A CONJUGACY THEOREM FOR SYMMETRIC SPACES

A.G HELMINCK, J. HILGERT, A. NEUMANN, AND G. ÓLAFSSON

Abstract. In this paper we prove that two Cartan subspaces of a semisimple symmetric pair (𝔤, τ) are conjugate under G = Int(𝔤) if and only if they are conjugate under (G^{τ})_{0}. Moreover we derive a double coset decomposition for G, which improves the results of Oshima and Matsuki.

1. Introduction

Let G_{ℂ} be a semisimple complex Lie group and G ⊆ G_{ℂ} a real form. Let 𝔱_{1}, … 𝔱_{n} be representatives of the G-conjugacy classes of Cartan subalgebras of 𝔤. Then the set

⋃_{i=1}^{n} GN_{G_{ℂ}}(𝔱_{i})G

contains an open and dense subset of G_{ℂ}. This was proved in [St86] using a result of result of Preiss-Rothschild ([PR72]) which says that two Cartan subalgebras in 𝔤 are conjugate under G_{ℂ} if and only if they are conjugate under G. In [HN93] and [La94] this double coset decomposition was a main tool for examining maximal subsemigroups of G_{ℂ} that contain G nonisolated.

In [NO97] this decomposition was generalized to noncompactly causal semisimple symmetric spaces G/H. It was shown there, that

⋃_{i=1}^{n} HN_{G}(𝔞_{i})H

contains an open and dense subset of G. Here the 𝔞_{1}, …, 𝔞_{n} are the representatives of H_{0}-conjugacy classes of Cartan subspaces in 𝔮, where H_{0} is the connected component of H containing the identity. The essential fact used in the proof is that in noncompactly causal symmetric spaces the G-conjugacy classes of Cartan subspaces are the same as the H_{0}-conjugacy classes. In [NO97] this was established by classifying the H_{0}-conjugacy classes and required detailed information on the special structure theory of noncompactly causal spaces.

In this paper we prove for general connected semisimple symmetric spaces G/H that two Cartan subspaces are conjugate under G if and only if they are conjugate under H. As a consequence of our theorem we can generalize the double coset decomposition to all semisimple symmetric spaces.

We would like to thank Karl-Hermann Neeb who spotted a gap in our original proof of Theorem 10 and helped us to fix it.

1991 Mathematics Subject Classification. 20G15, 20G20, 22E15, 22E46.

Key words and phrases. Symmetric spaces, Cartan subspaces.

The first author is partially supported by N.S.A. grant MDA904-97-1-0092.

The fourth author is supported by NSF grant DMS-9626541 and LEQSF grant (1996-99)-RD-A-12.

2. The Conjugacy Theorem

Definition 1. 1. A symmetric space is a triple (G, H, τ) consisting of a Lie group G, an involutive automorphism τ of G and a subgroup H with G_{0}^{τ} ⊆ H ⊆ G^{τ}.

2. A symmetric Lie algebra is a pair (𝔤, τ) of a Lie algebra 𝔤 and an involutive automorphism τ. We denote by 𝔥 respectively 𝔮 the +1 respectively −1 eigenspaces of τ.

3. We denote by 𝔤 the Lie algebra of G and we write τ also for the derivative of τ acting on 𝔤.

For the purpose of this paper we will assume that the group G is connected and semisimple. Denote the center of G by Z(G). We will assume that H/(H ∩ Z(G)) is noncompact. In particular τ is not a Cartan involution. We recall that for each such involution there exists a Cartan involution θ on 𝔤 that commutes with τ. We denote by 𝔨 (respectively 𝔭) the +1 (respectively −1) eigenspaces of θ. Then

𝔤 = 𝔥 ⊕ 𝔮 = 𝔨 ⊕ 𝔭
= 𝔥_{𝔨} ⊕ 𝔥_{𝔭} ⊕ 𝔮_{𝔨} ⊕ 𝔮_{𝔭}

where subscripts denote intersection, i.e. 𝔥_{𝔨} := 𝔥 ∩ 𝔨 and analogously in the other cases.

Definition 2. A subspace 𝔞 of 𝔤 is called a Cartan subspace, if:

1. 𝔞 is a maximal abelian subalgebra of 𝔮.

2. Every element of 𝔞 is semisimple.

Lemma 3. [Lo69, p.153, Theorem 2.1] Let θ_{1} and θ_{2} denote two Cartan involutions on 𝔤 commuting with τ. Then there exists an h ∈ H such that θ_{1} = hθ_{2}h^{−1}.

Lemma 4. [OM80, Lemma 5] Let 𝔞 be a Cartan subspace of 𝔤. Then there exists a Cartan involution θ on 𝔤, which commutes with τ and leaves 𝔞 invariant.

Lemma 5. Let 𝔞 be a Cartan subspace of 𝔤 and let θ be a Cartan involution commuting with τ. Then there exists an h ∈ H such that Ad(h)𝔞 is θ-invariant.

Proof. This follows from Lemma 3 and 4 .

Lemma 6. [Wr72, Lemma 1.1.3.7] Let K denote the analytic subgroup corresponding to 𝔨. Let Y′ = Ad(k exp X)Y, where k ∈ K and X ∈ 𝔭. If both Y and Y′ are eigenvectors of θ, we have [X, Y] = 0.

Lemma 7. Let 𝔞_{1} and 𝔞_{2} be two θ-invariant Cartan subspaces of 𝔤. Let G_{1} be a θ-invariant subgroup of G. Denote the Lie algebra of G_{1} by 𝔤_{1}. Assume that G_{1} = (G_{1} ∩ K) exp(𝔤_{1} ∩ 𝔭). Then the Cartan subspaces 𝔞_{1} and 𝔞_{2} are conjugate under G_{1} if and only if they are conjugate under G_{1} ∩ K.

Proof. One direction is obvious. Now assume that for some g ∈ G_{1} we have Ad(g)𝔞_{1} = 𝔞_{2}. Since the 𝔞_{i} are θ-invariant, we have 𝔞_{i} = (𝔞_{i} ∩ 𝔨) + (𝔞_{i} ∩ 𝔭). Further 𝔞_{i} ∩ 𝔨 (respectively 𝔞_{i} ∩ 𝔭) are exactly the X ∈ 𝔞_{i} for which adX has purely imaginary (respectively real) eigenvalues. This property is invariant under conjugation. Hence

Ad(g)(𝔞_{1} ∩ 𝔨) = 𝔞_{2} ∩ 𝔨 and Ad(g)(𝔞_{1} ∩ 𝔭) = 𝔞_{2} ∩ 𝔭 .

Write g = k exp Y with k ∈ G_{1} ∩ K and Y ∈ 𝔤_{1} ∩ 𝔭. Let X ∈ 𝔞_{1} ∩ 𝔨. Then Ad(k exp Y)X ∈ 𝔞_{2} ∩ 𝔨. By Lemma 6 we get that [Y, X] = 0. So Ad(k exp Y)X = Ad(k)X. It follows that

Ad(g)|_{𝔞_{1}∩𝔨} = Ad(k)|_{𝔞_{1}∩𝔨} .

In the same way we obtain Ad(g)|_{𝔞_{1}∩𝔭} = Ad(k)|_{𝔞_{1}∩𝔭}. Taken together this results in Ad(k)𝔞_{1} = Ad(g)𝔞_{1} = 𝔞_{2}.

Note that the hypothesis on the Cartan decomposition G_{1} = (G_{1}∩K) exp(𝔤_{1}∩𝔭) in Lemma 7 is satisfied if G_{1} has only finitely many connected components.

Lemma 8. Let 𝔞_{1} and 𝔞_{2} be two maximal abelian subalgebras of 𝔮_{𝔭}. Then they are conjugate under (H ∩ K)_{0}.

Proof. In a reductive Lie algebra two maximal abelian subalgebras of 𝔭 are conjugate under a maximal compact subgroup. The algebra 𝔥_{𝔨} ⊕ 𝔮_{𝔭} is a reductive Lie algebra. Hence all maximal abelian subspaces in 𝔮_{𝔭} are conjugate under the analytic subgroup corresponding to 𝔥_{𝔨}, which is (K ∩ H)_{0}.

Lemma 9. Let G_{0}^{τ} ⊆ H ⊆ G^{τ}. Then two θ-invariant Cartan subspaces 𝔞_{1} and 𝔞_{2} are conjugate under H ∩ K if and only if 𝔞_{1} ∩ 𝔨 and 𝔞_{2} ∩ 𝔨 are conjugate under H ∩ K.

Proof. The one direction is obvious. Now let us assume there exists a k ∈ H ∩ K such that Ad(k)(𝔞_{1} ∩ 𝔨) = 𝔞_{2} ∩ 𝔨. Then 𝔞_{3} := Ad(k)𝔞_{1} will also be a θ-stable Cartan subspace of 𝔤. Furthermore 𝔞_{2} and 𝔞_{3} both lie in the centralizer 𝔷_{𝔤}(𝔞_{2} ∩ 𝔨) of 𝔞_{2} ∩ 𝔨 in 𝔤. The subspaces 𝔞_{2} ∩ 𝔭 and 𝔞_{3} ∩ 𝔭 are both maximal abelian in 𝔷_{𝔤}(𝔞_{2} ∩ 𝔨) ∩ 𝔮_{𝔭}. Now the assertion follows if we apply Lemma 8 to 𝔷_{𝔤}(𝔞_{2} ∩ 𝔨).

Theorem 10. Suppose that (G, H, τ) is a semisimple symmetric space with G connected. Assume that H is connected. Let 𝔞_{1} and 𝔞_{2} be two Cartan subspaces of (𝔤, 𝔥) in 𝔮. Then 𝔞_{1} and 𝔞_{2} are conjugate under G if and only if they are conjugate under H.

Proof. One direction is obvious. To prove the other implication we first look at the case that G is simply connected. We assume that 𝔞_{1} and 𝔞_{2} are conjugate under G. With Lemma 5 we can assume that 𝔞_{1} and 𝔞_{2} are both θ-invariant. Then by applying Lemma 7 to G we can find a k ∈ K such that Ad(k)𝔞_{1} = 𝔞_{2}. In particular Ad(k)(𝔞_{1} ∩ 𝔨) = 𝔞_{2} ∩ 𝔨.

As 𝔨 is reductive we have 𝔨 = 𝔷(𝔨) ⊕ 𝔨′, where 𝔷(𝔨) is the center of 𝔨 and 𝔨′ = [𝔨, 𝔨]. Both 𝔷(𝔨) and 𝔨′ are τ-invariant. Since G is simply connected so is K. Therefore we have that K = V × K′, where V = exp 𝔷(𝔨) is a vector group and K′ = exp 𝔨′ is semisimple and hence compact. Because V acts trivially on 𝔨 we can find a k′ ∈ K′ such that Ad(k′)(𝔞_{1} ∩ 𝔨) = 𝔞_{2} ∩ 𝔨. We denote by 𝔞′_{i} the projection of 𝔞_{i} ∩ 𝔨 to 𝔨′. The 𝔞′_{i} are abelian and Ad(k′)𝔞′_{1} = 𝔞′_{2}.

We have K′ = (H ∩ K′)_{0} exp(𝔮 ∩ 𝔨′), cf., [He78, Lemma 6.3 and Theorem 6.7]. Thus we can write k′ = m exp Y with m ∈ (H ∩ K′)_{0} and Y ∈ 𝔮 ∩ 𝔨′. Since 𝔞′_{1} and 𝔞′_{2} both lie in 𝔮∩𝔨′ we have for U ∈ 𝔞′_{1} that Ad(k′)U = τ(−Ad(k′)U) = Ad(τ(k′))U and therefore exp(2Y) = τ(k′)^{−1}k′ ∈ Z_{K}(𝔞′_{1}). We claim that there exists an abelian subspace 𝔟 ⊆ 𝔮 ∩ 𝔨′ such that 𝔞′_{1} ⊆ 𝔟 and exp(2Y) ∈ exp(𝔟).

Denote by S the abelian group generated by exp(2Y) and exp(𝔞′_{1}). We claim that S ⊆ exp(𝔮 ∩ 𝔨′). To prove this we recall from the polar decomposition for compact symmetric spaces that exp(𝔮∩𝔨′) is precisely the connected component of e in Q := {k ∈ K′ | τ(k)^{−1} = k} (cf. [Lo69, Prop. IV.4.4]). Note that exp(n2Y) exp(X) ∈ Q for each n ∈ ℤ and X ∈ 𝔞′_{1} since exp(n2Y), exp(X) ∈ exp(𝔮 ∩ 𝔨′) ⊆ Q commute. Since exp(𝔞′_{1}) is connected and contains the identity we see that exp(n2Y) exp(X) is in the same connected component as exp(n2Y), i.e. in exp(𝔮 ∩ 𝔨′). Thus we obtain S ⊆ exp(𝔮 ∩ 𝔨′).

Since exp(𝔮∩𝔨′) is closed in K′ we obtain \overline{S} ⊆ exp(𝔮∩𝔨′). Thus \overline{S} is compact and \overline{S}/\overline{S}_{0} is finite. Since \overline{S}/\overline{S}_{0} is generated by exp(2Y)\overline{S}_{0} it is also cyclic. Therefore we can find an element x ∈ \overline{S} such that \{x^{n} : n ∈ ℕ\} lies dense in \overline{S}. We write x = exp Z for Z ∈ 𝔮 ∩ 𝔨′. Then B := \overline{exp(ℝZ)} is a connected abelian group lying in exp(𝔮 ∩ 𝔨′) and containing exp(2Y) and exp(𝔞′_{1}). Therefore 𝔟, the Lie algebra of B, is the desired subspace.

Let X ∈ 𝔟 be such that exp(2X) = exp(2Y) and let k_{1} := k′ exp(−X). Since B is abelian we have

k_{1}^{−1}τ(k_{1}) = exp(X) exp(−Y)m^{−1}m exp(−Y) exp(X)
= exp(−2Y) exp(2X) = e.

So k_{1} ∈ G^{τ} ∩ K. But G is simply connected, so by [Lo69, p. 337] we have that G^{τ} is connected. Therefore k_{1} lies in H ∩ K. Since [X, 𝔞′_{1}] = 0 we obtain Ad(k_{1})|𝔞′_{1} = Ad(k′)|𝔞′_{1}. Then obviously Ad(k_{1})(𝔞_{1} ∩ 𝔨) = Ad(k′)(𝔞_{1} ∩ 𝔨) = 𝔞_{2} ∩ 𝔨. Now Lemma 9 completes the proof.

If G is not simply connected we can look at the universal covering group G̃. Then Ad(G) = Ad(G̃) and Ad(G_{0}^{τ}) = Ad(G̃_{0}^{τ}) which completes the proof in the general case.

We would like to remark here that the classification of H_{0}-conjugacy classes of Cartan subspaces will be given in [Helm99]. Note the following simple consequence of our main theorem.

Corollary 11. Let G/H be a connected semisimple symmetric space and 𝔞 a θinvariant Cartan subspace of 𝔤. Then

H = N_{H∩K}(𝔞)H_{0}.

Proof. Note first that H = (H ∩ K) exp(𝔥 ∩ 𝔭). Pick an h ∈ H. With Theorem 10 we can find an h′ ∈ H_{0} such that Ad(h)𝔞 = Ad(h′)𝔞. Write h(h′)^{−1} = k exp(X) with k ∈ H ∩ K and X ∈ 𝔥 ∩ 𝔭. Then Lemma 6 shows that k ∈ N_{H∩K}(𝔞) which implies h ∈ N_{H∩K}(𝔞)H_{0}.

Using Lemma 5 we see that without the assumption that 𝔞 is θ-invariant we can still show that H = N_{H∩K}(𝔞)H_{0} holds for some subgroup K with maximal compactly embedded subalgebra.

3. The Double Coset Decomposition

Theorem 12. Let G/H be a connected semisimple symmetric space. Let 𝔞_{1}, … 𝔞_{n} be representatives of the H-conjugacy classes of Cartan subspaces. Then the set

⋃_{j=1}^{n} HN_{G}(𝔞_{j})H

contains an open and dense subset of G.

Proof. This proof is almost the same as in [HN93] or [NO97]. For g ∈ G we set g^{∗} := τ(g)^{−1}. Let A_{i} := Z_{G}(𝔞_{i}) and B_{i} := {g ∈ G : gg^{∗} ∈ A_{i}}. Denote by G^{reg} the set of regular elements in G. Then G′ := {g ∈ G : gg^{∗} ∈ G^{reg}} is open and dense in G and [OM80, Theorem 2.ii] tells us, that

G′ ⊆ ⋃_{i=1}^{n} HB_{i}H.

Let g ∈ B_{i}. Then Ad(gg^{∗})|_{𝔞_{i}} = id. We have Ad(g^{∗}) = τAd(g)^{−1}τ. Therefore Ad(g) commutes pointwise with τ on 𝔞_{i}. So Ad(g)𝔞_{i} is a Cartan subspace and therefore

Ad(h)Ad(g)𝔞_{i} = 𝔞_{j}

for some j ∈ {1, …, n} and h ∈ H. By Theorem 10 we have that i = j and thus hg ∈ N_{G}(𝔞_{i}). It follows that

HB_{i}H ⊆ HN_{G}(𝔞_{i})H

which proves the claim.

References

[He78] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York 1978

[Helm99] A. G. Helminck, Tori Invariant Under an Involutorial Automorphism III, Real groups. In preparation

[HN93] J. Hilgert, K.-H. Neeb, Lie Semigroups and Their Applications, Lecture Notes Math. 1552, Springer 1993

[La94] J. D. Lawson, Maximal Ol’shanskii Semigroups, Journal of Lie Theory, 4 (1994), 17-29 [Lo69] O. Loos, Symmetric Spaces, Benjamin, New York 1969

[NO97] A. Neumann, G. Ólafsson, Minimal and Maximal Semigroups Related to Causal Symmetric Spaces, submitted

[OM80] T. Oshima, T. Matsuki, Orbits on Affine Symmetric Spaces Under the Action of the Isotropy Subgroups, J. Math. Soc. Japan, 32 (1980), 399-414

[PR72] L. Preiss-Rothschild, Orbits in a Real Reductive Lie Algebra, Trans. AMS 168 (1972), 403–421

[St86] R. Stanton, Analytic extension of the holomorphic discrete series, Amer. J. Math¿ 108 (1986), 1411–1424

[Wr72] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer, Berlin and New York 1972

(A.H.) Dep. of Math. North Carolina State University, Raleigh, N.C., 27695

   E-mail address: loek@.math.ncsu.edu

(J.H.) Institut für Mathematik, Technische Universität Clausthal, Erzstr. 1, D-38678, Clausthal-Zellerfeld, Germany

Current address: Institut de Mathématiques, Analyse Algèbrique, Université Pierre et Marie Curie, 4, place Jussieu, F-75252 Paris, France

   E-mail address: hilgert@math.tu-clausthal.de

(A.N.) Dep. of Math. Louisiana State University, Baton Rouge, LA., 70803

   E-mail address: neumann@math.lsu.edu

(G.Ó.) Dep. of Math., Louisiana State University, Baton Rouge, LA., 70803

   E-mail address: olafsson@math.lsu.edu