Stability of Einstein metrics on symmetric spaces of compact type

We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces $\mathrm{SU}(n)$, $n\geq3$, and $E_6/F_4$. Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type.


Introduction
Let M be a closed manifold of dimension n > 2. It is a well-known fact (see [2]) that Einstein metrics are critical points of the total scalar curvature functional g → S(g) = M scal g vol g , also called the Einstein-Hilbert action, restricted to the space of Riemannian metrics of a fixed volume. In general, these critical points are neither maximal nor minimal. If we, however, restrict S to the set S of all Riemannian metrics on M of the same fixed volume that have constant scalar curvature, then some Einstein metrics are maximal, while others form saddle points. To examine this, one considers the second variation S ′′ Moreover, the smallest eigenvalue of ∆ L on trace-free symmetric 2-tensors has been computed in each case (see [3]). Among the spaces that possess infinitesimal deformations, we have ∆ L ≥ 2E on S 2 0 (M) on the spaces SU(n)/ SO(n), SU(2n)/ Sp(n) (n ≥ 3), SU(p + q)/S(U(p) × U(q)) (p ≥ q ≥ 2), which shows that they are linearly stable. However, this did not fully settle the stability problem on irreducible symmetric spaces of compact type. In particular, it had not been decided whether unstable directions exist on the spaces SU(n) (where n ≥ 3), E 6 /F 4 , F 4 / Spin(9), Sp(p + q)/(Sp(p) × Sp(q)) (where p ≥ q ≥ 2 or p = 2, q = 1).
In these cases, we know that ∆ L has eigenvalues smaller than 2E on the space of trace-free symmetric 2-tensors, but it had not been checked whether the corresponding eigentensors are also divergence-free. In a recent paper [14], U. Semmelmann and G. Weingart show the following results.
The current article finally resolves the question of stability for the last remaining cases by proving the following.
1.3 Theorem. The symmetric spaces SU(n), where n ≥ 3, as well as E 6 /F 4 are linearly stable.
Consider a manifold (M, g) that is a Riemannian product of Einstein manifolds. Then (M, g) is Einstein if and only if the factors have the same Einstein constant E. It turns out that if E > 0, then (M, g) is always unstable (see [12], Prop. 3.3.7). For example, if (M, g) is the Riemannian product of two Einstein manifolds (M n i i , g i ) (i = 1, 2) with the same Einstein constant, then an unstable direction is given by h := n 2 π * 1 g 1 − n 1 π * 2 g 2 , where π i : M → M i are the projections onto each factor, respectively. In particular, a product of symmetric spaces of compact type is always unstable since the factors have positive curvature. If we take (M, g) to be locally symmetric of compact type, we cannot in general conclude its instability from the instability of its universal cover (M ,g). The same holds for the existence of infinitesimal Einstein deformations. On the other hand, if (M ,g) is infinitesimally non-deformable (resp. stable), then the same follows for (M, g). In [11], N. Koiso has proved the infinitesimal non-deformability of a large class of such manifolds: 1.4 Theorem. Let (M, g) be a locally symmetric Einstein manifold of compact type. Let (M ,g) be its universal cover and (M,g) = N i=1 (M i , g i ) its decomposition into irreducible symmetric spaces.
2. If N = 2 and M i are neither of the spaces listed in Theorem 1.1, 1., nor G 2 or any Hermitian space except S 2 , then (M, g) is infinitesimally non-deformable.
3. If N ≥ 3 and M i are neither of the above nor S 2 , then (M, g) is infinitesimally non-deformable.
A closely related notion of stability arises in the study of the Ricci flow. The fixed points (modulo diffeomorphisms and scaling) of the Ricci flow are called Ricci solitons. The ν-entropy defined by G. Perelman is a quantity that increases monotonically under the Ricci flow. Its critical points are the shrinking gradient Ricci solitons, which include Einstein manifolds. An Einstein metric is called ν-linearly stable if the second variation of the ν-entropy is negative-semidefinite. H.-D. Cao, R. Hamilton and T. Ilmanen first studied the ν-linear stability of Einstein metrics (see [4]). It turns out that an Einstein metric is ν-linearly stable if and only if ∆ L ≥ 2E on tt-tensors and if the first nonzero eigenvalue of the ordinary Laplacian on functions is bounded below by 2E as well. In particular, ν-linear stability implies linear stability with respect to the Einstein-Hilbert action. In [3], the ν-linear stability of irreducible symmetric spaces of compact type is completely decided.
There is yet another notion of stability worth mentioning. It is motivated, for example, by the investigation of Anti-de Sitter product spacetimes and generalized Schwarzschild-Tangherlini spacetimes (see [5] or [8]). An Einstein manifold (M n , g) with Einstein constant E is called physically stable if on tt-tensors. This critical eigenvalue is significantly smaller than the one from stability with respect to the Einstein-Hilbert action, and even negative for n > 9. As it turns out, all irreducible symmetric spaces of compact type are physically stable (see [5]). If (M, g) is a product of at least two symmetric spaces of compact type, then the smallest eigenvalue of ∆ L on tt-tensors is actually equal to 0; hence (M, g) is physically stable if and only if n ≥ 9.
In Section 2, we fix the notation and definitions used throughout this work. In particular, we elaborate on the notion of stability of an Einstein metric. In Section 3, we recall some tools from the harmonic analysis of homogeneous spaces that are routinely employed. Furthermore, we prove a technical lemma that allows us to make explicit computations involving the divergence operator. A helpful formula for the dimension of tt-eigenspaces of the Lichnerowicz Laplacian is worked out in Section 4, generalizing a proposition of Koiso and utilizing properties of Killing vector fields on Einstein manifolds. Section 5 uses representation theory to determine the stability of SU(n), making use of the formula from Section 4; in Section 6, the same is done for E 6 /F 4 . A different approach for proving the stability of both spaces that involves explicit computations of the divergence operator can be found in the Appendix.

Preliminaries
Throughout what follows, let (M, g) be a compact, orientable Riemannian manifold. Let ∇ denote the Levi-Civita connection of g. The Riemannian curvature tensor, Ricci tensor and scalar curvature are in our convention given as scal := tr g Ric, respectively. 1 The action of the Riemannian curvature extends to an endomorphism on tensor bundles as where ∇ also denotes the induced connection on the respective tensor bundle. Furthermore, let S p (M) = Γ(Sym p T * M) for p ≥ 0. We denote by the divergence operator on symmetric tensors, given by The space of tt-tensors, i.e. trace-and divergence-free symmetric 2-tensors on M, is denoted by S 2 tt (M). Let δ * : S p (M) → S p+1 (M) be the formal adjoint 2 of the divergence operator. It can be written as where (e i ) is a local orthonormal basis of T M. Here, ⊙ denotes the (associative) symmetric product, defined by for α ∈ Sym k T , β ∈ Sym l T , where T is any vector space and the symmetrization map sym : T ⊗k → Sym k T is given by for X 1 , . . . , X k ∈ T . This is analogous to the definition of the wedge product via the alternation map. For tensors α, β of rank 1, we have It should be noted that δ * X ♭ = L X g for any vector field X ∈ X(M). Consequently, the kernel of δ * on Ω 1 (M) is (via the metric) isomorphic to the space of Killing vector fields on (M, g). More generally, symmetric tensors α ∈ S k (M) with δ * α = 0 are called Killing tensors of rank k, and δ * is sometimes called the Killing operator.
2.1 Definition. On tensors of any rank, the following operators are defined: 1. The curvature endomorphism q(R) is defined by where (e i ) is a local orthonormal basis of T M and the asterisk indicates the natural action of Λ 2 T ∼ = so(T ).
2. The Lichnerowicz Laplacian ∆ L is defined by Recall that on Ω p (M), p ≥ 0, this coincides with the Hodge Laplacian ∆.
On the space of Riemannian metrics on M, which is an open cone in S 2 (M), the total scalar curvature functional or Einstein-Hilbert action is given by for any Riemannian metric g on M. As mentioned earlier, if we restrict this functional to the space of metrics of a fixed total volume, then Einstein metrics are precisely the critical points of the restriction of S.
Let (M, g) be an Einstein manifold with Einstein constant E ∈ R, that is and suppose that (M, g) is not isometric to a standard round sphere. Denote It is well known (see [2]) that there is a decomposition of S 2 (M), which is orthogonal with respect to the second variation S ′′ g of the total scalar curvature functional, into the four summands S 2 (M) = Rg ⊕ C ∞ g (M)g ⊕ im δ * ⊕ S 2 tt (M). These correspond to infinitesimal changes in the metric by homothety, volume-preserving conformal scaling, the action of diffeomorphisms, and moving within S, respectively. The second variation S ′′ g is positive on C ∞ g (M)g, zero on im δ * and is given by where it has finite coindex and nullity; that is, the maximal subspace of S 2 tt (M) where S ′′ g is nonnegative is finite-dimensional. In fact, the null directions in S 2 tt (M) are precisely the infinitesimal Einstein deformations of g, i.e. infinitesimal deformations of g that preserve the Einstein property, the total volume and are orthogonal to the orbit of g under diffeomorphisms.

Invariant differential operators
Let G be a compact Lie group with Lie algebra g and K a closed subgroup such that (M = G/K, g) is a reductive Riemannian homogeneous space with K-invariant decomposition g = k⊕m, where k is the Lie algebra of K and m is the reductive complement which is canonically identified with the tangent space T o M at the base point o := eK ∈ M. Recall that for some representation ρ : K → Aut V , the left-regular representation on the space of K-equivariant smooth functions C ∞ (G, V ) K is defined as for any x ∈ G. If V can be expressed in terms of the isotropy representation m, then G × ρ V is a tensor bundle; for example, we have where Sym 2 0 , S 2 0 denotes the space of trace-free elements with respect to the metric. Note that the invariant Riemannian metric yields an equivalence between m and m * .
Suppose that V is a complex representation. Choose a maximal torus T inside G with Lie algebra t. Recall that up to equivalence, every irreducible finite-dimensional complex representation of G is characterized by its highest weight γ ∈ t * . By the Peter-Weyl theorem and Frobenius reciprocity (cf. [15]), the left-regular representation C ∞ (G, V ) K can be decomposed into irreducible summands as 3 where γ runs over all highest weights of G-representations and (V γ , ρ γ ) is the (up to equivalence) unique irreducible representation of G with highest weight γ. For any Since the Lichnerowicz Laplacian ∆ L on Γ(G× ρ V ) is a G-invariant differential operator, Schur's Lemma implies that on each of the isotypical subspaces In order to obtain the spectrum of ∆ L , one would have to find the eigenvalues of each L γ -a potentially very cumbersome task. We will shortly see that this matter is considerably simpler in the symmetric case.
Fix an Ad-invariant inner product ·, · g on the Lie algebra g. If we assume that G is semisimple, one such inner product is given by −B, where B is the Killing form on g, defined by Recall that for any representation π : G → Aut W , the Casimir operator Cas G π with respect to the chosen inner product is an equivariant endomorphism of W , defined as for any orthonormal basis (e i ) of g.
The following proposition combines two well-known results that allow us to compute the eigenvalues of ∆ L on compact symmetric spaces, the latter being a formula due to H. Freudenthal (cf. [6]).
3.1 Proposition. Let (M = G/K, g) be a compact Riemannian symmetric space where the Riemannian metric is induced by an Ad-invariant inner product ·, · g on g, and let ρ : K → Aut V be a representation.

On the left-regular representation
2. On each irreducible representation V γ , the Casimir eigenvalue is given by where δ g is the half-sum of positive roots and ·, · t * is the inner product on t * induced by the inner product on t ⊂ g.
3.2 Remark. The first statement is a consequence of a more general result. Let G be a compact Lie group and (M = G/K, g) be a reductive Riemannian homogeneous space. To the reductive decomposition corresponds a canonical G-invariant connection on M (also called the Ambrose-Singer connection), which we denote by∇. This connection in turn defines a curvature tensorR and an analogue to the Lichnerowicz Laplacian viā called the standard Laplacian of this connection (introduced in [13]). Then, in fact, ∆ = Cas G ℓ on Γ(G × ρ V ). The above statement follows when we note that on Riemannian symmetric spaces, the Ambrose-Singer connection coincides with the Levi-Civita connection.
According to (PW), we can write the complexified left-regular representation on tracefree symmetric 2-tensors as Recall that irreducible symmetric spaces of compact type can be endowed with a Riemannian metric induced by the Killing form (the so-called standard metric). In this case, the critical eigenvalue of ∆ L is 2E = 1. Supposing we have a representation V γ with subcritical Casimir eigenvalue Cas G γ < 1 occurring in this decomposition, it remains to check whether the tensors in the corresponding subspace are divergence-free. By Schur's Lemma, the G-invariant operator is constant on each irreducible subspace. This means that we can regard δ as a linear mapping δ : , the so-called prototypical differential operator associated to δ and V γ . For a further discussion of invariant differential operators on homogeneous spaces, we refer the reader to Section 2 of [14].
The following lemma is of use when we need to calculate δ explicitly. A derivation of essentially the same formula can also be found in [14], Section 2.
where ρ γ is the representation of G on V γ . The covariant derivative of h at the base point may be expressed by

tt-Eigenspaces of the Lichnerowicz Laplacian
We return to the general setting of a compact Einstein manifold (M, g). Define so that θα is precisely the trace-free part of δ * α ∈ S 2 (M). The kernel of this operator is (via the metric) isomorphic to the space of conformal Killing fields on (M, g), that is, the space of vector fields X ∈ X(M) such that L X g = f g for some f ∈ C ∞ (M). We thus call θ the conformal Killing operator.
The following lemma is a generalization of a proposition by Koiso [11,Prop. 3.3]. For the proof, we refer the reader to the Appendix.
4.1 Lemma. Let (M, g) be a compact Einstein manifold of dimension n ≥ 3. For any λ ∈ R, the dimension of the eigenspace of ∆ L to the eigenvalue λ on tt-tensors is given by At first glance, the third term on the right hand side of the above formula does not look very amenable to computation. However, matters are made easier if we observe the following properties of (conformal) Killing vector fields on Einstein manifolds, both of which are proven in the Appendix.
If we assume that (M, g) is not isometric to a standard sphere, we can immediately conclude that the intersection ker(∆ − λ) Ω 1 (M ) ∩ ker θ is trivial if λ = 2E. By virtue of Lemma 4.1, we obtain the following.
4.4 Corollary. Let (M, g) be a compact Einstein manifold that is not isometric to a standard round sphere, and let E be its Einstein constant. For any λ = 2E, the dimension of the eigenspace of ∆ L to the eigenvalue λ on tt-tensors is given by As in the proof of Lemma 4.1, the dimension formula essentially arises from the short exact sequence and the fact that the Laplacian commutes with every arrow. In the homogeneous case, we note that we have a short exact sequence of G-representations and use Frobenius reciprocity to arrive at the statement.
Let g and k denote the corresponding Lie algebras of G and K, respectively. We endow M with the standard metric g induced by the Killing form on g. Hence, M is Einstein with critical eigenvalue 2E = 1. The reductive decomposition of g with respect to g is given by The K-representations k,k and m are all equivalent. We denote by E = C n the standard representation of K.
5.1 Lemma. Let V γ be an irreducible complex representation of G with Cas G γ < 1 and Then V γ is equivalent to one of the G-representations E ⊗ E * and E * ⊗ E. In fact, Proof. Let t be the torus of diagonal matrices in k. The dual t * is generated by the weights ε 1 , . . . , ε n of the defining representation E. Explicitly, Fix the ordering on roots and weights such that the simple roots of k are given by The semigroup of dominant integral weights is then generated by the fundamental weights Let γ, γ ′ ∈ t * be two dominant integral weights. In particular, they satisfy γ, γ ′ t * ≥ 0.
Using Freudenthal's formula for the Casimir operator Cas K γ of a K-representation V γ , this implies the estimate In particular, we obtain Cas K γ ≥ r a r Cas K ωr ( * ) for γ = n−1 r=1 a r ω r . The Casimir eigenvalues of the fundamental representations are given as Cas K ωr = (n + 1)r(n − r) 2n 2 for r = 1, . . . , n − 1. Note that this expression is symmetric around r = n 2 and strictly increasing for r ≤ n 2 . Furthermore, we can compute that cf. table on p. 15 of [14]. Combining the above with inequality ( * ), we can deduce that if γ is a highest weight with Cas K γ < 1, then necessarily γ ∈ {0, ω 1 , ω n−1 , ω 2 , ω n−2 , ω 3 , ω n−3 if n=6,7

}.
These dominant integral weights are, respectively, highest weights of the representations The irreducible representations of G = K × K are precisely the tensor products of irreducible representations of K. Let γ, γ ′ be highest weights of K-representations such that Cas G (γ,γ ′ ) = Cas K γ + Cas K γ ′ < 1 holds. Assuming that γ, γ ′ = 0, we conclude that γ, γ ′ ∈ {ω 1 , ω n−1 }. This yields the four pairwise inequivalent G-representations E ⊗E, E ⊗E * , E * ⊗E and E * ⊗E * . Furthermore, in the case of γ = 0 or γ ′ = 0 we obtain the representations of K that were listed above, composed with the projection onto one factor, G → K : (k 1 , k 2 ) → k 1 or (k 1 , k 2 ) → k 2 , respectively. By restricting the mentioned G-representations to K via the embedding we again obtain the irreducible K-representations C, E, E * , Λ 2 E, Λ 2 E * , Λ 3 E, Λ 3 E * as well as the tensor product representations E ⊗ E, E ⊗ E * and E * ⊗ E * . The latter are not irreducible, but decompose into irreducible summands as follows: Here E ⊗ 0 E * is the set of trace-free elements of E ⊗ E * when regarded as n × n-matrices over C. As a representation of K, we have The K-representation Sym 2 k C ∼ = Sym 2 (E ⊗ 0 E * ) appears on one hand as a summand of On the other hand, the symmetric power of the tensor product is given by 4 The tensor products Sym 2 E ⊗ Sym 2 E * and Λ 2 E ⊗ Λ 2 E * can in turn be decomposed into By comparing summands, we see that Hence, the trace-free part is given by Now that we have decomposed the relevant representations into irreducible summands, we recognize that E ⊗ E * and E * ⊗ E are the only two of the specified subcritical representations of G that, after restriction to K, have a common summand with Sym 2 0 k C . In each case, the summand in question E ⊗ 0 E * ∼ = k C appears with multiplicity 1; hence we have dim Hom K (E ⊗ E * , Sym 2 0 k C ) = dim Hom K (E * ⊗ E, Sym 2 0 k C ) = 1. Moreover, both G-representations exhibit the same Casimir eigenvalue Cas G (ω 1 ,ω n−1 ) = Cas G (ω n−1 ,ω 1 ) = Cas K ω 1 + Cas K ω n−1 = (n − 1)(n + 1) n 2 .
According to Lemma 5.1, the only representations of G (up to equivalence) with subcritical Casimir eigenvalue that occur in decomposition (PW) of S 2 0 (M) C are E ⊗ E * and E * ⊗ E, and we have (recall that m ∼ = k), i.e. the summand occurs with multiplicity 1. It remains to check whether the tensors in the corresponding subspaces are divergence-free. Since meaning that both summands also occur in the left-regular representation Ω 1 (M) with the same multiplicity. It now follows from Corollary 4.4 that . Since this is the only subcritical eigenvalue on S 2 0 (M), we have shown the following. The exceptional Lie group E 6 can be realized as while F 4 is defined as the set of algebra automorphisms By complex-linearly extending linear automorphisms of H, one obtains the inclusion Aut R H ⊂ Aut C H C . In this sense, we have F 4 ⊂ E 6 . In fact, As a representation of E 6 , H C is irreducible. As an F 4 -representation, H decomposes into the irreducible summands where H 0 is the set of trace-free elements of H. An invariant inner product on H is defined by A, B := tr (A • B) for A, B ∈ H. An orthogonal basis of H (cf. Section 2.1 of [16]) is given by the matrices where x runs through the standard basis of O as a real vector space.
In this section, we consider the Riemannian symmetric space M = E 6 /F 4 equipped with the standard metric (hence with critical eigenvalue 2E = 1). The reductive decomposition of e 6 with respect to the standard metric is given by where m ∼ = H 0 as a representation of F 4 .
6.1 Lemma. Let V γ be an irreducible complex representation of E 6 with Cas E 6 γ < 1 and Then V γ is equivalent to one of the E 6 -representations H C and H C . In fact, and the Casimir eigenvalue is Cas G γ = 13 18 . Proof. We abstain from specifying a particular choice of simple root system and fundamental weights for E 6 and F 4 , since we are merely interested in the corresponding fundamental representations of the respective Lie group. Following the enumerative convention of Bourbaki (as used by the software package LiE), if we denote the fundamental weights of E 6 by ω 1 , . . . , ω 6 and of F 4 by η 1 , . . . , η 4 , then the associated representations are identified as where the number indicates the dimension.
As in the proof of Lemma 5.  [14]). Since 13 18 + 13 18 > 1, it follows that only the representations to the highest weights C, H C , H C come into question.
Consider now the F 4 -representation H C 0 ∼ = V η 4 . We obtain 5 the decomposition 18 , and since this is the only subcritical eigenvalue on S 2 0 (M), we have shown the following, which, together with Prop. 5.2, finishes the proof of the main theorem.
6.2 Proposition. The symmetric space E 6 /F 4 is linearly stable.

A.1. Proofs of general statements
Proof of Lemma 4.1. The following is a slightly generalized version of the proof of a result by N. Koiso [11,Prop. 3.3]. We first note that for α ∈ Ω 1 (M), h ∈ S 2 (M), so the formal adjoint of θ is given by We show that θ is overdetermined elliptic. The principal symbol of θ is Take an orthonormal basis (e i ) with respect to g of T p M and write For i, j = 1, . . . , n, it follows that ξ i α j + ξ j α i = 2 n ξ, α g δ ij and so ξ i α j = −ξ j α i if i = j, as well as ξ i α i = ξ j α j for any i, j. Then If α j = 0, this would imply that ξ 2 i + ξ 2 j = 0 and so ξ i = ξ j = 0, which contradicts the assumption that ξ = 0. Overall, we conclude that α = 0 and thus the injectivity is proven.
A.2. Alternative proof of the stability of SU(n) An alternative method of checking that the prototypical differential operators are injective is an explicit computation by means of Lemma 3.3. To do so, we first pick out an explicit element A ∈ Hom K (E ⊗ E * , Sym 2 0 m C ) and then proceed to compute the divergence on the corresponding subspace of S 2 0 (M).
A.1 Lemma. Let π : Sym 2 (E ⊗ E * ) → E ⊗ E * denote the mapping defined by where A, B ∈ E ⊗ E * are regarded as complex n × n-matrices. Then Moreover, the restriction π : Sym 2 0 (E ⊗ 0 E * ) → E ⊗ 0 E * is surjective, and W := ker π Sym 2 0 (E⊗ 0 E * ) Proof. The equivariance of π under the action of K follows from for any k ∈ K = SU(n) and A, B ∈ E ⊗ E * . Furthermore, we have where the last trace is taken with respect to the inner product on E ⊗ E * . This means that π(Sym 2 0 (E ⊗ E * )) ⊂ E ⊗ 0 E * . Next we want to show that π does not vanish when restricted to Sym 2 0 (E ⊗ 0 E * ). If we denote by E ij the n × n-matrix that has entry 1 at position (i, j) and 0 elsewhere, then we have for example E 21 , E 31 ∈ E ⊗ 0 E * and E 21 , E 31 = 0, so E 21 ⊙ E 31 ∈ Sym 2 0 (E ⊗ 0 E * ) and π(E 21 ⊙ E 31 ) = E 21 E 13 + E 31 E 12 = E 23 + E 32 = 0. Now, since E ⊗ 0 E * is irreducible, the mapping π : Sym 2 0 (E ⊗ 0 E * ) → E ⊗ 0 E * must be surjective. We have seen in the proof of Lemma 5.1 that E ⊗ 0 E * appears in the decomposition of Sym 2 0 (E ⊗ 0 E * ) with multiplicity 1; hence W := ker π Sym 2 0 (E⊗ 0 E * ) ⊥ must be the irreducible summand of Sym 2 0 (E ⊗ 0 E * ) that is equivalent to E ⊗ 0 E * . Alternative proof of Prop. 5.2. The properties of π from Lemma A.1 allow us to definẽ and extend it with zero to a mappingÃ ∈ Hom K (E ⊗ E * , Sym 2 0 (E ⊗ 0 E * )). Via the identification m C ∼ = E ⊗ 0 E * , this gives rise to a mapping From the equivariance of π W , the irreducibility of W ∼ = E ⊗ 0 E * and Schur's Lemma it follows that π W is unitary up to a positive constant, that is for all v, w ∈ W and some c > 0. Denote the tensor product representation of G on for F ∈ E ⊗ E * . Its differential is given by for X 1 , X 2 ∈ k. In particular, dρ(X, −X)F = XF + F X.
Let (e i ) be an orthonormal basis of m, e i = (f i , −f i ) with f i ∈ k. Under the identification m C ∼ = E ⊗ 0 E * , the invariant inner product changes by some positive constant factor, and e i is mapped to f i . Hence, (f i ) is an orthonormal basis of k ⊂ E ⊗ 0 E * up to a positive factor. Now, let X ∈ k and F ∈ E ⊗ E * . Using the formula from Lemma 3.3, it follows that for some c, c ′ > 0. Since the trivial summand of Sym 2 (E ⊗ 0 E * ) can only be mapped to the trivial summand of E ⊗ E * under the equivariant map π, we have π • pr Sym 2 0 (E⊗ 0 E * ) = pr E⊗ 0 E * • π on Sym 2 (E ⊗ 0 E * ), implying that Choose the (up to a positive factor) orthonormal basis (f i ) of k in such a way that . Then, and we obtain . This means that the linear mapping Hence, there are no tt-eigentensors for the subcritical Casimir eigenvalue. This proves the assertion.
A.3. Alternative proof of the stability of E 6 /F 4 As we did before in the situation of SU(n), we want to apply Lemma 3.3 to verify that the mappings δ : Hom F 4 (H C , Sym 2 0 m C ) → Hom F 4 (H C , m C ), δ : Hom F 4 (H C , Sym 2 0 m C ) → Hom F 4 (H C , m C ) are injective. Surprisingly, the computation works very similar to the SU(n) case. where the last trace is taken with respect to the inner product on H. This means that π(Sym 2 0 H) ⊂ H 0 .
Since H 0 is irreducible over F 4 , the mapping π : Sym 2 0 H 0 → H 0 must be surjective. From the proof of Lemma 6.1, we know that H 0 appears in the decomposition of Sym 2 0 H 0 with multiplicity 1; hence W := ker π Sym 2 0 H 0 ⊥ must be the irreducible summand of Sym 2 0 H 0 that is equivalent to H 0 . Alternative proof of Prop. 6.2. By Lemma A.2, we can define A := π −1 W ∈ Hom F 4 (H 0 , Sym 2 0 H 0 ), extend it with zero to H and then complex-linearly to a mapping A ∈ Hom F 4 (H C , Sym 2 0 H C 0 ). Again, we need that π W is unitary up to a positive constant, which follows by Schur's Lemma from the equivariance of π W and the irreducibility of W ∼ = H 0 . By Theorem 3.2.4 in [16], every element α ∈ e 6 ⊂ End C (H C ) can be written as α = β + iT • with unique elements β ∈ f 4 ⊂ e 6 and T ∈ H 0 . This corresponds to the F 4 -invariant decomposition Throughout what follows, we identify m ∼ = H 0 . If we denote the defining representation by ρ : E 6 → Aut H C , then in particular, dρ(X) = iX• for X ∈ m. Let (e i ) be an orthonormal basis of H 0 (again, under the identification m ∼ = H 0 , the invariant inner product changes at most by some positive constant factor), X ∈ m and F ∈ H C . Using Lemma 3.3, we thus obtain In particular, we have found Y ∈ m such that (δh) o (Y ) = 0, where h ∈ S 2 0 (M) is associated to F ⊗ A ∈ H C ⊗ Hom F 4 (H C , Sym 2 0 m C ). This means that the linear mapping δ : Hom F 4 (H C , Sym 2 0 m C ) → Hom F 4 (H C , m C ) is nonzero. The same argument works for the E 6 -representation H C , since we exclusively used real elements and automorphisms in the computation. In total, there are no tteigentensors for the subcritical Casimir eigenvalue, which proves the assertion.