The Line Bundles on Moduli Stacks

of Principal Bundles on a Curve

Indranil Biswas and Norbert Hoffmann^{1}

Received: August 6, 2009

Communicated by Ulf Rehmann

Abstract. Let G be an affine reductive algebraic group over an algebraically closed field k. We determine the Picard group of the moduli stacks of principal G–bundles on any smooth projective curve over k.

2010 Mathematics Subject Classification: 14C22, 14D20, 14H10 Keywords and Phrases: principal bundle, moduli stack, Picard group

1. Introduction

As long as moduli spaces of bundles on a smooth projective algebraic curve C have been studied, their Picard groups have attracted some interest. The first case was the coarse moduli scheme of semistable vector bundles with fixed determinant over a curve C of genus g_{C} ≥ 2. Seshadri proved that its Picard group is infinite cyclic in the coprime case [28]; Drézet and Narasimhan showed that this remains valid in the non–coprime case also [9]. The case of principal G–bundles over C for simply connected, almost simple groups G over the complex numbers has been studied intensively, motivated also by the relation to conformal field theory and the Verlinde formula [1, 12, 20]. Here Kumar and Narasimhan [19] showed that the Picard group of the coarse moduli scheme of semistable G–principal bundles over a curve C of genus g_{C} ≥ 2 embeds as a subgroup of finite index into the Picard group of the affine Grassmannian, which is canonically isomorphic to ℤ; this finite index was determined recently in [6]. Concerning the Picard group of the moduli stack M_{G} of principal G–bundles over a curve C of any genus g_{C} ≥ 0, Laszlo and Sorger [23, 30] showed that its canonical map to the Picard group ℤ of the affine Grassmannian is actually an isomorphism. Faltings [13] has generalised this result to positive characteristic, and in fact to arbitrary noetherian base scheme.

1The second author gratefully acknowledges the support of the SFB/TR 45 ”Perioden, Modulräume und Arithmetik algebraischer Varietäten”

If G is not simply connected, then the moduli stack M_{G} has several connected components which are indexed by π_{1}(G). For any d ∈ π_{1}(G), let M_{G}^{d} be the corresponding connected component of M_{G}. For semisimple, almost simple groups G over ℂ, the Picard group Pic(M_{G}^{d}) has been determined case by case by Beauville, Laszlo and Sorger [2, 22]. It is finitely generated, and its torsion part is a direct sum of 2g_{C} copies of π_{1}(G). Furthermore, its torsion–free part again embeds as a subgroup of finite index into the Picard group ℤ of the affine Grassmannian. Together with a general expression for this index, Teleman [31] also proved these statements, using topological and analytic methods. In this paper, we determine the Picard group Pic(M_{G}^{d}) for any reductive group G, working over an algebraically closed ground field k without any restriction on the characteristic of k (for all g_{C} ≥ 0). Endowing this group with a natural scheme structure, we prove that the resulting group scheme \underline{Pic}(M_{G}^{d}) over k contains, as an open subgroup, the scheme of homomorphisms from π_{1}(G) to the Jacobian J_{C}, with the quotient being a finitely generated free abelian group which we denote by NS(M_{G}^{d}) and call it the Néron–Severi group (see Theorem 5.3.1). We introduce this Néron–Severi group combinatorially in § 5.2; in particular, Proposition 5.2.11 describes it as follows: the group NS(M_{G}^{d}) contains a subgroup NS(M_{G^{ab}}) which depends only on the torus G^{ab} = G/[G , G]; the quotient is a group of Weyl–invariant symmetric bilinear forms on the root system of the semisimple part [G , G], subject to certain integrality conditions that generalise Teleman’s result in [31]. We also describe the maps of Picard groups induced by group homomorphisms G ⟶ H. An interesting effect appears for the inclusion ι_{G} : T_{G} ↪ G of a maximal torus, say for semisimple G: Here the induced map NS(M_{G}^{d}) ⟶ NS(M_{T_{G}}^{δ}) for a lift δ ∈ π_{1}(T_{G}) of d involves contracting each bilinear form in NS(M_{G}^{d}) to a linear form by means of δ (cf. Definition 4.3.5). In general, the map of Picard groups induced by a group homomorphism G ⟶ H is a combination of this effect and of more straightforward induced maps (cf. Definition 5.2.7 and Theorem 5.3.1.iv). In particular, these induced maps are different on different components of M_{G}, whereas the Picard groups Pic(M_{G}^{d}) themselves are essentially independent of d. Our proof is based on Faltings’ result in the simply connected case. To deduce the general case, the strategy of [2] and [22] is followed, meaning we “cover” the moduli stack M_{G}^{d} by a moduli stack of “twisted” bundles as in [2] under the universal cover of G, more precisely under an appropriate torus times the universal cover of the semisimple part [G , G]. To this “covering”, we apply Laszlo’s [22] method of descent for torsors under a group stack. To understand the relevant descent data, it turns out that we may restrict to the maximal torus T_{G} in G, roughly speaking because the pullback ι_{G}^{∗} is injective on the Picard groups of the moduli stacks. We briefly describe the structure of this paper. In Section 2, we recall the relevant moduli stacks and collect some basic facts. Section 3 deals with the case that G = T is a torus. Section 4 treats the “twisted” simply connected case as indicated above. In the final Section 5, we put everything together to prove our main theorem, namely Theorem 5.3.1. Each section begins with a slightly more detailed description of its contents. Our motivation for this work was to understand the existence of Poincaré families on the corresponding coarse moduli schemes, or in other words to decide whether these moduli stacks are neutral as gerbes over their coarse moduli schemes. The consequences for this question are spelled out in [4].

2. The stack of G–bundles and its Picard functor

Here we introduce the basic objects of this paper, namely the moduli stack of principal G–bundles on an algebraic curve and its Picard functor. The main purpose of this section is to fix some notation and terminology; along the way, we record a few basic facts for later use.

2.1. A Picard functor for algebraic stacks. Throughout this paper, we work over an algebraically closed field k. There is no restriction on the characteristic of k. We say that a stack X over k is algebraic if it is an Artin stack and also locally of finite type over k. Every algebraic stack X ≠ ∅ admits a point x_{0} : Spec(k) ⟶ X according to Hilbert’s Nullstellensatz. A 1–morphism Φ : X ⟶ Y of stacks is an equivalence if some 1–morphism Ψ : Y ⟶ X admits 2–isomorphisms Ψ ∘ Φ ≅ id_{X} and Φ ∘ Ψ ≅ id_{Y}. A diagram

\xymatrix{X \ar[r]^{A} \ar[d]_{Φ} & X′ \ar[d]^{Φ′} \\ Y \ar[r]^{B} & Y′}

of stacks and 1–morphisms is 2–commutative if a 2–isomorphism Φ′∘A ≅ B∘Φ is given. Such a 2–commutative diagram is 2–cartesian if the induced 1–morphism from X to the fibre product of stacks X′ ×_{Y′} Y is an equivalence. Let X and Y be algebraic stacks over k. As usual, we denote by Pic(X) the abelian group of isomorphism classes of line bundles L on X. If X ≠ ∅, then

pr_{2}^{∗} : Pic(Y) ⟶ Pic(X × Y)

is injective because x_{0}^{∗} : Pic(X × Y) ⟶ Pic(Y) is a left inverse of pr_{2}^{∗}.

Definition 2.1.1. The Picard functor \underline{Pic}(X) is the functor from schemes S of finite type over k to abelian groups that sends S to Pic(X × S)/ pr_{2}^{∗} Pic(S).

If \underline{Pic}(X) is representable, then we denote the representing scheme again by \underline{Pic}(X). If \underline{Pic}(X) is the constant sheaf given by an abelian group Λ, then we say that \underline{Pic}(X) is discrete and simply write \underline{Pic}(X) ≅ Λ. (Since the constant Zariski sheaf Λ is already an fppf sheaf, it is not necessary to specify the topology here.)

Lemma 2.1.2. Let X and Y be algebraic stacks over k with Γ(X , O_{X}) = k. i) The canonical map

pr_{2}^{∗} : Γ(Y, O_{Y}) ⟶ Γ(X × Y, O_{X ×Y})

is an isomorphism. ii) Let L ∈ Pic(X × Y) be given. If there is an atlas u : U ⟶ Y for which

u^{∗}L ∈ Pic(X × U) is trivial, then L ∈ pr_{2}^{∗} Pic(Y).

Proof. i) Since the question is local in Y, we may assume that Y = Spec(A) is an affine scheme over k. In this case, we have

Γ(X × Y, O_{X ×Y}) = Γ(X , (pr_{1})_{∗}O_{X ×Y}) = Γ(X , A ⊗_{k} O_{X}) = A = Γ(Y, O_{Y}).

ii) Choose a point x_{0} : Spec(k) ⟶ X. We claim that L is isomorphic to pr_{2}^{∗} L_{x_{0}} for L_{x_{0}} := x_{0}^{∗}L ∈ Pic(Y). More precisely there is a unique isomorphism L ≅ pr_{2}^{∗} L_{x_{0}} whose restriction to {x_{0}} × Y ≅ Y is the identity. To prove this, due to the uniqueness involved, this claim is local in Y. Hence we may assume Y = U, which by assumption means that L is trivial. In this case, statement (i) implies the claim. □

Corollary 2.1.3. For ν = 1, 2, let X_{ν} be an algebraic stack over k with Γ(X_{ν}, O_{X_{ν}}) = k. Let Φ : X_{1} ⟶ X_{2} be a 1–morphism such that the induced morphism of functors Φ^{∗} : \underline{Pic}(X_{2}) ⟶ \underline{Pic}(X_{1}) is injective. Then

Φ^{∗} : Pic(X_{2} × Y) ⟶ Pic(X_{1} × Y)

is injective for every algebraic stack Y over k.

Proof. Since Y is assumed to be locally of finite type over k, we can choose an atlas u : U ⟶ Y such that U is a disjoint union of schemes of finite type over k. Suppose that L ∈ Pic(X_{2} × Y) has trivial pullback Φ^{∗}L ∈ Pic(X_{1} × Y). Then (Φ × u)^{∗}L ∈ Pic(X_{1} × U) is also trivial. Using the assumption on Φ^{∗} it follows that u^{∗}L ∈ Pic(X_{2} × U) is trivial. Now apply Lemma 2.1.2(ii). □

We will also need the following stacky version of the standard see–saw principle.

Lemma 2.1.4. Let X and Y be two nonempty algebraic stacks over k. If \underline{Pic}(X) is discrete, and Γ(Y, O_{Y}) = k, then

pr_{1}^{∗} ⊕ pr_{2}^{∗} : \underline{Pic}(X) ⊕ \underline{Pic}(Y) ⟶ \underline{Pic}(X × Y)

is an isomorphism of functors.

Proof. Choose points x_{0} : Spec(k) ⟶ X and y_{0} : Spec(k) ⟶ Y. The morphism of functors pr_{1}^{∗} ⊕ pr_{2}^{∗} in question is injective, because

y_{0}^{∗} ⊕ x_{0}^{∗} : \underline{Pic}(X × Y) ⟶ \underline{Pic}(X) ⊕ \underline{Pic}(Y)

is a left inverse of it. Therefore, to prove the lemma it suffices to show that y_{0}^{∗} ⊕ x_{0}^{∗} is also injective. So let a scheme S of finite type over k be given, as well as a line bundle L on X × Y × S such that y_{0}^{∗}L is trivial in \underline{Pic}(X). We claim that L is isomorphic to the pullback of a line bundle on Y × S. To prove the claim, tensoring L with an appropriate line bundle on S if necessary, we may assume that y_{0}^{∗}L is trivial in Pic(X × S). By assumption, \underline{Pic}(X) ≅ Λ for some abelian group Λ. Sending any (y, s) : Spec(k) ⟶ Y × S to the isomorphism class of

(y, s)^{∗}(L) ∈ Pic(X)

we obtain a Zariski–locally constant map from the set of k–points in Y × S to Λ. This map vanishes on {y_{0}} × S, and hence it vanishes identically on Y × S because Y is connected. This means that u^{∗}L ∈ Pic(X × U) is trivial for any atlas u : U ⟶ Y × S. Now Lemma 2.1.2(ii) completes the proof of the claim. If moreover x_{0}^{∗}L is trivial in \underline{Pic}(Y), then L is even isomorphic to the pullback of a line bundle on S, and hence trivial in \underline{Pic}(X × Y). This proves the injectivity of y_{0}^{∗} ⊕ x_{0}^{∗}, and hence the lemma follows. □

2.2. Principal G–bundles over a curve. We fix an irreducible smooth projective curve C over the algebraically closed base field k. The genus of C will be denoted by g_{C}. Given a linear algebraic group G ↪ GL_{n}, we denote by

M_{G}

the moduli stack of principal G–bundles E on C. More precisely, M_{G} is given by the groupoid M_{G}(S) of principal G–bundles on S ×C for every k–scheme S. The stack M_{G} is known to be algebraic over k (see for example [23, Proposition 3.4], or [24, Théorème 4.6.2.1] together with [29, Lemma 4.8.1]). Given a morphism of linear algebraic groups φ : G ⟶ H, the extension of the structure group by φ defines a canonical 1–morphism

φ_{∗} : M_{G} ⟶ M_{H}

which more precisely sends a principal G–bundle E to the principal H–bundle

φ_{∗}E := E ×^{G} H := (E × G)/H,

following the convention that principal bundles carry a right group action. One has a canonical 2–isomorphism (ψ ∘ φ)_{∗} ≅ ψ_{∗} ∘ φ_{∗} whenever ψ : H ⟶ K is another morphism of linear algebraic groups.

Lemma 2.2.1. Suppose that the diagram of linear algebraic groups

\xymatrix{H \ar[r]^{ψ_{2}} \ar[d]_{ψ_{1}} & G_{2} \ar[d]^{φ_{2}} \\ G_{1} \ar[r]_{φ_{1}} & G}

is cartesian. Then the induced 2–commutative diagram of moduli stacks

\xymatrix{M_{H} \ar[r]^{(ψ_{2})_{∗}} \ar[d]_{(ψ_{1})_{∗}} & M_{G_{2}} \ar[d]^{(φ_{2})_{∗}} \\ M_{G_{1}} \ar[r]_{(φ_{1})_{∗}} & M_{G}}

is 2–cartesian.

Proof. The above 2–commutative diagram defines a 1–morphism

M_{H} ⟶ M_{G_{1}} ×_{M_{G}} M_{G_{2}}.

To construct an inverse, let E be a principal G–bundle on some k–scheme X. For ν = 1, 2, let E_{ν} be a principal G_{ν}–bundle on X together with an isomorphism E_{ν} ×^{G_{ν}} G ≅ E; note that the latter defines a map E_{ν} ⟶ E of schemes over X. Then G_{1} × G_{2} acts on E_{1} ×_{X} E_{2}, and the closed subgroup H ⊆ G_{1} × G_{2} preserves the closed subscheme

F := E_{1} ×_{E} E_{2} ⊆ E_{1} ×_{X} E_{2} .

This action turns F into a principal H–bundle. Thus we obtain in particular a 1–morphism

M_{G_{1}} ×_{M_{G}} M_{G_{2}} ⟶ M_{H}.

It is easy to check that this is the required inverse. □

Let Z be a closed subgroup in the center of G. Then the multiplication Z × G ⟶ G is a group homomorphism; we denote the induced 1–morphism by

⊗ : M_{Z} × M_{G} ⟶ M_{G}

and call it tensor product. In particular, tensoring with a principal Z–bundle ξ on C defines a 1–morphism which we denote by

(1)  t_{ξ} : M_{G} ⟶ M_{G}.

For commutative G, this tensor product makes M_{G} a group stack. Suppose now that G is reductive. We follow the convention that all reductive groups are smooth and connected. In particular, M_{G} is also smooth [3, 4.5.1], so its connected components and its irreducible components coincide; we denote this set of components by π_{0}(M_{G}). This set π_{0}(M_{G}) can be described as follows; cf. for example [15] or [16]. Let ι_{G} : T_{G} ↪ G be the inclusion of a maximal torus, with cocharacter group Λ_{T_{G}} := Hom(𝔾_{m}, T_{G}). Let Λ_{coroots} ⊆ Λ_{T_{G}} be the subgroup generated by the coroots of G. The Weyl group W of (G, T_{G}) acts on Λ_{T_{G}}. For every root α with corresponding coroot α^{∨}, the reflection s_{α} ∈ W acts on λ ∈ Λ_{T_{G}} by the formula s_{α}(λ) = λ − ⟨α, λ⟩α^{∨}. As the s_{α} generate W, this implies w(λ) − λ ∈ Λ_{coroots} for all w ∈ W and all λ ∈ Λ_{T_{G}}. Thus W acts trivially on Λ_{T_{G}}/Λ_{coroots}, so this quotient is, up to a canonical isomorphism, independent of the choice of T_{G}. We denote this quotient by π_{1}(G); if π_{1}(G) is trivial, then G is called simply connected. For k = ℂ, these definitions coincide with the usual notions for the topological space G(ℂ). Sending each line bundle on C to its degree we define an isomorphism π_{0}(M_{𝔾_{m}}) ⟶ ℤ, which induces an isomorphism π_{0}(M_{T_{G}}) ⟶ Λ_{T_{G}}. Its inverse, composed with the map

(ι_{G})_{∗} : π_{0}(M_{T_{G}}) ⟶ π_{0}(M_{G}) ,

is known to induce a canonical bijection

π_{1}(G) = Λ_{T_{G}}/Λ_{coroots} ⟶^{∼} π_{0}(M_{G}), cf. [10] and [16]. We denote by M_{G}^{d} the component of M_{G} given by d ∈ π_{1}(G).

Lemma 2.2.2. Let φ : G ↠ H be an epimorphism of reductive groups over k whose kernel is contained in the center of G. For each d ∈ π_{1}(G), the 1– morphism

φ_{∗} : M_{G}^{d} ⟶ M_{H}^{e}, e := φ_{∗}(d) ∈ π_{1}(H),

is faithfully flat.

Proof. Let T_{H} ⊆ H be the image of the maximal torus T_{G} ⊆ G. Let B_{G} ⊆ G be a Borel subgroup containing T_{G}; then

B_{H} := φ(B_{G}) ⊂ H

is a Borel subgroup of H containing T_{H}. For the moment, we denote

• by M_{T_{G}}^{d} ⊆ M_{T_{G}} and M_{B_{G}}^{d} ⊆ M_{B_{G}} the inverse images of M_{G}^{d} ⊆ M_{G}, and

• by M_{T_{H}}^{e} ⊆ M_{T_{H}} and M_{B_{H}}^{e} ⊆ M_{B_{H}} the inverse images of M_{H}^{e} ⊆ M_{H}. Let π_{G} : B_{G} ↠ T_{G} and π_{H} : B_{H} ↠ T_{H} denote the canonical surjections. Then

M_{B_{G}}^{d} = (π_{G})_{∗}^{−1}(M_{T_{G}}^{d})  and  M_{B_{H}}^{e} = (π_{H})_{∗}^{−1}(M_{T_{H}}^{e}),

because π_{0}(M_{T_{G}}) = π_{0}(M_{B_{G}}) and π_{0}(M_{T_{H}}) = π_{0}(M_{B_{H}}) according to the proof of [10, Proposition 5]. Applying Lemma 2.2.1 to the two cartesian squares

\xymatrix{T_{G} \ar[d]_{φ_{T}} & B_{G} \ar[d]^{φ_{B}} \ar@{^{(}->}[r] \ar@{->>}[l]_{π_{G}} & G \ar[d]^{φ} \\ T_{H} & B_{H} \ar@{^{(}->}[r] \ar@{->>}[l]_{π_{H}} & H}

of groups, we get two 2–cartesian squares

\xymatrix{M_{T_{G}}^{d} \ar[d]_{(φ_{T})_{∗}} & M_{B_{G}}^{d} \ar[l] \ar[r] \ar[d]^{(φ_{B})_{∗}} & M_{G}^{d} \ar[d]^{φ_{∗}} \\ M_{T_{H}}^{e} & M_{B_{H}}^{e} \ar[l] \ar[r] & M_{H}^{e}}

of moduli stacks. Since (φ_{T})_{∗} is faithfully flat, its pullback (φ_{B})_{∗} is so as well. This implies that φ_{∗} is also faithfully flat, as some open substack of M_{B_{H}}^{e} maps smoothly and surjectively onto M_{H}^{e}, according to [10, Propositions 1 and 2]. □

3. The case of torus

This section deals with the Picard functor of the moduli stack M_{G}^{0} in the special case where G = T is a torus. We explain in the second subsection that its description involves the character group Hom(T, 𝔾_{m}) and the Picard functor of its coarse moduli scheme, which is isomorphic to a product of copies of the Jacobian J_{C}. As a preparation, the first subsection deals with the Néron–Severi group of such products of principally polarised abelian varieties. A little care is required to keep everything functorial in T, since this functoriality will be needed later.

3.1. On principally polarised abelian varieties. Let A be an abelian variety over k, with dual abelian variety A^{∨} and Néron–Severi group

NS(A) := Pic(A)/A^{∨}(k) .

For a line bundle L on A, the standard morphism

ϕ_{L} : A ⟶ A^{∨}

sends a ∈ A(k) to τ_{a}^{∗}(L) ⊗ L^{dual} where τ_{a} : A ⟶ A is the translation by a. ϕ_{L} is a homomorphism by the theorem of the cube [27, §6]. Let a principal polarisation

ϕ : A ⟶^{∼} A^{∨}

be given. Let

c^{ϕ} : NS(A) ⟶ End A

be the injective homomorphism that sends the class of L to ϕ^{−1} ∘ ϕ_{L}. We denote by † : End A ⟶ End A the Rosati involution associated to ϕ; so by definition, it sends α : A ⟶ A to α^{†} := ϕ^{−1} ∘ α^{∨} ∘ ϕ.

Lemma 3.1.1. An endomorphism α ∈ End(A) is in the image of c^{ϕ} if and only if α^{†} = α.

Proof. If k = ℂ, this is contained in [21, Chapter 5, Proposition 2.1]. For polarisations of arbitrary degree, the analogous statement about End(A) ⊗ ℚ is shown in [27, p. 190]; its proof carries over to the situation of this lemma as follows. Let l be a prime number, l ≠ char(k), and let

e_{l} : T_{l}(A) × T_{l}(A^{∨}) ⟶ ℤ_{l}(1)

be the standard pairing between the Tate modules of A and A^{∨}, cf. [27, §20]. According to [27, §20, Theorem 2 and §23, Theorem 3], a homomorphism ψ : A ⟶ A^{∨} is of the form ψ = ϕ_{L} for some line bundle L on A if and only if

e_{l}(x, ψ_{∗}y) = −e_{l}(y, ψ_{∗}x) for all x, y ∈ T_{l}(A) .

In particular, this holds for ϕ. Hence the right hand side equals

−e_{l}(y, ψ_{∗}x) = −e_{l}(y, ϕ_{∗}ϕ_{∗}^{−1}ψ_{∗}x) = e_{l}(ϕ_{∗}^{−1}ψ_{∗}x, ϕ_{∗}y) = e_{l}(x, ψ_{∗}^{∨}(ϕ^{−1})_{∗}^{∨}ϕ_{∗}y) ,

where the last equality follows from [27, p. 186, equation (I)]. Since the pairing e_{l} is nondegenerate, it follows that ψ = ϕ_{L} holds for some L if and only if

ψ_{∗}y = ψ_{∗}^{∨}(ϕ^{−1})_{∗}^{∨}ϕ_{∗}y for all y ∈ T_{l}(A) ,

hence if and only if ψ = ψ^{∨} ∘ (ϕ^{−1})^{∨} ∘ ϕ. By definition of the Rosati involution †, the latter is equivalent to (ϕ^{−1} ∘ ψ)^{†} = ϕ^{−1} ∘ ψ. □

Let Λ be a finitely generated free abelian group. Let Λ ⊗ A denote the abelian variety over k with group of S–valued points Λ ⊗ A(S) for any k–scheme S.

Definition 3.1.2. The subgroup

Hom^{s}(Λ ⊗ Λ, End A) ⊆ Hom(Λ ⊗ Λ, End A)

consists of all b : Λ ⊗ Λ ⟶ End A with b(λ_{1} ⊗ λ_{2})^{†} = b(λ_{2} ⊗ λ_{1}) for all λ_{1}, λ_{2} ∈ Λ.

Corollary 3.1.3. There is a unique isomorphism

c_{Λ}^{ϕ} : NS(Λ ⊗ A) ⟶^{∼} Hom^{s}(Λ ⊗ Λ, End A)

which sends the class of each line bundle L on Λ ⊗ A to the linear map

c_{Λ}^{ϕ}(L) : Λ ⊗ Λ ⟶ End A

defined by sending λ_{1} ⊗ λ_{2} for λ_{1}, λ_{2} ∈ Λ to the composition

A ⟶^{λ_{1}⊗} Λ ⊗ A ⟶^{ϕ_{L}} (Λ ⊗ A)^{∨} ⟶^{(λ_{2}⊗ )^{∨}} A^{∨} ⟶^{ϕ^{−1}} A.

Proof. The uniqueness is clear. For the existence, we may then choose an isomorphism Λ ≅ ℤ^{r}; it yields an isomorphism Λ ⊗ A ≅ A^{r}. Let

ϕ^{r} = \underbrace{ϕ × ⋯ × ϕ}_{r factors} : A^{r} ⟶^{∼} (A^{∨})^{r} = (A^{r})^{∨}

be the diagonal principal polarisation on A^{r}. According to Lemma 3.1.1,

c^{ϕ^{r}} : NS(A^{r}) ⟶ End(A^{r})

is an isomorphism onto the Rosati–invariants. Under the standard isomorphisms

End(A^{r}) = Mat_{r×r}(End A) = Hom(ℤ^{r} ⊗ ℤ^{r}, End A),

the Rosati involution on End(A^{r}) corresponds to the involution (α_{ij}) ⟼ (α_{ji}^{†}) on Mat_{r×r}(End A), and hence the Rosati–invariant part of End(A^{r}) corresponds to Hom^{s}(ℤ^{r} ⊗ ℤ^{r}, End A). Thus we obtain an isomorphism

NS(Λ ⊗ A) ≅ NS(A^{r}) ⟶^{c^{ϕ^{r}}} Hom^{s}(ℤ^{r} ⊗ ℤ^{r}, End A) ≅ Hom^{s}(Λ ⊗ Λ, End A).

By construction, it maps the class of each line bundle L on Λ ⊗ A to the map c_{Λ}^{ϕ}(L) : Λ ⊗ Λ ⟶ End A prescribed above. □

3.2. Line bundles on M_{T}^{0} . Let T ≅ 𝔾_{m}^{r} be a torus over k. We will always denote by

Λ_{T} := Hom(𝔾_{m}, T)

the cocharacter lattice. We set in the previous subsection this finitely generated free abelian group and the Jacobian variety J_{C}, endowed with the principal polarisation ϕ_{Θ} : J_{C} ⟶^{∼} J_{C}^{∨} given by the autoduality of J_{C}. Recall that ϕ_{Θ} comes from a line bundle O(Θ) on J_{C} corresponding to a theta divisor Θ ⊆ J_{C}.

Definition 3.2.1. The finitely generated free abelian group

NS(M_{T}) := Hom(Λ_{T} , ℤ) ⊕ Hom^{s}(Λ_{T} ⊗ Λ_{T} , End J_{C})

is the Néron–Severi group of M_{T}.

For each finitely generated abelian group Λ, we denote by \underline{Hom}(Λ, J_{C}) the k– scheme of homomorphisms from Λ to J_{C}. If Λ ≅ ℤ^{r} × ℤ/n_{1} × ⋯ × ℤ/n_{s}, then

\underline{Hom}(Λ, J_{C}) ≅ J_{C}^{r} × J_{C}[n_{1}] × ⋯ × J_{C}[n_{s}]

where J_{C}[n] denotes the kernel of the map J_{C} ⟶ J_{C} defined by multiplication with n.

Proposition 3.2.2. i) The Picard functor \underline{Pic}(M_{T}^{0} ) is representable by

a scheme locally of finite type over k. ii) There is a canonical exact sequence of commutative group schemes

0 ⟶ \underline{Hom}(Λ_{T} , J_{C}) ⟶^{j_{T}} \underline{Pic}(M_{T}^{0} ) ⟶^{c_{T}} NS(M_{T}) ⟶ 0.

iii) Let ξ be a principal T –bundle of degree 0 ∈ Λ_{T} on C. Then the diagram

[figure: 0; \underline{Hom}(Λ_{T} , J_{C}); j_{T}; \underline{Pic}(M_{T}^{0} ); c_{T}; t_{ξ}^{∗}; NS(M_{T}); 0; 0; \underline{Hom}(Λ_{T} , J_{C}); j_{T}; \underline{Pic}(M_{T}^{0} ); c_{T}; NS(M_{T}); 0]

commutes.

Proof. Let a line bundle L on M_{T}^{0} be given. Consider the point in M_{T}^{0} given by a principal T –bundle ξ on C of degree 0 ∈ Λ_{T}. The fiber of L_{ξ} at this point is a 1-dimensional vector space L_{ξ}, endowed with a group homomorphism

w(L)_{ξ} : T = Aut(ξ) ⟶ Aut(L_{ξ}) = 𝔾_{m}

since L is a line bundle on the stack. As M_{T}^{0} is connected, the character w(L)_{ξ} is independent of ξ; we denote it by

w(L) : T ⟶ 𝔾_{m}

and call it the weight w(L) of L. Let

q : M_{T}^{0} ⟶ 𝔐_{T}^{0}

be the canonical morphism to the coarse moduli scheme 𝔐_{T}^{0} , which is an abelian variety canonically isomorphic to \underline{Hom}(Λ_{T} , J_{C}). Line bundles of weight 0 on M_{T}^{0} descend to 𝔐_{T}^{0} , so the sequence

0 ⟶ Pic(𝔐_{T}^{0} ) ⟶^{q^{∗}} Pic(M_{T}^{0} ) ⟶^{w} Hom(Λ_{T} , ℤ)

is exact. This extends for families. Since \underline{Pic}(A) is representable for any abelian variety A, the proof of (i) is now complete. Standard theory of abelian varieties and Corollary 3.1.3 together yield another short exact sequence

0 ⟶ \underline{Hom}(Λ_{T} , J_{C}) ⟶ \underline{Pic}(𝔐_{T}^{0} ) ⟶ Hom^{s}(Λ_{T} ⊗ Λ_{T} , End J_{C}) ⟶ 0 .

Given a character χ : T ⟶ 𝔾_{m} and p ∈ C(k), we denote by χ_{∗}L_{p}^{univ} the line bundle on M_{T}^{0} that associates to each T –bundle L on C the 𝔾_{m}–bundle χ_{∗}L_{p}.

Clearly, χ_{∗}L_{p}^{univ} has weight χ; in particular, it follows that w is surjective, so we get an exact sequence of discrete abelian groups

0 ⟶ Hom^{s}(Λ_{T} ⊗ Λ_{T} , End J_{C}) ⟶ \underline{Pic}(M_{T}^{0} )/\underline{Hom}(Λ_{T} , J_{C}) ⟶ Hom(Λ_{T} , ℤ) ⟶ 0 .

Since C is connected, the algebraic equivalence class of χ_{∗}L_{p}^{univ} does not depend on the choice of p; sending χ to the class of χ_{∗}L_{p}^{univ} thus defines a canonical splitting of the latter exact sequence. This proves (ii). Finally, it is standard that t_{ξ}^{∗} (see (1)) is the identity map on \underline{Pic}^{0}(𝔐_{T}^{0} ) = \underline{Hom}(Λ, J_{C}) (see [26, Proposition 9.2]), and t_{ξ}^{∗} induces the identity map on the discrete quotient \underline{Pic}(M_{T}^{0} )/\underline{Pic}^{0}(𝔐_{T}^{0} ) because ξ can be connected to the trivial T –bundle in M_{T}^{0} . □

Remark 3.2.3. The exact sequence in Proposition 3.2.2(ii) is functorial in T. More precisely, each homomorphism of tori φ : T ⟶ T′ induces a morphism of exact sequences

\xymatrix{0 \ar[r] & \underline{Hom}(Λ_{T′}, J_{C}) \ar[r]^{j_{T′}} \ar[d]^{φ^{∗}} & \underline{Pic}(M_{T′}^{0}) \ar[r]^{c_{T′}} \ar[d]^{φ^{∗}} & NS(M_{T′}) \ar[r] \ar[d]^{φ^{∗}} & 0 \\ 0 \ar[r] & \underline{Hom}(Λ_{T} , J_{C}) \ar[r]^{j_{T}} & \underline{Pic}(M_{T}^{0} ) \ar[r]^{c_{T}} & NS(M_{T}) \ar[r] & 0.}

Corollary 3.2.4. Let T_{1} and T_{2} be tori over k. Then

pr_{1}^{∗} ⊕ pr_{2}^{∗} : \underline{Pic}(M_{T_{1}}^{0}) ⊕ \underline{Pic}(M_{T_{2}}^{0}) ⟶ \underline{Pic}(M_{T_{1}×T_{2}}^{0})

is a closed immersion of commutative group schemes over k.

Proof. As before, let Λ_{T_{1}}, Λ_{T_{2}} and Λ_{T_{1}×T_{2}} denote the cocharacter lattices. Then

pr_{1}^{∗} ⊕ pr_{2}^{∗} : \underline{Hom}(Λ_{T_{1}}, J_{C}) ⊕ \underline{Hom}(Λ_{T_{2}}, J_{C}) ⟶ \underline{Hom}(Λ_{T_{1}×T_{2}}, J_{C})

is an isomorphism, and the homomorphism of discrete abelian groups

pr_{1}^{∗} ⊕ pr_{2}^{∗} : NS(M_{T_{1}}) ⊕ NS(M_{T_{2}}) ⟶ NS(M_{T_{1}×T_{2}})

is injective by Definition 3.2.1. □

4. The twisted simply connected case

Throughout most of this section, the reductive group G over k will be simply connected. Using the work of Faltings [13] on the Picard functor of M_{G}, we describe here the Picard functor of the twisted moduli stacks M_{Ĝ,L} introduced in [2]. In the case G = SL_{n}, these are moduli stacks of vector bundles with fixed determinant; their construction in general is recalled in Subsection 4.2 below. The result, proved in that subsection as Proposition 4.2.3, is essentially the same: for almost simple G, line bundles on M_{Ĝ,L} are classified by an integer, their so–called central charge. The main tool for that are as usual algebraic loop groups; what we need about them is collected in Subsection 4.1.

For later use, we need to keep track of the functoriality in G, in particular of the pullback to a maximal torus T_{G} in G. To state this more easily, we translate the central charge into a Weyl–invariant symmetric bilinear form on the cocharacter lattice of T_{G}, replacing each integer by the corresponding multiple of the basic inner product. This allows to describe the pullback to T_{G} in Proposition 4.4.7(iii). Along the way, we also consider the pullback along representations of G; these just correspond to the pullback of bilinear forms, which reformulates — and generalises to arbitrary characteristic — the usual multiplication by the Dynkin index [20]. Subsection 4.3 describes these pullback maps combinatorially in terms of the root system, and Subsection 4.4 proves that these combinatorial maps actually give the pullback of line bundles on these moduli stacks.

4.1. Loop groups. Let G be a reductive group over k. We denote

• by LG the algebraic loop group of G, meaning the group ind–scheme over k whose group of A–valued points for any k–algebra A is G(A((t))),

• by L^{+}G ⊆ LG the subgroup with A–valued points G(A[[t]]) ⊆ G(A((t))),

• and for n ≥ 1, by L^{≥n}G ⊆ L^{+}G the kernel of the reduction modulo t^{n}. Note that L^{+}G and L^{≥n}G are affine group schemes over k. The k-algebra corresponding to L^{≥n}G is the inductive limit over all N > n of the k–algebras corresponding to L^{≥n}G/L^{≥N}. A similar statement holds for L^{+}G. If X is anything defined over k, let X_{S} denote its pullback to a k–scheme S.

Lemma 4.1.1. Let S be a reduced scheme over k. For n ≥ 1, every morphism φ : (L^{≥n}G)_{S} ⟶ (𝔾_{m})_{S} of group schemes over S is trivial.

Proof. This follows from the fact that L^{≥n}G is pro–unipotent; more precisely: As S is reduced, the claim can be checked on geometric points Spec(k′) ⟶ S. Replacing k by the larger algebraically closed field k′ if necessary, we may thus assume S = Spec(k); then φ is a morphism L^{≥n}G ⟶ 𝔾_{m}. Since the k–algebra corresponding to 𝔾_{m} is finitely generated, it follows that φ factors through L^{≥n}G/L^{≥N}G for some N > n. Denoting by 𝔤 the Lie algebra of G, [8, II, §4, Theorem 3.5] provides an exact sequence

1 ⟶ L^{≥N}G ⟶ L^{≥N−1}G ⟶ 𝔤 ⟶ 1.

Thus the restriction of φ to L^{≥N−1}G induces a character on the additive group scheme underlying 𝔤. Hence this restriction has to vanish, so φ also factors through L^{≥n}G/L^{≥N−1}G. Iterating this argument shows that φ is trivial. □

Lemma 4.1.2. Suppose that the reductive group G is simply connected, in particular semisimple. If a central extension of group schemes over k

(2)  1 ⟶ 𝔾_{m} ⟶ H ⟶^{π} L^{+}G ⟶ 1

splits over L^{≥n}G for some n ≥ 1, then it splits over L^{+}G.

Proof. Let a splitting over L^{≥n}G be given, i. e. a homomorphism of group schemes σ : L^{≥n}G ⟶ H such that π ∘ σ = id. Given points h ∈ H(S) and g ∈ L^{≥n}G(S) for some k-scheme S, the two elements

h · σ(g) · h^{−1}  and  σ(π(h) · g · π(h)^{−1})

in H(S) have the same image under π, so their difference is an element in 𝔾_{m}(S), which we denote by φ_{h}(g). Sending h and g to h and φ_{h}(g) defines a morphism

φ : (L^{≥n}G)_{H} ⟶ (𝔾_{m})_{H}

of group schemes over H. Since L^{+}G/L^{≥1}G ≅ G and L^{≥N−1}G/L^{≥N} ≅ 𝔤 for N ≥ 2 are smooth, their successive extension L^{+}G/L^{≥N}G is also smooth. Thus the limit L^{+}G is reduced, so H is reduced as well. Using the previous lemma, it follows that φ is the constant map 1; in other words, σ commutes with conjugation. σ is a closed immersion because π ∘ σ is, so σ is an isomorphism onto a closed normal subgroup, and the quotient is a central extension

1 ⟶ 𝔾_{m} ⟶ H/σ(L^{≥n}G) ⟶ L^{+}G/L^{≥n}G ⟶ 1.

If n ≥ 2, then this restricts to a central extension of L^{≥n−1}G/L^{≥n}G ≅ 𝔤 by 𝔾_{m}. It can be shown that any such extension splits. (Indeed, the unipotent radical of the extension projects isomorphically to the quotient 𝔤. Note that the unipotent radical does not intersect the subgroup 𝔾_{m}, and the quotient by the subgroup generated by the unipotent radical and 𝔾_{m} is reductive, so this this reductive quotient being a quotient of 𝔤 is in fact trivial.) Therefore, the image of a section 𝔤 ⟶ H/σ(L^{≥n}G) has an inverse image in H which π maps isomorphically onto L^{≥n−1}G ⊆ L^{+}G. Hence the given central extension (2) splits over L^{≥n−1}G as well. Repeating this argument, we get a splitting over L^{≥1}G, and finally also over L^{+}G, because every central extension of L^{+}G/L^{≥1}G ≅ G by 𝔾_{m} splits as well, G being simply connected. (To prove the last assertion, for any extension G̃ of G by 𝔾_{m}, consider the commutator subgroup [G̃ , G̃] of G̃. It projects surjectively to the commutator subgroup of G which is G itself. Since [G̃ , G̃] is connected and reduced, and G is simply connected, this surjective morphism must be an isomorphism.) □

4.2. Descent from the affine Grassmannian. Let G be a reductive group over k. We denote by Gr_{G} the affine Grassmannian of G, i. e. the quotient LG/L^{+}G in the category of fppf–sheaves. Let Ô_{C,p} denote the completion of the local ring O_{C,p} of the scheme C in a point p ∈ C(k). Given a uniformising element z ∈ Ô_{C,p}, there is a standard 1–morphism

glue_{p,z} : Gr_{G} ⟶ M_{G}

that sends each coset f·L^{+}G to the trivial G–bundles over C\{p} and over Ô_{C,p}, glued by the automorphism f(z) of the trivial G–bundle over the intersection; cf. for example [23, Section 3], [13, Corollary 16], or [14, Proposition 3].

For the rest of this subsection, we assume that G is simply connected, hence semisimple. In this case, Gr_{G} is known to be an ind–scheme over k. More precisely, [13, Theorem 8] implies that Gr_{G} is an inductive limit of projective Schubert varieties over k, which are reduced and irreducible. Thus the canonical map

(3)  pr_{2}^{∗} : Γ(S, O_{S}) ⟶ Γ(Gr_{G} ×S, O_{Gr_{G} ×S})

is an isomorphism for every scheme S of finite type over k. Define the Picard functor \underline{Pic}(Gr_{G}) from schemes of finite type over k to abelian groups as in definition 2.1.1. The following theorem about it is proved in full generality in [13]. Over k = ℂ, the group Pic(Gr_{G}) is also determined in [25] as well as in [20], and Pic(M_{G}) is determined in [23] together with [30].

Theorem 4.2.1 (Faltings). Let G be simply connected and almost simple. i) \underline{Pic}(Gr_{G}) ≅ ℤ.

ii) glue_{p,z}^{∗} : \underline{Pic}(M_{G}) ⟶ \underline{Pic}(Gr_{G}) is an isomorphism of functors.

The purpose of this subsection is to carry part (ii) over to twisted moduli stacks in the sense of [2]; cf. also the first remark on page 67 of [13]. More precisely, let an exact sequence of reductive groups

(4)  1 ⟶ G ⟶ Ĝ ⟶^{dt} 𝔾_{m} ⟶ 1

be given, and a line bundle L on C. We denote by M_{Ĝ,L} the moduli stack of principal Ĝ–bundles E on C together with an isomorphism dt_{∗} E ≅ L; cf. section 2 of [2]. If for example the given exact sequence is

1 ⟶ SL_{n} ⟶ GL_{n} ⟶^{det} 𝔾_{m} ⟶ 1,

then M_{GL_{n},L} is the moduli stack of vector bundles with fixed determinant L. In general, the stack M_{Ĝ,L} comes with a 2–cartesian diagram

\xymatrix{M_{Ĝ,L} \ar[r] \ar[d] & MĜ \ar[d]^{_{dt_{∗}}} \\ Spec(k) \ar[r]^{L} & M_{𝔾_{m}}}

from which we see in particular that M_{Ĝ,L} is algebraic. It satisfies the following variant of the Drinfeld–Simpson uniformisation theorem [10, Theorem 3].

Lemma 4.2.2. Let a point p ∈ C(k) and a principal Ĝ–bundle E on C × S for some k–scheme S be given. Every trivialisation of the line bundle dt_{∗} E over (C \ {p}) × S can étale–locally in S be lifted to a trivialisation of E over (C \ {p}) × S.

Proof. The proof in [10] carries over to this situation as follows. Choose a maximal torus TĜ ⊆ Ĝ. Using [10, Theorem 1], we may assume that E comes from a principal TĜ–bundle; cf. the first paragraph in the proof of [10, Theorem 3]. Arguing as in the third paragraph of that proof, we may change this principal TĜ–bundle by the extension of 𝔾_{m}–bundles along coroots 𝔾_{m} ⟶ TĜ. Since simple coroots freely generate the kernel T_{G} of TĜ ↠ 𝔾_{m}, we can thus achieve that this TĜ–bundle is trivial over (C \ {p}) × S. Because 𝔾_{m} is a direct factor of TĜ, we can hence lift the given trivialisation to the TĜ–bundle, and hence also to E. □

Let d ∈ ℤ be the degree of L. Since dt in (4) maps the (reduced) identity component Z^{0} ≅ 𝔾_{m} of the center in Ĝ surjectively onto 𝔾_{m}, there is a Z^{0}– bundle ξ (of degree 0) on C with dt_{∗}(ξ)⊗O_{C}(dp) ≅ L; tensoring with it defines an equivalence

t_{ξ} : M_{Ĝ,O_{C}(dp)} ⟶^{∼} M_{Ĝ,L}.

Choose a homomorphism δ : 𝔾_{m} ⟶ Ĝ with dt ∘δ = d ∈ ℤ = Hom(𝔾_{m}, 𝔾_{m}). We denote by t^{δ} ∈ LĜ(k) the image of the tautological loop t ∈ L𝔾_{m}(k) under δ_{∗} : L𝔾_{m} ⟶ LĜ. The map

t^{δ} · : Gr_{G} ⟶ GrĜ

sends, for each point f in LG, the coset f · L^{+}G to the coset t^{δ}f · L^{+}Ĝ. Its composition Gr_{G} ⟶ MĜ with glue_{p,z} factors naturally through a 1–morphism

glue_{p,z,δ} : Gr_{G} ⟶ M_{Ĝ,O_{C}(dp)},

because dt_{∗} ∘(t^{δ} · ) : LG ⟶ LĜ ⟶ L𝔾_{m} is the constant map t^{d}, which via gluing yields the line bundle O_{C}(dp). Lemma 4.2.2 provides local sections of glue_{p,z,δ}. These show in particular that

glue_{p,z,δ}^{∗} : Γ(M_{Ĝ,O_{C}(dp)}, O_{M_{Ĝ,O_{C} (dp)}}) ⟶ Γ(Gr_{G}, O_{Gr_{G}})

is injective. Hence both spaces of sections contain only the constants, since Γ(Gr_{G}, O_{Gr_{G}}) = k by equation (3). Using the above equivalence t_{ξ}, this implies

(5)  Γ(M_{Ĝ,L}, O_{M_{Ĝ,L}}) = k .

Proposition 4.2.3. Let G be simply connected and almost simple. Then

glue_{p,z,δ}^{∗} : \underline{Pic}(M_{Ĝ,O_{C}(dp)}) ⟶ \underline{Pic}(Gr_{G})

is an isomorphism of functors.

Proof. LG acts on Gr_{G} by multiplication from the left. Embedding the k– algebra O_{C\p} := Γ(C \ {p}, O_{C}) into k((t)) via the Laurent development at p in the variable t = z, we denote by L_{C\p}G ⊆ LG the subgroup with A–valued points G(A ⊗_{k} O_{C\p}) ⊆ G(A((t))) for any k–algebra A. Consider the stack quotient L_{C\p}G\ Gr_{G}. The map glue_{p,z} descends to an equivalence

L_{C\p}G\ Gr_{G} ⟶^{∼} M_{G}

because the action of L_{C\p}G on Gr_{G} corresponds to changing trivialisations over C \ {p}; cf. for example [23, Theorem 1.3] or [13, Corollary 16]. More generally, consider the conjugate

L_{C\p}^{δ}G := t^{−δ} · L_{C\p}G · t^{δ} ⊆ LĜ, which is actually contained in LG since LG is normal in LĜ. Using Lemma 4.2.2, we see that the map glue_{p,z,δ} descends to an equivalence

L_{C\p}^{δ}G\ Gr_{G} ⟶^{∼} M_{Ĝ,O_{C}(dp)},

because the action of L_{C\p}^{δ}G on Gr_{G} again corresponds to changing trivialisations over C \ {p}. Let S be a scheme of finite type over k. Each line bundle on S × M_{Ĝ,L} with trivial pullback to S × Gr_{G} comes from a character (L_{C\p}^{δ}G)_{S} ⟶ (𝔾_{m})_{S}, since the map (3) is bijective. But L_{C\p}^{δ}G is isomorphic to L_{C\p}G, and every character (L_{C\p}G)_{S} ⟶ (𝔾_{m})_{S} is trivial according to [13, p. 66f.]. This already shows that the morphism of Picard functors glue_{p,z,δ}^{∗} is injective. The action of LG on Gr_{G} induces the trivial action on \underline{Pic}(Gr_{G}) ≅ ℤ, for example because it preserves ampleness, or alternatively because LG is connected. Let a line bundle L on Gr_{G} be given. We denote by Mum_{LG}(L) the Mumford group. So Mum_{LG}(L) is the functor from schemes of finite type over k to groups that sends S to the group of pairs (f, g) consisting of an element f ∈ LG(S) and an isomorphism g : f^{∗}L_{S} ⟶^{∼} L_{S} of line bundles on Gr_{G} ×S. If f = 1, then g ∈ 𝔾_{m}(S) due to the bijectivity of (3), while for arbitrary f ∈ LG(S), the line bundles L_{S} and f^{∗}L_{S} have the same image in \underline{Pic}(Gr_{G})(S), implying that L_{S} and f^{∗}L_{S} are Zariski–locally in S isomorphic. Consequently, we have a short exact sequence of sheaves in the Zariski topology

(6)  1 ⟶ 𝔾_{m} ⟶ Mum_{LG}(L) ⟶^{q} LG ⟶ 1 .

This central extension splits over L^{+}G ⊆ LG, because the restricted action of L^{+}G on Gr_{G} has a fixed point. We have to show that it also splits over L_{C\p}^{δ}G ⊆ LG. Note that L_{C\p}^{δ}G = γ(L_{C\p}G) for the automorphism γ of LG given by conjugation with t^{δ}. Hence it is equivalent to show that the central extension

(7)  1 ⟶ 𝔾_{m} ⟶ Mum_{LG}(L) ⟶^{γ^{−1}∘q} LG ⟶ 1

splits over L_{C\p}G. We know already that it splits over γ^{−1}(L^{+}G), in particular over L^{≥n}G for some n ≥ 1. Thus it also splits over L^{+}G, due to Lemma 4.1.2. Hence it comes from a line bundle on LG/L^{+}G = Gr_{G} (whose associated 𝔾_{m}–bundle has total space Mum_{LG}(L)/L^{+}G, where L^{+}G acts from the right via the splitting). According to Theorem 4.2.1(ii), this line bundle admits a L_{C\p}G–linearisation, and hence the extension (7) splits indeed over L_{C\p}G. Thus the extension (6) splits over L_{C\p}^{δ}G, so L admits an L_{C\p}^{δ}G–linearisation and consequently descends to M_{Ĝ,O_{C}(dp)}. This proves that glue_{p,z,δ}^{∗} is surjective as a homomorphism of Picard groups. Hence it is also surjective as a morphism of Picard functors, because \underline{Pic}(Gr_{G}) ≅ ℤ is discrete by Theorem 4.2.1(i). □

Remark 4.2.4. Put G^{ad} := G/Z, where Z ⊆ G denotes the center. Given a representation ρ : G^{ad} ⟶ SL(V), we denote its compositions with the canonical epimorphisms G ↠ G^{ad} and Ĝ ↠ G^{ad} also by ρ. Then the diagram

\xymatrix{\underline{Pic}(M_{SL(V)}) \ar[d]_{ρ^{∗}} \ar[r]^{^{glue{∗ \atop p,z}}} & \underline{Pic}(Gr_{SL(V)}) \ar[d]^{ρ^{∗}} \\ \underline{Pic}(M_{Ĝ,L}) \ar[r]^{(t_{ξ}∘glue_{p,z,δ})^{∗}} & \underline{Pic}(Gr_{G})}

commutes.

Proof. Let t^{ρ∘δ} ∈ L SL(V) denote the image of the canonical loop t ∈ L𝔾_{m} under the composition ρ∘δ : 𝔾_{m} ⟶ SL(V). Then the left part of the diagram

[figure: Gr_{G}; ^{glue^{p,z,δ}}; ρ_{∗}; t^{δ}·; M_{Ĝ,O_{C}(dp)}; t_{ξ}; M_{Ĝ,L}; _{Gr_{SL(V)}}; t^{ρ∘δ}·; _{GrĜ}; _{glue_{p,z}}; ρ_{∗}; MĜ; ρ_{∗}; t_{ξ}; MĜ; ρ_{∗}; _{Gr_{SL(V)}}; _{glue_{p,z}}; M_{SL(V)}; M_{SL(V)}]

commutes. The four remaining squares are 2–commutative by construction of the 1–morphisms glue_{p,z,δ}, glue_{p,z} and t_{ξ}. Applying \underline{Pic} to the exterior pentagon yields the required commutative square, as L SL(V) acts trivially on \underline{Pic}(Gr_{SL(V)}). □

4.3. Néron–Severi groups NS(M_{G}) for simply connected G. Let G be a reductive group over k; later in this subsection, we will assume that G is simply connected. Choose a maximal torus T_{G} ⊆ G, and let

(8)  Hom(Λ_{T_{G}} ⊗ Λ_{T_{G}}, ℤ)^{W}

denote the abelian group of bilinear forms b : Λ_{T_{G}} ⊗ Λ_{T_{G}} ⟶ ℤ that are invariant under the Weyl group W = W_{G} of (G, T_{G}). Up to a canonical isomorphism, the group (8) does not depend on the choice of T_{G}. More precisely, let T′_{G} ⊆ G be another maximal torus; then the conjugation γ_{g} : G ⟶ G with some g ∈ G(k) provides an isomorphism from T_{G} to T′_{G}, and the induced isomorphism from Hom(Λ_{T′_{G}} ⊗ Λ_{T′_{G}}, ℤ)^{W} to Hom(Λ_{T_{G}} ⊗ Λ_{T_{G}}, ℤ)^{W} does not depend on the choice of g. The group (8) is also functorial in G. More precisely, let φ : G ⟶ H be a homomorphism of reductive groups over k. Choose a maximal torus T_{H} ⊆ H containing φ(T_{G}).

Lemma 4.3.1. Let T′_{G} ⊆ G be another maximal torus, and let T′_{H} ⊆ H be a maximal torus containing φ(T′_{G}). For every g ∈ G(k) with T′_{G} = γ_{g}(T_{G}), there is an h ∈ H(k) with T′_{H} = γ_{h}(T_{H}) such that the following diagram commutes:

\xymatrix{T_{G} \ar[r]^{γ_{g}} \ar[d]^{φ} & T′_{G} \ar[d]^{φ} \\ T_{H} \ar[r]^{γ_{h}} & T′_{H}}

Proof. The diagram

[figure: T_{G}; φ; T_{G}; γ_{g}; φ; T′_{G}; φ; T_{H}; γ_{φ(g)}^{−1}(T′_{H}); γ_{φ(g)}; T′_{H}]

allows us to assume T′_{G} = T_{G} and g = 1 without loss of generality. Then T_{H} and T′_{H} are maximal tori in the centraliser of φ(T_{G}), which is reductive according to [17, 26.2. Corollary A]. Thus T′_{H} = γ_{h}(T_{H}) for an appropriate k–point h of this centraliser, and γ_{h} ∘ φ = φ on T_{G} by definition of the centraliser. □

Applying the lemma with T′_{G} = T_{G} and T′_{H} = T_{H}, we see that the pullback along φ_{∗} : Λ_{T_{G}} ⟶ Λ_{T_{H}} of a W_{H}–invariant form Λ_{T_{H}} ⊗ Λ_{T_{H}} ⟶ ℤ is W_{G}– invariant, so we get an induced map

(9)  φ^{∗} : Hom(Λ_{T_{H}} ⊗ Λ_{T_{H}} , ℤ)^{W_{H}} ⟶ Hom(Λ_{T_{G}} ⊗ Λ_{T_{G}}, ℤ)^{W_{G}}

which does not depend on the choice of T_{G} and T_{H} by the above lemma again. For the rest of this subsection, we assume that G and H are simply connected.

Definition 4.3.2. i) The Néron–Severi group NS(M_{G}) is the subgroup

NS(M_{G}) ⊆ Hom(Λ_{T_{G}} ⊗ Λ_{T_{G}}, ℤ)^{W}

of symmetric forms b : Λ_{T_{G}} ⊗ Λ_{T_{G}} ⟶ ℤ with b(λ ⊗ λ) ∈ 2ℤ for all λ ∈ Λ_{T_{G}}.

ii) Given a homomorphism φ : G ⟶ H, we denote by

φ^{∗} : NS(M_{H}) ⟶ NS(M_{G})

the restriction of the induced map φ^{∗} in (9).

Remarks 4.3.3. i) If G = G_{1} × G_{2} for simply connected groups G_{1} and G_{2}, then

NS(M_{G}) = NS(M_{G_{1}}) ⊕ NS(M_{G_{2}}),

since each element of Hom(Λ_{T_{G}} ⊗Λ_{T_{G}}, ℤ)^{W_{G}} vanishes on Λ_{T_{G_{1}}} ⊗Λ_{T_{G_{2}}} +Λ_{T_{G_{2}}} ⊗ Λ_{T_{G_{1}}} . ii) If on the other hand G is almost simple, then

NS(M_{G}) = ℤ · b_{G}

where the basic inner product b_{G} is the unique element of NS(M_{G}) that satisfies b_{G}(α^{∨}, α^{∨}) = 2 for all short coroots α^{∨} ∈ Λ_{T_{G}} of G. iii) Let G and H be almost simple. The Dynkin index d_{φ} ∈ ℤ of a homomorphism φ : G ⟶ H is defined by φ^{∗}(b_{H}) = d_{φ} · b_{G}, cf. [11, §2]. If φ is nontrivial, then d_{φ} > 0, since b_{G} and b_{H} are positive definite.

Let Z ⊆ G be the center. Then G^{ad} := G/Z contains T_{G^{ad}} := T_{G}/Z as a maximal torus, with cocharacter lattice Λ_{T_{G^{ad}}} ⊆ Λ_{T_{G}} ⊗ ℚ. We say that a homomorphism l : Λ ⟶ Λ′ between finitely generated free abelian groups Λ and Λ′ is integral on a subgroup Λ̃ ⊆ Λ ⊗ ℚ if its restriction to Λ ∩ Λ̃ admits a linear extension l̃: Λ̃ ⟶ Λ′. By abuse of language, we will not distinguish between l and its unique linear extension l̃.

Lemma 4.3.4. Every element b : Λ_{T_{G}} ⊗ Λ_{T_{G}} ⟶ ℤ of NS(M_{G}) is integral on Λ_{T_{G^{ad}}} ⊗ Λ_{T_{G}} and on Λ_{T_{G}} ⊗ Λ_{T_{G^{ad}}} .

Proof. Let α : Λ_{T_{G}} ⊗ ℚ ⟶ ℚ be a root of G, with corresponding coroot α^{∨} ∈ Λ_{T_{G}}. Lemme 2 in [5, Chapitre VI, §1] implies the formula

b(λ ⊗ α^{∨}) = α(λ) · b(α^{∨} ⊗ α^{∨})/2

for all λ ∈ Λ_{T_{G}}. Thus b( ⊗ α^{∨}) : Λ_{T_{G}} ⟶ ℤ is an integer multiple of α; hence it is integral on Λ_{T_{G^{ad}}} , the largest subgroup of Λ_{T_{G}} ⊗ ℚ on which all roots are integral. But the coroots α^{∨} generate Λ_{T_{G}}, as G is simply connected. □

Now let ι_{G} : T_{G} ↪ G denote the inclusion of the chosen maximal torus.

Definition 4.3.5. Given δ ∈ Λ_{T_{G^{ad}}} , the homomorphism

(ι_{G})^{NS,δ} : NS(M_{G}) ⟶ NS(M_{T_{G}})

sends b : Λ_{T_{G}} ⊗ Λ_{T_{G}} ⟶ ℤ to

b(−δ ⊗ ) : Λ_{T_{G}} ⟶ ℤ  and  id_{J_{C}} ·b : Λ_{T_{G}} ⊗ Λ_{T_{G}} ⟶ End J_{C}.

This map (ι_{G})^{NS,δ} is injective if g_{C} ≥ 1, because all multiples of id_{J_{C}} are then nonzero in End J_{C}. If g_{C} = 0, then End J_{C} = 0, but we still have the following

Lemma 4.3.6. Every coset d ∈ Λ_{T_{G^{ad}}} /Λ_{T_{G}} = π_{1}(G^{ad}) admits a lift δ ∈ Λ_{T_{G^{ad}}} such that the map (ι_{G})^{NS,δ} : NS(M_{G}) ⟶ NS(M_{T_{G}}) is injective.

Proof. Using Remark 4.3.3, we may assume that G is almost simple. In this case, (ι_{G})^{NS,δ} is injective whenever δ ≠ 0, because NS(M_{G}) is cyclic and its generator b_{G} : Λ_{T_{G}} ⊗ Λ_{T_{G}} ⟶ ℤ is as a bilinear form nondegenerate. □

Remark 4.3.7. Given φ : G ⟶ H, let ι_{H} : T_{H} ↪ H be a maximal torus with φ(T_{G}) ⊆ T_{H}. If δ ∈ Λ_{T_{G}}, or if more generally δ ∈ Λ_{T_{G^{ad}}} is mapped to Λ_{H^{ad}} by φ_{∗} : Λ_{T_{G}} ⊗ ℚ ⟶ Λ_{T_{H}} ⊗ ℚ, then the following diagram commutes:

\xymatrix{NS(M_{H}) \ar[r]^{(ι_{H})^{NS,φ∗δ}} \ar[d]^{φ_{∗}} & NS(M_{T_{H}}) \ar[d]^{φ_{∗}} \\ NS(M_{G}) \ar[r]^{(ι_{G})^{NS,δ}} & NS(M_{T_{G}})}

4.4. The pullback to torus bundles. Let L^{det} = L_{n}^{det} be determinant of cohomology line bundle [18] on M_{GL_{n}}, whose fibre at a vector bundle E on C is det H^{∗}(E) = det H^{0}(E) ⊗ det H^{1}(E)^{dual}.

Lemma 4.4.1. Let ξ be a line bundle of degree d on C. Then the composition

Pic(M_{𝔾_{m}}) ⟶^{t_{ξ}^{∗}} Pic(M_{𝔾_{m}}^{0}) ⟶^{c_{𝔾m}} NS(M_{𝔾_{m}}) = ℤ ⊕ End_{J_{C}}

maps L^{det} to 1 − g_{C} + d ∈ ℤ and − id_{J_{C}} ∈ End_{J_{C}}.

Proof. For any line bundle L on C and any point p ∈ C(k), we have a canonical exact sequence

0 ⟶ L(−p) ⟶ L ⟶ L_{p} ⟶ 0

of coherent sheaves on C. Varying L and taking the determinant of cohomology, we see that the two line bundles L^{det} and t_{O(−p)}^{∗}L^{det} on M_{𝔾_{m}}^{0} have the same image in the second summand End J_{C} of NS(M_{𝔾_{m}}). Thus the image of t_{ξ}^{∗}L^{det} in End J_{C} does not depend on ξ; this image is − id_{J_{C}} because the principal polarisation ϕ_{Θ} : J_{C} ⟶ J_{C}^{∨} is essentially given by the dual of the line bundle L^{det}. The weight of t_{ξ}^{∗}L^{det} at a line bundle L of degree 0 on C is the Euler characteristic of L ⊗ ξ, which is indeed 1 − g_{C} + d by Riemann–Roch theorem. □

Let ι : T_{SL_{n}} ↪ SL_{n} be the inclusion of the maximal torus T_{SL_{n}} := 𝔾_{m}^{n} ∩ SL_{n}, where 𝔾_{m}^{n} ⊆ GL_{n} as diagonal matrices. Then the cocharacter lattice Λ_{T_{SL_{n}}} is the group of all d = (d_{1}, …, d_{n}) ∈ ℤ^{n} with d_{1} + ⋯ + d_{n} = 0. The basic inner product b_{SL_{n}} : Λ_{T_{SL_{n}}} ⊗ Λ_{T_{SL_{n}}} ⟶ ℤ is the restriction of the standard scalar product on ℤ^{n}.

Corollary 4.4.2. Let ξ be a principal T_{SL_{n}}–bundle of degree d ∈ Λ_{T_{SL_{n}}} on C. Then the composition

Pic(M_{SL_{n}}) ⟶^{ι^{∗}} Pic(M_{T_{SL_{n}}}) ⟶^{t_{ξ}^{∗}} Pic(M_{T_{SL_{n}}}^{0} ) ⟶^{c_{T_{_{SL}_{n}}}} NS(M_{T_{SL_{n}}})

maps L^{det} to b_{SL_{n}}(d ⊗ ) : Λ_{T_{SL_{n}}} ⟶ ℤ and − id_{J_{C}} ·b_{SL_{n}} : Λ_{T_{SL_{n}}} ⊗ Λ_{T_{SL_{n}}} ⟶ End J_{C}.

Proof. Since the determinant of cohomology takes direct sums to tensor products, the pullback of L_{n}^{det} to M_{𝔾_{m}^{n}} is isomorphic to pr_{1}^{∗} L_{1}^{det} ⊗ ⋯ ⊗ pr_{n}^{∗} L_{1}^{det}, where pr_{ν} : 𝔾_{m}^{n} ↠ 𝔾_{m} is the projection onto the νth factor. Now use the previous lemma to compute the image of L_{n}^{det} in NS(M_{𝔾_{m}^{n}}) and then restrict to NS(M_{T_{SL_{n}}}). □

Corollary 4.4.3. If ρ : SL_{2} ⟶ SL(V) has Dynkin index d_{ρ}, then the pullback ρ^{∗} : Pic(M_{SL(V)}) ⟶ Pic(M_{SL_{2}}) maps L^{det} to (L_{2}^{det})^{⊗d_{ρ}}.

Proof. Let ι : T_{SL(V)} ↪ SL(V) be the inclusion of a maximal torus that contains the image of the standard torus T_{SL_{2}} ↪ SL_{2}. The diagram

\xymatrix{Pic(M_{SL(V)}) \ar[r]^{ι^{∗}} \ar[d]^{ρ^{∗}} & Pic(M_{T_{SL(V )}}) \ar[r]^{t_{ρ∗(ξ)}^{∗}} \ar[d]^{ρ^{∗}} & Pic(M_{T_{SL(V )}}^{0}) \ar[r]^{c_{T_{SL(V )}}} \ar[d]^{ρ^{∗}} & NS(M_{T_{SL(V )}}) \ar[d]^{ρ^{∗}} \\ Pic(M_{SL_{2}}) \ar[r]^{ι^{∗}} & Pic(M_{T_{SL_{2}}} ) \ar[r]^{t_{ξ}^{∗}} & Pic(M_{T_{SL_{2}}}^{0} ) \ar[r]^{c_{T_{_{SL}_{2}}}} & NS(M_{T_{SL_{2}}} )}

commutes for each principal T_{SL_{2}}–bundle ξ on C. We choose ξ in such a way that deg(ξ) ∈ Λ_{T_{SL_{2}}} ≅ ℤ is nonzero if g_{C} = 0. Then the composition

c_{T_{SL_{2}}} ∘ t_{ξ}^{∗} ∘ ι^{∗} : Pic(M_{SL_{2}}) ⟶ NS(M_{T_{SL_{2}}} )

of the lower row is injective according to Theorem 4.2.1 and Corollary 4.4.2. The latter moreover implies that the two elements ρ^{∗}(L^{det}) and (L_{2}^{det})^{⊗d_{ρ}} in Pic(M_{SL_{2}}) have the same image in NS(M_{T_{SL_{2}}} ). □

Now suppose that the reductive group G is simply connected and almost simple. We denote by O_{Gr_{G}}(1) the unique generator of Pic(Gr_{G}) that is ample on every closed subscheme, and by O_{Gr_{G}}(n) its nth tensor power for n ∈ ℤ. Over k = ℂ, the following is proved by a different method in section 5 of [20].

Proposition 4.4.4 (Kumar-Narasimhan-Ramanathan). If ρ : G ⟶ SL(V) has Dynkin index d_{ρ}, then ρ^{∗} : Pic(Gr_{SL(V)}) ⟶ Pic(Gr_{G}) maps O(1) to O_{Gr_{G}}(d_{ρ}).

Proof. Let φ : SL_{2} ⟶ G be given by a short coroot. Then d_{φ} = 1 by definition, and [13] implies that φ^{∗} : Pic(Gr_{G}) ⟶ Pic(Gr_{SL_{2}}) maps O(1) to O(1), for example because φ^{∗} : Pic(M_{G}) ⟶ Pic(M_{SL_{2}}) preserves central charges according to their definition [13, p. 59]. Hence it suffices to prove the claim for ρ ∘ φ instead of ρ. This case follows from Corollary 4.4.3, since glue_{p,z}^{∗}(L_{n}^{det}) ≅ O_{Gr_{_{SL}_{n}}} (−1). □

As in Subsection 4.2, we assume given an exact sequence of reductive groups

1 ⟶ G ⟶ Ĝ ⟶^{dt} 𝔾_{m} ⟶ 1

with G simply connected, and a line bundle L on C.

Corollary 4.4.5. Suppose that G is almost simple. Then the isomorphism

(t_{ξ} ∘ glue_{p,z,δ})^{∗} : \underline{Pic}(M_{Ĝ,L}) ⟶^{∼} \underline{Pic}(Gr_{G})

constructed in Subsection 4.2 does not depend on the choice of p, z, ξ or δ.

We say that a line bundle on M_{Ĝ,L} has central charge n ∈ ℤ if this isomorphism maps it to O_{Gr_{G}}(n); this is consistent with the standard central charge of line bundles on M_{G}, as defined for example in [13].

Proof. If ρ : G ⟶ SL(V) is a nontrivial representation, then d_{ρ} > 0, as explained in Remark 4.3.3(iii). Using Proposition 4.4.4, this implies that

ρ^{∗} : Pic(Gr_{SL(V)}) ⟶ Pic(Gr_{G})

is injective. Due to Remark 4.2.4, it thus suffices to check that

glue_{p,z}^{∗} : \underline{Pic}(M_{SL(V)}) ⟶^{∼} \underline{Pic}(Gr_{SL(V)})

does not depend on p or z. This is clear, since it maps L^{det} to O_{Gr_{SL(V )}}(−1). □

The chosen maximal torus ι_{G} : T_{G} ↪ G induces maximal tori ιĜ : TĜ ↪ Ĝ and ι_{G^{ad}} : T_{G^{ad}} ↪ G^{ad} compatible with the canonical maps G ↪ Ĝ ↠ G^{ad}. Given a principal TĜ–bundle ξ̂ on C and an isomorphism dt_{∗} ξ̂ ≅ L, the composition

M_{T_{G}}^{0} ⟶^{tξ̂} M_{TĜ} ⟶^{(ιĜ)_{∗}} MĜ

factors naturally through a 1–morphism

(10)  ιξ̂: M_{T_{G}}^{0} ⟶ M_{Ĝ,L}.

Remark 4.4.6. Given a representation ρ : G^{ad} ⟶ SL(V), let ι : T_{SL(V)} ↪ SL(V) be a maximal torus containing ρ(T_{G^{ad}}). Then the diagram

\xymatrix{M_{T_{G}}^{0} \ar[rr]^{ιξ̂} \ar[d]^{ρ_{∗}} &  & M_{Ĝ,L} \ar[d]^{ρ_{∗}} \\ M_{T_{SL(V )}}^{0} \ar[r]^{t_{ρ∗ξ̂}} & M_{T_{SL(V )}} \ar[r]^{ι_{∗}} & M_{SL(V)}}

is 2–commutative, by construction of ιξ̂.

Proposition 4.4.7. i) Γ(M_{Ĝ,L}, O_{M_{Ĝ,L}}) = k.

ii) There is a canonical isomorphism

c_{G} : \underline{Pic}(M_{Ĝ,L}) ⟶^{∼} NS(M_{G}).

iii) For all choices of ι_{G} : T_{G} ↪ G and of ξ̂, the diagram

\xymatrix{\underline{Pic}(M_{Ĝ,L}) \ar[r]^{ι∗ ; ξ̂} \ar[d]^{c_{G}} & \underline{Pic}(M_{T_{G}}^{0}) \ar[d]^{c_{T_{G}}} \\ NS(M_{G}) \ar[r]^{(ι_{G})^{NS,δ̄}} & NS(M_{T_{G}})}

commutes; here δ̄ ∈ Λ_{T_{G^{ad}}} denotes the image of δ̂:= deg ξ̂ ∈ Λ_{TĜ}.

Proof. We start with the special case that G is almost simple. Here part (i) of the proposition is just equation (5) from Subsection 4.2. We let c_{G} send the line bundle of central charge 1 to the basic inner product b_{G} ∈ NS(M_{G}). Due to Theorem 4.2.1(i), Proposition 4.2.3, Corollary 4.4.5 and Remark 4.3.3(ii), this defines a canonical isomorphism, and hence proves (ii). To see that the diagram in (iii) then commutes, we choose a nontrivial representation ρ : G^{ad} ⟶ SL(V). We note the functorialities, with respect to ρ, according to Remark 4.4.6, Remark 4.2.4, Proposition 4.4.4, Remark 4.3.7 and Remark 3.2.3. In view of these, comparing Corollary 4.4.2 and Definition 4.3.5 shows that the two images of ρ^{∗}L^{det} ∈ Pic(M_{Ĝ,L}) in NS(M_{T_{G}}) coincide. Since the former generates a subgroup of finite index and the latter is torsionfree, the diagram in (iii) commutes. For the general case, we use the unique decomposition

G = G_{1} × ⋯ × G_{r}

into simply connected and almost simple factors G_{i}. As Ĝ is generated by its center and G, every normal subgroup in G is still normal in Ĝ. Let Ĝ_{i} denote the quotient of Ĝ modulo the closed normal subgroup ∏_{j≠i} G_{j}; then

[figure: 0; G; _{pr_{i}}; Ĝ; 𝔾_{m}; 0; 0; G_{i}; Ĝ_{i}; _{dt_{i}}; 𝔾_{m}; 0]

is a morphism of short exact sequences. Since the resulting diagram

\xymatrix{Ĝ \ar[d]_{dt} \ar[r] & ∏_{i} Ĝ_{i} \ar[d]^{_{∏ dt_{i}}} \\ 𝔾_{m} \ar[r]^{diag} & 𝔾_{m}^{r}}

is cartesian, it induces an equivalence of moduli stacks

(11)  M_{Ĝ,L} ⟶^{∼} M_{Ĝ_{1},L} × ⋯ × M_{Ĝ_{r},L}

due to Lemma 2.2.1. We note that equation (5), Lemma 2.1.2(i), Lemma 2.1.4, Remark 4.3.3(i) and Corollary 3.2.4 ensure that various constructions are compatible with the products in (11). Therefore, the general case follows from the already treated almost simple case. □

5. The reductive case

In this section, we finally describe the Picard functor \underline{Pic}(M_{G}^{d}) for any reductive group G over k and any d ∈ π_{1}(G). We denote

• by ζ : Z^{0} ↪ G the (reduced) identity component of the center Z ⊆ G, and

• by π : G̃ ⟶ G the universal cover of G′ := [G , G] ⊆ G. Our strategy is to descend along the central isogeny

ζ · π : Z^{0} × G̃ ⟶ G,

applying the previous two sections to Z^{0} and to G̃, respectively. The 1– morphism of moduli stacks given by such a central isogeny is a torsor under a group stack; Subsection 5.1 explains descent of line bundles along such torsors, generalising the method introduced by Laszlo [22] for quotients of SL_{n}. In Subsection 5.2, we define combinatorially what will be the discrete torsionfree part of \underline{Pic}(M_{G}^{d}); finally, these Picard functors and their functoriality in G are described in Subsection 5.3. The following notation is used throughout this section. The reductive group G yields semisimple groups and central isogenies

G̃ ↠ G′ ↠ Ḡ := G/Z^{0} ↠ G^{ad} := G/Z.

We denote by d̄ ∈ π_{1}(Ḡ) ⊆ π_{1}(G^{ad}) the image of d ∈ π_{1}(G). The choice of a maximal torus ι_{G} : T_{G} ↪ G induces maximal tori and isogenies

TG̃ ↠ T_{G′} ↠ TḠ ↠ T_{G^{ad}}.

Their cocharacter lattices are hence subgroups of finite index

Λ_{TG̃} ↪ Λ_{T_{G′}} ↪ Λ_{TḠ} ↪ Λ_{T_{G^{ad}}} .

The central isogeny ζ · π makes Λ_{Z^{0}} ⊕ Λ_{TG̃} a subgroup of finite index in Λ_{T_{G}}.

5.1. Torsors under a group stack. All stacks in this subsection are stacks over k, and all morphisms are over k. Following [7, 22], we recall the notion of a torsor under a group stack. Let G be a group stack. We denote by 1 the unit object in G, and by g_{1} · g_{2} the image of two objects g_{1} and g_{2} under the multiplication 1–morphism G ×G ⟶ G.

Definition 5.1.1. An action of G on a 1–morphism of stacks Φ : X ⟶ Y consists of a 1–morphism

G × X ⟶ X , (g, x) ⟼ g · x,

and of three 2–morphisms, which assign to each k–scheme S and each object

x in X (S)  an isomorphism 1 · x ⟶^{∼} x in X (S),
(g, x) in (G × X)(S)  an isomorphism Φ(g · x) ⟶^{∼} Φ(x) in Y(S),
(g_{1}, g_{2}, x) in (G × G × X)(S) an isomorphism (g_{1} · g_{2}) · x ⟶^{∼} g_{1} · (g_{2} · x) in X (S).

These morphisms are required to satisfy the following five compatibility conditions: the two resulting isomorphisms

(g · 1) · x ⟶^{∼} g · x in X (S),
(1 · g) · x ⟶^{∼} g · x in X (S),
Φ(1 · x) ⟶^{∼} Φ(x) in Y(S),
Φ((g_{1} · g_{2}) · x) ⟶^{∼} Φ(x) in Y(S),
and (g_{1} · g_{2} · g_{3}) · x ⟶^{∼} g_{1} · (g_{2} · (g_{3} · x)) in X (S),

coincide for all k–schemes S and all objects g, g_{1}, g_{2}, g_{3} in G(S) and x in X (S).

Example 5.1.2. Let φ : G ⟶ H be a homomorphism of linear algebraic groups over k, and let Z be a closed subgroup in the center of G with Z ⊆ ker(φ). Then the group stack M_{Z} acts on the 1–morphism φ_{∗} : M_{G} ⟶ M_{H} via the tensor product ⊗ : M_{Z} × M_{G} ⟶ M_{G}.

From now on, we assume that the group stack G is algebraic.

Definition 5.1.3. A G–torsor is a faithfully flat 1–morphism of algebraic stacks Φ : X ⟶ Y together with an action of G on Φ such that the resulting 1–morphism

G × X ⟶ X ×_{Y} X , (g, x) ⟼ (g · x, x)

is an isomorphism.

Example 5.1.4. Suppose that φ : G ↠ H is a central isogeny of reductive groups with kernel μ. For each d ∈ π_{1}(G), the 1–morphism

(12)  φ_{∗} : M_{G}^{d} ⟶ M_{H}^{e}  e := φ_{∗}(d) ∈ π_{1}(H)

is a torsor under the group stack M_{μ}, for the action described in example 5.1.2.

Proof. The 1–morphism φ_{∗} is faithfully flat by Lemma 2.2.2. The 1–morphism

M_{μ} × M_{G} ⟶ M_{G} ×_{M_{H}} M_{G}, (L, E) ⟼ (L ⊗ E, E)

is an isomorphism due to Lemma 2.2.1. Since φ_{∗} : π_{1}(G) ⟶ π_{1}(H) is injective, M_{G}^{d} ⊆ M_{G} is the inverse image of M_{H}^{e} ⊆ M_{H} under φ_{∗}; hence the restriction

M_{μ} × M_{G}^{d} ⟶ M_{G}^{d} ×_{M_{H}^{e}} M_{G}^{d}

is an isomorphism as well. □

Definition 5.1.5. Let Φ_{ν} : X_{ν} ⟶ Y_{ν} be a G–torsor for ν = 1, 2. A morphism of G–torsors from Φ_{1} to Φ_{2} consists of two 1–morphisms

A : X_{1} ⟶ X_{2}  and  B : Y_{1} ⟶ Y_{2}

and of two 2–morphisms, which assign to each k–scheme S and each object

x in X_{1}(S)  an isomorphism Φ_{2}A(x) ⟶^{∼} BΦ_{1}(x) in Y_{2}(S),
(g, x) in (G × X_{1})(S) an isomorphism A(g · x) ⟶^{∼} g · A(x) in X_{2}(S).

These morphisms are required to satisfy the following three compatibility conditions: the two resulting isomorphisms

A(1 · x) ⟶^{∼} A(x) in X_{2}(S),
Φ_{2}A(g · x) ⟶^{∼} BΦ_{1}(x) in Y_{2}(S)
and A((g_{1} · g_{2}) · x) ⟶^{∼} g_{1} · (g_{2} · A(x)) in X_{2}(S)

coincide for all k–schemes S and all objects g, g_{1}, g_{2} in G(S) and x in X_{1}(S).

Example 5.1.6. Let a cartesian square of reductive groups over k

\xymatrix{G_{1} \ar[r]^{α} \ar[d]_{φ_{1}} & G_{2} \ar[d]^{φ_{2}} \\ H_{1} \ar[r]^{β} & H_{2}}

be given. Suppose that φ_{1} and φ_{2} are central isogenies, and denote their common kernel by μ. For each d_{1} ∈ π_{1}(G_{1}), the diagram

\xymatrix{M_{G_{1}}^{d_{1}} \ar[r]^{α_{∗}} \ar[d]_{(φ_{1})_{∗}} & M_{G_{2}}^{d_{2}} \ar[d]^{(φ_{2})_{∗}} & d_{2} := α_{∗}(d_{1}) ∈ π_{1}(G_{2}) \\ M_{H_{1}}^{e_{1}} \ar[r]^{β_{∗}} & M_{H_{2}}^{e_{2}} & e_{ν} := (φ_{ν})_{∗}(d_{ν}) ∈ π_{1}(H_{ν})}

is then a morphism of torsors under the group stack M_{μ}.

Proposition 5.1.7. Let a G–torsor Φ_{ν} : X_{ν} ⟶ Y_{ν} with Γ(X_{ν}, O_{X_{ν}}) = k be given for ν = 1, 2, together with a morphism of G–torsors

\xymatrix{X_{1} \ar[r]^{A} \ar[d]_{Φ_{1}} & X_{2} \ar[d]^{Φ_{2}} \\ Y_{1} \ar[r]^{B} & Y_{2}}

such that the induced morphism of Picard functors A^{∗} : \underline{Pic}(X_{2}) ⟶ \underline{Pic}(X_{1}) is injective. Then the diagram of Picard functors

\xymatrix{\underline{Pic}(X_{1}) & \underline{Pic}(X_{2}) \ar[l]_{A^{∗}} \\ \underline{Pic}(Y_{1}) \ar[u]^{Φ_{1}^{∗}} & \underline{Pic}(Y_{2}) \ar[u]_{Φ_{2}^{∗}} \ar[l]_{B^{∗}}}

is a pullback square.

Proof. The proof of [22, Theorem 5.7] generalises to this situation as follows. Let S be a scheme of finite type over k. For a line bundle L on S×X_{ν}, we denote by Lin^{G}(L) the set of its G–linearisations, cf. [22, Definition 2.8]. According to Lemma 2.1.2(i), each automorphism of L comes from Γ(S, O_{S}^{∗}) and hence respects each linearisation of L. Thus [22, Theorem 4.1] provides a canonical bijection between the set Lin^{G}(L) and the fibre of

Φ_{ν}^{∗} : Pic(S × Y_{ν}) ⟶ Pic(S × X_{ν})

over the isomorphism class of L. Let T be an algebraic stack over k. We denote for the moment by Pic(T ) the groupoid of line bundles on T and their isomorphisms. Lemma 2.1.2(i) and Corollary 2.1.3 show that the functor

A^{∗} : Pic(T × X_{2}) ⟶ Pic(T × X_{1}) is fully faithful for every T . We recall that an element in Lin^{G}(L) is an isomorphism in Pic(G × S × X_{ν}) between two pullbacks of L such that certain induced diagrams in Pic(S × X_{ν}) and in Pic(G × G × S × X_{ν}) commute. Thus it follows for all L ∈ Pic(S × X_{2}) that the canonical map

A^{∗} : Lin^{G}(L) ⟶ Lin^{G}(A^{∗}L)

is bijective. Hence the diagram of abelian groups

\xymatrix{Pic(S × X_{1}) & Pic(S × X_{2}) \ar[l]_{A^{∗}} \\ Pic(S × Y_{1}) \ar[u]^{Φ_{1}^{∗}} & Pic(S × Y_{2}) \ar[u]_{Φ_{2}^{∗}} \ar[l]_{B^{∗}}}

is a pullback square, as required. □

5.2. Néron–Severi groups NS(M_{G}^{d}) for reductive G.

Definition 5.2.1. The Néron–Severi group NS(M_{G}^{d}) is the subgroup

NS(M_{G}^{d}) ⊆ NS(M_{Z^{0}}) ⊕ NS(MG̃)

of all triples l_{Z} : Λ_{Z^{0}} ⟶ ℤ, b_{Z} : Λ_{Z^{0}}⊗Λ_{Z^{0}} ⟶ End J_{C} and b : Λ_{TG̃}⊗Λ_{TG̃} ⟶ ℤ with the following properties:

(1) For every lift δ̄ ∈ Λ_{TḠ} of d̄∈ π_{1}(Ḡ), the direct sum

l_{Z} ⊕ b(−δ̄ ⊗ ) : Λ_{Z^{0}} ⊕ Λ_{TG̃} ⟶ ℤ

is integral on Λ_{T_{G}}.

(2) The orthogonal direct sum

b_{Z} ⊥ (id_{J_{C}} ·b) : (Λ_{Z^{0}} ⊕ Λ_{TG̃}) ⊗ (Λ_{Z^{0}} ⊕ Λ_{TG̃}) ⟶ End J_{C}

is integral on Λ_{T_{G}} ⊗ Λ_{T_{G}}.

Lemma 5.2.2. If condition 1 above holds for one lift δ̄ ∈ Λ_{TḠ} of d̄ ∈ π_{1}(Ḡ), then it holds for every lift δ̄ ∈ Λ_{TḠ} of the same element d̄∈ π_{1}(Ḡ).

Proof. Any two lifts δ̄ of d̄differ by some element λ ∈ Λ_{TG̃}. Lemma 4.3.4 states in particular that

b(−λ ⊗ ) : Λ_{TG̃} ⟶ ℤ

is integral on Λ_{TḠ}, and hence admits an extension Λ_{T_{G}} ⟶ ℤ that vanishes on Λ_{Z^{0}}. □

Remark 5.2.3. If G is simply connected, then NS(M_{G}^{0}) coincides with the group NS(M_{G}) of definition 4.3.2. If G = T is a torus, then NS(M_{T}^{d}) coincides for all d ∈ π_{1}(T) with the group NS(M_{T}) of definition 3.2.1.

Remark 5.2.4. The Weyl group W of (G, T_{G}) acts trivially on NS(M_{G}^{d}). Hence the group NS(M_{G}^{d}) does not depend on the choice of T_{G}; cf. Subsection 4.3.

Definition 5.2.5. Given a lift δ ∈ Λ_{T_{G}} of d ∈ π_{1}(G), the homomorphism

(ι_{G})^{NS,δ} : NS(M_{G}^{d}) ⟶ NS(M_{T_{G}})

sends (l_{Z}, b_{Z}) ∈ NS(M_{Z^{0}}) and b ∈ NS(MG̃) to the pair

l_{Z} ⊕ b(−δ̄ ⊗ ) : Λ_{G} ⟶ ℤ and b_{Z} ⊥ (id_{J_{C}} ·b) : Λ_{T_{G}} ⊗ Λ_{T_{G}} ⟶ End J_{C}

where δ̄ ∈ Λ_{TḠ} denotes the image of δ.

Note that this definition agrees with the earlier definition 4.3.5 in the cases covered by both, namely G simply connected and δ ∈ Λ_{T_{G}}.

Lemma 5.2.6. Given a lift δ ∈ Λ_{T_{G}} of d ∈ π_{1}(G), the diagram

\xymatrix{NS(M_{G}^{d}) \ar[rr]^{(ι_{G})^{NS,δ}} \ar@{^{(}->}[d] &  & NS(M_{T_{G}}) \ar[d]^{(ζ·π)^{∗}} \\ NS(M_{Z^{0}}) ⊕ NS(MG̃) \ar[r]^{id ⊕(ιG̃)^{NS,δ̄}} & NS(M_{Z^{0}}) ⊕ NS(M_{TG̃}) \ar@{^{(}->}[r] & NS(M_{Z^{0}×TG̃})}

is a pullback square; here δ̄ ∈ Λ_{T_{G^{ad}}} again denotes the image of δ.

Proof. This follows directly from the definitions. □

Let e ∈ π_{1}(H) be the image of d ∈ π_{1}(G) under a homomorphism of reductive groups φ : G ⟶ H. φ induces a map φ : G̃ ⟶ H̃ between the universal covers of their commutator subgroups. If φ maps the identity component Z_{G}^{0} in the center Z_{G} of G to the center Z_{H} of H, then it induces an obvious pullback map

φ^{∗} : NS(M_{H}^{e}) ⟶ NS(M_{G}^{d})

which sends l_{Z}, b_{Z} and b simply to φ^{∗}l_{Z}, φ^{∗}b_{Z} and φ^{∗}b. This is a special case of the following map, which φ induces even without the hypothesis on the centers, and which also generalises the previous definition 5.2.5.

Definition 5.2.7. Choose a maximal torus ι_{H} : T_{H} ↪ H containing φ(T_{G}), and a lift δ ∈ Λ_{T_{G}} of d ∈ π_{1}(G); let η ∈ Λ_{T_{H}} be the image of δ. Then the map

φ^{NS,d} : NS(M_{H}^{e}) ⟶ NS(M_{G}^{d})

sends (l_{Z}, b_{Z}) ∈ NS(M_{Z_{H}^{0}} ) and b ∈ NS(MH̃) to the pullback along φ : Z_{G}^{0} ⟶ T_{H} of (ι_{H})^{NS,η}(l_{Z}, b_{Z}, b) ∈ NS(M_{T_{H}}), together with φ^{∗}b ∈ NS(MG̃).

Lemma 5.2.8. The map φ^{NS,d} does not depend on the choice of T_{G}, T_{H} or δ.

Proof. Let W_{G} denote the Weyl group of (G, T_{G}). It acts trivially on Λ_{Z_{G}^{0}}, and without nontrivial coinvariants on Λ_{TG̃}; these two observations imply

(13)  Hom(Λ_{TG̃} ⊗ Λ_{Z_{G}^{0}}, ℤ)^{W_{G}} = 0.

Lemma 4.3.4 states that b is integral on Λ_{TH̃} ⊗ Λ_{TH̄}; its composition with the canonical projection Λ_{T_{H}} ↠ Λ_{TH̄} is a Weyl–invariant map b_{r} : Λ_{TH̃} ⊗ Λ_{T_{H}} ⟶ ℤ. As explained in Subsection 4.3, Lemma 4.3.1 implies that φ^{∗}b_{r} : Λ_{TG̃} ⊗ Λ_{T_{G}} ⟶ ℤ is still Weyl–invariant; hence it vanishes on Λ_{TG̃} ⊗ Λ_{Z_{G}^{0}} by (13).

Any two lifts δ of d differ by some element λ ∈ Λ_{TG̃}; then the two images of (l_{Z}, b_{Z}, b) ∈ NS(M_{H}^{e}) in NS(M_{T_{H}}) differ, according to the proof of Lemma 5.2.2, only by b_{r}(−λ ⊗ ) : Λ_{T_{H}} ⟶ ℤ. Thus their compositions with φ : Λ_{Z_{G}^{0}} ⟶ Λ_{T_{H}} coincide by the previous paragraph. This shows that the two images of (l_{Z}, b_{Z}, b) have the same component in the direct summand Hom(Λ_{Z_{G}^{0}}, ℤ) of NS(M_{G}^{d}); since the other two components do not involve δ at all, the independence on δ follows. The independence on T_{G} and T_{H} is then a consequence of Lemma 4.3.1, since the Weyl groups W_{G} and W_{H} act trivially on NS(M_{G}^{d}) and on NS(M_{H}^{e}). □

Lemma 5.2.9. For all maximal tori ι_{G} : T_{G} ↪ G and ι_{H} : T_{H} ↪ H with φ(T_{G}) ⊆ T_{H}, and all lifts δ ∈ Λ_{T_{G}} of d ∈ π_{1}(G), the diagram

\xymatrix{NS(M_{H}^{e}) \ar[r]^{(ι_{H})^{NS,η}} \ar[d]^{φ^{NS,d}} & NS(M_{T_{H}}) \ar[d]^{φ^{∗}} \\ NS(M_{G}^{d}) \ar[r]^{(ι_{G})^{NS,δ}} & NS(M_{T_{G}})}

commutes, with η := φ_{∗}δ ∈ Λ_{T_{H}} and e := φ_{∗}d ∈ π_{1}(H) as in definition 5.2.7.

Proof. Given an element in NS(M_{H}^{e}), we have to compare its two images in NS(M_{T_{G}}). The definition 5.2.7 of φ^{NS,d} directly implies that both have the same pullback to NS(M_{Z_{G}^{0}}) and to NS(M_{TG̃}). Moreover, their components in the direct summand Hom^{s}(Λ_{T_{G}} ⊗ Λ_{T_{G}}, End J_{C}) of NS(M_{T_{G}}) are both Weyl– invariant due to Lemma 4.3.1; thus equation (13) above shows that these components vanish on Λ_{TG̃} ⊗Λ_{Z_{G}^{0}} and on Λ_{Z_{G}^{0}} ⊗Λ_{TG̃}. Hence two images in question even have the same pullback to NS(M_{Z_{G}^{0}×TG̃}). But Λ_{Z_{G}^{0}} ⊕Λ_{TG̃} has finite index in Λ_{T_{G}}. □

Corollary 5.2.10. Let ψ : H ⟶ K be another homomorphism of reductive groups, and put f := ψ_{∗}e ∈ π_{1}(K). Then

φ^{NS,d} ∘ ψ^{NS,e} = (ψ ∘ φ)^{NS,d} : NS(M_{K}^{f}) ⟶ NS(M_{G}^{d}).

Proof. According to the previous lemma, this equality holds after composition with (ι_{G})^{NS,δ} : NS(M_{G}^{d}) ⟶ NS(M_{T_{G}}) for any lift δ ∈ Λ_{T_{G}} of d. Due to the Lemma 4.3.6 and Lemma 5.2.6, there is a lift δ of d such that (ι_{G})^{NS,δ} is injective. □

We conclude this subsection with a more explicit description of NS(M_{G}^{d}). It turns out that genus g_{C} = 0 is special. This generalises the description obtained for k = ℂ and G semisimple by different methods in [31, Section V].

Proposition 5.2.11. Let q : G ↠ G/G′ =: G^{ab} denote the maximal abelian quotient of G. Then the sequence of abelian groups

0 ⟶ NS(M_{G^{ab}}) ⟶^{q^{∗}} NS(M_{G}^{d}) ⟶^{pr_{2}} NS(MG̃) is exact, and the image of the map pr_{2} in it consists of all forms b : Λ_{TG̃} ⊗ Λ_{TG̃} ⟶ ℤ in NS(MG̃) that are integral

• on Λ_{TḠ} ⊗ Λ_{T_{G′}}, if g_{C} ≥ 1;

• on (ℤδ̄) ⊗ Λ_{T_{G′}} for a lift δ̄ ∈ Λ_{TḠ} of d̄∈ π_{1}(Ḡ), if g_{C} = 0.

The condition does not depend on the choice of this lift δ̄, due to Lemma 4.3.4.

Proof. Since q : Z^{0} ⟶ G^{ab} is an isogeny, q^{∗} is injective; it clearly maps into the kernel of pr_{2}. Conversely, let (l_{Z}, b_{Z}, b) ∈ NS(M_{G}^{d}) be in the kernel of pr_{2}; this means b = 0. Then condition 1 in the definition 5.2.1 of NS(M_{G}^{d}) provides a map

l_{Z} ⊕ 0 : Λ_{T_{G}} ⟶ ℤ

which vanishes on Λ_{TG̃}, and hence also on Λ_{T_{G′}}; thus it is induced from a map on Λ_{T_{G}}/Λ_{T_{G′}} = Λ_{G^{ab}}. Similarly, condition 2 in the same definition provides a map b_{Z} ⊥ 0 on Λ_{T_{G}} ⊗ Λ_{T_{G}} which vanishes on Λ_{TG̃} ⊗ Λ_{T_{G}} + Λ_{T_{G}} ⊗ Λ_{TG̃}, and hence also on Λ_{T_{G′}} ⊗ Λ_{T_{G}} + Λ_{T_{G}} ⊗ Λ_{T_{G′}}; thus it is induced from a map on the quotient Λ_{G^{ab}} ⊗ Λ_{G^{ab}}. This proves the exactness. Now let b ∈ NS(M_{G}^{d}) be in the image of pr_{2}. Then b is integral on (ℤδ̄) ⊗ Λ_{G′} by condition 1 in definition 5.2.1. If g_{C} ≥ 1, then

· id_{J_{C}} : ℤ ⟶ End J_{C}

is injective with torsionfree cokernel; thus condition 2 in definition 5.2.1 implies that

0 ⊕ b : (Λ_{Z^{0}} ⊕ Λ_{TG̃}) ⊗ Λ_{TG̃} ⟶ ℤ

is integral on Λ_{T_{G}} ⊗ Λ_{T_{G′}} and hence, vanishing on Λ_{Z^{0}} ⊆ Λ_{T_{G}}, comes from a map on the quotient Λ_{TḠ} ⊗ Λ_{T_{G′}}. This shows that b satisfies the stated condition. Conversely, suppose that b ∈ NS(M_{G}^{d}) satisfies the stated condition. Then b is integral on (ℤδ̄) ⊗ Λ_{T_{G′}}; since Λ_{T_{G′}} ⊆ Λ_{T_{G}} is a direct summand,

b(−δ̄ ⊗ ) : Λ_{T_{G′}} ⟶ ℤ

can thus be extended to Λ_{T_{G}}. We restrict it to a map l_{Z} : Λ_{Z^{0}} ⟶ ℤ. In the case g_{C} = 0, the triple (l_{Z}, 0, b) is in NS(M_{G}^{d}) and hence an inverse image of b. It remains to consider g_{C} ≥ 1. Then b is by assumption integral on Λ_{TḠ} ⊗Λ_{T_{G′}}, so composing it with the canonical subjection Λ_{T_{G}} ↠ Λ_{TḠ} defines a linear map Λ_{T_{G}}⊗Λ_{T_{G′}} ⟶ ℤ. Since b is symmetric, this extends canonically to a symmetric linear map from

Λ_{T_{G}} ⊗ Λ_{T_{G′}} + Λ_{T_{G′}} ⊗ Λ_{T_{G}} ⊆ Λ_{T_{G}} ⊗ Λ_{T_{G}}

to ℤ. It can be extended further to a symmetric linear map from Λ_{T_{G}} ⊗ Λ_{T_{G}} to ℤ, because Λ_{T_{G′}} ⊆ Λ_{T_{G}} is a direct summand. Multiplying it with id_{J_{C}} and restricting to Λ_{Z^{0}} defines an element b_{Z} ∈ Hom^{s}(Λ_{Z^{0}} ⊗ Λ_{Z^{0}}, End J_{C}). By construction, the triple (l_{Z}, b_{Z}, b) is in NS(M_{G}^{d}) and hence an inverse image of b. □

In particular, the free abelian group NS(M_{G}^{d}) has rank

rk NS(M_{G}^{d}) = r + r · rk NS(J_{C}) + \frac{r(r − 1)}{2} · rk End(J_{C}) + s

if G^{ab} ≅ 𝔾_{m}^{r} is a torus of rank r, and G^{ad} contains s simple factors.

5.3. Proof of the main result.

Theorem 5.3.1. i) Γ(M_{G}^{d}, O_{M_{G}^{d}}) = k.

ii) The functor \underline{Pic}(M_{G}^{d}) is representable by a k–scheme locally of finite

type. iii) There is a canonical exact sequence

0 ⟶ \underline{Hom}(π_{1}(G), J_{C}) ⟶^{j_{G}} \underline{Pic}(M_{G}^{d}) ⟶^{c_{G}} NS(M_{G}^{d}) ⟶ 0

of commutative group schemes over k. iv) For every homomorphism of reductive groups φ : G ⟶ H, the diagram

\xymatrix{0 \ar[r] & \underline{Hom}(π_{1}(H), J_{C}) \ar[r]^{j_{H}} \ar[d]^{φ^{∗}} & \underline{Pic}(M_{H}^{e}) \ar[r]^{c_{H}} \ar[d]^{φ^{∗}} & NS(M_{H}^{e}) \ar[r] \ar[d]^{φ^{NS,d}} & 0 \\ 0 \ar[r] & \underline{Hom}(π_{1}(G), J_{C}) \ar[r]^{j_{G}} & \underline{Pic}(M_{G}^{d}) \ar[r]^{c_{G}} & NS(M_{G}^{d}) \ar[r] & 0}

commutes; here e := φ_{∗}(d) ∈ π_{1}(H).

Proof. We record for later use the commutative square of abelian groups

\xymatrix{π_{1}(G) & Λ_{T_{G}} \ar@{->>}[l]_{pr} \\ Λ_{Z^{0}} \ar[u]_{ζ_{∗}} & Λ_{Z^{0}×TG̃}. \ar[u]_{(ζ·π)_{∗}} \ar@{->>}[l]_{_{pr_{1}}}}

The mapping cone of this commutative square

(14)  0 ⟶ Λ_{Z^{0}} ⊕ Λ_{TG̃} ⟶ Λ_{Z^{0}} ⊕ Λ_{T_{G}} ⟶ π_{1}(G) ⟶ 0

is exact, because its subsequence 0 ⟶ Λ_{TG̃} ⟶ Λ_{T_{G}} ⟶ π_{1}(G) ⟶ 0 is exact, and the resulting sequence of quotients 0 ⟶ Λ_{Z^{0}} = Λ_{Z^{0}} ⟶ 0 ⟶ 0 is also exact.

Lemma 5.3.2. There is an exact sequence of reductive groups

(15)  1 ⟶ G̃ ⟶ Ĝ ⟶^{dt} 𝔾_{m} ⟶ 1

and an extension π̂ : Ĝ ⟶ G of π : G̃ ⟶ G such that π̂_{∗} : π_{1}(Ĝ) ⟶ π_{1}(G) maps 1 ∈ ℤ = π_{1}(𝔾_{m}) = π_{1}(Ĝ) to the given element d ∈ π_{1}(G).

Proof. We view the given d ∈ π_{1}(G) as a coset d ⊆ Λ_{T_{G}} modulo Λ_{coroots}. Let

Λ_{TĜ} ⊆ Λ_{T_{G}} ⊕ ℤ

be generated by Λ_{coroots} ⊕ 0 and (d, 1), and let

(π̂, dt) : Ĝ ⟶ G × 𝔾_{m}

be the reductive group with the same root system as G, whose maximal torus TĜ = π^{−1}(T_{G}) has cocharacter lattice Hom(𝔾_{m}, TĜ) = Λ_{TĜ}. As π_{∗} maps Λ_{TG̃} isomorphically onto Λ_{coroots}, we obtain an exact sequence

0 ⟶ Λ_{TG̃} ⟶^{π_{∗}} Λ_{TĜ} ⟶^{pr_{2}} ℤ ⟶ 0,

which yields the required exact sequence (15) of groups. By its construction, π̂_{∗} maps the canonical generator 1 ∈ π_{1}(𝔾_{m}) = π_{1}(Ĝ) to d ∈ π_{1}(G). □

Let μ denote the kernel of the central isogeny ζ · π : Z^{0} × G̃ ↠ G. Then

ψ : Z^{0} × Ĝ ⟶ G × 𝔾_{m}, (z^{0}, ĝ) ⟼ (ζ(z^{0}) · π̂(ĝ), dt(ĝ))

is by construction a central isogeny with kernel μ. Hence the induced 1– morphism

ψ_{∗} : M_{Z^{0}}^{0} × M1 ; Ĝ ⟶ M_{G}^{d} × M_{𝔾_{m}}^{1}

is faithfully flat by Lemma 2.2.2. Restricting to the point Spec(k) ⟶ M_{𝔾_{m}}^{1} given by a line bundle L of degree 1 on C, we get a faithfully flat 1–morphism

(ψ_{∗})_{L} : M_{Z^{0}}^{0} × M_{Ĝ,L} ⟶ M_{G}^{d}.

Since Γ(M_{Z^{0}}^{0} ×M_{Ĝ,L}, O) = k by Proposition 4.4.7(i) and Lemma 2.1.2(i), part (i) of the theorem follows. The group stack M_{μ} acts by tensor product on these two 1–morphisms ψ_{∗} and (ψ_{∗})_{L}, turning both into M_{μ}–torsors; cf. Example 5.1.4. The idea is to descend line bundles along the torsor (ψ_{∗})_{L}. We choose a principal TĜ–bundle ξ̂ on C together with an isomorphism of line bundles dt_{∗} ξ̂ ≅ L. Then ξ := π̂_{∗}(ξ̂) is a principal T_{G}–bundle on C; their degrees δ̂:= deg(ξ̂) ∈ Λ_{TĜ} and δ := deg(ξ) ∈ Λ_{T_{G}} are lifts of d ∈ π_{1}(G). The diagram

(16)  \xymatrix{Z^{0} × TĜ \ar[r]^{id ×ιĜ} \ar[d]^{ψ} & Z^{0} × Ĝ \ar[d]^{ψ} \\ T_{G} × 𝔾_{m} \ar[r]^{ι_{G}×id} & G × 𝔾_{m}}

of groups induces the right square in the 2–commutative diagram

(17) \xymatrix{M_{Z^{0}}^{0} × M_{TĜ}^{0} \ar[r]^{id ×tξ̂} \ar[d]^{ψ_{∗}} & M_{Z^{0}}^{0} × Mδ̂ ; TĜ \ar[r]^{(id ×ιĜ)_{∗}} \ar[d]^{ψ_{∗}} & M_{Z^{0}}^{0} × M1 ; Ĝ \ar[d]^{ψ_{∗}} \\ M_{T_{G}}^{0} × M_{𝔾_{m}}^{1} \ar[r]^{t_{ξ}×id} & M_{T_{G}}^{δ} × M_{𝔾_{m}}^{1} \ar[r]^{(ι_{G}×id)_{∗}} & M_{G}^{d} × M_{𝔾_{m}}^{1}}

of moduli stacks; note that tξ̂ and t_{ξ} are equivalences. Restricting the outer rectangle again to the point Spec(k) ⟶ M_{𝔾_{m}}^{1} given by L, we get the diagram

(18)  [figure: M_{Z^{0}×TG̃}^{0}; ∼; (ζ·π)_{∗}; M_{Z^{0}}^{0} × M_{TG̃}^{0}; id ×ιξ̂; ζ_{∗}⊗π_{∗}; M_{Z^{0}}^{0} × M_{Ĝ,L}; (ψ_{∗})_{L}; M_{T_{G}}^{0}; (ι_{G})_{∗}∘t_{ξ}; M_{G}^{d}]

containing an instance ιξ̂ of the 1–morphism (10) defined in Subsection 4.4. According to the Proposition 3.2.2 and Proposition 4.4.7,

ι∗ ; ξ̂: \underline{Pic}(M_{Ĝ,L}) ⟶ \underline{Pic}(M_{TG̃}^{0})

is a morphism of group schemes over k. This morphism is a closed immersion, according to Proposition 4.4.7(iii), if g_{C} ≥ 1 or if ξ̂ is chosen appropriately, as explained in Lemma 4.3.6; we assume this in the sequel. Using Lemma 2.1.4 and Corollary 3.2.4, it follows that

(id ×ιξ̂)^{∗} : \underline{Pic}(M_{Z^{0}}^{0}) ⊕ \underline{Pic}(M_{Ĝ,L}) ≅ \underline{Pic}(M_{Z^{0}}^{0} × M_{Ĝ,L}) ⟶ \underline{Pic}(M_{Z^{0}×TG̃}^{0})

is a closed immersion of group schemes over k as well. The group stack M_{μ} still acts by tensor product on the vertical 1–morphisms in (17) and in (18). Since the diagram (16) of groups is cartesian, (17) and (18) are morphisms of G–torsors; cf. Example 5.1.6. Proposition 5.1.7 applies to the latter morphism of torsors, yielding a cartesian square

(19)  \xymatrix{\underline{Pic}(M_{G}^{d}) \ar[d]^{ψ_{L}^{∗}} \ar[r]^{t_{ξ}^{∗}∘ι_{G}^{∗}} & \underline{Pic}(M_{T_{G}}^{0}) \ar[d]^{(ζ·π)^{∗}} \\ \underline{Pic}(M_{Z^{0}}^{0}) ⊕ \underline{Pic}(M_{Ĝ,L}) \ar[r]^{(id ×ιξ̂)^{∗}} & \underline{Pic}(M_{Z^{0}×TG̃}^{0})}

of Picard functors. Thus \underline{Pic}(M_{G}^{d}) is representable, and t_{ξ}^{∗} ∘ ι_{G}^{∗} is a closed immersion; this proves part (ii) of the theorem. The image of the mapping cone (14) under the exact functor \underline{Hom}( , J_{C}), and the mapping cones of the two cartesian squares given by diagram (19) and Lemma 5.2.6, are the columns of the commutative diagram

0

[figure]

0

[figure]

0

[figure]

\underline{Hom}(π_{1}(G), J_{C})

[figure]

\underline{Pic}(M_{G}^{d})

[figure]

NS(M_{G}^{d})

[figure]

0 [figure] \underline{Hom}(Λ_{Z^{0}}, J_{C}) ⊕

\underline{Hom}(Λ_{T_{G}}, J_{C})

j_{Z0} ⊕j_{TG} [figure]

[figure]

\underline{Pic}(M_{Z^{0}}^{0}) ; ⊕ ; \underline{Pic}(M_{Ĝ,L}) ; ⊕ ; \underline{Pic}(M_{T_{G}}^{0})

[figure]

c_{Z0} ⊕cG̃⊕c_{TG} [figure]

\begin{matrix}NS(M_{Z^{0}}) \\ ⊕ \\ NS(MG̃) \\ ⊕ \\ NS(M_{T_{G}}) \end{matrix}

[figure]

[figure] 0

0 [figure] \underline{Hom}(Λ_{Z^{0}×TG̃}, J_{C})

[figure]

j_{Z0×TG̃}

[figure] \underline{Pic}(M_{Z^{0}×TG̃}^{0})

c_{Z0×TG̃}

[figure] NS(M_{Z^{0}×TG̃}) [figure] 0

0

whose two rows are exact due to Proposition 3.2.2(ii) and Proposition 4.4.7(ii). Applying the snake lemma to this diagram, we get an exact sequence

0 ⟶ \underline{Hom}(π_{1}(G), J_{C}) ⟶^{j_{G}(ι_{G},δ)} \underline{Pic}(M_{G}^{d}) ⟶^{c_{G}(ι_{G},δ)} NS(M_{G}^{d}) ⟶ 0.

The image of j_{G}(ι_{G}, δ) and the kernel of c_{G}(ι_{G}, δ) are a priori independent of the choices made, since both are the largest quasicompact open subgroup in \underline{Pic}(M_{G}^{d}). If G is a torus and d = 0, then this is the exact sequence of Proposition 3.2.2; in general, the construction provides a morphism of exact sequences

(20)

\xymatrix{0 \ar[r] & \underline{Hom}(π_{1}(G), J_{C}) \ar[r]^{j_{G}(ι_{G},δ)} \ar[d]^{_{pr_{∗}}} & \underline{Pic}(M_{G}^{d}) \ar[d]^{t_{ξ}^{∗}∘ι_{G}^{∗}} \ar[r]^{c_{G}(ι_{G},δ)} & NS(M_{G}^{d}) \ar[r] \ar[d]^{(ι_{G})^{NS,δ}} & 0 \\ 0 \ar[r] & \underline{Hom}(Λ_{T_{G}}, J_{C}) \ar[r]^{j_{T_{G}}} & \underline{Pic}(M_{T_{G}}^{0}) \ar[r]^{c_{T_{G}}} & NS(M_{T_{G}}) \ar[r] & 0}

whose three vertical maps are all injective. Using Proposition 3.2.2(iii), this implies that j_{G}(ι_{G}, δ) and c_{G}(ι_{G}, δ) depend at most on the choice of ι_{G} : T_{G} ↪ G and of δ, but not on the choice of Ĝ, L or ξ̂; thus the notation. Together with the following two lemmas, this proves the remaining parts (iii) and (iv) of the theorem. □

Lemma 5.3.3. The above map j_{G}(ι_{G}, δ) : \underline{Hom}(π_{1}(G), J_{C}) ⟶ \underline{Pic}(M_{G}^{d}) i) does not depend on the lift δ ∈ Λ_{T_{G}} of d ∈ π_{1}(G), ii) does not depend on the maximal torus ι_{G} : T_{G} ↪ G, and iii) satisfies φ^{∗} ∘ j_{H} = j_{G} ∘ φ^{∗} : \underline{Hom}(π_{1}(H), J_{C}) ⟶ \underline{Pic}(M_{G}^{d}) for all

φ : G ⟶ H.

Proof. If G is a torus, then δ and ι_{G} are unique, so (i) and (ii) hold trivially. The claim is empty for g_{C} = 0, so we assume g_{C} ≥ 1. Then the above construction works for all lifts δ of d, because ι∗ ; ξ̂ is a closed immersion for all

ξ̂. Given φ : G ⟶ H and a maximal torus ι_{H} : T_{H} ↪ H with φ(T_{G}) ⊆ T_{H}, we again put e := φ_{∗}d ∈ π_{1}(H) and η := φ_{∗}δ ∈ Λ_{T_{H}}. Then the diagram

(21)  \xymatrix{\underline{Hom}(π_{1}(H), J_{C}) \ar[r]^{j_{H}(ι_{H},η)} \ar[d]^{φ^{∗}} & \underline{Pic}(M_{H}^{e}) \ar[d]^{φ^{∗}} \\ \underline{Hom}(π_{1}(G), J_{C}) \ar[r]^{j_{G}(ι_{G},δ)} & \underline{Pic}(M_{G}^{d})}

commutes, because it commutes after composition with the closed immersion

t_{ξ}^{∗} ∘ ι_{G}^{∗} : \underline{Pic}(M_{G}^{d}) ⟶ \underline{Pic}(M_{T_{G}}^{0})

from diagram (20), using Remark 3.2.3. In particular, (iii) follows from (i) and (ii).

i) For G = GL_{2}, it suffices to take φ = det : GL_{2} ⟶ 𝔾_{m} in the above diagram (21), since det_{∗} : π_{1}(GL_{2}) ⟶ π_{1}(𝔾_{m}) is an isomorphism. For G = PGL_{2}, it then suffices to take φ = pr : GL_{2} ↠ PGL_{2} in the same diagram (21), since pr_{∗} : π_{1}(GL_{2}) ⟶ π_{1}(PGL_{2}) is surjective. As (i) holds trivially for G = SL_{2}, and clearly holds for G × 𝔾_{m} if it holds for G, this proves (i) for all groups G of semisimple rank one. In the general case, let α^{∨} ∈ Λ_{T_{G}} be a coroot, and let φ : G_{α} ↪ G be the corresponding subgroup of semisimple rank one. Then the diagram (21) shows j_{G}(ι_{G}, δ) = j_{G}(ι_{G}, δ + α^{∨}), since φ_{∗} : π_{1}(G_{α}) ⟶ π_{1}(G) is surjective. This completes the proof of i, because any two lifts δ of d differ by a sum of coroots. ii) now follows from Weyl–invariance; cf. Subsection 4.3. □

Lemma 5.3.4. The above map c_{G}(ι_{G}, δ) : \underline{Pic}(M_{G}^{d}) ⟶ NS(M_{G}^{d})

i) does not depend on the lift δ ∈ Λ_{T_{G}} of d ∈ π_{1}(G), ii) does not depend on the maximal torus ι_{G} : T_{G} ↪ G, and iii) satisfies φ^{NS,d} ∘c_{H} = c_{G} ∘φ^{∗} : \underline{Pic}(M_{H}^{e}) ⟶ NS(M_{G}^{d}) for all φ : G ⟶

H.

Proof. If G is a torus, then δ and ι_{G} are unique; if G is simply connected, then c_{G}(ι_{G}, δ) coincides by construction with the isomorphism c_{G} of Proposition 4.4.7(ii). In both cases, (i) and (ii) follow, and we can use the notation c_{G} without ambiguity. Given a representation ρ : G ⟶ SL(V), the diagram

(22)  \xymatrix{\underline{Pic}(M_{SL(V)}) \ar[r]^{c_{SL(V )}} \ar[d]^{ρ^{∗}} & NS(M_{SL(V)}) \ar[d]^{ρ^{NS,d}} \\ \underline{Pic}(M_{G}^{d}) \ar[r]^{c_{G}(ι_{G},δ)} & NS(M_{G}^{d})}

Documenta Mathematica 15 (2010) 35–72

commutes, because it commutes after composition with the injective map

(ι_{G})^{NS,δ} : NS(M_{G}^{d}) ⟶ NS(M_{T_{G}})

from diagram (20), using Lemma 5.2.9, Corollary 4.4.2, Remark 3.2.3, and the 2–commutative squares

\xymatrix{M_{T_{G}}^{0} \ar[r]^{t_{ξ}} \ar[d]^{ρ_{∗}} & M_{T_{G}}^{δ} \ar[r]^{(ι_{G})_{∗}} \ar[d]^{ρ_{∗}} & M_{G}^{d} \ar[d]^{ρ_{∗}} \\ M_{T_{SL(V )}}^{0} \ar[r]^{t_{ρ∗ξ}} & M_{T_{SL(V )}}^{ρ_{∗}δ} \ar[r]^{ι_{∗}} & M_{SL(V)})}

in which ι : T_{SL(V)} ↪ SL(V) is a maximal torus containing ρ(T_{G}). Similarly, given a homomorphism χ : G ⟶ T to a torus T, the diagram

(23)  \xymatrix{\underline{Pic}(M_{T}^{χ_{∗}d}) \ar[r]^{c_{T}} \ar[d]^{χ^{∗}} & NS(M_{T}) \ar[d]^{χ^{∗}} \\ \underline{Pic}(M_{G}^{d}) \ar[r]^{c_{G}(ι_{G},δ)} & NS(M_{G}^{d})}

commutes, again because it commutes after composition with the same injective map (ι_{G})^{NS,δ} from diagram (20), using Lemma 5.2.9, Remark 3.2.3, and the 2–commutative squares

[figure: M_{T_{G}}^{0}; t_{ξ}; χ_{∗}; M_{T_{G}}^{δ}; (ι_{G})_{∗}; χ_{∗}; M_{G}^{d}; χ_{∗}; M_{T}^{0}; t_{χ∗ξ}; M_{T}^{χ_{∗}δ}; M_{T}^{χ_{∗}δ}.]

The two commutative diagrams (22) and (23) show that the restriction of c_{G}(ι_{G}, δ) to the images of all ρ^{∗} and all χ^{∗} in \underline{Pic}(M_{G}^{d}) modulo \underline{Hom}(π_{1}(G), J_{C}) does not depend on the choice of δ or ι_{G}. But these images generate a subgroup of finite index, according to Proposition 5.2.11 and Remark 4.3.3. Thus (i) and (ii) follow. The functoriality in (iii) is proved similarly; it suffices to apply these arguments to homomorphisms ρ : H ⟶ SL(V), χ : H ⟶ T and their compositions with φ : G ⟶ H, using Corollary 5.2.10. □

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Indranil Biswas

School of Mathematics

Tata Institute of Fundamental Research Homi Bhabha Road Mumbai 400005 India indranil@math.tifr.res.in Norbert Hoffmann Mathematisches Institut der Freien Universität Arnimallee 3 14195 Berlin Germany norbert.hoffmann@fuberlin.de