About \(G\)-bundles over elliptic curves. (English) Zbl 0901.14019

In this note, we study principal bundles over a complex elliptic curve \(X\) with reductive structure group \(G\). As in the vector bundle case, we first show that a non semistable bundle has a canonical semistable \(L\)-structure with \(L\) some Levi subgroup of \(G\) reducing the study of \(G\)-bundles to the study of semistable bundles. We then look at the coarse moduli space \(M_G\) of topologically trivial semistable bundles on \(X\) (there is not any stable topologically trivial \(G\)-bundle) and prove that it is isomorphic to the quotient \([\Gamma(T) \otimes_{\mathbb{Z}} X]/W\) where \(\Gamma(T)\) is the group of one parameter subgroups of a maximal torus \(T\) and \(W= N(G,T)/T\) is the Weyl group (theorem 4.16). Suppose that \(G\) is simple and simply connected and let \(\theta\) be the longest root (relative to some basis \((\alpha_1,\dots, \alpha_l)\) of the root system \(\Phi(G,T))\). The coroot \(\theta^\vee\) of \(\theta\) has a decomposition \(\theta^\vee= \sum_i g_i\alpha_i^\vee\) with \(g_i\) a positive integer. Using theorem 4.16 and Looijenga’s isomorphism \[ [\Gamma(T) \otimes_{\mathbb{Z}} X]/W \widetilde{\longrightarrow} \mathbb{P} (1,g_1,\dots, g_l), \] one gets that \(M_G\) is isomorphic to the weighted projective space \(\mathbb{P} (1,g_1,\dots, g_l)\), generalizing the well-known isomorphism \(M_{\text{SL}_{l+1}} \widetilde{\rightarrow} \mathbb{P}^l\) [see L. Tu, Adv. Math. 98, No. 1, 1-26 (1993; Zbl 0786.14021) for instance]. One recovers for instance the Verlinde formula in this case. We know that these results are certainly well-known from experts, but we were unable to find any reference in the literature, except of course when \(G\) is either SL or GL.


14H60 Vector bundles on curves and their moduli
14D22 Fine and coarse moduli spaces
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)


Zbl 0786.14021
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[1] [AB] , , The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond., A 308 (1982), 523-615. · Zbl 0509.14014
[2] [Be] , Conformal blocks, fusion rules and the Verlinde formula, Israel Math. Conf. Proc., 9 (1996), 75-96. · Zbl 0848.17024
[3] [BS] , , Chevalley’s theorem for complex crystallographic Coxeter groups, Funkt. Anal. i Ego Prilozheniya, 12 (1978), 79-80. · Zbl 0458.32017
[4] [BLS] , , , The Picard group of the moduli stack of G-bundles on a curve, preprint alg-geom/9608002, to appear in Compos. Math. · Zbl 0976.14024
[5] [Bo] , Groupes et algèbres de Lie, chap. 7, 8 (1990), Masson.
[6] [BG] , Conjugacy classes in loop groups and G-bundles on elliptic curves, Int. Math. Res. Not., 15 (1966), 733-752. · Zbl 0992.20034
[7] [D] Espaces projectifs anisotropes, Bull. Soc. Math. France, 103 (1975), 203-223. · Zbl 0314.14016
[8] [FMW] , , Vector bundles and F Theory, eprint hep-th 9701162. · Zbl 0919.14010
[9] [Hu] , Linear algebraic groups, GTM 21, Berlin, Heidelberg, New-York, Springer (1975). · Zbl 0325.20039
[10] [LS] , Picard group of the moduli stack of G-bundles, Ann. Scient. Éc. Norm. Sup., 4e série, 30 (1997), 499-525. · Zbl 0918.14004
[11] [LeP] , Fibrés vectoriels sur les courbes algébriques, Publ. Math. Univ. Paris 7, 35 (1995). · Zbl 0842.14025
[12] [Lo] , Root systems and elliptic curves, Invent. Math., 38 (1976), 17-32. · Zbl 0358.17016
[13] [Ra1] , Moduli for principal bundles over algebraic curves, I and II, Proc. Indian Acad. Sci. Math. Sci., 106 (1996), 301-328 and 421-449. · Zbl 0901.14007
[14] [Ra2] , Stable principal bundles on a compact Rieman surface, Math. Ann., 213 (1975), 129-152. · Zbl 0289.32020
[15] [S] , Cohomologie galoisienne, LNM 5 (1964). · Zbl 0128.26303
[16] [T] , Semistable bundles over an elliptic curve, Adv. Math., 98 (1993), 1-26. · Zbl 0786.14021
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