Equivalence of domains arising from duality of orbits on flag manifolds II

Matsuki, Toshihiko

doi:10.1090/S0002-9939-06-08406-1

Equivalence of domains arising from duality of orbits on flag manifolds II
HTML articles powered by AMS MathViewer

by Toshihiko Matsuki PDF

Proc. Amer. Math. Soc. 134 (2006), 3423-3428 Request permission

Abstract:

S. Gindikin and the author defined a $G_{\mathbb {R}}$-$K_{\mathbb {C}}$ invariant subset $C(S)$ of $G_{\mathbb {C}}$ for each $K_{\mathbb {C}}$-orbit $S$ on every flag manifold $G_{\mathbb {C}}/P$ and conjectured that the connected component $C(S)_0$ of the identity would be equal to the Akhiezer-Gindikin domain $D$ if $S$ is of nonholomorphic type. This conjecture was proved for closed $S$ in the works of J. A. Wolf, R. Zierau, G. Fels, A. Huckleberry and the author. It was also proved for open $S$ by the author. In this paper, we prove the conjecture for all the other orbits when $G_{\mathbb {R}}$ is of non-Hermitian type.

References

Kazuhiko Aomoto, On some double coset decompositions of complex semisimple Lie groups, J. Math. Soc. Japan 18 (1966), 1–44. MR 191994, DOI 10.2969/jmsj/01810001
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, DOI 10.1007/BF01453562
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, DOI 10.1007/s00208-003-0419-8
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, DOI 10.24033/bsmf.2481
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, DOI 10.1007/s00031-003-0725-y
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
Alan Huckleberry, On certain domains in cycle spaces of flag manifolds, Math. Ann. 323 (2002), no. 4, 797–810. MR 1924279, DOI 10.1007/s002080200326
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, DOI 10.2969/jmsj/03120331
Toshihiko Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), no. 2, 307–320. MR 665498
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, DOI 10.2969/aspm/01410541
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
Toshihiko Matsuki, Stein extensions of Riemann symmetric spaces and some generalization, J. Lie Theory 13 (2003), no. 2, 565–572. MR 2003160
Toshihiko Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds, Trans. Amer. Math. Soc. 358 (2006), no. 5, 2217–2245. MR 2197441, DOI 10.1090/S0002-9947-05-03824-9
T. Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds III, preprint (RT/0410302).
W. Rossmann, The structure of semisimple symmetric spaces, Canadian J. Math. 31 (1979), no. 1, 157–180. MR 518716, DOI 10.4153/CJM-1979-017-6
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, DOI 10.2969/aspm/00610525
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, DOI 10.1007/BF01389104
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 447645, DOI 10.2307/1970918
Joseph A. Wolf and Roger Zierau, Linear cycle spaces in flag domains, Math. Ann. 316 (2000), no. 3, 529–545. MR 1752783, DOI 10.1007/s002080050342
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