Using the complexi ty of cons t ruc t ion of the sequence (~Sf~ : a ~ ~/) (Sec. 6) and the re la t ion of analyt ic defin- abi l i ty with ~ -def inabi l i ty , we see that a and b a r e ~ in L[a, b].

Using the analogous c h a r a c t e r i s t i c of ~ - g e n e r i c points, the uniqueness of the ~ -gene r i c point a in 1 L[a, b], and the complex i ty of cons t ruc t ion of the sequence ( ~ : a ~ c0~ ~) (Sec. 4), we see that a is A n in L[a, b].

The proof of T h e o r e m 1 is comple te .

In conclusion, the author e x p r e s s e s grat i tude to V. A. Uspenski i for a s s i s t e n c e with the a r t i c l e and V. G. Kanovei for useful advice.

L I T E R A T U R E C I T E D

1. H. R o g e r s , J r . , Theory of H e c u r s i v e Functions and Effec t ive Computabi l i ty , McGraw-Hi l l , New York (1967).

2. G . E . Sacks , "Forc ing with pe r fec t c losed se t s , " P roc . Syrup. Pure Math., 1__33, No. 1, 331-357 (1971). 3. H . B . Jensen , "Definable se ts of min imal deg ree , " in: Math. Logic and Foundation of Set Theory , Nor th-

Holland, A m s t e r d a m (1970), pp. 122-128. 4. V . G . Kanovei , Definabil i ty with the Help of Degrees of Const ruct ib i l i ty . Invest igat ions in Set Theory and

Nonclass ica l Logics [in Russ ian] , Nauka, Moscow (1976). 5. B . L . Budinas, "Th ree l inear ly o rde r ed degrees of A~ n u m b e r s , " Vestn. Mosk. Gos. Univ., No. 5, 3-6

(1979). 6. V . G . Kanovei, "On nonempt iness of c l a s s e s in the ax iomat ic theory of s e t s , " Izv. Akad. Nauk SSSR,

Ser . Mat. , 42, No. 3, 550-579 (1978). 7. P . J . Cohen, Set Theory and Continuum Hypothes is , W. A. Benjamin, New York (1966). 8. K. Devlin, Aspec t s of Cons t ruc t ib i l i ty , S p r i n g e r - V e r l a g , Ber l in (1973). 9. R . B . J ensen and H. Johnsb ra t ea , "A new cons t ruc t ion of a nonconst ruct ib le A~ subse t of ~ ," Fundamen.

Math. , 8._!1, 279-290 (1974). 10. R . B . Jensen and K. Karp, " P r i m i t i v e r e c u r s i v e se t f imctions," P roc . Symp. Pure Math., 1.~3, No. 1, 143-

176 (1971). 11. T. Jech , Lec tu res in Set Theory with P a r t i c u l a r Emphas i s on the Method of Forc ing , Lect . Notes Math.,

Vol. 217, S p r i n g e r - V e r l a g , B e r l i n - N e w York (1971).

C O M P A C T H O M O G E N E O U S

F U N D A M E N T A L G R O U P

V. V. G o r b a t s e v i c h

S P A C E S W I T H S E M I S I M P L E

UDC 519.4

I n t r o d u c t i o n

In this paper we cons ider the p rope r t i e s of a c l a s s of compac t homogeneous spaces . If M is an a r b i t r a r y compac t homogeneous space , then for some finitely shee ted cover ing space M' we have the following d i a g r a m of f ibra t ions .

Mr

Mr -+ M' -+Ma

M,

Here M c ~ M' ~ Ma is the na tura l f ibra t ion for M [1], and the f iber M c is a homogeneous space c o r r e - sponding to a s e m i s i m p l e Lie group, while the base manifold M a is a sphe r i ca l [i .e. , ~i(Ma) = 0 for i -> 2]. The ve r t i ca l a r r o w s give the s t r u c t u r e bundle M r --- Ma ~ Ms for Ma; he re the f iber Mr is a so lvemani fo ld (i.e., the homogeneous space of a so lvable Lie group) and the base M s is d i f feomorphic to a local ly s y m m e t r i c R i e - mannian space of negat ive cu rva tu re [2]. The manifolds Mr and M s (called the solvable and s e m i s i m p l e c o m - ponents for M) a r e uniquely de te rmined by M up to a finite cove r , and the s imply connected component ~ is

Moscow Aviat ion Technology Inst i tu te , Moscow. T r a n s l a t e d f r o m Sibi rsk i i Matemat ichesk i i Zhurna l , Vol. 22, No. 1, pp. 47-67, J a n u a r y - F e b r u a r y , 1981. Original a r t i c l e submi t t ed Apr i l 19, 1979.

34 0037-4466/81/2201- 0034 $07.50 �9 1981 Plenum Publishing Corpora t ion

determined up to homotopy equivalence (Mc is the universal covering manifold from M c) [i, 2]. If one of the two components Mr or M s is trivial for M, we have some natural classes of compact homogeneous spaces: if

M c = * (point) then M = M a is an aspherical homogeneous space (concerning which~ cf. [2, 3]); when M s = *,

lh(M') is solvable (cf. [2, 4, 5] and elsewhere for this case). In this paper we consider the class of those com- pact M such that the solvable component M r is trivial. It turns out that such M are characterized by the fact that ~i (Iv[) has no infinite solvable normal subgroups (it is natural to call such groups '~i (M) semisimple - cf. Sec. 1 and [I]).

It follows from Theorem 2.2 proved in Sec. 2 that if M is compact and homogeneous and ~1 (M) is s e m i - s imple, then some semis imple Lie group acts t ransi t ively on M (although groups which are not semis imp!e may in general also act t ransi t ively on M). In this paper, we descr ibe in detail the t ransi t ive actions of semi - simple Lie groups on compact manifolds M with semis imple ~h(M) as well as the s t ruc tu re of alI compact homo- geneous M having a given semis imple ~I(M) = ,~.

If dim Ma = 2, then either ~(M) contains a subgroup of finite index isomorphic to Z 2 (andthen Mis finitely covered by an M' such that the base of the natural bundle for M' is the torus T2), or else, as shown in Theorem 5.1, Ma = Fg, an orientable surface of genus g _> 2, and ,xl(M ) is semis imple . The s t ruc ture of compact homo- geneous spaces M for which M a = Fg is studied in detail.

Some of the resul ts of this paper were stated in [1, 2].

As for notation and conventions, Lie groups are denoted by capital Latin le t te rs , their Lie algebras by the corresponding lower -case Gothic symbols . If H is a Lie group, then H 0 is the connected component con- taining the identity, and ~0(H) = H / H 0. The normal i ze r of a subgroup H in G is denoted by NG(H) , and the cen- ter of G by Z(G). We denote by Ad G :G --GL(g)the adjoint action of the Lie group G. Following [6, 7], we define a t -subgroup (subaigebra) to be a subgroup of G (subalgebra of ~ ) containing a maximal connected t r iangular subgroup T in G (subalgebra t in g ). A subgroup H of a Lie group G is said to be uniform if H is closed and G / H is compact; a discrete uniform subgroup is called a lattice. Fibrations of a manifold M over a base M ~ with fiber M' (smooth, locally trivial) are denoted by M' -- M -- M ", and ~I denotes the universal covering manifold of M.

The author is grateful to A. L. Onishehik for helpful discussions of the results in this paper.

I . R a d i c a l and S e m i s i r n p l i c i t y of a Group

In this section we consider a number of group-theoretic concepts and results which will then be used to study the fundamental group of a compact homogeneous space.

,When studying certain questions involving infinite groups [in particular, when studying ~I(M) for homo- geneous M], it turns out to be convenient and useful to take two groups [l s and i] 2 to be close in some sense if they satisfy one of the following two relations:

(i) ffl and [I 2 are commensurable (i.e., there exist subgroups fl~ of finite index in [li, i = i, 2 such that ?

[Ii is isomorphic to [I~).

(ii) There exists an epimorphism ~: [I s ~/]2 (or ~: [l 2 --fll) with finite kernel.

For example, if the manifolds M s and M 2 are finite covers of some Mo, then ~s(Ml) and ~I(M 2) are corn- P

mensurable. Furthermore, if M'-r. M-+ M" is a fibration such that M ~ is aspherical and M ' is connected, with ~I (M') finite (all these conditions hold, e.g., for the natural fibration M c -~ M ~ Ma of a compact homogeneous M, cf. [I]), then one sees easily that p. : ~i(M) -- ~1(Ma) is an epimorphism with finite kernel.

Relation (i) (commensurability) is an equivalence in the class of all groups, while condition (ii) is not. In the class of all groups, we consider the smallest equivalence relation containing relations (i) and (ii); we call it the weak commensurability relation and denote it by ~. It is not hard to give an explicit description of the rela- tion ~.

Proposition 1.1. H I ~ ]]2 if and only if the groups H i contain subgroups H~ of finite index in Hi, i = i , 2, ?

and there exist epimorphisms ~i : Hi -- H onto the same group [I such that the kernels ker ~i are finite.

Proof. We denote by ~ the relation between groups given in the statement of the proposition. It is clear

that ~ is reflexive and symmetric. It is not hard to verify that it is also transitive, and therefore ~ is an equiv- alence relation.

35

It is c l e a r f r o m the def ini t ion of ~ that it conta ins r e l a t ions (i) and (ii) and is gene ra t ed by them. But

then the min ima l i ty of the r e l a t i on ~ impl ies that it co inc ides with - , which proves the propos i t ion .

R e m a r k 1. Le t ill, 112 be two t o r s i o n - f r e e g roups . If H 1 ~ H2, then k e r r i = {e~ (the ~i a r e the e p i m o r - ph i sms in P r o p o s i t i o n 1.1), and we ge t the r e s u l t that weak c o m m e n s u r a b i l i t y is equivalent to c o m m e n s u r a b i l i t y fo r t o r s i o n - f r e e g roups .

2. If M c -* M' - - Ma is the na tu ra l f ib ra t ioa for a c o m p a c t homogeneous M (M' a finite c o v e r of lV[, Ma a s p h e r i c a l , and 7rl(M c) f in i te , cf. [1]), then c l e a r l y ~ri(M) ~ 7rl(Ma).

3. If iI~ ~ l ] 2 and ll 1 is finite then so is 1] 2.

We now turn to the not ion of the r a d i c a l of a g roup .

Definit ion. The r a d i c a l r(lI) of the g roup H is the l a r g e s t so lvable n o r m a l subgroup of H (if it exis ts ) .

We r e c a l l that a g roup is ca l led loca l ly so lvab le if e v e r y f ini tely g e n e r a t e d subgroup is so lvable . It is c l e a r that the p roduc t of al l so lvable n o r m a l subgroups of H is a loca l ly so lvable n o r m a l subgroup , which we cal l the loca l r a d i c a l of H and denote by lr(H). The fol lowing r e s u l t is obvious.

P r o p o s i t i o n 1.2. If e v e r y loca l ly so lvable n o r m a l subgroup of H is so lvab le , then the r ad i ca l r(H) exis t s and lr(1]) = r(l l) .

M a l ' t s e v [8] p roved that if e v e r y Abel ian subgroup F of 1] is a g roup of type A 3 (it is i m p o r t a n t fo r our p u r p o s e s that condi t ion A 3 is weake r than the condi t ion A 5 that F be f ini te ly genera ted) , then eve ry loca l ly so lvab le subgroup of I] is so lvable and r([I) ex i s t s . F r o m this we obtain an obvious c o r o l l a r y .

COROLLARY 1.3. If e v e r y Abel ian subgroup of i~ is f in i te ly g e n e r a t e d , then the r ad i ca l r(1]) exis ts (and is a po lycyc l i c g roup by [9]).

In this pape r we make r e p e a t e d use of the fol lowing r e s u l t (and e spec ia l ly P r o p o s i t i o n 1.6, which follows f r o m it).

THEOREM 1.4. If II 1 ~ II2, then r([i 1) ~ r(lI 2) [in pa r t i cu l a r , r(H l) exis ts if and only if r(1] 2) does] .

P roo f . By the defini t ion of weak c o m m e n s u r a b i l i t y , it su f f i ces in the p roof to cons ide r the two ca se s :

(i) H 1 and [12 a r e c o m m e n s u r a b l e .

(ii) T h e r e ex is t s an e p i m o r p h i s m ~ : 111 - - [12 with finite kerne l .

F o r c a s e (i), the p roof of T h e o r e m 1.4 follows eas i ly f r o m the fol lowing r e s u l t , which is m o r e p r e c i s e than T h e o r e m 1.4.

LEMMA 1.5. If [11 is a subgroup of finite index in i12, then r(1]l) exis ts if and only if r(ll 2) ex i s t s , and HI N r(1]2) is a n o r m a l subgroup of finite index in r(H1) [ andof finite index in r([12)].

P roof . We f i r s t a s s u m e that H 1 is n o r m a l in 112.

A s s u m e r(II 2) ex i s t s . We p rove that i r (ill) co inc ides with the p roduc t of those so lvable n o r m a l subgroups of [11 which a r e a l so n o r m a l in ~I 2 .

The index of 112 in 1] 2 is f ini te; let g,, . . . , g ~ H ~ be r e p r e s e n t a t i v e s for all the eose t c l a s s e s in ft2/[11. If F is a so lvable n o r m a l subg roup of H l, then [g~-l.F.g~} (t ~< i ~<n) is a finite se t of so lvable n o r m a l subgroups of H i (since II, <~ II2), and we wr i t e �9 for the i r product . Then -F is a so lvable n o r m a l subgroup of II 2 and F ~ F-~ 11 s. T h e r e f o r e , the p roduc t of al l so lvable n o r m a l subgroups F of the g roup 111 [equal to lr(IIi) ] coincides with the p roduc t of so lvable n o r m a l subgroups of the f o r m �9 which a r c a l so n o r m a l in [I 2 . But then it is c l e a r tha t lr(II~) ~ r(H2), and t h e r e f o r e lr(H l) is so lvable . Hence H i has a r ad i ca l r(ll l) [equal to lr(lI1)]. In addit ion, i t is c l e a r f r o m the p roo f that r(H,) = II1 N r(H~).

We now a s s u m e that r([1 l) ex i s t s . If F ' is a so lvable n o r m a l subgroup of H 2 then c l e a r l y F' N H, ~ r(II,). We c o n s i d e r lr(iI2) N H,. It is ea sy to show that lr(II2) NII, is a p roduc t of so lvable n o r m a l subgroups of Hi, and t h e r e f o r e lr(II~) n HI ~ ]r(II,) = r(II1). Hence lr(H2) N H1 is so lvable . The group It(1] 2) is loca l ly so lvab le , and lr(II2) NII, is a so lvable subgroup of finite index, so that lr(H 2) is a l so so lvable . But then lr(II=) = r(H~), i . e . ,

r(H 2) ex i s t s .

Thus , if [I s is n o r m a l in H2, L e m m a 1.5 is proved; m o r e o v e r , in this ca se r(II,) = H~ N r(H_~). We now c o n s i d e r the c a s e when 112 is an a r b i t r a r y subgroup of finite index in [I 2.

36

Putting l-l, = D ~-~ H~.~, it is easy to see that ~[~ is a normal subgroup of H2, and If: cl-i~ and the index @~II 2

of I~ I in Ill is finite (see, e.g., [9]). Since JI~ < I[,, H, < l]~, the existence of the radical for i~ll is equivalent (by the case already considered) to the existence of radicals for H i and /~2. In particular, r(~) exists if and only if r(II 2) exists. Next, the result proved above implies that r(Ht) ~ I~ ~ r(l~) = r(H~) D H~, whence by the obvious inclusion r(H,) ~ II~ fl r(ll~) we obtain that r(II~) fl Ill is a subgroup of finite index in r(Hi). The lemma is com- pletely proved.

Using Lemma 1.5, Theorem 1.4 for case (i) is now easily proved. As an equivalence relation, commen- surability is generated by the relation of inclusion of subgroups of finite index. But the statement of Theorem 1.4 holds for the latter relation, by Lemma 1.5, and therefore Theorem 1.4 is also true for commensurability.

We now turn to the proof of Theorem 1.4 in case (ii). Let ~ : ii~ ~ II~ be an epimorphism with finite ker- nel. If the group ke r ~ is solvable t hen r(Hl) = 9 -~ (r(H~)), and in pa r t i cu l a r r(IIl) and rg I 9 exis t s imul taneous ly and a re weakly c o m m e n s u r a b l e [since ke r (9 I~(~)t is finite], We now cons ider the case when ke r ~ neednot be so lvable .

Let C = r(ker ~), which exists because ker ~ is finite. Then C is characteristic in ker ~ and is therefor~ normal in I[ I. We put H I = H I/C. Then the epimorphism ~ splits as a composition of epimorphisms rl I -~ Ill ~ H~,

where ~ : HI-~ H'I is the natural epimorphism with kernel C and the kernel of ~' is equal to ker %0/C. Since ker is solvable, the foregoing implies that it suffices to prove Theorem 1.4 in case (ii) for ~'. However, it is clear that r (ker ~') = { e }; in particular, the center Z (ker ~0') is trivial. We have an extension {e}-+ker 9' -+ H~--+ H~-+ {el which is well known (see, e.g., [I0]) to be uniquely determined up to equivalence by the characteristic class r ~ II2(H~ Z(ker~')) -----{0}, whence c = 0, i.e., this extension splits. But then FI~ contains a subgroup of finite index isomorphic to H 2. This reduces case (ii) to the case (i) already considered. The proof of Theorem 1.4 is complete.

Definition. A group is called semisimple if its radical exists and is finite.

Theorem 1.4 obviously implies

Propos i t ion 1.6. If [l 1 ~ lI 2 and [11 is s e m i s i m p l e , then so is H 2,

In the r e s t of this work we cons ider for the mos t pa r t the fundamental group [1 = ~1 (M) of a homogeneous space M toge ther with its r ad ica l , the ex is tence of which we now prove.

We r eca l l that if D is a so lvable group, then its r ank is r ank D = ~ rankDUD~+~) where the D(i ) a r e the i - 0

t e r m s in the descending c o m m u t a t o r s e r i e s : D(0 ) = D, D(i+l ) = [D(i), D(i)] (which t e rmina te s s ince D is so lv - able) , and r ankD( i ) /D( i+ 9 is the o rd ina ry r ank for f initely genera ted Abel ian groups (equal to infinity if the Abel ian group is not f initely genera ted) .

P ropos i t ion 1.7. If M is a homogeneous space , then r( , h (M)) exis ts and is a polycycl ic group, and r ank r(Tr(M)) _< d imM.

Proof . The group '~I(M) contains a subgroup ~ of finite index i somorph ic to a d i s c r e t e subgroup of some connected Lie group F (cf. [2]). If F is l inear (i.e., has a faithful l inear r ep resen ta t ion ) , then ~ is a l so l inear and t he re fo re r(~) exis ts (the Zassenhaus t heo rem, see, e .g . , [11]). When F is a r b i t r a r y , we cons ider the adjoint r e p r e s e n t a t i o n Ad~ :Y-~ GL(f), whose kernel Z iF) is an Abel ian group. One sees eas i ly that r (~) ----z~ ~q Ad~: ~ (r (AdF (~))), ~ where the r ad ica l r(AdF(z0) of the group AdF(~) exis ts s ince AdF(~) is l inear . Thus r(,v) ex- i s t s , and s ince r ~ 7r I (M), by Theorem 1.4 r(~l(M)) exis ts a lso .

Since r(~l(M)) is so lvab le , the fact that it is polycycl ic and the inequali ty rankr(~l(M)) _ d i m M were proved in [12] (the polycycl ic i ty follows f r o m the fac t that eve ry subgroup of ~(M) is finitely genera ted , cf. [12]).

COROLLARY 1.8. Let M be compac t and homogeneous . Then:

(i) If M' is a finite cover ing of M, then r(=l(lVD) ~ r(=I(MD) ; in pa r t i cu l a r , =l(1Vl) is s e m i s i m p l e if and only if =I(M') is .

(ii) If M' is a finite cover ing of M and has a na tura l f ibra t ion with base Ma (cf. [1]), then r(=l(M)) r('~l(Ma)) and =l(Ma) is s e m i s i m p l e if and only if ~h(M) [or =I(M')] is .

37

2. Lie G r o u p s Which Ar e T r a n s i t i v e on C o m p a c t M a n i f o l d s

wi th a S e m i s i m p l e F u n d a m e n t a l G r o u p

In this and the following two sections, we consider some general properties of compact homogeneous spaces with semisimple fundamental groups. The description given in [2] of the topological structure of com- pact homogeneous spaces (cf. also the Introduction) and 1.8 imply:

Proposition 2.1. If M is compact and homogeneous, then ~I(M) is semisimple if and only if the solvable component M r of M is trivial.

It is clear from this that the compact homogeneous M with semisimple ~i(M) constitute a very large class of homogeneous spaces. In this section, we will study the structure of Lie groups which are transitive on such M.

If M = G/H is compact and ~I(M) is semisimple, then even if G acts effectively on M, G is not neces- sarily semisimple - see the examples in [13, 14] for the case of trivial ~I(M). However, we have the following result.

THEOREM 2.2. Let M = G/H be compact and ~I(M) semisimple. Then if S is a maximal semisimple subgroup of G, S is transitive on M.

Proof. By results in [I, 2], there exist an M' which is a finite covering of M and a Lie group G' tran- sitive on M' such that:

a) M' has a natural fibration ([1, Theorem I]).

b) The maximal semisimple subgroup S' of G' is locally isomorphic to S, and if M' = G ' /H ' then NG,(H ~) �9 S' = G' ([2, Theorem 1]).

We may take G and G' simply connected; then the action of S' on M' coincides with the natural action of S on M', and in particular S is transitive on M if and only if S' is transitive on M'. Since ~I(M') is semisimple (cf. 1.8), it is enough to prove Theorem 2.2 for M'. Moreover, by suitably choosing M' we can arrange that for some n -> 0 the group ~I(M') is isomorphic to a lattice in a connected Lie group F = (NG'(H~))0/H ~ • R n,

t and moreover the base M a of the natural fibration for M' is diffeomorphic to L F/F, where F is a lattice in F and L is a maximal compact subgroup of F (cf. [2]).

Let F = S I "R I be the Levy decomposition of F: R I the radical, S I the semlsimple part (i.e., a maximal semisimple subgroup of F). We consider the group F nB~ and prove that it is finite. As was noted in [2], the group F A L is normal in F, finite (since L is compact) , and F/F fl L is to r s ion- f ree [since F / F 0 L : ~1 (M~)). In par t icu lar , F is weakly commensurab le with ul(M~) and is therefore semis imple by 1.6 and 1.8. Since F D R, is a solvable normal subgroup of F, the semis impl ic i ty of F implies that 1" r, R~ is finite.

If C is a maximal connected compact normal subgroup of F, then C c L and we can drop f rom F to F 1 = F / C , where Ma = LI \F1/FI, and FI = F/F n C ~ F, and LI = L / C is a maximal compact subgroup of F1. Replacing F by F1, we may therefore assume that F has no connected compact normal subgroups. But then F ~R, is a lattice in R1 [11, 8.28]. Since we have proved that F N/~ is finite, the fact that it is uniform in R 1 implies that the radical R1 of the group F is compact . This proves that F is reductive and its radica l is compact (this will be used below, cf. 2.6).

Since L is a maximal compact subgrouP of F, we have L ~ R , . Consider the natural ep imorphism )~: (N G, r

(H~))0 x R n -- F with kernel H~. Then L = )~ (K' n (N~, (H0))0), where K' is a maximal compact subgroup of G'. G' may be assumed simply connected, and therefore K' is semis imple and can be chosen so that . K" ~ S ' . There - fore , ~(S' N (N~, (H~))0) ~ R1. F u r t h e r m o r e , if S" is the semis imple part of the Lie group (S'N (Nv, (H~))0, it is c lear that ~(S W) coincides with the s e m i s i m p l e part of the group F. But we then obtain that ~. (S'[~ (Na, (H~)) 0) = F, which is equivalent to S'.H'o ~ (NG,(Ho))o. We therefore get S ' . H~ = S ' . (S' . H~) = S ' - N G, (H~) = G' (the last equality by the choice of G'). This proves that S' �9 H~ = G' .

' = G ' S ' The equality S' �9 H 0 means that is t ransi t ive on the manifold G' / H~ which is the universal cover of M' = G' / H'. But it is then c lea r that S' is also t ransi t ive on M'. By what was said at the beginning of the proof, this implies that S, a maximal semis imple subgroup in the original group G, is also transi t ive on M.

The asse r t ion of Theo rem 2.2 can be viewed as a natural general izat ion of a well-known theorem of Mont- gomery : if a Lie group G is t ransi t ive on a compact manifold M with finite Ul(M), then a maximal compact semis imple subgroup of G is also t ransi t ive on M [15].

38

A Lie group O acting transitively on a manifold M is said to be irreducible on M if its action is locally effective and G contains no proper subgroups which are transitive on M. If M is connected (and we will not

consider disconnected M in this paper), any Lie group transitive and locally effective on M contains a subgroup

which is irreducible on M. A theorem of Montgomery says that only compact semisimple Lie groups can act

irreducibly on a compact manifold M with finite at(M). From 2.2 we obtain

COROLLARY 2.3. If M is compact, ~I(M) semisimple, and the Lie G is transitive and irreducible on M,

then G is semisimple.

Proposition 2.4. Let M be compact homogeneous and assume nl(M) is semisimple. Then any Lie group

G acting transitively and locally effectively on M contains a subgroup S such that S is transitive on IvI and

dimS -< 3/2m(m + I), where m = dimM.

Proof. By 2.2, a maximal semisimple subgroup S acts transitively on M, and the action of S on M is also

locally effective. Therefore, the assertion follows from the next lemma.

LEMMA 2.5. If 9 is a semisimple real Lie algebra and ~ a subalgebra not containing any ideals of g,

then dim ~ ~< 3/2k(k + I), where k = codim g~.

Proof of the Lemma. Although the proof can be given without leaving the framework of Lie algebra theory,

we pass to the corresponding Lie groups to make things simpler.

Let G be a simply connected Lie group corresponding to the Lie algebra ~, and H the connected subgroup

of G corresponding to the subalgebra ~ The conditions on [} imply that H contains no connected normal sub- groups of G. We consider the Lie group F : Z (G) �9 H obtained by taking the closure of the subgroup Z (G) �9 H in

G, and we show that G acts locally effectively on G/F.

Assume that F contains some connected normal subgroup Q of G. Since G is semisimple, so is Q. We

know that [~7, H] C H, i.e., H/H is Abelian. Since Z(G) is central, this implies that iF, F] = [Z(G).H, Z(G)H] =

[~, H]cH. Since QcF andQis s emisimple, 0:[(2, Q]c[F, F]cH, i.e., 0 mIf. But we then obtain from our

assumption concerning ~ that Q = {el. This proves that G acts locally effectively on G/F.

We put k ~ : codlin iF and prove that dim~ ~< 3/2k'(k'+ i). Let G* = Ad G (G) be the adjoint group of G and put

F* : Ad G (F). Then kerAds ~ Z(G) ~ F implies that G*/F* = G / F, and it is clear that G* acts effectively on

G*/F*. If K* is a maximal compact subgroup of G*, then K* also acts effectively on G*/F*, and therefore

dimK* _< :/2k'(k' + 1) (this fact is well known and most simply proved by considering a K*-invariant metric on G*/F*).

Since G* is an adjoint semisimple Lie group [i.e., Z(G*) = ~e)], the Lie algebra t* of the subgroup K* is a maximal compact subalgebra in ~. If ~ is an arbitrary real Lie algebra and ~ is a maximal compact sub- algebra, then all the pairs (~, t), have been enumerated, as is well known (see, e.g., Table I in [7]L It is not hard to see by direct computations, examining all such pairs (~, t), that we always have dim 5~3dimt. It is obvious that this same inequality will also hold for any semisimple Lie algebra ~. Therefore, in our case we ob ta in g ~ 3 dim k* ~ 3/2k'(k' :q- l ) .

S ince ~ ~ ~, we have k = codlin ~f~ /~ ' = codim ~ , and t h e r e f o r e dim ~ ~< 3/~k(k+ t). Thi s p r o v e s L e m m a 2 .5 , and wi th i t F r o p o s i t i o n 2.4.

R e m a r k s . 1. F o r a s e m i s i m p l e ~, the e q u a l i t y dim z = 3 dimt ho lds only i f ~ = 4- ~I (2, R). i = 1

2. It is c l e a r tha t the n u m b e r of s e m i s i m p l e L ie g r o u p s of f i x e d d i m e n s i o n i s f i n i t e , p r o v i d e d we take t h e m up to l o c a l i s o m o r p h i s m . Us ing 2 .5 , we g e t f r o m th is the e x i s t e n c e of a func t ion d(n) s u c h tha t i f ~(n) i s the s e t of a l l s e m i s i m p l e s i m p l y c o n n e c t e d L ie g r o u p s wh ich can a c t t r a n s i t i v e l y and l o c a l l y e f f e c t i v e l y on m a n i f o l d s of d i m e n s i o n <_ n, i t i s f i n i t e of c a r d i n a l i t y at m o s t d(n) (for a f i xed c o m p a c t m a n i f o l d M wi th f in i t e ~ ( M ) , an a n a l o g o u s r e s u l t was p r o v e d in [1 4]).

In p a r t i c u l a r , 2.2 and 2.5 i m p l y tha t the n u m b e r of s i m p l y c o n n e c t e d L ie g r o u p s w h i c h a c t i r r e d u c i b l y and t r a n s i t i v e l y on c o m p a c t m a n i f o l d s M wi th s e m i s i m p l e ~l(M), and s u c h tha t d i m M -< n, does not e x c e e d d(n).

We now c o n s i d e r in m o r e d e t a i l the p r o p e r t i e s of an i m p o r t a n t c l a s s of t r a n s i t i v e L ie g r o u p a c t i o n s on c o m p a c t m a n i f o l d s M wi th s e m i s i m p l e ~l(M). By 2.2 , i t would be p o s s i b l e in many c a s e s to l i m i t o u r s e l v e s - t o s tudy ing the a c t i o n s of s e m i s i m p l e L ie g r o u p s . H o w e v e r , i t t u r n s out tha t many of t h e i r p r o p e r t i e s a r e a l s o va l id in a m o r e g e n e r a l s i t u a t i o n , v i z . , fo r a c t i o n s and g r o u p s G s u c h tha t NG(H 0) .S = G, w h e r e M = G / H and S is the s e m A s i m p l e p a r t of G. A t r a n s i t i v e a c t i o n of G on M s u c h tha t N G (H 0) �9 S = G wi l l be c a l l e d r e g u l a r . I t

39

is c l e a r tha t a t r a n s i t i v e a c t i o n of a s e m i s i m p l e L ie g r o u p is a l w a y s r e g u l a r . By T h e o r e m 1 in [2], for e v e r y c o m p a c t h o m o g e n e o u s M t h e r e e x i s t s a f in i t e c o v e r i n g man i fo ld M' s u c h tha t t h e r e e x i s t s a r e g u l a r a c t i o n of s o m e Lie g r o u p on M ' .

In add i t i on to the ma in r e s u l t p r o v e d in T h e o r e m 2 .2 , the p r o o f in f ac t con ta ins a n u m b e r of r e s u l t s which we now c o n s i d e r .

P r o p o s i t i o n 2.6. L e t M = G / H be c o m p a c t , ~ (M) s e m i s i m p l e , G s i m p l y c o n n e c t e d , and l e t G ac t r e g u - l a r l y on M. Then i f I) = N G(H 0) we have

(i) I) has f i n i t e l y many c o n n e c t e d c o m p o n e n t s ;

(ii) I) 0 i s a u n i f o r m t - s u b g r o u p of G. If K i s a m a x i m a l c o m p a c t s u b g r o u p of G then P0" K = K" P0 = G;

(iii) I ) 0 / H 0 i s r e d u c t i v e wi th c o m p a c t r a d i c a l , and i t s s e m i s i m p l e p a r t has f in i te c e n t e r .

P r o o f . In the c o u r s e of the p r o o f of T h e o r e m 2.2 , i t was shown tha t the g roup F = I ) 0 / H 0 x R n has a c o m p a c t r a d i c a l . Th i s i m p l i e s in p a r t i c u l a r tha t n = 0. But n = vcd 7r0(i)) (cf. [217 and the cond i t ion vcd z;0(i)) = 0 i s e q u i v a l e n t to f i n i t e n e s s of 7r0(i)). Th is p r o v e s (i).

(ii) S ince P ~ I I , H i s u n i f o r m in G, and n0(i)) is f i n i t e , P0 is a l s o u n i f o r m in G. F u r t h e r , G / I ) is c o m - p a c t and s i m p l y c o n n e c t e d , and t h e r e f o r e K is t r a n s i t i v e on G / P 0 (a t h e o r e m of M o n t g o m e r y [15]), whence I) 0" K = K" I) 0 = G. The f ac t tha t I) 0 is a t - s u b g r o u p i s p r o v e d in [14].

(iii) I t fo l lows f r o m the c o m p a c t n e s s of the r a d i c a l of the g r o u p F and n = 0 tha t F = I ) 0 / H 0 is r e d u c t i v e .

L e t F = S l -R~; we now p r o v e tha t Z(S 1) is f in i te .

F o r s o m e f in i t e c o v e r i n g M' of M, the b a s e s p a c e of the n a t u r a l f i b r a t i o n has the f o r m Mo =L\F/F, w h e r e F is a l a t t i c e and L is a m a x i m a l c o m p a c t s u b g r o u p of F. S ince R1 is c o m p a c t , L ~R~, and t h e r e f o r e Ma = K~' \ S~/F,, w h e r e S~ = SI/SI~R1 i s l o c a l l y i s o m o r p h i c to S1, F, = F / F N R , is a l a t t i c e , and K1 = L / R 1 i s a m a x i - m a l c o m p a c t s u b g r o u p of S~. M o r e o v e r , lh(M) ~ F , but F ~ F~ (s ince F n R , is f in i t e ) , and t h e r e f o r e F 1 ,~l(M), whence i t fo l lows by 1.6 tha t F 1 is s e m i s i m p l e . We c o n s i d e r the s u b g r o u p F, NZ(S~) wh ich is c e n t r a l in F 1 and has f in i te index in Z (S~) ( see , e . g . , [4, 14]). S ince F 1 is s e m i s i m p l e , F,,Q Z (S~) i s f in i te and t h e r e - f o r e s o is Z(S]). But S, n RL is a l s o f in i te ( s ince R l i s c o m p a c t ) , and t h e r e f o r e Z(S1) i s a l s o f in i t e .

We note tha t if G is s e m i s i m p l e and M = G / H is c o m p a c t , then I ) 0 / H 0 i s r e d u c t i v e even wi thout the a s -

s u m p t i o n tha t 7h(M) is s e m i s i m p l e [16].

3. Semisimple Lie Groups Transitive on Compact

M a n i f o l d s w i t h S e m i s i m p l e ~ l ( M )

L e t N be a c o m p a c t h o m o g e n e o u s s p a c e wi th s e m i s i m p l e 7h(N). By 2.2 , we r e s t r i c t o u r s e l v e s to the c a s e when a L ie g r o u p t r a n s i t i v e on s u c h an N is s e m i s i m p l e . In th is s e c t i o n , for e a c h f ixed s e m i s i m p l e n = ~l iN) we f ind a l l the s i m p l y c o n n e c t e d s e m i s i m p l e L ie g r o u p s w h i c h a r e t r a n s i t i v e on those c o m p a c t man i fo lds M

for w h i c h 7h (M) ~ ~.

We beg in by c o n s i d e r i n g a s l i g h t l y m o r e g e n e r a l s i t u a t i o n . Le t M = G / H be c o m p a c t , hi(M) ~ ~, and l e t G be a s i m p l y c o n n e c t e d L ie g roup a c t i n g r e g u l a r l y on M ( i .e . , fo r now we do not r e q u i r e G to be s e m i s i m p l e ) . By 2.6 we m a y a s s u m e (by r e p l a c i n g M by a s u i t a b l e f in i te c o v e r i n g M') tha t i f I) = (NG(H0)) 0 then I) is a c o n - n e c t e d u n i f o r m t - s u b g r o u p of G con t a in ing H, and P / H 0 i s a r e d u c t i v e L ie g r o u p wi th c o m p a c t r a d i c a l and s e m i s i m p l e p a r t hav ing f in i t e c e n t e r . M o r e o v e r , M~= L\F/F (where Ma is the b a s e s p a c e of the n a t u r a l f i b r a - t ion fo r M), and F i s a l a t t i c e in F = P / H 0 w h i c h i s w e a k l y c o m m e n s u r a b l e w i th hi(M) and t h e r e f o r e wi th n

(cf. p r o o f of T h e o r e m 2.2) .

L e t F = S H" S k w h e r e S k i s a m a x i m a l c o n n e c t e d c o m p a c t n o r m a l s u b g r o u p in F (S k i s not n e c e s s a r i l y s e m i s i m p l e , but is r e d u c t i v e ) , and l e t S H be a s e m i s i m p l e n o r m a l s u b g r o u p which con ta in s no c o m p a c t f a c - t o r s . I t is c l e a r tha t F N S~ i s f i n i t e , and t h e r e f o r e the g r o u p F ' = F /F n Sh.is a l a t t i c e i n S'H ~ S H / S H ~ Sh, which

!

is w e a k l y c o m m e n s u r a t e wi th F (and t h e r e f o r e a l s o wi th ~). If K~t i s a m a x i m a l c o m p a c t s u b g r o u p of S H, then * = K ~ / Z (SH) ( s ince Z (S H) i s f in i te , i t i s c l e a r that Ma =K~\S '~ /F ' , s i n c e L ~ S , . Pu t t i ng S~{ = S H / Z ( S H ) , K H

~ * F* K ~ Z ( S ~ ) ) , F * ~ F'/F'(]Z(S'~), and so M~,~ KH\SH/F �9 M o r e o v e r , ~ F [s ince Z(SH) i s f in i te ] .

We ob ta in the r e s u l t t ha t up to a f in i t e c o v e r i n g , Ma can be e x p r e s s e d in the f o r m L\S*/D, w h e r e D is a l a t t i c e in the ad jo in t s e m i s i m p l e L ie g r o u p S* c on t a in ing no c o m p a c t f a c t o r s , and L i s a m a x i m a l c o m p a c t s u b - g r o u p of S*, w i th D ~ 7r. But such an S* i s un ique ly d e t e r m i n e d up to i s o m o r p h i s m by i t s l a t t i c e (cf. [17]), a l - though we note tha t in g e n e r a l the p o s i t i o n of D in S* i s not un ique ly d e t e r m i n e d (because of the p o s s i b l e

40

existence of factors of rank i). We will denote this S* by S*(~), and S H is a finite covering of S*(~). The above arguments prove the following result.

Proposition 3.1. Let M be compact homogeneous and let 31(M) be semisimple. Then up to isomorphism there exists a unique semisimple adjoint Lie group S* = S* (3 i (M)) determined by ~I(M) and without compact

fac tors , such that :

(i) ~l(M) is weakly commensurab le with some latt ice F* in S*.

(ii) If M = G / H , G acts regu la r ly on M (e.g., if G is semis imple) , and P = NG(H0) , then S* is finitely covered by a maximal semis imple normal subgroup containing no compact fac tors of the reductive

Lie group P0 / H0.

(iii) The base Ma of the natural fibration for M is diffeomorphic up to a finite covering to the manifold 31" = K*\S*,F*, where K* is a maximal compact subgroup of S*.

Y In particular, M a is diffeomorphic to a Riemannian locally symmetric space of negative curvature and

31 (M~) ~ ~i (M).

COROLLARY 3.2. Let M be compact homogeneous and 3t(M) be semisimple. Then ~I (M) is weakly com- mensurable with a lattice in a semisimple Lie group which has trivial center and contains no compact com- ponents.

In one particular case, this result has a converse: If F is a lattice in a semisimple Lie group G such that Z (G) is finite (G the universal covering group of G), then there exists a compact homogeneous space M such that ~i(M) ~ F (it suffices to take M = G / F ) . For arbitrary semisimple G, the co~verse is false (e.g., one can see that it is false for EIII by using the methods in Sec. 4).

We now consider in more detail the structure of the semisimple Lie groups G which act transitively on compact M such that ~I(M) ~ 3 (where ,v is the semisimple group fixed above).

If M = G / H then we put P = (NG(H0))0, so that P is a connected uniform t-subgroup of G and we can as- sume that H ~ P . By [7] we have P = W . Z ~ ' S ~ - N F , where F ~ I I , i s some sys t em of white roots in theSatake d iag ram of the a lgebra g (we will use the notation in [7]). Henceforth F (possibly with indices) will denote only root sys tems and not latt ices in Lie groups (in con t ras t to some of our ea r l i e r usages).

In the above decomposi t ion of P, NF is the unipotent radical in UF, S~ a semis imple Lie group without compact fac tors , E F a maximal compact factor in the semis imple par t of the group U F, where U F = Z F" E F"

I . . . .

S F �9 N F l s ' a s tandard parabohc subgroup of G containing P and corresponding to the root sy s t em F. Fur ther , Z F = Z F ' Z F is the connected component of the ~dent~ty of the center of the reduct ive part of U F and W is a closed connected subgroup of Z ~ . E F.

' = S(3) �9 S' , where S' and S(=) are normal subgroups of S~ and H0 = 2', By 2.6 and 3.1, we obtain that S F S(=) n S' d i sc re te , and S(3) is locally i somorphic to the group S*(3) in 3.1 [S(Tr) is a cover ing of S*(~)]. As is well known, the Satake d iag ram of the group S~ is the union of those connected components of the se t F U II0 (where go is the set of all black roots in the Satake d iag ram for ~i) which contain at least one white root. tt follows that the Satake d iagram for S(3) is also the union of cer ta in connected components of the set F U II0 which contain white roots (at least one white root). The Satake d iag ram for S (=) depends only on 3 - it coincides with the Satake d iagram 1](3) for the group S*(3). We denote the subset of roots of g which cor respond to S(~) by [I (Tr, G); it depends on the imbedding of S(3) in G.

We now consider n e c e s s a r y conditions which must be sat isf ied by the position of rl (~, G) in F U IIo in o rder for the corresponding subgroup S(Tr) and G to be obtained by the above construct ion f rom a compact homogeneous M with ~I(M) ~ ~.

We assume the Lie group G to be s imply connected, and if K is a maximal compact subgroup, then K" P = G [cf. 2.6 (ii)]. But then F = F(g), where F(~) is some subset of 1I t descr ibed in [18]. If g is s imple then g(~) -- ~, except for the cases g = s~ (p, p), s~(2, n) (n>~ 3), s~(n, R), ~*(4n),EIII, and E VII, in which F(~) con- s is ts of a single s imple root descr ibed in [18]. For a semis imple g the set F(~) is the union of the co r r e spond- ing sets for the s imple ideals of g. In this way F(g) is descr ibed completely , and we obtain one of the neces - s a r y conditions in which we are in teres ted , i .e. , F = F(g).

By 2.6 (iii), the center of the semis imple par t of the group P / H 0 is finite, and therefore Z(S(=)/S(=) n S') is also finite. This implies that the index of Z(S(=))AZ(S') in Z(S(3)) is finite. We prove that Z(S(,~)) is finite. To this end we will need some auxi l iary resu l t s .

41

LEMMA 3.3. Let UF be a standard parabolic subgroup of a simply connected semisimple Lie group, S

the semisimple part of U F. K S = S' �9 S", where S', S" are connected normal subgroups of S and dim(S' N S") = 0,

then :

(i) S' n S" = Z(S') N Z(S") is a finite group.

(ii) If G is simple then at least one of the groups Z(S'), Z (S") is finite.

n J~

Proof. Let G = • G~, where the G i are simple. Then Ur = • Urn, where the UFi are standard parabolic i = l i =1

subgroups of the simply connected Gi which correspond to the root subsystems Fi c (II, L, and U F~ = I'. In par- i~1

tieular, S = x Si, where S~ = S n G~ and S' - • (S' N Gi), S" = ~4 (S" n G~). This shows that to prove (i) it suffices i = l i - 1 i =z

to c o n s i d e r only the c a s e when G is s imple . But then (i) fol lows i m m e d i a t e l y f r o m (ii), and t h e r e f o r e we p rove (ii).

If Z (G) is finite then so are Z(S'), Z(S") [since finiteness of Z (G) is equivalent to the existence of a linear

representation for G with finite kernel]. It therefore remains to consider the case where G is simple, simply

connected, and Z (G) is infinite. But these conditions are possible only for g = 3,(p, q), ~0(2, n), 3~(n, R), ~0" (2n),

EIII, and EVIl.

(m 1 F i r s t le t g = 3ttip~ q). One s e e s ea s i l y tha t in this e a s e e v e r y ~'r has the f o r m . ~ (p~. q~) 4- ~ 3I (kj. C) . \ j = l

Since 3' n ~" = {0}, it is c l e a r tha t one of ~', ~" is a d i r e c t s u m of a l g e b r a s of the type .~I(k~, 12), s ay ~". But then it is c l e a r that Z (S') is f inite. F o r .g = 30 (2, n) we have ~r = {0}, 3I(2, R) , o r 30 (1, n). Since he re ~r is

' ~" is t r iv i a l and the a s s e r t i o n of the l e m m a is obvious . If g = 3~(n, R), then ~. = ~ (n t, N) 4- simple, one of ~,

I'~- ~l(kj, R)l. However, it is shown in [18] that all subgroups of the simply group cor- G = Sp(~--n,~-'~), connected j=l I

responding to 3l(kj, R), imbedded in the standard way have finite centers (this is nontrivial only if kj = 2), while subgroups corresponding to ~(nl, R), nt ~> i have infinite centers. As in the case of 3.(p, q), this easily im-

plies that Z(S') o r Z(S") is f inite. F o r g =3o*(2n)~r=~O*(2n~)4- " 3u*(2kj) , a l s o t h e f i n i t e n e s s o f Z ( S T) o r Z(SD

is p roved s i m i l a r l y . F o r g = E I I I , EVII~r is not s i m p l e only in the e a s e EVII, w h e n i t i s i s o m o r p h i c to ~o(1, 4)4- ~I(2, R). But i t is c l e a r h e r e tha t Z (S ~) o r Z (S ~) is f ini te . The l e m m a is p roved .

P r o p o s i t i o n 3.4. Le t Ur = Zr. Er .Sr 'Ur be a s t a n d a r d pa rabo l i c subgroup of the s e rn l s imp le s i m p l y c o n - n e t t e d Lie group G. Le t S = S ~ . S ' , w h e r e S = EF'S ) is the s e m t s i m p l e p a r t of U F and S ~, S" a r e n o r m a l s u b - g roups of S wi th dimiS' N S") = 0. Then S' N (S" . Zr. Nr) is f ini te .

P roo f . J u s t as in the p roo f of 3 .3 , we show that i t is su f f i c i en t to c o n s i d e r only the ease when G is s imp le . If in this e a s e Z(S ~) is f ini te , then P r o p o s i t i o n 3.4 is obvious [s ince S'.Cl (S" "Zr 'Nr) c Z ( S ' ) 1. We may t h e r e - fo re a s s u m e that Z(S ~) is infini te; but then by 3.3, Z (S ' ) is f inite. It is c l e a r f r o m this tha t S':a iS" "Zr "Nr) is f inite if and only if S.' n (Zr �9 Nr) is f ini te .

We c o n s i d e r the subg roup S'Cl Zr" Nr. We have Z F = Z~ ' . Z i , and a s s u m e in i t ia l ly tha t Z~" = {e}. Then Z p . N r = Zi~" N r and this subgroup is con ta ined in a m a x i m a l connec ted t r i a n g u l a r subg roup T of G. I t is known that I'~ Z iG)= {el (this fo l lows , e .g . , f r o m the Iwasawa d e c o m p o s i t i o n fo r G). The g roup Z(G)n Z(S') has finite index in Z(S ~) [s ince S' /Z(S ' )n Z(G) is c l e a r l y l i nea r and t h e r e f o r e has f ini te c e n t e r ] . T h e r e f o r e Z(S') N T is f inite (in fac t , Z(S ' ) ~ r = {e}). But then S ' ~ (S" �9 Z~- Nr) is a l s o finite.

It r e m a i n s to c o n s i d e r the c a s e when Z + =/= {e}. Since G is s imp le and Z (G) is inf ini te , i t follows e a s i l y f r o m [7] and the c l a s s i f i c a t i o n of the s i m p l e Lie a l g e b r a s tha t the c a s e we a r e in is poss ib le only if g = ~ ( p , q).

) Here ~r =~t (p~,q t ) 4- 4-~t(k~,12) , and s ince Z ( S ' ) i s infinite we m u s t have ~'~>~t(p~, q~), and ~(p , , q~)is s t a n - \~=~

dard ly imbedded in ~t(p, q). We show that S ' ~ (Zr "Nr) is finite in this c a s e a l so .

It was shown above that Z(G) ~ Z(S') has finite index in Z (S'), and t h e r e f o r e i t is enough for us to prove that Z(G) n Zr �9 N t i s f inite. Le t K be the connec ted subgroup of G c o r r e s p o n d i n g to a m a x i m a l c o m p a c t s u b a l - g e b r a ~ of g. Then K=Z(G) and K c a n be c h o s e n so that , K f l U r ~ Z +. Then, K~(Zr .Nr) = Z +, and t h e r e f o r e

Z (G) N Zr .Nr ~ K ~ Zr .Nr = ZF. Hence Z(G) ~ (Zr �9 Nr) is finite if S' ~ Z + is . A d i r e c t compu ta t i on shows tha t if is the connec ted subgroup of G c o r r e s p o n d i n g to a S tanda rd ly imbedded s u b a l g e b r a ~,(p,, q~), then S ~ Z + = {e}.

42

This implies that S' n Z~ is finite, and hence S' n (S" �9 Zr �9 Nr) is finite in the case g = z~t(p, q). Proposition 3.4 is proved.

We return to the study of semisimple Lie groups G which act transitively on compact manifolds M with

semisimple ~I (M).

By 3.4 the group Z(S(~)) nZ(S') is finite. But it was shown above that Z(S(~)) nz(s') has finite index in Z(S(~)), which implies that Z(S(,v)) is finite. But then since G is simply connected, and by properties of the sub- set F(~) (cf. [18]), we obtain that H(~, G) N F(~)=~.

THEOREM 3.5. Let ,7 be an arbitrary semisimple group. If G is a simply connected semisimpie Lie group, then G acts transitively and locally effectively on at least one compact manifold M for which 'h (M) ~ if and only if the following conditions hold:

(i) There exists a compact homogeneous space N such that ,v1(N) ~ 7r.

When this condition holds, the group ~ corresponds by 3.1 to a unique group S*(,v), the Satake diagram of which we denote by I[ (,v).

(ii) The Satake diagram of the group G contains a F c II~ such that N (~) coincides with the union of a cer- tain number of connected components of the set F U If0. We denote this union by H(~, G).

(iii) i[~ F(g) and H(~, g) n F(~) = ~ (for the description of F(g) cf. the above and [18]).

Proof. The necessity of conditions (ii), (iii) was proved above, and the necessity of (i) is obvious. Let us prove that these conditions are sufficient.

Let U F be a standard parabolic subgroup of G corresponding to the system of roots F in (iii). Since F ~ F(g), Ur is uniform in G (cf. [18]). We have Ur = Zr.Er.S'r.Nr, and by (ii) S~ = S(~r)"S', where S(Tr), S' are connected normal subgroups of S~, S(~) corresponds to the root system ll(Tr, G). It follows from H(~, G) N F(~) =

that Z(G) !I Z(S(.~)) is finite [18], and therefore Z(S(~)) is also easily seen to be finite. In particular, S(~) is a finite covering of S*(,v). By 3.1, S*(,7) contains a lattice D* such that D* ~ ~. But then S(~) also contains

a lattice D ~ ~. We put H' =Zr Er "D "5" .N~ and consider the homogeneous space M' = G'/H'. By construc- tion, H' is uniform in G. But U I. is uniform in G, and therefore M' is compact. Furthermore, since G is simply connected, ~i(M') = ~0(H'); but by construction r0(H') = D ~ ,% and therefore ~I(M') ~ ~. However, G can act on M' in a way that is not locally effective (because all the compact factors of the group G are con- tained in H'). Therefore, let E F = E~ x K', where K' is a maximal compact normal subgroup of G. Then we

t

putH= Zr. Er.D.S'.Nr, and it is clear that M = G/H is the desired homogeneous space for the Lie group G.

4. St'andard Compact Homogeneous M with Semisimple 7ri(M)

In this section we describe all possible transitive actions of' semisimple Lie groups on compact manifolds M with semisimple ~,(M) (Theorem 4.17. This is clearly equivalent to describing the associated stationary sub- groups. However, this description is extremely lengthy and tedious (which, to be sure, is due to the essence of the matter - the class of objects to be described is very large). It is therefore natural to select from all such homogeneous spaces the ones which are simplest in a certain sense. We call these spaces standard, and it is to them that this section is devoted.

THEOREM 4.1. Let M = G/H be compact, ~ (M) semisimple, and let the group G be simply connected and semisimple. Then there exist " a root system F~ HI and subsetH(~, G) c F U H0. satisfying conditions (ii), (iii) of Theorem 3.5. Moreover, H 0 = W.Z[-.S'.Nr, where S' is a normal subgroup of S~ such that S~ = S'. S(~t) [where S (~) corresponds to the subsystem H(~,G)], dim(S(~) N S') --0, and W is a connected uniform subgroup of

Z ~-E F. The group H/H 0 is weakly commensurable with a lattice in the Lie group S (n).NE r (W N Er).Z~/W = S(Tr) �9 K, where K = NE r (Wn Er).Z+/W :.is a compact Lie group.

Conversely, if F = W. Z~-S'. NF, where F and [l (~, G) satisfy (ii), (iii) in 3.5 and W, S' are as above, then there exists a uniform subgroup H of G such that H 0 = F and ,71 (G / H) ~ ~. Every such H is uniquely deter- mined by the lattice in NuF(F)/F.

Proof. Let M = G/H be compact, ~i(M) ~ ~, G simply connected and semisimple. Then the existence of F and II(~, G) follows from Theorem 3.5. If we put P = NG(H0) , then Po~ Ur (cf. [6]), and ,~0(P) is finite" (cf. 2.6). P0 is connected and uniform in G and is a t-subgroup, and clearly Pc ~ Zr-Nr. The group P0 / H0 is reduc- tire with compact radical, and its semisimple part is locally isomorphic to S(~). Writing S~ in the form S(~) * S', we obtain that H o ~ Zr.S'.Nr, and P0 ~ Zr-S~.Nr. It follows from P ~ S(~) that H 0 ----- W.Z~.S'.Nr, where

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W =Ho~(Er.Z+). But then NG(Ito)=N ~(W).Zr.S'r.Nr (cf. [6]). It is c l e a r tha t :V ~(W)~Z +. Since the ErZr F r z r

+ r a d i c a l of the g roup P 0 / H 0 i s c o m p a c t , Z+/Z{ N W is c o m p a c t , and t h e r e f o r e W is u n i f o r m in E F" Z F.

_u - - t

It is c l e a r t h a t NEr.Z~_ (W) = N~r (W • Er)- Z +, and t h e r e f o r e P0 = N z r (Er [~ W ) . Z r . Z r . S r . N r . But then

s i n c e r%(M) is i s o m o r p h i c to H / H 0 , i t c o n t a i n s a s u b g r o u p of f in i t e index i s o m o r p h i c to a l a t t i c e in the L ie g r o u p P n / H 0. But P0 / H 0 i s i s o m o r p h i c to the g roup S (~).NE r (W~ Er).Z+/W = S (n).K, w h e r e K=NE r (W n E r ) . Z + / W is a c o m p a c t L ie g roup .

C o n v e r s e l y , i f F = W.Zr.S ' .Nr, then c h o o s i n g a l a t t i c e D in S(lr) w e a k l y c o m m e n s u r a b l e wi th ,v (cf. 3.1) and put t ing H = D . F ~ S ( n ) . Zr �9 E r �9 S : �9 Nr, we ob ta in the d e s i r e d s t a t i o n a r y s u b g r o u p H.

R e m a r k s . 1. When 7r is f i n i t e , T h e o r e m 4.1 is the m a i n r e s u l t in [6], t o g e t h e r wi th the c o m p l e m e n t s in [18].

2. F i x i n g a s e m i s i m p l e ~ = 7rl(M) i s e q u i v a l e n t to f ix ing the b a s e s p a c e Ma of the n a t u r a l f i b r a t i o n fo r M (cf. [1, 2]), and t h e r e f o r e 4.1 can be r e g a r d e d as a d e s c r i p t i o n of the c o m p a c t h o m o g e n e o u s M wi th f ixed M a.

3. In g e n e r a l , the c o n n e c t e d A b e l i a n L ie g roup Z ) i s n e i t h e r c o m p a c t no r s i m p l y connec t ed . If T F is a m a x i m a l c o m p a c t s u b g r o u p of Z ~ , then the cond i t ion tha t a c o n n e c t e d s u b g r o u p W be u n i f o r m in Z ~ ' E r is ob - v i o u s l y e q u i v a l e n t to the cond i t i on W. Tr ~ Z +.

We now def ine a c l a s s of h o m o g e n e o u s s p a c e s M wi th ~I(M) ~ ~ and depend ing on a much s m a l l e r n u m b e r

of con t inuous p a r a m e t e r s than a r b i t r a r y M. t !

L e t F and rI(~, G) be the r o o t s y s t e m s in 3.5. C o n s i d e r Ur=Zr.Er.Sr .Nr, and le t S F = S(rr) . S ' . We c h o o s e a l a t t i c e in S (~) wh ich is w e a k l y c o m m e n s u r a b l e wi th ~ and put H = Z~. E~. (D �9 S ') �9 N~. Then M = G / H is c o m p a c t , ,vl(M) ~ 7r, and the r e s u l t i n g h o m o g e n e o u s s p a c e wi l l be deno ted by M(Tr, G, F , I1) and c a l l e d the s t a n - d a r d h o m o g e n e o u s s p a c e c o r r e s p o n d i n g to the f ixed re, G, F , and rl = rl(Tr, G). F o r f ixed 7r, G, F , and rl , th is s t a n d a r d s p a c e depends on D and the way i t is i m b e d d e d in S(~r). H o w e v e r , u s ing the r i g i d i t y t h e o r e m and s h a r p - en ings t h e r e o f in [17], i t i s not h a r d to show tha t a l l s u c h M(Tr, G, F , rl) b e c o m e d i f f e o m o r p h i c a f t e r p a s s i n g to a s u i t a b l e f in i te c o v e r i n g [in th is p r o c e s s a s u b g r o u p of f in i te index in D is d e f o r m e d i n s i d e S(Tr)]. I t is e a s i l y s e e n f r o m e x a m p l e s (cf. Sec . 5) tha t even i f G is s i m p l e , d i m G and dimM(,~, G, F , rl) can be a r b i t r a r i l y l a r g e

fo r f ixed 7r.

We c o n s i d e r the s t i l l m o r e s p e c i a l c l a s s of h o m o g e n e o u s s p a c e s de f ined as fo l lows . Le t S*(~) be the g r o u p in 3.1 and D be a l a t t i c e in S*(~r) w e a k l y c o m m e n s u r a b l e wi th ,v. We put M = S*(Tr) /D. One s e e s e a s i l y

tha t 7q(M) ~ ~r i f and only i f Z(S*(~)~) is f in i t e [ (S*(~---~is the u n i v e r s a l c o v e r i n g of S*(Tr)]. In this c a s e we denote M by M(~). If G is an a r b i t r a r y s e m i s i m p l e s i m p l y c o n n e c t e d L ie g r o u p and D is a l a t t i c e in G, M = G / D is a f in i t e c o v e r i n g of s o m e M(~) i f and only i f Z (G) is f in i t e . The m a n i f o l d s M(~) a r e s t a n d a r d wi th F = [I 1.

The nex t p r o p o s i t i o n fo l lows f r o m the d e s c r i p t i o n of a l l c o m p a c t h o m o g e n e o u s M wi th ~r 1 (M) ~ ~r ( T h e o r e m

4.1) , the de f in i t i on of s t a n d a r d h o m o g e n e o u s s p a c e s , and T h e o r e m 2.2.

P r o p o s i t i o n 4.2. L e t M = G / H be c o m p a c t h o m o g e n e o u s and ~r = ~I(M) be s e m i s i m p l e . Then:

(i) d i m M -> d imS*Or) , and d i m M = d imS*0r ) i f and only i f G is a f in i te c o v e r i n g of S*(~r) and M is d i f -

f e o m o r p h i c to s o m e M(rr).

(ii) d i m M -> d imM(r r , G, F , H), w h e r e F c 11~ is a r o o t s y s t e m c o r r e s p o n d i n g to the ac t ion on M of a m a x - i m a l s e m i s i m p l e s u b g r o u p of G (cf. 2.2 and 3.5). M o r e o v e r , d i m M = d i m M ( % G, F , 11) i f and only if

M is d i f f e o m o r p h i c to a s t a n d a r d s p a c e .

E x a m p l e . L e t 7r = rq(Fg), w h e r e Fg is an o r i e n t a b l e s u r f a c e of genus g -> 2. Then ~ i s s e m i s i m p l e and

one s e e s e a s i l y tha~ S*(~) = PSL(2 , R) , w h e r e rI (~) c o n s i s t s of a s i n g l e whi te roo t . But Z (s* (~)) : : Z is i n f in i t e ,

and t h e r e f o r e M(r0 does not e x i s t .

We now s tudy how the c o m p a c t h o m o g e n e o u s M wi th s e m i s i m p l e ~l(M) a r e r e l a t e d to the s t a n d a r d s p a c e s . L e t F and [I (~r, G) be the r o o t s y s t e m s c o r r e s p o n d i n g to M = G / H, w h e r e G is s e m i s i m p l e and ~r = gl (M) i s s e m i s i m p l e . Pu t t i ng F = H. Z ~ . E F , i t fo l lows e a s i l y f r o m 4.1 tha t F is a c l o s e d s u b g r o u p of G. It is c l e a r t ha t G / F is the s t a n d a r d h o m o g e n e o u s s p a c e M(rr, G, F , rI). We have the f i b r a t i o n F / H - - G / H = M - - G / F . I ts f i b e r i s the m a n i f o l d F / H , on w h i c h the c o m p a c t L ie g roup Q = E F" T r is e a s i l y s e e n to ac t t r a n s i t i v e l y , w h e r e T F i s a m a x i m a l c o m p a c t s u b g r o u p of Z ~ . If F / H = Q / C , then we ge t a f i b r a t i o n Q / C - - M - - M(Tr, G,

F , rl).

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We now clarify the relation of the standard spaces M(~r, G, F, fl) with a manifold of the type S*(~r)/D,

where D ~ n [in particular, with the manifolds M(~r)].

If Ur = Zr. Er.S'r. Nr is a standard parabolic subgroup of G corresponding to the root system F, then it

is clear that F~ Ur and /"0 = Zr Er.S'.Nr, where S~ = S(~) "S'. Consider the fibration UF/F--G/F--G/U F. Here the base G/UF is a compact simply connected homogeneous space which depends only on G and F; we

denote it by N(G, F). For the fiber UF/F we have Ur/F=S(:~)/S(~) ~F=S(:~)/D, where D =S(:~) AF is a lattice in S(~), and D ~ ~. Using 2.2 in addition, this proves

Proposition 4.3. Let M be a compact homogeneous space and let ~ = ~t(M) be semisimpie. Then for some

finite covering M' of M we have a diagram of fibrations

S (a)/D

O/C -~ M ' - ~ M (n, G. F, H)

:v (G~ F)

Here N(G, F) is a compact simply connected homogeneous space (depending only on G and F), D a lattice in S(Tr) [corresponding to rl(~, G) and a finite covering of S*(~)], and Q/C is a homogeneous space of the compact

Lie group Q = E F. T F (E F is semisimple, T F a torus). M(~, G, F, [l) is a standard homogeneous space, and up to diffeomorphism and finite coverings depends only on ~, G, F, and II = 11(~, G).

If Z(S~)) is finite then 7ri(S(,7)/D) ~ ,~ and S(,~)/D is a finite covering of some M(,7).

Using 4.3, we now obtain a description of the fibers M c of natural fibrations for compact homogeneous M with given semisimple ~ = ~I(M) (giving ~ is equivalent to fixing the base M a of the natural fibration, cf. [i]). As for the structure of the natural fibration itself (e.g., the structure group, its action on the fibers, etc.), it can also be described using 4.1 and 4.3, but in general the description is quite lengthy and tedious, and we do not present it here.

Let M = G / H be compact homogeneous and admit a natural fibration. Then the fiber of this fibration is Ma = K/L, where K is a maximal compact subgroup of G, assumed to be simply connected (then K is semi, simple), and L = H n K. If M c is the universal covering space of Mc then I~I c = K/L 0 and we have a fibration K n Ur/L~ 0 K/Lo -~ K/K n Ur. Because K. U F = G (cf. 2.6), the base of this fibration is diffeomorphic to the mani-

fold G/U F = N(G, F) introduced above. We now consider the fiber of this fibration for M c. Clearly, K fl Ur- Tr .K(.n)K' "Er, where T F is a maximal compact subgroup of Z~, K(~) maximal in S(~), and K' maximal in S'

Zr * Er, [F and ~I(,% G) are the root sys tems in 4.1]. Fur ther , by 4.1 we have [ I o = W . Z ~ . S ' . N r , where W ~ + whence it follows that K' ~ L0 ~ T~ �9 K ~ �9 E~. But then K h Ur/Lo ~ K(~) �9 T~ �9 E r / ~ where L = L 0 / K'. Since K(~), TF, E r are c o m p a c t , K ( ~ ) . T r . E r / L = K' (~) • 2 1 5 K'(~), T~, E~ are finite coverings of K(~), TF, and EF, respect ive ly , and L' is a closed subgroup of K' (~)• T~• This proves:

Proposi t ion 4.4. Let Mc be the fiber of the natural f ibration for a compact homogeneous M with s emi - j ,

simple ~t (M) = ~. For Mc we then have the fibration K (~)• Tr • E r / L ' - + M~--~ N (G~ F), where N(G, F) is a s imply connected homogeneous space depending only on G and F (where 1 ~ ~ II~ is in 4.1), K'(T0 is a compact Lie group which finitely covers K*(Tr) [a maximal compact subgroup of S*0r)], T~ is a torus (isomorphic to T F) , E ~ a finite covering of E F. Moreover, L'N (T~ • Er) has finite index in L', and up to a finite covering K'(rOXTr• r depends only on ~r, G, F, and H(~, G).

In concluding this section, we consider the structure of M c in a special case. We recall that a real semi- simple Lie group G is called normal if it possesses a Cartan subgroup which splits over IR. In each of the series A-G there exists exactly one normal simply connected group for each rank possible for the series.

Proposition 4.5. Let M = G/H be compact, ~I(M) semisimple, and let G = S • C be a semisimple Lie group acting locally effectively on M with S normal and C compact. Then if dimC > 0 and ~t(M) is infinite, IVI c is a direct product of two homogeneous spaces of positive dimension.

Proof. The normality of S implies that Z~ = {e} and E F = C (see, e.g., [7]). If K s is a maximal compact subgroup of S, then K = Ks • C is maximal compact in G. We have Mo = K/K fl Iio. Moreover, H o = Z r . S ' . N r .

(H0 [7 C) (cf. 4.1), and therefore H0 N K =K' X (H0 n C), where K' is a maximal compact subgroup of S'. But then K / K f l H o = K J K ' X C / C f l H o . Since G acts locally effectively on M, dim(C~Ho)<dimC, and therefore dimC/CN H0> 0. Furthermore, K s = K' would imply that K(~) = {e}, which is possible only if ~ is finite. Therefore, we also have dimKs/K > 0, and the decomposition zff~ = K~/K ~ XIC/C fl II o is nontrivial.

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5 . C o m p a c t H o m o g e n e o u s M f o r W h i c h d i m M a = 2

In th is s e c t i o n , we a p p l y the r e s u l t s in S e c s . 2 -4 c o n c e r n i n g c o m p a c t h o m o g e n e o u s s p a c e s M wi th an a r b i t r a r y s e m i s i m p l e ,~I(M) to the s tudy of t hose M for wh ich the b a s e s p a c e of the n a t u r a l f i b r a t i o n i s two- d i m e n s i o n a l . I t t u r n s out t ha t m o s t of t h e s e M have ~I(M) s e m i s i m p l e .

If d i m M a = 0, then ~I(M) i s f in i t e and M is the h o m o g e n e o u s s p a c e of s o m e c o m p a c t s e m i s i m p l e L ie g r o u p ( conce rn ing s u c h M, s e e , e . g . , [19]). If d i m M a = 1, then M a = S t and t h e r e e x i s t s a f in i te c o v e r i n g M' of M s u c h tha t the n a t u r a l f i b r a t i o n fo r M' i s t r i v i a l and M' = S ~ x Mc , w h e r e Mc is a s i m p l y c o n n e c t e d h o m o - geneous s p a c e [1]. Next , the p r o b l e m n a t u r a l l y a r i s e s of d e s c r i b i n g the M fo r wh ich d i m M a = 2, i . e . , Ma is a c l o s e d s u r f a c e .

P r o p o s i t i o n 5.1. H d i m M a = 2, then e i t h e r Ma is o r i e n t a b l e (and then i t is d i f f e o m o r p h i c to a s u r f a c e Fg of genus g >_ 1), o r Ma i s d i f f e o m o r p h i c to the K le in bo t t l e K 2.

P r o o f . By h y p o t h e s i s , M p o s s e s s e s a n a t u r a l f i b r a t i o n , i . e . , if M = G / H w h e r e G is s i m p l y c o n n e c t e d and K is an a r b i t r a r y m a x i m a l c o m p a c t s u b g r o u p of G (all s u c h K a r e con juga t e in G), K N H does not depend up to con juga t i on in H on the c h o i c e of K [1]. L e t P = H . (NG(H0)) 0. One s e e s wi thou t d i f f i cu l t y tha t P i s a c l o s e d s u b g r o u p of G. We show tha t up to c o n j u g a t i o n , P n K does not depend on the cho i ce of K, e i t h e r .

A l l the m a x i m a l c o m p a c t s u b g r o u p s of G a r e m u t u a l l y c o n j u g a t e ; l e t K' be one of t hem. In the p r o o f of T h e o r e m i in [1], one of the r e s u l t s p r o v e d a long the w a y is tha t the s u b g r o u p s K N P and K ' N P a r e con juga t e in P . But th is m e a n s tha t a l l the o r b i t s of the a c t i o n of K on G / P have the s a m e o r b i t t y p e , and t h e r e f o r e G / P a d m i t s a n a t u r a l f i b r a t i o n G/P-+ K~G/P, in w h i c h the b a s e is a c o m p a c t m a n i f o l d K\G/P.

We now c o n s i d e r the n a t u r a l m a p K\G/H ~ K \ G / P i n d u c e d by the f i b r a t i o n G / H - - G / P of the K - s p a c e G / H o v e r the K - s p a c e G / P . I t i s c l e a r tha t th is n a t u r a l map is a l s o a f i b r a t i o n , and the f i b e r is K n P/K N H. Since d i m K \ G / H = 2 , we have d imK\G/P ~ 2. We c o n s i d e r the c a s e s dimK\G/P = 2, 1, and 0 s e p a r a t e l y .

L e t d i m K \ G / P = 2 ; we then p r o v e tha t Ma = T 2 o r K 2. F i r s t we show tha t M a is f i n i t e l y c o v e r e d by the t o r u s T 2. By p a s s i n g f r o m M to a s u i t a b l e f in i t e c o v e r i n g M' (so tha t the c o r r e s p o n d i n g M~ is then a f in i te c o v e r i n g of Ma) , and r e p l a c i n g the g r o u p G if n e c e s s a r y , we can a s s u m e tha t G a c t s r e g u l a r l y on M, i . e . , G = NG(H 0) . S , w h e r e S is the s e m i s i m p l e p a r t of G. We t h e r e f o r e ge t tha t P0 "S = G. L e t / ~ b e a c o n n e c t e d s u b - g r o u p c o n t a i n i n g K and c o r r e s p o n d i n g to a m a x i m a l c o m p a c t s u b a l g e b r a T of ~ ( s ince G is s i m p l y c o n n e c t e d , we m a y a s s u m e tha t K ~ S). S ince P i s a t - s u b g r o u p [13], we have f r o m the I w a s a w a d e c o m p o s i t i o n fo r S tha t

P - K = S, and t h e r e f o r e /~. P = S - P = G, i . e . , K" P = G. But then K\G/P = K\I~/K N P, and s i n c e K \ /~ i s known v v

to be an A b e l i a n L ie g r o u p , K \ K / K N P is a t o r u s . This t o r u s is a f in i t e c o v e r i n g of M a.

This i m p l i e s in p a r t i c u l a r t ha t the E u l e r c h a r a c t e r i s t i c • is equa l to z e r o . But t h e r e e x i s t e x a c t l y two c l o s e d s u r f a c e s wi th z e r o E u l e r c h a r a c t e r i s t i c , T 2 and K 2, and t h e r e f o r e Ma = T 2 (we note tha t T 2 has genus g = 1, i . e . , T 2 = F 1) o r K 2.

If dim K\G/P = 1, then dim K N P/K N H ~ l a l s o , and t h e r e f o r e K\G/P = K N P/K N H = S 1. But we then ob t a in fo r M~ = K\G/H a f i b r a t i o n S 1 ~ Ma ~ S 1, w h i c h e a s i l y i m p l i e s t ha t )/(M a) = 0. But then as above we ge t Ma = T 2 o r K 2.

F i n a l l y , a s s u m e tha t dim K\G/P ~ O, i . e . , K" I ) = G. But then K\G/H = K N P \ P / H = K' \G' /D' , w h e r e G T = P o / H o i s a c o n n e c t e d L i e g r o u p ( s ince Ho<~P), D" =HNPo/Ho i s a l a t t i c e in G ' , and K" = K N P o / K N Ho i s a m a x i - m a l c o m p a c t s u b g r o u p of G ' . M o r e o v e r , dim K ' \ G ' / D ' ~ 2, and t h e r e f o r e d i m K ' \ G ' ~ 2. If G v i s s o l v a b l e then ul(M) and 7rl(M a) a r e a l s o s o l v a b l e . But then Ma = T 2 o r K 2, s i n c e t h e r e a r e no o t h e r a s p h e r i c a l c l o s e d s u r f a c e s w i th s o l v a b l e f u n d a m e n t a l g r o u p . T h e r e f o r e , i f G ' = S ~ "R v is a Lev i d e c o m p o s i t i o n fo r G ~ (R' the r a d i a e a l , S ' the s e m i s i m p l e p a r t ) , then we m a y a s s u m e tha t d i m S ' > 0. L e t S ' = S~I .S~ , w h e r e S~t, S~ a r e n o r m a l s u b g r o u p s wi th S~ c o m p a c t , and S~I i s w i thou t c o m p a c t f a c t o r s . I t fo l lows f r o m the c l a s s i f i c a t i o n o f s e m i s i m p l e L i e g r o u p s t ha t i f S~t ~ { e }, then dim S'H/S'H n K ~ 2, and e q u a l i t y i s p o s s i b l e only i f S~i i s l o c a l l y i s o m o r p h i c to SL(2 , R) .

If S~i = { e } then K' ~ S'~, and i t i s t h e r e f o r e e a s y to show tha t ul(Ma) i s s o l v a b l e (and then Ma = T 2 o r K 2, cf . above ) . If S~t ~ { e } , then by the f o r e g o i n g S~t i s l o c a l l y i s o m o r p h i c to SL(2 , R) , and t h e r e f o r e K' ~ S'h.R. But then we m a y a s s u m e tha t G ' = S~t , and s i n c e d i m K ' \ G ' = 2, K ' = SO(2). T h e r e f o r e , Z (G v) i s f in i t e ; but then Z(G') ~ K ' , and we m a y a s s u m e by r e p l a c i n g G ' by G ' / Z ( G i) t ha t G ' = P S L ( 2 , R) . We ge t t ha t M~= S0(2) \PSL(2, R)/D, but th i s m a n i f o l d i s w e l l known to be d i f f e o m o r p h i c to s o m e o r i e n t a b l e s u r f a c e F g , i . e . , Ma = F g .

We note tha t for g >- 2, u~(Fg) i s s e m i s i m p l e , and ul(T 2) and ul(K 2) a r e s o l v a b l e . The f a c t tha t T 2 and K 2 can a c t u a l l y be r e a l i z e d as the b a s e s p a c e s of a n a t u r a l f i b r a t i o n of s o m e c o m p a c t h o m o g e n e o u s s p a c e M c a u s e s no t r o u b l e , s i n c e t h e s e s p a c e s a r e t h e m s e l v e s h o m o g e n e o u s (and we can s i m p l y t ake M = T 2 o r K2). As fo r F g ,

46

fo r g ~> 2 they a r e not homogeneous [see , e .g . , 4.3; this fol lows m o r e s i m p l y f r o m the fac t tha t X(Fg) = 2 - 2g < 0 for g > 2]. Howeve r , in spi te of this we show that each Fg is the base space of a na tu ra l f ib ra t ion for s o m e homogeneous M.

Let G(n) = SL(n, R), F = {aT} a s y s t e m cons i s t i ng of a s ingle white roo t , and c o n s i d e r the a s s o c i a t e d s t a n d a r d pa rabo l i c subgroup UF. I t is ea sy to see that UF = SL(2, R) . R , where R is the r ad ica l .

Wr i t i ng Dg = ~l(Fg), then for e v e r y g -> 2 the g roup Dg is wel l known to be i s o m o r p h i c to a l a t t i ce in PSL(2, R). It tu rns out tha t Dg is a l so i s o m o r p h i c to a la t t ice in SL(2, R). Na tanson has i n f o r m e d the au thor tha t this fac t is ea s i ly de r ived f r o m T h e o r e m s 1 and 2 in [20]; ano the r p roo f of this r e s u l t is con ta ined in [21]. In what fo l lows, D wil l denote a la t t ice in SL(2, R) i s o m o r p h i c to one of the Dg.

Put t ing H ( n ) = D �9 R c G(n) = G and M(n) = G ( n ) / H ( n ) , M(n) is then c o m p a c t , s ince H(n) is u n i f o r m in UF, and U F is ea s i ly s e e n to be u n i f o r m in G. We have K = SO(n) is a m a x i m a l c o m p a c t subg roup of G, and then K fi H = so(n) N H = S0(2) fi D = {e}, s ince D = Dg is wi thout to r s ion . F o r n -> 3, SO(n) is s e m i s i m p l e , and t h e r e - f o r e s ince H n K = {e} we have a na tu ra l f ib ra t ion K / K fi H = SO(n) -~ G(n) /H(n) + K \ G / H = Mo. We have K \ G / H :~

Ur n H \ U r / H = S O ( 2 ) \ S L ( 2 , R ) / D = F g , i . e . , Ma = Fg. Consequen t ly , Fg is in fac t r e a l i z e d as the base space of a na tu ra l f ib ra t ion of c o m p a c t homogeneous M = M(n). We note tha t G(n) is s imp le and d imM(n) - - ~ as n - - oo

We c o n s i d e r the homogeneous space M(n) in m o r e deta i l and prove that the na tu ra l f ib ra t ion fo r M(n) is t r iv ia l . The f iber is Mc = SO(n) and the s t r u c t u r e g roup is Q = SO(n) [2]. T h e r e f o r e , the na tu ra l f ib ra t ion for M(n) is a p r inc ipa l SO(n)-bundle . We find the obs t ruc t ions to cons t ruc t i ng a s ec t ion fo r this bundle.

The s e c t i o n m a y be defined a r b i t r a r i l y on the z e r o - s k e l e t o n of the base M a = Fg ( a s sumed t r i angu lab le ) . The obs t ruc t i on to cont inuing the sec t ion to the o n e - d i m e n s i o n a l ske le ton is an e l e m e n t C[~)~ H ~ (Fg, ~0 (SO(n))L but ~0(SO(n)) = {e } and t h e r e f o r e c~ n) = 0, and the s ec t ion extends to the one - ske l e ton . The o b s t r u c t i o n to ex- tending the s ec t ion to the two- ske l e ton (i.e., to all of Ma s ince d i m M a = 2) is an e l emen t c(~),~ H ~ (Fg, n~(SO (n))) = Z~ (for n -> 3). We show that fo r n -> 3, c~ n) = 0.

The na tu ra l inc lus ions SL(n, R) c-~L(n § l, R) induce imbeddings /~ : M ( n ) C . M ( n + 1), which a r e c l e a r l y morph i s ms of na tu ra l f ib ra t ions :

, S o (n) in ----- so (n+d

I l M(n/ ~ MIn+d

It fol lows f r o m the fune to r i a l i ty of the e h a r a c t e r i s t i c e l a s s e(n) that ~ . (c~ ~+~) = ~,~' whe re ,~ : / /~ (F~, ~ (SO (n)) --~//~ (Fg, .~ (SO (n + '1)) is indueed by a h o m o m o r p h i s m aT(SO(n)) ~- ~(SOin + 1). We f i r s t c o n s i d e r the manifold M(2)~ Then it is known (see, e .g . , [5]) that c ~ = 2 (i -- g) ~ Z = H~ (F~, Z). But i2, is induced by talcing quot ients mod 2, so i2, (c~ 2)) = 0. But this means that c~ ~) = 0 and t h e r e f o r e c2(n~ = 0 a l so for al l n -> 3.

Thus fo r n -> 3, e (n) = 0, the na tu ra l f ib ra t ion has a see t ion , and s ince in the e a s e we c o n s i d e r , the l i b r a - t ion is a p r ine ipa l bundle, i t is t r iv ia l . In p a r t i c u l a r , M(n) is d i f f eomorph ic to Fg x SO(n).

We have p roved that SO(n) x Fg is h o m o g e n e o u s , a l though for g ~ 2 Fg is not homogeneous . In p a r t i c u l a r , 80(3) x Fg is homogeneous and t h e r e f o r e its twofold c o v e r i n g S a x Fg is a l so homogeneous . We note tha t the f ib ra t ion c o n s i d e r e d above is not a na tu ra l f ib ra t ion for M(2), s ince K = SO(2) is not s e m i s i m p l e . Since c~ ) ~ 0~ this f ib ra t ion is non t r iv ia l . M o r e o v e r , i t can be shown that S ~ x Fg is not homogeneous .

We wil l now c o n s i d e r only the e a s e when Ma = Fg fo r g -- 2; then 7rl(M) is s e m i s i m p l e and the r e s u l t s of Sees . 2 -4 apply (of. [21 for the ease when M a = ~ o r Kz).

Le t M = G / H be e o m p a c t and Try(M) ~ Dg. Then by 2.2 we may a s s u m e (3 s e m i s i m p l e . We apply 3.5 to this c a s e .

I t is c l e a r tha t S*(Dg) = PSL(2, I{) and rI(Dg) c o n s i s t s of a s ing le white roo t . By 3.5, a s e m i s i m p l e ~ imply connec ted Lie group G can a c t t r a n s i t i v e l y on s o m e M with ~l (M) ~ Dg if and only i f I I 1 conta ins a r o o t s u b s y s - t e m F s u c h that: a) rio u F conta ins an i so l a t ed white r o o t a [which c o r r e s p o n d s to H(D~, G)]; b) F ~ F(g)~ c) ~ F(~) [this is equ iva len t to the condi t ion ri(D~, G) N F(~) = ~ in 3.5].

47

By 4.1, H 0 = W.Zr.S'.Nr, where W is a connected uniform subgroup of EF'Z~. Let G = S • C, where

C is compact and S is semisimple without compact factors. We show that in this case, we can limit ourselves

to the case when S is simple.

Proposition 5.2. Let M = G/H be compact, ,~I(M) ~ Dg, where g -> 2, and let G be simply connected and

act irreducibly on M. Then G is semisimple and has the form G = S • C, where S is simple and C compact.

Proof. It follows from the irreducibility of G that G is semisimple (cf. 2.3). Let G = S • C, where C is

n

compact and S = • Si, where the S i are simple and noncompact. If F c HI is the root system corresponding /=I

to H (cf. 4.1) then F = ~ F~, where the II~c (rIi)~ are subsystems in the Satake diagrams of the groups S i. The i=l

7z

root ~ (see above) belongs to one of the Fi, say ~'~ Ft. But then it is clear that H o ~ x (Zri. Sri. Nri), in par-

titular H 0 contains a maximal connected triangular subgroup T' of the group S' = x S~. If K' is the connected i--2

subgroup of S' corresponding to a maximal compact subalgebra 7' in 6', then by the lwasawa decomposition we

have [('. T' = S', and therefore Ho /i' ~ S' But then H0 �9 (S, /~' �9 C) = G, whence it follows that the subgroup

S~ �9 K' �9 C of G is transitive on M. Since G acts irreducibly on IV[, this implies that n = 1 and S = S I is simple.

We note that the assertion in Proposition 5.2 is also valid in a more general situation. If M is compact,

homogeneous, and ~ = ~i(M) is semisimple, then we call the group ~ elementary if [I (,7) is connected [or equiv-

alently, S*(~) is simple]. One sees easily that 5.2 is valid not only for ~ ~ Dg but also for any elementary ~.

From 3.5 and 5.2 we obtain

COROLLARY 5.3. A simply connected Lie group is transitive and irreducible on at least one compact

homogeneous M with ~I(M) ~ Dg for g -> 2 if and only if G = S • C, where S is simple, C is compact, and there

exists a subsystem Fc 171 of the Satake diagram for the group S such that:

(i) 1 ~ U ri0 contains an isolated root (we denote it by ~).

(ii) F~F(~) (cf. [18] concerning (o F(g)).

(iii) j r r(g).

The simple Lie groups S (and also their Lie algebras) satisfying conditions (i)-(iii) in 5.3 for at least one F will be called admissible. Using the classification of all the simple Lie algebras, it is easy to pick out the admissible ones from their Satake diagrams: they are AI(~I(n, R)), BI(~o(p, 2n+l-p), 3~p~n) CI (~(n, R), n~3), Di (s0(p, 2n-p), 3 -< p -< n- I), El, EV, EVI, EVIII, EIX, FI, G (in the notation of Table 1 in [7]).

Starting with this list, it is possible using 4.1 to list in explicit form all the compact homogeneous M (up to finite coverings) for which ~i(M) ~ Dg, g -> 2. However, the resulting description is lengthy and cumber- some, and we do not give it here.

LITERATURE CITED

I. V.V. Gorbatsevich, "Topological properties of compact homogeneous spaces," DoM. Akad. Nauk SSSR, 239, No. 5, 1033-1036 (1978).

2. V.V. Gorbatsevich, "On the structure of compact homogeneous spaces," Dokl. Akad. Nauk SSSR, 249,

No. 2, 274-277 (1979). 3. V.V. Gorbatsevich, "On aspherical homogeneous spaces," Mat. Sb., i00, No. 2, 248-265 (1976). 4. V.V. Gorbatsevich, "On Lie groups transitive on compact solvemanifolds," Izv. Akad. Nauk SSSR, 4_~i,

No. 2, 285-307 (1977). 5. L. Auslander, L. Green, and F. Hahn, Flows on Homogeneous Spaces [Russian translation], Mir, Mos-

cow (1966). 6. A.L. Onishchik, "On Lie groups transitive on compact manifolds. If, n Mat. Sb., 74, No. 3, 398-416

(1967). 7. A.L. Onishchik, "Decompositions of reductive Lie groups," Mat. Sb., 8__00, No. 4, 553-599 (1969). 8. A.I. Mal'tsev, "On some classes of infinite solvable groups," Mat. Sb., 28, No. 3, 567-588 (1961). 9. M.I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow

(1972). i0. A.G. Kurosh, Group Theory, Chelsea Publ.

48

11. M. Raganuthan, Discrete Subgroups of Lie Groups [Russian translation], Mir, Moscow (1977).

12. G.D. Mostow, "On the fundamental group of a homogeneous space," Ann. Math., 6_66, No. 2, 249-255

(1957}. 13. A. L. Onishchik, "On Lie groups t r ans i t ive on compac t manifo lds ," Mat. Sb., 71, No. 4, 483-494 (1966)o 14. A. L. Onishchik, "Extension of t r ans i t ive t r a n s f o r m a t i o n g roups , " Izv. Vyssh. Uchebn. Zaved. , Mat.,

No. 3, 53-65 (1977). 15. D. Montgomery , "Simply connected homogeneous s p a c e s , " P roc . Am. Math. Soc . , )~ No. 4, 467-469

(1950). 16. M. Goto and H.-C. Wang, "Nondiscrete uniform subgroups of semisimple Lie groups," Math. Ann., 1_98,

No. 4, 259-286 (1972). 17. G. Mostow, "On the topology of homogeneous spaces of finite measure," in: Syrup. Mut. Ins. Naz. alia

Math., Vol. 16, London-New York (1975), pp. 375-398. 18. V.V. Gorbatsevich, "On a class of decompositions of semisimple Lie groups and algebras," Mat. Sb.,

9__55, No. 2, 294-304 (1974). 19. A.L. Onishchik, "Transitive compact transformation groups," Mat. Sb., 6__00, No. 4, 447-485 (1963). 20. S.M. Natanson, "Invariant direct Fuchsian groups," Usp. Mat. Nauk, 2__77, No. 4, 145-160 (1972). 21. F. Kamber and P. Tondeur, Flat MaI1ifolds, Springer-Verlag, New York (1968).

CARTAN CONJUGACY EQUIVALENCE PROBLEM

P. Ya. Grushko UDC 514.31

Introduction. It is known [i] that the structure function of a locally transitive geometric structure can be used effectively mainly when it is constant. When we go from automorphisms to conjugate automorphisms and from transitive to conjugate-transitive structures, we obtain a larger class. However, in this case the struc- ture function is no longer constant. In this paper we discuss a general theory of such structures, which in-

cludes a criterion for local equivalence in the involutive case and a theorem on the realization of these struc- tures. A new invariant, which we call a fan, is used to obtain these results. It is not defined for all struc- tures, but every locally conjugate-transitive structure has a fan, and therefore the possibility of constructing such a fan is a necessary condition for local conjugate transitivity.

Notgtion. We consider the category of finite dimensional analytic spaces with TM the tangent bundle,

~(M) the frame bundle, f* :TM I --TM 2 the differential of the map f: M I ~ M2, G the Lie algebra of the Lie group G. In addition, Hi,J(V, G) are the Spencer homology groups of bidegree i, j, with boundary operator 0 and cyclic sum operator ft. We use exponential notation to describe the action of a Lie group and its Lie alge- bra on tensors. For example, if G is a group of transformations of a vector space V, i.e., G = Aut V, G ~ End V~ then for F~ V* | G, (2 E A~V * | V ~we have:

PC(v) = Ad~ Fg-Lv, g ~ G, v ~ V,

F~(v) = [~, rv] - - r~v, ~ ~ G, v ~ V, Qg(vt, v2) = gQ(g-lvl, g-lye), vl, v~ ~ v, g E G,

Q~(vl, v2) = )~P(vt, v2) -- p(7~vl, v~) -- Q(vl, ~v2).

I. Concept of Fan

Definition i. A box �9 = {V, N, H, G, W, Z } is a collection consisting of a vector space Vandnotneces- sarily closed Lie subgroups N, H, G, W, Z in AutV such that H~N, G is a normal subgroup of N, W = HoG (=G" H since G is normal}, Z = H fl G. The groups N, H, G, W, Z are, respectively, called the base~ funda- mental, structure, auxiliary, and principal groups of the box ~P. Similar terminology is used for the corre- sponding Lie algebras.

A. A. Zhdanov State University, Irkutsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 22, No. i, pp. 68-80, January-February, 1981. Original article submitted February I0, 1979.

0037-4466/81/2201-0049507.50 �9 1981 Plenum Publishing Corporation 49