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.
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).
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.
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.
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
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
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 .
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.
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 .
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 .
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.
Bouthier-Ngo-Sakellaridis  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 .
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).
The following essentially says that there is no obstruction for deforming -bundles while fixing the induced formal arc.
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.
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.
On the formal arc space of a reductive monoid, Amer. J. Math. 138 (2016), no.1, 81--108.
Hankel transform, Langlands functoriality and functional equation of automorphic L-functions, http://math.uchicago.edu/~ngo/takagi.pdf.