Invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$

It is well known that the Einstein equation on a Riemannian flag manifold $(G/K,g)$ reduces to an algebraic system if $g$ is a $G$-invariant metric. In this paper we obtain explicitly new invariant Einstein metrics on generalized flag manifolds of $Sp(n)$ and $SO(2n)$; and we compute the Einstein system for generalized flag manifolds of type $Sp(n)$. We also consider the isometric problem for these Einstein metrics.


Introduction
A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor Ric(g) satisfies the Einstein equation Ric(g) = cg, for some real constant c. The study of Einstein manifold is related with several areas of mathematics and has important applications on physics.(see [5], for example).
Let G be a connected compact semisimple Lie group and G/K a flag manifold, where K is the centralizer of a torus in G. It is well known that the Einstein equation of a G-invariant (or simply invariant) metric g on a flag manifold G/K reduces to an (complicated in most cases) algebraic system. It is also known that G/K admits an invariant Kähler Einstein metric associated to the canonical complex structure, see [7]. The problem of determining invariant Einstein metrics non Kähler has been studied by several authors, see for example [2], [9], [12], [14] and [18].
In the algebraic Einstein system for flag manifolds, the number of unknowns is equal the number of equations and it is determined by the amount summands in the isotropy representation. In this sense, several authors have approached the problem of finding new Einstein metrics considering flag manifold with few isotropy summands, see [14], [10] and [3]. Recently Wang-Zhao obtained, in [18], new invariant Einstein metrics on certain generalized flag manifolds with six isotropy summands using a computational method.
Few authors have obtained new invariant Einstein metrics on generalized flag manifolds with many isotropy summands. For instance, Arvanitoyeorgos presented new Einstein metrics on generalized flag manifolds of type SU (n) and SO(2n), see [2]. In [14], Sakane obtained new invariant Einstein metrics on full flag manifolds of a classical Lie group.
Bohm-Wang-Ziller conjectured in [6] that if G/H is a compact homogeneous space whose isotropy representation consists of pairwise inequivalent irreducible summands, e.g. when rank G = rank H, then the algebraic Einstein equations have only finitely many real solutions. In particular, this problem is opened yet for flag manifolds.
This paper is organized as follows: In Section 2 we discuss the construction of flag manifolds of a complex simple Lie group, and we use Weyl basis to see these spaces as the quotient U/K Θ of a semisimple compact Lie group U ⊂ G modulo the centralizer K Θ of a torus in U . In Section 3 we recall the description of invariant metrics and its Ricci tensor on flag manifolds. The problem of isometric and non isometric metrics is also treated in the Section 4. In Section 5, we prove our results solving explicitly the algebraic Einstein system with a specific restriction condition on the invariant metrics.

Preliminaries
In this section we set up our notation and present the standard theory of partial (or generalized) flag manifolds associated with semisimple Lie algebras, see for example [15], [8], for similar description of flag manifolds.
Let g be a finite-dimensional semisimple complex Lie algebra and take a Lie group G with Lie algebra g. Let h be a Cartan subalgebra. We denote by R the system of roots of (g, h). A root α ∈ R is a linear functional on g. It uniquely determines an element H α ∈ h by the Riesz representation α(X) = B(X, H α ), X ∈ g, with respect to the Killing form B(·, ·) of g. The Lie algebra g has the following decomposition where g α is the one-dimensional root space corresponding to α. Besides the eigenvectors E α ∈ g α satisfy the following equation We fix a system Σ of simple roots of R and denote by R + and R − the corresponding set of positive and negative roots, respectively. Let Θ ⊂ Σ be a subset, define We denote by R M := R \ R Θ the complementary set of roots. Note that The partial flag manifold determined by the choice Θ ⊂ R is the homogeneous space F Θ = G/P Θ , where P Θ is the normalizer of p Θ in G. In the special case Θ = ∅, we obtain the full (or maximal) flag manifold For further use, to each α ∈ R M , define the following sets Now we will discuss the construction of any flag manifold as the quotient U/K Θ of a semisimple compact Lie group U ⊂ G modulo the centralizer K Θ of a torus in U . We fix once and for all a Weyl base of g which amounts to giving X α ∈ g α , H α ∈ h with α ∈ R, with the standard properties: The real numbers N α,β are non-zero if and only if α + β ∈ R. Besides that it satisfies We consider the following two-dimensional real spaces Let U = exp u be the compact real form of G corresponding to u. By the restriction of the action of G on F Θ , we can see that U acts transitively on F Θ then The tangent space of F Θ = U/K Θ at the origin o = eK Θ can be identified with the orthogonal complement (with respect to the Killing form) of k Θ in u On the other hand, there exists a nice way to decompose the tangent space m. It is known (see for example [1] or [17]) that F Θ is a reductive homogeneous space, this means that the adjoint representation of k Θ and K Θ leaves m invariant, i.e. ad(k Θ )m ⊂ m. Thus we can decompose m into a sum of irreducible ad(k Θ ) submodules m i of the module m: m = m 1 ⊕ · · · ⊕ m s . Now we will see how to obtain each irreducible ad(k Θ ) submodules m i . By complexifying the Lie algebra of K Θ we obtain be the intersection of the center of the subalgebra k C Θ with ih R . According to [2], we can write Let ih * R and t * be the dual vector space of ih R and t, respectively, and consider the map the set of positive t-roots. There exists a 1-1 correspondence between positive t-roots and irreducible submodules of the adjoint representation of k Θ , see [1]. This correspondence is given by Besides these submodules are inequivalents. Hence the tangent space can be decomposed as follows 3 Invariant metrics and Ricci tensor on F Θ A Riemannian invariant metric on F Θ is completely determined by a real inner product g (·, ·) on m = T o F Θ which is invariant by the adjoint action of k Θ . Besides that any real inner product ad(k Θ )-invariant on m has the form where m i = m ξi and λ i = λ ξi > 0 with ξ i ∈ R + t , for i = 1, . . . , s. So any invariant Riemannian metric on F Θ is determined by |R + t | positive parameters. We will call an inner product defined by (4) as an invariant metric on F Θ .
In a similar way, the Ricci tensor Ric g (·.·) of a invariant metric on F Θ depends on |R + t | parameters. Actually, it has the form Ric g (·, ·) = −r 1 λ 1 B (·, ·) | m1×m1 − · · · − r s λ s B (·, ·) | ms×ms where r i are constants. Thus an invariant metric g on F Θ is Einstein iff r 1 = · · · = r s . The next result shows a way to compute the components of the Ricci tensor by means of vectors of Weyl base.

Remark 3.2. Although
In [13], Park-Sakane computed the Ricci tensor in a similar way. In their formula appears the dimension d i of each irreducible submodules m i , while (equivalently) the equation (5) depends on the amounts of factors U (n i ) in the isotropy subgroup K. Actually Park-Sakane formula is very useful when one wants to describe the Ricci tensor on homogeneous spaces with few isotropy summands or maximal flag manifolds (see for example [18], [3] [14]). The advantage of using (5) is that we can examine at once the Einstein equation for different families of flag manifolds, of the same type, in terms of the size and the amounts of U (n)-factors in the isotropy subgroup K. We will use Proposition 3.1 to complete the list of the algebraic Einstein system for all generalized flag manifolds of classical Lie groups.

Isometric and non isometric metrics
We discuss the problem of determining if two invariant Einstein metrics on F Θ are isometric or non isometric. Let d i r i is the scalar curvature of g, V = V g /V B and V B denotes the volume of the normal metric induced by the negative of the Killing form in U (compact real form of G). We normalize V B = 1, then It is known that H g is a scale invariant under a common scaling of the parameter λ i (see [3] or [18]).
If g 1 and g 2 are two invariant Einstein metrics on F Θ are isometric then H g1 = H g2 . Thus if H g1 = H g2 then g 1 and g 2 are non isometric. In general, to determine if two invariant Einstein metrics are isometric is not a trivial problem (see for example [4]). Now we note that if g is an invariant Einstein metric then S g = c · d, where c is the Einstein constant from Ric g (·, ·) = cg(·, ·). Besides, if g has volume V g then g = 1 V 1/d g g has volume V g = 1 and in this case H g = cd, since Ric g (·, ·) = Ric g (·, ·) = cg(·, ·). So if g 1 and g 2 are two invariant Einstein metrics with different Einstein constants c 1 and c 2 , then g 1 and g 2 are non isometric.

Proof of Theorem A
In this section we consider flag manifolds of the form Sp(n)/U (n 1 ) × · · · × U (n s ), where n ≥ 3 and n = n i .
The next result was obtained in a different way in [11], we proved this theorem following the method used in [2] with the aim of introducing the notation.
Theorem 5.1. The set R t of t-roots corresponding to the flag manifolds Sp(n)/U (n 1 ) × · · · × U (n s ) is a system of roots of type C s . Proof. A Cartan subalgebra of sp(n, C) consists in taking matrices of the form where Λ = diag(ε 1 , . . . , ε n ), ε i ∈ C. Following the notation of [2], we will denote the linear functional h → ±2ε i and h → ±(ε i ± ε j ) by ±2ε i and ±(ε i ± ε j ) respectively. Thus the root system is The root system for the subalgebra k C Θ = sl (n 1 , C) × · · · × sl (n s , C) is given by Then ≤ n i } and the algebra t has the form t = Λ 0 0 −Λ with Λ = diag ε 1 n1 , . . . , ε 1 n1 , ε 2 n2 , . . . , ε 3 n3 , . . . , ε s ns , . . . , ε s ns . Here each ε i ni appears exactly n i times, i = 1, . . . , s. So restricting the roots of R M in t, and using the notation δ i = k(ε i a ), we obtain the t-root set: Note that k(ε i a + ε i b ) = k(2ε i a ), 1 ≤ i ≤ s. In particular, there exist s 2 positive t-roots.
Then the eigenvectors X α ∈ g α satisfying (3) are where E α denotes the canonical eigenvectors of g α . It is convenient to use the following notation An invariant metric on F C (n 1 , . . . , n s ) will be denoted by Considering short and long roots of sp(n), one can see that the square of structural constants are given by In the next table we compute R Θ (α) and R M (α) for each α ∈ R M .
Using Proposition 3.1 we obtain the following result.
In this way, we get that It is easy to see that if n > 2m then h > 0. Besides, these metrics are non isometric since c 1 = c 2 . If n = m we obtain the isotropy irreducible space Sp(n)/U (n) that (up to homotheties) admits a unique invariant metric which is Einstein, ( see 7.44, [5]).
Example 5.3. If we fix m = 2, then for each n ≥ 10 the flag manifold Sp(n)/U (2) s , n = 2s, admits at least two non Kähler (and non isometric) invariant Einstein metrics Corollary 5.4. The Einstein equations on the full flag manifold Sp(n)/U (1) n reduce to an algebraic system of n 2 equations and unknowns g ij , f ij , h i :