HOMOGENEOUS EINSTEIN METRICS ON G 2 /T

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141 Number 7 July 2013 Pages S (2013) Article electronically published on March HOMOGENEOUS EINSTEIN METRICS ON G 2 /T ANDREAS ARVANITOYEORGOS IOANNIS CHRYSIKOS AND YUSUKE SAKANE (Communicated by Lei Ni) Abstract. We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold M G 2 /T. By computing agröbner basis for a system of polynomials on six variables we prove that this manifold admits exactly two non-kähler invariant Einstein metrics. Thus G 2 /T turns out to be the first known example of an exceptional full flag manifold which admits a non-kähler and not normal homogeneous Einstein metric. Introduction A Riemannian manifold (Mg) is called Einstein if the metric g has constant Ricci curvature that is Ric g λg for some λ R whereric g is the Ricci tensor corresponding to g. The question whether M carries an Einstein metric and if so how many is a fundamental one in Riemannian geometry. A number of interesting results in geometry have been motivated and inspired by this hard problem. The Einstein equation is a nonlinear second order PDE and a good understanding of its solutions in the general case seems far from being attained. It becomes more manageable in the homogeneous setting. Most known examples of compact simply connected Einstein manifolds are homogeneous. In the homogeneous case the Einstein equation reduces to a system of algebraic equations for which we are looking for positive solutions. For some cases such solutions have been obtained explicitly. We refer to [14] and the references therein for more details on compact homogeneous Einstein manifolds. The low-dimensional cases were also examined in [5]. Let K be a compact connected and semisimple Lie group. A full flag manifold is a compact homogeneous space of the form K/T wheret is a maximal torus in K. It is known that K/T admits a unique (up to isometry) K-invariant Kähler-Einstein metric (cf. [13]). Non-Kähler homogeneous Einstein metrics on full flag manifolds corresponding to classical Lie groups have been studied by several authors (cf. [2] [16] [10]). Although various existence results of homogeneous Einstein metrics on these spaces have been obtained the classification of such metrics is a demanding task which remains widely open. In the present paper we study the classification problem of homogeneous Einstein metrics on the full flag manifold G 2 /T. The isotropy representation of this space decomposes into six inequivalent irreducible submodules. Received by the editors October Mathematics Subject Classification. Primary 53C25; Secondary 53C30. Key words and phrases. Homogeneous Einstein metric full flag manifold exceptional Lie group G 2. The third author was supported by Grant-in-Aid for Scientific Research (C) c 2013 American Mathematical Society

2 2486 A. ARVANITOYEORGOS I. CHRYSIKOS AND Y. SAKANE There are three (nonisomorphic) flag manifolds corresponding to the exceptional Lie group G 2 since there are exactly three different ways to paint black the simple roots in the Dynkin diagram of g 2 as shown in Figure 1 (for the classification of generalized flag manifolds in terms of painted Dynkin diagrams we refer to [1] or [6]). α 1 α > 2 G 2 (α 2 ) α 1 α > 2 G 2 (α 1 ) α 1 α > 2 G 2 /T Figure 1. The painted Dynkin diagrams corresponding to G 2 If we paint black one simple root in the Dynkin diagram of g 2 then we obtain a flag manifold of the form G 2 /U(2) with two or three isotropy summands depending on the height of this simple root. Recall that for g 2 we can choose a set of simple roots by Π M {α 1 α 2 } with (α 1 α 1 )3(α 2 α 2 ) so the highest root has the form α 2α 1 +3α 2 (see Section 5). Thus the flag manifold G 2 (α 2 ) in Figure 1 has two isotropy summands and U(2) is represented by the short root of g 2. For this space all G 2 -invariant Einstein metrics have been obtained explicitly in [3]. The second flag manifold G 2 (α 1 ) in Figure 1 has three isotropy summands and the isotropy group U(2) is represented by the long root of g 2. For this space the G 2 -invariant Einstein metrics were studied in [12] [2]. The full flag manifold G 2 /T wheret U(1) U(1) is a maximal torus in G 2 is obtained by painting black both simple roots in the Dynkin diagram of g 2. According to [19] a full flag manifold K/T is a normal homogeneous Einstein manifold if and only if all roots of K have the same length and in this case the normal metric of K/T is never Kähler. Therefore if K is an exceptional Lie group then K/T is a normal homogeneous Einstein manifold if and only if K {E 6 E 7 E 8 }sog 2 /T is not normal. Our main result is the following: Theorem A. The full flag manifold G 2 /T admits exactly three G 2 -invariant Einstein metrics (up to isometry). ThereisauniqueKähler-Einstein metric given (up to a scalar) by g ( ) and the other two are non-kähler. The approximate values of these invariant metrics are given in Theorem 4.1. As a consequence of Theorem A G 2 /T is the first known example of an exceptional full flag manifold which admits a non-kähler and not normal homogeneous Einstein metric. Also the present work on G 2 /T is the first attempt towards the classification of homogeneous Einstein metrics on generalized flag manifolds with six isotropy summands. The paper is organised as follows: In Section 2 we recall the Lie-theoretic description of a full flag manifold K/T of a compact and connected semisimple Lie group K and we study its isotropy representation. Next following [16] we describe the structure constants of K/T relative to the associated isotropy decomposition and we give the expression of the Ricci tensor of a K-invariant metric on K/T. In Section 3 we consider the exceptional full flag manifold G 2 /T andwegiveitslietheoretic description. Then we construct the Einstein equation for a G 2 -invariant Riemannian metric. In the last section we give the corresponding polynomial system and by computing a Gröbner basis for this system we prove Theorem A and obtain the full classification of homogeneous Einstein metrics on G 2 /T.

3 HOMOGENEOUS EINSTEIN METRICS ON G 2 /T Full flag manifolds Let K/T be a full flag manifold where T is a maximal torus of a compact semisimple Lie group K. We will give a characterization of K/T in terms of root system theory and we will describe some topics of the associated Kähler geometry. Then we study the isotropy representation of K/T and we give the expression of the Ricci tensor for a K-invariant metric on K/T Lie-theoretic description of K/T. Assume that dim R T rankg l. We denote by k t the Lie algebras of K and T respectively and by k C k ik t C t it the corresponding complexifications. Let t and t C be the dual spaces of t and t C respectively. The subalgebra t C is a Cartan subalgebra of the complex semisimple Lie algebra k C and thus we obtain the root space decomposition k C t C α R kα C where R is the root system of k C relative to t C and k α C are the corresponding root spaces. Recall that by C-linearity a root α R is completely determined by its restriction to either t or it. Since the Killing form B of k C is nondegenerate for any λ t C we define H λ it by the equation B(H λ H)λ(H) for all H t C. Let it denote the real linear subspace of t C which consists of all λ t C such that the restriction λ t has values in ir. Note that the restriction map λ λ it defines an isomorphism from it onto the real linear dual space it which allows us to identify these spaces. Then it is well known that R spans it and that R is a finite subset of it \{0}. Thusifα R thenα it. Let ( ) be the bilinear form on t C induced by the Killing form that is (λ μ) B(H λ H μ ) for any λ μ t C. Then since B is negative definite on t and positive definite on it the restriction of ( )onit is a positive definite inner product. The weight lattice of k C with respect to t C is given by Λ {λ it 2(λ α) (α α) Z for all α R}. Let Π {α 1...α l } be a simple root system of R and let R + be the set of all positive roots with respect to Π. Consider the fundamental weights corresponding to Π that is Λ 1...Λ l Λ such that (2.1) 2(Λ i α j ) (α j α j ) δ ij (1 i j l). Then {Λ 1...Λ l } forms a Z-basis for the weight lattice Λ and since it it it is it l i1 RΛ i. In the weight lattice Λ there is a distinguished subset Λ + given by Λ + {λ Λ (λ α) > 0 for any α R + }. One can see that Λ + Λ C(Π) where C(Π) is the fundamental Weyl chamber corresponding to Π given by C(Π) {λ it (λ α i ) > 0 for all α i Π}. The elements of Λ + are usually called dominant weights relative to R + and any dominant weight can be expressed as a linear combination of the fundamental weights with nonnegative coefficients. For example set δ 1 2 α R α + it. Then δ l i1 Λ i and thus δ Λ + (cf. [8 p. 168]). We now define the complex Lie subalgebras of k C by n α R + kα C and b t C n. One can easily show that n is a nilpotent ideal of k C and that b is a maximal solvable Lie subalgebra of k C ; i.e. it is a Borel subalgebra. Let K C denote the complex simply connected semisimple Lie group whose Lie algebra is k C. Then the connected subgroup B K C with Lie algebra b is a Borel subgroup of K C and K C /B K/T as C -manifolds.

4 2488 A. ARVANITOYEORGOS I. CHRYSIKOS AND Y. SAKANE Since K C is a complex Lie group and B a closed complex subgroup the quotient K C /B admits a K-invariant complex structure. Furthermore the K-invariant complex structures on K C /B K/T are in 1-1 correspondence with different choices of positive roots for k C (cf. [7]). Since the Weyl group W (R) of the root system of k C acts transitively on the sets of systems of positive roots all these complex structures are equivalent. Moreover the following holds: Theorem 2.1 ([7] [17]). There is a 1-1 correspondence between K-invariant Kähler metrics on K C /B and dominant weights in Λ +. In particular the K-invariant Kähler metric on K C /B corresponding to 2δ is a Kähler-Einstein metric. According to [7 p. 504] a full flag manifold admits a unique (up to equivalence) invariant complex structure hence a unique (up to scale) Kähler Einstein metric (cf. also [13]). This Kähler Einstein metric will be computed in Section The isotropy representation of K/T. We will now examine the isotropy representation of a full flag manifold K C /B K/T. Consider the reductive decomposition k t m of k with respect to the negative of the Killing form Q B( ) that is m t and Ad(T )m m. As usual we identify m T o (K/T) (where o et is the identity coset of K/T). Choose a Weyl basis {H α1...h αl } {E α k α C α R} with B(E αe α ) 1 [E α E α ] H α. Recall for later use that the root vectors satisfy [E α E β ] N αβ E α+β if α β α + β R and [E α E β ] 0 otherwise. The numbers N αβ R satisfy N αβ N βα N αβ N α β R if α β α + β R andn αβ 0if α β R α + β/ R. They can also be chosen so that N α β N αβ. Then the real subalgebra k is given by (2.2) k l RiH αj j1 α R + (RA α + RB α )t α R + (RA α + RB α ) where A α E α + E α and B α i(e α E α )(α R + ). Since t span R {ih αj 1 j l} the reductive decomposition g t m implies that (2.3) m T o (K/T) α R + (RA α + RB α ). Set m α RA α + RB α for any α R +. The linear space m α is an irreducible Ad(T )-module which does not depend on the choice of an ordering in R. Furthermore since the roots of k C with respect to t C are distinct and the root spaces are one-dimensional it is obvious that m α m β as Ad(T )-representations for any two roots α β R +. Thus by using (2.3) we obtain the following: Proposition 2.2. Let M K/T be a full flag manifold of a compact simple Lie group K. Then the isotropy representation of M decomposes into a direct sum of 2-dimensional pairwise inequivalent irreducible T -submodules m α as follows: (2.4) m α R + m α. The number of these submodules is equal to the cardinality R +.

5 HOMOGENEOUS EINSTEIN METRICS ON G 2 /T The Ricci tensor for a K-invariant metric on K/T. Since K/T is a reductive homogeneous space there is a natural 1-1 correspondence between K- invariant symmetric covariant 2-tensors on K/T and Ad(T )-invariant symmetric bilinear forms on m. For example in this correspondence a K-invariant Riemannian metric g on K/T corresponds to an Ad(T )-invariant inner product on m. In particular since m admits the decomposition (2.4) and the Ad(T )-submodules are mutually inequivalent the space of K-invariant Riemannian metrics on K/T is given by (2.5) {g }. α R + x α Q mα x α R + The K-invariant Kähler-Einstein metric on K C /B K/T corresponding to 2δ 2 l i1 Λ i is given by (2.6) g 2δ 2(Λ 1 + +Λ l α) Q mα. α R + Similarly the Ricci tensor Ric g of a K-invariant metric g on K/T as a K- invariant covariant 2-tensor will be described by an Ad(T )-invariant symmetric bilinear form on m given by Ric g r α x α Q mα α R + where r α (α R + ) are the components of the Ricci tensor on each submodule m α. Since m α m β for any α β R + itisric g (m α m β ) 0 (cf. [20]). There is a useful description of the components r α associated to the isotropy decomposition (2.4). Let K/L be a compact homogeneous space of a compact semisimple Lie group K whose isotropy representationm decomposesinto s pairwise inequivalent irreducible Ad(L)-submodules m i as m m 1 m s. Following [20] and [15] we choose a Q-orthonormal basis {e p } adapted to m s i1 m i. Let A r pq Q([e p e q ]e r )sothat[e p e q ] m γ Ar pqe r andset (2.7) [ k ij ] (A r pq) 2 ( Q([ep e q ]e r ) ) 2 where the sum [ ] is taken over all indices p q r with e p m i e q m j ande r m k. k The triples are called the structure constants of K/L with respect to the ij decomposition m s i1 m i and are symmetric to all three indices. For the case of a full flag manifold K/T we study its structure constants with respect to the Q-orthogonal decomposition m α R m αwhereq B( ) + and m α RA α + RB α. Note that we can rewrite the previous splitting as m m 1 m s wheres R +. Since B(E α E β ) δ αβ one can verify that the vectors A α and B α are such that B(A α A β ) B(B α B β ) 2δ αβ and B(A α B β ) 0. Therefore the set {X α A α / 2 Y α B α / 2 α R + } is a Q-orthonormal basis of each m α. We use the notation [ [ γ αβ] for α β γ R instead of k ij] for submodules mα m α m β m β andm γ m γ. Recall that if α β α+β R then[k α C kβ C ]kα+β C and B(k α C kβ C ) 0 (cf. [11 p. 168]). Since [ γ αβ] 0 if and only if Q([mα m β ] m γ ) 0 we can easily conclude that [ γ αβ] 0 if and only if the roots α β γ satisfy one of

6 2490 A. ARVANITOYEORGOS I. CHRYSIKOS AND Y. SAKANE the relations α + β γ 0α β + γ 0 α + β + γ 0. Notethatif [ γ αβ] 0 then we can rewrite [ ] [ γ αβ as γ ] α β with α β γ R + by rearranging roots and changing the sign of roots. By using the above notation it can be shown ([16 p. 75]) that for each α R + the Ricci component r α corresponding to the isotropy summand m α is given by (2.8) r α [ ] x α α 1 [ ] x γ γ. 2x α 8 x β x γ βγ 4 x α x β αβ βγ R + βγ R + Hence a K-invariant metric (2.5) on K/T is an Einstein metric with Einstein constant k if and only if it is a positive real solution of the system {r α k α R + }. Proposition 2.3. For a full flag manifold K/T the triples [ α+β α β] are given by [ ] α + β (2.9) 2N 2 αβ αβ. Proof. By definition (2.7) we see that [ ] α + β (B([X α X β ]X α+β )) 2 +(B([X α X β ]Y α+β )) 2 αβ +(B([Y α X β ]X α+β )) 2 +(B([Y α X β ]Y α+β )) 2 +(B([X α Y β ]X α+β )) 2 +(B([X α Y β ]Y α+β )) 2 +(B([Y α Y β ]X α+β )) 2 +(B([Y α Y β ]Y α+β )) 2. Since B(B α+β A α+β )0andB(A α+β A α+β )B(B α+β B α+β ) 2 a straightforward computation using the properties of the root vectors and the numbers N αβ gives that B([X α X β ]X α+β )1/(2 2)B(N αβ A α+β + N α β A α β A α+β ) N αβ / 2 B([Y α X β ]Y α+β )B([X α Y β ]Y α+β ) B([Y α Y β ]X α+β ) N αβ / 2 B([X α X β ]Y α+β )B([Y α X β ]X α+β ) B([X α Y β ]X α+β )B([Y α Y β ]Y α+β )0 and the result follows. Remark 2.4. Two roots α β R have the same length with respect to the Killing form B if and only if there is an element w of the Weyl group W (R) oftheroot system R such that β w(α) (see for example [18 p. 242]). Thus because of the invariance of the Killing form under W (R) it is obvious that for any element w W (R) wehavethat [ w(γ) ] [ γ w(α) w(β) αβ]. 3. The full flag manifold G 2 /T We now study the geometry of the full flag manifold G 2 /T wheret is a maximal torus of G 2. We start by describing its isotropy representation The decomposition of the isotropy representation of G 2 /T. The root system of the exceptional complex simple Lie algebra g 2 can be chosen to be R {±α 1 ±α 2 ±(α 1 + α 2 ) ±(α 1 +2α 2 ) ±(α 1 +3α 2 ) ±(2α 1 +3α 2 )}. We fix a system of simple roots to be Π {α 1 α 2 }. With respect to Π the positive roots are given by (3.1) R + {α 1 α 2 α 1 + α 2 α 1 +2α 2 α 1 +3α 2 2α 1 +3α 2 }.

7 HOMOGENEOUS EINSTEIN METRICS ON G 2 /T 2491 The highest root is α 2α 1 +3α 2 (see Figure 2). Also it is α 1 3 α 2 and the roots of g 2 make succesive angles of π/6. The Weyl group is generated by rotations of R 2 about the origin through an angle π/6 and reflections about the vertical axis. Figure 2. The root system of g 2 The full flag manifold G 2 /T is obtained by painting black both simple roots in the Dynkin diagram of g 2. Proposition 2.2 implies that the isotropy representation m of G 2 /T decomposes into six inequivalent irreducible ad(k)-submodules i.e. m m 1 m 2 m 3 m 4 m 5 m 6 where the submodules m i (1 i 6) are given by (3.2) m 1 m α1 m 2 m α2 m 3 m α1 +α 2 m 4 m α1 +2α 2 m 5 m α1 +3α 2 m 6 m 2α1 +3α Kähler-Einstein metrics. It is well known ([7 p. 504]) that a full flag manifold K/T admits W (K) /2 invariant complex structures (here W (K) istheweyl group of K) which are all equivalent under an automorphism of K. We now compute the unique Kähler Einstein metric which is compatible with the natural complex structure J nat that is the complex structure corresponding to the natural invariant ordering R + given by (3.1). From (2.5) a G 2 -invariant Riemannian metric on G 2 /T is given by (3.3) g x 1 Q m1 + + x 6 Q m6 wherewehavesetx 1 x α1 x 2 x α2 x 3 x α1 +α 2 x 4 x α1 +2α 2 x 5 x α1 +3α 2 x 6 x 2α1 +3α 2 andthem k (k 1...6) are given by (3.2). We will denote such metrics by g (x 1 x 2 x 3 x 4 x 5 x 6 ) R 6 +. Theorem 3.1. The full flag manifold G 2 /T admits six invariant Kähler Einstein metrics which are isometric to each other. The Kähler-Einstein metric g 2δ which is compatible with the natural invariant ordering J nat is given (up to a scale) by g 2δ ( ). Proof. According to the notation of Section 2.1 the weight δ for G 2 /T is given by δ 1 2 α R α 2 + i1 Λ i where Λ 1 and Λ 2 are the fundamental weights corresponding to the simple roots α 1 and α 2 respectively. In Figure 2 one can easily distinguish the long roots L 1 α 1 L 2 α 1 +3α 2 L 3 2α 1 +3α 2 }

8 2492 A. ARVANITOYEORGOS I. CHRYSIKOS AND Y. SAKANE from the short roots S 1 α 2 S 2 α 1 + α 2 S 3 α 1 +2α 2. Then L i 3 S j where 1 i j 3andi j are independent. We set (L i L i )3and(S i S i )1forany1 i 3. We denote the Kähler-Einstein metric g 2δ by g α1 Q m1 + g α2 Q m2 + g α1 +α 2 Q m3 + g α1 +2α 2 Q m4 + g α1 +3α 2 Q m5 + g 2α1 +3α 2 Q m6 which is compatible with the natural invariant complex structure J nat defined by the ordering R +. By using (2.1) and applying relation (2.6) we obtain the following values for the components g α (2δ α) α R + of this metric: g α1 2(Λ 1 α 1 )(α 1 α 1 )3 g α2 2(Λ 2 α 2 )(α 2 α 2 )1 g α1 +α 2 2(Λ 1 α 1 )+2(Λ 2 α 2 )4 g α1 +2α 2 2(Λ 1 α 1 )+4(Λ 2 α 2 )5 g α1 +3α 2 2(Λ 1 α 1 )+6(Λ 2 α 2 )6 g 2α1 +3α 2 4(Λ 1 α 1 )+6(Λ 2 α 2 ) Homogeneous Einstein metrics. We now proceed to the calculation of the Ricci tensor Ric g corresponding to a G 2 -invariant metric (3.3) on G 2 /T. Following the notation of Section 2.3 the tensor Ric g as a G 2 -invariant symmetric covariant 2-tensor on G 2 /T is given by Ric g r 1 x 1 Q m1 + + r 6 x 6 Q m6 where for simplicity we have set r 1 r α1 r 2 r α2 r 3 r α1 +α 2 r 4 r α1 +2α 2 r 5 r α1 +3α 2 and r 6 r 2α1 +3α 2. In order to apply (2.8) we first need to find the nonzero structure constants [ k ij] of G2 /T. By using (3.1) and (3.2) it follows that these are [ ] c 3 α1 + α 2 12 α 1 α 2 [ 3 12 [ ] c 6 2α1 +3α 2 15 α 1 α 1 +3α 2 ] [ ] c 4 α1 +2α 2 23 α 2 α 1 + α 2 [ ] [ 6 c [ 4 23 ] [ c 5 α1 +3α 2 24 α 2 α 1 +2α 2 ] 2α 1 +3α 2 α 1 + α 2 α 1 +2α 2 [ ] ] [ ] 5 24 By Remark 2.4 and the remarks on the notation for [ γ αβ] we obtain the following: Lemma 3.2. The triples c 3 12 c 5 24 and c 6 34 are equal. Proof. The Weyl group W (R) is generated by the reflections {s α1 s α2 }andwe have that s α1 (α 1 ) α 1 s α1 (α 2 )α 1 + α 2 s α2 (α 2 ) α 2 s α2 (α 1 )α 1 +3α 2. Now we see that s α2 (α 1 + α 2 )α 1 +2α 2 and hence we have that [ ] [ ] [ ] c 5 α1 +3α 2 s α2 (α 1 ) α 1 24 α 2 α 1 +2α 2 s α2 (α 2 ) s α2 (α 1 + α 2 ) α 2 α 1 + α 2 [ ] α1 + α 2 c 3 α 1 α We also see that s α1 (α 1 +3α 2 )2α 1 +3α 2 and s α1 (α 1 +2α 2 )α 1 +2α 2 and hence we have that [ ] [ c 6 2α 1 +3α 2 34 α 1 + α 2 α 1 +2α 2 s α1 (α 1 +3α 2 ) s α1 (α 2 ) s α1 (α 1 +2α 2 ) ] [ ] α1 +3α 2 c 5 α 2 α 1 +2α 24. 2

9 HOMOGENEOUS EINSTEIN METRICS ON G 2 /T 2493 For the calculation of the above triples we use Proposition 2.3 and the fact that (3.4) Nαβ 2 q(p +1) N αβ N α β Q(α α) 2 where p q are the largest nonnegative integers such that β + kα R with p k q. We first proceed to the calculation of c 3 12 [ α 1 +α 2 ] α 1 α 2. By using the relation (α 1 α 1 )3(α 2 α 2 ) and equation (3.4) we obtain that Nα 2 1 α 2 3Q(α 2 α 2 )/2 so Proposition 2.3 implies that c Q(α 2 α 2 ). The normalizing value Q(α 2 α 2 )is given by Q(α 2 α 2 )1/12 (cf. [8]); thus c /4. Similarly we obtain that [ ] c 4 α1 +2α Nα 2 α 2 α 1 + α 2 α 1 +α 2 1 [ ] 2 3 2α1 +3α 2 c6 15 2Nα 2 α 1 α 1 +3α 1 α 1 +3α The above computations combined with Lemma 3.2 give the following: Proposition 3.3. The nonzero triples [ k ij] of the full flag manifold G2 /T are given by [ ] [ ] [ ] 6 34 [ ] 6 1 [ ] and Therefore we obtain the following proposition for the Ricci tensor from (2.8): Proposition 3.4. The components r i (i 1...6) of the Ricci tensor associated to the G-invariant Riemannian metric g given by (3.3) are the following: r ( x ( x1 x 5 6 2x 1 16 x 2 x 3 x 1 x 3 x 1 x 2 16 x 5 x 6 x 1 x 6 x 1 x 5 r ( x2 x ( x2 x 3 4 2x 2 16 x 1 x 3 x 2 x 3 x 1 x 2 12 x 3 x 4 x 2 x 4 x 2 x ( x2 x x 4 x 5 x 2 x 5 x 2 x 4 r ( x ( x3 4 2x 3 16 x 1 x 2 x 1 x 3 x 2 x 3 12 x 2 x 4 x 3 x 4 x 2 x ( x3 x x 4 x 6 x 3 x 6 x 3 x 4 r ( x ( x4 5 2x 4 12 x 2 x 3 x 3 x 4 x 2 x 4 16 x 2 x 5 x 4 x 5 x 2 x ( x4 x x 3 x 6 x 4 x 6 x 3 x 4 r ( x5 x ( x5 4 2x 5 16 x 1 x 6 x 5 x 6 x 1 x 5 16 x 2 x 4 x 4 x 5 x 2 x 5 r ( x6 x ( x6 x x 6 16 x 1 x 5 x 5 x 6 x 1 x 6 16 x 3 x 4 x 4 x 6 x 3 x 6 A G 2 -invariant Riemannian metric on the full flag manifold G 2 /T is Einstein if and only if there is a positive constant k such that (3.5) r 1 k r 2 k r 3 k r 4 k r 5 k r 6 k where r i (i 1...6) are given in Proposition 3.4.

10 2494 A. ARVANITOYEORGOS I. CHRYSIKOS AND Y. SAKANE 4. Proof of Theorem A Note that the action of the Weyl group of g 2 on its root system (cf. Figure 2) induces an action on the components of the G 2 -invariant metric (3.3). In particular if (x 1 x 2 x 3 x 4 x 5 x 6 )(a 1 a 2 a 3 a 4 a 5 x 6 ) is a solution for the system of equations (3.5) then (x 1 x 2 x 3 x 4 x 5 x 6 )(a 5 a 2 a 4 a 3 a 1 a 6 ) is also a solution of the system (3.5). In fact if w is a reflection about 2α 1 +3α 2 in the root diagram of g 2 thenw(α 1 )α 1 +3α 2 w(α 1 +α 2 )α 1 +2α 2. This induces an interchange of x 1 with x 5 and x 3 with x 4 and keeps x 2 x 6 fixed. Similarly we see that (x 1 x 2 x 3 x 4 x 5 x 6 )(a 6 a 3 a 4 a 2 a 1 a 5 ) (x 1 x 2 x 3 x 4 x 5 x 6 )(a 1 a 3 a 2 a 4 a 6 a 5 ) (x 1 x 2 x 3 x 4 x 5 x 6 )(a 5 a 4 a 2 a 3 a 6 a 1 ) (x 1 x 2 x 3 x 4 x 5 x 6 )(a 6 a 4 a 3 a 2 a 5 a 1 ) are also solutions of system (3.5). These metrics are all isometric to each other. The above analysis using the Weyl group suggests splitting the study of solutions for the system (3.5) into two cases: Case A: (x 1 x 5 )(x 1 x 6 )(x 5 x 6 )0and Case B: (x 1 x 5 )(x 1 x 6 )(x 5 x 6 ) 0. Note that the system of equations (3.5) is equivalent to the equations (4.1) r 1 r 2 0 r 2 r 3 0 r 3 r 4 0 r 4 r 5 0 r 5 r 6 0. Moreover we normalize our equations by setting x 1 1. Then the system of equations (4.1) is equivalent to the equations (4.2) f 1 3x 2 2 x 3 x 6 6x 2 2 x 4 x 5 x 6 4x x 5 x 6 3x 2 x 3 x 4 x 5 +24x 2 x 3 x 4 x 5 x 6 3x 2 x 3 x 4 x x 2 x 3 x 4 +4x 2 3 x 5 x 6 +3x 3 x 2 4 x 6 24x 3 x 4 x 5 x 6 +3x 3 x 2 5 x 6 +4x 2 4 x 5 x 6 +6x 4 x 5 x 6 0 f 2 3x 2 2 x 3 x 6 +6x 2 2 x 4 x 5 x 6 +8x 2 2 x 5 x 6 3x 2 x 2 3 x 5 +3x 2 x 2 4 x 5 24x 2 x 4 x 5 x 6 +3x 2 x 5 x 2 6 6x 2 3 x 4 x 5 x 6 8x 2 3 x 5 x 6 3x 3 x 2 4 x 6 +24x 3 x 4 x 5 x 6 3x 3 x 2 5 x 6 0 f 3 3x 2 2 x 3 x 6 3x 2 2 x 4 x 5 x 6 +6x 2 x 2 3 x 5 24x 2 x 3 x 5 x 6 6x 2 x 2 4 x 5 +24x 2 x 4 x 5 x 6 +3x 2 3 x 4 x 5 x 6 +8x 2 3 x 5 x 6 3x 3 x 2 4 x 6 +3x 3 x 2 5 x 6 8x 2 4 x 5 x 6 3x 4 x 5 x 6 0 f 4 4x 2 2 x 5 x 6 3x 2 x 2 3 x 5 3x 2 x 3 x 4 x x 2 x 3 x 4 x 6 24x 2 x 3 x 4 x 6 +3x 2 x 3 x 4 +24x 2 x 3 x 5 x 6 +3x 2 x x 5 3x 2 x 5 x 6 4x 2 3 x 5 x 6 +6x 3 x 2 4 x 6 6x 3 x 2 5 x 6 +4x 2 4 x 5 x 6 0 f 5 x 2 2 x 3 x 6 + x 2 x 2 3 x 5 +2x 2 x 3 x 4 x x 2 x 3 x 4 x 5 2x 2 x 3 x 4 x 6 +8x 2 x 3 x 4 x 6 + x 2 x 2 4 x 5 x 5 x 2 6 x 3 x 2 4 x 6 + x 3 x 2 5 x 6 0 for solutions with x 2 x 3 x 4 x 5 x 6 0. To find nonzero solutions of equations (4.2) we compute a Gröbner basis (see for example [9]) by using algebraic manipulations in a computer system. Case A. We may assume that x 1 x 5 1. Ifx 6 1 we consider a polynomial ring R 1 Q[y x 2 x 3 x 4 x 5 x 6 ] and an ideal I 1 generated by {f 1 f 2 f 3 f 4 f 5 x 5 1 x 6 1 yx 2 x 3 x 4 x 5 x 6 1}.

11 HOMOGENEOUS EINSTEIN METRICS ON G 2 /T 2495 We take a lexicographic order > with y>x 2 >x 3 >x 4 >x 5 >x 6 for a monomial ordering on R 1.ThenaGröbner basis for the ideal I 1 is given by {x 6 1x x x 4 +9x 3 x 4 x 2 x x y}. Now the equation 15x x has no real solutions. Thus there are no Einstein metrics for this case. If x 6 1 we consider an ideal I 2 generated by {f 1 f 2 f 3 f 4 f 5 x 5 1 (x 6 1) yx 2 x 3 x 4 x 5 x 6 1}. We take a lexicographic order > with y>x 6 >x 5 >x 2 >x 3 >x 4 for a monomial ordering on R 1.Thenweseethatx 3 x 4 is an element of a Gröbner basis for the ideal I 2. Thus we obtain the following expression for the Ricci components in this case: r 1 r ( 1 16 r 2 r 3 r 4 r x x x 6 16 x 3 x 2 x 3 x 3 ( x2 x 2 ) ) x 6 16 ( x2 ) x 2 2 x x 2 x 3 x 2 x 2 x 3 ( x3 1 ) x 2 x 3 x 2 x 3 12x 2 x x 2 3 ) (x 6 2x6 + 1 ( x6 16 x 2 2 ). 3 x 6 Now the system of equations (3.5) is equivalent to the equations (4.3) r 1 r 2 r 2 r 3 r 3 r 6. Moreover we see that the system of equations (4.3) is equivalent to the equations (4.4) h 1 9x 2 2 x 3 4x 2 2 3x 2 x 3 2 x 6 +24x 2 x x x x 3 0 h 2 9x 2 2 x 3 +8x x 2 x 3 +3x 2 x 6 9x x 3 2 3x 3 0 h 3 3x 2 2 x 3 x 6 4x 2 2 x 6 3x 2 x 3 2 x x 2 x x 2 x 3 x 6 6x 2 x x 3 3 x 6 3x 3 x 6 0 for solutions with x 2 x 3 x 6 0. We consider a polynomial ring R 2 Q[y x 2 x 3 x 6 ] and an ideal I 3 generated by {h 1 h 2 h 3 yx 2 x 3 x 6 1}. We take a lexicographic order > with y>x 2 >x 3 >x 6 for a monomial ordering on R 2. Then a Gröbner basis for the ideal I 3 contains the following polynomials p 1 p 2 p 3 : p x x x x x x x x x x x x x x

12 2496 A. ARVANITOYEORGOS I. CHRYSIKOS AND Y. SAKANE p x x x x x x x x x x x x x x p x x x x x x x x x x x x x x By solving the equation p 1 0 numerically we obtain exactly two real solutions which are approximately given by x and x We substitute these values for x 6 into equations p 2 0andp 3 0andweobtain two positive solutions approximately given by x x and x x Moreover we obtain the value for k from (3.5). Thus we have the following:

13 HOMOGENEOUS EINSTEIN METRICS ON G 2 /T 2497 Theorem 4.1. The full flag manifold G 2 /T admits two non-kähler G 2 -invariant Einstein metrics. These metrics are given approximately as follows: x 1 1 x x 3 x x 5 1 x k x 1 1 x x 3 x x 5 1 x k Case B. We consider a polynomial ring R 1 Q[y x 2 x 3 x 4 x 5 x 6 ] and an ideal I 4 generated by {f 1 f 2 f 3 f 4 f 5 (x 5 1)(x 6 1) yx 2 x 3 x 4 x 5 x 6 1}. We take a lexicographic order > with y>x 2 >x 3 >x 4 >x 5 >x 6 for a monomial ordering on R 1.ThenaGröbner basis for the ideal I 4 contains a polynomial of the form (x 6 3)(x 6 2)(2x 6 3)(2x 6 1)(3x 6 2)(3x 6 1)q 1 (x 6 ) where q 1 (x 6 ) is an explicitly given polynomial of degree 84 with integer coefficients. For the case when (x 6 3)(x 6 2)(2x 6 3)(2x 6 1)(3x 6 2)(3x 6 1) 0 we first consider an ideal I 5 generated by {f 1 f 2 f 3 f 4 f 5 (x 6 3)yx 2 x 3 x 4 x 5 x 6 1}. We take a lexicographic order > with y>x 2 >x 3 >x 4 >x 5 >x 6 for a monomial ordering on R 1.ThenaGröbner basis for the ideal I 5 is given by {x 6 3x 5 2 3x 4 5 3x 3 4 3x y 9}. We also compute the Gröbner basis for other cases and we obtain the following sets of solutions for equations (4.2): (x 6 3x 5 2x 4 5 x x ) (x 3 6 2x 5 3x 4 5 x x ) 3 (x 6 3 x x x x ) (x x x x x ) 6 (x 6 2 x x x x ) (x x x x x ). 9 Note that these are the six Kähler-Einstein metrics of Theorem 3.1. Now by solving the equation q 1 (x 6 ) 0 numerically we obtain 14 positive solutions which are approximately given by x x x x x x x x x x x x x x To get the solutions of equations (4.2) for the variables x 2 x 3 x 4 x 5 correspondingtothesolutionx 6 we compute a Gröbner basis under the condition (x 6 3)(x 6 2)(2x 6 3)(2x 6 1)(3x 6 2)(3x 6 1) 0; that is we consider an ideal I 6 generated by {f 1 f 2 f 3 f 4 f 5 (x 6 3)(x 6 2)(2x 6 3)(2x 6 1)(3x 6 2)(3x 6 1) yx 2 x 3 x 4 x 5 x 6 1} and take a lexicographic order > with y>x 2 >x 3 >x 4 > x 5 >x 6 for a monomial ordering on R 1. Then q 1 iscontainedinthisgröbner basis and by examining the other elements of the obtained Gröbner basis we see that the other variables x 2 x 3 x 4 x 5 can be expressed by polynomials of x 6 with degree 83. Let x 2 q 2 (x 6 ) x 3 q 3 (x 6 ) x 4 q 4 (x 6 ) and x 5 q 5 (x 6 )bethese polynomials. Now we substitute these 14 values for x 6 into the expressions of the

14 2498 A. ARVANITOYEORGOS I. CHRYSIKOS AND Y. SAKANE polynomials q 2 q 3 q 4 q 5 of x 6 with degree 83. Then we get the following solutions which are approximately given by x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Weseethatatleastoneofthex i s in these solutions is negative. Thus there are no invariant Einstein metrics for these cases and this completes the proof of Theorem A. References 1. D. V. Alekseevsky and A. M. Perelomov: Invariant Kähler-Einstein metrics on compact homogeneous spaces Funct. Anal. Appl. 20 (1986) MR (88c:53049) 2. A. Arvanitoyeorgos: New invariant Einstein metrics on generalized flag manifolds Trans. Amer. Math. Soc. 337 (1993) MR (93h:53043) 3. A. Arvanitoyeorgos and I. Chrysikos: Invariant Einstein metrics on generalized flag manifolds with two isotropy summands J.Aust.Math.Soc.90 (2011) MR A. Arvanitoyeorgos and I. Chrysikos: Invariant Einstein metrics on generalized flag manifolds with four isotropy summands Ann. Glob. Anal. Geom. 37 (2010) MR (2011d:53083) 5. C. Böhm and M. Kerr: Low-dimensional homogeneous Einstein manifolds Trans. Amer. Math. Soc. 358 (2006) MR (2006g:53056) 6. M. Borderman M. Forger and H. Römer: Homogeneous Kähler manifolds: paving the way towards new supersymmetric sigma models Comm. Math. Phys. 102 (1986) MR (87c:53096) 7. A. Borel and F. Hirzebruch: Characteristic classes and homogeneous spaces I Amer.J.Math. 80 (1958) MR (21:1586) 8. N. Bourbaki: Groupes et algèbres de Lie Chapitres 4 5 et 6 Hermann Paris MR (39:1590) 9. D. Cox J. Little and D. O Shea: IdealsVarietiesandAlgorithms.AnIntroductiontoComputational Algebraic Geometry and Commutative Algebra Undergraduate Texts in Mathematics Springer-Verlag New York MR (93j:13031) 10. E. C. F. Dos Santos and C. J. C. Negreiros: Einstein metrics on flag manifolds Revista Della Unión Mathemática Argentina 47 (2006) MR (2008b:53061) 11. S. Helgason: Differential Geometry Lie Groups and Symmetric Spaces Academic Press New York MR (80k:53081) 12. M. Kimura: Homogeneous Einstein metrics on certain Kähler C-spaces Adv. Stud. Pure Math. 18-I Academic Press Boston MA MR (93b:53039) 13. J. L. Koszul: Sur la forme hermitienne canonique des espaces homogènes complexes Can.J. Math. 7 (1955) MR (17:1109a)

15 HOMOGENEOUS EINSTEIN METRICS ON G 2 /T Yu. G. Nikonorov E. D. Rodionov and V. V. Slavskiĭ: Geometry of homogeneous Riemannian manifolds J. Math. Sciences 146 (2007) MR (2011a:53083) 15. J-S. Park and Y. Sakane: Invariant Einstein metrics on certain homogeneous spaces Tokyo J. Math. 20 (1997) MR (98c:53058) 16. Y. Sakane: Homogeneous Einstein metrics on flag manifolds Lobachevskii J. Math. 4 (1999) MR (2000m:53069) 17. M. Takeuchi: Homogeneous Kähler submanifolds in complex projective spaces Japan. J. Math. 4 (1978) MR (80i:32059) 18. P. Tauvel and R. W. T. Yu: Lie Algebras and Algebraic Groups Springer Monographs in Mathematics Springer-Verlag Berlin MR (2006c:17001) 19. M. Wang and W. Ziller: On normal homogeneous Einstein manifolds Ann. Scient. Éc. Norm. Sup. 18 (1985) MR (87k:53113) 20. M. Wang and W. Ziller: Existence and non-existence of homogeneous Einstein metrics Invent. Math. 84 (1986) MR (87e:53081) Department of Mathematics University of Patras GR Rion Greece address: arvanito@math.upatras.gr Department of Mathematics and Statistics Masaryk University Kotlarska Brno Czech Republic address: xrysikos@master.math.upatras.gr Department of Pure and Applied Mathematics Graduate School of Information Science and Technology Osaka University Toyonaka Osaka Japan address: sakane@math.sci.osaka-u.ac.jp