COMPATIBILITY OF HECKE AND EXCURSION OPERATORS
TONY FENG
1.
Hecke operators as Excursion operators
Last time we saw how to canonically attach a Langlands parameter
b ` ) to a character ν
G(Q
σ : Gal(F /F ) →
on the algebra of excursion operators. We would like to know
that this correspondence satises the properties expected of the global Langlands
correspondence. The rst and foremost expected property is compatibility with the
Satake correspondence, i.e. that the Hecke eigenvalues (essentially) match up with
the eigenvalues of Frobenius.
In the construction of
σ
we arranged that the eigenvalues of the excursion oper-
ators should match up with the characteristic polynomials of some corresponding
Galois elements. Therefore, the desired compatibility will follow immediately from
knowing that the Hecke operator
TV,v
coincides with an appropriate excursion oper-
ator. Before we explain which one, we review Hecke operators.
1.1.
Review of Hecke operators.
Denition 1.1.
Fix a representation
V
of
b.
G
We dene the Hecke correspondence
(I)
ΓV,v to be the stack classifying modications of
EO 0
φ0
/ τ E0
O
κ
κ
E
where
by
φ
E−
→ τ E ∈ ShtN,∅
and
G-bundles
φ
/ τE
φ
E0 −
→ τ E 0 ∈ ShtN,∅ , and κ is a modication at v bounded
V.
Remark 1.2.
Where are actually most interested in the case where
representation
W
doesn't matter). In this case,
ShtN,∅
I = ∅ (so the
(which is BunG,N (Fq ) viewed
as a discrete stack). In fact, the general case easily reduces to this one, using the
factorization property of the ane Grassmannian.
Denition 1.3.
We dene
TV,v
to be the cohomological Hecke operator induced by
Γ∅V,v .
Date
: December 1, 2016.
1
2
TONY FENG
The compatibility theorem.
1.2.
We now want to realize
TV,v
as an excursion
SI,W,x,ξ,(γi )i∈I . We will take
• I = {1, 2}
b {1,2} ,
• W = V V ∗ as a representation of G
∗
• x = ev : 1 → V V ,
• ξ = coev : V V ∗ → 1,
v
• γ1 = Frobdeg
, γ2 = 1.
1
denote this excursion operator by SV,v .
operator
We
Theorem 1.4. We have TV,v = SV,v .
Remark 1.5. See Laorgue's paper for a more general statement.
It is indeed impor-
tant for Laorgue's argument, but not for the purposes of the present compatibility
assertion, to study the more general Hecke operators on
and
W,
≤µ
HN,I,W
for more general
I
because they are used to dene the Hecke-nite part of the intersection
cohomology.
I = ∅ is that TV,v is dened away
SV,v is dened away from (X − N )I .
Therefore the theorem can be used to give an extension of the Hecke operator TV,v
at v . (You can think of this as being an issue of having integral models v .) An
In particular, a phenomenon which is invisible for
from
(X − (N ∪ v))I , where N
is the level, while
Eichler-Shimura relation for the excursion operator is then used to show that the
property of being Hecke-nite implies stability by the partial Frobenius operators,
which was an important technical issue.
Exercise 1.6.
Check that Theorem 1.4 implies compatibility of the Langlands cor-
respondence with the Satake correspondence.
compatibility between
ν
and
2.
[This is a matter of digesting the
σ .]
Overview of the proof
We now survey the proof of Theorem 1.4. First of all, we simplify the situation by
assuming that
v
is a rational point, and that
V
is a miniscule representation. The
technical signicance of the latter assumption is that the correspondence Schubert
varieties are smooth.
The main thrust of this theorem is that the excursion operators, which are in
general built by a complicated procedure involving the Geometric Satake correspondence, have a very simple description in this special case.
That is, they can be
realized by an actual correspondence. Speaking very roughly, one can appreciate the
content of this claim as being like an instance of a Hodge Conjecture-type statement,
because it says that a map between cohomology can be realized geometrically.
We can view the excursion operators as being built out of three distinct steps:
(1) a creation operator,
(2) a Galois action, and
(3) an annihilation operator.
The main point is to see that the relevant creation and annihilation operators in this
special case are induced by correspondences. Let us elaborate on this point.
COMPATIBILITY OF HECKE AND EXCURSION OPERATORS
Denition 2.1.
We dene the creation operator
Cx#
3
to be the composition
χ−1
x
H{0},1 −
→ H{0},V V ∗ −−→ H{1,2},V V ∗ |∆(X−N ) .
We dene the annihilation operator
Cx[
(2.1)
to be the composition
χ
ξ
H{1,2},V V ∗ |∆(X−N ) −
→ H{0},V V ∗ →
− H{0},1
(2.2)
H{0},1 is really the constant local system over X , with ber H∅ , the excursion
operator SV,v is obtained by the restriction to ∆(v) ⊂ ∆(X − N ) of composing
Since
the creation operator with a partial Frobenius, and then applying the annihilation
operator.
The main point is to show that (2.1) and (2.2) can be realized by correspondences.
More precisely, let
({1},{2})
ZI,W
and
(I)
ZI,W = ShtI,W .
(I,{1},{2})
= ShtI∪{1,2},W V V ∗ |∆(v)
(Keep in mind the case
I = ∅,
in which case this is a discrete
stack.) We will exhibit closed substacks
({1},{2})
j1 : Y1 ,→ ZI,W
({1},{2})
j2 : Y2 ,→ ZI,W
possessing maps
α1 : Y1 ,→ ZI,W
α2 : Y2 ,→ ZI,W
such that (2.1) is realized by
(j1 )∗ ◦ α1∗
and (2.2) is realized by
(α2 )∗ ◦ j2∗ .
Since
Z itself, it is easy to modify one of the correspondences by
#
it to see that, for instance, (Frobv , 1)◦Cx is also realized by a correspondence. Then
partial Frobenius acts on
the composition will be is realized by the intersection of the two correspondences. A
tangent space calculation shows that this intersection is smooth. We won't delve into
these details; we will focus on the problem of describing the geometric interpretation
of creation and annihilation operators as correspondences.
3.
Since
Cx[
The annihilation operator as a correspondence
is induced by a maps of sheaves on the moduli stack of shtukas:
χ
ξ
FI∪{1,2},W V V ∗ |∆(X−N ) −
→ F{0},V V ∗ →
− F{0},1
which is pulled back from the Beilinson-Drinfeld Grassmannian, it suces to show
that the corresponding map at the level of Beilinson-Drinfeld Grassmannians is induced by a correspondence. Since we are localizing over a point
∆(v),
we only need
to show this for the ber, which is the usual ane Grassmannian. In other words,
we want to show that the map
ξ
S{0},V V ∗ →
− S{0},1
4
TONY FENG
can be realized by a correspondence, where
S{0},V V ∗ = ICGr{1},{2}
V V ∗
♠♠♠
TONY: [still need to understand why this is true...]
and
S{0},1 = ICGr1 = ICpt .
The thing ultimately comes down to understand the map
ξ
S{1,2},V V ∗ →
− S1
in the Geometric Satake correspondence. The point is then as follows. The space
GrV V ∗
parametrizes modications of lattices with invariant bounded by
V V ∗.
This admits a map
{1},{2} π
{1,2}
GrV V ∗ −
→ GrV V ∗
{1},{2}
(E0 99K E1 99K E2 ) 7→ (E0 99K E2 ). The space GrV V ∗
bration of GrV over GrV ∗ , and these are smooth because V is
sending a chain
being a
is smooth,
a miniscule
representation. We then claim:
π∗ ICGr{1},{2} ' SV V ∗ = ICGr{1,2} ⊕ δ
V V ∗
V ⊗V
where, by proper base change, we know that
δ
is a skyscraper sheaf at the point
corresponding to the trivial modication. Let us call this point
η[ .
δ.
v , and the ber over it
π∗ ICGr{1},{2} →
The Geometric Satake correspondence is then the projection map
We claim that it coincides with the correspondence induced by
η[ ,
i.e.
V V ∗
α∗ ◦ i∗
in
the diagram:
{1},{2}
GrV V ∗ o
η[
i
α
{1,2}
GrV V ∗ o
This tells us that the cycle in
{1},{2}
GrV V ∗
v
is the pre-image of the trivial modication
{1,2}
in GrV V ∗ . We will illustrate this claim with an example.
Example 3.1.
be the standard representation, with dominant
coweight
map in question is
Let G = GLn and V
λ = (1, 0, . . . , 0). Then the
Gr(1,0,...,0),(0,...,0,−1)
π
Gr(1,0,...,0,−1)
Gr(1,0,...,0),(0,...,0,−1) is a Pn−1 -bration over Pn−1 . It is smooth, so its
Q` [2(n − 1)]. It is smooth over the complement in Gr(1,0,...,0,−1) of the
We know that
IC sheaf is
COMPATIBILITY OF HECKE AND EXCURSION OPERATORS
5
point corresponding to the trivial bundle (the unique proper sub-Schubert variety).
Gr(1,0,...,0),(0,...,0,−1) o
Pn−1
π
Gr(1,0,...,0,−1) o
pt
By the general properties of IC sheaves, we have that
π∗ Q` [2(n − 1)] = ICGr(1,0,...,0,−1) ⊕ . . . .
Restricting to the complement of the singular point, this is an isomorphism so the
...
has to be supported on the singular point. Restricting to the singular point, we
see by cohomology and base change that
H ∗ (Pn−1 )[2(n − 1)] = H ∗ (pt, ICGr(1,0,...,0,−1) ) ⊕ H ∗ (pt, δ).
δ
By the general properties of IC sheaves,
in degree
grees.
0,
while
H ∗ (pt, ICGr(1,0,...,0,−1) )
is perverse and therefore concentrated
is concentrated in strictly negative de-
Therefore we see by proper base change that the lone contribution comes
H 2(n−1) (Pn−1 ). Similarly, the pushforward map on cohomology induced by
Pn−1 → pt maps the top degree cohomology isomorphically to Q` and kills the rest.
In other words, the projection to δ coincides with the correspondence induced by
n−1 .
the P
from
Now let's interpret this knowledge at the level of shtukas. Recall that
({1},{2})
ZI,W
parametrizes modications
> E1
φ1
φ2
E0
where the modications of
from
N ∪ v.
φ1 , φ2
φ3
E2
occur at
v,
/ τ E0
and the modications of
φ3
occur away
As such, we can uniquely ll in such a diagram to
φ1
E0
/ E0
2
? E1
φ2
E2
φ3
/
τφ
τE
τ
= E1
1
0
The calculation we just did shows that the closed substack
cut out by the condition that the modication
φ2 ◦ φ1
parametrizes
φ1
E0
/ E0
2
? E1
φ2
/ E2
∼
φ3
/ E0
τ
> E1
τφ
1
Y2 ⊂ Z {1},{2}
is then
actually be trivial, i.e.
Y2
6
TONY FENG
It should then come as no surprise that
φ1
E0
Y1
/ E0
2
? E1
φ2
E2
parametrizes
φ3
/ E0
∼
/ τ E1
>
τφ
1
Therefore, their intersection parametrizes modication diagrams
φ1
E0
/ E0
2
? E1
φ2
/
∼ E2
φ3
/ E0
∼
/ τ E1
>
τφ
1
which we can visibly contract to obtain the Hecke correspondence.