3. Geometric Reformulation.

Before describing what is known of the so-called Langlands-Drinfeld or
Geometric Langlands Correspondence, let us reformulate the Langlands
reciprocity, following the first three seconds of M. Kapranov's
"informal" talk at the Santa-Cruz conference.
Here we let X be a one-dimensional scheme (e.g. Spec Z or a Riemann
surface), K=the field of rational functions on X, and K_x the local
field corresponding to x in X. Class field theory concerns
one-dimensional representations of the Galois group of K_/K
(K_ denoting K bar, algebraic closure..), or in the local
case K_x_/K_x - but is more easily formulated (?) in terms of the Weil
group, a certain (Z-)extension of the Galois group (don't blame
me for gross inaccuracies..please..I don't actually understand this
stuff..)
Local class field theory then says W(K_x) abelianized is just
K_x* (the units), while the global theory computes W(X)=W(K)
abelianized as a familiar-looking object - 
K*\adeles*/O* completed - namely Gl1(K)\GL1_a/GL1(O^).
This is the same gadget as Pic X=H^1(X,O*) - 
K* corresponds to principal divisors, while Gl1_a/O^* can 
be identified with divisors (think Cartier divisors=section
of K*/O*, adeles here embodying the sheaf structure as opposed
to the globally-defined K*...)
The relation of this to class field theory as you may know it
(and unfortunately I don't) is that a point gives a Frobenius
element in the Galois group, while a principal divisor gives a trivial
element..

In any case the above lends itself to a pleasant nonabelian
generalization, which will give Langlands reciprocity of part 1 a
different interpretation. Namely, the (global) Langlands
associates to n-dimensional representations of W(X) (more 
generally G-valued representations) "automorphic representations"
of G - which are irreps of G_a entering into Fun(G(K)\G_a),
AND possessing an invariant vector with respect to G(O^) - 
namely an automorphic form f, which is thus a function
on G(K)\G_a/G(O^) (and an eigenfunction for Hecke operators).

Before stating the key geometric observation, I want to digress for a
second about the role of automorphic FORMS, as opposed to functions,
which I've been completely glossing over. As you know, automorphic forms
of weight k (say in the classical SL2 case) are NOT functions on the
modular curve SL2Z\SL2R/K, but rather k-forms - namely sections of the
canonical bundle raised to the kth power. This is crucial so as not to
obscure the analogy.. these correspond to induced representations from
nontrivial characters (via the Borel-Weil correspondence in some guise
or another - line bundles correspond to induced representations).
However, K (oops clash of notation - here I mean the canonical bundle -
let's say from now on K is the canonical bundle - I won't need the other
K any more), yes, K doesn't generate the Picard group of the modular
curve - there is a line bundle which is a square-root of K, whose
sections are automorphic forms of half-integral weight, the primary
example of which is the theta function (note that the modular curve is
P1, and by the classic formula K is -(n+1)=-2 times the tautological
bundle, so a square root of K generates..) The point is, we never were
really just talking about functions, but rather about sections of line
bundles. It turns out that people sometimes speak (according to Mackey!)
about modular forms of arbitrary real weights - defined by some cocycle,
interpolating between the integer weights. This should ring a bell maybe
- namely Beilinson-Bernstein introduced objects corresponding to
arbitrary cplex powers of line bundles, called TDOs (sheaves of
twisted differential operators..)   Hopefully this will make the
statement of geometric Langlands as having to do with D-modules slightly
less bizarre..

Now for the key geometric observation - this home of automorphic
functions, G(k)\G_a/G(O^) (excuse the varying notation - hopefully you
know what I mean..) has an interpretation as the first nonabelian
cohomology of our one-dim scheme X with coefficients in G, H1(X,G), or
in less fancy terminology, as the (moduli) space of principal G-bundles
(=G-vector bundles=G-torsors) on X..
One way to see this is to repeat the "argument" we gavve for the abelian
case - G_a/G(O^) corresponds to G-valued divisors, and then we mod out
by "principal" (globally trivialized) ones - so this should give
G-torsors, which ARE G-bundles.. If this doesn't convince you (which I'm
not saying it should), pick a point x on X. Any G-vector bundle V may be
trivialized on X\x - for k high enough, we may find a section of V which
has no zeros other than possibly a zero of order <=k at x (i.e. a
section of V tensor L^k, L the appropriate line bundle), which allows us
to reduce the rank of V by 1 and trivialize it inductively. Thus we may
describe V by an element of G(Laur), the Laurent-series-valued sections
of G around x - we think of Laur as the coordinate ring of a little
(formal) circle around x in X, and the element of G(Laur) describes the
clutching function for V - i.e. we trivialize V inside an outside a
formal neighborhood of x, and all the information is then contained in a
G-valued function on the (formal) circle around x encoding the gluing of
the two patches (you may also think of this as a Cech cohomology
construction.) Now we must quotient out by the changes of trivialization
on our neighborhood of x, and on its complement, and we have a model for
the moduli space of vector bundles. The coordinate ring of our formal
neighborhood is just O^, so we  must mod out G(Laur)/G(O^), and we must
mod out by G(O(X\x)), G with values in meromorphic functions with poles
only at x...    BUT x was completely arbitrary - doing this construction
for every x on our scheme (and waving our hands a bit) we realize the
moduli space of G-bundles on X as....G(k)\G_a/G(O^)  !!!

I'll let you ponder the above statement for a while, but in the
meanwhile let's phrase it more geometrically. From now on X is a
(complete) algebraic curve over the complex numbers, G is a complex
semisimple Lie group with Lie algebra g, etc. What is the notion of
adeles in this setting? We need to consider the values of G at each
point in the complete local field associatd to that point - the Laurent
series. But to interpret this we take a small (no need to be overly
formal here!) circle around x, and consider the space of
maps (in some good class)from this circle into G - namely the loop group LG of
G! Thus the G_adeles of X is the product of LG, indexed by all points of
X (and suitably restricted). Now this group has a canonical Borel
subgroup LG+, consisting of those loops which extend to the interior of
the discs (i.e. all negative Fourier coefficients vanish). The quotient
LG/LG+ is the (Sato) Grassmannian of LG (and of course has nothing to do
with x or X.) Now we have realized the moduli space of G-bundles on X as
the double coset space Ax\LG/LG+, where Ax (I don't know a universal
notation) consists of those maps from the little circle to G which
extend holomorphically to the complement of the disc in X. This is a
"Borel" of LG, which is assoicated specifically to the Riemann surface
X, and constitutes the "other half" of LG+, besides a deficit coming
from Riemann-Roch.

This construction is at the heart of the geometrical understanding of
conformal field theory, since it allows one to interpret, via a
Borel-Weil/ Beilinson-Bernstein construction, the representation theory
of LG as geometrically realized on Riemann surfaces, and in particular
enables one to prove geometrical results about Riemann surfaces using
the algebraic theory of Kac-Moody Lie algebras.

Well I think I'll try to keep these "parts" short, and move on now to
part 4 - the geometry of these moduli spaces. But before that let us
just state what the geometric Langlands program should do. We've found
the analogue of the double coset spaces in our geometric setting. As
you etale geometers know, the geometric analog of the Galois group of a
field is the fundamental group of a curve (having that field as rational
functions, or something.) Hence the geometric Langlands reciprocity will
be a correspondence between representations of the fundamental group
into G^L (the Langlands dual group of G) - namely flat G^L - bundles on the
curve; and certain "automorphic" representations of LG (or the adele
group), which are induced representations from one of the above
"Borels", and may be realized geometrically on M(G), the moduli space of
G-bundles on X. More precisely we will associate to G^L torsors on X,
D-modules on M(G).. but more of that later!