About \(G\)-bundles over elliptic curves
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 \textit{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.
principal bundles over a complex elliptic curve
scientific article; zbMATH DE number 1199190