j. differential geometry 56 (2000) 301-379

HOLOMORPHIC PRINCIPAL BUNDLES OVER ELLIPTIC CURVES II: THE PARABOLIC CONSTRUCTION

ROBERT FRIEDMAN & JOHN W. MORGAN

Abstract

This paper continues the study of holomorphic semistable principal Gbundles over an elliptic curve. In this paper, the moduli space of all such bundles is constructed by considering deformations of a minimally unstable G-bundle. The set of all such deformations can be described as the ℂ^{∗}- quotient of the cohomology group of a sheaf of unipotent groups, and we show that this quotient has the structure of a weighted projective space. We identify this weighted projective space with the moduli space of semistable G-bundles, giving a new proof of a theorem of Looijenga.

Introduction

Let E be a smooth elliptic curve and let G be a simple complex algebraic group of rank r. We shall always assume that π_{1}(G) is cyclic and that c is a generator. We shall freely identify c with the corresponding element in the center of the universal cover G̃ of G. A C^{∞} G-bundle ξ_{0} over E has a characteristic class c_{1}(ξ_{0}) ∈ H^{2}(E; π_{1}(G)) ≅ π_{1}(G) which determines ξ_{0} up to C^{∞} isomorphism. The goal of this paper is to continue the study, begun in [10], of the moduli space M(G, c) of semistable holomorphic G-bundles ξ with c_{1}(ξ) = c. In [10], this space was studied from the transcendental viewpoint of (0, 1)-connections using the results of Narasimhan-Seshadri and Ramanathan that in every S-equivalence class there is a unique representative whose holomorphic structure is given by a flat connection. This viewpoint, however, is not suitable for many questions, such as finding universal bundles, studying singular elliptic curves, or generalizing to families of elliptic curves. In this paper, which is largely independent of [10], we describe M(G, c) from an algebraic point of view. As we shall show in later papers, this construction is much better adapted for dealing with the questions described above.

Received August 30, 2000. The first author was partially supported by NSF grant DMS-99-70437, and the second author was partially supported by NSF grant DMS-97-04507.

Our motivation comes from the theory of deformations of singularities with a ℂ^{∗}-action. In this theory, the deformations corresponding to nonnegative weights are topologically equisingular. Thus, from the point of view of smoothings, it is the negative weight deformation space which is interesting. This space can be studied infinitesimally, by looking at the ℂ^{∗}-action on the Zariski tangent space to the deformation functor. There is also a globalization of this local description. For example, in the case of hypersurface simple elliptic singularities which are given as weighted cones over an elliptic curve E, the global moduli space, modulo the action of ℂ^{∗}, can (essentially) be identified with pairs (X, D), where X is a smooth del Pezzo surface or a del Pezzo surface with rational double points, and D is a hyperplane section of X isomorphic to E.

By analogy, our method is to describe M(G, c) as a certain set of deformations of a “singular” bundle ξ_{u}, where in this context singular means unstable. Of course, it is natural to require that ξ_{u} be minimally unstable, in the sense that all small deformations which are not roughly speaking topologically equivalent to ξ_{u} should be semistable. As we showed in [11], such minimally unstable bundles always exist, and, in most cases, are unique once we fix the determinant. There is a distinguished subgroup ℂ^{∗} in the automorphism group of ξ_{u}, and (with our conventions) it acts with nonnegative weights. The positive weight deformations of ξ_{u} correspond to semistable bundles.

More precisely, to every unstable bundle ξ_{u} there is associated a conjugacy class of parabolic subgroups of G. The parabolic subgroup corresponding to a minimally unstable ξ_{u} is a maximal parabolic subgroup. The conjugacy classes of maximal parabolic subgroups are indexed by the simple roots of G. Let P be such a maximal parabolic subgroup, with unipotent radical U and Levi factor L. Then L has a one-dimensional center, and hence there is a natural inclusion of ℂ^{∗} in L. Correspondingly there is a unique primitive dominant character χ_{0} of L. We consider semistable L-bundles η whose degree is −1 with respect to χ_{0}. In particular, this implies that P_{−}, the parabolic subgroup opposite to P, is a Harder-Narasimhan parabolic subgroup for the unstable G-bundle ξ_{u} = η ×_{L} G. The central subgroup ℂ^{∗} of L acts on the deformation space of η ×_{L} G, and we shall be interested in the positive weight deformations. As in the singularities case, there is a global interpretation of this infinitesimal picture: it is the space of isomorphisms classes of pairs (ξ, φ) consisting of a P -bundle ξ and an isomorphism φ: ξ/U → η. The isomorphism classes of such pairs (ξ, φ) are classified by the cohomology group H^{1}(E; U(η)), where U(η) is the sheaf of unipotent groups η ×_{L} U. In general, U(η) is not a sheaf of abelian groups, and so H^{1}(E; U(η)) is a priori simply a set. However, the fact that U(η) has a filtration whose successive quotients are vector groups implies, in our situation, that H^{1}(E; U(η)) carries in a functorial sense the structure of an affine space. The group ℂ^{∗} acts on this space, fixing the origin and such that the differential of the action has positive weights at the origin. We show that this action can be linearized. From this point of view, it is the nonabelian nature of U which allows there to be different weights for the ℂ^{∗}-action, so that the quotient (H^{1}(E; U(η))−{0})/ℂ^{∗} is a weighted projective space 𝕎ℙ(η), typically with distinct weights. In fact, choosing η to be minimally unstable, the weights are given as follows in case G is simply connected. Let Δ̃ be the extended set of simple roots for G, and for each α ∈ Δ̃, let α^{∨} be the corresponding dual coroot. Then there is a unique linear relation

∑_{α∈Δ̃} g_{α}α^{∨} = 0,

provided that we require that the coefficient of the coroot dual to the negative of the highest root is 1. Up to multiplying by a common factor, the weights of 𝕎ℙ(η) are then the integers g_{α}, counted with multiplicity. In particular, the dimension of 𝕎ℙ(η) is equal to the rank of G, which we know to be the dimension of M(G, 1). When G is nonsimply connected and we consider G-bundles whose first Chern class is a generator c ∈ π_{1}(G), there is a similar result where, again up to a common factor, the weights are given by the coroot integers g\overline{β} on the quotient of the extended Dynkin diagram of G by the action of the central element c in the universal covering group G̃ of G as described in [7].

Assuming that the bundle η ×_{L} G described above is minimally unstable, all points of 𝕎ℙ(η) correspond to semistable bundles. Thus, there is an induced morphism Ψ: 𝕎ℙ(η) → M(G, c). Now every morphism whose domain is a weighted projective space is either constant or finite, and in our case it is easy to see that the morphism is nonconstant. This already yields some information about L and η: assuming that the action of Aut_{L} η on 𝕎ℙ(η) is essentially effective, since Ψ is constant on the Aut_{L} η/ℂ^{∗}-orbits, it follows that ℂ^{∗} is the identity component of Aut_{L} η. To go further, we need to use the fact that E is an elliptic curve, which implies that the map Ψ: 𝕎ℙ(η) → M(G, c) is dominant, and more generally that the rational map from 𝕎ℙ(η) to M(G, c) is dominant for every maximal parabolic. This implies that dim 𝕎ℙ(η) ≥ dim M(G, c), with equality for the minimally unstable case. By contrast, for curves of genus at least 2, even for the case of SL_{2}(ℂ), the map 𝕎ℙ(η) → M(G, c) is typically not dominant, but rather maps 𝕎ℙ(η) to a proper subvariety of M(G, c).

To sum up, then, in the minimally unstable case, we have a weighted projective space 𝕎ℙ(η) and a finite morphism Ψ: 𝕎ℙ(η) → M(G, c). In fact, we show the following theorem:

Theorem. Suppose that the L-bundle η is minimally unstable. Then the map Ψ: 𝕎ℙ(η) → M(G, c) is an isomorphism.

Corollary. The moduli space M(G, c) is a weighted projective space, with weights g\overline{β}/n_{0}, where n_{0} is the gcd of the g\overline{β}. In particular, if G is simply connected, M(G, c) is a weighted projective space, with weights g_{α}, α ∈ Δ̃.

In the simply connected case, the corollary is due to Looijenga [17] (see also [5]). Note however that the theorem goes beyond an abstract description of M(G, c) as a weighted projective space: it identifies an algebraically defined moduli space, 𝕎ℙ(η), with a transcendentally defined moduli space which in some sense is obtained by taking the periods of a flat connection. We view this as a theorem of Torelli type in a nonlinear context.

The bundles produced by the parabolic construction, in addition to being semistable, are regular in the sense that their automorphism groups have minimal possible dimension. This is reminiscent of the Steinberg cross-section of regular elements for the map of G to its adjoint quotient and of the Kostant section of regular elements for the adjoint quotient of the Lie algebra 𝔤 of G. In fact, as we shall show in a future paper, the parabolic construction extends to the case of nodal curves of genus one and to cuspidal curves of genus one when G ≠ E_{8}. For nodal curves, the parabolic construction produces a weighted projective space and an open subset which is identified with a Steinberg-like crosssection of regular elements in each conjugacy class. For cuspidal curves (and G ≠ E_{8}) the weighted projective space contains an open subset producing a Kostant-like section of the adjoint quotient of 𝔤. Thus, in both cases, the parabolic construction yields a new approach to the proof of the existence of a section of regular elements for the adjoint quotient, and produces a natural compactification of the adjoint quotient which is a weighted projective space.

In case G = E_{8}, E_{7}, E_{6} (as well as D_{5} and A_{4}), the relationship between deformations of minimally unstable G-bundles and deformations of simple elliptic singularities goes far beyond a formal analogy. Indeed, this observation, which is connected to what is called in the physics literature F-theory, was a major motivation for us to study G-bundles over elliptic curves. This connection will be described elsewhere.

This paper is related to [11], where we enumerate the minimally unstable strata in the space of (0, 1)-connections on a G-bundle. While we make no use of these results, at least in the simply connected case, that paper helps explain the characterizing properties that a minimally unstable bundle ξ_{u} has to satisfy: it predicts, for example, that its Harder-Narasimhan parabolic is the maximal parabolic associated with what we call a special root and that the L-bundle has degree −1 with respect to the dominant character. But even without knowing that these bundles lie in minimally unstable strata in the Atiyah-Bott formalism, one can establish the isomorphism given in the main theorem above.

The contents of this paper are as follows. In Section 1, we collect together preliminary technical results. Many of these results concern numerical facts related to irreducible representations of the Levi factor of a maximal parabolic on the unipotent radical of that parabolic. These are used in Section 2 to compute the dimensions of various cohomology groups related to bundles over maximal parabolic subgroups of G, as well as to understand the weights of the ℂ^{∗}-action. These dimensions and weights could be computed by case-by-case checking of the root tables. We have tried instead to find classification-free arguments wherever possible. In calculating the ℂ^{∗}-weights, we make use of a property we call circular symmetry. This property was introduced by Witten and established for the coroot integers in [7]. Its name derives from the relation of this property to a symmetry statement for points placed on a circle according to these numbers, as described in [7, §3.8]. Here, we do not need this geometric interpretation. Rather we need to know only that (as was proved in [7]) the coroot integers and the coroot integers on the quotient diagram by the action of the center satisfy circular symmetry, and that numbers satisfying circular symmetry are completely determined by three pieces of information, which in our context are the dimension of the weighted projective space, the highest weight appearing in the ℂ^{∗}-action and the dual Coxeter number of the group. In the minimally unstable case, we are able to show that these invariants agree with the corresponding ones for the coroot integers and this agreement is what allows us to identify the ℂ^{∗}-weights with the coroot integers. We emphasize, however, that circular symmetry holds for all maximal parabolic subgroups, not just those which correspond to minimally unstable bundles. Unfortunately, our proof of this resorts in the end to case-by-case checking. It would be extremely illuminating to have a more conceptual understanding of the meaning of circular symmetry. In §1.5, we discuss the volume of the moduli space of flat connections on E and again relate it to the coroot integers. These two pieces of numerical information, the ℂ^{∗}-weights for 𝕎ℙ(η) and the volume of the space of flat connections, will turn out to be crucial in Section 5 for the proof that deg Ψ = 1.

In Section 2, we give a general description, for every maximal parabolic subgroup P, of the bundles η over the Levi factor L. We then compute the dimensions of the nonabelian cohomology space (or rather its tangent space at the origin) and the ℂ^{∗}-weights in terms of the numbers introduced in Section 1. In Section 3, we study the minimally unstable case in detail. As we have mentioned above, we expect from general principles that dim Aut_{L} η = 1 and that the cohomology dimension must be dim M(G, c)+1. We identify the simple roots in the minimally unstable case, verifying the above facts in a classification-free way in the simply connected case, and identify the ℂ^{∗}-weights with the coroot integers via circular symmetry. The description of the minimally unstable case is also given in [11], although the discussion here in the simply connected case is independent of that paper. In §3.4, we consider the non-simply connected case. Here we use the results of [11] as well as a case-by-case analysis to identify the minimally unstable bundles and to identify the cohomology dimension, the dimension of the automorphism group of the bundle, and the ℂ^{∗}-weights.

Section 4 is concerned with the nonabelian cohomology space, i.e., the affine space H^{1}(E; U(η)). We show that the ℂ^{∗}-action can be linearized and discuss the obstructions to the existence of a universal bundle. In §4.4, we show that, in the minimally unstable case, the points of H^{1}(E; U(η)) − {0} correspond to regular semistable bundles. This means that the algebraic families provided by the parabolic construction are different from the families provided by the space of flat connections. Every S-equivalence class of bundles has two extreme representatives which are unique up to isomorphism, the flat representative and the regular representative. On an open dense set of M(G, c) these representatives agree and, when they do, all bundles of the given S-equivalence class are isomorphic. But, along a codimension one subvariety of M(G, c), the regular representative does not have a flat connection. It turns out that, because the dimension of the automorphism group of regular representatives is constant, the regular representatives behave better in families.

Finally, in Section 5, we prove the main theorem by calculating the degree of Ψ. The crux of the argument is to study the determinant line bundle on M(G, c), which pulls back via Ψ^{∗} to the determinant line bundle on 𝕎ℙ(η). Thus, once we show that both determinant line bundles have the same top self-intersection, then it follows that the degree of Ψ is one. Some of the technical results concerning the nonabelian cohomology space and its interpretation are deferred to the appendix. We prove that the cohomology space naturally has the structure of an affine space and represents an appropriate functor.

The parabolic construction of semistable G-bundles was originally introduced and explained, for the simply connected case, in [12], in a paper written for an audience of physicists, as well as in the announcement [13]. It is a pleasure to thank Ed Witten for originally raising the questions which led to this work and for the insights he shared with us during the course of our joint work on these subjects. We would also like to thank A. Borel, P. Deligne, and W. Schmid for various helpful conversations and correspondence. Finally, during the preparation of this paper, S. Helmke and P. Slodowy sent us their preprint [16], which has a considerable overlap with the first part of this paper and which also analyzes the case where the automorphism group of η ×_{L} G is just slightly larger than in the minimally unstable case.

1. Preliminaries

1.1 Notation

Throughout this paper, R denotes a reduced and irreducible root system of rank r in a real vector space V, with Weyl group W = W (R), and Δ is a set of simple roots for R. Let R^{+} be the set of positive roots corresponding to the choice of Δ. There exists a W -invariant positive definite inner product ⟨·, ·⟩ on V and it is unique up to scalars.

Given a root α, there is an associated coroot α^{∨} ∈ V^{∗}. Using the inner product to identify V with V^{∗}, we have α^{∨} = 2α/⟨α, α⟩. As usual, we denote the Cartan integer α(β^{∨}) by n(α, β). The coroot lattice Λ is the lattice inside V^{∗} spanned by the coroots. Given α ∈ Δ, we have the fundamental weight ϖ_{α} ∈ V, which satisfies ϖ_{α}(β^{∨}) = δ_{αβ} for all α, β ∈ Δ. The fundamental coweights ϖ_{α}^{∨} ∈ V^{∗} are defined similarly. As usual, let ρ be the sum of the fundamental weights, so that ρ = ∑_{α∈Δ} ϖ_{α} = \frac{1}{2} ∑_{β∈R^{+}} β.

Let α̃ be the highest root of R^{+}. We have α̃ = ∑_{β∈Δ} h_{β}β, with h_{β} > 0. We set α_{0} = −α̃, Δ̃ = Δ ∪ {α_{0}}, and h_{α_{0}} = 1 so that ∑_{β∈Δ̃} h_{β}β = 0. The number h = 1 + ∑_{β∈Δ} h_{β} is the Coxeter number of R. Similarly, we have α^{∨} = ∑_{β∈Δ} g_{β}β^{∨} with g_{β} > 0. We set g_{α_{0}} = 1 so that ∑_{β∈Δ̃} g_{β}β^{∨} = 0. We call g = 1 + ∑_{β∈Δ} g_{β} = ∑_{β∈Δ̃} g_{β} the dual Coxeter number of R. An easy calculation then shows:

Lemma 1.1.1. In the above notation, we have

g_{α} = \frac{h_{α}⟨α, α⟩}{⟨α̃, α̃⟩}.

Thus g_{α}|h_{α}, and g_{α} = h_{α} if and only if α is a long root of R. □

Let Q ∈ Sym^{2} Λ^{∗} be the quadratic form defined by

Q = ∑_{α∈R}⟨α, ·⟩⟨α, ·⟩.

This form was first introduced by Looijenga in [17], where he showed:

Lemma 1.1.2. Let g be the dual Coxeter number. Then

Q = (2g)I_{0},

where I_{0} is the unique W -invariant quadratic form on Λ such that I_{0}(α^{∨}) = 2. For example, if R is simply laced, then I_{0} is the usual intersection form on Λ. □

Throughout this paper, we use the inner product on V^{∗} defined by I_{0} and the corresponding dual inner product on V. It has the property that all long roots have length 2.

Lemma 1.1.3. Let {α^{∗}}_{α∈Δ} be the dual basis to {α^{∨}}_{α∈Δ} with respect to I_{0}. Then α^{∗} = g_{α}ϖ_{α}^{∨}/h_{α}.

Proof. This is just the statement that

⟨α^{∨}, ϖ_{β}^{∨}⟩ = \frac{2}{⟨α, α⟩}α(ϖ_{β}^{∨}) = \frac{h_{α}}{g_{α}} δ_{αβ}. □

1.2 Structure of maximal parabolic subgroups

Let H ⊆ G be a Cartan subgroup, and let R ⊆ 𝔥^{∗} be the set of roots for the pair (G, H). We denote by 𝔤 the Lie algebra of G and by 𝔥 ⊆ 𝔤 the Cartan subalgebra of 𝔤 corresponding to H. Let R^{+} be a set of positive roots, and let Δ denote the corresponding set of simple roots. We shall also view the roots α ∈ R as characters α: H → ℂ^{∗} on H. There is a unique dual coroot α^{∨} ∈ 𝔥. We can view α^{∨} as defining a linear map ℂ → 𝔥. Exponentiating this map gives us a cocharacter of H, i.e., a one-parameter subgroup ℓ_{α}: ℂ^{∗} → H.

If P is a parabolic subgroup of G, then the unipotent radical U of P is a normal subgroup, and there is a semidirect product P = U · L, where L is a reductive subgroup of P, unique up to conjugation, called the Levi factor of P. If G̃ is the universal cover of G, then there is a one-to-one correspondence between parabolic subgroups of G and those of G̃, which associates to the subgroup P of G its preimage P̃ ⊆ G̃. Since a unipotent group is torsion-free, the unipotent radicals of P and P̃ are isomorphic, and the Levi factor L̃ of P̃ is the preimage of the Levi factor L of P. For the remainder of §1.2, unless otherwise stated, we assume that G is simply connected.

For α ∈ Δ, let P^{α} be the connected subgroup of G whose Lie algebra is spanned by 𝔥 and the root spaces 𝔤^{β}, where either β ∈ R^{+} or β lies in the span of Δ − {α}. Then P^{α} is a maximal parabolic subgroup of G, and every maximal parabolic subgroup P is conjugate to exactly one P^{α}, α ∈ Δ. Thus there are exactly r maximal parabolic subgroups of G up to conjugation. We denote the unipotent radical of P^{α} by U^{α} and its Levi factor by L^{α}. The torus H is a maximal torus of L^{α}. The semisimple part S^{α} of L^{α} (or equivalently the derived subgroup of L^{α}) has Lie algebra spanned by 𝔥′ and by the root spaces corresponding to the set of roots in the linear span of Δ − {α}, where 𝔥′ ⊆ 𝔥 is the subspace spanned by the coroots β^{∨} ∈ 𝔥 dual to the simple roots β ∈ Δ − {α}. A maximal torus H′ of S^{α} is given by the subtorus which is the image under exponentiation of 𝔥′ ⊆ 𝔥, and H′ = H ∩ S^{α}. The Dynkin diagram of S^{α} is the subdiagram of the Dynkin diagram of G obtained by deleting the vertex corresponding to α and the edges incident to this vertex. Since {α^{∨}}_{α∈Δ} is a basis for the coroot lattice Λ of G, the intersection of the coroot lattice Λ for G with 𝔥′ is exactly the coroot lattice for S^{α}. Since G is simply connected, S^{α} is also simply connected. Note that S^{α} is a semisimple group of rank r − 1. Clearly we have:

Lemma 1.2.1. Let Δ − {α} = ∐_{i=1}^{t} Δ_{i}, where each Δ_{i} defines a connected component of the Dynkin diagram of Δ − {α}. Then S^{α} = ∏_{i=1}^{t} S_{i}, where S_{i} is the simple and simply connected group corresponding to Δ_{i}. □

Definition 1.2.2. Let K(α) = ⋂_{β∈Δ−{α}} Ker β ⊆ Λ. Then K(α) is an infinite cyclic group. Let ζ_{α} = ∑_{β∈Δ} m_{β}β^{∨} be the generator of K(α) such that m_{α} > 0. It then follows that m_{β} > 0 for all β ∈ Δ. Define n_{α} = α(ζ_{α}), so that ζ_{α} = n_{α}ϖ_{α}^{∨}. Note that m_{α} = ϖ_{α}(ζ_{α}).

Lemma 1.2.3. Define the map φ_{α}: ℂ^{∗} → H by

φ_{α}(λ) = ∏_{β∈Δ} ℓ_{β} (λ^{m_{β}}) .

Then φ_{α} is an isomorphism from ℂ^{∗} to the identity component of the center of L^{α}. Moreover,

L^{α} = S^{α} ×_{ℤ/m_{α}ℤ} ℂ^{∗},

where 1 ∈ ℤ/m_{α}ℤ maps to e^{2πi/m_{α}} ∈ ℂ^{∗} and to the central element ∏_{i=1}^{t} γ_{i}^{−n(β_{i},α)} ∈ S^{α} = ∏_{i=1}^{t} S_{i}, where β_{i} ∈ Δ_{i} is the unique element for which n(β_{i}, α) ≠ 0 and γ_{i} = exp(2π\sqrt{−1}ϖ_{β_{i}}^{∨}).

Proof. With φ_{α} defined as above, φ_{α}(ℂ^{∗}) is in the kernel of all simple roots β distinct from α, and thus, since ζ_{α} is primitive, φ_{α} is an embedding of ℂ^{∗} into the center of L^{α}. Also, since S^{α} ∩ H = H′, if λ = e^{2πit}, then φ_{α}(λ) ∈ S^{α} if and only if ∑_{β∈Δ} tm_{β}β^{∨} ≡ 0 mod 𝔥′ + Λ, if and only if m_{α}t ∈ ℤ. Thus φ_{α}(ℂ^{∗})∩S^{α} is the cyclic subgroup of order m_{α} in ℂ^{∗}, and so

L^{α} = S^{α} ×_{ℤ/m_{α}ℤ} ℂ^{∗},

where the image of 1 ∈ ℤ/m_{α}ℤ in the first factor lies in the center of S^{α}, and corresponds to the element c = ζ_{α}/m_{α} − α^{∨}. To describe this central element, let β ∈ Δ be a root of S_{i}. Then β(c) = 0 if β ≠ β_{i} and β_{i}(c) = −n(β_{i}, α). Thus c is the central element of S_{i} given by exp(−2n(β_{i}, α)π\sqrt{−1}ϖ_{β_{i}}^{∨}). □

The next lemma gives a more precise description of the center of L^{α}.

Lemma 1.2.4. The center of L^{α} is generated by φ_{α} (ℂ^{∗}) and the center of G. The intersection of φ_{α} (ℂ^{∗}) with the center of G is a cyclic group of order n_{α}.

Proof. We have

Z(L^{α}) = Z(S^{α}) ×_{ℤ/m_{α}ℤ} ℂ^{∗}.

Then Z(G) is the subgroup of Z(L^{α}) which is in the kernel of the character α. The restriction of α to ℂ^{∗} ⊆ Z(L^{α}) is nontrivial, and hence surjective, and for an element of Z(S^{α}) ×_{ℤ/m_{α}ℤ} ℂ^{∗}, written as [z, ζ], we clearly have α([z, ζ]) = α(z)α(ζ). Thus, for each z ∈ Z(S^{α}) there is an element u ∈ Ker(α) of the form u = [z, ζ] ∈ Z(S^{α}) ×_{ℤ/m_{α}ℤ} ℂ^{∗}. This element u is in the center of G, since it is in the kernel of β for all β ≠ α, as well as in Ker α. It follows that an arbitrary [z, λ] ∈ Z(L^{α}) is of the form u · μ, where μ ∈ ℂ^{∗}, as claimed.

To see the second statement, note that α∘φ_{α}(λ) = λ^{n_{α}}. Since φ_{α}(λ) is in the kernel of all of the remaining roots, φ_{α}(λ) lies in the center of G if and only if λ^{n_{α}} = 1. □

Remark 1.2.5. Along similar lines, one can show that there is an exact sequence

{1} → ℤ/n_{α}ℤ → Z(G) → Z(S^{α})/(ℤ/m_{α}ℤ) → {1}.

In particular, m_{α}/n_{α} = #Z(S^{α})/#Z(G).

To state the next result, recall that a character χ: H → ℂ^{∗} is dominant if the corresponding linear function 𝔥 → ℂ is nonnegative on the positive coroots. A character of any subgroup containing H will be called dominant if its restriction to H is dominant.

Lemma 1.2.6. The group of characters of L^{α} is isomorphic to ℤ. There is a unique primitive dominant character χ_{0} : L^{α} → ℂ^{∗}, and χ_{0} ∘ φ_{α}(λ) = λ^{m_{α}}.

Proof. Let ϖ_{α} be the fundamental weight corresponding to the simple root α. The unique primitive dominant character of L^{α} is ϖ_{α}, viewed as a character on H, and ϖ_{α} ∘ φ_{α}(λ) = λ^{m_{α}}. □

If G is not simply connected, we continue to denote the parabolic subgroup associated to the simple root α by P^{α}, the unipotent radical of P^{α} by U^{α}, and the Levi factor of P^{α} by L^{α}. The map φ_{α}: ℂ^{∗} → L^{α} is defined to be embedding of ℂ^{∗} into to the center of L^{α} so that the composition of φ_{α} followed by the primitive dominant character of L^{α} is a positive character on ℂ^{∗}. Let φ̃_{α}: ℂ^{∗} → L̃^{α} be as given in Definition 1.2.2 for the simply connected group G̃ and α ∈ Δ. Clearly, φ_{α}(ℂ^{∗}) is the quotient of φ̃_{α}(ℂ^{∗}) by the finite subgroup φ̃_{α}(ℂ^{∗}) ∩ ⟨c⟩. It is still the case, of course, that the center of L^{α} is generated by the center of G and φ_{α}(ℂ^{∗}). The following lemma is then clear:

Lemma 1.2.7. Let n_{c,α} be the order of φ_{α}(ℂ^{∗}) ∩ Z(G). Then the order of φ̃_{α}(ℂ^{∗})∩⟨c⟩ is n_{α}/n_{c,α}, and the induced map from φ̃_{α}(ℂ^{∗}) ≅ ℂ^{∗} to φ_{α}(ℂ^{∗}) ≅ ℂ^{∗} is given by raising to the power n_{α}/n_{c,α}. □

1.3 The unipotent radical of a maximal parabolic subalgebra

In §1.3, we assume that G is simply connected. Fix the simple root α and consider the maximal parabolic subgroup P^{α}. Here we will describe the Lie algebra 𝔲 = 𝔲(α) of U^{α}. It is spanned by the root spaces 𝔤^{δ} such that, if δ = ∑_{β∈Δ} x_{β}β, then the coefficient of α in the sum is positive. The action of φ_{α}(ℂ^{∗}) on 𝔲 is given as follows. Let 𝔤^{δ} be a root space and let X ∈ 𝔤^{δ}. Then φ_{α}(λ)(X) = λ^{δ(ζ_{α})} · X. Of course, since ζ_{α} is in the kernel of all the simple roots except α, δ(ζ_{α}) = δ(ϖ_{α}^{∨})α(ζ_{α}) where δ(ϖ_{α}^{∨}) is the coefficient of α in the expression for δ as a sum of simple roots. In particular, using Lemma 1.4.5 below, the weights of the action of ℂ^{∗} on 𝔲 are n_{α}, 2n_{α}, …, h_{α} · n_{α} where n_{α} = α(ζ_{α}).

Let 𝔲^{k} be the sum of all such root spaces 𝔤^{β} where the coefficient of α in β is exactly k. The Lie algebra 𝔲 is a direct sum of the spaces 𝔲^{k} for k > 0. Each space 𝔲^{k} is an L^{α}-module, and as we shall see below it is in fact irreducible.

The top exterior power ⋀^{top} 𝔲 is a one-dimensional L^{α}-module, and as such it is given by a character χ_{+} of L. The next lemma identifies this character:

Lemma 1.3.1. Let d_{1}(α) = 2ρ(ζ_{α})/m_{α} = 2ρ(ϖ_{α}^{∨})/ϖ_{α}(ϖ_{α}^{∨}). In the above notation, χ_{+} = χ_{0}^{d_{1}(α)} where χ_{0} is the primitive dominant character.

Proof. Since χ_{0} is primitive χ_{+} = χ_{0}^{N_{+}} for some integer N_{+}. Since the character χ_{0} ∘ φ_{α} of ℂ^{∗} is given by λ ↦ λ^{m_{α}}, the character χ_{+} ∘ φ_{α} of ℂ^{∗} is given by λ ↦ λ^{m_{α}·N_{+}}. The action of φ_{α}(ℂ^{∗}) is diagonal with respect to the decomposition of 𝔲 as a sum of root spaces, and the character on the one-dimensional subspace spanned by a root δ is simply the restriction of δ to φ_{α}(ℂ^{∗}). The space 𝔲 is the subspace of 𝔤 spanned by the set of all roots δ = ∑_{β∈Δ} x_{β}β with the property that x_{α} > 0. Recall that φ_{α}(ℂ^{∗}) is in the kernel of all the simple roots except α. Thus, to compute the character of ℂ^{∗} given by the product of φ_{α}^{δ} over all δ such that the coefficient of α in δ is positive, we may as well take the product over all of the positive roots δ. In other words, the character of ℂ^{∗} which gives the degree of the top exterior power is simply the character φ_{α}^{∑_{δ∈R^{+}} δ}. We can rewrite this expression as φ_{α}^{2ρ}. Thus, the character that we are computing is φ_{α}^{2 ∑_{β∈Δ} ϖ_{β}}. Recalling that the embedding of φ_{α}: ℂ^{∗} → H is given by ∏_{β∈Δ} ℓ_{β}^{m_{β}}, we see that the character χ_{+} ∘ φ_{α} is given by raising to the power 2ρ(ζ_{α}). Thus N_{+} = 2ρ(ζ_{α})/m_{α} = d_{1}(α).

□

Remark 1.3.2. The proof above also shows that the integer d_{1}(α) is the degree of divisibility of the canonical bundle of the homogeneous space G/P^{α}.

Now we give a purely root theoretic formula for d_{1}(α).

Lemma 1.3.3. d_{1}(α) = ∑_{β(ϖ_{α}^{∨})>0} n(β, α), where the β in the sum range over R.

Proof. By definition, χ_{+} = ∑_{β(ϖ_{α}^{∨})>0} β as additive characters on 𝔥. By the previous lemma, χ_{+}(ζ_{α}/m_{α}) = d_{1}(α)χ_{0}(ζ_{α}/m_{α}) = d_{1}(α). On the other hand, ζ_{α}/m_{α} = α^{∨} + ν, where ν lies in the ℚ-span of the simple coroots β^{∨}, β ≠ α, and hence in the Lie algebra of the derived group S^{α}. Thus

χ_{+}(ζ_{α}/m_{α}) = χ_{+}(α^{∨}) = ∑_{\begin{matrix}β∈R \\ β(ϖ_{α}^{∨})>0 \end{matrix}} β(α^{∨}),

and so d_{1}(α) = ∑_{β(ϖ_{α}^{∨})>0} n(β, α). □

Similarly, we compute the character χ_{k} of L^{α} corresponding to ⋀^{top} 𝔲^{k}:

Lemma 1.3.4. For k > 0, let c(α, k) be the dimension of 𝔲^{k}, in other words the number of roots β such that the coefficient of α in β is k. Let i(α, k) = kc(α, k)/ϖ_{α}(ϖ_{α}^{∨}) = kn_{α}c(α, k)/m_{α}. Then the character χ_{k} of L^{α} corresponding to ⋀^{top} 𝔲^{k} is given by

χ_{k} = χ_{0}^{i(α,k)}.

Proof. As before, χ_{k} = χ_{0}^{N_{k}} for some integer N_{k}. To compute N_{k}, note that, if 𝔤^{β} ⊆ 𝔲^{k}, then φ_{α} acts on 𝔤^{β} via raising to the power β(ζ_{α}) = kn_{α}. Since χ_{0} ∘ φ_{α}(λ) = λ^{m_{α}}, we must have

m_{α}N_{k} = kn_{α} dim 𝔲^{k}.

Hence, N_{k} = kn_{α}c(α, k)/m_{α} = i(α, k). □

Remark 1.3.5. An argument very similar to the proof of Lemma 1.3.3 shows that

i(α, k) = ∑_{β(ϖ_{α}^{∨})=k} n(β, α).

1.4 Some lemmas on root systems

Fix α ∈ Δ. Our goal now is to analyze the L^{α}-representations 𝔲^{k}, and in particular the numbers d_{1}(α) and i(α, k) introduced above.

Definition 1.4.1. Fix a positive integer k, and consider the set

S(α, k) = {β ∈ R : β(ϖ_{α}^{∨}) = k}.

For k = 0, we define similarly

S^{+}(α, 0) = {β ∈ R^{+} : β(ϖ_{α}^{∨}) = 0}.

The latter is a set of positive roots for the root system R′ = Δ−{α}, i.e., for S^{α}. We define S^{−}(α, 0) similarly. A lowest root σ_{k}(α) for S(α, k) is a root σ_{k}(α) ∈ S(α, k) such that, for every β ∈ S(α, k), β − σ_{k}(α) is a sum (possibly empty) of simple roots. For example, σ_{1}(α) = α. A lowest root in S(α, k) is clearly unique, if it exists. A highest root λ_{k}(α) ∈ S(α, k) is defined similarly, and is also clearly unique if it exists.

The following is related to results of Borel-Tits (unpublished) as well as Azad-Barry-Seitz [3].

Proposition 1.4.2. In the above notation, if S(α, k) ≠ ∅, then lowest roots and highest roots always exist and are unique.

Proof. Let R(α, k) be the subset of R consisting of roots β such that k divides β(ϖ_{α}^{∨}). Clearly R(α, k) is again a root system. Let V be the real span of R and let V′ be the subspace of V spanned by Δ − {α}. Then clearly V′ ∩ R = V′ ∩ R(α, k) = R′ is the set of all roots which are linear combinations of elements of Δ − {α}. Thus R′ is a root system with simple roots Δ − {α}. By [8, VI §1, Proposition 24], since R′ ⊆ R(α, k), there exists a set of simple roots for R(α, k) containing Δ − {α}. In fact, the proof of this proposition shows that there are at least two different sets of simple roots, each of the form (Δ−{α})∪{β}. Then k|β(ϖ_{α}^{∨}), and since S(α, k) ≠ ∅, in fact β(ϖ_{α}^{∨}) = ±k. Suppose for example that β(ϖ_{α}^{∨}) = k. If γ ∈ S(α, k), then γ = ∑_{δ≠α} m_{δ}δ + mβ and γ(ϖ_{α}^{∨}) = β(ϖ_{α}^{∨}) = k. Thus m = 1 and m_{δ} ≥ 0 for all δ ∈ Δ − {α}, so that β is a lowest root for S(α, k). Suppose now that (Δ − {α}) ∪ {β′} is also a set of simple roots for R(α, k), where β′ ≠ β. It follows that ϖ_{α}^{∨}(β′) = −k. In this case, it is easy to check that −β′ is a highest root for S(α, k). □

Corollary 1.4.3. The L^{α}-modules 𝔲^{k} are irreducible. □

Remark 1.4.4. Using the Borel-de Siebenthal theorem [8, p. 229, ex. 4], the Dynkin diagram of R(α, k) is given abstractly as follows. Begin with the extended Dynkin diagram of R. There exists a root β such that h_{β} = k, and such that the Dynkin diagram for Δ̃ − {β} contains a root γ such that the diagram for Δ̃−{β, γ} is the same as the diagram for Δ − {α}. In practice, these properties uniquely determine β and γ. The Dynkin diagram for R(α, k) is then the Dynkin diagram of Δ̃ − {β}, and γ corresponds to λ_{k}(α).

Lemma 1.4.5. Let k be a positive integer. Then S(α, k) ≠ ∅ if and only if 1 ≤ k ≤ h_{α}.

Proof. Since α ∈ S(α, 1), S(α, 1) ≠ ∅. Suppose inductively we have shown that S(α, k) ≠ ∅ for all k, 1 ≤ k ≤ ℓ. If β ∈ Δ−α, then λ_{ℓ}(α)+β is not a root. If λ_{ℓ}(α) + α is not a root, then λ_{ℓ}(α) is the highest root and ℓ = h_{α}. Otherwise, λ_{ℓ}(α) + α ∈ S(α, ℓ + 1). □

For future reference, we record the following properties of σ_{k}(α) and λ_{k}(α):

Lemma 1.4.6. In the above notation, let R′ ⊆ R be the subroot system with simple roots Δ − {α}. Suppose that S(α, k) ≠ ∅, and let w′_{0} ∈ W (R′) ⊆ W (R) be the unique element such that w′_{0}(Δ − {α}) = −(Δ − {α}). Suppose that τ is the permutation of Δ − {α} induced by −w′_{0}, which we can also view as a permutation of Δ fixing α. Let τ act on V and V^{∗} in the natural way. Then:

(i) w′_{0}σ_{k}(α) = λ_{k}(α).

(ii) λ_{1}(α^{∨}) = (λ_{1}(α))^{∨}.

(iii) σ_{k}(α) = kλ_{1}(α) − τ(λ_{k}(α)) + kα. Likewise σ_{k}(α)^{∨} = k′λ_{1}(α^{∨}) − τ(λ_{k}(α)^{∨}) + k′α^{∨}, where k′ is the coefficient of α^{∨} in λ_{k}(α)^{∨}.

Proof. Clearly w′_{0}σ_{k}(α) ∈ S(α, k) has the property that

(Δ − {α}) ∪ {−w′_{0}σ_{k}(α)}

is a set of simple roots for R(α, k). By the proof of Proposition 1.4.2, w′_{0}σ_{k}(α) = λ_{k}(α), proving (i). To see (ii), note that α^{∨} = σ_{1}(α^{∨}), and thus λ_{1}(α^{∨}) = w′_{0}σ_{1}(α^{∨}) = w′_{0}(α^{∨}) = (w′_{0}α)^{∨} = (λ_{1}(α))^{∨}. To see (iii), write λ_{k}(α) = kα + λ′, where λ′ ∈ V′. Then

σ_{k}(α) = w′_{0}λ_{k}(α) = kw′_{0}α − τ(λ′) = kλ_{1}(α) − τ(λ′).

On the other hand, −τ(λ′) = kα − τ(λ_{k}(α)), and plugging this back in gives the first part of (iii). The second part is proved in a very similar way. □

Recall that, for k > 0, c(α, k) is the cardinality of S(α, k), in other words the number of β ∈ R such that β(ϖ_{α}^{∨}) = k, and that i(α, k) = kc(α, k)/ϖ_{α}(ϖ_{α}^{∨}).

Definition 1.4.7. Define

d_{k}(α) = ∑_{k|x} i(α, x) = ∑_{ℓ>0} i(α, ℓk).

In particular d_{1}(α) agrees with the previous definition. Of course, the d_{k}(α) determine i(α, k) via Moebius inversion. Since the i(α, k) are all positive integers, the d_{k}(α) are positive integers as well.

Lemma 1.4.8. With notation as above,

i(α, k) = \frac{h_{α}gkc(α, k)}{g_{α} ∑_{β∈R^{+}}(β(ϖ_{α}^{∨}))^{2}} ;

d_{1}(α) = 2ρ(ϖ_{α}^{∨})/ϖ_{α}(ϖ_{α}^{∨}) = \frac{h_{α}g ∑_{β∈R^{+}} β(ϖ_{α}^{∨})}{g_{α} ∑_{β∈R^{+}}(β(ϖ_{α}^{∨}))^{2}} .

Moreover ∑_{k>0} φ(k)d_{k}(α) = h_{α}g/g_{α}.

Proof. The first equality follows since, by Lemma 1.1.2 and Lemma 1.1.3,

2g (\frac{h_{α}}{g_{α}} ) ϖ_{α}(ϖ_{α}^{∨}) = 2g⟨ϖ_{α}^{∨}, ϖ_{α}^{∨}⟩ = 2gI_{0}(ϖ_{α}^{∨}) = 2 ∑_{β∈R^{+}}(β(ϖ_{α}^{∨}))^{2}.

The second follows since

∑_{k>0} kc(α, k) = ∑_{β∈R^{+}} β(ϖ_{α}^{∨}).

The third is an easy consequence of the fact that ∑_{k|x} φ(k) = x, using

∑_{k>0} k^{2}c(α, k) = ∑_{β∈R^{+}}(β(ϖ_{α}^{∨}))^{2}. □

Proposition 1.4.9. We have d_{1}(α) = ρ(λ_{1}(α^{∨})) + 1, where λ_{1}(α^{∨}) is the highest coroot such that the coefficient of α^{∨} in λ_{1}(α^{∨}) is 1.

Proof. By Lemma 1.3.3, it suffices to prove that

ρ(λ_{1}(α^{∨})) + 1 = ∑_{β(ϖ_{α}^{∨})>0} n(β, α).

By Lemma 1.4.6, since σ_{1}(α^{∨}) = α^{∨}, w′_{0}(α^{∨}) = λ_{1}(α^{∨}). Since w′_{0} exchanges positive and negative roots in R′, it follows that −w′_{0} is a permutation of S^{+}(α, 0). Clearly, given β ∈ S^{+}(α, 0),

β(λ_{1}(α^{∨})) = −(−w′_{0}(β)(α^{∨})).

Thus, since −w′_{0} permutes S^{+}(α, 0), we have

∑_{\begin{matrix}β∈S^{+}(α,0) \\ β(λ_{1}(α^{∨}))>0 \end{matrix}} β(λ_{1}(α^{∨})) + ∑_{\begin{matrix}β∈S^{+}(α,0) \\ β(α^{∨})<0 \end{matrix}} β(α^{∨}) = 0.

Next we claim:

Lemma 1.4.10.

− ∑_{\begin{matrix}β∈S^{+}(α,0) \\ β(α^{∨})<0 \end{matrix}} β(α^{∨}) + 2 = ∑_{β(ϖ_{α}^{∨})>0} β(α^{∨}) = ∑_{β(ϖ_{α}^{∨})>0} n(β, α).

Proof. Suppose that β ≠ ±α ∈ R, and consider the α-string defined by β, say β − qα, …, β + pα. Since p − q + 1 = −n(β, α) + 1, it is easy to see that ∑_{i=−q}^{p}(β + iα)(α^{∨}) = 0. If β is a negative root, then every root in the α-string is negative and thus none of them appears in the right hand side of the above equality. After reindexing, we can assume that β is the origin of the α-string. If β(α^{∨}) > 0, then (β + iα)(α^{∨}) > 0 for all i > 0, and so the total contribution to the right hand side from the sum over the α-string is zero. If β ∈ S^{+}(α, 0), then the contribution to the sum on the right hand side is ∑_{i≥1}(β + iα)(α^{∨}) = −β(α^{∨}). The remaining possibility for the right hand side is β = α, and in this case β(α^{∨}) = 2. Thus we see that the right hand side in Lemma 1.4.10 is equal to the left hand side. □

Lemma 1.4.11.

∑_{\begin{matrix}β∈S^{+}(α,0) \\ β(λ_{1}(α^{∨}))>0 \end{matrix}} β(λ_{1}(α^{∨})) = ρ(λ_{1}(α^{∨})) − 1.

Proof. First note that

2ρ(λ_{1}(α^{∨})) = ∑_{β∈R^{+}} β(λ_{1}(α^{∨})).

We consider as before the λ_{1}(α)-strings defined by a root β ≠ λ_{1}(α) which lie in R^{+}. If the origin of such a string lies in R^{+}, then so does every γ lying in the string, and the sum over all such γ of n(γ, λ_{1}(α)) is zero. Next we claim:

Claim 1.4.12. If a λ_{1}(α)-string meets R^{+} but is not contained in R^{+}, then either:

(i) The origin of the string lies in S^{−}(α, 0) and all other elements of the string lie in R^{+}.

(ii) The extremity of the string lies in S^{+}(α, 0), and all other elements lie in R^{−}.

Moreover, there is a length-preserving bijection between strings of types (i) and (ii) above.

Proof. First note that, if β ∈ S(α, 1), then β − 2λ_{1}(α) cannot be a root. For then β − λ_{1}(α) would also be a root, necessarily negative, and then 2λ_{1}(α) − β would be an element of S(α, 1) higher than λ_{1}(α). Thus, every λ_{1}(α)-string meeting R^{+} but not contained in it must either begin or terminate in S^{+}(α, 0) ∪ S^{−}(α, 0). Clearly, the only possibilities are (i) and (ii) above, and the bijection is given by sending the origin of a string of type (i) to its negative, which is the extremity of a string of type (ii). □

Returning to the proof of Lemma 1.4.11, the only nonzero contributions to the sum ∑_{β∈R^{+}} β(λ_{1}(α^{∨})) come from

(i) λ_{1}(α)-strings whose origin is γ = −β ∈ S^{−}(α, 0), and these contribute −γ(λ_{1}(α^{∨})) = β(λ_{1}(α^{∨}));

(ii) λ_{1}(α)-strings whose extremity is β ∈ S^{+}(α, 0), and these contribute β(λ_{1}(α^{∨}));

(iii) the root λ_{1}(α) and this contributes λ_{1}(α)(λ_{1}(α^{∨})) = 2.

Summing these up, we see that

2ρ(λ_{1}(α^{∨})) = ∑_{β∈R^{+}} β(λ_{1}(α^{∨})) = 2 ∑_{\begin{matrix}β∈S^{+}(α,0) \\ β(λ_{1}(α^{∨}))>0 \end{matrix}} β(λ_{1}(α^{∨})) + 2.

Dividing by 2 gives the final formula of Lemma 1.4.11. □

To complete the proof of Proposition 1.4.9, we have

∑_{β(ϖ_{α}^{∨})>0} n(β, α) = − ∑_{\begin{matrix}β∈S^{+}(α,0) \\ β(α^{∨})<0 \end{matrix}} β(α^{∨}) + 2

= ∑_{\begin{matrix}β∈S^{+}(α,0) \\ β(λ_{1}(α^{∨}))>0 \end{matrix}} β(λ_{1}(α^{∨})) + 2

= ρ(λ_{1}(α^{∨})) + 1,

as claimed. □

There is a generalization of the previous proposition to the computation of d_{k}(α) for every k > 0:

Proposition 1.4.13. Let

k′ = k⟨α, α⟩/⟨λ_{k}(α), λ_{k}(α)⟩

be the coefficient of α^{∨} in λ_{k}(α)^{∨}. Then

d_{1}(α) + d_{k}(α) = \frac{2}{k′} (ρ(λ_{k}(α)^{∨}) + 1).

Proof. In the notation of Proposition 1.4.2, set α_{k} = −λ_{k}(α) and let R_{k} = R(α, k), with simple roots (Δ−{α})∪{α_{k}}. Although R_{k} need not be irreducible, we can still define the integer d_{1}^{R_{k}}(α_{k}) with respect to the root system R_{k} as in Lemma 1.3.1. By Proposition 1.4.9, which holds even if R_{k} is reducible, d_{1}^{R_{k}}(α_{k}) = ρ_{k}(λ_{1}((α_{k})^{∨})) + 1, where ρ_{k} is the sum of the fundamental weights of the root system R_{k}. Applying Lemma 1.4.6, and using the notation introduced in its statement, we have

λ_{1}((α_{k})^{∨}) = λ_{1}(α_{k})^{∨} = w′_{0}(α_{k})^{∨} = (w′_{0}α_{k})^{∨} = −(σ_{k}(α))^{∨}
= −k′λ_{1}(α^{∨}) + τ(λ_{k}(α))^{∨}) − k′α^{∨},

where k′ = k⟨α, α⟩/⟨λ_{k}(α), λ_{k}(α)⟩ is the coefficient of α^{∨} in λ_{k}(α)^{∨}. Next we compute ρ_{k}. Write

(α_{k})^{∨} = ∑_{β∈Δ−{α}} c_{β}β^{∨} − k′α^{∨} = ∑_{β∈Δ} c_{β}β^{∨},

where c_{α} = −k′. Denote the fundamental weights of R_{k} by ϖ_{β}^{k}, β ≠ α, and ϖ_{α_{k}}. Then it is easy to check that

ϖ_{α_{k}} = −\frac{1}{k′} ϖ_{α};
ϖ_{β}^{k} = ϖ_{β} + \frac{c_{β}}{k′} ϖ_{α}  for β ≠ α.

Thus

ρ_{k} = ∑_{β≠α} ϖ_{β}^{k} + ϖ_{α_{k}} = ∑_{β≠α} ϖ_{β} + \frac{1}{k′} (∑_{β≠α} c_{β} − 1) ϖ_{α}

= ∑_{β} ϖ_{β} + \frac{1}{k′} (∑_{β} c_{β} − 1) ϖ_{α}

= ρ − \frac{1}{k′} (ρ(λ_{k}(α)^{∨}) + 1) ϖ_{α}.

Since

ϖ_{α}(−k′λ_{1}(α^{∨}) + τ(λ_{k}(α))^{∨}) − k′α^{∨}) = −k′

and ρ(τ(γ)) = ρ(γ) for all γ, we see that

d_{1}^{R_{k}}(α_{k}) = −k′(ρ(λ_{1}(α^{∨}) + 1) + ρ(λ_{k}(α)^{∨})) + 1 + ρ(λ_{k}(α)^{∨})) + 1
= −k′d_{1}(α) + 2(ρ(λ_{k}(α)^{∨})) + 1).

Now an argument similar to the calculation of ρ_{k} above shows that ϖ_{α_{k}}^{∨}, the fundamental coweight for R_{k} dual to α_{k}, is given by −(1/k)ϖ_{α}^{∨}. Also, c(α, nk) = c(α_{k}, n) for all n. Thus i(α_{k}, n) = k′i(α, nk) for all positive integers n. It follows that d_{1}^{R_{k}}(α_{k}) = k′d_{k}(α). Putting this together with the above gives k′d_{1}(α) + k′d_{k}(α) = 2(ρ(λ_{k}(α)^{∨})) + 1), which is the statement of the proposition. □

Corollary 1.4.14. d_{1}(α) + d_{h_{α}}(α) = \frac{2g}{g_{α}}.

Proof. For k = h_{α}, we have λ_{k}(α) = α̃ and k′ = kg_{α}/h_{α} = g_{α}, and the corollary is clear. □

To put the corollary in a more general context, we have the following definition which is taken from [7]:

Definition 1.4.15. For a sequence d_{1}, …, d_{N} of positive integers, let M = ∑_{k>0} φ(k)d_{k}. Let the Farey sequence

F_{N} = {0/1, 1/N, 1/(N − 1), … }

be the sequence of rational numbers between 0 and 1, written in lowest terms, whose denominator is at most N, written in increasing order. We say that the sequence {d_{k}} has the circular symmetry property with respect to N and M if, for all consecutive terms r/x and s/y in F_{N},

d_{x} + d_{y} = \frac{2M}{xy} .

The geometric meaning of this property is explained in [7].

Since every integer x, 1 ≤ k ≤ N, appears as a denominator of some element of F_{N}, the following is clear:

Lemma 1.4.16. Suppose that d_{1}, …, d_{N} and d′_{1}, …, d′_{N} are two sequences of positive integers which both satisfy the circular symmetry property with respect to N and M. If d_{1} = d′_{1}, then d_{x} = d′_{x} for all x, 1 ≤ k ≤ N. □

For the sequence {d_{k}(α)}, the largest integer N such that d_{N}(α) ≠ 0 is h_{α}, and ∑_{k>0} φ(k)d_{k}(α) = gh_{α}/g_{α}, by Lemma 1.4.8. Moreover d_{1}(α) is given by Proposition 1.4.9. Then Corollary 1.4.14 is the first case x = 1, y = h_{α} of the following:

Proposition 1.4.17. With notation as above, the sequence {d_{k}(α)} has the circular symmetry property with respect to h_{α} and gh_{α}/g_{α}.

Proof. If h_{α} ≤ 3, it is easy to check that the above conditions follow from Corollary 1.4.14 and the fact that ∑_{k>0} φ(k)d_{k}(α) = gh_{α}/g_{α}. The remaining cases are: F_{4} with α the root such that h_{α} = 4, E_{7} with α the root such that h_{α} = 4, or E_{8} with α a root such that h_{α} = 4, 5, 6. These cases may be checked by hand. □

It would be very interesting to find a more conceptual proof of Proposition 1.4.17.

1.5 The moduli space of semistable G-bundles

If ξ_{0} is a C^{∞} principal G-bundle over E, then there is a characteristic class c_{1}(ξ_{0}) ∈ H^{2}(E; π_{1}(G)) ≅ π_{1}(G) = ⟨c⟩. Let M(G, c) be the set of Sequivalence classes of holomorphic semistable G-bundles ξ over E with c_{1}(ξ) = c. Suppose that K is the compact form of G. Then ⟨c⟩ = π_{1}(K). Let K̃ be the universal covering group of K. By the theorem of Narasimhan-Seshadri and Ramanathan and [10, 5.8(i)], M(G, c) is homeomorphic to the the space

{(x, y) ∈ K̃ × K̃ : xyx^{−1}y^{−1} = c}/K̃,

where the action of K̃ is by simultaneous conjugation. By [7, 3.2], corresponding to c ∈ Z(G̃) there is an element w_{c} ∈ W and an affine isomorphism φ_{c} of V^{∗} which permutes the set Δ̃ and thus acts on the fundamental alcove A corresponding to the Weyl chamber defined by Δ.

Definition 1.5.1. The Weyl element w_{c} induces a permutation of the set Δ̃ which induces an automorphism of the extended Dynkin diagram of G. Thus g_{w_{c}(α)} = g_{α} for all α ∈ Δ̃. Let \overline{α} denote the w_{c}- orbit of α and let n\overline{α} be the cardinality of \overline{α}. Set g\overline{α} = n\overline{α}g_{α}, for any choice of α ∈ \overline{α}. Let n_{0} be the gcd of the integers g\overline{α}. Let Λ̃ be the free abelian group with basis Δ̃. Then w_{c} acts on Λ̃, preserving the relation ∑_{β∈Δ̃} g_{β}β^{∨} = 0, and so w_{c} acts on the quotient which is Λ. Define r_{c} by: r_{c} + 1 is the cardinality of Δ̃/w_{c}.

Let A^{c} be the fixed subspace of φ_{c}, acting on A. With V^{∗} = Λ ⊗ ℝ, let T = V^{∗}/Λ, so that T is a real torus of dimension r. Then W and w_{c} act on T. Let T_{0} = (T^{w_{c}})^{0} be the identity component of the group T^{w_{c}}. Thus r_{c} = dim T_{0}.

For an abelian group A and an automorphism σ of A, we denote as usual by A^{σ} the subgroup of invariants of σ and by A_{σ} the group A/ Im(Id −σ) of coinvariants of σ. The automorphism w_{c} acts on T = V^{∗}/Λ, and we can define T^{w_{c}} and T_{w_{c}} as before. Let T_{0} = (T^{w_{c}})^{0} be the identity component of the group T. There is an induced map from T_{0} to T_{w_{c}} and it is finite. We then have:

Theorem 1.5.2. Fix x_{0} ∈ A^{c} and y_{0} an element in the normalizer N_{T} (K̃) of T in K̃ which projects to w_{c} in W = N_{T} (K̃)/T. Then for s, t ∈ T_{0} the pair x = sx_{0}, y = ty_{0} satisfies [x, y] = c. We define a map T_{0} × T_{0} → M(G, c) by (s, t) ↦ [(sx_{0}, ty_{0})]. This map is finite and surjective. Its degree is

(r_{c})!\frac{det(I_{0}|Λ^{w_{c}})}{n_{0}} ∏_{\overline{α}} g\overline{α},

where the product is over the w_{c}-orbits of Δ̃.

Proof. By [7, Lemma 6.1.7], with K̃ and c as above, every pair (x, y) such that xyx^{−1}y^{−1} = c is conjugate to such a pair with x ∈ A^{c} and y ∈ T_{0} · w_{c}. This proves that the map is surjective. Clearly, it is finite-to-one. If x is in the interior of A^{c}, then it is regular, and the only further possible conjugation is via an element t ∈ T, which acts on y via t − w_{c}(t). Thus, a fundamental domain for the map T_{0} × T_{0} → M(G, c) is given by A^{c} × S, where S is a fundamental domain for the quotient map T_{0} → T_{w_{c}}. It follows that the degree of the map T_{0}×T_{0} → M(G, c) is the product of the degree of the map from T_{0} to T_{w_{c}} with the ratio vol(T_{0})/ vol(A^{c}), where volume is computed with respect to any Weyl invariant metric. We consider these two integers separately.

Lemma 1.5.3. Let \overline{α}^{∨} = ∑_{α∈\overline{α}} α^{∨}. Then

Λ^{w_{c}} ≅ ⨁_{\overline{α}} ℤ · \overline{α}^{∨}/ ∑_{\overline{α}} g_{α}\overline{α}^{∨}.

Moreover, the set \{\overline{α}^{∨} : \overline{α} ≠ \overline{α}_{0}\} is an integral basis for Λ^{w_{c}}. Finally, for each orbit \overline{α}, choose α ∈ \overline{α} and let e\overline{α} be the image of α in Λ_{w_{c}}. Then

Λ_{w_{c}} ≅ ⨁_{\overline{α}} ℤ · e\overline{α}/ ∑_{\overline{α}} g\overline{α}e\overline{α}.

Proof. There is an exact sequence

0 → ℤ → Λ̃ → Λ → 0,

where 1 ∈ ℤ ↦ ∑_{β∈Δ̃} g_{β}β^{∨}. The homomorphisms in this sequence are equivariant with respect to the action of w_{c}. Moreover, w_{c} acts on Λ̃ by a permutation of the basis. The proof of the lemma follows easily by considering the associated long exact sequence

0 → ℤ^{w_{c}} → (Λ̃)^{w_{c}} → Λ^{w_{c}} → ℤ_{w_{c}} → (Λ̃)_{w_{c}} → Λ_{w_{c}} → 0. □

Corollary 1.5.4. The torsion subgroup Tor Λ_{w_{c}} ≅ ℤ/n_{0}ℤ, and

Λ_{w_{c}}/ Tor Λ_{w_{c}} ≅ ⨁_{\overline{α}} ℤ · e\overline{α}/ ∑_{\overline{α}} \frac{g\overline{α}}{n_{0}} e\overline{α}. □

Lemma 1.5.5. The order of T^{w_{c}}/T_{0} is n_{0}. The natural map T_{0} → T_{w_{c}} is finite and surjective of degree ∏\overline{α} n\overline{α}/n_{0}.

Proof. Beginning with the short exact sequence of w_{c}-modules

0 → Λ → V → T → 0,

we get a long exact sequence

0 → Λ^{w_{c}} → V^{w_{c}} → T^{w_{c}} → Λ_{w_{c}} → V_{w_{c}} → T_{w_{c}} → 0.

The quotient V^{w_{c}}/Λ^{w_{c}} = T_{0}. Since T^{w_{c}}/T_{0} is a finite group and V_{w_{c}} is torsion free, the induced map T^{w_{c}}/T_{0} → Λ_{w_{c}} is an isomorphism from T^{w_{c}}/T_{0} to Tor Λ_{w_{c}}, and hence T^{w_{c}}/T_{0} ≅ ℤ/n_{0}ℤ. Moreover, it is clear that the degree of the map from T_{0} to T_{w_{c}} is the index of the image of Λ^{w_{c}} in Λ_{w_{c}}/ Tor Λ_{w_{c}}. By Lemma 1.5.3 and Corollary 1.5.4, it suffices to compute the order of the quotient of ⨁\overline{α} ℤ · e\overline{α} by the relations n\overline{α}e\overline{α} and ∑\overline{α} g\overline{α}e\overline{α}. Since g_{\overline{α}_{0}} = 1, it is clear that the quotient has order ∏\overline{α} n\overline{α}/n_{0}. □

Now we compute vol(T_{0})/ vol(A^{c}) using the volume determined by the inner product I_{0}. The alcove A has vertices equal to 0 and ϖ_{α}^{∨}/h_{α}, α ∈ Δ. By Lemma 1.1.3, the vertices are 0 and α^{∗}/g_{α}, where the α^{∗} are the dual basis with respect to I_{0} to the basis α^{∨} of Λ. Now we have the following elementary lemma, whose proof is left to the reader:

Lemma 1.5.6. Let A be a simplex in ℝ^{n} with vertices 0 = e_{0}, e_{1}, …, e_{n}. Let φ be an affine linear transformation of ℝ^{n} which acts via a permutation of the vertices of A. Suppose that the orbits of w on the vertices are o_{0}, …, o_{s}, with 0 ∈ o_{0}. If n_{o} is the order of the orbit o, set

v_{o} = \frac{1}{n_{o}} ∑_{e_{i}∈o} e_{i}.

Then the fixed set of A^{φ} for the action of φ on A is a simplex with vertices

v_{o_{0}}, v_{o_{1}} + v_{o_{0}}, …, v_{o_{s}} + v_{o_{0}}. □

Applying the lemma to A^{c}, we see that A^{c} is a translate of the simplex in (V^{∗})^{w_{c}} spanned by 0 and 1/g_{α}v\overline{α}, where v\overline{α} = 1/n\overline{α} ∑_{α∈\overline{α}} α^{∗} and α is any representative for \overline{α}. It follows by Lemma 1.5.3 that \{\overline{α}^{∨}; \overline{α} ≠ \overline{α}_{0}\} is an integral basis for Λ^{w_{c}}, where \overline{α}^{∨} = ∑_{α∈\overline{α}} α^{∨}. Since

I_{0}(v\overline{α}, \overline{β}^{∨}) = \begin{cases}0, & if \overline{α} ≠ \overline{β}; \\ 1, & if \overline{α} = \overline{β}, \end{cases}

we see that \{\overline{α}^{∨}\} and \{v\overline{α}\} are dual bases for the restriction of I_{0} to (V^{∗})^{w_{c}}. Now

vol(A^{c}) = \frac{vol(C_{1})}{(r_{c})! ∏\overline{α} g_{α}} ,

where C_{1} is the parallelepiped spanned by the basis \{v\overline{α}\} and as usual α is any representative for \overline{α}. On the other hand, vol(T_{0}) = vol(C_{2}), where C_{2} is the parallelepiped spanned by the dual basis \{\overline{α}^{∨}\}. Thus

\frac{vol(T_{0})}{vol(A^{c})} = (r_{c})! (∏_{\overline{α}} g_{α}) \frac{vol(C_{2})}{vol(C_{1})} = (r_{c})! (∏_{\overline{α}} g_{α}) det(I_{0}|Λ^{w_{c}}).

To complete the proof of Theorem 1.5.2, the degree in question is the product

(r_{c})! (∏_{\overline{α}} g_{α}) det(I_{0}|Λ^{w_{c}}) · (\frac{∏\overline{α} n\overline{α}}{n_{0}} ) = (r_{c})!\frac{det(I_{0}|Λ^{w_{c}})}{n_{0}} ∏_{\overline{α}} g\overline{α},

as claimed. □

2. Bundles over maximal parabolic subgroups

2.1 Description of bundles and their automorphisms

Fix α ∈ Δ. We consider L^{α}-bundles η over E such that c_{1}(η×_{L^{α}}G) = c, and will refer to such a bundle as an unliftable bundle of type c. The primitive dominant character χ_{0} of P^{α} lifts to a character on P̃^{α} which is a positive power of the primitive dominant character ϖ_{α} of P̃^{α}. We denote this power by o_{c,α}. Note that ϖ_{α}(c) is well-defined as an element of ℚ/ℤ and o_{c,α} is its order. In fact, we have:

Lemma 2.1.1. Let β be a root such that c ≡ ϖ_{β}^{∨} (mod Λ). Then o_{c,α}, the order of ϖ_{α}(c), is the order of ϖ_{α}(ϖ_{β}^{∨}) mod ℤ. □

In the notation of [7, §3.4], o_{c,α} = 1 if and only if α ∉ Δ(c). The Dynkin diagram for Δ(c) is a union of diagrams of A-type and is described in the tables at the end of [7].

If η is an L^{α}-bundle, then the character χ_{0} defines an associated line bundle η ×_{L^{α}} ℂ. This line bundle is the determinant of η, which we write as det η. Its degree is called the degree of η and is denoted deg η.

Let η → E be a principal L^{α}-bundle whose degree d is negative. We shall study the corresponding bundles η ×_{L^{α}} P^{α} and ξ = η ×_{L^{α}} G. Associated to η and the action of L^{α} on the Lie algebra 𝔤 there is the vector bundle η×_{L^{α}} 𝔤 = ad ξ. The Lie algebra 𝔤 decomposes under L^{α} as 𝔲⊕𝔩⊕𝔲_{−} where 𝔩 is the Lie algebra of L^{α}, 𝔲 is the subspace of 𝔤 on which φ_{α}(ℂ^{∗}) ⊆ L^{α} acts with positive weights, and 𝔲_{−} is the subspace of 𝔤 on which φ_{α}(ℂ^{∗}) ⊆ L^{α} acts with negative weights. Since the coefficients of ζ_{α} are nonnegative, 𝔲 is the Lie algebra of U^{α}, 𝔭 = 𝔩 ⊕ 𝔲 is the Lie algebra of P^{α}, and 𝔲_{−} is the orthogonal space to 𝔲 under the Killing form. Clearly:

Lemma 2.1.2. There is a direct sum decomposition

ad ξ = η ×_{L^{α}} 𝔤 = ad_{L^{α}} η ⊕ 𝔲(η) ⊕ 𝔲_{−}(η).

The action of the ℂ^{∗} ⊆ L^{α} is trivial on ad_{L^{α}} η and has positive (resp. negative) weights on 𝔲(η) (resp. 𝔲_{−}(η)). □

We now have:

Lemma 2.1.3. For every negative integer d, there exists a semistable L^{α}-bundle η over E of degree d. There is an unliftable semistable L^{α}-bundle η of type c if and only if deg η/o_{c,α} ≡ ϖ_{α}(c) mod ℤ. In particular, if η is unliftable of type c and degree −1, then −1/o_{c,α} ≡ ϖ_{α}(c) mod ℤ. For a semistable L^{α}-bundle η of degree d < 0, we have:

(i) The bundle η ×_{L^{α}} G is unstable.

(ii) The parabolic P_{−}^{α} opposite to P^{α} is a Harder-Narasimhan parabolic of ξ.

(iii) 𝔲(η) is a direct sum of semistable vector bundles of strictly negative degrees.

(iv) The Atiyah-Bott point of η as defined in [2, 10, 11] is given by

μ(η) = \frac{dζ_{α}}{o_{c,α}m_{α}} = \frac{dn_{α}}{o_{c,α}m_{α}} ϖ_{α}^{∨}.

Proof. The dominant character χ_{0} lifts to the character ϖ_{α}^{o_{c,α}} on L̃^{α}. By [11, Definition 2.1.1], μ(η) is the unique point μ in the center of 𝔩 such that o_{c,α}ϖ_{α}(μ) = d. Thus μ = dζ_{α}/o_{c,α}m_{α}, showing (iv). The congruence condition deg η/o_{c,α} ≡ ϖ_{α}(c) mod ℤ follows from [11, Lemma 2.1.2 (ii)]. Statement (iii) follows from the fact that the 𝔲^{k} are irreducible L-modules, by Corollary 1.4.3, and from [11, Lemma 2.2.1]. The remaining statements are clear. □

Our goal now is to study the spaces H^{1}(E; ad ξ) and H^{0}(E; ad ξ). We shall primarily be interested in the case where η is semistable. It is enough to study the spaces H^{i}(E; ad_{L^{α}} η), H^{i}(E; 𝔲(η)), and H^{i}(E; 𝔲_{−}(η)). Since L^{α} has a one-dimensional center, regardless of the choice of η we must always have dim H^{0}(ad_{L^{α}} η) ≥ 1, and thus, since deg ad_{L^{α}} η = 0, dim H^{1}(ad_{L^{α}} η) ≥ 1 as well by Riemann-Roch on E. More precisely, we have:

Lemma 2.1.4. Let Ŝ^{α} be the quotient of L^{α} by its center. Let η be a semistable L^{α}-bundle and let η̂ be the induced Ŝ^{α}-bundle. Let r(η̂) be the dimension of AutŜ^{α}(η̂). Then

dim H^{0}(E; ad_{L^{α}} η) = dim H^{1}(E; ad_{L^{α}} η) = 1 + r(η̂).

Proof. On the level of Lie algebras, there is a direct sum decomposition 𝔩 = ℂ ⊕ Lie(Ŝ^{α}), where ℂ = Lie(Z(L^{α})), and the proof follows. □

We turn next to the groups H^{i}(E; 𝔲(η)) and H^{i}(E; 𝔲_{−}(η)):

Lemma 2.1.5. Let η be a principal L^{α}-bundle of negative degree. Then

dim H^{1}(E; 𝔲(η)) ≥ − deg (𝔲(η)) ,

with equality holding if and only if H^{0}(E; 𝔲(η)) = 0. Likewise,

dim H^{0}(E; 𝔲_{−}(η)) ≥ deg (𝔲_{−}(η)) ,

with equality holding if and only if H^{1}(E; 𝔲_{−}(η)) = 0. Finally, if η is semistable, then H^{0}(E; 𝔲(η)) = H^{1}(E; 𝔲_{−}(η)) = 0.

Proof. The first two statements are immediate from Riemann-Roch on E. The final one follows from Statement (iii) of Lemma 2.1.3. □

From the decomposition 𝔭 = 𝔩 ⊕ 𝔲, it follows that H^{0}(E; 𝔲(η)) = 0 if and only if

dim Aut_{L^{α}} η = dim Aut_{P^{α}}(η ×_{L^{α}} P^{α}).

The vanishing of H^{1}(E; 𝔲_{−}(η)) says on the other hand that the map

H^{1}(E; ad_{P^{α}}(η ×_{L^{α}} P^{α})) → H^{1}(E; ad_{G}(η ×_{L^{α}} G))

is an isomorphism. In particular, every small deformation of the Gbundle η ×_{L^{α}} G arises from a small P^{α}-deformation of η ×_{L^{α}} P^{α}.

To complete the determination of H^{1}(E; 𝔲(η)), we must compute the degree of 𝔲(η).

Proposition 2.1.6. We have deg 𝔲(η) = (deg η)d_{1}(α)/o_{c,α}, where d_{1}(α) is the integer defined in Lemma 1.3.1 for G̃. Likewise, deg 𝔲_{−}(η) = −(deg η)d_{1}(α)/o_{c,α}.

Proof. We compute the degree of the line bundle ⋀^{top} 𝔲(η). Let χ̃_{0} be the dominant character for P̃^{α}. The line bundle ⋀^{top} 𝔲(η) is associated to η by the character χ_{+} : L^{α} → ℂ^{∗}. By Lemma 1.3.1, χ_{+} lifts to the character χ̃_{+} = χ_{0}^{d_{1}(α)} of L̃^{α}. Since χ_{0} lifts to χ_{0}^{o_{c,α}} on L̃^{α} and since the line bundle associated to η by the character χ_{0} has degree deg η, the degree of 𝔲(η) is (deg η) · d_{1}(α)/o_{c,α}. A similar argument (or duality) handles the case of 𝔲_{−}(η). □

Corollary 2.1.7. If η is semistable of negative degree, then

dim H^{1}(E; 𝔲(η)) = dim H^{0}(E; 𝔲_{−}(η)) = −(deg η)d_{1}(α)/o_{c,α}.

Thus

dim H^{0}(E; ad(η ×_{L^{α}} G)) = dim H^{1}(E; ad(η ×_{L^{α}} G))
= 1 + r(η̂) − (deg η)d_{1}(α)/o_{c,α}.

Proof. By (iii) of Lemma 2.1.3, 𝔲(η) is a direct sum of semistable vector bundles of negative degrees, so that H^{0}(E; 𝔲(η)) = 0. The proof now follows by combining Lemmas 2.1.4 and 2.1.5 with Proposition 2.1.6. □

2.2 The ℂ^{∗}-action in cohomology

The Lie algebra 𝔲 is a direct sum of the subspaces 𝔲^{k}, k > 0, where 𝔲^{k} is the sum of all the root spaces 𝔤^{β} where the coefficient of β is exactly k. By Proposition 1.4.2, 𝔲^{k} is an irreducible L^{α}-module. Thus 𝔲(η) is the direct sum of vector bundles 𝔲^{k}(η) associated to irreducible representations of L^{α}. By [21], 𝔲^{k}(η) is semistable. Our goal now is to study the action of ℂ^{∗} = φ_{α}(ℂ^{∗}) on 𝔲(η) and on H^{1}(E; 𝔲(η)). As we saw in Section 1.3, φ̃_{α}(ℂ^{∗}) ⊆ L̃^{α} acts on 𝔲^{k} with weight kn_{α}. Thus, by Lemma 1.2.7, φ_{α}(ℂ^{∗}) acts on 𝔲^{k} with weight kn_{c,α}.

Lemma 2.2.1. The degree deg 𝔲^{k}(η) is equal to (deg η)·i(α, k)/o_{c,α}, where

i(α, k) = kc(α, k)/ϖ_{α}(ϖ_{α}^{∨}) = kn_{α}c(α, k)/m_{α}.

The slope of 𝔲^{k}(η) is kϖ_{α}(μ(η))/ϖ_{α}(ϖ_{α}^{∨}), where μ(η) is the Atiyah-Bott point of η. Thus, if η is semistable of negative degree, then

dim H^{1}(E; 𝔲^{k}(η)) = −(deg η) · i(α, k)/o_{c,α}.

Proof. By Lemma 1.3.4 applied to G̃, the character of L^{α} defined by the determinant on 𝔲^{k} lifts to the character χ_{0}^{i(α,k)} on L̃^{α}. The degree of 𝔲^{k}(η) is thus (deg η) · i(α, k)/o_{c,α}. It follows that the slope of 𝔲^{k}(η) is kn_{α} deg η/o_{c,α}m_{α} = kϖ_{α}(μ)/ϖ_{α}(ϖ_{α}^{∨}). □

3. Special roots and the associated bundles

In §3.1–3.3, we assume that G is simply connected. We will defer the discussion of the non-simply connected case until §3.4.

3.1 Definition of special roots

Definition 3.1.1. A simple root α is special if

(i) The Dynkin diagram associated to Δ−{α} is a union of diagrams of A-type.

(ii) The simple root α meets each component of the Dynkin diagram associated to Δ − {α} at an end of the component.

(iii) The root α is a long root.

If R is of type A_{n}, then every simple root is special. All other irreducible root systems have a unique special simple root. It corresponds to the unique trivalent vertex if the Dynkin diagram is of type D_{n}, n ≥ 4 or E_{n}, n = 6, 7, 8. For R = C_{n}, n ≥ 2 and G_{2}, it is the long simple root. For R = B_{n}, n ≥ 2, and F_{4} it is the unique long simple root which is not orthogonal to a short simple root.

We shall investigate the structure of the group L^{α} and the space H^{1}(E; 𝔲(η)) more closely in case α is special.

Let α be special and let η be a semistable bundle over P^{α} of degree −1. By the results of [11], the unstable bundles η ×_{L^{α}} G are minimally unstable bundles, in the sense that every small deformation of such a bundle is either of the same type or semistable. Moreover, if G is not of A-type, then for every unstable G-bundle ξ, there is a small deformation of ξ to a bundle of the form η×_{L^{α}} G. Now for every unstable G-bundle ξ, there is the Harder-Narasimhan reduction to a parabolic subgroup, not necessarily maximal, and in fact ξ reduces to a bundle η the Levi factor L. Let P be the opposite parabolic to the Harder-Narasimhan parabolic and let 𝔲 be the Lie algebra of the unipotent radical of P. It is easy to see that the function ξ ↦ dim H^{1}(E; 𝔲(η)) is strictly decreasing for the Atiyah-Bott ordering, and hence attains its minimum in the case where P is a maximal parabolic corresponding to a special root and − deg η is minimal. We shall see this directly below.

3.2 Bundles associated to special roots

Our first lemma determines the structure of S^{α} and L^{α} in case α is special:

Lemma 3.2.1. Suppose that α is special, and let t be the number of components in the Dynkin diagram of S^{α}. Then there exist integers n_{i} ≥ 2 such that S^{α} ≅ ∏_{i=1}^{t} SL_{n_{i}}(ℂ) and m_{α} = lcm(n_{i}). Moreover,

L^{α} ≅ {(A_{1}, …, A_{t}) ∈ ∏_{i=1}^{t} GL_{n_{i}}(ℂ) : det A_{1} = ⋯ = det A_{t}} ,

in such a way that the primitive dominant character of L^{α} corresponds to the common value of the determinant.

Proof. It follows from the definition of a special root that S^{α} ≅ ∏_{i=1}^{t} SL_{n_{i}}(ℂ). By Lemma 1.2.3, there is an isomorphism

L^{α} ≅ S^{α} ×_{ℤ/m_{α}ℤ} ℂ^{∗},

where the image of 1 ∈ ℤ/m_{α}ℤ is mapped to e^{2πi/m_{α}} ∈ ℂ^{∗} and to e^{−2πi/n_{i}} Id ∈ SL_{n_{i}}(ℂ). From this, we must have m_{α} = lcm(n_{i}). The map from S^{α} × ℂ^{∗} to ∏_{i=1}^{t} GL_{n_{i}}(ℂ) which is the natural inclusion

∏_{i=1}^{t} SL_{n_{i}}(ℂ) ⊆ ∏_{i=1}^{t} GL_{n_{i}}(ℂ)

and which maps λ ∈ ℂ^{∗} to

(λ^{m_{α}/n_{1}} Id, …, λ^{m_{α}/n_{t}} Id)

then factors to give an induced homomorphism

S^{α} ×_{ℤ/m_{α}ℤ} ℂ^{∗} → ∏_{i=1}^{t} GL_{n_{i}}(ℂ).

It is clear from the construction that this induced homomorphism is injective and that its image is the subgroup of matrices of equal determinant. Let det L^{α} → ℂ^{∗} denote the value of any of these determinants under the inverse isomorphism. For λ ∈ ℂ^{∗}, we see that det ∘φ_{α} = λ^{m_{α}}, and hence det = χ_{0}. □

Lemma 3.2.2. If α is special, the positive integers n_{α} are equal to 1 except in the following cases:

(a) If G = SL_{n}(ℂ) and α corresponds to the k^{th} vertex in the usual ordering of the simple roots, then n_{α} = n/ gcd(k, n).

(b) If G is of type B_{n} and n is even, then n_{α} = 2.

(c) If G is of type C_{n}, then n_{α} = 2.

(d) If G is of type D_{n} and n is odd, then n_{α} = 2.

Proof. If the center of G is trivial, then ϖ_{α}^{∨} is a primitive element of Λ, and hence n_{α} = 1. This handles the cases E_{8}, F_{4}, G_{2}. Next suppose that R is simply laced and not of type A_{n}, so that the Dynkin diagram of R is a T_{p,q,r} diagram, with (p, q, r) = (2, 2, n) or (2, 3, s) with s = 3, 4, 5. Let N = (1/p+1/q+1/r−1)^{−1}, so that N = n, 6, 12, 30 in the respective cases above. In particular N ∈ ℤ. There exists a labeling of the roots as {α, β_{1}, …, β_{p−1}, γ_{1}, …, γ_{q−1}, δ_{1}, …, δ_{r−1}}, where β_{1}, γ_{1}, δ_{1} are ends of the diagram, ⟨β_{i}, β_{i+1}⟩ = −1, 1 ≤ i ≤ p − 2 and similarly for the γ_{j} and δ_{k}, and β_{p−1}, γ_{q−1}, δ_{r−1} meet α, such that

ϖ_{α}^{∨} = \frac{N}{p} ∑_{i=1}^{p−1} iβ_{i}^{∨} + \frac{N}{q} ∑_{j=1}^{q−1} jγ_{j}^{∨} + \frac{N}{r} ∑_{k=1}^{r−1} kδ_{k}^{∨} + Nα.

It follows that ϖ_{α}^{∨} is integral, and hence n_{α} = 1, unless (p, q, r) = (2, 2, n) and n is odd, in which case n_{α} = 2. A similar argument handles the case of A_{n}. In case B_{n}, if we number the roots as in [8] beginning at the long end of the Dynkin diagram, then α = α_{n−1} and

ϖ_{α_{n−1}}^{∨} = ∑_{i=1}^{n−1} iα_{i}^{∨} + \frac{n − 1}{2}α_{n}^{∨}.

Thus n_{α} = 1 if n is odd and 2 if n is even. Finally, for the case of C_{n}, again numbering the roots in order as in [8] beginning at a short root, so that α = α_{n}, we have

ϖ_{α_{n}}^{∨} = \frac{1}{2} ∑_{i=1}^{n} iα_{i}^{∨}.

Thus n_{α} = 2. □

We turn now to the existence of special bundles over L^{α}.

Proposition 3.2.3. Suppose that α is special. Then there is a unique principal L^{α}-bundle η_{0} over E with the following properties:

(i) det η_{0} = O_{E}(−p_{0}).

(ii) For 1 ≤ i ≤ t, if V_{i} is the vector bundle associated to the principal GL_{n_{i}}(ℂ)-bundle obtained from the composition of the inclusion L^{α} ⊆ ∏_{i=1}^{t} GL_{n_{i}}(ℂ) followed by projection onto the i^{th} factor, then each V_{i} is a stable vector bundle.

The automorphism group of η_{0}, as an L^{α}-bundle, is identified with the center of L^{α} which is of the form Z(S^{α}) ×_{ℤ/m_{α}ℤ}φ_{α}(ℂ^{∗}), acting by multiplication.

Proof. Recall that, for every d ≥ 1, there is a unique stable vector bundle W_{d} of rank d over E such that det W_{d} = O_{E}(p_{0}). Given the structure of L^{α} as in Lemma 3.2.1, it is clear that there is a unique principal L^{α}-bundle, up to isomorphism, satisfying (i) and (ii) above, with V_{i} = W_{n_{i}}^{∗} for every i. Since the vector bundles V_{i} in (i) are simple, the automorphism group of each of these is isomorphic to the center of GL_{n_{i}}(ℂ) acting by multiplication. It then follows that the L^{α}- automorphisms of η_{0} are given by the action of the center of L^{α} acting by multiplication. □

Definition 3.2.4. If η is an L^{α}-bundle which satisfies (ii) of the Proposition, together with the condition that deg η = −1, then η is the pullback of η_{0} under a translation map E → E. We call η a translate of η_{0}.

Let us describe the unstable bundle η_{0}×_{L^{α}} G for the classical groups. First, we need the following notation. As above, let W_{d} be the unique stable vector bundle over E of rank d and such that det W_{d} = O_{E}(p_{0}). Let θ_{i}, i = 1, 2, 3 be the three nontrivial line bundles of order two on E. Let

Q_{3} = θ_{1} ⊕ θ_{2} ⊕ θ_{3}

be the corresponding rank three vector bundle. Fix isomorphisms θ_{i} ⊗ θ_{i} → O_{E} and give Q_{3} the corresponding diagonal symmetric bilinear form. Define similarly

Q_{4} = O_{E} ⊕ θ_{1} ⊕ θ_{2} ⊕ θ_{3},

together with a similar choice of a diagonal symmetric bilinear form. We then have:

Proposition 3.2.5. With notation as above, and supposing that α is special, let η_{0} be the principal L^{α}-bundle constructed in Proposition 3.2.3. Then the vector bundle associated to η_{0} ×_{L^{α}} G under the standard representation of G is:

(i) W_{k}^{∗} ⊕ W_{n−k}, if G = SL_{n}(ℂ) and α is the root corresponding to the k^{th} vertex in the Dynkin diagram, ordered in the usual way.

(ii) W_{n}^{∗} ⊕ W_{n}, if G = Sp(2n), where each factor is isotropic and we choose an isomorphism W_{n}^{∗} → W_{n}^{∗}, unique up to a scalar, to define the alternating form on the direct sum.

(iii) W_{n−2}^{∗} ⊕ Q_{4} ⊕ W_{n−2}, if G = Spin(2n), where Q_{4} is given the form described above, W_{n−2}^{∗} and W_{n−2} are isotropic, we choose an isomorphism W_{n−2}^{∗} → W_{n−2}^{∗}, unique up to a scalar, to define the symmetric form on the direct sum W_{n−2}^{∗} ⊕ W_{n−2}, and Q_{4} is orthogonal to this direct sum.

(iv) W_{n−1}^{∗} ⊕ Q_{3} ⊕ W_{n−1}, if G = Spin(2n + 1), where Q_{3} is given the form described above, W_{n−1}^{∗} and W_{n−1} are isotropic, we choose an isomorphism W_{n−1}^{∗} → W_{n−1}^{∗}, unique up to a scalar, to define the symmetric form on the direct sum W_{n−1}^{∗} ⊕ W_{n−1}, and Q_{3} is orthogonal to this direct sum.

Proof. The cases G = SL_{n}(ℂ) and G = Sp(2n) follow easily from the explicit descriptions of the maximal parabolic subgroups and are left to the reader. In case G = Spin(2n), the corresponding maximal parabolic of SO(2n) is the set of g ∈ SO(2n) preserving an isotropic subspace of ℂ^{2n} of dimension n − 2. The corresponding Levi factor L is the subgroup of matrices in GL_{n−2}(ℂ) × GL_{2}(ℂ) × GL_{2}(ℂ) with equal determinant. If ρ_{1} is the representation of L induced by the standard representation of GL_{n−2}(ℂ) on ℂ^{n−2} and ρ_{2}, ρ_{3} are the two representations of L induced by the standard representations of the second and third factors of L on ℂ^{2}, then it is easy to check that the representation of L on ℂ^{2n} which is the restriction of the standard representation of Spin(2n) is just

ρ_{1} ⊕ ρ_{1}^{∗} ⊕ (ρ_{2} ⊗ ρ_{3}^{∗}).

The vector bundle associated to η_{0} is thus

W_{n−2}^{∗} ⊕ W_{n−2} ⊕ (W_{2} ⊗ W_{2}^{∗}).

Moreover this is an orthogonal direct sum with respect to the induced form and W_{n−2}^{∗} and W_{n−2} are isotropic subspaces. Furthermore, by a result of Atiyah [1], W_{2} ⊗ W_{2}^{∗} ≅ Q_{4}, and since the each line bundle summand of Q_{4} is not isomorphic to the dual of any other summand, the direct sum decomposition of Q_{4} must be orthogonal with respect to the quadratic form and thus as described above.

The case of Spin(2n + 1) is similar. □

3.3 Cohomology dimensions and weight spaces

We have seen that the bundles η_{0} are minimally unstable in the sense of deformation theory. Here we begin by showing that their deformation spaces have minimal dimension among all unstable bundles.

Theorem 3.3.1. Let η_{0} be the bundle described in Proposition 3.2.3. Then

dim H^{0}(E; ad(η_{0} ×_{L^{α}} G)) = dim H^{1}(E; ad(η_{0} ×_{L^{α}} G)) = r + 2.

If ξ is any unstable G-bundle, then dim H^{1}(E; ad ξ) ≥ r+2, with equality if and only if ξ is isomorphic to η ×_{L^{α}} G, where η is a translate of η_{0}. Finally, if U is the unipotent radical of a parabolic subgroup P of G with Levi factor L and η is a semistable L-bundle of negative degree, then

dim H^{1}(E; 𝔲(η)) ≥ r + 1,

with equality if and only if P is conjugate to P^{α}, where α is special, and η is a translate of η_{0}.

Proof. First, by Corollary 2.1.7,

dim H^{0}(E; ad(η_{0} ×_{L^{α}} G)) = dim H^{1}(E; ad(η_{0} ×_{L^{α}} G))
= 1 + r(η̂_{0}) + d_{1}(α) = 1 + d_{1}(α),

since dim AutŜ^{α} η̂_{0} = 0.

Next we show that d_{1}(α) has the following minimality property:

Lemma 3.3.2. If α is special, then d_{1}(α) = r + 1. If β is not special, then d_{1}(β) > r + 1.

Proof. To see that d_{1}(α) = r + 1, it suffices by Proposition 1.4.9 to show that λ_{1}(α^{∨}) = ∑_{β∈Δ} β^{∨}. First, by [8, cor. 3, p. 160], λ = ∑_{β∈Δ} β^{∨} is always a coroot. Clearly, n(λ^{∨}, β^{∨}) ≥ 0 for all β ∈ Δ−{α}, n(λ^{∨}, β^{∨}) > 0 if β ≠ α is an end of the Dynkin diagram and λ − β^{∨} can only be a coroot if β ≠ α is an end of the Dynkin diagram. These properties say that, if β ≠ α, then λ + β^{∨} is not a coroot. Thus λ = λ_{1}(α^{∨}).

Now suppose that R is not of A-type, and hence that R^{∨} is not of A-type. Then λ_{1}(α^{∨}) is not the highest coroot of R^{∨}. Thus there exists a simple root β such that λ_{1}(α^{∨}) + β^{∨} is again a coroot. By what we have just seen, we must have β = α. It follows that, for α ≠ β, λ_{1}(β^{∨}) is equal to λ_{1}(α^{∨}) + α^{∨} plus a sum of simple coroots. Hence d_{1}(β) = ρ(λ_{1}(β^{∨})) + 1 ≥ ρ(λ_{1}(α^{∨})) + 2 = r + 2. □

Thus, we have proved the first statement in the theorem. To see the second, first assume that the Harder-Narasimhan parabolic subgroup for ξ is maximal. In this case, we can assume that the Harder-Narasimhan parabolic for ξ is P_{−}^{β} for some β. Thus ξ is isomorphic to η ×_{L^{β}} G, where η is a semistable bundle of negative degree −n on L^{β}. Now

dim H^{1}(E; ad(η ×_{L^{β}} G)) = 1 + r(η̂) + nd_{1}(β) ≥ r + 2,

with equality holding if and only if r(η̂) = 0, n = 1, and β is special. In this last case, it follows from Definition 3.2.4 that η is a translate of η_{0}.

Now suppose that the Harder-Narasimhan parabolic for ξ is not maximal. There exists a maximal parabolic subgroup P^{β} such that ξ has a reduction to an L^{β}-bundle η, where η has degree −n < 0. By Lemma 2.1.3, η is unstable, for otherwise the Harder-Narasimhan parabolic for ξ would be P_{−}^{β}, which is maximal. Hence, the associated Ŝ^{β}-bundle η̂ is also unstable. Thus the vector bundle adŜ^{β} η̂ is unstable of degree zero, and hence contains a semistable summand of negative degree. It follows that dim H^{1}(E; adŜ^{β} η̂) ≥ 1. Applying Lemmas 2.1.4 and 2.1.5 and Proposition 2.1.6, we see that

h^{1}(E; ad ξ) = 1 + h^{1}(E; adŜ^{β} η̂) + h^{1}(E; 𝔲(η)) + h^{1}(E; 𝔲_{−}(η))
≥ 2 + nd_{1}(β) ≥ r + 3.

This completes the proof of the second statement in case the Harder-Narasimhan parabolic for ξ is not maximal. The proof of the final statement of the theorem is implicit in the above discussion. □

We turn now to the ℂ^{∗}-weights for the action of ℂ^{∗} on 𝔲(η_{0}).

Proposition 3.3.3. Suppose that α is special. Then the ℂ^{∗}-weights for the action of the center of L^{α} on H^{1}(E; 𝔲(η_{0})), with multiplicity, are the integers n_{α}g_{β}, β ∈ Δ̃.

Proof. The group ℂ^{∗} acts on H^{1}(E; 𝔲^{k}(η)) with weight kn_{α}. By Lemma 2.2.1,

dim H^{1}(E; 𝔲^{k}(η)) = −(deg η_{0}) · i(α, k) = i(α, k).

Thus, it suffices to show that i(α, k) = #{β ∈ Δ̃ : g_{β} = k}. If we define

i(k) = #{β ∈ Δ̃ : g_{β} = k};
d(k) = ∑_{k|x} i(x) = ∑_{ℓ≥1} i(ℓk),

then it clearly suffices to show that, for all k, d(k) = d_{k}(α), in the notation of Definition 1.4.7. By Proposition 1.4.17, the integers d_{k}(α) have the circular symmetry property with respect to g_{α} and g, since α is a long root, and by Lemma 3.3.2, d_{1}(α) = r + 1. By the proof of Theorem 3.8.7 in [7], the integers d(k) have the circular symmetry property with respect to N and g, where N = max{g_{β} : β ∈ Δ}. By Corollary 6.2.5 of [11], N = g_{α}. (We will give another proof of this fact in Part III.) Clearly d(1) = #Δ̃ = r+1. By Lemma 1.4.16, d(k) = d_{k}(α) for all k. □

3.4 The non-simply connected case

We now establish the analogues in the non-simply connected case of the previous results. While we believe there should be classification-free arguments for these results, we argue here in a case-by-case analysis.

Definition 3.4.1. Let o(c) denote the order of c ∈ π_{1}(G). A root α ∈ Δ is c-special if there exists an integer d < 0 such that:

(i) d/o_{c,α} ≡ ϖ_{α}(c) mod ℤ.

(ii) The Weyl orbit of the point μ_{c,α} = dζ_{α}/o_{c,α}m_{α} is minimal in the Atiyah-Bott ordering [11] among all Weyl orbits of nonzero points of Atiyah-Bott type for c.

Remark 3.4.2. a) The first condition means that that there is a holomorphic semistable L^{α}-bundle η with c_{1}(η ×_{L^{α}} G) = c and Atiyah-Bott point equal to μ_{c,α}, and the topological type of the L^{α}-bundle η is uniquely determined by μ_{c,α}. This topological type is specified by an element c_{1}(η) = γ ∈ π_{1}(L^{α}). The second condition means that the point μ_{c,α}, or the corresponding stratum of (0, 1)-connections, is minimally unstable in the sense of [11, Definition 6.1.1].

b) It is not always true that there is a unique c-special root. Uniqueness fails exactly when G = SL_{n}(ℂ)/⟨c⟩ and c does not generate the center, where there are n/o(c) special roots, and for G = SO(2n), where there are two special roots.

c) As defined here, c-special roots are certain simple roots. In [11] we used the simple roots to index strata of the space of (0, 1)-connections, by associating to α the stratum lying in the Lie algebra of the center of L^{α} and having the smallest possible positive value under the dominant character. The convention here differs by a sign from the one of [11]. This means that the image under the automorphism of the Dynkin diagram induced by −w_{0} of the roots which are c-special as defined here correspond to the roots indexing the minimally unstable strata in [11]. This automorphism sends c to c^{−1} and thus fixes the set of c-special roots if and only if c is of order 1 or 2. Otherwise, the c-special roots correspond to the roots indexing the minimally unstable strata for c^{−1} in [11].

Theorem 3.4.3. Let α be a c-special root for G, and let γ ∈ π_{1}(L^{α}) be the first Chern class of the L^{α}-bundle η corresponding to μ_{c,α}. Then:

(i) The integer d = −1 and o_{c,α} = o(c).

(ii) The adjoint quotient ad(L^{α}) = L^{α}/Z(L^{α}) is a product ∏_{i=1}^{k} Ŝ_{i}, where the Ŝ_{i} are simple groups of A-type.

(iii) Let γ̂ be the image of γ ∈ π_{1}(L^{α}) under the projection π_{1}(L^{α}) → π_{1}(∏_{i=1}^{k} Ŝ_{i}) = ∏_{i=1}^{k} π_{1}(Ŝ_{i}). For i = 1, …, k, the image of γ̂ in π_{1}(Ŝ_{i}) generates the cyclic group π_{1}(Ŝ_{i}).

(iv) d_{1}(α)/o_{c,α} = r_{c} + 1.

Proof. The minimally unstable points μ_{c,α} are listed in §6.3 of [11]. From this list, it is easy to check that d = −1 and o_{c,α} = o(c). To prove the remaining statements, we make a case-by-case analysis.

G̃ = SL_{n}(ℂ):

We choose an identification of Λ with

{(x_{1}, …, x_{n}) ∈ ℤ^{n} | ∑_{i=1}^{n} x_{i} = 0}

in such a way that Δ = {α_{1}, …, α_{n−1}} with α_{i} = e_{i} −e_{i+1} with e_{i} being the standard unit vector in the i^{th}-coordinate direction. We write c as the image of an element of the form

(m/n)(α_{1}^{∨} + 2α_{2}^{∨} + ⋯ + (n − 1)α_{n−1}^{∨})

for some 1 ≤ m < n. We factor m = m_{0} · ℓ where (m_{0}, n) = 1 and ℓ|n. Then c is an element of order f = n/ℓ. By [11, §6.3], the c-special roots are those α for which ϖ_{α}(c) ≡ −1/f (mod ℤ). There are exactly n/f such roots. Suppose α = α_{k}. Then

−k · m/n ≡ −1/f (mod ℤ).

In particular, (k, f) = 1. Since α is a special root for SL_{n}(ℂ), d_{1}(α) = n, and hence d_{1}(α)/f = n/f = ℓ. On the other hand, since ⟨c⟩ acts freely on the Dynkin diagram for G we see that r_{c} + 1 = ℓ. Let us consider the group L^{α}. By Lemma 3.2.1, L̃^{α} is isomorphic to the subgroup of GL_{k}(ℂ) × GL_{n−k}(ℂ) matrices of equal determinant. Hence

ad(L^{α}) = ad(L̃^{α}) = P GL_{k}(ℂ) × P GL_{n−k}(ℂ).

The map det: L̃^{α} → ℂ^{∗} induces an identification π_{1}(L̃^{α}) = ℤ and the projection L̃^{α} → ad(L^{α}) sends 1 ∈ ℤ to the element

(a_{k}, a_{n−k}) ∈ π_{1}(P GL_{k}(ℂ)) × π_{1}(P GL_{n−k}(ℂ))

where a_{k}, resp. a_{n−k} generates π_{1}(P GL_{k}(ℂ)), resp. π_{1}(P GL_{n−k}(ℂ)). Direct computation shows that π_{1}(L^{α}) = ℤ and that the natural map π_{1}(L̃^{α}) → π_{1}(L^{α}) is multiplication by f. Thus, we have an identification π_{1}(L^{α}) = ℤ[1/f]. Under this identification the element γ ∈ π_{1}(L^{α}) is −1/f. π_{1}(L^{α}) and projects to

γ̂ = (f^{−1}a_{k}, f^{−1}a_{n−k}) ∈ π_{1}(P GL_{k}(ℂ)) × π_{1}(P GL_{n−k}(ℂ)).

Since (k, f) = 1, the projection of this element to either factor generates that factor.

G = SO(2n + 1), n ≥ 3.

The c-special root is the unique short root α in the Dynkin diagram. Direct inspection shows that L^{α} = GL_{n}(ℂ), and that γ ∈ π_{1}(L^{α}) is a generator of the fundamental group. Thus, ad(L^{α}) = P GL_{n}(ℂ) and γ̂ generates π_{1}(P GL_{n}(ℂ)). Furthermore, d_{1}(α) = 2n so that d_{1}(α)/o(c) = n. Since c acts on the extended Dynkin diagram for G with one free orbit and n − 1 fixed points, we see that r_{c} + 1 = n = d_{1}(α)/o(c).

G = Sp(2n)/⟨c⟩, n ≥ 2:

Suppose first that n is odd. Then there is a unique c-special root, the unique long root α. In this case, L^{α} = GL_{n}(ℂ)/(ℤ/2ℤ) and γ generates the fundamental group of L^{α}. Hence, ad(L^{α}) = P GL_{n}(ℂ) and γ̂ is the square of a generator for this group. Since n is odd, γ̂ is a generator of π_{1}(P GL_{n}(ℂ)). Since α is special for the simply connected form of the group, d_{1}(α) = n+1. In this case the element c acts freely on the nodes of the extended Dynkin diagram so that r_{c}+1 = (n+1)/2 = d_{1}(α)/o(c).

Now suppose that n is even. Then there is a unique c-special root, the unique short simple root α which is not orthogonal to the unique long simple root. Direct computation shows that L̃^{α} is isomorphic to GL_{n−1}(ℂ)×SL_{2}(ℂ) and that c is the diagonal element (−1, −1). Thus, π_{1}(L^{α}) = ℤ and γ is a generator of this group. Furthermore, ad(L^{α}) = P GL_{n−1}(ℂ) × P GL_{2}(ℂ) and the map π_{1}(L^{α}) → π_{1}(ad(L^{α})) is onto. Thus, the image γ̂ of γ generates π_{1}(ad(L^{α})), and hence its projection to each factor generates the fundamental group of that factor. Lastly, direct computation shows that d_{1}(α) = n + 2. Since c acts on the extended Dynkin diagram for G with one fixed point and n/2 free orbits, we see that r_{c} + 1 = (n + 2)/2 = d_{1}(α)/o(c).

G = SO(2n), n ≥ 4:

For G̃ = Spin(2n) we identify Λ with the even integral lattice inside ℝ^{n}. Let e_{i} be the standard unit vector in the i^{th}-coordinate direction. Then Δ = {α_{1}, ⋯, α_{n−1}, α_{n}} where α_{i} = e_{i} − e_{i+1} for 1 ≤ i < n and α_{n} = e_{n−1} + e_{n}.

There are two c-special roots α_{n−1} = e_{n−1} − e_{n} and α_{n} = e_{n−1} + e_{n}. (Of course, these elements are interchanged by an outer automorphism of SO(2n).) Let α be one of the c-special roots. Then L^{α} = GL_{n}(ℂ) and γ is a generator of π_{1}(L^{α}) ≅ ℤ. Thus, ad(L^{α}) = P GL_{n}(ℂ) and γ̂ is a generator of this group. Direct computation shows that d_{1}(α) = 2(n−1). Since c acts on the Dynkin diagram for G with two free orbits and n − 1 fixed points, we see that r_{c} + 1 = n − 1 = d_{1}(α)/o(c).

G̃ = Spin(4n + 2), n ≥ 2 and c is an element of order 4:

There is one c-special root. It is the simple root α corresponding to the “ear” of the Dynkin diagram (i.e., either α_{n−1} or α_{n}) with the property that ϖ_{α}(c) = −1/4 (mod ℤ). In this case

L^{α} = GL_{2n+1}(ℂ)/(ℤ/2ℤ).

Hence π_{1}(L^{α}) = ℤ and γ is a generator. Under the projection to ad(L^{α}) = P GL_{2n+1}(ℂ) the image γ̂ of γ is the square of the usual generator. This is clearly still a generator. Lastly, as above d_{1}(α) = 2((2n + 1) − 1) = 4n whereas r_{c} + 1 = n. Thus, d_{1}(α)/o(c) = r_{c} + 1.

G̃ = Spin(4n), n ≥ 2 and c is an element of order two not contained in π_{1}(SO(4n)):

There is one c-special root. It is the simple root α_{n−3} corresponding to the node of the “long” arm of the Dynkin diagram next to the trivalent node. Thus, L̃^{α} is isomorphic to (SL_{2n−3}(ℂ)×SL_{4}(ℂ))×_{(ℤ/(4n−6)ℤ)} ℂ^{∗} where the cyclic group is embedded in the standard way in ℂ^{∗} and the usual generator maps to the standard generator of Z(SL_{2n−3}(ℂ)) and to the element of order 2 in Z(SL_{4}(ℂ)). Thus, we can identify L̃^{α} with GL_{2n−3}(ℂ) ×_{(ℤ/2ℤ)} SL_{4}(ℂ). The element c is the image of the element (a, b) where a and b are central elements of order 4 in GL_{2n−3}(ℂ) and SL_{4}(ℂ) under the inclusion of L̃^{α} ⊂ G̃. Thus, L^{α} is isomorphic GL_{2n−3}(ℂ) ×_{(ℤ/4ℤ)} SL_{4}(ℂ). Hence γ is a generator of π_{1}(L^{α}) = ℤ, the image γ̂ of γ is a generator for π_{1}(P GL_{2n−3}(ℂ)) × π_{1}(P GL_{4}(ℂ)), and hence the projection of γ̂ into either factor generates the fundamental group of that factor. Direct computation shows that d_{1}(α) = 2(n + 1). Since the action of c on the extended Dynkin diagram for G has one fixed point and n free orbits, we see that r_{c} + 1 = n + 1 = d_{1}(α)/o(c).

G = ad(E_{6}):

There is one c-special root. It is a simple root α corresponding to the node next to the trivalent node on one of the arms of length 3 with the property ϖ_{α}(c) = −1/3. In this case L̃^{α} is isomorphic to (SL_{5}(ℂ) × SL_{2}(ℂ)) ×_{(ℤ/10ℤ)} ℂ^{∗} where the element in ℤ/10ℤ that maps to exp(2πi/10) maps to the generator in Z(SL_{2}(ℂ)) and to the square of the usual generator in Z(SL_{5}(ℂ)). Hence, L^{α} is isomorphic to (SL_{5}(ℂ) × SL_{2}(ℂ)) ×_{(ℤ/10ℤ)} ℂ^{∗} where the element in ℤ/10ℤ that maps to exp(2πi/10) maps to the generator in Z(SL_{2}(ℂ)) and to the usual generator in Z(SL_{5}(ℂ)). Thus, ad(L^{α}) = P GL_{2}(ℂ) × P GL_{5}(ℂ), π_{1}(L^{α}) is cyclic and γ is a generator of this group. It follows that γ̂ ∈ π_{1}(P GL_{5}(ℂ) × P GL_{2}(ℂ)) generates and hence the image of γ̂ under projection to either factor is a generator of the fundamental group of that factor. Direct computation shows that d_{1}(α) = 9. Since the action of ⟨c⟩ on the extended Dynkin diagram of E_{6} has two free orbits and one fixed point, we see that r_{c} + 1 = 3 = d_{1}(α)/o(c).

G = ad(E_{7}):

There is one c-special root. It corresponds to the node of the Dynkin diagram next to the trivalent node on the arm of length 4. In this case L̃^{α} is isomorphic to (SL_{3}(ℂ) × SL_{5}(ℂ)) ×_{(ℤ/15ℤ)} ℂ^{∗}, where the element in ℤ/15ℤ that maps to exp(2πi/15) ∈ ℂ^{∗} maps to the usual generator of Z(SL_{3}(ℂ)) and the square of the usual generator of Z(SL_{5}(ℂ)). Thus, L^{α} = (SL_{3}(ℂ) × SL_{5}(ℂ)) ×_{(ℤ/15ℤ)} ℂ^{∗} where the element in ℤ/15ℤ that maps to exp(2πi/15) ∈ ℂ^{∗} maps to the inverse of the usual generator of Z(SL_{3}(ℂ)) and the inverse of the usual generator of Z(SL_{5}(ℂ)). Thus, ad(L^{α}) = P GL_{3}(ℂ) × P GL_{5}(ℂ), π_{1}(L^{α}) is isomorphic to ℤ and γ is a generator. Consequently, γ̂ is a generator of π_{1}(ad(L^{α})).

Direct computation shows that d_{1}(α) = 10. The action of c on the extended Dynkin diagram of E_{7} has two fixed points and 3 free orbits so that r_{c} + 1 = 5 = d_{1}(α)/o(c). □

Next we compute the integer n_{c,α} defined in Lemma 1.2.7:

Lemma 3.4.4. If α is c-special and c is nontrivial, then n_{c,α} = 1 except in the following cases:

(i) If G̃ = SL_{n}(ℂ), α corresponds to the k^{th} vertex in the usual ordering, and o(c) = d, then n_{c,α} = n/d · gcd(k, n).

(ii) If G̃ = Spin(2n) and c is of order 2, then n_{c,α} = 2.

Proof. If Z(G̃) is cyclic and c is a generator, then n_{c,α} = 1. The remaining cases are G̃ = SL_{n}(ℂ) and G̃ = Spin(2n), and these can be checked directly. □

As in the simply connected case, we have:

Lemma 3.4.5. There is a unique semistable L^{α}-bundle η_{0} with the following properties:

(i) c_{1}(η_{0} ×_{L^{α}} G) = c.

(ii) The Atiyah-Bott point of η_{0} is μ_{c,α}.

(iii) det η_{0} = O_{E}(−p_{0}). □

As before, a bundle η satisfying (i) and (ii) above is the pullback of η_{0} via a translation of E, and we will call such an η a translate of η_{0}. The following is then proved via arguments similar to those used in the proof of Theorem 3.3.1.

Theorem 3.4.6. Let η_{0} be the bundle described in Lemma 3.4.5. Then

dim H^{0}(E; ad(η_{0} ×_{L^{α}} G)) = dim H^{1}(E; ad(η_{0} ×_{L^{α}} G)) = r_{c} + 2.

If ξ is any unstable G-bundle which is C^{∞} isomorphic to η_{0} ×_{L^{α}} G, then dim H^{1}(E; ad ξ) ≥ r_{c} + 2, with equality if and only if ξ is isomorphic to η ×_{L^{α}} G for some translate η of η_{0}. □

Finally, we must determine the weights for the action of ℂ^{∗} on H^{1}(E; 𝔲(η_{0})):

Proposition 3.4.7. Suppose that α is c-special. Then the ℂ^{∗} weights for the action of \overline{φ}_{α}(ℂ^{∗}) on H^{1}(E; 𝔲(η_{0})), with multiplicity, are the integers n_{c,α}g\overline{β}/n_{0}, \overline{β} ∈ Δ̃/w_{c}.

Proof. The weight for the action of ℂ^{∗} on H^{1}(E; 𝔲^{k}(η_{0})) is kn_{c,α}. By Lemma 2.2.1, the dimension of H^{1}(E; 𝔲^{k}(η_{0})) is

i(α, k)/o_{c,α} = i(α, k)/o(c).

By Proposition 1.4.17, the integers d_{k}(α)/o(c) have the circular symmetry property with respect to h_{α} and gh_{α}/g_{α}o(c). By Theorem 3.4.3, d_{1}(α)/o(c) = r_{c} + 1. If we define

i_{c}(k) = #\{β ∈ Δ̃ : g\overline{β} = kn_{0}\}
d_{c}(k) = ∑_{k|x} i_{c}(x) = ∑_{ℓ≥1} i_{c}(ℓk),

then the integers d_{c}(k) satisfy: d_{c}(1) = r_{c} + 1, and the d_{c}(k) have the circular symmetry property with respect to N/n_{0} and g/n_{0}, where N is the maximum value of the g\overline{β}. It follows by inspection or from [7, Proposition 10.1.8] that n_{0} = o(c)g_{α}/h_{α}, i.e., h_{α}/g_{α}o(c) = 1/n_{0}. By inspection, h_{α} = N/n_{0}. Thus i(α, k)/o(c) = i_{c}(k), and the proof follows. □

4. The nonabelian cohomology space

4.1 The affine space and the universal bundle

Let α be an arbitrary simple root. We abbreviate P^{α} = P, L^{α} = L, and U^{α} = U. Let η be an unliftable semistable principal L-bundle of type c. There is the associated sheaf of (not necessarily abelian) groups U(η). The cohomology set H^{1}(E; U(η)) classifies pairs (ξ, φ), where ξ is a P -bundle and φ is an isomorphism from the induced bundle ξ/U on L to η. There is a marked point 0 ∈ H^{1}(E; U(η)), corresponding to the pair (η ×_{L} P, I), where I is the canonical identification of the bundle (η ×_{L} P)/U with η. There is a corresponding functor F from schemes to sets defined as follows: for a scheme of finite type over ℂ, F(S) is the set of isomorphism classes of pairs (Ξ, Φ), where Ξ is a P -bundle over E × S and Φ is an isomorphism from Ξ/U to π_{1}^{∗}η.

Lemma 4.1.1. The functor F is represented by an affine space.

Proof. Let U_{i} be the closed subgroup of U whose Lie algebra is ⨁_{k≥i} 𝔲^{k}. Then the filtration {U_{i}} is a decreasing filtration of U by normal, L-invariant subgroups such that U_{i}/U_{i+1} is in the center of U/U_{i+1} for every i, and U_{i}/U_{i+1} ≅ 𝔲^{i}. By Theorem A.2.2 of the appendix, it suffices to check that H^{0}(E; (U_{i}/U_{i+1})(η)) = H^{2}(E; (U_{i}/U_{i+1})(η)) = 0. The second statement is clear since dim E = 1, and the first follows from Lemma 2.1.5, which implies that H^{0}(E; 𝔲^{k}(η)) = 0 for every k > 0. □

Thus, there is a structure of an affine space on H^{1}(E; U(η)) and a universal pair (Ξ_{0}, Φ_{0}) over the scheme E × H^{1}(E; U(η)) which represents the functor F. We will somewhat carelessly identify Ξ_{0} with the associated G-bundle Ξ_{0} ×_{P} G.

We now identify φ_{α}(ℂ^{∗}) with ℂ^{∗}. Thus we have fixed the embedding of ℂ^{∗} in L. Since L acts on U, there are induced actions of ℂ^{∗} on U(η) and 𝔲(η), and hence on H^{1}(E; U(η)) and on H^{1}(E; 𝔲(η)). Viewing H^{1}(E; U(η)) as the set of pairs (ξ, φ) as above, the action of ℂ^{∗} is via the action of Aut L on the isomorphism φ, and this action fixes the origin in H^{1}(E; U(η)), i.e., the bundle η ×_{L} P. By Theorem A.2.2, the action of ℂ^{∗} lifts to an action on the universal principal bundle Ξ_{0} over E × H^{1}(E; U(η)). The first goal of this section is to prove that the action of φ_{α}(ℂ^{∗}) on H^{1}(E; U(η)) is linearizable, and in fact there is a ℂ^{∗}- equivariant isomorphism from H^{1}(E; U(η)) to H^{1}(E; 𝔲(η)). Thus the quotient (H^{1}(E; U(η)) − {0})/ℂ^{∗} is a weighted projective space 𝕎ℙ(η). In §4.3, we give a sufficient condition for the existence of universal bundles over E ×𝕎ℙ(η). Next we show that, in the case where α is c-special and η_{0} is the bundle of Proposition 3.2.3 or Lemma 3.4.5, the points of 𝕎ℙ(η_{0}) correspond to semistable bundles whose automorphism groups have minimal possible dimensions. In §4.5, we analyze the Kodaira-Spencer homomorphism. The results of §4.5 will not however be used in this paper. Finally, we discuss the singular locus of the weighted projective space and relate it to moduli spaces of bundles with a non-simply connected structure group.

4.2 Linearization of the action

We analyze the ℂ^{∗}-action on the affine space H^{1}(E; U(η)) more closely. Our goal will be to show that this action can be linearized and to calculate the ℂ^{∗}-weights.

Lemma 4.2.1. Every ℂ^{∗}-orbit in H^{1}(E; U(η)) contains the origin in its closure.

Proof. Let x ∈ H^{1}(E; U(η)) be represented by the 1-cocycle {u_{ij}}, where {Ω_{i}} is an open cover of E and u_{ij}: Ω_{i} ∩ Ω_{j} → U is a morphism. Then λ ∈ ℂ^{∗} ⊆ L acts on the cocycle {u_{ij}}. Define morphisms

ũ_{ij}(e, λ): (Ω_{i} ∩ Ω_{j}) × ℂ → U

as follows:

ũ_{ij}(e, λ) = \begin{cases}λ · u_{ij}(e), & if λ ≠ 0; \\ 1, & if λ = 0. \end{cases}

There is a ℂ^{∗}-equivariant morphism from the unipotent subgroup U to the affine space 𝔲 (see for example [6, Remark, p. 183]). Using this ℂ^{∗}- equivariant isomorphism, and the fact that all of the ℂ^{∗}-weights on 𝔲 are positive, it is easy to check that the ũ_{ij} are morphisms and so define a 1-cocycle for the sheaf U(π_{1}^{∗}η) over E × ℂ. Thus, they define a bundle Ξ over E ×ℂ, reducing to π_{1}^{∗}η mod U and such that Ξ|E ×{0} = η×_{L}P. By the functorial property of H^{1}(E; U(η)), there is a morphism from ℂ to H^{1}(E; U(η)) corresponding to Ξ. Clearly, the image of 0 ∈ ℂ is the origin of H^{1}(E; U(η)), and the image of ℂ^{∗} is exactly the ℂ^{∗}-orbit of the cocycle {u_{ij}}. This proves Lemma 4.2.1. □

Lemma 4.2.2. Let T be the tangent space of H^{1}(E; U(η)) at the fixed point 0, with the natural ℂ^{∗}-action. Then the Kodaira-Spencer homomorphism from T to H^{1}(E; ad(η ×_{L} G)) induced by the bundle Ξ_{0} is given by a ℂ^{∗}-equivariant isomorphism T → H^{1}(E; 𝔲(η)) followed by the inclusion of H^{1}(E; 𝔲(η)) as a direct summand in

H^{1}(E; ad(η ×_{L} G)) ≅ H^{1}(E; ad_{L} η) ⊕ H^{1}(E; 𝔲_{−}(η)) ⊕ H^{1}(E; 𝔲(η)).

Proof. Let ℂ[ε] denote the dual numbers. The space T is the set of maps from Spec ℂ[ε] to H^{1}(E; U(η)) such that the closed point is mapped to the origin. By the functorial interpretation of H^{1}(E; U(η)), such a morphism corresponds to a P -bundle Ξ over E ×Spec ℂ[ε], which is the pullback of Ξ_{0}, together with an isomorphism from Ξ/U to π_{1}^{∗}η, and such that Ξ restricts to η ×_{L} P over the closed point. The second condition says that Ξ is a first order deformation of the P -bundle η×_{L}P. Such deformations are classified by

H^{1}(E; ad_{P} (η ×_{L} P)) = H^{1}(E; ad_{L} η) ⊕ H^{1}(E; 𝔲(η)).

The first condition says that the corresponding first order deformation of the L-bundle η is trivial, or equivalently that the projection of the Kodaira-Spencer class of Ξ to H^{1}(E; ad_{L} η) is zero. Thus we have defined a canonical map from T to H^{1}(E; 𝔲(η)). Conversely, by reversing this construction, every element of H^{1}(E; 𝔲(η)) defines a first order deformation of η ×_{L} P which reduces to π_{1}^{∗}η mod U, so that in fact the map from T to H^{1}(E; 𝔲(η)) is an isomorphism. Since this isomorphism is canonical, it is easily seen to be ℂ^{∗}-equivariant. The last statement is clear by construction. □

Using the previous two lemmas, we show that the ℂ^{∗}-action on H^{1}(E; U(η)) can be linearized. There is the following general result about ℂ^{∗}-actions on an affine space.

Lemma 4.2.3. Let 𝔸^{n} be an affine space with a ℂ^{∗}-action, and suppose that 0 ∈ 𝔸^{n} is a fixed point for the action. Let T be the tangent space of 𝔸^{n} at the origin, together with the induced linear ℂ^{∗}-action on T. Further suppose that:

(i) Every ℂ^{∗}-orbit in 𝔸^{n} contains 0 in its closure.

(ii) All of the weights in the ℂ^{∗}-action on T are strictly positive.

Then there is a ℂ^{∗}-equivariant isomorphism from 𝔸^{n} to T. Hence, the ℂ^{∗}-action on 𝔸^{n} is linearizable, and the ℂ^{∗}-weights for this action are those for the action on T.

Proof. Let A = ℂ[z_{1}, …, z_{n}] be the affine coordinate ring of 𝔸^{n}, where 0 is defined by z_{1} = ⋯ = z_{n} = 0, and let x_{1}, …, x_{n} be a basis for the linear functions on T. The finite-dimensional subspace of A spanned by the z_{i} is contained in a finite-dimensional ℂ^{∗}-invariant subspace V of A, by the Cartier lemma [20, p. 25] (or by using the grading on A induced by the ℂ^{∗}-action). The map p ∈ A ↦ (dp)_{0} is a ℂ^{∗}-equivariant map from A to T^{∗}, and hence restricts to a ℂ^{∗}-equivariant map from V to T^{∗}. Choosing a ℂ^{∗}-equivariant splitting of the map V → T^{∗} defines a ℂ^{∗}-equivariant map T^{∗} → A and thus a ℂ^{∗}-equivariant homomorphism ℂ[x_{1}, …, x_{n}] → A. Let f : 𝔸^{n} → T be the corresponding morphism. By construction, f has an invertible differential at the origin and is ℂ^{∗}-equivariant. Thus, f is injective in a neighborhood Ω of the origin, and the image of f contains an open set Ω′ about the origin. Since the weights on T are positive, every point of T lies in the ℂ^{∗}-orbit of some point of Ω′. Thus f is surjective. Likewise, f is injective: if f(x_{1}) = f(x_{2}), then since the closures of the ℂ^{∗}-orbits of x_{1} and x_{2} contain the origin, and the weights of the action on the tangent space at 0 are all positive, it follows that there is a λ ∈ ℂ^{∗} such that λ·x_{1} and λ·x_{2} both lie in Ω. By assumption f(λ·x_{1}) = λ·f(x_{1}) = λ·f(x_{2}) = f(λ·x_{2}). But since f is injective on Ω, λ · x_{1} = λ · x_{2}, and hence x_{1} = x_{2}. It follows that f is a ℂ^{∗}-equivariant bijection from 𝔸^{n} to T and thus it is an isomorphism. □

Corollary 4.2.4. The ℂ^{∗}-equivariant morphism from H^{1}(E; U(η)) to H^{1}(E; 𝔲(η)) defined in Lemma 4.2.2 is an isomorphism. Hence, the ℂ^{∗}-action on H^{1}(E; U(η)) can be linearized, and (H^{1}(E; U(η)) − {0})/ℂ^{∗} is a weighted projective space 𝕎ℙ(η). □

In case α is c-special and η = η_{0}, we have calculated the corresponding weights of the weighted projective space in Proposition 3.3.3 and Proposition 3.4.7.

4.3 Existence of universal bundles on the weighted projective space

Next we discuss the existence of universal bundles over the ℂ^{∗}- quotient. It is easy to see that such bundles cannot exist at the orbifold singular points of the weighted projective space, essentially because there are no local sections from the weighted projective space back to the affine space at such points. We shall show that, away from such points, we can almost find a universal bundle. In particular, there is a universal adjoint bundle away from the orbifold singular points of the weighted projective space.

Recall that Z(G) ∩ φ_{α}(ℂ^{∗}) is a finite cyclic group which we have denoted ℤ/n_{c,α}ℤ.

Lemma 4.3.1. The subgroup ℤ/n_{c,α}ℤ of ℂ^{∗} acts trivially on H^{1}(E; U(η)), and the quotient group ℂ^{∗}/(ℤ/n_{c,α}ℤ) acts faithfully on H^{1}(E; U(η)).

Proof. The ℂ^{∗}-weights are of the form kn_{c,α} for 1 ≤ k ≤ h_{α}, and so the lemma is clear. □

The fact that ℤ/n_{c,α}ℤ acts trivially on H^{1}(E; U(η)) also follows from the fact that it is contained in Z(G). Note however that, if n_{c,α} > 1, then the associated action of ℤ/n_{c,α}ℤ on Ξ_{0} is not in general trivial, and in fact is just multiplication by the corresponding subgroup of the center of G.

Proposition 4.3.2. Let 𝕎ℙ(η) be the weighted projective space (H^{1}(E; U(η)) − {0})/ℂ^{∗}, and let 𝕎ℙ_{reg}(η) denote the open subset of 𝕎ℙ(η) which is the ℂ^{∗}-quotient of the set of points of H^{1}(E; U(η)) where ℂ^{∗}/(ℤ/n_{c,α}ℤ) acts freely. Let Ĝ be the quotient of G by the subgroup ℤ/n_{c,α}ℤ. Then the universal bundle Ξ_{0} over E × H^{1}(E; U(η)) induces a principal Ĝ-bundle over E × 𝕎ℙ_{reg}(η).

Proof. Let H^{1}(E; U(η))_{reg} be the set of points of H^{1}(E; U(η)) where ℂ^{∗}/(ℤ/n_{c,α}ℤ) acts freely and effectively. We have seen that there is a lifted action of ℂ^{∗} on Ξ_{0}|E × H^{1}(E; U(η))_{reg}, which in fact is free. The action of the isotropy group ℤ/n_{c,α}ℤ of a point in the base on the fiber is via multiplication by elements of the center of G, and thus there is an induced Ĝ-bundle on the ℂ^{∗}-quotient of H^{1}(E; U(η))_{reg} with the desired properties. □

It is easy to see that we could replace Ĝ-bundles with bundles over an appropriate conformal form G ×_{ℤ/n_{c,α}ℤ} ℂ^{∗} of the group.

If n_{c,α} = 1, then there is an induced G-bundle over E × 𝕎ℙ_{reg}(η). If n_{c,α} > 1, then it is easy to see that the corresponding Ĝ-bundle does not lift to a G-bundle. For example, in case G = SL_{n}(ℂ), the vector bundle over E×ℙ^{n−1} constructed by taking the k^{th} vertex of the Dynkin diagram is given as an extension

0 → π_{1}^{∗}W_{k}^{∗} ⊗ π_{2}^{∗}O_{ℙ^{n−1}}(1) → U → π_{1}^{∗}W_{n−k} → 0,

and no twist of this bundle by a line bundle will have trivial determinant. In a future paper, we shall discuss methods for constructing universal bundles in case n_{c,α} > 1 via spectral covers.

4.4 The case of a c-special root

Definition 4.4.1. Let ζ be a semistable G-bundle with c_{1}(ζ) = c. Then ζ is regular if dim Aut ζ = r_{c}. By [10, Corollary 6.3], every ζ is S-equivalent to a regular semistable G-bundle, which is unique up to isomorphism. Moreover, by [10, Corollary 6.3], if ζ is not regular, then dim Aut ζ ≥ r_{c} + 2.

Let η_{0} be the distinguished bundle defined in Proposition 3.2.3 and Lemma 3.4.5. We next show that the nonzero elements of H^{1}(E; U(η_{0})) correspond to regular semistable G-bundles.

Proposition 4.4.2. With η_{0} as above, for every x ∈ H^{1}(E; U(η_{0}))− {0}, let ξ_{x} denote the principal P -bundle Ξ|E × {x} induced by the restriction of Ξ to the slice over x. Then ξ_{x} ×_{G} P is a regular semistable G-bundle.

Proof. By Theorem 3.3.1 and Theorem 3.4.6, if ζ is unstable and c_{1}(ζ) = c, then dim Aut ζ ≥ r_{c} + 2. As we have noted above, the same holds for a semistable G-bundle ζ which is not regular. Thus, a G-bundle ζ is semistable and regular if and only if dim Aut ζ ≤ r_{c} + 1. To prove Proposition 4.4.2, we shall show that, for all x ≠ 0, dim Aut_{G}(ξ_{x}×_{P} G) ≤ r_{c} + 1.

Let ξ = ξ_{x}. We have the inclusion of the Lie algebra 𝔭 in 𝔤. Clearly, viewing 𝔤 as a representation of P, the vector bundle 𝔤(ξ) is the same as 𝔤(ξ ×_{P} G) = ad_{G}(ξ ×_{P} G). Moreover 𝔭(ξ) = ad_{P} ξ. Thus there is an exact sequence of vector bundles

0 → ad_{P} ξ → ad_{G}(ξ ×_{P} G) → (𝔤/𝔭)(ξ) → 0.

Now replacing ξ by η_{0} ×_{L} P gives the corresponding exact sequence

0 → ad_{P} (η_{0} ×_{L} P) → ad_{G}(η_{0} ×_{L} G) → 𝔲_{−}(η_{0}) → 0,

since (𝔤/𝔭)(η_{0} ×_{L} P) = 𝔲_{−}(η_{0}). Furthermore, by Corollary 2.1.7 and Lemma 3.3.2 in the simply connected case and Theorem 3.4.3 in the nonsimply connected case, H^{0}(E; 𝔲_{−}(η_{0})) has dimension r_{c} + 1. By semicontinuity, there is a neighborhood Ω of the origin in H^{1}(E; U(η_{0})) such that, if ξ = ξ_{x} corresponds to an x ∈ Ω, then dim H^{0}(E; (𝔤/𝔭)(ξ)) ≤ r_{c} + 1 as well. As every point of H^{1}(E; U(η_{0})) is ℂ^{∗}-equivalent to such an x, we must have dim H^{0}(E; (𝔤/𝔭)(ξ)) ≤ r_{c} + 1 for all possible ξ.

Next consider the exact sequence of Lie algebras

0 → 𝔲 → 𝔭 → 𝔩 → 0.

There is the associated bundle sequence

0 → 𝔲(ξ) → ad_{P} ξ → ad_{L} η_{0} → 0.

Since 𝔲(η_{0}) is a direct sum of semistable bundles of negative degrees, H^{0}(E; 𝔲(η_{0})) = 0. It follows as before from semicontinuity and ℂ^{∗}- equivariance that H^{0}(E; 𝔲(ξ)) = 0 for all ξ. So

H^{0}(E; ad_{P} ξ) ⊆ H^{0}(E; ad_{L} η_{0}) ≅ ℂ.

Thus

h^{0}(E; ad_{G}(ξ ×_{P} G)) ≤ h^{0}(E; (𝔤/𝔭)(ξ)) + h^{0}(E; ad_{P} ξ) ≤ r_{c} + 2,

with equality holding if and only if the map

H^{0}(E; ad_{P} ξ) → H^{0}(E; ad_{L} η_{0})

is surjective. This can only happen if the natural homomorphism Aut_{P} ξ → Aut_{L} η_{0} is surjective on the connected component of the identity, which would say that every λ ∈ ℂ^{∗} in Aut_{L} η_{0} lifts to an element of Aut_{P} ξ. But then x must be a fixed point for the ℂ^{∗}-action, and hence x is the origin. Conversely, if x ≠ 0, then H^{0}(E; ad_{P} ξ) = 0 and dim Aut_{G}(ξ ×_{P} G) ≤ r_{c} + 1, and as we have seen above, this statement implies Proposition 4.4.2. □

4.5 The Kodaira-Spencer homomorphism

For the moment, we return to the case of an arbitrary root α. Let x ∈ H^{1}(E; U(η)), and let ξ be the corresponding P -bundle. The bundle Ξ_{0} induces a Kodaira-Spencer homomorphism from the tangent space T_{x} of H^{1}(E; U(η)) at x to H^{1}(E; ad(ξ ×_{P} G)), and we wish to find some general circumstances where this map is surjective.

Theorem 4.5.1. Suppose that the differential of the action of Aut_{L} η on H^{1}(E; U(η)) at 1 ∈ Aut_{L} η and x ∈ H^{1}(E; U(η)) is an injective homomorphism H^{0}(E; ad_{L} η) → T_{x}. Then the Kodaira-Spencer homomorphism T_{x} → H^{1}(E; ad(ξ ×_{P} G)) is surjective.

Proof. As in the proof of Lemma 4.2.2, the tangent space T_{x} can be identified with bundles ξ_{ε} over E × Spec ℂ[ε] which restrict to ξ over the closed fiber and reduce mod U to π_{1}^{∗}η. If η is given by the 1-cocycle {ℓ_{ij}}, where the ℓ_{ij} take values in L, and ξ by the 1-cocycle {ℓ_{ij}u_{ij}}, where the u_{ij} take values in U, then it is easy to see that ξ_{ε} is given by a 1-cocycle {ℓ_{ij}(u_{ij} + εv_{ij})}, where the v_{ij} are also U-valued. Moreover w_{ij} = u_{ij}^{−1}v_{ij} defines an element of H^{1}(E; 𝔲(ξ)). In this way, we identify T_{x} with H^{1}(E; 𝔲(ξ)).

There is a long exact sequence

0 → 𝔲(ξ) → ad_{P} ξ → ad_{L} η → 0.

The natural map H^{1}(E; 𝔲(ξ)) → H^{1}(E; ad_{P} ξ) is the Kodaira-Spencer map for deformations of the P -bundle ξ, and its kernel is the image of the coboundary map δ: H^{0}(E; ad_{L} η) → H^{1}(E; 𝔲(ξ)). This kernel also contains the image of the differential of the action of Aut_{L} η on H^{1}(E; U(η)) at x, which has dimension equal to dim H^{0}(E; ad_{L} η). Thus it follows by hypothesis that δ is injective.

The Killing form identifies the vector bundle (𝔤/𝔭)(ξ) with the dual of 𝔲(ξ). In particular, (𝔤/𝔭)(ξ) has a filtration whose successive quotients are stable bundles of positive degrees. Hence H^{1}(E; (𝔤/𝔭)(ξ)) = 0 and the natural map H^{1}(E; ad_{P} ξ) → H^{1}(E; ad_{G}(ξ×_{P} G)) is surjective. Now consider the commutative diagram

H^{1}(E; 𝔲(ξ)) ⟶ H^{1}(E; ad_{G}(ξ ×_{P} G))
↓  ∥
H^{0}(E; (𝔤/𝔭)(ξ)) ⟶ H^{1}(E; ad_{P} ξ) ⟶ H^{1}(E; ad_{G}(ξ ×_{P} G)) ⟶ 0
↓  ↓
H^{1}(E; ad_{L} η)  H^{1}(E; ad_{L} η)

↓

0

where the middle row and column are exact. A diagram chase shows that, if the map H^{0}(E; (𝔤/𝔭)(ξ)) → H^{1}(E; ad_{L} η) is surjective, then so is the map

H^{1}(E; 𝔲(ξ)) → H^{1}(E; ad_{G}(ξ ×_{P} G)),

which is the statement of the theorem. The map H^{0}(E; (𝔤/𝔭)(ξ)) → H^{1}(E; ad_{L} η) is given by the composition

H^{0}(E; (𝔤/𝔭)(ξ)) → H^{1}(E; ad_{P} ξ) → H^{1}(E; ad_{L} η),

and the above sequence is Serre dual to the sequence

H^{1}(E; 𝔲(ξ)) ← H^{0}(E; (𝔤/𝔲)(ξ)) ← H^{0}(E; ad_{L} η).

By the naturality of the connecting homomorphisms associated to the following commutative diagram of short exact sequences

0 ⟶ 𝔲(ξ) ⟶  ad_{P} ξ  ⟶ ad_{L} ξ ⟶ 0
∥  ↓  ↓
0 ⟶ 𝔲(ξ) ⟶ ad_{G}(ξ ×_{P} G) ⟶ (𝔤/𝔲)(ξ) ⟶ 0,

the composition H^{0}(E; ad_{L} η) → H^{1}(E; 𝔲(ξ)) is just the homomorphism δ, which is injective by hypothesis. Hence by duality H^{0}(E; (𝔤/𝔭)(ξ)) → H^{1}(E; ad_{L} η) is surjective, which completes the proof. □

Corollary 4.5.2. If dim Aut_{L} η = 1, and x is not the origin of H^{1}(E; U(η)), then the Kodaira-Spencer homomorphism

T_{x} → H^{1}(E; ad(ξ ×_{P} G))

is surjective. If α is c-special, η = η_{0}, and x is not the origin, then the Kodaira-Spencer homomorphism induces an isomorphism from T_{x} modulo the tangent space to ℂ^{∗} · x to H^{1}(E; ad(ξ ×_{P} G)).

Proof. Since the ℂ^{∗}-action is via strictly positive weights, the differential of the action is injective at every nonzero point H^{1}(E; U(η)), so the hypothesis of the previous theorem is satisfied. This proves the first statement. To see the second statement, the induced map from T_{x} modulo the tangent space to ℂ^{∗} · x to H^{1}(E; ad(ξ ×_{P} G)) is surjective. The dimension of T_{x} modulo the tangent space to ℂ^{∗} · x is r_{c}. Since, for every ξ, dim H^{1}(E; ad(ξ ×_{P} G)) ≥ r_{c}, equality must hold, giving a new proof of Proposition 4.4.2, and the induced map from T_{x} modulo the tangent space to ℂ^{∗} · x to H^{1}(E; ad(ξ ×_{P} G)) is an isomorphism. □

This shows that the map Ψ: 𝕎ℙ(η_{0}) → M(G, c) is a local diffeomorphism over the smooth points of M(G, c). Moreover, giving M(G, c) and 𝕎ℙ(η_{0}) their natural orbifold structures, it follows from Corollary 4.5.2 that Ψ is an orbifold covering. Let d_{k}(𝕎ℙ(η_{0})) and d_{k}(M(G, c)) be the dimensions of the subspaces of 𝕎ℙ(η_{0}) and M(G, c) where k divides the order of the orbifold isotropy. The orbifold covering property implies that

d_{k}(𝕎ℙ(η_{0})) ≤ d_{k}(M(G, c)) for all k ≥ 1.

The results in [7] and [10] imply that d_{k}(M(G, c)) = d(k) in the notation of Proposition 3.3.3 in the simply connected case.

In Section 5 we shall prove that Ψ is an isomorphism. In proving this result we do not appeal to Corollary 4.5.2, but rather use the fact that both d_{k}(𝕎ℙ(η_{0})) and d_{k}(M(G, c)) satisfy circular symmetry. One can in fact turn this argument around. Using the above inequality and the fact that the sums of the weights for 𝕎ℙ(η_{0}) and M(G, c) add up to g, one can prove directly that d_{k}(𝕎ℙ(η_{0})) = d_{k}(M(G, c)) for all k ≥ 1, and hence apply the results of [7] to show that the d_{k}(𝕎ℙ(η_{0})) satisfy circular symmetry. In the simply connected case, this gives a classification-free proof of circular symmetry for the numbers d_{k}(α), where α is a special root.

4.6 The singular locus of the weighted projective space

The weighted projective space 𝕎ℙ(η) is naturally an orbifold. Its singular locus (as an orbifold) corresponds to the set of points in the affine space H^{1}(E; U(η)) whose isotropy group is larger than that of the generic point, i.e., is larger than ℤ/n_{c,α}ℤ. This will be the singular locus of 𝕎ℙ(η) as a variety provided that its codimension is at least two. If we choose a linear structure and a diagonal basis of H^{1}(E; U(η)), then the orbifold singular locus of 𝕎ℙ(η) is a union of weighted projective subspaces 𝕎ℙ(η)_{k} for k > 1, where 𝕎ℙ(η)_{k} is the subvariety of 𝕎ℙ(η) where all of the coordinates are equal to zero except for those for which the weights are divisible by kn_{c,α}. Our goal is to show that, when G is simply connected and α is special, each such subspace can be naturally identified the moduli space for a non-simply connected subgroup of G.

Given α ∈ Δ, fix an integer k > 1 such that k = β(ϖ_{α}^{∨}) for some root β. Recall from the proof of Proposition 1.4.2 that the set of all β ∈ R such that k|β(ϖ_{α}^{∨}) is a root system R(α, k). Moreover, Δ(α, k) = (Δ − {α}) ∪ {−λ_{k}(α)} is a set of simple roots for R(α, k). There is a semisimple subalgebra

𝔤(α, k) = 𝔥 ⊕ ⨁_{β∈R(α,k)} 𝔤^{β} ⊆ 𝔤.

Let G(α, k) be the corresponding closed connected subgroup of G. Of course, 𝔤(α, k) will not be simple in general. Let

Δ(α, k) = ∐_{i≥1} Δ(α, k)_{i},

where each subset Δ(α, k)_{i} corresponds to a connected component of the Dynkin diagram of R(α, k), and where −λ_{k}(α) ∈ Δ(α, k)_{1}. Since G(α, k) is semisimple, we can write

G(α, k) = (∏_{i} G_{i}) /F,

where each G_{i} is simple and simply connected and corresponds to the subset Δ(α, k)_{i} of Δ(α, k), and where F is finite. If G is simply connected, then F is cyclic of order k, generated by an element c_{k}.

Viewing −λ_{k}(α) as an element of Δ(α, k), i.e., a simple root for G(α, k), −λ_{k}(α) defines a maximal parabolic subgroup P (α, k) of G(α, k) contained in the maximal parabolic subgroup P^{α} = P of G determined by α. The parabolic subgroup P (α, k) is of the form (P_{1} × ∏_{i≥2} G_{i})/F, where P_{1} is the maximal parabolic subgroup in G_{1} corresponding to −λ_{k}(α). Clearly, the Levi factor L(α, k) of P (α, k) is just L^{α} = L = (L_{1} × ∏_{i≥2} G_{i})/F, where L_{1} is the Levi factor of P_{1}. The unipotent radical U(α, k) of P (α, k) has Lie algebra ⨁_{k|j} 𝔲^{j}. Let η be a semistable L-bundle of negative degree. The degree of η is of course independent of whether we view L as the Levi factor of P or of P (α, k), and we can define the cohomology set H^{1}(E; U(α, k)(η)). The inclusion of U(α, k) in U defines an Aut(L)-equivariant function H^{1}(E; U(α, k)(η)) → H^{1}(E; U(η)) on the level of cohomology sets, as well as a morphism between the corresponding functors. Since the two associated functors are both represented by affine spaces, the function H^{1}(E; U(α, k)(η)) → H^{1}(E; U(η)) corresponds to an Aut(L)-equivariant morphism of affine spaces. The geometric meaning of this morphism is as follows: let F_{i} be the projection of F to the factor G_{i}, so that there is a homomorphism from (L_{1} × ∏_{i≥2} G_{i})/F to (L_{1}/F_{1}) × ∏_{i≥2}(G_{i}/F_{i}) for i ≥ 2. The bundle η thus induces an (L_{1}/F_{1}) × ∏_{i≥2}(G_{i}/F_{i})-bundle ∏_{i} η_{i}, where η_{1} is an (L_{1}/F_{1})-bundle and the η_{i} are (G_{i}/F_{i})-bundles for i ≥ 2, Moreover η defines a canonical lifting of the (L_{1}/F_{1}) × ∏_{i≥2}(G_{i}/F_{i})-bundle ∏_{i} η_{i} to an (L_{1} × ∏_{i≥2} G_{i})/F -bundle. A class x in H^{1}(E; U(α, k)(η)) defines a lifting of η_{1} to a (P_{1}/F_{1})-bundle ξ_{1}. The lift η then defines a lift of the (P_{1}/F_{1}) × ∏_{i≥2}(G_{i}/F_{i})-bundle defined by ξ_{1} and the η_{i}, i ≥ 2, to a (P_{1} × ∏_{i≥2} G_{i})/F -bundle ξ′. Since P (α, k) = (P_{1} × ∏_{i≥2} G_{i})/F is a subgroup of P, we can form the associated bundle ξ = ξ′ ×_{P (α,k)} P, and this bundle is clearly the lift of η corresponding to the image of x in H^{1}(E; U(η)).

Proposition 4.6.1. Let η be a semistable L-bundle of negative degree −d. There are compatible linear structures on H^{1}(E; U(α, k)(η)) and on H^{1}(E; U(η)) so that the morphism

H^{1}(E; U(α, k)(η)) → H^{1}(E; U(η))

is a ℂ^{∗}-equivariant embedding of H^{1}(E; U(α, k)(η)) onto the linear subspace of H^{1}(E; U(η)) defined by the span of all of the eigenvectors of the ℂ^{∗}-action on H^{1}(E; U(η)) whose weights are divisible by kn_{c,α}d.

Proof. It is an elementary exercise to check that, if ℂ^{∗} acts linearly and with positive weights on two affine spaces 𝔸_{1} and 𝔸_{2} and if f: 𝔸_{1} → 𝔸_{2} is a ℂ^{∗}-equivariant morphism whose differential at the origin is injective, then there exist coordinates on 𝔸_{2} for which ℂ^{∗} acts linearly and such that f is a linear embedding, and the ℂ^{∗}-weights for the image of f can be determined from the differential of f at the origin. The differential of the morphism H^{1}(E; U(α, k)(η)) → H^{1}(E; U(η)) is given by the inclusion

⨁_{k|j} H^{1}(E; 𝔲^{j}(η)) → ⨁_{j>0} H^{1}(E; 𝔲^{j}(η)).

Hence the image of the differential of f at the origin is the span of the eigenvectors of the ℂ^{∗}-action on H^{1}(E; 𝔲(η)) whose weights are divisible by kn_{c,α}d. Thus, the same is true for the morphism f. □

We turn now to the case of a special root α. For simplicity, we assume that G is simply connected, so that the finite group F = π_{1}(G(α, k)) is generated by an element c_{k} of order k.

Proposition 4.6.2. Suppose that G is simply connected and that α is special. Let η_{0} be the L-bundle of Proposition 3.2.3. Then:

(i) For i > 1, Δ(α, k)_{i} is of type A and c_{k} projects to a generator of the corresponding fundamental group.

(ii) The (L_{1}/F_{1})-bundle η_{1} induced by η_{0} has the property that η_{1} ×_{L_{1}/F_{1}} (G_{1}/F_{1}) is a minimally unstable (G_{1}/F_{1})-bundle, and the root −λ_{k}(α) is a c_{k}-special simple root in Δ(α, k)_{1}.

Proof. Part (i) follows easily from the explicit description of the special root. To see Part (ii), it follows from Proposition 4.6.1 that, if x is a nonzero element of H^{1}(E; U(α, k)(η_{0})), then the image of x in H^{1}(E; U(η_{0})) is also nonzero. In particular, if ξ is the corresponding P -bundle, then ξ ×_{P} G is semistable. It is easy to check that, in this case, the (P_{1}/F_{1})-bundle corresponding to x is again semistable. Thus, η_{1} ×_{(P_{1}/F_{1})} (G_{1}/F_{1}) is a minimally unstable (G_{1}/F_{1})-bundle, and so −λ_{k}(α) is a c_{k}-special simple root for G_{1}/F_{1}. □

Of course, we could also check the above proposition by a case-by-case analysis. This also shows that the projection F → G_{1} is always an embedding of F into the center of G_{1}. For i ≥ 2, the factor G_{i} is of type A and F_{i} is the full center, and hence the bundle η_{i} is always rigid. On the other hand, the image F_{1} need not be the full center of G_{1}. For example, if G is of type E_{8} and k = 4, then G_{1} is of type A_{7}, and thus its center is isomorphic to ℤ/8ℤ, whereas F_{1} has order 4.

There is thus an induced morphism of weighted projective spaces. In terms of moduli spaces, if we grant Looijenga’s theorem (Theorem 5.1.1) in both the simply connected and the non-simply connected cases, this morphism identifies the sub-projective space 𝕎ℙ(η_{0})_{k} of 𝕎ℙ(η_{0}) corresponding to setting all the coordinates in the weight spaces with weights not divisible by kn_{α} equal to zero with the weighted projective space which is the moduli space of unliftable semistable (G_{1}/F_{1})-bundles. Of course, on the level of G-bundles, this shows that up to S-equivalence a bundle corresponding to a point of the moduli space lying in this subprojective space has a reduction of structure to a semistable unliftable (∏_{i} G_{i})/F -bundle, and conversely such a bundle defines a semistable G-bundle whose moduli point lies in this sub-projective space. In terms of the ℂ^{∗}-weights, the number of such weights divisible by kn_{α} can be related to the weights appearing for the appropriate non-simply connected form of a subgroup. For example, if G is of type E_{8}, the root system R(α, 2) is of type E_{7} × A_{1} and the ℂ^{∗}-weights divisible by 2 for G, in other words the g_{α} such that 2|g_{α}, are the weights occurring in the weighted projective space for the adjoint form of E_{7}. These are exactly the weights appearing in the quotient diagram for Ẽ_{7} modulo the action of the nontrivial element of the center, namely twice the weights for the group of type F_{4}: 2, 2, 4, 4, 6. In [7, §9], these quotient root systems appear in a different context, unrelated to the special roots, as certain root systems Φ(𝔱(k)) constructed on certain subtori of H. It would be nice to understand this somewhat mysterious connection more directly.

5. A new proof of Looijenga’s theorem

5.1 Statement of the theorem

Fix a c-special root α. We denote the corresponding parabolic subgroup simply by P, and similarly for L. We have defined the bundle η_{0} in Proposition 3.2.3 and Lemma 3.4.5. As in Section 1, let M(G, c) denote the coarse moduli space of semistable G-bundles ξ with c_{1}(ξ) = c, modulo S-equivalence. We have seen that there is a universal family of regular semistable G-bundles Ξ_{0} over E × (H^{1}(E; U(η_{0})) − {0}). Thus there is an induced morphism

Ψ̃: H^{1}(E; U(η_{0})) − {0} → M(G, c).

The morphism Ψ̃ is constant on ℂ^{∗}-orbits. Let 𝕎ℙ(η_{0}) be the weighted projective space which is the ℂ^{∗}-quotient of H^{1}(E; U(η_{0})) − {0} and let

Ψ: 𝕎ℙ(η_{0}) → M(G, c)

be the morphism induced by Ψ̃. We can then state our version of Looijenga’s theorem as follows:

Theorem 5.1.1. Let E be an elliptic curve, let G be a simple group and let c ∈ π_{1}(G) be a generator. The morphism Ψ: 𝕎ℙ(η_{0}) → M(G, c) defined above is an isomorphism.

Corollary 5.1.2. The moduli space M(G, c) is isomorphic to a weighted projective space with weights g\overline{β}/n_{0}, \overline{β} ∈ Δ̃/w_{c} as defined in Definition 1.5.1.

In particular, if G is simply connected, then we obtain a new proof of Looijenga’s theorem [17].

5.2 The classical cases

Let us first sketch the proof of the theorem above for the case of the classical groups, bearing in mind the description of the bundle η_{0} given in Proposition 3.2.5. In case G = SL_{n}(ℂ), the theorem asserts that every regular semistable vector bundle V of rank n and trivial determinant is S-equivalent to a unique extension

0 → W_{k}^{∗} → V → W_{n−k} → 0.

In this form, the theorem is proved in [14], Theorem 3.2(iv).

Let us next consider the case of the symplectic group. We shall show that the morphism Ψ has degree one in this case (as we shall see below, this implies that Ψ is an isomorphism). Let V be a generic regular semistable symplectic vector bundle, in other words a regular semistable vector bundle of rank 2n with a nondegenerate symplectic form A. Here generic will mean that V is a direct sum

⨁_{i=1}^{n}(λ_{i} ⊕ λ_{i}^{−1}),

where the λ_{i} are line bundles of degree zero, not of order 2, and such that, for i ≠ j, λ_{i} ≠ λ_{j}^{±1}. In this case, the symplectic form on V is an orthogonal sum of symplectic forms A_{i} on λ_{i} ⊕ λ_{i}^{−1}. The space of such forms which are nondegenerate corresponds to the choice of an isomorphism from λ_{i}^{−1} to itself, in other words to a nonzero multiple of A_{i}, and the group of symplectic automorphisms of A_{i} is also isomorphic to ℂ^{∗}. For each i, the space of surjections φ_{i}: W_{n}^{∗} → λ_{i}^{±1} is a ℂ^{∗}. Thus, the space of morphisms W_{n}^{∗} → λ_{i} ⊕ λ_{i}^{−1} is a ℂ^{∗} × ℂ^{∗}. It is clear that the pullback of A_{i} to W_{n} under such a morphism is a nonzero symplectic form, which we denote by B_{i}. Moreover, by varying the morphism from W_{n}^{∗} → λ_{i} ⊕ λ_{i}^{−1}, we exactly get all symplectic forms on W_{n}^{∗} of the form t_{i}B_{i}, where t_{i} ∈ ℂ^{∗}.

Now suppose we have found, for every i, a morphism f_{i}: W_{n}^{∗} → λ_{i} ⊕ λ_{i}^{−1} such that, if t_{i}B_{i} is the pulled back morphism, then t_{i} ≠ 0 for every i and such that ∑_{i=1}^{n} t_{i}B_{i} = 0. It follows that the morphism W_{n}^{∗} → λ_{i}^{±1} is nonzero for every i. By Proposition 3.6 of [14], the morphism W_{n}^{∗} → V embeds W_{n}^{∗} as an isotropic subbundle of V, and the quotient V/W_{n}^{∗} is isomorphic to W_{n}. This implies that we have reduced the structure group of V to a maximal parabolic subgroup P of Sp(2n) corresponding to the special root, and the corresponding L-bundle is η_{0}. Hence (V, A) is in the image of Ψ, and the statement that Ψ has degree 1 is the statement that the t_{i} above are uniquely determined up to multiplying by a fixed nonzero scalar.

Thus, we must find t_{i} such that ∑_{i=1}^{n} t_{i}B_{i} = 0 and show that the t_{i} are unique up to a scalar. To see this, note that the space of alternating forms on W_{n}^{∗} is H^{0}(E; ⋀^{2} W_{n}). One easily computes that deg ⋀^{2} W_{n} = n − 1. Since W_{n} is stable, ⋀^{2} W_{n} is a semistable vector bundle of positive degree. Thus, h^{1}(E; ⋀^{2} W_{n}) = 0, and so by Riemann-Roch, h^{0}(E; ⋀^{2} W_{n}) = n − 1. It follows that every collection of n forms B_{i} is linearly dependent, and so some linear combination of the B_{i} is zero. To prove uniqueness, and also to prove that the quotient is a W_{n}, it will suffice to show that no smaller linear combination is zero. Suppose, say, that ∑_{i=1}^{k} t_{i}B_{i} = 0, with k < n and t_{i} ≠ 0 for all i ≤ k. Consider the induced morphism from W_{n}^{∗} to ⨁_{i=1}^{k}(λ_{i} ⊕ λ_{i}^{−1}). If k > n/2, then by Proposition 3.6 of [14] there would exist an embedding of W_{n}^{∗} as an isotropic subbundle of a vector bundle of rank < 2n, which is impossible since the symplectic form on ⨁_{i=1}^{k}(λ_{i} ⊕ λ_{i}^{−1}) is nondegenerate. If k ≤ n/2, then in fact the argument of Proposition 3.6 of [14] shows that the image of W_{n}^{∗} would be a subbundle of rank equal to 2k, and hence that the symplectic form on ⨁_{i=1}^{k}(λ_{i}⊕λ_{i}^{−1}) is identically zero, which is again a contradiction. Thus no smaller linear combination of the B_{i} is zero, proving that Ψ has degree one. A similar argument shows that, even for a regular symplectic bundle (V, A) which is not generic in the above sense, there still exists an embedding of W_{n}^{∗} in V as a maximal isotropic subbundle, and this embedding is unique up to the automorphism group of (V, A). Thus, we can also show directly that Ψ is a bijection instead of merely having degree one in this case.

The case of Spin(n) is similar. By a generic Spin(2n)-bundle, we mean one whose associated SO(2n)-bundle V is of the form ⨁_{i=1}^{n}(λ_{i} ⊕ λ_{i}^{−1}), where the λ_{i} are line bundles of degree zero, not of order 2, and such that, for i ≠ j, λ_{i} ≠ λ_{j}^{±1}, and the quadratic form is the orthogonal sum of quadratic forms A_{i} on λ_{i} ⊕ λ_{i}^{−1}. We consider maps W_{n−2}^{∗} → V whose image is isotropic. Arguments as in the symplectic case show that there is an embedding ι of W_{n−2}^{∗} as an isotropic subbundle of V, and such that the projection to each summand λ_{i}^{±1} is nonzero, and such an embedding is unique up to orthogonal isomorphisms of V. Dually, we have a map V^{∗} ≅ V → W_{n−2}. Thus there is a complex W_{n−2}^{∗} ⟶^{ι} V ⟶^{ι^{∗}} W_{n−2}. The symmetric form identifies V/(W_{n−2}^{∗})^{⊥} with W_{n−2}. It is easy to check that the bundle Ker(ι^{∗})/ Im(ι) is a semistable SO(4)-bundle which does not lift to Spin(4), and hence is of the form W_{2} ⊗ W_{2}^{∗} ≅ Q_{4}. The filtration on V then reduces the structure group to the appropriate maximal parabolic subgroup as before. A very similar argument handles the case of SO(2n+1), using instead an isotropic subbundle isomorphic to W_{n−1}^{∗}, and showing that (W_{n−1}^{∗})^{⊥}/W_{n−1}^{∗} is isomorphic to ad W_{2}. Similar statements can also handle the case of unliftable bundles.

5.3 Proof of the main theorem: determinant line bundles

We turn to the proof of Theorem 5.1.1.

First, it is an elementary result that a morphism from a weighted projective space to a quasiprojective variety is either constant or finite. Indeed, since every weighted projective space has a finite ramified cover which is an ordinary projective space, this follows from the analogous and well-known result for ℙ^{n}. Thus, since 𝕎ℙ(η_{0}) and M(G, c) are normal, if we can show that Ψ has degree one, then it follows from Zariski’s main theorem that Ψ is an isomorphism.

To calculate the degree of Ψ we shall compare determinant line bundles on the two sides. The idea will be to find line bundles 𝔻_{1}, 𝔻_{2} on 𝕎ℙ(η_{0}) and M(G, c) respectively, such that [𝔻_{1}] = Ψ^{∗}[𝔻_{2}] in H^{2}(𝕎ℙ(η_{0}); ℚ) and such that ∫_{𝕎ℙ(η_{0})} c_{1}(𝔻_{1})^{r_{c}} = ∫_{M(G,c)} c_{1}(𝔻_{2})^{r_{c}} ≠ 0. Since on the other hand we have c_{1}(𝔻_{1})^{r_{c}} = deg Ψ·c_{1}(𝔻_{2})^{r_{c}}, it will then follow that deg Ψ = 1.

To define the line bundles 𝔻_{i} (which in fact will be a priori only ℚ-Cartier divisors), we recall the definition of the determinant line bundle on the moduli functor. First recall the definition of the moduli functor M itself: for a scheme S over ℂ, M(S) is the set of isomorphism classes of principal G-bundles Ξ over E×S such that Ξ|E×{x} is semistable for every x ∈ S. The moduli functor is coarsely represented by M(G, c). Given an element of M(S), corresponding to a principal G-bundle Ξ over E × S, we have the associated vector bundle ad Ξ over E × S, and thus we can form the determinant line bundle det Rπ_{2}_{∗} ad Ξ over S (see for example Chapter 5, Section 3 of [9]). Since π_{2} has relative dimension one, H^{i}(ad Ξ|E × {s}) = 0 and likewise R^{i}π_{2}_{∗} ad Ξ = 0 for i > 1. The fiber of det Rπ_{2}_{∗} ad Ξ over s ∈ S is then the complex line

⋀^{top} H^{0}(ad Ξ|E × {s}) ⊗ (⋀^{top} H^{1}(ad Ξ|E × {s}))^{−1} .

Here are some of the basic properties of this line bundle:

1. If R^{0}π_{2}_{∗} ad Ξ and R^{1}π_{2}_{∗} ad Ξ are locally free, for example if Ξ|E × {x} is regular for every x ∈ S, then

det Rπ_{2}_{∗} ad Ξ = (⋀^{top} R^{0}π_{2}_{∗} ad Ξ) ⊗ (⋀^{top} R^{1}π_{2}_{∗} ad Ξ)^{−1} .

2. Suppose that S is smooth, that λ is a line bundle of degree zero on E, and that

D_{λ} = { x ∈ S : h^{0}(E; (ad Ξ|E × {x}) ⊗ λ) ≠ 0}

is a hypersurface in S. Let Z_{i} be the irreducible components of D_{λ}, and let n_{i} be the length of the torsion sheaf R^{1}π_{2}_{∗}(ad Ξ ⊗ π_{1}^{∗}λ) at a generic point of Z_{i}. Then (see e.g., [9], Chapter 5, Corollary 1.2 and Proposition 3.9) there is a canonical section div of ( det Rπ_{2}_{∗}(ad Ξ ⊗ π_{1}^{∗}λ))^{−1} whose divisor of zeroes is ∑_{i} n_{i}Z_{i}, and hence

c_{1}(det Rπ_{2}_{∗} ad Ξ) = c_{1}(det Rπ_{2}_{∗}(ad Ξ ⊗ π_{1}^{∗}λ)) = − ∑_{i} n_{i}[Z_{i}].

By general results, there is an associated ℚ-divisor on M(G, c) (i.e., an element in Pic(M(G, c)) ⊗ ℚ, which we shall denote by 𝔻_{2}. As we shall see, 𝔻_{2} is in fact Cartier, in other words, a line bundle. We will find an analogous divisor 𝔻_{1} over 𝕎ℙ(η_{0}) and show that (1) ∫_{𝕎ℙ(η_{0})} c_{1}(𝔻_{1})^{r_{c}} = ∫_{M(G,c)} c_{1}(𝔻_{2})^{r_{c}} and (2) [𝔻_{1}] = Ψ^{∗}[𝔻_{2}]. This will prove Theorem 5.1.1.

5.4 The divisor on the weighted projective space

Consider the universal G-bundle Ξ_{0} over E × H^{1}(E; U(η_{0})). The action of ℂ^{∗} on Ξ_{0} gives a linearization of the action on ℂ^{∗} on the associated vector bundle ad Ξ_{0} and thus on the line bundle det Rπ_{2}_{∗} ad Ξ_{0}. Since H^{1}(E; U(η_{0})) is an affine space, det Rπ_{2}_{∗} ad Ξ_{0} is the trivial line bundle. Every linearization of the ℂ^{∗}-action on the trivial bundle is given by a character χ: ℂ^{∗} → ℂ^{∗} of the form χ(z) = z^{a} for a unique a ∈ ℤ. There is the corresponding coherent sheaf O_{𝕎ℙ(η_{0})}(a), which is given by viewing 𝕎ℙ(η_{0}) as Proj ℂ[z_{0}, …, z_{r_{c}}] with the appropriate grading. By a general result on weighted projective spaces [19], the coherent sheaf O_{𝕎ℙ(η_{0})}(a) is a line bundle if and only if the ℂ^{∗}-weights all divide a. Note that, on the open set 𝕎ℙ_{reg}(η_{0}) of free ℂ^{∗}-orbits, the vector bundle ad Ξ_{0} is defined, and in fact R^{0}π_{2}_{∗} ad Ξ_{0} and R^{1}π_{2}_{∗} ad Ξ_{0} are both locally free of rank r_{c}. Thus there is a well-defined line bundle (⋀^{r_{c}} R^{0}π_{2}_{∗} ad Ξ_{0}) ⊗ (⋀^{r_{c}} R^{1}π_{2}_{∗} ad Ξ_{0})^{−1}, and this line bundle clearly agrees with the restriction of O_{𝕎ℙ(η_{0})}(a) to 𝕎ℙ_{reg}(η_{0}). We next identify the integer a:

Lemma 5.4.1. The natural ℂ^{∗}-linearization on det Rπ_{2}_{∗} ad Ξ_{0} corresponds to the line bundle O_{𝕎ℙ(η_{0})}(−2gn_{c,α}/n_{0}).

Proof. We must show that the ℂ^{∗}-linearization on det Rπ_{2}_{∗} ad Ξ_{0} is given by the character which is raising to the power −2gn_{c,α}/n_{0}. To compute the ℂ^{∗}-linearization, it suffices to compute the action of ℂ^{∗} on the fiber of det Rπ_{2}_{∗} ad Ξ_{0} over the origin, which is a fixed point for the ℂ^{∗}-action on H^{1}(E; U(η_{0})). The fiber over 0 is canonically

⋀^{top} H^{0}(E; ad_{G}(η_{0} ×_{L} G)) ⊗ (⋀^{top} H^{1}(E; ad_{G}(η_{0} ×_{L} G)))^{−1} .

Now, by Lemma 2.1.2 and Lemma 2.1.5,

H^{0}(E; ad_{G}(η_{0} ×_{L} G)) = H^{0}(E; 𝔲_{−}(η_{0})) ⊕ H^{0}(E; ad_{L} η_{0}).

Since ℂ^{∗} is contained in the center of L, the action of ℂ^{∗} on H^{0}(E; ad_{L} η_{0}) is trivial. By Proposition 3.3.3 and Proposition 3.4.7, ℂ^{∗} acts on H^{0}(E; 𝔲_{−}(η_{0})) with weights −n_{c,α}g\overline{β}/n_{0}. Thus the action on ⋀^{top} H^{0}(E; ad_{G}(η_{0} ×_{L} G)) is via − ∑\overline{β} n_{c,α}g\overline{β}/n_{0} = −gn_{c,α}/n_{0}. A similar argument (or duality) handles the case of the ℂ^{∗}-action on ⋀^{top} H^{1}(E; ad_{G}(η_{0} ×_{L} G)). Putting these together, we get the power −2gn_{c,α}/n_{0}. □

Remark 5.4.2. In case 𝕎ℙ(η_{0}) is the weighted projective space arising from the moduli space of G-bundles, where G is simply connected, one can use the Kodaira-Spencer map to check that 𝔻_{1} = K_{𝕎ℙ(η_{0})}^{⊗2}. Now, by a standard fact about weighted projective spaces, K_{𝕎ℙ(η_{0})} = O_{𝕎ℙ(η_{0})}(−g), and thus K_{𝕎ℙ(η_{0})}^{⊗2} = O_{𝕎ℙ(η_{0})}(−2g).

Define 𝔻_{1} = O_{𝕎ℙ(η_{0})}(−2gn_{c,α}/n_{0}). We now compute the top selfintersection of c_{1}(𝔻_{1}):

Lemma 5.4.3. Let 𝕎ℙ^{r} be the weighted projective space which is the quotient of ℂ^{r+1}−{0} by the action of ℂ^{∗} acting with positive weights w_{0}, …, w_{r}, and let a be an integer such that w_{i}|a for every i. Then

∫_{𝕎ℙ_{r}} c_{1}(O_{𝕎ℙ}^{r}(a))^{r} = a^{r}d/(w_{0} ⋯ w_{r}),

where d = gcd{w_{0}, …, w_{r}}.

Proof. The morphism ℂ^{r+1} → ℂ^{r+1} defined by

(z_{0}, …, z_{r}) ↦ (z_{0}^{w_{0}}, …, z_{r}^{w_{r}})

is ℂ^{∗}-equivariant, where ℂ^{∗} acts with all weights equal to 1 on the domain and with weights w_{i} on the range. Thus there is an induced cover f: ℙ^{r} → 𝕎ℙ^{r}, and it is easy to check that the degree of this cover is w_{0} ⋯ w_{r}/d. There is always a natural inclusion f^{∗}O_{𝕎ℙ}^{r}(a) → O_{ℙ^{r}}(a), and one checks that this inclusion is an isomorphism if w_{i}|a for every a. In this case,

∫_{𝕎ℙ_{r}} c_{1}(O_{𝕎ℙ}^{r}(a))^{r} = ∫_{ℙ^{r}} c_{1}(O_{ℙ^{r}}(a))^{r}/ deg f = a^{r}d/(w_{0} ⋯ w_{r}).

This proves the formula of Lemma 5.4.3. □

Corollary 5.4.4. c_{1}(𝔻_{1})^{r_{c}} = (−2g)^{r_{c}}n_{0}/ ∏\overline{β} g\overline{β}.

Proof. This is immediate from Lemma 5.4.1 and Lemma 5.4.3, with w_{i} = n_{c,α}g\overline{β}/n_{0}, a = −2gn_{c,α}/n_{0}, and d = n_{c,α}. □

5.5 The divisor on the moduli space: the simply connected case

We turn now to the calculation of c_{1}(𝔻_{2})^{r}. In order to make the argument easier to follow, we begin by working out the simply connected case. Recall that we have the finite morphism E ⊗_{ℤ} Λ → M(G, c). Let 𝔻_{3} be the pullback of 𝔻_{2} to a divisor on E ⊗_{ℤ} Λ. Clearly, c_{1}(𝔻_{2})^{r} = c_{1}(𝔻_{3})^{r}/#(W). Thus, we shall begin by computing c_{1}(𝔻_{3})^{r}. For a point ρ ∈ E ⊗_{ℤ}Λ, there is an associated flat G-bundle ξ_{0}, and as we have seen in Lemma 3.1 of [10],

ad ξ_{0} ≅ O_{E}^{r} ⊕ ⨁_{α∈R} λ_{α(ρ)},

where λ_{α(ρ)} is the line bundle of degree zero corresponding to the point α(ρ) ∈ E ≅ Pic^{0} E, or equivalently is the line bundle associated to the flat U(1)-bundle whose holonomy is given by α(ρ). In particular, we see that for λ = O_{E}, the set D_{λ} defined in the discussion on determinant line bundles is all of E ⊗_{ℤ} Λ, whereas for a nontrivial line bundle λ of degree zero, D_{λ} is a hypersurface in E ⊗_{ℤ} Λ. In fact, D_{λ} is a union of distinct hypersurfaces D_{λ,α}, where if λ corresponds to the point e ∈ E, then

D_{λ,α} = { ρ ∈ E ⊗_{ℤ} Λ : α(ρ) = −e }.

Each D_{λ,α} is a union of translates of abelian subvarieties of E ⊗_{ℤ} Λ. In particular, the hypersurface D_{λ,α} is smooth. The next lemma says that every component of D_{λ,α} counts with multiplicity one in the expression for −c_{1}(𝔻_{3}).

Lemma 5.5.1. For λ a nontrivial line bundle of degree zero,

c_{1}(𝔻_{3}) = − ∑_{α∈R}[D_{λ,α}].

Proof. There is a universal G-bundle Ξ_{1} over E × (E ⊗_{ℤ} Λ), which in fact arises from a universal H-bundle, which we shall also denote by Ξ_{1}. One can describe Ξ_{1} as follows. An H-bundle over E × (E ⊗_{ℤ} Λ) is the same thing as an element of Pic(E × (E ⊗_{ℤ} Λ)) ⊗ Λ. The inclusion

Pic(E × E) ⊗ Λ^{∗} → Pic(E × (E ⊗_{ℤ} Λ))

induces an inclusion

Pic(E×E)⊗Λ^{∗}⊗Λ = Pic(E×E)⊗Hom(Λ, Λ) → Pic(E×(E⊗_{ℤ}Λ))⊗Λ,

and we take the image of the element P ⊗ Id. As vector bundles over E × (E ⊗_{ℤ} Λ),

ad Ξ_{1} = O_{E×(E⊗_{ℤ}Λ)}^{r} ⊕ ⨁_{α∈R} P_{α},

where P_{α} is the pullback to E ×(E ⊗_{ℤ}Λ) of the Poincaré bundle P over E × E, via the morphism induced from α from E ⊗_{ℤ} Λ to E. Thus, by functorial properties of determinant line bundles (cf. [9, Chapter 5, Proposition 3.8]), it will suffice to show that, over E × E,

det Rπ_{2}_{∗}(P ⊗ π_{1}^{∗}λ) = O_{E}(−e),

where as before λ = O_{E}(e − p_{0}). It is clear in any case that the inverse of det Rπ_{2}_{∗}(P ⊗ π_{1}^{∗}λ) is represented by an effective divisor supported at e, and the only question is the length of R^{1}π_{2}_{∗}(P ⊗ π_{1}^{∗}λ). A standard calculation using the Grothendieck-Riemann-Roch theorem shows that this length is one (cf. [9, Chapter 7, Lemma 1.6] for the case of the trivial line bundle). □

Next we identify the divisor ∑_{α∈R} D_{λ,α}. Using the identifications

H^{2}(E ⊗_{ℤ} Λ; ℤ) = ⋀^{2} H^{1}(E ⊗_{ℤ} Λ; ℤ) = ⋀^{2} (H^{1}(E; ℤ) ⊗ Λ^{∗})

there is an inclusion

⋀^{2} H^{1}(E; ℤ) ⊗ Sym^{2} Λ^{∗} ⊆ ⋀^{2} (H^{1}(E; ℤ) ⊗ Λ^{∗}) ,

and hence, since there is a canonical identification ⋀^{2} H^{1}(E; ℤ) ≅ ℤ there is a natural inclusion of Sym^{2} Λ^{∗} in H^{2}(E ⊗_{ℤ} Λ; ℤ). Let Q ∈ Sym^{2} Λ^{∗} be the quadratic form described in Section 1 defined by

Q = ∑_{α∈R}⟨α, ·⟩⟨α, ·⟩,

and let Q̂ be the corresponding element of H^{2}(E ⊗_{ℤ} Λ; ℤ). By Lemma 1.1.2, Q = (2g)I_{0}, where I_{0} is the unique W -invariant quadratic form on Λ such that I_{0}(α^{∨}, α^{∨}) = 2.

Lemma 5.5.2. −c_{1}(𝔻_{3}) = Q̂.

Proof. By Lemma 5.5.1, it clearly suffices to show that, for every α ∈ R, we have an equality (under the obvious identifications)

[D_{λ,α}] = ⟨α, ·⟩⟨α, ·⟩.

As such, this equality is a general fact about lattices Λ: suppose that Λ is a lattice and α: Λ → ℤ is a homomorphism. There is an associated morphism E ⊗_{ℤ} Λ → E which we shall denote by e_{α}. We can define the divisor D_{α} = e_{α}^{∗}(p) for p ∈ E, as well as the cohomology class ⟨α, ·⟩⟨α, ·⟩ ∈ Sym^{2} Λ^{∗} ⊂ H^{2}(E; ℤ). To prove Lemma 5.5.2, it is enough to prove:

Claim 5.5.3. The class of the divisor D_{α} = e_{α}^{∗}(p) is equal to ⟨α, ·⟩⟨α, ·⟩.

Proof of Claim 5.5.3. First assume that α is primitive. Then after a suitable choice of a basis of Λ we can assume that Λ ≅ ℤ^{r} and that α is projection onto the last factor. In this case, D_{α} = E^{r−1} × {p} and ⟨α, ·⟩⟨α, ·⟩ = π_{r}^{∗}(e∧f), where e∧f is a positive generator of H^{2}(E; ℤ) ≅ ℤ. Clearly, equality holds in this case. If α is not primitive, we can write α = nα_{0}, where n is a nonnegative integer and α_{0} is primitive. In this case, e_{α} factors as the morphism α_{0} followed by multiplication by n on E, and so D_{α} is cohomologous to n^{2} copies of D_{α_{0}}. Likewise

⟨α, ·⟩⟨α, ·⟩ = n^{2}⟨α_{0}, ·⟩⟨α_{0}, ·⟩,

and so the claim follows from the case where α is primitive. □

5.6 The divisor on the moduli space: the non-simply connected case

We now redo the above calculations to handle the non-simply connected case. We have the moduli space M(G, c) and the corresponding determinant line bundle as before, and we use the notation of §1.5. To make the calculation, we can pull the determinant line bundle up to the space E ⊗ Λ^{w_{c}} = T_{0} × T_{0}, where there is a universal flat bundle. Let 𝔻_{3} be the class of the determinant line bundle pulled back to E ⊗ Λ^{w_{c}}. As before, we have an inclusion Sym^{2}(Λ^{w_{c}})^{∗} → H^{2}(E ⊗ Λ^{w_{c}}; ℤ). Let Q_{0} be the element (2g)(I_{0}|Λ^{w_{c}}) ∈ Sym^{2}(Λ^{w_{c}})^{∗} and let Q̂_{0} be the corresponding element of H^{2}(E ⊗ Λ^{w_{c}}; ℤ). We have the following analogue of Lemma 5.5.2:

Lemma 5.6.1. −c_{1}(𝔻_{3}) = Q̂_{0}.

Proof. The proof is similar to that in the simply connected case for Lemmas 5.5.1 and 5.5.2, and we shall be a little sketchy. Suppose that ξ is a flat K-bundle corresponding to the c-pair (x, y). Let λ be a fixed, general line bundle of degree zero on E. We compute when ξ is in the support of (R det π_{2}_{∗}(ad Ξ ⊗ π_{1}^{∗}λ))^{−1} = −𝔻_{3}. As we have seen in Lemma 4.5 of [10],

ad ξ ≅ (𝔥^{w_{c}} ⊗ O_{E}) ⊕ V′_{0} ⊕ ⨁_{o}(V_{y,o} ⊗ L_{x,o}),

where the o are the orbits for the action of w_{c} on R. Here V′_{0} is a sum of certain torsion line bundles, L_{x,o} is a line bundle with holonomy α(x) for any fixed choice of α ∈ o, and V_{y,o} is the sum of the root spaces 𝔤^{α}, α ∈ o, with the action defined by y. It follows that V_{y,o} ⊗ L_{x,o} is a direct sum of distinct line bundles of degree zero. Given a w_{c}-orbit o, let α_{o} be a choice of α ∈ o. Next we construct a universal bundle Ξ_{1} as in the simply connected case, along the lines of [10, Lemma 5.21]. The construction outlined there shows that

ad Ξ_{1} = (𝔥^{w_{c}} ⊗ O_{E×(E⊗Λ^{wc})}) ⊕ π_{1}^{∗}V′_{0} ⊕ ⨁_{o}(V_{y_{0},o} ⊗ L_{x_{0},o} ⊗ P_{o}),

where π_{1} : E × (E ⊗ Λ^{w_{c}}) → E is the projection onto the first factor, (x_{0}, y_{0}) is a fixed c-pair and, as in the simply connected case, P_{o} is the pullback to E × (E ⊗ Λ^{w_{c}}) of the Poincaré bundle P on E × E via the morphism E ⊗Λ^{w_{c}} → E induced by α_{o}. The proof of Lemma 5.5.1 then shows that the divisor 𝔻_{3} is reduced.

For a general choice of λ, there exist c_{x} and c_{y} depending only on λ such that ξ is in the support of −𝔻_{3} if and only if there exists an o such that

α_{o}(x) = c_{x};
∑_{α∈o} α(y) = c_{y}.

Thus, in cohomology, c_{1}(𝔻_{3}) corresponds to the element

∑_{o} (α_{o} ⊗ ∑_{α∈o} α) ∈ (Λ^{w_{c}})^{∗} ⊗ (Λ^{w_{c}})^{∗}.

Now every α ∈ o has the same restriction to Λ^{w_{c}} as α_{o}. Thus the above sum become

∑_{o} d_{o}(α_{o} ⊗ α_{o}),

where d_{o} is the order of o. On the other hand, we have (2g)I_{0} = ∑_{α∈R}(α ⊗ α). Clearly, the restriction of this form to Λ^{w_{c}} is ∑_{o} d_{o}(α_{o} ⊗ α_{o}), and this completes the proof of Lemma 5.6.1. □

5.7 Completion of the proof of Theorem 5.1.1

We have identified a divisor 𝔻_{1} on 𝕎ℙ(η_{0}) and computed its top intersection. We have identified a divisor 𝔻_{2} on M(G, c), or rather its pullback to a divisor 𝔻_{3} on a finite cover of M(G, c). To find the top power of 𝔻_{3}, we use the next lemma:

Lemma 5.7.1. Let J be a quadratic form on Λ^{∗}, and view J as an element of H^{2}(E ⊗_{ℤ}Λ; ℤ) via the inclusion Sym^{2} Λ^{∗} ⊂ H^{2}(E; ℤ). If the rank of Λ is r, then the top power of J is (r!) det J.

Proof. Let Ω be the 2-form corresponding to J. First suppose that J is diagonalizable with respect to some ℤ-basis for Λ, corresponding to a given isomorphism Λ ≅ ℤ^{r}. Then Ω is of the form

∑_{i=1}^{r} a_{i}[(e ∧ f) ⊗ (π_{i})^{2},

where J = ∑_{i=1}^{r} a_{i}(π_{i})^{2}, say, π_{i}: Λ ≅ ℤ^{r} → ℤ is projection onto the i^{th} factor, and e ∧ f is the positive generator for H^{2}(E; ℤ). In this case, we can write Ω = ∑_{i=1}^{r} a_{i}(e_{i} ∧ f_{i}), where e_{i} ∧ f_{i} is the generator on the i^{th} factor of E ⊗_{ℤ} Λ ≅ E^{r}. Clearly

Ω^{r} = (r!)a_{1} ⋯ a_{r} = (r!) det J.

In general, the statement makes sense for ℚ-coefficients. Note that

dim_{ℚ} ⋀^{2r} (H^{1}(E; ℚ) ⊗ (Λ^{∗} ⊗ ℚ)) = 1,

and a basis element b is given by choosing the standard positive generator for H^{2}(E ⊗_{ℤ} Λ; ℤ), together with a ℤ-basis for Λ. Changing the ℤ-basis for Λ to some new ℚ-basis changes the element b by (det X)^{2}, where X is the change of basis matrix. In particular, if X has determinant 1, then b is unchanged. Now every quadratic form on Λ can be diagonalized via a ℚ-basis such that the change of basis matrix relating the new ℚ-basis to a ℤ-basis has determinant 1. Thus we may reduce to the case where J is diagonalizable, where we have already checked the result. □

Corollary 5.7.2. ∫_{M(G,c)} c_{1}(𝔻_{2})^{r_{c}} = (−2g)^{r_{c}}n_{0}/ ∏\overline{β} g\overline{β}.

Proof. Let e be the degree of the covering T_{0} × T_{0} → M(G, c). We see by Lemma 5.5.2 and Lemma 5.6.1 that it suffices to prove that

(2g)^{r_{c}}n_{0}/ ∏_{\overline{β}} g\overline{β} = (2g)^{r_{c}}(r_{c})! det(I_{0}|Λ^{w_{c}})/e,

which we can rewrite as

e = \frac{(r_{c})!}{n_{0}} det(I_{0}|Λ^{w_{c}}) ∏_{\overline{β}} g\overline{β}.

This is exactly the statement of Theorem 1.5.2. □

To complete the proof of Theorem 5.1.1, it suffices to show that Ψ^{∗}[𝔻_{2}] = [𝔻_{1}] in H^{2}(𝕎ℙ(η_{0}), ℚ). There is a Zariski open and dense subset M_{0} of M(G, c) consisting of semistable G-bundles for which the regular representative also carries a flat connection [10, Corollary 6.2]. Let M̃_{0} be the preimage of this subset in E ⊗ Λ^{w_{c}} = M̃.

Lemma 5.7.3. Let x ∈ M̃_{0}, and let ξ_{x} be the corresponding Gbundle. Then the tautological bundle Ξ_{1} constructed above identifies an analytic neighborhood of x ∈ M̃_{0} with the local semi-universal deformation space of ξ_{x}, which is locally universal.

Proof. By the definition of M̃_{0}, there is a unique representative up to isomorphism for the S-equivalence class of ξ_{x} and it is both flat and regular. Hence the map H^{1}(E; 𝔥^{w_{c}} ⊗ O_{E}) → H^{1}(E; ad ξ_{x}) is an isomorphism. Then the tautological bundle Ξ_{1} constructed above identifies an analytic neighborhood of x ∈ M̃_{0} with the local semi-universal deformation space of ξ_{x}. Since Lie Aut ξ_{x} = 𝔥^{w_{c}}, it acts trivially on H^{1}(E; ad ξ_{x}) and thus the local semi-universal deformation of ξ_{x} is in fact locally universal. □

(See [10, Theorem 6.12] for a more general result along these lines.) We now claim:

Lemma 5.7.4. Let X be irreducible, and let \overline{Ξ} → E×X be an ad Gbundle which lifts to a G-bundle on every slice E × {x}. Let f: X → M(G, c) be the corresponding morphism, and suppose that f(X)∩M_{0} ≠ ∅. Let 𝔻_{X} = det Rπ_{2}_{∗} ad \overline{Ξ}. Then [𝔻_{X}] = f^{∗}[𝔻_{2}] in H^{2}(X; ℚ).

Proof. Choose a component X̃ of the fiber product X ×_{M(G,c)} M̃, and let f̃: X̃ → M̃ be the induced map. It suffices to show that f̃^{∗}[𝔻_{3}] = [𝔻X̃] in the obvious notation. Since both sides are algebraic, it suffices to show the following: let Σ ⊆ X̃ be an irreducible curve such that f̃(Σ) ∩ M̃_{0} ≠ ∅. Then f̃_{∗}[Σ] · 𝔻_{3} = Σ · 𝔻X̃. Now choose a line bundle λ of degree zero on E such that, if div is the canonical section of det Rπ_{2}_{∗}(ad Ξ_{1} ⊗ π_{1}^{∗}λ)^{−1}, then div ∩f̃(Σ) ⊆ M̃_{0} is a finite set of points. Let p ∈ Σ be a point such that f̃(p) ∈ div ∩f(Σ). Since f̃(p) ∈ M̃_{0}, it follows from Lemma 5.7.3 that, in an analytic neighborhood Ω of p, the bundle ad \overline{Ξ} is pulled back via f̃ from ad Ξ_{1}. It follows that the line bundle ( det Rπ_{2}_{∗}(ad \overline{Ξ} ⊗ π_{1}^{∗}λ))^{−1} and its canonical section are also pulled back via f̃. Thus f̃_{∗}[Σ] · 𝔻_{3} = Σ · 𝔻X̃ as claimed. □

We cannot apply Lemma 5.7.4 directly to the morphism

Ψ: 𝕎ℙ(η_{0}) → M(G, c),

since there is no universal ad G-bundle over 𝕎ℙ(η_{0}). However, as in the proof of Lemma 5.4.3, there is a finite cover of 𝕎ℙ(η_{0}) by a projective space ℙ^{r_{c}} and an ad G-bundle over ℙ^{r_{c}}. Let Ψ̂: ℙ^{r_{c}} → M(G, c) be the induced morphism. It follows from the proof of Lemma 5.4.3 that O_{𝕎ℙ(η_{0})}(−2gn_{c,α}/n_{0}) pulls back to O_{ℙ^{rc}}(−2gn_{c,α}/n_{0}) and that this line bundle is the determinant line bundle on ℙ^{r_{c}}. By Lemma 5.7.4, Ψ̂^{∗}[𝔻_{2}] = c_{1}(O_{ℙ^{rc}}(−2gn_{c,α}/n_{0})), and hence Ψ^{∗}[𝔻_{2}] = [𝔻_{1}]. Together with Corollary 5.7.2 and Corollary 5.4.4, this completes the proof of Theorem 5.1.1.

Appendix: Nonabelian cohomology

We discuss the general formalism, due to Grothendieck [15], for deciding when a principal bundle with structure group a quotient group can be lifted back to the full group, and for classifying all such liftings. In general, the set of liftings (suitably interpreted) is given by a nonabelian cohomology set. We then go on to discuss circumstances under which this cohomology set has a natural scheme structure, and indeed represents an appropriate functor. The arguments here are modifications of arguments due to Deligne and Babbitt-Varadarajan [4], given in a somewhat different context.

A.1 Lifting

We begin with a very general discussion of nonabelian cohomology and liftings of principal bundles. First suppose that G is a linear algebraic group and that ξ is a principal G-bundle over X, where X is a scheme or analytic space. Here it is understood that there is some topology for which ξ is locally trivial, for example, Zariski, étale, or classical, and cohomology will always be computed with respect to the appropriate topology. Throughout this paper, we have always worked with holomorphic bundles and the classical topology. One basic result is that, if X is projective, there is a natural bijection between the set of isomorphism classes of principal holomorphic G-bundle over X in the classical topology and the set of principal G-bundles over X in the étale topology. This follows from the method of proof of Prop. 20 in GAGA [22] and the fact that, if G is a closed subgroup of GL(n), then the quotient GL(n)/G is quasiprojective and admits local cross-sections in the étale topology. Thus the set of isomorphism classes of holomorphic principal G-bundles over X is canonically identified with the set of principal G-bundles over X in the étale topology. However, when we discuss representable functors below and try to put a scheme structure on various cohomology sets, it will be convenient to use the étale topology. For most of the paper, we have only considered the case where X is a smooth projective curve, and the issue of the correct topology is not important. Indeed, it follows from [24] that, if X is a smooth curve and G is linear, then a locally trivial G-bundle in the étale topology is actually Zariski locally trivial. However, we will not use this fact. One fact about cohomology which we shall need is the following: if X is a scheme and V is a coherent sheaf on X, then H^{i}(X; V) computed for the étale topology is the usual sheaf cohomology computed in the Zariski topology [18], III (3.8). Of course, by GAGA, a similar statement holds in the classical topology for the analytic sheaf associated to V provided that X is projective.

If S is a scheme on which G acts, we can form the associated scheme ξ ×_{G} S (not to be confused with fiber product). It is fibered over X and the fibers are isomorphic to S. Denote the sheaf (of sets) of cross sections (regular, holomorphic, or étale, depending on the context) of ξ ×_{G} S by S(ξ). We will usually be interested in the case where S is itself an algebraic group and where G acts on S by homomorphisms. In this case, ξ ×_{G} S is a group scheme and S(ξ) is a sheaf of (not necessarily abelian) groups. For example, if S = G and the action is by conjugation, then G(ξ) is the automorphism group scheme of ξ and its global sections are the group Aut_{G} ξ. If S is a vector space and G acts on S linearly, then S(ξ) is the vector bundle associated to the corresponding representation on G. Given an algebraic group G and a space X, the sheaf of morphisms from X to G will be denoted \underline{G}. Here \underline{G} is a sheaf in the Zariski, étale, or classical topology, depending on the context.

Suppose that G is an algebraic group and that N is a closed normal subgroup. Let H = G/N, with π: G → H the induced morphism. Let X be a scheme, and let ξ_{0} be a principal H-bundle over X. Suppose that ξ is a principal G-bundle lifting ξ_{0}. Note that G acts on N by conjugation, so that N(ξ) is defined. If moreover N is abelian, then this action of G on N factors through an action of H on N, and so N(ξ_{0}) is defined.

The group H^{0}(X; H(ξ_{0})) acts on the cohomology set H^{1}(X; N(ξ)). We have the following general result [15] or [23]:

Lemma A.1.1. With the above notation, the set of all principal Gbundles lifting ξ_{0}, or in other words the fiber of the class [ξ_{0}] ∈ H^{1}(X; \underline{H}) under the natural map H^{1}(X; \underline{G}) → H^{1}(X; \underline{H}), may be identified with H^{1}(X; N(ξ))/H^{0}(X; H(ξ_{0})). □

We will also want a slight variant of the above:

Lemma A.1.2. In the notation of Lemma A.1.1, the set of all isomorphism classes of pairs (η, φ), where η is a principal G-bundle and φ is an isomorphism from η/N to ξ_{0}, can be identified with H^{1}(X; N(ξ)).

□

Note that H^{0}(X; H(ξ_{0})) is the group of global automorphisms of ξ_{0}, and this group acts naturally on the set of pairs (η, φ) as above. In fact, this action is the same as the coboundary action of H^{0}(X; H(ξ_{0})) on H^{1}(X; N(ξ)).

Next, we ask the bundle ξ_{0} lifts to an G-bundle. For example, suppose that G = N ⋊ H is a semidirect product of N and H. Then there is a natural lift of ξ_{0} given by the choice of an inclusion of H in G. In particular, the map H^{1}(X; \underline{G}) → H^{1}(X; \underline{H}) is surjective. In this case, we can see the identification of Lemma A.1.1 quite explicitly: suppose that ξ_{0} is defined by the cocycle {h_{ij}} with respect to some open cover {U_{i}} of X. Viewing the h_{ij} as taking values in G via the inclusion, it is easy to see that, if ξ is a G-bundle lifting ξ_{0} on H, then we can assume that ξ is given by transition functions of the form h_{ij}n_{ij}. In order to be a 1-cocycle, the n_{ij} must satisfy

(h_{jk}^{−1}n_{ij}h_{jk}) n_{jk} = n_{ik},

which says that {n_{ij}} defines an element of H^{1}(X; N(ξ)). If two such cocycles, say {h_{ij}n_{ij}} and {h′_{ij}n′_{ij}} define isomorphic G-bundles, then we can first arrange by a 1-coboundary that h_{ij} = h′_{ij}. In this case, if {h_{ij}n_{ij}} and {h_{ij}n′_{ij}} are cohomologous, then there exist h_{i} such that h_{i}^{−1}h_{ij}h_{j} = h_{ij}, so that {h_{i}} ∈ H^{0}(X; H(ξ_{0})), and moreover n′_{ij} = h_{j}^{−1}n_{ij}h_{j}, so that the cocycles {n_{ij}} and {n′_{ij}} differ by the action of H^{0}(X; H(ξ_{0})) on H^{1}(X; N(ξ)). Conversely, if {n_{ij}} and {n′_{ij}} differ by an element of H^{0}(X; H(ξ_{0})), then reversing the above argument shows that the corresponding G-bundles are isomorphic.

For another example of the surjectivity of the map H^{1}(X; \underline{G}) → H^{1}(X; \underline{H}), we have:

Lemma A.1.3. Suppose that N is abelian and that H^{2}(X; N(ξ_{0})) = 0, for example suppose that N is a vector space and that dim X = 1. Then the bundle ξ_{0} lifts to a G-bundle.

Proof. This follows from [23], Corollary to Prop. 41, p. 70 or [15]. □

One trivial observation which we shall often use is the following:

Lemma A.1.4. Suppose that we are given an exact sequence

1 → U → G → H → 1,

and that ξ_{0} is a principal H-bundle over X. Suppose that U_{0} is a closed subgroup of the center of U which is normal in G. Finally suppose that ξ̃_{0} is a lift of ξ_{0} to a principal G/U_{0}-bundle. Then the sheaf U_{0}(ξ_{0}) defined by the natural action of H on U_{0} is isomorphic to U_{0}(ξ̃_{0}).

Proof. The groups H and G/U_{0} act on U_{0} by conjugation, and the action of G/U_{0} factors through the projection to H. Thus, the sheaves U_{0}(ξ_{0}) and U_{0}(ξ̃_{0}) are identified as well. □

A.2 Representability

Let G = LU, where U is a normal subgroup of G and G is a semidirect product of U and L. Let ξ_{0} be a principal L-bundle. We want to find some circumstances under which the points of the cohomology set H^{1}(X; U(ξ_{0})) can be identified with the points in an affine space. In fact, it is important to prove a much stronger statement, that a certain functor corresponding to the cohomology set is representable by an affine space. For example, if U is a vector space on which L acts linearly, then H^{1}(X; U(ξ_{0})) is an ordinary sheaf cohomology group and thus is itself a vector space, and the corresponding affine space represents a functor. We will encounter nonabelian groups U which are unipotent. Thus, U has a filtration U_{N} ⊂ ⋯ ⊂ U_{1} = U by normal, L-invariant subgroups U_{i} with the property that U_{i} is contained in the inverse image in U of the center of U/U_{i+1}. The idea then, following the general lines of [4], will be to work inductively, starting with the case where U is a vector group. The inductive step depends on the following ([4], Lemma 2.5.3):

Lemma A.2.1. Let R be a ring. Suppose that F and G are two covariant functors from the category of R-algebras to sets and that φ: F → G is a morphism of functors, with the following property:

(i) G is represented by a polynomial algebra R[x_{1}, …, x_{n}] over the ring R.

(ii) For every R-algebra S and for every object ξ ∈ G(S), the functor F_{φ,ξ} from S-algebras to sets defined by

T ↦ φ(T)^{−1}(ξ′),

where ξ′ is the element of G(T) induced by ξ, is represented by a polynomial algebra S[y_{1}, …, y_{m}] over S.

Then the functor F is represented by R[x_{1}, …, x_{n}, y_{1}, …, y_{m}].

Proof. Let S = R[x_{1}, …, x_{n}] and let ξ ∈ G(S) correspond to the identity in Hom_{R}(S, S). For an R-algebra T, if η ∈ F(T), let ξ′ = φ(T)(η). Then there exists a unique homomorphism f: S → T corresponding to ξ′ ∈ G(T), so that T is an S-algebra. Now ξ′ is the image of ξ ∈ G(S) under f_{∗}, and the element η ∈ φ(T)^{−1}(ξ′) defines a unique homomorphism S[y_{1}, …, y_{m}] → T. Thus η defines a unique homomorphism S[y_{1}, …, y_{m}] = R[x_{1}, …, x_{n}, y_{1}, …, y_{m}] → T. Conversely, suppose given a homomorphism

f: R[x_{1}, …, x_{n}, y_{1}, …, y_{m}] = S[y_{1}, …, y_{m}] → T.

Then f induces a homomorphism R[x_{1}, …, x_{n}] = S → T and thus an element ξ′ ∈ G(T) induced by ξ, and the homomorphism f then gives an element η ∈ F(T) mapping to ξ′. Clearly these are inverse constructions. It follows that F is represented by R[x_{1}, …, x_{n}, y_{1}, …, y_{m}]. □

The proof shows more generally that, if G is represented by some R-algebra S, and, for ξ the object of G(S) corresponding to the identity, if the functor from S-algebras to sets defined by

T ↦ φ(T)^{−1}(ξ′),

where ξ′ is the element of G(T) induced by ξ, is represented by an S-algebra S̃, then S̃ represents F.

We shall apply Lemma A.2.1 as follows. First let us define the functor F from ℂ-algebras to sets corresponding to the group H^{1}(X; U(ξ_{0})). For a ℂ-algebra S, let F(S) be the set of isomorphism classes of pairs (Ξ, Φ), where Ξ is a principal LU-bundle over X×Spec S and Φ : Ξ/U → π_{1}^{∗}ξ_{0} is an isomorphism from the principal L-bundle over X × Spec S induced by Ξ to the pulled back bundle π_{1}^{∗}ξ_{0}. Thus,

F(S) = H^{1}(X × Spec S; U(π_{1}^{∗}(ξ_{0})).

Theorem A.2.2. Let G = LU be an algebraic group over ℂ, where U is a closed normal unipotent subgroup of G and G is isomorphic to the semidirect product of L and U. Let X be a projective scheme, let ξ_{0} be a principal L-bundle over X, and let U(ξ_{0}) be the corresponding sheaf of unipotent groups. Let {U_{i}}_{i=1}^{N} be a decreasing filtration of U by normal L-invariant subgroups such that, for every i, U_{i}/U_{i+1} is contained in the center of U/U_{i+1}. Suppose that, for every i,

H^{0}(X; (U_{i}/U_{i+1})(ξ_{0})) = H^{2}(X; (U_{i}/U_{i+1})(ξ_{0})) = 0.

Then:

(i) The cohomology set H^{1}(X; U(ξ_{0})) has the structure of affine nspace 𝔸^{n}. More precisely, there is a G-bundle Ξ_{0} over X × 𝔸^{n} and an isomorphism Φ_{0} : Ξ_{0}/U → π_{1}^{∗}ξ_{0} such that the pair (Ξ_{0}, Φ_{0}) represents the functor F defined above.

(ii) There is a natural action of the algebraic group Aut_{L} ξ_{0} on the affine n-space 𝔸^{n} representing H^{1}(X; U(ξ_{0})). This action lifts to an action on Ξ_{0}.

Proof. We claim that the functor F is representable by an affine space. The proof is by induction on the length of the filtration {U_{i}}. If this length is zero, then U = {0} and there is nothing to prove. Suppose that the claim has been verified for every group and filtration of length less than N, and let {U_{i}} be a filtration of U length exactly N satisfying the hypotheses of Theorem A.2.2. If U_{N} is the first term in {U_{i}}, then the filtration {U_{i}/U_{N}} of U/U_{N} has length N − 1. By induction, the functor G corresponding to LU/U_{N} is representable. Moreover, there is an obvious morphism of functors φ: F → G. Now suppose that S is a ℂ-algebra and that we are given an object ξ of G(S), in other words a pair (Ξ, Φ), where Ξ is a principal LU/U_{N}-bundle over X × Spec S and Φ : Ξ/U → π_{1}^{∗}ξ_{0} is an isomorphism from the principal L-bundle over X × Spec S induced by Ξ to π_{1}^{∗}ξ_{0}. We define the functor F_{φ,ξ} on S-algebras T as follows: F_{φ,ξ}(T) consists of isomorphism classes of pairs (P, Ψ), such that P is a principal LU-bundle over X × Spec T and Ψ is an isomorphism from the principal LU/U_{N}- bundle over X × Spec T induced by P to the pullback Ξ̃ = (Id ×f)^{∗}Ξ of Ξ to X × Spec T, where f: Spec T → Spec S is the morphism corresponding to the homomorphism S → T. There is a natural map F_{φ,ξ}(T) → φ(T)^{−1}(Ξ̃, (Id ×f)^{∗}Φ) ⊆ F(T). First we claim:

Claim A.2.3. There exists a a principal LU-bundle P over X × Spec T lifting the principal LU/U_{N}-bundle Ξ̃ = (Id ×f)^{∗}Ξ over X × Spec T. In other words, for all T, F_{φ,ξ}(T) ≠ ∅.

Proof. By Lemma A.1.3, the obstruction to finding such a lift lives in the group H^{2}(X ×Spec T; U_{N}(Ξ̃)). By Lemma A.1.4, we can identify U_{N}(Ξ̃) with the sheaf U_{N}(π_{1}^{∗}ξ_{0}), which is the pullback via π_{1}^{∗} of the vector bundle V = U_{N}(ξ_{0}) on X. Now since Spec T is affine,

H^{2}(X × Spec T; U_{N}(Ξ̃)) = H^{2}(X × Spec T; π_{1}^{∗}V) = H^{2}(X; V) ⊗_{ℂ} T.

Since H^{2}(X; V) = 0 by hypothesis, we can lift Ξ̃ to a bundle P. □

Once we know that there exists one lift P as in the claim, it follows from Lemma A.1.2 that the set of all such pairs (P, Ψ) is classified by H^{1}(X × Spec T; U_{N}(P)). Next we claim that the map F_{φ,ξ}(T) → F(T) is injective, and thus identifies F_{φ,ξ}(T) with the fiber φ(T)^{−1}(Ξ̃, (Id ×f)^{∗}Φ). To see this, it follows from the general formalism of nonabelian cohomology that there is a transitive action of H^{0}(X × Spec T; U/U_{N}(P)) on the fibers of the map from F_{φ,ξ}(T) = H^{1}(X × Spec T; U_{N}(P)) to F(T) = H^{1}(X × Spec T; U(π_{1}^{∗}ξ_{0})), which identifies the fibers with the coset space

H^{0}(X × Spec T; U/U_{N}(P))/ Im H^{0}(X × Spec T; U(P)).

In our case, the unipotent group H^{0}(X × Spec T; U/U_{N}(P)) is filtered, with successive quotients contained in

H^{0}(X × Spec T; (U_{i}/U_{i−1})(π_{1}^{∗}ξ_{0})) = 0.

Hence F_{φ,ξ}(T) → F(T) is injective.

Using Lemma A.1.4 and the fact that Spec T is affine,

H^{1}(X × Spec T; U_{N}(P)) = H^{1}(X × Spec T; π_{1}^{∗}V) = H^{1}(X; V) ⊗_{ℂ} T.

If e_{1}, …, e_{n} is a basis for H^{1}(X; V), with dual basis x_{1}, …, x_{n}, this says that F_{φ,ξ}(T) ≅ Hom_{S}(S[x_{1}, …, x_{n}], T). Thus F_{φ,ξ} is representable by an affine space over Spec S. Applying Lemma A.2.1 and induction, the functor F is then representable by an affine space, in other words there exists an affine space 𝔸^{n} = Spec ℂ[x_{1}, …, x_{n}] and an object (Ξ_{0}, Φ_{0}) ∈ F(ℂ[x_{1}, …, x_{n}]) such that the couple (ℂ[x_{1}, …, x_{n}], (Ξ_{0}, Φ_{0})) represents F.

The fact that the functor F defined above is representable implies in particular that there is a universal LU-bundle over X × 𝔸^{n}, where 𝔸^{n} is the affine space representing F. A formal reduction also shows that obvious extension of F to a functor from schemes of finite type over ℂ to sets is also representable by 𝔸^{n}.

Lastly we must analyze the action of Aut_{L} ξ_{0} on H^{1}(X; U(ξ_{0})).

Proposition A.2.4. Suppose that X is proper and that L is a linear algebraic group. Then Aut_{L} ξ_{0} is also a linear algebraic group, and the natural set-theoretic action of Aut_{L} ξ_{0} on H^{1}(X; U(ξ_{0})) is an algebraic action. Moreover, this action lifts to an action on Ξ_{0}.

Proof. Let A be the functor from ℂ-algebras to groups corresponding to Aut_{L} ξ_{0}: for a ℂ-algebra S, A(S) is the group of automorphisms of the pullback of ξ_{0} to X × Spec S. Since X is proper and L is affine, A is representable by a linear algebraic group scheme Aut_{L} ξ_{0} over ℂ. To see this, first assume that L = GL_{n}. If V is the vector bundle corresponding to ξ_{0}, Aut_{L} ξ_{0} is an affine open subset of the affine space H^{0}(X; End V) and we claim that this linear algebraic group represents the associated functor. Indeed, an automorphism of π_{1}^{∗}V is the same thing as a section of π_{1}^{∗}End V, in other words a Spec S-valued point φ of H^{0}(X; End V), such that the determinant of φ is an invertible element of S, and this is the same thing as a morphism from S to the Zariski open subset of H^{0}(X; End V) consisting of elements with nonzero determinant. In general, choose an embedding of L as a closed subgroup of GL_{n} for some n, defined by polynomials f_{i}. It is then straightforward to verify that A is representable by a closed subgroup of the corresponding group scheme for GL_{n}.

If F is the functor associated to H^{1}(X; U(ξ_{0})), there is an obvious morphism of functors A × F → F: if the points of F(S) correspond to pairs (Ξ, Φ), where Φ is an isomorphism from the principal L-bundle over X × Spec S induced by Ξ to π_{1}^{∗}ξ_{0}, then the automorphisms of π_{1}^{∗}ξ_{0} act by composition with Φ. Since A and F are representable by Aut_{L} ξ_{0} and by the affine coordinate ring of H^{1}(X; U(ξ_{0})) respectively, there is a corresponding morphism Aut_{L} ξ_{0} × H^{1}(X; U(ξ_{0})) → H^{1}(X; U(ξ_{0})), which is easily checked to give an action. It again follows formally by representability that this action lifts to an action on Ξ_{0}. □

This concludes the proof of Theorem A.2.2. □

Remark A.2.5. In the hypotheses of Theorem A.2.2, suppose we only assume that, for all i, H^{2}(X; (U_{i}/U_{i+1})(ξ_{0})) = 0 and that, for all i > 1, H^{1}(X; (U_{i}/U_{i+1})(ξ_{0})) = 0. Then, in the inductive construction of the proof, the fibers F_{φ,ξ}(T) are all a single point and thus the map F_{φ,ξ}(T) → F(T) is automatically injective. Thus the proof goes through in this case as well.

There is also a relative version of Theorem A.2.2.

Theorem A.2.6. Let G = LU and the filtration {U_{i}} be as in Theorem A.2.2. Suppose that π: Z → Spec R is a flat proper morphism, and that ξ_{0} is a principal L-bundle over Z, with V_{i} = (U_{i}/U_{i+1})(ξ_{0}) the vector bundle associated to the action of L on U_{i}/U_{i+1} and the principal L-bundle ξ_{0}. Suppose that H^{2}(π^{−1}(t); V_{i}|π^{−1}(t)) = 0 for every point t ∈ X, that H^{0}(π^{−1}(t); V_{i}|π^{−1}(t)) = 0 for every point t ∈ X, and that the R-module H^{1}(Z; V_{i}) is locally free and compatible with base change, in the sense that, for every homomorphism R → S of ℂ-algebras, with corresponding morphism f: Spec S → Spec R, we have

H^{1}(Z ×_{Spec R} Spec S; f^{∗}V_{i}) ≅ H^{1}(Z; V_{i}) ⊗_{R} S.

For example, if π has relative dimension one and, for every i, dim H^{1}(π^{−1}(t); V_{i}|π^{−1}(t)) is independent of t ∈ Spec R, then H^{1}(Z; V_{i}) is locally free and compatible with base change in the above sense. Then

(i) There exists a locally trivial bundle of affine spaces 𝔸 over Spec R, such that the set of sections of 𝔸 is isomorphic to the set H^{1}(Z; U(ξ_{0})).

(ii) There exists a universal bundle Ξ over Z ×_{Spec R} 𝔸 in the obvious sense.

(iii) The automorphism group scheme A of ξ_{0} acts on the bundle 𝔸 of affine spaces over Spec R, and this action lifts to an action on Ξ.

□

Remark A.2.7. One can also replace Spec R in the above statements by a scheme of finite type over ℂ. Moreover, in case R = ℂ[t_{1}, …, t_{j}], or more generally if every vector bundle over Spec R is trivial, then the inductive proof of Theorem A.2.2 shows that we can take 𝔸 = Spec R[x_{1}, …, x_{n}].

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