Modular geodesics and wedge domains in non-compactly causal symmetric spaces

We continue our investigation of the interplay between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory. We adopt the perspective that the geometric implementation of the modular group is given by the flow generated by an Euler element of the Lie algebra (an element defining a 3-grading). Since any Euler element of a semisimple Lie algebra specifies a canonical non-compactly causal symmetric space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M = G/H$$\end{document}M=G/H, we turn in this paper to the geometry of this flow. Our main results concern the positivity region W of the flow (the corresponding wedge region): If G has trivial center, then W is connected, it coincides with the so-called observer domain, specified by a trajectory of the modular flow which at the same time is a causal geodesic. It can also be characterized in terms of a geometric KMS condition, and it has a natural structure of an equivariant fiber bundle over a Riemannian symmetric space that exhibits it as a real form of the crown domain of G/K. Among the tools that we need for these results are two observations of independent interest: a polar decomposition of the positivity domain and a convexity theorem for G-translates of open H-orbits in the minimal flag manifold specified by the 3-grading.


Introduction
A new Lie theoretical approach to localization on spacetimes involved in Algebraic Quantum Field Theory (AQFT) has been introduced in the recent years by the authors and collaborators in a series of works, see [MN21,NÓ17,NÓ21,NÓ23a,NÓ23b,MNÓ23,Oeh22]. In the current paper we continue the investigation of the structure of wedge regions in non-compactly causal symmetric spaces, started in [NÓ23b]. First we briefly recall the motivation form AQFT, then we introduce tools and details to formulate our results.
Symmetric spaces are quotients M = G/H, where G is a Lie group, τ is an involutive automorphism of G and H ⊆ G τ is an open subgroup (cf. [Lo69]). A causal symmetric space carries a G-invariant field of pointed generating closed convex cones C m ⊆ T m (M ) in their tangent spaces. Typical examples are de Sitter space dS d ∼ = SO 1,d (R) e / SO 1,d−1 (R) e and anti-de Sitter space AdS d ∼ = SO 2,d−1 (R) e / SO 1,d−1 (R) e . These are Lorentzian, but we do not require our causal structure to come from a Lorentzian metric, which creates much more flexibility and a richer variety of geometries. Causal symmetric spaces permit to study causality aspects of spacetimes in a highly symmetric environment. Here we shall always assume that M is non-compactly causal in the sense that the causal curves define a global order structure with compact order intervals (they are called globally hyperbolic) and in this context one can also prove the existence of a global "time function" with group theoretic methods (see [Ne91]). We refer to the monograph [HÓ97] for more details and a complete exposition of the classification of irreducible causal symmetric spaces.
Recent interest in causal symmetric spaces in relation with representation theory arose from their role as analogs of spacetime manifolds in the context of Algebraic Quantum Field Theory in the sense of Haag-Kastler (BW) Bisognano-Wichmann property: Ω is separating for some "wedge region" W ⊆ M and there exists an element h ∈ g with ∆ −it/2π = U (exp th) for t ∈ R, where ∆ is the modular operator corresponding to (M(W ), Ω) in the sense of the Tomita-Takesaki Theorem ([BR87, Thm. 2.5.14]).
The (BW) property gives a geometrical meaning to the dynamics provided by the modular group (∆ it ) t∈R of the von Neumann algebra M(W ) associated to wedge regions with respect to the vacuum state specified by Ω. On Minkowski/de Sitter spacetime it provides an identification of the oneparameter group (Λ W (t)) t∈R of boosts in the Poincaré/Lorentz group with the Tomita-Takesaki modular operator: U (Λ W (t)) = ∆ −it/2π .
Here Λ W = gΛ W1 g −1 is a one-parameter group of boosts associated with W = g.W 1 , where W 1 = {x ∈ M : |x 0 | < x 1 } is the standard right wedge and Λ W1 (t) = (cosh(t)x 0 + sinh(t)x 1 , cosh(t)x 1 + sinh(t)x 0 , x 2 , . . . , x d ) describes the boosts associated to W 1 . The homogeneous spacetimes occurring naturally in AQFT are causal symmetric spaces associated to their symmetry groups (Minkowski spacetime for the Poincaré group, de Sitter space for the Lorentz group and anti-de Sitter space for SO 2,d (R)), and the localization in wedge regions is ruled by the acting group. The rich interplay between the geometric and algebraic objects in AQFT allowed a generalization of fundamental localization properties and the subsequent definition of fundamental models (second quantization fields), having as initial data a general Lie group with distinguished elements (Euler elements) in the Lie algebra. Given an AQFT on Minkowski spacetime M = R 1,d (or de Sitter spacetime dS d ⊆ R 1,d ), the Bisognano-Wichmann (BW) property allows an identification of geometric and algebraic objects in both free and interacting theories in all dimensions ([DM20, Mu01,BW76]). This plays a central role in many results in AQFT and is a building block of our discussion.
Nets of von Neumann algebras can be constructed on causal symmetric spaces with representation theoretic methods. We refer to [NÓ21] for left invariant nets on reductive Lie groups, to [Oeh21] for left invariant nets on non-reductive Lie groups, to [NÓ23a] for nets on compactly causal symmetric spaces, and to [FNÓ23] for nets on non-compactly causal symmetric spaces. These papers construct so-called one-particle nets on symmetric spaces from which nets of von Neumann algebras can be obtained by second quantization functors (cf. [CSL22]).
We know from [MN23] that, in the general context, the potential generators h ∈ g of the modular groups in (BW) are Euler elements, i.e., ad h defines a 3-grading where g λ (h) = ker(ad h − λ1).
This leads to the question how the existence and the choice of the Euler element affects the geometry of the associated symmetric space. The (BW) property establishes a one-to-one correspondence between "wedge regions" W ⊆ M and the associated Euler elements. So these fundamental localization regions can be determined in terms of Euler elements. This allowed the following generalization of nets of von Neumann algebras on Minkowski/de Sitter spacetime: • Given a Lie group G with Lie algebra g, then the couples (h, τ h ), where h ∈ g is an Euler element and τ h an involutive automorphism of G, inducing on g the involution τ h = e πi ad h , allow the definition of an ordered, G-covariant set of "abstract wedge regions" carrying also some locality information [MN21]. In particular, they encode the commutation relation property of the Tomita operators (modular operator and modular conjugation).
• In the setting we are going to consider here, given an (anti-)unitary representation (U, H) of G ⋊ {1, τ h }, the bosonic second quantization net of von Neumann algebras associated to wedge regions can be constructed on the associated Fock space and, in this sense, it defines a generalized AQFT. Here the net is constructed from the local one-particle spaces in H both abstract-algebraic, as in [MN21,BGL02], and through distribution vectors, as in [NÓ21,NÓ23a,FNÓ23]. In general, in particular when Z(G) = {e}, one may consider other second quantization schemes as in [GL95,CSL22]. This has not yet been done, see also [MN21].
• Causal symmetric spaces provide manifolds and a causal structure supporting nets of algebras.
Here the wedge regions can be defined as open subsets in several ways. The equivalence of various characterizations has been shown in [NÓ23a,NÓ23b]; see also the discussion below.
The whole picture complies with Minkowski, de Sitter and anti-de Sitter spacetimes and the associated free fields. A generalization of wedge regions of the Minkowski or de Sitter spacetime on general curved spacetimes have been proposed by many authors, see for instance [DLMM11] and references therein. In our framework, on non-compactly causal symmetric spaces, the rich geometric symmetries allow different characterizations of wedge regions, in particular in terms of positivity of the modular flow, or geometric KMS conditions and in terms of polar decompositions as described in [NÓ23b]. Some of them directly accord with the literature, for instance for positivity of the modular flow, see [CLRR22,Defin. 3.1] and in particular [NPT96] for the connection to thermodynamics on de Sitter space. To see how these definitions apply to wedges in de Sitter space, see [NÓ23b,App. D.3] and [BM96]. On the other hand, the definition of wedge regions investigated in [NÓ23b,MNÓ23] coincides, up to specification of a connected component (cf. [NÓ23b,Thm. 7.1]). In Theorem 7.1 we prove that the identification is actually complete for the adjoint groups since the wedge region defined in term of positivity of the modular flow is connected. This contrasts the situation for compactly causal symmetric spaces, where wedges are in general not connected, cf. anti-de Sitter spacetime [NÓ23b, Lemma 11.2]. Before we turn to the detailed formulation of our results, we recall some basic terminology concerning symmetric Lie algebras (see [NÓ23b] for more details).
• A causal symmetric Lie algebra is a triple (g, τ, C), where (g, τ ) is a symmetric Lie algebra and C ⊆ q is a pointed generating closed convex cone invariant under the group Inn g (h) = e ad h acting in q. We call (g, τ, C) compactly causal (cc) if C is elliptic in the sense that, for x ∈ C • (the interior of C in q), the operator ad x is semisimple with purely imaginary spectrum. We call (g, τ, C) non-compactly causal (ncc) if C is hyperbolic in the sense that, for x ∈ C • , the operator ad x is diagonalizable.
As explained in detail in [MNÓ23], Euler elements in reductive Lie algebras g lead naturally to ncc symmetric Lie algebras: For an Euler element h ∈ g, choose a Cartan involution θ of g with z(g) ⊆ g −θ such that θ(h) = −h. Then τ h := e πi ad h is an involutive automorphism of g commuting with θ, so that τ := τ h θ defines a symmetric Lie algebra (g, τ ) and there exists a pointed generating Inn g (h)-invariant hyperbolic cone C with h ∈ C • . Under the assumption that h = g τ contains no non-zero ideal of g, there is a unique minimal cone C min q (h) with this property. It is generated by the orbit Inn g (h)h ⊆ q.
Let (g, τ, C) be an ncc symmetric Lie algebra and (G, τ G , H) a corresponding symmetric Lie group, i.e., G is a connected Lie group, τ G an involutive automorphism of G integrating τ , and H ⊆ G τ G an open subgroup. If, in addition, Ad(H)C = C, then we call the quadruple (G, τ G , H, C) a causal symmetric Lie group. On M = G/H we then obtain the structure of a causal symmetric space, specified by the G-invariant field of open convex cones We further assume that defines on M a partial order, called the causal order on M . According to Lawson's Theorem ( [La94] and Theorem C.1), this is always the case if z(g) ⊆ q and exp | z(g) is injective. The second condition is always satisfied if G is simply connected.
For an Euler element h ∈ g we consider the associated modular flow on M = G/H, defined by α t (gH) = exp(th)gH. (1.3) We study orbits of this flow which are geodesics γ : R → M with respect to the symmetric space structure and causal in the sense that γ ′ (t) ∈ V + (γ(t)) for t ∈ R. We call them h-modular geodesics. All these are contained in the positivity domain of the vector field X M h generating the modular flow. We refer to [NÓ23b] for a detailed analysis of the latter domain in the special situations where the modular flow on M has fixed points, which is equivalent to the adjoint orbit O h = Inn(g)h intersecting h.
We show for ncc symmetric Lie algebras, which are direct sums of irreducible ones, that: • Causal modular geodesics exist if and only if the adjoint orbit O h = Ad(G)h ⊆ g intersects the interior of the cone C ⊆ q, and then the centralizer G h = {g ∈ G : Ad(g)h = h} of h acts transitively on the union of the corresponding curves (Proposition 3.2(c)).
• Suppose that the cone is maximal, i.e., C q = C max q (see (2.3) and [MNÓ23, §3.5.2] for details). Let q k = q ∩ k for a Cartan decomposition g = k ⊕ p with h ∈ q p and consider the domain • A key step in the proof of Theorem 5.7 is the following Convexity Theorem. Let be the "negative" parabolic subgroup of G specified by h and identity g 1 (h) with the open subset B := exp(g 1 (h)).eP − ⊆ G/P − . Then D := H.0 ⊆ B is an open convex subset, and for any g ∈ G with g.D ⊆ B, the subset g.D ⊆ B is convex (Theorem 4.5).
• In Section 6 we further show that, for C = C max q and g simple, that the real tube domain h + C • intersects the set E(g) of Euler elements in a connected subset (Theorem 6.1). As a consequence, we derive that In particular, only one conjugacy class of Euler elements possesses non-empty positivity regions.
• In Theorem 7.1 we show that the positivity domain W + M (h) is connected for G = Inn(g) and g simple, and this implies that W (γ) = W = W + M (h). From this in turn we derive that the stabilizer group G W = {g ∈ G : g.W = W } coincides with G h (Proposition 7.3), so that the wedge space W(M ) := {g.W : g ∈ G} of wedge regions in M can be identified, as a homogeneous G-space, with the adjoint orbit O h = Ad(G)h ∼ = G/G h . In particular W(M ) also is a symmetric space.
• Finally, we show in Theorem 8.2 that W coincides with the KMS wedge domain We conclude this introduction with some more motivation from AQFT. The analysis of the properties of the modular flow on symmetric spaces is also motivated by the investigation of energy inequalities in quantum and relativistic theories. In General Relativity, there exist many solutions to the Einstein equation that, for various reasons, may not be physical. Energy conditions such as the pointwise non-negativity of the energy density, which ensures that the gravity force is attractive, can be required to discard non-physical models ( [Wal84,Few12]). In quantum and relativistic theories, the energy conditions need to be rewritten. For instance, it is well known that the energy density at individual spacetime points is unbounded from below, even if the energy density integrated over a Cauchy surface is non-negative (see [FH05,Few12] and references therein).
Families of inequalities have been discussed in several models, employing different mathematical and physical approaches (see for instance [Few12,KS20,FLPW20,Ve00,KLL+18,MPV23]). In the recent years operator algebraic techniques have been very fruitful for the study of the energy inequalities because of the central role played by the modular hamiltonian in some of these energy conditions. This object corresponds to the logarithm of the modular group of a local algebra of a specific region, which in some cases can be identified with the generator of a one-parameter group of spacetime symmetries by the Bisognano-Wichmann property. In this regard, we cite-without entering into details-the ANEC (Averaged Null Energy Condition) and the QNEC (Quantum Null Energy Condition) and their relation with the Araki relative entropy, an important quantuminformation quantity, defined in terms of relative modular operators (see for instance [MTW22, Lo20, CLRR22, Lo19, CLR20, Ara76, LX18, LM23]). We stress that, in this analysis, the study of the modular flow on the manifold can be particularly relevant. Moreover, in order to find regions where to prove energy inequalities, one may also need to deform the modular flow ( [MTW22,CF20]). In our abstract context the Euler element specifies the flow that can be implemented by the modular operator, hence the modular Hamiltonian, when the Bisognano-Wichmann property holds. In particular, the identification of specific flows on symmetric spaces (modular flows), the characterization in terms of modular operators of covariant local subspaces attached to specific regions (wedges) deeply motivate an analysis of modular flows on non-compactly causal symmetric spaces pursued in our project.
In this respect, the wedge regions are the first fundamental open subsets of spacetime to be studied in detail. Following General Relativity (see for instance [CLRR22,DLMM11] and references therein), one can define them as an open connected, causally convex subregion W of a spacetime M , associated to a Killing flow Λ preserving W , which is timelike and time-oriented on W . On Minkowski spacetime the flow Λ, a one-parameter group of boosts, corresponds to the time-evolution of a uniformly accelerated observer moving within W . Then W is a horizon for this observer: he cannot send a signal outside W and receive it back. Then the vacuum state becomes a thermal state for the algebra of observables inside the wedge region W by the Bisognano-Wichmann property ( [GL03,Ha76,Lo97]). In our general context, we recover the definition (and equivalent ones) of wedge regions. Then, by the Bisognano-Wichmann property, the thermal property of the vacuum state holds when nets of algebras/standard subspaces are considered ([MNÓ23, NÓ23b, MN21]). In this paper we focus on the related properties of the wedge regions in non-compactly causal symmetric spaces.

Notation:
• If M is a topological space and m ∈ M , then M m denotes the connected component of M containing m. In particular we write e ∈ G for the identity element in the Lie group G and G e for its identity component.
• For x ∈ g, we write G x := {g ∈ G : Ad(g)x = x} for the stabilizer of x in the adjoint representation and G x e = (G x ) e for its identity component.
• For h ∈ g and λ ∈ R, we write g λ (h) := ker(ad h − λ1) for the corresponding eigenspace in the adjoint representation.
• If g is a Lie algebra, we write E(g) for the set of Euler elements h ∈ g, i.e., ad h is non-zero and diagonalizable with Spec(ad h) ⊆ {−1, 0, 1}. The corresponding involution is denoted τ h = e πi ad h .
• For a Lie subalgebra s ⊆ g, we write Inn g (s) = e ad s ⊆ Aut(g) for the subgroup generated by e ad s .
• For a convex cone C in a vector space V , we write C • := int C−C (C) for the relative interior of C in its span.
Acknowledgment: We thank J. Wolf for discussions concerning the Convexity Theorem 4.5.

Causal Euler elements and ncc symmetric spaces
In this section we recall some basic results on Euler elements and their relation with non-compactly causal symmetric spaces. Most of these statements are discussed in detail in [MNÓ23].
Recall from above that an Euler element in a Lie algebra g is an element h defining a 3-grading of g by g = g −1 ⊕ g 0 ⊕ g +1 with g j = ker(ad h − j1), j = −1, 0, 1. We write E(g) for the set of Euler elements in g. In this section we recall some results on from [MNÓ23] on Euler elements that are crucially used in the following.
Definition 2.1. Let g be a reductive Lie algebra. (a) A Cartan involution of g is an involutive automorphism θ for which z(g) ⊆ g −θ and g θ is maximal compactly embedded in the commutator algebra [g, g]. We then write, using the notation from the introduction, g = k ⊕ p with k = g θ and p = g −θ (b) If τ is another involution on g commuting with θ, h := g τ and q := g −τ , then we have (c) The Cartan dual of the symmetric Lie algebra (g, τ ) is the symmetric Lie algebra (g c , τ c ) with Note that g c = (g C ) τ where τ is the conjugate-linear extension of τ to g C ; in particular g c is a real form of g C .
Definition 2.2. Let (g, τ ) be a symmetric Lie algebra and h ∈ E(g) ∩ q. We say that h is causal if there exists an Inn g (h)-invariant closed pointed generating convex cone C in q with h ∈ C • . We write E c (q) ⊆ E(g) ∩ q for the set of causal Euler elements in q. Recall that the triple (g, τ, C) is ncc if C is hyperbolic.
Lemma 2.3. Let (g, τ, C) be a simple ncc symmetric Lie algebra and h ∈ q be a causal Euler element. Then the following assertions hold: (a) There exist closed convex pointed generating Inn g (h)-invariant cones such that h ∈ C min q (h) • and either If h is an Euler element in the reductive Lie algebra g and θ a Cartan involution with θ(h) = −h, τ := θτ h and z(g) ⊆ g −θ , then [MNÓ23,Thm. 4.2] implies that there exists an Inn g (h)-invariant pointed closed convex cone C ⊆ q with h ∈ C • , so that (g, τ, C) is ncc. Further, all ideals of g contained in g τ = h are compact. We have a decomposition where g s is the sum of all simple ideals not commuting with h (the strictly ncc part), g r is the sum of the center z(g) and all non-compact simple ideals commuting with h on which τ = θ (the non-compact Riemannian part), and g k is the sum of all simple compact ideals (they commute with h). All these ideals are invariant under θ and τ = τ h θ, so that we obtain decompositions where h r ⊕ h k is a compact ideal of h, g r = h r ⊕ q r is a Cartan decomposition and q p = q p,s ⊕ q r .
In particular q = q s ⊕ q r . Let p s : q → q s be the projection onto q s with kernel q r . Then [MNÓ23, Prop. B.4] implies that every Inn g (h)-invariant closed convex cone C satisfies C s := p s (C) = C ∩ q s and C • s = C • ∩ q s .
By Lemma 2.3(a) we obtain a pointed Inn g (h)-invariant cone C min qs (h) ⊆ q s , adapted to the decomposition into irreducible summands, whose dual cone C max qs (h) with respect to the Cartan-Killing form κ(x, y) = tr(ad x ad y) satisfies C min qs (h) ⊆ C max qs (h). Put Both cones are adapted to the decomposition of (g, τ ) into irreducible summands. Further, each pointed generating Inn g (h)-invariant cone C containing h satisfies Here the first inclusion is obvious, and the second one follows from the fact that h is also contained in the dual cone This leads to C min q (h) ⊆ C ⋆ , and thus to C ⊆ C min .5] for more details).
Lemma 2.4. If x ∈ (C max q ) • , then the centralizer z h (x) = h ∩ ker(ad x) is compactly embedded in g, i.e., consists of elliptic elements.
Proof. First we observe that the cone C max q is adapted to the decomposition g = (g k + g r ) + g s and so is the centralizer of x = x r + x s in h = (g k + k r ) + h s . Hence the assertion follows from the fact that g k + k r is compactly embedded and z hs (x) = z hs (x s ) is compactly embedded because the cone C max qs is pointed ([Ne00, Prop. V.5.11]).
Theorem 2.5. (Uniqueness of the causal involution) ([MNÓ23, Thm. 4.5]) Let (g, τ, C) be a semisimple ncc symmetric Lie algebra for which all ideals of g contained in h are compact, g s the sum of all non-Riemannian ideals, q s := g s ∩ q, C s := C ∩ q s , and θ a Cartan involution commuting with τ . Then the following assertions hold: (a) C • s ∩ q p contains a unique Euler element h, and this Euler element satisfies τ = τ h θ.
(c) For every Euler element h ∈ C • s , the involution τ τ h is Cartan.
Proposition 2.6. Let (G, τ G , H, C) be a connected semisimple ncc symmetric Lie group for which h = g τ contains no non-compact ideal of g (g = g r + g s ) and let h ∈ C • s (cf. Theorem 2.5) be a causal Euler element. Then the following assertions hold:   Definition 2.7. If g is a simple hermitian Lie algebra, θ a Cartan involution of g and a ⊆ p maximal abelian, then the restricted root system Σ(g, a) is either of type C r or BC r . In the first case, we say that g is of tube type.
Recall that if (g, τ ) is simple ncc, then either g c is simple hermitian or Proposition 2.8. ([MNÓ23, Lemma 5.1, Prop. 5.2]) Let (g, τ, C) be a simple ncc symmetric Lie algebra. Pick a causal Euler element h ∈ C • and t q ⊆ q k maximal abelian and set s := dim t q . Then the following assertions hold: (a) The Lie algebra l generated by h and t q is reductive.
and only if g c is of tube type.
(e) For x ∈ t q , we have ρ(ad x) = ρ(ad x| s ), where ρ denotes the spectral radius. With the basis Note that (c) implies that l is semisimple, i.e., h ∈ [l, l], if and only if g c is of tube type.

The positivity domain and modular geodesics
Let (G, τ G , H, C) be a connected semisimple causal symmetric Lie group with ncc symmetric Lie algebra (g, τ, C). We fix a causal Euler element h ∈ C • (Theorem 2.5) and write M = G/H for the associated symmetric space. One of our goals in this paper is to describe the structure of the positivity domain (Theorem 3.6). Some of the results in this section had been obtained in [NÓ23b] for the special case of ncc symmetric Lie algebras for which h contains an Euler element, whereas here we are dealing with general non-compactly causal symmetric Lie algebras.

Modular geodesics
In this subsection we introduce the concept of an h-modular geodesic in a non-compactly causal symmetric space M and discuss some of its immediate properties. We also show that, in compactly causal spaces, non-trivial causal modular geodesics do not exist. • We call a geodesic γ : • Let h ∈ g be an Euler element. The flow on M defined by is called the modular flow (associated to h). Its infinitesimal generator is denoted X M h ∈ V(M ).
Proposition 3.2. Suppose that (g, τ ) is a direct sum of irreducible ncc symmetric Lie algebras (g = g s ).
The following assertions hold for any Euler element h ∈ E(g) and the corresponding modular flow α t (m) = exp(th).m on M = G/H: The orbit under the modular flow is a causal geodesic if and only if m is contained in Then (3.2) implies that the orbit of m = gH under the modular flow is a geodesic. The causality is by definition equivalent to Ad Therefore ad x h + ad x q is the unique Jordan decomposition of ad x into elliptic and hyperbolic summand. As Ad(g) −1 h is an Euler element, the elliptic summand vanishes, and thus ad Choosing m as a base point, we may assume that m = eH, so that (a) implies that h ∈ C • ⊆ q is a causal Euler element. Pick a Cartan involution θ commuting τ which satisfies θ(h) = −h (cf. [KN96]), i.e., h ∈ q p . Then τ = τ h θ follows from Theorem 2.5(a). As (M c C ) m = G h e .m by Lemma B.2, the assertion now follows from g h = h k ⊕ q p . (c) The first assertion follows immediately from (a). For the second assertion, suppose that We record the following consequence of (3.2): Due to the hyperbolicity of Euler elements, modular causal geodesics do not exist for compactly causal symmetric spaces: Proposition 3.4. If M = G/H is a compactly causal symmetric space, then non-trivial causal modular geodesics do not exist.
Proof. If there exists a modular causal geodesic and (g, τ, C) is the infinitesimal data of M , then there exists a g ∈ G such that the Euler element h satisfies Ad(g) −1 h = x h + x q with x q ∈ C • and [x h , x q ] = 0 (Lemma B.1). As C is elliptic, x q is elliptic. Further the pointedness of C implies that x h ∈ ker(ad x q ) is elliptic. This implies that the Euler element Ad(g) −1 h is elliptic, a contradiction.

The fiber bundle structure of the positivity domain
The main result of this section is Theorem 3.6 in which we exhibit a natural bundle structure on the wedge domain W ⊆ M that is equivariant with respect to the connected group G h e , the base is the Riemannian symmetric space of this group, and the fiber is a bounded convex subset of q k .
h is the non-compact Riemannian symmetric space associated to the symmetric Lie algebra (h, θ).
Theorem 3.6. (Positivity Domain Theorem) Suppose that (G, τ G , C, H) is a connected semisimple non-compactly causal Lie group for which (g, τ ) contains no τ -invariant Riemannian ideals (g = g s ) and that h is a causal Euler element. Suppose that C : Then the following assertions hold: If x ∈ Ω q k , then ρ(ad x) < π/2, so that (3.5) implies that g = exp x satisfies If g ∈ G h satisfies g Exp eH (Ω q k ) ∩ W = ∅, then g.W = W follows from (3.7) and the fact that g permutes the connected components of W + M (h). Therefore (3.8), combined with (3.7), leads with and this entails Next we observe that the exponential map Exp eH : Exp eH (Ω q k ), and this completes the proof.
The surjectivity of Ψ follows from Theorem 3.6. As e whose fiber is the convex set Ω q k . Therefore it is homotopy equivalent to the base G h e /K h e , which is also contractible because the exponential map Exp eH : q p → G h e /K h e is a diffeomorphism. It therefore suffices to show that Ψ is a diffeomorphism. The proof of (a) shows already that its differential is everywhere surjective, hence invertible by equality of the dimensions of both spaces. So it suffices to check injectivity, i.e., that Exp := Exp eH : q → M satisfies (3.10) Step 1: Exp | Ωq k is injective. If Exp(x 1 ) = Exp(x 2 ), then applying the quadratic representation implies exp(2x 1 ) = exp(2x 2 ) in G. As x 1 and x 2 are both exp-regular, [HN12, Lemma 9.2.31] implies that [x 1 , x 2 ] = 0 and exp(2x 1 − 2x 2 ) = e.
We conclude that e 2 ad(x1−x2) = id g , and since the spectral radius of 2 ad(x 1 − x 2 ) is less than 2π, it follows that ad(x 1 − x 2 ) = 0, so that x 1 = x 2 .
Step 2: g. Exp(x 1 ) = Exp(x 2 ) with g ∈ G h e and x 1 , x 2 ∈ Ω q k implies g ∈ K h e . Applying the involution θ M , we see that g. Exp(x 1 ) is a fixed point, so that g. Exp(x 1 ) = θ(g). Exp(x 1 ) entails that θ(g) −1 g fixes m 1 := Exp(x 1 ). We now write g = k exp z in terms of the polar decomposition of G h e and obtain Applying the quadratic representation, we get which can be rewritten as exp(e 2 ad x1 2z) = exp(−2z).
Since ad z has real spectrum, so has e 2 ad x1 z. Therefore the same arguments as in Step 1 above imply that [z, e 2 ad x1 z] = 0 and exp(2e 2 ad x1 z + 2z) = e, and e 2 ad x1 z = −z. The vanishing h-component of this element is sinh(2 ad x 1 )z, and since ρ(2 ad x 1 ) < π, it follows that [x 1 , z] = 0. Now (3.11) leads to exp(4z) = e, and further to z = 0, because the exponential function on q p is injective. This proves that g = k ∈ K h e .
Step 3: From (3.10) we derive , so that Step 2 shows that k := g −1 2 g 1 ∈ K h e . We thus obtain and since Ad(k)x 1 ∈ Ω q k , we infer from Step 1 that Ad(k)x 1 = x 2 . This completes the proof.
Proof. For g ∈ G h and x ∈ q k : Exp eH (Ω q k ) and the polar decomposition G h e = K h e exp(q p ) = exp(q p )(H K ) e (Theorem3.6(b)) we derive that the fixed point set is The preceding corollary shows that the wedge domain W ⊆ M = G/H contains the symmetric subspace M h eH = Exp eH (q p ) as the fixed point set of an involution. Hence the description of W from Theorem 3.6 as W = G h e . Exp eH (Ω q k ) suggest to consider W as a real "crown domain" of the Riemannian symmetric space M h Remark 3.8. Theorem 3.6 has a trivial generalization to semisimple non-compactly causal Lie algebras of the form g = g k ⊕ g r ⊕ g s because then However, if g r = {0}, then C max q is not pointed, and there are many pointed invariant cones C, which are not maximal, for which the domain W + M (h) may have a more complicated structure.
Example 3.9. We consider the reductive Lie algebra Any Euler element in g is conjugate to some With the Euler element The group G := GL 2 (R) e acts by g.A := gAg ⊤ on the 3-dimensional space Sym 2 (R) of symmetric matrices and the stabilizer of I 1,−1 := 1 0 0 −1 is the subgroup H := SO 1,1 (R) with Lie algebra h. Therefore M := G.I 1,1 ∼ = G/H can be identified with the subspace Sym 1,1 (R) of indefinite symmetric matrices. Note that R × e 1 = Z(G) e acts by multiplication with λ 2 and that R × is a realization of 2-dimensional de Sitter space. Note that the determinant defines a quadratic form of signature (1, 2) on Sym 2 (R) which is invariant under the action of the subgroup For the Euler element h s := 1 We also note that the "semisimple part" of C m is for the positivity domain of the Euler element h with respect to the causal structure specified by the cone C m . Then Theorem 3.6 implies that By G h e -invariance, we have to determine when Exp eH (tz), |t| < π 2 , is contained in W (C m , h). For g = exp(tz) we have We then have We conclude that, for |t| < π 2 , the inclusion h z + cos(t)h s ∈ (C m ) • is equivalent to We thus obtain the condition |t| < arccos( √ m|λ + µ|).
We thus obtain for By the Hurwitz criterion, this matrix is positive semidefinite if and only if As these two conditions imply that x ±1 ≥ 0, we see that the canonical order on M corresponds to the cone C 1 , i.e., to m = 1. (c) For the modular vector field X h we have 2µd .
The positivity domain of X h depends on λ, and with this formula one can also determine the positivity domain quite directly for m = 1, where C 1 corresponds to the canonical order. (here the index s refers to "semisimple"). On g = gl n (R), we consider the Cartan involution given by θ(x) = −x ⊤ and write n = p + q with p, q > 0. Then are Euler elements and τ := τ h p θ leads to a non-compactly causal symmetric Lie algebra (g, τ, C), where To identify G/H in the boundary of the crown domain in G C /K C ∼ = G C .1 ∼ = Sym n (C) × , where G C acts on Sym n (C) by g.A := gAg ⊤ ([NÓ23b, Thm. 5.4]), we observe that exp(ith p ).1 = e 2ith p = cos(2th p ) + i sin(2th p ) = 1 p 0 0 (cos(2t) + i sin(2t))1 q , so that we obtain for t = π 2 the matrix exp πi 2 h p .1 = I p,q .
The G-orbit of this matrix is the open subset of symmetric matrices of signature (p, q). We have These matrices are never positive definite. So we have to take h s instead to find non-trivial positivity domains. For the case p = q = 1 and n = 2 this has been carried out in Example 3.9. We also write which is equivalent to λµ < 0.

The connected components of M h C
The main result in this section is Proposition 3.11 below on the subgroup H K of K h . We then discuss several examples to clarify the situation.
Proposition 3.11. (Connected components of M h C ) If G = Inn(g) and (g, τ ) is irreducible ncc with causal Euler element h, then π 0 (M h C ) ∼ = K h /H K contains at most two elements.
Proof. We recall from Proposition 3.
In view of Proposition 2.6(c), we have for G = Inn(g) , we know that π 0 (G h ) ∼ = π 0 (K h ) has at most two elements.
Example 3.12. (The inclusion H K ⊆ K h may be proper) We have G h = K h exp(q p ) and K τ G = K τ G h because K = G θ . Further H K ⊆ K h by Proposition 2.6(a), so that the equality H K = K h is equivalent to K h ⊆ H K . This may fail for two reasons. One is failure in the adjoint group Inn(g) (Proposition 3.11), and the other reason is that Z(G) may be non-trivial.
Assume that g is semisimple and (g, τ, C) ncc. Let G be a corresponding connected Lie group on which τ G exists (for τ = τ h θ) and H := G τ G e . For the connected group K := G θ , the intersection need not be contained in H K . This can be seen easily for g = sl 2 (R). For (3.13) For any connected Lie group G with Lie algebra g, the group K = G θ is connected 1-dimensional and As O h ∩ Rh = {±h}, this leaves two possibilities: In this case gH is a fixed point of the modular flow.
we obtain h = so k,k (R) for τ = θτ h . There exists a subalgebra s ∼ = sl 2 (R) k , where the sl 2 -factors correspond to the coordinates x j and x j+k for 1 ≤ j ≤ k. Accordingly, h = k j=1 h j , where the Euler elements h j in the sl 2 -factors are conjugate to Euler elements h ′ j in h. Therefore the "geodesic condition" is satisfied by all elements k j=1 h j ∈ O h , where h j is either h j or h ′ j . The following example shows that modular geodesics also exist in symmetric spaces without causal structure. They can be "space-like" rather than "time-like", resp., causal.
Example 3.14. The d-dimensional hyperbolic space carries a modular flow specified by any Euler element h ∈ q ⊆ so 1,d (R) (corresponding to a tangent vector of length 1). Every geodesic of H d is an orbit of the flow generated by an Euler element of so 1,d (R).
Remark 3.15. Let (g, τ, C) be a simple ncc symmetric Lie algebra. In general, we have for a causal Euler element h ∈ E(g) ∩ C • a proper inclusion

Open H-orbits in flag manifolds and a convexity theorem
In this section we prove a convexity theorem that is vital to derive the equality W = W (γ) in the next section. Here, as above, 1 (h))G h ⊆ G be the "negative" parabolic subgroup of G specified by h and identity g 1 (h) with the open subset B := exp(g 1 (h)).eP − ⊆ G/P − . Then D := H.0 ⊆ B is an open convex subset, and our convexity theorem (Theorem 4.5) asserts that, for any g ∈ G with g.D ⊆ B, the subset g.D ⊆ B is convex.
(b) g.D + ⊆ B is relatively compact if and only if g ∈ P + exp(y) for y ∈ n − with y < 1.
Proof. The condition g.eP − ∈ B is equivalent to g ∈ N + P − = N + G h N − = P + N − . Let y ∈ n − with g ∈ P + exp(y). Then the invariance of B under P + implies that g.D + ⊆ B is equivalent to exp(y).D + ⊆ B. so that, for r > 0, exp(y).x ∈ B is equivalent to exp(r −1 y).(rx) ∈ B. For x ∈ D g , we pick r > 1 with rx ∈ D g . Then r −1 y < 1 implies exp(r −1 .y). exp(rx) ∈ B, and thus exp(y). exp(x) ∈ B. This shows that exp(y).D + ⊆ B. (b) If y < 1, then the argument under (a) shows that exp(y).D + ⊆ P + .D + is relatively compact. Now we assume that y = 1. We show that this implies that exp(y).D + is unbounded. As above, we use the Spectral Theorem to write y = k j=1 β j θ(c j ) and observe that there exists an ℓ ∈ {1, . . . , k} with |b ℓ | = 1. For x = j α j c j ∈ D g we then obtain with (4.4) exp(y).x = k j=1 α j 1 + α j β j c j .
2 This theorem is stated for complex hermitian Jordan triple systems, but V = g 1 (h) is a real form of the complex JTS V C = g C,1 (h) on which we have an antilinear isomorphism σ with V = V σ C . Therefore the uniqueness in the spectral decomposition shows that, for x ∈ V , the corresponding spectral tripotents are contained in V .
3 As D + is invariant under the group (H K )e which acts linearly, and this group acts transitively on the set of all maximal flat subtriples of V ([Ro00, Lemma VI.3.1]), it suffices to shows that an element with a spectral resolution x = r j=1 x j c j is contained in Dg if and only if |x j | < 1 for every j. This follows easily from (4.5).
Theorem 4.5. (Convexity Theorem for conformal balls) If g ∈ G is such that g.D + ⊆ B, then gD + is convex. If g.D + is relatively compact in B, then there exists an element p ∈ P + with g.D + = p.D + , so that g.D + is an affine image of D + .
Proof. If g.D + ⊆ B is relatively compact, then Proposition 4.4(b) and its proof imply the existence of p ∈ P + with g.D + = p.D + . In particular g.D + is an affine image of D + and therefore convex.

The subset realization of the ordered space M = G/H
As before G is assumed to be a connected semisimple Lie group. To simplify the notation we write  Proof. This result was announced in [Ol81, Ol82] and a detailed proof was given in [HN95, Thm. VI.11] for the case where G ⊆ G C , G C is simply connected and H = G τ . In this case Ad(G τ ) preserves C max q (h), so that G τ ⊆ K h exp(h p ). Conversely, K h leaves D + invariant, so that we obtain G τ = H = K h exp(h p ) in this particular case.
To see that the lemma also holds in the general case, note that the center of G acts trivially on G/P − and that Z(G) ⊆ K h ⊆ H. Therefore the general assertion follows if the equality (4.9) holds at least for one connected Lie group G with Lie algebra g. Hence it follows from the special case discussed above.
We now use this to realize G/H as an ordered symmetric space as a set of subsets of M and describe the ordering in that realization. In this identification the set {x ∈ G/H : x ≥ eH} is mapped to {s −1 D + : s ∈ comp(D + )} and {x ∈ G/H : x ≤ eH} is mapped to {sD + : s ∈ comp(D + )}. In particular, gH ≥ eH is equivalent to D + ⊂ g.D + and eH ≥ gH to g.D + ⊂ D + .
Proof. This follows from Lemma 4.7.   Example 4.11. A special case of the above construction is the "complex case" where H is a connected semisimple Lie group of hermitian type contained in a complex Lie group G with Lie algebra h C = h ⊕ ih. Then G/H is a ncc symmetric space. Let θ H be a Cartan involution on H. Then θ H extends to a Cartan involution θ on G. Denote the corresponding maximal compact subgroup of G by K. Then H ∩ K is a maximal compact subgroup of H and the Riemannian symmetric space H/H ∩ K can be realized as complex symmetric bounded domain D + ⊆ G/P − . Let z 0 ∈ z(h ∩ k) be the element determining the complex structure on H/H ∩ K. Then h = −iz 0 is an Euler element in q = ih. Now (4.1) is the Harish-Chandra realization of H/H ∩ K as D + in G/P − (see [Sa80,p. 58] or [He78, Ch. VII] for details).
Suppose that the complex conjugation τ of g with respect to h integrates to an involution τ G on G. This is the case if G is simply connected or if G = Inn g. We then assume that H = G τ G e . If G is simply connected, then H = G τ G is connected and [HN95, Thm. VI.11] implies that H = G D+ , where G D+ is the stabilizer of the base point D + .
But in general, if G is not simply connected, then G D+ and G τ G may differ.
As an example, consider H = PSL 2 (R) ⊆ G = PSL 2 (C) ∼ = Inn(g) and note that τ G (g) = τ gτ in this case. Then which is not connected because it also contains the image of (the Riemann sphere) is the upper half plane and the stabilizer subgroup of D + is The reflections in GL 2 (R) exchange the two open H-orbits.
Remark 4.12. The flag manifolds M = G/P − ∼ = K/K h appearing in this section are compact symmetric spaces on which the maximal compactly embedded subgroup K ⊆ G acts by automorphisms. These spaces are called symmetric R-spaces. Defining a symmetric R-space as a compact symmetric space M which is a real flag manifold, Loos shows in [Lo85, Satz 1] that this implies the existence of an Euler element h ∈ E(g) such that M ∼ = G/P − , so that M ∼ = K/K ∩ P − = K/K h as a Riemannian symmetric space (see [MNÓ23, §7.2] for more details). If G is hermitian of tube type, then M ∼ = K/K h can be identified with theŠhilov boundary of the corresponding bounded symmetric domain D G ∼ = G/K, and this leads to a G-invariant causal structure on M. As dim Z(K) = 1, with respect to the K-action, we have a natural 1-parameter family of K-invariant Lorentzian structures on M. They correspond to K h -invariant Lorentzian forms on T eK h (M) ∼ = q k = z(k) ⊕ [h k , q k ] which are positive definite on z(k) and negative definite on its orthogonal space [h k , q k ].

Observer domains associated to modular geodesics
As the intersection of order convex subsets is order convex, we can defined the order convex hull the observer domain associated to γ. Note that this domain depends on the cone C ⊆ q specifying the order on M .
Lemma 5.2. The subset W (γ) has the following properties:  Proof. (a) To see that W (γ) is open, we first observe that γ(s) ∈ (↑γ(t)) • for t < s. For real numbers t j ∈ R with t 1 < t 2 < t 3 < t 4 , this implies that

This shows that W (γ) is open.
To see that W (γ) is connected, we recall that the order on M is globally hyperbolic, in particular all order intervals [x, y] are compact. As all elements z ∈ [x, y] lie on causal curves from x to y ([HN93, Thm. 4.29]), the order intervals are pathwise connected. As an increasing union of the order intervals [γ(−n), γ(n)], the wedge domain W (γ) is connected. (b) Order intervals are convex and directed unions of convex sets of convex. Therefore is convex, whence W (γ) = oconv(γ(R)).
From the fact that h is central in h k + q p , it easily follows that, in the symmetric space M h eH = Exp eH (q p ) the geodesic line γ(R) is cofinal in both directions because we have in q: For x ∈ q p , we thus find s, t ∈ R with x ∈ sh + C • and x ∈ st − C • . Then This completes the proof. (c) The modular group acts on B ∼ = N + .eP − ⊆ G/P − by exp(th).x = e t x. Therefore γ(t) = e t D + enlarges D + for t > 0 and shrinks D + for t < 0 (Theorem 4.5). As γ is strictly increasing, this implies that Here We claim that, for the modular geodesic γ(t) = cosh(t)e 1 + sinh(t)e 0 = e th e 1 , [NÓ23b]). As the right wedge W R ⊆ R 1,d is causally complete, we clearly have W (γ) ⊆ W R ∩ dS d = W dS d (h). For the converse inclusion, let x ∈ W R . We have to find a t ∈ R with x ≤ γ(t), i.e., Since β(γ(t), γ(t)) = −1, we obtain for the right hand side and if x 1 > |x 0 |, this expression is arbitrarily large for t → ∞. This shows that W R ⊆ ↓γ(R), and we likewise see that W R ⊆ ↑γ(R).
Proof. If gH ∈ W (γ) ⊆ G/H, then the corresponding subset g.D + ⊆ B is convex by Theorem 4.5, and it contains 0 by (5.1). Therefore the curve η : R → M, η(t) := exp(th)gH is increasing because t → e t g.D + is an increasing family of subsets of B. The invariance of the order thus implies that We also know that g.D + ∈ P + .D + (Theorem 4.5 and Lemma 5.2(c)), so that there exist g 1 ∈ G h and y ∈ g 1 (h) with g.H = g 1 exp(y).H. Thus and therefore Recall the definition of D g in (4.7). The condition eP − ∈ g.D + = g 1 exp(y).D + = g 1 . exp(y + D g )P − is equivalent to −y ∈ D g = −D g , showing that 5.2(c)). We therefore derive from (5.3) that h + D g ⊆ h + C max q , and since h ∈ C max,• q and D + is starlike with respect to 0, we obtain (5.5) We thus obtain Ad(g) −1 .h ∈ h + C max,• q , i.e., gH ∈ W + M (h). This shows that W (γ) ⊆ W + M (h), and the connectedness of W (γ) (Lemma 5.2(a)) yields W (γ) ⊆ W .
Remark 5.5. From (5.4) it follows that, as a subset of M , For the quotient map q : G → G/H, this means that This is a G h × H-invariant domain in G specified by its intersection with the abelian subgroup N + = exp(g 1 (h)); see [MNÓ23, Rem. 6.2].
Combined with Theorem 7.1 below, that asserts the connectedness of W + M (h), the following result implies that W + M (h) ⊆ W (γ). Proposition 5.6. If H K = K h and C = C max q , then W ⊆ W (γ).
Proof. As both sides are G h e -invariant (Lemma 5.2), the Positivity Domain Theorem (Theorem 3.6) implies that we have to verify the inclusion Invariance of both sides under (H K ) e and Ad((H K ) e )t q = q k further reduce the problem to the inclusion Exp eH (Ω tq ) ⊆ W (γ). (5.7) To this end, we use the Lie subalgebra l ⊆ g generated by h and t q (Proposition 2.8). Then [l, l] ∼ = sl 2 (R) s and t q ∼ = so 2 (R) s . This reduces the verification of the inclusion (5.7) to the case where g = sl 2 (R) s , h = so 1,1 (R) s and t q ∼ = so 2 (R) s .
Combining the preceding two propositions, we get the main result of this section. It shows that the observer domain W (γ) coincides with a connected component of the positivity domain W + M (h). This result provides two complementary perspectives on this domain.
Theorem 5.7. (Observer Domain Theorem) Let (g, τ, C) be a non-compactly causal semisimple symmetric Lie algebra with causal Euler element h ∈ C • ∩ q p with τ = τ h θ and let G be a connected Lie group with Lie algebra g and H : We can even extend this result to coverings: First we show that W ′ ⊆ M ′ is order convex. So let x ≤ y ≤ z in M ′ with x, z ∈ W ′ and let η : [0, 2] → M ′ be a causal curve with η(0) = x, η(1) = y, η(2) = z.
As W is contractible by Theorem3.6(b), it is in particular simply connected. Therefore q −1 (W ) is a disjoint union of open subsets (W ′ j ) j∈J mapped by q diffeomorphically onto W . By definition, W ′ is one such connected component, so that is a diffeomorphism. Therefore η is the unique continuous lift of q • η in M ′ , hence contained in W ′ . This implies that y ∈ W ′ , so that W ′ is order convex.
Remark 5.9. It is not clear to which extent W (γ) depends on the specific cone C. In particular it would be interesting to see if the minimal and maximal cones lead to the same domain W (γ). We have already seen that the positivity domain W + M (h) depends non-trivially on the cone C ([MNÓ23, Ex. 6.8]) so one may expect that this is also the case for W (γ). Proof. (a) The condition gH ∈ W + M (h) is equivalent to Ad(g) −1 h ∈ T C by (5.10), and this implies that so that Ad(τ (g)) −1 h ∈ T C , i.e., τ M (gH) ∈ W + M (h). As τ M is an involution, it follows that As τ (C) = −C, the involution τ M reverses the causal structure on M . Moreover, τ M (γ(t)) = γ(−t), so that We have seen above that, for the modular geodesic γ(t) = Exp eH (th) in M , we have W (γ) = W . The modular geodesic γ is a specific orbit of the modular flow inside W . Now we show that all other α-orbits in W lead to the same "observer domain". As β(R) ⊆ W (γ), the order convex hull W (β) of β(R) is contained in W (γ) = W . To verify the converse inclusion, let D ′′ ∈ W . Then 0 ∈ D ′′ , and since D ′ is bounded, there exists a t ∈ R with β(t) ⊆ D ′′ . Likewise the boundedness of D ′′ implies the existence of some s ∈ R with D ′′ ⊆ β(s).
Remark 5.12. A similar result also holds in Minkowski space. If x ∈ W R = {y ∈ R 1,d : y 1 > |y 0 |} and then any other element y ∈ W R satisfies y ∈ [β(t), β(s)] for suitable t < s, i.e., y − β(t) ∈ V + and β(s) − y ∈ V + . In fact, This shows that W (β) = W R for all integral curves of the modular flow in W R . because it preserves h and the causal structure. This is not desirable because we would prefer that τ h maps W + (h) to some "opposite" wedge region (cf. [MN21]). Possible ways to resolve this problem and ideas how to implement locality conditions on non-compactly causal symmetric spaces are briefly discussed in [MNÓ23,§4.3].

Existence of positivity domains for Euler elements
In this section we show that, for the maximal cone C = C max q and a simple Lie algebra g, the real tube domain T C = h + C • intersects the set E(g) of Euler elements in a connected subset (Theorem 6.1). This implies that, for an Euler element h ′ ∈ g, the positivity domain W + M (h ′ ) is non-empty if and only if h ′ and h are conjugate (Corollary 6.3). Hence there exists g 1 ∈ H e with g 1 .m − = eP + , and thus y := Ad(g 1 )x ∈ p + = g 1 (h) ⋊ g 0 (h). (6.2) Then y ∈ T C ∩ p + is an Euler element, and a similar argument shows that the vector field X G/P − y has a unique repelling fixed point m + ∈ D + . So m + = exp(−z)P − for some z ∈ g 1 (h), and exp(z).m + = eP − . Hence the base point eP − ∈ G/P − is a repelling fixed point of the Euler element y ′ := e ad z y ∈ g 0 (h), and eP + is an attracting fixed point in G/P + . The attracting and repelling properties of the fixed points imply that so that we also have g 0 (h) = [g 1 (h), g −1 (h)] ⊆ g 0 (y ′ ).
Conversely, we have seen in Proposition 5.4 that We finally obtain (6.1).
Remark 6.2. Note that the preceding proof is based on the natural embedding 4 This reference deals with bounded symmetric domains in complex spaces, but D can be embedded into such a domain D C by embedding g ֒→ g C ∼ = g c C . If C g c ⊆ g c is an invariant cone with C = g ∩ iC g c , then (g, τ, C) ֒→ (g C , τg C , iC g c ) is a causal embedding and  Proof. As X M h1 (g 1 H) ∈ C • g1H is equivalent to Ad(g 1 ) −1 h 1 ∈ h+C • by (see Lemma 3.3), Theorem 6.1 implies that h 1 = Ad(g)h ∈ O h for some g ∈ G. The relation W + M (h 1 ) = g.W + M (h) now follows directly from the definitions.
The preceding corollary shows that any wedge domain of the type W + Remark 6.5. (Extensions to the non-simple case) If (g, τ ) is a direct sum of irreducible ncc symmetric Lie algebra (g j , τ j ) and h = j h j accordingly, then (2.3)). Projecting to the ideals g j , we obtain with Theorem 6.1 for C = C max q (h) and hj ∩ T Cj and D g = j D gj imply (6.1) for this case. Note that the situation corresponds to g = g s (see (2.2)). In the general situation, where we assume only that all ideals of g contained in h are compact, we have where g k ⊆ h is compact, g r is a direct sum of Riemannian symmetric Lie algebras and g s is a direct sum of irreducible ncc symmetric Lie algebras. All Euler elements are contained in g r + g s . If g is only reductive, we assume z(g) ⊆ g −θ , so that z(g) ⊆ g r . Then h = h r + h s and We conclude that This shows that, for any Euler element k ∈ g with W + M (k) = ∅ we must have k s ∈ O hs , but there is no restriction on the Riemannian component k r ∈ E(g r ).
and this leads with Lemma 3.3 to .eH. Since G h has at most two connected components, this set is either connected or has two connected components ([MNÓ23, Thm. 7.8]). As G h = K h exp(q p ), we have G h = K h G h e , and Ad(K h ) preserves the open unit ball in g 1 (h). We thus derive from K h = H K : .eH, which is connected. W (γ) = {gD + : 0 ∈ g.D + , g.D + bounded in exp(g 1 (h)).P − } (Lemma 5.2(c)). Since exp(Rh) acts on exp(g 1 (h)) by dilations, it follows that Therefore gW (γ) = W (γ) for the action of g on G/H ⊆ P(G/P − ) implies that g preserves the intersection {eP − } of all subsets contained in W (γ). This shows that g fixes eP − , so that g ∈ P − . Next we recall that the involution τ M on M defined by τ M (gH) = τ (g)H leaves W (γ) invariant (Lemma 5.10), and this leads to The preceding proposition shows that the set W = W(M ) of wedge domains in M = G/H coincides with In particular, it is a symmetric space. Recall that, by Corollary 7.2, the observer domain coincides with the positivity domain W + M (h).

KMS wedge regions
With the structural results obtained so far, we have good control over the positivity domains W + M (h) in ncc symmetric spaces M = G/H. So one may wonder if they also have an interpretation in terms of a KMS like condition. In [NÓ23b], this has been shown for modular flows with fixed points, using such a fixed point as a base point. In this section we extend the characterization of the wedge domain W in terms of a geometric KMS condition to general ncc spaces.
To simplify references, we list our assumptions and the relevant notation below: • g is simple, • G = Inn(g) ⊆ G C = Inn(g C ) e (by (GP) and (Eff), [NÓ23b, Lemma 2.12]) • σ : G C → G C denotes the complex conjugation with respect to G.
h (H C ) = H C for the holomorphic involution of G C integrating the complex linear extension of τ .
• σ(H C ) = H C for the conjugation of G C with respect to G.
is a holomorphic involutive automorphism of G C inducing τ h on the Lie algebra g. Let Ξ := G. Exp eK (iΩ p ) ⊆ G C /K C be the crown of G/K. The involution τ h on G preserves K, hence induces an involution on G/K, and we extend it to an antiholomorphic involution τ h on G C /K C . The canonical map G× K iΩ p → Ξ is a diffeomorphism ([NÓ23b, Prop. 4.7]) and is an open subgroup of G θ C that coincides with the stabilizer G mK C . In this sense G C /H C ∼ = G C /K C , but with different base points m H := eH C and m K . Recall that τ = e πi ad h θ = e πi 2 ad h θe − πi so that K C and H C are exchanged by the order-4 automorphism κ G h and invariant under τ G h .
Therefore the antiholomorphic extension τ G h also preserves K C and induces on G C /K C ∼ = G C /H C an antiholomorphic involution τ h fixing the base point m K with stabilizer K C . Then may be different from m H .
Remark 8.1. The condition m H = m ′ H is equivalent to exp(πih) ∈ H C . Note that e πi ad h = τ h ∈ Aut(g C ) is an involution that commutes with τ , so that the choice of H C determines whether exp(πih) is contained in H C or not.
For z = πi 2 , we thus obtain α πi/2 (p) ∈ M τ h C . We conclude that This suggest to define a "polar wedge domain" as We actually know from Theorem 3.6, that this is the connected component To see that this domain is contained in the G h e -invariant domain W KMS ⊆ M , we thus have to show that, for x ∈ Ω hp , we have α it . Exp eK (ix) ∈ Ξ for |t| < π/2. Let t q ⊆ q k is a maximal abelian subspace (they are all conjugate under (H K ) e ). Then a h := iκ h (t q ) ⊆ h p is also maximal abelian and Ω hp = e ad h k .Ω a h . So it suffices to show that, for x ∈ Ω a h and |t| < π/2, we have α it . Exp eK (ix) ∈ Ξ. By Proposition 2.8, t q is contained in a τ -invariant subalgebra s ∼ = sl 2 (R) s , where Rh + s is generated by h and t q and h = h 0 + h 1 + · · · + h s , where h j , j = 1, . . . , s, is an Euler element in a simple ideal s j ∼ = sl 2 (R) of s. Then a h = iκ h (t q ) ⊆ a is spanned by s Euler elements x 1 , . . . , x s and Ω a h = s j=1 t j x j : (∀j) |t j | < π/2 . Let S := exp s and Ξ S := S. Exp(i(Ω p ∩ s)) ⊆ Ξ. Then the discussion in Remark D.1 implies that, for |t| < π/2 and x = j t j x j ∈ Ω a h , we have α it (Exp eK (ix)) ∈ Ξ S ⊆ Ξ.
The preceding proof implies in particular the following interesting observation: Corollary 8.3. For every m ∈ Ξ τ h , we have α it (m) ∈ Ξ for |t| < π/2, so that the orbit map α m extends to a holomorphic map S ±π/2 → Ξ.
Corollary 8.4. α πi 2 : W KMS → Ξ τ h is a diffeomorphism that induces an equivalence of fiber bundles Proof. Theorem 8.2 implies in particular that α πi 2 : W KMS → Ξ τ h is bijective. Since W KMS = W + M (h) eH is an open subset of M and Ξ τ h an open subset of M τ h C , it actually is a diffeomorphism. The second assertion follows from the fact that it commutes with the action of the subgroup G h e .

A Irreducible ncc symmetric Lie algebras
The following table lists all irreducible non-compactly causal symmetric Lie algebras (g, τ ) according to the following types: • Complex type: g = h C and τ is complex conjugation with respect to h. In this case g c ∼ = h ⊕2 , so that rk R (g c ) = 2 rk R (h).
• Split type (ST): τ = τ h1 for all h 1 ∈ h ∩ E(g) and rk R h = rk R g c : • Non-split type (NST): τ = τ h1 for all h 1 ∈ h ∩ E(g) and rk R h = rk R g c 2 : In the table we write r = rk R (g c ) and s = rk R (h). Further a ⊆ p is maximal abelian of dimension r. For root systems Σ(g, a) of type A n−1 , there are n − 1 Euler elements h 1 , . . . , h n−1 , but for the other root systems there are less; see [MN21,Thm. 3.10] for the concrete list. For 1 ≤ j < n we write j ′ := min(j, n − j). g g c = h + iq r h = g τ h θ s Σ(g, a) h g 1 (h) which leads to exp(2ty) = exp(tx) exp(−tτ (x)). Evaluating the derivative of this curve in the right trivialization of T (G), we get 2y = x + e t adx (−τ (x)) = x − e t adx (τ (x)) for all t ∈ R.
For t = 0 we get p q (x) = y, and taking derivatives in 0 shows that [x, τ (x)] = 0. If, conversely, this condition is satisfied, then x = x h + x q with x h ∈ h and x q ∈ q, where Let U ⊆ q be a 0-neighborhood for which Exp eH | U is a diffeomorphism onto an open subset of M and the spectral radius of ad y is smaller than π for y ∈ U . Then sinh(ad y) ad y : h → h is invertible. With the above formula, we thus conclude for y ∈ U that Exp m0 (g 0 .y) ∈ M x is equivalent to [y, x c ] = 0, which is equivalent to Ad(g 0 )y ∈ g x . This shows that M x is a submanifold of M .
As Exp m0 (g 0 .y) = exp(Ad(g 0 )y).m 0 ∈ exp(g x ).m 0 , it further follows that the the orbit of m 0 under the connected group G x e contains a neighborhood of m 0 . This shows that the orbits of G x e in M x are connected open subsets, hence coincide with its connected components. Remark B.3. For x ∈ q the centralizer g x is τ -invariant, so that g x = h x ⊕ q x and the dimension of the G x e -orbit through eH is dim q x . We have M x eH ∼ = G x e /(H ∩ G x e ), and Lemma B.2 shows that the geodesic Exp eH (Rx) is central in the symmetric space M x eH in the sense that its tangent space Rx is central in the Lie algebra g x (cf. [Lo69]).
Lemma B.4. For y ∈ q, the equality M y = G y .eH is equivalent to O y ∩ q = Ad(H)y. (B.2) Proof. As y ∈ q, the base point eH is contained in M y , and thus G y .eH ⊆ M y . So the equality M y = G y .eH means that M y ⊆ G y .eH, i.e., Ad(g) −1 y ∈ q implies gH ∈ G y .eH, resp., g ∈ G y H. This in turn is equivalent to Ad(g) −1 y ∈ Ad(H)y.

D de Sitter space
In this appendix we collect some concrete observations concerning de Sitter space dS d , which is an important example of a non-compactly causal symmetric space. Some facts on 2-dimensional de Sitter space are used in particular in some of our proofs to verify the corresponding assertions for g = sl 2 (R). In (d + 1)-dimensional Minkowski space R 1,d , we write the Lorentzian form as β(x, y) = x 0 y 0 − xy for x = (x 0 , x), y = (y 0 , y).
Comparing both expressions leads for h-modular geodesics to the conditions x 2 = · · · = x d = 0 and v = h.x = (x 1 , x 0 , 0, . . . , 0). Therefore exactly two orbits of the modular flow are timelike geodesics. If we also ask for the geodesic to be positive with respect to the causal structure, then x 1 > 0 determines the geodesic uniquely. We infer from [NÓ23b, Prop. D.3] that Exp e2 (te 1 ) = cos(t)e 2 + sin(t)e 1 is a closed space-like geodesics. For 0 < t < π, its values are contained in W + M (h), and this geodesic arc connects the two fixed points e 2 to −e 2 of the modular flow.
Note that, in g, we have τ h (h) = h and τ h (h d ) = −h d , so that h ∈ q p and h d ∈ h p .
We also note that, for 0 < t < π, the Lie algebra element Ad(g t )h 0 corresponds to Exp e2 (te 1 ) ∈ W + M (h 0 ). Note that g π ∈ K τ G h = K τ G .