0041-6932 0041-6932 S0041-69322008000200008 Argentina 00 12 2008 00 12 2008 49 2 67 80

Admissible restriction of holomorphic discrete series for exceptional groups

Jorge Vargas

To Mischa Cotlar with respect

Abstract. In this note, we give results about the restriction of a holomorphic discrete series of an exceptional simple Lie real group to a subgroup.

2000 Mathematics Subject Classification. Primary 22E46.
Key words and phrases. Holomorphic Discrete Series representations, branching laws.
Partially supported by FONCYT, CONICET, AgenciaCbaCiencia, SECYTUNC (Argentina), ICTP, TWAS (Italy)

1. Introduction

A basic problem in representation theory of Lie groups is to derive "branching laws". By this we mean, for a given unitary irreducible representation of an ambient group G , consider its restriction to a fixed subgroup H and find the decomposition as a direct integral, and in particular compute the multiplicity of each irreducible factor of the restriction. There is a vast literature on this subject, and here we just direct the reader's attention to the extensive reviews of [13], [14] and references therein. In this note, we consider a holomorphic discrete series of a connected simple exceptional Lie group, and determine whether or not it has an admissible restriction to a given closed connected reductive subgroup H ⊂ G . Let us recall that a unitary representation of a topological group H is admissible if it is a discrete Hilbert sum of irreducible unitary sub-representations and each irreducible summand occurs with finite multiplicity.

Holomorphic discrete series are associated to Hermitian symmetric spaces. We consider a Hermitian symmetric space G∕K , where G is a simple connected real Lie group G (which we shall assume for convenience, to minimize notations, with finite center), and K a maximal compact subgroup. For a Lie group we denote its Lie algebra by the corresponding German lower case letter. We write the Cartan decomposition of 𝔤 as 𝔤 = 𝔨 ⊕ 𝔰 . Thus 𝔰 , the tangent space of G ∕K at the origin, is provided with a complex structure J ∈ Hom ℝ(𝔰,𝔰) corresponding to a choice of square root i ∈ ℂ . To denote the complexification of a vector space, we add the subscript ℂ . We denote by 𝔰+ and 𝔰- the eigenspaces of J in 𝔰 ℂ with respective eigenvalues {+i, - i} : a linear form f ∈ Hom ℝ(𝔰,ℂ) is ℂ -linear if and only if its linear extension to 𝔰ℂ is zero on the subspace - 𝔰 . Moreover, + - 𝔰ℂ = 𝔰 ⊕ 𝔰 , is the decomposition of 𝔰ℂ as a direct sum of two irreducible K -modules, dual to each other.

]]> Recall (see [7]) that the center 𝔷 of 𝔨 is one dimensional, and that we can choose uniquely a basis (denoted by the same letter J ) of 𝔷 whose adjoint action in 𝔰 is the complex structure J of the tangent space at the origin of G ∕K . We write 𝔨ss = [𝔨,𝔨] . We have 𝔤 = 𝔨ss ⊕ ℝJ ⊕ 𝔰 , and 𝔤ℂ = 𝔨ssℂ ⊕ ℂJ ⊕ 𝔰+ ⊕ 𝔰- . Correspondingly, we have K = KssZ , where Z is isomorphic to SO (2,ℝ ) , and Kss ∩ Z is finite.

An irreducible unitary representation of G is called holomorphic if its underlying Harish-Chandra module has a non zero vector v which is annihilated by 𝔰- . An irreducible irreducible unitary representation of G is called a discrete series representation if its coefficients are square integrable on G with respect to a given Haar measure.

The exceptional connected simple Lie groups whose quotient by a maximal compact subgroup carries an invariant complex structure has been classified by E. Cartan. They are the connected groups with Lie algebras 𝔢6(-14) and 𝔢7(-25) . The respective complexified Cartan decompositions are :

𝔢6 = 𝔢6(-14)ℂ = 𝔰𝔬(10,ℂ ) + ℂJ + (𝔰+ ⊕ 𝔰- ).
Here, 𝔰± are the half spin 16 -dimensional representations.
𝔢7 = 𝔢7(- 25)ℂ = 𝔢6 + ℂJ + (ϖ1 ⊕ ϖ6 ).
Here, ϖ ⋆ are the two fundamental representations of dimension twenty seven of the complex simple algebra 𝔢6 .

In this paper, for 𝔤 = 𝔢6(-14) and 𝔤 = 𝔢7(- 25) we give list closed connected reductive subgroups H of G such that an holomorphic discrete series of G has an admissible restriction to H . In [6], we gave several results concerning restrictions of more general discrete series for more general reductive groups, in particular, we introduced a sufficient condition —we call it condition (C)— which implies admissibility of restriction, and allows to compute multiplicities of restrictions by mean of a Blattner-Kostant type formula involving a partition function. However, there exist many cases of admissibility where condition (C) is not satisfied —many examples are given in [6], all of them for compact groups H . One of our interests in studying precisely what happens for holomorphic discrete series of exceptional groups, besides our wish to understand the full picture, is to find other interesting examples. In particular, we give several non compact examples.

We would like to point out that in his Ph.D. thesis [21], S. Simondi has obtained the results on admissibility when rank of L is equal to rank of K, they follow from Theorem 1. His technique is different from the one is used in this note.

The author would like to express his gratitude to Michel Duflo for the enlighten comments on the topics of this note.

2. Some general results

2.1. A criterium for admissibility of restriction.

]]> We recall some results which we will use in our proofs. Let G be a connected simple Lie group with finite center, choose a maximal compact group K , and Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔰 . We denote by K ℂ the corresponding complex group. Let H be a closed connected reductive subgroup. We assume that L := H ∩ K is a maximal compact subgroup of H .

In [6], we prove a result which reduces the problem of admissibility of restriction of discrete series to the case of compact subgroups :

Proposition 1. Let (π, V) be a discrete series for G . Then its restriction to H is admissible if and only if its restriction to L is admissible.

There are many criteria for admissibility of the restriction to a subgroup of an irreducible unitary representation (π, V) of G (see e. g. [13], [14]). When the subgroup is compact, we will use a criterium in term of the associated variety which we explain. We denote by (π,Vf ) the representation of 𝔤C in the space of K -finite vectors of V . Vogan [26] defined the associated variety V (Vf) , which is a Zariski-closed K ℂ -invariant cone of dual 𝔰* ℂ of 𝔰 ℂ . Let us denote by ℂ [V (V )] f the ring of regular functions on V(Vf ) . The following criterium is known (see in particular Huang and Vogan [9], Kobayashi [12], Vergne [25]).

Proposition 2. Let (π,V ) be an irreducible unitary representation of G . Then its restriction to L is admissible if and only if Lℂ ℂ [V(Vf)] = ℂ , that is the only L ℂ invariant regular functions on V (Vf) are the constant.

Assume now that G ∕K is hermitian symmetric. The criterium is particularly pleasant for holomorphic discrete series (see [12], [6], [25]) :

Proposition 3. Let (π,V ) be a holomorphic discrete series of G . Then V (V ) f is the orthogonal of 𝔰- in 𝔰* ℂ . Thus its restriction to L is admissible if and only if + Lℂ S[𝔰 ] = ℂ .

The most obvious example of proposition 3 is the group K . The restriction to K of an holomorphic discrete series is admissible (in fact it is true for any unitary irreducible representation of G ), and we have also S [𝔰+ ]K ℂ = ℂ . Thus our problem of restriction is a particular case of a well known problem in invariant theory (see [22],[23]): Find pairs of connected reductive groups complex linear groups A ⊂ B ⊂ GL (𝔰+) such that S [𝔰+ ]A = S[𝔰+]B .

Remark 1. If L is semi-simple, the condition + L S[𝔰 ] ℂ = ℂ holds if and only if Lℂ has an open orbit in 𝔰+ .

The subgroups Z and Kss of K deserve a special attention. For completeness, we recall the following well known result (which can serve as an illustration of proposition 3)

]]> Proposition 4. Let (π,V ) be a holomorphic discrete series of G . Its restriction to Z (and also to any closed subgroup H ⊂ G which contains Z ) is admissible.

To study the restriction to Kss , recall that Hermitian symmetric spaces G ∕K are divided in two categories: the tube type, and the non tube type. One of the many equivalent definitions of tube type is (see [7]):

The Hermitian symmetric spaces G ∕K is of non tube type if and only if S [𝔰+ ]Kssℂ = ℂ .

We will also say that 𝔤 is of tube type. Hermitian symmetric spaces of tube type are related to simple Jordan algebras [7]; They are interesting because they have associated Zeta functions. However, from our point of view, non tube type is more interesting:

Proposition 5. Let (π,V ) be a holomorphic discrete series of G . Its restriction to Kss is admissible if and only if G ∕K is not of tube type.

The list of Hermitian symmetric spaces G ∕K of tube type is well known (see [7]). Among the two exceptional ones, 𝔢6(-14) is not of tube type, and 𝔢 7(-25) is of tube type. Thus we have the following preliminary results, which explains why the case 𝔢6(- 14) is richer.

Theorem 1. Let (π, V ) be a holomorphic discrete series of G with Lie algebra 𝔢 6(-14) . Its restriction to K ss is admissible.

Theorem 2. Let (π, V ) be a holomorphic discrete series of G with Lie algebra 𝔢7(- 25) . Its restriction to Kss (and to any of its closed subgroups L ) is not admissible.

2.2. Condition (C).

We recall what is condition (C) of [6] in the particular case of a holomorphic discrete series. We choose a Cartan subgroup T of K , and denote by Φ ⊂ i𝔱* be set the roots of T in 𝔤 ℂ . We choose a positive system Ψ ⊂ Φ such that the set of non compact roots Ψn is exactly the set of roots of T in + 𝔰 . We denote by * C ⊂ i𝔱 the closed convex pointed cone generated by Ψn .

]]> We assume that U := L ∩ T is a Cartan subgroup of L . Let 𝔲⊥ ⊂ 𝔱*ℂ be the orthogonal of 𝔲 . Here is condition (C):

(C ) : C ∩ 𝔲⊥ = {0}.

We rephrase condition (C). Let ⊤ C ⊂ 𝔱 the cone dual to * C∕i ⊂ 𝔱 ; It is a closed convex cone whose interior C⊤int contains J . Then condition (C) is equivalent to condition (C'):

′ ⊤ (C ) : Cint ∩ 𝔲 ⁄= {0 }.

Condition (C) depends only on the maximal torus* U of L . We have:

Theorem 3. Let (π, V ) be a holomorphic discrete series of G . Let U ⊂ T be a compact connected torus. Then the restriction of (π, V) to U is admissible if and only if condition (C) holds.

Proof. As a K ℂ -module, V is isomorphic to + F ⊗ S [𝔰 ] , where F is an irreducible representation of K ℂ . Thus, as a T module, it is a finite direct sum of ℂμ ⊗ S [𝔰+ ] , where ℂ μ is a one dimensional representation of T with weight μ ∈ i𝔱* . The weights of T in S [𝔰+ ] are exactly the weights of T contained in C , occurring with finite multiplicity. The theorem follows. □

If condition (C) is satisfied for a torus U ⊂ T , it is also satisfied for some one dimensional torus U1 ⊂ U . Then U1 satisfies condition (C) if and only if 𝔲1 has a basis B which belongs to ⊤ Cint . In particular, Z satisfies condition (C) (which is a way of proving proposition 4), and also all one dimensional torus U1 not to far away from Z .

On the other hand, it is easy to see that condition (C) is never satisfied for 𝔩 ⊂ 𝔨ss . Thus, for 𝔤 = 𝔢6(-14) , the group Kss is an easy example where there is admissibility and condition (C) does not hold.

2.3. Formulation of the problem.

]]> Let us explain more precisely what has to be done in general. We fix a compact connected semisimple group D ⊂ K ss with a Cartan subgroup A ⊂ T . Let B be the connected component group of the centralizer of D in T . Then AB is a Cartan subgroup of DB .

Consider a connected closed groups L ⊂ K such that Lss = D . Up to conjugation, it will be of the form L = BLD , where BL , the connected center of L , is a closed connected subgroup of B . Note that B contains the center Z of K . For clarity, we distinguish two cases.

First, assume there is admissible restriction of holomorphic discrete series of G to D — or equivalently, that 𝔰+ is a D -prehomogeneous space. Then there will be admissible restriction to any subgroup H containing D .

We assume now that the restriction of holomorphic discrete series of G to D is not admissible. Since 𝔟 contains J , the restriction of a holomorphic discrete series (π,V ) of G to the group BD is admissible. Let us choose a positive Weyl chamber * Γ ⊂ i(𝔞 + 𝔟) for the group BD . Let CD,V ⊂ Γ the set of highest weights of the irreducible representations of BD which occur in V , and CD ⊂ Γ the asymptotic cone of CD,V . It is known that C D is a closed convex polyhedral cone, independent of V , contained in the projection on * i(𝔞 + 𝔟) of the cone * C ⊂ i𝔱 . We identify the orthogonal ⊥ 𝔞 of 𝔟 in 𝔞 + 𝔟 to * 𝔟 . We consider the cone 𝔟 ⊥ * CD = CD ∩ 𝔞 ⊂ i𝔟 . The fact that the restriction of holomorphic discrete series of G to D is not admissible is equivalent to the fact that the cone C𝔟D is not reduced to {0 } . We consider its dual cone C 𝔟⊥ ⊂ 𝔟 D , and its interior C𝔟⊥ D,int . Note that J belongs to 𝔟⊥ CD,int .

Theorem 4. Let (π,V ) be a holomorphic discrete series of G . Suppose that its restriction to D is not admissible. Then the restriction of (π,V ) to L is admissible if and only if one of the two following equivalent conditions hold:

(CL ) : CD𝔟∩ 𝔟 ⊥L = {0}.

or

(C ′) : C𝔟⊤ ∩ 𝔟 ⁄= {0}. L D,int L

Thus, discrete series of G have admissible restriction to L if and only if L contains a closed subgroup L1 = B1D , where 𝔟1 ⊂ 𝔟 is a one-dimensional subspace which intersects C 𝔟⊤D,int .

Theorem 4 suggests a method to find all closed connected reductive groups H ⊂ G for which there is admissible restriction of holomorphic discrete series.

]]> 1. For each closed connected semisimple subgroup D ⊂ Kss , determine whether there is admissibility of restriction of holomorphic discrete series. This step is not too difficult, for instance this is never the case when 𝔤 is of tube type, and we will give below the complete answer for 𝔤 = 𝔢6(-14) .

2. When it is not the case, compute (with the notation as above) the algebra 𝔟 and the cone C𝔟D⊥ ⊂ 𝔟 . This will give the list of closed connected subgroup L ⊂ K such that Lss = D for which there is admissibility of restriction of holomorphic discrete series.

For each particular D , this is probably a feasible task, and we give several examples. However, we do not know an useful statement for all D .

3. Given L as in 2, list the closed connected reductive subgroups H ⊂ G such that H ∩ K = L .

2.4. Some cones.

We use the notations of the previous subsection. We assume moreover that T normalizes D , or, equivalently, that T = AB . This means that 𝔡ℂ is the sum of the root spaces for a certain subset ΦD of roots, and of the space 𝔞ℂ generated by the corresponding coroots. We give some bounds on the cone CD .

For this we need to recall some important facts proven in [20]. Let r be the real rank of 𝔤 . There exists a set {γ1,...,γr} ⊂ Ψn of pairwise strongly orthogonal roots such that the highest weights of the representations of K ℂ occurring in + S[𝔰 ] are exactly those which belong to the cone generated by γ1,γ1 + γ2,...,γ1 + γ2 + ⋅⋅⋅ + γr . We recall that γ1 is the highest weight of the K ℂ -module 𝔰+ , that γ2 is the maximal element (for a suitable order) among the roots orthogonal to γ1 , etc...

This means that C Kss is the cone generated by γ ,γ + γ ,...,γ + γ + ⋅⋅⋅ + γ 1 1 2 1 2 r . Moreover, CD is a polyhedral cone such that CKss ⊂ CD ⊂ C .

For later use, we introduce some related notations. We will label the simple compact roots as α 1 ,…, α d , and the unique simple non compact root will be denoted by β . Note that γ1 is the corresponding fundamental weight, and that γ1 = woβ , where wo is the longest element of the Weyl group of K ℂ .

2.5. Non compact H .

]]> For this subsection G denotes one of the groups E6(-14), E7 (-25). We fix a holomorphic discrete series representation (π,V ) for G. Then,

Theorem 5. For a maximal connected reductive subgroup H of G, (π,V ) restricted to H is admissible if and only if the center of K is a subgroup of H.

When H is so that (G, H ) a symmetric space, the theorem is a result of Kobayashi, [12], [15].

In [3] is shown that a maximal connected subgroup of G is either parabolic or reductive. For sake of completeness we list the maximal reductive subalgebras of 𝔤. The classification of maximal connected subgroups of G was completed by [16]. Some of the subalgebras has a compact abelian one dimensional factor which may not be the center of 𝔨.

|--------------------------|-------------------| |𝔢6(-14)--------------------|𝔢7(-25)-------------| |𝔰𝔬(10) ⊕-𝔷----------------|𝔢6 ⊕-𝔷-------------| |𝔰𝔬(2,8) ⊕-𝔰𝔬(2)-----------|𝔢6(-26) ⊕-ℝ---------| -𝔰𝔬⋆(10) ⊕-𝔰𝔬(2)------------𝔢6(-14) ⊕-𝔰𝔬-(2-)----| |𝔰𝔲(5,1) ⊕ 𝔰𝔩(2,ℝ ) |𝔰𝔬⋆(12) ⊕ 𝔰𝔲(2) | |--------------------------|--⋆----------------| |𝔰𝔲(2,4)-⊕-𝔰𝔲(2)-----------|𝔰𝔲-(8)-------------| |𝔣4(-20)--------------------|𝔰𝔲(6,2)------------| |𝔰𝔭(2,2)-------------------|𝔰𝔬(10,2) ⊕-𝔰𝔩(2,ℝ-)-| -𝔤2 ⊕-𝔰𝔲(2,1)---------------𝔤2 ⊕-𝔰𝔭(3,ℝ)-------| |𝔰𝔲(2,1) ⊕ 𝔰𝔲(2,1) ⊕ 𝔰𝔲(3) |𝔣4(-20) ⊕ 𝔰𝔩(2,ℝ ) | |--------------------------|-------------------| | |

Proof. The subgroups H listed on the first seven lines corresponds to symmetric pairs (G,H ). The result follows from Kobayashi [12]. Under his hypothesis, Kobayashi has shown that the multiplicity function is bounded. We do not know if this fact holds for other pairs (G, H ).

For 𝔤2 ⊕ 𝔰𝔲 (2,1) we have that the center of the maximal compact subgroup of e6(-14) is contained in 𝔰𝔲(2,1). Hence, owing to Proposition 4 there is admissible restriction to the subgroup. In fact, center of K is contained in SU(2,1). For this, we consider the usual imbedding Spin (7) × Spin (3) as a subgroup of Spin (10). Then, 𝔰+ restricted to Spin (7 ) is equivalent to twice the spin representation of Spin (7).
Counting dimensions, we get + 𝔰 restricted to G2 ⊂ Spin (7 ) is equivalent to twice V7 ⊕ ℂ. Here, V7 (resp. ℂ) is the seven dimensional (resp. one dimensional) irreducible representation for G2. It follows from a computation that the Cartan decomposition of 𝔰𝔲(2,1 ) is spin(3) ⊕ 𝔷 ⊕ 𝔭+ ⊕ 𝔭- where 𝔭+ is a subspace of the two copies of the trivial representation. From this we get that the center of 𝔨 is contained in 𝔰𝔲 (2, 1).

The maximal compact subgroup of G2 × Sp(3,ℝ ) is G2 × U (3). We show the center of U(3) is the center of K. In fact, in [16] is stated G2 × SU (3) is a maximal subgroup of E6. Hence, the projection of the center of U (3) on the direction of E 6 is trivial. Proposition 4 yields (π, V ) has admissible restriction to G2 × Sp (3,ℝ ).

]]> Next, we dealt with 𝔣4(-20) + 𝔰𝔩2 in 𝔢7(-25). Let β denote the noncompact simple root for the holomorphic system in e7(-25) and let α be the compact simple root adjacent to β. We claim that of 𝔰 𝔩2 ∩ 𝔨 is spanned by i(Λ β - Λα). The root system for the immersion of 𝔢 6(- 14) in 𝔢 7(-25) is spanned by β + α and the five compact simple roots different from α. Hence, i(Λ β - Λα) belongs to the centralizer of 𝔢6(-14). Since, [16], 𝔣4(-20) is a subalgebra of 𝔢6(-14), and the centralizer of 𝔣4(-20) in 𝔢7 is 𝔰𝔩2 [1], the claim follows. Thus, a maximal compact subgroup of F4(-20) × SL2 is Spin (9 ) × exp (ℝi (Λβ - Λ α)). Moreover, the 𝔰𝔬(2) factor of the immersion spin(10) ⊕ so(2) in 𝔢 6 is spanned by iΛ . α
Now, ϖ1 restricted to Spin (10) is equivalent to + 10 𝔰 ⊕ ℂ ⊕ ℂY β. Also, Λ α takes on the values 2,1,0 on each irreducible factor. Hence Λ β - Λ α takes on the values - 1,0,1. Therefore, Spin (9) × exp(ℝi(Λ β - Λα)) fix nonzero vectors in ϖ1. □

It may happen there is admissible restriction to a non compact subgroup which is not a maximal subgroup. In fact, we have,

Theorem 6. A holomorphic discrete series for E6 (- 14) has admissible restriction to any of the subgroups

SO ⋆(10), SO (2,8), SU (4,1) × SU (2).

We notice that none of the subgroups listed above contain the center of K. A consequence of Theorem 2 and Theorem 8 is

Theorem 7. If a holomorphic Discrete Series of an exceptional group has an admissible restriction to H, then center of L is a torus.

Theorem 7 does not hold for classical groups because Proposition 3 yields that holomorphic Discrete series for SU (2n, 1) has an admissible restriction to Sp(n ).

3. The case 𝔤 = 𝔢6(-14)

In this section, 𝔤 = 𝔢6(-14) . Following [8], we label the Dynkin diagram as follows.

]]>

PIC

The real rank is 2 . We have γ1 = α1 + 2α2 + 3α3 + 2α4 + 2 α5 + β , and γ2 = α1 + α2 + α3 + α4 + β .

We provide i𝔱* with the invariant scalar product for which (α, α) = 2 for each root.This scalar product produces an isomorphism λ → H λ from i𝔱* to i𝔱 , and H α is the coroot corresponding to α .

Let ϖβ be the fundamental weight corresponding to β . We have 1 ϖ β = 3(2α1 + 4α2 + 6α3 + 5α4 + 3α5 + 4β) . We note that iH ϖβ = J .

Since, by theorem 1, there is admissible restriction to Kss , we consider proper maximal subgroups of Kss . We show

Theorem 8. Let (π,V ) be a holomorphic discrete series representation for E6 (- 14) Then,

i) (π,V ) restricted to Kss = Spin (10) is admissible.

ii) Let U(5) → SO (10) denote the usual imbedding and let U(5) denote the analytic subgroup of Spin (10) associated to 𝔲 (5 ), then the restriction of π to U (5 ) is admissible.

iii) For any other maximal subgroup L of Kss, (π, V) restricted to L is not admissible.

]]> iv) Let L be a closed proper subgroup of U (5). Then π restricted to L is not an admissible representation.

Proof. To begin with, we recall the Cartan decomposition of 𝔢 = 𝔰𝔬(10) + 𝔰𝔬(2) ⊕ 𝔰+ ⊕ 𝔰- , 6(-14) where 𝔰+, 𝔰- are the two spin representations of 𝔰𝔬(10 ). Let SU (5 ) (resp. Z5) denote the simple factor of U (5) (resp. the center of U (5)).

Owing to Proposition 3, Theorem 8 follows from:

a) + Spin(10) S[𝔰 ] = ℂ, b) +U (5) S [𝔰 ] = ℂ,

c) + L S[𝔰 ] ⁄= ℂ for a maximal subgroup L of Kss not locally isomorphic to U (5).

d) For subgroup L ⊆ Z 1 5 and maximal subgroup L ⊆ SU (5), 2 or
L1 = {e} and L2 = SU (5). Then + L L S[𝔰 ] 1 2 ⁄= ℂ.

In [2], we find a proof of S[𝔰+]Spin(10) = ℂ. Thus a) follows.
To continue, we fix an orthogonal basis ε1,⋅ ⋅⋅ ,ε5 of i𝔱⋆ so that a system of positive compact roots is { εi ± εj,i < j} and the weights of the representation 𝔰+ are 1(± ε ± ε ± ε ± ε ± ε ) 2 1 2 3 4 5 with and odd number of + . The positive roots of 𝔲 (5 ) are {εi - εj,i < j}. Let J5 denotes an infinitesimal generator of Z5 chosen so that J5(ε1 + ⋅⋅⋅ε5) = i. The 𝔲(5)- module 𝔰+ decomposes as

V 1 ⊕ V 1 ⊕ V 1 . 2(ε1+ε2+ε3+ ε4+ε5) ε1- 2(ε1+ε2+ ε3+ε4+ε5) ε1+ε2+ ε3- 2(ε1+ε2+ ε3+ε4+ε5)
In [10] page 98, we find a proof that ]]> (SL (2m + 1 ),Λ1 (ℂ2m+1 )⋆ ⊕ Λ2 (ℂ2m+1 )) is a prehomogeneous space. Hence, SU (5)ℂ has an open orbit in
V ε1 ⊕ Vε1+ε2+ ε3
Therefore, U (5) ℂ has an open orbit in 𝔰+. Hence, S [𝔰+ ]U (5) = ℂ and b) follows. Since, SU (5) acts trivially on the factor 1 V 2(ε1+ε2+ε3+ ε4+ε5), it follows d ) for L1 = {e},L2 = SU (5). Since, an element of V1(ε+ ε+ε +ε +ε) 2 1 2 3 4 5 times un element of any of the other irreducible factors is invariant under Z5 we obtain d) for the other extreme case.

In order to show c) we list, up to conjugation, the maximal connected closed subgroups of SO (10). These subgroups have been classified by Dynkin in [5]. They are:

To continue, we assume S [𝔰+ ]L = ℂ for each maximal subgroup L of Spin (10) not locally isomorphic to U(5). From this we derive a contradiction.
As before, + + 𝔰 : Spin (10) - → Gl(𝔰 ) denote the half spin representation. To begin with, we consider ρ : L - → Spin (10 ) an irreducible, simple, maximal subgroup. Then (𝔰+ ∘ ρ, 𝔰+) decomposes as the sum irreducible L - modules

V1 ⊕ ⋅⋅⋅ ⊕ Vr.
We set ρj equal to the projection onto Vj followed by 𝔰+ ∘ ρ. Owing to our hypothesis, it follows S [V ]ρj(L ) = ℂ j for j = 1,⋅⋅⋅ ,r. In [2], [10], [11] [17] we find the list of triples (L,ρj,Vj) where: L is a simple algebraic group, ρj is an irreducible representation and ρ (L) S [Vj] j = ℂ. The list is:
n+1 n(2n+1) 2n (An,Λ1, ℂ ), (A2n,Λ2, ℂ ), (Cn, Λ1, ℂ ).
We first verify none of the Vj is equivalent to (Cn, Λ1). Since the ten dimensional irreducible representation of SL2 is symplectic, we have n ≥ 2. For L of type C ,n ≥ 2 n and r = 1 we obtain n = 8, a contradiction. For L of type Cn,n ≥ 2 and r ≥ 2 the symplectic form lead us to + L S[𝔰 ] ⁄= ℂ, another contradiction. ]]> For L of type An,n ≥ 2, if at least one Vj is equivalent to (A2k, Λ2), then n = 2k and k(2k + 1) ≤ 16, hence, L is one of A ,A . 2 4 SL (3) has two irreducible representations of dimension ten whose highest weight are (3,0,0) = 3Λ1, - wo (3Λ1), neither of these two representations are orthogonal [3]. SL (5) also has two ten dimensional representations of highest weight Λ2 or Λ3, [3] neither of them is orthogonal.
We are left to analyze the situation all V j are equivalent to (A ,Λ ),n ≥ 2. n 1 Since L is a subgroup of Spin (10) we have n ≤ 5. The case n even was analyzed in the previous paragraph. The ten dimensional irreducible representations of SL (4) have highest weight 2Λ1 or 2Λ3 they are not orthogonal. SL (6) has no irreducible representation of dimension ten.

To conclude the proof of c) we show

+ 𝔰𝔬(p)⊕ 𝔰𝔬(q) S[𝔰 ] ⁄= ℂ for p ≥ 1, q ≥ 1,p + q = 10.

We recall the following facts, for a proof, see [1], [3] Table 1.