Principal Gelfand pairs

Oksana Yakimova

Let X = G/K be a connected Riemannian homogeneous space of a real Lie group G. We assume that the action G : X of G on X is locally effective, i.e., K contains no non-trivial connected central subgroups of G. Denote by D(X)^{G} the algebra of G-invariant differential operators on X and by P(T^{∗}X)^{G} the algebra of G-invariant functions on T^{∗}X polynomial on fibres. It is well known that P(T^{∗}X)^{G} is a Poisson algebra, the Poisson bracket being induced by the commutator in D(X)^{G}.

The homogeneous space X is called commutative or the pair (G, K) is called a Gelfand pair if the following three equivalent conditions are satisfied:

1) the algebra of K-invariant measures on X with compact support is commutative with respect to the convolution;

2) the algebra D(X)^{G} is commutative;

3) the algebra P(T^{∗}X)^{G} is commutative with respect to the Poisson bracket.

The equivalence of the first two conditions was proved by Thomas [20] and Helgason [9], independently. Evidently, condition 3) is a consequence of 2). The inverse implication is recently proved by Rybnikov [19]. Commutative spaces can also be characterised by several other conditions. For instance, X is commutative if and only if the representation of G in L^{2}(X) has a simple spectrum, see [5].

Symmetric Riemannian homogeneous spaces introduced by Élie Cartan are commutative. The theory of symmetric spaces is well developed. Works of Élie Cartan and Helgason describe their structure and also deal with harmonic analysis on such manifolds. One can hope that some day commutative spaces will be as thoroughly studied as symmetric spaces.

We denote Lie algebras of Lie groups by corresponding small gothic letters, for example, 𝔤 = Lie G and 𝔨 = Lie K.

Definition 1. A real or complex linear Lie group with finitely many connected components is said to be reductive if it is completely reducible.

If X is a commutative homogeneous space of G, then, up to the local isomorphism, G has a factorisation G = N ⋋ L, where N is a nilpotent radical of G, K ⊂ L, L and K have the same invariants in ℝ[𝔫] and [[𝔫, 𝔫], 𝔫] = 0, see [21]. Without loss of generality we may assume that the center of L is compact and the commutator group L′ of L is a real form of some complex semisimple group. Hence, L is a reductive group. For a reductive Lie group F acting on a linear space V, we denote by F_{∗}(V) a generic stabiliser for this action. The stabiliser of a point y ∈ V is denoted by F_{y}. Set 𝔪 := 𝔩/𝔨. In section 1 we prove the following criterion of commutativity. Theorem 0.1. X = (N ⋋ L)/K is commutative if and only if all of the following three conditions hold: i) ℝ[𝔫]^{L} = ℝ[𝔫]^{K}; ii) for any point γ ∈ 𝔫^{∗} the homogeneous space L_{γ}/K_{γ} is commutative; iii) for any point β ∈ 𝔪^{∗} the homogeneous space (N ⋋ K_{β})/K_{β} is commutative.

Let F be a complex reductive Lie group and H ⊂ F a reductive subgroup.

Definition 2. An affine F -variety X is called spherical if a Borel subgroup B(F) ⊂ F has an open orbit in X. If X is a linear space and a spherical F -variety then it is called a spherical representation of F. If a homogeneous space F/H is spherical, then the pair (F, H) and the subgroup H are also called spherical.

Let G be a real form of a complex reductive group G(ℂ). Suppose K ⊂ G is a compact subgroup. We call the real homogeneous space G/K, the subgroup K and the pair (G, K) spherical if the complexification X(ℂ) = G(ℂ)/K(ℂ) is a spherical G(ℂ)-variety.

The commutative homogeneous spaces of reductive Lie groups are just the real forms of the spherical affine homogeneous spaces, see, for example, [21]. The theory of spherical homogeneous spaces is well developed, in particular, they are classified in [11], [6] and [15]. Note that [6] deals only with so-called principal homogeneous spaces (see Definition 5 in the third part of the present article). In [15] one type of non-principal spherical homogeneous spaces is described. The final classification is obtained in [23]. The real forms of homogeneous spherical spaces, i.e., commutative homogeneous spaces are explicitly described in [23].

Let X = (N ⋋ L)/K be a commutative homogeneous space. Denote by P the ineffective kernel of the action L : 𝔫. Evidently, P is a normal subgroup of L and G. Due to (i) we have L/P ⊂ O(𝔫). Hence the group L_{γ} is reductive for any γ ∈ 𝔫^{∗}, so in (ii) we can replace the word “commutative” by “spherical” and look up the given homogeneous space in the list.

Because the orbits of the compact group K in 𝔫 are separated by polynomial invariants, L and K have the same invariants in ℝ[𝔫] if and only if they have the same orbits. In other words, condition (i) means that there is a factorisation L = L_{∗}(𝔫)K or equivalently L/P = (L/P)_{∗}(𝔫)K. All nontrivial factorisations of compact groups into products of two subgroups are classified by Onishchik [16]. The classification results are also nicely presented in [17, Chapter 4].

Commutativity is a local property, i.e., it depends only on the pair of algebras (𝔤, 𝔨), see [21]. We assume that both G and K are connected, N is simply connected, L = Z(L) × L_{1} × ... × L_{m}, where Z(L) stands for the connected centre of L and L_{i} are connected noncommutative simple groups. We also assume that L_{i} are real forms of simply connected complex simple groups and the action of a factor group Z(L)/(Z(L) ∩ P^{0}) on 𝔫 is effective. It can happen, that for a given pair (𝔤, 𝔨), there is no effective pair (G, K) satisfying these assumptions, so we admit not only effective actions G : (G/K) but locally effective as well. In tables we write U_{n} instead of U_{1} × SU_{n} and sometimes SO_{n} instead of Spin_{n}.

Suppose X = (N ⋋ L)/K is commutative. Let 𝔷_{0} ⊂ [𝔫, 𝔫] be an L-invariant subspace and Z_{0} ⊂ N a corresponding connected subgroup. Then the homogeneous space X/Z_{0} = ((N/Z_{0}) ⋋ L)/K is also commutative, see [21]. The passage from X to X/Z_{0} is called a central reduction. A commutative homogeneous space is said to be maximal, if it can not be obtained by a non-trivial central reduction from a larger one. We consider only maximal commutative homogeneous spaces. All commutative homogeneous spaces could be obtained as their central reductions.

Definition 3. A homogeneous space G/K is called indecomposable if it cannot be represented as a product G_{1}/K_{1} × G_{2}/K_{2}, where G = G_{1} × G_{2}, K = K_{1} × K_{2} and K_{i} ⊂ G_{i}.

Obviously, G_{1}/K_{1} × G_{2}/K_{2} is commutative if and only if both spaces G_{1}/K_{1} and G_{2}/K_{2} are commutative. Hence, for the classification of commutative homogeneous spaces it is enough to describe indecomposable ones.

Denote by H_{n} the 2n + 1 dimensional Heisenberg group, i.e., 𝔥_{n} = Lie H_{n} ≅ ℂ^{n} ⊕ ℝ. Simply connected commutative groups are denoted by ℝ^{n} or ℂ^{n}. The simplest and most important results are obtained in case of simple L. Theorem 0.2. Suppose X = (N ⋋ L)/K is an indecomposable commutative space, where L is simple, L ≠ K and 𝔫 ≠ 0. Then X is one of the following eight spaces up to the local isomorphism.

(H_{2n} ⋋ SU_{2n})/Sp_{n}; (ℝ^{7} ⋋ SO_{7})/G_{2};  ((ℝ^{8} × ℝ^{2}) ⋋ SO_{8})/Spin_{7};
(ℂ^{2n} ⋋ SU_{2n})/Sp_{n}; (ℝ^{8} ⋋ Spin_{7})/Spin_{6}; (ℝ^{8} ⋋ SO_{8})/Spin_{7};
(ℝ^{2n} ⋋ SO_{2n})/SU_{n};  (ℝ^{8} ⋋ SO_{8})/(Sp_{2} × SU_{2});

First we describe commutative homogeneous spaces satisfying condition

(∗) L ≠ K and the action L : 𝔫 is locally effective, i.e., P is finite. Theorem 0.3. Suppose a commutative homogeneous space X = G/K satisfies condition (∗). Then any non-commutative normal subgroup of K different from SU_{2} is contained in some simple direct factor of L.

In the present work we impose on X two technical conditions of “principality” and “Sp_{1}- saturation”, see sections 3 and 5 for precise definitions and explanations. The first condition concerns the embedding of the connected centre of K into L and the action of the connected centre of L on 𝔫. The second one is essentially the same, but we consider normal subgroups of K and L isomorphic to Sp_{1}. Theorem 0.4. Let X = (N ⋋ L)/K be a maximal principal indecomposable commutative homogeneous space satisfying condition (∗). Suppose there is a simple normal subgroup L_{1}⊲L, such that L_{1} ≠ SU_{2} and L_{1} ⊄ K. Then either L is simple (and X is listed in Theorem 0.2) or X is locally isomorphic to one of the following two spaces: (H_{2n} ⋋ U_{2n})/(Sp_{n} · U_{1}), ((ℝ^{8} × ℝ^{2}) ⋋ (SO_{8} × SO_{2}))/(Spin_{7} × SO_{2}).

These additional spaces are essentially the same as the spaces from Theorem 0.2. All of them are listed in Table 2b (section 2).

A commutative homogeneous space (N ⋋L)/K is said to be of Heisenberg type if L = K. Recently commutative homogeneous spaces of Heisenberg type were intensively studied by several people, see, e.g., [3], [13], [21], [22].

The following theorem is the main result of this article. Theorem 0.5. Any indecomposable maximal principal Sp_{1}-saturated commutative homogeneous space belongs to the one of the following four classes:

1) affine spherical homogeneous spaces of reductive real Lie groups;

2) homogeneous spaces listed in Table 2b;

3) homogeneous spaces ((ℝ^{n} ⋋ SO_{n}) × SO_{n})/SO_{n}, ((H_{n} ⋋ U_{n}) × SU_{n})/U_{n}, where the normal subgroups SO_{n} and SU_{n} of K are diagonally embedded into SO_{n} × SO_{n} and SU_{n} × SU_{n}, respectively;

4) commutative homogeneous spaces of Heisenberg type.

Commutative homogeneous spaces of Heisenberg type are considered in the sixth section. In this case D(G/K)^{G} ≅ U(𝔫)^{K}, where U(𝔫) is the universal enveloping algebra of 𝔫. If 𝔫 is commutative then obviously G/K is also commutative and it is called a commutative space of Euclidian type. We assume that 𝔫 is not commutative.

Consider a homogeneous space (N ⋋ K)/K. Suppose 𝔫 is at most two-step nilpotent and [𝔫, 𝔫] is a trivial K-module. We can decompose 𝔫 into an K-invariant sum 𝔫 = (𝔴 ⊕ 𝔷) ⊕ V, where V is an abelian ideal and [𝔴, 𝔴] = 𝔷. Any point α ∈ 𝔷^{∗} determines a skew-symmetric form α̂ on 𝔴, namely α̂(ξ, η) = α([ξ, η]) for ξ, η ∈ 𝔴. The form α̂ is non-degenerate for a generic α. The complexification 𝔴(ℂ) is an orthogonal and a symplectic representation of K(ℂ) at the same time. Hence it is reducible 𝔴(ℂ) = W ⊕ W^{∗}. According to [3] and [25], (N ⋋ K)/K is commutative if and only if W is a spherical representation of the complexification K_{∗}(V)(ℂ) of K_{∗}(V). In the simplest situation when V = 0 the statement means that W is a spherical representation of K(ℂ).

Spherical representations of reductive Lie groups are classified by Kac [10] (irreducible representations), Brion [7], finally by Benson and Ratcliff [4] and Leahy [14], independently. Historical comments and the classification result can be found in [12]. The list of commutative homogeneous spaces (N ⋋ K)/K, where N is a product of several Heisenberg groups, is given in [3]. That article also claims to classify all commutative homogeneous spaces (N ⋋ K)/K with 𝔫 = (𝔴 ⊕ 𝔷) ⊕ V, where V is an abelian ideal and 𝔴 ⊕ 𝔷 is a direct sum of several Heisenberg algebras. The authors of [3] erroneously assume that if for a subgroup H with Lie H = 𝔴 ⊕ 𝔷 the homogeneous space (H ⋋ K)/K is commutative, then (N ⋋ K)/K is also commutative. The simplest counterexample is ((ℂ^{2} × H_{2}) ⋋ SU_{2})/SU_{2}. This space is not commutative according to [21], but (H_{2} ⋋ SU_{2})/SU_{2} is commutative.

Commutative homogeneous spaces of Heisenberg type with an irreducible action K : (𝔫/𝔫′) are classified in [21] and [22]. Generally, 𝔫 is a sum of a commutative ideal and algebras listed in [21, Table 3] and [22]. But the sum cannot be arbitrary.

The classification of saturated commutative homogeneous spaces of Heisenberg type was announced in [25]. The condition of saturation is a little bit stronger than both conditions of “principality” and “Sp_{1}-saturation”. We present the result of [25] in Table 4 and give a proof.

We will frequently use the following results of [21].

Proposition 1. [21, Corollaries to Proposition 10] Let G/K be commutative. Then

1) for any normal subgroup N ⊂ G the homogeneous space G/NK = (G/N)/(K/(N ∩ K)) is commutative;

2) for any compact subgroup F ⊂ G containing K the homogeneous space G/F is commutative;

3) for any subgroup F ⊂ G containing K the homogeneous space F/K is commutative.

We fix some additional notation.

G′ is the commutator group of G;

G(ℂ) is the complexification of a real group G;

B(G(ℂ)) and U(G(ℂ)) ⊂ B(G(ℂ)) are a Borel and a maximal unipotent subgroups of a reductive group G(ℂ);

X//G stands for the categorical quotient of an affine algebraic variety X by the action of a reductive group G.

Acknowledgments. I am grateful to E.B. Vinberg and D.I. Panyushev for helpful discussions and permanent attention to this work. Thanks are also due to A.L. Onishchik and D.N. Akhiezer for useful comments on the earlier version. This research was completed during my stay at the Max-Planck-Institut für Mathematik (Bonn). I would like to thank the Institut for hospitality and wonderful working conditions.

1. Commutativity criterion

Let U(𝔤) stand for the universal enveloping algebra of 𝔤. There is a natural filtration on U(𝔤):

U_{0}(𝔤) ⊂ U_{1}(𝔤) ⊂ … ⊂ U_{m}(𝔤) ⊂ …,

where U_{m}(𝔤) consists of all elements of order at most m.

Recall that the Poisson bracket on the symmetric algebra S(𝔤) = gr U(𝔤) is determined by the formula

{a + U_{n−1}(𝔤), b + U_{m−1}(𝔤)} = [a, b] + U_{n+m−2}(𝔤) ∀a ∈ U_{n}(𝔤), b ∈ U_{m}(𝔤).

Let X = G/K be a Riemannian homogeneous space. It is well known, see, for example, [21], that there is an isomorphism of the associated graded algebras:

gr U(𝔤)^{K}/(U(𝔤)𝔨)^{K} = gr D(X)^{G} = P(T^{∗}X)^{G} = S(𝔤/𝔨)^{K}.

The space (U(𝔤)𝔨)^{K} is an ideal of U(𝔤)^{K}, also (S(𝔤)𝔨)^{K} is a Poisson ideal of S(𝔤)^{K}. The well defined Poisson bracket on the factor S(𝔤)^{K}/(S(𝔤)𝔨)^{K} ≅ S(𝔤/𝔨)^{K} coincides up to a sign with the Poisson bracket on P(T^{∗}X)^{G}. In particular, X is commutative if and only if the Poisson algebra S(𝔤/𝔨)^{K} is commutative.

Suppose X = (N ⋋L)/K is commutative. Then, as proved in [21], the following condition holds

i) ℝ[𝔫]^{L} = ℝ[𝔫]^{K}. The orbits of a compact group are separated by polynomial invariants. Hence (i) is fulfilled if and only if L and K have the same orbits on 𝔫. There is a K-invariant positive-definite symmetric bilinear form on 𝔫 which is automatically L-invariant. In particular, vector spaces 𝔫 and 𝔫^{∗} are isomorphic as L-modules. Therefore, ad^{∗}(𝔨)γ = ad^{∗}(𝔩)γ for each point γ ∈ 𝔫^{∗} and hence 𝔩 = 𝔨 + 𝔩_{γ}. Moreover, the natural restriction

τ : 𝔩^{∗} ⟶ 𝔩_{γ}^{∗}

(which is also a homomorphism of L_{γ}-modules) determines an isomorphism of K_{γ}-modules (𝔩/𝔨)^{∗} and (𝔩_{γ}/𝔨_{γ})^{∗}.

Recall that 𝔤 = 𝔩 + 𝔫, where 𝔫 is a nilpotent ideal and 𝔩 is a reductive subalgebra. Let 𝔫̌ and 𝔩̌be commutative Lie algebras of dimensions dim 𝔫 and dim 𝔩, respectively. We determine new Lie algebras 𝔤̌_{1} = 𝔩 + 𝔫̌ and 𝔤̌_{2} = 𝔩̌⊕ 𝔫, where 𝔩̌, 𝔫̌ are commutative ideals and 𝔫̌ ≅ 𝔫 as an 𝔩-modules.

Denote by { , }_{𝔩} and { , }_{𝔫} the Poisson brackets on S(𝔤̌_{1}) and S(𝔤̌_{2}). There is a Kinvariant bi-grading S(𝔤) = ⨁ S^{n,l}(𝔤), where S^{n,l}(𝔤) = S^{n}(𝔫)S^{l}(𝔩). We identify elements of S(𝔤) with the corresponding elements of S(𝔤̌_{1}) and S(𝔤̌_{2}).

Lemma 1. We have

{ξ, η} = {ξ, η}_{𝔫} + {ξ, η}_{𝔩}, with{ξ, η}_{𝔫} ∈ S^{n+n′−1,l+l′}(𝔤), {ξ, η}_{𝔩} ∈ S^{n+n′,l+l′−1}(𝔤).

for any bi-homogeneous elements ξ ∈ S^{n,l}(𝔤), η ∈ S^{n′,l′}(𝔤). In other words, the Poisson bracket on S(𝔤) is a direct sum of the brackets { , }_{𝔫} and { , }_{𝔩}.

Proof. Recall that the Poisson bracket of bi-homogeneous elements ξ = ξ_{1}...ξ_{n}, η = η_{1}...η_{m} ∈ S(𝔤) is given by the formula

{ξ, η} = ∑_{i,j}[ξ_{i}, η_{j}]ξ_{1}...ξ̂_{i}...ξ_{n}η_{1}...η̂_{j}...η_{m}.  (1)

This expression for {ξ, η} contains summands of three different types, depending on whether ξ_{i} and η_{j} are elements of 𝔩 or 𝔫. Because [𝔩, 𝔫] ⊂ 𝔫 and 𝔩, 𝔫 are subalgebras, if ξ_{i}, η_{j} ∈ 𝔫, then [ξ_{i}, η_{j}] ∈ S^{n+n′−1,l+l′}(𝔤), otherwise [ξ_{i}, η_{j}] ∈ S^{n+n′,l+l′−1}(𝔤).

We suppose that 𝔨̌ is embedded into 𝔩̌ as a commutative subalgebra of dimension dim 𝔨. We also denote by { , }_{𝔫} and { , }_{𝔩} the corresponding Poisson brackets on the Poisson factors S(𝔤̌_{2}/𝔨̌)^{K} = S(𝔤̌_{2})^{K}/(S(𝔤̌_{2})𝔨̌)^{K} and S(𝔤̌_{1}/𝔨)^{K} = S(𝔤̌_{1})^{K}/(S(𝔤̌_{1})𝔨)^{K}, where the actions K : 𝔤̌_{i} are the same as K : 𝔤. We have {a, b}_{𝔩} ∈ S^{n+n′,l+l′−1}(𝔤/𝔨) and {a, b}_{𝔫} ∈ S^{n+n′−1,l+l′}(𝔤/𝔨) for any a ∈ S^{n,l}(𝔤/𝔨), b ∈ S^{n′,l′}(𝔤/𝔨) (a, b ∈ S(𝔤/𝔨)^{K}).

Lemma 2. The Poisson bracket on S(𝔤/𝔨)^{K} is of the form { , } = { , }_{𝔫} + { , }_{𝔩}.

Proof. This is a straightforward consequence of Lemma 1.

Corollary 1. Let G/K be a commutative homogeneous space and Ň a simply connected commutative Lie group with a Lie algebra 𝔫̌. Set Ǧ := Ň ⋋ L. Then Ǧ/K is also commutative.

Theorem 1. The homogeneous space X = (N ⋋ L)/K is commutative if and only if all of the following three conditions hold:

i) ℝ[𝔫]^{L} = ℝ[𝔫]^{K};

ii) for any point γ ∈ 𝔫^{∗} the homogeneous space L_{γ}/K_{γ} is commutative;

iii) for any point β ∈ (𝔩/𝔨)^{∗} the homogeneous space (N ⋋ K_{β})/K_{β} is commutative.

Remark 1. The statement of the theorem remains true if we replace arbitrary points by generic points in conditions (ii) and (iii).

Proof. As was already mentioned, Vinberg proved in [21] that the condition (i) holds for any commutative space. So let us assume that it is fulfilled.

Let γ be a point in 𝔫^{∗}. Recall that the K_{γ}-modules 𝔩/𝔨 and 𝔩_{γ}/𝔨_{γ} are isomorphic. Hence, S(𝔩/𝔨) is isomorphic to S(𝔩_{γ}/𝔨_{γ}) as a graded associative algebra and also as a K_{γ}-module.

Consider the homomorphism

φ_{γ} : S(𝔤/𝔨) ⟶ S(𝔤/𝔨)/(ξ − γ(ξ) : ξ ∈ 𝔫) = S(𝔩/𝔨) = S(𝔩_{γ}/𝔨_{γ}).

Evidently, φ_{γ}(S(𝔤/𝔨)^{K}) ⊂ S(𝔩_{γ}/𝔨_{γ})^{K_{γ}}.

Let ξ ∈ 𝔩_{γ}, η ∈ 𝔫. Then γ({ξ, η}) = γ([ξ, η]) = −[ad^{∗}(ξ)γ](η) = 0 = {ξ, γ(η)}.

It can easily be deduced from the above statement and from the formula (1), that for arbitrary bi-homogeneous elements a, b ∈ S(𝔤/𝔨)^{K}, which can be regarded as elements of S((𝔩_{γ} ⊕ 𝔫)/𝔨_{γ}), we have

φ_{γ}({a, b}_{𝔩}) = {φ_{γ}(a), φ_{γ}(b)},

where the second bracket is the Poisson bracket on S(𝔩_{γ}/𝔨_{γ})^{K_{γ}}. In other words, φ_{γ} is a homomorphism of the Poisson algebras S(𝔤̌_{1}/𝔨)^{K} and S(𝔩_{γ}/𝔨_{γ})^{K_{γ}}.

Recall that we denoted by 𝔪 the vector space (𝔩/𝔨). We repeat the procedure for the point β ∈ 𝔪^{∗}. Consider the homomorphism

φ_{β} : S(𝔤/𝔨) ⟶ S(𝔤/𝔨)/(ξ − β(ξ) : ξ ∈ 𝔪) = S(𝔫).

Clearly, φ_{β}(S(𝔤/𝔨)^{K}) ⊂ S(𝔫)^{K_{β}}. Note that φ_{β} is a homomorphism of Poisson algebras S(𝔤̌_{2}/𝔨̌)^{K} and S(𝔫)^{K_{β}}. For arbitrary bi-homogeneous elements a, b ∈ S(𝔤/𝔨)^{K} we have

φ_{β}({a, b}_{𝔫}) = {φ_{β}(a), φ_{β}(b)}.

Here the second bracket is a Poisson bracket on S(𝔫)^{K_{β}}.

Now we show that homomorphisms φ_{γ} and φ_{β} are surjective. We have S(𝔤) = ℝ[𝔤^{∗}], S(𝔤/𝔨)^{K} = ℝ[(𝔤/𝔨)^{∗}]^{K} = ℝ[(𝔤/𝔨)^{∗}//K] and S(𝔩_{γ}/𝔨_{γ})^{K_{γ}} = ℝ[𝔪^{∗}//K_{γ}], S(𝔫)^{K_{β}} = ℝ[𝔫^{∗}//K_{β}]. Note that

\begin{matrix}𝔪^{∗}//K_{γ} ≅ (Kγ × 𝔪^{∗})//K ⊂ (𝔤/𝔨)^{∗}//K; \\ 𝔫^{∗}//K_{β} ≅ (𝔫^{∗} × Kβ)//K ⊂ (𝔤/𝔨)^{∗}//K. \end{matrix}

Moreover, Kγ and Kβ are closed in 𝔫^{∗} and 𝔪^{∗}, respectively. Hence the subsets (Kγ⊕𝔪^{∗})//K and (𝔫^{∗} ⊕Kβ)//K are closed in (𝔤/𝔨)^{∗}//K. Thus, the restrictions ℝ[(𝔤/𝔨)^{∗}]^{K} → ℝ[Kγ ⊕𝔪^{∗}]^{K} and ℝ[(𝔤/𝔨)^{∗}]^{K} → ℝ[𝔫^{∗} ⊕ Kβ]^{K} are surjective. It is therefore proved that φ_{γ} and φ_{β} are surjective. (⟸) Suppose conditions (ii) and (iii) are satisfied. Clearly, X is commutative if and only if both Poisson brackets { , }_{𝔫} and { , }_{𝔩} equal zero on S(𝔤/𝔨)^{K}. If {a, b}_{𝔩} ≠ 0 for some elements a, b ∈ S(𝔤/𝔨)^{K} then there is a (generic) point γ ∈ 𝔫^{∗} such that φ_{γ}({a, b}_{𝔩}) ≠ 0. But φ_{γ}({a, b}_{𝔩}) = {φ_{γ}(a), φ_{γ}(b)} = 0. Analogously, if {a, b}_{𝔫} ≠ 0 for some elements a, b ∈ S(𝔤/𝔨)^{K} then there is a (generic) point β ∈ 𝔪^{∗} such that φ_{β}({a, b}_{𝔩}) ≠ 0. But φ_{β}({a, b}_{𝔩}) = {φ_{β}(a), φ_{β}(b)} = 0. (⟹) Suppose X is commutative. Then both Poisson brackets { , }_{𝔫} and { , }_{𝔩} vanish on S(𝔤/𝔨)^{K}. Hence {φ_{γ}(a), φ_{γ}(b)} = 0, {φ_{β}(a), φ_{β}(b)} = 0 for any a, b ∈ S(𝔤/𝔨)^{K}. The homomorphisms φ_{γ} and φ_{β} are surjective, so the Poisson algebras S(𝔩_{γ}/𝔨_{γ})^{K_{γ}} and S(𝔫)^{K_{β}} are commutative.

2. Properties of commutative spaces

Suppose X = G/K = (N ⋋ L)/K is a commutative homogeneous space. Denote by P the ineffective kernel of the action L : 𝔫. Note that P is a normal subgroup of L and G. Due to (i) we have L/P ⊂ O(𝔫). Hence, the stabiliser L_{γ} is reductive for any γ ∈ 𝔫^{∗} and the generic stabiliser L_{∗}(𝔫) is also reductive. Condition (i) holds if and only if L = L_{∗}(𝔫)K or equivalently L/P is a product of L_{∗}/P and K/(K ∩ P). All nontrivial factorisations of compact groups as products of two subgroups are classified by Onishchik [16]. The group L_{γ} is reductive, hence the homogeneous space L_{γ}/K_{γ} considered in (ii), is commutative if and only if it is spherical.

Definition 4. Let M, F, G, K be Lie groups, with F ⊂ M and K ⊂ G. The pair (M, F) is called an extension of (G, K) if

G ⫋ M, M = GF, K = F ∩ G.

The condition (i) means that (L, K) is an extension of (L_{∗}(𝔫), K_{∗}(𝔫)).

Evidently, G/P = N ⋋ (L/P) and (G/P)/[K/(K ∩ P)] is a commutative homogeneous space of G/P.

In this section we are interested in commutative spaces satisfying condition

(∗) L ≠ K and the action L : 𝔫 is locally effective, i.e., P is finite. In particular, this condition means that L is compact.

Lemma 3. Let a symmetric pair (M = F × F, F) with a simple compact group F be an extension of a spherical pair (G, H). Then G contains either F × {e} or {e} × F.

Proof. Let G_{1} and G_{2} be the images of the projections of G onto the first and the second factors respectively. The group G_{1} × G_{2} acts spherically on F ≅ M/F ≅ G/H. If neither G_{1} nor G_{2} equals F, then due to [2, Theorem 4] we have dim B(G_{i}(ℂ)) ≤ dim U(F (ℂ)). Hence, dim B((G_{1} × G_{2})(ℂ)) ≤ 2 dim U(F (ℂ)) < dim F (ℂ) and the action (G_{1} × G_{2}) : F cannot be spherical. Assume that G_{1} = F but F × {e} is not contained in G. Then G ≅ F and H = {e}. But the pair (F, {e}) cannot be spherical.

Lemma 4. Suppose a compact group F ⊂ Sp_{n} acts irreducibly on ℍ^{n} and F |_{ξℍ} = Sp_{1} for every ξ ∈ ℍ^{n}, ξ ≠ 0. Then F = Sp_{n}.

Proof. Let F (ℂ) ⊂ Sp_{2n}(ℂ) be the complexification of F. Then F (ℂ) acts on a generic subspace ℂ^{2} ⊂ ℂ^{2n} as SL_{2}(ℂ). Hence it acts on ℂ^{2n} locally transitively. It was proved by Panyushev [18] in a classification-free way, that F (ℂ) = Sp_{2n}(ℂ).

Lemma 5. Suppose 𝔩 ⊂ 𝔰𝔬(V) is a Lie algebra. Let 𝔩_{1} be a non-abelian simple ideal of 𝔩. Denote by π the projection onto 𝔩_{1}. If π(𝔩_{∗}(V)) = 𝔩_{1} and W_{1} is a non-trivial irreducible 𝔩-submodule of V that is also non-trivial as an 𝔩_{1}-module, then 𝔩_{1} = 𝔰𝔲_{2} and W_{1} is of the form ℍ^{1} ⊗_{ℍ} ℍ^{n}, where 𝔩 acts on ℍ^{n} as 𝔰𝔭_{n}.

Proof. Set 𝔩 = 𝔩_{1}⊕𝔩_{2}. We may assume that V = W_{1}. The vector space V can be decomposed into a tensor product V = V_{1,1} ⊗_{𝔻} V_{1}^{1} of 𝔩_{1} and 𝔩_{2}-modules, where 𝔻 is one of ℝ, ℂ or ℍ. Here 𝔩_{1} acts trivially on V_{1}^{1} and 𝔩_{2} acts trivially on V_{1,1}. Both actions 𝔩_{1} : V_{1,1} and 𝔩_{2} : V_{1}^{1} are irreducible.

Let x = x_{1,1} ⊗ x_{1}^{1} ∈ V be a non-zero decomposable vector. Because V_{1,1} is a nontrivial irreducible 𝔩_{1}-module, (𝔩_{1})_{x} ≠ 𝔩_{1}. We have 𝔩_{∗} ⊂ 𝔩_{x} up to conjugation. Evidently, 𝔩_{x} ⊂ 𝔫_{1}(x) ⊕ 𝔫_{2}(x), where 𝔫_{i}(x) = {ξ ∈ 𝔩_{i} : ξx ∈ 𝔻x}. Since 𝔩_{1} = π(𝔩_{∗}) ⊂ 𝔫_{1}(x), we have 𝔫_{1}(x) = 𝔩_{1}. Hence, 𝔻x_{1,1} is an 𝔩_{1}-invariant subspace of V_{1,1}. Thus V_{1,1} = 𝔻x_{1,1} and 𝔩_{1} ⊂ 𝔤𝔩_{1}(𝔻). If 𝔻 equals ℝ or ℂ, 𝔤𝔩_{1}(𝔻) is commutative. If 𝔻 = ℍ then 𝔩_{1}(x) ⊂ 𝔰𝔭_{1}. Thus we have shown that 𝔩_{1} = 𝔰𝔭_{1} = 𝔰𝔲_{2} and W_{1} = ℍ^{1} ⊗_{ℍ} ℍ^{n}. Moreover, 𝔩_{2}|_{ℍx} = 𝔰𝔭_{1}. To conclude, notice that 𝔩 has to act on ℍ^{n} as 𝔰𝔭_{n} by Lemma 4.

Set L_{∗} := L_{∗}(𝔫), K_{∗} := K_{∗}(𝔫). Recall that there is a factorisation L = Z(L)×L_{1}×...×L_{m}. Denote by π_{i} the projection onto L_{i}.

Theorem 2. Suppose a commutative homogeneous space X = G/K satisfies condition (∗). Then any non-commutative normal subgroup of K distinct from SU_{2} is contained in some simple factor of L.

Proof. Assume K_{1} is a normal subgroup of K having non-trivial projections onto, say, L_{1} and L_{2}. Consider the subgroup M = Z(L) × π_{1}(K) × π_{2}(K) × L_{3} × ... × L_{m}. Evidently, K ⊂ M. Without loss of generality we can replace L by M or better assume from the beginning that L_{i} = π_{i}(K) = π_{i}(K_{1}) ≅ K_{1} (i = 1, 2). Denote by π_{1,2} the projection onto L_{1} × L_{2}. According to condition (i), L_{1} × L_{2} = K_{1}π_{1,2}(L_{∗}). Recall that due to condition (ii), L_{∗}/K_{∗} is commutative, hence (L_{∗}, K_{∗}) is a spherical pair. The pair (π_{1,2}(L_{∗}), π_{1,2}(K_{∗})) is also spherical as a factor of a spherical pair. Clearly, π_{1,2}(K_{∗}) ⊂ π_{1,2}(K)∩π_{1,2}(L_{∗}). Thus the symmetric pair (L_{1}×L_{2}, K_{1}) is an extension of the spherical pair (π_{1,2}(L_{∗}), π_{1,2}(K)∩π_{1,2}(L_{∗})). By Lemma 3 the group π_{1,2}(L_{∗}) contains L_{1} or L_{2} (we can assume that it contains L_{1}). Then π_{1}(L_{∗}) = L_{1} and by Lemma 5 we have L_{1} = SU_{2}.

In Table 1 we present the list of all factorisations of compact simple Lie algebras obtained in [16]. Here 𝔤^{1}, 𝔤^{2} are subalgebras of 𝔤, 𝔤 = 𝔤^{1} + 𝔤^{2}, 𝔲 = 𝔤^{1} ∩ 𝔤^{2}. In all cases n > 1, φ^{1} and φ^{2} are the restrictions of the defining representation of the complexification 𝔤(ℂ) to 𝔤^{1}(ℂ) and 𝔤^{2}(ℂ) (whose highest weights are indicated), ϖ_{m} are the fundamental weights, I1 is the trivial representation.

Table 1.

𝔤  𝔤^{1}  φ^{1}  𝔤^{2}  φ^{2}  𝔲
𝔰𝔲_{2n}  𝔰𝔭_{n}  ϖ_{1}  𝔰𝔲_{2n−1}  ϖ_{1} + I1  𝔰𝔭_{n−1}
𝔰𝔲_{2n−1} ⊕ ℝ  𝔰𝔭_{n−1} ⊕ ℝ
𝔰𝔬_{2n+4} 𝔰𝔬_{2n+3} ϖ_{1} + I1 𝔰𝔲_{n+2} ϖ_{1} + ϖ_{n+1} 𝔰𝔲_{n+1}
𝔰𝔲_{n+2} ⊕ ℝ  𝔰𝔲_{n+1} ⊕ ℝ
𝔰𝔬_{4n} 𝔰𝔬_{4n−1} ϖ_{1} + I1 𝔰𝔭_{n} ϖ_{1} + ϖ_{1} 𝔰𝔭_{n−1}
𝔰𝔭_{n} ⊕ ℝ  𝔰𝔭_{n−1} ⊕ ℝ
𝔰𝔭_{n} ⊕ 𝔰𝔲_{2}  𝔰𝔭_{n−1} ⊕ 𝔰𝔲_{2}
𝔰𝔬_{16}  𝔰𝔬_{15} ϖ_{1} + I1  𝔰𝔬_{9}  ϖ_{4}  𝔰𝔬_{7}
𝔰𝔬_{8}  𝔰𝔬_{7}  ϖ_{3}  𝔰𝔬_{7}  ϖ_{1} + I1  G_{2}
𝔰𝔬_{7}  G_{2}  ϖ_{1}  𝔰𝔬_{5}  ϖ_{1} + I1 + I1  𝔰𝔲_{2}
𝔰𝔬_{5} ⊕ ℝ  𝔰𝔲_{2} ⊕ ℝ
𝔰𝔬_{6}  ϖ_{1} + I1  𝔰𝔲_{3}

Note that in Table 1 all algebras 𝔤^{1} are simple; if 𝔤^{2} is not simple, then 𝔤^{2} = 𝔤^{3} ⊕ 𝔞, where 𝔤^{3} is simple, 𝔞 = ℝ or 𝔞 = 𝔰𝔲_{2} and 𝔤 = 𝔤^{1} + 𝔤^{3}.

Suppose X = (N ⋋ L)/K is commutative and K ≠ L. Let L_{1} ≠ SU_{2} be a simple noncommutative normal subgroup of L such that π_{1}(K) ≠ L_{1} and L_{1} ⊄ P. Due to Lemma 5 and Theorem 1 there is a non-trivial factorisation L_{1} = π_{1}(K)π_{1}(L_{∗}). Because L_{1} is compact, this equality holds if and only if 𝔩_{1} = π_{1}(𝔨) + π_{1}(𝔩_{∗}). Note that if X satisfies condition (∗), then no simple factor L_{i} of L is contained in P. Besides, if X satisfies condition (∗) and L_{i} ≠ SU_{2}, then, by Theorem 2, π_{i}(K) ≠ L_{i} if and only if L_{i} ⊄ K.

Let X = (N ⋋ L)/K be commutative and L_{1} be a simple normal subgroup of L such that π_{1}(K) ≠ L_{1} and L_{1} ⊄ P. Let V̂ stand for the sum of those irreducible L-invariant subspaces of 𝔫, on which L_{1} acts non-trivially. Denote by P̂ the identity component of the ineffective kernel of L : V̂. Set L̂ = L/P̂. The group L̂ can be considered as the maximal connected subgroup of L, which acts on V̂ locally effectively. Denote by K̂ the image of the projection of K onto L̂, i.e., K̂ ≅ K/(K ∩ P̂). Clearly, V̂ need not be a subalgebra of 𝔫, but it can be considered as a factor algebra. Let 𝔞̂ stand for the maximal L-invariant subspace of 𝔫 on which L_{1} acts trivially. Evidently, 𝔞̂ is a subalgebra. Moreover, because different L-invariant summands of 𝔫 commute, see [21, Proposition 15], it is an ideal. We identify V̂ with 𝔫/𝔞̂.

Proposition 2. All triples (L̂, K̂, V̂) which can be obtained from a commutative space X as a result of the procedure described above are contained in Table 2b.

Proof. The homogeneous space (V̂ ⋋ L)/K is commutative. Let L = L_{1} × L^{1} × Z(L), where L^{1} is the product of all simple factors of L except L_{1}. The space V̂ can be represented as a sum

V̂ = (V_{1,1} ⊗_{𝔻_{1}} V_{1}^{1}) ⊕ … ⊕ (V_{1,p} ⊗_{𝔻_{p}} V_{p}^{1}),

where V_{1,i} are irreducible L_{1}-modules, V_{i}^{1} are L^{1}-modules, L_{1} acts trivially on each V_{i}^{1} and L^{1} acts trivially on each V_{1,i}. In each summand the tensor product is taken over the skew-field 𝔻_{i}, which equals ℝ, ℂ or ℍ depending on V_{1,i} and V_{i}^{1}.

Set V_{1} := V_{1,1} ⊕ … ⊕ V_{1,p}. Let x_{1} = x_{1,1} + … + x_{1,p} be a generic vector of V_{1}. Then there is a sum of decomposable vectors x := x_{1,1} ⊗ x_{1}^{1} + … + x_{1,p} ⊗ x_{p}^{1} ∈ V, such that x_{i}^{1} are linearly independent.

Note that (π_{1}(L_{x}), π_{1}(K_{x})) is a spherical pair as a factor of the spherical pair (L_{x}, K_{x}). Clearly, π_{1}(K_{x}) ⊂ π_{1}(L_{x}) ∩ π_{1}(K). According to condition (i), L_{1} = π_{1}(L_{x})π_{1}(K). Hence, the pair (L_{1}, π_{1}(K)) is an extension of the spherical pair (π_{1}(L_{x}), π_{1}(L_{x}) ∩ π_{1}(K)).

The generic stabiliser (L_{1})_{∗}(V_{1}) is defined up to conjugation. We may assume that (L_{1})_{∗}(V_{1}) = (L_{1})_{x_{1}} ⊂ π_{1}(L_{x}). Obviously, π_{1}(L_{x})x_{1} ⊂ 𝔻_{1}x_{1,1}+…+𝔻_{p}x_{1,p}. Hence, (L_{1})_{∗}(V_{1}) is a normal subgroup of π_{1}(L_{x}) and π_{1}(L_{x})/(L_{1})_{∗}(V_{1}) is locally isomorphic to a direct product of (SU_{2})^{k} and (U_{1})^{n}. Recall that there is a non-trivial factorisation L_{1} = π_{1}(L_{x})π_{1}(K). Looking in Table 1 one can see that (L_{1})_{∗}(V_{1})^{0} ≠ {E}, L_{1} = π_{1}(L_{x})(L_{1})_{∗}(V_{1}), and π_{1}(L_{x})/(L_{1})_{∗}(V_{1}) is locally isomorphic to SU_{2} or U_{1}, or trivial. To conclude the proof, we need the following lemma.

Lemma 6. All triples (L_{1}, π_{1}(K), V_{1}) which can be obtained as a result of the above procedure are contained in Table 2a.

Proof. As we already know, (L_{1}, π_{1}(K)) is an extension of a spherical pair (π_{1}(L_{x}), π_{1}(L_{x}) ∩ π_{1}(K)); in particular, it is spherical. The Lie algebra 𝔩_{1} is simple, hence this extension is contained in Table 1. Suppose 𝔩_{1} = 𝔤 for some 𝔤 from Table 1 and (𝔤, 𝔤_{i}) is an extension of a spherical pair (𝔤_{j}, 𝔲). Then (𝔩_{1})_{∗}(V_{1}) equals to one of the following three algebras: 𝔤_{j}, 𝔤′_{j}, 𝔤_{j}/𝔰𝔭_{1}. The last case is only possible if 𝔤_{j} = 𝔰𝔭_{n} ⊕ 𝔰𝔭_{1}. We have to check if any of these three algebras is a generic stabiliser for some representation of 𝔩_{1}. The representations of complex simple algebras with non-trivial generic stabiliser are classified by Élashvili [8]. We are interested in the real forms of the orthogonal representations with non-trivial generic stabiliser.

The algebra 𝔩_{1} is either 𝔰𝔲_{2n} or 𝔰𝔬_{m}. If 𝔩_{1} = 𝔰𝔲_{2n}, (𝔩_{1})_{∗}(V_{1}) have to be one of the following three algebras 𝔰𝔭_{n}, 𝔰𝔲_{2n−1}, 𝔲_{2n−1}. According to [8], V_{1} is either ℂ^{2n} or ℝ^{6} for 𝔩_{1} = 𝔰𝔲_{4}.

Suppose 𝔩_{1} = 𝔰𝔬_{m}. According to [8], if m > 15, then V_{1} is the sum of k copies of ℝ^{m} and (𝔩_{1})_{∗}(V_{1}) = 𝔰𝔬_{m−k}. Table 1 tell and Krämer’s classification [11] tell us that k = 1, so 𝔤_{j} = 𝔰𝔬_{m−1} and (𝔤_{j}, 𝔲) is spherical only in one case, namely (𝔰𝔬_{2n+3}, 𝔰𝔲_{n+1} ⊕ ℝ). For smaller m one has to check several cases by direct computations. The result is given in rows 2a, 2b, 4a and 4b of Table 2a.

Note that, we assume that a triple (L_{1}, π_{1}(K), W) is contained in Table 2a, if W is an L_{1}-invariant subspace of some V_{1} and (L_{1}, π_{1}(K), V_{1}) is a triple precisely listed in Table 2a.

Proceed to the proof of Proposition 2. As can be easily seen, Table 2a does not contain symplectic representations, i.e., those that could be a factor in a tensor product over ℍ. It contains only one Hermitian representation (in the first line).

Set W_{1} := V_{1,1}⊗V_{1}^{1}. Our next goal is to describe all possible W_{1}. The pair (L_{1}, π_{1}(K), V_{1,1}) is contained in Table 2a, i.e., V_{1,1} is an L_{1}-invariant subspace of V_{1}. In particular, the tensor product in W_{1} is taken over ℝ or ℂ.

Set n_{1} = dim V_{1,1}, n_{2} = dim V_{1}^{1} and s = min(n_{1}, n_{2}) (here we consider the dimensions over the same field as the tensor product in W_{1}.) Let us prove that up to conjugation

[π_{1}(L_{∗}(W_{1}))]′ = [(L_{1})_{ξ_{1}} ∩ ... ∩ (L_{1})_{ξ_{s}}]′,

where ξ_{i} ∈ V_{1,1} are linear independent generic vectors. In particular, this equality means that if dim V_{1,1} ≤ dim V_{1}^{1}, then [π_{1}(L_{∗}(W_{1}))]′ = {E}. Note that the right hand side is evidently a subset of the left hand side. So, it is enough to proof the inclusion “⊂”.

The group L_{1} is compact, so we can assume that L_{1} ⊂ O_{n_{1}} if 𝔻_{1} = ℝ or L_{1} ⊂ U_{n_{1}} if 𝔻_{1} = ℂ. Without loss of generality it can be also assumed that the considered action is L_{1} · O_{n_{2}} : V_{1,1} ⊗_{ℝ} V_{1}^{1} or L_{1} · U_{n_{2}} : V_{1,1} ⊗_{ℂ} V_{1}^{1}. The left hand side of the proving equality could become only larger after such replacement. The required inclusion can be deduced from a well known fact: if n ≤ m, then the generic stabilisers for O_{n} × O_{m} : ℝ^{n} ⊗ ℝ^{m} and U_{n} × U_{m} : ℂ^{n} ⊗ ℂ^{m} are O_{m−n} × (ℤ/2ℤ)^{n} and U_{m−n} × U_{1}^{n}, respectively.

Recall that there should be factorisations L_{1} = π_{1}(K̂)π_{1}(L̂_{∗}) and L_{1} = π_{1}(K̂)[π_{1}(L̂_{∗})]′. This imposes rather strong restrictions on s. Thus s = 1 in cases 1a, 1b, 2b, 3 and 4b; s < 3 in case 2a; s < 4 in case 4a. If V_{1} is irreducible L_{1}-module, then V̂ = W_{1}. Hence, V̂ = V_{1} in cases 1a, 1b, 2b, 3 and 4b. Note that in general

π_{1}(L̂_{∗}(V̂)) ⊂ π_{1}(L̂_{∗}(V_{1,1} ⊗ V_{1}^{1}) ∩ … ∩ L̂_{∗}(V_{1,p} ⊗ V_{p}^{1})).

So there are only a few possibilities for V̂ and hence for (L̂, K̂). Namely, p ≤ 2, (dim V_{1}^{1} + dim V_{2}^{1}) ≤ 2 in case 2a; and p ≤ 3, (dim V_{1}^{1} + dim V_{2}^{1} + dim V_{3}^{1}) ≤ 3 in case 4a. For each representation one can easily verify whether conditions (i) and (ii) hold. The result is shown in Table 2b.

Remark 2. In the first row of Table 2b V̂ is written as ℂ^{2n}(⊕ℝ). Of course, ℝ is a trivial L-module. So, formally it cannot be put into the table. But it is done here to indicate that in this case V̂ can be a non-commutative subspace of 𝔫.

Table 2a.

L_{1}  π_{1}(K)  V_{1}
1a SU_{2n}  Sp_{n}  ℂ^{2n}
1b SU_{4}  U_{3}  ℝ^{6}
2a SO_{7}  G_{2}  ℝ^{7} ⊕ ℝ^{7}
2b Spin_{7}  Spin_{6}  ℝ^{8}

Spin_{5} · U_{1}

3 SO_{2n} SU_{n} ℝ^{2n}

U_{n}

4a SO_{8} Spin_{7} ℝ^{8} ⊕ ℝ^{8} ⊕ ℝ^{8}
4b SO_{8} Sp_{2} × SU_{2}  ℝ^{8}

Table 2b.

L̂ K̂ V̂
1a (S)U_{2n}  Sp_{n}(·U_{1}) ℂ^{2n}(⊕ℝ)
1b  SU_{4}  U_{3}  ℝ^{6}
2a  SO_{7}  G_{2}  ℝ^{7}
2b  Spin_{7}  Spin_{6}  ℝ^{8}
3  SO_{2n}  U_{n}  ℝ^{2n}
4a SO_{8} × SO_{2} Spin_{7} × SO_{2} ℝ^{8} ⊗ ℝ^{2}
4b  SO_{8}  Spin_{7}  ℝ^{8} ⊗ ℝ^{2}
4c  SO_{8}  Spin_{7}  ℝ^{8}
4d  SO_{8}  Sp_{2} × SU_{2}  ℝ^{8}

So far nothing was said about the Lie algebra structure on V̂. Denote by 𝔫̂ the subalgebra of 𝔫 generated by V̂, i.e., 𝔫̂ = V̂ + [V̂ , V̂].

Suppose [V̂ , V̂] = 0. Then we have a commutative homogeneous space (N̂ ⋋L̂)/K̂, where N̂ is a simply connected commutative Lie group.

Suppose X = (N ⋋ L)/K is commutative, L_{1} ≠ SU_{2} is a simple normal subgroup of L, such that π_{1}(K) ≠ L_{1}, L_{1} ⊄ P and a triple (L̂, K̂, V̂) corresponds to L_{1} in the aforementioned way. By Proposition 2, (L̂, K̂, V̂) is contained in Table 2b.

Lemma 7. If (L̂, K̂, V̂) is a triple from Table 2b but not from the first row of it, then V̂ has to be a commutative subspace of 𝔫.

Proof. Assume that [V̂ , V̂] ≠ 0. There in an L-invariant surjection Λ^{2}V̂ ↦ [V̂ , V̂]. Because representation of L in Λ^{2}V̂ is completely reducible, [V̂ , V̂] can be regarded as an L-invariant subspace of Λ^{2}V̂. If [V̂ , V̂] is a non-trivial L_{1}-module, then it is an L̂-invariant subspace of V̂ (by definition of V̂). Recall that 𝔫 is nilpotent. Hence, [V̂ , V̂] ≠ V̂. In particular, if V̂ is an irreducible L̂-module, then L_{1} has to act on [V̂ , V̂] trivially. In all rows of Table 2b except 4b the representation L̂ : V̂ is irreducible. The space Λ^{2}V̂ contains non-trivial L̂-invariants only in cases 1a, 4a and 4b.

Consider case 4a. Here K̂_{∗}(𝔪) = SU_{4} × SO_{2} and ℝ^{8}⊗ℝ^{2} = ℂ^{4}⊗_{ℝ}ℂ ≅ ℂ^{4}⊕ℂ^{4} is a sum of two isomorphic K_{∗}(𝔪)-modules. According to [21, Proposition 15], these two submodules are commutative and commute with each other. Hence, [V̂ , V̂] = 0. In case 4b V̂ = ℝ^{8} ⊕ ℝ^{8} is a sum of two isomorpich L̂-modules. Hence, it is commutative.

The first case is different. It corresponds to six commutative spaces, namely L̂ can be SU_{2n} or U_{2n}. In the second case there are also two possibilities K̂ = Sp_{n} or K̂ = Sp_{n} × U_{1}. Independently 𝔫̂ can be either ℂ^{2n} or ℂ^{2n} ⊕ ℝ, with N̂ being commutative or the Heisenberg group H_{2n}.

So, the first row gives rise to six commutative spaces and each of the other to only one. In each case the commutativity can easily be proved by means of Theorem 1. Two cases are considered in detail.

Example 1. Let us prove that (H_{n} ⋋ U_{2n})/Sp_{n} is commutative. Since U_{2n} and Sp_{n} are transitive on the sphere in ℂ^{2n}, ℝ[ℂ^{2n}]^{U_{2n}} = ℝ[q] = ℝ[ℂ^{2n}]^{Sp^{n}}, where q is the invariant of degree 2.

The generic stabiliser for Sp_{n} : ℂ^{2n} is equal to Sp_{n−1}. The space U_{2n−1}/Sp_{n−1} is a compact real form of the complex spherical space GL_{2n−1}(ℂ)/Sp_{2n−2}(ℂ), and, hence, is commutative.

So, only (iii) is left. Here we have 𝔪 = 𝔲_{2n}/𝔰𝔭_{n} = ⋀^{2} ℂ^{2n}. It is a classical result that

K_{∗}(⋀ ℂ^{2n}) = \underbrace{SU_{2} × ... × SU_{2}}_{n}. As a K_{∗}(𝔪)-module 𝔫 = 𝔳_{1} ⊕ ... ⊕ 𝔳_{n} ⊕ ℝ, where 𝔳_{i} = ℂ^{2}

for every i. Each 𝔳_{i} is acted upon by its own SU_{2}. Note that [𝔳_{i}, 𝔳_{j}] = 0 for i ≠ j. For K_{∗}(𝔪)-invariants we have S(𝔫)^{K_{∗}(𝔪)} = ℝ[t_{1}, ...t_{n}, ξ], where t_{i} is the quadratic SU_{2} invariant in S^{2}(𝔳_{i}) and ξ ∈ 𝔥′_{n}. Evidently, t_{i} and t_{j} commute as elements of the Poisson algebra S(𝔫), and ξ lies in the centre of S(𝔫).

Example 2. The homogeneous space (ℝ_{2n} ⋋ SO_{2n})/U_{n} is also commutative. Here 𝔫 is commutative, so we do not need to check condition (iii). For (i) we have ℝ[ℝ^{2n}]^{SO^{2n}} = ℝ[q] = ℝ[ℝ^{2n}]^{U_{n}}. It can be easily seen that L_{∗} = SO_{2n−1} and K_{∗} = U_{n−1}. The corresponding homogeneous space SO_{2n−1}/U_{n−1} is spherical by the Krämer’s classification [11].

Note that the triple (SU_{4}, U_{3}, ℝ^{6}) is locally isomorphic to a triple from the row 3 of Table 2b with n=3.

Proposition 3. Suppose X = (N ⋋ L)/K is an indecomposable commutative space, 𝔫 ≠ 0, L is simple and L ≠ K. Then X is locally isomorphic to a homogeneous space from Table 2b.

Proof. The action L : 𝔫 is non-trivial, otherwise X = N × (L/K). Consider the triple (L, K, V̂) corresponding to L_{1} = L. By Proposition 2, it is contained in Table 2b. Assume that 𝔫̂ ≠ 𝔫 and let 𝔞 be an L-invariant complement of 𝔫̂ in 𝔫. By definition of V̂, L acts on 𝔞 trivially. Thus 𝔞 is an abelian ideal and X is decomposable X = ((N̂ ⋋ L)/K) × A, where A ⊂ N and Lie A = 𝔞.

Lemma 8. Let (N ⋋ L)/K be a commutative homogeneous space satisfying condition (∗) and K_{1} ≅ SU_{2} a normal subgroup of K. Then either K_{1} ⊂ SO_{8}, where SO_{8} is a simple direct factor of L contained in the row 4d of Table 2b, or K_{1} is the diagonal of a product of at most three simple direct factors of L isomorphic to SU_{2}.

Proof. Suppose π_{i}(K_{1}) ≠ {E} and L_{i} ≠ SU_{2}. Then π_{i}(K) ≠ L_{i}. Consider a triple (L̂, K̂, V̂) corresponding to L_{i}. By Proposition 2, it is contained in Table 2b. Note that K_{1} is a normal subgroup of K̂. Thus K̂ = Sp_{2} × SU_{2}, L̂ = SO_{8}. Assume that K_{1} has a non-trivial projections onto some other simple direct factor of L. Then the pair (SO_{8} × SU_{2}, Sp_{2} × SU_{2}) with SU_{2} embedded into SO_{8} as a centraliser of Sp_{2} and into SU_{2} isomorphically, should be spherical, but it is not. To conclude, note that the pair (SU_{2} × SU_{2} × SU_{2} × SU_{2}, SU_{2}) is not spherical either.

3. Principal commutative spaces

Let G/K = (N ⋋ L)/K be a commutative homogeneous space. Recall that by our assumptions L = Z(L)×L_{1}×… L_{m}, where Z(L) is the connected centre of L. The ineffective kernel of L : 𝔫 is denoted by P. Denote by Z(K) the connected centre of K. Decompose 𝔫/𝔫′ into a sum of irreducible L-invariant subspaces 𝔫/𝔫′ = 𝔴_{1} ⊕ ... ⊕ 𝔴_{p}.

Definition 5. Let us call a commutative homogeneous space G/K principal if P is semisimple, Z(K) = Z = Z(L)×(L_{1}∩Z)×...×(L_{m}∩Z) and Z(L) = C_{1}×...×C_{p}, where C_{i} ⊂ GL(𝔴_{i}).

The classification of commutative homogeneous spaces can be divided in two parts: the classification of principal commutative spaces and description of the possible centres of L and K in the general case. Note that Table 2b contained only two non-principal homogeneous spaces, namely (H_{2n} ⋋ U_{2n})/Sp_{n} and (ℂ^{2n} ⋋ U_{2n})/Sp_{n}.

Example 3. We have proved that the homogeneous spaces X = (H_{2n} ⋋ SU_{2n})/Sp_{n} and Y = (H_{2n}⋋U_{2n})/Sp_{n}, where N = H_{2n} is the Heisenberg group with Lie H_{2n} = 𝔥_{2n} = ℂ^{2n}⊕ℝ, are commutative. Denote by X^{m} = (H_{2n} ⋋ SU_{2n})^{m}/(Sp_{n})^{m} the product of m copies of X. Let L and K be the products of m copies of SU_{2n} and Sp_{n} respectively. Suppose F_{1} ⊂ F_{2} ⊂ (U_{1})^{m}. Then, evidently, ((H_{2n})^{m} ⋋ (F_{2} × L))/(F_{1} × K) is a commutative homogeneous space, in general indecomposable but non-principal.

The above example is rather simple, because there are no conditions on F_{1} and F_{2}. In other cases the situation is more difficult. For commutative homogeneous spaces of reductive Lie groups the description of possible centres of L and K is given in [23]. The same problem for commutative homogeneous spaces (H_{n} ⋋ K)/K, where K ⊂ U_{n} is solved in [14] and [4]. In the present article we concentrate on principal commutative spaces.

Theorem 3. Let X = (N ⋋ L)/K be a maximal indecomposable principal commutative homogeneous space satisfying condition (∗). Then either X is contained in Table 2b (and L′ is simple); or (L, K) is isomorphic to a product of pairs (SU_{2} × SU_{2} × SU_{2}, SU_{2}), (SU_{2} × SU_{2}, SU_{2}) or (SU_{2}, U_{1}) and a pair (K^{1}, K^{1}), where K^{1} is a compact Lie group.

Proof. Suppose there is a simple normal subgroup L_{i} ≠ SU_{2} of L, which is not contained in K. Then by Theorem 2 π_{i}(K) ≠ L_{i}. Consider the corresponding triple (L̂, K̂, 𝔫̂). It is a homogeneous space from Table 2b. Recall that by definition L̂ is the maximal connected subgroup of L acting on V̂ locally effectively, P̂ is the identity component of the ineffective kernel of L : V̂ and L = L̂·P̂. Because X is principal, Z(L) = Ĉ×C^{1}, where Ĉ = GL(V̂)∩C, hence Ĉ ⊂ L̂, and C^{1} ⊂ P̂. In particular, L = L̂ × P̂. Similarly, the connected centre Z = Z(K) is a product Z = Z(L) × Ẑ × Z^{1}, where Ẑ ⊂ L̂′ and Z^{1} ⊂ P̂′. According to Theorem 2 each normal subgroup K_{i} ≅ S̸U_{2} of K is contained in some simple direct factor of L, hence either in L̂ or in P̂. Suppose a normal subgroup K_{j} ≅ SU_{2} of K is not contained in any simple direct factor of L. Then, by Lemma 8, it is diagonal in a product of at most three direct factors of L isomorphic to SU_{2}. The group L̂ has no normal subgroups isomorphic to SU_{2}. Hence, K_{j} ⊂ P̂. Thus K̂ = L̂ ∩ K. Moreover, K = K̂ × F, where F ⊂ P̂. Recall that 𝔫 = V̂ ⊕ 𝔞̂, where 𝔞̂ is an ideal and L_{i} acts on 𝔞̂ trivially. Let Â ⊂ N be a corresponding connected subgroup. Note that either L̂ = L_{i} or L̂ = U_{1} × L_{i}. Anyway L̂ acts on 𝔞̂ trivially. Thus in case [V̂ , V̂] = 0, i.e., 𝔫̂ = V̂, we have obtained a decomposition X = ((N̂ ⋋ L̂)/K̂) × ((Â ⋋ P̂)/F). But X in indecomposable, hence X = (N̂ ⋋ L̂)/K̂ and it is contained in table 2b.

There is only one possibility for non-commutative 𝔫̂, namely L_{i} = SU_{2n}, 𝔫 = V̂ ⊕ 𝔷, where 𝔷 ∈ 𝔞̂ and 𝔷 ≅ ℝ is a trivial L-module. Let 𝔞 be an L-invariant complement of 𝔷 in 𝔞̂. If 𝔞 is a subalgebra (then it is an ideal), we again have a decomposition of X. Assume that 𝔷 ⊂ [𝔞, 𝔞]. We have an L-invariant decomposition 𝔫 = 𝔞 ⊕ 𝔷 ⊕ V̂, where [𝔞, 𝔞] ⊂ 𝔞 ⊕ 𝔷, [V̂ , V̂] = 𝔷 and [𝔷, 𝔫] = 0. Thus X is a central reduction of ((N̂ ⋋ L̂)/K̂) × ((Â ⋋ P̂)/F) by a one dimensional subgroup embedded diagonally into N̂′ × Â′. Hence X is not maximal.

We have proved that if there is a simple normal subgroup L_{i} ≠ SU_{2} of L, which is not contained in K, then X is a homogeneous space from Table 2. If this not the case the spherical pair (L, K) is a product of the “SU_{2}-pairs” and (K^{1}, K^{1}), where K^{1} contains the connected center of L and all its simple normal subgroups different from SU_{2}.

Note that the homogeneous space (N̂ ⋋ L̂)/K̂ corresponding to the first row of Table 2b is maximal if and only if 𝔫 = 𝔥_{2n}, and it is principal if and only if L̂ = SU_{2n}, K̂ = Sp_{n} or L̂ = U_{2n}, K̂ = U_{1} · Sp_{n}. Homogeneous spaces corresponding to other rows of Table 2b are maximal and principal.

Let X be a commutative homogeneous space. Denote by (L^{△}, K^{△}) a spherical subpair of (L, K) of the type (SU_{2} × SU_{2} × SU_{2}, SU_{2}), (SU_{2} × SU_{2}, SU_{2}) or (SU_{2}, U_{1}), and by π^{△} a projection onto L^{△}.

Lemma 9. If (L^{△}, K^{△}) = (SU_{2} × SU_{2} × SU_{2}, SU_{2}) or (L^{△}, K^{△}) = (SU_{2}, U_{1}) then π^{△}(L_{∗}) = L^{△}, if (L^{△}, K^{△}) = (SU_{2} × SU_{2}, SU_{2}) then π^{△}(L_{∗}) equals L^{△}, or SU_{2} × U_{1}.

Proof. The group SU_{2} has only trivial factorisations, besides, (π^{△}(L_{∗}), π^{△}(L_{∗}) ∩ K^{△}) is a spherical pair. In particular, π^{△}(L_{∗}) ∩ K^{△} is not empty. This reasoning explains the first and the second cases. It remains to observe that in the third case the group π^{△}(L_{∗}) can not be SU_{2} × SU_{2} × U_{1}, because the pair (SU_{2} × SU_{2} × U_{1}, U_{1}) is not spherical.

We complete our classification modulo a description of possible actions of normal subgroups of L isomorphic to SU_{2} on 𝔫. This description is a very intricate problem, which may be a subject of another article.

4. The ineffective kernel

Suppose X = (N ⋋ L)/K is a commutative homogeneous Riemannian space. Let P be the ineffective kernel of the action L : 𝔫. Then L can be decomposed as L = P ·L^{⋄}, where L^{⋄} is the maximal connected normal subgroup of L acting on 𝔫 locally effectively. We assume that G is not reductive, hence P ≠ L. From the classification of spherical pairs we know that any normal subgroup K_{1} of K not locally isomorphic to SU_{2} can have non-trivial projections only on two different simple factors of L.

Lemma 10. Let X be commutative. Suppose a normal subgroup K_{1} ≠ SU_{2} of K is not contained in either P or L^{⋄}. Then there are simple factors P_{1}, L_{1}^{⋄} of P, L^{⋄} such that K_{1} ⊂ P_{1} × L_{1}^{⋄}, P_{1} ≅ L_{1}^{⋄} ≅ K_{1}. Moreover, either K_{1} = SO_{n+1}, where n ≥ 4; or K_{1} = SU_{n+1}, where n ≥ 2.

Proof. It can be seen from the classification of spherical pairs that K_{1} ⊂ L_{i} × L_{j}. We can assume that K_{1} ⊂ P_{1} × L_{1}^{⋄}. The action K_{1} : 𝔫 is non-trivial, otherwise K_{1} would be a subgroup of P. Denote by π_{1}^{K} the projection onto K_{1} in K and by π_{1,1} the projection onto P_{1} × L_{1}^{⋄} in L. By Lemma 5, π_{1}^{K}(K_{∗}) ≠ K_{1}. Recall that (L_{∗}, K_{∗}) is spherical. Hence, the pair (π_{1,1}(L_{∗}), π_{1,1}(K_{∗})) is also spherical. Note that L_{∗} = P ·L_{∗}^{⋄}(𝔫). Hence, π_{1,1}(L_{∗}) = P_{1}×π_{1}^{⋄}(L_{∗}), where π_{1}^{⋄} is a projection onto L_{1}^{⋄} in L.

We claim that (K_{1} × π_{1}^{K}(K_{∗}), π_{1}^{K}(K_{∗})) is spherical. Without loss of generality, we can assume that P_{1} ≅ L_{1}^{⋄} ≅ K_{1}. If it is not the case, we replace L by a smaller subgroup containing K, namely each of P_{1} and L_{1}^{⋄} is replaced by a projection of K onto it. We illustrate the embedding π_{1,1}(K_{∗}) ⊂ π_{1,1}(L_{∗}) by the following diagram.

[figure: π_{1,1}(L_{∗}); ≅; K_{1}; ×; π_{1}^{⋄}(L_{∗}); π_{1,1}(K_{∗}); =; π_{1}^{K}(K_{∗})]

Because the pair (π_{1,1}(L_{∗}), π_{1,1}(K_{∗})) is spherical, (K_{1} × π_{1}^{K}(K_{∗}), π_{1}^{K}(K_{∗})) is also spherical. According to the classification of spherical pairs, there are only two possibilities: either K_{1} = SO_{n+1}, π_{1}^{K}(K_{∗}) = SO_{n}; or K_{1} = SU_{n+1}, π_{1}^{K}(K_{∗}) = U_{n}. Assume that P_{1} ≅ K_{1}. If L_{1}^{⋄} is larger than K_{1}, then (P_{1} ×L_{1}^{⋄}, π_{1,1}(K)) is one of the following three pairs: (SO_{n+1} ×SO_{n+2}, SO_{n+1}); (Sp_{2} × Sp_{m+2}, Sp_{2} × Sp_{m}); (SU_{n+1} × SU_{n+2}, U_{n+1}). Recall that L_{1}^{⋄} = π_{1}^{⋄}(L_{∗})π_{1}^{⋄}(K). The group Sp_{m+2} has no non-trivial factorisation, hence the second case is not possible. Also, we know that π_{1}^{⋄}(L_{∗}) ∩ π_{1}^{⋄}(K) contains SO_{n} or U_{n}, depending on K_{1}. Thus, only one possibility is left (K_{1} × L_{1}^{⋄}, π_{1,1}(K)) = (SU_{3} × SU_{4}, U_{3}). According to Table 1, π_{1}^{⋄}(L_{∗}) = Sp_{2}, Sp_{2} ∩ U_{3} = Sp_{1} × U_{1}. We have π_{1,1}(L_{∗}) = SU_{3} × Sp_{2}. The subgroup π_{1,1}(K_{∗}) is contained in Sp_{1} ×U_{1}, which is not spherical in SU_{3} ×Sp_{2}. Hence, in this case the pair (π_{1,1}(L_{∗}), π_{1,1}(K_{∗})) is not spherical. We illustrate this case by the following diagram.

[figure: π_{1,1}(L_{∗}); =; _{SU_{3}}; ×; _{Sp_{2}}; π_{1,1}(K_{∗}); ⊂; Sp_{1} × U_{1}]

Thus, we have proved that L_{1}^{⋄} ≅ K_{1}.

To conclude, we show that P_{1} ≅ K_{1}. Denote by π_{1}^{P} the projection onto P_{1}. We can decompose π_{1}^{P} (K) into a locally direct product π_{1}^{P} (K) ≅ F · K_{1}. The subgroups F · K_{1} and F ·π_{1}^{K}(K_{∗}) are spherical in P_{1}. Moreover, the pairs (P_{1} ×K_{1}, F ×K_{1}) and (P_{1} ×π_{1}^{K}(K_{∗}), F × π_{1}^{K}(K_{∗})) are also spherical. There are the same three possibilities for (P_{1} × K_{1}, F × K_{1}), in which P_{1} ≠ K_{1}, namely: (SU_{n+2} × SU_{n+1}, U_{n+1}), (Sp_{m+2} × Sp_{2}, Sp_{m} × Sp_{2}) and (SO_{n+2} × SO_{n+1}, SO_{n+1}). But even the pair (P_{1}, F · π_{1}^{K}(K_{∗})) is not spherical in any of these cases.

Example 4. We show that the homogeneous spaces ((ℝ^{n} ⋋ SO_{n}) × SO_{n})/SO_{n} and ((H_{n} ⋋ U_{n}) × SU_{n})/U_{n} are commutative. We have L_{∗} = SO_{n} × SO_{n−1} for the first space and L_{∗} = SU_{n} × U_{n−1} for the second one, K_{∗} is either SO_{n−1} or U_{n−1}. The stabiliser L_{∗} contains the first direct factor and K is the diagonal multiplied by the connected centre of L. Hence, L = L_{∗}K. According to [6] and [15], L_{∗}/K_{∗} is spherical. For the second space we have to check condition (iii) of Theorem 1. We have K_{∗}(𝔪) = (U_{1})^{n}. As a K_{∗}(𝔪)-module 𝔫 = 𝔳_{1} ⊕ ... ⊕ 𝔳_{n} ⊕ ℝ, where 𝔳_{i} = ℝ^{2} for every 1 ≤ i ≤ n. Each 𝔳_{i} is acted upon by its own U_{1}. Note that [𝔳_{i}, 𝔳_{j}] = 0 for i ≠ j. For K_{∗}(𝔪)-invariants we have S(𝔫)^{K_{∗}(𝔪)} = ℝ[t_{1}, ...t_{n}, ξ], where t_{i} is the quadratic U_{1} invariant in S^{2}(𝔳_{i}) and ξ ∈ 𝔫′. Evidently, t_{i} and t_{j} commute as elements of the Poisson algebra S(𝔫), and ξ lies in the centre of S(𝔫).

Note that the homogeneous space ((ℂ^{n} ⋋ U_{n}) × SU_{n})/U_{n} is a central reduction of ((H_{n} ⋋ U_{n}) × SU_{n})/U_{n} and it is not maximal. Theorem 4. Suppose X = (N ⋋ L)/K is a maximal principal indecomposable commutative homogeneous space. Then either X is one of the spaces ((ℝ^{n} ⋋ SO_{n}) × SO_{n})/SO_{n}, ((H_{n} ⋋ U_{n}) × SU_{n})/U_{n} or each non-commutative simple normal subgroup K_{1} ≠ SU_{2} of K is contained in either P or L^{⋄}. Proof. Essentially, this is a corollary of Lemma 10. Let K_{1} ≠ SU_{2} be a non-commutative simple normal subgroup of K that is not contained in either L^{⋄} or P. Then either K_{1} = SO_{n} or K_{1} = SU_{n} and there are P_{1} ≅ L_{1}^{⋄} ≅ K_{1} such that K_{1} ⊂ P_{1}×L_{1}^{⋄}. Besides, π_{1}^{K}(K_{∗}) = SO_{n−1} or π_{1}^{K}(K_{∗}) = U_{n−1}, depending on K_{1}. According to [8], L_{1}^{⋄} can act non-trivially only on its simplest module V (V = ℝ^{n} or V = ℂ^{n}) and V ⊂ 𝔫 is an L-invariant subspace. Set C_{V} = Z(L)∩GL(V). Because X is principal, C_{V} ⊂ K and if C_{V} is trivial, then π_{1}^{K}(K_{∗}) = (K_{1})_{∗}(V). For K_{1}^{⋄} = SU_{n} we have (SU_{n})_{∗}(ℂ^{n}) = SU_{n−1}, so C_{V} = U_{1}, (U_{n})_{∗}(ℂ^{n}) ≅ π_{1}^{K}(K_{∗}) = U_{n−1}. In case K_{1} = SO_{n} the group C_{V} is trivial.

Denote by 𝔫_{1} := V + [V, V] the Lie subalgebra generated by V, by N_{1} ⊂ N the corresponding connected subgroup. If K_{1} = SO_{n}, then (K_{1})_{∗}(𝔪) = (U_{1})^{[n/2]}. According to condition (iii) of Theorem 1, 𝔫_{1} has to be commutative, i.e, 𝔫_{1} = V. If K_{1} = SU_{n}, then 𝔫_{1} can be either a Heisenberg or a commutative algebra.

Assume that X ≠ ((N_{1} ⋋ (L_{1} × C_{V})) × P_{1})/(K_{1} × C_{V}). Then L = (L_{1} × C_{V} × P_{1}) × F, K = (K_{1} × C_{V}) × H, where H ⊂ F. Let 𝔞 be an L invariant complement of 𝔫_{1} in 𝔫. Recall that the actions (L_{1} × C_{V} × P_{1}) : 𝔞 and F : 𝔫_{1} are trivial.

To conclude repeat the proof of Theorem 3. If 𝔞 is a subalgebra (ideal), then X is decomposable. Assume that [𝔞, 𝔞] ⊄ 𝔞. The action L_{1} : 𝔞 is trivial, hence [𝔞, 𝔞] ∩ V = 0. Thus [V, V] ⊂ [𝔞, 𝔞] and X is not maximal.

Let K_{1} = SU_{2} be a normal subgroup of K. Suppose it has a non-trivial projections onto P_{1} and L_{1}^{⋄}. If L_{1}^{⋄} ≠ SU_{2}, then, as we can see from Table 2a, L_{1}^{⋄} ≅ Spin_{8}. But as was already mentioned, the pair (SU_{2} × Spin_{8}, SU_{2} × Sp_{2}) is not spherical. So L_{1}^{⋄} = SU_{2}.

If π_{1}^{K}(K_{∗}) ≠ K_{1}, i.e., π_{1}^{K}(K_{∗})^{0} = U_{1}, then K_{1} ⊂ P_{1} × L_{1}^{⋄} and P_{1} = SU_{2}. But if π_{1}^{K}(K_{∗}) = K_{1} (and this can be the case), then P_{1} can be larger and K_{1} can have a nontrivial projection onto some other simple factor P_{2} or L_{2}^{⋄} = SU_{2}. Example 5. Set P := Sp_{m} × Sp_{l}, L^{⋄} := Sp_{1} × Sp_{n}, K := Sp_{m−1} × Sp_{l−1} × Sp_{1} × Sp_{n} and take for N a commutative group ℍ^{n}. The inclusions and actions are illustrated by the following diagram.

[figure: _{Sp_{m}}; _{Sp_{l}}; _{Sp_{1}}; _{Sp_{n}}; Sp_{m−1}; _{Sp_{l−1}}; _{Sp_{1}}; _{Sp_{n}}; ℍ^{n}]

The homogeneous space (N ⋋(P ×L^{⋄}))/K is commutative. Here L_{∗} = Sp_{m}×Sp_{l}×Sp_{1}×Sp_{n−1} and K_{∗} = Sp_{m−1} × Sp_{l−1} × Sp_{1} × Sp_{n−1}.

To avoid complicated technical details concerning actions and inclusions of normal subgroup isomorphic to Sp_{1} we impose on X a condition of Sp_{1}-saturation.

5. Sp_{1}-saturated spaces

Let X = (N ⋋ L)/K be a commutative homogeneous space. Let L_{i} be a simple direct factor of L. By our assumptions L is a product L = Z(L) × L_{i} × L^{i}, where L^{i} contains all direct factor L_{j} with j ≠ i.

Definition 6. A commutative homogeneous space X is called Sp_{1}-saturated, if (1) any normal subgroup K_{1} ≅ SU_{2} of K is contained in either P or L^{⋄};

(2) if a simple direct factor L_{i} is not contained in P and π_{i}(L_{∗}) = L_{i}, then L_{i} ⊂ K;

(3) if there is an L-invariant subspace 𝔴_{j} ⊂ (𝔫/𝔫′) such that for some L_{i} the action L_{i} : 𝔴_{j} is non-trivial and the action Z(L) × L^{i} : 𝔴_{j} is irreducible, then L_{i} acts on (𝔫/𝔫′)/𝔴_{j} trivially.

Example 6. Set X = ((N ⋋ (Sp_{n} × Sp_{1})) × Sp_{1})/(Sp_{n} × Sp_{1}), where 𝔫 = ℍ^{n} ⊕ ℍ_{0} is a two-step nilpotent non-commutative Lie algebra with [ℍ^{n}, ℍ^{n}] = ℍ_{0}, ℍ_{0} is the space of purely imaginary quaternions, the normal subgroup Sp_{1} of K is the diagonal of the product Sp_{1} × Sp_{1}. Here ℍ^{n} = ℍ^{n} ⊗_{ℍ} ℍ, where Sp_{n} acts on ℍ^{n} and Sp_{1} acts on ℍ^{1}; ℍ_{0} ≅ 𝔰𝔭_{1} as an L-module, i.e., Sp_{n} acts on it trivially and Sp_{1} via adjoint representation.

Evidently, X = (N ⋋ L)/K is not Sp_{1}-saturated. We show that it is commutative. First we compute the generic stabiliser L_{∗}. Recall that (Sp_{n} × Sp_{1})_{∗}(ℍ^{n}) = Sp_{n−1} × Sp_{1}. Clearly L_{∗}(ℍ_{0}) = Sp_{n} × U_{1} × Sp_{1}, (Sp_{n} × U_{1})_{∗}(ℍ^{n}) = Sp_{n−1} × U_{1}. We have L_{∗} = Sp_{n−1} × U_{1} × Sp_{1}, K_{∗} = K ∩ L_{∗} = Sp_{n−1} × U_{1}, K_{∗}(𝔪) = Sp_{n} × ((Sp_{1})_{∗}(𝔰𝔭_{1})) = Sp_{n} × U_{1}. One can easily verify conditions (i) and (ii) of Theorem 1. Tables of [21] and [22] shows that (iii) is also satisfied.

The description of non Sp_{1}-saturated spaces seems to be an interesting problem. We are going to consider this problem in a forthcoming paper.

Let X be a non Sp_{1}-saturated commutative homogeneous space. It can be made Sp_{1}- saturated by enlarging K, L and possibly N too. For instance, if a simple factor Sp_{1} of K has non-trivial projections onto P and L^{⋄}, then we replace P by P ×Sp_{1} or P ×Sp_{1}×Sp_{1} (the second replacement is needed if Sp_{1} has non-trivial projections onto two simple factors of P). The group K is replaced by K ×Sp_{1}. Starting with the commutative spaces from Example 6 we construct an Sp_{1}-saturated commutative homogeneous space (Sp_{1} × Sp_{1}/Sp_{1}) × (N ⋋ K/K), where K = Sp_{1} × Sp_{n} and N is the same as before.

Example 7. Set L = K = Sp_{n} × Sp_{1} × Sp_{m}, 𝔫 = ℍ^{n} ⊕ ℍ^{m} ⊕ ℍ_{0}, where both algebras ℍ^{n} and ℍ^{m} are not commutative and [ℍ^{n}, ℍ^{n}] = [ℍ^{m}, ℍ^{m}] = ℍ_{0}. We have S(𝔤/𝔨)^{K} = S(𝔫)^{K} = ℝ[ξ_{1}, ξ_{2}, η], where ξ_{1} ∈ S^{2}(ℍ^{n})^{Sp^{n}}, ξ_{2} ∈ S^{2}(ℍ^{m})^{Sp^{m}}, η ∈ S^{2}(ℍ_{0})^{Sp^{1}}, so the corresponding homogeneous space (N ⋋ K)/K is commutative. The third condition of Definition 6 is not fulfilled. If we want to enlarge L, we also need to enlarge N. As a Sp_{1}-saturation we have a product of two commutative spaces (N_{i} ⋋ K_{i})/K_{i}, where 𝔫_{1} = ℍ^{n} ⊕ ℍ_{0}, K_{1} = Sp_{n} × Sp_{1}; 𝔫_{2} = ℍ^{m} ⊕ ℍ_{0}, K_{2} = Sp_{m} × Sp_{1}.

The procedure that is inverse to Sp_{1}-saturation can have steps of three different types. First, one simple factor Sp_{1} of K is replaced by U_{1}; second, two of three simple factors Sp_{1} of K are replaced by the diagonal of their product; third, several simple factors Sp_{1} of L are replaced by the diagonal of their product, K is replaced by the intersection of the former K and new L and probably N is decreased.

According to Lemma 9, if (L, K) contains any of the “SU_{2}” pairs condition (2) of Definition 6 is not satisfied. So an Sp_{1}-saturated maximal principal commutative homogeneous space is a product of the spaces from Table 2b, spaces of the type ((ℝ^{n} ⋋ SO_{n}) × SO_{n})/SO_{n}, ((ℂ^{n} ⋋ U_{n}) × SU_{n})/U_{n}, ((H_{n} ⋋ U_{n}) × SU_{n})/U_{n}, spherical homogeneous space of a reductive Lie group and a space of Heisenberg type. Below we prove a more precise statement.

6. Commutative homogeneous spaces of Heisenberg type

In this section we consider homogeneous spaces of the type (K ⋌ N)/K. In this case S(𝔤/𝔨)^{K} = S(𝔫)^{K}. We assume that 𝔫 is not commutative.

Let us decompose 𝔫/𝔫′ into a sum of irreducible K-invariant subspaces, namely 𝔫/𝔫′ = 𝔴_{1} ⊕ ... ⊕ 𝔴_{p}. As was proved in [21], if X is commutative then [𝔴_{i}, 𝔴_{j}] = 0 for i ≠ j, also [𝔴_{i}, 𝔴_{i}] = 0 if there is j ≠ i such that 𝔴_{i} ≅ 𝔴_{j} as a K-module. Denote by 𝔫_{i} := 𝔴_{i} ⊕ [𝔴_{i}, 𝔴_{i}] the subalgebra generated by 𝔴_{i}. Let 𝔳^{i} stand for a K-invariant complement of 𝔫_{i} in 𝔫 and set K^{i} := K_{∗}(𝔳^{i}).

Theorem 5. ([25, Theorem 1]) In the above notation, G/K is commutative if and only if each Poisson algebra S(𝔫_{i})^{K^{i}} is commutative.

Note that the irreducibility of 𝔴_{i} is not important here. The statement of the theorem remains true for any K-invariant subspace 𝔴 ⊂ 𝔫/𝔫′.

For convenience of the reader we present here the classification results of [21] and [22]. All maximal commutative homogeneous spaces of Heisenberg type with 𝔫/𝔫′ being an irreducible K-module and dim 𝔫′ > 1 are listed in Table 3. The following notation is used:

𝔫 = 𝔴 ⊕ 𝔷, where 𝔷 = 𝔫′ is the centre of 𝔫;

ℍ_{0} is the space of purely imaginary quaternions;

ℂ^{m} ⊗ ℍ^{n} is the tensor product over ℂ;

ℍ^{m} ⊗ ℍ^{n} is the tensor product over ℍ; HΛ^{2}𝔻^{n}, where 𝔻 = ℂ or ℍ, is the skew-Hermitian square of 𝔻; HS_{0}^{2}ℍ^{n} is the space of Hermitian quaternion matrices of order n with zero trace.

For all cases in Table 3 the commutation operation 𝔴 × 𝔴 ↦ 𝔷 is uniquely determined by the condition of K-equivariance. Notation (U_{1}·)F means that K can be either F or U_{1} · F. Cases in which U_{1} is necessary are indicated in the column “U_{1}”. Some spaces are not always maximal. This is indicated in the column “max”.

Table 3.

K  𝔴  𝔷  U_{1} max
1  SO_{n}  ℝ^{n}  Λ^{2}ℝ^{n} = 𝔰𝔬_{n}
2  Spin_{7}  ℝ^{8}  ℝ^{7}
3  G_{2}  ℝ^{7}  ℝ^{7}
4  (U_{1}·)SU_{n}  ℂ^{n}  Λ^{2}ℂ^{n} ⊕ ℝ
(n even)
5 (U_{1}·)SU_{n} ℂ^{n} Λ^{2}ℂ^{n}
(n odd)
6  U_{n}  ℂ^{n}  HΛ^{2}ℂ^{n} = 𝔲_{n}
7  (U_{1}·)Sp_{n}  ℍ^{n}  HS_{0}^{2}ℍ^{n} ⊕ ℍ_{0}
8  U_{1} · Spin_{7}  ℂ^{8}  ℝ^{7} ⊕ ℝ
9  Sp_{1} × Sp_{n}  ℍ^{n}  ℍ_{0} = 𝔰𝔭_{1}  n ⩾ 2
10  Sp_{2} × Sp_{n}  ℍ^{2} ⊗ ℍ^{n} HΛ^{2}ℍ^{2} = 𝔰𝔭_{2}
11 (U_{1}·)SU_{2} × SU_{n} ℂ^{2} ⊗ ℂ^{n} HΛ^{2}ℂ^{2} = 𝔲_{2} n = 2
12 U_{2} × Sp_{n} ℂ^{2} ⊗ ℍ^{n} HΛ^{2}ℂ^{2} = 𝔲_{2}

In general case, the classification of maximal principal Sp_{1}-saturated commutative spaces of Heisenberg type is being done in the following way. If (N ⋋ K)/K is a commutative homogeneous space of non-Euclidian type, then there is a non-commutative subspace 𝔴_{1} ⊂ (𝔫/𝔫′). Denote by K_{e} the maximal connected subgroup of K acting on 𝔴_{1} effectively and by π_{e} the projection onto K_{e} in K. Recall that 𝔫_{1} := 𝔴_{1} ⊕ [𝔴_{1}, 𝔴_{1}].

If 𝔫_{1} is not a Heisenberg algebra (i.e., dim 𝔫′_{1} > 1), then the pair (K_{e}, 𝔫_{1}) is a central reduction of some pair from Table 3. If 𝔫_{1} is a Heisenberg algebra, then (K_{e}, 𝔫_{1}) corresponds to an irreducible spherical representation K_{e}(ℂ) : W, in a sense that 𝔴_{1} is a K_{e}-invariant real form of W ⊕ W^{∗}. According to Kac’s list [10], there are 11 such cases.

For any commutative homogeneous space (N_{1} ⋋ K_{e})/K_{e}, where 𝔫_{1}/𝔫′_{1} = 𝔴_{1}, we have to find out if it arise in the aforementioned way from some larger commutative homogeneous space (Ñ ⋋ K)/K, and if so, list all of them. Note that K^{1} acts on 𝔫_{1} as π_{e}(K^{1}). Due to lemma 5 and the third condition of Definition 6, π_{e}(K^{1}) is a proper subgroup of K_{e}.

As was proved in [3], homogeneous space (N ⋋ F)/F, where 𝔫 = ℝ^{n} ⊕ Λ^{2}ℝ^{n}, is not commutative for any proper subgroup F ⊂ SO_{n}. So if 𝔫_{1} corresponds to the first row of Table 3, then, by Lemma 5, 𝔫 = 𝔫_{1}. Below we consider the remaining 22 cases.

We say that the action K : 𝔫 is commutative if the corresponding homogeneous space (N ⋋ K)/K is commutative.

Lemma 11. Suppose that K′_{e} is simple and K_{e} = K′_{e} × U_{1}, [𝔴_{1}, 𝔴_{1}] ≠ 0 and 𝔴_{1} = W ⊗ ℝ^{2}, where K′_{e} acts on W and U_{1} on ℝ^{2}. Let F be a proper subgroup of K′_{e}. If the action F : W is reducible then (N_{1} ⋋ (F × U_{1}))/(F × U_{1}) is not commutative.

Proof. We show that the action H = (SO_{n} × SO_{m}) × SO_{2} : 𝔫_{1} ≅ (ℝ^{n} ⊕ ℝ^{m}) ⊗ ℝ^{2} ⊕ [𝔴_{1}, 𝔴_{1}] cannot be commutative. Assume that it is commutative and apply Theorem 5. We have H_{∗}(ℝ^{n} ⊗ ℝ^{2})^{0} = SO_{n−1} × SO_{m}. The subspace ℝ^{m} ⊗ ℝ^{2} is a sum of two isomorphic SO_{n−1} × SO_{m}-modules. Hence, ℝ^{m} ⊗ ℝ^{2} is a commutative subalgebra of 𝔫_{1}. This can happen only if [𝔴_{1}, 𝔴_{1}] = 0.

Lemma 12. Let (N ⋋ K)/K be a commutative homogeneous space from Table 3, but not from the second, third or ninth row. Suppose a subgroup F ⊂ K acts on 𝔫/𝔫′ reducibly. Then (N ⋋ F)/F is not commutative.

Proof. Assume that (N ⋋ F)/F is commutative. Then due to Proposition 15 of [21] there are at list two subspaces V_{1}, V_{2} ⊂ 𝔫/𝔫′, such that V_{1} ⊕ V_{2} = 𝔫/𝔫′ and [V_{1}, V_{2}] = 0. Evidently, this is not true in cases 1, 4, 5, 6. For the same reason, in cases 10, 11 and 12 F contains the first simple factor of K, either Sp_{2} or SU_{2}.

In the seventh case F has to be a subgroup of Sp_{m} × Sp_{n−m}. Subspaces ℍ^{m} and ℍ^{n−m} do not commute with each other. For the eighth case we apply Lemma 11.

In case 10, we have F ⊂ Sp_{2} × Sp_{m} × Sp_{n−m}, F_{∗}(ℍ^{2} ⊗ ℍ^{m}) ⊂ Sp_{1} × Sp_{1} × Sp_{n}. The subspace ℍ^{2} ⊗ ℍ^{n−m} is a sum of two isomorphic F_{∗}(ℍ^{2} ⊗ ℍ^{m})-modules. Hence, ℍ^{2} ⊗ ℍ^{n−m} is a commutative subalgebra of 𝔫. But this is not the case.

In case 11, F is a subgroup of either SU_{2} × U_{m} × U_{n−m} or SU_{2} × Sp_{m/2} for even m. The first case is just the same as the previous one. For the second case note that as a SU_{2} × Sp_{m/2}-module ℂ^{2} ⊗ ℂ^{n} ≅ ℍ^{m/2} ⊕ ℍ^{m/2}. The 12-th case is exactly the same.

Corollary (of the proof). Let (N ⋋ K)/K be a non-Euclidian central reduction of a homogeneous space from row 11 or 12 of Table 3. Then (N ⋋ F)/F is not commutative for any proper subgroup of F ⊂ K acting on 𝔫/𝔫′ reducibly.

Lemma 13. Suppose π_{e}(K^{1}) = (U_{1})^{n} and a homogeneous space (N_{1} ⋋ K^{1})/K^{1} is commutative. Then 𝔫_{1} = ℝ^{2n} ⊕ ℝ, K_{e} = U_{n}.

Proof. An irreducible representation of U_{1} on a real vector space is either trivial (ℝ) or ℝ^{2}. If 𝔴_{1} is the sum of more than n K^{1}-invariant summands, then two of them are isomorphic and there is a non-zero η ∈ 𝔴_{1} such that [η, 𝔴_{1}] = 0. But K_{e}η = 𝔴_{1} ⊂ 𝔷(𝔫_{1}). By the same reason 𝔴_{1}^{(U_{1})^{n}} = 0. Because the action (U_{1})^{n} : 𝔴_{1} is locally effective, 𝔴_{1} = ℝ^{2n}. We have Λ^{2}ℝ^{2} = ℝ, hence K^{1} acts on 𝔫′_{1} trivially. Each element of K_{e} is contained in some maximal torus, that is up to conjugation in π_{e}(K^{1}). Hence K_{e} acts on 𝔫′_{1} trivially and K_{e} ⊂ U_{n}.

The group U_{n} has no proper subgroups of rank n acting on ℝ^{2n} irreducibly. Thus we have K_{e} = U_{n}, 𝔫′_{1} = (Λ^{2}ℝ^{n})^{U_{n}} ≅ ℝ.

From now on, let (N ⋋ K)/K be an indecomposable maximal Sp_{1}-saturated principal commutative space with 𝔫_{1} ≠ 𝔫. In particular, the connected centre Z(K_{e}) of K_{e} acts on 𝔳^{1} trivially. Let 𝔞 ⊂ 𝔫 be a K-invariant subalgebra. Clearly, if the action K : 𝔫 is commutative, then K : 𝔞 is also commutative. We assume that K : 𝔫 is not a “subaction” of some large commutative action.

Assume for the time being that K′_{e} is simple and denote it by K_{1}. Decompose 𝔫 into a sum of K-invariant subspaces 𝔫 = 𝔫_{1} ⊕V_{2} ⊗_{𝔻_{2}}V^{2} ⊕...⊕V_{q} ⊗_{𝔻_{q}} V^{q} ⊕V_{tr}, where V_{i} are pairwise non-isomorphic irreducible non-trivial K_{1}-modules, V_{tr} and V^{i} are trivial K_{1}-modules and all the other simple normal subgroups of K act on V_{i} trivially. First we have to describe possible V_{i}, then dimensions of V^{i}, afterwards the actions K : ⨁_{i=2}^{q} V_{i} ⊗ V^{i} and K : V_{tr}. Once

again we use Élashvili’s classification [8]. Lemma 13 tells us that the adjoint representation can be among V_{i} only for (K_{e}, 𝔫_{1}) = (U_{n}, ℂ^{n} ⊕ ℝ). Note that if V_{tr} is not contained in 𝔫′, then there is a simple factor of K acting non-trivially both on ⨁_{i=2}^{q} V_{i} ⊗ V^{i} and V_{tr}.

Recall that π_{e}(K^{1}) is contained in the product Z(K_{e})·N(ξ), where N(ξ) := {k ∈ K_{1}|kξ ∈ 𝔻_{i}ξ} is the “normaliser” of a generic vector ξ ∈ V_{i}. Suppose (K_{1})_{∗}(V_{i}) is trivial or finite. Then π_{e}(K^{1}) is finite for 𝔻_{i} = ℝ and commutative for 𝔻_{i} = ℂ. In the following lemma we prove that for 𝔻_{i} = ℍ the projection π_{e}(K^{1}) is also commutative.

Lemma 14. Let F ⊂ Sp_{n}, n ≥ 2 and (F_{∗}(ℍ^{n}))^{0} = E. Then the image of the generic stabiliser (F × Sp_{m})_{∗}(ℍ^{n} ⊗ ℍ^{m}) under the projection on F is commutative.

Proof. Assume that this is not the case, i.e., the image contains Sp_{1}. Then this is true not only for generic points, but for all of them, in particular, for decomposable vectors. In other words, the restriction F |_{ξℍ} for ξ ≠ 0, ξ ∈ ℍ^{n} contains Sp_{1}. Due to Lemma 5, we have F = Sp_{n}. But then F_{∗}(ℍ^{n}) = Sp_{n−1}.

Example 8. Let (K_{e}, 𝔫_{1}) be a pair from the second row of Table 3. We show that 𝔫 ⊂ 𝔫_{1} ⊕ ℝ^{7} ⊗ ℝ^{2}. All representations of Spin_{7} are orthogonal, so here all 𝔻_{i} equal ℝ. The group Spin_{7} has only three irreducible representations with infinite generic stabiliser, namely 𝔰𝔬_{7}, ℝ^{7} and ℝ^{8}. If V_{i} = ℝ^{8} for some i, then K^{1} has a non-zero invariant in 𝔴_{1}, which commute with 𝔴_{1}. This is a contradiction. Thus 𝔫 = 𝔫_{1} ⊕ ℝ^{7} ⊗ V^{2} ⊕ V_{tr}. If dim V^{2} ≥ 3, then π_{e}(K^{1}) ⊂ SU_{2} ×SU_{2}. But the action SU_{2} ×SU_{2} : (ℂ^{2} ⊕ℂ^{2})⊕ℝ^{7} is not commutative. In case dim V^{2} = 2 we have π_{e}(K^{1}) = Spin_{5} = Sp_{2}. The pair (Sp_{2}, ℍ^{2} ⊕ ℝ^{7}) is a central reduction of the pair from the 7-th row of Table 3 with n = 2 (K = Sp_{2}) by a subgroup corresponding to ℍ_{0} (here HS_{0}^{2}ℍ^{2} ≅ ℝ^{7} as a K-module).

We have seen that dim V^{2} ≤ 2, so no simple normal subgroup of K except Spin_{7} acts non-trivially on V_{2} ⊗ V^{2}. Assume that there is a non-commutative subspace 𝔴_{2} ⊂ 𝔫. It is either ℝ^{7} or ℝ^{7} ⊗ ℝ^{2}. The first case is not possible, because Λ^{2}ℝ^{7} ≅ 𝔰𝔬_{7} as a Spin_{7}-module. In the second case we apply Theorem 5 to 𝔴_{2} = ℝ^{7} ⊗ ℝ^{2}. Recall that, K = Spin_{7} or K = Spin_{7} ×SO_{2}, (Spin_{7})_{∗}(ℝ^{8} ⊕ℝ^{7}) ⊂ Spin_{6}. By Lemma 11, the action SO_{6} ×SO_{2} : 𝔴_{2} ⊕[𝔴_{2}, 𝔴_{2}] is commutative only if [𝔴_{2}, 𝔴_{2}] = 0. The corresponding commutative space is indicated in the 13-th row of Table 4.

Other cases are done in almost the same way. There are 7 cases in which K_{e} has two simple factors, 4 last rows in Table 3 and 3 irreducible spherical representations: (ℂ^{∗}·)SL_{m} × SL_{n} : ℂ^{m}⊗ℂ^{n}, (ℂ^{∗}·)SL_{n} × Sp_{4} : ℂ^{n}⊗ℂ^{4}, GL_{3} × Sp_{n} : ℂ^{3}⊗ℂ^{2n}. We have to carry the same procedure for both simple normal subgroups of K. Note that, if (K_{e}, 𝔫_{1}) is the pair from the 9-th row of Table 3, then due to the third condition of Definition 6, the normal subgroup Sp_{1} of K_{e} acts on 𝔳^{1} trivially.

For convenience of the reader, we list here all irreducible representations of 𝔰𝔲_{n} with nontrivial generic stabiliser. They are described by the highest weights of the complexifications V (ℂ) with respect to 𝔰𝔩_{n}.

Table A_{n−1}.
n representation (SU_{n})_{∗}(V)^{0}
R(ϖ_{1}) ⊕ R(ϖ_{1})^{∗}  SU_{n−1}
R(ϖ_{2}) ⊕ R(ϖ_{2})^{∗} (SU_{2})^{[n/2]}
R(2ϖ_{1}) ⊕ R(2ϖ_{1})^{∗} (U_{1})^{[n/2]}
R(ϖ_{1} + ϖ_{1}^{∗})  (U_{1})^{n−1}
4  R(ϖ_{2})  Sp_{2}
6  2R(ϖ_{3})  (U_{1})^{2}

Note that the action (SU_{n})_{∗}(V) : ℂ^{n} is irreducible only in one case, namely n = 4, V = R(φ_{2}) ≅ ℝ^{6}.

For almost all pairs from Table 3, the action K^{1} : 𝔴_{1} have to be irreducible. This leaves only a few possibilities for V_{i}. More precisely, there are seven spaces (N ⋋ K)/K with K_{e} ≠ K and non-trivial action K : 𝔫/𝔫′ (rows 2, 4, 5, 10, 11, 12 and 13 of Table 4).

If 𝔫′ is a trivial K-module, the calculations are even simpler. However, we have more such cases. Assume that 𝔫_{1} is not commutative. Recall that 𝔴_{1}(ℂ) = W_{1} ⊕ W_{1}^{∗} as a K-module. We have to check whether the action π_{e}(K^{1}) : W_{1} is spherical or not. We will consider one example in full details.

Lemma 15. Let (N ⋋K)/K, where K = SU_{2} ×F be a principal Sp_{1}-saturated commutative space. Suppose that the image of the projection of K_{∗}(𝔫) on SU_{2} contains U_{1}. Then there are three possibilities: K = SU_{2}, 𝔫 = ℝ^{3}; F = (S)U_{s} or F = (U_{1}·)Sp_{s/2}, 𝔫 = ℂ^{2} ⊗ ℂ^{s} ⊕ ℝ; F = (S)U_{4}, 𝔫 = (ℂ^{2} ⊗ ℂ^{4} ⊕ ℝ) ⊕ ℝ^{6}.

Proof. Note that the intersection of two distinct general subgroups U_{1} ⊂ SU_{2} is finite. Hence, 𝔫 contains only one irreducible K-invariant subspace on which SU_{2} acts non-trivially. The Lie algebra 𝔰𝔲_{2} has only two non-trivial representations with a non-trivial generic normaliser, namely ℝ^{3} and ℂ^{2}. The first one is orthogonal and can not be contained in 𝔫 in the form ℝ^{3} ⊗ ℝ^{s} for s > 1. Assume that ℂ^{2} ⊗ ℂ^{s} ⊂ 𝔫. The group F acts on a generic subspace ℂ^{2} ⊂ ℂ^{s} as U_{1} or as SU_{2}. It can be easily verified that the first case is not possible. In the second one F and U_{m} have the same orbits in ℂ^{m}, i.e., U_{m} = F U_{m−1}. According to the classification [16] , there are only 4 listed in the lemma possibilities for F. Suppose F acts (non-trivially) on some other K-invariant subspace V ⊂ 𝔫. Then ℂ^{s} is an irreducible F_{∗}(V)-module. This can only happen for s = 4, F = (S)U_{4} and V = ℝ^{6}.

Example 9. We describe all possible principal Sp_{1}-saturated maximal commutative pairs (K, 𝔫) for (K_{e}, 𝔫_{1}) = ((U_{1}·)SU_{n}, ℂ^{n} ⊕ ℝ).

First assume that n = 2. The action U_{1} · U_{1} : ℂ^{2} ⊕ ℝ is commutative. Hence we can "replace" SU_{2} by U_{1}. This yields the three possibilities from Lemma 15.

Suppose n > 2. From Table A_{n−1} we know that 𝔰𝔲_{n} has the following irreducible representations with non-trivial generic stabiliser: 𝔰𝔲_{n}, ℂ^{n}, Λ^{2}ℂ^{n} for n > 4 and ℝ^{6} for n = 4. All of them can occur in 𝔳^{1} as V_{i}. The first one is orthogonal, so for s > 1 the space 𝔰𝔲_{n} ⊗ ℝ^{s} could not be a subspace of 𝔳^{1}. Moreover, if V_{2} = 𝔰𝔲_{n}, then 𝔳^{1} = 𝔰𝔲_{n}, because a rank of any proper subalgebra 𝔣 ⊂ 𝔰𝔲_{n} is smaller then n − 1. The action U_{n} : 𝔫_{1} ⊕ 𝔰𝔲_{𝔫} is commutative. In the following we assume that 𝔰𝔲_{n} is not contained in 𝔳^{1}.

Consider the case n = 4. We have 𝔳^{1} = ℝ^{6}⊗ℝ^{s}⊕ℂ^{4}⊗V^{3}⊕V_{tr}. Note that (SU_{4})_{∗}(ℝ^{6}⊗ℝ^{3}) is finite, so s ≤ 2. Also (U_{1} · U_{4})_{∗}(ℝ^{6} ⊗ ℝ^{2} ⊕ ℂ^{4}) = (U_{1})^{3} and ℂ^{4} is not a spherical representation of (ℂ^{∗})^{3}. Hence, for s = 2 we have (K, 𝔫) = ((S)U_{4}(·SO_{2}), ℂ^{4} ⊕ ℝ ⊕ ℝ^{6} ⊗ ℝ^{2}). Another possible pair is (U_{4} · U_{1}, (ℂ^{4} ⊕ ℝ) ⊕ ℝ^{6} ⊕ (ℂ^{4} ⊕ ℝ)). The rest of this case is the same for general n and is dealt upon below.

Note that (SU_{n})_{∗}(Λ^{2}ℂ^{n} ⊕ Λ^{2}ℂ^{n}) = U_{1} and (SU_{n})_{∗}(Λ^{2}ℂ^{n} ⊕ ℂ^{n}) = {E}. Thus we have either 𝔫 = 𝔫_{1} ⊕ (Λ^{2}ℂ^{n} ⊕ ℝ) or 𝔫 = 𝔫_{1} ⊕ ℂ^{n} ⊗_{𝔻_{2}} V^{2} ⊕ V_{tr}. Here 𝔻_{2} equals ℂ or ℝ. If 𝔻_{2} = ℝ and dim V^{2} > 1, then π_{e}(K^{1}) is contained in U_{1} · U_{n−2} ⊂ U_{2} · U_{n−2}. Evidently, the U_{1} · U_{n−2}-module ℂ^{n} is not spherical. Hence, 𝔻_{2} = ℂ.

Similar to the proof of Lemma 15 we obtain that K_{∗}(V_{tr})/SU_{n} acts on V^{2} = ℂ^{s} as (S)U_{s} or Sp_{s/2} for even s. In the second case (for s ≥ 4) we have π_{e}(K^{1}) ⊂ (U_{1})^{2} · SU_{n−3}, but the action (ℂ^{∗})^{2} : ℂ^{3} is not spherical. Thus V_{tr} = ℝ is a trivial K-module and 𝔫 ⊂ 𝔫_{1}⊕ℂ^{2}⊗ℂ^{s}⊕ℝ.

In Table 4 the algebra 𝔫 is given in the following way. Each subspace in parentheses represent a subalgebra 𝔴_{i} ⊕[𝔴_{i}, 𝔴_{i}]. The spaces given outside parentheses are commutative. The action of K is uniquely determined by irreducibility and by Table 3.

Notation (SU_{n}, U_{n}, U_{1} · Sp_{n/2}) means that this normal subgroup of K can be equal either of these three groups. Symbol Sp_{n/2} has sense only for even n. For n odd, the group Sp_{n/2} does not exist, so it cannot be a subgroup of K.

Table 4.

K  𝔫max
1  U_{n}  (ℂ^{n} ⊕ ℝ) ⊕ 𝔰𝔲_{n}
2  U_{4}  (ℂ^{4} ⊕ Λ^{2}ℂ^{4} ⊕ ℝ) ⊕ ℝ^{6}
3  U_{1} · U_{n}  (ℂ^{n} ⊕ ℝ) ⊕ (Λ^{2}ℂ^{n} ⊕ ℝ)
4  SU_{4}  (ℂ^{4} ⊕ HS_{0}^{2}ℍ^{2} ⊕ ℝ) ⊕ ℝ^{6}
5  U_{2} · U_{4}  (ℂ^{2} ⊗ ℂ^{4} ⊕ HΛ^{2}ℂ^{2}) ⊕ ℝ^{6}
6  SU_{4} · U_{m}  (ℂ^{4} ⊗ ℂ^{m} ⊕ ℝ) ⊕ ℝ^{6}
7  U_{m} · U_{n}  (ℂ^{m} ⊗ ℂ^{n} ⊕ ℝ) ⊕ (ℂ^{m} ⊕ ℝ)
8  U_{m} · SU_{2} · U_{p}  (ℂ^{m} ⊗ ℂ^{2} ⊕ ℝ) ⊕ (ℂ^{2} ⊗ ℂ^{p} ⊕ ℝ)
9  U_{1} · U_{1} · Sp_{n}  (ℍ^{n} ⊕ ℝ) ⊕ (ℍ^{n} ⊕ ℝ)
10  Sp_{n} · Sp_{1} · (Sp_{1}, U_{1}, {E})  (ℍ^{n} ⊕ ℍ_{0}) ⊕ ℍ^{1} ⊗ ℍ^{n}
11  Sp_{n} · (Sp_{1}, U_{1}) · Sp_{m}  (ℍ^{n} ⊕ ℍ_{0}) ⊕ ℍ^{n} ⊗ ℍ^{m}
12  Sp_{n} · (Sp_{1}, U_{1}, {E})  (ℍ^{n} ⊕ ℍ_{0}) ⊕ HS_{0}^{2}ℍ^{n}
13  Spin_{7} · (SO_{2}, {E})  (ℝ^{8} ⊕ ℝ^{7}) ⊕ ℝ^{7} ⊗ ℝ^{2}
14  U_{1} · Spin_{7}  (ℂ^{7} ⊕ ℝ) ⊕ ℝ^{8}
15  U_{1} · U_{1} · Spin_{8}  (ℂ_{+}^{8} ⊕ ℝ) ⊕ (ℂ_{−}^{8} ⊕ ℝ)
16  U_{1} · Spin_{10}  (ℂ^{16} ⊕ ℝ) ⊕ ℝ^{10}
17  (SU_{n}, U_{n}, U_{1} · Sp_{n/2}) · SU_{2}  (ℂ^{n} ⊗ ℂ^{2} ⊕ ℝ) ⊕ 𝔰𝔲_{2}
18  (SU_{n}, U_{n}, U_{1} · Sp_{n/2}) · U_{2}  (ℂ^{n} ⊗ ℂ^{2} ⊕ ℝ) ⊕ (ℂ^{2} ⊕ ℝ)
19 (SU_{n}, U_{n}, U_{1}·Sp_{n/2})·SU_{2}·(SU_{n}, U_{n}, U_{1}·Sp_{n/2}) (ℂ^{n} ⊗ ℂ^{2} ⊕ ℝ) ⊕ (ℂ^{2} ⊗ ℂ^{n} ⊕ ℝ)
20  (SU_{n}, U_{n}, U_{1} · Sp_{n/2}) · SU_{2} · U_{4}  (ℂ^{n}⊗ℂ^{2}⊕ℝ)⊕(ℂ^{2}⊗ℂ^{4}⊕ℝ)⊕ℝ^{6}
21  U_{4} · U_{2}  ℝ^{6} ⊕ (ℂ^{4} ⊗ ℂ^{2} ⊕ ℝ) ⊕ 𝔰𝔲_{2}
22  U_{4} · U_{2} · U_{4}  ℝ^{6}⊕(ℂ^{4}⊗ℂ^{2}⊕ℝ)⊕(ℂ^{2}⊗ℂ^{4}⊕ℝ)⊕ℝ^{6}
23  U_{1} · U_{1} · SU_{4}  (ℂ^{4} ⊕ ℝ) ⊕ (ℂ^{4} ⊕ ℝ) ⊕ ℝ^{6}
24  (U_{1}·)SU_{4}(·SO_{2})  (ℂ^{4} ⊕ ℝ) ⊕ ℝ^{6} ⊗ ℝ^{2}

Theorem 6. All indecomposable Sp_{1}-saturated maximal principal commutative homogeneous spaces (N ⋋ K)/K with non-commutative 𝔫 and reducible 𝔫/𝔫′ are given in Table 4 in a sense that 𝔫 is a K-invariant subalgebra of 𝔫max .

Proof. We have seen above how one can prove that all such commutative spaces are contained in Table 4. Now we prove that all listed spaces are commutative. We do it using Theorem 5 and the list of the spherical representations given in [12]. It is proved in [3], [4] and [14] that spaces listed in rows 3, 7, 8, 9, 15, 18 and 19 are commutative.

Suppose 𝔫 contains a commutative K-invariant ideal ℝ^{6}. According to Theorem 5, K : 𝔫 is commutative if and only if K_{∗}(ℝ^{6}) : 𝔫/ℝ^{6} is commutative. For second, 4-th and 5-th rows of Table 4 pairs K_{∗}(ℝ^{6}) : 𝔫/ℝ^{6} are contained in Table 3, hence, commutative. For 6-th, 20-th and 23-d rows of Table 4 pairs K_{∗}(ℝ^{6}) : 𝔫/ℝ^{6} correspond to spherical representations listed in [4], [14]. Analogously, for the 22-d row pair K_{∗}(ℝ^{6} ⊕ ℝ^{6}) : 𝔫/(ℝ^{6} ⊕ ℝ^{6}) corresponds to a spherical representation.

Let (N ⋋ L)/K be a commutative homogeneous space. Then the action K : 𝔫 ⊕ (𝔩/𝔨), where 𝔩/𝔨 is a commutative ideal, is commutative. The pairs from the first and 12-th rows of Table 4 are obtained in such a way from commutative spaces ((H_{n} ⋋ U_{n}) × SU_{n})/U_{n} and (H_{2n} ⋋ U_{2n})/Sp_{n}. In case [𝔫, 𝔫] = 0, we obtain a commutative homogeneous space of Euclidian type. Thus essentially these are the only non-trivial examples given by this construction.

In the remaining eight cases we use Theorem 5. For instance, take the 11-th row with K = Sp_{n} × Sp_{m}. Here 𝔫 contains only one non-commutative subspace 𝔴_{1} ≅ ℍ^{n}. Set d = |n−m| and s = min(n, m). Then K^{1} = K_{∗}(ℍ^{n} ×ℍ^{m}) = (Sp_{1})^{d} ×Sp_{s}. Anyway K^{1}/(K^{1} ∩P_{1}) contains (Sp_{1})^{n}. To conclude note that the action Sp_{1} : (ℍ ⊕ ℍ_{0}) is commutative according to Table 3.

7. Conclusion

Theorem 7. Any maximal indecomposable principal Sp_{1}-saturated commutative homogeneous space belongs to the one of the following four classes:

1) affine spherical homogeneous spaces of reductive real Lie groups;

2) spaces corresponding to the rows of Table 2b;

3) homogeneous space ((ℝ^{n}⋋SO_{n})×SO_{n})/SO_{n}, ((H_{n}⋋U_{n})×SU_{n})/U_{n}, where the normal subgroups SO_{n} and SU_{n} of K are diagonally embedded into SO_{n} × SO_{n} and SU_{n} × SU_{n}, respectively;

4) commutative homogeneous spaces of Heisenberg type.

Proof. Let X = G/K be a commutative homogeneous space. If G is reductive, X belongs to the first class. If L = K then it is a space of Heisenberg type.

Assume that G is not reductive and L ≠ K. Suppose a simple factor K_{1} of K has non-trivial projections onto both P and L^{⋄}. Then due to condition (1) of the definition of Sp_{1}-saturated commutative spaces, K_{1} ≠ SU_{2}. By theorem 4, X belongs to the 3-d class. If all simple factors of K are contained in either P or L^{⋄}, then, because X is principal, P^{0}/(P^{0} ∩ K) is a factor of X. But X is indecomposable and G is not reductive, so P^{0} is trivial. Thus, X satisfies condition (∗). If there is a simple factor L_{1} of L such that π_{1}(L_{∗}) ≠ K and L_{1} ⫋ K, then, according to Theorem 3, X is contained in the second class.

If there is not any such factor, then also by Theorem 3, (L, K) is a product of pairs of the type (SU_{2} × SU_{2} × SU_{2}, SU_{2}), (SU_{2} × SU_{2}, SU_{2}) or (SU_{2}, U_{1}) and a pair (K^{1}, K^{1}), where K^{1} is a compact Lie group. But these pairs (except (K^{1}, K^{1})) are not allowed in Sp_{1}-saturated commutative space. The second condition of the definition of Sp_{1}-saturated commutative space contradicts the conditions of Lemma 8. Thus, L would be equal K, but this is not the case.

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