Journal of the Mathematical Society of Japan

A remark on Schubert cells and the duality of orbits on flag manifolds

Simon GINDIKIN and Toshihiko MATSUKI

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Abstract

It is known that the closure of an arbitrary K C -orbit on a flag manifold is expressed as a product of a closed K C -orbit and a Schubert cell ([M2], [Sp]). We already applied this fact to the duality of orbits on flag manifolds ([GM]). We refine here this result and give its new applications to the study of domains arising from the duality.

Article information

Source
J. Math. Soc. Japan, Volume 57, Number 1 (2005), 157-165.

Dates
First available in Project Euclid: 13 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1160745819

Digital Object Identifier
doi:10.2969/jmsj/1160745819

Mathematical Reviews number (MathSciNet)
MR2114726

Zentralblatt MATH identifier
1076.14067

Subjects
Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]
Secondary: 32M05: Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10]

Keywords
Schubert cell flag manifold

Citation

GINDIKIN, Simon; MATSUKI, Toshihiko. A remark on Schubert cells and the duality of orbits on flag manifolds. J. Math. Soc. Japan 57 (2005), no. 1, 157--165. doi:10.2969/jmsj/1160745819. https://projecteuclid.org/euclid.jmsj/1160745819


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References

  • D. N. Akhiezer and S. G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann., 286 (1990), 1–12.
  • L. Barchini, Stein extensions of real symmetric spaces and the geometry of the flag manifold, Math. Ann., 326 (2003), 331–346.
  • L. Barchini, S. G. Gindikin and H. W. Wang, The geometry of flag manifold and holomorphic extension of Szegö kernels for $SU(p,q)$, Pacific J. Math., 179 (1997), 201–220.
  • D. Burns, S. Halverscheid and R. Hind, The geometry of Grauert tubes and complexification of symmetric spaces, Duke Math. J., 118 (2003), 465–491.
  • G. Fels and A. Huckleberry, Characterization of cycle domains via Kobayashi hyperbolicity, preprint (AG/0204341).
  • S. Gindikin, Tube domains in Stein symmetric spaces, In: Positivity in Lie theory: Open problems, (eds. Hilgert, Lawson, Neeb and Vinberg), Walter de Gruyter, Berlin-New York, 1998, 81–98.
  • S. Gindikin and T. Matsuki, Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds, In: Transform. Groups, 8 (2003), 333–376.
  • A. Huckleberry, On certain domains in cycle spaces of flag manifolds, Math. Ann., 323 (2002), 797–810.
  • A. Huckleberry and J. A. Wolf, Schubert varieties and cycle spaces, Duke Math. J., 120 (2003), 229–249.
  • B. Krötz and R. Stanton, Holomorphic extension of a representation: (I) automorphic functions, Ann. of Math., 159 (2004), 641–724.
  • B. Krötz and R. Stanton, Holomorphic extensions of representations: (II) geometry and harmonic analysis, preprint.
  • T. Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J., 12 (1982), 307–320.
  • T. Matsuki, Closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups, In: Representation of Lie Groups, Kyoto, Hiroshima, 1986, (eds. K. Okamoto and T. Oshima), Adv. Stud. Pure Math., 14, Kinokuniya Company LTD., Tokyo, 1988, pp.,541–559.
  • 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), 59–67.
  • T. Matsuki, Stein extensions of Riemann symmetric spaces and some generalization, J. Lie Theory, 13 (2003), 563–570.
  • T. A. Springer, Some results on algebraic groups with involutions, In: Algebraic Groups and Related Topics, (ed. R. Hotta), Adv. Stud. Pure Math., 6, Kinokuniya Company LTD., Tokyo; North-Holland, Amsterdam-New York-Oxford, 1985, pp.,525–534.
  • R. O. Wells and J. A. Wolf, Poincaré series and automorphic cohomology on flag domains, Ann. of Math., 105 (1977), 397–448.
  • J. A. Wolf, The Stein condition for cycle spaces of open orbits on complex flag manifolds, Ann. of Math., 136 (1992), 541–555.
  • J. A. Wolf and R. Zierau, Linear cycle spaces in flag domains, Math. Ann., 316 (2000), 529–545.
  • J. A. Wolf and R. Zierau, A note on the linear cycle spaces for groups of Hermitian type, J. Lie Theory, 13 (2003), 189–191.