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!
(Updated: 03/22/2017: thank Tony Feng for helpful 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
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.
To measure how "perverse" these intersections are, we introduce the
perversity, which is a function
![$p: \mathbb{Z}_{\ge2}\rightarrow \mathbb{Z}_{\ge0}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_38370121_-4.gif)
such that
,
.
Given a perversity
![$p$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42628114_-3.gif)
, we define a
![$i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42169362_0.gif)
-chain
![$\sigma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_208953823_0.gif)
to be
-allowable if
![$$\dim (\sigma\cap X_{n-j})\le i-j+p(j).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_61571607_.gif)
We then define the
intersection homology groups
![$I_pH_i(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_3846389_-4.gif)
by considering only the
![$p$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42628114_-3.gif)
-allowable locally finite
![$i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42169362_0.gif)
-chains. When
![$p$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42628114_-3.gif)
is the middle perversity,
![$I_pH_i(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_3846389_-4.gif)
is often written as
![$IH_i(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_38927883_-4.gif)
for short. Similarly one can define the
intersection homology with compact support ![$I_pH_i^c(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_216553350_-4.gif)
by considering only the
![$p$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42628114_-3.gif)
-allowable finite
![$i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42169362_0.gif)
-chains.
09/17/2015
- All intersection homology groups are finitely generated, independent of the choice the stratification. It gives the usual homology groups for manifolds without singularities.
if
is connected.
- (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
.
Let
be the category of sheaves of
-vector spaces on
. Two important examples:
- The constant sheaf on
, denoted by
.
- 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.
Denote
![$I_pH^i(X)={}I_pH_{-i}(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_176324480_-4.gif)
. 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
(Deligne, Goresky-MacPherson)
Let
![$U_i=X-X_{n-i}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_110273780_-2.gif)
. Let
![$j_k: U_k\hookrightarrow U_{k+1}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_180624601_-3.gif)
(
![$k\ge2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_50624155_-3.gif)
). Let
![$p$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42628114_-3.gif)
be a perversity. Then in the bounded derived category
![$D^b(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_229350579_-4.gif)
, we have
![$$\mathcal{I}_p\mathcal{C}^\cdot \cong \cdots\tau_{\le p(3)-n}(R j_3)_*\tau_{\le p(2)-n}(Rj_2)_* \underline{\mathbb{Q}}_{U_2}[n].$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_109724454_.gif)
Here
![$\tau_{\le k}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_74104132_-4.gif)
is the usual truncation functor:
![$$\tau_{\le k} \mathcal{E}^\cdot=
\begin{cases}
\mathcal{E}^i & i\le k,\\
\ker(\mathcal{E}^k\rightarrow \mathcal{E}^{k+1}) & i=k,\\
0 & i>k.
\end{cases}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_4056317_.gif)
so that
![$\mathcal{H}^i(\mathcal{E}^\cdot)=\mathcal{H}^i(\tau_{\le k}(\mathcal{E}^\cdot))$](./latex/FundamentalLemma/latex2png-FundamentalLemma_123676017_-4.gif)
for
![$i\le k$](./latex/FundamentalLemma/latex2png-FundamentalLemma_12215166_-3.gif)
. Inductively, let
![$\mathbb{P}_2= \underline{\mathbb{Q}}_{U_2}[n]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_94362267_-9.gif)
and
![$$\mathbb{P}_{k+1}=\tau_{\le p(k)-n}(Rj_k)_* \mathbb{P}_k.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_32311577_.gif)
Then
09/24/2015
References:
- An introduction to perverse sheaves, Rietsch (brief)
- Intersection homology, edited by Borel (detailed)
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:
is constructible with respect to the given stratification.
in
.
- The stalk
if
and
.
- 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
.
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
.
Let
![$\mathcal{L}_{U_2}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_45096286_-4.gif)
be a local system on
![$U_2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_81497068_-2.gif)
and
![$\mathcal{L}^\vee_{U_2}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_113806053_-6.gif)
be the dual local system. Let
![$p$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42628114_-3.gif)
and
![$q$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42693650_-3.gif)
be complementary perversity (i.e.,
![$p+q$](./latex/FundamentalLemma/latex2png-FundamentalLemma_184251412_-3.gif)
is the top perversity). Then
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
.
Given a local system
![$\mathcal{L}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_123373566_0.gif)
on an open
![$U$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43480082_0.gif)
of real codimension at least 2, there exists a unique
![$\mathbb{P}\in D_c^b(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_210880121_-4.gif)
such that
![$\mathbb{P}|_U=\mathcal{L}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_181121100_-4.gif)
satisfying the axioms a, b, c2, c2' for any stratification
![$\sigma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_208953823_0.gif)
.
10/06/2015
The uniqueness follows easily from the following lemma.
Suppose
![$U\hookrightarrow X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_264258120_0.gif)
has codimension at least 2. If
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
is connected, then
![$U$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43480082_0.gif)
is connected. Moreover, the natural map
![$\pi_1(U,x)\rightarrow\pi_1(X,x)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_12085258_-4.gif)
is surjective. In particular, a local system on
![$U$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43480082_0.gif)
has
at most one extension to
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
.
The connectedness of
![$U$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43480082_0.gif)
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
).
A sheaf
![$\mathcal{E}\in D^b_c(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_119557709_-4.gif)
is
perverse if
![$$\dim_\mathbb{C}\supp\mathcal{H}^{-i}(\mathcal{E})\le i$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_253911619_.gif)
and
![$$\dim_\mathbb{C}\cosupp\mathcal{H}^{-i}(\mathcal{E})\le i.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_181421401_.gif)
The category of perverse sheaves is denoted by
![$\Perv(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_232097841_-4.gif)
.
is an abelian category (the heart of perverse
-structure on
defined by the support and cosupport condition), stable under Verdier dual.
- All simple objects are DGM complexes.
- Every object in
is a successive extension of simple objects.
A proper surjective morphism between two varieties
![$f: X\rightarrow Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_152188897_-3.gif)
is
small (resp.
semi-small) if
![$\codim\{y\in Y: \dim f^{-1}(y)\ge r\}>2r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_212721648_-4.gif)
(resp.
![$\ge2r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_261327182_-3.gif)
) for any
![$r\ge1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_50558626_-3.gif)
. In particular,
![$f$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41972754_-3.gif)
is a generically finite morphism. For example, when
![$\dim Y\le2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_234566799_-3.gif)
, small is equivalent to finite. When
![$\dim Y=3$](./latex/FundamentalLemma/latex2png-FundamentalLemma_47221655_0.gif)
, 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
![$\dim X\times_Y X=\dim X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_32758953_-2.gif)
and small if and only
![$\dim (X\times_Y X-\Delta_X)<\dim X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_144820370_-4.gif)
.
Small morphisms are compatible with IC sheaves.
If
![$f:X\rightarrow Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_112759265_-3.gif)
is small and has generic degree 1, then
![$f_! \mathcal{IC}_X=\mathcal{IC}_Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_188280432_-3.gif)
. In particular,
![$IH^i(Y)=H^i(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_25652946_-4.gif)
.
Consider the case
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
is smooth. Then
![$f_! \mathbb{Q}[n]|_U=\mathbb{Q}[n]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_40684349_-5.gif)
. So by Theorem
5 it suffices to check the support and cosupport axioms for
![$\mathcal{E}=f_! \mathbb{Q}[n]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_186951887_-5.gif)
. By base change,
![$\mathcal{H}^{-i}(\mathcal{E})_y= H^{-i}(X_y, \mathbb{Q}[n])=H^{-i+n}(X_y, \mathbb{Q})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_243337085_-5.gif)
. If
![$\mathcal{H}^{-i}(\mathcal{E})_y\ne0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_101773922_-4.gif)
, then
![$-i+n\le 2\dim X_y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_1264599_-4.gif)
, i.e.,
![$i\ge n-2\dim X_y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_125035672_-4.gif)
. 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:
Suppose
![$f: X\rightarrow Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_152188897_-3.gif)
is small.
![$\mathbb{P}(\mathcal{L})\in \Perv(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_132704654_-4.gif)
. Then
![$Rf_*(\mathbb{P}(\mathcal{L}))=\mathbb{P}(f_*\mathcal{L})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_87340639_-4.gif)
.
Suppose
![$f: X\rightarrow Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_152188897_-3.gif)
is a blow-up along a point
![$z\in Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_128213614_-1.gif)
with fiber
![$\mathbb{P}^{n-1}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_243914793_0.gif)
. Then
![$Rf_* \mathbb{Q}_X= \mathbb{Q}_Y\bigoplus \oplus_{i=0}^{n-1} \mathbb{Q}_z[-i]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_129966404_-5.gif)
. This is no longer perverse and the extra terms
![$\mathbb{Q}_z[-i]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_7215046_-5.gif)
measures the failure of
![$f$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41972754_-3.gif)
being small.
In general we have the following decomposition theorem.
(Beilinson-Bernstein-Deligne + Gaber)
If
![$f: X\rightarrow Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_152188897_-3.gif)
is proper and surjective, then in
![$D_c^b(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_232003363_-4.gif)
we have
![$Rf_*(\mathbb{Q}_X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_202600569_-4.gif)
is a direct sum of DGM complexes shifted by certain degrees:
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.
A
holomorphic symplectic variety (over
![$\mathbb{C})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_116819980_-4.gif)
) is a nonsingular variety such that there exists a closed holomorphic 2-form
![$\omega\in \Gamma(X,\Omega^{\otimes 2})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_92117903_-4.gif)
satisfying that
![$\omega^{\dim_\mathbb{C} X}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_115818506_0.gif)
is nonzero everywhere (i.e.,
![$\omega$](./latex/FundamentalLemma/latex2png-FundamentalLemma_248799715_0.gif)
is non-degenerate).
Let
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
be a nonsingular variety over
![$\mathbb{C}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_26235984_0.gif)
with
![$\dim_\mathbb{C}X=n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_68988635_-2.gif)
. Then the cotangent bundle
![$T^*X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_16413755_0.gif)
is naturally a holomorphic symplectic manifold. In fact, let
![$x_i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_164400108_-2.gif)
be the local coordinates on
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
and
![$y_i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_147622892_-3.gif)
be the dual basis of
![$dx_i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_164400070_-2.gif)
in the cotangent space, then
![$\omega=\sum_{i=1}^nd x_i\wedge dy_i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_73578127_-5.gif)
is a non-degenerate closed 2-form.
(Symplectic resolution)
Let
![$f: X\rightarrow Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_152188897_-3.gif)
be a proper surjective birational map. If
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
is holomorphic symplectic, then
![$f$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41972754_-3.gif)
is semi-small.
(Hilbert scheme of surfaces)
Let
![$S$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43349010_0.gif)
be a surface. Then symmetric power
![$S^{(n)}=S^n/S_n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_232507791_-4.gif)
becomes singular when some of the
![$n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42497042_0.gif)
-points on
![$S$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43349010_0.gif)
collide. However, the Hilbert scheme
![$S^{[n]}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_64352950_0.gif)
of
![$n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42497042_0.gif)
-points on
![$S$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43349010_0.gif)
(which remembers infinitesimal information of the collision as well) is nonsingular. Moreover, one can show that
![$S^{[n]}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_64352950_0.gif)
is a symplectic resolution of
![$S^{(n)}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_149556551_0.gif)
Notice when
![$S$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43349010_0.gif)
is a 3-fold, both the
![$S^{(n)}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_149556551_0.gif)
an
![$S^{[n]}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_64352950_0.gif)
are singular.
Let
be a semisimple connected linear group over
. Fix a Borel subgroup
. Let
be the nilpotent cone. Then
is singular.
Let
![$G=SL(V)=SL(n)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_193395771_-4.gif)
. Then
![$\mathfrak{g}=\mathfrak{sl}(V)=\{\phi\in \End(V): \tr(\phi)=0\}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_150321394_-4.gif)
. Define the invariant map given by the coefficients of the characteristic polynomial of
![$\phi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_205747260_-3.gif)
:
![$\mathfrak{g}\rightarrow \mathbb{A}^{n-1}, \phi\mapsto \tr(\wedge^i\phi)_{i=2}^n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_17200089_-4.gif)
. Then
![$\mathcal{N}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_121276414_-1.gif)
is the preimage of
![$0\in \mathbb{A}^{n-1}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_203101537_-1.gif)
, i.e., the set of all traceless matrices
![$\phi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_205747260_-3.gif)
such that
![$\phi^n=0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_170688601_-3.gif)
. When
![$n=2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_168064020_0.gif)
,
![$\mathcal{N}=\{\left(\begin{smallmatrix}x & y\\ z & -x\end{smallmatrix}\right): x^2-yz=0\}\subseteq \mathbb{A}^3$](./latex/FundamentalLemma/latex2png-FundamentalLemma_177724804_-4.gif)
, 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.
Let
![$\mathbb{B}=G/B$](./latex/FundamentalLemma/latex2png-FundamentalLemma_93153901_-4.gif)
be the flag variety, parametrizing all the Borel subgroups of
![$G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42562578_0.gif)
. Define
![$$\tilde{\mathcal{N}}=\{(\phi,b): \phi\in\rad(\Lie b)\}\subseteq \mathcal{N}\times \mathbb{B}.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108619753_.gif)
Then the natural map
![$\tilde{\mathcal{N}}\rightarrow\mathcal{N}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_21161913_-1.gif)
is an isomorphism on the regular nilpotent elements. It turns out that the natural map
![$\tilde{\mathcal{N}}\rightarrow\mathcal{N}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_21161913_-1.gif)
is a symplectic resolution, known as the
Springer resolution. In fact,
![$\tilde{\mathcal{N}}\cong T^*\mathbb{B}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_211007357_-1.gif)
. The fiber
![$\tilde{\mathcal{N}}_\phi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_57258718_-4.gif)
is called a
Springer fiber.
For
![$G=SL(2)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_189660985_-4.gif)
,
![$\tilde{\mathcal{N}}_{\left(\begin{smallmatrix}0 & 1\\0 &0\end{smallmatrix}\right) }=\mathrm{pt}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_88719404_-12.gif)
and
![$\tilde{\mathcal{N}}_0=\mathbb{P}^1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_154353603_-2.gif)
. The simple singularity of the 2-dimensional cone at 0 is resolved.
Using the fact that
, one can check that
is indeed semi-small. One can generalize the construction of
and obtain a small map
.
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 ![$$\tilde{\mathfrak{g}}_\phi=\{b \in \mathbb{B}: \phi\in b\}.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_179358544_.gif)
Let
![$G=SL(V)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_227409721_-4.gif)
. Let
![$0\subseteq V_1\subseteq\cdots\subseteq V_n=V$](./latex/FundamentalLemma/latex2png-FundamentalLemma_89700534_-3.gif)
be the complete flag associated to
![$b$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41710610_0.gif)
. Then for the regular element
![$\phi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_205747260_-3.gif)
, the Springer fiber satisfies
![$\phi V_i\subseteq V_{i-1}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_110400894_-3.gif)
while the Grothendieck-Springer fiber satisfies the weaker condition
![$\phi V_i\subseteq V_i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_22567487_-3.gif)
. For example, when
![$V$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43545618_0.gif)
is 2-dimensional,
![$\mathbb{B}\cong \mathbb{P}^1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_131312193_0.gif)
. Let
![$R= k[t]/t^2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_133626828_-5.gif)
, then
![$\tilde{\mathfrak{g}}_\phi(R)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_262998679_-4.gif)
has more than one points: it has an obvious point corresponding to
![$0\subseteq W=R \oplus 0\subseteq R \oplus R$](./latex/FundamentalLemma/latex2png-FundamentalLemma_104353869_-3.gif)
. It also has an extra
![$0\subseteq R(1,t) \subseteq R \oplus R$](./latex/FundamentalLemma/latex2png-FundamentalLemma_37813476_-4.gif)
because
![$t (1,t)=(t,0)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_93559907_-4.gif)
. We can compute the scheme-theoretical fixed points of
![$\phi=\left(\begin{smallmatrix}1 & 1 \\ 0 & 1\end{smallmatrix}\right) $](./latex/FundamentalLemma/latex2png-FundamentalLemma_83636806_-5.gif)
on
![$\mathbb{B}=\mathbb{P}^1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_131451435_0.gif)
: since
![$\phi(x,y)=(x+y,y)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_182897281_-4.gif)
, we have
![$xy=(x+y)y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108577765_-4.gif)
, hence
![$y^2=0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_167610711_-3.gif)
. Therefore
![$\tilde{\mathfrak{g}}_\phi=\Spec k[y]/y^2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_155216545_-5.gif)
.
Affine Springer fibers
Let
be a finite field. One can easily see that the number of
-points in the Springer fiber is the same as fixed points of
on the set
, which can be rewritten as a simple orbital integral. Moreover, the finite set
can be realized as the
-rational points
of the flag variety
. Now we want to upgrade this to an "affine" version, i.e., for local fields of equal characteristic.
Let
and
. We want to geometrize the infinite set
by an affine Grassmannian
. The analogue of the full flag variety
should be given by the affine flag variety whose
points gives
, where
is the Iwahori subgroup.
In terms of moduli interpretation, when
,
is the set of
-lattices in
, where the identity coset corresponds to the standard lattice
.
is the set of chain of lattices in
,
where
is length one
-module such that
.
Let
![$R$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43283474_0.gif)
be a
![$k$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42300434_0.gif)
-algebra. Let
![$R[ [t] ]=\varprojlim R \otimes \mathcal{O}_F/(t^n)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_230957158_-9.gif)
be the ring of power series in
![$R$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43283474_0.gif)
. Let
![$R((t))=R[ [t] ][1/t]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_82847633_-5.gif)
be the field of Laurent sereis. Due to the completion process,
![$R((t))$](./latex/FundamentalLemma/latex2png-FundamentalLemma_247176540_-4.gif)
is larger than the naive base change
![$R \otimes_k F^n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_209988008_-2.gif)
when
![$R$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43283474_0.gif)
is not finitely generated.
We define an
-family of lattices in
![$R((t))^n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_11419936_-4.gif)
is a finitely generated projective
![$R[ [t] ]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_7802510_-5.gif)
-submodule
![$\Lambda\subseteq R((t))^n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42912608_-4.gif)
such that
![$\Lambda \otimes_{\mathcal{O}_F}F=R((t))^n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_152106405_-4.gif)
. This is equivalent to the data
![$(\mathcal{E},\beta)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_121558609_-4.gif)
, where
![$\mathcal{E}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_130713598_0.gif)
is a vector bundle over
![$D_R$](./latex/FundamentalLemma/latex2png-FundamentalLemma_172258324_-2.gif)
of rank
![$n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42497042_0.gif)
and
![$\beta: \mathcal{E}|_{D_R^*}\cong \mathcal{E}_0|_{D_R^*}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_158249055_-6.gif)
is a trivialization of
![$\mathcal{E}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_130713598_0.gif)
over the punctured disk.
We define the functor
. Then
.
has a reasonable geometric structure (though infinite dimensional).
![$\Gr$](./latex/FundamentalLemma/latex2png-FundamentalLemma_12350484_-1.gif)
is represented by an ind-scheme, i.e.,
![$\Gr=\bigcup_{N\ge0} \Gr^{(N)}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_79715401_-7.gif)
, where each
![$\Gr^{(N)}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_209177529_-1.gif)
is a projective scheme of finite type and each
![$\Gr^{(N)}\hookrightarrow \Gr^{(N+1)}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_168514110_-1.gif)
is a closed immersion.
Here
consists of
-lattices
(projective as
-modules) such that
Due to this boundness, it can be viewed as the set of quotient
-modules of
(projective as
-modules). In other words, let
(so
). Then
is the set of quotient projective
-modules
of
, such that the
-action (given by a nilpotent operator
) satisfies
. Observe that
is nothing but a union of generalized version of Springer fibers
:
Here
is a Grassmannian of one-step flags (instead of full flags).
For
![$n=1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_167998484_-1.gif)
, we see
![$V_N= k^{2N}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_58288296_-2.gif)
. Let
![$\phi_N$](./latex/FundamentalLemma/latex2png-FundamentalLemma_121675294_-3.gif)
be a regular nilpotent operator on
![$V_N$](./latex/FundamentalLemma/latex2png-FundamentalLemma_62884844_-2.gif)
. When
![$N=1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_234654700_-1.gif)
, we have
![$$\Gr^{(N)}=\mathrm{pt}\coprod \Spec k[y]/y^2\coprod \mathrm{pt}.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_98721468_.gif)
In general
![$\Gr^{(N)}(k)=2N+1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_74185240_-4.gif)
(compare:
![$\Gr(k)= F^\times/ \mathcal{O}_F^\times\cong \mathbb{Z}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_227830823_-5.gif)
).
10/29/2015
More generally,
(Affine Grassmannians)
Suppose
![$k$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42300434_0.gif)
is algebraically closed. Let
![$G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42562578_0.gif)
be a group scheme over
![$\mathcal{O}_F$](./latex/FundamentalLemma/latex2png-FundamentalLemma_113482678_-2.gif)
. We define the functor
![$\Gr_G: \mathbf{Alg}_k\rightarrow \mathbf{Sets}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_135986765_-4.gif)
such that
![$\Gr_G(R)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42631147_-4.gif)
is the set of pairs
![$(\mathcal{E},\beta)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_121558609_-4.gif)
, where
![$\mathcal{E}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_130713598_0.gif)
is a
![$G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42562578_0.gif)
-torsor over
![$D_R$](./latex/FundamentalLemma/latex2png-FundamentalLemma_172258324_-2.gif)
and
![$\beta$](./latex/FundamentalLemma/latex2png-FundamentalLemma_33257137_-3.gif)
is a trivialization
![$\mathcal{E}|_{D_R^*}\cong \mathcal{E}_0\times_F D_R^*$](./latex/FundamentalLemma/latex2png-FundamentalLemma_29310075_-6.gif)
, where
![$\mathcal{E}_0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_114924470_-2.gif)
is the trivial
![$G_F$](./latex/FundamentalLemma/latex2png-FundamentalLemma_221803540_-2.gif)
-torsor. In other words,
![$\Gr_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_3765938_-2.gif)
is the moduli space of
![$G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42562578_0.gif)
-torsor together with a rigidification. This may remind you of the definition of a Rapoport-Zink space, which is the moduli space of certain
![$p$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42628114_-3.gif)
-divisible groups together with a rigidification.
Notice
. So
acts on
by
.
(Affine Springer fibers)
Let
![$\gamma\in G(F)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_166026343_-4.gif)
, we define
![$\tilde X_\gamma:=\Gr_{G}^\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_141717207_-5.gif)
to be the fixed points of
![$\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108290540_-3.gif)
and
![$X_\gamma:=\tilde X_\gamma^\mathrm{red}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_223810004_-6.gif)
. It turns out that
![$X_\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_105555692_-4.gif)
is a closed subscheme of
![$\Gr_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_3765938_-2.gif)
.
We provide two more analogous constructions.
(Affine Schubert varieties)
By the Bruhat decomposition (over
![$k$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42300434_0.gif)
)
![$$G=\coprod_{w\in W} BwB.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_102755619_.gif)
The orbits of
![$B$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42234898_0.gif)
on
![$G/B$](./latex/FundamentalLemma/latex2png-FundamentalLemma_171209748_-4.gif)
are parametrized by elements in the Weyl group
![$W$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43611154_0.gif)
. Let
![$$\inv: G/B\times G/B\rightarrow W,\quad ([g], [h])\mapsto B(g^{-1}h)B$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_47817986_.gif)
be the invariant map. Define the orbit associated to
![$w$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43086866_0.gif)
to be the
Schubert variety, so
![$$X_w=\inv^{-1}(w)\cap (\{1\}\times G/B).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_82563962_.gif)
These are locally closed subvariety of
![$G/B$](./latex/FundamentalLemma/latex2png-FundamentalLemma_171209748_-4.gif)
(defined by incidence relations) . Then
![$X_w$](./latex/FundamentalLemma/latex2png-FundamentalLemma_29264876_-2.gif)
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 ![$$X_w(R)=\{(\mathcal{E}, \beta): \forall\text{geometric points } x\in \Spec R, \inv_x(\mathcal{E},\beta)=w\}.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_24801478_.gif)
(Affine Deligne-Lusztig varieties)
Assume
![$k$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42300434_0.gif)
is finite. Define the
Deligne-Lusztig variety ![$X_w^\mathrm{DL}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_106424511_-4.gif)
to be the subvariety of
![$G/B$](./latex/FundamentalLemma/latex2png-FundamentalLemma_171209748_-4.gif)
given by
![$$X_{w}^\mathrm{DL}=\{ b\in G/B: \inv(b,\Frob(b))=w\}.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_29362039_.gif)
In other words, the Deligne-Lusztig variety sits in the Cartesian diagram
![$$\xymatrix{ X_w^\mathrm{DL} \ar[r] \ar[d] & \inv^{-1}(w) \ar[d]\\ G/B \ar[r]^-{(\Id, \Frob)} & G/B\times G/B.}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_127936342_.gif)
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.
When
![$X=\mathbb{G}_m$](./latex/FundamentalLemma/latex2png-FundamentalLemma_199762996_-2.gif)
, we have
![$\mathrm{L}^+X(R)=(R[ [t] ])^\times$](./latex/FundamentalLemma/latex2png-FundamentalLemma_140585346_-5.gif)
. Therefore
![$\mathrm{L}^+X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_15769292_0.gif)
is in fact represented by a scheme
![$\mathbb{G}_m\times \mathbb{A}^\infty$](./latex/FundamentalLemma/latex2png-FundamentalLemma_200932202_-2.gif)
, given by the leading coefficient
![$a_0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_14519276_-2.gif)
and rest of the coefficients
![$(a_i)_{i\ge1}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_61259246_-4.gif)
. The points
![$\mathrm{L}X(R)=R((t))^\times$](./latex/FundamentalLemma/latex2png-FundamentalLemma_160891929_-4.gif)
are more complicated: they are Laurent series of the form
![$$\sum_{i\gg-\infty} a_it^i: a_{i_0}\in R^\times, a_i \text{ nilpotent}, i< i_0.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_231364878_.gif)
When taking the reduced structure, we find that
![$\mathrm{L}X^\mathrm{red}=\mathrm{L}^+X\times \mathbb{Z}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_62700588_0.gif)
is an infinite copies of
![$\mathrm{L}^+X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_15769292_0.gif)
.
More generally, we have
is represented by a scheme. It is affine if
is affine.
is represented by an ind-scheme.
- The affine Grassmannian
(as the quotient sheave under the fpqc topology).
Orbital integrals
Suppose
![$F$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42497042_0.gif)
is a local field. For
![$\phi\in C_c^\infty( \mathfrak{g}(F))$](./latex/FundamentalLemma/latex2png-FundamentalLemma_23457909_-4.gif)
and
![$\gamma\in \mathfrak{g}(F)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_206447955_-4.gif)
. Define the orbital integral
![$$\Orb(\gamma, \phi)=\int_{G_\gamma(F)\backslash G(F)}\phi(g^{-1}\gamma g)dg.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_15267669_.gif)
Notice that the convergence of this orbital integral is already an issue.
Let us consider the case
![$G=SL_n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_37264580_-2.gif)
,
![$\mathfrak{g}=\mathfrak{sl}_n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_261079079_-4.gif)
, and
![$\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108290540_-3.gif)
is regular semisimple. In this case the centralizer
![$G_\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_105625324_-4.gif)
of
![$\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108290540_-3.gif)
is the diagonal torus
![$A\subseteq G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_206201690_-3.gif)
. Since
![$\phi(g^{-1}\gamma g)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_118449930_-4.gif)
is
![$K=GL_n(\mathcal{O}_F)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_267247150_-4.gif)
-bi-invariant and
![$K$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42824722_0.gif)
is compact, to show the convergence of the orbital integral, it suffices to show the convergence of
![$$\int_{A(F)\backslash GL_n(F)/K}\phi(g^{-1}\gamma g)dg.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_13314795_.gif)
By the Iwasawa decomposition
![$$G(F)=A(F) N(F)K,$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_16485804_.gif)
where
![$N$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43021330_0.gif)
is the unipotent radical of the Borel subgroup of
![$G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42562578_0.gif)
(i.e., group of upper triangular unipotent matrices for
![$G=GL_n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_37264579_-2.gif)
), we know that the convergence is equivalent to
![$$\int_{N(F)} \phi(n^{-1}\gamma n) dn.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_142054254_.gif)
The key observation is that
![$$n^{-1}\gamma n=n^{-1}(\gamma n\gamma^{-1})\gamma$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_235485941_.gif)
and one easily compute
![$\gamma n\gamma^{-1}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_12756412_-3.gif)
since each root group is an eigenvector under the adjoint action of the semisimple element
![$\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108290540_-3.gif)
. For example, when
![$G=SL_2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_35953860_-2.gif)
and
![$\gamma=\diag(\alpha_1,\alpha_2)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_222198159_-4.gif)
, we have
![$$n^{-1}(\gamma n\gamma^{-1})\gamma = \left(\begin{smallmatrix}1 & (\frac{\alpha_1}{\alpha_2}-1)n \\ & 1\end{smallmatrix}\right) \left(\begin{smallmatrix}\alpha_1 &\\ & \alpha_2\end{smallmatrix}\right).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_229313944_.gif)
Take
![$\phi=\mathbf{1}_{\mathfrak{g}(\mathcal{O}_F)}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_260482706_-6.gif)
, we find the integrand is nonzero only when the above matrix has
![$\mathcal{O}_F$](./latex/FundamentalLemma/latex2png-FundamentalLemma_113482678_-2.gif)
-entries. This means that
![$n\in (\alpha_1-\alpha_2)^{-1}\mathcal{O}_F$](./latex/FundamentalLemma/latex2png-FundamentalLemma_23608417_-4.gif)
, hence
![$n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42497042_0.gif)
is bounded and the integral converges.
Ngo showed that the action of
on
is not faithful. The action factors through the quotient
known as the local Picard group.
For
![$G=SL_2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_35953860_-2.gif)
,
![$\gamma=\diag(t, -t)\in \mathfrak{g}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43086393_-4.gif)
. Then
![$G_\gamma=\mathbb{G}_m \otimes_k F$](./latex/FundamentalLemma/latex2png-FundamentalLemma_258011512_-4.gif)
and
![$\mathcal{P}_\gamma=\mathbb{G}_m\times \mathbb{Z}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_225290972_-4.gif)
(i.e., the
![$\mathbb{A}^\infty$](./latex/FundamentalLemma/latex2png-FundamentalLemma_86685211_0.gif)
-part acts trivially on
![$X_\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_105555692_-4.gif)
).
(Goresky-Kottwitz-MacPherson, Ngo)
Assume
![$\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108290540_-3.gif)
is regular semisimple. Then
![$$[\mathcal{P}_\gamma\backslash X_\gamma](k)=({*})\cdot \mathrm{SO}_\gamma(\mathbf{1}_{\mathfrak{g}(\mathcal{O}_F)}).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_217130172_.gif)
Namely, the
![$k$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42300434_0.gif)
-points of the stacky quotient is in fact a stable orbital integral (i.e., a sum of orbital integrals over conjugacy classes
![$\gamma'$](./latex/FundamentalLemma/latex2png-FundamentalLemma_121891777_-3.gif)
which are stably conjugate to
![$\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108290540_-3.gif)
).
Instead of the loop group action, Kazhdan-Lusztig also considered the discrete action of the lattice
on the affine Springer fiber.
(Kazhdan-Lusztig)
![$\Lambda_\gamma\backslash X_\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_176127027_-4.gif)
is proper and of finite type.
![$X_\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_105555692_-4.gif)
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 ![$$\Bun_G(k)=G(F)\backslash G(\mathbb{A}_F)/\prod_xG(\mathcal{O}_{x}).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_168671797_.gif)
A pair
![$(\mathcal{E}, \phi: \mathcal{E}\rightarrow \mathcal{E}\otimes\Omega_X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_157375930_-4.gif)
is called a
Higgs bundle. The Hitchin moduli space
![$T^*\Bun_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_19692519_-2.gif)
is defined to be moduli space of Higgs bundles:
For the
Higgs field ![$\phi\in \Hom(\mathcal{E},\mathcal{E}\otimes \Omega_X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_73763515_-4.gif)
, define its invariants
![$a_i(\phi):=\tr\wedge^i\phi\in H^0(X, \Omega_X^{\otimes i})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_57337796_-5.gif)
. So
![$a_1(\phi)=\tr\phi\in H^0(X, \Omega_X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41787051_-4.gif)
and
![$a_n(\phi)=\det\phi\in H^0(X, \Omega_X^{\otimes n})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_243926055_-5.gif)
. We call the affine space
![$\bigoplus_{i=1}^n H^0(X, \Omega_X^{\otimes i})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_96477429_-5.gif)
the
Hitchin base. We have the invariant map to the Hitchin base
More generally, we can replace
![$\Omega_X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_259295139_-2.gif)
by any vector bundle
![$\mathcal{L}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_123373566_0.gif)
. The resulting moduli space of Higgs bundles
![$\mathcal{M}_{G,\mathcal{L}}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_54591101_-4.gif)
is called the
-twisted Hitchin moduli space. The affine space
![$\mathcal{A}_{\mathcal{L}}=\bigoplus_{i=1}^n H^0(X, \mathcal{L}^{\otimes i})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_111999196_-5.gif)
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,
Define
![$$\mathcal{A}_d^i=\{(\mathcal{L},s): \mathcal{L}\in \Pic^d(X), s\in H^0(X,\mathcal{L}^{\otimes i})\}.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_26211149_.gif)
The
universal Hitchin base is defined to be
![$$\mathcal{A}_d^1\times_{\Pic_X^d} \mathcal{A}_d^2 \cdots \times_{\Pic_X^d}\mathcal{A}_d^n,$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_160089081_.gif)
a family of Hitchin bases over
![$\Pic_X^d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_239634531_-4.gif)
.
The fibers
of the invariant map (Hitchin fibers)
are the global analogue of affine Springer fibers.
For
![$\gamma\in G(F)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_166026343_-4.gif)
with characteristic polynomial
![$\Char(\gamma)=P_a(T):=\sum_{i=0}^n(-1)^ia_i T^{n-i}\in F[T]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_257996032_-5.gif)
. Assume that
![$\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_108290540_-3.gif)
is elliptic (i.e.
![$P_a\in F[T]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_154240615_-5.gif)
is irreducible, equivalently
![$G_\gamma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_105625324_-4.gif)
is an anisotropic torus). Then
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.
Consider the total space of the line bundle
![$\mathcal{L}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_123373566_0.gif)
,
![$$\mathbb{L}=\Spec(\Sym\mathcal{L}^{\vee})=\Spec(\mathcal{O}_X \oplus \mathcal{L}^{\vee} \oplus \mathcal{L}^{\vee \otimes 2 }\oplus \cdots).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_27588323_.gif)
It is a
![$\mathbb{A}^1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_202132941_0.gif)
-fibration over
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
. This total space
![$\mathbb{L}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_35673168_0.gif)
sits in the projective bundle
![$$\pi : \mathbb{P}:=\mathbb{P}(\mathcal{O}_X \otimes \mathcal{L}^{\vee})\rightarrow X$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_64858747_.gif)
(so
![$\pi_*\mathcal{O}(1)=\mathcal{O}_X \otimes \mathcal{L}^\vee$](./latex/FundamentalLemma/latex2png-FundamentalLemma_144173358_-4.gif)
). We have two affine charts given by the two coordinates
![$x\in H^0(\mathbb{P},\mathcal{O}(1))$](./latex/FundamentalLemma/latex2png-FundamentalLemma_200306167_-4.gif)
and
![$y\in H^0(\mathbb{P}, \pi^*(\mathcal{L})(1)).$](./latex/FundamentalLemma/latex2png-FundamentalLemma_211668148_-4.gif)
Then
![$\mathbb{L}\subseteq \mathbb{P}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_22189963_-3.gif)
is given by the
![$x\ne0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_167933608_-4.gif)
. Let
![$$P_a(x,y)=\sum_{i=1}^n (-1)^ia_i x^i y^{n-i}\in H^0(\mathbb{P}, \pi^*\mathcal{L}^{\otimes n}(n)).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_102177636_.gif)
We define the
spectral curve ![$Y_a\subseteq \mathbb{L}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_30284416_-3.gif)
to be the zero locus of
![$P_a(x,y)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_245049649_-4.gif)
.
Suppose
![$Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43742226_0.gif)
is a reduced curve. The
compactified Picard (or Jacobian) stack ![$\overline{\Pic}_Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_208746635_-2.gif)
of
![$Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43742226_0.gif)
is defined to be the stack of torsion-free coherent sheaves of rank 1 on
![$Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43742226_0.gif)
. When
![$Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43742226_0.gif)
is smooth,
![$\overline{\Pic}_Y=\Pic_Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_176454748_-2.gif)
. 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
![$\Pic_Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_246702554_-2.gif)
is always open dense in
![$\overline{\Pic}_Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_208746635_-2.gif)
. Hence
![$\overline{\Pic}_Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_208746635_-2.gif)
is naturally a compactification of
![$\Pic_Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_246702554_-2.gif)
. Notice that in general
![$\overline{\Pic}_Y$](./latex/FundamentalLemma/latex2png-FundamentalLemma_208746635_-2.gif)
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.
For
![$a\in \mathcal{A}_\mathcal{L}^{\heartsuit}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_122036757_-5.gif)
, we have
![$\mathcal{M}_a=\overline{\Pic}_{Y_a}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_16501218_-4.gif)
.
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.
(Waldspurger)
Let
![$\pi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_12809236_0.gif)
be an automorphic cuspidal representation of
![$G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42562578_0.gif)
that unramified everywhere. Let
![$\phi\in \pi^K$](./latex/FundamentalLemma/latex2png-FundamentalLemma_194891539_-3.gif)
, where
![$K=\prod_{x\in |X|} G(\mathcal{O}_x)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_91230894_-8.gif)
(so
![$\phi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_205747260_-3.gif)
is unique up to scaling). Then up to some explicit constants we have an equality
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 ![$$\mathbb{J}(u,f)=\mathbb{I}(u,f).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_19559911_.gif)
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.
Let
![$\mathbf{d}=(d_{ij})_{1\le i,j\le2}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_16415396_-4.gif)
such that
![$d_{ij}\ge0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_187099533_-4.gif)
and
![$d_{11}+d_{22}=d_{12}+d_{21}=:d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_80345399_-2.gif)
. Define the moduli space of pairs of rank two vector bundles together with a morphism:
![$$\mathcal{N}_{\mathbf{d}}(S)=\{ (K_1 \oplus K_2, K_1' \oplus K_2', \phi_{ij}: K_i\rightarrow K_j'): \deg K_j'-\deg K_i=d_{ij}\}/\Pic_X ,$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_176908991_.gif)
where
![$K_i, K_j'$](./latex/FundamentalLemma/latex2png-FundamentalLemma_170219529_-6.gif)
are line bundles on
![$X\times S$](./latex/FundamentalLemma/latex2png-FundamentalLemma_818069_0.gif)
. For simplicity (since we only consider
regular semisimple orbits) we also impose the non degeneracy condition that
![$\phi_{ij}\ne0\in H^0( K_j' \otimes K_i^{\vee})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_20974699_-6.gif)
(which strictly speaking defines an open subset of
![$\mathcal{N}_\mathbf{d}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_207575004_-2.gif)
). Let
![$\mathcal{N}_d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_154297418_-2.gif)
be the union of all such
![$\mathcal{N}_{\mathbf{d}}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_20513930_-2.gif)
's with
![$d_{11}+d_{22}=d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_7333187_-2.gif)
.
Now we define an analogue of the invariant map to the Hitchin base and an analogue of Hitchin fibers.
Let
![$\hat X_d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_27756560_-2.gif)
be the the moduli space of pairs
![$(\mathcal{L}, s)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_149941431_-4.gif)
, where
![$\mathcal{L}\in \Pic_X^d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_141246196_-4.gif)
,
![$s\in H^0(\mathcal{L})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_266376874_-4.gif)
. Let
![$\mathcal{A}_d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_114138039_-2.gif)
be the moduli space of triples
![$(\mathcal{L},\alpha,\beta)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_217570356_-4.gif)
, where
![$\alpha,\beta\in H^0(\mathcal{L})$](./latex/FundamentalLemma/latex2png-FundamentalLemma_204108708_-4.gif)
. We have a natural map
![$\mathcal{A}_d\rightarrow \hat X_d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_147833093_-2.gif)
given by
![$(\mathcal{L},\alpha,\beta)\mapsto (\mathcal{L},\alpha-\beta)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_194454463_-4.gif)
.
We have an invariant map
, given by
,
and
. Let
be the fiber of this invariant map above
.
Let
![$D\in X_d(k)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_161416713_-4.gif)
be an effective divisor on
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
of degree
![$d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41841682_0.gif)
. Let
![$\mathcal{A}_D(k)\cong H^0(\mathcal{O}(D))$](./latex/FundamentalLemma/latex2png-FundamentalLemma_249815491_-4.gif)
(viewed as a
![$k$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42300434_0.gif)
-subspace of
![$F $](./latex/FundamentalLemma/latex2png-FundamentalLemma_142767122_0.gif)
) be the fiber of
![$\mathcal{A}_d\rightarrow \hat X_d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_147833093_-2.gif)
above
![$D$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42365970_0.gif)
. Then
11/19/2015
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
.
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).
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).
In order to geometrize the orbital integral
, we define an analogue of the space
.
Let
![$\nu: X'\rightarrow X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_241447685_0.gif)
be an etale double cover. Define
![$\mathcal{M}_d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_114138038_-2.gif)
to be the moduli space
![$\{(K, K', \phi): \deg K'-\deg K=d\}/\Pic_X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_217568632_-4.gif)
, where
![$K, K'\in\Pic_{X'}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_216621565_-3.gif)
and the map
![$\phi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_205747260_-3.gif)
is an element of
![$$\scriptstyle\Hom_{\mathcal{O}_X}(\nu_* K, \nu_* K')=\Hom_{\mathcal{O}_{X'}}(\nu^*\nu_*K, K')=\Hom_{\mathcal{O}_{X'}}(K, K')\oplus \Hom_{\mathcal{O}_{X'}}(\sigma^*K,K'),$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_70633208_.gif)
where
![$\sigma$](./latex/FundamentalLemma/latex2png-FundamentalLemma_208953823_0.gif)
is the nontrivial Galois involution. Since we only consider regular semisimple orbits, we further impose the non-degeneracy condition
![$\phi=(\alpha,\beta)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_113015310_-4.gif)
where
![$\alpha\ne0,\beta\ne0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_179213637_-4.gif)
.
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
: ![$$\xymatrix{\mathcal{M}_d \ar[r]^-\cong \ar[d] & X'_ d\times_{\Pic_{X}^d}X'_ d \ar[d] \\ \mathcal{A}_d \ar[r]^-\cong& X_d\times_{\Pic_X^d}X_d}.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_266241776_.gif)
Analogous to Theorem 19, we have
Let
![$D\in X_d(k)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_161416713_-4.gif)
be an effective divisor on
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
of degree
![$d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41841682_0.gif)
. Let
![$\mathcal{A}_D\cong H^0(\mathcal{O}(D))$](./latex/FundamentalLemma/latex2png-FundamentalLemma_211479560_-4.gif)
(viewed as a
![$k$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42300434_0.gif)
-subspace of
![$F $](./latex/FundamentalLemma/latex2png-FundamentalLemma_142767122_0.gif)
) be the fiber of
![$\mathcal{A}_d\rightarrow \hat X_d(k)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_19572895_-4.gif)
above
![$D$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42365970_0.gif)
. 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 ![$$\xymatrix{\mathcal{M}_d \ar[rd]_{\pi_\mathcal{M}} && \mathcal{N}_d \ar[ld]^{\pi_\mathcal{N}} \\ & \mathcal{A}_d&}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_25435713_.gif)
The identity
now becomes a statement purely about two perverse sheaves.
There is an isomorphism between perverse sheaves
This will follow from the even stronger claim.
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,
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.
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
.
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).
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: ![$$d_{11}+2d=\dim \mathcal{N}_{\mathbf{d},(0,D)} +2\dim(0\times X_d)>\dim \mathcal{A}_d=2d-(g-1).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_103319698_.gif)
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.
Let
![$\mathcal{E}:=Rf_{\mathcal{N}_d,!}\mathbb{L}_\mathbf{d}[\dim \mathcal{A}_d]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_8358575_-5.gif)
. Then
![$\dim\supp \mathcal{H}^{-i}(\mathcal{E})<i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_72300113_-4.gif)
.
Notice
![$H^*(\mathcal{N}_{\mathbf{d},(0,D)},\mathbb{L}_\mathbf{d})=H^*(X_{d_{11}},\mathbb{L}_{d_{11}})\otimes V$](./latex/FundamentalLemma/latex2png-FundamentalLemma_59880289_-6.gif)
, where
![$V$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43545618_0.gif)
from the finite part
![$\mathrm{add}^{-1}(D)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_177800983_-4.gif)
. The claim follows from the fact that
![$$H^*(X_{d},\mathbb{L}_{d})=(H^*(X,\mathbb{L})^{\otimes d})^{S_d}.$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_118304105_.gif)
By Remark
44, the right hand side has only one possibly nonzero term
![$(\wedge^d H^1(X,\mathbb{L}))[-d]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_2251459_-5.gif)
, which becomes zero when
![$d> 2(g-1)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_45840222_-4.gif)
.
¡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.
Consider
![$\mathcal{M}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_122324990_-1.gif)
defined over a finite field
![$k=\mathbb{F}_q$](./latex/FundamentalLemma/latex2png-FundamentalLemma_68934268_-4.gif)
(usually a certain moduli space). Suppose
![$C\rightarrow \mathcal{M}\times\mathcal{M}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_112113364_-1.gif)
is a correspondence. We define the the moduli space of shutaks associated to
![$C $](./latex/FundamentalLemma/latex2png-FundamentalLemma_139621394_0.gif)
to be the fiber product
![$$\xymatrix{\Sht_C \ar[r] \ar[d] & C \ar[d]\\ \mathcal{M} \ar[r]^-{(\Id,\Frob_q)} & \mathcal{M}\times \mathcal{M}}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_75894715_.gif)
More generally, suppose there are
![$r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42759186_0.gif)
correspondences
![$C_i\rightarrow \mathcal{M}\times \mathcal{M}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_53044595_-2.gif)
, we define
![$\Sht_{\{C_i\}}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_66106812_-6.gif)
to be the moduli space of shtukas associated to the convolution correspondence
Consider
![$G=GL_n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_37264579_-2.gif)
. For
![$\mathcal{M}=\Bun_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_3872936_-2.gif)
, we define the Hecke stack
![$\Hk_D\rightarrow \Bun_G\times \Bun_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_63872442_-2.gif)
to be the moduli space of arrows
![$\phi: \mathcal{E}\rightarrow \mathcal{E}'$](./latex/FundamentalLemma/latex2png-FundamentalLemma_236090766_-3.gif)
of vector bundles of rank
![$n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42497042_0.gif)
such that
![$\div\det\phi=D$](./latex/FundamentalLemma/latex2png-FundamentalLemma_152485918_-3.gif)
. Similarly define
![$\Hk_d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_154039986_-2.gif)
using the condition that
![$\deg\det\phi=d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_99696610_-3.gif)
.
Define
![$\Hk^+$](./latex/FundamentalLemma/latex2png-FundamentalLemma_151877298_-1.gif)
to be the Hecke stack of
upper (increasing) modifications, i.e.,
![$\phi: \mathcal{E}\rightarrow\mathcal{E}'$](./latex/FundamentalLemma/latex2png-FundamentalLemma_133873895_-3.gif)
(over
![$X\times S$](./latex/FundamentalLemma/latex2png-FundamentalLemma_818069_0.gif)
) is an injection such that
![$\coker\phi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_163370190_-3.gif)
is a line bundle on the graph of a marked point
![$S\rightarrow X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_110074769_0.gif)
. Similarly define
![$\Hk^-$](./latex/FundamentalLemma/latex2png-FundamentalLemma_152008370_-1.gif)
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.
Both
![$\overrightarrow{p},\overleftarrow{p}: \Hk_d\rightarrow\Bun_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_175333418_-3.gif)
are representable and proper. When
![$d=1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_226324_-1.gif)
, both have relative dimension
![$(n-1)+1=n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_126157901_-4.gif)
(Here 1 comes from the choice the location of modification and
![$n-1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_151221268_-1.gif)
comes from the choice of the modification with at a fixed location, i.e., a line in an
![$n$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42497042_0.gif)
-dimensional vector space).
Let
![$r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42759186_0.gif)
be an even integer. Let
![$\mu=(\mu_i)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_203656674_-4.gif)
be a
![$r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42759186_0.gif)
-tuple of signs. Let
![$\Sht_G^\mu$](./latex/FundamentalLemma/latex2png-FundamentalLemma_179750479_-5.gif)
be the moduli space of shtukas associated to the convolution of
![$\{\Hk^{\mu_i}\}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_195878954_-4.gif)
. In other words, an
![$S$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43349010_0.gif)
-point of
![$\Sht_G^\mu$](./latex/FundamentalLemma/latex2png-FundamentalLemma_179750479_-5.gif)
corresponds to a
![$r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_42759186_0.gif)
-tuple of modification of vector bundles
![$\mathcal{E}_0\rightarrow\mathcal{E}_1 \cdots \rightarrow\mathcal{E}_r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_203789490_-2.gif)
such that
![$\mathcal{E}_r\cong (\Id \otimes \Frob_S)^*\mathcal{E}_0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_122160741_-4.gif)
.
Recall that (Remark 28)
itself is only an Artin stack (which has a lot of automorphism). The moduli of shtukas has better properties.
(Drinfeld
, Varshavsky in general)
is a Deligne-Mumford stack, locally of finite type.
- The projection map
is separated, smooth of relative dimension
(in fact, an
-iterated
-bundle).
When
![$r=0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_235041812_0.gif)
, the Hecke stack simply consists of isomorphisms
![$\mathcal{E}\xrightarrow{\cong}\mathcal{E}'$](./latex/FundamentalLemma/latex2png-FundamentalLemma_53271984_0.gif)
. So
![$\Sht_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_231104983_-2.gif)
consists of vector bundles on
![$X\times S$](./latex/FundamentalLemma/latex2png-FundamentalLemma_818069_0.gif)
such that
![$\mathcal{E}\cong (\Id \times \Frob_S)^*\mathcal{E}1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_15484399_-4.gif)
, which must come from pullback of vector bundles on
![$X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_43676690_0.gif)
itself. Hence
![$\Sht_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_231104983_-2.gif)
is the discrete group
![$\Bun_G(k)=G(F)\backslash G(\mathbb{A})/\prod G(\mathcal{O}_x)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_129761559_-5.gif)
. This exactly puts us in the earlier situation of Waldspurger's formula when
![$G=GL_2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_35953859_-2.gif)
. 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
![$r=0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_235041812_0.gif)
marked points.
When
![$n=1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_167998484_-1.gif)
, we have
![$\Hk^\mu=\Pic_X\times X^r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_159781356_-2.gif)
given by the first line bundle
![$\mathcal{L}_0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_153510986_-2.gif)
and the location of modification
![$\{x_i\}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_149285480_-4.gif)
. So we have the fiber diagram
![$$\xymatrix{\Sht^\mu \ar[r] \ar[d]& X^r \ar[d] \\ \Pic X \ar[r]^-{\Id-\Frob} & \Pic_X^0}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_146492787_.gif)
Here the right vertical arrow is given by
![$(x_i)\mapsto \sum \mu_i x_i$](./latex/FundamentalLemma/latex2png-FundamentalLemma_134747040_-5.gif)
. In particular, considering the degree zero part (and rotating the previous diagram) we obtain the fiber diagram
![$$\xymatrix{\Sht^{\mu,0} \ar[r] \ar[d] &\Pic_X^0 \ar[d]\\ X^r \ar[r] &\Pic_X^0}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_159643958_.gif)
The right vertical arrow is exactly Lang's isogeny, whose kernel is the class group
![$\Pic_X^0(k)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_228624629_-4.gif)
of the function field
![$k(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_39095238_-4.gif)
. This is a generalization of unramified geometric class field theory: when
![$r=1$](./latex/FundamentalLemma/latex2png-FundamentalLemma_235107348_-1.gif)
, the etale map
![$\Sht^{\mu,d}\rightarrow X$](./latex/FundamentalLemma/latex2png-FundamentalLemma_103749419_0.gif)
has Galois group the class group
![$\Pic_X^0(k)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_228624629_-4.gif)
and realizes the Hilbert class field of
![$k(X)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_39095238_-4.gif)
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
.
Let
![$\mathcal{E}_0\rightarrow\mathcal{E}_1\rightarrow\cdots\mathcal{E}_r\cong(\Id\times\Frob)^*\mathcal{E}_0$](./latex/FundamentalLemma/latex2png-FundamentalLemma_168710926_-4.gif)
and
![$\mathcal{E}_0'\rightarrow\mathcal{E}_1'\rightarrow\cdots\mathcal{E}_r'\cong(\Id\times\Frob)^*\mathcal{E}_0'$](./latex/FundamentalLemma/latex2png-FundamentalLemma_134256635_-4.gif)
be two points in
![$\Sht_G^\mu$](./latex/FundamentalLemma/latex2png-FundamentalLemma_179750479_-5.gif)
. We define a degree
![$d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41841682_0.gif)
Hecke correspondence to be the collection of injections
![$\mathcal{E}_i\rightarrow\mathcal{E}_i'$](./latex/FundamentalLemma/latex2png-FundamentalLemma_85392206_-4.gif)
such that
![$\deg \mathcal{E}_i'-\deg\mathcal{ E}_i=d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_207620415_-4.gif)
and the natural diagram
![$$\xymatrix{\mathcal{E}_0 \ar[r] \ar@{^(->}[d] & \mathcal{E}_1 \ar[r] \ar@{^(->}[d] & \cdots \ar[r] & \mathcal{E}_r \ar@{^(->}[d]\\ \mathcal{E}_0' \ar[r] & \mathcal{E}_1' \ar[r] & \cdots \ar[r] & \mathcal{E}_r'}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_44653489_.gif)
commutes. The stack of such degree
![$d$](./latex/FundamentalLemma/latex2png-FundamentalLemma_41841682_0.gif)
Hecke correspondences on
![$\Sht_G^\mu$](./latex/FundamentalLemma/latex2png-FundamentalLemma_179750479_-5.gif)
is denoted by
![$\Hk_d^\mu$](./latex/FundamentalLemma/latex2png-FundamentalLemma_54541636_-5.gif)
.
We define a Hecke correspondence version of
![$\Sht_G^\mu$](./latex/FundamentalLemma/latex2png-FundamentalLemma_179750479_-5.gif)
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.
The map
![$$h: \mathcal{H}\rightarrow \Ch_{c,2r}(\Sht_G\times\Sht_G),\quad h_D\mapsto \Sht(h_D)$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_121972630_.gif)
is a ring homomorphism.
The map
![$\theta:\Sht_T\rightarrow\Sht_G$](./latex/FundamentalLemma/latex2png-FundamentalLemma_154777117_-2.gif)
induces a map
![$\theta:\Sht_T\rightarrow\Sht_G':=\Sht_G\times_{X^r}(X')^r$](./latex/FundamentalLemma/latex2png-FundamentalLemma_224393164_-4.gif)
(like Heegner points are imaginary quadratic points of modular curves). We define the
Heegner-Drinfeld cycle to be the direct image of
![$[\Sht_T^\mu]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_73099040_-5.gif)
in
![$\Ch_{c,r}(\Sht_G')$](./latex/FundamentalLemma/latex2png-FundamentalLemma_215945262_-4.gif)
under
![$\theta$](./latex/FundamentalLemma/latex2png-FundamentalLemma_33267234_0.gif)
.
Let
![$f\in \mathcal{H}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_124929961_-3.gif)
. Then
![$$\mathbb{I}_r(f)=\mathbb{J}_r(f).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_19531855_.gif)
Here
![$\mathbb{I}_r(f)$](./latex/FundamentalLemma/latex2png-FundamentalLemma_267227987_-4.gif)
is the intersection number of the Heegner-Drinfeld cycles
![$$(\Sht_T, h(f)\Sht_T)_{\Sht_G'}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_225669697_.gif)
and
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.
We have
![$V= \bigoplus_{\pi \text{ cusp}} V_\pi \bigoplus V_{\mathrm{Eis}}$](./latex/FundamentalLemma/latex2png-FundamentalLemma_75307669_-7.gif)
.
It then makes sense to talk about the
-isotypic component and using the Theorem 26 one can show that
For
![$\pi$](./latex/FundamentalLemma/latex2png-FundamentalLemma_12809236_0.gif)
an everywhere unramified cuspidal automorphic representation of
![$GL_2$](./latex/FundamentalLemma/latex2png-FundamentalLemma_35943482_-2.gif)
, 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 ![$$\xymatrix{([\Sht_T, \Sht(h_d)\Sht_T]) \ar[r] \ar[d] & \Sht(h_d) \ar[d] \\ \Sht_T\times \Sht_T \ar[r] & \Sht_G\times\Sht_G}$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_198988265_.gif)
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.
(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
):
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
![$$\mathbb{J}_r(h_D,s)=\sum_{a\in \mathcal{A}_D(k)} \sum_{\mathbf{d}}q^{(d_{12}-d_{21})s}\Tr(\Frob_a, (Rf_{\mathcal{N}_d,!}\mathbb{L}_\mathbf{d})_{\bar a}).$$](./latex/FundamentalLemma/latex2png-FundamentalLemma_181061708_.gif)
Therefore,
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
.
![$[\Hk_{\mathcal{M}_d}]$](./latex/FundamentalLemma/latex2png-FundamentalLemma_36806508_-5.gif)
acts on
![$K_i \boxtimes K_j$](./latex/FundamentalLemma/latex2png-FundamentalLemma_1441935_-4.gif)
by the constant
![$d-2j$](./latex/FundamentalLemma/latex2png-FundamentalLemma_261852103_-3.gif)
.