Lobachevskii Journal of Mathematics Vol. 4, 1999, 71–87 \begin{matrix}c \\ ◯ \end{matrix}Y. Sakane

Y. Sakane

HOMOGENEOUS EINSTEIN METRICS ON FLAG MANIFOLDS

(submitted by B.N.Shapukov)

Abstract. It is known that a flag manifold admits a Kähler-Einstein metric. We investigate K-invariant Einstein metrics on a flag manifold M = K/T which is not Kähler-Einstein. This problem has been studied by Alekseevsky and Arvanitoyeorgos in case of generalized flag manifolds. We give an explicit expression of Ricci tensor of a flag manifold K/T for the case of a classical simple Lie group and we present more new K-invariant Einstein metrics on a flag manifold K/T. We compute a Gröbner basis for a system of polynomials of multi-variables and show the existence of positive solutions for a system of algebraic equations to prove the existence of K-invariant Einstein metrics.

1. Introduction

A Riemannian manifold (M, g) is called an Einstein manifold if the Ricci tensor Ric(g) satisfies Ric(g) = λg for some λ ∈ ℝ.

Let K be a connected compact semi-simple Lie group, T a maximal torus of K and M = K/T a flag manifold. Then M admits a K-invariant Kähler-Einstein metric (cf. Borel-Hirzebruch[4]). In this paper we study non-Kähler K-invariant Einstein metrics on a flag manifold. This problem has been studied by Alekseevsky [2] and Arvanitoyeorgos [1] in more general case, that is, the case of generalized flag manifolds or Kähler C-spaces and they constructed new K-invariant Einstein metrics on some generalized flag manifolds. In section 3 we give a formula of Ricci tensor for a K-invariant metric on a flag manifold using the notation introduced by Wang-Ziller [7] and determine it explicitly in the case of classical Lie groups. Notice that we have to prove the existence of a positive solution for a system of algebraic equations to see that there exists an K-invariant Einstein metric. We compute a Gröbner basis (cf. [5]) of a system of algebraic equations using algebraic manipulations in a computer system and show more existence of new K-invariant Einstein metrics on flag manifolds. In case of SU(n+1)/T Senda [6] has proved existence of new SU(n+1)-invariant Einstein metrics.

2. Ricci tensor of a homogeneous manifold

Let (M, g) be a Riemannian manifold and I(M, g) the Lie group of all isometries of M. A Riemannian manifold (M, g) is said to be K-homogeneous if a Lie subgroup K of I(M, g) acts transitively on M. For a K-homogeneous Riemannian manifold (M, g), we fix a point o ∈ M and identify M with K/L where L is the isotropy subgroup of K at o. Let 𝔨 be the Lie algebra of K and 𝔩 the subalgebra corresponding to L. Take a vector subspace 𝔭 complement to 𝔩 in 𝔨 with Ad(L)𝔭 ⊂ 𝔭. Then there is a natural one-to-one correspondence between K-invariant symmetric covariant 2-tensors on M = K/L and Ad(L)- invariant symmetric bilinear forms on 𝔭. In this correspondence a K-invariant metric g corresponds to an Ad(L)-invariant inner product ⟨ , ⟩ on 𝔭.

For X ∈ 𝔨, we will denote by X_{𝔩} (resp. X_{𝔭}) the 𝔩-component (resp. 𝔭component) of X. Let ∇ be the Levi-Civita connection of (M, g) and let X^{∗} denote the Killing vector field associated to X ∈ 𝔨.

Then for X, Y ∈ 𝔭 we have

(∇_{X^{∗}}Y^{∗})_{o} = −\frac{1}{2}[X, Y]_{𝔭} + U(X, Y),

where U : 𝔭 × 𝔭 → 𝔭 is a symmetric bilinear form defined by

2⟨U(X, Y), W ⟩ = ⟨[W, X]_{𝔭}, Y ⟩ + ⟨X, [W, Y]_{𝔭}⟩.

We set Z = ∑_{i} U(X_{i}, X_{i}), where {X_{i}} is an orthonormal basis of 𝔭 with re-

spect to ⟨ , ⟩.

Now the Ricci tensor Ric(g) of K-homogeneous manifold (M, g) is given by

\begin{aligned}Ric(g)(X, X) = & − \frac{1}{2} ∑_{i} |[X, X_{i}]_{𝔭}|^{2} + \frac{1}{2}(X, X) \\ & + \frac{1}{4} ∑_{i, j} ⟨[X_{i}, X_{j}]_{𝔭}, X⟩^{2} − ⟨[Z, X]_{𝔭}, X⟩ \end{aligned}  (1)

where ( , ) is the negative of the Killing form of Lie algebra 𝔨. (cf. Besse[3] Corollary 7.38). Note that if K is a compact Lie group then we have Z = 0.

From now on let K be a connected compact semi-simple Lie group. Then ( , ) the negative of the Killing form of Lie algebra 𝔨 is an Ad(K)-invariant inner product on 𝔨. We take as 𝔭 a orthogonal complement of 𝔩 in 𝔨 with respect to ( , ). Then we have

𝔨 = 𝔩 ⊕ 𝔭, Ad(L)𝔭 ⊂ 𝔭.

Let 𝔭 = 𝔭_{1} ⊕ ⋯ ⊕ 𝔭_{q} be a decomposition into irreducible Ad(L)-modules 𝔭_{j} (j = 1, ⋯, q) and assume that Ad(L)-modules 𝔭_{j} are mutually non-equivalent.

We now introduce a notation [ \begin{matrix}k \\ ij \end{matrix} ] due to Wang-Ziller[7].

Let {e_{α}} be a ( , )-orthonormal basis adapted to the decomposition of 𝔭, that is, e_{α} ∈ 𝔭_{i} for some i, and α < β if i < j (with e_{α} ∈ 𝔭_{i} and e_{β} ∈ 𝔭_{j}). We

set A_{αβ}^{γ} = ([e_{α}, e_{β}], e_{γ}), so that [e_{α}, e_{β}] = ∑_{γ} A_{αβ}^{γ}e_{γ} and [ \begin{matrix}k \\ ij \end{matrix} ] = ∑(A_{αβ}^{γ})^{2},

where the sum is taken over all indices α, β, γ with e_{α} ∈ 𝔭_{i}, e_{β} ∈ 𝔭_{j}, e_{γ} ∈ 𝔭_{k}.

Then, [ \begin{matrix}k \\ ij \end{matrix} ] is independent of the ( , )-orthonormal bases chosen for 𝔭_{i}, 𝔭_{j},

𝔭_{k}, and

[ \begin{matrix}k \\ ij \end{matrix} ] = [ \begin{matrix}k \\ ji \end{matrix} ] = [ \begin{matrix}j \\ ki \end{matrix} ] .

Moreover we have relations between [ \begin{matrix}k \\ ij \end{matrix} ] and Casimir operator C_{𝔭_{i},( , )|𝔩} of

representation (Ad, 𝔭_{i}) of L. Notice that C_{𝔭_{i},( , )|𝔩} = − ∑_{i} ad(Z_{i})ad(Z_{i}), where

Z_{i} is an orthonormal basis of 𝔩 with respect to ( , )|𝔩. Since 𝔭_{i} is an irreducible Ad(L)-module, we have C_{𝔭_{i},( , )|𝔩} = c_{i} · id for some c_{i} ≥ 0.

Put d_{k} = dim 𝔭_{k} (k = 1, ⋯, ℓ).

Lemma 1. (Wang-Ziller[7]) ∑_{j,k} [ \begin{matrix}k \\ ij \end{matrix} ] = d_{i}(1 − 2c_{i}).

Since Ad(L)-modules 𝔭_{j} are mutually non-equivalent by our assumption, the space of all K-invariant symmetric covariant 2-tensors on M = K/L is given by

{w_{1} · ( , ) |_{𝔭_{1}} + ⋯ + w_{ℓ} · ( , ) |_{𝔭_{ℓ}} | w_{1}, ⋯, w_{ℓ} ∈ ℝ}  (2)

and the space of all K-invariant metrics on M = K/L is given by

{x_{1} · ( , ) |_{𝔭_{1}} + ⋯ + x_{ℓ} · ( , ) |_{𝔭_{ℓ}} | x_{1}, ⋯, x_{ℓ} ∈ ℝ^{+}} .  (3)

Note that the Ricci tensor of a K-invariant metric on M = K/L is a Kinvariant symmetric covariant 2-tensor and hence is of form (2).

Lemma 2. The components r_{k} of the Ricci tensor Ric(g) of a K-invariant metric g = x_{1} · ( , ) |_{𝔭_{1}} + ⋯ + x_{ℓ} · ( , ) |_{𝔭_{ℓ}} on M = K/L are given by

r_{k} = \frac{1}{2x_{k}} + \frac{1}{4d_{k}} ∑_{i,j=1}^{ℓ} \frac{x_{k}}{x_{i}x_{j}} [ \begin{matrix}k \\ ij \end{matrix} ] − \frac{1}{2d_{k}} ∑_{i,j=1}^{ℓ} \frac{x_{j}}{x_{k}x_{i}} [ \begin{matrix}j \\ ki \end{matrix} ] (k = 1, ⋯, ℓ). (4)

Proof. Let {e_{α}^{(k)}}_{α=1}^{d_{k}} be an orthonormal basis on 𝔭_{k} (k = 1, ⋯, ℓ) with

respect to the inner product ( , ). Put X_{α}^{(k)} = \frac{1}{\sqrt{x_{k}}} e_{α}^{(k)}. Then \{X_{α}^{(k)}\}_{α=1}^{d_{k}} is a

g-orthonormal basis on 𝔭_{k} (k = 1, ⋯, ℓ). Since the Ricci tensor Ric(g) of the metric g is given by (1), we have

r_{k} = Ric(g)(X_{α}^{(k)}, X_{α}^{(k)})
= −\frac{1}{2} ∑_{j,i} \frac{x_{j}}{x_{k}x_{i}} ∑_{s} ([e_{α}^{(k)}, e_{s}^{(i)}]_{𝔭_{j}} , [e_{α}^{(k)}, e_{s}^{(i)}]_{𝔭_{j}}) + \frac{1}{2x_{k}}

+\frac{1}{4} ∑_{j,i} \frac{x_{k}}{x_{j}x_{i}} ∑_{t,s} ([e_{t}^{(j)}, e_{s}^{(i)}]_{𝔭_{k}}, e_{α}^{(k)})^{2} .

By the definition of \binom{k}{ij} we see that

d_{k}r_{k} = ∑_{α=1}^{d_{k}} Ric(g)(X_{α}^{(k)}, X_{α}^{(k)}) = \frac{d_{k}}{2x_{k}} − \frac{1}{2} ∑_{j,i} \frac{x_{j}}{x_{k}x_{i}} \binom{j}{ki} + \frac{1}{4} ∑_{j,i} \frac{x_{k}}{x_{j}x_{i}} \binom{k}{ji}.

3. Einstein metrics on a flag manifold

From now on we consider a flag manifold K/T where T is a maximal torus of a compact semi-simple Lie group K. We denote by 𝔨^{ℂ} the complexification of 𝔨 and by 𝔱^{ℂ} the complexification of the Lie algebra 𝔱 of T. Then 𝔱^{ℂ} is a Cartan subalgebra of 𝔨^{ℂ} and we denote by Δ the root system of 𝔨^{ℂ} relative to 𝔱^{ℂ}. Thus we have a root decomposition:

𝔨^{ℂ} = 𝔱^{ℂ} + ∑_{α∈Δ} 𝔨_{α}^{ℂ}.

Let Π = {α_{1}, ⋯, α_{n}} be a fundamental system of Δ and {Λ_{1}, ⋯, Λ_{n}} the fundamental weights of 𝔨^{ℂ} corresponding to Π, that is,

\frac{2(Λ_{i}, α_{j})}{(α_{j}, α_{j})} = δ_{ij} (1 ≤ i, j ≤ n).

We define a Borel subalgebra 𝔟 of 𝔨^{ℂ} by

𝔟 = 𝔱^{ℂ} + ∑_{α∈Δ^{+}} 𝔨_{α}^{ℂ}

where Δ^{+} is the set of all positive roots relative to Π. Let K^{ℂ} be a simply connected complex semi-simple Lie group whose Lie algebra is 𝔨^{ℂ} and B the Borel subgroup of K^{ℂ} generated by 𝔟. Then the complex homogeneous manifold K^{ℂ}/B is a compact simply connected and K acts transitively on K^{ℂ}/B. Note also that T = G ∩ B is a connected closed subgroup of K and K^{ℂ}/B = K/T as C^{∞}-manifold and K^{ℂ}/B admits a K-invariant Kähler metric.

We identify a root of 𝔨^{ℂ} with an element of \sqrt{−1}𝔱 by the duality defined by the Killing form φ of 𝔨^{ℂ}, that is, we identify H_{α} with α, where H_{α} ∈ \sqrt{−1}𝔱 is

defined by φ(H_{α}, H) = α(H) for H ∈ 𝔱^{ℂ}. Thus we have \sqrt{−1}𝔱 = ∑_{i=1}^{n} ℝΛ_{i}. We

put 𝔠_{+} = \{λ ∈ \sqrt{−1}𝔱 | φ(λ, α_{i}) > 0, i = 1, ⋯, n\} and δ = ∑_{α∈Δ^{+}} α. Then we

have δ = Λ_{1} + ⋯ + Λ_{n} and thus δ ∈ 𝔠_{+}.

Lemma 3. (Borel-Hirzebruch[4]) There is a one-to-one correspondence between K-invariant Kähler metrics on K^{ℂ}/B and the vectors in 𝔠_{+}. Moreover, for a K-invariant Kähler metric on K^{ℂ}/B corresponding to λ ∈ 𝔠_{+} the Ricci tensor is also a K-invariant Kähler metric on K^{ℂ}/B which corresponds to δ. In particular, the K-invariant Kähler metric on K^{ℂ}/B corresponding to δ is a Kähler-Einstein metric.

Now we consider following problem: Is there a K-invariant Einstein metric on a flag manifold K/L which is not a Kähler-Einstein metric ?

This problem was studied by Alekseevsky [2] and Arvanitoyeorgos [1], in case of generalized flag manifolds. We apply our result in section 2 to obtain a formula of Ricci tensor in case of flag manifolds.

We take as 𝔭 a orthogonal complement of 𝔱 in 𝔨 with respect to ( , ). Then we have

𝔨 = 𝔱 ⊕ 𝔭, Ad(T)𝔭 ⊂ 𝔭.

Take a Weyl basis {α_{1}, …, α_{n}, E_{α} ∈ 𝔨_{α}^{ℂ}(α ∈ Δ)} with

\begin{aligned}[E_{α}, E_{−α}] & = −α (α ∈ Δ) \\ [E_{α}, E_{β}] & = \{ \begin{matrix}N_{α,β}E_{α+β} & if α + β ∈ Δ \\ 0 & if α + β ∉ Δ \end{matrix} \end{aligned}  (5)

where N_{α,β} = N_{−α,−β} ∈ ℝ. Then we have

𝔨 = 𝔱 + ∑_{α∈Δ^{+}}\{ℝ(E_{α} + E_{−α}) + ℝ\sqrt{−1}(E_{α} − E_{−α})\}.

Set 𝔭_{α} = ℝ(E_{α}+E_{−α})+ℝ\sqrt{−1}(E_{α}−E_{−α}) for α ∈ Δ^{+}. Since φ(E_{α}, E_{−α}) = −1, \{X_{α} = (E_{α} + E_{−α})/\sqrt{2}, Y_{α} = \sqrt{−1}(E_{α} − E_{−α})/\sqrt{2}\} is an orthonormal basis of 𝔭_{α}. Then 𝔭_{α} is an irreducible Ad(T)-module and does not depend on the choice of an order, that is, the choice of a fundamental system of Δ. Thus the complement subspace 𝔭 decomposes into 2-dimensional irreducible Ad(T)- modules 𝔭_{α} which are are mutually non-equivalent:

𝔭 = ∑ 𝔭_{α}.

Now the space of all K-invariant metrics on the flag manifold M = K/T is given by

{⟨ , ⟩ = ∑_{α∈Δ^{+}} x_{α}( , )_{|𝔭_{α}} | x_{α} ∈ ℝ^{+}} .

Notice that the K-invariant Kähler-Einstein metric on K^{ℂ}/B corresponding to δ is given by g_{δ} = ∑_{α∈Δ^{+}} φ(Λ_{1} +⋯+Λ_{n}, α)( , )_{|𝔭_{α}}. We write each positive root α uniquely as α = ∑_{i=1}^{ℓ} m_{i}(α)α_{i} where m(α_{i}) are non-negative integers. Then the K-invariant Kähler-Einstein metric g_{δ} on K^{ℂ}/B is given by

g_{δ} = ∑_{α∈Δ^{+}} \frac{1}{2} (∑_{i=1}^{n} m_{i}(α)φ(α_{i}, α_{i})) ( , )_{|𝔭_{α}}.  (6)

By Lemma 2, we see that the components r_{α} of the Ricci tensor Ric(g) of a K-invariant metric g = ∑_{α∈Δ^{+}} x_{α}( , )_{|𝔭_{α}} on M = K/T are given by

r_{α} = \frac{1}{2x_{α}} + \frac{1}{8} ∑_{β,γ∈Δ^{+}} \frac{x_{α}}{x_{β}x_{γ}} [ \begin{matrix}α \\ βγ \end{matrix} ] − \frac{1}{4} ∑_{β,γ∈Δ^{+}} \frac{x_{γ}}{x_{α}x_{β}} [ \begin{matrix}γ \\ αβ \end{matrix} ] .  (7)

Notice that if [ \begin{matrix}γ \\ αβ \end{matrix} ] is non-zero, we have either [ \begin{matrix}α + β \\ αβ \end{matrix} ] = 2(N_{α,β})^{2} or

[ \begin{matrix}α − β \\ αβ \end{matrix} ] = 2(N_{α,−β})^{2}.

For roots α, β ∈ Δ, let p, q be the largest non-negative integers such that β + kα is a root (−p ≤ k ≤ q). Then

(N_{α,β})^{2} = N_{α,β}N_{−α,−β} = \frac{(p + 1)q}{2} φ(α, α).  (8)

Notice that for roots α, β ∈ Δ, φ(α, α) = φ(β, β) if and only if β = w(α) for some element w of Weyl group W. Thus we see that for an element w of the Weyl group W we have

[ \begin{matrix}w(γ) \\ w(α)w(β) \end{matrix} ] = [ \begin{matrix}γ \\ αβ \end{matrix} ] .  (9)

Now we consider the case of classical simple Lie groups.

3.1. Case of K = SU(n + 1). The Killing form φ of 𝔰l(n + 1, ℂ) is

φ(X, Y) = 2(n + 1)trXY.

A root system for a Cartan subalgebra 𝔱^{ℂ} of 𝔰l(n + 1, ℂ) is given by

{ε_{i} − ε_{j} | 1 ≤ i, j ≤ n + 1, i ≠ j}

and positive roots are

{ε_{i} − ε_{j} | 1 ≤ i < j ≤ n + 1}.

The space of K-invariant metrics on M = K/T is given by

{∑_{i<j} x_{ij}( , )_{|𝔭_{ε_{i}−ε_{j}}} | x_{ij} ∈ ℝ^{+}} .

Notice that in this case every root has the same length and we have

φ(α, α) = \frac{1}{n + 1} for any root α. From (8) we see that

[ \begin{matrix}ε_{i} − ε_{j} \\ ε_{i} − ε_{k} ε_{k} − ε_{j} \end{matrix} ] = \frac{1}{n + 1} (k ≠ i, j)

otherwise 0.

Now, from (7), we see that

r_{ε_{i}−ε_{j}} = \frac{1}{2x_{ij}} + \frac{1}{4(n + 1)} ∑_{k≠i,j} (\frac{x_{ij}}{x_{ik}x_{kj}} − \frac{x_{ik}}{x_{ij}x_{kj}} − \frac{x_{jk}}{x_{ij}x_{ik}} ) ,  (10)

where we use conventions x_{pq} = x_{qp}.

Remark. Components of Ricci tensor have been obtained by Alekseevsky [2] and Arvanitoyeorgos [1] in more general case, that is, the case of generalized flag manifolds of SU(n + 1)-type by a different method.

We now discuss SU(n + 1)-invariant Einstein metric on a flag manifold SU(n + 1)/T. Notice that the bi-invariant metric is a SU(n + 1)-invariant Einstein metric in this case.

(i) Case of n = 2.

SU(3)-invariant Einstein metrics on SU(3)/T were studied by Alekseevsky [2]. In this case the equations of Einstein metrics are given by

\{ \begin{matrix}r_{12} & = \frac{1}{2x_{12}} + \frac{1}{12} (\frac{x_{12}}{x_{13}x_{23}} − \frac{x_{13}}{x_{12}x_{23}} − \frac{x_{23}}{x_{12}x_{13}} ) = k \\ r_{13} & = \frac{1}{2x_{13}} + \frac{1}{12} (\frac{x_{13}}{x_{12}x_{23}} − \frac{x_{12}}{x_{13}x_{23}} − \frac{x_{23}}{x_{13}x_{12}} ) = k \\ r_{23} & = \frac{1}{2x_{23}} + \frac{1}{12} (\frac{x_{23}}{x_{12}x_{13}} − \frac{x_{13}}{x_{23}x_{12}} − \frac{x_{12}}{x_{23}x_{13}} ) = k. \end{matrix}

We can solve these equations easily. SU(3)-invariant Einstein metrics are, up

to scalar multiple, the bi-invariant metric \{x_{12} = 1, x_{13} = 1, x_{23} = 1, k = \frac{5}{12}\}

or Kähler Einstein metrics \{x_{12} = 1, x_{13} = 1, x_{23} = 2, k = \frac{1}{3}\},

\{x_{12} = 1, x_{13} = 2, x_{23} = 1, k = \frac{1}{3}\} \{x_{12} = 2, x_{13} = 1, x_{23} = 1, k = \frac{1}{3}\}.

(ii) Case of x_{1i} = x_{12} (i = 2, ⋯, n) and x_{jk} = x_{23} (2 ≤ jk ≤ n).

In this case a SU(n + 1)-invariant Einstein metric on SU(n + 1)/T was discovered by Arvanitoyeorgos [1]. The equations for Einstein metrics are given by

\{ \begin{matrix}r_{12} & = \frac{1}{2x_{12}} + \frac{n − 1}{4(n + 1)} · \frac{−x_{23}}{x_{12}^{2}} = k \\ r_{23} & = \frac{1}{2x_{23}} + \frac{1}{4(n + 1)} (\frac{x_{23}}{x_{12}^{2}} − \frac{2}{x_{23}} − \frac{n − 2}{x_{23}}) = k. \end{matrix}

Solving a equation r_{12} = r_{23}, we obtain that ((n+2)x_{12} −nx_{23})(x_{12} −x_{23}) = 0. Thus a non-Kähler SU(n + 1)-invariant Einstein metric is given by {x_{1i} = n (2 ≤ i ≤ n), x_{jk} = n + 2 (2 ≤ j < k ≤ n), k = (n^{2} + n + 2)/(4(n + 1)n^{2})} for n ≥ 3.

(iii) Case of n = 2m − 1, (m > 2) and x_{ij} = a (1 ≤ i < j ≤ m), x_{ij} = b (i ≤ m < j), x_{ij} = c (m < i < j ≤ 2m)

In this case a SU(n + 1)-invariant Einstein metric on SU(n + 1)/T was discovered by Senda [6]. The equations for Einstein metrics are given by

\{ \begin{matrix}r_{12} & = \frac{1}{2x_{12}} + \frac{1}{8m} · (\frac{2 − m}{x_{12}} + m (−\frac{2}{x_{12}} + \frac{x_{12}}{x_{1n}^{2}} )) = k \\ r_{1n} & = \frac{1}{2x_{1n}} + \frac{1}{8m} · (\frac{(1 − m)x_{12}}{x_{1n}^{2}} + \frac{(1 − m)x_{n−1n}}{x_{1n}^{2}} ) = k \\ r_{n−1n} & = \frac{1}{2x_{n−1n}} + \frac{1}{8m} · (\frac{2 − m}{x_{n−1n}} + m (−\frac{2}{x_{n−1n}} + \frac{x_{n−1n}}{x_{1n}^{2}} )) = k. \end{matrix}

Solving the equation r_{12} = r_{n−1n}, we obtain that (x_{12} − x_{n−1n}) (x_{12} − 8 k x_{1n}^{2} + x_{n−1n}) = 0. Thus we obtain a non-Kähler SU(n + 1)-invariant Einstein metric given by {x_{ij} = m + 2 (1 ≤ i < j ≤ m), x_{ij} = m + 2 (m + 1 ≤ i < j), x_{ij} = 3m − 2 (i ≤ m < j), k = (5m^{2} − 5m + 2)/(4m(3m − 2)^{2})}.

(iv) Case of SU(4)-invariant Einstein metric on SU(4)/T.

This case was also considered by Senda[6]. We normalize our equations by setting k = 1. A system of equations are given by the following 6 equations:

f_{1} =x_{12}^{2}x_{13} x_{23} − x_{13} x_{14}^{2}x_{23} + x_{12}^{2}x_{14} x_{24} − x_{13}^{2}x_{14} x_{24}
+8 x_{13} x_{14} x_{23} x_{24} − 2 x_{12} x_{13} x_{14} x_{23} x_{24} − x_{14} x_{23}^{2}x_{24} − x_{13} x_{23} x_{24}^{2} = 0,
f_{2} =x_{12} x_{13}^{2}x_{23} − x_{12} x_{14}^{2}x_{23} − x_{12}^{2}x_{14} x_{34} + x_{13}^{2}x_{14} x_{34}
+8 x_{12} x_{14} x_{23} x_{34} − 2 x_{12} x_{13} x_{14} x_{23} x_{34} − x_{14} x_{23}^{2}x_{34} − x_{12} x_{23} x_{34}^{2} = 0,
f_{3} = − x_{12} x_{13}^{2}x_{24} + x_{12} x_{14}^{2}x_{24} − x_{12}^{2}x_{13} x_{34} + x_{13} x_{14}^{2}x_{34}
+8 x_{12} x_{13} x_{24} x_{34} − 2 x_{12} x_{13} x_{14} x_{24} x_{34} − x_{13} x_{24}^{2}x_{34} − x_{12} x_{24} x_{34}^{2} = 0,
f_{4} = − x_{12} x_{14} x_{23}^{2} + x_{12} x_{14} x_{24}^{2} − x_{12}^{2}x_{23} x_{34} + 8 x_{12} x_{14} x_{23} x_{34}
−x_{14}^{2}x_{23} x_{34} − 2 x_{12} x_{14} x_{23} x_{24} x_{34} + x_{23} x_{24}^{2}x_{34} − x_{12} x_{14} x_{34}^{2} = 0,
f_{5} = − x_{13} x_{14} x_{23}^{2} − x_{13}^{2}x_{23} x_{24} + 8 x_{13} x_{14} x_{23} x_{24} − x_{14}^{2}x_{23} x_{24}
−x_{13} x_{14} x_{24}^{2} − 2 x_{13} x_{14} x_{23} x_{24} x_{34} + x_{13} x_{14} x_{34}^{2} + x_{23} x_{24} x_{34}^{2} = 0,
f_{6} =x_{12} x_{13} x_{23}^{2} − x_{12} x_{13} x_{24}^{2} − x_{12}^{2}x_{24} x_{34} + 8 x_{12} x_{13} x_{24} x_{34}
−x_{13}^{2}x_{24} x_{34} − 2 x_{12} x_{13} x_{23} x_{24} x_{34} + x_{23}^{2}x_{24} x_{34} − x_{12} x_{13} x_{34}^{2} = 0.

To find non-zero solutions of these equations, we compute a Gröbner basis (cf. [5]) using algebraic manipulations in a computer system. We consider a polynomial ring R = ℚ[y, x_{34}, x_{24}, x_{23}, x_{14}, x_{13}, x_{12}] and an ideal I generated by

{f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, yx_{12}x_{13}x_{14}x_{23}x_{24}x_{34} − 1}.

We take a lexicographic order with y > x_{34} > x_{24} > x_{23} > x_{14} > x_{13} > x_{12} for a monomial ordering on R. Then we see that following polynomial h(x_{12}) is contained in the ideal I.

h(x_{12}) = (x_{12} − 2)(x_{12} − 3)^{3}(x_{12} − 4)(x_{12} − 6)(3 x_{12} − 7)(9 x_{12} − 35)(3 x_{12} − 8)

(9 x_{12}^{2} − 64 x_{12} + 128)

(9 x_{12}^{3} − 60 x_{12}^{2} + 176 x_{12} − 192)(3 x_{12}^{3} − 68 x_{12}^{2} + 464 x_{12} − 832)

(27 x_{12}^{6} − 360 x_{12}^{5} + 2040 x_{12}^{4} − 6528 x_{12}^{3} + 12608 x_{12}^{2} − 13824 x_{12} + 6656) = 0.

Moreover, we see that if {x_{12}, x_{13}, x_{14}, x_{23}, x_{24}, x_{34}} (x_{ij} > 0) satisfies the equations {f_{1} = 0, f_{2} = 0, f_{3} = 0, f_{4} = 0, f_{5} = 0, f_{6} = 0}, then either (x_{12} − 2)(x_{12} − 3)(x_{12} − 4)(x_{12} − 6)(3 x_{12} − 7)(9 x_{12} − 35)(3 x_{12} − 8) = 0 or there exist 2 pairs {x_{ij}, x_{kℓ}}, {x_{pq}, x_{rs}} such that x_{ij} = x_{kℓ} and x_{pq} = x_{rs}. In latter case, we may assume that x_{12} = x_{13}, x_{24} = x_{34} by a symmetric property of the system of equations. If x_{12} satisfies the equation

27 x_{12}^{6} − 360 x_{12}^{5} + 2040 x_{12}^{4} − 6528 x_{12}^{3} + 12608 x_{12}^{2} − 13824 x_{12} + 6656 = 0,

we get a Gröbner basis

{6656 − 13824 x_{12} + 12608 x_{12}^{2} − 6528 x_{12}^{3} + 2040 x_{12}^{4} − 360 x_{12}^{5} + 27 x_{12}^{6},

− x_{12} + x_{13},

− 49008 + 88768 x_{12} − 64144x_{12}^{2} + 25536x_{12}^{3} − 5643x_{12}^{4} + 513x_{12}^{5} + 96x_{14},

2496 − 4672 x_{12} + 3376 x_{12}^{2} − 1344 x_{12}^{3} + 297 x_{12}^{4} − 27 x_{12}^{5} + 32 x_{23},

− 3392 + 5264 x_{12} − 3672 x_{12}^{2} + 1428 x_{12}^{3} − 306 x_{12}^{4} + 27 x_{12}^{5} + 48 x_{24},

− 3392 + 5264 x_{12} − 3672 x_{12}^{2} + 1428 x_{12}^{3} − 306 x_{12}^{4} + 27 x_{12}^{5} + 48 x_{34}}.

We also see that x_{24} = x_{34}, x_{12} = x_{13} are the different solutions of the equation 6656 − 13824 x + 12608 x^{2} − 6528 x^{3} + 2040 x^{4} − 360 x^{5} + 27 x^{6} = 0, x_{14} is the positive solution of −192 + 176 x_{14} − 60 x_{14}^{2} + 9 x_{14}^{3} = 0 and x_{23} is the positive solution of −832 + 464 x_{23} − 68 x_{23}^{2} + 3 x_{23}^{3} = 0. If x_{12} satisfies the equation 9 x_{12}^{2}−64 x_{12}+128 = 0 then x_{12} is not a real number, and satisfies the equation 3 x_{12} − 8 = 0 then x_{23} satisfies the equation 9 x_{23}^{2} − 64 x_{23} + 128 = 0 and hence x_{23} is not a real number in this case. Other cases are either a Kähler Einstein metric or the case of (ii) above.

3.2. Case of K = SO(2n + 1). The Killing form φ of 𝔰𝔬(2n + 1, ℂ) is

φ(X, Y) = (2n − 1)trXY.

A root system for a Cartan subalgebra 𝔱^{ℂ} of 𝔰l(n + 1, ℂ) is given by

{ ±ε_{i} ± ε_{j}, ±ε_{k} | 1 ≤ i < j ≤ n, 1 ≤ k ≤ n }

and positive roots are

{ε_{i} − ε_{j}, ε_{i} + ε_{j}, ε_{k} | 1 ≤ i < j ≤ n, 1 ≤ k ≤ n }

The space of SO(2n + 1)-invariant metrics on M = SO(2n + 1)/T is given by

{∑_{i<j} x_{ij}( , )_{|𝔭_{ε_{i}−ε_{j}}} + ∑_{i<j} y_{ij}( , )_{|𝔭_{ε_{i}+ε_{j}}} + ∑_{i} z_{i}( , )_{|𝔭_{ε_{i}}} | x_{ij}, y_{ij}, z_{i} ∈ ℝ^{+}} .

Notice that φ(α, α) = \frac{1}{2n − 1} for α = ±ε_{i} ± ε_{j} and φ(α, α) = \frac{1}{2(2n − 1)} for

α = ±ε_{i}.

From (8) we see that

[ \begin{matrix}ε_{i} − ε_{k} \\ ε_{i} − ε_{j} ε_{j} − ε_{k} \end{matrix} ] = [ \begin{matrix}ε_{i} − ε_{k} \\ ε_{i} + ε_{j} ε_{j} + ε_{k} \end{matrix} ]

= [ \begin{matrix}ε_{j} \\ ε_{i} − ε_{j} ε_{i} \end{matrix} ] = [ \begin{matrix}ε_{j} \\ ε_{i} + ε_{j} ε_{i} \end{matrix} ] = \frac{1}{2n − 1}

otherwise 0.

Now, from (7), we see that

r_{ε_{i}−ε_{j}} = \frac{1}{2x_{ij}} + \frac{1}{4(2n − 1)} (∑_{k≠i,j} (\frac{x_{ij}}{x_{ik}x_{kj}} − \frac{x_{ik}}{x_{ij}x_{kj}} − \frac{x_{jk}}{x_{ij}x_{ik}} )

+ ∑_{k≠i,j} (\frac{x_{ij}}{y_{ik}y_{kj}} − \frac{y_{ik}}{x_{ij}y_{kj}} − \frac{y_{jk}}{x_{ij}y_{ik}} ) + (\frac{x_{ij}}{z_{i}z_{j}} − \frac{z_{i}}{x_{ij}z_{j}} − \frac{z_{j}}{x_{ij}z_{i}} ))

r_{ε_{i}+ε_{j}} = \frac{1}{2y_{ij}} + \frac{1}{4(2n − 1)} (∑_{k≠i,j} (\frac{y_{ij}}{x_{ik}y_{kj}} − \frac{x_{ik}}{y_{ij}y_{kj}} − \frac{y_{jk}}{y_{ij}x_{ik}} )

+ ∑_{k≠i,j} (\frac{y_{ij}}{y_{ik}y_{kj}} − \frac{y_{ik}}{y_{ij}y_{kj}} − \frac{y_{jk}}{y_{ij}y_{ik}} ) + (\frac{y_{ij}}{z_{i}z_{j}} − \frac{z_{i}}{y_{ij}z_{j}} − \frac{z_{j}}{y_{ij}z_{i}} ))

r_{ε_{i}} = \frac{1}{2z_{i}} + \frac{1}{4(2n − 1)} (∑_{k≠i} (\frac{z_{i}}{z_{k}x_{ki}} − \frac{x_{ki}}{z_{i}z_{k}} − \frac{z_{k}}{z_{i}x_{ki}} )

+ ∑_{k≠i} (\frac{z_{i}}{z_{k}y_{ki}} − \frac{y_{ki}}{z_{i}z_{k}} − \frac{z_{k}}{z_{i}y_{ki}} ))

where we use conventions x_{pq} = x_{qp}, y_{pq} = y_{qp}.

(i) Case of n = 2, that is, SO(5)-invariant Einstein metrics on SO(5)/T. In this case the equations of Einstein metrics are given by

r_{ε_{1}−ε_{2}} = \frac{1}{2 x_{12}} + \frac{1}{12}(\frac{x_{12}}{z_{1} z_{2}} − \frac{z_{1}}{x_{12} z_{2}} − \frac{z_{2}}{x_{12} z_{1}} ) = k
r_{ε_{1}+ε_{2}} = \frac{1}{2 y_{12}} + \frac{1}{12}(\frac{y_{12}}{z_{1} z_{2}} − \frac{z_{1}}{y_{12} z_{2}} − \frac{z_{2}}{y_{12} z_{1}} ) = k
r_{ε_{1}} = \frac{1}{2 z_{1}} + \frac{1}{12}(\frac{z_{1}}{x_{12} z_{2}} + \frac{z_{1}}{y_{12} z_{2}} − \frac{z_{2}}{x_{12} z_{1}} − \frac{z_{2}}{y_{12} z_{1}} − \frac{x_{12}}{z_{1} z_{2}} − \frac{y_{12}}{z_{1} z_{2}} ) = k
r_{ε_{2}} = \frac{1}{2 z_{2}} + \frac{1}{12}(\frac{z_{2}}{x_{12} z_{1}} + \frac{z_{2}}{y_{12} z_{1}} − \frac{x_{12}}{z_{1} z_{2}} − \frac{y_{12}}{z_{1} z_{2}} − \frac{z_{1}}{x_{12} z_{2}} − \frac{z_{1}}{y_{12} z_{2}} ) = k.

We normalize our equations by setting z_{1} = 1. A system of equations is given by the following 4 equations:

f_{1} = x_{12}^{2} − 1 + 6 z_{2} − 12 kx_{12}z_{2} − z_{2}^{2} = 0
f_{2} = y_{12}^{2} − 1 + 6 z_{2} − 12 ky_{12}z_{2} − z_{2}^{2} = 0
f_{3} = −x_{12}^{2}y_{12} − x_{12}y_{12}^{2} + x_{12} + y_{12} + 6 x_{12}y_{12}z_{2}
−12 kx_{12}y_{12}z_{2} − x_{12}z_{2}^{2} − y_{12}z_{2}^{2} = 0
f_{4} = −x_{12}^{2}y_{12} − x_{12}y_{12}^{2} + 6 x_{12}y_{12} − x_{12} − y_{12}
−12 kx_{12}y_{12}z_{2} + x_{12}z_{2}^{2} + y_{12}z_{2}^{2} = 0.

To find non-zero solutions of these equations, we compute a Gröbner basis using algebraic manipulations in a computer system. We consider a polynomial ring R = ℚ[k, z_{2}, y_{12}, x_{12}] and an ideal I generated by {f_{1}, f_{2}, f_{3}, f_{4}}. We take a lexicographic order with k > x_{12} > y_{12} > z_{2} for a monomial ordering on R. Then we see that following polynomial is contained in the ideal I.

(−3 + z_{2}) (−1 + z_{2}) (−1 + 3 z_{2}) (1 − 6 z_{2} + z_{2}^{2}) (5 − 14 z_{2} + 5 z_{2}^{2}) = 0.

If z_{2} satisfies the equation (1 − 6 z_{2} + z_{2}^{2}) = 0 then we see that y_{12} = 0. If z_{2} satisfies the equation (5 − 14 z_{2} + 5 z_{2}^{2}) = 0 then we see that

{x_{12} = y_{12}, −2 + 3 x_{12} − 2 z_{2} = 0, −19 + 36 k + 5 z_{2} = 0}.

Thus we get positive solutions

z_{1} = 1, x_{12} = y_{12} = \frac{24 + 4\sqrt{6}}{15}, z_{2} = \frac{7 + 2\sqrt{6}}{5}, k = \frac{6 − \sqrt{6}}{18}
z_{1} = 1, x_{12} = y_{12} = \frac{24 − 4\sqrt{6}}{15}, z_{2} = \frac{7 − 2\sqrt{6}}{5}, k = \frac{6 + \sqrt{6}}{18}.

The other cases are Kähler Einstein metrics given by

\{z_{1} = 1, x_{12} = 2, z_{2} = 3, y_{12} = 4, k = \frac{1}{6}\}
\{z_{1} = 1, x_{12} = 4, z_{2} = 3, y_{12} = 2, k = \frac{1}{6}\}
\{z_{1} = 3, x_{12} = 2, z_{2} = 1, y_{12} = 4, k = \frac{1}{6}\}
\{z_{1} = 3, x_{12} = 4, z_{2} = 1, y_{12} = 2, k = \frac{1}{6}\}.

(ii) Case of x_{ij} = x, y_{ij} = y, z_{i} = z. In this case the equations for Einstein metrics are given by

\{ \begin{matrix}f_{1} & = \frac{1}{2x} + \frac{1}{4(2n − 1)} · (\frac{−2}{x} + (n − 2) (\frac{−3}{x} + \frac{x}{y^{2}} ) + \frac{x}{z^{2}} ) − k = 0 \\ f_{2} & = \frac{1}{2y} + \frac{1}{4(2n − 1)} · (\frac{−2}{y} + (n − 2) (\frac{−1}{y} − \frac{x}{y^{2}} ) + \frac{y}{z^{2}} ) − k = 0 \\ f_{3} & = \frac{1}{2z} + \frac{1}{4(2n − 1)} · (n − 1) (\frac{−x}{z^{2}} + \frac{−y}{z^{2}} ) − k = 0. \end{matrix}

We normalize our equations by setting x = 1. A system of equations is given by the following 3 equations:

g_{1} = y^{2} − 2z^{2} + nz^{2} + 2y^{2}z^{2} + 4ky^{2}z^{2} + ny^{2}z^{2} − 8kny^{2}z^{2} = 0
g_{2} = y^{3} + 2z^{2} − nz^{2} − 2yz^{2} + 3nyz^{2} + 4ky^{2}z^{2} − 8kny^{2}z^{2} = 0
g_{3} = 1 − n + y − ny − 2z + 4nz + 4kz^{2} − 8knz^{2} = 0.

To find non-zero solutions of these equations, we compute a Gröbner basis using algebraic manipulations in a computer system. We consider a polynomial ring R = ℚ[k, y, z] and an ideal I generated by {g_{1}, g_{2}, g_{3}}. We take a lexicographic order with k > z > y for a monomial ordering on R. Then we see that following polynomial h(y) is contained in the ideal I.

h(y) = ((−n^{4} − 4n^{3} − 4n^{2})y^{4} + (2n^{4} + 4n^{3} − 8n^{2} − 12n + 8)y^{3}
+ (3n^{4} − 18n^{3} + 8n^{2})y^{2} + (−4n^{4} + 42n^{3} − 100n^{2} + 72n − 16)y
− (4n^{4} − 20n^{3} + 33n^{2} − 20n + 4)) (y − 1).

If y = 1 then we see that −1 + 2n + 2(1 − 2n)z + 2nz^{2} = 0 and this equation has no real solutions in this case. Consider the case

p_{1}(y) =(−n^{4} − 4n^{3} − 4n^{2})y^{4} + (2n^{4} + 4n^{3} − 8n^{2} − 12n + 8)y^{3}
+(3n^{4} − 18n^{3} + 8n^{2})y^{2} + (−4n^{4} + 42n^{3} − 100n^{2} + 72n − 16)y
−(4n^{4} − 20n^{3} + 33n^{2} − 20n + 4) = 0.

Note that

p_{1}(1) = −12 + 80 n − 137 n^{2} + 44 n^{3} − 4 n^{4}

= −144 − 268 (n − 6) − 209 (n − 6)^{2} − 52 (n − 6)^{3} − 4 (n − 6)^{4} < 0,

p_{1}(2 − 6/n) = \frac{1}{n^{3}} (−1728 − 864 n + 384 n^{2} + 316 n^{3} − 124 n^{4} + 7 n^{5})

= \frac{1}{n^{3}} (176337 + 321831 (n − 15) + 83454 (n − 15)^{2} + 8626 (n − 15)^{3}

+ 401 (n − 15)^{4} + 7 (n − 15)^{5}) > 0, (for n ≥ 15)

p_{1}(2) = 28 + 68 n − 329 n^{2} < 0.

Thus p_{1}(y) = 0 has solutions y_{0}, y_{1} with 1 < y_{0} < 2−6/n and 2−6/n < y_{1} < 2 for n ≥ 15.

Remark. For 14 ≥ n ≥ 12 we can also show that p_{1}(y) = 0 has solutions y_{0}, y_{1} with 1 < y_{0} < y_{1} < 2 by a computation.

Further, a Gröbner basis is given by

{p_{1} =(−n^{4} − 4n^{3} − 4n^{2}) y^{4} + (2n^{4} + 4n^{3} − 8n^{2} − 12n + 8) y^{3}
+(3n^{4} − 18n^{3} + 8n^{2}) y^{2} + (−4n^{4} + 42n^{3} − 100n^{2} + 72n − 16) y
−(4n^{4} − 20n^{3} + 33n^{2} − 20n + 4),
p_{2} =8 − 28 n + 26 n^{2} − 3 n^{3} − 2 n^{4} + (16 − 52 n + 40 n^{2} + 5 n^{3} − 3 n^{4}) y
+ (−8 + 12 n − 4 n^{3}) y^{2} + (4 n^{2} + 4 n^{3} + n^{4}) y^{3}
+ (−24 + 80 n − 74 n^{2} + 20 n^{3}) z,
p_{3} =104 − 692 n + 1734 n^{2} − 2015 n^{3} + 1084 n^{4} − 243 n^{5} + 14 n^{6}
+k (16 − 144 n + 516 n^{2} − 928 n^{3} + 864 n^{4} − 384 n^{5} + 64 n^{6})
+ (16 − 64 n + 12 n^{2} + 236 n^{3} − 316 n^{4} + 109 n^{5} − 8 n^{6}) y
+ (−48 + 176 n − 148 n^{2} − 68 n^{3} + 82 n^{4} + 7 n^{5} − 10 n^{6}) y^{2}
+ (24 n^{2} − 28 n^{3} − 30 n^{4} + 3 n^{5} + 4 n^{6}) y^{3}}.

By computing resultant of p_{1}, p_{2} with respect to y, we obtain that

q =(−2 + n) (−1 + 2 n)^{2} − 8 (−2 + n) (−1 + 2 n)^{2} z
+(−8 + 60 n − 98 n^{2} + 15 n^{3} + 12 n^{4}) z^{2}
−2 (2 + n) (−1 + 2 n) (−6 − 3 n + 4 n^{2}) z^{3} + n (2 + n)^{2} (−6 + 5 n) z^{4}.

Then it is easy to see that q = 0 has no negative solutions. Thus we see that from p_{2} = 0 the solutions z_{0}, z_{1} corresponding to y_{0}, y_{1} are positive. Now we claim that the solutions k_{0}, k_{1} corresponding to y_{0}, y_{1} obtained from p_{3} = 0 are also positive. Consider the following polynomial reduced from p_{3} = 0.

w(y) =104 − 692 n + 1734 n^{2} − 2015 n^{3} + 1084 n^{4} − 243 n^{5} + 14 n^{6}
+ (16 − 64 n + 12 n^{2} + 236 n^{3} − 316 n^{4} + 109 n^{5} − 8 n^{6}) y
+ (−48 + 176 n − 148 n^{2} − 68 n^{3} + 82 n^{4} + 7 n^{5} − 10 n^{6}) y^{2}
+ (24 n^{2} − 28 n^{3} − 30 n^{4} + 3 n^{5} + 4 n^{6}) y^{3}.

We claim that w(y) < 0 for 1 < y < 2. Note that

w(−2) = −120 + 140 n + 926 n^{2} − 2535 n^{3} + 2284 n^{4} − 457 n^{5} − 42 n^{6}
= −16476 − 62398 (−3 + n) − 72973 (−3 + n)^{2} − 38937 (−3 + n)^{3}
− 10241 (−3 + n)^{4} − 1213 (−3 + n)^{5} − 42 (−3 + n)^{6} < 0,
w(0) = (−2 + n)^{2} (−1 + 2 n) (−26 + 95 n − 90 n^{2} + 7 n^{3}) > 0
w(1) = 72 − 580 n + 1622 n^{2} − 1875 n^{3} + 820 n^{4} − 124 n^{5} < 0
w(2) = −56 − 116 n + 1358 n^{2} − 2039 n^{3} + 540 n^{4} + 27 n^{5} − 10 n^{6}
= −59864 − 224242 (−7 + n) − 150241 (−7 + n)^{2} − 42289 (−7 + n)^{3}
− 5865 (−7 + n)^{4} − 393 (−7 + n)^{5} − 10 (−7 + n)^{6} < 0
w(3) = −280 + 700 n + 1086 n^{2} − 2675 n^{3} + 64 n^{4} + 228 n^{5} + 8 n^{6} > 0.

Thus w(y) = 0 has solutions in intervals −2 < y < 0, 0 < y < 1, 2 < y < 3 for n ≥ 7. Since w(y) = 0 has just 3 roots, we get our claim. Therefore we have non-Kähler SO(2n + 1)-invariant Einstein metrics on the flag manifold SO(2n + 1)/T given by {k_{0}, y_{0}, z_{0}} and {k_{1}, y_{1}, z_{1}} for n ≥ 12.

3.3. Case of K = Sp(n). The Killing form φ of 𝔰𝔭(n, ℂ) is

φ(X, Y) = 2(n + 1)trXY.

A root system for a Cartan subalgebra 𝔱^{ℂ} of 𝔰p(n, ℂ) is given by

{±2ε_{k}, ±ε_{i} ± ε_{j} | 1 ≤ k ≤ n, 1 ≤ i < j ≤ n}

and positive roots are

{2ε_{k}, ε_{i} − ε_{j}, ε_{i} + ε_{j} | 1 ≤ k ≤ n, 1 ≤ i < j ≤ n}.

The space of Sp(n)-invariant metric on M = Sp(n)/T is given by

{∑_{i<j} x_{ij}( , )_{|𝔭_{ε_{i}−ε_{j}}} + ∑_{i<j} y_{ij}( , )_{|𝔭_{ε_{i}+ε_{j}}} + ∑_{i} z_{i}( , )_{|𝔭_{2ε_{i}}} | x_{ij}, y_{ij}, z_{i} ∈ ℝ^{+}} .

Notice that φ(α, α) = \frac{1}{2(n + 1)} for α = ±ε_{i} ± ε_{j} and φ(α, α) = \frac{1}{(n + 1)} for

α = ±2ε_{i}. From (8) we see that

[ \begin{matrix}ε_{i} − ε_{k} \\ ε_{i} − ε_{j} ε_{j} − ε_{k} \end{matrix} ] = [ \begin{matrix}ε_{i} + ε_{k} \\ ε_{i} − ε_{j} ε_{j} + ε_{k} \end{matrix} ] = \frac{1}{2(n + 1)}

[ \begin{matrix}ε_{i} + ε_{j} \\ ε_{i} − ε_{j} 2ε_{j} \end{matrix} ] = \frac{1}{(n + 1)}

otherwise 0.

Now, from (7), we see that

r_{ε_{i}−ε_{j}} = \frac{1}{2x_{ij}} + \frac{1}{8(n + 1)} (∑_{k≠i,j} (\frac{x_{ij}}{x_{ik}x_{kj}} − \frac{x_{ik}}{x_{ij}x_{kj}} − \frac{x_{kj}}{x_{ij}x_{ik}} )

+ ∑_{k≠i,j} (\frac{x_{ij}}{y_{ik}y_{kj}} − \frac{y_{ik}}{x_{ij}y_{kj}} − \frac{y_{kj}}{x_{ij}y_{ik}} ))

+ \frac{1}{4(n + 1)} ((\frac{x_{ij}}{z_{i}y_{ij}} − \frac{z_{i}}{x_{ij}y_{ij}} − \frac{y_{ij}}{x_{ij}z_{i}} ) + (\frac{x_{ij}}{z_{j}y_{ij}} − \frac{z_{j}}{x_{ij}y_{ij}} − \frac{y_{ij}}{x_{ij}z_{j}} ))

r_{ε_{i}+ε_{j}} = \frac{1}{2y_{ij}} + \frac{1}{8(n + 1)} (∑_{k≠i,j} (\frac{y_{ij}}{x_{ik}y_{kj}} − \frac{x_{ik}}{y_{ij}y_{kj}} − \frac{y_{kj}}{y_{ij}x_{ik}} )

+ ∑_{k≠i,j} (\frac{y_{ij}}{y_{ik}x_{kj}} − \frac{y_{ik}}{y_{ij}x_{kj}} − \frac{x_{kj}}{y_{ij}y_{ik}} ))

+ \frac{1}{4(n + 1)} ((\frac{y_{ij}}{z_{i}x_{ij}} − \frac{z_{i}}{x_{ij}y_{ij}} − \frac{x_{ij}}{y_{ij}z_{i}} ) + (\frac{y_{ij}}{z_{j}x_{ij}} − \frac{z_{j}}{x_{ij}y_{ij}} − \frac{x_{ij}}{y_{ij}z_{j}} ))

r_{2ε_{i}} = \frac{1}{2z_{i}} + \frac{1}{4(n + 1)} ∑_{k≠i} (\frac{z_{i}}{x_{ik}y_{ik}} − \frac{x_{ik}}{z_{i}y_{ik}} − \frac{y_{ik}}{z_{i}x_{ik}} )

where we use conventions x_{pq} = x_{qp}, y_{pq} = y_{qp}.

We consider the case of x_{ij} = y_{ij} = x, z_{i} = z. In this case the equations for Einstein metrics are given by

\{ \begin{matrix}f_{1} & = \frac{1}{2 x} − \frac{(n − 2)}{4 (n + 1)} \frac{1}{x} − \frac{1}{2 (n + 1)} \frac{z}{x^{2}} − k \\ f_{2} & = \frac{1}{2 z} + \frac{(n − 1)}{4 (n + 1)} (−\frac{2}{z} + \frac{z}{x^{2}} ) − k \end{matrix}

We normalize our equations by setting x = 1. A system of equations is given by the following 2 equations:

4 − 4k + n − 4kn − 2z = 0
4 − 4kz − 4knz − z^{2} + nz^{2} = 0.

Solving equations, we obtain that 4 − (4 + n)z + (n + 1)z^{2} = 0. Thus we get non-Kähler Sp(n)-invariant Einstein metrics given by

\{x = y = 1, z = \frac{4 + n − \sqrt{(n − 8) n}}{2 (n + 1)} k = \frac{4n + n^{2} + \sqrt{(n − 8)n}}{4(n + 1)^{2}}\} ,

\{x = y = 1, z = \frac{4 + n + \sqrt{(n − 8) n}}{2 (n + 1)} k = \frac{4n + n^{2} − \sqrt{(n − 8)n}}{4(n + 1)^{2}}\}

for n ≥ 8.

3.4. Case of K = SO(2n). The Killing form φ of 𝔰𝔭(n, ℂ) is

φ(X, Y) = 2(n − 1)trXY.

A root system for a Cartan subalgebra 𝔱^{ℂ} of 𝔰o(2n, ℂ) is given by

{±ε_{i} ± ε_{j} | 1 ≤ i < j ≤ n}

and positive roots are

{ε_{i} − ε_{j}, ε_{i} + ε_{j} | 1 ≤ i < j ≤ n}.

The space of SO(2n)-invariant metric on M = SO(2n)/T are given by

{⟨ , ⟩ = ∑_{i<j} x_{ij}( , )_{|𝔭_{ε_{i}−ε_{j}}} + ∑_{i<j} y_{ij}( , )_{|𝔭_{ε_{i}+ε_{j}}} | x_{ij}, y_{ij} ∈ ℝ^{+}} .

Notice that in this case every root has the same length and we have

φ(α, α) = \frac{1}{2(n − 1)}

for any root α. From (8) we see that

[ \begin{matrix}ε_{i} − ε_{k} \\ ε_{i} − ε_{j} ε_{j} − ε_{k} \end{matrix} ] = [ \begin{matrix}ε_{i} + ε_{k} \\ ε_{i} − ε_{j} ε_{j} + ε_{k} \end{matrix} ] = \frac{1}{2(n − 1)} (k ≠ i, j)

otherwise 0.

Now, from (7), we see that

r_{ε_{i}−ε_{j}} = \frac{1}{2x_{ij}} + \frac{1}{8(n − 1)} (∑_{k≠i,j} (\frac{x_{ij}}{x_{ik}x_{kj}} − \frac{x_{ik}}{x_{ij}x_{kj}} − \frac{x_{jk}}{x_{ij}x_{ik}} )

+ ∑_{k≠i,j} (\frac{x_{ij}}{y_{ik}y_{kj}} − \frac{y_{ik}}{x_{ij}y_{kj}} − \frac{y_{jk}}{x_{ij}y_{ik}} ))

r_{ε_{i}+ε_{j}} = \frac{1}{2y_{ij}} + \frac{1}{8(n − 1)} (∑_{k≠i,j} (\frac{y_{ij}}{x_{ik}y_{jk}} − \frac{x_{ik}}{y_{ij}y_{jk}} − \frac{y_{jk}}{y_{ij}x_{ik}} )

+ ∑_{k≠i,j} (\frac{y_{ij}}{y_{ik}x_{jk}} − \frac{y_{ik}}{y_{ij}x_{jk}} − \frac{x_{jk}}{y_{ij}y_{ik}} ))

where we use conventions x_{pq} = x_{qp}, y_{pq} = y_{qp}.

Remark. Components of Ricci tensor have been obtained by Arvanitoyeorgos [1] in more general case, that is, the case of generalized flag manifolds of SO(2n)-type by a different method.

87 HOMOGENEOUS EINSTEIN METRICS

We consider the case of x_{ij} = x, y_{ij} = y. In this case the equations for Einstein metrics are given by

\{ \begin{matrix}f_{1} & = \frac{1}{2 x} + \frac{(n − 2)}{8 (n − 1)} (\frac{−3}{x} + \frac{x}{y^{2}} ) − k \\ f_{2} & = \frac{1}{2 y} + \frac{(n − 2)}{4 (n − 1)} \frac{1}{2y} − k. \end{matrix}

Thus we have

\begin{matrix}− 2 x^{2} + n x^{2} + 2 y^{2} + n y^{2} + 8 k x y^{2} − 8 k n x y^{2} = 0 \\ 2 x − n x − 2 y + 2 n y + 4 k y^{2} − 4 k n y^{2} = 0. \end{matrix}

Solving equations, we obtain that (x − y)(3 (2 − n) x + (2 + n) y) = 0. If x = y then we see that the metric is bi-invariant. If (3 (2 − n) x+(2 + n) y) = 0 then we get a non-Kähler SO(2n)-invariant Einstein metric given by

x = n + 2, y = 3(n − 2), k = \frac{5n − 8}{36(n − 2)(n − 1)}.

References

[1] A. Arvanitoyeorgos, New invariant Einstein metrics on generalized flag manifolds, Trans. Amer. Math. Soc. 337 (1993), 981 – 995.

[2] D. V. Alekseevsky, Homogeneous Einstein metrics, Differential Geometry and its Applications. Communications. (Proceedings of Brno Conference), Univ. of J. E. Purkyne-Czechoslovakia, (1987), 1 – 12.

[3] A.L. Besse, “Einstein Manifolds”, Springer-Verlag 1987.

[4] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces I , Amer. J. Math. 80 (1958), 458–538.

[5] D. Cox, J. Little and D. O’Shea, “Ideals, Varieties, and Algorithms”, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag 1992.

[6] R. Senda, Improved Gröbner basis algorithms and its applications, Master thesis, Osaka University (1997), 1–39 (in Japanese).

[7] M. Wang and W. Ziller, Existence and non-existence of homogeneous Einstein metrics, Invent math. 84 (1986), 177 – 194.

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043 JAPAN E-mail address: sakane@math.sci.osaka-u.ac.jp

Received July 27, 1999