On compact homogeneous symplectic manifolds
Annales de l'Institut Fourier, Volume 30 (1980) no. 1, pp. 129-157.

In this paper the authors study compact homogeneous spaces G/K (of a Lie group G) on which there if defined a G-invariant symplectic form Ω. It is an important feature of the paper that very little is assumed concerning G and K. The essential assumptions are: (1) G is connected and (2) K is uniform (i.e., G/K is compact). Further, for convenience only and with no loss of generality, it is supposed that G is simply connected and K contains no connected normal subgroup of G, i.e., that G acts almost effectively on G/K. It is then shown that G=S×R, a direct product, where S is compact semi-simple and R is a semi-direct product AN of a connected abelian subgroup A and the maximal connected normal nilpotent group N, which is also abelian. Further K=(KS)×(KR) and S/(KS),R/KR) each have a natural symplectic structure. Some further results on R/(KR) are given together with an example which shows that R can actually possess this two step solvable structure, i.e., it need not be abelian, although R/(KR) is a torus topologically. Once it has been established that S is compact and S/(KS) symplectic, then the structure of S/(KS) is well-known from the work of others.

Dans cet article, les auteurs considèrent les espaces homogènes compacts d’un groupe de Lie G sur lesquels est donnée une structure symplectique G-invariante Ω. Un point important de l’article est le fait que les hypothèses sur G et K sont minimales : (1) G est connexe et (2) K est uniforme (i.e. G/K est compact). Pour faciliter l’exposition mais sans diminuer la généralité des conclusions, on a aussi supposé que G est simplement connexe et que K ne contient aucun sous-groupe invariant connexe de G, i.e. l’action de G sur G/K est presque effective. Alors, il est démontré que G=S×R, produit direct de S semi-simple et compact et R produit semi-direct AN d’un sous-groupe abélien A avec un sous-groupe invariant abélien N. De plus, K=(KS)×(KR) et les espaces homogènes S(KS) et R/(KR) héritent chacun d’une structure homogène symplectique. Quelques résultats supplémentaires sur R/(KR) sont donnés avec un exemple qui montre que R peut, en fait, être muni d’une structure de groupe résoluble non-abélien mais que R/(KR) doit être homéomorphe au tore T n . Quant à S/(KR), une fois établi que S est compact, semi-simple, la structure de cet espace est bien connue.

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     title = {On compact homogeneous symplectic manifolds},
     journal = {Annales de l'Institut Fourier},
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     number = {1},
     year = {1980},
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Zwart, P. B.; Boothby, William M. On compact homogeneous symplectic manifolds. Annales de l'Institut Fourier, Volume 30 (1980) no. 1, pp. 129-157. doi : 10.5802/aif.778. https://aif.centre-mersenne.org/articles/10.5802/aif.778/

[1] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1975. | MR | Zbl

[2] W.M. Boothby and H.C. Wang, On contact manifolds, Ann. of Math., Vol. 68, No. 3 (1958), 721-734. | MR | Zbl

[3] A. Borel, Kählerian coset spaces of semi-simple Lie groups, Proc. Nat. Acad. Sci. U.S.A., Vol. 40 (1954) 1147-1151. | MR | Zbl

[4] A. Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math., Vol. 72 (1960), 179-188. | MR | Zbl

[5] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., Vol. 63 (1948), 85-124. | MR | Zbl

[6] B.-Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Soc., 197 (1974), 145-159. | MR | Zbl

[7] C. Godbillon, Géométrie Différentiable et Mécanique Analytique, Hermann (Paris) 1969. | Zbl

[8] N. Jacobson, Lie Algebras, Wiley-Interscience, New York, 1962. | Zbl

[9] G. Hochschild, The structure of Lie Groups, Holden-Day, San Francisco, 1965. | MR | Zbl

[10] S. Kobayashi, and K. Nomizu, Foundations of Differential Geometry I, II, Interscience, New York, 1963. | Zbl

[11] B. Kostant, Quantization and unitary representation. Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics, 170, Springer, 1970. | MR | Zbl

[12] A. Lichnerowicz, Some Problems on Transformations of Riemannian and Kählerian Manifolds, Princeton, Mimeographed notes.

[13] A. Lichnerowicz, Théorème de réductivité sur des algèbres d'automorphismes, Rendiconti di Mathematic, (1-2), Vol. 22 (1963), 197-244. | Zbl

[14] Y. Matsushima, Sur les espaces homogènes kahlérians d'un groupe de Lie réductif, Nagoya Math. J., 11 (1957), 53-60. | MR | Zbl

[15] G.D. Mostow, Factor spaces of solvable groups, Ann. of Math., Vol. 60, (1954), 1-27. | MR | Zbl

[16] Malcev, On a class of homogeneous spaces, Izvestia Nauk, 13 (1949), 9-32, Amer. Math. Soc. Transl., No. 39 (1951).

[17] L. Pukanszky, Leçons sur la représentation des groupes, Soc. Math. de France Monographie, Dunod, Paris, 1967. | MR | Zbl

[18] J.-M. Souriau, Structure des systèmes dynamiques, Maîtrises de Mathématiques, Dunod, Paris, 1970. | MR | Zbl

[19] S. Sternberg, Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 212 (1975), 113-130. | MR | Zbl

[20] A. Weinstein, Lectures on Symplectic Manifolds, NSF-CBMS Regional Conference Monograph No. 29. | Zbl

[21] P. Zwart, Compact homogeneous spaces possessing invariant contact, symplectic, or cosymplectic structures, Ph.D. Thesis, Washington University, St. Louis, MO, 1965.

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