Journal of Lie Theory Volume 14 (2004) 569–581 ◯^{C}2004 Heldermann Verlag

On the Principal Bundles over a Flag Manifold

Hassan Azad and Indranil Biswas

Communicated by K.-H. Neeb

Abstract. Let P be a parabolic subgroup of a semisimple simply connected linear algebraic group G over ℂ and ρ an irreducible homomorphism from P to a complex reductive group H. We show that the associated principal H –bundle over G/P, associated for ρ to the principal P –bundle defined by the quotient map G ⟶ G/P, is stable. We describe the Harder–Narasimhan reduction of the G–bundle over G/P obtained using the composition P ⟶ L(P) ⟶ G, where L(P) is the Levi factor of P.

1. Introduction

Let G be a semisimple simply connected linear algebraic group over the field of complex numbers and P a proper parabolic subgroup of G. So G/P is an irreducible smooth projection manifold, and the projection of G to G/P defines a principal P –bundle over G/P. Let E denote this principal P –bundle over G/P.

Let H be a complex reductive linear algebraic group and

ρ : P ⟶ H

a homomorphism. The homomorphism ρ will be called irreducible if its image is not contained in a proper parabolic subgroup of H.

Let E(H) be the principal H –bundle over G/P obtained by extending the structure group of the P –bundle E using ρ.

We prove that the principal H –bundle E(H) over G/P is stable with respect to any polarization of G/P provided the homomorphism ρ is irreducible (Theorem 2.6).

We recall that the notion of a stable principal bundle was introduced by A. Ramanathan in [Ra1] generalizing the original notion of a stable vector bundle due to D. Mumford.

Fix T ⊂ B ⊂ P, where T is a maximal torus and B a Borel subgroup of G. Using the pair (B , T), the Levi quotient L(P) := P/R_{u}(P), where R_{u}(P) is the unipotent radical of P, gets identified with a subgroup of P. Let E(L(P)) be

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag the principal L(P)–bundle obtained by extending the structure group of E using the quotient homomorphism of P to L(P). Let E(L(P))(G) be the principal G– bundle over G/P obtained by extending the structure group of E(L(P)) using the inclusion of the copy of L(P) in G. This G–bundle E(L(P))(G) s not semistable. In Proposition 3.1 we construct the Harder–Narasimhan reduction of E(L(P))(G) for the polarization on G/P defined by ⋀^{top} T G/P.

In Section 2 we also consider the special case of G = SL(n, ℂ). The stability of vector bundles, associated to some naturally occurring L(P)–modules, over a flag variety for ℂ^{n} has been investigated (for example, the tangent bundle of a Grassmannian is stable).

2. Extension of structure group and stability

Let G be a semisimple simply connected linear algebraic group over ℂ and P ⊂ G a parabolic subgroup. A parabolic subgroup will always be assumed to be a proper subgroup. Let H be a connected reductive linear algebraic group over ℂ.

A homomorphism ρ : P ⟶ H is called irreducible if there is no parabolic subgroup of H that contains ρ(P).

Let R_{u}(P) be the unipotent radical of P. So the quotient group L(P) := P/R_{u}(P), which is called the Levi factor of P, is reductive (see [Bo], [Sp]). If T ⊂ B ⊂ P, where T is a maximal torus and B a Borel subgroup, then L(P) is identified with the T –invariant maximal reductive subgroup of P. Fix T and B as above. Henceforth L(P) will be considered both as a quotient group of P and a subgroup of P.

Lemma 2.1. Let ρ : P ⟶ H is an irreducible homomorphism. Then ρ(R_{u}(P)) = e.

Let ZL(P) (respectively, Z(H)) be the connected component of the center of L(P) (respectively, H) containing the identity element. The homomorphism L(P) ⟶ H induced by the irreducible homomorphism ρ takes ZL(P) into Z(H).

Proof. Assume that ρ(R_{u}(P)) ≠ e. Consider the unipotent subgroup U_{1} := ρ(R_{u}(P)) of H. The normalizer of U_{1} in H will be denoted by N_{1}. Inductively define U_{i+1}, i ≥ 1, to be the unipotent radical of N_{i}, and define N_{i}, i ≥ 1, to be the normalizer of U_{i} in H. So we have

N_{1} ⊂ N_{2} ⊂ N_{3} ⊂ ⋯ ⊂ H .

Let Q ⊂ H be the direct limit of the subgroups {N_{i}}. Note that Q is a proper subgroup of H (as U_{1} is nontrivial and H is reductive). Since Q, by construction, is the normalizer of its own unipotent radical, we conclude that Q is a parabolic subgroup of H.

Since R_{u}(P) is a normal subgroup of P, it follows immediately that ρ(P) ⊂ N_{1}. So we have ρ(P) ⊂ Q. This contradicts the assumption that ρ is irreducible, and hence ρ(R_{u}(P)) = e.

To prove the second part, consider the image of the torus ZL(P) in H; we will denote this image by Z′. If Z′ ⊊ Z(H), then the centralizer of the torus Z′ ⊂ H is a Levi subgroup of some parabolic subgroup Q ⊂ H. In that case we have ρ(P) ⊂ Q (since ρ(P) is contained in the centralizer). But this contradicts the assumption that ρ is irreducible. Hence Z′ ⊂ Z(H) and the proof of the lemma is complete.

Proposition 2.2. Let ρ : P ⟶ H be an irreducible homomorphism, and let 𝔥 denote the Lie algebra of H with 𝔷 ⊂ 𝔥 its center. The center 𝔷 coincides with the space of all invariants of 𝔥 for the adjoint action of P on it. If H is simple, then 𝔥 is an irreducible P –module.

Proof. Clearly 𝔷 is contained in 𝔥^{P}, the space of P –invariants.

Take any θ ∈ 𝔥^{P}. Let

θ = θ_{n} + θ_{s}

be the Jordan decomposition, where θ_{n} is nilpotent and θ_{s} is semisimple [Bo, page 83]. From the uniqueness of Jordan decomposition if follows immediately that both θ_{n} and θ_{s} are individually preserved by P. If

θ_{s} ∉ 𝔷

then the centralizer (in 𝔥) of θ_{s} is the Levi subalgebra of a parabolic subalgebra. In that case, ρ(P) ⊂ Q, where Q is the parabolic subgroup corresponding to a parabolic subalgebra of 𝔥 containing the centralizer of θ_{s}. This contradicts the given condition that ρ is irreducible. Therefore, θ_{s} ∈ 𝔷.

Assume that θ_{n} ≠ 0. Let U_{1} ⊂ H be the unipotent subgroup generated by θ_{n}. Setting U_{1} in the construction described in the proof of Lemma 2.1 we get a parabolic subgroup Q ⊂ H. The normalizer N_{1} of U_{1} in H (see the proof of Lemma 2.1) contains the subgroup of H that fixes θ_{n} by the adjoint action (as θ_{n} generates U_{1}). Therefore, we conclude that

ρ(P) ⊂ N_{1} ⊂ Q .

This contradicts the given condition that ρ is irreducible. Therefore, θ_{n} = 0, and hence 𝔥^{P} = 𝔷.

Let the group H be simple. Assume that the P –module 𝔥 is not irreducible. Let 0 ≠ V ⊊ 𝔥 be a nonzero proper subspace preserved by the adjoint action of P on 𝔥. Let Q ⊂ H be the subgroup that preserves V by the adjoint action. This Q is a parabolic subgroup of H. Since ρ(P) ⊂ Q, we conclude that no such V exists. This proves that 𝔥 is an irreducible P –module, and the proof of the proposition is complete.

The quotient G/P is a smooth complex projective variety. Fix an ample line bundle ζ on G/P. For any coherent sheaf F on G/P, the degree of F is defined as

degree(F) := ∫_{G/P} c_{1}(F)c_{1}(ζ)^{d−1} ,

where d is the complex dimension of G/P. For any coherent sheaf F′ defined on a nonempty Zariski open subset U′ ⊂ G/P with the codimension of the complement of U′ at least two, the direct image ι_{∗}F′ is a coherent sheaf on G/P, where ι is the inclusion map of U′. The degree of F′ is defined to be the degree of ι_{∗}F′.

Let H be a complex connected reductive algebraic group and E_{H} a holomorphic principal H –bundle over G/P. The H –bundle E_{H} is called stable (respectively, semistable) if for any reduction of structure group σ : U′ ⟶ E_{H}/Q of E_{H} to any maximal parabolic subgroup Q ⊂ H over a Zariski open subset U′ ⊂ G/P, with codim(G/P \ U′) ≥ 2, the following inequality

degree(ι_{∗}σ^{∗}T_{rel}) > 0

(respectively, degree(ι_{∗}σ^{∗}T_{rel}) ≥ 0) holds; here T_{rel} is the relative tangent bundle for the projection E_{H}/Q ⟶ G/P (see [Ra1]).

Let Z_{0} denote the connected component of the center of H containing the identity element. Let E_{Q} ⊂ E_{H} be a reduction of structure group of E_{H} over G/P to a parabolic subgroup Q ⊂ H. This reduction is called admissible if for every character χ of Q trivial on Z_{0}, the associated line bundle E_{Q}(χ) over G/P, associated to E_{Q} for χ, is of degree zero [Ra2, page 307, Definition 3.3].

A holomorphic principal H –bundle E_{H} over G/P is called polystable if either E_{H} is stable or there is a parabolic subgroup Q of H and a reduction

E_{L(Q)} ⊂ E_{H}

over G/P of structure group of E_{H} to the Levi factor L(Q) (the quotient L(Q) can be realized as a subgroup of Q) such that

1. the principal L(Q)–bundle E_{L(Q)} is stable;

2. the extension of structure group of E_{L(Q)} to Q, constructed using the inclusion of L(Q) in Q, is an admissible reduction of E_{H} to Q.

(See [Ra2], [RS] for the details.)

A stable H –bundle is polystable, and a polystable H –bundle is semistable. The following simple proposition gives a criterion for a polystable H –bundle to be stable.

Proposition 2.3. Let 𝔷 ⊂ 𝔥 be the center of the Lie algebra of H. A polystable H –bundle E_{H} over G/P is stable if and only if H^{0}(G/P, ad(E_{H})) ≅ 𝔷, where ad(E_{H}) is the adjoint vector bundle.

Proof. If E_{H} is stable then H^{0}(G/P, ad(E_{H})) ≅ 𝔷 [Ra1, page 136, Proposition 3.2]. On the other hand, if E_{H} is only polystable but not stable, then there is a reduction of structure group E_{L(Q)} ⊂ E_{H} to a Levi factor L(Q) of some parabolic subgroup Q ⊂ H. The center of L(Q) is contained in the automorphism group of E_{L(Q)}, and hence the center is contained in the automorphism group of E_{H}. But the dimension of the center of a Levi subgroup is more than dim 𝔷. This completes the proof of the proposition.

Note that the projection G ⟶ G/P defines a holomorphic principal P – bundle over G/P; this P –bundle will be denoted by E. Let

β : P ⟶ L(P) := P/R_{u}(P)  (1)

be the quotient map. Let E(L(P)) denote the principal L(P)–bundle obtained by extending the structure group of the principal P –bundle E using β in (1).

Lemma 2.4. The principal L(P)–bundle E(L(P)) over G/P is stable with respect to any polarization on G/P.

Proof. A principal L(P)–bundle is polystable if and only if admits an Einstein– Hermitian connection [RS], [AB]. We will prove that E(L(P)) is polystable by exhibiting an Einstein–Hermitian connection on it.

Fix an ample line bundle ζ on G/P to define degree of a sheaf. Since G is simply connected, the Picard group of G/P is identified with the group of characters of P. Let χ be the character of P that corresponds to ζ.

Fix a maximal compact subgroup K ⊂ G. Set

K(P) := K ∩ P .

Note that G/P = K/K(P) and K(P) projects isomorphically to a maximal compact subgroup of L(P). The maximal compact subgroup of L(P) defined by K(P) will be denoted by K(L(P)).

Consider the action of P on ℂ defined by the character χ (that gives the polarization ζ). Fix a Hermitian structure H_{χ} on ℂ fixed by the action of K(P); since K(P) is compact, such a Hermitian structure exists.

Since G/P = K/K(P) and ζ = (K ×ℂ)/K(P), the condition that K(P) preserves H_{χ} implies that H_{χ} induces a Hermitian structure on the line bundle ζ. The curvature of the corresponding Chern connection on H_{χ} defines a Kähler structure on G/P. This Kähler form will be denoted by Ω_{χ}. Note that Ω_{χ} is K –invariant form on G/P representing c_{1}(ζ).

Recall that E(L(P)) = (G × L(P))/P, where the action of any p ∈ P sends any (g , l) ∈ G × L(P) to (gp , β(p)^{−1}lβ(p)), with β defined in (1). Since the submanifold K × K(L(P)) ⊂ G × L(P) is K(P)–invariant, we have

E_{K}(L(P)) := (K × K(L(P)))/K(P) ⊂ (G × L(P))/K(P) = E(L(P)) . (2)

Note that E_{K}(L(P)) in (2) defines a reduction of structure group of the principal L(P)–bundle E(L(P)) to the the maximal compact subgroup K(L(P)). The action of G/P lifts naturally to E(L(P)) preserving the holomorphic structure; the action of G on G × L(P) defined by g ∘ (z , l) = (gz , l), where g, z ∈ G and l ∈ L(P), descends to an action of G on E(L(P)) = (G × L(P))/P. Furthermore, the action of K ⊂ G on E(L(P)) preserves E_{K}(L(P)) in (2).

A reduction of structure group to a maximal compact subgroup of a holomorphic principal bundle with a reductive group as a structure group has a unique connection known as the Chern connection which is compatible with the holomorphic structure as well as with the reduction (see [AB], [RS]). Let

Ω_{L(P)} ∈ C^{∞}(G/P, Ω_{G/P}^{1,1} (ad(E(L(P)))))  (3)

be the curvature of the Chern connection on E(L(P)) for the reduction of structure group to K(L(P)) in (2), where ad(E(L(P))) is the adjoint bundle. Since the action of K on E(L(P)) preserves the holomorphic structure as well as the reduction of structure group to K(L(P)) in (2), it follows that the action of K on Ω^{1,1}(ad(E(L(P)))) preserves the section Ω_{L(P)} in (3).

Let 𝔩(P) be the Lie algebra of L(P) and

𝔷(L) ⊂ 𝔩(P)

be the center of 𝔩(P). Since the adjoint action of L(P) on 𝔷(L) is trivial, we have

𝔷(L) ⊂ H^{0}(G/P, ad(E(L(P)))) .

Let

Λ_{χ} : Ω_{G/P}^{i,j} ⟶ Ω_{G/P}^{i−1,j−1}

be the adjoint operator of the multiplication operation by the Kähler form Ω_{χ} on G/P; recall that Ω_{χ} is the K –invariant form representing c_{1}(ζ). We will show that the Chern connection satisfies the Einstein–Hermitian condition which says that

Λ_{χ}Ω_{L(P)} ∈ 𝔷(L)  (4)

(we showed earlier that 𝔷(L) defines a subspace of the space of holomorphic sections of ad(E(L(P)))); it should be clarified that the condition in (4) says that there is a fixed element in 𝔷(L) independent of the point of G/P such that the section Λ_{χ}Ω_{L(P)} takes that value at any point of G/P (see [AB, page 220, Definition 3.2], [RS]).

Since both the Chern connection on E(L(P)) and the Kähler form Ω_{χ} on G/P are preserved by the action of K, it follows immediately that the section Λ_{χ}Ω_{L(P)} in (4) is preserved by the action of K on ad(E(L(P))).

The isotropy subgroup at eP ∈ G/P, for the action of K on G/P, is K(P). Since K(P) is a maximal compact subgroup of L(P), we have

(𝔩(P))^{K(P)} = 𝔷(L)

for the adjoint action of K(P) the Lie algebra of L(P); here (𝔩(P))^{K(P)} is the space of all K –invariants. Since K preserves Λ_{χ}Ω_{L(P)} we conclude that the evaluation

Λ_{χ}Ω_{L(P)}(eP) ∈ 𝔷(L) .

Now, since the action of K on G/P is transitive and Λ_{χ}Ω_{L(P)} is K –invariant, it follows that Ω_{L(P)} satisfies the Einstein–Hermitian condition stated in (4).

Therefore, the principal L(P)–bundle E(L(P)) over G/P is polystable [RS, page 24, Theorem 1], [AB, page 221, Theorem 3.7]. We will use the criterion in Proposition 2.3 to prove the E(L(P)) is stable. For that we need the following proposition.

Proposition 2.5. Let V be a nontrivial irreducible L(P)–module such that V ≅ V^{∗}. Let E_{V} = (E(L(P)) × V)/L(P) be the vector bundle over G/P associated to E(L(P)) for V . Then,

H^{0}(G/P, E_{V}) = 0 .

Proof. To prove the proposition, first note that the action of G on E(L(P)) induces an action of G on E_{V} lifting the action of G on G/P. Assume that H^{0}(G/P, E_{V}) ≠ 0. Take a nonzero holomorphic section s of E_{V} . Let W ⊂ E_{V} be the coherent subsheaf generated by all the translations of s by the elements of G. Since the action of G on G/P is transitive, W is a subbundle of E_{V} .

The fiber of E_{V} over eP ∈ G/P is naturally identified with V (send any v ∈ V to the element in (E_{V})_{eP} defined by (e , v)). The isotropy subgroup P of eP (for the action of G on G/P) acts on the fiber (E_{V})_{eP}; the action of P induces an action of L(P) on (E_{V})_{eP}, and the induced action coincides with the L(P)–module structure of V .

Since W is generated by all translates of a section, the subspace

W_{eP} ⊂ (E_{V})_{eP} = V

is left invariant by the action of L(P). This, in view of the given condition that V is an irreducible L(P)–module, implies that W = E_{V} . In particular, E_{V} is globally generated (generated by its global sections).

Since V ≅ V^{∗}, we have E_{V} ≅ E_{V}^{∗} . Hence the dual vector bundle E_{V}^{∗} is also globally generated.

We will now prove that E_{V} is a trivial vector bundle. Fix a point x_{0} ∈ G/P. Take holomorphic sections

v_{j} ∈ H^{0}(G/P, E_{V}) ,

j ∈ [1 , dim V], such that {v_{j}(x_{0})}_{j=1}^{dim V} is a basis of the fiber (E_{V})_{x_{0}} . Now consider the homomorphism from the trivial vector bundle

ψ : (G/P) × ℂ^{dim V} ⟶ E_{V}

defined by (z; c_{1} , ⋯, c_{dim V}) ⟼ ∑_{j=1}^{dim V} c_{j}v_{j}(z), where z ∈ G/P and v_{j} ∈ ℂ. This homomorphism ψ of vector bundles is an isomorphism over a Zariski open subset of G/P containing x_{0} (as it is an isomorphism over x_{0}). Now, if ψ is not an isomorphism everywhere, then the dual homomorphism

ψ^{∗} : E_{V}^{∗} ⟶ (G/P) × ℂ^{dim V}

makes E_{V}^{∗} a proper subsheaf of the vector bundle (G/P) × ℂ^{dim 𝔥_{i}}. Since all the global sections of a trivial vector bundle (G/P) × ℂ^{dim V} are constant sections, the dimension of the space of all global sections of any proper subsheaf of the vector bundle (G/P) × ℂ^{dim V} is less than dim V . This means that if ψ is not an isomorphism over G/P, then E_{V}^{∗} is not globally generated. This contradicts the earlier obtained conclusion that E_{V}^{∗} is globally generated. Therefore, ψ must be an isomorphism. Hence E_{V} is a trivial vector bundle.

Since V is a nontrivial L(P)–module, the associated vector bundle E_{V} is not trivial, contradicting the earlier observation. Therefore, we conclude that H^{0}(G/P, E_{V}) = 0. This completes the proof of the proposition.

Continuing with the proof Lemma 2.4, let

𝔩(P) ≅ 𝔷(L) ⊕ (⨁_{i=1}^{m} V_{i})

be a decomposition of the L(P)–module 𝔩(P) (module structure is defined by the adjoint action), where each L(P)–module V_{i} is nontrivial and irreducible and 𝔷(L) is the center. Note that as L(P) is reductive, we have 𝔩(P) ≅ 𝔩(P)^{∗} as L(P)–modules. Therefore, each L(P)–module V_{i} is self–dual.

The above decomposition of 𝔩(P) induces a decomposition

ad(E(L(P))) ≅ (G/P × 𝔷(L)) ⊕ (⨁_{i=1}^{m} E_{V_{i}}) ,

where E_{V_{i}} = (E(L(P)) × V_{i})/L(P) is the vector bundle associated to E(L(P)) for the L(P)–module V_{i}, and G/P × 𝔷(L) is the trivial vector bundle over G/P with fiber 𝔷(L).

From Proposition 2.5 it follows that

H^{0}(G/P, E_{V_{i}}) = 0 ,

and hence we have

H^{0}(G/P, ad(E(L(P)))) = 𝔷(L) .

Now using Proposition 2.3 we conclude that E(L(P)) is stable. This completes the proof of the lemma.

Fix an irreducible homomorphism

ρ : P ⟶ H ,

where H is reductive. Let E(H) := (G × H)/P be the principal H –bundle over G/P obtained by extending the structure group of the P –bundle E using ρ. (The action of any p ∈ P sends any (g , h) ∈ G × H to (gp , ρ(p^{−1})h).)

The following theorem, which follows from Lemma 2.4 and Proposition 2.2, was proved in [Rm] under the two assumptions that Pic(G/P) = ℤ and H = GL(n, ℂ) (see [Rm, page 168, Theorem 2]).

Theorem 2.6. Let ρ : P ⟶ H be an irreducible homomorphism. The associated principal H –bundle E(H) over G/P is stable with respect to any polarization on G/P.

Proof. From Lemma 2.1 it follows that ρ = ρ′ ∘ β, where

ρ′ : L(P) ⟶ H

is a homomorphism and β is defined in (1). Consequently, E(H) is identified with the principal H –bundle over G/P obtained by extending the structure group of E(L(P)) using ρ′.

Consider the connection on E(H) induced by the Einstein–Hermitian connection on E constructed in the proof of Lemma 2.4. If

Ω_{H} ∈ C^{∞}(G/P, Ω_{G/P}^{1,1} (ad(E(H))))

is the curvature of the induced connection, then

Λ_{χ}Ω_{H} = dρ′(Λ_{χ}Ω_{L(P)}) ∈ C^{∞}(G/P, ad(E(H))) ,  (5)

where Λ_{χ}Ω_{L(P)} is as in (4) and

dρ′ : 𝔩(P) ⟶ 𝔥  (6)

is the homomorphism of Lie algebras defined by ρ′ (here 𝔥 is the Lie algebra of H).

From the second part of Lemma 2.1 it follows that

dρ′(𝔷(L)) ⊂ 𝔷 ,

where 𝔷 (respectively, 𝔷(L)) is the center of 𝔥 (respectively, 𝔩(P)), and dρ′ is defined in (6). This and (5) together immediately imply that the connection on E(H) induced by the Einstein–Hermitian connection is also Einstein–Hermitian. Consequently, the H –bundle E(H) is polystable.

Consider the decomposition of the L(P)–module

𝔥 ≅ 𝔷 ⊕ (⨁_{i=1}^{n} V′_{i})

for the adjoint action, where each V′_{i} is an irreducible L(P)–module. From the first part of Proposition 2.2 it follows that each V′_{i} is nontrivial.

Since 𝔥^{∗} ≅ 𝔥, we have (V′_{i})^{∗} ≅ V′_{i} for each i ∈ [1 , n]. Now using Proposition 2.5 it follows that H^{0}(G/P, E_{V′_{i}}) = 0, i ∈ [1 , n], where E_{V′_{i}} is the vector bundle over G/P associated to the L(P)–bundle E(L(P)) for the L(P)– module V′_{i}. Consequently, we have

H^{0}(G/P, ad(E(H))) ≅ 𝔷 ,

where ad(E(H)) = (E(H) × 𝔥)/H is the adjoint bundle (note that ad(E(H)) ≅ ((G/P) × 𝔷) ⊕ (⨁_{i=1}^{n} E_{V′_{i}}). Finally, using Proposition 2.3 it follows that E(H) is polystable. This completes the proof of the theorem.

Note that if ρ : P ⟶ H is a homomorphism with the property that the homomorphism dρ′ (defined in (6)) takes the center 𝔷(L) to 𝔷, then the principal H –bundle E(H) is polystable; the assumption in Theorem 2.6 that ρ is irreducible was used only to prove that the polystable bundle E(H) is stable.

Let V be a complex vector space of dimension n. Let P ⊂ SL(V) be a parabolic subgroup. So there is a filtration

0 = V_{0} ⊂ V_{1} ⊂ ⋯ ⊂ V_{l−1} ⊂ V_{l} = V

of subspaces such that P is the space of all automorphisms T of V with T ∈ SL(V) and T (V_{i}) = V_{i} for all i ∈ [0 , l]. The Levi quotient of P is described as follows:

L(P) ⊂ ∏_{i=1}^{l} GL(V_{i}/V_{i−1})  (7)

is the subgroup defined by all ∏_{i=1}^{l} A_{i} ∈ ∏_{i=1}^{l} GL(V_{i}/V_{i−1}), where A_{i} ∈ GL(V_{i}/V_{i−1}), such that ∏_{i=1}^{l} det(A_{i}) = 1.

We use the convention that W^{⊗0} := ℂ and W^{⊗−j} = (W^{∗})^{⊗j} if j ≥ 1. Consider the L(P)–module

W_{a_{1},··· ,a_{l}} := ⨂_{i=1}^{l}(V_{i}/V_{i−1})^{⊗a_{i}} ,

where a_{i} ∈ ℤ. The center ZL(P) of L(P) acts as scalar multiplications on W_{a_{1},··· ,a_{l}}; in other words, ZL(P) is mapped to the center of GL(W_{a_{1},··· ,a_{l}}). Therefore, the vector bundle over G/P associated to the L(P)–module W_{a_{1},··· ,a_{l}} is polystable (with respect to any polarization on G/P).

If we have −1 ≤ a_{i} ≤ 1 for each i ∈ [1 , l], then the homomorphism

L(P) ⟶ GL(W_{a_{1},··· ,a_{l}})

defined using the L(P)–module structure is clearly irreducible. Therefore, from Theorem 2.6 we conclude that the vector bundle over G/P associated to the L(P)– module W_{a_{1},··· ,a_{l}}, where a_{i} ∈ {−1 , 0 , 1} for each i, is stable with respect to any polarization of G/P.

More generally, let W′_{i}, i ∈ [1 , l], be an irreducible GL(V_{i}/V_{i−1})–module. (For example, we can take W′_{i} = Sym^{k_{i}}(V_{i}/V_{i−1}).) So

W′ := ⨂_{i=1}^{l} W′_{i}

is a ∏_{i=1}^{l} GL(V_{i}/V_{i−1})–module, and using the inclusion in (7) W′ is a L(P)– module. Since each W′_{i} is an irreducible GL(V_{i}/V_{i−1})–module, it follows immediately that W′ is an irreducible L(P)–module. Therefore, Theorem 2.6 says that the vector bundle over G/P associated to the L(P)–module W′ is stable.

Consider the special case where l = 2. So P is a maximal parabolic subgroup and G/P is a Grassmannian. The vector bundle over G/P corresponding to the L(P)–module W_{−1,1} is the tangent bundle of the Grassmannian. Therefore, the tangent bundle of a Grassmannian is stable.

Remark 2.7. The connection on E(H) induced by the Einstein–Hermitian connection (constructed in the proof of Lemma 2.4) on the L(P)–bundle E(L(P)) can be described as follows. Let K(H) ⊂ H be a maximal compact subgroup such that ρ(K(P)) ⊂ K(H), where ρ is as in Theorem 2.6 and K(P) = K ∩ P as before. Since E(H) is the extension of structure group of E using ρ, and ρ is defined using a homomorphism from L(P) to H, it follows that the reduction of structure group E_{K}(L(P)) ⊂ E(L(P)) in (2) gives a reduction of structure group of E(H)

E(H)_{K(H)} ⊂ E(H)

to K(H) ⊂ H. This reduction is constructed using the natural inclusion

E(H)_{K(H)} := (E_{K}(L(P))×K(H))/K(L(P)) ⊂ (E(L(P))×H)/L(P) =: E(H) .

The connection on E(H) induced by the Einstein–Hermitian connection on E(L(P)) is identified with the Chern connection on E(H) corresponding to the above reduction of structure group E(H)_{K(H)} to the maximal compact subgroup. Indeed, this identification follows immediately by comparing this Chern connection with the construction of the Einstein–Hermitian connection on E(L(P)). As it was noted in the proof of Theorem 2.6, the induced connection on E(H) is the Einstein–Hermitian connection on it.

3. Harder–Narasimhan reduction

The top exterior power of the tangent bundle T G/P is an ample line bundle over G/P. In this section we fix the polarization on G/P defined by ⋀^{top} T G/P.

As before, let E be the principal P –bundle defined by the projection of G to G/P. The principal G–bundle E(G) over G/P obtained by extending the structure group of E using the inclusion P ↪ G is trivial. Indeed, E(G) has a natural section that sends any point gP ∈ G/P to the point in E(G) defined by (g , g^{−1}). Therefore, the G–bundle E(G) is trivial.

Now, let E_{L}(G) be the principal G–bundle obtained by extending the structure group of the L(P)–bundle E(L(P)) using the inclusion of L(P) in G. The G–bundle E_{L}(G) is not trivial, in fact, it is not even semistable (as it will be shown later). We will describe its Harder–Narasimhan reduction.

Let Q ⊂ G be the opposite parabolic of P. So the roots (with respect to the fixed pair (B , T)) corresponding to the Lie algebra of Q are dual to the roots corresponding to the Lie algebra of P. We have P ∩ Q = L(P), so both P and Q share a common Levi subgroup.

Let E_{L}(Q) be the principal Q–bundle over G/P obtained by extending the structure group of E(L(P)) using the inclusion of L(P) in Q. Since L(P) ⊂ Q ⊂ G, we have

E_{L}(Q) ⊂ E_{L}(G)

which defines a reduction of structure group of E_{L}(G) to Q.

Proposition 3.1. The reduction E_{L}(Q) ⊂ E_{L}(G) to Q is the Harder–Narasimhan reduction of the G–bundle E_{L}(G) with respect to the polarization on G/P defined by ⋀^{top} T G/P.

Proof. Let L(Q) := Q/R_{u}(Q) be the Levi quotient, where R_{u}(Q) is the unipotent radical of Q. Let E_{L}(Q)(L(Q)) be the principal L(Q)–bundle obtained by extending the structure group of E_{L}(Q) using the quotient map Q ⟶ L(Q). The first of the two conditions for a Harder–Narasimhan reduction says that E_{L}(Q)(L(Q)) should be semistable (see [AAB, page 694, Theorem 1]).

Since P and Q share a common Levi subgroup L(P), we have L(Q) ≅ L(P), and furthermore, E_{L}(Q)(L(Q)) is identified with E(L(P)) using the the isomorphism of L(Q) with L(P). From Theorem 2.6 we know that the principal L(P)–bundle E(L(P)) is semistable. Therefore, the principal L(Q)–bundle E_{L}(Q)(L(Q)) is semistable.

Let R_{n}(𝔮) be the Lie algebra of the unipotent radical R_{u}(Q) of Q. Consider the L(Q)–module R_{n}(𝔮)/[R_{n}(𝔮) , R_{n}(𝔮)]. The second and final condition in [AAB, page 694, Theorem 1] for a Harder–Narasimhan reduction says that for any irreducible L(Q)–submodule

V ⊂ \frac{R_{n}(𝔮)}{[R_{n}(𝔮) , R_{n}(𝔮)]} ,  (8)

the associated vector bundle over G/P

E_{L(Q)}(V) = (E_{L}(Q)(L(Q)) × V)/L(Q)  (9)

(associated to the principal L(Q)–bundle E_{L}(Q)(L(Q)) for the L(Q)–module V ) should be of positive degree.

To prove that the vector bundle E_{L(Q)}(V) in (9) is of positive degree, first note that R_{n}(𝔮) is identified with the quotient of the Lie algebra of G by the Lie algebra of P. This identification makes R_{n}(𝔮) a P –module. Furthermore, the vector bundle (E × R_{n}(𝔮))/P (associated to the principal P –bundle E for the P –module R_{n}(𝔮)) is identified with the (holomorphic) tangent bundle T G/P.

Since V in (8) is a quotient of the L(P)–module R_{n}(𝔮), we conclude that the vector bundle E_{L(Q)}(V) in (9) is a quotient of the tangent bundle T G/P.

The tangent bundle T G/P is polystable of positive degree. Indeed, G/P admits a Kähler–Einstein metric (see [AzBi] for an explicit construction of a Kähler–Einstein metric on G/P); the existence of a Kähler–Einstein metric on G/P implies that T G/P is polystable with respect to the polarization defined by ⋀^{top} T G/P. Therefore, any quotient bundle of T G/P, in particular V , is of positive degree. This completes the proof of the proposition.

Acknowledgements. We are grateful to the referee for providing detailed comments to improve the manuscript. We thank the International Centre for Theoretical Physics, Trieste, where the work was carried out, for hospitality. The second author thanks the International Mathematical Union for a travel support.

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Hassan Azad Department of Mathematical Sciences King Fahd University Dhahran 31261 Saudi Arabia hassanaz@kfupm.edu.sa Indranil Biswas School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400005 India indranil@math.tifr.res.in

Received July 17, 2003 and in final form November 11, 2003