Equivalence of domains arising from duality of orbits on flag manifolds

Matsuki, Toshihiko

doi:10.1090/S0002-9947-05-03824-9

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

Equivalence of domains arising from duality of orbits on flag manifolds

Author: Toshihiko Matsuki
Journal: Trans. Amer. Math. Soc. 358 (2006), 2217-2245
MSC (2000): Primary 14M15, 22E15, 22E46, 32M05
Posted: October 21, 2005
MathSciNet review: 2197441
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: S. Gindikin and the author defined a $G_{\mathbb R}$ - $K_{\mathbb C}$ invariant subset of $G_{\mathbb C}$ for each $K_{\mathbb C}$ -orbit on every flag manifold $G_{\mathbb C}/P$ and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of non-holomorphic type by computing many examples. In this paper, we first prove this conjecture for the open $K_{\mathbb C}$ -orbit on an ``arbitrary'' flag manifold generalizing the result of Barchini. This conjecture for closed was solved by J. A. Wolf and R. Zierau for Hermitian cases and by G. Fels and A. Huckleberry for non-Hermitian cases. We also deduce an alternative proof of this result for non-Hermitian cases.

[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. 1-3, 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/s00208-003-0419-8
[Be] Marcel Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85–177 (French). MR 0104763 (21 #3516)
[FH] G. Fels and A. Huckleberry, Characterization of cycle domains via Kobayashi hyperbolicity, preprint (AG/0204341).
[G] Simon Gindikin, Tube domains in Stein symmetric spaces, Positivity in Lie theory: open problems, de Gruyter Exp. Math., vol. 26, de Gruyter, Berlin, 1998, pp. 81–97. MR 1648697 (99i:32041)
[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/s00031-003-0725-y
[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
[HN] A. Huckleberry and B. Ntatin, Cycles in 𝐺-orbits in 𝐺^{ℂ}-flag manifolds, Manuscripta Math. 112 (2003), no. 4, 433–440. MR 2064652 (2005b:32046), http://dx.doi.org/10.1007/s00229-003-0405-1
[HW] Alan T. Huckleberry and Joseph A. Wolf, Schubert varieties and cycle spaces, Duke Math. J. 120 (2003), no. 2, 229–249. MR 2019975 (2004j:14056), http://dx.doi.org/10.1215/S0012-7094-03-12021-9
[KR] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 0311837 (47 #399)
[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, Double coset decompositions of reductive Lie groups arising from two involutions, J. Algebra 197 (1997), no. 1, 49–91. MR 1480777 (99a:22012), http://dx.doi.org/10.1006/jabr.1997.7123
[M6] Toshihiko Matsuki, Classification of two involutions on compact semisimple Lie groups and root systems, J. Lie Theory 12 (2002), no. 1, 41–68. MR 1885036 (2002k:22012)
[M7] Toshihiko Matsuki, Stein extensions of Riemann symmetric spaces and some generalization, J. Lie Theory 13 (2003), no. 2, 565–572. MR 2003160 (2004i:53062)
[M8] T. Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds II, preprint (RT/0309469).
[M9] T. Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds III, preprint (RT/0410302).
[MO] Toshihiko Matsuki and Toshio Ōshima, Embeddings of discrete series into principal series, The orbit method in representation theory (Copenhagen, 1988) Progr. Math., vol. 82, Birkhäuser Boston, Boston, MA, 1990, pp. 147–175. MR 1095345 (92d:22020)
[MUV] I. Mirković, T. Uzawa, and K. Vilonen, Matsuki correspondence for sheaves, Invent. Math. 109 (1992), no. 2, 231–245. MR 1172690 (93k:22011), http://dx.doi.org/10.1007/BF01232026
[O] A. L. Oniščik, Decompositions of reductive Lie groups, Mat. Sb. (N.S.) 80 (122) (1969), 553–599 (Russian). MR 0277660 (43 #3393)
[OM] Toshio Ōshima and Toshihiko Matsuki, Orbits on affine symmetric spaces under the action of the isotropy subgroups, J. Math. Soc. Japan 32 (1980), no. 2, 399–414. MR 567427 (81f:53043), http://dx.doi.org/10.2969/jmsj/03220399
[PR] Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263 (95b:11039)
[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/CJM-1979-017-6
[Se] Jirō Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), no. 1, 127–138. MR 867991 (88g:53053), http://dx.doi.org/10.2969/jmsj/03910127
[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, North-Holland, Amsterdam, 1985, pp. 525–543. MR 803346 (86m:20050)
[Su] Mitsuo Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Japan 11 (1959), 374–434. MR 0146305 (26 #3827)
[V] David A. Vogan, Irreducible characters of semisimple Lie groups. III. Proof of Kazhdan-Lusztig conjecture in the integral case, Invent. Math. 71 (1983), no. 2, 381–417. MR 689650 (84h:22036), http://dx.doi.org/10.1007/BF01389104
[Wh] Hassler Whitney, Elementary structure of real algebraic varieties, Ann. of Math. (2) 66 (1957), 545–556. MR 0095844 (20 #2342)
[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)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|