Journal of Lie Theory Volume 6 (1996) 147–163 C ◯1996 Heldermann Verlag

Criterion for proper actions on homogeneous spaces of reductive groups

Toshiyuki Kobayashi

Communicated by M. Moskowitz

Abstract. Let M be a manifold, on which a real reductive Lie group G acts transitively. The action of a discrete subgroup Γ on M is not always properly discontinuous. In this paper, we give a criterion for properly discontinuous actions, which generalizes our previous work [6] for an analogous problem in the continuous setting. Furthermore, we introduce the discontinuous dual ⋔(H:G) of a subset H of G, and prove a duality theorem that each subset H of G is uniquely determined by its discontinuous dual up to multiplication by compact subsets.

1. Introduction

This paper is a continuation of [6]. We shall give a criterion for the properly discontinuous action on a homogeneous space of a real reductive group.

Suppose a discrete group Γ acts on a manifold M properly discontinuously and freely. Then the coset space Γ\M carries naturally a manifold structure on the quotient topology. We are particularly interested in the setting where a Lie group G acts on M transitively and Γ is a discrete subgroup of G, so that any local G-invariant structure on M is inherited by the quotient manifold Γ\M.

Let us formulate our object of study in terms of groups. We denote by H the isotropy subgroup of G at a point o of M. A Clifford-Klein form of a homogeneous manifold M ≃ G/H is the double coset space Γ\G/H if Γ is a subgroup of a Lie group G acting properly discontinuously and freely on G/H. A typical example is a closed Riemann surface with genus ≥ 2, which is biholomorphic to a Clifford-Klein form of the Poincaré plane P SL(2, ℝ)/SO(2). Clifford-Klein forms have been studied extensively, in particular, for a Riemannian symmetric space G/H, namely, when H is a maximal compact subgroup of a real reductive Lie group G (e.g. [1, 2, 15, 14]). An important remark here is that if the isotropy subgroup H is compact then the action of a torsion free discrete subgroup Γ on G/H is automatically properly discontinuous (see §2.1

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag for the definition) and free. Hence, the study of Clifford-Klein forms of G/H with H compact is essentially equivalent to the study of discrete subgroups of G.

In this paper, we deal with a more general setting where H is a noncompact closed subgroup of a real reductive Lie group G. A distinguishing feature in our setting with H noncompact is that a discrete subgroup Γ of G does not always act properly discontinuously on G/H so that the double coset space Γ\G/H is not necessarily Hausdorff in the quotient topology. For instance, Calabi-Markus proved in [3] that any relativistic spherical space form with dimension ≥ 3 is noncompact and has a finite fundamental group, which is a consequence of the fact that no infinite discrete subgroup of O(n, 1) acts properly discontinuously on O(n, 1)/O(n − 1, 1). The Calabi-Markus phenomenon for a homogeneous space G/H has been studied in [3, 18, 19, 20, 12, 16, 6] where both G and H are reductive; and in [7, 9, 13] where G is a solvable Lie group. The method of “the continuous analog” (e.g. [6, 7, 9, 13]) has been a powerful tool in the study of properly discontinuous actions, in particular, for the Calabi-Markus phenomenon. However, for a further study of discontinuous groups and Clifford-Klein forms of homogeneous manifolds, we think it is important to obtain more straightforward methods for properly discontinuous actions without using the continuous analog (see [7], [10] for open problems and the survey of recent results on Clifford-Klein forms of homogeneous manifolds). This is the motivation for this paper. In fact, our result (see Theorem 3.4) plays a key role in the deformation of compact Clifford-Klein forms with indefinite metric in higher dimensions, where the action of the Zariski closure of a discrete subgroup Γ on G/H is not always proper even though that of Γ on G/H is properly discontinuous (see [11]).

Let us now state our results. Suppose that H and L are subsets of a locally compact topological group G. We write H ∼ L in G if and only if there exists a compact set S of G such that L ⊂ SHS^{−1} and H ⊂ SLS^{−1} , where SHS^{−1} := {ahb^{−1} ∈ G : a, b ∈ S, h ∈ H}. We write H ⋔ L in G if and only if SHS^{−1} ∩ L is relatively compact for any compact set S in G (see Definition 2.1.1). We note that if H is a closed subgroup and Γ is a discrete subgroup of G, then the action of Γ on the homogeneous manifold G/H is properly discontinuous if and only if H ⋔ Γ in G (see Observation 2.1.3). Now, let G be a real reductive linear Lie group and 𝔤 = 𝔨 + 𝔭 be a Cartan decomposition. We fix a maximally abelian subspace 𝔞 of 𝔭 and denote by W_{G} the Weyl group of the restricted root system Σ(𝔤, 𝔞). For a subset L of G we set 𝔞(L) := {X ∈ 𝔞 : exp X ∈ KLK} (see Definition 3.2). Then we shall prove:

Theorem 1.1. (See Theorem 3.4 and Theorem 5.6). Suppose H, L are subsets of a real reductive linear Lie group G.

1) H ⋔ L in G ⇔ 𝔞(H) ⋔ 𝔞(L) in 𝔞.

2) H ∼ L in G ⇔ 𝔞(H) ∼ 𝔞(L) in 𝔞.

If both H and L are connected closed subgroups which are reductive in G, then the part (1) yields our previous result (see Theorem 4.1 in [6]). The novelty here is that we allow L to be non-reductive, in particular, to be discrete. We note that we could assume one of H or L to be reductive without loss of generality if we replace the triplet (H, G, L) by (H × L, G × G, diag(G)).

The part (2) is proved by a “duality theorem” (see §5) asserting that a subset H of G is determined up to ∼ if we know the discontinuous dual of H in G (see §5.2), that is, all the subsets L ⊂ G with the property H ⋔ L in G.

In §6, we shall give computations of 𝔞(H) for some typical subsets H of G. For instance, we describe 𝔞(Γ) for an arbitrary cyclic Γ in Example 6.2; and we find a certain nilpotent subgroup N of G = GL(n, ℝ) such that 𝔞(N) coincides with that of a reductive subgroup O(p, n − p) in Example 6.5.

Theorem 1.1 leads to:

Corollary 1.2. (See Corollary 3.5). Suppose H is a closed subgroup of a real reductive linear Lie group G. Then the following conditions are equivalent: (1.2.1) The action of a discrete subgroup Γ on a homogeneous space G/H is properly discontinuous. (1.2.2) For any compact subset V of 𝔞, (𝔞(Γ) + V) ∩ 𝔞(H) is compact.

Here is a necessary condition for the Calabi-Markus phenomenon to occur:

Corollary 1.3. Suppose G is a real reductive linear Lie group and H is a closed subgroup. If 𝔞(H) is contained in a proper cone of 𝔞, then there exists an infinite discrete subgroup Γ of G acting freely and properly discontinuously on G/H.

Let us consider the reductive case, namely, H is a closed subgroup which is stable under a Cartan involution of G. Then rank_{ℝ}H < rank_{ℝ}G if and only if 𝔞(H) ≠ 𝔞, which is equivalent to the condition that 𝔞(H) is contained in a proper cone of 𝔞. Hence, Corollary 1.3 in the reductive case means that if rank_{ℝ}H < rank_{ℝ}G then there exists an infinite discrete subgroup Γ of G acting freely and properly discontinuously on G/H. This was the non-trivial implication of Corollary 4.4 in [6] (the criterion for the Calabi-Markus phenomenon in the reductive case).

The results of this paper were announced in the fifth workshop on Lie groups at Tottori and in the Lie group and representation theory seminar at University of Tokyo in September 1994 (see also [10] §2.11 and §3.9). After completing this work, the author received a preprint of Benoist about a result similar to Theorem 3.4. His proof is different from ours and he also found some applications such as the non-existence theorem of compact Clifford-Klein forms of SO(4n, ℂ)/SO(4n − 1, ℂ). (See Y. Benoist, Actions propres sur les espaces homogènes réductifs, Preprint.)

2. Preliminary Results

Suppose that H and L are subsets of a locally compact topological group G.

Definition 2.1.1. We denote by H ∼ L in G the existence of a compact subset S of G such that L ⊂ SHS^{−1} and H ⊂ SLS^{−1} . Then it is easy to see that the relation H ∼ L in G defines an equivalence relation. That is, the relation ∼ is reflexive, symmetric and transitive. We say the pair (H, L) is proper in G, denoted by H ⋔ L in G, if and only if SHS^{−1} ∩ L is relatively compact for any compact subset S in G.

Example 2.1.2. Let G be a finite dimensional vector space, and H and L subspaces of G. Then H ⋔ L in G if and only if H ∩ L = {0}, and H ∼ L in G if and only if H = L.

The above definitions are motivated by the following:

Observation 2.1.3. Let H and L be closed subgroups of G, and Γ a discrete subgroup of G.

1) The action of L on a homogeneous space G/H is proper if and only if L ⋔ H in G.

2) The action of Γ on a homogeneous space G/H is properly discontinuous if and only if Γ ⋔ H in G. Here we recall the definition of proper actions and properly discontinuous actions: Suppose that a locally compact topological group L acts continuously on a locally compact Hausdorff space X. This action is called proper if and only if L_{S} := {g ∈ L : g · S ∩ S ≠ Ø} is compact for every compact subset S in X. The action is called properly discontinuous if and only if L is discrete and acts properly on X.

We record some elementary properties of the relations ∼, ⋔ :

Lemma 2.2. Suppose G is a locally compact topological group and that H, H′, L are its subsets.

1) H ⋔ L in G if and only if L ⋔ H in G.

2) If H ∼ H′ and if H ⋔ L in G, then H′ ⋔ L in G.

3) H ⋔ L in G if and only if H ⋔ \overline{L} in G. Here \overline{L} denotes the closure of L in G. Proof. 1) Let S be a compact subset of G. If H ⋔ L in G, then SHS^{−1}∩L is relatively compact for each compact subset S in G. Therefore H ∩ S^{−1}LS ⊂ S^{−1}(SHS^{−1} ∩ L)S is also relatively compact. Hence L ⋔ H in G.

2) Take a compact subset T ⊂ G such that H′ ⊂ T HT^{−1} . If H ⋔ L, then L ∩ SH′S^{−1} ⊂ L ∩ (ST)H(ST)^{−1} is relatively compact, whence H′ ⋔ L.

3) Take a compact neighborhood V of the identity e in G. Then \overline{L} ⊂ LV^{−1} because xV ∩ L ≠ Ø for each x ∈ \overline{L}. Suppose H ⋔ L in G. Then for any compact subset S ⊂ G, S\overline{L}S^{−1} ∩ H ⊂ SLV^{−1}S^{−1} ∩ H ⊂ (SV)L(SV)^{−1} ∩ H is relatively compact. Hence H ⋔ \overline{L} in G.

2.3. The following lemma is immediate from Definition 2.1.1:

Lemma 2.3. Suppose G′ is a closed subgroup of a locally compact topological group G. Let H and L be subsets of G′.

1) If H ∼ L in G′, then H ∼ L in G.

2) If H ⋔ L in G, then H ⋔ L in G′.

3. Criterion for proper actions

Hereafter, we suppose that G is a real reductive linear Lie group. We assume G is contained in a connected complex Lie group G_{ℂ} (G is not necessarily connected, e.g. G = GL(n, ℝ)). We fix a maximal compact subgroup K of G. Let θ be the corresponding Cartan involution of G and we write 𝔤 = 𝔨 + 𝔭 for the corresponding Cartan decomposition of the Lie algebra 𝔤 of G. We fix a maximally abelian subspace 𝔞_{G} ≡ 𝔞 ⊂ 𝔭. All such subspaces are mutually conjugate by K. The dimension dim 𝔞 is called the real rank of G, denoted by rank_{ℝ}G. We write A for the connected subgroup of G having the Lie algebra 𝔞. Let M be the centralizer of 𝔞 in K, M′ the normalizer of 𝔞 in K. Then the finite group W_{G} := M′/M acts effectively on 𝔞 (also on A) as the Weyl group associated to the restricted root system Σ(𝔤, 𝔞) (see §4.1). We have a Cartan decomposition

(3.1.1)  G = KAK.

In (3.1.1), there is a unique element a(g) ∈ A up to conjugation by W_{G} such that g ∈ Ka(g)K for each g ∈ G.

Definition 3.2. For each subset L of G, we define:

\begin{aligned}A(L) & := A ∩ KLK = \{w · a(g) : w ∈ W_{G}, g ∈ L\} ⊂ A, \\ 𝔞(L) & := log A(L) ⊂ 𝔞. \end{aligned}

Here log: A → 𝔞 is the inverse of the diffeomorphism exp: 𝔞 → A.

We remark that the above notation is essentially the same with that in [6] (see also [8]) in the special case where L is a θ-stable connected subgroup of G (see Example 6.1). (To be more precise, they coincide as subsets of the set of equivalence classes 𝔞/W_{G}, the quotient of 𝔞 by the action of the Weyl group W_{G}.)

We list some elementary properties of A(L) and 𝔞(L) which are immediate from their definition:

Lemma 3.3. Suppose L is a subset of G.

1) 𝔞(L) is a W_{G} -invariant set.

2) L ∼ A(L) in G.

3) L is a closed set in G if and only if 𝔞(L) is closed in 𝔞.

4) If L′ ⊂ L are subsets of G such that \overline{L′} = L, then \overline{A(L′)} = A(L).

5) If L′, L are subsets of G such that KL′K = KLK, then A(L′) = A(L). Proof. (1), (3) and (5) are obvious.

(2) follows from L ⊂ KA(L)K and A(L) ⊂ KLK. Now we prove (4). As A(L) is closed by (3) and as A(L′) ⊂ A(L), we have \overline{A(L′)} ⊂ A(L). Conversely, as L′ = KA(L′)K, we have \overline{L′} = \overline{KA(L′)K} = K\overline{A(L′)}K. Hence, A(L) ⊂ KLK = K\overline{L′}K = K\overline{A(L′)}K, and we have A(L) ⊂ \overline{A(L′)}.

Theorem 3.4. Let H and L be subsets of a real reductive linear Lie group G. Then the following four conditions are equivalent: (3.4.1) H ⋔ L in G. (3.4.1)′ A(H) ⋔ A(L) in G. (3.4.2) 𝔞(H) ⋔ 𝔞(L) in 𝔞. (3.4.2)′ A(H) ⋔ A(L) in A.

Strategy of Proof. The equivalence (1) ⇔ (1)′ follows from Lemma 3.3 (2) and Lemma 2.2. The equivalence (2) ⇔ (2)′ is trivial, because exp: 𝔞 ≃ A is an isomorphism of Lie groups. The implication (1)′ ⇒ (2)′ follows from Lemma 2.3. The only non-trivial implication (2) ⇒ (1)′ will be proved after a sequence of lemmas in §4.

Corollary 3.5. Let G be a real reductive linear Lie group, and H, L closed subgroups of G. Then the following conditions are equivalent: (3.5.1) The action of L on a homogeneous space G/H is proper. (3.5.2) For any compact subset V of 𝔞, (𝔞(L) + V) ∩ 𝔞(H) is compact.

Moreover, if L is a discrete subgroup of G, then (3.5.2) is equivalent to (3.5.1)′ The action of L on a homogeneous space G/H is properly discontinuous.

4. Proof of Theorem 3.4

In this section we give a proof of the non-trivial implication (2)′ ⇒ (1)′ in Theorem 3.4. The idea of its proof is based on our previous technique in [6]. In particular, the first half of this section parallels [6] §3. However, in order to obtain sharper results such as Lemma 4.8, we need to deal with 𝔞 rather than with the unit sphere in 𝔞 and we have changed some notation of [6] (e.g. \frac{Y_{n}}{∥Y_{n}∥} here corresponds to Y_{n} in [6] §3). So, we have decided to give a self-contained proof here for the convenience of the reader.

Let G be a real reductive linear Lie group. Let 𝔤 = 𝔨 + 𝔭 be a Cartan decomposition, and 𝔷(𝔤) the center of 𝔤. We define ideals of 𝔤 by 𝔤′ := [𝔤, 𝔤] + (𝔷(𝔤) ∩ 𝔨), 𝔠_{𝔤} := 𝔷(𝔤) ∩ 𝔭. Then 𝔤 = 𝔤′ ⊕ 𝔠_{𝔤}. Let C_{G} := exp(𝔠_{𝔤}) (⊂ G), and let G′ be the (unique) normal subgroup of G with Lie algebra 𝔤′ such that G′ meets each connected component of G. Then G is isomorphic to the direct product group G′ × C_{G} ([17] Ch.1, §1.5). According to this decomposition, we write g = g′g^{c} ∈ G ≃ G′×C_{G}, X = X′+X^{c} ∈ 𝔤 ≃ 𝔤′⊕𝔠_{𝔤}. Analogous notation is used for other reductive subgroups. The simultaneous diagonalization of ad(𝔞)|_{𝔤} gives the direct sum decomposition

(4.1.1)  𝔤 = 𝔤(𝔞; 0) + ∑_{α∈Σ} 𝔤(𝔞; α).

Here

𝔤(𝔞; α) := {X ∈ 𝔤; [H, X] = α(H)X for any H ∈ 𝔞} for α ∈ 𝔞^{∗},
Σ ≡ Σ(𝔤, 𝔞) := {α ∈ 𝔞^{∗}; 𝔤(𝔞; α) ≠ 0} \ {0}.

For each α ∈ Σ ∪ {0}, we define the projection with respect to the direct sum (4.1.1) by

(4.1.2)  p_{α} : 𝔤 → 𝔤(𝔞; α).

For Y ∈ 𝔞 we define a parabolic subalgebra of 𝔤 by

𝔭(Y) = 𝔫(Y) + 𝔩(Y) := ∑_{α(Y)>0} 𝔤(𝔞; α) + ∑_{α(Y)=0} 𝔤(𝔞; α).

The corresponding parabolic subgroup of G is given by

P (Y) := N_{G}(𝔭(Y)) ≡ {g ∈ G : Ad(g)𝔭(Y) = 𝔭(Y)}

= N(Y) · L(Y) (Levi decomposition),

where N(Y) = exp(𝔫(Y)) and L(Y) = P (Y) ∩ θP (Y) = Z_{G}(Y) because G is contained in a connected complex reductive group G_{ℂ}.

Suppose 𝔞_{1} , 𝔞_{2} are W_{G} -invariant subsets of 𝔞, A_{j} := exp 𝔞_{j} ⊂ A (j = 1, 2). If A_{1} ⋔̸ A_{2} in G, namely, if there exists a compact subset S of G such that SA_{1}S^{−1} ∩ A_{2} is not relatively compact, then we find sequences

a_{n}, b_{n} ∈ S, Y_{n} ∈ 𝔞_{1}, Z_{n} ∈ 𝔞_{2} (n ∈ ℕ)

such that a_{n} exp(Y_{n})b_{n}^{−1} = exp(Z_{n}), and {Z_{n} : n ∈ ℕ} is not relatively compact. We fix a closed positive Weyl chamber \overline{𝔞_{+}} , i.e., the closure of a connected component of the subset 𝔞′ := {H ∈ 𝔞 : α(H) ≠ 0 for any α ∈ Σ}, and a W_{G} - invariant norm | · | on 𝔞. Replacing a_{n}, b_{n}, Y_{n}, Z_{n} by appropriate subsequences and taking conjugation by W_{G} (or by representatives in M′) if necessary, we have the following setting:

Setting 4.2. There are sequences a_{n}, b_{n} ∈ G; Y_{n}, Z_{n} ∈ 𝔞 (n ∈ ℕ), and there are a, b ∈ G, Y, Z ∈ \overline{𝔞_{+}} \ \{0\} such that

Y_{n} ∈ 𝔞_{1}, Z_{n} ∈ 𝔞_{2} (n ∈ ℕ), (4.2.1) a_{n} = exp(Z_{n}) b_{n} exp(−Y_{n}), (4.2.2)

(4.2.3) lim_{n→∞} |Y_{n}| = lim_{n→∞} |Z_{n}| = ∞, lim_{n→∞} \frac{Y_{n}}{|Y_{n}|} = Y, lim_{n→∞} \frac{Z_{n}}{|Z_{n}|} = Z,

lim_{n→∞} a_{n} = a, lim_{n→∞} b_{n} = b. (4.2.4)

For each α, β ∈ Σ(𝔤, 𝔞) and for any X_{α} ∈ 𝔤(𝔞; α), the equation (4.2.2) gives rise to a formula:

(4.2.5)  p_{β} (Ad(a_{n})X_{α}) = exp (β(Z_{n}) − α(Y_{n})) p_{β} (Ad(b_{n})X_{α}) .

This formula will play a crucial role later in the reduction from a non-commutative setting to an abelian setting.

Now we explain the remaining part of the proof of Theorem 3.4. We shall prove later in Lemma 4.8 that T := {Y_{n} − Z_{n} : n ∈ ℕ} ⊂ 𝔞 is a bounded subset. This means that 𝔞_{1} ∩ (𝔞_{2} + T) contains an unbounded set {Y_{n} : n ∈ ℕ}. Hence, 𝔞_{1} ⋔̸ 𝔞_{2} , which shows (2) ⇒ (1)′ of Theorem 3.4. The rest of this section is devoted to the proof of Lemma 4.8.

Lemma 4.3. In the setting (4.2), the sequence \frac{|Z_{n}|}{|Y_{n}|} is bounded away from 0 and ∞.

Proof. Assume there were a subsequence n(k), k = 1, 2, … such that

lim_{k→∞} \frac{|Z_{n(k)}|}{|Y_{n(k)}|} = ∞.

We take β ∈ Σ(𝔤, 𝔞) such that β(Z) > 0. For each α ∈ Σ(𝔤, 𝔞) ∪ {0}, we have

(4.3.1)  lim_{k→∞} exp (β(Z_{n(k)}) − α(Y_{n(k)})) = ∞.

Because the left side of (4.2.5) is a convergent sequence, (4.3.1) leads to

p_{β} (Ad(b)X_{α}) = lim_{k→∞} p_{β} (Ad(b_{n(k)})X_{α}) = 0.

Since this equation holds for any X_{α} ∈ 𝔤(𝔞; α) and for any α ∈ Σ(𝔤, 𝔞) ∪ {0}, we have p_{β}(Ad(b) 𝔤) = {0}, which contradicts the fact that Ad(b) 𝔤 = 𝔤. Thus, we have proved sup_{n} \frac{|Z_{n}|}{|Y_{n}|} < ∞. Because the role of Y_{n} and Z_{n} is symmetric, we

have similarly sup_{n} \frac{|Y_{n}|}{|Z_{n}|} < ∞. Hence \frac{|Z_{n}|}{|Y_{n}|} is bounded away from 0 and ∞.

Suppose we are in the setting of (4.2). It follows from Lemma 4.3 that we can find a subsequence n(k) such that lim_{k→∞} \frac{|Z_{n(k)}|}{|Y_{n(k)}|} = C for some positive constant C (we shall show that C is equal to 1 in the proof of Lemma 4.6).

Lemma 4.4. In the setting as above,

(4.4.1)  Ad(b) ( ∑_{α(Y)≤t} 𝔤(𝔞; α)) = ∑_{Cβ(Z)≤t} 𝔤(𝔞; β) for any t ∈ ℝ.

Proof. If Cβ(Z) > α(Y), then we have lim_{k→∞} exp (β(Z_{n(k)}) − α(Y_{n(k)})) = ∞. This implies p_{β} (Ad(b)X_{α}) = 0 from (4.2.5). Therefore we have

Ad(b) (X_{α}) ∈ ∑_{Cβ(Z)≤t} 𝔤(𝔞; β),

if X_{α} ∈ 𝔤(𝔞; α) such that α(Y) ≤ t. Hence we have shown (4.4.1).

We recall a basic structural result of a reductive Lie algebra.

Lemma 4.5. If b ∈ G and X, X′ ∈ \overline{𝔞_{+}} satisfy

(4.5.1)  Ad(b) ( ∑_{α(X)≤t} 𝔤(𝔞; α)) = ∑_{α(X′)≤t} 𝔤(𝔞; α) for any t ∈ ℝ,

then b ∈ P (−X) = P (−X′) and X − X′ ∈ 𝔠_{𝔤} (see §4.1 for the definition).

Proof. The first claim b ∈ P (−X) = P (−X′) follows from standard properties of parabolic subalgebras (see [17] Ch. 1), if we substitute t = 0 in (4.5.1).

Then it follows from b ∈ P (−X) and from (4.5.1) that (4.5.2)

∑_{α(X′)≤t} 𝔤(𝔞; α) = Ad(b) ( ∑_{α(X)≤t} 𝔤(𝔞; α)) ⊂ ∑_{α(X)≤t} 𝔤(𝔞; α) for any t ∈ ℝ.

We fix β ∈ Σ(𝔤; 𝔞) and ε > 0 and put t := β(X) − ε. Applying the projection p_{β} to (4.5.2), we have β(X) − ε < β(X′) because the right side vanishes. Because ε > 0 is arbitrary, The converse inequality β(X) ≥ β(X′) is proved in a similar way by using the equality (4.5.1) multiplied by Ad(b^{−1}). Therefore β(X) = β(X′) for any β ∈ Σ(𝔤, 𝔞). Hence X − X′ ∈ 𝔷(𝔤) ∩ 𝔞 = 𝔠_{𝔤}.

Lemma 4.6. In the setting (4.2), we have b ∈ P (−Z), a ∈ P (Z) and Y = Z.

Proof. The first statement b ∈ P (−Z) is deduced from Lemmas 4.4 and 4.5. Similarly, we have a ∈ P (Z). Now, let us prove Y = Z. Let C be an accumulation point of \frac{|Z_{n}|}{|Y_{n}|}. We take a subsequence n(k) such that lim_{k→∞} |\frac{Z_{n(k)}}{Y_{n(k)}} | = C. It follows from Lemma 4.4 and Lemma 4.5 that

(4.6.1)  Y − CZ ∈ 𝔠_{𝔤}.

According to the decomposition G ≃ G′ × C_{G}, g ↦ g′g^{c}, we have from (4.2.2) that

a_{n(k)}^{c} = exp(Z_{n(k)}^{c}) b_{n(k)}^{c} exp(−Y_{n(k)}^{c}) ∈ C_{G}.

Hence exp(Z_{n(k)}^{c} − Y_{n(k)}^{c}) = a_{n(k)}^{c}(b_{n(k)}^{c})^{−1} is a convergent sequence. By (4.2.3), we have

CZ − Y = lim_{k→∞} ( \frac{Z_{n(k)}}{\frac{1}{C} |Z_{n(k)}|} − \frac{Y_{n(k)}}{|Y_{n(k)}|}) = lim_{k→∞} \frac{Z_{n(k)} − Y_{n(k)}}{|Y_{n(k)}|} ,

and then

(CZ − Y)^{c} = lim_{k→∞} \frac{Z_{n(k)}^{c} − Y_{n(k)}^{c}}{|Y_{n(k)}|} = 0

because lim_{k→∞} |Y_{n(k)}| = ∞ and because Z_{n(k)}^{c} − Y_{n(k)}^{c} converges. Combined with (4.6.1), we have CZ − Y = 0. Because |Y | = |Z| = 1 and Y, Z ∈ \overline{𝔞_{+}} , we have C = 1. Hence, Y = Z.

Now, we prove a rank reduction lemma.

Lemma 4.7. In the setting (4.2), assume that there are sequences a_{n}, b_{n} ∈ G, Y_{n}, Z_{n} ∈ 𝔞 satisfying (4.2.1), (4.2.2), (4.2.3), and (4.2.4). Then we can find sequences A_{n}, B_{n} in L(Z) and elements A, B in L(Z) that satisfy (4.2.2) and (4.2.4) if we replace a_{n}, b_{n}, a, b by A_{n}, B_{n}, A, B, respectively.

Proof. First we note that b ∈ P (−Z) ⊂ P (−Z)N(Z) by Lemma 4.6. Because P (−Z)N(Z) is an open set of G, b_{n} ∈ P (−Z)N(Z) if n is large enough. According to the diffeomorphism

N(−Z) × N(Z) × L(Z) ≃ P (−Z)N(Z) ⊂ G,

we write b_{n} = B_{n}^{−}B_{n}^{+}B_{n}. The point here is that each sequence B_{n}^{±} ∈ N(±Z), B_{n} ∈ L(Z) converges as n → ∞, respectively. We put

A_{n}^{±} := exp(Z_{n})B_{n}^{±} exp(−Z_{n}), A_{n} := exp(Z_{n})B_{n} exp(−Y_{n}).

Then lim_{n→∞} A_{n}^{−} = 1 because B_{n}^{−} ∈ N(−Z) are bounded and because

lim_{n→∞} \frac{Z_{n}}{|Z_{n}|} = Z and lim_{n→∞} |Z_{n}| = ∞.

It follows from (4.2.2) and the definitions of A^{±}, B^{±}, A_{n} and B_{n} that

(A_{n}^{−})^{−1} a_{n} = (A_{n}^{−})^{−1} exp(Z_{n})B_{n}^{−}B_{n}^{+}B_{n} exp(−Y_{n}) = (A_{n}^{−})^{−1} A_{n}^{−}A_{n}^{+}A_{n} = A_{n}^{+}A_{n}.

The first term converges to a as n → ∞. Therefore the last term A_{n}^{+}A_{n} ∈ N(Z)L(Z) also converges as n → ∞. Because N(Z) × L(Z) ≃ P (Z) is a homeomorphism, A_{n}^{+} ∈ N(Z) and A_{n} ∈ L(Z) converge, respectively. In particular,

A_{n} = exp(Z_{n})B_{n} exp(−Y_{n})

converges. Hence the conditions (4.2.1), (4.2.2), (4.2.3) and (4.2.4) are also satisfied if we replace a_{n}, b_{n} ∈ G by A_{n}, B_{n} ∈ L(Z).

Let us complete the proof of Theorem 3.4 (2)⇒(1)′. What we want to prove now is:

Lemma 4.8. In the setting of (4.2), the sequence Y_{n} − Z_{n} forms a bounded subset in 𝔞.

Proof. We proceed by induction on rank_{ℝ}G. If rank_{ℝ}G = 0, that is, if 𝔞 = 0, there is nothing to prove. Assume now that we have proved the Lemma for real reductive groups with real rank < r. Suppose the setting (4.2) is satisfied for G with rank_{ℝ}G = r. It follows from Lemma 4.7 that we find sequences A_{n}, B_{n} ∈ L(Z), Y_{n} ∈ 𝔞_{1} , Z_{n} ∈ 𝔞_{2} satisfying the setting (4.2). We write the Lie group L(Z) which is a reductive subgroup, for simplicity, as L. Likewise we write 𝔩 := 𝔩(Z). With similar notation as in §4.1, we have

(4.8.1)  𝔩 = 𝔩′ ⊕ 𝔠_{𝔩}, L ≃ L′ × C_{L}.

According to the decomposition (4.8.1), we write Y_{n} = Y′_{n} +Y_{n}^{c}, Z_{n} = Z′_{n} +Z_{n}^{c} ∈ 𝔩(Z), A_{n} = A′_{n}A_{n}^{c}, B_{n} = B′_{n}B_{n}^{c} ∈ L(Z), respectively, with Y′_{n}, Z′_{n} ∈ 𝔩′, Y_{n}^{c}, Z_{n}^{c} ∈ 𝔠_{𝔩}, A′_{n}, B′_{n} ∈ L′ and A_{n}^{c}, B_{n}^{c} ∈ C_{L}. Here we note that A′_{n}, B′_{n} ∈ L′ and A_{n}^{c}, B_{n}^{c} ∈ C_{L} are convergent sequences, respectively. Then the equation A_{n} = exp(Z_{n})B_{n} exp(−Y_{n}) is decomposed as

A′_{n} = exp(Z′_{n})B′_{n} exp(−Y′_{n}) ∈ L′,
A_{n}^{c} = exp(Z_{n}^{c})B_{n}^{c} exp(−Y_{n}^{c}) ∈ C_{L}.

Then the set {Y_{n}^{c} − Z_{n}^{c} : n ∈ ℕ} is a bounded set in 𝔠_{𝔩} because C_{L} is an abelian Lie group and because both of the sequences A_{n}^{c}, B_{n}^{c} converge as n → ∞. Also the set {Y′_{n} − Z′_{n} : n ∈ ℕ} is a bounded set in 𝔩′ from the inductive assumption because rank_{ℝ}L′ = rank_{ℝ}L − dim 𝔠_{𝔩} < rank_{ℝ}L = rank_{ℝ}G = r. Therefore, {Y_{n} − Z_{n} : n ∈ ℕ} = {(Y′_{n} + Y_{n}^{c}) − (Z′_{n} + Z_{n}^{c}) : n ∈ ℕ} is also a bounded set in 𝔞, proving the Lemma.

5. Duality

Consider the following duality problems:

Question 5.1.

1) Is a subset H determined up to the equivalence relation ∼ if one knows all the subsets L with the property L ⋔ H in G?

2) Is a subgroup H determined up to the equivalence relation ∼ if one knows all the subgroups L with the property L ⋔ H in G?

In this section we give an affirmative answer to Question 5.1(1). The proof also leads us to affirmative answers to the following natural questions as we shall see in Theorem 5.6 (cf. Theorem 3.4 and Observation 2.1.3):

i) Does 𝔞(H) ∼ 𝔞(H′) in 𝔞 if H ∼ H′ in G?

ii) Does 𝔞(H) ∼ 𝔞(gHg^{−1}) in 𝔞 if g ∈ G?

Let P(G) denote the totality of subsets of a locally compact topological group G. For H ∈ P(G), we define the discontinuous dual of H by

(5.2.1)  ⋔ (H : G) := {L ⊂ G : L ⋔ H in G} ⊂ P(G).

Suppose 𝔞 is a finite dimensional vector space over ℝ and 𝔞_{1} is a subset. We write the multiplication of 𝔞 in an additive way. Then ⋔ (𝔞_{1} : 𝔞) is given by

⋔ (𝔞_{1} : 𝔞) = {𝔞_{2} ⊂ 𝔞 : 𝔞_{1} ⋔ 𝔞_{2} in 𝔞}
= {𝔞_{2} ⊂ 𝔞 : 𝔞_{1} ∩ (𝔞_{2} + V) is relatively compact

for any compact subset V ⊂ 𝔞}.

If a finite group W acts linearly on 𝔞, we set

P(𝔞)^{W} := {𝔞′ ∈ P(𝔞) : 𝔞′ is stable under the action of W } ,
⋔ (𝔞_{1} : 𝔞)^{W} := ⋔ (𝔞_{1} : 𝔞) ∩ P(𝔞)^{W} .

The following Lemma is obvious from the definition of the pitchfork symbol ⋔ ( : ) and from Lemma 2.2 (3):

Lemma 5.3.1. Suppose 𝔞_{1} ∈ P(𝔞)^{W} . Then 𝔞_{2} ∈ ⋔ (𝔞_{1} : 𝔞) if and only if W · \overline{𝔞_{2}} ∈ ⋔ (𝔞_{1} : 𝔞)^{W} .

Lemma 5.3.2. Retain the above setting. Suppose W is a finite group acting linearly on 𝔞.

1) Let 𝔞_{1}, 𝔞_{1}′ ∈ P(𝔞). Then ⋔ (𝔞_{1} : 𝔞) = ⋔ (𝔞_{1}′ : 𝔞) if and only if 𝔞_{1} ∼ 𝔞_{1}′ in 𝔞.

2) Let 𝔞_{1}, 𝔞_{1}′ ∈ P(𝔞)^{W} . Then ⋔ (𝔞_{1} : 𝔞)^{W} = ⋔ (𝔞_{1}′ : 𝔞)^{W} if and only if 𝔞_{1} ∼ 𝔞_{1}′ in 𝔞. Proof. 1) If 𝔞_{1} ∼ 𝔞_{1}′ in 𝔞, then ⋔ (𝔞_{1} : 𝔞) = ⋔ (𝔞_{1}′ : 𝔞) by Lemma 2.2 (2). Let us prove the converse implication. Assume 𝔞_{1} ≁ 𝔞_{1}′ in 𝔞. Let us show ⋔ (𝔞_{1} : 𝔞) ≠ ⋔ (𝔞_{1}′ : 𝔞). Without loss of generality, we may assume 𝔞_{1} ⊄ 𝔞_{1}′ +B_{n} for all n ∈ ℕ, where B_{n} = {X ∈ 𝔞 : |X| < n}. Then we find x_{n} ∈ 𝔞_{1} such that inf_{y∈𝔞_{1}′} |x_{n} − y| ≥ n for each n ∈ ℕ. We set 𝔞_{2} := {x_{j} ∈ 𝔞_{1} : j ∈ ℕ}. Then we have

𝔞_{2} ∩ (B_{n} + 𝔞_{1}′ + (−B_{n})) ⊂ {x_{j} : 1 ≤ j ≤ 2n} .

Therefore 𝔞_{2} ∈ ⋔ (𝔞_{1}′ : 𝔞). On the other hand, 𝔞_{2} ∉ ⋔ (𝔞_{1} : 𝔞) because 𝔞_{2} is a noncompact subset of 𝔞_{1} . Hence ⋔ (𝔞_{1} : 𝔞) ≠ ⋔ (𝔞_{1}′ : 𝔞).

2) The proof is similar. We only need to replace the definition of 𝔞_{2} in the above proof by 𝔞_{2} := {w · x_{j} : w ∈ W, j ∈ ℕ} ∈ P(𝔞)^{W} .

Now we return to our setting where G is a real reductive linear Lie group. Retain the notation of §3.1.

Lemma 5.4. Let G be a real reductive linear Lie group. Then the mapping

Ψ: P(G) → P(𝔞)^{W_{G}}, H ↦ 𝔞(H)

is well-defined and surjective. Proof. The mapping Ψ is well-defined because of Lemma 3.3 (1). If 𝔞′ ∈ P(𝔞)^{W_{G}}, then we put H := exp(𝔞′) ⊂ G. Then we have 𝔞(H) = 𝔞′, proving the surjectivity of Ψ.

Lemma 5.5. Suppose H is a subset of a real reductive linear Lie group G. Then we have

(5.5.1)  Ψ (⋔ (H : G)) = ⋔ (𝔞(H) : 𝔞)^{W_{G}},
(5.5.2)  Ψ^{−1} (⋔ (𝔞(H) : 𝔞)^{W_{G}}) = ⋔ (H : G).

Proof. The equivalence (1) ⇔ (2) in Theorem 3.4 implies the following relations:

\begin{aligned}Ψ (⋔ (H : G)) & ⊂ ⋔ (𝔞(H) : 𝔞)^{W_{G}} \\ Ψ^{−1} (⋔ (𝔞(H) : 𝔞)^{W_{G}}) & ⊂ ⋔ (H : G). \end{aligned}

Then we have ⋔ (H : G) ⊂ Ψ^{−1}Ψ(⋔ (H : G)) ⊂ Ψ^{−1}(⋔ (𝔞(H) : 𝔞)^{W_{G}}), showing (5.5.2). Because Ψ: P(G) → P(𝔞)^{W_{G}} is surjective, we have

⋔ (𝔞(H) : 𝔞)^{W_{G}} = ΨΨ^{−1}(⋔ (𝔞(H) : 𝔞)^{W_{G}}) ⊂ Ψ(⋔ (H : G)),

showing (5.5.1).

Now we are ready to prove a duality theorem:

Theorem 5.6. Let G be a real reductive linear Lie group, and H and H′ subsets. Then the following four conditions are equivalent: (5.6.1) H ∼ H′ in G. (5.6.2) 𝔞(H) ∼ 𝔞(H′) in 𝔞. (5.6.3) ⋔ (H : G) = ⋔ (H′ : G). (5.6.4) ⋔ (𝔞(H) : 𝔞)^{W_{G}} = ⋔ (𝔞(H′) : 𝔞)^{W_{G}}.

Proof. (2) ⇒ (1): Suppose H, H′ ∈ P(G) satisfy 𝔞(H) ∼ 𝔞(H′) in 𝔞, equivalently, A(H) ∼ A(H′) in A. Then it follows from Lemma 2.3 (1) that A(H) ∼ A(H′) in G. Since H ∼ A(H) and H′ ∼ A(H′) in G, we have H ∼ H′ in G.

(1) ⇒ (3): Suppose H ∼ H′ in G. Then we have ⋔ (H : G) = ⋔ (H′ : G) ⊂ P(G) from Lemma 2.2 (2).

(3) ⇒ (4): Applying Ψ to (5.6.3), we have ⋔ (𝔞(H) : 𝔞)^{W_{G}} = ⋔ (𝔞(H′) : 𝔞)^{W_{G}} by Lemma 5.5.

(4) ⇒ (2): This part is proved in Lemma 5.3.2.

The implication (1) ⇒ (2) in Theorem 5.6 gives a uniform estimate of the Cartan decomposition. It is also related to a uniform estimate of eigenvalues of matrices under perturbation, through the transform g ↦ gθ(g^{−1}), namely, kak′ ↦ ka^{2}k^{−1} for k, k′ ∈ K and a ∈ A. In this direction, there have been extensive studies by H. Weyl, P. Lax, and so on, with the following prototype of the inequalities due to H. Weyl: Let A, B be Hermitian matrices with eigenvalues α_{1} ≥ … ≥ α_{n} and β_{1} ≥ … ≥ β_{n}, respectively. Then we have

max |α_{k} − β_{k}| ≤ ∥A − B∥.

We end this section with a clarification of Theorem 3.4 and a part of Theorem 5.6 by the following reformulation:

Let [P(G)] be the set of equivalence class of P(G) defined by the equivalence relation in Definition 2.1.1. Similarly, [P(𝔞)^{W_{G}}] denotes P(𝔞)^{W_{G}}, where the role of G in Definition 2.1.1 is replaced by an abelian Lie group 𝔞. We fix a subset H of G. Theorem 3.4 and the equivalence of (5.6.1) and (5.6.2) in Theorem 5.6 can be stated in the well-definedness and the bijection of \overline{Ψ}: [P(G)] ⟶^{∼} [P(𝔞)^{W_{G}}] in the following:

The commutative diagram

Ψ: P(G) ↠ P(𝔞)^{W_{G}}
∪  ∪
Ψ: ⋔ (H : G) ↠ ⋔ (𝔞(H) : 𝔞)

induces

\overline{Ψ}: [P(G)] ⟶^{∼} [P(𝔞)^{W_{G}}]
∪  ∪
\overline{Ψ}: [⋔ (H : G)] ⟶^{∼} [⋔ (𝔞(H) : 𝔞)^{W_{G}}].

6. Examples

Example 6.1. If H is a subgroup subgroup of G. Assume that there exists a Cartan involution θ of G such that θH = H. Then H is a reductive subgroup. Let 𝔞_{H} be a maximally abelian subspace of 𝔥 ∩ 𝔭. We take a maximally abelian subspace 𝔞 of 𝔭 such that 𝔞_{H} ⊂ 𝔞. Then 𝔞(H) = W_{G} · 𝔞_{H} (cf. [6]).

Example 6.2. Let G be a real reductive linear Lie group, and Γ ⊂ G a cyclic group. Then there exist H, H′ ∈ 𝔞 such that

(6.2.1)  𝔞(Γ) ∼ W_{G} · {xH + sinh^{−1}(x)H′ : x ∈ ℤ} in 𝔞/W_{G}.

Proof. Let γ be a generator of Γ, γ = γ_{e}γ_{h}γ_{u} be its complete multiplicative Jordan decomposition (see [5] Chapter IX), where γ_{e}, γ_{h} and γ_{u} are elliptic, hyperbolic and unipotent, respectively, and all three commute. Because {γ_{e}^{n} : n ∈ ℤ} is relatively compact, we have Γ ∼ {γ_{h}^{n}γ_{u}^{n} : n ∈ ℤ} in G. It follows from Theorem 5.6 that

𝔞(Γ) ∼ 𝔞 ({γ_{h}^{n}γ_{u}^{n} : n ∈ ℤ}) in𝔞.

Thus we may and do assume γ_{e} = 1. Let L := Z_{G}(γ_{h}), which is reductive in G. We note that γ_{u} ∈ L. By a theorem of Jacobson-Morozov, there is a Lie group

homomorphism ψ: SL(2, ℝ) → L such that ψ ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) = γ_{u}. There is a Cartan

involution θ of G such that θψ(SL(2, ℝ)) = ψ(SL(2, ℝ)) (see [5], p.277), and such that θL = L. We write the corresponding Cartan decomposition 𝔤 = 𝔨 + 𝔭. Let H ∈ 𝔭∩𝔩 be the unique element such that γ_{h} = exp H. We take a maximally abelian subspace 𝔞 for L which contains H′ := dψ(diag(1, −1)) (∈ dψ(𝔰𝔩(2, ℝ)). Since H is contained in the center of 𝔩, we have H ∈ 𝔞. Then we have

γ_{h}^{n}γ_{u}^{n} ∈ (L ∩ K) · exp(nH + sinh^{−1}(n)H′) · (L ∩ K).

thanks to the following lemma. Hence, we have (6.2.1).

Lemma 6.3. Let G = SL(2, ℝ), K = SO(2). We take a maximally abelian subspace 𝔞 := {diag(y, −y) : y ∈ ℝ} ⊂ 𝔰𝔩(2, ℝ). Then the Cartan projection of

( \begin{matrix}1 & 2x \\ 0 & 1 \end{matrix} ) ∈ G amounts to

𝔞 \{( \begin{matrix}1 & 2x \\ 0 & 1 \end{matrix} )\} = \{( \begin{matrix}sinh^{−1} x & 0 \\ 0 & − sinh^{−1} x \end{matrix} ) , ( \begin{matrix}− sinh^{−1} x & 0 \\ 0 & sinh^{−1} x \end{matrix} )\} ⊂ 𝔞.

Proof. We put θ := −\frac{1}{2} tan^{−1}(\frac{1}{x}) ∈ (−\frac{π}{2} , \frac{π}{2} ) for x ≠ 0. Then we have 2x = −2 cot 2θ = tan θ − cot θ and so exp(sinh^{−1}(x)) = tan θ. Now, the Lemma is a direct consequence of the following formula

( \begin{matrix}1 & 2x \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}sin θ & − cos θ \\ cos θ & sin θ \end{matrix} ) ( \begin{matrix}tan θ & 0 \\ 0 & cot θ \end{matrix} ) ( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} )

Kobayashi 161

In the following two examples, we let 𝔽 = ℝ or ℂ and G := GL(m + n, 𝔽). We fix a maximally abelian subspace

𝔞_{G} := {diag(a_{1}, …, a_{m+n}) : a_{j} ∈ ℝ (1 ≤ j ≤ m + n)} ,

on which W_{G} ≃ 𝔖_{m+n} acts by permutation of coordinates. The following example gives a simpler proof of one of the main results in [4]. Example 6.4 (Friedland). Consider a subgroup H = GL(m, 𝔽) ≃ GL(m, 𝔽) × I_{n} of G = GL(m + n, 𝔽). Since H is reductive in G, it follows from Example 6.1 that

𝔞(H) = W_{G} · {diag(a_{1}, …, a_{m}, 0, …, 0) : a_{j} ∈ ℝ (1 ≤ j ≤ m)} .

In particular, a discrete subgroup Γ of G acts properly discontinuously on G/H = GL(m + n, 𝔽)/GL(m, 𝔽) if and only if 𝔞(Γ) ∩ {diag(a_{1}, …, a_{m}, b_{1}, …, b_{n}) : a_{j} ∈ ℝ (1 ≤ j ≤ m), |b_{j}| ≤ C (1 ≤ j ≤ n)} is a finite set for any C > 0. Example 6.5. Consider a subgroup N := exp(𝔫) of G = GL(m + n, 𝔽), where 𝔫 := {(X_{ij}) ∈ M(m + n, 𝔽) : X_{ij} = 0 if i ≥ m or j ≤ m} ⊂ 𝔤𝔩(m + n, 𝔽). Put k := min(m, n). In order to compute 𝔞(N), we define a k -dimensional subalgebra by

𝔫′ := {(X_{ij}) ∈ 𝔫 : X_{ij} = 0 if j − i ≠ m} ,

and define subgroups of G by N′ := exp(𝔫′) and L_{K} := U(m; 𝔽) × U(n; 𝔽). In view of 𝔫 ≃ M(m, n; 𝔽) = {AXB ∈ M(m, n; 𝔽) : A ∈ U(m; 𝔽), X ∈ 𝔫′, B ∈ U(n; 𝔽)} , we have KNK = KN′K and therefore 𝔞(N) = 𝔞(N′). By using Lemma 6.3, we conclude

𝔞(N) = W_{G} · {diag(a_{1}, −a_{1}, …, a_{k}, −a_{k}, 0, …, 0) : a_{j} ∈ ℝ (1 ≤ j ≤ k)} . Thus, a subgroup Γ of G acts properly discontinuously on G/N if and only if

𝔞(Γ) ∩ {diag(a_{1}, …, a_{2k}, b_{1}, …, b_{|n−m|}) : |a_{2i−1} + a_{2i}| ≤ C, |b_{j}| ≤ C (∀i, j)} is a finite set for any C > 0. It might be remarkable that 𝔞(N) (a nilpotent subgroup) coincides with 𝔞(U(m, n; 𝔽)) (a reductive subgroup).

References

[1] Borel, A., Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122.

[2] Borel, A. and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962), 485–535.

[3] Calabi, E. and L. Markus, Relativistic space forms, Ann. of Math. 75 (1962), 63–76.

[4] Friedland, S., Properly discontinuous groups on certain matrix homogeneous spaces, Preprint, 1994.

[5] Helgason, S., “Differential Geometry, Lie Groups and Symmetric Spaces,” Academic Press, New York etc., 1978.

[6] Kobayashi, T. Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), 249–263.

[7] —, Discontinuous groups acting on homogeneous spaces of reductive type, in: T. Kawazoe, T. Oshima and S. Sano, Eds., Proceedings of the Conference on Representation Theory of Lie Groups and Lie Algebras at Fuji-Kawaguchiko, held in 1990, World Scientific, Singapore etc., 1992, 59–75.

[8] —, A necessary condition for the existence of compact Clifford-Klein forms of homogeneous spaces of reductive type, Duke Math. J. 67 (1992), 653–664.

[9] —, On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geometry and Physics 12 (1993), 133–144.

[10] —, Discontinuous groups and Clifford-Klein forms of pseudo-Riemannian homogeneous manifolds, 70 pp., in: H. Schlichtkrull and B. Ørsted, Eds., Lecture Notes in the European School on Group Theory held at Sandbjerg Gods’ in August 1994, Algebraic and Analytic Methods in Representation Theory, Perspectives in Mathematics, Academic Press, New York etc., to appear.

[11] —, Deformation of compact Clifford-Klein forms of indefinite Riemannian homogeneous manifolds, Preprint, Institut Mittag-Leffler Report 32, (1995/96).

[12] Kulkarni, R. S., Proper actions and pseudo-Riemannian space forms, Advances in Math. 40 (1981), 10–51.

[13] Lipsman, R. L., Proper actions and a compactness condition, Journal of Lie Theory 5 (1995), 25–39.

[14] Margulis, G., “Discrete subgroups of semisimple Lie groups,” Ergebnisse der Math. 17, Springer Verlag, 1991.

[15] Mostow, G. D., and T. Tamagawa, On the compactness of arithmetically defined homogeneous spaces, Ann. of Math. 76 (1962), 446–463.

[16] Wallach, N. R., Two problems in the theory of automorphic forms, “Open Problems in Representation Theory,” Proceedings at Katata held in 1986, 1988, 39–40.

[17] Warner, G., “Harmonic Analysis on Semisimple Lie Groups I and II,” Springer Verlag, Berlin etc., 1972.

[18] Wolf, J. A., The Clifford-Klein space forms of indefinite metric, Ann. of Math. 75 (1962), 77–80.

[19] —, Isotropic manifolds of indefinite metric, Comment. Math. Helv. 39 (1964), 21-64.

[20] —, “Spaces of constant curvature,” 5-th edition, Publish or Perish, Inc., Wilmington, Delaware, 1984.

Toshiyuki Kobayashi Graduate School of Mathematical Sciences University of Tokyo Komaba, Meguro, Tokyo 153 Japan and Institut Mittag-Leffler The Royal Swedish Academy of Sciences Auravägen 17 S-182 62 Djursholm, Sweden toshi@ms.u-tokyo.ac.jp

Received December 28, 1995 and in final form June 29, 1996