Revista de la 95 . Uni6n Matematica Argentina Volumen 42, Nro. 1, 2000, �5-108

Flag Spaces for Reductive Lie Groups

Tim Bratten Facultad de Ciencias Exactas UNICEN Paraje Arroyo Seeo, (7000) Tandil, Argentina e-mail: [email protected]

Abstract

Let Go be a reductive Lie and suppose is the complexification K of a maximal compact of Go. In this study we define complex flag spaces for Go and chara:cterize the isotropy groups for corresponding Go and K-actions.

1 Introduction Suppose Go is a reductive of Harish-Chandra class [5, Section 3] and letG denote the corresponding complex adjoint group. be the complexification of Let K a of Go. By defi.bition, a complexflag spacefo r Go is a homogeneous, complex projective, algebraic G-space. A parabolic subgroup of G can be defined as any subgroup that contains a maximal connected solvable subgroup of G. From the work of Tits and Borel [6], olie knows th�t the complex flag spaceS Y for Go arethe sp�es of the form

Y = G/P where G is a parabolic subgroup. The groups Go and act on Y. P � K The purpose of this paper is to prove a basictechnical result characterizing certain the structure of the isotropy groups for the Go and K-actions in Y. Thischaracter­ ization is fundamental to the program of geometric construction of representations for Go in Y (consider for example [1, Proposition 1 and Lemma I], where the result is without proof, as well as [2] where the result is also but · a special, used used, in well-known context; also mention [3] and [4] where the result of this paper is needed). Althoughwe one an infinitesimal result [9], it seems a general also finds in proof of the structure and decomposition for the isotropy groups does not appear in the existing literature. The of this article is to that void. In particular, our aim fill result here is Theorem 4.4 in Section 4. main Actually, open the scope a little wider in this paper. Specifically, we prove our result thewe context of complex flagspaces for reductive Lie groups (not necessarily in Tim 96 Bratten of Harish-Chandra class). Although quite simple, our definition· and characteriza­ tion of complex flag spaces for reductive Lie groups also does not seem to appear anywhere in the current literature. This context requireSsome simple gen­ enlarged eralizations as well as consideration of spaces, parabolic sub oups, etc. for flag disconnected complex algebraica groups. gr Our paperis organizedas follows. The firstsection the introduction. the second is In sectionwe usher in the class reductive groups [7, e n review some of Lie D fini io 4.29] , key structure theory, and define a correspondingcomplex adjoint t group. In the third section introduce the complex algebraic groups and prove a simple lemma linear about thewe fixed point sets of involutions. We then consider the theory of flag spaces, parabolic , etc. for disconnected complex linear a.lgebraic groups. In the last section we define complex flag spaces for reductive Lie groups and prove our result about the structure of the isotropy groups. We conclude this introduction with a few remarks about our terminology. The nilradical of a Lie algebra is defined to be the radical of the corresponding derived algebra. Thus the nilradical the intersectionof the kernels of finite-dimensional is all irreducible representations [11, Theorem 3.8.1, Theorem3.14.1 and Theorem 3.16.2]. With this definition, the radical of a parabolic subgroup of a complex reductive group has Lie algebra the nilradical of the corresponding parabolic subal­ gebra [12, Section 1.2.2, page 55]. When Go is aLie group with Lie algebra go, then the derived subgroup of is the connected subgroup with Lie algebra. Go [go, go] . A finite-dimensional representation of Go always means a continuousrepresentation in a finite-dimensional complex . Suppose a complex Lie group with G is Lie algebra Then a conjugation of G a continuous involutive automorphism g. is whose derivative is conjugate linear on The conjugation is called compact its g. if fixedpoint set is a compact subset of form of means a closed subgroup G. A real G whose Lie algebra is a of g.

2 Reductive Lie Groups In this section we beginby defining reductive Lie groups and reviewing the Cartan decompositio,n. Next, review the well-known correspondence between complex reductive groups and compactwe Lie groups. We conclude the section by defining and considering what we the complex adjoint group for (in case of a call Go the Harish-Chandra class group, our definition coincides with the usual definition).

The Cartan Decomposition. Go be Lie group Lie algebra Then Let with go. Go is called a reductive Lie a group if:

o has finitely many componentsj (1) G connected (2) The Lie algebra of reductivej Go is (3) The derived subgroup of Go has . finite

Suppose is a reductiveLie group let Gif denotethe of Go and Go. By a classical result of G. Mostow [10, Section 3] , the fact tha.t Go hasfinitely many Flag spaces for reductive Lie groups 97

connected components j1nplies that each connected compact subgroup Mo maximal of G�d is contained in a maximal compact subgroup Ko of Go s�ch that

Ko GOo Mo and Ko . od = Go· n d = GO In addition, all compact subgroups of Go are conjugate by an element of m.a.xim.al° Gido · Let Ko be a maximal compact subgroup of Go and let denote the Lie algebra Jto of Ko ; We define a Gartan complement .60 to go to be an Ad(Ko)-invariant Jto in complement to go such that . Jto in

Cartancomplements to in go are knownto exist and any two are conjugate by an Jto automorphism of go that pointwise fixes Section 3] . Ino fact, the automorphism � be chosen the exponential of a derivation[5, of go. can as Suppose .60 is a Cartan complement to in go. Let E .6 and let g denote the Jto e 0 complex.ification of go. Then it is not hard toshow that adjoint map

is seinisimple. We also observe that the linear map

defined to have eigenvalues in and -1 in is an involutive automorphism of +1 Jto .60 90. is called a Gartan involution of 90 corresponding to Ko . 8 is This analysis of go descends to the group. In particular, one knows that ex.p(.60) a closed regular analytic submanifold of Go and that the exponential map

ex.p : .60-+ ex.p(.60) is an isomorphism of analytic varieties. Letting e E Go denote the identity, we have

Ko exp(.6o) {e} and o Ko . ex.p(.60) n = G =

a so-called Cartan decomposition of Go with respect to Ko . Thus the Cartan invo- ' lution of go descends to an automorphism of the group, by defining

8( . . for k E Ko and E .60. k ex.p(e» = k ex.p(-e) e

Complex Reductive Groups. A com�lex Lie group G is called a complex reductive group if G has finitely many connected components and if a maximal is compact subgroup of G a real formof G. Observe that a complex reductive group is a reductive Lie group. particular, G In Mo is maximalcompact subgroup of G �d is the Lie algebra of Mo then if So 1110 Mo · exp i111o) G = ( Tim Bratten 98 is a Cartan decomposition of G with respect to Mo . From the decomposition each finite-dimensional representa­ Cartan it follows that tion of lifts to holomorphic of there is a finer point Mo a representation G. But involved here: compatible that the group G carr ies a linear is uniquely e mi e homomorphism of d ter ned by th condition that any holomotphic G a complex. linear algebraic algebraic. the · holomorphic into group is in fact Hence representations of are t e e a natural equivalence between the G algebraic and h r is representation the finite-dimensional algebraic finite-dimensional theory of Mo and of G. On the other be a Lie group with Lie algepra .fto let be handlet Ko compact ahd K the complexification of the canonical morphism Ko . SinceKo is compact,

Ko --. K is injective the of Ko form o knows ha complex. Lie and is a real f K. One t t a group is the complexification of a compact form if only if the compact real and real form is a compact subgroup. a reductive group with maximal Thus K is complex maximalcompact subgroup corresponding decomposition Ko and Cartan ex.p ( ) K = Ko . i� . general, when is a complex. reductive group, note that open subgroup of In G we an the fixed point set of a conjugation is a reductive Lie group. More specifically, in G suppose

T:G--.G is a conjugation of C. Then there exists compact conjugation a such that "I:G -+ G' "IT = T"I. From this one can deduce the decomposition for the point of Cartan fixed set T (and hence Property from the Lie gr up) 1 definition of reductive o . The Complex Adjoint Group. p is reductive Lie group with Lie Su pose Go a algebra go. Let g enote the complexifi.cation go and Aut(g) d of let denote the complex. of g. Then adjoint action of on efines a morphism of the 9 9 d complex Lie algebras . Lie(Aut(g)). ad : 9 --. By definition, by Int the complex adjoint group of g, denoted (g), is the connected subgroup of Aut(g) with Lie algebra ad(g}. The adjoint action morphism of groups of Go on 9 defines a Lie Aut(g). Ad:Go --. We define group be the Zariski closure of the complex adjoint G of Go to Ad(Go) in Aut(g). Thus has finitely many components G connected and

Gid = Int (g). Flag for reductive Lie spaces groups 99

particular, is either a or a complex semisimple group with finitely In G . connected cOmponents. many Fix a maximal compact subgroup Ko of Go and let e be a corresponding Cartan . involution of go. We use the same letter to indicate the unique extension of e to Aut(g). Let T be the conjugation of g determined by go. Then T and e commute and normalizes the group Int(g). In addition each TOAd(g) T = Ad(g) and eoAd (g) = A (e(g)) for each E Go. 0 0 e d 9 Thus G is nornialized bY T and e, since Ad(Go) meets each connected component of G. Let

be the product of e and T. Then 'Y is a compact conjugation of G. Indeed, since Ad(Ko) meets each connected component of G, it follows that 'Y a Cartaninvolu­ is tion of G. We refer to . 'Y the associated compact conjugation of G. will as

3 Linear Algebraic Groups and Flag Spaces to associatea family of complex flagspaces to a given reductive Lie group, In need to consider flag spaces for disconnected linear algebraic groups. Although weour concepts. are simple generalizations from the connected they are general­ case, izations which we have not found in the literature. Since a fundament.al tool in the theory of geometric realization of representations is the canonical projection from a full flag space to a general flag space, require that flag spaces be con­ will nected. This requirement leads naturallywe to our definition of parabolic subgroup and Proposition 3.2.

Complex Linear Algebraic Groups. Let G be a complex linear . Then there exists a connected, simply connected, Zariski closed, nilpotent largest of called the unipotent radical U of A Levi factor L of G G. G is a complex subgroup such that

L·U=G and LnU={e}.

Levifactors exist anyand twoare conjugate under U. Indeed, a Levifactor of a G is complex reductive subgroup. In particular, Levi factors are Zariski closed maximal and G is a complex reductive group and only U = {e}. The decomposition of if if G the of a Levi factor with the unipotent radical is called a Levias decomposition of G. The following simple le will be utilized in the nextsection: mma Let G be a complex and suppose · Lemma 3.1 e:G-G 100 Tim Bratten

is a continuous automorphism. Assume Levi preserved und�r (). Let L is a factor Go denote the fixed point set of () in G, let Lo be · the fixed pointset in an let of () L d . Uo denote the fixed point set of () in the Lie algebra of the unipotent. radical U of u G. Then we have the following. exp(tJo) is a closed nilpotent s up of Go and (a) normal ubgro exp(tJo) exp : tJo - is an equivalence of analytic varieties.

L t Go be an open subgroup o Go and put Lo = nLo. Then Go a semidirect (b) e f Go is product of Lo with exp(Uo).

First observe that () stabilizes the unipotent cal is a con­ Proof: Since radi U nected, simplyconnected nilpotent complexu. group it follows that Lie

equivalence of complex algebraic varieties. Since Uo is a subalgebra of u, is an it follows from standard facts about simply connected nilpotent Lie groups that exp(uo) a closed nilpotent subgroup On the other hand, normalizes is of U. Go exp(Uo), since Go � GIJ. Therefore (a) folloWs, since Go contains exp(Uo). To establish (b) let UIJ be the fixed point set of () in U. From part (a) of the lemma, together with the equation

(}(exp(�») = exp((}(�» for � E u it follows that UIJ =exp(Uo). Thus, GIJ is a semidirect product of LIJ with exp(tJo). Therefore the result follows, since every open subgroup of GIJcontains L�d.exp(tJo). • Fix a complex linear algebraic group G with Lie algebra We define Flag Spaces. g. a flag space fo r G to be a connected, homogeneous, complex projective, algebraic G-space. By definition, a Borel subalgebra of a maximalsolvabl e subalgebraof More b g is g. generally, a parabolicsubalgebra of s complexsubalgebra that contains a Borel p g i a subalgebra of One has the following g. properties.

Any two Borel subalgebras of g are conjugate by element of Gid. (1) (2) The normalizer in Gid of a subalgebra anis. the connectedsubgroup parabolic p with Lie algebra p. (3) The G-homogeneous variety of Borel sub,algebrasof projective. X g is

The variety from item is cal1ed the full flag sp ace g (or We define X (3) of of G). of to be the normalizer of Borel b of g. Thus a G in G a subalge ra X=GjB Flag spaces fo r reductive Lie groups 101

(2) where B is a Borel subg;:oup of G. Properties (1) and imply that

. = G and B = B Gid n Gid Bid. Suppose b is a Borel subalgebra of g. Then property (1) implies that any parabolic g subalgebra of is conjugate, by an element of Gid, to a parabolic subalgebra con­ taining b. On the other hand, let p be a parabolic subalgebra of containing b and g. let be the normalizer of pin G. Applying property (1) to p and P, it follows that P any Borel subalgebra of contained in p is conjugate to by -element of g b an pid. Since the normalizer in Gid of b also normalizes p, one deduces that no two distinct parabolic subalgebras of g containing can be conjugate under Thus we obtain b Gid. the following additional property.

(4) Anyparabolic subalgebra of gisconjugate, under Gid, to a unique parabolic subalgebra containing b.

We define a parabolic gubgroup of to be any closed complex subgroup of G that G contains a Borel subgroup of G. Observethat the Lie algebraof a parabolic subgroup is parabolic. However, it is important to realize that the correspondence between parabolic subalgebras and parabolic subgroups that occurs in the connected need not occur in the disconnected case [consider the example later in this sectioncase,] . The following proposition shows that the connected, complex projective algebraic quotients of a complex linear algebraic group G are characterized by the parabolic subgroups of G. This generalizes the standard result ([6]) for the when G is connected. case

Proposition Let G be a complex linear algebraic group. 3.2 P is P is ( ) Suppose a parabolic subgroup ofG with Lie algebra p. Then the normalizera ofp in G. Suppose Y G/Q is a complex algebraic quotient of G. Then Y is connected (b) = and projective if and only if Q is a parabolic subgroup of G.

Proof: To establish (a) let B be a Borel subgroup of G contained in P and let NG (p) denote the normalizer of pinG. Suppose NG (p). Since G B· 9 = Gid can write E we . g = h b for some h Gi and some b E d E B. Thus normalizes p. By property (4) h E Therefore P. h pid. g E To establish (b) we first suppose Y = G/Q is a connected complex projective al­ gebraic quotient of G. Then Gid acts transitively on Y. Thus G Q . and = Qid it follows by the usual facts about connected linear algebraic groups and complete homogeneous spaces that Q n Gid is a parabolic subgroup of Qid. Thus the = Qid Lie algebra of s q Q is a parabolic ubalgebra of g. To see that Q contains a Borel subgroup of G, let b be a Borel subalgebraof contained in and let be the corre­ q B sponding Borel subgroup of G. Suppose b B. Then there exists g ' E Gid such that gQ. But this implies E bQ = Ad(b) Ad(g . q = )q 102 Tim Bratten

(4) li s in one shows that Q Therefore property imp 9 normalizes q. Arguing Ja), is the o e of e as n rmaliz r in G q. Thus b EQ. For the converse, let Q be a o c of et B be a Borel subgroup a par b li subgroup G and l of G contained in Q. Then part (a) of the proposition implies Q is Zariski closed in G. Let Y G / Q be the quotient and let X G B be the full flag space = variety = / for we projection G. Then have a G-equivariant algebraic 7r:X-+Y.

It follows immediately that Y is co ected andcomplete. Thus the result is proved, since a complete G-homogeneous algebraicnn variety is necessarilyprojecti ve .• When is a connected reductive group and a Borel Example. G complex b is subalgebra of then there is a well-known 1 - 1 correspondence between the G­ equivariant equivalence9 classes of flag spaces for G and the parabolic subaigebras of containing However, this need not be true when G is disconnected. 9 b. For example, suppose G is the automorphism group of the complex willfollow from our analysis that G not connected . Let = .5[(3, q (it is ) be the Cartan9 subalgebra of consisting of the diagonal matrices in let be ac and b Borel subalgebra of containing.9 If is a root ofc in we let U 9 denote the g, g � corresponding root subspace.9 Let c. anda be the two simple roots of 9 Then in b. there exist exactly four parabolic alsubalgebras a2 of containing the twoc nontrivial b, cases being: 9

Let be the complex automorphism of defined by {} 9 for E {}(�) =:: _�t � .5[(3, q

denotes the transpose of the matrix Then is (}-stable and acts - 1 on (�t �). {} the roots of in Let be an element from the normalizerc of in representingas g. 9 Gid the longest elementc In the of in Then a straightforwardc calculation g. shows that c

follows that neither one of the norma1izers or can contain a It Na (PU1) Na (PU2) Borel subgroup of Therefore there o two flag spaces for G: the full flag G. are nly space X and the trivi.& one point space.

4 Complex Flag Spaces and the Isotropy Groups this section o e of In ciat a family complex flag spaces to a reductive Lie group Go and characterizewe ass both the structure of otro groups the is py in Go well the structure of the isotropy groups for the natural action of the complexi.ficationas as of a maximal compactsubgroup of Go. T main result is 4.4. he of our study Theorem · Flag spaces for reductive Lie groups 103

Throughout th� in er a; ieductive Lie group Go with Lie algebra go ema we fix and ' complexified r�braa g We Jet denote complexconj ugation of determined Lie ..• the 9 by go. We a compact-subgroup1" Ko of Go, aswe1l a Cartan involution fix maximal determined by Ko . The group denotes the Complexificationas of Ko . The Lie (J K algebras of Ko and. K wiU be denoted Py and respectively. .!'to A, Complex Flag Spac,e.s for a Reductive Lie Group. Let G be the complex adjoint group of Go [Section 2]. We define a complexflag sPace lor Go to be a flag space of G. Y Suppose a complex flag space of Go.' Thus an Int(g)-conjugacy classes Y is Y is of 'parabolic subalge1:>ras of that remainsinvariant underthe adjoint actionQf Go. We 9 want to this correspondence . between points in arid parabolic subalgebras make Y of explicit. In particular, if Lie(G) denotes. the ,Lie algebra of G and if 9 T:g-+g a derivation of belonging to Lie(G), then there ·existsa unique [g,g] such is 9 e that E T = ad . ( Thus can (and will)identify Lie subalgebras of Lie(G) with Lie subalgebras of 9 containedwe in [g, g] . Let be the center of Then 3 g. =3 [g, g]. 9 , G) Fbr y. let Py denote the parabolic subgroup of G that stabilizes y. Then Y, Lie(Py)E a weparabolic subalgebraof [g, g] and is Lie(Py) py = 3 G) the unique parabolic subalgebraof containing Lie(Py). With this identification, is 9 for Go, the point . corresponds to the parabolic subalgebra Ad(g )py. 9 E 9 Y Observe that there a natural algebraic K-action on particular, the adjoint is Y. In action of Ko on determines an algebraic action of on (also referred to the 9 K 9 adjoint action) which, in turn, defines a morphism of algebraic groups as Ad :K-+G. We adopt the following convention for denoting the isotropy groups in Go, Ko and Y �. For y let Go[y], Ko [y] and Ky denote the corresponding stabilizers. In particular,E Go [y]we, Ko [y] and Ky are the normalizers .of py in each of the respective groups. We reserve the notation K[yJ to indicate the Zariski closure of Ko[Y] in Ky . Thus K[y] is the complexifidation of Ko [y].

The Structure of the. Isotropy Groups. Suppose p is a parabolic subalgebraof and let denote the of We define a Levifa ctor [of to be a complex 9 u nilr�cal p. . p subalgebra [ of such that ' p [nu= {O} and [+u=,p. Tiro Bratten 104 "

s Y for let denote the unipotent We fix complex flag pace For y Y Uy aof the parabolic subgroup Go. Eput we Then is the radical Py and Uy =Lie(Uy). Uy of there a correSpondencewe between Levi factors of Py and , nilradical I'y and is 1 - 1 Levi factorsof Specifically; if L factor of then I'Y. is a Levi Py Lie{L) '[ = 3 + is the corresponding Levi. factor of I'y. G [Section2] . We associatea specific Let"I be the associatedcompact conjugation of by way of the following lemma. Levifactor to each y E Y 4.1 Suppose and define Lemma y E Y '" . L = Py n "I(Py). Then L is a Levi factor of Py .

Proof: Zariski Since L is a closed subgroup of (,i invariant under "I (and thus a ,complex reductive group) it suffices to check that L contains amaximal compact subgroup of Py • Let Mo be a maximal compact subgroup of Py and let Fo be the fixed point set of in Since Fa is a compact subgroup of G there exists "I G. maximal G gMag-l Fa . 9 such that � On the other hand, Fa acts transitively on Y, thereE exists h E Fa such that hgPy -1h-1 = Py . It follows that hg E Py . Thereforeso g Fa hgMag-1h-1 = Py n is a maximal compact subgroup of contained in L. Py •

the "I-stable Levi factor of The Levi factor L of the previous le will be called Py • Define mma

1=3 + Lie(L) the -corresponding "I-stable Levi factor of In fact, using the same letter to I'Y. "I denote the conjugation of g defined by it follows that

Since it is known that the centralizer of in is trivial, it follows from the Levi I Uy decomposition for Py that is the normalizer in Py L of I, • A complexsubalgebra be called bistable t � g will if and O(t) = r{t) = t t. Let ma be the real form given bythe fixedpoint set of Observetha t is a � g "I. if t . bistablesubal gebra of then ma n is a real form of Since there exiflts g t t. a compact

) " J Flag spaces for reductive Lie groups 105

. Lie group with Lie algebra it follows that every subalgebra of o is reductive. Thus each bistable subalgebramo, of a .m 9 is complex We define a point y E Y to be special the corresponding parabolic subalgebra PlI if contains a bistable Cartan subalgebra of One knows that each point in Y is Go- g. in conjugate (and K-conjugate) to a special point [8]. Therefore, order to describe the stabilizers Go[Y] and KlI it suffices to consider special points. We a special fix point Let [ be the 'Y-stabl� Levi factor of PlI and put y E Y. j = [nT(£ ).

Thus j is the largest bistable subalgebra of PlI' We refer to j as the fullbistable subalgebra of PlI' Let Uy be the nilradical of PlI and define the following two complex subalgebras of PlI:

Suppose yE special. Then, wing the notations introduced above, Lemma 4.2 Y is we have the following. ( ) j El) [nU(T) and j El) [nu(O) are parabolic subalgebras of[ with Lem factor j and corresponda ing nilradicals [nu( and [ n u( respectively. 1') 0), (b) [=j El) [nU(T) El) [nu(o) . ( ) U(T) and u(O) are the nilradicals of plInT (PlI) and Py nO(PlI),respectively. e PlIn T(P ) and PlI (d) 1/ = jEl) U(T) n O(PlI)= j El)u(O) .

Proof: Observe that normalizes both and Let be a bistable j [nT( Uy) [n O(Uy). c Cartan subalgebra contained in Thus is a Cartan subalgebra of Let py. c j. :E+(c,j) be a positive system of roots for c in j and let :E(C,[nT(Uy)) and :E(c,[nO(Uy))denote the roots of c in [nT(Uy) and [nO( Uy),respectively. Observe that the conjugation 'Y TO acts as -I on the roots of c in It follows that = g.

are each positive system of roots for in [ and that c :E(c,[n T( Uy))= -:E( [nO( Uy)). c, Thus (a) and(b ) follow. We establish c d) for the Lie algebra The proof for is ( ) and ( PlIn T(Py). Py n O(PlI) identical. Observe that u( is a subalgebra ofPlI n T(PlI)normalized by j. From part 1') (a) it follows that [nT(Uy) is contained in the nilradical of a Borel subalgebra El) ull of Therefore is a nilpotent ideal of We can thus deduce the direct g. U(T) jEl)U(T) . sum decomposition

as well as the fact that u( iscontained in the derived algebra ofPlI n T(P1/)directly 1') from the root space decomposition for the adjoint action of c in P1/nT (py). Thus (c) and (d follow. - ) • Bratten Tim 106

descends level of the groups. particular, We now show that the above lemma to In suppose E Y is special and let L Py be the -y-stable Levi factor. Let y �

J = Lnr{L).

Observe that J is the largest subgroup of Py invariant under both r and O. Also observe that the corresponding Lie algebra

j Lie{J . = 3 $ ) is the full bistable subalgebra of py.

Lemma 4.3 Suppose y E Y is special. Then, using the notations from above we have the following. ( ) The unipotent radicals ofpy nr{Py) and py nO{py) are exp{u(r)) and exp(u(O)), respea ctively. a Levi factor of both and (b) J is Py n r(Py) Py n O(Py).

Proof: We establish (a) and (b) for the group py nr(Py). The proof for pynO{py) is identical. Since u{ r) is complex subalgebra of the nilradical of a Borel subalgebra of [g, g] it follows that a

exp(u(r)) G � is connected, simply connected, Zariski closed, nilpotent subgroup of G. addi­ a In tion, u{r) is the nilradical of py n r{py), so that exp(u{r)) is normal in Py r(Py). On the other hand, J is a complex reductive group since J is Zariski closedn in G and invariant under Therefore it follows from the previous lemma that -y.

J n p(u( = {e} ex r)) and that J.exp(u(r)) is an open subgroup of Py n r(Py). This establishes (a). To establish (b), let Q be a factor of containing Suppose lE Levi Py nr{Py) .]id . L, E Uy and l . normalizes the identity component Then centralizes the g = Jid. ugroup l l-l. Sinceu this last group contains a Cartan subgroupu of it follows . .]id . Cid that is trivial. Therefore Q is contained in L. Similarly one shows Q is contained in r(L)u . This proves (c) , .• Observe that J is the normalizer of j in each of the two groups Py n r(Py) and Py n O(Py).

We arenow ready to describe the stabilizers Go [y] and Ky• Let jo = go n j and

let u(r)o = go nu(r) . Then jo and u(r)o are real forms of j and u(r), respectively. Observe that n py n j is the set of -y-fixed vectors in jo well a real � = � form of j. follows from Lemma 4.3 that as as Jt n It

) Flag spaces for reductive Lie groups 107

We now show that this result to the groups. Let descends the normalizer of j in G y Jo = o[ ] recall Zariski closure of Ko y in K . and that K[y] is the [ ] y Theorem 4.4 Sup ose E Y is special. Then, using the notationsfrom above, we P y ha'IJe thefo llowing.. is (a) exp(u(r}o) a closed, connected, simply connected, nilpotent normal subgroup of and o y is a semidirectproduct of Jo G [Y] G [ ] with exp(u(r)o). o is reductive L.ie group with Lie algebrajOi mazimal compact subgroup (b) Jo a. Ko [y] and corresponding Cartan involution given by the restriction of() to Jo . exp(Jtnu«()) a closed, connected, simply connected, nilpotent normal subgroup (c) is of Ky and Ky is a semidirect product of K[y] with exp(Jtnu«() ). is the normalizer of j in (d) K[y] Ky Proof: Let J., be the fixed point set of r in J. Then J., is a reductive Lie group and a real form of J. Since J is the normalizer of j in Py n r(Py) and since J is r-stable, it follows that ( Ad Jo) = Ad(Go[Y]) n.J., is an open subgroup of J., . Using Lemma 3.1, it follows that Ad(Go[Y]) be written thesemidirect product � as d( y ) d ) A Go[ ] = A (Jo . exp(u(r)o). Therefore (a) follows, since exp(u(r)o)is a simplyconnected nilpotent Lie group and since the of the adjoint morphism is contained in Jo • Similarly, one checks that Ky is a semidirect product of the normalizer of j in Ky with exp(Jtn u«()). To establish (b), first note that J has Lie algebra we o

= Lie(J.,) jo 30 (9 Next, where = go is the Lie algebra of the kernel of adjoint morphism. 30 3 n we observethat [J�d ,J!.d] is a connectedsemisimple Lie groupwith finite centerand that the adjoint morphism defines a finite covering

,fad] _ [.f,.d , .f,.d ]. Ad : [,fad, Therefore �d ,Jbd and since and [J ] has finite center. Since Ad(Ko[y]) S; L n r(L) J ( therefore Jo ) ()-stable, it remains to show that is closed under the Gartan are Jo decomposition determined by(). Let .6o be the -J. clgenspace of () in go and suppose k k ' E K, E p( ) and that = ( ) E Jo • Then E Jo . Hence e ex .6o 9 exp e exp(2e) Ad(exp(2e» normaliZes py and j. Since �:g-g semisimple operator it follows that normalizes py and ;. Therefore is a Ad(exp(e» p( ) and k J • This establishes (b). ex e E Jo E o 108 Tim Bratten

(d), let a le, To establish N be the normalizet of j in KfI ' Observe that � Nis(J st b since show isaroouctive J is. Arguing as above, in the case ()f Jo , one caD. that N Lie group with Cartan decomposition

• N = Ko [y] exp(i(� n pfl))' proves the theorem. This •

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Recibido 19 de Noviembre de 1999 Modificada Version 15 de Febrero de 2001 Aceptado . 20 de Febrero de 2001