Wedge domains in non-compactly causal symmetric spaces

Abstract

This article is part of an ongoing project aiming at the connections between causal structures on homogeneous spaces, Algebraic Quantum Field Theory, modular theory of operator algebras and unitary representations of Lie groups. In this article we concentrate on non-compactly causal symmetric spaces G/H. This class contains de Sitter space but also other spaces with invariant partial ordering. The central ingredient is an Euler element h in the Lie algebra of ${{\mathfrak {g}}}$. We define three different kinds of wedge domains depending on h and the causal structure on G/H. Our main result is that the connected component containing the base point eH of these seemingly different domains all agree. Furthermore we discuss the connectedness of those wedge domains. We show that each of these spaces has a natural extension to a non-compactly causal symmetric space of the form $G_{{\mathbb {C}}}/G^c$ where $G^c$ is a certain real form of the complexification $G_{{\mathbb {C}}}$ of G. As $G_{{\mathbb {C}}}/G^c$ is non-compactly causal, it also contains three types of wedge domains. Our results says that the intersection of these domains with G/H agree with the wedge domains in G/H.

Appendices

Appendix

1.1 A Irreducible modular ncc symmetric Lie algebras

See Table 1.

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

B Some calculations in $\mathop {{\mathfrak {sl}}}\nolimits _2({{\mathbb {R}}})$

Arguments are often reduced to relatively simple $\mathop {{\mathfrak {sl}}}\nolimits _2({{\mathbb {R}}})$ calculations. We therefore collect the basic notations and calculations here in one place for reference. For ${{\mathfrak {g}}}= \mathop {{\mathfrak {sl}}}\nolimits _2({{\mathbb {R}}})$, we fix the Cartan involution $\theta (x) = - x^\top$, so that

$$\begin{aligned} {{\mathfrak {k}}}= \mathop {{\mathfrak {so}}}\nolimits _2({{\mathbb {R}}}) \quad \text{ and } \quad {{\mathfrak {p}}}= \{ x \in \mathop {{\mathfrak {sl}}}\nolimits _2({{\mathbb {R}}}) : x^\top = x \}.\end{aligned}$$

The basis elements

$$\begin{aligned} h^0 := \frac{1}{2} \begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix},\quad e^0 = \begin{pmatrix} 0 &{} 1 \\ 0 &{} 0 \end{pmatrix}\quad \text{ and } \quad f^0 = \begin{pmatrix} 0 &{} 0 \\ 1 &{} 0 \end{pmatrix} \end{aligned}$$

(B.1)

and

$$\begin{aligned} h^1 = \frac{1}{2} \begin{pmatrix} 0 &{} 1 \\ 1 &{} 0 \end{pmatrix} = \frac{1}{2}(e^0 + f^0),\quad e^1 = \frac{1}{2}\begin{pmatrix} -1 &{} 1 \\ -1 &{} 1 \end{pmatrix}\quad \text{ and } \quad f^1= \frac{1}{2} \begin{pmatrix} -1 &{} -1 \\ 1 &{} 1 \end{pmatrix} \end{aligned}$$

(B.2)

satisfy

$$\begin{aligned}{}[h^j, e^j] = e^j, \quad [h^j, f^j] = -f^j, \quad {[}e^j, f^j] = 2 h^j \quad \text{ and } \quad \theta (e^j) = -f^j. \end{aligned}$$

For the involution

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

we have

$$\begin{aligned} {{\mathfrak {h}}}= {{\mathfrak {g}}}^\tau = {{\mathbb {R}}}h^0, \quad {{\mathfrak {q}}}= {{\mathfrak {g}}}^{-\tau } = {{\mathbb {R}}}h^1 + {{\mathbb {R}}}(e^0 - f^0) \quad \text {and} \quad C = [0,\infty ) e^0 + [0,\infty ) f^0 \end{aligned}$$

is a hyperbolic $\mathop {\mathrm{Inn}}\nolimits ({{\mathfrak {h}}})$-invariant cone in ${{\mathfrak {q}}}$, containing $h^1$ as a causal Euler element. Further

$$\begin{aligned} C_+ = [0,\infty ) e^0 =\left\{ \begin{pmatrix} 0 &{} x\\ 0 &{} 0\end{pmatrix} : x\ge 0\right\} \quad \text{ and } \quad C_- = - [0,\infty ) f^0 =\left\{ \begin{pmatrix} 0 &{} 0\\ -y &{} 0\end{pmatrix}: y\ge 0\right\} \end{aligned}$$

lead to

$$\begin{aligned} C_+ + C_- = [0,\infty ) e^0 - [0,\infty ) f^0 =\left\{ \begin{pmatrix} 0 &{} x \\ -y &{} 0\end{pmatrix}: x,y\ge 0\right\} ,\end{aligned}$$

so that

$$\begin{aligned} (C_+ + C_-)^\pi \cap {{\mathbb {R}}}(e^0 - f^0) = \Big \{ t (e^0 - f^0) : 0< t < \frac{\pi }{2}\Big \}. \end{aligned}$$

(B.3)

The subspace ${{\mathfrak {t}}}_{{\mathfrak {q}}}:= {{\mathbb {R}}}(e^0 - f^0) = \mathop {{\mathfrak {so}}}\nolimits _2({{\mathbb {R}}})$ of ${{\mathfrak {q}}}$ is maximal elliptic. For

$$\begin{aligned} x_0 := \frac{\pi }{4} (e^0 - f^0) = \frac{\pi }{4} \begin{pmatrix} 0 &{} 1 \\ -1 &{} 0 \end{pmatrix} \in {{\mathfrak {t}}}_{{\mathfrak {q}}}\end{aligned}$$

and

$$\begin{aligned} g_0 := \exp (x_0) = \begin{pmatrix} \cos \big (\frac{\pi }{4}\big ) &{} \sin \big (\frac{\pi }{4}\big )\\ -\sin \big (\frac{\pi }{4}\big ) &{} \cos \big (\frac{\pi }{4}\big ) \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 &{} 1 \\ -1 &{} 1 \end{pmatrix},\end{aligned}$$

we then have

$$\begin{aligned} \mathop {\mathrm{Ad}}\nolimits (g_0) h^1 = h^0 \quad \text{ and } \quad \mathop {\mathrm{Ad}}\nolimits (g_0) h^0 = - h^1, \end{aligned}$$

(B.4)

More generally, we have for $t \in {{\mathbb {R}}}$

$$\begin{aligned} e^{t\mathop {\mathrm{ad}}\nolimits (e^0 - f^0)} h^1 = \cos (2t) h^1 + \sin (2t) h^0 \end{aligned}$$

(B.5)

because $[e^0 - f^0, h^1] = 2 h^0, \quad [e^0 - f^0, h^0] = -2 h^1.$

Now fix the Euler element

$$\begin{aligned} h = h^0 = \frac{1}{2} \begin{pmatrix} 1 &{} 0 \\ 0 &{} -1 \end{pmatrix} \quad \text{ and } \quad \tau _h = e^{\pi i \mathop {\mathrm{ad}}\nolimits h},\end{aligned}$$

and similarly drop the ${}^0$ in other places. Then we have

$$\begin{aligned} {{\mathfrak {g}}}_1(h) = {{\mathbb {R}}}e, \quad e = \begin{pmatrix} 0 &{} 1 \\ 0 &{} 0 \end{pmatrix} \quad \text{ and } \qquad {{\mathfrak {g}}}_{-1}(h) = {{\mathbb {R}}}f, \quad f = \begin{pmatrix} 0 &{} 0 \\ 1 &{} 0 \end{pmatrix}.\end{aligned}$$

Recall that

$$\begin{aligned}{}[h,e] = e, \qquad [h,f] = -f \quad \text{ and } \quad [e,f] = 2h. \end{aligned}$$

(B.6)

We have

$$\begin{aligned} (C_{{\mathfrak {g}}}^\circ )^{-\tau _h} = \underbrace{{{\mathbb {R}}}_+ e}_{C_+^\circ } - \underbrace{{{\mathbb {R}}}_+ f}_{C_-^\circ } \quad \text{ with } \quad C_+^\circ = {{\mathbb {R}}}_+ e, \quad C_-^\circ = {{\mathbb {R}}}_+ f.\end{aligned}$$

For $y = \lambda e + \mu f \in {{\mathfrak {g}}}^{-\tau _h}$, we have

$$\begin{aligned}{}[y,h] = -\lambda e + \mu f \quad \text{ and } \quad [y,[y,h]] = [\lambda e + \mu f, -\lambda e + \mu f] = \lambda \mu 2[e,f] = 4 \lambda \mu h.\end{aligned}$$

We define the function $S : {{\mathbb {C}}}\rightarrow {{\mathbb {C}}}$ by $S(z) = \sum _{k = 0}^\infty \frac{(-1)^k}{(2k+1)!} z^k$, so that $\sin (z) = z S(z^2).$ Then

$$\begin{aligned} \sin (\mathop {\mathrm{ad}}\nolimits y)h&= S((\mathop {\mathrm{ad}}\nolimits y)^2) [y,h] = S(4 \lambda \mu ) \cdot (-\lambda e+ \mu f). \end{aligned}$$

This element is contained in $-C_{{\mathfrak {g}}}^\circ$ if $\lambda , \mu > 0$ and $\sin (2\sqrt{\lambda \mu }) > 0$. This is satisfied for $0< 4\mu \lambda < \pi ^2$. Then y is hyperbolic with $\mathrm{Spec}(\mathop {\mathrm{ad}}\nolimits y) = \{ \pm 2 \sqrt{\lambda \mu }\} \subseteq (-\pi , \pi )$.

Example B.1

(cf. Example 6.2) Let G be a connected Lie group with Lie algebra ${{\mathfrak {g}}}$ and $h \in {{\mathfrak {g}}}$ an Euler element. In general $G^h$ may be much larger than $G^{\tau _h}$. This can be seen for $G={\widetilde{\mathop {\mathrm{SL}}\nolimits }}_2({{\mathbb {R}}})$ (the simply connected covering group of $\mathop {\mathrm{SL}}\nolimits _2({{\mathbb {R}}})$) and the Euler element $h = \mathop {\mathrm{diag}}\nolimits (1/2,-1/2)$. The group $G^h$ contains $Z(G) \cong {{\mathbb {Z}}}$ and $G^h_e = \exp ({{\mathbb {R}}}h)$, which shows that $\pi _0(G^h) \cong {{\mathbb {Z}}}$. The involution $\tau _h$ acts on $Z(G)\subseteq \exp (\mathop {{\mathfrak {so}}}\nolimits _2({{\mathbb {R}}}))$ by inversion (cf. [25, Ex. 2.10(d)]).

For $G = \mathop {\mathrm{SL}}\nolimits _2({{\mathbb {R}}})$ and the same Euler element h, the centralizer $G^h$ is the non-connected subgroup of diagonal matrices, which coincides with the fixed point group $G^{\tau _h}$ of the corresponding involution

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

On the other hand, if we consider the centerfree group $G=\mathop {\mathrm{SO}}\nolimits _{1,2}({{\mathbb {R}}})_e \cong \mathop {\mathrm{PSL}}\nolimits _2({{\mathbb {R}}})$, then $G^h=G^h_e$, but the subgroup $G^{\tau _h}$ has 2 connected components: $G^{\tau _h}= G^h\cup \theta G^h$, where $\theta$ is a Cartan involution with $\theta (h)=-h$.

C Polar maps

In this section we discuss polar maps associated to an involution on a symmetric space, resp., to a pair of commuting involutions on a Lie group. Key properties are collected in Lemma C.4. These results are used in particular in Sect. 4.2 to obtain a polar decomposition of the crown domain of the Riemannian symmetric space $M^r = G/K$ and in the characterization of this domain as a subset of the tube domain of the cone C (Theorem 4.10).

1.1 C.1 Some spectral theory

Let V be a finite dimensional real vector space and $A \in \mathop {\mathrm{End}}\nolimits (V)$.

Lemma C.1

$\ker \big (\frac{\sinh (A)}{A}\big ) = \bigoplus _{n \not =0} \ker (A^2 + n^2\pi ^2 \mathbf{1 })$.

Proof

We may w.l.o.g. assume that V is complex. Then $B := \sinh (A)/A$ is invertible on all generalized eigenspace corresponding to eigenvalues $\lambda \not = \pi n i$, $n\in {{\mathbb {Z}}}\setminus \{0\}$. We may therefore assume that V has only one eigenvalue $\lambda = n \pi i$, $n \not =0$. Then A is invertible, so that

$$\begin{aligned} \ker (B) = \ker (\sinh (A)) = \ker (e^A- e^{-A}) = \ker (e^{2A} - \mathbf{1 }).\end{aligned}$$

Writing $A = A_s + A_n$ for the Jordan decomposition of A, it follows that

$$\begin{aligned} e^{2A} = e^{2 A_s} e^{2 A_n} = e^{2 n \pi i} e^{2 A_n} = e^{2A_n}.\end{aligned}$$

As $\ker (e^{2A_n}-\mathbf{1 }) = \ker (A_n)$ follows from $2A_n = \log (e^{2A_n})$ as a polynomial in $e^{2A_n}-1$, we see that $\ker (B) = \ker (A_n)$ is the $\lambda$-eigenspace of A. $\square$

With similar arguments, or by replacing A by $A - \frac{\pi i}{2}\mathbf{1 }$, we get:

Lemma C.2

$\ker (\cosh (A)) = \ker (e^{-2A} + \mathbf{1 }) = \bigoplus _{n \in {{\mathbb {N}}}}\ker \Big (A^2 + \big (n + \frac{1}{2}\big )^2 \pi ^2 \mathbf{1 }\Big )$.

1.2 C.2 Fine points on polar maps

In this subsection, we consider two commuting involutions $\sigma$ and $\tau$ on a connected, not necessarily reductive, Lie group G and an open $\sigma$-invariant subgroup $H \subseteq G^\tau$. We shall study the polar map

$$\begin{aligned} \Phi : G^\sigma \times _{H^\sigma } {{\mathfrak {q}}}^{-\sigma } \rightarrow M, \quad [g,x] \mapsto g.\mathop {\mathrm{Exp}}\nolimits _{eH}(x) = g \exp (x)H\in G/H \end{aligned}$$

(C.1)

and its applications.

As H is invariant under $\tau$ and $\sigma$, both define commuting involutions on M and their fixed point manifolds intersect transversally in eH. The map

$$\begin{aligned} \Psi : G^\sigma \times _{H^\sigma } {{\mathfrak {q}}}^{-\sigma }\rightarrow N(G^\sigma /H^\sigma ), \quad \ [g,x] \mapsto g.x\end{aligned}$$

is a diffeomorphism onto the normal bundle $N(G^\sigma /H^\sigma )$ of the subspace $G^\sigma .eH \cong G^\sigma /H^\sigma$ and $\Phi = \mathop {\mathrm{Exp}}\nolimits \circ \Psi$, where $\mathop {\mathrm{Exp}}\nolimits : T(M) \rightarrow M$ is the exponential map.

First, we determine the regular points of $\Phi$. As $\Phi$ is $G^\sigma$-equivariant, it suffices to determine for which points [e, x] the tangent map $T_{[e,x]}(\Phi )$ is injective, hence bijective for dimensional reasons. In the following calculation, we shall use the formula

$$\begin{aligned} T_x(\mathop {\mathrm{Exp}}\nolimits _{eH})y = \exp (x).\frac{\sinh (\mathop {\mathrm{ad}}\nolimits x)}{\mathop {\mathrm{ad}}\nolimits x} y \quad \text{ for } \quad x,y \in {{\mathfrak {q}}}\end{aligned}$$

(C.2)

for the differential of $\mathop {\mathrm{Exp}}\nolimits$ [12, Lemma 4.6], where ${{\mathfrak {q}}}\rightarrow T_{\mathop {\mathrm{Exp}}\nolimits _{eH}(x)}, v \mapsto \exp x.v$, is the linear isomorphism induced by the action of $\exp x \in G$ on M. For $a \in {{\mathfrak {g}}}^\sigma , x,b \in {{\mathfrak {q}}}^{-\sigma }$, we obtain

$$\begin{aligned} T_{[e,x]}(\Phi )(a,b)&= a.\mathop {\mathrm{Exp}}\nolimits (x) + T_{x}(\mathop {\mathrm{Exp}}\nolimits _{eH})(b) \nonumber \\&= \exp (x).\Big (p_{{{\mathfrak {q}}}}(e^{-\mathop {\mathrm{ad}}\nolimits x} a) + \frac{\sinh (\mathop {\mathrm{ad}}\nolimits x)}{\mathop {\mathrm{ad}}\nolimits x} b\Big ) \end{aligned}$$

(C.3)

Note that $e^{-\mathop {\mathrm{ad}}\nolimits x}a = \cosh (\mathop {\mathrm{ad}}\nolimits x) a - \sinh (\mathop {\mathrm{ad}}\nolimits x) a$. If $a\in {{\mathfrak {h}}}^{\sigma }$ then $p_{{\mathfrak {q}}}(e^{-\mathop {\mathrm{ad}}\nolimits x}a) = - \sinh (\mathop {\mathrm{ad}}\nolimits x)a$, and if $a \in {{\mathfrak {q}}}^\sigma$, then $p_{{\mathfrak {q}}}(e^{-\mathop {\mathrm{ad}}\nolimits x}a) = \cosh (\mathop {\mathrm{ad}}\nolimits x)a$. Writing $a = a_{{\mathfrak {h}}}+ a_{{\mathfrak {q}}}$ with $a_{{\mathfrak {h}}}\in {{\mathfrak {h}}}^\sigma$ and $a_{{\mathfrak {q}}}\in {{\mathfrak {q}}}^\sigma$, we thus obtain

$$\begin{aligned} T_{[e,x]}(\Phi )(a,b) = \exp (x).\Big (\underbrace{\cosh (\mathop {\mathrm{ad}}\nolimits x)a_{{\mathfrak {q}}}}_{\in {{\mathfrak {q}}}^\sigma } + \underbrace{\frac{\sinh (\mathop {\mathrm{ad}}\nolimits x)}{\mathop {\mathrm{ad}}\nolimits x} b - \sinh (\mathop {\mathrm{ad}}\nolimits x)a_{{\mathfrak {h}}}}_{\in {{\mathfrak {q}}}^{-\sigma }} \Big ). \end{aligned}$$

(C.4)

The following lemma provides a characterization of the regular points.

Lemma C.3

For $x \in {{\mathfrak {q}}}$, the following assertions hold:

• (a) $\mathop {\mathrm{Exp}}\nolimits _{eH}$ is regular in x if and only if the map $\frac{\sinh (\mathop {\mathrm{ad}}\nolimits x)}{\mathop {\mathrm{ad}}\nolimits x} : {{\mathfrak {q}}}\rightarrow {{\mathfrak {q}}}$ is invertible, which is equivalent to

  $$\begin{aligned} \mathrm{Spec}(\mathop {\mathrm{ad}}\nolimits x\vert _{{{\mathfrak {q}}}_L}) \cap {{\mathbb {Z}}}\pi i \subseteq \{0\}, \quad \text{ where } \quad {{\mathfrak {q}}}_L := {{\mathfrak {q}}}+ [{{\mathfrak {q}}},{{\mathfrak {q}}}]. \end{aligned}$$

  (C.5)

• (b) If $\mathop {\mathrm{Exp}}\nolimits _{eH}\vert _{{{\mathfrak {q}}}^{-\sigma }}$ is regular in $x \in {{\mathfrak {q}}}^{-\sigma }$, then the polar map $\Phi$ in (C.2) is regular in [g, x] if and only if, in addition, $\cosh (\mathop {\mathrm{ad}}\nolimits x) : {{\mathfrak {q}}}^\sigma \rightarrow {{\mathfrak {q}}}^\sigma$ is invertible, which is equivalent to

  $$\begin{aligned} \mathrm{Spec}(\mathop {\mathrm{ad}}\nolimits x\vert _{{{\mathfrak {q}}}_L}) \cap \Big (\frac{\pi }{2} + {{\mathbb {Z}}}\pi \Big ) i = \emptyset . \end{aligned}$$

  (C.6)

Proof

(a) follows from the spectral theoretic description of the kernel of $\frac{\sinh (\mathop {\mathrm{ad}}\nolimits x)}{\mathop {\mathrm{ad}}\nolimits x}\big |_{{{\mathfrak {q}}}}$ as the intersection of ${{\mathfrak {q}}}$ with the sum of the eigenspaces of $\mathop {\mathrm{ad}}\nolimits x$ in ${{\mathfrak {g}}}_{{\mathbb {C}}}$ for the eigenvalues $\lambda \in \pi i {{\mathbb {Z}}}\setminus \{0\}$ (Lemma C.1).

(b) Suppose that the restriction of $\mathop {\mathrm{Exp}}\nolimits _{eH}$ to ${{\mathfrak {q}}}^{-\sigma }$ is regular, i.e., that $\frac{\sinh (\mathop {\mathrm{ad}}\nolimits x)}{\mathop {\mathrm{ad}}\nolimits x} : {{\mathfrak {q}}}^{-\sigma } \rightarrow {{\mathfrak {q}}}^{-\sigma }$ is invertible. Then (C.4) shows that $\Phi$ is regular in [e, x] if and only if $\cosh (\mathop {\mathrm{ad}}\nolimits x) : {{\mathfrak {q}}}^\sigma \rightarrow {{\mathfrak {q}}}^\sigma$ is invertible, and this is equivalent to the condition on $\mathrm{Spec}(\mathop {\mathrm{ad}}\nolimits x\vert _{{{\mathfrak {q}}}_L})$ stated in (b). $\square$

The following lemma contains a wealth of information on singular points of the polar map $\Phi$.

Lemma C.4

Let $\Omega \subseteq {{\mathfrak {q}}}^{-\sigma }$ be an open $H^\sigma$-invariant subset consisting of $\mathop {\mathrm{Exp}}\nolimits$-regular elliptic elements, and consider the polar map

$$\begin{aligned} \Phi : G^\sigma _e \times _{H^\sigma } \Omega \rightarrow M, \quad [g,y] \mapsto g.\mathop {\mathrm{Exp}}\nolimits _{eH}(y).\end{aligned}$$

Let $x \in \Omega$, write $m := \mathop {\mathrm{Exp}}\nolimits _{eH}(x) \in M = G/H$, ${\mathcal {O}}_m := G^\sigma _e.m$ for its orbit, and put

$$\begin{aligned} \sigma _x := e^{-2 \mathop {\mathrm{ad}}\nolimits x}, \qquad \zeta _x := e^{-\mathop {\mathrm{ad}}\nolimits x} \in \mathop {\mathrm{Aut}}\nolimits ({{\mathfrak {g}}}).\end{aligned}$$

Then the following assertions hold:

• (a) ${{\mathfrak {g}}}^{-\sigma _x} := \{ y \in {{\mathfrak {g}}}: \sigma _x(y) = - y \} = \ker (\cosh (\mathop {\mathrm{ad}}\nolimits x))$.

• (b) ${{\mathfrak {q}}}^{\sigma , -\sigma _x}$ complements the subspace $\exp (-x).\mathop {\mathrm{im}}\nolimits (T_{[e,x]}(\Phi )) = \cosh (\mathop {\mathrm{ad}}\nolimits x){{\mathfrak {q}}}^\sigma \oplus {{\mathfrak {q}}}^{-\sigma }$ in ${{\mathfrak {q}}}$.

• (c) The eigenspaces ${{\mathfrak {g}}}^{\pm \sigma _x}$ are $\tau$-invariant, and on the Lie subalgebra ${{\mathfrak {g}}}^{\sigma _x^2} = {{\mathfrak {g}}}^{\sigma _x} \oplus {{\mathfrak {g}}}^{-\sigma _x}$, the involution $\tau$ commutes with $\sigma _x$.

• (d) The eigenspaces ${{\mathfrak {g}}}^{\pm \sigma _x}$ are $\zeta _x$-invariant, on ${{\mathfrak {g}}}^{\sigma _x}$ the automorphisms $\zeta _x$ and $\tau$ commute, and on on ${{\mathfrak {g}}}^{-\sigma _x}$ the complex structure $\zeta _x$ and $\tau$ anticommute. In particular, we have

  $$\begin{aligned} \zeta _x({{\mathfrak {h}}}^{\sigma _x}) = {{\mathfrak {h}}}^{\sigma _x}, \quad \zeta _x({{\mathfrak {q}}}^{\sigma _x}) = {{\mathfrak {q}}}^{\sigma _x}, \quad \zeta _x({{\mathfrak {h}}}^{-\sigma _x}) = {{\mathfrak {q}}}^{-\sigma _x}, \quad \zeta _x({{\mathfrak {q}}}^{-\sigma _x}) = {{\mathfrak {h}}}^{-\sigma _x}.\end{aligned}$$

• (e) The eigenspaces ${{\mathfrak {g}}}^{\pm \sigma _x}$ are $\sigma$-invariant, and on the Lie subalgebra ${{\mathfrak {g}}}^{\sigma _x^2}$, the involution $\sigma$ commutes with $\sigma _x$. On ${{\mathfrak {g}}}^{\sigma _x}$, the automorphisms $\zeta _x$ and $\sigma$ commute, and on on ${{\mathfrak {g}}}^{-\sigma _x}$ the complex structure $\zeta _x$ anticommute with $\sigma$. In particular, we have

  $$\begin{aligned} \zeta _x({{\mathfrak {g}}}^{\sigma ,\sigma _x}) = {{\mathfrak {g}}}^{\sigma ,\sigma _x}, \quad \zeta _x({{\mathfrak {g}}}^{-\sigma ,\sigma _x}) = {{\mathfrak {g}}}^{-\sigma , \sigma _x}, \quad \zeta _x({{\mathfrak {g}}}^{\pm \sigma , -\sigma _x}) = {{\mathfrak {g}}}^{{\mp }\sigma , -\sigma _x}.\end{aligned}$$

• (f) The stabilizer Lie algebra of m in ${{\mathfrak {g}}}$ is

  $$\begin{aligned} {{\mathfrak {g}}}_m = {{\mathfrak {g}}}^{\tau \sigma _x}= \{ y \in {{\mathfrak {g}}}: \sigma _x(y) = \tau (y)\} \quad \text{ and } \quad {{\mathfrak {g}}}_m^{\sigma _x^2} = {{\mathfrak {h}}}^{\sigma _x} \oplus {{\mathfrak {q}}}^{-\sigma _x}.\end{aligned}$$

  The stabilizer Lie algebra in ${{\mathfrak {g}}}^\sigma$ is

  $$\begin{aligned} {{\mathfrak {g}}}_m^\sigma ={{\mathfrak {h}}}^{\sigma ,\sigma _x} \oplus {{\mathfrak {q}}}^{\sigma , -\sigma _x}. \end{aligned}$$

  (C.7)

  The stabilizer group $G_m$ acts on

  $$\begin{aligned} T_m(M) = \exp x.{{\mathfrak {q}}}\, by \, g.(\exp x.y) = \exp x.\big (\mathop {\mathrm{Ad}}\nolimits (\zeta _x^G(g))y\big ). \end{aligned}$$

• (g) The tangent space of the orbit ${\mathcal {O}}_m$ is

  $$\begin{aligned} T_m({\mathcal {O}}_m) = \exp (x).\big (\cosh (\mathop {\mathrm{ad}}\nolimits x){{\mathfrak {q}}}^\sigma + [x,{{\mathfrak {h}}}^\sigma ]\big ).\end{aligned}$$

• (h) Let ${{\mathfrak {q}}}_x := {{\mathfrak {q}}}^{\sigma ,-\sigma _x} + {{\mathfrak {q}}}^{-\sigma ,\sigma _x}$ and ${{\mathfrak {h}}}_x := [{{\mathfrak {q}}}_x,{{\mathfrak {q}}}_x]$. If the group $\mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}({{\mathfrak {h}}}_x)$ acts as a relatively compact group on ${{\mathfrak {q}}}_x$ and $\mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}({{\mathfrak {h}}}_x){{\mathfrak {q}}}^{-\sigma ,\sigma _x} = {{\mathfrak {q}}}_x$, then m is an interior point of $\mathop {\mathrm{im}}\nolimits (\Phi )$.

Proof

(a) follows directly from $2\cosh (\mathop {\mathrm{ad}}\nolimits x) = e^{\mathop {\mathrm{ad}}\nolimits x} + e^{-\mathop {\mathrm{ad}}\nolimits x}$.

(b) With (C.3) we see that the image of $T_{[e,x]}(\Phi )$ is the subspace

$$\begin{aligned} \exp (x).\Big (\underbrace{\cosh (\mathop {\mathrm{ad}}\nolimits x){{\mathfrak {q}}}^{\sigma }}_{\subseteq {{\mathfrak {q}}}^{\sigma }} + \underbrace{\frac{\sinh (\mathop {\mathrm{ad}}\nolimits x)}{\mathop {\mathrm{ad}}\nolimits x} {{\mathfrak {q}}}^{-\sigma }}_{\subseteq {{\mathfrak {q}}}^{-\sigma }}\Big ).\end{aligned}$$

As $\mathop {\mathrm{ad}}\nolimits x$ is semisimple, a complement of $\exp (-x).\mathop {\mathrm{im}}\nolimits (T_{[e,x]}(\Phi ))$ in ${{\mathfrak {q}}}$ is

$$\begin{aligned} \ker \big (\cosh (\mathop {\mathrm{ad}}\nolimits x)\vert _{{{\mathfrak {q}}}^{\sigma }}\big ) = {{\mathfrak {q}}}^{\sigma , -\sigma _x}. \end{aligned}$$

(C.8)

(c) In view of $\tau \sigma _x^2 \tau = \sigma _x^{-2}$, the fixed point space ${{\mathfrak {g}}}^{\sigma _x^2}$ is $\tau$-invariant. On this subspace $\sigma _x= \sigma _x^{-1} = \tau \sigma _x \tau$, so that $\tau$ preserves the two eigenspaces ${{\mathfrak {g}}}^{\pm \sigma _x}$ of $\sigma _x$ on ${{\mathfrak {g}}}^{\sigma _x^2}$.

(d) As $\zeta _x$ commutes with $\sigma _x$, the eigenspaces ${{\mathfrak {g}}}^{\pm \sigma _x}$ are $\zeta _x$-invariant. Further, (c) implies that, on ${{\mathfrak {g}}}^{\sigma _x}$, we have $\tau \zeta _x \tau = \zeta _x^{-1} = \zeta _x$.

(e) is shown with similar arguments as (c) and (d).

(f) In $M = G/H$, the point $m = \mathop {\mathrm{Exp}}\nolimits _{eH}(\exp x) = \exp x H$ is obtained by acting with $\exp x$ on the base point eH. Therefore its stabilizer group is $G_m = \exp x H \exp (-x)$ with the Lie algebra

$$\begin{aligned} {{\mathfrak {g}}}_m = e^{\mathop {\mathrm{ad}}\nolimits x} {{\mathfrak {h}}}= \mathop {\mathrm{Fix}}\nolimits (e^{\mathop {\mathrm{ad}}\nolimits x} \tau e^{-\mathop {\mathrm{ad}}\nolimits x}) = \mathop {\mathrm{Fix}}\nolimits (\tau e^{-2\mathop {\mathrm{ad}}\nolimits x}) = \mathop {\mathrm{Fix}}\nolimits (\tau \sigma _x).\end{aligned}$$

Now the $\tau$-invariance of ${{\mathfrak {g}}}^{\sigma _x^2}$ implies that

$$\begin{aligned} {{\mathfrak {g}}}^{\sigma _x^2}_m = {{\mathfrak {g}}}_m \cap {{\mathfrak {g}}}^{\sigma _x^2} = {{\mathfrak {g}}}^{\sigma _x,\tau } \oplus {{\mathfrak {g}}}^{-\sigma _x,-\tau } = {{\mathfrak {h}}}^{\sigma _x} \oplus {{\mathfrak {q}}}^{-\sigma _x}.\end{aligned}$$

To verify (C.7), let $y = y_{{\mathfrak {h}}}+ y_{{\mathfrak {q}}}\in {{\mathfrak {g}}}^\sigma$ with $y_{{\mathfrak {h}}}\in {{\mathfrak {h}}}^\sigma$ and $y_{{\mathfrak {q}}}\in {{\mathfrak {q}}}^\sigma$. Then the corresponding vector field $Y_M$ on M satisfies

$$\begin{aligned} Y_M(m)&= y.m = \exp (x).p_{{{\mathfrak {q}}}}(e^{-\mathop {\mathrm{ad}}\nolimits x}y) = \exp (x).\Big (\frac{1}{2}(e^{-\mathop {\mathrm{ad}}\nolimits x}y - \tau (e^{-\mathop {\mathrm{ad}}\nolimits x}y)\Big ) \nonumber \\&= \exp (x).\Big (\frac{1}{2}(e^{-\mathop {\mathrm{ad}}\nolimits x}y - e^{\mathop {\mathrm{ad}}\nolimits x}\tau (y))\Big ) = \exp (x).\big (\cosh (\mathop {\mathrm{ad}}\nolimits x)y_{{\mathfrak {q}}}- \sinh (\mathop {\mathrm{ad}}\nolimits x)y_{{\mathfrak {h}}}\big ). \end{aligned}$$

(C.9)

Therefore $Y_M(m) = 0$ is equivalent to

$$\begin{aligned} 0 = \underbrace{\cosh (\mathop {\mathrm{ad}}\nolimits x)y_{{\mathfrak {q}}}}_{\in {{\mathfrak {q}}}^{\sigma }} - \underbrace{\sinh (\mathop {\mathrm{ad}}\nolimits x) y_{{\mathfrak {h}}}}_{\in {{\mathfrak {q}}}^{-\sigma }}.\end{aligned}$$

Thus both summands have to vanish, which is equivalent to

$$\begin{aligned} e^{-2\mathop {\mathrm{ad}}\nolimits x} y_{{\mathfrak {q}}}= - y_{{\mathfrak {q}}}\quad \text{ and } \quad e^{-2\mathop {\mathrm{ad}}\nolimits x} y_{{\mathfrak {h}}}= y_{{\mathfrak {h}}}.\end{aligned}$$

This implies (C.7).

To complete the proof of (f), we note that, for $v \in {{\mathfrak {q}}}$ and $g \in G_m$, we have

$$\begin{aligned} g.(\exp (x).y) = \exp (x).\big (\mathop {\mathrm{Ad}}\nolimits (\zeta ^G_x(g))y\big ), \end{aligned}$$

(C.10)

where $\zeta ^G_x(G_m) = H$ acts on $T_{eH}(M) \cong {{\mathfrak {q}}}$ by the adjoint representation.

(g) From (C.9) it follows that

$$\begin{aligned} \exp (-x).T_m({\mathcal {O}}_m)&= \cosh (\mathop {\mathrm{ad}}\nolimits x) {{\mathfrak {q}}}^\sigma + \sinh (\mathop {\mathrm{ad}}\nolimits x) {{\mathfrak {h}}}^\sigma = \cosh (\mathop {\mathrm{ad}}\nolimits x) {{\mathfrak {q}}}^\sigma + \frac{\sinh (\mathop {\mathrm{ad}}\nolimits x)}{\mathop {\mathrm{ad}}\nolimits x} [x,{{\mathfrak {h}}}^\sigma ]\\&{\buildrel ! \over =} \cosh (\mathop {\mathrm{ad}}\nolimits x) {{\mathfrak {q}}}^\sigma + [x,{{\mathfrak {h}}}^\sigma ]. \end{aligned}$$

Here the last equality follows from $(\mathop {\mathrm{ad}}\nolimits x)^2 {{\mathfrak {h}}}^\sigma \subseteq {{\mathfrak {h}}}^\sigma$ and the $\mathop {\mathrm{Exp}}\nolimits$-regularity of x, which, by Lemma C.3, is equivalent to the invertibility of $\frac{\sinh (\mathop {\mathrm{ad}}\nolimits x}{\mathop {\mathrm{ad}}\nolimits x}$ on ${{\mathfrak {q}}}$.

(h) By (C.8), a natural complement of

$$\begin{aligned} \exp (-x).T_m({\mathcal {O}}_m) = \cosh (\mathop {\mathrm{ad}}\nolimits x) {{\mathfrak {q}}}^\sigma + [x,{{\mathfrak {h}}}^\sigma ] \end{aligned}$$

in ${{\mathfrak {q}}}$ is the subspace

$$\begin{aligned} {{\mathfrak {q}}}_x := {{\mathfrak {q}}}^{\sigma ,-\sigma _x} \oplus ({{\mathfrak {q}}}^{-\sigma } \cap \ker (\mathop {\mathrm{ad}}\nolimits x)) = {{\mathfrak {q}}}^{\sigma ,-\sigma _x} \oplus {{\mathfrak {q}}}^{-\sigma , \sigma _x},\end{aligned}$$

where the last equality follows from

$$\begin{aligned} {{\mathfrak {q}}}^{\sigma _x} = \bigoplus _{n \in {{\mathbb {Z}}}} \ker ( (\mathop {\mathrm{ad}}\nolimits x)^2 + \pi ^2 n^2 \mathop {\mathrm{id}}\nolimits _{{\mathfrak {q}}}) \quad \text{ and } \quad \mathrm{Spec}(\mathop {\mathrm{ad}}\nolimits x) \cap {{\mathbb {Z}}}\pi i \subseteq \{0\}.\end{aligned}$$

As x is $\mathop {\mathrm{Exp}}\nolimits _{eH}$-regular and

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits _{eH}(x + y) = \exp (x).\mathop {\mathrm{Exp}}\nolimits _m(y) \quad \text{ for } \quad y \in {{\mathfrak {q}}}^{-\sigma } \cap \ker (\mathop {\mathrm{ad}}\nolimits x) = {{\mathfrak {q}}}^{-\sigma ,\sigma _x},\end{aligned}$$

the subset

$$\begin{aligned} \Omega := \{ y \in {{\mathfrak {q}}}_x : \mathop {\mathrm{Exp}}\nolimits _m(\exp (x).y) \in \mathop {\mathrm{im}}\nolimits (\Phi ) \} \end{aligned}$$

contains a 0-neighborhood $U_0$ in ${{\mathfrak {q}}}^{-\sigma ,\sigma _x}$. Now our assumption $\mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}({{\mathfrak {h}}}_x) {{\mathfrak {q}}}^{-\sigma ,\sigma _x} ={{\mathfrak {q}}}_x$ and the relative compactness of the group $\mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}({{\mathfrak {h}}}_x)$ on ${{\mathfrak {q}}}_x$ imply that $U_1 := \mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}({{\mathfrak {h}}}_x) U_0$ is a 0-neighborhood in ${{\mathfrak {q}}}_x$.

Finally, we obtain from (C.7)

$$\begin{aligned} \zeta _x^{-1}({{\mathfrak {h}}}_x) \subseteq \zeta _x^{-1}({{\mathfrak {h}}}^{\sigma ,\sigma _x} + {{\mathfrak {h}}}^{-\sigma ,-\sigma _x}) \ {\buildrel {(d),(e)} \over \subseteq }\ {{\mathfrak {h}}}^{\sigma , \sigma _x} + {{\mathfrak {q}}}^{\sigma , -\sigma _x} \ {\buildrel (C.7) \over =}\ {{\mathfrak {g}}}_m^\sigma ,\end{aligned}$$

so that

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits _m(\exp (x).U_1) = \mathop {\mathrm{Exp}}\nolimits _m(\exp (x).\mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}({{\mathfrak {h}}}_x) U_0) = (G^\sigma _e)_m.\mathop {\mathrm{Exp}}\nolimits _m(\exp (x).U_0) \subseteq \mathop {\mathrm{im}}\nolimits (\Phi ).\end{aligned}$$

This means that $\Omega$ is a 0-neighborhood in ${{\mathfrak {q}}}_x$, and hence that $\mathop {\mathrm{im}}\nolimits (\Phi )$ is a neighborhood of m because the map

$$\begin{aligned} G^\sigma _e \times {{\mathfrak {q}}}_x \rightarrow M, \quad (g,y) \mapsto g.\mathop {\mathrm{Exp}}\nolimits _m(\exp (x).y) \end{aligned}$$

has surjective differential in (e, 0). $\square$

Remark C.5

If $\rho _i(\mathop {\mathrm{ad}}\nolimits x) < \pi /2$, then the polar map $\Phi$ in the preceding lemma is regular in [e, x] (Lemma C.3). In this case $\ker (\cosh (\mathop {\mathrm{ad}}\nolimits x)) = {{\mathfrak {g}}}^{-\sigma _x}$ is trivial, so that

$$\begin{aligned} {{\mathfrak {g}}}_m^\sigma = {{\mathfrak {h}}}^{\sigma ,\sigma _x} = {{\mathfrak {h}}}^\sigma \cap \ker (\mathop {\mathrm{ad}}\nolimits x) = {{\mathfrak {z}}}_{{{\mathfrak {h}}}^\sigma }(x)\end{aligned}$$

follows from Lemma C.4(f).

The next lemma is useful to verify condition (h) in the preceding lemma.

Lemma C.6

Suppose that $({{\mathfrak {g}}},\tau )$ is a reductive symmetric Lie algebra such that ${{\mathfrak {q}}}$ consist of elliptic elements, $\sigma \tau = \tau \sigma$, and that ${{\mathfrak {a}}}\subseteq {{\mathfrak {q}}}^{-\sigma }$ is a maximal abelian subspace in ${{\mathfrak {q}}}$. Then

$$\begin{aligned} {{\mathfrak {q}}}^\sigma =[{{\mathfrak {h}}}^{-\sigma }, {{\mathfrak {q}}}^{-\sigma }], \qquad \mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}([{{\mathfrak {q}}},{{\mathfrak {q}}}]){{\mathfrak {q}}}^{-\sigma } = {{\mathfrak {q}}},\end{aligned}$$

and the group $\mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}([{{\mathfrak {q}}},{{\mathfrak {q}}}])$ is compact.

Proof

Let ${{\mathfrak {a}}}\subset {{\mathfrak {q}}}$ be maximal abelian in ${{\mathfrak {q}}}$. Then ${{\mathfrak {a}}}$ contains ${{\mathfrak {z}}}({{\mathfrak {q}}})$. We then note that ${{\mathfrak {z}}}_{{\mathfrak {g}}}({{\mathfrak {a}}}) = {{\mathfrak {a}}}\oplus {{\mathfrak {z}}}_{{\mathfrak {h}}}({{\mathfrak {a}}})$ is $\tau$- and $\sigma$-invariant. Further ${{\mathfrak {g}}}= {{\mathfrak {z}}}_{{\mathfrak {g}}}({{\mathfrak {a}}}) \oplus [{{\mathfrak {a}}},{{\mathfrak {g}}}]$ implies that ${{\mathfrak {q}}}= {{\mathfrak {a}}}\oplus [{{\mathfrak {a}}},{{\mathfrak {h}}}].$ As

$$\begin{aligned}{}[{{\mathfrak {a}}},{{\mathfrak {h}}}] = [{{\mathfrak {a}}}, {{\mathfrak {h}}}^\sigma \oplus {{\mathfrak {h}}}^{-\sigma }] \subseteq {{\mathfrak {q}}}^{-\sigma } \oplus {{\mathfrak {q}}}^{\sigma },\end{aligned}$$

this leads to

$$\begin{aligned} {{\mathfrak {q}}}^{\sigma } = [{{\mathfrak {a}}}, {{\mathfrak {h}}}^{-\sigma }] \subseteq [{{\mathfrak {q}}}^{-\sigma }, {{\mathfrak {h}}}^{-\sigma }] \subseteq {{\mathfrak {q}}}^\sigma .\end{aligned}$$

The second assertion follows from $\mathop {\mathrm{Inn}}\nolimits _{{\mathfrak {g}}}([{{\mathfrak {q}}},{{\mathfrak {q}}}]){{\mathfrak {a}}}= {{\mathfrak {q}}}$, and the third from the fact that the reductive Lie algebra ${{\mathfrak {q}}}+ [{{\mathfrak {q}}},{{\mathfrak {q}}}]$ (it is an ideal of ${{\mathfrak {g}}}$) is compact. $\square$

1.3 C.3 Fibers of the polar map

For the polar map we have to analyze the relation

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits (x) = g.\mathop {\mathrm{Exp}}\nolimits (y).\end{aligned}$$

Applying the quadratic representation yields

$$\begin{aligned} \exp (2x) = g \exp (2y) g^\sharp = \exp (2 \mathop {\mathrm{Ad}}\nolimits (g)y) gg^\sharp .\end{aligned}$$

For $x,y \in {{\mathfrak {q}}}^{-\sigma }$ and $g \in G^\sigma$, we also have $gg^\sharp \in G^\sigma$, so that

$$\begin{aligned} \exp (4x) = \exp (2x) \sigma (\exp (2x))^{-1} = \exp (4 \mathop {\mathrm{Ad}}\nolimits (g)y).\end{aligned}$$

If x and y are sufficiently small (imaginary spectral radius $< \frac{\pi }{4}$), we thus obtain $\mathop {\mathrm{Ad}}\nolimits (g)y = x$, and thus $gg^\sharp = e$, i.e., $\tau (g) = g$.

See [16, Cor. 7.35] for similar arguments.

1.4 C.4 Fibers of $\mathop {\mathrm{Exp}}\nolimits$

Suppose that $x, y \in {{\mathfrak {q}}}$ have the same exponential image $\mathop {\mathrm{Exp}}\nolimits (x) = \mathop {\mathrm{Exp}}\nolimits (y)$ in M. We further assume that $\mathrm{Spec}(\mathop {\mathrm{ad}}\nolimits x) \cap i \pi {{\mathbb {Z}}}\subseteq \{0\}$, so that $\mathop {\mathrm{Exp}}\nolimits$ is regular in x. Then we obtain in G the identity

$$\begin{aligned} \exp (2x) = Q(\mathop {\mathrm{Exp}}\nolimits x) = Q(\mathop {\mathrm{Exp}}\nolimits y) = \exp (2y),\end{aligned}$$

and since $\mathrm{Spec}(\mathop {\mathrm{ad}}\nolimits (2x)) \cap 2 \pi i {{\mathbb {Z}}}\subseteq \{0\}$, $\exp$ is regular in x. Therefore [17, Lemma 9.2.31] implies that

$$\begin{aligned}{}[x,y]= 0 \quad \text{ and } \quad \exp (2x - 2y) = e.\end{aligned}$$

We conclude that $\exp (x-y) = \exp (y-x)$, which leads to

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits (y-x) = \tau _M(\mathop {\mathrm{Exp}}\nolimits (x-y)) = \exp (y-x)H = \exp (x-y)H = \mathop {\mathrm{Exp}}\nolimits (x-y),\end{aligned}$$

so that $\mathop {\mathrm{Exp}}\nolimits (y-x) \in M^\tau$, and $\mathop {\mathrm{Exp}}\nolimits ({{\mathbb {R}}}(x-y)) \subseteq M$ is a closed geodesic.

We also conclude that $x-y$ is elliptic with $\mathrm{Spec}(\mathop {\mathrm{ad}}\nolimits (x-y))\subseteq \pi i {{\mathbb {Z}}}$.

Lemma C.7

If $\rho _i(\mathop {\mathrm{ad}}\nolimits x), \rho _i(\mathop {\mathrm{ad}}\nolimits y) < \pi ,$ then $\exp (x) = \exp (y)$ implies $x-y \in {{\mathfrak {z}}}({{\mathfrak {g}}})$.

If, in addition, G is simply connected or ${{\mathfrak {g}}}$ is semisimple, then $x = y$.

Proof

The preceding discussion implies that $[x,y] = 0$. Now $\rho _i(\mathop {\mathrm{ad}}\nolimits (x-y)) < 2\pi$ leads to $\mathop {\mathrm{ad}}\nolimits (x-y) = 0$, i.e., to $x-y \in {{\mathfrak {z}}}({{\mathfrak {g}}})$. $\square$

D Quadrics as symmetric spaces

In this appendix we discuss an important example of a non-compactly causal symmetric spaces: d-dimensional de Sitter space $\mathop {\mathrm{dS}}\nolimits ^d$, realized as a hyperboloid in Minkowski space.

1.1 D.1 Quadrics as symmetric spaces

Definition D.1

[24] (a) Let M be a smooth manifold and

$$\begin{aligned} \mu : M \times M \rightarrow M, \quad (x,y) \mapsto x \cdot y =: s_x(y) \end{aligned}$$

be a smooth map with the following properties: each $s_x$ is an involution for which x is an isolated fixed point and

$$\begin{aligned} s_x(y \cdot z) = s_x(y)\cdot s_x(z) \quad \text{ for } \text{ all } \quad x,y \in M. \end{aligned}$$

(D.1)

Then we call $(M,\mu )$ a symmetric space.

(b) A morphism of symmetric spaces M and N is a smooth map $\varphi : M\rightarrow N$ such that $\varphi (x\cdot y)= \varphi (x)\cdot \varphi (y)$ for $x, y \in M$.

(c) A geodesic of a symmetric space is a morphism $\gamma : {{\mathbb {R}}}\rightarrow M$ of symmetric spaces, i.e.,

$$\begin{aligned} \gamma (2x-y) = \gamma (x) \cdot \gamma (y) \quad \text{ for } \quad x,y \in {{\mathbb {R}}}.\end{aligned}$$

Any geodesic is uniquely determined by $\gamma '(0) \in T_{\gamma (0)}(M)$, and, conversely, every $v \in T_p(M)$ generates a unique geodesic $\gamma _v$ with $\gamma _v(0) = p$ and $\gamma _v'(0) = v$. Accordingly, geodesics are encoded in the exponential functions

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits _p : T_p(M) \rightarrow M, \quad \mathop {\mathrm{Exp}}\nolimits _p(v) := \gamma _v(1).\end{aligned}$$

We then have

$$\begin{aligned} \gamma _v(t) = \mathop {\mathrm{Exp}}\nolimits _p(tv) \quad \text{ for } \quad t \in {{\mathbb {R}}}.\end{aligned}$$

Example D.2

Let $(V,\beta )$ be a finite dimensional real vector space, endowed with a non-degenerate symmetric bilinear form $\beta$. Then every anisotropic element $x \in V$ defines an involution

$$\begin{aligned} s_x(y) := -y + 2 \frac{\beta (x,y)}{\beta (x,x)} x \end{aligned}$$

fixing x and satisfying $\mathop {\mathrm{Fix}}\nolimits (-s_x) = x^\bot = \{ y \in V: \beta (x,y) = 0\}$.

For $c \in {{\mathbb {R}}}^\times$, let

$$\begin{aligned} Q_c := Q_c(V,\beta ) := \{ v \in V : \beta (v,v) = c\} \end{aligned}$$

denote the corresponding quadric in $(V,\beta )$. Then $(Q_c, \mu )$ with $\mu (x,y) = s_x(y)$ is a symmetric space (Definition D.1) and $\mathop {\mathrm{dim}}\nolimits Q_c = \mathop {\mathrm{dim}}\nolimits V -1$. Note that dilation by $r \in {{\mathbb {R}}}^\times$ is an isomorphism of symmetric spaces from $Q_c$ to $Q_{r^2c}$.

For $p \in Q_c$, the tangent space is $T_p(Q_c) = p^\bot = \{ v \in V : \beta (p,v) = 0\}$. To describe the exponential function of the symmetric space $Q_c$, we use the entire functions $C, S : {{\mathbb {C}}}\rightarrow {{\mathbb {C}}}$ defined by

$$\begin{aligned} C(z) := \sum _{k = 0}^\infty \frac{(-1)^k}{(2k)!} z^{k} \quad \text{ and } \quad S(z) := \sum _{k = 0}^\infty \frac{(-1)^k}{(2k+1)!} z^{k} \end{aligned}$$

(D.2)

which satisfy

$$\begin{aligned} \cos z = C(z^2) \quad \text{ and } \quad \sin z = z S(z^2) \quad \text{ for } \quad z \in {{\mathbb {C}}}\end{aligned}$$

(D.3)

and

$$\begin{aligned} \cosh z = C(-z^2) \quad \text{ and } \quad \sinh z = z S(-z^2) \quad \text{ for } \quad z \in {{\mathbb {C}}}. \end{aligned}$$

(D.4)

Note that

$$\begin{aligned} 1 = C(z)^2 + z S(z)^2 \quad \text{ for } \quad z \in {{\mathbb {C}}}\end{aligned}$$

(D.5)

follows from $1 = \cos ^2 z + \sin ^2 z$ and the surjectivity of the square map on ${{\mathbb {C}}}$.

Proposition D.3

The exponential function of the symmetric space $Q_c$ is given by

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits _p(v) = C\Big (\frac{\beta (v,v)}{\beta (p,p)}\Big ) p + S\Big (\frac{\beta (v,v)}{\beta (p,p)}\Big ) v \quad \text{ for } \quad p \in Q_c, v \in T_p(Q_c) = p^\bot . \end{aligned}$$

(D.6)

Proof

To verify this claim, abbreviate $\varepsilon := \frac{\beta (v,v)}{\beta (p,p)}$. By (D.5), on the right hand side of (D.6), the quadratic form $\beta (\cdot ,\cdot )$ has the value

$$\begin{aligned} C(\varepsilon )^2 \beta (p,p) + S(\varepsilon )^2 \beta (v,v) = \big (C(\varepsilon )^2 + S(\varepsilon )^2 \varepsilon \big ) \beta (p,p) = \beta (p,p) = c.\end{aligned}$$

Therefore the right hand side of (D.6) is contained in $Q_c$.

It remains to show that, for $\varepsilon \in \{-1,0,1\}$, the curve

$$\begin{aligned} \gamma _v(t) := C(t^2\varepsilon ) p + S(t^2\varepsilon ) t v, \end{aligned}$$

which satisfies $\gamma _v'(0) = v$, actually is a geodesic, i.e.,

$$\begin{aligned} \gamma _v(2t-s) = \gamma _v(t) \cdot \gamma _v(s)\quad \text{ for } \quad t,s \in {{\mathbb {R}}}.\end{aligned}$$

We consider three cases:

$\varepsilon = 0$: Then $\gamma _v(t) = p + tv$ and

$$\begin{aligned} \gamma _v(t) \cdot \gamma _v(s) = - \gamma _v(s) + \frac{2}{c} \beta (\gamma _v(t),\gamma _v(s)) \gamma _v(t) = - (p + sv) + 2(p + tv) = p + (2t-s)v.\end{aligned}$$

$\varepsilon = 1$: Then

$$\begin{aligned} \gamma _v(t) = \cos (t) p + \sin (t) v\end{aligned}$$

by (D.3), and

$$\begin{aligned} \gamma _v(t) \cdot \gamma _v(s)&= - \gamma _v(s) + \frac{2}{c} \beta (\gamma _v(t),\gamma _v(s)) \gamma _v(t) \\&= - (\cos (s) p + \sin (s) v) + 2(\cos (t)\cos (s) + \sin (t) \sin (s)) (\cos (t)p + \sin (t)v) \\&= (-\cos (s) + 2\cos (t)^2\cos (s) + 2\sin (t)\sin (s)\cos (t)) p \\&\quad + (-\sin (s) + 2\sin (t)\cos (t)\cos (s) +2 \sin (t)^2\sin (s)) v \\&= (\cos (s) - 2\sin (t)^2\cos (s) + 2\sin (t)\sin (s)\cos (t)) p \\&\quad + (-\sin (s) + 2\sin (t)\cos (t)\cos (s) +2 \sin (t)^2\sin (s)) v \\&= \cos (2t-s) p + \sin (2t-s) v = \gamma _v(2t-s). \end{aligned}$$

$\varepsilon = -1$: Then

$$\begin{aligned} \gamma _v(t) = \cosh (t) p + \sinh (t) v\end{aligned}$$

by (D.4), and

$$\begin{aligned} \gamma _v(t) \cdot \gamma _v(s)&= - \gamma _v(s) + \frac{2}{c} \beta (\gamma _v(t),\gamma _v(s)) \gamma _v(t) \\&= - (\cosh (s) p + \sinh (s) v) + 2(\cosh (t)\cosh (s) \\&\quad + \sinh (t) \sinh (s)) (\cosh (t)p + \sinh (t)v) \\&= (-\cosh (s) + 2\cosh (t)^2\cosh (s)\\&\quad + 2\sinh (t)\sinh (s)\cosh (t)) p \\&\quad + (-\sinh (s) + 2\sinh (t)\cosh (t)\cosh (s) +2 \sinh (t)^2\sinh (s)) v \\&= (\cosh (s) +2\sinh (t)^2\cosh (s) + 2\sinh (t)\sinh (s)\cosh (t)) p \\&\quad + (-\sinh (s) + 2\sinh (t)\cosh (t)\cosh (s) +2 \sinh (t)^2\sinh (s)) v \\&= \cosh (2t-s) p + \sinh (2t-s) v = \gamma _v(2t-s). \end{aligned}$$

$\square$

1.2 D.2 Minkowski space

On $V := {{\mathbb {R}}}^{1,d}$, we consider the Lorentzian form

$$\begin{aligned}{}[x,y] := x_0 y_0 - {\mathbf{x}}{\mathbf{y}},\end{aligned}$$

the future light cone

$$\begin{aligned} V_+ := \{ x = (x_0, {\mathbf{x}}) \in V : x_0> 0, x_0^2 - {\mathbf{x}}^2 > 0\}, \end{aligned}$$

and the tube domain

$$\begin{aligned} {\mathcal {T}}_V = V + i V_+ \subseteq V_{{\mathbb {C}}}.\end{aligned}$$

The standard boost vector field $X_h(v) = hv$ is defined by $h \in \mathop {{\mathfrak {so}}}\nolimits _{1,d}({{\mathbb {R}}})$, given by

$$\begin{aligned} hx = (x_1, x_0, 0,\ldots , 0). \end{aligned}$$

(D.7)

It generates the flow

$$\begin{aligned} \alpha _t(x) = e^{th} x = (\cosh t \cdot x_0 + \sinh t \cdot x_1, \cosh t \cdot x_1 + \sinh t \cdot x_0, x_2, \ldots , x_d)\end{aligned}$$

and defines the involution

$$\begin{aligned}\tau _h(x) := e^{\pi i h} x= (-x_0, -x_1, x_2, \ldots , x_d), \end{aligned}$$

that we extend to an antilinear involution $\overline{\tau }_h$ on $V_{{\mathbb {C}}}$. It also defines a Wick rotation

$$\begin{aligned} \kappa _h(x) = e^{-\frac{\pi i}{2} h} x = (-i x_1, -i x_0, x_2, \ldots , x_d) \end{aligned}$$

satisfying $\kappa _h^2 = \tau _h$.

Lemma D.4

The following subsets of V are equal:

• (a) The standard right wedge $W_R := \{ x \in V : x_1 > |x_0| \}$.

• (b) The positivity domain $W_V^+(h) := \{ v \in V : X_h(v) \in V_+ \}$ of $X_h(v) = hv$.

• (c) The KMS domain of $\alpha$: $\{ x \in V : (\forall t \in (0,\pi ))\ \alpha _{it}(x) \in {\mathcal {T}}_V\}$.

• (d) The Wick rotation $\kappa _h({\mathcal {T}}_V^{\overline{\tau }_h})$ of the fixed point set ${\mathcal {T}}_V^{\overline{\tau }_h} = {\mathcal {T}}_V \cap (V^{\tau _h} + i V^{-\tau _h})$ of the antiholomorphic involution $\overline{\tau }_h$ on ${\mathcal {T}}_V$.

Proof

The equality of $W_R$ and $W_V^+(h)$ follows immediately from (D.7). For $0< t < \pi$, we have

$$\begin{aligned} \mathop {\mathrm{Im}}\nolimits (\alpha _{it}(x))= & {} \mathop {\mathrm{Im}}\nolimits (\cos t \cdot x_0 + \sin t \cdot i x_1, \cos t \cdot x_1 + \sin t \cdot i x_0, x_2, \ldots , x_d)\\= & {} \sin t\cdot (x_1, x_0, 0,\ldots ,0).\end{aligned}$$

This implies the equality of the sets in (a) and (c). Finally, we observe that

$$\begin{aligned} {\mathcal {T}}_V^{\overline{\tau }_h} = (V + i V_+)^{\overline{\tau }_h} = V^{\tau _h} + i V_+^{-\tau _h} =\{ (ix_0, i x_1, x_2, \ldots , x_d) \in V : x_0 > |x_1|\}\end{aligned}$$

implies that $\kappa _h({\mathcal {T}}_V^{\overline{\tau }_h}) = W_R$. $\square$

Remark D.5

For the closed light cone $C := \overline{V_+}$ and the h-eigenspaces

$$\begin{aligned} V_{\pm 1}(h) = {{\mathbb {R}}}({\mathbf{e}}_1 \pm {\mathbf{e}}_0), \quad \text{ we } \text{ put } \quad C_\pm := \pm C \cap V_{\pm 1}(h) = [0,\infty ) ({\mathbf{e}}_1 \pm {\mathbf{e}}_0),\end{aligned}$$

so that we obtain the following description of the standard right wedge

$$\begin{aligned} W_R = W_V^+(h) = V_0(h) + C_+^\circ + C_-^\circ = V_0(h) + {{\mathbb {R}}}_+ ({\mathbf{e}}_1 + {\mathbf{e}}_0) + {{\mathbb {R}}}_+ ({\mathbf{e}}_1 - {\mathbf{e}}_0).\end{aligned}$$

1.3 D.3 De Sitter space $\mathop {\mathrm{dS}}\nolimits ^d, d \ge 2$

We write $G := \mathop {\mathrm{SO}}\nolimits _{1,d}({{\mathbb {R}}})^{\uparrow }$ for the connected Lorentz group acting on Minkowski space $(V,[\cdot ,\cdot ])$ and consider the de Sitter space

$$\begin{aligned} M := \mathop {\mathrm{dS}}\nolimits ^d := \{ x \in V : [x,x] = -1\} = G.{\mathbf{e}}_1.\end{aligned}$$

The purpose of this appendix is derive our main results for the example $\mathop {\mathrm{dS}}\nolimits ^d$ by direct calculations without the elaborate structure theory developed for the general case. By Proposition D.3, the exponential function of the symmetric space $\mathop {\mathrm{dS}}\nolimits ^d$ is given by

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits _p(v) = C(-[v,v]) p + S(-[v,v]) v,\end{aligned}$$

so that spacelike vectors v generate closed geodesics.

De Sitter space inherits the structure of a non-compactly causal symmetric space from the embedding into Minkowski space $(V,V_+)$. In particular, the positive cone in a point $x \in \mathop {\mathrm{dS}}\nolimits ^d$ is given by

$$\begin{aligned} V_+(x) := V_+ \cap T_x(\mathop {\mathrm{dS}}\nolimits ^d) = V_+ \cap x^\bot .\end{aligned}$$

We keep the notation h, $\alpha _t$ and $X_h$ from our discussion of Minkowski space in Subsection D.2. As $\mathop {\mathrm{dS}}\nolimits ^d$ is $\alpha$-invariant, the vector field $X_h$ is tangential to $\mathop {\mathrm{dS}}\nolimits ^d$, and Lemma D.4 immediately implies that its positivity domain is given by

$$\begin{aligned} W_{\mathop {\mathrm{dS}}\nolimits ^d}^+(h) := \{ x \in \mathop {\mathrm{dS}}\nolimits ^d : X_h(x) \in V_+(x) \} = W_R \cap \mathop {\mathrm{dS}}\nolimits ^d = \{ x \in \mathop {\mathrm{dS}}\nolimits ^d : x_1 > |x_0|\}.\end{aligned}$$

The centralizer of h in G is the subgroup

$$\begin{aligned} G^h = \exp ({{\mathbb {R}}}h) \mathop {\mathrm{SO}}\nolimits _{d-1}({{\mathbb {R}}}) \cong \mathop {\mathrm{SO}}\nolimits _{1,1}({{\mathbb {R}}})^{\uparrow }\times \mathop {\mathrm{SO}}\nolimits _{d-1}({{\mathbb {R}}}) \end{aligned}$$

(D.8)

(cf. [31, Lemma 4.12]).

Lemma D.6

For

$$\begin{aligned} V_+^\pi ({\mathbf{e}}_1) := \{ x \in T_{{\mathbf{e}}_1}(\mathop {\mathrm{dS}}\nolimits ^d) \cong {\mathbf{e}}_1^\bot : x_0 > 0, 0< [x,x] < \pi ^2 \}, \end{aligned}$$

we have

$$\begin{aligned} (V + i V_+) \cap \mathop {\mathrm{dS}}\nolimits ^d_{{\mathbb {C}}}= \{ z \in V_{{\mathbb {C}}}: [z,z] = -1, \mathop {\mathrm{Im}}\nolimits z \in V_+\} = G.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_1}(iV_+^\pi ({\mathbf{e}}_1)). \end{aligned}$$

(D.9)

Proof

First we observe that both sides of (D.9) are G-invariant.

“$\supseteq$”: We have to show that any $x \in V_+^\pi ({\mathbf{e}}_1)$ satisfies $\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_1}(ix) \in V + i V_+$. Since the orbit of x under $G_{{\mathbf{e}}_1}$ contains an element in ${{\mathbb {R}}}{\mathbf{e}}_0$, we may assume that $x = x_0 {\mathbf{e}}_0$ with $0< x_0 < \pi$. Then

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_1}(ix) = \mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_1}(ix_0 {\mathbf{e}}_0) = C(x_0^2) {\mathbf{e}}_1 + S(x_0^2) x_0 i {\mathbf{e}}_0 = \cos (x_0) {\mathbf{e}}_1 + \sin (x_0) i {\mathbf{e}}_0 \in V + i V_+ \end{aligned}$$

follows from $\sin (x_0) > 0$.

“$\subseteq$”: If $z = x + i y\in \mathop {\mathrm{dS}}\nolimits ^d_{{\mathbb {C}}}\cap (V + i V_+)$, then $[z,z] = -1$ and $y \in V_+$. Acting with G, we may thus assume that $y = y_0 {\mathbf{e}}_0$ with $y_0 > 0$. Then $x \in y^\bot = {\mathbf{e}}_0^\bot$ follows from $\mathop {\mathrm{Im}}\nolimits [z,z] = 0$, so that we may further assume that $x = x_1 {\mathbf{e}}_1$ for some $x_1 \ge 0$. Now $z = i y_0 {\mathbf{e}}_0 + x_1 {\mathbf{e}}_1 \in \mathop {\mathrm{dS}}\nolimits ^d_{{\mathbb {C}}}$ implies that

$$\begin{aligned} -1 = [z,z] = - y_0^2 - x_1^2.\end{aligned}$$

Hence there exists a $t \in (0,\pi )$ with $y_0 = \sin t$ and $x_1 = \cos t$. This leads to

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_1}(i t {\mathbf{e}}_0) = \cos (t) {\mathbf{e}}_1 + \sin (t) i {\mathbf{e}}_0 = x_1 {\mathbf{e}}_1 + y_0 i {\mathbf{e}}_0 = z. \end{aligned}$$

$\square$

Definition D.7

The complex manifold

$$\begin{aligned} {\mathcal {T}}_M = {\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d} := G.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_1}(i V_+^\pi ({\mathbf{e}}_1)) = \mathop {\mathrm{dS}}\nolimits ^d_{{\mathbb {C}}}\cap (V + i V_+) \end{aligned}$$

(D.10)

is called the tube domain of $\mathop {\mathrm{dS}}\nolimits ^d$. If coincides with $G.\mathop {\mathrm{Exp}}\nolimits _m(i V_+^\pi (m))$ for every $m \in \mathop {\mathrm{dS}}\nolimits ^d$.

Starting from the relation ${\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d} = G.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2}(i V_+^\pi ({\mathbf{e}}_2))$, the invariance of ${\mathbf{e}}_2$ under $\tau _h = e^{\pi i h}$ permits us to obtain a nice description of the fixed point set of $\overline{\tau }_h$ on ${\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}$.

Lemma D.8

${\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}^{\overline{\tau }_h} = G^h.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2}(i V_+^\pi ({\mathbf{e}}_2)^{-\tau _h}).$

Proof

We clearly have

$$\begin{aligned} {\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}^{\overline{\tau }_h} \supseteq G^{\tau _h}.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2}(i V_+^\pi ({\mathbf{e}}_2)^{-\tau _h}).\end{aligned}$$

We also note that

$$\begin{aligned} V_+^\pi ({\mathbf{e}}_2)^{-\tau _h}= & {} V_+^\pi ({\mathbf{e}}_2) \cap ({{\mathbb {R}}}{\mathbf{e}}_0 + {{\mathbb {R}}}{\mathbf{e}}_1)\\= & {} \{ x_0 {\mathbf{e}}_0 + x_1 {\mathbf{e}}_1 : x_0 > 0, 0< x_0^2 - x_1^2 < \pi ^2\}.\end{aligned}$$

Next we observe that

$$\begin{aligned} {\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}^{\overline{\tau }_h}= & {} (V + i V_+)^{\overline{\tau }_h} \cap \mathop {\mathrm{dS}}\nolimits ^d_{{\mathbb {C}}}= (V^{\tau _h} + i V_+^{-\tau _h}) \cap \mathop {\mathrm{dS}}\nolimits ^d_{{\mathbb {C}}}\\= & {} \{ (i x_0, i x_1, x_2, \ldots , x_d) \in \mathop {\mathrm{dS}}\nolimits ^d_{{\mathbb {C}}}: x_0 > |x_1|\}.\end{aligned}$$

Hence the elements $z \in {\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}^{\overline{\tau }_h}$ are of the form

$$\begin{aligned} z = (ix_0, ix_1, x_2, \ldots , x_d), \quad x_0 > |x_1|, \ x_0^2 - x_1^2 + x_2^2 + \cdots + x_d^2 = 1.\end{aligned}$$

Therefore the orbit of z under $G^h = \exp ({{\mathbb {R}}}h) \mathop {\mathrm{SO}}\nolimits _{d-1}({{\mathbb {R}}})$ (see (D.8)) contains an element of the form $y = (iy_0, 0, y_2,0,\ldots ,0)$ with $y_0 > 0$ and $y_0^2 + y_2^2 = 1$. Hence there exists a $t \in (0,\pi )$ with $y_0 = \sin (t)$ and $y_2 = \cos (t)$. Then

$$\begin{aligned} \mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2} (it{\mathbf{e}}_0) = \cos (t) {\mathbf{e}}_2 + \sin (t) i {\mathbf{e}}_0 = y_2 {\mathbf{e}}_2 + i y_0 {\mathbf{e}}_0 = y.\end{aligned}$$

This implies that

$$\begin{aligned} {\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}^{\overline{\tau }_h} \subseteq G^h.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2}(i (0,\pi ) {\mathbf{e}}_0) = G^h.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2}(i V_+^\pi ({\mathbf{e}}_2)^{-\tau _h}). \end{aligned}$$

$\square$

Proposition D.9

The following subsets of de Sitter space $M =\mathop {\mathrm{dS}}\nolimits ^d$ are equal:

• (a) $W_R \cap \mathop {\mathrm{dS}}\nolimits ^d = \{ x \in \mathop {\mathrm{dS}}\nolimits ^d : x_1 > |x_0|\}$.

• (b) $W_M^+(h) := \{ x \in \mathop {\mathrm{dS}}\nolimits ^d : X_h(x) \in V_+(x) \}$ (the positivity domain of $X_h$).

• (c) $W_M^{\mathrm{KMS}}(h) = \{ x \in \mathop {\mathrm{dS}}\nolimits ^d : (\forall t \in (0,\pi )) \ \alpha _{it}(x) \in {\mathcal {T}}_M\}$ (the KMS domain of $\alpha$).

• (d) $\kappa _h({\mathcal {T}}_M^{\overline{\tau }_h})$.

• (e) $W_M(h) = G^h.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2}((C_+^\circ + C_-^\circ )^\pi )$.

Proof

The equality of the sets under (a) and (b) follows from Lemma D.4.

As $\alpha _{it}(m) \in M_{{\mathbb {C}}}$ for every $t \in {{\mathbb {R}}}$ and $m \in M$, Lemma D.6 shows that the set under (c) coincides with $W_V^+(h) \cap M = W_M ^+(h)$.

From the proof of Lemma D.8, we recall that

$$\begin{aligned} {\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}^{\overline{\tau }_h} = \{ (i x_0, i x_1, x_2, \ldots , x_d) \in \mathop {\mathrm{dS}}\nolimits ^d_{{\mathbb {C}}}: x_0 > |x_1|\}.\end{aligned}$$

Now $\kappa _h(x) = (-i x_1, -i x_0, x_2, \ldots , x_d)$ implies

$$\begin{aligned} \kappa _h({\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}^{\overline{\tau }_h}) = \{ (x_0, x_1, x_2, \ldots , x_d) \in \mathop {\mathrm{dS}}\nolimits ^d : x_1 > |x_0|\} = W_R \cap \mathop {\mathrm{dS}}\nolimits ^d = W_M(h).\end{aligned}$$

Combining Lemma D.8 with (d), we finally obtain

$$\begin{aligned} W_M^+(h)&= \kappa _h({\mathcal {T}}_{\mathop {\mathrm{dS}}\nolimits ^d}^{\overline{\tau }_h}) = G^h.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2}( \kappa _h(iV_+^\pi ({\mathbf{e}}_2)^{-\tau _h})) = G^h.\mathop {\mathrm{Exp}}\nolimits _{{\mathbf{e}}_2}((C_+^\circ + C_-^\circ )^\pi ) \end{aligned}$$

for $C_\pm ^\circ = {{\mathbb {R}}}_+ ({\mathbf{e}}_1 \pm {\mathbf{e}}_0).$ $\square$

Remark D.10

For the $\alpha$-fixed base point ${\mathbf{e}}_2 \in \mathop {\mathrm{dS}}\nolimits ^d$ the cone

$$\begin{aligned} W_R({\mathbf{e}}_2) := W_R \cap T_{{\mathbf{e}}_2}(\mathop {\mathrm{dS}}\nolimits ^d) = C_+^\circ + C_-^\circ + T_{{\mathbf{e}}_2}(\mathop {\mathrm{dS}}\nolimits ^d)^{\tau _h}, \quad \text{ where } \quad C_\pm ^\circ = {{\mathbb {R}}}_+ ({\mathbf{e}}_1 \pm {\mathbf{e}}_0)\end{aligned}$$

is an infinitesimal version of the wedge domain in $M = \mathop {\mathrm{dS}}\nolimits ^d$.