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GENERALIZING THE SATAKE-BAILY-BORELCOMPACTIFICATION

COLLEEN ROBLES

Abstract. These are informal, expository notes on two generalizations (group-

theoretic and Hodge-theoretic) of the Satake-Baily-Borel compactification of a lo-

cally Hermitian symmetric space Γ\D. These two generalizations are recent, joint

work with Mark Green, Phillip Griffiths and Radu Laza, that is motivated by a

project to apply Hodge theory to study KSBA compactifications of moduli spaces

of algebraic surfaces of general type. The current draft is inchoate (in preparation

for a mini-course at Duke).

Contents

1. Introduction 1

Part A: Background 10

2. Linear algebra 10

3. Hodge theory 15

Part B: Group theoretic generalization 31

4. Overview 31

5. Basic structure 33

6. Boundary components induced by horizontal SL(2)s 38

References 49

1. Introduction

1.1. Objective. Let D ' G/G0 be a Mumford–Tate domain. Here, G = G(R)

denotes the Lie group consisting of the real points of a connected, reductive, linear

algebraic group G defined over Q, and G0 ⊂ G is a compact subgroup and the

Date: October 17, 2017.

Robles is partially supported by NSF grants DMS 1361120 and 1611939.1

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centralizer of circle S1 ↪→ G, [14]. Let Γ ⊂ G(Q) be an arithmetic group. In the case

that D is Hermitian symmetric,1 there are many compactifications of Γ\D; see [5]

for an excellent survey. Among these the Satake–Baily–Borel (SBB) compactification

Γ\D∗ is distinguished by the following properties: (i) it is a projective algebraic

variety [1] (a Shimura variety), and (ii) by Borel’s extension theorem [2], which asserts

that every holomorphic, locally liftable map (∆∗)r → Γ\D defined on a product of

punctured discs, extends across the punctures to ∆r → Γ\D∗.Both SBB and Borel’s extension theorem have been instrumental in applying

Hodge theory to study moduli spaces of K3 surfaces and abelian varieties, and their

compactifications. However, for most moduli spaces the domains arising are not

Hermitian, and it has been an open question since the 1960s to generalize SBB. (The

idea being that this will allow us to extend the application of Hodge theory to a larger

class of moduli spaces.)

This turns out to be a subtle and interesting problem. SBB is a (algebraic) group

theoretic construction with a robust Hodge theoretic interpretation. These two ‘faces’

of SBB suggest two generalizations: Let Φ : B → Γ\D be a holomorphic, horizontal,

locally liftable map defined on a smooth quasi-projective B with smooth projective

completion B such that B\B is a normal crossing divisor.

(1) The group theoretic generalization (GT-SBB) is a horizontal completion Γ\Dh

of an arithmetic quotient Γ\D of a Mumford-Tate domain. It has the same

‘good’ group theoretic properties as SBB (§4), and the closure of Φ(B) in Γ\Dh

is compact.

(2) The Hodge theoretic generalization (HT-SBB) is an algebraic compactification

Φ(B) of the image of Φ that is constructed by attaching limiting mixed Hodge

structures (modulo extension data), and that admits an extension Φe : B →Φ(B) of Φ.2 Here the monodromy group need not be arithmetic.

In the Hermitian case, these two constructions coincide. In general, they do not.

(Whence the subtlety and fun.) Indeed the objects have fundamentally different

natures: The completion Φ(B) of Φ(B) in (2) is a projective algebraic variety (and

1Equivalently, G0 is a maximal compact subgroup K of G.2Greg Pearlstein and Christian Schnell are working on a completion of the period map that

attaches the full limiting mixed Hodge structure.

INFORMAL COURSE NOTES 3

realizes Φ(B) as a quasi-projective variety). In contrast, if D does not fiber (anti-

)holomorphically over a Hermitian domain, then Γ\D carries no algebraic structure

[15]. Similarly, in the setting of (2), we have a generalization of Borel’s theorem:

there exists an extension Φe : B → Φ(B) of the period map. In the setting of (1),

there always exists an extension B → Γ\Dh when dim B = 1. But when dim B > 1,

the existence of the extension depends on representation theoretic properties of the

monodromy cones associated with points of B\B.3

GT-SBB may be viewed as a ‘meta-construction’ encoding the structure that is

universal among all instances of HT-SBB for a given period/Mumford-Tate domain.

This is made precise as follows: Given Φ : B → Γ\D, let Φ(B)h be the closure of

Φ(B) in Γ\Dh. Then there is a continuous surjection Φ(B)h → Φ(B) that restricts

to the identity on Φ(B).

The objective of the mini-course is to explain these two generalizations, and their

relationship, both to each other, and to other constructions.

1.2. Mumford–Tate groups and domains. Mumford–Tate groups are reductive

Q–algebraic groups G admitting a circle ϕ : S1 → G(R) = G with compact centralizer

G0. The full definition is given in §3.1.3; for now we give two examples in §§1.2.1 &

4.3. The associated Mumford–Tate domain is D = G/G0. Mumford–Tate domains

generalize Hermitian symmetric domains. The compact dual D of D is a rational

G(C)–homogeneous variety (a.k.a. a generalized flag domain) containing D as an

open G–orbit.

Let i =√−1. Fix a rational vector space V , an integer n ∈ Z and a nondegener-

ate, (−1)n-symmetric bilinear form Q : V × V → Q. For the purposes of these two

examples, let

G := Aut(V,Q)

be the automorphism group of Q. (In general, we will have G ⊂ Aut(V,Q).) The

group G can always be realized as a Mumford–Tate group. If the rational structure

on V = VZ ⊗Z Q is induced by a lattice VZ, then Γ = Aut(VZ, Q) is an arithmetic

subgroup.

3These properties always hold when D is Hermitian.

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1.2.1. A Hermitian example. Suppose that n = 1 so that Q is skew-symmetric. Then

dimV = 2g is even, and G ' Sp(2g,R). Define a Hermitian form Q∗ on VC by

Q∗(u, v) := iQ(u, v) .

Then the set

D := {E ∈ Gr(g, VC) |Q|E ≡ 0 , Q∗|E > 0}

of all g–dimensional linear subspaces of VC on which Q restricts to be zero (we say

E is Q–isotropic) and the Hermitian form to be positive definite is a Mumford–Tate

domain. The stabilizer

G0 ' U(g)

of a point E ∈ D is a maximal compact subgroup of G (D is Hermitian symmetric)

and may be realized as the centralizer of the circle ϕ : S1 → G defined by ϕ(z)v = z v

and ϕ(z)v = z−1v for all z ∈ S1 and v ∈ E.

In this case Γ = Aut(VZ, Q) ' Sp(2g,Z). The quotient Γ\D may be identified

with the moduli space Ag of principally polarized abelian varieties of dimension g,

and the SBB compactification Γ\D∗ = Ag ∪ Ag−1 ∪ · · · ∪ A1 ∪ A0.

The group G(C) ' Sp(2g,C) of complex points acts transitively on the compact

dual

D := {E ∈ Gr(g, VC) |Q|E ≡ 0}

Remark 1.2.1. Under the identification E = V 1,0, D is the period domain parameter-

izing effective, weight n = 1, Q–polarized Hodge structures on V . The compact dual

is the set of subspaces E ⊂ VC satisfying the first Hodge–Riemann bilinear relation,

but not the second.

Example 1.2.2 (A toy example). Suppose that g = 1. Fixing an identification V ' Q2,

and regarding v ∈ V as column vectors (v1, v2)>, we may take Q(u, v) to be the

determinant det(u, v). Then G = Sp(2,R) = SL(2,R). Then the compact dual D is

the complex projective line P1. We compute

Q∗(v, v) = i det(v, v) = i(v1v2 − v2v1

).

In order for E = span{v} to lie in D, we much have v1v2 6= 0. So without loss of

generality we may suppose that v = (1, τ)⊥. Then 0 < Q∗(v, v) = Im(τ) if and only


if τ lies in the upper half plane H ⊂ C ⊂ P1. The group G acts on H by projective

linear transformation(a b

c d

)[1

τ

]=

[a+ bτ

c+ dτ

]=

[1

c+dτa+bτ

](a.k.a. Mobius transformation τ 7→ c+dτ

a+bτin this context).

1.2.2. Running example: period domain for h = (a, b, a). Suppose that n = 2 so that

Q is symmetric. Then G ' O(b, 2a), where (b, 2a) is the signature of Q on VR. Define

a Hermitian form Q∗ on VC by

Q∗(u, v) := −Q(u, v) .

Then the set

D := {E ∈ Gr(a, VC)| Q|E ≡ 0 , Q∗|E > 0}

of all Q–isotropic, a–dimensional linear subspaces of VC on which the Hermitian form

restricts to be positive definite is a Mumford–Tate domain. Fix E ∈ D. Then

V 2,0 := E and V 2,0 ⊕ V 1,1 := E⊥ = {v ∈ VC | Q(e, v) = 0 , ∀ e ∈ E} ,

determines an Q∗–orthogonal decomposition

VC = V 2,0 ⊕ V 1,1 ⊕ V 0,2 .

The stabilizer

G0 ' O(b)× U(a)

of E ∈ D may be realized as the centralizer of the circle ϕ : S1 → G defined by

ϕ(z)v = zp−q v for all z ∈ S1 and v ∈ V p,q. Note that G0 is a maximal compact

subgroup of G (and D is Hermitian symmetric) if and only if a = 1.

The group G(C) ' O(2a+b,C) of complex points acts transitively on the compact

dual

D := {E ∈ Gr(g, VC) |Q|E ≡ 0} .

Remark 1.2.3. As in §1.2.1, D is the period domain parameterizing effective, weight

n = 2, Q–polarized Hodge structures on V with Hodge numbers h = (a, b, a). And

the compact dual parameterizes filtrations E ⊂ E⊥ ⊂ VC that satisfy the first Hodge–

Riemann bilinear relation.

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This example is one of the simplest non-Hermitian Mumford–Tate domains, and we

will use it throughout to illustrate the constructions.

1.3. Review of Satake-Baily-Borel. Satake’s construction produces a (finite) fam-

ily of compactifications. These are topological compactifications and the family ad-

mits a partial order. Baily–Borel showed that one of the minimal (with respect to the

partial order) compactifications admits the structure of a normal projective variety.

In this section we briefly survey the Satake compactifications from the perspective

of Borel–Ji’s “uniform approach” [5]. Parabolic subgroups play a fundamental role

in the construction; we begin by recalling those properties that will be necessary for

the construction.

1.3.1. Parabolic subgroups of G. In these notes parabolic subgroups P ⊂ G will arise

as stabilizers of filtrations

W−n ⊂ W1−n ⊂ · · · ⊂ Wn−1 ⊂ Wn = VR .

(For a general discussion of parabolic subgroups, consult any standard text on rep-

resentation theory.) Because P preserves W•, there is an induced action of P on the

graded quotients

GrW` := W`/W`−1 ;

the unipotent radical is the normal subgroup of U ⊂ P acting trivially on the GrW` .

Specifically,

U = {g ∈ P | (g − 1)(W`) ⊂ W`−1} .

The Levi quotient is the reductive subgroup P/U . A choice of maximal compact

subgroup K ⊂ G, with Cartan involution θK , determines a Levi factor (a lift of the

Levi quotient)

L := P ∩ θK(P ) ,

with the property that

P = U o L .

The Levi factor further decomposes

L = A × M


into a product of the R–split center A of L and a reductive subgroup M with compact

center. Intuitively, one should think of A as a maximal, connected abelian subgroup

of the center of L consisting of semisimple elements acting on VR by real eigenvalues.

The Langlands decomposition of P (with respect to K) is

(1.3.1) P = U × A × M .

Example 1.3.2 (The toy example). Returning to our Toy Example 1.2.2, modulo G–

conjugacy, there is only one nontrivial parabolic subgroup

P =

{(a 0

b 1/a

) ∣∣∣∣∣ a, b ∈ Ra 6= 0

}.

Taking K = SO(2), we have

U =

{(1 0

b 1

) ∣∣∣∣∣ b ∈ R

}, A =

{(et 0

0 e−t

) ∣∣∣∣∣ t ∈ R

},

and M = {±12}, with 12 the 2× 2 identity matrix.

Remark 1.3.3. Let g ⊃ p = u⊕a⊕m denote the Lie algebras of G ⊃ P = U×A×M .

The Lie algebra g admits a matrix representation with a block decomposition so that

p is the Lie algebra of lower block triangle matrices in g, u is the Lie algebra of

strictly lower block triangle matrices in g, l = a ⊕ m is the Lie algebra of diagonal

block matrices in g, and a is the Lie algebra of diagonal matrices in g.

1.3.2. Satake compactifications of D. The construction depends on an choice of domi-

nant integral weight µ of G. This weight determines a G–invariant set Pµ of parabolic

subgroups P ⊂ G. Recall that D = G/K and G = PK. Then the Langlands decom-

position (1.3.1) induces a horospherical decomposition

D = U × A × M/(M ∩K) .

The weight µ also determines a decomposition of M = Cµ ·C⊥µ as the product of two

normal subgroups, and from this we obtain the refined horospherical decomposition

D = U × A × (X ′µ,P ×Xµ,P ) .4

4In the GT-SBB we will similarly have a horospherical decomposition and a factorization of M .

However, in the general case the isotropy group G0 is not a maximal compact subgroup of G. And

8 ROBLES

The factor Xµ,P is a boundary component, and the Satake compactification of D is

the disjoint union

D?µ :=

∐P∈P

Xµ,P

The set P contains G, and Xµ,G = D. We call D the trivial boundary component. The

refined horospherical decomposition is used to define a notion of convergent sequences

on D? that gives D? the structure of a compact, Hausdorff space [4]. (Each point

x ∈ D? admits a neighborhood basis Ux so that each U ∈ Ux intersects Xµ,P is a

(possibly empty) generalized Siegel domain.) Furthermore, the action of G on D

extends to a continuous action on D? with only finitely many orbits.

Example 1.3.4 (The toy example). In the Toy Example 1.3.2 we have D = H. There

is only one Satake compactification (because rankG = 1). In this case both Xµ,P

and X ′µ,P are points. We have H = U × A. The maximal compact subgroup K =

SO(2) ⊃ M is the stabilizer of τ = i. Given u = u(b) ∈ U and a = a(t) ∈ A, as in

Example 1.3.2, the (refined) horospherical decomposition is

H = P · i = UA · i = {b+ i e−2t | b, t ∈ R} = U × A .

A sequence bj + i e−2tj ∈ H converges to the point xµ,P if and only if tj → −∞ and

e2tjbj → 0.

Remark 1.3.5. There are finitely many Satake compactifications D?µ of D. The dom-

inant integral weight can be expressed as a linear combination µ =∑µiωi of fun-

damental weights {ωi} with non-negative integral coefficients 0 ≤ µi ∈ Z. Two

dominant integral weights µ and ν determine the same compactification D?µ ' D?

ν if

and only if µ and ν have the same support in the sense that {i | µi > 0} = {i | νi > 0}.

1.3.3. Satake compactifications of Γ\D. The basic idea behind the compactification

of Γ\D is to restrict to the ‘rational boundary components,’ and then quotient by the

action of Γ.

For SBB we take µ to be the fundamental weight canonically associated with D

as a rational homogeneous variety. We now restrict to this choice of µ, and write

Pµ = P , Xµ,P = XP and D?µ = D?. A boundary component XP is rational if

this means that there will not be an induced factorization of M/(M ∩ K), and thus no refined

horospherical decomposition.


and only if the normalizer P is defined over Q.5 Then one defines PQ = {P ∈P | P is defined over Q}, and

D∗ :=∐P∈PQ

XP .

Endow D∗ ⊂ D? with the subspace topology, and Γ\D∗ with the quotient topology.

Then Γ\D∗ is SBB as a topological compactification. This space is then given the

structure of a normal projective variety by using automorphic forms to construct a

projective embedding [1].

1.4. Contents. As a point b ∈ B approaches b0 ∈ B\B the VHS Φ is asymptotically

approximated by a ‘nilpotent orbit’ [23]. This so-called Nilpotent Orbit Theorem is

the sine qua non when studying the asymptotics of Φ (a.k.a. degenerations of Hodge

structure): period maps are difficult-to-understand ‘transcendental’ objects; nilpotent

orbits are simple algebraic/Lie theoretic objects that capture the singularities that

arise as b→ b0. A nilpotent orbit is equivalent to a ‘polarized mixed Hodge structure’,

and we may think of the latter as a degeneration of Hodge structures. Each nilpotent

orbit is in turn approximated by an especially nice type of nilpotent orbit, a ‘horizontal

SL(2)’ [23, 10]; the latter correspond to ‘R–split polarized mixed Hodge structures’.

These objects will play a key role in both generalizations of SBB; the necessary

background material from representation theory and Hodge theory is reviewed in

Part A. Then the two generalizations, GT-SBB and HT-SBB, are presented in Parts

B and C, respectively.

5For general µ, rationality of P is a necessary, but not sufficient, condition for the boundary

component Xµ,P to be rational.

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Part A: Background

2. Linear algebra

Fix a real, finite dimensional vector space V and a nondegenerate, (skew-)symmetric

bilinear form Q : V× V→ R. Let End(V) be the set of R–linear maps V→ V, set

End(V,Q) := {ξ ∈ End(V) | Q(ξu, v) + Q(u, ξv) , ∀ u, v ∈ V} ,

and let A ⊂ End(V,Q) denote any reductive Lie subgroup.

2.1. Parabolic subgroups. Let K ⊂ G be a maximal parabolic subgroup, and let

P ⊂ G be a parabolic subgroup. The Iwasawa decomposition implies

G = PK ,

a fact that we will make frequent use of.

2.1.1. Properties of the Langlands decomposition. For later use we recall some prop-

erties of the Langlands decomposition (§1.3.1): We have U = exp(u) and A = exp(a).

The Levi L = A×M is the centralizer Z(A) of A. The intersection K∩P = K∩M =

ZK(A) is a maximal compact subgroup of both P and M . The group M (generally)

fails to be connected; it is generated by the connected identity component M o and

ZK(A) = M ∩K.

2.1.2. Relative Langlands decomposition. Given two parabolic subgroups P1, P2 ⊂ G

with Langlands decompositions Pi = Ui × Ai ×Mi, the relative Langlands decompo-

sition asserts that P1 ⊂ P2 if and only if there exists a parabolic subgroup P ′ ⊂ M2

so that P1 = U2 × A2 × P ′. Furthermore, if P ′ = U ′ × A′ ×M ′ is the Langlands

decomposition of P ′ with respect to K ∩MQ, then

U1 = U2 × U ′ , A1 = A2 × A′ and M1 = M ′ .


2.1.3. Rational Langlands decomposition.

Remark 2.1.1 (Rational versus real Langlands decompositions). Given a rational par-

abolic subgroup Q ⊂ G, let Q = Q(R) denote the Lie group of real points. In general

the rational Langlands decomposition Q = UQ×A′Q×M ′Q need not coincide with the

(real) Langlands decomposition Q = UQ×AQ×MQ. Specifically, while the unipotent

radicals UQ of the two decompositions agree, and for a suitable choice of K we can

arrange A′Q ×M ′Q = AQ ×MQ, in general A′Q ⊂ AQ. (The rational factor A′Q is the

connected identity component of SQ(R), where SQ is the split center of the Levi over

Q.) See [5] for further discussion. Nonetheless the identification A′Q×M ′Q = AQ×MQ

is real analytic, so that a(x, y)m(x, y) = a(x, y)m(x, y) determines real analytic func-

tions a(x, y) and m(x, y) taking values in AQ and MQ, respectively. Moreover, assum-

ing conditions (b) and (d) of Theorem 6.4.2, both exp(12

log(y)Y )a(x, y) and m(x, y)

converge (uniformly in x) to the identity as y →∞.

2.2. Grading elements. Grading elements are semisimple endomorphisms that pro-

vide a convenient framework to work with both parabolic subgroups and (mixed)

Hodge structures.

2.2.1. Definition. A grading element of A is any semisimple endomorphism E ∈ A

with respect to which A decomposes as a direct sum

(2.2.1a) A =⊕`∈Z

A`

of E–eigenspaces

(2.2.1b) A` = {ξ ∈ A | [E, ξ] = `ξ}

with integer eigenvalues ` ∈ Z.6 Note that

(2.2.2) E ∈ A0 .

6For more on grading elements see [7, §§3.1.2, 3.1.3, 3.2.7]. However, the reader should beware

that our notion of grading element is more general than that of [7] as we do not assume that g1

generates the nilpotent algebra g+ = ⊕`>0 g`.

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The Jacobi identity implies that the decomposition (2.2.1a) respects the Lie al-

gebra structure in the sense that

(2.2.3) [Ak,A`] ⊂ Ak+` .

In particular, A0 is a (reductive) subalgebra of A. From (2.2.1b) we see that E lies in

the center of A0.

Conversely, any direct sum decomposition (2.2.1a) satisfying (2.2.3), can be real-

ized as an eigenspace decomposition (2.2.1b) for some grading element E ∈ A. And

this grading element is unique modulo the center z of A. (Note that z ⊂ A0.) In

particular, if we let Ass = [A,A] denote the semisimple factor of the reductive A,

then A = Ass⊕ z, and we may uniquely specify the grading element by requiring that

it lie in Ass.

2.2.2. Grading elements and filtrations. The grading element E acts on V with rational

eigenvalues. Moreover, if V is irreducible as an A–module, then any two nontrivial

eigenvalues of E differ by an integer. If V = ⊕Vq is the E–eigenspace decomposition

of V, with Vq = {v ∈ V | E(v) = q v}, then we define a decreasing filtration W (E,V)

of V by

(2.2.4) Wr(E,V) =⊕q≤r

Vq .

When the representation V is clear or immaterial, we will write W (E,V) = W (E).

Note that

W`(E,A) =⊕m≤`

Am = {ξ ∈ A | ξ(Wq(E,V)) ⊂ Wq+` , ∀ q} .

It follows from (2.2.3) that W−`(E) is a subalgebra of A for all ` ≥ 0. Indeed,

W0(E,A) is the parabolic subalgebra of A preserving W•(E,V), and the unipotent

radical of W0(E,A) is W−1(E,A). The reductive subalgebra A0 is a Levi factor of the

parabolic W0(E,A).

Remark 2.2.5. Every parabolic subalgebra of A can be realized as W0(E,A) for some

(non-unique) E.


Conversely, suppose that W is any filtration of V with the property that the

subalgebra

pW := {ξ ∈ A | ξ(W`) ⊂ W` , ∀ `}

stabilizing W is parabolic. Fix a Levi factor l ⊂ pW . Then there exists a unique

grading element E = E(W, l) of A in the center of l so that W = W (E). This

grading element is obtained as follows: let a be the R–split center of l, and let

Σ = {α1, . . . , αs} ⊂ a∗ be the simple roots for the adjoint action of a on the unipo-

tent radical uW of pW . Let {E1, . . . , Es} be the dual basis of a, in the sense that

αi(Ej) = δij. Then the Ei are grading elements of A, and there exist unique integers

ni > 0 so that E =∑niE

i is the desired grading element, W = W (E).

Note that l is the centralizer of E in A. That is, give W , the choice of Levi factor

l ⊂ pW is equivalent to a choice of grading element E ∈ A with W = W (E).

2.3. Nilpotent endomorphisms.

2.3.1. Standard triples. A standard triple in A is a set of three elements {N+, Y,N} ⊂A such that

[Y,N+] = 2N+ , [N+, N ] = Y and [Y,N ] = −2N .

The elements N+, Y,N are, respectively, the nilpositive, neutral and nilnegative ele-

ments of the triple. They span a subalgebra of A that is isomorphic to sl(2). Moreover,

N+ is uniquely determined by N, Y .

Example 2.3.1. The matrices

(2.3.2) n+ =

(0 1

0 0

), y =

(1 0

0 −1

)and n =

(0 0

1 0

)form a standard triple in sl(2,R); while the matrices

(2.3.3) e = 12

(−i 1

1 i

), z =

(0 −i

i 0

)and e = 1

2

(i 1

1 −i

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

See Example 3.2.22 for examples of standard triples in the case that A = so(2a, b).

14 ROBLES

Example 2.3.4. The neutral element Y of a standard triple is an example of a grading

element (§2.2.1).7

2.3.2. Jacobson–Morosov filtrations. Every nilpotent endomorphism N : V → V de-

termines a unique increasing filtration

0 ( W−k(N) ⊂ W1−k(N) ⊂ · · · ⊂ Wk−1(N) ⊂ Wk(N) = V

of V, with k = max{` | N ` 6= 0}, and the properties that

N(W`(N)) ⊂ W`−2(N)

and the induced N ` : GrW` (N)→ GrW−`(N) is an isomorphism for all ` ≥ 0, where

GrW` (N) := W`(N)/W`−1(N) .

Moreover, if N lies in the Lie algebra End(V,Q) of infinitesimal automorphisms of Q,

then the filtration W (N) is Q–isotropic.

Given a nilpotent N ∈ A, the Jacobson–Morosov filtration W (N) admits the

following explicit description. The Jacobson–Morosov Theorem asserts that every

nilpotent N ∈ A may be realized as the nilnegative element of a standard triple in

A. Recall that the neutral element Y is a grading element (Example 2.3.4). The

Jacobson–Morosov filtration W (N) is the filtration W (Y ) given by (2.2.4).

More precisely, we have the following structure. The neutral element Y acts on

V by integer eigenvalues; let

(2.3.5) V` := {v ∈ V | Y (v) = `v}

denote the eigenspace for the eigenvalue ` ∈ Z. The Jacobi identity implies Y (V`) ⊂V`−2. In fact, if

P` := {v ∈ V` | N `+1v = 0} = kerN `+1 ∩ V` ,

with ` ≥ 0, then it is a classical result in the representation theory of sl(2) that

(2.3.6) V =⊕0 ≤ `

0 ≤ k ≤ `

NkP` .

7However, it is not the case that every grading element may be realized as the neutral element of

a standard triple.


Implicit in this direct sum decomposition is the statement that N ` : V` → V−` is an

isomorphism. The Jacobson–Morosov filtration W (N) is the increasing filtration

(2.3.7) W`(N) :=⊕m≤`

Vm .

While (2.3.5) depends on standard triple containing N as the nilnegative element,

the filtration (2.3.7) is independent of this choice. Note that

(2.3.8) GrW` (N) ' V` .

Finally we note that the Lie algebra pW ⊂ A stabilizing W = W (N) is the direct

sum

(2.3.9) pW =⊕`≥0

A−` , A` := {ξ ∈ A | [Y, ξ] = ` ξ} ,

of the nonpositive ad(Y )–eigenspaces.

3. Hodge theory

We tersely recall those aspects of Hodge theory that we will make use of. Good

references for this material include [8, 14, 23, 10].

3.1. Hodge structures. A brief review of Hodge theory follows; good sources for

this material include [8, 9, 20] and the references therein.

3.1.1. Definition. A (real) Hodge structure of weight n on a real vector space V is a

homomorphism ϕ : U(R) ' S1 → SL(V) ⊂ Aut(V) of R–algebraic groups such that

ϕ(−1) = (−1)n1. The associated Hodge decomposition VC = ⊕p+q=nVp,qϕ , is given by

Vp,qϕ = {v ∈ VC | ϕ(z)v = zp−qv , ∀ z ∈ S1}

and satisfies Vp,qϕ = Vq,pϕ . (Any of the integers p, q, n may be negative.) The corre-

sponding Hodge filtration

· · · ⊂ F p+1 ⊂ F p ⊂ F p−1 ⊂ · · ·

of VC = V⊗R C is given by

F pϕ =

⊕q≥p

Vq,n−qϕ .

16 ROBLES

The Hodge numbers h = (hp,q) and f = (fp) are

hp,q = dimC Vp,q and fp = dimC F

p .

A Hodge structure of weight n ≥ 0 is effective if Vp,q 6= 0 only when p, q ≥ 0.

Example 3.1.1. The Hodge Theorem asserts that the n-th cohomology group V =

Hn(X,R) of a compact Kahler manifold admits a Hodge structure of weight n, with

Vp,q = Hp,q(X) ⊂ Hn(X,C) the cohomology classes in represented by (p, q)–forms.

A polarization of a weight n Hodge structure (V, ϕ) is a nondegenerate, (−1)n–

symmetric bilinear form Q : V × V → R satisfying the Hodge–Riemann bilinear

relations :

Q(F p, F n−p+1) = 0 ,(3.1.2a)

Q(v, ϕ(i)v) > 0 ∀ 0 6= v ∈ VC .(3.1.2b)

Equivalently,

Q(Vp,q,Vr,s) = 0 if (p, q) 6= (s, r) ,(3.1.3a)

ip−qQ(v, v) > 0 for all 0 6= v ∈ Vp,q .(3.1.3b)

The period domain D = Dh,Q is the set of all Q–polarized Hodge structures on V with

Hodge numbers h. It is a homogeneous space with respect to the action of the real

automorphism group

G := Aut(V,Q) ,

and the isotropy group is compact. If n = 2k + 1 is odd, then G ' Sp(2g,R), where

dimV = 2g; the isotropy group G0 = StabG(ϕ) '∏k

p=0 U(hn−p,p). If n = 2k is even,

then G ' O(a, b), where a =∑hk+2p,k−2p and b =

∑hk+1+2p,k−1−2p; the isotropy

group is G0 ' O(k, k)×∏k−1

p=0 U(hn−p,p).

Example 3.1.4. Let X ⊂ Pm be a projective algebraic manifold of dimension d with

hyperplane class ω ∈ H2(X,Z). Given n ≤ d, the primitive cohomology

V = P n(X,R) := {α ∈ Hn(X,R) | ωd−n+1 ∧ α = 0}

inherits the weight n Hodge decomposition VC = ⊕p+q=nHp,q(X)∩VC from Hn(X,R).

The Hodge–Riemann bilinear relations assert that this Hodge structure is polarized

by Q(α, β) := (−1)n(n−1)∫Xα ∧ β ∧ ωd−n.


3.1.2. Compact dual of a period domain. The first Hodge–Riemann bilinear relation

(3.1.2a) asserts that the Hodge filtration F = (F p) is Q–isotropic; equivalently, the

Hodge filtration defines a point in the rational homogeneous variety (a.k.a. generalized

flag manifold/variety)

D := FlagQ(f ,VC)

of Q–isotropic filtrations F • = (F p) of VC; the variety D is known as the compact

dual (of the period domain D). The complex automorphism group

GC := Aut(VC,Q)

acts transitively on D, and contains the period domain D as an open subset (in the

analytic topology). In summary, the compact dual D parameterizes filtrations F of

VC satisfying the first Hodge–Riemann bilinear relation, and the period domain D

parameterizes filtrations satisfying both Hodge–Riemann bilinear relations.

3.1.3. Mumford–Tate groups and domains. Suppose that V = V ⊗QR =: VR admits an

underlying rational structure, and that Q is induced by a rational Q : V ×V → Q. The

Mumford–Tate group of ϕ ∈ D is the Q–algebraic closure Gϕ of ϕ(S1) in Aut(V,Q).

The group Gϕ is reductive and is precisely the subgroup of Aut(V,Q) stabilizing

the Hodge tensors of ϕ [14]; in particular, Gϕ is the symmetry group of ϕ. The

associated Mumford–Tate domain is Dϕ := Gϕ(R) · ϕ ⊂ D, with compact dual

Dϕ := Gϕ(C) · ϕ ⊂ D. Note that given ϕ′ ∈ Dϕ we have Gϕ′ ⊂ Gϕ and Dϕ′ ⊂ Dϕ.

For generic ϕ ∈ D, we have Gϕ = Aut(V,Q); in particular, period domains are

Mumford–Tate domains.

From this point on we assume that G is the Mumford–Tate group

of a weight n, Q–polarized Hodge structure ϕ0 on V with Hodge

numbers h = (hp,q), and that D is the associated Mumford–Tate domain.

This implies that D = G/G0, where G0 is the compact centralizer of ϕ0(S1).

By a Hodge domain we mean any domain of the form D = G/G0 without the

underlying rational structure. That is, G is (the Lie group of real points of) a reductive

R–algebraic group, and G0 is the compact centralizer of a circle ϕ : S → G.

The fact that G0 is the centralizer of a circle, implies that the Lie algebra g0

contains a Cartan subalgebra t of g. Furthermore, the fact that G0 is compact,

18 ROBLES

implies that t is a compact Cartan subalgebra.8 It follows that

(3.1.5) there exists a unique maximal compact subgroup K ⊂ G containing G0

Notice that both G0 ⊂ K ⊂ G are all of equal (real) rank, and that the center of G

has R–rank zero.

3.1.4. Induced Hodge structure. Suppose that ϕ is a (possibly unpolarized) Hodge

structure with Hodge decomposition VC = ⊕p+q=nVp,qϕ . There is an induced weight

zero Hodge structure on End(V) defined by

End(V)p,−pϕ := {ξ ∈ End(VC) | ξ(Vr,sϕ ) ⊂ Vr+p,s−pϕ , ∀ r, s} ;

equivalently,

F pϕ(End(V)) := {ξ ∈ End(VC) | ξ(F q

ϕ) ⊂ F p+qϕ , ∀ q} .

The Jacobi identity implies

(3.1.6)[End(V)p,−pϕ , End(V)q,−qϕ

]⊂ End(V)p+q,−p−qϕ .

It follows from §2.2.1 that there exists a grading element Eϕ ∈ i End(V) so that

End(V)p,−pϕ is the Eϕ–eigenspace for the eigenvalue p. One may show that

(3.1.7) spanR{i Eϕ} = dϕ(T1S1) ⊂ End(V,Q) ;

we think of i Eϕ as the derivative of the circle ϕ [21].

Notice that the parabolic subalgebra

(3.1.8)W0(−Eϕ,End(V)) = ⊕p≥0 End(V)p,−pϕ

is the Lie algebra of StabAut(VC)(Fϕ),

the stabilizer of the Hodge filtration.

If ϕ is Q–polarized, then Eϕ ∈ i End(V,Q), and End(V,Q) inherits the Hodge

structure with

End(V,Q)p,−pϕ = End(V)p,−pϕ ∩ End(VC,Q) .

Moreover, the Hodge structure on End(V,Q) is polarized by the Killing form.

8Every Hodge domain is a flag domain [13]; the converse is false.


More generally, if the grading element Eϕ preserves the complexification AC of a

reductive subalgebra A ⊂ End(V), then A inherits the Hodge structure with

Ap,−pϕ = End(VC)p,−pϕ ∩ AC .

And this Hodge structure is polarized by the Killing form if ϕ is Q–polarized and

A ⊂ End(V,Q).

We will be especially interested in the case that A is the Lie algebra g of G =

G(R). If ϕ ∈ D, then (3.1.7) implies Eϕ ∈ i g. Consequently we have an induced

Hodge decomposition

gC =⊕

gp,−pϕ .

3.1.5. Variation of Hodge structure. Given F ∈ D, there is an induced filtration

F pg := {ξ ∈ gC | ξ(F q) ⊂ F p+q , ∀ q}

of gC. Notice that F 0g is the Lie algebra of the stabilizer PF = StabG(C)(F ). So the

holomorphic tangent space is

TF D = gC/F0g .

As a G(C)–homogeneous bundle, the holomorphic tangent bundle is

TD = G(C) ×PF (gC/F0g ) .

Notice that F−1g /F 0

g is a subspace of gC/F0g . The horizontal subbundle

T hD := G(C) ×PF F−1g /F 0

g

is the holomorphic, G(C)–homogeneous subbundle with fibre

T hF D = F−1g /F 0

g .

A horizontal map is a holomorphic map f : M → D with the property that

df(TxM) ⊂ T hf(x)D .

Given a discrete subgroup Γ ⊂ G, let π : D → Γ\D denote the quotient. A map

f : M → Γ\D is locally liftable if every point x ∈M admits a neighborhood U with a

map f : U → D so that f |U = π ◦ f . Furthermore, the map f is said to be horizontal

if the lifts f are horizontal. A period map is a locally liftable, horizontal map

Φ : B → Γ\D .

20 ROBLES

Geometrically, period maps arise when considering a family {Xb}b∈B of polarized

algebraic manifolds: very roughly, Φ maps b ∈ B to the Hodge structure Hn(Xb,C) =

⊕p+q=nHp,q(Xb) [16, 17].

3.2. Mixed Hodge structures. Mixed Hodge structures are generalizations of Hodge

structures.

3.2.1. Definition. A (real) mixed Hodge structure (MHS) on V is given an increasing

filtration W = (W`) of V, and a decreasing filtration F = (F p) of VC with the property

that F induces a weight ` Hodge structure on the graded quotients

GrW` := W`/W`−1 .

Example 3.2.1. If X is a Kahler manifold of dimension d and

V = H(X,R) :=⊕n

Hn(X,R) ,

then W` = ⊕n≤`Hn(X,R) and F k = ⊕p≥kHp,•(X) defines a mixed Hodge structure

on V.

Example 3.2.2. Alternatively, if X is a Kahler manifold of dimension d and V =

H(X,R), then W` = ⊕n≥2d−`Hn(X,R) and F k = ⊕q≤d−kH•,q(X) defines a mixed

Hodge structure on V.

Example 3.2.3. Deligne [11] has shown that the cohomology Hn(X,Q) of an algebraic

variety X admits a (functorial) mixed Hodge structure. Here X need not be smooth

or closed. However, when X is smooth and closed, Deligne’s MHS is the (usual)

Hodge structure of Example 3.1.1. For an expository introduction to mixed Hodge

structures on algebraic varieties see [12]; for a thorough treatment see [20].

3.2.2. Deligne splitting. The Deligne splitting associated with a mixed Hodge struc-

ture (W,F ) is the unique decomposition

(3.2.4a) VC =⊕

Ip,q

with the properties that

(3.2.4b) F p =⊕p≥r

Ir,• , W` =⊕p+q≤`

Ip,q


and

(3.2.4c) Ip,q ≡ Iq,p mod⊕r < qs < p

Ir,s .

The MHS is R–split if Ip,q = Iq,p.

3.2.3. Induced mixed Hodge structure. A MHS (W,F ) on V induces a MHS on End(V)

by

F p End(V) := {ξ ∈ End(VC) | ξ(F a) ⊂ F a+p , ∀ a}

W` End(V) := {ξ ∈ End(V) | ξ(Wm) ⊂ W`+m , ∀ m} .

The Deligne splitting of this MHS is

(3.2.5) End(V)p,q = {ξ ∈ End(VC) | ξ(Ir,s) ⊂ Ip+r,q+s , ∀ r, s} .

The Jacobi identity implies

(3.2.6) [End(V)p,q , End(V)r,s] ⊂ End(V)p+r,q+s .

Consequently, §2.2.1 implies that there exist commuting grading elements YW,F ∈End(V) and EW,F ∈ End(VC) so that

(3.2.7) EW,F := p1 and YW,F := (p+ q)1 on End(VC)p,q .

(These grading elements act on Ip,q with the same eigenvalues.) If the MHS is R–split,

then YW,F ∈ End(V) is real.

If the grading elements preserve the complexification AC of a reductive subalgebra

A ⊂ End(V), then A inherits a MHS

WA,` = W` End(V) ∩ A and F pA = F p End(V) ∩ AC .

The Deligne splitting of this MHS is given by

Ap,q = End(V)p,q ∩ AC .

It is immediate that (3.2.5) and (3.2.6) hold with A in place of End(V). The MHS

(WA, FA) is R–split if (W,F ) is.

22 ROBLES

3.2.4. Polarized mixed Hodge structures. A polarized mixed Hodge structure on D

(PMHS) is a triple (W,F,N) such that:

(a) (W,F ) is a MHS on VR, and W` = W`−n(N);

(b) F ∈ D, N ∈ g, and N(F p) ⊂ F p−1 for all p;

(c) when restricted to the primitive subspace

PrimN` := kerN `+1 ⊂ GrWn+`

the weight n+ `, F–induced Hodge structure on GrWn+` is polarized by

QN` (·, ·) := Q(·, N `·) .9

In this situation we say that the MHS (W,F ) is polarizable, and is polarized by N .

Note also that the grading elements (3.2.7) satisfy

EW,F , Y ∈ gC ,

and Y ∈ g if (W,F ) is R–split. Since N determines W , we will often denote a PMHS

by (F,N). The Deligne splitting VC = ⊕Ip,q satisfies N(Ip,q) ⊂ Ip−1,q−1; that is,

(3.2.8) N ∈ g−1,−1W,F .

3.2.5. Nilpotent orbits. A (one-variable) nilpotent orbit on D is a map θ : H→ D on

the upper-half plan of the form

θ(z) = exp(zN) · F ,

and with the properties that N ∈ g and F ∈ D, it is horizontal in the sense that

N(F p) ⊂ F p−1 for all p, and θ(z) ∈ D when Im z � 0. A pair (F,N) defines a

nilpotent orbit if and only if (W,F,N) is a PMHS [10].

9The bilinear form QN` is (−1)n+`–symmetric and nondegenerate on GrWn+`. The F–induced

Hodge structure on GrWn+` satisfies the first Hodge–Riemann bilinear relation: it is QN` –isotropic.

However it does not, in general, satisfy the second Hodge–Riemann bilinear relation; the restriction

to PrimN` does.


3.2.6. Group action. Note that G acts on the set of R–split PMHS on D by

(3.2.9) g · (F,N) := (g · F,AdgN) .

Let

ΨD := {[F,N ] | (F,N) is an R–split PMHS}

denote the set of G–conjugacy classes

[F,N ] := {(g · F,AdgN) | g ∈ G}

of R–split PMHS.

The R–split PMHS on D are classified in [22]. The general classification is repre-

sentation theoretic in nature, formulated in terms of Levi subgroups, (distinguished)

grading elements and Weyl groups. However in the case that G = Aut(V,Q), so

that D is the period domain D, the classification may be given in terms of Hodge

diamonds (Corollary 3.2.29); specifically, the elements of ΨD are classified by the

possible Hodge diamonds.

3.2.7. Hodge diamonds. Given a MHS (W,F ), let VC = ⊕Ip,q be the Deligne splitting.

The Hodge diamond of (W,F ) is the function 3(W,F ) : Z× Z→ Z given by

3(W,F )(p, q) := dimC Ip,q .

Lemma 3.2.10 ([18]). The Hodge diamond 3 = 3(W,F,N) of a PMHS on a pe-

riod domain D parameterizing weight n Hodge structures with Hodge numbers h =

(hp,q)p+q=n satisfies the following four properties: The columns of the Hodge diamond

sum to the Hodge numbers

(3.2.11a)∑

p3(p, q) = hn−q,q .

The Hodge diamond is symmetric about the diagonal p = q:

(3.2.11b) 3(p, q) = 3(q, p) .

The Hodge diamond is symmetric about p+ q = n:

(3.2.11c) 3(p, q) = 3(n− q, n− p) .

24 ROBLES

The values 3(p, q) are non-increasing as one moves away from p + q = n along a(n

off) diagonal:

(3.2.11d) 3(p, q) ≥ 3(p+ 1, q + 1) for all p+ q ≥ n .

Note that the four conditions (3.2.11) imply that the Hodge diamond of a PMHS

“lies in” the square [0, n]× [0, n]; that is

3(p, q) 6= 0 implies 0 ≤ p, q ≤ n .

Given a PMHS (F,N), we will denote the Hodge diamond by 3(F,N). The

following proposition asserts that (i) every non-negative function satisfying (3.2.11)

may be realized as the Hodge diamond of an R–split PMHS, and (ii) the R–split

PMHS on D are classified, up to the action of G, by their Hodge diamonds.

Theorem 3.2.12 ([18]). Any function f : Z × Z → Z≥0 satisfying (3.2.11) may be

realized as the Hodge diamond 3(F,N) of an R–split polarized mixed Hodge structure

(F,N), N ∈ gR, on the period domain D. Moreover, 3(F1, N1) = 3(F2, N2) if and

only if (F2, N2) = (g · F1,AdgN1) for some g ∈ G.

Example 3.2.13 (Period domain with h = (a, b, a)). If D is the period domain param-

eterizing polarized Hodge structures with h = (a, b, a), then the Hodge diamonds 3r,s

are indexed by 0 ≤ r, s satisfying r+s ≤ a and r+ 2s ≤ b, [18]. They are represented

by

a− r − s

s

r

s

b− 2s

s

r

s

a− r − s

The corresponding equivalence classes in ΥD and ΨD are represented by the SL(2) of

Example 3.2.22 and the R–split PMHS of Example 3.2.30, respectively.

Example 3.2.14. Specializing Example 3.2.13 to a = 2, we see that the Hodge dia-

monds 3r,s are indexed by 0 ≤ r, s satisfying r + s ≤ 2 and r + 2s ≤ b. They are

represented as follows.


30,0 30,1 31,0

30,2 31,1 32,0

The Hodge diamonds for the induced PMHS on g are

30,0

1

2b

30,11

b− 1

b− 1

1

31,0

b

b

30,2

4

4

31,1

1

b− 1

b− 1

1

32,0

1

2b

Remark 3.2.15. From the Hodge diamonds above we can see that (i) the weight fil-

trations for 30,1 and 32,0 are G–inequivalent, while their stabilizers are G–congruent.

This is non-classical behavior: when D is Hermitian the weight filtrations are G–

equivalent if and only if their stabilizers are G–congruent.

3.2.8. Horizontal SL(2)–orbits. The R–split PMHS are in bijective correspondence

with horizontal SL(2)–orbits on D; the latter are defined as follows. Fix a Hodge

structure ϕ ∈ D, and recall the induced Hodge structure §3.1.4 on the Lie algebra g.

A horizontal SL(2) at ϕ is given by a representation

(3.2.16) η : SL(2,C) → G(C)

26 ROBLES

such that

(3.2.17a) η(SL(2,R)) ⊂ G

and

(3.2.17b) η∗(e) ∈ g1,−1ϕ , η∗(z) ∈ g0,0

ϕ , η∗(e) ∈ g−1,1ϕ .

We say that ϕ is the base point of the horizontal SL(2).

Remark 3.2.18. Note that a horizontal SL(2) includes the information of both the

representation (3.2.16) and the base point ϕ. We will denote a horizontal SL(2) by

(η, ϕ) when we wish explicitly specify the base point; and we will use the simpler

notation η when the base point is understood, or inessential.

Let hSL2(D) denote the set of horizontal SL(2)’s on D, and hSL2(ϕ) the set of

horizontal SL(2)’s with base point ϕ ∈ D. The standard triple of η is

(3.2.19) {N+η , Yη, Nη} := η∗{n+,y,n} ⊂ gR .

The standard triple determines η. In fact, {Yη, Nη} determine η (§2.3.1).

Definition 3.2.20. If {N+j , Yj, Nj} are the standard triples (3.2.19) underlying two

commuting SL(2)’s (η1, ϕ) and (η2, ϕ) with a common base point ϕ, then {N+1 +

N+2 , Y1 + Y2, N1 + N2} is a standard triple underlying a third SL(2) that is also

horizontal at ϕ. We denote this third SL(2) by η = η1 � η2. In a slight abuse

notation, we will also write η1 � η2 to indicate the pair of commuting SL(2). More

generally, η1�· · ·�ηa will denote either an a-tuple of pairwise commuting SL(2) with

common base point or the resulting SL(2). The set of all such tuples will be denoted

hSL2a(D), so that hSL21(D) = hSL2(D), and the set of all tuples based at ϕ will be

denoted hSL2a(ϕ).

Remark 3.2.21. If a > rankRg, then hSL2a(D) = ∅, [6].

Example 3.2.22 (Period domain with h = (a, b, a)). Recollect our Running Example

(§4.3), the period domainD parameterizingQ–polarized Hodge structures with Hodge

numbers h = (a, b, a). Fix 0 ≤ r, s satisfying r+s ≤ a and r+2s ≤ b. For convenience


we assume that b ≥ 2a. Then we may fix linearly independent {e1, . . . , e4a} ⊂ V so

that Q(ej, ek) = δ4a+1j+k . Given 1 ≤ j ≤ 4a, define j∗ = 4a+ 1− j. Then

F 2ϕ = H2,0

ϕ := spanC{

(ej − ej∗)− i(ea+j − e(a+j)∗) | 1 ≤ j ≤ a}

defines a point ϕ ∈ D of the period domain. Let ekj ∈ End(V ) denote the endomor-

phism mapping ek to ej and annihilating the vectors Q–orthogonal to ek∗ . Then

N =r∑j=1

(ej(a+j)∗ − e

a+jj∗ + e

(a+j)∗

j∗ − eja+j

)+

r+s∑k=r+1

(e

(a+k)∗

k∗ − eka+k

)

Y =r∑j=1

2(ejj − e

j∗

j∗

)+

r+s∑k=r+1

(ekk + e

(a+k)∗

(a+k)∗ − ek∗

k∗ − ea+ka+k

)

N+ =r∑j=1

(e

(a+j)∗

j − ej∗

a+j + ej∗

(a+j)∗ − ea+jj

)+

r+s∑k=r+1

(ek∗

(a+k)∗ − ea+kk

)defines a standard triple underlying a SL(2) ⊂ G that is horizontal at ϕ.

Note that G acts naturally on hSL2(D): if Cg : G → G denotes the conjugation

a 7→ gag−1, then the action is given by

(3.2.23) g · (η, ϕ) := (Cg ◦ η, g · ϕ)

defines an action of G on hSL2(D). Let

ΥD := G\hSL2(D)

denote the set of G–congruence classes. The bijection between R–split PMHS and

horizontal SL(2)s discussed in §3.2.9 is G–equivariant, and yields an identification

ΨD ' ΥD .

3.2.9. Equivalence of horizontal SL(2)’s and R–split PMHS. The bijective correspon-

dence between R–split PMHS and horizontal SL(2)–orbits on D is given as follows.

First suppose that η ∈ hSL2(ϕ); the associated R–split PMHS (Fη, Nη) on V is given

by (3.2.19) and (3.2.24) below. Recall the standard triple (3.2.19) of η. Setting

(3.2.24a) ρη := exp iπ4(N+

η +Nη) ∈ G(C) ,

28 ROBLES

the flag is given by

(3.2.24b) Fη := ρ−1η · ϕ ∈ ∂D .

Conversely, given an R–split MHS (W,F ), the Deligne splitting defines

(3.2.25) YW,F ∈ End(VR) by YW,F |Ip,q := (p+ q − n)1 .

If (W,F ) is polarized by N , then the elements N and YW,F are the nilnegative and

neutral elements, respectively, of a unique standard triple, and so span a unique

subalgebra

s(F,N) ' sl(2,R) .

The triple yields a representation η : SL(2,C)→ G(C) with N = Nη and YW,F = Yη.

The SL(2) is horizontal at the point

(3.2.26) ϕF,N := ρη · F ∈ D

given by (3.2.24). We say that η polarizes (W,F ).

As observed in [10, §3], we have

(3.2.27a) exp(iyN) · F = exp(−12

log(y)Y ) · ϕF,N ,

so that

(3.2.27b) ϕF,N = exp(iN) · F

and

(3.2.27c) limt→∞

exp(tY ) · ϕF,N = F .

Finally, it will be convenient to note that

(3.2.28) g · ϕF,N = ϕg·F,AdgN .

The above bijection between hSL2(D) and the set of R–split PMHS is equivariant

with respect to natural actions (3.2.9) and (3.2.23) of G. From Theorem 3.2.12 and

the equivalence of R–split PMHS with horizontal SL(2)’s (§3.2.9), we obtain

Corollary 3.2.29. The horizontal SL(2)’s on the period domain D are classified by

Hodge diamonds. That is, ΥD is in bijective correspondence with the set of possible

Hodge diamonds (which are characterized by (3.2.11)).


Example 3.2.30 (Period domain with h = (a, b, a)). Recall the horizontal SL(2) of

Example 3.2.22. The associated R–split PMHS (F,N) has Deligne splitting VC =

⊕ Ip,qF,N given by

I2,2F,N = spanC {ej | 1 ≤ j ≤ r} ,

I2,1F,N = spanC

{ek + ie(a+k)∗ | r + 1 ≤ k ≤ r + s

},

I2,0F,N = spanC

{(e` − e`∗)− i(ea+` − e(a+`)∗) | r + s+ 1 ≤ ` ≤ a

}.

(The remaining subspaces in the splitting are given by I1,2F,N = I2,1

F,N , I1,0 = N(I2,1F,N),

I0,2F,N = I2,0

F,N , I0,1F,N = I1,0

F,N , I0,0F,N = N2(I2,2

F,N) and I1,1F,N is the Q–orthogonal complement

of the above subspaces.)

3.2.10. Horizontal SL(2)’s for unpolarized Hodge structures. Grading elements (§2.2)

provide a convenient framework to extend the relationship between η ∈ hSL2(ϕ) and

the PMHS (Fη, Nη) to the unpolarized case (§3.2.10).

The boundary components Xνa constructed in this paper will admit surjections

onto flag domains Dνa parameterizing weight zero Hodge structures on semisimple Lie

algebras gνa−1 that will, in general, be unpolarized. By a weight zero Hodge structure

on a real, reductive Lie algebra g we mean a Hodge decomposition gC = ⊕ gp,−p with

the property that [gp,−p , gq,−q

]⊂ gp+q,−p−q .

From (3.2.17) we see that notion of a horizontal SL(2) does not require that

the Hodge structure on g be polarized; it is well-defined for any weight zero Hodge

decomposition of gC. Moreover, as in the polarized case (§3.2.9), the horizontal SL(2)

determines an R–split MHS on g.

Lemma 3.2.31. Given a reductive, real algebra group G with Lie algebra g, let

η : SL(2,C) → G(C) be horizontal with respect to a (possibly unpolarized) weight

zero Hodge structure gC = ⊕ gp,−p. Let ϕ denote the associated Hodge filtration of gC.

Then (3.2.19) and (3.2.24) define an R–split MHS (W (Nη), Fη) on g. The associated

grading elements (cf. §§3.1.4 & 3.2.3) are related by

(3.2.32) EW,F = Ad−1ρη (Eϕ) .

And (3.2.8), (3.2.26) and (3.2.27) hold.

30 ROBLES

In this situation we say that η polarizes (W (Nη), Fη). [Ugh. . . Bad terminology.]

Proof. Let {N+, Y,N} denote the standard triple (3.2.19), and set F = Fη and ρ = ρη.

Define

(3.2.33a) {E,Z,E} := η∗{e, z, e}

It is a computation to confirm

(3.2.33b) {E,Z,E} = Adρ{N+, Y,N} .

By definition Z ∈ g0,0; therefore, Eϕ and Z commute. Consequently, Ad−1ρ (Eϕ) and Y

commute. The simultaneous eigenspace decomposition

(3.2.34a) gC = ⊕ gp,qW,F

given by

(3.2.34b) gp,qW,F := {ξ ∈ gC | Ad−1ρ (Eϕ)(ξ) = p ξ , adY (ξ) = (p+ q) ξ}

splits the filtrations (W (N), F ) in the sense that

W`(N) =⊕p+q≤`

gp,qW,F and F =⊕r≥p

gr,•W,F .

So to prove that (W (N), F ) is a MHS it suffices to show that gp,qW,F = gq,pW,F . It will

then follow that (3.2.34) is the Deligne splitting of the MHS, and that the latter is

R–split; consequently (3.2.32) holds. Equivalently, it suffices to show

(3.2.35) EW,F = Y − EW,F .

From (2.3.2) and (2.3.3) we see that N+ + N = E + E. Moreover, E ∈ g1,−1 and

E ∈ g−1,1 imply [Eϕ,E] = E and [Eϕ,E] = −E. And we compute

EW,F = exp−iπ4(E + E) Eϕ = Eϕ − 1

2Z + i

2(E− E) = Eϕ − 1

2Z + 1

2Y .

Now (3.2.35) follows from the fact that both Eϕ and Z are imaginary.

Equation (3.2.8) now follows from (3.2.17b), the definition of a standard triple

(which asserts [Z,E] = −2E), and (3.2.33). Since Eϕ and EW,F determine ϕ and

F , respectively, it follows from (3.2.32) that (3.2.26) holds. Finally, (3.2.27) is a

computation. �


Part B: Group theoretic generalization

4. Overview

Our goal is to generalize SBB to obtain a ‘horizontal completion’ Γ\Dh of the

arithmetic quotient of a Mumford–Tate domain D.

4.1. Construction. We begin by constructing a real horizontal completion

DRh =

∐π∈Ph

Xπ .10

The boundary components Xπ are indexed by triples π = (Pπ,Yπ, gπ) consisting of

a parabolic subgroup Pπ ⊂ G, a Pπ–orbit Yπ ⊂ D and a semisimple ideal gπ ⊂ m.11

Any such triple yields a homogeneous space (Xπ, Gπ) by a general procedure that

is described in §5; briefly, the data determines a normal subgroup Zπ ⊂ Pπ, and

Gπ = Pπ/Zπ is a semisimple group acting transitively on Xπ = Zπ\Yπ. We will

find that the group G acts on DRh with only finitely many orbits, and that Pπ is the

normalizer of Xπ with respect to this action.

Given that we wish for Φ(B) to have compact closure in Γ\Dh, and that Φ is

asymptotically approximated by a nilpotent orbit [23] (which is equivalent to a polar-

ized mixed Hodge structure), it is to be expected that Hodge theory enters into the

construction of the boundary components. Indeed, the triples π ∈ Ph are determined

by equivalence classes νa of a–tuples of pairwise commuting horizontal SL(2)s. (The

latter are a special type of nilpotent orbit.) The corresponding triple π ∈ Ph will be

denoted π(νa) = (Pνa ,Yνa , gνa).12 When a = 1, the corresponding parabolic subgroup

Pν1 ⊂ G is the stabilizer of the associated weight filtration; when a > 1, the parabolic

admits a similar characterization. The corresponding Pνa–orbit Yνa ⊂ D is the set of

base points of a–tuples representing νa. The semisimple ideal gνa is constructed from

the Deligne splitting of a PMHS associated with the tuple.

10When D is Hermitian, we will have DRh = D?.

11The domain D is Hermitian if and only if Yπ = D.12A subtle point is that the assignment νa 7→ π(νa) is not injective. For example, when D is

Hermitian every triple π(νa) can be realized as π(µ1), for some equivalence class µ1 determined by

a single horizontal SL(2).

32 ROBLES

The topology on DRh is defined by using a refined Langlands decomposition and

certain semisimple endomorphisms Yνa , to specify a notion of convergent sequences;

this is done in such a way that neighborhood bases may be constructed from unions

of generalized Siegel sets. Up to this point, the construction is over R. To obtain

Dh =∐π∈PQ

h

Xπ ,

we restrict to the triples π = π(νa) with rationality properties. [Refine this state-

ment.] We then give Dh ⊂ DRh the subspace topology, and show that the quotient

topology on Γ\Dh gives the desired horizontal completion.

4.2. Properties. The key properties of the Satake construction that we wish for Dh

to have as well are:

(a) The closure of Φ(B) in Γ\Dh is compact.

(b) The group G acts continuously on DRh with compact isotropy and finitely many

orbits.

(c) The normalizer N (Xνa) = Pνa of a boundary component in DRh is a parabolic

subgroup Pνa ⊂ G.

(d) If the normalizer Pνa is defined over Q, the the image of Pνa ∩ Γ in Gνa is a

discrete subgroup.

The property (a) is motivated by the desire to extend the applications of Hodge

theory to study moduli of algebraic varieties (as discussed in §1.1). The properties

(b)–(d) ensure that we can apply reduction theory for Γ to obtain a quotient Γ\Dh

with the desired properties (eg. Hausdorff). These objectives will drive much of the

construction. Additional properties of the construction include the following:

(f) The space Γ\D is dense in Γ\Dh, and Γ\Dh is Hausdorff.

(g) There is a Gνa–equivariant double fibration

Xνa

Dνa Eνa .

δνa ενa

Both Dνa and Eνa are homogeneous spaces. The space Dνa is a flag domain

parameterizing (generally unpolarized) weight zero Hodge structures on the Lie


algebra gνa of the automorphism group Gνa . The ενa fibres are Hodge domains

(§3.1.3). When D is Hermitian , Eν1 is a point and δν1 is an isomorphism.

4.3. Running example: period domain for h = (a, b, a). Throughout we will

illustrate the theory with one of the simplest nonclassical period domains, the domain

D ' O(b, 2a)

O(b)× U(a)

parameterizing weight 2, polarized Hodge structures with Hodge numbers h = (a, b, a).

(The examples are listed in the table of contents.) For example, the boundary com-

ponents Xν ⊂ DR1 for ν = ν1 determined by a single horizontal SL(2) are

Xν =SL(2s,R)

U(s)× O(b′, 2a′ + r)

O(b′) × U(a′) × O(r),

with Gν = SL(2s,R)×O(b′, 2a′ + r), and

Dν =SL(2s,R)

Sp(2s,R)× O(b′, 2a′ + r)

O(b′, 2a′) × O(r)

Eν =SL(2s,R)

U(s)× O(b′, 2a′ + r)

O(b′, r) × U(a′);

see Examples ?? and 6.5.4 for further discussion. The boundary components Xν2 ⊂DR

2 are described in Example ??, and in the case that a = 2, the complete set of

boundary components is given in Example ??.

4.4. Outline. The general procedure underlying the construction of a boundary com-

ponent Xπ and automorphism group Gπ given the data π ∈ P is developed in §5.

The triples π = π(ν1) ∈ Ph determined by a single horizontal SL(2) are introduced

in §6. [. . . ]

5. Basic structure

The purpose of this section is to describe the basic algorithm underlying the con-

struction of the boundary components, and to describe their structure/properties. We

fix once and for all maximal compact subgroup K ⊂ G. Each boundary component

Xνa will be constructed from the following data:

• A parabolic subgroup Pνa ⊂ G, with Langlands decomposition Uνa × Aνa ×Mνa

(with respect to K).

34 ROBLES

• A Pνa–orbit Yνa ⊂ D.

• A semisimple ideal gνa ⊂ mνa .

To that end, we define P to be the set of all triples (P,Y , g) such that P is a parabolic

subgroup of G with Langlands decomposition U × A × M (with respect to K); g

is a semisimple ideal of the Lie algebra m of M ; and Y is a P–orbit in D. In

this section we describe how π ∈ P determines both a space Xπ and a semisimple

group Gπ acting transitively on Xπ, and establish basic properties for these objects.

In subsequent sections (§§6, ??&??) we will identify the set Ph ⊂ P indexing the

boundary components of DRh .

5.1. Definition. Given π = (P,Y , g) ∈ P , define Pπ = P , gπ = g and Yπ = Y . Let

Pπ = Uπ × Aπ ×Mπ denote the Langlands decomposition.

5.1.1. Automorphism group. Fix π = (P,Y , g) ∈ P . Let

Cπ := CMπ(gπ) = {g ∈Mπ | Adg(ξ) = ξ , ∀ ξ ∈ gπ}

be the centralizer of gπ in Mπ. Then Cπ is a normal, R–algebraic, reductive subgroup

of M . Furthermore

Zπ := Uπ × Aπ × Cπ

is a normal subgroup of Pπ. The automorphism group associated with this data is the

R–algebraic quotient group

Gπ := Pπ/Zπ .

Let

(5.1.1) pπ : Pπ � Gπ

denote the projection.

Lemma 5.1.2. The group Gπ is semisimple, and has Lie algebra isomorphic to gπ.

Proof. Observe that the Lie algebra g⊥π ⊂ mπ of Cπ is an ideal of mπ, and that

mπ = g⊥π ⊕ gπ is a Killing orthogonal decomposition. �


5.1.2. Boundary component. Define

Xπ := Zπ\Yπ .

Let

(5.1.3) qπ : Yπ � Xπ

denote the projection.

Lemma 5.1.4. The group Gπ acts transitively on Xπ with compact isotropy.

Proof. It is clear that Gπ acts transitively on Xπ.

Consider the set

(5.1.5) Z := {ϕ ∈ D | StabG(ϕ) ⊂ K} .

Recall that G = PπK. This implies that

(5.1.6) Zπ := Yπ ∩ Z 6= ∅ .

Fix ϕ0 ∈ Zπ. Next let

C⊥π = CMπ(g⊥π ) := {g ∈Mπ | Adg(ξ) = ξ , ∀ ξ ∈ g⊥π }

be the centralizer of g⊥π in Mπ. Then C⊥π is a normal subgroup of Mπ and

(5.1.7) Mπ = Cπ · C⊥π = C⊥π · Cπ .

Consequently,

(5.1.8) Xπ ' (Cπ ∩ C⊥π )\(C⊥π · ϕ0) .

As the the centralizer of Mπ,

(5.1.9) Cπ ∩ C⊥π is compact and contained in K.

The lemma follows. �

Note that (5.1.7) yields

(5.1.10) Gπ ' C⊥π /(Cπ ∩ C⊥π ) .

The fact that K ∩ C⊥π is a maximal compact subgroup of C⊥π implies

36 ROBLES

Lemma 5.1.11. The image Kπ := pπ(K∩Pπ) = (K∩Pπ)/(K∩Zπ) ' (K∩C⊥π )/(Cπ∩C⊥π ) is a maximal compact subgroup of Gπ.

Example 5.1.12. Let G0 = StabG(ϕ0). The domain D is Hermitian symmetric if and

only if K = G0; equivalently, Z = {ϕ0} consists of a single point.

Example 5.1.13 (Period domain with h = (a, b, a)). For the example of §4.3 we have

K = O(b)×O(2a), and Z = Gr(a, (H1,1ϕ0

)⊥) ' Gr(a,C2a) is the collection of all ϕ ∈ Dwith H1,1

ϕ = H1,1ϕ0

.

5.2. Fat cross-sections. Define

Xπ := C⊥π · Zπ ⊂ Yπ .

Lemma 5.2.1. The natural projection q′ : Xπ → Xπ is surjective with compact fibre

(Cπ ∩K) · ϕ0 through ϕ0.

We call Xπ a fat cross-section of the projection qπ : Yπ � Xπ.

Proof. First note that (3.1.5) implies that

(5.2.2) Z is a K–orbit.

We claim that

(5.2.3) Zπ is a (Pπ ∩K) = (Mπ ∩K)–orbit.

To see this, suppose that ϕ ∈ Zπ. Then ϕ = p·ϕ0 = k ·ϕ0, for some p ∈ Pπ and k ∈ K.

So p−1k ∈ G0. It follows from (5.1.5) and (5.2.2) that p ∈ K. Since Pπ∩K = Mπ∩K,

the claim follows.

Therefore Xπ = C⊥π (Mπ ∩K) ·ϕ0. Since C⊥π is a normal subgroup of Mπ, we have

Xπ = (Mπ ∩K)C⊥π ·ϕ0. It then follows from (5.1.8) that the projection q′ : Xπ → Xπ

is surjective. The fibre through ϕ0 is the orbit of ϕ0 under Cπ ∩((Mπ ∩K)C⊥π

). So

to complete the proof, it remains to show that this intersection is Cπ∩K. To see this,

notice that Mπ ∩K = (Cπ ∩K) · (C⊥π ∩K). Therefore, (Mπ ∩K)C⊥π = (Cπ ∩K)C⊥π

is a subgroup of Mπ. So Cπ ∩((Mπ ∩K)C⊥π

)= (Cπ ∩K)(Cπ ∩C⊥π ), and this is equal

to Cπ ∩K by (5.1.9). �

It follows from Lemma 5.1.11 and (5.2.2) that

(5.2.4) qπ(Zπ) is a Kπ–orbit.


5.3. Group actions. By construction there is a natural action of Pπ on Xπ. The

normal subgroup Zπ ⊂ Pπ is the centralizer of Xπ. The map qπ is Pπ–equivariant in

the sense that

qπ(g · y) = pπ(g) · qπ(y)

for all y ∈ Yπ and g ∈ Pπ. We claim that the action of Pπ on Xπ extends to an action

of G on the disjoint union X := tπ∈PXπ.

To see this, first notice g · (P,Y g) := (gPg−1 , g · Y , Adgg). Defines an action of

G on P . Since Pπ is equal to its normalizer in G, we see that Pπ is the stabilizer of π.

To define the G–action on X, fix g ∈ G and x ∈ Xπ. We may write x = Zπy for

some y ∈ Yπ. Notice that Zg·π = gZπg−1. So g · x := Zg·π(g · y) defines a point in

Xg·π. We leave it to the reader to verify that this is indeed a group action. Notice

that Pπ is the normalizer of Xπ with respect to this action. Further more the action

is compatible with the projections in the sense that

g · x = qg·π(g · q−1π (x)) ,

for all g ∈ G and x ∈ Xπ.

5.4. Reductions.

5.4.1. A reduction of Pπ is any parabolic subgroup Rπ ⊂ Pπ with the property that

pπ(Rπ) = Gπ, and that is minimal with this property. The relative Langlands de-

composition implies that reductions Rπ ⊂ Pπ are in bijection with minimal parabolic

subgroups Bπ ⊂ Cπ; specifically,

Rπ = Uπ × Aπ × (Bπ · C⊥π ) .

The relative Langlands decomposition also implies Zπ′ ⊂ Zπ whenever there exist

reductions Rπ ⊂ Rπ′ .

5.4.2. Since any two minimal parabolics B,B′ ⊂ Cπ are conjugate under the action of

the maximal compact subgroupK∩Cπ, it follows that any two reductionsRπ, R′π ⊂ Pπ

are conjugate under the action of K ∩ Zπ = K ∩ Cπ.

38 ROBLES

5.4.3. Suppose that Rπ ⊂ Q ⊂ Pπ and that Rπ = URπ × ARπ × MRπ and Q =

UQ × AQ × MQ are the Langlands decompositions. Then the relative Langlands

decomposition implies C⊥π ⊂MRπ ⊂MQ. Consequently,

X ⊂ MQ · Z .

6. Boundary components induced by horizontal SL(2)s

In §5 we described how a triple π = (P,Y , g) ∈ P determines a group Gπ and a

Gπ–homogeneous space Xπ. We now turn to identifying the subset Ph ⊂ P that will

index the boundary components Xνa ⊂ DRh . This is the point at which horizontal

SL(2) orbits enter the construction: each νa will be ‘polarized’ by an a–tuple of

commuting horizontal SL(2)’s. The parabolics Pνa will arise as the stabilizers of

associated weight filtrations, the set Yνa ⊂ D parameterizes the base points of such

tuples, and gνa is constructed from Deligne splittings associated with the tuples.

‘Key Lemmas I & II’ establish the necessary properties for gνa and Yνa , including

that π(νa) := (Pνa ,Yνa , gνa) is indeed an element of P . This section addresses the

case a = 1. (Sections ?? & ?? cover the cases a ≥ 2.)

6.1. Boundary components Xν. Fix ϕ0 ∈ Z. Let

G0 := StabG(ϕ0) ⊂ K

denote the stabilizer of ϕ0 in G.

Given η ∈ hSL2(D), let (Fη, Nη) denote the associated R–split PMHS (§3.2.9),

and let Pη ⊂ G be the parabolic subgroup stabilizing the weight filtration W (Nη).

Define ν = [η] to be the Pη–orbit of η, cf. (3.2.23), and notice that Pν := Pη is

well-defined.

Remark 6.1.1. The following four conditions are equivalent: the horizontal SL(2)

is trivial; the nilpotent Nη is zero; W (Nη) is the trivial filtration 0 = W−1(Nη) ⊂W1(Nη) = VR; the parabolic is Pη = G. In this case, we let ν0 denote the associated

G–orbit, set ΠR0 := {ν0}, and π(ν0) := (G,D, g) ∈ P . Notice that D = Xπ.

From this point on, we assume that all horizontal SL(2)s are nontrivial, unless

explicitly stated otherwise.


Define

ΠR1 := {[η] | η ∈ hSL2(D) is nontrivial} .

We say η ∈ hSL2(D) polarizes ν = [η] ∈ ΠR1 . We now proceed to explain how ν ∈ ΠR

1

determines a triple π(ν) = (Pν ,Yν , gν) ∈ Ph. The parabolic Pν is defined above. Set

(6.1.2) Yν := {ϕ ∈ D | ∃ η ∈ hSL2(ϕ) polarizing ν} .

It is clear from the definition that Yν is a Pν–orbit.

Given ν ∈ ΠR1 , let Pν = Uν × Aν ×Mν be the Langlands decomposition (with

respect to K). Let Lν = A×Mν denote the Levi factor.

Lemma 6.1.3 (Key Lemma I). Fix ν ∈ ΠR1 . (a) Given any η ∈ ν, let (W,F ) =

(W (Nη), Fη) be the associated R–split PMHS (§3.2.9). The Deligne splitting gC =

⊕ gp,qW,F satisfies

pν ⊗ C =⊕p+q≤0

gp,qW,F and uν ⊗ C =⊕p+q<0

gp,qW,F .

(b) There exists a choice (η, ϕ) ∈ hSL2(D) polarizing ν with base point ϕ ∈ Z. For

any such choice we have

lν ⊗ C =⊕

p+q=0 gp,qW,F , aν ⊗ C ⊂ g0,0

W,F ,

and gp,−pW,F ⊂ mν ⊗ C when p 6= 0 .

(c) The neutral element Yν := Yη is independent of our choice of η in part (b), lies

in aν, and has centralizer {g ∈ G | AdgYν = Yν} = Lν. (In fact, Yν is the grading

element E(W, lν) of §2.2.2.)

Proof of Lemma 6.1.3(a). The descriptions of pν and uν follow directly from the

definitions of these two algebras, and properties of the induced Deligne splitting

(§3.2.3). �

Proof of Lemma 6.1.3(b). Recall that the set

Zν := Yν ∩ Z = {ϕ ∈ Z | ∃ η ∈ hSL2(ϕ) polarizing ν}

is a non-empty Pν∩K = Mν∩K–orbit, cf. (5.1.6) and (5.2.3). Therefore, there exists

(η, ϕ) ∈ hSL2(D) polarizing ν and with ϕ ∈ Z.

40 ROBLES

Let {N+, Y,N} the standard triple underlying η. Then [22, Remark 4.21] asserts

that this standard triple is a Cayley triple with respect to K; in particular, θK(Y ) =

−Y . Consequently,

(6.1.4) lν = pν ∩ θK(pν) = {Y ∈ gC | [Y, ξ] = 0} =⊕p+q=0

gp,qW,F .

It now follows from (3.2.6), and the fact that aν is the split component of the center

of lν , that aν ⊗ C ⊂ g0,0W,F . Likewise the definition of mν as the Killing orthogonal

complement of aν in lν implies gp,−pW,F ⊂ mν for all p 6= 0. �

Proof of Lemma 6.1.3(c). Since Yη acts on gp,qW,F by the scalar p+q, it follows directly

from Lemma 6.1.3(b) that Yη ∈ aν and Lν is the centralizer of Yη. The set of all

(η, ϕ) ∈ ν with ϕ ∈ Z is a Pν ∩ K = Mν ∩ K–orbit. (This follows directly from

the argument establishing (5.2.3).) It then follows that Yη = Yη′ for all η, η′ ∈hSL2(ν, Z). �

Lemma 6.1.5 (Key Lemma II). Given η ∈ ν as in Lemma 6.1.3(b), let gν ⊂ lν

be the real form of the subalgebra gν,C ⊂ gC generated by ⊕p6=0 gp,−pW,F . Then gν is a

semisimple subalgebra of mν, an ideal of the Levi subalgebra lν, and is independent of

our choice of η ∈ ν.

Proof. The claim that gν is a semisimple ideal of lν follows from Lemmas ?? and

6.1.3(b). Any other choice (η′, ϕ′) ∈ ν is of the form g · (η, ϕ) for some g ∈ Pν . Since

both ϕ, ϕ′ ∈ Zν ⊂ Z, we see from (3.1.5) and (5.1.5) that gKg−1 = K. Since the

normalizer of K in G is K itself, it follows that g ∈ K. Thus g ∈Mν ∩K. The effect

of replacing η ∈ ν with η′ ∈ ν is to replace gν with Adggν . Since gν ⊂ mν is an ideal,

Ad(Mν) necessarily preserves gν . �

It follows from Lemma 6.1.5 that π(ν) := (Pν ,Yν , gν) ∈ P ; let

Gν = Pν/Zν and Xν = Zν\Yν

be the associated automorphism group and boundary component (§5.1). In order to

describe the boundary components Xν , it will be helpful to understand the stabilizer

of ϕ ∈ Zν in the Levi factor Lν .


Lemma 6.1.6. Given (η, ϕ) as in Lemma 6.1.3(b), let (W,F ) the be corresponding

MHS and let Y and N be the neutral and nilnegative elements of η. Then

StabLν (ϕ) = StabG(F ) ∩ StabG(N) ∩ StabG(Y )

= {g ∈ G | g · F = F , AdgN = N , AdgY = Y } .

Proof. By Lemma 6.1.3(c), we have StabG(Y ) = Lν . The containment

(6.1.7) StabG(F ) ∩ StabG(N) ∩ StabG(Y ) ⊂ StabLν (ϕ)

then follows directly from (3.2.27b).

To establish the reverse containment, first note that (3.2.27c) and Lemma 6.1.3(c)

yield

(6.1.8) StabLν (ϕ) ⊂ StabG(F ) ∩ StabG(Y ) = StabLν (F ) .

Second, from Lemma 6.1.3(b) and (??), we see that ⊕p≥0 gp,−pF,W is the Lie subalgebra

of lν ⊗C stabilizing F . Moreover, since (W,F ) is R–split, we see that g0,0F,W ∩ g is the

Lie algebra of lν stabilizing F . It follows that the stabilizer

(6.1.9) StabLν (F ) = {g ∈ G | g(Ip,qW,F ) = Ip,qW,F , ∀p, q}

of F in Lν is precisely the stabilizer of the Deligne splitting in G. Now (3.2.27b)

implies

exp(iAdgN) · F = exp(iN) · F

for any g ∈ StabLν (ϕ). Therefore, exp(−iN) exp(iAdgN) stabilizes F . On the other

hand (3.2.8) and (6.1.9) imply that both N and AdgN lie in g−1,−1F,W . Consequently,

exp(−iN) exp(iAdgN) must be the identity element; that is N = AdgN , and we

conclude that

(6.1.10) StabLν (ϕ) ⊂ StabG(N) .

The lemma now follows from (6.1.7), (6.1.8) and (6.1.10). �

It will be helpful to rephrase Lemma 6.1.6 as follows. Given N and Y as in the

lemma, let {N+, Y,N} be the associated standard triple (§2.3.1). The fact that Y

and N determine N+ implies that StabG(N) ∩ StabG(Y ) stabilizes the triple. Let

P` := {v ∈ V | Y v = `V , N+v = 0}

42 ROBLES

denote the associated primitive subspaces, and

(6.1.11) P` :=⊕p+q=`

P p,q , P p,q := P` ∩ Ip,qW,F

the corresponding Hodge decompositions. Then

VC =⊕a,`≥0

NaP` .

It then follows from (6.1.9) that g ∈ StabLν (ϕF,N) preserves the P p,q, and the full

action of g on VC is completely determined by its action on the P p,q. We have

(6.1.12) StabLν (ϕF,N) =

{g ∈ Aut(VR, Q)

∣∣∣∣∣ AdgY = Y , AdgN = N

g(P p,q) = P p,q ∀ p, q

}.

6.2. Example: period domains. Suppose that D is a period domain parameteriz-

ing effective, Q–polarized Hodge structures on V of weight n. Given ν ∈ ΠR1 , we will

see (Lemma 6.1.3) that ν admits a representative (W,F ) so that Lν is the centralizer

Lν = {g ∈ G | AdgY = Y } of the semisimple Y = YW,F ∈ g of (3.2.25). Equivalently,

Lν is the subgroup of G = Aut(V,Q) preserving the subspaces

I` := {v ∈ V | Y v = `v} ' GrW` := W`/W`−1 .

Consequently, Lν admits the following description. Observe that I` is a real form of

⊕p+q=` Ip,qF,W ⊂ VC. Moreover, Q(I`, Ik) = 0 unless k+ ` = 2n, and Q is nondegenerate

on I` + I2n−`. Consequently, the action of g ∈ Lν on I` determines the action of g on

I2n−`. Conversely, any element of Aut(I`), ` < n, uniquely determines an element of

Aut(I` + I2n−`, Q). Therefore,

Lν = Aut(In, Q) ×∏`<n

Aut(I`) ,

and

Mν = Aut(In, Q) ×∏`<n

Aut0(I`) ,

where Aut0(I`) ' SL(dim I`,R) is the set of automorphisms preserving the determi-

nant. Also, Aν ' (R>0)r where r = #{` < 0 | I` 6= 0}.The group Cν contains the factor Aut0(I`) ⊂ Mν , 0 ≤ ` < n, if and only if

I` = Im,m (` = 2m)}; likewise Cν contains the factor Aut(In, Q) if and only if In =


Im,m (n = 2m). Consequently, Gν is isomorphic to the product of those factors in

Mν with I` % Im,m.

We now restrict to the case that n = 2 for the remainder of the section, specializing

to our running example. Set

r := dimC I0,0W,F and s := dimC I

1,0W,F ,

so that the Hodge diamond of Example 3.2.11 corresponding to ν is 3r,s, and

a′ := a− r − s = dimC I2,0W,F and b′ + r := b− 2s = dimC I

1,1W,F .

Then I0 ' Rr, I1 ' R2s and I2 ' R2a′+b′+r. The discussion above yields

Lν = GL(r,R) × GL(2s,R) × O(b′, 2a′ + r) ,

Mν = SL(r,R) × SL(2s,R) × O(b′, 2a′ + r) ;

and Aν = (R>0)×2 if both r, s > 0, and Aν = R>0 if only one of r, s is positive.

Similarly,

Cν = SL(r,R) × O(b′, 2a′ + r)︸︷︷︸omit when a′ > 0

and

Gν = SL(2s,R) × O(b′, 2a′ + r)︸︷︷︸omit when a′ = 0

.

It follows from (6.1.12) that the stabilizer of ϕ in Mν is

StabMν (ϕ) ' U(s) × U(a′) × O(b′) × O(r) ,

so that

(6.2.1) Xν =SL(2s,R)

U(s)× O(b′, 2a′ + r)

O(b′) × U(a′) × O(r)︸︷︷︸omit when a′ = 0

.

The first factor of Xν fibres as

(6.2.2)

Sp(2s,R)

U(s)

SL(2s,R)

U(s)

SL(2s,R)

Sp(2s,R).

↪→

44 ROBLES

The base SL(2s,R)/Sp(2s,R) parameterizes nondegenerate, skew-symmetric bilinear

forms Q1 on GrW1 ' I1 ' R2s with detQ1 = 1; and the fibre Sp(2s,R)/U(s) param-

eterizes weight one, Q1–polarized Hodge structures on GrW1 . Likewise, the second

factor of Xν fibres as

(6.2.3)

O(b′, 2a′)

O(b′) × U(a′)

O(b′, 2a′ + r)

O(b′) × U(a′) × O(r)

O(b′, 2a′ + r)

O(b′, 2a′) × O(r).

↪→

The base O(b′, 2a′ + r)/(O(2a′, b′) × O(r)) parameterizes r–dimensional subspaces

V0 ⊂ GrW2 ' I2 ' R2a′+b′+r on which Q restricts to be negative definite; and the fibre

O(2a′, b′)/(U(a′)×O(b′)) parameterizes Q–polarized Hodge structures on V ⊥0 ⊂ GrW0

with Hodge numbers (a′, b′, a′).

It follows from the above description that our boundary components are the

“boundary bundles” of Cattani–Kaplan’s [?].

6.3. Convergence of sequences in D. Given a parabolic subgroup Q ⊂ G, let

Q = UQ × AQ ×MQ be the Langlands decomposition with respect to K. Then we

have a horospherical decomposition

(6.3.1) D ' UQ × AQ × (MQ · Z) .

Fix ν ∈ ΠR1 . Suppose Q ⊂ Pν contains a reduction Rν of Pν (§5.4). Let ∆(UQ, AQ)

denote roots of the adjoint action of aQ on uQ. Since AQ = exp(aQ) we may also

regard the roots as characters of AQ defined by aα := eα(log a) for all a ∈ A and

α ∈ ∆(UQ, AQ). Let Σ(UQ, AQ) ⊂ ∆(UQ, AQ) be the simple roots. Recall that, if

Q ⊂ P , then AP ⊂ AQ and we may identify Σ(UP , AP ) with a subset of Σ(UQ, AQ).

Definition 6.3.2 (Attaching DR1 to D). Fix a reduction Rν of Pν . Fix a sequences

kj ∈ K and yj ∈ D. Suppose that the image of kj in K/K ∩ Z(Xν) converges to the

identity coset. The sequence kjyj ∈ D converges to x∞ ∈ Xν if there exists a parabolic

subgroup Rν ⊂ Q ⊂ Pν so that, with respect to the horospherical decomposition

(6.3.1), we may write yj = ujajzj with uj ∈ UQ, aj ∈ AQ, zj ∈ MQ · Z satisfying the

following conditions:


(i) given α ∈ Σ(UQ, AQ), aαj is bounded from below;

(ii) if α ∈ Σ(Uν , Aν), then aαj → +∞;

(iii)α(log aj)

β(log aj)→ α(Yν)

β(Yν)for all α, β ∈ Σ(Uν , Aν);

(iv) a−1j ujaj → 1;

(v) there exists a pre-compact set O ⊂MQ ∩ Cν = CMQ(gν) so that zj ∈ O · Xν , for

j sufficiently large, and qν(zj)→ x∞.

Note that Q ⊂ Pν and condition (v) imply that yj ⊂ Q ·Zν ⊂ Yν . And §5.4.2 implies

that Definition 6.3.2 does not depend on the choice of reduction.

6.4. Extension of one-variable period maps. Later we will see that Definition

6.3.2 defines a topology on DRh with respect to which the group action is continuous.

Assuming this for the moment, the purpose of this section is to prove Proposition

6.4.1. Let Φ : H → D be a lift of a one-variable period map Φ : ∆∗ → Γ\Dwith unipotent monodromy. Schmid’s Nilpotent Orbit Theorem [23] asserts that Φ

is asymptotically approximated by a nilpotent orbit θ(z) = exp(zN) · F ′. The corre-

sponding PMHS (W (N), F ′) need not be R–split. However a corollary [23, (5.19)] of

the one-variable SL(2) Orbit Theorem associates to this PMHS a (η, ϕ) ∈ hSL2(D)

that is defined over Q. We let νΦ = [η] ∈ ΠR1 denote the equivalence class polarized

by this rational, horizontal SL(2).

Proposition 6.4.1. Let Φ : H→ D be the lift of a one-variable period map Φ : ∆∗ →Γ\D with unipotent monodromy. Then the limit limy→∞ Φ(x + iy) in DR

1 exists, lies

in the boundary component Xν, where ν = νΦ, and is independent of x.

We will see that the proposition follows from the following theorem of Schmid.

Theorem 6.4.2 (Schmid [23, (5.26)]). There exists a minimal rational parabolic

subgroup Q with real points Q := Q(R) ⊂ Pν and rational Langlands decomposition

Q = UQ×A′Q×M ′Q; and functions u(x, y), a(x, y), m(x, y) and k(x, y) taking values

in UQ, A′Q, M ′Q and Kϕ, respectively, and defined and real analytic on a set of the

form {(x, y) ∈ R2 | y > β} so that:

(a) For y > β, we have Φ(x+ iy) = k(x, y)u(x, y)a(x, y)m(x, y)ϕ.

46 ROBLES

(b) As y →∞, the limit of u(x, y), exp(12

log(y)Y )a(x, y), m(x, y) and k(x, y) exist

uniformly in x. Here Y = η∗(y) is the neutral element of the approximating

SL(2).

(c) In the case of u(x, y) this limit is a continuous function of x, taking value in UQ;

and

(d) the limits of exp(12

log(y)Y )a(x, y), m(x, y) and k(x, y) converge to the identity.

Remark 6.4.3. A careful reading of Schmid’s proof reveals that limy→∞ u(x, y) =

exp(xN)ζ for some ζ ∈ Uν ∩ Z(N); in particular, the limit lies in Uν .

Proof of Proposition 6.4.1. A priori, ϕ need not lie in Zν . However there exists g ∈ Pνso that g · ϕ ∈ Zν (cf. proof of Lemma 5.1.4). Then the Langlands decomposition of

Q = Q(R) with respect to our fixed K is Q = UQ × AQ ×MQ, where UQ = UQ(R)

and LQ = AQ ×MQ = gLQ(R)g−1. Consequently,

Φ(x+ iy) = g−1k(x, y) u(x, y) ga(x, y)m(x, y)g−1 · (g · ϕ) ,

where k(x, y) = gk(x, y)g−1 ∈ K and u(x, y) = gu(x, y)g−1 ∈ UQ.

As in Remark 2.1.1, ga(x, y)m(x, y)g−1 = a(x, y)m(x, y) determines real-analytic

functions a(x, y) and m(x, y) taking values in AQ and MQ respectively; we have

Φ(x+ iy) = g−1k(x, y) u(x, y) a(x, y)m(x, y) · (g · ϕ) .

These functions also have the property that both

(6.4.4)exp(1

2log(y)Y )a(x, y) and m(x, y) converge,

uniformly in x, to the identity as y →∞.

Now the parabolic Q may not satisfy the hypothesis of Definition 6.3.2; that is,

it need not be the case that MQ/(MQ ∩ Cν) = Gν . However, Pν trivially satisfies

the hypothesis of Definition 6.3.2; so we need to understand the Langlands decom-

position of u(x, y)a(x, y)m(x, y) with respect to P . Recall the relative Langlands

decomposition [5, §I.1.11] of Q ⊂ P : there exists a parabolic subgroup P ′ ⊂ MP

so that NQ = NP × NP ′ , AQ = AP × AP ′ and MP ′ = MQ. In particular, we may

define u(x, y) ∈ NP and u′(x, y) ∈ NP ′ ⊂ MP by u(x, y) = u(x, y)u′(x, y); likewise,

we define a(x, y) ∈ AP and a′(x, y) ∈ AP ′ ⊂ MP by a(x, y) = a(x, y)a′(x, y). Setting


m(x, y) := u′(x, y)a′(x, y)m(x, y), we have

u(x, y) a(x, y) m(x, y) = u(x, y) a(x, y) m(x, y) .

From Remark 6.4.3 we see that

(6.4.5) u(x, y) = u(x, y) and u′(x, u) = 1 .

Since Y ∈ AP , from (6.4.4) we deduce

limy→∞

exp(12

log(y)Y ) a(x, y) = 1 ,(6.4.6a)

limy→∞

a′(x, y) = 1 ,(6.4.6b)

uniformly in x. Equation (6.4.6a) implies both that

(6.4.7) a(x, y)α → +∞ , for all α ∈ Σ(UP , AP ) ,

and, taking Theorem 6.4.2(b)-(c) and u(x, y) = u(x, y) = gu(x, y)g−1 into account,

a(x, y)−1u(x, y)a(x, y) → 1 ,

so long as x is bounded. Finally we note (6.4.4), (6.4.5) and (6.4.6b) imply limy→∞ m(x, y) =

1, and this limit exists uniformly in x.

Given the continuity of the G action on DRh we have

limy→∞

Φ(x+ iy) = g−1 · limy→∞

k(x, y) u(x, y) a(x, y)m(x, y) · (g · ϕ) .

It now follows from Definition 6.3.2 that

limy→∞

Φ(x+ iy) = pν(g−1) · qν(gϕ) ∈ Xν .

�

6.5. Double fibration structure. In the case that D is a period domain parame-

terizing weight n = 2 polarized Hodge structures, we saw in §6.2 that the boundary

component Xν admits the structure of a fibre bundle εν : Xν → Eν . In fact, it does so

in two distinct ways, and this is true for any Mumford–Tate domain D and boundary

component Xν . Given η ∈ hSL2(ϕ) polarizing ν, let (F,N) be the corresponding

PMHS (§3.2.9), and set

Gν,F := pν(StabPν (F )) ⊂ Gν and Gν,N := pν(StabPν (N)) ⊂ Gν .

48 ROBLES

Then pν(Pν) = pν(Lν) and Lemma 6.1.6 imply that Gν,F ∩ Gν,N is contained in

the stabilizer of x = qν(ϕ) ∈ Xν in Gν ; Lemma 5.2.1 implies that equality holds.

Therefore,

(6.5.1) Xν = Gν/(Gν,F ∩Gν,N)

as a Gν–homogeneous space. Consequently, Xν admits a double fibration structure

Xν

Gν/Gν,F Dν Eν Gν/Gν,N .

δν εν

=: :=

Lemma 6.5.2. The εν–fibres are Hodge domains parameterizing Q(·, N `·)–polarized

Hodge structures on kerN `+1 ⊂ W`/W`−1.

Proof. The lemma is straightforward; see, for example, Kerr and Pearlstein’s [?]: the

εν–fibre Gν,N/(Gν,N ∩Gν,F ) is precisely the D(N) of [?]. �

Lemma 6.5.3. The base Dν is a flag domain parameterizing weight zero Hodge de-

compositions gν,C = ⊕ gp,−pν,F defined by gp,−pν,F := pν,∗(gp,−pW,F ).

Proof. Define gp,−pν,F = gν,C ∩ gp,−pW,F . (Note that gp,−pν,F = gp,−pW,F if p 6= 0, and g0,0W,F =

g0,0ν,F ⊕ g⊥ν,C.) The fact that (W,F ) is R–split implies that gν,C = ⊕gp,−pν,F is a Hodge

decomposition. Moreover, the restriction of Q to gν is nondegenerate (because gν is

semisimple). And while the Hodge decomposition is not Q–polarized in general, the

fact that gC = ⊕ gp,qW,F is the Deligne splitting of a polarizable MHS implies that the

Hermitian form Q(·, ·) is nondegenerate on gp,−pν,F .

Since the restriction of pν,∗ : pν → gν to gν is an isomorphism of real Lie algebras,

we see that gν,C = ⊕ gp,−pν,F is a Hodge decomposition. In a mild abuse of notation,

we continue to let Q denote the nondegenerate bilinear form on gν . To see that Dν

is a flag domain, it suffices to show that Dν is open in the Gν(C)–orbit Dν of the

corresponding Hodge flag. This follows from the fact that the Hermitian form Q(·, ·)is nondegenerate on gp,−pν,F . �

Example 6.5.4 (Period domain with h = (a, b, a)). Continuing with the case that D

is a period domain parameterizing weight n = 2 polarized Hodge structures (§6.2),


the δν–fibration is given by

{pt} SL(2s,R)

U(s)

SL(2s,R)

U(s)

↪→

on the first factor of (6.2.1), and by

O(b′, r)

O(b′) × O(r)

O(b′, 2a′ + r)

O(b′) × U(a′) × O(r)

O(b′, 2a′ + r)

O(b′, r) × U(a′)

↪→

on the second.

Remark 6.5.5 (Relationship between grading elements). Let Eψ ∈ i gν be the grading

element corresponding to the Hodge decomposition ψ := δν ◦ qν(ϕF,N), cf. §3.1.4.

Recall the grading element EW,F of §3.2.3. We have Eψ = pν,∗(EW,F ).

Remark 6.5.6. When D is Hermitian, we have Gν = Gν,N , so that Eν is a point, and

Xν = Dν (Proposition ??).

References

[1] W. L. Baily, Jr. and A. Borel. Compactification of arithmetic quotients of bounded symmetric

domains. Ann. of Math. (2), 84:442–528, 1966.

[2] Armand Borel. Some metric properties of arithmetic quotients of symmetric spaces and an

extension theorem. J. Differential Geometry, 6:543–560, 1972. Collection of articles dedicated

to S. S. Chern and D. C. Spencer on their sixtieth birthdays.

[3] Armand Borel. Linear algebraic groups, volume 126 of Graduate Texts in Mathematics. Springer-

Verlag, New York, second edition, 1991.

[4] Armand Borel and Lizhen Ji. Compactifications of locally symmetric spaces. J. Differential

Geom., 73(2):263–317, 2006.

[5] Armand Borel and Lizhen Ji. Compactifications of symmetric and locally symmetric spaces.

Mathematics: Theory & Applications. Birkhauser Boston, Inc., Boston, MA, 2006.

50 ROBLES

[6] P. Brosnan, G. Pearlstein, and C. Robles. Nilpotent cones and their representation theory,

volume 39 of Advanced Lectures in Mathematics (ALM), pages 151–206. Higher Education

Press and International Press, 2017. arXiv:1602.00249.

[7] Andreas Cap and Jan Slovak. Parabolic geometries. I, volume 154 of Mathematical Surveys and

Monographs. American Mathematical Society, Providence, RI, 2009. Background and general

theory.

[8] James Carlson, Stefan Muller-Stach, and Chris Peters. Period mappings and period domains,

volume 85 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cam-

bridge, 2003.

[9] Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Le Dung Trang, editors. Hodge theory,

volume 49 of Mathematical Notes. Princeton University Press, Princeton, NJ, 2014.

[10] Eduardo Cattani, Aroldo Kaplan, and Wilfried Schmid. Degeneration of Hodge structures. Ann.

of Math. (2), 123(3):457–535, 1986.

[11] Pierre Deligne. Theorie de Hodge. III. Inst. Hautes Etudes Sci. Publ. Math., (44):5–77, 1974.

[12] Alan H. Durfee. A naive guide to mixed Hodge theory. In Singularities, Part 1 (Arcata, Calif.,

1981), volume 40 of Proc. Sympos. Pure Math., pages 313–320. Amer. Math. Soc., Providence,

RI, 1983.

[13] Gregor Fels, Alan Huckleberry, and Joseph A. Wolf. Cycle spaces of flag domains, volume 245

of Progress in Mathematics. Birkhauser Boston Inc., Boston, MA, 2006. A complex geometric

viewpoint.

[14] Mark Green, Phillip Griffiths, and Matt Kerr. Mumford-Tate groups and domains: their geom-

etry and arithmetic, volume 183 of Annals of Mathematics Studies. Princeton University Press,

Princeton, NJ, 2012.

[15] Phillip Griffiths, Colleen Robles, and Domingo Toledo. Quotients of non-classical flag domains

are not algebraic. Algebr. Geom., 1(1):1–13, 2014.

[16] Phillip A. Griffiths. Periods of integrals on algebraic manifolds. I. Construction and properties

of the modular varieties. Amer. J. Math., 90:568–626, 1968.

[17] Phillip A. Griffiths. Periods of integrals on algebraic manifolds. II. Local study of the period

mapping. Amer. J. Math., 90:805–865, 1968.

[18] Matt Kerr, Gregory Pearlstein, and Colleen Robles. Polarized relations on horizontal sl(2)’s.

arXiv:1705.03117, 2017.

[19] Anthony W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathematics.

Birkhauser Boston Inc., Boston, MA, second edition, 2002.

[20] Chris A. M. Peters and Joseph H. M. Steenbrink. Mixed Hodge structures, volume 52 of Ergeb-

nisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathe-

matics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in

Mathematics]. Springer-Verlag, Berlin, 2008.


[21] C. Robles. Schubert varieties as variations of Hodge structure. Selecta Math. (N.S.), 20(3):719–

768, 2014.

[22] Colleen Robles. Classification of horizontal SL(2)s. Compos. Math., 152(5):918–954, 2016.

[23] Wilfried Schmid. Variation of Hodge structure: the singularities of the period mapping. Invent.

Math., 22:211–319, 1973.

[24] T. A. Springer. Linear algebraic groups, volume 9 of Progress in Mathematics. Birkhauser Boston

Inc., Boston, MA, second edition, 1998.

E-mail address: [email protected]

Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320

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