These are my live-TeXed notes for the course Math G6659: Langlands correspondence for general reductive groups over function fields taught by Michael Harris at Columbia, Spring 2016.
Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!
01/26/2016
This course will discuss one of the most exciting recent development in automorphic forms and number theory: the paper (still under revision) Chtoucas pour les groupes reductifs et parametrisation de Langlands globale of Vincent Lafforgue on the global Langlands correspondence over function fields. I have been thinking about his paper for a few years and more intensively in the past several months. The more I think about it the more I realize that my original intention was unrealistic due to the huge amount of technical material. I was trying the write up the notes for each lecture during the break, but every single day (at least 80 percent of the time) I realised the topic I was tring to discuss could well be a semester course in itself. To be more realistic, I will not try to explain the entire proof but rather to explain the background, present the framework of the Langlands correspondence and explain some highlights of the proof in details. We will also have five extra sessions on background of Langlands correspondence: including -groups, the Satake isomorphism and representations of adelic groups and so on (the notetaker will not attend these due to scheduling conflicts but refers to his other notes for the background.)
Let us begin with the standard function field set-up. Let be a prime power and
. Let
be a smooth irreducible projective curve and
its field of rational functions. Recall that the valuations on
corresponds bijectively to closed points of
over finite extensions of
. Let
a valuation of
and
be its completion. If
is the residue field, then
, with its ring of integers
. Notice there is only one kind of localization in the function field case, which simplifies things a bit (but not too much) compared to the number field case.
Let be a connected reductive group over
. For simplicity we assume
is coming from base extension from a split group over
. We almost always assume
is semisimple (e.g.,
,
,
,
,
,
,
,
,
), or otherwise
. Notice V. Lafforgue works in much more generality.
Recall that is the restricted product
with respect to
. Let
be an effective divisor on
. Let
be the open compact subgroup of level
. Let
be a coefficient ring (usually
,
, sometimes
). Recall:
The adelic group acts on
by right translation. The central question in the theory of automorphic forms is to decompose this space as
-representations.
Let be the space of cusp forms. If
is algebraically closed, then we have a decomposition
where
runs through (a countable set of) admissible irreducible representations of
and
is a non-negative integer. The central question is to determine the multiplicity
for any
. In particular, to determine which abstract representations
are automorphic (i.e.
). Fix
, the question becomes to determine
with
and
. The conjectural answer is provided by Langlands parameters:
The beginning of V. Lafforgue's theorem is:
Recall that the decomposition of is given by the action of the Hecke algebra
which is an algebra under convolution with identity
. The Hecke algebra
acts on
by convolution operators: for
,
,
The Hecke algebra decomposes as restricted product of local Hecke algebras
. The basic fact is that
It follows that is a commutative algebra and acts on
and decomposes it into a direct sum over
. Each
gives a collection of semisimple conjugacy classes
indexed by
.
V. Lafforgue defines a new commutative algebra of excursion operators acting on
. It contains the image of
and moreover connects to Galois representations: any Langlands parameter
is in fact a character of
!
More precisely, for any , V. Lafforgue associates a Langlands parameter
. When
,
is unramified at
and hence defines a conjugacy class
. The crucial property of the decomposition in Theorem 1 is
Blasius (for ) and Larsen (for
) have constructed examples of generic automorphic representations (over number fields) which are locally isomorphic but not globally. For
there are
orbits of generic characters and these automorphic representations can be distinguished by these different orbits. But for
(adjoint), there is only one orbit of generic characters and such examples of generic automorphic representations are unramified everywhere. The Whittaker functional
is nonzero on both factors and has 1-dimensional kernel. They can be detected by the global parameter (the character of the Hecke algebra extends to the excursion algebra in different ways). Question: can they be distinguished in a purely automorphic way (e.g., by non-Whittaker type of Fourier coefficients?)
01/28/2016
Today we will introduce the objects which V. Lafforgue uses to construct the global Langlands parameter. One novelty of his work is that he does not construct the global parameter directly but instead he constructs some combinatorial invariant involving the Galois group and . He then uses geometric invariant theory to show that the combinatorial invariant is equivalent to a global parameter.
When , any Galois representation
is uniquely determined by its trace function
. For general
, a global parameter
is not determined by any single invariant function. Instead, consider for any
,
When
,
can be identified as a function on the maximal torus invariant under the Weyl group. In general,
can be thought of as a generalized matrix coefficient, namely there exists a triple
where
It turns out any can be uniquely determined as a function on the space of triples
given by
where
. Our final goal is then to construct such functions on
using the geometry of the moduli space of shtukas.
Let be a character. It turns out (see next section) from this theorem that
is a
-valued
-pseudo-representation of
. Our next goal is to make the following theorem precise, hence reduce V. Lafforgue's construction of global Langlands parameters to Theorem 4.
02/02/2016
We observe that is invariant under
since the the cycle decomposition is invariant under conjugation and
is replaced by
under the action of
. We can extend
to a multi-linear map on
, then
is determined by its values on the subspace of symmetric tensors
by the invariance under
.
Since ,
is spanned by symmetric tensors of the form
. It remains to show that
for all
. It suffices to check semisimple
since these semisimple elements are Zariski dense in
. Define
We claim that
. Since the skew-symmetrization
maps
into
, we know
. The claim then implies that
as desired.
It remains to prove the claim. Choose a basis of
so that
is diagonalized to be
under this basis. Then
has a basis
where
runs all maps
and
Therefore
Notice that
if and only if
is constant on each cycle in the cycle decomposition of
. It follows that
which is equal to
by definition.
¡õ
For a character , Theorem 4 gives a continuous algebra homomorphism
By the face relation, for a map
, we have
By the degeneracy relation we have
where
is given by
It turns out the collection
gives a global Langlands parameter
.
02/04/2016
Now let us consider general . The construction of the global Langlands parameter does not follow directly from Theorem 6. We need more inputs from geometric invariant theory.
The following theorem constructs a global Langlands parameter from the collection
, which makes Theorem 5 more precise.
Choose ,
and a representative
so that
Let , we define
to be
such that
We need to verify the following:
Let . We will show that the
-tuple
is semisimple. In fact, the face relation implies that
lies over
. Theorem 5.2 of Richardson then implies that
has a Levi isomorphic to
. By (H1), they must have the same dimension. Hence
is semisimple and thus equal to
for some
. Therefore
. We can then define
, which proves (A).
Since , by (H2) we know that they must be equal. The uniqueness of
(B) then follows from the fact that
lies in the center of
. The degeneracy relation implies that
. Hence (C) follows from the uniqueness.
Notice takes value in a reductive group
, the center of
. To show (D), it suffices to show that for any
, the composition
is continuous. It follows from geometric invariant theory that the map
is surjective. If we lift
to
, then by the construction of
we know that
is equal to the map
which is continuous.
¡õ
The rest of the course will focus on proving Theorem 4, using the geometry of moduli spaces of shtukas.
02/09/2016
Our next goal is to explain that , the moduli space of shtukas of level
with no paws, is the discrete stack
. Since the IC sheaf on the discrete stack is simply the constant sheaf
, it follows that
whose Hecke finite is exactly the space of cusp forms (Remark 11).
(b) to (a): take .
(b) to (c): it follows from fpqc descent for isomorphisms.
(c) to (b): take .
¡õ
More generally, let be a
-bundle over
. If
is affine (or quasi-projective with a
-equivariant ample line bundle), then then quotient
always exists. This defines a functor
It is exact, commutes with direct sum, tensor product and sends the trivial representation to the trivial bundle.
02/16/2016
Now let us come back to the situation that is a smooth projective curve over
. Let
(resp.
) be the moduli stack of
bundles on
(resp.
) with a trivialization along
. We obtain the following corollary.
Let us consider the case . In this case
is the moduli stack of vector bundle of rank
with trivialization along
. Consider the tuples
, where
From this tuple we can define given by
. Then
. Since
is integral at
for almost all
, we know that
. Forgetting
and replacing
by the given trivialization
amounts to taking quotient by
on one side and by
on the other side. Conversely, an element
gives a projective module of rank
over each affine open of
(since projective modules over a Dedekind domain are equivalent to its local data) and they glue together to a rank
vector bundle on
. So we obtain Weil's uniformization
For general , we need the following theorem (Hasse principle):
Notice any -bundle over
becomes locally trivial by Lang's theorem and Hensel's lemma. This theorem implies that the generic fiber of any
-bundle over
is trivial as well. Hence we obtain the same uniformization for general
using Tannakian formalism (Theorem 9),
02/18/2016
Notice is not of finite type as one can easily see from the example of
. The Harder-Narasimhan (i.e. slope) filtration of vector bundle on curves naturally gives a stratification on
such that the strata with bounded slopes become finite type. For general
, we choose
a maximal torus and a Borel. Let
be the image of
in
. Let
, which one can think of as the "slopes" for a
-bundle.
Let us briefly sketch the proof of the algebraicity statement (c). Fix an ample line bundle on
. For fixed
, there exists an integer
such that for any
, and any
, the following (relative version of Serre's theorem, uniform in
) holds:
Moreover, for fixed and
, when
with sufficiently larger degree, the vector bundle
is a subbundle of
. Using the level
structure, we can then embed
into the moduli space classifying pairs
, where
is a subbundle of
of fixed rank and
is a locally free quotient of
of rank
and degree
. The latter moduli space is a generalized Grassmannian represented by a quasi-projective scheme (using Grothendieck's Quot scheme construction). The smoothness follows from the vanishing of
for curves.
03/01/2016
Let and
. Let
be a split group. Let
(resp.
) be the loop (resp. positive loop) group. Let
be the affine Grassmannian. All these are ind-schemes over
. We have
I was notified by X. Zhu during the weekend that there are some gaps in the literature on the foundation of affine Grassmannians. His recent PCMI notes Introduction to Affine Grassmannians filled the gaps.
The proof (we will follow Richarz's proof) of geometric Satake requires global input: the Beilinson-Drinfeld affine Grassmannian (a global analogue of the affine Grassmannian).
The similar proof as in the case of affine Grassmannians gives:
03/29/2016
In other words, a point in the Hecke stack is a length sequence of modifications of
-bundles and the
-th modification has prescribed location
. We can further restrict the order of poles for these modifications.
Similarly define .
We have a map analogous to the map ,
Now we can restate the geometric Satake correspondence for all possible parameters (Theorem 1.17 in V. Lafforgue's paper).
03/31/2016
The local system corresponds to a
-local system on the arithmetic etale fundamental group
, which is an extension of
by the geometric etale fundamental group. Drinfeld's lemma (see the next section) allows us to extend the action of
to
(the latter has
copies of
).
Taking and
be the trivial representation, we obtain the isomorphism
Taking
, we obtain the isomorphism
Now combining these two isomorphisms we can define the creation/annihilation operators.
04/07/2016
Reference: thesis of Eike Lau and L. Lafforgue.
Theorem 10 has the following consequence:
The hard part is to show the essential surjectivity. By fully faithfulness it suffices to deal with the case is affine. Let
be a compactification, let
be the normalization of
in the function field
. Because
is smooth, we know that
. So we are in a situation of a normal morphism between projective schemes, which on an open part becomes etale. Since
does not change the scheme (but only change the
-structure),
is the normalization of
in
. Since normalization is canonical, it follows that
. By Theorem ##VLgaloisdescent applying to the projective scheme
, we get
. Since
is etale, we know that
is also etale (by base change).
¡õ
04/26/2016
We now come to the last key point of this course, i.e., item e) in Theorem 4), which is Lemma 10.2/Prop. 6.2 in V. Lafforgue's paper:
Here is a rough strategy. Consider the Deligne-Mumford stack over
. Then one constructs two closed substack
(
) and together with morphisms
such that