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 group and suppose is the complexification K of a maximal compact subgroup 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 Lie group of Harish-Chandra class [5, Section 3] and letG denote the corresponding complex adjoint group. be the complexification of Let K a maximal compact subgroup 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 subgroups, 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 unipotent 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 vector space. 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 real form 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 center. finite Suppose is a reductiveLie group let Gif denotethe identity component 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 algebraic structure 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 representation theory 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 image 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. automorphism group of g. Then adjoint action of on efines a morphism of the 9 9 d complex Lie algebras . Lie(Aut(g)). ad : 9 --.