Polar Actions on Symmetric Spaces of Noncompact Type 343
is a Cartan decomp osition of the semisimple subalgebra g
Φ
and a
Φ
is a maximal
abelian subspace of b
Φ
.Ifwedefine
(g
Φ
)
0
=(g
Φ
∩k
0
) ⊕a
Φ
=(k
0
z
Φ
) ⊕a
Φ
,
then
g
Φ
=(g
Φ
)
0
⊕
α
∈Ψ
Φ
g
α
is the restricted root space decomposition of g
Φ
with respect to the maximal abelian
subspace a
Φ
of b
Φ
and Φ is a set of simple roots for Ψ
Φ
.
We will now explain how th ese algebraic concepts and constructions relate to
the geometry of Riemannian symmetric spaces of noncompact type. Let M = G/K
be the connected Riemannian symmetric space of noncompact type associated with
the Riemannian symmetric pair (g,k).TheAd(K)-invariant inner product ·,· on
p induces the Ad(G)-invariant Riemannian metric on M = G/K, G is the identity
component of the isometry group of M,andK is a maximal compact subgroup of G.
The Lie algebras of G and K coincide with g and k, respectively. We denote by o ∈M
the unique fixed point of K,thatis,o is the point in M for which the isotropy group
of G at o coincides with K. We identify the subspace p in the Cartan decomposition
g = k ⊕p with the tangent space T
o
M of M at o in the usual way. Th e rank rk(M) of
M is equal to r = |Λ|.
Let Exp : g → G be the Lie exponential map. Then
A = Exp(a) and N = Exp(n)
are simply connected closed subgroups of G with Lie algebras a and n, r espectively.
By construction, A is an abelian Lie group and N is a nilpotent Lie group. The orbit
A ·o is an r-dimensional Euclidean space E
r
which is embedded in M as a totally
geodesic submanifold, and the orbit N ·o is a horocycle in M. The Iwasawa decom-
position g = k ⊕a ⊕n of g indu ces an Iwasawa decomposition
G = KAN
of G. The solvable Lie group AN acts simply transitively on the symmetric space
M = G/K. As a consequence we can realize M = G/K as a solvable Lie group AN
equipped with a suitable left- invariant Riemann ian metric:
M = G/K = AN.
Let Φ be a subset of Λ.Then
A
Φ
= Exp(a
Φ
) and N
Φ
= Exp(n
Φ
)
are simply connected closed subgroups of G with Lie algebras a
Φ
and n
Φ
, respec-
tively. By construction, A
Φ
is an abelian Lie group and N
Φ
is a nilpotent Lie group.
The centralizer
L
Φ
= Z
G
(a
Φ
)={g ∈ G :Ad(g)H = H for all H ∈ a
Φ
}
344 Submanifolds and Holonomy
of a
Φ
in G is a reductive Lie group with Lie algebra l
Φ
and L
Φ
normalizes N
Φ
.The
subgroup
Q
Φ
= L
Φ
N
Φ
is the parabolic subgroup of G associated with Φ. Its Lie algebra is q
Φ
and Q
Φ
can
be characterized as the normalizer
Q
Φ
= N
G
(q
Φ
)={g ∈ G :Ad(g )q
Φ
⊂ q
Φ
}
of q
Φ
in G, which implies that Q
Φ
is a closed subgroup of G.
Let G
Φ
be the connected semisimple subgroup of G with Lie algebra g
Φ
.The
intersection K
Φ
= L
Φ
∩K is a maximal compact subgroup of L
Φ
with Lie algebra k
Φ
.
The adjoint group Ad(L
Φ
) normalizes g
Φ
, and consequently
M
Φ
= K
Φ
G
Φ
is a subgroup of L
Φ
. One can show that M
Φ
is a closed subgroup of L
Φ
, K
Φ
is a
maximal compact subgroup of M
Φ
, and the center Z
Φ
of M
Φ
is a compact subgroup
of K
Φ
. The Lie algebra of M
Φ
is m
Φ
and L
Φ
is isomorphic to the Lie group direct
product M
Φ
×A
Φ
:
L
Φ
= M
Φ
×A
Φ
.
The m ultiplication
M
Φ
×A
Φ
×N
Φ
→ Q
Φ
, (m,a,n) → man
is an analytic diffeomorphism and, via this diffeomorphism, the group structure on
Q
Φ
is given by
(m,a,n)(m
,a
,n
)=(mm
,aa
,(m
a
)
−1
n(m
a
)n
).
The parabolic subgroup Q
Φ
acts transitively on M and the isotropy group at o is K
Φ
,
that is, the symmetric space M is a homogeneous space of Q
Φ
, namely
M = Q
Φ
/K
Φ
.
Recall that g
Φ
=(g
Φ
∩k
Φ
)⊕b
Φ
is a Cartan decomposition of the semisimple Lie
algebra g
Φ
. This implies that
[b
Φ
,b
Φ
]=g
Φ
∩k
Φ
and G
Φ
is the connected closed subgroup of G with Lie algebra [b
Φ
,b
Φ
] ⊕b
Φ
.Since
b
Φ
is a Lie triple system in p, the orbit
B
Φ
= G
Φ
·o = G
Φ
/(G
Φ
∩K
Φ
)
of the G
Φ
-action on M containing o is a connected totally geodesic submanifold of M
with T
o
B
Φ
= b
Φ
.IfΦ = /0, then B
Φ
= {o},otherwiseB
Φ
is a Riemannian symmetric
Polar Actions on Symmetric Spaces of Noncompact Type 345
space of noncompact type and rank(B
Φ
)=|Φ|. The reductive Lie group M
Φ
also acts
transitively on B
Φ
,sothat
B
Φ
= M
Φ
·o = M
Φ
/K
Φ
,
but this action is not (almost) effective in general. The submanifold B
Φ
also appears
in the context of the maximal Satake compactification of M and is sometimes called
a boundary component of M (see, e.g., [48]).
Since a
Φ
is an abelian Lie triple system in p, the corresponding totally geodesic
submanifold of M is a Euclidean space
E
r−|Φ|
= A
Φ
·o.
Finally, p
Φ
= b
Φ
⊕a
Φ
is a Lie triple system in p and the corresponding totally
geodesic submanifold F
Φ
of M is the symmetric space
F
Φ
= L
Φ
·o = L
Φ
/K
Φ
=(M
Φ
×A
Φ
)/K
Φ
= B
Φ
×E
r−|Φ|
. (13.2)
The submanifolds F
Φ
and B
Φ
have a nice geometric interpretation. Choose Z ∈ a
such that
α
(Z)=0forall
α
∈ Φ and
α
(Z) > 0forall
α
∈ Λ Φ. Then consider the
geodesic
γ
Z
(t)=Exp(tZ)·o in M with
γ
Z
(0)=o and
˙
γ
Z
(0)=Z. The totally geodesic
submanifold F
Φ
is the union of all geodesics in M which are parallel to
γ
Z
and B
Φ
is the semisimple part of F
Φ
in the de Rham decomposition of F
Φ
(see, e.g., [118],
Proposition 2 .11.4 and Proposition 2.20.10).
The nilpotent Lie group N
Φ
acts freely on M so that we can identify N
Φ
with its
orbit through o,
N
Φ
= N
Φ
·o.
The following result is an important application of the Langlands decomposition
of a p arabolic subalgebra.
Theorem 13.2.2 (Horospherical decomposition of a symmetric space) The ana-
lytic diffeomorphism M
Φ
×A
Φ
×N
Φ
→ Q
Φ
induces an analytic diffeomorphism
B
Φ
×A
Φ
×N
Φ
→ M,(m ·o,a,n) → (man) ·o.
The action of Q
Φ
on M is given by
Q
Φ
×M → M, ((m,a,n),(m
·o,a
,n
)) → ((mm
) ·o,aa
,(m
a
)
−1
n(m
a
)n
).
The analytic diffeomorphism B
Φ
×A
Φ
×N
Φ
→ M is known as a horospherical
decompositio n of M. This concept generalizes horocyclic coordinates on the real
hyperbolic plane. We will now discuss these concepts in some examples.
Example 13.2.1 (The symmetric space SL
r+1
(R)/SO
r+1
) For the symmetric space
M = G/K = SL
r+1
(R)/SO
r+1
346 Submanifolds and Holonomy
we have rk(M)=r and dim M =
1
2
r(r + 3). We denote by GL
r+1
(R) the general
linear group of R
r+1
.Then
G = SL
r+1
(R)={A ∈ GL
r+1
(R) :det(A)=1}
and
K = SO
r+1
= {A ∈ SL
r+1
(R) : A
t
= A
−1
}
is a maximal compact subgroup of SL
r+1
(R). The Lie algebra of GL
r+1
(R) is
gl
r+1
(R)
∼
=
Mat
r+1,r+1
(R)
with Lie bracket [X ,Y ]=XY −YX.TheLiealgebrasofG = SL
r+1
(R) and K =
SO
r+1
are given by
g = sl
r+1
(R)={X ∈ gl
r+1
(R) :tr(X)=0}
and
k = so
r+1
= {X ∈ sl
r+1
(R) : X
t
= −X},
respectively. The Cartan involution
θ
∈ Aut(g) is given by
θ
: sl
r+1
(R) → sl
r+1
(R) , X →−X
t
.
The (−1)-eigenspace of
θ
is
p = {X ∈ sl
r+1
(R) : X
t
= X },
and so the Cartan decom position
g = sl
r+1
(R)=so
r+1
⊕p = k ⊕p
corresponds to the decomposition of a matrix X into its skewsymmetric and symmet-
ric part.
For a =(a
1
,...,a
r+1
) ∈ R
r+1
with a
1
+ ...+ a
r+1
= 0wedefine
Δ(a)=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a
1
0 ··· 00
0 a
2
.
.
.
00
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00
.
.
.
a
r
0
00··· 0 a
r+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
∈ sl
r+1
(R).
Then
a = {Δ(a) : a ∈ R
r+1
, a
1
+ ...+ a
r+1
= 0}
is a maximal abelian subspace of p. We put
e
1
= Δ(1,0,...,0),e
2
= Δ(0,1,0,...,0),...,e
r+1
= Δ(0,...,0,1)
Polar Actions on Symmetric Spaces of Noncompact Type 347
and denote by
ε
1
,...,
ε
r+1
∈ a
∗
their dual vectors. Then
ε
i
−
ε
j
for i = j and i, j ∈
{1,...,r + 1} is a restr icted root of g = sl
r+1
(R) with corresponding root space
g
ε
i
−
ε
j
= {xE
ij
: x ∈ R},
where E
ij
∈ gl
r+1
(R) is the matrix with (E
ij
)
ij
= 1 and zero everywhere else. Thus,
we have
Ψ = {
ε
i
−
ε
j
: i = j, i, j ∈{1,...,r + 1}}
and
g
0
= a.
The restricted root space decomposition of sl
r+1
(R) therefore is
sl
r+1
(R)=a ⊕
⎛
⎜
⎜
⎝
r+1
i, j=1
i= j
RE
ij
⎞
⎟
⎟
⎠
.
The set Λ = {
α
1
,...,
α
r
} with
α
i
=
ε
i
−
ε
i+1
, i ∈{1,...,r},
is a set of simple roots for Ψ and
Ψ
+
= {
ε
i
−
ε
j
: i < j, i, j ∈{1,...,r + 1}}
is the induced set of positive roots. The nilpotent Lie algebra n is
n =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 x
12
x
13
··· x
1,r+1
00x
23
··· x
2,r+1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00 0
.
.
.
x
r,r+1
00 0··· 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
: x
ij
∈ R
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
.
The Iwasawa decom position of sl
r+1
(R) therefore describes the unique decompo-
sition of a matrix in sl
r+1
(R) into the sum of a skewsymmetric matrix, a diagonal
matrix with trace zero, and a strictly upper triangular matrix. The subgroup N of
SL
r+1
(R) with Lie algebra n is
N =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 x
12
x
13
··· x
1,r+1
01x
23
··· x
2,r+1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00 0
.
.
.
x
r,r+1
00 0··· 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
: x
ij
∈ R
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
.
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