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]
Kazuhiko
Aomoto, On some double coset decompositions of complex semisimple
Lie groups, J. Math. Soc. Japan 18 (1966),
1–44. MR
0191994 (33 #221)
 [AG]
D.
N. Akhiezer and S.
G. Gindikin, On Stein extensions of real symmetric spaces,
Math. Ann. 286 (1990), no. 13, 1–12. MR 1032920
(91a:32047), http://dx.doi.org/10.1007/BF01453562
 [B]
L.
Barchini, Stein extensions of real symmetric spaces and the
geometry of the flag manifold, Math. Ann. 326 (2003),
no. 2, 331–346. MR 1990913
(2004d:22007), http://dx.doi.org/10.1007/s0020800304198
 [FH]
Gregor
Fels and Alan
Huckleberry, Characterization of cycle domains via Kobayashi
hyperbolicity, Bull. Soc. Math. France 133 (2005),
no. 1, 121–144 (English, with English and French summaries). MR 2145022
(2006j:32028)
 [GM1]
S.
Gindikin and T.
Matsuki, Stein extensions of Riemannian symmetric spaces and
dualities of orbits on flag manifolds, Transform. Groups
8 (2003), no. 4, 333–376. MR 2015255
(2005b:22017), http://dx.doi.org/10.1007/s000310030725y
 [GM2]
Simon
Gindikin and Toshihiko
Matsuki, A remark on Schubert cells and the duality of orbits on
flat manifolds, J. Math. Soc. Japan 57 (2005),
no. 1, 157–165. MR 2114726
(2005j:14070)
 [H]
Alan
Huckleberry, On certain domains in cycle spaces of flag
manifolds, Math. Ann. 323 (2002), no. 4,
797–810. MR 1924279
(2003g:32037), http://dx.doi.org/10.1007/s002080200326
 [M1]
Toshihiko
Matsuki, The orbits of affine symmetric spaces under the action of
minimal parabolic subgroups, J. Math. Soc. Japan 31
(1979), no. 2, 331–357. MR 527548
(81a:53049), http://dx.doi.org/10.2969/jmsj/03120331
 [M2]
Toshihiko
Matsuki, Orbits on affine symmetric spaces under the action of
parabolic subgroups, Hiroshima Math. J. 12 (1982),
no. 2, 307–320. MR 665498
(83k:53072)
 [M3]
Toshihiko
Matsuki, Closure relations for orbits on affine symmetric spaces
under the action of minimal parabolic subgroups, Representations of
Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14,
Academic Press, Boston, MA, 1988, pp. 541–559. MR 1039852
(91c:22014)
 [M4]
Toshihiko
Matsuki, Closure relations for orbits on affine symmetric spaces
under the action of parabolic subgroups. Intersections of associated
orbits, Hiroshima Math. J. 18 (1988), no. 1,
59–67. MR
935882 (89f:53073)
 [M5]
Toshihiko
Matsuki, Stein extensions of Riemann symmetric spaces and some
generalization, J. Lie Theory 13 (2003), no. 2,
565–572. MR 2003160
(2004i:53062)
 [M6]
Toshihiko
Matsuki, Equivalence of domains arising from
duality of orbits on flag manifolds, Trans.
Amer. Math. Soc. 358 (2006), no. 5, 2217–2245 (electronic). MR 2197441
(2006m:14066), http://dx.doi.org/10.1090/S0002994705038249
 [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), no. 1, 157–180. MR 518716
(81i:53042), http://dx.doi.org/10.4153/CJM19790176
 [Sp]
T.
A. Springer, Some results on algebraic groups with
involutions, Algebraic groups and related topics (Kyoto/Nagoya, 1983)
Adv. Stud. Pure Math., vol. 6, NorthHolland, Amsterdam, 1985,
pp. 525–543. MR 803346
(86m:20050)
 [V]
David
A. Vogan, Irreducible characters of semisimple Lie groups. III.
Proof of KazhdanLusztig conjecture in the integral case, Invent.
Math. 71 (1983), no. 2, 381–417. MR 689650
(84h:22036), http://dx.doi.org/10.1007/BF01389104
 [WW]
R.
O. Wells Jr. and Joseph
A. Wolf, Poincaré series and automorphic cohomology on flag
domains, Ann. of Math. (2) 105 (1977), no. 3,
397–448. MR 0447645
(56 #5955)
 [WZ1]
Joseph
A. Wolf and Roger
Zierau, Linear cycle spaces in flag domains, Math. Ann.
316 (2000), no. 3, 529–545. MR 1752783
(2001g:32054), http://dx.doi.org/10.1007/s002080050342
 [WZ2]
Joseph
A. Wolf and Roger
Zierau, A note on the linear cycle space for groups of Hermitian
type, J. Lie Theory 13 (2003), no. 1,
189–191. 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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
14M15,
22E15,
22E46,
32M05
Retrieve articles in all journals
with MSC (2000):
14M15,
22E15,
22E46,
32M05
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
