These are my live-TeXed notes for the course Math G6761: Perverse Sheaves and Fundamental Lemmas taught by Wei Zhang at Columbia, Fall 2015. The final part of the course discusses the recent breakthrough Shtukas and the Taylor expansion of L-functions by Zhiwei Yun and Wei Zhang.

Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!

09/15/2015

## Motivation

A fundamental lemma is an identity between orbital integrals on two different groups. For example, the endoscopic fundamental lemma arises from the stabilization of the Arthur-Selberg trace formula and endoscopic functoriality in the Langlands program. The Jacquet-Rallis fundamental lemma arises from the W. Zhang's relative trace formula approach to the Gan-Gross-Prasad conjecture for unitary groups. Both sides of the fundamental lemma can be thought of as counting the number of lattices satisfying certain properties.

In the equal characteristic case, the endoscopic fundamental lemma was proved by Ngo and the Jacquet-Rallis fundamental lemma was proved by Yun. These proofs use geometric methods (perverse sheaves) in an essential way. The advantage in the equal characteristic is one can endow the space of lattices in question with a geometric structure (e.g., the -rational points of an algebraic variety, usually realized as the fiber of an invariant map from a certain moduli space of vector bundles to an affine space). These -rational points then can be counted using Lefschetz trace formula.

In order to prove the identity between orbital integrals ("functions on orbits"), one instead proves the identity between perverse sheaves ("sheaves on orbits"). The miracle is that one can prove the identity between perverse sheaves by only verifying it over certain open dense subsets ("very regular" orbits), which can be easier. This reflects a uniqueness principle of orbital integrals: the "very regular" orbital integral determines the more degenerate ones. This principle, however, is hard to see directly from orbital integrals.

## Intersection homology

The idea of perverse sheaves begins from Goresky-MacPherson's theory of intersection homology, which is a homology theory for singular manifolds with an analogue of Poincare duality and Hodge decomposition. More precisely, we define

Definition 1 A Hausdorff space is a pseudo-manifold if it admits a stratification by closed subspaces such that
1. is dense in .
2. . (Notice this condition is automatic when is the complex points of an algebraic variety)
3. For each , and , there exists a neighborhood of such that there is a stratification-preserving homeomorphism . Here is a compact pseudo-manifold of dimension and is the open cone of (of dimension ).
Remark 1
1. We call the link of . It is a fact that (up to homeomorphism) depends only on , not on .
2. is called a distinguished neighborhood of . It is a fact that distinguished neighborhoods form a basis around .
Example 1 (complex points of the union of two lines ), or a pinched torus (complex points of a nodal curve)), is a pseudo-manifold. Near the singular point, looks like the open cone of two circles.
1. For , we have , , . The Poincare duality fails.
2. For a pinched torus, we have , , . The Hodge decomposition fails.
Remark 2 It is a theorem that the complex points of any quasi-projective varieties (with singularities) is a pseudo-manifold. This forms the main example considered in this course.

To get a homology theory for pseudo-manifold with desired properties, one needs to consider only the chains that has nice intersection properties with the stratification.

Definition 2 To measure how "perverse" these intersections are, we introduce the perversity, which is a function such that
1. ,
2. .
Example 2 There are three special perversity functions:
1. the zero perversity ,
2. the top perversity ,
3. the middle perversity .
Definition 3 Given a perversity , we define a -chain to be -allowable if We then define the intersection homology groups by considering only the -allowable locally finite -chains. When is the middle perversity, is often written as for short. Similarly one can define the intersection homology with compact support by considering only the -allowable finite -chains.
Remark 3 Notice the -allowable condition is most strict when . For the top perversity, the -allowable condition is nothing but the condition that , i.e., "most" of lies in the "non-singular" part .
Example 3 Suppose (of dimension ) has only one singular point . Then one can compute Using this we find that:
1. For , we have , , .
2. For a pinched torus, we have , , .

Notice that both Poincare duality and Hodge decomposition hold for !

09/17/2015

Theorem 1
1. All intersection homology groups are finitely generated, independent of the choice the stratification. It gives the usual homology groups for manifolds without singularities.
2. if is connected.
3. (intersection product) Suppose are perversities such that is still a perversity. There is an intersection product where is the real dimension of . When , we have a non-degenerate pairing In particular, when has even dimensional strata, we have the duality between and .
Remark 4 The intersection homology is a topological invariant but is not a homotopy invariant, unlike the usual homology theory. In the 70's, the only proof of the topological invariance is sheaf-theoretic.

Let be the category of sheaves of -vector spaces on . Two important examples:

Example 4
1. The constant sheaf on , denoted by .
2. Locally constant sheaves on . Recall that is locally constant if for any , there exists an open such that the restriction map is an isomorphism. Locally constant sheaves with stalks of finite rank are called local systems.

Repeating the construction of -allowable -chains for each open we obtain a sheaf . We define a complex of sheaves given by (so is concentrated in the negative degrees). In particular, is concentrated in negative degrees.

Theorem 2 Denote . Then there is an isomorphism

It turns out that the IC sheaf (as an object in the derived category ) is topologically invariant. This topological invariance on the level of sheaves is more flexible and easier to prove.

09/22/2015

Theorem 3 (Deligne, Goresky-MacPherson) Let . Let (). Let be a perversity. Then in the bounded derived category , we have Here is the usual truncation functor: so that for . Inductively, let and Then
Remark 5 When is a manifold, are all empty, so it follows from the theorem that . This is clear since there is a quasi-isomorphisms of complex sheaves and The quasi-isomorphism can be checked locally: the Borel-Moore homology (allowing infinite, locally finite chains) of an Euclidean space is nontrivial only in the top degree.
Proof To prove this theorem we only need to do local calculation for , where is compact and has dimension . It boils down to the following two parts:
1. Kunneth formula gives .
2. Suppose is a compact pseudo-manifold of dimension . Then

It then follows that inductively. ¡õ

09/24/2015

References:

• An introduction to perverse sheaves, Rietsch (brief)
• Intersection homology, edited by Borel (detailed)
Remark 6 The IC sheaf is constructible (not far from being a local system). More precisely, we say is constructible if is a local system for any . We say is constructible if is constructible for any (for the given stratification on ). The category of constructible sheaves (for a given stratification) is denoted by . We denoted by the larger category of sheaves which are constructible for some stratification. Notice that is locally constant (but not necessarily constant, depending on the link ).

By checking at stalks, one finds that the cohomology of IC sheaves has support of dimension in a particular range depending on the perversity, It turns out that we can characterize Deligne's sheaf using the following four axioms:

1. is constructible with respect to the given stratification.
2. in .
3. The stalk if and .
4. The attachment map (given by ) is a quasi-isomorphism up to degree .

Our next goal is to rephrase axiom d as an axiom c' similar to c for the dual of and use Deligne's construction to prove the topological invariance of IC sheaves.

## Operations on sheaves and duality

We would like to define a dualizing functor . We hope to satisfy the following properties. Suppose is a morphism. Then and Moreover, when is a point, we should have . Moreover, the natural transformation becomes a bi-duality when restricted the constructible sheaves .

Remark 7 Here is the pull-back with compact support satisfying the adjoint property with respect to (Verdier duality): However, in general is not the right derived functor of any functor . It can be defined as follows: for and open, we define where is any soft resolution of . Recall that a sheaf is soft if any of its section supported on a compact subset can be extended to a global section.
Remark 8 Suppose is an open immersion and is a closed immersion. We have a distinguished triangle By composing with (the structure morphism ), we obtain a long exact sequence in cohomology with compact support Notice by definition one can check that is right adjoint of , hence . By definition we also have . We have the following adjoint relations
Remark 9 Dually we have a distinguished triangle In this case can be actually defined as a derived functor of : . By composing with , we obtain a long exact sequence in cohomology The first term recovers the cohomology with support in .
Remark 10 (Purity) Suppose is a manifold, and . Then
Remark 11 Recall that for , we define where . Let be its derived functor and .

It turns out there exists a dualizing sheaf such that (at least for constructible sheaves) It turns out that one can define to be the sheaf of locally finite -chains . In particular, when is a manifold, is quasi-isomorphic to .

Remark 12 We get the following the relationship where . Moreover, for a morphism , we have
Remark 13 In the case of manifolds of dimension , we have for any local system .
Theorem 4 Let be a local system on and be the dual local system. Let and be complementary perversity (i.e., is the top perversity). Then
Remark 14 This theorem follows from the uniqueness of the characterizing properties of IC sheaves (see Theorem 5). When is a manifold, this theorem recovers the usual Poincare duality.

09/29/2015

## Reformulation and topological invariance

Using the theorem that and the properties of the dualizing functor. We find the axiom d for can be formulated as a similar axiom on the costalk .

c'. if for .

In fact, by Remark 9 we have the exact sequence One sees that axiom d is equivalent to Let and , then Using the purity (since is a manifold), we have . Hence axiom d is equivalent to axiom c'.

Now suppose we are in the case of middle perversity. Let and We can (by shifting) translate the axioms c and c' to the axioms which don't mention the stratification:

c2. .

c2'. .

Notice that a, b still depend on the stratification . For example, when is a manifold and is the trivial stratification. Then the sheaves satisfying a, b, c2, c2' are of the form for local systems on . The shift by the complex dimension ensures that the dualizing functor preserves .

Theorem 5 Given a local system on an open of real codimension at least 2, there exists a unique such that satisfying the axioms a, b, c2, c2' for any stratification .

10/06/2015

The uniqueness follows easily from the following lemma.

Lemma 1 Suppose has codimension at least 2. If is connected, then is connected. Moreover, the natural map is surjective. In particular, a local system on has at most one extension to .
Proof The connectedness of follows from the long exact sequence in cohomology with compact support. The surjectivity follows from the connectedness and that any path can be deformed in codimension 2. ¡õ

We can use Deligne's construction to prove the existence of inductively. Suppose is a local system on . Then there exists a maximal open such that extends to . We define to be the maximal open of such that is a local system. Refining this idea, we obtain a coarser stratification by also requiring is also a local system. Iterating Deligne's construction, we obtain a stratification given by . The stratification thus obtained is the "coarsest" in some sense. This allows us to remove the dependence on the stratification and to show the topological invariance of IC sheaves.

## Perverse sheaves

We now relax the axiom c2, c2' by allowing non-strict inequality. This modification is intended to include sheaves like for an IC sheaf on a closed (e.g., if ).

Definition 4 A sheaf is perverse if and The category of perverse sheaves is denoted by .
Remark 15 If is an irreducible local system on an open of . Then is called a Deligne-Goresky-MacPherson (DGM) complex, which is an simple object in .
Theorem 6
1. is an abelian category (the heart of perverse -structure on defined by the support and cosupport condition), stable under Verdier dual.
2. All simple objects are DGM complexes.
3. Every object in is a successive extension of simple objects.
Definition 5 A proper surjective morphism between two varieties is small (resp. semi-small) if (resp. ) for any . In particular, is a generically finite morphism. For example, when , small is equivalent to finite. When , small morphism can only have dimension 1 fibers above finitely many points. One can check that a morphism is semi-small if and only if and small if and only .

Small morphisms are compatible with IC sheaves.

Theorem 7 If is small and has generic degree 1, then . In particular, .
Remark 16 This is especially useful when is smooth (i.e., a small resolution of ).
Proof Consider the case is smooth. Then . So by Theorem 5 it suffices to check the support and cosupport axioms for . By base change, . If , then , i.e., . The support and cosupport axioms then follow exactly by the smallness assumption. ¡õ

10/08/2015

The above theorem can be generalized to the following:

Theorem 8 Suppose is small. . Then .
Remark 17 Notice can be viewed as a local system on a smaller open subset where is unramified. In terms of the monodromy representation, corresponds to the induction of of .
Remark 18 This is the key theorem used in the proof of fundamental lemma: one can check the identity between perverse sheaves on by throwing away bad fibers. Ngo called this "perverse continuation".
Example 5 Suppose is a blow-up along a point with fiber . Then . This is no longer perverse and the extra terms measures the failure of being small.

In general we have the following decomposition theorem.

Theorem 9 (Beilinson-Bernstein-Deligne + Gaber) If is proper and surjective, then in we have is a direct sum of DGM complexes shifted by certain degrees:
Remark 19 The proof is rather indirect: it relies on reducing to the positive characteristics and use Deligne's Weil II and also Gaber's improvement on purity. One can replace by more general complexes of "geometric origin" (coming from the pushforward of a constant sheaf): this contains local systems of finite monodromy and also IC sheaves, but is still quite restrictive.
Remark 20 The key issue is that the 's appearing in the decomposition are hard to determine. A famous theorem (the support theorem) of Ngo further determines the possible 's under more restrictions on .

10/27/2015

(I was out of town and missed the two lectures on Oct 20 and 22. Thank Pak-Hin Lee for sending his notes to me. )

## Springer fibers

Reference: PCMI 2015 lectures by Ngo, Yun and Zhu, transcription available on Tony Feng's website.

A important class of semi-small maps comes from holomorphic symplectic varieties.

Definition 6 A holomorphic symplectic variety (over ) is a nonsingular variety such that there exists a closed holomorphic 2-form satisfying that is nonzero everywhere (i.e., is non-degenerate).
Example 6 Let be a nonsingular variety over with . Then the cotangent bundle is naturally a holomorphic symplectic manifold. In fact, let be the local coordinates on and be the dual basis of in the cotangent space, then is a non-degenerate closed 2-form.
Theorem 10 (Symplectic resolution) Let be a proper surjective birational map. If is holomorphic symplectic, then is semi-small.
Example 7 (Hilbert scheme of surfaces) Let be a surface. Then symmetric power becomes singular when some of the -points on collide. However, the Hilbert scheme of -points on (which remembers infinitesimal information of the collision as well) is nonsingular. Moreover, one can show that is a symplectic resolution of Notice when is a 3-fold, both the an are singular.

Let be a semisimple connected linear group over . Fix a Borel subgroup . Let be the nilpotent cone. Then is singular.

Example 8 Let . Then . Define the invariant map given by the coefficients of the characteristic polynomial of : . Then is the preimage of , i.e., the set of all traceless matrices such that . When , , which is simply a 2-dimensional cone, with a simple singularity at 0.

In general the nilpotent cone may have very bad singularities away from the regular nilpotent elements. Springer found a systematic way of resolving the singularities.

Definition 7 Let be the flag variety, parametrizing all the Borel subgroups of . Define Then the natural map is an isomorphism on the regular nilpotent elements. It turns out that the natural map is a symplectic resolution, known as the Springer resolution. In fact, . The fiber is called a Springer fiber.
Example 9 For , and . The simple singularity of the 2-dimensional cone at 0 is resolved.
Remark 21 It follows that . Therefore has codimension , where is the rank of . One can also see this from that is the preimage at 0 of the invariant map , where is the Cartan subalgebra and is the Weyl group.

Using the fact that , one can check that is indeed semi-small. One can generalize the construction of and obtain a small map .

Definition 8 The Grothendieck-Springer fibration is defined to be

The Springer fiber is always reduced. However, for the Grothendieck-Springer resolution , the fibers may be non-reduced. Therefore we have a commutative (but not Cartesian) diagram Explicitly, while

Example 10 Let . Let be the complete flag associated to . Then for the regular element , the Springer fiber satisfies while the Grothendieck-Springer fiber satisfies the weaker condition . For example, when is 2-dimensional, . Let , then has more than one points: it has an obvious point corresponding to . It also has an extra because . We can compute the scheme-theoretical fixed points of on : since , we have , hence . Therefore .
Remark 22 In general, is the reduced scheme associated to .

## Affine Springer fibers

Let be a finite field. One can easily see that the number of -points in the Springer fiber is the same as fixed points of on the set , which can be rewritten as a simple orbital integral. Moreover, the finite set can be realized as the -rational points of the flag variety . Now we want to upgrade this to an "affine" version, i.e., for local fields of equal characteristic.

Let and . We want to geometrize the infinite set by an affine Grassmannian . The analogue of the full flag variety should be given by the affine flag variety whose points gives , where is the Iwahori subgroup.

In terms of moduli interpretation, when , is the set of -lattices in , where the identity coset corresponds to the standard lattice . is the set of chain of lattices in , where is length one -module such that .

Definition 9 Let be a -algebra. Let be the ring of power series in . Let be the field of Laurent sereis. Due to the completion process, is larger than the naive base change when is not finitely generated.
Remark 23 Notice geometrically is a formal disc (the formal completion of a curve over at a point) and is a punctured disk. In general, is the formal completion of a curve along the graph of (so the completion is only along the -direction).
Definition 10 We define an -family of lattices in is a finitely generated projective -submodule such that . This is equivalent to the data , where is a vector bundle over of rank and is a trivialization of over the punctured disk.

We define the functor . Then . has a reasonable geometric structure (though infinite dimensional).

Theorem 11 is represented by an ind-scheme, i.e., , where each is a projective scheme of finite type and each is a closed immersion.

Here consists of -lattices (projective as -modules) such that Due to this boundness, it can be viewed as the set of quotient -modules of (projective as -modules). In other words, let (so ). Then is the set of quotient projective -modules of , such that the -action (given by a nilpotent operator ) satisfies . Observe that is nothing but a union of generalized version of Springer fibers : Here is a Grassmannian of one-step flags (instead of full flags).

Example 11 For , we see . Let be a regular nilpotent operator on . When , we have In general (compare: ).
Remark 24 One can also try to geometrize and obtain the mixed characteristic affine Grassmannian. In terms of moduli, its -points should be given by the -lattices in . It turns out the Witt ring only behaves well when restricting to a subcategory of perfect -algebras (these algebras are large but still useful for many purposes, e.g., for answering topological questions). We now know that the mixed characteristic affine Grassmannian is an ind-projective algebraic space (union of projective schemes), after the recent work of X. Zhu, Scholze and Bhatt.

10/29/2015

More generally,

Definition 11 (Affine Grassmannians) Suppose is algebraically closed. Let be a group scheme over . We define the functor such that is the set of pairs , where is a -torsor over and is a trivialization , where is the trivial -torsor. In other words, is the moduli space of -torsor together with a rigidification. This may remind you of the definition of a Rapoport-Zink space, which is the moduli space of certain -divisible groups together with a rigidification.
Theorem 12 IF is a flat group scheme over . Then
1. The presheaf (under the fpqc topology) is an ind-scheme (ind of finite type).
2. If is reductive, then is ind-projective.
3. If is parabolic (i.e., a smooth group scheme whose general fiber is reductive but the special fiber may fail to be reductive, e.g., the Iwahori model of ), then is ind-projective.

Notice . So acts on by .

Definition 12 (Affine Springer fibers) Let , we define to be the fixed points of and . It turns out that is a closed subscheme of .
Remark 25 The number of the -points of the affine Springer fiber more or less gives the orbital integrals .
Remark 26 One can also define a Lie algebra version of affine Springer fibers. If and (not necessarily an automorphism), we define as the intersection of the graph of with the diagonal. Namely, sits in the Cartesian diagram

We provide two more analogous constructions.

Example 12 (Affine Schubert varieties) By the Bruhat decomposition (over ) The orbits of on are parametrized by elements in the Weyl group . Let be the invariant map. Define the orbit associated to to be the Schubert variety, so These are locally closed subvariety of (defined by incidence relations) . Then sits in the Cartesian diagram

One can analogously define affine Schubert varieties. By the Cartan decomposition where consists of the dominant co-characters of , we have an invariant map Then for , define the affine Schubert variety by

Example 13 (Affine Deligne-Lusztig varieties) Assume is finite. Define the Deligne-Lusztig variety to be the subvariety of given by In other words, the Deligne-Lusztig variety sits in the Cartesian diagram Deligne-Lusztig constructed all the irreducible representations of finite reductive groups in the cohomology (with local systems as coefficients) of Deligne-Lusztig varieties. The computation of the Deligne-Lusztig characters are naturally related to counting points of Springer fibers. Deligne-Lusztig varieties form one of the starting point of the geometric approach to representation theory initiated by Kazhdan and Lusztig.

One can then analogously define affine Deligne-Lusztig varieties using the point-wise condition. Affine Grassmannians and affine Deligne-Lusztig varieties are fundamental objects in geometric representation theory and in the study of local models of Shimura varieties.

11/05/2015

An alternative definition of affine Grassmannians uses loop spaces and arc spaces. Let be a field and . Let be a scheme. One would like to geometrize the sets and . We define the loop space functors Similarly we define the arc space (or positive loop space) functor These are presheaves under the fpqc topology.

Example 14 When , we have . Therefore is in fact represented by a scheme , given by the leading coefficient and rest of the coefficients . The points are more complicated: they are Laurent series of the form When taking the reduced structure, we find that is an infinite copies of .

More generally, we have

Theorem 13
1. is represented by a scheme. It is affine if is affine.
2. is represented by an ind-scheme.
3. The affine Grassmannian (as the quotient sheave under the fpqc topology).

## Orbital integrals

Definition 13 Suppose is a local field. For and . Define the orbital integral Notice that the convergence of this orbital integral is already an issue.
Example 15 Let us consider the case , , and is regular semisimple. In this case the centralizer of is the diagonal torus . Since is -bi-invariant and is compact, to show the convergence of the orbital integral, it suffices to show the convergence of By the Iwasawa decomposition where is the unipotent radical of the Borel subgroup of (i.e., group of upper triangular unipotent matrices for ), we know that the convergence is equivalent to The key observation is that and one easily compute since each root group is an eigenvector under the adjoint action of the semisimple element . For example, when and , we have Take , we find the integrand is nonzero only when the above matrix has -entries. This means that , hence is bounded and the integral converges.
Remark 27 In general, when , the above argument shows that the integral essentially counts the set where is the loop space of and is the affine Springer fiber. The last inequality is problematic in general since the rational points of the quotient is not necessarily the quotient of the rational points.

Ngo showed that the action of on is not faithful. The action factors through the quotient known as the local Picard group.

Example 16 For , . Then and (i.e., the -part acts trivially on ).
Theorem 14 (Goresky-Kottwitz-MacPherson, Ngo) Assume is regular semisimple. Then Namely, the -points of the stacky quotient is in fact a stable orbital integral (i.e., a sum of orbital integrals over conjugacy classes which are stably conjugate to ).

Instead of the loop group action, Kazhdan-Lusztig also considered the discrete action of the lattice on the affine Springer fiber.

Theorem 15 (Kazhdan-Lusztig) is proper and of finite type. is a finite dimensional ind-scheme and is locally of finite type.

11/10/2015

## Hitchin fibers

The quotient is a projective variety. It has singularities, but people still expect certain "purity" of its cohomology, which implies the fundamental lemma for regular semisimple elements in the maximal torus.

This purity is still unknown. To prove the fundamental lemma, one instead consider a global version of the affine Springer fibers. Suppose is a smooth projective curve over a finite field with . Consider . The global analogue of affine Grassmannian is , the moduli stack of rank vector bundles on . It represents the functor It has rational points

Remark 28 It turns out is an Artin stack, smooth over and is locally finite type. It has a stratification given by the instability and it is of finite type on any part with bounded instability. For , by the smoothness we have , so When is stable, . So the coarse moduli space of has dimension (at least for the stable part).
Remark 29 The the fiber of the tangent bundle of at is given by . Since is self-dual, we know that by Serre duality this is isomorphic to . Hence the fiber of the cotangent bundle at is given by the vector space .
Definition 14 A pair is called a Higgs bundle. The Hitchin moduli space is defined to be moduli space of Higgs bundles:
Definition 15 For the Higgs field , define its invariants . So and . We call the affine space the Hitchin base. We have the invariant map to the Hitchin base
Remark 30 Since by Riemann-Roch we have So the Hitchin base has dimension , which becomes the same dimension of the coarse moduli space of and hence half of the dimension of . This was one of Hitchin's key observations. Using this Hitchin was able to define certain integrable hamiltonian system, which has applications to differential geometry and physics.
Definition 16 More generally, we can replace by any vector bundle . The resulting moduli space of Higgs bundles is called the -twisted Hitchin moduli space. The affine space is called the Hitchin base.

When , then , which is a finite dimensional -subspace of . So one can view as a finite dimensional -subspace of an infinite dimensional -space . On the other hand, when varying (allowing more poles) these finite dimensional -subspaces will exhaust all elements of . More precisely, one can define a family version of the Hitchin base by considering Let be the open substack with . It turns out is the same as (the effective divisors of degree on ), hence is indeed a scheme. The complement of is isomorphic to (given by the zero section). More generally,

Definition 17 Define The universal Hitchin base is defined to be a family of Hitchin bases over .

The fibers of the invariant map (Hitchin fibers) are the global analogue of affine Springer fibers.

Theorem 16 For with characteristic polynomial . Assume that is elliptic (i.e. is irreducible, equivalently is an anisotropic torus). Then
Remark 31 Since the global orbital integral factors into the product of local orbital integrals, we would expect the Hitchin fiber "factors" as the product of affine Springer fibers. To do this one needs more explicit description of the Hitchin fiber. We will define a spectral curve such that , the compactified Picard variety of (at least when is reduced). This allows us to define a relative group scheme . Moreover, is in fact smooth or has only mild singularity over a large open of the Hitchin base. One can then prove a sheaf-theoretic version of the fundamental lemma on this large open and use perverse continuity to deduce the rest. This help us to avoid the difficulty in dealing with the complicated singularities of the affine Springer fibers.
Remark 32 Why does this globalization help to prove the fundamental lemma? Here is a more philosophical explanation. The fundamental lemma is easy if and is regular semisimple. However, these conditions are quite restrictive: i.e., the discriminant is required to be a unit. But globally, is a unit at almost all places. The idea is then to use the fact that the fundamental lemma is easy for all most all places to deduce the fundamental lemma at the remaining places.

11/12/2015

## Spectral curves

Today we will discuss a bit more on the spectral curve mentioned last time. Starting next time we will do a concrete example: use the perverse continuation principle to prove Waldspurger's theorem for central values of -functions on in the function field setting.

Definition 18 Consider the total space of the line bundle , It is a -fibration over . This total space sits in the projective bundle (so ). We have two affine charts given by the two coordinates and Then is given by the . Let We define the spectral curve to be the zero locus of .
Remark 33 Locally is a finite cover of degree given by the equation , where . The generic fiber of has function field . Hence the generic fiber is reduced if and only if has no repeated roots over , i.e., is regular semisimple. We define the locus (resp. ), where the spectral curve is reduced (resp. smooth).
Lemma 2 Suppose and .
1. is reduced (i.e., ) if and only if is regular semisimple.
2. is etale over if and only if , where is the discriminant section.
3. If is a multiplicity-free effective divisor, then is smooth (i.e., ).
Definition 19 Suppose is a reduced curve. The compactified Picard (or Jacobian) stack of is defined to be the stack of torsion-free coherent sheaves of rank 1 on . When is smooth, . By a theorem Altman-Iarrobino-Kleinman, for reduced curves with only planar singularities (which by definition is satisfied by the spectral curves), the usual Picard scheme is always open dense in . Hence is naturally a compactification of . Notice that in general may have singularities.

For a torsion-free coherent sheaf on , the pushforward under is a torsion-free sheaf of rank on the smooth curve , hence is indeed a vector bundle of rank . One can further construct an -linear endomorphism of with the given characteristic polynomial using the action of on . In this way one can describe a Hitchin fiber as the compactified Picard of the spectral curve.

Theorem 17 For , we have .

11/17/2015

## Waldspurger's formula via Jacquet's relative trace formula

Let be a quadratic extension of function fields, corresponding to an etale double cover of curves over a finite field . Let and be an anisotropic torus (with a fixed embedding ). Waldspurger's formula relates the toric automorphic period to central values of automorphic -functions on . We state a very special (unramified everywhere) case.

Theorem 18 (Waldspurger) Let be an automorphic cuspidal representation of that unramified everywhere. Let , where (so is unique up to scaling). Then up to some explicit constants we have an equality
Remark 34 Recall the construction of the -function on due to Hecke (and Jacquet-Langlands in modern language). Consider the diagonal torus . Then for , we have up to local factors Taking we know that Waldspurger's formula is equivalent to an equality of two different toric integrals here is the quadratic Hecke character on associated to the quadratic extension .

Now we use the well known procedure of relative trace formula to remove the dependence on the automorphic representations . Consider the distribution where and the kernel function is given by The kernel function has a spectral decomposition where runs over an orthonormal basis of level one cusp forms on . So we obtain the spectral decomposition where is the character determined by .

One can repeat the same story for the period on the anisotropic torus. Define Then similarly we have a spectral decomposition By the previous remark, Waldspurger's certainly implies the relative trace formula identity Conversely, using the linear independence of the automorphic representations, this identity is in fact also sufficient to prove Waldspurger's formula.

To prove this identity of two distributions, we use the geometric decomposition Notice the generic stabilizer is trivial and so the double integral is over and factors as a product of local orbital integrals. One has a similar geometric decomposition for .

We can parametrize the orbits and in a similar way. Consider the invariant map Then consists of exactly one orbit when . We call these regular semisimple and the corresponding orbital integral is automatically convergent (regularization process is needed for other ). Write It remains to compare the orbital integrals

Remark 35 Since we are only concerned with spherical Hecke algebra, this identity of orbital integral is nothing but the fundamental lemma in this setting. One can prove this fundamental lemma by explicit calculation on both sides (which is what Jacquet did). We are going to prove this fundamental lemma without doing explicit calculation (at least when is regular semisimple), using perverse continuation.
Remark 36 For an effective divisor on , let , where . As a consequence of Cantan decomposition, we have , where . So forms a basis of the spherical Hecke algebra .

## Geometrization for the split torus

Let us ignore the quadratic character for the moment. So where . It is now convenient to lift the situation to and consider for , where is the diagonal torus in .

In order to geometrize this orbital integral, we define an analogue of Hitchin moduli space.

Definition 20 Let such that and . Define the moduli space of pairs of rank two vector bundles together with a morphism: where are line bundles on . For simplicity (since we only consider regular semisimple orbits) we also impose the non degeneracy condition that (which strictly speaking defines an open subset of ). Let be the union of all such 's with .

Now we define an analogue of the invariant map to the Hitchin base and an analogue of Hitchin fibers.

Definition 21 Let be the the moduli space of pairs , where , . Let be the moduli space of triples , where . We have a natural map given by .

We have an invariant map , given by , and . Let be the fiber of this invariant map above .

Theorem 19 Let be an effective divisor on of degree . Let (viewed as a -subspace of ) be the fiber of above . Then

11/19/2015

Remark 37 Let be the Hecke stack, i.e., the moduli stack of morphism of vector bundles such that . Then is closely related to the fiber product Notice and is the moduli of pairs of line bundles on . Therefore the distribution can be thought of as an intersection number.

Now sending a point in to defines a map By the non-degeneracy assumption on , this induces an isomorphism Now let the moduli space of triples such that . The we have a commutative diagram Here the right vertical map is induced by the addition map . In this ways the analogue of Hitchin moduli space becomes a simple construction using symmetric powers of the curve .

Remark 38 By Riemann-Roch, when , we know the dimension of is . Moreover, in this case is smooth over and is a finite map. So we are in the simplest situation to use the machinery of perverse sheaves (Remark 16).

By the previous theorem, we would like to study Therefore we can forget about the orbital integrals and focus on the sheaf . At this stage one can also insert the character by taking a nontrivial local system on and then take . Here , is the local system on associated the the double cover and is the natural quotient map by .

Now it it remains to study the simpler object: where is the addition map. This is nothing but the push-forward of a local system under a finite map, a simplest example of a perverse sheaf (after shifting by the dimension).

• Since (the multiplicity free locus) is a Galois covering with Galois group , we know that is the middle extension (by the perverse continuation principle). Here and the local system on corresponds to the induced representation (of dimension ).
• To deal with the nontrivial coefficient, we need to go to the double covering to trivialize the local system. So we have a Galois covering which is Galois with Galois group . Here permutes in a natural way, in other words, is the wreath product . Let be the character that is nontrivial on the first factors and trivial on the last factors. The action of on has stabilizer exactly . Hence we can extend to . Then the local system on corresponds to the representation . It is irreducible of dimension (one check the irreducibility by computing the endormophism algebra to be a division algebra).
Remark 39 The group is also known as the Hyperoctahedral group (symmetry group of a hypercube), which is also the Weyl group of type BC.

11/24/2015

## Geometrization for the nonsplit torus

Today we will geometrize the distribution on the nonsplit torus as well and verify the identity for (at least for the regular semisimple orbits).

Remark 40 By a density argument, to prove the identity on the spectral side (equivalent to Waldspurger's theorem), it suffices to verify when is sufficiently large. The reason is that the action of the Hecke algebra on the space of automorphic forms has a large kernel and becomes finitely generated.

In order to geometrize the orbital integral , we define an analogue of the space .

Definition 22 Let be an etale double cover. Define to be the moduli space , where and the map is an element of where is the nontrivial Galois involution. Since we only consider regular semisimple orbits, we further impose the non-degeneracy condition where .

Now sending a point in to and , we obtain an isomorphism (by an analogue of Hilbert 90) where the map is induced by the norm map We also have the invariant map induced by the norm map :

Remark 41 By taking the split double cover , we recover the space defined last time. Since the isomorphism recovers the decomposition . In this case one can view and the invariant map is simply .
Remark 42 The non-split torus is easier for analysis since the adelic quotient is already compact. Geometrically this means there is no need to do compactification for the moduli spaces in question.

Analogous to Theorem 19, we have

Theorem 20 Let be an effective divisor on of degree . Let (viewed as a -subspace of ) be the fiber of above . Then

Similarly to the split case, we are now interested in the sheaf , where . When , is smooth, is a projective bundle and the norm map is smooth with kernel a Prym variety of dimension . Therefore is in fact smooth and hence is perverse (after shifting by the dimension). The local system underlying is the induced representation .

## Orbital integral identity for regular semisimple orbits

Now we have tow invariant maps with a common base

The identity now becomes a statement purely about two perverse sheaves.

Theorem 21 There is an isomorphism between perverse sheaves

This will follow from the even stronger claim.

Theorem 22 Let and . Then

By the perverse continuation principle, the proof of this theorem essentially boils down to representation theory of finite groups because the local system underlying both perverse sheaves have finite monodromy (trivialized after a finite covering). Namely,

Theorem 23

This is much simpler statement to prove! Notice that both sides have dimension . By Frobenius reciprocity, it remains to show that there is a -equivariant embedding which can be explicitly written down.

Remark 43 For non-regular semisimple orbits, one needs to consider the larger moduli space where the non-degeneracy condition is removed. In this case the invariant map is no longer small and one needs to verify the support condition directly and use the full strength of Deligne's uniqueness principle. But in the end, it boils down to the same finite group theoretic identity as above.

12/01/2015

## Orbital integral identity for non regular semisimple orbits

Now consider the case of non regular semisimple orbits, i.e., when the invariant . Let us only consider the case . The case corresponds to three -orbits, the identity orbit and two unipotent orbits represented by and . The case corresponds to one -orbit: the identity orbit (i.e., ) under the decomposition .

Now let us consider the moduli spaces for the non regular semisimple orbits. The moduli space for the nonsplit torus is again simpler. Let be the space as in Definition 22 but only requiring that are not zero simultaneously, i.e., . Our old nondegenerate moduli space is thus an open . By definition we have where is the closed locus where . Since , we have Now consider the invariant map When is sufficiently large, is smooth and is proper. By the same logic for the regular semisimple orbits, it remains to consider the norm map and check if is still perverse. Its restriction on is given by the norm map , whose fiber is certainly not finite (the kernel is the Prym variety of dimension ). But when is sufficiently large, this map is still small. In fact, the smallness in this case means , i.e, . Now by the perverse continuation principle for small maps, still decomposes as IC sheaves associated to the earlier finite group representation .

Remark 44 Suppose is a proper dominant map between varieties and is smooth. Then is perverse if and only if is semismall. Moreover this is the IC sheaf associated to the local system over the locus where is etale (i.e., there is only one irreducible component in its perverse sheaf decomposition) if and only if is small. The reason is that the nontrivial cohomology of the constant sheaf is accounted for by the top dimension of the bad fibers. Notice this only applies to the constant sheaf. For example, consider the local system associated to an etale double covering. Then is 0 when and when by Euler characteristic formula.

Now consider the moduli space for the split torus. The situation is slightly more complicated. In this case , which has infinitely many components (when , since there is only the zero section for a line bundle of negative degree).

Remark 45 This infiniteness corresponds is in fact natural from the point of view of orbital integrals. There is an integral (already familiar from Tate's thesis) appearing in the unipotent orbital integral: Though is infinite, the Tate integral essentially a finite sum. This is due to the unusual phenomenon for function fields that the Fourier transform of a compactly supported function is still compactly supported. So by the Poisson summation (ignoring the contribution of and ), we have It follows from Riemann-Roch that when has large enough degree both sides are identically zero!

Notice the the identity orbit gives no contribution to the orbital integral since we are inserting the nontrivial quadratic character . So we require the four sections in has at most one zero, which corresponds to the two unipotent orbits and . We impose further assumptions that if ; if ; if and if .

By these further assumptions if is nonempty, then . Again is smooth when is sufficiently larger and is proper. For a point in . Assume (so ), the fiber at is then The second term is finite (since the addition map is finite). Therefore the fiber has dimension . From this one can see that is no longer small:

Even though is no longer small, we can check that the sheaf in question still satisfies the strict support condition in Deligne's uniqueness principle.

Lemma 3 Let . Then .
Proof Notice , where from the finite part . The claim follows from the fact that By Remark 44, the right hand side has only one possibly nonzero term , which becomes zero when . ¡õ

Hence the orbital integral identity for non regular semisimple orbits follows by same finite group representations identity (Theorem 23)!

12/03/2015

## Moduli spaces of shtukas

In the final part of the course, we are going to generalize the previous trace formula identity to higher derivatives. For this we need to introduce the moduli space of shtukas. We begin with a rather general construction.

Definition 23 Consider defined over a finite field (usually a certain moduli space). Suppose is a correspondence. We define the the moduli space of shutaks associated to to be the fiber product More generally, suppose there are correspondences , we define to be the moduli space of shtukas associated to the convolution correspondence
Definition 24 Consider . For , we define the Hecke stack to be the moduli space of arrows of vector bundles of rank such that . Similarly define using the condition that .
Definition 25 Define to be the Hecke stack of upper (increasing) modifications, i.e., (over ) is an injection such that is a line bundle on the graph of a marked point . Similarly define to be the Hecke stack of lower (decreasing) modifications.

We have two natural projections and also a natural map given by the location of modification.

Lemma 4 Both are representable and proper. When , both have relative dimension (Here 1 comes from the choice the location of modification and comes from the choice of the modification with at a fixed location, i.e., a line in an -dimensional vector space).
Definition 26 Let be an even integer. Let be a -tuple of signs. Let be the moduli space of shtukas associated to the convolution of . In other words, an -point of corresponds to a -tuple of modification of vector bundles such that .
Remark 46 Notice is empty unless has the same number of plus and minus signs, since the degree is preserved under Frobenius.

Recall that (Remark 28) itself is only an Artin stack (which has a lot of automorphism). The moduli of shtukas has better properties.

Theorem 24 (Drinfeld , Varshavsky in general)
1. is a Deligne-Mumford stack, locally of finite type.
2. The projection map is separated, smooth of relative dimension (in fact, an -iterated -bundle).
Remark 47 To show the smoothness of , one uses the fact that the relative cotangent bundle of (to the first factor) restricts to the absolute cotangent bundle of (using that the differential of the Frobenius is zero).
Example 17 When , the Hecke stack simply consists of isomorphisms . So consists of vector bundles on such that , which must come from pullback of vector bundles on itself. Hence is the discrete group . This exactly puts us in the earlier situation of Waldspurger's formula when . From this point of view, the study of automorphic forms (over function fields) is nothing but the study of degree 0 cohomology of the moduli of shtukas with marked points.
Example 18 When , we have given by the first line bundle and the location of modification . So we have the fiber diagram Here the right vertical arrow is given by . In particular, considering the degree zero part (and rotating the previous diagram) we obtain the fiber diagram The right vertical arrow is exactly Lang's isogeny, whose kernel is the class group of the function field . This is a generalization of unramified geometric class field theory: when , the etale map has Galois group the class group and realizes the Hilbert class field of geometrically.

Next time we will introduce the Hecke algebra action on the moduli of shtukas and see how the equality of higher derivatives of -functions and intersection numbers of certain cycles on the moduli of shtukas becomes a refined structure on the perverse sheaves we constructed using Hitchin moduli spaces.

12/08/2015

## Heegner-Drinfeld cycles and higher derivatives

Let , with an embedding . Fix a -tuple of signs . We have an induced morphism where is the etale double cover. This induces a map of moduli of shtukas and we have commutative diagram Notice the right vertical arrow has relative dimension , whereas left right vertical arrow has relative dimension 0 (generically etale with Galois group the class group). Though is not of finite type due to the instability, we can still talk about intersection number since is a proper smooth Deligne-Mumford stack (at least after dividing by ).

Now let us define Hecke correspondence on .

Definition 27 Let and be two points in . We define a degree Hecke correspondence to be the collection of injections such that and the natural diagram commutes. The stack of such degree Hecke correspondences on is denoted by .
Remark 48 Remembering the first column and the last column gives a map and remembering the two rows gives a map So the Hecke correspondence is a hybrid version of and .
Definition 28 We define a Hecke correspondence version of by taking the fiber product

Then is indeed a correspondence on and thus defines a compactly supported cycle class of dimension In particular, acts on . One can similarly define a more refined correspondence for any effective divisor . Recall the spherical Hecke algebra is generated by , where runs over all effective divisors.

Theorem 25 The map is a ring homomorphism.
Remark 49 This can be proved easily on the generic fiber using the method of Drinfeld (in his work on the global Langlands for ). Extra work need to be done on the integral level (i.e., over the entire base ).
Definition 29 The map induces a map (like Heegner points are imaginary quadratic points of modular curves). We define the Heegner-Drinfeld cycle to be the direct image of in under .
Theorem 26 Let . Then Here is the intersection number of the Heegner-Drinfeld cycles and
Remark 50 We have already proved the case (see Example 17).

The ride hand side essentially corresponds to . After spectral decomposition it follows that the intersection number of the -isotypic component of the Heegner-Drinfeld cycle (turns out to be independent of the choice of ) is essentially the -th derivatives at the center. More precisely, even though that we don't yet know the action of on the entire Chow group is automorphic, we can consider the subspace of the Chow group generated by the Heegner-Drinfeld cycle. Let be its quotient by the kernel of the intersection pairing.

Theorem 27 We have .

It then makes sense to talk about the -isotypic component and using the Theorem 26 one can show that

Theorem 28 For an everywhere unramified cuspidal automorphic representation of , we have up to a simpler factor

## Orbital integral identity for higher derivatives

Our remaining goal is to prove that for sufficiently large, we have Notice the intersection number in question is given by the degree of the 0-dimensional scheme (in the proper intersection case) of the fiber product

The key observation is that this fiber product can be viewed in an alternative way involving the Hitchin moduli space . Look at the following commutative diagram:

Here all the vertical upward arrows are given by ). The bottom row shows the fiber products of the three columns and the right column shows the fiber product of the three rows. The intersection in question is the fiber product of the bottom row, which should also equal to the fiber product the right column! (Of course this needs extra work to check after defining the intersection number in the right way, like the change of order of integration). We denote this common fiber product by , which is a Hitchin version of moduli of shutaks. One can further decompose into pieces, i.e., the convolution of (consisting of only 2 by 2 diagram).

We have the following general Lefschetz trace formula for computing the intersection of a correspondence with the graph of the Frobenius morphism.

Theorem 29 (Lefschetz trace formula)

Let be the invariant map to the Hitchin base. Notice a correspondence over defines an endomorphism . One can refine the Lefschetz trace formula relative to (take ):

Theorem 30
Remark 51 Notice is independent of since any upper modification can also be realized as a lower modification (this is special to our rank 2 situation).

This reduces the intersection number of Heegner-Drinfeld cycles to the study of the action of on the cohomology the Hitchin moduli spaces, which one can then compare to the -th derivative of the orbital integral on the split torus!

12/10/2015

Theorem 31 Therefore,
Remark 52 Since . The extra factor should be .

Recall that where is a perverse sheaf on with generic rank . Now the final key thing is that each such perverse sheaf is an Hecke eigensheaf whose eigenvalue exactly matching up the extra factor in .

Theorem 32 acts on by the constant .
Proof Notice the Hecke stack consists of commutative diagrams of line bundles on Here is an upper modification (at the same location ), the vertical arrows are morphisms of -modules and is the nontrivial automorphism of . Notice these two diagrams are uniquely determined by the data . Moreover, must satisfy that in order to complete the right vertical arrow (and the latter necessarily has divisor ). Define the incidence stack Here the top map is given by and the left map given by . It then follows that the is given by the base change the incidence stack along the projection map . It remains to check that acts on by . Because of the perversity, we only need to check this on the generic fiber.

Now let . Let be the preimage of in . A basis of the dimensional space consists of monomials of the form . Moreover, the elements form the basis of the representation corresponding to , where is the number of minus signs. Then in particular behaves like a derivation. Now notice keeps and negates . It then follows that acts on by the scalar ! ¡õ