no. 2
About G-bundles over elliptic curves
Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 413-424.

Soit G un groupe algébrique complexe simple et simplement connexe, T un tore maximal et W le groupe de Weyl. On démontre que l’espace de modules grossier M G paramétrant les classes de S-équivalence de G-fibrés semi-stables sur une courbe elliptique X, est isomorphe à [Γ(T)⊗ Z X]/W. D’après un résultat de Looijenga, ceci prouve que M G est un espace projectif anistotrope.

Let G be a complex algebraic group, simple and simply connected, T a maximal torus and W the Weyl group. One shows that the coarse moduli space M G (X) parametrizing S-equivalence classes of semistable G-bundles over an elliptic curve X is isomorphic to [Γ(T)⊗ Z X]/W. By a result of Looijenga, this shows that M G (X) is a weighted projective space.

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     author = {Laszlo, Yves},
     title = {About $G$-bundles over elliptic curves},
     journal = {Annales de l'Institut Fourier},
     pages = {413--424},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {2},
     year = {1998},
     doi = {10.5802/aif.1623},
     mrnumber = {99c:14016},
     zbl = {0901.14019},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1623/}
}
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Laszlo, Yves. About $G$-bundles over elliptic curves. Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 413-424. doi : 10.5802/aif.1623. http://archive.numdam.org/articles/10.5802/aif.1623/

[AB] M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond., A 308 (1982), 523-615. | MR | Zbl

[Be] A. Beauville, Conformal blocks, fusion rules and the Verlinde formula, Israel Math. Conf. Proc., 9 (1996), 75-96. | MR | Zbl

[BS] I. N. Bernshtein, O. V. Shvartsman, Chevalley's theorem for complex crystallographic Coxeter groups, Funkt. Anal. i Ego Prilozheniya, 12 (1978), 79-80. | MR | Zbl

[BLS] A. Beauville, Y. Laszlo, C. Sorger, The Picard group of the moduli stack of G-bundles on a curve, preprint alg-geom/9608002, to appear in Compos. Math. | Zbl

[Bo] N. Bourbaki, Groupes et algèbres de Lie, chap. 7, 8 (1990), Masson.

[BG] V. Baranovsky, V. Ginzburg Conjugacy classes in loop groups and G-bundles on elliptic curves, Int. Math. Res. Not., 15 (1966), 733-752. | MR | Zbl

[D] C. Delorme Espaces projectifs anisotropes, Bull. Soc. Math. France, 103 (1975), 203-223. | Numdam | MR | Zbl

[FMW] R. Friedman, J. Morgan, E. Witten Vector bundles and F Theory, eprint hep-th 9701162. | Zbl

[Hu] J. E. Humphreys, Linear algebraic groups, GTM 21, Berlin, Heidelberg, New-York, Springer (1975). | MR | Zbl

[LS] Y. Laszlo, C. Sorger Picard group of the moduli stack of G-bundles, Ann. Scient. Éc. Norm. Sup., 4e série, 30 (1997), 499-525. | Numdam

[LeP] J. Le Potier, Fibrés vectoriels sur les courbes algébriques, Publ. Math. Univ. Paris 7, 35 (1995). | MR | Zbl

[Lo] E. Looijenga, Root systems and elliptic curves, Invent. Math., 38 (1976), 17-32. | MR | Zbl

[Ra1] A. Ramanathan, Moduli for principal bundles over algebraic curves, I and II, Proc. Indian Acad. Sci. Math. Sci., 106 (1996), 301-328 and 421-449. | MR | Zbl

[Ra2] A. Ramanathan, Stable principal bundles on a compact Rieman surface, Math. Ann., 213 (1975), 129-152. | MR | Zbl

[S] J.-P. Serre, Cohomologie galoisienne, LNM 5 (1964). | Zbl

[T] L. Tu, Semistable bundles over an elliptic curve, Adv. Math., 98 (1993), 1-26. | MR | Zbl

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