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{\bf Geometric Langlands  Part 6: Hecke Algebras revisited (6/96)}

Well, almost a year has passed since the last installment of this series of rants.. In the meanwhile I've 
learned a bit more about various topics related to the interplay of representation theory, algebraic 
geometry and mathematical physics - i.e. that vague conglomeration known as conformal field theory. So 
I figured maybe I'll inflict some more of this on whoever is willing to listen, namely you.
To begin with I'd like to talk for a while about some of the geometric and algebraic definitions of Hecke 
algebras (with probably little overlap with Langlands Part 2, though I haven't checked..) Again a 
disclaimer - I am not very familiar with the number theorist's Hecke operators and their in-depth study, 
so I will either skirt the topic or butcher it. My references will be some lectures of A. Beilinson on 
geometric Langlands (describing work of V. Ginzburg - Perverse Sheaves on a Loop Group and 
Langlands' Duality - available on the MSRI server, as well as unpublished work of Beilinson-Drinfeld), 
and A. Pressley and G. Segal's book Loop Groups.

OK with that ambitious plan, let's begin, following Beilinson's lecture. Here X is a compact (algebraic, 
complex) curve, and G is a (complex reductive) algebraic group. We use the unfortunate notation 
$Bun_G$ to denote the moduli space of G-bundles on X (actually we mean the moduli stack - for our 
current purposes it is better to have all G-bundles, and have to live on a stack, rather than throwing away 
unstable bundles and getting an actual space..) Given a point $x\in X$ and a bundle $F_1\in Bun_G$, we 
wish to consider modifications of the bundle at $x$, in other words bundles $F_2$ which differ from 
$F_1$ only at $x$. Thus we define the Hecke correspondence $Hecke_x$ at x to be the space of triples 
$(F_1,F_2,\nu)$ where $\nu$ is an isomorphism between the restrictions of our two bundles to the 
complement of $x$. 

Now what does this correspondence look like? Suppose we start with $F_1$ a trivialized bundle. 
Trivialize $F_2$ in some disc neighborhood of the point $x$. The comparison $\nu$ is now given by a 
function on the punctured disc with values in our group $G$.. This follows since in fact all $G$-bundles 
are trivial on the complement of $x$ so all the information is contained in clutching data on a punctured 
disc. However, we may change the trivialization of $F_2$ by multiplying by an arbitrary $G$-valued 
function on the (unpunctured) disc around $x$. Thus the fiber of the Hecke correspondence over the 
trivial bundle is the space $G(K_x)/G(O_x)$: this is the space of functions from the punctured disc to $G$ 
modulo those that extend to the puncture, i.e. the quotient of the $K_x$ valued points of $G$ by the 
$O_x$ points. (Recall $K=K_x$ is the function field of the punctured disc, namely rational functions, and 
$O=O_x$ is its local ring of functions defined at $0$.) The fiber over a different bundle $F_1$ is a version 
of the above space twisted by the left action of $G(O)$. Thus if we deal with functions, sheaves, etc. on 
$G(K)/G(O)$ which are $G(O)$-equivariant, then we may assume we're on the trivial bundle.. but we'll 
get back to that (much) later.

The space $Grass(G)=G(K_x)/G(O_x)$, known as the affine Grassmannian of $G$, is the geometric key 
to the entire representation theory of affine Lie algebras, the theory of KdV equations, etc. What does it 
look like? To understand $Grass(G)$ better, let us first note that if we pass from rational 
functions/polynomials to Laurent series/formal power series, in other words take the completion of our 
local field/ring, the quotient space remains the same because the completion is happening on the positive 
(=polynomial) side, by which we're modding out. This is telling us that in fact what we have in hand is 
not a space but a formal space - not a scheme but an inductive limit of schemes, aka an ind-scheme. The 
ind structure is expressed by looking at the finite dimensional pieces consisting of rational functions to 
$G$ with some bound $n$ on the order of the pole at $0$. 

Analytically $Grass(G)$ can be described (say in the $Gl_n$ case) as (a large piece in) the Grassmannian 
of all ``half-dimensional" subspaces in a (polarized) Hilbert space. This means that we decompose our 
space $H=H_+\oplus H_-$ and look at all subspaces whose projection onto $H_+$ has finite dimensional 
kernel and cokernel. For a beautiful geometric introduction to the study of representation theory of infinite 
dimensional Lie groups, with a major emphasis on this space, see the book Loop Groups by Pressley and 
Segal. In the persistent dichotomy between the compact-groups approach and the algebraic-groups 
approach to representation theory, this book lies heavily on the former side while my tendency is to 
support the latter, so I'm going to skimp on the details. The first point is the Iwasawa decomposition 
(``Birkhoff factorization") $G(K)=\Omega \times G(O)$ where $\Omega$ is the group of based loops, i.e. 
maps from $S^1$ to the compact form of $G$ taking $1$ to$1$. This is the infinite-dimensional analog of 
the usual Gram-Schmidt (Iwasawa) process, with groups replaced by their loop groups. The Birkhoff 
factorization implies that we have an identification $Grass(G)=\Omega$, so the Grassmannian is in fact a 
group!

The affine Grassmannian, as the fundamental homogeneous space of the loop group, possesses many 
properties analogous to those of classical Grassmannians and flag varieties. The Iwasawa decomposition 
is the statement that the compact form $K$ of $G$ acts transitively on the flag variety $G/B$ (and hence 
on the Grassmannians). Next comes the Bruhat decomposition: this states that the Borel subgroup $B$ of 
$G$  acts on $G/B$ with finitely many orbits labelled by the Weyl group $W$: $B\setminus G/B=W$. These orbits 
are known as Bruhat cells (their closures being Schubert cells), with the orbit of $1B\in G/B$ being a 
single point, while for $w \in W$ the ``long word" (a product of two-cycles in the $SL_n$ case) the orbit 
$BwB$ is dense. There is also an alternative way to view the Bruhat decomposition, which is to switch 
our point of view from our Borel $B$ (say the upper triangulars) to its opposite Borel $B_-$ (the lower 
triangulars), corresponding to the above ``long word". The set of orbits $B_- G B$ is again labeled by the 
Weyl group, except that now the orbit of the identity coset is dense and the orbit of the long word is a 
point. This new set of orbits is transverse to our old Bruhat decompositions, with cells of opposing 
dimensions meeting in single points. 

What is the analog of the Bruhat decomposition for the affine Grassmannian? The first thing that suggests 
that there is no single analog is the problem of the ``longest words" - in the affine Weyl group (which we 
have yet to define), there is no longest element in any sense, since it is infinite! It turns out that the two 
points of view on the Bruhat decomposition above give very distinct decompositions of $Grass(G)$. The 
first such decomposition is the set of orbits $G(O)\setminus G(K)/G(O)$, which we call the ``cells" of $Grass(G)$, 
and the second consists of $G(O_-)\setminus G(K)/G(O)$, where $G(O_-)$ consists of loops in $G$ which extend 
holomorphically to the {\em complement} of the unit disc - these will be known as the ``strata" of the 
Grassmannian. The cells are all finite dimensional, while the strata all have finite codimension! Both the 
cells and the strata are labeled by the same set, which we proceed to describe. 

Among all loops in $G$ we may single out those loops which are in fact homomorphisms from the circle 
into the maximal torus $T$ of $G$ - in other words one parameter subgroups of $T$. Such a 
homomorphism is determined by a map from the line to the Lie algebra $\cal t$ of $T$, satisfying an 
integrality condition which allows us to exponentiate it back to the group. In other words, the set of these 
special loops is labeled by the integral lattice, known as the coweight lattice (don't kill me if I've 
confused coweight and coroot though) of $G$ (or $T$). The dual lattice in the dual space $\cal t ^*$ 
parametrizes homomorphisms from the torus to $\Bbb C ^*$ (ignore my inconsistent uses of the circle 
and punctured plane!). These are just characters of the torus, also known as weights - hence the name 
weight lattice. The quotient of the dual to $\cal t$ by the weight lattice is known as the dual torus 
$T^{\vee}$ (pronounced ``T \v{C}ech"), and can also be described as the group of one-dimensional local 
systems on the torus (or $H^1(T, \Bbb C ^*)$). Starting from this dual torus and its weight/root lattices 
one may construct a group like we constructed $G$ from $T$ (!), and the resulting group is known as the 
Langlands dual Lie group $G^{\vee}$, which of course is important in the Langlands correspondence (by 
the principle that this name can't be appearing here for no reason!)

In any case, we have now singled out a discrete subset of the loop group labeled by the lattice of one 
parameter subgroups in $T$. The corresponding points on the Grassmannians will be the ``centers" of 
both our cells and strata. In other words we claim every cell and strata contains an almost unique element 
of the above set, and that cells and strata labeled by the same index meet only at the index points (The 
problem with uniqueness is that we've had to choose a maximal torus. Thus a given cell or stratum will 
actually contain $|W|$ points of the above set, coming from the outer automorphisms of the torus in $G$ - 
namely the Weyl group. Similarly the total intersection of a cell and a stratum contains these points for 
{\em any} maximal torus in $G$, and thus will look somewhat like a finite dimensional flag variety. 

In the $GL_n$ case, the description is quite concrete: we embed the Grassmannian of $G$ into the 
$GL_\infty$ Grassmannian $Gr$, which is the space of all half-dimensional subspaces in the space of 
Laurent series. This consists of all subspaces comparable with the space of formal power series, and is a 
projective limit of the half-dimensional Grassmannian of $GL_n$ as $n$ gets big. The space 
$Grass(GL_n)$ sits in here by lexicographically ordering the $n$ coordinates of a subspace, and thus 
passing from an $n$-tuple of Laurent series to a single one… Anyway, in $Gr$ sit special subspaces 
which possess a basis of monomials. These are labeled by the collection of exponents of the generating 
monomials. These must agree with the positive integers (the label of the power series) except for a finite 
discrepancy - we're allowed to miss a few positive integers and to add on a few negative ones. On $Gr$ 
there is an action of $\Bbb C^*$ which rescales the parameter of our power series. Our monomial points 
are all the fixed points of this action. It turns out we may define a decomposition of $Gr$ into finite 
dimensional cells by looking at the unstable manifolds of these fixed points (points which converge to a 
fixed point as $t \rightarrow \infty$), and similarly a stratification by finite-codimensional pieces which 
are stable manifolds of fixed points on $Gr$. Then the above decompositions of affine Grassmannians 
come from that of $Gr$ under the appropriate embedding. (For a more detailed explanation of these 
structures on affine homogeneous spaces, see the book by Pressley and Segal, or in more condensed form, 
the article Loop groups and equations of KdV type by Segal and Wilson.)

Before returning to Hecke operators, which we seem to have abandoned, I'd like to say a bit more about 
the different ``flag varieties" for loop groups, and the affine Weyl group. (For those who are impatient to 
return to Hecke, recall that the Hecke algebra of a Lie group was just a quantization of its Weyl group, a 
topic to which we can return once we know what affine Weyl groups are!) The first observation to make is 
that indeed the affine Grassmannian is more like a Grassmannian ($G$ modulo maximal parabolic) than 
a flag variety ($G$ modulo minimal parabolic). (p-adic theorists will recall that G(O) is in fact the 
maximal compact of G(K)..) To see this more explicitly, let's recall that $Grass(G)=G(K)/G(O)$ 
parametrizes bundles on the punctured disc, except that we haven't modded out by change of 
trivialization on the complement of the unit disc. This is determined by identifying the space of sections 
on the punctured disc with the space of G-Laurent series, and then specifying a subspace of sections which 
extend holomorphically to $0=x$. To obtain a flag variety, we note that we can rigidify our requirement 
for our section at $0$ just as one rigidifies from requiring just a subspace to requiring a full flag. In other 
words, we can look at the moduli space of bundles equipped with a flag in the fiber at $0$. A point in this 
moduli space is represented not by a subspace of the space of Laurent series but by a $G$-flag, consisting 
of sections lying in various pieces of the flag at $0$. In group theoretic terms, we introduce the Iwahori 
subgroup $Iw$ of $G(O)$, consisting of all loops in $G$ that are contained in the Borel of $G$ at $0$. 
The moduli space we introduced is $G(K)/Iw$, which is the ``real" flag variety of $G(K)$. 

The set $Iw\setminus G(K)/Iw$ of $Iw$ double cosets differs from the space $G(O)\setminus G(K)/G(O)$ in that now we 
have to take into account the Weyl group $W$ of $G$, represented as the set 
$B\setminus G/B$. In fact this space 
of double cosets in the affine flag variety is labeled by a group which is a semidirect product of the Weyl 
group of G and the coweight lattice of one-parameter subgroups in $T$ which we introduced before, and 
on which $W$ acts through its action on $T$. This group is known as the affine Weyl group of $G$, and 
occupies a spot in the representation theory of $G(K)$ closely analogous to that of $W$ for $G$. We may 
visualize the lattice directions as accounting for the loop direction in the loop group, while the ordinary 
Weyl group comes from finite dimensional geometry. To recap, the set of double cosets $Iw\setminus G(K)/Iw$ is 
given by the affine Weyl group, $Iw\setminus G(K)/G(O)$ is labeled by the affine Weyl group modulo the finite 
Weyl group, namely by the coweight lattice, while $G(O)\setminus G(K)/G(O)$ is given by the coweight lattice 
modulo the action of the Weyl group.

Now that we have described the set of $G(O)$ in the affine Grassmannian in terms of $G$, we notice 
something remarkable. Classical representation theory teaches us that irreducible finite dimensional 
representations of $G$ are determined uniquely by their highest weight, which is an element of a lattice 
(the weight lattice) in the dual to the maximal torus, defined up to a change of notion of ``highest", i.e. 
up to the action of the Weyl group (which permutes all the Borel subgroups containing a given maximal 
torus). In other words, except for some dualizing, the orbits on the Grassmannian are labeled by the same 
set as irreducible representations. To be more precise, we recall that the coweight lattice serves as the 
weight lattice for the dual torus. It is not hard to see that both the group $G$ and its Langlands dual group 
$G^{\vee}$ share the same Weyl group. So what we have shown is that finite dimensional irreps of 
$G^{\vee}$ are in one to one correspondence with $G(O)$ orbits on $Grass(G)$! This observation is the 
core of the classical Satake correspondence and the reason for the introduction of the Langlands dual 
group in representation theory. (Quick note: Langlands duality preserves types A,D,E and interchanges 
types B,C.)

Before proceeding with the affine Satake correspondence, let's introduce the Satake correspondence for 
algebraic groups $G$ (especially $GL_n$), where we replace the field of complex numbers with some 
finite field, so our sets become discrete and we can indeed sum over cosets, convolve functions and so on 
(see Langlands2). Again we consider the (discrete) moduli space $\cal M$of rank n vector bundles over a 
complete smooth curve $X$ defined over our field, and $x\in X$ a rational point. We repeat the notion of 
modification of a bundle in this context: given a bundle $V$ and a subspace $W$ of the fiber $V_x$, we 
construct the elementary transformation $A_x(V,W)$ to be the subsheaf of $V$ consisting of sections 
which at $x$ are contained in $W$. Now we consider the vector space of complex functions on $\cal M$ 
with finite support, $\Bbb C[\cal M]$. On this space we define the $k$-th Hecke operator 
$$T_{x,k}f(V)=\sum_{W\subset V_x, codim=k} f(A_x(V,W))$$ - in other words we add up the function 
on all $k$-th elementary transforms of $V$. The algebra of endomorphisms of $\Bbb C[\cal M]$ 
generated by the Hecke operators at $x$ is known as the Hecke algebra of $G=GL_n$. The Satake 
isomorphism then claims that the Hecke algebra is isomorphic to the algebra $\Bbb C[G]^G$ of class 
functions on $GL_n$.

In the complex curve case, we have to be more subtle in constructing the Satake correspondence. What we 
want to do is replace the finite sums above with some kind of convolution algebra of functions. Recall 
from Langlands2 that the Hecke algebra, over a finite field, can be thought of as the convolution algebra 
of $B$-invariant functions on $G/B$, or as functions on the finite set $B\setminus G/B$, which looked like the 
group algebra of the Weyl group with an interesting multiplication (as constructed in the above 
paragraph.) Now the algebra of functions on the double coset space is generated by the simple functions 
(delta functions) which are $1$ on a particular orbit and $0$ elsewhere. These are in one-to-one 
correspondence with orbits and hence with representations of the dual group. To extend this 
correspondence to all invariant functions we need to take sums and scalar products of functions - so on the 
representation side we must consider not only irreps but all representations, with the operations of direct 
sum and tensor powers. So now we see the Satake correspondence should be an equivalence between the 
some category of ``functions" on $G(O)\setminus G(K)/G(O)$ with convolution, and the tensor category of 
representations of the Langlands dual Lie group.

The question now is what is to take the place of the algebra of invariant functions on the weird, large ind-
stack $Grass(G)$. The answer has already been indicated in several of the previous rants - it is the notion 
of perverse sheaves, or (holonomic) $\cal D$-modules. There are two main ways to see a connection 
between these bizarre objects and functions. First of all, we may replace functions by the differential 
equations they satisfy, getting $\cal D$-modules. The importance of this step is evident already in fairly 
down-to-earth contexts, namely most interesting functions are not algebraic but do satisfy algebraic 
differential equations. The exponential function $e^{\lambda x}$ for example can be identified with the 
solution of the $\cal D$-module ${\cal D}/(\partial-\lambda)$, which is a perfectly algebraic object. Similar 
characterizations hold for hypergeometric, Bessel and many other functions. On the perverse sheaf side of 
things, there is a significantly more bizarre relation, known as Grothendieck's ``fonctions-faisceaux" 
correspondence for perverse sheaves over a finite field. First recall a perverse sheaf is basically a 
collection of local systems (flat bundles) on various subvarieties of our space, sitting in different degrees 
and with differentials between them. Irreducible perverse sheaves (ones which aren't direct sums) come 
from taking a subvariety (quasiprojective say), taking the constant $\Bbb C$ local system on it, and 
extending it to the entire space by the ``intermediate" or Deligne-Goresky-MacPherson extension, which 
picks up residual information on the closure of our subvariety encoding its singularity structure.. Anyway 
to a perverse sheaf over a finite field Grothendieck assigns the Euler characteristic associated to the 
Frobenius, in other words the alternating sum of the traces of Frobenius on the stalks of the cohomologies 
of the perverse sheaf at a point $x$ (this is a function of $x$). It turns out that interesting algebraic 
properties of this function encode geometric information about the sheaf..

Now we may finally introduce our class of geometric Hecke operators. Recall we constructed the Hecke 
correspondence ${\cal H}_x$ as a space of triples $(F_1,F_2,\nu)$ of two bundles and an isomorphism 
between them defined outside of $x$. Furthermore the fiber of the Hecke correspondence over the trivial 
bundle was identified with $Grass(G)$, and the other fibers came by twisting via the $G(O)$ action on the 
left. Given a sheaf on $Bun_G$, we may pull it back to ${\cal H}_x$, tensor with some sheaf upstairs, and 
then push our sheaf back down $Bun_G$ (though in fact what we get won't quite be a sheaf but rather an 
object in the derived category of sheaves..) This is the sheaf theoretic version of convolution of functions. 
To get Hecke operators we want to convolve with $G(O)$-equivariant (perverse) sheaves on $Grass(G)$. 
A theorem of G. Lusztig then says that the category of these sheaves is semisimple, and the irreducible 
objects are middle extensions of constant sheaves (delta functions) on the $G(O)$ orbits. Hence the 
irreducible objects are in one-to-one correspondence with orbits and thus with irreps of the Langlands dual 
group. The Satake correspondence, due to V. Ginzburg in this general setting, states that there exists a 
canonical equivalence of categories between the corresponding categories, that of equivariant perverse 
sheaves and that of representations, extending the above correspondence on irreducible objects.

So far we've only constructed the set of Hecke operators - we don't yet know how to compose them! 
Equivalently, we've compared the category of sheaves with that of representations, but don't yet have an 
analogue of the tensor product on the latter. There are many ways to construct this, which we'll mostly 
skip except to say that one method is to identify $Grass(G)$ with the group $\Omega$ as before and then 
to use the multiplication on $\Omega$ to define a convolution for sheaves on $Grass(G)$.. Also we have 
not yet said what happens when we allow the point $x$ to vary, which brings local systems on $X$ into 
the picture. The notion of a Hecke eigensheaf is also easy to state at this stage - it's a bundle that gets 
twisted by some local system under any Hecke operator.. Anyway we leave all this, and thus the actual 
precise statement of the Geometric Langlands conjecture, for later, as we wish to focus on the algebra of 
Hecke operators for now. So in part 7 we'll introduce yet another way to view the notion of Hecke 
algebras, as an algebra of intertwining operators (really this will be reminiscent of part 2). Then we'll see 
how an infinite dimensional version of this process, which goes through the notion of ``ghosts" and the 
BRST complex, produces $\cal W$-algebras and other exotic infinite dimensional creatures with beautiful 
ties to all the above…
See you then!
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