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!

[-] Contents

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:

Definition 1
We have the space of *automorphic forms of level with coefficient in * Notice these automorphic forms are automatically locally constant. The space is the fixed space of the space of all automorphic forms Here the subscript means uniformly locally constant functions.

Remark 1
When (or is not semisimple), we choose a discrete subgroup in the center so that is compact. All I am about to say will be true after this modification.

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:

Definition 2
Let be the Galois group of the maximal separable extension of unramified outside the support of . A *Langlands parameter* (unramified outside ) is a homomorphism , up to conjugacy by , where is the Langlands dual group of .

Remark 2
When , the conjecture boils down to class field theory. When , Laurent Lafforgue proved the conjecture and in this case the multiplicity of automorphic representations is always 1.

Remark 3
To be more rigorous, the source of a Langlands parameter should be the conjectural *Langlands group*. In the function field case, there is no issue of archimedean places and the conjectural Langlands group should be the motivic Galois group of the category of motives realizing the space of automorphic forms of level . This is the Galois group , at least after taking the -adic realization: the independence of for the Langlands parameters is still an issue.

The beginning of V. Lafforgue's theorem is:

Remark 4
The theorem as stated is meaningless so far. The main point is that this decomposition is compatible with the the local Langlands correspondence (Satake parameters). In particular, each is a sum of , i.e. this decomposition is an refinement of the decomposition (1).

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

Proposition 1 (Satake isomorphism) If , then

- is a commutative algebra over , isomorphic to a polynomial algebra.
- There is a canonical bijection between the characters with semisimple conjugacy classes in .

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

Remark 6
For a globally generic automorphic representation, let be the subset of -orbits of generic characters that is generic with respect to. If and are locally isomorphic everywhere but not globally, then the conjecture is that . The different way to extend from the Hecke algebra to the excursion algebra should be indexed by these subsets of generic characters. When is adjoint, one can compute that the set of generic character is a singleton and therefore the image of the Hecke algebra and the (semi-simplification of ) the excursion algebra should be the same.

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.

Definition 3
For and an algebraic dimensional representation , where , V. Lafforgue, following Varshavsky, associates a moduli stack , the moduli stack of *-shtukas of level and paws (or legs) bounded by *. It is a Deligne-Mumford (ind-)stack, hence can be thought of as a finite quotient of a scheme locally. For a test scheme , classifies the following data:

- An -tuple of -points on .
- A -torsor over .
- An isomorphism away from the union of the graphs of , where , such that the relative position of at each is bounded by the dominant weight of (a coweight of ).
- A trivialization of along , i.e., an isomorphism .

Remark 7
One can view a -torsor as a functor . When , this is the same as a vector bundle of rank . For general , this may be thought of as a vector bundle with extra structures.

Example 1
When , , consists of two points, and , a point of is exactly a Drinfeld shtukas with two legs with minuscule modification (a simple pole at and a simple zero at ).

Remark 8
Fixing a bound (dominant coweight of ) on the Harder-Narasimhan polygon gives an open substack of *finite type*, which is in fact represented by a scheme when is sufficiently large.

Definition 4
Sending each -shtukas to its paws defines a *paw morphism* . We define a sheaf on , whose stalks should be thought of as the middle intersection cohomology with compact support.

Theorem 3

- When is the trivial representation, is the constant discrete stack over . It follows that
- The map extends to an additive functor to .
- The stalk of at a (good) geometric point contains the subspace of Hecke finite elements.
- Each carries a monodromy action of .
- Given any morphism of finite sets of and elements. There is a natural projection and similarly . For any , composing with defines . There is a canonical isomorphism equivariant for the action of (which acts on the left hand side via ). Moreover, In fancier language, these data can be thought of as a morphism from the classifying stack of to the classifying stack of .
- There is a canonical action of the Hecke algebra on each and all morphisms in (e) commute with this action.

Remark 10
When , the space is known as the *vacuum space*, if one thinks of the paws as moving particles. The vacuum space has no Galois action but for larger , the space admits more and more Galois action. Drinfeld and L. Lafforgue computed by comparing the Grothendieck-Lefschetz trace formula and Arthur-Selberg trace formula. V. Lafforgue, however, does not compute these spaces at all (except checking some commutative diagrams).

Remark 11
Over function fields, the space of Hecke finite compactly supported functions of level (i.e., which spans a finite dimensional space under the unramified Hecke operators) is exactly the space of cups forms of level . On the one hand, cusp forms are compactly supported by a theorem of Harder (see Borel-Jacquet 5.2). The space of cups forms of level is finite dimensional, hence Hecke finite. On the other hand, let be a rational maximal parabolic subgroup of with Levi subgroup , we would like to show that We may assume is standard, so and . We may assume that after conjugation. Since is compactly supported, one can check that is also compactly supported. On the other hand, the Satake transform is given by integration on the unipotent radical of , hence factors through . Because is Hecke finite over and is a finite -algebra, it follows that is Hecke finite over . The center is infinite and preserves the space . Since is compactly supported, this contradicts the Hecke finiteness unless .

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

- ,
- an invariant vector of ,
- an invariant covector of , such that

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.

Definition 5
The invariant vector and covector can be thought of as a *creation operator* and an *annihilation operator* . The diagonal map then induces and . We define the *excursion operator* to be the composition: It does not depend on the choice of the triple representing . Let be the *excursion algebra* generated by the excursion operators.

Theorem 4

- The excursion algebra is commutative. For fixed , is an algebra morphism.
- (Face relations) satisfies natural relations subject to morphisms and .
- (Degeneracy relations) satisfies natural relations subject to multiplications on and .
- For fixed , the homomorphism is continuous under the the -adic topology.
- The unramified Hecke algebra . In fact, for , let , then , for .

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.

Remark 12
For , this is a theorem of R. Taylor. For general groups, it can be deduced from a theorem of Richardson, based on geometric invariant theory.

02/02/2016

Definition 6
Let be a topological ring. Let be a topological group with unit . An -valued *pseudo-representation* of of dimension is a continuous function such that

- (sometimes also requiring that is not a zero divisor in , not needed for V. Lafforgue's work).
- for any .
- is the smallest integer with the following property: let be the symmetric group on letters and be the sign character, then for all , the following holds: Here if is the is the cycle decomposition, has length and

Remark 13
As we will see, the identity in (c) comes from the fact -st exterior power of a -dimensional vector space is zero.

Theorem 6 (Taylor, Rouquier)

- Suppose is a continuous representation, then its character is a pseudo-representation of dimension .
- Conversely, if is an algebraically closed field of characteristic zero or , then any -dimensional pseudo-representation of is the trace of a semisimple representation of dimension , unique up to homeomorphism.

Remark 14
Recall that the Brauer-Nesbitt theorem implies that the semi-simplification of a representation is determined by its character. This is the best possible one can do because one can only pin down a representation up to semi-simplification from its character.

Remark 15
Here is how Theorem 6 is used in deformation theory of Galois representations. Suppose is a -adic ring with maximal ideal and fraction field . Suppose for any , we have a torsion representation with the compatibility . Then is a pseudo-representation. Hence Theorem 6 implies that actually comes from a genuine representation deforming all 's.

Proof
Let us show that the character of a representation satisfies the identity (2) in (c). Write to be the left hand side of (2). Writing as a quotient of two integral domains of characteristic zero, we reduce to the case that is an algebraically closed field of characteristic 0. Let and . We may reduce to the universal case and can be viewed as a function on .

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

Remark 16
When , the global Langlands parameter can be constructed using Theorem 6. Notice that determines all since all representations of can be constructed from the standard representation using tensor powers. It remains to check that satisfies the identity (2). In fact, for , one can compute that Taking , the degeneracy relation (and is an algebra homomorphism) implies that Now take and take skew-symmetrization, we obtain the identity (2) for by using the fact .

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.

Definition 7
We say an -tuple is *semisimple* if , the Zariski closure of the subgroup generated by in , is semisimple.

Theorem 7 (Richardson)
-orbit of (under conjugation) is Zariski closed in if and only if is semisimple.

Remark 17
Notice by geometric invariant theory, the geometric points of GIT quotient is in bijection with closed orbits of the reductive on the affine variety . Thus is in bijection with -conjugacy classes of semisimple -tuples in .

Definition 8
A homomorphism is *semisimple* if whenever the image of is contained in a parabolic , the also contained in the Levi of . In characteristic 0, this is the same as saying that the Zariski closure of is reductive.

The following theorem constructs a global Langlands parameter from the collection , which makes Theorem 5 more precise.

Theorem 8
Let be a homomorphism. Then there is a homomorphism , unique up to conjugacy, such that

- is continuous and takes values in , for a finite extension .
- is semisimple.
- corresponds to at unramified places under the Satake isomorphism.

Proof
For any -tuple , gives a homomorphism , i.e. a point in . Let be the corresponding semisimple conjugacy class in given by Theorem 7.

Choose , and a representative so that

- (H1) is of maximal dimension.
- (H2) The centralizer of (which is also the centralizer of the Zariski closure) is of the smallest dimension with smallest number of connected components.

Let , we define to be such that We need to verify the following:

- (A) Such exists.
- (B) Such is unique.
- (C) The map is a homomorphism.
- (D) The map is continuous.

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).

Proposition 2
Let be a scheme over . Let be an affine group scheme over . The following data (called a *-bundle*) are equivalent:

- A sheaf on the fpqc site of with a left action of such that is an isomorphism and there exists an fpqc cover with (local triviality in fpqc topology).
- A scheme with a left action of and an fpqc cover such that in a -equivariant way.
- An fpqc scheme with a left action of such that

Proof
(a) to (b): Take as in (a). Then implies that is trivial. The existence of the scheme then follows from fpqc descent for affine morphisms since is affine.

(b) to (a): take .

(b) to (c): it follows from fpqc descent for isomorphisms.

(c) to (b): take . ¡õ

Example 2
Let . Let be a -bundle over . Then the associated bunlde is a rank vector bundle. This gives a functor from -bundle on to rank vector bundle on . It is an equivalence of groupoids. The inverse functor sends a vector bundle on to the fpqc sheaf .

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.

Proof
Let us describe the inverse construction. Given , we would like to define a -bundle . We can extend to direct limits of finite dimensional represetations by Since each is a vector bundle on and flatness is preserved under direct limit, we know that is a flat -module. Take to be the regular representation, then is a -algebra, flat as an -module. Define the scheme . Then is flat over and admits a -action over . By the exactness of , the short exact sequence induces an exact sequence Because is a flat -module, we know that remains injective when reducing any maximal ideal. In particular, is not zero for all . Hence is surjective. Since for any representation , the -representation is isomorphic to , where is the vector space underlying with trivial -action. It follows from is a tensor functor that is a -equivariant isomorphism.
¡õ

02/16/2016

Theorem 10
Let be a projective scheme. Then the functor induces an equivalence between the category of coherent sheaves on and the category of pairs , where is a coherent sheaf on and . Here if is a scheme, then .

Proof
Let be a very ample line bundle on . The functor gives an equivalence between coherent sheaves on and the quotient category of graded modules of finite type over modulo the subcategory of eventually zero modules. We have a similar functor for , which also commutes with . Using this equivalence we are reduced to the case when is a point, which follows from the following lemma (Galois descent over finite fields).
¡õ

Lemma 1
Let be vector spaces over . Let be a -linear map and let be a -semi-linear map. Let . Then

- is injective if is injective;
- is bijective if and are bijective.

Proof

- Suppose is a -basis of , we need to show that are linearly independent over . Suppose there is a relation . Apply and use the definition of and the injectivity of , we obtain that . It follows that is a scalar multiple of . Hence all , by Hilbert 90, thus the relation must be trivial.
- It suffices to show that , if . Let be a closed subscheme defined by where are the matrices corresponding to . Using the addition on , this becomes an affine group scheme over . One can check that it is also etale by computing the relative differential. Let be its Zariski closure. In homogeneous coordinates we have So implies that all , i.e., . It follows that is in fact
*finite*over . But the fiber over is nothing but , hence the fibers have constant cardinality . ¡õ

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

- ,
- , where ,
- , where is the generic point of .

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):

Theorem 11 (Steinberg, Borel-Springer, Kottwitz)
Let be a split reductive connected group. Then Here

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.

Remark 21
For , if a vector bundle has slope corresponding to . Then one can see that the -structure of can be reduced to the parabolic subgroup associated to . Intuitively smaller parabolics correspond to more degenerate "cusps" of . In fact, one can use this correspondence to give a geometric proof of Remark 11.

Definition 9
Define to be the substack of such that for any dominant weight , any geometric point and any structure on , we have . Here a *-structure* on means a -bundle such that and is the line bundle associated to using the 1-dimensional representation of .

Remark 22
The Harder-Narasimhan stratification on parallels Arthur's truncation process in the trace formula. In fact a comparison between them already appeared in the work of L. Lafforgue.

Theorem 12

- is represented by an algebraic stack, locally of finite type.
- is an open substack.
- If is sufficiently large (relative to ), then is a smooth quasi-projective scheme.

Remark 23
All the statements are valid for general . In fact, the case implies the general case: one can embed into some to view -bundles as vector bundles with -structure and deduce the relative representativity of from that the quotient is affine (since is reductive). See Behrend's thesis, Sec 4.

Remark 24
The finite group acts on the level structure and hence acts on the stack . Hence (c) implies that the quotient stack is Deligne-Mumford and of finite type. This together with (b) implies (a).

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:

- The direct image is a vector bundle, where is the projection.
- .
- is generated by .

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.

Definition 10
Recall we have the Cartan decomposition It defines an invariant map For , we define to classify pairs , where is a -bundle on the disk and is a trivialization on the puncture disk such that the invariant . We endow it with the closed reduced subscheme structure. We further define

Proposition 3
The subscheme is a single -orbit. It is a smooth quasi-projective variety of dimension . Moreover, is a (possibly singular) projective variety.

Proof
The first claim follows from Cartan decomposition. We have It follows that The smoothness follows since it is an orbit of the group. The claim can be seen by doing long combinatorics or reducing to the case of inductively.
¡õ

Remark 25
The crucial idea is that the -IC sheaves on form a basis (when varying ) of the abelian category of -equivariant perverse sheaves on . Our next goal is to give a multiplicative structure (convolution product) on perverse sheaves and prove this product is commutative. This is the analogue of the commutativity of the spherical Hecke algebra. Under the sheaf-function dictionary this indeed recovers the convolution on the classical Hecke algebra. Moreover, taking (appropriate signed summation of) the cohomology provides a fiber functor. This gives a neutral Tannakian structure and hence we can abstractly identify with the category of representations of an affine group . The content of the *geometric Satake correspondence* is that is exactly the Langlands dual group . Under the sheaf-function dictionary, this indeed recovers the classical Satake isomorphism.

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).

Definition 11
We define the *Beilinson-Drinfeld affine Grassmannian* to be the (ind-)stack classifying the same data as the Hecke stack (see Definition 14) (with no level structure) plus a trivialization for the last -bundle along . Notice we recover the usual affine Grassmannian when there is only one point of modification (, ).

The similar proof as in the case of affine Grassmannians gives:

Definition 12
Let be the moduli space of relative effective Cartier divisors on . Then is represented by . Let be such a divisor. We write be the formal completion of along and . We define the *Beilinson-Drinfeld Grassmannian* It is an ind-scheme, ind-proper over .

Definition 13
We define (resp. ) to be the *global loop group* (resp. the *global positive loop group*), classifying pairs , where and (resp. ).

Theorem 14

- is represented by an ind-group scheme over . It classifies tuples , where , a -torsor on , a trivialization of away from , a trivialization along .
- is represented by an ind-group scheme over .
- The map given by forgetting is a right -torsor and induces an isomorphism of fpqc sheaves over ,

Remark 26
The proof of geometric Satake is discussed in the course. Since we will only need its statement (Theorem 15), we refer to Richarz's paper for the proof and do not reproduce the lectures here.

03/29/2016

Definition 14
Let be a finite set and be a partition. We define the *Hecke stack* to be the moduli stack whose -points are given by tuples consisting of:

- , .
- , .
- Isomorphisms compatible with the level structure .

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.

Definition 15
Let be an -tuple of dominant coweights of . We define to be the closed substack of such that for all dominant weights of , Here is the vector bundle associated to using the finite dimensional representation of of highest weight .

Definition 16
We define the *moduli stack of shtukas* to be the stack over classifying the data (a-c) together with an isomorphism preserving the level structure. Thus we have a Cartesian diagram

Similarly define .

Proposition 4
There is a canonical isomorphism Here parametrizes a bundle together with a trivialization along modulo the equivalence mod .

Remark 27
For an effective divisor on , (a finite dimensional quotient of two infinite dimensional objects). The quotient by concretely means forgetting the trivialization along . The morphism is the same as a -torsor over . In terms of moduli interpretation, the above isomorphism is simply reorganizing the same data.

Remark 28
If is sufficiently large relative to , then the action of on factors through a finite dimensional quotient.

Definition 17
We define be the reduced closed subscheme given by the Zariski closure of , where is the complement of all diagonals. Define be the inverse image of .

Definition 18
More generally for any a finite dimensional representation of , we define to be the union of where the representation appears in . Define using similarly.

We have a map analogous to the map ,

Proof
Use the fact that is smooth and the derivative of Frobenius is zero.
¡õ

Now we can restate the geometric Satake correspondence for all possible parameters (Theorem 1.17 in V. Lafforgue's paper).

Theorem 15
There is a canonical functor of tensor categories from the representations of to (universally locally acyclic equivariant perverse sheaves). The support of lies in .

- If is irreducible, then is the IC sheaf of .
- Suppose is a refinement of , this induces natural maps (and similarly for ). These maps turns out to be
*small*and hence maps IC sheaves to IC sheaves and to . - Suppose , where . Then , where .
- Suppose and . Let and via pullback. Let and . Then we have isomorphisms, functorial in , Here

Definition 20
Let be the paw morphism. We define Notice the result does not depend on the partition of , which is a consequence of the smallness in b).

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 ).

Proposition 6
Let . Let be a finite dimensional representation of . Then there is a canonical *coalescence isomorphism* .

Proof
It is enough to treat the case when is injective and surjective.

- Suppose is injective. For simplicity let us assume is irreducible. The shtukas in question involves no modification along (since is a trivial representation of ). So we have a canonical isomorphism (In particular, the is nothing but .)
- Suppose is surjective. Then It follows from d) of Theorem 15 that . The result then follows from proper base change. ¡õ

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.

Definition 21
Let be an invariant vector. Let be the corresponding morphism on cohomology. We define the *creation operator* to be to be the composition Similarly define the *annihilation operator* for an invariant covector.

Proposition 7

- The creation and annihilation operators commute when the paws are disjoint.
- The composition of two creation/annihilation operators correspond to disjoint set of paws is the creation/annihilation operators for their union.

04/07/2016

Reference: thesis of Eike Lau and L. Lafforgue.

Theorem 10 has the following consequence:

Corollary 2
Let be smooth of finite type. Let be algebraically closed. Let . Then the functor gives an equivalence between finite etale covers and finite etale covers together with an isomorphism .

Proof
View as relative spectrum . For any finite etale, we have (see Stacks project 50.79). So the isomorphism is equivalent to the isomorphism since . The same argument as in Theorem 10 gives that is fully faithful.

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

Definition 22
Let be the *fundamental groupoid* of , whose objects are geometric points of and morphisms are the isomorphisms . Here is the fiber functor at from the category of finite etale coverings of to sets.

Definition 23
A *representation of * is a functor from to sets, namely, a collection of sets together with for any . Notice the category of representations of is equivalent to the category of finite etale coverings of .

Theorem 16
Let be smooth schemes of finite type over a finite field . Let . Let . Let be the -th partial Frobenius morphism . Then

- let be a local system over open dense, equipped with isomorphism over such that commute and is the canonical isomorphism. Then there exists an open dense such that extends a local system to .
- we have an equivalence : namely, a representation of is the same as a finite etale covering plus compatible isomorphisms .

Proof

- Assume is the largest subset of to which extends. Let . Then for all . Say is a union of irreducible components. We will show that each for some , .
- First assume that . Let be an irreducible divisor. Let be the two projections. Suppose otherwise both projections are surjective. Let . Then . But is finite (only finitely many component), a contradiction. So every component of is either horizontal or vertical.
- For general , let . Let be the maximal subset such that . Then by the previous case. We may then replace by their subsets of pure codimension 1. Hence is closed of pure codimension 1 and invariant under the remaining partial Frobenius. Then we induct on to prove the claim.

- For simplicity let us assume all 's in the theorem are the same. Let be the geometric generic point. Now we apply Corollary 2 to , we know for each the finite etale cover for some . Thus factors through for each , hence factors through their intersection by part a). ¡õ

04/26/2016

Remark 29
Notice though each partial Frobenius does not act on directly, the action of on does extend to an action of . In fact, the first partial Frobenius which lies over the map . Because the cohomology does not depend on the partition of and the partial Frobenius induces an isomorphism of etale sites, we know that the induces an automorphism (it increases but does not effect the Hecke finite classes). The composition of the permutation maps to itself, and is exactly the absolute Frobenius. Since covers on , commutes with each other and is the absolute Frobenius, by Theorem 16 we know that the action of extends to an action of .

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:

Lemma 2
Let , an irreducible representation of . Let . Let have degree under . Then depends only on . In particular, the inertia acts trivially. Moreover, if , then

Here is a rough strategy. Consider the Deligne-Mumford stack over . Then one constructs two closed substack () and together with morphisms such that

- The first two stages of the excursion operator (creation and ) is realized by a cohomological correspondence supported on (after normalizing by a half-integral power of ).
- The last stage (annihilation) is realized by a cohomological correspondence supported on . Thus is a cohomological correspondence supported on . Now:
- The fiber product
*is*the usual etale Hecke correspondence , - equals to this fiber product.