Equivalence of domains arising from duality of orbits on flag manifolds II
Author:
Toshihiko Matsuki
Journal:
Proc. Amer. Math. Soc. 134 (2006), 34233428
MSC (2000):
Primary 14M15, 22E15, 22E46, 32M05
Published electronically:
May 31, 2006
MathSciNet review:
2240651
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: S. Gindikin and the author defined a  invariant subset of for each orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the AkhiezerGindikin domain if is of nonholomorphic type. This conjecture was proved for closed in the works of J. A. Wolf, R. Zierau, G. Fels, A. Huckleberry and the author. It was also proved for open by the author. In this paper, we prove the conjecture for all the other orbits when is of nonHermitian type.
 [A]
K. Aomoto, On some double coset decompositions of complex semisimple Lie groups, J. Math. Soc. Japan 18 (1966), 144. MR 0191994 (33:221)
 [AG]
D. N. Akhiezer and S. G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 112. MR 1032920 (91a:32047)
 [B]
L. Barchini, Stein extensions of real symmetric spaces and the geometry of the flag manifold, Math. Ann. 326 (2003), 331346. MR 1990913 (2004d:22007)
 [FH]
G. Fels and A. Huckleberry, Characterization of cycle domains via Kobayashi hyperbolicity, Bull. Soc. Math. France 133 (2005), 121144.MR 2145022
 [GM1]
S. Gindikin and T. Matsuki, Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds, Transform. Groups 8 (2003), 333376. MR 2015255 (2005b:22017)
 [GM2]
S. Gindikin and T. Matsuki, A remark on Schubert cells and the duality of orbits on flag manifolds, J. Math. Soc. Japan 57 (2005), 157165. MR 2114726 (2005j:14070)
 [H]
A. Huckleberry, On certain domains in cycle spaces of flag manifolds, Math. Ann. 323 (2002), 797810. MR 1924279 (2003g:32037)
 [M1]
T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331357. MR 0527548 (81a:53049)
 [M2]
T. Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), 307320. MR 0665498 (83k:53072)
 [M3]
T. Matsuki, Closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups, Adv. Stud. Pure Math. 14 (1988), 541559. MR 1039852 (91c:22014)
 [M4]
T. Matsuki, Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits, Hiroshima Math. J. 18 (1988), 5967.MR 0935882 (89f:53073)
 [M5]
T. Matsuki, Stein extensions of Riemann symmetric spaces and some generalization, J. Lie Theory 13 (2003), 563570. MR 2003160 (2004i:53062)
 [M6]
T. Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds, Trans. Amer. Math. Soc. 358 (2006), 22172245. MR 2197441
 [M7]
T. Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds III, preprint (RT/0410302).
 [R]
W. Rossmann, The structure of semisimple symmetric spaces, Canad. J. Math. 31 (1979), 157180. MR 0518716 (81i:53042)
 [Sp]
T. A. Springer, Some results on algebraic groups with involutions, Adv. Stud. Pure Math. 6 (1984), 525534. MR 0803346 (86m:20050)
 [V]
D. A. Vogan, Irreducible characters of semisimple Lie groups III, Invent. Math. 71 (1983), 381417. MR 0689650 (84h:22036)
 [WW]
R. O. Wells and J. A. Wolf, Poincaré series and automorphic cohomology on flag domains, Ann. of Math. 105 (1977), 397448. MR 0447645 (56:5955)
 [WZ1]
J. A. Wolf and R. Zierau, Linear cycle spaces in flag domains, Math. Ann. 316 (2000), 529545. MR 1752783 (2001g:32054)
 [WZ2]
J. A. Wolf and R. Zierau, A note on the linear cycle spaces for groups of Hermitian type, J. Lie Theory 13 (2003), 189191. MR 1958581 (2004a:22015)
 [A]
 K. Aomoto, On some double coset decompositions of complex semisimple Lie groups, J. Math. Soc. Japan 18 (1966), 144. MR 0191994 (33:221)
 [AG]
 D. N. Akhiezer and S. G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. 286 (1990), 112. MR 1032920 (91a:32047)
 [B]
 L. Barchini, Stein extensions of real symmetric spaces and the geometry of the flag manifold, Math. Ann. 326 (2003), 331346. MR 1990913 (2004d:22007)
 [FH]
 G. Fels and A. Huckleberry, Characterization of cycle domains via Kobayashi hyperbolicity, Bull. Soc. Math. France 133 (2005), 121144.MR 2145022
 [GM1]
 S. Gindikin and T. Matsuki, Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds, Transform. Groups 8 (2003), 333376. MR 2015255 (2005b:22017)
 [GM2]
 S. Gindikin and T. Matsuki, A remark on Schubert cells and the duality of orbits on flag manifolds, J. Math. Soc. Japan 57 (2005), 157165. MR 2114726 (2005j:14070)
 [H]
 A. Huckleberry, On certain domains in cycle spaces of flag manifolds, Math. Ann. 323 (2002), 797810. MR 1924279 (2003g:32037)
 [M1]
 T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331357. MR 0527548 (81a:53049)
 [M2]
 T. Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), 307320. MR 0665498 (83k:53072)
 [M3]
 T. Matsuki, Closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups, Adv. Stud. Pure Math. 14 (1988), 541559. MR 1039852 (91c:22014)
 [M4]
 T. Matsuki, Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits, Hiroshima Math. J. 18 (1988), 5967.MR 0935882 (89f:53073)
 [M5]
 T. Matsuki, Stein extensions of Riemann symmetric spaces and some generalization, J. Lie Theory 13 (2003), 563570. MR 2003160 (2004i:53062)
 [M6]
 T. Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds, Trans. Amer. Math. Soc. 358 (2006), 22172245. MR 2197441
 [M7]
 T. Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds III, preprint (RT/0410302).
 [R]
 W. Rossmann, The structure of semisimple symmetric spaces, Canad. J. Math. 31 (1979), 157180. MR 0518716 (81i:53042)
 [Sp]
 T. A. Springer, Some results on algebraic groups with involutions, Adv. Stud. Pure Math. 6 (1984), 525534. MR 0803346 (86m:20050)
 [V]
 D. A. Vogan, Irreducible characters of semisimple Lie groups III, Invent. Math. 71 (1983), 381417. MR 0689650 (84h:22036)
 [WW]
 R. O. Wells and J. A. Wolf, Poincaré series and automorphic cohomology on flag domains, Ann. of Math. 105 (1977), 397448. MR 0447645 (56:5955)
 [WZ1]
 J. A. Wolf and R. Zierau, Linear cycle spaces in flag domains, Math. Ann. 316 (2000), 529545. MR 1752783 (2001g:32054)
 [WZ2]
 J. A. Wolf and R. Zierau, A note on the linear cycle spaces for groups of Hermitian type, J. Lie Theory 13 (2003), 189191. MR 1958581 (2004a:22015)
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Additional Information
Toshihiko Matsuki
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 6068502, Japan
Email:
matsuki@math.kyotou.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993906084061
PII:
S 00029939(06)084061
Keywords:
Flag manifolds,
symmetric spaces,
Stein extensions
Received by editor(s):
January 20, 2004
Received by editor(s) in revised form:
June 29, 2005
Published electronically:
May 31, 2006
Communicated by:
Dan M. Barbasch
Article copyright:
© Copyright 2006 American Mathematical Society
