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Cayley Transforms and Orbit Structure in Complex Flag Manifolds

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Cayley Transforms and Orbit Structure in Complex Flag Manifolds Transformation Groups, Vol. 2, No. 4, 1997, pp. 391-405 (~ Birkh/iuser Boston (1997) CAYLEY TRANSFORMS AND ORBIT STRUCTURE IN COMPLEX FLAG MANIFOLDS J. A. WOLF* R. ZIERAU** Department of Mathematics Department of Mathematics University of California Oklahoma State University Berkeley, CA 94720-3840 Stillwater, Oklahoma 74074 U.S.A. U.S.A. j awolfQmath.berkeley.edu [email protected] Abstract. Let Z = G/Q be a complex flag manifold and Go a real form of G. Suppose that Go is the analytic automorphism group of an irreducible bounded symmetric domain and that some open G0-orbit on Z is a semisimple symmetric space. Then the G0-orbit structure of Z is described explicitly by the partial Cayley transforms of a certain hermitian symmetric sub-flag F C Z. This extends the results and simplifies the proof for the classical orbit structure description of [10] and [11], which applies when F = Z. Introduction Let Z = G/Q be a complex flag manifold where G is a complex simple Lie group and Q is a parabolic subgroup. Let Go C G be a real form of hermitian type. In other words Go corresponds to a bounded symmetric domain B = Go/Ko. Suppose that at least one open G0-orbit on Z is a semisimple symmetric space. The main result, Theorem 3.8, gives a detailed description of the G0- orbits in Z in terms of partial Cayley transforms. The orbits are given by DF,~ = Go(crc~z) where F and E are disjoint subsets of a certain set of strongly orthogonal roots, cr and c~ are the corresponding Cayley trans- forms, and z is a certain base point in Z. We give precise conditions for two orbits Dr,~, to be equal, for one to be contained in the closure of another, for one to be open, etc.. The case where Z is the compact hermitian symmetric space dual to Go/Ko was worked out directly by Wolf in [10] and [11], worked out from earlier results of Kors and Wolf [13] by Takeuchi [9] and again, later, by Lassalle [3]. The proof in our more general setting is independent of [10], *Research partially supported by N.S.F. Grant DMS 93 21285 **Research partially supported by N.S.F. Grant DMS 93 03224 and by hospi- tality of the MSRI and the Institute for Advanced Study Received February 23, 1996. Accepted July 21, 1997. Typeset by .AA4~S-TEX 392 J. A. WOLF AND R. ZIERAU [11] and [13]. It is self contained except for a few general results from [10] on open orbits in a flag manifold and a decomposition in [14]. Some points of the proof simplify the corresponding considerations in [10] and [11], as noted in Remarks 3.10 and 3.11. See Wolf [10] for the more general setting of G0-orbits on general complex flag manifolds G/Q with no symmetry requirements. See Matsuki [5] for the more general setting of H-orbits on G/Q where H is any subgroup that is, up to finite index, the fixed point set of an involutive automorphism of G. See Makarevi~ [4] for analogous considerations on symmetric R-spaces from the Jordan algebra viewpoint, where (although not formulated this way in [4]) primitive idempotents replace strongly orthogonal positive complementary roots as in Kor~nyi-Wolf [2] and Drucker [1]. And see Richardson and Springer [7], [8] for the the dual setting of K-orbits on G/B where B is a Borel subgroup of G. The results are much more complicated in these more general settings. The virtue of the present paper is that it picks out a few somewhat more general settings in which the results are extremely simple. Notation and some basic definitions are established in Section 1. Then in Section 2 we work out the geometric basis for the G0-orbit structure of Z. We first prove every open G0-orbit on Z is symmetric, and at least one such orbit D = Go/Lo takes part in a holomorphic double fibration D +- Go/(LonKo) ~ B. Fix that open orbit D C Z. We may assume that the base point z C Z is chosen so that the isotropy subgroup L0 C Go is the identity component of the fixed point set G~ of an involutive automorphism a that commutes with the Cartan involution 0. Then 0or is an involutive automorphism. Let M and M0 denote the identity components of its respective fixed point sets on G and Go. We then prove that Mo(z) is an irreducible bounded symmetric domain, that F = M(z) is an hermitian symmetric space and a complex sub-flag of Z, and that Mo(z) C F is the Borel embedding. In Section 3 we recall the partial Cayley transform theory [2], [13] for Mo(z) C F, and use it to specify the orbits Dr,~ = Go(crc~z) C Z. Then we state the main result as Theorem 3.8. The proof is carried out in Section 4 for open orbits and in Section 5 for orbits in general. It follows that the G0-orbit structure of Z is exactly the same as the M0-orbit structure of F. The appendix contains a distance formula that simplifies our arguments under certain circumstances and gives an interpretation of the results in Section 4 in terms of Riemannian distance in Z. 1. The double fibration Fix a connected simply connected complex simple Lie group G and a parabolic subgroup Q. This defines a connected irreducible complex flag manifold Z = G/Q. Let Go C G be arealform, let go C g be the cor- responding real form of the Lie algebra of G, and fix a Cartan involution 0 of Go and go. We extend 0 to a holomorphic automorphism of G and a CAYLEY TRANSFORMS AND ORBIT STRUCTURE 393 complex linear automorphism of 9, thus decomposing (1.1) 9 = ~ +s and 90 = [o + So, decomposition into • 1 eigenspaces of 0. Then the fixed point set Ko = Go~ is a maximal compact subgroup of Go, Ko has Lie algebra e0, and K = G e is the complexification of Ko. Ko is connected and is the Go-normalizer of ~0, and K is connected because G is connected and simply connected. The subspace 9u = eo + ~ So C 9 is a compact real form of 9. The corresponding real analytic subgroup G~ C G is a compact real form of G. We can view Z as the space of G-conjugates of q. Then gQ = z E Z = G/Q corresponds to Qz = Ad(g)Q = {g C G I g(z) = z} as well as its Lie algebra q~ = Ad(g)q. From this point on we assume that Go is of hermitian symmetric type, that is, (1.2) s = ~+ | ~_ where K0 acts irreducibly on each of s+ and s_ = ~++ where { ~ ~ denotes complex conjugation of 9 over 90. Set S+ = exp(~+). So S_ = S+ where g ~ ~ also denotes complex conjugation of G over Go. Then p = ~ + ~_ is a parabolic subalgebra of 9 , (1.3) P = KS_ is a parabolic subgroup of G and X = G/P is an hermitian symmetric flag manifold. Note that ~+ represents the holomorphic tangent space of X and s_ = ~++ represents the antiholomorphic tangent space. As above we can view X as the space of G-conjugates of p. Then Go/(Go N P) = Go/Ko = Ad(Go)p is an open subset of X and thus inherits an invariant complex structure. Define B = Go/Ko : (1.4) symmetric space Go/Ko with the complex structure from X. The distinction between s_ and s+ in (1.2) is made by a choice of positive root system A + -- A+(9, ~) for 9 relative to a Cartan subalgebra I] -= ~ C of 9- The choice is made so that ~+ is spanned by positive root spaces and consequently s_ is spanned by negative root spaces. Under our standing assumption (1.2), Theorems 2.12 and 4.5 of [10] say that Go(z) is an open Go-orbit if and only if q~ contains a compact Caftan subalgebra. Since all compact Caftan subalgebras are Go-conjugate we have (1.5) a G0-orbit in Z is open iff it contains some z such that ~ C qz. 394 J. A. WOLF AND R. ZIERAU Now fix an open orbit D = Go(Z) with z chosen as in (1.5). Write q for qz and decompose q in its Levi decomposition; q = [ + r_ with O C [ and r_ the nilradical of q. Since [J0 C t~0, complex conjugation acts on root spaces by ~ = g_~. Decomposing q into D-root spaces we see that q N~ = [. Thus Go(z) = Go/Lo with L0 = Go n Q and L0 is a connected real reductive group having Lie algebra [0 = [ n fl0- In the terminology of [10] we have shown that all open orbits for Go (satisfying (1.2)) are measurable. Similarly Lu = G~ n Q is connected and is the compact real form of L, and it has Lie algebra [u = IJu N [. 1.6. Definition. The open orbit D = Go(z) C Z is symmetric if it has the structure of a pseudoriemannian symmetric space for Go, that is, if L0 is open in the fixed point set G~ of an involutive automorphism 1 cr of Go. 1.7. Remark. Suppose D is a symmetric orbit in Z. As mentioned above, one may choose the base point z, and the Levi factor [ ofqz, so that [1 C [. It follows that the involutive automorphism ~ is conjugation by some element of H.
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