2. Double cosets and intertwiners.

Before interpreting Langlands reciprocity geometrically, we'll
digress a bit to talk about the interpretations of automorphic forms and
Hecke operators in terms of intertwiners for representations (which
might make the geometric analogy slightly clearer.) 
[Our main reference here is Mackey's "Unitary Group Representations in
Physics, Probability and Number Theory" and assorted shorter works.]

A central issue in representation theory is the construction of
operators that intertwine two given representations. Since by Schur's
lemma, irreducible representations don't intertwine with anything but
themselves (via scalar operators), finding the intertwiners between a
given representation and irreducible representations is equivalent to
decomposing said representation into irreducibles, the primary universal
concern of representation theorists. 

The simplest class of intertwining operators consists of the algebra
(which we denote R(V)) of operators intertwining a given representation
G with itself. First the Lie-algebra approach:
Suppose V is a representation of g, a semisimple Lie
algebra. Then Z, the center of the universal envelopping algebra U(g),
acts automatically as intertwiners of V. If V is irreducible, then Z
acts via a character x:Z-> C, called the central (or infinitesimal)
character of V. The determination of these and relation with the
highest-weight theory was centeral to the work of Harish-Chandra. In any
case, we know that Z is a polynomial algebra in l (=rank g) generators,
the "higher Casimirs", which correspond to the invariant differential
operators on the symmetric space G/K associated to g. Hence the first
step in the decomposition of a representation is to find the
simultaneous eigenspaces of these Casimirs, which corresponds
geometrically to finding eigenfunctions of the Laplacian and its
higher-rank analogs. The understanding of this step is the key to what
is known of the geometric Langlands program (where g is replaced by an
affine algebra.)

Given any other representation W, the algebra R(V) acts by
precomposition on the space of intertwiners R(V,W) between the two
representations, making the latter into an R(V) module. Hence the
importance of understanding and utilizing the structure of R(V)! When V
is an induced representation from some subgroup K to G, which we realize
geometrically as the space of functions on G/K, the corresponding
algebra is essentially what is known as the Hecke algebra H(G;K).
The meaning of this (or other) notation for Hecke algebras depends on
the context, so before returning to the case of automorphic
representations let's recount some of these contexts. 

Let's first define H(G;K) as C(K\G/K), the convolution algebra of
functions on G which are
K-bi-invariant. Such a function is thus constant on each K double coset,
and we get a basis for this algebra consisting of the characteristic
functions of the double cosets. Suppose first that G is finite and the
above makes perfect sense, and is indeed the interwining algebra R(V)
for V the induced representation for the trivial character of K. In this
setting it is easy to generalize this construction to get R(V,W), where
V is induced from character x1 of K1 and W from x2 on K2 - take
functions on G which transform via x1 under K1 (right-multiplication) and
x2 under K2 (from the left) - corresponding to the appropriate 
line bundles on G/K1 and K2\G instead of ordinary functions.

Another setting where one can make nice sense of this construction is
when G is locally compact and K is a compact open subgroup, and we take
H(G;K) to be compactly-supported C^infty functions on G. The
corresponding convolution algebra again has a basis of characteristic
functions of double cosets, and acts as intertwiners on C^infty(K\G),
the corresponding induced rep. This algebra can also be characterized as
the subalgebra of Funct(G) which acts on the K-fixed vectors in any
(smooth) G-representation, and in fact one has a resulting equivalence
between reps of Funct(G) (where we mean C^infty, cpt. support) and
reps of H(G;K), which is a much smaller, much simpler algebra. This
should remind the reader of the interpretation of automorphic forms as
K-fixed vectors where K is the subgroup G(Zp). In fact a standard usage
of the term "Hecke algebra" (though not precisely the one used in
modular form theory) - aka the Hecke-Iwahori algebra, is this group
H(G;K) where G is a p-adic group and K is its "Iwahori" subgroup, a
compact open subgroup slightly smaller than the above maximal compact.
In this case the Hecke algebra is almost abelian (in fact its large
abelian subgroup is itself often called Hecke algebra),and its basis is
indexed by a slight extension of the Weyl group..

A variation on the above case of Hecke algebras leads to the most common
usage in representation theory, where the Hecke algebras are known as
deformations of the Weyl group. Here our group K is the Borel subgroup
of G, a semisimple (algebraic) group, and the double cosets B\G/B are
indexed (by the Bruhat decomposition) by the elements of the Weyl group
W of G. The notion of K-fixed vectors in this case is taken up by the
highest-weight vectors. Alas the full algebra structure can't quite be
recovered naturally over C, due to problems of finiteness. Over a finite
field Fq, however, one can recover an algebra structure, and an explicit
presentation in terms of the generators which correspond to double
cosets (i.e. to elements of the Weyl group) and of course q. We may now
use this presentation to DEFINE the Hecke algebra for G any algebraic
group (in particular over C) as an algebra over Z[q] - i.e. a one
parameter deformation of the (group algebra of the) Weyl group, which
specializes to it at q=1... Now if this smacks to you of quantum groups,
you're right! (You may already have noticed something fishy when we
mentioned the appearance of a huge center above...)
This Hecke algebra, which has been used clasically to
encode more detailed information about Bruhat decompositions, to define
the Kazhdan-Lusztig polynomials, etc, is indeed in some sense a quantum
object. It plays for the quantum group Uq(g) a role analogous to some
aspects of the role of the Weyl group (though not all) - in particular
the Schur-Weyl reciprocity between representations of the Weyl group and
representations of G. This analogy is at the heart of the expectation,
critical to much of the interest in quantum groups (and in the geometric
Langlands correspondence), that they somehow are characteristic 0
analogs of algebraic groups over finite fields...

Finally to return to the case at hand, we'll sketch (following Mackey)
the interpretation of cusp forms, Eisenstein series etc. as certain
intertwining operators, through which point of view (due to Gelfand and
his school) representation theory takes over modular form theory and
leads to the aforedescribed Langlands program. First let's recall the
definitions of the principal and discrete series of representations of
SL2(R). An element of the principal series corresponds to the induced
representation from the upper-triangular subgroup of the character which
assigns to an upper diagonal matrix with a, 1/a on the diagonal the
number x(a), where x is a character on the real line. Thus we get a
one-parameter family of representations.
The parameter in this family may also be interpreted as an eigenvalue
for the hyperbolic Laplacian, belonging to the functions in the
representation space.  The discrete series coresponds to
square-integrable representations, and may be realized as 
"square-integrable k-forms on the upper half plane" - i.e.
functions on the upper half plane, square integrable wrt Poincare
metric to the kth, and transforming under SL2(R) with a 
"factor of automorphy" denominator to the (-2k)th power.

Now given a discrete subgroup (which should be gamma - but this is
ascii) J of SL2(R), we wish to understand the representation V(J) of SL2
induced from the trivial representation of J. By the general theory,
this representation will decompose as a direct sum of two parts, a
continuous part which is a direct integral of principal series
representations, and a discrete part - a discrete direct sum of other
representations. 
The continuous part, as it turns out, is the image of all intertwining
operators from a specific representation U to V(J). Here U is the
induced representation from N, the maximal unipotent subgroup of SL2
(upper-diagonals with 1 on the diagonal) to SL2 of the trivial
character. Inducing partway from N to B (all upper-triangulars) it
follows that U is the direct integral of ALL members of the principal
series..thus indeed giving all the continuous spectrum of V(J) through
intertwiners. To construct such intertwiners as before we pick an
element z of some J,N double coset and take the appropriate convolution,
which in this case reduces to the discrete sum Tz(f)(x)=
\Sum f(z^{-1} j x), where j varies over J/(J intersect the z-conjugate
of N)  ... I won't elaborate, but this can be worked out (or if
lazy like me check Mackey p.369!)  Since B normalizes N, it turns
out we just need to take one z from each J,B double coset, and in
particular those for which the above intersection is nontrivial, blah
blah blah (it's getting late!) The point is these z's correspond in a
one-to-one fashion with the cusps of J, and the corresponding
intertwiners is in fact an Eisenstein series!!

In this fashion Eisenstein series corresponding to cusps may be said to
account for the continuous part of the decomposition of V(J). In a
similar fashion cusp forms account for the presence of discrete series
representations in V(J). The definition of the discrete series above
probably looked suspicious already - a cusp form for J is a form which
transforms as a discrete-series form under J.. This can be fleshed out
to say that the dimension of the cusp forms of weight k wrt J equals the
multiplicity with which the kth discrete series representation appears
in V(J)! Cusp forms may be identified with the corresponding
intertwining operators. The role of the Hecke algebra now pops up
immediately - the Hecke algebra for J is an algebra of intertwining
operators for V(J) with itself. In fact they are the intertwining
operators corresponding to the J,J double cosets which contain only
finitely many left and right cosets (so there is no problem of
convergence / compactness/ etc to impede us.) This is a noncommutative
algebra in general, but contains a nice commutative subalgebra which is
the classical Hecke algebra, whose eigenforms have Euler product
expansions and so on.

As a final note on the decomposition of V(J), there may be components
appearing discretely in the decomposition which are NOT discrete series
representations ("discrete series" refers to discreteness in the regular
representation of SL2!) - namely components of the principal or
supplementary series can show up. These then correspond to
eigenfunctions of the hyperbolic Laplacian which are J-invariant -
namely nonanalytic cusp forms, known as the Maass waveforms or 
nonanalytic automorphic forms. All this picture can be put together
nicely as an explanation of the Selberg trace formula, as was done by
Gelfand etc - see Mackey or Gelfand- Graev - Piatetskii-Shapiro,
Representation Theory and Automorphic Functions. Mackey follows
the Gelfand school and interprets the Poisson and Radon transforms in a
similar fashion as intertwiners between the corresponding induced
representations.. but I'll leave that for another time!
- and now move towards the geometric interpretation of the Langlands
program and representation theory of affine algebras.