De Rham $2$-cohomology of real flag manifolds

Let $\mathbb{F}_{\Theta }=G/P_{\Theta }$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $% P_{\Theta }$. This is a closed subgroup of $G$ determined by a subset $% \Theta $ of simple restricted roots of $\mathfrak{g}=Lie(G)$. This paper computes the second de Rham cohomology group of $\mathbb{F}_\Theta$. We prove that it zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2(\mathbb{F}_\Theta,\mathbb{R})$ through the Weil construction of closed 2-forms as characteristic classes of principal fiber bundles. The starting point is the computation of the second homology group of $\mathbb{F}_{\Theta }$ with coefficients in a ring $R$.


Introduction
A real flag manifold is a homogeneous manifold F Θ = G/P Θ where G is a connected Lie group with Lie algebra g which is non-compact and semi-simple, and P Θ 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 flags was considered by Wiggerman [13] giving a continuation to the work of Ehresmann on real flags 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 [7], based on previous work of Bott and Samelson [2]. A different approach to study their homology is given by Rabelo and the second named author of this paper [10], focusing on the geometry involved in the cellular decomposition of the manifolds. Mare [8] considered the cohomology rings of a subfamily of real flags, namely, those having all roots of rank greater of equal than two. Real flags of split real forms are not part of the subfamily considered in [8].
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 flags 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, H 2 (F Θ , R). The description of H 2 (F Θ , 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, after developments in Sections 2 and 3 where we apply the tools of Rabelo-San Martin [10]. Mainly, we show that H 2 (F Θ , 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 rank α = dim g α + dim g 2α , where these subspaces are the root spaces associated to α and 2α (if it is a root).
It follows that H 2 dR (F Θ , R) = {0} unless the root system of the real Lie algebra contains roots having rank 2. Moreover, the amount of such roots gives the dimension of H 2 dR (F Θ , R) (see Theorem 5.1 below). These data can be read off from the classification table of real simple Lie algebras (see Warner [14], pages 30-32) and hence H 2 dR (F Θ , 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 F Θ satisfying H 2 dR (F Θ , R) = {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 −→ F Θ with structure group K Θ = P Θ ∩ K, where K is a maximal compact subgroup of G, and F Θ = G/P Θ = K/K Θ . To perform the Weil construction we choose in a standard way a left invariant connection ω in the principal bundle K −→ F Θ with curvature form Ω. Each adjoint invariant f in the dual k * Θ of the Lie algebra k Θ of K Θ yields an invariant closed 2-form f in K/K Θ . We prove that the 2-forms f exhaust the 2-cohomology classes by exhibiting a basis formed by characteristic classes which is dual to the Schubert cells that generate the second real homology.
To prove the surjectivity of the map f → f it is required a careful description of the center of K Θ which we provide at 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. 2. 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 AIII 1 and AIII 2 .
Acknowledgement. The authors express their gratitude to Lonardo Rabelo for careful reading a previous version of this manuscript and his useful suggestions.

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 [5], Helgason [4] and Warner [14]. 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 [10]. There is a summarized version of definitions and relevant results in [10], without proof, which are necessary to the computation of the second homology group of real flag manifolds. In addition, the section includes preliminary results that will be used in the next section.
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 + g α and g α 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 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 P Θ of p Θ in G is the standard parabolic subgroup associated to Θ. In this manner, each subset Θ ⊂ Σ defines the homogeneous manifold F Θ = G/P Θ ; these homogeneous manifolds are the object of study of this paper. Denote by b Θ the class of the identity in F Θ , that is , b Θ = eP Θ . For Θ = ∅ the index Θ is dropped to simplify notations and F is referred as the maximal flag manifold. When Θ = ∅, F Θ is called intermediate flag manifold.
Let W be the Weyl group associated to a. This is a finite group and is generated by reflections r α over the hyperplanes α (H) = 0 in a, with α a simple root and H a regular element. The length ℓ(w) of an element w ∈ W is the number of simple reflections in any reduced expression as a product w = r α 1 · · · r αt of reflections with α i ∈ Σ. Denote Π w = Π + ∩ wΠ − , the set of positive roots taken to negative by w −1 . For Θ ⊂ Σ, W Θ denotes the subgroup of W generated by r α 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 F Θ and the left N classes of the orbit M * b Θ give a decomposition of F Θ , known as the Bruhat decomposition: One should notice that N · wb Θ does not depend on the choice of the representative, namely, A cellular decomposition of the flag manifold F Θ is given by Schubert cells S Θ w = cl(N · wb Θ ) with w a representative of wW Θ ; here cl denotes closure. The dimension of the Schubert cell defined by wW Θ is For the maximal flag this formula reads dim S w = α∈Πw dim g α .
In what follows there is recalled the definition of the cellular complex and the boundary map giving the homology of F Θ with coefficients in a ring R. The cellular complex for the maximal flag is the model for the intermediate flag, so it is presented in a first instance. When Θ = ∅, a submodule of cells of the complex of the maximal flag with a slightly modified boundary map give the homology of F Θ . Since the paper focuses on second homology groups, only the chains and boundary maps up to dimension three are described thoroughly.
At this point we fix, once and for all, reduced expressions of the elements w in W, w = r α 1 · · · r αt , 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 F Θ [10]. So different choices of reduced expressions lead to different boundary maps, as it will become evident.

Maximal flags
Let C i be the R-module freely generated by S w , w ∈ W and dim S w = i, for i = 0, . . . , dim F Θ . Notice that C 0 = R since there is just one zero dimensional cell, namely the origin {b}.
Given w ∈ W with reduced expression w = r α 1 · · · r αt , the dimension of (see [13,Corollary 2.6]). In what follows, for a simple root α, we denote by rank α the dimension of S rα which is given by We note that g 2α = {0} (that is 2α / ∈ Π) if dim g α = 1 so that rank α = 2 if and only if dim g α = 2. Notice also that if w = r α 1 · · · r αt is a reduced expression then dim S w ≥ ℓ(w) = t if and only if rank α i = 1 for all i = 1, · · · , t.
• C 1 is the free module spanned by S rα with α ∈ Σ split .
Formula (2.1) implies in particular the following.
The boundary operator ∂ applied to a Schubert cell S w of dimension i, gives a linear combination of Schubert cells of dimension i − 1 (2.2) If i = 1 then ∂ : C 1 −→ C 0 is the zero map because there is just one 0dimensional cell. In general the coefficients c(w, w ′ ) are given in [10, Section 2]. These coefficients are always 0 or ±2 and they behave as follows (see also [12]) 1. Given w = r α 1 · · · r αt ∈ W, one has c(w, w ′ ) = 0 for any w ′ ∈ W which is not obtained from w by removing a reflection r α i . 2. If w = r α 1 · · · r α t−1 r αt and w ′ = r α 1 · · · r α t−1 then c(w, w ′ ) = 0. 3. Assume w = r α 1 · · · r αt and w ′ = r α 1 · · · r α i · · · r αt are reduced decompositions of w and w ′ respectively. If rank α i = 1 (that is, dim g α i = 1 and hence g 2α The choice of the sign ± in the formula of c(w, w ′ ) is given by the degree of a certain map depending on w and w ′ [10, 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.
From the items above one has that if w = r α 1 · · · r αt is a reduced expression, the coefficients c(w, w ′ ) are non-zero only when, by erasing a reflection r α i different from the last one, a reduced expression of a root w ′ with dim S w ′ = dim S w − 1 is obtained.
The next proposition gives the boundary of cells of dimension ≤ 3.

(2.5)
Proof: The boundary of S rα contains only the 0-cell hence ∂S rα = 0 regardless the dimension of S rα . For the cell S rαr β its boundary ∂S rαr β has no component in the direction of S rα which obtained by removing the last reflection r β from r α r β . If rank α = 2 then c (r α r β , r β ) = 0 because S r β has codimension 2 in S rαr β . On the other hand if rank β = 2 then c(r α r β , r β ) = is even. This proves the second statement. The last two statements follow from c (r α r β r γ , r α r β ) = 0 (to remove the last simple reflection) and c (r α r β r γ , r α r α ) = 0 since r 2 α = 1 and this is not a reduced expression.
By combining Propositions 2.1 and 2.3 we conclude that the second homology group of F is determined by the second homology group of the subcomplex spanned by the roots in Σ split and by the free module spanned by cells S rα with α ∈ Σ 2 as stated in the theorem below.
Let g split be the split real form whose Dynkin diagram is given by Σ split . Denote F split the maximal flag manifold corresponding to g split .
Theorem 2. 4. Let g be a non-compact simple Lie algebra and let Σ be a simple system of restricted roots. Let F be the maximal flag of g and let F split be as above. Then where the last sum extends to the set Σ 2 = {α ∈ Σ : rank α = 2}.
This theorem in particular holds for complex Lie algebras for which all the roots have multiplicity equal to two. In this case, Σ split is empty and the second homology is given by the free module spanned by the Schubert cells S rα with α ∈ Σ = Σ 2 .

Partial flag manifolds
The chain complex for the homology of an intermediate flag F Θ is built up with Schubert cells associated to minimal elements, which are introduced below.
Lemma 3.1 in [10] shows that for any w ∈ W there exists a unique w 1 ∈ wW Θ such that Π + ∩ w −1 1 Π − ∩ Θ = ∅. This means that the set of positive roots taken into a negative root by w 1 has no intersection with Θ . The element w 1 with this property is called minimal in the coset wW Θ (since it is actually minimal for the Bruhat-Chevalley order, see [10]). A minimal element w 1 satisfies dim S Θ w = dim S w 1 . Denote W min Θ the set of minimal elements in W with respect to Θ and let C Θ i be the free R module spanned by S w with w ∈ W min Θ and dim S w = i, for i = 0, . . . , dim F Θ . Next there are described the minimal elements giving cells of dimension up to three.
Proof: The only positive roots taken into a negative root by w = r α with α ∈ Σ are α and 2α, when this is a root. So Π + ∩ r α Π − ∩ Θ + = ∅ if and only if α / ∈ Θ. If w = r α r β , with α, β ∈ Σ, α = β then the positive roots taken to negative by r α r β are β, r β α and the multiples 2β and r β (2α) when they are actually roots. Hence by the above condition w is minimal if and only if β / ∈ Θ and r β α / ∈ Θ . The same way, if w = r α r β r γ with α = β = γ then The boundary map ∂ min is defined through the boundary map ∂ for the homology of the maximal flag. Given a Schubert cell S w with w ∈ W min Θ , ∂S w is a linear combination of Schubert cells in the maximal flag, which might not correspond to minimal elements.
Consider a Schubert cell S w , with w ∈ W min Θ . Let I w be the set of min- The homology of F Θ with coefficients in R is the homology of the complex (C Θ , ∂ min Θ ) [10, Theorem 3.4]. Using Proposition 2.3 one obtains the boundary map on the minimal cells up to dimension three. For instance ∂ min Θ S rα = 0 when α / ∈ Θ. Moreover, let S rαr β with β / ∈ Θ and r β α / ∈ Θ . Then ∂ min Θ S rαr β = 0 when α ∈ Σ 2 and β ∈ Σ split or vice-versa. Conversely, when α = β ∈ Σ split one has ∂ min Θ S rαr β = c(r α r β , r β )S r β and S r β ∈ C Θ 2 . An analogous result to Theorem 2.4 can be given in this case. Let Σ split and g split as in Subsection 2.1, and set Θ split = Θ ∩ Σ split . Let F Θ split be the flag manifold of g split associated to Θ split . Theorem 2. 6. Let g be a non-compact simple Lie algebra with simple system of restricted roots Σ and fix Θ ⊂ Σ. Let F Θ be the flag manifold of g determined by Θ and let F Θ split as above. Then where the last sum extends to the set Σ Θ 2 = {α ∈ Σ\Θ : rank α = 2}.

Remark:
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 g split as pointed out in [10, pg. 18]. Theorems 2.4 and 2.6 are particular instances of this fact.
3 Homology groups of flag manifolds of split real forms By Theorems 2.4 and 2.6 the second homology group of a flag manifold associated to a non-compact simple Lie algebra is determined once the homology of the split Lie algebras is understood. This is the focus of the present section: the computation of the second homology group H 2 (F Θ , R) of a real flag manifold F Θ associated to a split real simple Lie algebra g.
The homology groups are obtained trough the explicit computation of the coefficients c(w, w ′ ) in Proposition 2.3; these are either zero or ±2. This coefficient being zero or not depends on the links in the Dynkin diagram of g. Namely, c(w, w ′ ) depends on the links of the simple roots in the decomposition of w as product of simple reflections in relation with the retired root to obtain w ′ .
The maximal flag case (with g split) is treated first since it gives the model for the intermediate flag, by restricting the boundary map to minimal cells. Even though the homology of flag manifolds of type G 2 is treated in [10], they are considered here for the sake of completeness.
From now on we assume that the coefficient ring R has characteristic carR = 2. If carR = 2 then ∂ = 0 and the homology of any flag manifold is freely generated by the Schubert cells.

Maximal flags
The results below make reference to the lines in Dynkin diagrams associated to simple real Lie algebras g which are split. We treat G 2 separately so that in the following statements it is assumed that the diagrams have at most double links.

Proof: By Proposition 2.3 we have ∂S
Since g is a split real form dim g β = 1, thus c(r α r β , r β ) = 0 if and only if α ∨ , β is even, otherwise c(r α r β , r β ) = ±2 and ∂S rαr β = ±2S r β . 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.
If a diagram has only simple lines then ∂ : C 2 −→ C 1 is surjective. If a diagram has double lines then the image of ∂ : C 2 −→ C 1 is spanned S r β with β a short simple root, or β long simple root such that there exists α ∈ Σ simple linked to β.
Proof: Lemma 3.1 and the reasoning above show that the elements in the statement are indeed in the kernel of ∂ : C 2 −→ C 1 . One should see that these are the only generators. The boundary ∂S rαr β is a multiple of S r β hence an element of ker ∂ 2 is a sum of linear combinations of 2-cells of the form j∈J n j S rα j r β , with n j ∈ R for all j ∈ J.
In such a linear combination we can take α j not orthogonal to β for all j because ∂S rαr β = 0 if α, β = 0.
In a simply laced diagram a set of roots {α j , β} appearing in a linear combination as above α j , β = 0 occurs only in an A 3 configuration (with β the middle root) or if the roots are in a D 4 configuration as follows A linear combination n 1 S rα 1 r β + n 2 S rα 2 r β + n 3 S rα 3 r β ∈ ker ∂ associated to a D 4 configuration belongs to the span of the combinations S rαr β + η α,β,γ S rγ r β given by A 3 configurations where as above η α,β,γ = ±1 is the sign such that as we wanted to show. If the images ∂S rα 1 r β have two coincident signs and one opposite, a similar argument gives 3 i=1 n i S rα i r β as a combination of the elements in the kernel given by the A 3 subdiagrams.
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 ∂ : On the other hand for a linear combination of 2-dimensional cells in ker ∂ we can assume that it has the form j∈J n j S rα j r β with α j not orthogonal to β for all j. Again, the only possibilities is that β and α ′ i s are in a C 3 , A 3 or a D 4 configuration. The last case is in fact a linear combination of A 3 configurations so the result follows.
With the next proposition complete the computation of the kernel of ∂ : C 2 −→ C 1 by considering the G 2 Lie algebra.
Proposition 3. 5. If the diagram is of type G 2 then the kernel of ∂ : C 2 −→ C 1 is zero.
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 The number of summands equals the number of generators of ker(∂ : C 2 −→ C 1 ) given in Propositions 3.3 and 3.4.
Proof: If g is of type G 2 the kernel of ∂ is trivial as shown in Proposition 3. 5. If g is not of type G 2 , the generators of the kernel of ∂ : C 2 −→ C 1 were given in Propositions 3.3 and 3. 4. Each such generator has a ±2 multiple which is indeed a boundary. In fact, let S rαr β be a 2-cell such that ∂S rαr β = 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 ∂S r β rαr β = ±2S rαr β because of (1) in Proposition 3. 6.
Finally, each generator of the form S rαrγr β with α, β, γ ∈ Σ in an A 3 or C 3 diagram has a ±2 multiple in the image of ∂ : C 3 −→ C 2 as shown in Proposition 3.7.

Partial flag manifolds
At this point, all the relevant computations of kernel and images is done.
The key to deal with the homology in the intermediate case, Θ = ∅, is to determine when a cell S w in the kernel corresponds to a minimal element and, in the case there is a boundary in R · S w , if this boundary comes from a minimal cell. The last part of the section pursue this analysis. Proposition 3. 9. If a diagram has only single and double lines, the kernel of ∂ min Θ : C Θ 2 −→ C Θ 1 is spanned by the following elements: • S rαr β with α, β / ∈ Θ and α, β = 0, • S rαr β with β / ∈ Θ, α and β double linked and α short, • S rαr β + η * α,β,γ S rγ r β whenever α, β, γ form a * 3 diagram, in this order, with β / ∈ Θ.
Proof: By Lemma 2.5, any w = r α r β is a minimal element under the first two conditions, and also r α r β and r γ r β are minimal when they fit into an A 3 or C 3 diagram with β / ∈ Θ. Hence any element in the list above is indeed in C Θ 2 . Also, their image under ∂ in the maximal flag is zero, so ∂ min Θ is zero.
In the case of a G 2 diagram, nothing is left to prove since ker(∂ : C 2 −→ C 1 ) = {0}. Thus ker(∂ min Θ : C 2 −→ C 1 ) is zero independently of Θ. By Lemma 2.5, w = r α r β r γ is a minimal element in W if and only if γ, r γ β, r γ r β α are all outside Θ .
The propositions above together with the proof of Theorem 3.8 show that any 2-dimensional cell corresponding to a minimal element has a border in the free R module spanned by itself. This leads to the following conclusion.
The number of summands equals dim ker(∂ min

Classifications
Let F Θ be a flag manifold associated to a non-compact simple real Lie algebra g. Theorems 3.8, 3.11, 2.6 and 2.4 imply that the second homology group of F Θ is determined by the simple roots in Σ split and Σ 2 .
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. [14, pp. 30-32]). If for a Lie algebra g the sets Σ split and Σ 2 are empty (that is, if rank α ≥ 3 for every α ∈ Σ) then the second homology of any flag manifold associated to g is zero. The Lie algebras having roots with rank α ≤ 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 3.8, 3.11, 2.6 and 2.4. • H 2 (F Θ , R) = 0 if and only if g is of type G 2 for any Θ or if g is not of type G 2 and both Σ split and Σ 2 are contained in Θ. This is the case if g does not appear in Table 1.
• H 2 (F Θ , R) is non-zero and has only torsion components R/2R if and only if g is not of type G 2 and Σ 2 ⊂ Θ and Σ split is not contained in Θ. In particular, if g is a split real form, not of type G 2 .
• H 2 (F Θ , R) contains a free R module if and only if Σ 2 ∩ (Σ\Θ) = ∅. The rank of the module equals the cardinality of this intersection.
Notation: In Table 1, the simple roots of B l , C l and F 4 are labelled according to the following diagrams.

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 Lie algebra Dynkin diagram Table 1: Lie algebras having roots with rank ≤ 2 manifold F Θ , H 2 dR (F Θ , R) ≃ R s where s is the rank of the free Z-module in H 2 (F Θ , Z). Therefore the description of the homologies of the flag manifolds of Theorem 4.1 combined with Table 1 yields the following.
As particular cases, one obtains that H 2 dR (F Θ , R) = 0 for any flag manifold F Θ 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 AIII 1 , AIII 2 , DI 2 and EII. These 2-forms will be obtained by the Weil construction that provides closed differential forms as characteristic classes of principal bundles. In the next theorem we recall this construction for differential 2-forms Theorem 5. 1. (See Kobayashi-Nomizu [6], 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.
. Then the 2-form f • Ω is such that there exists a closed 2-form f on M with f • Ω = π * f . The de Rham cohomology class of f 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 = 0, is that l has non trivial center z (l), that is, l is not semi-simple and is = {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 nonidentity components of L. This construction yields a map z (l) → H 2 (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 F Θ = K/K Θ we will apply the Weil construction to the principal bundle K → K/K Θ . 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/K Θ .
For a flag manifold F Θ = K/K Θ associated to one of the Lie algebras AIII 1 , AIII 2 , DI 2 or EII we intend to prove that there are enough f ∈ k * Θ so that the 2-forms f exhaust the 2-cohomology. To this purpose it is required to describe the center of k Θ . When F Θ = 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.
Before starting let us write some notation and facts related to the Satake diagrams. Denote by g C the complexification of g and let u ⊂ g C be a compact real form of g C 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 g C 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 = τ σ 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 h C ⊂ g C . The Cartan subalgebra h decomposes as h = h k ⊕ a with h k ⊂ k. The Cartan subalgebras h and h C are invariant by σ, τ and θ = στ .
Denote by Π C the set of roots of (g C , h C ) and by Π the set of restricted roots of (g, a). Each α ∈ Π is the restriction to a of a root in Π C . For α ∈ h * C define H α ∈ h C by α (·) = H α , · and denote by h R the real subspace spanned by H α , α ∈ Π C . The roots are real on a and purely imaginary in h k so that a = h R ∩ h, h k = ih R ∩ h and h R = ih k ⊕ a. The last decomposition is orthogonal with respect to the Cartan-Killing inner product in h R because τ is an involutive isometry satisfying τ = −1 in ih k and τ = 1 in a.
A root α ∈ Π C is said to be imaginary if α • τ = −α. Denote by Π Im the set of imaginary roots. There are the following equivalent ways to define Π Im : Conversely if α is zero on a then H α is orthogonal to a in h R so that showing that α ∈ Π Im .

2.
α ∈ Π C is imaginary if and only if H α ∈ ih k because the decomposition h R = ih k ⊕ a is orthogonal. 3. Let H ∈ a be regular real, that is, β (H) = 0 for every β ∈ Π. Then α ∈ Π C is imaginary if and only if α (H) = 0. In fact, the roots in Π are the restrictions to a of the roots in Π C 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 Σ C ⊂ Π C be a simple system of roots such that α (H) ≥ 0 for every α ∈ Σ C . Equivalently Σ C 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 Σ C by painting black the imaginary roots in Σ C and by joining with a double arrow two roots α, β ∈ Σ C whose restrictions to a are equal.
The set Σ Im of imaginary roots in Σ C is given by If β ∈ Σ C \Σ Im then β (H) > 0. Hence a positive root γ is a linear combination of Σ Im if and only if γ (H) = 0 so that Π Im is the set of roots Σ Im spanned by the simple imaginary roots. The next proposition allows to reconstruct h k from the Satake diagram. For its statement we use the following notation • h Im is the subspace spanned by iH α with α ∈ Σ Im (or what is the same α ∈ Π Im ).
• Σ C,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 Σ C,arr of pairs of simple roots not linked to imaginary roots.
Σ arr and Σ ⊥ arr are the restrictions to a of the roots in Σ C,arr and Σ ⊥ C,arr respectively.
Proof: As mentioned above there is the orthogonal direct sum h R = ih k ⊕ a which implies that h Im ⊂ h k . Also, if α and β are simple roots linked by a double arrow then γ = α − β is zero on a which means that H γ is orthogonal to a so that iH γ ∈ h k . Hence h arr ⊂ h k . We have h Im ∩ h arr = {0} because Σ C is a basis. A dimension check shows that the sum is the whole space. In fact, dim h is the number of roots in Σ C (rank of g) while dim a (real rank of g) is the number of white roots not linked plus dim h arr which is half the number of white linked roots. Finally dim h Im is the number of black roots. Hence dim h k = dim h Im + dim h arr 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 g Im be the subalgebra of g C generated by the root spaces (g C ) α with α ∈ Π Im . Then g Im is a complex semi-simple Lie algebra whose Dynkin diagram corresponds to the simple roots Σ Im .
Put k Im = g Im ∩ u. Then k Im is a compact real form of g Im and therefore it is semi-simple. Moreover, h k is a Cartan subalgebra of k Im .
Proof: The first statement holds because Π Im is a root system generated by Σ Im which is a simple system of roots. Regarding to the subalgebra k Im it can be proved that (g C ) α is contained in k + ik if α is imaginary (see [11,Lemma 14.6]). Hence g Im ⊂ k + ik implying that k Im ⊂ k. By the Weyl construction of the compact real form applied simultaneously to g C and g Im it follows that the intersection k Im = g Im ∩ u is a compact real form of g Im . Finally, Σ Im is a simple system of roots of g Im so that h Im , which is spanned by iH α with α ∈ Σ Im , is a Cartan subalgebra of k Im . Now we combine the above pieces to write down the Lie algebra m from the Satake diagram of g. Proof: The centralizer of a in g C is z = h C ⊕ α∈Π Im (g C ) α . Thus m = z∩k = h k ⊕ k Im . Since h Im ⊂ k Im we have m = h ⊥ Im ⊕ k Im where h ⊥ Im is the orthogonal complement of h Im in h k . Any imaginary root is zero on h ⊥ Im so that this subspace commutes with k Im . This implies that z (m) = h ⊥ Im because k Im is semi-simple.
By Proposition 6.1 we have h k = h Im ⊕ h arr so that h ⊥ Im = {0} if and only if h arr = {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 g C ). By the above proposition z (m) = h ⊥ Im so we get the following case where z (m) is not trivial. Remark. For α, β ∈ Σ ⊥ C,arr and γ = α − β, iH γ ∈ h ⊥ Im = z(m). In fact, if δ is an imaginary root, then iH γ , iH δ = 0 since α and β are orthogonal (non-adjacent) to any imaginary root.
The diagrams AIII 1 , AIII 2 , DI 2 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.
In these diagrams a simple restricted root α ∈ Σ has rank α = 2 if and only if it is the restriction of γ, δ ∈ Σ ⊥ C,arr , that is, we have Σ 2 = Σ ⊥ arr . The diagrams AIII 2 , DI 2 and EII have no imaginary roots. Hence in these Lie algebras m = h k is abelian. For AIII 1 the imaginary simple roots form an A l Dynkin diagram. So that in AIII 1 we have k Im ≈ 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. 5. Let F = G/P = K/M be the maximal flag manifold associated to the real form g of the complex simple Lie algebra g C . Then the center z (m) of the Lie algebra m of M is non-trivial if g is of the type AIII 1 , AIII 2 , DI 2 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 e iH 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 e iH X = X because z (m) is contained h k 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 e iH z 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. 6. 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 g C then the subalgebra g C (α) generated by the root spaces (g C ) ±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. 7. 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 . If α denotes their common restrictions to a then rank α = 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 h α spanned over R by H α ∈ a and iH γ−δ ∈ h k is a Cartan subalgebra of g (α).

Characteristic classes
In this section we apply the Weil homomorphism to get representatives of the de Rham cohomology of the flag manifolds of the Lie algebras AIII 1 , AIII 2 , DI 2 and EII appearing in Theorem 5. 1.
Connections in the principal bundles K → F Θ = K/K Θ 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) = L q * (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 ω q X l (q) = P X ∈ 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 (L g * Hor = Hor, L * g ω = ω and L * g Ω = Ω).
Left invariant connections on the bundles K → F Θ = K/K Θ given by this proposition will be used to get characteristic classes on the flag manifolds F Θ .
We work out first the maximal flag manifolds F = K/M. Take the root space decomposition g = m ⊕ a ⊕ α∈Π g α and for a positive restricted root α ∈ Π + write k α = (D α + D −α ) ∩ k where D α = g α/2 + g α + g 2α . We have k = m ⊕ α∈Π + k α so that For any m ∈ M there is the invariance Ad (m) p = p because Ad (m) g α = g α for every root α. Hence p defines a left invariant connection in the bundle K → K/M. Its curvature form Ω is defined at the identity by the projection into m of the bracket. Since [g α , g β ] ⊂ g α+β it follows that Ω (X, Y ) = 0 if X ∈ g α , Y ∈ g β with α = β.
Hence Ω (at the origin) is the direct sum of 2-forms Ω α on the spaces k α : where Ω α is the projection into m of the bracket in k α . If Z ∈ z (m) then by the Weil homomorphism theorem the left invariant 2form Z, Ω (·, ·) in K is the pull back of a closed differential form in F = K/M that we denote by f Z . The tangent space at the origin identifies with p where f Z is given by .
We write f Z (S rα ) for the value of f Z in the Schubert cell S rα given by the duality between H 2 (F, R) and H 2 (F, R).
The following example of CP 1 ≈ S 2 will be used to compute the values of characteristic classes in the Schubert 2-cells.
Example: Let us look at the characteristic classes at the complex projective line CP 1 ≈ S 2 obtained by the action of SU (2) so that S 2 = SU (2) /T where T is the group of diagonal matrices diag{e it , e −it }, t ∈ R. The Lie algebra of T is t = {H t = diag{it, −it} : t ∈ R} and the subspace complements t in su (2). The bracket in p is given by Hence by taking the appropriate inner product in t we have that f H 1 is the only invariant 2-form in S 2 that in the origin satisfies f H 1 (X 1 , X i ) = 1. Thus f H 1 is the unique (up to scale) SU (2)-invariant volume form in S 2 . We can normalize the inner product in t so that the integral of f H 1 over S 2 equals to 1.
Now we can compute the basis of H 2 (F, R) dual to the Schubert cells spanning H 2 (F, R) when F is the maximal flag manifold of one of the Lie algebras AIII 1 , AIII 2 , DI 2 and EII. Recall that by Proposition 6.4 for these Lie algebras z (m) = h ⊥ Im = {0}. On the other hand, H 2 (F, R) is spanned by the 2-cells S rα ≈ S 2 with α running through Σ ⊥ arr as in Proposition 6.7. To get a basis of H 2 (F, R) dual to the basis of homology given by the Schubert cells we define the Weil map z (m) → H 2 (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 f iH γ−δ (S rα ) = 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 AIII 1 , AIII 2 , DI 2 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 h k given by where for Z ∈ h k , Z N = Z/ (Z, Z) c and we are using the following notation: 1.3]). Let φ α : su (2) → k (α) be an isomorphism assured by Proposition 6.7 and put We have Then Hence the restriction to S rα yields Since {Z 1 , . . . , Z k } is the basis dual to the basis {iH N γ 1 −δ 1 , . . . , iH N γ k −δ k } with respect to (·, ·) c it follows that f Z j S rα k = 1 if j = k and 0 otherwise so that {f Z 1 , . . . f Z k } is the basis dual to {S rα : α ∈ Σ ⊥ arr }, concluding the proof.
This result holds also for a partial flag manifold F Θ = K/K Θ by taking roots in Σ 2 = Σ ⊥ arr that are outside Θ. By Theorem 4.1 the 2-homology H 2 (F Θ , R) is generated by the Schubert cells S Θ rα with α ∈ Σ ⊥ arr \ Θ. On the other hand, let {Z 1 , . . . , Z k } be the dual basis as in the statement of the above theorem and take an index j such that both roots γ j and δ j restrict to α j ∈ Σ ⊥ arr \ Θ. To prove that forms f Z j corresponding to these indices form a dual basis in H 2 (F Θ , R) it remains to check that Z j belong to the center z (k Θ ) so that the forms f Z j are well defined in F Θ . Lemma 7. 3. Let Z j , j = 1, . . . , k be as in the above theorem the elements of the dual basis and take an index j that corresponds to α j ∈ Σ ⊥ arr \ Θ. Then Z j ∈ z (k Θ ).
Proof: By Proposition 6.3 we have Z j ∈ z (m) ⊂ m ⊂ k Θ . Denote by Θ C ⊂ Σ C the set of roots of the Satake diagram whose restrictions to a belong to Θ ∪ {0}. Take γ ∈ Θ C and let α ∈ Θ ∪ {0} be its restriction. We claim that γ (Z j ) = 0. There are the possibilities: 2. α has multiplicity 1 so that γ = α is not imaginary and not linked to another root of Σ C by a double arrow. Then H γ ∈ a and since iZ j ∈ a ⊥ we have γ (Z j ) = 0. 3. γ is linked to δ by a double arrow. By definition of the dual basis B ⊥ we have that Z j is orthogonal to iH γ−δ . On the other hand, H γ+δ ∈ a since α = (γ + δ) /2. Hence H γ+δ , Z j = 0 so that γ (Z j ) = 0 as claimed.
It follows that Z j centralizes the Lie algebra g C (Θ C ) generated by (g C ) γ , γ ∈ Θ C . Hence Z j centralizes k Θ = g C (Θ C ) ∩ k as well, concluding the proof.
The 2-forms {f Z 1 , . . . , f Z k } are zero on any k(α) with α not of rank 2. Therefore, if there is a closed form on H 2 (F Θ , R) which is non-degenerate on F Θ then Π\Θ ⊂ Σ ⊥ arr . This implies that Π\Θ consists of roots with double arrows in the Satake diagram. In this case, F Θ is a product of complex flag manifolds of the form SU(n)/T .

su (p, q)
In this section we present concrete realizations of the flag manifolds of the Lie algebras of types AIII 1 and AIII 2 . 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 C p+q with matrix That is, su (p, q) = {Z ∈ M p+q (C) : ZJ p,q + J p,q Z * = 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  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 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 The matrices in s are hermitian, that is, 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 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  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 Im D + T ) = 0, so the real part of D is arbitrary. The Cartan subalgebra h decomposes as with Λ and T are diagonal matrices such that tr(2Λ + T ) = 0. Therefore, dim h k = p + (q − p) − 1 = q − 1 is the rank of su (p, q) and dim a + dim h k = p + q − 1, which is the rank of sl (p + q, C). The Cartan subalgebra h C is the Lie algebra of complex diagonal matrices in sl (n, C), n = p + q.
In order to give a basis {Z 1 , . . . , Z p−1 } as in Theorem 7.2, we need first a basis B of h k determined by the roots in Σ ⊥ C,arr , Σ C,arr and Σ Im . So we proceed with the description of the root system.
The roots of h C are the linear functionals given by the differences of the diagonal coordinate functionals in h C . To simplify notations, for H = diag{a 1 , . . . , a n } ∈ h C write µ j (H) = a j if 1 ≤ j ≤ 2p and θ j (H) = a 2p+j if 1 ≤ j ≤ q − p.
As usual in sl (n, C), h R is the subspace of real diagonal matrices. The imaginary roots (annihilating on a) are θ j − θ k . In particular, there are no imaginary roots if p = q.
The Satake diagram is given by a simple system Σ C of h C such that the imaginary roots in Σ C span the set of all imaginary roots. Simple systems Σ C are written differently in the cases p < q and p = q. If p < q then It can be checked that these sets are in fact simple systems of roots with Dynkin diagram A l .
To write the restricted system determined by a we consider the parametrization by real matrices Λ = diag{a 1 , . . . , a p } in a way that a is constituted by the following matrices Denote λ i (Λ) = a i . 1. The simple roots θ j − θ j+1 are imaginary. 2. For j = 1, . . . , p − 1, the restrictions to a given by the simple roots µ j − µ j+1 and µ p+j+1 − µ p+j are equal to λ j − λ j+1 . 3. If p < q the roots µ p − θ 1 are θ q−p − µ 2p restrict to λ p . 4. If p = q the root µ p − µ 2p restricts to 2λ p .
The center z(m) is h ⊥ Im , the orthogonal of h Im in h k (see Proposition 6.4), so we have 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 iH µp−θ 1 −(θ q−p −µ 2p ) since {µ p − θ 1 , θ q−p − µ 2p } is the only pair of complex roots in Σ C,arr linked to imaginary roots. The matrices corresponding to H µp−θ 1 and H θ q−p −µ 2p are diagonal matrices with non zero entries being 1 and −1 in positions p, 2p + 1 and p + q, 2p, respectively. Then iH µp−θ 1 −(θ q−p −µ 2p ) = i(H µp−θ 1 − H θ q−p −µ 2p ).
The computations above account to 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 Z 1 , . . . , Z p−1 appearing in Theorem 7.2. These are, up to normalization, where E j = 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, {f Z 1 , . . . , f Z p−1 } is a basis of H 2 (F, R). For a partial flag manifold F Θ with Θ ⊂ Σ the 2-homology H 2 (F Θ , R) is spanned by the Schubert cells S rα 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 H 2 (F Θ , R) of the form {f Z j 1 , . . . , f Z js } where j 1 , . . . , j s 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.