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
2 ROBLES
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
INFORMAL COURSE NOTES 5
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
INFORMAL COURSE NOTES 7
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.
INFORMAL COURSE NOTES 9
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 ′ .
INFORMAL COURSE NOTES 11
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.
INFORMAL COURSE NOTES 13
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).
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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.
INFORMAL COURSE NOTES 15
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.
INFORMAL COURSE NOTES 17
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.
INFORMAL COURSE NOTES 19
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
INFORMAL COURSE NOTES 21
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.
INFORMAL COURSE NOTES 23
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.
INFORMAL COURSE NOTES 25
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
INFORMAL COURSE NOTES 27
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)).
INFORMAL COURSE NOTES 29
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. �
INFORMAL COURSE NOTES 31
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
INFORMAL COURSE NOTES 33
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. �
INFORMAL COURSE NOTES 35
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.
INFORMAL COURSE NOTES 37
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.
INFORMAL COURSE NOTES 39
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 .
INFORMAL COURSE NOTES 41
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 =
INFORMAL COURSE NOTES 43
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:
INFORMAL COURSE NOTES 45
(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
INFORMAL COURSE NOTES 47
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),
INFORMAL COURSE NOTES 49
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 ??).
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INFORMAL COURSE NOTES 51
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E-mail address: [email protected]
Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320
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