# Wedge domains in compactly causal symmetric spaces

Karl-Hermann Neeb and Gestur Ólafsson The research of K.-H. Neeb was partially
supported by DFG-grant NE 413/10-1. The research of G. Ólafsson was partially
supported by Simons grant 586106.

###### Abstract

Motivated by construction in Algebraic Quantum Field Theory we introduce wedge
domains in compactly causal symmetric spaces $M=G/H$, which includes in
particular anti de Sitter space in all dimensions and its coverings. Our wedge
domains generalize Rindler wedges in Minkowski space. The key geometric
structure we use is the modular flow on $M$ defined by an Euler element in the
Lie algebra of $G$. Our main geometric result asserts that three seemingly
different characterizations of these domains coincide: the positivity domain
of the modular vector field; the domain specified by a KMS like analytic
extension condition for the modular flow; and the domain specified by a polar
decomposition in terms of certain cones.

In the second half of the article we show that our wedge domains share
important properties with wedge domains in Minkowski space. If $G$ is
semisimple, there exist unitary representations $(U,\mathcal{H})$ of $G$ and
isotone covariant nets of real subspaces ${\sf
H}(\mathcal{O})\subseteq\mathcal{H}$, defined for any open subset
$\mathcal{O}\subseteq M$, which assign to connected components of the wedge
domains a standard subspace whose modular group corresponds to the modular
flow on $M$. This corresponds to the Bisognano–Wichmann property in Quantum
Field Theory. We also show that the set of $G$-translates of the connected
components of the wedge domain provides a geometric realization of the
abstract wedge space introduced by the first author and V. Morinelli.

Dedicated to the memory of Ottmar Loos

MSC 2010: Primary 22E45; Secondary 81R05, 81T05

###### Contents

  1. 1 Introduction
  2. 2 Causal symmetric Lie algebras
    1. 2.1 Embedding into group type
    2. 2.2 Hermitian simple Lie algebras
    3. 2.3 Reductive Lie algebras
    4. 2.4 The global setting for symmetric spaces
  3. 3 Fixed points of the modular flow in $M$
    1. 3.1 $\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$-orbits in $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$
    2. 3.2 Connected components of $M^{\alpha}$
  4. 4 Wedge domains in compactly causal symmetric spaces
    1. 4.1 Tube domains of compactly causal symmetric spaces
    2. 4.2 The modular flow and three types of wedge domains
  5. 5 Wedge domains in spaces of group type
  6. 6 Wedge domains in extendable causal symmetric spaces
  7. 7 Nets of standard subspaces
    1. 7.1 Standard subspaces
    2. 7.2 The BGL construction
    3. 7.3 Nets of real subspaces on homogeneous spaces
    4. 7.4 Connecting wedge domains and standard subspaces
  8. 8 Nets of standard subspaces on compactly causal symmetric spaces
    1. 8.1 $C_{\mathfrak{g}}$-positive representations and standard subspaces
    2. 8.2 The group case: biinvariant nets on Lie groups
    3. 8.3 The general case: Nets on compactly causal symmetric spaces
  9. 9 The wedge space as an ordered symmetric space
    1. 9.1 The order structure
    2. 9.2 The symmetric space structure
  10. 10 Open problems
  11. 11 Anti-de Sitter space $\mathop{{\rm AdS}}\nolimits^{d}$
    1. 11.1 Subgroup data
    2. 11.2 The tube domain
    3. 11.3 The wedge domains
  12. A Some facts on symmetric Lie groups
  13. B The topology on $\mathcal{H}^{\infty}$ and the space $\mathcal{H}^{-\infty}$

##  1 Introduction

Let $M=G/H$ be a homogeneous space of the Lie group $G$ and $(U,\mathcal{H})$
a unitary representation of $G$. Motivated by their occurrence in Algebraic
Quantum Field Theory (AQFT), in the sense of Haag–Kastler ([Ha96]), one
considers nets of real subspaces on $M$, i.e., to each open subset
$\mathcal{O}\subseteq M$, we assign a closed real subspace ${\sf
H}(\mathcal{O})\subseteq\mathcal{H}$ satisfying

  * •

Isotony: $\mathcal{O}_{1}\subseteq\mathcal{O}_{2}\Rightarrow{\sf
H}(\mathcal{O}_{1})\subseteq{\sf H}(\mathcal{O}_{2})$,

  * •

Equivariance: ${\sf H}(g\mathcal{O})=U(g){\sf H}(\mathcal{O})$ for $g\in G$.

If $(W(v))_{v\in\mathcal{H}}$ are the Weyl operators on bosonic Fock space
$\mathcal{F}_{+}(\mathcal{H})$, then we assign to each closed real subspace
${\tt E}\subseteq\mathcal{H}$ the von Neumann algebra $\mathcal{R}({\tt
E}):=W({\tt E})^{\prime\prime}$ ([Si74]). We thus obtain on $M$ the isotone
covariant net $(\mathcal{R}({\sf H}(\mathcal{O})))_{\mathcal{O}\subseteq M}$
of von Neumann algebras, where covariance refers to the canonical
representation of $G$ on $\mathcal{F}_{+}(\mathcal{H})$. This method has been
developed by Araki and Woods in the context of free bosonic quantum fields
([Ar63, Ar64, AW63, AW68]), but it can be modified to deal with fermionic Fock
spaces (cf. [EO73], [BJL02]) and there are variants for other statistics
(anyons) ([Schr97], [Le15, §3]).

We are interested in structures on homogeneous spaces which resemble space-
times in AQFT. In this context the von Neumann algebra associated to an open
subset $\mathcal{O}\subseteq M$ corresponds to observables measurable in the
“laboratory” $\mathcal{O}$ ([Ha96]). One often focuses on those von Neumann
algebras $\mathcal{R}({\tt V})$ for which the “vacuum vector”
$\Omega\in\mathcal{F}_{+}(\mathcal{H})$ is cyclic and separating, so that the
Tomita–Takesaki Theorem applies to $(\mathcal{R}({\tt V}),\Omega)$ ([BR87,
Thm. 2.5.14]). It is not hard to see that $\Omega$ is

  * •

cyclic for $\mathcal{R}({\tt V})$ if and only if ${\tt V}+i{\tt V}$ is dense
in $\mathcal{H}$,

  * •

separating for $\mathcal{R}({\tt V})$ if and only if ${\tt V}\cap i{\tt
V}=\\{0\\}$.

A closed real subspace ${\tt V}\subseteq\mathcal{H}$ with both properties is
called standard (cf. [Lo08] for the basic theory of standard subspaces). It is
therefore of particular interest to understand how to construct nets for which
the subspaces ${\sf H}(\mathcal{O})$ are standard. The main feature of
Tomita–Takesaki theory is that it provides a modular conjugation and a modular
automorphism group, and these can already be associated to a standard
subspaces ${\tt V}\subseteq\mathcal{H}$ (and pass through second quantization
to the von Neumann context). Concretely, we associate to ${\tt V}$ a pair of
modular objects $(\Delta_{\tt V},J_{\tt V})$:

  * •

the modular operator $\Delta_{\tt V}$ is a positive selfadjoint operator,

  * •

$J_{\tt V}$ is a conjugation (an antiunitary involution),

and these two operators satisfy the modular relation $J_{\tt V}\Delta_{\tt
V}J_{\tt V}=\Delta_{\tt V}^{-1}.$ The pair $(\Delta_{\tt V},J_{\tt V})$ is
obtained by the polar decomposition $\sigma_{\tt V}=J_{\tt V}\Delta_{\tt
V}^{1/2}$ of the closed operator

$\sigma_{\tt V}\colon{\tt V}+i{\tt V}\to\mathcal{H},\quad x+iy\mapsto x-iy$

with ${\tt V}=\mathop{{\rm Fix}}\nolimits(\sigma_{\tt V})$. Key properties of
these operators are that

$J_{\tt V}{\tt V}={\tt V}^{\prime}:=\\{w\in\mathcal{H}\colon(\forall v\in{\tt
V})\ \mathop{{\rm Im}}\nolimits\langle v,w\rangle=0\\}\quad\mbox{ and
}\quad\Delta_{\tt V}^{it}{\tt V}={\tt V}\quad\mbox{ for }\quad
t\in{\mathbb{R}}.$

So we obtain a one-parameter group of automorphisms of ${\tt V}$ (the modular
group) and a symmetry between ${\tt V}$ and its commutant ${\tt V}^{\prime}$,
implemented by $J_{\tt V}$.

The current interest in standard subspaces arose in the 1990s from the work of
Borchers and Wiesbrock ([Bo92, Wi93]). This led to the concept of modular
localization in Quantum Field Theory introduced by Brunetti, Guido and Longo
in [BGL02, BGL93]; see also [BDFS00] and [Le15, LL15] for important
applications of this technique.

For a net on a homogeneous space $G/H$, the domains $\mathcal{O}$ for which
${\sf H}(\mathcal{O})$ is standard are of particular relevance. Here one would
like to know when the modular group
$(\Delta_{\mathcal{O}}^{it})_{t\in{\mathbb{R}}}$ is “geometric” in the sense
that it is implemented by a one-parameter subgroup of $G$, hence corresponds
to a one-parameter group of symmetries of $M$ preserving the domain
$\mathcal{O}$. For the modular conjugation $J_{\mathcal{O}}$, we may likewise
ask for the existence of an involution $\tau^{M}$ on $M$ satisfying ${\sf
H}(\tau^{M}\mathcal{O})={\sf H}(\mathcal{O})^{\prime}$. Building on [NÓ21a],
which deals with the case $H=\\{e\\}$, i.e., left invariant nets on Lie
groups, we turn in this paper to nets on a class of symmetric spaces to which
the tools and methods developed in [NÓ21a] apply particularly well; see also
the companion paper [NÓ21b].

First we explain how to find natural standard subspaces for unitary
representations. So let $(U,\mathcal{H})$ be a unitary representation of $G$
which extends to an antiunitary representation of the extended group

$G_{\tau}=G\rtimes\\{\mathop{{\rm id}}\nolimits_{G},\tau^{G}\\},$

where $\tau^{G}$ is an involutive automorphism of $G$. Then $J:=U(\tau^{G})$
is a conjugation satisfying $JU(g)J=U(\tau^{G}(g))$ for $g\in G$. We then
obtain for each pair $(h,\tau^{G})$, for which $h$ is fixed by the Lie algebra
involution $\tau$ induced by $\tau^{G}$, a standard subspace ${\tt V}:={\tt
V}_{(h,\tau^{G},U)}=\mathop{{\rm Fix}}\nolimits(J_{\tt V}\Delta_{\tt
V}^{1/2})$, specified by

$J_{\tt V}=U(\tau^{G})\quad\mbox{ and }\quad\Delta_{{\tt V}}^{-it/2\pi}=U(\exp
th)\quad\mbox{ for }\quad t\in{\mathbb{R}}.$ (1)

This assignment is called the Brunetti–Guido–Longo (BGL) construction (see
[BGL02]). As a consequence, standard subspaces can be associated to
antiunitary representations in abundance, but only a few of them carry
interesting geometric information. In particular, we would like to understand
when a standard subspace of the form ${\tt V}_{(h,\tau^{G},U)}$ arises from a
net on some homogeneous space $G/H$.

A structural property with strong impact in this context is the spectrum
condition on the infinitesimal generators $\partial U(x)$ of the one-parameter
groups $(U(\exp tx))_{t\in{\mathbb{R}}}$. This is the requirement that the
closed convex cone

$C_{U}:=\\{x\in{\mathfrak{g}}\colon-i\partial U(x)\geq 0\\}$

(the positive cone of $U$) is “large” in the sense that the ideal
${\mathfrak{g}}_{C}:=C_{U}-C_{U}$ satisfies
${\mathfrak{g}}={\mathfrak{g}}_{C_{U}}+{\mathbb{R}}h$. Let ${\tt V}={\tt
V}_{(h,\tau^{G},U)}$ be a standard subspace obtained from the BGL construction
as in (1). Then it is a natural requirement that the semigroup

$S_{\tt V}=\\{g\in G\colon U(g){\tt V}\subseteq{\tt V}\\}$ (2)

of endomorphisms of ${\tt V}$ is large in the sense that its Lie wedge

$\mathop{\bf L{}}\nolimits(S_{\tt
V})=\\{x\in{\mathfrak{g}}\colon\exp({\mathbb{R}}_{+}x)\subseteq S_{\tt V}\\}$

(the set of infinitesimal generators of one-parameter subsemigroups of $S_{\tt
V}$) spans the Lie algebra ${\mathfrak{g}}$. In [Ne21] it is shown that if
$\mathop{\bf L{}}\nolimits(S_{\tt V})$ spans ${\mathfrak{g}}$, then
$\mathop{{\rm ad}}\nolimits h$ defines a $3$-grading in the sense that

${\mathfrak{g}}={\mathfrak{g}}_{1}(h)\oplus{\mathfrak{g}}_{0}(h)\oplus{\mathfrak{g}}_{-1}(h)\quad\mbox{
for }\quad{\mathfrak{g}}_{j}(h):=\ker(\mathop{{\rm ad}}\nolimits
h-j\mathop{{\rm id}}\nolimits_{\mathfrak{g}}),$ (3)

and

$\mathop{{\rm Ad}}\nolimits(\tau^{G})=\tau_{h},$ (4)

so that ${\mathfrak{g}}_{0}(h)={\mathfrak{g}}^{\tau_{h}}$ and
${\mathfrak{g}}_{1}(h)+{\mathfrak{g}}_{-1}(h)={\mathfrak{g}}^{-\tau_{h}}$. We
call elements $h$ with (3) Euler elements because they correspond to the Euler
vector field on an open subset of the homogeneous spaces with tangent space
${\mathfrak{g}}/({\mathfrak{g}}_{0}(h)+{\mathfrak{g}}_{-1}(h))$. Assuming that
$h$ is an Euler element and $\tau=\tau_{h}$, the semigroup $S_{\tt V}$ has
been completely determined in [Ne19]. To describe the structure of $S_{\tt
V}$, let

$C_{\pm}:=\pm C_{U}\cap{\mathfrak{g}}_{\pm 1}(h)\quad\mbox{ and write }\quad
G_{\tt V}=\\{g\in G\colon U(g){\tt V}={\tt V}\\}$

for the stabilizer group of ${\tt V}$ in $G$. Then

$S_{\tt V}=\exp(C_{+})G_{\tt V}\exp(C_{-})=G_{\tt V}\exp(C_{+}+C_{-}).$ (5)

Combining Euler elements with symmetric spaces leads to the following concept:
A modular causal symmetric Lie algebra is a quadruple
$({\mathfrak{g}},\tau,C,h)$, where

  * •

$({\mathfrak{g}},\tau)$ is a symmetric Lie algebra, i.e., ${\mathfrak{g}}$ is
a finite-dimensional real Lie algebra and $\tau$ an involutive automorphism of
${\mathfrak{g}}$.

  * •

$({\mathfrak{g}},\tau,C)$ is a causal symmetric Lie algebra, i.e.,
$C\subseteq{\mathfrak{g}}^{-\tau}$ is a pointed generating closed convex cone
invariant under the group $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})=\langle e^{\mathop{{\rm
ad}}\nolimits{\mathfrak{h}}}\rangle$.

  * •

$h\in{\mathfrak{g}}^{\tau}$ is an Euler element, i.e., $\mathop{{\rm
ad}}\nolimits h$ is diagonalizable with eigenvalues $\\{-1,0,1\\}$, and the
involution $\tau_{h}:=e^{\pi i\mathop{{\rm ad}}\nolimits h}\in\mathop{{\rm
Aut}}\nolimits({\mathfrak{g}})$ satisfies $\tau_{h}(C)=-C$.

A causal symmetric Lie algebra $({\mathfrak{g}},\tau,C)$ is called compactly
causal (cc for short) if the cone $C$ is elliptic, i.e., if its interior
consists of elements $x$ which are elliptic in the sense that $\mathop{{\rm
ad}}\nolimits x$ is semisimple with purely imaginary spectrum. Typical
examples arise from invariant pointed generating cones
$C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$. Then

$({\mathfrak{g}}\oplus{\mathfrak{g}},\tau_{\rm
flip},C_{\mathfrak{g}})\quad\mbox{ with }\quad\tau_{\rm flip}(x,y)=(y,x),$ (6)

is a compactly causal symmetric Lie algebra; called of group type (GT).
Another interesting class of compactly causal symmetric Lie algebras arises
from the involution $\tau_{h}=e^{\pi i\mathop{{\rm ad}}\nolimits h}$ defined
by an Euler element. If $C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ satisfies
$-\tau_{h}(C_{\mathfrak{g}})=C_{\mathfrak{g}}$, then
$({\mathfrak{g}},\tau_{h},C_{\mathfrak{g}}^{-\tau_{h}},h)$ is a modular
compactly causal symmetric Lie algebra; called of Cayley type (CT).

Irreducible compactly causal symmetric Lie algebras $({\mathfrak{g}},\tau,C)$
are either of group type with ${\mathfrak{g}}^{\tau}$ simple hermitian, or
${\mathfrak{g}}$ is simple hermitian, and
${\mathfrak{z}}({\mathfrak{k}})\subseteq{\mathfrak{g}}^{-\tau}$ holds for a
$\tau$-invariant Cartan decomposition
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$. We refer to [HÓ97] for a
classification. The additional assumption that ${\mathfrak{h}}$ contains an
Euler element leaves us with the following four types:

  * (GT)

Group type: ${\mathfrak{g}}={\mathfrak{h}}\oplus{\mathfrak{h}}$,
${\mathfrak{h}}$ simple hermitian of tube type

${\mathfrak{h}}=\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}}),\ \
\mathop{{\mathfrak{sp}}}\nolimits_{2r}({\mathbb{R}}),\ \
\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}}),\ \
\mathop{{\mathfrak{so}}}\nolimits^{*}(4r),\ \ {\mathfrak{e}}_{7(-25)}.$

  * (CT)

The Euler element $h$ is central in ${\mathfrak{h}}$ and $\tau=\tau_{h}$:

${\mathfrak{g}}$ | $\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}})$ | $\mathop{{\mathfrak{sp}}}\nolimits_{2r}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}}),d>2$ | $\mathop{{\mathfrak{so}}}\nolimits^{*}(4r)$ | ${\mathfrak{e}}_{7(-25)}$
---|---|---|---|---|---
${\mathfrak{h}}$ | ${\mathbb{R}}\oplus\mathop{{\mathfrak{sl}}}\nolimits_{r}({\mathbb{C}})$ | ${\mathbb{R}}\oplus\mathop{{\mathfrak{sl}}}\nolimits_{r}({\mathbb{R}})$ | ${\mathbb{R}}\oplus\mathop{{\mathfrak{so}}}\nolimits_{1,d-1}({\mathbb{R}})$ | ${\mathbb{R}}\oplus\mathop{{\mathfrak{sl}}}\nolimits_{r}({\mathbb{H}})$ | ${\mathbb{R}}\oplus{\mathfrak{e}}_{6(-26)}$

  * (ST)

Split types: $\tau\not=\tau_{h}$ and ${\rm rk}_{\mathbb{R}}{\mathfrak{h}}={\rm
rk}_{\mathbb{R}}{\mathfrak{g}}$:111The real rank $\mathop{{\rm
rank}}\nolimits_{\mathbb{R}}({\mathfrak{g}})$ of a reductive Lie algebra
${\mathfrak{g}}$ is the dimension of a maximal abelian $\mathop{{\rm
ad}}\nolimits$-diagonalizable subspace.

${\mathfrak{g}}$ | $\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{2,p+q}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits^{*}(4r)$ | ${\mathfrak{e}}_{7(-25)}$
---|---|---|---|---
${\mathfrak{h}}$ | $\mathop{{\mathfrak{so}}}\nolimits_{r,r}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{1,p}({\mathbb{R}})\oplus\mathop{{\mathfrak{so}}}\nolimits_{1,q}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{2r}({\mathbb{C}})$ | $\mathop{{\mathfrak{sl}}}\nolimits_{4}({\mathbb{H}})$

  * (NST)

Non-split types: $\tau\not=\tau_{h}$, $\mathop{{\rm
rank}}\nolimits_{\mathbb{R}}{\mathfrak{h}}=\frac{\mathop{{\rm
rank}}\nolimits_{\mathbb{R}}{\mathfrak{g}}}{2}$:

${\mathfrak{g}}$ | $\mathop{{\mathfrak{su}}}\nolimits_{2s,2s}({\mathbb{C}})$ | $\mathop{{\mathfrak{sp}}}\nolimits_{4s}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}})$
---|---|---|---
${\mathfrak{h}}$ | ${\mathfrak{u}}_{s,s}({\mathbb{H}})$ | $\mathop{{\mathfrak{sp}}}\nolimits_{2s}({\mathbb{C}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{1,d}({\mathbb{R}})$

We introduce “wedge domains” in compactly causal symmetric spaces $M=G/H$,
specified by a modular causal symmetric Lie algebra
$({\mathfrak{g}},\tau,C,h)$ and a connected Lie group $G$ with Lie algebra
${\mathfrak{g}}$ on which $\tau$ induces an involutive automorphism $\tau^{G}$
for which $H\subseteq G^{\tau^{G}}$ is an open subgroup. We then obtain a one-
parameter group $\alpha_{t}:=e^{t\mathop{{\rm ad}}\nolimits h}$ of
automorphisms of ${\mathfrak{g}}$, inducing automorphisms of $G$ and $M$.

Let $V_{+}(gH):=g.C^{0}\subseteq T_{gH}(M)$ denote the open cones defining the
causal structure on $M$ and let $X_{h}^{M}\in\mathcal{V}(M)$ be the modular
vector field defined by

$X_{h}^{M}(gH):=\frac{d}{dt}\Big{|}_{t=0}\alpha_{t}(g)H=\frac{d}{dt}\Big{|}_{t=0}\exp(th)gH.$
(7)

We introduce the following domains associated to this data:

  * •

The positivity domain of the vector filed $X_{h}^{M}$ in $M$:

$W_{M}^{+}(h):=\\{m\in M\colon X_{h}^{M}(m)\in V_{+}(m)\\}.$

  * •

The KMS wedge domain

$W_{M}^{\rm KMS}(h):=\\{m\in M\colon(\forall z\in\mathcal{S}_{\pi})\
\alpha_{z}(m)\in\mathcal{T}_{M}\\},$

where $(\alpha_{z})_{z\in{\mathbb{C}}}$ denotes the holomorphic extension of
the modular flow to the complex symmetric space $M_{\mathbb{C}}$ and

$\mathcal{T}_{M}:=G\times_{H}iC^{0}$

is the tube domain of $M$, a complex manifold containing $M\cong
G\times_{H}\\{0\\}$ in its “boundary”.

  * •

The polar wedge domain

$W_{M}(h):=\bigcup_{m\in M^{\alpha}}\mathop{{\rm
Exp}}\nolimits_{m}((C_{m}^{c})^{0}),$

where $M^{\alpha}$ is the submanifold of fixed points of the modular flow on
$M$,

$C_{m}:=\overline{V_{+}(m)}\subseteq T_{m}(M),$

and

$C_{m}^{c}:=C_{m}\cap T_{m}(M)_{1}-C_{m}\cap T_{m}(M)_{-1}\subseteq T_{m}(M),$

where $T_{m}(M)_{j}$, $j=-1,0,1$ denote the eigenspaces of the generator of
the modular flow on $T_{m}(M)$.

Our main geometric result, which is proved after various preparations in
Section 6 (Theorem 65), asserts that these three open subsets coincide:

$W_{M}^{+}(h)=W_{M}^{\rm KMS}(h)=W_{M}(h).$

For this result we assume that the causal symmetric Lie algebra
$({\mathfrak{g}},\tau,C)$ is extendable, i.e., there exists a pointed closed
convex invariant cone $C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ satisfying

$C_{\mathfrak{g}}\cap{\mathfrak{q}}=C\quad\mbox{ and
}\quad-\tau(C_{\mathfrak{g}})=C_{\mathfrak{g}}.$

If ${\mathfrak{g}}$ is reductive and ${\mathfrak{h}}$ contains no non-zero
ideals (which is a very natural assumption as these ideals act trivially on
$M$), then $({\mathfrak{g}},\tau,C)$ is extendable (Theorem 24), but there are
also interesting non-reductive examples. As we show in [NÓ21b], the companion
paper dealing with wedge domains in non-compactly causal symmetric spaces, the
geometry of wedge domains in a non-compactly causal space is more complicated
because these domains are generated from pieces of closed geodesics, the
corresponding polar map has singularities and it is defined only on a subset
of an open cone.

To connect this geometric insight with nets of standard subspaces, we show in
our second main result (Theorem 75) that for any antiunitary representation
$(U,\mathcal{H})$ of $G_{\tau_{h}}=G\rtimes\\{\mathbf{1},\tau_{h}^{G}\\}$
whose positive cone $C_{U}$ is pointed and generating and satisfies
$C=C_{U}\cap{\mathfrak{q}}$, every cyclic distribution vector
$\eta\in\mathcal{H}^{-\infty}$ fixed by $H$ and the conjugation
$J=U(\tau_{h}^{G})$ leads to a natural covariant isotone net ${\sf H}^{M}$ on
$M$ assigning to the connected component $W_{M}(h)_{eH}$ of the wedge domain
the standard subspace ${\tt V}_{(h,\tau^{G}_{h},U)}$. For classification
results concerning this class of representations, we refer to [KNÓ97].

Our third main result (Theorem 812) provides a concrete construction of such
representations and thus implies that such nets exist on all reductive
compactly causal modular symmetric spaces. Starting with a
$C_{\mathfrak{g}}$-positive antiunitary representation $(\rho,\mathcal{K})$ of
$G_{\tau_{h}}$ which is a finite sum of irreducible ones, we obtain on the
space $B_{2}(\mathcal{K})$ of Hilbert–Schmidt operators on $\mathcal{K}$ a
representation by $U(g)A:=\rho(g)A\rho(\tau^{G}(g))^{-1}$ and the trace
defines a distribution vector which generates a subspace $\mathcal{H}_{\rho}$,
so that Theorem 75 applies to $(U,\mathcal{H}_{\rho})$. We even obtain a
rather direct description of ${\tt V}$ as the closed real subspace generated
by $\rho(C^{\infty}_{c}(S_{{\tt V},e}^{0},{\mathbb{R}}))\subseteq
B_{1}(\mathcal{K})\subseteq B_{2}(\mathcal{K})$. Here we use that $\rho$ is a
trace class representation (see Section 8 for details).

Motivated by the analysis of abstract wedge spaces in [MN21], we prove in
Section 9 that the wedge space $\mathcal{W}(M,h)$ of all $G$-translates of the
connected component $W:=W_{M}(h)_{eH}$ actually is a non-compactly causal
parahermitian symmetric space. It is a covering of the adjoint orbit
$\mathcal{O}_{h}=\mathop{{\rm Ad}}\nolimits(G)h\subseteq{\mathfrak{g}}$. The
most difficult part of the proof is to get hold of the stabilizer group
$G_{W}=\\{g\in G\colon gW=W\\}$. It is an interesting open problem to connect
the covering theory and the corresponding twisted duality developed in [MN21]
to the geometry of coverings of compactly causal symmetric spaces $G/H$.

We conclude this paper with a section on the anti de Sitter space
$\mathop{{\rm AdS}}\nolimits^{d}$, the prototypical Lorentzian example of a
compactly causal symmetric space. This section can be read independently. For
this example we evaluate all geometric data explicitly and show directly that
the three types of wedge domains coincide, without referring to the elaborate
structure theory used in the rest of the paper.

Structure of this paper: We start in Section 2 by recalling basic facts on
causal symmetric spaces on the Lie algebra level (Subsections 2.1 to 2.3) and
also on the global level (Subsection 2.4).

In Subsection 3.1 we classify for a reductive compactly causal symmetric
reductive Lie algebra $({\mathfrak{g}},\tau)$ the orbits of $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ in the set of Euler elements in
${\mathfrak{h}}$. As this set describes the different choices of the Euler
element $h\in{\mathfrak{h}}$, this can be seen as a classification of all
“modular structures” on the associated symmetric spaces. In Subsection 3.2 we
show that the orbits of the centralizer group $G^{h}$ in $M^{\alpha}$ are open
and that this leads to a natural bijection between the orbit spaces
$M^{\alpha}/G^{h}$ and $(\mathcal{O}^{h}\cap{\mathfrak{h}})/H$. Actually both
sets correspond to the set of double cosets $G^{h}gH\subseteq G$ for which
$\mathop{{\rm Ad}}\nolimits(g)^{-1}h\in{\mathfrak{h}}$.

In Section 4 we introduce the three types of wedge domains in a compactly
causal symmetric space $M=G/H$. In our analysis of wedge domains we follow the
strategy to first study spaces of group type and then use embeddings into
these spaces to derive corresponding results in general. In Section 5 we start
with causal symmetric spaces of group type, i.e., pairs $(G,C_{\mathfrak{g}})$
of a connected Lie group $G$ and a pointed generating invariant cone
$C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$.

In Section 6 we turn to wedge domains in compactly causal symmetric spaces. In
this context we assume that $({\mathfrak{g}},\tau,C)$ is extendable in the
sense of Subsection 2.1, keeping in mind that this is always the case if
${\mathfrak{g}}$ is reductive. The main result of this section is Theorem 65,
asserting that the three wedge domains in $M$ coincide. This is first proved
for the special case $H=G^{\tau^{G}}$ in Theorem 63. The proof of this theorem
builds heavily on the group case (Theorem 52). We conclude this section with a
brief discussion of the assumption that the Euler element $h$ is contained in
${\mathfrak{h}}$, i.e., that the corresponding modular flow on $M$ has a fixed
point. Note that the definition of $W_{M}(h)$ makes no sense if
$M^{\alpha}=\emptyset$.

In Section 7 we turn to representation theoretic aspects of compactly causal
symmetric spaces. We start with introducing standard subspaces, the
Brunetti–Guido–Longo construction and recall the construction of nets of
closed real subspaces from distribution vectors, introduced in [NÓ21a]. In
Subsection 7.4 we describe in the general Theorem 75 how the methods from
[NÓ21a] can be used to construct covariant nets of standard subspaces on
compactly causal symmetric spaces. In Section 8 we construct such
representations explicitly in spaces of Hilbert–Schmidt operators
$\mathcal{H}_{\rho}\subseteq B_{2}(\mathcal{K})$, where $(\rho,\mathcal{K})$
is an antiunitary representation of $G_{\tau^{G}_{h}}$ which is a finite sum
of irreducible representations. This is done in three steps: First we recall
from [Ne00, Ne19] some results on the representations $(\rho,\mathcal{K})$ of
$G_{\tau_{h}}=G\rtimes\\{\mathbf{1},\tau^{G}_{h}\\}$, then we use these
representations to construct nets of standard subspaces on the symmetric space
of group type $G\cong(G\times G)/\Delta_{G}$, and finally we use the twisted
embedding $G\to G\times G,g\mapsto(g,\tau^{G}(g))$ to obtain by pullbacks
representations of $G_{\tau_{h}}$ that can be used to obtain with Theorem 75
nets of standard subspaces on $M=G/H$.

In Section 9 we return to a geometric topic. We show that, under rather
natural assumptions, the wedge space $\mathcal{W}(M,h)$ of all $G$-translates
of the connected component $W_{M}(h)_{eH}$ of the wedge domain in $M$ carries
the structure of an ordered symmetric space. We conclude this paper with a
short section discussing some open problems and two appendices on some useful
facts on symmetric Lie groups and on distribution vectors of unitary
representations.

Notation:

  * •

${\mathfrak{g}}$, ${\mathfrak{h}}$ and ${\mathfrak{k}}$ will denote Lie
algebras of the Lie groups $G$, $H$, $K$.

  * •

An element $h\in{\mathfrak{g}}$ is called an Euler element if $\mathop{{\rm
ad}}\nolimits h$ is non-zero and diagonalizable with ${\rm Spec}(\mathop{{\rm
ad}}\nolimits h)\subseteq\\{-1,0,1\\}$. The set of Euler elements in
${\mathfrak{g}}$ is denoted by $\mathcal{E}({\mathfrak{g}})$.

  * •

For $h\in\mathcal{E}({\mathfrak{g}})$ we write $\tau_{h}:=e^{\pi i\mathop{{\rm
ad}}\nolimits h}$ for the corresponding involution on ${\mathfrak{g}}$ and
note that $\kappa_{h}:=e^{-\frac{\pi i}{2}\mathop{{\rm ad}}\nolimits h}$ is an
automorphism of ${\mathfrak{g}}_{\mathbb{C}}$ of order $4$ with
$\kappa_{h}^{2}=\tau_{h}$ which implements the “Wick rotation” (partial Cayley
transform).

  * •

For $h\in{\mathfrak{g}}$, $\lambda\in{\mathbb{R}}$, and
${\mathfrak{l}}\subset{\mathfrak{g}}$ an $\mathop{{\rm ad}}\nolimits h$
invariant subspace we write ${\mathfrak{l}}_{\lambda}(h):=\ker(\mathop{{\rm
ad}}\nolimits h|_{{\mathfrak{l}}}-\lambda\mathbf{1})$ for the corresponding
eigenspace in the adjoint representation.

  * •

We write $e\in G$ for the identity element in the Lie group $G$, $G_{e}$ for
its identity component and $Z(G)$ for its center.

  * •

For the left and right translations on the tangent bundle $T(G)$, we write
$g.v$ and $v.g$ for $g\in G,v\in T(G)$, respectively.

  * •

For $x\in{\mathfrak{g}}$, we write $G^{x}:=\\{g\in G\colon\mathop{{\rm
Ad}}\nolimits(g)x=x\\}$ for the stabilizer of $x$ in the adjoint
representation and $G^{x}_{e}:=(G^{x})_{e}$ for its identity component.

  * •

If $\tau^{G}\in\mathop{{\rm Aut}}\nolimits(G)$ is an involution, then we write
$G_{\tau}:=G\rtimes\\{\mathop{{\rm id}}\nolimits_{G},\tau^{G}\\}$,
$g^{\sharp}:=\tau^{G}(g)^{-1}$, and $G^{\sharp}:=\\{g\in G\colon
g^{\sharp}=g\\}$.

  * •

For a Lie subalgebra ${\mathfrak{h}}\subseteq{\mathfrak{g}}$, we write
$\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})=\langle
e^{\mathop{{\rm ad}}\nolimits{\mathfrak{h}}}\rangle\subseteq\mathop{{\rm
Aut}}\nolimits({\mathfrak{g}})$ for the corresponding group of inner
automorphism and put $\mathop{{\rm Inn}}\nolimits({\mathfrak{g}}):=\langle
e^{\mathop{{\rm ad}}\nolimits{\mathfrak{g}}}\rangle$.

  * •

For a (continuous) unitary representation $(U,\mathcal{H})$ of a Lie group $G$
with Lie algebra ${\mathfrak{g}}$, we write $\partial U(x)$ for the skew-
adjoint infinitesimal generator of the unitary one-parameter group $U(\exp
tx)$, so that we have $U(\exp tx)=e^{t\partial U(x)}$ for $t\in{\mathbb{R}}$.
The positive cone of $U$ is the $\mathop{{\rm Ad}}\nolimits(G)$-invariant
closed convex cone

$C_{U}:=\\{x\in{\mathfrak{g}}\colon-i\partial U(x)\geq 0\\}.$ (8)

  * •

For a complex Hilbert space, we write $\mathop{{\rm
AU}}\nolimits(\mathcal{H})$ for the group of unitary and antiunitary
operators.

  * •

For a function $f\colon G\to{\mathbb{C}}$, we write $f^{\vee}(g):=f(g^{-1})$.

##  2 Causal symmetric Lie algebras

In this section we collect some basic facts and definitions concerning causal
symmetric spaces. Our main reference for these spaces is [HÓ97] for the
semisimple situation and [HN93] for the general case.

######  Definition 21.

For a symmetric Lie algebra $({\mathfrak{g}},\tau)$, we write

${\mathfrak{h}}:={\mathfrak{g}}^{\tau}=\\{x\in{\mathfrak{g}}\colon\tau(x)=x\\}\quad\mbox{
and
}\quad{\mathfrak{q}}:={\mathfrak{g}}^{-\tau}=\\{x\in{\mathfrak{g}}\colon\tau(x)=-x\\}.$

If ${\mathfrak{g}}$ is semisimple, then there exists a Cartan involution
$\theta$ commuting with $\tau$, and all these Cartan involutions are mutually
conjugate under $\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$
([KN96, Prop. I.5(iii)]). If $\theta$ is a Cartan involution commuting with
$\tau$, then we write ${\mathfrak{k}}:={\mathfrak{g}}^{\theta}$ and
${\mathfrak{p}}:={\mathfrak{g}}^{-\theta}$ for the $\theta$-eigenspaces. Now
${\mathfrak{g}}$ decomposes as follows into simultaneous
$(\tau,\theta)$-eigenspaces:

${\mathfrak{g}}={\mathfrak{h}}\oplus{\mathfrak{q}}={\mathfrak{k}}\oplus{\mathfrak{p}}={\mathfrak{h}}_{\mathfrak{k}}\oplus{\mathfrak{h}}_{\mathfrak{p}}\oplus{\mathfrak{q}}_{\mathfrak{k}}\oplus{\mathfrak{q}}_{\mathfrak{p}}$
(9)

with

${\mathfrak{h}}_{\mathfrak{k}}:={\mathfrak{h}}\cap{\mathfrak{k}},\quad{\mathfrak{h}}_{\mathfrak{p}}:={\mathfrak{h}}\cap{\mathfrak{p}},\quad{\mathfrak{q}}_{\mathfrak{k}}:={\mathfrak{q}}\cap{\mathfrak{k}},\quad{\mathfrak{q}}_{\mathfrak{p}}:={\mathfrak{q}}\cap{\mathfrak{p}}.$

We call $({\mathfrak{g}}^{c},\tau^{c})$ with
${\mathfrak{g}}^{c}={\mathfrak{h}}+i{\mathfrak{q}}$ and $\tau^{c}(x+iy)=x-iy$
for $x\in{\mathfrak{h}},y\in{\mathfrak{q}}$, the $c$-dual symmetric Lie
algebra.

######  Definition 22.

Let $({\mathfrak{g}},\tau)$ be a symmetric Lie algebra. 222 Here we extend
concepts typically used for irreducible semisimple symmetric Lie algebras to a
more general context. This is motivated by our heavy use of embeddings, which
is most nicely done in a suitable functorial framework.

  * (a)

A triple $({\mathfrak{g}},\tau_{h},h)$ is called a parahermitian symmetric Lie
algebra if $h\in\mathcal{E}({\mathfrak{g}})$ is an Euler element and
$\tau_{h}=e^{\pi i\mathop{{\rm ad}}\nolimits h}$.

  * (b)

A triple $({\mathfrak{g}},\tau,C)$ is called a compactly causal (cc) symmetric
Lie algebra if $C\subseteq{\mathfrak{q}}$ is a pointed generating elliptic
cone invariant under $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}}):=\langle e^{\mathop{{\rm
ad}}\nolimits{\mathfrak{h}}}\rangle$.

  * (c)

A triple $({\mathfrak{g}},\tau,C)$ is called a non-compactly causal (ncc)
symmetric Lie algebra if $C\subseteq{\mathfrak{q}}$ is a pointed generating
hyperbolic cone invariant under $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$, i.e., the interior of $C$
consists of $\mathop{{\rm ad}}\nolimits$-diagonalizable elements.

  * (d)

A triple $({\mathfrak{g}},\tau,C)$ is called a causal symmetric Lie algebra if
it is either a cc or an ncc symmetric Lie algebra.

  * (e)

A modular causal symmetric Lie algebra is a quadruple
$({\mathfrak{g}},\tau,C,h)$, where $({\mathfrak{g}},\tau,C)$ is a causal
symmetric Lie algebra, $h\in{\mathfrak{g}}^{\tau}$ is an Euler element, and
the involution $\tau_{h}=e^{\pi i\mathop{{\rm ad}}\nolimits h}\in\mathop{{\rm
Aut}}\nolimits({\mathfrak{g}})$ satisfies $\tau_{h}(C)=-C$.

Note that the $c$-dual of a cc symmetric Lie algebra $({\mathfrak{g}},\tau,C)$
is ncc and vice versa.

For the classification of $\mathop{{\rm
Inn}}\nolimits({\mathfrak{h}})$-invariant elliptic cones
$C\subseteq{\mathfrak{q}}$, i.e., causal structures on the symmetric Lie
algebra $({\mathfrak{g}},\tau)$, see Sections 4.4 and 4.5 in [HÓ97] and [KN96,
Thm. X.3]. Note that the classification in the cc case and ncc case are the
same via $c$-duality.

######  Remark 23.

Suppose that $C\subseteq{\mathfrak{q}}$ is a closed convex $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$-invariant cone with
$-\tau_{h}(C)=C$. Since $\mathop{{\rm ad}}\nolimits h$ preserves
${\mathfrak{q}}$, the space ${\mathfrak{q}}$ decomposes as

${\mathfrak{q}}={\mathfrak{q}}_{-1}(h)\oplus{\mathfrak{q}}_{0}(h)\oplus{\mathfrak{q}}_{1}(h).$

As $C$ is invariant under $e^{{\mathbb{R}}\mathop{{\rm ad}}\nolimits h}$ and
$-\tau_{h}$, we see that $x=x_{1}+x_{0}+x_{-1}\in C$ with
$x_{j}\in{\mathfrak{q}}_{j}(h)$ implies

$e^{t\mathop{{\rm ad}}\nolimits h}x=e^{t}x_{1}+x_{0}+e^{-t}x_{-1}\in C.$

This shows that $x_{\pm 1}=\lim_{t\to\infty}e^{-t}e^{\pm t\mathop{{\rm
ad}}\nolimits h}x\in C$. Therefore the two closed convex cones

$C_{\pm}:=\pm C\cap{\mathfrak{q}}_{\pm 1}(h)$ (10)

satisfy

$p_{{\mathfrak{q}}^{-\tau_{h}}}(C)=C^{-\tau_{h}}=C_{+}-C_{-}.$ (11)

Furthermore, the automorphism $\kappa_{h}=e^{-\frac{\pi i}{2}\mathop{{\rm
ad}}\nolimits h}\in\mathop{{\rm Aut}}\nolimits({\mathfrak{g}}_{\mathbb{C}})$
takes the form

$\kappa_{h}(z_{-1}+z_{0}+z_{1})=iz_{-1}+z_{0}-iz_{1}$

with respect to the $3$-grading defined by $\mathop{{\rm ad}}\nolimits h$ and
we have

$i\kappa_{h}(C^{-\tau_{h}})=C_{+}+C_{-}=:C^{c}$ (12)

###  2.1 Embedding into group type

In this section we discuss the embedding of compactly causal triples
$({\mathfrak{g}},\tau,C)$ into a natural causal triple of group type
$(({\mathfrak{g}}\oplus{\mathfrak{g}})_{C},\tau_{\rm flip},C_{\mathfrak{g}})$.
This embedding will play a crucial role in the following.

We call a compactly causal symmetric Lie algebra $({\mathfrak{g}},\tau,C)$
extendable if there exists a pointed closed convex invariant cone
$C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ satisfying

$C_{\mathfrak{g}}\cap{\mathfrak{q}}=C\quad\mbox{ and
}\quad-\tau(C_{\mathfrak{g}})=C_{\mathfrak{g}}.$

We then call $({\mathfrak{g}},\tau,C_{\mathfrak{g}})$ an extension of
$({\mathfrak{g}},\tau,C$). The following theorem ([KN96, Thm. X.7], [HÓ97,
Thm. 4.5.8], [Ó91, Thm. 7.8]) shows that, for reductive Lie algebras,
extensions always exist, even with generating cones $C_{\mathfrak{g}}$.

######  Theorem 24.

(Extension Theorem) Let $({\mathfrak{g}},\tau,C)$ be a reductive compactly
causal symmetric Lie algebra for which ${\mathfrak{h}}$ contains no non-zero
ideals of ${\mathfrak{g}}$. Then there exists a pointed generating invariant
closed convex cone $C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ with

$-\tau(C_{\mathfrak{g}})=C_{\mathfrak{g}}\quad\mbox{ and }\quad
C_{\mathfrak{g}}\cap{\mathfrak{q}}=C.$ (13)

Suppose that $({\mathfrak{g}},\tau,C_{\mathfrak{g}})$ is an extension of the
compactly causal symmetric Lie algebra $({\mathfrak{g}},\tau,C)$. If
$C_{\mathfrak{g}}$ is not generating, then

${\mathfrak{g}}_{C}:=C_{\mathfrak{g}}-C_{\mathfrak{g}}\trianglelefteq{\mathfrak{g}}$

is a $\tau$-invariant ideal containing $C-C={\mathfrak{q}}$, so that
${\mathfrak{g}}_{C}={\mathfrak{h}}_{C}\oplus{\mathfrak{q}}$ for
${\mathfrak{h}}_{C}={\mathfrak{h}}\cap{\mathfrak{g}}_{C}$. Then

$({\mathfrak{g}}\oplus{\mathfrak{g}})_{C}=\\{(x,y)\in{\mathfrak{g}}\oplus{\mathfrak{g}}\colon
x-y\in{\mathfrak{g}}_{C}\\}\quad\mbox{ with }\quad\tau_{e}(x,y)=(y,x)$

is a symmetric Lie algebra with

$({\mathfrak{g}}\oplus{\mathfrak{g}})_{C}^{\tau_{e}}=\\{(x,x)\colon
x\in{\mathfrak{g}}\\}\cong{\mathfrak{g}}\quad\text{and}\quad({\mathfrak{g}}\oplus{\mathfrak{g}})_{C}^{-\tau_{e}}=\\{(x,-x)\colon
x\in{\mathfrak{g}}_{C}\\}\cong{\mathfrak{g}}_{C}.$

We thus obtain an embedding of causal symmetric Lie algebras

$({\mathfrak{g}},\tau,C)\hookrightarrow(({\mathfrak{g}}\oplus{\mathfrak{g}})_{C},\tau_{e},C_{\mathfrak{g}}),\quad
x\mapsto(x,\tau x)\quad\mbox{ for }\quad x\in{\mathfrak{g}}.$ (14)

This embedding is also compatible with the corresponding Euler elements. Any
Euler element $h\in{\mathfrak{h}}={\mathfrak{g}}^{\tau}$ is mapped to the
Euler element $(h,h)\in({\mathfrak{g}}\oplus{\mathfrak{g}})_{C}^{\tau_{e}}$.

On the global level, the embedding (14) corresponds to the quadratic
representation of the symmetric space $G/H$ in the group $G_{C}$ with Lie
algebra ${\mathfrak{g}}_{C}$ (see (24) below). We shall use such embeddings to
derive properties of compactly causal symmetric spaces from those of spaces of
group type.

Since $\tau$ commutes with $\mathop{{\rm ad}}\nolimits h$ on ${\mathfrak{g}}$,
it preserves all eigenspaces ${\mathfrak{g}}_{j}(h)$, $j=-1,0,1$. We conclude
from (13) that the cones

$C_{{\mathfrak{g}},\pm}:=\pm C_{\mathfrak{g}}\cap{\mathfrak{g}}_{\pm 1}(h)$

satisfy $-\tau(C_{{\mathfrak{g}},\pm})=C_{{\mathfrak{g}},\pm}$, which shows in
particular that the cone

$C_{\mathfrak{g}}^{c}:=C_{{\mathfrak{g}},+}+C_{{\mathfrak{g}},-}$ (15)

is $-\tau$-invariant with

$(C_{\mathfrak{g}}^{c})^{-\tau}=(C_{{\mathfrak{g}},+}+C_{{\mathfrak{g}},-})^{-\tau}=C_{+}+C_{-}=C^{c}.$
(16)

######  Remark 25.

In [KN96, Ex. X.4] one finds examples of compactly causal symmetric Lie
algebras $({\mathfrak{g}},\tau,C)$ which are not extendable, but [KN96, Thm.
X.7] describes sufficient conditions for general Lie algebras that cover all
cases where $C_{\mathfrak{g}}$ is generating.

We also note that, if ${\mathfrak{g}}$ is not reductive and
$C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ is a pointed generating invariant
cone, then $C_{\mathfrak{g}}\cap{\mathfrak{z}}({\mathfrak{g}})\not=\\{0\\}$.
As $\tau_{h}$ fixes the center pointwise, this condition is incompatible with
$-\tau_{h}(C_{\mathfrak{g}})=C_{\mathfrak{g}}$. Therefore it would be too much
to hope for extensions of modular compactly causal symmetric Lie algebras
$({\mathfrak{g}},\tau,C,h)$ for which $C_{\mathfrak{g}}$ is also generating.
We refer to [Oeh20, Oeh20b] for more details on the structure of non-reductive
modular compactly causal symmetric Lie algebras $({\mathfrak{g}},\tau,C,h)$
and corresponding classification results.

######  Example 26.

(A non-reductive example; cf. [Ne19, Ex. 3.7]) We consider the Lie algebra

${\mathfrak{g}}={\mathfrak{hcsp}}(V,\omega):=\mathfrak{heis}(V,\omega)\rtimes{\mathfrak{csp}}(V,\omega),$

where $(V,\omega)$ is a symplectic vector space,
$\mathfrak{heis}(V,\omega)={\mathbb{R}}\oplus V$ is the corresponding
Heisenberg algebra with the bracket
$[(z,v),(z^{\prime},v^{\prime})]=(\omega(v,v^{\prime}),0)$, and

${\mathfrak{csp}}(V,\omega):=\mathfrak{sp}(V,\omega)\oplus{\mathbb{R}}\mathop{{\rm
id}}\nolimits_{V}$

is the conformal symplectic Lie algebra of $(V,\omega)$.

The hyperplane ideal
${\mathfrak{j}}:=\mathfrak{heis}(V,\omega)\rtimes\mathfrak{sp}(V,\omega)$ (the
Jacobi algebra) can be identified by the linear isomorphism

$\varphi\colon{\mathfrak{j}}\to\mathop{{\rm Pol}}\nolimits_{\leq
2}(V),\qquad\varphi(z,v,x)(\xi):=z+\omega(v,\xi)+\frac{1}{2}\omega(x\xi,\xi),\quad\xi\in
V$

with the Lie algebra of real polynomials $\mathop{{\rm Pol}}\nolimits_{\leq
2}(V)$ of degree $\leq 2$ on $V$, endowed with the Poisson bracket ([Ne00,
Prop. A.IV.15]). The set

$C:=\\{f\in\mathop{{\rm Pol}}\nolimits_{\leq 2}(V)\colon f\geq 0\\}$

is a pointed invariant cone in ${\mathfrak{g}}$ generating the ideal
${\mathfrak{g}}_{C}={\mathfrak{j}}$. The element $h_{0}:=\mathop{{\rm
id}}\nolimits_{V}$ defines a derivation on ${\mathfrak{j}}$ by $(\mathop{{\rm
ad}}\nolimits h_{0})(z,v,x)=(2z,v,0)$ for $z\in{\mathbb{R}},v\in
V,x\in\mathfrak{sp}(V,\omega)$. Any involution $\tau_{V}$ on $V$ satisfying
$\tau_{V}^{*}\omega=-\omega$ defines by

$\widetilde{\tau}_{V}(z,v,x):=(-z,\tau_{V}(v),\tau_{V}x\tau_{V})$ (17)

an involution on ${\mathfrak{g}}$ with $\widetilde{\tau}_{V}(h_{0})=h_{0}$,
and $-\widetilde{\tau}_{V}(C)=C$ follows from

$\varphi(\widetilde{\tau}_{V}(z,v,x))=-\varphi(z,v,x)\circ\tau_{V}.$

Considering $\tau_{V}$ as an element of $\mathfrak{sp}(V,\omega)$, we obtain
an Euler element

$h:={\textstyle{\frac{1}{2}}}(\mathop{{\rm
id}}\nolimits_{V}+\tau_{V})\in{\mathfrak{csp}}(V,\omega).$

Writing $V=V_{1}\oplus V_{-1}$ for the $\tau_{V}$-eigenspace decomposition,
the $\mathop{{\rm ad}}\nolimits h$-eigenspaces in ${\mathfrak{g}}$ are

${\mathfrak{g}}_{-1}=0\oplus
0\oplus\mathfrak{sp}(V,\omega)_{-1},\quad{\mathfrak{g}}_{0}=0\oplus
V_{-1}\oplus\mathfrak{sp}(V,\omega)_{0}\cong
V_{-1}\rtimes\mathop{{\mathfrak{gl}}}\nolimits(V_{-1}),\quad{\mathfrak{g}}_{1}={\mathbb{R}}\oplus
V_{1}\oplus\mathfrak{sp}(V,\omega)_{1}.$

The eigenspace ${\mathfrak{g}}_{1}$ can be identified with the space
$\mathop{{\rm Pol}}\nolimits_{\leq 2}(V_{-1})$ of polynomials of degree $\leq
2$ on $V_{-1}$ and

$C_{+}=C\cap{\mathfrak{g}}_{1}=\\{f\in\mathop{{\rm Pol}}\nolimits_{\leq
2}(V_{-1})\colon f\geq 0\\}.$

This cone is invariant under the natural action of the affine group

$G^{0}\cong\mathop{{\rm Aff}}\nolimits(V_{-1})_{e}\cong
V_{-1}\rtimes\mathop{{\rm GL}}\nolimits(V_{-1})_{e}$

whose Lie algebra is ${\mathfrak{g}}_{0}$. We also note that

${\mathfrak{g}}_{-1}=\mathfrak{sp}(V,\omega)_{-1}\cong\mathop{{\rm
Pol}}\nolimits_{2}(V_{1})\quad\mbox{ and }\quad
C_{-}=-C\cap{\mathfrak{g}}_{-1}=\\{f\in\mathop{{\rm
Pol}}\nolimits_{2}(V_{1})\colon f\leq 0\\}.$

Finally, we observe that $\tau_{h}=e^{\pi i\mathop{{\rm ad}}\nolimits
h}=(-\tau_{V})\,\widetilde{}$ implies in particular $-\tau_{h}(C)=C$, so that

$({\mathfrak{g}},\tau_{h},C^{c},h)\quad\mbox{ with }\quad
C^{c}=C_{+}-C_{-}=C^{-\tau_{h}}$

is a modular causal symmetric Lie algebra for which
$({\mathfrak{g}},\tau_{h},C^{c})$ is compactly causal and extendable to
$({\mathfrak{g}},\tau_{h},C)$.

###  2.2 Hermitian simple Lie algebras

A real simple Lie algebra ${\mathfrak{g}}$ contains a pointed generating
invariant cone $C_{\mathfrak{g}}$ if and only if ${\mathfrak{g}}$ is hermitian
([Vi80]). If this is the case, then there exist two such cones
$C_{\mathfrak{g}}^{\rm min}\subseteq C_{\mathfrak{g}}^{\rm max}$ with the
property that any other pointed generating invariant cone $C_{\mathfrak{g}}$
satisfies

$C_{\mathfrak{g}}^{\rm min}\subseteq C_{\mathfrak{g}}\subseteq
C_{\mathfrak{g}}^{\rm max}\quad\mbox{ or }\quad C_{\mathfrak{g}}^{\rm
min}\subseteq-C_{\mathfrak{g}}\subseteq C_{\mathfrak{g}}^{\rm max}.$ (18)

We start by collecting some information on the relevant class of simple Lie
algebras.

######  Proposition 27.

Let ${\mathfrak{g}}$ be a simple hermitian Lie algebra. Then ${\mathfrak{g}}$
contains an Euler element if and only if ${\mathfrak{g}}$ is of tube type. If
this is the case, then the following assertions hold:

  * (a)

$\mathop{{\rm Inn}}\nolimits({\mathfrak{g}})$ acts transitively on the set
$\mathcal{E}({\mathfrak{g}})$ of Euler elements.

  * (b)

If $({\mathfrak{g}},\tau,C)$ is compactly causal , then
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}\not=\emptyset$.

  * (c)

For every Euler element $h\in{\mathfrak{g}}$, the cone $C^{\rm
max}_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ satisfies $\tau_{h}(C^{\rm
max}_{\mathfrak{g}})=-C^{\rm max}_{\mathfrak{g}}$. In particular
$({\mathfrak{g}},\tau_{h})$ is compactly causal.

  * (d)

For any pointed generating invariant cone $C_{\mathfrak{g}}\subseteq C^{\rm
max}_{\mathfrak{g}}$, we have
$C_{\mathfrak{g}}\cap{\mathfrak{g}}^{-\tau_{h}}=C^{\rm
max}_{\mathfrak{g}}\cap{\mathfrak{g}}^{-\tau_{h}}$.

###### Proof.

(a) follows from [MN21, Prop. 3.11].

(b) Assume that $({\mathfrak{g}},\tau)$ is compactly causal. In [NÓ21b, App.
D] we show that $\tau$ preserves a $3$-grading of ${\mathfrak{g}}$, and this
means that the corresponding Euler element $h$ is fixed by $\tau$, i.e.,
$h\in{\mathfrak{h}}$.

(c) Since $\tau_{h}(C_{\mathfrak{g}}^{\rm max})$ is one of the two maximal
invariant cones $\pm C_{\mathfrak{g}}^{\rm max}$, (c) follows from [Oeh20,
Lemma 2.28].

(d) Clearly, $C^{\rm
max}_{\mathfrak{g}}\cap{\mathfrak{g}}^{-\tau_{h}}\supseteq
C_{\mathfrak{g}}\cap{\mathfrak{g}}^{-\tau_{h}}$ by (18). Writing
${\mathfrak{g}}^{-\tau_{h}}={\mathfrak{g}}_{1}(h)\oplus{\mathfrak{g}}_{-1}(h)$
and using the invariance under $e^{{\mathbb{R}}\mathop{{\rm ad}}\nolimits h}$,
it follows as in Remark 23 that

$C_{\mathfrak{g}}\cap{\mathfrak{g}}^{-\tau_{h}}=C_{{\mathfrak{g}},+}+-C_{{\mathfrak{g}},-},\quad\mbox{
where }\quad C_{{\mathfrak{g}},\pm}=\pm
C_{\mathfrak{g}}\cap{\mathfrak{g}}_{\pm 1}(h).$ (19)

The cones $C_{{\mathfrak{g}},\pm}$ and $C^{\rm max}_{{\mathfrak{g}},\pm}$
coincide by [Oeh20, Lemma 2.28], so that (d) follows from (19). ∎

######  Proposition 28.

Let ${\mathfrak{g}}$ be simple and $h\in\mathcal{E}({\mathfrak{g}})$. Then
$({\mathfrak{g}},\tau_{h})$ is causal if and only if it is of Cayley type,
i.e., ${\mathfrak{g}}$ is simple hermitian of tube type.

In this case any causal structure represented by a cone
$C\subseteq{\mathfrak{q}}={\mathfrak{g}}^{-\tau_{h}}$ leads to two cones
$C=C_{+}-C_{-}$ and $C^{c}=C_{+}+C_{-}$. One is hyperbolic and the other is
elliptic.

###### Proof.

We choose a Cartan involution $\theta$ with $\theta(h)=-h$.

If $({\mathfrak{g}},\tau_{h})$ is causal and $C\subseteq{\mathfrak{q}}$ is a
pointed generating $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$-invariant cone, then $C^{0}$
contains fixed points of the compact group $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}}_{\mathfrak{k}})$, so that
${\mathfrak{q}}^{{\mathfrak{h}}_{\mathfrak{k}}}\not=\\{0\\}$. Hence the
conclusion follows from [HÓ97, Thm. 1.3.11].

If, conversely, ${\mathfrak{g}}$ is simple hermitian and
$h\in\mathcal{E}({\mathfrak{g}})$, then Proposition 27(c) implies that
$({\mathfrak{g}},\tau_{h})$ is causal. ∎

###  2.3 Reductive Lie algebras

The last section was devoted to the semisimple case. In this section we
discuss the reductive case. We assume that ${\mathfrak{h}}$ does not contain
any non-zero ideal. This is no restriction because the integral subgroup
corresponding to such an ideal would act trivially on $G/H$ and thus can be
factorized out. This in particular implies that
${\mathfrak{z}}({\mathfrak{g}})\cap{\mathfrak{h}}=\\{0\\}$ because
${\mathfrak{z}}({\mathfrak{g}})\cap{\mathfrak{h}}$ is an ideal of
${\mathfrak{g}}$. Thus $H\cap Z(G)$ is discrete.

The following proposition extends Proposition 27(c) to the larger class of
reductive Lie algebras.

######  Proposition 29.

If $({\mathfrak{g}},\tau)$ is a reductive compactly causal symmetric Lie
algebra containing an Euler element and ${\mathfrak{h}}$ contains no non-zero
ideal, then $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}\not=\emptyset$.

###### Proof.

If $h\in{\mathfrak{g}}$ is an Euler element and $h=h_{0}+h_{1}$ with
$h_{0}\in{\mathfrak{z}}({\mathfrak{g}})$ and
$h_{1}\in[{\mathfrak{g}},{\mathfrak{g}}]$, then
$h_{1}\in\mathcal{E}([{\mathfrak{g}},{\mathfrak{g}}])$. As
$({\mathfrak{g}},\tau)$ is compactly causal and ${\mathfrak{h}}$ contains no
non-zero ideal, all simple ideals of ${\mathfrak{g}}$ are either hermitian or
compact. In fact, the Extension Theorem 24 implies the existence of a pointed
generating invariant cone in ${\mathfrak{g}}$, hence ${\mathfrak{g}}$ is
quasihermitian by [Ne00, Cor. VII.3.9], i.e.,
${\mathfrak{k}}={\mathfrak{z}}_{\mathfrak{g}}({\mathfrak{z}}({\mathfrak{k}}))$
holds for a maximal compactly embedded subalgebra. Since ${\mathfrak{k}}$ is
adapted to the decomposition into simple ideals ${\mathfrak{g}}_{j}$, the
ideal ${\mathfrak{g}}_{j}$ is compact if
${\mathfrak{z}}({\mathfrak{k}}_{j})=\\{0\\}$ and it is hermitian if
${\mathfrak{z}}({\mathfrak{k}}_{j})\not=\\{0\\}$ (by definition).

As $\mathop{{\rm ad}}\nolimits h_{1}$ is diagonalizable, $h_{1}$ is contained
in the sum of all hermitian simple ideals. Any such ideal is either
$\tau$-invariant, or corresponds to an irreducible summand of group type. For
group type Lie algebras $({\mathfrak{g}}\oplus{\mathfrak{g}},\tau_{\rm flip})$
(cf. (6)), the assertion is trivial because
$h=(h_{1},h_{2})\in\mathcal{E}({\mathfrak{g}}\oplus{\mathfrak{g}})$ implies
$(h_{j},h_{j})\in{\mathfrak{h}}$ for $j=1,2$, and at least one of these two
elements is non-central, hence an Euler element. For the $\tau$-invariant
ideals, the assertion follows from Proposition 27(b). ∎

Let $({\mathfrak{g}},\tau,C)$ be a reductive compactly causal symmetric Lie
algebra with ${\mathfrak{z}}({\mathfrak{g}})\subseteq{\mathfrak{q}}$ (which
follows from ${\mathfrak{h}}\cap{\mathfrak{z}}({\mathfrak{g}})=\\{0\\}$). Then
${\mathfrak{g}}={\mathfrak{z}}({\mathfrak{g}})\oplus\bigoplus_{j=1}^{n}{\mathfrak{g}}_{j}$,
where the ${\mathfrak{g}}_{j}$ are simple ideals that are either compact or
simple hermitian. If ${\mathfrak{g}}_{j}$ is compact, we put
$C_{{\mathfrak{g}}_{j}}^{\rm min}=\\{0\\}$ and $C_{{\mathfrak{g}}_{j}}^{\rm
max}={\mathfrak{g}}_{j}$. With (18) this leads to invariant cones

$C_{\mathfrak{g}}^{\rm min}:=\bigoplus_{j=1}^{n}C_{{\mathfrak{g}}_{j}}^{\rm
min}\subseteq C_{\mathfrak{g}}^{\rm
max}:={\mathfrak{z}}({\mathfrak{g}})\oplus\sum_{j=1}^{n}C_{{\mathfrak{g}}_{j}}^{\rm
max}.$ (20)

Every Euler element $h\in{\mathfrak{g}}$ decomposes as
$h=h_{0}+\sum_{j=1}^{n}h_{j}$ with $h_{0}\in{\mathfrak{z}}({\mathfrak{g}})$
and $h_{j}=0$ if ${\mathfrak{g}}_{j}$ is compact, and if $h_{j}\not=0$, then
it is an Euler element in ${\mathfrak{g}}_{j}$. From Proposition 27 we now
obtain:

######  Corollary 210.

Let ${\mathfrak{g}}$ be a reductive quasihermitian Lie algebra, i.e., all
simple ideals are compact or hermitian, and
$h\in\mathcal{E}({\mathfrak{g}})$.Then the following assertions hold:

  * (a)

If ${\mathfrak{g}}^{\tau_{h}}=\ker(\mathop{{\rm ad}}\nolimits h)$ contains no
simple hermitian ideal, then $\tau_{h}(C^{\rm max}_{\mathfrak{g}})=-C^{\rm
max}_{\mathfrak{g}}$.

  * (b)

For any pointed generating invariant cone $C_{\mathfrak{g}}\subseteq C^{\rm
max}_{\mathfrak{g}}$, we have
$C_{\mathfrak{g}}\cap{\mathfrak{g}}^{-\tau_{h}}=C^{\rm
max}_{\mathfrak{g}}\cap{\mathfrak{g}}^{-\tau_{h}}$.

###### Proof.

(a) Our assumption on $h$ means that $h_{j}\in\mathcal{E}({\mathfrak{g}}_{j})$
for every simple hermitian ideal ${\mathfrak{g}}_{j}$ because $h_{j}\not=0$.
Hence (a) follows immediately from (20) and the corresponding assertion for
hermitian simple Lie algebras (Proposition 27).

(b) As ${\mathfrak{g}}^{\tau_{h}}$ contains the center and all compact ideals,
${\mathfrak{g}}^{-\tau_{h}}$ is contained in the sum

${\mathfrak{g}}_{\rm
herm}:={\mathfrak{g}}_{j_{1}}\oplus\cdots\oplus{\mathfrak{g}}_{j_{k}}$

of all hermitian simple ideals. Further, the invariance of all ideals under
$\tau_{h}$ shows that

${\mathfrak{g}}^{-\tau_{h}}={\mathfrak{g}}_{j_{1}}^{-\tau_{h}}\oplus\cdots\oplus{\mathfrak{g}}_{j_{k}}^{-\tau_{h}}.$

The classification of invariant cones (see [Ne00, Thm. VIII.3.21] for more
details) implies that $C_{{\mathfrak{g}},{\rm
herm}}:=C_{\mathfrak{g}}\cap{\mathfrak{g}}_{\rm herm}$ is a pointed generating
invariant cone in ${\mathfrak{g}}_{\rm herm}$ with

$\sum_{\ell=1}^{k}C_{{\mathfrak{g}}_{j_{\ell}}}^{\rm min}\subseteq
C_{{\mathfrak{g}},{\rm herm}}\subseteq C_{{\mathfrak{g}},{\rm herm}}^{\rm
max}=\sum_{\ell=1}^{k}C_{{\mathfrak{g}}_{j_{\ell}}}^{\rm max}.$

Therefore Proposition 27(d) leads to

$C_{\mathfrak{g}}^{\rm max}\cap{\mathfrak{g}}^{-\tau_{h}}\supseteq
C_{\mathfrak{g}}^{-\tau_{h}}\supseteq\sum_{\ell=1}^{k}C_{{\mathfrak{g}}_{j_{\ell}}}^{\rm
min}\cap{\mathfrak{g}}_{j_{\ell}}^{-\tau_{h}}=\sum_{\ell=1}^{k}C_{{\mathfrak{g}}_{j_{\ell}}}^{\rm
max}\cap{\mathfrak{g}}_{j_{\ell}}^{-\tau_{h}}=C_{\mathfrak{g}}^{\rm
max}\cap{\mathfrak{g}}^{-\tau_{h}}.\qed$

###  2.4 The global setting for symmetric spaces

In the following $G$ denotes a connected real Lie group and $\eta_{G}\colon
G\to G_{\mathbb{C}}$ its universal complexification. We assume that
$\ker(\eta_{G})$ is discrete, which is always the case if ${\mathfrak{g}}$ is
semisimple (because $\ker\eta_{G}\subseteq Z(G)$), if $G$ is simply connected,
or if $G$ is a matrix group. The groups $G$ and $G_{\mathbb{C}}$ carry natural
structures of Loos symmetric spaces ([Lo69]), defined by

$g\bullet h:=s_{g}(h):=gh^{-1}g,$ (21)

Then automorphisms, antiautomorphisms and left and right translations define
automorphisms of the symmetric space $(G,\bullet)$. In particular, we obtain a
transitive action of the product group $G\times G$ on $G$ by automorphisms of
$(G,\bullet)$ via

$(g_{1},g_{2}).g:=g_{1}gg_{2}^{-1}.$

Then $T_{e}(G)\cong\\{(x,-x)\colon
x\in{\mathfrak{g}}\\}\subseteq{\mathfrak{g}}\oplus{\mathfrak{g}}$, and the
exponential function of the pointed symmetric space $(G,\bullet,e)$ is given
by

$\mathop{{\rm Exp}}\nolimits_{e}\colon T_{e}(G)\to
G,\quad(x,-x)\mapsto\exp(2x)=(\exp x,\exp-x).e.$ (22)

If $\tau^{G}$ is an involutive automorphism of $G$ and $H\subseteq
G^{\tau^{G}}$ an open subgroup, then

$M:=G/H,\qquad g_{1}H\bullet g_{2}H:=g_{1}\tau^{G}(g_{2})^{-1}g_{1}H$ (23)

is the corresponding symmetric space. Its quadratic representation is the map

$Q\colon M\to G,\quad Q(gH):=gg^{\sharp},\qquad g^{\sharp}:=\tau^{G}(g)^{-1}.$
(24)

It is a covering of the identity component $G^{\sharp}_{e}\cong
G/G^{\tau^{G}}$ of the symmetric subspace

$G^{\sharp}:=\\{g\in G\colon g^{\sharp}=g\\}\subseteq G$

(see Lemma A1 in Appendix A).

We write the natural action of $G$ on the tangent bundle $T(M)$ by
$(g,v)\mapsto g.v$ and identify the tangent space $T_{eH}(M)$ with
${\mathfrak{q}}$. If $({\mathfrak{g}},\tau,C)$ is a causal symmetric Lie
algebra for which $\mathop{{\rm Ad}}\nolimits(H)C=C$, then

$V_{+}(gH)=g.C^{0}\subseteq T_{gH}(M)$

defines on $M=G/H$ a $G$-invariant cone field, i.e., a causal structure.

  * •

We call the pair $(M,C)$ a causal symmetric space. Note that $H$ need not be
connected, so that the requirement $\mathop{{\rm Ad}}\nolimits(H)C=C$ is
stronger than the invariance under $\mathop{{\rm
Ad}}\nolimits(H_{e})=\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$.

  * •

We call $(M,C)$ compactly causal if $({\mathfrak{g}},\tau,C)$ is compactly
causal.

######  Example 211.

The following example shows that the causality of $G/H$ can depends on
$\pi_{0}(H)$. For that let $G=\mathop{{\rm Ad}}\nolimits(\mathop{{\rm
SL}}\nolimits_{2}({\mathbb{R}}))\cong\mathop{{\rm
SO}}\nolimits_{2,1}({\mathbb{R}})_{e}$. Let
$h=\frac{1}{2}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\in\mathfrak{sl}(2,{\mathbb{R}})$. Then $h$ is an Euler
element and the involution $\tau_{h}$ is given by

$\tau_{h}\begin{pmatrix}x&y\\\ z&-x\end{pmatrix}=\begin{pmatrix}x&-y\\\
-z&-x\end{pmatrix}.$

Hence

${\mathfrak{h}}={\mathbb{R}}h\quad\text{and}\quad{\mathfrak{q}}=\left\\{\begin{pmatrix}0&y\\\
z&0\end{pmatrix}\colon
y,z\in{\mathbb{R}}\right\\}\cong{\mathbb{R}}x_{+}+{\mathbb{R}}x_{-}\quad\mbox{
with }\quad x_{+}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix},x_{-}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}.$

We also have ${\mathfrak{g}}_{+1}(h)={\mathbb{R}}x_{+}$,
${\mathfrak{g}}_{-1}(h)={\mathbb{R}}x_{-}$ and
${\mathfrak{g}}_{0}(h)={\mathbb{R}}h$. The map $\theta(x)=-x^{\top}$ is a
Cartan involution commuting with $\tau_{h}$. Let

$C_{+}:=[0,\infty)x_{+}\subset{\mathfrak{g}}_{1}\quad\mbox{ and }\quad
C_{-}:=[0,\infty)x_{-}\subset{\mathfrak{g}}_{-}.$

Then $({\mathfrak{g}},\tau_{h},C_{+}-C_{-})$ is compactly causal and
$({\mathfrak{g}},\tau_{h},C^{c}=C_{+}+C_{-})$ is non-compactly causal. Let
$z=x_{+}-x_{-}\in{\mathfrak{k}}=\mathop{{\mathfrak{so}}}\nolimits_{2}({\mathbb{R}})$.
Then $\theta=e^{\pi\mathop{{\rm ad}}\nolimits z}\in G^{\tau_{h}}=:H$ as
$\theta$ commutes with $\tau_{h}$, but $\theta(C^{c})=-C^{c}$. As $H=(H\cap
K)H_{e}=\\{\mathbf{1},\theta\\}H_{e}$ it follows that $C$ is $H$-invariant but
$\mathop{{\rm Ad}}\nolimits(H)C^{c}\supseteq-C^{c}$, so that $C^{c}$ is not.
It is only invariant under the connected group $H_{e}$.

In general, if ${\mathfrak{g}}$ is simple hermitian of tube type with Euler
element $h$ and involution $\tau=\tau_{h}$, then if $C=C_{+}-C_{-}$ defines a
compactly causal structure, then the cone $C^{c}=C_{+}+C_{-}$ defines a non-
compactly causal structure on $({\mathfrak{g}},\tau_{h})$. As above we see
that $C^{c}$ is invariant under $\mathop{{\rm Ad}}\nolimits(G)^{\tau_{h}}_{e}$
but not under $\mathop{{\rm Ad}}\nolimits(G)^{\tau_{h}}$. On the other hand
$C$ is invariant under the full group $\mathop{{\rm
Ad}}\nolimits(G)^{\tau_{h}}$.

If $({\mathfrak{g}},\tau,C,h)$ is a modular compactly causal symmetric Lie
algebra, then we further assume that the involution $\tau_{h}\in\mathop{{\rm
Aut}}\nolimits({\mathfrak{g}})$ integrates to an involution $\tau_{h}^{G}$ of
$G$ satisfying

$\tau_{h}^{G}(H)=H,$ (25)

so that it induces an involution $\tau_{h}^{M}$ on $M$.

######  Remark 212.

(Integrability of $\tau_{h}$) If $G$ is a connected Lie group with Lie algebra
${\mathfrak{g}}$ and $h\in\mathcal{E}({\mathfrak{g}})$ an Euler element, then
$\tau_{h}$ does not always integrate to an automorphism of $G$.

Here is an example where ${\mathfrak{g}}$ is simple. Let $G:=\mathop{{\rm
SU}}\nolimits_{p,p}({\mathbb{C}})$ and $K:={\rm S}(\mathop{\rm
U{}}\nolimits_{p}({\mathbb{C}})\times\mathop{\rm
U{}}\nolimits_{p}({\mathbb{C}}))$ be the canonical maximal compact subgroup.
For the Euler element $h:=\frac{1}{2}\begin{pmatrix}0&\mathbf{1}_{p}\\\
\mathbf{1}_{p}&0\end{pmatrix}$, the corresponding involution $\tau_{h}$ acts
on ${\mathfrak{g}}$ by

$\tau_{h}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}d&c\\\
b&a\end{pmatrix},$

so that we have on the group level $\tau_{h}^{G}(k_{1},k_{2})=(k_{2},k_{1})$
on $K\subseteq\mathop{\rm U{}}\nolimits_{p}({\mathbb{C}})\times\mathop{\rm
U{}}\nolimits_{p}({\mathbb{C}})$. On

$Z(\widetilde{G})=\\{(n,z_{1},z_{2})\in{\mathbb{Z}}\times C_{p}\times
C_{p}\colon e^{\frac{2\pi in}{p}}z_{1}=z_{2}\\}\cong{\mathbb{Z}}\times C_{p}$

(see [Ti67, p. 28]), this leads to the non-trivial involution on
${\mathbb{Z}}\times C_{p}$, given by

$\tau_{h}^{\widetilde{G}}(n,z_{1})=(-n,\zeta_{p}^{n}z_{1}),\quad\mbox{ where
}\quad\zeta_{p}=e^{\frac{2\pi i}{p}}.$

It fixes the subgroup $\\{0\\}\times C_{p}$ pointwise and maps
$(-2,\zeta_{p})$ to its inverse. In particular, the cyclic subgroup
$\Gamma=\langle(1,1)\rangle$ is not invariant, so that $\tau_{h}$ does not
integrate to the group $\widetilde{G}/\Gamma$.

######  Remark 213.

(Symmetric space structures) Let $G$ be a connected Lie group and $H\subseteq
G$ be a closed subgroup with $\mathop{\bf L{}}\nolimits(H)={\mathfrak{h}}$,
where $({\mathfrak{g}},\tau)$ is a symmetric Lie algebra. Let $N\subseteq H$
be the largest normal subgroup of $G$, contained in $H$, i.e., the kernel of
the $G$-action on $M$. We put

$G_{1}:=G/N\quad\mbox{ and }\quad H_{1}:=H/N.$

(a) $M=G/H\cong G_{1}/H_{1}$ carries a symmetric space structure if and only
if $\tau$ integrates to an involution $\tau^{G_{1}}$ on $G_{1}$ leaving
$H_{1}$ invariant. Then $H_{1}\subseteq G_{1}^{\tau^{G_{1}}}$ is an open
subgroup because it preserves the base point and the corresponding involution
$\tau^{M}$ on $M$ is determined by its differential in the base point. If
$\tau$ integrates to an involution $\tau^{G}$ on $G$, then we need not have
$H\subseteq G^{\tau^{G}}$, we only have the weaker condition

$H\subseteq\\{g\in G\colon g\tau^{G}(g)^{-1}\in N\\}.$

(b) If ${\mathfrak{g}}$ is semisimple, then $Z(G)\cap H$ acts trivially on
$M$, hence is contained in $N$. It follows that, for $N=\\{e\\}$, the adjoint
action restricts to an injective representation of $H$.

If, in addition, ${\mathfrak{h}}$ contains no non-zero ideal of
${\mathfrak{g}}$, then $N$ is a discrete normal subgroup of the connected
group $G$, hence central, and thus

$N=H\cap Z(G).$ (26)

(c) From $\mathop{\bf L{}}\nolimits(H)={\mathfrak{h}}$ we obtain $\mathop{{\rm
Ad}}\nolimits(H){\mathfrak{h}}={\mathfrak{h}}$. If ${\mathfrak{g}}$ is
semisimple, so that ${\mathfrak{q}}={\mathfrak{h}}^{\bot}$ with respect to the
Cartan–Killing form implies $\mathop{{\rm
Ad}}\nolimits(H){\mathfrak{q}}={\mathfrak{q}}$. This already implies

$H\subseteq\mathop{{\rm Ad}}\nolimits^{-1}(\mathop{{\rm
Ad}}\nolimits(G)^{\tau})=\\{g\in G\colon gg^{\sharp}\in Z(G)\\}.$ (27)

(d) If $\tau^{G}(H)=H$ and $\mathop{{\rm Ad}}\nolimits$ is faithful on $H$,
then $\mathop{{\rm Ad}}\nolimits(H)\subseteq\mathop{{\rm
Ad}}\nolimits(G)^{\tau}$ implies $H\subseteq G^{\tau^{G}}$.

##  3 Fixed points of the modular flow in $M$

In Subsection 3.1 we classify for a reductive compactly causal symmetric
reductive Lie algebra $({\mathfrak{g}},\tau)$ the orbits of $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ in the set
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$ of Euler elements in
${\mathfrak{h}}$. As this set describes the choices of the Euler element
$h\in{\mathfrak{h}}$, this provides a classification of all modular structures
on the associated symmetric spaces. The classification is done by reduction to
the simple case.

On $M=G/H$, an element $m=gH$ is fixed by the modular flow

$\alpha_{t}^{M}(gH)=\exp(th)gH=g\exp(t\mathop{{\rm Ad}}\nolimits(g)^{-1}h)H$
(28)

if and only if $\mathop{{\rm Ad}}\nolimits(g)^{-1}h\in{\mathfrak{h}}$.
Therefore elements of the fixed point set $M^{\alpha}$ correspond to Euler
elements in $\mathcal{O}_{h}\cap{\mathfrak{h}}$, where
$\mathcal{O}_{h}=\mathop{{\rm Ad}}\nolimits(G)h$ is the adjoint orbit of $h$.

In Subsection 3.2 we show that the orbits of the group $G^{h}$ in $M^{\alpha}$
are open and that this leads to a natural bijection between the orbit spaces
$M^{\alpha}/G^{h}$ and $(\mathcal{O}^{h}\cap{\mathfrak{h}})/H$. Both sets
correspond to the set of double cosets $G^{h}gH\subseteq G$ for which
$\mathop{{\rm Ad}}\nolimits(g)^{-1}h\in{\mathfrak{h}}$.

###  3.1 $\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$-orbits
in $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$

Let $({\mathfrak{g}},\tau,C)$ be a reductive compactly causal symmetric Lie
algebra. Then $({\mathfrak{g}},\tau)$ decomposes as

$({\mathfrak{g}},\tau)\cong({\mathfrak{g}}_{0},\tau_{0})\oplus\bigoplus_{j=1}^{N}({\mathfrak{g}}_{j},\tau_{j}),$

where ${\mathfrak{g}}_{0}$ contains the center and all compact ideals and the
$\tau$-invariant ideals ${\mathfrak{g}}_{j}$, $j\geq 1$, are either simple
hermitian or irreducible of group type. This decomposition leads to a
corresponding decomposition for ${\mathfrak{h}}$:

${\mathfrak{h}}={\mathfrak{h}}_{0}\oplus\bigoplus_{j=1}^{N}{\mathfrak{h}}_{j},$

and an element $h\in{\mathfrak{g}}\setminus{\mathfrak{z}}({\mathfrak{g}})$ is
an Euler element in ${\mathfrak{g}}$ if and only if all its components
$h_{0},h_{1},\ldots,h_{N}$ either vanish or are Euler elements in
${\mathfrak{g}}_{j}$. In particular, if $h_{j}\not=0$, then
${\mathfrak{g}}_{j}$ is simple hermitian of tube type (Proposition 28).
Furthermore, $h_{0}\in{\mathfrak{z}}({\mathfrak{g}})$.

The group $\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ is a
product of the subgroups $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}}_{j})$, $j=0,\ldots,N$, acting on
the ideals ${\mathfrak{g}}_{j}$. This reduces the problem to classify
$\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$-orbits in the
subset $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$ to the case where
$({\mathfrak{g}},\tau)$ is irreducible. In the group case
${\mathfrak{g}}={\mathfrak{h}}\oplus{\mathfrak{h}}$, and $\mathop{{\rm
Inn}}\nolimits({\mathfrak{h}})$ acts transitively on
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}=\mathcal{E}({\mathfrak{h}})$
(Proposition 27). Therefore it suffices to analyze the case where
${\mathfrak{g}}$ is simple hermitian.

We assume for the moment that ${\mathfrak{g}}$ is simple hermitian and that
$({\mathfrak{g}},\tau,C)$ is compactly causal. We do not assume that
$\mathcal{E}({\mathfrak{g}})\not=\emptyset$. Let
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$ a Cartan decomposition
invariant under $\tau$. Then the compact causality of $({\mathfrak{g}},\tau)$
is equivalent to ${\mathfrak{z}}({\mathfrak{k}})\subseteq{\mathfrak{q}}$. Let
${\mathfrak{c}}\subseteq{\mathfrak{h}}_{\mathfrak{p}}$ be maximal abelian and
${\mathfrak{a}}\supseteq{\mathfrak{c}}$ maximal abelian in ${\mathfrak{p}}$,
so that

$s:=\mathop{{\rm dim}}\nolimits{\mathfrak{c}}=\mathop{{\rm
rank}}\nolimits_{\mathbb{R}}({\mathfrak{h}})\quad\mbox{ and }\quad
r:=\mathop{{\rm dim}}\nolimits{\mathfrak{a}}=\mathop{{\rm
rank}}\nolimits_{\mathbb{R}}({\mathfrak{g}}).$

If $h\in\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$, then $\tau$ preserves
the $3$-grading defined by $h$. On the euclidean simple Jordan algebra
$E:={\mathfrak{g}}_{1}(h)$ ([Oeh20, Rem. 2.24]), $\gamma:=-\tau|_{E}$ is a
Jordan automorphism ([Oeh20, Prop. 3.12]). If $\gamma=\mathop{{\rm
id}}\nolimits_{E}$, then $({\mathfrak{g}},\tau)$ is of Cayley type (CT) and
$\tau=\tau_{h}$, and if this is not the case, then $\gamma$ is non-trivial and
there are two cases. For split type (ST) we have $r=s$, and for non-split type
(NST) we have $r=2s$ (cf. [Ó91, §3], [NÓ21b, §D.2.2, Lemma D.5], [BH98]). If
$r=s$, then $\Delta({\mathfrak{g}},{\mathfrak{c}})\cong C_{r}$ by Moore’s
Theorem as ${\mathfrak{g}}$ is of tube type.

To understand the $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$-orbits in
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$, we have to compare the root
systems

$\Delta({\mathfrak{h}},{\mathfrak{c}})\subseteq\Delta({\mathfrak{g}},{\mathfrak{c}})$

and the inclusions
$\mathcal{W}({\mathfrak{h}},{\mathfrak{c}})\hookrightarrow\mathcal{W}({\mathfrak{g}},{\mathfrak{c}})$
of the corresponding Weyl groups. Any Euler element
$h\in\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$ is conjugate under
$\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ to one in
${\mathfrak{c}}$. The space $\mathcal{E}({\mathfrak{g}})$ consists of a single
$\mathop{{\rm Inn}}\nolimits({\mathfrak{g}})$-orbit (Proposition 27(a)), but
$\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ may not act
transitively on $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$ (see Theorem
32).

For the table below, we need the following two examples:

######  Example 31.

(a) Assume that $\tau=\tau_{h}$ and that $({\mathfrak{g}},\tau,C)$ is
compactly causal of Cayley type. Then ${\mathfrak{a}}={\mathfrak{c}}$, and the
root system $\Delta({\mathfrak{g}},{\mathfrak{a}})$ is of type $C_{r}$:

$\Delta({\mathfrak{g}},{\mathfrak{a}})\cong C_{r}=\\{\pm
2\varepsilon_{j},\pm\varepsilon_{i}\pm\varepsilon_{j}\colon 1\leq i\not=j\leq
r\\}$

and

$\Delta({\mathfrak{h}},{\mathfrak{a}})\cong
A_{r-1}=\\{\pm(\varepsilon_{i}-\varepsilon_{j})\colon 1\leq i\not=j\leq r\\}.$

The Weyl group $\mathcal{W}(C_{r})$ consists of all products of permutations
of $\\{1,\ldots,r\\}$ with sign changes. Its contains the Weyl group
$\mathcal{W}(A_{r-1})$ as the subgroup of permutations. Let $w_{0}=\mathbf{1}$
and $w_{j}(x_{1},\ldots,x_{r})=(-x_{1},\cdots,-x_{j},x_{j+1},\ldots,x_{r})$.
Then

$\mathcal{W}(A_{r-1})\backslash\mathcal{W}(C_{r})/\mathcal{W}(A_{r-1})\cong\\{w_{0},\ldots,w_{r}\\}.$

It follows that, up the conjugation by $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$, the Euler elements are given
by

$h_{k}=\frac{1}{2}(\underbrace{1,\ldots,1}_{k},-1,\ldots,-1)\quad\text{for}\
0\leq k\leq r.$

To see this, we first note that $h=(x_{1},\ldots,x_{r})\in{\mathfrak{a}}$ is
an Euler element if and only if $x_{j}\in\\{\frac{1}{2},-\frac{1}{2}\\}$.
Applying a suitable permutation we can always order the $x_{j}$ so that
$x_{1},\ldots,x_{k}=1/2$ and $x_{k+1},\ldots,x_{r}=-1/2$. Note that all of
those elements are conjugate under $\mathcal{W}(C_{r})$ as this group also
contains the sign changes.

(b) For
${\mathfrak{g}}=\mathop{{\mathfrak{su}}}\nolimits_{2s,2s}({\mathbb{C}})$ and
${\mathfrak{h}}={\mathfrak{u}}_{s,s}({\mathbb{H}})$ we have
$\Delta({\mathfrak{g}},{\mathfrak{a}})\cong C_{2s}$ and
$\Delta({\mathfrak{h}},{\mathfrak{c}})\cong C_{s}$. To determine
$\Delta({\mathfrak{g}},{\mathfrak{c}})$, we realize ${\mathfrak{g}}$ with
respect to the form defined by the hermitian $2\times 2$-block matrix

$B=\begin{pmatrix}0&\mathbf{1}_{2s}\\\
\mathbf{1}_{2s}&0\end{pmatrix}\quad\mbox{ as
}\quad{\mathfrak{g}}=\\{X\in\mathop{{\mathfrak{sl}}}\nolimits_{2r}({\mathbb{C}})\colon
X^{*}B=-BX\\}\cong\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}}).$

Then ${\mathfrak{a}}\subseteq{\mathfrak{g}}$ can be realized by diagonal
matrices of the form

$\mathop{{\rm diag}}\nolimits(x_{1},\ldots,x_{2s},-x_{1},\ldots,-x_{2s}),$

and ${\mathfrak{c}}\subseteq{\mathfrak{a}}$ corresponds to diagonal matrices
satisfying $x_{1}=x_{2},x_{3}=x_{4},\ldots,x_{2s-1}=x_{2s}$. In these
coordinates,

$\Delta({\mathfrak{g}},{\mathfrak{a}})\cong C_{2s}=\\{\pm
2\varepsilon_{j},\pm\varepsilon_{i}\pm\varepsilon_{j}\colon 1\leq i\not=j\leq
2s\\}\quad\mbox{ and }\quad\Delta({\mathfrak{g}},{\mathfrak{c}})\cong C_{s}.$

Restricting from ${\mathfrak{a}}$ to ${\mathfrak{c}}$, maps
$\varepsilon_{2j-1}$ and $\varepsilon_{2j}$ (in $C_{2s}$) to $\varepsilon_{j}$
(in $C_{s}$) for $j=1,\ldots,s$. Now it is easy to see that

$\Delta({\mathfrak{g}},{\mathfrak{c}})=\Delta({\mathfrak{h}},{\mathfrak{c}})\cup\\{0\\}.$

(c) For ${\mathfrak{g}}=\mathfrak{sp}_{4s}({\mathbb{R}})$ and
${\mathfrak{h}}=\mathfrak{sp}_{2s}({\mathbb{C}})$ we have exactly the same
pattern and therefore

$\Delta({\mathfrak{g}},{\mathfrak{c}})=\Delta({\mathfrak{h}},{\mathfrak{c}})\cup\\{0\\}.$

${\mathfrak{g}}$ | ${\mathfrak{g}}^{c}={\mathfrak{h}}+i{\mathfrak{q}}$ | ${\mathfrak{h}}$ | type | $\Delta({\mathfrak{g}},{\mathfrak{a}})$ | $\Delta({\mathfrak{g}},{\mathfrak{c}})$ | $\Delta({\mathfrak{h}},{\mathfrak{c}})$
---|---|---|---|---|---|---
$\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}})^{\oplus 2}$ |  $\mathop{{\mathfrak{sl}}}\nolimits_{2r}({\mathbb{C}})$ ( | $\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}})$ | (GT) | $C_{r}\oplus C_{r}$ | $C_{r}$ | $C_{r}$
$\mathfrak{sp}_{2r}({\mathbb{R}})^{\oplus 2}$ | $\mathfrak{sp}_{2r}({\mathbb{C}})$ | $\mathfrak{sp}_{2r}({\mathbb{R}})$ | (GT) | $C_{r}\oplus C_{r}$ | $C_{r}$ | $C_{r}$
$\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}})^{\oplus 2},d>2$ | $\mathop{{\mathfrak{so}}}\nolimits_{2+d}({\mathbb{C}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}})$ | (GT) | $C_{2}\oplus C_{2}$ | $C_{2}$ | $C_{2}$
$\mathop{{\mathfrak{so}}}\nolimits^{*}(4r)^{\oplus 2}$ | $\mathop{{\mathfrak{so}}}\nolimits_{4r}({\mathbb{C}})$ | $\mathop{{\mathfrak{so}}}\nolimits^{*}(4r)$ | (GT) | $C_{r}\oplus C_{r}$ | $C_{r}$ | $C_{r}$
$({\mathfrak{e}}_{7(-25)})^{\oplus 2}$ | ${\mathfrak{e}}_{7}({\mathbb{C}})$ | ${\mathfrak{e}}_{7(-25)}$ | (GT) | $C_{3}\oplus C_{3}$ | $C_{3}$ | $C_{3}$
$\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}})$ | $\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}})$ | ${\mathbb{R}}\oplus\mathop{{\mathfrak{sl}}}\nolimits_{r}({\mathbb{C}})$ | (CT) | $C_{r}$ | $C_{r}$ | $A_{r-1}$
$\mathfrak{sp}_{2r}({\mathbb{R}})$ | $\mathfrak{sp}_{2r}({\mathbb{R}})$ | ${\mathbb{R}}\oplus\mathop{{\mathfrak{sl}}}\nolimits_{r}({\mathbb{R}})$ | (CT) | $C_{r}$ | $C_{r}$ | $A_{r-1}$
$\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}}),d>2$ | $\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}})$ | ${\mathbb{R}}\oplus\mathop{{\mathfrak{so}}}\nolimits_{1,d-1}({\mathbb{R}})$ | (CT) | $C_{2}$ | $C_{2}$ | $A_{1}$
$\mathop{{\mathfrak{so}}}\nolimits^{*}(4r)$ | $\mathop{{\mathfrak{so}}}\nolimits^{*}(4r)$ | ${\mathbb{R}}\oplus\mathop{{\mathfrak{sl}}}\nolimits_{r}({\mathbb{H}})$ | (CT) | $C_{r}$ | $C_{r}$ | $A_{r-1}$
${\mathfrak{e}}_{7(-25)}$ | ${\mathfrak{e}}_{7(-25)}$ | ${\mathbb{R}}\oplus{\mathfrak{e}}_{6(-26)}$ | (CT) | $C_{3}$ | $C_{3}$ | $A_{2}$
$\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}})$ | $\mathop{{\mathfrak{sl}}}\nolimits_{2r}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{r,r}({\mathbb{R}})$ | (ST) | $C_{r}$ | $C_{r}$ | $D_{r}$
$\mathop{{\mathfrak{so}}}\nolimits^{*}(4r)$ | $\mathop{{\mathfrak{so}}}\nolimits_{2r,2r}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{2r}({\mathbb{C}})$ | (ST) | $C_{r}$ | $C_{r}$ | $D_{r}$
${\mathfrak{e}}_{7(-25)}$ | ${\mathfrak{e}}_{7}({\mathbb{R}})$ | $\mathop{{\mathfrak{sl}}}\nolimits_{4}({\mathbb{H}})$ | (ST) | $C_{3}$ | $C_{3}$ | $A_{3}=D_{3}$
$\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{p+1,q+1}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{1,p}({\mathbb{R}})\oplus\mathop{{\mathfrak{so}}}\nolimits_{1,q}({\mathbb{R}})$ | (ST) | $C_{2}$ | $C_{2}$ | $A_{1}\oplus A_{1}=D_{2}$
$d=p+q>2$ | $1<q<d-1$ |  |  |  |  |
$\mathop{{\mathfrak{su}}}\nolimits_{2s,2s}({\mathbb{C}})$ | $\mathop{{\mathfrak{sl}}}\nolimits_{2s}({\mathbb{H}})$ | ${\mathfrak{u}}_{s,s}({\mathbb{H}})$ | (NST) | $C_{2s}$ | $C_{s}$ | $C_{s}$
$\mathfrak{sp}_{4s}({\mathbb{R}})$ | ${\mathfrak{u}}_{s,s}({\mathbb{H}})$ | $\mathfrak{sp}_{2s}({\mathbb{C}})$ | (NST) | $C_{2s}$ | $C_{s}$ | $C_{s}$
$\mathop{{\mathfrak{so}}}\nolimits_{2,d}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{1,d+1}({\mathbb{R}})$ | $\mathop{{\mathfrak{so}}}\nolimits_{1,d}({\mathbb{R}})$ | (NST) |  | $A_{1}$ | $A_{1}$

Table 1: Irreducible compactly causal symmetric Lie algebras
$({\mathfrak{g}},\tau)$ with
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}\not=\emptyset$

From Table 1 we get the following classification theorem. Here we use that, as
in [MN21, Thm. 3.10], orbits of Euler elements can be classified by
representatives in a positive Weyl chamber in terms of a root basis. They
correspond to $3$-gradings of the root system
$\Delta({\mathfrak{g}},{\mathfrak{a}})$ for
$h\in{\mathfrak{a}}\cap\mathcal{E}({\mathfrak{g}})$ and to $3$-gradings of
$\Delta({\mathfrak{g}},{\mathfrak{c}})$ for
$h\in{\mathfrak{c}}\cap\mathcal{E}({\mathfrak{g}})$. We also recall from
Proposition 27 that the existence of an Euler element in a simple hermitian
Lie algebra ${\mathfrak{g}}$ implies that ${\mathfrak{g}}$ is of tube type.

######  Theorem 32.

(Classification of $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$-orbits in
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$) Let $({\mathfrak{g}},\tau)$
be an irreducible compactly causal symmetric Lie algebra. Then we have the
following situations:

  * (GT)

For group type
$\Delta({\mathfrak{g}},{\mathfrak{c}})=\Delta({\mathfrak{h}},{\mathfrak{c}})\cong
C_{r}$, and $\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ is
transitive on $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$.

  * (CT)

For Cayley type

$\Delta({\mathfrak{h}},{\mathfrak{c}})\cong A_{r-1}\subseteq
C_{r}=\Delta({\mathfrak{g}},{\mathfrak{c}}).$

In the canonical identification of ${\mathfrak{c}}$ with ${\mathbb{R}}^{r}$,
we have

$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{c}}=\\{{\textstyle{\frac{1}{2}}}(\pm
1,\cdots,\pm 1)\\}$

and the orbits of $\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$
in $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$ are represented by the
elements

$h_{k}={\textstyle{\frac{1}{2}}}(\underbrace{1,\ldots,1}_{k},-1,\ldots,-1),\quad
k=0,1,\ldots,r.$ (29)

  * (ST)

For split type

$\Delta({\mathfrak{h}},{\mathfrak{c}})\cong D_{r}\subseteq
C_{r}=\Delta({\mathfrak{g}},{\mathfrak{c}}),$

where we identify $D_{3}\cong A_{3}$ and $D_{2}\cong A_{1}\oplus A_{1}$. Then
$\mathop{{\rm Inn}}\nolimits({\mathfrak{g}})^{\tau}$ acts transitively on
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$ and its identity component
$\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ has two orbits
represented by $h_{r-1}$ and $h_{r}$.

  * (NST)

For non-split type

$\Delta({\mathfrak{h}},{\mathfrak{c}})=\Delta({\mathfrak{g}},{\mathfrak{c}})=C_{r}$

and $\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ acts
transitively on $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$.

###### Proof.

(GT) Here
$({\mathfrak{g}},\tau)\cong({\mathfrak{h}}\oplus{\mathfrak{h}},\tau_{\rm
flip})$, where ${\mathfrak{h}}$ is simple hermitian of tube type. Accordingly
${\mathfrak{a}}={\mathfrak{c}}\oplus{\mathfrak{c}}$, and the assertion
follows.

(CT) The first assertion follows from Table 1 and the remainder from the
discussion in Example 31(a).

(ST) For split type a look at Table 1 shows that

$\Delta({\mathfrak{h}},{\mathfrak{c}})\cong D_{r}\subseteq
C_{r}=\Delta({\mathfrak{g}},{\mathfrak{c}}),$

where we identify $D_{3}\cong A_{3}$ and $D_{2}\cong A_{1}\oplus A_{1}$. Then
$\mathcal{W}(D_{r})\trianglelefteq\mathcal{W}(C_{r})$ is an index $2$ subgroup
and $\mathcal{W}(C_{r})$ acts by automorphisms on $D_{r}$. Therefore
$\mathop{{\rm Inn}}\nolimits({\mathfrak{g}})^{\tau}$ induces on
${\mathfrak{c}}$ the full group $\mathcal{W}(C_{r})$. Hence the
$\mathcal{W}(D_{r})$-orbits in $\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{c}}$
are represented by $h_{r-1}$ and $h_{r}$, and both are conjugate under
$\mathcal{W}(C_{r})$.

(NST) Table 1 shows that, for non-split type, we have
$\Delta({\mathfrak{h}},{\mathfrak{c}})=\Delta({\mathfrak{g}},{\mathfrak{c}})=C_{r}.$
As $\mathcal{W}(C_{r})$ acts transitively on
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{c}}$, it follows that $\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ acts transitively on
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$. ∎

######  Corollary 33.

If $({\mathfrak{g}},\tau)$ is irreducible and not of Cayley type, then
$\mathop{{\rm Inn}}\nolimits({\mathfrak{g}})^{\tau}$ acts transitively on
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$.

###  3.2 Connected components of $M^{\alpha}$

Recall our global context from Subsection 2.4. Our goal in this section is to
prove the following proposition:

######  Proposition 34.

The orbits of the identity component $G^{h}_{e}$ on the fixed point set

$M^{\alpha}:=\\{gH\in M\colon\mathop{{\rm
Ad}}\nolimits(g)^{-1}h\in{\mathfrak{h}}\\}=\\{m\in M\colon X_{h}^{M}(m)=0\\}$

of the modular flow coincide with its connected components.

Example 36 below shows that, for Cayley type involutions on hermitian simple
Lie algebras, the connected components of $M^{\alpha}$ may have different
dimensions. Our discussion of de Sitter space in Subsection 11.1 reveals a
situation where $G^{h}$ does not preserve the connected components of
$M^{\alpha}$.

###### Proof.

Clearly, $M^{\alpha}$ is invariant under the action of the group $G^{h}$ which
commutes with $\alpha$. Therefore $G^{h}_{e}$ preserves all connected
components of $M^{\alpha}$. We now show that all $G^{h}_{e}$-orbits in
$M^{\alpha}$ are open, hence coincide with its connected components.

First we consider the base point $eH$ which is contained in $M^{\alpha}$
because $h\in{\mathfrak{h}}$. Using the exponential function $\mathop{{\rm
Exp}}\nolimits_{eH}\colon{\mathfrak{q}}\to M$ as a chart around $eH$, its
equivariance with respect to $\alpha$ and the one-parameter group
$e^{t\mathop{{\rm ad}}\nolimits h}$ on ${\mathfrak{q}}$, it follows that
$M^{\alpha}$ is a symmetric subspace of $M$ with
$T_{eH}(M^{\alpha})={\mathfrak{q}}_{0}(h)\subseteq{\mathfrak{g}}^{h}$.
Therefore $G^{h}_{e}.eH\supseteq\mathop{{\rm
Exp}}\nolimits_{eH}({\mathfrak{q}}_{0}(h))$ contains a neighborhood of $eH$ in
$M^{\alpha}$, and this implies that the $G^{h}_{e}$-orbit of $eH$ in
$M^{\alpha}$ is open.

Now let $m=gH\in M^{\alpha}$ be arbitrary. Then

$\alpha_{t}^{M}(gH)=\exp(th)gH=g\exp(t\mathop{{\rm Ad}}\nolimits(g)^{-1}h)H$

shows that $m\in M^{\alpha}$ is equivalent to $\mathop{{\rm
Ad}}\nolimits(g)^{-1}h\in{\mathfrak{h}}$ and

$\Phi\colon{\mathfrak{q}}\to M,\quad\Phi(x):=g\mathop{{\rm
Exp}}\nolimits_{eH}(x)$

defines a chart around $m$ satisfying

$\alpha_{t}\Phi(x)=\exp(th)g\mathop{{\rm
Exp}}\nolimits_{eH}(x)=\Phi(e^{t\mathop{{\rm Ad}}\nolimits(g)^{-1}h}x).$ (30)

Hence $\Phi$ maps

$\\{x\in{\mathfrak{q}}\colon[\mathop{{\rm
Ad}}\nolimits(g)^{-1}h,x]=0\\}={\mathfrak{q}}\cap\mathop{{\rm
Ad}}\nolimits(g)^{-1}{\mathfrak{g}}^{h}$ (31)

onto a neighborhood of $m$ in $M^{\alpha}$. Finally

$\Phi({\mathfrak{q}}\cap\mathop{{\rm
Ad}}\nolimits(g)^{-1}{\mathfrak{g}}^{h})\subseteq
g\exp({\mathfrak{q}}\cap\mathop{{\rm
Ad}}\nolimits(g)^{-1}{\mathfrak{g}}^{h})H\subseteq\exp({\mathfrak{g}}^{h})gH\subseteq
G^{h}_{e}.m$

shows that $G^{h}_{e}.m$ is open in $M^{\alpha}$. ∎

######  Example 35.

(The group case) In the group case the set $M^{\alpha}=G^{\alpha}$ is rather
simple to describe. Here $M=G$, $h=(h_{0},h_{0})$ and $\mathop{{\rm
Ad}}\nolimits(g_{1},g_{2})h=(\mathop{{\rm
Ad}}\nolimits(g_{1})h_{0},\mathop{{\rm
Ad}}\nolimits(g_{2})h_{0})\in{\mathfrak{h}}$ is equivalent to $\mathop{{\rm
Ad}}\nolimits(g_{1})h_{0}=\mathop{{\rm Ad}}\nolimits(g_{2})h_{0}$, i.e.,
$g_{1}^{-1}g_{2}\in G^{h_{0}}$. If this is the case, then $\mathop{{\rm
Ad}}\nolimits(g_{1},g_{2})h=\mathop{{\rm
Ad}}\nolimits(g_{1},g_{1})h\in\mathop{{\rm Ad}}\nolimits(\Delta_{G})h$.

We also note that $(G\times G)^{h}=G^{h_{0}}\times G^{h_{0}}$ acts
transitively on the submanifold $M^{\alpha}=G^{h_{0}}$.

######  Example 36.

(An example of Cayley type) As in Example 31, we consider the hermitian form
on ${\mathbb{C}}^{2r}$ defined by the block diagonal matrix
$B=\begin{pmatrix}0&\mathbf{1}\\\ \mathbf{1}&0\end{pmatrix}$, so that

${\mathfrak{g}}:=\\{X\in\mathop{{\mathfrak{sl}}}\nolimits_{2r}({\mathbb{C}})\colon
X^{*}B=-BX\\}\cong\mathop{{\mathfrak{su}}}\nolimits_{r,r}({\mathbb{C}})\quad\mbox{
and }\quad\tau\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}a&-b\\\
-c&d\end{pmatrix}.$

Then

${\mathfrak{h}}=\Big{\\{}\begin{pmatrix}X&0\\\
0&-X^{*}\end{pmatrix}\colon\mathop{{\rm
tr}}\nolimits(X)\in{\mathbb{R}}\Big{\\}}\cong{\mathbb{R}}\oplus\mathop{{\mathfrak{sl}}}\nolimits_{r}({\mathbb{C}}).$

The corresponding groups are

$G\cong\mathop{{\rm SU}}\nolimits_{r,r}({\mathbb{C}})\supseteq
H\cong\\{g\in\mathop{{\rm
GL}}\nolimits_{r}({\mathbb{C}})\colon\det(g)\in{\mathbb{R}}\\}.$

Then

${\mathfrak{a}}=\\{\mathop{{\rm
diag}}\nolimits(x_{1},\ldots,x_{r},-x_{1},\ldots,-x_{r}\\}\colon
x_{j}\in{\mathbb{R}}\\}\cong{\mathbb{R}}^{r},$

and in these coordinates, we obtain Euler elements

$h_{k}=\frac{1}{2}(\mathbf{1}_{k},-\mathbf{1}_{r-k},-\mathbf{1}_{k},\mathbf{1}_{r-k}),\quad
k=0,\ldots,r$

(Theorem 32). Then

$\mathop{{\rm dim}}\nolimits\mathcal{O}^{H}_{h_{k}}=\mathop{{\rm
dim}}\nolimits[h_{k},{\mathfrak{h}}]=\mathop{{\rm
dim}}\nolimits_{\mathbb{R}}(M_{k,r-k}({\mathbb{C}})\oplus
M_{r-k,k}({\mathbb{C}}))=2k(r-k)$

depends on $k$. For $k=0,r$, the orbit is trivial. This corresponds to the
fact that the base point $eH$ in $M=G/H\cong\mathcal{O}^{G}_{h}$ is an
isolated fixed point of the modular flow. The components

$M_{k}^{\alpha}:=\\{gH\colon\mathop{{\rm
Ad}}\nolimits(g)^{-1}h\in\mathcal{O}^{H}_{h_{k}}\\},\quad k=0,\ldots,r,$

are of different dimensions (for $r>1$).

######  Lemma 37.

We have a bijection

$\Gamma\colon M^{\alpha}/G^{h}\to(\mathcal{O}_{h}\cap{\mathfrak{h}})/H,\quad
G^{h}gH\mapsto\mathop{{\rm Ad}}\nolimits(H)\mathop{{\rm
Ad}}\nolimits(g)^{-1}h.$

###### Proof.

As $gH\in M^{\alpha}$ is equivalent to $\mathop{{\rm
Ad}}\nolimits(g)^{-1}h\in{\mathfrak{h}}$, we obtain a well-defined surjective
map sending the $G^{h}$-orbit $G^{h}gH$ to the $H$-orbit $\mathop{{\rm
Ad}}\nolimits(H)\mathop{{\rm
Ad}}\nolimits(g)^{-1}h\subseteq\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$.

If $g_{1}H,g_{2}H\in M^{\alpha}$ map to the same $H$-orbit in
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$, then there exists an element
$k\in H$ with $\mathop{{\rm Ad}}\nolimits(g_{1})^{-1}h=\mathop{{\rm
Ad}}\nolimits(k^{-1})\mathop{{\rm Ad}}\nolimits(g_{2})^{-1}h$, i.e.,
$g_{2}kg_{1}^{-1}\in G^{h}$. This implies that $g_{2}H=g_{2}kH\in
G^{h}g_{1}H.$ As the surjectivity of $\Gamma$ follows from the fact that
$gH\in M^{\alpha}$ is equivalent to $\mathop{{\rm
Ad}}\nolimits(g)^{-1}h\in{\mathfrak{h}}$, $\Gamma$ is bijective. ∎

######  Remark 38.

(a) If $H$ is connected, then the preceding lemma shows that the orbits of
$\mathop{{\rm Ad}}\nolimits(H)=\mathop{{\rm
Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$ in
$\mathcal{E}({\mathfrak{g}})\cap{\mathfrak{h}}$ are in one-to-one
correspondence with the $G^{h}$-orbits in $M^{\alpha}$ for $M=G/H$. Therefore
Theorem 32 provides a classification of $G^{h}$-orbits in $M^{\alpha}$
whenever $H$ is connected.

(b) As $G^{h}$-orbits in $M^{\alpha}$ are open by Proposition 34, we may also
think of $\pi_{0}(G^{h})$-orbits in $\pi_{0}(M^{\alpha})$ being classified by
$(\mathcal{O}_{h}\cap{\mathfrak{h}})/H$.

######  Remark 39.

The involution $\tau_{h}^{M}$ acts on the fixed point manifold $M^{\alpha}$,
but not necessarily trivially. A typical example is the group type space
$M=\widetilde{\mathop{{\rm SL}}}\nolimits_{2}({\mathbb{R}})$, where
$Z(G)\subseteq M^{\alpha}$ and $Z(G)\cap M^{\tau_{h}^{G}}=\\{e\\}$.

######  Remark 310.

($G^{h}$ and $G^{\tau^{G}}$ for simple Lie algebras) As $G^{h}=\\{g\in
G\colon\mathop{{\rm Ad}}\nolimits(g)h=h\\}$, we have

$\mathop{{\rm Ad}}\nolimits(G^{h})=\mathop{{\rm
Ad}}\nolimits(G)^{h}=\mathop{{\rm Ad}}\nolimits(G)^{\mathop{{\rm ad}}\nolimits
h}\subseteq\mathop{{\rm Ad}}\nolimits(G)^{\tau_{h}}.$

We claim that, if ${\mathfrak{g}}$ is simple, then $\mathop{{\rm
Ad}}\nolimits(G^{h})$ is of index $2$ in $\mathop{{\rm
Ad}}\nolimits(G)^{\tau_{h}}$. In fact, in this case
${\mathfrak{g}}^{-\tau_{h}}$ is a direct sum of two irreducible
${\mathfrak{g}}_{0}(h)$-modules ${\mathfrak{g}}_{\pm 1}(h)$ ([HÓ97, Cor.
1.3.13]), and any element $\mathop{{\rm Ad}}\nolimits(g)$ commuting with
$\tau_{h}$ either preserves both subspaces or exchanges them. If it preserves
both, then $\mathop{{\rm Ad}}\nolimits(g)$ commutes with $\mathop{{\rm
ad}}\nolimits h$, i.e., $g\in G^{h}$. If not, then $\mathop{{\rm
Ad}}\nolimits(g)h=-h$, and as such an element exists (Proposition 27(a)), the
index of $\mathop{{\rm Ad}}\nolimits(G^{h})$ in $\mathop{{\rm
Ad}}\nolimits(G)^{\tau^{G}}$ is two.

##  4 Wedge domains in compactly causal symmetric spaces

In this section we introduce the wedge domains $W_{M}^{+}(h)$, $W_{M}^{\rm
KMS}(h)$ and $W_{M}(h)$ in a compactly causal symmetric space $M=G/H$ with the
infinitesimal data $({\mathfrak{g}},\tau,C,h)$ (under the global assumption
from Subsection 2.4 that the universal complexification $\eta_{G}$ has
discrete kernel). One of our main results asserts that these three subsets
coincide, which is far from obvious from their definitions. Moreover, their
connected components are orbits of an open real Olshanski subsemigroup of $G$.
We therefore start recalling the construction of Olshanski subsemigroups.

###  4.1 Tube domains of compactly causal symmetric spaces

######  Definition 41.

(Complex Olshanski semigroups) Let $G$ be a connected Lie group with simply
connected covering group $q_{G}\colon\widetilde{G}\to G$. For a pointed closed
convex $\mathop{{\rm Ad}}\nolimits(G)$-invariant cone
${C_{\mathfrak{g}}\subseteq{\mathfrak{g}}}$, Lawson’s Theorem ([La94],[Ne00,
Thm. IX.1.10]) implies the existence of a semigroup
$\Gamma_{\widetilde{G}}(C_{\mathfrak{g}})$ which is a covering of the
subsemigroup $\eta_{\widetilde{G}}(\widetilde{G})\exp(iC_{\mathfrak{g}})$ of
the universal complexification $\widetilde{G}_{\mathbb{C}}$ (the simply
connected group with Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$). Then the
exponential function
$\exp\colon{\mathfrak{g}}+iC_{\mathfrak{g}}\to\widetilde{G}_{\mathbb{C}}$
lifts to an exponential function $\mathop{{\rm
Exp}}\nolimits\colon{\mathfrak{g}}+iC_{\mathfrak{g}}\to\Gamma_{\widetilde{G}}(C_{\mathfrak{g}})$,
and the polar map

$\widetilde{G}\times
C_{\mathfrak{g}}\to\Gamma_{\widetilde{G}}(C_{\mathfrak{g}}),\quad(g,x)\mapsto
g\mathop{{\rm Exp}}\nolimits(ix)$

is a homeomorphism. We now define the closed complex Olshanski semigroup
corresponding to the pair $(G,C_{\mathfrak{g}})$ by

$\Gamma_{G}(C_{\mathfrak{g}}):=\Gamma_{\widetilde{G}}(C_{\mathfrak{g}})/(\ker
q_{G}),$ (32)

where $q_{G}\colon\widetilde{G}\to G$ is the universal covering map (cf.
[Ne00, Thm. XI.1.12]). Then the polar map $G\times
C_{\mathfrak{g}}\to\Gamma_{G}(C_{\mathfrak{g}})$ is a homeomorphism, and if
$C_{\mathfrak{g}}$ has interior points, then it restricts to a diffeomorphism
from $G\times C_{\mathfrak{g}}^{0}$ onto the open subsemigroup

$\Gamma_{G}(C_{\mathfrak{g}}^{0}):=\Gamma_{\widetilde{G}}(C_{\mathfrak{g}}^{0})/(\ker
q_{G})\subseteq\Gamma_{G}(C_{\mathfrak{g}}),$ (33)

Complex Olshanski semigroups are non-abelian generalizations of complex tube
domains defined by open cones in real vector spaces; the domains
${\mathfrak{g}}+iC_{\mathfrak{g}}^{0}$ which are the tangent objects of the
open complex Olshanski semigroups $\Gamma_{G}(C_{\mathfrak{g}}^{0})$ are
typical examples. We now turn to similar objects for symmetric spaces, which
on the tangent level correspond to the tube domain
${\mathfrak{q}}+iC^{0}\subseteq{\mathfrak{q}}_{\mathbb{C}}$.

######  Definition 42.

(The complex tube domain of a compactly causal symmetric space) Let
$(G,\tau,H,C)$ be a compactly causal symmetric Lie group, i.e., $G$ is a
connected Lie group, $\tau^{G}$ an involutive automorphism of $G$, $H\subseteq
G^{\tau^{G}}$ an open subgroup, and $C\subseteq{\mathfrak{q}}$ a pointed
generating $\mathop{{\rm Ad}}\nolimits(H)$-invariant closed convex cone such
that $C^{0}$ consists of elliptic elements. We assume that the universal
complexification $\eta_{G}\colon G\to G_{\mathbb{C}}$ has discrete kernel
(which is the case if $G$ is simply connected or semisimple). The involution
$\tau^{G}\in\mathop{{\rm Aut}}\nolimits(G)$ induces by the universal property
a holomorphic automorphism $\tau^{G_{\mathbb{C}}}$ of $G_{\mathbb{C}}$ with
$\eta_{G}(H)\subseteq G_{\mathbb{C}}^{\tau^{G_{\mathbb{C}}}}$, and

$G_{\mathbb{C}}^{\sharp}=\\{g\in
G_{\mathbb{C}}\colon\tau^{G_{\mathbb{C}}}(g)=g^{-1}\\}$

is a complex symmetric subspace of $G_{\mathbb{C}}$.

We call the fiber product

$\mathcal{T}_{M}(C):=G\times_{H}iC^{0}$

the tube domain of the pair $(M,C)$. To obtain a complex manifold structure on
this domain, we observe that the quadratic representation $Q\colon M=G/H\to G$
extends to a map

$Q_{\mathcal{T}}\colon\mathcal{T}_{M}(C)=G\times_{H}iC^{0}\to
G_{\mathbb{C}}^{\sharp},\quad[g,ix]\mapsto g.\mathop{{\rm
Exp}}\nolimits(ix)=g\exp(2ix)g^{\sharp}$

which is a covering of an open subset. Therefore $\mathcal{T}_{M}(C)$ carries
a unique complex manifold structure for which $q_{\mathcal{T}}$ is
holomorphic. Clearly, $\mathcal{T}_{M}(C)\subseteq G\times_{H}iC$ contains $M$
as $G\times_{H}\\{0\\}$ in its “boundary” (cf. [HÓØ91],[KNÓ97], [Ne99]).

######  Remark 43.

(a) Suppose that $({\mathfrak{g}},\tau,C)$ is extendable and that
$C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ is a $-\tau$-invariant $\mathop{{\rm
Ad}}\nolimits(G)$-invariant pointed cone with
$C=C_{\mathfrak{g}}\cap{\mathfrak{q}}$ (see Subsection 2.1). Let
$G_{C}\trianglelefteq G$ be the normal integral subgroup with Lie algebra
${\mathfrak{g}}_{C}=C_{\mathfrak{g}}-C_{\mathfrak{g}}$. Then we have the
corresponding open complex Olshanski semigroup

$\mathcal{T}_{G_{C}}(C_{\mathfrak{g}}):=G_{C}\exp(iC_{\mathfrak{g}}^{0})$

and the quadratic representation $Q\colon M\to G_{C}$ extends to a map

$Q\colon\mathcal{T}_{M}(C)=G_{C}\times_{H}iC^{0}\to\mathcal{T}_{G_{C}}(C_{\mathfrak{g}}),\quad(g,ix)\mapsto
g\exp(2ix)g^{\sharp}.$

By Lemma A2 and Remark A3, its range is the identity component of the complex
submanifold $\mathcal{T}_{G_{C}}(C_{\mathfrak{g}})^{\sharp}$:

$Q(\mathcal{T}_{M}(C))=\mathcal{T}_{G_{C}}(C_{\mathfrak{g}})_{e}^{\sharp}=\\{ss^{\sharp}\colon
s\in\mathcal{T}_{G_{C}}(C_{\mathfrak{g}})\\}.$

Since $Q\colon\mathcal{T}_{M}(C)\to Q(\mathcal{T}_{M}(C))$ is a covering map
and the $G_{C}$-action on its range lifts to the $G_{C}$-action on $M$, the
holomorphic action of the semigroup $\mathcal{T}_{G_{C}}(C_{\mathfrak{g}})$ on
$Q(\mathcal{T}_{M}(C))$ by $s.m=sms^{\sharp}$ lifts to a holomorphic action

$\mathcal{T}_{G_{C}}(C_{\mathfrak{g}})\times\mathcal{T}_{M}(C)\to\mathcal{T}_{M}(C),$
(34)

extending the $G$-action.

(b) Let $\tau^{G_{\mathbb{C}}}\in\mathop{{\rm Aut}}\nolimits(G_{\mathbb{C}})$
be the holomorphic involution with
$\tau^{G_{\mathbb{C}}}\circ\eta_{G}=\eta_{G}\circ\tau^{G}$ and
$H_{\mathbb{C}}\subseteq G_{\mathbb{C}}^{\tau^{G_{\mathbb{C}}}}$ be an open
subgroup with $\eta_{G}^{-1}(H_{\mathbb{C}})=H$. Then we have a natural
embedding of $G/H$ as the $G$-orbit of the base point $eH_{\mathbb{C}}\in
M_{\mathbb{C}}:=G_{\mathbb{C}}/H_{\mathbb{C}}$, and

$\mathcal{T}_{M}(C)=G.\mathop{{\rm Exp}}\nolimits(iC^{0})$ (35)

can be identified with the orbit $\Gamma_{G}(C_{G}^{0}).eH_{\mathbb{C}}$ of
the base point under the action of the open complex Olshanski semigroup
$\mathcal{T}_{G}(C_{G})$, which is an open subset of $M_{\mathbb{C}}$ (cf.
[HÓØ91, Lemma 1.3]).

######  Example 44.

(a) Let $(G,\tau)$ be a connected symmetric Lie group and
$({\mathfrak{g}},\tau,C)$ be a corresponding compactly causal symmetric Lie
algebra. Suppose that $C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ is a pointed
$\mathop{{\rm Ad}}\nolimits(G)$-invariant generating cone satisfying
$-\tau(C_{\mathfrak{g}})=C_{\mathfrak{g}}$ and
$C:=C_{\mathfrak{g}}\cap{\mathfrak{q}}$ (cf. Theorem 24). Then we have a
corresponding open complex Olshanski semigroup
$S=\mathcal{T}_{G}(C_{\mathfrak{g}})$ (Definition 41) and thus an embedding

$\mathcal{T}_{G/G^{\tau^{G}}}(C)=G\times_{G^{\tau^{G}}}iC^{0}\cong\bigcup_{g\in
G}g\exp(iC^{0})g^{\sharp}\ {\buildrel!\over{=}}\ S^{\sharp}_{e}\hookrightarrow
S$

(Lemma A2(4)).

(b) Consider a symmetric space $M=(G,\bullet)$ of group type and an
$\mathop{{\rm Ad}}\nolimits(G)$-invariant cone
$C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$. The corresponding symmetric Lie
group is $G\times G$ with

$\tau^{G\times G}(g_{1},g_{2})=(g_{2},g_{1}),\quad(G\times G)^{\tau^{G\times
G}}=\Delta_{G},\quad{\mathfrak{q}}=\\{(x,-x)\colon x\in{\mathfrak{g}}\\},$

and the cone

$C=\\{(x,-x)\colon x\in C_{\mathfrak{g}}\\}=(C_{\mathfrak{g}}\oplus-
C_{\mathfrak{g}})\cap{\mathfrak{q}}$

is $\mathop{{\rm Ad}}\nolimits(\Delta_{G})$-invariant. We have the open
complex Olshanski semigroup

$\Gamma_{G\times G}(C_{\mathfrak{g}}\oplus-
C_{\mathfrak{g}})=\Gamma_{G}(C_{\mathfrak{g}})\times\Gamma_{G}(-C_{\mathfrak{g}}).$

Here $(g_{1},g_{2})^{\sharp}=(g_{2}^{-1},g_{1}^{-1})$ is the
$\sharp$-operation in $G\times G$, so that the $\sharp$-fixed points in the
open complex Olshanski semigroup

$S:=\Gamma_{G\times G}(C_{\mathfrak{g}}^{0}\oplus-C_{\mathfrak{g}}^{0})$

are the pairs $(s,s^{-1})$, $s\in\Gamma_{G}(C_{\mathfrak{g}}^{0})$. We thus
obtain

$\mathcal{T}_{G}(C)=(G\times G)\times_{\Delta_{G}}iC^{0}\cong G\times
iC^{0}\cong\Gamma_{G}(C_{\mathfrak{g}}^{0})=\\{(s,s^{-1})\colon
s\in\Gamma_{G}(C_{\mathfrak{g}}^{0})\\}\cong S^{\sharp}.$

As a complex manifold, this is a copy of the open complex Olshanski semigroup
$\Gamma_{G}(C_{\mathfrak{g}}^{0})$.

###  4.2 The modular flow and three types of wedge domains

We are now ready to introduce the three different type of wedge domains in
$M$. We already introduced the modular flow in (28) but repeat the definition
here:

######  Definition 45.

(The modular flow) The Euler element $h\in{\mathfrak{h}}$ defines an
${\mathbb{R}}$-action by automorphisms on $G$ via

$\alpha_{t}(g)=\exp(th)g\exp(-th),\qquad g\in G.$

Then $\alpha$ preserves all connected components of the subgroup
$G^{\tau^{G}}$, hence in particular $H$. Therefore $\alpha_{t}$ induces a flow

$\alpha_{t}^{M}(gH)=\exp(th)gH=g\exp(t\mathop{{\rm Ad}}\nolimits(g)^{-1}h)H$
(36)

on $M=G/H$. This flow is generated by the modular vector field

$X_{h}^{M}\in\mathcal{V}(M),\quad
X_{h}^{M}(m)=\frac{d}{dt}\Big{|}_{t=0}\alpha_{t}(m).$ (37)

The modular flow extends to ${\mathbb{R}}$-actions by holomorphic maps on the
complex tube domains $\mathcal{T}_{G}(C_{\mathfrak{g}})$ and on
$\mathcal{T}_{M}(C)$ via

$\alpha_{t}(g\mathop{{\rm Exp}}\nolimits(ix))=\alpha_{t}(g)\mathop{{\rm
Exp}}\nolimits(ie^{t\mathop{{\rm ad}}\nolimits h}x)\quad\mbox{ and
}\quad\alpha_{t}^{M}([g,ix]):=[\alpha_{t}(g),ie^{t\mathop{{\rm ad}}\nolimits
h}x].$

Their infinitesimal generators are denoted $X^{\mathcal{T}_{G}}_{h}$ and
$X^{\mathcal{T}_{M}}_{h}$, respectively.

On $G_{\mathbb{C}}$ we even obtain a holomorphic flow by
$\alpha_{z}(g)=\exp(zh)g\exp(-zh)$, $z\in{\mathbb{C}}$, but on
$\mathcal{T}_{G}(C_{\mathfrak{g}})$ and $\mathcal{T}_{M}(C)$, the real flow
does not extend to all of ${\mathbb{C}}$. However, for $s$ in the closed
complex semigroup $\Gamma_{G}(C_{\mathfrak{g}})$, we consider the orbit map

$\alpha^{s}\colon{\mathbb{R}}\to\Gamma_{G}(C_{\mathfrak{g}}),\quad
t\mapsto\alpha_{t}(s)$

and define

$\alpha_{x+iy}(s):=\alpha^{s}(x+iy)\quad\mbox{ for }\quad
x+iy\in{\mathbb{C}},$

whenever the maximal local flow of the vector field $iX^{\mathcal{T}_{M}}$
with initial value $\alpha_{x}(s)|_{s=0}$ is defined in $y\in{\mathbb{R}}$.
This implies that $\alpha^{s}$ extends to a continuous map on the closed strip
between ${\mathbb{R}}$ and ${\mathbb{R}}+iy$ which is holomorphic on the
interior in the sense that its composition with the natural map
$\Gamma_{G}(C_{\mathfrak{g}})\to G_{\mathbb{C}},g\mathop{{\rm
Exp}}\nolimits(ix)\mapsto\eta_{G}(g)\exp(ix)$ is a holomorphic
$G_{\mathbb{C}}$-valued map (cf. Definition 41).

Recall that we assume that $\tau$ integrates to an involution $\tau_{h}^{G}$
on $G$ which leaves $H$ invariant, so that it induces an involution
$\tau_{h}^{M}$ on $M=G/H$ (see the end of Subsection 2.4).

######  Definition 46.

We consider the following types of wedge domains: 333In complex analysis
“domains” are assumed connected. Here we use the term “domain” for an open
subset.

  * •

The positivity domain of the modular vector field $X_{h}^{M}$ in $M$ is

$W_{M}^{+}(h):=\\{m\in M\colon X_{h}^{M}(m)\in V_{+}(m)\\},$

where $V_{+}(m)\subseteq T_{m}(M)$ is the open cone corresponding to the
$G$-invariant cone field with $V_{+}(eH)=C^{0}$ in
$T_{eH}(M)\cong{\mathfrak{q}}$. If $M$ is Lorentzian and $V_{+}(m)$ is the
future light cone in $T_{m}(M)$, then this is the domain where the modular
vector field is future oriented timelike.

  * •

The KMS wedge domain is

$\displaystyle W_{M}^{\rm KMS}(h):=$ $\displaystyle\\{m\in M\colon(\forall
z\in\mathcal{S}_{\pi})\ \alpha_{z}(m)\in\mathcal{T}_{M}(C)\\}$
$\displaystyle=$ $\displaystyle\\{m\in M\colon(\forall y\in(0,\pi))\
\alpha_{iy}(m)\in\mathcal{T}_{M}(C)\\},$

where $\alpha_{z}(m)$ is assumed to be defined in $m$ in as in Definition 45,
and

$\mathcal{S}_{\pi}=\\{z\in{\mathbb{C}}\colon 0<\mathop{{\rm Im}}\nolimits
z<\pi\\}.$

This is the set of all points $m\in M$ whose $\alpha$-orbit map extends
analytically to a map $\mathcal{S}_{\pi}\to\mathcal{T}_{M}(C)$. Comparing
boundary values on ${\mathbb{R}}$ and $\pi i+{\mathbb{R}}$ resembles KMS
conditions; hence the name (see also [NÓØ21, App. A.2]).

  * •

Let $m\in M^{\alpha}$ be a fixed point of the modular flow and
$C_{m}=\overline{V_{+}(m)}\subseteq T_{m}(M)$. Then $\alpha$ induces a
$1$-parameter group of linear automorphisms on $T_{m}(M)$ whose infinitesimal
generator has the eigenvalues $-1,0,1$. We write $T_{m}(M)_{j}$, $j=-1,0,1$,
for the corresponding eigenspaces and consider the pointed cone

$C_{m}^{c}:=C_{m}\cap T_{m}(M)_{1}-C_{m}\cap T_{m}(M)_{-1}\subseteq T_{m}(M).$

Note that $C_{m}^{c}-C_{m}^{c}\subseteq T_{m}(M)$ is a is a vector space
complement of $T_{m}(M^{\alpha})=T_{m}(M)_{0}$.

Then the wedge domain in $m$ is defined as

$W_{M}(h)_{m}:=G^{h}_{e}.\mathop{{\rm Exp}}\nolimits_{m}(C_{m}^{c,0}),$

where $C_{m}^{c,0}:=(C_{m}^{c})^{0}$ is the relative interior of $C_{m}^{c}$
in $T_{m}(M)_{1}+T_{m}(M)_{-1}$ (cf. (10)) and $G^{h}_{e}=(G^{h})_{e}$ is the
connected subgroup with Lie algebra
${\mathfrak{g}}^{\tau_{h}}={\mathfrak{g}}_{0}(h)$. On the infinitesimal level,
this domain corresponds to the wedge $T_{m}(M)_{0}+C_{m}^{c}\subseteq
T_{m}(M)$.

Identifying $C$ with $C_{eH}\subseteq T_{eH}(M)\cong{\mathfrak{q}}$, the wedge
domain in the base point is

$W_{M}(H)_{eH}:=G^{h}_{e}.\mathop{{\rm
Exp}}\nolimits_{eH}((C_{+}+C_{-})^{0}),\quad\mbox{ where }\quad C_{\pm}=\pm
C\cap{\mathfrak{q}}_{\pm 1}(h).$

We define the polar wedge domain of $(M,h)$ as

$W_{M}(h):=\bigcup_{m\in M^{\alpha}}W_{M}(h)_{m}=\bigcup_{m\in
M^{\alpha}}\mathop{{\rm Exp}}\nolimits_{m}(C_{m}^{c,0}).$ (38)

######  Remark 47.

For $g\in G$ and $v\in T_{m}(M)$, the fact that $G$ acts by automorphisms on
the symmetric space $M$ implies that

$g.\mathop{{\rm Exp}}\nolimits_{m}(v)=\mathop{{\rm
Exp}}\nolimits_{gm}(g.v)\quad\mbox{ for }\quad g\in G,m\in M,v\in T_{m}(M).$
(39)

Now let $m=gH\in M^{\alpha}$ and identify $T_{eH}(M)$ with ${\mathfrak{q}}$.
For $h^{\prime}:=\mathop{{\rm Ad}}\nolimits(g)^{-1}h\in{\mathfrak{h}}$, we
then have

$C_{m}=g.C\quad\mbox{ and }\quad T(\alpha_{t})(g.x)=g.(e^{t\mathop{{\rm
ad}}\nolimits h^{\prime}}x)\quad\mbox{ for }\quad
t\in{\mathbb{R}},x\in{\mathfrak{q}},$

so that

$C_{m}^{c}=g.C^{c}(h^{\prime})\quad\mbox{ for }\quad
C^{c}(h^{\prime}):=C\cap{\mathfrak{q}}_{1}(h^{\prime})-C\cap{\mathfrak{q}}_{-1}(h^{\prime}).$
(40)

This leads to the relation

$\mathop{{\rm Exp}}\nolimits_{m}(C_{m}^{c,0})=\mathop{{\rm
Exp}}\nolimits_{gH}(g.C^{c}(h^{\prime})^{0})=g.\mathop{{\rm
Exp}}\nolimits_{eH}(C^{c}(h^{\prime})^{0}),$ (41)

which in turn entails with $G^{h}g=gG^{h^{\prime}}$

$\displaystyle W_{M}(h)_{gH}$ $\displaystyle=G^{h}_{e}.\mathop{{\rm
Exp}}\nolimits_{gH}(C_{m}^{c,0})=G^{h}_{e}g.\mathop{{\rm
Exp}}\nolimits_{eH}(C^{c}(h^{\prime})^{0})=gG^{h^{\prime}}_{e}.\mathop{{\rm
Exp}}\nolimits_{eH}(C^{c}(h^{\prime})^{0})$
$\displaystyle=g.W_{M}(h^{\prime})_{eh}\quad\mbox{ for }\quad gH\in
M^{\alpha}.$ (42)

Example 36 shows that the dimensions of the eigenspaces of $\mathop{{\rm
ad}}\nolimits h^{\prime}$ on ${\mathfrak{q}}$ are not always the same.

######  Remark 48.

(On the assumption $h\in{\mathfrak{h}}$)
(a) The domains $W_{M}^{+}(h)$ and $W_{M}^{\rm KMS}(h)$ are defined for any
Euler element $h\in\mathcal{E}({\mathfrak{g}})$. They only require the
corresponding flows in $M$ and $\mathcal{T}_{M}(C)$. However, these domains
may be trivial if $h\not\in{\mathfrak{h}}$ (cf. Proposition 66).

(b) If the vector field $X_{h}^{M}$ has a zero, then we may choose the base
point accordingly to obtain $h\in{\mathfrak{h}}$. An Euler element
$h\in\mathcal{E}({\mathfrak{g}})$ has this property if and only if its adjoint
orbit $\mathcal{O}_{h}$ intersects ${\mathfrak{h}}$ (cf. (36) in Definition
45).

For a compactly causal symmetric Lie algebra $({\mathfrak{g}},\tau,C)$, there
may be many different cones $C^{\prime}$ for which
$({\mathfrak{g}},\tau,C^{\prime})$ is compactly causal, but there is a rather
explicit classification of all these cones which is described in [NÓ21a, Thm.
3.6] in some detail in the $c$-dual context of non-compactly causal spaces. It
implies in particular that $C$ is contained in a uniquely determined maximal
$\mathop{{\rm Inn}}\nolimits_{\mathfrak{g}}({\mathfrak{h}})$-invariant
elliptic cone $C_{\mathfrak{q}}^{\rm max}$ (which need not be pointed; see
also (20) and the discussion in Subsection 2.3).

Recall the definition of $C_{\pm}:=(\pm C)\cap{\mathfrak{g}}_{\pm 1}$ from
Remark 23. The following proposition follows immediately by $c$-duality from
[NÓ21b, Prop. 3.8]. It has the interesting consequence that, if
${\mathfrak{g}}$ is reductive, then the cones $C_{\pm}$ remain the same when
we replace $C$ by the maximal cone $C^{\rm max}\subseteq{\mathfrak{q}}$
containing $C$ (see also Corollary 210(b) for spaces of group type).

######  Proposition 49.

Let $({\mathfrak{g}},\tau,C,h)$ be a reductive modular compactly causal
symmetric Lie algebra. Then

$C_{+}-C_{-}=C\cap{\mathfrak{q}}^{-\tau_{h}}=C^{\rm
max}\cap{\mathfrak{q}}^{-\tau_{h}}.$ (43)

This proposition implies in particular that the wedge domain $W_{M}(h)$
remains the same if we replace $C$ by $C^{\rm max}$. So it only depends on the
“direction” of the cone $C$, not on its specific shape. This has interesting
consequences for the global structure of $W_{M}(h)$.

######  Remark 410.

(Factorization of wedge domains) Let $({\mathfrak{g}},\tau,C)$ be a reductive
compactly causal symmetric Lie algebra. Then $({\mathfrak{g}},\tau)$
decomposes as

$({\mathfrak{g}},\tau)\cong({\mathfrak{g}}_{0},\tau_{0})\oplus\bigoplus_{j=1}^{N}({\mathfrak{g}}_{j},\tau_{j}),$

where ${\mathfrak{g}}_{0}$ contains the center and all compact ideals and the
$\tau$-invariant ideals ${\mathfrak{g}}_{j}$, $j\geq 1$, are either simple
hermitian or irreducible of group type (cf. [NÓ21a, Prop. 2.5] and Subsection
2.3). Decomposing
${\mathfrak{q}}={\mathfrak{q}}_{0}\oplus\bigoplus_{j=1}^{N}{\mathfrak{q}}_{j}$
accordingly,

$C^{\rm max}={\mathfrak{q}}_{0}+\sum_{j=1}^{N}C^{\rm max}_{j}\quad\mbox{ with
}\quad C_{j}^{\rm max}=C^{\rm max}\cap{\mathfrak{q}}_{j},$

where the cones $C_{j}^{\rm max}$ are pointed. If $M$ is the simply connected
symmetric space corresponding to $({\mathfrak{g}},\tau)$, it follows that

$M\cong M_{0}\times M_{1}\times\cdots\times M_{N},$ (44)

and

$W_{M}(h)=M_{0}\times\prod_{j=1}^{N}W_{M_{j}}(h_{j}),$ (45)

where $h=h_{0}+\sum_{j=1}^{N}h_{j}$ with $h_{j}\in{\mathfrak{g}}_{j}$ and
$h_{0}\in{\mathfrak{z}}({\mathfrak{g}})$.

We have a similar global decomposition if $G$ is semisimple with
$Z(G)=\\{e\\}$. Then

$G=\mathop{{\rm Aut}}\nolimits(G)_{e}\cong\prod_{j=0}^{N}\mathop{{\rm
Aut}}\nolimits({\mathfrak{g}}_{j}),\quad
H=G^{\tau}\cong\prod_{j=0}^{N}\mathop{{\rm
Inn}}\nolimits({\mathfrak{g}}_{j})^{\tau_{j}},\quad
M\cong\prod_{j=0}^{N}M_{j}.$

######  Example 411.

(The case of compact Lie algebras) If ${\mathfrak{g}}$ is a compact Lie
algebra, then $\mathop{{\rm ad}}\nolimits x$ has purely imaginary spectrum for
every $x\in{\mathfrak{g}}$, so that $\mathcal{E}({\mathfrak{g}})=\emptyset$.
However, we may consider central elements $h\in{\mathfrak{z}}({\mathfrak{g}})$
as a degenerate kind of Euler elements with $\mathop{{\rm ad}}\nolimits h=0$
and ${\mathfrak{g}}={\mathfrak{g}}_{0}(h)$.

There are many compactly causal symmetric spaces for which ${\mathfrak{g}}$ is
a compact Lie algebra. For instance, a compact Lie algebra ${\mathfrak{g}}$
contains pointed generating invariant cones $C$ if and only if
${\mathfrak{z}}({\mathfrak{g}})\not=\\{0\\}$, and then
${\mathfrak{z}}({\mathfrak{g}})\cap C^{0}\not=\emptyset$ holds for any such
cone. The Lie algebra

${\mathfrak{g}}={\mathfrak{u}}_{n}({\mathbb{C}})\quad\mbox{ with }\quad
C:=\\{X\in{\mathfrak{g}}\colon-iX\geq 0\\}$

is an important example. The corresponding causal group $(\mathop{\rm
U{}}\nolimits_{n}({\mathbb{C}}),C)$ is the conformal compactification of the
euclidean Jordan algebra $\mathop{{\rm Herm}}\nolimits_{n}({\mathbb{C}})$,
which, for $n=2$, is isomorphic to the $4$-dimensional Minkowski space
([FK94]). Involutions $\tau$ of ${\mathfrak{u}}_{n}({\mathbb{C}})$ with
$\tau(C)=-C$ are obtained from involutive automorphisms of
$\mathop{{\mathfrak{su}}}\nolimits_{n}({\mathbb{C}})$, extending by
$\tau(i\mathbf{1})=-i\mathbf{1}$.

For any compactly causal symmetric Lie algebra $({\mathfrak{g}},\tau,C)$ and
$h\in{\mathfrak{z}}({\mathfrak{g}})$, we have $C_{\pm}=\\{0\\}$, so that
$W_{M}(h)=\emptyset$. Further, the triviality of the modular flow implies
$W_{M}^{\rm KMS}(h)=\emptyset$, but if
$h\in{\mathfrak{z}}({\mathfrak{g}})^{-\tau}\cap C^{0}$, then $W_{M}^{+}(h)=M$.

##  5 Wedge domains in spaces of group type

For our analysis of wedge domains in compactly causal symmetric spaces, we
shall follow the strategy to first study spaces of group type $G\cong(G\times
G)/\Delta_{G}$, and then use embeddings into these spaces implemented by the
quadratic representation to derive corresponding results in general.

Let $(G,C)$ be a causal symmetric spaces of group type, i.e., $G$ is a
connected Lie group and $C\subseteq{\mathfrak{g}}$ a pointed generating
invariant cone. We further assume that the universal complexification
$\eta_{G}\colon G\to G_{\mathbb{C}}$ has discrete kernel. We consider $G$ as
the symmetric space $(G\times G)/\Delta_{G}$ with $\tau^{G\times
G}(g_{1},g_{2})=(g_{2},g_{1})$ and $G\times G$ acting by
$(g_{1},g_{2}).g=g_{1}gg_{2}^{-1}$. The associated causal symmetric Lie
algebra is

$({\mathfrak{g}}\oplus{\mathfrak{g}},\tau,C),\quad\mbox{ where
}\quad\tau(x,y)=(y,x),$

and

$C\subseteq{\mathfrak{g}}\cong{\mathfrak{q}}=\\{(x,-x)\colon
x\in{\mathfrak{g}}\\}$

is an $\mathop{{\rm Ad}}\nolimits(G)$-invariant pointed generating invariant
cone. The Euler element $h\in\mathcal{E}({\mathfrak{g}})$ defines the Euler
element $(h,h)\in{\mathfrak{g}}\times{\mathfrak{g}}$ which generates the flow
$\alpha_{t}(g)=\exp(th)g\exp(-th)$ on $G$. If $\tau_{h}^{G}$ is an involution
on $G$ integrating $\tau_{h}=e^{\pi i\mathop{{\rm ad}}\nolimits h}$ (cf.
Remark 212), then this involution extends to the involution
$\tau^{G}_{h}\times\tau_{h}^{G}$ on $G\times G$.

We recall the closed/open complex Olshanski semigroup $\Gamma_{G}(C)$ from
Definition 41. We start with the preparation of the analysis of wedge domains
in modular symmetric spaces of group type.

######  Proposition 51.

Let $G$ be a connected Lie group with Lie algebra ${\mathfrak{g}}$,
$h\in\mathcal{E}({\mathfrak{g}})$ an Euler element and
$C\subseteq{\mathfrak{g}}$ an $\mathop{{\rm Ad}}\nolimits(G)$-invariant
pointed closed convex cone such that:

  * (i)

$-\tau_{h}(C)=C$. 444If ${\mathfrak{g}}$ is not reductive and $C$ is pointed
and generating, then $C\cap{\mathfrak{z}}({\mathfrak{g}})\not=\\{0\\}$ ([Ne00,
Thm. VII.3.10]).
As $\tau_{h}|_{{\mathfrak{z}}({\mathfrak{g}})}=\mathop{{\rm
id}}\nolimits_{{\mathfrak{z}}({\mathfrak{g}})}$, the condition
$-\tau_{h}(C)=C$ can only be satisfied if $C$ is not generating (see Example
26).

  * (ii)

$\tau_{h}$ integrates to an automorphism $\tau_{h}^{G}$ of $G$.

  * (iii)

The kernel of the universal complexification $\eta_{G}\colon G\to
G_{\mathbb{C}}$ is discrete.

Let $g^{\sharp}=\tau(s)^{-1}$, $g\in G$. Then the following assertions hold:

  * (a)

Let $C^{c}:=C_{+}+C_{-}$ with $C_{\pm}:=\pm C\cap{\mathfrak{g}}_{\pm 1}(h)$.
Then the convex cones $C^{c},C_{+}$ and $C_{-}$ are $G^{h}$-invariant and the
subset

$\Gamma_{G^{h}}(C^{c}):=G^{h}\exp(C^{c})$ (46)

is a real Olshanski semigroup, associated to the symmetric Lie group
$(G,\tau_{h}^{G})$. In particular, the polar map $G^{h}\times
C^{c}\to\Gamma_{G^{h}}(C^{c})$ is a homeomorphism and $\Gamma_{G^{h}}(C^{c})$
is invariant under $s\mapsto s^{\sharp}$.

  * (b)

The semigroup

$S(C,h):=\\{g\in G\colon h-\mathop{{\rm Ad}}\nolimits(g)h\in C\\}$

is invariant under $s\mapsto s^{\sharp}$ and satisfies

$\displaystyle S(C,h)$
$\displaystyle=\exp(C_{+})G^{h}\exp(C_{-})=G^{h}\exp(C_{+})\exp(C_{-})$
$\displaystyle=\exp(C_{-})\exp(C_{+})G^{h}=\exp(C_{-})G^{h}\exp(C_{+})=\Gamma_{G^{h}}(C^{c}).$
(47)

If $C$ is generating,

$S(C,h)^{0}=\Gamma_{G^{h}}(C^{c,0})=S(C^{0},h):=\\{g\in G\colon h-\mathop{{\rm
Ad}}\nolimits(g)h\in C^{0}\\}.$ (48)

It is invariant under $g\mapsto g^{\sharp}$.

  * (c)

The subset of those elements $g\in G$ for which the orbit map

$\alpha^{g}\colon{\mathbb{R}}\to G,\quad\alpha^{g}(t)=\exp(th)g\exp(-th)$

extends to a continuous map
$\alpha^{g}\colon\overline{\mathcal{S}_{\pi}}\to\Gamma_{G}(C)$, which is
holomorphic on $\mathcal{S}_{\pi}$ when composed with the natural map
$\Gamma_{G}(C)\to G_{\mathbb{C}}$, coincides with the closed semigroup
$\Gamma_{G^{h}}(C^{c})$. If $C$ has interior points, then

$W_{G}^{\rm KMS}(h)=\Gamma_{G^{h}}(C^{c,0}).$ (49)

  * (d)

Let $X_{h}^{G}\in\mathcal{V}(G)$ be the vector field defined by
$X_{h}^{G}(g):=\frac{d}{dt}\big{|}_{t=0}\alpha_{t}(g).$ We write

$G\times TG\to TG,\quad(g,x)\mapsto g.x$

for left translation of vectors on $G$. Then

$S(C,h)=\\{g\in G\colon X_{h}^{G}(g)\in g.C\\},\quad\mbox{ and }\quad
S(C,h)^{0}=\\{g\in G\colon X_{h}^{G}(g)\in g.C^{0}\\}\ $ (50)

if $C$ is generating.

  * (e)

On $\Gamma_{G}(C)$ we consider the antiholomorphic involution defined by

$\overline{\tau}_{h}(g\exp(ix))=\tau_{h}^{G}(g)\exp(-i\tau_{h}(x)).$ (51)

Its set of fixed points coincides with

$\Gamma_{G}(C)^{\overline{\tau}_{h}}=G^{\tau_{h}^{G}}\exp(iC^{-\tau_{h}})$
(52)

and, if $C$ has interior points, then the interior of the identity component
$\Gamma_{G}(C)^{\overline{\tau}_{h}}_{e}$ satisfies

$S(C,h)_{e}^{0}=\alpha_{-\pi
i/2}\big{(}(\Gamma_{G}(C)^{\overline{\tau}_{h}})_{e}^{0}\big{)}.$

###### Proof.

Similar results are stated in [Ne19] for simply connected Lie groups $G$. If
$q_{G}\colon\widetilde{G}\to G$ is the universal covering group of $G$, they
apply to the group $\widetilde{G}$. It therefore suffices to derive everything
from [Ne19].

(a) That $C^{c},C_{+}$ and $C_{-}$ are $G^{h}$ invariant follows from
$\mathop{{\rm Ad}}\nolimits(G^{c}){\mathfrak{g}}_{j}(h)={\mathfrak{g}}_{j}(h)$
and the invariance of $C$. The remainder of (a) follows by applying [Ne19,
Prop. 2.6] to $\widetilde{G}$ because $\ker(q_{G})\subseteq\widetilde{G}^{h}$
and $G^{h}=q_{G}(\widetilde{G}^{h})$.

(b) First we note that for $g\in G$ we have

$-\tau(h-\mathop{{\rm Ad}}\nolimits(g^{\sharp})h)=\mathop{{\rm
Ad}}\nolimits(g^{-1})h-h=\mathop{{\rm Ad}}\nolimits(g^{-1})(h-\mathop{{\rm
Ad}}\nolimits(g)h)\in C$

As $C$ is $G$-invariant. Thus $h-\mathop{{\rm
Ad}}\nolimits(g^{\sharp})h\in-\tau(C)=C$. That the four middle sets in ((b))
agree now follows now from the $\mathop{{\rm Ad}}\nolimits(G^{h})$ invariance
of $C_{\pm}$.

For $\widetilde{S}(C,h):=\\{g\in\widetilde{G}\colon h-\mathop{{\rm
Ad}}\nolimits(g)h\in C\\}$ we have $\widetilde{S}(C,h)=q_{G}^{-1}(S(C,h))$, so
that the corresponding result for $\widetilde{G}$ ([Ne19, Thm. 2.16]) and the
invariance of $G^{h},\widetilde{S}(C,h)$ and
$\widetilde{\Gamma}_{G^{h}}(C^{c}):=\widetilde{G}^{h}\exp(C^{c})$ under
multiplication with the central subgroup $\ker(q_{G})$ show that it also holds
for $G$.

Assume, in addition, that $C$ is generating, so that $C^{0}\not=\emptyset$.
Then $S(C^{0},h)\subseteq S(C,h)$ is an open subsemigroup, hence contained in

$S(C,h)^{0}=G^{h}\exp(C^{c,0})=G^{h}\exp(C_{+}^{0})\exp(C_{-}^{0})=\exp(C_{-}^{0})\exp(C_{+}^{0})G^{h}.$

To show equality, write an element $g\in S(C,h)^{0}$ as

$g=\exp(x_{-})\exp(x_{+})g^{\prime}\quad\mbox{ with }\quad g^{\prime}\in
G^{h}_{C},x_{\pm}\in C_{\pm}^{0}.$

Then

$\displaystyle\mathop{{\rm Ad}}\nolimits(g)h$ $\displaystyle=e^{\mathop{{\rm
ad}}\nolimits x_{-}}e^{\mathop{{\rm ad}}\nolimits x_{+}}h=e^{\mathop{{\rm
ad}}\nolimits x_{-}}(h+[x_{+},h])=e^{\mathop{{\rm ad}}\nolimits
x_{-}}(h-x_{+})$
$\displaystyle=h-x_{+}+[x_{-},h]-[x_{-},x_{+}]-[x_{-},[x_{-},x_{+}]]$
$\displaystyle=h-\mathop{{\rm Ad}}\nolimits(\exp x_{-})(x_{+}-x_{-})\in
h-C^{0}$

as $C^{0}$ is $G$-invariant. It follows that $h-\mathop{{\rm
Ad}}\nolimits(g)h\in C^{0}$. and hence $g\in S(C^{0},h)$.

(c) For $\widetilde{G}$ and the closed cone, this is [Ne19, Thm. 2.21]. As
$\widetilde{G}^{h}$ and the specified subset of $G$ are
$\ker(q_{G})$-invariant, the assertion also holds for $G$.

For $z=x+iy\in{\mathfrak{g}}_{\mathbb{C}}$ with $x,y\in{\mathfrak{g}}$, we
write $x=\mathop{{\rm Re}}\nolimits z$ and $y=\mathop{{\rm Im}}\nolimits z$.
To verify (49), for $g=g_{0}\exp(x_{-}+x_{+})$ with $g_{0}\in G^{h}$,
$x_{\pm}\in C_{\pm}=C\cap{\mathfrak{g}}_{\pm 1}(h)$, and $0<t<\pi$, we note
that $\sin(t)>0$ and

$\displaystyle\alpha_{it}(g_{0}\exp(x_{+}+x_{-}))$
$\displaystyle=g_{0}\exp(e^{it}x_{+}+e^{-it}x_{-})$
$\displaystyle=g_{0}\exp((\cos t+i\sin t)x_{+}+(\cos t-i\sin t)x_{-}).$

Thus

$\mathop{{\rm Im}}\nolimits(e^{it}x_{+}e^{-it}x_{-})=\sin(t)(x_{+}-x_{-}).$

This element is contained in the interior $C^{0}$ if and only if $x_{\pm}\in
C_{\pm}^{0}$, and in this case

$\exp(e^{it}x_{+}+e^{-it}x_{-})\in\exp({\mathfrak{g}}+iC^{0})\subseteq\Gamma_{G}(C^{0}).$

We therefore obtain (49).

(d) With the notation for left and right translations by $G$ on $TG$ (see the
notation introduced in the introduction), we have

$X_{h}^{G}(g)=h.g-g.h=g.(\mathop{{\rm Ad}}\nolimits(g)^{-1}h-h).$

Hence (50) follows immediately from $\mathop{{\rm Ad}}\nolimits(g)C=C$, ((b))
and (48).

(e) Equation (52) follows from (51). We observe that

$\alpha_{\pi i/2}(S(C,h)_{e})=(G^{h})_{e}\exp\big{(}e^{\frac{\pi
i}{2}\mathop{{\rm ad}}\nolimits
h}(C_{+}+C_{-})\big{)}=(G^{h})_{e}\exp(i(C_{+}-C_{-})),$

so that $\alpha_{\pi i/2}(S(C,h)_{e})$ is the identity component of the fixed
point set $\Gamma_{G}(C)^{\overline{\tau}_{h}}$ in the closed Olshanski
semigroup $\Gamma_{G}(C)$. If $C$ has interior points, this argument shows
that $\alpha_{\pi i/2}(S(C,h)_{e}^{0})$ is the connected component of the
fixed point submanifold in the open semigroup $\Gamma_{G}(C^{0})$ whose
closure contains the identity. This proves (e). ∎

To provide a good context to deal also with group type spaces that are not
reductive, we have to deal with invariant cones $C\subseteq{\mathfrak{g}}$
which are not generating. This is mainly due to the fact that the requirement
of $C$ being generating is not compatible with $-\tau_{h}(C)=C$. As in
Subsection 2.1, this problem can be overcome by observing that

${\mathfrak{g}}_{C}=C-C\trianglelefteq{\mathfrak{g}}$

is an ideal of the Lie algebra ${\mathfrak{g}}$. We consider the corresponding
integral subgroup $G_{C}\trianglelefteq G$, endowed with its natural Lie group
structure (it is closed if $G$ is simply connected). Then

$(G\times G)_{C}:=\\{(g_{1},g_{2})\in G\colon g_{1}g_{2}^{-1}\in G_{C}\\}$

is a Lie group containing the diagonal $\Delta_{G}$ and isomorphic to
$G_{C}\rtimes G$, where $G$ acts by conjugation on $G_{C}$. Then

$G_{C}\cong(G\times G)_{C}/\Delta_{G}$

is a symmetric space on which

$V_{+}(g):=g.C^{0}$

defines a $(G\times G)_{C}$-invariant field of pointed open cones (a causal
structure). The corresponding symmetric Lie algebra is

$({\mathfrak{g}}\oplus{\mathfrak{g}})_{C}=\\{(x,y)\in{\mathfrak{g}}\oplus{\mathfrak{g}}\colon
x-y\in{\mathfrak{g}}_{C}\\}\quad\mbox{ with }\quad\tau(x,y)=(y,x).$

Here ${\mathfrak{q}}=\\{(x,-x)\colon
x\in{\mathfrak{g}}_{C}\\}\cong{\mathfrak{g}}_{C}$ contains the pointed
generating cone $\\{(x,-x)\colon x\in C\\}\cong C$. We now determine the wedge
domains in the causal symmetric space $M=G_{C}$ for the structure defined by
$(({\mathfrak{g}}\oplus{\mathfrak{g}})_{C},\tau,C)$.

We do not assume that $C$ is generating in ${\mathfrak{g}}$, but it is natural
to assume that the two cones $C_{\pm}$ generate ${\mathfrak{g}}_{\pm 1}(h)$ to
ensure that the semigroup $S(C,h)$ has interior points.

######  Theorem 52.

(Wedge domains in spaces of group type) Suppose that $C_{\pm}$ generate
${\mathfrak{g}}_{\pm 1}(h)$. Then the wedge domains in the compactly causal
symmetric space $(G_{C},C)$ corresponding to the modular compactly causal
symmetric Lie algebra
$(({\mathfrak{g}}\oplus{\mathfrak{g}})_{C},\tau,C,(h,h))$ are

$W_{G_{C}}^{+}(h)=W_{G_{C}}^{\rm
KMS}(h)=W_{G_{C}}(h)=G_{C}^{h}\exp(C_{+}^{0}+C_{-}^{0}).$

###### Proof.

The fixed point set of the modular flow in $M=G_{C}$ is the centralizer
$G_{C}^{h}$ of $G$. For a pair $(g_{1},g_{2})\in(G\times G)_{C}$, we have
$m:=(g_{1},g_{2}).e=g_{1}g_{2}^{-1}\in G_{C}^{h}$ if and only if $\mathop{{\rm
Ad}}\nolimits(g_{1})^{-1}h=\mathop{{\rm Ad}}\nolimits(g_{2})^{-1}h$, i.e.,
$\mathop{{\rm
Ad}}\nolimits(g_{1},g_{2})^{-1}(h,h)\in{\mathfrak{h}}=\Delta_{\mathfrak{g}}$.
The invariance of $C$ under $\mathop{{\rm Ad}}\nolimits(G)$ implies the
invariance of the cones $C_{\pm}=\pm C\cap{\mathfrak{g}}_{\pm 1}(h)$ under
$\mathop{{\rm Ad}}\nolimits(G^{h})$. Therefore
$C^{c}=C_{+}+C_{-}\subseteq{\mathfrak{g}}^{-\tau_{h}}$ is invariant under
$\mathop{{\rm Ad}}\nolimits(G^{h})$.

For $m=g_{1}g_{2}^{-1}\in G_{C}^{h}$, the modular flow acts on
$T_{m}(G_{C})=g_{1}.{\mathfrak{g}}_{C}.g_{2}^{-1}=m.{\mathfrak{g}}_{C}$ by

$\alpha_{t}(m.x)=m.e^{t\mathop{{\rm ad}}\nolimits h}x,$

so that $C_{m}^{c}:=m.C^{c}$ (here we use the dot notation for left and right
translations on $T(G)$). Therefore the polar wedge domain in $M=G_{C}$ is

$W_{G_{C}}(h)=\bigcup_{m\in G_{C}^{h}}m\exp(C^{c,0})=G_{C}^{h}\exp(C^{c,0}).$
(53)

Next we derive from Proposition 51(c) that

$W_{G_{C}}^{\rm KMS}(h)=G^{h}_{C}\exp(C_{-}^{0}+C_{+}^{0})=W_{G_{C}}(h).$ (54)

Finally, we observe that Proposition 51(a),(d) imply that

$W_{G_{C}}^{+}(h)=G_{C}^{h}\exp(C_{+}^{0}+C_{-}^{0})=W_{G_{C}}(h).$ (55)

The assertion now follows from (54) and (55). ∎

The open subset $W_{G_{C}}(h)$ neet not be connected, but its connected
components are parametrized by the group $\pi_{0}(G_{C}^{h})$ of connected
components of $G_{C}^{h}$.

######  Remark 53.

(Some information on $\pi_{0}(G^{h})$)

(a) As the example $G=\widetilde{\mathop{{\rm
SL}}}\nolimits_{2}({\mathbb{R}})$ with the Cayley type involution

$\tau\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}a&-b\\\
-c&d\end{pmatrix}$

shows, $G^{h}$ is in general not connected. It contains
$Z(G)\cong{\mathbb{Z}}$ and

$(G^{h})_{e}=\exp({\mathbb{R}}h)\quad\mbox{ for }\quad
h=\frac{1}{2}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

This shows that $\pi_{0}(G^{h})\cong{\mathbb{Z}}$. This example also shows
that, in general

$G^{h}\not\subseteq G^{\tau^{G}},$

because here $\tau^{G}$ acts on $Z(G)$ by inversion (cf. [MN21, Ex. 2.10(d)]).

(b) If $G$ is contained in $G_{\mathbb{C}}$, then $(G_{\mathbb{C}})^{h}$
coincides with the centralizer of the circle group $\overline{\exp(ih)}$,
hence is connected ([HN12, Cor. 14.3.10]). Therefore
$G^{h}=(G_{\mathbb{C}})^{h}\cap G$ is the fixed point group of the complex
conjugation (with respect to $G$) on the connected complex group
$(G_{\mathbb{C}})^{h}$. Thus $\pi_{0}(G^{h})$ is a finite elementary abelian
$2$-group, i.e., isomorphic to ${\mathbb{Z}}_{2}^{k}$ for some
$k\in{\mathbb{N}}_{0}$ ([Lo69, Thm. IV.3.4]).

##  6 Wedge domains in extendable causal symmetric spaces

In this section we turn to the wedge domains in compactly causal symmetric
spaces. We assume that $({\mathfrak{g}},\tau,C)$ is extendable in the sense of
Subsection 2.1, i.e., there exists a $G$-invariant convex cone
$C_{\mathfrak{g}}\subset{\mathfrak{g}}$ such that $-\tau
C_{\mathfrak{g}}=C_{\mathfrak{g}}$ and $C_{\mathfrak{g}}\cap{\mathfrak{q}}=C$.
This always holds if ${\mathfrak{g}}$ is reductive by the Extension Theorem
24. We write
${\mathfrak{g}}_{C}=C_{\mathfrak{g}}-C_{\mathfrak{g}}\trianglelefteq{\mathfrak{g}}$
for the ideal generated by $C_{\mathfrak{g}}$ and $G_{C}\trianglelefteq G$ for
the connected normal subgroup whose Lie algebra is ${\mathfrak{g}}_{C}$ (see
(24)).

The main result of this section is Theorem 65, asserting that all three wedge
domains in $M=G/H$ are the same. This is first proved for the special case
$H=G^{\tau^{G}}$ in Theorem 63. The proof of this theorem builds heavily on
the group case (Theorem 52). We conclude this section with a brief discussion
of the assumption that the Euler element $h$ is contained in ${\mathfrak{h}}$,
i.e., that the corresponding modular flow on $M$ has a fixed point.

The following proposition identifies the basic connected components of the
wedge domains in $G^{\sharp}_{e}$.

######  Proposition 61.

Let $(G,\tau^{G})$ be a connected symmetric Lie group corresponding to the
modular compactly causal symmetric Lie algebra $({\mathfrak{g}},\tau,C,h)$,

$M_{G}:=G/G^{\tau^{G}}\cong G^{\sharp}_{e}=(G_{C}^{\sharp})_{e},$

and let $C_{\mathfrak{g}}\subseteq{\mathfrak{g}}$ be a pointed invariant
closed convex cone with

$C_{\mathfrak{g}}\cap{\mathfrak{q}}=C\quad\mbox{ and
}\quad\quad-\tau(C_{\mathfrak{g}})=-\tau_{h}(C_{\mathfrak{g}})=C_{\mathfrak{g}}.$

For the real Olshanski semigroup
$S(C_{\mathfrak{g}},h)=G_{C}^{h}\exp(C_{\mathfrak{g}}^{c})$, the following
assertions hold:

  * (a)

The basic connected component of the wedge domains in $M_{G}$ coincide:

$W_{M_{G}}(h)_{e}=W_{M_{G}}^{+}(h)_{e}=W_{M_{G}}^{\rm KMS}(h)_{e},$

and this domain can be obtained from the semigroup
$S(C_{\mathfrak{g}},h)=G_{C}^{h}\exp(C_{\mathfrak{g}}^{c})$ as

$S(C_{\mathfrak{g}}^{0},h)^{\sharp}_{e}=\bigcup_{g\in(G_{C}^{h})_{e}}g\exp(C_{+}^{0}+C_{-}^{0})g^{\sharp}\cong(G_{C}^{h})_{e}\times_{(G_{C}^{h})_{e}^{\tau}}(C_{+}^{0}+C_{-}^{0}).$
(56)

  * (b)

$W_{M_{G}}(h)_{e}$ is the orbit $S(C_{\mathfrak{g}}^{0},h)_{e}.e$ of the base
point under the action of the open semigroup $S(C_{\mathfrak{g}}^{0},h)_{e}$
on ${M_{G}}=(G_{C}^{\sharp})_{e}$ by $g.x:=gxg^{\sharp}$.

###### Proof.

The flow $\alpha_{t}$ commutes with the involution $\tau$. If
$m\in{M_{G}}\cong(G_{C}^{\sharp})_{e}\subseteq G_{C}^{\sharp}$ is contained in
$W_{M_{G}}^{\rm KMS}(h)$ (Definition 46), then

$\alpha_{z}(m)\in\mathcal{T}_{M_{G}}(C)\subseteq\mathcal{T}_{G_{C}}(C_{\mathfrak{g}})=\Gamma_{G_{C}}(C_{\mathfrak{g}}^{0})\quad\mbox{
for }\quad z\in\mathcal{S}_{\pi}$

(cf. Remark 43), so that Theorem 52 implies that

$W_{M_{G}}^{\rm KMS}(h)\subseteq{M_{G}}\cap W_{G_{C}}^{\rm KMS}(h)={M_{G}}\cap
S(C_{\mathfrak{g}}^{0},h)\subseteq S(C_{\mathfrak{g}}^{0},h)^{\sharp},$ (57)

and thus

$\displaystyle W_{M_{G}}^{\rm KMS}(h)_{e}$ $\displaystyle\subseteq
S(C_{\mathfrak{g}}^{0},h)^{\sharp}_{e}=(G_{C}^{h})_{e}.\mathop{{\rm
Exp}}\nolimits((C_{\mathfrak{g}}^{c,0})^{-\tau})$
$\displaystyle{\buildrel\eqref{eq:doubledcone2}\over{=}}\
(G_{C}^{h})_{e}.\mathop{{\rm
Exp}}\nolimits((C_{+}+C_{-})^{0})=W_{M_{G}}(h)_{e}.$ (58)

The invariance of both sides under $(G_{C}^{h})_{e}$ reduces the verification
of the converse inclusion to showing that

$\mathop{{\rm Exp}}\nolimits(C_{+}^{0}+C_{-}^{0})\subseteq W_{M_{G}}^{\rm
KMS}(h).$ (59)

For $x_{\pm}\in C_{\pm}^{0}$ and $0<t<\pi$, we have $\sin t>0$ and hence

$\mathop{{\rm Im}}\nolimits(e^{it}x_{+}+e^{-it}x_{-})=\sin(t)(x_{+}-x_{-})\in
C^{0}.$

Now (59) follows from $\alpha_{it}\mathop{{\rm
Exp}}\nolimits(x_{+}+x_{-})=\mathop{{\rm
Exp}}\nolimits(e^{it}x_{+}+e^{-it}x_{-})$,
$\exp(e^{it}x_{+}+e^{-it}x_{-})\in\Gamma_{G_{C}}(C_{\mathfrak{g}}^{0})$ and

$\mathcal{T}_{M_{G}}(C)=\\{ss^{\sharp}\colon
s\in\Gamma_{G_{C}}(C_{\mathfrak{g}}^{0})\\}\subseteq\Gamma_{G_{C}}(C_{\mathfrak{g}}^{0})=\mathcal{T}_{G_{C}}(C_{\mathfrak{g}})$

(Remark 43(a)). We thus obtain

$W_{M_{G}}^{\rm KMS}(h)_{e}=(G_{C}^{h})_{e}.\mathop{{\rm
Exp}}\nolimits(C_{-}^{0}+C_{+}^{0})=W_{M_{G}}(h)_{e}.$ (60)

Next we observe that $C^{0}=C_{\mathfrak{g}}^{0}\cap{\mathfrak{q}}$ implies
that, for $m=gg^{\sharp}\in{M_{G}}=(G_{C}^{\sharp})_{e}\subseteq G_{C}$, we
have

$V^{M_{G}}_{+}(m)=g.C^{0}.g^{\sharp}=T_{m}({M_{G}})\cap
g.C_{\mathfrak{g}}^{0}.g^{\sharp}=T_{m}({M_{G}})\cap
V_{+}^{G_{C}}(gg^{\sharp}).$ (61)

In view of Theorem 52, this shows that

$W_{M_{G}}^{+}(h)=W_{G_{C}}^{+}(h)\cap{M_{G}}=S(C_{\mathfrak{g}},h)^{0}\cap{M_{G}}.$
(62)

The semigroup $S(C_{\mathfrak{g}},h)=G_{C}^{h}\exp(C_{\mathfrak{g}}^{c})$ is a
real Olshanski semigroup corresponding to the involution $\tau_{h}$ on
${\mathfrak{g}}$ (Proposition 51). Further, $\tau$ and $\tau_{h}$ commute and