Jean-Louis Milhorat
email: jean-louis.milhorat@univ-nantes.fr

THE FIRST EIGENVALUE OF THE DIRAC OPERATOR ON COMPACT OUTER SPIN SYMMETRIC SPACES

In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit expression of the first eigenvalue for "outer" compact spin symmetric spaces.

Introduction

It is well-known that symmetric spaces provide examples where the spectrum of Laplace or Dirac operators can be (theoretically) explicitly computed. However this explicit computation is far from being simple in general and only a few examples are known. On the other hand, several classical results in geometry involve the first (nonzero) eigenvalue of those spectra, so it seems interesting to get this eigenvalue without computing all the spectrum. In two previous papers (see [START_REF] Milhorat | The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces[END_REF] and [START_REF]A formula for the first eigenvalue of the Dirac operator on compact spin symmetric spaces[END_REF]), we stated a formula giving the square of the first eigenvalue of the Dirac operator of a spin compact symmetric space in terms of the algebraic data of the groups involved. However, this formula was based on a result of R. Parthasarathy, [START_REF] Parthasarathy | Dirac operator and the discrete series[END_REF], only valid for symmetric spaces of inner type. Recall that a symmetric space G/K is said to be of inner type if the involution characterizing it is given by an inner conjugation in the group G, or alternatively, by the fact that the groups G and K have same rank, i.e. own a common maximal torus. Otherwise, it is said to be outer, (see for instance Sec. 8.6 in [START_REF] Wolf | Spaces of constant curvature[END_REF] for details). The study of subgroups K of maximal rank in a compact Lie group was initiated by A. Borel and J. De Siebenthal in [START_REF] Borel | Les sous-groupes fermés de rang maximum des groupes de Lie clos[END_REF], with an explicit description for compact simple groups. In [START_REF] Murakami | On the automorphisms of a real semi-simple Lie algebra[END_REF] (see also [START_REF]Sur la classification des algèbres de Lie réelles et simples[END_REF]), S. Murakami gave a general method to study outer involutive automorphisms of compact simple Lie algebras. Using those results, the following complete list of irreducible compact simply-connected Riemannian symmetric spaces G/K of type I with rank K < rank G, can be obtained (see [START_REF]Sur la classification des algèbres de Lie réelles et simples[END_REF] or Sec. 8.12 in J. A. Wolf's book [START_REF] Wolf | Spaces of constant curvature[END_REF]) : SU(2m)/SO(2m) ; SU(2m + 1)/SO(2m + 1) ; SU(2m)/Sp(m) ; SO(2p + 2q + 2)/SO(2p + 1) × SO(2q + 1) ; E 6 /F 4 ; E 6 /Sp(4) .

It was proven by M. Cahen and S. Gutt in [START_REF] Cahen | Spin Structures on Compact Simply Connected Riemannian Symmetric Spaces[END_REF], that all symmetric spaces in that list, except SU(2m + 1)/SO(2m + 1), are spin.

In the present paper, the following explicit value for the square of the first eigenvalue of the Dirac operator is obtained for all those symmetric spaces G/K, endowed with the Riemannian metric induced by the Killing form of G sign-changed.

Symmetric space

Square of the first eigenvalue of the Dirac operator m even, 1 12 (m + 1)(4m -1) + 1 32 SU(2m) SO(2m) m odd, 1 12 (m + 1)(4m -1) + 1 32 1 -1 m 2 m even, 1 12 (m -1)(4m + 1) + 1 32 SU(2m) Sp(m) m odd ≥ 3, 1 12 (m -1)(4m + 1) + 1 32 1 -1 m 2 SO(2p+2q+2) SO(2p+1)×SO(2q+1) 1 16 (p+q) 8pq (2p + q + 1) + 4p (p + 1) + 4q (q + 1) + 1

p ≤ q E 6 F 4
277 72 E 6 Sp 4 529 72 2. Preliminaries for the proof 2.1. Spectrum of the Dirac operator on spin compact irreducible symmetric spaces. From now on, we consider a spin compact simply connected irreducible symmetric space G/K of "type I", where G is a simple compact and simply-connected Lie group and K is the connected subgroup formed by the fixed elements of an involution σ of G. This involution induces the Cartan decomposition of the Lie algebra g of G into

g = k ⊕ p ,
where k is the Lie algebra of K and p is the vector space {X ∈ g ; σ * • X = -X}. The symmetric space G/K is endowed with the Riemannian metric induced by the restriction to p of the Killing form B G of G sign-changed. The spin condition implies that the homomorphism

α : h ∈ K -→ Ad G (h) |p ∈ SO(p)
lifts to a homomorphism α : K -→ Spin(p) such that ξ • α = α where ξ is the two-fold covering Spin(p) → SO(p), [START_REF] Cahen | Spin Structures on Compact Simply Connected Riemannian Symmetric Spaces[END_REF]. Then the group K inherits a spin representation given by

ρ K : K α -→ Spin(p) ρ -→ GL C (Σ) ,
where ρ is the spinor representation in the complex spinor space Σ.

The Dirac operator has a real discrete spectrum, symmetric with respect to the origin. A real number λ belongs to the spectrum if and only if there exists an irreducible representation γ : G → GL C (V γ ) whose restriction Res G K (γ) to the subgroup K, contains in its decomposition into irreducible parts, a representation equivalent to some irreducible component of the decomposition of the spin representation ρ K of K. Then (1)

λ 2 = c γ + n/16 ,
where c γ is the Casimir eigenvalue of the irreducible representation γ (which only depends on the equivalence class of γ) and where n = dim(G/K), n/16 being Scal/8 for the choice of the metric (cf. [START_REF] Hijazi | A spinorial approach to Riemannian and conformal geometry[END_REF] or [START_REF] Ginoux | The Dirac spectrum[END_REF] for details).

Hence the first eigenvalue of the Dirac operator is given by the lowest c γ , among the irreducible representations γ ′ s of G such that Res G K (γ) contains an irreducible component of the spin representation ρ K of K. We will say for short that such an irreducible representation γ (or its highest weight) verifies the "spin condition". 2.2. Outer symmetric spaces. Let T be a maximal torus of G. Then T K := T ∩ K is a maximal torus of K. As it was already mentioned, the symmetric space G/K is said to be outer if the involution σ : G → G is not a conjugation in the group, and this is equivalent to the condition dim(T K ) < dim(T ). Let t and t K be the Lie algebras of T and T K , and let t 0 := t ∩ p. Note that t 0 is the orthogonal complement of t K in t for the scalar product -B G . Let g C and k C be the complexifications of the Lie algebras g and k, and , the Cextension of -B G to g C . The root decompositions of g C and k C under the respective actions of T and T K are given by (2)

g C = t C ⊕ N ⊕ i=1 g ±θi ,
where ±θ i are the G-roots, and

k C = t K,C ⊕ M ⊕ i=1 k ±θ ′ i ,
where ±θ ′ i are the K-roots. The set Φ of G-roots decomposes into the disjoint union

Φ = Φ 1 ∪ Φ 2 ∪ Φ 3 , where Φ 1 = {θ ∈ Φ, g θ ⊂ k C } , Φ 2 = {θ ∈ Φ, g θ ⊂ p C } , Φ 3 = Φ\(Φ 1 ∪ Φ 2 ) . Note that if θ ∈ Φ i , i = 1, 2, 3, then -θ ∈ Φ i , since g -θ = g θ .
We use the same notation for the involution σ * : g → g and its C-linear extension to g C . As σ * (t) = t, by means of the scalar product -B G , σ * |t induces an involution σ * : i t * → i t * . The scalar product on i t * induced by -B G is denoted by , . We consider for any G-root θ i , a basis E θi of the one-dimensional space g θi , and set

U θi := 1 2 (E θi + σ * E θi ) and V θi := 1 2 (E θi -σ * E θi ) . Then U θi ∈ k, V θi ∈ p, E θi = U θi + V θi , and
θ i ∈ Φ 3 ⇐⇒ (U θi = 0 and V θi = 0) .
In order to determine an expression of the weights of the spin representation ρ K : K → GL C (Σ), we will have to use the following properties of G-roots. For any root θ ∈ Φ, the restriction θ |tK is denoted θ ′ for short.

Lemma 2.1. R1:

∀α ∈ Φ 3 , ∃X ∈ t K such that α(X) = 0, hence α ′ is a (non zero) K-root. R2: ∀α ∈ Φ, α ∈ Φ 1 ∪ Φ 2 ⇐⇒ α |t0 = 0. Hence if α and β are two roots in Φ 1 such that β = ±α, then α ′ and β ′ are two K-roots such that β ′ = ±α ′ . R3: If α ∈ Φ 1 and β ∈ Φ 3 , then α ′ and β ′ are two K-roots such that β ′ = ±α ′ . R4: For all α ∈ Φ, σ * (α) is a root such that (1) if α ∈ Φ 1 ∪ Φ 2 , then σ * (α) = α, (2) if α ∈ Φ 3 , then σ * (α) = ±α, R5: For all α and β ∈ Φ 3 , β ′ = α ′ ⇐⇒ β = α or β = σ * (α).
Proof. The proofs are given in Appendix. Some of those results appear in some way in Chapter 3 of [START_REF] Burnstall | Twistor theory for Riemannian symmetric spaces[END_REF]. According to R4, we consider Φ

′ 3 = {γ 1 , . . . , γ p } ⊂ Φ 3 such that Φ 3 is the disjoint union Φ 3 = {γ 1 , . . . , γ p } ∪ {σ * (γ 1 ), . . . , σ * (γ p )} . Then Lemma 2.2. The set Φ K of K-roots under the action of T K is given by Φ K = {θ |tK ; θ ∈ Φ 1 ∪ Φ ′ 3 } . Proof.
By definition, the restriction to t K of a root in Φ 1 is a K-root, and by R1, the restriction to t K of a root in Φ ′ 3 is also a K-root. Conversely, let E ′ ̺ ∈ k C be a root-vector for a K-root ̺. As for any α ∈ Φ 3 , σ * (E α ) is a root-vector for the root σ * (α) by ( 11) and (12), we may write, according to (2),

E ′ ̺ = H + α∈Φ1 λ α E α + α∈Φ2 λ α E α + α∈Φ ′ 3 λ α E α + α∈Φ ′ 3 λ σ * (α) σ * (E α ) ,
where H ∈ t and λ α ∈ C,

= H + α∈Φ1 λ α E α + α∈Φ2 λ α E α + α∈Φ ′ 3 λ α + λ σ * (α) U α + α∈Φ ′ 3 λ α -λ σ * (α) V α . Since σ * (E ′ ̺ ) = E ′ ̺ , one gets H ∈ t K , ∀α ∈ Φ 2 , λ α = 0, and ∀α ∈ Φ ′ 3 , λ σ * (α) = λ α , hence E ′ ̺ = H + α∈Φ1 λ α E α + 2 α∈Φ ′ 3 λ α U α . Now, as for any X ∈ t K , [X, E ′ ̺ ] = ̺(X) E ′ ̺ , one gets ∀X ∈ t K , -̺(X) H + α∈Φ1 λ α (α(X) -ρ(X)) E α + 2 α∈Φ ′ 3 λ α (α(X) -ρ(X)) U α = 0 .
Hence H = 0, and, as there exists at least one λ α = 0, ̺ = α ′ . Note furthermore that by R2, R3 and R5, there exists only one such a λ α = 0. By the results R1 and R2, the restriction to t K of any G-root is nonzero. As t K can not be the finite union of the hyperplanes Ker(α ′ ) for α ∈ Φ, there exists X ∈ t K such that for any α ∈ Φ, α(X) = 0. This implies that X is regular in t, and also regular in t K , by the description of K-roots given in the above lemma. We define the set Φ + (resp. Φ + K ) of positive roots of G (resp. K) by the condition

θ ∈ Φ + (resp. Φ + K ) ⇐⇒ θ(X) > 0 . Note that as X ∈ t K , one has θ ∈ Φ + ⇐⇒ σ * (θ) ∈ Φ + ,
and that by the above considerations

(3) Φ + K = {θ |tK ; θ ∈ Φ + 1 ∪ Φ ′+ 3 } . Note furthermore that, by the description of positive G-roots, the half-sum δ G of the positive roots verifies σ * (δ G ) = δ G .
2.3. Weights of the spin representation of K. The decomposition of the spin representation depends on the parity of dim p C , so we begin by considering the even dimensional case.

2.3.1. First case : dim p C even. As it will be seen below, this amounts to suppose that dim t 0 is even. Setting dim t 0 = 2r 0 , we consider an orthonormal basis (T k ), k = 1, . . . 2r 0 , of t 0 , and construct the Witt basis (Z k , Z k ), k = 1, . . . r 0 , defined by

Z k := 1 2 (T 2k-1 + i T 2k ) , and
Z k := 1 2 (T 2k-1 -i T 2k ) .
For the C-linear extension , of the scalar product on g, one has

Z i , Z j = Z i , Z j = 0 , and Z i , Z j = 1 2 δ ij . Let Φ + 1 = {α 1 , . . . , α ℓ }, Φ + 2 = {β 1 , . . . , β p } and Φ ′+ 3 = {γ 1 , . . . , γ q }. Lemma 2.3. Vectors E θ , E -θ , θ = β 1 , . . . , β p , V θ , V -θ , θ = γ 1 .
. . , γ q , may be choosen such that, with Z i , Z i , i = 1, . . . , r 0 , they define a Witt basis of p C , in the sense that

E α , E β = E -α , E -β = E α , V β = E -α , V -β = V α , V β = V -α , V -β , = Z i , Z j = Z i , Z j = 0 , and (4) E α , E -β = 1 2 δ αβ , V α , V -β = 1 2 δ αβ , Z i , Z j = 1 2 δ ij .
Proof. Clearly, any such vectors span p C : any X ∈ p C may be written as

H + α∈Φ1 λ α E α + α∈Φ2 λ α E α + α∈Φ ′ 3 λ α + λ σ * (α) U α + α∈Φ ′ 3 λ α -λ σ * (α) V α ,
with H ∈ t C and λ α ∈ C. The condition σ * (X) = -X implies H ∈ t 0,C and ∀α ∈ Φ 1 , λ α = 0, and ∀α ∈ Φ ′ 3 , λ α = -λ σ * (α) , hence

X = H + α∈Φ + 2 λ α E α + α∈Φ + 2 λ -α E -α + 2 α∈Φ ′+ 3 λ α V α + 2 α∈Φ ′+ 3 λ -α V -α ,
so X is a linear combination of the considered vectors. On the other hand, it is well-known that, for any couple of roots (α, β) such that α + β = 0, g α , g β = 0. Hence

• if α and β ∈ Φ 2 are such that α + β = 0, then E α , E β = E -α , E -β = 0. • if α ∈ Φ 2 and β ∈ Φ 3 , then from E α , E β = 0, one obtains as E α , U β = 0, E α , V β = 0, and then E -α , V -β = 0. • if α ∈ Φ ′
3 and β ∈ Φ ′ 3 are such that α + β = 0, then α ′ + β ′ = 0 (otherwise by R5, β = σ * (-α), which is impossible), so U α , U β = 0, and then V α , V β = 0, (so V -α , V -β = 0 also).

Finally, as , is non-degenerate, (E α , E -α ) = 0, α ∈ Φ + 2 , and (V α , V -α ) = 0, α ∈ Φ ′+ 3 , and we may choose the basis (E α ) such that (4) is verified. All of those orthogonality relations imply that the considered vectors are linearly independent.

Since E -α belongs to g α , we may suppose that E -βj = E βj and V -γj = V γj . In the Clifford algebra Cℓ(p), let

w := E β1 • • • E βp • V γ1 • • • V γq • Z 1 • • • Z r0 .
For any

I = {i 1 , . . . , i a } ⊂ {1, . . . , p}, i 1 < i 2 < • • • < i a , (resp. J = {j 1 , . . . , j b } ⊂ {1, . . . , q}, j 1 < j 2 < • • • < j b ; resp. K = {k 1 , . . . , k c } ⊂ {1, . . . , r 0 }, k 1 < k 2 < • • • < k c ), we introduce the notation E I • V J • Z K := E βi 1 • • • E βi a • V γj 1 • • • V γj b • Z k1 • • • Z kc ,
setting E I (resp. V J , resp. Z K ) = 1, if I (resp. J, resp. K) = ∅. As I (resp. J, resp. K) runs through the set of subsets of {1, . . . , p}, (resp. {1, . . . , q}, resp. {1, . . . , r 0 }), the vectors E I •V J •Z K •w define a basis of the spinor space Σ := Cℓ(p)•w (cf. [START_REF] Hijazi | A spinorial approach to Riemannian and conformal geometry[END_REF]). Considering (5)

     X j = E βj + E βj , Y j = i E βj -E βj , 1 ≤ j ≤ p , and
     F j = V γj + V γj , G j = i V γj -V γj , 1 ≤ j ≤ q , one obtains an orthonormal basis (X i , Y i , F j , Y j , T k ) 1≤i≤p, 1≤j≤q, 1≤k≤2r0 of p.
Then for any X ∈ t K , one has

     [X, X j ] = -i β j (X) Y j , [X, Y j ] = i β j (X) X j , 1 ≤ j ≤ p , ;      [X, F j ] = -i γ j (X) G j , [X, G j ] = i γ j (X) F j , 1 ≤ j ≤ q ,
and [X,

T j ] = 0 , 1 ≤ j ≤ 2r 0 . So ∀X ∈ t K , α * (X) = -i p j=1 β j (X) X j ∧ Y j -i q j=1 γ j (X) F j ∧ G j , hence ∀X ∈ t K , α * (X) = - 1 2 i p j=1 β j (X) X j • Y j - 1 2 i q j=1 γ j (X) F j • G j .
Now it is easy to verify that

X k • Y k • E I • V J • Z K • w = -i E I • V J • Z K • w if k / ∈ I, i E I • V J • Z K • w if k ∈ I, F k • G k • E I • V J • Z K • w = -i E I • V J • Z K • w if k / ∈ J, i E I • V J • Z K • w if k ∈ J. Hence ∀X ∈ t K , ρ K * (X)(E I • V J • Z K • w) = 1 2   i∈I β i (X) - i / ∈I β i (X) + j∈J γ j (X) - j / ∈J γ j (X)   E I • V J • Z K • w . So the E I • V J • Z K • w are weight-vectors, and a µ ∈ i t *
K is a weight if and only if it can be expressed as

(6) 1 2 ±β ′ 1 ± • • • ± β ′ p ± γ ′ 1 ± • • • ± γ ′ q .
Note that as K runs through the set of subsets of {1, . . . , r 0 }, all of the E I •V J •Z K •w for a given I and J, belong to same weight-space, hence the multiplicity of a weight of the form (6) is 2 r0 times the number of ways in which it can be expressed in the given form.

Considering the volume element

ω := i p+q+r0 X 1 • Y 1 • • • X p • Y p • F 1 • G 1 • • • F q • G q • T 1 • T 2 • • • T 2r0-1 • T 2r0 , one has ω • E I • V J • Z K • w = (-1) #I+#J+#K E I • V J • Z K • w ,
hence the spinor space Σ decomposes into two irreducible components

Σ = Σ + ⊕ Σ -, where Σ + (resp. Σ -) is the linear span of the E I • V J • Z K • w 's such that #I + #J + #K is even (resp. odd).
2.3.2. Second case : dim p C odd. This corresponds to the case dim t 0 odd. Setting n = dim p, by the choice of an orthonormal basis such as (5), SO(p) is identified with SO(n), which is itself embedded in SO(n + 1) in such a way that SO(n) acts trivially on the last vector e n+1 of the standard basis of R n+1 . Setting dim t 0 = 2r 0 -1, r 0 ≥ 1, the spinor space can be described (cf. [START_REF] Hijazi | A spinorial approach to Riemannian and conformal geometry[END_REF]), as the space of positive spinors introduced above in the even-dimensional case :

Σ = Span{E I • V J • Z K • w, #I + #J + #K even}.
By the result above, the weights may be expressed as ( 6), but now, the multiplicity of such a weight is 2 r0-1 times the number of ways in which it can be expressed in the given form.

A remark on highest weights.

Lemma 2.4. Any highest weight of the spinor representation of K has necessarily the form

(7) 1 2 ±β ′ 1 ± • • • ± β ′ p + γ ′ 1 + • • • + γ ′ q .
Proof. Let µ be a weight such that for a j 0 ∈ {1, . . . , q}, γ ′ j0 appears with a minus sign in the expression of µ. Then µ + γ ′ j0 is also a weight, hence µ can not be dominant, since γ ′ j0 is a positive K-root.

3. The symmetric space SU(2m) SO(2m) .

In the following, M 2m (C) denotes the space of 2m × 2m matrices with complex coefficients, and (E ij ) its standard basis. Here

G = SU(2m) = {A ∈ M 2m (C), t AA = I 2m } , and
K = {A ∈ SU(2m) ; A = A} = SO(2m) .
The symmetric space structure is given by the involution σ : G → G, A → t A -1 . This involution induces the decomposition of the Lie algebra su 2m of SU(2m) into:

su 2m = so 2m ⊕ p ,
where so 2m is the Lie algebra {X ∈ su 2m ; t X = -X} of SO(2m), and

p = {X ∈ su 2m ; t X = X} .

Elements of p are traceless symmetric 2m × 2m matrices with coefficients in

i R, hence dim(p) = 2m 2 + m -1. Let T s =         e iθ1 . . . e iθ2m    ; θ i ∈ R , 2m i=1 θ i = 0   
  be the standard maximal torus of SU(2m). We consider a conjugate T of T s in G in such a way that T ∩ K is the standard maximal torus of K. For that, let A 0 ∈ SU(2m) be the 2 × 2 block diagonal matrix defined by

A 0 = √ 2 2    ✷ . . . ✷    , where ✷ = 1 i i 1 .
Then for any

H =    e iθ1 . . . e iθ2m    ∈ T s , A 0 HA -1 0 =    ✷ 1 . . . ✷ m   ,
where

✷ j = e 1 2 i (θ2j-1+θ2j ) cos 1 2 (θ 2j-1 -θ 2j ) sin 1 2 (θ 2j-1 -θ 2j ) -sin 1 2 (θ 2j-1 -θ 2j ) cos 1 2 (θ 2j-1 -θ 2j )
.

Hence A 0 HA -1 0 ∈ K ⇐⇒ A 0 HA -1 0 =    ✷ ′ 1 . . . ✷ ′ m   ,
where

✷ ′ j = cos(θ 2j ) -sin(θ 2j ) sin(θ 2j ) cos(θ 2j
) , so A 0 HA -1 0 ∈ K if and only if it belongs to the standard maximal torus T K of K. The Lie algebra t of T is defined by

t = m j=1 iγ 2j-1 (E 2j-1 2j-1 + E 2j 2j ) + γ 2j (E 2j 2j-1 -E 2j-1 2j ) ; γ j ∈ R , m j=1 γ 2j-1 = 0 , and t K =    m j=1 γ 2j (E 2j 2j-1 -E 2j-1 2j ) ; γ 2j ∈ R    , t 0 =    m j=1 iγ 2j-1 (E 2j-1 2j-1 + E 2j 2j ) , γ 2j-1 ∈ R , m j=1 γ 2j-1 = 0    .
It is useful to consider the elements x j , j = 1, . . . , 2m of i t * , defined by considering the restriction to t of the dual basis of the family of vectors i

(E 2j-1 2j-1 + E 2j 2j ), (E 2j 2j-1 -E 2j-1 2j ), j = 1, . . . , m : ∀H = m j=1 iγ 2j-1 (E 2j-1 2j-1 + E 2j 2j ) + γ 2j (E 2j 2j-1 -E 2j-1 2j ) ∈ t , x 2j-1 (H) = i γ 2j-1 and x 2j (H) = i γ 2j .
The scalar product on i t * is given by the scalar product on su 2m defined by

X, Y = - 1 2 ℜ Tr(XY ) = - 1 8m B(X, Y ) , X, Y ∈ su 2m .
Lemma 3.1. Each µ ∈ i t * may be uniquely written as

(8) µ = 2m i=1 µ i x i , where µ i ∈ R verify m i=1 µ 2i-1 = 0,
and, for any µ = 2m

i=1 µ i x i and µ ′ = 2m i=1 µ ′ i x i ∈ i t * , µ, µ ′ = 2m i=1 µ i µ ′ i .
Proof. As the x 2j-1 , j = 1, . . . , m -1 and x 2j , 1, . . . , m define a basis of t * , any µ ∈ t * is uniquely written as

µ = m-1 i=1 µ 2i-1 x 2i-1 + m i=1 µ 2i x 2j , µ i ∈ R. Now since x 2m-1 = - m-1 i=1 x 2i-1 , setting µ 2m-1 = -1 m m-1 i=1 µ 2j-1 , one gets µ = m-1 i=1 (µ 2i-1 + µ 2m-1 ) x 2i-1 + µ 2m-1 x 2m-1 + m i=1 µ 2i x 2i , so µ may be written as (8). Now if µ = 2m i=1 λ i x i , where λ i ∈ R verify m i=1 λ 2i-1 = 0, then µ = m-1 i=1 (λ 2i-1 -λ 2m-1 ) x 2i-1 + m i=1 λ 2i x 2i , and so µ 2i-1 = λ 2i-1 -λ 2m-1 , i = 1, . . . , m -1 , and µ 2i = λ 2i , i = 1, . . . , m . Then µ 2m-1 = -1 m m-1 i=1 µ 2i-1 = λ 2m-1 , hence µ 2i-1 +µ 2m-1 = µ 2i-1 +λ 2m-1 = λ 2i-1
, hence the unicity of the writting (8). The last result follows from the fact that the vectors E 2j-1 2j-1 + E 2j 2j , E 2j 2j-1 -E 2j-1 2j , j = 1, . . . , m, define an orthonormal set for the scalar product , . In the following, any µ ∈ i t * , of the form (8) will be denoted

µ = [µ 1 , µ 3 , . . . , µ 2m-1 ; µ 2 , µ 4 , . . . , µ 2m ] , m i=1 µ 2i-1 = 0 . Note that the x ′ 2i , 1 ≤ i ≤ m, define a basis of i t * K . Any λ ∈ i t * K of the form λ = m i=1 λ 2i x ′ 2i , λ 2i ∈ R, will be denoted λ = (λ 2 , λ 4 , . . . , λ 2m ) .
The involution σ * of it * induced by σ is defined by

[µ 1 , µ 3 , . . . , µ 2m-1 ; µ 2 , µ 4 , . . . , µ 2m ] σ * -→ [-µ 1 , -µ 3 , . . . , -µ 2m-1 ; µ 2 , µ 4 , . . . , µ 2m ] .
3.1. Sets of roots. Since the root-vectors relative to the standard torus T s are the E ij , the root-vectors relative to the torus T are given by the A 0 E ij A -1 0 . Explicitly, the roots are

( x 2i-1 -x 2j-1 ) + ( x 2i -x 2j ) , 1 ≤ i = j ≤ m , root-space : C A 0 E 2i-1 2j-1 A -1 0 , ( x 2i-1 -x 2j-1 ) -( x 2i -x 2j ) , 1 ≤ i = j ≤ m , root-space : C A 0 E 2i 2j A -1 0 , ( x 2i-1 -x 2j-1 ) + ( x 2i + x 2j ) , 1 ≤ i, j ≤ m , root-space : C A 0 E 2i-1 2j A -1 0 , ( x 2i-1 -x 2j-1 ) -( x 2i + x 2j ) , 1 ≤ i, j ≤ m , root-space : C A 0 E 2i 2j-1 A -1
0 . Note that the only roots that verify σ * (θ) = θ are ±2 x 2i , and, as they are not K-roots, one may conclude from lemma 2.1 that

Φ 1 = ∅ , Φ 2 = {±2 x 2i , i = 1, . . . , m} ,
and all other roots belong to Φ 3 . We choose positive roots such that

Φ + 2 = {2 x 2i , i = 1, . . . , m} , Φ ′+ 3 = ( x 2i-1 -x 2j-1 ) + ( x 2i -x 2j ) , 1 ≤ i < j ≤ m , ( x 2i-1 -x 2j-1 ) + ( x 2i + x 2j ) , 1 ≤ i < j ≤ m , , σ * (Φ ′+ 3 ) = -( x 2i-1 -x 2j-1 ) + ( x 2i -x 2j ) , 1 ≤ i < j ≤ m , -( x 2i-1 -x 2j-1 ) + ( x 2i + x 2j ) , 1 ≤ i < j ≤ m .
It is easy to verify that a system of simple roots is given by

β :=2 x 2m , γ i :=( x 2i-1 -x 2i+1 ) + ( x 2i -x 2(i+1) ) , i = 1, . . . , m -1 , σ * (γ i ) = -( x 2i-1 -x 2i+1 ) + ( x 2i -x 2(i+1) ) , i = 1, . . . , m -1 . Note that γ i , γ i = σ * (γ i ), σ * (γ i ) = β, β = 4 , i = 1, . . . , m -1 , γ i , γ i+1 = σ * (γ i ), σ * (γ i+1 ) = -2 , i = 1, . . . , m -2 , γ m-1 , β = σ * (γ m-1 ), β = -2, ,
all the other scalar products being zero. Hence

Lemma 3.2. A vector µ = [µ 1 , µ 3 , . . . , µ 2m-1 ; µ 2 , µ 4 , . . . , µ 2m ] ∈ i t * , m i=1 µ 2i-1 = 0, is a dominant weight if and only if µ 2i -µ 2(i+1) ∈ N , µ 2i-1 -µ 2i+1 ∈ Z ,
both simultaneously odd or even,

|µ 2i-1 -µ 2i+1 | ≤ µ 2i -µ 2(i+1) , 1 ≤ i ≤ m -1 , µ 2m ∈ N .
Proof. Note just that the conditions 2 γi,µ γi,γi ∈ N,

2 σ * (γi),µ γi,γi ∈ N, i = 1, . . . m -1, are equivalent to (µ 2i-1 -µ 2i+1 ) + (µ 2i -µ 2(i+1) ) 2 ∈ N ,
and

-(µ 2i-1 -µ 2i+1 ) + (µ 2i -µ 2(i+1) ) 2 ∈ N ,
which imply (and are equivalent to) the first three conditions of the lemma.

As it was remarked before, the positive K-roots are given by considering the restrictions to t K of the positive roots in Φ ′ 3 , so

Φ + K = x ′ 2i -x ′ 2j , x ′ 2i + x ′ 2j , 1 ≤ i < j ≤ m .
A set of simple roots is given by

θ ′ i = x ′ 2i -x ′ 2(i+1) , i = 1, . . . , m -1 , θ ′ m = x ′ 2(m-1) -x ′ 2m .
Note that

θ ′ i , θ ′ i = 2 , θ ′ i , θ ′ i+1 = -1 , 1 ≤ i ≤ m -2 , θ ′ m-2 , θ ′ m = -1 , θ ′ m-1 , θ ′ m = 0 ,
hence the following "classical" characterization :

Lemma 3.3. A vector λ = (λ 2 , λ 4 , . . . , λ 2m ) ∈ i t * K , is a dominant weight if and only if λ 2 ≥ λ 4 ≥ • • • ≥ λ 2(m-1) ≥ |λ 2m | ,
and the λ 2i are all simultaneously integers or half-integers.

Highest weights of the spin representation of K.

Proposition 3.4. The spin representation of K has two highest weights:

(m, m -1, . . . , 2, ±1) , both with multiplicity 2 [ m-1 2 ] . Proof. One has 1 2 γ∈Φ ′+ 3 γ ′ = 1≤i<j≤m x ′ 2i = m-1 i=1 (m -i) x ′ 2j .
Hence by the result of lemma 2.4, any highest weight λ has necessarily the form

λ = m i=1 (m -i ± 1) x ′ 2i .
But the dominance condition of lemma 3.3 implies then

λ = m-1 i=1 (m -i + 1) x ′ 2i ± x ′ 2m = (m, m -1, . . . , 2, ±1) .
Denote by λ ± those two dominant weights. Let δ K be the half-sum of the positive K-roots. One has δ K = (m -1, m -2, . . . , 1, 0), hence

λ + + δ K , λ + + δ K = λ -+ δ K , λ -+ δ K .
Hence the two weights are both highest weights, since otherwise, one of the two should be contained in the set of weights of an irreducible representation having the other one as highest weight, and the above equality should be impossible (cf. lemma C, 13.4 in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF]).

With the help of the Weyl dimension formula, it may be checked that any irreducible module with highest weight λ ± has dimension 2 (m-1)(m+1) . As we noticed it before, the multiplicity of each one of those two weights is at least 2

m-1 2 if m is odd, and 2 m-2 2 if m is even. Since if m is odd, 2 × 2 m-1 2 × 2 (m-1)(m+1) = 2 2m 2 +m-1 2 = dim(Σ) ,
and if m is even, 2 × 2 m-2 2 × 2 (m-1)(m+1) = 2 2m 2 +m-2 2 = dim(Σ) ,
one concludes that the multiplicity of each weight is exactly 2 [ m-1 2 ] .

3.3. The first eigenvalue of the Dirac operator. Recall that the first eigenvalue of the Dirac operator is given by the lowest Casimir eigenvalue c γ , among the irreducible representations γ of G verifying the "spin condition" : Res G K (γ) contains an irreducible component of the spin representation ρ K of K. By the Freudenthal formula, the Casimir eigenvalue of an irreducible representation with highest weight µ γ is given by

c γ = µ γ , µ γ + 2 δ G = µ γ + δ G 2 -δ G 2 .
Hence we look for G-dominant weights µ γ , verifying the spin condition and such that µ γ + δ G 2 is minimal. We first determine the G-weights µ (non necessarily dominant) for which

µ |tK = λ ± and µ + δ G 2 is minimal. Note first that µ = [µ 1 , µ 3 , . . . , µ 2m-1 ; µ 2 , µ 4 , . . . , µ 2m ], m i=1 µ 2i-1 = 0, is a G- weight if and only if µ 2i-1 -µ 2i+1 ∈ Z, i = 1, . . . m -1, µ 2i ∈ Z, i = 1, . . . ,
δ G = 2 1≤k<l≤m x 2k + m k=1 x 2k = 2 m k=1 m -k + 1 2 x 2k , = [0, 0, . . . , 0; (2m -1), (2m -3), . . . , 1] . Hence µ + δ G 2 is minimal if and only if m i=1 µ 2 2i-1 is minimal. Set µ 2i-1 -µ 2i+1 = k i , i = 1, . . . , m -1.
For any j = 1, . . . , m -1, one has

p j := j i=1 k i = µ 1 + j-1 i=1 µ 2i+1 - j-1 i=1 µ 2i+1 -µ 2j+1 , = µ 1 -µ 2j+1 . Using m i=1 µ 2i-1 = 0, one then gets m-1 j=1 p j = (m -1) µ 1 + µ 1 = m µ 1 , hence µ 1 = 1 m m-1 i=1 p i , and
µ 2j+1 = µ 1 -p j = 1 m m-1 i=1 p i -m p j .
The expression m i=1 µ 2 2i-1 is a polynomial F (p 1 , . . . , p m-1 ) of the variables p 1 , p 2 . . . , p m-1 . With the notation p = (p 1 , . . . , p m-1 ), one has

∂F ∂p i (p) = -2 µ 2i+1 .
With no surprise, F as only one critical point at (0, . . . , 0).

Now ∂ 2 F ∂p i ∂p j (0) = - 2 m (1 -m δ ij ) .
Denote by H the Hessian matrix of F at 0. It has for eigenvalues 2 m with multiplicity one and 2 with multiplicity m -2, and the following vectors define an orthogonal basis of eigenvectors.

v 1 = (1, 1, . . . , 1) , v i = (0, 0, . . . , 0, m -i, -1, -1, . . . , -1 m-i ) , 2 ≤ i ≤ m -1 .
Considering the orthonormal basis (ǫ i v i ) 1≤i≤m-1 , ǫ i = 1/ v i , and denoting by Q the orthogonal matrix of the change of basis, one gets

F (p) = 1 2 t pHp , = 1 m t p Q      1 0 0 0 0 m 0 0 . . . 0 0 0 m      t Q p , = 1 m ǫ 1 (m -1)k 1 + (m -2)k 2 + • • • + k m-1 2 + ǫ 2 (m -2)k 2 + (m -3)k 3 • • • + k m-1 2 + • • • + ǫ m-i i k m-i + • • • + k m-1 2 + • • • + ǫ m-2 2k m-2 + k m-1 2 + ǫ m-1 k m-1 2 .
The minimum is obtained if and only if all the squares are minimal. Hence, since the k i have to be odd integers, Lemma 3.5. The minimum is obtained only when

(k 1 , k 2 , . . . , k m-1 ) = (1, -1, 1, -1, . . .) or (-1, 1, -1, 1, . . .) .
Proof. Assume that all the squares are minimal. First, since k m-1 is an odd integer, k 2 m-1 is minimum only when k m-1 = ±1. Now, let us show by induction that k m-i = (-1) i+1 k m-1 , i = 1, . . . , m-1. Assuming that this is true for 1

≤ i ≤ 2j-1, j ≥ 1, one gets 2j k m-2j + (2j -1) k m-(2j-1) + • • • + k m-1 2 = 2j k m-2j + j k m-1 2 , = j 2 2 k m-2j + k m-1 2 .
Since 2 k m-2j + k m-1 is an odd integer, the above square is minimal if and only if

2 k m-2j + k m-1 2 = 1, which implies 4 k m-2j (k m-2j + k m-1 ) = 0 , hence k m-2j = -k m+1 = (-1) 2j+1 k m-1 .
Thus the result is also true for i = 2j, and one gets

(2j + 1) k m-(2j+1) + 2j k m-(2j-1) + • • • + k m-1 2 = (2j + 1) k m-(2j+1) -j k m-1 2 . If k m-1 = 1, then (2j + 1) k m-(2j+1) -j k m-1 ≥ j + 1 if k m-(2j+1) ≥ 1 and (2j + 1) k m-(2j+1) -j k m-1 ≤ -(3j + 1) if k m-(2j+1) ≤ -1, thus the above square is minimal if and only if k m-(2j+1) = 1 = k m-1 = (-1) 2j+2 k m-1 . If k m-1 = -1, then (2j + 1) k m-(2j+1) -j k m-1 ≥ 3j + 1 if k m-(2j+1) ≥ 1 and (2j + 1) k m-(2j+1) -j k m-1 ≤ -(j + 1) if k m-(2j+1) ≤ -1, thus the above square is minimal if and only if k m-(2j+1) = -1 = k m-1 = (-1) 2j+2 k m-1 .
Hence the G-weights µ for which µ |tK = λ ± and µ + δ G 2 is minimal are given by (1) If m is even m = 2p, (a) if (k 1 , k 2 , . . . , k m-1 ) = (1, -1, 1, -1, . . . , -1, 1), then since p 2i-1 = 1, i = 1, . . . , p, and p 2i = 0, i = 1, . . . , p -1,

µ ± = 1 2 , - 1 2 , . . . , 1 2 , - 1 2 ; m, m -1, . . . , 2, ±1 , (b) if (k 1 , k 2 , .
. . , k m-1 ) = (-1, 1, -1, 1, . . . , 1, -1), then since p 2i-1 = -1, i = 1, . . . , p, and p 2i = 0, i = 1, . . . , p -1,

µ ′ ± = - 1 2 , 1 2 , . . . , - 1 2 , 1 2 ; m, m -1, . . . , 2, ±1 . Note that µ ′ ± = σ * (µ ± ) and µ ± + δ G 2 = µ ′ ± + δ G 2 . (2) If m is odd, m = 2p + 1, (a) if (k 1 , k 2 , . . . , k m-1 ) = (1, -1, 1, -1, . . . , 1, -1), then since p 2i-1 = 1 and p 2i = 0, i = 1, . . . , p, µ ± = 1 2 - 1 2m , - 1 2 - 1 2m . . . , - 1 2 - 1 2m , 1 2 - 1 2m ; m, m -1, . . . , 2, ±1 , (b) if (k 1 , k 2 , . . . , k m-1 ) = (-1, 1, -1, 1, . . . , -1, 1), then since p 2i-1 = -1 and p 2i = 0, i = 1, . . . , p, µ ′ ± = - 1 2 + 1 2m , 1 2 + 1 2m , . . . , 1 2 + 1 2m , - 1 2 + 1 2m ; m, m -1, . . . , 2, ±1 ,
Here also note that µ

′ ± = σ * (µ ± ) and µ ± + δ G 2 = µ ′ ± + δ G 2 .
Note that, by lemma 3.2, the weights µ + and µ ′ + are G-dominant, whereas µ - (resp. µ ′ -) belongs to the orbit of µ + (resp. µ ′ + ) under the Weyl group, since µ -= σ β (µ + ), (resp. µ ′ -= σ β (µ ′ + )), where σ β is the reflexion across β ⊥ , β being the simple root 2 x 2m . In order to conclude, we first remark that the G-dominant weights µ + and µ -verify the spin condition. That follows from the following general result.

Lemma 3.6. Let µ γ be the highest weight of an irreducible representation γ of G. Then µ γ|tK is a K-dominant weight, and any irreducible representation of K having µ γ|tK as a highest weight is contained in the restriction Res G K (γ) of γ to K. Proof. By the result of lemma 3.2, a dominant G-weight

µ = [µ 1 , µ 3 , . . . , µ 2m-1 ; µ 2 , µ 4 , . . . , µ 2m ] , m i=1 µ 2i-1 = 0 , restricts to t K as µ |tK = m i=1 µ 2i x 2i = (µ 2 , µ 4 , . . . , µ 2m )
, where the µ 2i are nonnegative integers verifying the condition µ 2 ≥ µ 4 ≥ • • • ≥ µ 2m , hence, by the result of lemma 3.3, µ |tK is a K-dominant weight. Now let v γ be the 1 maximal vector of the representation γ. Since it is killed by the action of root-vectors corresponding to positive roots, it is in particular killed by the action of the E α and E σ * (α) , α ∈ Φ ′ 1 it is unique up to a scalar multiple.

Lemma 3.7. Let µ be a G-dominant weight. Let Π µ be the set of weights of any irreducible G-representation with highest weight µ. If λ ± is the restriction to t K of a weight λ ∈ Π γ and if µ + δ G 2 is minimal, then λ belongs to the orbit of µ under the Weyl group W G .

Proof. The weight λ lies in the orbit under the Weyl group W G of G of a dominant weight ̺ ∈ Π γ . By the result of lemma B, 13.3 in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF], λ + δ G 2 ≤ ̺ + δ G 2 , with equality only if λ = ̺, and by lemma C, 13.4 in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF], ̺+δ G 2 ≤ µ+δ G 2 , with equality only if ̺ = µ. Now as ̺ is a G-dominant weight which posses in its orbit a weight λ whose restriction to t K is λ ± , the minimality condition verified by µ on that sort of weights implies ̺ + δ G 2 = µ + δ G 2 , hence ̺ = µ, and λ belongs to the orbit of µ.

Lemma 3.8. If µ γ is a G-dominant weight verifying the spin condition, then

µ γ + δ G 2 ≥ µ + + δ G 2 .
Proof. As µ γ verifies the spin condition, there exists a weight λ ∈ Π γ such that λ |tK = λ ± . By the result of the above lemma, λ belongs to the orbit of µ γ under the Weyl group W G , so λ + δ G 2 ≤ µ γ + δ G 2 , with equality if and only if λ = µ γ . If λ = µ γ , then, by the above considerations µ γ = µ + or µ ′ + , and the result follows

since µ + + δ G 2 = µ ′ + + δ G 2 . If λ = µ γ ,
then by replacing λ by σ β (λ) if necessary, we may suppose that λ |tK = λ + . But then, by the above considerations,

µ + + δ G 2 ≤ λ + δ G 2 , and so µ + + δ G 2 < µ γ + δ G 2
. Finally, we may conclude that the square of the first eigenvalue of the Dirac operator is given by 1 8m µ + , µ + + 2δ G + 2m 2 + m -1 16 , hence the result.

The symmetric space SU(2m) Sp(m) .

Let J be the matrix in SU(2m) defined by

J := 0 -I m I m 0 .
Note that J 2 = -I 2m and J -1 = -J = t J.

The group Sp(m) is identified with the subgroup of SU(2m)

Sp(m) := {A ∈ SU(2m) ; t A J A = J} .
The symmetric space structure is given by the involution

σ : G → G , A → t J t A -1 J .
This involution induces the decomposition of the Lie algebra su 2m into:

su 2m = sp m ⊕ p ,
where sp 2m is the Lie algebra

{X ∈ su 2m ; t XJ = -JX} = {X ∈ su 2m ; JXJ = t X} ,
of Sp(m), and

p = {X ∈ su 2m ; t XJ = JX} = {X ∈ su 2m ; JXJ = -t X} .
Elements of p are matrices X of the form X = X 1 X 2 X 2 -X 1 where X 1 and X 2 are m×m matrices verifying t X 1 = -X 1 and t X 2 = -X 2 . hence dim(p) = 2m 2 -m-1.

Denoting by T the maximal standard torus of SU(2m), T ∩ K is the standard maximal torus of Sp(m) :

T K := T ∩ K =                                    e iβ1 . . . e iβm e -iβ1 . . . e -iβm             ; β j ∈ R                        .
The Lie algebra t of T is defined by

t =    2m j=1 iβ j E jj ; β j ∈ R , 2m j=1
β j = 0    ,
and

t K =    m j=1 iβ j (E jj -E m+j m+j ) ; β j ∈ R    , t 0 =    m j=1 iγ j (E jj + E m+j m+j ) ; γ j ∈ R , m j=1 γ j = 0    .
Let y j , 1 ≤ j ≤ 2m, be the vectors of i t * defined by considering the restriction to t of the dual basis of the family of vectors i E jj , 1 ≤ j ≤ 2m :

∀H = 2m j=1 iβ j E jj ∈ t , y j (H) = β j .
Any element µ ∈ i t * may be uniquely written as

µ = 2m j=1 µ j y j , 2m j=1 µ j = 0 ,
and is denoted µ = [µ 1 , . . . , µ 2m ] .
The scalar product on i t * considered here is given by the scalar product on su 2m defined by

X, Y = -ℜ Tr(XY ) = - 1 4m B(X, Y ) , X, Y ∈ su 2m . For µ = [µ 1 , . . . , µ 2m ] and µ ′ = [µ ′ 1 , . . . , µ ′ 2m ] ∈ i t * , µ, µ ′ = 2m j=1 µ j µ ′ j .
The involution σ * of it * induced by σ is defined by

µ = [µ 1 , . . . , µ 2m ] σ * -→ [-µ m+1 , -µ m+2 , . . . , -µ 2m , -µ 1 , -µ 2 , . . . , -µ m ] .
Note that the vectors x j = y ′ j := y j|tK , j = 1, . . . , m, define a basis of it * K . Any λ ∈ it * K of the form λ = m j=1 λ j x j is denoted λ = (λ 1 , . . . , λ m ) .

4.1. Sets of roots. The root-vectors of SU(2m) relative to the standard torus are the E ij . Explicitly the roots are

y k -y l , 1 ≤ k = l ≤ m , with root-vector E kl , y m+k -y m+l , 1 ≤ k = l ≤ m , with root-vector E m+k m+l , ±( y k -y m+l ) , 1 ≤ k, l ≤ m , with root-vector E k m+l (resp. E m+l k ).
Note that the only roots θ that verify σ * (θ) = θ are ±( y ky m+k ), and as E k m+k and E m+k k belong to sp m ⊗ C, those roots belong to Φ 1 . Hence

Φ 1 = {±( y k -y m+k ) , 1 ≤ k ≤ m} , Φ 2 = ∅ ,
and all other roots belong to Φ 3 . We choose positive roots such that

Φ + 1 = { y k -y m+k , 1 ≤ k ≤ m} , Φ ′+ 3 = y k -y l y k -y m+l , 1 ≤ k < l ≤ m , σ * Φ ′+ 3 = σ * ( y k -y l ) = -y m+k + y m+l σ * ( y k -y m+l ) = -y m+k + y l , 1 ≤ k < l ≤ m .
It is easy to verify that a system of simple roots is given by

α := y m -y 2m , γ k := y k -y k+1 , 1 ≤ k ≤ m -1 , σ * (γ k ) = -y m+k + y m+k+1 , 1 ≤ k ≤ m -1 . Note that α, α = γ k , γ k = σ * (γ k ), σ * (γ k ) = 2 , k = 1, . . . , m -1 , γ k , γ k+1 = σ * (γ k ), σ * (γ k+1 ) = -1 , k = 1, . . . , m -2 , γ m-1 , α = σ * (γ m-1 ), α = -1 ,
all the other scalar products being zero. Hence Lemma 4.1.

A vector µ = [µ 1 , . . . , µ 2m ] ∈ i t * , 2m j=1 µ j = 0, is a dominant weight if and only if µ k -µ k+1 ∈ N , µ m+k+1 -µ m+k ∈ N , k = 1, . . . , m -1 , µ m -µ 2m ∈ N .
By (3), the positive K-roots are the restrictions to t K of the positive G-roots in Φ 1 and Φ ′ 3 , hence, as y m+k|tK =y k|tK ,

Φ + K =    2 x k , 1 ≤ k ≤ m x k -x l , 1 ≤ k < l ≤ m x k + x l , 1 ≤ k < l ≤ m    .
A system of simple roots is given by

θ ′ k := x k -x k+1 , k = 1, . . . , m -1 , and θ ′ m := 2 x m . Note that, as x k = 1 2 ( y k -y m+k ), θ ′ k , θ ′ k = 1 , k = 1, . . . , m -1 , θ ′ m , θ ′ m = 2 , θ ′ k , θ ′ k+1 = -1/2 , k = 1, . . . , m -2 , θ ′ m-1 , θ ′ m = -1 , all the other scalar products being zero. Hence Lemma 4.2. A vector λ = (λ 1 , . . . λ m ) ∈ i t * K , is a dominant weight if and only if λ 1 ≥ λ 2 ≥ • • • ≥ λ m ≥ 0 ,
and all the λ i are integers.

4.2.

Highest weights of the spin representation of K. Since Φ 2 = ∅, we may conclude from lemma 2.4 : Lemma 4.3. The spin representation of K has only one highest weight

λ 0 := (m -1, m -2, . . . , 1, 0) , with multiplicity 2 [ m-1 2 ]
Proof. As we saw it before the multiplicity of λ 0 is at least 2 [ m-1 2 ] . But it may be checked, using the Weyl dimension formula, that any irreducible K-module with highest weight λ 0 has dimension 2 m(m-1) , and

2 [ m-1 2 ] × 2 m(m-1) = dim(Σ) , hence the result.
4.3. The first eigenvalue of the Dirac operator. We apply the same method as above, and begin by determining the G-weights µ such that µ |tK = λ 0 and

µ + δ G 2 is minimal. First a vector µ = [µ 1 , . . . , µ 2m ] ∈ it * , 2m k=1 µ k = 0, is a G-weight if and only if µ k -µ k+1 ∈ Z, k = 1, . . . , 2m -1. Such a vector µ verifies µ |tK = λ 0 if and only if µ k -µ m+k = m -k , k = 1, . . . , m .

The condition

2m k=1 µ k = 0 then implies that m k=1 µ k = m(m -1) 4 .
The half-sum of the positive G-roots is given by

δ G = m k=1 m -k + 1 2 ( y k -y m+k ) . (Note that σ * (δ G ) = δ G ). Hence µ + δ G = m k=1 µ k + m -k + 1 2 y k + m k=1 µ k -2(m -k) - 1 2 y m+k .
As

m k=1 µ k = m(m-1) 4 , µ + δ G 2 is minimal if and only if m k=1 (µ 2 k + k µ k ) is minimal.
Set µ iµ i+1 = k i , i = 1, . . . , m -1. For any j = 1, . . . , m -1, one has

p j := j i=1 k i = µ 1 -µ j+1 . Using m k=1 µ k = m(m-1) 4 , one then gets m-1 j=1 p j = (m -1) µ 1 + µ 1 - m(m -1) 4 = m µ 1 - m(m -1) 4 , hence µ 1 = m -1 4 + 1 m m-1 i=1 p i , and
µ j+1 = µ 1 -p j = m -1 4 + 1 m m-1 i=1 p i -m p j .
The expression

m k=1 (µ 2 k + k µ k ) is a polynomial F (p 1 , . . . , p m-1
) of the variables p 1 , p 2 . . . , p m-1 . With the notation p = (p 1 , . . . , p m-1 ), one has

∂F ∂p i (p) = -2µ i+1 + m -(i + 1) .
The function F has a unique critical point when

µ i = m-i 2 , i = 1, . . . , m, which is equivalent to p i = i 2 , or k i = 1 2 , i = 1, . . . , m -1. Let p 0 = 1 2 , 1, 3 2 , . . . m-1 2 . As ∂ 2 F ∂p i ∂p j (p 0 ) = - 2 m (1 -m δ ij ) ,
we may apply the result of the previous example. With the same notations,

F (p) -F (p 0 ) = 1 2 t (p -p 0 )H(p -p 0 ) , = 1 m t (p -p 0 ) Q      1 0 0 0 0 m 0 0 . . . 0 0 0 m      t Q (p -p 0 ) , = 1 m ǫ 1 (m -1)k 1 + (m -2)k 2 + • • • + k m-1 - m(m -1) 4 2 + ǫ 2 (m -2)k 2 + (m -3)k 3 • • • + k m-1 - (m -1)(m -2) 4 2 + • • • + ǫ m-i i k m-i + • • • + k m-1 - i(i + 1) 4 2 + • • • + ǫ m-2 2k m-2 + k m-1 - 3 2 2 + ǫ m-1 k m-1 - 1 2 2 .
The minimum is obtained if and only if all the squares are minimal. Hence, since the k i have to be integers, Lemma 4.4. The minimum is obtained only when (k 1 , k 2 , . . . , k m-1 ) = (1, 0, 1, 0, . . .) or (0, 1, 0, 1, . . .) .

Proof. Note that

F (p) -F (p 0 ) = 1 4m ǫ 1 ((m -1)(2k 1 -1) + (m -2)(2k 2 -1) + • • • + (2k m-1 -1)) 2 + 1 4 ǫ 2 ((m -2)(2k 2 -1) + (m -3)(2k 3 -1) • • • + (2k m-1 -1)) 2 + • • • + 1 4 ǫ m-i (i (2k m-i -1) + • • • + (2k m-1 -1)) 2 + • • • + 1 4 ǫ m-2 (2(2k m-2 -1) + (2k m-1 -1)) 2 + 1 4 ǫ m-1 (2k m-1 -1) 2 .
Since the 2k i -1 are odd integers, the result of lemma 3.5 shows that the minimum is obtained if and only if (2k 1 -1, 2k 2 -1, . . . , 2k m-1 -1) = (1, -1, 1, -1, . . .) or (-1, 1, -1, 1, . . .), hence if and only if (k 1 , k 2 , . . . , k m-1 ) = (1, 0, 1, 0, . . .) or (0, 1, 0, 1, . . .).

Hence the G-weights µ for which µ |tK = λ 0 and µ + δ G 2 is minimal are given by

(1) If m is even, m = 2p, 1 4m µ 0 , µ 0 + 2δ G + 2m 2 -m -1 16 ,
hence the result.

5. The symmetric space SO(2p + 2q + 2) SO(2p + 1) × SO(2q + 1)

, p ≤ q, p + q ≥ 1.

Let (e i ) 1≤i≤2p+2q+2 be the standard basis of R 2p+2q+2 . Let J be the diagonal matrix

J =     -I 2p 0 0 0 0 I 2q 0 0 0 0 -1 0 0 0 0 1     .
We consider the involution σ of the group SO(2p + 2q + 2) defined by

A -→ JAJ -1 .
Note that J is orthogonal but det(J) = -1 hence σ is not a conjugation in the group. The connected component of the subgroup of fixed points is isomorphic to SO(2p + 1) × SO(2q + 1). We choose to consider G = Spin(2p + 2q + 2) instead. We view Spin(2p + 1) and Spin(2q + 1) as subgroups of G by considering

Spin(2p + 1) = {v 1 • • • v 2k ; v i ∈ span{e 1 , . . . , e 2p , e 2p+2q+1 } ; v i = 1} , and
Spin(2q + 1) = {v 1 • • • v 2ℓ ; v i ∈ span{e 2p+1 , . . . , e 2p+2q , e 2p+2q+2 } ; v i = 1} .
Let K be the connected subgroup of G defined by the image of the morphism

Spin(2p + 1) × Spin(2q + 1) -→ Spin(2p + 2q + 2) ; (ϕ, ψ) -→ ϕ • ψ .
This group K is the connected component of the subgroup of fixed elements of the (outer) involution

σ : G -→ G ; ψ -→ ϕ 0 • ψ • ϕ -1 0 , where ϕ 0 := e 2p+1 • e 2p+2 • • • e 2p+2q • e 2p+2q+2 .
The decomposition of the Lie algebra spin 2p+2q+2 induced by the involution σ is given by spin 2p+2q+2 = (spin 2p+1 ⊕ spin 2q+1 ) ⊕ p , where p =        i∈{1,...,2p,2p+2q+1} j∈{2p+1,...,2p+2q,2p+2q+2}

β ij e i • e j ; β ij ∈ R        .
Thus dim(p) = (2p + 1)(2q + 1) .

Let T be the standard maximal torus of G :

T = p+q+1 k=1 (cos β k + sin β k e 2k-1 • e 2k ) ; β k ∈ R . Then t = p+q+1 k=1 β k e 2k-1 • e 2k ; β k ∈ R , and
t K =    p k=1 β k e 2k-1 • e 2k + p+q k=p+1 β k e 2k-1 • e 2k ; β k ∈ R    , t 0 = span{e 2p+2q+1 • e 2p+2q+2 } .
Let x k , k = 1, . . . , p + q + 1, be the basis of i t * defined by

x k (H) = 2i β k , for H = p+q+1 k=1 β k e 2k-1 • e 2k ∈ t .
Any element µ in i t * of the form

µ = p+q+1 k=1 µ k x k , µ k ∈ R , is denoted µ = (µ 1 , . . . , µ p+q+1 ) .
The scalar product on i t * considered here is given by the scalar product on the Lie algebra spin 2p+2q+2 defined by

X, Y = - 1 2 Tr (ξ * (X)ξ * (Y )) = - 1 4(p + q) B(X, Y ) , X, Y ∈ spin 2p+2q+2 ,
where ξ is the covering Spin(2p + 2q + 2) → SO(2p + 2q + 2). For any µ = (µ 1 , . . . , µ p+q+1 ) and any

µ ′ = (µ ′ 1 , . . . , µ ′ p+q+1 ) µ, µ ′ = p+q+1 k=1 µ k µ ′ k .
The involution σ * of i t * induced by σ is defined by µ = (µ 1 , . . . , µ p+q , µ p+q+1 )

σ *
-→ (µ 1 , . . . , µ p+q , -µ p+q+1 ) .

Note that the vectors x ′ k := x k|tK , k = 1, . . . , p + q, define a basis of i t * K . Any λ ∈ i t * K of the form λ = p+q k=1 λ k x ′ k is denoted λ = (λ 1 , . . . , λ p+q ) .

Sets of roots. Let

u k = 1 2 (e 2k-1 -i e 2k
) and v k = 1 2 (e 2k-1 + i e 2k ) , k = 1, . . . , p + q + 1 .

The G-roots are

x i + x j , 1 ≤ i < j ≤ p + q + 1 , with root-vector space C u i • u j , -( x i + x j ) , 1 ≤ i < j ≤ p + q + 1 , with root-vector space C v i • v j , x i -x j , 1 ≤ i < j ≤ p + q + 1 , with root-vector space C u i • v j , -( x i -x j ) , 1 ≤ i < j ≤ p + q + 1 , with root-vector space C v i • u j . Note that Φ 1 = {±( x i ± x j ) ; 1 ≤ i < j ≤ p , p + 1 ≤ i < j ≤ p + q} , Φ 2 = {±( x i ± x j ) ; 1 ≤ i ≤ p , p + 1 ≤ j ≤ p + q} , Φ 3 = {±( x i ± x p+q+1 ) ; 1 ≤ i ≤ p + q} .
We choose positive roots such that

Φ + 1 = x i -x j x i + x j ; 1 ≤ i < j ≤ p , p + 1 ≤ i < j ≤ p + q , Φ + 2 = x i -x j x i + x j ; 1 ≤ i ≤ p , p + 1 ≤ j ≤ p + q , Φ ′+ 3 = { x i -x p+q+1 ; 1 ≤ i ≤ p + q} , σ * (Φ ′+ 3 ) = { x i + x p+q+1 ; 1 ≤ i ≤ p + q} .
It is easy to see that a system of simple roots is given by

α k := x k -x k+1 , k = 1, . . . , p -1 , k = p + 1, . . . , p + q -1 , (∈ Φ + 1 ) , β := x p -x p+1 , (∈ Φ + 2 ) , γ := x p+q -x p+q+1 , (∈ Φ ′+ 3 ) , σ * (γ) := x p+q + x p+q+1 , (∈ σ * (Φ ′+ 3 )) . Note that α k , α k = β, β = γ, γ = σ * (γ), σ * (γ) = 2 , k = 1, . . . , p -1, p + 1, . . . , p + q -1 , α k , α k+1 = α p-1 , β = α p+1 , β = α p+q-1 , γ = α p+q-1 , σ * γ = -1 , k = 1, . . . , p -2, p + 1, . . . , p + q -2 ,
all the other scalar products being zero. Hence the following (classical) characterization.

Lemma 5.1. A vector µ = (µ 1 , . . . , µ p+q+1 ) ∈ i t * is G-dominant if and only if

µ 1 ≥ µ 2 ≥ • • • ≥ µ p+q ≥ |µ p+q+1 | ,
the µ i being all simultaneously integers or half-integers.

By (3), the positive K-roots are the restrictions to t K of the positive G-roots in Φ 1 and Φ ′ 3 , hence,

Φ + K = x ′ i -x ′ j x ′ i + x ′ j 1 ≤ i < j ≤ p , p + 1 ≤ i < j ≤ p + q , ; x ′ i , 1 ≤ i ≤ p + q .
A system of simple roots is given by

θ ′ k = x ′ k -x ′ k+1 , k = 1, . . . , p -1, p + 1, . . . p + q -1 , x ′ p and x ′ p+q . Note that θ ′ k , θ ′ k = 2 , k = 1, . . . , p -1, p + 1, . . . p + q -1 , x ′ p , x ′ p = x ′ p+q , x ′ p+q = 1 , θ ′ k , θ ′ k+1 = θ ′ p-1 , x ′ p = θ ′ p+q-1 , x ′ p+q = -1 , k = 1, . . . , p -2 , p + 1, . . . p + q -2 . Hence Lemma 5.2. A vector λ = (λ 1 , . . . , λ p+q ) ∈ i t * K is K-dominant if and only if λ 1 ≥ λ 2 ≥ • • • ≥ λ p ≥ 0 , and λ p+1 ≥ λ p+2 ≥ • • • ≥ λ p+q ≥ 0 ,
the λ i for i = 1, . . . , p (resp. for i = p + 1, . . . , p + q) being all simultaneously integers or half-integers.

5.2.

A characterization of the highest weights of the spin representation of K. The explicit determination of the highest weights is far from being simple. Some results on the decomposition of the spin representation for oriented grassmannians may be found in [START_REF] Klinker | The decomposition of the spinor bundle of Grassmann manifolds[END_REF]. By the result of lemma 2.4, any highest weight of the spin representation of K has the form 1 2

  β∈Φ + 2 ±β ′ + p+q i=1 x ′ i   .
Now, remark that the α ′ s in Φ 1 and the β ′ s in Φ 2 are respectively compact and non-compact roots relative to the maximal common torus T K of the groups G 1 = SO(2p + 2q) and K 1 = SO(2p) × SO(2q). Since G 1 /K 1 is an inner symmetric space, the results of R. Parthasarathy in [Par71] may be applied here. First the weights of the spin representation of K 1 are 1 2 [START_REF] Parthasarathy | Dirac operator and the discrete series[END_REF]. Now from lemma 2.2 in [START_REF] Parthasarathy | Dirac operator and the discrete series[END_REF] we may conclude Lemma 5.3. Any highest weight of the spin representation of K has necessarily the form

  β∈Φ + 2 ±β ′   , see Remark 2.1 in
w • δ G1 -δ K1 + 1 2 p+q i=1 x ′ i , w ∈ W 1 ,
where • δ G1 is the half-sum of the positive G 1 -roots, that are the α ′ s in Φ + 1 and the β ′ s in Φ + 2 , and whose set is denoted Φ + G1 , • δ K1 is the half-sum of the positive roots of K 1 , that are the α ′ s in Φ + 1 , and whose set is denoted Φ + K1 , • W 1 is the subset of the Weyl group W G1 of G 1 defined by

W 1 = {w ∈ W G1 ; w • Φ + G1 ⊃ Φ + K1 } . Proof. Let λ = 1 2 β∈Φ + 2 ε λβ β ′ + p+q i=1 x ′ i , ε λβ = ±1, be a highest weight of the spin representation of K. For any α ∈ Φ 1 , α ′ + 1 2 β∈Φ + 2 ε λβ β ′ = 1 2 β∈Φ + 2 ± β ′ ,
since otherwise λ + α ′ is a weight of the spin representation of K, contradicting the fact that λ is a highest weight. So 1 2 β∈Φ + 2 ε λβ β ′ is a highest weight of the spin representation of K 1 , hence of the form w • δ G1δ K1 , w ∈ W 1 , by the result of Parthasarathy. Now let λ be a highest weight of that sort. One has

δ G1 = p+q i=1 (p + q -i) x i , and
δ K1 = p i=1 (p -i) x i + p+q i=1 (p + q -i) x i .
On the other hand, the Weyl group W G1 acts on i t * K as (λ 1 , . . . , λ p+q ) → (ǫ 1 λ σ(1) , ǫ 2 λ σ(2) , . . . , ǫ p+q λ σ(p+q) ) ,

where σ ∈ S p+q , ǫ i = ±1, ǫ 1 • • • ǫ p+q = 1, see for instance [HBM+15]. So, for any w ∈ W G1 , w • δ G1 -δ K1 = p i=1 ǫ σ(i) (p + q -σ(i)) -(p -i) x i + p+q i=p+1 ǫ σ(i) (p + q -σ(i)) -(p + q -i) x i ,
where σ ∈ S p+q . But the dominance conditions verified by λ implies ǫ i = 1, i = 1, . . . , p -1, p + 1, . . . , p + q -1, and this is also true for ǫ p and ǫ p+q since ǫ p = ǫ p+q = -1 implies σ(p) = p + q = σ(p + q). Now the dominance conditions also imply that any highest weight λ of the spin representation of K has necessarily the form

(9) λ = p i=1 q + i -σ(i) + 1 2 x i + p+q i=p+1 i -σ(i) + 1 2 x i , where σ ∈ S p+q verifies σ(i) ≤ σ(i + 1) -1, 1 ≤ i ≤ p -1, p + 1 ≤ i ≤ p + q -1.
Note that this implies σ(i) ≥ i, 1 ≤ i ≤ p, and σ(i) ≤ i, p + 1 ≤ i ≤ p + q.

5.3. The first eigenvalue of the Dirac operator. We begin by determining necessary conditions for a G-weight µ = (µ 1 , . . . , µ p+q+1 ) in order that µ |tK = (µ 1 , . . . , µ p+q ) is a highest weight of the spin representation and µ + δ G 2 is minimal. Note first that a vector µ = (µ 1 , . . . , µ p+q+1 ) ∈ it * , is a G-weight if and only if all the µ i are all simultaneously integers or half-integers. So, by (9), for such a G-weight, µ |tK = (µ 1 , . . . , µ p+q ) is a highest weight of the spin representation only if µ p+q+1 is a half-integer. Furthermore, since δ G = (p + q, p + q -1, . . . , 1, 0) , the condition that µ + δ G 2 is minimal implies then µ p+q = ± 1 2 (with the same value in both cases). So let µ be such a weight. We first consider the case where µ is dominant.

First case

: µ is G-dominant. By (9), one has µ + δ G = p i=1 p + 2q + 3 2 -σ(i) x i + p+q i=p+1 p + q + 3 2 -σ(i) x i , where σ ∈ S p+q verifies σ(i) ≤ σ(i + 1) -1, 1 ≤ i ≤ p -1, p + 1 ≤ i ≤ p + q -1,
and also, as µ is G-dominant σ(p) ≤ σ(p + 1) + q -1.

Proof. As p is odd-dimensional, we have to use the description of spinors given in § 2.3.2. By the choice of an orthonormal basis of p such as (5), the group SO(p) is identified with SO(4pq + 2(p + q) + 1), which is itself embedded in the group SO(4pq + 2(p + q) + 2) in such a way that it acts trivially on the last vector of the standard basis of R 4pq+2(p+q)+2 . Let (Z, Z) be the Witt basis defined by the two orthonormal vectors on which SO(p) acts trivially. Using the notations of § 2.3, a basis of spinors is given by considering the vectors

E I • V J • w , if #I + #J is even, E I • V J • Z • w , if #I + #J is odd. Denoting for short E ij = E xi+ xj , E ′ ij = E xi-xj , 1 ≤ i ≤ p, p + 1 ≤ j ≤ p + q
, and V j = E xj -xp+q+1 , 1 ≤ j ≤ p + q, one gets from (6), that the spinor

v λ := 1≤i≤p p+1≤j≤p+q E ij • 1≤i≤p q+1≤j≤p+q E ′ ij • V 1 • • • V p+q • w , if (p + 1)(p + q) is even, := 1≤i≤p p+1≤j≤p+q E ij • 1≤i≤p q+1≤j≤p+q E ′ ij • V 1 • • • V p+q • Z • w , if (p + 1)(p + q) is odd,
is a weight-vector for the weight λ. Now, it may be checked that the root-vector u k • v k+1 , associated to the simple root

θ ′ k = x ′ k -x ′ k+1
, acts on spinors by a linear combination of

p+q j=p+1 E k+1 j • E kj , p+q j=p+1 E ′ k+1 j • E ′ kj , and V k+1 • V k , if 1 ≤ k ≤ p -1, p i=1 E ik+1 • E ik , p i=1 E ′ ik • E ′ ik+1 , and V k+1 • V k , if p + 1 ≤ k ≤ p + q -1.
So v λ is killed by the action of u k • v k+1 since all the E ij 's and all the V j 's occur in the expression of v λ , and either E ′ kj (resp. E ′ ik+1 ) occur in that expression or both E ′ kj and E ′ k+1 j , (resp. E ′ ik+1 and E ′ ik ) do not occur in the expression. In the same way, v λ is killed by the action of the root-vector u p •e 2p+2q+1 , associated to the simple root x ′ p , since that root-vector acts on spinors by a linear combinations of Finally, v λ is also killed by the action of the root-vector u p+q+1 • e 2p+2q+2 , associated to the simple root x ′ p+q , since that root-vector acts on spinors by a linear combinations of Thus v λ is a maximal vector, and so λ is a highest weight of the spin representation.

So µ ± = (p + 1 2 , . . . , p + 1 2 q , 1 2 , . . . , 1 2 p , ± 1 2 ) are two G-dominant weights verifying the spin condition and such that µ + δ G2 is minimal among all the G-dominant weights µ verifying the spin condition. Hence the square of the first eigenvalue λ of the Dirac operator is given by λ 2 = 1 4(p + q) µ + , µ + + 2δ G + (2p + 1)(2q + 1) 16 , = 1 16 (p + q) 8pq (2p + q + 1) + 4p (p + 1) + 4q (q + 1) + 1 .

Note that for p = 0, we retrieve 2 the value of the square of the first eigenvalue of the Dirac operator on the standard sphere S 2q+1 :

λ 2 = 1 4q 2q + 1 2 2 , cf. [Sul79].
6. The symmetric space E 6 F 4 .

We use here the results of Murakami [START_REF]Sur la classification des algèbres de Lie réelles et simples[END_REF] (for an outline, see for instance the section "non-inner involutions" in chapter 3 of [START_REF] Burnstall | Twistor theory for Riemannian symmetric spaces[END_REF]). Explicit computations can be made (and are obtained) with the help of the programs GAP3, [S+97], and LiE, [START_REF] Van Leeuwen | A Package for Lie Group Computations[END_REF]. First, the existence of non-inner involutions corresponds with the existence of nontrivial symmetries of the Dynkin diagram. The Dynkin diagram of E 6 is

θ 1 θ 3 θ 4 θ 5 θ 6 θ 2
There is only one non-trivial symmetry s given by s(θ 1 ) = θ 6 , s(θ 2 ) = θ 2 , s(θ 3 ) = θ 5 , s(θ 4 ) = θ 4 , s(θ 5 ) = θ 3 , s(θ 6 ) = θ 1 .

The symmetry s is extended by linearity to an involution σ * of i t * , which itself induces an involution σ * of t, by means of the scalar product (re-normalized here in such a way that all simple roots θ verify θ 2 = 2). Now, choosing a root-vector E θ for each simple root θ, σ * is extended to the span of these vectors by (10) σ * (E θ ) = E σ * (θ) .

Finally σ * can be uniquely extended to a non-inner involution of g (cf. § 14.2 in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF]). A first outer symmetric space structure is obtained by considering the connected subgroup K of E 6 whose Lie algebra is the set {X ∈ g ; σ * (X) = X}. It is a simple group, and a system of simple roots (relatively to the maximal torus T K = K ∩ T ) is obtained by considering the restriction to t K of the simple roots θ i , 1 ≤ i ≤ 6, (cf. Proposition 3.20 in [START_REF] Burnstall | Twistor theory for Riemannian symmetric spaces[END_REF]).

The group E 6 has 36 positive roots. The positive roots θ such that σ * (θ) = θ all belong to Φ + 1 by (10). The partition of the set of positive roots is given by are G-dominant. Hence we may conclude exactly as we did it in the first example with the result of lemma 3.8. We have first to note that, for the scalar product , K induced by the Killing form sign-changed, the "strange formula" of Freudenthal and de Vries, [START_REF] Freudenthal | Linear Lie Groups[END_REF], gives

δ G 2 K = dim(g) 24 = 13 4 ,
whereas for our choice of scalar product δ G 2 = 78, hence , K = 1 24 , . So, finally, the square of the first eigenvalue of the Dirac operator is given by By the result of Murakami, there also exists a complementary non-inner involution on the Lie algebra of E 6 , which is not conjugate to the above non-inner involution σ and is defined on the Lie algebra by σ ′ * = σ * • Ad exp(π ξ2) , where ξ 2 ∈ t is defined by θ j (ξ 2 ) = i δ j2 , cf. Theorem 3.25 in [START_REF] Burnstall | Twistor theory for Riemannian symmetric spaces[END_REF]. A new outer symmetric space structure is obtained by considering the connected subgroup K ′ of E 6 whose Lie algebra is the set {X ∈ g ; σ ′ * (X) = X}. Note that as ξ 2 ∈ t, σ ′ * = σ * . Lemma 7.3. Among the G-dominant weights µ verifying the spin condition, µ + δ G 2 is minimal if and only if µ = µ 1 or µ = µ 2 .

1 24 µ 1 + 2 δ G ,
Hence the square of the first eigenvalue of the Dirac operator is

1 24 µ 1 + 2 δ G ,

  m, and µ 2i-1µ 2i+1 , µ 2iµ 2(i+1) are both simultaneously even or odd, i = 1, . . . m -1.Let µ be such a G-weight. It verifies µ |tK = λ ± if and only if it has the formµ = [µ 1 , µ 3 , . . . , µ 2m-1 ; m, m -1, . . . , 2, ±1] ,where m i=1 µ 2i-1 = 0, and µ 2i-1µ 2i+1 are odd integers, i = 1, . . . m -1. The half-sum of the positive G-roots δ G is given by

3+

  , hence by the action of theU α = 1 2 (E α + σ * (E α )), α ∈ Φ ′ 3 + . Since the U α are root-vectors for the positive K-roots α ′ , α ∈ Φ ′ 3 + , (see (3)), v γ is a maximal vector of Res G K (γ)for the weight µ γ|tK . The conclusion now results from the following remark.

E

  pj • V j , p+q j=p+1 E ′ pj • V j , and V p • (e 2p+2q+1 • e 2p+2q+2 ) .

E

  i p+q • V i , p i=1 E ′ i p+q • V i , and V p+q • (e 2p+2q+1 • e 2p+2q+2 ) .

in that case, Φ 2 = ∅, so the spin representation of K has only one highest weight.

Note that µ ′ 0 = σ * (µ 0 ) and

. . , k m-1 ) = (1, 0, 1, 0, . . . , 0), since p 2i = i, and p 2i+1 = i+1,

(b) if (k 1 , k 2 , . . . , k m-1 ) = (0, 1, 0, 1, . . . , 1), since p 2i = p 2i+1 = i,

Note that µ ′ 0 = σ * (µ 0 ) and

Note that µ 0 and µ ′ 0 are G-dominant, hence we may conclude exactly as in the above case with the result of lemma 3.8 : the square of the first eigenvalue of the Dirac operator is given by So

But σ(p) can not be > q in that case. Indeed, if σ(p) > q, then, as we suppose p ≤ q, σ(1) > 1. So σ(p + 1), . . . σ(p + q) belong to the set {1, . . . , σ(1) -1, σ(p) + 1, . . . , p + q}, and then the conditions σ(i) ≤ σ(i + 1) -1, p + 1 ≤ i ≤ p + q -1, imply σ(p + 1) = 1. But that contradicts the condition σ(p) ≤ σ(p + 1) + q -1. Hence µ + δ G 2 is minimal if and only if

Now, as µ is conjugate under W G to one and only one G-dominant weight, one sees that since the Weyl group W G acts on i t * as (µ 1 , . . . , µ p+q+1 ) → (ǫ 1 µ σ(1) , ǫ 2 µ σ(2) , . . . , ǫ p+q+1 µ σ(p+q+1) ) ,

, µ is conjugate under the Weyl group to the G-dominant weight met in the above case

) .

Finally, we may conclude that Lemma 5.5. If a G-dominant weight µ verifies the spin condition and is such that

.

In order to conclude, we only have to verify that Lemma 5.6. The vector λ ∈ i t * K defined by

is a highest weight of the spin representation of K.

this is the "classical" Dynkin diagram of the group F 4 . In that basis, the Cartan matrix is

In the following, ω 1 , ω 2 , ω 3 , ω 4 are the fundamental weights associated with α 1 , α 2 , α 3 , α 4 .

6.1. Highest weights of the spin representation of K. Since Φ 2 = ∅, and dim t 0 = 2, we may conclude from lemma 2.4, Lemma 6.1. The spin representation of K has only one highest weight λ with multiplicity 2,

Any irreducible module with highest weight ω 3 + ω 4 has dimension 2 12 . Since dim t 0 = 2, one knows from § 2.3 that the multiplicity of the weight is at least 2. Now 2 × 2 12 = 2 13 = 2 dim(p)/2 = dim Σ, hence the result.

6.2. The first eigenvalue of the Dirac operator. As we did before, we first determine G-weights µ (non necessarily dominant) such that

the last condition is equivalent to k and l ∈ Z (resp. k = 0 or 1 and l = 0 or 1).

Viewing then µ + δ G 2 as a polynomial f (k, l) of the variables k and l, one gets

Hence among the G-weights µ such that

The partition of the set of positive roots is now given by

The set of positive K ′ -roots is

and a system of simple K ′ -roots is given by

(cf. Theorem 3.25 in [START_REF] Burnstall | Twistor theory for Riemannian symmetric spaces[END_REF]).

Note that here dim g = 78 , dim k ′ = 36 , dim p = 42 ,

One has

all the other scalar products being zero. The Cartan matrix is

This is the Dynkin diagram of Sp 4 .

7.1. Highest weights of the spin representation of K ′ . By the result of lemma 2.4, any highest weight of the spin representation of K has the form

Lemma 7.1. The spin representation of K ′ has three highest weights both of multiplicity 2 :

Proof. Note first that

Now, denoting by β 1 , . . . , β 8 the roots occurring in that order in the expression of Φ + 2 above, it is easily checked that

so λ 1 , λ 2 and λ 3 are actually weights of the spin representation of K ′ . It is also easily checked that they are also G-dominant. Indeed, considering the basis of fundamental weights

Note now that, for any i, j = 1, 2, 3, i = j, any relation of the form λ i ≺ λ j , where "≺" is the standard order on weights, is impossible, so λ i can not be an element of the set of weights of an irreducible module with highest weight λ j . This implies that λ 1 , λ 2 and λ 3 are actually highest weights of the spin representation of K ′ .

3 multiplying for instance the transpose of the Cartan matrix with the vector representing the weight in the basis

Since dim t 0 = 2, one knows from § 2.3 that the multiplicity of each such highest weight is at least 2. Now, denoting by n i the dimension of any irreducible module with highest weight λ i , i = 1, 2, 3, one gets with the help of the LiE program n 1 = 180224 , n 2 = 524288 and n 3 = 344064 .

Remark 7.2. Actually, the determination of the three possible highest weights was obtained by using the same argument as in § 5.2. Indeed the α's in Φ 1 and the β's in Φ 2 appear to be respectively compact and non-compact roots for the inner symmetric space SO(8)/(SO(4) × SO(4)), relatively to the standard torus of SO(8). Using the results of R. Parthasarathy in [START_REF] Parthasarathy | Dirac operator and the discrete series[END_REF], we obtain, with the help of the GAP3 program, the twelve highest weights of the spin representation for the symmetric space SO(8)/(SO(4) × SO(4)), providing a list of twelve candidates for the possible highest weights of the spin representation of K ′ . Finally, the dominance condition (for K ′ ) reduces that list to the three weights λ 1 , λ 2 and λ 3 .

7.2. The first eigenvalue of the Dirac operator. We determine G-weights µ (non necessarily dominant) such that µ |tK = λ 1 , λ 2 or λ 3 , and µ+δ G 2 is minimal. Let µ be such a weight. We first examine the case where it is G-dominant.

Thus setting Thus f (k, l) has only one critical point ( 1 2 , 1 2 ). Now

Hence among the dominant G-weights µ such that

Now let θ ∈ Φ 3 . By R2, there exists Y ∈ t 0 such that θ(Y ) = 0. Note that

Furthermore, as for any

This implies in particular that the decomposition is orthogonal, and θ + = θ |tK = θ ′ . Now let α and β ∈ Φ 3 such that β ′ = α ′ , with β = α and β = σ * (α). We first claim that α + β is not a root. Otherwise

But α ′ is a K-root (R1) and it is well known that α ′ and 2α ′ can not be both K-roots. Thus α + β is not a root, and so α, β ≥ 0, since otherwise α + β should be a root (see for instance 9.4. in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF]). One the other hand, αβ can not be a root by R1 since (αβ) |tK = 0. Thus α, β ≤ 0, since otherwise αβ should be a root. Finally, α, β = 0. Note that, as σ * (α) |tK = α |tK , the same arguments hold for σ * (α) instead of α. Hence σ * (α), β = 0, and so α + σ * (α), β = 0. Now as α + σ * (α) = 2 α + , this implies α + , β + = 0. But α + = α ′ = β ′ = β + , hence α ′ 2 = 0, which is impossible.