# Homogeneous codimension one foliations on reducible symmetric spaces of
noncompact type

Ivan Solonenko  Department of Mathematics, King’s College London, United
Kingdom<EMAIL_ADDRESS>
###### Abstract.

We extend the classification of homogeneous codimension-one foliations on
irreducible Riemannian symmetric spaces of noncompact type obtained by Berndt
and Tamaru in [4] to the reducible case, thus completing it for all noncompact
symmetric spaces.

##  1\. Introduction

A foliation $\mathcal{{F}}$ with connected properly embedded leaves on a
connected Riemannian manifold $M$ is called homogeneous if the subgroup of
isometries of $M$ preserving the leaves of $\mathcal{{F}}$ acts transitively
on each leaf. Given a connected Lie group $G$ acting properly and
isometrically on $M$ without singular orbits, the orbit foliation of $G$ is
homogeneous, and, by design, every homogeneous foliation arises in this way.
We say that $\mathcal{{F}}$ is polar (resp., hyperpolar) if it is the orbit
foliation of a polar (resp., hyperpolar) action on $M$. Perhaps the easiest
class of homogeneous foliations consists of foliations of codimension one.
These correspond to cohomogeneity-one actions and are always hyperpolar. Thus,
homogeneous codimension-one foliations are an interesting object of study in
the intersection of the theories of polar actions and Riemannian foliations.

In [4], Berndt and Tamaru invented two new methods for constructing
homogeneous codimension-one foliations on symmetric spaces of noncompact type.
One of them produces such a foliation – denoted by $\mathcal{{F}}_{\ell}$ –
for every one-dimensional linear subspace $\ell\subseteq\mathfrak{{a}}$, where
$\mathfrak{{a}}$ is a maximal abelian subspace in the $(-1)$-eigenspace of a
Cartan involution of the Lie algebra $\mathfrak{{g}}$ of the isometry group of
$M$. The other method produces a foliation $\mathcal{{F}}_{\upalpha_{i}}$ for
each of the simple restricted roots $\upalpha_{i}\in\mathfrak{{a}}^{*}$. These
two types of foliations are parametrized by
$\mathbb{{P}}\mathfrak{{a}}\simeq\mathbb{R}P^{r-1}$ (here
$r=\mathrm{rank}\hskip 1.00006pt(M)=\dim(\mathfrak{{a}})$) and $\left\\{\hskip
0.29999pt{\upalpha_{1},\ldots,\upalpha_{r}}\hskip 0.29999pt\right\\}$,
respectively. Restricting to the case when $M$ is irreducible, the authors
proved that every homogeneous codimension-one foliation on it arises in one of
these two ways – up to isometric congruence. Moreover, if we let
$\mathrm{DD}_{M}$ denote the Dynkin diagram of $M$, then its automorphism
group $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ acts naturally on
$\left\\{\hskip 0.29999pt{\upalpha_{1},\ldots,\upalpha_{r}}\hskip
0.29999pt\right\\}$ and $\mathfrak{{a}}$, and two foliations
$\mathcal{{F}}_{\ell}$ and $\mathcal{{F}}_{\ell^{\prime}}$ (resp.,
$\mathcal{{F}}_{\upalpha_{i}}$ and $\mathcal{{F}}_{\upalpha_{j}}$) are
congruent if and only if $\ell$ and $\ell^{\prime}$ (resp., $\upalpha_{i}$ and
$\upalpha_{j}$) lie in the same $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})$-orbit (whereas the foliations
$\mathcal{{F}}_{\ell}$ and $\mathcal{{F}}_{\upalpha_{i}}$ are never
congruent). This gives the complete classification of homogeneous codimension-
one foliations on irreducible symmetric spaces of noncompact type and shows
that their moduli space is isomorphic to

$(\mathbb{R}P^{r-1}\sqcup\left\\{\hskip 0.29999pt{1,\ldots,r}\hskip
0.29999pt\right\\})/\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD_{M}}).$

In particular, if $M$ is of rank one and thus isometric to a hyperbolic space
over $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$, it has only two
homogeneous codimension-one foliations up to congruence:
$\mathcal{{F}}_{\mathfrak{{a}}}$ and $\mathcal{{F}}_{\upalpha_{1}}$.

In their subsequent paper [3] with Díaz-Ramos, Berndt and Tamaru came up with
a geometric construction that allows one to produce homogeneous hyperpolar
foliations on any symmetric space of noncompact type from the foliations
$\mathcal{{F}}_{\upalpha_{1}}$ on hyperbolic spaces and foliations by parallel
affine subspaces on the Euclidean spaces. They went on to show that every
homogeneous hyperpolar foliation on $M$ arises via this construction up to
congruence. Notably, they do not impose the irreducibility assumption on $M$.
On the other hand, it is not generally known when two foliations obtained by
this construction are congruent to each other. When applied to foliations of
codimension one, their method produces precisely the foliations
$\mathcal{{F}}_{\ell}\,(\ell\subseteq\mathfrak{{a}})$ and
$\mathcal{{F}}_{\upalpha_{i}}\,(1\leqslant i\leqslant r)$ defined above.
Altogether, this implies that on any symmetric space of noncompact type, any
homogeneous codimension-one foliation is congruent to either some
$\mathcal{{F}}_{\ell}$ or some $\mathcal{{F}}_{\upalpha_{i}}$. Hence, to
complete the classification of such foliations, one just needs to tell when
two foliations of the form $\mathcal{{F}}_{\ell}$ (resp.,
$\mathcal{{F}}_{\upalpha_{i}}$) are congruent to each other.

In this paper, we accomplish this by showing that the classification in the
irreducible case works verbatim in the situation when $M$ is reducible:

######  Main Theorem.

Let $M$ be a symmetric space of noncompact type and rank $r$. Then, given one-
dimensional linear subspaces $\ell,\ell^{\prime}\subseteq\mathfrak{{a}}$
(resp., simple roots $\upalpha_{i},\upalpha_{j}\in\left\\{\hskip
0.29999pt{\upalpha_{1},\ldots,\upalpha_{r}}\hskip 0.29999pt\right\\}$), the
foliations $\mathcal{{F}}_{\ell}$ and $\mathcal{{F}}_{\ell^{\prime}}$ (resp.,
$\mathcal{{F}}_{\upalpha_{i}}$ and $\mathcal{{F}}_{\upalpha_{j}}$) are
isometrically congruent if and only if $\ell$ and $\ell^{\prime}$ (resp.,
$\upalpha_{i}$ and $\upalpha_{j}$) differ by some $P\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})$. Consequently, the moduli space of homogeneous
codimension-one foliations on $M$ is isomorphic to

$(\mathbb{R}P^{r-1}\sqcup\left\\{\hskip 0.29999pt{1,\ldots,r}\hskip
0.29999pt\right\\})/\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD_{M}}).$

One noteworthy caveat is that we make the routine assumption that the
Riemannian metric of $M$ is normalized in such a way that it agrees with the
Killing form of the Lie algebra of the isometry group. When $M$ is
irreducible, this assumption is relatively benign, for it amounts to simply
rescaling the Riemannian metric by a constant factor. It poses a nontrivial
restriction, however, when $M$ is reducible, because different de Rham factors
of $M$ may carry different normalization constants in general. Since the
authors of [3] make this assumption as well, it remains an open question how
much of the classification remains valid in the reducible case when the
Riemannian metric is normalized in a different way.

The paper is organized as follows. In Section 2 we review some rudiments of
the theories of symmetric spaces of noncompact type and parabolic subgroups
that will be necessary in the rest of paper. We also prove a somewhat folklore
result about the isometry group of a Riemannian manifold decomposed as a
Riemannian product. In Section 3 we first introduce the results of Berndt,
Tamaru, and Díaz-Ramos and then prove the Main Theorem. The paper is concluded
by Section 4, in which we give an alternative – more algebraic – proof of the
uniqueness result for foliations of the type $\mathcal{{F}}_{\ell}$.

Acknowledgments. I am grateful to my supervisor Jürgen Berndt for his careful
guidance and countless invaluable discussions on the topic.

##  2\. Prerequisites

The purpose of this section is to establish the notation and terminology
necessary to understand the classification results of Berndt, Tamaru, and
Díaz-Ramos as well as the rest of the paper. We also prove a useful result
pertaining to the isometry group of a Riemannian manifold decomposed as a
Riemannian product. We follow the notation established in [4] and [3] and
refer to the same articles as well as [9, 7, 10] for more detailed
expositions.

###  2.1. Polar and codimension-one foliations

Let $M$ be a connected Riemannian manifold, and let $\mathcal{{F}}$ be a
foliation on $M$ with connected properly embedded leaves. Then we call
$\mathcal{{F}}$ homogeneous if the group $I(M,\mathcal{{F}})=\left\\{\hskip
0.29999pt{\upvarphi\in I(M)\mid\upvarphi(F)=F\hskip 3.50006pt\text{for every
leaf}\hskip 3.50006ptF\in\mathcal{{F}}}\hskip 0.29999pt\right\\}$ acts
transitively on each leaf of $\mathcal{{F}}$. Since the leaves are properly
embedded, it follows that $I(M,\mathcal{{F}})$ is a closed Lie subgroup of
$I(M)$. Moreover, its identity component $I^{0}(M,\mathcal{{F}})$ also acts
transitively on each leaf because the leaves are connected. Conversely, given
a connected Lie group $G$ acting properly and isometrically on $M$ without
singular orbits, the leaves of the action form a homogeneous foliation. Note
that the leaves of a homogeneous foliation are equidistant from each other.

Assume that $M$ is complete. We say that a homogeneous foliations is polar if
it is the orbit foliation of a polar action on $M$. In other words,
$\mathcal{{F}}$ is polar if there exists a complete immersed submanifold
$\Upsigma\subseteq M$, called a section, that intersects all the leaves of
$\mathcal{{F}}$ and does so orthogonally. If exist, all sections are
isometrically congruent to each other and necessarily totally geodesic. Many
authors usually assume sections to be embedded or even properly embedded. A
homogeneous polar foliation is called hyperpolar if its sections are flat.
Homogeneous foliations of codimension one turn out to be automatically
hyperpolar. Indeed, given such a foliation $\mathcal{{F}}$, its sections
should be just geodesics normal to its leaves. One can show that any normal
geodesic intersects all the leaves, does so orthogonally, and its image is an
immersed submanifold, hence a section.

###  2.2. The Iwasawa and horospherical decompositions

Let $M$ be a symmetric space of noncompact type. We denote
$G=I^{0}(M),\widetilde{G}=I(M),$ and $\mathfrak{{g}}=\mathrm{Lie}\hskip
0.59998pt(G)$. Pick any point $o\in M$ and write
$\widetilde{K}=\widetilde{G}_{o}$ and $K=G_{o}=\widetilde{K}^{0}$ for the
isotropy groups at $o$ and $\mathfrak{{k}}=\mathrm{Lie}\hskip 0.59998pt(K)$
for the isotropy Lie subalgebra of $\mathfrak{{g}}$. We have the corresponding
Cartan decomposition $\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{p}}$ and
the Cartan involution $\uptheta\in\mathrm{Aut}\hskip
0.59998pt(\mathfrak{{g}})$ that has $\mathfrak{{k}}$ and $\mathfrak{{p}}$ as
its $(+1)$\- and $(-1)$-eigenspaces, respectively. We identify
$\mathfrak{{p}}\cong T_{o}M$ in the usual way. If we denote the Killing form
of $\mathfrak{{g}}$ by $B$, then $B_{\uptheta}(X,Y)=-B(X,\uptheta Y)$ is a
$\widetilde{K}$-invariant inner product on $\mathfrak{{g}}$. The form
$B_{\uptheta}$ coincides with $B$ on $\mathfrak{{p}}$ and equals $-B$ on
$\mathfrak{{k}}$. The Cartan decomposition is orthogonal with respect to both
$B$ and $B_{\uptheta}$. We stick with $B_{\uptheta}$ as our default inner
product on $\mathfrak{{g}}$. Whenever $U\subset V$ are subspaces of
$\mathfrak{{g}}$, we write $V\ominus U$ for the orthogonal complement of $U$
in $V$. We make a standard but pivotal assumption that the Riemannian metric
of $M$ at $o$ coincides with the restriction of $B_{\uptheta}$ (hence of $B$)
to $\mathfrak{{p}}\cong T_{o}M$. We will sometimes refer to this for brevity
as “the Riemannian metric on $M$ comes from the Killing form”.

Pick a maximal abelian subspace $\mathfrak{{a}}$ in $\mathfrak{{p}}$ and write
$r=\dim(\mathfrak{{a}})=\mathrm{rank}\hskip 1.00006pt(M)$. The connected Lie
subgroup $A\subset G$ with Lie algebra $\mathfrak{{a}}$ is closed and abelian
and its orbit $A\cdot o\simeq\mathbb{{E}}^{r}$ is a maximal flat in $M$. The
restriction of $B_{\uptheta}$ to $\mathfrak{{a}}$ is an inner product and thus
gives an isomorphism
$\mathfrak{{a}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{a}}^{*}$
and the corresponding inner product on $\mathfrak{{a}}^{*}$. We have the
restricted root system $\Upsigma\subset\mathfrak{{a}}^{*}$, which gives rise
to the restricted root space decomposition
$\mathfrak{{g}}=\mathfrak{{g}}_{0}\oplus\bigoplus_{\upalpha\in\Upsigma}\mathfrak{{g}}_{\upalpha}$.
Here $\mathfrak{{g}}_{0}=\mathfrak{{k}}_{0}\oplus\mathfrak{{a}}$, where
$\mathfrak{{k}}_{0}=Z_{\mathfrak{{k}}}(\mathfrak{{a}})=N_{\mathfrak{{k}}}(\mathfrak{{a}})$.
The summands of the restricted root space decomposition are pairwise
orthogonal with respect to $B_{\uptheta}$. Note that
$\uptheta\mathfrak{{g}}_{\upalpha}=\mathfrak{{g}}_{-\upalpha}$. Finally, we
make a choice of positive roots $\Upsigma^{+}\subset\Upsigma$ and write
$\Uplambda=\left\\{\hskip 0.29999pt{\upalpha_{1},\ldots,\upalpha_{r}}\hskip
0.29999pt\right\\}\subseteq\Upsigma^{+}$ for the set of simple roots. The sum
$\mathfrak{{n}}=\bigoplus_{\upalpha\in\Upsigma^{+}}\mathfrak{{g}}_{\upalpha}$
is a nilpotent subalgebra of $\mathfrak{{g}}$. We have a vector space
decomposition
$\mathfrak{{g}}=\mathfrak{{k}}\oplus\mathfrak{{a}}\oplus\mathfrak{{n}}$ called
the Iwasawa decomposition. The connected Lie subgroup of $G$ corresponding to
$\mathfrak{{n}}$ is closed and will be denoted by $N$. The multiplication
induces a diffeomorphism
$KAN\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}G$
called the global Iwasawa decomposition. Since $K$ is a maximal compact
subgroup of $G$, the subgroups $A$ and $N$ are simply connected. Their product
$S=AN$ is a simply connected closed solvable subgroup of $G$ (with Lie algebra
$\mathfrak{{s}}=\mathfrak{{a}}\oplus\mathfrak{{n}}$) that acts simply
transitively on $M$. In particular, $M$ can be realized, albeit not uniquely,
as a simply connected solvable Lie group with a suitable left-invariant
Riemannian metric. Every root $\upalpha$ can be carried to a vector
$H_{\upalpha}\in\mathfrak{{a}}$ along the isomorphism between $\mathfrak{{a}}$
and $\mathfrak{{a}}^{*}$. On the other hand, the basis
$\upalpha_{1},\ldots,\upalpha_{r}$ for $\mathfrak{{a}}^{*}$ gives the dual
basis for $\mathfrak{{a}}$, which we denote by
$H^{\upalpha_{1}},\ldots,H^{\upalpha_{r}}$. By definition, $\langle\hskip
1.00006pt{H^{\upalpha_{i}}}\hskip 1.00006pt|\hskip
1.00006pt{H_{\upalpha_{j}}}\hskip 1.00006pt\rangle\hskip
0.20004pt=\upalpha_{j}(H^{\upalpha_{i}})=\updelta_{ij}$.

A linear endomorphism $T\in\mathrm{End}(\mathfrak{{a}}^{*})$ is called an
automorphism of $\Upsigma$ if it maps $\Upsigma$ onto itself and preserves the
integers $n_{\upalpha\upbeta}=\frac{2\langle\hskip 0.70004pt{\upalpha}\hskip
0.70004pt|\hskip 0.70004pt{\upbeta}\hskip 0.70004pt\rangle\hskip
0.14003pt}{||\upbeta||^{2}}$, i.e.
$n_{F(\upalpha)F(\upbeta)}=n_{\upalpha\upbeta}$ for all
$\upalpha,\upbeta\in\Upsigma$. Such $T$ is necessarily a linear automorphism
and, since the inner product on $\mathfrak{{a}}^{*}$ comes from the Killing
form $B$, in fact an orthogonal transformation of $\mathfrak{{a}}^{*}$. The
group $\mathrm{Aut}\hskip 0.59998pt(\Upsigma)$ is a finite subgroup of
$\mathrm{O}(\mathfrak{{a}}^{*})$ and it contains the Weyl group
$\mathrm{{W}}(\Upsigma)$ as a normal subgroup. A choice of positive roots
allows to split the short exact sequence
$\mathrm{{W}}(\Upsigma)\hookrightarrow\mathrm{Aut}\hskip
0.59998pt(\Upsigma)\twoheadrightarrow\mathrm{Aut}\hskip
0.59998pt(\Upsigma)/\mathrm{{W}}(\Upsigma)$ as a semidirect product. Indeed,
having fixed $\Upsigma^{+}$, note that $\Uplambda$ gives rise to a Dynkin
diagram, which we denote by $\mathrm{DD}_{M}$. We treat it as a vertex-
weighted graph (with some edges oriented and doubled/tripled and some vertices
”marked” in case they correspond to a simple root whose double is also root)
where each vertex is assigned the multiplicity of the corresponding simple
root, i.e. the dimension of the corresponding root subspace. We will be
working with automorphisms of $\mathrm{DD}_{M}$, which we require by
definition to preserve the multiplicities of the vertices. Any such
automorphism $P\in\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ gives rise to
a linear automorphism of $\mathfrak{{a}}^{*}$ – denoted by the same letter –
simply by permuting the basis vectors in $\Uplambda$. One can easily see that
$P\in\mathrm{Aut}\hskip 0.59998pt(\Upsigma)$ and the map $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})\to\mathrm{Aut}\hskip 0.59998pt(\Upsigma)$ is an
injective group homomorphism. Moreover, its image and $\mathrm{{W}}(\Upsigma)$
do not intersect and their product is the whole automorphism group of
$\Upsigma$, hence $\mathrm{Aut}\hskip
0.59998pt(\Upsigma)\simeq\mathrm{{W}}(\Upsigma)\rtimes\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})$. Note that there may be elements in
$\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ interchanging some connected
components of $\mathrm{DD}_{M}$ – precisely when $M$ has isometric de Rham
factors or, equivalently, when $\mathfrak{{g}}$ has isomorphic simple ideals
(this is one of the several places where the assumption on the Riemannian
metric of $M$ plays an important role).

Now, we have a compact group $N_{\widetilde{K}}(\mathfrak{{a}})$ and its
representation $\uppsi\colon
N_{\widetilde{K}}(\mathfrak{{a}})\to\mathrm{O}(\mathfrak{{a}}^{*}),k\mapsto({\left.\kern-1.2pt\mathrm{Ad}\hskip
0.59998pt(k)\vphantom{\big{|}}\right|_{\mathfrak{{a}}}}^{*})^{-1}$. One
readily sees that $\mathrm{Ker}\hskip
0.39993pt(\uppsi)=Z_{\widetilde{K}}(\mathfrak{{a}})$ and $\hskip
0.80002pt\mathrm{Im}\hskip 0.39993pt(\uppsi)\subseteq\mathrm{Aut}\hskip
0.59998pt(\Upsigma)$, which implies that $\mathrm{Ad}\hskip 0.59998pt(k)$
permutes the restricted root subspaces: $\mathrm{Ad}\hskip
0.59998pt(k)\mathfrak{{g}}_{\upalpha}=\mathfrak{{g}}_{\uppsi(k)\upalpha}$.
Since the Riemannian metric on $M$ comes from the Killing form $B$, one can
show that the image of $\uppsi$ equals the whole automorphism group of
$\Upsigma$, hence
$N_{\widetilde{K}}(\mathfrak{{a}})/Z_{\widetilde{K}}(\mathfrak{{a}})\cong\mathrm{Aut}\hskip
0.59998pt(\Upsigma)$. We can also give a similar description of the subgroups
$\mathrm{{W}}(\Upsigma)$ and $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$
of $\mathrm{Aut}\hskip 0.59998pt(\Upsigma)$. Namely, consider the normalizers
$N_{K}(\mathfrak{{a}})$ and $N_{\widetilde{K}}(\mathfrak{{n}})$. These are
both subgroups of $N_{\widetilde{K}}(\mathfrak{{a}})$, and we have
$N_{K}(\mathfrak{{a}})\cap
Z_{\widetilde{K}}(\mathfrak{{a}})=Z_{K}(\mathfrak{{a}})$ and
$N_{\widetilde{K}}(\mathfrak{{n}})\supseteq
Z_{\widetilde{K}}(\mathfrak{{a}})$. It turns out that
$\uppsi(N_{K}(\mathfrak{{a}}))=\mathrm{{W}}(\Upsigma)$ and
$\uppsi(N_{\widetilde{K}}(\mathfrak{{n}}))=\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})$, so
$N_{K}(\mathfrak{{a}})/Z_{K}(\mathfrak{{a}})\cong\mathrm{{W}}(\Upsigma)$ and
$N_{\widetilde{K}}(\mathfrak{{n}})/Z_{\widetilde{K}}(\mathfrak{{a}})\cong\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})$. Note that all of these subgroups of
$\widetilde{K}$ share the same Lie algebra $\mathfrak{{k}}_{0}$.

In order to be able to describe the results of [3], we will need some basic
facts from the theory of parabolic subgroups, which we now briefly introduce.
Let $\Upphi$ be any subset of $\Uplambda$. Write $\Upsigma_{\Upphi}$ for the
root subsystem of $\Upsigma$ spanned by $\Upphi$ and let
$\Upsigma_{\Upphi}^{+}=\Upsigma_{\Upphi}\cap\Upsigma^{+}$ be the induced
choice of positive roots in $\Upsigma_{\Upphi}$. We start by dividing
$\mathfrak{{a}}$ into the orthogonal sum of two subspaces:
$\mathfrak{{a}}_{\Upphi}=\bigcap_{\upalpha\in\Upphi}\mathrm{Ker}\hskip
0.39993pt(\upalpha)=\bigoplus_{\upalpha\in\Uplambda\mathbin{\mathchoice{\hbox{
\leavevmode\hbox to5.5pt{\vbox to9.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.5pt\lower-1.8pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{} {}{{}}{}{}{}{}{{}}{}
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}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{3.5pt}{-1.5pt}\pgfsys@lineto{0.0pt}{7.25pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\hbox{
\leavevmode\hbox to5.5pt{\vbox to9.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.5pt\lower-1.8pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{} {}{{}}{}{}{}{}{{}}{}
{}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{3.5pt}{-1.5pt}\pgfsys@lineto{0.0pt}{7.25pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\hbox{
\leavevmode\hbox to2.95pt{\vbox to5.45pt{\pgfpicture\makeatletter\hbox{\hskip
0.72499pt\lower-0.22499pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.45pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{2.0pt}{0.0pt}\pgfsys@lineto{-0.5pt}{5.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\hskip 1.0pt}{\hbox{
\leavevmode\hbox to1.9pt{\vbox to3.4pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{1.5pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\Upphi}\mathbb{R}H^{\upalpha}$,
$\mathfrak{{a}}^{\Upphi}=\mathfrak{{a}}\ominus\mathfrak{{a}}_{\Upphi}=\bigoplus_{\upalpha\in\Upphi}\mathbb{R}H_{\upalpha}$.
Next, we introduce
$\mathfrak{{l}}_{\Upphi}=N_{\mathfrak{{g}}}(\mathfrak{{a}}_{\Upphi})=Z_{\mathfrak{{g}}}(\mathfrak{{a}}_{\Upphi})=\mathfrak{{g}}_{0}\oplus\bigoplus_{\upalpha\in\Upsigma_{\Upphi}}\mathfrak{{g}}_{\upalpha}$
and
$\mathfrak{{n}}_{\Upphi}=\bigoplus_{\upalpha\in\Upsigma^{+}\mathbin{\mathchoice{\hbox{
\leavevmode\hbox to5.5pt{\vbox to9.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.5pt\lower-1.8pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{} {}{{}}{}{}{}{}{{}}{}
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}{}\pgfsys@moveto{3.5pt}{-1.5pt}\pgfsys@lineto{0.0pt}{7.25pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\hbox{
\leavevmode\hbox to5.5pt{\vbox to9.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.5pt\lower-1.8pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{} {}{{}}{}{}{}{}{{}}{}
{}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{3.5pt}{-1.5pt}\pgfsys@lineto{0.0pt}{7.25pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{\hbox{
\leavevmode\hbox to2.95pt{\vbox to5.45pt{\pgfpicture\makeatletter\hbox{\hskip
0.72499pt\lower-0.22499pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.45pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{2.0pt}{0.0pt}\pgfsys@lineto{-0.5pt}{5.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\hskip 1.0pt}{\hbox{
\leavevmode\hbox to1.9pt{\vbox to3.4pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\pgfsys@roundcap\pgfsys@invoke{
}{}\pgfsys@moveto{1.5pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}}\Upsigma_{\Upphi}^{+}}\mathfrak{{g}}_{\upalpha}\subseteq\mathfrak{{n}}$,
which are reductive and nilpotent Lie subalgebras of $\mathfrak{{g}}$,
respectively. One has
$\mathfrak{{l}}_{\Upphi}\cap\mathfrak{{n}}_{\Upphi}=\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}$ and
$[\mathfrak{{l}}_{\Upphi},\mathfrak{{n}}_{\Upphi}]\subseteq\mathfrak{{n}}_{\Upphi}$.
The semidirect sum
$\mathfrak{{q}}_{\Upphi}=\mathfrak{{l}}_{\Upphi}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}} \end{picture}}}}}\mathfrak{{n}}_{\Upphi}$ is
a parabolic subalgebra of $\mathfrak{{g}}$. Of particular interest to us will
be the reductive subalgebra
$\mathfrak{{m}}_{\Upphi}=\mathfrak{{l}}_{\Upphi}\ominus\mathfrak{{a}}_{\Upphi}$
and its semisimple subalgebra
$\widetilde{\mathfrak{{g}}}_{\Upphi}=\mathfrak{{m}}_{\Upphi}\ominus
Z_{\mathfrak{{k}}_{0}}(\mathfrak{{b}}_{\Upphi})$, where
$\mathfrak{{b}}_{\Upphi}=\mathfrak{{m}}_{\Upphi}\cap\mathfrak{{p}}$. One can
show that $\widetilde{\mathfrak{{g}}}_{\Upphi}$ is generated by
$\mathfrak{{b}}_{\Upphi}$ or
$\bigoplus_{\upalpha\in\Upsigma_{\Upphi}}\mathfrak{{g}}_{\upalpha}$ as a Lie
algebra (see Subsection 2.4 and Remark 2.4 in [11] for more details on
$\widetilde{\mathfrak{{g}}}_{\Upphi}$). The decomposition
$\mathfrak{{q}}_{\Upphi}=\mathfrak{{m}}_{\Upphi}\oplus\mathfrak{{a}}_{\Upphi}\mathbin{\mathchoice{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\displaystyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\displaystyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[9.99997pt]{\raisebox{-0.83333pt}{\begin{picture}(1.0,1.0)\textstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.83333pt}{$\textstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[7.66663pt]{\raisebox{-0.58333pt}{\begin{picture}(1.0,1.0)\scriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.58333pt}{$\scriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}}
\end{picture}}}}{\makebox[6.11107pt]{\raisebox{-0.41666pt}{\begin{picture}(1.0,1.0)\scriptscriptstyle\displaystyle
3
\roundcap\put(0.5,0.5){\makebox(0.0,0.0)[]{\raisebox{0.41666pt}{$\scriptscriptstyle+$}}}
\put(0.5,0.5){\arc[270,90]{0.5}} \end{picture}}}}}\mathfrak{{n}}_{\Upphi}$ is
called the Langlands decomposition. Finally, define
$\mathfrak{{k}}_{\Upphi}=\mathfrak{{k}}\cap\mathfrak{{m}}_{\Upphi}$.

Now we look at the subgroups of $G$ and submanifolds of $M$ arising from these
Lie subalgebras. Let $\widetilde{G}_{\Upphi},A_{\Upphi},$ and $N_{\Upphi}$
stand for the connected Lie subgroups of $G$ corresponding to
$\widetilde{\mathfrak{{g}}}_{\Upphi},\mathfrak{{a}}_{\Upphi},$ and
$\mathfrak{{n}}_{\Upphi}$, respectively. They are all closed and thus
determine properly embedded submanifolds $\widetilde{G}_{\Upphi}\cdot
o=B_{\Upphi},A_{\Upphi}\cdot o,$ and $N_{\Upphi}\cdot o$, the first two of
which are totally geodesic with the corresponding Lie triple systems
$\mathfrak{{b}}_{\Upphi}$ and $\mathfrak{{a}}_{\Upphi}$. The submanifold
$B_{\Upphi}$ is called a boundary component of $M$ (the term coined in [5] in
the context of the maximal Satake compactification of $M$), and it is itself a
noncompact symmetric space of rank equal to $r_{\Upphi}=|\Upphi|$. The
subgroup $L_{\Upphi}=Z_{G}(\mathfrak{{a}}_{\Upphi})$ is closed and reductive
and has $\mathfrak{{l}}_{\Upphi}$ as its Lie algebra. The group
$K_{\Upphi}=K\cap L_{\Upphi}$ is a maximal compact subgroup of $L_{\Upphi}$
and has Lie algebra $\mathfrak{{k}}_{\Upphi}$. The product
$Q_{\Upphi}=L_{\Upphi}N_{\Upphi}$ is a parabolic subgroup of $G$; it is
closed, acts transitively on $M$, and has $\mathrm{Lie}\hskip
0.59998pt(Q_{\Upphi})=\mathfrak{{q}}_{\Upphi}$. Define also
$M_{\Upphi}=K_{\Upphi}\widetilde{G}_{\Upphi}\subseteq L_{\Upphi}$, which is a
closed reductive subgroup of $G$ with $\mathrm{Lie}\hskip
0.59998pt(M_{\Upphi})=\mathfrak{{m}}_{\Upphi}$. One has a direct product
decomposition $L_{\Upphi}=M_{\Upphi}\times A_{\Upphi}$, which induces the
global Langlands decomposition $Q_{\Upphi}=M_{\Upphi}\times A_{\Upphi}\ltimes
N_{\Upphi}$. The latter allows us to draw a commutative diagram:

$\textstyle{M_{\Upphi}\times A_{\Upphi}\times
N_{\Upphi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{Q_{\Upphi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{\Upphi}\times
A_{\Upphi}\times
N_{\Upphi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{Q_{\Upphi}/K_{\Upphi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{M.}$

The diffeomorphism $B_{\Upphi}\times A_{\Upphi}\times N_{\Upphi}\simeq M$ is
called a horospherical decomposition of $M$.

Of special interest to us will be subsets $\Upphi\subseteq\Uplambda$ such that
no two roots in $\Upphi$ are connected by an edge in the Dynkin diagram
$\mathrm{DD}_{M}$. In other words, all roots in $\Upphi$ should be mutually
orthogonal. Such subsets are called orthogonal in [3]. Fix one such subset
$\Upphi\subseteq\Uplambda$. For any
$\upalpha,\upbeta\in\Upphi,\upalpha\neq\upbeta,$ and any $k,l\neq 0$, we have
$\mathfrak{{g}}_{k\upalpha+l\upbeta}=\left\\{\hskip 0.29999pt{0}\hskip
0.29999pt\right\\}$ and hence
$[\mathfrak{{g}}_{k\upalpha},\mathfrak{{g}}_{l\upbeta}]=\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}$. Since
$\widetilde{\mathfrak{{g}}}_{\Upphi}$ is generated by
$\bigoplus_{\upalpha\in\Upsigma_{\Upphi}}\mathfrak{{g}}_{\upalpha}$, we have a
direct sum Lie algebra decomposition

$\widetilde{\mathfrak{{g}}}_{\Upphi}=\bigoplus_{\upalpha\in\Upphi}\widetilde{\mathfrak{{g}}}_{\\{\upalpha\\}}.$

Consequently, the multiplication map
$\prod_{\upalpha\in\Upphi}\widetilde{G}_{\\{\upalpha\\}}\twoheadrightarrow\widetilde{G}_{\Upphi}$
is a local isomorphism, and it passes to a Riemannian covering
$\prod_{\upalpha\in\Upphi}B_{\\{\upalpha\\}}\twoheadrightarrow B_{\Upphi}$.
Since $B_{\Upphi}$ is simply connected, we get an isometric decomposition

$B_{\Upphi}\cong\prod_{\upalpha\in\Upphi}B_{\\{\upalpha\\}}.$

Note that each $B_{\\{\upalpha\\}}$ is a noncompact symmetric space of rank 1
and thus is isometric to a hyperbolic space over a finite-dimensional real
normed division algebra.

###  2.3. Isometries of a Riemannian product

In order to tell foliations $\mathcal{{F}}_{\upalpha_{i}}$ apart, we will need
a certain structural result about the isometry group of a Riemannian manifold
decomposed as a Riemannian product. Although we will only work with the case
of the de Rham decomposition of a symmetric space of noncompact type, we
formulate and prove this result in a much more general setting. In the
following, whenever we say that a connected Riemannian manifold is
irreducible, we mean that it is not flat and its restricted holonomy
representation is irreducible. If the manifold is complete, this is equivalent
to requiring its universal Riemannian covering to be (non-flat and)
indecomposable as a nontrivial Riemannian product.

Let $M=M_{0}\times M_{1}^{l_{1}}\times\cdots\times M_{k}^{l_{k}}$ be a
Riemannian product, where all $M_{i}$’s are connected Riemannian manifolds,
$M_{0}$ is flat, $M_{i}$ is irreducible for $1\leqslant i\leqslant k$, and
$M_{i}^{l_{i}}$ simply means the product of $l_{i}$ copies of $M_{i}$. We also
assume that $M_{i}$ is not isometric to $M_{j}$ for $i\neq j$. Note that we
have a Lie subgroup111This is a Lie subgroup because the induced isometric
action $I(M_{0})\times I(M_{1})^{l_{1}}\times\cdots\times
I(M_{k})^{l_{k}}\curvearrowright M$ is smooth and effective. $I(M_{0})\times
I(M_{1})^{l_{1}}\times\cdots\times I(M_{k})^{l_{k}}\subseteq I(M)$. Let
$l=\sum_{i=1}^{k}l_{i}$, write $S_{l}$ for the symmetric group on $l$
elements, and let $S_{l}^{l_{1},\ldots,l_{k}}\cong S_{l_{1}}\times\cdots\times
S_{l_{k}}$ stand for the subgroup of elements that permute the first $l_{1}$
elements with each other, the next $l_{2}$ elements with each other, and so
on. Observe that we have an embedding
$S_{l}^{l_{1},\ldots,l_{k}}\hookrightarrow I(M)$ given by the rule
$\upsigma\cdot(p_{0},(p_{s}))=(p_{0},(p_{\upsigma(s)}))$ (to be precise, this
map is an antihomomorphism of groups).

######  Proposition 2.1.

The group $I(M)$ decomposes as a semidirect product of its subgroups

$\left[I(M_{0})\times I(M_{1})^{l_{1}}\times\cdots\times
I(M_{k})^{l_{k}}\right]\rtimes S_{l}^{l_{1},\ldots,l_{k}}=I(M).$

In particular, $I(M_{0})\times I(M_{1})^{l_{1}}\times\cdots\times
I(M_{k})^{l_{k}}$ is an open normal subgroup of $I(M)$. The corresponding
action of $S_{l}^{l_{1},\ldots,l_{k}}$ on it is given by
$\upsigma\cdot(g_{0},(g_{s}))=(g_{0},(g_{\upsigma(s)}))$.

One obvious example of such a decomposition is the de Rham decomposition of a
complete simply connected Riemannian manifold. Note that we do not require the
factors to be complete in the proposition.

###### Proof.

The subgroups $I(M_{0})\times I(M_{1})^{l_{1}}\times\cdots\times
I(M_{k})^{l_{k}}$ and $S_{l}^{l_{1},\ldots,l_{k}}$ of $I(M)$ clearly do not
intersect. So we only need to show that their product is the whole isometry
group. Let
$p=(p_{0},(p_{1}^{s})_{s=1}^{l_{1}},\ldots,(p_{k}^{s})_{s=1}^{l_{k}})\in M$
and consider the decomposition

$T_{p}M=T_{p_{0}}M_{0}\oplus\left(\bigoplus_{s=1}^{l_{1}}T_{p_{1}^{s}}M_{1}\right)\oplus\cdots\oplus\left(\bigoplus_{s=1}^{l_{k}}T_{p_{k}^{s}}M_{k}\right).$

Let $\mathrm{Hol}^{0}(M,p)$ stand for the restricted holonomy group of $M$ at
$p$, which is defined using the parallel transport along all contractible
piecewise-smooth loops based at $p$ and is the identity component of the full
holonomy group $\mathrm{Hol}(M,p)$. The decomposition of $T_{p}M$ above is
obviously orthogonal, the first summand $T_{p_{0}}M_{0}$ is the subspace of
invariants of $\mathrm{Hol}^{0}(M,p)$ in $T_{p}M$, and all the other summands
are irreducible $\mathrm{Hol}^{0}(M,p)$-subrepresentations in $T_{p}M$. A
decomposition of $T_{p}M$ satisfying these three properties is called a
canonical decomposition in [10]. Now, Theorem 5.4(4) in Chapter IV of the book
asserts that a canonical decomposition of $T_{p}M$ is unique up to reordering
of its factors. It is worth noting that the authors actually talk about
decompositions with respect to the full holonomy group, and Theorem 5.4(4)
requires $M$ to be simply connected. But note that the restricted holonomy
representation of $M$ is isomorphic to the full holonomy representation of its
universal Riemannian covering $\widetilde{M}$ (simply by lifting contractible
loops). So the uniqueness of a canonical
$\mathrm{Hol}(\widetilde{M},\widetilde{p})$-decomposition for $\widetilde{M}$
(here $\widetilde{p}\in\widetilde{M}$ is any point over $p$) translates into
the uniqueness of a canonical $\mathrm{Hol}^{0}(M,p)$-decomposition for $M$
(see [10, Ch. IV, Section 5] for more details). Let $g\in I(M)$ be any
isometry. It is not hard to show that the differential $dg$ must send
canonical decompositions of $T_{p}M$ to canonical decompositions of
$T_{g(p)}M$ (basically because isometries commute with parallel transport).
Let us write $M_{0,p}$ for
$M_{0}\times\\{((p_{1}^{s})_{s=1}^{l_{1}},\ldots,(p_{k}^{s})_{s=1}^{l_{k}})\\}$
and $M_{i,p}^{(j)}$ for
$\\{(p_{0},(p_{1}^{s})_{s=1}^{l_{1}},\ldots,(p_{i}^{s})_{s=1}^{j-1}))\\}\times
M_{i}\times\\{((p_{i}^{s})_{s=j+1}^{l_{i}},\ldots,(p_{k}^{s})_{s=1}^{l_{k}})\\}$
for any $i\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}$
and $j\in\left\\{\hskip 0.29999pt{1,\ldots,l_{i}}\hskip 0.29999pt\right\\}$.
These are totally geodesic submanifolds of $M$. Since isometries commute with
the exponential map and respect canonical decompositions, $g$ must send
$M_{0,p}$ onto $M_{0,g(p)}$ and $M_{i,p}^{(j)}$ onto
$M_{i,g(p)}^{(j^{\prime})}$ for some $j^{\prime}\in\left\\{\hskip
0.29999pt{1,\ldots,l_{i}}\hskip 0.29999pt\right\\}$. If we write
$j=\upsigma(j^{\prime})$, we obtain a permutation $\upsigma\in
S_{l_{1}}\times\cdots\times S_{l_{k}}\cong S_{l}^{l_{1},\ldots,l_{k}}$. We
also have isometries $M_{0}\cong M_{0,p}\xrightarrow{g}M_{0,g(p)}\cong M_{0}$
and $M_{i}\cong M_{i,p}^{(j)}\xrightarrow{g}M_{i,g(p)}^{(j^{\prime})}\cong
M_{i}$, which we denote by $g_{0}$ and $g_{i}^{(j^{\prime})}$, respectively.
By construction, the isometry
$(g_{0},(g_{1}^{(s)})_{s=1}^{l_{1}},\ldots,(g_{k}^{(s)})_{s=1}^{l_{k}})\circ\upsigma$
lies in the product of $I(M_{0})\times I(M_{1})^{l_{1}}\times\cdots\times
I(M_{k})^{l_{k}}$ and $S_{l}^{l_{1},\ldots,l_{k}}$ and coincides with $g$ on
$M_{0,p}\cup\bigcup_{\begin{subarray}{c}1\leqslant i\leqslant k\\\ 1\leqslant
j\leqslant l_{i}\end{subarray}}M_{i,p}^{(j)}$. But then the differentials of
these two isometries at $p$ must coincide as well. Since an isometry of a
connected Riemannian manifold is uniquely determined by its value at a point
and its differential at that point, the constructed isometry coincides with
$g$, which finishes the proof. ∎

######  Corollary 2.1.1.

We have as isomorphism $I^{0}(M_{0})\times
I^{0}(M_{1})^{l_{1}}\times\cdots\times I^{0}(M_{k})^{l_{k}}\cong I^{0}(M)$.

##  3\. Main result

In this section we will discuss the classification results of Berndt, Tamaru,
and Díaz-Ramos and prove the Main Theorem. Historically, their paper [3] about
homogeneous hyperpolar foliations came out after Berndt and Tamaru obtained
the classification of homogeneous codimension-one foliations in [4]. We will,
however, present the results of these two papers in an anachronistic manner,
as it makes the exposition clearer.

In [3], Berndt, Tamaru, and Díaz-Ramos devised a method allowing one to obtain
homogeneous hyperpolar foliations on an arbitrary symmetric space of
noncompact type from those on Euclidean and hyperbolic spaces and then showed
that the foliations obtained in this way exhaust the list of all homogeneous
hyperpolar foliations up to congruence. We begin by introducing their method.

Let $M$ be a symmetric space of noncompact type, and let the rest of the
notation be as in Subsection 2.2. Let $\Upphi\subseteq\Uplambda$ be an
orthogonal subset. Recall that we have a horospherical decomposition

$M\simeq B_{\Upphi}\times A_{\Upphi}\times
N_{\Upphi}\simeq\left(\prod_{\upalpha\in\Upphi}B_{\\{\upalpha\\}}\right)\times
A_{\Upphi}\times N_{\Upphi}.$ (1)

Now, each boundary component $B_{\\{\upalpha\\}}$ is a symmetric space of
noncompact type and rank 1, hence it is isometric to a hyperbolic space
$\mathbb{F}_{\upalpha}H^{n_{\upalpha}},\,\mathbb{F}_{\upalpha}\in\left\\{\hskip
0.29999pt{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}}\hskip
0.29999pt\right\\}$. Every nontrivial homogeneous hyperpolar foliation on this
space is of codimension one because its sections are totally geodesic and they
must be flat by definition, hence their dimension cannot exceed the rank of
the space. As $\mathbb{F}_{\upalpha}H^{n_{\upalpha}}$ is irreducible, the
classification of homogeneous codimension-one foliations obtained in [4] (to
be formulated below, see Theorem 3.3) tells us that there are exactly two such
foliations on this space up to congruence: one of them is a foliation by
horospheres all congruent to each other, while the other, which we
denote222Throughout the paper we commit the usual sin of not distinguishing
notationally between a congruence class of foliations and its specific
representatives. For example,
$\mathcal{F}_{\mathbb{F}_{\upalpha}}^{n_{\upalpha}}$ is really a congruence
class of homogeneous codimension-one foliations on
$\mathbb{F}_{\upalpha}H^{n_{\upalpha}}$. by
$\mathcal{F}_{\mathbb{F}_{\upalpha}}^{n_{\upalpha}}$, has a unique minimal
leaf. On the other hand, the orbit $A_{\Upphi}\cdot o$ is a simply connected
totally geodesic flat submanifold of $M$ of dimension $r-r_{\Upphi}$, so it is
isometric to $\mathbb{{E}}^{r-r_{\Upphi}}$. We can also think of it as
$\mathfrak{{a}}_{\Upphi}$ endowed with the Riemannian metric coming from its
inner product or the subgroup $A_{\Upphi}$ endowed with the left-invariant
metric corresponding to the inner product on $\mathfrak{{a}}_{\Upphi}$. We
have the isometries
$\exp_{A_{\Upphi}}\colon\mathfrak{{a}}_{\Upphi}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}A_{\Upphi}$
and
$A_{\Upphi}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}A_{\Upphi}\cdot
o,\,g\mapsto g\cdot o$. Every homogeneous hyperpolar (which is the same as
polar in this context) foliation on $\mathfrak{{a}}_{\Upphi}$ is by affine
subspaces parallel to a fixed linear subspace
$V\subseteq\mathfrak{{a}}_{\Upphi}$: $\mathcal{F}_{V}=\left\\{\hskip
0.29999pt{x+V\mid x\in V^{\perp}\subseteq\mathfrak{{a}}_{\Upphi}}\hskip
0.29999pt\right\\}$. We denote the corresponding foliation on $A_{\Upphi}$ (or
$A_{\Upphi}\cdot o$) by the same symbol $\mathcal{F}_{V}$.

Going back to the horospherical decomposition (1), consider the product
foliation

$\mathcal{F}_{\Upphi,V}=\left(\prod_{\upalpha\in\Upphi}\mathcal{F}_{\mathbb{F}_{\upalpha}}^{n_{\upalpha}}\right)\times\mathcal{F}_{V}\times
N_{\Upphi}$

on $M$, where the last factor is simply the trivial foliation on $N_{\Upphi}$
consisting of just one leaf. The main result of [3] is

######  Theorem 3.1 ([3], Th. 4.11).

Let $M$ be a symmetric space of noncompact type with all notation as above.
Then

  1. (1)

For every orthogonal subset $\Upphi\subseteq\Uplambda$ and linear subspace
$V\subseteq\mathfrak{{a}}_{\Upphi}$, $\mathcal{F}_{\Upphi,V}$ is a homogeneous
hyperpolar foliation on $M$.

  2. (2)

Every homogeneous hyperpolar foliation on $M$ is isometrically congruent to
$\mathcal{F}_{\Upphi,V}$ for some orthogonal subset $\Upphi\subseteq\Uplambda$
and linear subspace $V\subseteq\mathfrak{{a}}_{\Upphi}$.

In order to construct a Lie subgroup of $G$ whose orbits are the leaves of
$\mathcal{F}_{\Upphi,V}$, pick an $r_{\Upphi}$-dimensional linear subspace
$\ell_{\Upphi}\subset\mathfrak{{g}}$ such that
$\dim(\mathfrak{{g}}_{\upalpha}\cap\ell_{\Upphi})=1$ for each
$\upalpha\in\Upphi$ and consider the subspace

$\mathfrak{{s}}_{\Upphi,V}=(\mathfrak{{a}}^{\Upphi}\oplus
V)\oplus(\mathfrak{{n}}\ominus\ell_{\Upphi})\subseteq\mathfrak{{a}}\oplus\mathfrak{{n}}=\mathfrak{{s}}.$

One can easily check that this is a Lie subalgebra and the corresponding
connected Lie subgroup $S_{\Upphi,V}\subseteq G$ is closed and acts on $M$
without singular orbits. Its action turns out to be hyperpolar and its orbit
foliation is exactly $\mathcal{F}_{\Upphi,V}$. A different choice of
$\ell_{\Upphi}$ would result in a subgroup congruent to $S_{\Upphi,V}$ and
thus a foliation congruent to $\mathcal{F}_{\Upphi,V}$.

Notably, Theorem 3.1 allows $M$ to be reducible. On the other hand, the
assumption that the Riemannian metric on $M$ comes from the Killing form of
$\mathfrak{{g}}$ seems to play a crucial role in the proof of this theorem in
[3] and we do not know whether it can be safely removed. Note that this result
gives only partial classification of homogeneous hyperpolar foliations, for we
may have $\mathcal{F}_{\Upphi,V}$ congruent to
$\mathcal{F}_{\Upphi^{\prime},V^{\prime}}$ for
$(\Upphi,V)\neq(\Upphi^{\prime},V^{\prime})$, and we have no way of telling
when this is the case in general.

Let us apply Theorem 3.1 to homogeneous codimension-one foliations. Observe
that
$\mathrm{codim}(\mathcal{F}_{\Upphi,V})=|\Upphi|+\mathrm{codim}_{\mathfrak{{a}}_{\Upphi}}(V)$.
Therefore, there are two types of such foliations:

  1. (1)

$\mathcal{F}_{\ell}=\mathcal{F}_{\Upphi,V}$ with $\Upphi=\varnothing$ and
$V=\mathfrak{{a}}\ominus\ell$, where $\ell\subseteq\mathfrak{{a}}$ is a one-
dimensional linear subspace, and

  2. (2)

$\mathcal{F}_{\upalpha_{i}}=\mathcal{F}_{\Upphi,V}$ with333This foliation is
denoted by $\mathcal{F}_{i}$ in [4] but our notation will prove less ambiguous
in the reducible case so we stick with it. $\Upphi=\left\\{\hskip
0.29999pt{\upalpha_{i}}\hskip 0.29999pt\right\\}$ and
$V=\mathfrak{{a}}_{\Upphi}$.

The subalgebra $\mathfrak{{s}}_{\Upphi,V}$ (resp., subgroup $S_{\Upphi,V}$) in
this case will be denoted simply by $\mathfrak{{s}}_{\ell}$ (resp.,
$S_{\ell}$) or $\mathfrak{{s}}_{\upalpha_{i}}$ (resp., $S_{\upalpha_{i}}$).
Note that
$\mathfrak{{s}}_{\ell}=(\mathfrak{{a}}\ominus\ell)\oplus\mathfrak{{n}}$ and
$\mathfrak{{s}}_{\upalpha_{i}}=\mathfrak{{a}}\oplus(\mathfrak{{n}}\ominus\ell_{i})$,
where $\ell_{i}\subseteq\mathfrak{{g}}_{\upalpha_{i}}$ is a line. Berndt and
Tamaru showed in [4] that $\mathcal{F}_{\ell}$ has all its leaves
isometrically congruent to each other, while $\mathcal{F}_{\upalpha_{i}}$ has
a unique minimal leaf, namely $S_{\upalpha_{i}}\cdot o$. In particular,
$\mathcal{F}_{\ell}$ is never congruent to $\mathcal{F}_{\upalpha_{i}}$. It
means that in order to complete the classification of homogeneous codimension-
one foliations, one is left to solve the congruence problem on the parameter
space $\mathbb{P}\mathfrak{{a}}\sqcup\left\\{\hskip
0.29999pt{\upalpha_{1},\ldots,\upalpha_{r}}\hskip 0.29999pt\right\\}$, that
is, understand when $\mathcal{F}_{\ell}$ is congruent to
$\mathcal{F}_{\ell^{\prime}}$ and $\mathcal{F}_{\upalpha_{i}}$ to
$\mathcal{F}_{\upalpha_{j}}$. This was done in [4] in the irreducible case.
Before formulating this result, let us note that every $P\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})\subseteq\mathrm{Aut}\hskip 0.59998pt(\Upsigma)$
acts orthogonally on $\mathfrak{{a}}$ as
$\widehat{P}\vcentcolon=(P^{*})^{-1}$. The operator $\widehat{P}$ can be
explicitly described in terms of the basis
$H_{\upalpha_{1}},\ldots,H_{\upalpha_{r}}$ of $\mathfrak{{a}}$ by sending
$H_{\upalpha_{i}}$ to $H_{P(\upalpha_{i})}$, or, equivalently, in terms of the
basis $H^{\upalpha_{1}},\ldots,H^{\upalpha_{r}}$ by sending $H^{\upalpha_{i}}$
to $H^{P(\upalpha_{i})}$. If $k\in N_{\widetilde{K}}(\mathfrak{{a}})$ is such
that $\uppsi(k)=P$ (such $k$ must lie in
$N_{\widetilde{K}}(\mathfrak{{n}})\subseteq
N_{\widetilde{K}}(\mathfrak{{a}})$), then $\widehat{P}$ coincides with
${\left.\kern-1.2pt\mathrm{Ad}\hskip
0.59998pt(k)\vphantom{\big{|}}\right|_{\mathfrak{{a}}}}$. Altogether, we have
the actions of $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ on
$\mathfrak{{a}}$ and $\Uplambda$.

######  Proposition 3.2.

If $\ell,\ell^{\prime}\in\mathbb{P}\mathfrak{{a}}$ (respectively,
$\upalpha_{i},\upalpha_{j}\in\Uplambda$) lie in the same $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD_{M}})$-orbit, then the corresponding foliations
$\mathcal{F}_{\ell}$ and $\mathcal{F}_{\ell^{\prime}}$ (respectively,
$\mathcal{F}_{\upalpha_{i}}$ and $\mathcal{F}_{\upalpha_{j}}$) are
isometrically congruent.

###### Proof.

Given $P\in\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ sending $\ell$ onto
$\ell^{\prime}$, take any $k\in N_{\widetilde{K}}(\mathfrak{{n}})$ such that
$\uppsi(k)=P$. By design, $\mathrm{Ad}\hskip 0.59998pt(k)$ preserves both
$\mathfrak{{a}}$ and $\mathfrak{{n}}$ and thus restricts to an automorphism of
$\mathfrak{{s}}$ that sends $\ell$ onto $\ell^{\prime}$ and thus
$\mathfrak{{s}}_{\ell}=\mathfrak{{s}}\ominus\ell$ onto
$\mathfrak{{s}}_{\ell^{\prime}}=\mathfrak{{s}}\ominus\ell^{\prime}$.
Consequently, $k$ is the desired congruence between $\mathcal{F}_{\ell}$ and
$\mathcal{F}_{\ell^{\prime}}$. The argument for $\mathcal{F}_{\upalpha_{i}}$
and $\mathcal{F}_{\upalpha_{j}}$ is analogous. ∎

The converse is true provided that $M$ is irreducible:

######  Theorem 3.3 ([4], Th. 3.5 and 4.8).

Let $M$ be an irreducible symmetric space of noncompact type, and assume that
the foliations $\mathcal{F}_{\ell}$ and $\mathcal{F}_{\ell^{\prime}}$
(respectively, $\mathcal{F}_{\upalpha_{i}}$ and $\mathcal{F}_{\upalpha_{j}}$)
are isometrically congruent. Then there exists $P\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD_{M}})$ mapping $\ell$ onto $\ell^{\prime}$ (respectively,
$\upalpha_{i}$ to $\upalpha_{j}$). Consequently, the moduli space of
homogeneous codimension-one foliations on $M$ is isomorphic to

$(\mathbb{R}P^{r-1}\sqcup\left\\{\hskip 0.29999pt{1,\ldots,r}\hskip
0.29999pt\right\\})/\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD_{M}}).$

We will extend this result verbatim to the general (reducible) case. First, we
deal with the discrete part of the moduli space.

######  Proposition 3.4.

Let $M$ be a symmetric space of noncompact type. Assume that the foliations
$\mathcal{F}_{\upalpha_{i}}$ and $\mathcal{F}_{\upalpha_{j}}$ are
isometrically congruent. Then there exists $P\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD_{M}})$ mapping $\upalpha_{i}$ to $\upalpha_{j}$.

###### Proof.

Let $M=M_{1}\times\cdots\times M_{k}$ be the de Rham decomposition of $M$. We
write $G_{i}=I^{0}(M_{i})$ and $\mathfrak{{g}}_{i}=\mathrm{Lie}\hskip
0.59998pt(G_{i})$, so we have $G=G_{1}\times\cdots\times G_{k}$ and
$\mathfrak{{g}}=\mathfrak{{g}}_{1}\oplus\cdots\oplus\mathfrak{{g}}_{k}$. The
Cartan involution on $\mathfrak{{g}}$ preserves its decomposition into simple
ideals so we can write
$\mathfrak{{g}}_{i}=\mathfrak{{k}}_{i}\oplus\mathfrak{{p}}_{i}$. We also have
$\mathfrak{{a}}=\mathfrak{{a}}_{1}\oplus\cdots\oplus\mathfrak{{a}}_{k}$ and
$\Upsigma^{+}=\Upsigma_{1}^{+}\sqcup\cdots\sqcup\Upsigma_{k}^{+}$, where
$\mathfrak{{a}}_{i}=\mathfrak{{a}}\cap\mathfrak{{g}}_{i}$ and
$\Upsigma_{i}^{+}=\Upsigma^{+}\cap\Upsigma_{i}$. This induces the
corresponding decompositions
$\mathfrak{{s}}=\mathfrak{{s}}_{1}\oplus\cdots\oplus\mathfrak{{s}}_{k},\,S=S_{1}\times\cdots\times
S_{k},$ and $\Uplambda=\Uplambda_{1}\sqcup\cdots\sqcup\Uplambda_{k}$, where we
denote $\Uplambda_{l}=\left\\{\hskip
0.29999pt{\upalpha_{l}^{1},\ldots,\upalpha_{l}^{r_{l}}}\hskip
0.29999pt\right\\}$. To comply with this notation, we write $\upalpha_{i}^{p}$
and $\upalpha_{j}^{q}$ instead of $\upalpha_{i}$ and $\upalpha_{j}$. Pick some
lines $\ell_{i}^{p}\subseteq\mathfrak{{g}}_{\upalpha_{i}^{p}}$ and
$\ell_{j}^{q}\subseteq\mathfrak{{g}}_{\upalpha_{j}^{q}}$. We have the orbit
foliations of $S_{\upalpha_{i}^{p}}$ and $S_{\upalpha_{j}^{q}}$ as
representatives of $\mathcal{F}_{\upalpha_{i}^{p}}$ and
$\mathcal{F}_{\upalpha_{j}^{q}}$. Observe that

$\displaystyle S_{\upalpha_{i}^{p}}$ $\displaystyle=S_{1}\times\cdots\times
S_{i,\upalpha_{i}^{p}}\times\cdots\times S_{k},\;\text{so}$ $\displaystyle
S_{\upalpha_{i}^{p}}\cdot o$
$\displaystyle=M_{1}\times\cdots\times(S_{i,\upalpha_{i}^{p}}\cdot
o)\times\cdots\times M_{k},$ (2)

where $S_{i,\upalpha_{i}^{p}}$ is the connected Lie subgroup of $G_{i}$ with
Lie algebra
$\mathfrak{{s}}_{i,\upalpha_{i}^{p}}=\mathfrak{{s}}_{i}\ominus\ell_{i}^{p}$.
The same is true for $S_{\upalpha_{j}^{q}}$ and its orbit
$S_{\upalpha_{j}^{q}}\cdot o$. Let $g\in\widetilde{G}$ be a congruence
identifying the orbit foliations of $S_{\upalpha_{i}^{p}}$ and
$S_{\upalpha_{j}^{q}}$ and thus their minimal leaves
$S_{\upalpha_{i}^{p}}\cdot o$ and $S_{\upalpha_{j}^{q}}\cdot o$. Without loss
of generality, we may assume that $g$ fixes $o$. Looking at (2) and
Proposition 2.1, we see that $g$ must send $M_{i,o}=G_{i}\cdot o$ onto
$M_{j,o}=G_{j}\cdot o$ and thus provide a congruence between the orbit
foliations of $S_{i,\upalpha_{i}^{p}}$ on $M_{i,o}\cong M_{i}$ and
$S_{j,\upalpha_{j}^{q}}$ on $M_{j,o}\cong M_{j}$ (in particular, it follows
that $M_{i}$ and $M_{j}$ are isometric). By Theorem 3.3, there exists an
isomorphism $P$ between the Dynkin diagrams $\mathrm{DD}_{M_{i}}$ and
$\mathrm{DD}_{M_{j}}$ sending $\upalpha_{i}^{p}$ to $\upalpha_{j}^{q}$. Note
that these two Dynkin diagrams can be regarded as connected components of
$\mathrm{DD}_{M}$. Therefore, we can extend $P$ to $\mathrm{DD}_{M}$ by
letting it be $P^{-1}$ on $\mathrm{DD}_{M_{j}}$ and the identity on all the
components other than $\mathrm{DD}_{M_{i}}$ and $\mathrm{DD}_{M_{j}}$. This
gives an element of $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ sending
$\upalpha_{i}^{p}$ to $\upalpha_{j}^{q}$, as required. ∎

One would expect the congruence problem for $\mathcal{F}_{\ell}$’s to be
subtler in the reducible case, for $\ell$ does not have to be positioned
nicely with respect to the de Rham decomposition. In other words, if we write
$\mathfrak{{a}}=\mathfrak{{a}}_{1}\oplus\cdots\oplus\mathfrak{{a}}_{k}$ as we
did in the proof of Proposition 3.4, then $\ell$ does not have to be contained
in any of the summands. Nevertheless, this problem has recently been solved in
greater generality for all reducible spaces in a way that does not use Berndt
and Tamaru’s classification in the irreducible case. Namely, in their recent
paper [6], Domínguez-Vázquez and Sanmartín-López studied certain isoparametric
families of hypersurfaces in symmetric spaces of noncompact type constructed
in a way very similar to that for $\mathcal{{F}}_{\ell}$. Instead of taking
Lie subalgebras of the form $(\mathfrak{{a}}\ominus\ell)\oplus\mathfrak{{n}}$,
they consider those of the form $\mathfrak{{b}}\oplus\mathfrak{{n}}$, where
$\mathfrak{{b}}$ is a subspace of $\mathfrak{{a}}$ satisfying certain
properties. The following result solves the congruence problem for such
families:

######  Theorem 3.5 ([6], Th. B(iii)).

Let $M$ be a symmetric space of noncompact type, let
$\mathfrak{{b}},\mathfrak{{b}}^{\prime}\subseteq\mathfrak{{a}}$ be two linear
subspaces, and let $BN$ and $B^{\prime}N$ be the subgroups of $S$
corresponding to the Lie subalgebras $\mathfrak{{b}}\oplus\mathfrak{{n}}$ and
$\mathfrak{{b}}^{\prime}\oplus\mathfrak{{n}}$ of $\mathfrak{{s}}$,
respectively. Then the families of equidistant tubes (or equidistant
hypersurfaces in case
$\dim(\mathfrak{{b}})=\dim(\mathfrak{{b}}^{\prime})=r-1$) around the orbits
$BN\cdot o$ and $B^{\prime}N\cdot o$ are congruent to each other if and only
if there exists $P\in\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ whose
corresponding orthogonal transformation $\widehat{P}$ of $\mathfrak{{a}}$
sends $\mathfrak{{b}}$ onto $\mathfrak{{b}}^{\prime}$.

The way this result is formulated in the paper is slightly different. Instead
of $\widehat{P}$, the authors talk about ${\left.\kern-1.2pt\mathrm{Ad}\hskip
0.59998pt(k)\vphantom{\big{|}}\right|_{\mathfrak{{a}}}}$, where $k$ is an
element of $N_{\widetilde{K}}(\mathfrak{{a}})$ preserving $\left\\{\hskip
0.29999pt{H_{\upalpha}\mid\upalpha\in\Upsigma^{+}}\hskip 0.29999pt\right\\}$.
But note that this is the same as to ask that $k$ lies in
$N_{\widetilde{K}}(\mathfrak{{n}})$, and we know that the image of the latter
in $\mathrm{O}(\mathfrak{{a}}^{*})$ is precisely $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})$. Note that the proof of part (iii) of Theorem B in
[6] works without the restrictions imposed on $\mathfrak{{b}}$ and
$\mathfrak{{b}}^{\prime}$ at the beginning of the theorem.

When we have one-dimensional subspaces
$\ell,\ell^{\prime}\subseteq\mathfrak{{a}}$, Theorem 3.5 applied to
$\mathfrak{{a}}_{\ell}=\mathfrak{{a}}\ominus\ell$ and
$\mathfrak{{a}}_{\ell^{\prime}}=\mathfrak{{a}}\ominus\ell^{\prime}$ yields:

######  Proposition 3.6.

Let $M$ be a symmetric space of noncompact type, and let
$\ell,\ell^{\prime}\subseteq\mathfrak{{a}}$ be one-dimensional linear
subspaces. Then the foliations $\mathcal{{F}}_{\ell}$ and
$\mathcal{{F}}_{\ell^{\prime}}$ are isometrically congruent if and only if
there exists $P\in\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ whose
corresponding orthogonal transformation $\widehat{P}$ of $\mathfrak{{a}}$
sends $\ell$ onto $\ell^{\prime}$.

Combining Propositions 3.2, 3.4, and 3.6 yields a proof of the Main Theorem.

##  4\. Alternative proof of Proposition 3.6

In this section we provide an alternative proof of the more difficult part of
Proposition 3.6, namely that if the foliations $\mathcal{{F}}_{\ell}$ and
$\mathcal{{F}}_{\ell^{\prime}}$ are congruent, then $\ell$ and $\ell^{\prime}$
lie in the same $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$-orbit. The
difficulty here lies in the fact that if the congruence is realized by some
$k\in\widetilde{G}$ and even if we assume that $k$ preserves $o$, it may not
preserve $\mathfrak{{a}}$, let alone $\mathfrak{{n}}$. In their original proof
of this statement in the irreducible case, Berndt and Tamaru bypassed this
problem by establishing an isomorphism
$\mathfrak{{s}}_{\ell}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{s}}_{\ell^{\prime}}$
that respects certain natural gradings on these Lie algebras. The main
obstacle in this approach is that this isomorphism does not in general come
from $\mathrm{Ad}\hskip 0.59998pt(\widetilde{K})$ and is just an abstract Lie
algebra isomorphism. Yet, the authors managed to prove – purely algebraically
– that the existence of such an isomorphism implies that $\ell$ and
$\ell^{\prime}$ differ by $\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$ (see
pp. 9-20 in [4]). The proof is quite complicated and involves a case-by-case
consideration of all possible irreducible root systems from $A_{r}$ to $G_{2}$
and $(BC)_{r}$.

In their alternative approach taken in [6], Domínguez-Vázquez and Sanmartín-
López do not relinquish the congruence $k$ and instead use some subtle
algebro-geometric arguments and modify $k$ to preserve $\mathfrak{{a}}$ and
$\mathfrak{{n}}$, which gives them an element of
$N_{\widetilde{K}}(\mathfrak{{n}})$ (and thus $\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})$) sending $\ell$ onto $\ell^{\prime}$. This results
in a much shorter proof that works in the reducible case and applies to a more
general situation described in Theorem 3.5.

On the principle that two proofs rooted in different sets of ideas are always
better than one, we give a proof of Proposition 3.6 alternative to the one in
[6], which is mostly algebraic in nature and is in fact a direct extension of
the original proof by Berndt and Tamaru.

Before we begin, given $\ell\subseteq\mathfrak{{a}}$, note that the solvable
Lie algebra $\mathfrak{{s}}_{\ell}$ is naturally graded by the height
function:

$\mathfrak{{s}}_{\ell}=\bigoplus_{i=0}^{m}\mathfrak{{s}}_{\ell}^{i}\hskip
6.99997pt\text{with}\hskip
6.99997pt\mathfrak{{s}}_{\ell}^{0}=\mathfrak{{a}}_{\ell}=\mathfrak{{a}}\ominus\ell\hskip
6.99997pt\text{and}\hskip
6.99997pt\mathfrak{{s}}_{\ell}^{i}=\mathfrak{{n}}^{i}=\bigoplus_{\mathrm{ht}(\upalpha)=i}\mathfrak{{g}}_{\upalpha}\hskip
8.00003pt\text{for}\hskip 5.0pti\geqslant 1.$

In particular,
$\mathfrak{{s}}_{\ell}^{1}=\mathfrak{{n}}^{1}=\bigoplus_{\upalpha\in\Uplambda}\mathfrak{{g}}_{\upalpha}$.
If we denote $L^{k}=\bigoplus_{i=k}^{m}\mathfrak{{s}}_{\ell}^{i}$, it follows
from the properties of restricted roots that
$[\mathfrak{{s}}_{\ell},\mathfrak{{s}}_{\ell}]=L^{1}=\mathfrak{{n}}$ and
$[L^{1},L^{k}]=L^{k+1}$ for $k\geqslant 0$. Note that $\mathfrak{{s}}_{\ell}$
is completely solvable444Over $\mathbb{\mathbb{C}}$, the notions of
solvability and complete solvability coincide by Lie’s theorem., i.e.
$\mathrm{ad}\hskip
0.59998pt(\mathfrak{{s}}_{\ell})\subset\mathfrak{{gl}}(\mathfrak{{g}})$
consists of upper-triangular matrices in a suitable basis for
$\mathfrak{{s}}_{\ell}$. Indeed, first take a basis for
$\mathfrak{{s}}_{\ell}^{m}$, then for $\mathfrak{{s}}_{\ell}^{m-1}$, and so
on, and then combine all these bases together. Clearly, each
$\mathrm{ad}\hskip 0.59998pt(x),x\in\mathfrak{{s}}_{\ell}$, is upper-
triangular in the resulting basis for $\mathfrak{{s}}_{\ell}$.

###### Proof of Proposition 3.6.

We will be using the notation established in the proof of Proposition 3.4. Let
$\ell$ and $\ell^{\prime}$ be one-dimensional subspaces of $\mathfrak{{a}}$
such that the foliations $\mathcal{{F}}_{\ell}$ and
$\mathcal{{F}}_{\ell^{\prime}}$ are congruent by some $k\in\widetilde{G}$.
Since all the leaves of $\mathcal{{F}}_{\ell}$ are congruent, we may assume
that $k\in\widetilde{K}$. First, suppose that
$\ell\subseteq\mathfrak{{a}}_{i}$ for some $i\in\left\\{\hskip
0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}$. Borrowing the notation from
the proof of Proposition 2.1, we see that the leaf $S_{\ell}\cdot o$ contains
$M_{l,o}$ for all $l\neq i$. By Proposition 2.1, the leaf
$S_{\ell^{\prime}}\cdot o=k(S_{\ell}\cdot o)$ must then contain $M_{l,o}$ for
all $l\neq j$, where $j\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip
0.29999pt\right\\}$ is such that $k(M_{i,o})=M_{j,o}$. In particular, we must
have $\ell^{\prime}\subseteq\mathfrak{{a}}_{j}$ and $M_{i}\simeq M_{j}$.
Arguing in a similar vein as in the proof of Proposition 3.4, we see that
there is an isomorphism
$\mathrm{DD}_{M_{i}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathrm{DD}_{M_{j}}$
that extends to an automorphism $P\in\mathrm{Aut}\hskip
0.59998pt(\mathrm{DD}_{M})$ such that $\widehat{P}$ sends $\ell$ onto
$\ell^{\prime}$. Therefore, we may assume that $\ell$ (and hence
$\ell^{\prime}$) does not lie in any of the $\mathfrak{{a}}_{i}$’s.

In this case, one can show that $\mathfrak{{a}}_{\ell}$ and
$\mathfrak{{a}}_{\ell^{\prime}}$ are Cartan subalgebras of
$\mathfrak{{s}}_{\ell}$ and $\mathfrak{{s}}_{\ell^{\prime}}$, respectively
(see [4, Lemma 3.3]). Recall that both $S_{\ell}$ and $S_{\ell^{\prime}}$ are
connected and completely solvable (this just means that their Lie algebras are
completely solvable). By means of the congruence $k$, we can regard both of
these groups as Lie subgroups of $I(S_{\ell}\cdot o)$. It follows from [1]
that any two connected completely solvable transitive Lie groups of isometries
of a connected Riemannian manifold are conjugate in the isometry group of that
manifold. Therefore, $S_{\ell}$ and $S_{\ell^{\prime}}$ are isomorphic, hence
so are $\mathfrak{{s}}_{\ell}$ and $\mathfrak{{s}}_{\ell^{\prime}}$. Let
$F\colon\mathfrak{{s}}_{\ell}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathfrak{{s}}_{\ell^{\prime}}$
be an isomorphism.

The first step is to adjust $F$ to make it look nice. Observe that
$F(\mathfrak{{a}}_{\ell})$ is a Cartan subalgebra of
$\mathfrak{{s}}_{\ell^{\prime}}$. Every two Cartan subalgebras of a solvable
Lie algebra are conjugate by an inner automorphism, so we may assume
$F(\mathfrak{{a}}_{\ell})=\mathfrak{{a}}_{\ell^{\prime}}$. Note also that
$\mathfrak{{n}}=[\mathfrak{{s}}_{\ell},\mathfrak{{s}}_{\ell}]=[\mathfrak{{s}}_{\ell^{\prime}},\mathfrak{{s}}_{\ell^{\prime}}]$,
so $F(\mathfrak{{n}})=\mathfrak{{n}}$. Next we modify $F$ to make it into a
graded isomorphism. Writing $X^{i}$ for the $i$-th graded component of a
vector $X$, define a map
$\mathfrak{{s}}_{\ell}\to\mathfrak{{s}}_{\ell^{\prime}},X\mapsto\sum_{i=0}^{m}(F(X^{i}))^{i}$.
It is easy to check that this map is a graded Lie algebra isomorphism (see [4,
Theorem 3.4] for an argument). We continue to denote it by $F$.

The main idea of the proof is to use the commutator relations between
$\mathfrak{{a}}_{\ell}$ (or $\mathfrak{{a}}_{\ell^{\prime}}$) and
$\mathfrak{{n}}$ to show that $F$ must permute the restricted root subspaces
of $\mathfrak{{n}}$ in a way that induces an automorphism of
$\mathrm{Aut}\hskip 0.59998pt(\mathrm{DD}_{M})$. For each pair
$\upalpha,\upbeta\in\Uplambda,\upalpha\neq\upbeta$, let $L_{\upalpha\upbeta}$
stand for the hyperplane in $\mathfrak{{a}}$ consisting of the vectors $Z$
such that the eigenvalue of $\mathrm{ad}\hskip 0.59998pt(Z)$ on
$\mathfrak{{g}}_{\upalpha}$ coincides with that on $\mathfrak{{g}}_{\upbeta}$.
If we write $Z=\sum_{\upgamma\in\Uplambda}Z_{\upgamma}H^{\upgamma}$, then
$L_{\upalpha\upbeta}=\left\\{\hskip 0.29999pt{Z\in\mathfrak{{a}}\mid
Z_{\upalpha}=Z_{\upbeta}}\hskip
0.29999pt\right\\}=\mathfrak{{a}}\ominus\mathbb{R}(H_{\upalpha}-H_{\upbeta})$.

First, we consider the generic choice of $\ell$ such that
$\mathfrak{{a}}_{\ell}\neq L_{\upalpha\upbeta}$ for any pair
$\upalpha,\upbeta\in\Uplambda$. In this case,
$\bigcup_{\begin{subarray}{c}\upalpha,\upbeta\in\Uplambda\\\
\upalpha\neq\upbeta\end{subarray}}(\mathfrak{{a}}_{\ell}\cap
L_{\upalpha\upbeta})$ is the union of finitely many hyperplanes in
$\mathfrak{{a}}_{\ell}$, so its complement $\mathfrak{{a}}_{\ell}^{\circ}$ is
open and dense in $\mathfrak{{a}}_{\ell}$. Pick any
$Z\in\mathfrak{{a}}_{\ell}^{\circ}$. By design, all coordinates $Z_{\upgamma}$
of $Z$ with respect to the basis $(H^{\upgamma})_{\upgamma\in\Uplambda}$ of
$\mathfrak{{a}}$ are pairwise distinct. Since $Z_{\upgamma}$ is the eigenvalue
of $\mathrm{ad}\hskip 0.59998pt(Z)$ on $\mathfrak{{g}}_{\upgamma}$,
$\mathrm{ad}\hskip 0.59998pt(Z)$ has the maximal possible number ($=r$) of
distinct eigenvalues on $\mathfrak{{n}}^{1}$ among all vectors in
$\mathfrak{{a}}$. Now, given $X\in\mathfrak{{g}}_{\upgamma}$, one has

$[F(Z),F(X)]=F[Z,X]=F(Z_{\upgamma}X)=Z_{\upgamma}F(X),$

hence $F$ maps the eigenspaces of $\mathrm{ad}\hskip 0.59998pt(Z)$ in
$\mathfrak{{n}}^{1}$ onto the eigenspaces of $\mathrm{ad}\hskip
0.59998pt(F(Z))$ in $\mathfrak{{n}}^{1}$ corresponding to the same
eigenvalues. Since there are $r$ such eigenvalues, $F$ must permute the root
subspaces in $\mathfrak{{n}}^{1}$. Formally, there exists a bijection
$\widetilde{F}\colon\Uplambda\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\Uplambda$
such that

$F(\mathfrak{{g}}_{\upgamma})=\mathfrak{{g}}_{\widetilde{F}(\upgamma)}\;(\forall\;\upgamma\in\Uplambda)\hskip
5.0pt\text{and}\hskip
3.00003pt\;F(Z)=\sum_{\upgamma\in\Uplambda}Z_{\upgamma}H^{\widetilde{F}(\upgamma)}.$

We see that $\ell$ is generic in the sense explained above if and only if
$\ell^{\prime}$ is, and we also have
$F(\mathfrak{{a}}_{\ell}^{\circ})=\mathfrak{{a}}_{\ell^{\prime}}^{\circ}$.
Assume for a moment that $\widetilde{F}$ is an automorphism of
$\mathrm{DD}_{M}$. The corresponding orthogonal transformation
$T=\widehat{\widetilde{F}}$ of $\mathfrak{{a}}$ sends $H^{\upgamma}$ to
$H^{\widetilde{F}(\upgamma)}$. Therefore, we have
$T(Z)=\sum_{\upgamma\in\Uplambda}Z_{\upgamma}H^{\widetilde{F}(\upgamma)}$. We
conclude that the restrictions of $T$ and $F$ to
$\mathfrak{{a}}_{\ell}^{\circ}$ coincide. Since
$\mathfrak{{a}}_{\ell}^{\circ}$ is dense in $\mathfrak{{a}}_{\ell}$, these two
operators coincide on the whole $\mathfrak{{a}}_{\ell}$. But
$T\in\mathrm{O}(\mathfrak{{a}})$, which means that it sends
$\ell=\mathfrak{{a}}\ominus\mathfrak{{a}}_{\ell}$ onto $\mathfrak{{a}}\ominus
T(\mathfrak{{a}}_{\ell})=\mathfrak{{a}}\ominus\mathfrak{{a}}_{\ell^{\prime}}=\ell^{\prime}$,
which was to be proved. So we are left to show that $\widetilde{F}$ is indeed
an automorphism of $\mathrm{DD}_{M}$.

Suppose that $\upalpha,\upbeta\in\Uplambda$ are connected by an edge in
$\mathrm{DD}_{M}$, i.e. the angle between them is greater than
$\frac{\uppi}{2}$. This is equivalent to asking that
$\mathfrak{{g}}_{\upalpha+\upbeta}=[\mathfrak{{g}}_{\upalpha},\mathfrak{{g}}_{\upbeta}]\neq\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}$. But then

$\mathfrak{{g}}_{\widetilde{F}(\upalpha)+\widetilde{F}(\upbeta)}=[\mathfrak{{g}}_{\widetilde{F}(\upalpha)},\mathfrak{{g}}_{\widetilde{F}(\upbeta)}]=[F(\mathfrak{{g}}_{\upalpha}),F(\mathfrak{{g}}_{\upbeta})]=F([\mathfrak{{g}}_{\upalpha},\mathfrak{{g}}_{\upbeta}])=F(\mathfrak{{g}}_{\upalpha+\upbeta})\neq\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\},$ (3)

which means that $\widetilde{F}(\upalpha)$ and $\widetilde{F}(\upbeta)$ are
also connected by an edge in the Dynkin diagram. Applying the same argument to
$\widetilde{F}^{-1}$, we see that $\widetilde{F}$ preserves adjacency between
the vertices. Consequently, $\widetilde{F}$ permutes the connected components
of $\mathrm{DD}_{M}$: there exists $\upsigma\in S_{k}$ such that
$\widetilde{F}(\mathrm{DD}_{M_{i}})=\mathrm{DD}_{M_{\upsigma(i)}}$ for each
$i\in\left\\{\hskip 0.29999pt{1,\ldots,k}\hskip 0.29999pt\right\\}$. Note that
we then have $F(\mathfrak{{n}}^{1}_{i})=\mathfrak{{n}}^{1}_{\upsigma(i)}$ and,
since $\mathfrak{{n}}^{1}_{i}$ generates $\mathfrak{{n}}_{i}$ for each $i$,
$F(\mathfrak{{n}}_{i})=\mathfrak{{n}}_{\upsigma(i)}$. We will now show that
$\widetilde{F}$ provides an isomorphism between $\mathrm{DD}_{M_{i}}$ and
$\mathrm{DD}_{M_{\upsigma(i)}}$.

To begin with, note that $\mathrm{DD}_{M_{i}}$ and
$\mathrm{DD}_{M_{\upsigma(i)}}$ must have the same number of vertices, and
$\widetilde{F}$ preserves the degrees and multiplicities of the vertices.
Next, recall that any positive root $\upeta$ can be expressed as
$\upgamma_{l_{1}}+\upgamma_{l_{2}}+\cdots+\upgamma_{l_{s}}$, where each
summand is a simple root and each partial sum
$\upgamma_{l_{1}}+\cdots+\upgamma_{l_{t}},t\in\left\\{\hskip
0.29999pt{2,\ldots,t-1}\hskip 0.29999pt\right\\}$, is also a root. This,
together with a computation similar to (3), implies that a linear combination
$\sum_{j=1}^{r_{i}}n_{j}\upalpha_{i}^{j}$ is a root in $\Upsigma_{i}^{+}$ if
and only if $\sum_{j=1}^{r_{i}}n_{j}\widetilde{F}(\upalpha_{i}^{j})$ is a root
in $\Upsigma_{\upsigma(i)}^{+}$. Among other things, this implies that
$\\#\Upsigma_{i}=\\#\Upsigma_{\upsigma(i)}$ and that $\Upsigma_{i}$ is reduced
if and only if $\Upsigma_{\upsigma(i)}$ is. By looking at the list of the
Dynkin diagrams of all possible irreducible symmetric spaces of noncompact
type (see, for instance, [2, pp. 336-340]), one deduces that
$\mathrm{DD}_{M_{i}}$ and $\mathrm{DD}_{M_{\upsigma(i)}}$ must be isomorphic
(which is also the same as to say that $M_{i}$ and $M_{\upsigma(i)}$ are
isometric or that $\mathfrak{{g}}_{i}$ and $\mathfrak{{g}}_{\upsigma(i)}$ are
isomorphic) and one such isomorphism is given by $\widetilde{F}$, which
completes the proof for the generic choice of $\ell$.

######  Remark 4.1.

To tell the diagrams $B_{r}$ and $C_{r}$ apart, one might need to use the fact
that if we denote the two adjacent vertices of nonequal lengths by
$\upalpha_{r-1}$ and $\upalpha_{r}$, the sum $\upalpha_{r-1}+2\upalpha_{r}$ is
a root for $B_{r}$ but not for $C_{r}$.

We are left to consider the situation when
$\mathfrak{{a}}_{\ell}=L_{\upalpha\upbeta}$ for some
$\upalpha,\upbeta\in\Uplambda,\upalpha\neq\upbeta$, which simply means that
$\ell$ is spanned by $H_{\upalpha}-H_{\upbeta}$. Since we assume that $\ell$
does not lie in any of the $\mathfrak{{a}}_{l}$’s, the roots $\upalpha$ and
$\upbeta$ must lie in different components of $\mathrm{DD}_{M}$, i.e.
$\upalpha\in\Uplambda_{i},\upbeta\in\Uplambda_{j},i\neq j$. In this case, each
$L_{\upalpha^{\prime}\upbeta^{\prime}}$ with
$(\upalpha^{\prime},\upbeta^{\prime})\neq(\upalpha,\upbeta)$ intersects
$\mathfrak{{a}}_{\ell}$ by a hyperplane in $\mathfrak{{a}}_{\ell}$. Let
$\mathfrak{{a}}_{\ell}^{\circ}\subseteq\mathfrak{{a}}_{\ell}$ stand for the
complement to the union of all such hyperplanes (for all
$(\upalpha^{\prime},\upbeta^{\prime})\neq(\upalpha,\upbeta)$). For every
$Z\in\mathfrak{{a}}_{\ell}^{\circ}$, $\mathrm{ad}\hskip 0.59998pt(Z)$ has
$r-1$ distinct eigenvalues on $\mathfrak{{n}}^{1}$. The only two restricted
root subspaces in $\mathfrak{{n}}^{1}$ with the same eigenvalue (equal to
$Z_{\upalpha}=Z_{\upbeta}$) are $\mathfrak{{g}}_{\upalpha}$ and
$\mathfrak{{g}}_{\upbeta}$. As we have already seen, $\mathrm{ad}\hskip
0.59998pt(F(Z))$ must have the same eigenvalues on $\mathfrak{{n}}^{1}$ and
$F$ must send the eigenspaces of $\mathrm{ad}\hskip 0.59998pt(Z)$ onto the
corresponding eigenspaces of $\mathrm{ad}\hskip 0.59998pt(F(Z))$. This implies
that there exist some $\upalpha^{\prime},\upbeta^{\prime}\in\Uplambda$ such
that $\mathfrak{{a}}_{\ell^{\prime}}=L_{\upalpha^{\prime}\upbeta^{\prime}}$
and thus $\ell^{\prime}$ is spanned by
$H_{\upalpha^{\prime}}-H_{\upbeta^{\prime}}$. Since $\ell^{\prime}$ cannot lie
in any of the $\mathfrak{{a}}_{l}$’s, we have
$\upalpha^{\prime}\in\Uplambda_{i^{\prime}},\upbeta^{\prime}\in\Uplambda_{j^{\prime}},i^{\prime}\neq
j^{\prime}$. Similarly to what we did in the proof of Proposition 3.4, note
that $S_{\ell}\cdot o=(\prod_{l\neq i,j}M_{l})\times(S_{i}\times
S_{j})_{\ell}\cdot o$ (here $(S_{i}\times S_{j})_{\ell}$ is the connected Lie
subgroup of $G_{i}\times G_{j}$ corresponding to the Lie subalgebra
$(\mathfrak{{s}}_{i}\oplus\mathfrak{{s}}_{j})\ominus\ell$) and
$S_{\ell^{\prime}}\cdot o=(\prod_{l\neq
i^{\prime},j^{\prime}}M_{l})\times(S_{i^{\prime}}\times
S_{j^{\prime}})_{\ell^{\prime}}\cdot o$. Therefore, the congruence $k$ between
$\mathcal{{F}}_{\ell}$ and $\mathcal{{F}}_{\ell^{\prime}}$ must send $M_{i,o}$
onto either $M_{i^{\prime},o}$ or $M_{j^{\prime},o}$ and $M_{j,o}$ onto the
other of the two. Consequently (and slightly informally), the (unordered) pair
$(M_{i},M_{j})$ is isometric to the pair $(M_{i^{\prime}},M_{j^{\prime}})$.
This means that there exists an isomorphism of Dynkin diagrams
$\mathrm{DD}_{M_{i}}\sqcup\mathrm{DD}_{M_{j}}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\mathrm{DD}_{M_{i^{\prime}}}\sqcup\mathrm{DD}_{M_{j^{\prime}}}$,
which extends easily – just like we did at the end of the proof of Proposition
3.4 – to an automorphism $P$ of $\mathrm{DD}_{M}$. We will prove that all four
of the spaces $M_{i},M_{j},M_{i^{\prime}},M_{j^{\prime}}$ have rank 1 and are
thus isometric to hyperbolic spaces. If this is the case, then $P$ sends
$\Uplambda_{i}\sqcup\Uplambda_{j}=\left\\{\hskip
0.29999pt{\upalpha,\upbeta}\hskip 0.29999pt\right\\}$ onto
$\Uplambda_{i^{\prime}}\sqcup\Uplambda_{j^{\prime}}=\left\\{\hskip
0.29999pt{\upalpha^{\prime},\upbeta^{\prime}}\hskip 0.29999pt\right\\}$, hence
the corresponding orthogonal transformation $\widehat{P}$ of $\mathfrak{{a}}$
sends $H_{\upalpha}-H_{\upbeta}\in\ell$ to
$\pm(H_{\upalpha^{\prime}}-H_{\upbeta^{\prime}})\in\ell^{\prime}$, which will
complete the proof.

It follows from our argument involving the eigenspaces of $\mathrm{ad}\hskip
0.59998pt(Z)$ and $\mathrm{ad}\hskip 0.59998pt(F(Z))$ that $F$ establishes a
bijection

$\left\\{\hskip
0.29999pt{\mathfrak{{g}}_{\upalpha_{1}},\ldots,\widehat{\mathfrak{{g}}}_{\upalpha},\ldots,\widehat{\mathfrak{{g}}}_{\upbeta},\ldots,\mathfrak{{g}}_{\upalpha_{r}},\mathfrak{{g}}_{\upalpha}\oplus\mathfrak{{g}}_{\upbeta}}\hskip
0.29999pt\right\\}\mathrel{\ensurestackMath{\stackengine{}{\xrightarrow{\phantom{\sim}}}{\scriptstyle\\!\sim\,}{O}{c}{F}{F}{S}}}\left\\{\hskip
0.29999pt{\mathfrak{{g}}_{\upalpha_{1}},\ldots,\widehat{\mathfrak{{g}}}_{\upalpha^{\prime}},\ldots,\widehat{\mathfrak{{g}}}_{\upbeta^{\prime}},\ldots,\mathfrak{{g}}_{\upalpha_{r}},\mathfrak{{g}}_{\upalpha^{\prime}}\oplus\mathfrak{{g}}_{\upbeta^{\prime}}}\hskip
0.29999pt\right\\},$ (4)

sending subspaces on the left isomorphically onto the corresponding subspaces
on the right (here a hat over a subspace means that it is omitted from the
list). We have to consider two cases.

Case 1:
$F(\mathfrak{{g}}_{\upalpha}\oplus\mathfrak{{g}}_{\upbeta})=\mathfrak{{g}}_{\upgamma}$
for some $\upgamma\neq\upalpha^{\prime},\upbeta^{\prime}$. We first prove two
Lie-theoretic lemmas of separate interest.

Observe that the adjoint action of the compact Lie group
$Z_{\widetilde{K}}(\mathfrak{{a}})$ on $\mathfrak{{g}}$ preserves the
restricted root space decomposition. Thus, this group acts orthogonally on
each restricted root subspace.

######  Lemma 4.2.

For every $\upmu\in\Upsigma$ of multiplicity greater than one, the action
$Z_{\widetilde{K}}(\mathfrak{{a}})\curvearrowright\mathfrak{{g}}_{\upmu}$ is
transitive on the unit sphere in $\mathfrak{{g}}_{\upmu}$. It is even
transitive when restricted to the connected Lie subgroup $K_{0}\subseteq
Z_{\widetilde{K}}(\mathfrak{{a}})$ corresponding to the Lie subalgebra
$\mathfrak{{k}}_{0}$.

###### Proof.

If $\upmu$ is a simple root or the double of a simple root, this is the
content of Problem D.2 in [8, Ch. III] (see p. 585 for a solution). But it is
a well-known fact that every root in any root system is either a simple root
or the double of one under a suitable choice of a Weyl chamber. The second
statement follows trivially from the first one, because $K_{0}$ is the
connected component of the identity in $Z_{\widetilde{K}}(\mathfrak{{a}})$ and
the unit sphere in $\mathfrak{{g}}_{\upmu}$ is connected. ∎

######  Lemma 4.3.

Let $\upmu,\upnu\in\Upsigma$ be any two roots such that $\upmu+\upnu$ is also
a root (e.g. they can be simple roots connected by an edge in the Dynkin
diagram $\mathrm{DD}_{M}$). Then the pairing
$\mathfrak{{g}}_{\upmu}\times\mathfrak{{g}}_{\upnu}\to\mathfrak{{g}}_{\upmu+\upnu}$
given by the Lie bracket is nondegenerate555An analogous result in the complex
semisimple case is well known and is just a reformulation of the fact that
$[\mathfrak{{g}}_{\upmu},\mathfrak{{g}}_{\upnu}]=\mathfrak{{g}}_{\upmu+\upnu}$,
for all the root subspaces are one-dimensional over $\mathbb{C}$.. In other
words, for every $X\in\mathfrak{{g}}_{\upmu}$ (resp.,
$Y\in\mathfrak{{g}}_{\upnu}$), there exists $Y\in\mathfrak{{g}}_{\upnu}$
(resp., $X\in\mathfrak{{g}}_{\upmu}$) such that $[X,Y]\neq 0$.

###### Proof.

Assume the converse: there exists $X\in\mathfrak{{g}}_{\upmu}$ such that
$[X,\mathfrak{{g}}_{\upnu}]=\left\\{\hskip 0.29999pt{0}\hskip
0.29999pt\right\\}$. For each $k\in Z_{\widetilde{K}}(\mathfrak{{a}})$, we
have
$[kX,\mathfrak{{g}}_{\upnu}]=[kX,k(\mathfrak{{g}}_{\upnu})]=k[X,\mathfrak{{g}}_{\upnu}]=\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}$. But, according to Lemma 4.2, every
vector in $\mathfrak{{g}}_{\upmu}$ is a multiple of $kX$ for a suitable choice
of $k$. Therefore,
$[\mathfrak{{g}}_{\upmu},\mathfrak{{g}}_{\upnu}]=\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}$, which contradicts the standard fact
that
$[\mathfrak{{g}}_{\upmu},\mathfrak{{g}}_{\upnu}]=\mathfrak{{g}}_{\upmu+\upnu}\neq\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}$. ∎

Now, assume there is $\upgamma^{\prime}\in\Uplambda$ connected to $\upgamma$
by an edge in $\mathrm{DD}_{M}$. Note that
$F^{-1}(\mathfrak{{g}}_{\upgamma^{\prime}})$ must lie within a single
restricted root subspace, which we denote by
$\mathfrak{{g}}_{\upgamma^{\prime\prime}},\upgamma^{\prime\prime}\in\Uplambda$.
By Lemma 4.3, for any nonzero $X\in\mathfrak{{g}}_{\upalpha}$, there exists
$Y\in\mathfrak{{g}}_{\upgamma^{\prime}}$ with $[F(X),Y]\neq 0$, which implies
that $[X,F^{-1}(Y)]\neq 0$, hence $\upgamma^{\prime\prime}$ must be connected
to $\upalpha$ in $\mathrm{DD}_{M}$. But we can also apply this argument to a
nonzero $X\in\mathfrak{{g}}_{\upbeta}$ to deduce that
$\upgamma^{\prime\prime}$ must be connected to $\upbeta$ as well. Since
$\upalpha$ and $\upbeta$ lie in different connected components of
$\mathrm{DD}_{M}$, we arrive at a contradiction. Thus, the connected component
of $\mathrm{DD}_{M}$ containing $\upgamma$ consists of nothing but $\upgamma$.
But then the same must hold for $\upalpha$ and $\upbeta$:
$\Uplambda_{i}=\left\\{\hskip 0.29999pt{\upalpha}\hskip
0.29999pt\right\\},\Uplambda_{j}=\left\\{\hskip 0.29999pt{\upbeta}\hskip
0.29999pt\right\\}$. Indeed, otherwise we would have $\upgamma^{\prime}$
connected to, say, $\upalpha$, meaning that
$[\mathfrak{{g}}_{\upalpha},\mathfrak{{g}}_{\upgamma^{\prime}}]\neq\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}$, whereas
$F([\mathfrak{{g}}_{\upalpha},\mathfrak{{g}}_{\upgamma^{\prime}}])=[F(\mathfrak{{g}}_{\upalpha}),F(\mathfrak{{g}}_{\upgamma^{\prime}})]\subseteq[\mathfrak{{g}}_{\upgamma},F(\mathfrak{{g}}_{\upgamma^{\prime}})]=\left\\{\hskip
0.29999pt{0}\hskip 0.29999pt\right\\}$. We conclude that $\Uplambda_{i}$ and
$\Uplambda_{j}$ are singletons, i.e. $M_{i}$ and $M_{j}$ have rank 1. The same
considerations can be applied to $F^{-1}$ to deduce that $M_{i^{\prime}}$ and
$M_{j^{\prime}}$ are also of rank 1. This finishes the proof in Case 1.

Case 2:
$F(\mathfrak{{g}}_{\upalpha}\oplus\mathfrak{{g}}_{\upbeta})=\mathfrak{{g}}_{\upalpha^{\prime}}\oplus\mathfrak{{g}}_{\upbeta^{\prime}}$.
It suffices to show that $F(\mathfrak{{g}}_{\upalpha})$ is one of the
subspaces
$\mathfrak{{g}}_{\upalpha^{\prime}},\mathfrak{{g}}_{\upbeta^{\prime}}$, while
$F(\mathfrak{{g}}_{\upbeta})$ is the other one. Indeed, if this is the case,
then we get a permutation $\widetilde{F}$ of $\Uplambda$, which – as we have
already seen in the proof for a generic choice of $\ell$ – must be an
automorphism of $\mathrm{DD}_{M}$ whose corresponding orthogonal
transformation of $\mathfrak{{a}}$ maps $\ell$ onto $\ell^{\prime}$. Arguing
by contradiction, we may assume, without loss of generality, that the
projections of $F(\mathfrak{{g}}_{\upalpha})$ in both
$\mathfrak{{g}}_{\upalpha^{\prime}}$ and $\mathfrak{{g}}_{\upbeta^{\prime}}$
are nonzero. We also do not lose generality by assuming that the projection of
$F(\mathfrak{{g}}_{\upbeta})$ in $\mathfrak{{g}}_{\upbeta^{\prime}}$ is
nonzero. By invoking Lemma 4.3, we see, just as we did for $\upgamma$ in Case
1, that $\Uplambda_{j^{\prime}}=\left\\{\hskip
0.29999pt{\upbeta^{\prime}}\hskip 0.29999pt\right\\}$. There might be two
possibilities.

Subcase 2.1:
$F(\mathfrak{{g}}_{\upbeta})\subseteq\mathfrak{{g}}_{\upbeta^{\prime}}$. In
this case, we can write $\mathfrak{{g}}_{\upalpha}=U\oplus V$, where
$F(U)=\mathfrak{{g}}_{\upalpha^{\prime}}$ and $F(V)\oplus
F(\mathfrak{{g}}_{\upbeta})=\mathfrak{{g}}_{\upbeta^{\prime}}$. Mimicking the
arguments from the proof in Case 1, we immediately see that
$\Uplambda_{j}=\left\\{\hskip 0.29999pt{\upbeta}\hskip 0.29999pt\right\\}$. If
there exists $\upgamma\in\Uplambda$ connected to $\upalpha$ in
$\mathrm{DD}_{M}$, then, by Lemma 4.3, for any nonzero $X\in V$, there exists
$Y\in\mathfrak{{g}}_{\upgamma}$ such that $[X,Y]\neq 0$. But then
$[F(X),\mathfrak{{g}}_{\upbeta^{\prime}}]\neq\left\\{\hskip 0.29999pt{0}\hskip
0.29999pt\right\\}$, which implies that $F(\mathfrak{{g}}_{\upgamma})$ must be
a root subspace whose corresponding root is connected to $\upbeta$, which is a
contradiction. Therefore, $\Uplambda_{i}=\left\\{\hskip
0.29999pt{\upalpha}\hskip 0.29999pt\right\\}$. Arguing in the same fashion, we
deduce that $\Uplambda_{i^{\prime}}=\left\\{\hskip
0.29999pt{\upalpha^{\prime}}\hskip 0.29999pt\right\\}$, so all four of
$M_{i},M_{j},M_{i^{\prime}},M_{j^{\prime}}$ are of rank 1 and we are done.

Subcase 2.2: The projection of $F(\mathfrak{{g}}_{\upbeta})$ in
$\mathfrak{{g}}_{\upalpha^{\prime}}$ is nonzero. Just like we did for
$\Uplambda_{j^{\prime}}$, we see that $\Uplambda_{i^{\prime}}=\left\\{\hskip
0.29999pt{\upalpha^{\prime}}\hskip 0.29999pt\right\\}$. Arguing in a similar
manner, we deduce that $\Uplambda_{i}=\left\\{\hskip 0.29999pt{\upalpha}\hskip
0.29999pt\right\\}$ and $\Uplambda_{j}=\left\\{\hskip 0.29999pt{\upbeta}\hskip
0.29999pt\right\\}$. Once again, all four of
$M_{i},M_{j},M_{i^{\prime}},M_{j^{\prime}}$ are of rank 1. This completes the
proof of Subcase 2.2 and thus Proposition 3.6. ∎

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