Algebraic Loop Groups and Moduli Spaces of Bundles

Gerd Faltings Max-Planck-Institut für Mathematik Vivatsgasse 7, 53111 Bonn, Germany August 20, 2002

Abstract We study algebraic loop groups and affine Grassmannians in positive characteristic. The main results are normality of Schubert-varieties, the construction of line-bundles on the affine Grassmannian, and the proof that they induce line-bundles on the moduli-stack of torsors.

The paper has been accepted for publication by the Journal of the European Mathematical Society.

AMS-classification: 14D20

1 Introduction

The present note aims to explain some basic facts about algebraic loop groups. Some are well known in characteristic zero, but not so in positive characteristics, and the paper started as an attempt to check them in full generality. The topics are

• construction of loop groups and their central extensions,

• the infinite Grassmannians and flag spaces,

• the theory of Schubert varieties,

• integrable representations,

• moduli spaces and line-bundles on them

• the Suguwara construction. Most of the facts are neither surprising nor difficult to prove. Maybe the Suguwara construction in positive characteristics comes as a little surprise, because its usual definition involves complicated denominators. It should be mentioned that normality of Schubert varieties has been shown (at about the same time as the research for our paper) for the groups SL_{n} by Pappas and Rapoport, using a different method ([12]). Also I learned that P.Littelmann gave a general proof ([9]). In general there seems to be some confusion because several definitions of ”Schubert varieties” seem possible. Although logically it suffices to use ours I have tried to explain the relation to others as well as I can understand it. I thank V.Drinfeld for useful comments.

2 Construction of loop groups

Suppose G is a linear algebraic group defined over ℤ (or over any other base). For an indeterminate t R((t)) denotes the ring of Laurent-series.

Definition 1

• The positive loop group L^{+}G is the affine group scheme representing the

functor

L^{+}G(R) = G(R[[t]]).

• The strict positive loop group L^{++}(G) ⊂ L^{+}(G) is the kernel of the reduction map to G.

• The negative loop group L^{−}G is the ind-affine group scheme representing

L^{−}G(R) = G(R[t^{−1}]).

• L^{−−}(G) ⊆ L^{−}G is again the kernel of reduction.

• LG is the ind-affine group with

LG(R) = G(R((t)).

That these functors are indeed representable (or, for L^{−}G, L^{−−}G, LG, indrepresentable) by affine schemes is easily seen (see also [1] Prop.1.2). For example for H = SL_{d} LH(R) is the union of its subfunctors L_{n}H(R) consising of matrices of determinant 1

A = ∑_{i≥−n} A_{i} · t^{i},

with the A_{i} d×d matrices with coefficients in R. Obviously L_{n}H is representable by a closed subscheme of infinite affine space, and the inclusion L_{n}H ⊆ L_{n+1}H is a closed immersion. Also

L_{m}(H) · L_{n}(H) ⊆ L_{m+n}H

and (L_{n}H)^{−1} ⊆ L_{n(d−1)}H. These assertions also hold for proper closed G ⊆ H = SL_{d}.

Now recall the construction of the infinite Grassmannian. Consider (for H = SL_{d}) the functor D_{H} which associates to a ring R the set D_{H}(R) consisting of projective R[[t]]-modules L ⊂ R((t))^{d} with 1. t^{n} · R[[t]]^{d} ⊆ L ⊆ t^{−n} · R[[t]]^{d} for n ≫ 0

2. det(L) = R[[t]].

D_{H} is an ind-algebraic scheme, increasing union of the closed subschemes where the value of n in the first condition above is fixed. Also it admits a natural action by LH = LSL_{d}. This action is transitive (locally in the Zariski-topology of Spec(R) any L is free and the a transform of R[[t]]^{d}), and the stabiliser of the origin R[[t]]^{d} is L^{+}H.

One defines on D_{H} a very ample line bundle L = det_{R}(L)^{⊗−1}. In fact what is naturally defined is the ”difference”

det_{R}(L_{1}) ⊗ det_{R}(L_{2})^{⊗−1}.

For example choose an n with t^{n} · R[[t]]^{d} ⊆ L_{1,2} and consider the quotient of the R-determinants of the quotients L_{1,2}/t^{n} · R[[t]]^{d}. It is independant of n.

Fixing L_{1} and varying L_{2} = L then defines L. Because of this procedure the action of LH extends only to a projective action on L. However the stabiliser of a fixed lattice (like L^{+}(H)) acts honestly on L.

Dually this also holds for L^{−}H: Namely consider the submodule M = t^{−1} · R[t^{−1}]^{d} ⊂ R((t))^{d}. Then det_{R}(L)^{⊗−1} is equal to the determinant of cohomology of the complex

0 → L ⊕ M → R((t))^{d} → 0,

or more precisely to the determinant of the quotient complex obtained by dividing both terms by

t^{n} · R[[t]]^{d} ⊕ M.

Hence an action of the stabiliser L^{−}H of M. Also the actual determinant of the map above (i.e. of the quotient) defines a section

ϑ_{M} ∈ Γ(D_{H}, L)

invariant under L^{−}H. Namely choose an integer n such that L contains t^{n} · R[[t]]^{d}. Then the determinant of cohomology is equal to that of the complex

L/t^{n} · R[[t]]^{d} → R((t))^{d}/(t^{n} · R[[t]]^{d} ⊕ M),

and ϑ_{M} is the determinant of the boundary map in this complex.

(One can show that its transforms under LH define a projective embedding, but we do not need that.)

Now consider the complement D_{H} − Θ of the zero-set Θ of ϑ_{M}.

Lemma 2 The action of L^{−−}H on the origin L_{0} = R[[t]]^{d} defines an isomorphism

L^{−−}H ˜→D_{H} − Θ.

Proof Suppose L ∈ D_{H}(R) lies in the complement of Θ, that is L⊕M = R((t))^{d}. If {e_{1}, …, e_{d}} denotes the standard base of L_{0} = R[[t]]^{d}, then there exists a unique d × d-matrix

h ∈ 1 + t^{−1} · R[t^{−1}]^{d×d}

with h(e_{i}) ∈ L. We claim that these elements generate L, that is that L = h(L_{0}):

Namely choose a large integer n with t^{n} · h ∈ R[t]^{d×d} integral and such that t^{n} · L_{0} ⊆ t · L. Then it suffices that the h(e_{i}) generate L/t^{n} · L_{0} as R[[t]]-module. Each element in this quotient can be represented by an element f ∈ L ⊆ R((t))^{d} = L_{0} ⊕M such that its projection to L_{0} is a linear combination of t^{j} · e_{i} with j < n. However the projection of h(t^{j} · e_{i}) is of the form t^{j} · e_{i} modulo terms with lower t-powers. Thus by decreasing induction one can make the L_{0}-projection of f and thus also f itself vanish.

Now as L and L_{0} have the same R[[t]]-determinant the determinant of h must be a unit in R[[t]]. Especially it contains no negative t-powers, thus it is 1 and h ∈ L^{−−}H. It is clear that the map

h : D_{H} − Θ → L^{−−}H

is an inverse to the L^{−−}H-action on L_{0}.

It follows from the lemma that the multiplication map

L^{−−}H × L^{+}H → LH

is an open immersion. We claim that this also holds for certain subgroups.

Corollary 3 Suppose G ⊆ H is a closed subgroup, which is the stabiliser of an element e ∈ E for an algebraic representation of H = SL_{d} on a finitely generated projective R-module E. (For example this holds if G is defined and flat over the integers ℤ, and admits no non trivial characters G → 𝔾_{m}.)

Then the multiplication map

L^{−−}G × L^{+}G → LG

is an open immersion. An element g ∈ LG(R) lies in the image iff g(L_{0}) ∈ D_{H} − Θ.

Proof We obviously must show that if a product g = g_{−} · g_{+} with g_{−} ∈ L^{−−}H(R), g_{+} ∈ L^{+}H(R) lies in LG(R), then also both factors are in LG. But g_{−}^{−1}(e) = g_{+}(e) ∈ E((t)) lies both in e+t^{−1} ·E[t^{−1}] and in E[[t]], thus must be equal to e.

As a consequence one easily constructs the quotient D_{G} = LG/L^{+}G: It is an ind-scheme which has an open covering by translates of L^{−−}G. Also the map D_{G} → D_{H} is a closed immersion. The orbits of L^{−}G on D_{G} form an interesting ”stratification with strata of finite codimension”. As this is not used we refrain from elucidating what this should mean. However for completeness we list the orbits in a special case.

Namely assume that G is a connected split reductive algebraic group, and T ⊆ G a split maximal torus, Y = Hom(𝔾_{m}, T) its group of one parameter subgroups. Any ρ ∈ Y defines an element ρ(t) ∈ LT ⊆ LG. Also our base ring R is a field k.

Lemma 4 We have a disjoint union

LG(k) = ⋃_{ρ∈Y} L^{−}G(k)ρ(t)L^{+}G(k).

Proof. Any g ∈ LG(k) defines a G-torsor P on ℙ_{k}^{1} by gluing trivial torsors on ℙ^{1} − {∞} = 𝔸^{1} and on the formal completion along {∞}. The assertion amounts to the fact that this torsor reduces to the T -torsor defined by some ρ ∈ Y, or that it just reduces to a T -torsor.

We know that the associated group-scheme G_{P} contains a Borel, since this is true over 𝔸^{1} and the space of Borels is proper over ℙ^{1}. If we divide the Borel by its uniponent radical we obtain a torus, isomorphic to the constant torus T. The choice of a Borel reduces the structur group of P to the standard Borel B ⊂ G, and by pushout we obtain a T -torsor. This torsor is given by a one parameter group ρ ∈ Y. We claim that we can chose these data such that α(ρ) ≥ 0 for any positive root (occuring in the unipotent radical).

Namely it suffices to do this for simple roots. Number these as {α_{1}, ..., α_{l}}, and denote by ϵ_{i} the dual weights (that is dual weights on the maximal torus of the semisimple part, and trivial on the connected center). Some multiples of them are weights of T, so some power of the associated line-bundle L(ϵ_{i}) is defined on ℙ^{1}, and we can define its degree (a rational number) by linearity.

We define an operation which for a simple root α_{i} makes the degree of L(α_{i}) positive, while not changing the degrees of L(ϵ_{j}) for j ≠ i. If we reply it repeatedly as long as some L(α_{i}) has negative degree, we increase in each step the degree of some L(ϵ_{i}) by a positive integer. However the line-bundle L(2 ∑_{i} ϵ_{i}) is a sub-bundle of the vectorbundle E associated to the G-representation of highest weight 2 ∑_{i} ϵ_{i}, thus has its degree bounded independantly of our choice of a Borel. Thus the process must stop.

If for such a simple α_{i} α_{i}(ρ) is negative, we remedy this by change of Borels, without affecting the other ϵ_{j}(ρ)’s. Namely in G^{ad} we have a parabolic P_{i} (associated to α_{i}) and a quotient P_{α} ↠ P GL_{2}. The associated P GL_{2}-torsor is an Azumaya-algebra of the form End(E) for a vectorbundle E on ℙ^{1} of rank 2. The Borel in there corresponds to a quotient line bundle L, and the degree α_{i}(ρ) is the difference of the degree of the kernel and that of L, that is

α_{i}(ρ) = deg(E) − 2 · deg(L).

If this is negative we write E = O(a) ⊕ O(b) with a ≤ b, and replace L by the quotient O(a). Its normaliser defines the new Borel in G_{P}. As some power of L(ϵ_{j}) (j ≠ i) is defined by a character of P_{i} which is not changed by our operation, its degree remains the same.

Thus assume that ρ is dominant. Then the unipotent radical of the Borel (in G_{P}) has a filtration, with successive quotients isomorphic to twisted additive groups, namely to O(α(ρ)), α running through the positive roots. As these have trivial H^{1} the same is true for the whole unipotent radical. This means that the fibre over ρ of the pushforward

H^{1}(ℙ^{1}, B) → H^{1}(ℙ^{1}, T) = Y

is trivial, and our torsor is indeed defined by ρ.

Similarly one can consider double classes for the subgroups

B^{−} ⊂ L^{−}G, B^{+} ⊆ L^{+}G

which are the preimages of the Borels B^{±} ⊂ G. Now the double classes are indexed by Y ∝ W, the semidirect product of Y and the Weyl-group. If N^{−} ⊂ B^{−} denotes the preimage of the unipotent radical N^{−} ⊂ B^{−}, then the map

N^{−} × B^{+} → LG

is an open immersion.

Thus we define the affine flag-variety

X_{G} = LG/B^{+}.

The N^{−}-orbit trough the origin ∗ is an open subset of X_{G} isomorphic to N^{−}.

For the group H = SL_{d} X_{H} classifies R[[t]]-lattices L ⊂ R((t))^{d} with determinant R[[t]], together with a complete flag in L/t · L:

F_{0} = t · L ⊂ F_{1} ⊂ … ⊂ F_{d} = L

It admits L^{+}H- and L^{−}H-equivariant line-bundles L_{i} = det_{R}(F_{i})^{⊗−1}. Furthermore if we choose a complete flag in M = R[t^{−1}]^{d}

G^{0} = M ⊃ G^{1} ⊃ … ⊃ G^{d} = t^{−1} · M,

we obtain global sections

ϑ_{i} = det(F_{i} → R((t))^{d}/G^{i}) ∈ Γ(X_{G}, L_{i}).

These are invariant under N^{−}, T acts on them via the inverse of the i-th fundamental character ϵ_{i}, and they generate L_{i} on the ”big cell” N^{−}∗. By pullback via G → H we obtain such line-bundles and sections also over X_{G}.

3 Geometry of orbits

¿From now on we assume that G is split, simply connected, semi-simple and simple. That is G is the Chevalley group associated to an irreducible root system Φ, and Y is generated by the coroots H_{α}, with α running through the simple roots Δ ⊆ Φ^{+}. Unless otherwise mentioned we choose R = ℤ as basering. Let θ ∈ Φ^{+} denote the highest root.

We denote affine roots by n + α, with the corresponding root-vector t^{n} · X_{α} in the Lie-algebra L(g) = 𝔊((t)). Such a root is positive if either n > 0, or n = 0 and α ∈ Φ^{+}, otherwise negative. Thus the Lie-algebra of N^{−} is generated by root-vectors belonging to negative roots. Also number the simple roots of G as Δ = {α_{1}, …, α_{l}}, and denote by α_{0} = 1 − θ the remaining affine simple root. For each affine simple root we have an associated inclusion SL_{2} ⊂ LG which factors over a parabolic (or ”parahoric”?) group P_{i} ⊂ LG. that is P_{i} contains the Borel B^{+}. We denote by U_{i}^{+} ⊂ B^{+} the corresponding unipotent subgroup (≅ 𝔾_{a}), by U_{i}^{−} ⊂ N^{−} its opposite. By conjugation with W^{aff} we obtain an U_{α} for any affine root α.

The affine Weyl-group W^{aff} is a Coxeter group generated by the reflections {s_{i} | 0 ≤ i ≤ l}. For each w ∈ W^{aff} its length l(w) is the minimal number of factors in a product of s_{i}’s representing w. Any such product realising the minimum is called a reduced decomposition of w.

The Bruhat-order on W^{aff} is defined as follows: Suppose w = s_{1} ·s_{2} ⋯ s_{r} is a reduced decomposition. Then the elements w′ ≤ w are obtained by replacing some factors by 1 (after that one has to remove further pairs of factors to get a minimal representation). It is known that this is independant of the choice of reduced decomposition.

Now for a moment we work over an algebraically closed field k. It is known that W^{aff} parametrises the double classes of B^{+} in LG (Bruhat decomposition), that is the B^{+}-orbits in X_{G}. We have seen that the same holds for N^{−}-orbits. If we order orbits by inclusions of closures, this order will be the Bruhat-ordering respectively its inverse. To prove this we need some preparation:

Definition 5 For w ∈ W^{aff} the Schubert-variety C(w) ⊂ X_{G} is the closure of B^{+}w∗.

To describe C(w) recall that for each reduced decomposition w = s_{1} · s_{2} ⋯ s_{r} we obtain a B^{+}-equivariant (Demazure) resolution

D(w) → C(w)

(The notation does not quite reflect the complexity of the situation). It is defined as

D(w) = P_{1} ×^{B^{+}} P_{2} ×^{B^{+}} ⋯ ×^{B^{+}} P_{r}/B.

Here ”×^{B^{+}}” denotes the contraction. If w = w_{1} · s_{i} then D(w) is a fiber-bundle over D(w_{1}) with fibre P_{i}/B^{+} = ℙ^{1}. Thus by induction it is smooth, proper, and irreducible. We usually define subschemes of D(w), or more generally schemes mapping to D(w), by giving conditions on the ”factors” (which lie in P_{i}).

Also D(w) has a stratification where closed strata are defined by requiring that the some factors lie in B^{+}. The image in X_{G} of such a closed stratum contains the element w′ ∈ W^{aff} which is the product of the remaining s_{i}’s. Also from the theory of Tits-systems (see for example [11] II) one knows that only such elements of W^{aff} can lie in the image. The open stratum maps isomorphically onto B^{+}w∗. From this follows the assertion about the ordering on B^{+}-orbits, as well as the independance of the Bruhat-order from the choice of reduced decomposition.

To go further we need some notation: For an element w ∈ W^{aff} define subgroups N_{w}^{−} ⊆ N^{−}, N_{w}^{+} ⊆ N^{+} as

N_{w}^{−} = N^{−} ∩ wN^{−}w^{−1},
N_{w}^{+} = N^{+} ∩ wN^{−}w^{−1}.

One easily shows that

N^{−} ≅ N_{w}^{−} × ∏ U_{α},

where the product is over the negative affine roots α for which w^{−1}(α) is positive. Also N_{w}^{+} is the product of the their w-conjugates w^{−1}U_{α}w.

For example choose a strictly dominant one-parameter group ρ ∈ Y, and choose a very large positive constant c. This defines a linear form on affine roots by the rule

l(α) = l((m + β)) = m · c + β(ρ).

Thus α is positive iff l(α) is. Now N^{−} is obtained from N_{w}^{−} by adding affine roots α with l(α) < 0, l(w^{−1}(α)) > 0. Order these by the value l(α), and multiply from the right (with U_{α}) in that order.

For later use we also need the subgroup N^{−}(n) ⊂ N^{−} whose R-points are elements in L^{−}G(R) = G(R[t^{−1}]) which lie modulo t^{−n} in L^{−}T. Here n should be big enough so that N^{−}(n) ⊆ N_{w}^{−}. One then finds again

N_{w}^{−} ≅ N^{−}(n) × ∏ U_{α}^{−},

where the product is over the negative affine roots α = (−m + β) which occur in N_{w}^{−} but not in N^{−}(n).

Also one checks that

wN^{−}∗ ≅ N_{w}^{−} × N_{w}^{+}w ∗ .

Next we claim that the closure of an N^{−}-orbit N^{−}w′∗ meets C(w) iff it contains w iff w ≥ w′:

Firstly if w ≥ w′ consider the the first factor s_{i} missing in w′. Consider the affine line in D(w) which is U_{i}^{−} in the corresponding factor, and which has the same coordinates as W′ in all other positions. This maps to the N^{−}-orbit through w′∗ and contains a bigger w” ≤ w in its closure. Continue by induction.

Conversely assume the Bruhat cell C(w) meets the N^{−}-orbit through w′. There exists a 𝔾_{m}-action on X_{G} which contracts N^{−} and fixes all w∗: It is the product of a strictly dominant one-parameter group ρ : 𝔾_{m} → G, and a power of the 𝔾_{m}-action on LG which scales t (Virasoro-action via L_{0}), as in the linear form l used above. On the one hand this action respects the Bruhat-cell C(w), on the other the limit of an element in N^{−}w′∗ is w′. Thus w′ ≤ w, which is already close to the converse.

To derive this converse completely assume that C(w) meets the closure of the N^{−}-orbit through w′∗. We may replace w by a smaller element and assume that the closure meets B^{+}w∗. However this is contained in the open N^{−}-invariant set

N^{−}wN^{−}∗ = N^{−}N_{w}^{+}w∗ = N^{−}B^{+}w∗,

which must contain w′, and we already know that this means w′ ≤ w.

Finally this result implies that the order on N^{−}-orbits is indeed the inverse Bruhat-order.

Next choose a finite set of w ∈ W^{aff}. We assume that it contains with each w also any w′ ≤ w. This implies that the set

Ω = ⋃ wN^{−}∗ ⊂ X_{G}

is open and N^{−}-stable. We can chose an integer n such that for each chosen w and each negative affine root α = (−m + β) (β ∈ Φ) with m ≥ n w^{−1}(α) is also negative, that is N^{−}(n) ⊆ N_{w}^{−}.

Then

Lemma 6 There exists an open covering of Ω by N^{−}(n)-invariant open subsets equivariantly isomorphic to

N^{−}(n) × 𝔸^{r}.

Especially the quotient

N^{−}(n)\Ω

exists, is smooth, and covered by affine spaces.

Proof. We have

wN^{−}∗ ≅ N_{w}^{−} · N_{w}^{+}w∗ ≅ N^{−}(n) × 𝔸^{a} × 𝔸^{b}.

What this lemma tells us is that our X_{G} behaves very much like usual flag varieties. To wit:

Theorem 7 • Choose Ω, N^{−}(n) as before. The N^{−}-orbits in Ω are pullbacks of locally closed subschemes of N^{−}(n)\Ω isomorphic to affine spaces.

• These orbits are parametrised by the chosen set of w’s. The codimension of the w-orbit is l(w). Especially the orbits of codimension one correspond to the elementary reflections s_{i}. Call the corresponding orbit Θ_{i}.

• There exist on X_{G} line bundles L_{i} (0 ≤ i ≤ l). These are equivariant under L^{+}G and L^{−}G. They have a global section ϑ_{i} invariant under N^{−}, with divisor Θ_{i}, and transforming under T with the inverse of the fundamental character ϵ_{i} (1 ≤ i ≤ l) respectively trivially (i = 0).

• For each i consider the embedding

ℙ^{1} = P_{i}/B^{+} → X_{G}.

The degree of the pullback of L_{i} to it is 1 if i = j, 0 else.

Proof. The images in N^{−}(n)\Ω of orbits are isomorphic to N^{−}(n)\wN_{w}^{−}, thus affine spaces of codimension l(w). Furthermore the N^{−}-action on Ω is described by the groupoid

N^{−} × Ω → Ω × Ω.

Dividing by N^{−}(n) gives the smooth groupoid

N^{−}(n)\N^{−} ×^{N^{−}(n)} Ω → N^{−}(n)\Ω × N^{−}(n)\Ω.

As the orbits are stable under it so are their closures, and they define linebundles L_{i} with sections ϑ_{i} equivariant with respect to the groupoid. That means we get N^{−}-equivariant such objects on Ω. As Ω can be made arbitrarily big we obtain them on all of X_{G}.

To compute the degree of L_{i} on the j-th ℙ^{1}, we first note that for i ≠ j the projective line does not meet Θ_{i}, so the degree is zero. For i = j the intersection is transversal and the result is 1:

Let ℙ^{1} ⊂ X_{G} denote the projective line corresponding to SL_{i}. To compute the intersection multiplicity with Θ_{i} can use the open set

Ω = N^{−} · ℙ^{1} ≅ N_{s_{i}}^{−} × ℙ^{1}.

Then Θ_{i} = N_{s_{i}}^{−} × {∞} has degree 1.

For equivariance under L^{+}G we use that X_{G} is the inductive limit of Schubertcells C(w) which are flat, geometrically normal, and irreducible over ℤ and have a B^{+}-equivariant resolution D(w) → C(w). Especially over each field k

Γ(C(w) ⊗ k, O) = Γ(D(w) ⊗ k, O) = k.

This is shown in the next section, of course without making use of our present theorem. Furthermore one knows that line-bundles on D(w) are characterised by their degrees on certain B^{+}-invariant lines ℙ^{1}. By direct image this also holds for line-bundles on C(w), and the lines in question are the ℙ^{1}’s corresponding to simple affine roots α̃_{i} with s_{i} ≤ w (see [11], Ch.12, Prop. 6). Thus a line-bundle on C(w) is uniquely determined by the degree of its restriction to these lines.

A cofinal system of C(w)’s is G-invariant and thus L^{+}G-invariant. As the degrees on lines ℙ^{1} must be invariant (L^{+}G is connected) any line-bundle on these C(w) is equivariant under a central extension of L^{+}G and thus (uniquely) under L^{+}G as G is simply connected. Passing to the inductive limit it follows that any line-bundle on X_{G} is L^{+}G-equivariant.

Finally the actions of G and N^{−} on L_{i} must coincide on the intersection N^{−} because they can only differ by a character which must be trivial on that unipotent group. Thus equivariance under L^{−}G.

Remark. In fact these actions are induced from that of a central 𝔾_{m}- extension L̃G of LG. This follows because L^{+}(G) and L^{−}(G) generate LG (as sheaf in the Zariski-topology), and because over each algebraically closed field k all global sections Γ(X_{G} ⊗ k, O) = k are constant. Also note that equivariance is already known for the line-bundles obtained by pullback via representations G → H = SL_{d} (which will generate a subgroup of finite index in the Picardgroup).

Remark. An alternative construction of the L_{i} uses that the stack L^{−}G\X_{G} is the moduli stack of G-bundles on ℙ^{1}, and the construction of line-bundles on this moduli-stack, which however is not necessarily easier than our geometric approach.

Also for the construction of the L_{i} we could have replaced the groups N^{−}(n) by the congruence subgroups of L^{−}G, that is those elements which are the identity modulo t^{−n}.

4 Normality of Schubert-varieties

The main result is

Theorem 8 Each C(w) is normal, Cohen-Macaulay, and has rational singularities. Also they form an ind-scheme (using the Bruhat order) isomorphic to X_{G}.

Remark. In characteristic zero this is completely shown in [8]. We review these facts below after first establishing some properties of the Schubert-cells.

In positive characteristics most of it is in [11], Ch. 18. However as I could not quite follow the proof of lemma 134 (perhaps because of deficiencies in my French), I repeat the arguments. Note that in [11] there is no problem about normality of Schubert cells because the flag variety itself is defined as the inductive limit of normalised Schubert cells (in our terminology) C̃(w). What is new here is that C(w) itself is normal. Also Littelmann gives a representationtheoretic proof for normality. It is a little bit difficult to decide which definition of ”Schubert-cell” he uses, but it seems to be ours.

Begin of Proof. We assume that k has positive characteristics p > 0. By [11, Lemma 135] the variety D(w) is Frobenius-split, compatible with the inclusions D(w′) ⊂ D(w) for w′ < w: Namely the splitting is given by a global section of the (p − 1)-st power of the determinant of the logarithmic tangent bundle (vectorfields tangential to the boundary). However this determinant is the pullback of a line-bundle on X_{G} which is just the product of all L_{i}. As this product is generated near the origin by the product of θ_{i}’s we get a splitting.

Denote by C̃(w) the normalisation of C(w), so that π_{w} : D(w) → C(w) factors as

D(w) → C̃(w) → C(w).

The first map is a Stein-map, the second finite and birational. Furthermore the inclusions D(w′) ⊂ D(w) for w′ < w define maps C̃(w′) → C̃(w).

We have the following result, combining lemmas 32, 136, and assertion 6 in Th. 5. of [11]:

Lemma 9 a) All higher direct images R^{i}π_{w,∗}(O_{D(w)}) vanish, and the fibres of π_{w} are geometrically connected.

b) The normalisation C̃(w) → C(w) is a universal homeomorphism. c) For w′ < w the map

C̃(w′) → C̃(w)

is a closed immersion.

Proof. Use induction over l(w). If w = s_{i} · w′ with l(w′) = l(w) − 1 we can factor π_{w} as

D(w) = P_{i} ×^{B} D(w′) → P_{i} ×^{B} C(w′) → C(w).

The first map satisfies a) by induction. For the second consider the reduced P_{i}-stable subscheme

Z = ⋃ C(w”) ⊂ C(w′) ⊂ C(w),

where the union is over all w” < w′ with l(s_{i} · w”) < l(w”). Then our map is a stratawise isomorphism (thus a universal homeomorphism) over C(w) − Z. Over a point in z ∈ Z the fibre is isomorphic to ℙ^{1}.

This already implies geometric connectedness, which in turn trivially implies b). For c) then note that the maps C̃(w”) → C̃(w) are universally injective. As C̃(w′) and its image are compatibly Frobenius-split they must be closed immersions (the extension of function-fields is purely inseparable, thus trivial because of the splitting).

Finally for the direct images of π_{w} we now factor it as

D(w) = P_{i} ×^{B} D(w′) → P_{i} ×^{B} C̃(w′) → C̃(w),

and denote by Z̃ ⊂ C̃(w′) the reduced subscheme which is the union of all C̃(w”), for w” < w′ and l(s_{i} · w”) = l(w”) − 1. Then the map is isomorphic over the complement of Z̃, while for any z ∈ Z̃ the fiber is isomorphic to ℙ^{1}, as a scheme. To check that the higher direct images are trivial in such a z it suffices by the comparison theorem for proper maps that the formal completion of P_{i} ×^{B} C̃(w′) along the fibre ℙ^{1} has trivial cohomology. However this follows easily from the fact that globally generated sheaves on ℙ^{1} have trivial higher cohomology.

As remarked in [11, Th. 5] the proof of Ramanathan in [13] carries over, so also all higher ℝπ_{w,∗}^{i}(ω_{D(w)}) vanish,and C̃(w) is Cohen-Macaulay. Also we obtain flat models for the C̃(w) over the integers ℤ: Namely obviously the D(w) are naturally defined as smooth projective schemes over ℤ. Although we do yet not know that X_{G} is flat over ℤ, nevertheless we may define C̃(w) as the affine scheme over it whose algebra is π_{w,∗}(O_{D(w)}). The vanishing of higher direct images implies that this is flat over ℤ and commutes with base-change. Also for w′ ≤ w C̃(w′) is a closed subscheme of C̃(w), and the C̃(w)’s (indexed by W^{aff} with its Bruhat-order) form an ind-scheme C̃_{G} mapping to X_{G}. Also for each simple affine roots α_{i} there is a cofinal set of w’s with C(w) stable under the corresponding SL_{2}, that is with l(s_{i} · w) = l(w) − 1. It follows that the corresponding SL_{2} also operates on C̃_{G}.

We shall finish the proof of the theorem by showing that it is a closed subscheme of X_{G}. We already know that this holds in characteristic zero, by [8]. As his definition of Schubert-varieties is also a little bit complicated we repeat his basic argument here:

Choose a line-bundle L on X_{G} which is equivariant under a central extension of LG and has positive degree on each line ℙ^{1} corresponding to a simple affine root. Such an L can be obtained by pullback via an embediing G ⊆ SL_{d}. Form (over the base ℚ) the inductive limits

E(L) = Lim_{→}Γ(C(w), L)^{dual}

and

Ẽ(L) = Lim_{→}Γ(C̃(w), L)^{dual}.

These are both integrable modules (over the Lie-algebra) with a highest weight vector of multiplicity one. Thus they are irreducible and coincide, so (by duality) L(λ̃) has the same global section over X_{G} as over the inductive limit of all C̃(w). As the transition maps are surjective for the latter it follows that also for each individual w the global sections of L over C(w) and C̃(w) coincide. As L can be chosen such that is very ample on C(w) the assertion follows.

Also for each n the multiplication of sections defines a homomorphism

E(L^{⊗n}) → E(L)^{⊗n},

which is injective because of irreducibility. It follows that for each w Γ(C(w), L)^{⊗n} surjects onto Γ(C(w), L^{⊗n}). As for big n the latter space defines a projective embedding this holds true already for Γ(C(w), L), and the global sections Γ(X_{G}, L) (equal to the dual of E(L)) define a projective embedding of X_{G}. Thus our X_{G} coincides with the one defined in [8] via such a projective embedding. This finishes our excursion into characteristic zero.

For the general case consider the formal completion N̂^{−} of N^{−}. It is the indscheme whose R-valued points are elements of G(R[t^{−1}]) which are ≡ 1 modulo a nilpotent ideal of R, and whose constant term lies in N^{−}. It is defined over the integers ℤ. We claim that it is indrepresented by a power-series ring R = ℤ[[u_{i}]] in infinitely many variables {u_{j}|i = 1, 2, … }. That is if we call ideal J ⊂ R open if it contains almost all of the {u_{j}}, and some power of the remaining ones, then R represents the ind-scheme

Lim_{→}Spec(R/J).

Lemma 10 a) N̂^{−} is indrepresented by a ring of formal power-series.

b) For any g ∈ N̂^{−}(R) the image of g in LG(R) lies in the subgroup generated by the images of elements of SL_{2}(R), under the maps SL_{2} → LG associated to the simple affine roots.

Proof. Assertion a) follows easily from the fact that N̂^{−} is the product of the completions of L^{−−}T and of formal additive groups associated to negative affine roots. For b) the subgroup of LG(R) generated by the images of SL_{2}(R) contains the affine Weyl-group and thus all the additive groups U_{α}(R) corresponding to affine roots. Also L^{−−}T (R) is generated by the images of one-parameter groups H_{α}, hence we can reduce to G = SL_{2}. There everything comes down to the famous all important formula (u a unit)

(\begin{matrix}u & 0 \\ 0 & u^{−1} \end{matrix}) = (\begin{matrix}1 & u \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ −u^{−1} & 1 \end{matrix}) (\begin{matrix}1 & u \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

Now finally we can finish the proof of the theorem. Consider a torsion-free ℤalgebra R and an element g ∈ N̂^{−}(R), or equivalently its image z = g∗ ∈ X_{G}(R). As g lies in the subgroup of LG generated by SL_{2}(R)’s and as these operate on C̃_{G}, z lifts to a point z̃ ∈ C̃_{G}(R), that is to a z̃ ∈ C̃(w̃). As C̃(w̃) → C(w̃) is a universal homeomorphism this point also lies in the formal completion at the origin. Equivalently, if we denote by A the topological ℤ-algebra which is the projective limit (with surjective transition maps) of the formal completions A of the local rings of C̃(w) at the origin, the map R → R lifts to a continuous A → R. Furthermore this lift is unique, as R is torsion-free and C̃(w)_{ℚ} → C(w)_{ℚ} is an isomorphism. Thus in the limit the natural map R → A admits a section A → R.

We claim that the composition A → R → A is equal to the identity as well. For this denote by A the ring of the completion of some C̃(w). Lifting the origin to a ℤ-valued point of D(w) defines an injection A ⊆ B, with B the ring of the formal completion at the lift which is a ring of formal powerseries over ℤ. Now it suffices if the two endomorphisms of A become equal after composing A → A → B → B_{n}, where B_{n} is the quotient of B modulo some power of its defining ideal. However these two compositions correspond to B_{n}-valued points of C̃(w) which become equal in C(w). But then they are actually equal, as B_{n} is torsion-free and the map C̃(w)_{ℚ} → C(w)_{ℚ} is an isomorphism.

We derive that R ≅ A, and that the normalisation C̃(w) → C(w) is an isomorphism near the origin, because it induces a surjection on completed local rings.

Finally over any algebraically closed field k the locus where C(w) is not normal is Zariski-closed and stable under B^{+}. If it is non-empty it must contain a B^{+}-fixedpoint, which must be ∗.

Remark. The result implies normality of Schubert-varieties also in incomplete flag-varieties, like D_{G}. Namely their preimage in X_{G} is also a Schubertvariety.

Corollary 11 The closed immersion C̃_{G} ⊆ X_{G} is an isomorphism.

Proof. We must show that for any finitely generated ℤ-algebra R any element x ∈ X_{G}(R) lies in some C(w)(R). Equivalently pullback by x gives decreasing sequence of ideals I_{w} ⊂ R, and we must show that some I_{w} = (0).

We show this for any noetherian ring R. The annihilators of the ideals I_{w} form an increasing sequence which stabilises. If they stabilise to a proper ideal localise in a minimal prime over it. So we may assume that R is local, and the I_{w} are modules of finite length, for large w. Then they must also stabilise to a non-zero subideal of finite length. Dividing by a suitable power of the maximal ideal 𝔪 we then can assume that R is artinian.

Now we use that C̃_{G} is invariant under SL_{2}’s. If k = R/𝔪 denotes the residue-field the induced x(k) ∈ X_{G}(k) lies in the orbit of the origin ∗, under the group generated by SL_{2}(k)’s. Elements of this group lift to SL_{2}(R), so we may assume that x(k) = ∗. But then we have seen in the proof of the previous lemma that the group generated by SL_{2}(R)’s transforms x into the origin ∗.

Corollary 12 The Picard-group of rigidified line-bundles on X_{G} is (over any connected base) isomorphic to ℤ^{l+1}, with the isomorphism defined by degrees on the lines ℙ^{1} corresponding to simple affine roots. The Picard-group of D_{G} is ℤ.

Proof. The degree-map is surjective (by theorem 7). If L has trivial degree on each ℙ^{1} its pullback to each D(w) is trivial, as D(w) is a successice ℙ^{1}-fibration and elements of its Picard-group are determined by their degrees on fibres. It follows that L is trivial on each C(w) and also on X_{G}.

The assertion for D_{G} follows by pullback.

Another application is the construction of integrable representations. Namely consider an ”ample” line-bundle L on X_{G} (the restriction to each C(w) is ample), that is L is the tensor product of strictly positive powers of the fundamental L_{i}. Then by Frobenius-splitting in characteristic p for w′ ≤ w the restriction map Γ(C(w), L) → Γ(C(w′), L) is surjective, that these modules are torsion-free, and that all higher cohomology vanishes. We then form the module (with discrete topology)

E(L) = Lim_{→}Γ(C(w), L)^{dual}.

It is an inductive limit of finite free ℤ-modules, with split injective transition maps, and a suitable central extension L̃G of LG will act on it. Also it admits a canonical B^{+}-eigenvector e(L), the ”highest weight” vector, given by evaluation at ∗. It generates E(L) as module over the L̃G. More precisely each Γ(C(w), L)^{dual} is generated over ℤ by B^{+}w · e(L) (as B^{+}w · ∗ is dense in C(w)). The dual Γ(X̃_{G}, L(λ̃)) is the projective limit of free ℤ-modules, with surjective transition maps. It is naturally a topological module, and E(λ̃) is its continuous dual. Finally the ϑ’s define an N^{−}-invariant element of the dual, which takes the value 1 on e(L).

Some more details: The class of the central extension associated to L is linear in L, and trivial if L is defined by a character of B^{+} (induced from a character of T). It follows that there exists one central extension

𝔾_{m} → L̃G → LG

which operates on all L’s. The preimage of T in it is a torus T̃ of dimension l+1, and the one-parameter group H_{1,−θ} + H_{θ} defines an isomorphism between 𝔾_{m} and the kernel of T̃ → T. Thus T̃ is the product of one-parameter subgroups corresponding to simple affine roots, the L’s are parametrised by characters λ̃ : T̃ → 𝔾_{m}, and for each simple affine root α λ̃(H_{α}) is the degree of L(λ̃) on the associated ℙ^{1} ⊂ X_{G}. For dominant λ̃ (that is λ̃(H_{α}) ≥ 0 for each affine simple α) the cohomology H^{i}(X_{G}, L(λ̃)) vanishes in positive degrees, while in degree 0 we obtain the dual of the integrable module E(L(λ̃)) = E(λ̃). E(λ̃) contains a canonical B̃^{+}-eigenvector e(λ̃) of T̃-weight λ̃, which generates it as L̃G-module. Furthermore LG, L̃G, X_{G} and all L(λ̃) admit a 𝔾_{m}-action. A unit u acts by sending t to u · t. This action is expanding on L^{−}G, so it follows that the global sections Γ(X_{G}, L(λ̃)) have all weights ≥ 0, und dually E(λ̃) has weights ≤ 0 (the dual is the product of its 𝔾_{m}-eigenspaces). Furthermore the elements of weight 0 are the global sections Γ(G/B^{+}, L(λ)), respectively their duals. Finally for affine roots (n + α) the L_{0}-weight is n. Thus it is natural to speak of extended affine weights (n + λ̃). These additional gradings should not be confused with those defined by the kernel 𝔾_{m} of L̃G → LG: These are on E(λ̃) always equal to c, and on its dual to −c.

We denote these weights L_{0}-weights, since our 𝔾_{m} will appear as a subgroup of the Virasoro-group. Each L_{0}-weight in E(λ̃) has finite multiplicity, that is the eigenspace is a finite free ℤ-module. Dually Γ(X_{G}, L(λ̃)) is the product of their duals, and has the product topology. It is naturally a topological module which is the projective limit of discrete finitely generated free modules. Similarly after base-extension to an arbitrary ground-ring R. Note that the construction of E(λ̃) commutes with basechange, while for the dual we need to use the completed tensorproduct. It is precisely to avoid such topological difficulties that we prefer to use E(λ̃).

Finally we return to loop-groups.

Proposition 13 E(λ̃) contains a maximal proper submodule, with irreducible quotient F (λ̃). The image f(λ̃) of e(λ̃) is the unique non-trivial B^{+}-eigenvector in F (λ̃), and it generates it as N^{−}-module.

Also the L̃G-submodule of Γ(X_{G}, L(λ̃)) generated by ϑ(λ̃) is topologically irreducible, and already generated by the B^{+}-transforms of ϑ(λ̃). It is the dual of F (λ̃).

Proof. e(λ̃) is the only T̃-eigenvector of conformal weight 0 in E(λ̃). Thus ony L̃G-submodule of not containing it is annihilated by ϑ(λ̃). Hence the perpendicular of the closed subspace generated by ϑ(λ̃) is the maximal submodule not containing e(λ̃). In the quotient F (λ̃) any non-trivial submodule contains f(λ̃), and thus must coincide with F (λ̃) as f(λ̃) is a generator.

Now the L̃G-submodule generated by another B^{+}-eigenvector f ≠ 0 coincides with the N^{−}-submodule generated by it: Namely for any linear form v in the dual the function v(g(f)) defines an element of Γ(X_{G}, L(λ̃)) (λ̃ the associated character). This vanishes identically if it vanishes on the dense orbit N^{−} · ∗. By irreducibility this must be all of F (λ̃), and by considering weights it follows that f must coincide with f(λ̃).

Finally the dual of F (λ̃) is topologically irreducible, thus generated by ϑ(λ̃). Also since this element is N^{−}-invariant the dual is already topologically generated by its B^{+}-transforms, by a similar reasoning to the above.

5 Virasoro-action

We define the Virasoro-group Vir as a central extension automorphism indgroupscheme Aut of automorphisms of the punctured formel disk. For each R Aut(R) is the set of continuous R-automorphisms of R((t)). Such a automorphism ϕ is determined by ϕ(t), which must be a Laurent-series

ϕ(t) = ∑ a_{n} · t^{n},

with a_{1} invertible, a_{n} nilpotent for n ≤ 0, and almost all a_{n} with negative index n vanish. It is obvious that this is representable by an affine ind-scheme.

The subgroup Aut^{+} ⊂ Aut is the subgroup of automorphisms of R[[t]] respecting the ideal (t). It is the subscheme defined by a_{n} = 0 for n ≤ 0, and is an honest scheme, the product of 𝔾_{m} and an infinite number of copies of 𝔸^{1}. It contains as subgroup the Borel B_{V} ⊂ P GL_{2} of automorphisms of ℙ^{1} respecting {∞} (Recall that t is a local coordinate at ∞). Although we make no use of it this is the place to mention the subgroup { \atop P GL_{2}} ⊂ Aut (an ind-scheme), the formal completion of P GL_{2} along B_{V}.

Another important subgroup Aut^{−} ⊂ Aut is the group of automorphisms of the affine line ℙ^{1} − {∞} respecting {0}. Such an automorphism ϕ sends t^{−1} to a finite sum

ϕ(t^{−1}) = ∑_{n≥1} b_{n} · t^{−n},

with b_{1} invertible, b_{n} nilpotent for n ≥ 2, and almost all of them = 0. It is the semidirect product of 𝔾_{m} (giving L_{0}-weights) and the normal subgroup defined by b_{1} = 1. The latter is a formal group, ind-represented by a powerseries-ring in infinitely many variables. In terms of the coordinates a_{n} it is defined by a_{1} = 1, a_{n} = 0 for n ≥ 2. Finally

Aut = Aut^{−} · Aut^{+},

and

Aut^{−} ∩ Aut^{+} = 𝔾_{m}.

Obviously Aut acts (by change of variables) on LG, fixing G. Also Aut^{±} respects B^{±}. The main result of this section is that this action extends to an action on L̃G, and to a projective action on each irreducible integrable module F (λ̃). The corresponding central extension of Aut will be called Vir (Virasorogroup).

Theorem 14 a) Aut^{+} acts on E(λ̃) such that the action fixes the highest weight-vector e(λ̃), and intertwines the original with the conjugated B^{+}-action.

b) Aut^{−} acts on the closed submodule of Γ(X_{G}, L(λ̃)) generated by the thetafunction θ(λ̃), and intertwines the original with the conjugated B^{−}-action.

c) These two actions coincide on 𝔾_{m}. Furthermore, over an algebraically closed field k, they they induce a projective action of Aut on F (λ̃) which intertwines the action of L̃G with a conjugate action.

Proof. We already know the action of the group 𝔾_{m} fixing e(λ̃) and ϑ(λ̃).

For the rest of the proof we use that all these facts are already known in characteristic 0, thanks to the Suguwara-construction ([7, ex. 18.8], [16, Lemma 1.2.2.]): This gives in characteristic 0 an action of the Lie-algebra of a central extension of Aut on the integrable modules and thus on X_{G}. However apart from the L_{0}-factor 𝔾_{m} Aut^{±} is prounipotent, respectively a formal group, and the actions integrate (for Aut^{+} one also needs that L_{0}-weights are bounded above). Note however that we have normalised in such a way that e(λ̃) becomes L_{0}-invariant, which does not hold for the Sugawara-formula. The two actions differ by chosing different lifts of Aut^{±} into the central extension Vir.

Also Aut^{±}is flat over ℤ, that is is an inductive limit of affines Spec(R) with R torsion-free. It then suffices to construct functorial actions of Aut^{±}(R) on E(λ̃) ⊗ R, respectively the closed subspace of sections generated by ϑ(λ̃), for such torsion-free rings R.

To prove a) use that (R torsion-free) the action of Aut^{+}(R) on E(λ̃) ⊗ R_{ℚ} respects E(λ̃) ⊗ R, because it is generated by L̃G · e(λ̃).

For b) use the ”dual” argument with ϑ(λ̃).

Finally for c) we may assume that k has characteristic p > 0. Then the action of Aut^{+} on E(λ̃) stabilises e(λ̃) and must respect the unique irreducible quotient F (λ̃). Its dual is the unique irreducible quotient of the subspace of Γ(X_{G}, L(λ̃)) generated by ϑ(λ̃), and is respected by Aut^{−}. As both actions coincide on the intersection we obtain a map from Aut = Aut^{+} · Aut^{−} into the automorphisms of F (λ̃), which intertwines the L̃G-action with its conjugate. By irreducibility this defines a projective representation of Aut, or a representation of a central extension (which splits as a map of schemes) Vir → Aut.

Remarks. a) One can construct Vir-stable lattices in E(λ̃)_{ℚ}, for example the Aut^{−}-span of E(λ̃).

b) The Aut^{+}-action can also be defined via its geometric action on X_{G}. It follows that it respects the comultiplication-map

E(λ̃ + μ̃) → E(λ̃) ⊗ E(μ̃),

dual to multiplication on global sections. Also for weights λ̃ with λ = 0 (that is parametrised by the central charge c) the geometric action of { \atop P GL}_{2} on D_{G} defines an action on E(λ̃).

6 Moduli-stacks of G-bundles

For the beginning G can be again an arbitrary reductive connected groupscheme, but at the end we shall restrict to simple, semisimple, and simply connected G’s. Suppose C → S is a smooth projective curve, with geometrically irreducible fibres of constant genus g. The moduli-stack M_{G} classifies principal G-bundles P on base-changes C ×_{S} S′.

It is shown in [5] (th.1) that one can find a Borel B ⊂ G_{P}. Thus the structuregroup reduces to B ⊂ G, and we call B ”generic” if for each simple positive root α the associated line-bundle L(−α) is generated by its global sections and has trivial first cohomology. For example this holds if deg(L(α)) ≤ −2g. That there exists a ”generic” Borel follows also from [5].

Remark. If G_{P} contains a Borel after basechange S′ → S, it defines a classifying map

S′ ×_{S} C → (G/B)_{P} .

This map is smooth if S′ is the classifying space of ”generic” Borel’s, by infinitesimal calculations as in [5] (H^{1}(C, Lie(G_{P})/Lie(B)) is globally generated).

Now assume that C → S has a section. We denote its image by {∞} ⊂ C, and let C^{0} = C − {∞} denote the complement.

Proposition 15 Suppose G is semisimple. For any G-torsor P on C, there exists a flat finitely presented covering S′ ↠ S such that the pullback of P to C^{0} ×_{S} S′ is trivial. Moreover if G is simply connected we may use an étale covering.

Proof. See [5] (th.3).

As an application note that the sections of G over C^{0} ×_{S} S′ are representable by an ind-group-scheme R_{C^{0}/S}(G) (Weil- restriction). That one obtains an ind-groupscheme is an easy generalisation of the discussion after definition 1. R_{C^{0}/S}(G) is the inductive limit of its closed subschemes where one restricts the pole-order at infinity of an element of G and of its inverse.

Also suppose we have chosen an isomorphism between the formal completion of C along {∞} and Spf(O_{S}[[t]]). This is possible locally in the Zariski-topology of S. Recall that M_{G} denotes the algebraic ind-stack of principal G-torsors on C (in either flat, smooth or étale topology).

Corollary 16 Suppose G is semisimple. Then M_{G} = R_{C^{0}/S}(G)\D_{G}. More precisely D_{G} is an R_{C^{0}/S}(G)-torsor over M_{G}, in the flat topology (étale if G is simply connected).

Proof. We show that D_{G} represents the functor ”G-bundles trivialised on C^{0}”.

Firstly if we have a trivialised G-bundle P we can (flat locally) trivialise its restriction to {∞} and also Spf(O_{S}[[t]]). Then we obtain by projective limits also a trivialisation on the pullback to Spec(O_{S}[[t]]), and on Spec(O_{S}((t))) the two trivialisations differ by an element of g ∈ LG(O_{S}), so that g · ∗ ∈ D_{G}(O_{S}) is welldefined.

Conversely any point in D_{G} = LG/L^{+}G lifts Zariski-locally to g ∈ LG. This g defines a G-torsor on C, by gluing the trivial torsors on C^{0} and Spec(O_{S}[[t]]). For example for G-module E (over ℤ) we obtain an associated sheaf E_{G} such that E_{G}|C^{0} = E ⊗ O, and for an affine neighbourhood U of {∞} a section e ∈ Γ(U ∩ C^{0}, E_{G}) extends to all of U iff g^{−1} · e ∈ E[[t]]. (That this gluing works is easily seen for noetherian rings, which suffices for us since everything can be defined over finitely generated ℤ-algebras, and D_{G} is the inductive limit of schemes of finite type over ℤ. For non-Noetherian bases this is still true, but requires more thought. See [1], [2] for details)

For G = GL_{d} this suffices to construct a torsor. In general we have defined a tensor-functor from finitely generated torsion-free ℤ-G-modules to vectorbundles on C, and extends by inductive limit to all torsion-free G-modules. Especially for a closed embedding G ⊆ H = GL_{d} the ideal defining G in O_{H} is stable under G acting by left translations. Thus in the H-torsor P_{H} (with right H-action) it defines a closed subscheme P_{G} ⊆ P_{H}, stable under G-multiplication from the right. One checks that it is faithfully flat and a G-torsor, and that it defines the tensorfunctor. Thus our inverse map.

7 Linebundles on M_{G}

In this section S denotes a noetherian base-scheme, C → S a smooth projective geometrically connected curve, with a section {∞}, and G a semisimple simply connected split groupscheme. We want to compute the group of linebundles L on M_{G} which are rigidified, that is trivialised over the origin (which corresponds to the trivial torsor). By the previous this is the same as R_{C^{0}/S}(G)-equivariant linebundles on D_{G}, trivialised at the origin. These have as obvious invariant their central charge, that is the degree on the projective line ℙ^{1} ⊂ D_{G} corresponding to the affine root α_{0} = 1−θ. We claim that the central charge classifies linebundles:

Theorem 17 Suppose S is a connected noetherian base-scheme. Then the central charge defines an isomorphism from the group of rigidified linebundles on M_{G} to ℤ.

Remarks. a) In characteristic zero this has been shown by C. Sorger ([14]), following [10], [3]. Because of uniqueness his line-bundles must coincide with ours. Alternatively once the line-bundles are known to exist the arguments of [6] prove (in characteristic zero) the Verlinde-formula for them, so their global sections coincide with the spaces of conformal blocks. As Sorger’s construction starts from these he gets the same bundles as we do.

b) It would be nice to remove the hypotheses ”noetherian”.

c) For any representation E of G one obtains an L as determinant of cohomology. Its central charge can be computed as c = tr_{E}(H_{θ}^{2})/2. Also if E is orthogonal it has a square-root, the ”Pfaffian”. These suffice to construct all central charges for classical groups and for G_{2}. However for E_{8} there remains a factor 30.

Proof We first construct a linebundle with central charge 1. Denote by {α_{1}, …, α_{l}} the simple roots of G (semisimple, simple, and simply connected). We choose an invariant product on the cocharacters of the maximal torus T such that

< H_{α_{i}}, H_{α_{i}} >= 2 · a_{i},

with a_{i} = 1 for long roots α_{i}, and a_{i} = 2 or a_{i} = 3 for short roots. Also we define a symmetric matrix a_{ij} =< H_{α_{i}}, H_{α_{j}} >. If w_{0} ∈ W denotes the longest element in the Weyl-group, we define an involution on simple roots by the rule α_{i^{∗}} = −w_{0}(α_{i}).

Next we need some geometry on the space X = G/B, B = T · N^{+} the Borel formed with the positive roots. On X we have canonical G-equivariant line-bundles L(λ) corresponding to weights λ : T → 𝔾_{m}. The space of global sections Γ(X, L(λ)) is identified with the space of regular functions f on G satisfying f(g · b) = λ(b) · f(g). Evaluation at the origin defines a linear form l_{λ} on Γ(X, L(λ), which is a B-eigenvector (in the dual) of weight λ. Also it contains a highest weight-vector v_{λ} of weight −w_{0}(λ), normalised so that l_{λ}(w_{0}(v_{λ})) = 1. Denote its zero-divisor by V (v_{λ}). Especially important are the fundamental weights ϵ_{i} with ϵ_{i}(H_{α_{j}}) = δ_{ij}.

Next consider the product X × X, with the diagonal action of G. The G-orbits in it are indexed by the Weyl-group. The function

d_{i}(g_{1}, g_{2}) = l_{ϵ_{i}}(g_{2}^{−1}g_{1}(v_{ϵ_{i}}))

defines a G-invariant global section

d_{i} ∈ Γ(X × X, L(ϵ_{i^{∗}}, ϵ_{i})).

We denote its divisor by D_{i} ⊂ X × X. It is the closure of the G-orbit through (w_{0}, s_{i}). Furthermore if W_{≠i} ⊂ W denotes the subgroup generated by reflections s_{j} with j ≠ i, d_{i} takes the value 1 on (w_{0}, W_{≠i}).

Next we define a G-invariant section

e_{i} ∈ Γ(D_{i}, L(ϵ_{i^{∗}}, ϵ_{i} − α_{i}))

by the function

e_{i}(g_{1}, g_{2}) = l_{ϵ_{i}}(s_{i}g_{2}^{−1}g_{1}(v_{ϵ_{i}})).

(one checks without much difficulty that this is well-defined on D_{i}). It is normalised by the rule e_{i}(w_{0}, W_{≠i} · s_{i}) = 1. Note that

ϵ_{i} − α_{i} = −ϵ_{i} + ∑_{j≠i} α_{j}(H_{α_{i}}) · ϵ_{j}.

Similarly we can define

ẽ_{i} ∈ Γ(D_{i}, L(ϵ_{i^{∗}} − α_{i^{∗}}, ϵ_{i}))

by

ẽ_{i}(g_{1}, g_{2}) = l_{ϵ_{i}}(g_{2}^{−1}g_{1}s_{i^{∗}}(v_{ϵ_{i}})),

and ẽ_{i}(w_{0}, s_{i} · W_{≠i}) = 1. Finally one has the following equality (both sides are G-invariant and coincide on (w_{0}, s_{i})):

(e_{i} · ẽ_{i})^{a_{i}} = ∏_{j≠i}(s_{j}|D_{i})^{−a_{ij}},

as section of

⨂_{i≠j} L(ϵ_{j^{∗}}, ϵ_{j})^{−a_{ij}}.

The divisor of e_{i} is contained in the union of the closures of the G-orbits through (w_{0}, s_{i}s_{j}), where j ≠ i and s_{i} and s_{j} do not commute (These are the only possible codimension-2 orbits in D_{i} not excluded by the previous). Similarly for ẽ_{i} we get the orbits through (w_{0}, s_{j}s_{i}). From the equation above one sees that the multiplicities of these orbits are equal to −a_{ij}/a_{i} times the multiplicities in D_{i} ∩ D_{j}.

Finally we shall need a T -invariant deformation. Consider the strictly dominant one-parameter group ρ : 𝔾_{m} → T with α_{i}(ρ) = 1, 1 ≤ i ≤ l. It acts on X ×X via (ρ(t), 1). We show that the limits lim_{t→0,∞} of our objects exist, that is we obtain sections of line-bundles on (X × X) × ℙ^{1}. We only consider t → ∞, the other limit being similar. Also note that the conjugate (ρ(t), 1)(G) ⊂ G × G extends to a flat subgroup, whose fibre at {∞} is the product of the diagonal T with N^{+} × N^{−}. The limits of our sections will be invariant under this group, which will determine them up to a constant factor.

Lemma 18 (ρ(t), 1)(d_{i}) extends to a global section of L(ϵ_{i^{∗}}, ϵ_{i}) ⊠ O(2ϵ_{i}(ρ)) on X × X × ℙ^{1}. Its divisor D_{i} is flat over ℙ^{1}, and regular outside a closed subset of codimension ≥ 2 in each fibre. Its intersection with X × X × {∞} has two irreducible components, namely V (v_{ϵ_{i^{∗}}}) × X and X × w_{0}(V (v_{ϵ_{i}})). Similarly the intersection with X × X × {0} has two components w_{0}(V (v_{ϵ_{i^{∗}}})) × X and X × V (v_{ϵ_{i}}).

Proof. In the limit

lim_{t→∞}t^{−ϵ_{i^{∗}} (ρ)}(ρ(t), 1)(d_{i}) = v_{ϵ_{i^{∗}}} ⊠ w_{0}(v_{ϵ_{i}}) :

Namely firstly the limit exists and has first component v_{ϵ_{i^{∗}}}. Next it is N^{+}×N^{−}- invariant, and takes value 1 on (w_{0}, 1). This (and the analogous assertion at {0}) implies everything except regularity in relative codimension ≤ 1.

That could only fail over {0} or {∞}. Over {∞} this can only happen if D_{i} is not regular along the generic point of V (v_{ϵ_{i^{∗}}})×w_{0}(V (v_{ϵ_{i}})), the intersection of the two irreducible components. One point in this intersection is (w_{0}, s_{i}). But

t^{−ϵ_{i^{∗}} (ρ)}(ρ(t), 1)(d_{i})(w_{0}, s_{i}) = t^{−ϵ_{i^{∗}} (ρ)}l_{ϵ_{i}}(s_{i}w_{0}ρ(t)s_{i}(v_{ϵ_{i}})) = t^{−α_{i}(ρ)} = t^{−1},

so the equation for D_{i} has order one on the constant section (w_{0}, s_{i}). This ends the proof of the lemma.

Similarly the limit

lim_{t→∞}t^{−ϵ_{i^{∗}} (ρ)}(ρ(t), 1)(e_{i})

exists, vanishes on the first component and is equal to (v_{ϵ_{i^{∗}}} ⊠ s_{i}w_{0}(v_{ϵ_{i}}) on the second. However if we remove this component from the divisor both restrictions become non-zero.

Note that after this operation we obtain a reflexive sheaf on D_{i}, which will be locally free outside a closed subset of relative codimension ≥ 2. Especially it makes sense to restrict to the fibre at {∞}. Also the restrictions of the linebundle (better:reflexive sheaf) to the components of the fibre will change by the normal-bundle, so we obtain L(ϵ_{i^{∗}}) ⊠ ∏_{j≠i} L(ϵ_{j})^{−α_{i}(H_{α_{j}})}, and O ⊠ L_{i}.

Lemma 19 t^{−ϵ_{i^{∗}} (ρ)}(ρ(t), 1)(e_{i}) extends to a global section of

L(ϵ_{i^{∗}}, ϵ_{i} − α_{i}) ⊠ O(2ϵ_{i}(ρ))(−(w_{0}(V (v_{ϵ_{i^{∗}}})) × X × {0})) − (V (v_{ϵ_{i^{∗}}}) × X × {∞})))

on D_{i}. At {∞} its restriction to the two components are non-zero, and proportional to the products of the unique N^{±}-invariant sections.

Proof. We only have to show that the restrictions do not vanish. For this compute the value on (w_{0}, s_{i}) (which we did already in the previous proof).

This lemma will be applied to the product of four copies of X. We consider the pullback of the situation of the previous lemma (divisor with section of a line-bundle over it) once via the coordinates (1, 3) and (2, 4), and once via (1, 4) and (2, 3). After the degeneration (t → ∞) we shall obtain in totality the same (four) irreducible components, and the same sections of line-bundles over them.

Now finally we can construct our line-bundle L with central charge 1. It suffices to do this for curves over a regular base-scheme S (compatible with base-change), since the moduli-stack of curves is smooth. We have to associate to any G-torsor P on C → S a line-bundle L(P) on S. We first do this if G_{P} admits a ”generic” Borel, and then use descent to show that the construction is independant of it.

We use the Borel as follows: The G-equivariant line-bundles L(ϵ_{i}) on X define line-bundles on the twist X_{P} (a scheme over C). The Borel gives a section of X_{P}, and by pullback we obtain line-bundles L_{i} on C.

Next recall ([4, 6.1]) that from two line-bundles A and B on C one can construct a line-bundle < A, B > on the base S (assumed regular), as follows: Choose isomorphisms α : O_{C}(A) ≅ A, β : O_{C}(B) ≅ B, for two coprime divisors A and B on C. Then we can form the intersection-cycle A·B and its pushforward π_{∗}(A·B) on S. (For an irreducible cycle Z ⊂ C, its direct image π_{∗}(Z) vanishes if Z is not generically finite over its image π(Z). Otherwise π_{∗}(Z) = [k(Z) : k(π(Z)] · π(Z).) Then α and β define an isomorphism

< α, β >: O_{S}(π_{∗}(A · B)) ≅< A, B > .

Multiplying α by a local function f on S multiplies < α, β > by f^{deg(B)}, and similar for β. More generally an isomorphism between O_{C}(A)’s or O_{C}(B)’s is given by a rational function, and induces an isomorphism on the bracket. From this it follows that the bracket is well-defined. It is bilinear and symmetric in A and B, with no difficulty about signs except that the symmetry on < A, A > is equal to (−1)^{deg(A)}. Also if A is finite over S, < O(A), B > is isomorphic to the norm of B|A.

More generally one can define the meromorphic section < α, β > if β is only a generator of B in the generic points of the irreducible components of the divisor A of α. Also the construction < A, B > behaves well under base-change S′ → S. The same holds for < α, β > if the basechange is smooth.

Now, going back to torsors, we choose a covering S′ → S and a Borel in G_{P} ×_{S} S′ such that the associated map C ×_{S} S′ → X_{P} is smooth. For example S′ might classify ”generic” Borel’s. Then define the line-bundle L(P) on S′ as

L(P) = − ∑_{i} a_{i} < L_{i}, L_{i} > − ∑_{i<j} a_{ij} < L_{i}, L_{j} > .

(For simplicity we have switched to additive notation.)

If S” = S′ ×_{S} S′, we define an isomorphism

ϕ : pr_{1}^{∗}(L(P)) ≅ pr_{2}^{∗}(L(P)),

as follows:

As the divisors D_{i} ⊂ X × X as well as the sections d_{i}, e_{i} are all G-invariant, they define divisors respectively sections of line-bundles on (X × X)_{P}, and by smooth pullback on C ×_{S} S”. Especially we pullback the linear combination (using again additive notation)

∑_{i} a_{i} < d_{i}, e_{i} > + ∑_{i<j} a_{ij} < d_{i}, d_{j} > .

It defines a meromorphic section of

∑_{i} < L_{i^{∗}}^{(1)}+L_{i}^{(2)}, a_{i}(L_{i^{∗}}^{(1)}−L_{i}^{(2)})−∑_{j≠i} a_{ij}L_{j}^{(2)} > + ∑_{i<j} a_{ij} < L_{i^{∗}}^{(1)}+L_{i}^{(2)}, L_{j^{∗}}^{(1)}+L_{j}^{(2)} > .

One notes that the ”mixed terms” < L_{i^{∗}}^{(1)}, L_{j}^{(2)} > cancel because of symmetry (of a_{ij} and <, >). What remains is

∑_{i} a_{i}(< L_{i}^{(1)}, L_{i}^{(1)} > − < L_{i}^{(2)}, L_{i}^{(2)} >)+∑_{i<j} a_{ij}(< L_{i}^{(1)}, L_{j}^{(1)} > − < L_{i}^{(2)}, L_{j}^{(2)} >)

= pr_{2}^{∗}(L(P)) − pr_{1}^{∗}(L(P)).

Furthermore the divisor of this meromorphic section is the projection of the sum of the divisors of a_{i} · e_{i} (on D_{i}) and of a_{ij}(D_{i} · D_{j}). We claim that this sum vanishes:

This can be checked on X × X. However then all these divisors are linear combinations of the closures of G-orbits through (w_{0}, s_{i}s_{j}), for different (and non-commuting) reflections s_{i},s_{j}. The multiplicities have been computed before, and they cancel indeed.

Hence the conclusion is that we have defined an isomorphism

ϕ : pr_{1}^{∗}(L(P)) ≅ pr_{2}^{∗}(L(P)).

Before constructing a descent-datum out of ϕ we need some transitivity. For this consider the product S^{(4)} of 4 copies of S′, classifying a 4-tuple of generic Borel’s. On it we have various isomorphisms

ϕ_{ab} : pr_{a}^{∗}(L(P)) ≅ pr_{b}^{∗}(L(P)).

Lemma 20

ϕ_{13} · ϕ_{24} = ϕ_{14} · ϕ_{23}.

Proof. It suffices to check after faithfully flat base-change, so we may assume that G_{P} admits a Borel (not to be confused with the four Borel’s classified by S^{(4)}). Furthermore over S × 𝔸^{1} we can deform P into a T -torsor. As the desired equality holds up to an invertible function (which is constant on the second factor 𝔸^{1}) we may thus assume that P is actually a T -torsor. Thus all T -invariant constructions on X and its powers make sense on X_{P}, and we can use our previous deformation-procedure. Also it now suffices to check over a generic point of S^{(4)}, that is we may assume that C and P are defined over a field k, and our base is a field K ⊃ k. Furthermore the k-map C ⊗_{k} K → X_{P}^{4} is generic over k, that is dominant and its image avoids all subvarieties of codimension ≥ 2 which are defined over k.

Next we perform our deformation-procedure on X^{4}×ℙ^{1}, that is we transform over 𝔾_{m} with (ρ(t), ρ(t), 1, 1), and take the closure. Then d_{i,ab} = pr_{a,b}^{∗}(d_{i}) (a = 1, 2, b = 3, 4) extend to sections of

L(ϵ_{i^{∗}}, ϵ_{i})^{(a,b)} ⊠ O(2ϵ_{i}(ρ)),

with divisor D_{i,ab}. Furthermore the section e_{i,ab} gives a global section of the restriction to D_{i,ab} of

L(ϵ_{i^{∗}}, ϵ_{i} + α_{i})^{(a,b)} ⊠ O(2ϵ_{i}(ρ)),

which vanishes on pr_{b}^{−1}(w_{0}(V (v_{ϵ_{i^{∗}}})) × {0} and pr_{b}^{−1}((V (v_{ϵ_{i^{∗}}}) × {∞}. These are Cartier-divisors on D_{1,ab} outside a k-defined closed subset of sufficiently high codimension, thus after pullback to (C × ℙ^{1}) ⊗_{k} K we can substract them, and the e_{i,ab} become sections of a certain line-bundle on the pullback of D_{i,ab}. This pullback will be finite and flat over ℙ^{1} ⊗_{k} K, and thus we may form the norm of these sections, which become global sections of a certain line-bundle on ℙ^{1} ⊗_{k} K. This is the basic idea, although its execution will turn out a little different. Namely we do not modify the line-bundles and work instead with meromorphic sections.

We repeat the same with d_{j,ab}|D_{i,ab}, only that now there is no need to substract any divisors over {0} or {∞}. Finally we multiply these with a_{i} respectively with a_{ij}, take the product of these sections once for the pairs (a, b) = (1, 3), (2, 4), and once for (a, b) = (1, 4), (2, 3), and then form the quotient ϕ_{13} · ϕ_{24}/ϕ_{14} · ϕ_{23}. It defines a regular section of the trivial line-bundle on 𝔾_{m} ⊗_{k} K, that is a function. We claim that this function is regular with value 1 in {0, ∞}. This implies our claim (evaluate at t = 1).

Let us check at {t = ∞}: Over the formal completion of ℙ^{1} ⊗_{k} K at {t = ∞} the divisors D_{i,ab} are the disjoint sum of two disjoint divisors specialising to the two components over {t = ∞}. Modify the sections e_{i,ab} by multiplying them by t^{ϵ_{i^{∗}}} on the the part specialising to pr_{b}^{−1}(V (v_{ϵ_{i^{∗}}})×X). We thus get new sections ẽ_{i,ab} which are regular and everywhere (on the formal completion) nonzero. Thus if we replace in the definition of ϕ_{ab} each e_{i,ab} by ẽ_{i,ab}, the result will be a regular nonzero section of the corresponding constant line-bundle. It differs from the original ϕ_{ab} by some power of t which is easily computable from the degrees of the divisors pr_{b}^{−1}(w_{0}(V (v_{ϵ_{i^{∗}}})). It follows that these t-powers cancel in ϕ_{13} · ϕ_{24}/ϕ_{14} · ϕ_{23}. Hence this function is also regular at {t = ∞}.

To see that it takes the value 1 we use the modified ẽ_{i}. Consider each summand < d_{i}, ẽ_{i} > respectively < d_{i}, d_{j} > separately. In each case the restriction to {∞} of the divisors D_{i,ab} have two irreducible components which are pullbacks of the divisors V (v_{ϵ_{i^{∗}}}) respectively V (w_{0}(v_{ϵ_{i}})) on the factors. Thus we obtain the product (over 1 ≤ a ≤ 4) of (the pullbacks of) the N^{±}-invariant meromorphic section of a certain line-bundle (the same in both cases) on the a-th factor. These are only determined up to a factor, but these ambiguities cancel out.

Thus the quotient is 1 at {∞},and the lemma is shown.

Now going back to S” = S′×_{S}S′ the restriction of ϕ to the diagonal S′ defines an automorphism of L(P). Multiply (say from the left) ϕ with its inverse to obtain a new isomorphism ϕ^{∗} between pr_{1}^{∗}(L(P) and pr_{1}^{∗}(L(P). I claim that ϕ^{∗} is a descent-datum:

We have to show (with hopefully obvious notation) that

ϕ_{13}^{∗} = ϕ_{12}^{∗} · ϕ_{23}^{∗},

which means

ϕ_{13} · ϕ_{22} = ϕ_{12} · ϕ_{23},

and this follows from the lemma by setting ”2 = 4”, that is restricting to a partial diagonal.

So finally we obtain our L(P) on S. That it is independant from the choice of S′ is clear (for two of them consider their disjoint union). Also we can rigidify it by substracting its value on the trivial torsor. It remains to show that L(P) has central charge 1. It suffices to do this over a field k, so we can forget about rigidification.

To compute the central charge we form the G-torsor P over C × ℙ^{1} obtained from the projective line ℙ^{1} ⊂ D_{G} corresponding to α̃_{0} = 1 − θ, and try to compute the degree of the line-bundle L(P) on ℙ^{1}. Evidently this depends only on the pullback of L(P) to the moduli-space of SL_{2}-torsors.

On this space we have a wellknown line-bundle of central charge 1: Namely an SL_{2}-torsor over C → S is a rank-2-bundle E with trivial determinant. Then det(H^{∗}(C, O^{2})) ⊗ det(H^{∗}(C, E))^{⊗−1} is the desired line-bundle on S. To see that its central-charge is indeed 1 one computes for the SL_{2}-torsor on C × ℙ^{1} mentioned in the beginning:

It is given a sub-bundle E of the constant bundle O_{C} ⊕O_{C}(∞), with quotient concentrated on {∞} × ℙ^{1} and isomorphic to O(1). Thus the determinant of cohomology of E has degree −1.

Finally we finish by showing that the restriction of L(P) to SL_{2}-torsors is isomorphic to the determinant of cohomology. For this we use that line-bundles are uniquely determined by their central charge, that is the second half of the theorem (which we show next, of course without using the following arguments).

Thus the Picard-group is infinite cyclic, and it suffices if both line-bundles have the same non-zero degree on one curve.

To construct such an example consider an SL_{2} torsor on C × ℙ^{1} of the form E = M ⊕ M^{⊗−1}, with M a very ample line-bundle on C × ℙ^{1}. For its determinant of cohomology we obtain − < M, M >, whose degree is the negative self-intersection of M (and non-zero).

The computation of the degree of L(P) is more complicated. We know that the structure-group of P reduces to B (even to T), and thus by pullback from X_{P} we obtain as before a candidate L(P)^{naiv}. We do not yet know that it coincides with L(P) because our Borel will not be ”generic”. However we easily construct a covering S′ → S = ℙ^{1} as needed for the definition of L(P), as follows:

For each positive root α of G the line-bundle L_{α} is a positive power of M. Its global sections (over C ×ℙ^{1}) are affine spaces, and the product of these affine spaces is isomorphic to the space of global sections of the unipotent N_{P}^{−} opposite to the unipotent radical of B_{P}. Conjugating B_{P} by these sections defines the desired family of ”generic” Borel’s over the product S′ of S and this affine space.

Thus over S′ L(P) is given by our formula. Pulling back via the zero-section we derive that this also holds over S. Note that rigidification does not change this picture because C is constant.

Now finally the map SL_{2} → G induces an embedding ℙ^{1} ⊆ X, and the pullback of L(λ) under it is O(λ(H_{θ})). It follows that L(P) becomes a multiple of < M, M >, where the factor is the negative of

∑_{i} a_{i}ϵ_{i}(H_{θ})^{2} + ∑_{i<j} a_{ij}ϵ_{i}(H_{θ})ϵ_{i}(H_{θ})

= 1/2 ∑_{i,j} < H_{α_{i}}, H_{α_{j}} > ·ϵ_{i}(H_{θ})ϵ_{i}(H_{θ}).

But this quadratic form is W -invariant, so it takes the same value on H_{θ} as on H_{α_{i}}, for any long simple root α_{i}. And finally this value is 1.

All in all we have finished the constructive part of the proof, and can pass to uniqueness.

Suppose a line-bundle L on M_{G} = R_{C^{0}/S}(G)\D_{G} has trivial central charge, that is comes from an operation of R_{C^{0}/S}(G) on the trivial line-bundle on D_{G}. It then corresponds to a character of R_{C^{0}/S}(G), and we have to show that all such characters are trivial. We do this following an argument communicated to me by A.Beilinson, who (as I remember) attributes it to V.G.Drinfeld.

As we are working over a noetherian base and as R_{C^{0}/S}(G) is inductive limit of algebraic schemes over this base, it suffices to show that the character is trivial on any point with values in an artinian local ring R, with algebraically closed residue-field k. We also may assume that our base is S = Spec(R).

For any finite set T ⊂ C of k-points containing {∞} we may form the open affine subscheme C − T ⊂ C. As G-torsors are étale locally trivial on C_{T} the moduli-stack M_{G} can be represented as

M_{G} = R_{C−T/S}(G)\ ∏_{x∈T} D_{G,x}.

The product is over copies of D_{G} indexed by x ∈ T, and one has to choose local parameters at all x ∈ T.

Thus L defines an R_{C−T/S}(G)-equivariant line-bundle on the product. The isomorphism class of this line-bundle (forgetting equivariance) is given by a tuple of central charges c(x) ∈ ℤ. These do not change if we enlarge T, thus define a function from C(k) to ℤ, which extends to a linear function on divisors. I claim that this function vanishes on principal divisors:

This can be checked over the base k. Assume div(f) = ∑_{x∈T} n(x)x is a principal divisor. Define two homomorphisms SL_{2} → R_{C−T/S}(G) as follows: One is the constant homomorphism defined by the highest root θ, the other one is its twist by f (with root-vectors fX_{−θ}, f^{−1}X_{θ}). They coincide on diagonal matrices, that is they share the same one-parameter group H_{θ} : 𝔾_{m} → R_{C−T/S}(G). However if we choose the unique lifts (of SL_{2}’s) into the product of central extensions L̃G_{x}, the two characters differ by central characters n(x), and the actions of the diagonal matrices on the fibre of L at the origine differ by ∑_{x∈T} n(x)c(x). But these two actions are induced from the same action of R_{C−T/S}(G), thus must coincide, and the sum vanishes.

So finally our map c(x) factors over the Picard-group of C. As the Jacobian of C is divisible it must be trivial on divisors of degree 0. As it is also trivial on ∞ all c(x) vanish.

It follows (now again over R) that L is defined by an action of R_{C−T/S}(G) on the trivial bundle, that is by a character. That is the original character of R_{C^{0}/S}(G) extends to all R_{C−T/S}(G). It is necessarily trivial on the constants G, and also on all unipotents R_{C−T/S}(𝔾_{a}) parametrised by roots, as these lie in the commutator.

To show that it is trivial on an arbitrary g ∈ R_{C−T/S}(G)(R) we denote by K the function-field of C ⊗_{R} k, and note that G(K) is generated by its unipotent subgroups. Thus the image of g in G(K) can be written as the product of unipotent elements. These lie in some R_{C−T′/S}(G)(k) (T′ sufficiently big) and lift to R_{C−T′/S}(G)(R). Hence we may assume that g reduces to 1 modulo the maximal ideal of R, that is it lies in the formal group. But by the usual infinitesimal arguments (as in lemma 10 b)) this is contained in the subgroup generated by unipotents.

This finishes the proof of the theorem.

Remark. Everything works for twisted inner forms, and I think also for outer forms.

Remark. Suppose P is a G-torsor in C. Then the center Z(G) of G acts trivially on the fibre of L in P. Namely Z(G) ⊂ R_{C^{0}/S}(G) acts trivially on D_{G}, so its action on the line-bundle L is given by a character. Considering the fibre in the origin one sees that this character extends to G, thus must be trivial. However other automorphisms may act non-trivially, even for stable torsors.

This happens if the centralisers of elements of G are not simply connected. The simplest example occurs for Spin(6) (where L is the Pfaffian). As a consequence L does not descend to a line-bundle on the coarse moduli-space of stable Gtorsors.

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