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.
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 [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].
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.
[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.