Equivariant realizations of Hermitian symmetric space of noncompact type

Abstract

Let $M=G/K$ be a Hermitian symmetric space of noncompact type. We provide a way of constructing K-equivariant embeddings from M to its tangent space $T_oM$ at the origin by using the polarity of the K-action. As an application, we reconstruct the K-equivariant holomorphic embedding so called the Harish-Chandra realization and the K-equivariant symplectomorphism constructed by Di Scala-Loi and Roos under appropriate identifications of spaces. Moreover, we characterize the holomorphic/symplectic embedding of M by means of the polarity of the K-action. Furthermore, we show a special class of totally geodesic submanifolds in M is realized as either linear subspaces or bounded domains of linear subspaces in $T_oM$ by the K-equivariant embeddings. We also construct a K-equivariant holomorphic/symplectic embedding of an open dense subset of the compact dual $M^*$ into its tangent space at the origin as a dual of the holomorphic/symplectic embedding of M.

A Relation to known results

In this appendix, we show relations between our results and some known results. We confirm that, under appropriate identifications of spaces, $\Psi$ and $\Phi$ constructed in Theorem 1.3 coincide with the Harish-Chandra realization [8] and the symplectomorphism given by Di Scala-Loi [5] and Roos [16], respectively. Moreover, we mention the relation to the result by Loi-Mossa [12]. Throughout this appendix, we assume $M=G/K$ is a HSSNT of rank r and denote the Cartan decomposition by ${\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}$. We follow the notations given in Sect. 2. In the following, we denote the complexification of a real vector space V by $V_{{\mathbb {C}}}=V\oplus \sqrt{-1}V$.

1.1 A.1 Harish-Chandra realization

First, we briefly recall the Harish-Chandra realization of HSSNT. Throughout this subsection, we use several results proved in [9, Section 7 of Ch. VIII]. See also [2, Section 2] and [10] for a nice summary.

Let ${\mathfrak {g}}_{{\mathbb {C}}}$ be the complexification of ${\mathfrak {g}}$. Take a maximal abelian subalgebra ${\mathfrak {t}}$ of ${\mathfrak {k}}$. Then, the center ${\mathfrak {c}}({\mathfrak {k}})$ of ${\mathfrak {k}}$ is contained in ${\mathfrak {t}}$ and ${\mathfrak {t}}$ is a maximal abelian subalgebra of ${\mathfrak {g}}$. Moreover, ${\mathfrak {t}}_{{\mathbb {C}}}$ is a Cartan subalgebra of ${\mathfrak {g}}_{\mathbb {C}}$ (see Proof of Theorem 7.1 in [9, Ch. VIII]), and we have a root space decomposition with respect to ${\mathfrak {t}}_{\mathbb {C}}$:

$$\begin{aligned} {\mathfrak {g}}_{\mathbb {C}}={\mathfrak {t}}_{\mathbb {C}}\oplus \bigoplus _{\chi \in \Delta }{\mathfrak {g}}_{\chi }, \end{aligned}$$

where $\Delta$ is the set of non-zero roots, and ${\mathfrak {g}}_{\chi }:=\{Z\in {\mathfrak {g}}_{\mathbb {C}}: (\mathrm{ad}T)Z=\chi (T)Z\ \forall T\in {\mathfrak {t}}_{\mathbb {C}}\}$ is the root space. Note that $\mathrm{dim}_{{\mathbb {C}}}{\mathfrak {g}}_ {\chi }=1$ for any $\chi \in \Delta$ (see [9, Section 4 of Ch. III]). Since B is non-degenerate on the Cartan subalgebra ${\mathfrak {t}}_{\mathbb {C}}$, there exists a unique vector $T_{\chi }$ which is a dual of $\chi$ with respect to B. Then, we have $[{\mathfrak {g}}_{\chi }, {\mathfrak {g}}_{-\chi }]={\mathbb {C}} T_{\chi }$, and there exists a basis $E_{\chi }$ of ${\mathfrak {g}}_{\chi }$ so that

$$\begin{aligned} E_{\chi }-E_{-\chi },\quad \sqrt{-1}(E_{\chi }+E_{-\chi })\in {\mathfrak {k}}\oplus \sqrt{-1}{\mathfrak {p}},\quad [E_{\chi }, E_{-\chi }]=\frac{2T_{\chi }}{\chi (T_{\chi })}. \end{aligned}$$

Since each $\chi \in \Delta$ is real valued on $\sqrt{-1}{\mathfrak {t}}$, we define an ordering of $\Delta$ associated with the ordered real vector space $(\sqrt{-1}{\mathfrak {t}})^*$. Moreover, we may assume that the ordering is compatible with $\sqrt{-1}{\mathfrak {c}}({\mathfrak {k}})\subset \sqrt{-1}{\mathfrak {t}}$, that is, $\alpha |_{\sqrt{-1}{\mathfrak {t}}}$ is positive whenever $\alpha |_{\sqrt{-1}{\mathfrak {c}}({\mathfrak {k}})}$ is positive. We denote the set of positive roots by $\Delta _+$.

For each $\chi \in \Delta$, the root space ${\mathfrak {g}}_{\chi }$ is contained in either ${\mathfrak {k}}_{{\mathbb {C}}}$ or ${\mathfrak {p}}_{\mathbb {C}}$. We put $\Delta _{{\mathfrak {p}}}:=\{\chi \in \Delta :\ {\mathfrak {g}}_{\chi }\subset {\mathfrak {p}}_{{\mathbb {C}}}\}$, and the element in $\Delta _{{\mathfrak {p}}}$ is so called noncompact root (otherwise, we say compact root). We set $Q_+:=\Delta _+\cap \Delta _{{\mathfrak {p}}}$, and put

$$\begin{aligned} {\mathfrak {p}}_+=\bigoplus _{\chi \in Q_+}{\mathfrak {g}}_{\chi },\quad {\mathfrak {p}}_-=\bigoplus _{-\chi \in Q_+}{\mathfrak {g}}_{\chi }. \end{aligned}$$

Then, it turns out that ${\mathfrak {p}}_+$ and ${\mathfrak {p}}_-$ are abelian subalgebras of ${\mathfrak {g}}^{{\mathbb {C}}}$ and we have

$$\begin{aligned} {[}{\mathfrak {k}}_{\mathbb {C}}, {\mathfrak {p}}_+]\subset {\mathfrak {p}}_+,\quad [{\mathfrak {k}}_{\mathbb {C}}, {\mathfrak {p}}_-]\subset {\mathfrak {p}}_-,\quad {\mathfrak {p}}_{\mathbb {C}}={\mathfrak {p}}_+\oplus {\mathfrak {p}}_{-} \end{aligned}$$

(46)

([9, Proposition 7.2 in Ch. VIII]).

Let $\mathbf{G}$ be a simply connected Lie group with Lie algebra ${\mathfrak {g}}_{{\mathbb {C}}}$ as a real Lie algebra. We denote the connected Lie subgroups of $\mathbf{G}$ with Lie algebras ${\mathfrak {k}}_{{\mathbb {C}}}$, ${\mathfrak {p}}_{+}$ and ${\mathfrak {p}}_{-}$ by $\mathbf{K}, P_+$ and $P_{-}$, respectively. It turns out that $\mathbf{K}P_{+}$ is a subgroup of $\mathbf{G}$ by (46), and furthermore, it is closed in $\mathbf{G}$. Since $\mathbf{G}$, $\mathbf{K}$ and $P_+$ are complex Lie groups, the coset space $\mathbf{G}/\mathbf{K}P_+$ inherits a $\mathbf{G}$-invariant complex structure. Moreover, a natural correspondence

$$\begin{aligned} f: M^*=G^*/K\rightarrow \mathbf{G}/\mathbf{K}P_+,\quad g^*K\mapsto g^*\mathbf{K}P_+ \end{aligned}$$

gives rise to a holomorphic diffeomorphism from the compact dual $M^*$ of M, and it also follows that $M=G/K$ is holomorphically embedded into $\mathbf{G}/\mathbf{K}P_+\simeq M^*$ as a G-orbit through the origin, namely, the map

$$\begin{aligned} b: M=G/K\rightarrow \mathbf{G}/\mathbf{K}P_+,\quad gK\mapsto g\mathbf{K}P_+ \end{aligned}$$

is a holomorphic embedding onto an open subset in $\mathbf{G}/\mathbf{K}P_+$( [9, Proposition 7.14 in Ch.VIII]). The map b is referred as Borel embedding of M. By using the identification $f: M^*\simeq \mathbf{G}/\mathbf{K}P_+$, we say also the map $f^{-1}\circ b: M\rightarrow M^*$ the Borel embedding.

On the other hand, the map

$$\begin{aligned} \xi : {\mathfrak {p}}_{-}\rightarrow \mathbf{G}/\mathbf{K}P_+,\quad X\rightarrow (\exp X)\mathbf{K}P_+ \end{aligned}$$

gives a holomorphic diffeomorphism onto an open dense subset U in $\mathbf{G}/\mathbf{K}P_+$ such that $b(M)\subset U=\xi ({\mathfrak {p}}_-)$ ( [9, Theorem 7.16 in Ch. VIII]). By denoting $f^{-1}(U)=(M^*)^o\subset M^*$, we obtain two holomorphic embeddings

$$\begin{aligned} \psi :=\xi ^{-1}\circ b: M\rightarrow {\mathfrak {p}}_-,\quad \psi ^*:=\xi ^{-1}\circ f|_{(M^*)^o}: (M^*)^o\rightarrow {\mathfrak {p}}_{-}. \end{aligned}$$

Then, the image $D_{-}:=\psi (M)$ is a bounded domain in ${\mathfrak {p}}_{-}$, and the map $\psi$ is so called the Harish-Chandra realization of M.

We shall give more precise description of $\psi$ and $D_{-}$. It is known that there exists a subset $\{\mu _1,\ldots , \mu _r\}$ of $Q_+$, where r is the rank of M, such that they consist of strongly orthogonal roots, i.e., $\mu _i\pm \mu _j\notin \Delta$ for any $i,j=1,\ldots , r$. We simply denote the vectors $E_{\pm \mu _i}$ and $T_{\mu _i}$ by $E_{\pm i}$ and $T_i$, respectively. We put

$$\begin{aligned} {\mathfrak {a}}:=\bigoplus _{i=1}^r {\mathbb {R}}V_i,\quad \mathrm{where}\ V_i:=E_i+E_{-i}. \end{aligned}$$

Then, ${\mathfrak {a}}$ becomes a maximal abelian subspace of ${\mathfrak {p}}$ ( [9, Corollary 7.6 in Ch. VIII]). Moreover, the complex Lie subalgebra ${\mathfrak {l}}_i:={\mathbb {C}} E_{i}\oplus {\mathbb {C}} E_{-i}\oplus {\mathbb {C}} [E_i, E_{-i}]$ is isomorphic to ${{\mathfrak {s}}}{{\mathfrak {l}}}(2,{\mathbb {C}})$, and we have a decomposition

$$\begin{aligned} \exp (x_iV_i)=\exp ((\tanh x_i)E_{-i})\exp ((\log \cosh x_i)[E_i, E_{-i}])\exp ( (\tanh x_i)E_{i}) \end{aligned}$$

(47)

for any $x_i\in {\mathbb {R}}$ and each $i=1,\ldots , r$ (see [9, Lemma 7.11 in Ch. VIII]). In particular, for any element $v=\sum _{i=1}^r x_i V_i\in {\mathfrak {a}}$, $\exp v$ is expressed by $\exp v=\exp v_{-}\exp w\exp v_{+}$ with $v_{-}\in {\mathfrak {p}}_{-}$, $w\in {\mathfrak {k}}^{{\mathbb {C}}}$ and $v_{+}\in {\mathfrak {p}}_{+}$, and $v_{-}$ is given by

$$\begin{aligned} v_-=\sum _{i=1}^r (\tanh x_i) E_{-i}\in {\mathfrak {p}}_{-}. \end{aligned}$$

Thus, we have $\psi (\exp v \cdot o)=\xi ^{-1}((\exp v_{-}) \cdot o^*)=v_{-}=\sum _{i=1}^r (\tanh x_i) E_{-i}.$ Since $\psi$ is K-equivariant, the Harish-Chandra realization is expressed by

$$\begin{aligned}&\psi (k\exp v \cdot o)=\mathrm{Ad}(k)\sum _{i=1}^r (\tanh x_i) E_{-i} \end{aligned}$$

for $k\in K$, $v=\sum _{i=1}^r a_i V_i\in {\mathfrak {a}}$. In particular, we have

$$\begin{aligned}&D_-=\mathrm{Ad}(K)(\square _{-}),\ \mathrm{where}\ \square _{-}=\left\{ \sum _{i=1}^ry_i E_{-i}: |y_i|<1\right\} . \end{aligned}$$

Now, we adapt the above description to our description given in the present paper. Recall that the complex structure $J_o$ on ${\mathfrak {p}}$ is defined by $J_o=\mathrm{ad}(\zeta )|_{{\mathfrak {p}}}$ for some element $\zeta \in \mathfrak {c(k)}\subset {\mathfrak {t}}_{\mathbb {C}}$. We may assume that $\mu _i(\zeta )=-\sqrt{-1}$ so that

$$\begin{aligned} J_oE_i=-\sqrt{-1}E_i\quad \mathrm{and}\quad J_oE_{-i}=\sqrt{-1}E_{-i} \end{aligned}$$

([9, Corollary 7.13 in Ch.VIII]).

We first show the following correspondence:

Lemma A.1

The basis $\{V_i\}_{i=1}^r$ coincides with the set of normalized root vectors $\{{\widetilde{H}}_i\}_{i=1}^r$ given in Sect. 2.4, up to sign of each vector.

Proof

By Proposition 5.4, it suffices to show that $\{V_i\}_{i=1}^r$ satisfies the bracket relations (43). Indeed, since $J_oV_i=\mathrm{ad}(\zeta )(E_i+E_{-i})=-\sqrt{-1}(E_i-E_{-i})$, we see

$$\begin{aligned} {[}V_i, J_oV_i]=[E_i+E_{-i},-\sqrt{-1}(E_i-E_{-i})]=2\sqrt{-1}[E_i, E_{-i}]=\frac{4\sqrt{-1}}{\mu _i(T_i)}T_i. \end{aligned}$$

By using the fact $T_i\in {\mathfrak {t}}$, we see

$$\begin{aligned} {[}[V_i, J_oV_i], J_oV_i]&=\frac{4}{\mu _i(T_i)}[T_i,E_i-E_{-i}]=4V_i,\\ {[}[V_i, J_oV_i], V_i]&=\frac{4\sqrt{-1}}{\mu _i(T_i)}[T_i,E_i+E_{-i}]=-4J_oV_i \end{aligned}$$

for any $i=1,\ldots ,r$. Otherwise, we easily see that $[[V_i, J_oV_j], J_oV_k]=[[V_i, J_oV_j], V_k]=0$ by the strong orthogonality of roots. This proves $\{V_i\}_{i=1}^r$ satisfies (43) as required. $\square$

Next lemma gives an natural isomorphism between ${\mathfrak {p}}$ and ${\mathfrak {p}}_{-}$ (resp. ${\mathfrak {p}}^*=\sqrt{-1}{\mathfrak {p}}$ and ${\mathfrak {p}}_{-}$):

Lemma A.2

The differential $d\psi : {\mathfrak {p}}\rightarrow {\mathfrak {p}}_-$ at $o\in M$ and $d\psi ^*: {\mathfrak {p}}^*\rightarrow {\mathfrak {p}}_{-}$ at $o^*\in (M^*)^o$ are linear isomorphisms, and they are explicitly given by

$$\begin{aligned} d\psi (X)=\frac{1}{2}(X-\sqrt{-1}J_oX)\quad \mathrm{and}\quad d\psi ^*(X^*)=\frac{1}{2}(X^*-\sqrt{-1}J_oX^*) \end{aligned}$$

for $X\in {\mathfrak {p}}$, where $X^*:=\sqrt{-1}X$. In particular we have $d\psi ^*(\sqrt{-1}X)=\sqrt{-1}d\psi (X)$.

See [9, Corollary 7.13 in Ch.VIII] for a proof. Note that the proof of the formula for $d\psi ^*$ is given by a similar argument for $d\psi$.

We also define a linear isomorphism between ${\mathfrak {p}}$ and ${\mathfrak {p}}^*$ by

$$\begin{aligned} \iota :=(d\psi ^*)^{-1}\circ d\psi : {\mathfrak {p}}\rightarrow {\mathfrak {p}}^*,\quad X\mapsto -J_oX^*. \end{aligned}$$

The following proposition shows that the holomorphic diffeomorphism $\Psi$ (resp. $\Psi ^*$) coincides with $\psi$ (resp. $\psi ^*$) under the identification $d\psi : {\mathfrak {p}}\rightarrow {\mathfrak {p}}_{-}$ (resp. $d\psi ^*: {\mathfrak {p}}^* \rightarrow {\mathfrak {p}}_{-}$).

Proposition A.3

We have the following:

• (i) Let $\Psi :M\rightarrow {\mathfrak {p}}$ be the map constructed in Theorem 1.3 and $\psi : M\rightarrow {\mathfrak {p}}_-$ the Harish-Chandra realization. Then, we have $\Psi =d\psi ^{-1}\circ \psi$.

• (ii) Let $\Psi ^*: (M^*)^o\rightarrow {\mathfrak {p}}^*$ be the dual map of $\Psi$ given in Theorem 1.4. Then, we have $\Psi ^*=(d\psi ^*)^{-1}\circ \psi ^*$.

In particular, the Borel embedding $f^{-1}\circ b: M\rightarrow M^*$ is given by

$$\begin{aligned} f^{-1}\circ b=(\psi ^*)^{-1}\circ \psi =(\Psi ^*)^{-1}\circ \iota \circ \Psi , \end{aligned}$$

namely, we have the following commutative diagram:

Proof

• (i) It is sufficient to show that $d\psi \circ \Psi |_{A}=\psi |_{A}$ for $A=\mathrm{Exp}_o{\mathfrak {a}}$. We consider the maximal abelian subspace ${\mathfrak {a}}$ spanned by $\{V_i\}_{i=1}^r$. By Lemma A.1, we may assume $V_i$ coincides with either ${\widetilde{H}}_i$ or $-{\widetilde{H}}_i$ for each i. We use the notation $V_i=\epsilon _i{\widetilde{H}}_i$, where $\epsilon _i=1$ or $-1$. Then, any element v in ${\mathfrak {a}}$ is expressed by $v=\sum _{i=1}^r x_i V_i=\sum _{i=1}^r (\epsilon _i x_i ){\widetilde{H}}_i$, and hence, by using (24), we see

  $$\begin{aligned} d\psi \circ \Psi (\mathrm{Exp}_ov)=d\psi \left( \sum _{i=1}^r \tanh (\epsilon _i x_i) {\widetilde{H}}_i\right) =d\psi \left( \sum _{i=1}^r (\tanh x_i) V_i\right) =\sum _{i=1}^r (\tanh x_i) E_{-i}, \end{aligned}$$

  where we used the fact $d\psi (V_i)=E_{-i}$. This proves (i).

• (ii) We shall show $d\psi ^*\circ \Psi ^*|_{A^*}=\psi ^*|_{A^*}$ for $A^*=\mathrm{Exp}_o^*{\mathfrak {a}}^*$, where ${\mathfrak {a}}^*=\sqrt{-1}{\mathfrak {a}}$. Since $v^*\in {\mathfrak {a}}^*$ is expressed by $v^*=\sqrt{-1}\sum _{i=1}^r x_iV_i$, (47) implies that

  $$\begin{aligned} \psi ^*(\exp v^*\cdot o^*)=\sqrt{-1}\sum _{i=1}^r (\tan x_i)E_{-i} \end{aligned}$$

  if $\exp v^*\cdot o^*\in (M^*)^o$. On the other hand, we have $v^*=\sum _{i=1}^r(\epsilon _i x_i){\widetilde{H}}_i^*$ with ${\widetilde{H}}_i^*=\sqrt{-1}{\widetilde{H}}_i$, and hence,

  $$\begin{aligned} d\psi ^*\circ \Psi ^*(\mathrm{Exp}_{o^*}^* v^*)= & {} d\psi ^* \left( \sum _{i=1}^r \tan (\epsilon _i x_i) {\widetilde{H}}_i^*\right) =d\psi ^* \left( \sum _{i=1}^r (\tan x_i) V_i^*\right) \\= & {} \sqrt{-1}\sum _{i=1}^r (\tan x_i)E_{-i} \end{aligned}$$

  since $d\psi ^*(V_i^*)=\sqrt{-1}d\psi (V_i)=\sqrt{-1}E_{-i}$. This proves (ii).

Then, one easily checks that the above commutative diagram holds. This proves the proposition. $\square$

Remark A.4

In Remark 4.10, we define a holomorphic embedding $h: M\rightarrow (M^*)^o$ by $h:=(\Psi ^*)^{-1}\circ \sqrt{-1}\circ \Psi$. It is easy to see that we have $f^{-1}\circ b=F\circ h$, where $F:=(\Psi ^*)^{-1}\circ (-J_o)\circ \Psi ^*: (M^*)^o\rightarrow (M^*)^o$ is a holomorphic diffeomorphism on $(M^*)^o$. Note that the difference between h and $f^{-1}\circ b$ is caused by the difference of identifications between ${\mathfrak {p}}$ and ${\mathfrak {p}}^*$.

1.2 A.2 Di Scala-Loi-Roos’s formula

In this subsection, we consider the symplectomorphism given by Di Scala-Loi and Roos. See [5, 16] for details. Let M be a HSSNT. It is known that M is associated with a Hermitian positive Jordan triple system $(T_oM, \{, ,\})$, which is defined by

$$\begin{aligned} \{u,v,w\}=-\frac{1}{2}(R_o(u,v)w+J_oR_o(u, J_ov)w) \end{aligned}$$

for $u,v,w\in T_oM$, where $R_o$ and $J_o$ are the curvature tensor and the complex structure of M at the origin, respectively. The Bergman operator B is defined by

$$\begin{aligned} B(u,v)=\mathrm{Id}- D(u,v)+Q(u)Q(v), \end{aligned}$$

where D is an operator on ${\mathfrak {p}}$ defined by $D(u,v)(w)=\{u,v, w\}$ and Q is the quadratic representation which is defined by $2Q(u)(v)=\{u,v,u\}$.

In the following, we identify $T_oM$ with ${\mathfrak {p}}$, and we regard ${\mathfrak {p}}$ as a Jordan triple system. Moreover, we identify M with a bounded domain $D=\Psi (M)\subset {\mathfrak {p}}$ by the map $\Psi : M\rightarrow {\mathfrak {p}}$. Note that $\Psi$ is regarded as the Harish-Chandra realization (Proposition A.3). In [5], Di-Scala-Loi proved that the following map

$$\begin{aligned} {\widetilde{\Phi }}: D\rightarrow {\mathfrak {p}},\quad {\widetilde{\Phi }}(z):=B(z,z)^{-\frac{1}{4}}z, \end{aligned}$$

(48)

becomes a K-equivariant symplectomorphism between $(D, \Psi ^{-1}\omega )$ and $({\mathfrak {p}}, \omega _o)$. Our purpose of this subsection is to show the following relation:

Proposition A.5

Let $\Phi : M\rightarrow {\mathfrak {p}}$ be the symplectomorphism constructed in Theorem 1.3 and ${\widetilde{\Phi }}: D\rightarrow {\mathfrak {p}}$ the map given by (48). Then, we have $\Phi \circ \Psi ^{-1}={\widetilde{\Phi }}$.

To prove this, we recall some basic notions in Jordan triple systems. An element $c\in {\mathfrak {p}}$ is said to be tripotent if $\{c, c, c\}=2c$, and we say two elements $c_1, c_2\in {\mathfrak {p}}$ are orthogonal if $D(c_1,c_2)=0$. It is known that any element $z\in {\mathfrak {p}}$ has a decomposition

$$\begin{aligned} z=\lambda _1c_1+\cdots +\lambda _pc_p \end{aligned}$$

so called the spectral decomposition, where $\lambda _1>\lambda _2>\cdots>\lambda _p>0$ and $\{c_1,\ldots , c_p\}$ is a set of mutually orthogonal tripotents. In terms of the restricted root system, the set of orthogonal tripotents coincides with the set of strongly orthogonal root vectors (up to sign of each vector):

Lemma A.6

Let z be any element in ${\mathfrak {p}}$, ${\mathfrak {a}}$ a maximal abelian subspace in ${\mathfrak {p}}$ containing z, and $\{{\widetilde{H}}_i\}_{i=1}^r$ the set of normalized root vectors with respect to ${\mathfrak {a}}$ given in Sect. 2.4. Then, any subset of $\{{\widetilde{H}}_i\}_{i=1}^r$ becomes a set of mutually orthogonal tripotents.

Proof

First, we show ${\widetilde{H}}_i$ is a tripotent for each i. Since the curvature tensor $R_o$ at the origin is given by $R_o(u,v)w=-[[u,v],w]$, we see by (12) that

$$\begin{aligned} \{{\widetilde{H}}_i, {\widetilde{H}}_i, {\widetilde{H}}_i\}=\frac{1}{2}J_o[[{\widetilde{H}}_i, J_o{\widetilde{H}}_i], {\widetilde{H}}_i]=J_o[[{\widetilde{H}}_i, X_i^{{\mathfrak {p}}}], {\widetilde{H}}_i]=2J_o[X_i^{{\mathfrak {k}}}, {\widetilde{H}}_i]=-4J_oX_i^{\mathfrak {p}}=2{\widetilde{H}}_i, \end{aligned}$$

as required. Next, for any $i\ne j$, we have $[{\widetilde{H}}_i, {\widetilde{H}}_j]=0$ and $[{\widetilde{H}}_i, J_o{\widetilde{H}}_j]=2[{\widetilde{H}}_i, X_j^{{\mathfrak {p}}}]=2\gamma _j({\widetilde{H}}_i)X_i^{{\mathfrak {k}}}=0$ since ${\mathfrak {a}}$ is abelian and $\{{\widetilde{H}}_i\}_{i=1}^r$ is an orthogonal basis. Therefore, we obtain $D({\widetilde{H}}_i, {\widetilde{H}}_j)=0$ for any $i\ne j$, and $\{{\widetilde{H}}_i\}_{i=1}^r$ is a set of mutually orthogonal tripotents. $\square$

We give a proof of Proposition A.5.

Proof of Proposition A.5

Fix arbitrary $z\in {\mathfrak {p}}$. We may assume z is contained in a maximal abelian subspace ${\mathfrak {a}}$. Let $\{{\widetilde{H}}_i\}_{i=1}^r$ be the strongly orthogonal root vectors with respect to ${\mathfrak {a}}$. As described in [5, Equation (26) in Section 6], the symplectomorphism ${\widetilde{\Phi }}$ is expressed by

$$\begin{aligned} {\widetilde{\Phi }}(z)=\sum _{i=1}^p\frac{\lambda _i}{\sqrt{1-\lambda _i^2}}c_i \end{aligned}$$

with respect to the spectral decomposition $z=\sum _{i=1}^p\lambda _ic_i$. By Lemma A.6, the spectral decomposition can be written by $z=\sum _{i=1}^p\epsilon _i\lambda _i{\widetilde{H}}_i$, where $\epsilon _i$ is equal to either 1 or $-1$ which is determined by the relation $c_i=\epsilon _i{\widetilde{H}}_i$. By putting $x_i=\epsilon _i\lambda _i$, we have $z=\sum _{i=1}^p x_i{\widetilde{H}}_i$ and ${\widetilde{\Phi }}$ is expressed by

$$\begin{aligned} {\widetilde{\Phi }}(z)=\sum _{i=1}^p\frac{x_i}{\sqrt{1-x_i^2}}{\widetilde{H}}_i. \end{aligned}$$

Thus, by using the formulas in (24), we see

$$\begin{aligned} \Phi \circ \Psi ^{-1}(z)=\sum _{i=1}^p \sinh (\tanh ^{-1}x_i){\widetilde{H}}_i=\sum _{i=1}^p\frac{x_i}{\sqrt{1-x_i^2}}{\widetilde{H}}_i={\widetilde{\Phi }}(z), \end{aligned}$$

as required. $\square$

1.3 A.3 Loi-Mossa’s diastatic exponential map

Regarding M as a bounded domain D via $\Psi$ and ${\mathfrak {p}}$ as the associated Jordan triple system, Loi-Mossa [12] defined the following map $\mathrm{DE}_o:{\mathfrak {p}}\rightarrow D$ ( [12, eq. (11)]):

$$\begin{aligned} \mathrm{DE}_o(z):=\sum _{i=1}^p \sqrt{1-e^{-\lambda _i^2}}\cdot c_i \end{aligned}$$

with respect to the spectral decomposition $z=\sum _{i=1}^p \lambda _ic_i\in D$. Note that $\mathrm{DE}_o: {\mathfrak {p}}\rightarrow D$ is a diffeomorphism, and Loi-Mossa proved that $\mathrm{DE}_o$ is the diastatic exponential map for $D\simeq M$ at the origin $o\in D$ ( [12, Theorem 1]). We denote the inverse map by $\mathrm{DL}_o: D\rightarrow {\mathfrak {p}}$, which is given by

$$\begin{aligned} \mathrm{DL}_o(z)=\sum _{i=1}^p \sqrt{-\log (1-\lambda _i^2)}\cdot c_i \end{aligned}$$

for $z=\sum _{i=1}^p \lambda _ic_i\in D$. Our aim here is to confirm that $\mathrm{DL}_o$ is also constructed by a strongly diagonal realization $\Omega _{\eta }$:

Proposition A.7

Define an injective odd function $\eta :{\mathbb {R}}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \eta (x):= {\left\{ \begin{array}{ll} \sqrt{\log \cosh ^2x} &{} x\ge 0\\ -\sqrt{\log \cosh ^2x} &{} x<0\\ \end{array}\right. }. \end{aligned}$$

(49)

Then, we have $\Omega _{\eta }\circ \Psi ^{-1}=\mathrm{DL}_o.$

Proof

The spectral decomposition is written by $z=\sum _{i=1}^p \epsilon _i\lambda _i {\widetilde{H}}_i$, where $\epsilon _i=1$ or $-1$ and is determined by the relation $c_i=\epsilon _i{\widetilde{H}}_i$. We put $x_i=\epsilon _i\lambda _i$ so that $z=\sum _{i=1}^p x_i{\widetilde{H}}_i$. Then we have

$$\begin{aligned} \mathrm{DL}_o(z)=\sum _{i=1}^p \sqrt{-\log (1-x_i^2)}\cdot \frac{x_i}{|x_i|}{\widetilde{H}}_i \end{aligned}$$

with $\mathrm{DL}_o(0)=0$. On the other hand, we see

$$\begin{aligned} \Omega _{\eta }\circ \Psi ^{-1}(z)=\sum _{i=1}^p \eta (\tanh ^{-1}x_i){\widetilde{H}}_i=\sum _{i=1}^p \sqrt{-\log (1-x_i^2)}\cdot \frac{x_i}{|x_i|}{\widetilde{H}}_i \end{aligned}$$

since

$$\begin{aligned} \eta (x)=\sqrt{\log (\cosh ^2x)}\cdot \frac{x}{|x|}=\sqrt{-\log (1-\tanh ^2x)}\cdot \frac{x}{|x|} \end{aligned}$$

with $\eta (0)=0$. This proves the lemma. $\square$

We shall consider the dual map of $\Omega _{\eta }$. The dual function $\eta ^*: (-\pi /2, \pi /2)\rightarrow {\mathbb {R}}$ of (49) is given by

$$\begin{aligned} \eta ^*(x):= {\left\{ \begin{array}{ll} \sqrt{-\log \cos ^2x} &{} x\in [0, \pi /2)\\ -\sqrt{-\log \cos ^2x} &{} x\in (-\pi /2, 0)\\ \end{array}\right. } \end{aligned}$$

with $\eta (0)=0$. This function is formally obtained by $\eta ^*(x)=-\sqrt{-1}\cdot \eta (\sqrt{-1}x).$ Then, the dual map $\Omega _{\eta ^*}^*$ is a map from $(M^*)^o=M^*\setminus \mathrm{Cut}_o(M^*)$ onto ${\mathfrak {p}}^*=\sqrt{-1}{\mathfrak {p}}$.

Let us identify $(M^*)^o$ with ${\mathfrak {p}}^*$ by the dual map $\Psi ^*$ of $\Psi$. We put $\mathrm{DL}_o^*:=\Omega _{\eta ^*}^*\circ (\Psi ^*)^{-1}: {\mathfrak {p}}^*\rightarrow {\mathfrak {p}}^*$. Then, for any $z=\sum _{i=1}^r x_i{\widetilde{H}}_i^*\in {\mathfrak {p}}^*$, we obtain

$$\begin{aligned} \mathrm{DL}_o^*(z)=\sum _{i=1}^r \eta (\tan ^{-1}x_i){\widetilde{H}}_i^*=\sum _{i=1}^r \sqrt{\log (1+x^2_i)}\cdot \frac{x_i}{|x_i|} {\widetilde{H}}_i^* \end{aligned}$$

with $\eta ^*(0)=0$ since

$$\begin{aligned} \eta ^*(x)=\sqrt{-\log (\cos ^2x)}\cdot \frac{x}{|x|}=\sqrt{\log (1+\tan ^2x)}\cdot \frac{x}{|x|}. \end{aligned}$$

By using the spectral decomposition, we see

$$\begin{aligned} \mathrm{DL}_o^*(z)=\sum _{i=1}^p \sqrt{\log (1+\lambda ^2_i)}\cdot c_i, \end{aligned}$$

for $z=\sum _{i=1}^p \lambda _i c_i$, and the inverse map is given by

$$\begin{aligned} \mathrm{DE}_o^*(z)=\sum _{i=1}^p \sqrt{e^{\lambda _i^2}-1}\cdot c_i. \end{aligned}$$

This recovers the map defined in [12, eq. (17)]. Loi-Mossa proved that $\mathrm{DE}_o^*(z)$ is the diastatic exponential map for the compact dual $M^*$. See [12, Theorem 2].