A NECESSARY CONDITION FOR THE EXISTENCE OF

COMPACT CLIFFORD–KLEIN FORMS OF

HOMOGENEOUS SPACES OF REDUCTIVE TYPE

Toshiyuki KOBAYASHI*

Institute for Advanced Study, School of Mathematics

Princeton, NJ 08540, USA

toshi@math.ias.edu

and

Department of Mathematics, University of Tokyo

Komaba, Meguro, Tokyo, 153, Japan

toshi@tansei.cc.u-tokyo.ac.jp

Abstract. A compact Clifford-Klein form of a homogeneous space G/H is a compact quotient Γ\G/H, where Γ is a subgroup of G acting properly discontinuously and freely on G/H. In this paper we find a new necessary condition for the existence of compact Clifford-Klein forms of a homogeneous space of reductive type in terms of Lie algebras. As an application, we prove that a semisimple orbit with a compact Clifford-Klein form carries a G-invariant complex structure and that a semisimple symmetric space with a compact Clifford-Klein form has a basic associated symmetric pair among its H_{ε} family in the sense of Oshima-Sekiguchi.

Key words and phrases. Clifford-Klein form, uniform lattice, reductive Lie group, homogeneous space, semisimple symmetric space. *Partially supported by the Inamori foundation and DMS-9100383.

Typeset by AMS-TEX

§1. Introduction and statement of results

A compact Clifford-Klein form of a homogeneous space G/H is the quotient Γ\G/H, if Γ is a subgroup of G acting properly discontinuously and freely on G/H so that Γ\G/H is compact in the quotient topology. Our concern here is with the following:

Problem 1.1. Find a criterion for the existence for of compact Clifford-Klein forms of a homogeneous space G/H.

In this paper we treat the case where G/H is of reductive type (see §2 for definition). Semisimple orbits in 𝔤 = Lie(G) and semisimple symmetric spaces are typical examples. A distinguishing feature in our setting is that H is noncompact, and consequently the action of a discrete subgroup of G on G/H is not automatically properly discontinuous. So far, the following facts on discontinuous groups have been known:

Fact 1.2 (see §2 for notation). Let G/H be a homogeneous space of reductive type.

1) (Calabi-Markus phenomenon; [C-M; Wo1; Wo2; Ku; Ko1]) Only finite subgroups of G can act properly discontinuously on G/H iff ℝ-rank G = ℝ-rank H.

2) (A necessary condition; [Ko1] Proposition 4.10) If rank G = rank H and if c-rank G > c-rank H, then G/H does not have a compact Clifford-Klein form.

3) (A sufficient condition; [Bo-HC;Bo;Ku;Ko1]) If there exists a subgroup G′ reductive in G such that 𝔞(G′) ∩ W_{G} · 𝔞(H) = {0} and d(G′) + d(H) = d(G), then G/H has a compact Clifford-Klein form.

An existence result of compact Clifford-Klein forms (or noncompact ones of finite volume) of Riemannian symmetric spaces [Bo-HC; Bo] has opened a theory of Eisenstein series in harmonic analysis on square integrable functions over the double coset space Γ\G/H, a geometric construction of (Harish-Chandra’s) discrete series in [A-S], a construction of non-zero harmonic forms related to discrete series for semisimple homogeneous spaces in [T-W], and so on. It is natural to expect that an existence result for pseudo-Riemannian homogeneous spaces could open a theory on harmonic analysis on such nice double coset spaces.

Conversely, a non-existence result of compact Clifford-Klein forms is also interesting. In fact, in a special case where G is a semisimple Lie group with ℝ-rank G ≥ 2 and H is a semisimple part of a Levi part of a maximal parabolic subgroup of G, a non-existence result would simplify greatly a proof of a recent result on Anosov flows in [B-F-L], as Prof. Benoist informed the author.

Here are our results on a necessary condition for the existence of compact Clifford-Klein forms:

Theorem 1.3 (see §3). If a semisimple orbit Λ (⊂ 𝔤) has a compact Clifford-Klein form, then Λ carries an invariant complex structure.

In fact Λ is isomorphic to an elliptic orbit (Theorem (1.3)′ in §3). In particular, via a G-invariant non-degenerate bilinear form on 𝔤, Λ carries a G-invariant (pseudo- )Kähler structure which is compatible with the symplectic structure coming from the identification with a coadjoint orbit. It should be in sharp contrast that a hyperbolic orbit does not admit an infinite discontinuous group (Calabi-Markus phenomenon, Fact (1.2)(1)).

Theorem 1.4 (see §4). If a semisimple symmetric space G/H has a compact Clifford-Klein form, then the associated symmetric pair (𝔤, 𝔥^{a}) is basic in the H_{ε}-family through it in the sense of Oshima-Sekiguchi [O-S2].

These results are deduced from the following general theorem (Theorem (1.5)). Here 𝔞(G′) and 𝔞(H) denote subspaces of a maximally split abelian subspace 𝔞_{G} for G, which are uniquely defined up to the conjugacy of the Weyl group W_{G}. The numbers d(G′) and d(H) denote the dimensions of the Riemannian symmetric spaces associated to G′ and H respectively. See the first part of §2 for precise definitions.

Theorem 1.5. Let G/H be a homogeneous space of reductive type. If G/H has a compact Clifford-Klein form, then there does not exist a closed subgroup G′ reductive in G satisfying the following two conditions:

𝔞(G′) ⊂ 𝔞(H) in 𝔞_{G}/ ∼ W_{G}, (1.5)(a) d(G′) > d(H). (1.5)(b)

(We mean by (1.5)(a) that 𝔞(G′) is contained in W_{G} · 𝔞(H), the W_{G} orbit of 𝔞(H).)

The novelty in Theorem (1.5) is that it is applicable without any rank assumption. (cf. Fact (1.2)(1) and (1.2)(2)). The following examples reveal the applications and limitations of our results.

Example 1.6. The simplest choice G′ := G in Theorem (1.5) yields that a noncompact homogeneous space G/H of reductive type admits a compact Clifford-Klein form only if ℝ-rank G > ℝ-rank H. Alternatively, this follows from Fact (1.2)(1).

Example 1.7. For the existence of a compact Clifford-Klein form of an indefinite Grassmann manifold G/H = SO(i + j, k + l)/SO(i, k) × SO(j, l) (i ≤ j, k, l), it is necessary that i = jkl = 0 or 0 = i < l ≤ j − k from Theorem (1.5). It is also necessary from Fact (1.2)(2) that a parity condition jkl ≡ 0 mod 2, and conversely it is sufficient from Fact (1.2)(3) that i = l = 0 or (i, j, k, l) = (0, 2n, 1, 1), (0, 4n, 1, 3) or (0, 4, 2, 1).

Example 1.8. G/H = SO^{∗}(2n)/U(l, n − l) does not admit a compact Clifford-Klein form if 3l ≤ 2n ≤ 6l and if n ≥ 3. It admits compact Clifford-Klein forms if (n, l) = (4, 1), (4, 3), (2, 1), l = 0 or l = n.

Example 1.9. Suppose that G_{ℂ}/H_{ℂ} is a complex irreducible semisimple symmetric space. Then G_{ℂ}/H_{ℂ} does not have compact Clifford-Klein forms unless G_{ℂ}/H_{ℂ} is locally isomorphic to either a group manifold or one of SO(2n + 2, ℂ)/SO(2n + 1, ℂ), SL(2n, ℂ)/Sp(n, ℂ) (n ≥ 2) and E_{6,ℂ}/F_{4,ℂ}. We are left with the latter three cases.

In §2 we explain basic notations and give a proof of Theorem (1.5), which is direct from the estimate of cohomological dimensions in [Ko1]. For an actual application, it is important to find G′ in Theorem (1.5). In §3 (resp. §4) we do this systematically in the case of semisimple orbits (resp. semisimple symmetric spaces), and in §5 wo do this in the spirit of case by case so that we cover Examples from (1.6) to (1.9). The reader can go into any of §3, 4 and 5 after the main theorem in §2 according to his (her) interest.

§2. Preliminaries and proof of Theorem (1.5)

Let us fix notation. If G is a Lie group, then we denote by 𝔤 the Lie algebra of G and by 𝔤_{ℂ} the complexification of 𝔤. Analogous notation will be applied to Lie groups denoted by Roman uppercase letters. Suppose that G is a connected real reductive linear group and that H is a closed subgroup of G with finitely many connected components. If there exists a Cartan involution θ of G such that θH = H, then H is called reductive in G and G/H is called a homogeneous space of reductive type. If a homogeneous space G/H is of reductive type, then G/H carries a (pseudo-)Riemannian metric via a G-invariant non-degenerate bilinear form on 𝔤. Semisimple symmetric space (§4) and semisimple orbits in 𝔤 under the adjoint action (§3) are typical and important examples of homogeneous spaces of reductive type.

Suppose that G/H is a homogeneous space of reductive type. Fix a Cartan involution θ of G such that θH = H. We put K = {g ∈ G : θg = g}, a maximal compact subgroup of G. Let 𝔤 = 𝔨 + 𝔭 be a Cartan decomposition corresponding to θ. Fix a maximally abelian subspace 𝔞_{G} ⊂ 𝔭. 𝔞_{G} is called a maximally split abelian subspace for G. We write W_{G} for the Weyl group associated to the restricted root system Σ(𝔤, 𝔞_{G}), and we set ℝ-rank G := dim 𝔞_{G} (≤ rank G ≥) c-rank G := rank K, and d(G) := dim G/K = dim 𝔭. If H is reductive in G, then H is also a reductive Lie group. We take a maximally split abelian subspace 𝔞_{H} (⊂ 𝔭) for H. Then there exists an element g of G such that Ad(g)𝔞_{H} ⊂ 𝔞_{G}. Put 𝔞(H) := Ad(g)𝔞_{H} (⊂ 𝔞_{G}), which is uniquely defined up to the conjugacy of W_{G}. This notation is convenient when we compare different reductive subgroups of G at the same time.

Let Γ be an abstract group, and ℝ[Γ] the corresponding group ring over a real field ℝ. We recall that the cohomological dimension of Γ over ℝ denoted by cd_{ℝ}(Γ) is the projective dimension of ℝ as a left ℝ[Γ]-module. Equivalently,

cd_{ℝ}(Γ) = sup {n ∈ ℕ : H^{n}(Γ; A) ≠ 0 for some left ℝ[Γ]-module A} .

See [Ser] or [Bi] for general results on cohomological dimensions. In this paper we need:

Fact 2.1 ([Ko1], Corollary 5.5). Suppose that a discrete subgroup Γ of G acts properly discontinuously on a homogeneous space G/H of reductive type.

1) If Γ is virtually torsionless, then we have cd_{ℝ} Γ ≤ d(G) − d(H).

2) If Γ\G/H is compact, then Γ is virtually torsionless and we have cd_{ℝ} Γ = d(G) − d(H).

Lemma 2.2. Suppose that H and G′ are subgroups reductive in a connected real reductive linear group G satisfying (1.5)(a). If a discrete subgroup Γ acts properly discontinuously on G/H, then it also acts properly discontinuously on G/G′.

Proof. From the assumption (1.5)(a), we find elements g_{1}, g_{2} of G such that G′ ⊂ (g_{1}Kg_{2})H(g_{1}Kg_{2})^{−1} by a Cartan decomposition for G′ and H.

For a subset S of M on which Γ acts, we define a subset of Γ by Γ(S; M) := {γ ∈ Γ : γ · S ∩ S ≠ ∅}. For any compact set C of G, we have

Γ(CG′ mod G′; G/G′) = {γ ∈ Γ : γCG′ ∩ CG′ ≠ ∅inG/G′}
= Γ ∩ CG′C^{−1}
⊂ Γ ∩ C(g_{1}Kg_{2})H(g_{1}Kg_{2})^{−1}C^{−1}
= Γ(Cg_{1}Kg_{2} mod H; G/H).

The right side is a finite set because Cg_{1}Kg_{2} mod H is compact in G/H and because Γ acts on G/H properly discontinuously. Thus Γ(CG′ mod G′; G/G′) is finite and therefore the action of Γ on G/G′ is also properly discontinuous. □

Proof of Theorem (1.5). Suppose that there were Γ ⊂ G, acting properly discontinuously and freely on G/H such that Γ\G/H is compact. From Fact (2.1)(2), Γ is virtually torsionless and the cohomological dimension is given by

cd_{ℝ}(Γ) = d(G) − d(H).

On the other hand, since Γ acts on G/H properly discontinuously, so the condition (1.5)(a) implies that Γ acts on G/G′ also properly discontinuously by Lemma (2.2). Using Fact (2.1)(1), we have

cd_{ℝ}(Γ) ≤ d(G) − d(G′).

Thus we have d(G′) ≤ d(H), which contradicts (1.5)(b). □

§3. Semisimple orbits

Let us apply Theorem (1.5) to semisimple orbits of the adjoint action. First we recall some standard notation. Let G be a connected real linear reductive Lie group and X be an element of its Lie algebra 𝔤. X is called a semisimple element in 𝔤 if ad X: 𝔤 → 𝔤 is semisimple. X is called elliptic if X is a semisimple element and if any eigenvalue of ad X is purely imaginary. Denote the adjoint orbit through X ∈ 𝔤 by G · X := {Ad(g)X : g ∈ G} ⊂ 𝔤. Then G · X ≃ G/Z_{G}(X) as a homogeneous space of G, where Z_{G}(X) := {g ∈ G : Ad(g)X = X}. The orbit G · X is called a semisimple (resp. elliptic) orbit if X is a semisimple (resp. elliptic) element. If X is a semisimple element, then G/Z_{G}(X) is a homogeneous space of reductive type. As an application of Theorem (1.5) we have

Theorem (1.3)′. In the above setting, suppose that X is a semisimple element. If G · X ≃ G/Z_{G}(X) has a compact Clifford-Klein form, then there exists an elliptic element X_{1} ∈ 𝔤 such that Z_{G}(X) = Z_{G}(X_{1}).

Theorem (1.3)′ implies Theorem (1.3). Indeed, since a G-invariant complex structure on G/Z_{G}(X_{1}) comes from a (generalized) Borel embedding G · X_{1} ↪ G_{U} · X_{1} where G_{U} is a compact real form of a complexification G_{ℂ} of G, so the orbit G · X carries a G-invariant complex structure.

Remark 3.1. Fact (1.2)(1) implies that G itself never carries a complex Lie group structure if G · X ≃ G/Z_{G}(X) has a compact Clifford-Klein form. (Here we exclude the trivial case where G/Z_{G}(X) is compact itself.) In fact, ℝ-rank Z_{G}(X) = ℝ-rank G if G is a complex reductive Lie group and if X is a semisimple element of 𝔤.

Remark 3.2. The conclusion of Theorem (1.3)′ implies c-rank G = c-rank Z_{G}(X). Alternatively, this condition is also derived from Fact (1.2)(2) since rank G = rank Z_{G}(X).

Example 3.3. Fact (1.2)(3) guarantees the existence of compact Clifford-Klein forms of the following homogeneous spaces of reductive type which are obtained as elliptic orbits: SO(2n, 2)/U(n, 1), SU(2n, 2)/U(2n, 1) (n ∈ ℕ), and SO(4, 3)/SO(4, 1)×SO(2).

Remark 3.4. In general, it may happen that there are an elliptic element X_{1} and a non-elliptic element X such that Z_{G}(X) = Z_{G}(X_{1}). For example, it is the case if G is a complex Lie group and X_{1} is an elliptic element and if we put X := \sqrt{−1}X_{1}. In Theorem (1.3)′ we do not claim X itself is elliptic.

Proof of Theorem (1.3)′. There exists a Cartan involution θ of G which stabilizes Z_{G}(X). Take a θ-stable Cartan subalgebra 𝔧 of Z_{𝔤}(X), which is also a Cartan subalgebra of 𝔤 ([War] Proposition 1.4.3.2). We write 𝔧 = 𝔱 + 𝔞 := 𝔧 ∩ 𝔨 + 𝔧 ∩ 𝔭. As X is contained in the center of Z_{𝔤}(X), so X ∈ 𝔧. We write X = X_{1} + X_{2} corresponding to the direct sum decomposition 𝔧 = 𝔱 + 𝔞. Then X_{1} is elliptic. Let H := Z_{G}(X) and G′ := Z_{G}(X_{1}). It follows from the following Lemma (3.5)(1) and (2) that H ⊂ G′ and W_{G} · 𝔞(G′) = W_{G} · 𝔞(H). Thus Theorem (1.3)′ follows from Theorem (1.5) and from the following Lemma (3.5)(3). □

Lemma 3.5. Retain notation in the above proof. (So 𝔧 is a θ-stable Cartan subalgebra of 𝔤 and X = X_{1} + X_{2} ∈ 𝔧 = 𝔱 + 𝔞.) Then we have

1) Z_{G}(X) = Z_{G}(X_{1}) ∩ Z_{G}(X_{2}),

2) ℝ-rank Z_{G}(X) = ℝ-rank Z_{G}(X_{1}),

3) either Z_{G}(X) = Z_{G}(X_{1}) or d(Z_{G}(X)) < d(Z_{G}(X_{1})).

Proof of Lemma (3.5). 1) Since Z_{G}(X) is θ-stable, we have Z_{G}(X) = θ(Z_{G}(X)) = Z_{G}(θX). Thus, Z_{G}(X) = Z_{G}(X)∩Z_{G}(θX) = Z_{G}(X + θX)∩Z_{G}(X − θX) = Z_{G}(X_{1})∩ Z_{G}(X_{2}).

2) We observe [X_{1}, X_{2}] = 0, especially, X_{2} ∈ 𝔭 ∩ Z_{𝔤}(X_{1}). Take a maximally split abelian subspace 𝔞_{1} of Z_{𝔤}(X_{1}) such that X_{2} ∈ 𝔞_{1}. As [𝔞_{1}, X_{2}] = {0}, we have 𝔞_{1} ⊂ Z_{𝔤}(X_{1}) ∩ Z_{𝔤}(X_{2}) = Z_{𝔤}(X), which shows that 𝔞_{1} is also a maximally split abelian subspace of Z_{𝔤}(X). Hence, ℝ-rank Z_{G}(X) = ℝ-rank Z_{G}(X_{1}).

3) First we note that Z_{G}(X_{1}) is connected. Indeed, there is a Cartan decomposition Z_{G}(X_{1}) = exp (𝔭 ∩ Z_{𝔤}(X_{1})) Z_{K}(X_{1}) and Z_{K}(X_{1}) is a connected subgroup ([He] Chapter 7, Corollary 2.8). Suppose Z_{G}(X) ⫋ Z_{G}(X_{1}). Since Z_{G}(X_{1}) is connected, we have Z_{𝔤}(X) ⫋ Z_{𝔤}(X_{1}). In view of 𝔧_{ℂ} ⊂ Z_{𝔤_{ℂ}}(X) ⫋ Z_{𝔤_{ℂ}}(X_{1}), we find a root α ∈ Δ(Z_{𝔤_{ℂ}}(X_{1}), 𝔧_{ℂ}) \ Δ(Z_{𝔤_{ℂ}}(X), 𝔧_{ℂ}). If we write α = α_{1} + α_{2} corresponding to the direct sum decomposition 𝔧_{ℂ}^{∗} = 𝔱_{ℂ}^{∗} + 𝔞_{ℂ}^{∗}, we have α_{1}(X_{1}) = 0, α(X) = α_{1}(X_{1}) + α_{2}(X_{2}) ≠ 0. Fix a nonzero element Y ∈ 𝔤_{ℂ}(𝔧_{ℂ}; α) and set Z := Y − θY ∈ 𝔭_{ℂ}. Since α ≠ θα, Y and θY are linearly independent. Now we have [X_{1}, Z] = α_{1}(X_{1})Z = 0, and [X, Z] = α_{2}(X_{2})(Y + θY) ≠ 0. Thus Z ∈ 𝔭_{ℂ} ∩ (Z_{𝔤_{ℂ}}(X_{1}) \ Z_{𝔤_{ℂ}}(X)). Hence d(Z_{G}(X_{1})) − d(Z_{G}(X)) = dim (𝔭_{ℂ} ∩ Z_{𝔤_{ℂ}}(X_{1})) − dim (𝔭_{ℂ} ∩ Z_{𝔤_{ℂ}}(X)) > 0. □

§4. Semisimple symmetric spaces

We recall some basic facts about semisimple symmetric spaces and give a brief review of the notion of ε-families introduced by T.Oshima and J.Sekiguchi [O-S2] for the benefit of the reader. Let 𝔤 be a semisimple Lie algebra over ℝ, σ be an involution of 𝔤, θ be a Cartan involution of 𝔤 commuting with σ. Then σθ is also an involution of 𝔤. Let 𝔤 = 𝔨 + 𝔭 = 𝔥 + 𝔮 = 𝔥^{a} + 𝔮^{a} be direct sum decompositions corresponding to θ, σ and σθ respectively. (𝔤, 𝔥^{a}) is called the associated symmetric pair of (𝔤, 𝔥). Note that 𝔥^{a} = 𝔨 ∩ 𝔥 + 𝔭 ∩ 𝔮 and that (𝔥^{a})^{a} = 𝔥. Fix a maximally abelian subspace 𝔞_{𝔭,𝔮} of 𝔭 ∩ 𝔮. Then Σ(𝔞_{𝔭,𝔮}) ≡ Σ(𝔤, 𝔞_{𝔭,𝔮}) satisfies the axiom of root system (see [Ro] Theorem 5, [O-S2] Theorem 2.11) and is called the restricted root system of (𝔤, 𝔥). As σθ ≡ id on 𝔞_{𝔭,𝔮}, we have a direct sum decomposition of the root space 𝔤(𝔞_{𝔭,𝔮}, λ) = (𝔤(𝔞_{𝔭,𝔮}, λ) ∩ 𝔥^{a}) + (𝔤(𝔞_{𝔭,𝔮}, λ) ∩ 𝔮^{a}). We define a map

(4.1)  (m^{+}, m^{−}): Σ(𝔞_{𝔭,𝔮}) → ℕ × ℕ,

by m^{+}(λ) := dim (𝔤(𝔞_{𝔭,𝔮}, λ) ∩ 𝔥^{a}), m^{−}(λ) := dim (𝔤(𝔞_{𝔭,𝔮}, λ) ∩ 𝔮^{a}). Note that if (𝔤, 𝔥) is a Riemannian symmetric pair, then 𝔥^{a} = 𝔤 and m^{−} ≡ 0. A map ε: Σ(𝔞_{𝔭,𝔮}) ∪ {0} → {1, −1} is called a signature of Σ(𝔞_{𝔭,𝔮}) if ε satisfies ε(α + β) = ε(α)ε(β) for any α, β, α + β ∈ Σ(𝔞_{𝔭,𝔮}), ε(−α) = ε(α) for any α ∈ Σ(𝔞_{𝔭,𝔮}) and ε(0) = 1 (see [O-S2] (1.9.3.1)). Note that a signature is determined by its restriction to Ψ, a fundamental system for Σ(𝔞_{𝔭,𝔮}) and that any map Ψ → {1, −1} is uniquely extended to a signature.

To a signature ε of Σ(𝔞_{𝔭,𝔮}), we associate an involution σ_{ε} of 𝔤 defined by σ_{ε}(X) := ε(λ)σ(X) if X ∈ 𝔤(𝔞_{𝔭,𝔮}; λ), λ ∈ Σ(𝔞_{𝔭,𝔮}) ∪ {0}. Then σ_{ε} defines a symmetric pair (𝔤, 𝔥_{ε}). The set

F ((𝔤, 𝔥)) := {(𝔤, 𝔥_{ε}) : ε is a signature of Σ(𝔞_{𝔭,𝔮})}

is called an ε-family (or H_{ε}-family) of symmetric pairs ([O-S2] §6). Among an ε-family, there is a distinguished symmetric pair called basic characterized by,

m^{+}(λ) ≥ m^{−}(λ) for any λ ∈ Σ(𝔞_{𝔭,𝔮}) such that \frac{λ}{2} ∉ Σ(𝔞_{𝔭,𝔮}).

It is known that there exists a basic symmetric pair of F = F ((𝔤, 𝔥)) unique up to isomorphisms ([O-S2] Proposition 6.5). For example, {(𝔰𝔩(n, ℝ), 𝔰𝔬(p, n − p)) : 1 ≤ p ≤ n} is an ε-family with (𝔰𝔩(n, ℝ), 𝔰𝔬(n)) basic. If an ε-family F contains a Riemannian symmetric pair, then F is a K_{ε}-family in the sense of [O-S1]. Typical examples of basic symmetric pairs are (𝔤, 𝔨) (Riemannian symmetric pair), (𝔤, 𝔤) (trivial case), (𝔤 + 𝔤, diag 𝔤) (group manifold), (𝔤_{ℂ}, 𝔤) and (𝔲(p, q; 𝔽), 𝔲(m; 𝔽) + 𝔲(p − m, q; 𝔽)) (𝔽 = ℝ, ℂ, ℍ), whose associated symmetric pairs are (𝔤, 𝔤), (𝔤, 𝔨), (𝔤 + 𝔤, diag 𝔤), (𝔤_{ℂ}, 𝔨_{ℂ}) and (𝔲(p, q; 𝔽), 𝔲(m, q; 𝔽) + 𝔲(p − m; 𝔽)), respectively. On the other hand, G/H = U(i + j, k + l; 𝔽)/U(i, k; 𝔽) × U(j, l; 𝔽) is not basic if ijkl ≠ 0. Now we can deduce Theorem (1.4) from Theorem (1.5) with “H”= H_{ε}^{a} and “G′”= H^{a}, if we use (1) and (2) of the following lemma.

Lemma 4.2. With notations as above, let (𝔤, 𝔥) be basic in the ε-family F = F ((𝔤, 𝔥)) and (𝔤, 𝔥_{ε}) be not basic in F. Then we have

1) W_{G} · 𝔞(H^{a}) = W_{G} · 𝔞(H_{ε}^{a}).

2) d(H^{a}) > d(H_{ε}^{a}).

3) ℝ-rank(H) = ℝ-rank(G/H^{a}) ≤ ℝ-rank(G/H_{ε}^{a}) = ℝ-rank(H_{ε}).

Proof. 1) Clear because 𝔞_{𝔭,𝔮} is a maximally split abelian subspace for both H_{ε}^{a} and H^{a} from the definition of an ε-family.

2) We denote the map (4.1), by (m^{+}, m^{−}) for (𝔤, 𝔥) and by (m_{ε}^{+}, m_{ε}^{−}) for (𝔤, 𝔥_{ε}). From definition, we have

m_{ε}^{+}(λ) = \{ \begin{matrix}m^{+}(λ) & if ε(λ) = 1, \\ m^{−}(λ) & if ε(λ) = −1, \end{matrix}
m_{ε}^{−}(λ) = \{ \begin{matrix}m^{−}(λ) & if ε(λ) = 1, \\ m^{+}(λ) & if ε(λ) = −1. \end{matrix}

From the assumption that (𝔤, 𝔥_{ε}) is not isomorphic to a basic pair (𝔤, 𝔥), it follows that there exists λ ∈ Σ(𝔞_{𝔭,𝔮}) such that

(4.3)  m^{+}(λ) > m^{−}(λ) and ε(λ) = −1.

We define a linear map

ψ: 𝔤 ∋ X ↦ X − θX ∈ 𝔭.

Since ψσ = σψ, we have ψ (𝔤(𝔞_{𝔭,𝔮}, λ) ∩ 𝔥^{a}) ⊂ 𝔭∩𝔥^{a} = 𝔭∩𝔮 for λ ∈ Σ(𝔞_{𝔭,𝔮}). Fix a positive system Σ^{+} ≡ Σ^{+}(𝔞_{𝔭,𝔮}) ⊂ Σ(𝔞_{𝔭,𝔮}). Then ψ: ∑_{λ∈Σ^{+}} (𝔤(𝔞_{𝔭,𝔮}, λ) ∩ 𝔥^{a}) → 𝔭 ∩ 𝔮 is injective and 𝔭 ∩ 𝔮 = 𝔞 ⊕ ∑_{λ∈Σ^{+}} ψ (𝔤(𝔞_{𝔭,𝔮}, λ) ∩ 𝔥^{a}) is a direct sum decomposition. Thus we have d(H^{a}) = dim(𝔭 ∩ 𝔥^{a}) = dim(𝔭 ∩ 𝔮) = dim 𝔞_{𝔭,𝔮} + ∑_{λ∈Σ^{+}} m^{+}(λ). Similarly, d(H_{ε}^{a}) = dim 𝔞_{𝔭,𝔮} + ∑_{λ∈Σ^{+}} m_{ε}^{+}(λ) = dim 𝔞_{𝔭,𝔮} + ∑_{λ∈Σ^{+}, ε(λ)=1} m^{+}(λ) + ∑_{λ∈Σ^{+}, ε(λ)=−1} m^{−}(λ). From (4.2), we have d(H^{a}) − d(H_{ε}^{a}) = ∑_{λ∈Σ^{+}, ε(λ)=−1} (m^{+}(λ) − m^{−}(λ)) > 0.

3) The two equalities are obvious from definition. The middle inequality can be proved based on the classification of symmetric pairs due to Berger [Be]. This inequality is obvious if H is compact (i.e. F ((𝔤, 𝔥)) is a K_{ε}-family in the sense of [O-S1]), because H^{a} = G in this case. If H is noncompact, we can check them by using Table V in [O-S2]. We omit details since we do not use this result in this paper. □

Here is a list of an irreducible semisimple symmetric pair G/H_{ε}^{a} on which an infinite discontinuous group of G acts properly discontinuously from Fact (1.2)(1), but which does not admit a compact Clifford-Klein form from Theorem (1.4). Since the conclusion holds for any finitely covering group of G and for any (sub)group of finitely many connected components with the same Lie algebra, so we write Table (4.4) in terms of Lie algebras. In this Table, we have omitted the cases where ℝ-rank G = ℝ-rank H_{ε}^{a} (see Fact (1.2)(1)) and so H^{a} is necessarily noncompact. We also omitted an indefinite Grassmann manifold G/H = U(i + j, k + l; 𝔽)/U(i, k; 𝔽) × U(j, l; 𝔽), (𝔽 = ℝ, ℂ or ℍ), which we shall treat in §5.2 (see also Example (1.7)).

G/H_{ε}^{a} does not have a compact Clifford-Klein form

𝔤  𝔥_{ε}^{a}  𝔥^{a}
𝔰𝔩(2l, ℝ)  𝔰𝔬(l, l)  𝔰𝔭(l, ℝ)
𝔰𝔲^{∗}(4l)  𝔰𝔬^{∗}(4l)  𝔰𝔭(l, l)
𝔰𝔲(2l, 2l)  𝔰𝔬^{∗}(4l)  𝔰𝔭(l, l)
𝔰𝔭(2l, ℝ)  𝔲(l, l)  𝔰𝔭(l, ℂ)
𝔰𝔬(2l, 2l)  𝔰𝔬(2l, ℂ)  𝔲(l, l)
𝔰𝔬^{∗}(4l + 4)  𝔰𝔬^{∗}(4p + 2) + 𝔰𝔬^{∗}(4l − 4p + 2)  𝔰𝔬^{∗}(2) + 𝔰𝔬^{∗}(4l + 2)
𝔢_{6(6)}  𝔰𝔭(4, ℝ)  𝔣_{4(4)}
𝔢_{6(2)}  𝔰𝔲(4, 2) + 𝔰𝔲(2)  𝔰𝔬^{∗}(10) + \sqrt{−1}ℝ
𝔢_{7(7)}  𝔰𝔲(4, 4)  𝔢_{6(2)} + \sqrt{−1}ℝ
𝔢_{7(7)}  𝔰𝔲^{∗}(8)  𝔰𝔬^{∗}(12) + 𝔰𝔲(2)
𝔢_{7(−5)}  𝔰𝔲(6, 2)  𝔢_{6(−14)} + \sqrt{−1}ℝ
𝔢_{7(−25)}  𝔰𝔲(6, 2)  𝔢_{6(−14)} + \sqrt{−1}ℝ
𝔢_{8(8)}  𝔰𝔬^{∗}(16)  𝔢_{7(−5)} + 𝔰𝔲(2)
𝔰𝔩(2l, ℂ)  𝔰𝔬(2l, ℂ)  𝔰𝔭(l, ℂ)
𝔰𝔬(2l + 2, ℂ) 𝔰𝔬(2p + 1, ℂ) + 𝔰𝔬(2l − 2p + 1, ℂ)  𝔰𝔬(2l + 1, ℂ)
𝔢_{6,ℂ} 𝔰𝔭(4, ℂ) 𝔣_{4,ℂ}

Table 4.4.

Let us explain how to obtain the above table. For example, the first row in Table (4.4) is obtained from Tables in [O-S2] as follows: From the second block in Table V (p.471) there, F = \{(𝔤, 𝔥) = (𝔰𝔩(2l, ℝ), 𝔰𝔩(l, ℂ) + \sqrt{−1}ℝ), (𝔤, 𝔥_{ε}) = (𝔰𝔩(2l, ℝ), 𝔰𝔩(l, ℝ) + 𝔰𝔩(l, ℝ) + ℝ)} is an ε-family with (𝔤, 𝔥) basic. Th associated symmetric pairs are given by (𝔤, 𝔥^{a}) = (𝔰𝔩(2l, ℝ), 𝔰𝔭(l, ℝ)), (𝔤, 𝔥_{ε}^{a}) = (𝔰𝔩(2l, ℝ), 𝔰𝔬(l, l)) from (1.16) and from A_{l,j}^{1} of Table I (p.441) there, respectively. Other rows are similarly obtained. (Here we point out some minor misprints in [O-S2]: 𝔥 of D_{l,A}^{1} in Table I (p.441) should read 𝔰𝔬(l, ℂ). The integer l + 1 should read l in the pair (𝔰𝔩(l + 1, ℝ), 𝔰𝔩(p, ℝ) + 𝔰𝔩(l − p + 1, ℝ) + ℝ) and its dual (𝔰𝔲(p, l − p + 1), 𝔰𝔬(p, l − p + 1)) in Table V (p.471). It would be a compatible notation to have a border line separating between the ninth and tenth block in p.473. The Dynkin diagram of Ψ(𝔞) of the restricted root system of (𝔰𝔩(2l, ℂ), 𝔰𝔩(2l, ℝ)) (p.475) should read the type B_{l}. Correspondingly, the border line between (𝔰𝔩(2l, ℂ), 𝔰𝔲^{∗}(2l)) and (𝔰𝔩(2l, ℂ), 𝔰𝔩(2l, ℝ)) should be eliminated so that it indicates that these two constitute a single ε-family. Also the next Dynkin diagram of the type C_{l} should read only for (𝔰𝔩(2l + 1, ℂ), 𝔰𝔩(2l + 1, ℝ)).)

§5. Examples

In §4 and §5 we have applied Theorem (1.5) systematically to typical families of homogeneous spaces of reductive type. There is still a room for applications of Theorem (1.5) by a suitable choice of G′. For example, we can show that SL(n, ℂ)/SO(n, ℂ) does not have a compact Clifford-Klein form (see Table (5.3)). But this fact is derived from a choice of G′ which is different from the one in the proof of Theorem (1.4). In this section we illustrate such applications of Theorem (1.5).

5.1. Calabi-Markus phenomenon (see Example (1.6)). It is obviously necessary for the existence of a compact Clifford-Klein formof a noncompact homogeneous space of reductive type G/H that G/H admits an infinite discontinuous group, equivalently ℝ-rank G > ℝ-rank H, from Fact (1.2)(1). This is alternatively derived from the choice G′ := G in Theorem (1.5). Indeed, if ℝ-rank G = ℝ-rank H, then Theorem (1.5) implies the first equivalence of the following:

G/H has a compact Clifford-Klein form ⟺ d(G) = d(H) ⟺ G/H is compact.

The second equivalence is derived immediately from a K-equivariant fiber bundle structure G/H ≃ K/H ∩ K ×_{H∩K} 𝔭/𝔥 ∩ 𝔭 and from the definition that d(G) = dim 𝔭, d(H) = dim(𝔥 ∩ 𝔭).

5.2. An indefinite Grassmann manifold (see Example (1.7)). Let G/H = U(i + j, k + l; 𝔽)/U(i, k; 𝔽) × U(j, l; 𝔽), where 𝔽 = ℝ, ℂ or ℍ and i ≤ j, k, l. Here, we use the following notation: U(p, q; ℝ) = O(p, q), U(p, q; ℂ) = U(p, q), and U(p, q; ℍ) = Sp(p, q) (a group of rank p + q). Then Theorem (1.4) implies that if G/H has a compact Clifford-Klein form, then G/H is compact (i = j = 0 or i = k = 0), or H is compact (i = l = 0), or 0 = i < l ≤ j − k. On the other hand, we can obtain the same conclusion by a different choice of G′ from the one in a proof of Theorem (1.4) as follows: Put G_{1} := U(i + t, k + l; 𝔽), G_{2} = U(i + j, i + t; 𝔽), where t := min(j, l). Then W_{G} · 𝔞(H) = W_{G} · 𝔞(G_{1}) = W_{G} · 𝔞(G_{2}). If G/H has a compact Clifford-Klein form, then Theorem (1.5) with G′ = G_{1} or with G′ = G_{2} implies that

d(H) = d(ik + jl) ≥ d(G_{1}) = d(i + t)(k + l),
d(H) = d(ik + jl) ≥ d(G_{2}) = d(i + j)(i + t),

where d = 1, 2 or 4 according as 𝔽 = ℝ, ℂ or ℍ. By an elementary arithmetic we conclude that t = l and that i = jkl = 0 or 0 = i < l ≤ j − k.

5.3. Some table

Here is a table of some other examples of homogeneous spaces G/H of reductive type without a compact Clifford-Klein form. The choice of G′ in Theorem (1.5) is not unique and depends on parameters involved. We write only one typical choice of G′ for each (G, H) in Table (5.3) for simplicity. (See remarks after Table (5.3).)

G/H does not have a compact Clifford-Klein form

𝔤  𝔥  range of parameters  𝔤′
𝔤(n)  𝔤(p) + 𝔤(q) p + q ≤ n, pq > 0  𝔤(p + q)
𝔰𝔩(n, ℝ)  𝔰𝔭(l, ℝ)  0 < 2l ≤ n − 2  𝔰𝔬(l, n − l)
𝔰𝔩(n, ℂ)  𝔰𝔭(l, ℂ)  0 < 2l ≤ n − 1  𝔲(l, n − l)
𝔰𝔩(n, ℂ)  𝔰𝔬(l, ℂ)  0 < l ≤ n  𝔲([\frac{l}{2}], n − [\frac{l}{2}])
𝔰𝔩(2l, ℂ)  𝔲(l, l)  0 < l  𝔰𝔭(l, ℂ)
𝔰𝔬^{∗}(2n) 𝔲(l, n − l) 3l ≤ 2n ≤ 6l, n ≥ 3  𝔰𝔬^{∗}(4l + 2)
𝔰𝔭(n, ℝ)  𝔰𝔭(l, ℂ)  0 < 2l ≤ n  𝔲(l, n − l)
𝔰𝔲^{∗}(2n)  𝔰𝔬^{∗}(2l)  1 < l ≤ n  𝔰𝔭([\frac{l}{2}], n − [\frac{l}{2}])

Table 5.3.

We give here some remarks on Table (5.3).

1) In the first row, 𝔤(n) stands for one of the following classical Lie algebras:

𝔤𝔩(n, ℝ), 𝔤𝔩(n, ℂ), 𝔰𝔬^{∗}(2n), 𝔰𝔬(n, ℂ), 𝔰𝔭(n, ℝ) and 𝔰𝔭(n, ℂ).

If 𝔤(n) = 𝔰𝔬^{∗}(2n) or 𝔰𝔬(n, ℂ) and if both p and q are odd integers, then we must assume furthermore p, q ≥ 3 in the above table. In this case, we need also a modification of the choice of 𝔤′, that is, we take 𝔤′ = 𝔤(p + q − 1) instead of 𝔤′ = 𝔤(p + q).

2) In the sixth row, (𝔤, 𝔥) = (𝔰𝔬^{∗}(2n), 𝔲(l, n − l)), the choice 𝔤′ = 𝔰𝔬^{∗}(4l + 2) is valid for 2l < n. If 3l ≤ 2n < 4l or n = 2l, then we take 𝔤′ = 𝔰𝔬^{∗}(4n − 4l + 2) or 𝔤′ = 𝔰𝔬^{∗}(2n), respectively.

3) In the seventh row, (𝔤, 𝔥) = (𝔰𝔭(n, ℝ), 𝔰𝔭(l, ℂ)), the choice 𝔲(l, n − l) is valid for 0 < 2l < n. The non-existence of a compact Clifford-Klein form of G/H = Sp(2l, ℝ)/Sp(l, ℂ) is derived not from Theorem (1.5) but from Fact (1.2)(2) (see Example (4.11) in [Ko1]).

⟨Acknowledgement⟩ The author is grateful to Professors Flensted-Jensen, ’Olaffson, Ørsted and Schlichtkrull for their hospitality and helpful conversations during his stay in Denmark. Thanks are also due to Professors Benoist, Borel, Matumoto, Oshima and

Wallach for their interests and comments in this work.

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