ISOTROPY IRREDUCIBLE SPACES AND s-REPRESENTATIONS

ERNST HEINTZE AND WOLFGANG ZILLER Dedicated to the memory of Franco Tricerri

Abstract. A simple description of isotropy irreducible spaces G/K is given when G is a classical compact Lie group and K is compact and connected.

1. Introduction

A homogeneous space G/K is called isotropy irreducible if the action of the isotropy group K on the tangent space is irreducible. Classical examples are the irreducible symmetric spaces, whose isotropy representations are also called s-representations. More generally an s-representation is the isotropy representation of a semi-simple symmetric space. The non-symmetric isotropy irreducible spaces with K compact and connected have been classified by Manturov [Ma1],[Ma2],[Ma3] and Wolf [Wo], see also Krämer [Kr]. It turns out that G must be a simple compact Lie group. Looking at Wolf’s list C.T.C. Wall had remarked a striking connection between the isotropy irreducible spaces and symmetric spaces in case G is classical, i.e. SO(n), SU(n) or Sp(n) . A conceptual proof of this fact had been given by Wang and Ziller [WZ]. There are two directions in their proof. In one direction, one needs to show that to each irreducible symmetric space there is a naturally associated isotropy irreducible quotient of SO(n), SU(n) or Sp(n) . The proof of this fact in [WZ] is fairly straightforward. In the other direction, one needs to construct the symmetric space from an isotropy irreducible quotient of SO(n), SU(n) or Sp(n) . The proof of this part in [WZ], although conceptual, involves some tedious calculations and is rather long. The aim of this note is to give a simpler proof of this by constructing the symmetric space in a very direct way. The Theorem we prove is the following.

Theorem. Let K be a compact, connected subgroup of G = SO(n), SU(n) or Sp(n) such that G/K is isotropy irreducible but not symmetric (i.e. (G, K) not a symmetric pair). Let K̃ := K for G = SO(n) , K · U(1) for G = SU(n) , and K · Sp(1) for G = Sp(n) respectively. Then the natural induced representation of K̃ on ℝ^{n} , ℂ^{n} = ℝ^{2n} or ℍ^{n} = ℝ^{4n} is an s-representation unless (G, K) = (SO(7), G_{2}) .

The second author is partially supported by a grant from the National Science Foundation.

The theorem follows from Theorems 3.1,4.1,5.1 below which contains more precise information. In the case of G = SO(n), the proof is particularly simple and uses as the only non trivial ingredient the classification of transitive linear actions on spheres. In case G = SU(n) and G = Sp(n) it also uses a result from [WZ], which is easy in the case of G = SU(n) but more involved if G = Sp(n) .

The idea of the proof is the following:

We put 𝔤̃ := 𝔨̃+ V , where V = ℝ^{n}, ℂ^{n} or ℍ^{n} corresponding to G = SO(n), SU(n) and Sp(n) and try to make 𝔤̃ into a Lie algebra which extends 𝔨̃ , such that [V, V] ⊂ 𝔨̃ . Then (𝔤̃, 𝔨̃) is a symmetric pair and the action of K̃ on V is the isotropy action of a symmetric space. There is a natural way to do that, see e.g. [K], the only problem being the verification of the Jacobi identity for elements x, y, z ∈ V . This can be viewed as the vanishing of a certain symmetric mapping J : Λ^{2}V → Λ^{2}V . One of our main observations is that on 𝔨̃ ⊂ 𝔰𝔬(V) ≃ Λ^{2}V , J is always a multiple of the identity. This part of the paper is completely independent of the rest and might also in other situations be helpful to prove that a given representation is an s-representation (cf. [EH]).

2. Construction of J

Let K̃ be a compact connected subgroup of SO(n) which acts irreducibly on ℝ^{n} . Assume n > 1 , i.e. K̃ ≠ {e} . Let 𝔨̃ be the Lie algebra of K̃ and put 𝔤̃ := 𝔨̃ + V as vector spaces, where V = ℝ^{n} . We extend the bracket on 𝔨̃ to a skewsymmetric bilinear map on 𝔤̃ with [V, V] ⊂ 𝔨̃ by

[A, x] := A x ,
< A, [x, y] >𝔨̃ :=< Ax, y >_{V}

for all A ∈ 𝔨̃ , x, y ∈ V , where <, >_{V} is the standard inner product on V and <, >𝔨̃ is at first an arbitrary Ad(K̃)-invariant inner product on 𝔨̃ which will be specified later to be the negative of the ”Killingform” on 𝔤̃ , restricted to 𝔨̃ . The Killingform of 𝔤̃ is defined as usual by B𝔤̃(X, Y) := tr𝔤̃adX adY where adX(Z) := [X, Z] for all X, Y, Z ∈ 𝔤̃ . Note that B𝔤̃ is negative definite on 𝔨̃ since K̃ acts effectively and orthogonally on V. The so defined bracket on 𝔤̃ is K̃ equivariant, where K̃ acts on 𝔨̃ by the adjoint representation and on V through the embedding K̃ ⊂ SO(n) . This implies that this action of K̃ on 𝔤̃ leaves invariant the bilinear form B𝔤̃.

By differentiating the identity [φ_{t}X, φ_{t}Y] = φ_{t}[X, Y] where φ_{t} is 1-parameter group in K̃ , it follows that the Jacobi identity for 𝔤̃ is satisfied for any three elements in 𝔤̃ if at least one of them lies in 𝔨̃ . Therefore 𝔤̃ is a Lie algebra if and only if the Jacobi identiy holds for all elements in V . Let J_{1}(x, y, z) := [x, [y, z]] + [y, [z, x]] + [z, [x, y]] for x, y, z ∈ V . Then J_{1} is alternating and (x, y, z, u) →< J_{1}(x, y, z), u > is a 4-form on V , which we describe equivalently by a symmetric endomorphism

J : Λ^{2}V → Λ^{2}V

with < J x ∧ y, z ∧ u >=< J_{1}(x, y, z), u > .

That J corresponds to a 4-form is equivalent to < J x ∧ y, x ∧ y >= 0 for all x, y ∈ V which shows in particular that the trace of J vanishes. By construction, J is equivariant with respect to the natural action of K̃ on Λ^{2}V .

Lemma 2.1. The action of K̃ on ℝ^{n} is an s-representation if J = 0 .

Proof. As remarked above 𝔤̃ is a Lie algebra if J vanishes. Actually 𝔤̃ is then a compact semi-simple Lie algebra since the inner product on 𝔤̃ defined by <, >𝔨̃ and <, >_{V} with 𝔨̃⊥V is a biinvariant one and 𝔤̃ has no center since 𝔨̃ acts effectively on V with no 1-dimensional invariant subspaces. Furthermore 𝔤̃ and 𝔨̃ have no ideals in common since such an ideal of 𝔨̃ would act trivially on V. Since by definition [V, V] ⊂ 𝔨̃ , (𝔤̃, 𝔨̃) is thus an irreducible symmetric pair.

Let G̃ = Int(𝔤̃). Then K̃ is canonically embedded in G̃ and G̃/K̃ is an irreducible symmetric space whose isotropy representation is the K̃ action on ℝ^{n}.

We identify 𝔰𝔬(V) with Λ^{2}V by identifying A ∈ 𝔰𝔬(V) with \frac{1}{2} ∑_{i=1}^{n}e_{i} ∧ Ae_{i} . Then

< A, x ∧ y >_{Λ^{2}V} =< Ax, y >_{V} . Note that the identification is an isometry if we endow 𝔰𝔬(V) with the inner product < A, B >_{𝔰𝔬(V)}:= −\frac{1}{2}tr AB .

Lemma 2.2. There exists λ ∈ ℝ with J A = λA for all A ∈ 𝔨̃ ⊂ Λ^{2}V = 𝔰𝔬(V) if and only if the inner product on 𝔨̃ is proportional to B𝔤̃ (restricted to 𝔨̃) .

Proof. J A = λA for all A ∈ 𝔨̃ is equivalent to

< A, J x ∧ y >_{Λ^{2}V} = λ < Ax, y >_{V} for all A ∈ 𝔨̃ , x, y ∈ V .

From the definition of J we get J x ∧ y = −(ad[x, y] − [adx, ady])_{/_{V}} : V → V . Since the Jacobi identity holds whenever one element lies in 𝔨̃ we have (ad[x, y] − [adx, ady])_{/𝔨̃} = 0 .

Thus

< A, J x ∧ y >_{Λ^{2}V} = \frac{1}{2}tr_{V} A ∘ (ad[x, y] − [adx, ady])_{/_{V}}

= \frac{1}{2}tr𝔤̃adA ∘ (ad[x, y] − [adx, ady])

= \frac{1}{2}B𝔤̃(A, [x, y]) − \frac{1}{2}tr𝔤̃ad A ∘ [adx, ady]

Now tr𝔤̃adA ∘ [adx, ady] = tr𝔤̃[adA, adx] ∘ ady = tr𝔤̃adAx ∘ ady = B𝔤̃(Ax, y) , since [adA, adx] = adAx again by the fact that the Jacobi identity holds whenever one element lies in 𝔨̃ . Since B𝔤̃ is K̃-invariant and K̃ acts irreducibly on V it follows from Schur’s lemma that B𝔤̃/_{V} = μ· <, >_{V} for some real number μ. Therefore J = λ · id on 𝔨̃ is equivalent to

(λ + \frac{μ}{2}) < Ax, y >_{V} = \frac{1}{2}B𝔤̃(A, [x, y]) ,

i.e. to

(2λ + μ) < A, [x, y] >𝔨̃= B𝔤̃(A, [x, y])

for all A ∈ 𝔨̃ , x, y ∈ V . Since < A, [x, y] >𝔨̃=< Ax, y >_{V} there is no A ∈ 𝔨̃ perpendicular to all the brackets [x, y] with x, y ∈ V . Hence these brackets span 𝔨̃ and V is not abelian. Finally 2λ+μ ≠ 0 because B𝔤̃_{/𝔨̃} < 0 and the lemma follows.

For the rest of the paper we therefore use −B𝔤̃ restricted to 𝔨̃ as the inner product on 𝔨̃ . We decompose Λ^{2}V = 𝔰𝔬(V) orthogonally into 𝔨̃ + 𝔨̃^{⊥} .

Lemma 2.3. K̃ does not act transitively on S^{n−1} ⊂ ℝ^{n} if and only if 𝔨̃^{⊥} ⊂ Λ^{2}V contains a non zero decomposable element.

Proof. x ∧ y ∈ 𝔨̃^{⊥} if and only if < A, x ∧ y >_{Λ^{2}V} =< Ax, y >_{V} = 0 for all A ∈ 𝔨̃ , i.e. if y ∈ ν_{x} , where ν_{x} is the normal space of the K̃ orbit through x at x . Now ν_{x} contains an element linearly independent from x if and only if K̃ does not act transitively on S^{n−1} .

3. G = SO(n)

Theorem 3.1. Let SO(n)/K be an isotropy irreducible homogeneous space with K connected and compact. Then the representation of K on ℝ^{n} is an s-representation unless (SO(n), K) = (SO(n), SO(n − 1)) , (SO(7), G_{2}) or (SO(8), Spin(7)) .

Remark. SO(7)/G_{2} and SO(8)/Spin(7) = (SO(8)/(±Id))/SO(7) are, for any invariant metric, both isometric to ℝP^{7} with its constant curvature metric. But notice that (SO(7), G_{2}) is not a symmetric pair, whereas (SO(8), Spin(7)) is as well as (SO(n), SO(n − 1)) .

Proof. Let V = ℝ^{n} . Then K acts orthogonally on V . We first claim that K is a maximal connected subgroup of SO(n) . Indeed, if K ⊂ H ⊂ SO(n) , then 𝔨 ⊂ 𝔥 ⊂ 𝔰𝔬(n) and Ad(K) leaves 𝔥 invariant, since H is a subgroup. It now follows that we may assume that K acts irreducibly on V since if V = V_{1} ⊕ V_{2} is left invariant with dim V_{i} ≥ 1 , then K ⊂ SO(V_{1}) × SO(V_{2}) ⊂ SO(V) and by maximality K = SO(V_{1}) × SO(V_{2}) . The representation of K on V is then the isotropy representation of the symmetric space S^{dim V_{1}} × S^{dim V_{2}}. Hence K acts as an s-representation on ℝ^{n} unless dim V_{1} = 1 or dim V_{2} = 1 , i.e. K ≃ SO(n − 1) . Notice that S^{1} is not a semisimple symmetric space.

We now can apply the results from Section 2. We decompose Λ^{2}V = 𝔰𝔬(V) orthogonally into 𝔨 + 𝔨^{⊥} . By assumption K acts irreducibly on 𝔨^{⊥} . By lemma 2.2, J = λid on 𝔨 = 𝔨̃ for some λ ∈ ℝ and hence leaves 𝔨^{⊥} invariant. Since J is Kequivariant, J = μid on 𝔨^{⊥} for some μ ∈ ℝ by Schur’s Lemma. If K does not act transitively on S^{n−1} ⊂ V then there exists a non-trivial decomposable element x ∧ y in 𝔨^{⊥} which implies μ = 0 since < J x ∧ y , x ∧ y >= 0 . But then also λ = 0 since trJ = 0 and thus J = 0 in this case which implies that the action of K on ℝ^{n} is an s-representation. If K does act transitively on S^{n−1} ⊂ V then it follows from the classification of such actions (see [Be, 7.13]) and the fact that K must be maximal in SO(n) that K ⊂ SO(n) must be one of the following

1) U(n) ⊂ SO(n)
2) Sp(n)Sp(1) ⊂ SO(4n)
3) Spin(9) ⊂ SO(16)
4) Spin(7) ⊂ SO(8)
5) G_{2} ⊂ SO(7)

The first three are induced by isotropy representations of rank 1 symmetric spaces and hence the Theorem follows.

4. G = SU(n)

Theorem 4.1. Let SU(n)/K be an isotropy irreducible homogeneous space with K compact and connected. Then the action of K on ℂ^{n} can be extended to an action of K · U(1) on ℂ^{n} by (k, e^{iϑ})v = (kv) · e^{iϑ} and the underlying real representation on ℝ^{2n} is the isotropy representation of a hermitian symmetric space unless K = Sp(n) acting on ℂ^{2n} with n > 1 .

Remark. Notice that the exception SU(2n)/Sp(n) is a symmetric pair.

Proof. As in theorem 1 it follows that K is maximal in SU(n) and that if K is reducible as a complex representation on ℂ^{n} then K = S(U(p) × U(q)) ⊂ SU(p + q) and the representation of K·U(1) is then the isotropy representation of the symmetric space ℂP^{p} × ℂP^{q} . But it can also happen that K acts irreducibly as a complex representation on ℂ^{n} but not as a real representation on

ℝ^{2n} . This only occurs if there exists a real representaion of K on ℝ^{n} whose complexification is the given representation on ℂ^{n} . But then K ⊂ SO(n) ⊂ SU(n) and by maximality K = SO(n) . Then K̃ = SO(n)U(1) acts on ℂ^{n} and is the isotropy representation of the symmetric space SO(n + 2)/SO(n)SO(2) .

Hence we can assume that K acts irreducibly on ℝ^{2n} . Let K̃ = K · U(1) act on ℂ^{n} as above.

First we decompose Λ^{2}V = 𝔰𝔬(2n) = 𝔲(n)⊕𝔪_{2} , where 𝔲(n) are the skew hermitian matrices and 𝔪_{2} the hermitian matrices in 𝔰𝔬(2n). The complex structure on V which is given by multiplication with i induces a map I on Λ^{2}V by I(v∧w) = iv∧iw . Notice that under the identification Λ^{2}V = 𝔰𝔬(V) this becomes the map I(A) = −iAi . Hence 𝔲(n) and 𝔪_{2} are precisely the +1 and −1 eigenspaces of I. Furthermore 𝔰𝔲(n) = 𝔨 ⊕ 𝔪_{1} and 𝔨 acts irreducibly on 𝔪_{1} by assumption. Thus 𝔲(n) = 𝔨̃ ⊕ 𝔪_{1} and hence Λ^{2}V = 𝔨̃ ⊕ 𝔪_{1} ⊕ 𝔪_{2} . In [WZ, p.317] it was shown that K must be a simple Lie group unless SU(n)/K = SU(p · q)/SU(p)SU(q) with the embedding given by the tensor product representation. But in this case the representation of K · U(1) is precisely the isotropy representation of the hermitian symmetric space SU(p + q)/S(U(p) × U(q)) . Hence we can assume that K is simple. It was then shown in [WZ, Proof 4.8] that K̃ acts irreducibly on 𝔪_{2} (Since 𝔪_{2} = Λ^{2}ℂ^{n} the action of K is precisely the induced one on Λ^{2}ℂ^{n}) . Now K̃ acts irreducibly on 𝔪_{1} and 𝔪_{2} .

Lemma 2.2 implies that J (𝔨̃) ⊂ 𝔨̃ and J_{|𝔨̃} = λid and hence J (𝔪_{1} +𝔪_{2}) ⊂ 𝔪_{1} +𝔪_{2} . We claim that also J (𝔪_{i}) ⊂ 𝔪_{i} . Indeed since J is invariant under K̃ it must commute with I and hence J preserves the eigenspaces 𝔲(n) and 𝔪_{2} of I . By Schur’s Lemma we get

J_{|𝔨̃} = λ · id , J_{|𝔪_{1}} = μ · id , J_{|𝔪_{2}} = ν · id .

We first assume again that K̃ does not act transitively on S^{2n−1} ⊂ V . Then as before it follows that there exists a non zero decomposable element x ∧ y ∈ 𝔪_{1} ⊕ 𝔪_{2} .

Notice that 𝔨̃ = 𝔨 ⊕ ℝ and ℝ is generated by the complex structure A_{0}x = ix . Hence 0 =< x ∧ y, A_{0} >=< i · x, y > . We can then decompose x ∧ y into the

𝔪_{1} ⊂ 𝔲(n) and 𝔪_{2} components (the ±1 eigenspaces of I)

x ∧ y = \frac{1}{2}(x ∧ y + ix ∧ iy) + \frac{1}{2}(x ∧ y − ix ∧ iy) .

Since y⊥ix these components have the same length and since < J (x∧y), x∧y >= 0 it follows that μ + ν = 0 .

Now consider an arbitrary x ∧ y⊥A_{0} i.e. y⊥ix. Let x ∧ y = a + b + c with a ∈ 𝔨̃, b ∈ 𝔪_{1}, c ∈ 𝔪_{2} . As before it follows that a+b ∈ 𝔲(n) and c have the same length and hence it follows from 0 =< J (x ∧ y), x ∧ y > that 0 = λ∥a∥^{2} + μ∥b∥^{2} + ν∥c∥^{2} = λ∥a∥^{2} + μ(∥b∥^{2} − ∥c∥^{2}) = (λ − μ)∥a∥^{2} . λ = μ(≠ 0) is not possible since 𝔲(n) = 𝔨̃+ 𝔪_{1} contains decomposable elements of the form x ∧ ix .

Therefore a = 0 , which means that x ∧ y⊥𝔨̃ for any y⊥ix . In particular, if A ∈ 𝔨 , then 0 =< A, x ∧ y >=< Ax, y > as long as y⊥ix . Hence Ax = f(x) · ix which implies that A is a multiple of A_{0} . Hence 𝔨 = 0, which is a contradiction. Finally if K̃ acts transitively on S^{2n−1} ⊂ V , it follows from the classification that K̃ = U(n) on ℂ^{2n} or K̃ = Sp(n) · U(1) on ℂ^{2n} . In the first case SU(n)/K is a point and in the second case SU(n)/K = SU(2n)/Sp(n) is a symmetric space, but Sp(n)U(1) acting on ℂ^{2n} is not an s-representation unless n = 1.

5. G = Sp(n)

Theorem 5.1. Let Sp(n)/K be an isotropy irreducible homogenous space with K connected and compact. Then the action of K on ℍ^{n} can be extended to an action of K · Sp(1) on ℍ^{n} by (k, q)v = (k · v)q and the underlying real representation on ℝ^{4n} is the isotropy representation of a quaternionic symmetric space unless K = Sp(p) × Sp(q) in Sp(p + q) .

Remark. For the exception Sp(n)/K is again a symmetric space.

Proof. As before it follows that K is maximal in Sp(n) and that if K acts reducibly as a quaternionic representation on V = ℍ^{n} , then K = Sp(p) × Sp(q) ⊂ Sp(p + q). But K can act irreducibly on ℍ^{n} and reducibly on ℂ^{2n} . This can only happen if the representation of K on ℂ^{2n} is of the form σ + σ^{∗} where σ is an irreducible complex representation. But then K ⊂ U(n) ⊂ Sp(n) and by maximality K = U(n) in which case the action of K̃ = U(n)Sp(1) = U(n)SU(2) is the isotropy representation of the symmetric space SU(n + 2)/S(U(n)U(2)).

Finally, if the representation of K on ℍ^{n} and ℂ^{2n} are both irreducible, then the underlying representation of K on ℝ^{4n} is also irreducible. Hence we can again assume that K acts irreducibly on ℝ^{4n}.

Since ℍ is not commutative, we have to be careful with our identification. We regard ℍ^{n} as a vector space over ℍ from the right since then a matrix A ∈ Sp(n) , which acts on vectors in ℍ^{n} in the normal way A·v = ∑ a_{ij}v_{j} becomes an ℍ linear map and the ℍ linear maps in SO(4n) are then precisely the elements in Sp(n) . Sp(n)Sp(1) also acts on ℍ^{n} via (A, q)v = A(v) · q and hence Sp(n)Sp(1) ⊂ S0(4n) . Let K̃ = KSp(1) which acts on ℍ^{n} as above. By assumption we have 𝔰𝔭(n) = 𝔨 ⊕ 𝔪_{1} and Sp(n) acts irreducibly on 𝔪_{1} . Furthermore 𝔰𝔬(4n) = 𝔰𝔭(n)⊕𝔰𝔭(1)⊕𝔪_{2} = 𝔨̃⊕𝔪_{1}⊕𝔪_{2} . In [WZ, p.310] it was shown that K is a simple Lie group unless K = SO(n)Sp(1) in which case K̃ = SO(n)Sp(1)Sp(1) = SO(n)SO(4) is the isotropy representaion of the Grassmannian SO(n + 4)/SO(n)SO(4) . The isotropy representation of SO(4n)/Sp(n)Sp(1) on 𝔪_{2} is the irreducible representation(Λ^{2}ν_{n} − id) ⊗ Ad_{Sp(1)} where ν_{n} is the standard representation of Sp(n) on ℂ^{2n} and in [WZ, 4.4(b)] it was shown that 𝔪_{2} remains irreducible under the action of K̃ .

In 𝔰𝔭(1) we have the elements i, j, k , which regarded as maps V → V are simply right multiplications by i, j, k . These induce maps I, J, K on Λ^{2}V by I(v ∧ w) = v · i ∧ w · i , etc.. They satisfy I^{2} = J^{2} = K^{2} = Id , I · J = J · I = K , etc. and act on A ∈ 𝔰𝔬(V) by I(A)(v) = −A(vi)i .

We denote by E_{I}^{±}, E_{J}^{±}, E_{K}^{±} the ±1 eigenspaces of these linear maps in Λ^{2}V .

Notice that E_{I}^{+} ∩ E_{J}^{+} = 𝔰𝔭(n) since any element in E_{I}^{+} , regarded as a linear map V → V is invariant under right multiplication by i and invarience under I and J implies invariance under K .

We claim that we get a decomposition

Λ^{2}V = E_{I}^{+} ∩ E_{J}^{+} ⊕ E_{I}^{+} ∩ E_{J}^{−} ⊕ E_{I}^{−} ⊕ E_{J}^{+} ⊕ E_{I}^{−} ⊕ E_{J}^{−} .

Indeed these subspaces are clearly disjoint and x ∧ y ∈ Λ^{2}V can be written

x ∧ y = \frac{1}{4}(x ∧ y + xi ∧ yi + xj ∧ yj + xk ∧ yk)

+ \frac{1}{4}(x ∧ y + xi ∧ yi − xj ∧ yj − xk ∧ yk)

+ \frac{1}{4}(x ∧ y − xi ∧ yi + xj ∧ yj − xk ∧ yk)

+ \frac{1}{4}(x ∧ y − xi ∧ yi − xi ∧ yi + xk ∧ yk)

and each summand lies in the respective subspace. Notice that the i, j, k ∈ 𝔰𝔭(1) lie in E_{I}^{+} ∩ E_{J}^{−}, E_{I}^{−} ∩ E_{J}^{+}, E_{I}^{−} ⊕ E_{J}^{−} respectively.

Since Sp(n) commutes with the right action of Sp(1) on V , the action of K on Λ^{2}V commutes with I, J, K and hence leaves the above decomposition invariant. Furtermore J also commutes with I, J, K and hence also leaves the above decomposition invariant. J also preserves 𝔨̃ and J_{|𝔨̃} = λ·Id . Since J preserves E_{I}^{+}∩E_{J}^{+} = 𝔰𝔭(n) ⊃ 𝔨 we also have J (𝔨) ⊂ 𝔨 and J (i) = λi, J (j) = λj, J (k) = λk. Since we have 𝔪_{1} ⊂ 𝔰𝔭(n) it follows that J (𝔪_{i}) ⊂ 𝔪_{i} and we assume J_{|𝔪_{1}} = μ · id J_{|𝔪_{2}} = ν · id .

Next, notice that the last 3 summands in the above decomposition of Λ^{2}V are isomorphic to each other. E.g. A → A · k is the isomorphism from E_{I}^{+} ∩ E_{J}^{−} to E_{I}^{−} ∩ E_{J}^{+} since I(Ak) = −iAki = iAik = −Ak since A ∈ E_{I}^{+} and J(Ak) = −jAkj = jAjk = Ak since A ∈ E_{J}^{−} and A → Aj is the isomorphism from E_{I}^{+} ∩ E_{J}^{−} = E_{I}^{+} ∩ E_{K}^{−} to E_{I}^{−} ∩E_{J}^{−} = E_{I}^{−} ∩E_{K}^{+} . Hence 𝔪_{2} decomposes into equivalent representations under K, 𝔪_{2}^{i}, i = 1, 2, 3 , one in each of these summands. Summarising we have the following description of the eigenspaces of J :

E_{I}^{+} ∩ E_{J}^{+} = 𝔨 + 𝔪_{1} with J_{|𝔨} = λId J_{|𝔪_{1}} = μId
E_{I}^{+} ∩ E_{J}^{−} = i + 𝔪_{2}^{1} with J (i) = λi J_{|𝔪_{2}^{1}} = νId
E_{I}^{−} ∩ E_{J}^{+} = j + 𝔪_{2}^{2} with J (j) = λj  J_{|𝔪_{2}^{2}} = νId.
E_{I}^{−} ∩ E_{J}^{−} = k + 𝔪_{2}^{3} with J (k) = λk J_{|𝔪_{2}^{3}} = νId

The remaining argument is now similar to the proof in the unitary case.

We first assume that K̃ does not act transitively on S^{4n−1} ⊂ V . Then there exists a non zero decomposable element x ∧ y ∈ 𝔪_{1} + 𝔪_{2} . Since x ∧ y⊥𝔨̃ we also have x ∧ y⊥𝔰𝔭(1) . But this is equivalent to < x, yi >=< x, yj >=< x, yk > . This then implies that the above decomposition of x ∧ y into its 4 components alle have the same length and hence < J (x ∧ y), x ∧ y >= 0 implies

μ + 3ν = 0 .

Next we consider any x ∧ y⊥𝔰𝔭(1) or equivalently y⊥xi, xj, xk . Then x ∧ y = v_{1} + v_{2} + v_{3} + v_{4} where v_{i} are the 4 components of x ∧ y and v_{1} = v′_{1} + v′′_{1} where v′_{1} ∈ 𝔨 and v′′_{1} ∈ 𝔪_{1} . As before, all v_{i} have the same length. Now < J (x ∧ y), x ∧ y >= 0 implies

λ∥v′_{1}∥^{2} + μ∥v′′_{1}∥^{2} + 3ν∥v_{2}∥^{2} = λ∥v′_{1}∥^{2} + μ(∥v′′_{1}∥^{2} − ∥v_{2}∥^{2}) = (λ − μ)∥v′_{1}∥^{2} .

Hence there are 2 possibilities. Either λ = μ or v′_{1} = 0 . If λ = μ = −3ν we get a contradiction to trJ = 0 : trJ = λ(2n^{2} + n + 3 − \frac{1}{3} · 3(2n^{2} − n − 1)) = λ(2n + 4) ≠ 0 unless λ = μ = ν = 0 . If v′_{1} = 0 then any x ∧ y⊥𝔰𝔭(1) satisfies x ∧ y⊥𝔨 . But then A ∈ 𝔨 satisfies A⊥x ∧ y or < Ax, y >= 0 whenever y⊥xi, xj, xk . Hence Ax lies in the span of xi, xj , and xk for all x . We claim that it must then be of the form Ax = q · x for some q ∈ Im ℍ . Indeed, since A ∈ 𝔨 A is linear over ℍ , acting from the right. The matrix of A is then given by A(e_{i}) = _{∑j}e_{j}a_{ji} so that Av = Σa_{ij}v_{j} . Since A preserves quarternionic lines A(e_{i}) = e_{i}q_{i} for some q_{i} ∈ Im ℍ . But in order for A to preserve the quaternionic line through Σe_{i} we need that alle q_{i} are the same. Hence A(v) = q · v for some q ∈ Im ℍ . But the quaternionic line through v and q · v are not identical for all v . Hence no such A exists or 𝔨 = 0 which is a contradiction.

Finally we assume that K̃ acts transitively on S^{4n−1} . But from the classification it follows that this can only happen if K̃ = Sp(n)Sp(1) which is again an s-representation.

References

[Be] Besse, A. , Einstein manifolds, Ergeb. Math. Grenzgeb. 3. Folge, Bd. 10, Berlin Heidelberg New York: Springer 1987.

[EH] Eschenburg, J., Heintze, E. , Polar representations and symmetric spaces ,Preprint.

[K] Kostant, B. , On invariant skew tensors, Proc. Natl. Acad. Sci. 42 , 148–151 (1956).

[Kr] Krämer, M. , Eine Klassifikation bestimmter Untergruppen kompakter zusammenh angender Liegruppen, Commun. Algebra 3, 691 – 737 (1975).

[Ma1] Manturov, O.V. , Homogeneous asymmetric Riemannian spaces with an irreducible group of motions, Dokl. Akad. Nauk SSSR 141, 792 – 795 (1961).

[Ma2] Manturov, O.V. , Riemannian spaces with orthogonal and symplectic groups of motions and an irreducible group of rotations, Dokl. Akad. Nauk. SSSR 141, 1034 – 1037 (1961).

[Ma3] Manturov, O.V. , Homogeneous Riemannian manifolds with irreducible isotropy group , Tr. Semin. Vector Tensor Anal. 13, 68 – 145 (1966).

[Wo] Wolf, J.A. , The geometry and structure of isotropy irreducible homogeneous spaces , Acta Math. 120, 59 – 148 (1968); correction, Acta Math. 152, 141 – 142 (1984).

[WZ] Wang, M., Ziller, W. , Symmetric spaces and strongly isotropy irreducible spaces , Math. Ann. 296, 285 – 326 (1993).

Institut für Mathematik, Universität Augsburg, D-86135 Augsburg, Germany E-mail address: Heintze@uni-augsburg.de

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 U.S.A.

E-mail address: wziller@math.upenn.edu