These are notes for two talks in the Beyond Endoscopy Learning Seminar at Columbia, Spring 2018. Our main references are [1] and [2].

Recall that two key constructions are required in the Braverman-Kazhdan program for proving analytic continuation and functional equations for general Langlands -function . One is a suitable space of *Schwartz functions* at each local place, containing a distinguished function encoding the unramified local -factor (known as the *basic function* after Sakellaridis). The other is a generalized Fourier transform (known as the *Hankel transform* after Ngo) preserving the Schwartz space and the basic function. With a global Poisson summation formula, one should be able to establish the desired analytic properties of in a way analogous to Godement-Jacquet theory for standard -function on . Our goal today is to discuss the basic function and to explain its an algebro-geometric interpretation due to Bouthier-Ngo-Sakellaridis, using the -monoid appeared in previous talks and its arc space.

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Let be a non-archimedean local field. Let be a split reductive group over . Let be its dual group. Let be the spherical Hecke algebra. Recall that the classical Satake transform induces an algebra isomorphism onto the -invariants An unramified representation of corresponds to a 1-dimensional character of , given by its action on the spherical vector Langlands noticed that is the coordinate ring of the variety , so a 1-dimensional character corresponds to a point , i.e., a semisimple conjugacy class in . In this we obtain a bijection between unramified representations of and the Satake (or rather, Langlands) parameters. The Satake transform is then characterized by the identity Also notice that the target of the Satake isomorphism can be identified with the representation ring of , and thus with the -invariant regular functions on (via the trace map).

Remark 1
Notice the Satake isomorphism is of combinatorial nature: both the source and the target of depends only on the root datum of and the size of the residue field . In fact, the Satake isomorphism can be defined over .

The importance of the Satake parameter is due to its key role in defining the unramified local -factor . Let be an irreducible representation. Recall by definition Now if we have a diagonal matrix , then Therefore To remove the dependence on , we are motivated to introduce the following definition.

Definition 1
We define to be the inverse under the Satake transform of the function (so for any given . Define the *basic function* to be When , the sum is locally finite and makes sense as a function on .

Even though each is compactly supported (with support lies in the -double cosets indexed by dominant coweights of corresponding to weights of ), the support gets larger when increases and is not longer compactly supported. Moreover, the values of on each -double cosets can be written down in terms of representation theory (related to Kazhdan-Lusztig polynomials) and thus involve quite complicated combinatorial quantities.

Example 1
Take , and . Since , both the source and target of the Satake isomorphism are identified as functions on . The Satake transform sends the characteristic function to . So and the basic function is given by (always viewed as a function on ). This generalizes to the standard representation of , in which case (this is already a nontrivial computation) and hence .

Example 2
Take , and (i.e., ). Then has dimension given by , whose trace is . So the corresponding basic function is given by This is no longer the characteristic function of any set. More generally, take and (). Then So the value of encodes partition numbers, and can not have simple formula. We also see that the support of is contained in the cone generated by the weights of .

Example 3
Take and . Then computing amounts to decomposing into irreducibles, again this is a difficult combinatoric problem. In fact, we have Here the multiplicity , and is the number of partitions of into at most parts, having largest part at most .

I hope these examples illustrate that writing down an explicit formula for the basic function is quite hopeless in general (but see Wen-Wei Li's paper). Instead we would like to focus on finding some natural algebro-geometric object which encodes these combinatoric information. This is the main motivation to introduce the -monoid.

Let be a split reductive group over a field (later will be the residue field of the local field ). Assume has a nontrivial map to , denoted by . Assume is semisimple and simply-connected. Our first goal is to construct Vinberg's *universal monoid* . It is a normal affine variety fitting into a commutative diagram This monoid is universal in the sense that every reductive monoid with derived group equal to can be obtained by base change from (in fact the construction of will only depend on ).

Let be a maximal torus of . Let . Let be the semisimple rank of . Let be the set of fundamental weights of (dual to the coroots). Let be the fundamental representation of associated to . We extend from to by Here is the longest element in the Weyl group. We also extend the simple roots from to by These extensions together give a homomorphism

Example 4
Consider . Then , , . We have , , , , . So So In other words, this is a monoid in defined by the equation (which is smooth).

Now let be an irreducible representation. Let be a maximal torus in the adjoint group of . The highest weight of defines a cocharacter , hence a cocharacter of . We identify using a choice of simple roots. Then can be extended to a morphism of monoids

Definition 3
The *-monoid* is defined by base changing the universal monoid along . So we have a commutative diagram

Example 5
Again take , . Then . So we have Notice that the unit group is when is odd and when is even (the derived group is in both cases). Notice that this monoid is singular at the origin when , which reflects the fact that the basic function are more complicated than the case.

Remark 2
Assuming that is the identity map (e.g., in the previous example) ensures that the unit group of is and we obtain a commutative diagram The construction of can be characterized in terms of toric varieties: it is the unique reductive monoid with unit group such that the closure of any maximal torus in is the toric variety associated to and the cone generated by the weights of .

Directly comes from the construction of one sees that is exactly supported on the -double cosets associated dominant weights generated by the weights of . So the basic function can be viewed as a function on . Now take . Then we have the advantage of endowing an algebro-geometric structure over the residue field .

Definition 4
Let be an algebraic variety over a field . We define its *-th jet space * to be the functor sending a -algebra to the set . If is affine, then is also representable by an affine -scheme of finite type. In particular, consists of order- arcs in . When we exactly recover the tangent bundle of . For more general , contains information about the singularities of .

Example 7
Notice that if is defined by , then is defined by the equation with extra variables . Take . Then is given by One can find exactly has irreducible components, each isomorphic to given by the first of the -coordinates are 0 and first of the -coordinates equal to zero, where . The component with maps to the line , and the component with maps to the other line . All the rest components maps to the singularity (the origin).

Example 8
Take . Then has one irreducible component of dimension which dominates , and has one extra component of the same dimension mapping to the origin when .

If is smooth, then the natural map is smooth and surjective. In general, if is not smooth, then may fail to be surjective, and the transition maps can be rather complicated.

Definition 5
We define the (formal) *arc space* to be . In particular, , which consists of (formal) arcs of (here is the formal disc).

Again if is smooth then is formally smooth and surjective. A theorem of John Nash says that the inverse image of in has only finitely many irreducible components, each corresponds to a component in the inverse image of in any resolution of singularities of .

Definition 6
Let be a smooth open dense subvariety. We define to be the space of *non-degenerate arcs* in . Namely for a -algebra , consists of arcs such that inverse image is open in and surjects to . In particular, we have

If one has a -adic sheaf on , then taking the Frobenius trace gives us a function (if is a complex, then take alternating trace on the cohomology groups). Similarly, if we only have a sheaf on , we can still obtain a function on . When specializing to and , we can obtain a function on as desired. Our next goal is then to construct a canonical sheaf on , whose associated function gives the basic function .

If is a variety over , there is a canonical sheaf associated to , i.e., its IC sheaf which generalizes the constant sheaf and encodes the singularities of .

Definition 7
Let be a smooth open dense subvariety. We define to be the middle extension of the constant sheaf on (so a complex of sheaves in the derived category of ). It is independent of the choice of and measures the singularities of along the boundary. The shift serves as the dualizing sheaf for the Poincare(-Verdier) duality for singular varieties, and is a basic example of a perverse sheaf.

However, because the arc space is infinite type over , there is no good theory of IC sheaves/perverse sheaves on . Fortunately, the singularities of have a finite dimensional model.

Definition 8
A *finite dimensional formal model* of at is a formal scheme (the subscript means taking formal completion), where is a finite type -scheme and a point such that

Theorem 1 (Drinfeld (2002), generalizing Grinberg—Kazhdan (2000) for ) Finite dimensional model exists at each point .

Bouthier-Ngo-Sakellaridis [1] show that the stalk of the IC sheaf of does not depend on the choice of the finite dimensional formal model . It now makes sense to define the IC function on the non-degenerate arcs by It is a numerical invariant encoding the singularities of . By taking and , we obtain

Now we can state the main theorem of [1].

Example 9
When and , we have . So we recover The shift matches with the Godement-Jacquet zeta integral as well.

To prove the main theorem, we need a concrete construction of the finite dimensional model of at non-degenerate arcs. To do so we make use of a global smooth projective curve . From now on, let (with left and right -actions).

Definition 9
Recall that an -point of the quotient stack consists of a principal -bundle over together with a -equivariant map . Consider the stack , whose -points consists of maps , namely a principal -bundle over together a -equivariant homomorphism . We now add the non-degeneracy and define to be the open substack of such that factors through for a open subset . Then one can show that is an algebraic space locally of finite type.

Definition 10
To relate to , we fix a -point . Define to be the stack classifying a point together with , a trivialization of on the formal disc . Then we have a canonical projection which is a torsor under , hence is formally smooth. On the other hand, given a point , we obtain an arc by the composite map Moreover, this arc is non-degenerate (by the non-degenerate requirement when defining ). Thus we obtain a morphism

The following essentially says that there is no obstruction for deforming -bundles while fixing the induced formal arc.

Proposition 1
Let . Let be a point such that lies in the smooth locus of , and such that its image in is (such always exists by Beauville-Laszlo patching the trivial -bundle). Then is formally smooth.

It follows that and hence can serve as a finite dimensional formal model of at . In particular, we obtain

Let . From the fixed map one naturally associates to a line bundle on . Using the trivialization of induced from , we also obtain a generic section of this line bundle, hence a divisor on .

Let and be the substack whose associated divisor is . By the Beauville-Laszlo patching, the data of a -bundle and a trivialization away from is the same as giving -bundles on the formal disc together with a trivialization on the punctured formal disc . Then we obtain a map into the affine Grassmannians (whose -points are ) at 's, Moreover, a trivialization of actually comes from a -equivariant map if and only if has invariant for each . Thus we obtain an isomorphism Notice each term on the right is indeed a projective variety (a Schubert variety), which models singularity of when . Varying , we obtain an isomorphism Using this isomorphism and a fixed , we can choose the point explicitly corresponding to a point such that is the -component of .

Now recall the geometric Satake correspondence.

Theorem 3 (Mirkovic-Vilonen (2007))
Let be the IC sheaf of the Schubert variety shifted by its dimension . Then the map gives an equivalence of tensor categories between the finite dimensional representations of and -equivariant perverse sheaves on (the tensor structure given the convolution product).

Bouthier-Ngo-Sakellaridis show that (The symmetric power essentially comes from looking at the map ). Hence by the geometric Satake we have The main theorem now follows by taking the -component.

[1]On the formal arc space of a reductive monoid, Amer. J. Math. 138 (2016), no.1, 81--108.

[2]Hankel transform, Langlands functoriality and functional equation of automorphic L-functions, http://math.uchicago.edu/~ngo/takagi.pdf.