 2-formb andτ : End(E) → End(E) is given byx 7→ −x′. On the other hand, every maximal abelianand ad-nilpotent subalgebra ofsl (E) which is also contained insl (E)−τ for τ as in4.9 isequivalent to someL(N ) as above.

Note that the complex nilpotent groupNC , corresponding tonC ∼= n × n ⊂ sl(E) ×sl (E) ∼= (sl (E))C does not have an open orbit inIP(E⊕E), but the subgroup correspondingto the following subalgebra does

(n × n)⊕ C(id,− id) = (vnil)C ⊕ (v red)C ⊂ Csl(E⊕E)(vred) ⊂ sl(E⊕ E) .

MANSAs in sl(m,C) for low values of m. There exist exactly 1,1,1,2,3 equivalenceclasses of qualifying MANSAs insl(m,C) for m = 1, 2, 3, 4, 5 (see also the more detaileddescription at the end of the following Section 7). With our construction of nilpotent alge-bras out of cubic formsc in caseIF = C (compare Proposition 9.36) it follows that there areinfinitely many equivalence classes of qualifying MANSAs insl(m,C) for everym ≥ 8.

7. Normal forms for equations

Every local tube realizationTF = V + iF ⊂ E := V ⊕ iV of Sp,q is characterizedby a qualifying MASAv ⊂ su(p, q). In addition, the baseF of the tube can always bechosen to be a closed (real-analytic) hypersurface in the real vector spaceV , see [10]. In thefollowing we want to find canonical real-analytic real valued functionsψ onV with dψ 6= 0everywhere such thatF = {x ∈ V : ψ(x) = ψ(0)}0, where the upper index0 means to takethe connected component containing the origin. For this we consider theD-invariantD(v)of v = v red⊕ vnil, see (4.14), and start with the special case thatD(v) ∈ J = K ∪ L.The general case withD(v) ∈ F(J) arbitrary then is obtained by putting these specialequations together.

1. CaseD(v ) ∈ K: Let j := (p, q) andn := p + q − 1. Furthermore letVj := IRp+q

with coordinates(x0, x1, . . . , xn) and define the linear formλj on Vj by λj(x) = (p +q)x0. Also, identifyVj := IRn with coordinates(x1, . . . , xn) in the obvious way with the

hyperplane{x ∈ Vj : λj(x) = 0}. We defineΨj as the set of all real-analytic functions

ψ(x) = ex0f(x1, . . . , xn) on Vj wheref is an extended real nil-polynomial onVj and the

second derivative ofψ at the origin ofVj has type(p, q), compare the Appendix for thenotion of a nil-polynomial.

It is clear that for every realt > 0 and everyg ∈ GL(Vj) with ex0f(x) also thefunction t ex0f(g(x)) is contained inΨj . Furthermore,Ψjopp = −Ψj is evident for theoppositejopp = (q, p). In particular,Ψ(1,0) = {tex0 : t > 0}, Ψ(1,1) = {tex0x1 : t 6= 0}andΨ(2,1) is the orbit ofψ = ex0(x2 + x21) under the groupIR∗ ×GL(2, IR).

2. CaseD(v ) ∈ L: Let j := m for some integerm ≥ 1 and putn := m − 1. ConsiderVm := Cm with complex coordinates(z0, z1, . . . , zn) as real vector space and define thelinear formλm on Vm by λm(z) := m(z0 + z0). Furthermore, considerWm := Cn

with coordinates(z1, . . . , zn) in the obvious way as linear subspace ofVm. We defineΨm


as the set of all real-analytic functionsψ(z) = Re(ez0f(z1, . . . , zn)

)on Vj wheref is an

extended complex nil-polynomial onWm. Then the second derivative of everyψ ∈ Ψm atthe origin has type(m,m).

The groupC∗× GL(Wm) acts in a canonical way on the extended complex nil-polynomials onWm and thus also onΨm. In particular,Ψm is the orbit of the functionsRe(ez0), Re(ez0z1) and Re(ez0(z2 + z21)) for m = 1, 2, 3 respectively.

3. CaseD(v ) arbitrary: ThenD := D(v) ∈ Dp,q for integersp, q ≥ 1 with p + q ≥ 3,and there exists a unique sum representationD =

∑α∈A jα with (jα)α∈A a finite family

in J = K ∪ L. Put VD :=⊕

α∈A Vjα and define the linear formλD on VD by (xjα) 7→∑α λjα(xjα). Furthermore, letΨD be the space of all functions

ψ : (xjα)α 7−→∑

α

ψα(xjα)

on VD ,where(ψα)α∈A is an arbitrary family of functionsψα ∈ Ψjα . The second derivativeof everyψ ∈ ΨD at the origin then has type(p, q).

The relevance of the vector spacesVD with linear formλD and function spaceΨD

is the following: Consider inVD the hyperplaneV := {x ∈ VD : λD(x) = 0}. Then foreveryψ ∈ ΨD the analytic hypersurface

(7.1) F :={x ∈ V : ψ(x) = ψ(0)

}0

is the base of a local tube realization ofSp,q with D-invariantD, and every local tuberealization ofSp,q with D-invariantD occurs this way up to affine equivalence. Indeed,every local tube realization ofSp,q is associated with a qualifying MASAv ⊂ su (p, q).In particular, forE := Cp+q the complexificationvC := v ⊕ iv ⊂ sl (E) has an openorbit O in the projective spaceIP(E) that is the image of the locally biholomorphic mapϕ : vC → IP(E), ξ 7→ exp(ξ)a, wherea is a suitable point inIP(E). Then every connectedcomponentM of ϕ−1(Sp,q) is a closed tube submanifold ofvC , lets take the one thatcontains the origin. Then the baseF := M ∩ i v of the tube manifoldM has (7.1) asdefining equation if we putV := i v ⊂ sl (E) andψ := ϕ|V . Now consider the extended

spaceV := IR· id ⊕ iv ⊂ gl(E) and lettr be the trace functional onV . Also extendψ toV by t· id⊕x 7→ etψ(x) and denote the extension by the same symbolψ.Now theD-invariantD(v) and the corresponding decomposition ofv red in Lemma 4.10comes into play. We can identifyV with VD andtr with λD in a canonical way. Sinceψ isdefined in terms ofexp it is compatible with the decomposition in Lemma 4.10 and we onlyhave to discuss defining equations for the case that there is only one summand in (4.7), thatis, thatD(v) ∈ J :

1. CaseD(v) ∈ K: Let D(v) := (p, q). In casepq = 0 we haveV = 0 and we have upto a positive factorψ(x) = (p − q)ex on V = IR. We therefore assumep, q ≥ 1 in thefollowing. But thenN := V = i v is an associative commutative nilpotent real subalgebraof End(E) with 1-dimensional annihilatorA, compare the Appendix. Choose a pointingωonN such that the associated symmetric 2-formh(x, y) = ω(xy) has type(p, q) on N 0.Then as base for a tube realization associated withv we can take

F = {x ∈ N : f(x) = 0} with f(x) := h(expx/2, exp x/2) = ω(expx) .

But f is an extended real nil-polynomial onN andF is a smooth algebraic hypersurface.


2. CaseD(v ) ∈ L: Let D(v) := m and putn := m − 1. Thenv = IR· id+N ⊂gl(m,C) ⊂ u(m,m), whereN ⊂ sl (m,C) is a complex MANSA and at the sametime a commutative associative nilpotent complex subalgebra ofEnd(Cm). In casem = 1

we haveN = 0 andψ(z) = Re(ez) on V = C. Let us therefore assumem > 1 in thefollowing. ThenN has annihilatorA of complex dimension 1. Letω be a pointing onN .Then the real symmetric 2-formh(z, w) = Reω(zw) has type(m,m) on the complexunital extensionN 0 = C· id+N ⊂ End(Cm). We have to consider the complexificationvC = v ⊕ iv and to restrict the exponential mapping toiv . For this, we may identify theIR-linear spaceV := iv with iIR· id+N ⊂ N 0 and get as base for a tube realizationassociated tov the hypersurface

F = {w ∈ iIR· id+N : h(expw/2, expw/2) = 0} .

Writing w = is· id+z with s ∈ IR, z ∈ N we have

h(expw/2, expw/2) = Re(eisf(z)) for f(z) := ω(exp z) ,

that is,f is an extended complex nil-polynomial onN ∼= Cn, andF is affinely equivalentto the non-algebraic hypersurface

{(s, z) ∈ IR⊕ Cn : Re(eisf(z1, . . . , zn)

)= 0}0 .

7.2 Local tube realizations corresponding to Cartan subalgebras ofsu (p, q): By theabove we knowΨj for all j ∈ {(1, 0), (0, 1),1} and thus we can explicitly write down thenormal form equations of every CSA insu (p, q): For fixedp, q ≥ 1 and everyℓ ≥ 0 withℓ ≤ min(p, q) consider the CSAℓh as defined in Section 4. Then we have

D := D(ℓh ) = (p − ℓ)·(1, 0) + (q − ℓ)·(0, 1) + ℓ·1.With d := p+ q − 2ℓ then

VD ={(z, t) ∈ Cℓ ⊕ IRd :

ℓ∑

k=1

(zk + zk) +

d∑

k=1

tk = 0}

and as tube base we can take a connected component of the set ofall (z, t) ∈ VD satisfying

(7.3)ℓ∑

k=1

Re(ezk) +

p−ℓ∑

k=1

etk =

q−ℓ∑

k=1

etp−ℓ+k .

7.4 Comparing with the equations of Isaev-Mishchenko:It is easy to write down explicitlyΨj for all j = (p, q) with q ≤ 2, getting back the

classifications in [5] and [11]. In the following we compare the equations obtained in [11]with ours, whereD is the correspondingD-invariant andn = p :types 1), 4), 5):D = s·(1, 0)+(n− s, 2). The corresponding MANSA insu (n− s, 2) hasnil-index 2,3,4 respectively.type 2): D = s·(1, 0) + (0, 1) + (n− s, 1).type 3): D = s·(1, 0) + (n− s− 1, 1) + 1,type 6): D = (n− 2)·(1, 0) + 2.type 7): D = s·(1, 0) + (t, 1) + (n− s− t, 1).types 8), 9), 10):These types correspond to the three Cartan subalgebras ofsu (n, 2) andare affinely equivalent to the equations (7.3) forℓ = 0, 2, 1 respectively.


8. Some ExamplesWe will give applications of Proposition 9.36 in the real as well as in the complex

case. We start with the real version.

8.1 Examples obtained from real cubic formsLetW be a real vector space of dimension2n andq a quadratic form of type(n, n) onW . Then there exists a decompositionW =W ′ ⊕W ′′ into totally isotropic linear subspaces. Let furthermorec be a cubic form onW ′

and define the functionf onV :=W ⊕ IR by

(8.2) f(x, y, t) := t+ q(x+ y) + c(x) for all t ∈ IR, x ∈W ′, y ∈W ′′ .

Thenf is an extended nil-polynomial onV and the hypersurfaceF := {v ∈ V : f(v) = 0}in V is the base of a local tube realization forSp,p with p = n + 1. On the other hand,Fis affinely homogeneous and also the complementV \F is affinely homogeneous, comparethe end of the Appendix. The complementV \F decomposes into two affinely homogeneousdomainsD±, the tube domains over these domains are complex affinely homogeneous do-mains inV C = V ⊕ iV . With (9.44) we see that there exists a real

(n3

)-parameter family of

cubic forms onW ′ leading to pairwise affinely inequivalent examples (notice that property(∗) of Proposition 9.43 is satisfied for allα(tj) in (9.44) nearc0). In particular this showsthat there are infinitely many affinely non-equivalent local tube realizations forS4,4.

8.3 Examples obtained from complex cubic formsWe start with a more general situa-tion: Suppose thatV is a complex vector space andf : V → C is a holomorphic submersionwith f(0) = 0. Then

F := {z ∈ V : f(z) = 0}0

is a complex hypersurface inV , the complementD := V \F is a domain inV and

(8.4) F :={(t, z) ∈ IR⊕ V : Re

(eitf(z)

)= 0

}0

is a real hypersurface inIR ⊕ V . With respect to the canonical projectionpr : F → V ,(t, z) 7→ z, the surfaceF is a covering over the domainD and a trivial real line bundleoverH. In fact,F := pr−1(F ) is a connected real hypersurface inF , while the open subsetD := pr−1(D) in F in general is not connected.Now assume thatV =W ⊕C and thatf is an extended complex nil-polynomial of degree≤ 3 on V as considered in Proposition 9.36. As a consequence of Proposition 9.22 thegroupA := {g ∈ Aff(V ) : f ◦ g = f} acts transitively on every level setf−1(c) inV . Also, for everys ∈ C the linear transformationθs := es idW ′ ⊕ e2s idW ′′ ⊕ e3s idCsatisfiesf ◦ θs = e3sf , see also (9.41). The groupC × A acts by the affine transformations

(t, z) 7→(t− 3 Im(s) , θs g(z)

), s ∈ C, g ∈ A,

onF and has precisely three orbits there – the closed orbitF and the two connected com-ponentsD± of the domainD. Also, the translation(t, z) 7→ (t + π, z) interchanges thesetwo domainsD+, D− in F .Putting things together we got the following: LetW be a complex vector space of dimen-sion 2n andq a non-degenerate quadratic form onW . Then there exists a decompositionW = W ′ ⊕W ′′ into totally isotropic linear subspaces. Let furthermorec be a cubic formonW ′ and define the functionf onV := W ⊕ C by (8.2) withIR replaced byC. Thenfis an extended complex nil-polynomial andF as defined in (8.4) is the base of a local tuberealizationM ⊂ UC = U ⊕ iU for Sp,p, whereU := IR ⊕ V andp = 2(n + 1). Further-more, the real hypersurfaceM in UC contains an affinely homogeneous domain. For everyn ≥ 3 we get a complex

(n3

)-parameter family of pairwise affinely inequivalent examples.


9. Appendix – Nilpotent commutative algebras

In this Appendix, all occurring algebras are either associative or Lie. Throughout,IF is anarbitrary base field of characteristic zero. For every associative algebraA, everyx ∈ A andevery integerk ≥ 1 we put

(9.1) x(k) :=1

k!xk and x(0) := 11 if A has a unit11 .

We collect several purely algebraic statements that are used in the paper and might be ofindependent interest. Some of them are probably known to theexperts. Since we could notfind a reference in the literature we state it here. Recall e.g. our convention thatEnd(E) istheassociativeendomorphism algebra whilegl(E) is the same space, but endowed with thecorresponding Lie product. We start with a standard definition.

9.2 Definition. Let N be an commutative associative algebra overIF and define the idealsN k ⊂ N inductively byN 1 = N andN k+1 = 〈NN k〉. ThenN is callednilpotent ifN k+1 = 0 for somek ≥ 0, and the minimalk with this property is called thenil-index ofN . Furthermore,A := Ann(N ) := {x ∈ N : xN = 0} is called theannihilatorof N .

The general embedded caseIn the following, letE be a vector space of finite dimensionm ≥ 2 overIF. For every

subalgebraN ⊂ End(E) define the following characteristic subspaces ofE:

(9.3) B := BN := 〈N (v) : v ∈ E〉 and K := KN := {v ∈ E : N (v) = 0} .

9.4 Proposition. Suppose thatN is maximal among all commutative and nilpotent subal-gebras ofEnd(E). Then

(i) 0 6= KN ⊂ BN 6= E. Also, KN = BN holds if and only ifA = Ann(N ) hasnil-index 1.

(ii) N ⊕ IF· id is maximal among all commutative subalgebras ofEnd(E).(iii) N is irreducible onE, i.e., for everyN -invariant decompositionE = E

′ ⊕ E′′ either

E′ = 0 or E

′′ = 0.

9.5 N -adapted decompositions and matrix presentations.For everyN satisfying theassumptions in Proposition 9.4 we select subspacesE1,E2 of E such thatE1 ⊕ BN = E

andE2 ⊕KN = BN . Then, forE3 := KN , we have the decomposition

(9.6) E = E1 ⊕ E2 ⊕ E3 with dj := dimEj for j = 1, 2, 3 ,

that we also call anN -adapted decomposition. Everyx ∈ End(E) can be written as3× 3-matrix (xjk) with xjk ∈ Hom(Ek,Ej). With πjk : End(E) → Hom(Ek,Ej) we denotethe projectionx 7→ xjk.

We call two subalgebrasN ⊂ End(E) andN ′ ⊂ End(E′) conjugateif there existsan invertibleΨ ∈ Hom(E,E′) such thatN ′ = Ψ ◦ N ◦Ψ−1. One of our goals is to decideunder which conditions two isomorphic subalgebras (isomorphic as abstractIF-algebras) arealready conjugate. In general, there exist isomorphic subalgebras which are not conjugate.

It is obvious that a nilpotent subalgebraN ⊂ End(E) contains only nilpotent endo-morphisms. By a theorem of Engel the converse is also true: A subalgebraN ⊂ End(E)consisting of nilpotent endomorphisms only is nilpotent and there is a full flag

F1 ⊂ F2 ⊂ · · · ⊂ Fm = E, dimFk = k,

which is stable underN , i.e., with respect to a suitable basis ofE the algebraN consists ofstrictly lower-triangular matrices inIFm×m. With this notation we can state


9.7 Proposition. Suppose thatN ⊂ End(E) satisfies the assumptions of9.4 and has nil-indexν. For a fixedN -adapted decompositionE = E1 ⊕ E2 ⊕ E3 and all1 ≤ j, k ≤ 3 putNjk := πjk(N ). Then:

(i) There exists a linear bijectionJ : N21 → N32 and a linear mapN : N21 → N22 suchthat

N =

0 0 0y N(y) 0t J(y) 0

: y ∈ N21, t ∈ Hom(E1,E3)

.

(ii) A := {x ∈ N : x21 = 0} ∼= Hom(E1,E3) is the annihilator ofN .(iii) N21 × N21 → N31, (x, y) 7→ J(x) ◦ y, is a non-degenerate symmetric 2-form (in

fact, is equivalent to the restriction of the formb defined in(9.12)after the obviousidentifications).

(iv) N22 is a nilpotent commutative subalgebra ofEnd(E2)with nil-index≤ max(ν − 2, 1).

(v) E2 = 〈N23(E1)〉 and{z ∈ E2 : y(z) = 0 for all y ∈ N32} = 0.(vi) d1d3 + ⌈d2/µ⌉ ≤ dimN ≤ [m2/4] for m := d1 + d2 + d3 = dimE and

µ := min(d1, d3). In particular, ifd1 = 1 thendimN = d2 + d3 = m− 1.

9.8 Remark. The linear mapsN andJ in (i) above satisfy for allx, y ∈ E the relations:(a) N(x)k = 0 for some integerk,(b) N(x)y = N(y)x, J(x)y = J(y)x and J(x)N(y) = J(y)N(x),(c) N(N(x)y) = N(x)N(y).

On the other hand, let a vector spaceW over IF be given. Every pairN : W → End(W),J : W

≈→ W∗ of linear maps satisfying (a) - (c) gives rise by (i) above to acommutative

maximal nilpotent subalgebraN ⊂ End(E) with E = IF ⊕W ⊕ IF, i.e.,E1 = IF = E3,E2 = W andN has 1-dimensional annihilator. In particular,N ≡ 0 andJ given by anysymmetric and non-degenerate scalar product onW trivially satisfy (a) - (c) and define amaximal nilpotent subalgebraN with dimN = dimE − 1 and nil-index2. In this caseN/A is the zero product algebra.

9.9 Remark. The upper bound in inequality (vi) is sharp as the nilpotent subalgebra

{x ∈ End(E) : xjk = 0 if (j, k) 6= (3, 1)}

for d2 = d3 = ⌈m/2⌉ shows. It is much harder to find better lower bounds fordimN , notto speak of sharp ones. For infinitely many values ofm (starting withm = 14) there existmaximal commutative and nilpotent subalgebras ofEnd(IFm) with dimN < m − 1, see[18].

Abstract commutative nilpotent algebras

The left-regular representation of a nilpotent (associative) algebra is not faithful, con-trary to the case of any unital algebra. For every nilpotent algebraM denote byM0 :=IF·11⊕M its unital extension. Such extensions of nilpotent algebras are precisely thoseIF-algebras, which contain a maximal ideal of codimension 1 consisting of nilpotent elementsonly. Denote byL : M0 → End(M0) the corresponding left regular representation. It isobvious that the algebrasM andL(M) ⊂ End(M0) are isomorphic viaL.

9.10 Proposition. LetM be a commutative nilpotentIF-algebra of finite dimension. Then(i) L(M) is maximal among all commutative nilpotent subalgebras ofEnd(M0) and

consists entirely of nilpotent endomorphisms. Furthermore,KL(M) = Ann(M) andBL(M) = M. The image of the unital extension,L(M0) is maximal among allcommutative subalgebras ofEnd(M0).


(ii) A maximal commutative nilpotent subalgebraN ⊂ End(E) is conjugate to theimageL(M) of some commutative nilpotent algebraM as above if and only ifcodimE BN = 1, see(9.3) for the notation.

(iii) Let N ⊂ End(E) andM ⊂ End(F) be two maximal commutative nilpotent subal-gebras. IfcodimE BN = codimF BM = 1 thenM andN are conjugate (in the abovedefined sense) if and only ifM andN are isomorphic as abstract algebras. There ex-ist non-conjugate subalgebrasN ,M ⊂ End(E) with codimE BN > 1, which areisomorphic as abstract algebras.

9.11 Associated 2-forms.Let N be an commutative and nilpotentIF-algebra andA :=Ann(N ) its annihilator. On the unital extensionN 0 = IF·11⊕N fix a projectionπ = π2 ∈End(N 0) with rangeπ(N ) = A andπ(11) = 0. Then

(9.12) bπ : N 0 ×N 0 → A, (x, y) 7−→ π(xy)

defines anA-valued symmetric 2-form. Clearly, the restriction ofbπ toN factorizes throughN/A×N/A and we write alsobπ for the correspondingA-valued 2-form onN/A. Fur-thermore,π determines the decomposition

(9.13)N 0 = N1 ⊕N2 ⊕N3 with

N1 = IF·11, N2 = N ∩ kerπ ∼= N/A and N3 = A = Ann(N ) .

The canonical isomorphism∼= in (9.13) makesN2 to an algebra that we denote byN π2 .

In terms ofπ and the algebra structure onN the product onN π2 is given by(x, y) 7→

(id−π)(xy) for x, y ∈ N π2 .

9.14 Lemma. Let N 6= 0 be a commutative nilpotent associativeIF-algebra with annihi-latorA := Ann(N ) and letL : N 0 → End(N 0) be the left regular representation of theunital extensionN 0. Fix a projectionπ on N 0 with rangeA as above and letbπ be theassociated 2-form(9.12). Then

(i) The decomposition(9.13) is anL(N )-adapted decomposition ofN 0.(ii) bπ is nondegenerate.

(iii) bπ is associative, or equivalently, everyL(y) ∈ End(N 0) is bπ-selfadjoint.(iv) The subspacesN1 andN3 arebπ-isotropic while the subspaces(N1 ⊕ N3) andN2

arebπ-orthogonal to each other. Consequently, the restriction of bπ to the algebraN π

2∼= N/A is a non-degenerate associativeA-valued 2-form.

(v) In the particular caseIF = IR anddimA = 1 the following holds after fixing a linearisomorphismψ : A → IR: The type(p, q) of the real 2-formψ ◦ bπ onN 0 does notdepend on the choice of the projectionπ.

Commutative nilpotent algebras consisting of selfadjointendomorphisms

As indicated in the main part of this paper the classificationof the various local tube re-alizations ofSp,q is equivalent to the classification of maximal abelian subalgebrasv insu(p, q) = su(E, h), contained in the(−1)-eigenspace of the involutionτ of the Lie alge-brasu(p, q), up to conjugation by elements fromG := NSL(E)(su (p, q)), compare 3.4.

Every such abelian subalgebrav admits a unique decomposition into its ad-reduc-tive and its ad-nilpotent part,v = v red ⊕ vnil . Also, we have the further decompositionvnil =

⊕ n , where the building blocksn j are various ad-nilpotent abelian subalge-

brasn maximal insu(p, q) or sl (m,C). In turn, such Lie algebras (also contained inFix(−τ) ) are in a 1-1-correspondence to associative commutative and nilpotent subalge-bras ofEnd(V), that consist of selfadjoint endomorphisms with respect toa symmetric


non-degenerate 2-formh onV over IF = IR or IF = C. This is one motivation to investi-gate a symmetric version of maximal commutative and nilpotent subalgebras inEnd(E).

In this subsection letE be an arbitrary vector space of finite dimension overIF andh : E×E → IF a symmetric non-degenerate 2-form. With S(E, h) ⊂ End(E) we denote thelinear subspace of all operatorsa that are selfadjoint with respect toh (that is,h(ax, y) =h(x, ay) for all x, y ∈ E). GivenV ⊂ E we write V

⊥ for the orthogonal complementwith respect toh. Note that in generalV ∩ V

⊥ 6= 0. For (maximal) nilpotent commutativesubalgebras inEnd(E), contained in S(E, h), there areN -adapted decompositions whichare in addition related to the 2-formh:

9.15 Lemma.Leth : E×E → IF be a non-degenerate symmetric 2-form andN ⊂ End(E)a maximal nilpotent commutative subalgebra, contained inS(E, h). Let KN , BN be thecharacteristic subspaces as defined in9.3. Then:

(i) KN = B⊥N .

(ii) dimAnn(N ) = 1(iii) There exists anN -adapted decompositionE = E1 ⊕ E2 ⊕ E3 such that

(a) dimE1 = dimE3 = 1.(b) h

∣∣E3

= 0, h∣∣E1

= 0, and the pairingh : E1 × E3 → IF as well as the restriction

h∣∣E2

are non-degenerate.

(c) E2 = (E1 ⊕ E3)⊥.

9.16 Definition. LetN ⊂ End(E) be as in the previous Lemma. We call every decomposi-tionE = E1 ⊕E2 ⊕E3 which satisfies the condition (ii) in Lemma 9.15 an(N , h)-adapteddecomposition of E.

In the following letN ⊂ S(E, h) be a maximal nilpotent commutative subalgebra ofEnd(E) andA := Ann(N ) its annihilator (which is 1-dimensional according to the abovelemma). Next, we relate the 2-formbπ : N 0 × N 0 → A := Ann(N ) to the symmetric2-form h : E × E → IF. Recall that the choice of the projectionπ : N → A is equivalentto the choice of a linear subspaceN2 ⊂ N with N = N2 ⊕ A. It is easy to see that every(N , h)-adapted decomposition ofE gives rise to the complementary subspaceN2 := {n ∈N : n(E1) ⊂ E2}, i.e.,N2 ⊕A = N . It turns out to be more subtle to prove the oppositestatement as it involves the solution of certain quadratic equations inE.

9.17 Proposition. LetN ⊂ S(E, h) be a maximal nilpotent commutative subalgebra. Then:(i) For every linear subspaceN2 ⊂ N satisfyingN = N2 ⊕A, there exists an(N , h)-

adapted decompositionE = E1 ⊕ E2 ⊕ E3 with

N2 = {n ∈ N : n(E1) ⊂ E2} .

(ii) ForE = E1⊕E2⊕E3 as in(i) choose generatorse1 ∈ E1, e3 ∈ E3 with h(e1, e3) = 1and defineκ : A → IF by n(e1) = κ(n)e3 for all n ∈ A. Let π : N 0 → A theprojection corresponding with kernelN1 ⊕ N andκ ◦ bπ : N 0 × N 0 → IF thecorresponding nondegenerate symmetric 2-form. Then the map

N 0 → E, m 7→ m(e1)

is an isometry between(N 0, κ ◦ bπ) and(E, h) which respects the decompositionsN 0 = IF·11⊕N2 ⊕A andE = E1 ⊕ E2 ⊕ E3

Proposition 9.17 is the main ingredient in the proof of the following theorem, which can beconsidered as a symmetric version of Proposition 9.10:


9.18 Theorem. Let h andh′ be two non-degenerate symmetric forms onE andE′ respec-tively over IF , and letN ⊂ End(E), N ′ ⊂ End(E′) be two maximal nilpotent andcommutative subalgebras.

(i) Assume, that in additionN ⊂ S(E, h) andN ′ ⊂ S(E′, h′) holds. ThenN andN ′ areisomorphic asIF-algebras if and only if there exists an isometryΨ : (E, h) → (E′, h′)with N ′ = Ψ ◦ N ◦Ψ−1.

(ii) In particular, ifE = E′ andh = h′, two such subalgebrasN ,N ′ contained inS(E, h)

are isomorphic if and only if they are conjugate by an elementin SO(E, h).

In caseIF = IR,C the above theorem has the following application for the classifica-tion of maximal abelian subalgebras ofsu(p, q) and sl(m,C). Let su (p, q) ∼= su (E, h)and τ : su(p, q) → su(p, q) be as in 3.4, induced by a conjugationτ : E → E. Re-call that we writeV = E

τ for the real points with respect toτ . Note thatsu(p, q)τ ∼=so (p, q) andSU(p, q)τ ∼= SO(p, q) ∼= SO(V, h|V). Furthersl(m,C)τ ∼= so (m,C), i.e.,τ : sl (m,C) → sl(m,C) is induced by a symmetric non-degenerate 2-formhτ onE.

9.19 Corollary. Two maximal abelian Lie subalgebrasv1, v2 in su(E, h) respectivelysl (E) consisting of nilpotent elements and contained insu(E, h)−τ ∼= iS(V, h)0 respec-tively sl (E)−τ ∼= S(E, hτ )0 are conjugate underSO(V, h) respectively underSO(E, hτ ) ifand only if the corresponding associative algebrasiv1, iv2 in End(V) (respectivelyv1 andv2 in End(E)) are isomorphic asIF-algebras.

Some affinely homogeneous surfaces

LetN 6= 0 be a commutative associative nilpotent algebra overIF of finite dimensionwith nil-index ν. For everyk ≥ 1 choose a linear subspaceVk ⊂ N k with N k = Vk ⊕N k+1. ThenN =

⊕k≥1 Vk with Vν = N ν , and we write everyx ∈ N in the form

x =∑ν

k=1 xk with xk ∈ Vk. With π : N → N ν we denote the canonical projectionx 7→ xν . As before we denote byN 0 = IF·11 ⊕N the unital extension ofN and extendπlinearly toN 0 by requiringπ(11) = 0.

Denote byP the space of all polynomial mapsp : N → N of the form

(9.20) x 7−→∑

pi1i2...irxi1xi2 · · · xir ,

where the integersr ≥ 1 and 1 ≤ i1 ≤ . . . ≤ ir satisfy∑r

j=1 ij ≤ ν and the coefficientspi1i2...ir are fromN 0. It is clear that with respect to compositionP is a unital algebra overN 0. We are mainly interested in polynomialsp ∈ P that are invertible inP , that is, wherefor every1 ≤ j ≤ ν the coefficient pj in front of the linear monomialxj is invertible inN 0.

For everyp ∈ P the compositionf := π ◦p is a polynomial mapN → N ν of degree≤ ν. Furthermore, in casep is invertible, the algebraic subvariety

(9.21) F := {x ∈ N : f (x) = 0}

is smooth, in fact, is the graph of a polynomial mapker(π) → N ν . Denote byAff(N ) thegroup of all affine automorphisms ofN .

9.22 Proposition. Suppose thatN has nil-indexν ≤ 4 and thatp ∈ P is invertible. Inaddition assume that

(i) p12 is invertible inN 0 if ν = 3,(ii) VjVk ⊂ Vj+k for all j, k and thatp112, p13 are invertible inN 0 if ν = 4.

Then forf := π ◦ p the groupA := {g ∈ Aff(N ) : f ◦ g = f } acts transitively on everytranslated subvarietyc+ F = f −1(c), c ∈ N ν .


Proof. We assumeν = 4, the casesν < 4 are similar but easier. For everyk = 2, 3, 4 denotebyLk ⊂ End(N ) the subspace of nilpotent transformationsx 7→ α1x1, x 7→ α2x1+α1x2,x 7→ α3x1 + α2x2 + α1x3 respectively with arbitrary coefficientsαj ∈ Vj .Now fix an arbitrary pointa ∈ F and denote byτ ∈ Aff(N ) the translationx 7→ x+ a. Asimple computation shows

f ◦ τ(x) = f (x) + x21R2(x) + x1R3(x) +R4(x)

for suitableRk ∈ Lk. Thenρ := id−p−1112R2 ∈ GL(N ) is unipotent and satisfies

f ◦ τ ◦ ρ (x) = f (x) + x1S3(x) + S4(x)

for suitableSk ∈ Lk. Furtherσ := id−p−113 S3 ∈ GL(N ) satisfies

f ◦ τ ◦ ρ ◦ σ(x) = f (x) + T4(x)

for a suitableT4 ∈ L4. Finally, g(x) = τ ◦ ρ ◦ σ(x− p−14 T4(x)) defines an elementg ∈ A

with g(c) = c+ a for all c ∈ N ν .

9.23 Remark. The proof of Proposition 9.22 also works for fieldsIF of arbitrary character-istic. Special polynomialsp ∈ P can be defined in the following way: Let

(9.24) Φ :=∞∑

k=1

ckTk ∈ IF[[T ]]

be an arbitrary formal power series overIF with vanishing constant term. SinceN is nilpo-tentp(x) := Φ(x) =

∑ckx

k defines a polynomial mapp ∈ P . Clearly,p is invertible ifand only ifc1 6= 0. Furthermore, if we assume thatIF has characteristic0, then invertibilityof p12 is equivalent toc2 6= 0 and invertibility ofp13p112 is equivalent toc2c3 6= 0. For sim-plicity we may add a constant termc0 to the formal power seriesΦ (which will not count) ifwe at the same time extend the projectionπ fromN to its unital extensionN 0 = IF·11⊕Nby requiringπ(11) = 0. Later we are mainly interested in the case whereΦ = exp is theusual exponential series.

The conditions (i), (ii) in Proposition 9.22 cannot be omitted: As a simple examplewith ν = 3 consider the 3-dimensional cyclic algebraN with basise1, e2 = e21, e3 = e31satisfyinge41 = 0. Then, identifyingN with IF3 in the obvious way, we get forΦ = T +T 3

in (9.24) thatf (x) = x3 + x31 on IF3. In this case, the groupA does not act transitively onF = f −1(0) in general. In fact, in caseIF = IR the affine groupAff(F ) = {g ∈ Aff(N ) :g(F ) = F} has precisely two orbits inF - the line IRe2 ⊂ F and its complement inF .Indeed, the subgroup of all(x1, x2, x3) 7→ (tx1, x2 + s, t3x3) with s ∈ IR, t ∈ IR∗ actstransitively on the complement.

Nil-polynomials

In this subsection letN be an arbitrary commutative associative nilpotent algebraoffinite dimension overIF with annihilatorA of dimension1 and nil-indexν. Let us call everylinear formω onN with ω(A) = IF apointingof N . Also, N with a fixed pointing is calleda pointed algebra,a PANA for short. Two PANAs(N , ω), (N , ω) are calledisomorphicifthere is an algebra isomorphismg : N → N with ω = ω ◦ g. As before with projectionswe always consider every pointingω onN linearly extended toN 0 by requiringω(11) = 0.

For every vector spaceV of finite dimension we denote byIF[V ] the algebra of all(IF-valued) polynomials onV .


9.25 Definition. f ∈ IF[W ] is called anil-polynomialonW if there exists a PANA(N , ω)and a linear isomorphismϕ : W → ker(ω) ⊂ N such thatf = ω ◦ exp ◦ϕ. Two nil-polynomialsf ∈ IF[W ], f ∈ IF[W ] are calledequivalentif there existst ∈ GL(IF) ∼= IF∗

and a linear isomorphismg : W → W with f = t ◦ f ◦ g−1.

9.26 Definition. In caseV 6= 0 we callf ∈ IF[V ] anextended nil-polynomialonV if thereexists a PANA(N , ω) and a linear isomorphismϕ : V → N with f = ω ◦ exp ◦ϕ. In caseV = 0 every constant inIF∗ is called an extended nil-polynomial onV .

Nil-polynomials on vector spacesW of dimensionn and extended nil-polynomialson vector spacesV of dimensionn + 1 correspond to each other. Indeed, every extendednil-polynomial onV is a sumf =

∑k>0 f[k] of homogeneous partsf[k] of degreek.

Furthermore,V = W ⊕ A with W = ker f[1] andA = {y ∈ V : f[2](x + y) = 0 ∀x ∈V } ∼= IF. Then the restriction off toW is a nil-polynomial onW and every nil-polynomialonW occurs this way. For our applications in Section 7 we need extended nil-polynomials.In the following we consider only nil-polynomials for simplicity.

By definition, every equivalence class of nil-polynomials in IF[W ] is an orbit of thegroupGL(IF)×GL(W ) acting in the obvious way onIF[W ]. For every pair of nil-polyno-mialsP ∈ IF[W ], P ∈ IF[W ] we get a new nil-polynomialP ⊕ P ∈ IF[W ⊕ W ] by setting(P ⊕ P )(x, x) := P (x) + P (x) for all x ∈W andx∈W .

Fix a nil-polynomialf ∈ IF[W ] in the following. Then we have the expansionf =∑k≥2 f[k] into homogeneous parts. Notice thatf[2] is a non-degenerate quadratic form on

W . For everyk ≥ 2 there is a unique symmetrick-form ωk onW with

(9.27) ωk(x, . . . , x) = k! f[k](x)

for all x ∈ W . Usingf[2] andf[3] we define a commutative (not necessarily associative)product(x, y) 7→ x·y onW by

(9.28) ω2(x·y, z) = ω3(x, y, z) for all z ∈W

and also a commutative product onW ⊕ IF by

(9.29) (x, s)(y, t) := (x·y, ω2(x, y)) .

Then, if f = ω ◦ exp ◦ϕ for a PANA (N , ω) with kernelK = ker(ω) and linear isomor-phismϕ : W → K we haveωk(x1, . . . , xk) = ω((ϕx1)(ϕx2) · · · (ϕxk)) for all k ≥ 2andx1, . . . , xk ∈ W . For the annihilatorA of N there is a unique linear isomorphismψ : IF → A such thatπ = ψ ◦ ω is the canonical projectionK ⊕ A → A. With theseingredients we have9.30 Lemma. With respect to the product(9.29)the linear map

W ⊕ IF → N , (x, s) 7→ ϕ(x) + ψ(s) ,

is an isomorphism of algebras. In particular,W with productx·y is isomorphic to the nilpo-tent algebraN/A and has nil-indexν−1.

Proof. For allx, y ∈W we have

(ϕ(x) + ψ(s))(ϕ(y) + ψ(t)) = (N −A) +A with

N := ϕ(x)ϕ(y) ∈ N and A := π(ϕ(x)ϕ(y)) = ψ(ω2(x, y)) ∈ A.

It remains to showN −A = ϕ(x·y). But this follows from

N −A ∈ K and ω(ϕ(x·y)ϕ(z)

)= ω

(ϕ(x)ϕ(y)ϕ(z)

)= ω

((N −A)ϕ(z)

)

for all z ∈W .


9.31 Corollary. Every nil-polynomialf onW is uniquely determined by its quadratic andcubic term,f[2] andf[3]. In fact, the otherf[k] are recursively determined by

(9.32) ωk+1(x0, x1, . . . , xk) = ωk(x0·x1, x2, . . . , xk)

for all k ≥ 2 and allx0, . . . , xk ∈W , where the symmetricωk are determined by(9.27).

Another application of (9.30) is the following

9.33 Proposition.Let (N , ω), (N , ω) be arbitrary PANAs and letf ∈ IF[W ], f ∈ IF[W ] beassociated nil-polynomials respectively. Then, iff andf are equivalent as nil-polynomials,alsoN andN are isomorphic as algebras.

Proof. Write f = t ◦ f ◦ g−1 as in Definition 9.25 and define the products· and∼· onW

andW as in (9.28). With respect to these productsg : W → W is an algebra isomorphism.

As in (9.29) the products· and∼· extend to the algebrasW ⊕ IF ∼= N andW ⊕ IF ∼= N .

Finally g ⊕ id gives an algebra isomorphism between them.

9.34 Lemma. For every nil-polynomialf onW the cubic termc := f[3] is trace-free withrespect to the quadratic termq := f[2], see[7] p. 20 for this notion of trace,

Proof. Let f ∈ IF[W ] be given by the PANA(N , ω) with nil-index ν. Without loss ofgenerality we assume thatW = ker(ω). Choose a basise1, . . . , en of W and a mappingα : {1, . . . , n} → {1, . . . , ν−1} such that{ei : α(i) = ℓ} is a basis ofN ℓ/N ℓ+1 forℓ = 1, . . . , ν−1. With respect to this basis the formsq, c are given by the tensorsgij =ω2(ei, ej) andhijk = ω3(ei, ej , ek). It is clear thatgij = 0 holds if α(i) + α(j) > ν.Sinceα is surjective, this impliesgij = 0 if α(i) + α(j) < ν, where(gij) = (gij)

−1. Onthe other hand,hijk = 0 if α(i) + α(j) ≥ ν, proving the claim.

Corollary 9.31 suggests the following question: Given a non-degenerate quadraticform q and a cubic formc onW . When does there exist a nil-polynomialf ∈ IF[W ] withf[2] = q andf[3] = c ? Usingq, c we can define as above fork = 2, 3 the symmetrick-linear formωk onW and with it the commutative productx·y onW . A necessary andsufficient condition for a positive answer is thatW with this product is a nilpotent andassociative algebra. As a consequence we get for every fixed non-degenerate quadratic formq onW the following structural information on the space of all nil-polynomialsf onWwith f[2] = q: Denote byC the set of all cubic forms onW . ThenC is a linear space ofdimension

(n+23

), n = dimW , and

(9.35) Cq := {c ∈ C : ∃ nil-polynomialf onW with f[2] = q, f[3] = c}

is an algebraic subset. The orthogonal group O(q) = {g ∈ GL(W ) : q ◦ g = q} acts fromthe right onCq. The O(q)-orbits inCq are in 1-1-correspondence to the equivalence classesof nil-polynomialsf onW with f[2] = q.

Examples of nil-polynomials of degree3 can be constructed in the following way.

9.36 Proposition. LetW be anIF-vector space of finite dimension andq a non-degeneratequadratic form onW . Suppose furthermore thatW =W1 ⊕W2 for totally isotropic linearsubspacesWk and thatc is a cubic form onW1. Then, if we extendc toW by c(x+ y) =c(x) for all x ∈ W1, y ∈ W2, the sumf := q + c is a nil-polynomial onW . In particular,c ∈ Cq.Everyg ∈ GL(W1) extends to anh ∈ O(q) ⊂ GL(W ) in such a way that withc := c ◦ galsoq + c = f ◦ h is a nil-polynomial onW .

Proof. ω3(x, y, t) = 0 for all t ∈ W2 impliesW1·W1 ⊂ W2 andW1·W2 = 0, that is,(x·y)·z = 0 for all x, y, z ∈W .


Now fix g ∈ GL(W1). There exists a uniqueg♯ ∈ GL(W2) with ω2(gx, y) = ω2(x, g♯y)

for all x ∈W1 andy ∈W2. But thenh := g × (g♯)−1 ∈ O(q) does the job.

Let IF ⊂ IK be a field extension and consider every polynomialf onV in the canon-ical way as polynomialf onV ⊗IF IK. Then withf alsof is a nil-polynomial. In general,for non-equivalent nil-polynomialsf , g on V the nil-polynomialsf , g may be equivalent.We use these extensions in caseIR ⊂ C.

Graded PANAs

Let (N , ω) with annihilatorA be a PANA in the following. A grading then is a de-composition

(9.37) N =⊕

k>0

Nk , NjNk ⊂ Nj+k .

ClearlyA = Nd for d := max{k : Nk 6= 0} andν = d/m is the nil-index ofN , wherem := min{k : Nk 6= 0}. Without loss of generality we assume thatW :=

⊕k<dNk is the

kernel ofω.

For the corresponding nil-polynomialf = ω ◦ exp ∈ IF[W ] andℓ := d− 1 we thenhave

(9.38) f =ν∑

k=2

f[k] with f[k](x) =1

k!

( ∑

j1+...+jk=d

xj1xj2 · · · xjk)

for all x = (x1, . . . , xℓ) ∈ N1 ⊕ · · · ⊕ Nℓ = W ⊂ N , where every indexjℓ in (9.38) ispositive. If we put, using (9.1),

‖µ‖ = µ1 + 2µ2 + . . . + ℓµℓ and x(µ) = x(µ1)1 x

(µ2)2 · · · x(µℓ)

ℓ

for every multi-indexµ ∈ INℓ and everyx = (x1, . . . , xℓ), we can rewrite (9.38) as

(9.39) f(x) =∑

‖µ‖=d

x(µ) .

We consider an example.

9.40 Cyclic PANAs For fixed integerν ≥ 1 let N be thecyclic algebra of nil-indexνover IF, that is, there is an elementξ ∈ N such that the powersξk, 1 ≤ k ≤ ν, forma basis ofN and ξν+1 = 0. ThenN is a graded algebra with respect toNk := IF ξk

for all k > 0 in (9.37) and becomes a PANA with respect to the pointingω uniquelydetermined byω(ξk) = δk,ν for all k. For n := ν − 1 we identify IFn andker(ω) via(x1, . . . , xn) =

∑xkξ

k. The corresponding nil-polynomialf ∈ IF[x1, . . . , xn] will thenbe called acyclic nil-polynomial,see also Table 1. In caseIF = IR the quadratic formf[2]has type(⌈n

2⌉, ⌊n

2⌋).

f[2] f[3] f[4] f[5] f[6]

0 0 0 0 0

x(2)1 0 0 0 0

x1x2 x(3)1 0 0 0

x1x3 + x(2)2 x

(2)1 x2 x

(4)1 0 0

x1x4 + x2x3 x1x(2)2 + x

(2)1 x3 x

(3)1 x2 x

(5)1 0

x1x5 + x2x4 + x(2)3 x1x2x3 + x

(2)1 x4 + x

(3)2 x

(2)1 x

(2)2 + x

(3)1 x3 x

(4)1 x2 x

(6)1

Table 1: Cyclic nil-polynomialsf ∈ IF[x1, . . . , xn−1] for 1 ≤ n ≤ 6, wherey(k) := yk/(k!)


For graded PANAsN =⊕

k>0Nk with annihilatorA = Nd we have for everys ∈ IF∗ the algebra automorphism

(9.41) θs :=⊕

k>0

sk id|Nk∈ Aut(N ) .

As a consequence, ift ∈ IF∗ admits ad th root in IF, the pointingsω andtω differ by anautomorphism ofN .

We mention that the PANAN associated with the nil-polynomialf = ω + q + c

considered in Proposition 9.36 also has a grading: Indeed, put N1 := W1, N2 := W2 andA := N3 := IF with products given byxy := x·y if x, y ∈ W1 andxy := ω2(x, y) ifx ∈W1, y ∈W2. Using this we can improve the second statement in 9.36.

PANAs N admitting a grading enjoy a special property: It is easy to see as a con-sequence of (9.39) that for every associated nil-polynomial f ∈ IF[x1, . . . , xn] there existlinear formsλ1, . . . , λn on IFn with

(9.42) f =n∑

k=1

λk ∂f/∂xk .

9.43 Proposition. With the notation of Proposition9.36assume that the cubic formc onW1 has the following property:(∗) z = 0 is the only elementz ∈W1 with c(x+ z) = c(x) for all x ∈W1.The graded PANAN = W1 ⊕W2 ⊕ IF with product(x, y) 7→ xy corresponding to thenil-polynomial f = q + c onW then satisfiesN 2 = W2 ⊕ IF as a consequence of(∗).Furthermore, if c is a second cubic form onW1 with nil-polynomial f = q+ c and PANAN =W1 ⊕W2 ⊕ IF with appropriate product, the following conditions are equivalent.

(i) The nil-polynomialsf , f are equivalent.(ii) The algebrasN , N are isomorphic as abstract algebras.

(iii) f = f ◦ g for someg ∈ GL(W1).

Proof. Let ω : W ⊕ IF → IF be the canonical projection. Thenω is a pointing forN . Asin (9.27) define theωk for the nil-polynomialf = f[2] + f[3] onW . We have to show that{x·y : x, y ∈ W1} spansW2. If not, there exists a vectorz 6= 0 in W1 with ω2(x·y, z) = 0for all x, y ∈ W1. But thenω3(x, y, z) = 0 for all x, y ∈ W1 impliesc(x + z) = c(x) forall x ∈W1, a contradiction.

(i) ⇐⇒ (ii) This follows immediately from Definition 9.26.

(iii) =⇒ (i) This follows immediately from the second claim in Proposition 9.36.

(ii) =⇒ (iii) Let h : N → N be an algebra isomorphism. Thenh(W2⊕IF) ⊂ N 2 ⊂ (W2⊕IF) as a consequence of(∗), that is, there is ag ∈ GL(W1) with h(x) ≡ g(x)modN 2 forall x ∈ W1. By the second statement in Proposition 9.36 we may assumeg = id withoutloss of generality. But thenc(x) = c(h(x)) = c(x) for all x ∈W1 implies f = f .

Suppose thatW1∼= IFm with coordinates(x1, . . . , xm) has dimensionm > 0 in

Proposition 9.43. ThenW ∼= IF2m with coordinates(x1, . . . , xm, y1, . . . , ym) and we mayassumeq(x, y) = x1y1+ . . .+xmym. As already mentioned, the linear spaceC of all cubicformsc onW1 has dimension

(m+23

). The subsetC∗ of all c ∈ C satisfying the condition

(∗) in Proposition 9.43 is Zariski open and dense inC . The groupGL(W1) acts onC∗

from the right and has dimensionm2 overIF. The difference of dimensions is(m3

). But this

number is also the cardinality of the subsetJ ⊂ IN3, consisting of all triplesj = (j1, j2, j3)with 1 ≤ j1 < j2 < j3 ≤ m. Consider the affine map

(9.44) α : IFJ → C , (tj) 7→ c0 +∑

j∈J

tjcj ,


wherec0 := x31 + . . . + x3m ∈ C∗ andcj := xj1xj2xj3 ∈ C for all j ∈ J .

In caseIF = IR or IF = C, for a suitable neighbourhoodU of 0 ∈ IFJ the mapα : U → C∗ intersects allGL(n, IF)-orbits inC∗ transversally. Indeed, since all partialderivatives ofc0 are monomials containing a square, the tangent space atc0 of itsGL(n, IF)-orbit is transversal to the linear subspace〈cj : j ∈ J〉 of C . In particular, in casem ≥ 3 thereis a family of dimension

(m3

)≥ 1 overIF (= IR orC) of pairwise differentGL(n, IF)-orbits

and thus of non-equivalent nil-polynomials of degree3 onW . Notice that in casem = 3the mappingα in (9.44) reduces to

α : IF → C , t 7→ x31 + x32 + x33 + tx1x2x3 .

We apply Proposition 9.22 to the examples considered in Proposition 9.36. HereV =W1⊕W2⊕ IF, q is a quadratic form onW :=W1⊕W2 andc is a cubic form onW1. Withthe extended nil-polynomialf(x, y, t) = t+ q(x + y) + c(x) onW1 ⊕W2 ⊕ IF considerthe hypersurface

F := {z ∈ V : f(z) = 0}and identifyIF with the line{0} ⊕ IF in W ⊕ IF. Then the affine groupAff(F ) is transitiveonF by Proposition 9.22, and every orbit inV intersects the lineIF. For everys ∈ IF∗ thetransformationθs, see (9.41), satisfiesf ◦ θs = s3h for everys ∈ IF∗, that is,θs ∈ Aff(F ).As a consequence, the number ofAff(F )-orbits inV is bounded by the number of(IF∗)3-orbits inIF. In particular, if(IF∗)3 = IF∗, then there are precisely twoAff(F )-orbits inV ,namelyF and its complement. This situation occurs, for instance, for IF = IR and also forIF = C. In any case,F is the only Zariski closedAff(F )-orbit in V , and every other orbitis Zariski dense.

Nil-polynomials of degree 4

The method in Proposition 9.36 can be generalized to get nil-polynomials of higherdegree, say of degree 4 for simplicity. Throughout the subsection we use the notation (9.1).

Let W = W1 ⊕W2 ⊕ W3 be a vector space withW1 = IFn, W2 = IFm and letq be a fixed non-degenerate quadratic form onW in the following. Assume thatW1, W3

are totally isotropic and thatW1 ⊕W3, W2 are orthogonal with respect toq. ThenW hasdimension2n+m, and without loss of generality we assume that

q(y) =m∑

k=1

εky(2)k for suitable εk ∈ IF∗ and all y ∈W2 .

As before letC be the space of all cubic forms onW . Our aim is to find cubic formsc ∈ Cq

that are the cubic part of a nil-polynomial of degree 4.

Denote byC ′ the space of all cubic formsc on W1 ⊕ W2 such thatc(x + y) isquadratic inx ∈W1 and linear iny ∈W2, or equivalently, which are of the form

c(x+ y) =1

2

m∑

k=1

n∑

i,j=1

cijkxixjyk for all x ∈W1, y ∈W2

with suitable coefficientscijk = cjik ∈ IF. Extending everyc ∈ C ′ trivially to a cubic formonW we considerC ′ as subset ofC .

For fixedc ∈ C ′ the symmetric 2- and 3-linear formsω2, ω3 onW are defined byω2(x, x) = 2q(x) andω3(x, x, x) = 6c(x) for all x ∈ W . With the commutative product


x·y onW , see (9.28), define in addition also thek-linear formsωk by (9.32) for allk ≥ 4.Then, for everyx, y ∈ W1 the identityω2(x·y, t) = ω3(x, y, t) = 0 for all t ∈ W1 ⊕W3

impliesx·y ∈ W2, that isW1·W1 ⊂ W2. In the same wayω2(x·y, t) = 0 for all x ∈ W1,y ∈W2 andt ∈W2⊕W3 impliesW1·W2 ⊂W3. AlsoWj ·Wk = 0 follows for all j, k withj + k ≥ 4. Thereforec belongs toCq if and only if (a·b)·c = a·(b·c) for all a, b, c ∈W1.

In terms of the standard basise1, . . . , em of W2 = IFm we have

a·b =m∑

k=1

( n∑

i,j=1

ε−1k cijkaibj

)ek for all a, b ∈W1

and thus withΘi,j,r,s :=∑m

k=1 ε−1k cijkcrsk we get the identity

ω2

((a·b)·c, t

)= ω3(a·b, c, t) = ω2(a·b, c·t)

=n∑

i,j,r,s=1

Θi,j,r,s aibjcrts for all a, b, c, t ∈W1 .

Therefore,

(9.45) A := C ′ ∩ Cq = {c ∈ C ′ : Θi,j,r,s is symmetric in i, r} .

Notice that the condition in (9.45) implies thatΘi,j,r,s is symmetric in all indices.A is a ra-tional subvariety of the linear spaceC ′, it consists of all thosec for which the correspondingproductx·y onW is associative.

For everyc ∈ A the corresponding nil-polynomialf onW has the form

f = f[2] + f[3] + f[4] with f[2] = q, f[3] = c and

f[4](z) =1

12q(x·x) for all z = (x, y, t) ∈W1 ⊕W2 ⊕W3 .

The groupΓ := GL(W1) × O(q|W2) ⊂ GL(W1 ⊕W2) acts onC ′ by c 7→ c ◦ γ−1 for

everyγ ∈ Γ. Furthermore,(g, h) 7→ (g, h, (g♯)−1) embedsΓ into O(q), compare the proofof Proposition 9.36. As a consequence, the subvarietyA ⊂ C ′ is invariant underΓ.

For everyc ∈ A the corresponding nil-polynomial comes from a graded PANA withnil-index 4, providedc 6= 0. Indeed, putW4 := IF and endowW ⊕W4 with the product(9.29). It is obvious that the linear span ofW1·W1 in W2 has dimension≤

(n+12

).

Let us consider the special casen = 2 with m =(n+12

)= 3 in more detail. For

simplicity we assume that for suitable coordinates(x1, x2) of W1, (y1, y2, y3) of W2 and(z1, z2) of W3 the quadratic formq is given by

(9.46) q = x1z1 + x2z2 + y(2)1 + y

(2)2 + εy

(2)3 for fixed ε ∈ IF∗

(in caseIF = IR,C this is not a real restriction). For everyt ∈ IF consider the cubic form

ct := (x(2)1 + x

(2)2 )y1 + x1x2y2 + tx

(2)2 y3

onW1 ⊕W2. A simple computation reveals that everyct is contained inA = C ′ ∩ Cq. Thecorresponding nil-polynomial (depending on the choice ofε) then is

ft = q + ct + dt with dt := x(4)1 + x

(2)1 x

(2)2 + (1 + ε−1t2)x

(4)2 .


In addition we putf∞ := q + x

(2)2 y3 + ε−1x

(4)2

(a smash product with the cyclic nil-polynomial of degree 4,see Table 1), where∞ in theprojective lineIP1(IF) = IF ∪ {∞} is the point at infinity. Notice that fort ∈ IF∗ thenil-polynomialsft andf1/t := q + t−1ct + t−2dt are equivalent.

It is obvious thatft is equivalent tof−t for everyt ∈ IP1(IF). Also, for every cubicterm ct with t ∈ IF∗ the setW1·W1 spansW2. For t = 0,∞ the linear span ofW1·W1

in W2 has dimension2, 1 respectively. For everyt ∈ IF∗ an invariant ofdt is the numberφ(t) := g2(dt)

3/g3(dt)2 = ε2t−4(4+ε−1t2)3 ∈ IF, whereg2, g3 are the classical invariants

of binary quartics, compare [19] p. 27. Since every fiber ofφ : IF∗ → IF contains at most 6elements we conclude

9.47 Proposition. For every FieldIF and every fixedε ∈ IF∗ the set of all equivalenceclasses given by all nil-polynomialsft, t ∈ IF, has the same cardinality asIF and, in partic-ular, is infinite.

Remarks1. In caseIF = Q is the rational field there are infinitely many choices ofε ∈ Q∗

leading to pairwise non-equivalent quadratic formsq in (9.46). For each such choice thereis an infinite number of pairwise non-equivalent nil-polynomialsft of degree 4 overQ.

2. In caseIF = IR is the real field there are essentially the two choicesε = ±1. In caseε = 1 the formq has type(5, 2) and all nil-polynomialsft with 0 ≤ t ≤

√8 are pairwise

non-equivalent. In caseε = −1 the formq has type(4, 3) and allft with 0 ≤ t ≤ ∞ arepairwise non-equivalent.

3. Nil-polynomials of degree≥ 5 can be constructed just as in the case of degrees 3 and 4as before. As an example we briefly touch the case of degree 5: Fix a vector spaceW offinite dimension overIF together with a non-degenerate quadratic formq onW . Assumefurthermore that there is a direct sum decompositionW = W1 ⊕ W2 ⊕ W3 ⊕ W4 intototally isotropic subspaces such thatW1⊕W4 andW2⊕W3 are orthogonal. Then considera cubic formc onW that can be written as a sumc = c ′ + c ′′ of cubic forms with thefollowing properties:c ′ is a cubic form onW1 ⊕W3 (trivially extended toW ) that is linearin the variables ofW3 while c ′′ is a cubic form onW1 ⊕W2 that is linear in the variablesof W1. Denote byx·y the commutative product onW determined byq andc . If we putWk := 0 for all k > 4 we haveWj ·Wk ⊂ Wj+k for all j, k. Therefore,c ∈ Cq if and onlyif the productx·y onW is associative, see (9.35) for the notation. This is true without anyassumption ifc = c ′ or c = c ′′. ButW1·W1 = 0 in the first andW2·W2 = 0 in the lattercase. On the other hand, the nil-polynomial associated toc ∈ Cq has degree 5 ifW2·W2

spansW4 andW1·W1 6= 0.

Affine homogeneity

In this subsection letIF be eitherIR orC. Also letN 6= 0 be an arbitrary commutativeassociative nilpotent algebra of finite dimension overIF. In addition we assume that thereexists aZ-gradation

N =⊕

k>0

Nk , NjNk ⊂ Nj+k .

Let d := max{k : Nk 6= 0} and denote byπ : N → Nd the canonical projection withkernelK :=

⊕k<dNk. We do not require thatNk 6= 0 for all 1 ≤ k ≤ d nor thatNd is

the annihilator or has dimension 1. Extendingπ linearly toN 0 by π(11) = 0 we have thepolynomial mapf := π ◦ exp : N → Nd. The submanifoldF := f −1(0) then is the graphof a polynomial mapK → Nd andK = T0F is the tangent space toF at the origin. We


are interested in the affine groupAff(F ) = {g ∈ Aff(N ) : g(F ) = F} and its subgroupA = A(f ) := {g ∈ Aff(N ) : f ◦ g = f }.

Every pointx ∈ N has a unique representationx = x1 + . . . + xd with xk ∈ Nk.Consider onN the linear spana of all nilpotent affine vector fields of the form

(d− j)αj∂/∂xj −d−j∑

k=1

kαjxk ∂/∂xj+kwith 1 ≤ j < d and αj ∈ Nj .

As an example, in caseN ∼= IF4 is the cyclic PANA of nil-index 4, see Table 1, we haved = 4,f (x) = x4 + x1x3 + x

(2)2 + x

(2)1 x2 + x

(4)1 , anda is the linear span of the vector fields

3∂/∂x1 − x1 ∂/∂x2− 2x2 ∂/∂x3

− 3x3 ∂/∂x4

2∂/∂x2 − x1 ∂/∂x3− 2x2 ∂/∂x4

∂/∂x3− x1 ∂/∂x4

.

With some computation we get:

9.48 Lemma. a is a nilpotent Lie algebra and the evaluation mapεa : a → N , ξ 7→ ξa, isinjective for everya ∈ N . In particular, all orbits inN of the nilpotent subgroupexp(a) ⊂Aff(N ) have the same dimension.

9.49 Proposition. a f = 0, that is,exp(a) ⊂ A(f). In particular,A(f ) acts transitively oneveryc+ F = f −1(c), c ∈ Nd.

Proof. Putξ := (d− j)α∂/∂xj − ∑d−jk=1 kαxk ∂/∂xj+k

for fixed1 ≤ j < d andα ∈ Nj .Then

ξf =∑

cναx(ν1)1 x

(ν2)2 · · · x(νd)d ,

where the sum is taken over all multi indicesν ∈ INd with ν1 + 2ν2 + . . . + dνd = d − jandcν certain rational coefficients. Now fix such a multi indexν. For simpler notation weputx(−1) := 0 for everyx ∈ N . Then we have

cναx(ν1)1 x

(ν2)2 · · · x

(νd)d

= (d− j)α∂/∂xj

(x(ν1)1 · · ·x

(νj+1)j · · · x

(νd)d

)−

d−j∑

k=1

kαxk ∂/∂xj+k

(x(ν1)1 · · ·x

(νk−1)k

· · ·x(νj+k+1)j+k

· · · x(νd)d

)

=

(d− j −

d−j∑

k=1

kνk)αx

(ν1)1 x

(ν2)2 · · · x

(νd)d

= 0

sinceνk = 0 for k > d− j.

Next we specialize to the case whereNd has dimension 1, that is,F is a hypersurfacein N (we still do not require thatNd is the annihilator ofN although contained in it). Foreverys ∈ IF∗ we have the semi-simple linear transformation

θs :=⊕

k>0

sk id|Nk∈ GL(F )

satisfying f ◦ θs = sdf . As a consequence we have that the groupAff(F ) has at most3 orbits inN . In cased odd or IF = C this group has only two orbits inN , the closedhypersurfaceF and the open complementN\F . As in the subsection 8.3 we get in caseIF = C further affinely homogeneous real surfaces.


References

1. Baouendi, M.S., Jacobowitz, H., Treves F.: On the analyticity of CR mappings. Ann. of Math.122 (1985),365-400.

2. Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.:Real Submanifolds in Complex Spaces and Their Mappings.Princeton Math. Series47, Princeton Univ. Press, 1998.

3. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta. Math.133(1974), 219-271.

4. Courter, R.C.: Maximal commutative subalgebras ofKn at exponent three. Linear Algebra and Appl.6(1973), 1-11.

5. Dadok, J., Yang, P.: Automorphisms of tube domains and spherical hypersurfaces. Amer. J. Math.107(1985),999-1013.

6. Eastwood, M.G.: Moduli of isolated hypersurface singularities. Asian J. Math.8 (2004), 305-314.

7. Eastwood, M.G., Ezhov, V.V.: On affine normal forms and a classification of homogeneous surfacesin affinethree-space. Geom. Dedicata77 (1999), 11-69.

8. Fels, G.: Locally homogeneous finitely nondegenerate CR-manifolds. Math. Res. Lett.14 (2007), 693-922.

9. Fels, G., Kaup, W.: Classification of Levi degenerate homogeneous CR-manifolds of dimension 5. Acta Math.201 (2008), 1-82.

10. Fels, G., Kaup, W.: Local tube realizations of CR-manifolds and maximal abelian subalgebras. Ann. ScuolaNorm. Sup. Pisa. To appear

11. Isaev, A.V., Mishchenko, M.A.: Classification of spherical tube hypersurfaces that have one minus in the Levisignature form. Math. USSR-Izv.33 (1989), 441-472.

12. Isaev, A.V.: Classification of spherical tube hypersurfaces that have two minuses in the Levi signature form.Math. Notes46 (1989), 517-523.

13. Isaev, A.V.: Global properties of spherical tube hypersurfaces. Indiana Univ. Math.42 (1993), 179-213.

14. Isaev, A.V.:Rigid and Tube Spherical Hypersurfaces.Book manuscript in preparation.

15. Kaup, W.: On the holomorphic structure ofG-orbits in compact hermitian symmetric spaces. Math. Z.249(2005), 797-816.

16. Kaup, W., Zaitsev, D.: On local CR-transformations of Levi degenerate group orbits in compact Hermitiansymmetric spaces. J. Eur. Math. Soc.8 (2006), 465-490.

17. Knapp, Anthony W.: Lie groups beyond an introduction. 2nd ed. Progress in Mathematics140, Birkhauser,Boston 2002.

18. Laffay, Th.J.: The Minimal Dimension of Maximal Commutative Subalgebras of Full Matrix Algebras. LinearAlgebra and Appl.71 (1985), 199-212.

19. Mukai, S.: An introduction to invariants and moduli. Cambridge Univ. Press 2003

20. Tanaka, N.: On the pseudo-conformal gem,ometry of hypersurfaces of the space ofn complex variables. J.Math. Soc. Japan14 (1962), 397-429.

21. Warner, G.: Harmonic analysis on semi-simple Lie groupsI. Die Grundlehren der mathematischen Wis-senschaften. Band188. Berlin-Heidelberg-New York: Springer 1972.

G. Felse-mail:[email protected]

W. Kaupe-mail:[email protected]

Mathematisches Institut, Universitat Tubingen,Auf der Morgenstelle 10,72076 Tubingen, Germany

tube realizations of hyperquadrics

Documents

Transcript of tube realizations of hyperquadrics