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},
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     pages = {129--157},
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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. http://www.numdam.org/articles/10.5802/aif.778/

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