Invariant Einstein metrics on generalized flag manifolds with two isotropy summands
aa r X i v : . [ m a t h . DG ] F e b INVARIANT EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDSWITH TWO ISOTROPY SUMMANDS
ANDREAS ARVANITOYEORGOS AND IOANNIS CHRYSIKOS
Abstract.
Let M = G/K be a generalized flag manifold, that is the adjoint orbit of a compactsemisimple Lie group G . We use the variational approach to find invariant Einstein metrics forall flag manifolds with two isotropy summands. We also determine the nature of these Einsteinmetrics as critical points of the scalar curvature functional under fixed volume.2000 Mathematics Subject Classification.
Primary 53C25; Secondary 53C30, 22E46
Keywords : Einstein manifold, homogeneous space, generalized flag manifold, isotropy represen-tation, highest weight, Weyl’s formula, bordered Hessian.
Introduction
A Riemannian metric g on a manifold M is called Einstein if the Ricci curvature is a constantmultiple of the metric, i.e. Ric g = c · g , for some c ∈ R ( Einstein equation ). Einstein metricsform a special class of metrics on a given manifold M (cf. [Bes]), and the existence question isa fundamental problem in Riemannian geometry. The Einstein equation is a non-linear secondorder system of partial differential equations and general existence results are difficult to obtain.However, if the Riemannian manifold ( M, g ) is compact, then an old result of Hilbert statesthat g is an Einstein metric if and only if g is a critical point of the scalar curvature functional T : M → R given by T ( g ) = R M S( g ) d vol g , on the set M of Riemannian metrics of unit volume.This suggests a variational approach to finding Einstein metrics which, in the homogeneous case,has lead to several important existence and non-existence results mainly from the works of M.Wang, W. Ziller and C. B¨ohm ([WZ2], [BWZ], [B¨om]).A usual strategy for constructing examples of Einstein metrics is to employ symmetry in orderto reduce the Einstein equation into a more manageable system of equations. An important caseis when M is a homogeneous space, i.e. when a Lie group G acts transitively on M . Many ofthe known examples of compact simply connected Einstein manifolds are homogeneous.If M = G/K is a homogeneous space with
G, K compact Lie groups, then we can use thevariational approach to find Einstein metrics. In this case the G -invariant Einstein metrics on M are precisely the critical points of T restricted to M G , the set of G -invariant metrics ofvolume 1. An alternative method is the direct computation of the Ricci curvature. For bothcases, since we are searching for G -invariant metrics on M , the Einstein equation reduces to asystem of non-linear algebraic equations which, in some cases, can be solved explicity. Howevera general classification of all homogeneous spaces that admit an Einstein metric, as well as thecomplete description of all G -invariant Einstein metrics on a given homogeneous Riemannianspace ( M = G/K, g ) is difficult. For a detailed exposition on homogeneous Einstein manifoldswe refer to Besse’s book [Bes], and for more recent results to the surveys [LWa] and [NRS].An important class of homogeneous manifolds consists of the adjoint orbits of compact con-nected semisimple Lie groups, also known as generalized flag manifolds . Let G be a compact,connected and semisimple Lie group and let Ad : G → Aut( g ) be the adjoint representationof G , where g denotes its Lie algebra. A generalized flag manifold is a homogeneous space M = G/K such that the isotropy group K is the centralizer C ( S ) of a torus S in G . Thiscondition can be reformulated as follows: M is the orbit of an element γ o ∈ g under the action The authors were partially supported by the C. Carath´eodory grant of the adjoint representation of G , i.e. M = { Ad( g ) γ o : g ∈ G } ⊂ g . In fact, it can be shownthat the stabilizer of this action K = { g ∈ G : Ad( g ) γ o = γ o } is the centralizer of the torus S γ o = { exp tγ o : t ∈ R } ⊂ G generated by the one-parameter subgroup exp tγ o of G . In partic-ular, K is connected and the element γ o belongs to the center of the Lie algebra k of K (cf.[Bes]). If S γ o = T is a maximal torus in G , then K = C ( S γ o ) = T and M = G/T is called a fullflag manifold . Generalized flag manifolds have been classified in [BFR] using the notion of the painted Dynkin diagrams . There is an infinite family for each of the classical Lie groups, and afinite number for each of the exceptional Lie groups (see also [AlA]).Generalized flag manifolds have a rich complex geometry. It is known that they admit a finitenumber of invariant complex structures, and that there is a one-to-one correspondence betweeninvariant complex structures (up to a sign) and invariant K¨ahler-Einstein metrics (up to a scale)(cf. [AlP], [BFR]). The problem of finding (non K¨ahler) Einstein metrics on generalized flagmanifolds has been first studied by D. V. Alekseevsky in [Al1]. For some of these spaces thestandard metric is Einstein since they appear in the work of M. Wang and W. Ziller ([WZ1]),where they classified all normal homogeneous Einstein manifolds. In [Kim] M. Kimura usingthe variational method of [WZ2] found all G -invariant Einstein metrics for all flag manifoldsfor which the isotropy representation decomposes into three inequivalent irreducible summands.In [Sak] Y. Sakane gave an explicit expression for the Ricci tensor of full flag manifolds ofclassical Lie groups, and by use of Gr¨obner bases theory he proved the excistence of invariantnon K¨ahler-Einstein metrics on certain full flag manifolds. Finally, in [Arv] the first author foundnew G -invariant Einstein metrics on certain generalized flag manifolds with four inequivalentirreducible summands, by using a Lie theoretic description of the Ricci tensor.In the present article we study generalized flag manifolds for which the isotropy representationdecomposes into two inequivalent irreducible submodules. Any such space admits a unique G -invariant complex structure ([Nis]) and thus a unique K¨ahler-Einstein metric. The authorsclassified these spaces in a recent paper [ArC] and proved that any such flag manifold is localyisomorphic to one of the spaces listed in Table 1. Table 1.
The generalized flag manifolds with two isotropy summands. B ( ℓ, m ) = SO (2 ℓ + 1) /U ( ℓ − m ) × SO (2 m + 1) ( ℓ > , m ≥ , ℓ − m = 1) C ( ℓ, m ) = Sp ( ℓ ) /U ( ℓ − m ) × Sp ( m ) ( ℓ > , m > D ( ℓ, m ) = SO (2 ℓ ) /U ( ℓ − m ) × SO (2 m ) ( ℓ > , m > , ℓ − m = 1) G /U (2) ( U (2) is represented by the short root of G ) F /SO (7) × U (1) F /Sp (3) × U (1) E /SU (6) × U (1) E /SU (2) × SU (5) × U (1) E /SU (7) × U (1) E /SU (2) × SO (10) × U (1) E /SO (12) × U (1) E /E × U (1) E /SO (14) × U (1)In a recent work [DKe], W. Dickinson and M. Kerr, classified all simply connencted homoge-neous spaces M = G/H , where G is a simple Lie group, H is a connected and closed subgroup,and the isotropy representation decomposes into two irreducible summands. By using Theorem(3.1) of [WZ2] they counted the number of Einstein metrics. In the present paper by use ofthe variational method we find explicity the Einstein metrics for the flag manifolds presentedin Table 1. Next, by using the bordered Hessian we examine the nature of these critical points(minima or maxima). After the present work has been completed, the authors were informed byY. Sakane that solutions of Einstein equation have also been obtained in an unpublished workof I. Ohmura [Ohm], using the method of Riemannian submersions (cf. [Bes]). nvariant Einstein metrics on generalized flag manifolds with two isotropy summands 3 The paper is organized as follows: In Section 1 we recall some facts about compact homoge-neous spaces. In Section 2 we study the structure of a generalized flag manifold M = G/K ofa compact semisimple Lie group G . In Section 3 we use representation theory to compute thedimensions of the irreducible submodules of the isotropy representation corresponding to theflag manifolds presented in Table 1. In Section 4, we solve the Einstein equation by using thevariational approach of [WZ2] and prove the following: Theorem A.
Let M = G/K be a generalized flag manifold with two isotropy summands.Then M admits precisely two G -invariant Einstein metrics. One is K¨ahler and the other one isnon-Kahler. These Einstein metrics are given explicity in Theorem 2. From the above theorem we exclude the following Hermitian symmetric spaces for which thestandard metric is the unique (up to a scalar) G -invariant Einstein metric. Table 2.
Exceptions of the classification.Space Case Hermitian Symmetric Space B ( ℓ, m ) m = ℓ − SO (2 ℓ + 1) /U (1) × SO (2 ℓ − C ( ℓ, m ) m = 0 Sp ( ℓ ) /U ( ℓ ) B ( ℓ, m ) m = ℓ − SO (2 ℓ ) /U (1) × SO (2 ℓ − m = 0 SO (2 ℓ ) /U ( ℓ )In Section 5 we compute the bordered Hessian of the scalar curvature functional with theconstraint condition of volume 1 and characterize the nature of the solutions obtained in TheoremA. In particular we show the following: Theorem B.
Let M = G/K be a generalized flag manifold with two isotropy summands.Then the two G -invariant Einstein metrics on M given in Theorem A, are both local minima ofthe scalar curvature functional on the space M G . Acknowlegments
The second author wishes to thank Professor Yusuke Sakane for several useful discussionsduring his visit at the University of Patras.1.
Preliminaries
A Riemannian manifold (
M, g ) is G -homogeneous if there is a closed subgroup G of Isom( M, g )such that for any p, q ∈ M , there exists g ∈ G such that gp = q . Let K = { g ∈ G : gp = p } bethe isotropy subgroup corresponding to p . Note that K is compact since K ⊂ O ( T p M ), where T p M is the tangent space of M at p . Via the map g gp we identify the manifolds M ∼ = G/K .Let M = G/K be a homogeneous space, where G is a compact, connected and semisimpleLie group and K is a closed subgroup of G . Let o = eK be the identity coset of G/K . Severalgeometrical questions about M can be reformulated in terms of the pair ( G, K ) and then interms of the corresponding Lie algebras ( g , k ). In fact, there exists a one-to-one correspondencebetween G -invariant tensor fields of type ( p, q ) on M and tensors of the same type on the tangentspace T o M which are invariant under the isotropy representation χ : K → Aut( T o M ) of K on T o M (cf. [KoN]). For instance, left-invariant metrics on a Lie group are determined by an innerproduct on its Lie algebra, and G -invariant Riemannian metrics g on M = G/K are determinedby an inner product on g / k ∼ = T o ( M ), with the additional requirement that the inner product isAd( K )-invariant.Since the Lie group G is semisimple and compact, the Ad( K )-invariant Killing form B ( X, Y ) =tr(ad( X ) ◦ ad( Y )) of g is non-degenerate and negative definite. Let g = k ⊕ m be the orthog-onal decomposition of g with respect to − B . This is a reductive decomposition of g , that isAd( K ) m ⊂ m , and the tangent space T o M is identified with m . Then Ad G (cid:12)(cid:12) K = Ad K ⊕ χ , whereAd G and Ad K are the adjoint representations of G and K respectively. It follows that theisotropy representation χ is equivalent to the adjoint representation of K restricted on m , i.e. Andreas Arvanitoyeorgos and Ioannis Chrysikos χ ( K ) = Ad K (cid:12)(cid:12) m . Therefore, a G -invariant metric on G/K is determined by an Ad( K )-invariantinner product h· , ·i on m .Let Q ( . , . ) be an Ad( K )-invariant inner product on m . Consider the following Q -orthogonalAd( K )-invariant decomposition of m into its Ad( K )-irreducible submodules, that is m = m ⊕ · · · ⊕ m q . (1)By using (1) we can parametrize the space of G -invariant metrics on M so that any G -invariantmetric g on M = G/K is determined by an inner product on m of the form h , i = x Q | m + · · · + x q Q | m q , (2)where x i > i . Such a metric is called diagonal since h , i is diagonal with respectto Q . If m i and m j are pairwise inequivalent representations, then the decomposition (1) isunique up to order. But if the modules m i , m j are equivalent for some i and j , then h m i , m j i does not necessarily vanish. For the examples in the present work we always have m i ≇ m j for i = j (as Ad( K )-submodules). Also, the dimensions d i = dim m i are independent of the chosendecomposition.Since G is compact and K is a closed subgroup of G the homogeneous Riemannian manifold( M = G/K, g ) is compact. Let S ( g ) denotes the scalar curvature of the metric g . By a theoremof Bochner [Boc] S ( g ) in non-negative, and is zero if and only if the metric is flat [AlK]. Thuswe are interested only in homogeneous Einstein metrics with positive scalar curvature, which isequivalent to c >
0, where Ric g = c · g . The Einstein metrics are the critical points of the totalscalar curvature functional T ( g ) = Z M S ( g ) d vol g on the space M of Riemannian metrics of volume one. Recall that the space M has a naturalRiemannian metric, the L metric, which is given by k h k g = R M g ( h, h ) d vol g , where h is asymmetric 2-tensor (considered as a tangent vector at the metric g ), and d vol g is the volumeelement of g . Let M G ⊂ M denote the set of all G -invariant metrics of volume one on M ,equipped with the restriction of the L metric of M . This is also a Riemannian manifold,and since the isotropy representation of M = G/K consists of pairwise inequivalent irreduciblerepresentations, ( M G , L ) is flat with dimension equal to the number of irreducible summands(cf. [BWZ], p.693). Notice that on M G we have T ( g ) = S ( g ). The critical points of therestriction S (cid:12)(cid:12) M G : M G → R are precisely the G -invariant Einstein metrics of volume one (cf.[Bes], p.121). In Section 4 we will use this variational approach to find Einstein metrics.For a fixed Q -orthogonal Ad( K )-invariant decomposition (1), the scalar curvature of themetric (2) has a particularly simple expression, as shown in [WZ2]. Let { X α } be a Q -orthogonalbasis adapted to the decomposition of g , i.e. X α ∈ m i for some i , and α < β if i < j with X α ∈ m i and X β ∈ m j . Set A γαβ = Q ([ X α , X β ] , X γ ) so that [ X α , X β ] = P γ A γαβ X γ , and [ ijk ] = P ( A γαβ ) ,where the sum is taken over all indices α, β, γ with X α ∈ m i , X β ∈ m j and X γ ∈ m k . Notice that[ ijk ] is indepedent of the Q -orthogonal bases chosen for m i , m j and m k , but it depends on thechoise of the decomposition of m . Also, [ ijk ] is nonnegative and symmetric in all three entries.The set { X α / √ x i : X α ∈ m i } is a h , i -orthogonal basis of m . Then the scalar curvature of h , i is given by S = 12 q X i =1 d i b i x i − X i,j,k [ ijk ] x k x i x j , (3)where d i = dim m i , and b i is defined by − B (cid:12)(cid:12) m i = b i Q (cid:12)(cid:12) m i for all i = 1 , . . . , q .2. The structure of flag manifolds
Let G be a compact connected semisimple Lie group. We denote by g the corresponding Liealgebra and by g C its complexification. We choose a maximal torus T in G , and let h be the Lie nvariant Einstein metrics on generalized flag manifolds with two isotropy summands 5 algebra of T . The complexification h C is a Cartan subalgebra of g C . We denote by R ⊂ ( h C ) ∗ the root system of g C relative to h C , and we consider the root space decomposition g C = h C ⊕ X α ∈ R g C α , where by g C α = C E α we denote the 1-dimensional root spaces.Let Π = { α , . . . , α ℓ } (dim h C = ℓ ) be a fundamental system of R . We fix a lexicographicordering on ( h C ) ∗ and we denote by R + the set of positive roots. It is well known that for any α ∈ R we can choose root vectors E α ∈ g C α such that B ( E α , E − α ) = − E α , E − α ] = − H α ,where H α ∈ h C is determined by the equation B ( H, H α ) = α ( H ), for all H ∈ h C . By usingthe last equation we obtain a natural isomorphism between h C and the dual space ( h C ) ∗ . Thenormalized root vectors E α satisfy the relation[ E α , E β ] = (cid:26) N α,β E α + β , if α, β, α + β ∈ R , if α, β ∈ R, α + β / ∈ R where N α,β = N − α, − β ∈ R ( α, β ∈ R ). Then we obtain that (cf. [Hel]) g = h ⊕ X α ∈ R + ( R A α + R B α ) , where A α = E α + E − α , B α = √− E α − E − α ) , α ∈ R + . The complex conjugation τ on g C withrespect to the compact real form g satisfies the relations τ ( E α ) = E − α and τ ( E − α ) = E α .We now assume that G is simple. Let Π K be a subset of Π and setΠ M = Π \ Π K = { α i , . . . , α i m } , (1 ≤ i ≤ · · · ≤ i m ≤ ℓ ) . Let R K = R ∩ h Π K i , R + K = R + ∩ h Π K i , R + M = R + \ R + K , (4)where h Π K i denotes the set of roots generated by Π K . Then p = h C ⊕ X α ∈ R K g C α ⊕ X α ∈ R + M g C α (5)is a parabolic subalgebra of g C (cf. [Al2]).Let G C be the simply connected complex simple Lie group whose Lie algebra is g C and P theparabolic subgroup of G C generated by p . The homogeneous space G C /P is called generalizedflag manifold (or K¨ahler C -space ) and is a compact, simply connected complex manifold onwhich G acts transitively. Note that K = G ∩ P is a connected and closed subgroup of G .The canonical embedding G → G C gives a diffeomorphism of a compact homogeneous space M = G/K to a simply connected complex homogeneous space G C /P , i.e. G C /P ∼ = G/K and M admits a G -invariant K¨ahler metric (cf. [BoH]). The intersection k = p ∩ g ⊂ g is theLie subalgebra corresponding to K , given by k = h ⊕ P α ∈ R + K ( R A α + R B α ). By using (5) weeasily obtain the direct decomposition p = k C ⊕ n , where k C is the complexification of k and n = P α ∈ R + M g C α is the nilradical of p .Let m be the linear subspace of g defined as: m = X α ∈ R + M ( R A α + R B α ) . Then with respect to the Killing form B we obtain the reductive decomposition g = k ⊕ m of g with [ k , m ] ⊂ m . We define a complex structure J on m ∼ = T o M by J A α = B α , J B α = − A α ( α ∈ R + M ) . This gives a G -invariant complex structure on M = G/K and coincides with the canonicalstructure induced from the complex homogeneous space G C /P . Andreas Arvanitoyeorgos and Ioannis Chrysikos
In the following we assume that Π K = Π − { α i o } , that is Π M = { α i o } . For a non-negativeinteger n , we set R + ( α i , n ) = (cid:8) α ∈ R + : α = ℓ X i =1 m j α j ∈ R + , m i o = n (cid:9) , and define Ad( K )-invariant subspaces m n of g by m n = P α ∈ R + ( α i ,n ) ( R A α + R B α ). Put q =max (cid:8) m i o : α = P ℓj =1 m j α j ∈ R + (cid:9) . Then we obtain the decomposition m = q X n =1 m n , (6)and R + M = S qn =1 R + ( α i , n ) (cf. [Ith]). We set m = k . Then for n, m ∈ { , . . . , q } the followingare true: [ k , m n ] ⊂ m n , [ m n , m m ] ⊂ m n + m + m | n − m | , [ m n , m n ] ⊂ k ⊕ m n . (7)Note that m n are irreducible as Ad( K )-modules and are inequivalent to each other. Thus(6) defines an irreducible decomposition of m and according to (2) the space of G -invariantRiemannian metrics on M = G/K is given by n x ( − B ) (cid:12)(cid:12) m + · · · + x q ( − B ) (cid:12)(cid:12) m q : x > , . . . , x q > o . The following theorem describes the K¨ahler-Einstein metrics on the flag manifold M = G/K . Theorem 1. [BoH]
Let M = G C /P = G/K be a generalized flag manifold and let B be theKilling form of g . We assume that g admits a reductive decomposition given by (6). Then M admits a G -invariant K¨ahler-Einstein metric defined by g ( q X n =1 X n , q X n =1 Y n ) = q X n =1 n ( − B ( X n , Y n )) , ( X n , Y n ∈ m n ) . Let m C and m C n be the complexifications of m and m n respectively. These are complex linearsubspaces of g C and we have that m C = P α ∈ R + M ( C E α + C E − α ) and m C n = m + n ⊕ m − n , where m ± n = P α ∈ R + ( α i ,n ) C E ± α . Also k C = h C ⊕ P α ∈ R + K ( C E α + C E − α ) , and this is a complex reductiveLie algebra. Thus we obtain the decomposition k C = k C s ⊕ z ( k C ), where k C s = [ k C , k C ] denotes thesemisimple part of k C , and z ( k C ) its center. By [Kim], it is known that each m ± n is a complexirreducible ad g C ( k C s )-submodule of k C s , where by ad g C we denote the adjoint representation of g C .This fact is equivalent with the irreducibility of the real ad g ( k )-submodules m n of k .3. Irreducible submodules
Flag manifolds with two isotropy summands.
In this paper we are interested in flagmanifolds M = G/K for which the isotropy representation χ : K → Aut( m ) decomposes intoexactly two real irreducible inequivalent submodules, that is m = m ⊕ m . In order to obtainsuch flag manifolds it is sufficient to set R + ( α i , n ) = 0 for n ≥ µ = P ℓi =1 m i α i be the highest root of R , that is the unique root such that for any otherroot α = P ℓi =1 c i α i we have c i ≤ m i ( i = 1 , . . . , ℓ ). The positive coefficients m i ∈ Z are called heights of the simple roots α i . In [ArC] the authors proved that generalized flag manifolds with m = m ⊕ m are in one-to-one correspondence with the sets Π K = Π − { α i o } such that thesimple root α i o has height two, that is m i o = 2. We can present all these suitable pairs (Π , Π K )graphically, by painting black the simple root α i o in the Dynkin diagram of G . The subdiagramof white roots determines the semisimple part of the Lie algebra of K . These diagrams are givenin Table 3. nvariant Einstein metrics on generalized flag manifolds with two isotropy summands 7 Table 3.
Painted Dynkin diagrams of M = G/K such that m = m ⊕ m G (Π , Π K ) K dim m dim m B ℓ ❝ ❝ . . . (2 ≤ p ≤ ℓ ) s p . . . ❝ ℓ − > ❝ ℓ U ( p ) × SO (2( ℓ − p ) + 1) 2 p (2( ℓ − p ) + 1) p ( p − C ℓ ❝ ❝ . . . (1 ≤ p ≤ ℓ − s p . . . ❝ ℓ − < ❝ ℓ U ( p ) × Sp ( ℓ − p ) 4 p ( ℓ − p ) p ( p + 1) D ℓ ❝ ❝ . . . (2 ≤ p ≤ ℓ − s p . . . ❝ ✟❍ ❝❝ ℓ − ℓ U ( p ) × SO (2( ℓ − p )) 4 p ( ℓ − p ) p ( p − E ❝ s ❝❝ ❝ ❝ SU (5) × SU (2) × U (1) 4010 ❝ ❝ ❝s ❝ ❝ SU (6) × U (1) 402 E ❝ ❝ ❝❝ ❝ s ❝ SO (10) × SU (2) × U (1) 6420 s ❝ ❝❝ ❝ ❝ ❝ SO (12) × U (1) 642 ❝ ❝ ❝s ❝ ❝ ❝ SU (7) × U (1) 7014 E ❝ ❝ ❝❝ ❝ ❝ ❝ s E × U (1) 1122 s ❝ ❝❝ ❝ ❝ ❝ ❝ SO (14) × U (1) 12828 F ❝ ❝ > ❝ s SO (7) × U (1) 1614 s ❝ > ❝ ❝ Sp (3) × U (1) 282 G s > ❝ U (2) 823.2. The classical flag manifolds.
For the flag manifolds B ( ℓ, m ) , C ( ℓ, m ) and D ( ℓ, m ) thesimplest method to compute the dimensions of the irreducible submodules is the straightforwadcomputation of the isotropy representation. We will describe only the case of the flag manifold Andreas Arvanitoyeorgos and Ioannis Chrysikos C ( ℓ, m ) = Sp ( ℓ ) × U ( ℓ − m ) × Sp ( m ). Results for the spaces B ( ℓ, m ) and D ( ℓ, m ) are obtainedby a similar procedure.Set ℓ − m = p . Let µ p : U ( p ) → Aut( C p ) and ν ℓ : Sp ( ℓ ) → Aut( C ℓ ) be the standardrepresentations of the Lie groups U ( p ) and Sp ( ℓ ) respectively. It is known (cf. [WZ1]) thatAd U ( n ) ⊗ C = µ p ⊗ C ¯ µ p and Ad Sp ( ℓ ) ⊗ C = S ν ℓ , where S is the second symmetric power of C ℓ .Then Ad Sp ( ℓ ) ⊗ C (cid:12)(cid:12) U ( p ) × Sp ( m ) = S ( ν ℓ (cid:12)(cid:12) U ( p ) × Sp ( m ) ) = S ( µ p ⊕ ¯ µ p ⊕ ν m )= S µ p ⊕ S ¯ µ p ⊕ S ν m ⊕ ( µ p ⊗ ¯ µ p ) ⊕ ( µ p ⊗ ν m ) ⊕ (¯ µ p ⊗ ν m ) . The term S ν m corresponds to the complexified adjoint representation of Sp ( m ) and theterm µ p ⊗ ¯ µ p corresponds to the complexified adjoint representation of U ( p ). Therefore thecomplexified isotropy representation of C ( ℓ, m ) is given by ( µ p ⊗ ν m ) ⊕ (¯ µ p ⊗ ν m ) ⊕ S µ p ⊕ S ¯ µ p . This is the direct sum of four complex ad( k C )-invariant inequivalent submodules of dimension2 pm , 2 pm , (cid:0) p +12 (cid:1) and (cid:0) p +12 (cid:1) , respectively. The representations ¯ µ p ⊗ ν m and µ p ⊗ ν m are conjugateto each other and the same holds for the pair S ¯ µ p and S µ p . Thus m decomposes into a directsum of two real irreducible submodules m , m of dimensions 4 p ( ℓ − p ) and p ( p + 1) respectively.3.3. Review of Representation Theory.
In order to compute the dimensions of m , m forthe exceptional flag manifolds, we need to review some facts from representation theory ofcomplex semisimple Lie algebras and fix notation. We will first determine the irreducible (finite-dimensional) representations of a complex semisimple Lie algebra, by use of the highest weight .Let g be a complex semisimple Lie algebra of rank ℓ , h a Cartan subalgebra of g , and g = h ⊕ P α ∈ R g α the corresponding root space decomposition. Let ρ : g → End( V ) be a (finite-dimensional) representation on the (complex) vector space V . Then there is a decomposition V = L λ V λ , where V λ denotes the subspace of V defined by V λ = (cid:8) u ∈ V : ρ ( H ) u = λ ( H ) u for all H ∈ h (cid:9) . If V λ = 0, then the linear form λ ∈ h ∗ is called a weight of ρ and the eigenspace V λ is called the weight space . The fact that V is finite dimensional implies that there is only a finite number ofweights. It is well known that every weight is real-valued on the real form h R and is algebraicallyintegral, that is 2 ( λ,α )( α,α ) ∈ Z for all α ∈ R . This property follows by restricting ρ to copies of sl (2 , C ) lying in g and then using the representation theory of sl (2 , C ).Let Π = { α , . . . , α ℓ } be a system of simple roots for R . Then the elements { Λ , . . . , Λ ℓ } ,(Λ i ∈ h ∗ ) defined by i ,α j )( α j ,α j ) = δ ij , ( i, j = 1 , . . . , ℓ ) are called the fundamental weights . Under theidentification of h and h ∗ via the Killing form, for any root α ∈ h ∗ we consider the correspondingcoroot h α = H α ( H α ,H α ) = α ( α,α ) ∈ h . For the simple roots α i we set h i = h α i = α i ( α i ,α i ) . Then thefundamental weights Λ i satisfy Λ i ( h j ) = δ ij , so they form a basis of h ∗ dual to the basis { h i } (with respect to the inner product ( , ) on h ∗ ). The lattice of all integer combinations of thefundamental weights is called the weight lattice Λ of g . An important weight is the sum of thefundamental weights δ = P ℓi =1 Λ i . This is also equal to half the sum of the positive roots of R .We shall now give the relationship between the fundamental weights and the simple roots.Since the fundamental weights form a basis of h ∗ , there exist c ij ∈ C such that α i = P ℓj =1 c ij Λ j .An easy calculation shows that α i ( h j ) = c ij , therefore c ij = α i ( h j ) = α i (cid:16) α j ( α j , α j ) (cid:17) = (cid:16) α i , α j ( α j , α j ) (cid:17) = 2( α i , α j )( α j , α j ) = a ji , where a ji is the transpose of the Cartan matrix A = ( a ij ) = (cid:0) α i ,α j )( α i ,α i ) (cid:1) of g (cf. [Knp]). Thus α i = P ℓj =1 a ji Λ j , and the matrix expressing the simple roots as linear combinations of thefundamental weights is the transpose of the Cartan matrix. In particular, we note that all nvariant Einstein metrics on generalized flag manifolds with two isotropy summands 9 simple roots are integral combinations of fundamental weights and thus the lattice generated bythe root system R is contained in Λ.Since g is a complex semisimple Lie algebra it is well known that every finite dimensionalrepresentation of g is a direct sum of irreducible (sub)representations. Therefore, in order tostudy ρ it is sufficient to study the irreducible representations of g . Fix a lexicographic ordering R + on R (or equivalently, a positive Weyl chamber of g ). This induces a partial ordering on allpossible weights: λ > µ if λ − µ is a sum of positive roots. The maximal weight λ with respectto this ordering is called the highest weight . Each (finite-dimensional) irreducible representationof g contains a highest weight λ and will denote this representation by ρ λ . The highest weight λ characterizes completely ρ λ since it determines all of its properties, such as dimension. Now,a weight λ is called dominant if ( λ, α ) ≥
0, for all simple roots α , i.e. if λ lies in the closureof the Weyl chamber corresponding to R + . For example, δ is a dominant weight. We denotethe collection of all dominants weights by Λ + . Equivalently, we can say that an element λ ∈ h ∗ lies in Λ + if and only if λ ( h i ) ∈ Z and λ ( h i ) ≥ i = 1 , . . . , ℓ . Dominant weights areimportant since any such weight arises as the highest weight of an irreducible representation of g . More specifically, apart from equivalence, irreducible finite-dimensional representations ρ ofa semisimple Lie algebra g are in one-to-one correspondence with the dominant weights λ ∈ Λ + .The correspondence is that λ is the highest weight of ρ , i.e. ρ ∼ = ρ λ . The dimension of anirreducible representation of g is given by the following formula due to H. Weyl. Proposition 1. ( [Knp] ) Let ρ λ be an irreducible representation of a complex semisimple Liealgebra g with highest weight λ . Then dim C ρ λ = Y α ∈ R + (cid:16) λ, α )( δ, α ) (cid:17) . Exceptional flag manifolds.
We will use Proposition 1 to compute the dimensions ofthe irreducible Ad( K )-submodules m and m for the exceptional flag manifolds presented inTable 3. We will only treat two cases, and the other are similar. For the root systems of theexceptional Lie algebras we use the notation from [AlA]. Case of G . We fix a system of simple roots to be Π = { α , α } = { e − e , − e } , andlet R + = { α , α , α + α , α + 2 α , α + 3 α , α + 3 α } . It is ( α , α ) = 2 and ( α , α ) = .The maximal root is expressed in terms of simple roots as µ = 2 α + 3 α . The Cartan matrix A = ( a ij ) of G is given by (recall that the Cartan matrix depends on the enumeration of Π) A = (cid:18) (cid:19) Consider the painted Dynkin diagram s α > ❝ α . It determines the generalized flag manifold G /U (2), where U (2) is represented by the short root α . Thus R + K = { α } . The highestweights of the irreducible Ad( K )-submodules m and m are given by λ = α + 3 α and λ = µ respectively. By using the transpose of the Cartan matrix, we obtain that α = 2Λ − , α = − Λ + 3Λ , where Λ , Λ are the fundamental weights of G . Thus λ = − Λ + 3Λ and λ = Λ . Now wecan use Weyl’s formula, and obtain that dim C m = (cid:0) ) = 4 and dim C m = 1, thereforedim R m = 8 and dim R m = 2. Case of F . Let Π = { α = e − e , α = e − e , α = e , α = ( e − e − e − e ) } be a system of simple roots. Recall that F contains long and short roots. It is ( α , α ) =( α , α ) = 2 and ( α , α ) = ( α , α ) = 1. The maximal root is expressed in terms of simpleroots as µ = 2 α + 3 α + 4 α + 2 α . The Cartan matrix of F is given by A = Let { Λ , Λ , Λ , Λ } be the fundamental weights of F . Then by using the transpose of the abovematrix we obtain that α = 2Λ − Λ ,α = − Λ + 2Λ − ,α = − Λ + 2Λ − Λ ,α = − Λ + 2Λ . Consider the painted Dynkin diagram ❝ α ❝ α > ❝ α s α It detemines the generalized flag manifold M = G/K = F /SO (7) × U (1). The semisimple partof the isotropy subalgebra k C is the complex Lie algebra so (7 , C ) and its root system is generatedby the set Π K = { α , α , α } . In particular, we obtain that R + K = { α , α , α , α + α , α + α , α + 2 α , α + α + α , α + α + 2 α , α + 2 α + 2 α } . The highest weight of the irreducible Ad( K )-submodule m is given by λ = α + 2 α + 3 α + α ,and for m the corresponding highest weight is equal to the highest root, i.e. λ = µ . Byusing the above expressions of simple roots in terms of the fundamental weights we obtain that λ = Λ − Λ and λ = Λ .Now we use Weyl’s formula. Let α = P i =1 c i α i be a postive root of R + K . Since (Λ i , α j ) = 0if and only if i = j , then for the submodule m we have that( λ , α ) = (Λ − Λ , α ) = X i =1 c i (Λ − Λ , α i )= c (Λ − Λ , α ) + c (Λ − Λ , α ) + c (Λ − Λ , α )= c (Λ , α ) = c ( α , α )2 = c , where c ∈ { , , } . For the weight δ K = Λ + Λ + Λ and for α ∈ R + K , we obtain that( δ K , α ) = (Λ + Λ + Λ , α ) = X i =1 c i (Λ + Λ + Λ , α i )= c (Λ + Λ + Λ , α ) + c (Λ + Λ + Λ , α ) + c (Λ + Λ + Λ , α )= c (Λ , α ) + c (Λ , α ) + c (Λ , α ) = c ( α , α )2 + c ( α , α )2 + c ( α , α )2= c + c + c . From Weyl’s formula it follows thatdim C m = Y α ∈ R + K (cid:16) λ , α )( δ K , α ) (cid:17) = (1 + 1 / / /
21 + 1 / /
21 + 1 + 1 / , so dim R m = 16.For the irreducible submodule m the calculations are simpler. Since λ = Λ , for any positiveroot α = P i =1 c i α i ∈ R + K it is( λ , α ) = (Λ , α ) = X i =1 c i (Λ , α i ) = c (Λ , α ) = c ( α , α )2 = c , nvariant Einstein metrics on generalized flag manifolds with two isotropy summands 11 where c ∈ { , } . Thus,dim C m = Y α ∈ R + K (cid:16) λ , α )( δ K , α ) (cid:17) = (1 + 11 )(1 + 11 + 1 )(1 + 11 + 1 + 1 / , so dim R m = 14.We remark that it is also possible to use Proposition 1 for the flag manifolds of a classical Liegroup, but the computations are more complicated.4. Invariant Einstein metrics on flag manifolds
Let ( M = G/K, g ) be a Riemannian generalized flag manifold with m = m ⊕ m . In thissection we find the G -invariant Einstein metrics of M by use of the variational method. Firstwe will compute the scalar curvature of a G -invariant metric on M by use of formula (3). Since G is simple we have b i = 1. Thus, according to the relation (2), any G -invariant metric g on M is determined by two positive parameters x , x and has the form h , i = x ( − B ) | m + x ( − B ) | m . (8) Proposition 2.
Let M = G/K be a generalized flag manifold with two isotropy summands andlet g be a G -invariant Riemannian metric on M given by (8). Then the scalar curvature of g isgiven by: S ( g ) = 12 (cid:0) d x + d x (cid:1) − (cid:0) t x x + 2 t x (cid:1) , where t = [112] = 0 .Proof. We set d = dim m , and d = dim m . In order to use (3) we need to find the triples[ ijk ], where i, j, k ∈ { , } . Since [ ijk ] is symmetric in all three entries it is [111] = [222] = 0.Relations (7) imply that[ m , m ] ⊂ m , [ m , m ] ⊂ m ⊕ k , [ m , m ] ⊂ k , so a straightforward computation gives that[221] = [212] = [122] = 0 , thus the only non-zero triples are [112] = [211] = [121]. The result now follows. (cid:3) Let V ( g ) = x d x d be the volume of a G -invariant metric g on M given by (8). In order todetermine the G -invariant Einstein metrics of M subject to the constrained condition V = 1,we need to study the critical points of the restricted scalar curvature S (cid:12)(cid:12) M G . According to theLagrange multipliers method a metric g = ( x , x ) ∈ M G is a critical point of S (cid:12)(cid:12) M G if and onlyif it satisfies the equation ∇ S ( g ) = c ∇ V ( g ) , where ∇ denotes the gradient field and c is the Einstein constant. Note that since the irreduciblesubmodules m and m are inequivalent, the space M G is a 2-dimensional flat Riemannianmanifold, i.e. every point of M G has a neighborhood locally isometric to an open set in R . Theorem 2.
Let M = G/K be a generalized flag manifold with two isotropy summands, i.e. m = m ⊕ m with d i = dim m i ( i = 1 , . Then M admits two G -invariant Einstein metrics. Oneis K¨ahler given by x = 1 , x = 2 , and the other is non K¨ahler given by x = 1 , x = 4 d d + 2 d . Proof.
Set ˜ S = S − c ( x d x d − ∂ ˜ S∂c = 0. Thus a G -invariantEinstein metric of volume one is a solution of the system ∂ ˜ S∂x = 0 , ∂ ˜ S∂x = 0 , which is equivalent to − d x + tx x − cd x d − x d = 0 t − d x − t x − cd x d x d − = 0 ) (9)System (9) reduces to the following polynomial equation2 td x − d d x − td x + 2 d d x x − td x = 0 . (10)Next, we need to find the number t = [112]. By Theorem 1, the space M = G/K admits aunique K¨ahler-Einstein metric given by x = 1 , x = 2, so substituting these values to (10) weobtain the equation 2 td − d d − td + 4 d d − td = 0 , from which t = d d d + 4 d . We substitute this number to equation (10) and normalize x = 1, toobtain the equation d d ( x − (cid:0) d x + 2 d ( x − (cid:1) = 0 , whose solutions are x = 2 and x = 4 d d + 2 d . The first solution determines the K¨ahler-Einsteinmetric, and the second solution determines the non K¨ahler-Einstein metric on M . (cid:3) Example 1.
Consider the family C ( ℓ, m ) and set m = ℓ −
1. Then we obtain the generalizedflag manifold M = Sp ( ℓ ) /U (1) × Sp ( ℓ −
1) which is the complex projective space C P ℓ − . Thepainted Dynkin diagram is given by s ❝ . . . ❝ ℓ − < ❝ ℓ Form Table 3 we have that d = dim m = 4( ℓ −
1) and d = dim m = 2. Any Sp ( ℓ )-invariantmetric h , i on C P ℓ − is determined by two positive parameters x , x , so it is given by h , i = ( − B ) (cid:12)(cid:12) m + x x ( − B ) (cid:12)(cid:12) m , where B is the Killing form of Sp ( ℓ ). From Theorem 2 we obtain that the (non K¨ahler) Sp ( ℓ )-invariant Einstein metric on C P ℓ − is given by h , i = 1( − B ) (cid:12)(cid:12) m + ℓ ( − B ) (cid:12)(cid:12) m .Note that the same result has also been obtained by W. Ziller [Zil] by using the methodof Riemannian submersions. He proved that the complex projective space C P n +1 = Sp ( n +1) /U (1) × Sp ( n ) admits two Sp ( n + 1)-invariant Einstein metrics explicity given by h , i =1( − B ) (cid:12)(cid:12) m + 2 p ( − B ) (cid:12)(cid:12) m , where p = 1 or p = n +1 . The value p = 1 gives the K¨ahler-Einsteinmetric, and the value p = n +1 gives the non K¨ahler-Einstein metric.5. Characterization of the constrained critical points of S We will use a well known criterion (of second order partial derivatives) for minima and maximaof smooth functions to show that both Einstein metrics obtained in Theorem 2 are local minima nvariant Einstein metrics on generalized flag manifolds with two isotropy summands 13 of S (cid:12)(cid:12) M G . In particular, we use the bordered Hessian H of S ( g ) restricted to the space M G of G -invariant metrics with volume one. This is the 3 × H = − ∂V∂x − ∂V∂x − ∂V∂x ∂ ˜ S∂x ∂ ˜ S∂x ∂x − ∂V∂x ∂ ˜ S∂x ∂x ∂ ˜ S∂x where ˜ S = S − c ( x d x d − S ( x , x ) − cV ( x , x ).The critical points of S (cid:12)(cid:12) M G are characterized as follows: Let H ( g ) be the value of H at acritical point g ∈ M G , and let (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) denote its determinant. If (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) > g is a localmaximum of S (cid:12)(cid:12) M G , and if (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) < g is a local minimum of S (cid:12)(cid:12) M G . If (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) = 0 then g is a saddle point (cf. [MTr]). Theorem 3.
Let M = G/K be a generalized flag manifold with m = m ⊕ m , and let d i = dim m i ( i = 1 , . Then the G -invariant Einstein metrics of M given in Theorem 2 are both local minimaof the scalar curvature functional S restricted to the space of G -invariant metrics of volume one M G .Proof. The volume of g is V = x d x d , so − ∂V∂x = − d x d − x d , − ∂V∂x = − d x d x d − . From equations (9) we obtain that ∂ ˜ S∂x = d x − tx x − cd ( d − x d − x d ,∂ ˜ S∂x = d − tx − cd ( d − x d x d − ,∂ ˜ S∂x ∂x = t x − cd d x d − x d − , where t = d d d + 4 d .We first examine the critical point g = (1 , M . Acomputation gives that (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) = − ( d + d ) d d d − (cid:16) d d + 4 d + c d (cid:17) . (11)Since the Einstein constant c and the dimensions d , d are positive real numbers, it is (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) < S (cid:12)(cid:12) M G .For the second critical point g = (1 , d d + 2 d ), we obtain that (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) = − d (cid:0) d d + 2 d (cid:1) d − (cid:16) d d + 5 d d + 6 d d + 2 d ( d + 2 d )( d + 4 d ) + cd (cid:0) d d + 2 d (cid:1) d ( d + d ) (cid:17) , (12)so (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) <
0, and the non K¨ahler-Einstein metric is also a local minimum of S (cid:12)(cid:12) M G . (cid:3) Example 2.
Consider the space E /SU (5) × SU (2) × U (1). According to Table 3, it is d =dim m = 40 and d = dim m = 10, therefore t = 5. By Proposition 1, the scalar curvature isgiven by S = 20 x + 52 x − x x . From (11) and for the K¨ahler-Einstein metric g = (1 ,
2) we obtain that (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) = − c ) < , so g is a local minimum of S (cid:12)(cid:12) M G . The non K¨ahler-Einstein metric is given by g = (1 , / (cid:12)(cid:12) H ( g ) (cid:12)(cid:12) = − / − (53687091200000 c ) / < , so it is also a local minimum of S (cid:12)(cid:12) M G . References [AlA] D. V. Alekseevsky and A. Arvanitoyeorgos:
Riemannian flag manifolds with homogeneous geodesics , Trans.Amer. Math. Soc. 359 (8) (2007) 3769-3789.[AlK] D. V. Alekseevsky and B. N. Kimelfeld:
Structure of homogeneous Riemann spaces with zero Ricci curvature ,Func. Anal. Appl. 9 (1975) 97-102.[AlP] D. V. Alekseevsky and A. M. Perelomov:
Invariant K¨ahler-Einstein metrics on compact homogeneousspaces , Funct. Anal. Appl. 20 (3) (1986) 171-182.[Al1] D. V. Alekseevsky:
Homogeneous Einstein metrics , in: Differential Geometry and its Applications (Proc-cedings of the Conference), Univ. of. J. E. Purkyne, Chechoslovakia (1987) 1-21.[Al2] D. V. Alekseevsky:
Flag manifolds , in: Sbornik Radova, 11 Jugoslav, Seminr. Beograd 6(14) (1997) 1-35.[Arv] A. Arvanitoyeorgos:
New invariant Einstein metrics on generalized flag manifolds , Trans. Amer. Math.Soc. 337 (1993) 981–995.[ArC] A. Arvanitoyeorgos and I. Chrysikos:
Motion of charged particles and homogeneous geodesics in K¨ahler C -spaces with two isotropy summands , (preprint).[Bes] A. L. Besse: Einstein Manifolds , Springer-Verlag, Berlin, 1986.[Boc] S. Bochner:
Curvature and Betti numbers , Ann. Math. 49 (1948) 379-390.[B¨om] C. B¨ohm:
Homogeneous Einstein metrics and simplicial complexes , J. Diff. Geom. 67 (2004) 79-165.[BWZ] C. B¨ohm, M. Wang and W. Ziller:
A variational approach for compact homogeneous Einstein manifolds ,Geom. Funct. Anal. 14 (2004) (4) 681-733.[BFR] M. Bordeman, M. Forger and H. R¨omer:
Homogeneous K¨ahler manifolds: paving the way towards newsupersymmetric sigma models , Comm. in Math. Phys. 102 (1986) 604-647.[BoH] A. Borel and F. Hirzebruch:
Characteristics classes and homogeneous spaces I , Amer. J. Math. 80 (1958)458-538.[DKe] W. Dickinson and M. M. Kerr:
The geometry of compact homogeneous spaces with two isotropy summands ,Ann. of Global Analysis and Geometry, (to appear).[Hel] S. Helgason:
Differential Geometry, Lie Groups and Symmetric Spaces , Academic Press, New York 1978[Ith] M. Itoh:
On curvature properties of K¨ahler C -spaces , J. Math. Soc. Japan. Vol. 30 No.1 (1978) 39–71.[Kim] M. Kimura: Homogeneous Einstein metrics on certain K¨ahler C-spaces , Adv. Stud. Pure. Math. 18-I (1990)303-320.[Knp] A. W. Knapp:
Lie Groups Beyond an Introduction , Progress of mathematics, v. 140 Birkh¨auser, Boston1996.[KoN] S. Kobayashi and K. Nomizu:
Foundations of Differential Geometry Vol II , Wiley - Interscience, NewYork, 1969.[LWa] C. LeBrun and M. Wang (editors):
Surveys in Differential Geometry Volume VI: Essays on EinsteinManifolds , International Press, 1999.[MTr] J. E. Marsden and A.J. Tromba:
Vector Calculus, fifth edition , W. H. Freeman and Company, New York,2003.[NRS] Yu. G. Nikonorov, E. D. Rodionov and V. V. Slavskii:
Geometry of homogeneous Riemannian manifolds ,Journal of Mathematics Sciences, 146 (6) (2007) 6313-6390.[Nis] M. Nishiyama:
Classification of invariant complex structures on irreducible compact simply connected cosetspaces , Osaka J. Math. 21 (1984) 39-58.[Ohm] I. Ohmura:
On Einstein metrics on certain homogeneous spaces , Master Thesis (in Japanese), GraduateSchool of Science, Osaka Univ. 1987, (unpublished).[Sak] Y. Sakane:
Homogeneous Einstein metrics on flag manifolds , Lobachevskii Jour. Math. 4 (1999) 71-87. nvariant Einstein metrics on generalized flag manifolds with two isotropy summands 15 [WZ1] M. Wang and W. Ziller:
On normal homogeneous Einstein manifolds , Ann. Sci. Ec. Norm. Sup. 18 (4)(1985) 563-633.[WZ2] M. Wang and W. Ziller:
Existence and non-excistence of Homogeneous Einstein metrics , Inventiones Math84 (1986) 177-194.[Zil] W. Ziller:
Homogeneous Einstein metrics on spheres and projective spaces , Math. Ann. 259 (1982) 351-358.
University of Patras, Department of Mathematics, GR-26500 Rion, Greece
E-mail address : [email protected] University of Patras, Department of Mathematics, GR-26500 Rion, Greece
E-mail address ::