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!
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?)
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.
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 .
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.
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.
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),
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.
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:
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).
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.
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
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