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Geometric Langlands Part 1 - Overview of ``classical" Langlands reciprocity
(revised version)

This rambling essay is meant to serve as a general introduction to the Langlands program, and to set up 
some background for the later rambling essays on its geometric interpretations (and generally things I'm 
interested in..) There are several good survey articles on the Langlands program, so I won't get into too 
much detail - e.g. S. Gelbart, An Elementary Introduction to the Langlands Program, Bull AMS 10/2, 
1984 (also reprinted in a volume of BAMS reprints on Auomorphic Forms and Representation Theory, ed. 
By Paul Sally). I also have a review article by R. Murty called ``A Motivated Introduction to the 
Langlands Program" but can't find where it's from right now; then there's a lot of writing by George 
Mackey, the prime example being his article ``Harmonic Analysis as the Exploitation of Symmetry", in 
BAMS around '82 or in an AMS volume of his collected surveys, ``The Development of 
(Non)commutative Harmonic Analysis" or something of that sort - most recommended! D. Mumford's 
Tata Lectures on Theta (I) are another wonderful source of background not only on $\theta$ but on 
modular forms, Hecke operators and a bit leading up to Langlands. Finally I'll be following some lectures 
of Edward Frenkel - a review of Langlands is found in his address at the 1994 International Congress of 
Mathematical Physics, or on the net on q-alg, as Affine algebras, Langlands duality and Bethe Ansatz. 

In broadest outlines, the Langlands program seeks to find a functorial correspondence between two classes 
of objects: representations of Galois groups and ``automorphic forms" (in some generalized setting). We 
won't have much to say about the former, which encode a great amount of number theoretic/algebraic 
information. Basically the way to study Galois groups, or groups in general, is through the study of 
naturally occurring representations, starting from small ones where not too much complication shows up 
and building up. The study of one-dimensional representations, or characters, of Galois groups is the 
classical subject of Class Field Theory (CFT, to be confused with Conformal Field Theory and Harvard's 
famous Chinese Food Truck), which we should return to later in a more geometric setting. (The classic 
setting, with which I'm not very comfortable, classifies abelian extensions of nice fields..) While we're at 
it we should mention that the geometric role of Galois groups is as fundamental groups, thanks as always 
to Grothendieck - field extensions (unramified) correspond to covering spaces (unbranched). Thus Galois 
representations correspond (via monodromy) to representations of fundamental groups, in other words to 
flat vector bundles! 

The automorphic, or modular, part of the picture is much more involved (at the definitional level), both 
for the ordinary case and for the geometric case, which we will get to. Let us start by making the 
transition from the standard notion of a modular form to its representation theoretic manifestation. A 
modular function, in its simplest guise, is a holomorphic function on the upper half plane which is 
invariant under $Sl_2(\Bbb Z)$, acting on the upper half plane by M\"{o}bius transformations. Now the 
upper half plane comes with an action of $G=Sl_2(\Bbb R)$, and the stabilizer of a point is the group 
$K=SO_2$ of rotations, which is the maximal compact subgroup of $G$. Thus we may think of a modular 
function as a function on $Sl_2(\Bbb Z) \setminus G/K$, which is just the punctured sphere with two 
weird (``orbifold") points coming from the fixed points of $Sl_2(\Bbb Z)$ on the upper half plane. 
Slightly more generally, modular {\em forms} are allowed to be not quite invariant under $Sl_2(\Bbb Z)$ 
but rather pick up a ``factor of automorphy" of a certain specified form when they transform. This 
translates into the modular form being not a function but rather a section of some line bundle (actually a 
half-integral power of the canonical bundle, hence a form) on the modular curve above. The next generalization is to 
say that we only require nice transformation properties under some subgroup of the modular group (hence 
modular forms of different levels), which means that we look at covers of the above basic modular curve. 
Finally, it turns out that we get interesting modular form-type objects of necessity in number theory (e.g. 
for the Selberg trace formula) if we relax the condition of holomorphicity. We just need to require that our 
modular form be an eigenfunction of the (hyperbolic) Laplace operator on the upper half plane
rather than a solution. (This gives rise to the Maass non-analytic forms..)

The notion of automorphic forms comes from carrying out the above steps for a general (reductive) Lie 
group $G$, of rank $l$ (the dimension of a maximal torus - for $Sl_2$ $l=1$). The analog of the upper 
half plane is the symmetric space $G/K$, where $K$ is the maximal compact subgroup of $G$. This 
space has many nice properties, metric and otherwise, and in particular it comes with $l$ invariant 
differential operators, which commute with each other. For $G$ simple, the Laplacian is the only second-
order invariant operator, and is generally the most fundamental. Now look at the representation of $G$ on 
the space $L^2(G/K)$ of square integrable functions on the symmetric space and decompose this 
representation into irreducibles. The $G$-invariance guarantees that any of our differential operators will 
act as a scalar on any of these irreducibles. In fact the commutative algebra of invariant differential 
operators is isomorphic to the center of the universal enveloping algebra of $G$ (the Laplacian 
corresponds to the Casimir..) What this means is that decomposing $L^2$ into common eigenspaces of all 
the invariant operators is very close to decomposing into irreducibles! This gives a representation theoretic 
meaning to eigenfunctions of Laplace operators. Thus it turns out that the natural requirement to put on 
an automorphic function is that it be a function on $G/K$ which is a common eigenfunction of all 
invariant operators. The automorphy requirement is then that our function be almost invariant with 
respect to a discrete subgroup $\Gamma \subset G$, so that it descends to a section of some line bundle on 
a finite volume quotient $\Gamma \setminus G/K$. (We will consistently ignore mentioning the 
cuspidality conditions on hour our functions behave at the cusps of this quotient.) For some purposes it is 
more convenient to consider automorphic forms as living on $\Gamma \setminus G$. Also instead of  
actually considering the automorphic form, we shift attention to the representation. Thus we consider 
subrepresentations of the regular rep of $G$ on $\Gamma \setminus G$, which contain a vector invariant 
under $K$. Irreducibility of the representation then corresponds to the condition that this vector is an 
eigenfunction of all the invariant differential operators on $G/K$. 

Now all this data is only one small part of the picture - this is the archimedean theory! We are really 
interested in arithmetic, and we know that in arithmetic one attempts to solve problems over the rationals 
by completing them at various places and considering all that local information together. What we have 
described so far is the data for the place at infinity, i.e. over the reals. Consider a prime $p$, and consider 
the ring $O$ of $p$-adic integers and its quotient field $K$ (which from now on will no longer mean the 
maximal compact of $G$!). (We will refer only to the rationals, but they can be replaced by any other 
global field and all its places.) Our group $G$ (for example $Sl_2$) is now an algebraic group, and we 
may consider its points over the $p$-adic field, $G(K)$, as the substitute for the real group $G(\Bbb R)$ 
we had before. Now the maximal compact subgroup of $G(K)$ is $G(O)$, so the analog of the symmetric 
space here is the space $G(K)/G(O)$, to which we'll come back repeatedly in later installments (the $p$-
adic nature of $K$  is not terribly important for some purposes - rather the fact that it's a local field, with 
local ring $O$ is the key..) What should replace our discrete group $Sl_2(\Bbb Z)$ and its array of 
subgroups of various level? It turns out that on the other side of $G(O)$, which are elements of $G(K)$ 
regular at $p$, we should consider a subgroup $H$ (for lack of imagination on my part) of elements which 
are ``regular at infinity", in other words have only {\em negative} powers of $p$.. (This will make sense 
when we have a geometric interpretation of our double coset space as a moduli space of bundles on an 
algebraic curve, where $G(O)$ encodes change of trivialization around the point $p$ and $H$ will be the 
global object, change of trivializations outside..) 
In any case it turns out that what we are really studying now is functions on $H\setminus G(K)/ G(O)$. 

What is the $p$-analog of the action of the algebra of invariant differential operators? It turns out that this 
is precisely the action of the Hecke algebra! We can ask our automorphic form to be an eigenform of all 
the Hecke operators, and this in turn guarantees the irreducibility of the corresponding representation.. 
We will return to Hecke algebras in Parts 2, 6 and 7. For now, we note that we want it to be an algebra 
which acts on the space of $G(O)$ invariant vectors in the representations we consider - even better, an 
algebra which acts on the space of $G(O)$ invariants in {\em any} reasonable representation of $G(K)$. 
To achieve this we define the Hecke algebra as the convolution algebra of $G(O)$-invariant functions on 
$G(K)/G(O)$, which as a space is the same as functions on $G(O)\setminus G(K)/G(O)$. In our ``good", 
automorphic representations  there is a one-dimensional space of $G(O)$ invariants, and so the Hecke 
algebra acts here as a character - the analog of the eigenfunctions of differential operators.

Now comes some magic (which will be addressed in Part 6), known as the Satake isomorphism, proven in 
this generality by Langlands (aka the local Langlands correspondence.) To our good representations we 
have assigned a character of the Hecke algebra, and now to this one assigns something remarkable - a 
semisimple (i.e. diagonalizable) conjugacy class in a complex Lie group.. (Complex numbers come in 
since we are, or should be, considering complex valued functions on all our spaces.) This Lie group is 
$G^{\vee}$, the Langlands dual Lie group, which is defined as the group whose maximal torus is the 
{\em dual} to the maximal torus of $G$.. In very vague terms, let's look at the Lie algebra of $G^{\vee}$ 
and the action of conjugation, i.e. the adjoint action of the group. Then the orbit of a semisimple element 
intersects the set of diagonal elements (a Cartan subalgebra) in an orbit of $W$, the Weyl group. So from 
semisimple conjugacy classes we pass to $W$ orbits on the Cartan, the Lie algebra of a maximal torus. 
But the Cartan of $G^{\vee}$ is by definition the dual to that of $G$, so we get $W$ orbits on the dual 
space to the Cartan of $G$. But elements of the dual space to the Cartan are the same as weights, or 
characters of diagonal elements on representations, and the classical theory of highest weights says that 
representations are in one-one correspondence with $W$-orbits of their highest weights.. so we have 
explained in a tenuous fashion this mysterious link, the Satake correspondence - again, see Part 6 for 
more.. The local Langlands correspondence now states that good, automorphic representations of $G(K)$ 
are in one-to-one correspondence with these Langlands parameters, the conjugacy classes.
 

Great! So we now know the local Langlands correspondence.. what happens next? As usual, the hardest 
part is the passage from local to global! We want to put all our local fields together in some coherent 
fashion, and read off of that a description of Galois representations over the rationals. The way in which 
we do this is to consider the ad\`{e}le group of $G$. These are the points $G(\Bbb A)$ of $G$ over the 
ring $\Bbb A$ of ad\`{e}les, the restricted product of the field $K_p=\Bbb Q_p$ over all primes $p$, 
including the prime at infinity (where we get $\Bbb R$) - an infinite product of fields with the restriction 
that all but finitely many components of any given element will be ``unramified", i.e. in $O_p=\Bbb Z_p$ 
(respectively the integers). Thus $G(\Bbb A)$ is the restricted product of the groups $G(K_p)$ for all $p$ 
with respect to the subgroups $G(O_p)$. The analog of the discrete group $Sl_2(\Bbb Z)$ or our group 
$H$ will now be the group $G$ over the rational ad\`{e}les, the diagonal
embedding of $\Bbb Q$ itself at 
all primes. Finally the maximal compact subgroup is again the product of $G(O_p)$.. 

An automorphic representation of $G(\Bbb A)$ is a representation occurring in the space $L^2(G(\Bbb Q) 
\setminus G(\Bbb A))$, which (in the unramified case) contains a vector invariant under the maximal 
compact subgroup, the product of $G(O_p)$. It turns out that automorphic representations are in fact 
tensor products of good representations of the local groups at each prime $p$. We may now ask, when 
does a collection of local representations tensor together to give a good global representation? Langlands 
formulates his conjectural solution in terms of the local Langlands parameters, $\sigma_p$, which are 
conjugacy classes in the same group $G^{\vee}(\Bbb C)$. Recall that at each prime $p$ there is a 
distinguished element in the Galois group of the residue field $\Bbb F_p$, namely the Frobenius $F_p$. 
The Frobenius may be lifted to a distinguished conjugacy class in the unramified Galois group of the local 
field $\Bbb Q_p$, also denoted $F_p$.

\begin{conj}{Langlands} The collection $\sigma_p$ corresponds to a global automorphic 
representation if and only if there exists a representation of 
the unramified Galois group $$\rho:Gal(\bar{ 
\Bbb Q}/\Bbb Q) \rightarrow G^{\vee}(\Bbb C)$$ such that $\sigma_p$ is the conjugacy class of 
$\rho(F_p)$ in $G^{\vee}$ for all $p$.
\end{conj}

This is our desired Langlands correspondence between Galois representations and automorphic forms. It 
says that these two sides correspond via the local parameters of each, namely the Frobenii and the 
Langlands conjugacy classes. In fact both of these collections of local parameters are put in a standard 
way into a generating series - a sum over $p$ of the reciprocal of a simple expression with $p$ and the 
parameter. These are the L-series which number theorists are so familiar with, and the Langlands 
correspondence says our two very different classes of functions correspond via an equality of their L-
series! On the modular side one can prove various properties like analytic continuation and functional 
equation, and many interesting number theoretic problems can be stated in terms of L-series of Galois 
representations (generalizing L-series of elliptic curves etc.), so the correspondence would enable one to 
prove many desired properties by comparison with a naturally analytical or geometrical object.
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