Variations of Hodge structure and orbits in flag varieties

HODGE THEORY AND REAL ORBITS IN FLAG VARIETIES

arXiv:1407.4507v1 [math.AG] 16 Jul 2014

MATT KERR AND COLLEEN ROBLES Abstract. Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford–Tate domains, arise as open GR –orbits in flag varieties G/P (homogeneous with respect to a complex, semisimple Lie group G). We investigate Hodge– theoretic aspects of the geometry and representation theory associated with flag varieties. More precisely, we prove that smooth Schubert variations of Hodge structure are necessarily homogeneous, relate the Griffiths–Yukawa coupling to the variety of lines on G/P ֒→ PV (under a minimal homogeneous embedding), construct a large class of polarized (or “Hodge–theoretically accessible”) GR –orbits in G/P and compute the boundary components of the polarizing nilpotent orbits, use these boundary components to define “enhanced SL2 –orbits”, and initiate a project to express the homology classes of these homogeneous submanifolds of G/P in terms of Schubert classes.

Contents 1. Introduction 2. Preliminaries 3. Lines on flag varieties 4. Smooth Schubert VHS 5. Griffiths–Yukawa couplings and lines on G/P 6. Construction of limiting mixed Hodge structures 7. Enhanced SL2 -orbits and adjoint varieties Appendix A. Examples in codimension one Appendix B. Computation of [X(N )] in OG(2, C2r+1 ) Appendix C. Dynkin diagrams References

1 7 16 22 27 37 52 60 72 80 80

1. Introduction This is the first in a planned series of papers that studies the relationship between the Hodge theory of Mumford–Tate domains D = GR /R and the projective geometry and GR – ˇ = G/P . It continues lines of thought initiated in orbit structure of their compact duals D the first author’s 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, Mumford–Tate domains, Mumford–Tate groups, boundary components, variations of Hodge structure, Schubert varieties, Griffiths–Yukawa coupling. 1

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author’s work [43,45] on Schubert varieties and the infinitesimal period relation (a.k.a. Griffiths’s transversality condition). Our purpose is twofold: to use the representation theory of ˇ to better understand the asymptotics and differential invariants G and the geometry of D of variations of Hodge structure; and to give Hodge–theoretic interpretations of familiar ˇ including GR –orbits, Schubert varieties and their classes, and the geometric objects on D, variety of minimal degree rational curves on G/P . Throughout, the fundamental adjoint varieties (Section 2.5) will serve as illustrative examples. Background. Mumford–Tate domains are classifying spaces for Hodge structures on a Q-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 nonconstant homomorphism ϕ : S 1 → Aut(VR , Q) √ v ) > 0, for all v ∈ VC \ {0}. Let GQ denote the of real algebraic groups with Q(v, ϕ( −1)¯ smallest Q–algebraic subgroup of Aut(VQ , Q) with GR ⊇ ϕ(S 1 ). This is the Mumford–Tate group of ϕ. The real form GR is known to be reductive [20]. This group acts on ϕ by conjugation, and the orbit dfn ∼ GR /R D = GR · ϕ = is a Mumford–Tate domain. The stabilizer R is compact, and its Lie algebra r contains a compact Cartan subalgebra t of gR [24]. The Mumford–Tate domain is an analytic open subset of its compact dual, the projective homogeneous variety ˇ dfn D = G · Fϕ• ∼ = G/P . Here, G is the complexification of GR , and the action of G (and GR ) is by left translation ˇ on the associated Hodge flag Fϕ• ∈ D. One way to construct Mumford–Tate 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 grading element which is odd (respectively, even) on the noncompact (respectively, compact) roots, then ϕ(z) = e2 log(z)E defines a weight zero Hodge structure (HS) on gQ = Lie(GQ ) that is polarized by −(·, ·), where (·, ·) is the Killing form on gQ . Moreover, ϕ may lift to a HS on a representation VQ . In either case, a sufficiently general GR -conjugate of this HS has Mumford–Tate group GQ . In case ϕ lifts, the Hodge decomposition VC = ⊕V p,q is the E–eigenspace decomposition; the grading element acts on V p,q by the scalar (p − q)/2. Then ˇ = G/P is identified ϕ is identified with the Hodge flag F p = ⊕a≥p V a,b , the compact dual D • with the G–flag variety containing F , and the Mumford–Tate domain D = GR /R with the ˇ The Lie algebra of P is p = g≥0 = ⊕p≥0 gp , where gp = gp,−p . open GR –orbit of F • in D. The Lie algebra of R is a real form of g0 ; we shall write dfn

G0 = RC . For the purposes of this introduction, we shall assume that D has a base point o = ϕ defined over Q (cf. Definition (6.17)). Running Example 1. A case in point, which we shall use as a running example throughout this introduction, is the Q–form of the exceptional, rank two group G2 constructed in


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 = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ,

g−p = g−p,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 2,0

on the standard, 7–dimensional representation with Hodge numbers (2, 3, 2). The compact dual is the 5–dimensional adjoint variety in Pg. Recall that a variation of Hodge structure V over a complex manifold S (in the classical sense) consists of a Q-split local system V, and a filtration of V ⊗ OS by holomorphic subbundles F • whose fibre-wise restrictions are Hodge flags, and which satisfy the infinitesimal period relation ∇(F • ) ⊂ F •−1 ⊗ Ω1S for the flat connection ∇ on V annihilating V. The asymptotics of V over a punctured disk ∆∗ ⊂ S are captured by the (limiting) mixed Hodge dfn ˜ • structure (Vlim , Flim , W• ) = (V, F • , W• (N ))|s=0 , where N is the monodromy logarithm of ˜ V|∆∗ and V = exp(− log(s)/2πi)V. If GQ contains the Mumford–Tate groups of the fibres of V, the possible limiting MHS ˜ ) of [31]. Moreover, V induces a period are classified by the boundary components B(N mapping Φ : S → Γ\D, with Γ the monodromy group. We may lift Φ|∆∗ to the upper– e : H → D, and take the na¨ıve limit limIm(τ )→∞ Φ(τ e ) ∈ D, ˇ which lies in the half plane Φ ˇ This boundary breaks into finitely many GR –orbits, and analytic boundary bd(D) ⊂ D. those accessible by such Hodge–theoretic limits were termed polarizable in [32].(1) Schubert varieties and variations of Hodge structure. In this paper, the term variation of Hodge structure (VHS) is used primarily in a different sense. The infinitesimal period relation (IPR) forces the differential of a period map Φ to lie in the horizontal disˇ A subvariety (or submanifold) Y ⊂ D ˇ is horizontal if its tangent space tribution T 1 ⊆ T D. 1 Ty Y ⊂ Ty is contained in the horizontal distribution at every smooth point y ∈ Y . For ˇ In particular, we will us a VHS shall simply be a horizontal subvariety/submanifold of D. ˇ call a horizontal Schubert variety X ⊂ D a Schubert VHS. The Schubert VHS encode a great deal of information about the IPR and period mappings, cf. [45]. For example, there exists a VHS of dimension d if and only if there exists a Schubert VHS of dimension d. In particular, the rank of the differential dΦ of a period mapping is bounded above by ˇ is a Schubert VHS}. max{dim X | X ⊂ D Running Example 2. The horizontal distribution is a contact 4–plane distribution. Modulo ˇ of dimension d, for 0 ≤ d ≤ 5. the action of G2 , there is a unique Schubert variety Xd ⊂ D 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 ˇ Any such Schubert to translation) the compact dual of a Mumford–Tate subdomain of D? variety is necessarily homogeneous and therefore smooth. The majority of Schubert varieties (1)This notion of a “polarized” orbit is distinct from J. Wolf’s in [47, Definition 9.1] where the orbit ˇ is polarized if it realizes the minimal homogeneous CR–structure on the homogeneous manifold GR · x ⊂ D GR /StabGR (x), cf. [1, Remark 5.5].

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ˇ need not be homogeneous. are singular. Moreover, a smooth Schubert variety X ⊂ D Nevertheless, we show that a smooth Schubert VHS is necessarily a homogeneous embedded Hermitian symmetric space (Theorem 4.4); moreover, with an assumption on the “Q-types” of GQ and o ∈ D, these are all Mumford–Tate subdomains (Corollary 4.5). These results invite 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 result of J. Hong and N. Mok [27] which asserts that the smooth Schubert varieties in G/P , with P a maximal parabolic subgroup associated with a non–short simple root, are necessarily homogeneous. The argument underlying this beautiful result is in turn based J.-M. Hwang and Mok’s program to study projective varieties by considering the variety of minimal degree rational curves passing through a point [29]. Running Example 3. The 2–dimensional Schubert VHS X2 cannot be Mumford–Tate. This can be seen at once (and was first observed by the authors) by looking at its tangent space, ˇ In fact, it turns out that X2 is which cannot be that of a reductive group orbit in D. singular, and therefore not homogeneous. (The 1–dimensional Schubert VHS X1 is a P1 , and therefore homogeneous.) ˇ and the Griffiths–Yukawa coupling. When D ˇ is the minimal homogeneous Lines on D ˇ are lines P1 ⊂ PV (degree embedding G/P ֒→ PV , the minimal degree rational curves on D ˜ ˇ ˇ one). Let Co ⊂ PTo D be the variety of tangent directions to lines passing though o ∈ D. 1 ˇ Of particular interest here is the (We will often think of Cõ as the set of lines o ∈ P ⊂ D.) set dfn Co = Cõ ∩ PT 1 of directions horizontal with respect to the IPR: If P is a maximal parabolic, then the variety [ dfn (1.1) X = P1 P1 ∈Co

swept out by these lines (which is a cone over Co with vertex o) is a (typically singular) Schubert variety. When P is a maximal parabolic associated to a non-short root, we have Co = Cõ ˇ is an adjoint variety and X is horizontal, cf. [38, Proposition 2.11] and Lemma 3.3. When D (V = g), Co realizes the homogeneous Legendrian varieties studied by J.M. Landsberg and L. Manivel [36, 37]. Running Example 4. In our G2 example, X = X2 is the famous twisted cubic cone known ´ Cartan, cf. Section 5. to E. In this paper we will discuss two Hodge–theoretic characterizations of Co . The first is in terms of the Griffiths–Yukawa coupling, a differential invariant of VHS’s. (The second is quite distinct, and will be discussed below.) When the kernel of the Griffiths–Yukawa coupling is nonempty, it necessarily contains Co (Theorem 5.8). Moreover, for certain compact duals (including the An , E6 , E7 , E8 , F4 , and G2 adjoint varieties, cf. Lemma 5.13 and Section 5.3) equality holds (Co is the kernel of the Griffiths–Yukawa coupling) for some Hodge representation of weight two (Theorem 5.11).


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Polarized orbits. The polarized orbits, and the corresponding limiting mixed Hodge structures, describe the asymptotics of a variation of Hodge structure on a Mumford–Tate doˇ main. More generally, one may take the na¨ıve limit of nilpotent orbit exp(zN )F • on D. In this paper our emphasis is on representation theoretic descriptions of polarized orbits and mixed Hodge structures associated with nilpotent orbits, with the motivation that such descriptions may guide the construction of geometric/motivic examples.(2) Running Example 5. The boundary bd(D) contains a unique real-codimension one GR – orbit, which is polarizable. The weight-graded pieces of the corresponding limiting mixed Hodge structures have dim Gr3W Vlim = dim Gr1W Vlim = 2 and dim Gr2W Vlim = 3 (with Hodge numbers (1, 1, 1)). Additional detail on this example may be found in [32, Section 6.1.3]. ˇ Section 6 gives a systematic construction of polarizable boundary strata in bd(D) ⊂ D via distinguished “Q-Matsuki”(3) points on them and sets B = {β1 , . . . , βs } ⊂ h∗ of strongly orthogonal 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 parameterized by the construction of Section 6, and the parameterization is essentially that given by the Harish–Chandra compactification of D. While notationally dense, the construction is very natural, and we will briefly summarize it here. With each root βj ∈ B is associated a Cayley transform cβj ∈ G, and we let cB denote the composite/product. From the Q–Matsuki point o and cB we obtain a Q–Matsuki point ˇ with corresponding Hodge flag F • , a “weight filtration” W•B , and nilpotent cone oB ∈ D, B dfn

σB = {t1 N1B + · · · + ts NsB | ti ≥ 0} with NjB ∈ gR and NjB FBp ⊂ FBp−1 , for all p. These objects have the properties that the nilpotent cone σB underlies a multiple variable nilpotent orbit (z 1 , . . . , z s ) 7→ exp(z 1 N1B + · · · z s NsB ) · FB• , W•B = W• (σB ), and (FB• , W•B ) is a (Q–split) Q–MHS on g. In particular, the MHS (FB• , W•B ) ˜ B ), and the orbit belongs to the boundary component B(σ dfn

OB = GR · oB is 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 are its 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 c , o , F • , B B B • p,q W•B , σB , OB , gp,q are denoted c , o , F , W , σ , O , g , respectively. s s s s • s s s B

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Boundary components. The construction also a yields a semisimple element YB ∈ gR which grades the weight filtration. The Mumford–Tate group of (FB• , W•B ) is contained in dfn

Defining(6) the set

GB = StabG {YB } ∩ StabG {N1B } ∩ · · · ∩ StabG {NsB } . dfn ˇ dfn DB = GB,R · FB• ⊂ GB · FB• = D B, dfn

˜ B) B(σB ) = eC σB \ B(σ

of nilpotent orbits fibres naturally over DB . Passing to the quotient by the intersection Γ of an arithmetic subgroup of GQ with Stab(σB ), the surjection Γ\B(σB ) ։ Γ\DB has the structure 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 p,q ˜ components gp,q B in the Deligne splitting gC = ⊕gB and the structure of the B(σB ). For example, when P is a maximal parabolic there is exactly one orbit O1 ⊂ bd(D) of real codimension one (Proposition 6.56); it is of the form OB with B = {β1 }, and the corresponding boundary components B(σB ) = B(N ) are completely worked out in Appendix A. ˇ B . For the fundamental adjoint varieties, the Deligne biAdjoint varieties and Co ≃ D 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 Co ˇ B . Meanwhile, by Proposition 7.15, the Γ\B(N ) realize the above reappear in the guise of D intermediate Jacobian families associated to the Friedman–Laza [22] weight-three maximal Hermitian VHS’s of Calabi-Yau type! Running Example 6. Γ\B(N ) is just an elliptic modular surface. (The Friedman–Laza variation is the symmetric cube of a VHS of type (1, 1).) Enhanced horizontal SL2 –orbits and homology classes. Another motivation for the construction of Section 6 goes back to a basic question of Borel and Haefliger [8] on smoothability of cohomology classes in G/P . By [45, Theorem 4.1], the invariant characteristic ˇ is generated by classes of Schubert VHS. Moreover, the homology class cohomology of D ˇ may be expressed as a linear combination of Schubert VHS of any horizontal cycle in D classes [43, Theorem 4.7]. Applying a VHS twist to Borel and Haefliger’s question, we ask: when can the class (or a multiple thereof) of a singular Schubert VHS, such as (1.1), admit a smooth, horizontal, algebraic representative? One natural source of smooth subvarieties (in fact, Mumford–Tate subdomains) which are often horizontal, are the “enhanced multivariable SL2 -orbits” X(σB ) which arise as follows. The strongly orthogonal roots B above β determine a collection {sl2 j | βj ∈ B} of commuting sl2 ’s. Letting SLB 2 (C) ⊂ GC be the β1 βs B connected Lie subgroup with Lie algebra sl2 (C) = sl2 (C) × · · · × sl2 (C), the enhanced SL2 –orbit is dfn ˇ, X(σB ) = SLB (C) · GB,C · oB ⊂ D 2

(6)Here one should really think of G as acting on the associated graded of the MHS (F • , W B ). Since B • B

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


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ˇ is a fundamental adjoint variety, the enhanced SL2 -orbit cf. Section 7.1. In the case that D attached to O1 is a VHS and a cylinder on Co ; that is, X(σB ) = X(N ) ∼ = Co × P 1 .

Recalling that X is a cone over Co , cf. (1.1), we conclude this article by testing the irresistible hypothesis 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 salient is the systematic computation of the cohomology classes of smooth VHS (and the horizontal X(σB ) in particular). We will undertake this in a sequel (by a different method than those of Section 7.6 and Appendix B). It is also important to generalize the construction of Section 6 in order to parameterize all polarizable orbits; this work will appear in [44], and we expect to use it to tie together combinatorial structures related to nilpotent cones and boundary orbits. Finally, one reason for classifying Hodge–theoretic phenomena is to predict algebrogeometric ones. Turning one last time to the running example, suppose we have a 2parameter family of algebraic surfaces π : X → S for which (a subquotient of) R2 π∗ Q underlies a maximal VHS with Hodge numbers (2, 3, 2) and Mumford–Tate group G2 . Then one can ask what sort of degeneration produces a point in the “elliptic modular surface” boundary component Γ\B(N ), and what sort of geometry produces a subvariation whose period map image lies in the “cubic cone” Griffiths–Yukawa kernel. Now since one expects such 2-parameter VHS to arise from certain elliptic surfaces (with internal fibration over an elliptic curve, cf. [30]), neither question looks too difficult. With some optimism, one might imagine asking the analogous questions for all the fundamental adjoint varieties. Acknowledgements. The authors thank P. Griffiths and J.M. Landsberg for helpful discussions, 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 undertaken while Robles was a member of the Institute for Advanced Study; she thanks the institute for a wonderful working environment and the Robert and Luisa Fernholz Foundation for financial support. 2. Preliminaries In this section we set notation and review the necessary background material from Hodge theory and representation theory. 2.1. Flag varieties. Let G be a connected, complex semisimple Lie group, and let P ⊂ G be a parabolic subgroup. The homogeneous manifold G/P admits the structure of a rational homogeneous 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 set of roots ∆ = ∆(g, h) ⊂ h∗ . Given a root α ∈ ∆, let gα ⊂ g denote the root space. Given a subspace s ⊂ g, let dfn ∆(s) = {α ∈ ∆ | gα ⊂ s} .

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The choice of Borel determines positive roots ∆+ = ∆(b) = {α ∈ ∆ | gα ⊂ b}. Let S = {α1 , . . . , αr } denote the simple roots, and set (2.1)

dfn

I = I(p) = {i | g−αi 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 {ω1 , . . . , ωr } denote the fundamental weights of g and let V be the irreducible g– representation of highest weight X dfn (2.2) µ = µI = ωi . i∈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 weight line in V . The G–orbit G · o ⊂ PV is the minimal homogeneous embedding of G/P as a rational homogeneous variety. Remark 2.3 (Non-minimal embeddings).P More generally, suppose that V is the irreducible G–representation of highest weight µ ˜ = i∈I ai ωi with 0 < ai ∈ Z. Again, the G–orbit 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 Veronese re-embedding vd (Pn ) ⊂ P Symd Cn+1 of Pn is if minimal if and only if d = 1. (Here V = Symd Cn+1 has highest weight dω1 .) The rational homogeneous variety G/P is sometimes indicated by circling the nodes of the Dynkin diagram (Appendix C) corresponding to the index set I(p). 2.2. Flag domains and Mumford–Tate domains. Let GR be a (connected) real form of G. There are only finitely many GR –orbits on G/P . An open GR –orbit D = GR /R is a flag domain. The stabilizer R ⊂ GR is the centralizer of a torus T ′ ⊂ GR [21, Corollary 2.2.3]. The flag variety ˇ dfn D = G/P is the compact dual of the flag domain. We will be interested in the case that D admits the structure of a Mumford–Tate domain. Mumford–Tate domains are generalizations of period domains, and as such arise as the classification spaces of polarized Hodge structures (possibly with additional structure); see [24] for a thorough treatment. When D admits the structure of a Mumford–Tate domain the stabilizer R is compact, and as a consequence there exists a compact maximal torus T ⊂ GR such that T ′ ⊂ T ⊂ R [24]. We will assume this to be the case throughout. In the Mumford–Tate 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 assume this only where relevant. In a few places (mostly limited to Section 6), we shall make the stronger assumption that GQ is a Mumford–Tate–Chevalley group (Definition 6.9).


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2.3. Grading elements. Let {S1 , . . . , Sr } be the basis of h dual to the simple roots. A grading element is any member of spanZ {S1 , . . . , Sr }; these are precisely the elements T ∈ h of the Cartan subalgebra with the property that α(T) ∈ Z for all roots α ∈ ∆. Since T is semisimple (as an element of h), the Lie algebra g admits a direct sum decomposition M (2.4a) g = gℓ ℓ∈Z

into T–eigenspaces (2.4b)

dfn

gℓ = {ξ ∈ g | [T, ξ] = ℓξ} .

In terms of root spaces, we have

gℓ = (2.4c)

M

α(T)=ℓ

g0 = h ⊕

gα , M

ℓ 6= 0 , gα .

α(T)=0

The T–eigenspace decomposition is a graded Lie algebra decomposition in the sense that h i (2.5) gℓ , gm ⊂ gℓ+m ,

a straightforward consequence of the Jacobi identity. It follows that g0 is a Lie subalgebra of g (in fact, reductive), and each gℓ is a g0 –module. The Lie algebra (2.6)

p = pT = g0 ⊕ g+

is the parabolic subalgebra determined by the grading element T. See [45, Section 2.2] for details. 2.3.1. Minimal grading elements. Two distinct grading elements may determine the same parabolic p. As a trivial example of this, given a grading element T and 0 < d ∈ Z, both T and dT determine the same parabolic. Among those grading elements T determining the same parabolic (2.6), only one will have the property that g±1 generates g± as an algebra. That canonical grading element is defined as follows. Given a parabolic p subalgebra (and choice of Cartan and Borel h ⊂ b ⊂ p), the grading element associated to p is X dfn (2.7) E = Si . i∈I(p)

The reductive subalgebra g0 = g0ss ⊕ z has center z = spanC {Si | i ∈ I} and semisimple subalgebra g0ss = [g0 , g0 ]. A set of simple roots for g0ss is given by S(g0 ) = {αj | j 6∈ I}. 2.3.2. E–eigenspace decompositions. Any g–representation U admits a E–eigenspace decomposition M U = Uq , q∈Q

see [45, Section 2.2]. In the case that U = g, we recover (2.4c). Again, the Jacobi identity implies (2.8)

gℓ (U q ) ⊂ U q+ℓ .

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In particular, each E–eigenspace U q is a g0 –module. P i If V is the irreducible g–module of highest weight µ = µ ωi , then the E–eigenspace decomposition is of the form V = V m ⊕ V m−1 ⊕ V m−2 ⊕ · · · . Moreover, if µi = 0 when i 6∈ I, then V m is the (one–dimensional) highest weight line, cf. [45, Lemma 6.3]. If µi > 0 if and only if i ∈ I, then the G–orbit of [V m ] ∈ PV is a homogeneous embedding of G/P ; the embedding is minimal if µi = 1 for all i ∈ I. 2.4. Variations of Hodge structure. This section is a very brief review of the infinitesimal period relation on G/P and the solutions of this differential system, the variations of Hodge structure. The reader interested in greater detail is encouraged to consult [45] and the references therein. The (holomorphic) tangent bundle of ˇ dfn D = G/P is the G–homogeneous vector bundle ˇ = G ×P (g/p) . TD

By (2.5) and (2.6), the quotient g≥−1 /p is a p–module. Therefore, dfn

T 1 = G ×P (g≥−1 /p) ˇ This subbundle is the infinitesimal defines a homogeneous, holomorphic subbundle of T D. 1 ˇ This is the case when G/P is period relation (IPR). The IPR is trivial if T = T D. Hermitian 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 submanifold Z ⊂ G/P with the property that Tz Z ⊂ Tz1 for all z ∈ Z, or any irreducible subvariety ˇ with the property that Ty Y ⊂ T 1 for all smooth points y ∈ Y . Alternatively, we Y ⊂D y shall sometimes refer to such subvarieties as horizontal. (2.9)

2.5. Adjoint varieties. Throughout the paper we will illustrate the ideas and results with adjoint varieties. From the Hodge–theoretic perspective, these are the simplest varieties for which the infinitesimal period relation is nontrivial: the adjoint varieties are precisely those G/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 α ˜ ∈ ∆+ α ˜ be the highest root. See Table 2.1. Recall g is one-dimensional, and therefore a line in ˇ is the G–orbit of o. g; let o ∈ Pg be the corresponding point. Then the adjoint variety D ˇ = G/P , if T is the grading element associated to P then Table 2.1 yields Writing D where g2 = gα˜ .

g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ,

Remark 2.10. The adjoint varieties are precisely the compact, simply connected, homogeneous complex contact manifolds [6]. (7)We shall have use for the more standard meaning of VHS (over an arbitrary complex manifold, and

allowing monodromy) only in the beginning of Section 5, and in Proposition 7.15.


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Table 2.1. The highest root α ˜ of G. G

α ˜

Ar Br Cr Dr E6 E7 E8 F4 G2

α1 + · · · + αr = ω1 + ωr α1 + 2(α2 + · · · + αr ) = ω2 2(α1 + · · · + αr−1 ) + αr = 2ω1 α1 + 2(α2 + · · · + αr−2 ) + αr−1 + αr = ω2 α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 = ω2 2α1 + 2α2 + 3α3 + 4α4 + 3α5 + 2α6 + α7 = ω1 2α1 + 3α2 + 4α3 + 6α4 + 5α5 + 4α6 + 3α7 + 2α8 = ω8 2α1 + 3α2 + 4α3 + 2α4 = ω1 3α1 + 2α2 = ω2

Example 2.11. (a) If g = slr+1 C, then the adjoint variety is Flag(1, r, Cr+1 ) is the (partial) flag variety of lines in hyperplanes. (b) If g = som C, then the adjoint variety is the orthogonal grassmannian OG(2, Cm ) of 2–planes that are ν–isotropic for a nondegenerate symmetric bilinear form ν. (c) If g = sp2r C, then the adjoint variety is the second Veronese re-embedding v2 (P2r−1 ) ⊂ P Sym2 C2r . Assume that G 6= Ar , Cr so that the adjoint representation g is fundamental. Equivalently, α ˜ = ωi (cf. Table 2.1), g is the irreducible g–representation of highest weight ωi and P is maximal; we have I(p) = {i} and (2.12)

α ˜ − αj is a root if and only if j = i.

ˇ and N = dim Pg, and degree d = Table 2.2. The dimensions n = dim D ˇ deg D for the exceptional adjoint varieties. G E6 E7 E8 F4 G2

n d N 21 151,164 77 33 141,430,680 132 57 126,937,516,885,200 247 15 4,992 51 5 18 13

2.6. Schubert varieties. In the case that G is classical (one of SLn C, SOn C or Sp2n C), the Schubert varieties of a flag variety G/P may be described geometrically as degeneracy loci, cf. Example 2.17. However, these descriptions, aside from not generalizing easily to the exceptional groups, are type dependent. We will utilize a representation theoretic description 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 references therein.

12

KERR AND ROBLES

Given a simple root αi ∈ S, let (i) ∈ Aut(h∗ ) denote the corresponding simple reflection. The Weyl group W ⊂ Aut(h∗ ) of g is the group generated by the simple reflections {(i) | αi ∈ S}. A composition of simple reflections (i1 ) ◦ (i2 ) ◦ · · · ◦ (it ), which are understood to act on the left, is written (i1 i2 · · · it ) ∈ W. The length of a Weyl group element w is the minimal number dfn |w| = min{ℓ | w = (i1 i2 · · · iℓ )}

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/P decomposes into a finite number of B–orbits [ G/P = Bw−1 o Wp w∈Wp \W

which are indexed by the right cosets Wp \W. The B–Schubert varieties of G/P are the Zariski closures dfn Xw = Bw−1 o . Remark 2.13. Observe that the stabilizer Pw of Xw in G contains B, and is therefore a parabolic subgroup of G. More generally, any G–translate gXw is a Schubert variety (of type Wp w). Define a partial order on Wp \W by defining Wp w ≤ Wp v 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 wXw is the Zariski closure of Nw · o, where Nw ⊂ G is a unipotent subgroup with nilpotent Lie algebra M dfn (2.14) nw = g−α ⊂ g− α∈∆(w)

given by dfn

∆(w) = ∆+ ∩ w(∆− ) .

(2.15)

Moreover, Nw · o is an affine cell isomorphic to nw , and dim Xw = dim nw = |∆(w)|. Indeed To (wXw ) = nw . For any w ∈ Wpmin we have (2.16)

|w| = |∆(w)| = dim Xw .

(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 ν be a nondegenerate, symmetric bilinear form on Cm , and let dfn

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


13

be the orthogonal grassmannian of ν–isotropic 2–planes in Cm . Fix a basis {e1 , . . . , em } of Cm with the property that m+1 ν(ea , eb ) = δa+b . Then dfn

Fp = spanC {e1 , . . . , ep } ⊂ Cm

defines a flag F• with the property that

ν(Fp , Fm−p ) = 0 .

(2.18)

Any flag F• satisfying (2.18) is called ν–isotropic. Given 1 ≤ a < b ≤ m with a + b 6= m + 1 and a, b 6= r + 1, define (2.19a)

dfn

Xa,b (F• ) = {E ∈ OG(2, Cm ) | dim E ∩ Fa ≥ 1 , E ⊂ Fb } .

If m = 2r +1, then every Schubert variety of OG(2, C2r+1 ) is G–congruent to one of (2.19a). If m = 2r is even, define ˜ r = span {e1 , . . . , er−1 , er+1 } . F C ˜ r ⊂ Fr+1 ⊂ · · · is also a ν–isotropic flag. Set Note that F1 ⊂ · · · ⊂ Fr−1 ⊂ F ˜ a,r (F• ) (2.19b) X

dfn

˜ r,b (F• ) (2.19c) X

dfn

=

=

˜r} , {E ∈ OG(2, Cm ) | dim E ∩ Fa ≥ 1 , E ⊂ F

˜ r ≥ 1 , E ⊂ Fb } , {E ∈ OG(2, Cm ) | dim E ∩ F

a ≤ r − 1, b ≥ r + 2.

Every Schubert variety of OG(2, C2r ) is G–congruent to one of (2.19). 2.7. Schubert VHS. The condition that the Schubert variety Xw be a VHS is equivalent to ∆(w) ⊂ ∆(g1 ), where ∆(w) is given by (2.15), cf. [45, Theorem 3.8]. A convenient way to test for this condition is as follows. Let dfn

̺ =

r X

ωi =

i=1

1 2

X

α

α∈∆+

be the sum of the fundamental weights (which is also half the sum of the positive roots). Define X dfn (2.20) ̺w = ̺ − w(̺) = α. α∈∆(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 Hodge structure if and only if ̺w (E) = |w|. See [45, Section 3] for details. Let dfn

Wvhs = {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, ˇ ⊂ Pg is the adjoint variety with G = G2 is the exceptional Lie group of rank two, D 2 ˇ I(p) = {2} and E = S . Modulo the action of G, there is a single Schubert variety Xwd ⊂ D of dimension d, 0 ≤ d ≤ 5. The variety Xwd is horizontal (a VHS) if and only if d ≤ 2. We have w1 = (2) with ∆(w1 ) = {σ2 }, and w2 = (12) with ∆(w2 ) = {σ2 , σ1 + σ2 }. As will ˇ and therefore homogeneous, while Xw is a be discussed later, Xw1 = P1 is a line on D, 2 singular cone over the Veronese re-embedding v3 (P1 ) ⊂ 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 )⊥ . ˜ a,r (F• ), are all horizontal. (b) The Schubert varieties X ˜ r,b (F• ) are horizontal. (c) None of the Schubert varieties X Corollary 2.23 (Maximal Schubert VHS in OG(2, Cm )). The maximal (with respect to containment) Schubert VHS in SO(2, Cm ), where m ∈ {2r, 2r + 1}, are: the Xa,m−a (F• ), ˜ r−1,r (F• ), if m = 2r. Each of these maximal Schubert VHS is of with 1 ≤ a ≤ r − 1; and X dimension m − 4. Proof of Lemma 2.22. By definition, a Schubert variety Xw ⊂ OG(2, Cm ) is horizontal (equivalently, satisfies Griffiths’s transversality condition) if and only if (2.24)

dfn

dE|Nw ·o ⊂ E ⊥ = {v ∈ Cm | ν(E, v) = 0}

on the Schubert cell Nw · o.

(a) It is straight–forward to see that the condition Fa ⊂ (Fb )⊥ implies Xa,b (F• ) is horizontal. 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 dfn

E(t) = spanC {eb + tem+1−a , ea − tem+1−b } ∈ 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. ˜ a,r (F• ) are defined for a < r. (b) As noted in Example 2.17, the Schubert varieties X ˜ r = (F ˜ r )⊥ , it is immediate that the X ˜ a,r (F• ) are horizontal. Since Fa ⊂ F

˜ r,b (F• ) are defined for r + 2 ≤ b, cf. Example 2.17. (c) Recall that the Schubert varieties X It follows that dfn ˜ r,b (F• ) . E(t) = spanC {eb + ter , er+1 − tem+1−b } ∈ X

(As above, E(t) lies in the Schubert cell.) As (2.24) clearly fails for E(t), we see that ˜ r,b (F• ) is not horizontal. This establishes assertion (c) of the lemma. X


15

2.8. Nilpotent orbits. Let gR ⊆ End(VR , Q) be the Lie algebra of GR . A (n–variable) ˇ the Ni ∈ gR nilpotent orbit on D consists of a tuple (F • ; N1 , . . . , Nn ) such that F • ∈ D, p p−1 m ˇ , and the holomorphic map ψ : C → D defined by commute and Ni F ⊂ F ψ(z 1 , . . . , z n ) = exp(z i Ni )F •

(2.25)

has the property that ψ(z) ∈ D for Im(z i ) ≫ 0. We shall use the term σ-nilpotent orbit to ˇ The associated (open) nilpotent cone is refer to the submanifold eC σ F • ⊂ D. σ = {ti Ni | ti > 0} .

(2.26)

Given a nilpotent N ∈ gR such that N k+1 = 0, there exists a unique increasing filtration W0 (N ) ⊂ W1 (N ) ⊂ · · · ⊂ W2k (N ) of VR with the properties that and the induced

N Wℓ (N ) ⊂ Wℓ−2 (N )

N ℓ : Grk+ℓ W• (N ) → Grk−ℓ W• (N ) is an isomorphism for all ℓ ≤ k. Above, Grm W• (N ) = Wm (N )/Wm−1 (N ). Moreover, Qℓ (u, v) = Q(u, N ℓ v)

defines a nondegenerate (−1)k+ℓ –symmetric bilinear form on Grk+ℓ W• (N ). Define Grk+ℓ W• (N )prim = ker {N ℓ+1 : Grk+ℓ W• (N ) → Grk−ℓ−2 W• (N )} ,

for all ℓ ≥ 0. A limiting mixed Hodge structure (or polarized mixed Hodge structure) on D ˇ N ∈ gR and N (F p ) ⊂ F p−1 , the filtration is given by a pair (F • , N )(8) such that F • ∈ D, • F induces a weight m Hodge structure on Grm W• (N ) for all m, and the Hodge structure on Grk+ℓ (W• (N ))prim is polarized by Qℓ for all ℓ ≥ 0. The notions of nilpotent orbit and limiting mixed Hodge structure are closely related. Indeed, they are equivalent when n = 1. ˇ be a Mumford–Tate domain (and Theorem 2.27 (Cattani, Kaplan, Schmid). Let D ⊂ D compact dual) for a Hodge representation of GR . (a) A pair (F • ; N ) forms a one–variable nilpotent orbit if and only if it forms a limiting mixed Hodge structure, [15, Corollary 3.13] and [46, Theorem 6.16]. (b) Given an n–variable nilpotent orbit (F • ; N1 , . . . , Nn ), the weight filtration W• (N ) does not depend on the choice of N ∈ σ [14, Theorem 3.3]. Let W• (σ) denote this common weight filtration. The Deligne bigrading [15, 19] (2.28a)

VC =

M

I p,q

associated with the limiting mixed Hodge structure is given by X F q−j ∩ Wp+q−j−1 . (2.28b) I p,q = F p ∩ Wp+q ∩ F q ∩ Wp+q + j≥1

(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 M M (2.29) Fp = I r,• and Wℓ (σ) = I p,q , r≥p

p+q≤ℓ

and I p,q = I q,p

mod

M

I r,s .

r 0} .

22

KERR AND ROBLES

In particular, the roots of ∆(v) are precisely those positive roots α = mi αi such that mi = 0 and mj > 0 for some j ∈ I(q). Informally, the root α does not involve the simple root αi , but does involve some simple root adjacent to αi . Whence the reflection (i)α = α − 2

(α, αi ) αi (αi , αi )

has the property that the integer −2(α, αi )/(αi , αi ) ≥ 1 for every α ∈ ∆(v). It now follows from the second equality of (2.20) that ̺w (E) > |w| if and only if αi is short. 4. Smooth Schubert VHS A homogeneously embedded, homogeneous submanifold of G/P is a submanifold of the form 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 to G′ /P ′ , where P ′ = G′ ∩ P . Distinguished amongst the homogeneously embedded, homogeneous submanifolds of G/P are the rational homogeneous subvarieties X(D′ ) corresponding to subdiagrams D′ of the Dynkin diagram D of g. The correspondence is as follows: Identifying the nodes of D with the simple roots S of g, to any subdiagram we may associate a semisimple Lie subalgebra g′ ⊂ g by defining g′ to be the subalgebra generated by the root spaces {g±α | α ∈ D′ }. Note that the Dynkin diagram of g′ is naturally identified with the subdiagram D′ . Let G′ ⊂ G be the corresponding closed, connected semisimple Lie subgroup. For such G′ , the subgroup P ′ = G′ ∩ P is a parabolic subgroup. The G′ –orbit of P ∈ G/P is isomorphic to G′ /P ′ and is the homogeneously embedded, rational homogeneous subvariety X(D′ ) ⊂ G/P corresponding to subdiagram D′ ⊂ D. P Define the index set J = {j | gαj 6⊂ g′ }. Let F = j∈J Sj be the corresponding grading element (Section 2.3), and let g = ⊕gℓ be the F–eigenspace decomposition (2.4). Then g′ = [g0 , g0 ] is the semisimple component of the reductive subalgebra g0 . Let Q = PJ be the corresponding parabolic subgroup of G (with Lie algebra q = g0 ⊕ g+ .) It follows from the 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 the discussion above, that a Schubert variety Xw is of the form X(D′ ) if and only if ∆(w) = ∆(g′+ ), where g′ = ⊕g′ℓ is the E–eigenspace decomposition of g′ and ∆(w) is the set (2.15). By Section 2.7, X(D′ ) is a VHS if and only if g′ = g′1 ⊕ g′0 ⊕ g′−1 ; equivalently, X(D′ ) is Hermitian symmetric. The main result of this section is


23

ˇ = G/P be a Schubert variation of Hodge structure. If X is Theorem 4.4. Let X ⊂ D smooth, then X is a product of homogeneously embedded, rational homogeneous subvarieties X(D′ ) ⊂ G/P corresponding to subdiagrams D′ ⊂ D. Moreover, each X(D′ ) is Hermitian symmetric. Since a homogeneously embedded Hermitian symmetric space is necessarily smooth, Theorem 4.4 is equivalent to: A Schubert VHS is smooth if and only if it is a homogeneously embedded Hermitian symmetric space. Theorem 4.4 is proved in Sections 4.1–4.5. ˇ is the compact dual of a Mumford–Tate domain D, Friedman and Laza have When D shown the following [22, Theorem 1]: if Y ⊂ D is horizontal and semi-algebraic (that is, a ˇ with strongly connected component of the intersection with D of a closed subvariety of D), quasi-projective image in an arithmetic-group quotient Γ \ D, then Y is a horizontal (hence Hermitian) Mumford–Tate subdomain. Theorem 4.4 says that if Y is horizontal and “semismooth-Schubert”, then Y is Hermitian; so it is a rather different result from [22], without any reference to Mumford–Tate domains or quotients. However, we do have the ˇ is the compact dual of a Mumford–Tate–Chevalley domain D Corollary 4.5. Suppose D ˇ (Definition 6.17), X ⊂ D is a smooth Schubert VHS, and Y is a nonempty connected component of X ∩ D. Then some GR -translate of Y is a (necessarily horizontal, Hermitian symmetric) connected Mumford–Tate subdomain of D. Proof. Translate Y to pass through the distinguished base point o0 of D (Section 6.4), with corresponding grading element T and Hodge structure ϕ0 : S 1 → GR . By Theorem 4.4, we have Y = (G′R )◦ · o0 .(10) Since g′ = g′1 ⊕ g′0 ⊕ g′−1 decomposes into a direct sum of h ∩ g′ and a subset of h–root spaces, it follows that g′ is stable under T ∈ h. Therefore, ϕ0 (S 1 ) 1 ′ ˜ ′ dfn normalizes G′R , and we set G R = ϕ0 (S ) · GR . By Remarks 6.10 and 6.18, this is underlain ′ ˜ . by a Q-algebraic group G Q Now since ad(T)|g′−1 = −1g′−1 , the (complexified) Q–Lie-algebra-closure of a very general ad(g′R )–translate of T contains g′−1 = ˜g′−1 . Therefore, so does the C–Lie algebra of the ˜ ′′ of a very general G ˜ ′ -conjugate of ϕ0 . It follows that Y = (G ˜ ′′ )◦ ·o0 , Mumford–Tate group G Q R R ˜ ′′ is the Mumford–Tate group of a very general point (i.e., Hodge structure) in Y ; where G Q that is, Y is a connected Mumford–Tate subdomain. ˇ′ ⊂ D ˇ Remark 4.6. (i) The converse of Corollary 4.5 fails dramatically: the compact dual D ′ of a horizontal (Hermitian) Mumford–Tate subdomain D ⊂ D need not be a Schubert ˇ ′ is a maximal integral of the IPR. A large class of examples illustrating variety, even when D this, called enhanced SL2 -orbits, is constructed in Section 6; the simplest one is probably the 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) Mumford–Tate ¯ domains, to an extent which should be describable in terms of the action of Gal (Q/Q) in the Dynkin diagram D. Example 4.7 (Symplectic Grassmannians). In this example we will (a) illustrate the construction of X(D′ ) from D′ ⊂ D, and (b) observe that not every smooth Schubert variety (10)Here the superscript ◦ denotes the connected identity component.

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KERR AND ROBLES

of a rational homogeneous variety is homogeneous; in particular, the assumption that X be a VHS in Theorem 4.4 is essential. Let ν be a non-degenerate, skew-symmetric bilinear form on C2r . Fix i ≤ r. The symplectic grassmannian SG(i, C2r ) = E ∈ Gr(i, C2r ) | ν|E = 0

ˇ = G/P , where G = Aut(C2r , Q) = Sp2r C and P = Pi is a rational homogeneous variety D is the maximal parabolic subgroup associated to the i–th simple root. The Dynkin diagram of G (containing r nodes) is r r ♣ ♣ ♣ r r h r . Fix a Q–isotropic flag F 1 ⊂ F 2 ⊂ · · · ⊂ F 2r = C2r ; here F d is a complex linear subspace of dimension d and ν(F d , F 2r−d ) = 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′ = {αa+1 , . . . , αb−1 }; we have g′ ≃ slb−a C and Xa,b ≃ Gr(i − a, Cb−a ); these Schubert varieties are VHS. In the second case (b = 2r − a), the subdiagram D′ corresponds to the simple roots S′ = {αa−1 , . . . , αr }; we have g′ ≃ sp2(r−a) C and Xa,b = SG(i − a, C2(r−a) ); these Xa,b are VHS if and only if i = r (equivalently, the symplectic grassmannian is a Hermitian symmetric Lagrangian grassmannian). ˇ = SG(4, C16 ); that is, i = 4 and r = 8. The encircled subdiagram We illustrate this for D r

corresponds to g′ = sl6 C and

☛ r ✡

r

r❡ r

r

✟ r r ✠h

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

corresponds to g′ = sp12 C and

r

☛ r ✡

r❡ r

r

✟ rh r ✠

X2,14 = {E ∈ SG(4, C16 ) | F 2 ⊂ E} = SG(2, (F 2 )⊥ /F 2 ) ≃ SG(2, C12 ) . ˇ is a Hermitian symmetric Returning to the general case, we have i = r if and only if D 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 submanifold associated 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. ˇ = G/P with P a Remark 4.8 (Maximal parabolic associated to non–short root). If D maximal parabolic corresponding to a non–short simple root αi , then all smooth Schubert ˇ are homogeneously embedded, rational homogeneous varieties X(D′ ) correvarieties of D sponding to connected subdiagrams D′ ⊂ D containing the i–th node, cf. [27, Proposition 3.7]. For example, the smooth Schubert varieties in the Grassmannian Gr(k, Cn ) are all homogeneous; likewise, the smooth Schubert varieties of the adjoint varieties (Section 2.5) of G = Br , Dr , Er , F4 , G2 are all homogeneous.


25

4.1. Proof of Theorem 4.4: outline. The proposition is proved in three steps. First we reduce 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 is not 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 Schubert VHS. Let w ∈ W p be the associated Weyl group element (Section 2.6), so that X = Xw = Bw−1 o. The condition that the Schubert variety be a VHS is equivalent to ∆(w) ⊂ ∆(g1 ) (Section 2.7). By definition α ∈ ∆(g1 ) if and only if α(E) = 1. Equivalently, α(Si ) = 1 for exactly one i ∈ I, and α(Sj ) = 0 for all other j ∈ I. Therefore, we have a disjoint union G ∆(w) = ∆i (w) , i∈I

where

dfn

∆i (w) = {α ∈ ∆(w) | α(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 ϕ lifts it is a consequence of the fact that X is a VHS, cf. [45, Appendix D].) Whence Y X = Xi . i∈I

Let D denote the Dynkin diagram of G. Given i ∈ I, let D\{I\{i}} denote the subdiagram of D obtained by removing all nodes corresponding to j ∈ I\{i} (and their adjacent edges). 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 to ˇi ⊂ D ˇ be the Gi –orbit of o. Then D ˇ i ≃ Gi /(Gi ∩ P ) is rational homogeneous Di . Let D ˇ i. variety containing Xi as a Schubert subvariety. Moreover, Xi is a VHS for the IPR on D Finally, note that Gi ∩ P is a maximal parabolic subgroup of Gi – the corresponding index set 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 αi denote the associated simple root. 4.3. The case that αi is not a short root. Suppose that αi is not a short root of ˇ = G/Pi is the homogeneously G. By [27, Proposition 3.7], the Schubert variety X ⊂ D embedded, rational homogeneous subvariety corresponding to a subdiagram of D. ˇ = G/Pi is one of the following 4.4. The case that αi is a short root. In this case D five rational homogeneous varieties: ◦ Let ν be a nondegenerate symmetric bilinear form on C2r+1 . Then the orthogonal grassmannian OG(r, C2r+1 ) = E ∈ Gr(r, C2r+1 ) | ν|E = 0

of maximal, ν–isotropic subspaces is the rational homogeneous variety Br /Pr = Spin 2r+1 C/Pr , where Pr is the maximal parabolic associated to the short simple root αr .

26

KERR AND ROBLES

◦ Let ν be a nondegenerate skew-symmetric bilinear form on C2r . Then the symplectic grassmannian SG(i, C2r ) = E ∈ Gr(r, C2r+1 ) | ν|E = 0 of i–dimensional ν–isotropic subspaces is the rational homogeneous variety Cr /Pi = Sp2r C/Pi , where Pi is the maximal parabolic associated to the simple root αi . The simple root is short if and only if i < r. ◦ The exceptional F4 /P3 , F4 /P4 or G2 /P1 . Unfortunately, the Hong–Mok argument does not appear to extend to SG(i, C2r ), F4 /P3 ˇ above, Theorem 4.4 follows or G2 /P1 . Instead, we will see that, for each of the five D from either (i) the Brion–Polo classification of smooth minuscule Schubert varieties, or (ii) existing descriptions of the Schubert VHS, or a combination of both. ˇ = OG(r, C2r+1 ). In this case, D ˇ is minuscule, cf. [5]. Brion and 4.4.1. The case that D Polo have shown that the smooth Schubert varieties of minuscule G/P are homogeneous submanifolds. More precisely, let Q ⊃ B denote the stabilizer of X, cf. Remark 2.13. Then X = 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 algebra q0 , then Q ≃ Q0 ×q+ = Q0 ×exp(q+ ), by [12, Theorem 3.1.3]. Therefore, X = Q·o = Q0 ·o = Q0ss · o, where Q0ss is the closed semisimple Lie subgroup with Lie algebra q0ss = [q0 , q0 ]. The semisimple Q0ss is the subgroup of G associated to the subdiagram D′ = {α ∈ D | gα ∈ q0 } (Section 2.3). Thus, X = X(D′ ) is the homogeneously embedded, rational homogeneous subvariety associated to a subdiagram D′ ⊂ D.

ˇ = SG(i, C2r ). Recall that i < r. In this case, the marked Dynkin 4.4.2. The case that D diagram associated to SG(i, C2r ) is 1r

2r ♣ ♣ ♣ r

ir❡ r ♣ ♣ ♣ r

r h rr

It follows from the proof of [45, Proposition 3.11](11) that there is a unique maximal Schubert variation of Hodge structure; it is the homogeneous submanifold Z ≃ Gr(i, Cr ) associated to the circled subdiagram ✞r ✝

r ♣ ♣ ♣ r

r❡ r ♣ ♣ ♣ r

r☎ h 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 homogeneous subvarieties corresponding to connected subdiagrams of the the marked Dynkin diagram for Gr(i, Cr ) that contain the i–th node, cf. [5, Section 9.3]. (Alternatively, this follows from the Brion–Polo result, since Gr(i, Cr ) is minuscule, or Remark 4.8.) ˇ = F4 /P3 . By [45, Example 5.16], the only Schubert variations of 4.4.3. The case that D ˇ and) the P1 ⊂ P2 corresponding to the two subdiaHodge structure are (the trivial o ∈ D grams s

si

☛ ✟ s❣ s ✡ ✠

s

si

☛ s❣ ✡

s

✟ ✠

Therefore, any smooth Schubert VHS in F4 /P3 is a homogeneously embedded, rational homogeneous 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.


27

ˇ = F4 /P4 . By [45, Example 5.17], the only Schubert variations of 4.4.4. The case that D ˇ and) the P1 ⊂ P2 corresponding to the two subdiaHodge structure are (the trivial o ∈ D grams s

☛ ✟ si s s❣ ✡ ✠

s

si

☛ s ✡

✟ s❣ ✠

Therefore, any smooth Schubert VHS in F4 /P4 is a homogeneously embedded, rational homogeneous subvariety corresponding to a subdiagram of D. ˇ = G2 /P1 . This variety is the quadric hypersurface Q5 ⊂ P6 . By [45, 4.4.5. The case D ˇ and) the smooth P1 ⊂ D ˇ Example 5.30], the only Schubert VHS are (the trivial o ∈ D which is the homogeneous submanifold corresponding to the circled subdiagram ☛ ✟ s❢ h s ✡ ✠

Therefore, any smooth Schubert VHS in G2 /P1 is a homogeneously embedded, rational homogeneous subvariety corresponding to a subdiagram of D. 4.5. Fini. We have shown, in Sections 4.2–4.4, that given any rational homogeneous variety ˇ = G/P (here the parabolic is arbitrary – P need not be maximal) any smooth Schubert D ˇ is a product homogeneously embedded, rational homogeneous subvarieties VHS X ⊂ D Xi = X(Di ) corresponding to a subdiagram of Di ⊂ D. The condition that X (and therefore each Xi ) be a VHS forces Xi to be Hermitian symmetric. This completes the proof 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 was used 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. Griffiths–Yukawa couplings and lines on G/P A historically important special instance of the double fibrations introduced in Section ´ Cartan’s two realizations of G2 as a transformation group of 3 arose in connection with E. a complex 5-manifold preserving a nontrivial distribution [11, 13]. Indeed, they are related by the correspondence:

G2 /B ✠

(5.1)

G2 /P1

❅ ❘ ❅

G2 /P2

Each of the varieties of (5.1) may be realized as the compact dual of a Mumford–Tate domain with bracket-generating horizontal distribution [24, §IV.F]. This section is motivated by an ˇ = G2 /P2 and its observation about the Hodge theory associated to the adjoint variety D contact (hyperplane) distribution, which we now briefly explain. Suppose V = ⊕nj=0 V n−j,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 Gauss–Manin connection, the IPR forces the composition D

OU (V n,0 ) ֒→ OU (V) → OU (V) ։ OU (V 0,n )

to be OS -linear (so that only the symbol of D matters). One therefore obtains a well-defined linear mapping Symn Ts S → Hom(Vsn,0 , Vs0,n ) ∼ = (Vs0,n )⊗2

(5.2)

for each s ∈ S, which we shall call the Griffiths–Yukawa coupling. When V comes from a family of varieties, it may be of particular interest to study the geometry of subfamilies for which (5.2) vanishes. Recent work of Katz [30] suggests that motives with Hodge numbers (2, 3, 2) and Mumford–Tate group G2 , hence admitting a period map into a quotient of ˇ arise from certain families of elliptically fibered surfaces. We D = G2 (R)/(P2 ∩ P2 )(R) ⊂ D, shall not pursue the (algebro-)geometric angle here, but instead look at what representation theory 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 Symn g−1 → Hom(V n,0 , V 0,n )

(5.3)

which pulls back to (5.2) under the period map. If g = g2 and T is the adjoint grading element, which is defined by α1 (T) = 0 and α2 (T) = 1 on the simple roots, and V is the 7-dimensional irreducible representation (with n = 2 and V 2,0 = Che1 , e2 i), one may ask in which horizontal tangent directions (5.3) vanishes. Writing dfn

ξ = ξ0 x−α2 + ξ1 x−α1 −α2 + ξ2 x−2α1 −α2 + ξ3 x−3α1 −α2 ∈ g−1 , and {e∗1 , e∗2 } for the dual basis of V 0,2 , a straightforward computation (with appropriate normalizations) gives −2ξ1 ξ3 + 2ξ22 ξ1 ξ2 − ξ0 ξ3 2 (5.4) . e∗ [ξ ]e = ξ1 ξ2 − ξ0 ξ3 −2ξ0 ξ2 + 2ξ12 The common vanishing locus of the matrix entries in (5.4) yields a twisted cubic curve ˇ this recovers the field of cubic cones v3 (P1 ) ⊂ Pg−1 , cf. [25]. Upon varying o ∈ D (or D), −1 ˇ in the contact planes go ⊂ To D preserved by G2 , cf. [11]. ˇ in each cubic curve are precisely Furthermore, it is known that the lines ℓ ⊂ g−1 ⊂ To D ˇ under its minimal homogeneous embedding in Pg. To the lines through o contained in D, see this, we apply (adξ)2 to the highest weight vector v = x3α1 +2α2 ∈ g2 to check vanishing of the second fundamental form: (adξ)2 v = (−2ξ0 ξ2 + 2ξ12 )xα1 + (ξ0 ξ3 − ξ1 ξ2 )H3α1 +2α2 + (−2ξ1 ξ3 + 2ξ22 )x−α1 ∈ g0 . Hence we get an identification of the variety of horizontal lines Co (cf. (3.1)) with the “variety of Yukawa vanishing directions”, which provides a homogeneous-space description of the latter and a Hodge–theoretic interpretation of Co . Moreover, the argument produces equations for both as projective varieties in Pg−1 . The purpose of this section is to discern the degree to which both phenomena generalize.


29

ˇ = G/P . (We do not assume that 5.1. Yukawa vanishes along horizontal lines. Fix D P is maximal here.) Let E be the grading element (2.7) associated with the parabolic p, and let V be the irreducible G–module giving the minimal homogeneous embedding G/P ֒→ PV (Section 2.4). Fix a G–module U , and suppose that the E–eigenspace decomposition is of the form (5.5)

U = U m/2 ⊕ U m/2−1 ⊕ · · · ⊕ U 1−m/2 ⊕ U −m/2 ,

with 0 ≤ m ∈ Z, U m/2 6= 0 and (U a )∗ ≃ U −a as g0 –modules.

Remark 5.6. In general, the E–eigenvalues of U are rational. However, if G is Q–algebraic and E is the grading element Tϕ associated to a Hodge representation (G, UQ , ϕ), cf. [45, Section 2.3], then the E–eigenspace decomposition of U = UQ ⊗ C is of the form (5.5).

Any element of Symm (g−1 ) naturally defines a G0 –module map U m/2 → U −m/2 . Explicitly, given ξ ∈ g−1 and u ∈ U m/2 , the relation (2.8) implies ξ m (u) ∈ U −m/2 . Thus, we have a G0 –module map Ψ : Symm (g−1 ) → Hom(U m/2 , U −m/2 ) , which is the Griffiths–Yukawa coupling from above. Define (5.7a)

dfn

YU = {[ξ] ∈ P(g−1 ) | Ψ(ξ m ) = 0} .

In particular, given 0 6= ξ ∈ g−1 , (5.7b)

[ξ] ∈ YU

if and only if

ξ m |U m/2 = 0 .

Observe that YU is a closed, G0 –invariant subvariety of Pg−1 . Our first main goal in this section is to understand the relationship between YU and Cõ . Theorem 5.8. If the the kernel of the Griffiths–Yukawa coupling is nonempty, then it contains the (tangent directions to) lines through o = P/P in the minimal homogeneous embedding G/P ֒→ PV that are VHS. That is, if YU 6= ∅, then (5.9)

Co ⊂ Y U .

The theorem is proved in Section 5.4. The equations cutting out Co and YU are given in Section 5.2. Example 5.19 is a case in which containment is strict in (5.9). A large class of examples for which equality holds is the following. Take V = g, ˇ so that P 2.5). The associated grading element is P D ⊂i Pg is the adjoint variety of G (Section E = i∈I S where the set I is defined by α ˜ = i∈I ωi , cf. Table 2.1. Further assume that m = 2; equivalently, the E–eigenspace decomposition (5.5) of U is

(5.10)

U = U 1 ⊕ U 0 ⊕ U −1 .

Theorem 5.11. Let V = g, so that G/P ֒→ Pg is the adjoint variety of G. Assume this homogeneous embedding of G/P is minimal. Suppose that m = 2, so that (5.10) holds. If U 1 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 for the simple Lie algebra g = sp2r C — it is the second Veronese embedding v2 (P2r−1 ) ⊂ P Sym2 C2r . As such it contains no lines.

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KERR AND ROBLES

Example 5.17 is a case in which m = 2 and Co = YU despite the fact that the hypotheses of 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 element associated 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. Let U be an irreducible representation of highest weight λ. Then λ(E) = 1 if and only if λ is among those listed below. V (a) g = slr+1 C and E = S1 + Sr : λ = ωi , for any 1 ≤ i ≤ r; in this case U = i Cr+1 . (b) g = so2r+1 C and E = S2 : either λ = ω1 , in which case U = C2r+1 is the standard representation; or λ = ωr , in which case U is the spin representation. (c) g = sp2r C and E = S1 : λ = ωi , for any 1 ≤ i ≤ r. (d) g = so2r C and E = S2 : either λ = ω1 , in which case U = C2r is the standard representation; or λ = ωr−1 , ωr , in which case U is one of the spin representations. (e) g = e6 and E = S2 : λ = ω1 , ω6 . g = e7 and E = S1 : λ = ω7 . (In the case that g = e8 and E = S8 , we have λ(E) > 1 for all λ.) (f) g = f4 and E = S1 : λ = ω4 . (g) g = g2 and E = S2 : λ = ω1 and U = C7 is the standard representation. In each of these cases, we also have λ∗ (E) = 1. In Section 5.3 we determine to which cases in Lemma 5.13 Theorem 5.11 applies. (By Remark 5.12, the theorem can not be applied to the symplectic adjoint variety.) Proof. As a highest weight λ is of the form λ = λi ωi with 0 ≤ λi ωi . The tables of [9] express the ωi as linear combinations of the simple roots αj . The lemma follows. 5.2. Equations cutting out Co and YU . We begin with YU . Let Ψ∗ : Hom(U −m/2 , U m/2 ) → Symm (g−1 )∗

denote the dual map. To be explicit: let ν ⊗ u ∈ U m/2 ⊗ (U −m/2 )∗ = Hom(U −m/2 , U m/2 ), then the polynomial Φ∗ (ν ⊗ u) ∈ Symm (g−1 )∗ is given by Φ∗ (ν ⊗ u)(ξ) = ν(ξ m (u)), for any ξ ∈ g−1 . The image (5.14)

im Ψ∗ ⊂ Symm (g−1 )∗ is the set of polynomials defining YU ⊂ Pg−1 .

Turning to Co , the following lemma characterizes Co as the set of lines in g−1 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 be ˇ Then [ξ] ∈ Co if and only if ξ 2 (v) = 0. a highest weight vector. Let ξ ∈ g−1 ⊂ g− ≃ To D. Before proving the lemma, we note the following corollary which describes the equations cutting out the lines Co . Recall the E–eigenspace decomposition V = V h ⊕ V h−1 ⊕ V h−2 ⊕ · · ·


31

of Section 2.3.2, and that V h is one–dimensional. Fix a highest weight vector 0 6= v ∈ V h . Given ν ∈ (V h−2 )∗ define pν ∈ Sym2 (g−1 )∗ by pν (ξ) = ν(ξ 2 v). This defines a g0 –module map Φ∗ : (V h−2 )∗ ⊗ V h → Sym2 (g−1 )∗ . By Lemma 5.15, this map has the property that the image im Φ∗ ⊂ Sym2 (g−1 )∗ is the set of polynomials defining Co ⊂ Pg−1 .

(5.16)

In particular, if G/P is one of the fundamental adjoint varieties, then Co is the variety of lines through a fixed point o (cf. 3.16); and (5.16) gives equations for Co (12) as a projective variety. Proof of Lemma 5.15. Define T = spanC {v, (g− )v} ⊂ V . Then T is the embedded tangent ˇ ⊂ V over D ˇ at v. (Note that T depends only on the highest weight space to the cone C(D) h line o = V , not on our choice of v ∈ o.) ˇ is The unique P1 ⊂ PV containing o and satisfying ξ ∈ To P1 is P spanC {v, ξ(v)}. Since D 1 ˇ if and only cut out by degree two equations [5, Section 2.10], the line P is contained in D ˇ if the line osculates with D to second order; equivalently, the second fundamental form F 2 vanishes at ξ. That is, Co = P{ξ ∈ g−1 | F 2 (ξ) = 0} . Since F 2 (ξ) = ξ 2 (v)/T ∈ V /T , the vanishing F 2 (ξ) = 0 is equivalent to ξ 2 (v) ∈ T . If ξ 2 (v) = 0, then it is immediate that F 2 (ξ) = 0. Whence, [ξ] ∈ Co . Conversely, suppose that [ξ] ∈ Co . Then [ξ] is necessarily contained in one of the Co,i of (3.7). Since Co,i is the G0 –orbit of g−αi ∈ Pg−1 , we may assume without loss of generality that ξ ∈ g−αi . Let µ ˜ denote the highest weight of V . Then ξ 2 (v) is a weight vector of V for the weight µ ˜ − 2αi . Since [ξ] ∈ Co , and this variety is the zero locus of the second fundamental form, ξ 2 (v) is necessarily of the form av + ζ(v) for some a ∈ C and ζ ∈ g− . P β Write ζ = ζ with ζ β ∈ g−β . Each ζ β (v) is a weight vector of V for the weight µ ˜ − β. Therefore, the condition that ξ 2 (v) = av + ζ(v) be a weight vector for µ ˜ − 2αi forces a = 0 and all but at most one ζ β (v) to vanish. If ζ β (v) 6= 0, then µ ˜ − 2αi = µ ˜ − β forcing β = 2αi . 2 This is not possible, since 2αi is not a root. Therefore, ξ (v) = 0. 5.3. Examples. Example 5.17 (Lemma 5.13(a)). Let W = Cr+1 be the standard representation of g = slr+1 C. Let E = S1 + Sr . Then the E–eigenspace decomposition of W is W = W 1 ⊕ W 0 ⊕ W −1 ,

with dimC W ±1 = 1 .

Whence the E–eigenspace decomposition of U =

Vi

U = U 1 ⊕ U 0 ⊕ U −1

W is given by

Vi−1

U1 = W 1 ⊗ ( U0 =

W1 ⊗ (

Vi−1

U −1 = (

W 0) ,

Vi−2

W 0 ) ⊗ W −1

W 0 ) ⊗ W −1 .

(12)and for Y , when Theorem 5.11 applies U

⊕

Vi

W0 ,

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KERR AND ROBLES

The Levi factor is

g0 = C2 ⊕ slr−1 C

with C2 = span{S1 , Sr }. We claim that U 1 is not a faithful representation of g0 . To see this, first note that the central element S1 acts on W 1 and W 0 ⊕ W −1 by the eigenvalues r/(r + 1) and −1/(r + 1), respectively; the central element Sr acts on W 1 ⊕ W 0 and W −1 by the eigenvalues 1/(r + 1) and −r/(r+1), respectively. It follows that S1 acts on U 1 by the eigenvalue (r+1−i)/(r+1), and Sr acts by the eigenvalue i/(r + 1). Therefore, iS1 − (r + 1 − i)Sr ∈ C2 acts trivially on U 1 . Whence g0 does not act faithfully on U 1 , and we may not apply Theorem 5.11. Nonetheless, we claim that Co = Y U .

(5.18)

To see this, fix a basis {e0 , e1 , . . . , er } of W so that W 1 = span{e0 } ,

W 0 = span{e1 , . . . , er−1 }

and W −1 = span{er } .

Let {e0 , . . . , er } be the dual basis of W ∗ , and {eij = ej ⊗ ei | 0 ≤ i, j ≤ r} the induced basis of End(W ). Then g−1 = span{e0a , ear+1 | 1 ≤ a ≤ r − 1} and U 1 = span{e0 ∧ ea1 ∧ · · · ∧ eai−1 | 1 ≤ a1 , . . . , ai−1 ≤ r − 1} . The action of ξ = ξ b e0b + ξb ebr+1 ∈ g−1 on u = e0 ∧ ea1 ∧ · · · ∧ eai−1 ∈ U 1 is given by ξ(u) = ξ b eb ∧ ea1 ∧ · · · ∧ eai−1 − and

i−1 X

ξ 2 (u) = 2

s=1

(−1)s ξas e0 ∧ er+1 ∧ ea1 ∧ · · · ebas · · · ∧ eai−1

i−1 X X

b6∈{at } s=1

+(−1)i

i−1 X s=1

i

−(−1) Whence

ξ b ξas ea1 ∧ · · · eâs · · · ∧ eai−1 ∧ eb ∧ er+1 ξas ξ as ea1 ∧ · · · eai−1 ∧ er+1

X

b6∈{at }

ξb ξ b ea1 ∧ · · · eai−1 ∧ er+1 .

YU = {ξ a = 0 | 1 ≤ a ≤ r − 1} ⊔ {ξa = 0 | 1 ≤ a ≤ r − 1} . Next, we determine Co by applying Lemma 5.15 to the highest root vector v = er+1 0 . We find r−1 r−1 X X ξ a ξa (e00 + er+1 ) + ξ b ξc ecb . ξ 2 (v) = r+1 a=1

Whence

a,b=1

Co = {ξ a = 0 | 1 ≤ a ≤ r − 1} ⊔ {ξa = 0 | 1 ≤ a ≤ r − 1}


33

is a disjoint union Pr−2 ⊔ Pr−2 of linear spaces, and (5.18) follows. Example 5.19 (Lemma 5.13(b,d) with the standard representation). Let g = som C and U = Cm . Take the parabolic p determined by the grading element E = S2 , cf. (2.6). Then ˇ = G/P is the adjoint variety. Note that D g ≃

V2

U,

as a g–module. The E–eigenspace decomposition of U is U = U 1 ⊕ U 0 ⊕ U −1 .

(5.20)

Moreover, U is a real Hodge representation with Hodge numbers h = (2, m − 4, 2). In this case g0 ≃ C ⊕ sl2 C ⊕ som−4 C,

(5.21)

with center C = span{E}, and semisimple factor g0ss = [g0 , g0 ] = sl2 C ⊕ som−4 C. As a g0 –module U 1 ≃ C2 . In particular, the factor som−4 C ⊂ g0ss acts trivially on U 1 . Therefore g0 –representation on U 1 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 , . . . , em−2 } and {em−1 , em } of U 1 , U 0 and U −1 , respectively, which satisfy ν(ej , ek ) = 1 if j + k = m + 1 and zero otherwise. Following the standard classification of complex simple Lie algebras, we will assume that m ≥ 7. By definition [ξ] ∈ YU if and only if ξ 2 U 1 = 0, cf. (5.7); equivalently, [ξ] ∈ YU

(5.22)

if and only if

0 = ξ 2 ea

for a = 1, 2 .

At this point, making use of a coarse dimension count, we see that Co 6= YU . Note that dim Pg−1 = 2m − 9. The characterization (5.22) imposes four constrains on YU , implying dim YU ≥ 2m − 13. On the other hand, from Table 3.2 we see that dim Co = m − 7. 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 = em−1 − em 2 . 1 Let ξ ∈ g−1 . By Lemma 5.15, the point [ξ] ∈ Pg−1 lies in Co if and only if 0 = ξ 2 (v) = (ξ 2 e1 ) ⊗ em−1 + 2(ξe1 ) ⊗ (ξem−1 ) + e1 ⊗ (ξ 2 em−1 ) − (ξ 2 e2 ) ⊗ em − 2(ξe2 ) ⊗ (ξem ) − e2 ⊗ (ξ 2 em ) .

Thus, (5.23)

[ξ] ∈ Co

if and only if

(

0 = ξ 2 ea , ξ 2 es , 0 = (ξe1 ) ⊗

a = 1, 2 , s = m − 1, m ,

(ξem−1 )

− (ξe2 ) ⊗ (ξem ) .

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KERR AND ROBLES

We will see that the four equations ξ 2 es = 0 of (5.23) are redundant – they are satisfied by YU – and Co is cut out from YU by the 21 (m − 4)(m − 5) equations (ξe1 ) ⊗ (ξem−1 ) − (ξe2 ) ⊗ (ξem ) = 0. Write ξ as m+1−k m+1−s k ξ = ξaj (eaj − em+1−j m+1−a ) = ξm+1−s (em+1−k − es ) .

The sums above are over 3 ≤ j, k ≤ m − 2. Then

ξea = ξaj ej = ξam+1−j em+1−j ,

(5.24) and

ξ 2 ea = −

X

ξam+1−j ξbj em+1−b .

b,j

Here the sum is over 1 ≤ b ≤ 2 and 3 ≤ j ≤ m − 2. Therefore, X (5.25) [ξ] ∈ YU if and only if 0 = ξam+1−j ξbj for a = 1, 2 . j

We compute

(5.26) and

ξ es = − ξ 2 es = −

X

m+1−j j ξm+1−s e

j

X

m+1−j j b ξm+1−s ξb e

j,s

Substituting a for m + 1 − s, we see from (5.25) that (5.27)

[ξ] ∈ Y

implies

Finally, from (5.24) and (5.26), we compute (ξe1 ) ⊗ (ξem−1 ) − (ξe2 ) ⊗ (ξem ) = Whence (5.28)

(ξe1 ) ⊗ (ξem−1 ) − (ξe2 ) ⊗ (ξem ) = 0

ξ 2 es = 0 . X j . ξ2 ξ1k − ξ1j ξ2k em+1−k j j,k

if and only if

ξ2j ξ1k − ξ1j ξ2k = 0

for all 3 ≤ j, k ≤ m−2. It follows from (5.22), (5.23) and (5.27) that (5.28) are the equations cutting Co out from YU .

Example 5.29 (Lemma 5.13(b,d) with a spin representation). We have (5.21). If U is a spin representation of g, then U 1 is a spin representation of the factor som−4 C ⊂ g0 , while sl2 C ⊂ g0 acts trivially. The latter implies U 1 is not a faithful g0 representation – Theorem 5.11 does not apply. Example 5.30 (Lemma 5.13(e) with g = e6 ). Let U be the irreducible e6 (C)–representation of highest weight ω1 . Then U∗ is the irreducible representation of highest weight ω6 . In particular, U 6≃ U∗ , so that U is complex. Whence there exists a real form UR of U = U⊕U∗ . Moreover, E = S2 induces the structure of a Hodge representation of GR = E II on UR with Hodge numbers h = (12, 30, 12). In this case g0 = C ⊕ sl6 C; here C = span{E} is the center of g0 , and sl6 C the semisimple factor [g0 , g0 ]. We have U 1 = C6 ⊕ (C6 )∗ as a sl6 C–module. This representation is faithful, and Theorem 5.11 applies.


35

Example 5.31 (Lemma 5.13(e) with g = e7 ). The irreducible g–module U of highest weight ω7 is self–dual and of dimension 56. As a gR –representation U is quaternionic: there is a real form UR of U = U ⊕ U∗ with the property that UR is an irreducible gR –representation. Moreover, the grading element E = S1 induces a Hodge structure on UR with Hodge numbers h = (24, 64, 24). In this case g0 = C ⊕ so12 C with center C = span{E}. Additionally, U 1 ≃ C12 ⊕ C12 is two copies of the standard representation of g0ss = so12 C, and is therefore a faithful g0 –representation. Thus Theorem 5.11 applies. Example 5.32 (Lemma 5.13(f)). The irreducible g–module U of highest weight ω4 is self– dual and of dimension 26. As a gR –representation U is real: there is a real form UR of U . Moreover, the grading element E = S1 induces a Hodge structure on UR with Hodge numbers h = (6, 14, 6). In this case g0 = C ⊕ sp6 C, and U 1 = C6 is the standard representation of the semisimple factor g0ss = sp6 C. 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 ˇ = G/P is the adjoint variety, and let U = C7 be the standard representation. The that D E–eigenspace decomposition of U is of the form (5.20), and U is a real Hodge representation with Hodge numbers h = (2, 3, 2). In this case g0 = C ⊕ sl2 C , with C = span{E} the center and g0ss = [g0 , g0 ] = sl2 C the semisimple factor. Moreover, U 1 ≃ C2 as a g0ss –representation. Since the center acts on U 1 by the eigenvalue 1, it follows that U 1 is a faithful representation of g0 . Whence Theorem 5.11 yields Co = Y U .

(5.34)

We finish this example by showing how (5.34) may be obtained (independently of Theorem 5.11) by working with the equations of Section 5.2, but without the explicit computations at the beginning of the section. We anticipate that this approach may be used to determine 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 decomposes V2

U = U ⊕ g2

as a g2 –module. Given a basis {e1 , e2 } of U 1 , the product v = e1 ∧ e2

spans the highest weight (root) line of g2 ⊂

V2

U . Given ξ ∈ g−1 , notice that

0 = ξ 2 (v) = (ξ 2 e1 ) ∧ e2 + 2(ξe1 ) ∧ (ξe2 ) + e1 ∧ (ξ 2 e2 ) .

From (5.7) and Lemma 5.15 we see that

Co = {[ξ] ∈ YU | (ξe1 ) ∧ (ξe2 ) = 0} .

As ξe1 and ξe2 both lie in U 0 and the later is of dimension three, we see that the condition that [ξ] ∈ YU lie in Co is at most three additional conditions on ξ. It may be the case that that these conditions are empty: that is, every [ξ] ∈ YU satisfies (ξe1 ) ∧ (ξe2 ) = 0;

36

KERR AND ROBLES

equivalently, (5.34) holds. To see that this is indeed the case (without invoking Theorem 5.11) 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 g0 –invariant subspaces of Sym2 (g−1 )∗ . We have g0 = gl2 C, g0ss = sl2 C and g−1 = Sym3 C2 as a g0ss –module. Whence (5.35)

Sym2 (g−1 )∗ = Sym2 C2 ⊕ Sym6 C2 .

If Co ( YU , then 0 = 6 I2 (YU ) ( I2 (Co ). Given (5.35) this forces I2 (Co ) = Sym2 (g−1 )∗ ; equivalently, Co = ∅, a contradiction. This establishes (5.34). 5.4. Proof of Theorem 5.8. Let I = {i1 , . . . , iℓ } be the index set (2.1) associated with the parabolic subalgebra p. The proof will proceed in two steps. We begin with the case that p is a maximal parabolic (equivalently |I| = 1); this forms the basis for the general argument. ˇ = G/P and that P is a maximal parabolic. If YU 6= ∅, then Lemma 5.36. Suppose that D YU is connected and Co ⊂ Y U .

Proof. The parabolic P is maximal if and only if g−1 is an irreducible g0 –module. In this case, Co ⊂ Pg−1 is the unique closed G0 –orbit (Section 3.1). It is clear from the definition (5.7) that YU is closed and preserved under the action of G0 . Therefore, each connected component of YU contains a closed G0 –orbit. We now turn to the case that |I| > 1. First, we review the relationship between the lines on G/P ֒→ PV and G/Pi ֒→ PVωi , i ∈ I. The lines Co on the former are described in Section 3.2. For the latter, given i ∈ I, let Pi ⊂ G be the maximal parabolic associated with ˇ i ⊂ PVω be the image of the minimal homogeneous embedding the index set {i}. Let D i ˇ i ) be the lines on D ˇ i passing through oi = Pi /Pi . Then the Co,i G/Pi ֒→ PVωi , and let Coi (D of (3.5) is ˇ i ) , ξ ∈ pj ∀j ∈ I \ {i} . (5.37) Co,i = [ξ] ∈ Pg−1 | [ξ] ∈ Coi (D P Recall that E = i∈I Si , and U m/2 is the eigenspace of the largest E–eigenvalue m/2 on U . Let mi /2 be the largest Si –eigenvalue of U , and let Ui ⊂ U be the corresponding eigenspace. Then \ X Ui . m = mi and U m/2 = i∈I

i∈I

ˇ i ). Then (5.7) and Lemma (5.36) imply that ξ mi | = 0. Since mi ≤ m and Let [ξ] ∈ Coi (D Ui U m/2 ⊂ Ui , it follows that ξ m |U m/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) To that end, let [ξ] ∈ YU . Equivalently, (5.39)

Y U ⊂ Co .

ξ 2 (u) = 0

for all

u ∈ U1 .


37

Fix a highest root vector 0 6= v ∈ gα˜ . Given Lemma 5.15, to establish (5.38) it suffices to show that ad2ξ (v) = 0 .

(5.40) As an operator on U

ad2ξ (v) = ξξv − 2ξvξ + vξξ . As an element of g−1 , the endomorphism ξ lowers eigenvalues by one; that is ξ maps U a into U a−1 . Likewise, as an element of g2 = gα˜ , the root vector v maps U a into U a+2 . It then follows from (5.10) that (5.41)

ad2ξ (v) U 1 = vξξ|U 1

(5.39)

=

0.

On the other hand ad2ξ (v) ∈ g0 . By assumption U 1 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 GR –orbits; each orbit ˇ where N is an element of will contain the image of a na¨ıve limit mapping Φ∞ : B(N) → D, a nilpotent cone σ (cf. Section 7.1). ˇ = G/P (Section 2.1), and let E be the associated Fix a rational homogeneous variety D grading element (2.7). Associated with E is a rational form gQ of g (Section 6.3). The ˇ as rational form is equipped with a Hodge structure that realizes an open GR –orbit D ⊂ D a Mumford–Tate domain (Section 6.4). We will construct a sequence of GR –orbits {Oj }sj=0 ˇ such that O0 is the open orbit D and in D (6.1)

Oj+1 ⊂ ∂Oj .

We will see that these orbits are rational, polarized and (weakly) cuspidal in the sense of [32] (Remarks 6.40 and 6.41). The orbits arise as follows. We begin with the identity ˇ By definition O0 = D is the GR –orbit of o0 , cf. Section 6.4. We will coset o0 = P/P ∈ D. then define oj = gj (o0 ), where gj ∈ G and cj = Adgj is a composition of Cayley transforms (Section 6.6). These Cayley transforms are determined by a suitable set (6.29) of strongly orthogonal noncompact roots. Remark 6.2 (Closed orbits). In order for Os to be the unique closed orbit, it is necessary (but not sufficient) that s be the real rank of gR . Remark 6.3 (Maximal parabolics). In the case that P is maximal (Section 6.9) we will see that, with a few exceptions, there exists a sequence {Oj }sj=0 with s equal the real rank of gR (Lemma 6.60). The construction is given in Section 6.8. It is first necessary to review the representation theory underlying the construction.

38

KERR AND ROBLES

6.2. The elements Hα ∈ h. Given a root α, define Hα ∈ h by Hα ∈ [gα , g−α ] and α(Hα ) = 2. Then (6.4)

2

(β, α) = β(Hα ) ∈ Z (α, α)

for all α, β ∈ ∆ ,

cf. [23].(13) In particular, Hα is a grading element. Indeed, if 0 ≤ mα , nα ∈ Z are defined by the conditions that β − mα α , . . . , β + n α α

is the α–string through β, then mα − nα = β(Hα ), cf. [34, Proposition 2.29]. Moreover, if rα denotes the the reflection in the root α, then rα (β) = β − β(Hα ) α .

(6.5)

6.3. A rational form. Given a grading element T, we may define an integral structure on g as follows. The

{Hαi

Fix a Cartan subalgebra h ⊂ g.

| 1 ≤ i ≤ r} span h. Complete this to a Chevalley basis {xα | α ∈ ∆(h)} ∪ {Hα1 , . . . , Hαr }

of g, cf. [28, Section 25.2]. That is, xα ∈ gα and the following properties hold: the Killing form satisfies (Hα , Hα ) > 0 and (xα , x−α ) > 0 for all α ∈ ∆; and the Lie bracket satisfies ◦ [xα , x−α ] = Hα ∈ spanZ {Hα1 , . . . , Hαr }, for all α ∈ ∆(h), ◦ [Hβ , xα ] ∈ Zxα , for all α, β ∈ ∆(h), ◦ if α, β, α + β ∈ ∆ and [xα , xβ ] = cα,β xα+β , then c−α,−β = −cα,β ∈ Z.

We emphasize that the structure coefficients with respect to the Chevalley basis are all integers. Fix dfn

i =

√

−1 ,

and set hj

dfn

uα

dfn

vα

=

=

dfn

=

i Hαj , ( xα − x−α

(

i(xα

if α(T) is even,

x−α )

if α(T) is odd;

i(xα + x−α )

if α(T) is even,

xα

if α(T) is odd.

−

+

x−α

Define (6.6)

gZ = kZ ⊕ k⊥ Z

(13)The reader consulting other references should beware that the H of [16, 34] is not our Hα . To be α 2 precise, Hα = (α,α) Hα is the Zα of [16] and the Hα′ of [34]. However, our Hα is the Hα of [23], and the hα of [28]. (Ha ha!)


39

by (6.7)

kZ k⊥ Z

dfn

=

dfn

=

spanZ {hj | 1 ≤ j ≤ r} ∪ spanZ {uα , v α | α(T) even} , spanZ {uα , v α | α(T) odd} .

It is a straightforward exercise to confirm that [gZ , gZ ] ⊂ gZ and g = gZ ⊗ C; that is, gZ is an integral form of g. It follows that the Killing form B : gZ × gZ → Z is defined over Z. Likewise, ⊥ ⊥ [kZ , kZ ] ⊂ kZ , [kZ , k⊥ and [k⊥ Z ] ⊂ kZ Z , kZ ] ⊂ kZ ,

and one may confirm that the Killing form is negative definitive on kZ and positive definite on k⊥ Z , so that (6.6) is a Cartan decomposition. Let θ : gZ → gZ denote the corresponding Cartan involution; that is, θ|kZ =

(6.8)

1 and

θ|k⊥ = −1 . Z

dfn

Definition 6.9. Let GQ be the Q–form of GR with Lie algebra gQ = gZ ⊗ Q. When a Q-algebraic group arises in this way (from the above construction), we will call it a Mumford–Tate–Chevalley (MTC) group. Remark 6.10. In general, if GQ is a (Q-algebraic) Mumford–Tate group of a Q-Hodge structure underlying a semisimple real Lie group GR , it need not be MTC. Indeed, the above construction implies that MTC groups are split over Q(i).(14) Unlike Q–Chevalley groups, they need not be split over Q, since GR need not be split over R; for instance, apply the construction to SU(2, 1). However, we do have that gα ⊕ g−α is defined over Q for each α ∈ ∆. A Cartan subalgebra of gR is given (over Z) by (6.11)

dfn

hZ = h ∩ gZ = spanZ {i H1 , . . . , i Hr } ⊂ kZ .

From αi (Sj ) = δji and αi (Hβ ) ∈ Z, we see that the Si are Q–linear combinations of the Hj . Therefore, the i Si are defined over Q. In particular, the rational form is (6.12)

hQ = spanQ {i S1 , . . . , i Sr } ⊂ kQ .

Note that the real form dfn

t = hR

is compact.

Let GR ⊂ G be the real form of G with Lie algebra gR . For later use, we introduce the subalgebra (6.13)

dfn

slα2 (Z) = spanZ {uα , i Hα , v α } ⊂ gZ .

Let (6.14)

SLα2 (R) ⊂ GR

and SLα2 (C) ⊂ G

be the connected Lie subgroups with Lie algebras slα2 (R) and slα2 (C), respectively. (14)Of course, the construction could be modified to replace Q(i) by any imaginary quadratic field.

40

KERR AND ROBLES

Example 6.15 (g = sl2 C). Consider the case that g = sl2 C, and let T = S1 . We may take the Chevalley basis to be 0 1 1 0 0 0 xα = , Hα = , x−α = . 0 0 0 −1 1 0 Then the rational form (6.6) is spanned by 0 1 1 0 α 1 u = i , h = i , −1 0 0 −1

v

a

=

0 1 1 0

.

Observe that the real form defined by (6.6) is gR = su(1, 1) = X ∈ sl2 C | X t Q + QX = 0 , 1 0 where Q = . This is the Lie algebra of the group 0 −1 GR = SU(1, 1) = {A ∈ SL2 C | A∗ QA = Q} .

preserving the Hermitian form z¯t Qw.

6.4. Hodge structure on gZ . If g = ⊕gp is the T–eigenspace decomposition (2.4), then (6.7) yields (6.16)

dfn

k = kZ ⊗ C = geven

dfn

odd and k⊥ = k⊥ . Z ⊗C = g

Since −(u, θu) > 0 for all 0 6= u ∈ g, it follows that gp,−p = gp defines a weight zero, −(·, ·)–polarized Hodge structure on g. (We regard T as an infinitesimal Hodge structure; it is the (rescaled) derivative of a Hodge structure ϕ0 : S 1 → Ad(GR ), cf. [45, Section 2.3].) The corresponding Hodge flag F • is given by M Fp = gq,−q . q≥p

•

Note that P = StabG (F ). This allows us to identify with the flag F • with the point dfn

o0 = P/P ∈ G/P . Define ˇ dfn D = G/P , and let dfn

D = GR · o0 . ˇ Then D is open in D, and therefore a flag domain (Section 2.2). Definition 6.17. A Mumford–Tate domain arising in this way is called a Mumford–Tate– Chevalley (MTC) domain. Remark 6.18. From the Hodge–theoretic perspective, D is the Mumford–Tate domain parameterizing weight zero Hodge structures on gQ that are polarized by −(·, ·) and have Mumford–Tate group contained in GQ . By construction, this GQ is a MTC group. Moreover, the grading element T associated with o0 is defined over Q(i) and is purely imaginary. The Mumford–Tate group of o0 is therefore a 1-torus with real points ϕ0 (S 1 ).


41

In general, a Mumford–Tate domain (determined by Hodge–theoretic data) which is isomorphic to D as a GR -homogeneous space, need not be MTC: this is a property which reflects the arithmetic of the Q-Hodge tensors. The MTC case may be thought of as “maximizing” the density of Mumford–Tate subdomains in D. Example 6.19 (g = sl2 C). This is a continuation of Example 6.15. We have G = SL2 C and fix the parabolic subgroup a b P = a, b ∈ C , a 6= 0 . 0 1/a

ˇ be the complex projective line CP1 . Then the identity coset P/P corresponds to the Let D point o0 = (1 : 0) ∈ CP1 . The GR = SU(1, 1)–orbit D = {(1 : z) ∈ CP1 | |z| < 1} of o0 is naturally identified with the unit disc in C. 6.5. Standard triples. Let gk be a Lie algebra defined over a field in gk is a set of three elements {N + , Y, N } ⊂ gk such that [Y, N + ] = 2 N + ,

[N + , N ] = Y

k. A standard triple

and [Y, N ] = −2 N .

Note that {N + , Y, N } span a three–dimensional semisimple subalgebra (TDS), necessarily isomorphic to sl2 . We call Y the neutral element, N the nilnegative element and N + the nilpositive element, respectively, of the standard triple. Example 6.20. The matrices 0 1 + (6.21) N = , 0 0

Y =

1 0 0 −1

and N =

form a standard triple in sl2 R; while the matrices i 1 0 i + 1 , Y = (6.22) N = 2 1 −i −i 0

form a standard triple in su(1, 1).

and N =

0 0 1 0

1 2

−i 1 1 i

6.6. Cayley transforms. This is a brief review of Cayley transforms; see [34, Chapter VI.7] for details. Let gR = gZ ⊗ R be the real form of g constructed in Section 6.3. By (6.12), all roots are imaginary. Fix a noncompact positive root α. That is, gα ⊂ k⊥ ; equivalently, α(E) is odd. Then xα = x−α . The Cayley transform associated to α is dfn (6.23) cα = Ad exp π4 (x−α − xα ) ∈ Ad(G) . Since Ad(g) is a Lie algebra automorphism for any g ∈ G, it follows that cα is a Lie algebra automorphism. Therefore, h′ = c(h)

42

KERR AND ROBLES

is a θ–stable Cartan subalgebra. The root spaces of h′ are the Cayley transforms ′ gα = c(gα ) of the root spaces of h. Likewise, ′ Hβ = cα (Hβ ). In particular, ′ α

H

= c(Hα ) = xα + x−α = v α ∈ k⊥ Z .

The real form h′R = h′ ∩ gR of h′ is ′

(6.24)

dfn

hR = c(h) ∩ gR = ker {α : t → i R} ⊕ spanR {′ Hα } .

Additionally, we have cα (xα − x−α ) = xα − x−α

and cα (xα + x−α ) = −Hα .

Therefore, the root space ′ g−α = c(g−α ) is spanned by dfn

−α i 2 (x

y −α = i cα (x−α ) =

(6.25)

Likewise, ′ gα is spanned by y α = −i cα (xα ) =

(6.26)

i 2

− xα − H α ) ∈ g Q .

x−α − xα + Hα

∈ gQ .

From [xα , x−α ] = Hα and cα (Hα ) = [cα (xα ), cα (x−α )], we see that

{y α , ′ Hα , y −α } is a standard triple defined over Q,

(6.27)

cf. Section 6.5. Recall (6.13) and note that slα2 (Q) = spanQ {y α , ′ Hα , y −α } . Example 6.28 (g = sl2 C). This is a continuation of Examples 6.15 and 6.19. We have 1 1 i , cα = √ i 1 2

and

y

α

i = 2

1 −1 1 −1

,

′ α

H

=

0 1 1 0

,

y

−α

i = 2

−1 −1 1 1

.

6.7. The weight zero α–primitive subspace. Let dfn

Γ0α = {ζ ∈ g | [slα2 , ζ] = 0}

be the trivial isotypic component in the slα2 –module decomposition of g; this is the weight zero, α–primitive subspace of g. Note that the Cayley transform cα restricts to the identity on Γ0α . The Jacobi identity implies Γ0α is a subalgebra of g. More precisely, M Γ0α = ker{ α|h } ⊕ gβ , β∈∆(Γ0α )

where ∆(Γ0α ) = {β ∈ ∆ | β ± α 6∈ ∆}

is the set of roots that are strongly orthogonal to α. In particular, Γ0α is a reductive subalgebra defined over Q.


Let

43

dfn

Γα = [Γ0α , Γ0α ] be the semisimple component, and let G ⊂ G be the connected Lie subgroup with Lie algebra Γα . Note that P = G ∩ P is a parabolic subgroup, and let ˇ dfn D = G/P

ˇ be the identity coset. be the associated rational homogeneous variety. Let o = P/P ∈ D The real form Γα (R) ∩ h ⊂ ker{ α|t } of the Cartan subalgebra is contained in t, and therefore is compact. Whence, ˇ the G(R)–orbit D of o is open in D. 6.8. The construction. Recall the Cartan subalgebra h ⊂ g with compact real form t of Section 6.3. Fix a set (6.29a)

B = {β1 , . . . , βs } ⊂ h∗

of strongly orthogonal noncompact roots with the property that (6.29b)

βj (E) = 1 .

Given 1 ≤ j ≤ s, let cβj be the Cayley transformation associated to βj (Section 6.6). Set dfn

cj = cβj ◦ · · · ◦ cβ1 .

(Strong orthogonality implies that the roots {βj+1 , . . . , βs } are still imaginary and noncompact with respect to hj . Whence, cj is well-defined.) By (6.23), the Cayley transform cα may be written as cα = Adgα where gα = exp π4 (x−α − xα ) ∈ G. Let gj = gβj · · · gβ1 , so that cj = Adgj . It will be convenient to also set g0 = 1 and c0 = Ad1 = 1. Recall the weight zero polarized Hodge structure of Section 6.4. This Hodge structure is ˇ Let O0 = D be the the GR –orbit naturally identified with the identity coset o0 = P/P ∈ D. of o0 . Given 1 ≤ j ≤ s, set dfn ˇ, oj = gj (o0 ) ∈ D ˇ be the GR –orbit of oj . The containment (6.1) follows from this construction, and let Oj ⊂ D cf. [32]. Remark 6.30 (Hermitian symmetric case). Note that cj is the Cayley transform cb of [3, Section 1.6]. Therefore, in the case that D is Hermitian symmetric, the Oj account for all the GR –orbits in the boundary of D. That is, [ D = Oj . j

Example 6.31 (g = sl2 C). This is a continuation of Examples 6.15, 6.19 and 6.28. We have and

o1 = (1 : i) ∈ CP1 ,

O1 = {(1 : z) | |z| = 1} is naturally identified with the unit circle in C. If we define cos π4 t i sin π4 t dfn cα (t) = Ad exp π4 t(x−α − xα ) = , i sin π4 t cos π4 t

44

KERR AND ROBLES

then the Cayley transform is cα = cα (1). Moreover, the curve cα (t) · o0 is contained in the unit disc O0 for all 0 ≤ t < 1. Associated with the Matsuki point oj are two filtrations (6.37) of g, which we now describe. Set h0 = h and define dfn hj = cj (h) . Then dimR (hj ∩ k⊥ R ) = j. Thus

s ≤ rankR gR .

(6.32)

Recall the elements Hα ⊂ hj of Section 6.2,(15) and define dfn

Yj = Hβ 1 + · · · + Hβ j .

(6.33)

Let g = ⊕ j gp be the E–eigenspace(16) decomposition (2.4), and let g = ⊕ j gℓ be the Yj – eigenspace decomposition. That is, (6.34)

jg

(6.35)

p

j gℓ

dfn

{ξ ∈ g | [E, ξ] = p ξ} ,

=

dfn

{ξ ∈ g | [Yj , ξ] = ℓ ξ} .

=

By (6.4), the Yj –eigenvalues are integers. As commuting semisimple endomorphisms, E and Yj are simultaneously diagonalizable; therefore, (6.36a)

jg

p,q dfn

=

jg

p

∩ j gp+q = {ξ ∈ g | [E, ξ] = pξ , [Yj , ξ] = (p + q)ξ}

defines a direct sum decomposition (6.36b)

g =

M

jg

p,q

.

By the Jacobi identity, this is a bigraded Lie algebra decomposition; that is, [j gp,q , j gr,s ] ⊂ Define filtrations (6.37)

jF

p dfn

=

M q≥p

jg

q

and

j Wℓ

jg

dfn

=

p+r,q+s

M

p+q≤ℓ

.

jg

p,q

=

M

j gk

.

k≤ℓ

Note that StabG (j F • ) = gj P gj−1 = StabG (oj ), so that there is a natural identification of ˇ the flag j F • with the Matsuki point oj ∈ D. Theorem 6.38. For each 0 ≤ j ≤ s, there exists a j–dimensional, rational nilpotent cone σj ⊂ gQ such that {exp(C σj ) · oj } ∈ B(σj ) is a nilpotent orbit.(17) The weight filtration W• (σj ) is the filtration j W• of (6.37). The cone σj−1 is a face of σj , for all 1 ≤ j ≤ s. Remark 6.39. As a corollary to Theorem 6.38, (gQ , j F • , j W• ) is a (limit) Q–mixed Hodge structure. (15)Properly speaking, Hβ = Hβ depends on h . Precisely, if Hβ is defined with respect to h ⊂ b ⊂ p, then j 0 j

Hβj = cj (Hβ0 ). We have elected to drop the subscript j to keep the notation clean and simply write Hβ ∈ hj . (16)Again, E = c (E ), where E is the grading element associated to h ⊂ b ⊂ p. j 0 0 (17)See Section 7.1 for the definition of B(σ).


45

Remark 6.40 (Rational and polarized). In the language of [32, Section 5], Theorem 6.38 implies that the orbit Oj is polarized and contained in the nilpotent closure of the open GR –orbit D = O0 . Moreover, from σj ⊂ gQ it follows that j W• is defined over Q, so that Oj is rational, cf. [32, Definition 5.5]. Recall the Yj –eigenspace j g0 defined in (6.35). The Jacobi identity implies that j g0 is a ss Lie subalgebra of g. In fact, j g0 is reductive. Let j gss 0 = [j g0 , j g0 ]; then j g0 is semisimple, ss and j g0 = j g0 ⊕ z, where z is the center of j g0 . Following [32, Definition 5.9], we say that the GR –orbit Oj is cuspidal if j gss 0 (R) contains a compact Cartan subalgebra; that is, a Cartan subalgebra on which the Killing form restricts to be negative definite. Remark 6.41 (Weakly cuspidal). The orbit Oj is cuspidal in the following weak sense. In the course of the proof we will show that there exist semisimple subgroups Gs ⊂ · · · ⊂ G1 ⊂ G0 = G with the properties that: ˇ j = Gj · oj is a rational homogeneous variety containing the point oj+1 . ◦ The orbit D ˇ j and equals D ˇ j ∩ Oj ; and oj+1 ∈ ∂Dj . ◦ The orbit Dj = Gj,R · oj is open in D ˇ j ∩ Oj+1 is contained in the boundary of Dj and is ◦ The orbit Cj+1 = Gj,R · oj+1 ⊂ D ˇ cuspidal in Dj . (This will follow from Lemma 6.51 below.) Note that Dj+1 ⊂ Cj+1 ⊂ ∂Dj .

Remark 6.42 (Sub–Hodge structures). The subdomains {Dj }sj=0 may be described in greater detail. In the course of the proof we will show that: ◦ The Lie algebra of Gj is \ (6.43) Γj = Γβ i , 1≤k≤j

where the Γβi are as in Section 6.7. ˇ j−1 , for all 1 ≤ j ≤ s, cf. (6.67). ◦ The set {exp(C Nj ) · oj } is a nilpotent orbit in D These two observations, and results of [31], yield a natural identification of the open Gj,R – ˇ j as the Mumford–Tate domain of a Hodge structure ϕj,• , which we now orbit Dj ⊂ D describe. Let W• (Nj ) denote the weight filtration on Γj−1 , with the convention that Γ0 = g. Let Gr• (W• (Nj )) denote the associated graded and choose 0 6= Nj+ ∈ gβj . Then, given ℓ ≥ 0, the filtration j F • induces a weight ℓ Hodge structure ϕj,ℓ on ker{(Nj+ )ℓ+1 : Grℓ (W• (Nj )) → Gr−ℓ−2 (W• (Nj ))} that is polarized by −(·, Njℓ ·).(18) Theorem 6.38 and the lemmas that follow are proved in Section 6.10. The first three lemmas codify some distinguished properties of the bigrading (6.36) and the Cartan subalgebra hj . Lemma 6.44. The bigrading (6.36) satisfies the symmetries (6.45a)

p,q

=

jg

θ(j gp,q ) =

jg

jg

q,p

and (6.45b)

−q,−p

.

(18)In the notation of [31], the domain D is denoted D(N ), and the Hodge structure ϕ j j j,• by ϕsplit .

46

KERR AND ROBLES

Remark 6.46. By (6.37), (6.36) is the Deligne bigrading (2.28) on the Q-MHS (gQ , j F • , j W• ). It follows from (6.45a) that this MHS is R-split, but in fact more is true. Since Yj ∈ gQ , (gQ , j F • , j W• ) is Q-split; and by (6.43), the other MHS in its Gj,R -orbit remain Q-split. Lemma 6.47. For E and Hβ defined with respect to the Cartan subalgebra hj , we have E + E = Yj . As a consequence, j gp,q = j gp ∩ j gq . Lemma 6.48. The roots α ∈ ∆(g, hj ) of the Cartan subalgebra hj satisfy (6.49)

α = −α +

j X

α(Hβi ) βi .

i=1

Consider Lemma 6.48 in the case that j = 1. Then (6.5) and (6.49) yield α = −rβ1 (α). More generally, if j > 1, then the strong orthogonality of the roots B implies α ¯ = −rβj · · · rβ1 (α). The following lemma will be invoked in the proof of Theorem 6.38, which is by induction. 1 Lemma 6.50. There exists N1 ∈ g−β such that {exp(C N1 ) · o1 } ∈ B(N1 ) is a nilpotent Q orbit. (In particular, O1 is rational and polarized.)

Lemma 6.51. The GR –orbit O1 is cuspidal. Lemma 6.52. If the first root β1 ∈ B is simple, then the GR –orbit O1 has codimension one ˇ in D. Appendix A presents a thorough study of these codimension one boundary orbits in the case that P is a maximal parabolic. Remark 6.53 (codim O1 ). The codimension of O1 is the number of roots in ∆(+, +) = {α ∈ ∆ | α(E) , α ¯ (E) ≥ 1} ,

cf. [32, Proposition 4.1]. By Lemma 6.48 and β¯1 = β1 ∈ ∆(+, +).

∆(+, +) = {α ∈ ∆ | 1 ≤ α(E) < α(Hβ1 )} ,

Example 6.54 (codim O1 > 1). In the case that β1 is not simple there exist examples with codimR O1 > 1. Let P the maximal parabolic corresponding to a simple root αi ; equivalently, E = Si , cf. Section 2.1. In general it seems easiest to find examples in which codim O1 > 1 when the root system ∆ contains roots of distinct lengths. This is the case in each of the examples that follow, where αi is a long root and β1 is a short root. ˇ = IG(2, C7 ) is the isotropic grassmannian. (a) Suppose that g = so7 C and i = 2. Then D β 1 2 1 If β1 = α2 + α3 , then H = −2S + S . We have ∆(+, +) = {(010) , (011) , (012)}, where (a1 a2 a3 ) denotes the root α = aj αj (for example, (011) = α2 + α3 ). Thus codim O1 = 3. (Likewise, the reader may confirm that β1 = (111) yields codim O = 3.) ˇ = LG(3, C6 ) is the (Hermitian symmetric) (b) Suppose that g = sp6 C and i = 3. Then D Lagrangian grassmannian. If β1 = α2 + α3 , then Hβ1 = −S1 + 2S3 . We have ∆(+, +) = {(001) , (011) , (021)}. (Likewise, β1 = (111) also yields codim O1 = 3.)


47

ˇ ⊂ P g is the (c) Suppose that g is the exceptional Lie group f4 and i = 1. Then D β 1 4 adjoint variety of g (Section 2.5). If β1 = (1110), then H 1 = 2S − S . So ∆(+, +) = {(1000) , (1100) , (1110) , (1120) , (1220)}, and codim O1 = 5. ˇ ⊂ P g is the adjoint (d) Suppose that g is the exceptional Lie group g2 and i = 2. Again, D β 1 1 variety of g. If β1 = 2α1 + α2 , then H = S . So ∆(+, +) = {(21) , (31) , (32)}, and codim O1 = 3. Remark 6.55 (Relationship to work of Kerr–Pearlstein). Theorem 6.38 is closely related to the discussion and results of [32]. Indeed, Lemma 6.47 implies that the bigrading (6.36) is that of [32, Lemma 3.2]. One may also view oj = j F • as the image of a Q-split point in a boundary component under the na¨ıve limit map, as we explain briefly in Section 7.1. 6.9. The case that P is a maximal parabolic. Assume that G is simple and P is a maximal parabolic. Then the grading element (2.7) is of the form E = Si . Proposition 6.56. Let P be a maximal parabolic in a simple Lie group G. Fix a real form GR and an open GR –orbit D ⊂ G/P admitting the structure of a Mumford–Tate domain. ˇ contains a unique codimension one GR –orbit. Then bd(D) ⊂ D The proposition is proved in Section 6.10. A generalization of Proposition 6.56 holds for arbitrary (not necessarily maximal) parabolics: the number of codimension one orbits in bd(D) is |I(p)|, cf. [44]. Remark 6.57. That an open GR –orbit admit the structure of a Mumford–Tate domain is a strong constraint. Given o ∈ D of this type, with corresponding grading element E ∈ h ⊂ g0 , we have (6.58)

E(∆c ) ⊂ 2Z

and

E(∆n ) ⊂ 2Z + 1 ,

where ∆c and ∆n denote the compact and noncompact roots, respectively [24]. Applying the complex Weyl group W (g, h) to E yields points w(o) in each open orbit; more precisely, it produces a bijection between double cosets in W (gR , hR ) \ W (g, h)/W (p, h) and open orbits [32, Theorem 4.5(ii)]. The orbit containing w(o) is a Mumford–Tate domain if and only if w(E) satisfies (6.58). In fact, D is often unique (or nearly so) and Proposition 6.56 often allows us to identify it from orbit incidence data. Remark 6.59. Assume D admits a Mumford–Tate domain structure. Extrapolating from the G2 examples in [32, Section 6.1.3], one might hope that bd(D) contains at most one GR – orbit in each codimension, and that products of the cβj reach every orbit in bd(D). That this is false can be seen in the case of the F4 -adjoint variety. Running kgp and kgporder in ATLAS [2] and interpreting the results via Matsuki duality [42, Section 6.6], we find 70 orbits (all with codimension ≤ 15) in bd(D). As noted in (6.32), the number of elements s in the set (6.29) is bounded above by the real rank of gR . Lemma 6.60. (a) There exists a choice of B that both maximizes s and contains the simple root αi .

48

KERR AND ROBLES

(b) Excepting the pairs (6.61)

(g, i) = (e7 , 5) , (e8 ; 2, 5, 6) , (f4 , 2) , (g2 , 1) ,

there exists a set (6.29) with s equal to the real rank of gR . Proof. For most pairs (g, i) the sequence {βj }sj=1 is exhibited in [34, Appendix C.3-4]. The remainder may be identified from inspection of the root system. Corollary 6.62. Excepting the pairs (6.61), the Cartan subalgebra hs has maximally noncompact real form hs ∩ gR . Remark 6.63 (Exceptional cases). For each of the exceptional cases of (6.61) we may choose the set {βj }sj=1 as follows. Below we express the root β = bi αi as (b1 · · · br ). ◦ B(e7 , 5) = {(04 102 ) , (03 13 0) , (012 2102 ) , (015 0) , (01014 ) , (02 15 )}. Here s = 6, while rankR gR = 7. ◦ B(e8 , 2) = {(0106 ) , (012 2103 ) , (13 212 02 ) , (12 23 102 ) , (12 22 13 0) , (13 22 12 0) , (012 23 10)}. Here s = 7, while rankR gR = 8. ◦ B(e8 , 5) = {(04 103 ) , (03 13 02 ) , (012 2103 ) , (015 02 ) , (01014 0) , (02 15 0)}. Here s = 6, while rankR gR = 8. ◦ B(e8 , 6) = {(05 102 ) , (04 13 0) , (03 15 ) , (012 22 102 ) , (012 213 0) , (017 )}. Here s = 6, while rankR gR = 8. ◦ B(f4 , 2) = {(0100), (1110), (0120)}. Here s = 3, while rankR gR = 4. ◦ B(g2 , 1) = {(10)}. Here s = 1, while rankR gR = 2.

6.10. Proofs. 1 Proof of Lemma 6.50. Let N1 be the nilpotent element y −β1 ∈ ′ g−β of (6.25). By [46, Q Theorem 6.16], it suffices to show that exp(i t N1 ) · o1 ∈ D for all t > 0. ˇ = CP1 (of Examples 6.15, 6.19, 6.28 We may reduce to the case that g = sl2 C and D and 6.31) as follows. Set β = β1 and recall the Lie subalgebra slβ2 ⊂ g of (6.13) and the Lie subgroup SLβ2 (C) ⊂ G of (6.14). The Cayley transform cβ may be viewed as an element of ˇ containing o1 . We may identify o0 with Ad(SLβ2 (C)). The SLβ2 (C)–orbit of o0 is a CP1 ⊂ D 1 the point (1 : 0) ∈ CP of Example 6.19, and o1 with the point (1 : i) ∈ CP1 of Example 6.31. Moreover, slβ2 (R) is isomorphic with the Lie algebra su(1, 1) of Example 6.15, and we may identify D ∩ CP1 , which is the SLβ2 (R)–orbit of o0 , with the unit disc as in Example 6.19. Under these identifications, 1 1 −β 1 , iy = 2 −1 −1

and

exp(i t N1 ) · o1 = exp

1/2 1/2 −1/2 −1/2

1 i

=

1 + 12 t + 21 i t − 12 t + i(1 − 12 t)

!

.


Whence exp(i t N1 ) · o1 is contained in the unit disc D ∩ CP1 ⊂ D.

49

Proof of Theorem 6.38. Throughout the proof it will be helpful to keep in mind that strong orthogonality of the roots B implies (6.64)

the Cayley transform cβj restricts to the identity on slβ2 i for all j 6= i.

Whence, it follows from (6.13) and (6.27) that (6.65)

the cj (slβ2 i ) are defined over Q for all i, j.

The proof of the theorem is by induction. The inductive hypotheses. Assume the theorem holds for j, and that (6.66)

σj = spanQ>0 {N1 , . . . , Nj }

i with Ni ∈ g−β Q

for all 1 ≤ i ≤ j .

Suppose furthermore that a semisimple group Gj ⊂ G, with Lie algebra Γj ⊂ g, and parabolic subgroup Pj = Gj (C) ∩ P are given with the following properties: (a) The Lie algebra Γj is defined over Q. The roots of Γj are precisely those that are strongly orthogonal to the {β1 , . . . , βj }; equivalently, (6.43) holds.

Remark. Observe that this hypothesis on the roots of Γj implies that the subalgebra cj (slβ2 i ) = slβ2 i is contained in Γj for all i > j. (b) The parabolic Pj = StabGj (C) (oj ), so that Gj (C)/Pj is naturally identified with the ˇ j = Gj (C) · oj . The real orbit Dj = Gj (R) · oj is open in D ˇj. complex orbit D (c) For all i ≥ j + 1 the Cayley transform cβi lies in Ad(Gj ). ˇ j . By Remark. It follows from these last two hypotheses that oj+1 = gβj+1 (oj ) ∈ D (6.1), the orbit Cj+1 = Gj (R) · oj+1 is contained in the topological boundary ∂Dj . The base case. Set σ0 = {0}. It follows from Section 6.4 that the theorem holds for j = 0. The hypothesis (6.66) is trivial. Likewise, setting G0 = G, Γ0 = g, and P0 = P , the ˇ 0 = D, ˇ D0 = D and C1 = O1 .) hypotheses (a-c) are immediate. (Here, D The induction: part 1. First note that (6.64) implies that the nilpotent cone σj = cβj+1 (σj ) is unchanged under the next Cayley transform. By (6.65) the root space j+1 g−βj+1 is defined over Q. By Lemma 6.50, we may choose −β ˇj. Nj+1 ∈ j+1 gQ j+1 so that {exp(i yNj+1 ) · oj+1 } ∈ Bj (Nj+1 ) is a nilpotent orbit in D Equivalently, (6.67)

exp(i yNj+1 ) · oj+1 ∈ Dj

for all y > 0 .

By the hypothesis (a), Nj+1 ⊂ Γj so that exp(C Nj+1 ) ⊂ Gj (C). Moreover, the hypotheses (6.66) and (a) imply that exp(σj ) commutes with Gj . Whence, given any N ∈ σj and

50

KERR AND ROBLES

y > 0,

Setting

exp(i (N + yNj+1 )) · oj+1 = ⊂ = = ⊂

exp(i N ) exp(i y Nj+1 ) · oj+1 exp(i N ) Dj exp(i N ) Gj (R) · oj Gj (R) exp(i N ) · oj Gj (R) D = D .

σj+1 = spanQ>0 {σj , Nj+1 } ,

we have shown that {exp(C σj+1 ) · oj+1 } ∈ B(σj+1 ) is a nilpotent orbit; this establishes the first two (of three) claims of Theorem 6.38 for j + 1. Also, we see that (6.66) holds for j + 1. Given i ≤ j + 1, complete the pair Hβi , Ni ∈ gQ to a standard triple (Section 6.5) {Ni+ , Hβi , Ni } with Ni+ ∈ gβQi . To see that the third claim W• (σj+1 ) = j+1 W• holds it suffices to observe that the strong orthogonality of the roots implies + N + = N1+ + · · · + Nj+1 ,

Yj+1 = Hβ1 + · · · + Hβj+1 , N = N1 + · · · + Nj+1

is a standard triple in gQ .

The induction: part 2. It remains to show that the additional inductive hypotheses (a-c) hold for j + 1. Let Γj+1 ⊂ Γj be the Lie algebra Γβj+1 of Section 6.7. Then hypothesis (a) is immediate. Likewise, let Gj+1 ⊂ Gj and Pj+1 ⊂ Pj be the corresponding G and P, ˇ respectively. Recall that ˇ j+1 be the corresponding D ⊂ D, respectively; and let Dj+1 ⊂ D ˇ j+1 ; see Section 6.7. Thus the hypothesis (b) holds. Dj+1 is open in D Let i ≥ j + 2. A priori the Cayley transformation cβi lies in Ad(G). However, strong orthogonality implies that B\{β1 , . . . , βj+1 } ⊂ ∆(Γj+1 ) and slβ2 i = cj+1 (slβ2 i ) ⊂ Γj+1 .

Therefore, the Cayley transformation cβi lies in Ad(Gj+1 ). Whence hypothesis (c) holds. Proof of Lemma 6.44. By construction the Cartan subalgebra h = h0 is preserved by conjugation and Cartan involution (Section 6.3). Since the Cayley transform preserves these properties, the Cartan subalgebra hj is also preserved by conjugation and the Cartan involution. For any such Cartan subalgebra the roots satisfy θα = −α ¯. Proof of Lemma 6.47. Define dfn

tj = hj ∩ kR dfn

and

dfn

aj = hj ∩ k⊥ R .

Then the real form hj (R) = hj ∩ gR is tj ⊕ aj . In analogy with (6.24), tj =

j \

i=1

ker{βi : hR → C} and aj = spanR {Hβ1 , . . . , Hβj } .


51

Moreover, (6.68)

the roots of hj are R–valued on aj , and i R–valued on tj .

Since α(E) ∈ R, we see from (6.68) that E ∈ itj ⊕ aj . Therefore, E + E ∈ aj . Thus E + E ∈ spanR {Hβ1 , . . . , Hβj } .

Since the roots βi are real, (6.29b) and (6.69) yield βi (E + E) = βi (E) + βi (E) = 2.

¯ ¯ for any Proof of Lemma 6.48. By definition, the conjugate root β¯ is given by β(ξ) = β(ξ) ξ ∈ hj . Whence (6.49) holds when restricted to tj . It remains to show that the equation holds when restricted to aj . Since the latter is spanned by the {Hβi }ji=1 , it suffices to show that (6.49) holds when evaluated on any of these Hβi . Strong orthogonality of the roots βi ∈ B and (6.4) imply that (6.69)

2

(βi , βk ) = βi (Hβk ) = 2δik , (βk , βk )

for any 1 ≤ i, k ≤ s .

It follows that (6.49) holds when evaluated on the {Hβk }jk=1 .

Proof of Lemma 6.51. From the definition (6.35) we see that h1 is a Cartan subalgebra of g0 , and 1 Hβ1 is contained in the center z of 1 g0 . It then follows from (6.24) that h1 ∩ 1 gss 0 . is a compact Cartan subalgebra of 1 gss 0 Proof of Lemma 6.52. For the duration of the proof, let α denote the simple root β1 ∈ B. Set gp,q = 1 gp,q , and o = o1 and O = O1 . By [32, Proposition 4.1], the (real) codimension of O is the number of roots in We will show that (6.70)

∆(+, +) = {γ ∈ ∆ | γ(E) , γ¯ (E) > 0} . ∆(+, +) = {α} ;

whence codimR O1 = 1, establishing the lemma. By Remark 6.53, α ∈ ∆(+, +). Now suppose that γ ∈ ∆(+, +). The two inequalities γ(E), γ¯ (E) > 0 imply that both γ and γ¯ are positive roots. But from Lemma 6.48, we see that γ¯ = −γ + ℓα, where ℓ = γ(Hα ). In particular, if γ is not a multiple of the simple root α, then γ¯ is a negative root. Whence γ = α. Proof of Proposition 6.56. By [32, Proposition 4.14], the codimension one GR –orbits in bd(GR · o0 ) are of the form GR · (gβ · o0 ), with β ∈ ∆(g+ ). In fact, we must have β ∈ ∆(g1 ) in order not to violate [32, Corollary 4.4]. So let β be a (noncompact, positive) root with β(E) = 1. Recall E = Si . We will show that β belongs to the orbit of αi under the subgroup W (g0 , h) ⊂ W (gR , hR ) of the real Weyl group. It will follow that gβ · o0 and gαi · o0 belong to the same GR –orbit, giving the desired uniqueness. Acting by W (g0 , h) if necessary, we may assume (β, αj ) ≤ 0 for all j 6= i; equivalently,

(6.71)

αj (Hβ ) ≤ 0 .

(That is, use the Weyl group of G0 to reflect the orthogonal projection of β to ker(E) into the negative Weyl chamber.) Set B = {β} and gp,q = 1 gp,q for the bigrading associated to

52

KERR AND ROBLES

o1 = gβ · o0 . For any root α ∈ ∆(g, h1 ), write gp(α),q(α) ⊃ gα . By Lemma 6.47, p(α) + q(α) = α(Hβ ), and of course p(β) = 1 = q(β). Now for any positive root α, we have X α = p(α)αi + mj (α)αj j6=i

where mj (α) ≥ 0. Since p(β) = 1 and β is positive, X p(αi ) + q(αi ) = αi (Hβ ) = β(Hβ ) − mj (β)αj (Hβ ) j6=i

But p(αi ) = αi (E) = αi ≥ 2 unless β = αi .

(Si )

(6.71)

≥

β(Hβ ) = 2 .

= 1, and so q(αi ) ≥ 1. By Remark 6.53, codim(GR · o1 ) is now

7. Enhanced SL2 -orbits and adjoint varieties Following a brief review of boundary components of Mumford–Tate domains (Section 7.1), we begin in Section 7.2 by using the last section’s construction to produce a class of Mumford–Tate subdomains which are usually not Schubert varieties, hence will be interesting to compare with them in cohomology. (We only take this up in a limited way here, in Section 7.6 for the G2 adjoint variety.) The remainder of this section is devoted to working out these subdomains and boundary components for the fundamental adjoint varieties, where they have striking relationships to both the varieties of lines studied in Sections 3 and 5 and the Calabi-Yau VHS studied in [22]. 7.1. Boundary components and the naive limit map. First we review the definitions ˇ and pairwise and recall some results from [31, 32]. Fix a Mumford–Tate domain D ⊂ D dfn commuting nilpotents N1 , . . . , Nm ∈ gQ . Let σ = Q≥0 hN1 , . . . , Nm i ⊂ gQ be the rational dfn dfn nilpotent cone, with interior σ ◦ = Q>0 hN1 , . . . , Nm i. By Theorem 2.27(b), W• (σ) = W• (N ) is independent of N ∈ σ ◦ . Regarding(19) dfn ˜ ˇ | (F • ; N1 , . . . , Nm ) is a nilpotent orbit} B(σ) = {F • ∈ D

as a set of (limiting) Q-mixed Hodge structures (VQ , F • , W• (σ)), it makes sense to consider its generic Mumford–Tate group (or that of some subset). The boundary component associated to σ is defined by dfn ˜ B(σ) = B(σ)/ exp(C σ) ;

the points of B(σ) are σ-nilpotent orbits, or equivalently LMHS up to “reparametrization” (i.e. modulo eC σ ). When m = 1, we shall replace σ everywhere in our notation by N1 ; e.g. ˜ 1 ). By [32, Remark 5.6], we have B(σ) ˜ ˜ ). B(N = ∩N ∈σ◦ B(N Let D(σ) be the Mumford–Tate domain parameterizing the split polarized Hodge struc˜ ture ⊕ℓ (Grℓ W• (σ), F • ). We have a natural map ρ˜ : B(σ) ։ D(σ) given by taking the ˜ associated graded Hodge structure. A choice of Q-split base point in B(σ) gives rise to a ˜ section D(σ) ֒→ B(σ) of ρ, and we shall sometimes blur the distinction between D(σ) and its image. (19)See Section 2.8 for relevant definitions


53

Regarding N1 , . . . , Nm as elements of End(g), let W• (σ)g be the corresponding weight filtration of g. Let Z(σ) ⊂ G be the centralizer of σ, and let z(σ) denote the Lie algebra of Z(σ). Note that z(σ) ⊂ W0 (σ)g .

(7.1)

Let W• (σ)z = z(σ) ∩ W• (σ)g be the induced filtration of z(σ). Then M dfn z(σ) ≃ z−ℓ (σ) , where z−ℓ (σ) = W−ℓ (σ)z /W−ℓ−1 (σ)z . ℓ≥0

There is a natural tower of fibrations (factoring ρ˜) (1)

(k)

ρσ

ρσ

(7.2) (k)

B(σ) ։ · · · ։ B(σ)(k) ։ B(σ)(k−1) ։ · · · ։ B(σ)(1) ։ D(σ)

with ρσ –fibre F(k) through F • (mod eC σ ) equal to (7.3)

F(k) =

z−k (σ) , F 0 z−k (σ)

for k 6= 2 ,

and F(2) =

z−2 (σ) . C σ ⊕ F 0 z−2 (σ)

(cf. [31, §7]). dfn Now let σ = σs = spanQ≥0 {N1 , . . . , Ns } be the cone of Theorem 6.38, with Nj ∈ s g−βj , (20) ˜ and regard s F • (cf. (6.37)) as a Q-split base point in B(σ). The Lie algebra gB(σ) of the generic Mumford–Tate group GB(σ) of D(σ) may be described as follows (cf. [31, Sections P 4–5]). Let E and Y = sj=1 Hβj be the grading elements of (6.34)j=s and (6.35)j=s . Set ¯) (which is equal to i(2E − Y) by Lemma 6.47). Then gB(σ) is the Q–Lie algebraic φ = i(E − E closure in z0 (σ) = ∩sj=1 Γ0βj of Z0 (σ)R · Cφ . A consequence of our construction (and Remark 6.18) is that φ ∈ igR , and the roots of hs in z0 (σ) come in conjugate pairs defined over Q(i). Therefore, gB(σ) is the (C–)Lie algebraic closure of X Cφ + gβ , β ∈ ∆(Γs ) β(φ) 6= 0

and so gB(σ) ⊆ Γs + Cφ while gss B(σ) ⊆ Γs . In most cases of interest we will have equality. In any case(21) we have

D(σ) = Gs,R · s F • ∼ = Gs,R /Gs,R ∩ P (where Lie(Gs ) = Γs ), and this gives an open subset in the homogeneous variety dfn ˇ D(σ) = Gs · s F • ∼ = Gs /Gs ∩ P.

ˇ ˇ by identifying it with D ˇ s dfn ˇ = Gs · os ⊂ D Now we could regard D(σ) as a subvariety of D (cf. Remark 6.41). However, for some purposes it is better to regard it as a separate variety ˇ in a different way: and map it into D (20)Though o ∈ D ˇ corresponds to this filtration, here we shall write s F • for the point of (D(σ) ⊂) B(σ) ˜ s

ˇ and os for the point of D. • (21)The discrepancy between g B(σ) and Γs + Cφ lies in ker(ad φ), and therefore stabilizes s F .

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KERR AND ROBLES

Definition 7.4. Following [25, 32],(22) the naive limit map Φσ∞ : B(σ) → bd(D) sends F • 7→ limIm(z)→∞ ezN F • , which is independent of the choice of N ∈ σ ◦ , cf. [32, Remark 5.6]). ˆ Though Φσ∞ is not isomorphic onto its image (denoted B(σ) in [32]), its restriction to D(σ) is a Gs,R –equivariant isomorphism, which extends to a Gs –equivariant embedding of ˇ ˇ However, the image of this embedding is not D ˇ s. D(σ) into D. ˇ To present Ds as the image of a naive limit map, recall the commuting standard triples dfn {Nj+ , Hβj , Nj } from the proof of Theorem 6.38, and put σ ˜ = Q≥0 h−N1+ , . . . , −Ns+ i. Setting dfn dfn ˜ b dfn = ⊕p+q≤b s ˜gp,q , and reasoning as in the proof gp,q = s g−q,−p, s F˜ a = ⊕p≥a; q∈Z s ˜gp,q , s W s˜ of [32, Theorem 5.15], one shows that: ˜ • = W• (˜ ˜ dfn ◦ sW σ )g is graded by Y = −Hβ1 − · · · − Hβs ; ◦ (s F˜ • ; −N1+ , . . . , −Ns+ ) is a nilpotent orbit, and (s F˜ • , W• (˜ σ )g ) is Q-split; and σ ˜ • • ◦ Φ∞ sends s F˜ to s F . ˇ σ ) are the Gs,R and Gs –orbits of s F˜ • , respectively, and Φσ˜ We conclude that D(˜ σ ) and D(˜ ∞ ˇ σ ) to D ˇ s. restricts to a Gs -equivariant isomorphism from D(˜ 7.2. Construction of the enhanced SL2 -orbits. The above discussion gave rise to identiˇ we shall now “interpolate” ˇ ˇ s and Φσ (D(σ))Zar of D; fications of D(σ) with the two subsets D ∞ ˜ them. Via the section D(σ) ֒→ B(σ) ։ B(σ); each point g · s F • ∈ D(σ), with g ∈ Gs,R , ˇ with Q-split LMHS. Taking the Zariski closure produces a σ–nilpotent orbit eC σ g · os ∈ D of the union of these, we have the Definition 7.5. The enhanced (multivariable) SL2 -orbit associated to (os , σ) is dfn

X(σ) = eC σ Gs · os

Zar

˜ s · os , = G

where ˜ s dfn G = Gs × (

s Y

β

SL2 j ) .

j=1

˜ Note that X(σ) does not identify with a subset of B(σ), though eC σ Gs · os does. Proposition 7.6. The enhanced SL2 –orbit X(σ) is the compact dual of a Mumford–Tate subdomain Y (σ) ⊂ D; in particular, it is smooth. (22)The term reduced limiting period mapping is used in [25, App. to §10], but we prefer to reserve the

term “period mapping” for Hodge–theoretic classifying maps associated to families of motives. They are “naive” because they replace the limit MHS of a nilpotent orbit by its (much coarser) actual limit, which students in Hodge theory are trained to not take.


55

Proof. The homogeneous description in Definition 7.5 already implies smoothness. Since os is Q-split (a fortiori R-split) [32, Lemma 5.14] yields a diagram ≃ ˇ ˇ D(σ) × (P1 )×s → X(σ) ⊂ D ∪ ∪ ∪ ≃ ×s → Y (σ) ⊂ D D(σ) × H

in which the bottom row is given by

P (g · s F • ; νz1 , . . . , zs ) 7→ exp( sj Nj )g · s F • .

ˇ Moreover, it is clear that (Note that the P1 factors need not be minimally embedded in D.) β the generic Mumford–Tate group of Y (σ) is GB(σ) × (×sj=1 SL2 j ), which acts transitively on Y (σ). Lemma 7.7. The enhanced SL2 –orbit X(σ) is horizontal (and Hermitian) if and only if −1 ≤ E(α) ≤ 1 for every α ∈ ∆ strongly orthogonal to {β1 , . . . , βs }. Proof. The following are equivalent: ◦ X(σ) is horizontal. ◦ Y (σ) is horizontal. ˜ s )). ◦ −1 ≤ E(α) ≤ 1 for all α ∈ ∆(Lie(G ◦ −1 ≤ E(α) ≤ 1 for all α ∈ ∆(Lie(Gs )) = ∆(Γs ) = ∩j ∆(Γβj ), cf. Section 6.7.

In Appendix A we determine the bigradings {s gp,q } for s = 1 and P a maximal parabolic. The condition of the lemma fails in many cases, but holds for a sizable subset which includes all the adjoint varieties (including type A, where the parabolic is not maximal). We will elaborate on these cases in Sections 7.3-7.5. One reason for studying the X(σ) is to express classes of (effective linear combinations of) Schubert varieties in terms of “simpler” objects. Of particular interest is the horizontal case, where [X(σ)] can be written in terms of classes of Schubert VHS [45]. Here “simpler” might mean that X(σ) is smooth while the Schubert varieties [resp. VHS] of a given dimension are singular; or it could simply be the case that X(σ) (note its P1 factors) is decomposable while the {Xw } are not. Along these lines we have the easy

ˇ = G/P with P a maximal parabolic corresponding to a non-short Proposition 7.8. If D simple root, then the {X(σ)} of dimension at least 2 are never Schubert varieties.

Proof. By [27, Proposition 3.7], the smooth Schubert varieties are of the form X(D ′ ) for D ′ ⊂ D a connected subdiagram of the Dynkin diagram (cf. Remark 4.8). So they do not have P1 factors. ˇ = 7.3. Fundamental adjoint varieties I: codimension one orbits. In the case that D G/P is a fundamental adjoint variety (Section 2.5), there is a unique codimension-one GR – orbit O1 ∈ bd(D) by Proposition 6.56. If αi is the simple root associated to the maximal parabolic P , then taking B = {β1 = αi } yields a distinguished base point o1 ∈ O1 via the construction of Section 6.8.

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Figure 7.1. The bigrading g = ⊕ 1 gp,q for fundamental adjoint varieties. r

q ✻

r

❡ r

❡ r

✛

❡ r

✐ ❡ r ❡ r

r

❡ r

❡ r

❄

r

r ✲p r

We claim that the bigradings g = ⊕1 gp,q associated to o1 exhibit a uniform appearance for all the adjoint varieties, as indicated in Figure 7.1. There, the nodes • indicate a gp,q of dimension one, the once-circled nodes indicate dimension a, and the twice-circled node indicates dimension b, where the values of a, b are as given in Table 7.1. Table 7.1. Dimensions of the gp,q in Figure 7.1. G

Br (r ≥ 3)

r 1 r❡ 2r − 4 ✐ 2r2 − 11r + 18 r❡

Dr (r ≥ 4) 1

2r − 5

2r 2 − 13r + 24

E6

E7

E8

F4

G2

1

1

1

1

1

9

15

27

6

1

18

37

80

10

2

dfn

Writing gp,q for 1 gp,q and hp,q = dimC gp,q , we recall that gp,q = gp ∩ gp+q , where the grading g = ⊕gp [resp. g = ⊕gℓ ] is induced by E = Si [resp. Hβ1 = Hαi ]. By Lemma 6.44 we have hp,q = hq,p = h−q,−p = h−p,−q . To verify Figure 7.1 we shall also need Lemma 7.10 below, which gives dim(gℓ ) = dim(g−ℓ ). It also implies w(gp,q ) = w(gp ∩ gp+q ) = g−(p+q) ∩ g−p = g−p−q,q ,

where w ∈ W is as in Lemma 7.10, and hence the additional symmetry hp,q = h−p−q,q of the Hodge–Deligne numbers. From Section 2.5, we know that dim(g−2 ) = dim(g2 ) = 1 and dim(gj ) = 0 for |j| > 2. Hence dim(g−2 ) = dim(g2 ) = 1 (and dim(gℓ ) = 0 for |ℓ| > 2), so that hp,q = hq,p implies g−2 = g−1,−1 , g2 = g1,1 , h1,1 = h−1,−1 = 1. The “extra symmetry” now gives h−2,1 = h2,−1 = h−1,2 = h1,−2 = 1, as well as h1,0 = h−1,0 = h0,1 = h0,−1 = h−1,1 = h1,−1 =: a. Finally, to obtain the dimensions in Table 7.1 we solve ˇ = n, 2a + 3 = dimC g− = dimC D 6a + b + 6 = dimC g = N + 1 . using Table 2.2. 7.4. A curious symmetry. The main result of this section is Lemma 7.10, which asserts that (in the case of i associated to a fundamental adjoint representation) the decompositions (6.34)j=1 and (6.35)j=1 are congruent under the action of the Weyl group.


57

The grading element corresponding to P is Si . By Table 2.1, the largest Si –eigenvalue on g is α ˜ (Si ) = 2. Therefore, the graded decomposition (6.34)j=1 of g into Si –eigenspaces is g = g2 ⊕ g1 ⊕ g0 ⊕ g−1 ⊕ g−2 .

As discussed in Section 2.3.2, g2 is one-dimensional; indeed, (7.9)

g±2 = g±α˜ .

Let W ⊂ Aut(Λrt ) ⊂ Aut(h∗ ) denote the Weyl group of g. Recall the Hαi –eigenspace decomposition (6.35)j=1 , and the filtrations (6.37)j=1 . Lemma 7.10. There exists w ∈ W such that w(gℓ ) = g−ℓ . In particular, w(Wℓ ) = F −ℓ .

˜∈ Corollary 7.11. The filtration F • is the weight filtration associated to a nilpotent 0 6= N α ˜ g . Before proving Lemma 7.10 we first establish Lemma 7.12. ˜ ∈ [gα˜ , g−α˜ ] ⊂ h by α ˜) = 2. Then S ˜ = Si . Lemma 7.12. Define S ˜ (S

˜ ∈ gα˜ and N ˜ − ∈ g−α˜ such that [N ˜, N ˜ −] = S ˜. Proof of Lemma 7.12. Let 0 6= ξ ∈ gαj . Fix N Then ˜, N ˜− , ξ ˜, ξ = N ˜) ξ = S αj (S ˜,ξ , N ˜ − + ξ, N ˜− , N ˜ . = N

˜ , ξ] = 0. Similarly, the bracket Since α ˜ is a highest root, α ˜ + αj is not a root. Therefore, [N − ˜ [ξ, N ] is nonzero if and only if −α ˜ + αj is a root; equivalently, α ˜ − αj is a root. The latter ˜) = 0 if j 6= i. It follows that S ˜ is is equivalent to g−αj 6⊂ p; that is, j = i. Therefore, αj (S i ˜ ˜ a multiple of Si . Since both α ˜ (Si ) and α ˜ (S) equal 2, it must be the case that S = S . Proof of Lemma 7.10. Note that αi is not a short root. Since the highest root α ˜ of g is also not a short root, it follows αi and α ˜ have the same length. Therefore, there exists a Weyl group element w ∈ W mapping w(−αi ) = α ˜ , cf. [28, Lemma 10.4.C]. Therefore, w(Hi ) ∈ w g−αi , gαi = wg−αi , wgαi = gα˜ , g−α˜ .

˜ = Si . Moreover, By Lemma 7.12, w(Hi ) is necessarily a multiple of S 2 = αi (Hi ) = −(w−1 α ˜ )(Hi ) = −α ˜ (wHi )

forces w(Hi ) = −Si . Given ξ ∈ gℓ , we have

ℓ w(ξ) = w(ℓξ) = w[Hi , ξ] = −[Si , w(ξ)] .

Thus w(ξ) ∈ g−ℓ , and we conclude w(gℓ ) = g−ℓ .

Table 7.2 expresses the Hαi in terms of the grading elements {Sj }.

Remark 7.13. For the grading g = ⊕gp associated to a fundamental adjoint variety, we know that g−1 is an irreducible g0 –module. It will be important in the sequel that the symmetry in Lemma 7.10 identifies g−1 as a g0 -module with g1 as a g0 –module.

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Table 7.2. The grading element H. G D4 Br , D r E6 E7 E8 F4 G2

H 1

2

−S + 2S − S3 − S4 −S1 + 2S2 − S3 2S2 − S4 2S1 − S3 −S7 + 2S8 2S1 − S2 −S1 + 2S2

7.5. Fundamental adjoint varieties II: boundary components and enhanced SL2 orbits. We continue with the notation of Section 7.3, and write N ∈ g−1,−1 ∩ gQ for the nil-negative element of the standard triple associated to β1 = αi . In this section we briefly describe X(N ) and B(N ). In each case it is clear from the description in Section 7.1 that gss B(N ) is the semisimple Lie subalgebra of g whose roots are the ones strongly orthogonal to the αi . Since g1 ∼ = (23) ss ss g−1 = z−1 (N ) is a faithful representation of Gss , and φ ∈ g , G is the generic B(N ) B(N ) B(N ) Mumford–Tate group of the Hodge structures on g1 parameterized by D(N ). Moreover, the α ˜ 2,−1 ⊂ g presents D(N ˇ ) as a subvariety of Pg1 . Since action of Gss 1 B(N ) on the line g = g the image of α ˜ under w (cf. Section 7.4) is the highest g0 -weight of g−1 , we obtain ˇ ) ⊂ Pg1 is isomorphic to the variety of lines Proposition 7.14. The compact dual D(N −1 ˇ ˇ )∼ Co ⊂ Pg . We therefore have (in D) X(N ) ∼ = P1 × D(N = P 1 × Co . ˇ This Proof. We only need to check that the P1 factor of X(N ) is minimally embedded in D. α i ˜ is also the highest weight of g. follows from α ˜ (H ) = 1, since α ρ

Turning to the boundary component B(N ) ։ D(N ), we may think of points in D(N ) as dfn Hodge flags F • on the Tate twist UN = g1 (−1) = g−1 (−2) (weight 3 HS). From (7.3) we see at once that the fibres of ρ take the form UN /F 2 UN . Now the object that provides a partial compactification of the quotient of D by a neat arithmetic subgroup G of GQ , is the quotient B(N ) of B(N ) by G ∩ Z(N ), cf. [31, Section 7]. ρ¯

Proposition 7.15. For each of the fundamental adjoint varieties, B(N ) ։ D(N ) is a family of intermediate Jacobians associated to a variation of Hodge structure (with Hodge numbers (1, a, a, 1)) over a Shimura variety. The corresponding variations UN → D(N ) recover (with one exception) the list of weight 3 maximal Hermitian VHS of Calabi-Yau type from [22]. Proof. Immediate from comparison with [22] and the fact that Gss B(N ) is the Mumford–Tate group of UN . (23)Gss

B(N) is the group Gs (s = 1) of Section 7.2, but we will not write G1 due to notational conflict with g1 (which is obviously not its Lie algebra).


59

Table 7.3. D(N ) and X(N ) for the fundamental adjoint varieties g gR ss gB(N ),R D(N ) X(N ) dim(X(N ))

so(n + 4) (n ≥ 3) so(4, n)

so(2, n − 2) ⊕ su(1, 1)

H × IVn−2

e6

e7

e8

f4

g2

E II su(3, 3)

E VI so∗ (12)

E IX E VII

FI sp(3, R)

G su(1, 1)

I3,3

II6

E VII

III3

H

P1 × P1 × Qn−2

P1 × Gr(3, C6 )

P 1 × S6

P1 × E7 /P7

P1 × LG(3, C6 )

P1 × v3 (P1 )

n

10

16

28

7

2

Remark 7.16. We refer to Corollary 2.29 and Theorem 6.7 of [22] for further description of these maximal variations. The missing case, labeled I1,n in [22, Corollary 2.29], arises in the same way from a boundary component of the (non-fundamental) adjoint variety for An+2 . ˇ ) yield the homogeneous Legendrian varieties With this addition, the compact duals D(N as presented in [22, Theorem 6.7]; whereas the varieties of lines Co realize them (as in [37]) as the subadjoint varieties. The Weyl flip of Section 7.4 toggles between these realizations. To the authors, this suggests comparing the homology classes of X(N ) and the Schubert variety Cone(Co ), cf. Section 7.6. The data for all the fundamental adjoint varieties is summarized in Table 7.3, where we note that dim(X(N )) = a + 1. The type of D(N ) as a Hermitian symmetric domain comes from [22]; otherwise, see Appendix A for more details. ˇ Proposition 7.17. The X(N ) of Table 7.3 are all maximal horizontal subvarieties of D. Proof. By [45, Corollary 3.13], the dimension of a horizontal manifold is bounded by the ˇ So Corollary 2.23 establishes the proposition maximal dimension of the Schubert VHS in D. for g = so(n+4). Likewise [45, Example 5.9, and Corollaries 5.13 and 5.29] yield Proposition 7.17 for the exceptional g = e6 , f4 , g2 . Computing as in [45, Section 5], one may confirm that the maximal Schubert VHS in ˇ = E7 (C)/P1 are all of dimension 16. Indeed, they are the Xw the E7 –adjoint variety D with ∆(w) = {α ∈ ∆ | α(S1 ) = 1 , α(Tw ) ≤ 0} given by (the first column of) Table 7.4. ˇ = E8 (C)/P8 are all of Similarly, the maximal Schubert VHS in the E8 –adjoint variety D 8 dimension 28; they are the Xw with ∆(w) = {α ∈ ∆ | α(S ) = 1 , α(Tw ) ≤ 0} given by (the second column of) Table 7.4. 7.6. Computing [X(N )]: a simple example. One purpose of introducing the X(σ) was ˇ Z). In particular, for the to produce smooth algebraic representatives of classes in H∗ (D, G2 and F4 adjoint varieties, none of the maximal Schubert VHS are smooth. So it seems natural to conclude this paper by computing [X(N )] for the G2 -adjoint variety. ˇ Z) = Z[X], Recall the Schubert variety X of (3.2). Because dimC X(N ) = 2 and H4 (D, the class [X(N )] will necessarily be a multiple of [X]. We will determine this multiple. We begin with the description of X as a Tits transform. Let Q ⊂ G2 be the maximal parabolic associated to the first simple root, so that G2 /Q is a five dimensional quadric, and the minimal homogeneous embedding lies in PVω1 . Then G2 /Q parameterizes a uniruling ˇ by P1 s, cf. Table 3.1. Let G′ be the connected simple subgroup of G2 associated to of D

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Table 7.4. The maximal Schubert VHS in the E7 and E8 –adjoint varieties The grading element Tw e7 e8 −S1 + S3 S2 − S8 −S1 + S5 S5 − 2S8 1 3 6 2 −2S + S + S S + S6 − 3S8 −3S1 + S3 + S5 + S7 S2 + S5 + S7 − 5S8 −2S1 + S4 + S7 S4 + S7 − 4S8 1 2 7 −S + S + S S3 + S7 − 3S8 S7 S1 + S7 − 2S8 S7 − S8 .

the simple root α1 , and let Σ ⊂ G2 /Q be the G′ –orbit of the identity coset Q/Q. By (3.8a) and Lemma 3.13, the variety X is the Tits transform T (Σ). If η is the fundamental weight of G′ , then ω1 restricts to η on g′ . It follows that Σ is a minimal homogeneous embedding of P1 . Let Σ(N ) denote the Gss B(N ) (C)–orbit of Q/Q in G2 /Q. Then X(N ) is the Tits transform T (Σ(N )). Note that the simple root 2α1 + α2 of Gss B(N ) is image of α1 under the Weyl group ss element w = (12) ∈ W. It follows that GB(N ) = Adw (G′ ). As a consequence we may identify η with the fundamental weight of Gss B(N ) . Claim. The restriction of ω1 to Gss B(N ) is 2η. It follows that Σ(N ) is the second Veronese embedding of P1 . Since dimC Σ(N ) = 1 and H2 (G2 /Q, Z) = Z[Σ], we may conclude that [Σ(N )] = 2 [Σ] . It follows from [17, Lemmas 3.11 and 3.13] that [X(N )] = 2 [X] . α α −α where α = 2α + α . The Proof of claim. The Lie algebra gss 1 2 B(N ) (C) is g ⊕ C H ⊕ g α fundamental weight η is defined by η(H ) = 1. On the other hand Hα = S1 , as an element of the Cartan subalgebra of g2 , and ω1 = 2α1 + α2 . Therefore ω1 (Hα ) = 2, and the claim follows.

ˇ The analogous computation of [X(N )] in the case of the SO(2r + 1, C)–adjoint variety D is worked out in Appendix B. The systematic computation of [X(σ)] will be taken up in a subsequent paper, using different methods. Appendix A. Examples in codimension one Let G be simple and P a maximal parabolic with associated grading element E = Si . Assume that the first root α1 ∈ B is the simple root αi (Section 6.9). Then the GR –orbit


61

O1 has codimension one (Lemma 6.52). In this section we will determine the associated bigradings gp,q =

(A.1)

1g

p,q

for each pair (g, i). Setting N = N1 ∈ g−αi ⊂ g−1,−1 , we will also identify the boundary ˜ )/exp(CN ), and the Mumford–Tate domain D(N ) for the split component B(N ) = B(N Hodge structure. Remark A.2 (The Hermitian symmetric case). From these examples we will see that if G/Pi is Hermitian symmetric, then the bigrading is of the form depicted in Figure A.1.b. A.1. Type A. Suppose that g = slr+1 C. Then gR = su(i, r + 1 − i), the roots of g are ∆ = ±{αjk | 1 ≤ j ≤ k ≤ r} ,

(A.3) where

dfn

αjk = αj + · · · + αk , and H So, we have (A.4)

i

  2 S1 − S2 , −Si−1 + 2 Si − Si+1 , =  −Sr−1 + 2 Sr , gp,q = 0

if i = 1 , if 1 < i < r , if i = r .

if |p| > 1 or |q| > 1.

If i ∈ {1, r}, then we also have g1,−1 = g−1,1 = 0. All other gp,q are nonzero. These observations are represented in Figure A.1, where each node in the pq–plane represents a Figure A.1. The nonzero gp,q . q

q

s

s✻ s s ✲ s p

s s

s✻ s s ✲ s p

s

s

s

s

(A.1.A)

s

(A.1.B)

non-trivial gp,q . (Figure A.1.A represents the case i ∈ {1, r}, and Figure A.1.B represents the case 1 < i < r.) We now identify the Mumford–Tate domain D(N ) and boundary component B(N ). We will assume that 1 < i < r, leaving the other two cases (which are symmetric) to the reader. In particular, the bigrading gp,q is of the form depicted in Figure A.1.B. In this example, √ φ = − −1(Si−1 + Si+1 ). From Figure A.1.B and Lemma 6.52, we see that g0 = z0 (N )⊕hHi i, so that ∆(z0 (N )) = ± {αjk | i + 2 ≤ j ≤ k ≤ r , 1 ≤ j ≤ k ≤ i − 2 , or 1 ≤ j < i < k ≤ r} .

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To determine gB(N ) , observe that given φ acts on gα ⊂ z0 (N ) by a nonzero scalar if and √ only if j < i < k (in which case that scalar is ±2 −1). So, gB(N ) is the Q–Lie algebra closure of the subspace M g±αjk = z0 (N ) ∩ g−1,1 ⊕ g1,−1 . j 3 (Section B.7.2), and compute n1,2r , n2,2r−1 and n3,2r−2 (Sections B.7.1, B.7.3 and B.7.4). B.7.1. Computation of n1,2r . The integer n1,2r is the number of points in the intersection of X(N ) with a general Schubert variety X2,2r+1 representing the Poincaré dual x∗1,2r . From (B.11) we see that ) ( hui ∈ Q2 , hvi ∈ Q2r−5 N N . X(N ) ∩ X2,2r+1 = hu, vi dim(hu, vi ∩ F2 ) ≥ 1

Observe that the set of all one-dimensional subspaces contained in an intersection of the form hu, vi ∩ F2 is in bijection with in the intersection of PF2 with the join dfn

2r−5 ). J1 = J(Q2N , QN

Therefore, the intersection (B.11) is nonempty (and n1,2r is nonzero) if and only if the intersection P(F2 ) ∩ J1 is nonempty. 2r−5 ) are contained in the quadric Q2r−1 ⊂ P2r . By Note that both P(F2 ) and J(Q2N , QN Kleiman’s transversality theorem [33, Corollary 4], for a general choice of F2 this intersection is proper. Since P1 = P(F2 ) and J1 are of complimentary dimensions in Q2r−1 , we see that the intersection is a (nonzero) finite number of points each with multiplicity one. B.7.2. Computation of na,b . The computation of the coefficients na,b is similar to that of n1,2r . We proceed as follows. Recall that the Schubert varieties of Q2r−1 are dfn

Xa (F• ) = {hui ∈ Q2r−1 | u ∈ Fa } = Q2r−1 ∩ PFa ,

a 6= r + 1 .

Again, for notational convenience, we will abbreviate Xa (F• ) as Xa . If a ≤ r, then Xa = PFa has dimension a − 1; if a ≥ r + 2, then dim Xa = a − 2. By Kleiman’s transversality 2r−5 and theorem, we may choose the F• so that the Schubert varieties Xa intersect Q2N , QN 2r−1 m m a J1 properly in Q . In particular, as dim Xa = a − 2 and QN ∩ Xa = QN ∩ PF , we have the following.

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KERR AND ROBLES

(a) The intersection Q2N ∩ PFa is empty if a < 2r − 1; consists of two points if a = 2r − 1, and is one-dimensional quadric if a = 2r. (b) Assume r ≥ 5. If r + 2 ≤ a, then the intersection is a quadric of dimension a − 6. (c) If a ≤ r, then Xa ∩ J1 has dimension a − 2. (In particular, X1 ∩ J1 is empty.) Assume that r ≥ 5. The case that r = 3 is treated in Section B.8; we leave = 4 as an exercise for the reader. Following Section B.6 we assume that a+b = 2r +1 and proceed to compute na,b , keeping in mind that 1 ≤ a ≤ r and r + 2 ≤ b ≤ 2r + 1. As noted in (a) above, if b + 1 < 2r − 1, then Q2N ∩ Fb+1 is empty. It follows from (B.11) that X(N ) ∩ Xa+1,b+1 is empty and na,b = 0

if b ≤ 2r − 3 (equivalently, a > 3) .

So it remains to determine n3,2r−2 and n2,2r−1 .

B.7.3. Computation of n2,2r−1 . In this case, A2 = Q2N ∩ PFb+1 is a one–dimensional quadric 2r−5 ∩ PFb+1 is a quadric of dimension 2r − 6 by (b). By (B.11), the by (a), and B2 = QN points of X(N ) ∩ Xa+1,b+1 are in bijection with the intersection of PFa+1 with the join J2 = J(A2 , B2 ). 2r−4 in Fb+1 = F2r . We claim The two quadrics A2 and B2 span linear spaces C3A and CB 1 that F is transverse to 2r−4 . U2 = C3A ⊕ CB This is a consequence of (c). In particular, suppose that 0 6= u+v ∈ F1 , and that u ∈ C3A and 2r−3 2r−4 are ν–orthogonal, and C3A ⊂ C4N and . Since F1 is a ν–isotropic line, C4N and CN v ∈ CB 2r−5 2r−3 2r−4 . It follows that hu + vi = PF1 ∈ J1 . we see that hui ∈ Q4N and hvi ∈ QN ⊂ CN CB This contradicts (c), and establishes the claim. It follows from the claim that F2r = U2 ⊕ F1 . In particular, Fa+1 ∩ U2 is a ν–isotropic 2–plane. The Schubert variety X2r is cone over a smooth quadric Q2r−3 with vertex the point PF1 . Therefore, the intersection of X2r with U2 is a smooth quadric Q2r−3 . Note that P1 = P(Fa+1 ∩ U2 ) and J2 both lie in Q2r−3 and have complimentary dimension. By Kleiman’s transversality, the general translate of P1 by the symmetry group SO(2r − 1) = {g ∈ SO(2r + 1, C) | g(F1 ) = F1 and g(F2r ) = F2r } of Q2r−3 is transverse to J2 . Thus, the intersection P1 ∩ J2 = P(F3 ) ∩ J2 is a set of points (each of multiplicity one) of cardinality equal to the degree of J2 . Remark B.12. By Kleiman’s transversality we may choose F2 so that P(U2 ∩ F2 ) ∩ J2 = P(F2 ) ∩ J2 is empty.

B.7.4. Computation of n3,2r−2 . In this case, A3 = Q2N ∩ PFb+1 consists of two points by 2r−5 ∩ PFb+1 is a quadric of dimension 2r − 7 by (b). By (B.11), the points (a), and B3 = QN of X(N ) ∩ Xa+1,b+1 are in bijection with the intersection of PFa+1 with the join J3 of A3 and B3 . 2r−5 in Fb+1 = F2r−1 . We claim The two quadrics A3 and B3 span linear spaces C2A and CB that F2 is transverse to 2r−5 . U3 = C2A ⊕ CB This is a consequence of Remark B.12. In particular, suppose that 0 6= u + v ∈ F2 , and 2r−4 2r−5 are ν–orthogonal, and C2A ⊂ C3A and . Since C3A and CB that u ∈ C2A and v ∈ CB


79

2r−4 2r−5 we see that u ∈ A2 and v ∈ B2 . It follows that hu + vi ∈ J2 . This implies ⊂ CA CB 2 PF ∩ J2 is nonempty, contradicting Remark B.12. It follows from the claim that F2r−1 = U3 ⊕ F2 . In particular, Fa+1 ∩ U3 is a ν– isotropic 2–plane. The Schubert variety X2r−1 is cone over a smooth quadric Q2r−5 with vertex the line PF2 . Therefore, the intersection of X2r−1 with U3 is a smooth quadric Q2r−5 . Note that P1 = P(Fa+1 ∩ U3 ) and J3 both lie in Q2r−5 and have complimentary dimension. By Kleiman’s transversality, the general translate of P1 by the symmetry group SO(2r − 3) = {g ∈ SO(2r + 1, C) | g(F2 ) = F2 and g(F2r−1 ) = F2r−1 } of Q2r−5 is transverse to J3 . Thus, the intersection P1 ∩ J3 = P(F4 ) ∩ J3 is a set of points (each of multiplicity one) of cardinality equal to the degree of J3 .

B.8. The case r = 3. In Section B.7.2 we restricted attention to the case that r ≥ 5. As promised we consider here the case that r = 3. Specializing the work of Section B.6 and B.7 to r = 3 we see that dim X(N ) = 2r − 3 = 3 , and there are two Schubert varieties X = X1,6 and Y = X2,5 of dimension three. The elements of Wpmin indexing x1,6 and x2,5 are (231) and (232), respectively. Therefore, [X(N )] = n1,6 x1,6 + n2,5 x2,5 for integers n1,6 , n2,5 ≥ 0. The computation of n1,2r in Section B.7.1 holds for all r ≥ 3. However, we will compute the coefficients n1,6 and n2,5 by the BGG polynomials and divided difference operators of [4], rather than intersection theory approach of Section B.7. We find (B.13)

[X(N )] = 2 x1,6 + 2 x2,5 .

As a consistency check, we finish this section with some elementary observations on the degrees of the varieties involved. The Schubert varieties X1,6 and X2,5 are of degrees four and two, respectively (in the minimal homogeneous embedding G/P ֒→ Pg). Therefore, deg X(N ) = 4 n1,6 + 2 n2,5

(B.13)

=

12 .

On the other hand, we may compute deg X(N ) as follows. Set τ1 = α2 . Recall (A.5) that the simple roots of gss B(N ) are τ2 = α2 + 2α3 and τ3 = α1 + α2 + α3 . Note that the roots τ1 , τ2 , τ3 are all pairwise strongly orthogonal, and therefore form a root subsystem. Let g(N ) be the associated semisimple subalgebra of g; then strong orthogonality implies g(N ) ≃ sl2 C ⊕ sl2 C ⊕ sl2 C. Since N ∈ g−α2 , it follows that X(N ) is a G(N )–orbit. If η1 , η2 , η3 are the fundamental weights of g(N ), then the fundamental weight ω2 of g restricts to η1 + η2 + 2η3 on g(N ). It follows that X(N ) ≃ P1 × P1 × v2 (P1 ) , and by [7, Theorem 24.10] deg X(N ) = 12 . This completes the consistency check.

80

KERR AND ROBLES

Appendix C. Dynkin diagrams For the readers convenience we include in Figure C.1 the Dynkin diagrams of the complex simple Lie algebras. Recall that: each node corresponds to a simple root αi ∈ S; two nodes are connected if and only if hαi , αj i 6= 0 and in this case the number if edges is |αi |2 /|αj |2 ≥ 1 (that is, i, j are ordered so that the inequality holds). Below, if G = Br , then r ≥ 3; and if G = Dr , then r ≥ 4. Figure C.1. Dynkin diagrams for complex simple Lie algebras. 1r

2r

r♣ ♣ ♣ r

r −r 1 r r A = sl r r+1 C

1r

2r

r♣ ♣ ♣ r

r −r 1 r r B = so r 2r+1 C i

1r

2r

1r

2r

r −r 1 r r C = sp C r h 2r rr− 1 ✏ 2 r♣ ♣ ♣ r r−✏ r ✏ P P r D = so C

r♣ ♣ ♣ r

Pr

r2 1r

3r

4r

r

5r

6r

5r

6r

7r

E7

5r

6r

7r

8r

2r

E6

r2 1r

3r

4r

r2 1r

3r

4r

1r

2r

3 i r

4r F 4

1r

E8 2 h r G2

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Variations of Hodge structure and orbits in flag varieties

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