Classification of irreducible symmetric spaces which admit
standard compact Clifford–Klein forms
By Koichi TOJO
Graduate School of Mathematical Science, The University of Tokyo,
3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
(Communicated by Masaki K ASHIWARA, M. J.A., Jan. 15, 2019)
Abstract: We give a cl assification o f irr educible symmetric s paces which admit standard
compact Clifford–Klein forms. The method uses r epresentation theory over the real number field
and the criterion for properness and cocompactness of the action on homogeneous spaces due to
T. Kobayashi.
Key words: Clifford–Klein form; tangential homogeneous space; symmetric space;
properness criterion; discontinuous group; real representation.
1. Introduction and main theorem. Let
G be a Lie group, H a closed subgroup of G and a
discrete subgroup of G.If acts on a homogeneous
space G=H properly discontinuously and freely,
then the double coset space nG=H has a natural
manifold structure. In this case, the double coset
space n G=H with the manifold structure is called
a Clifford–Klein form of G=H and is called a
discontinuous group for G=H.
Inthelate1980s,asystematicstudyof
Clifford–Klein forms for non-Riemannian homoge-
neous spaces was initiated by T. Kobayashi. One of
important problems in the field is the following:
Problem 1.1 ([Ko89], [Ko01, Problem B]).
Which homogeneous space G=H admits a compact
Clifford–Klein form?
Our interest is in the case where G=H is of
reductive type,namely,whereG H are both real
reductive linear Lie groups. In this case, G=H
carries naturally a pseudo-Riemannian structure for
which G acts isometrically.
Not every homogeneous space G=H carries a
large discontinuous group when H is noncompact
because the action of an isometric discrete group
is not necessarily properly discontinuous. A useful
method to construct large discontinuous groups is
to use continuous analogue of discontinuous groups.
Definition 1.2 (standard Clifford–Klein form
[KK16, Definition 1.4]). Let G=H be a homoge-
neous space of reductive type and a discontinuous
group for G=H.AClifford–Kleinform
nG=H is
called standar d if there exists a reductive subgroup
L containing and acting on G=H properly.
Some homogeneous spaces admit compact
Clifford–Klein forms by using continuous analogue
as follows:
Fact 1.3 ([Ko89]). Le t G=H be a homoge-
neous space of reductive type. If there exists a
reductive subgroup L of G acting on G=H properly
and cocompactly, then G=H admits a standard
compact Clifford–Klein form nG=H by taking any
torsion-free uniform lattice of L.
Thus Fact 1.3 gives an affirmative answer t o
Problem 1.1 for G=H admitting such a reductive
subgroup L. Conversely, t he following conjecture
was proposed by T. Kobayashi.
Conjecture 1.4 ([Ko01, Conjecture 4 .3],
[KY05, Conjecture 3.3.10]). Let G=H be a homo-
geneous space of reductive type. If G=H admits a
compact Clifford–Klein form, then G=H admits a
standard compact Clifford–Klein form.
No counterexample to C onjecture 1.4 has been
known so far. (We remark that not all compact
Clifford–Klein forms are standard: it may happen
that a deformation of a standard compact Clifford–
Klein form yields a nonstandard compact Clifford–
Klein form [ G85], [Ko98].) On the other hand, many
evidences supporting the conjecture have been
obtained by T. Kobayashi, K. Ono, R. J. Zimmer,
R. Lipsman, Y. Benoist, F. Labourie, K. Corlette,
S. Mozes, G. A. Margulis, H. Oh, D. Witte,
doi: 10.3792/pjaa.95.11
#2019 The Japan Academy
2010 Mathematics Subj ect Classification. Primary 22F30,
57S30; Secondary 22E46, 53C30, 53C35.
No. 2] Proc. J apan Acad., 95,Ser.A(2019) 11
T. Yoshino, T. Okuda, N. Tholozan and Y. Morita
among others, see [Ko89], [KO90], [Ko92a],
[Ko92b],[Z94],[Li95],[Bn96],[LMZ95],[C94],[Ma97],
[OW00], [KY05], [Ok13], [Th], [Mo15].
IfConjecture1.4wereprovedtobetrue,the
answer to the following question would give a
solution to Problem 1.1.
Question 1.5. Classify homogeneous spaces
G=H of reductive type which admit standard
compact Clifford–Klein forms.
The goal of this paper is to give an answer to
Question 1.5 for irreducible symmetric spaces G=H.
InthecasewhereG=H is a symmetric space,
T. Kobayashi discovered that 5 series and 7 spor adic
types of non-Riemannian symmetric spaces admit
compact Clifford–Klein forms by using Fact 1.3.
Fact 1.6 ([KY05, Corollary 3.3.7]). Sym-
metric spaces G=H in Table I admit standard
compact Clifford–Klein forms. Here n ¼ 1; 2; .
From now on, we assume that G is a linear
noncompact semisimple Lie group. For a Cartan
involution of G,wewriteg ¼ k þ p for the
corresponding Cartan decomposition of the Lie
algebra g of G and set K :¼ G
¼fg 2 G : ðgÞ¼
gg.ThenK is a maximal compact subgroup of G
and G=K is a Riemannian symmetric space.
Theorem 1.7. Let G be a linear noncompact
semisimple Lie group and G=H an irreducible
symmetric space. If G=H admits a standard com-
pact Clifford–K lein form, then G=H is locally
isomorphic to one of the following:
. a Riemannian symmetric space G=K,
. a group manifold G
0
G
0
= diag G
0
,
. one o f the h omo geneo us sp aces in Table I.
2. Outline of pro of of Theore m 1 .7.
Among all irreducible symmetric spaces [Br57], we
prove that there are not many candidates for
irreducible symmetric spaces G=H that admit
standard compact Clifford–Klein forms other t han
those listed in Theorem 1.7. This statement is
formulated as follows:
Proposition 2.1. Let G be a simple Lie
group and G=H a symmetric space with noncompact
H.IfG=H admits a standard compact Clifford–
Klein form, then G=H is locally isomorphic to one of
the homogeneous spaces listed in Table I or in the
following symmetric spaces:
SOðp; q þ 1 Þ=SOðp; qÞð1 q<HRðpÞÞ;
SOðp; q þ 1 Þ=SOðp; 1ÞSOðqÞð1 q<HRðpÞÞ;
SU ð2p; 2qÞ=Spðp; qÞð1 q pÞ;
E
6ð14Þ
=F
4ð20Þ
:
Here HRðpÞ denotes the Hurwitz–Radon number
defined by
HRðpÞ :¼ 8 þ 2
;
where 2 Z
0
and 2f0; 1; 2; 3g are determined by
the following equation
p ¼ 2
4þ
ðodd numberÞ:
The proof of Proposition 2.1 uses Facts 2.2 and
2.3 bel o w.
Fact 2.2 ([Ko89, Example 4.11]). The sym-
metric space Spð2n; RÞ=Spðn; CÞ does not admit
compact Clifford–Klein forms for any positive
integer n.
Fact 2.3 ([KY05]). Le t G=H be a homoge-
neous space of reductive type. If G=H admits a
standard compact Clifford–Klein form, then its
tangential homogeneous space G
=H
admits a
compact Cliffor d–Kle in form. Here G
and H
are
theCartanmotiongroupsofG and H, respectively,
defined by using a Cartan involution of G such
that ðHÞ¼H as follows:
G
:¼ K n p;
H
:¼ðK \ HÞ n ðp \ hÞ:
For tangential homogeneous spaces, a cri terion
for the existence of compact Clifford–Klein forms
was obtained in [KY05].
For the three families of symmetric spaces in
Proposition 2.1 except for E
6ð14Þ
=F
4ð20Þ
,themeth-
od of our proof for Theorem 1.7 is the following:
Table I. Symmetric spac es G=H that admit proper and
cocompact actions of reductive subgroups L
G=H L
1 SU ð2; 2nÞ=Spð1;nÞ Uð1; 2nÞ
2 SU ð2; 2nÞ=Uð1; 2nÞ Spð1;nÞ
3 SOð2; 2nÞ=Uð1;nÞ SOð1; 2nÞ
4 SOð2; 2nÞ=SOð1; 2nÞ Uð1;nÞ
5 SOð4; 4nÞ=SOð3; 4nÞ
Spð1;nÞ
6 SOð4; 4Þ=SOð4; 1ÞSOð3Þ Spinð4; 3Þ
7 SOð4; 3Þ=SOð4; 1ÞSOð2Þ G
2ð2Þ
8 SOð8; 8Þ=SOð7; 8Þ Spinð1; 8Þ
9 SOð8; CÞ=SOð7; CÞ Spinð1; 7Þ
10 SOð8; CÞ=SOð7; 1Þ Spinð7; CÞ
11 SO
ð8Þ=Uð3; 1Þ Spinð1; 6Þ
12 SO
ð8Þ=SO
ð6ÞSO
ð2Þ Spinð1; 6Þ
12 K. TOJO [Vol. 95(A),
(A) Kobayashi’s criterion f or a reductive subgroup
L to act properly on G=H [Ko89],
(B) Kobayashi’s criterion for a reductive s ubgroup
L to act cocompactly on G=H [Ko89],
(C) a n upper bound of the dimension of represen-
tations of the ‘‘primary simple factor’’ of L,
(D) a generalization of Iwahori’s criterion for finite
dimensional representations of real semisimple
Lie algebras to admit certain structures (see
Proposition 2.9).
The step (A) concerns the criterion for properness
oftheactioninthesettingthatG is a linear
reductive Lie group and H, L are reductive
subgroups of G.
Fact 2.4 (properness criteri on, [ Ko89, Theo-
rem 4.1]). We fix a Cartan involution of G and
take a maximally split abelian subspace a of g.Take
Cartan involutions
1
and
2
of G such that
1
ðHÞ¼
H and
2
ðLÞ¼L. Take maximal abelian subspaces
a
0
H
h
1
and a
0
L
l
2
. Then we can and do
take S
1
;S
2
2 IntðgÞ such that a
H
:¼ S
1
ða
0
H
Þ; a
L
:¼
S
2
ða
0
L
Þa. Then the following two conditions on
the triple G, H and L are equivalent:
(i) the natural action of L on G=H is proper,
(ii) a
H
\ a
L
¼f0g modulo W -actions.
Here W ¼ W ðg; aÞ is the Weyl group coming from
the restricte d root syst em of g with the maximally
split abelian subspace a of g.
Remark 2.5 ([Ko89, Corollary 4.2]). If the
L-action on G=H is proper, then the following
inequality holds:
rank
R
L rank
R
G rank
R
H:
For the step (B), to state the crit erion of the
cocompactness, let us recall the noncompact di-
mension dðGÞ of a linear reductive Lie group G,
whichisdefinedby
dðGÞ :¼ dim G=K ¼ dim p:
Fact 2.6 (cocompactness criterion, [Ko89,
Theorem 4.7]). In the same setting, under the
assumption that the L-action on G=H is proper, the
following two conditions on the triple G, H and L
are equival ent:
(i) LnG=H is compact,
(ii) dðGÞ¼dðLÞþdðHÞ.
Owing to Facts 2.4 and 2.6, Question 1.5 is
reduced to classifying G=H that admits a reductive
subgroup L of G such that
a
L
\ a
H
¼f0g modulo W-actions
dðLÞ¼dðGÞdðHÞ:
Suppose there exists such a subgroup L.Letn be
the dimension of the natural representation of G.
We write : l ! slðn; CÞ for the differential of the
composition L ! G GLðn; CÞ. We shall find a
numerical necessary condition for the pair (l, ),
namely, find an upper bound of the dimension of
representations of the ‘‘primary simple factor’’ of L.
Definition 2.7. Let l be a reductive Lie
algebra and l ¼ z l
ss
a Levi decompositi on, wher e
z is the center of l and l
ss
is the semisimple Lie
subalgebra. Suppose
l
ss
:¼ l
1
l
k
is the decomposition into noncom pact simple ideals
labeled as follows:
dðL
i
Þ
rank
R
L
i
dðL
iþ1
Þ
rank
R
L
iþ1
ði ¼ 1; ;k 1Þ:
We call l
1
the primary simple factor of l.
We use the primary simple factor l
1
,when
rank
R
G rank
R
H 2 because l is not necessar ily
simple in this case. Then we also use the following
inequality:
Lemma 2.8.
dðL
ss
Þ
rank
R
L
ss
dðL
1
Þ
rank
R
L
1
:
The step (C) concerns an inequality for the
primary simpl e factor l
1
and any irreducible com-
ponent of the restriction j
l
1
.Weillustratethe
case G=H ¼ SUð2p; 2qÞ=Spðp; qÞ.Inthiscase,we
get the following inequalities.
dim dim
dðL
ss
Þ
rank
R
L
ss
dðL
1
Þ
rank
R
L
1
:
These inequalities give strong constraints about
possible pair (, l
1
) (see Table II), which is useful
for the classification of the triple G, H and L in
Question 1.5.
For the last step (D), we prepar e som e nota-
tions. Let g be a real semisimple Lie algebra. We
denote by IrrðgÞ the set of equivalence classes of
irreducible finite dimensional representations of g.
Then we define a subset Irr
c
ðgÞ of IrrðgÞ by
Irr
c
ðgÞ :¼f 2 IrrðgÞ : ’ g;
where
is the complex conjugate representation of
No. 2] Classification of irreducible symmetric spaces which admit standard compact Clifford–Klein forms 13
. For a representation : g ! slðn; CÞ and 2
IrrðgÞ,wewritem
:¼ dim Hom
g
ð; Þ.Weusethe
following map given in [I59, §9]:
in dex : Irr
c
ðgÞ!f1g:
We apply these notations to real semisimple Lie
algebras g
C
instead of g,whereg
C
is a complex
semisimple Lie algebra and is a real s tructure on
g
C
and write index
2f1g for 2 Irr
c
ðg
C
Þ.
Proposition 2.9. Let g
C
be a semisimple Lie
algebra over C, a real structu re on g
C
and
: g
C
! slðn; CÞ a representation. Let m be one of
the fo llowing Lie alg ebras slðn; RÞ, su
ð2mÞ, soðn; CÞ
and spðm; CÞ,wherem :¼
1
2
n for n even. Then
the following two conditions on g
C
, and are
equivalent:
(i) there exists 2 Intðslðn; CÞÞ such that
ððg
C
ÞÞ m,
(ii) ’
as a representation of g
C
and
ð" index
Þ
m
¼ 1 for any 2 Irr
c
ðg
C
Þ.
Here, " 2f1g and are given as follows:
m "
slðn; RÞ +1
su
ð2mÞ1
soðn; C Þ +1
spðm; CÞ1
Here is a compac t real structure on g
C
.
The last step by using Proposition 2.9 gives
further constraints on possible t riples G, H and L in
Question 1.5.
Detailed proof will appe ar els ewher e.
Acknowledgements. The author would like
to thank his supervisor Dr. Taro Yoshino for many
constructive comments. The author would like to
thank Dr. Takayuki Okuda for helpful suggestions
on exceptional Lie algebras. This work was sup-
ported by the Program for Leading Graduate
Schools, M EXT, Japan.
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No. 2] Classification of irreducible symmetric spaces which admit standard compact Clifford–Klein forms 15