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Classification of irreducible symmetric spaces which admit standard compact Clifford--Klein forms.

1. Introduction and main theorem. Let

G be a Lie group, H a closed subgroup of G and [GAMMA] a discrete subgroup of G. If [GAMMA] acts on a homogeneous space G/H properly discontinuously and freely, then the double coset space [GAMMA]\G/H has a natural manifold structure. In this case, the double coset space [GAMMA]\G/H with the manifold structure is called a Clifford Klein form of G/H and [GAMMA] is called a discontinuous group for G/H.

In the late 1980s, a systematic study of Clifford Klein forms for non-Riemannian homogeneous 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, where G [contains] 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 homogeneous space of reductive type and [GAMMA] a discontinuous group for G/H. A Clifford Klein form [GAMMA]\G/H is called standard if there exists a reductive subgroup L containing [GAMMA] and acting on G/H properly.

Some homogeneous spaces admit compact Clifford Klein forms by using continuous analogue as follows:

Fact 1.3 ([Ko89]). Let G/H be a homogeneous 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 [GAMMA]\G/H by taking any torsion-free uniform lattice [GAMMA] of L.

Thus Fact 1.3 gives an affirmative answer to Problem 1.1 for G/H admitting such a reductive subgroup L. Conversely, the following conjecture was proposed by T. Kobayashi.

Conjecture 1.4 ([Ko01, Conjecture 4.3], [KY05, Conjecture 3.3.10]). Let G/H be a homogeneous 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 Conjecture 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, T. Yoshino, T. Okuda, N. Tholozan and Y. Morita among others, see [Ko89], [K090], [Ko92a], [Ko92b], [Z94], [Li95], [Bn96], [LMZ95], [C94], [Ma97], [0W00], [KY05], [Ok13], [Th], [Mo15].

If Conjecture 1.4 were proved to be true, 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.

In the case where G/H is a symmetric space, T. Kobayashi discovered that 5 series and 7 sporadic types of non-Riemannian symmetric spaces admit compact Clifford Klein forms by using Fact 1.3.

Fact 1.6 ([KY05, Corollary 3.3.7]). Symmetric spaces G/H in Table I admit standard compact Clifford Klein forms. Here n = 1, 2, ***.

From now on, we assume that G is a linear noncompact semisimple Lie group. For a Cartan involution [theta] of G, we write g = l + p for the corresponding Cartan decomposition of the Lie algebra g of G and set K := [G.sup.[theta]] = {g [member of] G : [theta](g) = g}. Then K 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 compact Clifford Klein form, then G/H is locally isomorphic to one of the following:

* a Riemannian symmetric space G/K,

* a group manifold G' x G'/ diag G',

* one of the homogeneous spaces in Table I.

2. Outline of proof of Theorem 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 than 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. If G/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:
* SO(p,q + 1)/SO(p,q)           (1 [less than or equal to] q < HR(p)),

* SO(p, q + 1)/SO(p, 1)x SO(q)  (1 [less than or equal to] q < HR(p)),

* SU(2p, 2q)/Sp(p,q)            (1 [less than or equal to]
                                    q [less than or equal to] p),

* [E.sub.6(-14)]/[F.sub.4(-20)].


Here HR(p) denotes the Hurwitz Radon number defined by

HR(p) := 8[alpha] + [2.sup.[beta]],

where [alpha] [member of] [Z.sub.[less than or equal to]0] and [beta] [member of] {0,1, 2, 3} are determined by the following equation

p = [2.sup.4[alpha]+[beta]] x (odd number).

The proof of Proposition 2.1 uses Facts 2.2 and 2.3 below.

Fact 2.2 ([Ko89, Example 4.11]). The symmetric space Sp(2n, R)/Sp(n, C) does not admit compact Clifford Klein forms for any positive integer n.

Fact 2.3 ([KY05]). Let G/H be a homogeneous space of reductive type. If G/H admits a standard compact Clifford Klein form, then its tangential homogeneous space [G.sub.[theta]]/[H.sub.[theta]] admits a compact Clifford Klein form. Here [G.sub.[theta]] and [H.sub.[theta]] are the Cartan motion groups of G and H, respectively, defined by using a Cartan involution [theta] of G such that [theta](H) = H as follows:

[G.sub.[theta]] := K [??] p,

[H.sub.[theta]] := (K [intersection] H) [??] (p [intersection] h).

For tangential homogeneous spaces, a criterion 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.sub.6(-14)]/[F.sub.4(-20)], the method of our proof for Theorem 1.7 is the following:

(A) Kobayashi's criterion for a reductive subgroup L to act properly on G/H [Ko89],

(B) Kobayashi's criterion for a reductive subgroup L to act cocompactly on G/H [Ko89],

(C) an upper bound of the dimension of representations 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 of the action in the setting that G is a linear reductive Lie group and H, L are reductive subgroups of G.

Fact 2.4 (properness criterion, [Ko89, Theorem 4.1]). We fix a Cartan involution of G and take a maximally split abelian subspace a of g. Take Cartan involutions [[theta].sub.1] and [[theta].sub.2] of G such that [[theta].sub.1](H) = H and [[theta].sub.2](L) = L. Take maximal abelian subspaces [mathematical expression not reproducible]. Then we can and do take [S.sub.1];[S.sub.2] [member of] Int(g) such that [alpha]H := [S.sub.1]([a'.sub.H]), [a.sub.L] := S2([a'.sub.L]) [subset] 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.sub.H] [intersection] [a.sub.L] = {0} modulo W-actions.

Here W = W(g, a) is the Weyl group coming from the restricted root system 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.sub.R] L [less than or equal to] [rank.sub.R] G - [rank.sub.R] H.

For the step (B), to state the criterion of the cocompactness, let us recall the noncompact dimension d(G) of a linear reductive Lie group G, which is defined by

d(G) := dim G/K = 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 equivalent:

(i) L\G/H is compact,

(ii) d(G) = d(L) + d(H).

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

[mathematical expression not reproducible].

Suppose there exists such a subgroup L. Let n be the dimension of the natural representation of G. We write [rho] : l [right arrow] sl(n, C) for the differential of the composition L [right arrow] G [subset] GL(n, C). We shall find a numerical necessary condition for the pair (l, [rho]), 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 = z [direct sum] [l.sup.ss] a Levi decomposition, where Z is the center of l and [l.sup.ss] is the semisimple Lie subalgebra. Suppose

[l.sup.ss] := [l.sub.1] [direct sum] ... [direct sum] [l.sub.k]

is the decomposition into noncompact simple ideals labeled as follows:

d([L.sub.i])/[rank.sub.R][L.sub.i] [greater than or equal to] [d([L.sub.i+1])/[rank.sub.R][L.sub.i+1]] (i = 1, ..., k - 1).

We call [l.sub.1] the primary simple factor of l.

We use the primary simple factor [l.sub.1], when [rank.sub.R]G - [rank.sub.R] H [greater than or equal to] 2 because l is not necessarily simple in this case. Then we also use the following inequality:

Lemma 2.8.

d([L.sup.ss])/[rank.sub.R][L.sup.ss] [less than or equal to] d([L.sub.1])/[rank.sub.R] [L.sub.1].

The step (C) concerns an inequality for the primary simple factor [l.sub.1] and any irreducible component [pi] of the restriction [mathematical expression not reproducible]. We illustrate the case G/H = SU(2p, 2q)/Sp(p, q). In this case, we get the following inequalities.

dim [pi] [less than or equal to] dim [rho] [less than or equal to] d([L.sup.ss])/[rank.sub.R][L.sup.ss] [less than or equal to] d([L.sub.1])/[rank.sub.R][L.sub.1].

These inequalities give strong constraints about possible pair ([pi], [l.sub.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 prepare some notations. Let g be a real semisimple Lie algebra. We denote by Irr(g) the set of equivalence classes of irreducible finite dimensional representations of g. Then we define a subset [Irr.sup.c](g) of Irr(g) by

[Irr.sup.c](g) :={[rho] [member of] Irr(g) : [bar.[rho]] [congruent to] [rho]},

where [bar.[rho]] is the complex conjugate representation of [rho]. For a representation [rho] : g [right arrow] sl(n, C) and [pi] [member of] Irr(g), we write [m.sub.[pi]] := dim [Hom.sub.g] ([pi], [rho]). We use the following map given in [I59, [section]9]:

index : [Irr.sup.c](g) [right arrow] {[+ or -]1}.

We apply these notations to real semisimple Lie algebras [g.sup.[sigma].sub.C] instead of g, where [g.sub.C] is a complex semisimple Lie algebra and a is a real structure on gC and write [index.sub.[sigma]][pi] [member of] {[+ or -]1} for n 2 [Irr.sup.c]([g.sup.[sigma].sub.C]).

Proposition 2.9. Let [g.sub.C] be a semisimple Lie algebra over C, [tau] a real structure on [g.sub.C] and [rho] : [g.sup.[tau].sub.C] [right arrow] sl(n, C) a representation. Let m be one of the following Lie algebras sl(n, R), su*(2m), so(n, C) and sp(m, C), where m := [1/2]n for n even. Then the following two conditions on [g.sub.C], [tau] and [rho] are equivalent:

(i) there exists [alpha] [member of] Int(sl(n, C)) such that [alpha]([rho]([g.sup.[tau].sub.C])) [subset] m

(ii) [rho] [congruent to] [bar.[rho]] as a representation of [g.sup.[sigma].sub.C] and [mathematical expression not reproducible] for any [pi] [member of] [Irr.sup.c]([g.sup.[sigma].sub.C]).

Here, [epsilon] [member of] {[+ or -]1} and a are given as follows:
m                    [epsilon]          [sigma]

sl(n, R)                + 1              [tau]
su*(2m)                 -- 1              [tau]
so(n, C)                + 1             [theta]
sp(m, C)                -- 1             [theta]


Here [theta] is a compact real structure on [g.sub.C].

The last step by using Proposition 2.9 gives further constraints on possible triples G, H and L in Question 1.5.

Detailed proof will appear elsewhere.

doi: 10.3792/pjaa.95.11

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 supported by the Program for Leading Graduate Schools, MEXT, Japan.

References

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[Br57] M. Berger, Les espaces symetriques noncompacts, Ann. Sci. Ecole Norm. Sup. (3) 74 (1957), 85 177.

[C94] K. Corlette, Harmonic maps, rigidity, and Hodge theory, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), 465 471, Birkhauser, Basel, 1995.

[G85] W. M. Goldman, Nonstandard Lorentz space forms, J. Differential Geom. 21 (1985), no. 2, 301 308.

[I59] N. Iwahori, On real irreducible representations of Lie algebras, Nagoya Math. J. 14 (1959), 59 83.

[KK16] F. Kassel and T. Kobayashi, Poincare series for non-Riemannian locally symmetric spaces, Adv. Math. 287 (2016), 123 236.

[Ko89] T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), no. 2, 249 263.

[Ko92a] T. Kobayashi, A necessary condition for the existence of compact Clifford Klein forms of homogeneous spaces of reductive type, Duke Math. J. 67 (1992), no. 3, 653 664.

[Ko92b] T. Kobayashi, Discontinuous groups acting on homogeneous spaces of reductive type, in Representation theory of Lie groups and Lie algebras (Fuji-Kawaguchiko, 1990), 59 75, World Sci. Publ., River Edge, NJ, 1992.

[Ko98] T. Kobayashi, Deformation of compact Clifford Klein forms of indefiniteRiemannian homogeneous manifolds, Math. Ann. 310 (1998), no. 3, 395 409.

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[KO90] T. Kobayashi and K. Ono, Note on Hirzebruch's proportionality principle, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 1, 71 87.

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[LMZ95] F. Labourie, S. Mozes and R. J. Zimmer, On manifolds locally modelled on nonRiemannian homogeneous spaces, Geom. Funct. Anal. 5 (1995), no. 6, 955 965.

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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 KASHIWARA, M.J.A., Jan. 15, 2019)

2010 Mathematics Subject Classification. Primary 22F30, 57S30; Secondary 22E46, 53C30, 53C35.
Table I. Symmetric spaces G/H that admit proper and
cocompact actions of reductive subgroups L

                  G/H                   L

1          SU(2, 2n)/Sp(1,n)         U(1, 2n)
2          SU(2, 2n)/U(1,2n)         Sp(1,n)
3           SO(2,2n)/U(1,n)         SO(1, 2n)
4         SO(2, 2n)/SO(1, 2n)         U(1,n)
5          SO(4,4n)/SO(3,4n)         Sp(1,n)
6        SO(4,4)/SO(4,1)x SO(3)     Spin(4,3)
7        SO(4,3)/SO(4,1)x SO(2)       G2(2)
8           SO(8,8)/SO(7,8)         Spin(1,8)
9          SO(8, C)/SO(7, C)        Spin(1, 7)
10          SO(8, C)/SO(7,1)        Spin(7, C)
11           SO*(8)/U (3,1)         Spin(1,6)
12       SO*(8)/SO*(6) x SO*(2)     Spin(1,6)

Table II. Possible pairs of a simple Lie algebra [l.sub.1] and its
irreducible representation [pi] satisfying the above inequality.
Here, [pi] is the standard representation of each l1.

[l.sub.1]                                   dim [pi]    d([L.sub.1])/
                                                         [rank.sub.R]
                                                          [L.sub.1]

sl(n, C) (n [greater than or equal to] 2)       n           n + 1
su*(2n) (n [greater than or equal to] 2)       2n           2n + 1
su(k,l) (2 [less than or equal to] k          k + l           2k
   [greater than or equal to] l [greater
   than or equal to] 1)
so(2n + 1, C) (n > 2)                        2n + 1         2n + 1
so*(4n + 2) (n [greater than or equal        4n + 2         4n + 2
   to] 1)
sp(n, C) (n [greater than or equal to] 2)      2n           2n + 1
sp(k,l) (k [greater than or equal to] l     2(k + l)          4k
   [greater than or equal to] 1)
[g.sup.C.sub.2]                                 7             7
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