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Finiteness of Orbit Structure for Real Flag Manifolds

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Finiteness of Orbit Structure for Real Flag Manifolds JOSEPH A. WOLF* FINITENESS OF ORBIT STRUCTURE FOR REAL FLAG MANIFOLDS ABSTRACT. Let G be a reductive real Lie group, a an involutive automorphism of (7, and L = G u the fixed point set of a. It is shown that G has only finitely many L-conjugacy classes of parabolic subgroups, so if P is a parabolic subgroup of G then there are only finitely many L-orbits on the real flag manifold G/P. This is done by showing that G has only finitely many L-conjugacy classes of a-stable Cartan subgroups. These results extend known facts for the case where G is a complex group and L is a real form of G. Key words and phrases: flag manifold, reductive Lie group, semisimple Lie group, para- bolic subgroup, Caftan subgroup, Cartan subalgebra. AMS Subject Classification (1970) Primary: 22-50, 22-70, 53-66 Secondary: 14-45, 57-47 1. INTRODUCTION One knows [4] that there are only finitely many conjugacy classes of Cartan subalgebras in a reductive real Lie algebra. If G is a complex reductive Lie group and L is a real form, it follows [8] that there are only finitely many L-eonjugacy classes of parabolic subgroups of G. In particular, if X= G/P is a complex flag manifold of G, then [8] there are only finitely many L-orbits on X. Here we extend these results to the case where G is a reductive real Lie group and L is the fixed point set of an involutive automorphism, and we indicate the scope of applicability of the extension. 2. CONJUGACY OF CARTAN SUBALGEBRAS If G is a Lie group then Go denotes its identity component, g denotes its Lie algebra, and Int (g) denotes the inner automorphism group {Ad (g):g e Go) of g. THEOREM 1. Let g be a reductive real Lie algebra, a an involutive auto- morphism of g, and 1= g~ its fixed point set. Let L o denote the analytic sub- group oflnt (g ) for I. Then there are only finitely many Lo-conjugacy classes of a-stable Cartan subalgebras of g. Proof. It suffices to consider the case where g is semisimple and has no proper a-stable ideal. We now assume that, and we fix the notation g = I +m where rrt= {x~g:tr(x)= -x}. * Research partially supported by NSF Grant GP-16651. Geometriae Dedicata 3 (1974) 377-384. All Rights Reserved Copyright © 1974 by D. Reidel Publishing Company, Dordrecht-Holland 378 JOSEPH A. WOLF Case 1: Int(g) is compact. This case follows directly from Kostant's proof [4] that g* = I +ira has only finitely many conjugacy classes of Cartan subal- gebras. Kostant shows that every Cartan subalgebra of g* is conjugate to a a-stable one and that g* has only finitely many Lo-conjugacy classes of a-stable Cartan subalgebras D*. Then ~)* ~ D = ([9* n I) + i(D* n ira) gives our finiteness assertion. Case 2: Int(g) is complex and a is conjugate-linear. Then I is a real form of g, and our assertion is Kostant's result [4] on finiteness of the number of conjugacy classes of Cartan subalgebras of I. Case 3: Int (g) is complex and a is complex-linear. This is the nontrivial case. Fix a Cartan involution 8 of g that commutes with a. Its fixed point set go is a a-stable compact real form of g, so go=iO+m 0 where 1°=Ing° and m °=mc~g°. Let Lg denote the analytic subgroup of Lo for [o. Case 1 tells us that gO has only finitely many Lo°-conjugacy classes of a-stable Caftan subalgebras. Since gO is a real form of g, this says that g has only finitely many Lo°-conjugacy classes of 0-stable a-stable Cartan subalgebras. L~-conjugacy implies Lo- conjugacy. Now we need only show that every Lo-conjugacy class of a-stable Caftan subalgebras of g contains a 0-stable algebra. Let ~ be an Lo-conjugacy class of a-stable Cartan subalgebras of g. We first consider the situation if D~¢' then D:m i.e. [)c~I=0. Fix t)~ and let [)~ be its real form spanned by the roots. If ~b is an b-root then a*q~= -~b because Dcrrt. Thus g has a a-stable compact real form g' with i D# = g'. Let 0' denote the Cartan involution of g that is complex con- jugation over g'. Evidently 0' (t))= t). Thus, to exhibit a 8-stable element of it suffices to prove that 0 and 0' are L0-conjugate, and for this we need only check that g'= gO' is Lo-conjugate to gO. As g' is a-stable, I °' = I c~ g' is a compact real form of I, hence is Lo-con- jugate to I °. Thus we may assume I°'= I °. That assumption made, g'= = I ° +m °' just as gO= i0 +m o. Since I ° is a real form of I, I°c 0 °r and 00' is complex linear, we now have I ~ goo'. (=8 '-1.8 is a complex linear automorphism of g and trivial on [, so (Ira commutes with every element of Loire. If ~1,~= + 1 then 0 = + 0' on m, so m °' = rrt e or m °" =im °. If ra °' = m ° then g' = go as desired. If m °" = ira ° then g'= I ° +ira °, which is noncompact, contradicting the fact that 8' is a Cartan involution. Now we assume ~1,,~ + 1. g is simple because it has no proper ORBIT STRUCTURE FOR REAL FLAG MANIFOLDS 379 a-stable ideal, so now (go, p) is an irreducible hermitian symmetric pair, and re=m+ +m_ direct sum of I-invariant I-irreducible subspaces that are interchanged by 0 and also by 8'. Thus we have a complex number g with ~1~+=~ and ~l~.=~. Let x± era± and compute x+ +x_ = 02(x+ +x_) = (o'¢)'(x+ + x_) = O'~(O'~x_ + O'~x+) = + Thus ~ = 1 and ~ = Ad (z) where z is central in L~. Now Om°" = O' Ad (z) m °' = • = 0'11l °" = 1110', SO m e' = (m"" n m °) + (m °' c~ ira"). The second summand vanishes because g' is compact. We conclude g'= ga. In summary: if JY is an Lo-conjugacy class of a-stable Caftan subalgebras of g, and if be.~ implies I~=m, we have shown that ~Y' contains a P-stable algebra. Now let .Yg be any Lo-conjugacy class of a-stable Cartan subalgebras of g. Let ~)~ ~¢g and split [~ = (b n I)+ (I~ nm). Passing to the derived algebra of the g-centralizer of ~ n I, we reduce to the case just considered, obtaining a P-stable Lo-conjugate of ~). This completes the argument for Case 3. Case 4: the general case. Let n=dimg, r=rank g and N=(n) - 1. Use Pliicker coordinates to view the Grassmannian of r-dimensional subspaces of gc as a subvariety of the complex projective space PN(C). Let Loc denote the analytic subgroup of Int(g c) with Lie algebra I c. Now every LC-eon - jugacy class of Cartan subalgebras of gc is an LC-orbit on pN (C) under the rational representation A' (Adln t (gc)lLc). If O is such an orbit, then [6, Lemma 1.1 ] ~ c~ P N(R) is a finite union of Lo-orbits. Case 4 now follows from Cases 2 and 3. Q.E.D. 3. CONJUGACY OF PARABOLIC SUBGROUPS If G is a connected complex semisimple Lie group with Lie algebra g, and if P is a complex Lie subgroup with Lie algebra p, then one has equivalent conditions (i) G/P is compact, (ii) G/P is a compact simply connected kaehler manifold, 380 JOSEPH A. WOLF (iii) G/P is a complex projective variety, (iv) G/P is a closed G-orbit in a projective representation. Under those circumstances, P is a parabolic subgroup of G, p is a parabolic subalgebra of g, and one can see that P= {#~G:Ad(g) p =p}. Let G be a complex Lie group, g its Lie algebra, ~ the solvable radical of fl, and n:g ~ fl/~ the projection. The parabolic subalgebras of g are the n -1 (q) where q is a parabolic subalgebra of g[~. The parabolic subgroups of G are the normalizers P= {g~G:Ad(g) p =p} where p is a parabolic subal- gebra of g; then p is the Lie algebra of P. Let G be a real Lie group and fl its Lie algebra. Then parabolic subalgebra of g means a subalgebra p = g c~ q where q is a parabolic subalgebra of gc stable under complex conjugation over ft. The parabolic subgroups of G are the normalizers P = {g ~ G: Ad (g) p = p } where p is a parabolic subalgebra of fl, and then P has Lie algebra p. THEOREM 2. Let G be a real Lie group, tr an involutive automorphism of G, and L= G° its fixed point set. Then there are only finitely many Lo-con- jugacy classes of parabolic subgroups of G. Proof. We will show that g has only finitely many Lo-eonjugacy classes of parabolic subalgebras. For this we may assume G connected and simply connected. Now G has ¢-stable Levi decomposition GI" S where S is the solvable radical and we may replace G by its semisimple quotient G/S ~- Gt. Thus we may assume g semisimple. Let p be a parabolic subalgebra of g. Then p c~ a (p) is a-stable and con- tains a Cartan subalgebra of g, so it contains a a-stable Cartan subalgebra D of g.
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