The line bundles on moduli stacks of principal bundles on a curve

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.


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 [26]; Drézet and Narasimhan showed that this remains valid in the non-coprime case also [8].
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 [1,11,18].Here Kumar and Narasimhan [17] 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 [5].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 [21,28] showed that its canonical map to the Picard group Z of the affine Grassmannian is actually an isomorphism.Faltings [12] has generalised this result to positive characteristic, and in fact to arbitrary noetherian base scheme.
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 d G be the corresponding connected component of M G .For semisimple, almost simple groups G over C, the Picard group Pic(M d G ) has been determined case by case by Beauville, Laszlo and Sorger [2,20].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 [29] also proved these statements, using topological and analytic methods.
In this paper, we determine the Picard group Pic(M d G ) 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 Pic(M d G ) 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 d G ) 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 d G ) contains a subgroup NS(M G ab ) which depends only on the torus G ab = G/[G , G]; the quotient is a group of Weylinvariant symmetric bilinear forms on the root system of the semisimple part [G , G], subject to certain integrality conditions that generalise Teleman's result in [29].
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 d G ) −→ NS(M δ TG ) for a lift δ ∈ π 1 (T G ) of d involves contracting each bilinear form in NS(M d G ) 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 d G ) 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 [20] is followed, meaning we "cover" the moduli stack M d G 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 [20] 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 will be spelled out in a subsequent paper.

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 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 Definition 2.1.1.The Picard functor 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 Pic(X ) is representable, then we denote the representing scheme again by Pic(X ).If Pic(X ) is the constant sheaf given by an abelian group Λ, then we say that Pic(X ) is discrete and simply write 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 we may assume that Y = Spec(A) is an affine scheme over k.In this case, we have More precisely there is a unique isomorphism L ∼ = pr * 2 L x0 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.
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 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 Pic(X ) is discrete, and 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 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, 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 Pic(Y), then L is even isomorphic to the pullback of a line bundle on S, and hence trivial in 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 [21,Proposition 3.4], or [22, Théorème 4.6.2.1] together with [27,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 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 Proof.The above 2-commutative diagram defines a 1-morphism 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 × Gν 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 This action turns F into a principal H-bundle.Thus we obtain in particular a 1-morphism 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 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 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: Let ι G : T G ֒→ G be the inclusion of a maximal torus, with cocharacter group Λ TG := Hom(G m , T G ). Let Λ coroots ⊆ Λ TG be the subgroup generated by the coroots of G.The Weyl group of (G, T G ) acts trivially on Λ TG /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 Gm ) −→ Z, which induces an isomorphism π 0 (M TG ) −→ Λ TG .Its inverse, composed with the map cf. [9] and [14]

The case of torus
This section deals with the Picard functor of the moduli stack M 0 G 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 For a line bundle L on A, the standard morphism For polarisations of arbitrary degree, the analogous statement about End(A) ⊗ Q is shown in [25, 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 be the standard pairing between the Tate modules of A and A ∨ , cf. [25, §20].
According to [25, §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 , where the last equality follows from [25, 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 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.
which sends the class of each line bundle L on Λ ⊗ A to the linear map 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 be the diagonal principal polarisation on A r .According to Lemma 3.1.1, is an isomorphism onto the Rosati-invariants.Under the standard isomorphisms 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 By construction, it maps the class of each line bundle L on Λ ⊗ A to the map c φ Λ (L) : Λ ⊗ Λ −→ End A prescribed above.

Line bundles on M 0
T .Let T ∼ = G r m 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 standard principal polarisation φ Θ : For each finitely generated abelian group Λ, we denote by Hom(Λ, J C ) the kscheme of homomorphisms from Λ to i) The Picard functor Pic(M 0 T ) is representable by a scheme locally of finite type over k. ii) There is a canonical exact sequence of commutative group schemes Proof.Given a line bundle L on M 0 T , the automorphism group T of each point in M 0 T acts on the corresponding fibre of L, so we obtain a character w(L) : T be the canonical morphism to the coarse moduli scheme M 0 T , which is an abelian variety canonically isomorphic to Hom(Λ T , J C ). Line bundles of weight 0 on M 0 T descend to M 0 T , so the sequence is exact.This extends for families.Since Pic(A) is representable for any abelian variety A, the proof of (i) is now complete.Standard theory of abelian varieties and Corollary 3.1.3together yield another short exact sequence Clearly, χ * L univ p has weight χ; in particular, it follows that w is surjective, so we get an exact sequence of discrete abelian groups does not depend on the choice of p; sending χ to the class of χ * L univ p 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 Pic 0 (M 0 T ) = Hom(Λ, J C ) (see [24,Proposition 9.2]), and t * ξ induces the identity map on the discrete quotient Pic(M 0 T )/Pic 0 (M 0 T ) because ξ can be connected to the trivial T -bundle in M 0 T .
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 Proof.As before, let Λ T1 , Λ T2 and Λ T1×T2 denote the cocharacter lattices.Then ) is an isomorphism, and the homomorphism of discrete abelian groups

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 [12] on the Picard functor of M G , we describe here the Picard functor of the twisted moduli stacks M b G,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 b G,L are classified by an integer, their socalled 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 [18].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.

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))), • 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. 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 Since the k-algebra corresponding to G m is finitely generated, it follows that ϕ factors through L ≥n G/L ≥N for some N > n.Denoting by g the Lie algebra of G, [7, II, §4, Theorem 3.5] provides an exact sequence Here ϕ restricts to a character of g, which has to vanish; thus ϕ also factors through L ≥n G/L ≥N −1 .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 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 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 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 If n ≥ 2, then this restricts to a central extension of 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 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, [12,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 ) is an isomorphism for every scheme S of finite type over k.
Define the Picard functor 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 [12].Over k = C, the group Pic(Gr G ) is also determined in [23] as well as in [18], and Pic(M G ) is determined in [21] together with [28].
Theorem 4.2.1 (Faltings).Let G be simply connected and almost simple.
) 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 [12].More precisely, let an exact sequence of reductive groups be given, and a line bundle L on C. We denote by M b G,L the moduli stack of principal G-bundles E on C together with an isomorphism dt * E ∼ = L; cf.section 2 of [2].If for example the given exact sequence is then M GLn,L is the moduli stack of vector bundles with fixed determinant L.
In general, the stack  C\p G is isomorphic to L C\p G, and every character (L C\p G) S −→ (G m ) S is trivial according to [12, 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 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 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 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 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 δ C\p G ⊆ LG.

Note that L δ
C\p 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 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 δ C\p G, so L admits an L δ C\p G-linearisation and consequently descends to M b G,OC (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 Pic(Gr G ) ∼ = Z is discrete by Theorem 4.2.1(i).
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 Pic to the exterior pentagon yields the required commutative square, as L SL(V ) acts trivially on Pic(Gr SL(V ) ).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 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 ).
) such that the following diagram commutes: 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.ii) If on the other hand G is almost simple, then 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 TG ⊗ Q.
We say that a homomorphism l : Λ −→ Λ ′ between finitely generated free abelian groups Λ and Λ ′ is integral on a subgroup Λ ⊆ Λ ⊗ Q if its restriction to Λ ∩ Λ admits a linear extension l : Λ −→ Λ ′ .By abuse of language, we will not distinguish between l and its unique linear extension l.
for all λ ∈ Λ TG .Thus b( ⊗ α ∨ ) : Λ TG −→ Z is an integer multiple of α; hence it is integral on Λ T G ad , the largest subgroup of Λ TG ⊗ Q on which all roots are integral.But the coroots α ∨ generate Λ TG , as G is simply connected.Now let ι G : T G ֒→ G denote the inclusion of the chosen maximal torus.
, because all multiples of id JC are then nonzero in End J C .If g C = 0, then End J C = 0, but we still have the following 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 TG ⊗ Λ TG −→ Z is as a bilinear form nondegenerate. be determinant of cohomology line bundle [16] on M GL n , whose fibre at a vector bundle 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 0 Gm have the same image in the second summand End J C of NS(M Gm ).Thus the image of t * ξ L det in End J C does not depend on ξ; this image is − id JC 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.
Proof.Since the determinant of cohomology takes direct sums to tensor products, the pullback of ֒→ SL(V ) be the inclusion of a maximal torus that contains the image of the standard torus commutes for each principal T SL2 -bundle ξ on C. We choose ξ in such a way that deg(ξ 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 det 2 ) ⊗dρ in Pic(M SL 2 ) have the same image in NS(M TSL 2 ).Now suppose that the reductive group G is simply connected and almost simple.We denote by O GrG (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 [18].As in Subsection 4.2, we assume given an exact sequence of reductive groups with G simply connected, and a line bundle L on C.
Corollary 4.4.5.Suppose that G is almost simple.Then the isomorphism 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 b G,L has central charge n ∈ Z if this isomorphism maps it to O GrG (n); this is consistent with the standard central charge of line bundles on M G , as defined for example in [12].
iii) For all choices of ι G : T G ֒→ G and of ξ, the diagram 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. For the general case, we use the unique decomposition into simply connected and almost simple factors G i .As G is generated by its center and G, every normal subgroup in G is still normal in G. Let G i denote the quotient of G modulo the closed normal subgroup j =i G j ; then 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 3.2.4ensure that various constructions are compatible with the products in (11).Therefore, the general case follows from the already treated almost simple case.

The reductive case
In this section, we finally describe the Picard functor Pic(M d G ) for any reductive group G over k and any d ∈ π 1 (G).We denote Our strategy is to descend along the central isogeny applying the previous two sections to Z 0 and to G, 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 [20] for quotients of SL n .In Subsection 5.2, we define combinatorially what will be the discrete torsionfree part of Pic(M d G ); 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 Their cocharacter lattices are hence subgroups of finite index 5.1.Torsors under a group stack.All stacks in this subsection are stacks over k, and all morphisms are over k.Following [6,20], 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.
and of three 2-morphisms, which assign to each k-scheme S and each object These morphisms are required to satisfy the following five compatibility conditions: the two resulting isomorphisms 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 * : From now on, we assume that the group stack G is algebraic.
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 * : 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 and of two 2-morphisms, which assign to each k-scheme S and each object These morphisms are required to satisfy the following three compatibility conditions: the two resulting isomorphisms and A(( 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 be given.Suppose that ϕ 1 and ϕ 2 are central isogenies, and denote their common kernel by µ.For each is then a morphism of torsors under the group stack M µ .
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.[20,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 [20,Theorem 4.1] provides a canonical bijection between the set Lin G (L) and the fibre of 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 2.1.3show that the functor 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 is a pullback square, as required.

Néron-Severi groups NS(M
with the following properties: (1) For every lift δ ∈ Λ T Ḡ of d ∈ π 1 ( Ḡ), the direct sum ) is a pullback square; here δ ∈ Λ T G ad again denotes the image of δ.
Proof.This follows directly from the definitions.
The condition does not depend on the choice of this lift δ, 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 can thus be extended to Λ TG .We restrict it to a map l Z : Λ Z 0 −→ Z.In the case is representable by a k-scheme locally of finite type.iii) There is a canonical exact sequence Proof.We record for later use the commutative square of abelian groups The mapping cone of this commutative square There is an exact sequence of reductive groups is by construction a central isogeny with kernel µ.Hence the induced 1-morphism Gm is faithfully flat by Lemma 2.2.2.Restricting to the point Spec(k) −→ M 1 Gm given by a line bundle L of degree 1 on C, we get a faithfully flat 1-morphism Gm given by L, we get the diagram ) is a closed immersion of group schemes over k as well.
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 semisimple rank one.
ii) now follows from Weyl-invariance; cf.Subsection 4.3.T .The 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 Pic(M d G ) modulo 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.
where τ a : A −→ A is the translation by a. φ L is a homomorphism by the theorem of the cube [25, §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 [19, Chapter 5, Proposition 2.1].

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.
by G m splits as well, G being simply connected.(To prove the last assertion, for any extension G of G by G m , consider the commutator subgroup [ G , G] of G.It projects surjectively to the commutator subgroup of G which is G itself.Since [ G , G] 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.Given a point p ∈ C(k) and a uniformising element z ∈ O 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 O C,p , glued by the automorphism f (z) of the trivial G-bundle over the intersection; cf. for example [21, Section 3], [12, Corollary 16], or [13, Proposition 3].
from which we see in particular that M b G,L is algebraic.It satisfies the following variant of the Drinfeld-Simpson uniformisation theorem [9, Theorem 3].Lemma 4.2.2.Let a point p ∈ C(k) and a principal G-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 [9] carries over to this situation as follows.Choose a maximal torus T b G ⊆ G. Using [9, Theorem 1], we may assume that E comes from a principal T b G -bundle; cf. the first paragraph in the proof of [9, Theorem 3].Arguing as in the third paragraph of that proof, we may change this principal T b G -bundle by the extension of G m -bundles along coroots G m −→ T b G .Since simple coroots freely generate the kernel T G of T b G ։ G m , we can thus achieve that this T b G -bundle is trivial over (C \ {p}) × S. Because G m is a direct factor of T b G , we can hence lift the given trivialisation to the T b G -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 G 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 : Choose a homomorphism δ : G m −→ G with dt •δ = d ∈ Z = Hom(G m , G m ).We denote by t δ ∈ L G(k) the image of the tautological loop t ∈ LG m (k) under δ * : LG m −→ L G. The map t δ • : Gr G −→ Gr b G sends, for each point f in LG, the coset f • L + G to the coset t δ f • L + G. Its composition Gr G −→ M b G with glue p,z factors naturally through a 1-morphism glue p,z,δ : Gr G −→ M b G,O C (dp) , because dt * •(t δ • ) : LG −→ L G −→ 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 b G,OC (dp) , O M b G,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 GrG ) = k by equation (3).Using the above equivalence t ξ , this implies ) = k .Proposition 4.2.3.Let G be simply connected and almost simple.Then glue * p,z,δ : Pic(M b G,OC (dp) ) −→ 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. The map glue p,z is a torsor under L C\p G; cf. for example [21, Theorem 1.3] or [12, Corollary 16].More generally, Lemma 4.2.2 implies that glue p,z,δ is a torsor under the conjugate because the action of L δ C\p G corresponds to changing trivialisations over C \ {p}.Let S be a scheme of finite type over k.Each line bundle on S × M b G,L with trivial pullback to S × Gr G comes from a character (L δ C\p G) S −→ (G m ) S , since the map (3) is bijective.But L δ

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 G ։ G ad also by ρ.Then the diagram Pic

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 TG ⊗ Λ TG , Z) W denote the abelian group of bilinear forms b : Λ TG ⊗ Λ TG −→ Z that are invariant under the Weyl group W = W G of (G, T G ).

Lemma 4 . 3 . 1 .
Let T ′ G ⊆ G be another maximal torus, and let 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 [15, 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 we denote byϕ * : NS(M H ) −→ NS(M G )the restriction of the induced map ϕ * in(9).

Lemma 4 . 4 . 1 .
Let ξ be a line bundle of degree d on C. Then the composition Let ι : T SL n ֒→ SL n be the inclusion of the maximal torus T SL n := G n m ∩ SL n , where G n m ⊆ GL n as diagonal matrices.Then the cocharacter lattice Λ TSL 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 TSL n ⊗ Λ TSL n −→ Z is the restriction of the standard scalar product on Z n .Corollary 4.4.2.Let ξ be a principal T SLn -bundle of degree d ∈ Λ TSL n on C. Then the composition where pr ν : G n m ։ G m is the projection onto the νth factor.Now use the previous lemma to compute the image of L det n in NS(M G n m ) and then restrict to NS(M TSL n ).Corollary 4.4.3.If ρ : SL 2 −→ SL(V ) has Dynkin index d ρ , then the pullback ρ

Remark 4 . 4 . 6 .
as explained in Remark 4.3.3(iii).Using Proposition 4.4.4,this implies thatρ * : Pic(Gr SL(V ) ) −→ Pic(Gr G ) is injective.Due to Remark 4.2.4, it thus suffices to check that glue * p,z : Pic(M SL(V ) ) ∼ −→ 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 torusι G : T G ֒→ G induces maximal tori ι b G : T b G ֒→ G and ι G ad : T G ad ֒→ G ad compatible with the canonical maps G ֒→ G ։ G ad .Given a principal T b G -bundle ξ on C and an isomorphism dt * ξ ∼ = L, the composition 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 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 b G,L ) in NS(M TG ) coincide.Since the former generates a subgroup of finite index and the latter is torsionfree, the diagram in (iii) commutes.

Lemma 5 . 2 . 2 .Definition 5 . 2 . 5 .
If condition 1 above holds for one lift δ ∈ Λ T Ḡ of d ∈ π 1 ( Ḡ), then it holds for every lift δ ∈ Λ T Ḡ of the same element d ∈ π 1 ( Ḡ). Proof.Any two lifts δ of d differ by some element λ ∈ Λ T e G .Lemma 4.3.4states in particular that b(−λ ⊗ ) : Λ T e G −→ Z is integral on Λ T Ḡ , and hence admits an extension Λ TG −→ Z that vanishes on Λ Z 0 .Remark 5.2.3.If G is simply connected, then NS(M 0 G ) coincides with the group NS(M G ) of definition 4.3.2.If G = T is a torus, then NS(M d T ) 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 d G ). Hence the group NS(M d G ) does not depend on the choice of T G ; cf.Subsection 4.3.Given a lift δ ∈ Λ TG of d ∈ π 1 (G), the homomorphism (ι G ) NS,δ : NS(M d G ) −→ NS(M TG ) sends (l Z , b Z ) ∈ NS(M Z 0 ) and b ∈ NS(M e G ) to the pair l Z ⊕ b(− δ ⊗ ) : Λ G −→ Z and b Z ⊥ (id JC •b) : Λ TG ⊗ Λ TG −→ End J C where δ ∈ Λ T Ḡ 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 TG .Lemma 5.2.6.Given a lift δ ∈ Λ TG of d ∈ π 1 (G), the diagram

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 : G −→ H between the universal covers of their commutator subgroups.If ϕ maps the identity component Z 0 G in the center Z G of G to the center Z H of H, then it induces an obvious pullback map ϕ * : NS(M e H ) −→ NS(M d G ) 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 TG of d ∈ π 1 (G); let η ∈ Λ TH be the image of δ.Then the map ϕ NS,d : NS(M e H ) −→ NS(M d G ) sends (l Z , b Z ) ∈ NS(M Z 0 H ) and b ∈ NS(M e H ) to the pullback along ϕ :Z 0 G −→ T H of (ι H ) NS,η (l Z , b Z , b) ∈ NS(M TH ), together with ϕ * b ∈ NS(M e G ).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 0 G , and without nontrivial coinvariants on Λ T e G ; these two observations imply(13) Hom(Λ T e G ⊗ Λ Z 0 G , Z) WG = 0. Lemma 4.3.4states that b is integral on Λ T f H ⊗ Λ T H ; its composition with the canonical projection Λ TH ։ Λ T H is a Weyl-invariant map b r : Λ T f H ⊗ Λ TH −→ Z.As explained in Subsection 4.3, Lemma 4.3.1 implies that ϕ * b r : Λ T e G ⊗ Λ TG −→ Z is still Weyl-invariant; hence it vanishes on Λ T e G ⊗ Λ Z 0G by(13).Any two lifts δ of d differ by some element λ ∈ Λ T e G ; then the two images of (l Z , b Z , b) ∈ NS(M e H ) in NS(M TH ) according to the proof of Lemma 5.2.2,only by b r (−λ ⊗ ) : Λ TH −→ Z. Thus their compositions with ϕ : Λ Z 0 G −→ Λ TH 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 0 G , Z) of NS(M d G ); 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 d G ) and on NS(M e H ). 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 TG of d ∈ π 1 (G), the diagram NS(M e H ) / NS(M TG ) commutes, with η := ϕ * δ ∈ Λ TH and e := ϕ * d ∈ π 1 (H) as in definition 5.2.7.Proof.Given an element in NS(M e H ), we have to compare its two images in NS(M TG ).The definition 5.2.7 of ϕ NS,d implies that both have the same pullback to NS(M Z 0 G ) to NS(M T e G ).Moreover, their components in the direct summand Hom s (Λ TG ⊗ Λ TG , End J C ) of NS(M TG ) are both Weyl-invariant due to Lemma 4.3.1;thus equation (13) above shows that these components vanish on Λ T e G ⊗ Λ Z 0 G and on Λ Z 0 G ⊗ Λ T e G .Hence two images in question even have the same pullback to NS(M Z 0 G ×T e G ).But Λ Z 0 G ⊕ Λ T e G has finite index in Λ TG .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 f K ) −→ NS(M d G ). Proof.According to the previous lemma, this equality holds after composition with (ι G ) NS,δ : NS(M d G ) −→ NS(M TG ) for any lift δ ∈ Λ TG 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 d G ).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 [29, 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 ) exact, and the image of the map pr 2 in it consists of all forms b : Λ 2 .Conversely, let (l Z , b Z , b) ∈ NS(M d G ) be in the kernel of pr 2 ; this means b = 0. Then condition 1 in the definition 5.2.1 of NS(M d G ) provides a map l Z ⊕ 0 : Λ TG −→ Z which vanishes on Λ T e G , and hence also on Λ T G ′ ; thus it is induced from a map on Λ TG /Λ T G ′ = Λ G ab .Similarly, condition 2 in the same definition provides a map b Z ⊥ 0 on Λ TG ⊗ Λ TG which vanishes on Λ T e G ⊗ Λ TG + Λ TG ⊗ Λ T e G , and hence also on Λ T G ′ ⊗Λ TG +Λ TG ⊗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 d G ) be in the image of pr 2 .Then b is integral on (Z δ) ⊗ Λ G ′ by condition 1 in definition 5.2.1.If g C ≥ 1, then • id JC : 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 T e G ) ⊗ Λ T e G −→ Z is integral on Λ TG ⊗ Λ T G ′ and hence, vanishing on Λ Z 0 ⊆ Λ TG , comes from a map on the quotient Λ T Ḡ ⊗ Λ T G ′ .This shows that b satisfies the stated condition.Conversely, suppose that b ∈ NS(M d G ) satisfies the stated condition.Then b is integral on

3 .
It can be extended further to a symmetric linear map from Λ TG ⊗ Λ TG to Z, because Λ T G ′ ⊆ Λ TG is a direct summand.Multiplying it with id JC 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 d G ) and hence an inverse image of b.In particular, the free abelian group NS(M d G ) has rank rk NS(M d G ) = r + r • rk NS(J C ) + r(r − 1) 2 • rk End(J C ) + s if G ab ∼ = G r m is a torus of rank r, and G ad contains s simple factors.5.Proof of the main result.Theorem 5.3.1

Proof. 2 −
We view the given d ∈ π 1 (G) as a coset d ⊆ Λ TG modulo Λ coroots .Let Λ T b G ⊆ Λ TG ⊕ Z be generated by Λ coroots ⊕ 0 and (d, 1), and let ( π, dt) : G −→ G × G m be the reductive group with the same root system as G, whose maximal torus T b G = π −1 (T G ) has cocharacter lattice Hom(G m , T b G ) = Λ T b G .As π * maps Λ T e G isomorphically onto Λ coroots , we obtain an exact sequence 0 −→ Λ T → Z −→ 0, which yields the required exact sequence (15) of groups.By its construction, π * maps the canonical generator 1 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 T b G -bundle ξ on C together with an isomorphism of line bundles dt * ξ ∼ = L. Then ξ := π * ( ξ) is a principal T G -bundle on C; their degrees δ := deg( ξ) ∈ Λ T Gm (ιG×id) * / / M d G × M 1 Gm of moduli stacks; note that t b ξ t ξ are equivalences.Restricting the outer rectangle again to the point Spec(k) −→ M 1

ξ
of the 1-morphism(10) defined in Subsection 4.4.According to the Proposition 3.2.2 and Proposition 4.4.7, of group schemes over k.This morphism is a closed immersion, according to Proposition 4.4.7(iii),if g C ≥ 1 or if ξ is chosen appropriately, as explained in Lemma 4.3.6;we assume this in the sequel.Using Lemma 2.1.4and Corollary 3.2.4,it follows that (id ×ι b ξ )

Pic/Lemma 5 . 3 . 3 .
functors.Thus Pic(M d G ) 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 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 Z 0 , J C ) ⊕ Hom(Λ TG , J C )j Z 0 ⊕jT G / /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 −→ Hom(π 1 (G), J C ) jG(ιG,δ) −−−−−→ Pic(M d G ) cG(ιG,δ) −−−−−→ NS(M d G ) −→ 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 Pic(M d G ).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) 0 / / Hom(π 1 (G), J C ) jG(ιG,δ) / NS(M TG ) / / 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 G, L or ξ; thus the notation.Together with the following two lemmas, this proves the remaining parts (iii) and (iv) of the theorem.The above map jG (ι G , δ) : Hom(π 1 (G), J C ) −→ Pic(M d G ) i) does not depend on the lift δ ∈ Λ TG 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 * : Hom(π 1 (H), J C ) −→ Pic(M d G ) 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 * b ξ is a closed immersion for all ξ.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 TH .Then the diagram(21) Hom(π 1 (H), J C )jH (ιH ,η) 1 (G), J C ) jG(ιG,δ) / / Pic(M d G )commutes, because it commutes after composition with the closed immersiont * ξ • ι * G : Pic(M d G ) −→ Pic(M 0 TG )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.

Lemma 5 . 3 . 4 .T
The above map c G (ι G , δ) :Pic(M d G ) −→ NS(M d G ) i) does not depend on the lift δ ∈ Λ TG 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 * : Pic(M e H ) −→ NS(M d G ) 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 )it commutes after composition with the injective map (ι G ) NS,δ : NS(M d G ) −→ NS(M TG ) from diagram (20), using Lemma 5.2.9, Corollary 4.4.2,Remark 3.2.3, and the 2SL(V )t ρ * ξ / / M ρ * δ T SL(V ) 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 ,δ)/ / NS(M d G ) commutes, 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 . We denote by M d G the component of M G given by d ∈ π 1 (G).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 G : B G ։ T G and π H : B H ։ T H denote the canonical surjections.Then