Notes on "Einstein metrics on compact simple Lie groups attached to standard triples"

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by  Huibin Chen

NOTES ON “EINSTEIN METRICS ON COMPACT SIMPLE LIE GROUPS
ATTACHED TO STANDARD TRIPLES”
HUIBINCHENANDZHIQICHEN
7
1
0 Abstract. Inthepaper“EinsteinmetricsoncompactsimpleLiegroupsattachedtostandardtriples”,
2 theauthorsintroducedthedefinitionofstandardtriplesandprovedthateverycompactsimpleLiegroup
Gattachedtoastandardtriple(G,K,H)admitsaleft-invariantEinsteinmetricwhichisnotnaturally
n reductive except the standard triple (Sp(4),2Sp(2),4Sp(1)). For the triple (Sp(4),2Sp(2),4Sp(1)), we
a findthereexistsaninvolutionpairofsp(4)suchthat4sp(1)isthefixedpointofthepair,andthengive
J thedecompositionofsp(4)asadirectsumofirreduciblead(4sp(1))-modules. ButSp(4)/4Sp(1)isnota

6 generalizedWallachspace. Furthermorewegiveleft-invariantEinsteinmetricsonSp(4)whicharenon-
naturallyreductiveandAd(4Sp(1))-invariant. Forthegeneralcase(Sp(2n1n2),2Sp(n1n2),2n2Sp(n1)),

] there exist 2n2−1 involutions of sp(2n1n2) such that 2n2sp(n1)) is the fixed point of these 2n2−1
G
involutions,anditfollowsthedecompositionofsp(2n1n2)asadirectsumofirreduciblead(2n2sp(n1))-
D modules. In order to give new non-naturally reductive and Ad(2n2Sp(n1)))-invariant Einstein metrics
on Sp(2n1n2), we prove a general result, i.e. Sp(2k+l) admits at least two non-naturally reductive
.
h EinsteinmetricswhichareAd(Sp(k)×Sp(k)×Sp(l))-invariant ifk<l. Itimpliesthat everycompact
t simpleLiegroupSp(n)forn≥4admitsatleast2[n−1]non-naturallyreductiveleft-invariantEinstein
a 3
metrics.
m
[
1. Introduction
1
v
3 A Riemannian manifold (M,g) is called Einstein if there exists a constant λ R such that the Ricci
∈
1
7 tensor r with respect to g satisfies r =λg. See Besse’s book [4], the papers of Jensen [11] and Wang and
1
0 Ziller [17] for more details. For Lie groups,D’Atri and Ziller prove in [10] that every compactsimple Lie
.
1 groupexceptSO(3)admitsatleasttwoleft-invariantEinsteinmetricswhicharenaturallyreductive. But
0
7 they also mention in [10] that it is difficult to give non-naturally reductive Einstein metrics on compact
1
v: simple Lie groups.
i
X Recently, there are some studies on non-naturally reductive Einstein metrics on compact simple Lie
r
a groups. Mori gives in [12] a class of non-naturally reductive left-invariant Einstein metrics on compact
simple Lie groups SU(n) for n 6. After that, Arvanitoyeorgos,Mori and Sakane prove the existence of
≥
non-naturally reductive Einstein metrics on SO(n)(n 11), Sp(n)(n 3), E , E and E . Then Chen
6 7 8
≥ ≥
and Liang obtain in [7] a non-naturally reductive Einstein metric on the compact simple Lie group F .
4
Recently, Chrysikos and Sakane find new non-naturally reductive Einstein metrics on exceptional Lie
groups in [8], especially they give the first non-naturally reductive Einstein metric on G .
2
In [19], Yan and Deng study non-naturally reductive Einstein metrics on a compact Lie group G
associated with a triple (G,K,H). Here G is a compact simple Lie group, K is a closed subgroup of G
such that G/K is a compact irreducible symmetric space, and H is a closed subgroup of K. Denote the
Lie algebrasofG,K,H by g,k,h respectively. Let B,B ,B be the negative of the Killing forms of g,k,h
k h
1
2 HUIBINCHENANDZHIQICHEN
respectively. Consider the B-orthogonaldecomposition
g=k p=h u p (1.1)
⊕ ⊕ ⊕
and the left-invariant metrics on G determined by the ad(h)-invariant metric on g of the form
, =B +xB +yB , x,y R+. (1.2)
h u p
h· ·i | | | ∈
A triple (G,K,H) (or (g,k,h)) is called a basic triple in [19] if
(1) B =c B , B =c B for some c >0,c >0, and
k 1 k h 2 h 1 2
| |
(2) (ad2(h )) = λ id, (ad2(h )) = λ id for some λ > 0,λ > 0. Here h is a B-
i i |u − u i i |p − p u p { i}
P P
orthonormalbasis of h.
Abasictriple (G,K,H)is calledstandardif the standardhomogenousmetricg onG/H is Einstein. In
B
fact,the classificationofthe homogeneousspacewhosestandardhomogeneousmetric isEinsteinisgiven
by Wang and Ziller in [16] . Thus it is easy to give a complete classification of standard triples based
on [16, 18]. Furthermore in [19], Yan and Deng classify standard triples and prove that every compact
simple Lie groupG attachedto a standardtriple (G,K,H) admits a left-invariantEinsteinmetric ofthe
form (1.2) which is non-naturally reductive except the standard triple (Sp(4),2Sp(2),4Sp(1)).
Ingeneral,fora basic (evenstandard)triple (G,K,H), the decomposition(1.1) is notnecessaryto be
a direct sum of irreducible ad(h)-modules. That is, the metric of the form (1.2) is only a special class of
left-invariant metrics on G which are also Ad(H)-invariant. In order to obtain non-naturally reductive
left-invariantEinsteinmetrics on Sp(4) whichare Ad(4Sp(1))-invariant,we need to discuss the structure
of sp(4) as a direct sum of irreducible ad(4sp(1))-modules. Firstly, we have the following theorem.
Theorem 1.1. For the standard triple (Sp(4),2Sp(2),4Sp(1)), there exists an involution pair (θ,τ) of
sp(4) such that θτ =τθ and 4sp(1)= x sp(4)θ(x)=x,τ(x)=x .
{ ∈ | }
It follows the decomposition of sp(4) as a direct sum of irreducible ad(4sp(1))-modules. Although
4sp(1) is the fixed point of an involution pair of sp(4), Sp(4)/4Sp(1) is not a generalized Wallach space,
which is also called a three-locally-symmetric space. The notation of a generalized Wallach space is
introduced by Nikonorov in [13]. A compact homogeneous space G/H with a semisimple connected Lie
group G and a connected Lie subgroup H is a generalized Wallach space if m is the direct sum of three
irreducible adh-modules pairwise orthogonalwith respect to B, i.e.
m=m m m ,
1 2 3
⊕ ⊕
with [m ,m ] h for any i 1,2,3 . Here B is the negative of Killing from of g, h is the Lie algebra
i i
⊂ ∈ { }
of H, and m is the orthogonal complement of h in g with respect to B. The classification of generalized
NOTES ON “EINSTEIN METRICS ON COMPACT SIMPLE LIE GROUPS ATTACHED TO STANDARD TRIPLES” 3
WallachspacesisgivenbyNikonorovin[14]. In[5],Chen,KangandLiangprovethattheclassificationof
generalizedWallach spaces is equivalent with the classificationof involution pairs of compact Lie groups
satisfying certain conditions, and then give the classification for compact simple Lie groups.
In[2],Arvanitoyeorgos,SakaneandStatha study non-naturallyreductiveEinsteinmetrics onSp(k +
1
k +k ) which are Ad(Sp(k ) Sp(k ) Sp(k ))-invariant. In particular, they prove that Sp(k +2)
2 3 1 2 3 1
× ×
admitsanon-naturallyreductiveEinsteinmetricwhichisAd(Sp(k ) Sp(1) Sp(1))-invariant,andgive
1
× ×
three non-naturally reductive Einstein metrics for k = 2. Based on the decomposition of sp(4) as a
1
direct sum of irreducible ad(4sp(1))-modules and the results in [2], we have the following theorem.
Theorem1.2. ThecompactsimpleLiegroupSp(4)admitsnon-naturallyreductiveleft-invariantEinstein
metrics which are Ad(4Sp(1))-invariant.
Forthestandardtriple(Sp(2n n ),2Sp(n n ),2n Sp(n ))exceptthecase(Sp(4),2Sp(2),4Sp(1)),Yan
1 2 1 2 2 1
andDengprovein[19]thatSp(2n n )attachedtothetriple(Sp(2n n ),2Sp(n n ),2n Sp(n ))admitsa
1 2 1 2 1 2 2 1
non-naturallyreductiveleft-invariantEinsteinmetricoftheform(1.2)whichisAd(2n Sp(n ))-invariant.
2 1
Furthermore, they obtain an interesting result on the number of Einstein metrics.
Lemma 1.3 ([19]). For any integer n=pl11pl22···plss with pi prime and pi 6=pj, Sp(2n) admits at least
(l +1)(l +1) (l +1) 1
1 2 s
··· −
non-equivalent non-naturally reductive Einstein metrics.
In this paper, we character the structure of sp(2n n ) as a direct sum of irreducible ad(2n sp(n )))-
1 2 2 1
modules by the following theorem.
Theorem1.4. Forthestandardtriple(Sp(2n n ),2Sp(n n ),2n Sp(n )),thereexistsinvolutionsθ ,1
1 2 1 2 2 1 i
≤
i 2n 1ofsp(2n n )suchthatθ θ =θ θ and2n sp(n )= x sp(2n n )θ (x)=x,1 i 2n 1 .
2 1 2 i j j i 2 1 1 2 i 2
≤ − { ∈ | ≤ ≤ − }
It follows from Theorem 1.4 that a B-orthogonal decomposition of sp(2n n ) as a direct sum of irre-
1 2
duciblead(2n sp(n ))-modules. Herewepointoutthat,fromTheorem1.4,wegetanotherdecomposition
2 1
of sp(4) by three involutions of sp(4) directly, which is in essential the same as that from Theorem 1.1.
Inordertogivenew non-naturallyreductiveEinsteinmetricsonSp(2n n )whichareAd(2n Sp(n ))-
1 2 2 1
invariant, we consider another class of left-invariant metrics on Sp(2n n ) which are Ad(2n Sp(n ))-
1 2 2 1
invariant different from those given in [19]. The metrics correspond to the decomposition of Sp(2n n )
1 2
undertwospecialinvolutionsfromθ ,1 i 2n 1whosefixedpointissp(n ) sp(n ) sp(2(n 1)n ).
i 2 1 1 2 1
≤ ≤ − ⊕ ⊕ −
More general, we prove the following theorem.
4 HUIBINCHENANDZHIQICHEN
Theorem 1.5. Forgiven positive integers k andl withk <l, Sp(2k+l)admits at leasttwonon-naturally
reductive left-invariant Einstein metrics which are Ad(Sp(k) Sp(k) Sp(l))-invariant.
× ×
In particular, in Theorem 1.5, let n =k and take l =2(n 1)k, we have at least two non-naturally
1 2
−
reductive left-invariantEinsteinmetrics onSp(2n n )which areAd(Sp(n ) Sp(n ) Sp(2(n 1)n )).
1 2 1 1 2 1
× × −
In particular, they are Ad(2n Sp(n ))-invariant.
2 1
Comparingto Lemma 1.3,we getthe lowerboundofthe number ofnon-naturallyEinsteinmetrics on
Sp(n) for n 4 from Theorem 1.5.
≥
Theorem 1.6. For everyn 4,thecompact simpleLiegroupSp(n)admits at least2[n−1] non-naturally
≥ 3
reductive left-invariant Einstein metrics.
The paper is organized as follows. In section 2, we list some preliminaries which is necessary for this
paper. In section 3, we recall the results on the theory of involutions on compact simple Lie groups,
and then decompose sp(4) as a direct sum of irreducible ad(4sp(1))-modules based on the above theory.
That is, we prove theorem 1.1. Then in section 4, we prove Theorem 1.2 by the studies on Sp(n) given
in [2]. In section 5, we decompose sp(2n n ) as a direct sum of irreducible ad(2n sp(n )))-modules
1 2 2 1
corresponding to 2n 1 involutions of sp(2n n ). In particular, we prove Theorem 1.4. Furthermore,
2 1 2
−
we prove Theorems 1.5 and 1.6 in section 6.
2. Preliminaries: naturally reductive metrics and the Ricci tensor
LetGbeacompactsimpleLiegroupandletK beaconnectedclosedsubgroupofGwithLiealgebras
g and k respectively. Let g=k m be the B-orthogonaldecomposition. Here [k,m] m. Assume that m
⊕ ⊂
can be decomposed into mutually non-equivalent irreducible Ad(K)-modules:
m=m m .
1 q
⊕···⊕
Let k = k k k , where k = Z(k) is the center of k and every k is simple for i = 1, ,p.
0 1 p 0 i
⊕ ⊕···⊕ ···
It is well-known that there exists a one-to-one corresponding between G-invariant metrics on G/K and
Ad(K)-invariant inner products on m. The G-invariant metric , on M = G/K is called naturally
h· ·i
reductive if
[X,Y] ,Z + Y,[X,Z] =0, X,Y,Z m.
m m
h i h i ∀ ∈
In [10], D’Atri and Ziller give a sufficient and necessary condition for a left-invariant metric on a
compact simple Lie group to be naturally reductive.
Lemma 2.1 ([10]). For any inner product b on k , every left-invariant metric on G with the form
0
, =u b +u B + +u B +xB , (u ,u , ,u ,x R+)
h· ·i 0 |k0 1 |k1 ··· p |kp |m 0 1 ··· p ∈
NOTES ON “EINSTEIN METRICS ON COMPACT SIMPLE LIE GROUPS ATTACHED TO STANDARD TRIPLES” 5
is naturally reductive with respect to the action (g,k)y =gyk−1 of G K. Conversely, if a left-invariant
×
metric , on a compact simple Lie group G is naturally reductive, then there exists a closed subgroup
h· ·i
K of G such that , can be written as above.
h· ·i
Now we have a B-orthogonaldecomposition of g which is Ad(K)-invariant:
g=k k k m m =k k k k k ,
0 1 p 1 q 0 1 p p+1 p+q
⊕ ⊕···⊕ ⊕ ⊕···⊕ ⊕ ⊕···⊕ ⊕ ⊕···⊕
where k = m for 1 i q. Assume that dimk 1. Consider the following left-invariant metric on
p+i i 0
≤ ≤ ≤
G which is Ad(K)-invariant:
, =x B +x B + +x B , (2.1)
h· ·i 0· |k0 1· |k1 ··· p+q · |kp+q
where x R+ for i=1, ,p+q.
i
∈ ···
Let d = dimk and let ei di be a B-orthonormal basis of k . Denote Aγ = B([ei,ej],ek), i.e.
i i { α}α=1 i α,β α β γ
[ei,ej]= Aγ ek, and define
α β γ α,β γ
P
(ijk):= (Aγ )2,
α,β
X
where the sum is takenoverallindices α,β,γ with ei k ,ej k ,ek k . Then(ijk) is independent of
α ∈ i β ∈ j γ ∈ k
the choice for the B-orthonormal basis of k ,k ,k , and (ijk) = (jik) = (jki). Furthermore in [1], there
i j k
are fundamentalformulae for the Ricci tensor for compactLie groupsandcompact homogeneousspaces.
Here we just give the formula for Lie groups.
Lemma 2.2. [1,15] Let G be a compact connected simple Lie group endowed with a left-invariant metric
, of the form (2.1). Then the components r ,r , ,r of the Ricci tensor associated to , are:
0 1 p+q
h· ·i ··· h· ·i
1 1 x 1 x
k j
r = + (kji) (jki), (k =0,1, ,p+q).
k
2x 4d x x − 2d x x ···
k k Xj,i j i k Xj,i k i
Here, the sums are taken over all i=0,1, ,p+q. In particular for each k, (jki)=d .
··· i,j k
P
3. The decomposition of sp(4) as a direct sum of irreducible ad(4sp(1))-modules.
Thissectionistocharacterthedecompositionofsp(4)asadirectsumofirreduciblead(4sp(1))-modules
according to the standard triple (Sp(4),2Sp(2),4Sp(1)). Here, Sp(4)/2Sp(2) is a compact irreducible
symmetric space. That is, there is an involution θ of sp(4) such that
2sp(2)= x sp(4)θ(x)=x , p= x sp(4)θ(x)= x , sp(4)=2sp(2) p.
{ ∈ | } { ∈ | − } ⊕
In general, let G be a compact simple connected Lie group with the Lie algebra g and let θ be an
involution of G. Then for the involuation θ, we have a decomposition,
g=k p,
⊕
6 HUIBINCHENANDZHIQICHEN
where k = X gθ(X) = X and p = X gθ(X) = X . Cartan and Gantmacher made great
{ ∈ | } { ∈ | − }
attributions on the classificationof involutions on compact Lie groups. Let t be a Cartan subalgebra of
1
k and let t be a Cartan subalgebra of g containing t .
1
Lemma 3.1 (GantmacherTheorem). With the above notations, θ is conjugate with θ eadH under Autg,
0
where H t and θ is an involution which keeps the Dynkin diagram invariant.
1 0
∈
Let Π= α ,...,α be a fundamental system of t and φ= n m α be the maximal root respec-
{ 1 n} i=1 i i
P
tively. Let α′ = 1(α +θ (α )). Then Π′ = α′,...,α′ consisting different elements in α′,...,α′ is
i 2 i 0 i { 1 l} { 1 n}
a fundamental system of g , where g = X gθ (X) = X . Denote by φ′ = l m′α′ the maximal
0 0 { ∈ | 0 } i=1 i i
P
root of g respectively. Furthermore we have
0
Lemma 3.2 ([20]). If H =0, then for some i, we can take H satisfying
6
α′ =α ; H,α′ =π√ 1; H,α′ =0, j =i. (3.1)
i i h ii − h ji ∀ 6
Here m′ =1 or m′ =2.
i i
Moreover, k is described as follows.
Lemma 3.3 ([20]). Let the notations be as above. Assume that α satisfies the identity (3.1).
i
(1) If θ = Id and m = 1, then Π α is a fundamental system of k, and φ and α are the
0 i i i
−{ } −
highest weights of admCk corresponding to the fundamental system.
(2) If θ =Idand m =2, then Π α φ is a fundamentalsystem of k, and α is thehighest
0 i i i
−{ }∪{− } −
weight of admCk corresponding to the fundamental system.
(3) If θ =Id, then Π′ α′ β is a fundamental system of k, and α is the highest weight of
0 6 −{ i}∪{ 0} − i
admCk corresponding to the fundamental system.
Remark 3.4. In Lemma 3.3, the dimension of C(k), i.e. the center of k, is 1 for case (1); 0 for cases (2)
and (3), β0 in case (3) is the highest weight of admCk for θ =θ0 corresponding to Π′.
Let α ,α ,α ,α be a fundamental system of sp(4) such that the Dynkin diagram is
1 2 3 4
{ }
α1 α2 α3 α4
❝ ❝❝ ❝< ❝
Let φ = 2α +2α +2α +α . By the above lemmas, the irreducible symmetric space Sp(4)/2Sp(2)
1 2 3 4
corresponds to the involution θ =eadH of sp(4) defined by
H,α =π√ 1; H,α =0,j =1,3,4.
2 j
h i − h i
Furthermore the Dynkin diagram of 2sp(2) is
NOTES ON “EINSTEIN METRICS ON COMPACT SIMPLE LIE GROUPS ATTACHED TO STANDARD TRIPLES” 7
−φ α1 α3 α4
❝ >❝❝ ❝< ❝
Consider the involution τ1 =eadH1 of the first sp(2) with the Dynkin diagram
−φ α1
❝ >❝❝
defined by
H ,α =π√ 1; H , φ =0.
1 1 1
h i − h − i
Let φ = φ+2α = 2α 2α α . The Dynkin diagram of x sp(2)τ1(x)=x is
1 1 2 3 4
− − − − { ∈ | }
−φ −φ1
❝ ❝❝
Denotebyh thefirstsp(1)intheabovediagramandh thesecondsp(1). Thenu = x sp(2)τ1(x)=
1 2 1
{ ∈ |
x isanirreduciblead(h1)-moduleandsp(2)=h1 h2 u1. Similarly,considertheinvolutionτ2 =eadH2
− } ⊕ ⊕
of the second sp(2) with the Dynkin diagram
α3 α4
❝< ❝
defined by
H ,α =π√ 1; H ,α =0.
2 3 2 4
h i − h i
Let φ =2α +α . The Dynkin diagram of x sp(2)τ2(x)=x is
2 3 4
{ ∈ | }
−φ2 α4
❝ ❝
Denotebyh thefirstsp(1)intheabovediagramandh thesecondsp(1). Thenu = x sp(2)τ2(x)=
3 4 2
{ ∈ |
x is an irreducible ad(h )-module and sp(2)=h h u . Thus τ2sp(2) =τ1 τ2 is an involution of
2 3 4 2
− } ⊕ ⊕ ⊕
2sp(2) with the decomposition
2sp(2)=h h h h u=h h h h u u .
1 2 3 4 1 2 3 4 1 2
⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
Here u and u are irreducible ad(4sp(1))-modules. But it is unclear for p as an ad(4sp(1))-module. In
1 2
order to do this, we study the extension of the involution τ2sp(2) of 2sp(2) to sp(4).
Thetheoryontheextensionofinvolutionsofktogcanbefoundin[3],whichisdifferentinthemethod
from that in [20]. There are also some related discussion in [5, 6, 7, 9]. In the following we first give
the theory in general cases. Now for any involution τk of k, we can write τk = τkeadHk, where τk is an
0 0
involutionon k which keeps the Dynkin diagramof k invariant, Hk t and τk(Hk)=Hk. Since eadHk is
∈ 1 0
an inner-automorphism, naturally we can extend eadHk to an automorphism of g. Moreover,
Lemma 3.5 ([20]). The involution τk can be extended to an automorphism of g if and only if τk keeps
0 0
the weight system of admCk invariant.
If C(k)=0, then dimC(k)=1. Thus τk(Z)=Z or τk(Z)= Z for any Z C(k).
6 0 0 − ∈
8 HUIBINCHENANDZHIQICHEN
Lemma 3.6 ([20]). Assume that C(k)=0 and τk(Z)=Z for any Z C(k). If τk can be extended to an
6 0 ∈
automorphism of g, then τk can be extended to an involution of g.
Lemma 3.7 ([20]). Assume that C(k) = 0, or C(k) = 0 but τk(Z) = Z for any Z C(k). If τ is an
6 0 − ∈
automorphism of g extending an involution τk of k, then τ2 =Id or τ2 =θ. Furthermore, the following
conditions are equivalent:
(1) There exists an automorphism τ of g extending τk which is an involution.
(2) Every automorphism τ of g extending τk is an involution.
Lemma 3.8 ([20]). Let τ be the automorphism of g extending the involution τk on k. Then τ2 = Id
0 0 0
except g=Ai and n is even. For eadHk, we have:
n
(1) If θ =Id, then the natural extension of eadHk is an involution.
0
6
(2) Assume that θ = Id. Let α′ ,...,α′ be the roots satisfying α′,H = 0. Then the natural
0 i1 ik h j i 6
extension of eadHk is an involution if and only if ik m′ is even.
j=i1 j
P
Lemma 3.9 ([3, 20]). If τ is an involution of g extending an involution τk on k, then every extension of
τk is an involution of k, which is equivalent with τ or τθ.
Now we go back to discuss the above case for the standard triple (Sp(4),2Sp(2),4Sp(1)). By (2) of
Lemma 3.8, the natural extension τ of τ2sp(2) is an involution of sp(4). In fact, for the fundamental
system α ,α ,α ,α of sp(4), the involution τ =eadH of sp(4) is defined by
1 2 3 4
{ }
H,α = H,α =π√ 1; H,α = H,α =0.
1 3 2 4
h i h i − h i h i
Clearly, θτ =τθ. That is, theorem 1.1 holds.
Let p = x pτ(x) = x , and p = x pτ(x) = x . By the above discussion, we have the
1 2
{ ∈ | } { ∈ | − }
following decomposition:
sp(4)=h h h h u u p p .
1 2 3 4 1 2 1 2
⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
Consideranotherfundamentalsystem α , α α ,α +α +α ,α ofsp(4)withthe Dynkindiagram
2 1 2 1 2 3 4
{ − − }
α2 −α1−α2 α1+α2+α3 α4
❝ ❝ ❝< ❝
The maximal root is φ . The Dynkin diagram of h p = x sp(4)τ(x)=x is
1 1
− ⊕ { ∈ | }
−φ1 α2 α1+α2+α3 α4
❝ >❝ ❝< ❝
Consider the restriction of the involution θ of sp(4) on h p . Similar to the discussion for u,
1
⊕
p =p1 p2
1 1⊕ 1
NOTES ON “EINSTEIN METRICS ON COMPACT SIMPLE LIE GROUPS ATTACHED TO STANDARD TRIPLES” 9
as the direct sum of irreducible ad(4sp(1))-modules. For p , consider the fundamental system α +
2 1
{
α , α ,α +α ,α of sp(4) with the Dynkin diagram
2 2 2 3 4
− }
α1+α2 −α2 α2+α3 α4
❝ ❝ ❝< ❝
The maximal root is φ. The Dynkin diagram of h p = x sp(4)θτ(x)=x is
2
⊕ { ∈ | }
−φ α1+α2 α2+α3 α4
❝ >❝ ❝< ❝
Consider the restriction of the involution θ of sp(4) on h p . Similar to the discussion for u,
2
⊕
p =p1 p2
2 2⊕ 2
as the direct sum of irreducible ad(4sp(1))-modules. Up to now, we have the decomposition
sp(4)=h h h h u u p1 p2 p1 p2
1⊕ 2⊕ 3⊕ 4⊕ 1⊕ 2⊕ 1⊕ 1⊕ 2⊕ 2
as a direct sum of irreducible ad(4sp(1))-modules.
4. Non-naturally reductive Einstein metrics on Sp(4) which are Ad(4Sp(1))-invariant
For the standard triple (Sp(4),2Sp(2),4Sp(1), we have the following B-orthogonal decomposition of
sp(4) as irreducible ad(4sp(1))-modules:
sp(4)=h h h h u u p1 p2 p1 p2.
1⊕ 2⊕ 3⊕ 4⊕ 1⊕ 2⊕ 1⊕ 1⊕ 2⊕ 2
Consider the left-invariant metric , on Sp(4) which is Ad(H)-invariant of the form
h· ·i
x B x B x B x B x B x B x B x B x B x B , (4.1)
1 |h1 ⊕ 2 |h2 ⊕ 3 |h3 ⊕ 4 |h4 ⊕ 5 |u1 ⊕ 6 |u2 ⊕ 7 |p11 ⊕ 8 |p21 ⊕ 9 |p12 ⊕ 10 |p22
In [19], the authors show that there are no non-naturally reductive Einstein metrics which satisfy
x =x =x =x , x =x , x =x =x =x .
1 2 3 4 5 6 7 8 9 10
In the following, we consider the metrics on Sp(4) of the form (4.1) satisfying
x =x =x , x =x , x =x .
3 4 6 7 10 8 9
That is, we can consider the following B-orthogonaldecomposition
sp(4)=h h h h u u p1 p2 p2 p1 .
1⊕ 2⊕{ 3⊕ 4⊕ 2}⊕ 1⊕{ 1⊕ 2}⊕{ 1⊕ 2}
The abovedecompositioncorrespondsto anotherinvolutionpair(θ,σ) satisfyingθσ =σθ, whereθ isthe
involution in section 3 and σ =eadH is the involution defined by
H,α =π√ 1; H,α =0,j =2,3,4.
1 j
h i − h i
Here σ is the natural extension of the involution eadH′ on k defined by
H′,α =π√ 1; H′, φ = H′,α = H′,α =0.
1 3 4
h i − h − i h i h i
10 HUIBINCHENANDZHIQICHEN
We can check that h h h h u =sp(1) sp(1) sp(2)= x sp(4)θ(x)=σ(x)=x , and
1 2 3 4 2
⊕ ⊕{ ⊕ ⊕ } ⊕ ⊕ { ∈ | }
Sp(4)/Sp(1) Sp(1) Sp(2) is a generalized Wallach space.
× ×
In general, for any 1 k k k , Sp(k +k +k )/Sp(k ) Sp(k ) Sp(k ) is a generalized
1 2 3 1 2 3 1 2 3
≤ ≤ ≤ × ×
Wallach space [5, 14]. Let α ,α , ,α be a fundamental system of sp(k +k +k ) with the
{ 1 2 ··· k1+k2+k3} 1 2 3
Dynkin diagram
α1 α2 αk1+k2+k3
❝ ❝❝ ❝< ❝
The generalized Wallach space Sp(k +k +k )/Sp(k ) Sp(k ) Sp(k ) corresponds to the involution
1 2 3 1 2 3
× ×
pair (θ,τ) of sp(k +k +k ), where θ =eadH determined by
1 2 3
H,α =π√ 1; H,α =0,j =k +k ,
h k1+k2i − h ji 6 1 2
and τ =eadH′ determined by
H′,α =π√ 1; H′,α =0,j =k .
h k1i − h ji 6 1
Then we have the decomposition of sp(k +k +k ) as irreducible ad(sp(k ) sp(k ) sp(k ))-modules:
1 2 3 1 2 3
⊕ ⊕
sp(k +k +k )=sp(k ) sp(k ) sp(k ) m m m ,
1 2 3 1 2 3 1 2 3
⊕ ⊕ ⊕ ⊕ ⊕
where m = xθ(x) = τ(x) = x , m = xθ(x) = τ(x) = x , m = xθ(x) = τ(x) = x .
1 2 3
{ | − } { | − } { | − − }
Consider the left-invariantmetrics on Sp(k +k +k ) determined by the Ad(Sp(k ) Sp(k ) Sp(k ))-
1 2 3 1 2 3
× ×
invariant inner product on sp(k +k +k ) given by
1 2 3
, =y B y B y B y B y B y B . (4.2)
h· ·i 1 |sp(k1)⊕ 2 |sp(k2)⊕ 3 |sp(k3)⊕ 4 |m1 ⊕ 5 |m2 ⊕ 6 |m3
By the theory given in Lemma 2.1,
Lemma 4.1 ([2]). If a left-invariant metric of the form (4.2) is naturally reductive with respect to
Sp(k +k +k ) L for some closed subgroup L of Sp(k +k +k ), then one of the following holds:
1 2 3 1 2 3
×
(1) y =y =y , y =y .
1 2 4 5 6
(2) y =y =y , y =y .
2 3 6 4 5
(3) y =y =y , y =y .
1 3 5 4 6
(4) y =y =y .
4 5 6
Conversely, if one of the above conditions holds, then the metric of the form (4.2) is naturally reductive
with respect to Sp(k +k +k ) L for some closed subgroup L of Sp(k +k +k ).
1 2 3 1 2 3
×
Using the formula in Lemma 2.2, we have the following lemma.