Cayley Transforms and Orbit Structure in Complex Flag Manifolds

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.

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    15 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us