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HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES

MATT KERR AND COLLEEN ROBLES

Abstract. Period domains, the classifying spaces for (pure, polarized) Hodge structures,and more generally MumfordTate domains, arise as open GRorbits in flag varieties G/P(homogeneous with respect to a complex, semisimple Lie group G). We investigate Hodgetheoretic aspects of the geometry and representation theory associated with flag vari-eties. More precisely, we prove that smooth Schubert variations of Hodge structure arenecessarily homogeneous, relate the GriffithsYukawa coupling to the variety of lines onG/P PV (under a minimal homogeneous embedding), construct a large class of polar-ized (or Hodgetheoretically accessible) GRorbits in G/P and compute the boundarycomponents of the polarizing nilpotent orbits, use these boundary components to defineenhanced SL2orbits, and initiate a project to express the homology classes of thesehomogeneous submanifolds of G/P in terms of Schubert classes.

Contents

1. Introduction 12. Preliminaries 73. Lines on flag varieties 164. Smooth Schubert VHS 225. GriffithsYukawa couplings and lines on G/P 276. Construction of limiting mixed Hodge structures 377. Enhanced SL2-orbits and adjoint varieties 52Appendix A. Examples in codimension one 60Appendix B. Computation of [X(N)] in OG(2,C2r+1) 72Appendix C. Dynkin diagrams 80References 80

1. Introduction

This is the first in a planned series of papers that studies the relationship between theHodge theory of MumfordTate domains D = GR/R and the projective geometry and GRorbit structure of their compact duals D = G/P . It continues lines of thought initiated inthe first authors work with Pearlstein [31,32] on boundary components and in the second

Date: July 18, 2014.2010 Mathematics Subject Classification. 14D07, 32G20. 14M15. 14C30.Key words and phrases. Hodge theory, period domains, MumfordTate domains, MumfordTate groups,

boundary components, variations of Hodge structure, Schubert varieties, GriffithsYukawa coupling.

1

2 KERR AND ROBLES

authors work [43,45] on Schubert varieties and the infinitesimal period relation (a.k.a. Grif-fithss transversality condition). Our purpose is twofold: to use the representation theory ofG and the geometry of D to better understand the asymptotics and differential invariantsof variations of Hodge structure; and to give Hodgetheoretic interpretations of familiargeometric objects on D, including GRorbits, Schubert varieties and their classes, and thevariety of minimal degree rational curves on G/P . Throughout, the fundamental adjointvarieties (Section 2.5) will serve as illustrative examples.

Background. MumfordTate domains are classifying spaces for Hodge structures on aQ-vector space VQ with fixed Hodge tensors, including a polarizing form Q (V )2.Recall how these arise: a (pure) Hodge structure polarized by Q, is given by a nonconstanthomomorphism

: S1 Aut(VR, Q)of real algebraic groups with Q(v, (

1)v) > 0, for all v VC \ {0}. Let GQ denote thesmallest Qalgebraic subgroup of Aut(VQ, Q) with GR (S1). This is the MumfordTategroup of . The real form GR is known to be reductive [20]. This group acts on byconjugation, and the orbit

Ddfn= GR = GR/R

is a MumfordTate domain. The stabilizer R is compact, and its Lie algebra r contains acompact Cartan subalgebra t of gR [24]. The MumfordTate domain is an analytic opensubset of its compact dual, the projective homogeneous variety

Ddfn= G F = G/P .

Here, G is the complexification of GR, and the action of G (and GR) is by left translationon the associated Hodge flag F D.One way to construct MumfordTate domains is to begin with a simple adjoint group

GQ, for which GR contains a compact maximal torus T . If E it = iLie(T ) is a gradingelement which is odd (respectively, even) on the noncompact (respectively, compact) roots,

then (z) = e2 log(z)E defines a weight zero Hodge structure (HS) on gQ = Lie(GQ) thatis polarized by (, ), where (, ) is the Killing form on gQ. Moreover, may lift to aHS on a representation VQ. In either case, a sufficiently general GR-conjugate of this HShas MumfordTate group GQ. In case lifts, the Hodge decomposition VC = V p,q is theEeigenspace decomposition; the grading element acts on V p,q by the scalar (pq)/2. Then is identified with the Hodge flag F p = apV a,b, the compact dual D = G/P is identifiedwith the Gflag variety containing F , and the MumfordTate domain D = GR/R with theopen GRorbit of F

in D. The Lie algebra of P is p = g0 = p0gp, where gp = gp,p.The Lie algebra of R is a real form of g0; we shall write

G0dfn= RC .

For the purposes of this introduction, we shall assume that D has a base point o = definedover Q (cf. Definition (6.17)).

Running Example 1. A case in point, which we shall use as a running example throughoutthis introduction, is the Qform of the exceptional, rank two group G2 constructed in

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 3

Section 6.3. The grading element S2 tQ(1) (dual to the second simple root) induces (i)a weight zero, (, ) polarized Hodge decomposition of the complex Lie algebra

g = g2 g1 g0 g1 g2 , gp = gp,p ,with Hodge numbers (1, 4, 4, 4, 1); and (ii) a weight two polarized Hodge decomposition

V = V 0,2 V 1,1 V 2,0on the standard, 7dimensional representation with Hodge numbers (2, 3, 2). The compactdual is the 5dimensional adjoint variety in Pg.

Recall that a variation of Hodge structure V over a complex manifold S (in the classicalsense) consists of a Q-split local system V, and a filtration of VOS by holomorphic sub-bundles F whose fibre-wise restrictions are Hodge flags, and which satisfy the infinitesimalperiod relation (F) F1 1S for the flat connection on V annihilating V. Theasymptotics of V over a punctured disk S are captured by the (limiting) mixed Hodgestructure (Vlim, F

lim,W)

dfn= (V,F,W(N))|s=0, where N is the monodromy logarithm of

V| and V = exp( log(s)/2i)V.If GQ contains the MumfordTate groups of the fibres of V, the possible limiting MHS

are classified by the boundary components B(N) of [31]. Moreover, V induces a periodmapping : S \D, with the monodromy group. We may lift | to the upperhalf plane : H D, and take the nave limit limIm() () D, which lies in theanalytic boundary bd(D) D. This boundary breaks into finitely many GRorbits, andthose accessible by such Hodgetheoretic limits were termed polarizable in [32].(1)

Schubert varieties and variations of Hodge structure. In this paper, the term vari-ation of Hodge structure (VHS) is used primarily in a different sense. The infinitesimalperiod relation (IPR) forces the differential of a period map to lie in the horizontal dis-tribution T 1 TD. A subvariety (or submanifold) Y D is horizontal if its tangent spaceTyY T 1y is contained in the horizontal distribution at every smooth point y Y . Forus a VHS shall simply be a horizontal subvariety/submanifold of D. In particular, we willcall a horizontal Schubert variety X D a Schubert VHS. The Schubert VHS encode agreat deal of information about the IPR and period mappings, cf. [45]. For example, thereexists a VHS of dimension d if and only if there exists a Schubert VHS of dimension d.In particular, the rank of the differential d of a period mapping is bounded above bymax{dimX | X D is a Schubert VHS}.Running Example 2. The horizontal distribution is a contact 4plane distribution. Modulothe action of G2, there is a unique Schubert variety Xd D of dimension d, for 0 d 5.The Schubert variety is a VHS if and only if d 2. See [45] for additional detail.One of the motivating questions behind this work is: when can a Schubert VHS be (up

to translation) the compact dual of a MumfordTate subdomain of D? Any such Schubertvariety is necessarily homogeneous and therefore smooth. The majority of Schubert varieties

(1)This notion of a polarized orbit is distinct from J. Wolfs in [47, Definition 9.1] where the orbitGR x D is polarized if it realizes the minimal homogeneous CRstructure on the homogeneous manifoldGR/StabGR (x), cf. [1, Remark 5.5].

4 KERR AND ROBLES

are singular. Moreover, a smooth Schubert variety X D need not be homogeneous.Nevertheless, we show that a smooth Schubert VHS is necessarily a homogeneous embeddedHermitian symmetric space (Theorem 4.4); moreover, with an assumption on the Q-typesof GQ and o D, these are all MumfordTate subdomains (Corollary 4.5). These resultsinvite comparison with a similar (but quite distinct) theorem of Friedman and Laza [22];see the remarks after Theorem 4.4. The proof of Theorem 4.4 makes use of a recent resultof J. Hong and N. Mok [27] which asserts that the smooth Schubert varieties in G/P , withP a maximal parabolic subgroup associated with a nonshort simple root, are necessarilyhomogeneous. The argument underlying this beautiful result is in turn based J.-M. Hwangand Moks program to study projective varieties by considering the variety of minimal degreerational curves passing through a point [29].

Running Example 3. The 2dimensional Schubert VHS X2 cannot be MumfordTate. Thiscan be seen at once (and was first observed by the authors) by looking at its tangent space,which cannot be that of a reductive group orbit in D. In fact, it turns out that X2 issingular, and therefore not homogeneous. (The 1dimensional Schubert VHS X1 is a P

1,and therefore homogeneous.)

Lines on D and the GriffithsYukawa coupling. When D is the minimal homogeneousembedding G/P PV , the minimal degree rational curves on D are lines P1 PV (degreeone). Let Co PToD be the variety of tangent directions to lines passing though o D.(We will often think of Co as the set of lines o P1 D.) Of particular interest here is theset

Co dfn= Co PT 1of directions horizontal with respect to the IPR: If P is a maximal parabolic, then thevariety

(1.1) Xdfn=

P1Co

P1

swept out by these lines (which is a cone over Co with vertex o) is a (typically singular) Schu-bert variety. When P is a maximal parabolic associated to a non-short root, we have Co = Coand X is horizontal, cf. [38, Proposition 2.11] and Lemma 3.3. When D is an adjoint variety(V = g), Co realizes the homogeneous Legendrian varieties studied by J.M. Landsberg andL. Manivel [36, 37].

Running Example 4. In our G2 example, X = X2 is the famous twisted cubic cone knownto E. Cartan, cf. Section 5.

In this paper we will discuss two Hodgetheoretic characterizations of Co. The first isin terms of the GriffithsYukawa coupling, a differential invariant of VHSs. (The secondis quite distinct, and will be discussed below.) When the kernel of the GriffithsYukawacoupling is nonempty, it necessarily contains Co (Theorem 5.8). Moreover, for certain com-pact duals (including the An, E6, E7, E8, F4, and G2 adjoint varieties, cf. Lemma 5.13and Section 5.3) equality holds (Co is the kernel of the GriffithsYukawa coupling) for someHodge representation of weight two (Theorem 5.11).

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 5

Polarized orbits. The polarized orbits, and the corresponding limiting mixed Hodge struc-tures, describe the asymptotics of a variation of Hodge structure on a MumfordTate do-main. More generally, one may take the nave limit of nilpotent orbit exp(zN)F on D.In this paper our emphasis is on representation theoretic descriptions of polarized orbitsand mixed Hodge structures associated with nilpotent orbits, with the motivation that suchdescriptions may guide the construction of geometric/motivic examples.(2)

Running Example 5. The boundary bd(D) contains a unique real-codimension one GRorbit, which is polarizable. The weight-graded pieces of the corresponding limiting mixedHodge structures have dim GrW3 Vlim = dim Gr

W1 Vlim = 2 and dim Gr

W2 Vlim = 3 (with

Hodge numbers (1, 1, 1)). Additional detail on this example may be found in [32, Section6.1.3].

Section 6 gives a systematic construction of polarizable boundary strata in bd(D) Dvia distinguished Q-Matsuki(3) points on them and sets B = {1, . . . , s} h of stronglyorthogonal roots(4), culminating in Theorem 6.38.

Remark. In the case that D is Hermitian symmetric, all the boundary orbits O bd(D)are polarizable. From [21, Theorem 3.2.1] it may be seen that they are all parameterizedby the construction of Section 6, and the parameterization is essentially that given by theHarishChandra compactification of D.

While notationally dense, the construction is very natural, and we will briefly summarize ithere. With each root j B is associated a Cayley transform cj G, and we let cB denotethe composite/product. From the QMatsuki point o and cB we obtain a QMatsuki pointoB D, with corresponding Hodge flag F B, a weight filtration WB , and nilpotent cone

Bdfn= {t1NB1 + + tsNBs | ti 0}

with NBj gR and NBj F pB F p1B , for all p. These objects have the properties that thenilpotent cone B underlies a multiple variable nilpotent orbit

(z1, . . . , zs) 7 exp(z1NB1 + zsNBs ) F B ,WB =W(B), and (F B,W

B ) is a (Qsplit) QMHS on g. In particular, the MHS (F

B,W

B )

belongs to the boundary component B(B), and the orbit

OB dfn= GR oBis polarizable.(5)

(2)See [26] for more on this line of thought.(3)The reason for this terminology is that the Cartan subalgebra h g corresponding to o is assumed

Q-rational (hence stable under complex conjugation) and stable under the Cartan involution; hence so areits Cayley transforms.

(4)The set B is constrained by an additional condition imposed by the IPR.(5)The notation of this introduction differs slightly from that of Section 6, where the terms cB, oB, F

B ,

WB , B, OB, gp,qB are denoted cs, os, sF , sW, s, Os, sgp,q, respectively.

6 KERR AND ROBLES

Boundary components. The construction also a yields a semisimple element YB gR whichgrades the weight filtration. The MumfordTate group of (F B,W

B ) is contained in

GBdfn= StabG{YB} StabG{NB1 } StabG{NBs } .

Defining(6)

DBdfn= GB,R F B GB F B dfn= DB ,

the set

B(B)dfn= eC B \ B(B)

of nilpotent orbits fibres naturally over DB. Passing to the quotient by the intersection of an arithmetic subgroup of GQ with Stab(B), the surjection \B(B) \DB has thestructure of an iterated fibration by generalized intermediate Jacobians.Part of the value of this construction is that it allows us to compute the ranks of the

components gp,qB in the Deligne splitting gC = gp,qB and the structure of the B(B). Forexample, when P is a maximal parabolic there is exactly one orbit O1 bd(D) of real codi-mension one (Proposition 6.56); it is of the form OB with B = {1}, and the correspondingboundary components B(B) = B(N) are completely worked out in Appendix A.

Adjoint varieties and Co DB. For the fundamental adjoint varieties, the Deligne bi-grading takes a very intriguing form (Figure 7.1): an element of the Weyl group of (g, h)exchanges the Hodge and weight gradings, and the homogeneous Legendrian varieties Coabove reappear in the guise of DB. Meanwhile, by Proposition 7.15, the \B(N) realize theintermediate Jacobian families associated to the FriedmanLaza [22] weight-three maximalHermitian VHSs of Calabi-Yau type!

Running Example 6. \B(N) is just an elliptic modular surface. (The FriedmanLazavariation is the symmetric cube of a VHS of type (1, 1).)

Enhanced horizontal SL2orbits and homology classes. Another motivation for theconstruction of Section 6 goes back to a basic question of Borel and Haefliger [8] on smootha-bility of cohomology classes in G/P . By [45, Theorem 4.1], the invariant characteristiccohomology of D is generated by classes of Schubert VHS. Moreover, the homology classof any horizontal cycle in D may be expressed as a linear combination of Schubert VHSclasses [43, Theorem 4.7]. Applying a VHS twist to Borel and Haefligers question, we ask:when can the class (or a multiple thereof) of a singular Schubert VHS, such as (1.1), admita smooth, horizontal, algebraic representative? One natural source of smooth subvarieties(in fact, MumfordTate subdomains) which are often horizontal, are the enhanced multi-variable SL2-orbits X(B) which arise as follows. The strongly orthogonal roots B abovedetermine a collection {slj2 | j B} of commuting sl2s. Letting SLB2 (C) GC be theconnected Lie subgroup with Lie algebra slB2 (C) = sl

12 (C) sls2 (C), the enhanced

SL2orbit is

X(B)dfn= SLB2 (C) GB,C oB D ,

(6)Here one should really think of GB as acting on the associated graded of the MHS (FB ,W

B ). Since

this is a direct sum of Hodge structures, DB is a MumfordTate domain in the usual sense.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 7

cf. Section 7.1. In the case that D is a fundamental adjoint variety, the enhanced SL2-orbitattached to O1 is a VHS and a cylinder on Co; that is,

X(B) = X(N) = Co P1 .Recalling that X is a cone over Co, cf. (1.1), we conclude this article by testing the irresistiblehypothesis that [X(N)] = [X].

Running Example 7. We will show that [X(N)] = 2[X] in Section 7.6.

The case that G = SO(2r + 1,C) is worked out in Appendix B.

Future work. Of the many directions one could pursue from here, among the more salientis the systematic computation of the cohomology classes of smooth VHS (and the horizontalX(B) in particular). We will undertake this in a sequel (by a different method than thoseof Section 7.6 and Appendix B). It is also important to generalize the construction ofSection 6 in order to parameterize all polarizable orbits; this work will appear in [44], andwe expect to use it to tie together combinatorial structures related to nilpotent cones andboundary orbits.Finally, one reason for classifying Hodgetheoretic phenomena is to predict algebro-

geometric ones. Turning one last time to the running example, suppose we have a 2-parameter family of algebraic surfaces : X S for which (a subquotient of) R2Qunderlies a maximal VHS with Hodge numbers (2, 3, 2) and MumfordTate group G2. Thenone can ask what sort of degeneration produces a point in the elliptic modular surfaceboundary component \B(N), and what sort of geometry produces a subvariation whoseperiod map image lies in the cubic cone GriffithsYukawa kernel. Now since one expectssuch 2-parameter VHS to arise from certain elliptic surfaces (with internal fibration over anelliptic curve, cf. [30]), neither question looks too difficult. With some optimism, one mightimagine asking the analogous questions for all the fundamental adjoint varieties.

Acknowledgements. The authors thank P. Griffiths and J.M. Landsberg for helpful dis-cussions, and the NSF for partial support under the aegis of grants DMS-1068974, 1259024,and 1361147 (Kerr) and DMS-1309238 and 1361120 (Robles). This work was undertakenwhile Robles was a member of the Institute for Advanced Study; she thanks the institutefor a wonderful working environment and the Robert and Luisa Fernholz Foundation forfinancial support.

2. Preliminaries

In this section we set notation and review the necessary background material from Hodgetheory and representation theory.

2.1. Flag varieties. Let G be a connected, complex semisimple Lie group, and let P Gbe a parabolic subgroup. The homogeneous manifold G/P admits the structure of a rationalhomogeneous variety as follows. Fix a choice of Cartan and Borel subgroups H B P .Let h b p g be the associated Lie algebras. The choice of Cartan determines a setof roots = (g, h) h. Given a root , let g g denote the root space. Given asubspace s g, let

(s)dfn= { | g s} .

8 KERR AND ROBLES

The choice of Borel determines positive roots + = (b) = { | g b}. LetS = {1, . . . , r} denote the simple roots, and set(2.1) I = I(p)

dfn= {i | gi 6 p} .

Note that

I(b) = {1, . . . , r} ,and I = {i} consists of single element if and only if p is a maximal parabolic.Let {1, . . . , r} denote the fundamental weights of g and let V be the irreducible g

representation of highest weight

(2.2) = Idfn=iI

i .

Assume that the representation g End(V ) integrates to a representation G Aut(V )of G. (This is always the case if G is simply connected.) Let o PV be the highest weightline in V . The Gorbit G o PV is the minimal homogeneous embedding of G/P as arational homogeneous variety.

Remark 2.3 (Non-minimal embeddings). More generally, suppose that V is the irreducibleGrepresentation of highest weight =

iI a

i i with 0 < ai Z. Again, the Gorbit of

the highest weight line o PV is a homogeneous embedding of G/P . We write G/P PV .The embedding is minimal if and only if ai = 1 for all i I. For example, the Veronesere-embedding vd(P

n) P SymdCn+1 of Pn is if minimal if and only if d = 1. (Here V =SymdCn+1 has highest weight d1.)

The rational homogeneous variety G/P is sometimes indicated by circling the nodes ofthe Dynkin diagram (Appendix C) corresponding to the index set I(p).

2.2. Flag domains and MumfordTate domains. Let GR be a (connected) real formof G. There are only finitely many GRorbits on G/P . An open GRorbit

D = GR/R

is a flag domain. The stabilizer R GR is the centralizer of a torus T GR [21, Corollary2.2.3]. The flag variety

Ddfn= G/P

is the compact dual of the flag domain.We will be interested in the case that D admits the structure of a MumfordTate domain.

MumfordTate domains are generalizations of period domains, and as such arise as theclassification spaces of polarized Hodge structures (possibly with additional structure); see[24] for a thorough treatment. When D admits the structure of a MumfordTate domainthe stabilizer R is compact, and as a consequence there exists a compact maximal torusT GR such that T T R [24]. We will assume this to be the case throughout.In the MumfordTate domain case, G also has an underlying Q-algebraic group GQ,

whose groups of real and complex points recover GR and G, respectively. We will assumethis only where relevant. In a few places (mostly limited to Section 6), we shall make thestronger assumption that GQ is a MumfordTateChevalley group (Definition 6.9).

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 9

2.3. Grading elements. Let {S1, . . . , Sr} be the basis of h dual to the simple roots. Agrading element is any member of spanZ{S1, . . . , Sr}; these are precisely the elements T hof the Cartan subalgebra with the property that (T) Z for all roots . Since T issemisimple (as an element of h), the Lie algebra g admits a direct sum decomposition

(2.4a) g =Z

g

into Teigenspaces

(2.4b) gdfn= { g | [T, ] = } .

In terms of root spaces, we have

(2.4c)

g =

(T)=

g , 6= 0 ,

g0 = h

(T)=0

g .

The Teigenspace decomposition is a graded Lie algebra decomposition in the sense that

(2.5)[g, gm

] g+m ,

a straightforward consequence of the Jacobi identity. It follows that g0 is a Lie subalgebraof g (in fact, reductive), and each g is a g0module. The Lie algebra

(2.6) p = pT = g0 g+

is the parabolic subalgebra determined by the grading element T. See [45, Section 2.2] fordetails.

2.3.1. Minimal grading elements. Two distinct grading elements may determine the sameparabolic p. As a trivial example of this, given a grading element T and 0 < d Z, bothT and dT determine the same parabolic. Among those grading elements T determining thesame parabolic (2.6), only one will have the property that g1 generates g as an algebra.That canonical grading element is defined as follows. Given a parabolic p subalgebra (andchoice of Cartan and Borel h b p), the grading element associated to p is(2.7) E

dfn=

iI(p)

Si .

The reductive subalgebra g0 = g0ss z has center z = spanC{Si | i I} and semisimplesubalgebra g0ss = [g

0, g0]. A set of simple roots for g0ss is given by S(g0) = {j | j 6 I}.2.3.2. Eeigenspace decompositions. Any grepresentation U admits a Eeigenspace decom-position

U =qQ

U q ,

see [45, Section 2.2]. In the case that U = g, we recover (2.4c). Again, the Jacobi identityimplies

(2.8) g(U q) U q+ .

10 KERR AND ROBLES

In particular,

each Eeigenspace U q is a g0module.

If V is the irreducible gmodule of highest weight =

ii, then the Eeigenspacedecomposition is of the form V = V m V m1 V m2 . Moreover, if i = 0 wheni 6 I, then V m is the (onedimensional) highest weight line, cf. [45, Lemma 6.3]. If i > 0if and only if i I, then the Gorbit of [V m] PV is a homogeneous embedding of G/P ;the embedding is minimal if i = 1 for all i I.2.4. Variations of Hodge structure. This section is a very brief review of the infinites-imal period relation on G/P and the solutions of this differential system, the variations ofHodge structure. The reader interested in greater detail is encouraged to consult [45] andthe references therein.The (holomorphic) tangent bundle of

Ddfn= G/P

is the Ghomogeneous vector bundle

TD = GP (g/p) .By (2.5) and (2.6), the quotient g1/p is a pmodule. Therefore,

(2.9) T 1dfn= GP (g1/p)

defines a homogeneous, holomorphic subbundle of TD. This subbundle is the infinitesimalperiod relation (IPR). The IPR is trivial if T 1 = TD. This is the case when G/P isHermitian symmetric.For the purposes of this article, a variation of Hodge structure (VHS)(7) is a solution

(or integral manifold) of the IPR; that is, a VHS is any connected complex submanifoldZ G/P with the property that TzZ T 1z for all z Z, or any irreducible subvarietyY D with the property that TyY T 1y for all smooth points y Y . Alternatively, weshall sometimes refer to such subvarieties as horizontal.

2.5. Adjoint varieties. Throughout the paper we will illustrate the ideas and results withadjoint varieties. From the Hodgetheoretic perspective, these are the simplest varieties forwhich the infinitesimal period relation is nontrivial: the adjoint varieties are precisely thoseG/P for which the IPR (2.9) is a contact distribution.Let G Aut(g) be the adjoint group of a complex simple Lie algebra g, and let +

be the highest root. See Table 2.1. Recall g is one-dimensional, and therefore a line ing; let o Pg be the corresponding point. Then the adjoint variety D is the Gorbit of o.Writing D = G/P , if T is the grading element associated to P then Table 2.1 yields

g = g2 g1 g0 g1 g2 ,where g2 = g.

Remark 2.10. The adjoint varieties are precisely the compact, simply connected, homoge-neous complex contact manifolds [6].

(7)We shall have use for the more standard meaning of VHS (over an arbitrary complex manifold, andallowing monodromy) only in the beginning of Section 5, and in Proposition 7.15.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 11

Table 2.1. The highest root of G.

G

Ar 1 + + r = 1 + rBr 1 + 2(2 + + r) = 2Cr 2(1 + + r1) + r = 21Dr 1 + 2(2 + + r2) + r1 + r = 2E6 1 + 22 + 23 + 34 + 25 + 6 = 2

E7 21 + 22 + 33 + 44 + 35 + 26 + 7 = 1

E8 21 + 32 + 43 + 64 + 55 + 46 + 37 + 28 = 8

F4 21 + 32 + 43 + 24 = 1

G2 31 + 22 = 2

Example 2.11. (a) If g = slr+1C, then the adjoint variety is Flag(1, r,Cr+1) is the (partial)

flag variety of lines in hyperplanes.

(b) If g = somC, then the adjoint variety is the orthogonal grassmannian OG(2,Cm) of

2planes that are isotropic for a nondegenerate symmetric bilinear form .

(c) If g = sp2rC, then the adjoint variety is the second Veronese re-embedding v2(P2r1)

P Sym2C2r.

Assume that G 6= Ar, Cr so that the adjoint representation g is fundamental. Equiva-lently, = i (cf. Table 2.1), g is the irreducible grepresentation of highest weight i andP is maximal; we have I(p) = {i} and(2.12) j is a root if and only if j = i.

Table 2.2. The dimensions n = dim D and N = dimPg, and degree d =deg D for the exceptional adjoint varieties.

G n d NE6 21 151,164 77E7 33 141,430,680 132E8 57 126,937,516,885,200 247F4 15 4,992 51G2 5 18 13

2.6. Schubert varieties. In the case that G is classical (one of SLnC, SOnC or Sp2nC),the Schubert varieties of a flag variety G/P may be described geometrically as degeneracyloci, cf. Example 2.17. However, these descriptions, aside from not generalizing easilyto the exceptional groups, are type dependent. We will utilize a representation theoreticdescription of the Schubert varieties that allows a uniform treatment across all flag varieties.This section does little more than establish notation for our discussion of Schubert varieties.The reader interested in greater detail is encouraged to consult [45] and the referencestherein.

12 KERR AND ROBLES

Given a simple root i S, let (i) Aut(h) denote the corresponding simple reflection.TheWeyl group W Aut(h) of g is the group generated by the simple reflections {(i) | i S}. A composition of simple reflections (i1) (i2) (it), which are understood to act onthe left, is written (i1i2 it) W. The length of a Weyl group element w is the minimalnumber

|w| dfn= min{ | w = (i1i2 i)}of simple reflections necessary to represent w.Let Wp W be the subgroup generated by the simple reflections {(i) | i 6 I}. Then Wp

is naturally identified with the Weyl group of g0ss. The rational homogeneous variety G/Pdecomposes into a finite number of Borbits

G/P =

WpwWp\WBw1o

which are indexed by the right cosets Wp\W. The BSchubert varieties of G/P are theZariski closures

Xwdfn= Bw1o .

Remark 2.13. Observe that the stabilizer Pw of Xw in G contains B, and is therefore aparabolic subgroup of G.

More generally, any Gtranslate gXw is a Schubert variety (of type Wpw). Define a partialorder on Wp\W by defining Wpw Wpv if Xw Xv ; then Wp\W is the Hasse poset.Each right coset Wp\W admits a unique representative of minimal length; let

Wpmin Wp\W

be the set of minimal length representatives. Given w Wpmin, the Schubert variety wXwis the Zariski closure of Nw o, where Nw G is a unipotent subgroup with nilpotent Liealgebra

(2.14) nwdfn=

(w)

g g

given by

(2.15) (w)dfn= + w() .

Moreover, Nw o is an affine cell isomorphic to nw, and dimXw = dim nw = |(w)|. IndeedTo(wXw) = nw .

For any w Wpmin we have(2.16) |w| = |(w)| = dimXw .(The first equality holds for all w W, cf. [12, Proposition 3.2.14(3)].)Example 2.17 (Schubert varieties in OG(2,Cm)). Let be a nondegenerate, symmetricbilinear form on Cm, and let

OG(2,Cm)dfn= {E Gr(2,Cm) | |E = 0}

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 13

be the orthogonal grassmannian of isotropic 2planes in Cm. Fix a basis {e1, . . . , em} ofCm with the property that

(ea, eb) = m+1a+b .

Then

Fpdfn= spanC{e1, . . . , ep} Cm

defines a flag F with the property that

(2.18) (Fp,Fmp) = 0 .

Any flag F satisfying (2.18) is called isotropic.Given 1 a < b m with a+ b 6= m+ 1 and a, b 6= r + 1, define

(2.19a) Xa,b(F)

dfn= {E OG(2,Cm) | dimE Fa 1 , E Fb} .

If m = 2r+1, then every Schubert variety of OG(2,C2r+1) is Gcongruent to one of (2.19a).If m = 2r is even, define

Fr = spanC{e1, . . . , er1 , er+1} .Note that F1 Fr1 Fr Fr+1 is also a isotropic flag. Set

Xa,r(F)

dfn= {E OG(2,Cm) | dimE Fa 1 , E Fr} , a r 1,(2.19b)

Xr,b(F)

dfn= {E OG(2,Cm) | dimE Fr 1 , E Fb} , b r + 2 .(2.19c)

Every Schubert variety of OG(2,C2r) is Gcongruent to one of (2.19).

2.7. Schubert VHS. The condition that the Schubert variety Xw be a VHS is equivalentto (w) (g1), where (w) is given by (2.15), cf. [45, Theorem 3.8]. A convenient wayto test for this condition is as follows. Let

dfn=

ri=1

i =12

+

be the sum of the fundamental weights (which is also half the sum of the positive roots).Define

(2.20) wdfn= w() =

(w)

.

(See [35, (5.10.1)] for the second equality.) Then

|w| w(E) Z ,and equality holds if and only if (w) (g1); equivalently, Xw is a variation of Hodgestructure if and only if w(E) = |w|. See [45, Section 3] for details. Let

Wvhsdfn= {w Wpmin | w(E) = |w|}

be the set indexing the Schubert variations of Hodge structure.

14 KERR AND ROBLES

Example 2.21. This is a continuation of the introductory Running Example. In particular,G = G2 is the exceptional Lie group of rank two, D Pg is the adjoint variety withI(p) = {2} and E = S2. Modulo the action of G, there is a single Schubert variety Xwd Dof dimension d, 0 d 5. The variety Xwd is horizontal (a VHS) if and only if d 2.We have w1 = (2) with (w1) = {2}, and w2 = (12) with (w2) = {2, 1 + 2}. As willbe discussed later, Xw1 = P

1 is a line on D, and therefore homogeneous, while Xw2 is asingular cone over the Veronese re-embedding v3(P

1) P3.Lemma 2.22 and Corollary 2.23 are continuations of Example 2.17.

Lemma 2.22 (Schubert VHS in OG(2,Cm)). Recall the Schubert varieties (2.19) of OG(2,Cm).

(a) The Schubert variety Xa,b(F) is horizontal if and only if Fa (Fb).

(b) The Schubert varieties Xa,r(F), are all horizontal.

(c) None of the Schubert varieties Xr,b(F) are horizontal.

Corollary 2.23 (Maximal Schubert VHS in OG(2,Cm)). The maximal (with respect tocontainment) Schubert VHS in SO(2,Cm), where m {2r, 2r + 1}, are: the Xa,ma(F),with 1 a r 1; and Xr1,r(F), if m = 2r. Each of these maximal Schubert VHS is ofdimension m 4.Proof of Lemma 2.22. By definition, a Schubert variety Xw OG(2,Cm) is horizontal(equivalently, satisfies Griffithss transversality condition) if and only if

(2.24) dE|Nwo Edfn= {v Cm | (E, v) = 0}

on the Schubert cell Nw o.(a) It is straightforward to see that the condition Fa (Fb) implies Xa,b(F) is hori-

zontal. Suppose that Fa 6 (Fb); equivalently, a+b > m. Then the condition a+b 6= m+1(see Example 2.17) forces a+ b m+ 2. It follows that

E(t)dfn= spanC{eb + tem+1a , ea tem+1b} Xa,b(F) .

(In fact, E(t) lies in the Schubert cell.) As (2.24) clearly fails for E(t), we see that Xa,b(F)

is not horizontal. This establishes assertion (a) of the lemma.

(b) As noted in Example 2.17, the Schubert varieties Xa,r(F) are defined for a < r.

Since Fa Fr = (Fr), it is immediate that the Xa,r(F) are horizontal.(c) Recall that the Schubert varieties Xr,b(F

) are defined for r+2 b, cf. Example 2.17.It follows that

E(t)dfn= spanC{eb + ter , er+1 tem+1b} Xr,b(F) .

(As above, E(t) lies in the Schubert cell.) As (2.24) clearly fails for E(t), we see that

Xr,b(F) is not horizontal. This establishes assertion (c) of the lemma.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 15

2.8. Nilpotent orbits. Let gR End(VR, Q) be the Lie algebra of GR. A (nvariable)nilpotent orbit on D consists of a tuple (F ;N1, . . . , Nn) such that F

D, the Ni gRcommute and NiF

p F p1, and the holomorphic map : Cm D defined by(2.25) (z1, . . . , zn) = exp(ziNi)F

has the property that (z) D for Im(zi) 0. We shall use the term -nilpotent orbit torefer to the submanifold eCF D. The associated (open) nilpotent cone is(2.26) = {tiNi | ti > 0} .Given a nilpotent N gR such that Nk+1 = 0, there exists a unique increasing filtration

W0(N) W1(N) W2k(N) of VR with the properties thatN W(N) W2(N)

and the induced

N : Grk+W(N) GrkW(N)is an isomorphism for all k. Above, GrmW(N) =Wm(N)/Wm1(N). Moreover,

Q(u, v) = Q(u,Nv)

defines a nondegenerate (1)k+symmetric bilinear form on Grk+W(N).Define

Grk+W(N)prim = ker {N +1 : Grk+W(N) Grk2W(N)} ,for all 0. A limiting mixed Hodge structure (or polarized mixed Hodge structure) on Dis given by a pair (F , N)(8) such that F D, N gR and N(F p) F p1, the filtrationF induces a weight m Hodge structure on GrmW(N) for all m, and the Hodge structureon Grk+(W(N))prim is polarized by Q for all 0. The notions of nilpotent orbit andlimiting mixed Hodge structure are closely related. Indeed, they are equivalent when n = 1.

Theorem 2.27 (Cattani, Kaplan, Schmid). Let D D be a MumfordTate domain (andcompact dual) for a Hodge representation of GR.

(a) A pair (F ;N) forms a onevariable nilpotent orbit if and only if it forms a limitingmixed Hodge structure, [15, Corollary 3.13] and [46, Theorem 6.16].

(b) Given an nvariable nilpotent orbit (F ;N1, . . . , Nn), the weight filtration W(N) doesnot depend on the choice of N [14, Theorem 3.3]. Let W() denote this commonweight filtration.

The Deligne bigrading [15, 19]

(2.28a) VC =

Ip,q

associated with the limiting mixed Hodge structure is given by

(2.28b) Ip,q = F p Wp+q (F q Wp+q +

j1

F qj Wp+qj1).

(8)Of course, the actual (real) mixed Hodge structure is the pair (F ,W(N)). If V and g admit com-patible Q-rational structures, and N gQ, then (VQ, F ,W(N)) is a Q-mixed Hodge structure.

16 KERR AND ROBLES

It is the unique bigrading of VC with the properties that

(2.29) F p =rp

Ir, and W() =p+q

Ip,q ,

and

Ip,q = Iq,p mod

r

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 17

3.1. The case that P is maximal. We now recall two properties of Co in the case thatP is maximal (equivalently, I = {i} and E = Si). The results that follow are due to [38],where Co and Co are discussed for arbitrary (not necessarily maximal) P .(a) Since P is maximal, g1 is an irreducible g0module with highest weight line gi ,

cf. [48, Theorem 8.13.3]. The variety of lines Co Pg1 is the G0orbit of this highestweight line, cf. [38, Theorem 4.3]. In particular, Co is a rational homogeneous variety;indeed,

Co G0/(G0 Q) ,where Q B is the parabolic subgroup defined by

(3.4) I(q) = {j | gj 6 q} dfn= {j | i, j 6= 0} .That is, the simple roots indexed by I(q) are those adjacent to i in the Dynkindiagram of g; cf. [38, Proposition 2.5]. With only a few exceptions, Co is a G0cominuscule variety; equivalently, Co G0/(G0Q) admits the structure of a compactHermitian symmetric space, cf. [38, Proposition 2.11].

(b) If the simple root i associated with the maximal parabolic P is not short, then

Co = Co, cf. [38, Theorem 4.8.1]. If the simple root is short, then Co is the union oftwo Porbits, and open orbit and its boundary Co.

3.2. The case of a general parabolic. If we drop the assumption that P is maximal,then Co may be described as follows. Let I be the index set (2.1) corresponding to p. Giveni I, let

g1i = { g1 | [Si, ] = } .Each g1i is an irreducible G

0module with highest weight line gi , and

g0 =iI

g1i

is a G0module decomposition, cf. [48, Theorem 8.13.3]. Let Co,i Pg1i be the G0orbitof the highest weight line gi . Then the variety of lines Co Pg1 is the disjoint union(3.5) Co =

iI

Co,i .

As in the case that P is maximal (Section 3.1), the assertions here are established in [38,Section 4].

3.3. Uniruling of D. For ease of exposition we continue with the assumption that P ismaximal. However, analogous statements follow for unirulings on general G/P . For themore general statements, it is convenient to use Tits correspondences, which are brieflyreviewed in Section 3.4.Given x = go D, with g G, let Cx = gCo denote the corresponding set of lines through

x. (It is an exercise to show that Cx is well-defined; that is, Cx does not depend on ourchoice of g yielding x = go.) Then

C dfn= {P1 | P1 Cx , x D} =gG

g Co .

18 KERR AND ROBLES

forms a uniruling of D. (As will be shown in Corollary 3.15, this uniruling is parameterizedby G/Q that is, C G/Q.)Remark 3.6. More generally, the set of all lines on G/P is C = gG g Co.(a) It follows from definition (3.1) and the homogeneity of the IPR, that

(3.7) C = {P1 C | P1 is a VHS}is precisely the set of lines on G/P that are integrals of the IPR.

(b) As noted in Section 3.1(b), if the simple root associated to the maximal parabolic P

is not short, then C = C consists of a single Gorbit. If the simple root is short, thenC consists of two Gorbits, an open orbit and its boundary C, cf. [38, Theorem 4.3].

3.4. Tits correspondences. Tits correspondences describe homogeneous unirulings of arational homogeneous variety G/P by homogeneously embedded, rational homogeneoussubvarieties G/P ; these unirulings may be used to clarify the geometry of G/P . Thematerial in Sections 3.63.7 is taken from [17, 18]. Given two standard parabolics P andQ = PJ , the intersection PQ is also a standard parabolic. (Note that I(pq) = I(p)I(q).)There is a natural double fibration, called the Tits correspondence, given by the diagram inFigure 3.1; here the maps and are the natural projections. Given a subset G/Q,

Figure 3.1. Tits correspondence

G/P

G/(P Q)

G/Q

the Tits transform is T () := (1()).3.5. Tits transform of a point. The Tits transform of a point y G/Q will play acrucial role in our discussion of G/P and its Schubert varieties. What follows is a brief

review of [39, 2.7.1].(9)The Tits transform T (Q/Q) of the point Q/Q G/Q is the Gorbit G(P/P ) G/P ,

where G is the semisimple subgroup of G whose Dynkin diagram D is obtained from thediagram D of G by removing the nodes I(q)\I(p). Therefore, T (Q/Q) G/P , whereP = GP is the parabolic subgroup of G with index set (2.1) given by I(p) = I(p)\I(q).Since any point y G/Q is of the form y = gQ/Q for some g G, and T (gQ/Q) =

gT (Q/Q), it follows that T (y) G/P and

(3.8a)G/P is uniruled by subvarieties isomorphic to G/P ,

and the uniruling is parameterized by G/Q.

Moreover, G/(P Q) is the incidence space for this uniruling. Precisely,(3.8b) G/(P Q) = {(x,) G/P G/Q | x } .

(9)There is a typo in [39, 2.7.1]; on page 87, S and S should be swapped. This is corrected for in thediscussion above.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 19

Remark 3.9 (P1unirulings of G/P for maximal P ). In the case that P G is a maximalparabolic (Section 2.1), there is a unique Ghomogeneous variety G/Q parameterizing auniruling of G/P by lines P1; it may be identified by inspection of the Dynkin diagram D ofg as follows. The maximality of P is equivalent to I(p) = {i} for some i. In order to obtaina uniruling by P1s, we must choose Q (equivalently, the index set I(q)) so that G/P = P1.To that end, let J = {j 6= i | (i, j) 6= 0} index the nodes in the Dynkin diagram that areadjacent to the ith node. Then G/P P1 (equivalently, G/Q parameterizes a unirulingof G/P by P1s) if and only if J I(q). When I(q) = J we say that G/Q is the smallestrational Ghomogeneous variety parameterizing a uniruling of G/P by lines P1.Examples 3.10 and 3.11 below identify the varieties G/Q parameterizing lines on the

fundamental adjoint varieties (Section 2.5).

Example 3.10 (Unirulings of the orthogonal adjoint varieties). Fix a nondegenerate sym-metric bilinear form on Cn, n 7. Let G = Aut(Cn, )0 = SO(Cn) denote the identitycomponent of the orthogonal group. The adjoint variety (Section 2.5)

G/P2 = OG(2,Cn) = {E Gr(2,Cn) | |E = 0}

is the set of isotropic 2-planes. The partial flag variety

G/P1,3 = Flag(1, 3,Cn) = {F 1 OG(1,Cn)OG(3,Cn) | F 1 F 3}

parameterizes a uniruling of the adjoint variety G/P2 by P1s. Given one such flag F 1 F 3

the corresponding line is {E OG(2,Cn) | F 1 E F 3}.In this case D = D\{2} so that g = sl2C son4C. Additionally, I(q) = I(p) = {1, 3}.

Example 3.11 (Unirulings of the exceptional adjoint varieties). LetG/P Pg be the adjointvariety (Section 2.5) of an exceptional simple Lie group G. The rational homogeneousvariety G/Q parameterizing a uniruling of G/P by lines P1 is given in Table 3.1. In the

Table 3.1. Lines on exceptional adjoint varieties

Adjoint variety G/P E6/P2 E7/P1 E8/P8 F4/P1 G2/P2

Variety G/Q of lines E6/P4 E7/P3 E8/P7 F4/P2 G2/P1

table the subscript i in Pi indicates the set I = {i} of (2.1). For example, G/P = E6/P2indicates that g is the exceptional simple Lie algebra of rank six and and I(p) = {2}.3.6. Tits transform of a Schubert variety. The Tits transform preserves Schubertvarieties; that is, if Y G/Q is a Schubert variety, then so is X = T (Y ) = G/P . Moreover,X may be determined as follows. Recall (Section 2.6) that Wqmin is the set of (unique)minimal length representatives of the right cosets Wq\W. Each right coset also admits aunique representative of maximal length; let Wqmax W denote this set of representatives.If w0 is the longest element of Wq, then W

qmax = {w0w | w Wqmin}. We have Wq\W

Wqmin Wqmax. A proof of the following wellknown lemma is given in [17].

Lemma 3.12. Let w Wqmax, and let Yw G/Q denote the Schubert variety indexed bythe coset Wqw. Then the Tits transformation T (Yw) is the Schubert variety Xw G/Pindexed by the coset Wpw.

20 KERR AND ROBLES

3.7. Lines through a point. The following, first observed in [18], is an immediate conse-quence of (3.8).

Lemma 3.13. Suppose that G/Q parameterizes a uniruling of D = G/P by P1s. Let = (1(o)) G/Q denote the image of o D under the Tits transform, and letT () = (1()) D denote the image of under the Tits transform back to D. Then is the set of lines in D that are parameterized by G/Q and pass through o D. Likewise,

T () =

oP1G/QP1

is the subset of D swept out by these lines, and is naturally identified with a cone C() over with vertex o.

Remark 3.14. By Lemma 3.12, the variety T () is a Schubert variety Xw G/P .Corollary 3.15 (Maximal parabolics). Let P be a maximal parabolic and let Q be theparabolic of Section 3.1(a). Then the uniruling C of G/P (Section 3.3) is is precisely theuniruling G/Q obtained through the Tits correspondence. In particular, the variety ofLemma 3.13 is the variety Co of (3.1), and the variety T () of Lemma 3.13 is the varietyX of (3.2).

Proof. By the maximality of P we have I(p) = {i}, cf. Section 2.1. Recall the Levi subgroupG0 G whose Lie algebra g0 is the zeroeigenspace of the grading element E = Si associatedwith the parabolic p, cf. Section 2.1. Then the simple roots of the semisimple Lie algebrag0ss = [g

0, g0] are S\{i}, the simple roots of g minus the ith. As discussed in Section3.1(a), we have Co = G0/(G0 Q). It follows that Co = G0ss/(G0ss Q), where G0ss G0 isthe semisimple subgroup with Lie algebra g0ss. Note the index set I(g

0ss q) associated with

the parabolic G0ss Q by (2.1) is I(q).As discussed in Section 3.1(a), the set I(q) indexes those nodes of the Dynkin diagram

that are are adjacent to the ith node. By Remark 3.9, G/Q is the minimal rational Ghomogeneous variety parameterizing a uniruling of G/P by lines P1s. The Tits transform = T (P/P ) G/Q is of the form G/P . From the descriptions of G and P in Section 3.5and the discussion of Remark 3.9 we see that G = G0ss and P = G0ssQ. Thus, Co = . Remark 3.16 (Adjoint varieties). In Sections 7.5-7.6, we will be interested in the varietyX = T () in the case that G/P is a fundamental adjoint variety. In those cases, P is amaximal parabolic, so that Corollary 3.15 applies: the = Co are listed in Table 3.2. In

Table 3.2. The set = Co for the fundamental adjoint varieties

G/P OG(2,Cn) E6/P2 E7/P1 E8/P8 F4/P1 G2/P2

Co P1 Qn6 Gr(3,C6) S6 E7/P7 LG(3,C6) v3(P1)

the table S6 is a Spinor variety, one of the two connected components of the orthogonalgrassmannian OG(6,C12).Moreover, for each of the fundamental adjoint varieties, the simple root i associated

with the maximal parabolic P = Pi is not short. Whence Lemma 3.3 applies and the varietyX swept out by lines passing through a fixed point is a VHS.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 21

Proof of Lemma 3.3. Part (a) of the lemma follows directly from Lemma 3.13 and Corollary3.15. The argument for part (b) is based on the Tits transform recipe given by Lemma 3.12and the characterization of Schubert VHS discussed in Section 2.7. We will make use(without explicit mention) of observations made in the proof of Corollary 3.15.To begin, let P G be a maximal parabolic with index set I(p) = {i}, cf. Section 2.1.

Let G/Q parameterize the uniruling of D = G/P by P1s. The right coset indexing theSchubert variety o = P/P D is Wpw0, where w0 is the longest word of the Weyl groupWp of g

0. By Lemma 3.12, the Schubert variety = T (P/P ) is indexed by the coset Wqw0.Let w1 Wqmax be the longest representative of Wqw0 = Wqw1. Again, Lemma 3.12 impliesT () is the Schubert variety indexed by the right coset Wpw1. By Corollary 3.15, this isthe Schubert variety X. Let w Wpmin be a shortest representative of Wpw1 = Wpw. Asdiscussed in Section 2.7, the Schubert variety X is a VHS if and only if w(E) = dimX. Sothe substance of the proof is to compute the integer w(E). Note that E = S

i, cf. (2.7).Let v Wqmin be the shortest Weyl element indexing the Schubert variety . Let w0 Wq

be the longest element of the Weyl subgroup, so that w0v = w1 is the longest elementindexing . Then

Wqw0 = Wqv = Wqw0v ,

and Wpw0v indexes X = T (). We claim that

(3.17) Wpw0 = Wp(i) ,

where (i) W denotes the reflection associated with the simple root i. It follows from(3.17) that

(i)v = w Wpminis the shortest Weyl group element indexing X.

Proof of (3.17). Observe that Wp is generated by the simple reflections {(j) | j 6= i}.Likewise, Wq is generated by the simple reflections {(j) | j 6= i 1, i + 1}. In particular,any element u of Wq may be written as u

(i) with u Wp. The claim follows. We return to the proof of Lemma 3.3(b). From the discussion of Section 2.6 (specifically

(2.16)) we see that

|v| = |(v)| = dim and 1 + |v| = |w| = |(w)| = dimX .The argument establishing [12, Proposition 3.2.14(5)] yields

(w) = {i} (i)(v) .Since v indexes a homogeneous Schubert variety = G/P , we see from the discussion ofSection 2.6 that

(v) = { (g0) | (F) > 0} ,where

F =iI(q)

Si

is the grading element (2.7) associated with the parabolic q. Since g0 is the zeroeigenspaceof E = Si, this may be rewritten as

(v) = { | (E) = 0 , (F) > 0} .

22 KERR AND ROBLES

In particular, the roots of (v) are precisely those positive roots = mii such thatmi = 0

and mj > 0 for some j I(q). Informally, the root does not involve the simple root i,but does involve some simple root adjacent to i. Whence the reflection

(i) = 2 (,i)(i, i)

i

has the property that the integer 2(,i)/(i, i) 1 for every (v). It now followsfrom the second equality of (2.20) that w(E) > |w| if and only if i is short.

4. Smooth Schubert VHS

A homogeneously embedded, homogeneous submanifold of G/P is a submanifold of theform

g {aP G/P | a G} = g G/P ,where g G and G is a closed Lie subgroup of G. Such a submanifold is isomorphic toG/P , where P = G P .Distinguished amongst the homogeneously embedded, homogeneous submanifolds ofG/P

are the rational homogeneous subvarieties X(D) corresponding to subdiagrams D of theDynkin diagram D of g. The correspondence is as follows: Identifying the nodes of D withthe simple roots S of g, to any subdiagram we may associate a semisimple Lie subalgebrag g by defining g to be the subalgebra generated by the root spaces {g | D}.Note that the Dynkin diagram of g is naturally identified with the subdiagram D. LetG G be the corresponding closed, connected semisimple Lie subgroup. For such G, thesubgroup P = G P is a parabolic subgroup. The Gorbit of P G/P is isomorphic toG/P and is the homogeneously embedded, rational homogeneous subvariety X(D) G/Pcorresponding to subdiagram D D.Define the index set J = {j | gj 6 g}. Let F =jJ Sj be the corresponding grading

element (Section 2.3), and let g = g be the Feigenspace decomposition (2.4). Theng = [g0, g0] is the semisimple component of the reductive subalgebra g0. Let Q = PJ bethe corresponding parabolic subgroup of G (with Lie algebra q = g0 g+.) It follows fromthe discussion of Section 3.5 that

(4.1) X(D) G/P is the Tits transformation T (Q/Q) of the point in Q/Q G/Q.This, with Lemma 3.12, yields the following

Lemma 4.2. The homogeneously embedded, rational homogeneous subvarieties X(D) G/P corresponding to subdiagrams D D are smooth Schubert varieties of G/P .Remark 4.3. Let E be the grading element (2.7) associated with p. It follows from thediscussion above, that a Schubert variety Xw is of the form X(D

) if and only if (w) =(g+), where g = g is the Eeigenspace decomposition of g and (w) is the set (2.15).By Section 2.7, X(D) is a VHS if and only if g = g1 g0 g1; equivalently, X(D) isHermitian symmetric.

The main result of this section is

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 23

Theorem 4.4. Let X D = G/P be a Schubert variation of Hodge structure. If X issmooth, then X is a product of homogeneously embedded, rational homogeneous subvarietiesX(D) G/P corresponding to subdiagrams D D. Moreover, each X(D) is Hermitiansymmetric.

Since a homogeneously embedded Hermitian symmetric space is necessarily smooth, The-orem 4.4 is equivalent to: A Schubert VHS is smooth if and only if it is a homogeneouslyembedded Hermitian symmetric space.Theorem 4.4 is proved in Sections 4.14.5.

When D is the compact dual of a MumfordTate domain D, Friedman and Laza haveshown the following [22, Theorem 1]: if Y D is horizontal and semi-algebraic (that is, aconnected component of the intersection with D of a closed subvariety of D), with stronglyquasi-projective image in an arithmetic-group quotient \D, then Y is a horizontal (henceHermitian) MumfordTate subdomain. Theorem 4.4 says that if Y is horizontal and semi-smooth-Schubert, then Y is Hermitian; so it is a rather different result from [22], withoutany reference to MumfordTate domains or quotients. However, we do have the

Corollary 4.5. Suppose D is the compact dual of a MumfordTateChevalley domain D(Definition 6.17), X D is a smooth Schubert VHS, and Y is a nonempty connectedcomponent of X D. Then some GR-translate of Y is a (necessarily horizontal, Hermitiansymmetric) connected MumfordTate subdomain of D.

Proof. Translate Y to pass through the distinguished base point o0 of D (Section 6.4), withcorresponding grading element T and Hodge structure 0 : S

1 GR. By Theorem 4.4, wehave Y = (GR)

o0.(10) Since g = g1 g0 g1 decomposes into a direct sum of h gand a subset of hroot spaces, it follows that g is stable under T h. Therefore, 0(S1)normalizes GR, and we set G

R

dfn= 0(S

1) GR. By Remarks 6.10 and 6.18, this is underlainby a Q-algebraic group GQ.Now since ad(T)|g1 = 1g1 , the (complexified) QLie-algebra-closure of a very general

ad(gR)translate of T contains g1 = g

1. Therefore, so does the CLie algebra of the

MumfordTate group GQ of a very general GR-conjugate of 0. It follows that Y = (G

R)o0,

where GQ is the MumfordTate group of a very general point (i.e., Hodge structure) in Y ;that is, Y is a connected MumfordTate subdomain.

Remark 4.6. (i) The converse of Corollary 4.5 fails dramatically: the compact dual D Dof a horizontal (Hermitian) MumfordTate subdomain D D need not be a Schubertvariety, even when D is a maximal integral of the IPR. A large class of examples illustratingthis, called enhanced SL2-orbits, is constructed in Section 6; the simplest one is probablythe SL2 SL2 subdomain in the G2-adjoint variety (see Sections 7.5-7.6).(ii) We expect that Corollary 4.5 itself fails for more general (non-MTC) MumfordTate

domains, to an extent which should be describable in terms of the action of Gal(Q/Q) inthe Dynkin diagram D.

Example 4.7 (Symplectic Grassmannians). In this example we will (a) illustrate the con-struction of X(D) from D D, and (b) observe that not every smooth Schubert variety

(10)Here the superscript denotes the connected identity component.

24 KERR AND ROBLES

of a rational homogeneous variety is homogeneous; in particular, the assumption that X bea VHS in Theorem 4.4 is essential.Let be a non-degenerate, skew-symmetric bilinear form on C2r. Fix i r. The

symplectic grassmannian

SG(i,C2r) ={E Gr(i,C2r) | |E = 0

}is a rational homogeneous variety D = G/P , where G = Aut(C2r, Q) = Sp2rC and P = Piis the maximal parabolic subgroup associated to the ith simple root. The Dynkin diagramof G (containing r nodes) is r r r r r .Fix a Qisotropic flag F 1 F 2 F 2r = C2r; here F d is a complex linear subspace

of dimension d and (F d, F 2rd) = 0. Given 0 a < i < b 2r a,Xa,b = {E SG(i,C2r) | F a E F b}

is a Schubert variety. The variety Xa,b is the homogeneous submanifold X(D) associated

to a Dynkin subdiagram D D if and only if either b r, or b = 2r a. In the first case(b r), the subdiagram D corresponds to the simple roots S = {a+1, . . . , b1}; we haveg slbaC and Xa,b Gr(i a,Cba); these Schubert varieties are VHS. In the secondcase (b = 2r a), the subdiagram D corresponds to the simple roots S = {a1, . . . , r};we have g sp2(ra)C and Xa,b = SG(i a,C2(ra)); these Xa,b are VHS if and onlyif i = r (equivalently, the symplectic grassmannian is a Hermitian symmetric Lagrangiangrassmannian).We illustrate this for D = SG(4,C16); that is, i = 4 and r = 8. The encircled subdiagram

r r r r r r r r

corresponds to g = sl6C and

X1,7 = {E SG(4,C16) | F 1 E F 7} = Gr(3, F 7/F 1) Gr(3,C6) .Likewise, the encircled subdiagram

r r r r r r r r

corresponds to g = sp12C and

X2,14 = {E SG(4,C16) | F 2 E} = SG(2, (F 2)/F 2) SG(2,C12) .Returning to the general case, we have i = r if and only if D is a Hermitian symmetric

space. In this case, the IPR is trivial so that every Schubert variety is a Schubert VHS.Moreover, a Schubert variety is smooth if and only if it is the homogenous submanifoldassociated to a Dynkin subdiagram, cf. [10].In contrast to the Hermitian symmetric case, if i < r and b = 2n a 1, then Xa,b is

smooth, but not homogeneous, cf. [41]. These Schubert varieties are not VHS.

Remark 4.8 (Maximal parabolic associated to nonshort root). If D = G/P with P amaximal parabolic corresponding to a nonshort simple root i, then all smooth Schubertvarieties of D are homogeneously embedded, rational homogeneous varieties X(D) corre-sponding to connected subdiagrams D D containing the ith node, cf. [27, Proposition3.7]. For example, the smooth Schubert varieties in the Grassmannian Gr(k,Cn) are allhomogeneous; likewise, the smooth Schubert varieties of the adjoint varieties (Section 2.5)of G = Br,Dr, Er, F4, G2 are all homogeneous.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 25

4.1. Proof of Theorem 4.4: outline. The proposition is proved in three steps. First wereduce to the case that P is maximal (Section 4.2). Then, the result [27, Proposition 3.7]of Hong and Mok establishes the proposition in the case that the associated simple root isnot short (Section 4.3). Third, we address the short root case (Section 4.4).

4.2. Reduce to the case that P is maximal parabolic. Suppose that X is a SchubertVHS. Let w W p be the associated Weyl group element (Section 2.6), so that X = Xw =Bw1o. The condition that the Schubert variety be a VHS is equivalent to (w) (g1)(Section 2.7). By definition (g1) if and only if (E) = 1. Equivalently, (Si) = 1 forexactly one i I, and (Sj) = 0 for all other j I. Therefore, we have a disjoint union

(w) =iI

i(w) ,

where

i(w)dfn= { (w) | (Si) = 1} .

It is straightforward to confirm that both i(w) and +\i(w) are closed. Therefore, i(w)

determines a Schubert variety Xi X, cf. [45, Remark 3.7(c)]. (In general, i(w) (w)does not imply Xi X. In case lifts it is a consequence of the fact that X is a VHS,cf. [45, Appendix D].) Whence

X =iI

Xi .

Let D denote the Dynkin diagram of G. Given i I, let D\{I\{i}} denote the subdia-gram of D obtained by removing all nodes corresponding to j I\{i} (and their adjacentedges). Let Di denote the connected component of D\{I\{i}} containing the i-th node.Let Gi G denote the closed, connected semisimple Lie subgroup of G corresponding toDi. Let Di D be the Giorbit of o. Then Di Gi/(Gi P ) is rational homogeneousvariety containing Xi as a Schubert subvariety. Moreover, Xi is a VHS for the IPR on Di.Finally, note that Gi P is a maximal parabolic subgroup of Gi the corresponding indexset is just {i}. Since X is smooth if and only if each Xi is smooth, to prove the proposition,it suffices to show that Xi is a homogeneously embedded Hermitian symmetric space.This reduces the proof of the proposition to the case that P is a maximal parabolic,

which we now assume. Let i denote the associated simple root.

4.3. The case that i is not a short root. Suppose that i is not a short root ofG. By [27, Proposition 3.7], the Schubert variety X D = G/Pi is the homogeneouslyembedded, rational homogeneous subvariety corresponding to a subdiagram of D.

4.4. The case that i is a short root. In this case D = G/Pi is one of the followingfive rational homogeneous varieties:

Let be a nondegenerate symmetric bilinear form on C2r+1. Then the orthogonal grass-mannian

OG(r,C2r+1) ={E Gr(r,C2r+1) | |E = 0

}of maximal, isotropic subspaces is the rational homogeneous variety Br/Pr = Spin2r+1C/Pr,where Pr is the maximal parabolic associated to the short simple root r.

26 KERR AND ROBLES

Let be a nondegenerate skew-symmetric bilinear form on C2r. Then the symplecticgrassmannian

SG(i,C2r) ={E Gr(r,C2r+1) | |E = 0

}of idimensional isotropic subspaces is the rational homogeneous variety Cr/Pi =Sp2rC/Pi, where Pi is the maximal parabolic associated to the simple root i. Thesimple root is short if and only if i < r.

The exceptional F4/P3, F4/P4 or G2/P1.

Unfortunately, the HongMok argument does not appear to extend to SG(i,C2r), F4/P3or G2/P1. Instead, we will see that, for each of the five D above, Theorem 4.4 followsfrom either (i) the BrionPolo classification of smooth minuscule Schubert varieties, or (ii)existing descriptions of the Schubert VHS, or a combination of both.

4.4.1. The case that D = OG(r,C2r+1). In this case, D is minuscule, cf. [5]. Brion andPolo have shown that the smooth Schubert varieties of minuscule G/P are homogeneoussubmanifolds. More precisely, let Q B denote the stabilizer of X, cf. Remark 2.13. ThenX = Q o by [10, Proposition 3.3(a)].Let q = q0 q+ be the graded decomposition (2.6) associated to associated to the

parabolic q (and choices q b h). If Q0 Q is the closed Lie subgroup with Lie algebraq0, then Q Q0q+ = Q0exp(q+), by [12, Theorem 3.1.3]. Therefore, X = Qo = Q0o =Q0ss o, where Q0ss is the closed semisimple Lie subgroup with Lie algebra q0ss = [q0, q0]. Thesemisimple Q0ss is the subgroup of G associated to the subdiagram D

= { D | g q0}(Section 2.3). Thus, X = X(D) is the homogeneously embedded, rational homogeneoussubvariety associated to a subdiagram D D.4.4.2. The case that D = SG(i,C2r). Recall that i < r. In this case, the marked Dynkindiagram associated to SG(i,C2r) is

r r r r r r r r1 2 i r

It follows from the proof of [45, Proposition 3.11](11) that there is a unique maximal Schubertvariation of Hodge structure; it is the homogeneous submanifold Z Gr(i,Cr) associatedto the circled subdiagram

r r r r r r r r

So, any smooth Schubert VHS in SG(i,C2r) is a smooth Schubert variety of Gr(i,Cr).These are well-known to be precisely the homogeneously embedded, rational homogeneoussubvarieties corresponding to connected subdiagrams of the the marked Dynkin diagramfor Gr(i,Cr) that contain the ith node, cf. [5, Section 9.3]. (Alternatively, this followsfrom the BrionPolo result, since Gr(i,Cr) is minuscule, or Remark 4.8.)

4.4.3. The case that D = F4/P3. By [45, Example 5.16], the only Schubert variations ofHodge structure are (the trivial o D and) the P1 P2 corresponding to the two subdia-grams

s s s s

s s s s

Therefore, any smooth Schubert VHS in F4/P3 is a homogeneously embedded, rationalhomogeneous subvariety corresponding to a subdiagram of D.

(11)See, in particular, Step 3 of Section 7.3 in [45]; in the case under consideration, t = 1 and it = i.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 27

4.4.4. The case that D = F4/P4. By [45, Example 5.17], the only Schubert variations ofHodge structure are (the trivial o D and) the P1 P2 corresponding to the two subdia-grams

s s s s

s s s s

Therefore, any smooth Schubert VHS in F4/P4 is a homogeneously embedded, rationalhomogeneous subvariety corresponding to a subdiagram of D.

4.4.5. The case D = G2/P1. This variety is the quadric hypersurface Q5 P6. By [45,Example 5.30], the only Schubert VHS are (the trivial o D and) the smooth P1 Dwhich is the homogeneous submanifold corresponding to the circled subdiagram

s s

Therefore, any smooth Schubert VHS in G2/P1 is a homogeneously embedded, rationalhomogeneous subvariety corresponding to a subdiagram of D.

4.5. Fini. We have shown, in Sections 4.24.4, that given any rational homogeneous varietyD = G/P (here the parabolic is arbitrary P need not be maximal) any smooth SchubertVHS X D is a product homogeneously embedded, rational homogeneous subvarietiesXi = X(Di) corresponding to a subdiagram of Di D. The condition that X (andtherefore each Xi) be a VHS forces Xi to be Hermitian symmetric. This completes theproof of Theorem 4.4.

Remark 4.9. It is possible that [10, Theorem 2.6] can be used to prove Theorem 4.4, as it wasused to establish the homogeneity of minuscule and cominuscule Schubert varieties in [10].The arguments of [10] are representation theoretic in nature; in the proof of Theorem 4.4,we elected for the more geometric approach of [27].

5. GriffithsYukawa couplings and lines on G/P

A historically important special instance of the double fibrations introduced in Section3 arose in connection with E. Cartans two realizations of G2 as a transformation group ofa complex 5-manifold preserving a nontrivial distribution [11,13]. Indeed, they are relatedby the correspondence:

(5.1) G2/P1 G2/P2

G2/B

Each of the varieties of (5.1) may be realized as the compact dual of a MumfordTate domainwith bracket-generating horizontal distribution [24, IV.F]. This section is motivated by anobservation about the Hodge theory associated to the adjoint variety D = G2/P2 and itscontact (hyperplane) distribution, which we now briefly explain.Suppose V = nj=0Vnj,j is a variation of Hodge structure (in the classical sense) over a

complex manifold S, and D a holomorphic differential operator of order n in a neighborhood

28 KERR AND ROBLES

U S. Letting D act on OU (V) via the GaussManin connection, the IPR forces thecomposition

OU (Vn,0) OU (V) D OU (V) OU (V0,n)to be OS-linear (so that only the symbol of D matters). One therefore obtains a well-definedlinear mapping

(5.2) SymnTsS Hom(V n,0s , V 0,ns ) = (V 0,ns )2

for each s S, which we shall call the GriffithsYukawa coupling. When V comes from afamily of varieties, it may be of particular interest to study the geometry of subfamilies forwhich (5.2) vanishes. Recent work of Katz [30] suggests that motives with Hodge numbers(2, 3, 2) and MumfordTate group G2, hence admitting a period map into a quotient ofD = G2(R)/(P2P2)(R) D, arise from certain families of elliptically fibered surfaces. Weshall not pursue the (algebro-)geometric angle here, but instead look at what representationtheory can tell us.Fixing a point o D with stabilizer P G and a compact Cartan t gR p determines

the grading element T. The action of g on V defines a map

(5.3) Symng1 Hom(V n,0, V 0,n)which pulls back to (5.2) under the period map. If g = g2 and T is the adjoint gradingelement, which is defined by 1(T) = 0 and 2(T) = 1 on the simple roots, and V is the7-dimensional irreducible representation (with n = 2 and V 2,0 = Ce1, e2), one may ask inwhich horizontal tangent directions (5.3) vanishes. Writing

dfn= 0x

2 + 1x12 + 2x212 + 3x312 g1 ,and {e1, e2} for the dual basis of V 0,2, a straightforward computation (with appropriatenormalizations) gives

(5.4) e [2]e =

( 213 + 222 12 0312 03 202 + 221

).

The common vanishing locus of the matrix entries in (5.4) yields a twisted cubic curvev3(P

1) Pg1, cf. [25]. Upon varying o D (or D), this recovers the field of cubic conesin the contact planes g1o ToD preserved by G2, cf. [11].Furthermore, it is known that the lines g1 ToD in each cubic curve are precisely

the lines through o contained in D, under its minimal homogeneous embedding in Pg. Tosee this, we apply (ad)2 to the highest weight vector v = x31+22 g2 to check vanishingof the second fundamental form:

(ad)2v = (202 + 221)x1 + (03 12)H31+22 + (213 + 222)x1 g0 .Hence we get an identification of the variety of horizontal lines Co (cf. (3.1)) with thevariety of Yukawa vanishing directions, which provides a homogeneous-space descriptionof the latter and a Hodgetheoretic interpretation of Co. Moreover, the argument producesequations for both as projective varieties in Pg1. The purpose of this section is to discernthe degree to which both phenomena generalize.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 29

5.1. Yukawa vanishes along horizontal lines. Fix D = G/P . (We do not assume thatP is maximal here.) Let E be the grading element (2.7) associated with the parabolic p, andlet V be the irreducible Gmodule giving the minimal homogeneous embedding G/P PV(Section 2.4).Fix a Gmodule U , and suppose that the Eeigenspace decomposition is of the form

(5.5) U = Um/2 Um/21 U1m/2 Um/2 ,with 0 m Z, Um/2 6= 0 and (Ua) Ua as g0modules.Remark 5.6. In general, the Eeigenvalues of U are rational. However, if G is Qalgebraicand E is the grading element T associated to a Hodge representation (G,UQ, ), cf. [45,Section 2.3], then the Eeigenspace decomposition of U = UQ C is of the form (5.5).Any element of Symm(g1) naturally defines a G0module map Um/2 Um/2. Explic-

itly, given g1 and u Um/2, the relation (2.8) implies m(u) Um/2. Thus, we havea G0module map

: Symm(g1) Hom(Um/2, Um/2) ,which is the GriffithsYukawa coupling from above. Define

(5.7a) YU dfn= {[] P(g1) | (m) = 0} .In particular, given 0 6= g1,(5.7b) [] YU if and only if m|Um/2 = 0 .Observe that YU is a closed, G0invariant subvariety of Pg1.Our first main goal in this section is to understand the relationship between YU and Co.

Theorem 5.8. If the the kernel of the GriffithsYukawa coupling is nonempty, then itcontains the (tangent directions to) lines through o = P/P in the minimal homogeneousembedding G/P PV that are VHS. That is, if YU 6= , then(5.9) Co YU .The theorem is proved in Section 5.4. The equations cutting out Co and YU are given inSection 5.2. Example 5.19 is a case in which containment is strict in (5.9). A large class ofexamples for which equality holds is the following. Take

V = g ,

so that D Pg is the adjoint variety of G (Section 2.5). The associated grading element isE =

iI S

i where the set I is defined by =

iI i, cf. Table 2.1. Further assume thatm = 2; equivalently, the Eeigenspace decomposition (5.5) of U is

(5.10) U = U1 U0 U1 .Theorem 5.11. Let V = g, so that G/P Pg is the adjoint variety of G. Assume thishomogeneous embedding of G/P is minimal. Suppose that m = 2, so that (5.10) holds. IfU1 is a faithful representation of the Levi factor g0, then Co = YU .Remark 5.12. The adjoint variety fails to be a minimal homogeneous embedding only forthe simple Lie algebra g = sp2rC it is the second Veronese embedding v2(P

2r1) P Sym2C2r. As such it contains no lines.

30 KERR AND ROBLES

Example 5.17 is a case in which m = 2 and Co = YU despite the fact that the hypothesesof Theorem 5.11 fail.Theorem 5.11 is proved in Section 5.5. Given the theorem, it is interesting to identify

those irreducible representations U of G for which (5.10) holds when E is the grading elementassociated to an adjoint variety. These representations are listed in

Lemma 5.13. Let G/P Pg be an adjoint variety with associated grading element E. LetU be an irreducible representation of highest weight . Then (E) = 1 if and only if isamong those listed below.

(a) g = slr+1C and E = S1 + Sr: = i, for any 1 i r; in this case U = iCr+1.

(b) g = so2r+1C and E = S2: either = 1, in which case U = C

2r+1 is the standardrepresentation; or = r, in which case U is the spin representation.

(c) g = sp2rC and E = S1: = i, for any 1 i r.

(d) g = so2rC and E = S2: either = 1, in which case U = C

2r is the standardrepresentation; or = r1 , r, in which case U is one of the spin representations.

(e) g = e6 and E = S2: = 1 , 6.

g = e7 and E = S1: = 7.

(In the case that g = e8 and E = S8, we have (E) > 1 for all .)

(f) g = f4 and E = S1: = 4.

(g) g = g2 and E = S2: = 1 and U = C

7 is the standard representation.

In each of these cases, we also have (E) = 1.

In Section 5.3 we determine to which cases in Lemma 5.13 Theorem 5.11 applies. (ByRemark 5.12, the theorem can not be applied to the symplectic adjoint variety.)

Proof. As a highest weight is of the form = ii with 0 ii. The tables of [9] expressthe i as linear combinations of the simple roots j. The lemma follows.

5.2. Equations cutting out Co and YU . We begin with YU . Let : Hom(Um/2, Um/2) Symm(g1)

denote the dual map. To be explicit: let u Um/2 (Um/2) = Hom(Um/2, Um/2),then the polynomial ( u) Symm(g1) is given by ( u)() = (m(u)), for any g1. The image(5.14) im Symm(g1) is the set of polynomials defining YU Pg1.Turning to Co, the following lemma characterizes Co as the set of lines in g1 whose

iterated action annihilates the highest weight line of V .

Lemma 5.15. Let G/P PV be the minimal homogeneous embedding. Let 0 6= v V bea highest weight vector. Let g1 g ToD. Then [] Co if and only if 2(v) = 0.Before proving the lemma, we note the following corollary which describes the equationscutting out the lines Co. Recall the Eeigenspace decomposition

V = V h V h1 V h2

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 31

of Section 2.3.2, and that V h is onedimensional. Fix a highest weight vector 0 6= v V h.Given (V h2) define p Sym2(g1) by p() = (2v). This defines a g0modulemap

: (V h2) V h Sym2(g1) .By Lemma 5.15, this map has the property that the image

(5.16) im Sym2(g1) is the set of polynomials defining Co Pg1.In particular, if G/P is one of the fundamental adjoint varieties, then Co is the variety oflines through a fixed point o (cf. 3.16); and (5.16) gives equations for Co(12) as a projectivevariety.

Proof of Lemma 5.15. Define T = spanC{v, (g)v} V . Then T is the embedded tangentspace to the cone C(D) V over D at v. (Note that T depends only on the highest weightline o = V h, not on our choice of v o.)The unique P1 PV containing o and satisfying ToP1 is P spanC{v, (v)}. Since D is

cut out by degree two equations [5, Section 2.10], the line P1 is contained in D if and onlyif the line osculates with D to second order; equivalently, the second fundamental form F 2

vanishes at . That is,Co = P{ g1 | F 2() = 0} .

Since F 2() = 2(v)/T V/T , the vanishing F 2() = 0 is equivalent to 2(v) T .If 2(v) = 0, then it is immediate that F 2() = 0. Whence, [] Co.Conversely, suppose that [] Co. Then [] is necessarily contained in one of the Co,i of

(3.7). Since Co,i is the G0orbit of gi Pg1, we may assume without loss of generalitythat gi . Let denote the highest weight of V . Then 2(v) is a weight vector ofV for the weight 2i. Since [] Co, and this variety is the zero locus of the secondfundamental form, 2(v) is necessarily of the form av + (v) for some a C and g.Write =

with g. Each (v) is a weight vector of V for the weight .

Therefore, the condition that 2(v) = av + (v) be a weight vector for 2i forces a = 0and all but at most one (v) to vanish. If (v) 6= 0, then 2i = forcing = 2i.This is not possible, since 2i is not a root. Therefore,

2(v) = 0.

5.3. Examples.

Example 5.17 (Lemma 5.13(a)). Let W = Cr+1 be the standard representation of g =slr+1C. Let E = S

1 + Sr. Then the Eeigenspace decomposition of W is

W = W 1 W 0 W1 , with dimCW1 = 1 .Whence the Eeigenspace decomposition

U = U1 U0 U1of U =

iW is given byU1 = W 1 (i1W 0) ,U0 =

(W 1 (i2W 0) W1) iW 0 ,

U1 = (i1W 0) W1 .

(12)and for YU , when Theorem 5.11 applies

32 KERR AND ROBLES

The Levi factor isg0 = C2 slr1C

with C2 = span{S1, Sr}.We claim that U1 is not a faithful representation of g0. To see this, first note that the

central element S1 acts on W 1 and W 0W1 by the eigenvalues r/(r+1) and 1/(r+1),respectively; the central element Sr acts on W 1W 0 and W1 by the eigenvalues 1/(r+1)and r/(r+1), respectively. It follows that S1 acts on U1 by the eigenvalue (r+1i)/(r+1),and Sr acts by the eigenvalue i/(r+1). Therefore, iS1 (r+1 i)Sr C2 acts trivially onU1. Whence g0 does not act faithfully on U1, and we may not apply Theorem 5.11.Nonetheless, we claim that

(5.18) Co = YU .To see this, fix a basis {e0, e1, . . . , er} of W so that

W 1 = span{e0} , W 0 = span{e1, . . . , er1} and W1 = span{er} .Let {e0, . . . , er} be the dual basis of W , and {eij = ej ei | 0 i, j r} the induced basisof End(W ). Then

g1 = span{e0a , ear+1 | 1 a r 1}and

U1 = span{e0 ea1 eai1 | 1 a1, . . . , ai1 r 1} .The action of = be0b + be

br+1 g1 on u = e0 ea1 eai1 U1 is given by

(u) =(b eb ea1 eai1

)

i1s=1

(1)s as e0 er+1 (ea1 eas eai1

)and

2(u) = 2

b6{at}

i1s=1

basea1 eas eai1 eb er+1

+(1)ii1s=1

asasea1 eai1 er+1

(1)i

b6{at}b

bea1 eai1 er+1 .

WhenceYU = {a = 0 | 1 a r 1} {a = 0 | 1 a r 1} .

Next, we determine Co by applying Lemma 5.15 to the highest root vector v = er+10 . Wefind

2(v) =r1a=1

aa (e00 + e

r+1r+1) +

r1a,b=1

bc ecb .

WhenceCo = {a = 0 | 1 a r 1} {a = 0 | 1 a r 1}

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 33

is a disjoint union Pr2 Pr2 of linear spaces, and (5.18) follows.Example 5.19 (Lemma 5.13(b,d) with the standard representation). Let g = somC andU = Cm. Take the parabolic p determined by the grading element E = S2, cf. (2.6). ThenD = G/P is the adjoint variety. Note that

g 2U ,as a gmodule. The Eeigenspace decomposition of U is

(5.20) U = U1 U0 U1 .Moreover, U is a real Hodge representation with Hodge numbers h = (2,m 4, 2).In this case

(5.21) g0 C sl2C som4C,with center C = span{E}, and semisimple factor g0ss = [g0, g0] = sl2C som4C. As ag0module U1 C2. In particular, the factor som4C g0ss acts trivially on U1. Thereforeg0representation on U1 is not faithful, and we may not apply Theorem 5.11.We proceed to investigate the question of whether or not equality holds in (5.9) by

hand. Fix bases {e1, e2}, {e3, . . . , em2} and {em1, em} of U1, U0 and U1, respectively,which satisfy (ej , ek) = 1 if j + k = m + 1 and zero otherwise. Following the standardclassification of complex simple Lie algebras, we will assume that

m 7 .By definition [] YU if and only if 2

U1

= 0, cf. (5.7); equivalently,

(5.22) [] YU if and only if 0 = 2ea for a = 1, 2 .At this point, making use of a coarse dimension count, we see that

Co 6= YU .Note that dimPg1 = 2m 9. The characterization (5.22) imposes four constrains on YU ,implying dimYU 2m13. On the other hand, from Table 3.2 we see that dim Co = m7.We finish this example by identifying the additional equations that cut Co out from YU .

The highest root line of V = g is spanned by

v = em11 em2 .Let g1. By Lemma 5.15, the point [] Pg1 lies in Co if and only if

0 = 2(v) = (2e1) em1 + 2(e1) (em1) + e1 (2em1) (2e2) em 2(e2) (em) e2 (2em) .

Thus,

(5.23) [] Co if and only if{

0 = 2ea , 2es , a = 1, 2 , s = m 1,m ,

0 = (e1) (em1) (e2) (em) .

34 KERR AND ROBLES

We will see that the four equations 2es = 0 of (5.23) are redundant they are satisfiedby YU and Co is cut out from YU by the 12(m 4)(m 5) equations (e1) (em1) (e2) (em) = 0. Write as

= ja (eaj em+1jm+1a) = m+1km+1s (em+1sm+1k eks) .

The sums above are over 3 j, k m 2. Then(5.24) ea =

ja ej =

m+1ja em+1j ,

and2ea =

b,j

m+1ja jb em+1b .

Here the sum is over 1 b 2 and 3 j m 2. Therefore,(5.25) [] YU if and only if 0 =

j

m+1ja jb for a = 1, 2 .

We compute

(5.26) es = j

m+1jm+1s ej

and2es =

j,s

m+1jm+1s jb e

b

Substituting a for m+ 1 s, we see from (5.25) that(5.27) [] Y implies 2es = 0 .Finally, from (5.24) and (5.26), we compute

(e1) (em1) (e2) (em) =j,k

(j2

k1 j1 k2

)em+1kj .

Whence

(5.28) (e1) (em1) (e2) (em) = 0 if and only if j2 k1 j1 k2 = 0for all 3 j, k m2. It follows from (5.22), (5.23) and (5.27) that (5.28) are the equationscutting Co out from YU .Example 5.29 (Lemma 5.13(b,d) with a spin representation). We have (5.21). If U is aspin representation of g, then U1 is a spin representation of the factor som4C g0, whilesl2C g0 acts trivially. The latter implies U1 is not a faithful g0 representation Theorem5.11 does not apply.

Example 5.30 (Lemma 5.13(e) with g = e6). Let U be the irreducible e6(C)representationof highest weight 1. Then U

is the irreducible representation of highest weight 6. Inparticular, U 6 U, so that U is complex. Whence there exists a real form UR of U = UU.Moreover, E = S2 induces the structure of a Hodge representation of GR = E II on UR withHodge numbers h = (12, 30, 12).In this case g0 = C sl6C; here C = span{E} is the center of g0, and sl6C the semisimple

factor [g0, g0]. We have U1 = C6 (C6) as a sl6Cmodule. This representation is faithful,and Theorem 5.11 applies.

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 35

Example 5.31 (Lemma 5.13(e) with g = e7). The irreducible gmodule U of highest weight7 is selfdual and of dimension 56. As a gRrepresentation U is quaternionic: there is areal form UR of U = UU with the property that UR is an irreducible gRrepresentation.Moreover, the grading element E = S1 induces a Hodge structure on UR with Hodge numbersh = (24, 64, 24).In this case g0 = C so12C with center C = span{E}. Additionally, U1 C12 C12

is two copies of the standard representation of g0ss = so12C, and is therefore a faithfulg0representation. Thus Theorem 5.11 applies.

Example 5.32 (Lemma 5.13(f)). The irreducible gmodule U of highest weight 4 is selfdual and of dimension 26. As a gRrepresentation U is real: there is a real form UR ofU . Moreover, the grading element E = S1 induces a Hodge structure on UR with Hodgenumbers h = (6, 14, 6).In this case g0 = Csp6C, and U1 = C6 is the standard representation of the semisimple

factor g0ss = sp6C. Whence Theorem 5.11 applies.

Finally, we return to the example motivating this section:

Example 5.33 (Lemma 5.13(g)). Let g be the exceptional Lie algebra g2. Take E = S2, so

that D = G/P is the adjoint variety, and let U = C7 be the standard representation. TheEeigenspace decomposition of U is of the form (5.20), and U is a real Hodge representationwith Hodge numbers h = (2, 3, 2).In this case

g0 = C sl2C ,with C = span{E} the center and g0ss = [g0, g0] = sl2C the semisimple factor. Moreover,U1 C2 as a g0ssrepresentation. Since the center acts on U1 by the eigenvalue 1, it followsthat U1 is a faithful representation of g0. Whence Theorem 5.11 yields

(5.34) Co = YU .We finish this example by showing how (5.34) may be obtained (independently of The-

orem 5.11) by working with the equations of Section 5.2, but without the explicit compu-tations at the beginning of the section. We anticipate that this approach may be used todetermine equality (or its failure) in (5.9) in cases to which Theorem 5.11 does not apply.The second exterior power of the standard representation decomposes2U = U g2

as a g2module. Given a basis {e1, e2} of U1, the productv = e1 e2

spans the highest weight (root) line of g2 2U . Given g1, notice that0 = 2(v) = (2e1) e2 + 2(e1) (e2) + e1 (2e2) .

From (5.7) and Lemma 5.15 we see that

Co = {[] YU | (e1) (e2) = 0} .As e1 and e2 both lie in U

0 and the later is of dimension three, we see that the conditionthat [] YU lie in Co is at most three additional conditions on . It may be the casethat that these conditions are empty: that is, every [] YU satisfies (e1) (e2) = 0;

36 KERR AND ROBLES

equivalently, (5.34) holds. To see that this is indeed the case (without invoking Theorem5.11) we proceed as follows.Both YU and Co are cut out by quadrics; let I2(YU ) and I2(Co) be the quadrics vanishing

on YU and Co, respectively. Then both I2(YU ) and I2(Co) are g0invariant subspaces ofSym2(g1). We have g0 = gl2C, g0ss = sl2C and g1 = Sym

3C2 as a g0ssmodule. Whence

(5.35) Sym2(g1) = Sym2C2 Sym6C2 .If Co ( YU , then 0 6= I2(YU ) ( I2(Co). Given (5.35) this forces I2(Co) = Sym2(g1);equivalently, Co = , a contradiction. This establishes (5.34).5.4. Proof of Theorem 5.8. Let I = {i1, . . . , i} be the index set (2.1) associated withthe parabolic subalgebra p. The proof will proceed in two steps. We begin with the casethat p is a maximal parabolic (equivalently |I| = 1); this forms the basis for the generalargument.

Lemma 5.36. Suppose that D = G/P and that P is a maximal parabolic. If YU 6= , thenYU is connected and

Co YU .Proof. The parabolic P is maximal if and only if g1 is an irreducible g0module. In thiscase, Co Pg1 is the unique closed G0orbit (Section 3.1). It is clear from the definition(5.7) that YU is closed and preserved under the action of G0. Therefore, each connectedcomponent of YU contains a closed G0orbit. We now turn to the case that |I| > 1. First, we review the relationship between the

lines on G/P PV and G/Pi PVi , i I. The lines Co on the former are described inSection 3.2. For the latter, given i I, let Pi G be the maximal parabolic associated withthe index set {i}. Let Di PVi be the image of the minimal homogeneous embeddingG/Pi PVi , and let Coi(Di) be the lines on Di passing through oi = Pi/Pi. Then the Co,iof (3.5) is

(5.37) Co,i ={[] Pg1 | [] Coi(Di) , pj j I \ {i}

}.

Recall that E =

iI Si, and Um/2 is the eigenspace of the largest Eeigenvalue m/2

on U . Let mi/2 be the largest Sieigenvalue of U , and let Ui U be the corresponding

eigenspace. Then

m =iI

mi and Um/2 =

iI

Ui .

Let [] Coi(Di). Then (5.7) and Lemma (5.36) imply that mi |Ui = 0. Since mi m andUm/2 Ui, it follows that m|Um/2 = 0. Theorem 5.8 now follows from (3.5), (5.7) and(5.37).

5.5. Proof of Theorem 5.11. Given Theorem 5.8 it suffices to show that

(5.38) YU Co .To that end, let [] YU . Equivalently,(5.39) 2(u) = 0 for all u U1 .

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES 37

Fix a highest root vector 0 6= v g. Given Lemma 5.15, to establish (5.38) it suffices toshow that

(5.40) ad2(v) = 0 .

As an operator on U

ad2(v) = v 2v + v .As an element of g1, the endomorphism lowers eigenvalues by one; that is maps Uainto Ua1. Likewise, as an element of g2 = g, the root vector v maps Ua into Ua+2. Itthen follows from (5.10) that

(5.41) ad2(v)U1

= v|U1(5.39)= 0 .

On the other hand ad2(v) g0. By assumption U1 is a faithful representation of g0. So(5.41) holds if and only if (5.40) holds.

6. Construction of limiting mixed Hodge structures

6.1. Overview. In this section we construct a large class of polarized GRorbi