DE(g) where E is the universal G-bundle on MG;X . Remark that the bundle

E(g) comes with a natural quadratic form given by the Cartan-Killing form.

Hence the choice of a theta-characteristic denes, by the above, a square

1=2

root !MG;X () of !MG;X .

6.6. The pfaan divisor. It may seem easier to construct the pfaan

bundle looking at it from a divisorial point of view using smoothness of

MSOr ;X . Suppose F is a vector bundle on XS equipped with a non de-

generate quadratic form with values in !X . Suppose moreover that S is

smooth and that there are points s; t 2 S such that H 0 (X; Fs ) = 0 and

H 0(X; Ft ) 6= 0. We know from section 6.1.4 that if K 0 ! K 1 represents the

cohomology of F, then DF is the line bundle associated to the divisor de-

ned by det(

) = 0. Now, locally

may be represented by a skew-symmetric

matrix , so we may take its pfaan. This denes a local equation for a

divisor, which will be called the pfaan divisor, hence, by smoothness of S,

our pfaan line bundle. Of course the preceding sentence has to be made

rigorous, but this may seem easier than the (rather formal) considerations

of ([14], x7) which lead to Theorem 6.3.1.

However7 , even if the hypothesis of smoothness is satised for MSOr ;X ,

this approach fails to dene line bundles P for all theta-characteristics .

The point is that the hypotheses of the existence of s 2 S such that

0 (X; Fs ) = 0 is not always satised: the equation det(

) = 0 may not

H

dene a divisor (but the whole space).

In order to see this, consider the component MSOr ;X of MSOr ;X , contain-

0

ing the trivial SOr -bundle. Actually we haven0 t seen yet that the connected

components of MG;X are parameterized by 1 (G) (i.e. by their topological

type), but for the moment let0 s just use that MSOr ;X has two components:

MSOr0;X and MSOr1;X . They are distinguished by the second Stiefel-Whitney

class

w2 : He1t(X; SOr ) ! He2t (X; Z=2Z) = Z=2Z:

(6.6 a)

Let r 3 and (F; ) be the universal quadratic bundle over MSOr ;X X.

0

For a theta-characteristic, consider the substack of section 6.1.4.

7 There are of course many other reasons to prefer to construct a line bundle directly

and not as a line bundle associated to a divisor.

Moduli of G-bundles 33

6.6.1. Proposition. The substack of MSOr0;X is a divisor if and only

if r or are even.

Proof. We start with a useful lemma.

6.6.2. Lemma. Let A = (E; q) be an SOr -bundle, r 3 and be a theta-

characteristic. Then

w2 (A) = h0 (E

) + rh0 () mod 2:

(6.6 b)

Proof. Indeed, by Riemann0 s invariance mod 2 theorem, the right-hand side

0

of (6.6 b), denoted w2 (A) in the following, is constant over the 2 connected

components of MSOr ;X . Because (6.6 b) is true at the trivial SOr -bundle

0 0

T, it is enough to prove that w2 is not constant. As w2 (T ) = 0, we have to

construct an SOr -bundle A such that w2 (A) 6= 0. In order to do this, let

0

L; M be points of order 2 of the jacobian, such that for the Weil pairing we

have < L; M >= 1. The choice of a trivialization of their squares denes a

non degenerated quadratic form on

E = (L

M) L M (r 3)OX

0

hence an SOr -bundle A. By [17], we know that we have w2 (A) =< L; M >=

1 which proves (6.6 b).

Now choose an ineective theta-characteristic 0 and set L = 0

1 . If

r is even, there exists a SOr -bundle A = (E; q) such that H 0(E

) = 0 and

w2(A) = 0 (choose E = rL with the obvious quadratic form and use (6.6 b)).

If r is odd and is even, there exists a SOr -bundle A = (E; q) such that

H 0(E

) = 0 and w2 (A) = 0 (by Lemma 1.5 of [1], there is an SL2 -bundle

F on X such that H 0 (X; ad(F )

) = 0, then choose E = ad(F ) (r 3)L

with the obvious quadratic form). If r and are odd, then H 0 (E

) is odd

for all A 2 MSOr ;X .

0

7. Affine Lie algebras and groups

In the following sections I suppose k = C . In order to study the innite

Grassmannian I need some basic material on (ane) Lie algebras which I

will recall brie

y. I start xing the notations I will use in the rest of these

notes. The reader who is not very familiar with Lie algebras may have a

closer look at [8].

34

7.1. Basic notations.

7.1.1. Lie groups. From here to the end of these notes, G will be a simple

(not necessarily simply connected) complex algebraic group. Let G ! G be

e

the universal cover of G; its kernel is a subgroup of the center Z(G) G,

e e

canonically isomorphic to 1 (G). We will denote the adjoint group G=Z(G)

e e

by G. We will x a Cartan subgroup H G (and denote by H and H its e

inverse image in G and G respectively) as well as a Borel subgroup B G

e

(and denote by B and B its inverse image in G and G respectively).

e e

7.1.2. Lie algebras. Let g = Lie(G), b = Lie(B) and h = Lie(H). By the

roots of g we understand the set R of linear forms on h such that g = fX 2

g=[X; H] = (H) 8H 2 hg is non trivial. We have the root decomposition

g = h g :

2R

Let = f1 ; :::; r g be the basis of R dened by B and be the correspond-

ing highest root; we denote 0 = . Let ( ; ) be the Cartan-Killing form,

normalized such that (; ) = 2. Using ( ; ) we will identify h and h in the

sequel. The coroots of g are the elements of h dened by _ = (;) ; they

2

form the dual root system R_ .

Let Q(R) and Q(R_ ) be the root and coroot lattices with basis given

by f1 ; :::; r g and f_ ; :::; _ g respectively . We denote P(R) and P(R_ )

r

1

the weight and coweight lattices, i.e. the lattices dual to Q(R_ ) and Q(R)

respectively. They have basis given by the fundamental weights $i and

coweights $i_ dened by

< $i ; _ >=< $i_ ; j >= ij :

j

Note that Q(R_ ) Q(R) and P(R_ ) P(R) and that we have equality if

all roots are of equal length, i.e. if we are in the A-D-E case.

7.1.3. Representations. We denote by P+ P(R) the set of dominant weights

and by f$1 ; : : : ; $r g the fundamental weights. The set P+ is in bijection

with the set of simple g-modules; denote by L() the g-module associated

to the dominant weight .

7.1.4. The center. We will identify the quotient P(R_ )=Q(R_ ) with Z(G)e

through the exponential map; its Pontrjagin dual Hom(Z(G); C ) identies

e

to P(R)=Q(R). Recall from ([Bourbaki], VIII, SS3, prop. 8) that a system

of representatives of P(R_ )=Q(R_ ) is given by the miniscule coweights of

Moduli of G-bundles 35

R : these are exactly the fundamental coweights $j_; corresponding to the

roots j 2 having coecient 1 when writing

X

= ni i :

i 2

We will denote J(G) = fi 2 f1; : : : ; rg=ni = 1g and J0 (G) = J(G) [ f0g.

e e e

Then the set J0 (G) has a natural group structure provided by the group

e

structure on P(R_ )=Q(R_ ) which we will denote additively. Recall, for

further reference, that the miniscule coweights are given by

Br Cr Dr

Ar E6 E7 E8 F4 G2

Type of g (r 2) (r 2) (r 3)

f1; : : : ; rg f1g frg f1; r 1; rg f1; 6g f7g ; ; ;

J

For j 2 J0 (G) we will denote the corresponding element of Z(G), 1 (G),

e e

P(R)=Q(R) or P(R_ )=Q(R_ ) under the above identications by zj , j , $j ,

$j_ or wj respectively.

The subgroup of Z(G) corresponding to 1 (G) will be denoted by Z, the

e

corresponding subgroup of J0 (G) by J0 and the lattice generated by Q(R_ )

e

and $j for j 2 J0 by J (R_ ).

_

7.1.5. Dynkin diagrams. Associated to the Lie algebra g is its Cartan matrix

A with coecients aij = hi; _ i; i; j = 1; : : : ; r: This matrix is invertible;

j

its determinant is the connection index Ic, i.e. the index of the root lattice

Q(R) in P(R). The associated Dynkin diagram is constructed as follows.

The nodes are the simple roots i 2 ; the nodes i and j are connected

by maxfjaij j; jaji jg lines. Moreover these lines are labeled by \>" if aij 6= 0

and jaij j > jaji j. These diagrams have various interpretations: the i-th node

may be seen representing i or $i or _ or been labeled, for example by

i

_ dened by _ = Pr c_ _ . Note that if is

the dual Coxeter numbers ci P

r $ , then h; _i=1 iPr c_ . The number

i

i = i=1 i

the half sum of the roots = i=1 i

_ = 1 + h; _ i is called the dual Coxeter number. The possible Dynkin

h

diagrams, as well as their connection indexes and dual Coxeter numbers are

resumed in table A.

36

7.2. Ane Lie algebras and groups.

7.2.1. Loop algebras and central extensions. Let Lg = g

C C ((z)) be the

loop algebra of g. It has a canonical 2-cocycle dened by

g : (X

f; Y

g) 7! (X; Y ) Res(gdf);

(7.2 a) z=0

hence a central extension

0 ! C ! Lg ! Lg ! 0

(7.2 b) c

In other words, this means that on the level of vector spaces Lg = C c Lg

c

with Lie bracket given by (c central):

[X

f; Y

g] = [X; Y ]

fg + (X; Y ) Res(gdf):

(7.2 c) z=0

In the following, we denote X[f] the element X

f of Lg; if f = z n it is also

denoted by X(n). The Lie algebra Lg has several subalgebras which will be

important for us. Dene

L+g = g

C C [[z]], L>0 g = g

C zC [[z]], L<0 g = g

C z 1C [z 1 ]

These are in fact subalgebras of Lg.

c

7.2.2. Irreducible and integrable representations. In nature, representations

of Lg are projective; this is why we look at (true) representations of Lg.c

Fix an integer `. Call a representation of Lg of level `, if the center acts

c

by multiplication by `. In order to construct such representations we start

with a nite dimensional representation L(), which we may view as an

L+ g-module by evaluation. As the cocycle (7.2 a) is trivial over L+g the

central extension L+ g obtained by restriction from (7.2 b) splits. Hence we

d

may consider L() as an L+g-module of level ` by letting the center act by

d

multiplication by `; denote this module L` (). Now consider the generalized

Verma module:

c

M` () = IndLg g L` () = U(Lg)

U(L+g) L` ()

d

c

d

L+

In the case when ` is not the critical level h_ (the dual Coxeter number),

M`() has a unique irreducible quotient H`(). Moreover, if ` (; )

then H` () has an important niteness condition: for all X 2 g and all

f 2 C ((z)) the element X[f] acts locally nilpotent on H`(), i.e. for all

u 2 H` () there is N such that X[f]N :u = 0. Such Lg-modules are called

c

Moduli of G-bundles 37

integrable and it can be shown that all irreducible integrable Lg-modules

c

arise in this way. In view of this it is convenient to dene the following set.

P` = f 2 P(R)=(; i ) 0 for i 2 I and (; ) `g:

In the rest of this subsection we restrict to positive8 `. Actually if 2 P`

then H` () is the quotient of H` () by the sub-module Z (`) generated

by X ( 1)`+1 (;)

v , where v is a highest weight vector of L(). By

Poincar-Birkho-Witt, M (`) = U(L<0 g)

C L . It follows that we have

e

the exact sequence

0 ! Z (`) ! U(L<0 g)

C L` () ! H` () ! 0:

(7.2 d)

In other words:

[H` ()]L>0 g = L`() = fv 2 H` ()=L>0 g:v = 0g

(7.2 e)

H`() is generated by L() over L<0 g with only one relation:

(7.2 f)

X ( 1)`+1 (;)

v

= 0:

7.3. Loop groups and central extensions. We already have dened the

loop groups LG and L+ G in 5.1. The Lie algebra of LG is Lg, as the kernel

of the homomorphism LG(R["]) ! LG(R) is Lg(R) = g

C R((z)). For the

same reason we have Lie(L+ G) = L+g.

7.3.1. The adjoint action. Let H be an innite dimensional vector space

over C . We dene the C -space End(H) by R 7! End(H

C R), the C -group

GL(H) as the group of its units and PGL(H) by GL(H)=Gm . The C -group

LG acts on Lg by the adjoint action. We dene the adjoint action of LG on

e e

Lg as follows:

c

d

Ad(

):(0 ; s) = Ad(

):0 ; s + Res(

1 dz

; 0 )

z=0

where

2 LG(R), = (0 ; s) 2 Lg(R) and ( ; ) is the R((z))-bilinear

e c

extension of the Cartan-Killing form. Consider an integral highest weight

representation : Lg ! End(H). The basic result we will use in the sequel

c

is the following:

h_ ; I will come back to this later. See also

8 The interesting case for us is actually ` =

Frenkel0 s lectures.

38

7.3.2. Proposition. (Faltings) Let R be a C -algebra,

2 LG(R). Locally

e

over Spec(R), there is an automorphism u

of HR = H

C R, unique up to

R, such that

()

HR HR

G

(7.3 a) u

u

(Ad(

):) G

HR HR

for any 2 Lg(R).

c

Again, the important fact here is that we work over any C -algebra (and

not only over C .) The above proposition is proved in ([2], App. A) in the

case SLr ; its generalization to G is straightforward.

7.3.3. Integration. An immediate corollary of the above proposition is that

the representation may be \integrated" to a (unique) algebraic projective

representation of LG, i.e. that there is a morphism of C -groups : LG !

e e

PGL(H) whose derivate coincides with up to homothety. Indeed, thanks

to the unicity property the automorphisms u associated locally to

glue

together to dene an element (

) 2 PGL(H)(R) and still because of the

unicity property, denes a morphism of C -groups. The assertion on the

derivative is a consequence of (7.3 a).

7.3.4. Central extensions. We are now looking for a central extension of LG e

such that its derivative is the canonical central extension (7.2 b). In order to

do this, we apply the above to the basic representation H1 (0) of Lg. Consider

c

the central extension

1 ! Gm ! GL(H1 (0)) ! PGL(H1 (0)) ! 1:

(7.3 b)

Then the pullback of (7.3 b) to LG is convenient: it denes a central exten-

e

sion to which we refer to as the canonical central extension of LG: e

1 ! Gm ! LG ! LG ! 1

d

(7.3 c) e e

What happens if we restrict to L+ G ? e

7.3.5. Lemma. The extension (7.3 c) splits canonically over L+G. e

Proof. ([14], 4.9) By construction of (7.3 c), it is enough to prove that the

representation : L+ G ! End(H1 (0)) integrates to a representation :

e

L+g ! GL(H1 (0)). This will follow from the fact that in the case

2

Moduli of G-bundles 39

L+ G(R) we can normalize the automorphism u of Proposition 7.3.2. Indeed,

e

as L(0) = [H1 (0)]L+ g by (7.2 e), it follows from (7.3 a) that u maps L(0)R to

L(0)R . Now L(0)R is a free R-module of rank one, hence we may choose u

(in a unique way) such that it induces the identity on L(0)R .

8. The infinite Grassmannian

We will now study in some more detail the innite Grassmannian QG for

connected reductive groups over the complex numbers.

8.1. Ind-schemes. The category of C -spaces is closed under direct limits.

A C -space (resp. C -group) will be called a (strict) ind-scheme (resp. ind-

group) if it is the direct limit of a directed system of quasi-compact C -schemes

(Z )2I such that all the maps i; : Z ! Z are closed embeddings.

Remark that an ind-group is in general not a direct limit of a directed system

of groups. Any property P of schemes which is stable under passage to

closed subschemes make sense for ind-schemes: We say that Z satises the

ind-property P if each Z does. In particular we may dene Z to be of

ind-nite type or ind-proper.

An ind-scheme is integral (resp. reduced, irreducible) if it is the direct

limit of an increasing sequence of integral (resp. reduced, irreducible) C -

schemes.

A C -space Z is formally smooth if for every C -algebra R and for every

nilpotent ideal I R the map Z(Spec(R)) ! Z(Spec(R=I)) is surjective.

If R is an ind-scheme of ind-nite type, then formal smoothness is a local

property9 .

8.1.1. Lemma. Let Z be an ind-scheme, direct limit of an increasing se-

quence of C -schemes. Then the following is true (see [2], 6.3 for a proof).

(i) If Z is reduced and is the direct limit of an increasing sequence of C -

schemes (Zn ) then Z = lim(Zn ).

!

(ii) If Z is covered by reduced open sub-ind-schemes, Z is reduced.

(iii) The ind-scheme Z is integral if and only if Z is reduced and irreducible.

(iv) If U is a C -scheme and U Z is integral, Z is integral.

9 Formal smoothness is a weak property in our innite dimensional setup. For instance,

we will see that LGLr is formally smooth but not reduced.

40

8.2. The ind-structure of loop groups. Ind-schemes and ind-groups will

be important for us as the C -groups LG, LX G will be ind-groups, as well

as QG which will be an ind-scheme. Actually L+ G is an innite product of

ane schemes.

8.2.1. Lemma. The C -group L+GLr is represented by

1

Y

GLr (C ) Mr (C )

1

where Mr (C ) is the set of r r-matrices with entries in C .

C -algebra R the set GLr (R[[z]])

Proof. This follows from the fact that for any P

consists of the matrices of the form A(z) = 1 An z n with A0 2 GLr (R)

n=0

and An 2 Mr (R) for n 1.

Consider more generally the sub-C -space LGL(N) of LGLr dened for any

r

(N) (R) of matrices A(z) such that both A(z) and

C -algebra R by the set GLr

A(z) 1 have poles of order N. Of course, by denition LGL(0) = L+ GLr .

r

8.2.2. Lemma. The C -space LGL(N) is representable as an ane scheme.

r

Proof. If M (N) (R) is the set of r r-matrices with coecients in R, then

the corresponding C -space M(N) is represented by the ane scheme

1

Y

Mr (C )

n= N

Now remark that LGL(N) is represented by the closed ane subscheme of

r

M (N) M(N) of pairs of matrices (A(z); B(z)) such that A(z)B(z) = I.

8.2.3. Corollary. The C -group LGLr is an ind-group of ind-nite type,

direct limit of the increasing sequence of schemes (LGL(N) )N0

r

For a general reductive group choose an embedding G GLr . Then the

ind-structure of LGLr induces an ind-structure on LG.

8.3. The ind-structure of the innite Grassmannian. The following

theorem describes the ind-structure of QG

8.3.1. Theorem. Let G be a connected reductive complex group. Then

(i) The C -space QG is an ind-scheme, ind-proper of ind-nite type.

(ii) The projection : LG ! QG admits, locally for the Zariski topology, a

section.

Moduli of G-bundles 41

(iii) The ind-scheme QG is formally smooth.

(iv) The ind-scheme QG is reduced if and only if Hom(G; Gm ) = 0.

Proof. It follows from corollary 8.2.3 that QGLr is an ind-scheme of ind-nite

type: if Q(N)r = LGL(N) =L+ GLr ,

r

GL

QGLr = lim Q(N)r

! GL

In order to see that QGLr is ind-proper we use the following lattice approach

to QGLr (see [2],x2). For any C -algebra R we dene a lattice in R((z)))r to

be a sub-R[[z]]-module W of R((z)))r such that

zN R[[z]]r W z N R[[z]]r

for some integer N and such that W=z N R[[z]]r is projective.

8.3.2. Proposition. The C -space QGLr represents the functor which asso-

ciates to any ane C -scheme U = Spec(R) the set of lattices W R((z)))r .

Proof. This is a consequence of Proposition (5.3.2). Indeed if E is a vector

bundle over XU together with a trivialization : EjXU ! U C r , we get

by restriction an isomorphism : R((z))r ! H 0 (DU ; EjDU ). The inverse

image W of H 0 (DU ; EjDU ) is a lattice in R((z))r . On the other hand, given

a lattice W R((z))r we get a vector bundle EW on X by gluing the trivial

bundle over XU with the bundle on DR associated to the R[[z]]-module W;

the gluing isomorphism is given by W R[[z]] R((z)) ! R((z))r coming from

the inclusion W R((z))r . By denition, EW comes with a trivialization

W and it is easy to see that both constructions are inverse to each other.

It follows from the above that QGLr is ind-proper as the latter functor is.

Let us look at the special case of G = SLr . Then we obtain special

lattices, i.e. the lattices such that the projective module W=z N R[[z]]r is of

rank Nr. Let FN be the complex vector space z N C [[z]]r =z N C [[z]]r . Then

dim(FN ) = 2Nr. Multiplication by z induces a (nilpotent) endomorphism

N of FN . If follows that 1 + N is an automorphism of FN , hence we get

an automorphism of Grass(Nr; 2Nr); denote by VN its xed points. Then

it is easy to see from the above proposition that the C -space LSL(N) is r

isomorphic to the projective variety VN .

Once we know (i) for GLr it follows for a general reductive group after

choosing an embedding G GLr from the following lemma.

8.3.3. Lemma. Suppose G H is an inclusion of ane algebraic groups

such that H=G is ane and such that QH is an ind-scheme of ind-nite

42

type. Then QG is also an ind-scheme of ind-nite type and the morphism

QG ! QH is a closed embedding. In particular, if QH is ind-proper, QG is.

Proof. Left to the reader.

In order to see (ii), one reduces (use that is LG-invariant) to show that

admits a section over a Zariski open neighborhood of [e] 2 QG , which follows

<0 G is the C -group dened by R 7! G z 1 R[z 1 ],

from the fact that if L

the multiplication map

: L<0 G L+G ! LG

(8.3 a)

is an open immersion. The last statement is proved in ([2], 1.11) using that

QG may be seen as parameterizing G-bundles over P1 and that if E is a

G-bundle over P1 for some U then the set of points u 2 U such that Eu is

U

trivial is an open subset of U as H 1 (P1 ; O

g) = 0.

Formal smoothness of QG follows from formal smoothness of LG and

the above, so it remains to prove (iv). Actually to see that QG is reduced

whenever Hom(G; G m ) = 0 is quite delicate. It is proved in ([2],6.4) for SLr

where it is deduced from the corresponding statement for LP1G and a direct

calculation. For general G it is proved in ([14], 4.6) where it is deduced

from a theorem of Safarevi [24]. I will not enter into the proof here, let

c

me just say why QG m is not reduced. Actually, as L+ G m is reduced it is

equivalent to show that LG m is not reduced. Consider : G m ! G m dened

by (x) = xn and the induced morphism LG m ! LG m . Then the image is

not contained in (LG m )red , hence LG m is not reduced.

8.4. The connected components of the innite Grassmannian. Sup-

pose in this subsection that G is semi-simple and recall the notations of

section 7.1.

8.4.1. Lemma. ([4], 1.2)

(i) The group 0 (LG) is canonically isomorphic to 1 (G).

(ii) The projection : LG ! QG induces a bijection 0 (LG) ! 0 (QG ).

Each connected component of QG is isomorphic to QG . e

Proof. By [21], there exists a nite family of homomorphisms x : G a ! G e

such that for any extension K of C , the subgroups x (K) generate G(K). e

Since the ind-group G a (C ((z))) is connected, it follows that LG is connected.

e

In the general case, consider the exact sequence 1 ! 1 (G) ! G ! G ! 1

e

as an exact sequence of tale sheaves on D := Spec C((z)). Since H 1 (D ; G)

e e

Moduli of G-bundles 43

is trivial [22], it gives rise to an exact sequence of C -groups

1 ! LG=1 (G) ! LG ! H 1 (D ; 1 (G)) ! 1

(8.4 a) e

Then (i) follows from the connectedness of LG and the canonical isomor-

e

phism H 1 (D ; 1 (G)) ! 1 (G) (Puiseux theorem).

To prove (ii), we rst observe that the group L+ G is connected: for any

2 L+ G(C ), the map F

: G A 1 ! L+G dened by F

(g; t) = g 1

(tz)

satises F

(

(0); 0) = 1 and F

(1; 1) =

, hence connects

to the origin.

Therefore the canonical map 0 (LG) ! 0 (QG ) is bijective. Moreover it

follows from (8.4 a) that (LG)o is isomorphic to LG=1 (G).

e

9. The ind-group of loops coming from the open curve

Let G be a connected simple complex group, X be a connected smooth

projective complex curve. Recall the notations of 7.1.

9.1. The simply connected case.

9.1.1. Proposition. ([14],5.1) The ind-group LX G is integral.

e

Proof. To see that LX G is reduced, consider the morphism : QG ! MG;X ,

e e

e

which we know to be locally trivial for the tale topology by the uniformiza-

e

tion theorem 5.1.1. Hence, locally for the tale topology, is U LX G ! U.

e e

Now use that QG is reduced (Theorem 8.3.1) and Lemma 8.1.1 (iv).

e

To prove that LX G is irreducible it is enough, as connected ind-groups

e

are irreducible by Proposition 3 of [24], to show that LX G is connected. The

e

idea of its proof is due to V. Drinfeld: consider distinct points p1 ; : : : ; pi of X

which are all distinct from p. Dene Xi = X -fp; p1 ; : : : ; pi g and, for every

ane k-scheme U = Spec(R), dene Xi;U = Xi k U. Denote by AXi;U the

C -algebra (Xi;U ; OXi;U ) and by Li G the C -group R 7! G(AXi;U ). As LX G,

Xe e e

the C -group LiX G is an ind-group. The natural inclusion AXi;R AXi+1;R

e

denes a closed immersion f : LiX G ! Li+1 G.

e e

X

9.1.2. Lemma. ([14], 5.3) The map f : LiX G ! Li+1G denes a bijection

e

Xe

e

0 (LiX G) ! 0 (Li+1 G):

(9.1 a) e

X

Once we know the lemma, we do the following. Let g 2 LX G(C ) and let

e

K be the eld of rational functions on X. Using the fact (cf. [21]) that G(K)

e

is generated by the standard unipotent subgroups U (K), 2 , we may

Q

suppose that g is of the form j2J exp(fj nj ) where the nj are nilpotent

44

elements of g and fj 2 K. Let fp1 ; : : : ; pi g be the poles of the functions

fj ; j 2 J. Then the morphism

!LiX G

A1 e

Y

t 7! exp(tfj nj )

j2J

is a path from g to 1 in LiX G. By the above, the morphism 0 (LX G) !

e e

0(LiX G) is bijective which shows that g and 1 are in the same connected

e

component of LX G, hence LX G is connected.

e e

9.1.3. Corollary. ([14],5.2) Every character : LX G ! Gm is trivial.

e

Proof. The dierential of , considered as a function on LX G, is everywhere

e

vanishing. Indeed, since is a group morphism, this means that the deduced

Lie algebra morphism g

AX ! C is zero (with AX = O(X )). The derived

algebra [g

AX ; g

AX ] is [g; g]

AX and therefore equal to g

AX (as

g is simple). Therefore any Lie algebra morphism g

AX ! k is trivial.

As LX G is integral we can write LX G as the direct limit of an increasing

e e

sequence of integral varieties Vn . The restriction of to Vn has again zero

derivative and is therefore constant. For large n, the varieties Vn contain 1.

This implies jVn = 1 and we are done.

9.2. The general case.

9.2.1. Lemma. ([4], 1.2)

(i) The group 0 (LX G) is canonically isomorphic to H 1 (X; 1 (G)).

(ii) The group LX G is contained in the neutral component (LG)o of LG.

Proof. Consider the cohomology exact sequence on X associated to the

exact sequence 1 ! 1 (G) ! G ! G ! 1. As H 1 (X ; G) is trivial, we get

e e

the following exact sequence of C -groups

1 ! LX G=1 (G) ! LX G ! H 1 (X ; 1 (G)) ! 1

(9.2 a) e

Now using that the restriction H 1 (X; 1 (G)) ! H 1 (X ; 1 (G)) is bijective

and that LX G is connected by 9.1.1 we get (i).

e

It follows from (8.4 a) and (9.2 a) that (ii) is equivalent to claim that

the restriction map H 1 (X ; 1 (G)) ! H 1 (D ; 1 (G)) is zero. But this is a

Moduli of G-bundles 45

consequence of the commutative diagram of restriction maps

H 1(X; 1 (G)) G H 1 (X ; 1 (G))

H 1 (D ; 1 (G))

H 1 (D; 1 (G)) G

and the vanishing of H 1 (D; 1 (G)).

9.2.2. Corollary. There is a canonical bijection 0(MG;X ) ! 1(G).

Proof. This follows from the uniformization theorem and Lemma 8.4.1, (i),

(ii) and Lemma 9.2.1 (iv).

10. The line bundles on the moduli stack of G-bundles

Let G be a connected simple complex group, X be a connected smooth

projective complex curve. Recall the notations of 7.1.

10.1. The line bundles on the innite Grassmannian.

10.2. A natural line bundle. Consider the canonical central extension

(7.3 c) of LG and its restriction to L+ G. Then we may write

e e

d

QG = LG [

(10.2 a) e e +e

LG

By Lemma 7.3.5 we have a canonical character

[

: L+ G ! G m L+ G p1 G m ;

e!

(10.2 b) e

hence a line bundle L 1 on the homogeneous space QG .

e

10.2.1. A line in the innite Grassmannian. Consider the morphism of C -

groups ' : SL2 ! LSL2 dened by (for R a C -algebra)

: SL2 (R) !SL2 R((z))

a b 7 ! d cz 1

cd bz a

and moreover the morphism of C -groups : LSL2 ! LG deduced from the

e

map SL2 ! G associated to the highest root . Let

e

' = : SL2 ! LG:

(10.2 c) e

The Borel subgroup B2 SL2 of upper triangular matrices maps to L+ G by

e

construction, hence we get a morphism ' : P1 ! QG . An easy calculation

e

C

46

shows that the derivative Lie(') maps the standard sl2 -triplet fe; f; hg =

fX ; X ; H g to the sl2 -triplet fX

z; X

z 1 ; H g of Lg.

10.2.2. Proposition.

(i) The pullback denes an isomorphism ' : Pic(QG ) ! Pic(P1 ) C

(ii) We have ' (L ) = OP1 (1), i.e. Pic(QG ) = ZL

C

Proof. (i) follows from [12]. In order to prove (ii), we use that the restriction

of (7.3 c) to SL2 splits, hence ' lifts to a morphism ' : SL2 ! LG and all

d e

e

[

we have to do is to calculate the character of B2 ! L+ G ! Gm . For this it

e

is enough to calculate the character of B2 on the SL2 -module generated by

v0. By (7.2 f) this is the standard representation, so we are done.

In the following we denote, in view of the above, L by OQG (1).

10.3. Linearized line bundles on the innite Grassmannian. Con-

sider the group PicLX G (QG ) of LX G-linearized line bundles on QG . Re-

call that a LX G-linearization of L is an isomorphism m L ! pr2 L, where

m : LX G QG ! QG is the action of LX G on QG , satisfying the usual

cocycle condition. It follows from the section on stacks that

10.3.1. Proposition. The map : QG ! MG;X induces an isomorphism

: Pic(MG;X ) ! PicLX G(QG ):

Hence, once we know PicLX G (QG ), we know Pic(MG;X ).

10.4. The case of simply connected groups. In order to determine the

group PicLX G (QG ), consider the forgetful morphism f : PicLX G (QG ) !

ee ee

Pic(QG ).

e

10.4.1. Proposition. The map f : PicLX G(QG) ! Pic(QG ) is injective.

ee e

Proof. The kernel of this morphism consists of the LX G-linearizations of the

e

trivial bundle. Any two such trivializations dier by an automorphism of

pr2 OQG that is by an invertible function on LX GQG. Since QG is integral,

e e

e

it is the direct limit of the integral projective varieties and this function is

the pullback of an invertible function f on LX G. The cocycle conditions

e

on the linearizations imply that f is a character, hence f = 1 by Corollary

9.1.3.

Once we know that f is injective, we may ask whether f is surjective, i.e.

whether OQG (1) admits an LX G-linearization.

e

e

Moduli of G-bundles 47

10.4.2. Lemma. The line bundle OQG (1) admits an LX G-linearization if

e

e

and only if the restriction of the central extension (7.3 c) to LX G splits.

e

Proof. Let MumLX G (OQG (1)) be the Mumford group of OQG (1) under the

ee e

action of LX G on QG . This is the group of pairs (f; g) with g 2 LX G

e

e e

and f : g OQG (1) ! OQG (1). As QG is direct limit of integral projective

e

e e

schemes, we get a central extension

1 ! Gm ! MumLX G (OQG (1)) ! LX G ! 1:

(10.4 a) ee

In this setup, an LX G-linearization of OQG (1) corresponds to a splitting

e

of (10.4 a). Such a construction works in general10 and is functorial. Now

d

observe that LG is MumLG (OQG (1)). It follows that the extension (10.4 a)

e

e

is the pullback to LX G of (7.3 c), which proves the lemma.

e

Now our question of the surjectivity of f has a positive answer in view of

the following.

10.4.3. Theorem. The restriction of (7.3 c) to LX G splits.

Proof. Consider the inclusion i : LX G ,! LG. The map Lie(i) : LX g ,! Lg

e e

sends X

f to X

f=0 where f=0 is the Laurent development of f at x0 . By

b b

the residue theorem the cocycle (7.2 a) is trivial over LX g, hence LX g may be

seen as a subalgebra of Lg. Consider the basic highest weight representation

c

H1(0) of level one of Lg and take coinvariants:

c

B = [H1 (0)]LX g = H1(0)=LX g:H1 (0):

The crucial fact11 I will use is that B 6= 0.

Remark that the commutativity of (7.3 a) implies that for

2 LX G(R)

the associated automorphism u

of H maps coinvariants to coinvariants. We

get a morphism of C -groups : LX G ! PGL(B) hence we may consider

the diagram

Gm

1 1

LX G

G G[ G G

LG e

Xe

GL(B) PGL(B)

Gm

1 1

G G G G

10 The reader may consider to dene GL2 as the Mumford group of OP under the

1(1)

action of PGL2 on P 1.

11 This follows from the decomposition formulas of conformal eld theory where B is

seen as a space of conformal blocks (see [20]).

48

By construction, the central extension of LX G above coincides with the cen-

e

tral extension obtained by restriction of (7.3 c) to LX G. By denition of

e

B, the derivative of is trivial. As LX G is an integral ind-group by propo-

e

sition 9.1.1 it follows that has to be the constant map identity. Indeed,

write LX G as the direct limit of integral schemes Vn and remark that

has to be constant on Vn ; for large n, as Vn contains 1, this constant is

(1) = 1. So being the identity, factors through Gm which gives the

^

desired splitting.

10.4.4. Corollary. The line bundle OQG (1) descends to a line bundle de-

e

noted OMG;X (1) on MG;X . Moreover

e

e

Pic(MG;X ) = OMG;X (1)Z

(10.4 b) e e

10.5. The case of the special linear group. Now that we know that

Pic(MG;X ) = OMG;X (1)Z we may ask what happens to our determinant

e e

bundle D.

10.5.1. Lemma. Let D be the determinant line bundle on MSLr ;X . Then

D = OMSLr ;X (1)

Proof. Consider the morphism ' of 10.2.1:

'

QSLr

P1 q G

C

q

qqq5

MSLr ;X

Using

, we get a family E of SLr -bundles parameterized by P1 and, by the

C

above, we have to show that the determinant line bundle of this family is

OP1 (1). By denition of ' it is enough to treat the rank 2 case in which

C

this family is easily identied: if we think of QSL2 as parameterizing special

lattices as in Proposition 8.3.2 and the remarks following it. Then E[a:c] is

dened by the inclusion

1

W = bz cza (C [[z]] C [[z]]) ,! C ((z)) C ((z)):

d

As the lattice

1 C [[z]] C [[z]] ,! C ((z)) C ((z))

V =z

denes the rank 2-bundle F = OX (p) OX , we may view, via the inclusion

W V , the family E[a:c] as the kernel of the morphism F ! C p which maps

Moduli of G-bundles 49

the local sections (z 1 f; g) to af(p) cg(p). But then it is easy to see that

DE = OP1 (1) ([1], 3.4).

C

10.6. The Dynkin index. Let : G ! SLr be a representation. By

e

extension of structure group we get a morphism of stacks f : MG;X !e

MSLr ;X , hence by pullback

f : Pic(MSLr ;X ) ! Pic(MG;X ):

e

As we have seen, both groups are canonically isomorphic to Z, so f is an

injection. The index d of f is called the Dynkin index of . It has been

introduced to the theory of G-bundles over curves by Kumar, Narasimhan

and Ramanathan [12].

This index may be calculated as follows. Looking at the commutative

diagram

e

f

QG QSLr

e G

f G

MG;X MSLr ;X

e

e

we see that f (OQSLr (1)) = OQG (d ). As the canonical central extension

e

(7.3 c) is MumLSLr (OQSLr (1)), by functoriality of the Mumford group the

restriction of (7.3 c) to LG under L : LG ! LSLr denes the Mumford

e e

g

group LG = MumLG (OQG (d )). Looking at the dierentials we see that if

ee

e

we restrict the canonical central extension (7.2 b) to Lg

G Lg

0 G0

G Lg

GC f

L

Lslr

0 G0

Lsl

GC Gd G

r

all we have to do is to determine the extension Lg, i.e. calculate its cocycle.

f

10.6.1. Lemma. Let : g ! sl(V ) be a representation of g and consider

the central extension obtained by restriction of (7.2 b) to Lg. Then, if V =

P

n e is the formal character of V , its cocycle is given by

1 X n (H )2

(10.6 a) g

2

where g is the cocycle of (7.2 a).

50

Proof. By denition the cocycle is given by Tr((X )(X )) g , so all we

have to do is to calculate this number. For this, decompose the sl2 ()-module

V as i V (di ) , where V (di ) is the standard irreducible sl2 -module with highest

weight di . As usual, we may realize V (di ) as the vector space of homogeneous

polynomials in 2 variables x and y of degree di . Then X acts as x@=@y, and

X as y@=@x. Using the basis xl ydi l ; l = 0; : : : ; di of V (di ), we see

di

XX

k(di + 1 k):

Tr((X )(X )) =

i k=0

P

The formal character of the sl2 ()-module V (d) is d ed k where is

k=0

1 . Therefore we are reduced to prove

the positive root of sl2 () and = 2

the equality

d d d

1 X (d k )(H )2 = 1 X(d 2k)2

X

k(d + 1 k) = 2 2 k=0

k=0 k=0

which is easy.

Dene the Dynkin index dg of g itself by gcd(d ) where runs over all

representations of g. The Dynkin indices of the fundamental and the ad-

joint representations, as well as of g itself are listed in Table B. If is a

representation of g, we denote by D the pullback of the determinant bundle

under the morphism MG;X ! MSLr ;X . Let Picdet (MG;X ) be the subgroup

e e

generated by the D , where runs over all representations of g.

10.6.2. Corollary. The index of Picdet (MG;X ) in Pic(MG;X ) is dg .

e e

If G is of type B,D or G2 , choosing a theta-characteristic denes a

e

square-root P of the determinant bundle D = D$1 (see section 6). As

the Picard group is Z for simply connected groups we see that P does not

depend on in this case, hence we may denote it simply by P. Looking at

Table B, we see that dg is 2 in the B,D or G2 , hence

10.6.3. Corollary. Suppose G is of type B,D or G2. Then

e

Pic(MG;X ) = ZP

e

In particular, in the B,D or G2 case there are no other line bundles than

(powers of) the determinant and the pfaan line bundles. 1

We saw in 6.5 that !MG;X = DAd admits a square-root !MG;X (). Again,

1 2

e e

in the simply connected case, this square root does not depend on . Looking

at the Dynkin index of the adjoint representation in Table B, we see

Moduli of G-bundles 51

1

10.6.4. Corollary. Let !MG;X be the dualizing sheaf, !MG;X its canonical

2

e e

(G is simply connected) square root. Then

1

!MG;X = OMG;X ( h_ )

(10.6 b) 2

e

e

where h_ is the dual Coxeter number of g.

10.7. The non simply connected case. In the non simply connected

case, MG;X acquires 1 (G) connected components. I will restrict12 myself

here, for simplicity of the notations, to the component containing the trivial

bundle M0 . G;X

10.7.1. The basic index. We start by dening a number which will be useful

in the sequel. Dene the basic index `b (G) of G to be the smallest posi-

tive integer such that `b ($j ; $j 0 ) is an integer for all j; j 0 2 J0 (recall the

__

notations of 7.1). An easy calculation (see [23], Proposition 2.6.3), shows

that this number is given by Table C. In order to state the next theorem

correctly, I have to modify one of these numbers13: dene `b (SO ) = 2 if

4m

m is even.

If A is a nite abelian group, denote A^ = Hom(A; G m ) its Pontrjagin

dual.

10.7.2. Theorem. ([4]) Suppose g(X) 1.

(i) Let Pict (M0 ) be the torsion subgroup of Pic(M0 ). Then we have

G;X G;X

the canonical isomorphism

Pict (M0 ) ! H 1 (X; 1 (G))^

G;X

(ii) The quotient Pic(M0 )= Pict (M0 ) is innite cyclic. For its positive

G;X G;X

generator L we have

f L = OMG;X (`b )

e

where : G ! G and f : MG;X ! M0 is the morphism dened by

e

e

G;X

extension of the structure group.

Proof. It follows from the proof of Proposition 10.4.1 that the kernel of the

forgetful map f : PicLX G (QG ) ! Pic(QG ) identies to the character group

ee e

12 This is not really a restriction: actually the result is the same for the other

components.

13 This is related to the fact that the center of Spin is Z2 Z2 which has a non trivial

4m

central extension.

52

X(LX G) of LX G. Then (i) follows from (9.2 a) and the fact that LX G has

e

no non trivial characters by Corollary 9.1.3.

I will not prove (ii) here. Actually one shows, using central extensions of

LG, that an obstruction to the existence of L is that the following pairing

(recall the notations of section 7.1.4)

c : Z Z !G m

__

(zj ; zj0 ) 7 !e2i($j ;$j0 )

is trivial (see also [18], 4.6.3).

Once we know that we can0 t do better than `b , we have to show that

OMG;X (`b) actually descends. This may be easy, as for G = PGLr , where a

e

pfaan of DAd is convenient (just look at the numbers of Tables B and C)

or more complicated, as for SLr =s with s j r (see [13]).

10.8. The case of the special orthogonal group. We close the section

by looking in more detail at G = SOr . According to Theorem 10.7.2, there

is a canonical exact sequence

0 ! J2 ! Pic(MSOr ;X ) ! Z ! 0 ;

(10.8 a) 0

where the torsion free quotient is generated by any of the P 0 s.

Denote by (X) Pic(X) the subgroup of Pic(X) generated by the theta-

characteristics; it is an extension of Z by J2 .

10.8.1. Proposition. The map 7! P denes an isomorphism

P : (X) ! Pic(MSOr0;X ) ;

(10.8 b)

which coincides with on J2 .

This means that we have a canonical isomorphism of extensions

G (X)

G J2

0 G0

GZ

P

J2 G Pic(MSOr ;X )

0 0

G GZ G

0

Proof. ([4], 5.2)

Moduli of G-bundles 53

TABLE A

Type Dual Coxeter (Coxeter) numbers h_

Ic

:::: :::::

È É È É È É È É È É

Ar r+1 r+1

1 1 1 1 1

:::: :::::

È É È É È É È É È É

Br 2 2r 1

12

1 2 2 2

:::: :::::

È É È É È É È É È É

Cr r+1

2

12 12 12 12 1

1

m È É

mmm

È É m m m

: : :: : : : : :: :

È É È É È É

Dr 4 2r 2

È É

1 2 2 2

1

È É 2

È É È É È É È É È É

E6 3 12

1 2 3 2 1

È É 2

È É È É È É È É È É È É

E7 2 18

2 3 4 3 2 1

È É 3

È É È É È É È É È É È É È É

E8 1 30

2 4 6 5 4 3 2

È É È É È É È É

F4 1 4

24 12

2 3

È É È É

G2 1 9

13 2

54

TABLE B

Type Dynkin index dAd dg

:: :::::: :

È É È É È É È É È É

Ar 2r + 2 1

1 1

1 1 1 r r

r r r

r r

2 1

0 1 2

:: :::::: :

È É È É È É È É È É

Br 4r 2 2

1

1 1 1 2r

2r 2r 2r r 2

2r

2 2 2 2

2

0 1 2

:: :::::: :

È É È É È É È É È É

Cr 2r + 2 1

2r 2 2r 2

2r 2

2r 2

2r 2

r 2 2

r 1 2

0 1 2

2r 2 2r 2r

r4 r3

0

r 3

2

m È É

mmm

m

È É m m

:::::::::::

È É È É È É

Dr 2r 2

4r 4 2

2r 2 2r 2 2r 2

È É

2

2 2 2 r3

0 1 2

2r 3

È É 24

È É È É È É È É È É

E6 24 6

6 150 1800 150 6

È É 360

È É È É È É È É È É È É

E7 36 12

36 4680 297000 17160 648 12

È É 85500

5292000 141605100

È É È É È É È É È É È É È É

E8 60 60

1500 8345660400 1778400 14700 60

È É È É È É È É

F4 8 6

18 882 126 6

È É È É

G2 18 2

2 8

Moduli of G-bundles 55

TABLE C

Type Z(G) Z `b G = G=Z

e e

Ar Zr+1 Zr+1 PGLr+1

r+1

smallest k s.t. k(r+1)r 2 Z

Zs; sj(r + 1) s2

Z2 Z2 SO2r+1

Br 1

Z2 Z2 PSp2r

Cr 1 for r even, 2 for r odd

Z2 Z2 Z0 SO4m

D2m 1

2

Z+ SO+

1 for m even, 2 for m odd

2 4m

Z2 SO4m

1 for m even, 2 for m odd

Z2 Z2 PSO4m

2

Z4 Z2 SO4m+2

D2m+1 1

Z4 PSO4m+2

4

Z3 Z3 PE6

E6 3

Z2 Z2 PE7

E7 2

56

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