Chapter 13
Polar Actions on Symmetric Spaces of
Noncompact Type
In this chapter we discuss polar actions on Riemannian symmetric spaces of noncom-
pact type. Compared with the compact case, the situation is much more involved. For
example, on any Riemannian symmetric space of noncompact type and rank greater
than one there exist polar actions which are not hyperpolar, which is not true in the
compact case. As in the compact case we will distinguish between rank one and
higher rank. The theory for polar actions on Riemannian symmetric spaces of non-
compact type has been developed to a large extent by the Berndt and Tamaru in a
series of papers. The theoretical background for it includes the structure theory of
parabolic subalgebras of semisimple Lie algebras and horospherical decompositions
of Riemannian symmetric spaces of noncompact type. We will explain all this in the
first two sections of th is chapter.
The isometry group of a Riemannian symmetric space of noncompact type is a
noncompact real semisimple Lie group. It is therefore natural to expect that the well-
developed structure theory of real semisimple Lie groups and Lie algebras is useful in
the context of polar actions on these spaces. In Section 13.1 we explain the restricted
root space decomposition of a noncompact real semisimple Lie algebra and how this
leads to a Dynkin diagram for each Riemannian symmetric space of noncompact
type. For every irreducible Riemannian symmetric space of noncompact type we list
the corresponding restricted root system, Dynkin diagram and multiplicities of the
simple roots.
In Section 13.2 we discuss the construction and classification of parabolic sub al-
gebras of noncompact real semisimple Lie algebras. We explain the geometry behind
the Chevalley decomposition and the Langlands decomposition of a parabolic suba l-
gebra and how this leads to a horospherical decomposition of the symmetric space.
These horospherical decompositions play an important part in the construction of po-
lar actions on Riemannian symmetric spaces of noncompact type, for which there is
no analogue in the compact case. We discuss this explicitly for the symmetric spaces
SL
r+1
(R)/SO
r+1
, G
2
2
/SO
4
and G
2
(C)/G
2
.
In Section 13.3 we prove an algebraic criterion for the polarity of an action with-
out singular orbits on a Riemannian symmetric space of noncompact type. We use
this criterion to show that the nilpotent group in the Chevalley decomposition of
a parabolic subgroup acts polarly on a Riemannian symmetric space of noncom-
pact type. This action is not hyperpolar unless the parabolic subgroup is minimal
333
334 Submanifolds and Holonomy
parabolic. These examples show that the assumption o f compactness in Theorem
12.2.1 is essential.
In Section 13.4 we present the classificationbyBerndt,D´ıaz-Ramos and Tamaru
of hyperpolar actions without singular orbits on Riemannian symmetric spaces of
noncompact type. A remarkable feature o f this classification is that any correspond-
ing foliation by orbits can be construc ted from elementary fo liations on Euclidean
spaces and hyperbolic spaces, using the horospherical decomposition. We discuss
this explicitly for the symmetric space SL
r+1
(R)/SO
r+1
. We then focus on the spe-
cial case of cohomogeneity one and discuss the geometry o f the foliations arising in
the classification. Another remarkable feature in this special case is that the parame-
ter space of all homogeneous codimension one foliations depends only on the rank of
the symmetric space and on possible duality and triality principles on the symmetric
space. We will see that on any hyperbolic space there are exactly two congruence
classes of homogeneous codimension foliations.
In Section 13.5 we present an overview about the classification of polar actions on
hyperbolic spaces. On real hyperbolic spaces such actions were classified by Wu up
to orbit equivalence, and on complex hyperbolic spaces by D´ıaz-Ramos, Dom´ınguez-
V´azquez, and Kollross. For the complex hyperbolic space we will discuss in more
detail the special case of cohomogeneity one and the special case of the complex hy-
perbolic plane. The classification of polar actions on quaternionic hyperbolic spaces
is still an open problem, even for the special case of cohomogeneity one. We will
present explicit examples of cohomogeneity one actions on quaternionic hyperbolic
spaces and discuss the special case of the qua ternionic hyperbolic plane. Finally, we
discuss the classification by Berndt and Tamaru of cohomogeneity one actions on the
Cayley hyperbolic plane up to orbit equivalence.
In Section 13.6 we present the work by Berndt and Tamaru on the construction
and classification o f cohomogeneity one actions, up to orbit equivalence, on Rie-
mannian symmetric spaces of noncompact type and higher rank. There are two parts
to it, one related to reductive subgroups and the other one related to parabolic sub-
groups. In the reductive case this leads to cohomogeneity one actions with a totally
geodesic singular orbit and we will explain how to find all totally geodesic submani-
folds arising as such a singular orbit. In the parabolic case there are two construction
methods, both taking into account horospherical decompositions. We will explain
both construction methods and present examples and the main result by Berndt and
Tamaru.
In Section 13.7 we briefly discuss the classification problem for hypersurfaces
with constant principal curvatures in hyperbolic spaces. In real hyperbolic spaces
there is a complete classification due to
´
Elie Cartan, but in the other hyperbolic spaces
this is still an open problem.
Polar Actions on Symmetric Spaces of Noncompact Type 335
13.1 Dynkin diagrams of symmetric spaces of noncompact type
In this section we describe how to assign a Dynkin diagram to a Riemannian sym-
metric space of noncompact type. Let M = G/K be a Riemannian symmetric space
of noncompact type, where G = I
o
(M) and K is the isotropy group of G at some point
o ∈ M. We denote by r the r ank of M.TheLiealgebrag of G is a real semisimple
Lie algebra. Let g = k ⊕p be a Cartan decomposition of g with corresponding Cartan
involution
θ
: g = k ⊕p → g = k ⊕p , X +Y → X −Y.
We denote by B the Killing form on g given by
B : g ×g → R , (X,Y ) → tr(ad(X) ◦ad(Y )).
Then
·,· : g ×g → R , (X ,Y) →−B(X,
θ
Y )
defines a positive definite inner product on g. For linear subspaces V,W of g with
V ⊂W we use the notation
W V = {w ∈W : w, v = 0forallv ∈V }.
Let a be a maximal abelian subspace of p and denote by a
∗
the dual space of a.
Note that the dimension of a coincid es with the rank r of M. For each
α
∈ a
∗
we
define
g
α
= {X ∈ g : [H,X]=
α
(H)X for all H ∈ a}.
If
α
= 0andg
α
= {0},then
α
is a restricted root and g
α
is a restricted root space of
g with respect to a. The dimension m
α
= dimg
α
is called the multiplicity of the root
α
.For
α
= 0wehave
g
0
= k
0
⊕a,
where
k
0
= {X ∈ k : [H,X]=0forallH ∈ a} = Z
k
(a)
is the centralizer of a in k.
Let Ψ ⊂ a
∗
be the set of restricted roots o f g with respect to a. The direct sum
decomposition
g = g
0
⊕
α
∈Ψ
g
α
is the restricted root space decomposition of g with respect to a. For each
α
∈ Ψ the
root vector H
α
∈ a is defined by
α
(H)=H
α
,H for all H ∈ a .
The inner product ·,·induces a canonical isomorphism from a onto a
∗
by assigning
336 Submanifolds and Holonomy
to each H ∈ a the one-form H,· ∈ a
∗
. This isomorphism induces an inner product
on a
∗
, which we will also denote by ·,·, satisfying
α
,
β
= H
α
,H
β
for all
α
,
β
∈ Ψ.
We always consider a
∗
to be equipped with this inner product.
A subset Λ = {
α
1
,...,
α
r
}⊂Ψ is called a set of simple roots of Ψ if every
α
∈Ψ
can be written in the form
α
=
r
∑
i=1
c
i
α
i
with some integers c
1
,...,c
r
∈ Z such that c
1
,...,c
r
are either all nonpositive or
all nonnegative. A set of simple roots of Ψ always exists, and it is unique up to a
transformation in the Weyl group of Ψ.TheWeyl group of Ψ is the subgroup of
orthogonal transformations of a
∗
which is generated by the reflections
s
α
: a
∗
→ a
∗
, x → x −2
x,
α
|
α
|
2
α
,
α
∈ Ψ.
The set
Ψ
+
= {
α
∈Ψ :
α
= c
1
α
1
+ ···+ c
r
α
r
, c
1
,...,c
r
≥ 0}
is called the set of positive restricted roots of Ψ with respect to Λ. For the purpose of
consistency, our choice of simple roots will be the one used in [174].
We now assign to the symmetric space M a diagram consisting of vertices, lines
and arrows. Consider a set Λ = {
α
1
,...,
α
r
} of simple roots of Ψ. To each simple
root
α
i
∈Λ we assign a vertex which we denote b y
if 2
α
i
/∈Ψ and by
if 2
α
i
∈Ψ.
One can show that the angle between two simple roots in Λ is one of the following
four angles:
π
2
,
π
3
,
π
4
,
π
6
.
We connect the vertices corresponding to simple roots
α
i
and
α
j
, i = j,by0,1,2or
3 lines if the angle between
α
i
and
α
j
is
π
2
,
π
3
,
π
4
or
π
6
respectively. Moreover, if the
vertices corresponding to
α
i
and
α
j
are connected by at least one line and
α
i
,
α
i
>
α
j
,
α
j
, we draw an arrow from the vertex
α
i
to the vertex
α
j
. The resulting object
is called the Dynkin diagram associated with M.
We now list the Dynkin diagrams for the irreducible Riemannian symmetric
spaces of noncompact type. In this case the root system is of type (A
r
), (B
r
), (C
r
),
(D
r
), (E
6
), (E
7
), (E
8
), (F
4
), (G
2
) or (BC
r
). We view a root system as a subset of some
Euclidean vector space V .By
δ
we denote the highest root. We also list for each sym-
metric space M the multiplicities (m
α
1
,...,m
α
r
) of the simple roots
α
1
,...,
α
r
and
the centralizer k
0
of a in k (see also [314]).
(A
r
) V = {v ∈ R
r+1
: v,e
1
+ ...+ e
r+1
= 0}, r ≥ 1;
Ψ = {e
i
−e
j
: i = j}; Ψ
+
= {e
i
−e
j
: i < j};
α
1
= e
1
−e
2
,...,
α
r
= e
r
−e
r+1
;
Polar Actions on Symmetric Spaces of Noncompact Type 337
δ
=
α
1
+ ...+
α
r
= e
1
−e
r+1
;
α
1
α
2
α
r−1
α
r
M = SL
r+1
(R)/SO
r+1
: (1,...,1); k
0
= {0};
M = SL
r+1
(C)/SU
r+1
: (2,...,2); k
0
=(u
1
)
r
;
M = SU
∗
2r+2
/Sp
r+1
: (4,...,4); k
0
=(sp
1
)
r+1
;
M = E
−26
6
/F
4
: (8,8); k
0
= so
8
;
M = SO
o
1,n+1
/SO
n+1
(n ≥ 2): (n); k
0
= so
n
;
(B
r
) V = R
r
, r ≥ 2;
Ψ = {±e
i
±e
j
: i < j}∪{±e
i
}; Ψ
+
= {e
i
±e
j
: i < j}∪{e
i
};
α
1
= e
1
−e
2
,...,
α
r−1
= e
r−1
−e
r
,
α
r
= e
r
;
δ
=
α
1
+ 2
α
2
+ ...+ 2
α
r
= e
1
+ e
2
;
α
1
α
2
α
r−2
α
r−1
α
r
+3
M = SO
2r+1
(C)/SO
2r+1
: (2,...,2,2); k
0
=(u
1
)
r
;
M = SO
o
r,r+n
/SO
r
SO
r+n
(n ≥ 1): (1,...,1,n); k
0
= so
n
;
(C
r
) V = R
r
, r ≥ 3;
Ψ = {±e
i
±e
j
: i < j}∪{±2e
i
}; Ψ
+
= {e
i
±e
j
: i < j}∪{2e
i
};
α
1
= e
1
−e
2
,...,
α
r−1
= e
r−1
−e
r
,
α
r
= 2e
r
;
δ
= 2
α
1
+ ...+ 2
α
r−1
+
α
r
= 2e
1
;
α
1
α
2
α
r−2
α
r−1
α
r
ks
M = Sp
r
(R)/U
r
: (1,...,1,1); k
0
= {0};
M = Sp
r
(C)/Sp
r
: (2,...,2,2); k
0
=(u
1
)
r
;
M = Sp
r,r
/Sp
r
Sp
r
: (4,...,4,3); k
0
=(sp
1
)
r
;
M = SU
r,r
/S(U
r
U
r
): (2,...,2,1); k
0
=(u
1
)
r−1
;
M = SO
∗
4r
/U
2r
: (4,...,4,1); k
0
=(su
2
)
r
;
M = E
−25
7
/E
6
U
1
: (8,8,1); k
0
= so
8
;
(D
r
) V = R
r
, r ≥ 4;
Ψ = {±e
i
±e
j
: i < j}; Ψ
+
= {e
i
±e
j
: i < j};
α
1
= e
1
−e
2
,...,
α
r−1
= e
r−1
−e
r
,
α
r
= e
r−1
+ e
r
;
δ
=
α
1
+ 2
α
2
+ ...+ 2
α
r−2
+
α
r−1
+
α
r
= e
1
+ e
2
;
α
1
α
2
α
r−3
α
r−2
α
r−1
α
r
o
o
o
o
o
o
O
O
O
O
O
O
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