De Rham 2-Cohomology of Real Flag Manifolds
Let FΘ = G/PΘ be a flag manifold associated to a non-compact real simple Lie group G and the parabolic subgroup PΘ. This is a closed subgroup of G determined by a subset Θ of simple restricted roots of g = Lie(G). This paper computes the second de Rham cohomology group of FΘ. We prove that it is zer...
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| Cite this: | De Rham 2-Cohomology of Real Flag Manifolds / V. del Barco, L.A.B. San Martin // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 12 назв. — англ. |
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| citation_txt | De Rham 2-Cohomology of Real Flag Manifolds / V. del Barco, L.A.B. San Martin // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 12 назв. — англ. |
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| description | Let FΘ = G/PΘ be a flag manifold associated to a non-compact real simple Lie group G and the parabolic subgroup PΘ. This is a closed subgroup of G determined by a subset Θ of simple restricted roots of g = Lie(G). This paper computes the second de Rham cohomology group of FΘ. We prove that it is zero in general, with some rare exceptions. When it is non-zero, we give a basis of H²(FΘ, ℝ) through the Weil construction of closed 2-forms as characteristic forms of principal fiber bundles. The starting point is the computation of the second homology group of FΘ with coefficients in a ring R.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 051, 23 pages
De Rham 2-Cohomology of Real Flag Manifolds
Viviana DEL BARCO †‡ and Luiz Antonio Barrera SAN MARTIN ‡
† UNR-CONICET, Rosario, Argentina
E-mail: delbarc@fceia.unr.edu.ar
URL: http://www.fceia.unr.edu.ar/~delbarc/
‡ IMECC-UNICAMP, Campinas, Brazil
E-mail: smartin@ime.unicamp.br
Received January 08, 2019, in final form June 25, 2019; Published online July 05, 2019
https://doi.org/10.3842/SIGMA.2019.051
Abstract. Let FΘ = G/PΘ be a flag manifold associated to a non-compact real simple
Lie group G and the parabolic subgroup PΘ. This is a closed subgroup of G determined
by a subset Θ of simple restricted roots of g = Lie(G). This paper computes the second
de Rham cohomology group of FΘ. We prove that it is zero in general, with some rare
exceptions. When it is non-zero, we give a basis of H2(FΘ,R) through the Weil construction
of closed 2-forms as characteristic forms of principal fiber bundles. The starting point is the
computation of the second homology group of FΘ with coefficients in a ring R.
Key words: flag manifold; cellular homology; Schubert cell; de Rham cohomology; charac-
teristic classes
2010 Mathematics Subject Classification: 57T15; 14M15
1 Introduction
A real flag manifold is a homogeneous manifold FΘ = G/PΘ where G is a connected Lie group
with Lie algebra g which is non-compact and semi-simple, and PΘ is a parabolic subgroup of G.
Real grassmannians and projective spaces belong to the family of real flag manifolds.
Topological properties of these manifolds have been of interest for several authors. The
fundamental group of real flag manifolds was considered by Wiggerman [12] giving a continuation
to the work of Ehresmann on real flag manifolds with G = SL(n,R). The integral homology of
real flag manifolds has been studied by Kocherlakota who gives an algorithm for its computation
through Morse homology [6], based on previous work of Bott and Samelson [1]. A different
approach to study their homology is given by Rabelo and the second named author of this
paper [8], focusing on the geometry involved in the cellular decomposition of the manifolds.
Mare [7] considered the cohomology rings of a subfamily of real flag manifolds, namely, those
having all roots of rank greater of equal than two. Real flag manifolds of split real forms are
not part of the subfamily considered in [7].
The present paper focuses on the computation of the second de Rham cohomology group
of real flag manifolds. Our motivation comes from symplectic geometry; it is well known that
the annihilation of the second de Rham cohomology group is an obstruction to the existence of
symplectic structures on compact manifolds. It is worth stressing that we deal with real flag
manifolds associated to any non-compact real form G of complex simple Lie groups, and any
parabolic subgroup, without restrictions.
To obtain the cohomology groups, we start with the explicit computation of second homology
groups with coefficients on a ring R, H2(FΘ, R). The description of H2(FΘ, R) does not follow
directly from the works of Kocherlakota and Rabelo–San Martin. Instead it requires to work
over the root systems. The classification of the homology groups is achieved in Theorem 4.1,
mailto:delbarc@fceia.unr.edu.ar
http://www.fceia.unr.edu.ar/~delbarc/
mailto:smartin@ime.unicamp.br
https://doi.org/10.3842/SIGMA.2019.051
2 V. del Barco and L.A.B. San Martin
after developments in Sections 2 and 3 where we apply the tools of Rabelo–San Martin [8].
Mainly, we show that H2(FΘ, R) is a torsion group, except when there are roots of rank 2 in the
system of restricted roots of the real flag. The rank of a root α is rankα = dim gα + dim g2α,
where these subspaces are the root spaces associated to α and 2α (if it is a root).
It follows that H2
dR(FΘ,R) = {0} unless the root system of the real Lie algebra contains
roots having rank 2. Moreover, the number of such roots gives the dimension of H2
dR(FΘ,R)
(see Theorem 5.1 below). These data can be read off from the classification table of real simple
Lie algebras (see Warner [11, pp. 30–32]) and hence H2
dR(FΘ,R) can be completely determined.
We summarize the computation of the second de Rham cohomologies in Theorem 4.1.
Once we have the classification of the real flag manifolds FΘ satisfying H2
dR(FΘ,R) 6= {0} we
search for differential 2-forms representing a basis of these non-trivial spaces.
We get such a basis by applying the Weil construction to the principal fiber bundle π :
K −→ FΘ with structure group KΘ = PΘ ∩K, where K is a maximal compact subgroup of G,
and FΘ = G/PΘ = K/KΘ. To perform the Weil construction we choose in a standard way
a left invariant connection ω in the principal bundle K −→ FΘ with curvature form Ω. Each
adjoint invariant f in the dual k∗Θ of the Lie algebra kΘ of KΘ yields an invariant closed 2-form f̃
in K/KΘ. We prove that the 2-forms f̃ exhaust the 2-cohomology classes by exhibiting a basis
formed by characteristic forms which is dual to the Schubert cells that generate the second real
homology.
To prove the surjectivity of the map f → f̃ it is required a careful description of the center
of KΘ which we provide in Section 6 by looking first at the center of the M group where M = K∅
is the isotropy group of the maximal flag manifold. In Section 7 we apply the previous results to
get the desired dual bases of differential 2-forms in Theorem 7.3. In Section 8 we illustrate our
results with concrete computations in the flag manifolds of the real simple Lie algebras su(p, q),
p ≤ q, that are real forms of sl(p+ q,C) and realize the Lie algebras of types AIII1 and AIII2.
As consequence of our results, we obtain that a real flag manifold does not carry symplectic
structures, unless its corresponding root system contains roots of rank 2. Moreover, if the system
contains roots of rank 2, the only case where the manifold is symplectic is when the real flag
manifold is actually the product of complex flag manifolds of the form SU(n)/T .
2 Cellular decomposition and boundary maps
This section aims to fix notations and to introduce the preliminaries for the rest of the paper.
For the classical theory the reader is referred to the books of Knapp [4], Helgason [3] and
Warner [11]. The treatment of the cellular decomposition and the boundary maps of real flag
manifolds follows the presentation given in the work of L. Rabelo and L. San Martin [8].
Let g be a non-compact real simple Lie algebra and let g = k⊕ s be a Cartan decomposition.
Let a be a maximal abelian subalgebra of s and denote Π the set of restricted roots of the
pair (g, a). Let Σ be a subset of simple roots and denote Π± the set of positive and negative
roots, respectively. The Iwasawa decomposition of g is given by g = k⊕ a⊕ n with n =
∑
α∈Π+
gα
and gα the root space corresponding to α.
Given a simple Lie group G with Lie algebra g, denote K, A and N the connected subgroups
corresponding to the Lie subalgebras k, a and n, respectively.
To a subset of simple roots Θ ⊂ Π there is associated the parabolic subalgebra
pΘ = a⊕m⊕
∑
α∈Π+
gα ⊕
∑
α∈〈Θ〉−
gα,
where m is the centralizer of a in k and 〈Θ〉− is the set of negative roots generated by Θ. The
minimal parabolic subalgebra p is obtained for Θ = ∅. The normalizer PΘ of pΘ in G is the
De Rham 2-Cohomology of Real Flag Manifolds 3
standard parabolic subgroup associated to Θ. In this manner, each subset Θ ⊂ Σ defines the
homogeneous manifold FΘ = G/PΘ; these homogeneous manifolds are the object of study of
this paper. Denote by bΘ the class of the identity in FΘ, that is, bΘ = ePΘ. For Θ = ∅ the
index Θ is dropped to simplify notations and F is referred as the maximal flag manifold, also
called full or complete flag manifold. When Θ 6= ∅, FΘ is called partial flag manifold.
Let W be the Weyl group associated to a. This is a finite group and is generated by reflec-
tions rα over the hyperplanes α(H) = 0 in a, with α a simple root and H a regular element. The
length `(w) of an element w ∈ W is the number of simple reflections in any reduced expression
as a product w = rα1 · · · rαt of reflections with αi ∈ Σ. Denote Πw = Π+ ∩ wΠ−, the set of
positive roots taken to negative by w−1. For Θ ⊂ Σ, WΘ denotes the subgroup of W generated
by rα with α ∈ Θ.
The Weyl group is isomorphic the quotient M∗/M of the normalizer M∗ of a in K over the
respective centralizer M . It acts, through M∗, on FΘ and the left N classes of the orbit M∗bΘ
give a decomposition of FΘ, known as the Bruhat decomposition:
FΘ =
∐
wWΘ∈W/WΘ
N · wbΘ.
One should notice that N · wbΘ does not depend on the choice of the representative, namely,
N · w1bΘ = N · w2bΘ whenever w1WΘ = w2WΘ.
A cellular decomposition of the flag manifold FΘ is given by Schubert cells SΘ
w = cl(N ·wbΘ)
with w a representative of wWΘ; here cl denotes closure. The dimension of the Schubert cell
defined by wWΘ is
dimSΘ
w =
∑
α∈Πw\〈Θ〉+
dim gα.
For the maximal flag this formula reads dimSw =
∑
α∈Πw
dim gα. If w ∈ W has reduced expression
w = rα1 · · · rαt , then
dimSw =
t∑
i=1
dim(gαi + g2αi) (2.1)
(see [12, Corollary 2.6]). In particular, dimSw = `(w) for all w ∈ W if g is a split real form. For
a simple root α, we denote
rankα := dim gα + dim g2α.
Notice that rankα = dimSrα . Moreover, g2α = {0} if dim gα = 1 so that rankα = 2 if and only
if dim gα = 2. If w = rα1 · · · rαt is a reduced expression then dimSw ≥ `(w) = t if and only if
rankαi = 1 for all i = 1, . . . , t.
At this point we fix, once and for all, reduced expressions of the elements w in W, w =
rα1 · · · rαt , as a product of simple reflections. Such decompositions determine the characteristic
maps which describe how Schubert cells are glued to give the cellular decomposition of FΘ [8].
In what follows we recall the definition of the cellular complex and the boundary map giving
the homology of FΘ with coefficients in a ring R. We give explicit formulas for the boundary
maps up to dimension 3 which will be used in the next section.
Let Ci be the R-module freely generated by Sw, w ∈ W and dimSw = i, for i = 0, . . . ,dimF.
Notice that C0 = R since there is just one zero-dimensional cell, namely the origin {b}. Define
Σsplit = {α ∈ Σ: rankα = dim gα = 1},
Σ2 = {α ∈ Σ: rankα = 2}.
4 V. del Barco and L.A.B. San Martin
Proposition 2.1.
• C1 is the free module spanned by Srα with α ∈ Σsplit.
• C2 is the free module spanned by Srαrβ with α 6= β ∈ Σsplit and by Srα with α ∈ Σ2.
• C3 is the free module spanned by Srαrβrγ with β 6= α 6= γ ∈ Σsplit, by Srαrβ with α ∈ Σ2
and β ∈ Σsplit, or vice-versa, and by Srα, with α ∈ Σ and rankα = 3.
Proof. By definition Srα , Srαrβ , Srαrβrγ are of dimensions 1, 2 and 3 respectively, when α, β, γ ∈
Σsplit and γ 6= α 6= β and dimSrα = 2 if rankα = dim gα = 2. Similarly one obtains dimSrα = 3
if and only if 2α is not a root and dim gα = 3 or dim gα = 2 and dim g2α = 1 since 2α is not
a root if dim gα = 1.
Given w = rαrβ, dimSw = 3 only if dim(gα + g2α) = 2 and dim(gβ + g2β) = 1, or vice-versa.
But, dim(gα + g2α) = 2 implies dim gα = 2 and 2α is not a root. Thus dimSw = 3 implies
α ∈ Σ2 and β ∈ Σsplit or the inverse situation for α and β. �
The chain complex of the partial flag manifold FΘ is constituted by minimal Schubert cells.
For any w ∈ W there exists a unique w1 ∈ wWΘ such that Π+∩w−1
1 Π−∩〈Θ〉 = ∅ [8, Lemma 3.1].
Such element is called minimal in wWΘ and satisfies dimSΘ
w = dimSw1 . Denote Wmin
Θ the set
of minimal elements in W with respect to Θ. For i = 0, . . . ,dimFΘ let CΘ
i ⊂ Ci be the free R
module spanned by Sw with w ∈ Wmin
Θ and dimSw = i. Next we describe the minimal elements
giving cells of dimension ≤ 3.
Lemma 2.2.
• w = rα is minimal if and only if α /∈ Θ,
• w = rαrβ is minimal if and only if β /∈ Θ and rβα /∈ 〈Θ〉,
• w = rαrβrγ is minimal if and only if γ /∈ Θ and rγβ, rγrβα /∈ 〈Θ〉.
Proof. The only positive roots taken into a negative root by w = rα with α ∈ Σ are α and 2α,
when this is a root. So Π+ ∩ rαΠ− ∩ 〈Θ〉+ = ∅ if and only if α /∈ Θ. If w = rαrβ, with α, β ∈ Σ,
α 6= β then the positive roots taken to negative by rαrβ are β, rβα and the multiples 2β and
rβ(2α) when they are actually roots. Hence w is minimal if and only if β /∈ Θ and rβα /∈ 〈Θ〉.
Similarly, for w = rαrβrγ with α 6= β 6= γ we have
wΠ+ ∩Π− = {γ, 2γ, rγβ, 2rγβ, rγrβα, 2rγrβα} ∩Π+.
Thus w = rαrβrγ is minimal if γ, rγβ, rγrβα are not in 〈Θ〉. �
The boundary operator ∂ applied to a Schubert cell Sw of dimension i, gives a linear combi-
nation of Schubert cells of dimension i− 1
∂Sw =
∑
w′
c(w,w′)Sw′ . (2.2)
The coefficients c(w,w′) are given in [8, Section 2]. These coefficients are always 0 or ±2 and
they behave as follows (see also [10]). For α ∈ Σ we denote α∨ := 2α
〈α,α〉 , so that 〈α∨, β〉 is an
integer for β ∈ Σ (see for instance [3]).
1. Given w = rα1 · · · rαt ∈ W, one has c(w,w′) = 0 for any w′ ∈ W which is not obtained
from w by removing a reflection rαi .
2. If w = rα1 · · · rαt−1rαt and w′ = rα1 · · · rαt−1 then c(w,w′) = 0.
De Rham 2-Cohomology of Real Flag Manifolds 5
3. Assume w = rα1 · · · rαt and w′ = rα1 · · · r̂αi · · · rαt are reduced decompositions of w and w′
respectively. If rankαi = 1 then c(w,w′) = ±
(
1− (−1)σ(w,w′)
)
where
σ(w,w′) =
∑
β∈Πu
〈α∨i , β〉dim gβ, u = rαi+1 · · · rαt . (2.3)
Otherwise, c(w,w′) = 0.
Notice that if w = rα1 · · · rαt is a reduced expression, the coefficients c(w,w′) are non-zero
only when, by erasing a reflection rαi with i 6= t, a reduced expression of a root w′ with
dimSw′ = dimSw − 1 is obtained.
The choice of the sign ± in c(w,w′) is given by the degree of a certain map depending on
w and w′ [8, Theorem 2.6] and it can be determined precisely. The computations here will not
need the exact number, so the ± will be present along the paper.
The next proposition gives the boundary of cells in Ci for i ≤ 3.
Proposition 2.3. Given simple roots α, β, γ there are the following expressions for the boundary
operator ∂:
1. ∂Srα = 0.
2. If α 6= β then ∂Srαrβ = c(rαrβ, rβ)Srβ . In particular, if Srαrβ is a 3-cell then ∂Srαrβ = 0.
(This is the case when α ∈ Σ2 and β ∈ Σsplit or vice-versa.)
3. If α 6= β then
∂Srαrβrα = c(rαrβrα, rβrα)Srαrβ . (2.4)
4. If γ 6= α and α, β, γ ∈ Σsplit then
∂Srαrβrγ = c(rαrβrγ , rαrγ)Srαrγ + c(rαrβrγ , rβrγ)Srβrγ . (2.5)
Proof. The boundary of Srα contains only the 0-cell hence ∂Srα = 0 regardless the dimension
of Srα . For the cell Srαrβ its boundary ∂Srαrβ has no component in the direction of Srα which
obtained by removing the last reflection rβ from rαrβ. If rankα = 2 then c(rαrβ, rβ) = 0
because Srβ has codimension 2 in Srαrβ . On the other hand if rankβ = 2 then c(rαrβ, rβ) =
±
(
1 − (−1)σ(rαrβ ,rβ)
)
= 0 since σ(rαrβ, rβ) = 〈α∨, β〉 dim gβ is even. This proves the second
statement. The last two statements follow from the fact that c(rαrβrγ , rαrβ) = 0 (removal of
the last reflection) and c(rαrβrγ , rαrα) = 0 since r2
α = 1 is not a reduced expression. �
The boundary map ∂min
Θ : CΘ
i −→ CΘ
i−1 is defined through the boundary map ∂ for the homol-
ogy of the maximal flag. Given Sw, with w ∈ Wmin
Θ , let Iw be the set of minimal elements w′
such that dimSw′ = dimSw − 1. Define
∂min
Θ Sw =
∑
w′∈Iw
c(w,w′)Sw′ .
The homology of FΘ with coefficients in R is the homology of the complex
(
CΘ, ∂min
Θ
)
[8, Theo-
rem 3.4]. Proposition 2.3 gives the boundary map of minimal cells up to dimension three.
Propositions 2.1 and 2.3 and Lemma 2.2 account to a description of the second homology
group of a real flag manifold in terms of its split part and the roots of rank 2 in the system. In
the text below we include the maximal flag case F as the particular instance FΘ with Θ = ∅.
Denote gsplit the split real form whose Dynkin diagram is given by Σsplit. Set Θsplit = Θ∩Σsplit
and ΣΘ
2 = {α ∈ Σ\Θ: rankα = 2} and let FΘsplit
be the flag manifold of gsplit associated to Θsplit.
6 V. del Barco and L.A.B. San Martin
Theorem 2.4. Let g be a non-compact simple Lie algebra with simple system of restricted
roots Σ and fix Θ ⊂ Σ. Let FΘ be the flag manifold of g determined by Θ and let FΘsplit
be as
above. Then
H2(FΘ, R) = H2(FΘsplit
, R)⊕
∑
α∈ΣΘ
2
R · Srα .
Remark 2.5. The boundary operator ∂Θ of the flag manifolds of a non-compact real simple
Lie algebra g is basically determined by the homology of its split part gsplit as pointed out in [8,
p. 18]. Theorem 2.4 is a particular instance of this fact.
3 Homology groups of flag manifolds of split real forms
In this section we compute the second homology groups of real flag manifolds associated to split
real simple Lie algebras. We obtain these homology groups trough the explicit computation of
the coefficients c(w,w′). These coefficients are either zero or ±2, so ∂ = 0 if the characteristic
of R is charR = 2. Thus in the sequel we assume charR 6= 2.
The maximal flag case (with g split) is treated first since it gives the model for the partial
flag by restriction of the boundary map to the set of minimal cells. Even though the homology
of flag manifolds of type G2 is treated in [8], we consider those here for the sake of completeness
of the presentation.
3.1 Maximal flag manifolds
The results below make reference to the lines in Dynkin diagrams associated to simple real Lie
algebras g which are split.
Lemma 3.1. Let α, β be simple roots with α 6= β. Then ∂Srαrβ = ±2Srβ if and only if α and β
are either simple linked, or double linked and α is a long root.
Proof. By Proposition 2.3 we have ∂Srαrβ = c(rαrβ, rβ)Srβ where
c(rαrβ, rβ) = ±
(
1− (−1)〈α
∨,β〉dim gβ
)
. (3.1)
Since g is a split real form dim gβ = 1, thus c(rαrβ, rβ) = 0 if and only if 〈α∨, β〉 is even,
otherwise c(rαrβ, rβ) = ±2 and ∂Srαrβ = ±2Srβ . We have 〈α∨, β〉 is even only when 〈α∨, β〉 = 0
or when α and β are linked by a double line with α the short root. �
Corollary 3.2. If a diagram has only simple lines then ∂ : C2 −→ C1 is surjective. If a diagram
has double lines then the image of ∂ : C2 −→ C1 is spanned Srβ with β a short simple root, or β
long simple root such that there exists α ∈ Σ simple linked to β.
Three simple roots α, β, γ are said to be in an A3 configuration (in this order) if they are
linked as followse
α
e
β
e
γ
In this case, ∂Srαrβ = ±2Srβ and ∂Srγrβ = ±2Srβ , so there is a (unique) choice of a sign
ηα,β,γ = ±1 such that ∂(Srαrβ + ηα,β,γSrγrβ ) = 0. Therefore, each A3 configuration gives an
element in the kernel of ∂ : C2 −→ C1.
Proposition 3.3. If a diagram has only simple lines the kernel of ∂ = ∂2 : C2 −→ C1 is spanned
by the following elements:
• Srαrβ with 〈α, β〉 = 0 and
De Rham 2-Cohomology of Real Flag Manifolds 7
• Srαrβ + ηα,β,γSrγrβ with α, β, γ in an A3 configuration (in this order) and ηα,β,γ = ±1
the sign such that ∂(Srαrβ + ηα,β,γSrγrβ ) = 0.
Proof. Lemma 3.1 and the reasoning above show that the elements in the statement are indeed
in the kernel of ∂ : C2 −→ C1. One should see that these are the only generators.
The boundary ∂Srαrβ is a multiple of Srβ hence an element of ker ∂2 is a sum of linear
combinations of 2-cells of the form
∑
j∈J
nj Srαj rβ , with nj ∈ R for all j ∈ J.
In such a linear combination we can take αj not orthogonal to β for all j because ∂Srαrβ = 0 if
〈α, β〉 = 0.
In a simply laced diagram a set of roots {αj , β} appearing in a linear combination as above
〈αj , β〉 6= 0 occurs only in an A3 configuration (with β the middle root) or if the roots are in
a D4 configuration as follows
eα2
e
α1
e
β
e
α3
A linear combination n1Srα1rβ
+n2Srα2rβ
+n3Srα3rβ
∈ ker ∂ associated to a D4 configuration
belongs to the span of the combinations Srαrβ + ηα,β,γSrγrβ given by A3 configurations where
as above ηα,β,γ = ±1 is the sign such that ∂(Srαrβ + ηα,β,γSrγrβ ) = 0. In fact, suppose first that
∂Srαirβ = ∂Srαj rβ for all i 6= j, that is, ηα1,β,α2 = ηα1,β,α3 = ηα2,β,α3 = −1. Then
∂
(
n1Srα1rβ
+ n2Srα2rβ
+ n3Srα3rβ
)
= 0
is the same as 2(n1 + n2 + n3) = 0 so that n3 = −(n1 + n2) because charR 6= 2. Hence
3∑
i=1
niSrαirβ = n1(Srα1rβ
− Srα3rβ
) + n2(Srα2rβ
− Srα3rβ
)
= n1(Srα1rβ
+ ηα1,β,α3Srα3rβ
) + n2(Srα2rβ
+ ηα2,β,α3Srα3rβ
)
as we wanted to show. If the images ∂Srα1rβ
have two coincident signs and one opposite,
a similar argument gives
3∑
i=1
niSrαirβ as a combination of the elements in the kernel given by
the A3 subdiagrams. �
In a diagram with double lines, another configuration becomes relevant. The roots α, β, γ
are said to be in a C3 configuration if they are linked as followse
α
e
β
�
A
e
γ
As in an A3 configuration, ∂Srαrβ = ±2Srβ and ∂Srγrβ = ±2Srβ because β is the short root
in the double link with γ. Hence there is a (unique) sign ηα,β,γ = ±1 such that ∂(Srαrβ +
ηα,β,γSrγrβ ) = 0.
Proposition 3.4. If a diagram has double lines, then the kernel of ∂ : C2 −→ C1 is spanned
by Srαrβ with 〈α, β〉 = 0 and Srαrβ + ηα,β,γSrγrβ with α, β, γ in an A3 configuration, as in the
previous proposition together with the following generators:
• Srαrβ with α 6= β double linked and α the short root and
8 V. del Barco and L.A.B. San Martin
• Srαrβ + µα,β,γSrγrβ with α, β, γ in a C3 configuration (in this order) and µα,β,γ = ±1 the
sign such that ∂(Srαrβ + ηα,β,γSrγrβ ) = 0.
Proof. The proof of this proposition is the same as the previous one. The elements in the
previous proposition and those in the statement belong to the kernel of ∂ : C2 −→ C1.
On the other hand for a linear combination of 2-dimensional cells in ker ∂ we can assume that
it has the form
∑
j∈J
njSrαj rβ with αj not orthogonal to β for all j. Again, the only possibilities is
that β and α′is are in a C3, A3 or a D4 configuration. The last case is in fact a linear combination
of A3 configurations so the result follows. �
With the next proposition complete the computation of the kernel of ∂ : C2 −→ C1 by con-
sidering the Lie algebra G2.
Proposition 3.5. If the diagram is of type G2 then the kernel of ∂ : C2 −→ C1 is zero.
Proof. Recall that G2 has two simple roots α1, α2 linked by a triple line. Thus Srα1rα2
and
Srα2rα1
span C2. These roots satisfy 〈α∨1 , α2〉 = −1 and 〈α∨2 , α1〉 = −3. Then by (2.3),
σ(rαirαj , rαj ) = 〈α∨i , αj〉 is odd for 1 ≤ i 6= j ≤ 2and thus c(rαirαj , rαj ) = ±2. Therefore
∂Srα1rα2
= ±2Srα2
and ∂Srα2rα1
= ±2Srα1
. �
Next we focus on the computation of the boundaries of the 3-dimensional cells, that is, the
image of ∂ : C3 −→ C2. Such cells are of the form Srαrβrγ with α 6= β 6= γ (α = γ is allowed)
whose boundaries are given by Proposition 2.3:
∂Srαrβrγ = c(rαrβrγ , rαrγ)Srαrγ + c(rαrβrγ , rβrγ)Srβrγ
with c(rαrβrγ , rαrγ) = 0 in the case γ = α. The coefficients have the form ±(1−(−1)σ), where σ
is as in (2.3). Namely,
• σ(rαrβrγ , rαrγ) = 〈β∨, γ〉, for γ 6= α, since the only positive root taken to negative by rγ
is γ itself. Thus
c(rαrβrγ , rαrγ) = ±
(
1− (−1)〈β
∨,γ〉). (3.2)
• σ(rαrβrγ , rβrγ) = 〈α∨, β〉+ 〈α∨, rβγ〉 since the positive roots taken to negative by rγrβ =
(rβrγ)−1 are β and rβγ. Therefore,
c(rαrβrγ , rβrγ) = ±
(
1− (−1)〈α
∨,β〉+〈α∨,rβγ〉). (3.3)
Proposition 3.6. Let α, β, γ ∈ Σ.
1. If α and β are linked by a double line and α is a short root then
∂Srβrαrβ = ±2Srαrβ .
2. If 〈α, β〉 = 0, β is long and γ is such that 〈γ∨, α〉 = −1 then
∂Srγrαrβ = ±2Srαrβ .
3. If 〈α, β〉 = 0 and 〈γ∨, β〉 = −1 then
∂Srαrγrβ = ±2Srαrβ .
De Rham 2-Cohomology of Real Flag Manifolds 9
Proof. If α and β are as in (1) then 〈β∨, α〉 = −1 and 〈α∨, β〉 = −2. Hence
σ(rβrαrβ, rαrβ) = 〈β∨, α〉+ 〈β∨, rαβ〉 = 〈β∨, α〉+ 〈β∨, β + 2α〉
= 3〈β∨, α〉+ 〈β∨, β〉 = −1,
which is odd. Therefore c(rβrαrβ, rαrβ) = ±2 showing the first assertion.
For the second item, 〈α∨, β〉 = 0 implies ∂Srγrαrβ = c(rγrαrβ, rαrβ)Srαrβ which is non-zero
since
σ(rγrαrβ, rαrβ) = 〈γ∨, α〉+ 〈γ∨, β〉 = −1 + 〈γ∨, β〉
is odd.
For the last assertion, the assumption 〈γ∨, β〉 = −1 implies that removing from rαrγrβ the
reflection rγ one has
c(rαrγrβ, rαrβ) = ±2
by (2.3). On the other hand, the exponent reached when removing the first reflection rα is
σ(rαrγrβ, rγrβ) = 〈α∨, γ + rγβ〉 = 〈α, 2γ + β〉 = 2〈α, γ〉,
which is even. Hence c(rαrγrβ, rγrβ) = 0 and ∂Srαrγrβ = ±2Srαrβ . �
Proposition 3.7. Let α, β, γ ∈ Σ in an A3 or a C3 configuration. Then
∂Srαrγrβ = ±2
(
Srαrβ + ηα,β,γSrγrβ
)
,
where ηα,β,γ is the sign such that ∂(Srαrβ + ηα,β,γSrγrβ ) = 0.
Proof. By hypothesis, the roots verify 〈γ∨, β〉 = −1 = 〈α∨, β〉 and 〈α, γ〉 = 0. These conditions
give c(rαrγrβ, rαrβ) = ±2. In addition, σ(rαrγrβ, rγrβ) = 〈α∨, γ + rγβ〉 = 〈α∨, β〉 = −1 so
c(rαrγrβ, rγrβ) = ±2. This proves ∂Srαrγrβ = ±2(Srαrβ + νSrγrβ ), for some ν = ±1. The fact
that ∂2 = 0 implies ∂(Srαrβ + νSrγrβ ) = 0 which forces ν = ηα,β,γ , and the choice of the sign is
the right one. �
The results in this section yield the following expressions for the 2-homology of the maximal
flag manifolds.
Theorem 3.8. Let F be the maximal flag manifold of a split real form g. Then
H2(F, R) = 0 if g is of type G2 and
H2(F, R) = R/2R⊕ · · · ⊕R/2R otherwise.
The number of summands equals the number of generators of ker(∂ : C2 −→ C1) given in Propo-
sitions 3.3 and 3.4.
Proof. If g is of type G2 the kernel of ∂ is trivial as shown in Proposition 3.5. If g is not of
type G2, the generators of the kernel of ∂ : C2 −→ C1 were given in Propositions 3.3 and 3.4.
Each such generator has a ±2 multiple which is indeed a boundary. In fact, let Srαrβ be a 2-cell
such that ∂Srαrβ = 0. Then either 〈α, β〉 = 0 or α and β are double linked with α short.
In the double linked case with α the short root we have ∂Srβrαrβ = ±2Srαrβ because of (1)
in Proposition 3.6.
On the other hand for 〈α, β〉 = 0 suppose first that g is of type Cl and β = αl. Then there
exists γ ∈ Σ such that 〈γ∨, α〉 = −1 so that (2) of Proposition 3.6 gives ∂Srγrαrβ = ±2Srαrβ .
If g is not of type Cl or β 6= αl in the Cl case there always exists γ ∈ Σ with 〈γ∨, β〉 = −1.
Item (3) in the same proposition gives ∂Srαrγrβ = ±2Srαrβ .
Finally, each generator of the form Srαrγrβ with α, β, γ ∈ Σ in an A3 or C3 diagram has a ±2
multiple in the image of ∂ : C3 −→ C2 as shown in Proposition 3.7. �
10 V. del Barco and L.A.B. San Martin
3.2 Partial flag manifolds
At this point, all the relevant computations of kernel and images of the boundary map is done.
To deal with the homology of the partial case, Θ 6= ∅, we need to determine when does a cell Sw
in the kernel corresponds to a minimal element and when does a boundary in R ·Sw is the image
of a minimal cell.
Proposition 3.9. If a diagram has only single and double lines, the kernel of ∂min
Θ : CΘ
2 −→ CΘ
1
is spanned by the following elements:
• Srαrβ with α, β /∈ Θ and 〈α, β〉 = 0,
• Srαrβ with β /∈ Θ, α and β double linked and α short,
• Srαrβ + η∗α,β,γSrγrβ whenever α, β, γ form a ∗3 diagram, in this order, with β /∈ Θ.
Proof. By Lemma 2.2, any w = rαrβ is a minimal element under the first two conditions, and
also rαrβ and rγrβ are minimal when they fit into an A3 or C3 diagram with β /∈ Θ. Hence any
element in the list above is indeed in CΘ
2 . Also, their image under ∂ in the maximal flag is zero,
so ∂min
Θ is zero. �
In the case of a G2 diagram, ker(∂ : C2 −→ C1) = {0} and thus ker(∂min
Θ : C2 −→ C1) is zero
independently of Θ.
By Lemma 2.2, w = rαrβrγ is a minimal element in W if and only if γ, rγβ, rγrβα are all
outside 〈Θ〉.
Proposition 3.10. For any β /∈ Θ, the following cells are in CΘ
3 :
• Srβrαrβ with α, β double linked and α short;
• Srαrγrβ with α, β, γ in an A3 or C3 diagram (in this order);
• Srαrγrβ with α /∈ Θ, β 6= α and 〈γ, β〉 6= 0;
• Srγrαrβ with α /∈ Θ and 〈α, γ〉 6= 0.
Proof. Let w = rβrαrβ with α and β double linked, α short and β /∈ Θ. Then β, rβα = α+ β
and rβrαβ = β + 2α are all outside 〈Θ〉. Hence Srβrαrβ ∈ CΘ
3 .
Let α, β, γ be in ∗3 configuration (in this order) and let w = rαrγrβ. Then rβγ = γ+〈β∨, γ〉β
and rβrγα = rβα = β + α have non-zero component in β, so w is minimal.
If β 6= α, α, β /∈ Θ and 〈γ, β〉 6= 0 then rβγ has non-zero component in β and rβrγα =
α−〈γ∨, α〉− 〈β∨, rγα〉β which has non-zero component in α. Therefore w = rαrγrβ is minimal.
The proof of the last item follows in a similar way. �
The propositions above together with the proof of Theorem 3.8 show that if Sw is a 2-di-
mensional cell corresponding w ∈ Wmin
Θ then ±2Sw is a border. This leads to the following
conclusion.
Theorem 3.11. Let FΘ be a partial flag manifold of a split real form g. Then for any Θ
H2(FΘ, R) = 0 if g is of type G2 and
H2(FΘ, R) = R/2R⊕ · · · ⊕R/2R otherwise.
The number of summands equals dim ker
(
∂min
Θ : CΘ
2 −→ CΘ
1
)
.
De Rham 2-Cohomology of Real Flag Manifolds 11
4 Classifications
Let FΘ be a flag manifold associated to a non-compact simple real Lie algebra g. Theo-
rems 2.4, 3.8 and 3.11 imply that the second homology group of FΘ is determined by the simple
roots in Σsplit and Σ2.
The multiplicities (and hence the ranks) of the restricted roots of simple real Lie algebras
can be read from the classification table of the real forms (see, e.g., [11, pp. 30–32]). If for a Lie
algebra g the sets Σsplit and Σ2 are empty (that is, if rankα ≥ 3 for every α ∈ Σ) then the
second homology of any flag manifold associated to g is zero. The Lie algebras having roots
with rankα ≤ 2 are listed in Table 1 below. The homology of the flag manifolds associated to
the Lie algebras in the list is given in the following theorem that summarizes Theorems 2.4, 3.8
and 3.11.
Theorem 4.1. Let FΘ be the flag manifold associated to a non-compact simple real Lie algebra g
and to the subset Θ ⊂ Σ of restricted simple roots. If charR 6= 2 then
• H2(FΘ, R) = 0 if and only if g is of type G2 for any Θ or if g is not of type G2 and both
Σsplit and Σ2 are contained in Θ. This is the case if g does not appear in Table 1.
• H2(FΘ, R) is non-zero and has only torsion components R/2R if and only if g is not of
type G2, Σ2 ⊂ Θ and Σsplit is not contained in Θ. In particular, if g is a split real form,
not of type G2.
• H2(FΘ, R) contains a free R module if and only if Σ2∩(Σ\Θ) 6= ∅. The rank of the module
equals the cardinality of this intersection.
Notation. In Table 1, the simple roots of Bl, Cl and F4 are labelled according to the following
diagrams
De Rham 2-Cohomology of Real Flag Manifolds 11
4 Classifications
Let FΘ be a flag manifold associated to a non-compact simple real Lie algebra g. Theo-
rems 2.4, 3.8 and 3.11 imply that the second homology group of FΘ is determined by the simple
roots in Σsplit and Σ2.
The multiplicities (and hence the ranks) of the restricted roots for the simple real Lie algebras
can be read from the classification table of the real forms (see, e.g., [13, pp. 30–32]). If for a Lie
algebra g the sets Σsplit and Σ2 are empty (that is, if rankα ≥ 3 for every α ∈ Σ) then the
second homology of any flag manifold associated to g is zero. The Lie algebras having roots
with rankα ≤ 2 are listed in Table 1 below. The homology of the flag manifolds associated to
the Lie algebras in the list is given in the following theorem that summarizes Theorems 2.4, 3.8
and 3.11.
Theorem 4.1. Let FΘ be the flag manifold associated to a non-compact simple real Lie algebra g
and to the subset Θ ⊂ Σ of restricted simple roots. If charR 6= 2 then
• H2(FΘ, R) = 0 if and only if g is of type G2 for any Θ or if g is not of type G2 and both
Σsplit and Σ2 are contained in Θ. This is the case if g does not appear in Table 1.
• H2(FΘ, R) is non-zero and has only torsion components R/2R if and only if g is not of
type G2, Σ2 ⊂ Θ and Σsplit is not contained in Θ. In particular, if g is a split real form,
not of type G2.
• H2(FΘ, R) contains a free R module if and only if Σ2∩(Σ\Θ) 6= ∅. The rank of the module
equals the cardinality of this intersection.
Notation. In Table 1, the simple roots of Bl, Cl and F4 are labelled according to the following
diagrams
Bl, l ≥ 2 e e . . . e eA
�α1 α2 αl−1 αl
Cl, l ≥ 3 e e . . . e�
A
e
α1 α2 αl−1 αl
F4
e
α1
e
α2
e
α3
A
�
e
α4
5 De Rham cohomology
The computation of the second de Rham cohomology group of real flag manifolds was the first
motivation of this work. This cohomology group can be obtained from the homology groups
studied along this paper. In fact, by the Universal Coefficient Theorem (see for instance [3]) we
have, for a real flag manifold FΘ, H2
dR(FΘ,R) ' Rs where s is the rank of the free Z-module
in H2(FΘ,Z). Therefore the description of the homologies of the flag manifolds of Theorem 4.1
combined with Table 1 yield the following.
Theorem 5.1. Let FΘ be the flag manifold associated to a non-compact simple real Lie algebra g
and to the subset Θ ⊂ Σ of restricted simple roots. Then H2
dR(FΘ,R) = 0 unless g is of types
AIII1, AIII2, DI2, EII and Σ2∩(Σ\Θ) 6= ∅. In that case H2
dR(FΘ,R) ' Rs with s = |Σ2∩(Σ\Θ)|.
As particular cases, one obtains that H2
dR(FΘ,R) = 0 for any flag manifold FΘ of a split real
form.
The objective now is to get invariant differential forms that represent the 2-cohomologies
of the flag manifolds of the Lie algebras AIII1, AIII2, DI2 and EII. These 2-forms will be
obtained by the Weil construction that provides closed differential forms as characteristic forms
of principal bundles. In the next theorem we recall this construction for differential 2-forms
5 De Rham cohomology
The computation of the second de Rham cohomology group of real flag manifolds was the first
motivation of this work. This cohomology group can be obtained from the homology groups
studied along this paper. In fact, by the universal coefficient theorem (see for instance [2]) we
have, for a real flag manifold FΘ, H2
dR(FΘ,R) ' Rs where s is the rank of the free Z-module
in H2(FΘ,Z). Therefore the description of the homologies of the flag manifolds of Theorem 4.1
combined with Table 1 yield the following.
Theorem 5.1. Let FΘ be the flag manifold associated to a non-compact simple real Lie algebra g
and to the subset Θ ⊂ Σ of restricted simple roots. Then H2
dR(FΘ,R) = 0 unless g is of types
AIII1, AIII2, DI2, EII and Σ2∩(Σ\Θ) 6= ∅. In that case H2
dR(FΘ,R) ' Rs with s = |Σ2∩(Σ\Θ)|.
As particular cases, one obtains that H2
dR(FΘ,R) = 0 for any flag manifold FΘ of a split real
form.
The objective now is to get invariant differential forms that represent the 2-cohomologies
of the flag manifolds of the Lie algebras AIII1, AIII2, DI2 and EII. These 2-forms will be
obtained by the Weil construction that provides closed differential forms as characteristic forms
of principal bundles. In the next theorem we recall this construction for differential 2-forms
12 V. del Barco and L.A.B. San Martin
Lie algebra Dynkin diagram Σsplit Σ2
AI Al Σ —
AIII1 Bl — {α1, . . . , αl−1}
AIII2 Cl {αl} {α1, . . . , αl−1}
BI1 Bl Σ —
BI2 Bl {α1, . . . , αl−1} —
CI Cl Σ —
DI1 Bl {α1, . . . , αl−1} —
DI2 Bl {α1, . . . , αl−1} {αl}
DI3 Dl Σ —
DIII1 Cl {αl} —
EI E6 Σ —
EII F4 {α1, α2} {α3, α4}
EV E7 Σ —
EVI F4 {α1, α2} —
EVII C3 {α1}
EVIII E8 Σ —
EIX F4 {α1, α2} —
FI F4 Σ —
G G2 Σ —
Table 1. Lie algebras having roots with rank ≤ 2.
Theorem 5.2 (see Kobayashi–Nomizu [5, Chapter XII]). Let π : Q→M be a principal bundle
with structural group L having Lie algebra l. Endow Q with a connection form ω whose curvature
2-form (with values in l) is Ω. Take f ∈ l∗ which is L-invariant (that is, f ◦ Ad(g) = f for all
g ∈ L). Then the 2-form f ◦Ω is such that there exists a closed 2-form f̃ on M with f ◦Ω = π∗f̃ .
The de Rham cohomology class of f̃ remains the same if the connection is changed.
As a complement to this theorem we note that if L is compact then its Lie algebra l is
reductive and a necessary condition for the existence of an invariant f ∈ l∗, f 6= 0, is that l has
non trivial center z(l), that is, l is not semi-simple and is 6= {0}. If furthermore L is connected
then this condition is also sufficient and an invariant f is given by f(·) = 〈X, ·〉 with X ∈ z(l)
where 〈·, ·〉 is an invariant inner product in l. If L is not connected the invariant f ∈ l∗ are
given by f(·) = 〈X, ·〉 as well with X ∈ z(l) fixed by the non-identity components of L. This
construction yields a map z(l)→ H2(M,R) that depends in an isomorphic way on the invariant
inner product 〈·, ·〉 in l. (This map can be defined also via a negative definite form like −〈·, ·〉.)
Given a flag manifold FΘ = K/KΘ we will apply the Weil construction to the principal
bundle K → K/KΘ. For this bundle in which the total space is the Lie group K we can take
left invariant connections yielding invariant differential forms in the base space K/KΘ.
For a flag manifold FΘ = K/KΘ associated to one of the Lie algebras AIII1, AIII2, DI2 or EII
we intend to prove that there are enough f ∈ k∗Θ so that the 2-forms f̃ exhaust the 2-cohomology.
To this purpose it is required to describe the center of kΘ. When FΘ = F is a maximal flag
manifold then F = K/M where M is the centralizer of a in K. In the next section we discuss
the center z(m) of the Lie algebra m of M .
6 M -group and Satake diagrams
In this section we obtain preparatory results that will allow us, in the next section, to get the
characteristic forms in the flag manifolds. Its purpose is to see how the Lie algebra m of M and
De Rham 2-Cohomology of Real Flag Manifolds 13
its center z(m) can be read off from the Satake diagram of the real form g. Before starting let us
write some notation and facts related to the Satake diagrams. Denote by gC the complexification
of g and let u ⊂ gC be a compact real form of gC adapted to g, that is, g = k ⊕ s is a Cartan
decomposition where k = g ∩ u and s = g ∩ iu. Denote by σ and τ the conjugations in gC with
respect to u and g respectively. To say that u is adapted to g is the same as saying that these
conjugations commute and the Cartan involution θ = στ = τσ is an automorphism leaving
invariant both g and u. The subalgebra k is the fixed point set of θ implying that τ = −1 on ik.
Starting with the maximal abelian subspace a ⊂ s let h ⊃ a be a Cartan subalgebra of g
and complexify it to the Cartan subalgebra hC ⊂ gC. The Cartan subalgebra h decomposes as
h = hk ⊕ a with hk ⊂ k. The Cartan subalgebras h and hC are invariant by σ, τ and θ = στ .
Denote by ΠC the set of roots of (gC, hC) and by Π the set of restricted roots of (g, a). Each
α ∈ Π is the restriction to a of a root in ΠC. For α ∈ h∗C define Hα ∈ hC by α(·) = 〈Hα, ·〉 and
denote by hR the real subspace spanned by Hα, α ∈ ΠC. The roots are real on a and purely
imaginary in hk so that a = hR ∩ h, hk = ihR ∩ h and hR = ihk ⊕ a. The last decomposition
is orthogonal with respect to the Cartan–Killing inner product in hR because τ is an involutive
isometry satisfying τ = −1 in ihk and τ = 1 in a.
A root α ∈ ΠC is said to be imaginary if α ◦ τ = −α. Denote by ΠIm the set of imaginary
roots. There are the following equivalent ways to define ΠIm:
1. A root α ∈ ΠC is imaginary if and only if it annihilates on a. In fact, if H ∈ a then
τ(H) = H so that α(H) = α(τ(H)) = −α(H). Conversely if α is zero on a then Hα is
orthogonal to a in hR so that Hα ∈ ihk therefore
α ◦ τ(H) = 〈Hα, τH〉 = 〈τHα, H〉 = −α(H)
showing that α ∈ ΠIm.
2. α ∈ ΠC is imaginary if and only if Hα ∈ ihk because the decomposition hR = ihk ⊕ a is
orthogonal.
3. Let H ∈ a be regular real, that is, β(H) 6= 0 for every β ∈ Π. Then α ∈ ΠC is imaginary
if and only if α(H) = 0. In fact, the roots in Π are the restrictions to a of the roots in ΠC
and α is imaginary if and only if it annihilates on a. For this characterization the choice
of the regular element H is immaterial.
To get the Satake diagram take a regular real H ∈ a and let ΣC ⊂ ΠC be a simple system
of roots such that α(H) ≥ 0 for every α ∈ ΣC. Equivalently ΣC is the simple system of roots
associated to a Weyl chamber h+
R containing H in its closure. The Satake diagram is obtained
from the Dynkin diagram of ΣC by painting black the imaginary roots in ΣC and by joining
with a double arrow two roots α, β ∈ ΣC whose restrictions to a are equal.
The set ΣIm of imaginary roots in ΣC is given by
ΣIm = {α ∈ ΣC : α(H) = 0}.
If β ∈ ΣC \ ΣIm then β(H) > 0. Hence a positive root γ is a linear combination of ΣIm if and
only if γ(H) = 0 so that ΠIm is the set of roots 〈ΣIm〉 spanned by the simple imaginary roots.
The next proposition allows to reconstruct hk from the Satake diagram. For its statement
we use the following notation
• hIm is the subspace spanned by iHα with α ∈ ΣIm (or what is the same α ∈ ΠIm).
• ΣC,arr is the union of pairs of simple roots in a Satake diagram that are linked by a double
arrow. Σ⊥C,arr is the subset of ΣC,arr of pairs of simple roots not linked to imaginary roots.
Σarr and Σ⊥arr are the restrictions to a of the roots in ΣC,arr and Σ⊥C,arr respectively.
14 V. del Barco and L.A.B. San Martin
• harr is the subspace spanned by iHγ with γ running through the set of differences γ = α−β
with {α, β} ∈ ΣC,arr.
Proposition 6.1. hk = hIm ⊕ harr.
Proof. As mentioned above there is the orthogonal direct sum hR = ihk ⊕ a which implies that
hIm ⊂ hk. Also, if α and β are simple roots linked by a double arrow then γ = α − β is zero
on a which means that Hγ is orthogonal to a so that iHγ ∈ hk. Hence harr ⊂ hk. We have
hIm ∩ harr = {0} because ΣC is a basis. A dimension check shows that the sum is the whole
space. In fact, dim h is the number of roots in ΣC (rank of g) while dim a (real rank of g) is the
number of white roots not linked plus dim harr which is half the number of white linked roots.
Finally dim hIm is the number of black roots. Hence dim hk = dim hIm + dim harr concluding the
proof. �
The abelian subalgebra is one of the pieces of m. The other piece is the subalgebra generated
by the imaginary roots which are described next.
Proposition 6.2. Let gIm be the subalgebra of gC generated by the root spaces (gC)α with
α ∈ ΠIm. Then gIm is a complex semi-simple Lie algebra whose Dynkin diagram corresponds
to the simple roots ΣIm.
Put kIm = gIm ∩ u. Then kIm is a compact real form of gIm and therefore it is semi-simple.
Moreover, hk is a Cartan subalgebra of kIm.
Proof. The first statement holds because ΠIm is a root system generated by ΣIm which is
a simple system of roots. Regarding to the subalgebra kIm it can be proved that (gC)α is
contained in k + ik if α is imaginary (see [9, Lemma 14.6]). Hence gIm ⊂ k + ik implying that
kIm ⊂ k. By the Weyl construction of the compact real form applied simultaneously to gC
and gIm it follows that the intersection kIm = gIm ∩ u is a compact real form of gIm. Finally,
ΣIm is a simple system of roots of gIm so that hIm, which is spanned by iHα with α ∈ ΣIm, is
a Cartan subalgebra of kIm. �
Now we combine the above pieces to write down the Lie algebra m from the Satake diagram
of g.
Proposition 6.3. m = kIm ⊕ z(m) with kIm semi-simple and z(m) the orthogonal complement
of hIm in hk.
Proof. The centralizer of a in gC is z = hC ⊕
∑
α∈ΠIm
(gC)α. Thus m = z ∩ k = hk ⊕ kIm. Since
hIm ⊂ kIm we have m = h⊥Im ⊕ kIm where h⊥Im is the orthogonal complement of hIm in hk. Any
imaginary root is zero on h⊥Im so that this subspace commutes with kIm. This implies that
z(m) = h⊥Im because kIm is semi-simple. �
By Proposition 6.1 we have hk = hIm ⊕ harr so that h⊥Im 6= {0} if and only if harr 6= {0} which
means that the Satake diagram of g has double arrows. Diagrams with arrows are called outer
diagrams (because θ is an outer automorphism of gC). By the above proposition z(m) = h⊥Im so
we get the following case where z(m) is not trivial.
Proposition 6.4. m has non-trivial center if and only if the Satake diagram of g is outer and
z(m) = h⊥Im.
Remark 6.5. For α, β ∈ Σ⊥C,arr and γ = α− β, iHγ ∈ h⊥Im = z(m). In fact, if δ is an imaginary
root, then 〈iHγ , iHδ〉 = 0 since α and β are orthogonal (non-adjacent) to any imaginary root.
De Rham 2-Cohomology of Real Flag Manifolds 15
The diagrams AIII1, AIII2, DI2 and EII appearing in Theorem 5.1 whose Lie algebras have
flag manifolds with non-trivial 2-cohomology are outer diagrams. We reproduce them below.
De Rham 2-Cohomology of Real Flag Manifolds 15
The diagrams AIII1, AIII2, DI2 and EII appearing in Theorem 5.1 whose Lie algebras have
flag manifolds with non-trivial 2-cohomology are outer diagrams. We reproduce them below.
AIII1
e e e u
u
ue e e
?
6
?
6
?
6
AIII2 e e e e
e e e e e
�
�
Z
Z
?
6
?
6
?
6
?
6
DI2
e e e e e,,
l
l
e
e?
6
EII e e,,
l
l
e,, e
e
l
l e
6
?
6
?
In these diagrams a simple restricted root α ∈ Σ has rankα = 2 if and only if it is the
restriction of γ, δ ∈ Σ⊥C,arr, that is, we have Σ2 = Σ⊥arr.
The diagrams AIII2, DI2 and EII have no imaginary roots. Hence in these Lie algebras
m = hk is abelian. For AIII1 the imaginary simple roots form an Al Dynkin diagram. So that
in AIII1 we have kIm ≈ su(k) some k, that is, m = su(k) ⊕ z(m) where dim z(m) is the number
of pairs of roots linked double arrows and equals the real rank of the Lie algebra.
Now we use the well known fact that every connected component of M contains an element
that commutes with z(m) to conclude that z(m) is fixed by the whole group M .
Proposition 6.6. Let F = G/P = K/M be the maximal flag manifold associated to the real
form g of the complex simple Lie algebra gC. Then the center z(m) of the Lie algebra m of M is
non-trivial if g is of the type AIII1, AIII2, DI2 or EII. In these cases M centralizes z(m).
Proof. It remains to check only the last statement. For the realization F = G/P we can
take any connected Lie group G with Lie algebra g. If G is complexifiable then any connected
component of M has an element of the form eiH with H ∈ a (see Knapp [5, Section VII.5] for
the precise statement of this result and the notion of complexifiable group). If X ∈ z(m) then
Ad(eiH)X = X because z(m) is contained hk that commutes with a. So that M fixes z(m).
In general, we can take the adjoint representation and find elements in the several connected
components of M that have the form eiHz with z in the center of G. Hence the result follows
as well. �
To conclude this section we describe subalgebras of the real forms that are associated to
rank 2 roots and are isomorphic to sl(2,C). The following lemma can be checked by looking at
the table of Satake diagrams.
Lemma 6.7. Let α and β be simple roots in a Satake diagram that are linked by a double arrow.
Then 〈α, β〉 = 0.
If α is a root of a complex simple Lie algebra gC then the subalgebra gC (α) generated by the
root spaces (gC)±α is isomorphic to sl (2,C). We prove next that the same happens to certain
rank 2 roots of a real form.
Proposition 6.8. Let γ and δ be simple roots in the Satake diagram of real form g. Suppose that
γ and δ are linked by a double arrow and not linked to imaginary roots, that is, γ, δ ∈ Σ⊥C,arr.
In these diagrams a simple restricted root α ∈ Σ has rankα = 2 if and only if it is the
restriction of γ, δ ∈ Σ⊥C,arr, that is, we have Σ2 = Σ⊥arr.
The diagrams AIII2, DI2 and EII have no imaginary roots. Hence in these Lie algebras
m = hk is abelian. For AIII1 the imaginary simple roots form an Al Dynkin diagram. So that
in AIII1 we have kIm ≈ su(k) some k, that is, m = su(k) ⊕ z(m) where dim z(m) is the number
of pairs of roots linked double arrows and equals the real rank of the Lie algebra.
Now we use the well known fact that every connected component of M contains an element
that commutes with z(m) to conclude that z(m) is fixed by the whole group M .
Proposition 6.6. Let F = G/P = K/M be the maximal flag manifold associated to the real
form g of the complex simple Lie algebra gC. Then the center z(m) of the Lie algebra m of M is
non-trivial if g is of the type AIII1, AIII2, DI2 or EII. In these cases M centralizes z(m).
Proof. It remains to check only the last statement. For the realization F = G/P we can
take any connected Lie group G with Lie algebra g. If G is complexifiable then any connected
component of M has an element of the form eiH with H ∈ a (see Knapp [4, Section VII.5] for
the precise statement of this result and the notion of complexifiable group). If X ∈ z(m) then
Ad
(
eiH
)
X = X because z(m) is contained hk that commutes with a. So that M fixes z(m).
In general, we can take the adjoint representation and find elements in the several connected
components of M that have the form eiHz with z in the center of G. Hence the result follows
as well. �
To conclude this section we describe subalgebras of the real forms that are associated to
rank 2 roots and are isomorphic to sl(2,C). The following lemma can be checked by looking at
the table of Satake diagrams.
Lemma 6.7. Let α and β be simple roots in a Satake diagram that are linked by a double arrow.
Then 〈α, β〉 = 0.
If α is a root of a complex simple Lie algebra gC then the subalgebra gC(α) generated by the
root spaces (gC)±α is isomorphic to sl(2,C). We prove next that the same happens to certain
rank 2 roots of a real form.
Proposition 6.8. Let γ and δ be simple roots in the Satake diagram of real form g. Suppose
that γ and δ are linked by a double arrow and not linked to imaginary roots, that is, γ, δ ∈ Σ⊥C,arr.
16 V. del Barco and L.A.B. San Martin
If α denotes their common restrictions to a then rankα = 2. Let g(α) be the subalgebra of g
generated by the root spaces g±α. Then g(α) is isomorphic to the realification sl(2,C)R of sl(2,C).
The subspace hα spanned over R by Hα ∈ a and iHγ−δ ∈ hk is a Cartan subalgebra of g(α).
Proof. We have gα = ((gC)γ + (gC)δ)∩ g and the same for −α. By assumption γ and δ are the
only roots restricting to α hence g(α) = gC(γ, δ)∩g where gC(γ, δ) is the Lie algebra generated by
(gC)±γ and (gC)±δ. By the previous lemma 〈γ, δ〉 = 0 which implies that gC(γ, δ) = gC(γ)⊕gC(δ)
with [gC(γ), gC(δ)] = 0 so that gC(γ, δ) is isomorphic to the direct sum sl(2,C)⊕ sl(2,C). Hence
g(α) = (gC(γ)⊕gC(δ))∩g and gC(γ)⊕gC(δ) is the complexification of g(α). It follows that g(α)
is isomorphic to sl(2,C)R because this is the only non-compact real form of sl(2,C)⊕ sl(2,C).
The subspace hγ,δ = spanC{Hγ , Hδ} is a Cartan subalgebra of gC(γ)⊕ gC(δ). Hence
hα = hγ,δ ∩ g = hγ,δ ∩ (hk ⊕ a)
is a Cartan subalgebra of g(α). �
7 Characteristic forms
In this section we apply the Weil homomorphism to get representatives of the de Rham cohomo-
logy of the flag manifolds of the Lie algebras AIII1, AIII2, DI2 and EII appearing in Theorem 5.1.
Connections in the principal bundles K → FΘ = K/KΘ will be obtained from the following
general standard construction.
Proposition 7.1. Let Q be a Lie group and L ⊂ Q a closed subgroup with Lie algebras q and l
respectively. Suppose there exists a subspace p ⊂ q with q = l⊕ p and Ad(g)p = p for all g ∈ L.
Then
1. The left translations Hor(q) = Lq∗(p), q ∈ Q, are horizontal spaces for a connection in the
principal bundle π : Q→ Q/L.
2. The connection form ω is given by ωq
(
X l(q)
)
= PX ∈ l where X l is the left invariant
vector field defined by X ∈ q and P : q → l is the projection against the decomposition
q = l⊕ p.
3. The curvature form Ω is completely determined by its restriction to p (at the origin) which
is given by Ω
(
X l, Y l
)
= P [X,Y ].
The horizontal distribution Hor as well as ω and Ω are invariant by left translations (Lg∗Hor
= Hor, L∗gω = ω and L∗gΩ = Ω).
Left invariant connections on the bundles K → FΘ = K/KΘ given by this proposition will
be used to get characteristic forms on the flag manifolds FΘ.
We work out first the maximal flag manifolds F = K/M . Take the root space decomposition
g = m⊕ a⊕
∑
α∈Π
gα
and for a positive restricted root α ∈ Π+ write kα = (Dα+D−α)∩ k where Dα = gα/2 +gα+g2α.
We have k = m⊕ ∑
α∈Π+
kα so that
p =
∑
α∈Π+
kα
complements m in k. For any m ∈M there is the invariance Ad(m)p = p because Ad(m)gα = gα
for every root α. Hence p defines a left invariant connection in the bundle K → K/M . Its
De Rham 2-Cohomology of Real Flag Manifolds 17
curvature form Ω is defined at the identity by the projection into m of the bracket. Since
[gα, gβ] ⊂ gα+β it follows that Ω(X,Y ) = 0 if X ∈ gα, Y ∈ gβ with α 6= β. Hence Ω (at the
origin) is the direct sum of 2-forms Ωα on the spaces kα:
Ω =
∑
α∈Π+
Ωα,
where Ωα is the projection into m of the bracket in kα.
If Z ∈ z(m) then by the Weil homomorphism theorem the left invariant 2-form 〈Z,Ω(·, ·)〉
in K is the pull back of a closed differential form in F = K/M that we denote by fZ . The
tangent space at the origin identifies with p where fZ is given by
fZ(X,Y ) =
∑
α∈Π+
〈Z,Ωα(X,Y )〉.
We write fZ(Srα) for the value of fZ in the Schubert cell Srα given by the duality between
H2(F,R) and H2(F,R).
The following example of CP 1 ≈ S2 will be used to compute the values of characteristic forms
in the Schubert 2-cells.
Example 7.2. Let us look at the characteristic forms at the complex projective line CP 1 ≈ S2
obtained by the action of SU(2) so that S2 = SU(2)/T where T is the group of diagonal matrices
diag
{
eit, e−it
}
, t ∈ R. The Lie algebra of T is t = {Ht = diag{it,−it} : t ∈ R} and the subspace
p =
{
Xz =
(
0 z
−z 0
)
: z ∈ C
}
complements t in su(2). The bracket in p is given by
[Xz, Xw] =
(
−zw + zw 0
0 zw − zw
)
= 2
(
−i Im zw 0
0 i Im zw
)
.
Hence by taking the appropriate inner product in t we have that fH1 is the only invariant 2-
form in S2 that in the origin satisfies fH1(X1, Xi) = 1. Thus fH1 is the unique (up to scale)
SU(2)-invariant volume form in S2. We can normalize the inner product in t so that the integral
of fH1 over S2 equals to 1.
Now we can compute the basis of H2(F,R) dual to the Schubert cells spanning H2(F,R)
when F is the maximal flag manifold of one of the Lie algebras AIII1, AIII2, DI2 and EII. Recall
that by Proposition 6.4 for these Lie algebras z(m) = h⊥Im 6= {0}. On the other hand, H2(F,R)
is spanned by the 2-cells Srα ≈ S2 with α running through Σ⊥arr as in Proposition 6.8.
To get a basis of H2(F,R) dual to the basis of homology given by the Schubert cells we
define the Weil map z(m) → H2(F,R) by means of a suitable normalization (·, ·)c = c〈·, ·〉 of
an invariant inner product in k. The normalizing constant c is chosen so that fiHγ−δ(Srα) = 1
for every simple root α ∈ Σ⊥arr where γ, δ ∈ Σ⊥C,arr are such that α = γ|a = δ|a. The choice of c
is possible because the Satake diagrams AIII1, AIII2, DI2 and EII have only simple links and
hence the roots are all of the same length.
To state the next theorem we take the basis B of hk given by
B =
{
iHN
γ1−δ1 , . . . , iH
N
γk−δk
}
∪
{
iHN
γ−δ
}
∪
{
iHN
µ1
, . . . , iHN
µs
}
,
where for Z ∈ hk, Z
N = Z/(Z,Z)c and we are using the following notation:
1. {µ1, . . . , µs} are the imaginary simple roots so that {iHµ1 , . . . , iHµs} is a basis of hIm.
18 V. del Barco and L.A.B. San Martin
2. γj , δj , j = 1, . . . , k are the pairs of roots in Σ⊥C,arr, that is, γj is linked to δj by a double
arrow and both are not linked to imaginary roots.
3. γ, δ is the only pair of roots in ΣC,arr that are linked to imaginary roots (which occurs
only in AIII1).
Now let B⊥ be the dual basis of B (with respect to the normalized form (·, ·)c) and denote
by {Z1, . . . , Zk} the first k elements of B⊥ that correspond to the roots in Σ⊥arr.
Theorem 7.3. Let {Z1, . . . , Zk} be as above. Then {fZ1 , . . . , fZk} is the basis of H2(F,R) dual
to the basis {Srα : α ∈ Σ⊥arr} of H2(F,R).
Proof. Take α ∈ Σ⊥arr and let γ, δ ∈ Σ⊥C,arr be such that α = γ|a = δ|a. By Proposition 6.8 we
have g(α) ≈ sl(2,C). Put G(α) = 〈exp g(α)〉, k(α) = g(α)∩k and K(α) = 〈exp k(α)〉 = G(α)∩K.
We have Srα = G(α)x0 = K(α)x0 where x0 is the origin of the flag manifold F (see [8, Proposition
1.3]). Let φα : su(2)→ k(α) be an isomorphism assured by Proposition 6.8 and put
Xα = φα
(
0 1
−1 0
)
, Yα = φα
(
0 i
i 0
)
.
We have
[Xα, Yα] = 2φα
(
i 0
0 −i
)
= 2iHγ−δ
so that Ω(Xα, Yα) = 2iHγ−δ. Now take iH ∈ spanR B. Then
fiH(Xα, Yα) = (iH,Ω(Xα, Yα))c = (iH, 2iHγ−δ)c.
In particular fiHγ−δ(Xα, Yα) = 2(iHγ−δ, iHγ−δ)c so that
fiH(Xα, Yα) =
(iH, iHγ−δ)c
(iHγ−δ, iHγ−δ)c
fiHγ−δ(Xα, Yα) =
(
iH, iHN
γ−δ
)
c
fiHγ−δ(Xα, Yα).
Hence the restriction to Srα yields
fiH(Srα) =
(
iH, iHN
γ−δ
)
c
fiHγ−δ(Srα) =
(
iH, iHN
γ−δ
)
c
.
Since {Z1, . . . , Zk} is the basis dual to the basis
{
iHN
γ1−δ1 , . . . , iH
N
γk−δk
}
with respect to (·, ·)c it
follows that fZj (Srαk ) = 1 if j = k and 0 otherwise so that {fZ1 , . . . , fZk} is the basis dual to
{Srα : α ∈ Σ⊥arr}, concluding the proof. �
This result holds also for a partial flag manifold FΘ = K/KΘ by taking roots in Σ2 = Σ⊥arr
that are outside Θ. By Theorem 4.1 the 2-homology H2(FΘ,R) is generated by the Schubert
cells SΘ
rα with α ∈ Σ⊥arr \ Θ. On the other hand, let {Z1, . . . , Zk} be the dual basis as in the
statement of the above theorem and take an index j such that both roots γj and δj restrict
to αj ∈ Σ⊥arr \ Θ. To prove that forms fZj corresponding to these indices form a dual basis
in H2(FΘ,R) it remains to check that Zj belong to the center z(kΘ) so that the forms fZj are
well defined in FΘ.
Lemma 7.4. Let Zj, j = 1, . . . , k be as in the above theorem the elements of the dual basis and
take an index j that corresponds to αj ∈ Σ⊥arr \Θ. Then Zj ∈ z(kΘ).
Proof. By Proposition 6.3 we have Zj ∈ z(m) ⊂ m ⊂ kΘ. Denote by ΘC ⊂ ΣC the set of roots of
the Satake diagram whose restrictions to a belong to Θ∪{0}. Take γ ∈ ΘC and let α ∈ Θ∪{0}
be its restriction. We claim that γ(Zj) = 0. There are the possibilities:
De Rham 2-Cohomology of Real Flag Manifolds 19
1. α = 0, that is, γ is imaginary. Then γ(Zj) = 0 because Zj is orthogonal to iHγ .
2. α has multiplicity 1 so that γ = α is not imaginary and not linked to another root of ΣC
by a double arrow. Then Hγ ∈ a and since iZj ∈ a⊥ we have γ(Zj) = 0.
3. γ is linked to δ by a double arrow. By definition of the dual basis B⊥ we have that Zj
is orthogonal to iHγ−δ. On the other hand, Hγ+δ ∈ a since α = (γ + δ)/2. Hence
〈Hγ+δ, Zj〉 = 0 so that γ(Zj) = 0 as claimed.
It follows that Zj centralizes the Lie algebra gC(ΘC) generated by (gC)γ , γ ∈ ΘC. Hence Zj
centralizes kΘ = gC(ΘC) ∩ k as well, concluding the proof. �
The 2-forms {fZ1 , . . . , fZk} are zero on any k(α) with α not of rank 2. Therefore, if there is
a closed form on H2(FΘ,R) which is non-degenerate on FΘ then Π\Θ ⊂ Σ⊥arr. This implies that
Π\Θ consists of roots with double arrows in the Satake diagram. In this case, FΘ is a product
of complex flag manifolds of the form SU(n)/T .
8 su(p, q)
In this section we present concrete realizations of the flag manifolds of the Lie algebras of types
AIII1 and AIII2. These correspond to su(p, q) with p ≤ q which are non-compact real forms of
sl(p+ q,C).
The Lie algebras su(p, q), p ≤ q are constituted by zero trace matrices which are skew-
hermitian with respect to the hermitian form in Cp+q with matrix
Jp,q =
0 1p×p 0
1p×p 0 0
0 0 1q−p × 1q−p
.
That is, su(p, q) = {Z ∈Mp+q(C) : ZJp,q+Jp,qZ
∗ = 0, trZ = 0} where Z∗ denotes the transpose
conjugate. Therefore, su(p, q) is the Lie algebra of (p + q) × (p + q) matrices of the following
form
A B −Y ∗
C −A∗ −X∗
X Y Z
,
A ∈ gl(p,C), B, C ∈ u(p),
Z ∈ u(q − p), tr(2 ImA+ Z) = 0.
(8.1)
The Lie algebra su(p, q) can also be realized as the set of skew-hermitian matrices with respect
to the bilinear form with matrix
Ip,q =
(
1p×p 0
0 −1q×q
)
.
By using either realization one can see that the complexified Lie algebra su(p, q)C is sl(p+ q,C).
From the second realization it is clear that in the Cartan decomposition g = k⊕s, k is isomorphic
to (u(p)⊕u(q))/ tr, that is, zero trace matrices given by diagonal block matrices with two diagonal
elements in u(p) e u(q), respectively. In the first realization (8.1), k is described as follows
k =
A B −X∗
B A −Y ∗
X Y Z
: A,B ∈ u(p), Z ∈ u(q − p), tr(2 ImA+ Z) = 0
.
The matrices in s are hermitian, that is,
s =
A −B 0
B −A 0
0 0 0
: A = A∗, B ∈ u(p)
.
20 V. del Barco and L.A.B. San Martin
Under these choices for the Cartan decomposition, the maximal abelian subalgebra a of s is the
subspace of diagonal matrices in s, so one has
a =
Λ 0 0
0 −Λ 0
0 0 0
: Λ = diag{a1, . . . , ap}, aj ∈ R
.
Here one sees that the real rank of su(p, q) is p if p ≤ q. A Cartan subalgebra h containing a is
given by diagonal matrices of the form
D 0 0
0 −D 0
0 0 iT
,
where D has complex entries while T has real entries so that iT has pure imaginary entries. The
zero trace condition reads as tr(2 ImD+ T ) = 0, so the real part of D is arbitrary. The Cartan
subalgebra h decomposes as h = hk ⊕ a where hk is the set of diagonal matrices in k, that is,
iΛ 0 0
0 iΛ 0
0 0 iT
with Λ and T are diagonal matrices such that tr(2Λ+T ) = 0. Therefore, dim hk = p+(q−p)−1 =
q− 1 is the rank of su(p, q) and dim a+ dim hk = p+ q− 1, which is the rank of sl(p+ q,C). The
Cartan subalgebra hC is the Lie algebra of complex diagonal matrices in sl(n,C), n = p+ q.
In order to give a basis {Z1, . . . , Zp−1} as in Theorem 7.3, we need first a basis B of hk
determined by the roots in Σ⊥C,arr, ΣC,arr and ΣIm. So we proceed with the description of the
root system.
The roots of hC are the linear functionals given by the differences of the diagonal coordinate
functionals in hC. To simplify notations, for H = diag{a1, . . . , an} ∈ hC write µj(H) = aj if
1 ≤ j ≤ 2p and θj(H) = a2p+j if 1 ≤ j ≤ q − p.
As usual in sl(n,C), hR is the subspace of real diagonal matrices. The imaginary roots
(annihilating on a) are θj − θk. In particular, there are no imaginary roots if p = q.
The Satake diagram is given by a simple system ΣC of hC such that the imaginary roots
in ΣC span the set of all imaginary roots. Simple systems ΣC are written differently in the cases
p < q and p = q. If p < q then
ΣC = {µ1 − µ2, . . . , µp−1 − µp} ∪ {µp − θ1}
∪ {θ1 − θ2, . . . , θq−p−1 − θq−p} ∪ {θq−p − µ2p} ∪ {µ2p − µ2p−1, . . . , µp+2 − µp+1},
while for p = q we have
ΣC = {µ1 − µ2, . . . , µp−1 − µp} ∪ {µp − µ2p} ∪ {µ2p − µ2p−1, . . . , µp+2 − µp+1}.
It can be checked that these sets are in fact simple systems of roots with Dynkin diagram Al.
To write the restricted system determined by a we consider the parametrization by real
matrices Λ = diag{a1, . . . , ap} in a way that a is constituted by the following matrices
Λ 0 0
0 −Λ 0
0 0 0
.
Denote λj(Λ) = aj .
De Rham 2-Cohomology of Real Flag Manifolds 21
1. The simple roots θj − θj+1 are imaginary.
2. For j = 1, . . . , p−1, the restrictions to a given by the simple roots µj−µj+1 and µp+j+1−
µp+j are equal to λj − λj+1.
3. If p < q the roots µp − θ1 are θq−p − µ2p restrict to λp.
4. If p = q the root µp − µ2p restricts to 2λp.
The simple system Σ obtained by restriction of ΣC is
Σ = {λ1 − λ2, . . . , λp−1 − λp, λp} for p < q, and
Σ = {λ1 − λ2, . . . , λp−1 − λp, 2λp} for p = q.
The Satake diagram of ΣC and the Dynkin diagram of Σ in the case p < q are
De Rham 2-Cohomology of Real Flag Manifolds 21
1. The simple roots θj − θj+1 are imaginary.
2. For j = 1, . . . , p−1, the restrictions to a given by the simple roots µj−µj+1 and µp+j+1−
µp+j are equal to λj − λj+1.
3. If p < q the roots µp − θ1 are θq−p − µ2p restrict to λp.
4. If p = q the root µp − µ2p restricts to 2λp.
The simple system Σ obtained by restriction of ΣC is
Σ = {λ1 − λ2, . . . , λp−1 − λp, λp} for p < q, and
Σ = {λ1 − λ2, . . . , λp−1 − λp, 2λp} for p = q.
The Satake diagram of ΣC and the Dynkin diagram of Σ in the case p < q are
AIII1
e e e u
u
ue e e
?
6
?
6
?
6
Bp e e . . . e eA
�
The corresponding diagrams of ΣC and Σ in the case p = q are
AIII2 e e e e
e e e e e
�
�
Z
Z
?
6
?
6
?
6
?
6
Cp e e e�
A
e
The first elements in the basis B correspond to a basis of harr, and more precisely of h⊥arr if
p < q. We shall describe ΣC,arr and Σ⊥C,arr.
The roots in the Satake diagram of ΣC linked by double arrows are given by the following
pairs: if p < q
ΣC,arr = {µ1 − µ2, µp+2 − µp+1} ∪ {µ2 − µ3, µp+3 − µp+2} ∪ · · ·
· · · ∪ {µp−1 − µp, µ2p − µ2p−1} ∪ {µp − θ1, θq−p − µ2p},
while for p = q,
ΣC,arr = {µ1 − µ2, µp+2 − µp+1} ∪ {µ2 − µ3, µp+3 − µp+2} ∪ · · ·
· · · ∪ {µp−1 − µp, µ2p − µ2p−1}.
When p < q, the set of imaginary roots is ΣIm = {θ1 − θ2, . . . , θq−p−1 − θq−p}, so the subset of
pairs of roots which are orthogonal to ΣIm is Σ⊥C,arr = ΣC,arr \ {µp − θ1, θq−p − µ2p}. If p = q
then ΣC,arr = Σ⊥C,arr since there are no imaginary roots. We give a basis of h⊥arr to fill B which
is valid in both cases.
Up to normalization, Hµj−µj+1 is given by the diagonal matrix Λ with non-zero entries 1 and
−1 in positions j and j+1 respectively, j = 1, . . . , 2p−1. Given the pair of roots γj = µj−µj+1,
δj = µp+j+1−µp+j in Σ⊥C,arr, the elements spanning h⊥arr are iHδ−γ = iHµj−µj+1−(µp+j+1−µp+j) =
iHµj−µj+1 + iHµp+j−µp+j+1 . That is, for p ≤ q,
h⊥arr =
iD 0 0
0 iD 0
0 0 0
: trD = 0
. (8.2)
The corresponding diagrams of ΣC and Σ in the case p = q are
De Rham 2-Cohomology of Real Flag Manifolds 21
1. The simple roots θj − θj+1 are imaginary.
2. For j = 1, . . . , p−1, the restrictions to a given by the simple roots µj−µj+1 and µp+j+1−
µp+j are equal to λj − λj+1.
3. If p < q the roots µp − θ1 are θq−p − µ2p restrict to λp.
4. If p = q the root µp − µ2p restricts to 2λp.
The simple system Σ obtained by restriction of ΣC is
Σ = {λ1 − λ2, . . . , λp−1 − λp, λp} for p < q, and
Σ = {λ1 − λ2, . . . , λp−1 − λp, 2λp} for p = q.
The Satake diagram of ΣC and the Dynkin diagram of Σ in the case p < q are
AIII1
e e e u
u
ue e e
?
6
?
6
?
6
Bp e e . . . e eA
�
The corresponding diagrams of ΣC and Σ in the case p = q are
AIII2 e e e e
e e e e e
�
�
Z
Z
?
6
?
6
?
6
?
6
Cp e e e�
A
e
The first elements in the basis B correspond to a basis of harr, and more precisely of h⊥arr if
p < q. We shall describe ΣC,arr and Σ⊥C,arr.
The roots in the Satake diagram of ΣC linked by double arrows are given by the following
pairs: if p < q
ΣC,arr = {µ1 − µ2, µp+2 − µp+1} ∪ {µ2 − µ3, µp+3 − µp+2} ∪ · · ·
· · · ∪ {µp−1 − µp, µ2p − µ2p−1} ∪ {µp − θ1, θq−p − µ2p},
while for p = q,
ΣC,arr = {µ1 − µ2, µp+2 − µp+1} ∪ {µ2 − µ3, µp+3 − µp+2} ∪ · · ·
· · · ∪ {µp−1 − µp, µ2p − µ2p−1}.
When p < q, the set of imaginary roots is ΣIm = {θ1 − θ2, . . . , θq−p−1 − θq−p}, so the subset of
pairs of roots which are orthogonal to ΣIm is Σ⊥C,arr = ΣC,arr \ {µp − θ1, θq−p − µ2p}. If p = q
then ΣC,arr = Σ⊥C,arr since there are no imaginary roots. We give a basis of h⊥arr to fill B which
is valid in both cases.
Up to normalization, Hµj−µj+1 is given by the diagonal matrix Λ with non-zero entries 1 and
−1 in positions j and j+1 respectively, j = 1, . . . , 2p−1. Given the pair of roots γj = µj−µj+1,
δj = µp+j+1−µp+j in Σ⊥C,arr, the elements spanning h⊥arr are iHδ−γ = iHµj−µj+1−(µp+j+1−µp+j) =
iHµj−µj+1 + iHµp+j−µp+j+1 . That is, for p ≤ q,
h⊥arr =
iD 0 0
0 iD 0
0 0 0
: trD = 0
. (8.2)
The first elements in the basis B correspond to a basis of harr, and more precisely of h⊥arr if
p < q. We shall describe ΣC,arr and Σ⊥C,arr.
The roots in the Satake diagram of ΣC linked by double arrows are given by the following
pairs: if p < q
ΣC,arr = {µ1 − µ2, µp+2 − µp+1} ∪ {µ2 − µ3, µp+3 − µp+2} ∪ · · ·
· · · ∪ {µp−1 − µp, µ2p − µ2p−1} ∪ {µp − θ1, θq−p − µ2p},
while for p = q,
ΣC,arr = {µ1 − µ2, µp+2 − µp+1} ∪ {µ2 − µ3, µp+3 − µp+2} ∪· · ·∪ {µp−1− µp, µ2p − µ2p−1}.
When p < q, the set of imaginary roots is ΣIm = {θ1 − θ2, . . . , θq−p−1 − θq−p}, so the subset of
pairs of roots which are orthogonal to ΣIm is Σ⊥C,arr = ΣC,arr \ {µp − θ1, θq−p − µ2p}. If p = q
then ΣC,arr = Σ⊥C,arr since there are no imaginary roots. We give a basis of h⊥arr to fill B which
is valid in both cases.
Up to normalization, Hµj−µj+1 is given by the diagonal matrix Λ with non-zero entries 1 and
−1 in positions j and j+1 respectively, j = 1, . . . , 2p−1. Given the pair of roots γj = µj−µj+1,
δj = µp+j+1−µp+j in Σ⊥C,arr, the elements spanning h⊥arr are iHδ−γ = iHµj−µj+1−(µp+j+1−µp+j) =
iHµj−µj+1 + iHµp+j−µp+j+1 . That is, for p ≤ q,
h⊥arr =
iD 0 0
0 iD 0
0 0 0
: trD = 0
. (8.2)
The first p− 1 elements in the basis B are multiples of
iHµj−µj+1+µp+j−µp+j+1 =
iDj,j+1 0 0
0 iDj,j+1 0
0 0 0
22 V. del Barco and L.A.B. San Martin
with Dj,j+1 = diag{0, . . . , 1j ,−1j+1, . . . , 0}. In the case p = q these elements are enough to
complete B since there are no imaginary roots.
Suppose that p < q, that is, we are in the AIII1 case. The last elements of B constitute
a basis of hIm. We have ΣIm = {θ1 − θ2, . . . , θq−p−1 − θq−p}, therefore
kIm =
0 0 0
0 0 0
0 0 Z
: Z ∈ su(q − p)
,
hIm =
0 0 0
0 0 0
0 0 iT
: trT = 0
,
with T diagonal with real entries. Up to normalization iHθj−θj+1
∈ hIm is given by a matrix as
above with T diagonal with 1 in position j and −1 in position j + 1, for j = 1, . . . , q − p− 1.
The center z(m) is h⊥Im, the orthogonal of hIm in hk (see Proposition 6.4), so we have
h⊥Im =
iD 0 0
0 iD 0
0 0 ia Id
: 2 trD + (q − p)a = 0
.
One last element in B is missing since the inclusion h⊥arr ⊂ h⊥Im is strict and of codimension one.
The remaining element is a non zero multiple of iHµp−θ1−(θq−p−µ2p) since {µp− θ1, θq−p−µ2p} is
the only pair of complex roots in ΣC,arr linked to imaginary roots. The matrices corresponding
to Hµp−θ1 and Hθq−p−µ2p are diagonal matrices with non zero entries being 1 and −1 in positions
p, 2p+ 1 and p+ q, 2p, respectively. Then iHµp−θ1−(θq−p−µ2p) = i(Hµp−θ1 −Hθq−p−µ2p).
The computations above account to
B =
{
iHN
µj−µj+1+µp+j−µp+j+1
}p−1
j=1
∪
{
iHN
µp−θ1+µ2p−θq−p
}
∪
{
iHN
θj−θj+1
}q−p−1
j=1
.
Let B⊥ be the dual basis of B with respect to the Cartan–Killing form in sl(p + q,C). The
first p− 1 elements of B⊥ are the elements Z1, . . . , Zp−1 appearing in Theorem 7.3. These are,
up to normalization,
Zj =
iEj 0 0
0 iEj 0
0 0 −ia Id
, j = 1, . . . , p− 1,
where Ej = diag{b, . . . , b,−a, . . . ,−a}, the last b is in position j, and a, b ∈ R verify 2j − a(p+
q) = 0 and b + a = 1, that is, a = 2j/(p + q), b = (p + q − 2j)/(p + q). Through the Weil
construction, {fZ1 , . . . , fZp−1} is a basis of H2(F,R).
For a partial flag manifold FΘ with Θ ⊂ Σ the 2-homology H2(FΘ,R) is spanned by the
Schubert cells Srα with α running through the rank 2 simple roots in Σ outside Θ. Hence as in
the case of the maximal flag manifold we get a basis of H2(FΘ,R) of the form {fZj1 , . . . , fZjs}
where j1, . . . , js are the indices corresponding to the rank 2 roots in Σ \ Θ (long roots if p < q
and short roots if p = q). This basis is dual to the Schubert cells.
Acknowledgements
V. del Barco supported by FAPESP grants 2015/23896-5 and 2017/13725-4. L.A.B. San Martin
supported by CNPq grant 476024/2012-9 and FAPESP grant 2012/18780-0. The authors express
their gratitude to Lonardo Rabelo for careful reading a previous version of this manuscript and
his useful suggestions.
De Rham 2-Cohomology of Real Flag Manifolds 23
References
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964–1029.
[2] Hatcher A., Algebraic topology, Cambridge University Press, Cambridge, 2002.
[3] Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics,
Vol. 80, Academic Press, Inc., New York – London, 1978.
[4] Knapp A.W., Lie groups beyond an introduction, Progress in Mathematics, Vol. 140, Birkhäuser Boston,
Inc., Boston, MA, 1996.
[5] Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. II, Interscience Tracts in Pure and
Applied Mathematics, Vol. 15, Interscience Publishers John Wiley & Sons, Inc., New York – London –
Sydney, 1969.
[6] Kocherlakota R.R., Integral homology of real flag manifolds and loop spaces of symmetric spaces, Adv.
Math. 110 (1995), 1–46.
[7] Mare A.-L., Equivariant cohomology of real flag manifolds, Differential Geom. Appl. 24 (2006), 223–229,
arXiv:math.DG/0404369.
[8] Rabelo L., San Martin L.A.B., Cellular homology of real flag manifolds, arXiv:1810.00934.
[9] San Martin L.A.B., Álgebras de Lie, 2nd ed., UNICAMP, Campinas, 2010.
[10] Silva J.L., Rabelo L., Half-shifted Young diagrams and homology of real Grassmannians, arXiv:1604.02177.
[11] Warner G., Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wis-
senschaften, Vol. 188, Springer-Verlag, New York – Heidelberg, 1972.
[12] Wiggerman M., The fundamental group of a real flag manifold, Indag. Math. (N.S.) 9 (1998), 141–153.
https://doi.org/10.2307/2372843
https://doi.org/10.1007/978-1-4757-2453-0
https://doi.org/10.1007/978-1-4757-2453-0
https://doi.org/10.1006/aima.1995.1001
https://doi.org/10.1006/aima.1995.1001
https://doi.org/10.1016/j.difgeo.2005.09.006
https://arxiv.org/abs/math.DG/0404369
https://arxiv.org/abs/1810.00934
https://arxiv.org/abs/1604.02177
https://doi.org/10.1007/978-3-642-50275-0
https://doi.org/10.1016/S0019-3577(97)87572-6
1 Introduction
2 Cellular decomposition and boundary maps
3 Homology groups of flag manifolds of split real forms
3.1 Maximal flag manifolds
3.2 Partial flag manifolds
4 Classifications
5 De Rham cohomology
6 M-group and Satake diagrams
7 Characteristic forms
8 su(p,q)
References
|
| id | nasplib_isofts_kiev_ua-123456789-210244 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:53Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | del Barco, V. San Martín, L.A.B. 2025-12-04T13:11:30Z 2019 De Rham 2-Cohomology of Real Flag Manifolds / V. del Barco, L.A.B. San Martin // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 57T15; 14M15 arXiv: 1811.07854 https://nasplib.isofts.kiev.ua/handle/123456789/210244 https://doi.org/10.3842/SIGMA.2019.051 Let FΘ = G/PΘ be a flag manifold associated to a non-compact real simple Lie group G and the parabolic subgroup PΘ. This is a closed subgroup of G determined by a subset Θ of simple restricted roots of g = Lie(G). This paper computes the second de Rham cohomology group of FΘ. We prove that it is zero in general, with some rare exceptions. When it is non-zero, we give a basis of H²(FΘ, ℝ) through the Weil construction of closed 2-forms as characteristic forms of principal fiber bundles. The starting point is the computation of the second homology group of FΘ with coefficients in a ring R. V. del Barco supported by FAPESP grants 2015/23896-5 and 2017/13725-4. L.A.B. San Martin supported by CNPq grant 476024/2012-9 and FAPESP grant 2012/18780-0. The authors express their gratitude to Lonardo Rabelo for careful reading of a previous version of this manuscript and his useful suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications De Rham 2-Cohomology of Real Flag Manifolds Article published earlier |
| spellingShingle | De Rham 2-Cohomology of Real Flag Manifolds del Barco, V. San Martín, L.A.B. |
| title | De Rham 2-Cohomology of Real Flag Manifolds |
| title_full | De Rham 2-Cohomology of Real Flag Manifolds |
| title_fullStr | De Rham 2-Cohomology of Real Flag Manifolds |
| title_full_unstemmed | De Rham 2-Cohomology of Real Flag Manifolds |
| title_short | De Rham 2-Cohomology of Real Flag Manifolds |
| title_sort | de rham 2-cohomology of real flag manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210244 |
| work_keys_str_mv | AT delbarcov derham2cohomologyofrealflagmanifolds AT sanmartinlab derham2cohomologyofrealflagmanifolds |