Source: http://www.math.columbia.edu/~chaoli/doc/VincentLafforgue.html
Timestamp: 2019-04-21 22:50:36+00:00

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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.
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.
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.
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.
Theorem 1There is a canonical decomposition where runs over Langlands parameters.
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).
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 !
Theorem 2 corresponds to under the Satake isomorphism.
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.
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.
An -tuple of -points on .
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.
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 9 All one really needs is the special case .
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 .
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.
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.
Theorem 5 Any -pseudo-representation comes from a unique semisimple global Langlands parameter .
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.
(sometimes also requiring that is not a zero divisor in , not needed for V. Lafforgue's work).
Remark 13 As we will see, the identity in (c) comes from the fact -st exterior power of a -dimensional vector space is zero.
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.
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 .
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.
is continuous and takes values in , for a finite extension .
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.
(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.
(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.
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).
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.
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.
Remark 18 If is smooth, then any -bundle is automatically locally trivial in the etale topology.
Remark 19 If is a curve, then any -bundle is automatically locally trivial in the Zariski topology.
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.
Theorem 9 The category of -bundles on is equivalent to the category of exact tensor functors .
Example 3 The trival -bundle corresponds to the functor .
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 .
is bijective if and are bijective.
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.
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.
Remark 20 This nothing but , the moduli of shtukas with no paws.
, where is the generic point of .
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.
is represented by an algebraic stack, locally of finite type.
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).
The direct image is a vector bundle, where is the projection.
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.
Proposition 3 The subscheme is a single -orbit. It is a smooth quasi-projective variety of dimension . Moreover, is a (possibly singular) projective variety.
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 ( , ).
Theorem 13 is represented by an ind-scheme, ind-projective and of finite type over .
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. ).
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 .
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.
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 .
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.
Proposition 5 This map is smooth of dimension equal to .
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 .
Definition 19 If is irreducible. We set over to be the IC sheaf of . More generally, we define .
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).
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.
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.
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.
Reference: thesis of Eike Lau and L. Lafforgue.
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.
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 .
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 .
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.
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 .
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 ).
equals to this fiber product.
Last Update: 04/26/2016. Copyright © 2015 - 2016, Chao Li.

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