Journal of Lie Theory Volume 10 (2000) 127–146 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag Berlin

Noncompact, almost simple groups operating on locally compact, connected translation planes

Harald Löwe

Communicated by J. Hilgert

Abstract. Let 𝔼 be a locally compact, connected translation plane. The aim of this paper is a detailed investigation of noncompact, connected, almost simple subgroups of the point stabilizer 𝔾_{0} of 𝔼. It turns out that the only possible groups Δ of this kind are 2–fold covering groups of PSO_{m}(ℝ, 1) for 3 ≤ m ≤ 10. Moreover, the nontrivial central element of Δ is the reflection at 0. Furthermore, we will show the existence of an orbit S (called the weight sphere of Δ) homeomorphic to an (m − 2)–sphere on which the action of Δ is equivalent to the natural action of PSO_{m}(ℝ, 1) on 𝕊_{m−2} . This weight sphere S is characterized as the set of those lines in L_{0} whose stabilizer in Δ is a minimal parabolic subgroup of Δ.

As a by–product we prove that a semisimple subgroup of 𝔾_{0} always has real rank 0 or 1.

1. Introduction

Locally Compact, Connected Translation Planes. A projective plane is called topological if its point space and its line space are endowed with Hausdorff topologies such that the operations of joining points and intersecting lines are continuous. As in the theory of topological groups one obtains the nicest results under the additional assumption that these topologies are locally compact and connected. Examples are the projective planes over ℝ, ℂ, ℍ (quaternions) and 𝕆 (octonions), the so–called “classical planes”. For a detailed introduction we refer the reader to [10].

We are mainly interested in a special kind of locally compact, connected projective planes: Translation planes are distinguished by the existence of a line L_{∞} such that the automorphisms fixing exactly the points on L_{∞} form a group which is transitive on the set of points outside L_{∞}. This group is called the translation group with translation axis L_{∞}. The point space of a locally compact, connected translation plane is a manifold of dimension n ∈ {2, 4, 8, 16}; moreover, the projective lines (regarded as subspaces of the point set) are spheres of dimension l = n/2 (cf. [10, 64.1]). In the sequel we refer to n as the dimension of the plane.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

For a non–classical locally compact, connected translation plane one knows that every (continuous) automorphism leaves the translation axis L_{∞} invariant ([10, 64.4(c)]). Therefore, we can pass to the corresponding affine plane without losing automorphisms. The advantage of this procedure is a linearization: The affine point space P is in a natural way a right vector space of even dimension over the kernel 𝕂 of a coordinatizing quasifield and, moreover, 𝕂 is a topological field isomorphic to ℝ, ℂ or ℍ ([10, 42.6]). The translation group coincides with the group of vector translations x ↦ x+v of P. The line pencil L_{0} = {L ∈ L | 0 ∈ L} (where L denotes the set of affine lines) consists of vector subspaces of P of dimension (dim_{𝕂} P)/2. The other lines are exactly the affine cosets of the elements of L_{0}. It follows immediately that a locally compact, connected translation plane of dimension 2 is isomorphic to the real plane.

The group 𝔾 of all (continuous) automorphisms of (P, L) is a Lie group with respect to the compact–open topology ([10, 44.6]). Clearly, 𝔾 is a semidirect product of the stabilizer 𝔾_{0} of the origin 0 and the translation group. All elements of 𝔾_{0} are semilinear maps of the 𝕂–vector space P. The kernel 𝕂 is encoded in 𝔾_{0}: The normal subgroup 𝔾_{[0,L_{∞}]} (consisting of all those elements of 𝔾_{0} which leave the translation axis pointwise fixed) is precisely the group {x ↦ xa | a ∈ 𝕂, a ≠ 0}. The connected component of 𝔾_{0} is an almost direct product of the connected component of the so–called reduced stabilizer ^{1}

S𝔾_{0} = {γ ∈ 𝔾_{0} | γ is 𝕂–linear and det_{𝕂}γ = 1}

and the connected component of 𝔾_{[0,L_{∞}]}, see the considerations in [10, 81c].

Remark 1.1. According to [10, 42.6], the group 𝔾_{[0,L_{∞}]} is isomorphic to ℝ^{×}, ℂ^{×} or ℍ^{×} = ℝ_{pos} × Spin_{3}ℝ. We conclude that a noncompact, connected, almost simple subgroup of 𝔾_{0} is contained in the reduced stabilizer S𝔾_{0}.

We shall interpret a locally compact, connected affine translation plane 𝔼 = (P, L) as follows: Since 𝕂 can be considered as a real vector space, P is a real vector space, too. The set L_{0} consists of vector subspaces of dimension l = (dim P)/2. It is easy to see that L_{0} has to be a spread, i.e. L_{0} covers P and any two distinct elements K, L ∈ L_{0} satisfy P = K ⊕ L. The topology of L_{0} as a subspace of L and the topology of L_{0} as a subspace of the Grassmannian manifold G_{l}(P) of all l–dimensional vector subspaces of P coincide ([10, 64.4 (a)])^{2}. The reduced stabilizer S𝔾_{0} is a closed subgroup of SL(P) ([10, 44.6]).

The results given in this paper rest on a deep theorem due to Hähl which describes the structure of closed subgroups of S𝔾_{0} fixing two different lines in L_{0}, see [10, 81.8]:

Theorem 1.2. Let Γ ≤ S𝔾_{0} be a closed, connected subgroup which fixes two distinct lines W, S ∈ L_{0}. Consider the normal subgroup

K = {γ ∈ Γ | | det(γ|_{W})| = | det(γ|_{S})| = 1},

1If 𝕂 = ℍ, then the plane is isomorphic to the classical plane over ℍ ([2, Thm.1]); in this case the determinant det_{ℍ} γ is understood to be the real determinant of γ and hence the reduced stabilizer equals SL_{2}ℍ in its usual operation on the affine quaternion plane.

2Conversely, if S ⊆ G_{l}(P) is a compact spread, then S defines a locally compact, connected translation plane ([10, 64.4(d)]).

where det denotes the real determinant.

Then K is the largest compact subgroup of Γ. If Γ is not compact, then Γ is a direct product Γ = K × Y of K and a closed one–dimensional subgroup Y isomorphic to ℝ. We observe that K is connected.

Moreover, for any closed one–dimensional subgroup Y isomorphic to ℝ, there is an isomorphism ρ : ℝ → Y having the following property: For t → −∞, the maps ρ(t) converge to the constant map L_{0} \ {S} → {W } uniformly on each compact subset of L_{0} \ {S}. For t → ∞, an analogous convergence property holds with the rôles of W and S interchanged.

Remark. (a) Let Γ ≤ S𝔾_{0} as above. Then a closed one–dimensional subgroup Y ≤ Γ isomorphic to ℝ will be called a compression subgroup of Γ.

(b) If Y ≤ Γ is a compression subgroup, then for every isomorphism ρ : ℝ → Y we have either

lim_{t→∞} ρ(t)(L) = W and lim_{t→−∞} ρ(t)(L) = S for all L ∈ L_{0} \ {W, S}, or

lim_{t→∞} ρ(t)(L) = S and lim_{t→−∞} ρ(t)(L) = W for all L ∈ L_{0} \ {W, S}.

Notation. Throughout this paper, let 𝔼 = (P, L) be a locally compact, connected affine translation plane of dimension n = 2l (where l ∈ {1, 2, 4, 8}) with reduced stabilizer S𝔾_{0}.

For a Lie group G, we denote its neutral element by e. Moreover, G^{e} is the connected component of e in G and T_{e}G is the Lie algebra of G. If 𝔥 ≤ T_{e}G is a subalgebra, then we write exp 𝔥 for the connected Lie subgroup of G with Lie algebra 𝔥.

If V is a real vector space of finite dimension, then GL(V) will denote the Lie group of invertible linear maps of V . As usual, SL(V) is the group of linear maps of V with determinant 1. Notice that the Lie algebra 𝔤𝔩(V) of GL(V) consists of all linear maps of V . The Lie algebra 𝔰𝔩(V) of SL(V) consists of all linear maps with vanishing trace.

A diagonal matrix with diagonal entries ξ_{1}, …, ξ_{n} will be abbreviated by diag (ξ_{1}, …, ξ_{n}).

2. Results

The aim of this and two subsequent papers^{3} is to investigate the structure of a noncompact, connected subgroup Δ of the reduced stabilizer of a locally compact, connected translation plane. The first step in this direction deals with almost simple groups and shows that the possibilities are very restricted in this case:

Theorem A. Consider a connected, almost simple subgroup Δ of 𝔾_{0}. Let U ≠ {0} be a Δ–irreducible subspace of the point space P. If Δ is not compact, then only the following possibilities can occur:

3Löwe, H.: Noncompact subgroups of the reduced stabilizer of a locally compact, connected translation plane and Parabolic collineation groups of locally compact, connected translation planes, in preparation

(1) Δ = SL_{2}ℝ and dim U is even. If n = 2, then 𝔼 is isomorphic to the real affine plane. For n = 4 all possible planes are known, see [10, 73.22]

(2) Δ = SL_{2}ℂ and dim U is divisible by 4. If n = 4, then 𝔼 is isomorphic to the complex plane. If n = 8, then dim U = 4 and all possible planes are known, see [3].

(3) Δ = U_{2}(ℍ, 1) and dim U = 8. If n = 8, then ℍ is the quaternion plane.

(4) Δ = SL_{2}ℍ and dim U = 8. If n = 8, then ℍ is the quaternion plane^{4}.

(5) Δ = Spin_{k}(ℝ, 1) for some 7 ≤ k ≤ 10 and dim U = 16. In this case 𝔼 is isomorphic to the octonion plane.

In any case Δ is a 2–fold covering group of the connected component PSO_{m}^{e}(ℝ, 1) of PSO_{m}(ℝ, 1) for some^{5} m, 3 ≤ m ≤ 10 and its nontrivial central element acts as the reflection x ↦ −x at the origin.

The proof is surprisingly easy if one uses real Cartan subgroups of Δ. Roughly speaking a real Cartan subgroup C is a maximal Ad–diagonalizable subgroup of Δ. It turns out that C fixes exactly two lines W, S ∈ L_{0} and, moreover, is a compression subgroup of Δ_{W,S}. According to (1.2) the dimension of C (called the real rank of Δ) equals 1. The assertions can now be obtained by studying representations of groups of real rank 1.

We point out that each group Δ occuring in Theorem A acts on a classical plane. If Δ is not SL_{2}ℝ or SL_{2}ℂ, then the representation of Δ on the point space P of an arbitrary plane is equivalent to the representation of Δ on the classical plane of dimension dim P. It remains an open problem whether this is true for Δ = SL_{2}ℂ. For Δ = SL_{2}ℝ, the analogous statement is not true: Betten constructed an example of a 4–dimensional translation plane on which SL_{2}ℝ acts in its 4–dimensional irreducible representation, cf. [1].

However, there is another analogy between the classical and the non– classical case: A noncompact, almost simple subgroup Δ of the stabilizer of a classical plane has exactly one closed orbit S on L_{0}. This orbit is homeomorphic to a sphere and can be characterized as the set of lines in L_{0} which are fixed by some parabolic subgroup of Δ. From this point of view there is no difference between the classical and the non–classical case:

Theorem B. Let Δ ≤ 𝔾_{0} be a connected, noncompact, almost simple group. Then Δ has precisely one closed orbit S in L_{0}. Let 3 ≤ m ≤ 10 be the integer such that Δ is a 2–fold covering group of the connected component PSO_{m}^{e}(ℝ, 1) of PSO_{m}(ℝ, 1). (The existence of m is guaranteed by Theorem A). Then S satisfies the following assertions:

(1) S is homeomorphic to a sphere of dimension m − 2.

4For n = 16, the classification of all possible planes admitting a group isomorphic to SL_{2}ℍ will be given in another paper. It turns out that — except in the case of the octonion plane — the automorphism groups of these planes have dimension at most 35.

5In particular, m = 3 for Δ = SL_{2}ℝ, m = 4 for Δ = SL_{2}ℂ, m = 5 for Δ = U_{2}(ℍ, 1), and m = 6 for Δ = SL_{2}ℍ.

(2) The action of Δ on S is equivalent to the natural action of PSO_{m}^{e}(ℝ, 1) = Δ/{±e} on the sphere

𝕊_{m−2} = {ℝ · (x_{1}, x_{2}, …, x_{m−1}, 1) | x_{1}^{2} + x_{2}^{2} + … + x_{m−1}^{2} = 1}

contained in the real projective (m − 1)–space. In particular, Δ is doubly transitive on S for m = 3 and triply transitive for m ≥ 4.

(3) If m ≩ l/2 (where l is the dimension of a line), then 𝔼 is isomorphic to the classical translation plane of dimension n = 2l and thus Δ is a subgroup of Spin_{l+2}(ℝ, 1).

Consider a line L ∈ S. Then L is fixed by some parabolic subgroup Π of Δ, see (7.3), and thus is fixed by the real Cartan subgroup C ≤ Π, too. It turns out that L can be considered as the sum of those real weight spaces of Δ belonging to positive real weights^{6}. For this reason we call L a weight line and S — being the set of all weight lines — the weight sphere of Δ.

Organization of this Paper. We start with a short introduction to semisimple real Lie algebras.

In Section 4 we investigate arbitrary diagonalizable subgroups of S𝔾_{0}. The aim is to show that these groups have dimension 1 and fix exactly two lines in L_{0}. These lines can be recovered from the “real weight spaces” of the group. The latter is a consequence of a general result concerning arbitrary compression subgroups of S𝔾_{0}, cf. (4.1).

Since a real Cartan subgroup of a noncompact, semisimple group Δ ≤ 𝔾_{0} is diagonalizable on P and contained in S𝔾_{0}, we apply these results in Section 5 and obtain the main tools for the proof of Theorem A. The proof itself will be given at the end of Section 6, which is devoted to the investigation of groups of real rank 1.

The action of a noncompact, connected, almost simple group on L_{0} is the subject of the last chapter, where we will prove the assertion concerning the weight sphere stated in Theorem B.

3. Semisimple Real Lie Algebras

For basic facts concerning semisimple real Lie algebras we refer to Knapp [6]. Throughout this section let 𝔤 be a semisimple real Lie algebra.

A subalgebra 𝔞 ≤ 𝔤 is called a real Cartan subalgebra if 𝔞 is contained in the Cartan complement 𝔭 of some Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭 of 𝔤 and if it is maximal among the subalgebras of 𝔤 contained in 𝔭.

Since [𝔭; 𝔭] is a subspace of 𝔨, every real Cartan subalgebra of 𝔤 is abelian. If 𝔤 = 𝔨 ⊕ 𝔭 is a fixed Cartan decomposition, then a real Cartan subalgebra 𝔞 contained in 𝔭 is often called a maximal abelian subspace of 𝔭 in the literature.

We remark that 𝔞 is a real Cartan subalgebra if, and only if, 𝔞 is the abelian part of some Iwasawa decomposition of 𝔤. (This follows directly from the construction of the Iwasawa decomposition.) Moreover, real Cartan subalgebras are characterized as the maximal ad–diagonalizable subalgebras of 𝔤.

6For real weights and real weight spaces see the Sect. 3.

As a direct consequence of [6, 6.19] and [6, 6.51] we obtain the following result:

Proposition 3.1. Any two real Cartan subalgebras of 𝔤 are conjugate by an inner automorphism of 𝔤.

In particular, any two real Cartan subalgebras of 𝔤 have same dimension. We refer to this dimension as the real rank rk_{ℝ}𝔤 of 𝔤. Notice that the real rank of 𝔤 equals 0 if, and only if, 𝔤 is a compact (semisimple) Lie algebra.

Proposition 3.2. [9, Sec. 4.1] Let 𝔤_{1} and 𝔤_{2} be semisimple Lie algebras. Then rk_{ℝ}(𝔤_{1} ⊕ 𝔤_{2}) = rk_{ℝ}𝔤_{1} + rk_{ℝ}𝔤_{2}.

We shall now study faithful representations φ : 𝔤 → 𝔤𝔩(V) of a semisimple real Lie algebra 𝔤 on a finite dimensional real vector space V . Fix a real Cartan subalgebra 𝔞 of 𝔤. A real weight of φ (with respect to 𝔞) is a linear map ω : 𝔞 → ℝ such that the corresponding real weight space

V_{ω} := {x ∈ V | φ(A)(x) = ω(A) · x for all A ∈ 𝔞}

contains nonzero elements.

Remark 3.3. Suppose (in addition to the preceding assumptions) that rk_{ℝ}𝔤 = 1. Then 𝔞 = ℝ · X holds for every X ∈ 𝔞 \ {0}. Consequently, a real weight ω : 𝔞 → ℝ of the representation φ is uniquely determined by the eigenvalue ω(X) of φ(X). Moreover, the real weight space V_{ω} and the eigenspace of φ(X) belonging to ω(X) coincide. We will identify ω and ω(X) in this situation: A real weight of φ with respect to X is an eigenvalue of φ(X).

The real weights of φ are intimately related to the weights of the complexification φ_{ℂ} : 𝔤_{ℂ} → 𝔤𝔩(V_{ℂ}): According to [9, Chap.4, Sec.4.1], we find a Cartan subalgebra 𝔥 of 𝔤_{ℂ} containing 𝔞. Then every weight λ : 𝔥 → ℂ of φ_{ℂ} is real on 𝔞, cp. the proof of Prop. 4.3 in [9, Chap.4]. Since φ_{ℂ}(𝔥) is diagonalizable on V_{ℂ}, we obtain that φ(𝔞) is diagonalizable on V , i.e. φ(𝔞) consists of diagonal matrices with real entries with respect to a properly chosen basis of V . Thus, the vector space V is a direct sum of the real weight spaces of φ. One easily derives that

V_{ω} = V ∩ ⨁_{λ∈Δ;λ|_{𝔞}=ω}(V_{ℂ})_{λ}

holds for every real weight ω, where Δ is the set of weights of φ_{ℂ} and where (V_{ℂ})_{λ} is the weight space corresponding to the weight λ ∈ Δ. It follows that if ω is a real weight, then −ω is a real weight, too, and the dimensions of the corresponding real weight spaces V_{ω} and V_{−ω} coincide. We record these facts:

Proposition 3.4. Let 𝔤 be a semisimple real Lie algebra and let 𝔞 ≤ 𝔤 be a real Cartan subalgebra. Consider a representation φ of 𝔤 on some finite dimensional real vector space V . Let Λ be the set of real weights of φ with respect to 𝔞.

(a) φ(𝔞) is diagonalizable on V . Therefore, V = ⨁_{ω∈Λ} V_{ω} is the direct sum of real

weight spaces.

(b) If ω is a real weight, then −ω is a real weight, too. Moreover, dim V_{ω} = dim V_{−ω} holds for every ω ∈ Λ.

We should finally introduce the global counterpart to real Cartan subalgebras: Let Δ be a semisimple Lie group (preferably connected). A connected subgroup C ≤ Δ is called a real Cartan subgroup if, and only if, its Lie algebra T_{e}C is a real Cartan subalgebra of T_{e}Δ. The real rank rk_{ℝ}Δ of Δ is defined to be the real rank of T_{e}Δ.

We mention here that a real Cartan subgroup C ≤ Δ is always closed in Δ: Since ad T_{e}C is diagonalizable on T_{e}Δ, the group Ad C is diagonalizable, too, and hence Ad C is closed in SL(T_{e}Δ). This proves our claim.

4. Compression Subgroups and Diagonalizable Subgroups of the Reduced Stabilizer

Let 𝔼 = (P, L) be a locally compact, connected affine translation plane of dimension n = 2l and denote its reduced stabilizer by S𝔾_{0}.

Let Φ ≤ S𝔾_{W,S} be a compression subgroup with respect to the lines W, S ∈ L_{0}, W ≠ S. Note that dim Φ = 1. Choose a generator T ∈ 𝔤𝔩(P) of the Lie algebra of Φ. Then ρ : ℝ → Φ; r ↦ exp(rT) is a parametrization of Φ. In view of (1.2), we have that

lim_{r→∞} ρ(r)(L) = S and lim_{r→−∞} ρ(r)(L) = W

holds for every L ∈ L_{0} \ {W, S}. Moreover, since W and S are Φ–invariant, T fixes both W and S. Thus, T may be written as T = T_{W} + T_{S}, where T_{W} = T |_{W} and T_{S} = T |_{S} are endomorphisms of W and S, respectively.

Let α_{k} + β_{k}i, (k = 1, …, s) be the eigenvalues of T with β_{k} ≥ 0 and let P_{k} be the generalized eigenspace of α_{k} + β_{k}i. In other words, if d_{k} denotes the multiplicity of the eigenvalue α_{k} + β_{k}i, then P_{k} = ker(T − α_{k} · 1l)^{d_{k}} for β_{k} = 0 and P_{k} = ker(T^{2} − 2α_{k}T + (α_{k}^{2} + β_{k}^{2})1l)^{d_{k}} for β_{k} ≠ 0. Notice that α_{k} + β_{k}i is an eigenvalue of T_{W} [of T_{S}] if, and only if, P_{k} ∩ W ≠ {0} [P_{k} ∩ S ≠ {0}], and that every eigenvalue of T_{W} [of T_{S}] is also an eigenvalue of T.

Proposition 4.1. We consider a compression subgroup Φ ≤ S𝔾_{W,S} as above. Choose an index k_{0} such that α_{k_{0}} = max{α_{k} | P_{k} ∩ W ≠ {0}}. Then

W = ⨁_{k; α_{k}≤α_{k_{0}}} P_{k} und S = ⨁_{k; α_{k}≩α_{k_{0}}} P_{k}.

In particular, the real part of any eigenvalue of T_{W} is strictly smaller than the real part of every eigenvalue of T_{S}. Moreover, if α_{k} + β_{k}i is an eigenvalue of T_{W} [of T_{S}], then P_{k} is a subspace of W [of S].

Remark. We point out that we can recover W and S from the compression subgroup Φ.

Proof. (1) Let α_{k} +β_{k}i and α_{j} +β_{j}i be eigenvalues of T_{W} and T_{S}, respectively. First, we will show that α_{k} ≨ α_{j}.

(2) If β_{k} ≠ 0, then there exists a 2–dimensional subspace E_{k} of W such that

T_{E_{k}} = ( \begin{matrix}α_{k} & −β_{k} \\ β_{k} & α_{k} \end{matrix} ) .

Therefore, we obtain the following equation:

ρ(r)|_{E_{k}} = exp(rT)|_{E_{k}} = exp(rT_{E_{k}}) = e^{α_{k}r} · C_{k}(r),

where

C_{k}(r) = ( \begin{matrix}cos(β_{k}r) & − sin(β_{k}r) \\ sin(β_{k}r) & cos(β_{k}r) \end{matrix} )

is an element of the orthogonal group SO(E_{k}). If β_{k} = 0, then there exists a one–dimensional subspace E_{k} of W with

ρ(r)|_{E_{k}} = e^{α_{k}r} · C_{k}(r), where C_{k}(r) ∈ SO(E_{k}) = {1l}.

Analogously, there exists a subspace E_{j} of S such that

ρ(r)|_{E_{j}} = e^{α_{j}r} · C_{j}(r), where C_{j}(r) ∈ SO(E_{j}).

We choose an increasing sequence r_{ν} of real numbers with lim_{ν→∞} r_{ν} = ∞ such that the sequence (C_{k}(r_{ν}), C_{j}(r_{ν})) possesses a limit (D_{k}, D_{j}) in the compact set SO(E_{k}) × SO(E_{j}).

(3) We claim that α_{j} ≩ α_{k}. Aiming at a contradiction, we assume α_{j}−α_{k} ≤ 0 and infer

lim_{ν→∞} e^{(α_{j}−α_{k})r_{ν}} = ε = \{ \begin{matrix}0 & if α_{j} − α_{k} ≨ 0 \\ 1 & if α_{k} = α_{j} \end{matrix} .

Now, we choose x ∈ E_{k} \ {0} and y ∈ E_{j} \ {0}. Then the line L = 0 ∨ (x + y) is different from W and S and therefore lim_{r→∞} ρ(r)(L) = S. In contradiction to this fact, we have that the limit

lim_{ν→∞} ρ(r_{ν})(ℝ · (x + y)) = lim_{ν→∞} ℝ · (e^{α_{k}r_{ν}}C_{k}(r_{ν})(x) + e^{α_{j}r_{ν}}C_{j}(r_{ν})(y))
= lim_{ν→∞} ℝ · (C_{k}(r_{ν})(x) + e^{(α_{j}−α_{k})r_{ν}}C_{j}(r_{ν})(y))
= ℝ · (D_{k}(x) + ε · D_{j}(y))

of the one–dimensional subspaces ρ(r_{ν})(ℝ · (x + y)) of ρ(r)(L) is not contained in S since y ≠ 0.

(4) In view of (3), the real part of any eigenvalue of T_{W} is strictly smaller than the real part of every eigenvalue of T_{S}. By the definition of k_{0}, we obtain that α_{k} + β_{k}i is an eigenvalue of T_{W} if, and only if, α_{k} ≤ α_{k_{0}} . If W_{k} denotes the generalized eigenspace of T_{W} with respect to α_{k} + β_{k}i, then

W = ⨁_{k; α_{k}≤α_{k_{0}}} W_{k} ≤ ⨁_{k; α_{k}≤α_{k_{0}}} P_{k},

since W_{k} is a subspace of P_{k} for every k. Analogously, we have

S = ⨁_{j; α_{j}≩α_{k_{0}}} S_{j} ≤ ⨁_{j; α_{j}≩α_{k_{0}}} P_{j},

where S_{j} denotes the generalized eigenspace of T_{S} with respect to the eigenvalue α_{j} + β_{j}i (with α_{j} ≩ α_{k_{0}} ). From

W ⊕ S = P = ⨁_{k} P_{k}

one derives easily that P_{k} = W_{k} holds for every k with α_{k} ≤ α_{k_{0}} and that S_{j} = P_{j} holds for every j with α_{j} ≩ α_{k_{0}} . This finishes the proof of the proposition.

As a next step, we apply this proposition to diagonalizable subgroups of the reduced stabilizer. To this end, we have to show first that such a group is a compression subgroup:

Proposition 4.2. Let Θ ≤ S𝔾_{0} be a diagonalizable, connected subgroup. Then there exist exactly two distinct Θ–invariant lines W, S ∈ L_{0} and Θ ≤ S𝔾_{W,S} is a compression subgroup. In particular, dim Θ = 1 and we find a parametrization ρ : ℝ → Θ such that

lim_{t→∞} ρ(t)(L) = S and lim_{t→−∞} ρ(t)(L) = W

holds for every L ∈ L_{0} \ {W, S}.

Proof. Clearly, Θ is closed in S𝔾_{0} and isomorphic to ℝ^{m} for some m. The point space P = ℝ^{n} is a direct sum P = X_{1} ⊕ … ⊕ X_{n} of one–dimensional Θ– invariant subspaces. The one–dimensional subspace X_{1} is contained in some line W ∈ L_{0}. For ϑ ∈ Θ, the lines W and ϑ(W) intersect in X_{1} = ϑ(X_{1}) and hence are equal. Consequently, Θ leaves W invariant. Since dim W = l, there exists an index 1 ≤ j ≤ n = 2l such that X_{j} is not a subspace of W. Let S ∈ L_{0} \ {W } be the line containing X_{j}. By the same arguments as above, S is Θ–invariant, too.

We have proved that Θ ≅ ℝ^{m} is a closed subgroup of S𝔾_{W,S}. According to (1.2), the codimension of a maximal compact subgroup of Θ is at most 1. Consequently, m = 1 and Θ ≅ ℝ is a compression subgroup of S𝔾_{W,S}. The remaining assertions follow from (1.2).

Since Θ fixes the lines W, S ∈ L_{0}, we infer that Θ is diagonalizable on both W and S. We choose bases {w_{1}, …, w_{l}} and {s_{1}, …, s_{l}} of W and S, respectively, such that the parametrization ρ described in (4.2) is given by

ρ(t) = diag(e^{λ_{1}t}, … e^{λ_{l}t}, e^{μ_{1}t}, … e^{μ_{l}t}) for t ∈ ℝ

with respect to the basis {w_{1}, …, w_{l}, s_{1}, …, s_{l}} of P. From (4.1) we conclude that λ_{i} ≨ μ_{j} holds for every pair i, j (with 1 ≤ i, j ≤ l). In other words, we have just shown the following:

Proposition 4.3. Let X = {x_{1}, …, x_{n}} be a basis of P and let

ξ_{1} ≤ … ≤ ξ_{n}

be real numbers. For t ∈ ℝ, define the linear map ρ(t) ∈ GL(P) by

ρ(t) = diag(e^{ξ_{1}t}, …, e^{ξ_{n}t}),

where the matrix is given with respect to the basis X.

Suppose that Θ = {ρ(t) | t ∈ ℝ} is a subgroup of the reduced stabilizer S𝔾_{0}. Then the subspaces W and S spanned by {x_{1}, …, x_{l}} and {x_{l+1}, …, x_{n}}, respectively, are Θ–invariant lines. Moreover, ξ_{l} is strictly smaller than ξ_{l+1}, i.e. ξ_{i} ≨ ξ_{l+j} holds for every pair i, j (with 1 ≤ i, j ≤ l).

5. Semisimple Subgroups of 𝔾_{0}

Throughout this section let Δ be a connected, semisimple Lie group and let φ : Δ → 𝔾_{0} be a Lie homomorphism with discrete kernel. In particular, φ is a representation of Δ on the vector space P = ℝ^{n}.

Let 𝔤 = T_{e}Δ be the Lie algebra of Δ. By φ_{⋆} we denote the derivative of φ. Notice that φ_{⋆} : 𝔤 → 𝔤𝔩(P) is a faithful representation of the semisimple Lie algebra 𝔤.

Proposition 5.1. Let Δ be a connected, semisimple Lie group and let φ : Δ → 𝔾_{0} be a Lie homomorphism with discrete kernel. Then the real rank rk_{ℝ}Δ equals 0 or 1. In particular, Δ is either a compact group or Δ is an almost direct product of an almost simple group of real rank 1 and a compact group.

Moreover, if C ≤ Δ is a real Cartan subgroup, then its image φ(C) is contained in the reduced stabilizer S𝔾_{0}.

Proof. Recall that Δ is a compact group if, and only if, rk_{ℝ}Δ = 0. Therefore, it sufficies to consider the case that rk_{ℝ}Δ ≥ 1. Let C ≤ Δ be a real Cartan subgroup. Then C is contained in some noncompact, almost simple normal subgroup Δ_{1} of Δ. The image φ(Δ_{1}) is contained in the reduced stabilizer S𝔾_{0}, see (1.1). Moreover, φ(C) is diagonalizable on P, cf. (3.4). Using (4.2) we derive that rk_{ℝ}Δ = dim C is at most 1. According to (3.2), at most one of the almost simple factors of Δ has real rank 1 while the others have real rank 0 and thus are compact. This finishes the proof.

Corollary 5.2. Let ℙ be a locally compact, connected projective translation plane. If the group Γ of all continuous collineations of ℙ contains a semisimple group of real rank at least 2, then ℙ is a classical plane and Γ does not fix any point or line.

Remark. The connected components Aut(P_{2}𝔽)^{e} = PSL_{3}𝔽 (where 𝔽 = ℝ, ℂ or ℍ) and Aut(P_{2}𝕆)^{e} = E_{6}(−26) of the automorphism groups of the classical projective planes are almost simple Lie groups of real rank 2.

Proof. Let Δ ≤ Γ be a semisimple group. If ℙ is not classical, then Γ leaves the translation axis invariant ([10, 64.4(c)]) and thus operates on the corresponding locally compact, connected affine translation plane. Being an affine group, Γ is a semidirect product Γ = Γ_{0} ⋉ ℝ^{n} of the stabilizer Γ_{0} and the translation group. We conclude that every Levi complement of Γ is conjugate to a subgroup of Γ_{0}. Without loss of generality, we therefore may assume that the semisimple group Δ is a subgroup of Γ_{0}. Applying (5.1) shows that rk_{ℝ}Δ equals 0 or 1.

We restrict our attention to the noncompact case:

Proposition 5.3. Let Δ be a noncompact, connected, semisimple Lie group and let φ : Δ → 𝔾_{0} be a Lie homomorphism with discrete kernel. Notice that rk_{ℝ}Δ = 1, cf. (5.1). Fix a real Cartan subalgebra 𝔞 of the Lie algebra 𝔤 = T_{e}Δ and an element X ∈ 𝔞 \ {0}. Let Ω be the set of real weights of φ_{⋆} with respect to X (cp. (3.3)). As usual, P_{ω} denotes the real weight space of φ_{⋆} belonging to ω ∈ Ω. Then the following statements hold:

(a) 0 is not a real weight of φ_{⋆}.

(b) Both vector subspaces

L_{+} = ⨁_{ω∈Ω; ω>0} P_{ω} and L_{−} = ⨁_{ω∈Ω; ω<0} P_{ω}

of P are lines.

Remark. Since dim 𝔞 = 1, the set {L_{+}, L_{−}} ⊆ L_{0} depends only on the choice of 𝔞, not on the choice of X ∈ 𝔞.

Proof. By (5.1), the real Cartan subgroup φ(exp 𝔞) is contained in the reduced stabilizer S𝔾_{0}. In view of (3.4) there exists a basis X = {x_{1}, …, x_{n}} of P such that φ(exp(tX)) is expressed as the matrix

exp(tX) = diag(e^{ω_{1}t}, …, e^{ω_{n}t})

with respect to X, where ω_{1} ≤ ω_{2} ≤ … ≤ ω_{n} are the real weights of φ_{⋆} (i.e. the eigenvalues of φ_{⋆}(X)). By (4.3), the vector subspaces

W = span{x_{1}, …, x_{l}} and S = span{x_{l+1}, …, x_{n}}

are lines and, moreover, ω_{1} ≤ … ≤ ω_{l} ≨ ω_{l+1} ≤ … ≤ ω_{n}.

Assume that ω_{l} ≥ 0. We rewrite W and S, respectively, as a direct sum

W = P_{ω_{l}} ⊕ ⨁_{ω∈Ω; ω≨ω_{l}} P_{ω} and S = ⨁_{ω∈Ω; ω≩ω_{l}} P_{ω}

of real weight spaces. If ω ≩ ω_{l} ≥ 0 is a real weight, then −ω ≨ 0 ≤ ω_{l} is a real weight, too, and dim P_{−ω} = dim P_{ω}, cf. (3.4). Contrary to dim W = dim S = l, this implies that

U = ⨁_{ω∈Ω; ω≩ω_{l}} P_{−ω}

would be proper subspace of W with dim U = dim S. We conclude that ω_{l} ≨ 0. Similary one proves ω_{l+1} ≩ 0. In particular, 0 is not a weight and L_{+} = S and L_{−} = W are lines.

Corollary 5.4. Let Δ ≤ 𝔾_{0} be a noncompact, semisimple group. Then Δ cannot act trivially on a nonzero subspace of P.

Proof. If Δ did operate trivially on a subspace U ≠ 0, then 0 would be a weight of Δ, contrary to (5.3).

Definition 5.5. Let Δ ≤ 𝔾_{0} be a (connected) semisimple Lie group of real rank 1. Then L ∈ L_{0} is called a weight line with respect to Δ if L is fixed by some real Cartan subgroup C of Δ. We refer to the set of all weight lines as the weight sphere of Δ.

6. Proof of Theorem A

Let Δ be a noncompact, connected, almost simple subgroup of the stabilizer 𝔾_{0}. Then Δ is contained in the reduced stabilizer S𝔾_{0}, cf. (1.1). Let 𝔤 be the Lie algebra of Δ and let U ≠ {0} be a Δ–irreducible subspace of P.

The real rank of Δ equals 1 by (5.1) and hence 𝔤 is isomorphic to an algebra of the following list, see for example [8, p. 312]:

𝔰𝔬_{m}(ℝ, 1), where m ≥ 3,

𝔰𝔲_{m}(ℂ, 1), where m ≥ 3,

𝔲_{m}(ℍ, 1), where m ≥ 3,

the exceptional Lie algebra 𝔣_{4}(−20).

Remark. We have isomorphisms 𝔰𝔬_{3}(ℝ, 1) ≅ 𝔰𝔩_{2}(ℝ), 𝔰𝔬_{4}(ℝ, 1) ≅ 𝔰𝔩_{2}(ℂ), 𝔰𝔬_{5}(ℝ, 1) ≅ 𝔲_{2}(ℍ, 1) and 𝔰𝔬_{6}(ℝ, 1) ≅ 𝔰𝔩_{2}(ℍ).

Lemma 6.1. The stabilizer 𝔾_{0} does not contain a subgroup which is locally isomorphic to SU_{3}(ℂ, 1).

Proof. Assume that Δ ≤ 𝔾_{0} is locally isomorphic to SU_{3}(ℂ, 1). The almost simple group Δ acts completely reducible on P, i.e. P = U_{1} ⊕ … ⊕ U_{m} is a direct sum of Δ–irreducible subspaces U_{i} ≤ P.

Recall that dim U_{i} = 1 is impossible by (5.4) and that dim P ∈ {2, 4, 8, 16}. Checking the list of irreducible representations of SU_{3}(ℂ, 1), cp. [10, Sect. 95], yields dim U_{i} ∈ {6, 8, 12}. From dim U_{1} + … + dim U_{m} = dim P ∈ {2, 4, 8, 16} we obtain dim U_{i} = 8 as the only possibility.

On the other hand, dim U_{i} = 8 implies that Δ|_{U_{i}} realizes the adjoint representation of SU_{3}(ℂ, 1), which has 0 as a real weight in contradiction to (5.3).

Together with some well known bounds for the dimension of automorphism groups of locally compact, connected translation planes, this lemma proves the following result:

Proposition 6.2. Let Δ ≤ 𝔾_{0} be a connected, almost simple group. If Δ is not compact, then Δ is locally isomorphic to one of the following groups:

1. SL_{2}𝔽, where 𝔽 = ℝ, ℂ or ℍ, or

2. U_{2}(ℍ, 1), or

3. Spin_{m}(ℝ, 1) with 7 ≤ m ≤ 10.

Remark. Each of the groups listed above operates on the octonion plane. Thus, our result is the best possible.

Proof. Recall that the subgroup Δ of 𝔾_{0} does not contain a subgroup locally isomorphic to SU_{3}(ℂ, 1) by (6.1). Consequently, Δ is not locally isomorphic to SU_{m}(ℂ, 1) or U_{m}(ℍ, 1) for any m ≥ 3.

Moreover, the largest possible dimension of the stabilizer 𝔾_{0} of a locally compact, connected affine translation plane equals^{7} dim(ℝ_{pos}×Spin_{10}(ℝ, 1)) = 46, cf. [10, 81.9, 82.27]. Thus, Δ is not locally isomorphic to SO_{m}(ℝ, 1) for any m ≥ 11 nor is it locally isomorphic to the exceptional group F_{4}(−20).

Excluding these groups from the list of simple groups of real rank 1 (cp. the beginning of this section) finishes the proof.

We should note a corollary which is interesting in its own:

Corollary 6.3. Let Γ ≤ 𝔾_{0} be a closed subgroup and let Δ_{1}, Δ_{2} ≤ Γ be connected, almost simple, noncompact subgroups. If dim Δ_{1} = dim Δ_{2}, then Δ_{1} and Δ_{2} are conjugate in Γ.

Proof. Up to conjugation, we may assume that both Δ_{1} and Δ_{2} are contained in the same Levi complement of Γ. Therefore, it suffices to consider the case that Γ is semisimple and connected. Since Γ cannot be a compact group, Γ is isomorphic to an almost direct product Γ = Δ × C, where Δ is a simple group of real rank 1 and where C is a compact group, cf. (5.1). Obviously, both Δ_{1} and Δ_{2} are subgroups of Δ. Moreover, by (6.2), the group Δ is locally isomorphic to SO_{m}(ℝ, 1) for some m. Now the assertion can be easily derived from the following fact:

If 𝔥 is a noncompact, simple subalgebra of the Lie algebra

𝔰𝔬_{m}(ℝ, 1) = {A ∈ 𝔤𝔩_{m}ℝ | I_{m}AI_{m} = −A^{tr}}

(where I_{m} denotes the (m × m)–diagonal matrix diag(1, …, 1, −1)), then 𝔥 is conjugate to the subalgebra

𝔲_{l} = \{( \begin{matrix}0 \\ & B \end{matrix} ) ∈ 𝔤𝔩_{m}ℝ | I_{l+1}BI_{l+1} = −B^{tr}\} ≅ 𝔰𝔬_{l}(ℝ, 1)

of 𝔰𝔬_{m}(ℝ, 1) for some l.

For the proof of this claim, let 𝔰𝔬_{m}(ℝ, 1) = 𝔨⊕𝔭 be a Cartan decomposition of 𝔰𝔬_{m}(ℝ, 1) (where 𝔨 is the compact part) such that 𝔥 = (𝔨 ∩ 𝔥) ⊕ (𝔭 ∩ 𝔥) is a Cartan decomposition of 𝔥. (The existence of such a decomposition is guaranteed by a theorem of Mostow, cp. [7, p.53].) Note that the dimension l = dim(𝔭∩𝔥) ≠ 0 since 𝔥 is noncompact.

By [6, 6.19], the subspace 𝔭 is conjugate by an inner automorphism of 𝔰𝔬_{m}(ℝ, 1) to the noncompact part

𝔮 = \{( \begin{matrix}0 & x \\ x^{tr} & 0 \end{matrix} ) | x ∈ ℝ^{m−1}\}

of the standard Cartan decomposition of 𝔰𝔬_{m}(ℝ, 1). Moreover, the group

exp ad \{( \begin{matrix}C \\ & 0 \end{matrix} ) | C ∈ 𝔰𝔬_{m−1}ℝ\}

7 ℝ_{pos} × Spin_{10}(ℝ, 1) is the stabilizer of the origin of the affine octonion plane.

operates transitively on the set of l–dimensional subspaces of 𝔮. Consequently, we may assume that

𝔭_{𝔥} = 𝔭 ∩ 𝔥 = \{( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & y \\ 0 & y^{tr} & 0 \end{matrix} ) | y ∈ ℝ^{l}\} .

In particular, we obtain 𝔥 = [𝔭_{𝔥}; 𝔭_{𝔥}] ⊕ 𝔭_{𝔥} = 𝔲_{l}.

We will now investigate each of the possible groups appearing in (6.2) separately.

Convention 6.4. We will use the following notation in (6.5) – (6.8): Let Δ be a connected, almost simple Lie group of real rank 1 which operates on P as a group of automorphisms. To be more precise, let φ : Δ → GL(P) be a representation with discrete kernel such that φ(Δ) ≤ 𝔾_{0} and thus φ(Δ) ≤ S𝔾_{0} by (1.1). We will study a nonzero φ–irreducible subspace U ≤ P. By π : Δ → GL(U) we denote the corresponding irreducible representation defined by π(δ) = φ(δ)|_{U}. The kernel of π then is the discrete normal subgroup K(π).

We start with the smallest group SL_{2}ℝ:

Proposition 6.5. Let Δ = SL_{2}ℝ and retain the notation of (6.4). Then the dimension dim U is even and the kernel of π is trivial. In particular, φ(Δ) is isomorphic to SL_{2}ℝ and the central involution σ of

Δ = SL_{2}ℝ acts as the reflection φ(σ) : P → P; x ↦ −x.

Proof. Let m = dim U. Consider the real Cartan subgroup

C = \{( \begin{matrix}e^{r} & 0 \\ 0 & e^{−r} \end{matrix} ) | r ∈ ℝ\}

of Δ = SL_{2}ℝ. According to [5, thm.12 on p.85], π(C) is given by

π(C) = {diag(e^{(−m+1)r}, e^{(−m+3)r}, …, e^{(m−3)r}, e^{(m−1)r}) | r ∈ ℝ}

with respect to some basis of U. Since 0 is not a real weight of π by (5.3), we conclude that m = dim U is even. An even–dimensional irreducible representation of SL_{2}ℝ always has trivial kernel and its centralizer is always ℝ, see the table in [10, Sect. 95]. Thus, for every φ–irreducible subspace U of P we have that φ(Δ)|_{U} ≅ SL_{2}ℝ and that φ(σ)|_{U} = −id.

Proposition 6.6. Let Δ = SL_{2}ℂ and retain the notation of (6.4).

Then dim U ∈ {4, 8, 12, 16}, the kernel of π is always trivial and the representation π of Δ on U is complex linear. In particular, φ(Δ) is isomorphic to SL_{2}ℂ.

Proof. The representations of SL_{2}ℂ in dimension 4, 8, 12 and 16 are exactly those representations of dimension at most 16 with trivial kernel, see [10, Sect. 95]. Thus, we have to exclude all representations with nontrivial kernel, i.e we have to prove that Δ|_{U} is not isomorphic to PSL_{2}ℂ. But this is clear, since Δ|_{U} cannot contain a group isomorphic to PSL_{2}ℝ by (6.5).

Proposition 6.7. Let Δ = U_{2}(ℍ, 1) and retain the notation of (6.4).

Then dim U = 8 and the representation π of Δ on U is equivalent to the usual representation of U_{2}(ℍ, 1) on ℍ^{2}.

If dim P = 8, then P is isomorphic to the quaternion plane.

Proof. Since Δ cannot act trivially on a subspace of P, the dimension dim P ∈ {2, 4, 8, 16} has to be a sum of dimensions of irreducible representations of U_{2}(ℍ, 1). Irreducible representations of U_{2}(ℍ, 1) in dimension at most 16 have dimension 5, 8, 10 or 14, see [10, Sect. 95]. By simple combinatorics, we infer that only the (unique) 8–dimensional representation is possible. This proves the first part of our assertion.

If dim P = 8, then 𝔼 is the quaternion plane, see [10, 82.25].

Proposition 6.8. Let Δ = SL_{2}ℍ and retain the notation of (6.4).

(a) For dim P ≤ 8, the only possibility is the natural action of Δ = SL_{2}ℍ on the quaternion plane.

(b) If dim P = 16, then φ = π_{1} × π_{2}, where π_{1} is the natural representation of Δ = SL_{2}ℍ on ℍ^{2} and where π_{2} is the contragredient representation on ℍ^{2}.

In particular, we can identify P and the 4–dimensional right quaternal vector space ℍ^{2} × ℍ^{2} in such a way that φ(Δ) is given by

φ(Δ) = \{( \begin{matrix}A \\ & A^{⋆} \end{matrix} ) | A ∈ SL_{2}ℍ\} ,

where A^{⋆} = (Ā^{tr})^{−1} for a matrix A ∈ SL_{2}ℍ.

Proof. If dim P ≤ 8, then P is isomorphic to the quaternion plane and Δ = SL_{2}ℍ acts in the usual way, cp. [10, 82.25]. It remains to treat the case dim P = 16. By [10, Sect. 95], a φ–irreducible subspace U has dimension 6, 8, 10 or 15. If dim U ∈ {5, 10, 15}, then the kernel of the representation π equals the center of Δ. But this implies that φ(Δ)|_{U} ≅ PSL_{2}ℍ would contain a subgroup isomorphic to PSL_{2}ℝ in contradiction to (6.5).

Consequently, P is a direct sum of two φ–irreducible 8–dimensional subspaces U_{1} and U_{2}. Let π_{1} and π_{2} denote the representations of Δ = SL_{2}ℍ on U_{1} and U_{2}, respectively. Notice that π_{i} either is the natural representation of SL_{2}ℍ on ℍ^{2}, or is the contragredient one.

If π_{1} and π_{2} are equivalent, then we can identify P = U_{1} ⊕U_{2} with ℍ^{2} ×ℍ^{2} such that φ(Δ) becomes the group

φ(Δ) = \{( \begin{matrix}A \\ & A \end{matrix} ) | A ∈ SL_{2}ℍ\} .

In this case the weight lines belonging to the real Cartan subgroup

φ(C) = {diag(e^{r}, e^{−r}, e^{r}, e^{−r}) | r ∈ ℝ}

are W = ℍ × 0 × ℍ × 0 and S = 0 × ℍ × 0 × ℍ. Obviously, the compact group

K = {diag(a, 1, a, 1) | a ∈ ℍ, ∥a∥ = 1} ≅ Spin_{3}ℝ

leaves W invariant and fixes S pointwise. But this is impossible by [2].

Thus, π_{1} and π_{2} are inequivalent , i.e. π_{1} realizes (without loss of generality) the natural representation of SL_{2}ℍ and π_{2} realizes the contragredient one. This proves part (b).

Proposition 6.9. Let Δ ≤ 𝔾_{0} be locally isomorphic to Spin_{m}(ℝ, 1) for some m, 7 ≤ m ≤ 10. Then (P, L) is isomorphic to the octonion plane and Δ is isomorphic to Spin_{m}(ℝ, 1) in its usual embedding in Spin_{10}(ℝ, 1). In particular, Δ operates irreducibly on P.

Proof. Δ contains a group Ξ locally isomorphic to Spin_{7}(ℝ, 1). Thus, (P, L) is isomorphic to the octonion plane by [4] and Δ is a subgroup of the reduced stabilizer S𝔾_{0} = Spin_{10}(ℝ, 1) of the octonion plane. By (6.3), we infer that Δ is conjugate to the group Spin_{m}(ℝ, 1) in its usual embedding in Spin_{10}(ℝ, 1) for some m, 7 ≤ m ≤ 10.

Remark. The proof of Theorem A can be pieced together from Propositions 6.2 – 6.9 and the remark at the beginning of Sect. 6.

7. Proof of Theorem B

Let Δ be a noncompact, connected, almost simple subgroup of the stabilizer 𝔾_{0} and let S ⊆ L_{0} be the weight sphere of Δ. From Theorem A we know that Δ is a 2–fold covering group of PSO_{m}(ℝ, 1) for some m, 3 ≤ m ≤ 10. The Lie algebra 𝔤 = T_{e}Δ therefore is given by

𝔤 = 𝔰𝔬_{m}(ℝ, 1) = {A ∈ 𝔤𝔩_{m}ℝ | I · A · I = −A^{tr}},

where I denotes the matrix I = diag(1, …, 1, −1). In order to avoid confusion we define the faithful representation φ_{⋆} : 𝔤 → 𝔤𝔩(P) as the derivative of the natural embedding φ = id : Δ → 𝔾_{0}.

Lemma 7.1. Δ leaves S invariant.

Proof. A line L ∈ L_{0} is an element of S if, and only if, L is fixed by some real Cartan subgroup C of Δ. For δ ∈ Δ, the conjugate δCδ^{−1} is a real Cartan subgroup fixing δ(L) and thus δ(L) is an element of L again.

Before discussing the action of Δ on S, we have a closer look at the Lie algebra 𝔤 = 𝔰𝔬_{m}(ℝ, 1). For this purpose we fix the real Cartan subalgebra

𝔞 = ℝ · X ≤ 𝔤 = 𝔰𝔬_{m}(ℝ, 1), where X = ( \begin{matrix}0 \\ & . & . & . \\ & & & & 0 \\ & & & & & 0 & 1 \\ & & & & & 1 & 0 \end{matrix} )

of 𝔤. The corresponding Cartan involution is

ι : 𝔰𝔬_{m}(ℝ, 1) → 𝔰𝔬_{m}(ℝ, 1); Y ↦ I · Y · I^{−1},

i.e. 𝔞 is a maximal abelian subspace of the Cartan complement 𝔭 of the Cartan decomposition defined by ι, i.e.

𝔰𝔬_{m}(ℝ, 1) = 𝔨 ⊕ 𝔭 with 𝔨 = \{( \begin{matrix}A \\ & 0 \end{matrix} ) | A ∈ 𝔰𝔬_{m−1}ℝ\}
and 𝔭 = \{( \begin{matrix}0 & x \\ x^{tr} & 0 \end{matrix} ) | x ∈ ℝ^{m−1}\} .

The restricted roots^{8} with respect to X are 0, 1 and −1. We write 𝔤_{λ} for the restricted root space belonging to the restricted root λ and obtain

𝔤_{1} = \{Y ∈ 𝔤 | (ad X)(Y) = Y \} = \{( \begin{matrix}0 & −x & x \\ x^{tr} & 0 & 0 \\ x^{tr} & 0 & 0 \end{matrix} ) | x ∈ ℝ^{m−2}\} .

In fact, 𝔤_{1} is an abelian subalgebra of 𝔤 since [𝔤_{1}, 𝔤_{1}] ⊆ 𝔤_{2} = {0}. Notice that 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔤_{1} and 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔤_{−1} are the Iwasawa decompositions of 𝔤 defined by 𝔞.

In what follows we need the notion of parabolic subgroups of an almost simple Lie group. We will introduce this concept for the groups considered here only, for the general case we refer to [12, Chap.1.2].

We start with the Iwasawa decomposition Δ = K ·C ·N_{1}, where K = exp 𝔨, N_{1} = exp 𝔤_{1} and where C denotes the real Cartan subgroup C = exp 𝔞 of Δ. Let M and M^{⋆} be the centralizer and the normalizer, respectively, of C in K. Then Π = M · C · N_{1} is a closed subgroup of Δ, which we call the standard minimal parabolic subgroup of Δ. We emphasize the fact that Π is uniquely determined by the generator X of 𝔞. If we replace X by an arbitrary element Y generating some other real Cartan subalgebra of 𝔤, then we obtain another closed subgroup Π_{Y} of Δ by the same construction. These subgroups are called the minimal parabolic subgroups of Δ. Since every two real Cartan subalgebras of 𝔤 are conjugate, we see from the construction that every minimal parabolic subgroup is conjugate to a minimal parabolic subgroup defined by an element X′ of 𝔞. Obviously, Π_{(rY)} = Π_{Y} holds for every positive real number r and thus X′ can be taken as X or −X. In order to see that Π = Π_{X} and Π_{−X} are conjugate, too, we use the Weyl group W (Δ, C) = M^{⋆}/M of Δ. In our situation, this Weyl group is a group of order 2: For every w ∈ M^{⋆}, the automorphism Ad w leaves invariant the one–dimensional algebra 𝔞 and therefore (Ad w)|_{𝔞} = ±id_{𝔞}. Moreover, w induces the identity on 𝔞 if, and only if, w is an element of the centralizer M of C in K. Consequently, M^{⋆} · M^{⋆} ⊆ M and thus W (Δ, C) = M^{⋆}/M is a group of order 2 (since the Weyl group of a noncompact, almost simple group never is trivial, see the considerations in Warner [12, Chap. 1.2]). Given an element w ∈ M^{⋆} \ M, we conclude from (Ad w)(X) = −X that wΠ_{X}w^{−1} = Π_{−X}.

8A restricted root is a real weight of the adjoint representation; the corresponding real weight spaces are called restricted root spaces.

We should add another interesting conclusion: The set of orbits of Δ on the set of all parabolic subgroups (i.e. subgroups containing a minimal parabolic subgroup) of Δ is in one–to–one correspondence with the elements of the Weyl group, see [12, Thm.1.2.1.1]. In our situation, |W (Δ, C)| = 2 implies that every proper subgroup of Δ containing a minimal parabolic subgroup is a minimal parabolic subgroup itself.

Let us return to geometry. According to (5.3) the real Cartan subgroup C of Δ fixes exactly the weight lines

W = ⨁_{λ∈Λ,λ<0} P_{λ} and S = ⨁_{λ∈Λ,λ>0} P_{λ}.

Here, Λ is the set of real weights of φ_{⋆} with respect to X (cp. (3.3)) and P_{λ} is the real weight space belonging to λ ∈ Λ. We claim

Lemma 7.2. The stabilizer Δ_{S} of S equals the minimal parabolic subgroup Π = M · C · N_{1} and fixes no line except S in L_{0}.

Moreover, if w represents the non–trivial element of the Weyl group W (Δ, C), then w interchanges W and S.

Proof. Let λ′ be the maximal element of Λ. For Y ∈ 𝔤_{1} and v ∈ P_{λ′}, the equation

φ_{⋆}(Y)(v) = [φ_{⋆}(X); φ_{⋆}(Y)](v)
= φ_{⋆}(X)φ_{⋆}(Y)(v) − φ_{⋆}(Y)φ_{⋆}(X)(v)
= φ_{⋆}(X)(φ_{⋆}(Y)(v)) − λ′ · φ_{⋆}(Y)(v)

shows that φ_{⋆}(X)(φ_{⋆}(Y)(v)) = (1 + λ′) · φ_{⋆}(Y)(v). Consequently, φ_{⋆}(Y)(v) = 0 since λ′ is a maximal real weight. Therefore, the group N_{1} = exp 𝔤_{1} operates trivially on the subspace P_{λ′} of S and hence leaves S invariant.

Thus, N_{1} is a subgroup of Δ_{S}. Moreover, M centralizes C ≤ Δ_{S} and thus fixes S, too. This shows that Δ_{S} contains the minimal parabolic subgroup Π and thus is a parabolic subgroup of Δ. The Weyl group W (Δ, C) has order 2 and thus either Δ_{S} = Π or Δ_{S} = Δ. The latter case cannot occur, otherwise Δ leaves S invariant and realizes on S a representation having only positive weights contradicting (3.4). Therefore, Δ_{S} = Π and, analogously, Δ_{W} = Π_{−X}. Since Π ≠ Π_{−X}, the line W is not invariant under Π. Moreover, C ≤ Π fixes no lines except W and S. From these considerations we infer that Π cannot fix a line in L_{0} \ {S}.

Now consider an element w ∈ M^{⋆} representing the non–trivial element of the Weyl group W (Δ, C). Then Ad w induces the map Y ↦ −Y on 𝔞. Consequently, w interchanges the real weight spaces P_{λ} and P_{−λ} for every real weight λ. This proves the remaining assertion.

Corollary 7.3. The stabilizer of a weight line L ∈ S is a minimal parabolic subgroup of Δ. Conversely, a minimal parabolic subgroup of Δ fixes exactly one line L ∈ L_{0} and, moreover, L is a weight line.

Proof. Let C′ be a real Cartan subgroup of Δ which fixes L. Then C′ is conjugate to C by an element γ ∈ Δ, see (3.1), and C fixes γ(L). Without loss of generality we assume γ(L) = S. Consequently, Δ_{L} = γ^{−1}Δ_{S}γ = γ^{−1}Πγ is conjugate to Π and thus is a minimal parabolic subgroup of Δ.

Conversely, a minimal parabolic subgroup is conjugate to Π = Δ_{S} by an element, say, δ ∈ Δ and thus equals the stabilizer of δ(S) ∈ S.

Since any two minimal parabolic subgroups of Δ are conjugate to each other, we immediately obtain:

Corollary 7.4. Δ operates transitively on its weight sphere S.

Thus, the weight sphere S is a homogeneous space Δ/Π, where Π = Δ_{S} is a minimal parabolic subgroup of Δ. Moreover, the central element −e ∈ Δ operates trivially on L_{0}, i.e. S can be considered as a homogeneous space of Δ/ ± e ≅ PSO_{m}(ℝ, 1) modulo the minimal parabolic subgroup Π′ = Π/ ± e. In fact, this homogeneous space may also be derived from the natural action of PSO_{m}(ℝ, 1) on the corresponding projective quadric, which is an (m − 2)–sphere. Hence 𝕊_{m−2} ≈ PSO_{m}(ℝ, 1)/Π′ ≈ S and the action of Δ on S is equivalent to the natural action of PSO_{m}(ℝ, 1) on 𝕊_{m−2}, i.e. the assertions stated in part (1) and (2) of Theorem B are proved. Part (3) of Theorem B is a consequence of Theorem A.

It remains to show that no orbit except S is closed in L_{0}. Consider a line L ∈ L_{0} \ S. Fix a real Cartan subgroup C ≤ Δ. Then C fixes two lines W, S ∈ S. Since C ≤ 𝔾_{W,S} is a compression subgroup, we can apply (1.2) and find a parametrization ρ : ℝ → C such that lim_{t→∞} ρ(t)(L) = S. The closure of the orbit Δ(L) thus contains S ∈ S = Δ(S). We conclude that Δ(L) is not closed in L_{0}.

References

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Harald Löwe Technische Universität Braunschweig Institut für Analysis Abteilung für Topologie und Grundlagen der Analysis Pockelsstr. 14 38106 Braunschweig, Germany harald@geometrie.tu-bs.de

Received October 10, 1998 and in final form Mai 26, 1999