On the stack of semistable \(G\)-bundles over an elliptic curve (Q289849)

Let \(G\) be a reductive group and \(T\subset B\) be a maximal torus and a Borel subgroup. Let \(\mathrm{Bun}_{G}\) be the moduli stack of principal \(G\)-bundles over an elliptic curve (over an algebraically closed field) and \(\mathrm{Bun}_{G}^{\star}\) be the substack with the property \(\star\), e.g.\ semistable, degree, regular. In [Compos. Math. 151, No. 8, 1568--1584 (2015; Zbl 1325.22011)] \textit{D. Ben-Zvi} and \textit{D. Nadler} show that the induction map \(\mathrm{Bun}_{B}^{\deg 0,\mathrm{ss}}\to \mathrm{Bun}_{G}^{\deg 0,\mathrm{ss}}\), is a small map with Galois group isomorphic to the Weyl group \(W_{G}\) of \(G\). The paper under review generalises this result to parabolic bundles: Lemma 2.12. For any given degree \(\check\lambda\in \check\Lambda_{G,G}\), there is the unique minimal parabolic \(P\) with the same slope and same degree \(\check\lambda_{P}\). Theorem 3.2. The induction map \(\mathrm{Bun}^{\check\lambda_{P},\mathrm{ss}}_{P}\to \mathrm{Bun}^{\check\lambda,\mathrm{ss}}_{G}\) is a small map, where \(P,\check\lambda_{P}\) are as in the lemma. Moreover it is a Galois covering over the regular locus with Galois group isomorphic to the relative Weyl group \(W_{M,G}=N_{G}(M)/M\), where \(M\) denotes the Levi of \(P\). The lemma is irrelevant to bundles, which becomes clearer after the later formulation of slope and degree. The first assertion of the theorem remains true for arbitrary complete curves, and in the case the second assertion is adjusted to birational map instead of Galois covering (Remark 3.3). The motivation of the theorem arises from geometric Eisenstein series and character sheaves as the author writes in Introduction. Let us give definitions of several notions used in the statements, degree, slope, semistability, small and regular in order. Let \(\check\Lambda_{G},\Lambda_{G}\) be the coweight, weight lattice of \(G\) respectively. Let \(\check\Lambda_{G,G}\) be the quotient of \(\check\Lambda_{G}\) by the coroot sublattice. This can be generalised to \(\check\Lambda_{G,P}\), where \(P\) is a parabolic containing \(B\), as follows: Let \(M\) be the Levi of \(P\) with \(T\subset M\subset P\). Set \(\check\Lambda_{G,P}:=\check\Lambda_{G}/\)the coroot lattice of \(M\), which immediately equals \(\check\Lambda_{M,M}\). Here we recall that \(\check\Lambda_{G,P}\) bijectively corresponds to the connected components of \(\mathrm{Bun}_{P}\). Thus the quotient \(\pi: \check\Lambda_{G,P}=\check\Lambda_{M,M}\to \check\Lambda_{G,G}\) gives the correspondence of connected components under the pull-back-push-forward via \(\mathrm{Bun}_{M}\mathop{q}\limits{\leftarrow} \mathrm{Bun}_{P}\mathop{p}\limits{\to} \mathrm{Bun}_{G}\) (induced by the group homomorphisms \(M\twoheadleftarrow P\hookrightarrow G\)). So the restriction induction map \(\mathrm{Bun}_{P}^{\check\lambda_{P}}\to \mathrm{Bun}_{G}^{\check\lambda}\) makes sense only when \(\pi(\check\lambda_{P})=\check\lambda\), which we called same degree in the statement of the lemma. The slope and semistability are formulated due to \textit{S. Schieder} [Sel. Math., New Ser. 21, No. 3, 763--831 (2015; Zbl 1341.14006)] (in fact this semistability is a reformulation of Ramanathan's): We notice first that the \(\mathbb{Q}\)-tensor \(\check\Lambda_{G,G}^{\mathbb{Q}}=\check\Lambda_{Z(G)}^{\mathbb{Q}}\), where \(Z(G)\) denotes the centre of \(G\). The \textit{slope function} is the natural composite \(\phi_{P}: \check\Lambda_{G,P}\to \check\Lambda_{G,P}^{\mathbb{Q}}=\check\Lambda_{Z(M)}^{\mathbb{Q}}\hookrightarrow \check\Lambda_{G}^{\mathbb{Q}}\). For two degrees \(\check\lambda\in \check\Lambda_{G,G}\) and \(\check\lambda_{P}\in \check\Lambda_{G,P}\), their slopes are same if \(\phi_{G}(\check\lambda)=\phi_{P}(\check\lambda_{P})\) (called \textit{admissible} in the paper under review). Now a \(G\)-bundle \(F_{G}\) of degree \(\check\lambda\in \check\Lambda_{G,G}\) is \textit{semistable} if \(\phi_{G}(\check\lambda)\geq\phi_{P}(\check\lambda_{P})\) for any parabolic reduction \(F_{P}\) of degree \(\check\lambda_{P}\in \check\Lambda_{G,P}\). Note that \(\alpha\geq0\) in \(\check\Lambda_{G}^{\mathbb{Q}}\) amounts to \(\alpha\) lies the \(\mathbb{Q}_{\geq0}\)-span of the positive coweights. A \textit{small} map \(X\to Y\) is a proper morphism satisfying that the fibre-product \(X\times_{Y}X\) has only the irreducible components of dimension \(\leq\dim X\) (\textit{semismall}) and that every maximal dimensional component dominates \(Y\). An \(M\)-bundle \(F_{M}\) (resp.\ \(P\)-bundle \(F_{P}\)) is \textit{regular} if the associated vector bundle \(F_{M}\times_{M}\mathfrak{g}/\mathfrak{p}\) (resp.\ \(F_{P}\times_{P}\mathfrak{g}/\mathfrak{p}\)) has vanishing sheaf cohomology \(H^{i}=0\), \(i\geq0\). We point out quite briefly several essential steps in the proof of the theorem. Recall first a result of \textit{Y. Laszlo} [Ann. Inst. Fourier 48, No. 2, 413--424 (1998; Zbl 0901.14019)] that \(q: \mathrm{Bun}_{P}^{\check\lambda_{P},\mathrm{ss, reg}}\cong \mathrm{Bun}_{M}^{\check\lambda_{M},\mathrm{ss, reg}}\) with the inverse is the induction map by \(M\hookrightarrow P\). Next the relative Weyl group \(W_{M,G}\) acts freely on \(\mathrm{Bun}_{M}^{\check\lambda_{P},\mathrm{ss, reg}}\) while it acts trivially on \(\mathrm{Bun}_{G}^{\check\lambda}\), where \(\check\lambda_{P},\check\lambda\) are given as in the statement. Thirdly the étaleness of \(p\) over the regular semistable locus comes from the long exact sequence of sheaf cohomology induced from \(0\to F_{P}\times_{P}\mathfrak{p} \to F_{P}\times_{P}\mathfrak{g} \to F_{P}\times_{P}\mathfrak{g}/\mathfrak{p} \to 0\). Finally the proof of properness of \(p\) over the semistable locus makes use of Drinfeld's compactification \(\overline{\mathrm{Bun}}_{P}^{\check\lambda_{P}}\) of \(\mathrm{Bun}_{P}^{\check\lambda_{P}}\) due to \textit{A. Braverman} and \textit{D. Gaitsgory} [Invent. Math. 150, No. 2, 287--384 (2002; Zbl 1046.11048)]. \(\overline{\mathrm{Bun}}_{P}^{\check\lambda_{P}}\) parametrises \((F_{G},F_{M/[M,M]},\kappa)\) where \(\kappa: F_{M/[M,M]}\times_{M/[M,M]}V \to F_{G}\times_{G}V\) is an injective sheaf homomorphism (here \(V\) is the highest weight \(\lambda\) simple \(G\)-module). Hence the projection \(\overline{\mathrm{Bun}}_{P}^{\check\lambda_{P},\mathrm{ss}}\to \mathrm{Bun}_{G}^{\check\lambda,\mathrm{ss}}\) is proper. By semistability, the component \(\kappa\) involved in \(\overline{\mathrm{Bun}}_{P}^{\check\lambda_{P},\mathrm{ss}}\) is always an isomorphism. Thus the properness follows as the substack \(\mathrm{Bun}_{P}^{\check\lambda_{P}}\) is the locus in the Drinfeld's compactification where \(\kappa\) is an isomorphism.