Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

Simon Gindikin^{∗} and Bernhard Krötz^{†}

Abstract

In this paper we define a distinguished boundary for the complex crowns Ξ ⊆ G_{ℂ}/K_{ℂ} of non-compact Riemannian symmetric spaces G/K. The basic result is that affine symmetric spaces of G can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

Introduction

Let X = G/K be a semisimple non-compact Riemannian symmetric space. We may assume that G is semisimple with finite center. We write G = NAK for an Iwasawa decomposition of G. By our assumption, G sits in its universal complexification G_{ℂ} and so X ⊆ X_{ℂ}: = G_{ℂ}/K_{ℂ}. Note that G does not act properly on X_{ℂ}. The complex crown Ξ ⊆ X_{ℂ} of X (cf. [Gi98]) was first considered in [AkGi90]. It can be defined by

Ξ: = G exp(iΩ)K_{ℂ}/K_{ℂ}

where Ω is a polyhedral convex domain in 𝔞: = Lie(A) defined by

Ω: = \{X ∈ 𝔞: (∀α ∈ Σ) |α(X)| < \frac{π}{2} \}.

Here Σ denotes the restricted root system with respect to 𝔞. Note that G acts properly on Ξ (cf. [AkGi90]).

The domain Ξ is interesting in several ways. It accumulates many crucial geometrical and analytical properties of X. For example, it was shown in [GiMa01] that Ξ parametrizes the compact cycles in complex flag domains. From the point of harmonic analysis the domain Ξ is universal in the sense that all eigenfunctions on X for the algebra 𝔻(X) of G-invariant differential operators extend holomorphically to Ξ (cf. [KrSt01a]). It was conjectured in [AkGi90] that the domain Ξ is Stein; it was proved for different cases by different authors during the last year (cf. [GiKr01] for references and a more detailed account on the domain Ξ).

∗ Supported in part by the NSF-grant DMS-0070816 and the MSRI † Supported in part by the NSF-grant DMS-0097314 and the MSRI

2 Simon Gindikin^{∗} and Bernhard Krötz^{†}

In the case of groups G of Hermitian type, X is is a Hermitian symmetric space and Ξ is G-equivariantly biholomorphic to X × \overline{X}, where \overline{X} refers to X equipped with the opposite complex structure (cf. [BHH01], [GiMa01] or [KrSt01b]). This example can be seen in a more general framework. Suppose that X is the real form a Hermitian symmetric space. This means that there is a Hermitian group S containing G with maximal compact subgroup U ⊇ K such that the inclusion G/K ↪ S/U realizes X as a totally real submanifold of S/U. For example, if G is Hermitian we take S = G × G and U = K × K. The existence of S/U is guaranteed in many cases. In particular, S/U exists for all classical groups G in the sense that we have to replace G sometimes with G × ℝ (e.g. Sl(n, ℝ) with Gl(n, ℝ)). By the results of [BHH01] or [KrSt01b] there exists a generic G-invariant subdomain Ξ_{0} of Ξ which is G-biholomorphic to S/U. Moreover, equality Ξ_{0} = Ξ holds if and only if Σ is of type C_{n} or BC_{n} for n ≥ 2 or G = SO(1, n) with S = SO(2, n). For the cases where Ξ_{0} ⊊ Ξ, for example if G = SO(p, q) (p, q > 2) the geometric structure of Ξ becomes very complicated and especially complicated can be the boundary of Ξ.

In this paper we start to investigate the boundaries of Ξ. Write ∂Ξ for the boundary of Ξ in X_{ℂ}. Apparently, the boundary will be a union of G-orbits, but the stratification of these orbits can be intricate. Firstly, not all orbits on the boundary intersect exp(i𝔞)K_{ℂ}/K_{ℂ} – but in this paper we focus only on such orbits. Moreover, we are only interested in very special orbits of such type which are in some sense minimal. Let us define the distinguished boundary ∂_{d}Ξ of Ξ as the union of the following G-orbits in ∂Ξ:

∂_{d}Ξ: = G exp(i∂_{e}Ω)K_{ℂ}/K_{ℂ},

where ∂_{e}Ω the set of extreme points in the polyhedral compact convex set \overline{Ω}. Note that

∂_{e}Ω = W(Y_{1}) ⨿ … ⨿ W(Y_{n})

is a finite union of Weyl group orbits, where W is the Weyl group of Σ.

The distinguished boundary is a geometrically complicated object. Usually it is a disconnected set. Nevertheless, we show that it is minimal from some analytical points of view and that it features properties expected from a Shilov boundary. In particular,

Theorem A. Write A(Ξ) for the algebra of bounded holomorphic functions which continuously extend to the boundary of Ξ. Then

(∀f ∈ A(Ξ)) sup_{z∈Ξ} |f(z)| = sup_{z∈∂_{d}Ξ} |f(z)|.

Further ∂_{d}Ξ ⊆ ∂Ξ is minimal in a certain plurisubharmonic sense as explained in Section 1.

In above mentioned cases when Ξ_{0} = Ξ the situation is simpler. We can realize the Hermitian domain D = S/U via the Cartan embedding in the compact Hermitian symmetric Y space dual to S/U. Here it has a compact boundary and a compact Shilov boundary ∂_{c}D. Further, in this situation the Stein manifold X_{ℂ} can be realized as a Zariski open part of Y which will contain Ξ biholomorphically equivalent to D and ∂_{d}Ξ will be only a Zariski open part of the compact manifold ∂_{c}D. Let us emphasize that the Shilov boundary of D essentially depends on the realization of D.

We describe now the distinguished boundary ∂_{d}Ξ in more detail. Write z′_{j}: = exp(iY_{j})K_{ℂ} ∈ ∂_{d}Ξ for all 1 ≤ j ≤ n. Denote by H_{j} the isotropy subgroup of G in z′_{j}. Then it follows under the assumption that G_{ℂ} is simply connected that we have a G-isomorphism

∂_{d}Ξ ≃ G/H_{1} ⨿ … ⨿ G/H_{n}.

In [Gi98] it was conjectured that non-compactly causal symmetric spaces appear in the “Shilov boundary” of the complex crowns Ξ. We establish this conjecture (in a more exact form); namely we prove:

Theorem B. If one of the components G/H_{j} in ∂_{d}Ξ is a symmetric space, then it is a noncompactly causal symmetric space. Moreover, every non-compactly causal symmetric space occurs as a component of the distinguished boundary of some complex crown Ξ.

Let us say a few words about the motivation of this conjecture. On Riemannian symmetric spaces we have an elliptic analysis and on non-compactly causal symmetric spaces we have a hyperbolic analysis. It is known in mathematical physics that in many important cases elliptic and hyperbolic theories can be “connected” through complex domains (Laplacians and wave equations, Euclidean and Minkowski field theories etc). Theorem B implies a connection of Riemannian and non-compactly causal symmetric spaces through the complex crowns Ξ. It shows that the phenomenon described above has a non-trivial generalization to symmetric spaces.

Write 𝔤: = Lie(G) for the Lie algebra G. We call 𝔤 non-compactly causal if 𝔤 admits an involution such that (𝔤, τ) is a non-compactly causal symmetric Lie algebra. Theorem B then tells us that 𝔤 has to be non-compactly causal in order for ∂_{d}Ξ to contain symmetric spaces. The class of non-compactly causal Lie algebras is very rich; for example it contains all classical Lie algebras except 𝔰𝔲(p, q) for p ≠ q, 𝔰𝔭(p, q) for p ≠ q and 𝔰𝔬^{∗}(2n) for n odd (cf. [HiÓl96] or the table in Remark 3.3(a) below).

In Theorem 3.25 we give a complete classification of the distinguished boundary ∂_{d}Ξ for 𝔤 non-compactly causal. It turns out that ∂_{d}Ξ is a union of non-compactly causal symmetric spcaes except for 𝔤 = 𝔰𝔬(p, q) with 3 ≤ p < q or 𝔤 = 𝔰𝔬(2n + 3, ℂ), where ∂_{d}Ξ ≅ G/H_{1} ⨿ G/H_{2} with G/H_{1} non-compactly causal and H_{2} a non-symmetric subgroup of G. Another consequence of the classification is the following result:

Theorem C. Let 𝔤 be a non-compactly causal Lie algebra and ∂_{d}Ξ ≃ ∐_{j=1}^{n} G/H_{j} the decomposition of ∂_{d}Ξ into G-orbits. Then a boundary component G/H_{j} of ∂_{d}Ξ is totally real if and only if G/H_{j} is symmetric.

In Sections 4 and 5 we compare the dinstinguished boundary of Ξ with the distinguished boundary of Ξ_{0} for all those groups G for which Ξ_{0} exists and Ξ_{0} ⊊ Ξ. These are precisely the structure groups of Euclidean Jordan algebras and the special orthogonal groups G = SO(p, q) and G = SO(n, ℂ).

This paper serves as the geometric foundation for the forthcoming work of the authors with Gestur Ólafsson towards the definition of a Hardy space on Ξ which realizes the most-continuous part L^{2}(G/H)_{mc} of L^{2}(G/H) for a non-compactly causal symmetric space G/H (cf. [GKÓ01]).

It is our pleasure to thank the MSRI, Berkeley, for its hospitality during the Integral geometry program where this work was accomplished. We are grateful to Gestur Ólafsson for going over the manuscript and his worthy suggestions.

1. Notation

Let G be a semisimple Lie group sitting inside its universal complexification G_{ℂ}. We denote by 𝔤 and 𝔤_{ℂ} the Lie algebras of G and G_{ℂ}, respectively. Let K < G be a maximal compact subgroup and 𝔨 its Lie algebra.

Let 𝔤 = 𝔨 ⊕ 𝔭 be the Cartan decomposition attached to 𝔨. Take 𝔞 ⊆ 𝔭 a maximal abelian subspace and let Σ = Σ(𝔤, 𝔞) ⊆ 𝔞^{∗} be the corresponding root system. Related to this root system is the root space decomposition according to the simultaneous eigenvalues of ad(H), H ∈ 𝔞 :

𝔤 = 𝔞 ⊕ 𝔪 ⊕ ⨁_{α∈Σ} 𝔤^{α},

here 𝔪 = 𝔷_{𝔨}(𝔞) and 𝔤^{α} = {X ∈ 𝔤: (∀H ∈ 𝔞) [H, X] = α(H)X}. For the choice of a positive system Σ^{+} ⊆ Σ one obtains the nilpotent Lie algebra 𝔫 = ⨁_{α∈Σ^{+}} 𝔤^{α}. Then one has the Iwasawa decomposition on the Lie algebra level

𝔤 = 𝔫 ⊕ 𝔞 ⊕ 𝔨.

We write A, N for the analytic subgroups of G corresponding to 𝔞 and 𝔫. For these choices one has for G the Iwasawa decomposition, namely, the multiplication map

N × A × K → G, (n, a, k) ↦ nak

In particular, every element g ∈ G can be written uniquely as g = n(g)a(g)κ(g) with each of the maps κ(g) ∈ K, a(g) ∈ A, n(g) ∈ N depending analytically on g ∈ G. The last piece of structure theory we shall recall is the little Weyl group. We denote by W = N_{K}(𝔞)/Z_{K}(𝔞) the Weyl group of Σ(𝔞, 𝔤).

Next we define a domain using the restricted roots. We set

Ω = \{X ∈ 𝔞: (∀α ∈ Σ) |α(X)| < \frac{π}{2} \}.

Clearly, Ω is convex and W -invariant.

If L is a Lie group, then we write L_{0} for the connected component containing 1. If L is a group and σ is an involution on L, then we write L^{σ} for the σ -fixed points in L. Similarily, if 𝔩 is a Lie algebra and σ an involution on 𝔩, then we write 𝔩^{σ} for the subspace of 𝔩 which is pointwise fixed under σ.

The domain Ξ

Write A_{ℂ}, N_{ℂ} and K_{ℂ} for the complexifications of the groups A, N and K realized in G_{ℂ}. Define the domain

Ξ: = G exp(iΩ)K_{ℂ}/K_{ℂ}

which will be in the center of attention throughout this paper. Write ∂Ξ for the topological boundary of Ξ in G_{ℂ}/K_{ℂ}. Recall from [AkGi90] that the piece of the boundary of Ξ which intersects exp(i𝔞)K_{ℂ}/K_{ℂ} is given through

∂_{a}Ξ: = G exp(i∂Ω)K_{ℂ}/K_{ℂ}.

Note the following properties of Ξ:

· Ξ is open in G_{ℂ}/K_{ℂ} (cf. [AkGi90]). · Ξ is connected and G-invariant. · G acts properly on Ξ (cf. [AkGi90]). · The domain Ξ is Stein (special cases of [BHH01, Th. 10] or [GiKr01, Th. 3.4]) · One has Ξ ⊆ N_{ℂ}A exp(iΩ)K_{ℂ}/K_{ℂ} (special case of [GiKr01, Lemma 3.1])

2. The distinguished boundary of Ξ

Write \overline{Ξ} for the closure of Ξ in G_{ℂ}/K_{ℂ}. Note that \overline{Ξ} = Ξ ⨿ ∂Ξ. Since \overline{Ξ} is not compact, there is no standard definition of a Shilov boundary for Ξ. However, it is possible to define a natural distinguished boundary ∂_{d}Ξ which features many properties of a Shilov boundary as we will explain below.

We write A(Ξ) for the algebra of bounded continuous functions on \overline{Ξ} which are holomorphic when restricted to Ξ:

A(Ξ): = \{f ∈ C(\overline{Ξ}): f |_{Ξ} ∈ O(Ξ), ∥f∥: = sup_{z∈Ξ} |f(z)| < ∞\}.

It is easy to check that A(Ξ) equipped with the supremum norm is a commutative Banach algebra with identity.

There is a natural action of G on A(Ξ) by left translation in the arguments:

(2.1)  G × A(Ξ) → A(Ξ), (g, f) ↦ λ_{g}(f); λ_{g}(f)(z) = f(g^{−1}z).

The domain Ξ is special in the sense that it admits a maximal abelian flat subdomain:

T_{Ω} = A exp(iΩ) ⊆ A_{ℂ}.

Write \overline{T_{Ω}} = A exp(i\overline{Ω}) for the closure of T_{Ω} in A_{ℂ}. Note that T_{Ω} is biholomorphic to the tube domain 𝔞 + iΩ ⊆ 𝔞_{ℂ} over Ω via the mapping

𝔞 + iΩ → T_{Ω}, Z ↦ exp(Z).

Observe that \overline{Ω} is a compact polyhedron in 𝔞. Denote by ∂_{e}Ω the extreme points of \overline{Ω}. Notice that ∂_{e}Ω is a finite W -invariant set, hence a finite union of W -orbits:

∂_{e}Ω = W(Y_{1}) ⨿ … ⨿ W(Y_{n}).

Remark 2.1. If the restricted root system Σ is of type C_{n}, BC_{n} then ∂_{e}Ω consists of a single W -orbit (see our discussion in Section 3).

Set

∂_{d}T_{Ω} = A exp(i∂_{e}Ω).

Similarily as before we define the Banach algebra A(T_{Ω}). Then we have the following elementary lemma:

Lemma 2.2. For all f ∈ A(T_{Ω}) we have

sup_{z∈T_{Ω}} |f(z)| = sup_{z∈∂_{d}T_{Ω}} |f(z)|.

Proof. Recall that T_{Ω} is isomorphic to the tube domain 𝔞 + iΩ with polyhedral bounded base Ω. For dim 𝔞 = 1 this lemma is just the Phragmen-Lindelöf theorem. An easy iteration of this argument gives a proof in the general case.

Note that the mapping

\overline{T_{Ω}} ↪ \overline{Ξ}, a ↦ aK_{ℂ}

defines a continuous embedding which is holomorphic when restricted to T_{Ω} . If x_{0}: = K_{ℂ} denotes the base point in G_{ℂ}/K_{ℂ}, then

(2.2)  GT_{Ω}(x_{0}) = Ξ, G\overline{T_{Ω}}(x_{0}) ⊆ \overline{Ξ}, and G∂T_{Ω}(x_{0}) ⊆ ∂_{d}Ξ.

Finally we define

∂_{d}Ξ = G exp(i∂_{e}Ω)K_{ℂ}/K_{ℂ} ⊆ ∂Ξ.

We call ∂_{d}Ξ the distinguished boundary of Ξ in G_{ℂ}/K_{ℂ}. This notion is justified by the following result:

Theorem 2.3. For all f ∈ A(Ξ) we have

sup_{z∈Ξ} |f(z)| = sup_{z∈∂_{d}Ξ} |f(z)|.

Proof. Recall from (2.1) the natural action of G on A(Ξ). With Lemma 2.2 and (2.2) we therefore obtain that

sup_{z∈Ξ} |f(z)| = sup_{g∈G} sup_{z∈T_{Ω}} |f(g^{−1}z(x_{0}))| = sup_{g∈G} sup_{z∈T_{Ω}} |λ_{g}(f)(z(x_{0}))|
= sup_{g∈G} sup_{z∈∂_{d}T_{Ω}} |λ_{g}(f)(z(x_{0}))| = sup_{g∈G} sup_{z∈∂_{d}T_{Ω}} |f(g^{−1}z(x_{0}))|
= sup_{z∈∂_{d}Ξ} |f(z)|.

This proves the theorem.

Remark 2.4. If ∂_{d}Ξ is connected, i.e., if ∂_{e}Ω consists of a single W -orbit (cf. Remark 2.1), then ∂_{d}Ξ is in fact minimal in the sense of Theorem 2.3. Hence it makes perfect sense to call ∂_{d}Ξ the Shilov boundary of Ξ in this case.

Minimality of the distinguished boundary

The whole boundary ∂Ξ is a rather complicated stratified G-space. Also the piece ∂_{a}Ξ is only a very small subset in ∂Ξ.

Because of the complicated structure of ∂Ξ we were not able to prove that the distinguished boundary ∂_{d}Ξ is minimal in ∂Ξ in the sense of Theorem 2.3. However, we can obtain the minimality of ∂_{d}Ξ if we replace the algebra A(Ξ) by the Banach algebra P(Ξ) of bounded continuous plurisubharmonic functions on Ξ:

P(Ξ): = {f ∈ C(Ξ): f plurisubharmonic, ∥f∥: = sup_{z∈Ξ} |f(z)| < ∞}.

For 0 ≤ t < 1 define

∂_{d,t}Ξ = G exp(it∂_{e}Ω)K_{ℂ}/K_{ℂ}.

and note that lim_{t→1} ∂_{d,t}Ξ = ∂_{d}Ξ set-theoretically.

Theorem 2.4. The following assertions hold:

(i) For all f ∈ P(Ξ) we have

sup_{z∈Ξ} |f(z)| = lim_{t→1} ( sup_{z∈∂_{d,t}Ξ} |f(z)|) .

(ii) ∂_{d}Ξ is minimal in ∂_{a}Ξ with respect to the property in (i).

Proof. (i) This is proved in the same way as Theorem 2.3. (ii) Recall the partition ∂_{e}Ω = W(Y_{1}) ⨿ … ⨿ W(Y_{n}). Further set ∂_{d,t,j}Ξ: = G exp(itY_{j})K_{ℂ}/K_{ℂ} for every 1 ≤ j ≤ n and 0 ≤ t < 1.

Suppose that ∂_{d}Ξ is not minimal with respect to the property in (i). Since a minimal set is necessarily G-invariant, we hence find an Y_{j} such that

(2.3)  lim_{t→1} ( sup_{z∈∂_{d,t,j}Ξ} |f(z)|) ≤ lim_{t→1} max_{k≠j} ( sup_{z∈∂_{d,t,k}Ξ} |f(z)|)

for all f ∈ P(Ξ).

To conclude the proof of (ii) we have to recall some facts from [GiKr01, Sect. 3]. There exists a unique holomorphic surjective mapping

Ξ → T_{Ω} ⊆ A_{ℂ}, z ↦ a(z)

such that z ∈ N_{ℂ}a(z)K_{ℂ}/K_{ℂ}. Furthermore if z = g exp(iX)K_{ℂ} for g ∈ G and X ∈ Ω, then

Im log a(z) ⊆ conv(W(X)).

Recall that T_{Ω} is biholomorphic to 𝔞 + iΩ. Thus we can find an A-invariant F ∈ P(T_{Ω}) which peaks at exp(iY_{j}). Now the function

f: Ξ → ℂ, z ↦ F (a(z))

is a well defined element in P(Ξ). By construction f does not satisfy (2.3), concluding the proof of (ii).

3. Determination of the distinguished boundary

In this section we will show that the distinguished boundary ∂_{d}Ξ can only be a union of symmetric spaces under the geometric assumption that G is a non-compactly causal group. Further we will determine the distinguished boundary for all non-compactly causal groups.

We first have to recall some facts on causal symmetric Lie algebras.

Causal symmetric Lie algebras

The standard reference for the facts collected below is the book [HiÓl96].

As before 𝔤 denotes a semisimple real Lie algebra and θ a Cartan involution on 𝔤 with Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭. Let τ: 𝔰 → 𝔰 be an involution on 𝔤 of which we may assume to commute with θ. Write 𝔤 = 𝔥 ⊕ 𝔮 for the τ -eigenspace decomposition corresponding to the τ -eigenvalues +1 and −1.

The symmetric Lie algebra (𝔤, τ) is called irreducible if the only τ -invariant ideals of 𝔤 are {0} and 𝔤.

Remark 3.1. If (𝔤, τ) is irreducible, then 𝔤 is either simple or 𝔤 = 𝔥 ⊕ 𝔥 with 𝔥 simple and τ is the flip τ(X, Y) = (Y, X) (“group case”).

If (𝔤, τ) is a symmetric Lie algebra, then we define its c-dual by 𝔤^{c} = 𝔥 ⊕ i𝔮. Call the restriction of the complex linear extension of τ to 𝔤^{c} also by τ. Then (𝔤^{c}, τ) is called the symmetric Lie algebra c-dual to (𝔤, τ).

Definition 3.2. Let (𝔤, τ) be an irreducible semisimple symmetric Lie algebra.

(a) We call (𝔤, τ) compactly causal if 𝔷(𝔨) ∩ 𝔮 ≠ {0}.

(b) We call (𝔤, τ) non-compactly causal if its c-dual is compactly causal, or, equivalently, if there exists an 0 ≠ Y_{0} ∈ 𝔮 ∩ 𝔭 which is fixed under 𝔥 ∩ 𝔨.

(c) We call (𝔤, τ) of Cayley type if it is both compactly and non-compactly causal.

Remark 3.3. (a) Non-compactly causal symmetric Lie algebras are classified. The complete list is as follows (cf. [HiÓl96, Th. 3.2.8]):

\begin{matrix}𝔤 & 𝔥 \\ 𝔰𝔭(n, ℝ) & 𝔤𝔩(n, ℝ) \\ 𝔰𝔲(n, n) & 𝔰𝔩(n, ℂ) × ℝ \\ 𝔰𝔬^{∗}(4n) & 𝔰𝔩(n, ℍ) × ℝ \\ 𝔰𝔬(p, q) & 𝔰𝔬(1, p − 1) × 𝔰𝔬(1, q − 1) \\ 𝔰𝔬(n, n) & 𝔰𝔬(n, ℂ) \\ 𝔰𝔭(n, n) & 𝔰𝔭(n, ℂ) \\ 𝔰𝔩(n, ℝ) & 𝔰𝔬(q, n − q) (1 ≤ q < n) \\ 𝔰𝔩(n, ℍ) & 𝔰𝔭(q, n − q) (1 ≤ q < n) \\ 𝔰𝔬(2n, ℂ) & 𝔰𝔬^{∗}(2n) \\ 𝔰𝔬(n + 2, ℂ) & 𝔰𝔬(2, n) \\ 𝔰𝔭(n, ℂ) & 𝔰𝔭(n, ℝ) \\ 𝔰𝔩(n, ℂ) & 𝔰𝔲(q, n − q) (1 ≤ q < n) \\ 𝔢_{6(6)} & 𝔰𝔭(2, 2) \\ 𝔢_{6(−26)} & 𝔣_{4(−20)} \\ 𝔢_{6} & 𝔢_{6(−14)} \\ 𝔢_{7(7)} & 𝔰𝔲^{∗}(8) \\ 𝔢_{7(−25)} & 𝔢_{6(−26)} × ℝ \\ 𝔢_{7} & 𝔢_{7(−25)} \end{matrix}

(b) If (𝔤, τ) is of Cayley type, then 𝔤 has to be simple hermitian and of tube type. Conversely, if 𝔤 is simple hermitian and of tube type, then there exists up to conjugation only one involution τ on 𝔤 (the square of the Cayley transform) turning (𝔤, τ) into a Cayley type symmetric Lie algebra (cf. [HiÓl96]). The Cayley type spaces are the following:

\begin{matrix}𝔤 & 𝔥 \\ 𝔰𝔭(n, ℝ) & 𝔤𝔩(n, ℝ) \\ 𝔰𝔲(n, n) & 𝔰𝔩(n, ℂ) × ℝ \\ 𝔰𝔬^{∗}(4n) & 𝔰𝔩(n, ℍ) × ℝ \\ 𝔰𝔬(2, n) & 𝔰𝔬(1, n − 1) × ℝ \\ 𝔢_{7(−25)} & 𝔢_{6(−26)} × ℝ \end{matrix}

We call a Lie algebra 𝔤 non-compactly causal if there exists an involution τ on 𝔤 turning (𝔤, τ) into a non-compactly causal symmetric Lie algebra. A Lie group G is called non-compactly causal if its Lie algebra 𝔤 is non-compactly causal.

General results on G-orbits in the distinguished boundary

Let ∂_{d}Ξ = G exp(i∂_{e}Ω)K_{ℂ}/K_{ℂ} be the distinguished boundary of Ξ. Let
∂_{e}Ω = W(Y_{1}) ⨿ … ⨿ W(Y_{n})

be the partition of Weyl group orbits with Y_{j} ∈ ∂_{e}Ω. Set z_{j}: = exp(iY_{j}) and z′_{j}: = z_{j}K_{ℂ} ∈ ∂_{d}Ξ. Notice that

(3.1)  ∂_{d}Ξ = ⋃_{j=1}^{n} G(z′_{j}) .

If we write G_{z′_{j}} for the isotropy subgroup of G in z′_{j}, then we have G(z′_{j}) ≃ G/G_{z′_{j}}. We also write θ for the holomorphic extension of the Cartan involution from G to G_{ℂ}. If we assume that G_{ℂ}^{θ} = K_{ℂ} (this is always satisfied if G_{ℂ} is simply connected), then we

can characterize G_{z′_{j}} through the following equivalences:

(3.2)  \begin{aligned}g ∈ G_{z′_{j}} & ⟺ z_{j}^{−1}gz_{j} ∈ K_{ℂ} \\ & ⟺ (z_{j}^{−1}gz_{j})θ(z_{j}^{−1}gz_{j})^{−1} = 1 \\ & ⟺ gz_{j}^{2}θ(g)^{−1} = z_{j}^{2} \\ & ⟺ z_{j}^{2}θ(g)z_{j}^{−2} = g. \end{aligned}

Note that we always have (G_{ℂ}^{θ})_{0} = K_{ℂ}. If G_{ℂ}^{θ} ≠ K_{ℂ}, then the computation (3.2) still gives us the connected component (G_{z′_{j}})_{0} of G_{z′_{j}}, namely

(G_{z′_{j}})_{0} = {g ∈ G: z_{j}^{2}θ(g)z_{j}^{−2} = g}_{0}.

Summarizing our discussions from above we have proved:

Lemma 3.4. Let Y ∈ ∂_{e}Ω and set z: = exp(iY), z′: = zK_{ℂ} ∈ ∂_{d}Ξ. Then the following assertions hold:

(i) The connected component of the isotropy subgroup in z′ is given by

(G_{z′})_{0} = {g ∈ G: z^{2}θ(g)z^{−2} = g}_{0}.

(ii) If G_{ℂ} is simply connected, then

G_{z′} = {g ∈ G: z^{2}θ(g)z^{−2} = g}.

(iii) If G_{ℂ} is simply connected and if τ(g) = z^{2}θ(g)z^{−2} is an involution on G, then

g ∈ G_{z′} ⟺ g ∈ G^{τ} .

In general we cannot expect that the automorphism τ in Lemma 3.4(iii) preserves G. This can only happen under the geometric assumption that G is a non-compactly causal group.

Theorem 3.5. Let Y ∈ ∂_{e}Ω and z: = exp(iY). Consider the automorphism

τ: G_{ℂ} → G_{ℂ}, g ↦ z^{2}θ(g)z^{−2}.

Then the following assertions hold:

(i) If τ: = dτ(1) preserves 𝔤, then τ is an involution and (𝔤, τ) is a non-compactly causal symmetric Lie algebra.

(ii) If τ preserves G, then G/G^{τ} is non-compactly causal and we have a G-isomorphism

G/G^{τ} ≃ G(z′) ⊆ ∂_{d}Ξ.

Proof. (following a suggestion of Gestur Ólafsson) (i) Suppose that τ defines an automorphism of 𝔤. Note that τ(X) = e^{i2 ad Y} (X) for X ∈ 𝔤.

Since Y ∈ \overline{Ω} we have Spec ad(Y) ⊆ [−\frac{π}{2} , \frac{π}{2} ]. Further the fact that τ(X) ∈ 𝔤 for all X ∈ 𝔤 means precisely that Spec(ad Y) ⊆ ℤ\frac{π}{2} . Hence we obtain that Spec(ad Y) = \{−\frac{π}{2} , 0, \frac{π}{2} \} by the symmetry of the spectrum. In particular, τ is an involution on 𝔤 and it follows from [HiÓl96, Th. 3.2.4] that (𝔤, τ) is non-compactly causal. (ii) This follows from (i).

We conclude our general discussion on G-orbits in ∂_{d}Ξ addressing the question when the union in (3.1) is actually disjoint.

We now simplify notation and set H_{j}: = G_{z′_{j}} and denote by 𝔥_{j} the Lie algebra of H_{j}.

Theorem 3.6. Suppose that G_{ℂ} is simply connected. Then we have G(z′_{j}) = G(z′_{k}) if and only if j = k. In particular,

∂_{d}Ξ = ∐_{j=1}^{n} G(z′_{j}) ≃ ∐_{j=1}^{n} G/H_{j}

Proof. Write σ for the complex conjugation in G_{ℂ} with respect to the real form G. Notice that G(z′_{j}) = G(z′_{k}) means that there exists a g ∈ G and a k ∈ K_{ℂ} such that

(1)  gz_{j}k = z_{k} .

Then it follows

σ(gkz_{j})^{−1}gkz_{j} = σ(z_{k})^{−1}z_{k}

which is equivalent to σ(k)^{−1}z_{j}^{2}k = z_{k}^{2} or

(2)  z_{j}^{2}k = σ(k)z_{k}^{2} .

Write 𝔲 = 𝔨 ⊕ i𝔭 and U for the analytic subgroup of G_{ℂ} with Lie algebra 𝔲. Note that U is a maximal compact subgroup of G_{ℂ} which is simply connected. Now write k = k_{0} exp(Z) for some k_{0} ∈ K and Z ∈ i𝔨. Then it follows from (2) that

z_{j}^{2}k_{0} exp(Z) = k_{0}z_{k}^{2} exp(− Ad(z_{k}^{−2})Z)

and so z_{j}^{2}k_{0} = k_{0}z_{k}^{2} by the polar decomposition G_{ℂ} ≃ U × i𝔲 of G_{ℂ}. Thus we obtain

(3)  z_{j}^{2} = k_{0}z_{k}^{2}k_{0}^{−1} .

Notice that (3) is actually an identity in the simply connected compact group U. Since z_{l}^{2} = exp(i2Y_{j}) and Y_{l} ∈ \overline{Ω} for every 1 ≤ l ≤ n, it hence follows from [Hel78, Th. 8.6(ii)] that Y_{j} is conjugate to Y_{k} under W. Hence Y_{j} = Y_{k} since W(Y_{j}) ⨿ W(Y_{k}) for k ≠ j by definition.

In view of Theorem 3.5, we can only expect that symmetric spaces are contained in the distinguished boundary ∂_{d}Ξ if G is a non-compactly causal group. Before we start to classify the distinguished boundaries related to all such groups we first discuss the case where G is an arbitrary complex semisimple Lie group.

The case where G is complex

We will recall some material on complexifications of semisimple complex Lie algebras and certain complex homogeneous spaces.

Let 𝔤 be a complex semisimple Lie algebra and 𝔨 < 𝔤 a maximal compact subalgebra. Then 𝔨 is a compact real form of 𝔤 and 𝔤 = 𝔨 + i𝔨 is a Cartan decomposition of 𝔤. We write X ↦ \overline{X} for the complex conjugation of 𝔤 with respect to the real form 𝔨.

Write 𝔤_{ℝ} for 𝔤 considered as a real Lie algebra and define 𝔤_{ℂ} as the complexification of 𝔤_{ℝ}. Denote by J the multiplication with i in 𝔤_{ℂ}. Finally we write 𝔤^{opp} for 𝔤 equipped with the opposite complex structure.

The mapping

𝔤_{ℂ} → 𝔤 ⊕ 𝔤^{opp}, X + JY ↦ (X + iY, X − iY)

is an isomorphism of the complex Lie algebra (𝔤_{ℂ}, J) with 𝔤 ⊕ 𝔤^{opp} . Now

𝔤^{opp} → 𝔤, X ↦ \overline{X}

establishes a complex Lie algebra isomorphism. Hence we get the comlex Lie algebra isomorphism

𝔤_{ℂ} → 𝔤 ⊕ 𝔤, X + JY ↦ (X + iY, \overline{X} + i\overline{Y}).

Note the following realizations of subalgebras in 𝔤_{ℂ} inside of 𝔤 ⊕ 𝔤:

𝔤 ≃ \{(X, \overline{X}): X ∈ 𝔤\}
𝔨_{ℂ} ≃ Δ(𝔤): = {(X, X): X ∈ 𝔤}
𝔞_{ℂ} ≃ {(X, −X): X ∈ 𝔞 + i𝔞 ⊆ 𝔤}

Let now G be a Lie group with Lie algebra 𝔤. Then our observations from above imply the isomorphism

G_{ℂ}/K_{ℂ} ≃ (G × G)/Δ(G)

with Δ(G) the diagonal subgroup. Further we have the canonical isomorphism

G × G/Δ(G) → G, (g, h)Δ(G) ↦ gh^{−1}.

Hence we have G_{ℂ}/K_{ℂ} ≃ G. Note that the morphism of A_{ℂ} into G_{ℂ}/K_{ℂ} = G is given by

A_{ℂ} → G, a ↦ a^{2},

the square mapping. Further the action of G on G_{ℂ}/K_{ℂ} = G is given by

G × G_{ℂ}/K_{ℂ} → G_{ℂ}/K_{ℂ}, (g, x) ↦ gx\overline{g}^{−1},

where we identified G_{ℂ}/K_{ℂ} with G.

The summary of our discussion is now:

Proposition 3.7. Assume that G is a semisimple complex Lie group. Then G_{ℂ}/K_{ℂ} can be canonically identified with G. Moreover, if Y ∈ ∂_{e}Ω, z = exp(iY) and z′ = zK_{ℂ}, then the isotropy subgroup of G in z′ is given by

G_{z′} = \{g ∈ G: z^{2}\overline{g}z^{−2} = g\},

where g ↦ \overline{g} denotes the conjugation in G with respect to the compact real form K of G.

From now on we will assume that 𝔤 is a non-compactly causal Lie algebra. Depending on the type of the restricted root system Σ we are are now going to determine the distinguished boundary of Ξ.

Note: Unless not otherwise stated we will assume that G_{ℂ} is simply connected.

The cases with restricted root system of type C_{n}

Assume that 𝔤 is a non-compactly causal Lie algebra with restricted root system of type C_{n}. According to the list in Remark 3.3(a) this means that 𝔤 is one of the following list:

𝔰𝔭(n, ℝ) 𝔰𝔭(n, n) 𝔰𝔭(n, ℂ) 𝔰𝔲(n, n) 𝔰𝔬^{∗}(4n) 𝔰𝔬(2, n) 𝔢_{7(−25)}.

Note that for all these cases there exists up to conjugation only one involution τ on 𝔤 which turns (𝔤, τ) into a non-compactly causal symmetric Lie algebra.

On the group level we take G to be the real form of a simply connected Lie group G_{ℂ} with Lie algebra 𝔤_{ℂ}.

The restricted root system Σ is of type C_{n}, say

Σ = \{\frac{1}{2}(±ε_{i} ± ε_{j}): 1 ≤ i, j ≤ n\}\\{0\}.

Define Y_{j} ∈ 𝔞 by ε_{i}(Y_{j}) = \frac{π}{2} δ_{ij}. Then

Ω = ⨁_{j=1}^{n}] − 1, 1[Y_{j}

and

∂_{e}Ω = {±Y_{1} ± … ± Y_{n}}.

Define

Y_{0}: = Y_{1} + … + Y_{n}

and note that

∂_{e}Ω = W(Y_{0}),

since W consists of all permutations and sign changes. Observe that

Spec(ad Y_{0}) = \{−\frac{π}{2} , 0, \frac{π}{2} \}.

We have ∂_{d}Ξ = G exp(iY_{0})K_{ℂ}/K_{ℂ}. Set z_{0}: = exp(iY_{0}). Then ∂_{d}Ξ = G(z_{0}) and so ∂_{d}Ξ ≃ G/G_{z′_{0}} with z′_{0} = z_{0}K_{ℂ} ∈ ∂_{d}Ξ. Note that

τ: G_{ℂ} → G_{ℂ}, g ↦ z_{0}^{2}θ(g)z_{0}^{−2}

restricts to an involution on G since Spec(ad Y_{0}) = \{−\frac{π}{2} , 0, \frac{π}{2} \}. Thus we obtain from Lemma 3.4 and Theorem 3.5 that:

Proposition 3.8. Let G be a non-compactly causal group sitting inside a simply connected complex Lie group G_{ℂ}. Suppose that the restricted root system Σ is of type C_{n}. Then the distinguished boundary of Ξ is G-isomorphic to the (up to isomorphism unique) non-compactly causal symmetric space G/H associated to G:

∂_{d}Ξ ≃ G/H.

Examples with restricted root system of type A_{n}

The example of G = Sl(n, ℝ). Here we will encounter the situation that the distinguished boundary is not connected and a union of different non-compactly causal symmetric spaces. The example features the situation well where the restricted root system is of type A_{n}.

We introduce some notation. Let 𝔤 = 𝔰𝔩(n, ℝ) and choose 𝔨 = 𝔰𝔬(n, ℝ) as a maximal compact subalgebra of 𝔤. Write 𝔤 = 𝔨 ⊕ 𝔭 for the corresponding Cartan decomposition. For 1 ≤ j ≤ n we introduce the diagonal matrix e_{j}: = (δ_{ij})_{i,j}. Then

𝔞: = {∑_{j=1}^{n} x_{j}e_{j}: x_{j} ∈ ℝ, ∑_{j=1}^{n} x_{j} = 0}

is a maximal abelian subspace in 𝔭. Define linear functionals ε_{j} on the diagonal matrices by ε_{j}(e_{i}) = δ_{ij}. Then

Σ = {ε_{i} − ε_{j}: i ≠ j}

and W is the permutation group of {e_{1}, …, e_{n}}.

For every 1 ≤ q ≤ n − 1 we define elements Y_{q} ∈ 𝔞 by

Y_{q}: = \frac{π}{2} ( ∑_{j=1}^{q} e_{j} − \frac{q}{n} ∑_{j=1}^{n} e_{j}).

Then it follows from [KrSt01a, Lemma 2.1] that

(3.3)  ∂_{e}Ω = ∐_{q=1}^{n−1} W(Y_{q}).

The union in (3.3) disjoint. For every 1 ≤ q ≤ n − 1 set z_{q}: = exp(iY_{q}) and denote by H_{q} < G the stabilizer in G of the element z′_{q} = z_{q}K_{ℂ} ∈ ∂_{d}Ξ. Since G_{ℂ} = Sl(n, ℂ) is simply connected, it follows from Theorem 3.6 that

∂_{d}Ξ ≃ ∐_{q=1}^{n−1} G/H_{q} .

It remains to compute the isotropy subgroups H_{q}. We have

z_{q} = e^{−i \frac{π}{2} \frac{q}{n}} diag(i, …, i, 1, …, 1)

with q -times i on the diagonal. From (3.2) we obtain the following equivalences:

g ∈ H_{q} ⟺ z_{q}^{−1}gz_{q} ∈ K_{ℂ}
⟺ (z_{q}^{−1}gz_{q})(z_{q}^{−1}gz_{q})^{t} = 1
⟺ gz_{q}^{2}g^{t} = z_{q}^{2}
⟺ g diag(−1, …, −1, 1, …, 1)g^{t} = diag(−1, …, −1, 1, … 1)
⟺ g ∈ SO(q, n − q).

Alltogether we thus have shown:

Proposition 3.9. For G = Sl(n, ℝ) we have the following G-isomorphism of the distinguished boundary:

∂_{d}Ξ ≃ ∐_{q=1}^{n−1} Sl(n, ℝ)/ SO(q, n − q).

The example of G = Sl(n, ℂ). This is very similar to the case of G = Sl(n, ℝ). We will use our results on complex groups. Let G = Sl(n, ℂ) and K = SU(n). As before we can take 𝔞 to be the diagonal matrices in 𝔤. Since the root system is the same as for 𝔤 = 𝔰𝔩(n, ℝ) and since G_{ℂ} ≃ G × G is simply connected, we conclude that

∂_{d}Ξ = ∐_{q=1}^{n−1} G(z′_{q})

with z′_{q} = exp(iY_{q})K_{ℂ} and Y_{q} as before.

Now we are going to use the identification of G_{ℂ}/K_{ℂ} with G as explained before Proposition 3.7. Then z′_{q} ∈ G_{ℂ}/K_{ℂ} becomes identified with the point z_{q}^{2}: = exp(i2Y_{q}) in G. We have

z_{q}^{2} = c diag(1, …, 1, −1, …, −1)

with c a constant and q times +1 on the diagonal. Further G acts on G_{ℂ}/K_{ℂ} = G by g(x) = gx\overline{g}^{t} where \overline{g} denotes the complex conjugation with respect to Sl(n, ℝ). Hence Proposition 3.7 implies that G_{z′_{q}} ≃ SU(q, n − p). Thus we have proved:

Proposition 3.10. For G = Sl(n, ℂ) we have the following G-isomorphism of the distinguished boundary:

∂_{d}Ξ ≃ ∐_{q=1}^{n−1} Sl(n, ℂ)/ SU(q, n − q).

The example of G = Sl(n, ℍ). Considerations very similar to the two other A_{n} -cases treated before yield the following result:

Proposition 3.11. For G = Sl(n, ℍ) we have the following G-isomorphism of the distinguished boundary:

∂_{d}Ξ ≃ ∐_{q=1}^{n−1} Sl(n, ℍ)/ Sp(q, n − q).

Examples with restricted root systems of type B_{n} or D_{n}

The classical Lie algebras which have restricted root system of type B_{n} or D_{n} are 𝔰𝔬(p, q) and 𝔰𝔬(n, ℂ). Before we can treat these cases we first have to recall some facts of root systems of type B_{n} and D_{n}.

Root systems of type B_{n} and D_{n}. In the standard notation the root systems of type B_{n} and D_{n} are given by

B_{n} = {±ε_{i} ± ε_{j}: 1 ≤ i ≠ j ≤ n} ⨿ {±ε_{i}: 1 ≤ i ≤ n}

and

D_{n} = {±ε_{i} ± ε_{j}: 1 ≤ i ≠ j ≤ n}.

As usual we write e_{1}, …, e_{n} for the dual basis to ε_{1}, …, ε_{n}. If Σ is a root system, then we also write Ω = Ω(Σ) to make the relation to Σ clear whenever ambigious.

Lemma 3.12. For the root systems B_{n} and D_{n} the following assertions hold:

(i) We have Ω(B_{n}) = Ω(D_{n}). In particular, ∂_{e}Ω(B_{n}) = ∂_{e}Ω(D_{n}). (ii) For Σ = D_{n} and n ≥ 3 we have

∂_{e}Ω = W(Y_{1}) ⨿ W(Y_{2}) ⨿ W(Y_{3})

a disjoint union with Y_{1} = \frac{π}{2} e_{1} , Y_{2} = \frac{π}{4} (e_{1} + … + e_{n}) and Y_{3} = \frac{π}{4} (e_{1} + … + e_{n−1} − e_{n}). (iii) For Σ = B_{2} we have ∂_{e}Ω = W(Y_{1}) and for n ≥ 3 we have

∂_{e}Ω = W(Y_{1}) ⨿ W(Y_{2})

a disjoint union with Y_{1} and Y_{2} as in (i). Proof. [KrSt01a, Sect. 2].

The example of G locally SO(2n, ℂ) for n ≥ 3. In this case the restricted root system is of type D_{n}. Hence Lemma 3.12(ii) implies that

∂_{e}Ω = W(Y_{1}) ⨿ W(Y_{2}) ⨿ W(Y_{3})

consists of three Weyl group orbits. Note that Y_{2} and Y_{3} are conjugate under an outer isomorphism κ induced from an outer isomorphism of the Dynkin diagram.

As a maximal abelian subspace 𝔞 ⊆ 𝔭 we choose

𝔞: = \{( \begin{matrix}( & \begin{matrix}0 & it_{1} \\ −it_{1} & 0 \end{matrix} & ) \\ & & & . & . & . \\ & & & & & & ( & \begin{matrix}0 & it_{n} \\ −it_{n} & 0 \end{matrix} & ) \end{matrix} ): t_{1}, …, t_{n} ∈ ℝ\}.

To compute the isotropy Lie algebras 𝔥_{j} we may assume for a moment that G = SO(2n, ℂ) (cf. Lemma 3.4(i)). With z_{j}: = exp(iY_{j}) and z′_{j} = z_{j}K_{ℂ} for j = 1, 2, 3, we then get

z_{1}^{2} = diag(−1, −1, 1, …, 1), z_{2}^{2} = ( \begin{matrix}( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & & . & . & . \\ & & & & & & ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \end{matrix} )

and z_{3}^{2} = κ(z_{2}^{2}). From Proposition 3.7 we have

G_{z′_{j}} = \{g ∈ G: z_{j}^{2}\overline{g}z_{j}^{−2} = g\}

and so

G_{z′_{1}} ≃ SO(2, 2n − 2) and G_{z′_{2}} ≃ G_{z′_{3}} ≃ SO^{∗}(2n).

Thus we have proved:

Proposition 3.13. For G locally SO(2n, ℂ) with n ≥ 3 we have the following G-isomorphism of the distinguished boundary:

∂_{d}Ξ ≃ G/H_{1} ⨿ G/H_{2} ⨿ G/H_{2}

with 𝔥_{1} = 𝔰𝔬(2, 2n − 2) and 𝔥_{2} = 𝔰𝔬^{∗}(2n).

The example of G locally SO(2n + 1, ℂ). In this case the restricted root system is of type B_{n} and Lemma 3.12(iii) implies that

∂_{e}Ω = W(Y_{1}) ⨿ W(Y_{2})

where the second term in the union is not existent for n = 2. As before we now obtain that:

Proposition 3.14. For G locally SO(2n + 1, ℂ) we have the following G-isomorphism of the distinguished boundary:

(i) For n ≥ 3 one has ∂_{d}Ξ ≃ G/H_{1} ⨿ G/H_{2} with

𝔥_{1} = 𝔰𝔬(2, 2n − 1)  and  𝔥_{2} = 𝔰𝔬^{∗}(2n).

In particular, the boundary component G/H_{2} is not a symmetric space. (ii) For n = 2 one has ∂_{d}Ξ ≃ G/H with 𝔥: = Lie(H) = 𝔰𝔬(2, 3).

Some general remarks on 𝔤 = 𝔰𝔬(p, q) for 0 < p ≤ q. An appropriate maximal abelian subspace of 𝔭 in 𝔤 = 𝔰𝔬(p, q) is

𝔞: = \{( \begin{matrix}0_{pp} & I_{t_{1},...,t_{p}} \\ I_{t_{1},...,t_{p}}^{t} & 0_{qq} \end{matrix} ): t_{1}, …, t_{p} ∈ ℝ\},

where

I_{t_{1},...,t_{p}} = ( \begin{matrix}& & & t_{1} \\ 0_{p,p−q} & & ... \\ & t_{p} \end{matrix} ) ∈ M(p × q; ℝ).

Let n = p + q. For all 1 ≤ i, j ≤ n define E_{ij} ∈ M_{n}(ℝ) by E_{ij} = (δ_{ki}δ_{jl})_{k,l}. Then 𝔞 = ⨁_{j=1}^{p} ℝe_{j} with

e_{j} = E_{j,n+1−j} + E_{n+1−j,j}.

Define ε_{j} ∈ 𝔞^{∗} by ε_{j}(e_{k}): = δ_{jk}. Then the root system Σ = Σ(𝔞, 𝔤) is given by

Σ = \{ \begin{matrix}\{±ε_{i} ± ε_{j}: 1 ≤ i ≠ j ≤ p\} ⨿ \{±ε_{i}: 1 ≤ i ≤ p\} & for 1 < p < q, \\ \{±ε_{i} ± ε_{j}: 1 ≤ i ≠ j ≤ p\} & for 1 < p = q, \\ \{±ε_{1}\} & for p = 1. \end{matrix}

The restricted root system is hence of type B_{p}, D_{p} or A_{1} .

The example of G locally SO(n, n) for n ≥ 3. Here the root system is of type D_{n} and we have

∂_{e}Ω = W(Y_{1}) ⨿ W(Y_{2}) ⨿ W(Y_{3}).

To compute the isotropy subalgebras 𝔥_{j} we may assume for a moment that G = SO(n, n). In the coordinates introduced before we then have

z_{1}^{2} = diag(−1, 1, …, 1, −1), z_{2}^{2} = \{( \begin{matrix}0_{nn} & iI_{1,...,1} \\ −iI_{1,...,1} & 0_{nn} \end{matrix} )

and z_{3}^{2} = κ(z_{2}^{2}). From Lemma 3.4 we thus obtain that

(G_{z′_{1}})_{0} = SO_{e}(1, n − 1) × SO_{e}(1, n − 1) and (G_{z′_{2}})_{0} = SO(n, ℂ).

In view of Theorem 3.6 we thus we have shown that:

Proposition 3.15. For G locally SO(n, n) and n ≥ 3 we have the following G-isomorphism of the distinguished boundary: ∂_{d}Ξ ≃ G/H_{1} ⨿ G/H_{2} ⨿ G/H_{2} with

𝔥_{1} = 𝔰𝔬(1, n − 1) × 𝔰𝔬(1, n − 1)  and  𝔥_{2} = 𝔰𝔬(n, ℂ).

The example of G locally SO(p, q) for 1 ≤ p < q. Here the restricted root system is of type B_{p} for p ≥ 2 and A_{1} for p = 1. Considerations very similar to as before lead us to the following result:

Proposition 3.16. For G locally SO(p, q) and 1 ≤ p < q the distinguished boundary ∂_{d}Ξ is described as follows:

(i) For p = 1 one has ∂_{d}Ξ ≃ G/H with 𝔥 = 𝔰𝔬(1, q − 1). (ii) For p = 2 one has ∂_{d}Ξ ≃ G/H with 𝔥 = 𝔰𝔬(1, q − 1) × ℝ. (iii) For p ≥ 3 one has ∂_{d}Ξ ≃ G/H_{1} ⨿ G/H_{2} with

𝔥_{1} = 𝔰𝔬(1, p − 1) × 𝔰𝔬(1, q − 1)  and  𝔥_{2} = 𝔰𝔬(p, ℂ) × 𝔰𝔬(q − p).

In particular, the boundary component G/H_{2} is not a symmetric space.

Exceptional cases

We first have to recall some facts about the root systems E_{6} and E_{7} . Our references herefore are the tables from [Kn96]. Write e_{1}, …, e_{n} for the standard basis in Euclidean space ℝ^{n} and write ⟨·, ·⟩ for the standard inner product.

The system E_{6} . Let

V: = {v ∈ ℝ^{8}: ⟨v, e_{6} − e_{7}⟩ = ⟨v, e_{7} + e_{8}⟩ = 0}.

Then the root system E_{6} can be defined by

Σ: = \{±e_{i} ± e_{j}: 1 ≤ j < i ≤ 5\} ⨿ \{\frac{1}{2} ∑_{j=1}^{8}(−1)^{n(j)}e_{j} ∈ V: ∑_{j=1}^{8} n(j) even\}.

A positive system is given by

Σ^{+} = \{e_{i} ± e_{j}: 1 ≤ j < i ≤ 5\} ⨿ \{\frac{1}{2}(e_{8} − e_{7} − e_{6} + ∑_{j=1}^{5}(−1)^{n(j)}e_{j}): ∑_{j=1}^{5} n(j) even\}.

Note that ⟨α, α⟩ = 2 for all α ∈ Σ, i.e., all roots have the same length. A basis for Σ^{+} is given by

Π = \{α_{1}, …, α_{6}\} = \{\frac{1}{2}(e_{8} −e_{7} −e_{6} −e_{5} −e_{4} −e_{3} −e_{2} +e_{1}), e_{1} +e_{2}, e_{2} −e_{1}, e_{3} −e_{2}, e_{4} −e_{3}, e_{5} −e_{4}\}.

A simple calculation then shows that the fundamental weights are given by

\begin{aligned}ω_{1} & = \frac{2}{3}(e_{8} − e_{7} − e_{6}) \\ ω_{2} & = \frac{1}{2}(e_{1} + e_{2} + e_{3} + e_{4} + e_{5}) + \frac{1}{2}(e_{8} − e_{7} − e_{6}) \\ ω_{3} & = \frac{1}{2}(−e_{1} + e_{2} + e_{3} + e_{4} + e_{5}) + \frac{5}{6}(e_{8} − e_{7} − e_{6}) \\ ω_{4} & = (e_{3} + e_{4} + e_{5}) + (e_{8} − e_{7} − e_{6}) \\ ω_{5} & = (e_{4} + e_{5}) + \frac{2}{3}(e_{8} − e_{7} − e_{6}) \\ ω_{6} & = e_{5} + \frac{1}{3}(e_{8} − e_{7} − e_{6}). \end{aligned}

Note that ω_{1}, …, ω_{6} is also the dual basis of Π since ⟨α, α⟩ = 2 for all α ∈ Σ.

The highest root β in Σ^{+} can be writtem as

β = α_{1} + 2α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + α_{6}.

In the following we set 𝔞: = V and we identify 𝔞 with 𝔞^{∗} via the inner product on 𝔞. If E ⊆ 𝔞 is a closed convex set, then we write Ext(E) for the extreme points of E.

Lemma 3.17. Let Σ be an irreducible abstract root system in the Euclidean space 𝔞. Let Π = {α_{1}, …, α_{n}} be a basis of Σ and ω_{1}, …, ω_{n} its dual basis. Further let β = ∑_{j=1}^{n} m_{j}α_{j} be the highest root with respect to Π. Further write C for the closed Weyl chamber with respect to Π. Then the following assertions hold:

(i) Ω is W -invariant. (ii) Ext(\overline{Ω}) ∩ C ⊆ Ext(\overline{Ω} ∩ C). (iii) We have the following inclusion

Ext(\overline{Ω}) ⊆ W(\frac{π}{2} \{\frac{ω_{1}}{m_{1}} , …, \frac{ω_{n}}{m_{n}} \}).

Proof. (i) and (ii) are trivial. (iii) First note that (i) and (ii) imply that

Ext(\overline{Ω}) ⊆ W(Ext(\overline{Ω}) ∩ C).

Thus it is sufficient to show that

Ext(\overline{Ω}) ∩ C ⊆ \frac{π}{2} \{\frac{ω_{1}}{m_{1}} , …, \frac{ω_{n}}{m_{n}} \}.

Note that

C = {x ∈ 𝔞: (∀j)⟨x, α_{j}⟩ ≥ 0} = ⨁_{j=1}^{n} ℝ^{+}ω_{j}.

Thus we get from m_{j} ∈ ℕ for all j that

∂Ω ∩ C = \{x = ∑_{j=1}^{n} λ_{j}ω_{j}: λ_{j} ≥ 0, ∑_{j=1}^{n} m_{j}λ_{j} = \frac{π}{2} \}.

From this the assertion follows.

For any α ∈ 𝔞 we denote by s_{α} the reflection with respect to the hyperplane α^{⊥}.

Lemma 3.18. For the root system E_{6} we have

∂_{e}Ω = Ext(\overline{Ω}) = W(\frac{π}{2} ω_{1}) ⨿ W(\frac{π}{2} ω_{6}) .

Proof. First recall that E_{6} admits a nontrivial outer automorphims κ, the reflection of the Dynkin diagram. Note that κ leaves \overline{Ω} invariant, κ(α_{1}) = α_{6} and κ(α_{3}) = α_{5} .

In view of Lemma 3.17(iii), it is hence sufficient to show that \frac{ω_{2}}{2} , \frac{ω_{4}}{3} , \frac{ω_{5}}{2} are not extremal in \overline{Ω}. Now observe that

\begin{aligned}\frac{ω_{5}}{2} & = \frac{1}{2}(e_{4} + e_{5}) + \frac{1}{3}(e_{8} − e_{7} − e_{6}) = \frac{1}{2}(ω_{6}) + \frac{1}{2}(s_{e_{4}−e_{5}}ω_{6}) \\ \frac{ω_{4}}{3} & = \frac{1}{3}(e_{3} + e_{4} + e_{5}) + \frac{1}{3}(e_{8} − e_{7} − e_{6}) = \frac{2}{3}(\frac{ω_{5}}{2} ) + \frac{1}{3}(s_{e_{3}−e_{5}}ω_{6}). \end{aligned}

This shows that both \frac{ω_{5}}{2} and \frac{ω_{4}}{3} are non-trivial convex combinations of elements in \overline{Ω}, hence are not extremal. It remains to show that \frac{ω_{2}}{2} is not extremal. From the definition of Σ it is easy to check that \frac{ω_{2}}{2} + t(e_{1} − e_{2}) ∈ \overline{Ω} for t ∈ ℝ and |t| small. Thus \frac{ω_{2}}{2} is not extremal, concluding the proof of the lemma.

The system E_{7} . Let

V: = {v ∈ ℝ^{8}: ⟨v, e_{7} + e_{8}⟩ = 0}.

The root system E_{7} is defined by

Σ: = \{±e_{i} ± e_{j}: 1 ≤ i < j ≤ 6\} ⨿ \{±(e_{7} − e_{8})\} ⨿ \{\frac{1}{2} ∑_{j=1}^{8}: (−1)^{n(j)}e_{j} ∈ V: ∑_{j=1}^{8} n(j) even\}.

Further a positive system is given by

\begin{matrix}Σ^{+}: = & \{e_{i} ± e_{j}: 1 ≤ j < i ≤ 6\} ⨿ \{e_{8} − e_{7}\} \\ ⨿ \{\frac{1}{2}(e_{8} − e_{7} + ∑_{j=1}^{6}(−1)^{n(j)}e_{j}): ∑_{j=1}^{6} n(j) odd\}. \end{matrix}

Note that ⟨α, α⟩ = 2 for all α ∈ Σ, i.e., all roots have the same length. A basis for Σ^{+} is given by

\begin{matrix}Π = \{α_{1}, …, α_{6}, α_{7}\} \\ = \{\frac{1}{2}(e_{8} − e_{7} − e_{6} − e_{5} − e_{4} − e_{3} − e_{2} + e_{1}), e_{1} + e_{2}, e_{2} − e_{1}, e_{3} − e_{2}, e_{4} − e_{3}, e_{5} − e_{4}, e_{6} − e_{5}\}. \end{matrix}

The fundamental weights are:

\begin{aligned}ω_{1} & = (e_{8} − e_{7}) \\ ω_{2} & = \frac{1}{2}(e_{1} + e_{2} + e_{3} + e_{4} + e_{5} + e_{6}) + (e_{8} − e_{7}) \\ ω_{3} & = \frac{1}{2}(−e_{1} + e_{2} + e_{3} + e_{4} + e_{5} + e_{6}) + \frac{3}{2}(e_{8} − e_{7}) \\ ω_{4} & = (e_{3} + e_{4} + e_{5} + e_{6}) + 2(e_{8} − e_{7}) \\ ω_{5} & = (e_{4} + e_{5} + e_{6}) + \frac{3}{2}(e_{8} − e_{7}) \\ ω_{6} & = (e_{5} + e_{6}) + (e_{8} − e_{7}) \\ ω_{7} & = e_{6} + \frac{1}{2}(e_{8} − e_{7}). \end{aligned}

The highest root β in Σ^{+} has the expression

β = 2α_{1} + 2α_{2} + 3α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7}.

Lemma 3.19. For a root system of type E_{7} we have

∂_{e}Ω = Ext(\overline{Ω}) = W(\frac{π}{2} ω_{7}).

Proof. In view of Lemma 3.17(iii), it is sufficient to show that \frac{ω_{1}}{2} , \frac{ω_{2}}{2} , \frac{ω_{3}}{2} , \frac{ω_{4}}{4} , \frac{ω_{5}}{3} and \frac{ω_{6}}{2} are not extremal in \overline{Ω}. Simple verification shows that

\begin{aligned}\frac{ω_{1}}{2} & = \frac{1}{2}(ω_{7}) + \frac{1}{2}(s_{e_{5}−e_{6}}s_{e_{5}+e_{6}}ω_{7}) \\ \frac{ω_{3}}{3} & = \frac{2}{3}(\frac{ω_{2}}{2} ) + \frac{1}{3}(s_{e_{5}−e_{6}}s_{e_{5}+e_{6}}ω_{7}) \\ \frac{ω_{4}}{4} & = \frac{3}{4}(\frac{ω_{5}}{3} ) + \frac{1}{4}(s_{e_{3}−e_{6}}ω_{7}) \\ \frac{ω_{5}}{3} & = \frac{1}{3}(ω_{7}) + \frac{2}{3}(\frac{ω_{6}}{2} ) \\ \frac{ω_{6}}{2} & = \frac{1}{2}(ω_{7}) + \frac{1}{2}(s_{e_{5}−e_{6}}ω_{7}). \end{aligned}

Finally, it is easy to show that \frac{ω_{3}}{2} + t(e_{1} − e_{2}) ∈ \overline{Ω} for t ∈ ℝ small.

Having this information we can now handle all the exceptional non-compactly causal Lie algebras. We write E_{6} and E_{7} for simply connected complex Lie groups with complex exceptional Lie algebra 𝔢_{6} resp. 𝔢_{7} . By E_{6(∗)} and E_{7(∗)} we denote the real forms of E_{6} and E_{7} with real forms 𝔢_{6(∗)} , resp. 𝔢_{7(∗)} .

The case of E_{6(6)} . Let𝔤 = 𝔢_{6(6)} . Note that 𝔤 is a normal real form of 𝔤_{ℂ}, i.e., 𝔞 is a Cartan algebra of 𝔤. In particular Σ is of type E_{6} .

From Lemma 3.18 we obtain that

∂_{e}Ω = W(\frac{π}{2} ω_{1}) ⨿ W(\frac{π}{2} ω_{6}).

From the structure of Σ (in particular from the formula for the highest root) we obtain that

Spec(ad \frac{π}{2} ω_{1}) = Spec(ad \frac{π}{2} ω_{6}) = \{−\frac{π}{2} , 0, \frac{π}{2} \}.

Hence the prescriptions

σ_{1}(X) = Ad(exp(iπω_{1})(X)  and  σ_{2}(X) = Ad(exp(iπω_{6}))(X)

define two involutions on 𝔤. Then τ_{1}: = σ_{1} ∘ θ and τ_{2}: = σ_{2} ∘ θ are involutions on 𝔤. Note that both involutions are conjugate under the outer automorphism κ.

Recall from the list in Remark 3.3(a) that there is up to conjugation a unique involution τ on 𝔤 which turns (𝔤, τ) into a non-compactly causal symmetric Lie algebra. Thus Lemma 3.4, Theorem 3.5 and Theorem 3.6 imply that:

Proposition 3.20. If 𝔤 = 𝔢_{6(6)} , then

∂_{d}Ξ ≃ E_{6(6)}/E_{6(6)}^{τ_{1}} ⨿ E_{6(6)}/E_{6(6)}^{τ_{2}}

with 𝔤^{τ_{1}} ≃ 𝔤^{τ_{2}} ≃ 𝔰𝔭(2, 2).

The case of E_{6} . This case is completely analogous to the e_{6(6)} -case. One obtains that:

Proposition 3.21. If 𝔤 = 𝔢_{6} , then

∂_{d}Ξ ≃ E_{6}/E_{6(−14)} ⨿ E_{6}/E_{6(−14)}

The case of E_{6(−26)} . Let 𝔤 = 𝔢_{6(−26)} . In this case Σ is of type A_{2} and we can use our results from our discussions from the root systems A_{n}. In particular we have

∂_{e}Ω = W(Y_{1}) ⨿ W(Y_{2}).

Note that Y_{1} and Y_{2} are conjugate under the outer isomorphism of the Dynkin diagram. Hence we obtain two conjugate involutions on 𝔤 by

τ_{j}(X) = (Ad(exp(i2Y_{j})) ∘ θ)(X)

for j = 1, 2. From Lemma 3.4, Theorem 3.5 and Theorem 3.6 we thus obtain that:

Proposition 3.22. If 𝔤 = 𝔢_{6(−26)} , then

∂_{d}Ξ ≃ E_{6(−26)}/E_{6(−26)}^{τ_{1}} ⨿ E_{6(−26)}/E_{6(−26)}^{τ_{2}}

with 𝔤^{τ_{1}} ≃ 𝔤^{τ_{2}} ≃ 𝔣_{4(−20)} .

The case of E_{7(7)} . Let 𝔤 = 𝔢_{7(7)} . Note that 𝔤 is a normal real form of 𝔤_{ℂ}, i.e., 𝔞 is a Cartan algebra of 𝔤. In particular the restricted root system is of type E_{7} .

By Lemma 3.19 we have ∂_{e}Ω = W(\frac{π}{2} ω_{7}). As before we conclude that the prescription

τ(X) = (Ad(exp(iπω_{7})) ∘ θ)(X)

defines an involution on 𝔤. Hence Lemma 3.4, Theorem 3.5 and the list in Remark 3.3(a) imply:

Proposition 3.23. If 𝔤 = 𝔢_{7(7)} , then

∂_{d}Ξ ≃ E_{7(7)}/E_{7(7)}^{τ},

where 𝔤^{τ} ≃ 𝔰𝔲^{∗}(8).

The case of E_{7} . Let 𝔤 = 𝔢_{7} . This case is completely analogous to the 𝔢_{7(7)} -case and one obtains:

Proposition 3.24. If 𝔤 = 𝔢_{7} , then

∂_{d}Ξ ≃ E_{7}/E_{7(−25)}.

Classification of distinguished boundaries

Taking all our results from the previous discussions together we have proved the following classification result:

Theorem 3.25. Let 𝔤 be a non-compactly causal Lie algebra. Asuume that G is a connected Lie group with Lie algebra 𝔤 sitting in a simply connected complexification G_{ℂ}.

(i) If G is locally a classical group, then the situation is as follows: (a) If G is not a special orthogonal group, then

∂_{d}Ξ ≃ ∐_{j=1}^{m} G/H_{j}

where every G/H_{j} is a non-compactly causal symmetric space. With 𝔥_{j}: = Lie(H_{j}) we have:

\begin{matrix}𝔤 & ∐_{j=1}^{m} 𝔥_{j} \\ 𝔰𝔭(n, ℝ) & 𝔤𝔩(n, ℝ) \\ 𝔰𝔭(n, ℂ) & 𝔰𝔭(n, ℝ) \\ 𝔰𝔭(n, n) & 𝔰𝔭(n, ℂ) \\ 𝔰𝔲(n, n) & 𝔰𝔩(n, ℂ) × ℝ \\ 𝔰𝔬^{∗}(4n) & 𝔰𝔩(n, ℍ) × ℝ \\ 𝔰𝔩(n, ℝ) & ∐_{q=1}^{n−1} 𝔰𝔬(q, n − q) \\ 𝔰𝔩(n, ℂ) & ∐_{q=1}^{n−1} 𝔰𝔲(q, n − q) \\ 𝔰𝔩(n, ℍ) & ∐_{q=1}^{n−1} 𝔰𝔭(q, n − q) \end{matrix}

(b) If G is locally SO(n, n) or SO(2n, ℂ) for n ≥ 3, then

∂_{d}Ξ ≃ G/H_{1} ⨿ G/H_{2} ⨿ G/H_{2}

with G/H_{1} and G/H_{2} non-compactly causal symmetric spaces and:

\begin{matrix}𝔤 & 𝔥_{1} & 𝔥_{2} \\ 𝔰𝔬(n, n) & 𝔰𝔬(n − 1, 1) × 𝔰𝔬(n − 1, 1) & 𝔰𝔬(n, ℂ) \\ 𝔰𝔬(2n, ℂ) & 𝔰𝔬(2, 2n − 2) & 𝔰𝔬^{∗}(2n) \end{matrix}

(c) If G is locally SO(p, q) for 1 ≤ p < q or SO(2n + 1, ℂ), then for p, n ≥ 3 one has

∂_{d}Ξ ≃ G/H_{1} ⨿ G/H_{2}

with G/H_{1} non-compactly causal and G/H_{2} a homogeneous but non-symmetric space:

\begin{matrix}𝔤 & 𝔥_{1} & 𝔥_{2} \\ 𝔰𝔬(p, q) & 𝔰𝔬(p − 1, 1) × 𝔰𝔬(q − 1, 1) & 𝔰𝔬(p, ℂ) × 𝔰𝔬(q − p) \\ 𝔰𝔬(2n + 1, ℂ) & 𝔰𝔬(2, 2n − 1) & 𝔰𝔬^{∗}(2n) \end{matrix}

In the low-dimensional cases p = 1, 2 and n = 2 one has

∂_{d}Ξ ≃ G/H

with G/H non-compactly causal and:

\begin{matrix}𝔤 & 𝔥 \\ 𝔰𝔬(1, q) & 𝔰𝔬(1, q − 1) \\ 𝔰𝔬(2, q) & 𝔰𝔬(1, q − 1) × ℝ \\ 𝔰𝔬(5, ℂ) & 𝔰𝔬(2, 3) \end{matrix}

(ii) For the exceptional cases with 𝔤_{ℂ} ≃ 𝔢_{6} or 𝔤_{ℂ} ≃ 𝔢_{6} ⊕ 𝔢_{6} we have ∂_{d}Ξ ≃ G/H ⨿ G/H with G/H non-compactly causal. If 𝔥 = Lie(H), then:

\begin{matrix}𝔤 & 𝔥 \\ 𝔢_{6(6)} & 𝔰𝔭(2, 2) \\ 𝔢_{6(−26)} & 𝔣_{4(−20)} \\ 𝔢_{6} & 𝔢_{6(−14)} \end{matrix}

(iii) For the exceptional cases with 𝔤_{ℂ} ≃ 𝔢_{7} or 𝔤_{ℂ} ≃ 𝔢_{7} ⊕ 𝔢_{7} the distinguished boundary ∂_{d}Ξ ≃ G/H is connected. Further G/H is non-compactly causal and we have:

\begin{matrix}𝔤 & 𝔥 \\ 𝔢_{7(7)} & 𝔰𝔲^{∗}(8) \\ 𝔢_{7(−25)} & 𝔢_{6(−26)} × ℝ \\ 𝔢_{7} & 𝔢_{7(−25)} \end{matrix}

From the classification we obtain the following important result:

Theorem 3.26. Let 𝔤 be a non-compactly causal Lie algebra and ∂_{d}Ξ ≃ ∐_{j=1}^{n} G/H_{j} the decomposition of ∂_{d}Ξ into G-orbits. Then the following assertions hold:

(i) If G/H is a non-compactly causal symmetric space, then G/H appears locally as a component of ∂_{d}Ξ.

(ii) A boundary component G/H_{j} of ∂_{d}Ξ is totally real if and only if G/H_{j} is symmetric.

Remark 3.27. In the situation of Theorem 3.25 there is a boundary component of ∂_{d}Ξ with complex dimensions if and only if G is locally SO(p, q) for 2 < p < q or SO(2n+1, ℂ) for n > 2. In these cases we have ∂_{d}Ξ = G/H_{1} ⨿ G/H_{2} with G/H_{1} totally real and G/H_{2} a component admitting complex submanifolds of G_{ℂ}/K_{ℂ}.

4. Relations to Jordan algebras – the cases with A_{n}-type root systems

We first recall some standard facts of Jordan algebras and explain some results from [KrSt01b].

Let V be an Euclidean Jordan algebra. For x ∈ V we define L(x) ∈ End(V) by L(x)y = xy, y ∈ V . We denote by e the identity element of V . The symmetric cone W ⊆ V associated to V can be defined as

W: = int{x^{2}: x ∈ V },

where int(·) denotes the interior of (·). We write G: = Aut(W)_{0} for the connected component of the automorphism group of W which contains the identity. Recall that G is a reductive subgroup of Gl(V). The isotropy group in e

K: = G_{e}: = {g ∈ G: g(e) = e}

is a maximal compact subgroup of G. The mapping

G/K → W, gK ↦ g(e)

is a homeomorphism.

We write V_{ℂ} = V + iV for the complexification of V and define the tube domain

S_{W}: = V + iW ⊆ V_{ℂ}.

Let S denote the connected component of the complex automorphism group of S_{W} which contains the identity. Write U: = S_{ie} for the stabilizer of ie ∈ S_{W} . Then U is a maximal compact subgroup of S extending K and the mapping

S/U → S_{W} , gU ↦ g(ie)

is a biholomorphism of the Hermitian symmetric space S/U onto S_{W} . Note that G/K ⊆ S/U.

For the convenience of the reader we list here all irreducible Euclidean Jordan algebras and their associated groups G and S (cf. [FaKo94, p. 213]).

V  G  S
Symm(n, ℝ)  Gl(n, ℝ)_{+}  Sp(n, ℝ)
Herm(n, ℂ)  Sl(n, ℂ) × ℝ^{+}  SU(n, n)
Herm(n, ℍ)  Sl(n, ℍ) × ℝ^{+}  SO^{∗}(4n)
ℝ × ℝ^{n}  SO(1, n) × ℝ^{+}  SO(2, n + 1)
Herm(3, 𝕆)  E_{6(−26)} × ℝ^{+}  E_{7(−25)}

Let c_{1}, …, c_{n} be a Jordan frame (cf. [FaKo94, p. 44]) of V and set V^{0}: = ⨁_{j=1}^{n} ℝc_{j}. Recall that K(V^{0}) = V (cf. [FaKo94, Cor. IV.2.7])

The choice 𝔞: = ⨁_{j=1}^{n} ℝL(c_{j}) defines a maximal abelian hyperbolic subspace orthogonal to 𝔨. As the table shows, the root system Σ = Σ(𝔤, 𝔞) is classical and of type A_{n−1} . If we define ε_{j} ∈ 𝔞^{∗} by ε_{j}(L(c_{i})) = δ_{ij}, then we have

Σ = \{\frac{1}{2}(ε_{i} − ε_{j}): i ≠ j\}

Complex crowns of Riemannian symmetric spaces 25

(cf. [FaKo94, Prop. VI.3.3]). Thus

Ω = {x = ∑_{j=1}^{n} x_{j}L(c_{j}): x_{j} ∈ ℝ, |x_{i} − x_{j}| < π}.

In particular, for

Ω_{0}: = ⨁_{j=1}^{n}] − \frac{π}{2} , \frac{π}{2} [L(c_{j})

we have Ω_{0} ⊆ Ω. Note that the domain

Ξ_{0}: = G exp(iΩ_{0})K_{ℂ}/K_{ℂ} ⊆ G_{ℂ}/K_{ℂ}

is a G-invariant open subdomain of Ξ (cf. [KrSt01, Lemma 1.4]). Theorem 4.1. The mapping

Ξ_{0} → S_{W} , gK_{ℂ} ↦ g(ie)

is a biholomorphism. Proof. [KrSt01b, Th. 2.2.].

The distinguished boundary of Ξ_{0}

As in Section 2 we prove the following result: Proposition 4.2. The distinguished boundary of Ξ_{0} in G_{ℂ}/K_{ℂ} is given by

∂_{d}Ξ_{0} = G exp(i∂_{e}𝔞_{1})K_{ℂ}/K_{ℂ}

with

∂_{e}Ω_{0} = \{\frac{π}{2} (±L(c_{1}) ± … ± L(c_{n}))\}.

Recall that the Weyl group W is isomorphic to S_{n} and acts as the full permutation group of L(c_{1}), …, L(c_{n}). Thus we immediately obtain that: Lemma 4.3. We have the disjoint union

∂_{e}Ω_{0} = ∐_{p=0}^{n} W(Y_{p})

with

Y_{p}: = \frac{π}{2} (L(c_{1}) + … + L(c_{p}) − L(c_{p+1}) − … − L(c_{n})).

Recall that G_{ℂ}/K_{ℂ} sits canonically in V_{ℂ} via the embedding

G_{ℂ}/K_{ℂ} → V_{ℂ}, gK_{ℂ} ↦ g(ie).

In the sequel we identify G_{ℂ}/K_{ℂ} as a subset of V_{ℂ}.

Set z_{p}: = exp(iY_{p})K_{ℂ} ∈ V_{ℂ}. Then we have:

Lemma 4.4. For all 0 ≤ p ≤ n we have that

z_{p} = c_{1} + … + c_{p} − c_{p+1} − … − c_{n}.

Write G_{p} for the stabilizer of z_{p} in G.

Example 4.5. We take V = Symm(n, ℝ) with G = GL(n, ℝ)_{+} . The action of G is given by

G × V → V, (g, X) ↦ gXg^{t}.

Further we have that

z_{p} = diag(1, … 1, −1, …, −1)

with p-times +1 on the diagonal. Thus we see that

(∀0 ≤ p ≤ n) G_{p} = SO(p, n − p).

In particular G_{0} = G_{n} = SO(n, ℝ) is compact and all other G_{p} are non-compact. Write V_{p} in V for the symmetric matrices with signature (p, n − p). Further we write V^{reg} for the invertible matrices in V and set V_{p}^{reg}: = V_{p} ∩ V^{reg} . Then we have

V^{reg} = ∐_{p=0}^{n} V_{p}^{reg}

and

V_{p}^{reg} = G(z_{p}) ≃ G/G_{p}.

In particular we see that ∂_{d}Ξ_{0} is a Zariski open subset of V .

We are now going to generalize the results in Example 4.5 to arbitrary irreducible Euclidean Jordan algebras.

Let V be an irreducible Euclidean Jordan algebra. Write det(x) for the Jordan algebra determinant of V and note that det is a polynomial function on V . Define the subset of regular elements of V by

V^{reg}: = {x ∈ V: det x ≠ 0}

and note that V^{reg} is a Zariski open subset of V .

For every x ∈ V we define a real polynomial

f(λ, x): = det(λe − x).

Note that for fixed x, the polynomial f(λ, x) is completely reducible over ℝ. If x is regular, then it has degree n and all roots are non-zero. If f(λ) is a polynomial, then we define its signature sgn f to be the number of its positive roots. For every 0 ≤ p ≤ n now set

V_{p}: = {x ∈ V: sgn f(·, x) = p}.

Note that

V^{reg} = ∐_{p=0}^{n} V_{p}^{reg}

is a disjoint decomposition in cones. The only convex cones in this decomposition are V_{0}^{reg} = −W and V_{n}^{reg} = W.

Observe that z_{p} ∈ V_{p} for all 0 ≤ p ≤ n.

Proposition 4.6. For every 0 ≤ p ≤ n the set V_{p}^{reg} is G-invariant and G acts transitively on it. Hence the map

G/G_{p} → V_{p}^{reg}, gG_{p} ↦ g(z_{p})

is an isomorphism.

Proof. First we show that V_{p}^{reg} is invariant under K. For that fix x ∈ V_{p}^{reg} and k ∈ K. Recall that k(e) = e and that K acts on V by Jordan algebra automorphisms. We have

f(λ, k(x)) = det(λe − k(x)) = det(k(λ(e) − x)) = det(λe − x)

where in the last equality we used the fact that det k(y) = det y for all y ∈ V and k ∈ K. It is straightforward to check that

V_{p}^{reg} ∩ V^{0} = A(z_{p}).

Since V_{p}^{reg} is K -invariant and G = KAK the assertions of the proposition now follow.

Corollary 4.7. The distinguished boundary ∂_{d}Ξ_{0} ,realized in V_{ℂ}, is a Zariski open subset of V .

Next we compute the isotropy groups G_{p} for 0 ≤ p ≤ n.

Proposition 4.8. For an irreducible Euclidean Jordan algebra V the isotropy groups G_{p} for 0 ≤ p ≤ n are given as follows:

(i) For the classical matrix Jordan algebras one has:

V  G  G_{p}
Symm(n, ℝ)  Gl(n, ℝ)_{+}  SO(p, n − p)
Herm(n, ℂ)  Sl(n, ℂ) × ℝ^{+}  SU(p, n − p)
Herm(n, ℍ)  Sl(n, ℍ) × ℝ^{+}  Sp(p, n − p)

(ii) For V = ℝ × ℝ^{n} and G = SO(1, n) × ℝ^{+} one has:

(a) G_{p} = SO(n) for p = 0, 2.

(b) G_{p} = SO(1, n − 1) for p = 1. (iii) For V = Herm(3, 𝕆) and E_{6(−26)} × ℝ^{+} one has:

(a) G_{p} = K = F_{4(−52)} for p = 0, 3.

(b) G_{p} ≃ F_{4(−20)} for p = 1, 2.

Proof. (i) For all classical matrix algebras this is the same computation as in Example 4.5. (ii) Straightforward calculation similar to the one in (i). (iii) While (a) is clear, (b) is shown as in Proposition 3.22.

Corollary 4.9. The distinguished boundary ∂_{d}Ξ_{0} is G-isomorphic to ∂_{d}Ξ_{0} = ∐_{p=0}^{n} G/G_{p} where

(i) G/G_{p} is a Riemannian symmetric space for p = 0, n. (ii) G/G_{p} is a non-compactly causal symmetric space for p ≠ 0, n.

Remark 4.10. (a) Comparing Corollary 4.9 to our results for the distinguished boundary of the bigger domain Ξ we see that the distinguished boundary of Ξ_{0} is the distinguished boundary of Ξ plus two copies of the Riemannian symmetric space. Also observe that every boundary component of ∂_{d}Ξ_{0} is totally real.

(b) The observation in (a) fits into the following general philosphy: Non-compactly causal symmetric Lie algebras (𝔤, τ) are the class of symmetric Lie algebras which are closest to noncompactly Riemannian symmetric algebras (𝔤, θ). In fact these two classes make up the class of symmetric Lie algebras (𝔤, τ) where 𝔮 admits a non-trivial open Ad(H)-invariant hyperbolic convex set. These Lie algebras were subject of systematic study in [KrNe96].

5. Further results on the special orthogonal groups

The examples discussed in Section 4 fit into the broader context of comparing Ξ with symmetric spaces of Hermitian type S/U which contain G/K as a totally real submanifold (cf. [KrSt01b]).

In this section we will focus on the special orthogonal groups G = SO_{e}(p, q) and G = SO(n, ℂ) which in some sense are the classical groups with the most complicated structure of Ξ (cf. Theorem 3.25).

Our choice of the maximal compact subgroup K < G is as in Section 3. If G = SO_{e}(p, q), then we take S = SU(p, q) and U = S(U(p) × U(q)) and if G = SO(n, ℂ), then we choose S = SO^{∗}(2n) and U = U(n). The embedding

(5.1)  G/K ↪ S/U, gK ↦ gU

realizes G/K as a totally real submanifold of the symmetric space of Hermitian type S/U.

Let 𝔞 ⊆ 𝔭 be a maximal abelian subspace as in Section 3. Write Σ̂: = Σ(𝔞, 𝔰) for the doubly restricted root system of 𝔰 with respect to 𝔞. Notice that Σ̂ is of type C_{n} if 𝔤 = 𝔰𝔬(n, n) or 𝔤 = 𝔰𝔬(2n, ℂ) and of type BC_{n} otherwise. In the notation of Section 3 we have:

Σ̂ = \{ \begin{matrix}\{±ε_{i} ± ε_{j}: 1 ≤ i, j ≤ n\}\\{0\} & for 𝔤 = 𝔰𝔬(n, n), 𝔰𝔬(2n, ℂ), \\ \{±ε_{i} ± ε_{j}: 1 ≤ i, j ≤ p\}\\{0\} ⨿ \{±ε_{i}: 1 ≤ i ≤ p\} & for 𝔤 = 𝔰𝔬(p, q), 𝔰𝔬(2p + 1, ℂ) , (p < q). \end{matrix}

Define Ω_{0}: = \{X ∈ 𝔞: (∀α ∈ Σ̂) |α(X)| < \frac{π}{2} \}. Note that Ω_{0} ⊆ Ω since Σ̂ ⊇ Σ. Define

Ξ_{0}: = G exp(iΩ_{0})K_{ℂ}/K_{ℂ}

and notice that Ξ_{0} is a G-invariant open subdomain of Ξ. Finally define the distinguished boundary of Ξ_{0} by ∂_{d}Ξ_{0}: = G exp(i∂_{e}Ω_{0})K_{ℂ}/K_{ℂ}.

Write 𝔲 for the Lie algebra of U and 𝔰_{ℂ} = 𝔭^{+} ⊕ 𝔲_{ℂ} ⊕ 𝔭^{−} for the Harish-Chandra decomposition of 𝔰_{ℂ}. Let D ⊆ 𝔭^{+} be the Harish-Chandra realization of S/U as a bounded symmetric domain. The embedding in (5.1) extends to an embedding

(5.2)  G_{ℂ}/K_{ℂ} ↪ S_{ℂ}/U_{ℂ}P^{−}

and it follows from [KrSt01b, Th. 2.5] that the image of Ξ_{0} under (5.2) is precisely S/U ≃ D, i.e., we have a G-equivariant biholomorphism

Ξ_{0} ≃ S/U.

Theorem 5.1. Let G = SO_{e}(p, q) or G = SO(n, ℂ). Then the distinguished boundary ∂_{d}Ξ_{0} is given as follows:

(i) If G = SO_{e}(n, n) or G = SO(2n, ℂ), then ∂_{d}Ξ_{0} = G/H ⨿ G/H with 𝔥: = Lie(H) given by

𝔥 = \{ \begin{matrix}𝔰𝔬(n, ℂ) & for 𝔤 = 𝔰𝔬(n, n) \\ 𝔰𝔬^{∗}(2n) & for 𝔤 = 𝔰𝔬(2n, ℂ). \end{matrix}

(ii) If G = SO_{e}(p, q) for p < q or G = SO(2n + 1, ℂ), then ∂_{d}Ξ_{0} = G/H with 𝔥: = Lie(H) given by

𝔥 = \{ \begin{matrix}𝔰𝔬(p, ℂ) ⊕ 𝔰𝔬(q − p) & for 𝔤 = 𝔰𝔬(p, q) \\ 𝔰𝔬^{∗}(2n) & for 𝔤 = 𝔰𝔬(2n + 1, ℂ). \end{matrix}

Proof. (i) Keep the notation of Lemma 3.12. Then it follows from the structure of Σ̂ that

∂_{e}Ω_{0} = W(Y_{2}) ⨿ W(Y_{3}).

Now the assertion follows from the computations before Proposition 3.13 and Proposition 3.15. (ii) This is analogous to (i).

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Simon Gindikin Department of Mathematics Rutgers University New Brunswick, NJ 08903 USA

gindikin@math.rutgers.edu

Bernhard Krötz The Ohio State University Department of Mathematics 231 West 18th Avenue Columbus, OH 43210–1174 USA

kroetz@math.ohio-state.edu