## Poisson structures on affine spaces and flag varieties. II

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- by K. R. Goodearl and M. Yakimov PDF
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## Abstract:

The standard Poisson structures on the flag varieties $G/P$ of a complex reductive algebraic group $G$ are investigated. It is shown that the orbits of symplectic leaves in $G/P$ under a fixed maximal torus of $G$ are smooth irreducible locally closed subvarieties of $G/P$, isomorphic to intersections of dual Schubert cells in the full flag variety $G/B$ of $G$, and their Zariski closures are explicitly computed. Two different proofs of the former result are presented. The first is in the framework of Poisson homogeneous spaces, and the second one uses an idea of weak splittings of surjective Poisson submersions, based on the notion of Poisson–Dirac submanifolds. For a parabolic subgroup $P$ with abelian unipotent radical (in which case $G/P$ is a Hermitian symmetric space of compact type), it is shown that all orbits of the standard Levi factor $L$ of $P$ on $G/P$ are complete Poisson subvarieties which are quotients of $L$, equipped with the standard Poisson structure. Moreover, it is proved that the Poisson structure on $G/P$ vanishes at all special base points for the $L$-orbits on $G/P$ constructed by Richardson, Röhrle, and Steinberg.## References

- N. Bourbaki,
*Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées*, Hermann, Paris, 1975 (French). Actualités Sci. Indust., No. 1364. MR**0453824** - M. Brion and V. Lakshmibai,
*A geometric approach to standard monomial theory*, Represent. Theory**7**(2003), 651–680. MR**2017071**, DOI 10.1090/S1088-4165-03-00211-5 - K. A. Brown, K. R. Goodearl, and M. Yakimov,
*Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space*, Adv. Math.**206**(2006), no. 2, 567–629. MR**2263715**, DOI 10.1016/j.aim.2005.10.004 - Roger W. Carter,
*Finite groups of Lie type*, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR**1266626** - Marius Crainic and Rui Loja Fernandes,
*Integrability of Poisson brackets*, J. Differential Geom.**66**(2004), no. 1, 71–137. MR**2128714** - Vinay V. Deodhar,
*On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells*, Invent. Math.**79**(1985), no. 3, 499–511. MR**782232**, DOI 10.1007/BF01388520 - V. G. Drinfel′d,
*On Poisson homogeneous spaces of Poisson-Lie groups*, Teoret. Mat. Fiz.**95**(1993), no. 2, 226–227 (English, with English and Russian summaries); English transl., Theoret. and Math. Phys.**95**(1993), no. 2, 524–525. MR**1243249**, DOI 10.1007/BF01017137 - P. Foth and J.-H. Lu,
*A Poisson structure on compact symmetric spaces*, Comm. Math. Phys.**251**(2004), no. 3, 557–566. MR**2102330**, DOI 10.1007/s00220-004-1178-4 - William Fulton and Joe Harris,
*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249**, DOI 10.1007/978-1-4612-0979-9 - K. R. Goodearl,
*Prime spectra of quantized coordinate rings*, Interactions between ring theory and representations of algebras (Murcia), Lecture Notes in Pure and Appl. Math., vol. 210, Dekker, New York, 2000, pp. 205–237. MR**1759846** - Atsushi Kamita,
*Quantum deformations of certain prehomogeneous vector spaces. III*, Hiroshima Math. J.**30**(2000), no. 1, 79–115. MR**1753385** - David Kazhdan and George Lusztig,
*Representations of Coxeter groups and Hecke algebras*, Invent. Math.**53**(1979), no. 2, 165–184. MR**560412**, DOI 10.1007/BF01390031 - Jiang-Hua Lu and Milen Yakimov,
*Partitions of the wonderful group compactification*, Transform. Groups**12**(2007), no. 4, 695–723. MR**2365441**, DOI 10.1007/s00031-007-0062-7 - G. Lusztig,
*Total positivity in partial flag manifolds*, Represent. Theory**2**(1998), 70–78. MR**1606402**, DOI 10.1090/S1088-4165-98-00046-6 - Iris Muller, Hubert Rubenthaler, and Gérard Schiffmann,
*Structure des espaces préhomogènes associés à certaines algèbres de Lie graduées*, Math. Ann.**274**(1986), no. 1, 95–123 (French). MR**834108**, DOI 10.1007/BF01458019 - Ian M. Musson,
*Ring-theoretic properties of the coordinate rings of quantum symplectic and Euclidean space*, Ring theory (Granville, OH, 1992) World Sci. Publ., River Edge, NJ, 1993, pp. 248–258. MR**1344235** - Masatoshi Noumi,
*Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces*, Adv. Math.**123**(1996), no. 1, 16–77. MR**1413836**, DOI 10.1006/aima.1996.0066 - N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev,
*Quantization of Lie groups and Lie algebras*, Algebra i Analiz**1**(1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J.**1**(1990), no. 1, 193–225. MR**1015339** - Roger Richardson, Gerhard Röhrle, and Robert Steinberg,
*Parabolic subgroups with abelian unipotent radical*, Invent. Math.**110**(1992), no. 3, 649–671. MR**1189494**, DOI 10.1007/BF01231348 - K. Rietsch,
*Closure relations for totally nonnegative cells in $G/P$*, Math. Res. Lett.**13**(2006), no. 5-6, 775–786. MR**2280774**, DOI 10.4310/MRL.2006.v13.n5.a8 - T. A. Springer,
*Intersection cohomology of $B\times B$-orbit closures in group compactifications*, J. Algebra**258**(2002), no. 1, 71–111. With an appendix by Wilberd van der Kallen; Special issue in celebration of Claudio Procesi’s 60th birthday. MR**1958898**, DOI 10.1016/S0021-8693(02)00515-X - Elisabetta Strickland,
*Classical invariant theory for the quantum symplectic group*, Adv. Math.**123**(1996), no. 1, 78–90. MR**1413837**, DOI 10.1006/aima.1996.0067 - Masaru Takeuchi,
*On orbits in a compact hermitian symmetric space*, Amer. J. Math.**90**(1968), 657–680. MR**245827**, DOI 10.2307/2373477 - Joseph A. Wolf,
*Fine structure of Hermitian symmetric spaces*, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and App. Math., Vol. 8, Dekker, New York, 1972, pp. 271–357. MR**0404716** - Joseph A. Wolf,
*Classification and Fourier inversion for parabolic subgroups with square integrable nilradical*, Mem. Amer. Math. Soc.**22**(1979), no. 225, iii+166. MR**546511**, DOI 10.1090/memo/0225 - Ping Xu,
*Dirac submanifolds and Poisson involutions*, Ann. Sci. École Norm. Sup. (4)**36**(2003), no. 3, 403–430 (English, with English and French summaries). MR**1977824**, DOI 10.1016/S0012-9593(03)00013-2

## Additional Information

**K. R. Goodearl**- Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
- MR Author ID: 75245
- Email: goodearl@math.ucsb.edu
**M. Yakimov**- Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
- MR Author ID: 611410
- Email: yakimov@math.ucsb.edu
- Received by editor(s): June 13, 2007
- Published electronically: June 19, 2009
- Additional Notes: The research of the first author was partially supported by National Science Foundation grant DMS-0401558.

The research of the second author was partially supported by National Science Foundation grant DMS-0406057 and an Alfred P. Sloan research fellowship. - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**361**(2009), 5753-5780 - MSC (2000): Primary 14M15; Secondary 53D17, 14L30, 17B20, 17B63, 53C35
- DOI: https://doi.org/10.1090/S0002-9947-09-04654-6
- MathSciNet review: 2529913

Dedicated: Dedicated to the memory of our colleague Xu-Dong Liu (1962-2005)